Alphabeta MathHow 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.
Multiplication is associative
Statement
For all : .
Facts & Assumptions
Given: multiplication and addition (Multiplication of natural numbers, Addition of natural numbers).
Left distributivity (Distributivity and the successor law for multiplication).
The induction principle (The principle of mathematical induction).
Proof
Base : and , so the two sides are equal.
Inductive hypothesis: .
Step: , using the multiplication recursion, the hypothesis, left distributivity [L1], and .
By induction [L2], for all , hence for all .
Depends on
Used by
- For K≤ H≤ G with G finite, [G:K]=[G:H][H:K] Corollary
- Exponentiation of natural numbers, mⁿ, and its agreement with the integer power in ℝ Definition
- The factorial n! and the falling factorial n^k̲, defined by recursion in ℕ Definition
- A finite sum is unchanged by a permutation of its index range: ∑_k<n a_π(k) = ∑_k<n aₖ for every bijection π : n → n Lemma
- Laws of finite sums and products in ℕ, and ι(∑_k<n aₖ) = ∑_k<n ι(aₖ) Lemma
- C(n, k) k! (n-k)! = n! for k ≤ n; hence C(n, k) k! = n^k̲, the quotient n!/(k!(n-k)!) is a natural number, and C(n, k) = C(n, n-k) Theorem
- ℕ × ℕ ≈ ℕ Theorem
- The integers form a commutative ring Theorem
- The integers form a totally ordered ring Theorem
- The multinomial coefficient equals n!/∏_i<m kᵢ!, and (x₀+…+xₘ₋₁)ⁿ = ∑ iotaC(n, k)∏_i<m xᵢ^kᵢ in ℝ Theorem
Dependency tree · two levels
12 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.
Sources
- T. Tao, Analysis I, 3rd ed., §2.1-2.3 (Peano axioms, recursion, arithmetic) (standard reference, not scraped)
- Peano axioms (Wikipedia) (standard reference, not scraped)
- W. Aitken, MATH 378 Ch. 1: The Peano Axioms (CSU San Marcos) (standard reference, not scraped)
- Mathematics 144: Set Theory (UC Riverside lecture notes) (standard reference, not scraped)