LogicInt
plain-language theorem explainer
LogicInt re-exports the Grothendieck completion of logic naturals under addition to bridge logical integers with the phase-budget surface in number theory. Researchers modeling integer operations from logic foundations for use in finite phase arguments would cite this alias. The declaration is realized as a direct one-line abbreviation with no independent proof content.
Claim. Define the logic integers as the Grothendieck completion of the logic natural numbers under addition, constructed as the quotient of pairs of logic naturals by the equivalence relation identifying $(a,b)$ with $(c,d)$ when $a + d = b + c$.
background
The module transfers the recovered integer ledger from logic to the existing integer-ledger phase-budget surface in number theory. The integers recovered from logic are the Grothendieck completion of the naturals recovered from logic under addition. This is realized as a quotient of pairs of such naturals by the standard equivalence making the structure into an abelian group. The declaration depends on the upstream definition in the integers-from-logic module, which supplies the quotient construction and the ring operations used downstream.
proof idea
This is a one-line wrapper that aliases the upstream definition of the Grothendieck completion of logic naturals from the integers-from-logic module.
why it matters
The alias supports the transfer of integer operations into number theoretic contexts, feeding the downstream proofs of ring properties including associativity, commutativity, and distributivity on the logic integers via reduction to the standard integers using the transfer principle. It fills the role of making the logic-derived integers available on the phase-budget surface, consistent with the framework's recovery of number theory from logical foundations. The workhorse transfer principle states that an equation in these integers holds if and only if it holds under the map to the standard integers.
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