-- Hoogle documentation, generated by Haddock
-- See Hoogle, http://www.haskell.org/hoogle/


-- | A library for formulating and solving math programs.
--   
--   Please see the README on GitHub at
--   <a>https://github.com/prsteele/math-programming#readme</a>
@package math-programming
@version 0.4.0

module Math.Programming.Types

-- | A convient shorthand for the type of linear expressions used in a
--   given model.
type Expr m = LinearExpression (Numeric m) (Variable m)

-- | A monad for formulating and solving linear programs.
--   
--   We manipulate linear programs and their settings using the
--   <tt>Mutable</tt> typeclass.
class (Monad m, Show (Numeric m), RealFrac (Numeric m)) => LPMonad m where {
    
    -- | The numeric type used in the model.
    type family Numeric m :: *;
    
    -- | The type of variables in the model. <a>LPMonad</a> treats these as
    --   opaque values, but instances may expose more details.
    data family Variable m :: *;
    
    -- | The type of constraints in the model. <a>LPMonad</a> treats these as
    --   opaque values, but instances may expose more details.
    data family Constraint m :: *;
    
    -- | The type of objectives in the model. <a>LPMonad</a> treats these as
    --   opaque values, but instances may expose more details.
    data family Objective m :: *;
}

-- | Create a new decision variable in the model.
--   
--   This variable will be initialized to be a non-negative continuous
--   variable.
addVariable :: LPMonad m => m (Variable m)

-- | Remove a decision variable from the model.
--   
--   The variable cannot be used after being deleted.
removeVariable :: LPMonad m => Variable m -> m ()

-- | Get the name of the variable.
getVariableName :: LPMonad m => Variable m -> m String

-- | Set the name of the variable.
setVariableName :: LPMonad m => Variable m -> String -> m ()

-- | Get the allowed values of a variable.
getVariableBounds :: LPMonad m => Variable m -> m (Bounds (Numeric m))

-- | Constrain a variable to take on certain values.
setVariableBounds :: LPMonad m => Variable m -> Bounds (Numeric m) -> m ()

-- | Get the value of a variable in the current solution.
getVariableValue :: LPMonad m => Variable m -> m (Numeric m)

-- | Add a constraint to the model represented by an inequality.
addConstraint :: LPMonad m => Inequality (LinearExpression (Numeric m) (Variable m)) -> m (Constraint m)

-- | Remove a constraint from the model.
--   
--   The constraint cannot used after being deleted.
removeConstraint :: LPMonad m => Constraint m -> m ()

-- | Get the name of the constraint.
getConstraintName :: LPMonad m => Constraint m -> m String

-- | Set the name of the constraint.
setConstraintName :: LPMonad m => Constraint m -> String -> m ()

-- | Get the value of the dual variable associated with the constraint in
--   the current solution.
--   
--   This value has no meaning if the current solution is not an LP
--   solution.
getDualValue :: LPMonad m => Constraint m -> m (Numeric m)

-- | Add a constraint to the model represented by an inequality.
addObjective :: LPMonad m => LinearExpression (Numeric m) (Variable m) -> m (Objective m)

-- | Get the name of the objective.
getObjectiveName :: LPMonad m => Objective m -> m String

-- | Set the name of the objective.
setObjectiveName :: LPMonad m => Objective m -> String -> m ()

-- | Whether the objective is to be minimized or maximized.
getObjectiveSense :: LPMonad m => Objective m -> m Sense

-- | Set whether the objective is to be minimized or maximized.
setObjectiveSense :: LPMonad m => Objective m -> Sense -> m ()

-- | Get the value of the objective in the current solution.
getObjectiveValue :: LPMonad m => Objective m -> m (Numeric m)

-- | Get the number of seconds the solver is allowed to run before halting.
getTimeout :: LPMonad m => m Double

-- | Set the number of seconds the solver is allowed to run before halting.
setTimeout :: LPMonad m => Double -> m ()

-- | Optimize the continuous relaxation of the model.
optimizeLP :: LPMonad m => m SolutionStatus

