Analytic construction propositions, part 4
Definitionerdos390_remaining_analytic_propositions_004analytic-number-theoryerdos-390
Named propositions for the remaining analytic and finite construction estimates in the Erdős 390 central-anchor proof. Each name expands to the complete quantified statement, including every hypothesis and universe parameter. These definitions introduce no assertion of the propositions; their proofs are submitted separately. This part includes the following source declarations: forall_accuracy_exists_cutoff_eventually_canonical_fullSlowRow_sub_reference, exists_uniform_canonical_tiltedLaw_nuisance_valuation_rate_on_effectiveBall, exists_uniform_canonical_cellMediumLaw_tilted_valuation_prefix_rate_unrestricted, exists_uniform_canonical_cellMediumLaw_valuation_mean_profiles_rate_unrestricted, exists_uniform_movingPrefix_divInd_centered_bound, exists_uniform_rawCell_valuation_prefix_bound, exists_uniform_rawCell_tilted_valuation_prefix_rate_unrestricted, exists_uniform_fixedFinite_rawCell_tilted_valuation_mean_agreement_rate, bankPaperCanonicalActualSelectorMassEstimate_of_topFrozenBalancedInitial, bankPaperCanonicalBalancedRawSmoothRowDiscrepancy_isBigO, bankPaperCanonicalSectionNineFinalPayload_of_roundedSelector_guardedSlack, bankPaperCanonicalSectionNinePostHeight_sourceFirstPreselectedPostHfitInput, eventually_bankPaperCanonicalTopFrozenRoundedSelectorSourceState_of_residualInputs, eventually_bankPaperCanonicalTopFrozenRoundedSourceResidualInputsAt_of_coherentTarget, eventually_canonicalDistributedSectionNineCleanListLower_absorbed, eventually_roughCanonicalBalancedNonsmoothBounds, eventually_roughCanonicalUniformGuardedPostchargeCorrectionDensityBound, eventually_roughCanonicalUniformRawRowCorrectionDensityBound, exists_bankPaperCanonicalSectionNinePostHeight_sourceFirstCutoffAwareAnalyticCompletion, exists_bankPaperCanonicalSectionNinePostHeight_sourceFirstCutoffAwareDistributedTerminal.
Definition code
import Definitions.Def_erdos390_analytic_foundations_001
import Definitions.Def_erdos390_analytic_foundations_002
import Definitions.Def_erdos390_analytic_foundations_003
import Definitions.Def_erdos390_analytic_foundations_004
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
import Definitions.Def_erdos390_analytic_foundations_010
import Definitions.Def_erdos390_dickman_step_function
import Definitions.Def_erdos390_final_payload_cutoff_certificates
import Definitions.Def_erdos390_source_bank_interface
import Mathlib
import Theorems.Thm_Erdos390_Full_DickmanBasic_kernel_product_bound
import Theorems.Thm_Erdos390_Full_DickmanBasic_kernel_secondDerivative_first_bound
import Theorems.Thm_Erdos390_Full_DickmanBasic_rho_pos_on_zero_five
import Theorems.Thm_MediumPNT
universe u_1 u_2
set_option maxRecDepth 10000
set_option maxHeartbeats 0
noncomputable def Erdos390.RemainingAnalyticGoal004_001 : Prop :=
∀ (r : Real),
@LT.lt.{0} Real Real.instLT (@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real Real.instZero)) r →
@Exists.{1} Nat fun (W₀ : Nat) =>
∀ (W : Nat),
@LE.le.{0} Nat instLENat W₀ W →
∀ {delta eta : Real}
(hdelta :
@LT.lt.{0} Real Real.instLT (@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real Real.instZero))
delta)
(M : Erdos390.Full.RegularRelativeMesh.Mesh delta eta) {Head : Type u_1} [inst : Fintype.{u_1} Head]
[inst_1 : DecidableEq.{u_1 + 1} Head] [inst_2 : Nonempty.{u_1 + 1} Head]
(Phead : Head → Erdos390.Full.HeadPattern.Pattern),
(∀ (h : Head) (p : Nat),
Iff
(@Membership.mem.{0, 0} Nat (Finset.{0} Nat)
(@SetLike.instMembership.{0, 0} (Finset.{0} Nat) Nat (@Finset.instSetLike.{0} Nat)) (Phead h).primes
p)
(And (Nat.Prime p) (@LE.le.{0} Nat instLENat p W))) →
∀ (I : Erdos390.Full.PaperGuardCensus.PhysicalIntervals) (Cmax : Real),
(∀ (sigma : Erdos390.Full.PaperBridgeFit.PhysicalSign),
@LE.le.{0} Real Real.instLE
(@OfNat.ofNat.{0} Real (nat_lit 1) (@One.toOfNat1.{0} Real Real.instOne)) (I.lower sigma)) →
(∀ (sigma : Erdos390.Full.PaperBridgeFit.PhysicalSign),
@LE.le.{0} Real Real.instLE (I.upper sigma) Cmax) →
∀ (Cprom Cbank : Nat) (ledger : (n : Nat) → Erdos390.Full.PaperGuardCensus.Ledger n Cprom Cbank)
(Acoef Aphys : Real),
@LE.le.{0} Real Real.instLE
(@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real Real.instZero)) Acoef →
@LE.le.{0} Real Real.instLE
(@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real Real.instZero)) Aphys →
@Filter.Eventually.{0} Nat
(fun (n : Nat) =>
∀
(B :
@Erdos390.Full.PaperBridgeFit.BridgeData.{u_1, 0} Head
(Fin
(@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat)
(@Erdos390.Full.RegularRelativeMesh.Mesh.cellCount delta eta M)
(@OfNat.ofNat.{0} Nat (nat_lit 1) (instOfNatNat (nat_lit 1)))))
inst inst_1
(Fin.fintype
(@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat)
(@Erdos390.Full.RegularRelativeMesh.Mesh.cellCount delta eta M)
(@OfNat.ofNat.{0} Nat (nat_lit 1) (instOfNatNat (nat_lit 1)))))
(instDecidableEqFin
(@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat)
(@Erdos390.Full.RegularRelativeMesh.Mesh.cellCount delta eta M)
(@OfNat.ofNat.{0} Nat (nat_lit 1) (instOfNatNat (nat_lit 1)))))),
@Eq.{1} Nat
(@Erdos390.Full.PaperBridgeFit.StructuredSampleData.n.{u_1} Head inst
(@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{u_1, 0} Head
(Fin
(@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat)
(@Erdos390.Full.RegularRelativeMesh.Mesh.cellCount delta eta M)
(@OfNat.ofNat.{0} Nat (nat_lit 1) (instOfNatNat (nat_lit 1)))))
inst inst_1
(Fin.fintype
(@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat)
(@Erdos390.Full.RegularRelativeMesh.Mesh.cellCount delta eta M)
(@OfNat.ofNat.{0} Nat (nat_lit 1) (instOfNatNat (nat_lit 1)))))
(instDecidableEqFin
(@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat)
(@Erdos390.Full.RegularRelativeMesh.Mesh.cellCount delta eta M)
(@OfNat.ofNat.{0} Nat (nat_lit 1) (instOfNatNat (nat_lit 1)))))
B))
n →
@Eq.{1} Nat
(@Erdos390.Full.PaperBridgeFit.StructuredSampleData.W.{u_1} Head inst
(@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{u_1, 0} Head
(Fin
(@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat)
(@Erdos390.Full.RegularRelativeMesh.Mesh.cellCount delta eta M)
(@OfNat.ofNat.{0} Nat (nat_lit 1) (instOfNatNat (nat_lit 1)))))
inst inst_1
(Fin.fintype
(@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat)
(@Erdos390.Full.RegularRelativeMesh.Mesh.cellCount delta eta M)
(@OfNat.ofNat.{0} Nat (nat_lit 1) (instOfNatNat (nat_lit 1)))))
(instDecidableEqFin
(@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat)
(@Erdos390.Full.RegularRelativeMesh.Mesh.cellCount delta eta M)
(@OfNat.ofNat.{0} Nat (nat_lit 1) (instOfNatNat (nat_lit 1)))))
B))
W →
∀
(hsep :
@LT.lt.{0} Nat instLTNat
(Erdos390.Full.ArithmeticModel.physicalBound
(I.upper Erdos390.Full.PaperBridgeFit.PhysicalSign.minus)
(@Erdos390.Full.PaperBridgeFit.StructuredSampleData.n.{u_1} Head inst
(@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{u_1, 0} Head
(Fin
(@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat)
(@Erdos390.Full.RegularRelativeMesh.Mesh.cellCount delta eta M)
(@OfNat.ofNat.{0} Nat (nat_lit 1) (instOfNatNat (nat_lit 1)))))
inst inst_1
(Fin.fintype
(@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat)
(@Erdos390.Full.RegularRelativeMesh.Mesh.cellCount delta eta M)
(@OfNat.ofNat.{0} Nat (nat_lit 1) (instOfNatNat (nat_lit 1)))))
(instDecidableEqFin
(@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat)
(@Erdos390.Full.RegularRelativeMesh.Mesh.cellCount delta eta M)
(@OfNat.ofNat.{0} Nat (nat_lit 1) (instOfNatNat (nat_lit 1)))))
B)))
(Erdos390.Full.ArithmeticModel.physicalBound
(I.lower Erdos390.Full.PaperBridgeFit.PhysicalSign.plus)
(@Erdos390.Full.PaperBridgeFit.StructuredSampleData.n.{u_1} Head inst
(@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{u_1, 0} Head
(Fin
(@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat)
(@Erdos390.Full.RegularRelativeMesh.Mesh.cellCount delta eta M)
(@OfNat.ofNat.{0} Nat (nat_lit 1) (instOfNatNat (nat_lit 1)))))
inst inst_1
(Fin.fintype
(@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat)
(@Erdos390.Full.RegularRelativeMesh.Mesh.cellCount delta eta M)
(@OfNat.ofNat.{0} Nat (nat_lit 1) (instOfNatNat (nat_lit 1)))))
(instDecidableEqFin
(@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat)
(@Erdos390.Full.RegularRelativeMesh.Mesh.cellCount delta eta M)
(@OfNat.ofNat.{0} Nat (nat_lit 1) (instOfNatNat (nat_lit 1)))))
B))))
(hremaining :
∀ (c : Erdos390.Full.PaperBridgeFit.Cell.{u_1} Head),
@Finset.Nonempty.{0} Nat
(@SDiff.sdiff.{0} (Finset.{0} Nat)
(@Finset.instSDiff.{0} Nat instDecidableEqNat)
(@Erdos390.Full.PaperGuardCensus.rawCell.{u_1} Head Phead I
(@Erdos390.Full.PaperBridgeFit.StructuredSampleData.n.{u_1} Head inst
(@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{u_1, 0} Head
(Fin
(@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat)
(@Erdos390.Full.RegularRelativeMesh.Mesh.cellCount delta eta M)
(@OfNat.ofNat.{0} Nat (nat_lit 1) (instOfNatNat (nat_lit 1)))))
inst inst_1
(Fin.fintype
(@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat)
(@Erdos390.Full.RegularRelativeMesh.Mesh.cellCount delta eta M)
(@OfNat.ofNat.{0} Nat (nat_lit 1) (instOfNatNat (nat_lit 1)))))
(instDecidableEqFin
(@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat)
(@Erdos390.Full.RegularRelativeMesh.Mesh.cellCount delta eta M)
(@OfNat.ofNat.{0} Nat (nat_lit 1) (instOfNatNat (nat_lit 1)))))
B))
c)
(@Erdos390.Full.PaperGuardCensus.Ledger.guards
(@Erdos390.Full.PaperBridgeFit.StructuredSampleData.n.{u_1} Head inst
(@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{u_1, 0} Head
(Fin
(@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat)
(@Erdos390.Full.RegularRelativeMesh.Mesh.cellCount delta eta M)
(@OfNat.ofNat.{0} Nat (nat_lit 1) (instOfNatNat (nat_lit 1)))))
inst inst_1
(Fin.fintype
(@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat)
(@Erdos390.Full.RegularRelativeMesh.Mesh.cellCount delta eta M)
(@OfNat.ofNat.{0} Nat (nat_lit 1) (instOfNatNat (nat_lit 1)))))
(instDecidableEqFin
(@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat)
(@Erdos390.Full.RegularRelativeMesh.Mesh.cellCount delta eta M)
(@OfNat.ofNat.{0} Nat (nat_lit 1) (instOfNatNat (nat_lit 1)))))
B))
Cprom Cbank
(ledger
(@Erdos390.Full.PaperBridgeFit.StructuredSampleData.n.{u_1} Head inst
(@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{u_1, 0} Head
(Fin
(@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat)
(@Erdos390.Full.RegularRelativeMesh.Mesh.cellCount delta eta
M)
(@OfNat.ofNat.{0} Nat (nat_lit 1)
(instOfNatNat (nat_lit 1)))))
inst inst_1
(Fin.fintype
(@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat)
(@Erdos390.Full.RegularRelativeMesh.Mesh.cellCount delta eta
M)
(@OfNat.ofNat.{0} Nat (nat_lit 1)
(instOfNatNat (nat_lit 1)))))
(instDecidableEqFin
(@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat)
(@Erdos390.Full.RegularRelativeMesh.Mesh.cellCount delta eta
M)
(@OfNat.ofNat.{0} Nat (nat_lit 1)
(instOfNatNat (nat_lit 1)))))
B)))))),
@Eq.{u_1 + 1} (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.{u_1} Head inst)
(@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{u_1, 0} Head
(Fin
(@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat)
(@Erdos390.Full.RegularRelativeMesh.Mesh.cellCount delta eta M)
(@OfNat.ofNat.{0} Nat (nat_lit 1) (instOfNatNat (nat_lit 1)))))
inst inst_1
(Fin.fintype
(@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat)
(@Erdos390.Full.RegularRelativeMesh.Mesh.cellCount delta eta M)
(@OfNat.ofNat.{0} Nat (nat_lit 1) (instOfNatNat (nat_lit 1)))))
(instDecidableEqFin
(@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat)
(@Erdos390.Full.RegularRelativeMesh.Mesh.cellCount delta eta M)
(@OfNat.ofNat.{0} Nat (nat_lit 1) (instOfNatNat (nat_lit 1)))))
B)
(@Erdos390.Full.PaperGuardCensus.canonicalSampleData.{u_1} Head inst Phead I
(@Erdos390.Full.PaperBridgeFit.StructuredSampleData.n.{u_1} Head inst
(@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{u_1, 0} Head
(Fin
(@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat)
(@Erdos390.Full.RegularRelativeMesh.Mesh.cellCount delta eta M)
(@OfNat.ofNat.{0} Nat (nat_lit 1) (instOfNatNat (nat_lit 1)))))
inst inst_1
(Fin.fintype
(@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat)
(@Erdos390.Full.RegularRelativeMesh.Mesh.cellCount delta eta M)
(@OfNat.ofNat.{0} Nat (nat_lit 1) (instOfNatNat (nat_lit 1)))))
(instDecidableEqFin
(@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat)
(@Erdos390.Full.RegularRelativeMesh.Mesh.cellCount delta eta M)
(@OfNat.ofNat.{0} Nat (nat_lit 1) (instOfNatNat (nat_lit 1)))))
B))
(@Erdos390.Full.PaperBridgeFit.StructuredSampleData.W.{u_1} Head inst
(@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{u_1, 0} Head
(Fin
(@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat)
(@Erdos390.Full.RegularRelativeMesh.Mesh.cellCount delta eta M)
(@OfNat.ofNat.{0} Nat (nat_lit 1) (instOfNatNat (nat_lit 1)))))
inst inst_1
(Fin.fintype
(@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat)
(@Erdos390.Full.RegularRelativeMesh.Mesh.cellCount delta eta M)
(@OfNat.ofNat.{0} Nat (nat_lit 1) (instOfNatNat (nat_lit 1)))))
(instDecidableEqFin
(@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat)
(@Erdos390.Full.RegularRelativeMesh.Mesh.cellCount delta eta M)
(@OfNat.ofNat.{0} Nat (nat_lit 1) (instOfNatNat (nat_lit 1)))))
B))
Cprom Cbank
(ledger
(@Erdos390.Full.PaperBridgeFit.StructuredSampleData.n.{u_1} Head inst
(@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{u_1, 0} Head
(Fin
(@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat)
(@Erdos390.Full.RegularRelativeMesh.Mesh.cellCount delta eta M)
(@OfNat.ofNat.{0} Nat (nat_lit 1) (instOfNatNat (nat_lit 1)))))
inst inst_1
(Fin.fintype
(@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat)
(@Erdos390.Full.RegularRelativeMesh.Mesh.cellCount delta eta M)
(@OfNat.ofNat.{0} Nat (nat_lit 1) (instOfNatNat (nat_lit 1)))))
(instDecidableEqFin
(@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat)
(@Erdos390.Full.RegularRelativeMesh.Mesh.cellCount delta eta M)
(@OfNat.ofNat.{0} Nat (nat_lit 1) (instOfNatNat (nat_lit 1)))))
B)))
hsep hremaining) →
(@Exists.{0}
(@Ne.{1} Nat
(@Erdos390.Full.PaperBridgeFit.StructuredSampleData.W.{u_1} Head inst
(@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{u_1, 0} Head
(Fin
(@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat)
(@Erdos390.Full.RegularRelativeMesh.Mesh.cellCount delta eta M)
(@OfNat.ofNat.{0} Nat (nat_lit 1) (instOfNatNat (nat_lit 1)))))
inst inst_1
(Fin.fintype
(@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat)
(@Erdos390.Full.RegularRelativeMesh.Mesh.cellCount delta eta M)
(@OfNat.ofNat.{0} Nat (nat_lit 1) (instOfNatNat (nat_lit 1)))))
(instDecidableEqFin
(@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat)
(@Erdos390.Full.RegularRelativeMesh.Mesh.cellCount delta eta M)
(@OfNat.ofNat.{0} Nat (nat_lit 1) (instOfNatNat (nat_lit 1)))))
B))
(@OfNat.ofNat.{0} Nat (nat_lit 0) (instOfNatNat (nat_lit 0))))
fun
(hWne :
@Ne.{1} Nat
(@Erdos390.Full.PaperBridgeFit.StructuredSampleData.W.{u_1} Head inst
(@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{u_1, 0} Head
(Fin
(@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat)
(@Erdos390.Full.RegularRelativeMesh.Mesh.cellCount delta eta
M)
(@OfNat.ofNat.{0} Nat (nat_lit 1)
(instOfNatNat (nat_lit 1)))))
inst inst_1
(Fin.fintype
(@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat)
(@Erdos390.Full.RegularRelativeMesh.Mesh.cellCount delta eta
M)
(@OfNat.ofNat.{0} Nat (nat_lit 1)
(instOfNatNat (nat_lit 1)))))
(instDecidableEqFin
(@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat)
(@Erdos390.Full.RegularRelativeMesh.Mesh.cellCount delta eta
M)
(@OfNat.ofNat.{0} Nat (nat_lit 1)
(instOfNatNat (nat_lit 1)))))
B))
(@OfNat.ofNat.{0} Nat (nat_lit 0) (instOfNatNat (nat_lit 0)))) =>
@Exists.{0}
(@Erdos390.Full.RegularMeshPrimeCutoffs.ScaleSeparation delta eta M
(@Erdos390.Full.PaperBridgeFit.StructuredSampleData.n.{u_1} Head inst
(@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{u_1, 0} Head
(Fin
(@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat)
(@Erdos390.Full.RegularRelativeMesh.Mesh.cellCount delta eta M)
(@OfNat.ofNat.{0} Nat (nat_lit 1) (instOfNatNat (nat_lit 1)))))
inst inst_1
(Fin.fintype
(@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat)
(@Erdos390.Full.RegularRelativeMesh.Mesh.cellCount delta eta M)
(@OfNat.ofNat.{0} Nat (nat_lit 1) (instOfNatNat (nat_lit 1)))))
(instDecidableEqFin
(@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat)
(@Erdos390.Full.RegularRelativeMesh.Mesh.cellCount delta eta M)
(@OfNat.ofNat.{0} Nat (nat_lit 1) (instOfNatNat (nat_lit 1)))))
B))
(@Erdos390.Full.PaperBridgeFit.StructuredSampleData.W.{u_1} Head inst
(@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{u_1, 0} Head
(Fin
(@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat)
(@Erdos390.Full.RegularRelativeMesh.Mesh.cellCount delta eta M)
(@OfNat.ofNat.{0} Nat (nat_lit 1) (instOfNatNat (nat_lit 1)))))
inst inst_1
(Fin.fintype
(@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat)
(@Erdos390.Full.RegularRelativeMesh.Mesh.cellCount delta eta M)
(@OfNat.ofNat.{0} Nat (nat_lit 1) (instOfNatNat (nat_lit 1)))))
(instDecidableEqFin
(@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat)
(@Erdos390.Full.RegularRelativeMesh.Mesh.cellCount delta eta M)
(@OfNat.ofNat.{0} Nat (nat_lit 1) (instOfNatNat (nat_lit 1)))))
B)))
fun
(S :
@Erdos390.Full.RegularMeshPrimeCutoffs.ScaleSeparation delta eta M
(@Erdos390.Full.PaperBridgeFit.StructuredSampleData.n.{u_1} Head
inst
(@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{u_1, 0} Head
(Fin
(@HAdd.hAdd.{0, 0, 0} Nat Nat Nat
(@instHAdd.{0} Nat instAddNat)
(@Erdos390.Full.RegularRelativeMesh.Mesh.cellCount delta eta
M)
(@OfNat.ofNat.{0} Nat (nat_lit 1)
(instOfNatNat (nat_lit 1)))))
inst inst_1
(Fin.fintype
(@HAdd.hAdd.{0, 0, 0} Nat Nat Nat
(@instHAdd.{0} Nat instAddNat)
(@Erdos390.Full.RegularRelativeMesh.Mesh.cellCount delta eta
M)
(@OfNat.ofNat.{0} Nat (nat_lit 1)
(instOfNatNat (nat_lit 1)))))
(instDecidableEqFin
(@HAdd.hAdd.{0, 0, 0} Nat Nat Nat
(@instHAdd.{0} Nat instAddNat)
(@Erdos390.Full.RegularRelativeMesh.Mesh.cellCount delta eta
M)
(@OfNat.ofNat.{0} Nat (nat_lit 1)
(instOfNatNat (nat_lit 1)))))
B))
(@Erdos390.Full.PaperBridgeFit.StructuredSampleData.W.{u_1} Head
inst
(@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{u_1, 0} Head
(Fin
(@HAdd.hAdd.{0, 0, 0} Nat Nat Nat
(@instHAdd.{0} Nat instAddNat)
(@Erdos390.Full.RegularRelativeMesh.Mesh.cellCount delta eta
M)
(@OfNat.ofNat.{0} Nat (nat_lit 1)
(instOfNatNat (nat_lit 1)))))
inst inst_1
(Fin.fintype
(@HAdd.hAdd.{0, 0, 0} Nat Nat Nat
(@instHAdd.{0} Nat instAddNat)
(@Erdos390.Full.RegularRelativeMesh.Mesh.cellCount delta eta
M)
(@OfNat.ofNat.{0} Nat (nat_lit 1)
(instOfNatNat (nat_lit 1)))))
(instDecidableEqFin
(@HAdd.hAdd.{0, 0, 0} Nat Nat Nat
(@instHAdd.{0} Nat instAddNat)
(@Erdos390.Full.RegularRelativeMesh.Mesh.cellCount delta eta
M)
(@OfNat.ofNat.{0} Nat (nat_lit 1)
(instOfNatNat (nat_lit 1)))))
B))) =>
@Eq.{1}
(@Erdos390.Full.ArithmeticBandGeometry.Partition.{0}
(@Erdos390.Full.PaperBridgeFit.StructuredSampleData.n.{u_1} Head inst
(@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{u_1, 0} Head
(Fin
(@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat)
(@Erdos390.Full.RegularRelativeMesh.Mesh.cellCount delta eta
M)
(@OfNat.ofNat.{0} Nat (nat_lit 1)
(instOfNatNat (nat_lit 1)))))
inst inst_1
(Fin.fintype
(@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat)
(@Erdos390.Full.RegularRelativeMesh.Mesh.cellCount delta eta
M)
(@OfNat.ofNat.{0} Nat (nat_lit 1)
(instOfNatNat (nat_lit 1)))))
(instDecidableEqFin
(@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat)
(@Erdos390.Full.RegularRelativeMesh.Mesh.cellCount delta eta
M)
(@OfNat.ofNat.{0} Nat (nat_lit 1)
(instOfNatNat (nat_lit 1)))))
B))
(@Erdos390.Full.PaperBridgeFit.StructuredSampleData.W.{u_1} Head inst
(@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{u_1, 0} Head
(Fin
(@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat)
(@Erdos390.Full.RegularRelativeMesh.Mesh.cellCount delta eta
M)
(@OfNat.ofNat.{0} Nat (nat_lit 1)
(instOfNatNat (nat_lit 1)))))
inst inst_1
(Fin.fintype
(@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat)
(@Erdos390.Full.RegularRelativeMesh.Mesh.cellCount delta eta
M)
(@OfNat.ofNat.{0} Nat (nat_lit 1)
(instOfNatNat (nat_lit 1)))))
(instDecidableEqFin
(@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat)
(@Erdos390.Full.RegularRelativeMesh.Mesh.cellCount delta eta
M)
(@OfNat.ofNat.{0} Nat (nat_lit 1)
(instOfNatNat (nat_lit 1)))))
B))
(Fin
(@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat)
(@Erdos390.Full.RegularRelativeMesh.Mesh.cellCount delta eta M)
(@OfNat.ofNat.{0} Nat (nat_lit 1) (instOfNatNat (nat_lit 1)))))
(Fin.fintype
(@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat)
(@Erdos390.Full.RegularRelativeMesh.Mesh.cellCount delta eta M)
(@OfNat.ofNat.{0} Nat (nat_lit 1) (instOfNatNat (nat_lit 1))))))
(@Erdos390.Full.PaperBridgeFit.BridgeData.partition.{u_1, 0} Head
(Fin
(@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat)
(@Erdos390.Full.RegularRelativeMesh.Mesh.cellCount delta eta M)
(@OfNat.ofNat.{0} Nat (nat_lit 1) (instOfNatNat (nat_lit 1)))))
inst inst_1
(Fin.fintype
(@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat)
(@Erdos390.Full.RegularRelativeMesh.Mesh.cellCount delta eta M)
(@OfNat.ofNat.{0} Nat (nat_lit 1) (instOfNatNat (nat_lit 1)))))
(instDecidableEqFin
(@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat)
(@Erdos390.Full.RegularRelativeMesh.Mesh.cellCount delta eta M)
(@OfNat.ofNat.{0} Nat (nat_lit 1) (instOfNatNat (nat_lit 1)))))
B)
(@Erdos390.Full.RegularMeshPrimeCutoffs.Mesh.canonicalPartition delta
eta M
(@Erdos390.Full.PaperBridgeFit.StructuredSampleData.n.{u_1} Head inst
(@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{u_1, 0} Head
(Fin
(@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat)
(@Erdos390.Full.RegularRelativeMesh.Mesh.cellCount delta eta
M)
(@OfNat.ofNat.{0} Nat (nat_lit 1)
(instOfNatNat (nat_lit 1)))))
inst inst_1
(Fin.fintype
(@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat)
(@Erdos390.Full.RegularRelativeMesh.Mesh.cellCount delta eta
M)
(@OfNat.ofNat.{0} Nat (nat_lit 1)
(instOfNatNat (nat_lit 1)))))
(instDecidableEqFin
(@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat)
(@Erdos390.Full.RegularRelativeMesh.Mesh.cellCount delta eta
M)
(@OfNat.ofNat.{0} Nat (nat_lit 1)
(instOfNatNat (nat_lit 1)))))
B))
(@Erdos390.Full.PaperBridgeFit.StructuredSampleData.W.{u_1} Head inst
(@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{u_1, 0} Head
(Fin
(@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat)
(@Erdos390.Full.RegularRelativeMesh.Mesh.cellCount delta eta
M)
(@OfNat.ofNat.{0} Nat (nat_lit 1)
(instOfNatNat (nat_lit 1)))))
inst inst_1
(Fin.fintype
(@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat)
(@Erdos390.Full.RegularRelativeMesh.Mesh.cellCount delta eta
M)
(@OfNat.ofNat.{0} Nat (nat_lit 1)
(instOfNatNat (nat_lit 1)))))
(instDecidableEqFin
(@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat)
(@Erdos390.Full.RegularRelativeMesh.Mesh.cellCount delta eta
M)
(@OfNat.ofNat.{0} Nat (nat_lit 1)
(instOfNatNat (nat_lit 1)))))
B))
hdelta
(@Erdos390.Full.PaperBridgeFit.BridgeData.n_gt_one.{u_1, 0} Head
(Fin
(@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat)
(@Erdos390.Full.RegularRelativeMesh.Mesh.cellCount delta eta M)
(@OfNat.ofNat.{0} Nat (nat_lit 1) (instOfNatNat (nat_lit 1)))))
inst inst_1
(Fin.fintype
(@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat)
(@Erdos390.Full.RegularRelativeMesh.Mesh.cellCount delta eta M)
(@OfNat.ofNat.{0} Nat (nat_lit 1) (instOfNatNat (nat_lit 1)))))
(instDecidableEqFin
(@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat)
(@Erdos390.Full.RegularRelativeMesh.Mesh.cellCount delta eta M)
(@OfNat.ofNat.{0} Nat (nat_lit 1) (instOfNatNat (nat_lit 1)))))
B)
hWne S)) →
@Eq.{1} Real
(@Erdos390.Full.PaperBridgeFit.BridgeData.w.{u_1, 0} Head
(Fin
(@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat)
