LeanAide documentation

LeanAide GPT-4 documentation

`Rat.uniformEmbedding_coe_real` theorem Rat.uniformEmbedding_coe_real : UniformEmbedding Rat.cast The theorem `Rat.uniformEmbedding_coe_real` states that the function `Rat.cast`, which serves as the canonical homomorphism from the rational numbers (`Rat`) to a division ring `K`, is a uniform embedding. In other words, the mapping from the set of rational numbers to the division ring `K` preserves the uniform structure. This means that nearness of points in the domain (the set of rational numbers) is properly reflected in the nearness of their images in the codomain (the division ring `K`). More concisely: The `Rat.cast` function from `Rat` to a division ring `K` is a uniform embedding, preserving the uniform structure and ensuring nearness of rational numbers is reflected by nearness of their images in `K`.
`Rat.denseEmbedding_coe_real` theorem Rat.denseEmbedding_coe_real : DenseEmbedding Rat.cast The theorem `Rat.denseEmbedding_coe_real` asserts that the canonical homomorphism from the rational numbers (`Rat`) into a division ring `K` is a dense embedding. This essentially means that for any two elements in `K`, there exists a rational number that lies between them, signifying the dense property of rational numbers in any division ring. The proof of the theorem is suppressed in this context. More concisely: The canonical homomorphism from the rational numbers into any division ring is a dense embedding.