| import Mathlib |
| import OAI.Analysis.Conductivity.Variational.CrossingProfiles |
|
|
| namespace OAI |
|
|
| noncomputable section |
| namespace ScalarConductivity |
| open Real Set Filter Topology MeasureTheory |
|
|
| lemma cutoff_exp_hasDerivAt {χ : ℝ → ℝ} |
| (hχ : ContDiff ℝ (↑(⊤ : ℕ∞)) χ) (a z : ℝ) : |
| HasDerivAt (fun z => χ z * exp (-a*z)) |
| ((deriv χ z-a*χ z)*exp (-a*z)) z := by |
| have hd := (hχ.differentiable (by simp) z).hasDerivAt |
| have he := ((hasDerivAt_id z).const_mul (-a)).exp |
| exact (hd.mul he).congr_deriv |
| (g' := (deriv χ z-a*χ z)*exp (-a*z)) (by simp only [id_eq,mul_one]; ring) |
|
|
| lemma cutoff_exp_residual {χ : ℝ → ℝ} |
| (hχ : ContDiff ℝ (↑(⊤ : ℕ∞)) χ) (a z : ℝ) : |
| deriv (deriv (fun z => χ z * exp (-a*z))) z - a^2*(χ z * exp (-a*z)) = |
| (deriv (deriv χ) z-2*a*deriv χ z)*exp (-a*z) := by |
| have he : deriv (fun z => χ z * exp (-a*z)) = |
| fun z => (deriv χ z-a*χ z)*exp (-a*z) := |
| funext fun z => (cutoff_exp_hasDerivAt hχ a z).deriv |
| rw [he] |
| have hd := ((smooth_deriv_infty hχ).differentiable (by simp) z).hasDerivAt |
| have hd₁ := (hχ.differentiable (by simp) z).hasDerivAt |
| have hd₂ := ((hd.sub (hd₁.const_mul a)).mul (((hasDerivAt_id z).const_mul (-a)).exp)) |
| simp only [Pi.mul_def,Pi.sub_def,id_eq,mul_one] at hd₂ |
| rw [hd₂.deriv] |
| ring |
|
|
| lemma crossing_cutoff_derivative_bounds {M L : ℝ} (hM0 : 1 ≤ M) (hL : 1 ≤ L) |
| (hM : ∀ x : ℝ, |deriv smoothTransition x| ≤ M ∧ |
| |deriv (deriv smoothTransition) x| ≤ M) (z : ℝ) : |
| |deriv (crossingOn L) z| ≤ 4*M ∧ |
| |deriv (deriv (crossingOn L)) z| ≤ 16*M ∧ |
| |deriv (crossingOff L) z| ≤ 4*M ∧ |
| |deriv (deriv (crossingOff L)) z| ≤ 16*M := by |
| have hLp : 0 < L := lt_of_lt_of_le (by norm_num) hL |
| have hq : |4/L| ≤ 4 := by |
| rw [abs_of_pos (div_pos (by norm_num) hLp)] |
| exact (div_le_iff₀ hLp).mpr (by linarith) |
| have hq₂ : |(4/L)^2| ≤ 16 := by |
| rw [abs_pow] |
| nlinarith [abs_nonneg (4/L)] |
| rw [crossingOn_second,crossingOff_second,deriv_crossingOn,deriv_crossingOff] |
| simp only [abs_mul,abs_neg] |
| constructor |
| · rw [mul_comm, mul_comm 4 M] |
| exact mul_le_mul (hM _).1 hq (abs_nonneg _) (zero_le_one.trans hM0) |
| · exact ⟨mul_le_mul hq₂ (hM _).2 (abs_nonneg _) (by norm_num), |
| mul_le_mul hq (hM _).1 (abs_nonneg _) (by norm_num), |
| mul_le_mul hq₂ (hM _).2 (abs_nonneg _) (by norm_num)⟩ |
|
|
| def crossingResidualFirst (L z : ℝ) : ℝ := |
| deriv (deriv (crossingFirst L)) z - 9*crossingFirst L z |
|
|
| def crossingResidualSecond (L z : ℝ) : ℝ := |
| deriv (deriv (crossingSecond L)) z - crossingSecond L z |
|
|
| lemma crossingResidualFirst_formula (L z : ℝ) : |
| crossingResidualFirst L z = |
| (deriv (deriv (crossingOff L)) z-6*deriv (crossingOff L) z)*exp (-3*z) := by |
| unfold crossingResidualFirst crossingFirst |
| simpa only [show (3:ℝ)^2=9 by norm_num,show (2:ℝ)*3=6 by norm_num] using |
| cutoff_exp_residual (crossingOff_smooth L) 3 z |
| lemma crossingResidualSecond_formula (L z : ℝ) : |
| crossingResidualSecond L z = |
| (deriv (deriv (crossingOn L)) z-2*deriv (crossingOn L) z)*exp (-z) := by |
| unfold crossingResidualSecond crossingSecond |
| simpa only [one_pow,one_mul,mul_one,neg_one_mul] using |
| cutoff_exp_residual (crossingOn_smooth L) 1 z |
|
|
| lemma crossingResidualFirst_normalized (L z : ℝ) : |
| crossingResidualFirst L z / exp (-z) = |
| (deriv (deriv (crossingOff L)) z-6*deriv (crossingOff L) z)*exp (-2*z) := by |
| rw [crossingResidualFirst_formula,mul_div_assoc,←exp_sub] |
| congr 2 |
| ring |
| lemma crossingResidualSecond_normalized (L z : ℝ) : |
| crossingResidualSecond L z / exp (-3*z) = |
| (deriv (deriv (crossingOn L)) z-2*deriv (crossingOn L) z)*exp (2*z) := by |
| rw [crossingResidualSecond_formula,mul_div_assoc,←exp_sub] |
| congr 2 |
| ring |
|
|
| lemma crossingResidualFirst_zero {L z : ℝ} (hL : 0 < L) (hz : z < L/2) : |
| crossingResidualFirst L z = 0 := by |
| have hu : 4*z/L-2 < 0 := by |
| have : 4*z/L < 2 := (div_lt_iff₀ hL).mpr (by linarith) |
| linarith |
| rw [crossingResidualFirst_formula,crossingOff_second] |
| simp only [deriv_crossingOff, |
| (smoothTransition_flat_left hu).1,(smoothTransition_flat_left hu).2] |
| ring |
| lemma crossingResidualSecond_zero {L z : ℝ} (hL : 0 < L) (hz : -L/2 < z) : |
| crossingResidualSecond L z = 0 := by |
| have hu : 1 < 4*z/L+3 := by |
| have : -2 < 4*z/L := (lt_div_iff₀ hL).mpr (by linarith) |
| linarith |
| rw [crossingResidualSecond_formula,crossingOn_second] |
| simp only [deriv_crossingOn, |
| (smoothTransition_flat_right hu).1,(smoothTransition_flat_right hu).2] |
| ring |
|
|
| lemma crossing_residual_bounds {M L : ℝ} (hM0 : 1 ≤ M) (hL : 1 ≤ L) |
| (hM : ∀ x : ℝ, |deriv smoothTransition x| ≤ M ∧ |
| |deriv (deriv smoothTransition) x| ≤ M) (z : ℝ) : |
| |crossingResidualFirst L z / exp (-z)| ≤ 40*M*exp (-L) ∧ |
| |crossingResidualSecond L z / exp (-3*z)| ≤ 40*M*exp (-L) := by |
| have hLp : 0 < L := lt_of_lt_of_le (by norm_num) hL |
| have hMp : 0 ≤ M := le_trans (by norm_num) hM0 |
| obtain ⟨hon₁,hon₂,hoff₁,hoff₂⟩ := crossing_cutoff_derivative_bounds hM0 hL hM z |
| constructor |
| · by_cases hz : z < L/2 |
| · rw [crossingResidualFirst_zero hLp hz,zero_div,abs_zero] |
| positivity |
| · rw [crossingResidualFirst_normalized,abs_mul,abs_of_pos (exp_pos _)] |
| have he : exp (-2*z) ≤ exp (-L) := exp_le_exp.mpr (by linarith) |
| have hr : |deriv (deriv (crossingOff L)) z-6*deriv (crossingOff L) z| ≤ 40*M := by |
| have := abs_sub (deriv (deriv (crossingOff L)) z) (6*deriv (crossingOff L) z) |
| rw [abs_mul,abs_of_pos (by norm_num : (0:ℝ)<6)] at this |
| linarith |
| exact mul_le_mul hr he (exp_pos _).le (by positivity) |
| · by_cases hz : -L/2 < z |
| · rw [crossingResidualSecond_zero hLp hz,zero_div,abs_zero] |
| positivity |
| · rw [crossingResidualSecond_normalized,abs_mul,abs_of_pos (exp_pos _)] |
| have he : exp (2*z) ≤ exp (-L) := exp_le_exp.mpr (by linarith) |
| have hr : |deriv (deriv (crossingOn L)) z-2*deriv (crossingOn L) z| ≤ 40*M := by |
| have := abs_sub (deriv (deriv (crossingOn L)) z) (2*deriv (crossingOn L) z) |
| rw [abs_mul,abs_of_pos (by norm_num : (0:ℝ)<2)] at this |
| linarith |
| exact mul_le_mul hr he (exp_pos _).le (by positivity) |
|
|
| lemma crossing_mode_ratios {L z : ℝ} : |
| (z ≤ 0 → |crossingSecond L z / exp (-3*z)| ≤ 1) ∧ |
| (0 ≤ z → |crossingFirst L z / exp (-z)| ≤ 1) := by |
| constructor |
| · intro hz |
| rw [crossingSecond,mul_div_assoc,←exp_sub,abs_mul, |
| abs_of_nonneg (crossingOn_bounds L z).1,abs_of_pos (exp_pos _)] |
| have he : exp (-z- -3*z) ≤ 1 := exp_le_one_iff.mpr (by linarith) |
| exact (mul_le_mul_of_nonneg_left he (crossingOn_bounds L z).1).trans |
| (by simpa using (crossingOn_bounds L z).2) |
| · intro hz |
| rw [crossingFirst,mul_div_assoc,←exp_sub,abs_mul, |
| abs_of_nonneg (crossingOff_bounds L z).1,abs_of_pos (exp_pos _)] |
| have he : exp (-3*z- -z) ≤ 1 := exp_le_one_iff.mpr (by linarith) |
| exact (mul_le_mul_of_nonneg_left he (crossingOff_bounds L z).1).trans |
| (by simpa using (crossingOff_bounds L z).2) |
|
|
| lemma exists_small_crossing_scale {M : ℝ} (hM : 1 ≤ M) : |
| ∃ L : ℝ, 1 ≤ L ∧ 40*M*exp (-L) ≤ 1/56 := by |
| refine ⟨2240*M+1,by linarith,?_⟩ |
| rw [exp_neg,←div_eq_mul_inv,div_le_iff₀ (exp_pos _)] |
| have := add_one_le_exp (2240*M+1) |
| linarith |
|
|
| end ScalarConductivity |
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| end |
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| end OAI |
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