| import OAI.Geometry.CylinderCovering.Ruled.CellsDefinitions |
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| namespace OAI |
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| noncomputable section |
| open Set MeasureTheory |
| open |
| namespace |
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| lemma ℝ |
| exact |
| ((isClosed_le |
| (isClosed_le |
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| lemma ℝ |
| exact |
| ((isOpen_lt |
| (isOpen_lt |
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| lemma ℝ |
| closedCell ⊆ |
| rintro ⟨⟩ ⟨⟩ |
| exact ⟨⟨⟩ |
| ⟨(abs_le.mp ⟩⟩ |
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| lemma ℝ |
| isCompact_Icc.of_isClosed_subset |
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| lemma ℝ ⊆ |
| rintro ⟨⟩ |
| exact ⟨⟩ |
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| lemma ℝ |
| (closedCell_isCompact |
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| lemma ℝ |
| dilation |
| ext |
| constructor |
| · rintro ⟨⟨⟩⟩ |
| dsimp |
| rw |
| refine ⟨ |
| mul_le_mul_of_nonneg_left ⟩ |
| calc || || |
| rw |
| abs_mul, abs_of_nonneg |
| _ ≤ m^2*B |
| · intro |
| refine ⟨/m,p.2/⟩ |
| · rcases ⟨⟩ |
| dsimp |
| rw |
| refine ⟨₀ |
| (div_le_iff₀ hm).mpr ⟩ |
| rw |
| apply ₀ |
| simpa |
| · ext ;> dsimp [dilation] <;> field_simp |
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| lemma dilation_openCell (m A B : ℝ) (hm : 0 < m) : |
| dilation m '' openCell A B = openCell (m*A) (m^2*B) := by |
| ext p |
| constructor |
| · rintro ⟨q,⟨hx,hy,hxy⟩,rfl⟩ |
| dsimp [openCell, dilation] at * |
| rw [abs_mul, abs_of_pos hm, abs_mul, abs_of_pos hm] |
| refine ⟨mul_lt_mul_of_pos_left hx hm, |
| mul_lt_mul_of_pos_left hy hm, ?_⟩ |
| calc |m*q.1*(m*q.2)| = m^2*|q.1*q.2| := by |
| rw [show m*q.1*(m*q.2)=m^2*(q.1*q.2) by ring, |
| abs_mul, abs_of_nonneg (sq_nonneg m)] |
| _ < m^2*B := mul_lt_mul_of_pos_left hxy (sq_pos_of_pos hm) |
| · intro hp |
| refine ⟨(p.1/m,p.2/m), ?_, ?_⟩ |
| · rcases hp with ⟨hx,hy,hxy⟩ |
| dsimp [openCell] |
| rw [abs_div,abs_div,abs_of_pos hm] |
| refine ⟨(div_lt_iff₀ hm).mpr (by simpa [mul_comm] using hx), |
| (div_lt_iff₀ hm).mpr (by simpa [mul_comm] using hy), ?_⟩ |
| rw [div_mul_div_comm,abs_div,abs_of_pos (mul_pos hm hm)] |
| apply (div_lt_iff₀ (mul_pos hm hm)).mpr |
| simpa [pow_two,mul_comm] using hxy |
| · ext <;> dsimp [dilation] <;> field_simp |
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| lemma logarithmicCell_eq_dilation (m : ℝ) (hm : 0 < m) : |
| logarithmicCell m = dilation m '' parameterCell (1/m^2) := by |
| rw [parameterCell, dilation_closedCell m 1 (1/m^2) hm] |
| simp [logarithmicCell, ne_of_gt hm] |
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| lemma logarithmicIntercept_eq_dilation (m gamma : ℝ) (hm : 0 < m) : |
| logarithmicIntercept m gamma = dilation m '' interceptCell (1/m^2) gamma := by |
| rw [interceptCell, dilation_openCell m 3 ((1+gamma)*(1/m^2)) hm] |
