shP_pos (KFUU §2b: the layouts of the uniform reversal)
ProvedNLogN.KFUU.shP_posThe following holds (Lean statement, in the scope of module Solutions.KFUURev2):
theorem NLogN.KFUU.shP_pos (u : ℕ) (bs : List Bool) : ∀ a ∈ shP u bs, 1 ≤ a
Context. The lemma belongs to the part KFUU §2b: the layouts of the uniform reversal: The axes are L = range n with sizes 2^(u + b_j) (bs = [b_0, …]), the inner block is the
atom cid = n of size cz: mzB u n cz, bgB bs. Phase 1 runs through lay1 i
(i = n, …, 0): layA (L.take i) ++ mains (L.drop i) ++ bitsR (L.drop i) ++ [2n].
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 linter.unnecessarySeqFocus false set_option maxRecDepth 20000 open NLogN open NLogN.KFUU open IntMul KIL KFC KFT KFU AModel AModelL12
theorem NLogN.KFUU.shP_pos (u : ℕ) (bs : List Bool) : ∀ a ∈ shP u bs, 1 ≤ a := by sorry