TheoremStatement: Literature-sourcedProof: AI-adaptedprecheck 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.

The reals form a field

Statement

$R = C / N$ (The real numbers) is a field.

Facts & Assumptions

Given: Classes $[(a_{n})], [(b_{n})] \in R$ .

[L1]

$N$ is an ideal of $C$ (Null sequences form an ideal).

[L2]

$C$ is a commutative ring with $1$ (Cauchy sequences form a commutative ring).

[L3]

Maximality construction: for non-null $(a_{n})$ there is Cauchy $(b_{n})$ with $(a_{n} b_{n}) - 1$ null (The null ideal is maximal).

[L4]

The constant sequence $1$ is not a null sequence (its terms stay at $1$ ), so $N$ is a proper ideal and $\hat{1} \neq \hat{0}$ (Null sequence).

Proof

technique · direct

1.1L1L2

Operations on classes via representatives are well defined: for $z, w \in N$ , $((a + z) + (b + w)) - (a + b) = z + w \in N$ and $(a + z) (b + w) - a b = a w + z b + z w \in N$ , since ideals absorb products and sums.

2.1step 1.1L1L2L4

The ring axioms descend to the quotient because the operations are well defined and $C$ is a ring, verified on representatives; $\hat{1} \neq \hat{0}$ since the constant sequence $1$ is not null.

2.2step 1.1L3

Inverses: a nonzero class has a non-null representative $(a_{n})$ ; taking $(b_{n})$ from the maximality construction, $(a_{n}) (b_{n}) - 1 \in N$ , so $[(b_{n})]$ is a multiplicative inverse of $[(a_{n})]$ .

3.1step 2.1step 2.2∎

$R$ is a commutative ring with $1 \neq 0$ in which every nonzero element is invertible: a field.

The reals form a field

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources