CorollaryStatement: Literature-sourcedProof: AI-adaptedprecheck passjudge pass (openai/gpt-5.4)audited 2026-07-25

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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 reverse triangle inequality

Statement

Let $F$ be an ordered field (Ordered field) and let $x, y \in F$ . Then

$∣ ∣ x ∣ - ∣ y ∣ ∣ \leq ∣ x - y ∣ .$

Facts & Assumptions

Given: An ordered field $F$ and elements $x, y \in F$ .

[L1]

Triangle inequality: $∣ a + b ∣ \leq ∣ a ∣ + ∣ b ∣$ for all $a, b \in F$ (The triangle inequality).

[L2]

For every $u \in F$ , $∣ - u ∣ = ∣ u ∣$ , and $∣ w ∣$ equals $w$ or $- w$ (Basic properties of the absolute value).

Proof

technique · direct

1.1L1

Applying [L1] with $a = x - y$ and $b = y$ gives $∣ x ∣ = ∣ (x - y) + y ∣ \leq ∣ x - y ∣ + ∣ y ∣$ , so $∣ x ∣ - ∣ y ∣ \leq ∣ x - y ∣$ .

1.2L1L2

Applying [L1] with $a = y - x$ and $b = x$ , and using $∣ y - x ∣ = ∣ x - y ∣$ from [L2], gives $∣ y ∣ \leq ∣ y - x ∣ + ∣ x ∣ = ∣ x - y ∣ + ∣ x ∣$ , so $∣ y ∣ - ∣ x ∣ \leq ∣ x - y ∣$ .

2.1step 1.1step 1.2L2algebra∎

By [L2] the value $∣ ∣ x ∣ - ∣ y ∣ ∣$ equals $∣ x ∣ - ∣ y ∣$ or $- (∣ x ∣ - ∣ y ∣) = ∣ y ∣ - ∣ x ∣$ , and each of these is $\leq ∣ x - y ∣$ by steps 1.1 and 1.2, so $∣ ∣ x ∣ - ∣ y ∣ ∣ \leq ∣ x - y ∣$ .

The reverse triangle inequality

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources