ng_exists (KFY 19: the cost of the two passes for `n` (natural-number part))
ProvedNLogN.KFY.ng_existsThe following holds (Lean statement, in the scope of module Solutions.KFY19N):
theorem NLogN.KFY.ng_exists : ∃ n₂, ∀ n, IntMul.OpenAI.Glue.n0 ≤ n → n₂ ≤ n → NG n
Context. The lemma belongs to the part KFY 19: the cost of the two passes for n (natural-number part).
Formalization Note. This lemma belongs to layer 5 of the Lean formalization (integer-multiplication-kappa campaign) of the OpenAI preprint Integer multiplication below : the uniform item-language machine (register programs over the NLogN multitape framework) assembling the Proposition 14 / Lemma 13 / Proposition 18 pieces, the resampling and chirp stages and the cost recurrences into one fixed machine, culminating in NLogN.KFTop.kappa_bound : IntMul.KappaBound (1 / 2 ^ 182) (the bound of Def_IntMul_MultitapeModel). All names refer to the definition bundles up to IntMul_Kappa_K_Defs33.
-- OpenAI-preprint / κ-improvement development (integer-multiplication-kappa), generated by skeleton subtraction. import Definitions.Def_IntMul_Kappa_K_Defs33 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 maxRecDepth 100000 set_option exponentiation.threshold 2048 open NLogN open NLogN.KFY open IntMul NLogN IntMul.OpenAI.Glue NLogN.GL NLogN.AX NLogN.Rsp
theorem NLogN.KFY.ng_exists : ∃ n₂, ∀ n, IntMul.OpenAI.Glue.n0 ≤ n → n₂ ≤ n → NG n := by sorry