1. 1
2. 2
3. 3
4. 4
5. 5
6. 6
7. 7
8. 8
9. 9
10. 10
11. 11
12. 12
13. 13
14. 14
15. 15
16. 16
17. 17
18. 18
19. 19
20. 20
21. 21
22. 22
23. 23
24. 24
25. 25
26. 26
27. 27
28. 28
29. 29
30. 30
31. 31
32. 32
33. 33
34. 34
35. 35
36. 36
37. 37
38. 38
39. 39
40. 40
41. 41
42. 42
43. 43
44. 44
45. 45
46. 46
47. 47
48. 48
49. 49
50. 50
51. 51
52. 52
53. 53
54. 54
55. 55
56. 56
57. 57
58. 58
59. 59
60. 60
61. 61
62. 62
63. 63
64. 64
65. 65
66. 66
67. 67
68. 68
69. 69
70. 70
71. 71
72. 72
73. 73
74. 74
75. 75
76. 76

import Mathlib.Tactic
import Mathlib.Data.Real.Basic

namespace C06S02

structure AddGroup₁ (α : Type*) where
  add : α → α → α
  zero : α
  neg : α → α
  add_assoc : ∀ x y z : α, add (add x y) z = add x (add y z)
  add_zero : ∀ x : α, add x zero = x
  zero_add : ∀ x : α, add x zero = x
  add_left_neg : ∀ x : α, add (neg x) x = zero

@[ext]
structure Point where
  x : ℝ
  y : ℝ
  z : ℝ

namespace Point

def add (a b : Point) : Point :=
  ⟨a.x + b.x, a.y + b.y, a.z + b.z⟩

def neg (a : Point) : Point :=
  ⟨-a.x, -a.y, -a.z⟩

def zero : Point :=
  ⟨0, 0, 0⟩

def addGroupPoint : AddGroup₁ Point where
  add := Point.add
  zero := Point.zero
  neg := Point.neg
  add_assoc := by simp [Point.add, add_assoc]
  add_zero := by simp [Point.add, Point.zero]
  zero_add := by simp [Point.add, Point.zero]
  add_left_neg := by simp [Point.add, Point.neg, Point.zero]

end Point

class AddGroup₂ (α : Type*) where
  add : α → α → α
  zero : α
  neg : α → α
  add_assoc : ∀ x y z : α, add (add x y) z = add x (add y z)
  add_zero : ∀ x : α, add x zero = x
  zero_add : ∀ x : α, add x zero = x
  add_left_neg : ∀ x : α, add (neg x) x = zero

instance hasAddAddGroup₂ {α : Type*} [AddGroup₂ α] : Add α :=
  ⟨AddGroup₂.add⟩

instance hasZeroAddGroup₂ {α : Type*} [AddGroup₂ α] : Zero α :=
  ⟨AddGroup₂.zero⟩

instance hasNegAddGroup₂ {α : Type*} [AddGroup₂ α] : Neg α :=
  ⟨AddGroup₂.neg⟩

instance : AddGroup₂ Point where
  add := Point.add
  zero := Point.zero
  neg := Point.neg
  add_assoc := by simp [Point.add, add_assoc]
  add_zero := by simp [Point.add, Point.zero]
  zero_add := by simp [Point.add, Point.zero]
  add_left_neg := by simp [Point.add, Point.neg, Point.zero]

section
variable (x y : Point)

#check x + -y + 0

end