| import OAI.NumberTheory.Catalan.Determinants.FixedClearedColumnsAll |
| import OAI.NumberTheory.Catalan.Polynomial.FixedBaseCertificateTransfer |
|
|
| namespace OAI |
|
|
| section |
|
|
| noncomputable section |
| open scoped BigOperators |
| namespace InternalCatalan |
|
|
| def fixedIntegerBaseCanonical (r : Fin 49) (k : Fin 48) : ℤ := |
| 2 * (∑ i ∈ Finset.range 65, |
| (rowP 1 r.val).coeff i * fixedClearedMoment i k) - |
| 3 * (∑ i ∈ Finset.range 65, |
| (rowD 1 r.val).coeff i * fixedClearedZeta i k) |
|
|
| private theorem integerFractionSum |
| (s : Finset ℕ) (p a : ℕ → ℤ) (f : ℕ → ℚ) (d : ℚ) |
| (hf : ∀ i ∈ s, (p i : ℚ) * f i = ((p i * a i : ℤ) : ℚ) / d) : |
| (∑ i ∈ s, (p i : ℚ) * f i) = ((∑ i ∈ s, p i * a i : ℤ) : ℚ) / d := by |
| rw [Int.cast_sum, div_eq_mul_inv, Finset.sum_mul] |
| apply Finset.sum_congr rfl |
| intro i hi |
| simpa only [div_eq_mul_inv] using hf i hi |
|
|
| theorem fixedRowP_filtered_integer_fraction (r : Fin 49) (k : Fin 48) : |
| (∑ i ∈ Finset.range 65, ((rowP 1 r.val).coeff i : ℚ) * |
| fixedFilteredMomentRat i k.val) = |
| ((∑ i ∈ Finset.range 65, |
| (rowP 1 r.val).coeff i * fixedClearedMoment i k : ℤ) : ℚ) / |
| 427020799961118095468762173535463688638296295357889841725440000 := by |
| apply integerFractionSum |
| intro i hi |
| have hhi : i < 65 := Finset.mem_range.mp hi |
| by_cases hlo : 18 ≤ i |
| · rw [fixedFilteredMoment_cleared i k hlo hhi, Int.cast_mul] |
| ring |
| · have hz := fixed_rowP_coeff_eq_zero_below18 r (i := i) (by omega) |
| simp [hz] |
|
|
| theorem fixedRowD_filtered_integer_fraction (r : Fin 49) (k : Fin 48) : |
| (∑ i ∈ Finset.range 65, ((rowD 1 r.val).coeff i : ℚ) * |
| fixedFilteredZetaRat i k.val) = |
| ((∑ i ∈ Finset.range 65, |
| (rowD 1 r.val).coeff i * fixedClearedZeta i k : ℤ) : ℚ) / |
| 427020799961118095468762173535463688638296295357889841725440000 := by |
| apply integerFractionSum |
| intro i hi |
| have hhi : i < 65 := Finset.mem_range.mp hi |
| by_cases hlo : 19 ≤ i |
| · rw [fixedFilteredZeta_cleared i k hlo hhi, Int.cast_mul] |
| ring |
| · have hz := fixed_rowD_coeff_eq_zero_below19 r (i := i) (by omega) |
| simp [hz] |
|
|
| theorem fixedBaseEntry_integer_contraction (r : Fin 49) (k : Fin 48) : |
| fixedBaseEntryRat r k = |
| ((∑ i ∈ Finset.range 65, |
| (rowP 1 r.val).coeff i * fixedClearedMoment i k : ℤ) : ℚ) / |
| 427020799961118095468762173535463688638296295357889841725440000 - |
| (3 / 2 : ℚ) * |
| (((∑ i ∈ Finset.range 65, |
| (rowD 1 r.val).coeff i * fixedClearedZeta i k : ℤ) : ℚ) / |
| 427020799961118095468762173535463688638296295357889841725440000) := by |
| rw [fixedBaseEntry_dot_filtered, |
| fixedRowP_filtered_integer_fraction, fixedRowD_filtered_integer_fraction] |
|
|
| theorem fixedIntegerBaseCanonical_cast (r : Fin 49) (k : Fin 48) : |
| (fixedIntegerBaseCanonical r k : ℚ) = |
| (854041599922236190937524347070927377276592590715779683450880000 : ℚ) * |
| fixedBaseEntryRat r k := by |
| rw [fixedBaseEntry_integer_contraction] |
| simp only [fixedIntegerBaseCanonical, Int.cast_sub, Int.cast_mul, Int.cast_ofNat] |
| ring |
|
|
| theorem fixedCanonicalMatrix_integer_lift (sigma : ℤ) : |
| (fixedIntegerMatrix fixedIntegerBaseCanonical sigma).map |
| (fun x : ℤ => (x : ℚ)) = |
| (854041599922236190937524347070927377276592590715779683450880000 : ℚ) • |
| fixedMatrix (sigma : ℚ) := |
| fixedIntegerMatrix_cast_of_base_lift fixedIntegerBaseCanonical |
| 854041599922236190937524347070927377276592590715779683450880000 sigma |
| fixedIntegerBaseCanonical_cast |
|
|
| end InternalCatalan |
|
|
| end |
|
|
| end |
|
|
| end OAI |
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|
