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
99. 99
100. 100
101. 101
102. 102
103. 103
104. 104
105. 105
106. 106
107. 107
108. 108
109. 109
110. 110
111. 111
112. 112
113. 113
114. 114
115. 115
116. 116
117. 117
118. 118
119. 119
120. 120
121. 121
122. 122
123. 123
124. 124
125. 125
126. 126
127. 127
128. 128
129. 129
130. 130
131. 131
132. 132
133. 133
134. 134
135. 135
136. 136
137. 137
138. 138
139. 139
140. 140
141. 141
142. 142
143. 143
144. 144
145. 145
146. 146
147. 147
148. 148
149. 149
150. 150
151. 151
152. 152
153. 153
154. 154
155. 155
156. 156
157. 157
158. 158
159. 159
160. 160
161. 161

import data.real.basic

section
variables {x y : ℝ}

namespace my_abs

theorem le_abs_self (x : ℝ) : x ≤ abs x :=
begin
  cases le_or_gt 0 x with h h,
  { rw abs_of_nonneg h },
  rw abs_of_neg h,
  linarith
end

theorem neg_le_abs_self (x : ℝ) : -x ≤ abs x :=
begin
  cases le_or_gt 0 x with h h,
  { rw abs_of_nonneg h,
    linarith },
  rw abs_of_neg h
end

theorem abs_add (x y : ℝ) : abs (x + y) ≤ abs x + abs y :=
begin
  cases le_or_gt 0 (x + y) with h h,
  { rw abs_of_nonneg h,
    linarith [le_abs_self x, le_abs_self y] },
  rw abs_of_neg h,
  linarith [neg_le_abs_self x, neg_le_abs_self y]
end

theorem lt_abs : x < abs y ↔ x < y ∨ x < -y :=
begin
  cases le_or_gt 0 y with h h,
  { rw abs_of_nonneg h,
    split,
    { intro h', left, exact h' },
    intro h',
    cases h' with h' h',
    { exact h' },
    linarith },
  rw abs_of_neg h,
  split,
  { intro h', right, exact h' },
  intro h',
  cases h' with h' h',
  { linarith },
  exact h'
end

theorem abs_lt : abs x < y ↔ - y < x ∧ x < y :=
begin
  cases le_or_gt 0 x with h h,
  { rw abs_of_nonneg h,
    split,
    { intro h',
      split,
      { linarith },
      exact h' },
    intro h',
    cases h' with h1 h2,
    exact h2 },
  rw abs_of_neg h,
  split,
  { intro h',
    split,
    { linarith },
    linarith },
  intro h',
  linarith
end

end my_abs
end

example {z : ℝ} (h : ∃ x y, z = x^2 + y^2 ∨ z = x^2 + y^2 + 1) :
  z ≥ 0 :=
by { rcases h with ⟨x, y, rfl | rfl⟩; linarith [sq_nonneg x, sq_nonneg y] }

example {x : ℝ} (h : x^2 = 1) : x = 1 ∨ x = -1 :=
begin
  have h' : x^2 - 1 = 0,
  { rw [h, sub_self] },
  have h'' : (x + 1) * (x - 1) = 0,
  { rw ← h',
    ring },
  cases eq_zero_or_eq_zero_of_mul_eq_zero h'' with h1 h1,
  { right,
    exact eq_neg_iff_add_eq_zero.mpr h1 },
  left,
  exact eq_of_sub_eq_zero h1
end

example {x y : ℝ} (h : x^2 = y^2) : x = y ∨ x = -y :=
begin
  have h' : x^2 - y^2 = 0,
  { rw [h, sub_self] },
  have h'' : (x + y) * (x - y) = 0,
  { rw ← h',
    ring },
  cases eq_zero_or_eq_zero_of_mul_eq_zero h'' with h1 h1,
  { right,
    exact eq_neg_iff_add_eq_zero.mpr h1 },
  left,
  exact eq_of_sub_eq_zero h1
end

section
variables {R : Type*} [comm_ring R] [is_domain R]
variables (x y : R)

example (h : x^2 = 1) : x = 1 ∨ x = -1 :=
begin
  have h' : x^2 - 1 = 0,
  { rw [h, sub_self] },
  have h'' : (x + 1) * (x - 1) = 0,
  { rw ← h',
    ring },
  cases eq_zero_or_eq_zero_of_mul_eq_zero h'' with h1 h1,
  { right,
    exact eq_neg_iff_add_eq_zero.mpr h1 },
  left,
  exact eq_of_sub_eq_zero h1
end

example (h : x^2 = y^2) : x = y ∨ x = -y :=
begin
  have h' : x^2 - y^2 = 0,
  { rw [h, sub_self] },
  have h'' : (x + y) * (x - y) = 0,
  { rw ← h',
    ring },
  cases eq_zero_or_eq_zero_of_mul_eq_zero h'' with h1 h1,
  { right,
    exact eq_neg_iff_add_eq_zero.mpr h1 },
  left,
  exact eq_of_sub_eq_zero h1
end

end

section
open_locale classical

example (P Q : Prop) : (P → Q) ↔ ¬ P ∨ Q :=
begin
  split,
  { intro h,
    by_cases h' : P,
    { right,
      exact h h'},
    left,
    exact h' },
  rintros (h | h),
  { intro h',
    exact absurd h' h },
  intro _,
  exact h
end

end