| import OAI.NumberTheory.Ostmann.Conclusion.BadFraction |
| import OAI.NumberTheory.Ostmann.Conclusion.FinalRateAbsorptionScales |
| import OAI.NumberTheory.Ostmann.Conclusion.SymmetrizedNorm |
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|
| namespace OAI |
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|
| noncomputable section |
| namespace Ostmann.Conclusion |
| open Filter |
| open scoped Topology |
|
|
| theorem final_energy_absorption_of_bounds |
| (R0 V E F A D B r m : ℝ) (hR0 : 0 ≤ R0) (hF : 0 ≤ F) (hr : 0 ≤ r) (hm : 0 ≤ m) |
| (hcard : 2*V+1 ≤ 3*Real.exp E) |
| (hbad : E+A+D ≤ -(2*B+3)*r*m) |
| (hpre : 3*R0*(F+1) ≤ Real.exp (r*m)) : |
| R0*(2*V+1)*(F*Real.exp A*Real.exp D+Real.exp (-(E+(2*B+3)*r*m))) ≤ |
| Real.exp (-(2*B+1)*r*m) := by |
| have hid1 : Real.exp E*(F*Real.exp A*Real.exp D)=F*Real.exp (E+A+D) := by |
| rw [Real.exp_add,Real.exp_add] |
| ring |
| have hid2 : Real.exp E*Real.exp (-(E+(2*B+3)*r*m))=Real.exp (-(2*B+3)*r*m) := by |
| rw [← Real.exp_add] |
| congr 1 |
| ring |
| calc |
| _ ≤ R0*(3*Real.exp E)*(F*Real.exp A*Real.exp D+Real.exp (-(E+(2*B+3)*r*m))) := |
| mul_le_mul_of_nonneg_right (mul_le_mul_of_nonneg_left hcard hR0) (by positivity) |
| _ = 3*R0*(F*Real.exp (E+A+D)+Real.exp (-(2*B+3)*r*m)) := by |
| calc |
| _ = 3*R0*(Real.exp E*(F*Real.exp A*Real.exp D)+ |
| Real.exp E*Real.exp (-(E+(2*B+3)*r*m))) := by ring |
| _ = _ := by rw [hid1,hid2] |
| _ ≤ 3*R0*(F*Real.exp (-(2*B+3)*r*m)+Real.exp (-(2*B+3)*r*m)) := |
| mul_le_mul_of_nonneg_left |
| (add_le_add (mul_le_mul_of_nonneg_left (Real.exp_le_exp.mpr hbad) hF) le_rfl) |
| (by positivity) |
| _ = (3*R0*(F+1))*Real.exp (-(2*B+3)*r*m) := by ring |
| _ ≤ Real.exp (r*m)*Real.exp (-(2*B+3)*r*m) := |
| mul_le_mul_of_nonneg_right hpre (Real.exp_pos _).le |
| _ = Real.exp (-(2*B+2)*r*m) := by |
| rw [← Real.exp_add] |
| congr 1 |
| ring |
| _ ≤ _ := Real.exp_le_exp.mpr (by nlinarith [mul_nonneg hr hm]) |
|
|
| theorem eventually_finalRate_absorption (Bs BD Bz B R0 C : ℝ) |
| (hBs : 0 ≤ Bs) (hBD : 0 ≤ BD) (hBz : 0 ≤ Bz) (hR0 : 0 < R0) |
| {k : ℕ} (hk : 2 ≤ k) |
| (hrate : finalRate BD Bs Bz (C+2) k < -2*B-4) : |
| ∀ᶠ L : ℝ in atTop, |
| R0*(2*(frequencyBound Bs BD Bz k L k : ℝ)+1)* |
| (((2 : ℝ)^k)^(2*(2^k))* |
| Real.exp ((2-(3/4 : ℝ)*Real.log ((2 : ℝ)^k))*(2 : ℝ)^k*(bulkSize k L : ℝ))* |
| Real.exp ((2 : ℝ)^k*(initialGap Bs k L+C*(bulkSize k L : ℝ))+(bulkSize k L : ℝ))+ |
| Real.exp (-(frequencyBudget Bs BD Bz k L k+(2*B+3)*(2 : ℝ)^k*(bulkSize k L : ℝ)))) ≤ |
| Real.exp (-(2*B+1)*(2 : ℝ)^k*(bulkSize k L : ℝ)) := by |
| have hk1 : 1 ≤ k := by omega |
| filter_upwards [eventually_final_bad_exponent hBs hBD hBz B C hk1 hrate, eventually_final_prefactor R0 hk1] with L hbad hpre |
| exact final_energy_absorption_of_bounds R0 (frequencyBound Bs BD Bz k L k) |
| (frequencyBudget Bs BD Bz k L k) (((2 : ℝ)^k)^(2*(2^k))) |
| ((2-(3/4 : ℝ)*Real.log ((2 : ℝ)^k))*(2 : ℝ)^k*(bulkSize k L : ℝ)) |
| ((2 : ℝ)^k*(initialGap Bs k L+C*(bulkSize k L : ℝ))+(bulkSize k L : ℝ)) |
| B ((2 : ℝ)^k) (bulkSize k L) hR0.le (by positivity) (by positivity) (by positivity) |
| (frequency_card_le_exp hBs hBD hBz hk1 L k) hbad hpre |
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| end Ostmann.Conclusion |
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| end |
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| end OAI |
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