Safe Haskell	None
Language	Haskell2010

Data.Type.Nat.LE

Contents

Relation
Decidability
Lemmas

Description

Less-than-or-equal relation for (unary) natural numbers Nat.

There are at least three ways to encode this relation.

\(zero : 0 \le m\) and \(succ : n \le m \to 1 + n \le 1 + m\) (this module),
\(refl : n \le n \) and \(step : n \le m \to n \le 1 + m\) (Data.Type.Nat.LE.ReflStep),
\(ex : \exists p. n + p \equiv m \) (tricky in Haskell).

Depending on a situation, usage ergonomics are different.

Synopsis

class LE n m where
- leProof :: LEProof n m
data LEProof n m where
- LEZero :: LEProof Z m
- LESucc :: LEProof n m -> LEProof (S n) (S m)
withLEProof :: LEProof n m -> (LE n m => r) -> r
decideLE :: forall n m. (SNatI n, SNatI m) => Dec (LEProof n m)
leZero :: LEProof Z n
leSucc :: LEProof n m -> LEProof (S n) (S m)
leRefl :: forall n. SNatI n => LEProof n n
leStep :: LEProof n m -> LEProof n (S m)
leAsym :: LEProof n m -> LEProof m n -> n :~: m
leTrans :: LEProof n m -> LEProof m p -> LEProof n p
leSwap :: forall n m. (SNatI n, SNatI m) => Neg (LEProof n m) -> LEProof (S m) n
leSwap' :: LEProof n m -> LEProof (S m) n -> void
leStepL :: LEProof (S n) m -> LEProof n m
lePred :: LEProof (S n) (S m) -> LEProof n m
proofZeroLEZero :: LEProof n Z -> n :~: Z

Relation

class LE n m where Source #

Total order of Nat, less-than-or-Equal-to, \( \le \).

Methods

leProof :: LEProof n m Source #

Instances

LE Z m Source #
Instance details Defined in Data.Type.Nat.LE Methods leProof :: LEProof Z m Source #
(m ~ S m', LE n m') => LE (S n) m Source #
Instance details Defined in Data.Type.Nat.LE Methods leProof :: LEProof (S n) m Source #

data LEProof n m where Source #

An evidence of \(n \le m\). zero+succ definition.

Constructors

LEZero :: LEProof Z m
LESucc :: LEProof n m -> LEProof (S n) (S m)

Instances

Eq (LEProof n m) Source #	`LEProof` values are unique (not `Boring` though!).
Instance details Defined in Data.Type.Nat.LE Methods (==) :: LEProof n m -> LEProof n m -> Bool # (/=) :: LEProof n m -> LEProof n m -> Bool #
Ord (LEProof n m) Source #
Instance details Defined in Data.Type.Nat.LE Methods compare :: LEProof n m -> LEProof n m -> Ordering # (<) :: LEProof n m -> LEProof n m -> Bool # (<=) :: LEProof n m -> LEProof n m -> Bool # (>) :: LEProof n m -> LEProof n m -> Bool # (>=) :: LEProof n m -> LEProof n m -> Bool # max :: LEProof n m -> LEProof n m -> LEProof n m # min :: LEProof n m -> LEProof n m -> LEProof n m #
Show (LEProof n m) Source #
Instance details Defined in Data.Type.Nat.LE Methods showsPrec :: Int -> LEProof n m -> ShowS # show :: LEProof n m -> String # showList :: [LEProof n m] -> ShowS #
(SNatI n, SNatI m) => Decidable (LEProof n m) Source #
Instance details Defined in Data.Type.Nat.LE Methods decide :: Dec (LEProof n m) #

withLEProof :: LEProof n m -> (LE n m => r) -> r Source #

Constructor LE dictionary from LEProof.

Decidability

decideLE :: forall n m. (SNatI n, SNatI m) => Dec (LEProof n m) Source #

Find the LEProof n m, i.e. compare numbers.

Lemmas

Constructor like

leZero :: LEProof Z n Source #

\(\forall n : \mathbb{N}, 0 \le n \)

leSucc :: LEProof n m -> LEProof (S n) (S m) Source #

\(\forall n\, m : \mathbb{N}, n \le m \to 1 + n \le 1 + m \)

leRefl :: forall n. SNatI n => LEProof n n Source #

\(\forall n : \mathbb{N}, n \le n \)

leStep :: LEProof n m -> LEProof n (S m) Source #

\(\forall n\, m : \mathbb{N}, n \le m \to n \le 1 + m \)

Partial order

leAsym :: LEProof n m -> LEProof m n -> n :~: m Source #

\(\forall n\, m : \mathbb{N}, n \le m \to m \le n \to n \equiv m \)

leTrans :: LEProof n m -> LEProof m p -> LEProof n p Source #

\(\forall n\, m\, p : \mathbb{N}, n \le m \to m \le p \to n \le p \)

Total order

leSwap :: forall n m. (SNatI n, SNatI m) => Neg (LEProof n m) -> LEProof (S m) n Source #

\(\forall n\, m : \mathbb{N}, \neg (n \le m) \to 1 + m \le n \)

leSwap' :: LEProof n m -> LEProof (S m) n -> void Source #

\(\forall n\, m : \mathbb{N}, n \le m \to \neg (1 + m \le n) \)

>>> leProof :: LEProof Nat2 Nat3
LESucc (LESucc LEZero)

>>> leSwap (leSwap' (leProof :: LEProof Nat2 Nat3))
LESucc (LESucc (LESucc LEZero))

>>> lePred (leSwap (leSwap' (leProof :: LEProof Nat2 Nat3)))
LESucc (LESucc LEZero)

More

leStepL :: LEProof (S n) m -> LEProof n m Source #

\(\forall n\, m : \mathbb{N}, 1 + n \le m \to n \le m \)

lePred :: LEProof (S n) (S m) -> LEProof n m Source #

\(\forall n\, m : \mathbb{N}, 1 + n \le 1 + m \to n \le m \)

proofZeroLEZero :: LEProof n Z -> n :~: Z Source #

\(\forall n\ : \mathbb{N}, n \le 0 \to n \equiv 0 \)