{-# language DataKinds #-}
{-# language ExplicitForAll #-}
{-# language KindSignatures #-}
{-# language TypeFamilies #-}
{-# language TypeOperators #-}
module Arithmetic.Lte
(
zero
, reflexive
, substituteL
, substituteR
, incrementL
, incrementR
, decrementL
, decrementR
, weakenL
, weakenR
, transitive
, plus
, fromStrict
, constant
) where
import Arithmetic.Unsafe (type (<)(Lt),type (:=:)(Eq))
import Arithmetic.Unsafe (type (<=)(Lte))
import GHC.TypeNats (CmpNat,type (+))
import qualified GHC.TypeNats as GHC
substituteL :: (b :=: c) -> (b <= a) -> (c <= a)
substituteL :: (b :=: c) -> (b <= a) -> c <= a
substituteL b :=: c
Eq b <= a
Lte = c <= a
forall (a :: Nat) (b :: Nat). a <= b
Lte
substituteR :: (b :=: c) -> (a <= b) -> (a <= c)
substituteR :: (b :=: c) -> (a <= b) -> a <= c
substituteR b :=: c
Eq a <= b
Lte = a <= c
forall (a :: Nat) (b :: Nat). a <= b
Lte
plus :: (a <= b) -> (c <= d) -> (a + c <= b + d)
plus :: (a <= b) -> (c <= d) -> (a + c) <= (b + d)
plus a <= b
Lte c <= d
Lte = (a + c) <= (b + d)
forall (a :: Nat) (b :: Nat). a <= b
Lte
transitive :: (a <= b) -> (b <= c) -> (a <= c)
transitive :: (a <= b) -> (b <= c) -> a <= c
transitive a <= b
Lte b <= c
Lte = a <= c
forall (a :: Nat) (b :: Nat). a <= b
Lte
reflexive :: a <= a
reflexive :: a <= a
reflexive = a <= a
forall (a :: Nat) (b :: Nat). a <= b
Lte
incrementL :: forall (c :: GHC.Nat) (a :: GHC.Nat) (b :: GHC.Nat).
(a <= b) -> (c + a <= c + b)
incrementL :: (a <= b) -> (c + a) <= (c + b)
incrementL a <= b
Lte = (c + a) <= (c + b)
forall (a :: Nat) (b :: Nat). a <= b
Lte
incrementR :: forall (c :: GHC.Nat) (a :: GHC.Nat) (b :: GHC.Nat).
(a <= b) -> (a + c <= b + c)
incrementR :: (a <= b) -> (a + c) <= (b + c)
incrementR a <= b
Lte = (a + c) <= (b + c)
forall (a :: Nat) (b :: Nat). a <= b
Lte
weakenL :: forall (c :: GHC.Nat) (a :: GHC.Nat) (b :: GHC.Nat).
(a <= b) -> (a <= c + b)
weakenL :: (a <= b) -> a <= (c + b)
weakenL a <= b
Lte = a <= (c + b)
forall (a :: Nat) (b :: Nat). a <= b
Lte
weakenR :: forall (c :: GHC.Nat) (a :: GHC.Nat) (b :: GHC.Nat).
(a <= b) -> (a <= b + c)
weakenR :: (a <= b) -> a <= (b + c)
weakenR a <= b
Lte = a <= (b + c)
forall (a :: Nat) (b :: Nat). a <= b
Lte
decrementL :: forall (c :: GHC.Nat) (a :: GHC.Nat) (b :: GHC.Nat).
(c + a <= c + b) -> (a <= b)
decrementL :: ((c + a) <= (c + b)) -> a <= b
decrementL (c + a) <= (c + b)
Lte = a <= b
forall (a :: Nat) (b :: Nat). a <= b
Lte
decrementR :: forall (c :: GHC.Nat) (a :: GHC.Nat) (b :: GHC.Nat).
(a + c <= b + c) -> (a <= b)
decrementR :: ((a + c) <= (b + c)) -> a <= b
decrementR (a + c) <= (b + c)
Lte = a <= b
forall (a :: Nat) (b :: Nat). a <= b
Lte
fromStrict :: (a < b) -> (a <= b)
fromStrict :: (a < b) -> a <= b
fromStrict a < b
Lt = a <= b
forall (a :: Nat) (b :: Nat). a <= b
Lte
zero :: 0 <= a
zero :: 0 <= a
zero = 0 <= a
forall (a :: Nat) (b :: Nat). a <= b
Lte
constant :: forall a b. (IsLte (CmpNat a b) ~ 'True) => (a <= b)
constant :: a <= b
constant = a <= b
forall (a :: Nat) (b :: Nat). a <= b
Lte
type family IsLte (o :: Ordering) :: Bool where
IsLte 'GT = 'False
IsLte 'LT = 'True
IsLte 'EQ = 'True