Round-six lemma `phaseP`
ProvedJ6Seg.ATab.Rn.phasePAn auxiliary lemma of the round-six development, published as a separate node because its proof is long or shared.
The following holds (Lean statement, in the scope of module Solutions.J6AILTab8):
theorem J6Seg.ATab.Rn.phaseP {W p s t α pkA pkC bA bC : ℕ} {v1 : Vr} {c1 : Cs}
(hv : V1 p s t α pkA pkC bA bC v1) (hBv : BV W v1) (hB : PBnd W p t α pkA pkC bA bC)
(hc : c1 cLC = []) :
Rn J6Seg.ATab.phaseP v1 c1 (fun v' => V2 p s t α (pkA * p + pkC) (bA * p + bC) v' ∧ BV W v') c1
(cPhaseP W p (pkA * p + pkC) (bA * p + bC))
Context. A lines, table set-up, part 8: the parameter phase (run)
Rn.phase: from the registers of parA the phase pushes the kerRegs values onto
45 and |T| onto 42 and leaves the loop registers LV with 10 = nc.
Formalization Note. Part of the Lean formalization of the round-six κ-improvement (integer-multiplication-kappa campaign, target IntMul.Kappa.jain_round6_framework). The statement is copied verbatim from the Lean development; the platform node carries the same declaration name. Definitions it mentions live in the IntMul_Kappa_R6_Defs* bundles (on top of the published κ bundles IntMul_Kappa_K_Defs*).
-- Round-six κ development (integer-multiplication-kappa), generated by skeleton subtraction. import Definitions.Def_IntMul_Kappa_R6_Defs69 set_option autoImplicit false set_option relaxedAutoImplicit false set_option maxHeartbeats 1000000 open J6Seg J6Seg.ATab open IntMul NLogN NLogN.KIL NLogN.RS J6Seg.PJ IntMul.HvdH Function
theorem J6Seg.ATab.Rn.phaseP {W p s t α pkA pkC bA bC : ℕ} {v1 : Vr} {c1 : Cs}
(hv : V1 p s t α pkA pkC bA bC v1) (hBv : BV W v1) (hB : PBnd W p t α pkA pkC bA bC)
(hc : c1 cLC = []) :
Rn J6Seg.ATab.phaseP v1 c1 (fun v' => V2 p s t α (pkA * p + pkC) (bA * p + bC) v' ∧ BV W v') c1
(cPhaseP W p (pkA * p + pkC) (bA * p + bC)) := by sorry