(fmod VERTEX is ---> fth VERTEX 
 pr BOOL . 
 pr INT .
 op no : -> Nat .
 sort Vertex .
 op _~_ : Vertex Vertex -> Bool [comm] .
 vars I J : Vertex .
 eq  I ~ I = true .
 ceq I = J if I ~ J [nonexec]. 
endfm)

(fmod VTXSET is
 pr VERTEX .
 sort VtxSet .
 subsorts Vertex < VtxSet .
 op empty : -> VtxSet [ctor].
 op _U_ : Vertex VtxSet -> VtxSet [ctor assoc comm].
 op _U_ : VtxSet VtxSet -> VtxSet [assoc comm].
 vars A B : VtxSet .
 vars I J : Vertex . 
 eq (A U A) = A .
 ---
 op _in_ : Vertex VtxSet -> Bool .
 eq  I in empty   = false .
 ceq I in (J U A) = true   if I = J .
 ceq I in (J U A) = I in A if (I ~ J) = false .
---
 op card : VtxSet -> Nat .
 eq card(empty) = 0 .
 eq card(I U A) = 1 + card(A).
---
 op _<_ : VtxSet VtxSet -> Bool .
 eq  empty < B = true .
 ceq I U A < B = A < B if I in B .
 ceq I U A < B = false  if I in B = false .
---
 op _<>_ : VtxSet VtxSet -> Bool [comm].
 eq A <> A = true .
 ceq A <> B = true if A < B and B < A .
 ceq A <> B = false if not A < B .
 ceq A <> B = false if not B < A .
endfm)



(fmod GRAPH is
 pr VTXSET .
 sorts Edge Graph .
 op <_`,_> : Vertex Vertex -> Edge [ctor].
 op nil : -> Graph [ctor].
 op _;_ : Edge Graph -> Graph [ctor].
 ---
 vars A B C : Vertex .
 var G : Graph .
 ---
 op mcc : Vertex Graph -> VtxSet .
 eq  mcc(A,nil) = A .
 ceq mcc(A,< B,C > ; G) = mcc(B,G) U mcc(C,G) if mcc(A,G) = mcc(B,G) [metadata "CA-1"].
 ceq mcc(A,< B,C > ; G) = mcc(B,G) U mcc(C,G) if mcc(A,G) = mcc(C,G) [metadata "CA-2"].
 ceq mcc(A,< B,C > ; G) = mcc(A,G) if mcc(A,G) <> mcc(B,G) = false /\ 
                                      mcc(A,G) <> mcc(C,G) = false   [metadata "CA-3"].
 ---
 op nocycle : Graph -> Bool .
 eq  nocycle(nil) = true .
 ceq nocycle(< A,B > ; G) = false      if mcc(A,G) = mcc(B,G)         [metadata "CA-1"].
 ceq nocycle(< A,B > ; G) = nocycle(G) if mcc(A,G) <> mcc(B,G) = false[metadata "CA-2"].
 ---
 op nomcc : Graph -> Int .
 eq nomcc(nil) = no .
 ceq nomcc(< A,B > ; G) = nomcc(G)     if mcc(A,G) = mcc(B,G)          [metadata "CA-1"]. 
 ceq nomcc(< A,B > ; G) = nomcc(G) - 1 if mcc(A,G) <> mcc(B,G) = false [metadata "CA-2"].
 ---
 op mktree : Graph -> Graph .
 eq  mktree(nil) = nil .
 ceq mktree(< A,B > ; G) = mktree(G)           if mcc(A,G) = mcc(B,G)          [metadata "CA-1"].
 ceq mktree(< A,B > ; G) = < A,B > ; mktree(G) if mcc(A,G) <> mcc(B,G) = false [metadata "CA-1"].
endfm)