fromReal
fromReal supplies the forward transport that embeds any Mathlib real into the recovered real line LogicReal built from logic-derived rationals. Foundation researchers cite it whenever they need to move a standard analysis result into the Recognition Science setting for later transport back via toReal. The definition is a one-line wrapper that invokes the symmetric comparison equivalence on the Bourbaki completion.
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background
LogicReal is the wrapper structure around Mathlib's Bourbaki completion of ℚ, using the equivalence LogicRat ≃ ℚ recovered earlier in RationalsFromLogic. The wrapper isolates the completion so that algebraic instances on the standard reals are not globally polluted while still allowing reuse of Mathlib's completed theorems. The module RealsFromLogic therefore adopts a transport-first design: toReal and fromReal move data between the recovered line and the standard line, after which order and field operations are pulled back along toReal.
proof idea
The definition is a one-line wrapper that applies the symmetric form of CompareReals.compareEquiv to the input real, then packages the result inside the LogicReal constructor.
why it matters in Recognition Science
fromReal completes the bidirectional transport pair in the reals-recovery layer and is invoked by every transcendental definition (expL, logL, coshL, cosL, piL) as well as by fromComplex. It therefore lets the Recognition framework import standard real analysis results without re-proving them inside the logic-derived setting. The step sits between the LogicRat ≃ ℚ equivalence and the later phi-ladder constructions that rely on real-valued functions.
scope and limits
- Does not define addition or multiplication on LogicReal.
- Does not prove any continuity or measurability property of the map.
- Does not reference the J-cost function or the forcing chain T0-T8.
Lean usage
def fromComplex (z : ℂ) : LogicComplex where re := fromReal z.re im := fromReal z.imformal statement (Lean)
76noncomputable def fromReal (x : ℝ) : LogicReal :=
proof body
Definition body.
77 ⟨CompareReals.compareEquiv.symm x⟩
78