This book can be seen either as a text on theorem proving that uses techniques from general algebra, or else as a text on general algebra illustrated and made concrete by practical exercises in theorem proving. The book considers several different logical systems, including first-order logic, Horn clause logic, equational logic, and first-order logic with equality. Similarly, several different proof paradigms are considered. However, we do emphasize equational logic, and for simplicity we use only the OBJ3 software system, though it is used in a rather flexible manner. We do not pursue the lofty goal of mechanizing proofs like those of which mathematicians are justly so proud; instead, we seek to take steps towards providing mechanical assistance for proofs that are useful for computer scientists in developing software and hardware. This more modest goal has the advantage of both being achievable and having practical benefits. The following topics are covered: many-sorted signature, algebra and homomorphism; term algebra and substitution; equation and satisfaction; conditional equations; equational deduction and its completeness; deduction for conditional equations; the theorem of constants; interpretation and equivalence of theories; term rewriting, termination, confluence and normal form; abstract rewrite systems; standard models, abstract data types, initiality, and induction; rewriting and deduction modulo equations; first-order logic, models, and proof planning; second-order algebra; order-sorted algebra and rewriting; modules; unification and completion; and hidden algebra. In parallel with these are a gradual introduction to OBJ3, applications to group theory, various abstract data types (such as number systems, lists, and stacks), propositional calculus, hardware verification, the {\lambda}-calculus, correctness of functional programs, and other topics.

中文翻译：

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定理证明和代数

本书既可以看作是使用通用代数技术的关于定理证明的课本，也可以看作是通过定理证明中的实际练习说明并具体化的关于通用代数的课文。该书考虑了几种不同的逻辑系统，包括一阶逻辑，Horn子句逻辑，等式逻辑和具有相等性的一阶逻辑。类似地，考虑了几种不同的证明范例。但是，我们确实强调方程式逻辑，为简单起见，我们仅使用OBJ3软件系统，尽管使用方式相当灵活。我们不追求机械化证明的崇高目标，就像数学家为此感到骄傲的那样。取而代之的是，我们寻求采取措施，为证明提供机械帮助，这对计算机科学家在开发软件和硬件时很有用。这个更适度的目标具有既可以实现又具有实际好处的优点。涵盖以下主题：多种签名，代数和同态；术语代数和替换；方程式和满意度；条件方程；方程式推导及其完备性；条件方程式的推论；常数定理；理论的解释和对等；术语重写，终止，汇合和正常形式；抽象重写系统；标准模型，抽象数据类型，初始性和归纳；重写和推导模方程；一阶逻辑，模型和证明计划；二阶代数 排序排序的代数和重写；模块；统一与完成；和隐藏的代数。与此并行的是对OBJ3的逐步介绍，对群体理论的应用，