| import OAI.NumberTheory.Jacobsthal.Estimates.GateIterates |
| |
| namespace OAI |
|
|
| namespace Erdos970 |
| open scoped _root_.Erdos970 |
|
|
| section |
|
|
| open _root_.MeasureTheory _root_.Set |
| namespace ErdosContinuousOmission |
| open ErdosContinuousBoundary NumberTheoryLean.FinitePathGeometry |
|
|
| theorem gate_truncatedReference (N : ℕ) (i : Side) (r b : ℝ) : |
| gate (truncatedReference N) i r b= |
| ∑ n ∈ Finset.range N,(-1:ℝ)^n*boundaryTerm (n+1) i r b := by |
| change gate (fun j s t => ∑ n ∈ Finset.range N,(-1:ℝ)^n*boundaryTerm n j s t) i r b=_ |
| rw [gate_sum N (fun n j s t => (-1:ℝ)^n*boundaryTerm n j s t) |
| (fun n j => continuous_const.mul (boundaryTerm_continuous n j))] |
| apply Finset.sum_congr rfl |
| intro n _hn |
| rw [gate_mul] |
| rfl |
|
|
| theorem truncatedReference_recurrence (N : ℕ) (i : Side) (r b : ℝ) : |
| truncatedReference (N+1) i r b=1-gate (truncatedReference N) i r b := by |
| have hfirst : truncatedReference (N+1) i r b= |
| (∑ n ∈ Finset.range N,(-1:ℝ)^(n+1)*boundaryTerm (n+1) i r b)+1 := by |
| rw [truncatedReference,Finset.sum_range_succ'] |
| simp only [pow_zero,show boundaryTerm 0 i r b=(1:ℝ) by rfl,one_mul] |
| have hs (n : ℕ) : (-1:ℝ)^(n+1)*boundaryTerm (n+1) i r b= |
| -((-1:ℝ)^n*boundaryTerm (n+1) i r b) := by rw [pow_succ]; ring |
| rw [hfirst] |
| simp_rw [hs] |
| rw [Finset.sum_neg_distrib,← gate_truncatedReference] |
| ring |
| |
| theorem boundaryReference_recurrence (i : Side) (r b : ℝ) : |
| boundaryReference i r b=1-gate boundaryReference i r b := by |
| let N := referenceDepth r |
| have hN : 2 ≤ N := (referenceDepth_bounds r).1 |
| have hrN : r ≤ (N:ℝ)-1 := (referenceDepth_bounds r).2 |
| have hq : boundaryReference i r b=truncatedReference (N+1) i r b := |
| boundaryReference_eq_truncated (N+1) (by omega) i (by push_cast; linarith) |
| have hchild : gate boundaryReference i r b=gate (truncatedReference N) i r b := by |
| unfold gate |
| apply intervalIntegral.integral_congr |
| intro x hx |
| rw [uIcc_of_le (upperCutoff_bounds i r b).1] at hx |
| dsimp only |
| rw [boundaryReference_eq_truncated N hN i.flip (by linarith [hx.1] : r-x ≤ (N:ℝ)-1)] |
| rw [hq,truncatedReference_recurrence,← hchild] |
| |
| end ErdosContinuousOmission |
| |
| end |
| |
| end Erdos970 |
| |
| end OAI |
| |
