Source support identifies old and post-height head moments

Proved

Erdos390.WholePaper.BankPaperRealization.bankPaperCanonicalSectionNinePostHeight_headMoments_eq_roundedSource_of_sourceState

by doctosil · Sep 17, 2026 · Mathlib c5ea003 (Lean v4.30.0)

analytic-number-theoryerdos-390erdos390-source-construction

Fix a head-simplex bridge whose pattern has the specified prime set and exponent, containing every prime at most $W$ , and whose active values lie in the complete guarded smooth row. Let the post-height baseline target inputs have literal physical endpoints and head targets equal, at every prime at most $W$ , to the factorization of the selector-tail target. Suppose the old top-frozen rounded selector has a selector source state: feasibility, integral rows, and valuation deficit supported on the prime band. Then the fresh post-height seed and old rounded seed have the same head moments:

\sum_m s_{\rm post}(m)v_q(\mathrm{value}(m))=\sum_m s_{\rm rounded}(m)v_q(\mathrm{value}(m))\qquad(q\le W\text{ prime}).

This verifies the head-moment condition required when replacing the rounded source by the fresh post-height seed.

Preamble

import Mathlib
import Definitions.Def_erdos390_source_bank_interface
import Definitions.Def_erdos390_dickman_step_function
import Theorems.Thm_MediumPNT
import Definitions.Def_erdos390_analytic_foundations_001
import Theorems.Thm_Erdos390_Full_DickmanBasic_kernel_product_bound
import Theorems.Thm_Erdos390_Full_DickmanBasic_rho_pos_on_zero_five
import Theorems.Thm_Erdos390_Full_DickmanBasic_kernel_secondDerivative_first_bound
import Definitions.Def_erdos390_analytic_foundations_002
import Definitions.Def_erdos390_analytic_foundations_005
import Definitions.Def_erdos390_analytic_foundations_006
import Definitions.Def_erdos390_analytic_foundations_007
import Definitions.Def_erdos390_analytic_foundations_008
import Definitions.Def_erdos390_analytic_foundations_009

universe u_1

Formal statement

theorem Erdos390.WholePaper.BankPaperRealization.bankPaperCanonicalSectionNinePostHeight_headMoments_eq_roundedSource_of_sourceState :
∀ {P : Finset.{0} Nat} {Band : Type u_1} [inst : Fintype.{u_1} Band] [inst_1 : DecidableEq.{u_1 + 1} Band]
  (B :
    @Erdos390.Full.PaperBridgeFit.BridgeData.{0, u_1} (Erdos390.Full.PaperHeadSimplex.Tag P) Band
      (@instFintypeOption.{0}
        (@Subtype.{1} Nat fun (p : Nat) =>
          @Membership.mem.{0, 0} Nat (Finset.{0} Nat)
            (@SetLike.instMembership.{0, 0} (Finset.{0} Nat) Nat (@Finset.instSetLike.{0} Nat)) P p)
        (@Finset.Subtype.fintype.{0} Nat P))
      (fun (a b : Erdos390.Full.PaperHeadSimplex.Tag P) =>
        @Option.instDecidableEq.{0}
          (@Subtype.{1} Nat fun (p : Nat) =>
            @Membership.mem.{0, 0} Nat (Finset.{0} Nat)
              (@SetLike.instMembership.{0, 0} (Finset.{0} Nat) Nat (@Finset.instSetLike.{0} Nat)) P p)
          (fun
              (a b :
                @Subtype.{1} Nat fun (p : Nat) =>
                  @Membership.mem.{0, 0} Nat (Finset.{0} Nat)
                    (@SetLike.instMembership.{0, 0} (Finset.{0} Nat) Nat (@Finset.instSetLike.{0} Nat)) P p) =>
            @Subtype.instDecidableEq.{1} Nat
              (@Membership.mem.{0, 0} Nat (Finset.{0} Nat)
                (@SetLike.instMembership.{0, 0} (Finset.{0} Nat) Nat (@Finset.instSetLike.{0} Nat)) P)
              instDecidableEqNat a b)
          a b)
      inst inst_1)
  {c : Real} {depth K : Nat}
  (R :
    Erdos390.WholePaper.BankPaperRealization
      (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.n.{0} (Erdos390.Full.PaperHeadSimplex.Tag P)
        (@instFintypeOption.{0}
          (@Subtype.{1} Nat fun (p : Nat) =>
            @Membership.mem.{0, 0} Nat (Finset.{0} Nat)
              (@SetLike.instMembership.{0, 0} (Finset.{0} Nat) Nat (@Finset.instSetLike.{0} Nat)) P p)
          (@Finset.Subtype.fintype.{0} Nat P))
        (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, u_1} (Erdos390.Full.PaperHeadSimplex.Tag P) Band
          (@instFintypeOption.{0}
            (@Subtype.{1} Nat fun (p : Nat) =>
              @Membership.mem.{0, 0} Nat (Finset.{0} Nat)
                (@SetLike.instMembership.{0, 0} (Finset.{0} Nat) Nat (@Finset.instSetLike.{0} Nat)) P p)
            (@Finset.Subtype.fintype.{0} Nat P))
          (fun (a b : Erdos390.Full.PaperHeadSimplex.Tag P) =>
            @Option.instDecidableEq.{0}
              (@Subtype.{1} Nat fun (p : Nat) =>
                @Membership.mem.{0, 0} Nat (Finset.{0} Nat)
                  (@SetLike.instMembership.{0, 0} (Finset.{0} Nat) Nat (@Finset.instSetLike.{0} Nat)) P p)
              (fun
                  (a b :
                    @Subtype.{1} Nat fun (p : Nat) =>
                      @Membership.mem.{0, 0} Nat (Finset.{0} Nat)
                        (@SetLike.instMembership.{0, 0} (Finset.{0} Nat) Nat (@Finset.instSetLike.{0} Nat)) P p) =>
                @Subtype.instDecidableEq.{1} Nat
                  (@Membership.mem.{0, 0} Nat (Finset.{0} Nat)
                    (@SetLike.instMembership.{0, 0} (Finset.{0} Nat) Nat (@Finset.instSetLike.{0} Nat)) P)
                  instDecidableEqNat a b)
              a b)
          inst inst_1 B))
      (Erdos390.WholePaper.upperEndpoint
        (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.n.{0} (Erdos390.Full.PaperHeadSimplex.Tag P)
          (@instFintypeOption.{0}
            (@Subtype.{1} Nat fun (p : Nat) =>
              @Membership.mem.{0, 0} Nat (Finset.{0} Nat)
                (@SetLike.instMembership.{0, 0} (Finset.{0} Nat) Nat (@Finset.instSetLike.{0} Nat)) P p)
            (@Finset.Subtype.fintype.{0} Nat P))
          (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, u_1} (Erdos390.Full.PaperHeadSimplex.Tag P) Band
            (@instFintypeOption.{0}
              (@Subtype.{1} Nat fun (p : Nat) =>
                @Membership.mem.{0, 0} Nat (Finset.{0} Nat)
                  (@SetLike.instMembership.{0, 0} (Finset.{0} Nat) Nat (@Finset.instSetLike.{0} Nat)) P p)
              (@Finset.Subtype.fintype.{0} Nat P))
            (fun (a b : Erdos390.Full.PaperHeadSimplex.Tag P) =>
              @Option.instDecidableEq.{0}
                (@Subtype.{1} Nat fun (p : Nat) =>
                  @Membership.mem.{0, 0} Nat (Finset.{0} Nat)
                    (@SetLike.instMembership.{0, 0} (Finset.{0} Nat) Nat (@Finset.instSetLike.{0} Nat)) P p)
                (fun
                    (a b :
                      @Subtype.{1} Nat fun (p : Nat) =>
                        @Membership.mem.{0, 0} Nat (Finset.{0} Nat)
                          (@SetLike.instMembership.{0, 0} (Finset.{0} Nat) Nat (@Finset.instSetLike.{0} Nat)) P p) =>
                  @Subtype.instDecidableEq.{1} Nat
                    (@Membership.mem.{0, 0} Nat (Finset.{0} Nat)
                      (@SetLike.instMembership.{0, 0} (Finset.{0} Nat) Nat (@Finset.instSetLike.{0} Nat)) P)
                    instDecidableEqNat a b)
                a b)
            inst inst_1 B))
        (Erdos390.WholePaper.upperTailLength c
          (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.n.{0} (Erdos390.Full.PaperHeadSimplex.Tag P)
            (@instFintypeOption.{0}
              (@Subtype.{1} Nat fun (p : Nat) =>
                @Membership.mem.{0, 0} Nat (Finset.{0} Nat)
                  (@SetLike.instMembership.{0, 0} (Finset.{0} Nat) Nat (@Finset.instSetLike.{0} Nat)) P p)
              (@Finset.Subtype.fintype.{0} Nat P))
            (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, u_1} (Erdos390.Full.PaperHeadSimplex.Tag P) Band
              (@instFintypeOption.{0}
                (@Subtype.{1} Nat fun (p : Nat) =>
                  @Membership.mem.{0, 0} Nat (Finset.{0} Nat)
                    (@SetLike.instMembership.{0, 0} (Finset.{0} Nat) Nat (@Finset.instSetLike.{0} Nat)) P p)
                (@Finset.Subtype.fintype.{0} Nat P))
              (fun (a b : Erdos390.Full.PaperHeadSimplex.Tag P) =>
                @Option.instDecidableEq.{0}
                  (@Subtype.{1} Nat fun (p : Nat) =>
                    @Membership.mem.{0, 0} Nat (Finset.{0} Nat)
                      (@SetLike.instMembership.{0, 0} (Finset.{0} Nat) Nat (@Finset.instSetLike.{0} Nat)) P p)
                  (fun
                      (a b :
                        @Subtype.{1} Nat fun (p : Nat) =>
                          @Membership.mem.{0, 0} Nat (Finset.{0} Nat)
                            (@SetLike.instMembership.{0, 0} (Finset.{0} Nat) Nat (@Finset.instSetLike.{0} Nat)) P p) =>
                    @Subtype.instDecidableEq.{1} Nat
                      (@Membership.mem.{0, 0} Nat (Finset.{0} Nat)
                        (@SetLike.instMembership.{0, 0} (Finset.{0} Nat) Nat (@Finset.instSetLike.{0} Nat)) P)
                      instDecidableEqNat a b)
                  a b)
              inst inst_1 B)))))
  (certificate :
    Erdos390.WholePaper.GuardedCentralAnchorCertificate c depth
      (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.n.{0} (Erdos390.Full.PaperHeadSimplex.Tag P)
        (@instFintypeOption.{0}
          (@Subtype.{1} Nat fun (p : Nat) =>
            @Membership.mem.{0, 0} Nat (Finset.{0} Nat)
              (@SetLike.instMembership.{0, 0} (Finset.{0} Nat) Nat (@Finset.instSetLike.{0} Nat)) P p)
          (@Finset.Subtype.fintype.{0} Nat P))
        (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, u_1} (Erdos390.Full.PaperHeadSimplex.Tag P) Band
          (@instFintypeOption.{0}
            (@Subtype.{1} Nat fun (p : Nat) =>
              @Membership.mem.{0, 0} Nat (Finset.{0} Nat)
                (@SetLike.instMembership.{0, 0} (Finset.{0} Nat) Nat (@Finset.instSetLike.{0} Nat)) P p)
            (@Finset.Subtype.fintype.{0} Nat P))
          (fun (a b : Erdos390.Full.PaperHeadSimplex.Tag P) =>
            @Option.instDecidableEq.{0}
              (@Subtype.{1} Nat fun (p : Nat) =>
                @Membership.mem.{0, 0} Nat (Finset.{0} Nat)
                  (@SetLike.instMembership.{0, 0} (Finset.{0} Nat) Nat (@Finset.instSetLike.{0} Nat)) P p)
              (fun
                  (a b :
                    @Subtype.{1} Nat fun (p : Nat) =>
                      @Membership.mem.{0, 0} Nat (Finset.{0} Nat)
                        (@SetLike.instMembership.{0, 0} (Finset.{0} Nat) Nat (@Finset.instSetLike.{0} Nat)) P p) =>
                @Subtype.instDecidableEq.{1} Nat
                  (@Membership.mem.{0, 0} Nat (Finset.{0} Nat)
                    (@SetLike.instMembership.{0, 0} (Finset.{0} Nat) Nat (@Finset.instSetLike.{0} Nat)) P)
                  instDecidableEqNat a b)
              a b)
          inst inst_1 B))
      (@Erdos390.WholePaper.BankPaperRealization.anchorGuardLeftCore
        (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.n.{0} (Erdos390.Full.PaperHeadSimplex.Tag P)
          (@instFintypeOption.{0}
            (@Subtype.{1} Nat fun (p : Nat) =>
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                (@SetLike.instMembership.{0, 0} (Finset.{0} Nat) Nat (@Finset.instSetLike.{0} Nat)) P p)
            (@Finset.Subtype.fintype.{0} Nat P))
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                @Membership.mem.{0, 0} Nat (Finset.{0} Nat)
                  (@SetLike.instMembership.{0, 0} (Finset.{0} Nat) Nat (@Finset.instSetLike.{0} Nat)) P p)
              (@Finset.Subtype.fintype.{0} Nat P))
            (fun (a b : Erdos390.Full.PaperHeadSimplex.Tag P) =>
              @Option.instDecidableEq.{0}
                (@Subtype.{1} Nat fun (p : Nat) =>
                  @Membership.mem.{0, 0} Nat (Finset.{0} Nat)
                    (@SetLike.instMembership.{0, 0} (Finset.{0} Nat) Nat (@Finset.instSetLike.{0} Nat)) P p)
                (fun
                    (a b :
                      @Subtype.{1} Nat fun (p : Nat) =>
                        @Membership.mem.{0, 0} Nat (Finset.{0} Nat)
                          (@SetLike.instMembership.{0, 0} (Finset.{0} Nat) Nat (@Finset.instSetLike.{0} Nat)) P p) =>
                  @Subtype.instDecidableEq.{1} Nat
                    (@Membership.mem.{0, 0} Nat (Finset.{0} Nat)
                      (@SetLike.instMembership.{0, 0} (Finset.{0} Nat) Nat (@Finset.instSetLike.{0} Nat)) P)
