Type-checkers are used to verify certain attributes of programs are correct. Avoiding bugs in type-checkers is especially important when accepting faulty programs has serious real-world consequences. Correct-by-construction programming aims to prevent such bugs by embedding proofs of a program's correctness within the program itself. However, this approach introduces different challenges, such as added complexity. Whether the benefits of correct-by-construction type-checkers outweigh these challenges for more complex language features remains uncertain. This paper investigates correct-by-construction type-checking for a toy language with polymorphic algebraic data types and pattern matching. We do this by implementing a type-checker in the dependently typed language Agda. We show that this approach guarantees a type-checker that does not accept ill-typed terms. Furthermore, we reflect on the challenges of this approach and argue that this approach should be used when a guarantee of correctness is required.