We describe formalization of the Poincaré disc model of hyperbolic geometry within the Isabelle/HOL proof assistant. The model is defined within the complex projective line $$\mathbb {C}{}P^1$$ C P 1 and is shown to satisfy Tarski’s axioms except for Euclid’s axiom—it is shown to satisfy it’s negation, and, moreover, to satisfy the existence of limiting parallels axiom.

中文翻译：

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双曲几何庞加莱圆盘模型的形式化

我们描述了 Isabelle/HOL 证明助手中双曲几何的庞加莱圆盘模型的形式化。该模型在复射影线 $$\mathbb {C}{}P^1$$ CP 1 内定义，并证明满足塔斯基公理，但欧几里德公理除外——证明它满足它的否定，此外，满足极限平行公理的存在性。