Source preparation realizes the actual block-encoding endpoints

Proved

ThreeSum.LightBlockSourcePreparation.meets

by wurtle · Oct 14, 2026 · Mathlib 0df444a (Lean v4.33.1)

3sumalgorithmsfine-grained-complexitythree-sum-1962word-ram

Fix local parameter $h$ , block count $J$ , and an occurrence vector $o$ selecting one actual scheduled map per block. Let $S_i$ ( $i<n$ ) be arbitrary subsets of the $J\times512h$ coordinates, stored as flattened masks. Memory also stores the exact occurrence indices, schedule coefficient rows, candidate indices, and both candidate histograms. Place these inputs before the two coefficient arrays, then the left and right endpoint arrays, flags, bit scratch, row scratch, and arithmetic scratch with the separations in the preparation layout.

Assume the specified preparation procedures, $S\le W$ , depth allowance $J512h+5$ , and its exact space and word budgets with local prime $p_h$ , histogram bound $\binom{512h}{h}$ , and local bucket cap. The outputs satisfy

\mathrm{raw}_\ell[i]=E_\ell(o,S_i),\quad\mathrm{raw}_r[i]=E_r(o,S_i),\quad\mathrm{flag}_\ell[i]=\mathbf1_{\mathrm{LeftAllowed}(o,S_i)},\quad\mathrm{flag}_r[i]=\mathbf1_{\mathrm{RightAllowed}(o,S_i)}.

Endpoints are positional sums of raw local hashes; allowance requires balanced rows and bounded buckets in each block. Execution meets the preparation budget and preserves the input prefix.

This identifies machine outputs with the original block-encoding definitions.

Preamble

/-
Copyright (c) 2026 Anthropic, PBC. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
SPDX-License-Identifier: Apache-2.0
-/

import Aesop
import Definitions.Def_APSPSource_RemainingDefinitions
import Definitions.Def_APSPSource_ThreeSumApsp_Lang_Logic
import Definitions.Def_APSPSource_ThreeSumApsp_Lang_Regions
import Definitions.Def_APSPSource_ThreeSumApsp_Lang_Syntax
import Definitions.Def_ThreeSum1962Source_Bundle01
import Definitions.Def_ThreeSum1962Source_Bundle02
import Definitions.Def_ThreeSum1962Source_Bundle03
import Definitions.Def_ThreeSum1962Source_Bundle04
import Definitions.Def_ThreeSum1962Source_Bundle05
import Definitions.Def_ThreeSum1962Source_Bundle06
import Definitions.Def_ThreeSum1962Source_Bundle07
import Definitions.Def_ThreeSum1962Source_Bundle09
import Definitions.Def_ThreeSum1962Source_Bundle10
import Definitions.Def_ThreeSum1962Source_Bundle11
import Lean.Elab.Tactic.Omega
import Mathlib.Algebra.BigOperators.Fin
import Mathlib.Algebra.BigOperators.Group.Finset.Basic
import Mathlib.Algebra.BigOperators.Group.Finset.Piecewise
import Mathlib.Algebra.BigOperators.Group.Finset.Powerset
import Mathlib.Algebra.BigOperators.Group.Finset.Sigma
import Mathlib.Algebra.BigOperators.Ring.Finset
import Mathlib.Algebra.Field.ZMod
import Mathlib.Algebra.Group.Equiv.Basic
import Mathlib.Algebra.Group.Int.Defs
import Mathlib.Algebra.GroupWithZero.Nat
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Algebra.Order.BigOperators.Group.List
import Mathlib.Algebra.Order.BigOperators.Group.Multiset
import Mathlib.Algebra.Order.Chebyshev
import Mathlib.Algebra.Order.Field.Basic
import Mathlib.Algebra.Order.Floor.Div
import Mathlib.Algebra.Order.Group.Abs
import Mathlib.Algebra.Order.Group.Int
import Mathlib.Algebra.Order.Group.Nat
import Mathlib.Algebra.Order.Group.Unbundled.Abs
import Mathlib.Algebra.Order.Ring.Abs
import Mathlib.Algebra.Order.Ring.Int
import Mathlib.Algebra.Order.Ring.Nat
import Mathlib.Algebra.Order.Ring.Pow
import Mathlib.Algebra.Polynomial.Basic
import Mathlib.Algebra.Polynomial.Coeff
import Mathlib.Algebra.Ring.Nat
import Mathlib.Combinatorics.Colex
import Mathlib.Data.Bool.Basic
import Mathlib.Data.Fin.Embedding
import Mathlib.Data.Fin.VecNotation
import Mathlib.Data.Finset.Card
import Mathlib.Data.Finset.Powerset
import Mathlib.Data.Finset.Prod
import Mathlib.Data.Finset.Range
import Mathlib.Data.Finset.Sigma
import Mathlib.Data.Finset.Sort
import Mathlib.Data.Fintype.BigOperators
import Mathlib.Data.Fintype.Card
import Mathlib.Data.Fintype.EquivFin
import Mathlib.Data.Fintype.Pi
import Mathlib.Data.Fintype.Prod
import Mathlib.Data.Int.Cast.Lemmas
import Mathlib.Data.Int.ModEq
import Mathlib.Data.Int.Notation
import Mathlib.Data.Int.SuccPred
import Mathlib.Data.List.Basic
import Mathlib.Data.List.GetD
import Mathlib.Data.List.MinMax
import Mathlib.Data.List.Nodup
import Mathlib.Data.List.OfFn
import Mathlib.Data.List.Pi
import Mathlib.Data.List.Range
import Mathlib.Data.Nat.Cast.Order.Ring
import Mathlib.Data.Nat.Choose.Basic
import Mathlib.Data.Nat.Choose.Bounds
import Mathlib.Data.Nat.Choose.Central
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.Data.Nat.Count
import Mathlib.Data.Nat.Log
import Mathlib.Data.Nat.Notation
import Mathlib.Data.Nat.Size
import Mathlib.Data.Nat.Sqrt
import Mathlib.Data.Nat.SuccPred
import Mathlib.Data.Real.Basic
import Mathlib.Data.ZMod.Basic
import Mathlib.GroupTheory.Index
import Mathlib.LinearAlgebra.Dual.Lemmas
import Mathlib.LinearAlgebra.Matrix.Defs
import Mathlib.LinearAlgebra.Matrix.Dual
import Mathlib.LinearAlgebra.Matrix.ToLin
import Mathlib.Logic.Equiv.Fin.Basic
import Mathlib.Logic.Function.Basic
import Mathlib.NumberTheory.Bertrand
import Mathlib.Order.ConditionallyCompleteLattice.Basic
import Mathlib.RingTheory.Coprime.Lemmas
import Mathlib.Tactic
import Mathlib.Tactic.Abel
import Mathlib.Tactic.ByContra
import Mathlib.Tactic.FieldSimp
import Mathlib.Tactic.FinCases
import Mathlib.Tactic.GCongr
import Mathlib.Tactic.Linarith
import Mathlib.Tactic.NormNum
import Mathlib.Tactic.Positivity
import Mathlib.Tactic.Push
import Mathlib.Tactic.Ring
import Mathlib.Tactic.SplitIfs
import Mathlib.Tactic.Tauto
import Theorems.Thm_ThreeSum_BlockIndex_maskHash_left
import Theorems.Thm_ThreeSum_BlockIndex_maskHash_right
import Theorems.Thm_ThreeSum_LightLocalCoverSchedule_indices_lt
import Theorems.Thm_ThreeSum_LightLocalCoverSchedule_indices_map
import Theorems.Thm_ThreeSum_LightSourcePreparation_Procs_meets
import Theorems.Thm_ThreeSum_LocalMaskHashes_indexed_bucket_count
import Theorems.Thm_ThreeSum_MaskErase_eraseMask_maskOf
import Theorems.Thm_ThreeSum_OneHotMask_eval_maskOf


set_option autoImplicit false
set_option relaxedAutoImplicit false
set_option Elab.async false
namespace Light
end Light
namespace ThreeSum.BlockCoefficientRows
end ThreeSum.BlockCoefficientRows
namespace ThreeSum.BlockEncoding
end ThreeSum.BlockEncoding
namespace ThreeSum.BlockIndex
end ThreeSum.BlockIndex
namespace ThreeSum.BlockRows
end ThreeSum.BlockRows
namespace ThreeSum.BlockSourceFlags
end ThreeSum.BlockSourceFlags
namespace ThreeSum.CyclicFold
end ThreeSum.CyclicFold
namespace ThreeSum.GlobalEncoding
end ThreeSum.GlobalEncoding
namespace ThreeSum.Karatsuba
end ThreeSum.Karatsuba
namespace ThreeSum.LayerCounts
end ThreeSum.LayerCounts
namespace ThreeSum.LightHistogram
end ThreeSum.LightHistogram
namespace ThreeSum.LightResidues
end ThreeSum.LightResidues
namespace ThreeSum.LightSourceRowScan
end ThreeSum.LightSourceRowScan
namespace ThreeSum.LightSubsetAddress
end ThreeSum.LightSubsetAddress
namespace ThreeSum.LocalCoverBounds
end ThreeSum.LocalCoverBounds
namespace ThreeSum.LocalGraph
end ThreeSum.LocalGraph
namespace ThreeSum.LocalHashMoments
end ThreeSum.LocalHashMoments
namespace ThreeSum.LocalMaskGraph
end ThreeSum.LocalMaskGraph
namespace ThreeSum.MaskDecode
end ThreeSum.MaskDecode
namespace ThreeSum.MaskErase
end ThreeSum.MaskErase
namespace ThreeSum.SourceFlagProduct
end ThreeSum.SourceFlagProduct
namespace ThreeSum.SubsetAddress
end ThreeSum.SubsetAddress

Formal statement

theorem ThreeSum.LightBlockSourcePreparation.meets : ∀ {lim : Light.Limits} {P : Light.Program} {d pMain : Nat} (C : ThreeSum.LightSourcePreparation.Procs P pMain)
  {μ : Nat → Int}
  {h J occ maps candidates histL histR masks n coeff0 coeff1 rawLeft rawRight flags bitBase rowOut work : Nat}
  (o : ThreeSum.BlockEncoding.Occurrences h J)
  (sets :
    Nat →
      Finset.{0}
        (ThreeSum.BlockRows.Coord J
          (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat)
            (@OfNat.ofNat.{0} Nat (nat_lit 512) (instOfNatNat (nat_lit 512))) h)))
  (hw : @LE.le.{0} Int Int.instLEInt (@Nat.cast.{0} Int instNatCastInt lim.space) lim.word)
  (hd :
    @LE.le.{0} Nat instLENat
      (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat)
        (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) d
          (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat) J
            (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat)
              (@OfNat.ofNat.{0} Nat (nat_lit 512) (instOfNatNat (nat_lit 512))) h)))
        (@OfNat.ofNat.{0} Nat (nat_lit 5) (instOfNatNat (nat_lit 5))))
      lim.depth)
  (hocc : @LE.le.{0} Nat instLENat (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) occ J) coeff0)
  (hmaps :
    @LE.le.{0} Nat instLENat
      (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) maps
        (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat)
          (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat)
            (@OfNat.ofNat.{0} Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))
            (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat)
              (@OfNat.ofNat.{0} Nat (nat_lit 512) (instOfNatNat (nat_lit 512))) h))
          (ThreeSum.GlobalEncoding.scheduleSize h)))
      coeff0)
  (hcandidates :
    @LE.le.{0} Nat instLENat
      (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) candidates
        (ThreeSum.GlobalEncoding.scheduleSize h))
      coeff0)
  (hleft :
    @LE.le.{0} Nat instLENat
      (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) histL
        (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat) (ThreeSum.LightLocalCoverSchedule.rows h)
          (ThreeSum.LocalCoverBounds.localPrime h)))
      coeff0)
  (hright :
    @LE.le.{0} Nat instLENat
      (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) histR
        (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat) (ThreeSum.LightLocalCoverSchedule.rows h)
          (ThreeSum.LocalCoverBounds.localPrime h)))
      coeff0)
  (hmasks : @LE.le.{0} Nat instLENat (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) masks n) coeff0)
  (hc0 :
    @LE.le.{0} Nat instLENat
      (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) coeff0
        (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat) J
          (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat)
            (@OfNat.ofNat.{0} Nat (nat_lit 512) (instOfNatNat (nat_lit 512))) h)))
      coeff1)
  (hc1 :
    @LE.le.{0} Nat instLENat
      (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) coeff1
        (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat) J
          (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat)
            (@OfNat.ofNat.{0} Nat (nat_lit 512) (instOfNatNat (nat_lit 512))) h)))
      rawLeft)
  (hrawLeft :
    @LE.le.{0} Nat instLENat (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) rawLeft n) rawRight)
  (hrawRight :
    @LE.le.{0} Nat instLENat (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) rawRight n) flags)
  (hflags :
    @LE.le.{0} Nat instLENat
      (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat)
        (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) flags
          (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat)
            (@OfNat.ofNat.{0} Nat (nat_lit 2) (instOfNatNat (nat_lit 2))) n))
        (@OfNat.ofNat.{0} Nat (nat_lit 2) (instOfNatNat (nat_lit 2))))
      bitBase)
  (hbits :
    @LE.le.{0} Nat instLENat
      (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) bitBase
        (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat) J
          (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat)
            (@OfNat.ofNat.{0} Nat (nat_lit 512) (instOfNatNat (nat_lit 512))) h)))
      rowOut)
  (hrowOut :
    @LE.le.{0} Nat instLENat
      (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) rowOut
        (@OfNat.ofNat.{0} Nat (nat_lit 2) (instOfNatNat (nat_lit 2))))
      work)
  (hspace :
    @LE.le.{0} Nat instLENat
      (ThreeSum.LightSourcePreparation.space work J
        (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat)
          (@OfNat.ofNat.{0} Nat (nat_lit 512) (instOfNatNat (nat_lit 512))) h)
        (ThreeSum.LocalCoverBounds.localPrime h))
      lim.space)
  (hword :
    @LE.le.{0} Int Int.instLEInt
      (@Nat.cast.{0} Int instNatCastInt
        (ThreeSum.LightSourcePreparation.word J
