| import Mathlib |
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| namespace OAI |
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| section |
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| section |
| noncomputable section |
| open Filter MeasureTheory |
| open scoped Topology |
| namespace HardSphere.Bridge |
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| lemma power_div_identity {D : ℝ} (hD : 0 < D) (p : ℝ) : |
| D^p/D = D^(p-1) := by |
| rw [Real.rpow_sub hD,Real.rpow_one] |
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| lemma source_geometry_eventually {A b t : ℝ} (hA : 0 < A) (hb : 0 < b) (ht : 0 < t) : |
| ∀ᶠ D : ℝ in atTop, |
| 1 ≤ D ∧ A/D ≤ 1 ∧ (A/D)^2 ≤ t ∧ |
| (A/D)*D^(1/12:ℝ) ≤ 1 ∧ 2*A^2 ≤ D ∧ 8*A^2 ≤ D ∧ |
| 26*(A/D) ≤ t ∧ (13+Real.sqrt (2*t)*D^(1/12:ℝ))*(A/D) ≤ t ∧ |
| (D⁻¹/t)*13 ≤ (b*D^(-(1/3):ℝ)/8)/12 ∧ |
| 3*(1+D^(1/12:ℝ)/6) ≤ D^(1/12:ℝ) ∧ |
| A/D+2*(b*D^(-(1/3):ℝ)/8) < b*D^(-(1/3):ℝ) := by |
| have h1 : Tendsto (fun D : ℝ ↦ A/D) atTop (𝓝 0) := by |
| simpa [div_eq_mul_inv] using tendsto_inv_atTop_zero.const_mul A |
| have h2 : Tendsto (fun D : ℝ ↦ A*D^(-(11/12):ℝ)) atTop (𝓝 0) := by |
| simpa using (tendsto_rpow_neg_atTop (by norm_num : (0:ℝ) < 11/12)).const_mul A |
| have h3 : Tendsto (fun D : ℝ ↦ 13*(A/D)+Real.sqrt (2*t)*(A*D^(-(11/12):ℝ))) atTop (𝓝 0) := by |
| simpa using (h1.const_mul 13).add (h2.const_mul (Real.sqrt (2*t))) |
| have h4 : Tendsto (fun D : ℝ ↦ D^(-(2/3):ℝ)) atTop (𝓝 0) := |
| tendsto_rpow_neg_atTop (by norm_num) |
| have h5 : Tendsto (fun D : ℝ ↦ D^(1/12:ℝ)) atTop atTop := |
| tendsto_rpow_atTop (by norm_num) |
| filter_upwards [eventually_ge_atTop (1:ℝ),h1.eventually (gt_mem_nhds (by norm_num : (0:ℝ) < 1)), |
| (h1.pow 2).eventually (gt_mem_nhds (by simpa using ht)), |
| h2.eventually (gt_mem_nhds (by norm_num : (0:ℝ) < 1)), |
| eventually_ge_atTop (8*A^2), |
| (h1.const_mul 26).eventually (gt_mem_nhds (by simpa using ht)), |
| h3.eventually (gt_mem_nhds ht), |
| h4.eventually (gt_mem_nhds (by positivity : (0:ℝ) < b*t/(96*13))), |
| h5.eventually (eventually_ge_atTop (6:ℝ)), |
| h4.eventually (gt_mem_nhds (by positivity : (0:ℝ) < 3*b/(4*A)))] with D hD hd1 hd2 hd3 hd4 hd5 hd6 hd7 hd8 hd9 |
| have hDp : 0 < D := zero_lt_one.trans_le hD |
| have hp : (A/D)*D^(1/12:ℝ)=A*D^(-(11/12):ℝ) := by |
| calc |
| _ = A*(D^(1/12:ℝ)/D) := by ring |
| _ = _ := by rw [power_div_identity hDp]; norm_num |
| have hq : D⁻¹ = D^(-(2/3):ℝ)*D^(-(1/3):ℝ) := by |
| rw [←Real.rpow_add hDp] |
| norm_num [Real.rpow_neg_one] |
| refine ⟨hD,hd1.le,hd2.le,by rw [hp]; exact hd3.le,by nlinarith,hd4,hd5.le,?_,?_,by linarith,?_⟩ |
| · have he : (13+Real.sqrt (2*t)*D^(1/12:ℝ))*(A/D)= |
| 13*(A/D)+Real.sqrt (2*t)*(A*D^(-(11/12):ℝ)) := by rw [←hp]; ring |
| rw [he] |
| exact hd6.le |
| · have hh := (lt_div_iff₀ (by norm_num : (0:ℝ) < 96*13)).mp hd7 |
| rw [hq] |
| rw [div_mul_eq_mul_div] |
| apply (div_le_iff₀ ht).mpr |
| have hmul := mul_le_mul_of_nonneg_right hh.le (by positivity : 0 ≤ D^(-(1/3):ℝ)) |
| nlinarith |
| · have hh := (lt_div_iff₀ (by positivity : (0:ℝ) < 4*A)).mp hd9 |
| have hmul := mul_lt_mul_of_pos_right hh (by positivity : 0 < D^(-(1/3):ℝ)) |
| rw [div_eq_mul_inv,hq] |
| nlinarith |
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|
