Helper definitions and instances for Ordering

def Ordering.Compares {α : Type u_1} [LT α] : Ordering → α → α → Prop

Compares o a b means that a and b have the ordering relation o between them, assuming that the relation a < b is defined.

@[simp]

theorem Ordering.compares_lt {α : Type u_1} [LT α] (a b : α) : lt.Compares a b = (a < b)

@[simp]

theorem Ordering.compares_eq {α : Type u_1} [LT α] (a b : α) : eq.Compares a b = (a = b)

@[simp]

theorem Ordering.compares_gt {α : Type u_1} [LT α] (a b : α) : gt.Compares a b = (a > b)

@[macro_inline]

def Ordering.dthen (o : Ordering) : (o = eq → Ordering) → Ordering

o₁.dthen fun h => o₂(h) is like o₁.then o₂ but o₂ is allowed to depend on h : o₁ = .eq.

def cmpUsing {α : Type u} (lt : α → α → Prop) [DecidableRel lt] (a b : α) : Ordering

Lift a decidable relation to an Ordering, assuming that incomparable terms are Ordering.eq.

def cmp {α : Type u} [LT α] [DecidableLT α] (a b : α) : Ordering

Construct an Ordering from a type with a decidable LT instance, assuming that incomparable terms are Ordering.eq.