                    instDecidableEqNat a b)
                a b)
            inst inst_1 B))
        (Erdos390.WholePaper.upperEndpoint
          (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.n.{0} (Erdos390.Full.PaperHeadSimplex.Tag P)
            (@instFintypeOption.{0}
              (@Subtype.{1} Nat fun (p : Nat) =>
                @Membership.mem.{0, 0} Nat (Finset.{0} Nat)
                  (@SetLike.instMembership.{0, 0} (Finset.{0} Nat) Nat (@Finset.instSetLike.{0} Nat)) P p)
              (@Finset.Subtype.fintype.{0} Nat P))
            (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, u_1} (Erdos390.Full.PaperHeadSimplex.Tag P) Band
              (@instFintypeOption.{0}
                (@Subtype.{1} Nat fun (p : Nat) =>
                  @Membership.mem.{0, 0} Nat (Finset.{0} Nat)
                    (@SetLike.instMembership.{0, 0} (Finset.{0} Nat) Nat (@Finset.instSetLike.{0} Nat)) P p)
                (@Finset.Subtype.fintype.{0} Nat P))
              (fun (a b : Erdos390.Full.PaperHeadSimplex.Tag P) =>
                @Option.instDecidableEq.{0}
                  (@Subtype.{1} Nat fun (p : Nat) =>
                    @Membership.mem.{0, 0} Nat (Finset.{0} Nat)
                      (@SetLike.instMembership.{0, 0} (Finset.{0} Nat) Nat (@Finset.instSetLike.{0} Nat)) P p)
                  (fun
                      (a b :
                        @Subtype.{1} Nat fun (p : Nat) =>
                          @Membership.mem.{0, 0} Nat (Finset.{0} Nat)
                            (@SetLike.instMembership.{0, 0} (Finset.{0} Nat) Nat (@Finset.instSetLike.{0} Nat)) P p) =>
                    @Subtype.instDecidableEq.{1} Nat
                      (@Membership.mem.{0, 0} Nat (Finset.{0} Nat)
                        (@SetLike.instMembership.{0, 0} (Finset.{0} Nat) Nat (@Finset.instSetLike.{0} Nat)) P)
                      instDecidableEqNat a b)
                  a b)
              inst inst_1 B))
          (Erdos390.WholePaper.upperTailLength c
            (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.n.{0} (Erdos390.Full.PaperHeadSimplex.Tag P)
              (@instFintypeOption.{0}
                (@Subtype.{1} Nat fun (p : Nat) =>
                  @Membership.mem.{0, 0} Nat (Finset.{0} Nat)
                    (@SetLike.instMembership.{0, 0} (Finset.{0} Nat) Nat (@Finset.instSetLike.{0} Nat)) P p)
                (@Finset.Subtype.fintype.{0} Nat P))
              (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, u_1} (Erdos390.Full.PaperHeadSimplex.Tag P) Band
                (@instFintypeOption.{0}
                  (@Subtype.{1} Nat fun (p : Nat) =>
                    @Membership.mem.{0, 0} Nat (Finset.{0} Nat)
                      (@SetLike.instMembership.{0, 0} (Finset.{0} Nat) Nat (@Finset.instSetLike.{0} Nat)) P p)
                  (@Finset.Subtype.fintype.{0} Nat P))
                (fun (a b : Erdos390.Full.PaperHeadSimplex.Tag P) =>
                  @Option.instDecidableEq.{0}
                    (@Subtype.{1} Nat fun (p : Nat) =>
                      @Membership.mem.{0, 0} Nat (Finset.{0} Nat)
                        (@SetLike.instMembership.{0, 0} (Finset.{0} Nat) Nat (@Finset.instSetLike.{0} Nat)) P p)
                    (fun
                        (a b :
                          @Subtype.{1} Nat fun (p : Nat) =>
                            @Membership.mem.{0, 0} Nat (Finset.{0} Nat)
                              (@SetLike.instMembership.{0, 0} (Finset.{0} Nat) Nat (@Finset.instSetLike.{0} Nat)) P
                              p) =>
                      @Subtype.instDecidableEq.{1} Nat
                        (@Membership.mem.{0, 0} Nat (Finset.{0} Nat)
                          (@SetLike.instMembership.{0, 0} (Finset.{0} Nat) Nat (@Finset.instSetLike.{0} Nat)) P)
                        instDecidableEqNat a b)
                    a b)
                inst inst_1 B))))
        R)
      (@Erdos390.WholePaper.BankPaperRealization.anchorGuardRightCore
        (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.n.{0} (Erdos390.Full.PaperHeadSimplex.Tag P)
          (@instFintypeOption.{0}
            (@Subtype.{1} Nat fun (p : Nat) =>
              @Membership.mem.{0, 0} Nat (Finset.{0} Nat)
                (@SetLike.instMembership.{0, 0} (Finset.{0} Nat) Nat (@Finset.instSetLike.{0} Nat)) P p)
            (@Finset.Subtype.fintype.{0} Nat P))
          (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, u_1} (Erdos390.Full.PaperHeadSimplex.Tag P) Band
            (@instFintypeOption.{0}
              (@Subtype.{1} Nat fun (p : Nat) =>
                @Membership.mem.{0, 0} Nat (Finset.{0} Nat)
                  (@SetLike.instMembership.{0, 0} (Finset.{0} Nat) Nat (@Finset.instSetLike.{0} Nat)) P p)
              (@Finset.Subtype.fintype.{0} Nat P))
            (fun (a b : Erdos390.Full.PaperHeadSimplex.Tag P) =>
              @Option.instDecidableEq.{0}
                (@Subtype.{1} Nat fun (p : Nat) =>
                  @Membership.mem.{0, 0} Nat (Finset.{0} Nat)
                    (@SetLike.instMembership.{0, 0} (Finset.{0} Nat) Nat (@Finset.instSetLike.{0} Nat)) P p)
                (fun
                    (a b :
                      @Subtype.{1} Nat fun (p : Nat) =>
                        @Membership.mem.{0, 0} Nat (Finset.{0} Nat)
                          (@SetLike.instMembership.{0, 0} (Finset.{0} Nat) Nat (@Finset.instSetLike.{0} Nat)) P p) =>
                  @Subtype.instDecidableEq.{1} Nat
                    (@Membership.mem.{0, 0} Nat (Finset.{0} Nat)
                      (@SetLike.instMembership.{0, 0} (Finset.{0} Nat) Nat (@Finset.instSetLike.{0} Nat)) P)
                    instDecidableEqNat a b)
                a b)
            inst inst_1 B))
        (Erdos390.WholePaper.upperEndpoint
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            (@instFintypeOption.{0}
              (@Subtype.{1} Nat fun (p : Nat) =>
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                  (fun
                      (a b :
                        @Subtype.{1} Nat fun (p : Nat) =>
                          @Membership.mem.{0, 0} Nat (Finset.{0} Nat)
                            (@SetLike.instMembership.{0, 0} (Finset.{0} Nat) Nat (@Finset.instSetLike.{0} Nat)) P p) =>
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                        (@SetLike.instMembership.{0, 0} (Finset.{0} Nat) Nat (@Finset.instSetLike.{0} Nat)) P)
                      instDecidableEqNat a b)
                  a b)
              inst inst_1 B))
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                        (@SetLike.instMembership.{0, 0} (Finset.{0} Nat) Nat (@Finset.instSetLike.{0} Nat)) P p)
                    (fun
                        (a b :
                          @Subtype.{1} Nat fun (p : Nat) =>
                            @Membership.mem.{0, 0} Nat (Finset.{0} Nat)
                              (@SetLike.instMembership.{0, 0} (Finset.{0} Nat) Nat (@Finset.instSetLike.{0} Nat)) P
                              p) =>
                      @Subtype.instDecidableEq.{1} Nat
                        (@Membership.mem.{0, 0} Nat (Finset.{0} Nat)
                          (@SetLike.instMembership.{0, 0} (Finset.{0} Nat) Nat (@Finset.instSetLike.{0} Nat)) P)
                        instDecidableEqNat a b)
                    a b)
                inst inst_1 B))))
        R)
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            (fun (a b : Erdos390.Full.PaperHeadSimplex.Tag P) =>
              @Option.instDecidableEq.{0}
                (@Subtype.{1} Nat fun (p : Nat) =>
                  @Membership.mem.{0, 0} Nat (Finset.{0} Nat)
                    (@SetLike.instMembership.{0, 0} (Finset.{0} Nat) Nat (@Finset.instSetLike.{0} Nat)) P p)
                (fun
                    (a b :
                      @Subtype.{1} Nat fun (p : Nat) =>
                        @Membership.mem.{0, 0} Nat (Finset.{0} Nat)
                          (@SetLike.instMembership.{0, 0} (Finset.{0} Nat) Nat (@Finset.instSetLike.{0} Nat)) P p) =>
                  @Subtype.instDecidableEq.{1} Nat
                    (@Membership.mem.{0, 0} Nat (Finset.{0} Nat)
                      (@SetLike.instMembership.{0, 0} (Finset.{0} Nat) Nat (@Finset.instSetLike.{0} Nat)) P)
                    instDecidableEqNat a b)
                a b)
            inst inst_1 B))
        (Erdos390.WholePaper.upperEndpoint
          (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.n.{0} (Erdos390.Full.PaperHeadSimplex.Tag P)
            (@instFintypeOption.{0}
              (@Subtype.{1} Nat fun (p : Nat) =>
                @Membership.mem.{0, 0} Nat (Finset.{0} Nat)
                  (@SetLike.instMembership.{0, 0} (Finset.{0} Nat) Nat (@Finset.instSetLike.{0} Nat)) P p)
              (@Finset.Subtype.fintype.{0} Nat P))
            (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, u_1} (Erdos390.Full.PaperHeadSimplex.Tag P) Band
              (@instFintypeOption.{0}
                (@Subtype.{1} Nat fun (p : Nat) =>
                  @Membership.mem.{0, 0} Nat (Finset.{0} Nat)
                    (@SetLike.instMembership.{0, 0} (Finset.{0} Nat) Nat (@Finset.instSetLike.{0} Nat)) P p)
                (@Finset.Subtype.fintype.{0} Nat P))
              (fun (a b : Erdos390.Full.PaperHeadSimplex.Tag P) =>
                @Option.instDecidableEq.{0}
                  (@Subtype.{1} Nat fun (p : Nat) =>
                    @Membership.mem.{0, 0} Nat (Finset.{0} Nat)
                      (@SetLike.instMembership.{0, 0} (Finset.{0} Nat) Nat (@Finset.instSetLike.{0} Nat)) P p)
                  (fun
                      (a b :
                        @Subtype.{1} Nat fun (p : Nat) =>
                          @Membership.mem.{0, 0} Nat (Finset.{0} Nat)
                            (@SetLike.instMembership.{0, 0} (Finset.{0} Nat) Nat (@Finset.instSetLike.{0} Nat)) P p) =>
                    @Subtype.instDecidableEq.{1} Nat
                      (@Membership.mem.{0, 0} Nat (Finset.{0} Nat)
                        (@SetLike.instMembership.{0, 0} (Finset.{0} Nat) Nat (@Finset.instSetLike.{0} Nat)) P)
                      instDecidableEqNat a b)
                  a b)
              inst inst_1 B))
          (Erdos390.WholePaper.upperTailLength c
            (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.n.{0} (Erdos390.Full.PaperHeadSimplex.Tag P)
              (@instFintypeOption.{0}
                (@Subtype.{1} Nat fun (p : Nat) =>
                  @Membership.mem.{0, 0} Nat (Finset.{0} Nat)
                    (@SetLike.instMembership.{0, 0} (Finset.{0} Nat) Nat (@Finset.instSetLike.{0} Nat)) P p)
                (@Finset.Subtype.fintype.{0} Nat P))
              (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, u_1} (Erdos390.Full.PaperHeadSimplex.Tag P) Band
                (@instFintypeOption.{0}
                  (@Subtype.{1} Nat fun (p : Nat) =>
                    @Membership.mem.{0, 0} Nat (Finset.{0} Nat)
                      (@SetLike.instMembership.{0, 0} (Finset.{0} Nat) Nat (@Finset.instSetLike.{0} Nat)) P p)
                  (@Finset.Subtype.fintype.{0} Nat P))
                (fun (a b : Erdos390.Full.PaperHeadSimplex.Tag P) =>
                  @Option.instDecidableEq.{0}
                    (@Subtype.{1} Nat fun (p : Nat) =>
                      @Membership.mem.{0, 0} Nat (Finset.{0} Nat)
                        (@SetLike.instMembership.{0, 0} (Finset.{0} Nat) Nat (@Finset.instSetLike.{0} Nat)) P p)
                    (fun
                        (a b :
                          @Subtype.{1} Nat fun (p : Nat) =>
                            @Membership.mem.{0, 0} Nat (Finset.{0} Nat)
                              (@SetLike.instMembership.{0, 0} (Finset.{0} Nat) Nat (@Finset.instSetLike.{0} Nat)) P
                              p) =>
                      @Subtype.instDecidableEq.{1} Nat
                        (@Membership.mem.{0, 0} Nat (Finset.{0} Nat)
                          (@SetLike.instMembership.{0, 0} (Finset.{0} Nat) Nat (@Finset.instSetLike.{0} Nat)) P)
                        instDecidableEqNat a b)
                    a b)
                inst inst_1 B))))
        R depth))
  (Tsource :
    @Erdos390.Full.PaperGuardCensus.BarycentricTarget.{0} (Erdos390.Full.PaperHeadSimplex.Tag P)
      (@instFintypeOption.{0}
        (@Subtype.{1} Nat fun (p : Nat) =>
          @Membership.mem.{0, 0} Nat (Finset.{0} Nat)
            (@SetLike.instMembership.{0, 0} (Finset.{0} Nat) Nat (@Finset.instSetLike.{0} Nat)) P p)
        (@Finset.Subtype.fintype.{0} Nat P))
      (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, u_1} (Erdos390.Full.PaperHeadSimplex.Tag P) Band
        (@instFintypeOption.{0}
          (@Subtype.{1} Nat fun (p : Nat) =>
            @Membership.mem.{0, 0} Nat (Finset.{0} Nat)
              (@SetLike.instMembership.{0, 0} (Finset.{0} Nat) Nat (@Finset.instSetLike.{0} Nat)) P p)
          (@Finset.Subtype.fintype.{0} Nat P))
        (fun (a b : Erdos390.Full.PaperHeadSimplex.Tag P) =>
          @Option.instDecidableEq.{0}
            (@Subtype.{1} Nat fun (p : Nat) =>
              @Membership.mem.{0, 0} Nat (Finset.{0} Nat)
                (@SetLike.instMembership.{0, 0} (Finset.{0} Nat) Nat (@Finset.instSetLike.{0} Nat)) P p)
            (fun
                (a b :
                  @Subtype.{1} Nat fun (p : Nat) =>
                    @Membership.mem.{0, 0} Nat (Finset.{0} Nat)
                      (@SetLike.instMembership.{0, 0} (Finset.{0} Nat) Nat (@Finset.instSetLike.{0} Nat)) P p) =>
              @Subtype.instDecidableEq.{1} Nat
                (@Membership.mem.{0, 0} Nat (Finset.{0} Nat)
                  (@SetLike.instMembership.{0, 0} (Finset.{0} Nat) Nat (@Finset.instSetLike.{0} Nat)) P)
                instDecidableEqNat a b)
            a b)
        inst inst_1 B))
  (I : Erdos390.Full.PaperGuardCensus.PhysicalIntervals)
  (hlo :
    ∀ (sigma : Erdos390.Full.PaperBridgeFit.PhysicalSign),
      @Eq.{1} Nat
        (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.lo.{0} (Erdos390.Full.PaperHeadSimplex.Tag P)
          (@instFintypeOption.{0}
            (@Subtype.{1} Nat fun (p : Nat) =>
              @Membership.mem.{0, 0} Nat (Finset.{0} Nat)
                (@SetLike.instMembership.{0, 0} (Finset.{0} Nat) Nat (@Finset.instSetLike.{0} Nat)) P p)
            (@Finset.Subtype.fintype.{0} Nat P))
          (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, u_1} (Erdos390.Full.PaperHeadSimplex.Tag P) Band
            (@instFintypeOption.{0}
              (@Subtype.{1} Nat fun (p : Nat) =>
                @Membership.mem.{0, 0} Nat (Finset.{0} Nat)
                  (@SetLike.instMembership.{0, 0} (Finset.{0} Nat) Nat (@Finset.instSetLike.{0} Nat)) P p)
              (@Finset.Subtype.fintype.{0} Nat P))
            (fun (a b : Erdos390.Full.PaperHeadSimplex.Tag P) =>
              @Option.instDecidableEq.{0}
                (@Subtype.{1} Nat fun (p : Nat) =>
                  @Membership.mem.{0, 0} Nat (Finset.{0} Nat)
                    (@SetLike.instMembership.{0, 0} (Finset.{0} Nat) Nat (@Finset.instSetLike.{0} Nat)) P p)
                (fun
                    (a b :
                      @Subtype.{1} Nat fun (p : Nat) =>
                        @Membership.mem.{0, 0} Nat (Finset.{0} Nat)
                          (@SetLike.instMembership.{0, 0} (Finset.{0} Nat) Nat (@Finset.instSetLike.{0} Nat)) P p) =>
                  @Subtype.instDecidableEq.{1} Nat
                    (@Membership.mem.{0, 0} Nat (Finset.{0} Nat)
                      (@SetLike.instMembership.{0, 0} (Finset.{0} Nat) Nat (@Finset.instSetLike.{0} Nat)) P)
                    instDecidableEqNat a b)
                a b)
            inst inst_1 B)
          sigma)
        (Erdos390.Full.ArithmeticModel.physicalBound (I.lower sigma)
          (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.n.{0} (Erdos390.Full.PaperHeadSimplex.Tag P)
            (@instFintypeOption.{0}
              (@Subtype.{1} Nat fun (p : Nat) =>
                @Membership.mem.{0, 0} Nat (Finset.{0} Nat)
                  (@SetLike.instMembership.{0, 0} (Finset.{0} Nat) Nat (@Finset.instSetLike.{0} Nat)) P p)
              (@Finset.Subtype.fintype.{0} Nat P))
            (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, u_1} (Erdos390.Full.PaperHeadSimplex.Tag P) Band
              (@instFintypeOption.{0}
                (@Subtype.{1} Nat fun (p : Nat) =>
                  @Membership.mem.{0, 0} Nat (Finset.{0} Nat)
                    (@SetLike.instMembership.{0, 0} (Finset.{0} Nat) Nat (@Finset.instSetLike.{0} Nat)) P p)
                (@Finset.Subtype.fintype.{0} Nat P))
              (fun (a b : Erdos390.Full.PaperHeadSimplex.Tag P) =>
                @Option.instDecidableEq.{0}
                  (@Subtype.{1} Nat fun (p : Nat) =>
                    @Membership.mem.{0, 0} Nat (Finset.{0} Nat)
                      (@SetLike.instMembership.{0, 0} (Finset.{0} Nat) Nat (@Finset.instSetLike.{0} Nat)) P p)
                  (fun
                      (a b :
                        @Subtype.{1} Nat fun (p : Nat) =>
                          @Membership.mem.{0, 0} Nat (Finset.{0} Nat)
                            (@SetLike.instMembership.{0, 0} (Finset.{0} Nat) Nat (@Finset.instSetLike.{0} Nat)) P p) =>
                    @Subtype.instDecidableEq.{1} Nat
                      (@Membership.mem.{0, 0} Nat (Finset.{0} Nat)
                        (@SetLike.instMembership.{0, 0} (Finset.{0} Nat) Nat (@Finset.instSetLike.{0} Nat)) P)
                      instDecidableEqNat a b)
                  a b)
              inst inst_1 B))))
  (hhi :
    ∀ (sigma : Erdos390.Full.PaperBridgeFit.PhysicalSign),
      @Eq.{1} Nat
        (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.hi.{0} (Erdos390.Full.PaperHeadSimplex.Tag P)
          (@instFintypeOption.{0}
            (@Subtype.{1} Nat fun (p : Nat) =>
              @Membership.mem.{0, 0} Nat (Finset.{0} Nat)
                (@SetLike.instMembership.{0, 0} (Finset.{0} Nat) Nat (@Finset.instSetLike.{0} Nat)) P p)
            (@Finset.Subtype.fintype.{0} Nat P))
          (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, u_1} (Erdos390.Full.PaperHeadSimplex.Tag P) Band
            (@instFintypeOption.{0}
              (@Subtype.{1} Nat fun (p : Nat) =>
                @Membership.mem.{0, 0} Nat (Finset.{0} Nat)
                  (@SetLike.instMembership.{0, 0} (Finset.{0} Nat) Nat (@Finset.instSetLike.{0} Nat)) P p)
              (@Finset.Subtype.fintype.{0} Nat P))
            (fun (a b : Erdos390.Full.PaperHeadSimplex.Tag P) =>
              @Option.instDecidableEq.{0}
                (@Subtype.{1} Nat fun (p : Nat) =>
                  @Membership.mem.{0, 0} Nat (Finset.{0} Nat)
                    (@SetLike.instMembership.{0, 0} (Finset.{0} Nat) Nat (@Finset.instSetLike.{0} Nat)) P p)
                (fun
                    (a b :
                      @Subtype.{1} Nat fun (p : Nat) =>
                        @Membership.mem.{0, 0} Nat (Finset.{0} Nat)
                          (@SetLike.instMembership.{0, 0} (Finset.{0} Nat) Nat (@Finset.instSetLike.{0} Nat)) P p) =>
                  @Subtype.instDecidableEq.{1} Nat
                    (@Membership.mem.{0, 0} Nat (Finset.{0} Nat)
                      (@SetLike.instMembership.{0, 0} (Finset.{0} Nat) Nat (@Finset.instSetLike.{0} Nat)) P)
                    instDecidableEqNat a b)
                a b)
            inst inst_1 B)
          sigma)
        (Erdos390.Full.ArithmeticModel.physicalBound (I.upper sigma)
          (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.n.{0} (Erdos390.Full.PaperHeadSimplex.Tag P)
            (@instFintypeOption.{0}
              (@Subtype.{1} Nat fun (p : Nat) =>
                @Membership.mem.{0, 0} Nat (Finset.{0} Nat)
                  (@SetLike.instMembership.{0, 0} (Finset.{0} Nat) Nat (@Finset.instSetLike.{0} Nat)) P p)
              (@Finset.Subtype.fintype.{0} Nat P))
            (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, u_1} (Erdos390.Full.PaperHeadSimplex.Tag P) Band
              (@instFintypeOption.{0}
                (@Subtype.{1} Nat fun (p : Nat) =>
                  @Membership.mem.{0, 0} Nat (Finset.{0} Nat)
                    (@SetLike.instMembership.{0, 0} (Finset.{0} Nat) Nat (@Finset.instSetLike.{0} Nat)) P p)
                (@Finset.Subtype.fintype.{0} Nat P))
