Alphabeta Math
LemmaStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck 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.

Cancellation for multiplication by a nonzero factor

Statement

For all m,n,k∈Nm,n,k\in\mathbb{N} with k≠0k\ne 0: if m⋅k=n⋅km\cdot k=n\cdot k then m=nm=n.

Facts & Assumptions

Given: multiplication defined by m⋅0=0m\cdot 0=0 and m⋅σ(j)=m⋅j+mm\cdot\sigma(j)=m\cdot j+m; the order m≤n  ⟺  ∃j (m+j=n)m\le n\iff\exists j\ (m+j=n), with m<nm<n meaning m≤nm\le n and m≠nm\ne n.

[L1]

Trichotomy: for all m,n∈Nm,n\in\mathbb{N} exactly one of m<nm<n, m=nm=n, m>nm>n holds (Trichotomy of the order on N\mathbb{N}).

[L2]

Strict monotonicity: if k≠0k\ne 0 and m<nm<n then m⋅k<n⋅km\cdot k<n\cdot k (Order is compatible with multiplication).

Proof

technique · contradiction
1.1

Suppose m⋅k=n⋅km\cdot k=n\cdot k with k≠0k\ne 0 but m≠nm\ne n.

assume-contra
2.1

By trichotomy [L1], m≠nm\ne n forces m<nm<n or n<mn<m.

step 1.1L1
3.1

If m<nm<n then m⋅k<n⋅km\cdot k<n\cdot k by [L2] (since k≠0k\ne 0); if n<mn<m then n⋅k<m⋅kn\cdot k<m\cdot k by [L2]; either way m⋅k≠n⋅km\cdot k\ne n\cdot k.

step 2.1L2
4.1

This contradicts m⋅k=n⋅km\cdot k=n\cdot k, so the assumption m≠nm\ne n is impossible and m=nm=n.

step 1.1step 3.1discharge-contradiction∎

Depends on

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 27 results over 13 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