Allocating the temporaries (round six, `sImplQ_padRole`)

Proved

IntMul.Kappa.R6.IL3H.sImplQ_padRole

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

complexity-theoryinteger-multiplicationkappa-improvement

Allocating the temporaries.

The following holds (Lean statement, in the scope of module Solutions.R6IL3Alloc):

theorem IntMul.Kappa.R6.IL3H.sImplQ_padRole {lo hi r : ℕ} {st : Fin m × κ → ℕ} {n : ℕ}
    (hs : SLaws st n) (ha : ∀ i, 2 * st i ∣ n) {Tm : Butterfly.Op (Fin m × κ)}
    {A : ROp (Fin (W + T)) m κ}
    (hA : ∀ (v w' : Fin W) (x y : Addr m κ),
      A (Fin.castAdd T v, x) (Fin.castAdd T w', y) = if v = w' then Tm x y else 0)
    {σ : Fin HS → Option ℤ} (hlt : ∀ i v, σ i = some v → i.val < 56) (hoff : OffG σ)
    {xL xQ : Fin HS} (hL : σ xL = some ((2 * (W * n * r) : ℕ) : ℤ))
    (hQ : σ xQ = some ((2 * (T * n * r) : ℕ) : ℤ))
    {U : IStmt HS Pp} {e dw : ℕ} {cost : ℕ → ℕ}
    (h : RImplQ I iprog (fun s => Agree σ s) lo hi r n st A U e dw cost) (Z : ℕ)
    (hZ : zsz ((2 * (T * n * r) : ℕ) : ℤ) ≤ Z) :
    SImplQ I iprog (fun s => Agree σ s) lo hi r st (W * n) Tm (padCU xL xQ U) e dw
      (fun w => 400 * (cPadC (2 * (W * n * r)) (2 * (T * n * r)) Z w dw (fun _ => 0) + 0) +
        cost w)

Context. Round six, item language (I3): allocating the temporaries

The complex network acts on W data/scratch roles and T temporaries (Fin (W + T), data first). At a node the stream holds the W data roles (W n records); padCU xL xQ U appends T n zero records (the clean temporaries), runs the role program U on W + T roles, and drops the temporaries again. Only the data block of the network operator matters (DataCorrect), since the temporaries enter as zeros and leave unread.

flatR_eq_of_getD, exists_flatR, flatR_append, flatR_padX, flatR_take_castAdd;
roleOpF_padX: the data rows of roleOpF on zero-padded temporaries;
sImplQ_padRole: padCU xL xQ U implements the data block Tm on W n records.

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_K_Defs33
import Definitions.Def_IntMul_Kappa_R6_Defs1
import Mathlib
import Definitions.Def_IntMul_HvdH_M_Defs8
import Definitions.Def_IntMul_MultitapeModel

set_option autoImplicit false
set_option relaxedAutoImplicit false
set_option maxHeartbeats 1000000

open IntMul IntMul.Kappa IntMul.Kappa.R6
open IntMul NLogN NLogN.KIL NLogN.KFD NLogN.KFB NLogN.KFG NLogN.KFH NLogN.KFN NLogN.KFW Complex P8Lift Butterfly
variable {Pp : ℕ} {I : List Sym} {iprog : Fin Pp → IStmt HS Pp}
variable {W T m : ℕ} {κ : Type} [Fintype κ] [DecidableEq κ]

Formal statement

theorem IntMul.Kappa.R6.IL3H.sImplQ_padRole {lo hi r : ℕ} {st : Fin m × κ → ℕ} {n : ℕ}
    (hs : SLaws st n) (ha : ∀ i, 2 * st i ∣ n) {Tm : Butterfly.Op (Fin m × κ)}
    {A : ROp (Fin (W + T)) m κ}
    (hA : ∀ (v w' : Fin W) (x y : Addr m κ),
      A (Fin.castAdd T v, x) (Fin.castAdd T w', y) = if v = w' then Tm x y else 0)
    {σ : Fin HS → Option ℤ} (hlt : ∀ i v, σ i = some v → i.val < 56) (hoff : OffG σ)
    {xL xQ : Fin HS} (hL : σ xL = some ((2 * (W * n * r) : ℕ) : ℤ))
    (hQ : σ xQ = some ((2 * (T * n * r) : ℕ) : ℤ))
    {U : IStmt HS Pp} {e dw : ℕ} {cost : ℕ → ℕ}
    (h : RImplQ I iprog (fun s => Agree σ s) lo hi r n st A U e dw cost) (Z : ℕ)
    (hZ : zsz ((2 * (T * n * r) : ℕ) : ℤ) ≤ Z) :
    SImplQ I iprog (fun s => Agree σ s) lo hi r st (W * n) Tm (padCU xL xQ U) e dw
      (fun w => 400 * (cPadC (2 * (W * n * r)) (2 * (T * n * r)) Z w dw (fun _ => 0) + 0) +
        cost w) := 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 `IntMul.Kappa.R6.IL3H.sImplQ_padRole`, Solutions/R6IL3Alloc.lean (lines 177–311) of the round-six development.

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