Exact incidence coverage of the host reporting frontier

Proved

ThreeSum.ReportingHostFrontier.mem_frontier_iff

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

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

Fix natural block parameters $h,J$ and finite original-index types with decidable equality. Let $I$ be a finite set of raw items, each containing an original index triple, a retained choice in each block, and a signed-lift label. A wanted label $w$ records the original first index, lift label and global output. Write $\omega(r)$ for the label of an item and $v(r)$ for its length- $J$ word of left-subset choices.

The host frontier uses the actual ordered child enumeration and exact raw prefix counts. For every $k\le J$ and list $u$ ,

u\in\operatorname{frontier}(I,w,k) \iff \exists r\in I:\ \omega(r)=w\ \land\ \operatorname{take}_k(v(r))=u.

This identifies the prefixes retained by the host with exactly those supported by an original raw incidence.

Preamble

import Aesop
import Definitions.Def_APSPSource_RemainingDefinitions
import Definitions.Def_APSPSource_ThreeSumApsp_Lang_Logic
import Definitions.Def_APSPSource_ThreeSumApsp_Lang_Regions
import Definitions.Def_ThreeSum1962Source_Bundle01
import Definitions.Def_ThreeSum1962Source_Bundle04
import Definitions.Def_ThreeSum1962Source_Bundle07
import Definitions.Def_ThreeSum1962Source_Bundle08
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.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.Pow
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.Max
import Mathlib.Data.Finset.Powerset
import Mathlib.Data.Finset.Prod
import Mathlib.Data.Finset.Sigma
import Mathlib.Data.Finset.Sort
import Mathlib.Data.Fintype.BigOperators
import Mathlib.Data.Fintype.Card
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.Sublists
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.Notation
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.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_RawReporting_partitionWord_length
import Theorems.Thm_ThreeSum_RawReporting_word_children
import Theorems.Thm_ThreeSum_ReportingHostMasks_localChildren_mem
import Theorems.Thm_ThreeSum_Reporting_mem_frontier_iff


set_option autoImplicit false
set_option relaxedAutoImplicit false
set_option Elab.async false
namespace ThreeSum.BlockForced
end ThreeSum.BlockForced
namespace ThreeSum.GlobalEncoding
end ThreeSum.GlobalEncoding
namespace ThreeSum.LocalCoverBounds
end ThreeSum.LocalCoverBounds
namespace ThreeSum.RawIncidences
end ThreeSum.RawIncidences
namespace ThreeSum.RawReporting
end ThreeSum.RawReporting
namespace ThreeSum.ReportingHostMasks
end ThreeSum.ReportingHostMasks

universe u_1 u_2 u_3

Formal statement

theorem ThreeSum.ReportingHostFrontier.mem_frontier_iff : ∀ {α : Type u_1} {β : Type u_2} {γ : Type u_3} [Fintype.{u_1} α] [Fintype.{u_2} β] [Fintype.{u_3} γ]
  [inst : DecidableEq.{u_1 + 1} α] [DecidableEq.{u_2 + 1} β] [DecidableEq.{u_3 + 1} γ] {h J k : Nat}
  (I : Finset.{max (max u_3 u_2) u_1} (ThreeSum.RawIncidences.RawItem.{u_1, u_2, u_3} h J α β γ))
  (w : ThreeSum.RawReporting.Wanted.{u_1} h J α) (hk : @LE.le.{0} Nat instLENat k J)
  (pre :
    List.{0}
      (Finset.{0}
        (Fin
          (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat)
            (@OfNat.ofNat.{0} Nat (nat_lit 512) (instOfNatNat (nat_lit 512))) h)))),
  Iff
    (@Membership.mem.{0, 0}
      (List.{0}
        (Finset.{0}
          (Fin
            (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat)
              (@OfNat.ofNat.{0} Nat (nat_lit 512) (instOfNatNat (nat_lit 512))) h))))
      (List.{0}
        (List.{0}
          (Finset.{0}
            (Fin
              (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat)
                (@OfNat.ofNat.{0} Nat (nat_lit 512) (instOfNatNat (nat_lit 512))) h)))))
      (@List.instMembership.{0}
        (List.{0}
          (Finset.{0}
            (Fin
              (@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.ReportingHostFrontier.frontier.{u_1, u_2, u_3} α β γ inst h J I w k) pre)
    (∃ (r : ThreeSum.RawIncidences.RawItem.{u_1, u_2, u_3} h J α β γ),
      And
        (@Membership.mem.{max (max u_1 u_2) u_3, max (max u_1 u_2) u_3}
          (ThreeSum.RawIncidences.RawItem.{u_1, u_2, u_3} h J α β γ)
          (Finset.{max (max u_3 u_2) u_1} (ThreeSum.RawIncidences.RawItem.{u_1, u_2, u_3} h J α β γ))
          (@SetLike.instMembership.{max (max u_1 u_2) u_3, max (max u_1 u_2) u_3}
            (Finset.{max (max u_3 u_2) u_1} (ThreeSum.RawIncidences.RawItem.{u_1, u_2, u_3} h J α β γ))
            (ThreeSum.RawIncidences.RawItem.{u_1, u_2, u_3} h J α β γ)
            (@Finset.instSetLike.{max (max u_1 u_2) u_3} (ThreeSum.RawIncidences.RawItem.{u_1, u_2, u_3} h J α β γ)))
          I r)
        (And
          (@Eq.{u_1 + 1} (ThreeSum.RawReporting.Wanted.{u_1} h J α)
            (@ThreeSum.RawReporting.wantedOf.{u_1, u_2, u_3} h J α β γ r) w)
          (@Eq.{1}
            (List.{0}
              (Finset.{0}
                (Fin
                  (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat)
                    (@OfNat.ofNat.{0} Nat (nat_lit 512) (instOfNatNat (nat_lit 512))) h))))
            (@List.take.{0}
              (Finset.{0}
                (Fin
                  (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat)
                    (@OfNat.ofNat.{0} Nat (nat_lit 512) (instOfNatNat (nat_lit 512))) h)))
              k (@ThreeSum.RawReporting.word.{u_1, u_2, u_3} h J α β γ r))
            pre))) := by
  sorry

Source

https://github.com/c0ldzy17/three-sum-lean/blob/2b639655e07421b9ac743de9090459bab473a7c0/Submission/ThreeSum/ReportingHostFrontier.lean#L55-L68

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