| have h : m^2*((1+gamma)*(1/m^2)) = 1+gamma := by field_simp |
| rw [h] |
| simp only [logarithmicIntercept,mul_comm] |
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| lemma opposite_expansion_bounds {z s₁ s₂ tau : ℝ} |
| (hz : |z| ≤ 1) (hs₁ : |s₁| ≤ 1) (hs₂ : |s₂| ≤ 1) |
| (hs : |s₁*s₂| ≤ tau) : |
| |(1+z)*s₁| ≤ 2 ∧ |(1-z)*s₂| ≤ 2 ∧ |
| |((1+z)*s₁)*((1-z)*s₂)| ≤ tau := by |
| have hz' := abs_le.mp hz |
| have hplus : |1+z| ≤ 2 := by rw [abs_le]; constructor <;> linarith |
| have hminus : |1-z| ≤ 2 := by rw [abs_le]; constructor <;> linarith |
| have hcoef : 0 ≤ 1-z^2 := by nlinarith [mul_nonneg (by linarith : 0 ≤ 1-z) (by linarith : 0 ≤ 1+z)] |
| refine ⟨?_,?_,?_⟩ |
| · rw [abs_mul] |
| nlinarith [mul_le_mul hplus hs₁ (abs_nonneg s₁) (by norm_num : (0:ℝ) ≤ 2)] |
| · rw [abs_mul] |
| nlinarith [mul_le_mul hminus hs₂ (abs_nonneg s₂) (by norm_num : (0:ℝ) ≤ 2)] |
| · rw [show ((1+z)*s₁)*((1-z)*s₂)=(1-z^2)*(s₁*s₂) by ring, |
| abs_mul, abs_of_nonneg hcoef] |
| nlinarith [mul_nonneg (sq_nonneg z) (abs_nonneg (s₁*s₂))] |
| |
| lemma perturbed_product_bounds {u₁ u₂ e₁ e₂ eps tau gamma : ℝ} |
| (hu₁ : |u₁| ≤ 2) (hu₂ : |u₂| ≤ 2) (hu : |u₁*u₂| ≤ tau) |
| (he₁ : |e₁| ≤ eps) (he₂ : |e₂| ≤ eps) |
| (heps : eps < 1/2) (hgap : eps < gamma*tau/5) : |
| |u₁+e₁| < 3 ∧ |u₂+e₂| < 3 ∧ |
| |(u₁+e₁)*(u₂+e₂)| < (1+gamma)*tau := by |
| have heps0 : 0 ≤ eps := (abs_nonneg e₁).trans he₁ |
| have h₁ : |u₁*e₂| ≤ 2*eps := by |
| rw [abs_mul] |
| exact mul_le_mul hu₁ he₂ (abs_nonneg e₂) (by norm_num) |
| have h₂ : |e₁*u₂| ≤ 2*eps := by |
| rw [abs_mul] |
| nlinarith [mul_le_mul he₁ hu₂ (abs_nonneg u₂) heps0] |
| have h₃ : |e₁*e₂| ≤ eps^2 := by |
| rw [abs_mul,pow_two] |
| exact mul_le_mul he₁ he₂ (abs_nonneg e₂) heps0 |
| refine ⟨lt_of_le_of_lt (abs_add_le u₁ e₁) (by linarith), |
| lt_of_le_of_lt (abs_add_le u₂ e₂) (by linarith), ?_⟩ |
| have hsum : |(u₁+e₁)*(u₂+e₂)| ≤ tau+4*eps+eps^2 := by |
| calc |(u₁+e₁)*(u₂+e₂)| |
| = |((u₁*u₂+u₁*e₂)+e₁*u₂)+e₁*e₂| := by congr 1; ring |
| _ ≤ |u₁*u₂+u₁*e₂|+|e₁*u₂|+|e₁*e₂| := by |
| have h := abs_add_le (u₁*u₂+u₁*e₂) (e₁*u₂) |
| have h' := abs_add_le ((u₁*u₂+u₁*e₂)+e₁*u₂) (e₁*e₂) |
| linarith |
| _ ≤ (|u₁*u₂|+|u₁*e₂|)+|e₁*u₂|+|e₁*e₂| := by |
| gcongr; exact abs_add_le _ _ |
| _ ≤ tau+4*eps+eps^2 := by linarith |
| have hsq : eps^2 ≤ eps := by nlinarith |
| nlinarith |
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| lemma expanded_parameterCell_mem_intercept {z tau gamma eps : ℝ} {s e : Plane} |
| (hz : |z| ≤ 1) (hs : s ∈ parameterCell tau) |
| (he₁ : |e.1| ≤ eps) (he₂ : |e.2| ≤ eps) |
| (heps : eps < 1/2) (hgap : eps < gamma*tau/5) : |
| ((1+z)*s.1+e.1, (1-z)*s.2+e.2) ∈ interceptCell tau gamma := by |
| rcases opposite_expansion_bounds hz hs.1 hs.2.1 hs.2.2 with ⟨h₁,h₂,h₃⟩ |
| exact perturbed_product_bounds h₁ h₂ h₃ he₁ he₂ heps hgap |
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| lemma quadrant_integral (A B : ℝ) (hA : 0 < A) (hB : 0 < B) : |
| (∫ _ in (0:ℝ)..(B/A), A) + (∫ x in (B/A)..A, B/x) = |
| B*(1+Real.log (A^2/B)) := by |
| have hBA : 0 < B/A := div_pos hB hA |
| have hquot : A/(B/A) = A^2/B := by field_simp |
| have hf : (fun x : ℝ => B/x) = (fun x : ℝ => B*x⁻¹) := by |
| ext x; exact div_eq_mul_inv B x |
| rw [hf, intervalIntegral.integral_const_mul, integral_inv_of_pos hBA hA, |
| intervalIntegral.integral_const, hquot] |
| simp only [sub_zero,smul_eq_mul] |
| field_simp |
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| lemma areaRatio_exp (gamma t : ℝ) (hg : 0 < 1+gamma) (ht : 0 ≤ t) : |
| areaRatio (Real.exp (-t)) gamma = |
| (1+gamma) + (1+gamma)*Real.log (9/(1+gamma))/(1+t) := by |
| have hc : 1+gamma ≠ 0 := ne_of_gt hg |
| have hlog₁ : Real.log (1 / Real.exp (-t)) = t := by |
| rw [Real.log_div (by norm_num) (Real.exp_ne_zero _), Real.log_one, Real.log_exp] |
| ring |
| have hlog₂ : Real.log (9/((1+gamma)*Real.exp (-t))) = |
| Real.log (9/(1+gamma))+t := by |
| rw [Real.log_div (by norm_num) (mul_ne_zero hc (Real.exp_ne_zero _)), |
| Real.log_mul hc (Real.exp_ne_zero _), Real.log_exp, |
| Real.log_div (by norm_num) hc] |
| ring |
| unfold areaRatio |
| rw [hlog₁,hlog₂] |
| have hden : 1+t ≠ 0 := by linarith |
| field_simp |
| ring |
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| lemma areaRatio_exp_tendsto (gamma : ℝ) (hg : 0 < 1+gamma) : |
| Tendsto (fun t : ℝ => areaRatio (Real.exp (-t)) gamma) |
| atTop (𝓝 (1+gamma)) := by |
| have hinv : Tendsto (fun t : ℝ => (1+t)⁻¹) atTop (𝓝 0) := |
| tendsto_inv_atTop_zero.comp (tendsto_atTop_add_const_left atTop 1 tendsto_id) |
| have hlim : Tendsto (fun t : ℝ => |
| (1+gamma)+(1+gamma)*Real.log (9/(1+gamma))/(1+t)) |
| atTop (𝓝 (1+gamma)) := by |
| simpa [div_eq_mul_inv] using |
| (tendsto_const_nhds.add (tendsto_const_nhds.mul hinv)) |
| apply hlim.congr' |
| filter_upwards [eventually_ge_atTop (0:ℝ)] with t ht |
| exact (areaRatio_exp gamma t hg ht).symm |
| |
| lemma areaRatio_tendsto_zero (gamma : ℝ) (hg : 0 ≤ gamma) : |
| Tendsto (fun tau : ℝ => areaRatio tau gamma) (𝓝[>] 0) (𝓝 (1+gamma)) := by |
| have hc : 0 < 1+gamma := by linarith |
| have hlog : Tendsto (fun tau : ℝ => -(Real.log tau)) (𝓝[>] 0) atTop := |
| tendsto_neg_atBot_atTop.comp Real.tendsto_log_nhdsGT_zero |
| have hlim := (areaRatio_exp_tendsto gamma hc).comp hlog |
| apply hlim.congr' |
| filter_upwards [self_mem_nhdsWithin] with tau htau |
| simp only [Function.comp_apply,neg_neg,Real.exp_log htau] |
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| lemma exists_parameter_after_padding {gamma eta : ℝ} |
| (hg : 0 ≤ gamma) (hge : gamma < eta) : |
| ∃ tau : ℝ, 0 < tau ∧ tau < 1 ∧ (1+gamma)*tau < 1 ∧ |
| areaRatio tau gamma < 1+eta := by |
| have hc : 0 < 1+gamma := by linarith |
| have hr := (areaRatio_exp_tendsto gamma hc).eventually_lt_const (by linarith : 1+gamma < 1+eta) |
| have ht := Real.tendsto_exp_neg_atTop_nhds_zero.eventually_lt_const |
| (show (0:ℝ) < 1 by norm_num) |
| have hp : Tendsto (fun t:ℝ => (1+gamma)*Real.exp (-t)) atTop (𝓝 0) := by |
| simpa using tendsto_const_nhds.mul Real.tendsto_exp_neg_atTop_nhds_zero |
| have hpt := hp.eventually_lt_const (show (0:ℝ) < 1 by norm_num) |
| obtain ⟨t,ht, hpt,hr⟩ := (ht.and (hpt.and hr)).exists |
| exact ⟨Real.exp (-t), Real.exp_pos _, ht,hpt,hr⟩ |
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| lemma areaRatio_continuousAt_padding_zero (tau : ℝ) (htau : 0 < tau) : |
| ContinuousAt (areaRatio tau) 0 := by |
| unfold areaRatio |
| have hn : (1+(0:ℝ))*tau ≠ 0 := by positivity |
| have hn' : (9:ℝ)/((1+(0:ℝ))*tau) ≠ 0 := div_ne_zero (by norm_num) hn |
| apply ContinuousAt.div_const |
| apply ContinuousAt.mul |
| · fun_prop |
| · apply ContinuousAt.const_add |
| have hd : ContinuousAt (fun gamma : ℝ => (9:ℝ)/((1+gamma)*tau)) 0 := by |
| apply ContinuousAt.div |
| · exact continuousAt_const |
| · fun_prop |
| · exact hn |
| exact hd.log hn' |
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| lemma exists_padding_after_parameter {tau eta : ℝ} |
| (htau : 0 < tau) (htau1 : tau < 1) |
| (hr : areaRatio tau 0 < 1+eta) : |
| ∃ gamma : ℝ, 0 < gamma ∧ (1+gamma)*tau < 1 ∧ |
| areaRatio tau gamma < 1+eta := by |
| have hR := (areaRatio_continuousAt_padding_zero tau htau).tendsto.eventually_lt_const hr |
| have hP : Tendsto (fun gamma:ℝ => (1+gamma)*tau) (𝓝 0) (𝓝 tau) := by |
| have hc : ContinuousAt (fun gamma : ℝ => (1+gamma)*tau) 0 := by fun_prop |
| simpa using hc.tendsto |
| have hP' := hP.eventually_lt_const htau1 |
| have hboth : ∀ᶠ gamma in 𝓝[>] (0:ℝ), |
| (1+gamma)*tau < 1 ∧ areaRatio tau gamma < 1+eta := |
| (hP'.and hR).filter_mono nhdsWithin_le_nhds |
| have hpos : ∀ᶠ gamma in 𝓝[>] (0:ℝ), 0 < gamma := self_mem_nhdsWithin |
| obtain ⟨gamma,hgamma,hrest⟩ := (hpos.and hboth).exists |
| exact ⟨gamma,hgamma,hrest⟩ |
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| lemma exists_logarithmic_parameters {eta : ℝ} (heta : 0 < eta) : |
| ∃ m : ℝ, 1 < m ∧ ∃ gamma : ℝ, 0 < gamma ∧ |
| (1+gamma)*(1/m^2) < 1 ∧ areaRatio (1/m^2) gamma < 1+eta := by |
| obtain ⟨tau,htau,htau1,_,hr⟩ := |
| exists_parameter_after_padding (gamma:=0) (by norm_num) heta |
| obtain ⟨gamma,hgamma,hpad,hratio⟩ := exists_padding_after_parameter htau htau1 hr |
| have hsqrt : 0 < Real.sqrt tau := Real.sqrt_pos.2 htau |
| have hsqrt1 : Real.sqrt tau < 1 := by |
| nlinarith [Real.sq_sqrt htau.le, Real.sqrt_nonneg tau] |
| have hm : 1 < 1/Real.sqrt tau := (lt_div_iff₀ hsqrt).mpr (by simpa using hsqrt1) |
| have ht : 1/(1/Real.sqrt tau)^2 = tau := by |
| calc 1/(1/Real.sqrt tau)^2 = (Real.sqrt tau)^2 := by field_simp |
| _ = tau := Real.sq_sqrt htau.le |
| refine ⟨1/Real.sqrt tau,hm,gamma,hgamma,?_,?_⟩ |
| · simpa only [ht] using hpad |
| · simpa only [ht] using hratio |
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| end ProductCells |
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| end |
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| end OAI |
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