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f89a930 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 | 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
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