| import OAI.Geometry.CylinderCovering.Beams.sDefinitions |
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|
| namespace OAI |
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|
| noncomputable section |
| namespace ShortenedBeam |
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|
| theorem leftCoordinates_zero_iff (ha : a ≤ 2) : |
| leftCoordinates a x y 0 ↔ -1 ≤ x ∧ -1 ≤ y ∧ x+y ≤ -a := by |
| constructor |
| · rintro ⟨p,h,hp0,hp1,hh0,hh1,hx,hy⟩ |
| simp only [zero_mul, add_zero] at hx hy |
| refine ⟨?_,?_,?_⟩ |
| · nlinarith [mul_nonneg hh0 hp0] |
| · nlinarith [mul_nonneg hh0 (sub_nonneg.mpr hp1)] |
| · nlinarith |
| · rintro ⟨hx,hy,hxy⟩ |
| let h : ℝ := x+y+2 |
| have hh0 : 0 ≤ h := by dsimp [h]; linarith |
| have hh1 : h ≤ 2-a := by dsimp [h]; linarith |
| by_cases hh : h=0 |
| · have hxe : x = -1 := by dsimp [h] at hh; linarith |
| have hye : y = -1 := by dsimp [h] at hh; linarith |
| refine ⟨0,0,by norm_num,by norm_num,by norm_num,by linarith,?_,?_⟩ |
| · simp [hxe] |
| · simp [hye] |
| · have hhp : 0 < h := lt_of_le_of_ne hh0 (Ne.symm hh) |
| refine ⟨(x+1)/h,h,div_nonneg (by linarith) hh0,?_,hh0,hh1,?_,?_⟩ |
| · apply (div_le_one hhp).mpr |
| dsimp [h] |
| linarith |
| · simp only [zero_mul, add_zero] |
| field_simp |
| ring |
| · simp only [zero_mul, add_zero] |
| field_simp |
| dsimp [h] |
| ring |
|
|
| theorem reflected_leftCoordinates_zero_iff (ha : a ≤ 2) : |
| leftCoordinates a (-x) (-y) 0 ↔ x ≤ 1 ∧ y ≤ 1 ∧ a ≤ x+y := by |
| rw [leftCoordinates_zero_iff ha] |
| constructor <;> rintro ⟨h1,h2,h3⟩ <;> constructor |
| all_goals first | linarith | exact ⟨by linarith,by linarith⟩ |
| |
| theorem point_injective_three |
| (h : point x y t = point x' y' t') : x=x' ∧ y=y' ∧ t=t' := by |
| have hx := congrArg (fun z : Space => z 0) h |
| have hy := congrArg (fun z : Space => z 1) h |
| have ht := congrArg (fun z : Space => z 2) h |
| change x / Real.sqrt 2 = x' / Real.sqrt 2 at hx |
| change y / Real.sqrt 2 = y' / Real.sqrt 2 at hy |
| change t=t' at ht |
| have hs : Real.sqrt (2:ℝ) ≠ 0 := by positivity |
| exact ⟨(div_left_inj' hs).mp hx,(div_left_inj' hs).mp hy,ht⟩ |
| |
| theorem point_zero_mem_beamMinus_iff {a x y : ℝ} (ha : a ≤ 2) : |
| point x y 0 ∈ beamMinus a ↔ -1 ≤ x ∧ -1 ≤ y ∧ x+y ≤ -a := by |
| constructor |
| · rintro ⟨x',y',t,ht,hc,heq⟩ |
| obtain ⟨hx,hy,hz⟩ := point_injective_three heq |
| subst x' |
| subst y' |
| subst t |
| exact (leftCoordinates_zero_iff ha).mp hc |
| · intro h |
| exact ⟨x,y,0,by norm_num,(leftCoordinates_zero_iff ha).mpr h,rfl⟩ |
| |
| theorem point_zero_mem_beamPlus_iff {a x y : ℝ} (ha : a ≤ 2) : |
| point x y 0 ∈ beamPlus a ↔ x ≤ 1 ∧ y ≤ 1 ∧ a ≤ x+y := by |
| constructor |
| · rintro ⟨x',y',t,ht,hc,heq⟩ |
| obtain ⟨hx,hy,hz⟩ := point_injective_three heq |
| subst x' |
| subst y' |
| subst t |
| exact (reflected_leftCoordinates_zero_iff ha).mp hc |
| · intro h |
| exact ⟨x,y,0,by norm_num,(reflected_leftCoordinates_zero_iff ha).mpr h,rfl⟩ |
| |
| theorem point_zero_outside_beams_iff {a x y : ℝ} (ha : a ≤ 2) |
| (hx : |x| ≤ 1) (hy : |y| ≤ 1) : |
| (point x y 0 ∉ beamMinus a ∧ point x y 0 ∉ beamPlus a) ↔ |x+y| < a := by |
| rw [point_zero_mem_beamMinus_iff ha,point_zero_mem_beamPlus_iff ha] |
| have hxl : -1 ≤ x := (abs_le.mp hx).1 |
| have hxu : x ≤ 1 := (abs_le.mp hx).2 |
| have hyl : -1 ≤ y := (abs_le.mp hy).1 |
| have hyu : y ≤ 1 := (abs_le.mp hy).2 |
| simp only [hxl,hxu,hyl,hyu,true_and,not_le,abs_lt] |
| |
| end ShortenedBeam |
| |
| end |
| |
| end OAI |
| |
