Alphabeta Math

Roots, Rational Powers, and Classical Inequalities

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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.

✓ 17 results · all verified · 12 also independently AI-judged
Every result on this page is machine-checked by a proof checker and read in full by a delegated reviewing agent on the owner's instruction; the judge is an additional, independent cross-model AI review of the proofs. The 5 not AI-judged were verified by owner audit (typically over a confirmed judge false positive), not failures.
1 · Prerequisites
Real Analysis
Set Theory
2 · Logical flowchart

3 · Items

Definition

Finite sums and finite products, by recursion

Lemma

Laws of finite sums and finite products

Definition

Integer powers aᵐ

Lemma

Laws of integer exponents

Lemma

Monotonicity of x ↦ xⁿ and of n ↦ aⁿ

Lemma

Bernoulli's inequality (1+x)ⁿ ≥ 1 + nx

Lemma

Factorisation of bⁿ - aⁿ, and the resulting Lipschitz estimate

Theorem

Existence and uniqueness of n-th roots: a unique a^1/n ≥ 0 with (a^1/n)ⁿ = a

Definition

Rational powers aʳ of a positive base

Lemma

Rational powers do not depend on the representative

Lemma

Laws of rational exponents

Lemma

Monotonicity of r ↦ aʳ and of a ↦ aʳ

Lemma

Triangle inequality for finite sums

Theorem

The arithmetic mean, geometric mean inequality

Theorem

Weighted AM-GM inequality with rational weights

Theorem

Young's inequality for products (rational conjugate exponents)

Theorem

The Cauchy-Schwarz inequality for finite sums

Theorem

Hölder's inequality for finite sums (rational exponents)

Theorem

Minkowski's inequality for finite sums (rational exponent)

Remark

Why real exponents are deferred on the rational-powers page

False statement

FALSE: every real number has a real square root

False statement

FALSE: a^m/n := (a^1/n)ᵐ extends to negative bases

Sources