Theorem Proving
Theorem proving is the use of mathematical logic and computational methods to verify mathematical statements and solve complex problems. It combines formal logic, mathematics, and computer science to create rigorous proofs of mathematical theorems.
Approaches
Automated Theorem Proving (ATP)
- Fully automated systems that find proofs without human intervention
- Uses resolution, tableaux methods, and term rewriting
- Suitable for well-defined problem domains
- Examples: Vampire, E Theorem Prover, Z3
Interactive Theorem Proving (ITP)
- Human-guided proof development
- Provides mechanized verification of each step
- Allows complex mathematical reasoning
- Examples: Coq, Isabelle/HOL, Lean
SAT/SMT Solving
- Boolean satisfiability (SAT) and Satisfiability Modulo Theories (SMT)
- Efficient automated reasoning for specific domains
- Used in software verification and hardware design
- Examples: Z3, CVC4, Yices
Proof Assistants
- Coq - Formal proof management system
- Isabelle - Generic proof assistant
- Lean - Modern theorem prover and programming language
- Agda - Dependently typed functional language
- HOL Light - Interactive theorem prover
Automated Provers
- Vampire - State-of-the-art ATP system
- E Prover - Equational theorem prover
- Z3 - Microsoft's SMT solver
- CVC4 - SMT solver for multiple theories
Advanced Techniques
E-graphs and Equality Saturation
- Efficient representation of equivalent expressions
- Used in program optimization and synthesis
- Tools and implementations:
Rewrite Systems
- Term rewriting for equational reasoning
- Completion procedures
- Critical pair computation
- Termination analysis
Applications
Software Verification
- Program correctness proofs
- Security protocol verification
- Compiler verification
- Operating system verification
Mathematics
- Formal mathematics
- Complex theorem verification
- Mathematical knowledge management
- Computer-checked proofs
Hardware Design
- Circuit verification
- Hardware description languages
- Microprocessor verification
- Protocol verification
Learning Resources
Tutorials & Courses
Communities
Research Papers