(@Erdos390.Full.RegularRelativeMesh.Mesh.cellCount delta eta M)
(@OfNat.ofNat.{0} Nat (nat_lit 1) (instOfNatNat (nat_lit 1)))))
inst inst_1
(Fin.fintype
(@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat)
(@Erdos390.Full.RegularRelativeMesh.Mesh.cellCount delta eta M)
(@OfNat.ofNat.{0} Nat (nat_lit 1) (instOfNatNat (nat_lit 1)))))
(instDecidableEqFin
(@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat)
(@Erdos390.Full.RegularRelativeMesh.Mesh.cellCount delta eta M)
(@OfNat.ofNat.{0} Nat (nat_lit 1) (instOfNatNat (nat_lit 1)))))
B)
(@HAdd.hAdd.{0, 0, 0} Real Real Real (@instHAdd.{0} Real Real.instAdd)
delta eta) →
∀
(xi :
@Erdos390.Full.PaperBridgeFit.BridgeData.ParamSpace.{u_1, 0} Head
(Fin
(@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat)
(@Erdos390.Full.RegularRelativeMesh.Mesh.cellCount delta eta M)
(@OfNat.ofNat.{0} Nat (nat_lit 1) (instOfNatNat (nat_lit 1)))))
inst inst_1
(Fin.fintype
(@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat)
(@Erdos390.Full.RegularRelativeMesh.Mesh.cellCount delta eta M)
(@OfNat.ofNat.{0} Nat (nat_lit 1) (instOfNatNat (nat_lit 1)))))
(instDecidableEqFin
(@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat)
(@Erdos390.Full.RegularRelativeMesh.Mesh.cellCount delta eta M)
(@OfNat.ofNat.{0} Nat (nat_lit 1) (instOfNatNat (nat_lit 1)))))
B),
(∀
(p :
Erdos390.Full.ArithmeticBandGeometry.BandPrime
(@Erdos390.Full.PaperBridgeFit.StructuredSampleData.n.{u_1} Head
inst
(@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{u_1, 0}
Head
(Fin
(@HAdd.hAdd.{0, 0, 0} Nat Nat Nat
(@instHAdd.{0} Nat instAddNat)
(@Erdos390.Full.RegularRelativeMesh.Mesh.cellCount delta
eta M)
(@OfNat.ofNat.{0} Nat (nat_lit 1)
(instOfNatNat (nat_lit 1)))))
inst inst_1
(Fin.fintype
(@HAdd.hAdd.{0, 0, 0} Nat Nat Nat
(@instHAdd.{0} Nat instAddNat)
(@Erdos390.Full.RegularRelativeMesh.Mesh.cellCount delta
eta M)
(@OfNat.ofNat.{0} Nat (nat_lit 1)
(instOfNatNat (nat_lit 1)))))
(instDecidableEqFin
(@HAdd.hAdd.{0, 0, 0} Nat Nat Nat
(@instHAdd.{0} Nat instAddNat)
(@Erdos390.Full.RegularRelativeMesh.Mesh.cellCount delta
eta M)
(@OfNat.ofNat.{0} Nat (nat_lit 1)
(instOfNatNat (nat_lit 1)))))
B))
(@Erdos390.Full.PaperBridgeFit.StructuredSampleData.W.{u_1} Head
inst
(@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{u_1, 0}
Head
(Fin
(@HAdd.hAdd.{0, 0, 0} Nat Nat Nat
(@instHAdd.{0} Nat instAddNat)
(@Erdos390.Full.RegularRelativeMesh.Mesh.cellCount delta
eta M)
(@OfNat.ofNat.{0} Nat (nat_lit 1)
(instOfNatNat (nat_lit 1)))))
inst inst_1
(Fin.fintype
(@HAdd.hAdd.{0, 0, 0} Nat Nat Nat
(@instHAdd.{0} Nat instAddNat)
(@Erdos390.Full.RegularRelativeMesh.Mesh.cellCount delta
eta M)
(@OfNat.ofNat.{0} Nat (nat_lit 1)
(instOfNatNat (nat_lit 1)))))
(instDecidableEqFin
(@HAdd.hAdd.{0, 0, 0} Nat Nat Nat
(@instHAdd.{0} Nat instAddNat)
(@Erdos390.Full.RegularRelativeMesh.Mesh.cellCount delta
eta M)
(@OfNat.ofNat.{0} Nat (nat_lit 1)
(instOfNatNat (nat_lit 1)))))
B))),
@LE.le.{0} Real Real.instLE
(@abs.{0} Real Real.lattice Real.instAddGroup
(@Erdos390.Full.PaperBridgeFit.BridgeData.effectivePrimeCoefficient.{u_1,
0}
Head
(Fin
(@HAdd.hAdd.{0, 0, 0} Nat Nat Nat
(@instHAdd.{0} Nat instAddNat)
(@Erdos390.Full.RegularRelativeMesh.Mesh.cellCount delta eta
M)
(@OfNat.ofNat.{0} Nat (nat_lit 1)
(instOfNatNat (nat_lit 1)))))
inst inst_1
(Fin.fintype
(@HAdd.hAdd.{0, 0, 0} Nat Nat Nat
(@instHAdd.{0} Nat instAddNat)
(@Erdos390.Full.RegularRelativeMesh.Mesh.cellCount delta eta
M)
(@OfNat.ofNat.{0} Nat (nat_lit 1)
(instOfNatNat (nat_lit 1)))))
(instDecidableEqFin
(@HAdd.hAdd.{0, 0, 0} Nat Nat Nat
(@instHAdd.{0} Nat instAddNat)
(@Erdos390.Full.RegularRelativeMesh.Mesh.cellCount delta eta
M)
(@OfNat.ofNat.{0} Nat (nat_lit 1)
(instOfNatNat (nat_lit 1)))))
B xi p))
Acoef) →
@LE.le.{0}
((fun
(x :
@Erdos390.Full.PaperBridgeFit.BridgeData.Coord.{u_1, 0} Head
(Fin
(@HAdd.hAdd.{0, 0, 0} Nat Nat Nat
(@instHAdd.{0} Nat instAddNat)
(@Erdos390.Full.RegularRelativeMesh.Mesh.cellCount
delta eta M)
(@OfNat.ofNat.{0} Nat (nat_lit 1)
(instOfNatNat (nat_lit 1)))))
inst inst_1
(Fin.fintype
(@HAdd.hAdd.{0, 0, 0} Nat Nat Nat
(@instHAdd.{0} Nat instAddNat)
(@Erdos390.Full.RegularRelativeMesh.Mesh.cellCount
delta eta M)
(@OfNat.ofNat.{0} Nat (nat_lit 1)
(instOfNatNat (nat_lit 1)))))
(instDecidableEqFin
(@HAdd.hAdd.{0, 0, 0} Nat Nat Nat
(@instHAdd.{0} Nat instAddNat)
(@Erdos390.Full.RegularRelativeMesh.Mesh.cellCount
delta eta M)
(@OfNat.ofNat.{0} Nat (nat_lit 1)
(instOfNatNat (nat_lit 1)))))
B) =>
Real)
(@Erdos390.Full.PaperBridgeFit.MomentCoord.physical.{0, u_1}
(@Erdos390.Full.PaperBridgeFit.BridgeData.GaugeIndex.{u_1, 0}
Head
(Fin
(@HAdd.hAdd.{0, 0, 0} Nat Nat Nat
(@instHAdd.{0} Nat instAddNat)
(@Erdos390.Full.RegularRelativeMesh.Mesh.cellCount delta
eta M)
(@OfNat.ofNat.{0} Nat (nat_lit 1)
(instOfNatNat (nat_lit 1)))))
inst inst_1
(Fin.fintype
(@HAdd.hAdd.{0, 0, 0} Nat Nat Nat
(@instHAdd.{0} Nat instAddNat)
(@Erdos390.Full.RegularRelativeMesh.Mesh.cellCount delta
eta M)
(@OfNat.ofNat.{0} Nat (nat_lit 1)
(instOfNatNat (nat_lit 1)))))
(instDecidableEqFin
(@HAdd.hAdd.{0, 0, 0} Nat Nat Nat
(@instHAdd.{0} Nat instAddNat)
(@Erdos390.Full.RegularRelativeMesh.Mesh.cellCount delta
eta M)
(@OfNat.ofNat.{0} Nat (nat_lit 1)
(instOfNatNat (nat_lit 1)))))
B)
(@Erdos390.Full.PaperBridgeFit.BridgeData.HeadIndex.{u_1, 0}
Head
(Fin
(@HAdd.hAdd.{0, 0, 0} Nat Nat Nat
(@instHAdd.{0} Nat instAddNat)
(@Erdos390.Full.RegularRelativeMesh.Mesh.cellCount delta
eta M)
(@OfNat.ofNat.{0} Nat (nat_lit 1)
(instOfNatNat (nat_lit 1)))))
inst inst_1
(Fin.fintype
(@HAdd.hAdd.{0, 0, 0} Nat Nat Nat
(@instHAdd.{0} Nat instAddNat)
(@Erdos390.Full.RegularRelativeMesh.Mesh.cellCount delta
eta M)
(@OfNat.ofNat.{0} Nat (nat_lit 1)
(instOfNatNat (nat_lit 1)))))
(instDecidableEqFin
(@HAdd.hAdd.{0, 0, 0} Nat Nat Nat
(@instHAdd.{0} Nat instAddNat)
(@Erdos390.Full.RegularRelativeMesh.Mesh.cellCount delta
eta M)
(@OfNat.ofNat.{0} Nat (nat_lit 1)
(instOfNatNat (nat_lit 1)))))
B)))
Real.instLE
(@abs.{0}
((fun
(x :
@Erdos390.Full.PaperBridgeFit.BridgeData.Coord.{u_1, 0}
Head
(Fin
(@HAdd.hAdd.{0, 0, 0} Nat Nat Nat
(@instHAdd.{0} Nat instAddNat)
(@Erdos390.Full.RegularRelativeMesh.Mesh.cellCount
delta eta M)
(@OfNat.ofNat.{0} Nat (nat_lit 1)
(instOfNatNat (nat_lit 1)))))
inst inst_1
(Fin.fintype
(@HAdd.hAdd.{0, 0, 0} Nat Nat Nat
(@instHAdd.{0} Nat instAddNat)
(@Erdos390.Full.RegularRelativeMesh.Mesh.cellCount
delta eta M)
(@OfNat.ofNat.{0} Nat (nat_lit 1)
(instOfNatNat (nat_lit 1)))))
(instDecidableEqFin
(@HAdd.hAdd.{0, 0, 0} Nat Nat Nat
(@instHAdd.{0} Nat instAddNat)
(@Erdos390.Full.RegularRelativeMesh.Mesh.cellCount
delta eta M)
(@OfNat.ofNat.{0} Nat (nat_lit 1)
(instOfNatNat (nat_lit 1)))))
B) =>
Real)
(@Erdos390.Full.PaperBridgeFit.MomentCoord.physical.{0, u_1}
(@Erdos390.Full.PaperBridgeFit.BridgeData.GaugeIndex.{u_1, 0}
Head
(Fin
(@HAdd.hAdd.{0, 0, 0} Nat Nat Nat
(@instHAdd.{0} Nat instAddNat)
(@Erdos390.Full.RegularRelativeMesh.Mesh.cellCount delta
eta M)
(@OfNat.ofNat.{0} Nat (nat_lit 1)
(instOfNatNat (nat_lit 1)))))
inst inst_1
(Fin.fintype
(@HAdd.hAdd.{0, 0, 0} Nat Nat Nat
(@instHAdd.{0} Nat instAddNat)
(@Erdos390.Full.RegularRelativeMesh.Mesh.cellCount delta
eta M)
(@OfNat.ofNat.{0} Nat (nat_lit 1)
(instOfNatNat (nat_lit 1)))))
(instDecidableEqFin
(@HAdd.hAdd.{0, 0, 0} Nat Nat Nat
(@instHAdd.{0} Nat instAddNat)
(@Erdos390.Full.RegularRelativeMesh.Mesh.cellCount delta
eta M)
(@OfNat.ofNat.{0} Nat (nat_lit 1)
(instOfNatNat (nat_lit 1)))))
B)
(@Erdos390.Full.PaperBridgeFit.BridgeData.HeadIndex.{u_1, 0}
Head
(Fin
(@HAdd.hAdd.{0, 0, 0} Nat Nat Nat
(@instHAdd.{0} Nat instAddNat)
(@Erdos390.Full.RegularRelativeMesh.Mesh.cellCount delta
eta M)
(@OfNat.ofNat.{0} Nat (nat_lit 1)
(instOfNatNat (nat_lit 1)))))
inst inst_1
(Fin.fintype
(@HAdd.hAdd.{0, 0, 0} Nat Nat Nat
(@instHAdd.{0} Nat instAddNat)
(@Erdos390.Full.RegularRelativeMesh.Mesh.cellCount delta
eta M)
(@OfNat.ofNat.{0} Nat (nat_lit 1)
(instOfNatNat (nat_lit 1)))))
(instDecidableEqFin
(@HAdd.hAdd.{0, 0, 0} Nat Nat Nat
(@instHAdd.{0} Nat instAddNat)
(@Erdos390.Full.RegularRelativeMesh.Mesh.cellCount delta
eta M)
(@OfNat.ofNat.{0} Nat (nat_lit 1)
(instOfNatNat (nat_lit 1)))))
B)))
Real.lattice Real.instAddGroup
(@WithLp.ofLp.{u_1}
(@OfNat.ofNat.{0} ENNReal (nat_lit 2)
(@instOfNatAtLeastTwo.{0} ENNReal (nat_lit 2)
(@AddMonoidWithOne.toNatCast.{0} ENNReal
(@AddCommMonoidWithOne.toAddMonoidWithOne.{0} ENNReal
ENNReal.instAddCommMonoidWithOne))
(@Nat.instAtLeastTwoHAddOfNat
(@OfNat.ofNat.{0} Nat (nat_lit 1)
(instOfNatNat (nat_lit 1)))
(@Nat.instNeZeroSucc
(@OfNat.ofNat.{0} Nat (nat_lit 0)
(instOfNatNat (nat_lit 0)))))))
((i :
@Erdos390.Full.PaperBridgeFit.BridgeData.Coord.{u_1, 0} Head
(Fin
(@HAdd.hAdd.{0, 0, 0} Nat Nat Nat
(@instHAdd.{0} Nat instAddNat)
(@Erdos390.Full.RegularRelativeMesh.Mesh.cellCount
delta eta M)
(@OfNat.ofNat.{0} Nat (nat_lit 1)
(instOfNatNat (nat_lit 1)))))
inst inst_1
(Fin.fintype
(@HAdd.hAdd.{0, 0, 0} Nat Nat Nat
(@instHAdd.{0} Nat instAddNat)
(@Erdos390.Full.RegularRelativeMesh.Mesh.cellCount
delta eta M)
(@OfNat.ofNat.{0} Nat (nat_lit 1)
(instOfNatNat (nat_lit 1)))))
(instDecidableEqFin
(@HAdd.hAdd.{0, 0, 0} Nat Nat Nat
(@instHAdd.{0} Nat instAddNat)
(@Erdos390.Full.RegularRelativeMesh.Mesh.cellCount
delta eta M)
(@OfNat.ofNat.{0} Nat (nat_lit 1)
(instOfNatNat (nat_lit 1)))))
B) →
(fun
(x :
@Erdos390.Full.PaperBridgeFit.BridgeData.Coord.{u_1,
0}
Head
(Fin
(@HAdd.hAdd.{0, 0, 0} Nat Nat Nat
(@instHAdd.{0} Nat instAddNat)
(@Erdos390.Full.RegularRelativeMesh.Mesh.cellCount
delta eta M)
(@OfNat.ofNat.{0} Nat (nat_lit 1)
(instOfNatNat (nat_lit 1)))))
inst inst_1
(Fin.fintype
(@HAdd.hAdd.{0, 0, 0} Nat Nat Nat
(@instHAdd.{0} Nat instAddNat)
(@Erdos390.Full.RegularRelativeMesh.Mesh.cellCount
delta eta M)
(@OfNat.ofNat.{0} Nat (nat_lit 1)
(instOfNatNat (nat_lit 1)))))
(instDecidableEqFin
(@HAdd.hAdd.{0, 0, 0} Nat Nat Nat
(@instHAdd.{0} Nat instAddNat)
(@Erdos390.Full.RegularRelativeMesh.Mesh.cellCount
delta eta M)
(@OfNat.ofNat.{0} Nat (nat_lit 1)
(instOfNatNat (nat_lit 1)))))
B) =>
Real)
i)
xi
(@Erdos390.Full.PaperBridgeFit.MomentCoord.physical.{0, u_1}
(@Erdos390.Full.PaperBridgeFit.BridgeData.GaugeIndex.{u_1, 0}
Head
(Fin
(@HAdd.hAdd.{0, 0, 0} Nat Nat Nat
(@instHAdd.{0} Nat instAddNat)
(@Erdos390.Full.RegularRelativeMesh.Mesh.cellCount delta
eta M)
(@OfNat.ofNat.{0} Nat (nat_lit 1)
(instOfNatNat (nat_lit 1)))))
inst inst_1
(Fin.fintype
(@HAdd.hAdd.{0, 0, 0} Nat Nat Nat
(@instHAdd.{0} Nat instAddNat)
(@Erdos390.Full.RegularRelativeMesh.Mesh.cellCount delta
eta M)
(@OfNat.ofNat.{0} Nat (nat_lit 1)
(instOfNatNat (nat_lit 1)))))
(instDecidableEqFin
(@HAdd.hAdd.{0, 0, 0} Nat Nat Nat
(@instHAdd.{0} Nat instAddNat)
(@Erdos390.Full.RegularRelativeMesh.Mesh.cellCount delta
eta M)
(@OfNat.ofNat.{0} Nat (nat_lit 1)
(instOfNatNat (nat_lit 1)))))
B)
(@Erdos390.Full.PaperBridgeFit.BridgeData.HeadIndex.{u_1, 0}
Head
(Fin
(@HAdd.hAdd.{0, 0, 0} Nat Nat Nat
(@instHAdd.{0} Nat instAddNat)
(@Erdos390.Full.RegularRelativeMesh.Mesh.cellCount delta
eta M)
(@OfNat.ofNat.{0} Nat (nat_lit 1)
(instOfNatNat (nat_lit 1)))))
inst inst_1
(Fin.fintype
(@HAdd.hAdd.{0, 0, 0} Nat Nat Nat
(@instHAdd.{0} Nat instAddNat)
(@Erdos390.Full.RegularRelativeMesh.Mesh.cellCount delta
eta M)
(@OfNat.ofNat.{0} Nat (nat_lit 1)
(instOfNatNat (nat_lit 1)))))
(instDecidableEqFin
(@HAdd.hAdd.{0, 0, 0} Nat Nat Nat
(@instHAdd.{0} Nat instAddNat)
(@Erdos390.Full.RegularRelativeMesh.Mesh.cellCount delta
eta M)
(@OfNat.ofNat.{0} Nat (nat_lit 1)
(instOfNatNat (nat_lit 1)))))
B))))
Aphys →
∀
(i :
Fin
(@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat)
(@Erdos390.Full.RegularRelativeMesh.Mesh.cellCount delta eta
M)
(@OfNat.ofNat.{0} Nat (nat_lit 1)
(instOfNatNat (nat_lit 1))))),
@LE.le.{0} Real Real.instLE
(@abs.{0} Real Real.lattice Real.instAddGroup
(@HSub.hSub.{0, 0, 0} Real Real Real
(@instHSub.{0} Real Real.instSub)
(@Erdos390.Full.PaperBridgeFit.BridgeData.normalizedBandCovarianceRow.{u_1,
0}
Head
(Fin
(@HAdd.hAdd.{0, 0, 0} Nat Nat Nat
(@instHAdd.{0} Nat instAddNat)
(@Erdos390.Full.RegularRelativeMesh.Mesh.cellCount delta
eta M)
(@OfNat.ofNat.{0} Nat (nat_lit 1)
(instOfNatNat (nat_lit 1)))))
inst inst_1
(Fin.fintype
(@HAdd.hAdd.{0, 0, 0} Nat Nat Nat
(@instHAdd.{0} Nat instAddNat)
(@Erdos390.Full.RegularRelativeMesh.Mesh.cellCount delta
eta M)
(@OfNat.ofNat.{0} Nat (nat_lit 1)
(instOfNatNat (nat_lit 1)))))
(instDecidableEqFin
(@HAdd.hAdd.{0, 0, 0} Nat Nat Nat
(@instHAdd.{0} Nat instAddNat)
(@Erdos390.Full.RegularRelativeMesh.Mesh.cellCount delta
eta M)
(@OfNat.ofNat.{0} Nat (nat_lit 1)
(instOfNatNat (nat_lit 1)))))
B inst_2 xi
(@Erdos390.Full.PaperBridgeFit.BridgeData.slowScore.{u_1, 0}
Head
(Fin
(@HAdd.hAdd.{0, 0, 0} Nat Nat Nat
(@instHAdd.{0} Nat instAddNat)
(@Erdos390.Full.RegularRelativeMesh.Mesh.cellCount
delta eta M)
(@OfNat.ofNat.{0} Nat (nat_lit 1)
(instOfNatNat (nat_lit 1)))))
inst inst_1
(Fin.fintype
(@HAdd.hAdd.{0, 0, 0} Nat Nat Nat
(@instHAdd.{0} Nat instAddNat)
(@Erdos390.Full.RegularRelativeMesh.Mesh.cellCount
delta eta M)
(@OfNat.ofNat.{0} Nat (nat_lit 1)
(instOfNatNat (nat_lit 1)))))
(instDecidableEqFin
(@HAdd.hAdd.{0, 0, 0} Nat Nat Nat
(@instHAdd.{0} Nat instAddNat)
(@Erdos390.Full.RegularRelativeMesh.Mesh.cellCount
delta eta M)
(@OfNat.ofNat.{0} Nat (nat_lit 1)
(instOfNatNat (nat_lit 1)))))
B)
i)
(@Erdos390.Full.PaperBridgeFit.BridgeData.referenceSlowRow.{u_1,
0}
Head
(Fin
(@HAdd.hAdd.{0, 0, 0} Nat Nat Nat
(@instHAdd.{0} Nat instAddNat)
(@Erdos390.Full.RegularRelativeMesh.Mesh.cellCount delta
eta M)
(@OfNat.ofNat.{0} Nat (nat_lit 1)
(instOfNatNat (nat_lit 1)))))
inst inst_1
(Fin.fintype
(@HAdd.hAdd.{0, 0, 0} Nat Nat Nat
(@instHAdd.{0} Nat instAddNat)
(@Erdos390.Full.RegularRelativeMesh.Mesh.cellCount delta
eta M)
(@OfNat.ofNat.{0} Nat (nat_lit 1)
(instOfNatNat (nat_lit 1)))))
(instDecidableEqFin
(@HAdd.hAdd.{0, 0, 0} Nat Nat Nat
(@instHAdd.{0} Nat instAddNat)
(@Erdos390.Full.RegularRelativeMesh.Mesh.cellCount delta
eta M)
(@OfNat.ofNat.{0} Nat (nat_lit 1)
(instOfNatNat (nat_lit 1)))))
B i)))
(@HMul.hMul.{0, 0, 0} Real Real Real
(@instHMul.{0} Real Real.instMul)
(@HMul.hMul.{0, 0, 0} Real Real Real
(@instHMul.{0} Real Real.instMul) r
(@Erdos390.Full.PaperBridgeFit.BridgeData.w.{u_1, 0} Head
(Fin
(@HAdd.hAdd.{0, 0, 0} Nat Nat Nat
(@instHAdd.{0} Nat instAddNat)
(@Erdos390.Full.RegularRelativeMesh.Mesh.cellCount delta
eta M)
(@OfNat.ofNat.{0} Nat (nat_lit 1)
(instOfNatNat (nat_lit 1)))))
inst inst_1
(Fin.fintype
(@HAdd.hAdd.{0, 0, 0} Nat Nat Nat
(@instHAdd.{0} Nat instAddNat)
(@Erdos390.Full.RegularRelativeMesh.Mesh.cellCount delta
eta M)
(@OfNat.ofNat.{0} Nat (nat_lit 1)
(instOfNatNat (nat_lit 1)))))
(instDecidableEqFin
(@HAdd.hAdd.{0, 0, 0} Nat Nat Nat
(@instHAdd.{0} Nat instAddNat)
(@Erdos390.Full.RegularRelativeMesh.Mesh.cellCount delta
eta M)
(@OfNat.ofNat.{0} Nat (nat_lit 1)
(instOfNatNat (nat_lit 1)))))
B))
(@Erdos390.Full.PaperBridgeFit.BridgeData.bandCenter.{u_1, 0}
Head
(Fin
(@HAdd.hAdd.{0, 0, 0} Nat Nat Nat
(@instHAdd.{0} Nat instAddNat)
(@Erdos390.Full.RegularRelativeMesh.Mesh.cellCount delta
eta M)
(@OfNat.ofNat.{0} Nat (nat_lit 1)
(instOfNatNat (nat_lit 1)))))
inst inst_1
(Fin.fintype
(@HAdd.hAdd.{0, 0, 0} Nat Nat Nat
(@instHAdd.{0} Nat instAddNat)
(@Erdos390.Full.RegularRelativeMesh.Mesh.cellCount delta
eta M)
(@OfNat.ofNat.{0} Nat (nat_lit 1)
(instOfNatNat (nat_lit 1)))))
(instDecidableEqFin
(@HAdd.hAdd.{0, 0, 0} Nat Nat Nat
(@instHAdd.{0} Nat instAddNat)
(@Erdos390.Full.RegularRelativeMesh.Mesh.cellCount delta
eta M)
(@OfNat.ofNat.{0} Nat (nat_lit 1)
(instOfNatNat (nat_lit 1)))))
B i)))
(@Filter.atTop.{0} Nat Nat.instPreorder)
noncomputable def Erdos390.RemainingAnalyticGoal004_002 : Prop :=
∀ {Head : Type u_1} [inst : Fintype.{u_1} Head] [inst_1 : DecidableEq.{u_1 + 1} Head] [inst_2 : Nonempty.{u_1 + 1} Head] (P : Head → Erdos390.Full.HeadPattern.Pattern) (I : Erdos390.Full.PaperGuardCensus.PhysicalIntervals) (U : Real), @LE.le.{0} Real _ (@OfNat.ofNat.{0} Real (nat_lit 1) _) U → (∀ (sigma : Erdos390.Full.PaperBridgeFit.PhysicalSign), @LE.le.{0} Real _ (@OfNat.ofNat.{0} Real (nat_lit 1) _) (I.lower sigma)) → (∀ (sigma : Erdos390.Full.PaperBridgeFit.PhysicalSign), @LE.le.{0} Real _ (I.upper sigma) U) → ∀ (Cprom Cbank : Nat) (G : (n : Nat) → Erdos390.Full.PaperGuardCensus.Ledger n Cprom Cbank) (W : Nat), @LT.lt.{0} Nat _ (@OfNat.ofNat.{0} Nat (nat_lit 1) _) W → (∀ (h : Head) (p : Nat), @Membership.mem.{0, 0} Nat (Finset.{0} Nat) _ (P h).primes p → @LE.le.{0} Nat _ p W) → ∀ (a : NNReal), @Exists.{1} (Nat → Real) fun (epsilon : Nat → Real) => And (∀ (n : Nat), @LE.le.{0} Real _ (@OfNat.ofNat.{0} Real (nat_lit 0) _) (epsilon n)) (And (@Filter.Tendsto.{0, 0} Nat Real (fun (n : Nat) => @HMul.hMul.{0, 0, 0} Real Real Real _ (epsilon n) (Real.log (Erdos390.Full.Scale.L n))) (@Filter.atTop.{0} Nat _) (@nhds.{0} Real _ (@OfNat.ofNat.{0} Real (nat_lit 0) _))) (@Exists.{1} Nat fun (N₀ : Nat) => ∀ {Band : Type u_2} [inst_3 : Fintype.{u_2} Band] [inst_4 : DecidableEq.{u_2 + 1} Band] (B : @Erdos390.Full.PaperBridgeFit.BridgeData.{u_1, u_2} Head Band _ _ _ _) (z : @Subtype.{u_1 + 1} (@Erdos390.Full.PaperBridgeFit.BridgeData.EffectiveAmbient.{u_1, u_2} Head Band _ _ _ _ B) fun (x : @Erdos390.Full.PaperBridgeFit.BridgeData.EffectiveAmbient.{u_1, u_2} Head Band _ _ _ _ B) => @Membership.mem.{u_1, u_1} (@Erdos390.Full.PaperBridgeFit.BridgeData.EffectiveAmbient.{u_1, u_2} Head Band _ _ _ _ B) (@Submodule.{0, u_1} Real (@Erdos390.Full.PaperBridgeFit.BridgeData.EffectiveAmbient.{u_1, u_2} Head Band _ _ _ _ B) _ _ _) _ (@Erdos390.Full.PaperBridgeFit.BridgeData.EffectiveParamSpace.{u_1, u_2} Head Band _ _ _ _ B) x), @Membership.mem.{u_1, u_1} (@Subtype.{u_1 + 1} (@Erdos390.Full.PaperBridgeFit.BridgeData.EffectiveAmbient.{u_1, u_2} Head Band _ _ _ _ B) fun (x : @Erdos390.Full.PaperBridgeFit.BridgeData.EffectiveAmbient.{u_1, u_2} Head Band _ _ _ _ B) => @Membership.mem.{u_1, u_1} (@Erdos390.Full.PaperBridgeFit.BridgeData.EffectiveAmbient.{u_1, u_2} Head Band _ _ _ _ B) (@Submodule.{0, u_1} Real (@Erdos390.Full.PaperBridgeFit.BridgeData.EffectiveAmbient.{u_1, u_2} Head Band _ _ _ _ B) _ _ _) _ (@Erdos390.Full.PaperBridgeFit.BridgeData.EffectiveParamSpace.{u_1, u_2} Head Band _ _ _ _ B) x) (Set.{u_1} (@Subtype.{u_1 + 1} (@Erdos390.Full.PaperBridgeFit.BridgeData.EffectiveAmbient.{u_1, u_2} Head Band _ _ _ _ B) fun (x : @Erdos390.Full.PaperBridgeFit.BridgeData.EffectiveAmbient.{u_1, u_2} Head Band _ _ _ _ B) => @Membership.mem.{u_1, u_1} (@Erdos390.Full.PaperBridgeFit.BridgeData.EffectiveAmbient.{u_1, u_2} Head Band _ _ _ _ B) (@Submodule.{0, u_1} Real (@Erdos390.Full.PaperBridgeFit.BridgeData.EffectiveAmbient.{u_1, u_2} Head Band _ _ _ _ B) _ _ _) _ (@Erdos390.Full.PaperBridgeFit.BridgeData.EffectiveParamSpace.{u_1, u_2} Head Band _ _ _ _ B) x)) _ (@Metric.closedBall.{u_1} (@Subtype.{u_1 + 1} (@Erdos390.Full.PaperBridgeFit.BridgeData.EffectiveAmbient.{u_1, u_2} Head Band _ _ _ _ B) fun (x : @Erdos390.Full.PaperBridgeFit.BridgeData.EffectiveAmbient.{u_1, u_2} Head Band _ _ _ _ B) => @Membership.mem.{u_1, u_1} (@Erdos390.Full.PaperBridgeFit.BridgeData.EffectiveAmbient.{u_1, u_2} Head Band _ _ _ _ B) (@Submodule.{0, u_1} Real (@Erdos390.Full.PaperBridgeFit.BridgeData.EffectiveAmbient.{u_1, u_2} Head Band _ _ _ _ B) _ _ _) _ (@Erdos390.Full.PaperBridgeFit.BridgeData.EffectiveParamSpace.{u_1, u_2} Head Band _ _ _ _ B) x) _ (@OfNat.ofNat.{u_1} (@Subtype.{u_1 + 1} (@Erdos390.Full.PaperBridgeFit.BridgeData.EffectiveAmbient.{u_1, u_2} Head Band _ _ _ _ B) fun (x : @Erdos390.Full.PaperBridgeFit.BridgeData.EffectiveAmbient.{u_1, u_2} Head Band _ _ _ _ B) => @Membership.mem.{u_1, u_1} (@Erdos390.Full.PaperBridgeFit.BridgeData.EffectiveAmbient.{u_1, u_2} Head Band _ _ _ _ B) (@Submodule.{0, u_1} Real (@Erdos390.Full.PaperBridgeFit.BridgeData.EffectiveAmbient.{u_1, u_2} Head Band _ _ _ _ B) _ _ _) _ (@Erdos390.Full.PaperBridgeFit.BridgeData.EffectiveParamSpace.{u_1, u_2} Head Band _ _ _ _ B) x) (nat_lit 0) _) (↑a : Real)) z → @LE.le.{0} Nat _ N₀ (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.n.{u_1} Head _ (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{u_1, u_2} Head Band _ _ _ _ B)) → @Eq.{1} Nat (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.W.{u_1} Head _ (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{u_1, u_2} Head Band _ _ _ _ B)) W → ∀ (hsep : @LT.lt.{0} Nat _ (Erdos390.Full.ArithmeticModel.physicalBound (I.upper Erdos390.Full.PaperBridgeFit.PhysicalSign.minus) (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.n.{u_1} Head _ (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{u_1, u_2} Head Band _ _ _ _ B))) (Erdos390.Full.ArithmeticModel.physicalBound (I.lower Erdos390.Full.PaperBridgeFit.PhysicalSign.plus) (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.n.{u_1} Head _ (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{u_1, u_2} Head Band _ _ _ _ B)))) (hremaining : ∀ (c : Erdos390.Full.PaperBridgeFit.Cell.{u_1} Head), @Finset.Nonempty.{0} Nat (@SDiff.sdiff.{0} (Finset.{0} Nat) _ (@Erdos390.Full.PaperGuardCensus.rawCell.{u_1} Head P I (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.n.{u_1} Head _ (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{u_1, u_2} Head Band _ _ _ _ B)) c) (@Erdos390.Full.PaperGuardCensus.Ledger.guards (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.n.{u_1} Head _ (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{u_1, u_2} Head Band _ _ _ _ B)) Cprom Cbank (G (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.n.{u_1} Head _ (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{u_1, u_2} Head Band _ _ _ _ B)))))), @Eq.{u_1 + 1} (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.{u_1} Head _) (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{u_1, u_2} Head Band _ _ _ _ B) (@Erdos390.Full.PaperGuardCensus.canonicalSampleData.{u_1} Head _ P I (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.n.{u_1} Head _ (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{u_1, u_2} Head Band _ _ _ _ B)) (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.W.{u_1} Head _ (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{u_1, u_2} Head Band _ _ _ _ B)) Cprom Cbank (G (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.n.{u_1} Head _ (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{u_1, u_2} Head Band _ _ _ _ B))) hsep hremaining) → ∀ (c : Erdos390.Full.PaperBridgeFit.NuisanceCoord.{u_1} (@Erdos390.Full.PaperBridgeFit.BridgeData.HeadIndex.{u_1, u_2} Head Band _ _ _ _ B)) (p : Erdos390.Full.ArithmeticBandGeometry.BandPrime (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.n.{u_1} Head _ (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{u_1, u_2} Head Band _ _ _ _ B)) (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.W.{u_1} Head _ (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{u_1, u_2} Head Band _ _ _ _ B))), @LE.le.{0} Real _ (@abs.{0} Real _ _ (@Erdos390.Full.FiniteProbability.covariance.{u_1} (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.Sample.{u_1} Head _ (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{u_1, u_2} Head Band _ _ _ _ B)) _ (@Erdos390.Full.PaperBridgeFit.BridgeData.tiltedLaw.{u_1, u_2} Head Band _ _ _ _ B _ (@DFunLike.coe.{max (u_1 + 1) (u_2 + 1), u_1 + 1, max (u_1 + 1) (u_2 + 1)} (@ContinuousLinearEquiv.{0, 0, u_1, max u_1 u_2} Real Real _ _ (@RingHom.id.{0} Real _) (@RingHom.id.{0} Real _) _ _ (@Subtype.{u_1 + 1} (@Erdos390.Full.PaperBridgeFit.BridgeData.EffectiveAmbient.{u_1, u_2} Head Band _ _ _ _ B) fun (x : @Erdos390.Full.PaperBridgeFit.BridgeData.EffectiveAmbient.{u_1, u_2} Head Band _ _ _ _ B) => @Membership.mem.{u_1, u_1} (@Erdos390.Full.PaperBridgeFit.BridgeData.EffectiveAmbient.{u_1, u_2} Head Band _ _ _ _ B) (@Submodule.{0, u_1} Real (@Erdos390.Full.PaperBridgeFit.BridgeData.EffectiveAmbient.{u_1, u_2} Head Band _ _ _ _ B) _ _ _) _ (@Erdos390.Full.PaperBridgeFit.BridgeData.EffectiveParamSpace.{u_1, u_2} Head Band _ _ _ _ B) x) _ _ (@Erdos390.Full.PaperBridgeFit.BridgeData.ParamSpace.{u_1, u_2} Head Band _ _ _ _ B) _ _ _ _) (@Subtype.{u_1 + 1} (@Erdos390.Full.PaperBridgeFit.BridgeData.EffectiveAmbient.{u_1, u_2} Head Band _ _ _ _ B) fun (x : @Erdos390.Full.PaperBridgeFit.BridgeData.EffectiveAmbient.{u_1, u_2} Head Band _ _ _ _ B) => @Membership.mem.{u_1, u_1} (@Erdos390.Full.PaperBridgeFit.BridgeData.EffectiveAmbient.{u_1, u_2} Head Band _ _ _ _ B) (@Submodule.{0, u_1} Real (@Erdos390.Full.PaperBridgeFit.BridgeData.EffectiveAmbient.{u_1, u_2} Head Band _ _ _ _ B) _ _ _) _ (@Erdos390.Full.PaperBridgeFit.BridgeData.EffectiveParamSpace.{u_1, u_2} Head Band _ _ _ _ B) x) (fun (x : @Subtype.{u_1 + 1} (@Erdos390.Full.PaperBridgeFit.BridgeData.EffectiveAmbient.{u_1, u_2} Head Band _ _ _ _ B) fun (x : @Erdos390.Full.PaperBridgeFit.BridgeData.EffectiveAmbient.{u_1, u_2} Head Band _ _ _ _ B) => @Membership.mem.{u_1, u_1} (@Erdos390.Full.PaperBridgeFit.BridgeData.EffectiveAmbient.{u_1, u_2} Head Band _ _ _ _ B) (@Submodule.{0, u_1} Real (@Erdos390.Full.PaperBridgeFit.BridgeData.EffectiveAmbient.{u_1, u_2} Head Band _ _ _ _ B) _ _ _) _ (@Erdos390.Full.PaperBridgeFit.BridgeData.EffectiveParamSpace.{u_1, u_2} Head Band _ _ _ _ B) x) => @Erdos390.Full.PaperBridgeFit.BridgeData.ParamSpace.{u_1, u_2} Head Band _ _ _ _ B) _ (@Erdos390.Full.PaperBridgeFit.BridgeData.effectiveParamEquiv.{u_1, u_2} Head Band _ _ _ _ B) z)) (fun (m : @Erdos390.Full.PaperBridgeFit.StructuredSampleData.Sample.{u_1} Head _ (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{u_1, u_2} Head Band _ _ _ _ B)) => @WithLp.ofLp.{u_1} (@OfNat.ofNat.{0} ENNReal (nat_lit 2) _) ((i : Erdos390.Full.PaperBridgeFit.NuisanceCoord.{u_1} (@Erdos390.Full.PaperBridgeFit.BridgeData.HeadIndex.{u_1, u_2} Head Band _ _ _ _ B)) → (fun (x : Erdos390.Full.PaperBridgeFit.NuisanceCoord.{u_1} (@Erdos390.Full.PaperBridgeFit.BridgeData.HeadIndex.{u_1, u_2} Head Band _ _ _ _ B)) => Real) i) (@Erdos390.Full.PaperBridgeFit.BridgeData.nuisanceStatistic.{u_1, u_2} Head Band _ _ _ _ B _ m) c) fun (m : @Erdos390.Full.PaperBridgeFit.StructuredSampleData.Sample.{u_1} Head _ (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{u_1, u_2} Head Band _ _ _ _ B)) => Erdos390.Full.ArithmeticModel.valuation (@Subtype.val.{1} Nat (fun (p : Nat) => @Membership.mem.{0, 0} Nat (Finset.{0} Nat) _ (Erdos390.Full.ArithmeticModel.primeBand (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.n.{u_1} Head _ (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{u_1, u_2} Head Band _ _ _ _ B)) (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.W.{u_1} Head _ (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{u_1, u_2} Head Band _ _ _ _ B))) p) p) (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.value.{u_1} Head _ (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{u_1, u_2} Head Band _ _ _ _ B) m))) (@HMul.hMul.{0, 0, 0} Real Real Real _ (epsilon (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.n.{u_1} Head _ (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{u_1, u_2} Head Band _ _ _ _ B))) (@HDiv.hDiv.{0, 0, 0} Real Real Real _ (@OfNat.ofNat.{0} Real (nat_lit 1) _) (@Nat.cast.{0} Real _ (@Subtype.val.{1} Nat (fun (p : Nat) => @Membership.mem.{0, 0} Nat (Finset.{0} Nat) _ (Erdos390.Full.ArithmeticModel.primeBand (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.n.{u_1} Head _ (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{u_1, u_2} Head Band _ _ _ _ B)) (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.W.{u_1} Head _ (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{u_1, u_2} Head Band _ _ _ _ B))) p) p))))))
noncomputable def Erdos390.RemainingAnalyticGoal004_003 : Prop :=
∀ {Head : Type u_1} [inst : Fintype.{u_1} Head] [inst_1 : DecidableEq.{u_1 + 1} Head] [inst_2 : Nonempty.{u_1 + 1} Head]
(P : Head → Erdos390.Full.HeadPattern.Pattern) (I : Erdos390.Full.PaperGuardCensus.PhysicalIntervals)
(Cprom Cbank : Nat) (G : (n : Nat) → Erdos390.Full.PaperGuardCensus.Ledger n Cprom Cbank) (W : Nat),
@LT.lt.{0} Nat instLTNat (@OfNat.ofNat.{0} Nat (nat_lit 1) (instOfNatNat (nat_lit 1))) W →
(∀ (h : Head) (q : Nat),
@Membership.mem.{0, 0} Nat (Finset.{0} Nat)
(@SetLike.instMembership.{0, 0} (Finset.{0} Nat) Nat (@Finset.instSetLike.{0} Nat)) (P h).primes q →
@LE.le.{0} Nat instLENat q W) →
∀ (Acoef : Real),
@LE.le.{0} Real Real.instLE (@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real Real.instZero)) Acoef →
@Exists.{1} (Nat → Real) fun (epsilon : Nat → Real) =>
And
(∀ (n : Nat),
@LE.le.{0} Real Real.instLE (@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real Real.instZero))
(epsilon n))
(And
(@Filter.Tendsto.{0, 0} Nat Real
(fun (n : Nat) =>
@HMul.hMul.{0, 0, 0} Real Real Real (@instHMul.{0} Real Real.instMul) (epsilon n)
(Real.log (Erdos390.Full.Scale.L n)))
(@Filter.atTop.{0} Nat Nat.instPreorder)
(@nhds.{0} Real
(@UniformSpace.toTopologicalSpace.{0} Real
(@PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))
(@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real Real.instZero))))
(@Exists.{1} Nat fun (N₀ : Nat) =>
∀ {Band : Type u_2} [inst_3 : Fintype.{u_2} Band] [inst_4 : DecidableEq.{u_2 + 1} Band]
(B : @Erdos390.Full.PaperBridgeFit.BridgeData.{u_1, u_2} Head Band inst inst_1 inst_3 inst_4)
(xi :
@Erdos390.Full.PaperBridgeFit.BridgeData.ParamSpace.{u_1, u_2} Head Band inst inst_1 inst_3 inst_4
B),
@LE.le.{0} Nat instLENat N₀
(@Erdos390.Full.PaperBridgeFit.StructuredSampleData.n.{u_1} Head inst
(@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{u_1, u_2} Head Band inst inst_1 inst_3
inst_4 B)) →
@Eq.{1} Nat
(@Erdos390.Full.PaperBridgeFit.StructuredSampleData.W.{u_1} Head inst
(@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{u_1, u_2} Head Band inst inst_1 inst_3
inst_4 B))
W →
(∀
(p :
Erdos390.Full.ArithmeticBandGeometry.BandPrime
(@Erdos390.Full.PaperBridgeFit.StructuredSampleData.n.{u_1} Head inst
(@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{u_1, u_2} Head Band inst inst_1
inst_3 inst_4 B))
(@Erdos390.Full.PaperBridgeFit.StructuredSampleData.W.{u_1} Head inst
(@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{u_1, u_2} Head Band inst inst_1
inst_3 inst_4 B))),
@LE.le.{0} Real Real.instLE
(@abs.{0} Real Real.lattice Real.instAddGroup
(@Erdos390.Full.PaperBridgeFit.BridgeData.effectivePrimeCoefficient.{u_1, u_2} Head Band
inst inst_1 inst_3 inst_4 B xi p))
Acoef) →
∀
(hsep :
@LT.lt.{0} Nat instLTNat
(Erdos390.Full.ArithmeticModel.physicalBound
(I.upper Erdos390.Full.PaperBridgeFit.PhysicalSign.minus)
(@Erdos390.Full.PaperBridgeFit.StructuredSampleData.n.{u_1} Head inst
(@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{u_1, u_2} Head Band inst
inst_1 inst_3 inst_4 B)))
(Erdos390.Full.ArithmeticModel.physicalBound
(I.lower Erdos390.Full.PaperBridgeFit.PhysicalSign.plus)
(@Erdos390.Full.PaperBridgeFit.StructuredSampleData.n.{u_1} Head inst
(@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{u_1, u_2} Head Band inst
inst_1 inst_3 inst_4 B))))
(hremaining :
∀ (c : Erdos390.Full.PaperBridgeFit.Cell.{u_1} Head),
@Finset.Nonempty.{0} Nat
(@SDiff.sdiff.{0} (Finset.{0} Nat) (@Finset.instSDiff.{0} Nat instDecidableEqNat)
(@Erdos390.Full.PaperGuardCensus.rawCell.{u_1} Head P I
(@Erdos390.Full.PaperBridgeFit.StructuredSampleData.n.{u_1} Head inst
(@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{u_1, u_2} Head Band inst
inst_1 inst_3 inst_4 B))
c)
(@Erdos390.Full.PaperGuardCensus.Ledger.guards
(@Erdos390.Full.PaperBridgeFit.StructuredSampleData.n.{u_1} Head inst
(@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{u_1, u_2} Head Band inst
inst_1 inst_3 inst_4 B))
Cprom Cbank
(G
(@Erdos390.Full.PaperBridgeFit.StructuredSampleData.n.{u_1} Head inst
(@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{u_1, u_2} Head Band inst
inst_1 inst_3 inst_4 B)))))),
@Eq.{u_1 + 1} (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.{u_1} Head inst)
(@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{u_1, u_2} Head Band inst inst_1
inst_3 inst_4 B)
(@Erdos390.Full.PaperGuardCensus.canonicalSampleData.{u_1} Head inst P I
(@Erdos390.Full.PaperBridgeFit.StructuredSampleData.n.{u_1} Head inst
(@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{u_1, u_2} Head Band inst
inst_1 inst_3 inst_4 B))
(@Erdos390.Full.PaperBridgeFit.StructuredSampleData.W.{u_1} Head inst
(@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{u_1, u_2} Head Band inst
inst_1 inst_3 inst_4 B))
Cprom Cbank
(G
(@Erdos390.Full.PaperBridgeFit.StructuredSampleData.n.{u_1} Head inst
(@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{u_1, u_2} Head Band inst
inst_1 inst_3 inst_4 B)))
hsep hremaining) →
∀
(p :
Erdos390.Full.ArithmeticBandGeometry.BandPrime
(@Erdos390.Full.PaperBridgeFit.StructuredSampleData.n.{u_1} Head inst
(@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{u_1, u_2} Head Band inst
inst_1 inst_3 inst_4 B))
(@Erdos390.Full.PaperBridgeFit.StructuredSampleData.W.{u_1} Head inst
(@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{u_1, u_2} Head Band inst
inst_1 inst_3 inst_4 B)))
(c : Erdos390.Full.PaperBridgeFit.Cell.{u_1} Head) (k : Nat),
@LE.le.{0} Real Real.instLE
(@abs.{0} Real Real.lattice Real.instAddGroup
(@Erdos390.Full.FiniteProbability.covariance.{0}
(@Erdos390.Full.PaperBridgeFit.StructuredSampleData.SampleAt.{u_1} Head inst
(@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{u_1, u_2} Head Band inst
inst_1 inst_3 inst_4 B)
c)
(@Erdos390.Full.PaperBridgeFit.StructuredSampleData.sampleAtFintype.{u_1} Head
inst
(@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{u_1, u_2} Head Band inst
inst_1 inst_3 inst_4 B)
c)
(@Erdos390.Full.PaperBridgeFit.BridgeData.cellMediumLaw.{u_1, u_2} Head Band inst
inst_1 inst_3 inst_4 B inst_2 xi c)
(fun
(m :
@Erdos390.Full.PaperBridgeFit.StructuredSampleData.SampleAt.{u_1} Head inst
(@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{u_1, u_2} Head Band
inst inst_1 inst_3 inst_4 B)
c) =>
Erdos390.Full.ArithmeticModel.valuation
(@Subtype.val.{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))
(Erdos390.Full.ArithmeticModel.primeBand
(@Erdos390.Full.PaperBridgeFit.StructuredSampleData.n.{u_1} Head inst
(@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{u_1, u_2} Head
Band inst inst_1 inst_3 inst_4 B))
(@Erdos390.Full.PaperBridgeFit.StructuredSampleData.W.{u_1} Head inst
(@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{u_1, u_2} Head
Band inst inst_1 inst_3 inst_4 B)))
p)
p)
(@Subtype.val.{1} Nat
(fun (m : Nat) =>
@Membership.mem.{0, 0} Nat (Finset.{0} Nat)
(@SetLike.instMembership.{0, 0} (Finset.{0} Nat) Nat
(@Finset.instSetLike.{0} Nat))
(@Erdos390.Full.PaperBridgeFit.StructuredSampleData.cellFinset.{u_1}
Head inst
(@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{u_1, u_2} Head
Band inst inst_1 inst_3 inst_4 B)
c)
m)
m))
fun
(m :
@Erdos390.Full.PaperBridgeFit.StructuredSampleData.SampleAt.{u_1} Head inst
(@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{u_1, u_2} Head Band
inst inst_1 inst_3 inst_4 B)
c) =>
@ite.{1} Real
(@LE.le.{0} Nat instLENat
(@Subtype.val.{1} Nat
(fun (m : Nat) =>
@Membership.mem.{0, 0} Nat (Finset.{0} Nat)
(@SetLike.instMembership.{0, 0} (Finset.{0} Nat) Nat
(@Finset.instSetLike.{0} Nat))
(@Erdos390.Full.PaperBridgeFit.StructuredSampleData.cellFinset.{u_1}
Head inst
(@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{u_1, u_2} Head
Band inst inst_1 inst_3 inst_4 B)
c)
m)
m)
k)
((@Subtype.val.{1} Nat
(fun (m : Nat) =>
@Membership.mem.{0, 0} Nat (Finset.{0} Nat)
(@SetLike.instMembership.{0, 0} (Finset.{0} Nat) Nat
(@Finset.instSetLike.{0} Nat))
(@Erdos390.Full.PaperBridgeFit.StructuredSampleData.cellFinset.{u_1}
Head inst
(@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{u_1, u_2} Head
Band inst inst_1 inst_3 inst_4 B)
c)
m)
m).decLe
k)
(@OfNat.ofNat.{0} Real (nat_lit 1) (@One.toOfNat1.{0} Real Real.instOne))
(@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real Real.instZero))))
(@HDiv.hDiv.{0, 0, 0} Real Real Real
(@instHDiv.{0} Real (@DivInvMonoid.toDiv.{0} Real Real.instDivInvMonoid))
(epsilon
(@Erdos390.Full.PaperBridgeFit.StructuredSampleData.n.{u_1} Head inst
(@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{u_1, u_2} Head Band inst
inst_1 inst_3 inst_4 B)))
(@Nat.cast.{0} Real Real.instNatCast
(@Subtype.val.{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))
(Erdos390.Full.ArithmeticModel.primeBand
(@Erdos390.Full.PaperBridgeFit.StructuredSampleData.n.{u_1} Head inst
(@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{u_1, u_2} Head
Band inst inst_1 inst_3 inst_4 B))
(@Erdos390.Full.PaperBridgeFit.StructuredSampleData.W.{u_1} Head inst
(@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{u_1, u_2} Head
Band inst inst_1 inst_3 inst_4 B)))
p)
p)))))
noncomputable def Erdos390.RemainingAnalyticGoal004_004 : Prop :=
∀ {Head : Type u_1} [inst : Fintype.{u_1} Head] [inst_1 : DecidableEq.{u_1 + 1} Head] [inst_2 : Nonempty.{u_1 + 1} Head]
(P : Head → Erdos390.Full.HeadPattern.Pattern) (I : Erdos390.Full.PaperGuardCensus.PhysicalIntervals)
(Cprom Cbank : Nat) (G : (n : Nat) → Erdos390.Full.PaperGuardCensus.Ledger n Cprom Cbank) (W : Nat),
@LT.lt.{0} Nat instLTNat (@OfNat.ofNat.{0} Nat (nat_lit 1) (instOfNatNat (nat_lit 1))) W →
(∀ (h : Head) (p : Nat),
@Membership.mem.{0, 0} Nat (Finset.{0} Nat)
(@SetLike.instMembership.{0, 0} (Finset.{0} Nat) Nat (@Finset.instSetLike.{0} Nat)) (P h).primes p →
@LE.le.{0} Nat instLENat p W) →
∀ (Acoef : Real),
@LE.le.{0} Real Real.instLE (@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real Real.instZero)) Acoef →
@Exists.{1} Real fun (Aval : Real) =>
And
(@LE.le.{0} Real Real.instLE (@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real Real.instZero))
Aval)
(@Exists.{1} (Nat → Real) fun (epsilon : Nat → Real) =>
And
(∀ (n : Nat),
@LE.le.{0} Real Real.instLE
(@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real Real.instZero)) (epsilon n))
(And
(@Filter.Tendsto.{0, 0} Nat Real
(fun (n : Nat) =>
@HMul.hMul.{0, 0, 0} Real Real Real (@instHMul.{0} Real Real.instMul) (epsilon n)
(Real.log (Erdos390.Full.Scale.L n)))
(@Filter.atTop.{0} Nat Nat.instPreorder)
(@nhds.{0} Real
(@UniformSpace.toTopologicalSpace.{0} Real
(@PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))
(@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real Real.instZero))))
(@Exists.{1} Nat fun (N₀ : Nat) =>
∀ {Band : Type u_2} [inst_3 : Fintype.{u_2} Band] [inst_4 : DecidableEq.{u_2 + 1} Band]
(B : @Erdos390.Full.PaperBridgeFit.BridgeData.{u_1, u_2} Head Band inst inst_1 inst_3 inst_4)
(xi :
@Erdos390.Full.PaperBridgeFit.BridgeData.ParamSpace.{u_1, u_2} Head Band inst inst_1 inst_3
inst_4 B),
@LE.le.{0} Nat instLENat N₀
(@Erdos390.Full.PaperBridgeFit.StructuredSampleData.n.{u_1} Head inst
(@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{u_1, u_2} Head Band inst inst_1
inst_3 inst_4 B)) →
@Eq.{1} Nat
(@Erdos390.Full.PaperBridgeFit.StructuredSampleData.W.{u_1} Head inst
(@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{u_1, u_2} Head Band inst inst_1
inst_3 inst_4 B))
W →
(∀
(p :
Erdos390.Full.ArithmeticBandGeometry.BandPrime
(@Erdos390.Full.PaperBridgeFit.StructuredSampleData.n.{u_1} Head inst
(@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{u_1, u_2} Head Band inst
inst_1 inst_3 inst_4 B))
(@Erdos390.Full.PaperBridgeFit.StructuredSampleData.W.{u_1} Head inst
(@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{u_1, u_2} Head Band inst
inst_1 inst_3 inst_4 B))),
@LE.le.{0} Real Real.instLE
(@abs.{0} Real Real.lattice Real.instAddGroup
(@Erdos390.Full.PaperBridgeFit.BridgeData.effectivePrimeCoefficient.{u_1, u_2} Head
Band inst inst_1 inst_3 inst_4 B xi p))
Acoef) →
∀
(hsep :
@LT.lt.{0} Nat instLTNat
(Erdos390.Full.ArithmeticModel.physicalBound
(I.upper Erdos390.Full.PaperBridgeFit.PhysicalSign.minus)
(@Erdos390.Full.PaperBridgeFit.StructuredSampleData.n.{u_1} Head inst
(@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{u_1, u_2} Head Band inst
inst_1 inst_3 inst_4 B)))
(Erdos390.Full.ArithmeticModel.physicalBound
(I.lower Erdos390.Full.PaperBridgeFit.PhysicalSign.plus)
(@Erdos390.Full.PaperBridgeFit.StructuredSampleData.n.{u_1} Head inst
(@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{u_1, u_2} Head Band inst
inst_1 inst_3 inst_4 B))))
(hremaining :
∀ (c : Erdos390.Full.PaperBridgeFit.Cell.{u_1} Head),
@Finset.Nonempty.{0} Nat
(@SDiff.sdiff.{0} (Finset.{0} Nat) (@Finset.instSDiff.{0} Nat instDecidableEqNat)
(@Erdos390.Full.PaperGuardCensus.rawCell.{u_1} Head P I
(@Erdos390.Full.PaperBridgeFit.StructuredSampleData.n.{u_1} Head inst
(@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{u_1, u_2} Head Band
inst inst_1 inst_3 inst_4 B))
c)
(@Erdos390.Full.PaperGuardCensus.Ledger.guards
(@Erdos390.Full.PaperBridgeFit.StructuredSampleData.n.{u_1} Head inst
(@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{u_1, u_2} Head Band
inst inst_1 inst_3 inst_4 B))
Cprom Cbank
(G
(@Erdos390.Full.PaperBridgeFit.StructuredSampleData.n.{u_1} Head inst
(@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{u_1, u_2} Head Band
inst inst_1 inst_3 inst_4 B)))))),
@Eq.{u_1 + 1} (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.{u_1} Head inst)
(@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{u_1, u_2} Head Band inst
inst_1 inst_3 inst_4 B)
(@Erdos390.Full.PaperGuardCensus.canonicalSampleData.{u_1} Head inst P I
(@Erdos390.Full.PaperBridgeFit.StructuredSampleData.n.{u_1} Head inst
(@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{u_1, u_2} Head Band inst
inst_1 inst_3 inst_4 B))
(@Erdos390.Full.PaperBridgeFit.StructuredSampleData.W.{u_1} Head inst
(@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{u_1, u_2} Head Band inst
inst_1 inst_3 inst_4 B))
Cprom Cbank
(G
(@Erdos390.Full.PaperBridgeFit.StructuredSampleData.n.{u_1} Head inst
(@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{u_1, u_2} Head Band inst
inst_1 inst_3 inst_4 B)))
hsep hremaining) →
∀
(p :
Erdos390.Full.ArithmeticBandGeometry.BandPrime
(@Erdos390.Full.PaperBridgeFit.StructuredSampleData.n.{u_1} Head inst
(@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{u_1, u_2} Head Band inst
inst_1 inst_3 inst_4 B))
(@Erdos390.Full.PaperBridgeFit.StructuredSampleData.W.{u_1} Head inst
(@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{u_1, u_2} Head Band inst
inst_1 inst_3 inst_4 B))),
And
(∀ (c : Erdos390.Full.PaperBridgeFit.Cell.{u_1} Head),
@LE.le.{0} Real Real.instLE
(@Erdos390.Full.FiniteProbability.expect.{0}
(@Erdos390.Full.PaperBridgeFit.StructuredSampleData.SampleAt.{u_1} Head inst
(@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{u_1, u_2} Head Band
inst inst_1 inst_3 inst_4 B)
c)
(@Erdos390.Full.PaperBridgeFit.StructuredSampleData.sampleAtFintype.{u_1}
Head inst
(@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{u_1, u_2} Head Band
inst inst_1 inst_3 inst_4 B)
c)
(@Erdos390.Full.PaperBridgeFit.BridgeData.cellMediumLaw.{u_1, u_2} Head Band
inst inst_1 inst_3 inst_4 B inst_2 xi c)
fun
(m :
@Erdos390.Full.PaperBridgeFit.StructuredSampleData.SampleAt.{u_1} Head
inst
(@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{u_1, u_2} Head
Band inst inst_1 inst_3 inst_4 B)
c) =>
Erdos390.Full.ArithmeticModel.valuation
(@Subtype.val.{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))
(Erdos390.Full.ArithmeticModel.primeBand
(@Erdos390.Full.PaperBridgeFit.StructuredSampleData.n.{u_1} Head
inst
(@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{u_1, u_2}
Head Band inst inst_1 inst_3 inst_4 B))
(@Erdos390.Full.PaperBridgeFit.StructuredSampleData.W.{u_1} Head
inst
(@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{u_1, u_2}
Head Band inst inst_1 inst_3 inst_4 B)))
p)
p)
(@Subtype.val.{1} Nat
(fun (m : Nat) =>
@Membership.mem.{0, 0} Nat (Finset.{0} Nat)
(@SetLike.instMembership.{0, 0} (Finset.{0} Nat) Nat
(@Finset.instSetLike.{0} Nat))
(@Erdos390.Full.PaperBridgeFit.StructuredSampleData.cellFinset.{u_1}
Head inst
(@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{u_1, u_2}
Head Band inst inst_1 inst_3 inst_4 B)
c)
m)
m))
(@HDiv.hDiv.{0, 0, 0} Real Real Real
(@instHDiv.{0} Real (@DivInvMonoid.toDiv.{0} Real Real.instDivInvMonoid))
Aval
(@Nat.cast.{0} Real Real.instNatCast
(@Subtype.val.{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))
(Erdos390.Full.ArithmeticModel.primeBand
(@Erdos390.Full.PaperBridgeFit.StructuredSampleData.n.{u_1} Head
inst
(@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{u_1, u_2}
Head Band inst inst_1 inst_3 inst_4 B))
(@Erdos390.Full.PaperBridgeFit.StructuredSampleData.W.{u_1} Head
inst
(@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{u_1, u_2}
Head Band inst inst_1 inst_3 inst_4 B)))
p)
p))))
(∀ (c c' : Erdos390.Full.PaperBridgeFit.Cell.{u_1} Head),
@LE.le.{0} Real Real.instLE
(@abs.{0} Real Real.lattice Real.instAddGroup
(@HSub.hSub.{0, 0, 0} Real Real Real (@instHSub.{0} Real Real.instSub)
(@Erdos390.Full.FiniteProbability.expect.{0}
(@Erdos390.Full.PaperBridgeFit.StructuredSampleData.SampleAt.{u_1} Head
inst
(@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{u_1, u_2} Head
Band inst inst_1 inst_3 inst_4 B)
c)
(@Erdos390.Full.PaperBridgeFit.StructuredSampleData.sampleAtFintype.{u_1}
Head inst
(@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{u_1, u_2} Head
Band inst inst_1 inst_3 inst_4 B)
c)
(@Erdos390.Full.PaperBridgeFit.BridgeData.cellMediumLaw.{u_1, u_2} Head
Band inst inst_1 inst_3 inst_4 B inst_2 xi c)
fun
(m :
@Erdos390.Full.PaperBridgeFit.StructuredSampleData.SampleAt.{u_1}
Head inst
(@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{u_1, u_2}
Head Band inst inst_1 inst_3 inst_4 B)
c) =>
Erdos390.Full.ArithmeticModel.valuation
(@Subtype.val.{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))
(Erdos390.Full.ArithmeticModel.primeBand
(@Erdos390.Full.PaperBridgeFit.StructuredSampleData.n.{u_1}
Head inst
(@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{u_1,
u_2}
Head Band inst inst_1 inst_3 inst_4 B))
(@Erdos390.Full.PaperBridgeFit.StructuredSampleData.W.{u_1}
Head inst
(@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{u_1,
u_2}
Head Band inst inst_1 inst_3 inst_4 B)))
p)
p)
(@Subtype.val.{1} Nat
(fun (m : Nat) =>
@Membership.mem.{0, 0} Nat (Finset.{0} Nat)
(@SetLike.instMembership.{0, 0} (Finset.{0} Nat) Nat
(@Finset.instSetLike.{0} Nat))
(@Erdos390.Full.PaperBridgeFit.StructuredSampleData.cellFinset.{u_1}
Head inst
(@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{u_1,
u_2}
Head Band inst inst_1 inst_3 inst_4 B)
c)
m)
m))
(@Erdos390.Full.FiniteProbability.expect.{0}
(@Erdos390.Full.PaperBridgeFit.StructuredSampleData.SampleAt.{u_1} Head
inst
(@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{u_1, u_2} Head
Band inst inst_1 inst_3 inst_4 B)
c')
(@Erdos390.Full.PaperBridgeFit.StructuredSampleData.sampleAtFintype.{u_1}
Head inst
(@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{u_1, u_2} Head
Band inst inst_1 inst_3 inst_4 B)
c')
(@Erdos390.Full.PaperBridgeFit.BridgeData.cellMediumLaw.{u_1, u_2} Head
Band inst inst_1 inst_3 inst_4 B inst_2 xi c')
fun
(m :
@Erdos390.Full.PaperBridgeFit.StructuredSampleData.SampleAt.{u_1}
Head inst
(@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{u_1, u_2}
Head Band inst inst_1 inst_3 inst_4 B)
c') =>
Erdos390.Full.ArithmeticModel.valuation
(@Subtype.val.{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))
(Erdos390.Full.ArithmeticModel.primeBand
(@Erdos390.Full.PaperBridgeFit.StructuredSampleData.n.{u_1}
Head inst
(@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{u_1,
u_2}
Head Band inst inst_1 inst_3 inst_4 B))
(@Erdos390.Full.PaperBridgeFit.StructuredSampleData.W.{u_1}
Head inst
(@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{u_1,
u_2}
Head Band inst inst_1 inst_3 inst_4 B)))
p)
p)
(@Subtype.val.{1} Nat
(fun (m : Nat) =>
@Membership.mem.{0, 0} Nat (Finset.{0} Nat)
(@SetLike.instMembership.{0, 0} (Finset.{0} Nat) Nat
(@Finset.instSetLike.{0} Nat))
(@Erdos390.Full.PaperBridgeFit.StructuredSampleData.cellFinset.{u_1}
Head inst
(@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{u_1,
u_2}
Head Band inst inst_1 inst_3 inst_4 B)
c')
m)
m))))
(@HDiv.hDiv.{0, 0, 0} Real Real Real
(@instHDiv.{0} Real (@DivInvMonoid.toDiv.{0} Real Real.instDivInvMonoid))
(epsilon
(@Erdos390.Full.PaperBridgeFit.StructuredSampleData.n.{u_1} Head inst
(@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{u_1, u_2} Head
Band inst inst_1 inst_3 inst_4 B)))
(@Nat.cast.{0} Real Real.instNatCast
