content: {-# OPTIONS_GHC -Werror=incomplete-patterns #-}
  {-# LANGUAGE FlexibleInstances #-}
  {-# LANGUAGE MultiParamTypeClasses #-}
  {-# LANGUAGE FunctionalDependencies #-}
  {-# LANGUAGE TypeOperators #-}
  
  module PredicateSl where
  
  import Sl
  
  -- Universal Quantifier Introduction (UI)
  class Ui p where
    ui :: (a -> p) -> (forall x. a -> p)
  instance Ui p where
    ui f = f
  
  -- Universal Quantifier Elimination (UE)
  class Ue p a where
    ue :: (forall x. a -> p) -> a -> p
  instance Ue p a where
    ue f x = f x
  
  -- Existential Quantifier Introduction (EI)
  class Ei a p where
    ei :: a -> (exists x. p)
  instance Ei a p where
    ei x = undefined  -- Placeholder for actual implementation
  
  -- Existential Quantifier Elimination (EE)
  class Ee p b where
    ee :: (exists x. p) -> (forall x. p -> b) -> b
  instance Ee p b where
    ee ex f = undefined  -- Placeholder for actual implementation
  
  -- Predicate (PRED)
  class Pred a b where
    pred :: (a -> b) -> (a -> b)
  instance Pred a b where
    pred f = f
  
  -- Example predicates
  data Person
  data Loves
  data Food
  
  -- Example of using predicates
  johnLovesPizza :: Person -> Loves -> Food -> Bool
  johnLovesPizza john loves pizza = True  -- Example predicate logic
  
  -- Example universal quantifier usage
  everyoneLovesPizza :: (Person -> Loves -> Food -> Bool) -> (forall x. Person -> Loves -> Food -> Bool)
  everyoneLovesPizza f = ui f
  
  -- Example existential quantifier usage
  someoneLovesPizza :: (Person -> Loves -> Food -> Bool) -> (exists x. Person -> Loves -> Food -> Bool)
  someoneLovesPizza f = ei (johnLovesPizza undefined undefined undefined)
  
  main :: IO ()
  main = putStrLn "Predicate Logic Proof Checks"
  
  