| import OAI.MathematicalPhysics.Transonic.Profile.PhysicalBounds |
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| namespace OAI |
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| section |
| noncomputable section |
| namespace SepticProfile.PhysicalExterior |
| open Set |
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| lemma chart_antitone {y U V : ℝ} (hy : |y|<1) |
| (hU : 0<1-y*U) (hV : 0<1-y*V) (hUV : U≤V) : |
| velocityToU y V≤velocityToU y U := by |
| rw [velocityToU,velocityToU,div_le_div_iff₀ hV hU] |
| have hs : 0≤1-y^2 := by have := abs_lt.mp hy;nlinarith |
| nlinarith [mul_nonneg hs (sub_nonneg.mpr hUV)] |
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| lemma chart_first {q y U : ℝ} (hd : 1-y*U≠0) : |
| 1-q*y+(q-y)*velocityToU y U=(1-y^2)*(1-q*U)/(1-y*U) := by |
| unfold velocityToU |
| field_simp |
| ring |
| lemma chart_second {q y U : ℝ} (hd : 1-y*U≠0) : |
| 1+q*y-(y+q)*velocityToU y U=(1-y^2)*(1+q*U)/(1-y*U) := by |
| unfold velocityToU |
| field_simp |
| ring |
| lemma chart_critical {ell beta y U : ℝ} (hd : 1-y*U≠0) : |
| beta*ell*(1-(velocityToU y U)^2)*y- |
| 3*(velocityToU y U)*(1-y*velocityToU y U)= |
| (1-y^2)/(1-y*U)^2*(beta*ell*y*(1-U^2)-3*(y-U)) := by |
| unfold velocityToU |
| field_simp |
| ring |
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| lemma normalized_critical {beta r z p : ℝ} (hr : r*SepticProfile.q beta=3*sonicSpeed) : |
| beta*ell*(r*z)*(1-(sonicSpeed*p)^2)-3*(r*z-sonicSpeed*p)= |
| 3*sonicSpeed*(p-z)+(beta*r)*z*(1-p^2) := by |
| have hc : ell*sonicSpeed^2=1 := by rw [sonicSpeed_sq];norm_num [ell] |
| have hq : SepticProfile.q beta=3-beta*(ell-1) := rfl |
| rw [hq] at hr |
| linear_combination -(z)*hr - beta*r*z*p^2*hc |
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| lemma critical_exit {beta r z p : ℝ} (hbeta : 1<beta) |
| (hbeta1 : beta*(ell+Real.sqrt ell)<3) |
| (hr : r=sonicRadius beta) |
| (hy : |r*z|<1) (hU : |sonicSpeed*p|<1) |
| (hexit : 0<p-z+(3/7*(1-SepticProfile.kappa beta))*z*(1-p^2)) : |
| 0<beta*ell*(1-(velocityToU (r*z) (sonicSpeed*p))^2)*(r*z)- |
| 3*(velocityToU (r*z) (sonicSpeed*p))*(1-(r*z)*velocityToU (r*z) (sonicSpeed*p)) := by |
| have hp := source_parameter_bounds hbeta hbeta1 |
| have hc : 0<sonicSpeed := hp.2.1 |
| have hq : 0<SepticProfile.q beta := hp.2.2.2.1 |
| have hrr : r*SepticProfile.q beta=3*sonicSpeed := by |
| rw [hr,sonicRadius_eq,div_mul_cancel₀ _ (ne_of_gt hq)] |
| have hd : 0<1-(r*z)*(sonicSpeed*p) := by |
| have hh : |(r*z)*(sonicSpeed*p)|<1 := by |
| rw [abs_mul] |
| exact (mul_le_mul_of_nonneg_right hy.le (abs_nonneg _)).trans_lt (by simpa using hU) |
| linarith [(abs_lt.mp hh).2] |
| rw [chart_critical (ne_of_gt hd),normalized_critical hrr] |
| have hb : beta*r=(3*sonicSpeed)*(3/7*(1-SepticProfile.kappa beta)) := by |
| rw [hr,sonicRadius_eq,SepticProfile.kappa] |
| field_simp [ne_of_gt hq] |
| unfold SepticProfile.q ell |
| ring |
| rw [hb] |
| have hs : 0<1-(r*z)^2 := by have := abs_lt.mp hy;nlinarith |
| apply mul_pos (div_pos hs (sq_pos_of_pos hd)) |
| have hh := mul_pos (mul_pos (by norm_num : (0:ℝ)<3) hc) hexit |
| nlinarith |
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| lemma sonic_exit_factors {r z p : ℝ} (hy : |r*z|<1) |
| (hU : |sonicSpeed*p|<1) (hp : 1<p) : |
| (1-Real.sqrt ell*(r*z)+(Real.sqrt ell-r*z)*velocityToU (r*z) (sonicSpeed*p)<0) ∧ |
| (0<1+Real.sqrt ell*(r*z)-(r*z+Real.sqrt ell)*velocityToU (r*z) (sonicSpeed*p)) := by |
| have hc : Real.sqrt ell*sonicSpeed=1 := by |
| unfold sonicSpeed |
| exact mul_one_div_cancel (ne_of_gt (Real.sqrt_pos.mpr (by norm_num [ell]))) |
| have hd : 0<1-(r*z)*(sonicSpeed*p) := by |
| have hh : |(r*z)*(sonicSpeed*p)|<1 := by |
| rw [abs_mul] |
| exact (mul_le_mul_of_nonneg_right hy.le (abs_nonneg _)).trans_lt (by simpa using hU) |
| linarith [(abs_lt.mp hh).2] |
| rw [chart_first (ne_of_gt hd),chart_second (ne_of_gt hd),←mul_assoc,hc,one_mul] |
| have hs : 0<1-(r*z)^2 := by have := abs_lt.mp hy;nlinarith |
| exact ⟨div_neg_of_neg_of_pos (mul_neg_of_pos_of_neg hs (by linarith)) hd, |
| div_pos (mul_pos hs (by linarith)) hd⟩ |
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| end SepticProfile.PhysicalExterior |
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| end |
| end |
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| end OAI |
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