(@Subtype.val.{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))
(Erdos390.Full.ArithmeticModel.primeBand
(@Erdos390.Full.PaperBridgeFit.StructuredSampleData.n.{u_1} Head
inst
(@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{u_1, u_2}
Head Band inst inst_1 inst_3 inst_4 B))
(@Erdos390.Full.PaperBridgeFit.StructuredSampleData.W.{u_1} Head
inst
(@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{u_1, u_2}
Head Band inst inst_1 inst_3 inst_4 B)))
p)
p)))))))
noncomputable def Erdos390.RemainingAnalyticGoal004_005 : Prop :=
∀ (P : Erdos390.Full.HeadPattern.Pattern) {A C : Real},
LT.lt.{0} (0 : Real) A →
LT.lt.{0} A C →
∃ K,
And (LT.lt.{0} (0 : Real) K)
(∃ N₀,
∀ {n k d : Nat},
LE.le.{0} N₀ n →
LT.lt.{0} (Erdos390.Full.ArithmeticModel.physicalBound A n) k →
LE.le.{0} k (Erdos390.Full.ArithmeticModel.physicalBound C n) →
LT.lt.{0} (0 : Nat) d →
LE.le.{0} d (HPow.hPow.{0, 0, 0} (Erdos390.Full.ArithmeticModel.yNat n) (4 : Nat)) →
Membership.mem.{0, 0}
(Nat.smoothNumbers (HAdd.hAdd.{0, 0, 0} (Erdos390.Full.ArithmeticModel.yNat n) (1 : Nat)))
d →
Nat.Coprime d (Erdos390.Full.HeadPattern.Pattern.modulus P) →
have S :=
Erdos390.Full.StructuredCells.structuredCell P
(Erdos390.Full.ArithmeticModel.physicalBound A n)
(Erdos390.Full.ArithmeticModel.physicalBound C n)
(Erdos390.Full.ArithmeticModel.yNat n);
LE.le.{0}
(abs.{0}
(HSub.hSub.{0, 0, 0}
(Erdos390.Full.DivisibilityMomentBounds.uniformAverage S fun m =>
HMul.hMul.{0, 0, 0} (Erdos390.Full.ArithmeticModel.divInd d m)
(ite.{1} (LE.le.{0} m k) (1 : Real) (0 : Real)))
(HMul.hMul.{0, 0, 0}
(Erdos390.Full.DivisibilityMomentBounds.uniformAverage S
(Erdos390.Full.ArithmeticModel.divInd d))
(Erdos390.Full.DivisibilityMomentBounds.uniformAverage S fun m =>
ite.{1} (LE.le.{0} m k) (1 : Real) (0 : Real)))))
(HDiv.hDiv.{0, 0, 0} K (HMul.hMul.{0, 0, 0} (↑d) (Erdos390.Full.Scale.L n))))
noncomputable def Erdos390.RemainingAnalyticGoal004_006 : Prop :=
∀ (H : Erdos390.Full.HeadPattern.Pattern) {A C : Real},
LT.lt.{0} (0 : Real) A →
LT.lt.{0} A C →
LT.lt.{0} (0 : Real) C →
∀ (W : Nat),
LT.lt.{0} (1 : Nat) W →
(∀ (q : Nat), Membership.mem.{0, 0} (Erdos390.Full.HeadPattern.Pattern.primes H) q → LE.le.{0} q W) →
∃ (K : Real),
And (LT.lt.{0} (0 : Real) K)
(∃ (G : Real),
And (LT.lt.{0} (0 : Real) G)
(∃ (N₀ : Nat),
∀ {n k p : Nat},
LE.le.{0} N₀ n →
LT.lt.{0} (Erdos390.Full.ArithmeticModel.physicalBound A n) k →
LE.le.{0} k (Erdos390.Full.ArithmeticModel.physicalBound C n) →
Membership.mem.{0, 0} (Erdos390.Full.ArithmeticModel.primeBand n W) p →
∀
(hS :
Finset.Nonempty.{0}
(Erdos390.Full.StructuredCells.structuredCell H
(Erdos390.Full.ArithmeticModel.physicalBound A n)
(Erdos390.Full.ArithmeticModel.physicalBound C n)
(Erdos390.Full.ArithmeticModel.yNat n))),
LE.le.{0}
(abs.{0}
(Erdos390.Full.FiniteProbability.covariance.{0}
(Erdos390.Full.FiniteProbability.uniformOnFinset.{0}
(Erdos390.Full.StructuredCells.structuredCell H
(Erdos390.Full.ArithmeticModel.physicalBound A n)
(Erdos390.Full.ArithmeticModel.physicalBound C n)
(Erdos390.Full.ArithmeticModel.yNat n))
hS)
(fun
(m :
↥(Erdos390.Full.StructuredCells.structuredCell H
(Erdos390.Full.ArithmeticModel.physicalBound A n)
(Erdos390.Full.ArithmeticModel.physicalBound C n)
(Erdos390.Full.ArithmeticModel.yNat n))) =>
Erdos390.Full.ArithmeticModel.valuation p (↑m : Nat))
fun
(m :
↥(Erdos390.Full.StructuredCells.structuredCell H
(Erdos390.Full.ArithmeticModel.physicalBound A n)
(Erdos390.Full.ArithmeticModel.physicalBound C n)
(Erdos390.Full.ArithmeticModel.yNat n))) =>
ite.{1} (LE.le.{0} (↑m : Nat) k) (1 : Real) (0 : Real)))
(HAdd.hAdd.{0, 0, 0}
(HMul.hMul.{0, 0, 0}
(HDiv.hDiv.{0, 0, 0} (HMul.hMul.{0, 0, 0} (2 : Real) K)
(Erdos390.Full.Scale.L n))
(HDiv.hDiv.{0, 0, 0} (1 : Real) (↑p : Real)))
(HDiv.hDiv.{0, 0, 0}
(HMul.hMul.{0, 0, 0} (HMul.hMul.{0, 0, 0} (4 : Real) G)
(HPow.hPow.{0, 0, 0}
(HMul.hMul.{0, 0, 0}
(Erdos390.Full.PaperPrimePowerTailLedger.cutoffScale W)
(Erdos390.Full.Scale.L n))
(2 : Nat)))
(HMul.hMul.{0, 0, 0} (HPow.hPow.{0, 0, 0} (↑p : Real) (2 : Nat))
(HPow.hPow.{0, 0, 0} (↑(Erdos390.Full.ArithmeticModel.yNat n) : Real)
(2 : Nat)))))))
noncomputable def Erdos390.RemainingAnalyticGoal004_007 : Prop :=
∀ (H : Erdos390.Full.HeadPattern.Pattern) {A C : Real},
@LT.lt.{0} Real Real.instLT (@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real Real.instZero)) A →
@LT.lt.{0} Real Real.instLT A C →
@LT.lt.{0} Real Real.instLT (@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real Real.instZero)) C →
∀ (B : Real) (W : Nat),
@LE.le.{0} Real Real.instLE (@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real Real.instZero)) B →
@LT.lt.{0} Nat instLTNat (@OfNat.ofNat.{0} Nat (nat_lit 1) (instOfNatNat (nat_lit 1))) W →
(∀ (q : Nat),
@Membership.mem.{0, 0} Nat (Finset.{0} Nat)
(@SetLike.instMembership.{0, 0} (Finset.{0} Nat) Nat (@Finset.instSetLike.{0} Nat)) H.primes q →
@LE.le.{0} Nat instLENat q W) →
@Exists.{1} (Nat → Real) fun (epsilon : Nat → Real) =>
And
(@Filter.Tendsto.{0, 0} Nat Real
(fun (n : Nat) =>
@HMul.hMul.{0, 0, 0} Real Real Real (@instHMul.{0} Real Real.instMul) (epsilon n)
(Real.log (Erdos390.Full.Scale.L n)))
(@Filter.atTop.{0} Nat Nat.instPreorder)
(@nhds.{0} Real
(@UniformSpace.toTopologicalSpace.{0} Real
(@PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))
(@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real Real.instZero))))
(@Exists.{1} Nat fun (N₀ : Nat) =>
∀ {n k p : Nat} (eta : Nat → Real),
@LE.le.{0} Nat instLENat N₀ n →
@LT.lt.{0} Nat instLTNat (Erdos390.Full.ArithmeticModel.physicalBound A n) k →
@LE.le.{0} Nat instLENat k (Erdos390.Full.ArithmeticModel.physicalBound C n) →
@Membership.mem.{0, 0} Nat (Finset.{0} Nat)
(@SetLike.instMembership.{0, 0} (Finset.{0} Nat) Nat (@Finset.instSetLike.{0} Nat))
(Erdos390.Full.ArithmeticModel.primeBand n W) p →
(∀ (q : Nat),
@Membership.mem.{0, 0} Nat (Finset.{0} Nat)
(@SetLike.instMembership.{0, 0} (Finset.{0} Nat) Nat
(@Finset.instSetLike.{0} Nat))
(Erdos390.Full.ArithmeticModel.primeBand n W) q →
@LE.le.{0} Real Real.instLE (@abs.{0} Real Real.lattice Real.instAddGroup (eta q))
B) →
let S :=
Erdos390.Full.StructuredCells.structuredCell H
(Erdos390.Full.ArithmeticModel.physicalBound A n)
(Erdos390.Full.ArithmeticModel.physicalBound C n)
(Erdos390.Full.ArithmeticModel.yNat n);
∀ (hS : @Finset.Nonempty.{0} Nat S),
@LE.le.{0} Real Real.instLE
(@abs.{0} Real Real.lattice Real.instAddGroup
(@Erdos390.Full.FiniteProbability.covariance.{0}
(@Subtype.{1} Nat fun (x : Nat) =>
@Membership.mem.{0, 0} Nat (Finset.{0} Nat)
(@SetLike.instMembership.{0, 0} (Finset.{0} Nat) Nat
(@Finset.instSetLike.{0} Nat))
S x)
(@Finset.Subtype.fintype.{0} Nat S)
(@Erdos390.Full.FiniteProbability.exponentialTilt.{0}
(@Subtype.{1} Nat fun (x : Nat) =>
@Membership.mem.{0, 0} Nat (Finset.{0} Nat)
(@SetLike.instMembership.{0, 0} (Finset.{0} Nat) Nat
(@Finset.instSetLike.{0} Nat))
S x)
(@Finset.Subtype.fintype.{0} Nat S)
(@Erdos390.Full.FiniteProbability.uniformOnFinset.{0} Nat S hS)
fun
(m :
@Subtype.{1} Nat fun (x : Nat) =>
@Membership.mem.{0, 0} Nat (Finset.{0} Nat)
(@SetLike.instMembership.{0, 0} (Finset.{0} Nat) Nat
(@Finset.instSetLike.{0} Nat))
S x) =>
Erdos390.Full.ValuationScoreDomination.valuationScore
(Erdos390.Full.ArithmeticModel.primeBand n W) eta
(Erdos390.Full.Scale.L n)
(@Subtype.val.{1} Nat
(fun (x : Nat) =>
@Membership.mem.{0, 0} Nat (Finset.{0} Nat)
(@SetLike.instMembership.{0, 0} (Finset.{0} Nat) Nat
(@Finset.instSetLike.{0} Nat))
S x)
m))
(fun
(m :
@Subtype.{1} Nat fun (x : Nat) =>
@Membership.mem.{0, 0} Nat (Finset.{0} Nat)
(@SetLike.instMembership.{0, 0} (Finset.{0} Nat) Nat
(@Finset.instSetLike.{0} Nat))
S x) =>
Erdos390.Full.ArithmeticModel.valuation p
(@Subtype.val.{1} Nat
(fun (x : Nat) =>
@Membership.mem.{0, 0} Nat (Finset.{0} Nat)
(@SetLike.instMembership.{0, 0} (Finset.{0} Nat) Nat
(@Finset.instSetLike.{0} Nat))
S x)
m))
fun
(m :
@Subtype.{1} Nat fun (x : Nat) =>
@Membership.mem.{0, 0} Nat (Finset.{0} Nat)
(@SetLike.instMembership.{0, 0} (Finset.{0} Nat) Nat
(@Finset.instSetLike.{0} Nat))
S x) =>
@ite.{1} Real
(@LE.le.{0} Nat instLENat
(@Subtype.val.{1} Nat
(fun (x : Nat) =>
@Membership.mem.{0, 0} Nat (Finset.{0} Nat)
(@SetLike.instMembership.{0, 0} (Finset.{0} Nat) Nat
(@Finset.instSetLike.{0} Nat))
S x)
m)
k)
((@Subtype.val.{1} Nat
(fun (x : Nat) =>
@Membership.mem.{0, 0} Nat (Finset.{0} Nat)
(@SetLike.instMembership.{0, 0} (Finset.{0} Nat) Nat
(@Finset.instSetLike.{0} Nat))
S x)
m).decLe
k)
(@OfNat.ofNat.{0} Real (nat_lit 1) (@One.toOfNat1.{0} Real Real.instOne))
(@OfNat.ofNat.{0} Real (nat_lit 0)
(@Zero.toOfNat0.{0} Real Real.instZero))))
(@HDiv.hDiv.{0, 0, 0} Real Real Real
(@instHDiv.{0} Real (@DivInvMonoid.toDiv.{0} Real Real.instDivInvMonoid))
(epsilon n) (@Nat.cast.{0} Real Real.instNatCast p)))
noncomputable def Erdos390.RemainingAnalyticGoal004_008 : Prop :=
∀ {Cell : Type u_1} [Fintype.{u_1} Cell] (H : Cell → Erdos390.Full.HeadPattern.Pattern) (A C : Cell → Real)
(Cmax : Real),
(∀ (c : Cell), LT.lt.{0} (0 : Real) (A c)) →
(∀ (c : Cell), LT.lt.{0} (A c) (C c)) →
(∀ (c : Cell), LT.lt.{0} (0 : Real) (C c)) →
(∀ (c : Cell), LE.le.{0} (C c) Cmax) →
∀ (W : Nat),
LT.lt.{0} (1 : Nat) W →
(∀ (c : Cell) (p : Nat),
Membership.mem.{0, 0} (Erdos390.Full.HeadPattern.Pattern.primes (H c)) p → LE.le.{0} p W) →
∀ (B : Real),
LE.le.{0} (0 : Real) B →
∃ (epsilon : Nat → Real),
And (∀ (n : Nat), LE.le.{0} (0 : Real) (epsilon n))
(And
(Filter.Tendsto.{0, 0}
(fun (n : Nat) => HMul.hMul.{0, 0, 0} (epsilon n) (Real.log (Erdos390.Full.Scale.L n)))
Filter.atTop.{0} (nhds.{0} (0 : Real)))
(∃ (N₀ : Nat),
∀ {n p : Nat} (eta : Nat → Real),
LE.le.{0} N₀ n →
Membership.mem.{0, 0} (Erdos390.Full.ArithmeticModel.primeBand n W) p →
(∀ (z : Nat),
Membership.mem.{0, 0} (Erdos390.Full.ArithmeticModel.primeBand n W) z →
LE.le.{0} (abs.{0} (eta z)) B) →
have S := fun (c : Cell) =>
Erdos390.Full.StructuredCells.structuredCell (H c)
(Erdos390.Full.ArithmeticModel.physicalBound (A c) n)
(Erdos390.Full.ArithmeticModel.physicalBound (C c) n)
(Erdos390.Full.ArithmeticModel.yNat n);
∀ (hS : ∀ (c : Cell), Finset.Nonempty.{0} (S c)) (c c' : Cell),
LE.le.{0}
(abs.{0}
(HSub.hSub.{0, 0, 0}
(Erdos390.Full.FiniteProbability.expect.{0}
(Erdos390.Full.FiniteProbability.exponentialTilt.{0}
(Erdos390.Full.FiniteProbability.uniformOnFinset.{0} (S c) (hS c))
fun (m : ↥(S c)) =>
Erdos390.Full.ValuationScoreDomination.valuationScore
(Erdos390.Full.ArithmeticModel.primeBand n W) eta
(Erdos390.Full.Scale.L n) ↑m)
fun (m : ↥(S c)) => Erdos390.Full.ArithmeticModel.valuation p ↑m)
(Erdos390.Full.FiniteProbability.expect.{0}
(Erdos390.Full.FiniteProbability.exponentialTilt.{0}
(Erdos390.Full.FiniteProbability.uniformOnFinset.{0} (S c') (hS c'))
fun (m : ↥(S c')) =>
Erdos390.Full.ValuationScoreDomination.valuationScore
(Erdos390.Full.ArithmeticModel.primeBand n W) eta
(Erdos390.Full.Scale.L n) ↑m)
fun (m : ↥(S c')) => Erdos390.Full.ArithmeticModel.valuation p ↑m)))
(HDiv.hDiv.{0, 0, 0} (epsilon n) ↑p)))
noncomputable def Erdos390.RemainingAnalyticGoal004_009 : Prop :=
∀ (depth W K0 poolMinimum : Nat) {c deltaStar beta : Real},
LT.lt.{0} (0 : Real) c →
LT.lt.{0} (0 : Real) deltaStar →
LT.lt.{0} deltaStar (1 : Real) →
∀ (fixed bankBase candidates : Nat → Finset.{0} Nat) (initialSelector : Nat → Nat → Real),
(∀ᶠ (n : Nat) in Filter.atTop.{0},
Erdos390.WholePaper.BankPaperRealization.BankPaperCanonicalTopFrozenBalancedInitialRealization depth W K0
n c deltaStar beta (fixed n) (bankBase n) (candidates n) (initialSelector n)) →
Erdos390.WholePaper.BankPaperCanonicalActualSelectorMassEstimate c fixed bankBase candidates initialSelector
noncomputable def Erdos390.RemainingAnalyticGoal004_010 : Prop :=
∀ (W K0 : Nat) {c beta : Real},
LT.lt.{0} (0 : Real) c →
Asymptotics.IsBigO.{0, 0, 0} Filter.atTop.{0}
(fun n =>
Erdos390.WholePaper.BankPaperRealization.roughCanonicalRawRowDiscrepancy n
(Erdos390.WholePaper.upperTailLength c n) (HAdd.hAdd.{0, 0, 0} K0 (1 : Nat)) (1 : Nat)
(Erdos390.WholePaper.BankPaperRealization.bankPaperCanonicalBalancedRawWeight W K0 c beta n))
fun n => HDiv.hDiv.{0, 0, 0} (Erdos390.WholePaper.secondOrderScale n) (Erdos390.Full.Scale.L n)
noncomputable def Erdos390.RemainingAnalyticGoal004_011 : Prop :=
∀ {delta eta : Real} (M : Erdos390.Full.RegularRelativeMesh.Mesh delta eta) {P : Finset.{0} Nat} (B : @Erdos390.Full.PaperBridgeFit.BridgeData.{0, 0} (Erdos390.Full.PaperHeadSimplex.Tag P) (@Erdos390.WholePaper.BankPaperCanonicalExponentBand delta eta M) _ _ _ _) {c : Real} {depth K0 : Nat} (R : Erdos390.WholePaper.BankPaperRealization (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.n.{0} (Erdos390.Full.PaperHeadSimplex.Tag P) _ (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, 0} (Erdos390.Full.PaperHeadSimplex.Tag P) (@Erdos390.WholePaper.BankPaperCanonicalExponentBand delta eta M) _ _ _ _ B)) (Erdos390.WholePaper.upperEndpoint (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.n.{0} (Erdos390.Full.PaperHeadSimplex.Tag P) _ (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, 0} (Erdos390.Full.PaperHeadSimplex.Tag P) (@Erdos390.WholePaper.BankPaperCanonicalExponentBand delta eta M) _ _ _ _ B)) (Erdos390.WholePaper.upperTailLength c (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.n.{0} (Erdos390.Full.PaperHeadSimplex.Tag P) _ (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, 0} (Erdos390.Full.PaperHeadSimplex.Tag P) (@Erdos390.WholePaper.BankPaperCanonicalExponentBand delta eta M) _ _ _ _ B))))) (certificate : Erdos390.WholePaper.GuardedCentralAnchorCertificate c depth (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.n.{0} (Erdos390.Full.PaperHeadSimplex.Tag P) _ (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, 0} (Erdos390.Full.PaperHeadSimplex.Tag P) (@Erdos390.WholePaper.BankPaperCanonicalExponentBand delta eta M) _ _ _ _ B)) (@Erdos390.WholePaper.BankPaperRealization.anchorGuardLeftCore (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.n.{0} (Erdos390.Full.PaperHeadSimplex.Tag P) _ (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, 0} (Erdos390.Full.PaperHeadSimplex.Tag P) (@Erdos390.WholePaper.BankPaperCanonicalExponentBand delta eta M) _ _ _ _ B)) (Erdos390.WholePaper.upperEndpoint (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.n.{0} (Erdos390.Full.PaperHeadSimplex.Tag P) _ (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, 0} (Erdos390.Full.PaperHeadSimplex.Tag P) (@Erdos390.WholePaper.BankPaperCanonicalExponentBand delta eta M) _ _ _ _ B)) (Erdos390.WholePaper.upperTailLength c (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.n.{0} (Erdos390.Full.PaperHeadSimplex.Tag P) _ (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, 0} (Erdos390.Full.PaperHeadSimplex.Tag P) (@Erdos390.WholePaper.BankPaperCanonicalExponentBand delta eta M) _ _ _ _ B)))) R) (@Erdos390.WholePaper.BankPaperRealization.anchorGuardRightCore (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.n.{0} (Erdos390.Full.PaperHeadSimplex.Tag P) _ (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, 0} (Erdos390.Full.PaperHeadSimplex.Tag P) (@Erdos390.WholePaper.BankPaperCanonicalExponentBand delta eta M) _ _ _ _ B)) (Erdos390.WholePaper.upperEndpoint (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.n.{0} (Erdos390.Full.PaperHeadSimplex.Tag P) _ (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, 0} (Erdos390.Full.PaperHeadSimplex.Tag P) (@Erdos390.WholePaper.BankPaperCanonicalExponentBand delta eta M) _ _ _ _ B)) (Erdos390.WholePaper.upperTailLength c (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.n.{0} (Erdos390.Full.PaperHeadSimplex.Tag P) _ (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, 0} (Erdos390.Full.PaperHeadSimplex.Tag P) (@Erdos390.WholePaper.BankPaperCanonicalExponentBand delta eta M) _ _ _ _ B)))) R) (@Erdos390.WholePaper.BankPaperRealization.centralChangedMarkers (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.n.{0} (Erdos390.Full.PaperHeadSimplex.Tag P) _ (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, 0} (Erdos390.Full.PaperHeadSimplex.Tag P) (@Erdos390.WholePaper.BankPaperCanonicalExponentBand delta eta M) _ _ _ _ B)) (Erdos390.WholePaper.upperEndpoint (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.n.{0} (Erdos390.Full.PaperHeadSimplex.Tag P) _ (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, 0} (Erdos390.Full.PaperHeadSimplex.Tag P) (@Erdos390.WholePaper.BankPaperCanonicalExponentBand delta eta M) _ _ _ _ B)) (Erdos390.WholePaper.upperTailLength c (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.n.{0} (Erdos390.Full.PaperHeadSimplex.Tag P) _ (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, 0} (Erdos390.Full.PaperHeadSimplex.Tag P) (@Erdos390.WholePaper.BankPaperCanonicalExponentBand delta eta M) _ _ _ _ B)))) R depth)) (deltaStar alpha beta sigma : Real) (_radius : NNReal) (Cpost : Real) (endpoint : Nat → Real) (hdelta : @LT.lt.{0} Real _ (@OfNat.ofNat.{0} Real (nat_lit 0) _) delta) (hWtwo : @LE.le.{0} Nat _ (@OfNat.ofNat.{0} Nat (nat_lit 2) _) (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.W.{0} (Erdos390.Full.PaperHeadSimplex.Tag P) _ (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, 0} (Erdos390.Full.PaperHeadSimplex.Tag P) (@Erdos390.WholePaper.BankPaperCanonicalExponentBand delta eta M) _ _ _ _ B))) (S : @Erdos390.Full.RegularMeshPrimeCutoffs.ScaleSeparation delta eta M (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.n.{0} (Erdos390.Full.PaperHeadSimplex.Tag P) _ (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, 0} (Erdos390.Full.PaperHeadSimplex.Tag P) (@Erdos390.WholePaper.BankPaperCanonicalExponentBand delta eta M) _ _ _ _ B)) (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.W.{0} (Erdos390.Full.PaperHeadSimplex.Tag P) _ (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, 0} (Erdos390.Full.PaperHeadSimplex.Tag P) (@Erdos390.WholePaper.BankPaperCanonicalExponentBand delta eta M) _ _ _ _ B))) (hpartition : @Eq.{1} (@Erdos390.Full.ArithmeticBandGeometry.Partition.{0} (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.n.{0} (Erdos390.Full.PaperHeadSimplex.Tag P) _ 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(@Erdos390.WholePaper.BankPaperCanonicalExponentBand delta eta M) _ _ _ _ B)) (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.W.{0} (Erdos390.Full.PaperHeadSimplex.Tag P) _ (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, 0} (Erdos390.Full.PaperHeadSimplex.Tag P) (@Erdos390.WholePaper.BankPaperCanonicalExponentBand delta eta M) _ _ _ _ B))) (@Erdos390.WholePaper.BankPaperCanonicalExponentBand delta eta M) _ _ (@Erdos390.WholePaper.bankPaperCanonicalActualP87PointwiseUpper.{0, 0} (Erdos390.Full.PaperHeadSimplex.Tag P) (@Erdos390.WholePaper.BankPaperCanonicalExponentBand delta eta M) _ _ _ _ B (@Erdos390.Full.PaperBridgeFit.BridgeData.q.{0, 0} (Erdos390.Full.PaperHeadSimplex.Tag P) (@Erdos390.WholePaper.BankPaperCanonicalExponentBand delta eta M) _ _ _ _ B) Cpost) (@Erdos390.Full.ArithmeticBandGeometry.Partition.band.{0} (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.n.{0} (Erdos390.Full.PaperHeadSimplex.Tag P) _ (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, 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(@Erdos390.Full.PaperBridgeFit.StructuredSampleData.W.{0} (Erdos390.Full.PaperHeadSimplex.Tag P) _ (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, 0} (Erdos390.Full.PaperHeadSimplex.Tag P) (@Erdos390.WholePaper.BankPaperCanonicalExponentBand delta eta M) _ _ _ _ B)) hdelta (@Erdos390.Full.PaperBridgeFit.BridgeData.n_gt_one.{0, 0} (Erdos390.Full.PaperHeadSimplex.Tag P) (@Erdos390.WholePaper.BankPaperCanonicalExponentBand delta eta M) _ _ _ _ B) (fun (a : @Eq.{1} Nat (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.W.{0} (Erdos390.Full.PaperHeadSimplex.Tag P) _ (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, 0} (Erdos390.Full.PaperHeadSimplex.Tag P) (@Erdos390.WholePaper.BankPaperCanonicalExponentBand delta eta M) _ _ _ _ B)) (@OfNat.ofNat.{0} Nat (nat_lit 0) _)) => @Erdos390.WholePaper.BankPaperRealization.bankPaperCanonicalSectionNineFinalPayload_of_roundedSelector_guardedSlack._proof_1 delta eta M P B hWtwo a) S rho)) endpoint) (Hslack : @Erdos390.WholePaper.BankPaperRealization.BankPaperCanonicalGuardedEndpointSlackConstruction c depth (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.n.{0} (Erdos390.Full.PaperHeadSimplex.Tag P) _ (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, 0} (Erdos390.Full.PaperHeadSimplex.Tag P) (@Erdos390.WholePaper.BankPaperCanonicalExponentBand delta eta M) _ _ _ _ B)) R certificate deltaStar (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.W.{0} (Erdos390.Full.PaperHeadSimplex.Tag P) _ (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, 0} (Erdos390.Full.PaperHeadSimplex.Tag P) (@Erdos390.WholePaper.BankPaperCanonicalExponentBand delta eta M) _ _ _ _ B)) (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat _ K0 (@OfNat.ofNat.{0} Nat (nat_lit 1) _)) alpha beta (@Erdos390.Full.PaperBridgeFit.BridgeData.L.{0, 0} (Erdos390.Full.PaperHeadSimplex.Tag P) (@Erdos390.WholePaper.BankPaperCanonicalExponentBand delta eta M) _ _ _ _ B) sigma endpoint) (hKh : @LE.le.{0} Nat _ (@HMul.hMul.{0, 0, 0} Nat Nat Nat _ (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat _ K0 (@OfNat.ofNat.{0} Nat (nat_lit 1) _)) (Erdos390.WholePaper.upperTailLength c (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.n.{0} (Erdos390.Full.PaperHeadSimplex.Tag P) _ (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, 0} (Erdos390.Full.PaperHeadSimplex.Tag P) (@Erdos390.WholePaper.BankPaperCanonicalExponentBand delta eta M) _ _ _ _ B)))) (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.n.{0} (Erdos390.Full.PaperHeadSimplex.Tag P) _ (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, 0} (Erdos390.Full.PaperHeadSimplex.Tag P) (@Erdos390.WholePaper.BankPaperCanonicalExponentBand delta eta M) _ _ _ _ B))) (hdensity : @LT.lt.{0} Real _ (@OfNat.ofNat.{0} Real (nat_lit 0) _) density) (hsigma : @LT.lt.{0} Real _ (@OfNat.ofNat.{0} Real (nat_lit 0) _) sigma) (hscale : @Eq.{1} Real (@HMul.hMul.{0, 0, 0} Real Real Real _ (@Erdos390.Full.PaperBridgeFit.BridgeData.L.{0, 0} (Erdos390.Full.PaperHeadSimplex.Tag P) (@Erdos390.WholePaper.BankPaperCanonicalExponentBand delta eta M) _ _ _ _ B) N) (@Nat.cast.{0} Real _ (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.n.{0} (Erdos390.Full.PaperHeadSimplex.Tag P) _ (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, 0} (Erdos390.Full.PaperHeadSimplex.Tag P) (@Erdos390.WholePaper.BankPaperCanonicalExponentBand delta eta M) _ _ _ _ B)))) (hchargeDvd : @Dvd.dvd.{0} Nat _ (@Erdos390.WholePaper.BankPaperRealization.selectorTailCharge (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.n.{0} (Erdos390.Full.PaperHeadSimplex.Tag P) _ (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, 0} (Erdos390.Full.PaperHeadSimplex.Tag P) (@Erdos390.WholePaper.BankPaperCanonicalExponentBand delta eta M) _ _ _ _ B)) (Erdos390.WholePaper.upperEndpoint (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.n.{0} (Erdos390.Full.PaperHeadSimplex.Tag P) _ (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, 0} (Erdos390.Full.PaperHeadSimplex.Tag P) (@Erdos390.WholePaper.BankPaperCanonicalExponentBand delta eta M) _ _ _ _ B)) (Erdos390.WholePaper.upperTailLength c (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.n.{0} (Erdos390.Full.PaperHeadSimplex.Tag P) _ (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, 0} (Erdos390.Full.PaperHeadSimplex.Tag P) (@Erdos390.WholePaper.BankPaperCanonicalExponentBand delta eta M) _ _ _ _ B)))) R (@Erdos390.WholePaper.BankPaperRealization.paperFixedExceptionalFactors (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.n.{0} (Erdos390.Full.PaperHeadSimplex.Tag P) _ (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, 0} (Erdos390.Full.PaperHeadSimplex.Tag P) (@Erdos390.WholePaper.BankPaperCanonicalExponentBand delta eta M) _ _ _ _ B)) (Erdos390.WholePaper.upperTailLength c (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.n.