Equational reasoning arises in many areas of mathematics and computer science. It is a cornerstone of algebraic reasoning and results essential in tasks of specification and verification in functional programming, where a program is mainly a set of equations. The usual manipulation of identities while conducting informal proofs obviates many intermediate steps that are neccesary while developing them using a formal system, such as the equationally complete Birkhoff calculus |${\mathcal{B}}$|⁠. This deductive system does not fit in the common manner of doing mathematical proofs, and it is not compatible with the mechanisms of proof assistants. The aim of this work is to provide a deductive system |${\mathcal{B}}^{\textrm{GOAL}}$| for equality, equivalent to |${\mathcal{B}}$| but suitable for constructing equational proofs in a backward fashion. This feature makes it adequate for interactive proof-search in the approach of proof assistants. This will be achieved by turning |${\mathcal{B}}^{\textrm{GOAL}}$| into a transition system of formal tactics in the style of Edinburgh LCF, such transformation allows us to give a direct formal definition of backward proof in equational logic.

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方程式推理的交互式证明搜索

方程式推理出现在数学和计算机科学的许多领域。它是代数推理的基石，并且是函数式编程中规范和验证任务必不可少的结果，其中程序主要是一组方程。在进行非正式证明时对身份的通常操纵消除了使用正式系统开发身份时必须采取的许多中间步骤，例如方程式上完整的Birkhoff微积分| $ {\ mathcal {B}} $ |⁠。这种演绎系统不适合进行数学证明的通用方式，并且与证明助手的机制不兼容。这项工作的目的是提供一个演绎系统| $ {\ mathcal {B}} ^ {\ textrm {GOAL}} $ | 为了平等，等于| $ {\ mathcal {B}} $ | 但适用于以反向方式构造方程式证明。此功能使其足以用于以证明助手的方式进行交互式证明搜索。这可以通过| $ {\ mathcal {B}} ^ {\ textrm {GOAL}} $ |来实现。转变为爱丁堡LCF风格的形式策略过渡系统，这种转变使我们能够在方程逻辑中给出对后向证明的直接形式定义。