PRP0011canonicalv1Membership Decidability
The tau-membership relation is decidable: given A, B in tau-Idx, we can computably determine whether A in_tau B.
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Membership Decidability
The tau-membership relation is decidable: given A, B in tau-Idx, we can computably determine whether A in_tau B.
Membership Decidability
Summary
The tau-membership relation is decidable: given A, B in tau-Idx, we can computably determine whether A in_tau B.
Statement
%
\label{prop:membership-decidable}
The $\tau$-membership relation is \textbf{decidable}:
given $\underline{a}, \underline{b} \in \tau\text{-Idx}$,
there is a computable procedure that determines
whether $\underline{a} \in_\tau \underline{b}$.
Proof / Justification
$\underline{a} \in_\tau \underline{b}$ iff $\underline{a} \mid \underline{b}$
iff $\underline{b} \bmod \underline{a} = \underline{0}$.
The modular reduction $\underline{b} \bmod \underline{a}$
is computable on $\tau$-Idx
(it requires only the Euclidean division algorithm,
which uses only the earned arithmetic of Part~III).
Source Context
- Registry source:
book-01.jsonlline 78 - Manuscript source:
2nd-edition/book-i-categorical-foundations/02_mainmatter/part08/ch32-membership-divisibility.texlines 168-175
Lean / Formalization Notes
- Formalization:
formalized - Module:
TauLib.BookI.Sets.Membership - Name:
Tau.Sets.instDecidableTauMem
Dependencies
- Canonical: I.D31, I.D19a
Related Results
Generated by later projection phases.
Related Publications
Generated by later projection phases.
Revision Notes
- 2026-04-24: Initial pilot migration.
Identifiers
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I.P10membership-decidabilityprop:membership-decidableRelease lines
corpus_v3_workingcorpus_v2Relations
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Appears in (1)
Downstream uses (computed) (2)
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