-- | A (mixed) integer program.
--   
--   In addition to the methods of the <a>LPMonad</a> class, this monad
--   supports constraining variables to be either continuous or discrete.
class (LPMonad m) => IPMonad m

-- | Optimize the mixed-integer program.
optimizeIP :: IPMonad m => m SolutionStatus

-- | Get the domain of a variable.
getVariableDomain :: IPMonad m => Variable m -> m Domain

-- | Set the domain of a variable.
setVariableDomain :: IPMonad m => Variable m -> Domain -> m ()

-- | Get the allowed relative gap between LP and IP solutions.
getRelativeMIPGap :: IPMonad m => m Double

-- | Set the allowed relative gap between LP and IP solutions.
setRelativeMIPGap :: IPMonad m => Double -> m ()

-- | Whether a math program is minimizing or maximizing its objective.
data Sense
Minimization :: Sense
Maximization :: Sense

-- | The outcome of an optimization.
data SolutionStatus

-- | An optimal solution has been found.
Optimal :: SolutionStatus

-- | A feasible solution has been found. The result may or may not be
--   optimal.
Feasible :: SolutionStatus

-- | The model has been proven to be infeasible.
Infeasible :: SolutionStatus

-- | The model has been proven to be unbounded.
Unbounded :: SolutionStatus

-- | An error was encountered during the solve. Instance-specific methods
--   should be used to determine what occurred.
Error :: SolutionStatus

-- | An interval of the real numbers.
data Bounds b

-- | The non-negative reals.
NonNegativeReals :: Bounds b

-- | The non-positive reals.
NonPositiveReals :: Bounds b

-- | Any closed interval of the reals.
Interval :: b -> b -> Bounds b

-- | Any real number.
Free :: Bounds b

-- | The type of values that a variable can take on.
--   
--   Note that the <tt>Integer</tt> constructor does not interfere with the
--   <tt>Integer</tt> type, as the <tt>Integer</tt> type does not define a
--   constuctor of the same name. The ambiguity is unfortunate, but other
--   natural nomenclature such as <tt>Integral</tt> are similarly
--   conflicted.
data Domain

-- | The variable lies in the real numbers
Continuous :: Domain

-- | The variable lies in the integers
Integer :: Domain

-- | The variable lies in the set <tt>{0, 1}</tt>.
Binary :: Domain
class Nameable m a
getName :: Nameable m a => a -> m String
setName :: Nameable m a => a -> String -> m ()

-- | A linear expression containing symbolic variables of type <tt>b</tt>
--   and numeric coefficients of type <tt>a</tt>.
--   
--   Using <a>String</a>s to denote variables and <a>Double</a>s as our
--   numeric type, we could express <i>3 x + 2 y + 1</i> as
--   
--   <pre>
--   LinearExpression [(3, "x"), (2, "y")] 1
--   </pre>
data LinearExpression a b
LinearExpression :: [(a, b)] -> a -> LinearExpression a b

