| import OAI.Analysis.HenonEmden.Endpoints |
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| namespace OAI |
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| section LayerRecovery |
| open MeasureTheory Set Filter Topology |
| namespace HenonLaneEmden.WeightedLayers |
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| lemma source_root {w u q : ℝ} (hw : 0 < w) (hu : 0 < u) (hq : 0 < q) : |
| (w*u^q)^(1/q) = w^(1/q)*u := by |
| rw [Real.mul_rpow hw.le (Real.rpow_nonneg hu.le _), ← Real.rpow_mul hu.le] |
| have h : q*(1/q) = 1 := by field_simp |
| rw [h, Real.rpow_one] |
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| lemma weighted_layer_identity {w u q : ℝ} (hw : 0 < w) (hu : 0 < u) (hq : 0 < q) : |
| (∫ t in 0..w*u^q, (u-(t/w)^(1/q))) = w*u^(q+1)/(q+1) := by |
| have hq' : 0 < 1/q := one_div_pos.mpr hq |
| have hf : 0 < w*u^q := mul_pos hw (Real.rpow_pos_of_pos hu _) |
| have hr : IntervalIntegrable (fun t : ℝ => t^(1/q)) volume 0 (w*u^q) := |
| intervalIntegral.intervalIntegrable_rpow' (by linarith) |
| have heq : (∫ t in 0..w*u^q, (u-(t/w)^(1/q))) = |
| (∫ t in 0..w*u^q, (u-t^(1/q)/w^(1/q))) := by |
| apply intervalIntegral.integral_congr |
| intro t ht |
| rw [uIcc_of_le hf.le] at ht |
| dsimp only |
| rw [Real.div_rpow ht.1 hw.le] |
| rw [heq, intervalIntegral.integral_sub (continuous_const.intervalIntegrable ..) (hr.div_const _), |
| intervalIntegral.integral_const, intervalIntegral.integral_div, |
| integral_rpow (Or.inl (by linarith : -1 < 1/q)), |
| Real.zero_rpow (ne_of_gt (by linarith : 0 < 1/q+1)), sub_zero, |
| sub_zero, smul_eq_mul] |
| rw [Real.rpow_add hf, Real.rpow_one, source_root hw hu hq, |
| Real.rpow_add hu, Real.rpow_one] |
| have hw' : w^(1/q) ≠ 0 := ne_of_gt (Real.rpow_pos_of_pos hw _) |
| have hq0 : q ≠ 0 := ne_of_gt hq |
| have hq1 : q+1 ≠ 0 := by linarith |
| field_simp [hw', hq0, hq1] |
| ring |
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| lemma layer_root_bounds {w u q t : ℝ} (hw : 0 < w) (hu : 0 < u) (hq : 0 < q) |
| (ht : 0 ≤ t) (htf : t ≤ w*u^q) : |
| 0 ≤ (t/w)^(1/q) ∧ (t/w)^(1/q) ≤ u := by |
| constructor |
| · exact Real.rpow_nonneg (div_nonneg ht hw.le) _ |
| · have htw : t/w ≤ u^q := (div_le_iff₀ hw).mpr (by simpa [mul_comm] using htf) |
| have hroot := Real.rpow_le_rpow (div_nonneg ht hw.le) htw (one_div_nonneg.mpr hq.le) |
| rw [← Real.rpow_mul hu.le] at hroot |
| have h : q*(1/q) = 1 := by field_simp |
| rwa [h, Real.rpow_one] at hroot |
|
|
| /-- The exact scalar recovery inequality before integration. -/ |
| lemma pressure_recovery_pointwise {h χ u τ endpoint error : ℝ} |
| (hχ : 0 ≤ χ) (hχh : χ ≤ h) (hτ : 0 ≤ τ) |
| (hend : |endpoint-τ| ≤ error*u) : |
| h*(u-τ)-χ*(u-endpoint) ≤ (h-χ)*u+error*h*u := by |
| have hhu : 0 ≤ error*u := le_trans (abs_nonneg _) hend |
| have he := (abs_le.mp hend).2 |
| have hmul := mul_le_mul_of_nonneg_left he hχ |
| have hmul' := mul_le_mul_of_nonneg_right hχh hhu |
| have htail := mul_nonneg (sub_nonneg.mpr hχh) hτ |
| nlinarith |
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| end HenonLaneEmden.WeightedLayers |
| end LayerRecovery |
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| end OAI |
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