vol_append (Array model, Stage 2: runs of pivots as single interchanges of merged fields)
ProvedAModel.vol_appendThe following holds (Lean statement, in the scope of module Solutions.AModelRuns):
theorem AModel.vol_append (A B : Shape) : AModel.vol (A ++ B) = AModel.vol A * AModel.vol B
Context. The lemma belongs to the part Array model, Stage 2: runs of pivots as single interchanges of merged fields: Builds on Solutions/AModel.lean (whose definitions are unchanged).
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 open AModel
theorem AModel.vol_append (A B : Shape) : AModel.vol (A ++ B) = AModel.vol A * AModel.vol B := by sorry