              (fun (a b : Erdos390.Full.PaperHeadSimplex.Tag P) =>
                @Option.instDecidableEq.{0}
                  (@Subtype.{1} Nat fun (p : Nat) =>
                    @Membership.mem.{0, 0} Nat (Finset.{0} Nat)
                      (@SetLike.instMembership.{0, 0} (Finset.{0} Nat) Nat (@Finset.instSetLike.{0} Nat)) P p)
                  (fun
                      (a b :
                        @Subtype.{1} Nat fun (p : Nat) =>
                          @Membership.mem.{0, 0} Nat (Finset.{0} Nat)
                            (@SetLike.instMembership.{0, 0} (Finset.{0} Nat) Nat (@Finset.instSetLike.{0} Nat)) P p) =>
                    @Subtype.instDecidableEq.{1} Nat
                      (@Membership.mem.{0, 0} Nat (Finset.{0} Nat)
                        (@SetLike.instMembership.{0, 0} (Finset.{0} Nat) Nat (@Finset.instSetLike.{0} Nat)) P)
                      instDecidableEqNat a b)
                  a b)
              inst inst_1 B))))
  {q0 A0 : Real} {d : Int} {exponent : Nat}
  {activeHeadTarget :
    (@Subtype.{1} Nat fun (p : Nat) =>
        @Membership.mem.{0, 0} Nat (Finset.{0} Nat)
          (@SetLike.instMembership.{0, 0} (Finset.{0} Nat) Nat (@Finset.instSetLike.{0} Nat)) P p) →
      Real}
  (H :
    @Erdos390.WholePaper.BankPaperCanonicalSectionNinePostHeightBaselineTargetInputs.{u_1} P Band inst inst_1 B I q0 A0
      d exponent activeHeadTarget)
  (deltaStar betaProt alpha beta qTilde : Real)
  (hprime :
    ∀ (p : Nat),
      @Membership.mem.{0, 0} Nat (Finset.{0} Nat)
          (@SetLike.instMembership.{0, 0} (Finset.{0} Nat) Nat (@Finset.instSetLike.{0} Nat)) P p →
        Nat.Prime p),
  @Eq.{1} (Erdos390.Full.PaperHeadSimplex.Tag P → Erdos390.Full.HeadPattern.Pattern)
      (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.pattern.{0} (Erdos390.Full.PaperHeadSimplex.Tag P)
        (@instFintypeOption.{0}
          (@Subtype.{1} Nat fun (p : Nat) =>
            @Membership.mem.{0, 0} Nat (Finset.{0} Nat)
              (@SetLike.instMembership.{0, 0} (Finset.{0} Nat) Nat (@Finset.instSetLike.{0} Nat)) P p)
          (@Finset.Subtype.fintype.{0} Nat P))
        (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, u_1} (Erdos390.Full.PaperHeadSimplex.Tag P) Band
          (@instFintypeOption.{0}
            (@Subtype.{1} Nat fun (p : Nat) =>
              @Membership.mem.{0, 0} Nat (Finset.{0} Nat)
                (@SetLike.instMembership.{0, 0} (Finset.{0} Nat) Nat (@Finset.instSetLike.{0} Nat)) P p)
            (@Finset.Subtype.fintype.{0} Nat P))
          (fun (a b : Erdos390.Full.PaperHeadSimplex.Tag P) =>
            @Option.instDecidableEq.{0}
              (@Subtype.{1} Nat fun (p : Nat) =>
                @Membership.mem.{0, 0} Nat (Finset.{0} Nat)
                  (@SetLike.instMembership.{0, 0} (Finset.{0} Nat) Nat (@Finset.instSetLike.{0} Nat)) P p)
              (fun
                  (a b :
                    @Subtype.{1} Nat fun (p : Nat) =>
                      @Membership.mem.{0, 0} Nat (Finset.{0} Nat)
                        (@SetLike.instMembership.{0, 0} (Finset.{0} Nat) Nat (@Finset.instSetLike.{0} Nat)) P p) =>
                @Subtype.instDecidableEq.{1} Nat
                  (@Membership.mem.{0, 0} Nat (Finset.{0} Nat)
                    (@SetLike.instMembership.{0, 0} (Finset.{0} Nat) Nat (@Finset.instSetLike.{0} Nat)) P)
                  instDecidableEqNat a b)
              a b)
          inst inst_1 B))
      (Erdos390.Full.PaperHeadSimplex.pattern P hprime exponent) →
    @HasSubset.Subset.{0} (Finset.{0} Nat) (@Finset.instHasSubset.{0} Nat)
        (@Erdos390.WholePaper.bankPaperCanonicalStructuredActiveValues.{0} (Erdos390.Full.PaperHeadSimplex.Tag P)
          (@instFintypeOption.{0}
            (@Subtype.{1} Nat fun (p : Nat) =>
              @Membership.mem.{0, 0} Nat (Finset.{0} Nat)
                (@SetLike.instMembership.{0, 0} (Finset.{0} Nat) Nat (@Finset.instSetLike.{0} Nat)) P p)
            (@Finset.Subtype.fintype.{0} Nat P))
          (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, u_1} (Erdos390.Full.PaperHeadSimplex.Tag P) Band
            (@instFintypeOption.{0}
              (@Subtype.{1} Nat fun (p : Nat) =>
                @Membership.mem.{0, 0} Nat (Finset.{0} Nat)
                  (@SetLike.instMembership.{0, 0} (Finset.{0} Nat) Nat (@Finset.instSetLike.{0} Nat)) P p)
              (@Finset.Subtype.fintype.{0} Nat P))
            (fun (a b : Erdos390.Full.PaperHeadSimplex.Tag P) =>
              @Option.instDecidableEq.{0}
                (@Subtype.{1} Nat fun (p : Nat) =>
                  @Membership.mem.{0, 0} Nat (Finset.{0} Nat)
                    (@SetLike.instMembership.{0, 0} (Finset.{0} Nat) Nat (@Finset.instSetLike.{0} Nat)) P p)
                (fun
                    (a b :
                      @Subtype.{1} Nat fun (p : Nat) =>
                        @Membership.mem.{0, 0} Nat (Finset.{0} Nat)
                          (@SetLike.instMembership.{0, 0} (Finset.{0} Nat) Nat (@Finset.instSetLike.{0} Nat)) P p) =>
                  @Subtype.instDecidableEq.{1} Nat
                    (@Membership.mem.{0, 0} Nat (Finset.{0} Nat)
                      (@SetLike.instMembership.{0, 0} (Finset.{0} Nat) Nat (@Finset.instSetLike.{0} Nat)) P)
                    instDecidableEqNat a b)
                a b)
            inst inst_1 B))
        (@Erdos390.WholePaper.BankPaperRealization.roughCanonicalGuardedRow c depth
          (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.n.{0} (Erdos390.Full.PaperHeadSimplex.Tag P)
            (@instFintypeOption.{0}
              (@Subtype.{1} Nat fun (p : Nat) =>
                @Membership.mem.{0, 0} Nat (Finset.{0} Nat)
                  (@SetLike.instMembership.{0, 0} (Finset.{0} Nat) Nat (@Finset.instSetLike.{0} Nat)) P p)
              (@Finset.Subtype.fintype.{0} Nat P))
            (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, u_1} (Erdos390.Full.PaperHeadSimplex.Tag P) Band
              (@instFintypeOption.{0}
                (@Subtype.{1} Nat fun (p : Nat) =>
                  @Membership.mem.{0, 0} Nat (Finset.{0} Nat)
                    (@SetLike.instMembership.{0, 0} (Finset.{0} Nat) Nat (@Finset.instSetLike.{0} Nat)) P p)
                (@Finset.Subtype.fintype.{0} Nat P))
              (fun (a b : Erdos390.Full.PaperHeadSimplex.Tag P) =>
                @Option.instDecidableEq.{0}
                  (@Subtype.{1} Nat fun (p : Nat) =>
                    @Membership.mem.{0, 0} Nat (Finset.{0} Nat)
                      (@SetLike.instMembership.{0, 0} (Finset.{0} Nat) Nat (@Finset.instSetLike.{0} Nat)) P p)
                  (fun
                      (a b :
                        @Subtype.{1} Nat fun (p : Nat) =>
                          @Membership.mem.{0, 0} Nat (Finset.{0} Nat)
                            (@SetLike.instMembership.{0, 0} (Finset.{0} Nat) Nat (@Finset.instSetLike.{0} Nat)) P p) =>
                    @Subtype.instDecidableEq.{1} Nat
                      (@Membership.mem.{0, 0} Nat (Finset.{0} Nat)
                        (@SetLike.instMembership.{0, 0} (Finset.{0} Nat) Nat (@Finset.instSetLike.{0} Nat)) P)
                      instDecidableEqNat a b)
                  a b)
              inst inst_1 B))
          (Erdos390.WholePaper.upperTailLength c
            (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.n.{0} (Erdos390.Full.PaperHeadSimplex.Tag P)
              (@instFintypeOption.{0}
                (@Subtype.{1} Nat fun (p : Nat) =>
                  @Membership.mem.{0, 0} Nat (Finset.{0} Nat)
                    (@SetLike.instMembership.{0, 0} (Finset.{0} Nat) Nat (@Finset.instSetLike.{0} Nat)) P p)
                (@Finset.Subtype.fintype.{0} Nat P))
              (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, u_1} (Erdos390.Full.PaperHeadSimplex.Tag P) Band
                (@instFintypeOption.{0}
                  (@Subtype.{1} Nat fun (p : Nat) =>
                    @Membership.mem.{0, 0} Nat (Finset.{0} Nat)
                      (@SetLike.instMembership.{0, 0} (Finset.{0} Nat) Nat (@Finset.instSetLike.{0} Nat)) P p)
                  (@Finset.Subtype.fintype.{0} Nat P))
                (fun (a b : Erdos390.Full.PaperHeadSimplex.Tag P) =>
                  @Option.instDecidableEq.{0}
                    (@Subtype.{1} Nat fun (p : Nat) =>
                      @Membership.mem.{0, 0} Nat (Finset.{0} Nat)
                        (@SetLike.instMembership.{0, 0} (Finset.{0} Nat) Nat (@Finset.instSetLike.{0} Nat)) P p)
                    (fun
                        (a b :
                          @Subtype.{1} Nat fun (p : Nat) =>
                            @Membership.mem.{0, 0} Nat (Finset.{0} Nat)
                              (@SetLike.instMembership.{0, 0} (Finset.{0} Nat) Nat (@Finset.instSetLike.{0} Nat)) P
                              p) =>
                      @Subtype.instDecidableEq.{1} Nat
                        (@Membership.mem.{0, 0} Nat (Finset.{0} Nat)
                          (@SetLike.instMembership.{0, 0} (Finset.{0} Nat) Nat (@Finset.instSetLike.{0} Nat)) P)
                        instDecidableEqNat a b)
                    a b)
                inst inst_1 B)))
          (@Erdos390.WholePaper.BankPaperRealization.anchorGuardLeftCore
            (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.n.{0} (Erdos390.Full.PaperHeadSimplex.Tag P)
              (@instFintypeOption.{0}
                (@Subtype.{1} Nat fun (p : Nat) =>
                  @Membership.mem.{0, 0} Nat (Finset.{0} Nat)
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                              (@SetLike.instMembership.{0, 0} (Finset.{0} Nat) Nat (@Finset.instSetLike.{0} Nat)) P p)
                          (@Finset.Subtype.fintype.{0} Nat P))
                        (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, u_1}
                          (Erdos390.Full.PaperHeadSimplex.Tag P) Band
                          (@instFintypeOption.{0}
                            (@Subtype.{1} Nat fun (p : Nat) =>
                              @Membership.mem.{0, 0} Nat (Finset.{0} Nat)
                                (@SetLike.instMembership.{0, 0} (Finset.{0} Nat) Nat (@Finset.instSetLike.{0} Nat)) P p)
