Pros of Coq-HoTT

• Incorporates Homotopy Type Theory, providing a foundation for univalent mathematics
• Offers advanced features for working with higher-dimensional structures
• Enables more intuitive reasoning about equality and equivalence

Cons of Coq-HoTT

• Smaller community and fewer resources compared to the main Coq repository
• May have compatibility issues with some existing Coq libraries
• Steeper learning curve for those not familiar with HoTT concepts

Code Comparison

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.

Coq-HoTT:

Inductive hProp : Type :=
  | hcontr : forall A : Type, (A -> A -> A) -> hProp
  | hprop : forall A : Type, (A -> A -> Type) -> hProp.

Definition isContr (A : Type) : Type :=
  {x : A & forall y : A, x = y}.

The main difference lies in the focus on homotopy type theory concepts in Coq-HoTT, such as h-levels and contractibility, while standard Coq deals with more traditional type theory constructs.

Pros of Agda

• More flexible syntax and notation, allowing for Unicode characters and mixfix operators
• Stronger support for dependent types and type-level programming
• Integrated development environment with interactive editing features

Cons of Agda

• Smaller community and ecosystem compared to Coq
• Less extensive standard library and fewer tactics for proof automation
• Steeper learning curve for beginners due to its more advanced type system

Code Comparison

Coq example:

Theorem plus_comm : forall n m : nat,
  n + m = m + n.
Proof.
  intros n m. induction n as [| n' IHn'].
  - simpl. rewrite <- plus_n_O. reflexivity.
  - simpl. rewrite -> IHn'. rewrite -> plus_n_Sm. reflexivity.
Qed.

Agda example:

+-comm : ∀ m n → m + n ≡ n + m
+-comm zero n = sym (+-identity n)
+-comm (suc m) n =
  begin
    suc m + n
  ≡⟨⟩
    suc (m + n)
  ≡⟨ cong suc (+-comm m n) ⟩
    suc (n + m)
  ≡⟨ sym (+-suc n m) ⟩
    n + suc m
  ∎

Pros of Lean3

• Faster proof checking and compilation times
• More intuitive syntax for mathematicians
• Better metaprogramming capabilities

Cons of Lean3

• Smaller ecosystem and library of formalized mathematics
• Less mature and stable compared to Coq
• Fewer learning resources and documentation available

Code Comparison

Coq example:

Theorem add_comm : forall n m : nat,
  n + m = m + n.
Proof.
  intros n m. induction n as [| n' IHn'].
  - simpl. rewrite -> plus_n_O. reflexivity.
  - simpl. rewrite -> IHn'. rewrite -> plus_n_Sm. reflexivity.
Qed.

Lean3 example:

theorem add_comm (n m : ℕ) : n + m = m + n :=
begin
  induction n with n' ih,
  { simp },
  { simp [ih, nat.add_succ] }
end

Both examples prove the commutativity of addition for natural numbers. Lean3's syntax is more concise and uses tactics that are closer to mathematical reasoning. Coq's proof is more verbose but provides explicit steps in the proof construction.

Pros of Idris-dev

• More focused on general-purpose programming with dependent types
• Easier learning curve for programmers familiar with Haskell-like syntax
• Better support for effects and IO operations

Cons of Idris-dev

• Smaller community and ecosystem compared to Coq
• Less mature and stable, with more frequent breaking changes
• Fewer advanced proof automation tools

Code Comparison

Idris-dev:

vect : Nat -> Type -> Type
vect Z a = ()
vect (S k) a = (a, vect k a)

Coq:

Inductive vect (A : Type) : nat -> Type :=
  | nil : vect A 0
  | cons : forall n, A -> vect A n -> vect A (S n).

Both examples define a vector type, but Idris uses a more concise syntax with pattern matching on the length parameter, while Coq uses a more explicit inductive definition.

Idris-dev is designed for general-purpose programming with dependent types, making it more accessible for developers coming from functional programming backgrounds. It offers better support for effects and IO operations, which can be beneficial for practical applications.

On the other hand, Coq has a larger community, more stability, and advanced proof automation tools, making it more suitable for formal verification and theorem proving tasks. The choice between the two depends on the specific use case and the user's background.

Pros of OCaml

• More widely used in industry and practical applications
• Faster execution speed for general-purpose programming
• Simpler learning curve for developers familiar with functional programming

Cons of OCaml

• Less powerful for formal verification and theorem proving
• Limited support for dependent types compared to Coq
• Smaller ecosystem for proof-related libraries and tools

Code Comparison

OCaml:

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

let result = factorial 5

Coq:

Fixpoint factorial (n : nat) : nat :=
  match n with
  | 0 => 1
  | S n' => n * factorial n'
  end.

Definition result := factorial 5.

Summary

OCaml is a general-purpose functional programming language, while Coq is a proof assistant and dependently typed language. OCaml excels in practical software development, offering better performance and a more familiar syntax for many developers. Coq, on the other hand, provides superior capabilities for formal verification and theorem proving, with a focus on mathematical correctness. The code comparison illustrates the syntactical differences, with Coq's approach being more formal and precise, reflecting its emphasis on provable correctness.