File size: 3,461 Bytes
785c727 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 | import OAI.Geometry.CylinderCovering.Beams.sDefinitions
namespace OAI
noncomputable section
namespace ShortenedBeam
theorem leftCoordinates_zero_iff {a x y : ℝ} (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 {a x y : ℝ} (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 {x y t x' y' t' : ℝ}
(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
|