                            (@Finset.Subtype.fintype.{0} Nat P))
                          (fun (a b : Erdos390.Full.PaperHeadSimplex.Tag P) =>
                            @Option.instDecidableEq.{0}
                              (@Subtype.{1} Nat fun (p : Nat) =>
                                @Membership.mem.{0, 0} Nat (Finset.{0} Nat)
                                  (@SetLike.instMembership.{0, 0} (Finset.{0} Nat) Nat (@Finset.instSetLike.{0} Nat)) P
                                  p)
                              (fun
                                  (a b :
                                    @Subtype.{1} Nat fun (p : Nat) =>
                                      @Membership.mem.{0, 0} Nat (Finset.{0} Nat)
                                        (@SetLike.instMembership.{0, 0} (Finset.{0} Nat) Nat
                                          (@Finset.instSetLike.{0} Nat))
                                        P p) =>
                                @Subtype.instDecidableEq.{1} Nat
                                  (@Membership.mem.{0, 0} Nat (Finset.{0} Nat)
                                    (@SetLike.instMembership.{0, 0} (Finset.{0} Nat) Nat (@Finset.instSetLike.{0} Nat))
                                    P)
                                  instDecidableEqNat a b)
                              a b)
                          inst inst_1 B))))
                  R depth)
                R certificate deltaStar K)
              (@Erdos390.WholePaper.BankPaperRealization.bankPaperCanonicalTopFrozenRoundedSourceSelector.{u_1} P Band
                inst inst_1 B c depth K R certificate Tsource deltaStar betaProt alpha beta qTilde) →
            ∀ (q : Nat),
              Nat.Prime q →
                @LE.le.{0} Nat instLENat q
                    (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.W.{0} (Erdos390.Full.PaperHeadSimplex.Tag P)
                      (@instFintypeOption.{0}
                        (@Subtype.{1} Nat fun (p : Nat) =>
                          @Membership.mem.{0, 0} Nat (Finset.{0} Nat)
                            (@SetLike.instMembership.{0, 0} (Finset.{0} Nat) Nat (@Finset.instSetLike.{0} Nat)) P p)
                        (@Finset.Subtype.fintype.{0} Nat P))
                      (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, u_1}
                        (Erdos390.Full.PaperHeadSimplex.Tag P) Band
                        (@instFintypeOption.{0}
                          (@Subtype.{1} Nat fun (p : Nat) =>
                            @Membership.mem.{0, 0} Nat (Finset.{0} Nat)
                              (@SetLike.instMembership.{0, 0} (Finset.{0} Nat) Nat (@Finset.instSetLike.{0} Nat)) P p)
                          (@Finset.Subtype.fintype.{0} Nat P))
                        (fun (a b : Erdos390.Full.PaperHeadSimplex.Tag P) =>
                          @Option.instDecidableEq.{0}
                            (@Subtype.{1} Nat fun (p : Nat) =>
                              @Membership.mem.{0, 0} Nat (Finset.{0} Nat)
                                (@SetLike.instMembership.{0, 0} (Finset.{0} Nat) Nat (@Finset.instSetLike.{0} Nat)) P p)
                            (fun
                                (a b :
                                  @Subtype.{1} Nat fun (p : Nat) =>
                                    @Membership.mem.{0, 0} Nat (Finset.{0} Nat)
                                      (@SetLike.instMembership.{0, 0} (Finset.{0} Nat) Nat
                                        (@Finset.instSetLike.{0} Nat))
                                      P p) =>
                              @Subtype.instDecidableEq.{1} Nat
                                (@Membership.mem.{0, 0} Nat (Finset.{0} Nat)
                                  (@SetLike.instMembership.{0, 0} (Finset.{0} Nat) Nat (@Finset.instSetLike.{0} Nat)) P)
                                instDecidableEqNat a b)
                            a b)
                        inst inst_1 B)) →
                  @Eq.{1} Real
                    (@Finset.sum.{0, 0}
                      (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.Sample.{0}
                        (Erdos390.Full.PaperHeadSimplex.Tag P)
                        (@instFintypeOption.{0}
                          (@Subtype.{1} Nat fun (p : Nat) =>
                            @Membership.mem.{0, 0} Nat (Finset.{0} Nat)
                              (@SetLike.instMembership.{0, 0} (Finset.{0} Nat) Nat (@Finset.instSetLike.{0} Nat)) P p)
                          (@Finset.Subtype.fintype.{0} Nat P))
                        (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, u_1}
                          (Erdos390.Full.PaperHeadSimplex.Tag P) Band
                          (@instFintypeOption.{0}
                            (@Subtype.{1} Nat fun (p : Nat) =>
                              @Membership.mem.{0, 0} Nat (Finset.{0} Nat)
                                (@SetLike.instMembership.{0, 0} (Finset.{0} Nat) Nat (@Finset.instSetLike.{0} Nat)) P p)
                            (@Finset.Subtype.fintype.{0} Nat P))
                          (fun (a b : Erdos390.Full.PaperHeadSimplex.Tag P) =>
                            @Option.instDecidableEq.{0}
                              (@Subtype.{1} Nat fun (p : Nat) =>
                                @Membership.mem.{0, 0} Nat (Finset.{0} Nat)
                                  (@SetLike.instMembership.{0, 0} (Finset.{0} Nat) Nat (@Finset.instSetLike.{0} Nat)) P
                                  p)
                              (fun
                                  (a b :
                                    @Subtype.{1} Nat fun (p : Nat) =>
                                      @Membership.mem.{0, 0} Nat (Finset.{0} Nat)
                                        (@SetLike.instMembership.{0, 0} (Finset.{0} Nat) Nat
                                          (@Finset.instSetLike.{0} Nat))
                                        P p) =>
                                @Subtype.instDecidableEq.{1} Nat
                                  (@Membership.mem.{0, 0} Nat (Finset.{0} Nat)
                                    (@SetLike.instMembership.{0, 0} (Finset.{0} Nat) Nat (@Finset.instSetLike.{0} Nat))
                                    P)
                                  instDecidableEqNat a b)
                              a b)
                          inst inst_1 B))
                      Real Real.instAddCommMonoid
                      (@Finset.univ.{0}
                        (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.Sample.{0}
                          (Erdos390.Full.PaperHeadSimplex.Tag P)
                          (@instFintypeOption.{0}
                            (@Subtype.{1} Nat fun (p : Nat) =>
                              @Membership.mem.{0, 0} Nat (Finset.{0} Nat)
                                (@SetLike.instMembership.{0, 0} (Finset.{0} Nat) Nat (@Finset.instSetLike.{0} Nat)) P p)
                            (@Finset.Subtype.fintype.{0} Nat P))
                          (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, u_1}
                            (Erdos390.Full.PaperHeadSimplex.Tag P) Band
                            (@instFintypeOption.{0}
                              (@Subtype.{1} Nat fun (p : Nat) =>
                                @Membership.mem.{0, 0} Nat (Finset.{0} Nat)
                                  (@SetLike.instMembership.{0, 0} (Finset.{0} Nat) Nat (@Finset.instSetLike.{0} Nat)) P
                                  p)
                              (@Finset.Subtype.fintype.{0} Nat P))
                            (fun (a b : Erdos390.Full.PaperHeadSimplex.Tag P) =>
                              @Option.instDecidableEq.{0}
                                (@Subtype.{1} Nat fun (p : Nat) =>
                                  @Membership.mem.{0, 0} Nat (Finset.{0} Nat)
                                    (@SetLike.instMembership.{0, 0} (Finset.{0} Nat) Nat (@Finset.instSetLike.{0} Nat))
                                    P p)
                                (fun
                                    (a b :
                                      @Subtype.{1} Nat fun (p : Nat) =>
                                        @Membership.mem.{0, 0} Nat (Finset.{0} Nat)
                                          (@SetLike.instMembership.{0, 0} (Finset.{0} Nat) Nat
                                            (@Finset.instSetLike.{0} Nat))
                                          P p) =>
                                  @Subtype.instDecidableEq.{1} Nat
                                    (@Membership.mem.{0, 0} Nat (Finset.{0} Nat)
                                      (@SetLike.instMembership.{0, 0} (Finset.{0} Nat) Nat
                                        (@Finset.instSetLike.{0} Nat))
                                      P)
                                    instDecidableEqNat a b)
                                a b)
                            inst inst_1 B))
                        (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.sampleFintype.{0}
                          (Erdos390.Full.PaperHeadSimplex.Tag P)
                          (@instFintypeOption.{0}
                            (@Subtype.{1} Nat fun (p : Nat) =>
                              @Membership.mem.{0, 0} Nat (Finset.{0} Nat)
                                (@SetLike.instMembership.{0, 0} (Finset.{0} Nat) Nat (@Finset.instSetLike.{0} Nat)) P p)
                            (@Finset.Subtype.fintype.{0} Nat P))
                          (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, u_1}
                            (Erdos390.Full.PaperHeadSimplex.Tag P) Band
                            (@instFintypeOption.{0}
                              (@Subtype.{1} Nat fun (p : Nat) =>
                                @Membership.mem.{0, 0} Nat (Finset.{0} Nat)
                                  (@SetLike.instMembership.{0, 0} (Finset.{0} Nat) Nat (@Finset.instSetLike.{0} Nat)) P
                                  p)
                              (@Finset.Subtype.fintype.{0} Nat P))
                            (fun (a b : Erdos390.Full.PaperHeadSimplex.Tag P) =>
                              @Option.instDecidableEq.{0}
