fin-0.1.1: Nat and Fin: peano naturals and finite numbers

Safe HaskellNone
LanguageHaskell2010

Data.Type.Nat.LE.ReflStep

Contents

Synopsis

Relation

data LEProof n m where Source #

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

Constructors

LERefl :: LEProof n n 
LEStep :: LEProof n m -> LEProof n (S m) 
Instances
Eq (LEProof n m) Source #

LEProof values are unique (not Boring though!).

Instance details

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) Source # 
Instance details

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) Source # 
Instance details

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) Source # 
Instance details

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 Source #

Convert from zero+succ to refl+step definition.

Inverse of toZeroSucc.

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

Convert refl+step to zero+succ definition.

Inverse of fromZeroSucc.

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 :: forall n. SNatI n => 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 :: 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 #

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

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 #

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