| import OAI.MathematicalPhysics.Transonic.Exterior.Continuation |
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| namespace OAI |
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| section |
| noncomputable section |
| namespace SepticProfile.AxisBarriers |
|
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| def D (sigma z u : ℝ) : ℝ := z*(1-sigma*z^2)*(1-u^2) |
| def N (kappa z u : ℝ) : ℝ := (1-(3/5:ℝ)*u^2)*(kappa*z*(1-u^2)+3*(z-u)) |
| def lower (z : ℝ) : ℝ := (749/1000)*z |
| def polyDen (x : ℝ) : ℝ := 1-(41/200)*x-(347/5000)*x^2 |
| def upper (z : ℝ) : ℝ := (3/4)*z/polyDen (z^2) |
| def upperDeriv (z : ℝ) : ℝ := (3/4)*(1+(41/200)*z^2+(1041/5000)*z^4)/(polyDen (z^2))^2 |
|
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| def residualPoly (x : ℝ) : ℝ := |
| 9/1000+(220413/1600000)*x-(23123019/64000000)*x^2+ |
| (245617149/1280000000)*x^3+(1242882171/32000000000)*x^4- |
| (95591862237/20000000000000)*x^5-(784539168171/500000000000000)*x^6- |
| (43494981843/625000000000000)*x^7 |
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| lemma radius_bounds {z : ℝ} (hz : z ∈ Set.Icc 0 (9/10)) : |
| 0≤z^2 ∧ z^2≤81/100 ∧ (z^2)^2≤6561/10000 := by |
| have h1 : 0≤z*(9/10-z) := mul_nonneg hz.1 (sub_nonneg.mpr hz.2) |
| have h2 : z^2≤81/100 := by nlinarith [hz.2] |
| exact ⟨sq_nonneg z,h2,by nlinarith [mul_nonneg (sq_nonneg z) (sub_nonneg.mpr h2)]⟩ |
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| lemma polyDen_pos {z : ℝ} (hz : z ∈ Set.Icc 0 (9/10)) : |
| 78/100 < polyDen (z^2) := by |
| have h := radius_bounds hz |
| dsimp [polyDen] |
| nlinarith [h.2.1,h.2.2] |
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|
| lemma barriers_in_unit {z : ℝ} (hz : z ∈ Set.Icc 0 (9/10)) : |
| 0≤lower z ∧ lower z<upper z+(if z=0 then 1 else 0) ∧ |
| 0≤upper z ∧ upper z<1 := by |
| have hz0 := hz.1 |
| have hd := polyDen_pos hz |
| have hd0 : 0 < polyDen (z^2) := by linarith |
| have hx := radius_bounds hz |
| have hdu : polyDen (z^2)≤1 := by dsimp [polyDen]; nlinarith [hx.1,sq_nonneg (z^2)] |
| refine ⟨by dsimp [lower]; positivity,?_,by dsimp [upper]; positivity,?_⟩ |
| · by_cases h0 : z=0 |
| · simp [h0,lower,upper,polyDen] |
| · rw [ite_eq_right h0,add_zero] |
| have hzp : 0<z := lt_of_le_of_ne hz.1 (Ne.symm h0) |
| dsimp [lower,upper] |
| rw [lt_div_iff₀ hd0] |
| nlinarith |
| · dsimp [upper] |
| rw [div_lt_one hd0] |
| linarith [hz.2] |
|
|
| lemma upperDeriv_pos {z : ℝ} (hz : z ∈ Set.Icc 0 (9/10)) : 0<upperDeriv z := by |
| have hd := polyDen_pos hz |
| dsimp [upperDeriv] |
| positivity |
|
|
| lemma upper_hasDeriv {z : ℝ} (hz : z ∈ Set.Icc 0 (9/10)) : HasDerivAt upper (upperDeriv z) z := by |
| have hd : polyDen (z^2) ≠ 0 := by linarith [polyDen_pos hz] |
| have hg : HasDerivAt (fun z : ℝ => polyDen (z^2)) |
| (-(41/100)*z-(347/1250)*z^3) z := by |
| convert (((hasDerivAt_const z (1:ℝ)).sub (((hasDerivAt_id z).pow 2).const_mul (41/200))).sub |
| ((((hasDerivAt_id z).pow 2).pow 2).const_mul (347/5000))) using 1 <;> |
| first | rfl | (dsimp [polyDen]; ring) |
| convert (((hasDerivAt_id z).const_mul (3/4)).div hg hd) using 1 <;> |
| first | rfl | (dsimp [upper,upperDeriv,polyDen]; ring) |
|
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| lemma residualPoly_pos {x : ℝ} (hx : x ∈ Set.Icc 0 (81/100)) : 0<residualPoly x := by |
| have he : residualPoly x = (1/100000:ℝ) + (899000000000/22876792454961:ℝ)*x^0*(81/100-x)^7 + |
| (17451408125000/22876792454961:ℝ)*x^1*(81/100-x)^6 + |
| (82832988404375/30502389939948:ℝ)*x^2*(81/100-x)^5 + |
| (5792980533458545/1464114717117504:ℝ)*x^3*(81/100-x)^4 + |
| (1921567051918001291/732057358558752000:ℝ)*x^4*(81/100-x)^3 + |
| (10162415437554180665821/15251194969974000000000:ℝ)*x^5*(81/100-x)^2 + |
| (745226816895942492518749/114383962274805000000000000:ℝ)*x^6*(81/100-x)^1 + |
| (3066525437817139243066501/7148997642175312500000000000:ℝ)*x^7*(81/100-x)^0 := by |
| dsimp [residualPoly] |
| ring |
| rw [he] |
| have h : 0≤81/100-x := sub_nonneg.mpr hx.2 |
| have : 0≤x := hx.1 |
| positivity |
|
|
| lemma upper_residual_identity {z : ℝ} (hd : polyDen (z^2) ≠ 0) : |
| D (993/1000) z (upper z)*upperDeriv z-N 0 z (upper z) = |
| z^3*residualPoly (z^2)/(polyDen (z^2))^4 := by |
| dsimp [D,N,upper,upperDeriv] |
| field_simp |
| dsimp [polyDen,residualPoly] |
| ring |
|
|
| lemma upper_strict_barrier {sigma kappa z : ℝ} |
| (hs : sigma≤993/1000) (hk : kappa≤0) (hz : z ∈ Set.Ioc 0 (9/10)) : |
| N kappa z (upper z)<D sigma z (upper z)*upperDeriv z := by |
| have hzp := hz.1 |
| have hzc : z ∈ Set.Icc 0 (9/10) := ⟨le_of_lt hz.1,hz.2⟩ |
| have hu := barriers_in_unit hzc |
| have hd := polyDen_pos hzc |
| have hurd := upperDeriv_pos hzc |
| have hup : 0<1-(upper z)^2 := by nlinarith [hu.2.2.1,hu.2.2.2] |
| have huc : 0<1-(3/5:ℝ)*(upper z)^2 := by nlinarith [sq_nonneg (upper z)] |
| have hpatch1 : D (993/1000) z (upper z)≤D sigma z (upper z) := by |
| have hs0 : 0≤993/1000-sigma := sub_nonneg.mpr hs |
| have hm : 0≤z*(993/1000-sigma)*z^2*(1-(upper z)^2) := by positivity |
| dsimp [D] |
| nlinarith only [hm] |
| have hpatch2 : N kappa z (upper z)≤N 0 z (upper z) := by |
| have hm : (1-(3/5:ℝ)*(upper z)^2)*kappa*z*(1-(upper z)^2)≤0 := by |
| exact mul_nonpos_of_nonpos_of_nonneg |
| (mul_nonpos_of_nonpos_of_nonneg (mul_nonpos_of_nonneg_of_nonpos (le_of_lt huc) hk) (le_of_lt hz.1)) |
| (le_of_lt hup) |
| dsimp [N] |
| nlinarith only [hm] |
| have hbase : 0<D (993/1000) z (upper z)*upperDeriv z-N 0 z (upper z) := by |
| rw [upper_residual_identity (by linarith)] |
| have hp := residualPoly_pos ⟨(radius_bounds hzc).1,(radius_bounds hzc).2.1⟩ |
| positivity |
| have hm := mul_le_mul_of_nonneg_right hpatch1 (le_of_lt hurd) |
| linarith |
|
|
| lemma lower_strict_barrier {sigma kappa z : ℝ} |
| (hs : (991/1000:ℝ) ≤ sigma) (hk : -(1/500:ℝ)≤kappa) (hz : z ∈ Set.Ioc 0 (9/10)) : |
| D sigma z (lower z)*(749/1000)<N kappa z (lower z) := by |
| have hzp := hz.1 |
| have hzc : z ∈ Set.Icc 0 (9/10) := ⟨le_of_lt hz.1,hz.2⟩ |
| have hl0 : 0≤lower z := by dsimp [lower]; positivity |
| have hl1 : lower z<1 := by dsimp [lower]; linarith [hz.2] |
| have hup : 0<1-(lower z)^2 := by nlinarith |
| have huc : 0<1-(3/5:ℝ)*(lower z)^2 := by nlinarith [sq_nonneg (lower z)] |
| have hpatch1 : D sigma z (lower z)≤D (991/1000) z (lower z) := by |
| have hs0 : 0≤ sigma-991/1000 := sub_nonneg.mpr hs |
| have hm : 0≤z*(sigma-991/1000)*z^2*(1-(lower z)^2) := by positivity |
| dsimp [D] |
| nlinarith only [hm] |
| have hpatch2 : N (-(1/500)) z (lower z)≤N kappa z (lower z) := by |
| have hk0 : 0≤kappa+1/500 := by linarith |
| have hm : 0≤(1-(3/5:ℝ)*(lower z)^2)*(kappa+1/500)*z*(1-(lower z)^2) := by positivity |
| dsimp [N] |
| nlinarith only [hm] |
| have he : N (-(1/500)) z (lower z)-D (991/1000) z (lower z)*(749/1000) = |
| z*((1/500)+(2276959251/2500000000)*z^2- |
| (1041964269513503/2500000000000000)*(z^2)^2) := by |
| dsimp [N,D,lower] |
| ring |
| have hbase : 0<N (-(1/500)) z (lower z)-D (991/1000) z (lower z)*(749/1000) := by |
| rw [he] |
| apply mul_pos hz.1 |
| have hx := radius_bounds hzc |
| have hm := mul_nonneg hx.1 (sub_nonneg.mpr hx.2.1) |
| nlinarith |
| have hm := mul_le_mul_of_nonneg_right hpatch1 (by norm_num : (0:ℝ)≤749/1000) |
| linarith |
|
|
| end SepticProfile.AxisBarriers |
|
|
| namespace SepticProfile.AxisBarriers |
| open Set RegularContinuation |
|
|
| lemma upper_lt_nine_tenths {z : ℝ} (hz : z ∈ Icc 0 (9/10)) : upper z<9/10 := by |
| have hd := polyDen_pos hz |
| dsimp [upper] |
| rw [div_lt_iff₀ (by linarith : 0<polyDen (z^2))] |
| linarith [hz.2] |
|
|
| lemma D_pos {sigma z u : ℝ} (hs : sigma≤993/1000) |
| (hz : z ∈ Ioc 0 (9/10)) (hu : u ∈ Icc 0 (9/10)) : 0<D sigma z u := by |
| have hz2 := radius_bounds ⟨le_of_lt hz.1,hz.2⟩ |
| have hu2 := radius_bounds hu |
| have hm := mul_le_mul_of_nonneg_right hs (sq_nonneg z) |
| have h1 : 0<1-sigma*z^2 := by nlinarith [hz2.2.1] |
| have h2 : 0<1-u^2 := by linarith [hu2.2.1] |
| exact mul_pos (mul_pos hz.1 h1) h2 |
|
|
| lemma field_contDiffOn {sigma kappa a b : ℝ} (hs : sigma≤993/1000) |
| (ha : 0<a) (hb : b≤9/10) : |
| ContDiffOn ℝ 1 (fun p : ℝ×ℝ => N kappa p.1 p.2/D sigma p.1 p.2) |
| (Icc a b ×ˢ Icc 0 (9/10)) := by |
| have hN : ContDiff ℝ 1 (fun p : ℝ×ℝ => N kappa p.1 p.2) := by unfold N; fun_prop |
| have hD : ContDiff ℝ 1 (fun p : ℝ×ℝ => D sigma p.1 p.2) := by unfold D; fun_prop |
| exact hN.contDiffOn.div hD.contDiffOn (fun p hp => ne_of_gt (D_pos hs |
| ⟨lt_of_lt_of_le ha hp.1.1,le_trans hp.1.2 hb⟩ hp.2)) |
|
|
| / |
| to any b≤.9 while retaining the two explicit axis barriers. The clamping device |
| is not a replacement equation: the conclusion includes its complete removal. -/ |
| theorem exists_between_barriers {sigma kappa a b u0 : ℝ} |
| (hslo : (991/1000:ℝ) ≤ sigma) (hshi : sigma≤993/1000) |
| (hklo : -(1/500:ℝ)≤kappa) (hkhi : kappa≤0) |
| (ha : 0<a) (hab : a≤b) (hb : b≤9/10) |
| (hu0 : u0 ∈ Icc (lower a) (upper a)) : |
| ∃ u : ℝ → ℝ, u a=u0 ∧ ContinuousOn u (Icc a b) ∧ |
| (∀ t ∈ Icc a b, u t ∈ Icc (lower t) (upper t)) ∧ |
| ∀ t ∈ Icc a b, HasDerivWithinAt u |
| (N kappa t (u t)/D sigma t (u t)) (Icc a b) t := by |
| let f : ℝ×ℝ → ℝ := fun p => N kappa p.1 p.2/D sigma p.1 p.2 |
| obtain ⟨u,hu,huc,hud⟩ := exists_clamped_solution hab (by norm_num : (0:ℝ)≤9/10) |
| (field_contDiffOn hshi ha hb (kappa := kappa)) u0 |
| have ht0 {t : ℝ} (ht : t ∈ Icc a b) : t ∈ Icc 0 (9/10) := |
| ⟨le_trans (le_of_lt ha) ht.1,le_trans ht.2 hb⟩ |
| have htpos {t : ℝ} (ht : t ∈ Icc a b) : t ∈ Ioc 0 (9/10) := |
| ⟨lt_of_lt_of_le ha ht.1,le_trans ht.2 hb⟩ |
| have hlowD (t : ℝ) : HasDerivAt lower (749/1000) t := by |
| convert (hasDerivAt_id t).const_mul (749/1000:ℝ) using 1 <;> first | rfl | simp |
| have hright (t : ℝ) (ht : t ∈ Ico a b) : HasDerivWithinAt u |
| (f (t,clamp 0 (9/10) (u t))) (Ici t) t := by |
| apply (hud t ⟨ht.1,le_of_lt ht.2⟩).mono_of_mem_nhdsWithin |
| exact Filter.mem_of_superset (Icc_mem_nhdsGE ht.2) (Icc_subset_Icc_left ht.1) |
| have hupper : ∀ t ∈ Icc a b, u t≤upper t := by |
| apply image_le_of_deriv_right_lt_deriv_boundary' huc hright |
| (B' := upperDeriv) (by simpa [hu] using hu0.2) |
| (fun t ht => (upper_hasDeriv (ht0 ht)).continuousAt.continuousWithinAt) |
| (fun t ht => (upper_hasDeriv (ht0 ⟨ht.1,le_of_lt ht.2⟩)).hasDerivWithinAt) |
| intro t ht he |
| have htz := ht0 ⟨ht.1,le_of_lt ht.2⟩ |
| have htu : upper t ∈ Icc 0 (9/10) := |
| ⟨(barriers_in_unit htz).2.2.1,le_of_lt (upper_lt_nine_tenths htz)⟩ |
| rw [he,clamp_eq htu] |
| dsimp [f] |
| exact (div_lt_iff₀ (D_pos hshi (htpos ⟨ht.1,le_of_lt ht.2⟩) htu)).mpr |
| (by simpa [mul_comm] using upper_strict_barrier hshi hkhi (htpos ⟨ht.1,le_of_lt ht.2⟩)) |
| have hlower : ∀ t ∈ Icc a b, lower t≤u t := by |
| have hh := image_le_of_deriv_right_lt_deriv_boundary' |
| huc.neg (fun t ht => (hright t ht).neg) |
| (B := fun t => -lower t) (B' := fun _ => -(749/1000:ℝ)) |
| (by change -u a ≤ -lower a; rw [hu]; linarith [hu0.1]) |
| (fun t _ => (hlowD t).neg.continuousAt.continuousWithinAt) |
| (fun t _ => (hlowD t).neg.hasDerivWithinAt) |
| have hh' : ∀ t ∈ Icc a b, -u t≤ -lower t := by |
| apply hh |
| intro t ht he |
| have htz := ht0 ⟨ht.1,le_of_lt ht.2⟩ |
| have htu : lower t ∈ Icc 0 (9/10) := by |
| refine ⟨(barriers_in_unit htz).1,?_⟩ |
| dsimp [lower] |
| linarith [htz.2] |
| have he' : u t=lower t := neg_inj.mp he |
| rw [he',clamp_eq htu] |
| dsimp [f] |
| have hstrict := (lt_div_iff₀ (D_pos hshi (htpos ⟨ht.1,le_of_lt ht.2⟩) htu)).mpr |
| (by simpa [mul_comm] using lower_strict_barrier hslo hklo (htpos ⟨ht.1,le_of_lt ht.2⟩)) |
| linarith |
| intro t ht |
| linarith [hh' t ht] |
| refine ⟨u,hu,huc,fun t ht => ⟨hlower t ht,hupper t ht⟩,?_⟩ |
| intro t ht |
| have htu : u t ∈ Icc 0 (9/10) := |
| ⟨le_trans (barriers_in_unit (ht0 ht)).1 (hlower t ht), |
| le_trans (hupper t ht) (le_of_lt (upper_lt_nine_tenths (ht0 ht)))⟩ |
| simpa only [clamp_eq htu] using hud t ht |
|
|
| end SepticProfile.AxisBarriers |
|
|
| end |
| end |
|
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| end OAI |
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