Constructing and retaining both block-source zeta tables

Proved

ThreeSum.LightBlockSourceRetainedHost.Procs.meets

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

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

Let $h,J,c,p$ be natural numbers, $\delta$ an integer, and $q$ a global output. Put $W=512hJ$ and $K=\sum_{i\le c}\binom Wi$ , with $c\le W$ . Two arrays bijectively enumerate finite labeled inputs $X,Y$ , preserving their integer values. Assume their exact residue histograms, the occurrence indices of $q$ , and the local cover data are stored before the output region. Let $f,g$ be the corresponding left and shifted right block-source functions: admissibility-gated endpoint residue counts.

Assume the specified enumeration, preparation, and transform procedures, the ordered buffer layout, the retained-host space and word bounds, $S\le\Omega$ , and $d+W+9\le\Delta$ , where $S,\Omega,\Delta$ are the space, word magnitude, and depth limits. The call retains

\nu(b_L+jK+\rho(T))=Z_jf(T),\qquad \nu(b_R+jK+\rho(T))=Z_jg(T)\quad(j\le W,\ |T|\le c).

Here $\rho$ is compact subset rank and $Z_j$ sums over deletions among the last $j$ coordinates. It meets the retained-host time budget and preserves every cell before the mask-output base.

This constructs reusable source transforms from the finite inputs and stored cover.

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 Definitions.Def_ThreeSum1962Source_Bundle12
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.Fin
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_BlockSourceCoverInput_Input_of_kept
import Theorems.Thm_ThreeSum_LightBlockSourcePreparation_meets
import Theorems.Thm_ThreeSum_LightBlockSourceRetainedStages_meets_left
import Theorems.Thm_ThreeSum_LightBlockSourceRetainedStages_meets_right
import Theorems.Thm_ThreeSum_LightDownsetEnumerationSetup_meets
import Theorems.Thm_ThreeSum_SubsetAddress_layerRank_lt


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.BlockSourceEncoding
end ThreeSum.BlockSourceEncoding
namespace ThreeSum.BlockSourceFlags
end ThreeSum.BlockSourceFlags
namespace ThreeSum.GlobalEncoding
end ThreeSum.GlobalEncoding
namespace ThreeSum.LightBlockSourceBuild
end ThreeSum.LightBlockSourceBuild
namespace ThreeSum.LightPascal
end ThreeSum.LightPascal
namespace ThreeSum.LightSourceResidues
end ThreeSum.LightSourceResidues
namespace ThreeSum.LightSubsetAddress
end ThreeSum.LightSubsetAddress
namespace ThreeSum.LocalCoverBounds
end ThreeSum.LocalCoverBounds
namespace ThreeSum.LocalHashMoments
end ThreeSum.LocalHashMoments
namespace ThreeSum.SubsetAddress
end ThreeSum.SubsetAddress
namespace ThreeSumApsp
end ThreeSumApsp

universe u_1 u_2

Formal statement

theorem ThreeSum.LightBlockSourceRetainedHost.Procs.meets : ∀ {lim : Light.Limits} {P : Light.Program} {d : Nat} {β : Type u_1} {γ : Type u_2} [Fintype.{u_1} β] [Fintype.{u_2} γ]
  [DecidableEq.{u_1 + 1} β] [DecidableEq.{u_2 + 1} γ] {pMain : Nat}
  (C : ThreeSum.LightBlockSourceRetainedHost.Procs P pMain) {μ : Nat → Int}
  {h J cut p NL NR occ maps candidates histL histR sourceHistL sourceHistR masks coeff0 coeff1 rawLeft rawRight flags
    baseL baseR maskBase bitBase rowOut work : Nat}
  {shift : Int} (B : Finset.{u_1} β) (b : β → Int) (A : Finset.{u_2} γ) (a : γ → Int)
  (q : ThreeSum.GlobalEncoding.GlobalOutput h J) (vL vR : Nat → Int) (eL : Equiv.{1, u_1 + 1} (Fin NL) ↥B)
  (eR : Equiv.{1, u_2 + 1} (Fin NR) ↥A)
  (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 9) (instOfNatNat (nat_lit 9))))
      lim.depth)
  (hcut :
    @LE.le.{0} Nat instLENat cut
      (@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)))
  (hocc : @LE.le.{0} Nat instLENat (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) occ J) masks)
  (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)))
      masks)
  (hcandidates :
    @LE.le.{0} Nat instLENat
      (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) candidates
        (ThreeSum.GlobalEncoding.scheduleSize h))
      masks)
  (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)))
      masks)
  (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)))
      masks)
  (hsourceHistL :
    @LE.le.{0} Nat instLENat
      (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) sourceHistL
        (ThreeSum.BlockSourceEncoding.modulus h J))
      masks)
  (hsourceHistR :
    @LE.le.{0} Nat instLENat
      (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) sourceHistR
        (ThreeSum.BlockSourceEncoding.modulus h J))
      masks)
  (hmasks :
    @LE.le.{0} Nat instLENat
      (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) masks
        (ThreeSum.SubsetAddress.downsetCap
          (@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))
          cut))
      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
        (ThreeSum.SubsetAddress.downsetCap
          (@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))
          cut))
      rawRight)
  (hrawRight :
    @LE.le.{0} Nat instLENat
      (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) rawRight
        (ThreeSum.SubsetAddress.downsetCap
          (@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))
          cut))
      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)))
            (ThreeSum.SubsetAddress.downsetCap
              (@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))
              cut)))
        (@OfNat.ofNat.{0} Nat (nat_lit 2) (instOfNatNat (nat_lit 2))))
      baseL)
  (hbaseL :
    @LE.le.{0} Nat instLENat
      (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) baseL
        (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat)
          (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat)
            (@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 1) (instOfNatNat (nat_lit 1))))
          (ThreeSum.SubsetAddress.downsetCap
            (@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))
            cut)))
      baseR)
  (hbaseR :
    @LE.le.{0} Nat instLENat
      (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) baseR
        (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat)
          (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat)
            (@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 1) (instOfNatNat (nat_lit 1))))
          (ThreeSum.SubsetAddress.downsetCap
            (@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))
            cut)))
      maskBase)
  (hmaskBase :
    @LE.le.{0} Nat instLENat
      (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) maskBase
        (ThreeSum.SubsetAddress.downsetCap
          (@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))
          cut))
      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.LightBlockSourceRetainedHost.space work h J) lim.space)
  (hword :
    @LE.le.{0} Int Int.instLEInt
      (@Nat.cast.{0} Int instNatCastInt (ThreeSum.LightBlockSourceRetainedHost.word NL NR h J)) lim.word)
  (hinputL :
    ∀ (i : Fin NL),
      @Eq.{1} Int (vL (@Fin.val NL i))
        (b
          (@Subtype.val.{u_1 + 1} β
            (fun (x : β) =>
              @Membership.mem.{u_1, u_1} β (Finset.{u_1} β)
                (@SetLike.instMembership.{u_1, u_1} (Finset.{u_1} β) β (@Finset.instSetLike.{u_1} β)) B x)
            (@DFunLike.coe.{u_1 + 1, 1, u_1 + 1} (Equiv.{1, u_1 + 1} (Fin NL) ↥B) (Fin NL) (fun (x : Fin NL) => ↥B)
              (@EquivLike.toFunLike.{u_1 + 1, 1, u_1 + 1} (Equiv.{1, u_1 + 1} (Fin NL) ↥B) (Fin NL) (↥B)
                (@Equiv.instEquivLike.{1, u_1 + 1} (Fin NL) ↥B))
              eL i))))
  (hinputR :
    ∀ (i : Fin NR),
      @Eq.{1} Int (vR (@Fin.val NR i))
        (a
          (@Subtype.val.{u_2 + 1} γ
            (fun (x : γ) =>
              @Membership.mem.{u_2, u_2} γ (Finset.{u_2} γ)
                (@SetLike.instMembership.{u_2, u_2} (Finset.{u_2} γ) γ (@Finset.instSetLike.{u_2} γ)) A x)
            (@DFunLike.coe.{u_2 + 1, 1, u_2 + 1} (Equiv.{1, u_2 + 1} (Fin NR) ↥A) (Fin NR) (fun (x : Fin NR) => ↥A)
              (@EquivLike.toFunLike.{u_2 + 1, 1, u_2 + 1} (Equiv.{1, u_2 + 1} (Fin NR) ↥A) (Fin NR) (↥A)
                (@Equiv.instEquivLike.{1, u_2 + 1} (Fin NR) ↥A))
              eR i))))
  (hhistReadL :
    ∀ (r : Nat),
      @LT.lt.{0} Nat instLTNat r (ThreeSum.BlockSourceEncoding.modulus h J) →
        @Eq.{1} Int (μ (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) sourceHistL r))
          (@Nat.cast.{0} Int instNatCastInt
            (ThreeSum.LightBlockSourceRetainedStages.histogram p (ThreeSum.BlockSourceEncoding.modulus h J)
              (@OfNat.ofNat.{0} Int (nat_lit 0) (@instOfNat (nat_lit 0))) vL NL r)))
  (hhistReadR :
    ∀ (r : Nat),
      @LT.lt.{0} Nat instLTNat r (ThreeSum.BlockSourceEncoding.modulus h J) →
        @Eq.{1} Int (μ (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) sourceHistR r))
          (@Nat.cast.{0} Int instNatCastInt
            (ThreeSum.LightBlockSourceRetainedStages.histogram p (ThreeSum.BlockSourceEncoding.modulus h J) shift vR NR
              r)))
  (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 (@ThreeSum.BlockEncoding.outputOccurrences h J q) j)))
  (hcover : ThreeSum.BlockSourceCoverInput.Input μ h maps candidates histL histR),
  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 cut)
                (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt (ThreeSum.BlockSourceEncoding.modulus h J))
                  (@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 sourceHistL)
                              (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt sourceHistR)
                                (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt masks)
                                  (@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 baseL)
                                              (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt baseR)
                                                (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt maskBase)
                                                  (@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.LightBlockSourceRetainedHost.time h J cut) fun (x : Int) (ν : Nat → Int) =>
    And
      (@ThreeSum.LightZetaStages.StoredThrough
        (@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))
        ν baseL cut (@ThreeSum.BlockIndexedQuery.leftSource.{u_1} β h J (@Nat.cast.{0} Int instNatCastInt p) B b q)
        (@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)))
      (And
        (@ThreeSum.LightZetaStages.StoredThrough
          (@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))
          ν baseR cut
          (@ThreeSum.BlockIndexedQuery.rightSource.{u_2} γ h J (@Nat.cast.{0} Int instNatCastInt p) shift A a q)
          (@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)))
        (Light.Kept μ ν masks)) := by
  sorry

Source

https://github.com/c0ldzy17/three-sum-lean/blob/2b639655e07421b9ac743de9090459bab473a7c0/Submission/ThreeSum/LightBlockSourceRetainedHost.lean#L66-L217

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