left_cosets
std.algebra.left_cosets · Level L3The left cosets gH of a subgroup H of a finite group: each element g is labelled by the smallest element of gH, so two elements share a label exactly when they lie in the same coset; and the index [G : H], the number of cosets. Lagrange: the cosets partition G and each has |H| elements, so |G| = [G : H]·|H|.
Signature
left_cosets(T: i64[n, n], H: f64[n]) → f64[n], 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 smallest g·h over h in H, for each gin exact rational arithmetic, on all 40 test cases. - All 172 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.
- 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
sha256:9a9aa1f6333586c21e175bfb0ae96d73863e53bb0ca2da0f6e31936d0804bb55The semantic hash of the graph. It changes when the program changes, and never when only its documentation does.
Control handle
- Symbol
- Ω:std.algebra.left_cosets · Ω:left_cosets
- Pins
sha256:4a5b2c21eac78e32cc31e76854bf9407db7852b064e53aacf221685c75c5016dthis graph alone- Evidence
sha256:ea2f46abce9b8d2870c25c2f0d4b5ab57aa7aafcc1f478d305497462ffae435fthe 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.