-- | Non-strict inequalities.
data Inequality a
Inequality :: Ordering -> a -> a -> Inequality a
instance GHC.Show.Show a => GHC.Show.Show (Math.Programming.Types.Inequality a)
instance GHC.Read.Read a => GHC.Read.Read (Math.Programming.Types.Inequality a)
instance (GHC.Show.Show a, GHC.Show.Show b) => GHC.Show.Show (Math.Programming.Types.LinearExpression a b)
instance (GHC.Read.Read a, GHC.Read.Read b) => GHC.Read.Read (Math.Programming.Types.LinearExpression a b)
instance GHC.Show.Show Math.Programming.Types.Domain
instance GHC.Read.Read Math.Programming.Types.Domain
instance GHC.Show.Show b => GHC.Show.Show (Math.Programming.Types.Bounds b)
instance GHC.Read.Read b => GHC.Read.Read (Math.Programming.Types.Bounds b)
instance GHC.Show.Show Math.Programming.Types.SolutionStatus
instance GHC.Read.Read Math.Programming.Types.SolutionStatus
instance GHC.Classes.Ord Math.Programming.Types.SolutionStatus
instance GHC.Classes.Eq Math.Programming.Types.SolutionStatus
instance GHC.Show.Show Math.Programming.Types.Sense
instance GHC.Read.Read Math.Programming.Types.Sense
instance GHC.Classes.Ord Math.Programming.Types.Sense
instance GHC.Classes.Eq Math.Programming.Types.Sense
instance Math.Programming.Types.LPMonad m => Math.Programming.Types.Nameable m (Math.Programming.Types.Variable m)
instance Math.Programming.Types.LPMonad m => Math.Programming.Types.Nameable m (Math.Programming.Types.Constraint m)
instance Math.Programming.Types.LPMonad m => Math.Programming.Types.Nameable m (Math.Programming.Types.Objective m)
instance GHC.Base.Functor Math.Programming.Types.Inequality
instance Data.Foldable.Foldable Math.Programming.Types.Inequality
instance Data.Traversable.Traversable Math.Programming.Types.Inequality
instance GHC.Num.Num a => GHC.Base.Semigroup (Math.Programming.Types.LinearExpression a b)
instance GHC.Num.Num a => GHC.Base.Monoid (Math.Programming.Types.LinearExpression a b)
instance GHC.Base.Functor (Math.Programming.Types.LinearExpression a)
instance Data.Bifunctor.Bifunctor Math.Programming.Types.LinearExpression
instance Data.Foldable.Foldable (Math.Programming.Types.LinearExpression a)
instance Data.Traversable.Traversable (Math.Programming.Types.LinearExpression a)

module Math.Programming.Dsl

-- | Create an objective to be minimized.
minimize :: LPMonad m => Expr m -> m (Objective m)

-- | Create an objective to be maximized.
maximize :: LPMonad m => Expr m -> m (Objective m)

-- | Get the value of a linear expression in the current solution.
evalExpr :: LPMonad m => Expr m -> m (Numeric m)

-- | Create a new free variable.
free :: LPMonad m => m (Variable m)

-- | Create a new non-negative variable.
nonNeg :: LPMonad m => m (Variable m)

-- | Create a new non-positive variable.
nonPos :: LPMonad m => m (Variable m)

-- | Create a new variable bounded between two values.
bounded :: LPMonad m => Numeric m -> Numeric m -> m (Variable m)

-- | Constrain a variable to take on certain values.
--   
--   This function is designed to be used as an infix operator, e.g.
--   
--   <pre>
--   <a>addVariable</a> `<a>within</a>` <a>NonNegativeReals</a>
--   </pre>
within :: LPMonad m => m (Variable m) -> Bounds (Numeric m) -> m (Variable m)

-- | Create an integer-valued variable.
integer :: IPMonad m => m (Variable m)

-- | Create a binary variable.
binary :: IPMonad m => m (Variable m)

-- | Create an integer-value variable that takes on non-negative values.
nonNegInteger :: IPMonad m => m (Variable m)

-- | Create an integer-value variable that takes on non-positive values.
nonPosInteger :: IPMonad m => m (Variable m)

-- | Set the type of a variable.
--   
--   This function is designed to be used as an infix operator, e.g.
--   
--   <pre>
--   <a>addVariable</a> `<a>asKind</a>` <a>Binary</a>
--   </pre>
asKind :: IPMonad m => m (Variable m) -> Domain -> m (Variable m)

-- | Name a variable, constraint, or objective.
--   
--   This function is designed to be used as an infix operator, e.g.
--   
--   <pre>
--   <a>free</a> `<a>named</a>` <a>X_1</a>
--   </pre>
named :: (Monad m, Nameable m a) => m a -> String -> m a

