Agda-2.5.2: A dependently typed functional programming language and proof assistant

Safe HaskellNone
LanguageHaskell2010

Agda.Termination.Order

Contents

Description

An Abstract domain of relative sizes, i.e., differences between size of formal function parameter and function argument in recursive call; used in the termination checker.

Synopsis

Structural orderings

data Order #

In the paper referred to above, there is an order R with Unknown <= Le <= Lt.

This is generalized to Unknown <= 'Decr k' where Decr 1 replaces Lt and Decr 0 replaces Le. A negative decrease means an increase. The generalization allows the termination checker to record an increase by 1 which can be compensated by a following decrease by 2 which results in an overall decrease.

However, the termination checker of the paper itself terminates because there are only finitely many different call-matrices. To maintain termination of the terminator we set a cutoff point which determines how high the termination checker can count. This value should be set by a global or file-wise option.

See Call for more information.

TODO: document orders which are call-matrices themselves.

Constructors

Decr !Int

Decrease of callee argument wrt. caller parameter.

Unknown

No relation, infinite increase, or increase beyond termination depth.

Mat !(Matrix Int Order)

Matrix-shaped order, currently UNUSED.

Instances

Eq Order # 

Methods

(==) :: Order -> Order -> Bool #

(/=) :: Order -> Order -> Bool #

Ord Order # 

Methods

compare :: Order -> Order -> Ordering #

(<) :: Order -> Order -> Bool #

(<=) :: Order -> Order -> Bool #

(>) :: Order -> Order -> Bool #

(>=) :: Order -> Order -> Bool #

max :: Order -> Order -> Order #

min :: Order -> Order -> Order #

Show Order # 

Methods

showsPrec :: Int -> Order -> ShowS #

show :: Order -> String #

showList :: [Order] -> ShowS #

Pretty Order # 

Methods

pretty :: Order -> Doc #

prettyPrec :: Int -> Order -> Doc #

Pretty CallMatrix # 
PartialOrd Order #

Information order: Unknown is least information. The more we decrease, the more information we have.

When having comparable call-matrices, we keep the lesser one. Call graph completion works toward losing the good calls, tending towards Unknown (the least information).

HasZero Order # 

Methods

zeroElement :: Order #

NotWorse Order #

It does not get worse then `increase'. If we are still decreasing, it can get worse: less decreasing.

Methods

notWorse :: Order -> Order -> Bool #

CallComb CallMatrix #

Call matrix multiplication.

f --(m1)--> g --(m2)--> h is combined to f --(m2 mul m1)--> h

Note the reversed order of multiplication: The matrix c1 of the second call g-->h in the sequence f-->g-->h is multiplied with the matrix c2 of the first call.

Preconditions: m1 has dimensions ar(g) × ar(f). m2 has dimensions ar(h) × ar(g).

Postcondition: m1 >*< m2 has dimensions ar(h) × ar(f).

NotWorse (CallMatrix' Order) # 
Diagonal (CallMatrixAug cinfo) Order # 

Methods

diagonal :: CallMatrixAug cinfo -> [Order] #

decr :: (?cutoff :: CutOff) => Int -> Order #

Smart constructor for Decr k :: Order which cuts off too big values.

Possible values for k: - ?cutoff <= k <= ?cutoff + 1.

increase :: Int -> Order -> Order #

Raw increase which does not cut off.

decrease :: Int -> Order -> Order #

Raw decrease which does not cut off.

(.*.) :: (?cutoff :: CutOff) => Order -> Order -> Order #

Multiplication of Orders. (Corresponds to sequential composition.)

supremum :: (?cutoff :: CutOff) => [Order] -> Order #

The supremum of a (possibly empty) list of Orders. More information (i.e., more decrease) is bigger. Unknown is no information, thus, smallest.

infimum :: (?cutoff :: CutOff) => [Order] -> Order #

The infimum of a (non empty) list of Orders. Gets the worst information. Unknown is the least element, thus, dominant.

orderSemiring :: (?cutoff :: CutOff) => Semiring Order #

We use a record for semiring instead of a type class since implicit arguments cannot occur in instance constraints, like instance (?cutoff :: Int) => SemiRing Order.

le :: Order #

le, lt, decreasing, unknown: for backwards compatibility, and for external use.

orderMat :: Matrix Int Order -> Order #

Smart constructor for matrix shaped orders, avoiding empty and singleton matrices.

collapseO :: (?cutoff :: CutOff) => Order -> Order #

isDecr :: Order -> Bool #

Matrix-shaped order is decreasing if any diagonal element is decreasing.

class NotWorse a where #

A partial order, aimed at deciding whether a call graph gets worse during the completion.

Minimal complete definition

notWorse

Methods

notWorse :: a -> a -> Bool #

Instances

NotWorse Order #

It does not get worse then `increase'. If we are still decreasing, it can get worse: less decreasing.

Methods

notWorse :: Order -> Order -> Bool #

NotWorse (CallMatrixAug cinfo) # 

Methods

notWorse :: CallMatrixAug cinfo -> CallMatrixAug cinfo -> Bool #

NotWorse (CallMatrix' Order) # 
(Ord i, HasZero o, NotWorse o) => NotWorse (Matrix i o) #

We assume the matrices have the same dimension.

Methods

notWorse :: Matrix i o -> Matrix i o -> Bool #

isOrder :: (?cutoff :: CutOff) => Order -> Bool #