Nuprl - Proof: dfs-df-traversal 2 2 2 1 2 1

(44steps total) PrintForm Definitions Lemmas graph 1 3 Sections Graphs Doc

1. the_graph: Graph
2. the_obj: GraphObject(the_graph)
3. s1: traversal(the_graph)
4. s2: traversal(the_graph)
5. i1: Vertices(the_graph)
6. (inr(i1) s1)
7. (inl(i1) s1)
8. paren(Vertices(the_graph);s2)
9. l_disjoint(Vertices(the_graph)+Vertices(the_graph);s2;s1)
10. no_repeats(Vertices(the_graph)+Vertices(the_graph);s2)
11. j:Vertices(the_graph). (inr(j) s2) i1-the_graph- > *j
12. j:Vertices(the_graph). i1-the_graph- > j j = i1 (inl(j) s2) (inl(j) s1) (inr(j) s1)
13. df-traversal(the_graph;s1)
14. j:Vertices(the_graph). (inr(j) s1) (inl(j) s1) j-the_graph- > *i1
15. df-traversal(the_graph;s2 @ [inr(i1) / s1])
16. i: Vertices(the_graph)
17. s2@0: traversal(the_graph)
18. j:Vertices(the_graph). i-the_graph- > *j non-trivial-loop(the_graph;j)
19. j: Vertices(the_graph)
20. i-the_graph- > *j
21. inl(i1) = inl(i) Vertices(the_graph)+Vertices(the_graph)
22. (s2 @ [inr(i1) / s1]) = s2@0 traversal(the_graph)

(inr(j) s2@0)

By: RevHypSubst -1 0 THEN Thin -1 THEN Thin -5 THEN AssertBY (i = i1) (ObviousFrom [-1]) THEN Thin -2 THEN HypSubst -1 -2 THEN HypSubst -1 -4 THEN Thin -1 THEN Thin -4

Generated subgoal:

1 16. j:Vertices(the_graph). i1-the_graph- > *j non-trivial-loop(the_graph;j)
17. j: Vertices(the_graph)
18. i1-the_graph- > *j
(inr(j) s2 @ [inr(i1) / s1]) 20 steps

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(44steps total) PrintForm Definitions Lemmas graph 1 3 Sections Graphs Doc