| import Mathlib |
| import OAI.NumberTheory.Jacobsthal.Probability.FirstHitKernels |
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| namespace OAI |
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| namespace Erdos970 |
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| section |
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| namespace ErdosEvenThreshold |
| open Set NumberTheoryLean.FinitePathGeometry |
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| noncomputable def thresholdSlack (R t : ℝ) : ℝ := |
| nextGap R t-max (2*nextExponent R t) (nextExponent R t+2) |
| noncomputable def thresholdStrip (R d : ℝ) : Set ℝ := |
| {t | 2 ≤ t ∧ 1 < nextExponent R t ∧ 0 ≤ thresholdSlack R t ∧ thresholdSlack R t ≤ d} |
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| theorem gap_eq_exponent_mul (R t : ℝ) : nextGap R t=nextExponent R t*t := by |
| unfold nextGap nextExponent |
| ring |
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| theorem exponent_mul_add_one {R t : ℝ} (ht : 0 < t+1) : nextExponent R t*(t+1)=R := by |
| unfold nextExponent |
| field_simp |
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| theorem strip_parameters {R d t : ℝ} (hd : d ≤ 1) (ht : t ∈ thresholdStrip R d) : |
| 4 < R ∧ t ≤ 4 := by |
| obtain ⟨ht2,hx,hlo,hhi⟩ := ht |
| have hcons := gap_exponent_conservation (r:=R) (show 0 < t by linarith) |
| have hm := gap_eq_exponent_mul R t |
| have hlow := le_max_right (2*nextExponent R t) (nextExponent R t+2) |
| have hmax : max (2*nextExponent R t) (nextExponent R t+2) ≤ 3*nextExponent R t := |
| max_le (by linarith) (by linarith) |
| unfold thresholdSlack at hlo hhi |
| constructor |
| · linarith |
| · have hprod : nextExponent R t*t ≤ nextExponent R t*4 := by nlinarith |
| exact le_of_mul_le_mul_left hprod (by linarith : 0 < nextExponent R t) |
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|
| noncomputable def firstUpper (R d : ℝ) : ℝ := 2+4*d/R |
| noncomputable def secondLower (R : ℝ) : ℝ := (R+2)/(R-2) |
| noncomputable def secondUpper (R d : ℝ) : ℝ := (R+2+d)/(R-2-d) |
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| theorem strip_mem_cover {R d t : ℝ} (hd0 : 0 ≤ d) (hd1 : d ≤ 1) |
| (ht : t ∈ thresholdStrip R d) : |
| t ∈ Icc 2 (firstUpper R d) ∪ Icc (secondLower R) (secondUpper R d) := by |
| have hR := (strip_parameters hd1 ht).1 |
| have hR0 : 0 < R := by linarith |
| obtain ⟨ht2,hx,hlo,hhi⟩ := ht |
| have htpos : 0 < t+1 := by linarith |
| have hxpos : 0 < nextExponent R t := by linarith |
| have hm := gap_eq_exponent_mul R t |
| have hxR := exponent_mul_add_one (R:=R) htpos |
| unfold thresholdSlack at hlo hhi |
| by_cases hb : nextExponent R t+2 ≤ 2*nextExponent R t |
| · rw [max_eq_left hb] at hlo hhi |
| have hs : nextExponent R t*(t-2) ≤ d := by nlinarith |
| have ht3 : t ≤ 3 := by |
| have hh : nextExponent R t*t ≤ nextExponent R t*3 := by nlinarith |
| exact le_of_mul_le_mul_left hh hxpos |
| have hbound : t-2 ≤ 4*d/R := by |
| apply (le_div_iff₀ hR0).mpr |
| calc |
| (t-2)*R=(nextExponent R t*(t-2))*(t+1) := by |
| conv_lhs => rw [← hxR] |
| ring |
| _ ≤ d*(t+1) := mul_le_mul_of_nonneg_right hs htpos.le |
| _ ≤ d*4 := mul_le_mul_of_nonneg_left (by linarith : t+1 ≤ 4) hd0 |
| _ = _ := by ring |
| exact Or.inl ⟨ht2,by unfold firstUpper; linarith⟩ |
| · have hb' : 2*nextExponent R t ≤ nextExponent R t+2 := (lt_of_not_ge hb).le |
| rw [max_eq_right hb'] at hlo hhi |
| have he : (nextGap R t-nextExponent R t)*(t+1)=R*(t-1) := by |
| rw [hm] |
| calc |
| _ = (nextExponent R t*(t+1))*(t-1) := by ring |
| _ = _ := by rw [hxR] |
| have hlow : 2*(t+1) ≤ (nextGap R t-nextExponent R t)*(t+1) := |
| mul_le_mul_of_nonneg_right (by linarith) htpos.le |
| have hhigh : (nextGap R t-nextExponent R t)*(t+1) ≤ (2+d)*(t+1) := |
| mul_le_mul_of_nonneg_right (by linarith) htpos.le |
| rw [he] at hlow hhigh |
| apply Or.inr |
| constructor |
| · apply (div_le_iff₀ (show 0 < R-2 by linarith)).mpr |
| nlinarith |
| · apply (le_div_iff₀ (show 0 < R-2-d by linarith)).mpr |
| nlinarith |
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|
| end ErdosEvenThreshold |
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| end |
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| end Erdos970 |
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| end OAI |
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