# Documentation

## Init.Data.Ord

inductive Ordering :
Instances For
Equations
• x.instDecidableEq y = if h : x.toCtorIdx = y.toCtorIdx then else

Swaps less and greater ordering results

Equations
Instances For
@[macro_inline]

If o₁ and o₂ are Ordering, then o₁.then o₂ returns o₁ unless it is .eq, in which case it returns o₂. Additionally, it has "short-circuiting" semantics similar to boolean x && y: if o₁ is not .eq then the expression for o₂ is not evaluated. This is a useful primitive for constructing lexicographic comparator functions:

structure Person where
name : String
age : Nat

instance : Ord Person where
compare a b := (compare a.name b.name).then (compare b.age a.age)


This example will sort people first by name (in ascending order) and will sort people with the same name by age (in descending order). (If all fields are sorted ascending and in the same order as they are listed in the structure, you can also use deriving Ord on the structure definition for the same effect.)

Equations
• x✝.then x = match x✝, x with | Ordering.eq, f => f | o, x => o
Instances For

Check whether the ordering is 'equal'.

Equations
Instances For

Check whether the ordering is 'not equal'.

Equations
Instances For

Check whether the ordering is 'less than or equal to'.

Equations
Instances For

Check whether the ordering is 'less than'.

Equations
Instances For

Check whether the ordering is 'greater than'.

Equations
Instances For

Check whether the ordering is 'greater than or equal'.

Equations
Instances For
@[inline]
def compareOfLessAndEq {α : Type u_1} (x : α) (y : α) [LT α] [Decidable (x < y)] [] :

Yields an Ordering s.t. x < y corresponds to Ordering.lt / Ordering.gt and x = y corresponds to Ordering.eq.

Equations
Instances For
@[inline]
def compareOfLessAndBEq {α : Type u_1} (x : α) (y : α) [LT α] [Decidable (x < y)] [BEq α] :

Yields an Ordering s.t. x < y corresponds to Ordering.lt / Ordering.gt and x == y corresponds to Ordering.eq.

Equations
Instances For
@[inline]
def compareLex {α : Sort u_1} {β : Sort u_2} (cmp₁ : αβOrdering) (cmp₂ : αβOrdering) (a : α) (b : β) :

Compare a and b lexicographically by cmp₁ and cmp₂. a and b are first compared by cmp₁. If this returns 'equal', a and b are compared by cmp₂ to break the tie.

Equations
• compareLex cmp₁ cmp₂ a b = (cmp₁ a b).then (cmp₂ a b)
Instances For
class Ord (α : Type u) :

Ord α provides a computable total order on α, in terms of the compare : α → α → Ordering function.

Typically instances will be transitive, reflexive, and antisymmetric, but this is not enforced by the typeclass.

There is a derive handler, so appending deriving Ord to an inductive type or structure will attempt to create an Ord instance.

• compare : ααOrdering

Compare two elements in α using the comparator contained in an [Ord α] instance.

Instances
@[inline]
def compareOn {β : Type u_1} {α : Sort u_2} [ord : Ord β] (f : αβ) (x : α) (y : α) :

Compare x and y by comparing f x and f y.

Equations
Instances For
instance instOrdNat :
Equations
instance instOrdInt :
Equations
instance instOrdBool :
Equations
instance instOrdString :
Equations
instance instOrdFin (n : Nat) :
Ord (Fin n)
Equations
• = { compare := fun (x y : Fin n) => compare x y }
instance instOrdUInt8 :
Equations
instance instOrdUInt16 :
Equations
instance instOrdUInt32 :
Equations
instance instOrdUInt64 :
Equations
instance instOrdUSize :
Equations
instance instOrdChar :
Equations
instance instOrdOption {α : Type u_1} [Ord α] :
Ord ()
Equations
• One or more equations did not get rendered due to their size.
def lexOrd {α : Type u_1} {β : Type u_2} [Ord α] [Ord β] :
Ord (α × β)

The lexicographic order on pairs.

Equations
Instances For
def ltOfOrd {α : Type u_1} [Ord α] :
LT α
Equations
• ltOfOrd = { lt := fun (a b : α) => }
Instances For
instance instDecidableRelLt {α : Type u_1} [Ord α] :
Equations
def leOfOrd {α : Type u_1} [Ord α] :
LE α
Equations
Instances For
instance instDecidableRelLe {α : Type u_1} [Ord α] :
Equations
def Ord.toBEq {α : Type u_1} (ord : Ord α) :
BEq α

Derive a BEq instance from an Ord instance.

Equations
• ord.toBEq = { beq := fun (x y : α) => }
Instances For
def Ord.toLT {α : Type u_1} :
Ord αLT α

Derive an LT instance from an Ord instance.

Equations
• x.toLT = ltOfOrd
Instances For
def Ord.toLE {α : Type u_1} :
Ord αLE α

Derive an LE instance from an Ord instance.

Equations
• x.toLE = leOfOrd
Instances For
def Ord.opposite {α : Type u_1} (ord : Ord α) :
Ord α

Invert the order of an Ord instance.

Equations
• ord.opposite = { compare := fun (x y : α) => compare y x }
Instances For
def Ord.on {β : Type u_1} {α : Type u_2} :
Ord β(αβ)Ord α

ord.on f compares x and y by comparing f x and f y according to ord.

Equations
• x.on f = { compare := }
Instances For
def Ord.lex {α : Type u_1} {β : Type u_2} :
Ord αOrd βOrd (α × β)

Derive the lexicographic order on products α × β from orders for α and β.

Equations
• x✝.lex x = lexOrd
Instances For
def Ord.lex' {α : Type u_1} (ord₁ : Ord α) (ord₂ : Ord α) :
Ord α

Create an order which compares elements first by ord₁ and then, if this returns 'equal', by ord₂.

Equations
• ord₁.lex' ord₂ = { compare := compareLex compare compare }
Instances For
def Ord.arrayOrd {α : Type u_1} [a : Ord α] :
Ord ()

Creates an order which compares elements of an Array in lexicographic order.

Equations
• Ord.arrayOrd = { compare := fun (x y : ) => let x_1 := a.toLT; let x_2 := a.toBEq; compareOfLessAndBEq x.toList y.toList }
Instances For