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def LE.ofOrd (α : Type u) [Ord α] : LE α

Creates an LE α instance from an Ord α instance.

OrientedOrd α must be satisfied so that the resulting LE α instance faithfully represents the Ord α instance.

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def DecidableLE.ofOrd (α : Type u) [LE α] [Ord α] [Std.LawfulOrderOrd α] : DecidableLE α

Creates an DecidableLE α instance using a well-behaved Ord α instance.

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def LT.ofOrd (α : Type u) [Ord α] : LT α

Creates an LT α instance from an Ord α instance.

OrientedOrd α must be satisfied so that the resulting LT α instance faithfully represents the Ord α instance.

theorem Std.isLE_compare {α : Type u} [Ord α] [LE α] [LawfulOrderOrd α] {a b : α} : (compare a b).isLE = true ↔ a ≤ b

theorem Std.isGE_compare {α : Type u} [Ord α] [LE α] [LawfulOrderOrd α] {a b : α} : (compare a b).isGE = true ↔ b ≤ a

theorem Std.compare_eq_lt {α : Type u} [Ord α] [LT α] [LE α] [LawfulOrderOrd α] [LawfulOrderLT α] {a b : α} : compare a b = Ordering.lt ↔ a < b

theorem Std.compare_eq_gt {α : Type u} [Ord α] [LT α] [LE α] [LawfulOrderOrd α] [LawfulOrderLT α] {a b : α} : compare a b = Ordering.gt ↔ b < a

theorem Std.compare_eq_eq {α : Type u} [Ord α] [BEq α] [LE α] [LawfulOrderOrd α] [LawfulOrderBEq α] {a b : α} : compare a b = Ordering.eq ↔ (a == b) = true

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def DecidableLT.ofOrd (α : Type u) [LE α] [LT α] [Ord α] [Std.LawfulOrderOrd α] [Std.LawfulOrderLT α] : DecidableLT α

Creates a DecidableLT α instance using a well-behaved Ord α instance.

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def BEq.ofOrd (α : Type u) [Ord α] : BEq α

Creates a BEq α instance from an Ord α instance.

instance Std.instLawfulOrderOrd_ofOrd (α : Type u) [Ord α] [OrientedOrd α] : LawfulOrderOrd α

The LE α instance obtained from an Ord α instance is compatible with said Ord α instance if compare is oriented, i.e., compare a b = .lt ↔ compare b a = .gt.

instance Std.instLawfulOrderLT_ofOrd {α : Type u} [Ord α] [LE α] [LawfulOrderOrd α] : LawfulOrderLT α

The LT α instance obtained from an Ord α instance is compatible with the LE α instance instance if Ord α is compatible with it.

instance Std.instLawfulOrderBEq_ofOrd {α : Type u} [Ord α] [LE α] [LawfulOrderOrd α] : LawfulOrderBEq α

The BEq α instance obtained from an Ord α instance is compatible with the LE α instance instance if Ord α is compatible with it.