          (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat)
            (@OfNat.ofNat.{0} Nat (nat_lit 512) (instOfNatNat (nat_lit 512))) h)
          (ThreeSum.LocalCoverBounds.localPrime h) (ThreeSum.LocalCoverBounds.localPrime h)
          ((@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat)
                (@OfNat.ofNat.{0} Nat (nat_lit 512) (instOfNatNat (nat_lit 512))) h).choose
            h)
          (ThreeSum.LocalCoverBounds.localBucketCap h)))
      lim.word)
  (hoccRead :
    ∀ (j : Nat),
      @LT.lt.{0} Nat instLTNat j J →
        @Eq.{1} Int (μ (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) occ j))
          (@Nat.cast.{0} Int instNatCastInt (@ThreeSum.BlockCoefficientRows.occurrenceIndices h J o j)))
  (hcandidateRead :
    ∀ (s : ThreeSum.GlobalEncoding.Occurrence h),
      @Eq.{1} Int
        (μ
          (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) candidates
            (@Fin.val (ThreeSum.GlobalEncoding.scheduleSize h) s)))
        (@Nat.cast.{0} Int instNatCastInt (ThreeSum.LightLocalCoverSchedule.candidateAt h s)))
  (hread0 :
    ∀ (s : Nat),
      @LT.lt.{0} Nat instLTNat s (ThreeSum.GlobalEncoding.scheduleSize h) →
        ∀ (a : Nat),
          @LT.lt.{0} Nat instLTNat a
              (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat)
                (@OfNat.ofNat.{0} Nat (nat_lit 512) (instOfNatNat (nat_lit 512))) h) →
            @Eq.{1} Int
              (μ
                (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat)
                  (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) maps
                    (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat)
                      (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat)
                        (@OfNat.ofNat.{0} Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))
                        (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat)
                          (@OfNat.ofNat.{0} Nat (nat_lit 512) (instOfNatNat (nat_lit 512))) h))
                      s))
                  a))
              (@Nat.cast.{0} Int instNatCastInt (ThreeSum.BlockCoefficientRows.mapRow h Bool.false s a)))
  (hread1 :
    ∀ (s : Nat),
      @LT.lt.{0} Nat instLTNat s (ThreeSum.GlobalEncoding.scheduleSize h) →
        ∀ (a : Nat),
          @LT.lt.{0} Nat instLTNat a
              (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat)
                (@OfNat.ofNat.{0} Nat (nat_lit 512) (instOfNatNat (nat_lit 512))) h) →
            @Eq.{1} Int
              (μ
                (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat)
                  (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat)
                    (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) maps
                      (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat)
                        (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat)
                          (@OfNat.ofNat.{0} Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))
                          (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat)
                            (@OfNat.ofNat.{0} Nat (nat_lit 512) (instOfNatNat (nat_lit 512))) h))
                        s))
                    (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat)
                      (@OfNat.ofNat.{0} Nat (nat_lit 512) (instOfNatNat (nat_lit 512))) h))
                  a))
              (@Nat.cast.{0} Int instNatCastInt (ThreeSum.BlockCoefficientRows.mapRow h Bool.true s a)))
  (hcountL :
    ∀ (k : Nat),
      @LT.lt.{0} Nat instLTNat k (ThreeSum.LightLocalCoverSchedule.rows h) →
        ∀ (r : Nat),
          @LT.lt.{0} Nat instLTNat r (ThreeSum.LocalCoverBounds.localPrime h) →
            @Eq.{1} Int
              (μ
                (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat)
                  (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) histL
                    (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat) k
                      (ThreeSum.LocalCoverBounds.localPrime h)))
                  r))
              (@Nat.cast.{0} Int instNatCastInt (ThreeSum.BlockSourceFlags.leftCount h k r)))
  (hcountR :
    ∀ (k : Nat),
      @LT.lt.{0} Nat instLTNat k (ThreeSum.LightLocalCoverSchedule.rows h) →
        ∀ (r : Nat),
          @LT.lt.{0} Nat instLTNat r (ThreeSum.LocalCoverBounds.localPrime h) →
            @Eq.{1} Int
              (μ
                (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat)
                  (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) histR
                    (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat) k
                      (ThreeSum.LocalCoverBounds.localPrime h)))
                  r))
              (@Nat.cast.{0} Int instNatCastInt (ThreeSum.BlockSourceFlags.rightCount h k r)))
  (hmaskRead :
    ∀ (i : Nat),
      @LT.lt.{0} Nat instLTNat i n →