-- | Retrieve the name of a variable, constraint, or objective.
nameOf :: (Monad m, Nameable m a) => a -> m String
(#+@) :: Num a => a -> b -> LinearExpression a b
infixl 6 #+@
(#+.) :: Num a => a -> LinearExpression a b -> LinearExpression a b
infixl 6 #+.
(@+#) :: Num a => b -> a -> LinearExpression a b
infixl 6 @+#
(@+@) :: Num a => b -> b -> LinearExpression a b
infixl 6 @+@
(@+.) :: Num a => b -> LinearExpression a b -> LinearExpression a b
infixl 6 @+.
(.+#) :: Num a => LinearExpression a b -> a -> LinearExpression a b
infixl 6 .+#
(.+@) :: Num a => LinearExpression a b -> b -> LinearExpression a b
infixl 6 .+@
(.+.) :: Num a => LinearExpression a b -> LinearExpression a b -> LinearExpression a b
infixl 6 .+.
(#-@) :: Num a => a -> b -> LinearExpression a b
infixl 6 #-@
(#-.) :: Num a => a -> LinearExpression a b -> LinearExpression a b
infixl 6 #-.
(@-#) :: Num a => b -> a -> LinearExpression a b
infixl 6 @-#
(@-@) :: Num a => b -> b -> LinearExpression a b
infixl 6 @-@
(@-.) :: Num a => b -> LinearExpression a b -> LinearExpression a b
infixl 6 @-.
(.-#) :: Num a => LinearExpression a b -> a -> LinearExpression a b
infixl 6 .-#
(.-@) :: Num a => LinearExpression a b -> b -> LinearExpression a b
infixl 6 .-@
(.-.) :: Num a => LinearExpression a b -> LinearExpression a b -> LinearExpression a b
infixl 6 .-.
(#*.) :: Num a => a -> LinearExpression a b -> LinearExpression a b
infixl 7 #*.
(.*#) :: Num a => LinearExpression a b -> a -> LinearExpression a b
infixl 7 .*#
(#*@) :: Num a => a -> b -> LinearExpression a b
infixl 7 #*@
(@*#) :: Num a => b -> a -> LinearExpression a b
infixl 7 @*#
(@/#) :: Fractional a => b -> a -> LinearExpression a b
infixl 7 @/#
(./#) :: Fractional a => LinearExpression a b -> a -> LinearExpression a b
infixl 7 ./#

-- | Combine equivalent terms by summing their coefficients.
simplify :: (Ord b, Num a) => LinearExpression a b -> LinearExpression a b

-- | Reduce an expression to its value.
eval :: Num a => LinearExpression a a -> a

-- | Construct an expression representing a variable.
var :: Num a => b -> LinearExpression a b

-- | Construct an expression representing a constant.
con :: Num a => a -> LinearExpression a b

-- | Construct an expression by summing expressions.
exprSum :: Num a => [LinearExpression a b] -> LinearExpression a b

-- | Construct an expression by summing variables.
varSum :: Num a => [b] -> LinearExpression a b
(#<=@) :: LPMonad m => Numeric m -> Variable m -> m (Constraint m)
infix 4 #<=@
(#<=.) :: LPMonad m => Numeric m -> Expr m -> m (Constraint m)
infix 4 #<=.
(@<=#) :: LPMonad m => Variable m -> Numeric m -> m (Constraint m)
infix 4 @<=#
(@<=@) :: LPMonad m => Variable m -> Variable m -> m (Constraint m)
infix 4 @<=@
(@<=.) :: LPMonad m => Variable m -> Expr m -> m (Constraint m)
infix 4 @<=.
(.<=#) :: LPMonad m => Expr m -> Numeric m -> m (Constraint m)
infix 4 .<=#
(.<=@) :: LPMonad m => Expr m -> Variable m -> m (Constraint m)
infix 4 .<=@
(.<=.) :: LPMonad m => Expr m -> Expr m -> m (Constraint m)
infix 4 .<=.
(#>=@) :: LPMonad m => Numeric m -> Variable m -> m (Constraint m)
infix 4 #>=@
(#>=.) :: LPMonad m => Numeric m -> Expr m -> m (Constraint m)
infix 4 #>=.
(@>=#) :: LPMonad m => Variable m -> Numeric m -> m (Constraint m)
infix 4 @>=#
(@>=@) :: LPMonad m => Variable m -> Variable m -> m (Constraint m)
infix 4 @>=@
(@>=.) :: LPMonad m => Variable m -> Expr m -> m (Constraint m)
infix 4 @>=.
(.>=#) :: LPMonad m => Expr m -> Numeric m -> m (Constraint m)
infix 4 .>=#
(.>=@) :: LPMonad m => Expr m -> Variable m -> m (Constraint m)
infix 4 .>=@
(.>=.) :: LPMonad m => Expr m -> Expr m -> m (Constraint m)
infix 4 .>=.
(#==@) :: LPMonad m => Numeric m -> Variable m -> m (Constraint m)
infix 4 #==@
(#==.) :: LPMonad m => Numeric m -> Expr m -> m (Constraint m)
infix 4 #==.
(@==#) :: LPMonad m => Variable m -> Numeric m -> m (Constraint m)
infix 4 @==#
(@==@) :: LPMonad m => Variable m -> Variable m -> m (Constraint m)
infix 4 @==@
(@==.) :: LPMonad m => Variable m -> Expr m -> m (Constraint m)
infix 4 @==.
(.==#) :: LPMonad m => Expr m -> Numeric m -> m (Constraint m)
infix 4 .==#
(.==@) :: LPMonad m => Expr m -> Variable m -> m (Constraint m)
infix 4 .==@
(.==.) :: LPMonad m => Expr m -> Expr m -> m (Constraint m)
infix 4 .==.


