ring_axioms

std.algebra.ring_axioms · Level L3

The axioms of a finite ring with 1, given by its addition table Ad and multiplication table Mu, each checked over every pair or triple, as 1 or 0: addition is an abelian group; multiplication is closed and associative; it distributes over addition on both sides, a(b + c) = ab + ac and (a + b)c = ac + bc; there is a multiplicative identity 1; and 1 ≠ 0. Also the zero and the one. Multiplication need not commute.

(R, +) abelian; (ab)c = a(bc); a(b + c) = ab + ac; (a + b)c = ac + bc; 1 ≠ 0

Signature

ring_axioms(Ad: i64[n, n], Mu: i64[n, n]) → f64[], f64[], f64[], f64[], f64[], f64[], f64[]

Structure

The function as NOVA stores it: one box per input, operation and output, and arrows that carry values. A double border marks another library function this one runs — called once, or by Scan once per element; select it to open that function.

input
operation
constant
call
output

Verification

Signature proven by NOVA’s shape solver, for every size.
Equal to the reference every pair and triple, in Python in exact rational arithmetic, on all 40 test cases.
All 280 float64 results are exact: the error is zero.
Interpreter and NumPy backend return bit-identical results.
At float32 too: the 40 test cases rounded to float32, and 40 of them exact, with zero error.

Accuracy in detail

correctly rounded (the float64 nearest the exact value): 100%
bit-equal to the NumPy formula in float64: 100%
largest error, in units in the last place: 0

Identity

Calls

group_axioms is_abelian

Called by

is_field is_integral_domain is_ring ring_properties

sha256:4b1a54291ac8a8eae8a5666217f8a6eadbb11c7a7bd3d5b544ce69cda5f9ba31

The semantic hash of the graph. It changes when the program changes, and never when only its documentation does.

Control handle

Symbol: Ω:std.algebra.ring_axioms · Ω:ring_axioms
Pins: sha256:db107bb454f58ad0ae2f17296041226b236ae75a41182c8869183533029f57f2this graph and the 3 it reaches through calls
Evidence: sha256:04a5fbe81469e51f3be23237837a553c6555a2d3d09d153963e055b7b2b509bethe hash of its verification record
Needs: no capability: a pure function

Through NOVA’s control layer, the symbol launches this function only while the program still matches what it pins: a change to this graph, or to any graph it reaches, needs a migration first.