                                (@Subtype.{1} Nat fun (p : Nat) =>
                                  @Membership.mem.{0, 0} Nat (Finset.{0} Nat)
                                    (@SetLike.instMembership.{0, 0} (Finset.{0} Nat) Nat (@Finset.instSetLike.{0} Nat))
                                    P p)
                                (fun
                                    (a b :
                                      @Subtype.{1} Nat fun (p : Nat) =>
                                        @Membership.mem.{0, 0} Nat (Finset.{0} Nat)
                                          (@SetLike.instMembership.{0, 0} (Finset.{0} Nat) Nat
                                            (@Finset.instSetLike.{0} Nat))
                                          P p) =>
                                  @Subtype.instDecidableEq.{1} Nat
                                    (@Membership.mem.{0, 0} Nat (Finset.{0} Nat)
                                      (@SetLike.instMembership.{0, 0} (Finset.{0} Nat) Nat
                                        (@Finset.instSetLike.{0} Nat))
                                      P)
                                    instDecidableEqNat a b)
                                a b)
                            inst inst_1 B)))
                      fun
                        (m :
                          @Erdos390.Full.PaperBridgeFit.StructuredSampleData.Sample.{0}
                            (Erdos390.Full.PaperHeadSimplex.Tag P)
                            (@instFintypeOption.{0}
                              (@Subtype.{1} Nat fun (p : Nat) =>
                                @Membership.mem.{0, 0} Nat (Finset.{0} Nat)
                                  (@SetLike.instMembership.{0, 0} (Finset.{0} Nat) Nat (@Finset.instSetLike.{0} Nat)) P
                                  p)
                              (@Finset.Subtype.fintype.{0} Nat P))
                            (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, u_1}
                              (Erdos390.Full.PaperHeadSimplex.Tag P) Band
                              (@instFintypeOption.{0}
                                (@Subtype.{1} Nat fun (p : Nat) =>
                                  @Membership.mem.{0, 0} Nat (Finset.{0} Nat)
                                    (@SetLike.instMembership.{0, 0} (Finset.{0} Nat) Nat (@Finset.instSetLike.{0} Nat))
                                    P p)
                                (@Finset.Subtype.fintype.{0} Nat P))
                              (fun (a b : Erdos390.Full.PaperHeadSimplex.Tag P) =>
                                @Option.instDecidableEq.{0}
                                  (@Subtype.{1} Nat fun (p : Nat) =>
                                    @Membership.mem.{0, 0} Nat (Finset.{0} Nat)
                                      (@SetLike.instMembership.{0, 0} (Finset.{0} Nat) Nat
                                        (@Finset.instSetLike.{0} Nat))
                                      P p)
                                  (fun
                                      (a b :
                                        @Subtype.{1} Nat fun (p : Nat) =>
                                          @Membership.mem.{0, 0} Nat (Finset.{0} Nat)
                                            (@SetLike.instMembership.{0, 0} (Finset.{0} Nat) Nat
                                              (@Finset.instSetLike.{0} Nat))
                                            P p) =>
                                    @Subtype.instDecidableEq.{1} Nat
                                      (@Membership.mem.{0, 0} Nat (Finset.{0} Nat)
                                        (@SetLike.instMembership.{0, 0} (Finset.{0} Nat) Nat
                                          (@Finset.instSetLike.{0} Nat))
                                        P)
                                      instDecidableEqNat a b)
                                  a b)
                              inst inst_1 B)) =>
                      @HMul.hMul.{0, 0, 0} Real Real Real (@instHMul.{0} Real Real.instMul)
                        (@Erdos390.WholePaper.bankPaperCanonicalSectionNinePostHeightActiveSeed.{u_1} P Band inst inst_1
                          B I hlo hhi q0 A0 d exponent activeHeadTarget H m)
                        (Erdos390.Full.ArithmeticModel.valuation q
                          (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.value.{0}
                            (Erdos390.Full.PaperHeadSimplex.Tag P)
                            (@instFintypeOption.{0}
                              (@Subtype.{1} Nat fun (p : Nat) =>
                                @Membership.mem.{0, 0} Nat (Finset.{0} Nat)
                                  (@SetLike.instMembership.{0, 0} (Finset.{0} Nat) Nat (@Finset.instSetLike.{0} Nat)) P
                                  p)
                              (@Finset.Subtype.fintype.{0} Nat P))
                            (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, u_1}
                              (Erdos390.Full.PaperHeadSimplex.Tag P) Band
                              (@instFintypeOption.{0}
                                (@Subtype.{1} Nat fun (p : Nat) =>
                                  @Membership.mem.{0, 0} Nat (Finset.{0} Nat)
                                    (@SetLike.instMembership.{0, 0} (Finset.{0} Nat) Nat (@Finset.instSetLike.{0} Nat))
                                    P p)
                                (@Finset.Subtype.fintype.{0} Nat P))
                              (fun (a b : Erdos390.Full.PaperHeadSimplex.Tag P) =>
                                @Option.instDecidableEq.{0}
                                  (@Subtype.{1} Nat fun (p : Nat) =>
                                    @Membership.mem.{0, 0} Nat (Finset.{0} Nat)
                                      (@SetLike.instMembership.{0, 0} (Finset.{0} Nat) Nat
                                        (@Finset.instSetLike.{0} Nat))
                                      P p)
                                  (fun
                                      (a b :
                                        @Subtype.{1} Nat fun (p : Nat) =>
                                          @Membership.mem.{0, 0} Nat (Finset.{0} Nat)
                                            (@SetLike.instMembership.{0, 0} (Finset.{0} Nat) Nat
                                              (@Finset.instSetLike.{0} Nat))
                                            P p) =>
                                    @Subtype.instDecidableEq.{1} Nat
                                      (@Membership.mem.{0, 0} Nat (Finset.{0} Nat)
                                        (@SetLike.instMembership.{0, 0} (Finset.{0} Nat) Nat
                                          (@Finset.instSetLike.{0} Nat))
                                        P)
                                      instDecidableEqNat a b)
                                  a b)
                              inst inst_1 B)
                            m)))
                    (@Finset.sum.{0, 0}
                      (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.Sample.{0}
                        (Erdos390.Full.PaperHeadSimplex.Tag P)
                        (@instFintypeOption.{0}
                          (@Subtype.{1} Nat fun (p : Nat) =>
                            @Membership.mem.{0, 0} Nat (Finset.{0} Nat)
                              (@SetLike.instMembership.{0, 0} (Finset.{0} Nat) Nat (@Finset.instSetLike.{0} Nat)) P p)
                          (@Finset.Subtype.fintype.{0} Nat P))
                        (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, u_1}
                          (Erdos390.Full.PaperHeadSimplex.Tag P) Band
                          (@instFintypeOption.{0}
                            (@Subtype.{1} Nat fun (p : Nat) =>
                              @Membership.mem.{0, 0} Nat (Finset.{0} Nat)
                                (@SetLike.instMembership.{0, 0} (Finset.{0} Nat) Nat (@Finset.instSetLike.{0} Nat)) P p)
                            (@Finset.Subtype.fintype.{0} Nat P))
                          (fun (a b : Erdos390.Full.PaperHeadSimplex.Tag P) =>
                            @Option.instDecidableEq.{0}
                              (@Subtype.{1} Nat fun (p : Nat) =>
                                @Membership.mem.{0, 0} Nat (Finset.{0} Nat)
                                  (@SetLike.instMembership.{0, 0} (Finset.{0} Nat) Nat (@Finset.instSetLike.{0} Nat)) P
                                  p)
                              (fun
                                  (a b :
                                    @Subtype.{1} Nat fun (p : Nat) =>
                                      @Membership.mem.{0, 0} Nat (Finset.{0} Nat)
                                        (@SetLike.instMembership.{0, 0} (Finset.{0} Nat) Nat
                                          (@Finset.instSetLike.{0} Nat))
                                        P p) =>
                                @Subtype.instDecidableEq.{1} Nat
                                  (@Membership.mem.{0, 0} Nat (Finset.{0} Nat)
                                    (@SetLike.instMembership.{0, 0} (Finset.{0} Nat) Nat (@Finset.instSetLike.{0} Nat))
                                    P)
                                  instDecidableEqNat a b)
                              a b)
                          inst inst_1 B))
                      Real Real.instAddCommMonoid
                      (@Finset.univ.{0}
                        (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.Sample.{0}
                          (Erdos390.Full.PaperHeadSimplex.Tag P)
                          (@instFintypeOption.{0}
                            (@Subtype.{1} Nat fun (p : Nat) =>
                              @Membership.mem.{0, 0} Nat (Finset.{0} Nat)
                                (@SetLike.instMembership.{0, 0} (Finset.{0} Nat) Nat (@Finset.instSetLike.{0} Nat)) P p)
                            (@Finset.Subtype.fintype.{0} Nat P))
                          (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, u_1}
                            (Erdos390.Full.PaperHeadSimplex.Tag P) Band
                            (@instFintypeOption.{0}
                              (@Subtype.{1} Nat fun (p : Nat) =>
