Mixed radix: the digits cut by digI recombine to x mod 2^{bM}
ProvedNLogN.KFT.valN_chkNMixed radix: the digits cut by digI recombine to x mod 2^{bM}.
The following holds (Lean statement, in the scope of module Solutions.KFTOut):
theorem NLogN.KFT.valN_chkN (b : ℕ) : ∀ M x : ℕ, valN b (chkN b M x) = x % 2 ^ (b * M)
Context. The lemma belongs to the part KFT §3: linear cost bounds, the bin_{2n} write, and carries-then-write.
Formalization Note. This lemma belongs to the item-language (IL) layer of the integer-multiplication-kappa campaign: tape-level programs (register programs over 64 item stacks of the NLogN multitape framework, definition bundles IntMul_HvdH_M_Defs1–IntMul_HvdH_M_Defs8) that implement the data-movement primitives of the OpenAI preprint Integer multiplication below (field moves, row split/merge, shifts, the uniform Proposition 11 interchange, Lemma 12 with fine -per-element costs, Gaussian record primitives, Proposition 14 pieces) with exact list denotations from the array-model bundles IntMul_Kappa_K_Defs* and explicit cost bounds. All names refer to the definition bundles up to IntMul_Kappa_K_Defs18.
-- OpenAI-preprint / κ-improvement development (integer-multiplication-kappa), generated by skeleton subtraction.
import Definitions.Def_IntMul_Kappa_K_Defs18
set_option autoImplicit false
set_option relaxedAutoImplicit false
set_option maxHeartbeats 1000000
set_option linter.unusedSimpArgs false
set_option maxRecDepth 20000
open NLogN
open NLogN.KFT
open IntMul KFC KFB
variable {Pp : ℕ} {I : List Sym} {iprog : Fin Pp → IStmt HS Pp}
theorem NLogN.KFT.valN_chkN (b : ℕ) : ∀ M x : ℕ, valN b (chkN b M x) = x % 2 ^ (b * M) := by sorry