Derivations for Proving General Equalities or Statements

For a 23-year-old engaged with 'Derivations for Proving General Equalities or Statements,' the ultimate developmental leverage lies in mastering formal verification systems. At this age, the individual is ready to move beyond foundational proof techniques to the rigorous construction and machine-checking of complex derivations, reflecting a deep engagement with logical and mathematical structures.

Our choice, the Lean 4 Formal Proof Assistant Environment (VS Code Integrated), directly addresses three core developmental principles for this stage:

1. Deepening Formal System Mastery: Lean 4 is a state-of-the-art interactive theorem prover rooted in dependent type theory. It allows for the construction of proofs with absolute logical certainty, forcing the user to understand every minute step of a derivation. This pushes beyond mere symbolic manipulation to a profound understanding of how proofs are built from first principles, fostering a mastery of formal systems crucial for advanced mathematics, computer science, and logic.
2. Bridging Theory and Application: Lean 4 is not just a theoretical tool; it's actively used in cutting-edge mathematical formalization (e.g., Mathlib, which formalizes vast tracts of undergraduate and graduate-level mathematics). By engaging with Lean, the 23-year-old applies abstract derivation skills to real-world (albeit academic/research) problems, connecting theoretical knowledge with practical formalization challenges.
3. Metacognitive Development & Problem-Solving Strategies: The strictness of a proof assistant like Lean 4 provides immediate, unambiguous feedback on the correctness of each step in a derivation. This process inherently encourages metacognition, requiring the user to reflect on their proof strategies, anticipate logical requirements, identify flaws, and refine their problem-solving approaches for constructing valid and elegant formal arguments.

Implementation Protocol for a 23-year-old:

1. Setup & Basic Tutorial (Weeks 1-2): Install VS Code and the Lean 4 extension. Work through the 'Theorem Proving in Lean 4' online book, focusing on the basics of dependent type theory, tactics, and simple propositional and predicate logic derivations.
2. Formalizing Elementary Statements (Weeks 3-6): Use Lean 4 to formalize and prove familiar general equalities from elementary algebra, set theory, or number theory (e.g., properties of addition/multiplication, set identities like De Morgan's Laws). Focus on constructing direct, step-by-step proofs.
3. Engaging with Mathlib (Weeks 7-12+): Explore Mathlib, Lean's vast mathematics library. Attempt to understand existing formalizations, and then try to contribute by formalizing small, unproven statements or proving lemmas within a chosen mathematical domain. This advanced engagement fosters collaborative learning and exposes the individual to sophisticated proof techniques and mathematical structures. Regular participation in the Lean community forums or Discord can provide invaluable peer support and guidance.