function_properties

std.sets.function_properties · Level L3

For a function f : A → B on finite sets, each as 1 or 0: injective (every fibre f⁻¹(b) has at most one element), surjective (every b is hit, the range equals the codomain), bijective (both); and the range f[A] as a mask over B. On finite sets an injection A → B shows |A| ≤ |B|, a surjection |A| ≥ |B|, a bijection |A| = |B|; equal sizes do not make a given f a bijection.

|f⁻¹(b)| ≤ 1; f[A] = B

Signature

function_properties(f: i64[n], cod: f64[k]) → f64[], f64[], f64[], f64[k]

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 the set of values of f, against 0 … k − 1 in exact rational arithmetic, on all 40 test cases.
All 246 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

—

Called by

inverse_function

sha256:3f06379f90ccc2a812c2dcb8abda6443559e872fc0cf6791bb0d80c82ce4415e

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.function_properties · Ω:function_properties
Pins: sha256:786f61075456a0dc43fa8c81fcbf2321baed362090e94a703a292f8e04c652dfthis graph alone
Evidence: sha256:ab0594d1a6b44c03fa5edf4e27af9a27c282ca784a97bcabbd1269d4be4beaf3the 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.