| lemma source_trial_geometry {A b t : ℝ} (hA : 0 < A) (hb : 0 < b) (ht : 0 < t) : |
| ∀ᶠ D : ℝ in atTop, ∀α : ℝ,0 < α → α ≤ A → |
| let r := α/D |
| let n := ⌈t/r^2⌉₊ |
| 1 ≤ D ∧ 0 < n ∧ r ≤ 1 ∧ r*D^(1/12:ℝ) ≤ 1 ∧ |
| t/(n:ℝ) ≤ r^2 ∧ (n:ℝ)*r^2 ≤ 2*t ∧ |
| 0 ≤ D⁻¹/t ∧ D⁻¹/t ≤ 1 ∧ |
| 2*(t/(n:ℝ)) ≤ D⁻¹ ∧ 8*t ≤ (n:ℝ)*D⁻¹ ∧ |
| 2*13/(n:ℝ) ≤ r ∧ (13+Real.sqrt (2*t)*D^(1/12:ℝ))/(n:ℝ) ≤ r ∧ |
| (D⁻¹/t)*13 ≤ (b*D^(-(1/3):ℝ)/8)/12 ∧ |
| 3*(1+D^(1/12:ℝ)/6) ≤ D^(1/12:ℝ) ∧ |
| r+2*(b*D^(-(1/3):ℝ)/8) < b*D^(-(1/3):ℝ) := by |
| filter_upwards [source_geometry_eventually hA hb ht,eventually_ge_atTop t⁻¹] with D h hd |
| rcases h with ⟨hD,hA1,hAsq,hAR,hD2,hD8,h26,hQ,hwin,hrelax,hiso⟩ |
| intro α hα hαA |
| dsimp only |
| let r := α/D |
| let n := ⌈t/r^2⌉₊ |
| have hDp : 0 < D := zero_lt_one.trans_le hD |
| have hr : 0 < r := by dsimp [r]; positivity |
| have hrA : r ≤ A/D := div_le_div_of_nonneg_right hαA hDp.le |
| have hn : 0 < n := Nat.ceil_pos.mpr (by positivity : 0 < t/r^2) |
| have hnp : 0 < (n:ℝ) := by exact_mod_cast hn |
| have hnr : t ≤ (n:ℝ)*r^2 := by |
| have hh := (div_le_iff₀ (sq_pos_of_pos hr)).mp (Nat.le_ceil (t/r^2)) |
| exact hh |
| have hτ : t/(n:ℝ) ≤ r^2 := (div_le_iff₀ hnp).mpr (by nlinarith [hnr]) |
| have hrs : r^2 ≤ t := (pow_le_pow_left₀ hr.le hrA 2).trans hAsq |
| have hup : (n:ℝ)*r^2 ≤ 2*t := by |
| have hh := mul_le_mul_of_nonneg_right (Nat.ceil_lt_add_one (by positivity : 0 ≤ t/r^2)).le (sq_nonneg r) |
| have he : (t/r^2+1)*r^2=t+r^2 := by field_simp |
| rw [he] at hh |
| nlinarith |
| have hsq : r^2 ≤ D⁻¹/8 := by |
| calc |
| _ ≤ (A/D)^2 := pow_le_pow_left₀ hr.le hrA 2 |
| _ ≤ (1/8)/D := by |
| apply (le_div_iff₀ hDp).mpr |
| have he : (A/D)^2*D=A^2/D := by field_simp |
| rw [he] |
| exact (div_le_iff₀ hDp).mpr (by linarith) |
| _ = _ := by ring |
| have htime : D⁻¹/t ≤ 1 := by |
| apply (div_le_iff₀ ht).mpr |
| simp only [one_mul] |
| rw [←one_div] |
| apply (div_le_iff₀ hDp).mpr |
| have hh := mul_le_mul_of_nonneg_left hd ht.le |
| simpa [mul_inv_cancel₀ ht.ne'] using hh |
| have hsmall : r*D^(1/12:ℝ) ≤ 1 := (mul_le_mul_of_nonneg_right hrA (by positivity)).trans hAR |
| have hdr : 26/(n:ℝ) ≤ r := by |
| apply (div_le_iff₀ hnp).mpr |
| have hh : 26*r ≤ t := (mul_le_mul_of_nonneg_left hrA (by norm_num)).trans h26 |
| nlinarith |
| have hdr' : (13+Real.sqrt (2*t)*D^(1/12:ℝ))/(n:ℝ) ≤ r := by |
| apply (div_le_iff₀ hnp).mpr |
| have hq : 0 ≤ 13+Real.sqrt (2*t)*D^(1/12:ℝ) := by positivity |
| have hh := (mul_le_mul_of_nonneg_left hrA hq).trans hQ |
| nlinarith |
| refine ⟨hD,hn,hrA.trans hA1,hsmall,hτ,hup,by positivity,htime,?_,?_,?_,hdr',hwin,hrelax,?_⟩ |
| · change 2*(t/(n:ℝ)) ≤ D⁻¹ |
| nlinarith [inv_pos.mpr hDp] |
| · change 8*t ≤ (n:ℝ)*D⁻¹ |
| have hh := mul_le_mul_of_nonneg_left hsq hnp.le |
| nlinarith |
| · change 2*13/(n:ℝ) ≤ r |
| convert hdr using 1 |
| norm_num |
| · change r+2*(b*D^(-(1/3):ℝ)/8) < b*D^(-(1/3):ℝ) |
| linarith |
|
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| end HardSphere.Bridge |
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| end |
| end |
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| end |
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| end OAI |
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