ofRat_add
plain-language theorem explainer
The embedding of PRC rationals into the classical real boundary preserves addition. Anyone assembling the K4.14 real-completion boundary certificate cites this as the additive homomorphism clause. The proof unfolds the embedding to the Mathlib cast of the rational display and rewrites by the already-proved addition law on that display.
Claim. For any two PRC rationals $a$ and $b$, the real-boundary embedding satisfies $\iota(a+b)=\iota(a)+\iota(b)$, where $\iota$ sends a PRC rational to its conservative rational display cast into $\mathbb{R}$.
background
PRC rationals (K4.8) are nonzero-denominator ratio-orbit quotient classes, identified by cross-multiplication. Each carries a conservative rational display into Mathlib's $\mathbb{Q}$.
The real boundary carrier is simply $\mathbb{R}$. The embedding (K4.14/A5) maps a PRC rational through that display and the standard cast $\mathbb{Q}\to\mathbb{R}$. Upstream, the display itself is already an additive homomorphism: the sum of two PRC rationals displays as the sum of their displays.
This module sits at the honest classical completion layer of the Primitive Recognition Calculus stack, above internal Cauchy and null-distance constructions, and tags the Mathlib reals as the available boundary rather than rebuilding them.
proof idea
Unfold the embedding definition so both sides become casts of rational displays into $\mathbb{R}$. Rewrite the left-hand display via the upstream addition law on the rational display, which equates the display of a sum with the sum of the displays. Finish with norm_num, which discharges equality of real casts of equal rationals. Short tactic proof, three steps, no case splits.
why it matters
This lemma is the additive clause of the K4.14 real-completion boundary certificate. That certificate packages existence of the classical real boundary, existence of the rational embedding, and the facts that the embedding preserves addition and multiplication. Downstream, the boundary certificate plugs this result directly into its preserves_add field.
Preserving ring operations under the rational embedding is the minimal interface that lets later modules treat the boundary as a genuine ordered field extension of the PRC rationals, without smuggling structure through axioms. Sibling lemmas handle multiplication, negation, and inversion; together they close the algebraic part of the K4.14/A5 boundary claim.
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