First we define the general category of graphs.

(1) -> <spad>

)abbrev category GRAPHS GraphCategory
GraphCategory(nodes:Type, edges:Type): Category == with
  source:edges->nodes
  target:edges->nodes</spad>

Compiling FriCAS source code from file 
      /var/lib/zope2.10/instance/axiom-wiki/var/LatexWiki/7569958594964485335-25px001.spad
      using old system compiler.
   GRAPHS abbreviates category GraphCategory 
------------------------------------------------------------------------
   initializing NRLIB GRAPHS for GraphCategory 
   compiling into NRLIB GRAPHS 

;;;     ***       |GraphCategory| REDEFINED
Time: 0 SEC.

   finalizing NRLIB GRAPHS 
   Processing GraphCategory for Browser database:
--->-->GraphCategory(constructor): Not documented!!!!
--->-->GraphCategory((source (nodes edges))): Not documented!!!!
--->-->GraphCategory((target (nodes edges))): Not documented!!!!
--->-->GraphCategory(): Missing Description
; compiling file "/var/aw/var/LatexWiki/GRAPHS.NRLIB/GRAPHS.lsp" (written 01 NOV 2024 12:11:00 AM):

; wrote /var/aw/var/LatexWiki/GRAPHS.NRLIB/GRAPHS.fasl
; compilation finished in 0:00:00.008
------------------------------------------------------------------------
   GraphCategory is now explicitly exposed in frame initial 
   GraphCategory will be automatically loaded when needed from 
      /var/aw/var/LatexWiki/GRAPHS.NRLIB/GRAPHS

Now we define finite graphs as follows:

)abbrev domain FGRAPHS FiniteGraph

FiniteGraph(nodes: BasicType): exports == implementation where
  edges ==> Record(source:nodes,target:nodes)
  exports == GraphCategory(nodes,edges) with
    new: () -> %
    addNode: (%,List nodes) -> List nodes
    addNode: (%,nodes) -> List nodes
    addEdge: (%,nodes,nodes) -> edges
    edgeList: (%) -> List edges
    nodeList: (%) -> List nodes

  implementation == add
    Rep == Record(node: List nodes, edge: List edges)

    new:% == 
      n:List(nodes):=[]
      e:List(edges):=[]
      [n,e]$Rep pretend %

    addNode(g:%,n:List nodes):List nodes ==
      G:Rep:=g pretend Rep;
      if #G.node=0 then
        G.node:=n
      else
        concat!(G.node,n)
      n

    addNode(g:%,n:nodes):List nodes == addNode(g,[n])

    addEdge(g:%,src:nodes,tar:nodes):edges ==
      G:Rep:=g pretend Rep;
      if #G.edge=0 then
        G.edge:=[[src,tar]$edges]
      else
        concat!(G.edge,[[src,tar]$edges])
      [src,tar]$edges

    edgeList(g:%):List edges ==
      G:Rep:=g pretend Rep;
      G.edge
    nodeList(g:%):List nodes ==
      G:Rep:=g pretend Rep;
      G.node

    source(ed:edges):nodes == ed.source
    target(ed:edges):nodes == ed.target

   Compiling FriCAS source code from file 
      /var/lib/zope2.10/instance/axiom-wiki/var/LatexWiki/3160513407949145091-25px002.spad
      using old system compiler.
   FGRAPHS abbreviates domain FiniteGraph 
------------------------------------------------------------------------
   initializing NRLIB FGRAPHS for FiniteGraph 
   compiling into NRLIB FGRAPHS 
************* USER ERROR **********
available signatures for Rep: 
    NONE
NEED Rep: () -> ?
****** comp fails at level 1 with expression: ******
((DEF (|Rep|) (NIL)
  (|Record| (|:| |node| (|List| |nodes|))
            (|:| |edge|
                 (|List|
                  (|Record| (|:| |source| |nodes|) (|:| |target| |nodes|)))))))
****** level 1  ******
$x:= (DEF (Rep) (NIL) (Record (: node (List nodes)) (: edge (List (Record (: source nodes) (: target nodes))))))
$m:= $EmptyMode
$f:=
((((~= #) (= #) (|$DomainsInScope| # # #) (|target| #) ...)))

   >> Apparent user error:
   unspecified error

Example 1: create a simple finite graph:

g:FiniteGraph(INT)

g:=new()

   There are 1 exposed and 8 unexposed library operations named new 
      having 0 argument(s) but none was determined to be applicable. 
      Use HyperDoc Browse, or issue
                               )display op new
      to learn more about the available operations. Perhaps 
      package-calling the operation or using coercions on the arguments
      will allow you to apply the operation.

   Cannot find a no-argument definition or library operation named new 
      .