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LemmaStatement: Literature-sourcedProof: AI-adaptedSession-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.

Addition is associative

Statement

For all a,b,c∈Na, b, c \in \mathbb{N}: (a+b)+c=a+(b+c)(a + b) + c = a + (b + c).

Facts & Assumptions

Given: Addition with m+0=mm + 0 = m and m+σ(n)=σ(m+n)m + \sigma(n) = \sigma(m + n) (Addition of natural numbers).

[L1]

Induction principle (The principle of mathematical induction).

Proof

technique · induction
1.1

Base c=0c = 0: (a+b)+0=a+b(a + b) + 0 = a + b and a+(b+0)=a+ba + (b + 0) = a + b, so they are equal.

base
1.2

Inductive hypothesis: (a+b)+c=a+(b+c)(a + b) + c = a + (b + c).

ih
2.1

Step: (a+b)+σ(c)=σ((a+b)+c)=σ(a+(b+c))=a+σ(b+c)=a+(b+σ(c))(a + b) + \sigma(c) = \sigma((a + b) + c) = \sigma(a + (b + c)) = a + \sigma(b + c) = a + (b + \sigma(c)), using m+σ(n)=σ(m+n)m + \sigma(n) = \sigma(m + n) twice and the hypothesis.

step 1.2
3.1

By induction [L1], (a+b)+c=a+(b+c)(a + b) + c = a + (b + c) for all cc, hence for all a,b,c∈Na, b, c \in \mathbb{N}.

step 1.1step 2.1discharge-induction∎

Depends on

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 15 results over 9 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