-- | A library for modeling and solving linear and integer programs.
--   
--   This library is merely a frontend to various solver backends. At the
--   time this was written, the only known supported backend is
--   <a>GLPK</a>.
module Math.Programming

-- | A monad for formulating and solving linear programs.
--   
--   We manipulate linear programs and their settings using the
--   <tt>Mutable</tt> typeclass.
class (Monad m, Show (Numeric m), RealFrac (Numeric m)) => LPMonad m where {
    
    -- | The numeric type used in the model.
    type family Numeric m :: *;
    
    -- | The type of variables in the model. <a>LPMonad</a> treats these as
    --   opaque values, but instances may expose more details.
    data family Variable m :: *;
    
    -- | The type of constraints in the model. <a>LPMonad</a> treats these as
    --   opaque values, but instances may expose more details.
    data family Constraint m :: *;
    
    -- | The type of objectives in the model. <a>LPMonad</a> treats these as
    --   opaque values, but instances may expose more details.
    data family Objective m :: *;
}

-- | Create a new decision variable in the model.
--   
--   This variable will be initialized to be a non-negative continuous
--   variable.
addVariable :: LPMonad m => m (Variable m)

-- | Remove a decision variable from the model.
--   
--   The variable cannot be used after being deleted.
removeVariable :: LPMonad m => Variable m -> m ()

-- | Get the name of the variable.
getVariableName :: LPMonad m => Variable m -> m String

-- | Set the name of the variable.
setVariableName :: LPMonad m => Variable m -> String -> m ()

-- | Get the allowed values of a variable.
getVariableBounds :: LPMonad m => Variable m -> m (Bounds (Numeric m))

-- | Constrain a variable to take on certain values.
setVariableBounds :: LPMonad m => Variable m -> Bounds (Numeric m) -> m ()

-- | Get the value of a variable in the current solution.
getVariableValue :: LPMonad m => Variable m -> m (Numeric m)

-- | Add a constraint to the model represented by an inequality.
addConstraint :: LPMonad m => Inequality (LinearExpression (Numeric m) (Variable m)) -> m (Constraint m)

-- | Remove a constraint from the model.
--   
--   The constraint cannot used after being deleted.
removeConstraint :: LPMonad m => Constraint m -> m ()

-- | Get the name of the constraint.
getConstraintName :: LPMonad m => Constraint m -> m String

-- | Set the name of the constraint.
setConstraintName :: LPMonad m => Constraint m -> String -> m ()

-- | Get the value of the dual variable associated with the constraint in
--   the current solution.
--   
--   This value has no meaning if the current solution is not an LP
--   solution.
getDualValue :: LPMonad m => Constraint m -> m (Numeric m)