{0} (Erdos390.Full.PaperHeadSimplex.Tag P) _ (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, 0} (Erdos390.Full.PaperHeadSimplex.Tag P) (@Erdos390.WholePaper.BankPaperCanonicalExponentBand delta eta M) _ _ _ _ B))) R deltaStar)) (@Erdos390.WholePaper.GuardedCentralAnchorCertificate.prechargedTailTarget c depth (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.n.{0} (Erdos390.Full.PaperHeadSimplex.Tag P) _ (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, 0} (Erdos390.Full.PaperHeadSimplex.Tag P) (@Erdos390.WholePaper.BankPaperCanonicalExponentBand delta eta M) _ _ _ _ B)) (@Erdos390.WholePaper.BankPaperRealization.anchorGuardLeftCore (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.n.{0} (Erdos390.Full.PaperHeadSimplex.Tag P) _ (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, 0} (Erdos390.Full.PaperHeadSimplex.Tag P) (@Erdos390.WholePaper.BankPaperCanonicalExponentBand delta eta M) _ _ _ _ B)) (Erdos390.WholePaper.upperEndpoint (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.n.{0} (Erdos390.Full.PaperHeadSimplex.Tag P) _ (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, 0} (Erdos390.Full.PaperHeadSimplex.Tag P) (@Erdos390.WholePaper.BankPaperCanonicalExponentBand delta eta M) _ _ _ _ B)) (Erdos390.WholePaper.upperTailLength c (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.n.{0} (Erdos390.Full.PaperHeadSimplex.Tag P) _ (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, 0} (Erdos390.Full.PaperHeadSimplex.Tag P) (@Erdos390.WholePaper.BankPaperCanonicalExponentBand delta eta M) _ _ _ _ B)))) R) (@Erdos390.WholePaper.BankPaperRealization.anchorGuardRightCore (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.n.{0} (Erdos390.Full.PaperHeadSimplex.Tag P) _ (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, 0} (Erdos390.Full.PaperHeadSimplex.Tag P) (@Erdos390.WholePaper.BankPaperCanonicalExponentBand delta eta M) _ _ _ _ B)) (Erdos390.WholePaper.upperEndpoint (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.n.{0} (Erdos390.Full.PaperHeadSimplex.Tag P) _ (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, 0} (Erdos390.Full.PaperHeadSimplex.Tag P) (@Erdos390.WholePaper.BankPaperCanonicalExponentBand delta eta M) _ _ _ _ B)) (Erdos390.WholePaper.upperTailLength c (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.n.{0} (Erdos390.Full.PaperHeadSimplex.Tag P) _ (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, 0} (Erdos390.Full.PaperHeadSimplex.Tag P) (@Erdos390.WholePaper.BankPaperCanonicalExponentBand delta eta M) _ _ _ _ B)))) R) (@Erdos390.WholePaper.BankPaperRealization.centralChangedMarkers (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.n.{0} (Erdos390.Full.PaperHeadSimplex.Tag P) _ (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, 0} (Erdos390.Full.PaperHeadSimplex.Tag P) (@Erdos390.WholePaper.BankPaperCanonicalExponentBand delta eta M) _ _ _ _ B)) (Erdos390.WholePaper.upperEndpoint (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.n.{0} (Erdos390.Full.PaperHeadSimplex.Tag P) _ (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, 0} (Erdos390.Full.PaperHeadSimplex.Tag P) (@Erdos390.WholePaper.BankPaperCanonicalExponentBand delta eta M) _ _ _ _ B)) (Erdos390.WholePaper.upperTailLength c (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.n.{0} (Erdos390.Full.PaperHeadSimplex.Tag P) _ (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, 0} (Erdos390.Full.PaperHeadSimplex.Tag P) (@Erdos390.WholePaper.BankPaperCanonicalExponentBand delta eta M) _ _ _ _ B)))) R depth) certificate)) (hfixedComponents : @Erdos390.WholePaper.BankPaperRealization.BankPaperCanonicalSectionNineFixedComponentCollisionFree c (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.n.{0} (Erdos390.Full.PaperHeadSimplex.Tag P) _ (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, 0} (Erdos390.Full.PaperHeadSimplex.Tag P) (@Erdos390.WholePaper.BankPaperCanonicalExponentBand delta eta M) _ _ _ _ B)) R deltaStar) (hmain : @LE.le.{0} Real _ (Erdos390.WholePaper.tangentDistributedPaperMainBudget (Erdos390.WholePaper.bankPaperCanonicalRatioCellTrafficConstant rho) (Erdos390.WholePaper.bankPaperCanonicalRatioCellIncidentConstant rho) tangentConstant (@HAdd.hAdd.{0, 0, 0} Real Real Real _ delta (@Erdos390.Full.RegularRelativeMesh.Mesh.ratio delta eta M)) sigma) (@HDiv.hDiv.{0, 0, 0} Real Real Real _ (@HPow.hPow.{0, 0, 0} Real Nat Real _ density (@OfNat.ofNat.{0} Nat (nat_lit 2) _)) (@OfNat.ofNat.{0} Real (nat_lit 48) _))) (herror : @LE.le.{0} Real _ (Erdos390.WholePaper.tangentDistributedPaperErrorBudget (@Erdos390.WholePaper.bankPaperCanonicalRatioCellTrafficErrorCoefficient delta eta M (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.n.{0} (Erdos390.Full.PaperHeadSimplex.Tag P) _ (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, 0} (Erdos390.Full.PaperHeadSimplex.Tag P) (@Erdos390.WholePaper.BankPaperCanonicalExponentBand delta eta M) _ _ _ _ B)) (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.W.{0} (Erdos390.Full.PaperHeadSimplex.Tag P) _ (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, 0} (Erdos390.Full.PaperHeadSimplex.Tag P) (@Erdos390.WholePaper.BankPaperCanonicalExponentBand delta eta M) _ _ _ _ B)) rho tangentConstant) (Erdos390.WholePaper.bankPaperCanonicalRatioCellIncidentErrorCoefficient (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.W.{0} (Erdos390.Full.PaperHeadSimplex.Tag P) _ (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, 0} (Erdos390.Full.PaperHeadSimplex.Tag P) (@Erdos390.WholePaper.BankPaperCanonicalExponentBand delta eta M) _ _ _ _ B)) (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.n.{0} (Erdos390.Full.PaperHeadSimplex.Tag P) _ (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, 0} (Erdos390.Full.PaperHeadSimplex.Tag P) (@Erdos390.WholePaper.BankPaperCanonicalExponentBand delta eta M) _ _ _ _ B)) rho tangentConstant) sigma) (@HDiv.hDiv.{0, 0, 0} Real Real Real _ (@HPow.hPow.{0, 0, 0} Real Nat Real _ density (@OfNat.ofNat.{0} Nat (nat_lit 2) _)) (@OfNat.ofNat.{0} Real (nat_lit 96) _))) (hceiling : @LE.le.{0} Real _ (Erdos390.WholePaper.tangentDistributedPaperCeilingBudget (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.n.{0} (Erdos390.Full.PaperHeadSimplex.Tag P) _ (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, 0} (Erdos390.Full.PaperHeadSimplex.Tag P) (@Erdos390.WholePaper.BankPaperCanonicalExponentBand delta eta M) _ _ _ _ B)) (Erdos390.Full.ArithmeticModel.yNat (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.n.{0} (Erdos390.Full.PaperHeadSimplex.Tag P) _ (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, 0} (Erdos390.Full.PaperHeadSimplex.Tag P) (@Erdos390.WholePaper.BankPaperCanonicalExponentBand delta eta M) _ _ _ _ B))) (@Erdos390.WholePaper.tangentDistributedSupportCount.{0} (Erdos390.WholePaper.BankPaperCanonicalTangentPrime (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.n.{0} (Erdos390.Full.PaperHeadSimplex.Tag P) _ (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, 0} (Erdos390.Full.PaperHeadSimplex.Tag P) (@Erdos390.WholePaper.BankPaperCanonicalExponentBand delta eta M) _ _ _ _ B)) (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.W.{0} (Erdos390.Full.PaperHeadSimplex.Tag P) _ (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, 0} (Erdos390.Full.PaperHeadSimplex.Tag P) (@Erdos390.WholePaper.BankPaperCanonicalExponentBand delta eta M) _ _ _ _ B))) _)) (@HDiv.hDiv.{0, 0, 0} Real Real Real _ (@HPow.hPow.{0, 0, 0} Real Nat Real _ density (@OfNat.ofNat.{0} Nat (nat_lit 2) _)) (@OfNat.ofNat.{0} Real (nat_lit 96) _)))
(hlowerPos :
let flow := @Erdos390.WholePaper.BankPaperRealization.bankPaperCanonicalSectionNineRatioCellFlow delta eta M c depth (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.n.{0} (Erdos390.Full.PaperHeadSimplex.Tag P) _ (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, 0} (Erdos390.Full.PaperHeadSimplex.Tag P) (@Erdos390.WholePaper.BankPaperCanonicalExponentBand delta eta M) _ _ _ _ B)) (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.W.{0} (Erdos390.Full.PaperHeadSimplex.Tag P) _ (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, 0} (Erdos390.Full.PaperHeadSimplex.Tag P) (@Erdos390.WholePaper.BankPaperCanonicalExponentBand delta eta M) _ _ _ _ B)) R certificate deltaStar (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat _ K0 (@OfNat.ofNat.{0} Nat (nat_lit 1) _)) hdelta (@Erdos390.Full.PaperBridgeFit.BridgeData.n_gt_one.{0, 0} (Erdos390.Full.PaperHeadSimplex.Tag P) (@Erdos390.WholePaper.BankPaperCanonicalExponentBand delta eta M) _ _ _ _ B) (fun (a : @Eq.{1} Nat (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.W.{0} (Erdos390.Full.PaperHeadSimplex.Tag P) _ (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, 0} (Erdos390.Full.PaperHeadSimplex.Tag P) (@Erdos390.WholePaper.BankPaperCanonicalExponentBand delta eta M) _ _ _ _ B)) (@OfNat.ofNat.{0} Nat (nat_lit 0) _)) => @Erdos390.WholePaper.BankPaperRealization.bankPaperCanonicalSectionNineFinalPayload_of_roundedSelector_guardedSlack._proof_2 delta eta M P B c depth K0 R certificate deltaStar hWtwo a) S rho endpoint;
∀ (request : @Erdos390.WholePaper.BankPaperCanonicalDistributedTangentSplitRequest (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.n.{0} (Erdos390.Full.PaperHeadSimplex.Tag P) _ (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, 0} (Erdos390.Full.PaperHeadSimplex.Tag P) (@Erdos390.WholePaper.BankPaperCanonicalExponentBand delta eta M) _ _ _ _ B)) (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.W.{0} (Erdos390.Full.PaperHeadSimplex.Tag P) _ (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, 0} (Erdos390.Full.PaperHeadSimplex.Tag P) (@Erdos390.WholePaper.BankPaperCanonicalExponentBand delta eta M) _ _ _ _ B)) flow (@Erdos390.Full.PaperBridgeFit.BridgeData.L.{0, 0} (Erdos390.Full.PaperHeadSimplex.Tag P) (@Erdos390.WholePaper.BankPaperCanonicalExponentBand delta eta M) _ _ _ _ B) sigma), @LT.lt.{0} Nat _ (@OfNat.ofNat.{0} Nat (nat_lit 0) _) (@Erdos390.WholePaper.bankPaperCanonicalDistributedTangentLowerCard (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.n.{0} (Erdos390.Full.PaperHeadSimplex.Tag P) _ (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, 0} (Erdos390.Full.PaperHeadSimplex.Tag P) (@Erdos390.WholePaper.BankPaperCanonicalExponentBand delta eta M) _ _ _ _ B)) (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.W.{0} (Erdos390.Full.PaperHeadSimplex.Tag P) _ (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, 0} (Erdos390.Full.PaperHeadSimplex.Tag P) (@Erdos390.WholePaper.BankPaperCanonicalExponentBand delta eta M) _ _ _ _ B)) flow (@Erdos390.Full.PaperBridgeFit.BridgeData.L.{0, 0} (Erdos390.Full.PaperHeadSimplex.Tag P) (@Erdos390.WholePaper.BankPaperCanonicalExponentBand delta eta M) _ _ _ _ B) sigma density request))
(hlower :
let flow := @Erdos390.WholePaper.BankPaperRealization.bankPaperCanonicalSectionNineRatioCellFlow delta eta M c depth (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.n.{0} (Erdos390.Full.PaperHeadSimplex.Tag P) _ (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, 0} (Erdos390.Full.PaperHeadSimplex.Tag P) (@Erdos390.WholePaper.BankPaperCanonicalExponentBand delta eta M) _ _ _ _ B)) (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.W.{0} (Erdos390.Full.PaperHeadSimplex.Tag P) _ (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, 0} (Erdos390.Full.PaperHeadSimplex.Tag P) (@Erdos390.WholePaper.BankPaperCanonicalExponentBand delta eta M) _ _ _ _ B)) R certificate deltaStar (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat _ K0 (@OfNat.ofNat.{0} Nat (nat_lit 1) _)) hdelta (@Erdos390.Full.PaperBridgeFit.BridgeData.n_gt_one.{0, 0} (Erdos390.Full.PaperHeadSimplex.Tag P) (@Erdos390.WholePaper.BankPaperCanonicalExponentBand delta eta M) _ _ _ _ B) (fun (a : @Eq.{1} Nat (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.W.{0} (Erdos390.Full.PaperHeadSimplex.Tag P) _ (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, 0} (Erdos390.Full.PaperHeadSimplex.Tag P) (@Erdos390.WholePaper.BankPaperCanonicalExponentBand delta eta M) _ _ _ _ B)) (@OfNat.ofNat.{0} Nat (nat_lit 0) _)) => @Erdos390.WholePaper.BankPaperRealization.bankPaperCanonicalSectionNineFinalPayload_of_roundedSelector_guardedSlack._proof_2 delta eta M P B c depth K0 R certificate deltaStar hWtwo a) S rho endpoint;
∀ (request : @Erdos390.WholePaper.BankPaperCanonicalDistributedTangentSplitRequest (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.n.{0} (Erdos390.Full.PaperHeadSimplex.Tag P) _ (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, 0} (Erdos390.Full.PaperHeadSimplex.Tag P) (@Erdos390.WholePaper.BankPaperCanonicalExponentBand delta eta M) _ _ _ _ B)) (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.W.{0} (Erdos390.Full.PaperHeadSimplex.Tag P) _ (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, 0} (Erdos390.Full.PaperHeadSimplex.Tag P) (@Erdos390.WholePaper.BankPaperCanonicalExponentBand delta eta M) _ _ _ _ B)) flow (@Erdos390.Full.PaperBridgeFit.BridgeData.L.{0, 0} (Erdos390.Full.PaperHeadSimplex.Tag P) (@Erdos390.WholePaper.BankPaperCanonicalExponentBand delta eta M) _ _ _ _ B) sigma), @LE.le.{0} Nat _ (@Erdos390.WholePaper.bankPaperCanonicalDistributedTangentLowerCard (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.n.{0} (Erdos390.Full.PaperHeadSimplex.Tag P) _ (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, 0} (Erdos390.Full.PaperHeadSimplex.Tag P) (@Erdos390.WholePaper.BankPaperCanonicalExponentBand delta eta M) _ _ _ _ B)) (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.W.{0} (Erdos390.Full.PaperHeadSimplex.Tag P) _ (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, 0} (Erdos390.Full.PaperHeadSimplex.Tag P) (@Erdos390.WholePaper.BankPaperCanonicalExponentBand delta eta M) _ _ _ _ B)) flow (@Erdos390.Full.PaperBridgeFit.BridgeData.L.{0, 0} (Erdos390.Full.PaperHeadSimplex.Tag P) (@Erdos390.WholePaper.BankPaperCanonicalExponentBand delta eta M) _ _ _ _ B) sigma density request) (@Finset.card.{0} Nat (@Erdos390.WholePaper.tangentSplitCleanMultiplierLists.{0} (Prod.{0, 0} (Erdos390.WholePaper.BankPaperCanonicalTangentPrime (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.n.{0} (Erdos390.Full.PaperHeadSimplex.Tag P) _ (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, 0} (Erdos390.Full.PaperHeadSimplex.Tag P) (@Erdos390.WholePaper.BankPaperCanonicalExponentBand delta eta M) _ _ _ _ B)) (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.W.{0} (Erdos390.Full.PaperHeadSimplex.Tag P) _ (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, 0} (Erdos390.Full.PaperHeadSimplex.Tag P) (@Erdos390.WholePaper.BankPaperCanonicalExponentBand delta eta M) _ _ _ _ B))) (Erdos390.WholePaper.BankPaperCanonicalTangentPrime (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.n.{0} (Erdos390.Full.PaperHeadSimplex.Tag P) _ (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, 0} (Erdos390.Full.PaperHeadSimplex.Tag P) (@Erdos390.WholePaper.BankPaperCanonicalExponentBand delta eta M) _ _ _ _ B)) (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.W.{0} (Erdos390.Full.PaperHeadSimplex.Tag P) _ (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, 0} (Erdos390.Full.PaperHeadSimplex.Tag P) (@Erdos390.WholePaper.BankPaperCanonicalExponentBand delta eta M) _ _ _ _ B)))) (@Erdos390.WholePaper.tangentPositiveFlowEdges.{0} (Erdos390.WholePaper.BankPaperCanonicalTangentPrime (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.n.{0} (Erdos390.Full.PaperHeadSimplex.Tag P) _ (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, 0} (Erdos390.Full.PaperHeadSimplex.Tag P) (@Erdos390.WholePaper.BankPaperCanonicalExponentBand delta eta M) _ _ _ _ B)) (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.W.{0} (Erdos390.Full.PaperHeadSimplex.Tag P) _ (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, 0} (Erdos390.Full.PaperHeadSimplex.Tag P) (@Erdos390.WholePaper.BankPaperCanonicalExponentBand delta eta M) _ _ _ _ B))) _ _ flow) (@Erdos390.WholePaper.tangentStarEdgeSource.{0} (Erdos390.WholePaper.BankPaperCanonicalTangentPrime (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.n.{0} (Erdos390.Full.PaperHeadSimplex.Tag P) _ (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, 0} (Erdos390.Full.PaperHeadSimplex.Tag P) 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(@Erdos390.WholePaper.tangentStarEdgeTarget.{0} (Erdos390.WholePaper.BankPaperCanonicalTangentPrime (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.n.{0} (Erdos390.Full.PaperHeadSimplex.Tag P) _ (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, 0} (Erdos390.Full.PaperHeadSimplex.Tag P) (@Erdos390.WholePaper.BankPaperCanonicalExponentBand delta eta M) _ _ _ _ B)) (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.W.{0} (Erdos390.Full.PaperHeadSimplex.Tag P) _ (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, 0} (Erdos390.Full.PaperHeadSimplex.Tag P) (@Erdos390.WholePaper.BankPaperCanonicalExponentBand delta eta M) _ _ _ _ B))) (@Erdos390.WholePaper.bankPaperCanonicalTangentPrimeLabel (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.n.{0} (Erdos390.Full.PaperHeadSimplex.Tag P) _ (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, 0} (Erdos390.Full.PaperHeadSimplex.Tag P) (@Erdos390.WholePaper.BankPaperCanonicalExponentBand delta eta M) _ _ _ _ B)) (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.W.{0} (Erdos390.Full.PaperHeadSimplex.Tag P) _ (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, 0} (Erdos390.Full.PaperHeadSimplex.Tag P) (@Erdos390.WholePaper.BankPaperCanonicalExponentBand delta eta M) _ _ _ _ B)))) (@Erdos390.Full.PaperBridgeFit.BridgeData.L.{0, 0} (Erdos390.Full.PaperHeadSimplex.Tag P) (@Erdos390.WholePaper.BankPaperCanonicalExponentBand delta eta M) _ _ _ _ B) sigma (fun (edge : Prod.{0, 0} (Erdos390.WholePaper.BankPaperCanonicalTangentPrime (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.n.{0} (Erdos390.Full.PaperHeadSimplex.Tag P) _ (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, 0} (Erdos390.Full.PaperHeadSimplex.Tag P) (@Erdos390.WholePaper.BankPaperCanonicalExponentBand delta eta M) _ _ _ _ B)) (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.W.{0} (Erdos390.Full.PaperHeadSimplex.Tag P) _ (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, 0} (Erdos390.Full.PaperHeadSimplex.Tag P) (@Erdos390.WholePaper.BankPaperCanonicalExponentBand delta eta M) _ _ _ _ B))) (Erdos390.WholePaper.BankPaperCanonicalTangentPrime (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.n.{0} (Erdos390.Full.PaperHeadSimplex.Tag P) _ (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, 0} (Erdos390.Full.PaperHeadSimplex.Tag P) (@Erdos390.WholePaper.BankPaperCanonicalExponentBand delta eta M) _ _ _ _ B)) (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.W.{0} (Erdos390.Full.PaperHeadSimplex.Tag P) _ (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, 0} (Erdos390.Full.PaperHeadSimplex.Tag P) (@Erdos390.WholePaper.BankPaperCanonicalExponentBand delta eta M) _ _ _ _ B)))) => flow (@Prod.fst.{0, 0} (Erdos390.WholePaper.BankPaperCanonicalTangentPrime (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.n.{0} (Erdos390.Full.PaperHeadSimplex.Tag P) _ (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, 0} (Erdos390.Full.PaperHeadSimplex.Tag P) 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(Erdos390.WholePaper.BankPaperCanonicalTangentPrime (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.n.{0} (Erdos390.Full.PaperHeadSimplex.Tag P) _ (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, 0} (Erdos390.Full.PaperHeadSimplex.Tag P) (@Erdos390.WholePaper.BankPaperCanonicalExponentBand delta eta M) _ _ _ _ B)) (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.W.{0} (Erdos390.Full.PaperHeadSimplex.Tag P) _ (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, 0} (Erdos390.Full.PaperHeadSimplex.Tag P) (@Erdos390.WholePaper.BankPaperCanonicalExponentBand delta eta M) _ _ _ _ B))) (Erdos390.WholePaper.BankPaperCanonicalTangentPrime (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.n.{0} (Erdos390.Full.PaperHeadSimplex.Tag P) _ (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, 0} (Erdos390.Full.PaperHeadSimplex.Tag P) (@Erdos390.WholePaper.BankPaperCanonicalExponentBand delta eta M) _ _ _ _ B)) (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.W.{0} (Erdos390.Full.PaperHeadSimplex.Tag P) _ (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, 0} (Erdos390.Full.PaperHeadSimplex.Tag P) (@Erdos390.WholePaper.BankPaperCanonicalExponentBand delta eta M) _ _ _ _ B))) edge)) (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.n.{0} (Erdos390.Full.PaperHeadSimplex.Tag P) _ (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, 0} (Erdos390.Full.PaperHeadSimplex.Tag P) (@Erdos390.WholePaper.BankPaperCanonicalExponentBand delta eta M) _ _ _ _ B)) (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat _ K0 (@OfNat.ofNat.{0} Nat (nat_lit 1) _)) (Erdos390.WholePaper.upperTailLength c (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.n.{0} (Erdos390.Full.PaperHeadSimplex.Tag P) _ (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, 0} (Erdos390.Full.PaperHeadSimplex.Tag P) (@Erdos390.WholePaper.BankPaperCanonicalExponentBand delta eta M) _ _ _ _ B))) (Erdos390.WholePaper.roughHeadModulus (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.W.{0} 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(Erdos390.Full.PaperHeadSimplex.Tag P) _ (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, 0} (Erdos390.Full.PaperHeadSimplex.Tag P) (@Erdos390.WholePaper.BankPaperCanonicalExponentBand delta eta M) _ _ _ _ B)) (Erdos390.WholePaper.upperEndpoint (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.n.{0} (Erdos390.Full.PaperHeadSimplex.Tag P) _ (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, 0} (Erdos390.Full.PaperHeadSimplex.Tag P) (@Erdos390.WholePaper.BankPaperCanonicalExponentBand delta eta M) _ _ _ _ B)) (Erdos390.WholePaper.upperTailLength c (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.n.{0} (Erdos390.Full.PaperHeadSimplex.Tag P) _ (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, 0} (Erdos390.Full.PaperHeadSimplex.Tag P) (@Erdos390.WholePaper.BankPaperCanonicalExponentBand delta eta M) _ _ _ _ B)))) (@Erdos390.WholePaper.BankPaperRealization.anchorGuardLeftCore (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.n.{0} (Erdos390.Full.PaperHeadSimplex.Tag P) _ (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, 0} (Erdos390.Full.PaperHeadSimplex.Tag P) (@Erdos390.WholePaper.BankPaperCanonicalExponentBand delta eta M) _ _ _ _ B)) (Erdos390.WholePaper.upperEndpoint (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.n.{0} (Erdos390.Full.PaperHeadSimplex.Tag P) _ (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, 0} (Erdos390.Full.PaperHeadSimplex.Tag P) (@Erdos390.WholePaper.BankPaperCanonicalExponentBand delta eta M) _ _ _ _ B)) (Erdos390.WholePaper.upperTailLength c (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.n.{0} (Erdos390.Full.PaperHeadSimplex.Tag P) _ (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, 0} (Erdos390.Full.PaperHeadSimplex.Tag P) (@Erdos390.WholePaper.BankPaperCanonicalExponentBand delta eta M) _ _ _ _ B)))) R) (@Erdos390.WholePaper.BankPaperRealization.anchorGuardRightCore (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.n.{0} (Erdos390.Full.PaperHeadSimplex.Tag P) _ (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, 0} (Erdos390.Full.PaperHeadSimplex.Tag P) (@Erdos390.WholePaper.BankPaperCanonicalExponentBand delta eta M) _ _ _ _ B)) (Erdos390.WholePaper.upperEndpoint (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.n.{0} (Erdos390.Full.PaperHeadSimplex.Tag P) _ (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, 0} (Erdos390.Full.PaperHeadSimplex.Tag P) (@Erdos390.WholePaper.BankPaperCanonicalExponentBand delta eta M) _ _ _ _ B)) (Erdos390.WholePaper.upperTailLength c (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.n.{0} (Erdos390.Full.PaperHeadSimplex.Tag P) _ (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, 0} (Erdos390.Full.PaperHeadSimplex.Tag P) (@Erdos390.WholePaper.BankPaperCanonicalExponentBand delta eta M) _ _ _ _ B)))) R) (@Erdos390.WholePaper.BankPaperRealization.centralChangedMarkers (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.n.{0} (Erdos390.Full.PaperHeadSimplex.Tag P) _ (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, 0} (Erdos390.Full.PaperHeadSimplex.Tag P) (@Erdos390.WholePaper.BankPaperCanonicalExponentBand delta eta M) _ _ _ _ B)) (Erdos390.WholePaper.upperEndpoint (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.n.{0} (Erdos390.Full.PaperHeadSimplex.Tag P) _ (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, 0} (Erdos390.Full.PaperHeadSimplex.Tag P) (@Erdos390.WholePaper.BankPaperCanonicalExponentBand delta eta M) _ _ _ _ B)) (Erdos390.WholePaper.upperTailLength c (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.n.{0} (Erdos390.Full.PaperHeadSimplex.Tag P) _ (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, 0} (Erdos390.Full.PaperHeadSimplex.Tag P) (@Erdos390.WholePaper.BankPaperCanonicalExponentBand delta eta M) _ _ _ _ B)))) R depth) R certificate (@Erdos390.WholePaper.BankPaperRealization.paperFixedExceptionalFactors (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.n.{0} (Erdos390.Full.PaperHeadSimplex.Tag P) _ (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, 0} (Erdos390.Full.PaperHeadSimplex.Tag P) (@Erdos390.WholePaper.BankPaperCanonicalExponentBand delta eta M) _ _ _ _ B)) (Erdos390.WholePaper.upperTailLength c (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.n.{0} (Erdos390.Full.PaperHeadSimplex.Tag P) _ (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, 0} (Erdos390.Full.PaperHeadSimplex.Tag P) (@Erdos390.WholePaper.BankPaperCanonicalExponentBand delta eta M) _ _ _ _ B))) R deltaStar)) request)))
(hlowerScale :
let flow := @Erdos390.WholePaper.BankPaperRealization.bankPaperCanonicalSectionNineRatioCellFlow delta eta M c depth (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.n.{0} (Erdos390.Full.PaperHeadSimplex.Tag P) _ (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, 0} (Erdos390.Full.PaperHeadSimplex.Tag P) (@Erdos390.WholePaper.BankPaperCanonicalExponentBand delta eta M) _ _ _ _ B)) (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.W.{0} (Erdos390.Full.PaperHeadSimplex.Tag P) _ (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, 0} (Erdos390.Full.PaperHeadSimplex.Tag P) (@Erdos390.WholePaper.BankPaperCanonicalExponentBand delta eta M) _ _ _ _ B)) R certificate deltaStar (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat _ K0 (@OfNat.ofNat.{0} Nat (nat_lit 1) _)) hdelta (@Erdos390.Full.PaperBridgeFit.BridgeData.n_gt_one.{0, 0} (Erdos390.Full.PaperHeadSimplex.Tag P) (@Erdos390.WholePaper.BankPaperCanonicalExponentBand delta eta M) _ _ _ _ B) (fun (a : @Eq.{1} Nat (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.W.{0} (Erdos390.Full.PaperHeadSimplex.Tag P) _ (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, 0} (Erdos390.Full.PaperHeadSimplex.Tag P) (@Erdos390.WholePaper.BankPaperCanonicalExponentBand delta eta M) _ _ _ _ B)) (@OfNat.ofNat.{0} Nat (nat_lit 0) _)) => @Erdos390.WholePaper.BankPaperRealization.bankPaperCanonicalSectionNineFinalPayload_of_roundedSelector_guardedSlack._proof_2 delta eta M P B c depth K0 R certificate deltaStar hWtwo a) S rho endpoint;
∀ (request : @Erdos390.WholePaper.BankPaperCanonicalDistributedTangentSplitRequest (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.n.{0} (Erdos390.Full.PaperHeadSimplex.Tag P) _ (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, 0} (Erdos390.Full.PaperHeadSimplex.Tag P) (@Erdos390.WholePaper.BankPaperCanonicalExponentBand delta eta M) _ _ _ _ B)) (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.W.{0} (Erdos390.Full.PaperHeadSimplex.Tag P) _ (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, 0} (Erdos390.Full.PaperHeadSimplex.Tag P) (@Erdos390.WholePaper.BankPaperCanonicalExponentBand delta eta M) _ _ _ _ B)) flow (@Erdos390.Full.PaperBridgeFit.BridgeData.L.{0, 0} (Erdos390.Full.PaperHeadSimplex.Tag P) (@Erdos390.WholePaper.BankPaperCanonicalExponentBand delta eta M) _ _ _ _ B) sigma) (side : Erdos390.WholePaper.TangentEndpointSide), @LE.le.{0} Real _ (@HMul.hMul.{0, 0, 0} Real Real Real _ density (@Nat.cast.{0} Real _ (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.n.{0} (Erdos390.Full.PaperHeadSimplex.Tag P) _ (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, 0} (Erdos390.Full.PaperHeadSimplex.Tag P) (@Erdos390.WholePaper.BankPaperCanonicalExponentBand delta eta M) _ _ _ _ B)))) (@HMul.hMul.{0, 0, 0} Real Real Real _ (@Nat.cast.{0} Real _ (@Erdos390.WholePaper.bankPaperCanonicalDistributedTangentLowerCard (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.n.{0} (Erdos390.Full.PaperHeadSimplex.Tag P) _ (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, 0} (Erdos390.Full.PaperHeadSimplex.Tag P) (@Erdos390.WholePaper.BankPaperCanonicalExponentBand delta eta M) _ _ _ _ B)) (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.W.{0} (Erdos390.Full.PaperHeadSimplex.Tag P) _ (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, 0} (Erdos390.Full.PaperHeadSimplex.Tag P) (@Erdos390.WholePaper.BankPaperCanonicalExponentBand delta eta M) _ _ _ _ B)) flow (@Erdos390.Full.PaperBridgeFit.BridgeData.L.{0, 0} (Erdos390.Full.PaperHeadSimplex.Tag P) (@Erdos390.WholePaper.BankPaperCanonicalExponentBand delta eta M) _ _ _ _ B) sigma density request)) (@Nat.cast.{0} Real _ (@Erdos390.WholePaper.tangentEndpointLabel.{0} (@Erdos390.WholePaper.BankPaperCanonicalDistributedTangentSplitRequest (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.n.{0} (Erdos390.Full.PaperHeadSimplex.Tag P) _ (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, 0} (Erdos390.Full.PaperHeadSimplex.Tag P) (@Erdos390.WholePaper.BankPaperCanonicalExponentBand delta eta M) _ _ _ _ B)) (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.W.{0} (Erdos390.Full.PaperHeadSimplex.Tag P) _ (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, 0} (Erdos390.Full.PaperHeadSimplex.Tag P) (@Erdos390.WholePaper.BankPaperCanonicalExponentBand delta eta M) _ _ _ _ B)) flow (@Erdos390.Full.PaperBridgeFit.BridgeData.L.{0, 0} (Erdos390.Full.PaperHeadSimplex.Tag P) (@Erdos390.WholePaper.BankPaperCanonicalExponentBand delta eta M) _ _ _ _ B) sigma) (@Erdos390.WholePaper.bankPaperCanonicalDistributedTangentRequestSource (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.n.{0} (Erdos390.Full.PaperHeadSimplex.Tag P) _ (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, 0} (Erdos390.Full.PaperHeadSimplex.Tag P) (@Erdos390.WholePaper.BankPaperCanonicalExponentBand delta eta M) _ _ _ _ B)) (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.W.{0} (Erdos390.Full.PaperHeadSimplex.Tag P) _ (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, 0} (Erdos390.Full.PaperHeadSimplex.Tag P) (@Erdos390.WholePaper.BankPaperCanonicalExponentBand delta eta M) _ _ _ _ B)) flow (@Erdos390.Full.PaperBridgeFit.BridgeData.L.{0, 0} (Erdos390.Full.PaperHeadSimplex.Tag P) (@Erdos390.WholePaper.BankPaperCanonicalExponentBand delta eta M) _ _ _ _ B) sigma) (@Erdos390.WholePaper.bankPaperCanonicalDistributedTangentRequestTarget (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.n.{0} (Erdos390.Full.PaperHeadSimplex.Tag P) _ (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, 0} (Erdos390.Full.PaperHeadSimplex.Tag P) (@Erdos390.WholePaper.BankPaperCanonicalExponentBand delta eta M) _ _ _ _ B)) (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.W.{0} (Erdos390.Full.PaperHeadSimplex.Tag P) _ (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, 0} (Erdos390.Full.PaperHeadSimplex.Tag P) (@Erdos390.WholePaper.BankPaperCanonicalExponentBand delta eta M) _ _ _ _ B)) flow (@Erdos390.Full.PaperBridgeFit.BridgeData.L.{0, 0} (Erdos390.Full.PaperHeadSimplex.Tag P) (@Erdos390.WholePaper.BankPaperCanonicalExponentBand delta eta M) _ _ _ _ B) sigma) side request)))),
And (@Erdos390.WholePaper.BankPaperRealization.BankPaperCanonicalSectionNineFinalPayload c depth (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.n.{0} (Erdos390.Full.PaperHeadSimplex.Tag P) _ (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, 0} (Erdos390.Full.PaperHeadSimplex.Tag P) (@Erdos390.WholePaper.BankPaperCanonicalExponentBand delta eta M) _ _ _ _ B)) (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.W.{0} (Erdos390.Full.PaperHeadSimplex.Tag P) _ (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, 0} (Erdos390.Full.PaperHeadSimplex.Tag P) (@Erdos390.WholePaper.BankPaperCanonicalExponentBand delta eta M) _ _ _ _ B)) (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat _ K0 (@OfNat.ofNat.{0} Nat (nat_lit 1) _)) R certificate deltaStar endpoint (@Erdos390.WholePaper.BankPaperRealization.bankPaperCanonicalSectionNineRatioCellFlow delta eta M c depth (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.n.{0} (Erdos390.Full.PaperHeadSimplex.Tag P) _ (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, 0} (Erdos390.Full.PaperHeadSimplex.Tag P) (@Erdos390.WholePaper.BankPaperCanonicalExponentBand delta eta M) _ _ _ _ B)) (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.W.{0} (Erdos390.Full.PaperHeadSimplex.Tag P) _ (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, 0} (Erdos390.Full.PaperHeadSimplex.Tag P) (@Erdos390.WholePaper.BankPaperCanonicalExponentBand delta eta M) _ _ _ _ B)) R certificate deltaStar (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat _ K0 (@OfNat.ofNat.{0} Nat (nat_lit 1) _)) hdelta (@Erdos390.Full.PaperBridgeFit.BridgeData.n_gt_one.{0, 0} (Erdos390.Full.PaperHeadSimplex.Tag P) (@Erdos390.WholePaper.BankPaperCanonicalExponentBand delta eta M) _ _ _ _ B) (fun (a : @Eq.{1} Nat (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.W.{0} (Erdos390.Full.PaperHeadSimplex.Tag P) _ (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, 0} (Erdos390.Full.PaperHeadSimplex.Tag P) (@Erdos390.WholePaper.BankPaperCanonicalExponentBand delta eta M) _ _ _ _ B)) (@OfNat.ofNat.{0} Nat (nat_lit 0) _)) => @Erdos390.WholePaper.BankPaperRealization.bankPaperCanonicalSectionNineFinalPayload_of_roundedSelector_guardedSlack._proof_2 delta eta M P B c depth K0 R certificate deltaStar hWtwo a) S rho endpoint) (@Erdos390.Full.PaperBridgeFit.BridgeData.L.{0, 0} (Erdos390.Full.PaperHeadSimplex.Tag P) (@Erdos390.WholePaper.BankPaperCanonicalExponentBand delta eta M) _ _ _ _ B) sigma) (@Erdos390.WholePaper.TangentRatioCellGeometry.{0, 0} (Erdos390.WholePaper.BankPaperCanonicalTangentPrime (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.n.{0} (Erdos390.Full.PaperHeadSimplex.Tag P) _ (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, 0} (Erdos390.Full.PaperHeadSimplex.Tag P) (@Erdos390.WholePaper.BankPaperCanonicalExponentBand delta eta M) _ _ _ _ B)) (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.W.{0} (Erdos390.Full.PaperHeadSimplex.Tag P) _ (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, 0} (Erdos390.Full.PaperHeadSimplex.Tag P) (@Erdos390.WholePaper.BankPaperCanonicalExponentBand delta eta M) _ _ _ _ B))) (@Erdos390.WholePaper.BankPaperCanonicalExponentBand delta eta M) (@Erdos390.WholePaper.bankPaperCanonicalTangentPrimeLabel (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.n.{0} (Erdos390.Full.PaperHeadSimplex.Tag P) _ (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, 0} (Erdos390.Full.PaperHeadSimplex.Tag P) (@Erdos390.WholePaper.BankPaperCanonicalExponentBand delta eta M) _ _ _ _ B)) (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.W.{0} (Erdos390.Full.PaperHeadSimplex.Tag P) _ (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, 0} (Erdos390.Full.PaperHeadSimplex.Tag P) (@Erdos390.WholePaper.BankPaperCanonicalExponentBand delta eta M) _ _ _ _ B))) (@Erdos390.WholePaper.bankPaperCanonicalExponentBandOf delta eta M (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.n.{0} (Erdos390.Full.PaperHeadSimplex.Tag P) _ (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, 0} (Erdos390.Full.PaperHeadSimplex.Tag P) (@Erdos390.WholePaper.BankPaperCanonicalExponentBand delta eta M) _ _ _ _ B)) (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.W.{0} (Erdos390.Full.PaperHeadSimplex.Tag P) _ (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, 0} (Erdos390.Full.PaperHeadSimplex.Tag P) (@Erdos390.WholePaper.BankPaperCanonicalExponentBand delta eta M) _ _ _ _ B)) hdelta (@Erdos390.Full.PaperBridgeFit.BridgeData.n_gt_one.{0, 0} (Erdos390.Full.PaperHeadSimplex.Tag P) (@Erdos390.WholePaper.BankPaperCanonicalExponentBand delta eta M) _ _ _ _ B) (fun (a : @Eq.{1} Nat (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.W.{0} (Erdos390.Full.PaperHeadSimplex.Tag P) _ (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, 0} (Erdos390.Full.PaperHeadSimplex.Tag P) (@Erdos390.WholePaper.BankPaperCanonicalExponentBand delta eta M) _ _ _ _ B)) (@OfNat.ofNat.{0} Nat (nat_lit 0) _)) => @Erdos390.WholePaper.BankPaperRealization.bankPaperCanonicalSectionNineFinalPayload_of_roundedSelector_guardedSlack._proof_2 delta eta M P B c depth K0 R certificate deltaStar hWtwo a) S) (@Erdos390.WholePaper.bankPaperCanonicalRatioCellIndex delta eta M (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.n.{0} (Erdos390.Full.PaperHeadSimplex.Tag P) _ (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, 0} (Erdos390.Full.PaperHeadSimplex.Tag P) (@Erdos390.WholePaper.BankPaperCanonicalExponentBand delta eta M) _ _ _ _ B)) (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.W.{0} (Erdos390.Full.PaperHeadSimplex.Tag P) _ (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, 0} (Erdos390.Full.PaperHeadSimplex.Tag P) (@Erdos390.WholePaper.BankPaperCanonicalExponentBand delta eta M) _ _ _ _ B)) hdelta (@Erdos390.Full.PaperBridgeFit.BridgeData.n_gt_one.{0, 0} (Erdos390.Full.PaperHeadSimplex.Tag P) (@Erdos390.WholePaper.BankPaperCanonicalExponentBand delta eta M) _ _ _ _ B) (fun (a : @Eq.{1} Nat (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.W.{0} (Erdos390.Full.PaperHeadSimplex.Tag P) _ (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, 0} (Erdos390.Full.PaperHeadSimplex.Tag P) (@Erdos390.WholePaper.BankPaperCanonicalExponentBand delta eta M) _ _ _ _ B)) (@OfNat.ofNat.{0} Nat (nat_lit 0) _)) => @Erdos390.WholePaper.BankPaperRealization.bankPaperCanonicalSectionNineFinalPayload_of_roundedSelector_guardedSlack._proof_2 delta eta M P B c depth K0 R certificate deltaStar hWtwo a) S rho) r0)
noncomputable def Erdos390.RemainingAnalyticGoal004_012 : Prop :=
∀ {delta eta : Real} (M : Erdos390.Full.RegularRelativeMesh.Mesh delta eta) {P : Finset.{0} Nat}
(Patterns : Erdos390.Full.PaperHeadSimplex.Tag P → Erdos390.Full.HeadPattern.Pattern)
(I : Erdos390.Full.PaperGuardCensus.PhysicalIntervals) (Cprom Cbank : Nat)
(G : (n : Nat) → Erdos390.Full.PaperGuardCensus.Ledger n Cprom Cbank) {c : Real} (depth W K0 E Ntail : Nat)
(F : Erdos390.WholePaper.BankPaperCanonicalGuardedTailFamily c depth Ntail)
(deltaStar betaProt betaAct mu sourceMarginFloor headMarginFloor physicalEtaFloor postMarginFloor Cmass density :
Real)
(logY Lambda0 mFrozen qTilde : Nat → Real) (rho sigma Cinitial : Real) (radius : NNReal) (CP87 : Real)
(hdelta :
@LT.lt.{0} Real Real.instLT (@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real Real.instZero)) delta)
(B :
Nat →
@Erdos390.Full.PaperBridgeFit.BridgeData.{0, 0} (Erdos390.Full.PaperHeadSimplex.Tag P)
(@Erdos390.WholePaper.BankPaperCanonicalExponentBand delta eta M)
(@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)
(Fin.fintype
(@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat)
(@Erdos390.Full.RegularRelativeMesh.Mesh.cellCount delta eta M)
(@OfNat.ofNat.{0} Nat (nat_lit 1) (instOfNatNat (nat_lit 1)))))
(instDecidableEqFin
(@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat)
(@Erdos390.Full.RegularRelativeMesh.Mesh.cellCount delta eta M)
(@OfNat.ofNat.{0} Nat (nat_lit 1) (instOfNatNat (nat_lit 1)))))),
@Erdos390.WholePaper.BankPaperRealization.BankPaperCanonicalSectionNinePostHeightEventualCoherentBridgeSourceObligation
P Patterns I Cprom Cbank G c depth W K0 E Ntail F deltaStar betaProt betaAct mu sourceMarginFloor headMarginFloor
physicalEtaFloor postMarginFloor Cmass density logY Lambda0 mFrozen qTilde delta eta M hdelta B →
@Erdos390.WholePaper.BankPaperRealization.BankPaperCanonicalSectionNinePostHeightSourceFirstPreMeshPlacedSelectorCallback
P Patterns I Cprom Cbank G c depth W K0 deltaStar betaProt betaAct mu logY Lambda0 mFrozen qTilde Cinitial →
@Erdos390.WholePaper.BankPaperRealization.BankPaperCanonicalSectionNinePostHeightSourceFirstSpecializedEventualLocalP87Callback
delta eta M P Patterns I Cprom Cbank G c depth W K0 deltaStar betaProt betaAct postMarginFloor Cmass density
(Erdos390.WholePaper.bankPaperCanonicalSmoothFinalActiveMassFamily mu logY Lambda0 mFrozen qTilde) Cinitial
radius CP87 hdelta →
@LE.le.{0} Real Real.instLE (@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real Real.instZero))
Cinitial →
@LT.lt.{0} Real Real.instLT (@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real Real.instZero))
(↑radius : Real) →
@LE.le.{0} Real Real.instLE (@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real Real.instZero))
CP87 →
@LT.lt.{0} Real Real.instLT (@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real Real.instZero))
c →
@LE.le.{0} Nat instLENat (@OfNat.ofNat.{0} Nat (nat_lit 2) (instOfNatNat (nat_lit 2))) W →
@LT.lt.{0} Real Real.instLT
(@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real Real.instZero)) deltaStar →
@LE.le.{0} Real Real.instLE
(@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real Real.instZero)) betaProt →
@LT.lt.{0} Real Real.instLT
(@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real Real.instZero)) betaAct →
@LE.le.{0} Real Real.instLE
(@HAdd.hAdd.{0, 0, 0} Real Real Real (@instHAdd.{0} Real Real.instAdd) betaProt betaAct)
(@HDiv.hDiv.{0, 0, 0} Real Real Real
(@instHDiv.{0} Real (@DivInvMonoid.toDiv.{0} Real Real.instDivInvMonoid)) c
(Erdos390.WholePaper.roughHeadDensity W)) →
@LE.le.{0} Real Real.instLE
(@HDiv.hDiv.{0, 0, 0} Real Real Real
(@instHDiv.{0} Real (@DivInvMonoid.toDiv.{0} Real Real.instDivInvMonoid))
(@OfNat.ofNat.{0} Real (nat_lit 1) (@One.toOfNat1.{0} Real Real.instOne))
(Erdos390.WholePaper.roughHeadDensity W))
(@Nat.cast.{0} Real Real.instNatCast
(@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) K0
(@OfNat.ofNat.{0} Nat (nat_lit 1) (instOfNatNat (nat_lit 1))))) →
@LE.le.{0} Real Real.instLE
(@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real Real.instZero)) Cmass →
@LT.lt.{0} Real Real.instLT
(@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real Real.instZero)) density →
@LT.lt.{0} Real Real.instLT
(@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real Real.instZero)) sigma →
@LE.le.{0} Real Real.instLE sigma betaProt →
@LE.le.{0} Nat instLENat
Erdos390.Full.RegularMeshPrimeCutoffs.canonicalActualMomentCutoff W →
@Erdos390.WholePaper.BankPaperCanonicalSectionNineTopFrozenSynchronizedPostHfitInput
delta eta M P B c depth K0 deltaStar rho sigma CP87 hdelta
noncomputable def Erdos390.RemainingAnalyticGoal004_013 : Prop :=
∀ {P : Finset.{0} Nat} {Band : Type u_1} [inst : Fintype.{u_1} Band] [inst_1 : DecidableEq.{u_1 + 1} Band]
(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)
(E : Nat),
@LT.lt.{0} Nat instLTNat (@OfNat.ofNat.{0} Nat (nat_lit 0) (instOfNatNat (nat_lit 0))) E →
∀ (I : Erdos390.Full.PaperGuardCensus.PhysicalIntervals),
(∀ (sigma : Erdos390.Full.PaperBridgeFit.PhysicalSign),
@LE.le.{0} Real Real.instLE (@OfNat.ofNat.{0} Real (nat_lit 1) (@One.toOfNat1.{0} Real Real.instOne))
(I.lower sigma)) →
(∀ (sigma : Erdos390.Full.PaperBridgeFit.PhysicalSign),
@LT.lt.{0} Real Real.instLT (I.upper sigma)
(@OfNat.ofNat.{0} Real (nat_lit 2)
(@instOfNatAtLeastTwo.{0} Real (nat_lit 2) Real.instNatCast
(@Nat.instAtLeastTwoHAddOfNat (@OfNat.ofNat.{0} Nat (nat_lit 1) (instOfNatNat (nat_lit 1)))
(@Nat.instNeZeroSucc (@OfNat.ofNat.{0} Nat (nat_lit 0) (instOfNatNat (nat_lit 0)))))))) →
∀ (Cprom Cbank : Nat) (ledger : (n : Nat) → Erdos390.Full.PaperGuardCensus.Ledger n Cprom Cbank)
(depth W K : Nat) {c betaAct marginFloor deltaStar betaProt : Real},
@LT.lt.{0} Real Real.instLT (@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real Real.instZero)) c →
@LT.lt.{0} Real Real.instLT (@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real Real.instZero))
betaAct →
@LT.lt.{0} Real Real.instLT (@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real Real.instZero))
marginFloor →
@LE.le.{0} Real Real.instLE
(@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real Real.instZero)) betaProt →
@HasSubset.Subset.{0} (Finset.{0} Nat) (@Finset.instHasSubset.{0} Nat)
(Erdos390.WholePaper.primesUpTo W) P →
∀ (mFrozen qTilde A0 : Nat → Real),
Erdos390.WholePaper.BankPaperCanonicalSectionEightAnalyticLedger
(fun (n : Nat) =>
Erdos390.WholePaper.bankPaperCanonicalRawSmoothBaseMass W n
(Erdos390.WholePaper.upperTailLength c n) K betaAct)
qTilde A0 →
@Filter.Eventually.{0} Nat
(fun (n : Nat) =>
∀
(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),
@Eq.{1} Nat
(@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))
n →
@Eq.{1} Nat
(@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))
W →
∀
(hsep :
@LT.lt.{0} Nat instLTNat
(Erdos390.Full.ArithmeticModel.physicalBound
(I.upper Erdos390.Full.PaperBridgeFit.PhysicalSign.minus)
(@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.Full.ArithmeticModel.physicalBound
(I.lower Erdos390.Full.PaperBridgeFit.PhysicalSign.plus)
(@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))))
(hremaining :
∀
(cell :
Erdos390.Full.PaperBridgeFit.Cell.{0}
(Erdos390.Full.PaperHeadSimplex.Tag P)),
@Finset.Nonempty.{0} Nat
(@SDiff.sdiff.{0} (Finset.{0} Nat)
(@Finset.instSDiff.{0} Nat instDecidableEqNat)
(@Erdos390.Full.PaperGuardCensus.rawCell.{0}
(Erdos390.Full.PaperHeadSimplex.Tag P)
(Erdos390.Full.PaperHeadSimplex.pattern P hprime E) I
(@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))
cell)
(@Erdos390.Full.PaperGuardCensus.Ledger.guards
(@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))
Cprom Cbank
(ledger
(@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)))))),
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(@Erdos390.Full.PaperBridgeFit.StructuredSampleData.{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}
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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.Full.PaperGuardCensus.canonicalSampleData.{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.PaperHeadSimplex.pattern P hprime E) I
(@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.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))
Cprom Cbank
(ledger
(@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)))
hsep hremaining) →
∀
(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))
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P p)
(@Finset.Subtype.fintype.{0} Nat P))
(fun (a b : Erdos390.Full.PaperHeadSimplex.Tag P) =>
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(@SetLike.instMembership.{0, 0} (Finset.{0} Nat) Nat
(@Finset.instSetLike.{0} Nat))
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(@SetLike.instMembership.{0, 0} (Finset.{0} Nat) Nat
(@Finset.instSetLike.{0} Nat))
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(@Membership.mem.{0, 0} Nat (Finset.{0} Nat)
(@SetLike.instMembership.{0, 0} (Finset.{0} Nat) Nat
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instDecidableEqNat a b)
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P p)
(@Finset.Subtype.fintype.{0} Nat P))
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(@Finset.instSetLike.{0} Nat))
P p)
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(fun (a b : Erdos390.Full.PaperHeadSimplex.Tag P) =>
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(@SetLike.instMembership.{0, 0} (Finset.{0} Nat) Nat
(@Finset.instSetLike.{0} Nat))
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(@SetLike.instMembership.{0, 0} (Finset.{0} Nat) Nat
(@Finset.instSetLike.{0} Nat))
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(@SetLike.instMembership.{0, 0} (Finset.{0} Nat) Nat
(@Finset.instSetLike.{0} Nat))
P)
instDecidableEqNat a b)
a b)
