pigeonhole

std.sets.pigeonhole · Level L3

The pigeonhole principle on a function f : A → B between finite sets: whether two elements share an image, as 1 or 0, and the first such pair a₁ < a₂ (0, 0 when there is none); and the largest load, the size of the biggest fibre. When |A| > |B| a collision must exist, and the largest load is at least ⌈|A|/|B|⌉; |A| ≤ |B| does not make f injective.

|A| > |B| ⇒ ∃a₁ ≠ a₂: f(a₁) = f(a₂); max load ≥ ⌈|A|/|B|⌉

Signature

pigeonhole(f: i64[n], cod: f64[k]) → 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 the first equal pair in order, and the largest fibre 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:d7c6d5926722f3e58c93d63949269b14e88959c1b02ecf989429d94b1fea980b

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.pigeonhole · Ω:pigeonhole
Pins: sha256:50baecb6556d8301c2e0e9ac925035217ce2d44ab06800887063e2743ef07438this graph alone
Evidence: sha256:1909e2458125d326014ee123330728f63dd2272c89713562b6dae11f58deb48athe 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.