Agda-2.5.1.2: A dependently typed functional programming language and proof assistant

Agda.TypeChecking.SizedTypes.WarshallSolver

Synopsis

# Documentation

type Graph r f a = Graph (Node r f) (Node r f) a Source #

type Edge' r f a = Edge (Node r f) (Node r f) a Source #

type Key r f = Edge' r f () Source #

type Nodes r f = Nodes (Node r f) Source #

src :: Edge s t e -> s Source #

dest :: Edge s t e -> t Source #

lookupEdge :: (Ord s, Ord t) => Graph s t e -> s -> t -> Maybe e Source #

graphToList :: Graph s t e -> [Edge s t e] Source #

graphFromList :: (Ord s, Ord t) => [Edge s t e] -> Graph s t e Source #

insertEdge :: (Ord s, Ord t, MeetSemiLattice e, Top e) => Edge s t e -> Graph s t e -> Graph s t e Source #

outgoing :: (Ord r, Ord f) => Graph r f a -> Node r f -> [Edge' r f a] Source #

Compute list of edges that start in a given node.

incoming :: (Ord r, Ord f) => Graph r f a -> Node r f -> [Edge' r f a] Source #

Compute list of edges that target a given node.

Note: expensive for unidirectional graph representations.

setFoldl :: (b -> a -> b) -> b -> Set a -> b Source #

Set.foldl does not exist in legacy versions of the containers package.

transClos :: forall n a. (Ord n, Dioid a) => Graph n n a -> Graph n n a Source #

Floyd-Warshall algorithm.

# Edge weights

data Weight Source #

Constructors

 Offset Offset Infinity

Instances

 Source # MethodstoEnum :: Int -> Weight #enumFrom :: Weight -> [Weight] #enumFromThen :: Weight -> Weight -> [Weight] #enumFromTo :: Weight -> Weight -> [Weight] #enumFromThenTo :: Weight -> Weight -> Weight -> [Weight] # Source # Methods(==) :: Weight -> Weight -> Bool #(/=) :: Weight -> Weight -> Bool # Source # Partial implementation of Num. Methods(+) :: Weight -> Weight -> Weight #(-) :: Weight -> Weight -> Weight #(*) :: Weight -> Weight -> Weight #abs :: Weight -> Weight # Source # Methods(<) :: Weight -> Weight -> Bool #(<=) :: Weight -> Weight -> Bool #(>) :: Weight -> Weight -> Bool #(>=) :: Weight -> Weight -> Bool #max :: Weight -> Weight -> Weight #min :: Weight -> Weight -> Weight # Source # MethodsshowsPrec :: Int -> Weight -> ShowS #showList :: [Weight] -> ShowS # Source # Methodsshrink :: Weight -> [Weight] # Source # Methods Source # Methods Source # Methods Source # Methods Source # Methods Source # Methods Plus (SizeExpr' r f) Weight (SizeExpr' r f) Source # Methodsplus :: SizeExpr' r f -> Weight -> SizeExpr' r f Source #

class Negative a where Source #

Test for negativity, used to detect negative cycles.

Minimal complete definition

negative

Methods

negative :: a -> Bool Source #

Instances

 Source # Methods Source # Methods Source # Methods Source # Methods (Ord r, Ord f, Negative a) => Negative (Graphs r f a) Source # Methodsnegative :: Graphs r f a -> Bool Source # Negative a => Negative (Edge' r f a) Source # An edge is negative if its label is. Methodsnegative :: Edge' r f a -> Bool Source # (Ord r, Ord f, Negative a) => Negative (Graph r f a) Source # A graph is negative if it contains a negative loop (diagonal edge). Makes sense on transitive graphs. Methodsnegative :: Graph r f a -> Bool Source #

# Edge labels

data Label Source #

Going from Lt to Le is pred, going from Le to Lt is succ.

