Agda-2.2.6: A dependently typed functional programming language and proof assistant

Agda.Termination.CallGraph

Description

Call graphs and related concepts, more or less as defined in "A Predicative Analysis of Structural Recursion" by Andreas Abel and Thorsten Altenkirch.

Synopsis

# Structural orderings

data Order Source

The order called R in the paper referred to above. Note that `Unknown <= Le <= Lt`.

See `Call` for more information.

TODO: document orders which are call-matrices themselves.

Constructors

 Lt Le Unknown Mat (Matrix Integer Order)

Instances

 Eq Order Ord Order Show Order Arbitrary Order CoArbitrary Order

Multiplication of `Order`s. (Corresponds to sequential composition.)

supremum :: [Order] -> OrderSource

The supremum of a (possibly empty) list of `Order`s.

# Call matrices

type Index = IntegerSource

Call matrix indices.

newtype CallMatrix Source

Call matrices. Note the call matrix invariant (`callMatrixInvariant`).

Constructors

 CallMatrix Fieldsmat :: Matrix Index Order

(>*<) :: Call -> Call -> CallSource

`Call` combination.

Precondition: see `<*>`; furthermore the `source` of the first argument should be equal to the `target` of the second one.

In a call matrix at most one element per row may be different from `Unknown`.

# Calls

data Call Source

This datatype encodes information about a single recursive function application. The columns of the call matrix stand for `source` function arguments (patterns); the first argument has index 0, the second 1, and so on. The rows of the matrix stand for `target` function arguments. Element `(i, j)` in the matrix should be computed as follows:

• `Lt` (less than) if the `j`-th argument to the `target` function is structurally strictly smaller than the `i`-th pattern.
• `Le` (less than or equal) if the `j`-th argument to the `target` function is structurally smaller than the `i`-th pattern.
• `Unknown` otherwise.

The structural ordering used is defined in the paper referred to above.

Constructors

 Call Fieldssource :: IndexThe function making the call. target :: IndexThe function being called. cm :: CallMatrixThe call matrix describing the call.

Instances

 Eq Call Ord Call Show Call Arbitrary Call CoArbitrary Call

`Call` invariant.

# Call graphs

data CallGraph meta Source

A call graph is a set of calls. Every call also has some associated meta information, which should be `Monoid`al so that the meta information for different calls can be combined when the calls are combined.

Instances

 Eq meta => Eq (CallGraph meta) Show meta => Show (CallGraph meta)

`CallGraph` invariant.

fromList :: Monoid meta => [(Call, meta)] -> CallGraph metaSource

Converts a list of calls with associated meta information to a call graph.

toList :: CallGraph meta -> [(Call, meta)]Source

Converts a call graph to a list of calls with associated meta information.

empty :: CallGraph metaSource

Creates an empty call graph.

union :: Monoid meta => CallGraph meta -> CallGraph meta -> CallGraph metaSource

Takes the union of two call graphs.

insert :: Monoid meta => Call -> meta -> CallGraph meta -> CallGraph metaSource

Inserts a call into a call graph.

complete :: Monoid meta => CallGraph meta -> CallGraph metaSource

`complete cs` completes the call graph `cs`. A call graph is complete if it contains all indirect calls; if `f -> g` and ```g -> h``` are present in the graph, then `f -> h` should also be present.

showBehaviour :: Show meta => CallGraph meta -> StringSource

Displays the recursion behaviour corresponding to a call graph.