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
77. 77
78. 78
79. 79
80. 80
81. 81
82. 82
83. 83
84. 84
85. 85
86. 86
87. 87
88. 88
89. 89
90. 90
91. 91
92. 92
93. 93
94. 94
95. 95
96. 96
97. 97
98. 98

import Mathlib.Algebra.BigOperators.Ring
import Mathlib.Data.Real.Basic

noncomputable section

@[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⟩

protected theorem add_assoc (a b c : Point) : (a.add b).add c = a.add (b.add c) := by
  simp [add, add_assoc]

def smul (r : ℝ) (a : Point) : Point :=
  ⟨r * a.x, r * a.y, r * a.z⟩

theorem smul_distrib (r : ℝ) (a b : Point) :
    (smul r a).add (smul r b) = smul r (a.add b) := by
  simp [add, smul, mul_add]

end Point

structure StandardTwoSimplex where
  x : ℝ
  y : ℝ
  z : ℝ
  x_nonneg : 0 ≤ x
  y_nonneg : 0 ≤ y
  z_nonneg : 0 ≤ z
  sum_eq : x + y + z = 1

namespace StandardTwoSimplex

noncomputable section

def weightedAverage (lambda : Real) (lambda_nonneg : 0 ≤ lambda) (lambda_le : lambda ≤ 1)
  (a b : StandardTwoSimplex) : StandardTwoSimplex
where
  x := lambda * a.x + (1 - lambda) * b.x
  y := lambda * a.y + (1 - lambda) * b.y
  z := lambda * a.z + (1 - lambda) * b.z
  x_nonneg := add_nonneg (mul_nonneg lambda_nonneg a.x_nonneg) (mul_nonneg (by linarith) b.x_nonneg)
  y_nonneg := add_nonneg (mul_nonneg lambda_nonneg a.y_nonneg) (mul_nonneg (by linarith) b.y_nonneg)
  z_nonneg := add_nonneg (mul_nonneg lambda_nonneg a.z_nonneg) (mul_nonneg (by linarith) b.z_nonneg)
  sum_eq := by
    trans (a.x + a.y + a.z) * lambda + (b.x + b.y + b.z) * (1 - lambda)
    · ring
    simp [a.sum_eq, b.sum_eq]

end

end StandardTwoSimplex

open BigOperators

structure StandardSimplex (n : ℕ) where
  V : Fin n → ℝ
  NonNeg : ∀ i : Fin n, 0 ≤ V i
  sum_eq_one : (∑ i, V i) = 1

namespace StandardSimplex

def midpoint (n : ℕ) (a b : StandardSimplex n) : StandardSimplex n
    where
  V i := (a.V i + b.V i) / 2
  NonNeg := by
    intro i
    apply div_nonneg
    · linarith [a.NonNeg i, b.NonNeg i]
    norm_num
  sum_eq_one := by
    simp [div_eq_mul_inv, ← Finset.sum_mul, Finset.sum_add_distrib,
      a.sum_eq_one, b.sum_eq_one]
    field_simp

end StandardSimplex

namespace StandardSimplex

def weightedAverage {n : ℕ} (lambda : Real) (lambda_nonneg : 0 ≤ lambda) (lambda_le : lambda ≤ 1)
    (a b : StandardSimplex n) : StandardSimplex n
    where
  V i := lambda * a.V i + (1 - lambda) * b.V i
  NonNeg i :=
    add_nonneg (mul_nonneg lambda_nonneg (a.NonNeg i)) (mul_nonneg (by linarith) (b.NonNeg i))
  sum_eq_one := by
    trans (lambda * ∑ i, a.V i) + (1 - lambda) * ∑ i, b.V i
    · rw [Finset.sum_add_distrib, Finset.mul_sum, Finset.mul_sum]
    simp [a.sum_eq_one, b.sum_eq_one]

end StandardSimplex