English
In the piecewise-constant input scenario, curryFinFinset_symm_apply_piecewise_const gives equality with the original map when inputs are constant on blocks.
Русский
При разбиении входа на константные блоки, curryFinFinset_symm_apply_piecewise_const возвращает равенство с исходной картой, когда вход константен на каждом блоке.
LaTeX
$$$\\text{curryFinFinset}_{hk,hl}^{-1} f\\; (s\\!,\\text{piecewise constant}) = f(\\text{constants})$$$
Lean4
/-- If `s : Finset (Fin n)` is a finite set of cardinality `k` and its complement has cardinality
`l`, then the space of multilinear maps on `fun i : Fin n => M'` is isomorphic to the space of
multilinear maps on `fun i : Fin k => M'` taking values in the space of multilinear maps
on `fun i : Fin l => M'`. -/
def curryFinFinset {k l n : ℕ} {s : Finset (Fin n)} (hk : #s = k) (hl : #sᶜ = l) :
MultilinearMap R (fun _ : Fin n => M') M₂ ≃ₗ[R]
MultilinearMap R (fun _ : Fin k => M') (MultilinearMap R (fun _ : Fin l => M') M₂) :=
(domDomCongrLinearEquiv R R M' M₂ (finSumEquivOfFinset hk hl).symm).trans currySumEquiv
