THM0018canonicalv1No Additive Inverse
For n > 0, no m in tau-Idx satisfies n + m = 0. The absence of additive inverses is structural: rho generates positive depths only.
Payload
No Additive Inverse
For n > 0, no m in tau-Idx satisfies n + m = 0. The absence of additive inverses is structural: rho generates positive depths only.
No Additive Inverse
Summary
For n > 0, no m in tau-Idx satisfies n + m = 0. The absence of additive inverses is structural: rho generates positive depths only.
Statement
%
\label{thm:no-additive-inverse}
For any $\underline{n} > \underline{0}$ in $\tau$-Idx,
there exists no $\underline{m} \in \tau$-Idx such that
$\underline{n} + \underline{m} = \underline{0}$.
Proof / Justification
Immediate from Proposition~\ref{prop:sum-zero-iff}:
if $\underline{n} + \underline{m} = \underline{0}$,
then $\underline{n} = \underline{0}$,
contradicting $\underline{n} > \underline{0}$.
Source Context
- Registry source:
book-01.jsonlline 97 - Manuscript source:
2nd-edition/book-i-categorical-foundations/02_mainmatter/part03/ch11-swap-add-mul.texlines 285-290
Lean / Formalization Notes
- Formalization:
formalized - Module:
TauLib.BookI.Denotation.Structural - Name:
Tau.Denotation.tauIdx_no_additive_inverse
Dependencies
- Canonical: I.D10, I.P15
Related Results
Generated by later projection phases.
Related Publications
Generated by later projection phases.
Revision Notes
- 2026-04-24: Initial pilot migration.
Identifiers
Aliases & legacy IDs
I.T14no-additive-inversethm:no-additive-inverseRelease lines
corpus_v3_workingcorpus_v2Relations
Appears in (1)
Sources
Version & History
Status disclaimer
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