The call loop (round six, `cLoopR_sp`)

Proved

J6Seg.PreT.cLoopR_sp

by xuanji · Oct 20, 2026 · Mathlib 0df444a (Lean v4.33.1)

complexity-theoryinteger-multiplicationkappa-improvement

The call loop (general register sequence): W k the registers before step k, W' k after the call of step k (which sets 11 := f k).

The following holds (Lean statement, in the scope of module Solutions.J6SegILPre6):

theorem J6Seg.PreT.cLoopR_sp {r : Fin HS} (hr : r ∈ rg) (hr12 : r ≠ 12) (s : IState HS) (v : Fin HS → ℤ) (c : Fin HS → List ℤ) {O : Fin HS} (hO : O ∉ rg)
    (hO5 : O.val < 56) (hOC : O ≠ Cu) (hc : c Cu = []) {cl : RS} (hw : Cu ∉ cl.writes)
    {A N : ℕ} (hA : 2 ≤ A) (hv : VB v A) (h24 : v r = N) (hN : N ≤ 2 ^ A) (f : ℕ → ℤ)
    (W W' : ℕ → Fin HS → ℤ) (hW0 : W 0 = Function.update v 12 0)
    (hWs : ∀ k, W (k + 1) = Function.update (W' k) 12 (k + 1))
    (hW11 : ∀ k < N, W' k 11 = f k) (hW12 : ∀ k < N, W' k 12 = k)
    (hWV : ∀ k < N, VB (W' k) A) {Cc : ℕ}
    (hcl : ∀ k < N, ∀ cc : Fin HS → List ℤ,
      Sp cl (encK s (J6Seg.PreT.RF (W k) cc)) (encK s (J6Seg.PreT.RF (W' k) cc)) Cc) :
    Sp (cLoopR O cl r) (encK s (J6Seg.PreT.RF v c)) (encK s (J6Seg.PreT.RF (W N) (Function.update c O (rv f N ++ c O))))
      (J6Seg.PreT.cE (.lit 0) (BW A) + (2 * 2000 * (BW A + 2) + N * (Cc + J6Seg.PreT.cE (.reg 11) (BW A) +
        J6Seg.PreT.cE (.add (.reg 12) (.lit 1)) (BW A) + 8000 * (BW A + 2) + 30) + 32))

Context. B-line IL, pre-tables (B5), part 6: the call loop

cLoop O cl = U := 0; repeat N = v 24 times { cl; push X on O; U := U + 1 } where the call cl sets X = v 11 := f U. cLoop_sp: it pushes f 0, …, f (N-1) onto O (so O gains rv f N); the registers end as wS v f N (only 11, 12 change).

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*).

Preamble

-- Round-six κ development (integer-multiplication-kappa), generated by skeleton subtraction.

import Definitions.Def_IntMul_Kappa_R6_Defs68

set_option autoImplicit false
set_option relaxedAutoImplicit false
set_option maxHeartbeats 1000000

open J6Seg J6Seg.PreT
open IntMul NLogN NLogN.KIL NLogN.RS IntMul.HvdH

Formal statement

theorem J6Seg.PreT.cLoopR_sp {r : Fin HS} (hr : r ∈ rg) (hr12 : r ≠ 12) (s : IState HS) (v : Fin HS → ℤ) (c : Fin HS → List ℤ) {O : Fin HS} (hO : O ∉ rg)
    (hO5 : O.val < 56) (hOC : O ≠ Cu) (hc : c Cu = []) {cl : RS} (hw : Cu ∉ cl.writes)
    {A N : ℕ} (hA : 2 ≤ A) (hv : VB v A) (h24 : v r = N) (hN : N ≤ 2 ^ A) (f : ℕ → ℤ)
    (W W' : ℕ → Fin HS → ℤ) (hW0 : W 0 = Function.update v 12 0)
    (hWs : ∀ k, W (k + 1) = Function.update (W' k) 12 (k + 1))
    (hW11 : ∀ k < N, W' k 11 = f k) (hW12 : ∀ k < N, W' k 12 = k)
    (hWV : ∀ k < N, VB (W' k) A) {Cc : ℕ}
    (hcl : ∀ k < N, ∀ cc : Fin HS → List ℤ,
      Sp cl (encK s (J6Seg.PreT.RF (W k) cc)) (encK s (J6Seg.PreT.RF (W' k) cc)) Cc) :
    Sp (cLoopR O cl r) (encK s (J6Seg.PreT.RF v c)) (encK s (J6Seg.PreT.RF (W N) (Function.update c O (rv f N ++ c O))))
      (J6Seg.PreT.cE (.lit 0) (BW A) + (2 * 2000 * (BW A + 2) + N * (Cc + J6Seg.PreT.cE (.reg 11) (BW A) +
        J6Seg.PreT.cE (.add (.reg 12) (.lit 1)) (BW A) + 8000 * (BW A + 2) + 30) + 32)) := by sorry

Source

S. Jain, round-six networks for the integer-multiplication κ-improvement campaign (https://github.com/Swapnil-jain/integer-mult-kappa, commit 5868dc7) and D. Colkitt, integer-mult-bounds (https://github.com/CrocSwap/integer-mult-bounds); target `IntMul.Kappa.jain_round6_framework`. Lean declaration `J6Seg.PreT.cLoopR_sp`, Solutions/J6SegILPre6.lean (lines 80–143) of the round-six development.

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