PropositionStatement: AI-adaptedProof: AI-generatedSession-authored (Fable 5 assisted)precheck 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.

When $ab < b$ for positive $a, b$

Statement

Let $F$ be an ordered field and let $a, b \in F$ with $a > 0$ and $b > 0$ . Then:

$0 < ab < b \iff 0 < a < 1.$

Facts & Assumptions

Given: Elements $a, b$ of an ordered field $F$ with $a > 0$ and $b > 0$ .

[L1]

If $x > 0$ and $y > 0$ then $xy > 0$ : the positive cone is closed under multiplication (Ordered field).

[L2]

$x < y \iff x - y < 0 \iff y - x > 0$ (Ordered field).

[L3]

For $b > 0$ : $bz < 0 \iff z < 0$ ; the sign of $bz$ equals the sign of $z$ (Sign rules for products and monotonicity of multiplication).

[L4]

By distributivity and $b \cdot (-1) = -b$ , one has $ab - b = b(a - 1)$ (Sign rules for products: $(-a)b = -(ab)$ and $(-a)(-b) = ab$ ).

Proof

technique · direct

1.1

Since $a > 0$ and $b > 0$ , [L1] gives $ab > 0$ .

1.2

By [L4], $ab - b = b(a - 1)$ .

2.1

By [L2], $ab < b \iff ab - b < 0$ , and by step 1.2 this is $b(a - 1) < 0$ .

L2step 1.2

2.2

By [L3] with $z = a - 1$ , we have $b(a - 1) < 0 \iff a - 1 < 0$ , and by [L2] this is $a < 1$ .

L3L2step 1.2

3.1

Chaining, $ab < b \iff a < 1$ ; since $a > 0$ is given and $ab > 0$ by step 1.1, this is exactly $0 < ab < b \iff 0 < a < 1$ .

givenstep 1.1step 2.1step 2.2∎

Depends on

Used by

sup(0,1) = 1 and inf(0,1) = 0, with neither attained Example

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 5 results over 4 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

W. Rudin, Principles of Mathematical Analysis, 3rd ed., Ch. 1 (standard reference, not scraped)
M. Spivak, Calculus, 4th ed., Ch. 1 (standard reference, not scraped)

When ab<bab < bab<b for positive a,ba, ba,b