-- | Add a constraint to the model represented by an inequality.
addObjective :: LPMonad m => LinearExpression (Numeric m) (Variable m) -> m (Objective m)

-- | Get the name of the objective.
getObjectiveName :: LPMonad m => Objective m -> m String

-- | Set the name of the objective.
setObjectiveName :: LPMonad m => Objective m -> String -> m ()

-- | Whether the objective is to be minimized or maximized.
getObjectiveSense :: LPMonad m => Objective m -> m Sense

-- | Set whether the objective is to be minimized or maximized.
setObjectiveSense :: LPMonad m => Objective m -> Sense -> m ()

-- | Get the value of the objective in the current solution.
getObjectiveValue :: LPMonad m => Objective m -> m (Numeric m)

-- | Get the number of seconds the solver is allowed to run before halting.
getTimeout :: LPMonad m => m Double

-- | Set the number of seconds the solver is allowed to run before halting.
setTimeout :: LPMonad m => Double -> m ()

-- | Optimize the continuous relaxation of the model.
optimizeLP :: LPMonad m => m SolutionStatus

-- | A convient shorthand for the type of linear expressions used in a
--   given model.
type Expr m = LinearExpression (Numeric m) (Variable m)

-- | An interval of the real numbers.
data Bounds b

-- | The non-negative reals.
NonNegativeReals :: Bounds b

-- | The non-positive reals.
NonPositiveReals :: Bounds b

-- | Any closed interval of the reals.
Interval :: b -> b -> Bounds b

-- | Any real number.
Free :: Bounds b

-- | The outcome of an optimization.
data SolutionStatus

-- | An optimal solution has been found.
Optimal :: SolutionStatus

-- | A feasible solution has been found. The result may or may not be
--   optimal.
Feasible :: SolutionStatus

-- | The model has been proven to be infeasible.
Infeasible :: SolutionStatus

-- | The model has been proven to be unbounded.
Unbounded :: SolutionStatus

-- | An error was encountered during the solve. Instance-specific methods
--   should be used to determine what occurred.
Error :: SolutionStatus

-- | Whether a math program is minimizing or maximizing its objective.
data Sense
Minimization :: Sense
Maximization :: Sense

-- | A (mixed) integer program.
--   
--   In addition to the methods of the <a>LPMonad</a> class, this monad
--   supports constraining variables to be either continuous or discrete.
class (LPMonad m) => IPMonad m

-- | Optimize the mixed-integer program.
optimizeIP :: IPMonad m => m SolutionStatus

-- | Get the domain of a variable.
getVariableDomain :: IPMonad m => Variable m -> m Domain

-- | Set the domain of a variable.
setVariableDomain :: IPMonad m => Variable m -> Domain -> m ()

-- | Get the allowed relative gap between LP and IP solutions.
getRelativeMIPGap :: IPMonad m => m Double

-- | Set the allowed relative gap between LP and IP solutions.
setRelativeMIPGap :: IPMonad m => Double -> m ()

-- | The type of values that a variable can take on.
--   
--   Note that the <tt>Integer</tt> constructor does not interfere with the
--   <tt>Integer</tt> type, as the <tt>Integer</tt> type does not define a
--   constuctor of the same name. The ambiguity is unfortunate, but other
--   natural nomenclature such as <tt>Integral</tt> are similarly
--   conflicted.
data Domain

-- | The variable lies in the real numbers
Continuous :: Domain

-- | The variable lies in the integers
Integer :: Domain

-- | The variable lies in the set <tt>{0, 1}</tt>.
Binary :: Domain

-- | Create a new free variable.
free :: LPMonad m => m (Variable m)

-- | Create a new non-negative variable.
nonNeg :: LPMonad m => m (Variable m)

-- | Create a new non-positive variable.
nonPos :: LPMonad m => m (Variable m)

-- | Create a new variable bounded between two values.
bounded :: LPMonad m => Numeric m -> Numeric m -> m (Variable m)