        @Eq.{1} Int (μ (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) masks i))
          (@Nat.cast.{0} Int instNatCastInt
            (@ThreeSum.SubsetAddress.subsetMask
              (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat) J
                (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat)
                  (@OfNat.ofNat.{0} Nat (nat_lit 512) (instOfNatNat (nat_lit 512))) h))
              (@ThreeSum.BlockIndex.indexSet J
                (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat)
                  (@OfNat.ofNat.{0} Nat (nat_lit 512) (instOfNatNat (nat_lit 512))) h)
                (sets i))))),
  Light.Meets lim P pMain d
    (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt J)
      (@List.cons.{0} Int
        (@HMul.hMul.{0, 0, 0} Int Int Int (@instHMul.{0} Int Int.instMul)
          (@OfNat.ofNat.{0} Int (nat_lit 512) (@instOfNat (nat_lit 512))) (@Nat.cast.{0} Int instNatCastInt h))
        (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt h)
          (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt (ThreeSum.LocalCoverBounds.localPrime h))
            (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt (ThreeSum.LocalCoverBounds.localBucketCap h))
              (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt occ)
                (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt maps)
                  (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt candidates)
                    (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt histL)
                      (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt histR)
                        (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt masks)
                          (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt n)
                            (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt coeff0)
                              (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt coeff1)
                                (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt rawLeft)
                                  (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt rawRight)
                                    (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt flags)
                                      (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt bitBase)
                                        (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt rowOut)
                                          (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt work)
                                            (@List.nil.{0} Int)))))))))))))))))))))
    μ
    (ThreeSum.LightSourcePreparation.time n J
      (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat)
        (@OfNat.ofNat.{0} Nat (nat_lit 512) (instOfNatNat (nat_lit 512))) h)
      (ThreeSum.LocalCoverBounds.localPrime h))
    fun (x : Int) (ν : Nat → Int) =>
    And
      (∀ (i : Nat),
        @LT.lt.{0} Nat instLTNat i n →
          @Eq.{1} Int (ν (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) rawLeft i))
            (@ThreeSum.BlockEncoding.endpointLeft h J o (sets i)))
      (And
        (∀ (i : Nat),
          @LT.lt.{0} Nat instLTNat i n →
            @Eq.{1} Int (ν (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) rawRight i))
              (@ThreeSum.BlockEncoding.endpointRight h J o (sets i)))
        (And
          (∀ (i : Nat),
            @LT.lt.{0} Nat instLTNat i n →
              @Eq.{1} Int (ν (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) flags i))
                (@Nat.cast.{0} Int instNatCastInt
                  (@ite.{1} Nat (@ThreeSum.BlockEncoding.LeftAllowed h J o (sets i))
                    (@ThreeSum.BlockEncoding.instDecidableLeftAllowed h J o (sets i))
                    (@OfNat.ofNat.{0} Nat (nat_lit 1) (instOfNatNat (nat_lit 1)))
                    (@OfNat.ofNat.{0} Nat (nat_lit 0) (instOfNatNat (nat_lit 0))))))
          (And
            (∀ (i : Nat),
              @LT.lt.{0} Nat instLTNat i n →
                @Eq.{1} Int
                  (ν
                    (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat)
                      (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) flags n) i))
                  (@Nat.cast.{0} Int instNatCastInt
                    (@ite.{1} Nat (@ThreeSum.BlockEncoding.RightAllowed h J o (sets i))
                      (@ThreeSum.BlockEncoding.instDecidableRightAllowed h J o (sets i))
                      (@OfNat.ofNat.{0} Nat (nat_lit 1) (instOfNatNat (nat_lit 1)))
                      (@OfNat.ofNat.{0} Nat (nat_lit 0) (instOfNatNat (nat_lit 0))))))
            (Light.Kept μ ν coeff0)))) := by
  sorry

Source

https://github.com/c0ldzy17/three-sum-lean/blob/2b639655e07421b9ac743de9090459bab473a7c0/Submission/ThreeSum/LightBlockSourcePreparation.lean#L20-L89

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