Inductive and coinductive structures are everywhere in mathematics and computer science. The induction principle is well known and fully exploited to reason about inductive structures like natural numbers and finite lists. To prove theorems about coinductive structures such as infinite streams and infinite trees we can appeal to bisimulation or the coinduction principle. Pure inductive and coinductive types however are not the only data structures we are interested to reason about. In this paper we present a calculus to prove theorems about mutually defined inductive and coinductive data types. Derivations are carried out in an infinitary sequent calculus for first order intuitionistic multiplicative additive linear logic with fixed points. We enforce a condition on these derivations to ensure their cut elimination property and thus validity. Our calculus is designed to reason about linear properties but we also allow appealing to first order theories such as arithmetic, by adding an adjoint downgrade modality. We show the strength of our calculus by proving several theorems on (mutual) inductive and coinductive data types.

中文翻译：

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具有最小和最大不动点的一阶线性逻辑中的循环证明

归纳和共归纳结构在数学和计算机科学中无处不在。归纳原理是众所周知的，并被充分利用来推理自然数和有限列表等归纳结构。为了证明关于无限流和无限树等共导结构的定理，我们可以求助于互模拟或共导原理。然而，纯归纳和共归纳类型并不是我们感兴趣的唯一数据结构。在本文中，我们提出了一个微积分来证明关于相互定义的归纳和共同归纳数据类型的定理。推导是在具有不动点的一阶直觉乘法加法线性逻辑的无限连续演算中进行的。我们对这些推导执行一个条件，以确保它们的切割消除特性和有效性。我们的微积分旨在推理线性属性，但我们也允许通过添加伴随降级模态来吸引一阶理论，例如算术。我们通过证明（互）归纳和共同归纳数据类型的几个定理来展示我们微积分的力量。