| import OAI.Geometry.CylinderCovering.Beams.sDefinitions |
| import OAI.Geometry.CylinderCovering.Beams.Algebra |
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| namespace OAI |
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| namespace ShortenedBeam |
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|
| theorem radial_representation_nonneg_shift {U Y k : ℝ} |
| (hU : 0 ≤ U) (hY : 0 ≤ Y) (hk : 0 ≤ k) : |
| ∃ p h : ℝ, 0 ≤ p ∧ p ≤ 1 ∧ 0 ≤ h ∧ |
| U = h*p-k*p^2 ∧ Y = h*(1-p)+k*(1-p)^2 := by |
| by_cases hUzero : U = 0 |
| · subst U |
| by_cases hYk : k ≤ Y |
| · refine ⟨0, Y-k, by norm_num, by norm_num, sub_nonneg.mpr hYk, ?_, ?_⟩ |
| · ring |
| · ring |
| · have hkpos : 0 < k := lt_of_le_of_lt hY (lt_of_not_ge hYk) |
| have hkne : k ≠ 0 := ne_of_gt hkpos |
| have hYdiv : 0 ≤ Y/k := div_nonneg hY hk |
| have hYdiv' : Y/k ≤ 1 := (div_le_one hkpos).2 (le_of_lt (lt_of_not_ge hYk)) |
| refine ⟨1-Y/k, k-Y, by linarith, by linarith, by linarith, ?_, ?_⟩ |
| · field_simp |
| ring |
| · field_simp |
| ring |
| · have hUpos : 0 < U := lt_of_le_of_ne hU (Ne.symm hUzero) |
| let F : ℝ → ℝ := fun p => U*(1-p)-Y*p+k*p*(1-p) |
| have hF : ContinuousOn F (Set.Icc 0 1) := by |
| dsimp [F] |
| fun_prop |
| have hzero : (0 : ℝ) ∈ Set.Icc (F 1) (F 0) := by |
| dsimp [F] |
| constructor <;> nlinarith |
| obtain ⟨p, hp, hFp⟩ := intermediate_value_Icc' (by norm_num : (0:ℝ) ≤ 1) hF hzero |
| have hproot : U*(1-p)-Y*p+k*p*(1-p) = 0 := hFp |
| have hpne : p ≠ 0 := by |
| intro hpzero |
| subst p |
| norm_num at hproot |
| exact hUzero hproot |
| have hppos : 0 < p := lt_of_le_of_ne hp.1 (Ne.symm hpne) |
| let h : ℝ := U+Y+k*(2*p-1) |
| have heqU : U = h*p-k*p^2 := by |
| dsimp [h] |
| nlinarith [hproot] |
| have heqY : Y = h*(1-p)+k*(1-p)^2 := by |
| dsimp [h] |
| nlinarith [hproot] |
| have hhnonneg : 0 ≤ h := by |
| have hkp : 0 ≤ k*p^2 := mul_nonneg hk (sq_nonneg p) |
| have hhp : 0 ≤ h*p := by linarith |
| exact nonneg_of_mul_nonneg_left hhp hppos |
| exact ⟨p, h, hp.1, hp.2, hhnonneg, heqU, heqY⟩ |
|
|
| theorem radial_representation {U Y k : ℝ} (hU : 0 ≤ U) (hY : 0 ≤ Y) : |
| ∃ p h : ℝ, 0 ≤ p ∧ p ≤ 1 ∧ 0 ≤ h ∧ |
| U = h*p-k*p^2 ∧ Y = h*(1-p)+k*(1-p)^2 := by |
| by_cases hk : 0 ≤ k |
| · exact radial_representation_nonneg_shift hU hY hk |
| · obtain ⟨p, h, hp0, hp1, hh, hYeq, hUeq⟩ := |
| radial_representation_nonneg_shift hY hU (by linarith : 0 ≤ -k) |
| refine ⟨1-p, h, by linarith, by linarith, hh, ?_, ?_⟩ |
| · nlinarith [hUeq] |
| · nlinarith [hYeq] |
|
|
| theorem coordinate_representation {x y t : ℝ} (a : ℝ) |
| (hx : |x| ≤ 1-t) (hy : |y| ≤ 1+t) : |
| ∃ p h : ℝ, 0 ≤ p ∧ p ≤ 1 ∧ 0 ≤ h ∧ |
| x = -1+h*p+t*(1-a*(2-a)*p^2) ∧ |
| y = -1+h*(1-p)+t*(-1+a*(2-a)*(1-p)^2) := by |
| have hU : 0 ≤ x+1-t := by |
| have := (abs_le.mp hx).1 |
| linarith |
| have hY : 0 ≤ y+1+t := by |
| have := (abs_le.mp hy).1 |
| linarith |
| obtain ⟨p, h, hp0, hp1, hh, hxeq, hyeq⟩ := |
| radial_representation (k := a*(2-a)*t) hU hY |
| refine ⟨p, h, hp0, hp1, hh, ?_, ?_⟩ |
| · nlinarith only [hxeq] |
| · nlinarith only [hyeq] |
|
|
| theorem transverse_bounds {x y t : ℝ} |
| (hx : |x| ≤ 1-t) (hy : |y| ≤ 1+t) : |
| |(x-y)/2| ≤ 1 ∧ |(x-y)/2| ≤ 1-|t|+|x+y|/2 := by |
| have habs : |(x-y)/2| = |x-y|/2 := by |
| rw [abs_div] |
| norm_num |
| constructor |
| · rw [habs] |
| have := abs_sub x y |
| linarith |
| · by_cases ht : 0 ≤ t |
| · rw [abs_of_nonneg ht] |
| have heq : (x-y)/2 = x-(x+y)/2 := by ring |
| rw [heq] |
| have htri := abs_sub x ((x+y)/2) |
| have hsum : |(x+y)/2| = |x+y|/2 := by |
| rw [abs_div] |
| norm_num |
| rw [hsum] at htri |
| linarith |
| · have ht' : t ≤ 0 := le_of_not_ge ht |
| rw [abs_of_nonpos ht'] |
| have heq : (x-y)/2 = (x+y)/2-y := by ring |
| rw [heq] |
| have htri := abs_sub ((x+y)/2) y |
| have hsum : |(x+y)/2| = |x+y|/2 := by |
| rw [abs_div] |
| norm_num |
| rw [hsum] at htri |
| linarith |
|
|
| theorem outside_left_representation {a x y t : ℝ} |
| (hx : |x| ≤ 1-t) (hy : |y| ≤ 1+t) |
| (hout : ¬ leftCoordinates a x y t) : |
| ∃ q d : ℝ, |q| ≤ 1 ∧ 0 < d ∧ |
| x+y = -a-2*a*t*((2-a)*q/2)+d ∧ |
| (x-y)/2 = t+(2-a)*q/2+d*q/2-a*t*((2-a)*(1+q^2)/4) := by |
| obtain ⟨p, h, hp0, hp1, hh, hxeq, hyeq⟩ := coordinate_representation a hx hy |
| have hhlarge : 2-a < h := by |
| by_contra hle |
| exact hout ⟨p, h, hp0, hp1, hh, le_of_not_gt hle, hxeq, hyeq⟩ |
| refine ⟨2*p-1, h-(2-a), abs_le.mpr ⟨by linarith, by linarith⟩, |
| sub_pos.mpr hhlarge, ?_, ?_⟩ |
| · nlinarith only [hxeq, hyeq] |
| · nlinarith only [hxeq, hyeq] |
|
|
| theorem coordinate_coverage {a x y t : ℝ} |
| (ha : 0 ≤ a) (ha' : a ≤ 1/2) (ht : |t| ≤ 1) |
| (hx : |x| ≤ 1-t) (hy : |y| ≤ 1+t) : |
| leftCoordinates a x y t ∨ leftCoordinates a (-x) (-y) t ∨ |
| |x+y+drift a t*((x-y)/2)| ≤ width a t := by |
| by_cases hminus : leftCoordinates a x y t |
| · exact Or.inl hminus |
| by_cases hplus : leftCoordinates a (-x) (-y) t |
| · exact Or.inr (Or.inl hplus) |
| refine Or.inr (Or.inr ?_) |
| obtain ⟨q,d,hq,hd,hs,hl⟩ := outside_left_representation hx hy hminus |
| obtain ⟨q',d',hq',hd',hs',hl'⟩ := outside_left_representation |
| (by simpa only [abs_neg] using hx) (by simpa only [abs_neg] using hy) hplus |
| have hN : 2-a ≠ 0 := by linarith |
| have hform (r : ℝ) : |
| (2-a)*(1+r^2)/4 = (2-a)/4+((2-a)*r/2)^2/(2-a) := by |
| field_simp |
| ring |
| rw [hform] at hl hl' |
| have hsref : -(x+y) = -a-2*a*t*((2-a)*q'/2)+d' := by |
| linarith only [hs'] |
| have hlref : -((x-y)/2) = |
| t+(2-a)*q'/2+d'*q'/2-a*t*((2-a)/4+((2-a)*q'/2)^2/(2-a)) := by |
| linarith only [hl'] |
| exact CylinderCovering.BeamAlgebra.explicit_strip_bound ha ha' ht |
| (transverse_bounds hx hy).1 (transverse_bounds hx hy).2 |
| hd.le hd'.le hq hq' rfl rfl hs hsref hl hlref |
|
|
| theorem tetrahedron_covered {a : ℝ} (ha : 0 ≤ a) (ha' : a ≤ 1/2) : |
| tetrahedron ⊆ beamMinus a ∪ beamPlus a ∪ strip a := by |
| rintro z ⟨x,y,t,ht,hx,hy,rfl⟩ |
| rcases coordinate_coverage ha ha' ht hx hy with hm | hp | hs |
| · exact Or.inl (Or.inl ⟨x,y,t,ht,hm,rfl⟩) |
| · exact Or.inl (Or.inr ⟨x,y,t,ht,hp,rfl⟩) |
| · exact Or.inr ⟨x,y,t,ht,hx,hy,hs,rfl⟩ |
|
|
| theorem beamMinus_isCompact (a : ℝ) : IsCompact (beamMinus a) := by |
| let D : Set (ℝ × ℝ × ℝ) := |
| Set.Icc 0 1 ×ˢ (Set.Icc 0 (2-a) ×ˢ Set.Icc (-1) 1) |
| let F : (ℝ × ℝ × ℝ) → Space := fun z => |
| point (-1+z.2.1*z.1+z.2.2*(1-a*(2-a)*z.1^2)) |
| (-1+z.2.1*(1-z.1)+z.2.2*(-1+a*(2-a)*(1-z.1)^2)) z.2.2 |
| have hD : IsCompact D := isCompact_Icc.prod (isCompact_Icc.prod isCompact_Icc) |
| have hF : Continuous F := by |
| dsimp [F, point] |
| fun_prop |
| have heq : F '' D = beamMinus a := by |
| ext z |
| constructor |
| · rintro ⟨⟨p,h,t⟩, hpht, rfl⟩ |
| rcases hpht with ⟨hp, hh, ht⟩ |
| refine ⟨_, _, t, abs_le.mpr ht, ?_, rfl⟩ |
| exact ⟨p, h, hp.1, hp.2, hh.1, hh.2, rfl, rfl⟩ |
| · rintro ⟨x,y,t,ht,⟨p,h,hp0,hp1,hh0,hh1,hx,hy⟩,hz⟩ |
| refine ⟨(p,h,t), ⟨⟨hp0,hp1⟩,⟨hh0,hh1⟩,abs_le.mp ht⟩, ?_⟩ |
| dsimp [F] |
| rw [← hx, ← hy] |
| exact hz.symm |
| rw [← heq] |
| exact hD.image hF |
|
|
| theorem beamPlus_isCompact (a : ℝ) : IsCompact (beamPlus a) := by |
| let D : Set (ℝ × ℝ × ℝ) := |
| Set.Icc 0 1 ×ˢ (Set.Icc 0 (2-a) ×ˢ Set.Icc (-1) 1) |
| let F : (ℝ × ℝ × ℝ) → Space := fun z => |
| point (-(-1+z.2.1*z.1+z.2.2*(1-a*(2-a)*z.1^2))) |
| (-(-1+z.2.1*(1-z.1)+z.2.2*(-1+a*(2-a)*(1-z.1)^2))) z.2.2 |
| have hD : IsCompact D := isCompact_Icc.prod (isCompact_Icc.prod isCompact_Icc) |
| have hF : Continuous F := by |
| dsimp [F, point] |
| fun_prop |
| have heq : F '' D = beamPlus a := by |
| ext z |
| constructor |
| · rintro ⟨⟨p,h,t⟩, hpht, rfl⟩ |
| rcases hpht with ⟨hp, hh, ht⟩ |
| refine ⟨_, _, t, abs_le.mpr ht, ?_, rfl⟩ |
| refine ⟨p, h, hp.1, hp.2, hh.1, hh.2, ?_, ?_⟩ <;> simp |
| · rintro ⟨x,y,t,ht,⟨p,h,hp0,hp1,hh0,hh1,hx,hy⟩,hz⟩ |
| refine ⟨(p,h,t), ⟨⟨hp0,hp1⟩,⟨hh0,hh1⟩,abs_le.mp ht⟩, ?_⟩ |
| dsimp [F] |
| rw [← hx, ← hy, neg_neg, neg_neg] |
| exact hz.symm |
| rw [← heq] |
| exact hD.image hF |
|
|
| end ShortenedBeam |
|
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| end OAI |
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|
