We provide a fine-grained analysis on the relation between König's lemma, weak König's lemma, and the decidable fan theorem in the context of constructive reverse mathematics. In particular, we show that double negated variants of König's lemma and weak König's lemma are equivalent to double negated variants of the general decidable fan theorem and the binary decidable fan theorem, respectively, over a nearly intuitionistic system containing a weak countable choice only. This implies that the general decidable fan theorem is not equivalent to the binary decidable fan theorem in the absence of a strong countable choice. In addition, we introduce a principle which fills the gap between König's lemma and weak König's lemma, and show that König's lemma is equivalent to weak König's lemma plus this principle. We also solve several problems arising from the investigation.

中文翻译：

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König 引理、弱 König 引理和可判定扇定理

我们在构造逆向数学的背景下对 König 引理、弱 König 引理和可判定扇定理之间的关系进行了细粒度分析。特别是，我们表明，在仅包含弱可数选择的近直觉系统上，柯尼希引理和弱柯尼希引理的双重否定变体分别等效于一般可判定扇形定理和二元可判定扇形定理的双重否定变体。这意味着在没有强可数选择的情况下，一般可判定扇形定理不等同于二元可判定扇形定理。此外，我们引入了一个原理来填补 König 引理和弱 König 引理之间的空白，并证明 König 引理等价于弱 König 引理加上这个原理。