B1 (KFM S9: the cost of the source and inverse lists)
ProvedNLogN.KFM.CostEnv.B1The following holds (Lean statement, in the scope of module Solutions.KFMS9):
theorem NLogN.KFM.CostEnv.B1 {A B : ℕ} (h : CostEnv A B) : 1 ≤ B
Context. The lemma belongs to the part KFM S9: the cost of the source and inverse lists: Write K = 10^7 (B+2)^2. Assume every atom is at most A, with A^k < 2^B for k ≤ 8.
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 open NLogN open NLogN.KFM open IntMul RS
theorem NLogN.KFM.CostEnv.B1 {A B : ℕ} (h : CostEnv A B) : 1 ≤ B := by sorry