                                @Membership.mem.{0, 0} Nat (Finset.{0} Nat)
                                  (@SetLike.instMembership.{0, 0} (Finset.{0} Nat) Nat (@Finset.instSetLike.{0} Nat)) P
                                  p)
                              (@Finset.Subtype.fintype.{0} Nat P))
                            (fun (a b : Erdos390.Full.PaperHeadSimplex.Tag P) =>
                              @Option.instDecidableEq.{0}
                                (@Subtype.{1} Nat fun (p : Nat) =>
                                  @Membership.mem.{0, 0} Nat (Finset.{0} Nat)
                                    (@SetLike.instMembership.{0, 0} (Finset.{0} Nat) Nat (@Finset.instSetLike.{0} Nat))
                                    P p)
                                (fun
                                    (a b :
                                      @Subtype.{1} Nat fun (p : Nat) =>
                                        @Membership.mem.{0, 0} Nat (Finset.{0} Nat)
                                          (@SetLike.instMembership.{0, 0} (Finset.{0} Nat) Nat
                                            (@Finset.instSetLike.{0} Nat))
                                          P p) =>
                                  @Subtype.instDecidableEq.{1} Nat
                                    (@Membership.mem.{0, 0} Nat (Finset.{0} Nat)
                                      (@SetLike.instMembership.{0, 0} (Finset.{0} Nat) Nat
                                        (@Finset.instSetLike.{0} Nat))
                                      P)
                                    instDecidableEqNat a b)
                                a b)
                            inst inst_1 B))
                        (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.sampleFintype.{0}
                          (Erdos390.Full.PaperHeadSimplex.Tag P)
                          (@instFintypeOption.{0}
                            (@Subtype.{1} Nat fun (p : Nat) =>
                              @Membership.mem.{0, 0} Nat (Finset.{0} Nat)
                                (@SetLike.instMembership.{0, 0} (Finset.{0} Nat) Nat (@Finset.instSetLike.{0} Nat)) P p)
                            (@Finset.Subtype.fintype.{0} Nat P))
                          (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, u_1}
                            (Erdos390.Full.PaperHeadSimplex.Tag P) Band
                            (@instFintypeOption.{0}
                              (@Subtype.{1} Nat fun (p : Nat) =>
                                @Membership.mem.{0, 0} Nat (Finset.{0} Nat)
                                  (@SetLike.instMembership.{0, 0} (Finset.{0} Nat) Nat (@Finset.instSetLike.{0} Nat)) P
                                  p)
                              (@Finset.Subtype.fintype.{0} Nat P))
                            (fun (a b : Erdos390.Full.PaperHeadSimplex.Tag P) =>
                              @Option.instDecidableEq.{0}
                                (@Subtype.{1} Nat fun (p : Nat) =>
                                  @Membership.mem.{0, 0} Nat (Finset.{0} Nat)
                                    (@SetLike.instMembership.{0, 0} (Finset.{0} Nat) Nat (@Finset.instSetLike.{0} Nat))
                                    P p)
                                (fun
                                    (a b :
                                      @Subtype.{1} Nat fun (p : Nat) =>
                                        @Membership.mem.{0, 0} Nat (Finset.{0} Nat)
                                          (@SetLike.instMembership.{0, 0} (Finset.{0} Nat) Nat
                                            (@Finset.instSetLike.{0} Nat))
                                          P p) =>
                                  @Subtype.instDecidableEq.{1} Nat
                                    (@Membership.mem.{0, 0} Nat (Finset.{0} Nat)
                                      (@SetLike.instMembership.{0, 0} (Finset.{0} Nat) Nat
                                        (@Finset.instSetLike.{0} Nat))
                                      P)
                                    instDecidableEqNat a b)
                                a b)
                            inst inst_1 B)))
                      fun
                        (m :
                          @Erdos390.Full.PaperBridgeFit.StructuredSampleData.Sample.{0}
                            (Erdos390.Full.PaperHeadSimplex.Tag P)
                            (@instFintypeOption.{0}
                              (@Subtype.{1} Nat fun (p : Nat) =>
                                @Membership.mem.{0, 0} Nat (Finset.{0} Nat)
                                  (@SetLike.instMembership.{0, 0} (Finset.{0} Nat) Nat (@Finset.instSetLike.{0} Nat)) P
                                  p)
                              (@Finset.Subtype.fintype.{0} Nat P))
                            (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, u_1}
                              (Erdos390.Full.PaperHeadSimplex.Tag P) Band
                              (@instFintypeOption.{0}
                                (@Subtype.{1} Nat fun (p : Nat) =>
                                  @Membership.mem.{0, 0} Nat (Finset.{0} Nat)
                                    (@SetLike.instMembership.{0, 0} (Finset.{0} Nat) Nat (@Finset.instSetLike.{0} Nat))
                                    P p)
                                (@Finset.Subtype.fintype.{0} Nat P))
                              (fun (a b : Erdos390.Full.PaperHeadSimplex.Tag P) =>
                                @Option.instDecidableEq.{0}
                                  (@Subtype.{1} Nat fun (p : Nat) =>
                                    @Membership.mem.{0, 0} Nat (Finset.{0} Nat)
                                      (@SetLike.instMembership.{0, 0} (Finset.{0} Nat) Nat
                                        (@Finset.instSetLike.{0} Nat))
                                      P p)
                                  (fun
                                      (a b :
                                        @Subtype.{1} Nat fun (p : Nat) =>
                                          @Membership.mem.{0, 0} Nat (Finset.{0} Nat)
                                            (@SetLike.instMembership.{0, 0} (Finset.{0} Nat) Nat
                                              (@Finset.instSetLike.{0} Nat))
                                            P p) =>
                                    @Subtype.instDecidableEq.{1} Nat
                                      (@Membership.mem.{0, 0} Nat (Finset.{0} Nat)
                                        (@SetLike.instMembership.{0, 0} (Finset.{0} Nat) Nat
                                          (@Finset.instSetLike.{0} Nat))
                                        P)
                                      instDecidableEqNat a b)
                                  a b)
                              inst inst_1 B)) =>
                      @HMul.hMul.{0, 0, 0} Real Real Real (@instHMul.{0} Real Real.instMul)
                        (@Erdos390.WholePaper.BankPaperRealization.bankPaperCanonicalTopFrozenRoundedActiveSeed.{u_1} P
                          Band inst inst_1 B c depth K R certificate Tsource deltaStar betaProt alpha qTilde m)
                        (Erdos390.Full.ArithmeticModel.valuation q
                          (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.value.{0}
                            (Erdos390.Full.PaperHeadSimplex.Tag P)
                            (@instFintypeOption.{0}
                              (@Subtype.{1} Nat fun (p : Nat) =>
                                @Membership.mem.{0, 0} Nat (Finset.{0} Nat)
                                  (@SetLike.instMembership.{0, 0} (Finset.{0} Nat) Nat (@Finset.instSetLike.{0} Nat)) P
                                  p)
                              (@Finset.Subtype.fintype.{0} Nat P))
                            (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, u_1}
                              (Erdos390.Full.PaperHeadSimplex.Tag P) Band
                              (@instFintypeOption.{0}
                                (@Subtype.{1} Nat fun (p : Nat) =>
                                  @Membership.mem.{0, 0} Nat (Finset.{0} Nat)
                                    (@SetLike.instMembership.{0, 0} (Finset.{0} Nat) Nat (@Finset.instSetLike.{0} Nat))
                                    P p)
                                (@Finset.Subtype.fintype.{0} Nat P))
                              (fun (a b : Erdos390.Full.PaperHeadSimplex.Tag P) =>
                                @Option.instDecidableEq.{0}
                                  (@Subtype.{1} Nat fun (p : Nat) =>
                                    @Membership.mem.{0, 0} Nat (Finset.{0} Nat)
                                      (@SetLike.instMembership.{0, 0} (Finset.{0} Nat) Nat
                                        (@Finset.instSetLike.{0} Nat))
                                      P p)
                                  (fun
                                      (a b :
                                        @Subtype.{1} Nat fun (p : Nat) =>
                                          @Membership.mem.{0, 0} Nat (Finset.{0} Nat)
                                            (@SetLike.instMembership.{0, 0} (Finset.{0} Nat) Nat
                                              (@Finset.instSetLike.{0} Nat))
                                            P p) =>
                                    @Subtype.instDecidableEq.{1} Nat
                                      (@Membership.mem.{0, 0} Nat (Finset.{0} Nat)
                                        (@SetLike.instMembership.{0, 0} (Finset.{0} Nat) Nat
                                          (@Finset.instSetLike.{0} Nat))
                                        P)
                                      instDecidableEqNat a b)
                                  a b)
                              inst inst_1 B)
                            m))) := by sorry

Source

https://github.com/ShouqiaoW/erdos/blob/61325b10bbdc29f4fb5e0618b414b9f2189333ad/390/lean/Erdos390/WholePaper/BankPaperCanonicalSectionNinePostHeightSourcePrebridgeConnector.lean#L354-L528

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