nP_runs (KFM U5: `n` in binary from the ruler `2^n`, in linear time)
ProvedNLogN.KFM.nP_runsThe following holds (Lean statement, in the scope of module Solutions.KFMU5):
theorem NLogN.KFM.nP_runs (s : IState HS) (n : ℕ) (LN : List Item) (hN : s.G gN = .int ((2 : ℤ) ^ n) :: LN) :
IRuns I iprog nP s (s.upd gJ (.int (n : ℤ) :: s.G gJ)) (10 ^ 6 * (n + 1))
Context. The lemma belongs to the part KFM U5: n in binary from the ruler 2^n, in linear time.
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_Defs22.
-- OpenAI-preprint / κ-improvement development (integer-multiplication-kappa), generated by skeleton subtraction.
import Definitions.Def_IntMul_Kappa_K_Defs22
set_option autoImplicit false
set_option relaxedAutoImplicit false
set_option maxHeartbeats 1000000
set_option linter.unusedSimpArgs false
set_option linter.unusedVariables false
set_option linter.unusedSectionVars false
set_option linter.unnecessarySeqFocus false
set_option maxRecDepth 20000
open NLogN
open NLogN.KFM
open IntMul KFC KFB KFT
variable {P : ℕ} {I : List Sym} {iprog : Fin P → IStmt HS P}
theorem NLogN.KFM.nP_runs (s : IState HS) (n : ℕ) (LN : List Item) (hN : s.G gN = .int ((2 : ℤ) ^ n) :: LN) :
IRuns I iprog nP s (s.upd gJ (.int (n : ℤ) :: s.G gJ)) (10 ^ 6 * (n + 1)) := by sorry