first_isomorphism

std.algebra.first_isomorphism · Level L3

The first isomorphism theorem on a finite example, as 1 or 0: f : G → K is a homomorphism, the induced map gH ↦ f(g) on the cosets of H = ker f gives the same value for every representative, and it is a bijection onto the image, so G/ker f ≅ im f. Also |G/ker f| and |im f|. The image need not be all of K.

G/ker f ≅ im f

Signature

first_isomorphism(TG: i64[n, n], TK: i64[m, m], f: i64[n]) → 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 induced map on kernel cosets: well defined, and as many cosets as image elements in exact rational arithmetic, on all 40 test cases.
All 120 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

hom_kernel_image left_cosets

Called by

—

sha256:31f3a058a45554fc0c0a5af1b1e29fbbe1acc1603e276d55d6718cfe11a15e37

The semantic hash of the graph. It changes when the program changes, and never when only its documentation does.

Control handle

Symbol: Ω:std.algebra.first_isomorphism · Ω:first_isomorphism
Pins: sha256:c0782ea5fdede85ee2130e44ef005893ed3ebe68fc4264c78932978fe5c90878this graph and the 3 it reaches through calls
Evidence: sha256:b401ffaa0cfc217d05166f80963eac0e7e02a4d6329b009b92633c3a5e8b209fthe 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.