Data.Type.Nat.LE.ReflStep

Relation

data LEProof n m where

An evidence of \(n \le m\). refl+step definition.

Constructors

LERefl :: LEProof n n

LEStep :: LEProof n m -> LEProof n ('S m)

Instances

Category LEProof

The other variant (LEPRoof) isn't Category, because leRefl requires SNat evidence.

Defined in Data.Type.Nat.LE.ReflStep

Methods

id :: forall (a :: k). LEProof a a

(.) :: forall (b :: k) (c :: k) (a :: k). LEProof b c -> LEProof a b -> LEProof a c

Eq (LEProof n m)

LEProof values are unique (not Boring though!).

Defined in Data.Type.Nat.LE.ReflStep

Methods

(==) :: LEProof n m -> LEProof n m -> Bool

(/=) :: LEProof n m -> LEProof n m -> Bool

Ord (LEProof n m)

Defined in Data.Type.Nat.LE.ReflStep

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)

Defined in Data.Type.Nat.LE.ReflStep

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)

Defined in Data.Type.Nat.LE.ReflStep

Methods

decide :: Dec (LEProof n m)

fromZeroSucc :: forall n m. SNatI m => LEProof n m -> LEProof n m

Convert from zero+succ to refl+step definition.

Inverse of toZeroSucc.

toZeroSucc :: SNatI n => LEProof n m -> LEProof n m

Convert refl+step to zero+succ definition.

Inverse of fromZeroSucc.

Decidability

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

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

Lemmas

Constructor like

leZero :: forall n. SNatI n => LEProof 'Z n

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

leSucc :: LEProof n m -> LEProof ('S n) ('S m)

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

leRefl :: LEProof n n

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

leStep :: LEProof n m -> LEProof n ('S m)

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

Partial order

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

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

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

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

Total order

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

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

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

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

More

leStepL :: LEProof ('S n) m -> LEProof n m

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

lePred :: LEProof ('S n) ('S m) -> LEProof n m

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

proofZeroLEZero :: LEProof n 'Z -> n :~: 'Z

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