Pros of Agda

• Agda has a more advanced type system, allowing for more expressive and precise specifications.
• Agda's type system supports dependent types, which can be used to encode complex properties and invariants.
• Agda has a more mature and well-documented standard library, providing a wide range of functionality.

Cons of Agda

• Agda has a steeper learning curve compared to Idris, especially for users new to dependent type systems.
• The Agda ecosystem is smaller than Idris, with fewer available libraries and tools.
• Agda's compiler and tooling are generally less user-friendly and more complex than Idris.

Code Comparison

Idris-dev:

data Nat : Type where
  Z : Nat
  S : Nat -> Nat

plus : Nat -> Nat -> Nat
plus Z     y = y
plus (S x) y = S (plus x y)

Agda:

data Nat : Set where
  zero : Nat
  suc : Nat → Nat

_+_ : Nat → Nat → Nat
zero + n = n
suc m + n = suc (m + n)

The code snippets above show the definition of natural numbers and the addition operation in both Idris and Agda. The main differences are the syntax and the use of dependent types in Agda, which allows for more expressive type definitions.

Pros of Coq

• Coq has a larger and more established user community compared to Idris, with a wealth of resources and support available.
• Coq's proof assistant features are more mature and robust, making it a popular choice for formal verification and theorem proving.
• Coq has a broader range of libraries and tools available, covering a wide variety of domains and use cases.

Cons of Coq

• Coq's syntax and learning curve can be more complex and challenging for beginners compared to Idris.
• Coq's development and maintenance are primarily driven by a large, academic community, which may result in a slower pace of innovation and feature updates.
• Coq's performance can be less efficient than Idris for certain types of computations, especially in the context of programming language implementation.

Code Comparison

Idris-dev (Idris):

data Nat = Z | S Nat

plus : Nat -> Nat -> Nat
plus Z     y = y
plus (S x) y = S (plus x y)

Coq:

Inductive nat : Type :=
  | O : nat
  | S : nat -> nat.

Fixpoint plus (n m : nat) : nat :=
  match n with
  | O => m
  | S p => S (plus p m)
  end.

The code snippets above demonstrate the definition of natural numbers and the implementation of addition in both Idris and Coq. While the overall functionality is similar, the syntax and language-specific features differ between the two.

Pros of Lean3

• Lean3 is a powerful and expressive theorem prover, allowing for the formal verification of complex mathematical and software systems.
• The Lean3 ecosystem includes a rich set of libraries and tools, making it a versatile choice for a wide range of applications.
• Lean3 has a strong focus on automation and proof automation, which can significantly improve the productivity of users.

Cons of Lean3

• Lean3 has a steeper learning curve compared to Idris-dev, requiring a deeper understanding of formal logic and theorem proving.
• The Lean3 community is smaller than the Idris-dev community, which may result in fewer available resources and support.
• Lean3 is primarily focused on theorem proving, while Idris-dev has a broader focus on functional programming and language design.

Code Comparison

Idris-dev:

data Nat = Z | S Nat

plus : Nat -> Nat -> Nat
plus Z     y = y
plus (S x) y = S (plus x y)

Lean3:

inductive Nat : Type
| zero : Nat
| succ : Nat → Nat

def plus : Nat → Nat → Nat
| zero   y := y
| (succ x) y := succ (plus x y)

The code snippets above demonstrate the definition of natural numbers and the implementation of addition in both Idris-dev and Lean3. While the overall approach is similar, the syntax and specific constructs used in each language differ, reflecting their respective design choices and priorities.

Pros of Coq-HoTT

• Coq-HoTT provides a powerful and expressive type theory for formalizing mathematics and computer science concepts.
• The project has a strong focus on Homotopy Type Theory (HoTT), which offers a novel approach to reasoning about higher-dimensional structures.
• Coq-HoTT benefits from the robust and well-established Coq proof assistant, which has a large and active community.

Cons of Coq-HoTT

• Coq-HoTT may have a steeper learning curve compared to Idris-dev, as it requires familiarity with both Coq and HoTT.
• The project may have a narrower focus on HoTT, which may not be as applicable to a broader range of programming tasks as Idris-dev.
• Coq-HoTT may have a smaller community and fewer resources available compared to the Idris-dev project.

Code Comparison

Idris-dev (Idris):

data Nat = Z | S Nat

plus : Nat -> Nat -> Nat
plus Z     y = y
plus (S x) y = S (plus x y)

Coq-HoTT (Coq):

Inductive nat : Type :=
  | O : nat
  | S : nat -> nat.

Fixpoint plus (n m : nat) : nat :=
  match n with
  | O => m
  | S p => S (plus p m)
  end.

Pros of OCaml

• Mature and Widely Used: OCaml has been around since the 1990s and has a large and active community, with a wealth of libraries and tools available.
• Powerful Type System: OCaml's type system is highly expressive, allowing for advanced type-level programming and ensuring a high degree of safety and correctness.
• Efficient Compilation: OCaml code is compiled to efficient native code, making it a good choice for performance-critical applications.

Cons of OCaml

• Steeper Learning Curve: Compared to Idris, OCaml has a steeper learning curve, especially for developers coming from more mainstream languages like Java or Python.
• Limited Dependent Types: While OCaml has a powerful type system, it lacks the full dependent type capabilities of Idris, which can be useful for certain types of formal verification and program synthesis.
• Smaller Ecosystem: The Idris ecosystem, while growing, is smaller than the OCaml ecosystem, with fewer libraries and tools available.

Code Comparison

Here's a simple example of a function that computes the factorial of a number in both OCaml and Idris:

OCaml:

let rec factorial n =
  if n = 0 then 1 else n * factorial (n - 1)

Idris:

factorial : Nat -> Nat
factorial 0 = 1
factorial n = n * factorial (n - 1)

The OCaml version uses a standard recursive definition, while the Idris version leverages the built-in Nat type and pattern matching to achieve the same functionality.

Pros of purescript/purescript

• Mature and Stable: PureScript has been around for a longer time than Idris, with a more established ecosystem and a larger community.
• Extensive Documentation: PureScript has excellent documentation, making it easier for newcomers to get started and learn the language.
• Powerful Type System: PureScript's type system is highly expressive, allowing for advanced type-level programming.

Cons of purescript/purescript

• Steeper Learning Curve: PureScript has a steeper learning curve compared to Idris, especially for those new to functional programming.
• Limited Dependent Types: While PureScript has a powerful type system, it lacks the full dependent type capabilities of Idris.
• Smaller Community: PureScript has a smaller community compared to Idris, which may result in fewer resources and support available.

Code Comparison

Here's a brief code comparison between Idris and PureScript:

Idris:

data List a = Nil | (::) a (List a)

length : List a -> Nat
length Nil = 0
length (x :: xs) = 1 + length xs

PureScript:

data List a = Nil | Cons a (List a)

length :: forall a. List a -> Int
length Nil = 0
length (Cons _ xs) = 1 + length xs

The code snippets demonstrate the similar structure of the List data type and the length function in both Idris and PureScript. The main differences are the syntax and the use of Nat (natural numbers) in Idris versus Int in PureScript.