X --(R,n)--> Y means X (R) Y + n. [ ... if n positive and X + (-n) (R) Y if n negative. ]

Constructors

 Label Fieldslcmp :: Cmp loffset :: Offset LInf Nodes not connected.

Instances

 Source # Methods(==) :: Label -> Label -> Bool #(/=) :: Label -> Label -> Bool # Source # Methods(<) :: Label -> Label -> Bool #(<=) :: Label -> Label -> Bool #(>) :: Label -> Label -> Bool #(>=) :: Label -> Label -> Bool #max :: Label -> Label -> Label #min :: Label -> Label -> Label # Source # MethodsshowsPrec :: Int -> Label -> ShowS #show :: Label -> String #showList :: [Label] -> ShowS # Source # Methodsshrink :: Label -> [Label] # Source # Methods Source # Methods Source # Methods Source # Methods Source # Methods (Ord r, Ord f) => SetToInfty f (ConGraph r f) Source # MethodssetToInfty :: [f] -> ConGraph r f -> ConGraph r f Source # Plus (SizeExpr' r f) Label (SizeExpr' r f) Source # Methodsplus :: SizeExpr' r f -> Label -> SizeExpr' r f Source #

Convert a label to a weight, decrementing in case of Lt.

# Graphs

## Nodes

data Node rigid flex Source #

Constructors

 NodeZero NodeInfty NodeRigid rigid NodeFlex flex

Instances

 (Ord r, Ord f) => SetToInfty f (ConGraph r f) Source # MethodssetToInfty :: [f] -> ConGraph r f -> ConGraph r f Source # Eq f => SetToInfty f (Node r f) Source # MethodssetToInfty :: [f] -> Node r f -> Node r f Source # Eq f => SetToInfty f (Edge' r f a) Source # MethodssetToInfty :: [f] -> Edge' r f a -> Edge' r f a Source # (Eq flex, Eq rigid) => Eq (Node rigid flex) Source # Methods(==) :: Node rigid flex -> Node rigid flex -> Bool #(/=) :: Node rigid flex -> Node rigid flex -> Bool # (Ord flex, Ord rigid) => Ord (Node rigid flex) Source # Methodscompare :: Node rigid flex -> Node rigid flex -> Ordering #(<) :: Node rigid flex -> Node rigid flex -> Bool #(<=) :: Node rigid flex -> Node rigid flex -> Bool #(>) :: Node rigid flex -> Node rigid flex -> Bool #(>=) :: Node rigid flex -> Node rigid flex -> Bool #max :: Node rigid flex -> Node rigid flex -> Node rigid flex #min :: Node rigid flex -> Node rigid flex -> Node rigid flex # (Show rigid, Show flex) => Show (Node rigid flex) Source # MethodsshowsPrec :: Int -> Node rigid flex -> ShowS #show :: Node rigid flex -> String #showList :: [Node rigid flex] -> ShowS # (Ord r, Ord f, Dioid a) => Dioid (Edge' r f a) Source # Methodscompose :: Edge' r f a -> Edge' r f a -> Edge' r f a Source #unitCompose :: Edge' r f a Source # (Ord r, Ord f, MeetSemiLattice a) => MeetSemiLattice (Edge' r f a) Source # Methodsmeet :: Edge' r f a -> Edge' r f a -> Edge' r f a Source # (Ord r, Ord f, Top a) => Top (Edge' r f a) Source # Methodstop :: Edge' r f a Source #isTop :: Edge' r f a -> Bool Source # (Ord r, Ord f, Negative a) => Negative (Graphs r f a) Source # Methodsnegative :: Graphs r f a -> Bool Source # Negative a => Negative (Edge' r f a) Source # An edge is negative if its label is. Methodsnegative :: Edge' r f a -> Bool Source # (Ord r, Ord f, Negative a) => Negative (Graph r f a) Source # A graph is negative if it contains a negative loop (diagonal edge). Makes sense on transitive graphs. Methodsnegative :: Graph r f a -> Bool Source #

isFlexNode :: Node rigid flex -> Maybe flex Source #

isZeroNode :: Node rigid flex -> Bool Source #

isInftyNode :: Node rigid flex -> Bool Source #

nodeToSizeExpr :: Node rigid flex -> SizeExpr' rigid flex Source #

## Graphs

type Graphs r f a = [Graph r f a] Source #

A graph forest.

