metric_axioms

std.sets.metric_axioms · Level L3

The axioms of a metric on a finite set, given as its full distance table D, each checked over every pair or triple, as 1 or 0: d(x, y) ≥ 0; d(x, y) = 0 exactly when x = y; d(x, y) = d(y, x); and the triangle inequality d(x, z) ≤ d(x, y) + d(y, z), checked for every y at once as d ≤ the min-plus square of d. A finite metric space is complete: below its smallest positive distance a Cauchy sequence is eventually constant.

d ≥ 0; d(x, y) = 0 ⇔ x = y; d(x, y) = d(y, x); d(x, z) ≤ d(x, y) + d(y, z)

Signature

metric_axioms(D: f64[n, n]) → 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 exact arithmetic in exact rational arithmetic, on all 40 test cases.
All 160 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

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Called by

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sha256:2a711d5876c8ead0ebf35bc787cfee1aa17233aa06afcda21476af18068edf04

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

Control handle

Symbol: Ω:std.sets.metric_axioms · Ω:metric_axioms
Pins: sha256:273f03bc89e4ad0b1dc6b5537a9dab24f01f08f63da9946e13b94725b8750370this graph alone
Evidence: sha256:45ec7921c9d74f05b6e138a32f7c61e675aa22473aa2c2ae0de0893b0178a969the 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.