LemmaStatement: AI-adaptedProof: AI-generatedprecheck passverified 2026-08-02 (claude-opus-5)

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 integers embed in the rationals

Statement

The map $j (k) = [(k, 1)]$ is injective and preserves addition, multiplication, and order. Composing with The naturals embed in the integers embeds $N$ in $Q$ ; we write $k$ for $j (k)$ throughout.

Facts & Assumptions

Given: The map $j : Z \to Q$ , $j (k) = [(k, 1)]$ .

[L1]

Ring arithmetic in $Z$ (The integers form a commutative ring).

[L2]

The order on $Q$ (Order on the rationals), whose defining inequality is read in the order on $Z$ (Order on the integers).

[L3]

Equality in $Q$ : $[(a, b)] = [(c, d)]$ iff $a d = c b$ (The rationals as equivalence classes of pairs of integers).

Proof

technique · direct

1.1L3L1

Injectivity: $(k, 1) \sim (m, 1)$ means $k \cdot 1 = m \cdot 1$ , i.e. $k = m$ .

1.2L1

Addition: $j (k) + j (m) = [(k \cdot 1 + m \cdot 1, 1)] = j (k + m)$ .

1.3L1

Multiplication: $j (k) j (m) = [(k m, 1)] = j (k m)$ .

1.4L1L2

Order: denominators are $1 > 0$ , so $j (k) \leq j (m)$ reads $k \cdot 1 \leq m \cdot 1$ , i.e. $k \leq m$ .

2.1step 1.1step 1.2step 1.3step 1.4∎

$j$ embeds $Z$ into $Q$ , preserving arithmetic and order.

The integers embed in the rationals

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources