toReal_piL
plain-language theorem explainer
The transport lemma equates the recovered constant piL on LogicReal to Mathlib's Real.pi under the toReal map. Analysts working inside the Recognition Science foundation cite it when reducing identities involving π to standard real analysis. The proof is a direct one-line wrapper applying the general toReal_fromReal transport theorem.
Claim. Let $pi_L$ denote the element of LogicReal obtained by transporting the standard real $pi$ via fromReal. The transport map satisfies $toReal(pi_L) = pi$.
background
The module defines transcendental functions on the recovered real line LogicReal by transport through the equivalence to Mathlib's ℝ. The constant piL is introduced as fromReal Real.pi. The map toReal sends a LogicReal value x to CompareReals.compareEquiv x.val, serving as the inverse transport that recovers the standard real line.
proof idea
The proof is a one-line wrapper that applies the theorem toReal_fromReal to the definition of piL.
why it matters
This lemma completes the transport for the constant π inside the LogicRealTranscendentals module. It supports the program of recovering standard analysis on the LogicReal line defined from logic primitives, ensuring identities involving π reduce to the standard real line without leaving the foundation layer.
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