| | Some definitions of interest. |
|
| 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 |
|
| append | Def as @ bs == Case of as; nil  bs ; a.as' [a / (as' @ bs)] (recursive) |
| | | Thm* T:Type, as,bs:T List. (as @ bs) T List |
|
| connect | Def x-the_graph- > *y == p:Vertices(the_graph) List. path(the_graph;p) & p[0] = x & last(p) = y |
| | | Thm* For any graph
x,y:V. x-the_graph- > *y Prop |
|
| dfs | Def dfs(the_obj;s;i) == if member-paren(x,y.the_obj.eq(x,y);i;s) s else [inl(i) / (the_obj.eacc(( s',j. dfs(the_obj;s';j)),[inr(i) / s],i))] fi (recursive) |
| | | Thm* For any graph
the_obj:GraphObject(the_graph), s:Traversal, i:V. dfs(the_obj;s;i) Traversal |
|
| 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'} |
|
| traversal | Def traversal(G) == (Vertices(G)+Vertices(G)) List |
| | | Thm* For any graph
Traversal Type |
|
| gr_v | Def Vertices(t) == 1of(t) |
| | | Thm* t:Graph. Vertices(t) Type |
|
| graph | Def Graph == v:Typee:Type(evv)Top |
| | | Thm* Graph Type{i'} |
|
| l_disjoint | Def l_disjoint(T;l1;l2) == x:T.  ((x l1) & (x l2)) |
| | | Thm* T:Type, l,l':T List. l_disjoint(T;l;l') Prop |
|
| l_member | Def (x l) == i: . i < ||l|| & x = l[i] T |
| | | Thm* T:Type, x:T, l:T List. (x l) Prop |
|
| 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 |
|
| not | Def A == A False |
| | | Thm* A:Prop. (A) Prop |
