LemmaStatement: AI-adaptedProof: AI-generatedprecheck passjudge pass (openai/gpt-5.4)audited 2026-07-25

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.

Order is preserved by adding a constant and by adding inequalities

Let $F$ be an ordered field (Ordered field) with positive cone $P$ , and let $a, b, c, d \in F$ .

Given: An ordered field $F$ with positive cone $P$ , and elements $a, b, c, d \in F$ .

[L1]

For $x, y \in F$ , the relation $x < y$ means $y - x \in P$ (Ordered field).

[L2]

$P$ is closed under addition: if $u, v \in P$ then $u + v \in P$ (axiom O2 of Ordered field).

technique · direct

1.1assume-hypL1

Assume $a < b$ ; by the definition of the order this means $b - a \in P$ .

1.2algebra

For every $c \in F$ the field identities give $(b + c) - (a + c) = b - a$ .

1.3assume-hypL1

Assume moreover $c < d$ ; by the definition of the order this means $d - c \in P$ .

1.4algebra

The field identities give $(b + d) - (a + c) = (b - a) + (d - c)$ .

2.1step 1.1step 1.2L1

Hence $(b + c) - (a + c) = b - a \in P$ , which is exactly $a + c < b + c$ , proving claim 1.

2.2step 1.1step 1.3L2

Since $b - a \in P$ and $d - c \in P$ , closure under addition gives $(b - a) + (d - c) \in P$ .

3.1step 1.4step 2.2L1∎

Therefore $(b + d) - (a + c) = (b - a) + (d - c) \in P$ , which is exactly $a + c < b + d$ , proving claim 2.

Depends on