-- | Constrain a variable to take on certain values.
--   
--   This function is designed to be used as an infix operator, e.g.
--   
--   <pre>
--   <a>addVariable</a> `<a>within</a>` <a>NonNegativeReals</a>
--   </pre>
within :: LPMonad m => m (Variable m) -> Bounds (Numeric m) -> m (Variable m)

-- | Create an integer-valued variable.
integer :: IPMonad m => m (Variable m)

-- | Create a binary variable.
binary :: IPMonad m => m (Variable m)

-- | Create an integer-value variable that takes on non-negative values.
nonNegInteger :: IPMonad m => m (Variable m)

-- | Create an integer-value variable that takes on non-positive values.
nonPosInteger :: IPMonad m => m (Variable m)

-- | Set the type of a variable.
--   
--   This function is designed to be used as an infix operator, e.g.
--   
--   <pre>
--   <a>addVariable</a> `<a>asKind</a>` <a>Binary</a>
--   </pre>
asKind :: IPMonad m => m (Variable m) -> Domain -> m (Variable m)

-- | A linear expression containing symbolic variables of type <tt>b</tt>
--   and numeric coefficients of type <tt>a</tt>.
--   
--   Using <a>String</a>s to denote variables and <a>Double</a>s as our
--   numeric type, we could express <i>3 x + 2 y + 1</i> as
--   
--   <pre>
--   LinearExpression [(3, "x"), (2, "y")] 1
--   </pre>
data LinearExpression a b
LinearExpression :: [(a, b)] -> a -> LinearExpression a b

-- | Reduce an expression to its value.
eval :: Num a => LinearExpression a a -> a

-- | Combine equivalent terms by summing their coefficients.
simplify :: (Ord b, Num a) => LinearExpression a b -> LinearExpression a b

-- | Construct an expression representing a variable.
var :: Num a => b -> LinearExpression a b

-- | Construct an expression representing a constant.
con :: Num a => a -> LinearExpression a b

-- | Construct an expression by summing expressions.
exprSum :: Num a => [LinearExpression a b] -> LinearExpression a b

-- | Construct an expression by summing variables.
varSum :: Num a => [b] -> LinearExpression a b
(.+.) :: Num a => LinearExpression a b -> LinearExpression a b -> LinearExpression a b
infixl 6 .+.
(@+@) :: Num a => b -> b -> LinearExpression a b
infixl 6 @+@
(.+@) :: Num a => LinearExpression a b -> b -> LinearExpression a b
infixl 6 .+@
(@+.) :: Num a => b -> LinearExpression a b -> LinearExpression a b
infixl 6 @+.
(@+#) :: Num a => b -> a -> LinearExpression a b
infixl 6 @+#
(#+@) :: Num a => a -> b -> LinearExpression a b
infixl 6 #+@
(#+.) :: Num a => a -> LinearExpression a b -> LinearExpression a b
infixl 6 #+.
(.+#) :: Num a => LinearExpression a b -> a -> LinearExpression a b
infixl 6 .+#
(.-.) :: Num a => LinearExpression a b -> LinearExpression a b -> LinearExpression a b
infixl 6 .-.
(@-@) :: Num a => b -> b -> LinearExpression a b
infixl 6 @-@
(.-@) :: Num a => LinearExpression a b -> b -> LinearExpression a b
infixl 6 .-@
(@-.) :: Num a => b -> LinearExpression a b -> LinearExpression a b
infixl 6 @-.
(@-#) :: Num a => b -> a -> LinearExpression a b
infixl 6 @-#
(#-@) :: Num a => a -> b -> LinearExpression a b
infixl 6 #-@
(#-.) :: Num a => a -> LinearExpression a b -> LinearExpression a b
infixl 6 #-.
(.-#) :: Num a => LinearExpression a b -> a -> LinearExpression a b
infixl 6 .-#
(#*@) :: Num a => a -> b -> LinearExpression a b
infixl 7 #*@
(@*#) :: Num a => b -> a -> LinearExpression a b
infixl 7 @*#
(#*.) :: Num a => a -> LinearExpression a b -> LinearExpression a b
infixl 7 #*.
(.*#) :: Num a => LinearExpression a b -> a -> LinearExpression a b
infixl 7 .*#
(@/#) :: Fractional a => b -> a -> LinearExpression a b
infixl 7 @/#
(./#) :: Fractional a => LinearExpression a b -> a -> LinearExpression a b
infixl 7 ./#

