counting_rules

std.sets.counting_rules · Level L2

The two counting principles for finite choices given the number of options in each part: the sum rule, Σ counts, for choosing one option from mutually exclusive categories; and the product rule, Π counts, for a sequence of stages whose option counts do not depend on earlier choices. Overlapping categories must be corrected by inclusion-exclusion first.

Σ nᵢ, Π nᵢ

Signature

counting_rules(counts: f64[k]) → 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 the sum and the product in exact rational arithmetic, on all 40 test cases.
All 80 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

product_step

Called by

—

sha256:6040a33fcf44c3de814e23f05096294e1ed36fbec78ee3a8bd997a16481b6d66

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.counting_rules · Ω:counting_rules
Pins: sha256:95c53574f4b9970e157a9d196f3388a10e4c01e5f817dc2ee95d8e7879af37e3this graph and the 1 it reaches through calls
Evidence: sha256:449143c6bed01d0c81d1471061292685429ac84be9c8b6704d12788f322c5e9ethe 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.