File size: 3,018 Bytes
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 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 | import OAI.NumberTheory.Jacobsthal.Estimates.BoundaryGateSets
namespace OAI
namespace Erdos970
open scoped _root_.Erdos970
section
open _root_.MeasureTheory _root_.Set _root_.Filter
open scoped Topology
namespace ErdosContinuousBoundary
open NumberTheoryLean.FinitePathGeometry
noncomputable def truncatedReference (N : ℕ) (i : Side) (r b : ℝ) : ℝ :=
∑ n ∈ Finset.range N,(-1:ℝ)^n*boundaryTerm n i r b
theorem boundaryTerm_zero_of_index (n : ℕ) (hn : 2 ≤ n) (i : Side) {r b : ℝ}
(hr : r ≤ (n:ℝ)-1) : boundaryTerm n i r b=0 := by
have he : n=(n-2)+2 := by omega
rw [he]
apply boundaryTerm_depth_zero
have hc : ((n-2:ℕ):ℝ)=(n:ℝ)-2 := by rw [Nat.cast_sub hn]; norm_num
linarith
theorem truncatedReference_stable {N M : ℕ} (hN : 2 ≤ N) (hNM : N ≤ M)
(i : Side) {r b : ℝ} (hr : r ≤ (N:ℝ)-1) :
truncatedReference N i r b=truncatedReference M i r b := by
unfold truncatedReference
apply Finset.sum_subset (Finset.range_mono hNM)
intro n _hnM hnN
have hNn : N ≤ n := le_of_not_gt (by simpa only [Finset.mem_range] using hnN)
have hrn : r ≤ (n:ℝ)-1 := hr.trans (sub_le_sub_right (by exact_mod_cast hNn) 1)
rw [boundaryTerm_zero_of_index n (hN.trans hNn) i hrn,mul_zero]
noncomputable def referenceDepth (r : ℝ) : ℕ := ⌈max r 0⌉₊+3
theorem referenceDepth_bounds (r : ℝ) : 2 ≤ referenceDepth r ∧ r ≤ (referenceDepth r : ℝ)-1 := by
have h := Nat.le_ceil (max r 0)
have hr : r ≤ max r 0 := le_max_left _ _
unfold referenceDepth
constructor
· omega
· push_cast
linarith
noncomputable def boundaryReference (i : Side) (r b : ℝ) : ℝ :=
truncatedReference (referenceDepth r) i r b
theorem boundaryReference_eq_truncated (N : ℕ) (hN : 2 ≤ N) (i : Side)
{r b : ℝ} (hr : r ≤ (N:ℝ)-1) : boundaryReference i r b=truncatedReference N i r b := by
unfold boundaryReference
rcases le_total (referenceDepth r) N with h | h
· exact truncatedReference_stable (referenceDepth_bounds r).1 h i (referenceDepth_bounds r).2
· exact (truncatedReference_stable hN h i hr).symm
theorem truncatedReference_continuous (N : ℕ) (i : Side) :
Continuous (fun p : ℝ×ℝ => truncatedReference N i p.1 p.2) := by
apply continuous_finsetSum
intro n _hn
exact continuous_const.mul (boundaryTerm_continuous n i)
theorem boundaryReference_continuous (i : Side) :
Continuous (fun p : ℝ×ℝ => boundaryReference i p.1 p.2) := by
apply continuous_iff_continuousAt.mpr
intro p
let N := referenceDepth (p.1+1)
have hc := (truncatedReference_continuous N i).continuousAt (x:=p)
apply hc.congr_of_eventuallyEq
have he : ∀ᶠ q : ℝ×ℝ in 𝓝 p,q.1 < p.1+1 :=
continuous_fst.continuousAt.preimage_mem_nhds (Iio_mem_nhds (by linarith : p.1 < p.1+1))
filter_upwards [he] with q hq
exact boundaryReference_eq_truncated N (referenceDepth_bounds (p.1+1)).1 i
(hq.le.trans (referenceDepth_bounds (p.1+1)).2)
end ErdosContinuousBoundary
end
end Erdos970
end OAI
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