| | Some definitions of interest. |
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| topsort | Def topsort(G) == mapoutl(depthfirst(G)) |
| | | Thm* For any graph
 the_obj:GraphObject(the_graph). topsort(the_obj)  V List |
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| depthfirst | Def depthfirst(the_obj) == the_obj.vacc(( s,i. dfs(the_obj;s;i)),nil) |
| | | Thm* For any graph
the_obj:GraphObject(the_graph). depthfirst(the_obj) Traversal |
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| depthfirst-traversal | Def depthfirst-traversal(the_graph;s) == i:Vertices(the_graph), s1,s2:traversal(the_graph). (j:Vertices(the_graph). i-the_graph- > *j    non-trivial-loop(the_graph;j)) s = (s1 @ [inl(i)] @ s2) traversal(the_graph) (j:Vertices(the_graph). j = i i-the_graph- > *j (inl(j) s2)) |
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| df-traversal | Def df-traversal(G;s) == (i:Vertices(G), s1,s2:traversal(G). s = (s1 @ [inr(i)] @ s2) traversal(G) (j:Vertices(G). (inr(j) s2) (inl(j) s2) j-G- > *i)) & (i:Vertices(G), s1,s2:traversal(G). (j:Vertices(G). i-G- > *j non-trivial-loop(G;j)) s = (s1 @ [inl(i)] @ s2) traversal(G) (j:Vertices(G). i-G- > *j (inr(j) s2))) |
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| graphobj | Def GraphObject(the_graph) == eq:Vertices(the_graph)  Vertices(the_graph)  (x,y:Vertices(the_graph).  (eq(x,y))  x = y)(eacc:( T:Type. (TVertices(the_graph)T)TVertices(the_graph)T)(T:Type, s:T, x:Vertices(the_graph), f:(TVertices(the_graph)T).  L:Vertices(the_graph) List. (y:Vertices(the_graph). x-the_graph- > y (y L)) & eacc(f,s,x) = list_accum(s',x'.f(s',x');s;L))(vacc:(T:Type. (TVertices(the_graph)T)TT)(T:Type, s:T, f:(TVertices(the_graph)T). L:Vertices(the_graph) List. no_repeats(Vertices(the_graph);L) & (y:Vertices(the_graph). (y L)) & vacc(f,s) = list_accum(s',x'.f(s',x');s;L))Top)) |
| | | Thm* the_graph:Graph. GraphObject(the_graph) Type{i'} |
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| non-trivial-loop-free | Def non-trivial-loop-free(G) == i:Vertices(G). non-trivial-loop(G;i) |
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| topsorted | Def topsorted(the_graph;s) == i,j:Vertices(the_graph). i-the_graph- > *j i = j i before j s |
| | | Thm* For any graph
s:V List. topsorted(the_graph;s) Prop |
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| gr_v | Def Vertices(t) == 1of(t) |
| | | Thm* t:Graph. Vertices(t) Type |
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| graph | Def Graph == v:Typee:Type(evv)Top |
| | | Thm* Graph Type{i'} |
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| l_member | Def (x l) == i: . i < ||l|| & x = l[i] T |
| | | Thm* T:Type, x:T, l:T List. (x l) Prop |
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| no_repeats | Def no_repeats(T;l) == i,j:. i < ||l|| j < ||l|| i = j l[i] = l[j] T |
| | | Thm* T:Type, l:T List. no_repeats(T;l) Prop |
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| paren | Def paren(T;s) == s = nil (T+T) List  (t:T, s':(T+T) List. s = ([inl(t)] @ s' @ [inr(t)]) & paren(T;s')) (s',s'':(T+T) List. ||s'|| < ||s|| & ||s''|| < ||s|| & s = (s' @ s'') & paren(T;s') & paren(T;s'')) (recursive) |
| | | Thm* T:Type, s:(T+T) List. paren(T;s) Prop |
