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Formalizing Geometric Algebra in Lean
Advances in Applied Clifford Algebras ( IF 1.5 ) Pub Date : 2022-04-18 , DOI: 10.1007/s00006-021-01164-1
Eric Wieser , Utensil Song

This paper explores formalizing Geometric (or Clifford) algebras into the Lean 3 theorem prover, building upon the substantial body of work that is the Lean mathematics library, mathlib. As we use Lean source code to demonstrate many of our ideas, we include a brief introduction to the Lean language targeted at a reader with no prior experience with Lean or theorem provers in general. We formalize the multivectors as the quotient of the tensor algebra by a suitable relation, which provides the ring structure automatically, then go on to establish the universal property of the Clifford algebra. We show that this is quite different to the approach taken by existing formalizations of Geometric algebra in other theorem provers; most notably, our approach does not require a choice of basis. We go on to show how operations and structure such as involutions, versors, and the \(\mathbb {Z}_2\)-grading can be defined using the universal property alone, and how to recover an induction principle from the universal property suitable for proving statements about these definitions. We outline the steps needed to formalize the wedge product and \(\mathbb {N}\)-grading, and some of the gaps in mathlib that currently make this challenging.



中文翻译:

在精益中形式化几何代数

本文探讨了将几何(或 Clifford)代数形式化为精益 3 定理证明器,建立在精益数学库mathlib的大量工作之上. 当我们使用精益源代码来展示我们的许多想法时,我们会简要介绍精益语言,以针对一般没有精益或定理证明者经验的读者。我们通过适当的关系将多向量形式化为张量代数的商,自动提供环结构,然后继续建立克利福德代数的普遍性质。我们表明,这与其他定理证明中几何代数的现有形式化所采用的方法完全不同;最值得注意的是,我们的方法不需要选择基础。我们继续展示如何操作和结构,例如对合、versors 和\(\mathbb {Z}_2\)-分级可以单独使用通用属性来定义,以及如何从适合证明关于这些定义的陈述的通用属性中恢复归纳原理。我们概述了形式化楔积和\(\mathbb {N}\)分级所需的步骤,以及目前在mathlib中造成这一挑战的一些差距。

更新日期:2022-04-18
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