The rules of integer arithmetic
Everything you learned in school arithmetic — a+b = b+a, a(b+c) = ab+ac, there's a 0 and a 1, every number has a negative — is bundled into one formal statement: the integers form a 'commutative ring'. Naming the bundle lets every later proof simply cite 'the ring axioms' instead of re-deriving grade-school algebra.
It is the grammar of arithmetic. You rarely notice grammar while speaking, but every sentence depends on it — and every theorem here silently depends on these rules.
Why does d | a and d | b imply d | (a+b)? Write a = dk, b = dm; then a+b = dk+dm = d(k+m) — that last step IS distributivity, a ring axiom.
These axioms are the shared foundation of algebra, cryptography, coding theory, and computer arithmetic. Chips are verified against exactly these laws.
Level 1 The precise statement
The integers under + and * form a commutative ring: both operations are associative and commutative, * distributes over + (a*(b+c) = a*b + a*c), with identities 0 and 1.
Level 3 What it stands on (0 direct)
Nothing — this is foundational. It is one of the roots of the graph.
Level 4 The verified record
This page is generated from a machine-checked node. The kernel confirms its dependencies resolve, nothing is circular, and it grounds in axioms (foundation: peano). The content hash below makes tampering evident.
14 downstream results would collapse with it. See the blast radius on the graph →