-- | Non-strict inequalities.
data Inequality a
Inequality :: Ordering -> a -> a -> Inequality a
(#<=@) :: LPMonad m => Numeric m -> Variable m -> m (Constraint m)
infix 4 #<=@
(#<=.) :: LPMonad m => Numeric m -> Expr m -> m (Constraint m)
infix 4 #<=.
(@<=#) :: LPMonad m => Variable m -> Numeric m -> m (Constraint m)
infix 4 @<=#
(@<=@) :: LPMonad m => Variable m -> Variable m -> m (Constraint m)
infix 4 @<=@
(@<=.) :: LPMonad m => Variable m -> Expr m -> m (Constraint m)
infix 4 @<=.
(.<=#) :: LPMonad m => Expr m -> Numeric m -> m (Constraint m)
infix 4 .<=#
(.<=@) :: LPMonad m => Expr m -> Variable m -> m (Constraint m)
infix 4 .<=@
(.<=.) :: LPMonad m => Expr m -> Expr m -> m (Constraint m)
infix 4 .<=.
(#>=@) :: LPMonad m => Numeric m -> Variable m -> m (Constraint m)
infix 4 #>=@
(#>=.) :: LPMonad m => Numeric m -> Expr m -> m (Constraint m)
infix 4 #>=.
(@>=#) :: LPMonad m => Variable m -> Numeric m -> m (Constraint m)
infix 4 @>=#
(@>=@) :: LPMonad m => Variable m -> Variable m -> m (Constraint m)
infix 4 @>=@
(@>=.) :: LPMonad m => Variable m -> Expr m -> m (Constraint m)
infix 4 @>=.
(.>=#) :: LPMonad m => Expr m -> Numeric m -> m (Constraint m)
infix 4 .>=#
(.>=@) :: LPMonad m => Expr m -> Variable m -> m (Constraint m)
infix 4 .>=@
(.>=.) :: LPMonad m => Expr m -> Expr m -> m (Constraint m)
infix 4 .>=.
(#==@) :: LPMonad m => Numeric m -> Variable m -> m (Constraint m)
infix 4 #==@
(#==.) :: LPMonad m => Numeric m -> Expr m -> m (Constraint m)
infix 4 #==.
(@==#) :: LPMonad m => Variable m -> Numeric m -> m (Constraint m)
infix 4 @==#
(@==@) :: LPMonad m => Variable m -> Variable m -> m (Constraint m)
infix 4 @==@
(@==.) :: LPMonad m => Variable m -> Expr m -> m (Constraint m)
infix 4 @==.
(.==#) :: LPMonad m => Expr m -> Numeric m -> m (Constraint m)
infix 4 .==#
(.==@) :: LPMonad m => Expr m -> Variable m -> m (Constraint m)
infix 4 .==@
(.==.) :: LPMonad m => Expr m -> Expr m -> m (Constraint m)
infix 4 .==.

-- | Create an objective to be minimized.
minimize :: LPMonad m => Expr m -> m (Objective m)

-- | Create an objective to be maximized.
maximize :: LPMonad m => Expr m -> m (Objective m)

-- | Get the value of a linear expression in the current solution.
evalExpr :: LPMonad m => Expr m -> m (Numeric m)

-- | Name a variable, constraint, or objective.
--   
--   This function is designed to be used as an infix operator, e.g.
--   
--   <pre>
--   <a>free</a> `<a>named</a>` <a>X_1</a>
--   </pre>
named :: (Monad m, Nameable m a) => m a -> String -> m a

-- | Retrieve the name of a variable, constraint, or objective.
nameOf :: (Monad m, Nameable m a) => a -> m String