Is D1Tuple — finite A5 closed interval excluded
ProvedD1Bound5.IsD1Tuple.finiteA5_closed_interval_excludeddiophantine-tuplesnumber-theory
The whole closed A5 window is excluded for an actual full-ring subtuple. This also supplies A5 after selecting the b,s,c,d subtuple.
Preamble
import Lean.Elab.Tactic.Omega import Mathlib.Algebra.BigOperators.Fin import Mathlib.Algebra.BigOperators.Group.Finset.Sigma import Mathlib.Algebra.BigOperators.NatAntidiagonal import Mathlib.Algebra.CharZero.Infinite import Mathlib.Algebra.Polynomial.BigOperators import Mathlib.Algebra.Polynomial.Degree.Operations import Mathlib.Algebra.Polynomial.Div import Mathlib.Algebra.Polynomial.PartialFractions import Mathlib.Algebra.Polynomial.Reverse import Mathlib.Algebra.Polynomial.RingDivision import Mathlib.Algebra.Polynomial.Roots import Mathlib.Algebra.QuadraticAlgebra.Basic import Mathlib.Algebra.Ring.Basic import Mathlib.Algebra.Ring.Equiv import Mathlib.Analysis.Analytic.Binomial import Mathlib.Analysis.Complex.Basic import Mathlib.Analysis.Complex.Norm import Mathlib.Analysis.Normed.Ring.Basic import Mathlib.Analysis.Normed.Ring.InfiniteSum import Mathlib.Analysis.SpecialFunctions.Gaussian.GaussianIntegral import Mathlib.Analysis.SpecialFunctions.Pow.Real import Mathlib.Analysis.SpecificLimits.Normed import Mathlib.Combinatorics.Enumerative.Catalan.Basic import Mathlib.Data.Finset.Card import Mathlib.Data.Finset.Insert import Mathlib.Data.Int.GCD import Mathlib.Data.Sym.Basic import Mathlib.Data.Sym.Card import Mathlib.LinearAlgebra.Dimension.Free import Mathlib.LinearAlgebra.Lagrange import Mathlib.LinearAlgebra.Matrix.Determinant.Basic import Mathlib.LinearAlgebra.Matrix.NonsingularInverse import Mathlib.LinearAlgebra.Matrix.Polynomial import Mathlib.NumberTheory.NumberField.InfinitePlace.TotallyRealComplex import Mathlib.NumberTheory.NumberField.Norm import Mathlib.RingTheory.Binomial import Mathlib.RingTheory.MvPolynomial.Symmetric.Defs import Mathlib.RingTheory.PowerSeries.Binomial import Mathlib.RingTheory.PowerSeries.Trunc import Mathlib.Tactic import Mathlib.Tactic.Convert import Mathlib.Tactic.FieldSimp import Mathlib.Tactic.FinCases import Mathlib.Tactic.GCongr import Mathlib.Tactic.Linarith import Mathlib.Tactic.LinearCombination import Mathlib.Tactic.NormNum import Mathlib.Tactic.Positivity import Mathlib.Tactic.Ring import Definitions.Def_D1Bound5_IsD1Tuple
Formal statement
section
open scoped NumberField
open Module
namespace D1Bound5
theorem _root_.D1Bound5.IsD1Tuple.finiteA5_closed_interval_excluded
{K : Type*} [Field K] [NumberField K] [NumberField.IsTotallyComplex K]
(hK : finrank ℚ K = 2) (phi : K →+* ℂ)
(t : Fin 4 → 𝓞 K) (ht : IsD1Tuple t)
(hXY : ‖phi (t 0 : K)‖ ≤ ‖phi (t 1 : K)‖)
(hXC : 1000 ≤ ‖phi (t 0 : K)‖*‖phi (t 2 : K)‖)
(hC : 200*‖phi (t 1 : K)‖ ≤ ‖phi (t 2 : K)‖)
(hCD : ‖phi (t 2 : K)‖ ≤ ‖phi (t 3 : K)‖)
(hleft : 4*3113013^2*‖phi (t 1 : K)‖^24*‖phi (t 2 : K)‖^11 /
(1-1/4000 : ℝ) ≤ ‖phi (t 3 : K)‖)
(hright : ‖phi (t 3 : K)‖ ≤
1024*‖phi (t 0 : K)‖^12*‖phi (t 2 : K)‖^17 /
(245025*‖phi (t 1 : K)‖^10*(1+1/4000 : ℝ))) : False := by sorry
end D1Bound5
end
Source
D1Bound5/FiniteA5Interval.lean, lines 35-100, SHA-256 7c480f02b242d1351f6ce77e68792a0f2857b694eab69bf511843e090929a61f. Project: https://github.com/tomislav-math/d1-bound5-lean