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.
Order is compatible with addition
Statement
For all : ; and (Order on the natural numbers).
Facts & Assumptions
Given: The order (Order on the natural numbers) and addition with , (Addition of natural numbers).
Addition is associative and commutative (Addition is associative, Addition is commutative).
Cancellation: (Addition is cancellative).
Proof
Forward: if , say , then , so .
Backward: if , say , then , so by cancellation, giving .
The strict forms follow: means and , while by cancellation, so .
Depends on
Used by
- A progressive map with no fixed point, on a poset that is not chain-complete Counterexample
- Isomorphism does not make an inclusion elementary Counterexample
- In a finite set with a symmetric irreflexive relation and at least two elements, two elements have equally many neighbours Example
- The subsets of ℕ containing a tail form the Fréchet filter, and it is proper and not an ultrafilter Example
- FALSE: for all sets A and B with B having at least two elements, A × B is strictly larger than A False statement
- Convergence depends only on the tail Lemma
- Finite sums re-indexed along an injection, with a zero term deleted, and concatenated; and the closure properties of linear independence: an independent list is injective and never 0_V, its sublists are independent, a list is independent exactly when it is injective with linearly independent image, and every subset of a linearly independent set is linearly independent Lemma
- For positive integers a and b: a ∣ b if and only if vₚ(a) ≤ vₚ(b) for every prime p Lemma
- Laws of finite sums and products in ℕ, and ι(∑_k<n aₖ) = ∑_k<n ι(aₖ) Lemma
- A subset of a finite set is finite, with | B| ≤ | A|, and equality holds if and only if B = A Theorem
- For all m and n there is a list of mn pairwise distinct reals with no strictly increasing sublist of length m+1 and no strictly decreasing sublist of length n+1 Theorem
- For m ≥ 1 the number of weak compositions of n into m parts is C(n+m-1, m-1), and the number of compositions is C(n-1, m-1) for n ≥ 1 Theorem
- Generalised associativity: in a monoid the product of a finite list does not depend on the bracketing, and in a commutative monoid it does not depend on the order of the factors either Theorem
- ℕ × ℕ ≈ ℕ Theorem
- Stay-put elimination with an explicit simulation interface Theorem
- The arithmetic mean, geometric mean inequality Theorem
- The integers form a totally ordered ring Theorem
- The sum rule: a finite disjoint union is finite with | A ∪ B| = | A| + | B| and |⋃_i ∈ I Aᵢ| = ∑_i ∈ I| Aᵢ|, and a sum over a finite index set splits along a partition 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)