OAI/NumberTheory/JointDickman/Probability/CandidateFairSplit.lean

GitHub source file

4 declarations, 0 contain sorry, 0 declared as axioms

1. theorem OAI.JointDickman.candidateCoefficientFactor_retention line 11
2. theorem OAI.JointDickman.candidateSiteKernel_retention_sum line 39
3. theorem OAI.JointDickman.candidateCoefficientFactor_fair line 67

   Source documentation

   The two regularity indicators are the only restrictions in the arithmetic-to-fair normalization of an endpoint and its remainder.

   theorem candidateCoefficientFactor_fair {B L M : ℕ} {τ C : ℝ}
       (hB : 1 < B) (χ : BlockCandidateIndex M → ℝ) (i t : Fin M)
       {S R A D : Finset ℕ} (hS : S ⊆ auxiliaryPrimes B) (hR : R ⊆ auxiliaryPrimes B)
       (hAS : A ⊆ S) (hDR : D ⊆ R) :
       candidateCoefficientFactor B L τ C χ ((i,t),(A,D))*
         regularResidueWeight B L τ C (S \ A)*regularResidueWeight B L τ C (R \ D) =
       (B : ℝ)*independentRootMean B L τ C*subsetRetentionMass S A*subsetRetentionMass R D*
         (if (RegularPrimeSet B L τ C A ∧ RegularPrimeSet B L τ C (S \ A)) ∧
             (RegularPrimeSet B L τ C D ∧ RegularPrimeSet B L τ C (R \ D)) then
           regularCoefficientWeight B L τ C (candidateQuotient ((i,t),(A,D)))*χ ((i,t),(A,D))
         else 0) := by
   …

   Mentions: OAI.JointDickman.BlockCandidateIndex, OAI.JointDickman.auxiliaryPrimes, OAI.JointDickman.candidateCoefficientFactor, OAI.JointDickman.regularResidueWeight, OAI.JointDickman.independentRootMean, OAI.JointDickman.subsetRetentionMass, OAI.JointDickman.RegularPrimeSet, OAI.JointDickman.regularCoefficientWeight, OAI.JointDickman.candidateQuotient

4. theorem OAI.JointDickman.candidateSiteKernel_fair_sum line 100

   Source documentation

   Exact candidate kernel on the ambient prime powerset. This form keeps all restrictions visible before estimating the discarded regularity events.

   theorem candidateSiteKernel_fair_sum {B L T H M : ℕ} {τ C : ℝ}
       (hB : 1 < B) (χ : BlockCandidateIndex M → ℝ) (i t : Fin M) (hit : i < t)
       {S R : Finset ℕ} (hS : S ⊆ auxiliaryPrimes B) (hR : R ⊆ auxiliaryPrimes B) :
       candidateSiteKernel B L T H M τ C χ i t S R =
         (B : ℝ)*independentRootMean B L τ C*
           ∑ A ∈ (auxiliaryPrimes B).powerset, ∑ D ∈ (auxiliaryPrimes B).powerset,
             subsetRetentionMass S A*subsetRetentionMass R D*
               (if BlockCandidateAdmissible B L T H τ C ((i,t),(A,D)) ∧
                   (RegularPrimeSet B L τ C A ∧ RegularPrimeSet B L τ C (S \ A)) ∧
                   (RegularPrimeSet B L τ C D ∧ RegularPrimeSet B L τ C (R \ D)) then
                 regularCoefficientWeight B L τ C (candidateQuotient ((i,t),(A,D)))*χ ((i,t),(A,D))
               else 0) := by
   …

   Mentions: OAI.JointDickman.BlockCandidateIndex, OAI.JointDickman.auxiliaryPrimes, OAI.JointDickman.candidateSiteKernel, OAI.JointDickman.independentRootMean, OAI.JointDickman.subsetRetentionMass, OAI.JointDickman.BlockCandidateAdmissible, OAI.JointDickman.RegularPrimeSet, OAI.JointDickman.regularCoefficientWeight, OAI.JointDickman.candidateQuotient