Alphabeta Math
LemmaStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-11
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  • AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
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Coefficientwise sums and convolution products of finitely supported sequences are finitely supported

Statement

If a,b ⁣:N→Ra,b\colon\mathbb N\to R have finite support, then their coefficientwise sum and convolution product have finite support.

Facts & Assumptions

Given: A commutative ring RR and finitely supported coefficient sequences a,b∈R[x]a,b\in R[x].

[L1]

A coefficient sequence has finite support when it vanishes beyond some natural-number bound; addition is coefficientwise and multiplication is convolution (The polynomial ring over a commutative ring as finitely supported coefficient sequences with convolution).

Proof

technique · direct
1.1

Choose M,N∈NM,N\in\mathbb N such that ai=0a_i=0 for i>Mi>M and bj=0b_j=0 for j>Nj>N.

givenL1choose
2.1

If k>max⁡{M,N}k>\max\{M,N\} then (a+b)k=0(a+b)_k=0, and if k>M+Nk>M+N then every pair i+j=ki+j=k has i>Mi>M or j>Nj>N, so every summand aibja_i b_j in (ab)k(ab)_k is zero; hence both sequences have finite support.

step 1.1L1algebra∎

Depends on

Used by

Cited to discharge well-definedness by The polynomial ring over a commutative ring as finitely supported coefficient sequences with convolution.

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 11 results over 6 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources