feat(library/algebra): missing theorems
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@ -135,6 +135,7 @@ section
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theorem not_lt_of_ge (H : a ≥ b) : ¬ a < b :=
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assume H1 : a < b,
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lt.irrefl _ (lt_of_le_of_lt H H1)
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end
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structure strong_order_pair [class] (A : Type) extends weak_order A, has_lt A :=
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@ -245,6 +246,26 @@ section
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(assume H', absurd (H' ▸ !le.refl) H)
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(assume H', H')
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theorem lt_iff_not_ge : a < b ↔ ¬ a ≥ b :=
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iff.intro
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(suppose a < b, not_le_of_gt this)
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(suppose ¬ a ≥ b, lt_of_not_ge this)
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theorem le_iff_not_gt : a ≤ b ↔ ¬ a > b :=
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iff.intro
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(suppose a ≤ b, not_lt_of_ge this)
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(suppose ¬ a > b, le_of_not_gt this)
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theorem gt_iff_not_le : a > b ↔ ¬ a ≤ b :=
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iff.intro
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(suppose a > b, not_le_of_gt this)
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(suppose ¬ a ≤ b, lt_of_not_ge this)
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theorem ge_iff_not_lt : a ≥ b ↔ ¬ a < b :=
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iff.intro
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(suppose a ≥ b, not_lt_of_ge this)
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(suppose ¬ a < b, le_of_not_gt this)
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theorem lt_or_ge : a < b ∨ a ≥ b :=
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lt.by_cases
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(assume H1 : a < b, or.inl H1)
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@ -588,4 +588,18 @@ section discrete_linear_ordered_field
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have b = 0, from eq_of_le_of_ge (le_of_not_gt Hgt) Hb,
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by rewrite [this, div_zero]; apply le.refl
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theorem div_le_div_of_le_of_nonneg (H : a ≤ b) (Hc : 0 ≤ c) : a / c ≤ b / c :=
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begin
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cases lt_or_eq_of_le Hc with Hlt Heq,
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apply div_le_div_of_le_of_pos H Hlt,
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rewrite [-Heq, 2 div_zero]
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end
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theorem div_le_div_of_le_of_nonpos (H : b ≤ a) (Hc : c ≤ 0) : a / c ≤ b / c :=
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begin
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cases lt_or_eq_of_le Hc with Hlt Heq,
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apply div_le_div_of_le_of_neg H Hlt,
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rewrite [Heq, 2 div_zero]
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end
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end discrete_linear_ordered_field
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