Pros of Agda

• Dependent Types: Agda is a dependently-typed programming language, which allows for more expressive and precise type systems compared to Coq-HoTT.
• Powerful Type System: Agda's type system is highly expressive, enabling the construction of complex mathematical structures and proofs.
• Interactive Development: Agda provides a powerful interactive development environment, allowing for a more seamless and iterative development process.

Cons of Agda

• Steeper Learning Curve: Agda has a steeper learning curve compared to Coq-HoTT, particularly for those new to dependently-typed programming.
• Ecosystem: The Agda ecosystem is generally smaller than the Coq ecosystem, with fewer available libraries and tools.
• Proof Automation: Coq-HoTT may have more advanced proof automation features compared to Agda, which can make certain proofs easier to construct.

Code Comparison

Agda:

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

_+_ : Nat → Nat → Nat
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.

Pros of Coq

• Mature and well-established proof assistant with a large user community and extensive documentation.
• Supports a wide range of mathematical and computer science domains, including type theory, functional programming, and formal verification.
• Provides a powerful and expressive language for writing and verifying proofs.

Cons of Coq-HoTT

• Coq-HoTT is a specialized extension of Coq that focuses on Homotopy Type Theory (HoTT), which may not be as widely used or supported as the core Coq system.
• The learning curve for Coq-HoTT may be steeper than for the standard Coq, as it introduces additional concepts and features related to HoTT.
• The development and maintenance of Coq-HoTT may not be as active or well-resourced as the main Coq project.

Code Comparison

Coq:

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

Coq-HoTT:

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

The code for the plus_comm theorem is essentially the same in both Coq and Coq-HoTT, as the core Coq language and proof tactics are shared between the two.

Pros of Idris-dev

• Dependent Types: Idris is a dependently-typed programming language, which allows for more expressive and type-safe code compared to Coq-HoTT.
• Ease of Use: Idris has a more user-friendly syntax and development experience compared to the more academic-oriented Coq-HoTT.
• Active Development: Idris-dev has a larger and more active community, with more frequent updates and a wider range of available libraries.

Cons of Idris-dev

• Proof Automation: Coq-HoTT has more advanced proof automation features, which can make it easier to work with complex mathematical proofs.
• Formal Verification: Coq-HoTT is more widely used and accepted for formal verification of critical systems, due to its strong mathematical foundations.
• Ecosystem: Coq-HoTT has a larger and more established ecosystem, with a wider range of available libraries and tools.

Code Comparison

Idris-dev (Idris):

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)

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.