toComplex_add
plain-language theorem explainer
The transport from recovered complex numbers to Mathlib complexes preserves addition. Workers establishing equivalences between logic-derived carriers and standard analysis cite this when moving sums across the map. The proof is a one-line simp wrapper that unfolds the addition definitions on pairs.
Claim. Let $z, w$ be recovered complex numbers. The transport map $toComplex$ satisfies $toComplex(z + w) = toComplex(z) + toComplex(w)$, where $toComplex$ sends a pair of recovered reals to the corresponding element of $mathbb{C}$ by applying the real transport to each component.
background
LogicComplex is the structure of pairs of recovered real numbers, with fields re and im. The function toComplex converts such a pair to Mathlib's complex line by applying toReal to each component. This module builds the carrier for complex numbers over the recovered reals and proves its equivalence to standard complexes without redeveloping analysis.
proof idea
This is a one-line simp wrapper that applies the simplifier to HAdd.hAdd and Add.add, unfolding the addition operation on both the recovered and standard sides.
why it matters
The result confirms that the transport respects addition, a prerequisite for using the equivalence in algebraic settings. It supports the construction of complex numbers from recovered reals in the Recognition Science framework, where such carriers enable later transport of operations. No downstream uses are recorded, so it functions as a basic compatibility property.
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