add_eq
plain-language theorem explainer
On the primitive-recognition integers (signed-orbit quotient classes), the infix sum coincides definitionally with the named addition operation. Anyone simplifying arithmetic on these integers will cite it as a simp bridge. The proof is reflexivity: the notation is just sugar for that operation.
Claim. For any primitive-recognition integers $a,b$ (signed-orbit quotient classes), the notation $a+b$ equals the underlying addition map $\mathrm{add}(a,b)$.
background
Primitive Recognition Calculus builds integers before classical $\mathbb{Z}$. A PRC integer is a quotient of signed orbits by the internal balanced-length relation; the map into ordinary $\mathbb{Z}$ is a later display theorem, not part of the type itself.
Addition is defined on those quotient classes (via orbit-level arithmetic imported from Orbit and OrbitArithmetic). Lean then equips the type with the usual + notation. This lemma records that the notation and the named operation are the same object, so simp and rewriting can move freely between them.
The sibling rational construction (nonzero-denominator ratio-orbit quotients) is parallel but not used here; the present statement is purely about the integer layer.
proof idea
One-line reflexivity proof. The instance that installs + on PRC integers sets it equal to the named add, so a + b = add a b holds by rfl with no further lemmas.
why it matters
Small infrastructure in the K4.8 integer layer of Primitive Recognition Calculus. Without a simp bridge between notation and the named operation, later arithmetic identities on signed-orbit integers become noisy. No downstream theorems currently list this declaration as a direct dependency; it is local hygiene for the integer API rather than a forcing-chain step (T0–T8) or a physical constant identity.
It does not touch the Recognition Composition Law, phi-ladder masses, or the eight-tick octave. Its role is to keep the pre-classical integer arithmetic readable while the quotient construction stays primary.
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