Unrolling a linear level recursion

Proved

R6.level_unroll

by xuanji · Oct 19, 2026 · Mathlib 0df444a (Lean v4.33.1)

complexity-theoryinteger-multiplicationkappa-improvement

Let $s \ge 2$ , $E \ge 0$ , and $D : \mathbb N \to \mathbb R$ with $D(0) \ge 0$ and $D(k+1) \le E + s\,D(k)$ for all $k$ . Then

D(k) \le (E + D(0))\,(2 s^{k} - 1) \qquad \text{for all } k.

In Lean:

theorem R6.level_unroll (E : ℝ) (s : ℕ) (D : ℕ → ℝ) (hs : 2 ≤ s) (hE : 0 ≤ E) (hD0 : 0 ≤ D 0)
    (hstep : ∀ k, D (k + 1) ≤ E + s * D k) :
    ∀ k, D k ≤ (E + D 0) * (2 * (s : ℝ) ^ k - 1)

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 Mathlib

set_option autoImplicit false
set_option relaxedAutoImplicit false
set_option maxHeartbeats 1000000

open Real

Formal statement

theorem R6.level_unroll (E : ℝ) (s : ℕ) (D : ℕ → ℝ) (hs : 2 ≤ s) (hE : 0 ≤ E) (hD0 : 0 ≤ D 0)
    (hstep : ∀ k, D (k + 1) ≤ E + s * D k) :
    ∀ k, D k ≤ (E + D 0) * (2 * (s : ℝ) ^ k - 1) := 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 `R6.level_unroll`, Solutions/R6Guard.lean (lines 241–253) of the round-six development.

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