mentions :: (Ord r, Ord f) => Node r f -> Graphs r f a -> (Graphs r f a, Graphs r f a) Source #

Split a list of graphs gs into those that mention node n and those that do not. If n is zero or infinity, we regard it as "not mentioned".

addEdge :: (Ord r, Ord f, MeetSemiLattice a, Top a) => Edge' r f a -> Graphs r f a -> Graphs r f a Source #

Add an edge to a graph forest. Graphs that share a node with the edge are joined.

reflClos :: (Ord r, Ord f, Dioid a) => Set (Node r f) -> Graph r f a -> Graph r f a Source #

Reflexive closure. Add edges 0 -> n -> n -> oo for all nodes n.

implies :: (Ord r, Ord f, Show r, Show f, Show a, Top a, Ord a, Negative a) => Graph r f a -> Graph r f a -> Bool Source #

h implies g if any edge in g between rigids and constants is implied by a corresponding edge in h, which means that the edge in g carries at most the information of the one in h.

Application: Constraint implication: Constraints are compatible with hypotheses.

nodeFromSizeExpr :: SizeExpr' rigid flex -> (Node rigid flex, Offset) Source #

edgeFromConstraint :: Constraint' rigid flex -> LabelledEdge rigid flex Source #

graphFromConstraints :: (Ord rigid, Ord flex) => [Constraint' rigid flex] -> Graph rigid flex Label Source #

Build a graph from list of simplified constraints.

graphsFromConstraints :: (Ord rigid, Ord flex) => [Constraint' rigid flex] -> Graphs rigid flex Label Source #

Build a graph from list of simplified constraints.

type HypGraph r f = Graph r f Label Source #

hypGraph :: (Ord rigid, Ord flex, Show rigid, Show flex) => Set rigid -> [Hyp' rigid flex] -> Either String (HypGraph rigid flex) Source #

hypConn :: (Ord r, Ord f) => HypGraph r f -> Node r f -> Node r f -> Label Source #

simplifyWithHypotheses :: (Ord rigid, Ord flex, Show rigid, Show flex) => HypGraph rigid flex -> [Constraint' rigid flex] -> Either String [Constraint' rigid flex] Source #

type ConGraph r f = Graph r f Label Source #

constraintGraph :: (Ord r, Ord f, Show r, Show f) => [Constraint' r f] -> HypGraph r f -> Either String (ConGraph r f) Source #

type ConGraphs r f = Graphs r f Label Source #

constraintGraphs :: (Ord r, Ord f, Show r, Show f) => [Constraint' r f] -> HypGraph r f -> Either String ([f], ConGraphs r f) Source #

infinityFlexs :: (Ord r, Ord f) => ConGraph r f -> ([f], ConGraph r f) Source #

If we have an edge X + n <= X (with n >= 0), we must set X = oo.

class SetToInfty f a where Source #

Minimal complete definition

setToInfty

Methods

setToInfty :: [f] -> a -> a Source #

Instances

 (Ord r, Ord f) => SetToInfty f (ConGraph r f) Source # MethodssetToInfty :: [f] -> ConGraph r f -> ConGraph r f Source # Eq f => SetToInfty f (Node r f) Source # MethodssetToInfty :: [f] -> Node r f -> Node r f Source # Eq f => SetToInfty f (Edge' r f a) Source # MethodssetToInfty :: [f] -> Edge' r f a -> Edge' r f a Source #

# Compute solution from constraint graph.

type Bound r f = Map f (Set (SizeExpr' r f)) Source #

Lower or upper bound for a flexible variable

data Bounds r f Source #

Constructors

 Bounds FieldslowerBounds :: Bound r f upperBounds :: Bound r f mustBeFinite :: Set f

edgeToLowerBound :: LabelledEdge r f -> Maybe (f, SizeExpr' r f) Source #

Compute a lower bound for a flexible from an edge.

