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Question-1. How does Agda support constructive mathematics?
Answer-1: Every proof corresponds to a constructible object.
Question-2. How do you work with lists in Agda?
Answer-2: Define data List (A : Set) : Set where [] : List A; _::_ : A ? List A ? List A.
Question-3. What is an erased argument in Agda?
Answer-3: Arguments marked with @0 are erased at runtime for efficiency.
Question-4. How do you define a higher-order function in Agda?
Answer-4: By taking a function as an argument, e.g., map : (A ? B) ? List A ? List B.
Question-5. What is Agda?s role in formal verification?
Answer-5: It is used to verify mathematical proofs and software correctness.
Question-6. What are interactive proofs in Agda?
Answer-6: Step-by-step proof development using interactive editing.
Question-7. How do you handle dependent pattern matching?
Answer-7: Using pattern matching inside dependent functions.
Question-8. What is a homotopy type theory (HoTT) in Agda?
Answer-8: A framework extending type theory with higher-dimensional structures.
Question-9. How do you define a monoid in Agda?
Answer-9: Using a record, e.g., record Monoid (A : Set) : Set where field unit : A; op : A ? A ? A.
Question-10. How does Agda handle implicit conversions?
Answer-10: Using instance arguments and type inference.
Question-11. Can Agda compile to executable programs?
Answer-11: No, it is mainly for proofs, but it can generate Haskell code.
Question-12. What is {-# TERMINATING #-} used for?
Answer-12: It tells Agda to skip termination checking for a function.
Question-13. How do you use dependent pairs in Agda?
Answer-13: Using ? types or records.
Question-14. What is the role of the module system in Agda?
Answer-14: It helps structure large proofs and programs.
Question-15. How do you use interactive theorem proving in Agda?
Answer-15: By filling in terms interactively with a proof assistant.
Question-16. What is a category in Agda?
Answer-16: It represents a mathematical category with objects and morphisms.
Question-17. What are coinductive types in Agda?
Answer-17: Types that allow infinite structures like streams.
Question-18. How does Agda handle side effects?
Answer-18: Agda does not have built-in side effects, but monads can be used.
Question-19. What is {-# OPTIONS --safe #-} in Agda?
Answer-19: It ensures that only safe features of Agda are used.
Question-20. Can Agda be used for practical programming?
Answer-20: It is mainly for theorem proving, but it can model programs.
Question-21. How do you write a proof of a + b = b + a in Agda?
Answer-21: Using induction on a or b.
Question-22. Where can you learn more about Agda?
Answer-22: The official site: https://agda.readthedocs.io.
Question-23. What is Agda?
Answer-23: Agda is a dependently typed functional programming language and proof assistant.
Question-24. What makes Agda unique?
Answer-24: It features dependent types, allowing proofs and programs to be written together.
Question-25. What is the file extension for Agda source files?
Answer-25: .agda
Question-26. How do you declare a function in Agda?
Answer-26: f : Type followed by f = expression, e.g., add : Nat ? Nat ? Nat; add x y = x + y.
Question-27. What is a dependent type?
Answer-27: A type that depends on a value, enabling more precise type checking.
Question-28. How do you define a simple data type in Agda?
Answer-28: Use data, e.g., data Bool : Set where true : Bool; false : Bool.
Question-29. What is Set in Agda?
Answer-29: Set is the type of types, representing the universe of types.
Question-30. How do you define a recursive function in Agda?
Answer-30: Define a function that calls itself, e.g., fact : Nat ? Nat; fact 0 = 1; fact (suc n) = (suc n) * fact n.
Question-31. What is a proof in Agda?
Answer-31: A proof is a term that inhabits a type that represents a proposition.
Question-32. What is pattern matching in Agda?
Answer-32: It is a way to define functions by cases using constructors, e.g., not : Bool ? Bool; not true = false; not false = true.
Question-33. How does Agda handle infinite data structures?
Answer-33: It allows coinductive types using codata.
Question-34. What is the difference between Type and Set?
Answer-34: Set is used for types, while Type is not explicitly defined in Agda.
Question-35. What is an inductive type in Agda?
Answer-35: A type defined by cases, e.g., data Nat : Set where zero : Nat; suc : Nat ? Nat.
Question-36. How does Agda support theorem proving?
Answer-36: By writing dependent types that encode logical statements and constructing values to prove them.
Question-37. What is a sigma type in Agda?
Answer-37: A dependent pair type, written as ? A B or {x : A} ? B x.
Question-38. What is a pi type in Agda?
Answer-38: A dependent function type, written as (x : A) ? B x.
Question-39. What is implicit argument notation in Agda?
Answer-39: Arguments wrapped in curly braces {} are inferred automatically.
Question-40. How do you define a module in Agda?
Answer-40: Use module Name where, followed by definitions.
Question-41. How do you import a module in Agda?
Answer-41: Use open import ModuleName.
Question-42. What are universe levels in Agda?
Answer-42: Set is indexed as Set0, Set1, ... to prevent paradoxes.
Question-43. What is an Agda record?
Answer-43: A structure that groups related values, e.g., record Point : Set where field x y : Nat.
Question-44. What is termination checking in Agda?
Answer-44: Agda ensures recursive functions terminate to maintain soundness.
Question-45. What is totality in Agda?
Answer-45: Every function must be defined for all possible inputs without infinite recursion.
Question-46. How does Agda prevent infinite loops?
Answer-46: By enforcing termination checks on recursive functions.
Question-47. What is a parameterized data type in Agda?
Answer-47: A data type that takes parameters, e.g., data Vec (A : Set) : Nat ? Set where.
Question-48. How do you define equality in Agda?
Answer-48: Using data _?_ {A : Set} (x : A) : A ? Set where refl : x ? x.
Question-49. What is the Curry-Howard correspondence?
Answer-49: It relates types to propositions and programs to proofs.
Question-50. How do you prove an equality statement in Agda?
Answer-50: By constructing a value of type x ? y, usually using refl.
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