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T) →
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(@Erdos390.WholePaper.BankPaperRealization.bankPaperCanonicalTopFrozenSmoothFrozenMass.{u_1}
P Band inst inst_1 B c depth K R certificate deltaStar betaProt
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@Erdos390.WholePaper.BankPaperRealization.BankPaperCanonicalTopFrozenRoundedSourceResidualInputsAt.{u_1}
P Band inst inst_1 B c depth K R certificate fixed T deltaStar
betaProt alpha betaTotal (qTilde n) →
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(@SetLike.instMembership.{0, 0} (Finset.{0} Nat)
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(@Finset.Subtype.fintype.{0} Nat P))
(fun (a b : Erdos390.Full.PaperHeadSimplex.Tag P) =>
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(@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 fixed
(@Erdos390.WholePaper.BankPaperRealization.roughCanonicalGuardedCandidateSet
c depth
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(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))
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(@Erdos390.Full.PaperBridgeFit.StructuredSampleData.n.{0}
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(@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))
(@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)
(@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) =>
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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))
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(@Erdos390.Full.PaperBridgeFit.StructuredSampleData.n.{0}
(Erdos390.Full.PaperHeadSimplex.Tag P)
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(@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)
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(@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))
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(@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
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(@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}
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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)
(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.centralChangedMarkers
(@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))))
R depth)
R certificate deltaStar K)
(@Erdos390.WholePaper.BankPaperRealization.bankPaperCanonicalTopFrozenRoundedSourceSelector.{u_1}
P Band inst inst_1 B c depth K R certificate T deltaStar
betaProt alpha betaTotal (qTilde n)))
(@Filter.atTop.{0} Nat Nat.instPreorder)
noncomputable def Erdos390.RemainingAnalyticGoal004_014 : Prop :=
∀ {P : Finset.{0} Nat} {Band : Type u_1} [inst : Fintype.{u_1} Band] [inst_1 : DecidableEq.{u_1 + 1} Band] (Patterns : Erdos390.Full.PaperHeadSimplex.Tag P → Erdos390.Full.HeadPattern.Pattern) (I : Erdos390.Full.PaperGuardCensus.PhysicalIntervals) (Cprom Cbank : Nat) (G : (n : Nat) → Erdos390.Full.PaperGuardCensus.Ledger n Cprom Cbank) (depth W K0 : Nat) {c deltaStar betaProt betaAct sigma : Real}, @LT.lt.{0} Real _ (@OfNat.ofNat.{0} Real (nat_lit 0) _) c → @LT.lt.{0} Real _ (@OfNat.ofNat.{0} Real (nat_lit 0) _) deltaStar → @LE.le.{0} Real _ (@OfNat.ofNat.{0} Real (nat_lit 0) _) betaProt → @LT.lt.{0} Real _ (@OfNat.ofNat.{0} Real (nat_lit 0) _) betaAct → @LE.le.{0} Real _ (@OfNat.ofNat.{0} Real (nat_lit 0) _) sigma → @LE.le.{0} Real _ sigma betaProt → @LE.le.{0} Real _ (@HAdd.hAdd.{0, 0, 0} Real Real Real _ betaProt betaAct) (@HDiv.hDiv.{0, 0, 0} Real Real Real _ c (Erdos390.WholePaper.roughHeadDensity W)) → @LE.le.{0} Real _ (@HDiv.hDiv.{0, 0, 0} Real Real Real _ (@OfNat.ofNat.{0} Real (nat_lit 1) _) (Erdos390.WholePaper.roughHeadDensity W)) (@Nat.cast.{0} Real _ (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat _ K0 (@OfNat.ofNat.{0} Nat (nat_lit 1) _))) → @LE.le.{0} Nat _ (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat _ (@HMul.hMul.{0, 0, 0} Nat Nat Nat _ (@OfNat.ofNat.{0} Nat (nat_lit 2) _) depth) (@OfNat.ofNat.{0} Nat (nat_lit 1) _)) W → @Filter.Eventually.{0} Nat (fun (n : Nat) => ∀ (B : @Erdos390.Full.PaperBridgeFit.BridgeData.{0, u_1} (Erdos390.Full.PaperHeadSimplex.Tag P) Band _ _ _ _), @Eq.{1} Nat (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.n.{0} (Erdos390.Full.PaperHeadSimplex.Tag P) _ (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, u_1} (Erdos390.Full.PaperHeadSimplex.Tag P) Band _ _ _ _ B)) n → @Eq.{1} Nat (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.W.{0} (Erdos390.Full.PaperHeadSimplex.Tag P) _ (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, u_1} (Erdos390.Full.PaperHeadSimplex.Tag P) Band _ _ _ _ B)) W → ∀ (hcanonicalSep : @LT.lt.{0} Nat _ (Erdos390.Full.ArithmeticModel.physicalBound (I.upper Erdos390.Full.PaperBridgeFit.PhysicalSign.minus) (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.n.{0} (Erdos390.Full.PaperHeadSimplex.Tag P) _ (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, u_1} (Erdos390.Full.PaperHeadSimplex.Tag P) Band _ _ _ _ B))) (Erdos390.Full.ArithmeticModel.physicalBound (I.lower Erdos390.Full.PaperBridgeFit.PhysicalSign.plus) (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.n.{0} (Erdos390.Full.PaperHeadSimplex.Tag P) _ (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, u_1} (Erdos390.Full.PaperHeadSimplex.Tag P) Band _ _ _ _ B)))) (hremaining : ∀ (cell : Erdos390.Full.PaperBridgeFit.Cell.{0} (Erdos390.Full.PaperHeadSimplex.Tag P)), @Finset.Nonempty.{0} Nat (@SDiff.sdiff.{0} (Finset.{0} Nat) _ (@Erdos390.Full.PaperGuardCensus.rawCell.{0} (Erdos390.Full.PaperHeadSimplex.Tag P) Patterns I (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.n.{0} (Erdos390.Full.PaperHeadSimplex.Tag P) _ (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, u_1} (Erdos390.Full.PaperHeadSimplex.Tag P) Band _ _ _ _ B)) cell) (@Erdos390.Full.PaperGuardCensus.Ledger.guards (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.n.{0} (Erdos390.Full.PaperHeadSimplex.Tag P) _ (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, u_1} (Erdos390.Full.PaperHeadSimplex.Tag P) Band _ _ _ _ B)) Cprom Cbank (G (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.n.{0} (Erdos390.Full.PaperHeadSimplex.Tag P) _ (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, u_1} (Erdos390.Full.PaperHeadSimplex.Tag P) Band _ _ _ _ B)))))), @Eq.{1} (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.{0} (Erdos390.Full.PaperHeadSimplex.Tag P) _) (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, u_1} (Erdos390.Full.PaperHeadSimplex.Tag P) Band _ _ _ _ B) (@Erdos390.Full.PaperGuardCensus.canonicalSampleData.{0} (Erdos390.Full.PaperHeadSimplex.Tag P) _ Patterns I (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.n.{0} (Erdos390.Full.PaperHeadSimplex.Tag P) _ (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, u_1} (Erdos390.Full.PaperHeadSimplex.Tag P) Band _ _ _ _ B)) (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.W.{0} (Erdos390.Full.PaperHeadSimplex.Tag P) _ (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, u_1} (Erdos390.Full.PaperHeadSimplex.Tag P) Band _ _ _ _ B)) Cprom Cbank (G (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.n.{0} (Erdos390.Full.PaperHeadSimplex.Tag P) _ (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, u_1} (Erdos390.Full.PaperHeadSimplex.Tag P) Band _ _ _ _ B))) hcanonicalSep hremaining) → ∀ (R : Erdos390.WholePaper.BankPaperRealization (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.n.{0} (Erdos390.Full.PaperHeadSimplex.Tag P) _ (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, u_1} (Erdos390.Full.PaperHeadSimplex.Tag P) Band _ _ _ _ B)) (Erdos390.WholePaper.upperEndpoint (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.n.{0} (Erdos390.Full.PaperHeadSimplex.Tag P) _ (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, u_1} (Erdos390.Full.PaperHeadSimplex.Tag P) Band _ _ _ _ B)) (Erdos390.WholePaper.upperTailLength c (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.n.{0} (Erdos390.Full.PaperHeadSimplex.Tag P) _ (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, u_1} (Erdos390.Full.PaperHeadSimplex.Tag P) Band _ _ _ _ B))))) (certificate : Erdos390.WholePaper.GuardedCentralAnchorCertificate c depth (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.n.{0} (Erdos390.Full.PaperHeadSimplex.Tag P) _ (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, u_1} (Erdos390.Full.PaperHeadSimplex.Tag P) Band _ _ _ _ B)) (@Erdos390.WholePaper.BankPaperRealization.anchorGuardLeftCore (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.n.{0} (Erdos390.Full.PaperHeadSimplex.Tag P) _ (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, u_1} (Erdos390.Full.PaperHeadSimplex.Tag P) Band _ _ _ _ B)) (Erdos390.WholePaper.upperEndpoint (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.n.{0} (Erdos390.Full.PaperHeadSimplex.Tag P) _ (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, u_1} (Erdos390.Full.PaperHeadSimplex.Tag P) Band _ _ _ _ B)) (Erdos390.WholePaper.upperTailLength c (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.n.{0} (Erdos390.Full.PaperHeadSimplex.Tag P) _ (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, u_1} (Erdos390.Full.PaperHeadSimplex.Tag P) Band _ _ _ _ B)))) R) (@Erdos390.WholePaper.BankPaperRealization.anchorGuardRightCore (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.n.{0} (Erdos390.Full.PaperHeadSimplex.Tag P) _ (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, u_1} (Erdos390.Full.PaperHeadSimplex.Tag P) Band _ _ _ _ B)) (Erdos390.WholePaper.upperEndpoint (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.n.{0} (Erdos390.Full.PaperHeadSimplex.Tag P) _ (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, u_1} (Erdos390.Full.PaperHeadSimplex.Tag P) Band _ _ _ _ B)) (Erdos390.WholePaper.upperTailLength c (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.n.{0} (Erdos390.Full.PaperHeadSimplex.Tag P) _ (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, u_1} (Erdos390.Full.PaperHeadSimplex.Tag P) Band _ _ _ _ B)))) R) (@Erdos390.WholePaper.BankPaperRealization.centralChangedMarkers (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.n.{0} (Erdos390.Full.PaperHeadSimplex.Tag P) _ (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, u_1} (Erdos390.Full.PaperHeadSimplex.Tag P) Band _ _ _ _ B)) (Erdos390.WholePaper.upperEndpoint (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.n.{0} (Erdos390.Full.PaperHeadSimplex.Tag P) _ (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, u_1} (Erdos390.Full.PaperHeadSimplex.Tag P) Band _ _ _ _ B)) (Erdos390.WholePaper.upperTailLength c (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.n.{0} (Erdos390.Full.PaperHeadSimplex.Tag P) _ (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, u_1} (Erdos390.Full.PaperHeadSimplex.Tag P) Band _ _ _ _ B)))) R depth)) (T : @Erdos390.Full.PaperGuardCensus.BarycentricTarget.{0} (Erdos390.Full.PaperHeadSimplex.Tag P) _ (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, u_1} (Erdos390.Full.PaperHeadSimplex.Tag P) Band _ _ _ _ B)) (qTilde : Real), @Eq.{1} Real qTilde (@Erdos390.WholePaper.bankPaperCanonicalGuardedSmoothBaseMass c depth (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.n.{0} (Erdos390.Full.PaperHeadSimplex.Tag P) _ (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, u_1} (Erdos390.Full.PaperHeadSimplex.Tag P) Band _ _ _ _ B)) (Erdos390.WholePaper.upperTailLength c (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.n.{0} (Erdos390.Full.PaperHeadSimplex.Tag P) _ (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, u_1} (Erdos390.Full.PaperHeadSimplex.Tag P) Band _ _ _ _ B))) (@Erdos390.WholePaper.BankPaperRealization.anchorGuardLeftCore (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.n.{0} (Erdos390.Full.PaperHeadSimplex.Tag P) _ (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, u_1} (Erdos390.Full.PaperHeadSimplex.Tag P) Band _ _ _ _ B)) (Erdos390.WholePaper.upperEndpoint (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.n.{0} (Erdos390.Full.PaperHeadSimplex.Tag P) _ (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, u_1} (Erdos390.Full.PaperHeadSimplex.Tag P) Band _ _ _ _ B)) (Erdos390.WholePaper.upperTailLength c (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.n.{0} (Erdos390.Full.PaperHeadSimplex.Tag P) _ (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, u_1} (Erdos390.Full.PaperHeadSimplex.Tag P) Band _ _ _ _ B)))) R) (@Erdos390.WholePaper.BankPaperRealization.anchorGuardRightCore (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.n.{0} (Erdos390.Full.PaperHeadSimplex.Tag P) _ (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, u_1} (Erdos390.Full.PaperHeadSimplex.Tag P) Band _ _ _ _ B)) (Erdos390.WholePaper.upperEndpoint (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.n.{0} (Erdos390.Full.PaperHeadSimplex.Tag P) _ (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, u_1} (Erdos390.Full.PaperHeadSimplex.Tag P) Band _ _ _ _ B)) (Erdos390.WholePaper.upperTailLength c (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.n.{0} (Erdos390.Full.PaperHeadSimplex.Tag P) _ (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, u_1} (Erdos390.Full.PaperHeadSimplex.Tag P) Band _ _ _ _ B)))) R) (@Erdos390.WholePaper.BankPaperRealization.centralChangedMarkers (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.n.{0} (Erdos390.Full.PaperHeadSimplex.Tag P) _ (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, u_1} (Erdos390.Full.PaperHeadSimplex.Tag P) Band _ _ _ _ B)) (Erdos390.WholePaper.upperEndpoint (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.n.{0} (Erdos390.Full.PaperHeadSimplex.Tag P) _ (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, u_1} (Erdos390.Full.PaperHeadSimplex.Tag P) Band _ _ _ _ B)) (Erdos390.WholePaper.upperTailLength c (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.n.{0} (Erdos390.Full.PaperHeadSimplex.Tag P) _ (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, u_1} (Erdos390.Full.PaperHeadSimplex.Tag P) Band _ _ _ _ B)))) R depth) R certificate deltaStar (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.W.{0} (Erdos390.Full.PaperHeadSimplex.Tag P) _ (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, u_1} (Erdos390.Full.PaperHeadSimplex.Tag P) Band _ _ _ _ B)) (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat _ K0 (@OfNat.ofNat.{0} Nat (nat_lit 1) _)) betaAct) → ∀ (hprime : ∀ (p : Nat), @Membership.mem.{0, 0} Nat (Finset.{0} Nat) _ P p → Nat.Prime p) (Rhead : Erdos390.Full.PaperGuardCensus.HeadSimplexReserve P) (Kphysical : Erdos390.Full.PaperGuardCensus.PhysicalInterpolationTarget I) (hlo : ∀ (sign : Erdos390.Full.PaperBridgeFit.PhysicalSign), @Eq.{1} Nat (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.lo.{0} (Erdos390.Full.PaperHeadSimplex.Tag P) _ (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, u_1} (Erdos390.Full.PaperHeadSimplex.Tag P) Band _ _ _ _ B) sign) (Erdos390.Full.ArithmeticModel.physicalBound (I.lower sign) (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.n.{0} (Erdos390.Full.PaperHeadSimplex.Tag P) _ (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, u_1} (Erdos390.Full.PaperHeadSimplex.Tag P) Band _ _ _ _ B)))) (hhi : ∀ (sign : Erdos390.Full.PaperBridgeFit.PhysicalSign), @Eq.{1} Nat (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.hi.{0} (Erdos390.Full.PaperHeadSimplex.Tag P) _ (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, u_1} (Erdos390.Full.PaperHeadSimplex.Tag P) Band _ _ _ _ B) sign) (Erdos390.Full.ArithmeticModel.physicalBound (I.upper sign) (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.n.{0} (Erdos390.Full.PaperHeadSimplex.Tag P) _ (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, u_1} (Erdos390.Full.PaperHeadSimplex.Tag P) Band _ _ _ _ B)))), @Eq.{1} (@Erdos390.Full.PaperGuardCensus.BarycentricTarget.{0} (Erdos390.Full.PaperHeadSimplex.Tag P) _ (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, u_1} (Erdos390.Full.PaperHeadSimplex.Tag P) Band _ _ _ _ B)) T (@Erdos390.Full.PaperBridgeFit.BridgeData.barycentricTargetOfPaperData.{u_1} P Band _ _ B I hlo hhi Rhead Kphysical) → @Eq.{1} Real qTilde (@Erdos390.Full.PaperGuardCensus.HeadSimplexReserve.activeMass P Rhead) → @Eq.{1} (Erdos390.Full.PaperHeadSimplex.Tag P → Erdos390.Full.HeadPattern.Pattern) (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.pattern.{0} (Erdos390.Full.PaperHeadSimplex.Tag P) _ (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, u_1} (Erdos390.Full.PaperHeadSimplex.Tag P) Band _ _ _ _ B)) (Erdos390.Full.PaperHeadSimplex.pattern P hprime (@Erdos390.Full.PaperGuardCensus.HeadSimplexReserve.exponent P Rhead)) → @HasSubset.Subset.{0} (Finset.{0} Nat) _ (Erdos390.WholePaper.primesUpTo (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.W.{0} (Erdos390.Full.PaperHeadSimplex.Tag P) _ (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, u_1} (Erdos390.Full.PaperHeadSimplex.Tag P) Band _ _ _ _ B))) P → @Erdos390.Full.PaperBridgeFit.StructuredSampleData.HeadPatternsSeparated.{0} (Erdos390.Full.PaperHeadSimplex.Tag P) _ (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, u_1} (Erdos390.Full.PaperHeadSimplex.Tag P) Band _ _ _ _ B) → (∀ (sign : Erdos390.Full.PaperBridgeFit.PhysicalSign), @LE.le.{0} Real _ (@OfNat.ofNat.{0} Real (nat_lit 1) _) (I.lower sign)) → (∀ (sign : Erdos390.Full.PaperBridgeFit.PhysicalSign), @LE.le.{0} Real _ (I.upper sign) (@OfNat.ofNat.{0} Real (nat_lit 2) _)) → (∀ (sign : Erdos390.Full.PaperBridgeFit.PhysicalSign), @LE.le.{0} Nat _ (Erdos390.Full.ArithmeticModel.physicalBound (I.upper sign) (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.n.{0} (Erdos390.Full.PaperHeadSimplex.Tag P) _ (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, u_1} (Erdos390.Full.PaperHeadSimplex.Tag P) Band _ _ _ _ B))) (@HSub.hSub.{0, 0, 0} Nat Nat Nat _ (@HMul.hMul.{0, 0, 0} Nat Nat Nat _ (@OfNat.ofNat.{0} Nat (nat_lit 2) _) (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.n.{0} (Erdos390.Full.PaperHeadSimplex.Tag P) _ (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, u_1} (Erdos390.Full.PaperHeadSimplex.Tag P) Band _ _ _ _ B))) (@HMul.hMul.{0, 0, 0} Nat Nat Nat _ (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat _ K0 (@OfNat.ofNat.{0} Nat (nat_lit 1) _)) (Erdos390.WholePaper.upperTailLength c (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.n.{0} (Erdos390.Full.PaperHeadSimplex.Tag P) _ (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, u_1} (Erdos390.Full.PaperHeadSimplex.Tag P) Band _ _ _ _ B)))))) → @LE.le.{0} Nat _ (@HMul.hMul.{0, 0, 0} Nat Nat Nat _ (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat _ K0 (@OfNat.ofNat.{0} Nat (nat_lit 1) _)) (Erdos390.WholePaper.upperTailLength c (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.n.{0} (Erdos390.Full.PaperHeadSimplex.Tag P) _ (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, u_1} (Erdos390.Full.PaperHeadSimplex.Tag P) Band _ _ _ _ B)))) (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.n.{0} (Erdos390.Full.PaperHeadSimplex.Tag P) _ (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, u_1} (Erdos390.Full.PaperHeadSimplex.Tag P) Band _ _ _ _ B)) → (∀ (m : @Erdos390.Full.PaperBridgeFit.StructuredSampleData.Sample.{0} (Erdos390.Full.PaperHeadSimplex.Tag P) _ (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, u_1} (Erdos390.Full.PaperHeadSimplex.Tag P) Band _ _ _ _ B)), Not (@Membership.mem.{0, 0} Nat (Finset.{0} Nat) _ (@Erdos390.WholePaper.BankPaperRealization.roughCanonicalGuardSet c depth (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.n.{0} (Erdos390.Full.PaperHeadSimplex.Tag P) _ (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, u_1} (Erdos390.Full.PaperHeadSimplex.Tag P) Band _ _ _ _ B)) (Erdos390.WholePaper.upperTailLength c (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.n.{0} (Erdos390.Full.PaperHeadSimplex.Tag P) _ (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, u_1} (Erdos390.Full.PaperHeadSimplex.Tag P) Band _ _ _ _ B))) (@Erdos390.WholePaper.BankPaperRealization.anchorGuardLeftCore (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.n.{0} (Erdos390.Full.PaperHeadSimplex.Tag P) _ (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, u_1} (Erdos390.Full.PaperHeadSimplex.Tag P) Band _ _ _ _ B)) (Erdos390.WholePaper.upperEndpoint (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.n.{0} (Erdos390.Full.PaperHeadSimplex.Tag P) _ (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, u_1} (Erdos390.Full.PaperHeadSimplex.Tag P) Band _ _ _ _ B)) (Erdos390.WholePaper.upperTailLength c (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.n.{0} (Erdos390.Full.PaperHeadSimplex.Tag P) _ (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, u_1} (Erdos390.Full.PaperHeadSimplex.Tag P) Band _ _ _ _ B)))) R) (@Erdos390.WholePaper.BankPaperRealization.anchorGuardRightCore (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.n.{0} (Erdos390.Full.PaperHeadSimplex.Tag P) _ (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, u_1} (Erdos390.Full.PaperHeadSimplex.Tag P) Band _ _ _ _ B)) (Erdos390.WholePaper.upperEndpoint (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.n.{0} (Erdos390.Full.PaperHeadSimplex.Tag P) _ (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, u_1} (Erdos390.Full.PaperHeadSimplex.Tag P) Band _ _ _ _ B)) (Erdos390.WholePaper.upperTailLength c (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.n.{0} (Erdos390.Full.PaperHeadSimplex.Tag P) _ (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, u_1} (Erdos390.Full.PaperHeadSimplex.Tag P) Band _ _ _ _ B)))) R) (@Erdos390.WholePaper.BankPaperRealization.centralChangedMarkers (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.n.{0} (Erdos390.Full.PaperHeadSimplex.Tag P) _ (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, u_1} (Erdos390.Full.PaperHeadSimplex.Tag P) Band _ _ _ _ B)) (Erdos390.WholePaper.upperEndpoint (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.n.{0} (Erdos390.Full.PaperHeadSimplex.Tag P) _ (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, u_1} (Erdos390.Full.PaperHeadSimplex.Tag P) Band _ _ _ _ B)) (Erdos390.WholePaper.upperTailLength c (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.n.{0} (Erdos390.Full.PaperHeadSimplex.Tag P) _ (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, u_1} (Erdos390.Full.PaperHeadSimplex.Tag P) Band _ _ _ _ B)))) R depth) R certificate deltaStar) (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.value.{0} (Erdos390.Full.PaperHeadSimplex.Tag P) _ (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, u_1} (Erdos390.Full.PaperHeadSimplex.Tag P) Band _ _ _ _ B) m))) → (∀ (p : @Subtype.{1} Nat fun (p : Nat) => @Membership.mem.{0, 0} Nat (Finset.{0} Nat) _ P p), @LE.le.{0} Nat _ (@Subtype.val.{1} Nat (fun (p : Nat) => @Membership.mem.{0, 0} Nat (Finset.{0} Nat) _ P p) p) (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.W.{0} (Erdos390.Full.PaperHeadSimplex.Tag P) _ (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, u_1} (Erdos390.Full.PaperHeadSimplex.Tag P) Band _ _ _ _ B)) → @Eq.{1} Real (@Erdos390.Full.PaperGuardCensus.HeadSimplexReserve.target P Rhead p) (@Nat.cast.{0} Real _ (@DFunLike.coe.{1, 1, 1} (@Finsupp.{0, 0} Nat Nat _) Nat (fun (x : Nat) => Nat) _ (@Erdos390.WholePaper.GuardedCentralAnchorCertificate.selectorTailTarget c depth (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.n.{0} (Erdos390.Full.PaperHeadSimplex.Tag P) _ (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, u_1} (Erdos390.Full.PaperHeadSimplex.Tag P) Band _ _ _ _ B)) (@Erdos390.WholePaper.BankPaperRealization.anchorGuardLeftCore (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.n.{0} (Erdos390.Full.PaperHeadSimplex.Tag P) _ (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, u_1} (Erdos390.Full.PaperHeadSimplex.Tag P) Band _ _ _ _ B)) (Erdos390.WholePaper.upperEndpoint (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.n.{0} (Erdos390.Full.PaperHeadSimplex.Tag P) _ (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, u_1} (Erdos390.Full.PaperHeadSimplex.Tag P) Band _ _ _ _ B)) (Erdos390.WholePaper.upperTailLength c (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.n.{0} (Erdos390.Full.PaperHeadSimplex.Tag P) _ (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, u_1} (Erdos390.Full.PaperHeadSimplex.Tag P) Band _ _ _ _ B)))) R) (@Erdos390.WholePaper.BankPaperRealization.anchorGuardRightCore (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.n.{0} (Erdos390.Full.PaperHeadSimplex.Tag P) _ (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, u_1} (Erdos390.Full.PaperHeadSimplex.Tag P) Band _ _ _ _ B)) (Erdos390.WholePaper.upperEndpoint (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.n.{0} (Erdos390.Full.PaperHeadSimplex.Tag P) _ (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, u_1} (Erdos390.Full.PaperHeadSimplex.Tag P) Band _ _ _ _ B)) (Erdos390.WholePaper.upperTailLength c (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.n.{0} (Erdos390.Full.PaperHeadSimplex.Tag P) _ (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, u_1} (Erdos390.Full.PaperHeadSimplex.Tag P) Band _ _ _ _ B)))) R) (@Erdos390.WholePaper.BankPaperRealization.centralChangedMarkers (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.n.{0} (Erdos390.Full.PaperHeadSimplex.Tag P) _ (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, u_1} (Erdos390.Full.PaperHeadSimplex.Tag P) Band _ _ _ _ B)) (Erdos390.WholePaper.upperEndpoint (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.n.{0} (Erdos390.Full.PaperHeadSimplex.Tag P) _ (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, u_1} (Erdos390.Full.PaperHeadSimplex.Tag P) Band _ _ _ _ B)) (Erdos390.WholePaper.upperTailLength c (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.n.{0} (Erdos390.Full.PaperHeadSimplex.Tag P) _ (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, u_1} (Erdos390.Full.PaperHeadSimplex.Tag P) Band _ _ _ _ B)))) R depth) certificate R (@Erdos390.WholePaper.BankPaperRealization.paperFixedExceptionalFactors (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.n.{0} (Erdos390.Full.PaperHeadSimplex.Tag P) _ (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, u_1} (Erdos390.Full.PaperHeadSimplex.Tag P) Band _ _ _ _ B)) (Erdos390.WholePaper.upperTailLength c (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.n.{0} (Erdos390.Full.PaperHeadSimplex.Tag P) _ (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, u_1} (Erdos390.Full.PaperHeadSimplex.Tag P) Band _ _ _ _ B))) R deltaStar)).factorization (@Subtype.val.{1} Nat (fun (p : Nat) => @Membership.mem.{0, 0} Nat (Finset.{0} Nat) _ P p) p)))) → @Dvd.dvd.{0} Nat _ (@Erdos390.WholePaper.BankPaperRealization.selectorTailCharge (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.n.{0} (Erdos390.Full.PaperHeadSimplex.Tag P) _ (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, u_1} (Erdos390.Full.PaperHeadSimplex.Tag P) Band _ _ _ _ B)) (Erdos390.WholePaper.upperEndpoint (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.n.{0} (Erdos390.Full.PaperHeadSimplex.Tag P) _ (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, u_1} (Erdos390.Full.PaperHeadSimplex.Tag P) Band _ _ _ _ B)) (Erdos390.WholePaper.upperTailLength c (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.n.{0} (Erdos390.Full.PaperHeadSimplex.Tag P) _ (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, u_1} (Erdos390.Full.PaperHeadSimplex.Tag P) Band _ _ _ _ B)))) R (@Erdos390.WholePaper.BankPaperRealization.paperFixedExceptionalFactors (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.n.{0} (Erdos390.Full.PaperHeadSimplex.Tag P) _ (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, u_1} (Erdos390.Full.PaperHeadSimplex.Tag P) Band _ _ _ _ B)) (Erdos390.WholePaper.upperTailLength c (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.n.{0} (Erdos390.Full.PaperHeadSimplex.Tag P) _ (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, u_1} (Erdos390.Full.PaperHeadSimplex.Tag P) Band _ _ _ _ B))) R deltaStar)) (@Erdos390.WholePaper.GuardedCentralAnchorCertificate.prechargedTailTarget c depth (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.n.{0} (Erdos390.Full.PaperHeadSimplex.Tag P) _ (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, u_1} (Erdos390.Full.PaperHeadSimplex.Tag P) Band _ _ _ _ B)) (@Erdos390.WholePaper.BankPaperRealization.anchorGuardLeftCore (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.n.{0} (Erdos390.Full.PaperHeadSimplex.Tag P) _ (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, u_1} (Erdos390.Full.PaperHeadSimplex.Tag P) Band _ _ _ _ B)) (Erdos390.WholePaper.upperEndpoint (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.n.{0} (Erdos390.Full.PaperHeadSimplex.Tag P) _ (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, u_1} (Erdos390.Full.PaperHeadSimplex.Tag P) Band _ _ _ _ B)) (Erdos390.WholePaper.upperTailLength c (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.n.{0} (Erdos390.Full.PaperHeadSimplex.Tag P) _ (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, u_1} (Erdos390.Full.PaperHeadSimplex.Tag P) Band _ _ _ _ B)))) R) (@Erdos390.WholePaper.BankPaperRealization.anchorGuardRightCore (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.n.{0} (Erdos390.Full.PaperHeadSimplex.Tag P) _ (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, u_1} (Erdos390.Full.PaperHeadSimplex.Tag P) Band _ _ _ _ B)) (Erdos390.WholePaper.upperEndpoint (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.n.{0} (Erdos390.Full.PaperHeadSimplex.Tag P) _ (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, u_1} (Erdos390.Full.PaperHeadSimplex.Tag P) Band _ _ _ _ B)) (Erdos390.WholePaper.upperTailLength c (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.n.{0} (Erdos390.Full.PaperHeadSimplex.Tag P) _ (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, u_1} (Erdos390.Full.PaperHeadSimplex.Tag P) Band _ _ _ _ B)))) R) (@Erdos390.WholePaper.BankPaperRealization.centralChangedMarkers (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.n.{0} (Erdos390.Full.PaperHeadSimplex.Tag P) _ (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, u_1} (Erdos390.Full.PaperHeadSimplex.Tag P) Band _ _ _ _ B)) (Erdos390.WholePaper.upperEndpoint (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.n.{0} (Erdos390.Full.PaperHeadSimplex.Tag P) _ (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, u_1} (Erdos390.Full.PaperHeadSimplex.Tag P) Band _ _ _ _ B)) (Erdos390.WholePaper.upperTailLength c (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.n.{0} (Erdos390.Full.PaperHeadSimplex.Tag P) _ (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, u_1} (Erdos390.Full.PaperHeadSimplex.Tag P) Band _ _ _ _ B)))) R depth) certificate) → @Erdos390.WholePaper.BankPaperRealization.BankPaperCanonicalTopFrozenRoundedSourceResidualInputsAt.{u_1} P Band _ _ B c depth (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat _ K0 (@OfNat.ofNat.{0} Nat (nat_lit 1) _)) R certificate (@Erdos390.WholePaper.BankPaperRealization.paperFixedExceptionalFactors (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.n.{0} (Erdos390.Full.PaperHeadSimplex.Tag P) _ (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, u_1} (Erdos390.Full.PaperHeadSimplex.Tag P) Band _ _ _ _ B)) (Erdos390.WholePaper.upperTailLength c (@Erdos390.Full.PaperBridgeFit.StructuredSampleData.n.{0} (Erdos390.Full.PaperHeadSimplex.Tag P) _ (@Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{0, u_1} (Erdos390.Full.PaperHeadSimplex.Tag P) Band _ _ _ _ B))) R deltaStar) T deltaStar betaProt (@Erdos390.WholePaper.BankPaperRealization.bankPaperCanonicalPostHfitBalancedAlpha.{0, u_1} (Erdos390.Full.PaperHeadSimplex.Tag P) Band _ _ _ _ B c K0 betaProt betaAct) (@HAdd.hAdd.{0, 0, 0} Real Real Real _ betaProt betaAct) qTilde) (@Filter.atTop.{0} Nat _)
noncomputable def Erdos390.RemainingAnalyticGoal004_015 : Prop :=
∀ (W K : Nat) {c r0 deltaStar : Real},
@LT.lt.{0} Real Real.instLT (@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real Real.instZero)) c →
@LT.lt.{0} Real Real.instLT (@OfNat.ofNat.{0} Real (nat_lit 1) (@One.toOfNat1.{0} Real Real.instOne)) r0 →
@LT.lt.{0} Real Real.instLT r0
(@HDiv.hDiv.{0, 0, 0} Real Real Real (@instHDiv.{0} Real (@DivInvMonoid.toDiv.{0} Real Real.instDivInvMonoid))
(@OfNat.ofNat.{0} Real (nat_lit 3)
(@instOfNatAtLeastTwo.{0} Real (nat_lit 3) Real.instNatCast
(@Nat.instAtLeastTwoHAddOfNat (@OfNat.ofNat.{0} Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))
(@Nat.instNeZeroSucc (@OfNat.ofNat.{0} Nat (nat_lit 1) (instOfNatNat (nat_lit 1)))))))
(@OfNat.ofNat.{0} Real (nat_lit 2)
(@instOfNatAtLeastTwo.{0} Real (nat_lit 2) Real.instNatCast
(@Nat.instAtLeastTwoHAddOfNat (@OfNat.ofNat.{0} Nat (nat_lit 1) (instOfNatNat (nat_lit 1)))
(@Nat.instNeZeroSucc (@OfNat.ofNat.{0} Nat (nat_lit 0) (instOfNatNat (nat_lit 0)))))))) →
@LT.lt.{0} Real Real.instLT (@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real Real.instZero))
deltaStar →
@LT.lt.{0} Real Real.instLT deltaStar
(@HDiv.hDiv.{0, 0, 0} Real Real Real
(@instHDiv.{0} Real (@DivInvMonoid.toDiv.{0} Real Real.instDivInvMonoid))
(@OfNat.ofNat.{0} Real (nat_lit 1) (@One.toOfNat1.{0} Real Real.instOne))
(@OfNat.ofNat.{0} Real (nat_lit 18)
(@instOfNatAtLeastTwo.{0} Real (nat_lit 18) Real.instNatCast
(@Nat.instAtLeastTwoHAddOfNat (@OfNat.ofNat.{0} Nat (nat_lit 17) (instOfNatNat (nat_lit 17)))
(@Nat.instNeZeroSucc (@OfNat.ofNat.{0} Nat (nat_lit 16) (instOfNatNat (nat_lit 16)))))))) →
@LT.lt.{0} Real Real.instLT
(@HMul.hMul.{0, 0, 0} Real Real Real (@instHMul.{0} Real Real.instMul)
(@HMul.hMul.{0, 0, 0} Real Real Real (@instHMul.{0} Real Real.instMul)
(@OfNat.ofNat.{0} Real (nat_lit 80)
(@instOfNatAtLeastTwo.{0} Real (nat_lit 80) Real.instNatCast
(@Nat.instAtLeastTwoHAddOfNat (@OfNat.ofNat.{0} Nat (nat_lit 79) (instOfNatNat (nat_lit 79)))
(@Nat.instNeZeroSucc (@OfNat.ofNat.{0} Nat (nat_lit 78) (instOfNatNat (nat_lit 78)))))))
Erdos390.WholePaper.tangentSelbergCanonicalMainConstant)
deltaStar)
(Erdos390.WholePaper.tangentPaperHeadGap W r0) →
@Filter.Eventually.{0} Nat
(fun (n : Nat) =>
∀ (depth : Nat) (left right : Nat → Nat) (changed : Finset.{0} Nat)
(R :
Erdos390.WholePaper.BankPaperRealization n
(Erdos390.WholePaper.upperEndpoint n (Erdos390.WholePaper.upperTailLength c n)))
(certificate : Erdos390.WholePaper.GuardedCentralAnchorCertificate c depth n left right changed)
(fixedExceptional : Finset.{0} Nat) {Band : Type u_1} [inst : Fintype.{u_1} Band]
[inst_1 : DecidableEq.{u_1 + 1} Band] (lastCell : Band → Nat)
(residual : Erdos390.WholePaper.BankPaperCanonicalTangentPrime n W → Real)
(bandOf : Erdos390.WholePaper.BankPaperCanonicalTangentPrime n W → Band)
(cellIndex : Erdos390.WholePaper.BankPaperCanonicalTangentPrime n W → Nat) (L sigma : Real),
@HasSubset.Subset.{0} (Finset.{0} Nat) (@Finset.instHasSubset.{0} Nat) fixedExceptional
(@Finset.Ioc.{0} Nat Nat.instPreorder Nat.instLocallyFiniteOrder
(@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat)
(@OfNat.ofNat.{0} Nat (nat_lit 2) (instOfNatNat (nat_lit 2))) n)
(Erdos390.WholePaper.upperEndpoint n (Erdos390.WholePaper.upperTailLength c n))) →
@LE.le.{0} Nat instLENat (@OfNat.ofNat.{0} Nat (nat_lit 2) (instOfNatNat (nat_lit 2))) W →
@LE.le.{0} Nat instLENat
(@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat)
(@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat)
(@OfNat.ofNat.{0} Nat (nat_lit 2) (instOfNatNat (nat_lit 2))) depth)
(@OfNat.ofNat.{0} Nat (nat_lit 1) (instOfNatNat (nat_lit 1))))
W →
@LT.lt.{0} Nat instLTNat (Erdos390.Full.ArithmeticModel.yNat n)
(Erdos390.WholePaper.centralAnchorCutoff depth n) →
@Erdos390.WholePaper.TangentRatioCellGeometry.{0, u_1}
(Erdos390.WholePaper.BankPaperCanonicalTangentPrime n W) Band
(@Erdos390.WholePaper.bankPaperCanonicalTangentPrimeLabel n W) bandOf cellIndex r0 →
And
(∀
(request :
@Erdos390.WholePaper.BankPaperCanonicalDistributedTangentSplitRequest n W
(@Erdos390.WholePaper.tangentRatioCellEarthmoverFlow.{0, u_1}
(Erdos390.WholePaper.BankPaperCanonicalTangentPrime n W) Band
(@Finset.Subtype.fintype.{0} Nat (Erdos390.Full.ArithmeticModel.primeBand n W))
inst inst_1 lastCell residual bandOf cellIndex)
L sigma),
@LT.lt.{0} Nat instLTNat (@OfNat.ofNat.{0} Nat (nat_lit 0) (instOfNatNat (nat_lit 0)))
(@Erdos390.WholePaper.bankPaperCanonicalDistributedTangentLowerCard n W
(@Erdos390.WholePaper.tangentRatioCellEarthmoverFlow.{0, u_1}
(Erdos390.WholePaper.BankPaperCanonicalTangentPrime n W) Band
(@Finset.Subtype.fintype.{0} Nat (Erdos390.Full.ArithmeticModel.primeBand n W))
inst inst_1 lastCell residual bandOf cellIndex)
L sigma (Erdos390.WholePaper.tangentPaperCleanListDensity W r0) request))
(And
(∀
(request :
@Erdos390.WholePaper.BankPaperCanonicalDistributedTangentSplitRequest n W
(@Erdos390.WholePaper.tangentRatioCellEarthmoverFlow.{0, u_1}
(Erdos390.WholePaper.BankPaperCanonicalTangentPrime n W) Band
(@Finset.Subtype.fintype.{0} Nat
(Erdos390.Full.ArithmeticModel.primeBand n W))
inst inst_1 lastCell residual bandOf cellIndex)
L sigma),
@LE.le.{0} Nat instLENat
(@Erdos390.WholePaper.bankPaperCanonicalDistributedTangentLowerCard n W
(@Erdos390.WholePaper.tangentRatioCellEarthmoverFlow.{0, u_1}
(Erdos390.WholePaper.BankPaperCanonicalTangentPrime n W) Band
(@Finset.Subtype.fintype.{0} Nat
(Erdos390.Full.ArithmeticModel.primeBand n W))
inst inst_1 lastCell residual bandOf cellIndex)
L sigma (Erdos390.WholePaper.tangentPaperCleanListDensity W r0) request)
(@Finset.card.{0} Nat
(@Erdos390.WholePaper.tangentSplitCleanMultiplierLists.{0}
(Prod.{0, 0} (Erdos390.WholePaper.BankPaperCanonicalTangentPrime n W)
(Erdos390.WholePaper.BankPaperCanonicalTangentPrime n W))
(@Erdos390.WholePaper.tangentPositiveFlowEdges.{0}
(Erdos390.WholePaper.BankPaperCanonicalTangentPrime n W)
(@Finset.Subtype.fintype.{0} Nat
(Erdos390.Full.ArithmeticModel.primeBand n W))
(fun (a b : Erdos390.WholePaper.BankPaperCanonicalTangentPrime n W) =>
@Subtype.instDecidableEq.{1} Nat
(@Membership.mem.{0, 0} Nat (Finset.{0} Nat)
(@SetLike.instMembership.{0, 0} (Finset.{0} Nat) Nat
(@Finset.instSetLike.{0} Nat))
(Erdos390.Full.ArithmeticModel.primeBand n W))
instDecidableEqNat a b)
(@Erdos390.WholePaper.tangentRatioCellEarthmoverFlow.{0, u_1}
(Erdos390.WholePaper.BankPaperCanonicalTangentPrime n W) Band
(@Finset.Subtype.fintype.{0} Nat
(Erdos390.Full.ArithmeticModel.primeBand n W))
inst inst_1 lastCell residual bandOf cellIndex))
(@Erdos390.WholePaper.tangentStarEdgeSource.{0}
(Erdos390.WholePaper.BankPaperCanonicalTangentPrime n W)
(@Erdos390.WholePaper.bankPaperCanonicalTangentPrimeLabel n W))
(@Erdos390.WholePaper.tangentStarEdgeTarget.{0}
(Erdos390.WholePaper.BankPaperCanonicalTangentPrime n W)
(@Erdos390.WholePaper.bankPaperCanonicalTangentPrimeLabel n W))
L sigma
(fun
(edge :
Prod.{0, 0} (Erdos390.WholePaper.BankPaperCanonicalTangentPrime n W)
(Erdos390.WholePaper.BankPaperCanonicalTangentPrime n W)) =>
@Erdos390.WholePaper.tangentRatioCellEarthmoverFlow.{0, u_1}
(Erdos390.WholePaper.BankPaperCanonicalTangentPrime n W) Band
(@Finset.Subtype.fintype.{0} Nat
(Erdos390.Full.ArithmeticModel.primeBand n W))
inst inst_1 lastCell residual bandOf cellIndex
(@Prod.fst.{0, 0} (Erdos390.WholePaper.BankPaperCanonicalTangentPrime n W)
(Erdos390.WholePaper.BankPaperCanonicalTangentPrime n W) edge)
(@Prod.snd.{0, 0} (Erdos390.WholePaper.BankPaperCanonicalTangentPrime n W)
(Erdos390.WholePaper.BankPaperCanonicalTangentPrime n W) edge))
n K (Erdos390.WholePaper.upperTailLength c n)
(Erdos390.WholePaper.roughHeadModulus W)
(Erdos390.WholePaper.tangentPaperExceptionalCutoff deltaStar n)
(Erdos390.Full.ArithmeticModel.yNat n)
(@Erdos390.WholePaper.BankPaperRealization.tangentPaperDedicatedRows n
(Erdos390.WholePaper.upperEndpoint n
(Erdos390.WholePaper.upperTailLength c n))
R)
(@Erdos390.WholePaper.BankPaperRealization.tangentPaperNumericalGuardSet c
depth n
(Erdos390.WholePaper.upperEndpoint n
(Erdos390.WholePaper.upperTailLength c n))
left right changed R certificate fixedExceptional)
request)))
(∀
(request :
@Erdos390.WholePaper.BankPaperCanonicalDistributedTangentSplitRequest n W
(@Erdos390.WholePaper.tangentRatioCellEarthmoverFlow.{0, u_1}
(Erdos390.WholePaper.BankPaperCanonicalTangentPrime n W) Band
(@Finset.Subtype.fintype.{0} Nat
(Erdos390.Full.ArithmeticModel.primeBand n W))
inst inst_1 lastCell residual bandOf cellIndex)
L sigma)
(side : Erdos390.WholePaper.TangentEndpointSide),
@LE.le.{0} Real Real.instLE
(@HMul.hMul.{0, 0, 0} Real Real Real (@instHMul.{0} Real Real.instMul)
(Erdos390.WholePaper.tangentPaperCleanListDensity W r0)
(@Nat.cast.{0} Real Real.instNatCast n))
(@HMul.hMul.{0, 0, 0} Real Real Real (@instHMul.{0} Real Real.instMul)
(@Nat.cast.{0} Real Real.instNatCast
(@Erdos390.WholePaper.bankPaperCanonicalDistributedTangentLowerCard n W
(@Erdos390.WholePaper.tangentRatioCellEarthmoverFlow.{0, u_1}
(Erdos390.WholePaper.BankPaperCanonicalTangentPrime n W) Band
(@Finset.Subtype.fintype.{0} Nat
(Erdos390.Full.ArithmeticModel.primeBand n W))
inst inst_1 lastCell residual bandOf cellIndex)
L sigma (Erdos390.WholePaper.tangentPaperCleanListDensity W r0) request))
(@Nat.cast.{0} Real Real.instNatCast
(@Erdos390.WholePaper.tangentEndpointLabel.{0}
(@Erdos390.WholePaper.BankPaperCanonicalDistributedTangentSplitRequest n W
(@Erdos390.WholePaper.tangentRatioCellEarthmoverFlow.{0, u_1}
(Erdos390.WholePaper.BankPaperCanonicalTangentPrime n W) Band
(@Finset.Subtype.fintype.{0} Nat
(Erdos390.Full.ArithmeticModel.primeBand n W))
inst inst_1 lastCell residual bandOf cellIndex)
L sigma)
(@Erdos390.WholePaper.bankPaperCanonicalDistributedTangentRequestSource n W
(@Erdos390.WholePaper.tangentRatioCellEarthmoverFlow.{0, u_1}
(Erdos390.WholePaper.BankPaperCanonicalTangentPrime n W) Band
(@Finset.Subtype.fintype.{0} Nat
(Erdos390.Full.ArithmeticModel.primeBand n W))
inst inst_1 lastCell residual bandOf cellIndex)
L sigma)
(@Erdos390.WholePaper.bankPaperCanonicalDistributedTangentRequestTarget n W
(@Erdos390.WholePaper.tangentRatioCellEarthmoverFlow.{0, u_1}
(Erdos390.WholePaper.BankPaperCanonicalTangentPrime n W) Band
(@Finset.Subtype.fintype.{0} Nat
(Erdos390.Full.ArithmeticModel.primeBand n W))
inst inst_1 lastCell residual bandOf cellIndex)
L sigma)
side request))))))
(@Filter.atTop.{0} Nat Nat.instPreorder)
noncomputable def Erdos390.RemainingAnalyticGoal004_016 : Prop :=
∀ (W K0 : Nat) {c deltaStar betaProt betaAct sigma : Real},
LT.lt.{0} (0 : Real) c →
LT.lt.{0} (0 : Real) deltaStar →
LT.lt.{0} (0 : Real) betaAct →
LE.le.{0} sigma betaProt →
∀ᶠ (n : Nat) in Filter.atTop.{0},
∀ (depth : Nat)
(R :
Erdos390.WholePaper.BankPaperRealization n
(Erdos390.WholePaper.upperEndpoint n (Erdos390.WholePaper.upperTailLength c n)))
(certificate :
Erdos390.WholePaper.GuardedCentralAnchorCertificate c depth n
(Erdos390.WholePaper.BankPaperRealization.anchorGuardLeftCore R)
(Erdos390.WholePaper.BankPaperRealization.anchorGuardRightCore R)
(Erdos390.WholePaper.BankPaperRealization.centralChangedMarkers R depth)),
LE.le.{0} (Erdos390.WholePaper.centralAnchorCutoffThreshold depth) n →
LT.lt.{0} (Erdos390.Full.ArithmeticModel.yNat n) (Erdos390.WholePaper.centralAnchorCutoff depth n) →
Erdos390.WholePaper.BankPaperRealization.RoughCanonicalBalancedNonsmoothBounds R certificate deltaStar
W K0 betaProt betaAct sigma
noncomputable def Erdos390.RemainingAnalyticGoal004_017 : Prop :=
∀ (W K0 : Nat) {c beta deltaStar : Real},
LT.lt.{0} (0 : Real) c →
LT.lt.{0} (0 : Real) deltaStar →
∀ᶠ (n : Nat) in Filter.atTop.{0},
∀ (depth : Nat)
(R :
Erdos390.WholePaper.BankPaperRealization n
(Erdos390.WholePaper.upperEndpoint n (Erdos390.WholePaper.upperTailLength c n)))
(certificate :
Erdos390.WholePaper.GuardedCentralAnchorCertificate c depth n
(Erdos390.WholePaper.BankPaperRealization.anchorGuardLeftCore R)
(Erdos390.WholePaper.BankPaperRealization.anchorGuardRightCore R)
(Erdos390.WholePaper.BankPaperRealization.centralChangedMarkers R depth)),
LE.le.{0} (Erdos390.WholePaper.centralAnchorCutoffThreshold depth) n →
LT.lt.{0} (Erdos390.Full.ArithmeticModel.yNat n) (Erdos390.WholePaper.centralAnchorCutoff depth n) →
∀ (label : Nat),
Membership.mem.{0, 0}
(Erdos390.WholePaper.BankPaperRealization.roughCanonicalActiveRawCorrectionLabels n
(Erdos390.WholePaper.upperTailLength c n) (HAdd.hAdd.{0, 0, 0} K0 (1 : Nat)) deltaStar)
label →
LE.le.{0}
(abs.{0}
(Erdos390.WholePaper.BankPaperRealization.roughCanonicalGuardedPostchargeCorrectionDensity R
certificate deltaStar W (HAdd.hAdd.{0, 0, 0} K0 (1 : Nat)) label
(Erdos390.WholePaper.roughHeadBalancedAlpha W n (Erdos390.WholePaper.upperTailLength c n)
(HAdd.hAdd.{0, 0, 0} K0 (1 : Nat)) beta (Erdos390.Full.Scale.L n))
beta (Erdos390.Full.Scale.L n)))
(HDiv.hDiv.{0, 0, 0}
(Erdos390.WholePaper.BankPaperRealization.roughCanonicalUniformGuardedPostchargeCorrectionDensityConstant
W K0 c beta)
(HMul.hMul.{0, 0, 0} (4 : Real) (HPow.hPow.{0, 0, 0} (Erdos390.Full.Scale.L n) (2 : Nat))))
noncomputable def Erdos390.RemainingAnalyticGoal004_018 : Prop :=
∀ (W K0 : Nat) {c beta deltaStar : Real},
LT.lt.{0} (0 : Real) c →
LT.lt.{0} (0 : Real) deltaStar →
∀ᶠ (n : Nat) in Filter.atTop.{0},
Erdos390.WholePaper.BankPaperRealization.RoughCanonicalUniformRawRowCorrectionDensityBound W n
(Erdos390.WholePaper.upperTailLength c n) (HAdd.hAdd.{0, 0, 0} K0 (1 : Nat)) deltaStar
(Erdos390.WholePaper.roughHeadBalancedAlpha W n (Erdos390.WholePaper.upperTailLength c n)
(HAdd.hAdd.{0, 0, 0} K0 (1 : Nat)) beta (Erdos390.Full.Scale.L n))
beta (Erdos390.Full.Scale.L n)
(HDiv.hDiv.{0, 0, 0}
(Erdos390.WholePaper.BankPaperRealization.roughCanonicalUniformRawRowCorrectionDensityConstant W K0 c beta)
(HMul.hMul.{0, 0, 0} (4 : Real) (HPow.hPow.{0, 0, 0} (Erdos390.Full.Scale.L n) (2 : Nat))))
noncomputable def Erdos390.RemainingAnalyticGoal004_019 : Prop :=
∀ {c : Real}, @LT.lt.{0} Real _ Erdos390.WholePaper.C0 c → ∀ (depth : Nat), @LE.le.{0} Nat _ (@OfNat.ofNat.{0} Nat (nat_lit 201) _) depth → @Exists.{1} Nat fun (W0 : Nat) => ∀ (W : Nat), @LE.le.{0} Nat _ W0 W → @LE.le.{0} Nat _ (@OfNat.ofNat.{0} Nat (nat_lit 2) _) W → @LE.le.{0} Nat _ (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat _ (@HMul.hMul.{0, 0, 0} Nat Nat Nat _ (@OfNat.ofNat.{0} Nat (nat_lit 2) _) depth) (@OfNat.ofNat.{0} Nat (nat_lit 1) _)) W → @LE.le.{0} Nat _ Erdos390.Full.PrimeBandQuadrature.fullReciprocalSumUniformCutoff W → @LE.le.{0} Nat _ Erdos390.Full.RegularMeshPrimeCutoffs.canonicalActualMomentCutoff W → ∀ (r0 deltaStar : Real), @LT.lt.{0} Real _ (@OfNat.ofNat.{0} Real (nat_lit 1) _) r0 → @LT.lt.{0} Real _ r0 (@HDiv.hDiv.{0, 0, 0} Real Real Real _ (@OfNat.ofNat.{0} Real (nat_lit 3) _) (@OfNat.ofNat.{0} Real (nat_lit 2) _)) → Erdos390.WholePaper.IsPaperCombinedTangentDeltaStar c W r0 deltaStar → Erdos390.WholePaper.BankPaperCombinedChargeTerminalAtDepth c deltaStar depth → Erdos390.WholePaper.BankPaperCanonicalSectionNineTopFrozenSynchronizedAnalyticCompletion c depth W r0 deltaStar
noncomputable def Erdos390.RemainingAnalyticGoal004_020 : Prop :=
∀ {c : Real}, @LT.lt.{0} Real _ Erdos390.WholePaper.C0 c → @Exists.{1} Nat fun (depth : Nat) => @Exists.{1} Nat fun (W : Nat) => @Exists.{1} Real fun (r0 : Real) => @Exists.{1} Real fun (deltaStar : Real) => And (@LE.le.{0} Nat _ (@OfNat.ofNat.{0} Nat (nat_lit 201) _) depth) (And (@LE.le.{0} Nat _ (@OfNat.ofNat.{0} Nat (nat_lit 2) _) W) (And (@LE.le.{0} Nat _ (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat _ (@HMul.hMul.{0, 0, 0} Nat Nat Nat _ (@OfNat.ofNat.{0} Nat (nat_lit 2) _) depth) (@OfNat.ofNat.{0} Nat (nat_lit 1) _)) W) (And (@LE.le.{0} Nat _ Erdos390.Full.PrimeBandQuadrature.fullReciprocalSumUniformCutoff W) (And (@LE.le.{0} Nat _ Erdos390.Full.RegularMeshPrimeCutoffs.canonicalActualMomentCutoff W) (And (@LT.lt.{0} Real _ (@OfNat.ofNat.{0} Real (nat_lit 1) _) r0) (And (@LT.lt.{0} Real _ r0 (@HDiv.hDiv.{0, 0, 0} Real Real Real _ (@OfNat.ofNat.{0} Real (nat_lit 3) _) (@OfNat.ofNat.{0} Real (nat_lit 2) _))) (And (Erdos390.WholePaper.IsPaperCombinedTangentDeltaStar c W r0 deltaStar) (And (Erdos390.WholePaper.BankPaperCombinedChargeTerminalAtDepth c deltaStar depth) (Erdos390.WholePaper.BankPaperCanonicalDistributedSectionNineTerminalAtDepth c deltaStar depth)))))))))
Source