inverse_function

std.sets.inverse_function · Level L3

The inverse g : B → A of a function f on finite sets, by swapping the pairs of its graph, and whether f is a bijection, as 1 or 0. Only for a bijection is g a function with g(f(a)) = a and f(g(b)) = b; otherwise the swapped table is not an inverse, and g is the first preimage, or 0 where there is none.

g(f(a)) = a, f(g(b)) = b

Signature

inverse_function(f: i64[n], cod: f64[k]) → f64[], i64[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 first a with f(a) = b in exact rational arithmetic, on all 40 test cases.
All 184 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

function_properties

Called by

—

sha256:9e24b294b85be1049b6d78231df67e8b29f998a9028980b49fd41b65bd8dda49

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.inverse_function · Ω:inverse_function
Pins: sha256:f3c2c919ebe2aa6db22c957f84e160cc0032e00bdecb9e5271fc2015d3d8ea07this graph and the 1 it reaches through calls
Evidence: sha256:1d565ecd9deb798cdf3867373ddf33f5517bd3186466e2d7c7337e070d7798bethe 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.