Agda.Termination.Order

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.

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

Mat !(Matrix Int Order)

Matrix-shaped order, currently UNUSED.

Instances

Eq Order

Ord Order

Show Order

Arbitrary Order

Arbitrary CallMatrix

CoArbitrary Order

HasZero Order

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).

Pretty Order

Pretty CallMatrix

NotWorse Order

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

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

Ord i => NotWorse (Matrix i Order)

We assume the matrices have the same dimension.

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. Unknown is the least element, thus, dominant.

orderSemiring :: [cutoff :: CutOff] => Semiring Order

le :: Order

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

lt :: Order

unknown :: Order

orderMat :: Matrix Int Order -> Order

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

collapseO :: [cutoff :: CutOff] => Order -> Order

decreasing :: Order -> Bool

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.

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.

NotWorse (CallMatrixAug cinfo)

NotWorse (CallMatrix' Order)

Ord i => NotWorse (Matrix i Order)

We assume the matrices have the same dimension.

tests :: IO Bool