Pros of Idris2

• More focus on practical programming with dependent types
• Better support for effects and resource management
• Faster compilation times and improved performance

Cons of Idris2

• Smaller community and ecosystem compared to Agda
• Less mature tooling and IDE support
• Fewer learning resources and academic papers

Code Comparison

Idris2:

data Vec : Nat -> Type -> Type where
  Nil  : Vec Z a
  (::) : a -> Vec n a -> Vec (S n) a

append : Vec n a -> Vec m a -> Vec (n + m) a
append Nil ys = ys
append (x :: xs) ys = x :: append xs ys

Agda:

data Vec (A : Set) : ℕ → Set where
  []  : Vec A zero
  _∷_ : ∀ {n} → A → Vec A n → Vec A (suc n)

_++_ : ∀ {A m n} → Vec A m → Vec A n → Vec A (m + n)
[] ++ ys = ys
(x ∷ xs) ++ ys = x ∷ (xs ++ ys)

Both examples demonstrate vector concatenation with length preservation, showcasing dependent types. Idris2's syntax is more similar to Haskell, while Agda's is more explicit about universe levels and uses Unicode symbols.

Pros of Coq

• More mature ecosystem with extensive libraries and tactics
• Stronger emphasis on automation and proof assistance
• Better support for program extraction and code generation

Cons of Coq

• Steeper learning curve due to more complex syntax and concepts
• Less intuitive syntax for dependent types compared to Agda
• Slower compilation times for large projects

Code Comparison

Agda example:

data ℕ : Set where
  zero : ℕ
  suc  : ℕ → ℕ

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

Coq example:

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

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

Both examples define natural numbers and addition, but Agda's syntax is more concise and readable. Coq's syntax is more verbose but offers more flexibility in defining recursive functions. Agda's pattern matching is more intuitive, while Coq's requires explicit recursion using Fixpoint.

Pros of Coq-HoTT

• Built on the well-established Coq proof assistant, benefiting from its mature ecosystem and tooling
• Focuses specifically on Homotopy Type Theory, providing specialized libraries and tactics
• Supports classical mathematics alongside constructive approaches

Cons of Coq-HoTT

• Steeper learning curve due to Coq's syntax and proof style
• Less intuitive syntax for dependent types compared to Agda
• Smaller community and fewer learning resources specific to HoTT in Coq

Code Comparison

Agda:

data ℕ : Set where
  zero : ℕ
  suc  : ℕ → ℕ

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

Coq-HoTT:

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.

Both repositories provide foundations for formal verification and proof assistants. Agda offers a more intuitive syntax for dependent types and a gentler learning curve, while Coq-HoTT leverages Coq's established ecosystem and provides specialized support for Homotopy Type Theory. The choice between them depends on the specific needs of the project and the user's familiarity with each system.

Pros of Lean4

• Faster proof checking and compilation times
• More extensive metaprogramming capabilities
• Better integration with mainstream programming languages

Cons of Lean4

• Smaller community and ecosystem compared to Agda
• Less mature documentation and learning resources
• Fewer libraries and formalized mathematics available

Code Comparison

Lean4:

def factorial : Nat → Nat
  | 0 => 1
  | n + 1 => (n + 1) * factorial n

#eval factorial 5

Agda:

factorial : ℕ → ℕ
factorial zero = 1
factorial (suc n) = suc n * factorial n

_ : factorial 5 ≡ 120
_ = refl

Both Lean4 and Agda are dependently typed programming languages and proof assistants. Lean4 focuses on efficiency and metaprogramming, while Agda emphasizes expressiveness and flexibility in its type system. Lean4's syntax is more similar to mainstream programming languages, potentially making it easier for newcomers to adopt. Agda, on the other hand, has a larger community and more extensive libraries, particularly in formalized mathematics. The code comparison shows similar implementations of the factorial function, highlighting the syntactic differences between the two languages.

Pros of OCaml

• More mature and widely adopted in industry and academia
• Faster compilation and execution times
• Extensive standard library and package ecosystem

Cons of OCaml

• Less expressive type system compared to Agda's dependent types
• Limited support for theorem proving and formal verification
• Lacks some advanced features like holes and interactive development

Code Comparison

Agda:

data ℕ : Set where
  zero : ℕ
  suc  : ℕ → ℕ

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

OCaml:

type nat = Zero | Succ of nat

let rec add a b =
  match a with
  | Zero -> b
  | Succ n -> Succ (add n b)

The Agda code demonstrates its dependent type system and pattern matching on constructors, while OCaml shows its simpler type system and pattern matching syntax. Agda's code is more concise and allows for more advanced type-level programming, whereas OCaml's version is more straightforward but less expressive in terms of types.

Pros of Rust

• Larger community and more widespread adoption
• Focuses on systems programming and performance
• More mature ecosystem with a wider range of libraries and tools

Cons of Rust

• Steeper learning curve due to complex ownership and borrowing concepts
• Less emphasis on formal verification compared to Agda
• May be overkill for simple projects or rapid prototyping

Code Comparison

Rust example:

fn main() {
    let x = 5;
    println!("The value of x is: {}", x);
}

Agda example:

module Main where

open import IO

main : IO ()
main = putStrLn "Hello, world!"

Key Differences

• Rust is a systems programming language focused on safety and concurrency, while Agda is a dependently typed functional programming language and proof assistant.
• Rust has a larger user base and more industry adoption, whereas Agda is primarily used in academic and research settings.
• Agda provides stronger guarantees through its type system and is better suited for formal verification, while Rust offers a balance between safety and performance.
• Rust's syntax is more familiar to developers coming from C-like languages, while Agda's syntax is closer to functional languages like Haskell.