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formal note
Formal ideas / 2 min read
Theorem Proving
A comprehensive guide to automated and interactive theorem proving, tools, and methodologies.
proof / graph / notation
reading surface
Formal systems
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
Permalink to ApproachesAutomated Theorem Proving (ATP)
Permalink to 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)
Permalink to 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
Permalink to 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
Tools & Systems
Permalink to Tools & SystemsProof Assistants
Permalink to 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
Permalink to 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
Permalink to Advanced TechniquesE-graphs and Equality Saturation
Permalink to E-graphs and Equality Saturation- Efficient representation of equivalent expressions
- Used in program optimization and synthesis
- Tools and implementations:
- egg - Fast and flexible e-graph library
- egglog - Combines Datalog with equality saturation
- equality saturation - Foundational technique
Rewrite Systems
Permalink to Rewrite Systems- Term rewriting for equational reasoning
- Completion procedures
- Critical pair computation
- Termination analysis
Applications
Permalink to ApplicationsSoftware Verification
Permalink to Software Verification- Program correctness proofs
- Security protocol verification
- Compiler verification
- Operating system verification
Mathematics
Permalink to Mathematics- Formal mathematics
- Complex theorem verification
- Mathematical knowledge management
- Computer-checked proofs
Hardware Design
Permalink to Hardware Design- Circuit verification
- Hardware description languages
- Microprocessor verification
- Protocol verification
Learning Resources
Permalink to Learning ResourcesTutorials & Courses
Permalink to Tutorials & Courses- Software Foundations - Introduction to Coq
- Certified Programming with Dependent Types - Advanced Coq
- Theorem Proving in Lean - Lean tutorial
- Interactive Theorem Proving Course - Comprehensive course