ofRatCore
plain-language theorem explainer
ofRatCore coerces a standard rational into the recovered real line by wrapping the Bourbaki completion map. Researchers reconstructing the continuum from logic axioms cite it when lifting rational data into LogicReal. The definition is a one-line wrapper that applies the completion coefficient to the input rational cast as CompareReals.Q.
Claim. For a rational number $q$, the map sends $q$ to the element of the logic real line obtained by applying the Bourbaki completion coefficient to the rational viewed in the uniform space of rationals.
background
The module recovers the real numbers from the Law-of-Logic rational layer. LogicReal is the Cauchy completion of the recovered rationals, realized through the canonical completion of ℚ and the equivalence LogicRat ≃ ℚ. The wrapper prevents global instance pollution on Completion ℚ while still letting us reuse Mathlib's completed real line. This follows the from theorem in PrimitiveDistinction, which derives four structural conditions plus three definitional facts from seven independent axioms, and the Completion structure that packages integer counts for the kernel types.
proof idea
The definition is a one-line wrapper that applies Completion.coe' to the rational cast as CompareReals.Q and packages the result inside the LogicReal structure.
why it matters
This declaration supplies the core embedding used by ofLogicRat to lift recovered rationals and by toReal_ofRatCore to establish agreement with the standard real line. It fills the step from LogicRat ≃ ℚ to Completion ℚ ≃ ℝ in the recovery chain described in the module documentation. The construction supports the transport-first API in which algebra and order on LogicReal are defined by pulling back along toReal.
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