| | Some definitions of interest. |
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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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| 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 |
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| 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 |
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| non-trivial-loop | Def non-trivial-loop(G;i) == j:Vertices(G). j = i & i-G- > *j & j-G- > *i |
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| 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 |
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| decidable | Def Dec(P) == P P |
| | | Thm* A:Prop. Dec(A) Prop |
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| traversal | Def traversal(G) == (Vertices(G)+Vertices(G)) List |
| | | Thm* For any graph
Traversal Type |
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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:Type e:Type(e  vv)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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| not | Def A == A False |
| | | Thm* A:Prop. (A) Prop |
