Alphabeta Math
LemmaStatement: AI-adaptedProof: AI-generatedSession-authored (Fable 5 assisted)precheck passjudge pass (openai/gpt-5.4)audited 2026-07-24
How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

  • Literature-sourced — the exact statement appears in a cited source; only wording and notation differ.
  • AI-adapted — a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
  • AI-generated — a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Rational arithmetic is well defined

Statement

Sum, product, and negation of Arithmetic on the rationals are independent of the chosen representatives.

Facts & Assumptions

Given: Pairs with (a,b)∼(a′,b′)(a,b) \sim (a',b') and (c,d)∼(c′,d′)(c,d) \sim (c',d') (The rationals as equivalence classes of pairs of integers), all second components nonzero.

[L1]

Z\mathbb{Z} is a commutative ring (The integers form a commutative ring).

Proof

technique · direct
1.1

By hypothesis ab′=a′bab' = a'b and cd′=c′dcd' = c'd.

given
2.1

Product: (ac)(b′d′)=(ab′)(cd′)=(a′b)(c′d)=(a′c′)(bd)(ac)(b'd') = (ab')(cd') = (a'b)(c'd) = (a'c')(bd), so (ac, bd)∼(a′c′, b′d′)(ac,\, bd) \sim (a'c',\, b'd').

step 1.1L1
2.2

Sum: (ad+cb)(b′d′)=(ab′)(dd′)+(cd′)(bb′)=(a′b)(dd′)+(c′d)(bb′)=(a′d′+c′b′)(bd)(ad + cb)(b'd') = (ab')(dd') + (cd')(bb') = (a'b)(dd') + (c'd)(bb') = (a'd' + c'b')(bd), so (ad+cb, bd)∼(a′d′+c′b′, b′d′)(ad+cb,\, bd) \sim (a'd'+c'b',\, b'd').

step 1.1L1
2.3

Negation: from ab′=a′bab' = a'b, (−a)b′=(−a′)b(-a)b' = (-a')b, so (−a,b)∼(−a′,b′)(-a, b) \sim (-a', b').

step 1.1L1
3.1

All three operations are independent of representatives.

step 2.1step 2.2step 2.3∎

Depends on

Used by

Cited to discharge well-definedness by Arithmetic on the rationals.

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 21 results over 11 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