openAI-math-dataset / lean /OAI /Geometry /IsometricImmersion /Comparison /ComparisonEnergyScaling.lean
| import OAI.Geometry.IsometricImmersion.Comparison.ComparisonSourceBounds | |
| import OAI.Geometry.IsometricImmersion.Comparison.FiniteApproximationJets | |
| namespace OAI | |
| noncomputable section | |
| open Set Filter | |
| open scoped ContDiff Topology | |
| namespace SmoothLocal.Pulse | |
| open SmoothLocal.Geometry SmoothLocal.HighEquation | |
| def comparisonLowerSourceCoefficient (A Cfirst Ctest Ccompare Cres E Q a : ℝ) | |
| (ha : 0 < a) (delta : ℝ) : ℝ := | |
| 2*A*(testDensityErrorCoefficient Ctest a ha delta+Ccompare*E)+Cfirst*Q+Cres | |
| theorem comparisonSourceBudget_le_leading | |
| {A Cfirst Ctest Ccompare Cres E Q a delta tau epsilon epsilonQ : ℝ} | |
| (ha : 0 < a) (N : ℕ) (hA : 0 ≤ A) (hCf : 0 ≤ Cfirst) | |
| (hCc : 0 ≤ Ccompare) (hCr : 0 ≤ Cres) (ht : 1 ≤ tau) | |
| (he : epsilon ≤ E*tau/tau^N) (heQ : epsilonQ ≤ Q*tau/tau^N) | |
| (hlarge : comparisonLowerSourceCoefficient A Cfirst Ctest Ccompare Cres E Q a ha delta ≤ tau) : | |
| comparisonSourceBudget A Cfirst Ctest Ccompare Cres a ha N delta tau epsilon epsilonQ ≤ | |
| (2*A*pulseLeadingBound a ha+1)*tau^2/tau^N := by | |
| have ht0 : 0 < tau := zero_lt_one.trans_le ht | |
| have hp : 0 < tau^N := pow_pos ht0 N | |
| have hscale : 0 ≤ tau/tau^N := div_nonneg ht0.le hp.le | |
| have hrem : qForcingRemainderBudget A Cfirst Ctest Ccompare a ha N delta tau epsilon epsilonQ ≤ | |
| (2*A*(testDensityErrorCoefficient Ctest a ha delta+Ccompare*E)+Cfirst*Q)*tau/tau^N := by | |
| have hb := add_le_add | |
| (mul_le_mul_of_nonneg_left | |
| (add_le_add (le_refl (testDensityErrorCoefficient Ctest a ha delta*tau/tau^N)) | |
| (mul_le_mul_of_nonneg_left he hCc)) | |
| (mul_nonneg (by norm_num : (0 : ℝ) ≤ 2) hA)) | |
| (mul_le_mul_of_nonneg_left heQ hCf) | |
| change qForcingRemainderBudget A Cfirst Ctest Ccompare a ha N delta tau epsilon epsilonQ ≤ _ at hb | |
| convert hb using 1 | |
| ring | |
| have hres : Cres/tau^N ≤ Cres*tau/tau^N := by | |
| apply div_le_div_of_nonneg_right _ hp.le | |
| exact le_mul_of_one_le_right hCr ht | |
| calc | |
| _ ≤ 2*A*(pulseLeadingBound a ha*tau^2/tau^N)+ | |
| (2*A*(testDensityErrorCoefficient Ctest a ha delta+Ccompare*E)+Cfirst*Q)*tau/tau^N+ | |
| Cres*tau/tau^N := add_le_add (add_le_add le_rfl hrem) hres | |
| _ = 2*A*(pulseLeadingBound a ha*tau^2/tau^N)+ | |
| comparisonLowerSourceCoefficient A Cfirst Ctest Ccompare Cres E Q a ha delta*(tau/tau^N) := by | |
| unfold comparisonLowerSourceCoefficient | |
| ring | |
| _ ≤ 2*A*(pulseLeadingBound a ha*tau^2/tau^N)+tau*(tau/tau^N) := | |
| add_le_add le_rfl (mul_le_mul_of_nonneg_right hlarge hscale) | |
| _ = _ := by ring | |
| theorem approximationAccuracy_le_fixed_power {tau N : ℕ} (ht : 1 ≤ tau) (hN : N ≤ tau) : | |
| metricApproximationAccuracy tau ≤ 1/(tau : ℝ)^N := by | |
| have htR : (1 : ℝ) ≤ tau := by exact_mod_cast ht | |
| unfold metricApproximationAccuracy | |
| exact one_div_le_one_div_of_le (pow_pos (zero_lt_one.trans_le htR) N) | |
| (pow_le_pow_right₀ htR hN) | |
| theorem fixed_scaled_approximation_le_source_scale {tau N : ℕ} {E : ℝ} | |
| (hE : 0 ≤ E) (ht : 1 ≤ tau) (hN : N ≤ tau) : | |
| E*metricApproximationAccuracy tau ≤ E*(tau : ℝ)/(tau : ℝ)^N := by | |
| have htR : (1 : ℝ) ≤ tau := by exact_mod_cast ht | |
| have hp : 0 < (tau : ℝ)^N := pow_pos (zero_lt_one.trans_le htR) N | |
| calc | |
| _ ≤ E*(1/(tau : ℝ)^N) := mul_le_mul_of_nonneg_left (approximationAccuracy_le_fixed_power ht hN) hE | |
| _ = E/(tau : ℝ)^N := by ring | |
| _ ≤ _ := div_le_div_of_nonneg_right (le_mul_of_one_le_right hE htR) hp.le | |
| theorem comparison_exponential_tendsto_one (Lambda delta : ℝ) : | |
| Tendsto (fun tau : ℝ => Real.exp (Lambda*(2*delta/tau))) atTop (𝓝 1) := by | |
| have hzero : Tendsto (fun tau : ℝ => Lambda*(2*delta/tau)) atTop (𝓝 0) := by | |
| simpa only [id_eq, mul_div_assoc] using | |
| (tendsto_id.const_div_atTop (Lambda*(2*delta))) | |
| have he : Tendsto Real.exp (𝓝 (0 : ℝ)) (𝓝 (Real.exp 0)) := Real.continuous_exp.continuousAt.tendsto | |
| rw [Real.exp_zero] at he | |
| exact he.comp hzero | |
| theorem comparison_exponential_le_fixed {Lambda delta tau : ℝ} | |
| (ht : 0 < tau) (hlarge : Lambda*(2*delta) ≤ tau) : | |
| Real.exp (Lambda*(2*delta/tau)) ≤ Real.exp 1 := by | |
| apply Real.exp_le_exp.mpr | |
| rw [← mul_div_assoc] | |
| exact (div_le_one ht).mpr hlarge | |
| def comparisonGradientConstant (s0 width A a : ℝ) (ha : 0 < a) : ℝ := | |
| 2*Real.exp 1*Real.sqrt (2*(1+1/s0))*Real.sqrt (2*width)*(2*A*pulseLeadingBound a ha+1) | |
| theorem comparison_gradient_scaling | |
| {s0 width A a delta tau Lambda F : ℝ} (ha : 0 < a) (N : ℕ) | |
| (hdelta : 0 ≤ delta) (ht : 0 < tau) (hF : 0 ≤ F) | |
| (hsource : F ≤ (2*A*pulseLeadingBound a ha+1)*tau^2/tau^N) | |
| (hexp : Real.exp (Lambda*(2*delta/tau)) ≤ Real.exp 1) : | |
| Real.sqrt (2*(1+1/s0))* | |
| (Real.exp (Lambda*(2*delta/tau))*F*Real.sqrt (2*width)*(2*delta/tau)) ≤ | |
| comparisonGradientConstant s0 width A a ha*delta*tau/tau^N := by | |
| have h1 := mul_le_mul hexp hsource hF (Real.exp_pos 1).le | |
| have h2 := mul_le_mul_of_nonneg_right h1 (Real.sqrt_nonneg (2*width)) | |
| have h3 := mul_le_mul_of_nonneg_right h2 | |
| (div_nonneg (mul_nonneg (by norm_num : (0 : ℝ) ≤ 2) hdelta) ht.le) | |
| have h4 := mul_le_mul_of_nonneg_left h3 (Real.sqrt_nonneg (2*(1+1/s0))) | |
| convert h4 using 1 | |
| unfold comparisonGradientConstant | |
| field_simp [ht.ne'] | |
| end SmoothLocal.Pulse | |
| end | |
| end OAI | |