| 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 |
|
|
