LNAI 2741 Proof Search and Proof Check for Equational and Inductive Theorems 1st Edition by Eric Deplagne, Claude Kirchner, Hélène Kirchner, Quang Huy Nguyen ISBN 9783540450856

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ISBN 10: 
ISBN 13: 9783540450856 
Author: Eric Deplagne, Claude Kirchner, Hélène Kirchner, Quang Huy Nguyen

This paper presents on-going researches on theoretical and practical issues of combining rewriting based automated theorem proving and user-guided proof development, with the strong constraint of safe cooperation of both. In practice, we instantiate the theoretical study on the Coq proof assistant and the ELAN rewriting based system, focusing first on equational and then on inductive proofs. Different concepts, especially rewriting calculus and deduction modulo, contribute to define and to relate proof search, proof representation and proof check.

LNAI 2741 Proof Search and Proof Check for Equational and Inductive Theorems 1st Edition Table of contents:

Chapter 1: Introduction to Automated Proof Search

• What is Automated Proof Search?
• Role of Proof Search in Logic and Mathematics
• Types of Proofs: Direct, Indirect, Constructive, and Non-Constructive
• Proof Search in the Context of Equational and Inductive Theorems

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Chapter 2: Equational Theorems

• Basics of Equational Logic
• Solving Equational Theorems: Methods and Strategies
• Rewriting Systems and Termination Proofs
• Case Studies in Equational Theorem Proving

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Chapter 3: Inductive Theorems

• What Are Inductive Theorems?
• Inductive Reasoning in Mathematical Proofs
• Mathematical Induction: Techniques and Challenges
• Inductive Proof Search Algorithms
• Case Studies in Inductive Theorem Proving

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Chapter 4: Proof Search in Equational Logic

• Overview of Algorithms for Equational Proof Search
• Techniques for Equational Unification
• Proof Search Using Rewriting Systems
• Examples and Case Studies in Equational Proof Search

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Chapter 5: Proof Checking and Verification

• What is Proof Checking?
• Proof Checking Algorithms and Tools
• The Role of Proof Checking in Automated Theorem Proving
• Verifying Equational and Inductive Proofs

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Chapter 6: Proof Search for Inductive Theorems

• Special Techniques for Inductive Proof Search
• Structural Induction and Recursive Definitions
• Optimization of Inductive Proof Search Algorithms
• Examples in Inductive Theorem Proving

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Chapter 7: Proof Search Strategies and Techniques

• Depth-First vs. Breadth-First Search
• Backtracking and Branch-and-Bound Methods
• Search Space Pruning and Heuristics
• Comparing Different Proof Search Strategies
• Integrating Proof Search with Machine Learning

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Chapter 8: Tools and Frameworks for Proof Search and Checking

• Overview of Automated Theorem Proving Tools
• Using Theorem Proving Systems for Equational and Inductive Proofs
• Proof Assistants and Proof Checkers in Practice
• Case Studies: HOL, Coq, and Other Proof Tools

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Chapter 9: Theoretical Aspects of Proof Search

• Computational Complexity of Proof Search
• Soundness and Completeness in Automated Proofs
• Logical Foundations of Proof Search
• Challenges in Scaling Proof Search for Large Theorems

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Chapter 10: Applications of Proof Search and Proof Checking

• Proof Search in Software Verification
• Application in Hardware Design and Verification
• Formal Methods and Their Role in Software Engineering
• Applications to Cryptography and Security Protocols

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Chapter 11: Future Directions and Open Problems

• Advances in Automated Proof Search and Checking
• Integrating Proof Search with Artificial Intelligence
• Challenges in Proving Large-Scale Theorems
• Future Research in Inductive and Equational Theorem Proving

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