HvdH machine lemma AX.axOut_tg (One resampling axis, part 1: `axisP` with separate input / output boun)
ProvedNLogN.AX.axOut_tgThe following holds (Lean statement, in the scope of module NLogN.HAx1):
theorem NLogN.AX.axOut_tg (ax : GL.Axis M Ctx B) (rest : List ℕ) (hm : ax.m = rest.prod)
(L : List (ℤ × ℤ)) : tensorGo [(ax.n', ax.f)] (ax.n :: rest) L = axOut ax L
Context. The lemma belongs to the part One resampling axis, part 1: axisP with separate input / output bounds: GL.axisP_all needs the slice map to preserve one bound GB B. The resampling maps grow
their entries, so here AxReady2 ax Bo s Q asks instead that the slice map sends GB B
inputs (of length n) to GB Bo outputs (of length n'). The program is unchanged
(GL.axisP ax).
Formalization Note. This is a lemma of the Lean development NLogN, a multitape-Turing-machine implementation of the Harvey–van der Hoeven integer multiplication algorithm, used in the proof of NLogN.FIN.theorem_1_1_nlogn : IntMul.MulTimeBound fun n => n * Real.log n. All names refer to the definition bundles IntMul_HvdH_M_Defs1–IntMul_HvdH_M_Defs8 (namespace NLogN), built on the platform model Def_IntMul_MultitapeModel and the published reference Def_IntMul_HvdH_MachineRef. The machine layer: IState HS is a state of 64 item stacks; RM programs have a semantics p.run s, a definedness predicate p.ok M s and a cost p.cost M s relative to a multiplication spec M : MulSpec; zsz z is the bit length of .
-- Harvey–van der Hoeven multitape-TM development (NLogN), generated by skeleton subtraction.
import Definitions.Def_IntMul_HvdH_M_Defs8
set_option autoImplicit false
set_option relaxedAutoImplicit false
set_option maxHeartbeats 1000000
set_option linter.unusedSimpArgs false
open NLogN
open NLogN.AX
open IntMul NLogN IntMul.HvdH.Ref NLogN.GL
variable {M : MulSpec} {Ctx : IState HS → Prop} {B : ℕ}
theorem NLogN.AX.axOut_tg (ax : GL.Axis M Ctx B) (rest : List ℕ) (hm : ax.m = rest.prod)
(L : List (ℤ × ℤ)) : tensorGo [(ax.n', ax.f)] (ax.n :: rest) L = axOut ax L := by sorry