edgeToUpperBound :: LabelledEdge r f -> Maybe (f, Cmp, SizeExpr' r f) Source #

Compute an upper bound for a flexible from an edge.

graphToLowerBounds :: (Ord r, Ord f) => [LabelledEdge r f] -> Bound r f Source #

Compute the lower bounds for all flexibles in a graph.

graphToUpperBounds :: (Ord r, Ord f) => [LabelledEdge r f] -> (Bound r f, Set f) Source #

Compute the upper bounds for all flexibles in a graph.

bounds :: (Ord r, Ord f) => ConGraph r f -> Bounds r f Source #

Compute the bounds for all flexibles in a graph.

smallest :: (Ord r, Ord f) => HypGraph r f -> [Node r f] -> [Node r f] Source #

Compute the relative minima in a set of nodes (those that do not have a predecessor in the set).

largest :: (Ord r, Ord f) => HypGraph r f -> [Node r f] -> [Node r f] Source #

Compute the relative maxima in a set of nodes (those that do not have a successor in the set).

commonSuccs :: (Ord r, Ord f) => Graph r f a -> [Node r f] -> Map (Node r f) [Edge' r f a] Source #

Given source nodes n1,n2,... find all target nodes m1,m2, such that for all j, there are edges n_i --l_ij--> m_j for all i. Return these edges as a map from target notes to a list of edges. We assume the graph is reflexive-transitive.

commonPreds :: (Ord r, Ord f) => Graph r f a -> [Node r f] -> Map (Node r f) [Edge' r f a] Source #

Given target nodes m1,m2,... find all source nodes n1,n2, such that for all j, there are edges n_i --l_ij--> m_j for all i. Return these edges as a map from target notes to a list of edges. We assume the graph is reflexive-transitive.

lub' :: forall r f. (Ord r, Ord f, Show r, Show f) => HypGraph r f -> (Node r f, Offset) -> (Node r f, Offset) -> Maybe (SizeExpr' r f) Source #

Compute the sup of two different rigids or a rigid and a constant.

glb' :: forall r f. (Ord r, Ord f, Show r, Show f) => HypGraph r f -> (Node r f, Offset) -> (Node r f, Offset) -> Maybe (SizeExpr' r f) Source #

Compute the inf of two different rigids or a rigid and a constant.

lub :: (Ord r, Ord f, Show r, Show f) => HypGraph r f -> SizeExpr' r f -> SizeExpr' r f -> Maybe (SizeExpr' r f) Source #

Compute the least upper bound (sup).

glb :: (Ord r, Ord f, Show r, Show f) => HypGraph r f -> SizeExpr' r f -> SizeExpr' r f -> Maybe (SizeExpr' r f) Source #

Compute the greatest lower bound (inf) of size expressions relative to a hypotheses graph.

findRigidBelow :: (Ord r, Ord f) => HypGraph r f -> SizeExpr' r f -> Maybe (SizeExpr' r f) Source #

solveGraph :: (Ord r, Ord f, Show r, Show f) => Polarities f -> HypGraph r f -> ConGraph r f -> Either String (Solution r f) Source #

solveGraphs :: (Ord r, Ord f, Show r, Show f) => Polarities f -> HypGraph r f -> ConGraphs r f -> Either String (Solution r f) Source #

Solve a forest of constraint graphs relative to a hypotheses graph. Concatenate individual solutions.

# Verify solution

verifySolution :: (Ord r, Ord f, Show r, Show f) => HypGraph r f -> [Constraint' r f] -> Solution r f -> Either String () Source #

Check that after substitution of the solution, constraints are implied by hypotheses.

# Tests

testLub :: (Show f, Ord f) => Maybe (SizeExpr' [Char] f) Source #