A Comprehensive Formalization of Propositional Logic in Coq: Deduction Systems, Meta-Theorems, and Automation Tactics

The increasing significance of theorem proving-based formalization in mathematics and computer science highlights the necessity for formalizing foundational mathematical theories. In this work, we employ Coq interactive prover to methodically formalize language, semantics, syntax propositional logic, a fundamental aspect reasoning proof construction. We construct four Hilbert-style axiom systems natural deduction system establish their equivalences through meticulous proofs. Moreover, provide formal proofs essential meta-theorems including Deduction Theorem, Soundness Completeness Compactness Theorem. Importantly, present an exhaustive Theorem paper. To bolster also concepts related mappings countability, deliver Cantor–Bernstein–Schröder theorem. Additionally, devise automated tactics explicitly designed logic inference delineated study, enabling automatic verification all tautologies, internal theorems, majority syntactic semantic inferences within system. This research contributes versatile reusable library presenting solid foundation numerous applications mathematics, such as accurate expression properties software programs digital circuits. work holds particular importance domains formalization, hardware security, enhancing comprehension principles logical reasoning.