@@ -17,41 +17,40 @@ namespace Smt.Reconstruct.Real
1717variable {a b c d x₁ x₂ y₁ y₂ : Real}
1818
1919theorem sum_ub₁ (h₁ : a < b) (h₂ : c < d) : a + c < b + d := by
20- have r₁ : a + c < a + d := add_lt_add_left h₂ a
21- have r₂ : a + d < b + d := add_lt_add_right h₁ d
20+ have r₁ : a + c < a + d := add_lt_add_right h₂ a
21+ have r₂ : a + d < b + d := add_lt_add_left h₁ d
2222 exact lt_trans r₁ r₂
2323
2424theorem sum_ub₂ (h₁ : a < b) (h₂ : c ≤ d) : a + c < b + d := by
25- have r₁ : a + c ≤ a + d := add_le_add_left h₂ a
26- have r₂ : a + d < b + d := add_lt_add_right h₁ d
25+ have r₁ : a + c ≤ a + d := add_le_add_right h₂ a
26+ have r₂ : a + d < b + d := add_lt_add_left h₁ d
2727 exact lt_of_le_of_lt r₁ r₂
2828
2929theorem sum_ub₃ (h₁ : a < b) (h₂ : c = d) : a + c < b + d := by
3030 rewrite [h₂]
31- exact add_lt_add_right h₁ d
31+ exact add_lt_add_left h₁ d
3232
3333theorem sum_ub₄ (h₁ : a ≤ b) (h₂ : c < d) : a + c < b + d := by
34- have r₁ : a + c < a + d := add_lt_add_left h₂ a
35- have r₂ : a + d ≤ b + d := add_le_add_right h₁ d
34+ have r₁ : a + c < a + d := add_lt_add_right h₂ a
35+ have r₂ : a + d ≤ b + d := add_le_add_left h₁ d
3636 exact lt_of_lt_of_le r₁ r₂
3737
3838theorem sum_ub₅ (h₁ : a ≤ b) (h₂ : c ≤ d) : a + c ≤ b + d := by
39- have r₁ : a + c ≤ a + d := add_le_add_left h₂ a
40- have r₂ : a + d ≤ b + d := add_le_add_right h₁ d
39+ have r₁ : a + c ≤ a + d := add_le_add_right h₂ a
40+ have r₂ : a + d ≤ b + d := add_le_add_left h₁ d
4141 exact le_trans r₁ r₂
4242
4343theorem sum_ub₆ (h₁ : a ≤ b) (h₂ : c = d) : a + c ≤ b + d := by
4444 rewrite [h₂]
45- exact add_le_add_right h₁ d
45+ exact add_le_add_left h₁ d
4646
4747theorem sum_ub₇ (h₁ : a = b) (h₂ : c < d) : a + c < b + d := by
4848 rewrite [h₁]
49- exact add_lt_add_left h₂ b
49+ exact add_lt_add_right h₂ b
5050
5151theorem sum_ub₈ (h₁ : a = b) (h₂ : c ≤ d) : a + c ≤ b + d := by
5252 rewrite [h₁]
53- exact add_le_add_left h₂ b
54-
53+ exact add_le_add_right h₂ b
5554theorem sum_ub₉ (h₁ : a = b) (h₂ : c = d) : a + c = b + d := by
5655 rw [h₁, h₂]
5756
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