Nuprl - Proof: dfsl-properties 2 1 3 2 1 2 1 1

(158steps total) PrintForm Definitions Lemmas graph 1 3 Sections Graphs Doc

At: dfsl-properties 2 1 3 2 1 2 1 1

1. the_graph: Graph
2. the_obj: GraphObject(the_graph)
3. L: Vertices(the_graph) List
4. L1: Vertices(the_graph) List
5. s: traversal(the_graph)
6. i: Vertices(the_graph)
7. paren(Vertices(the_graph);s)
8. no_repeats(Vertices(the_graph)+Vertices(the_graph);s)
9. df-traversal(the_graph;s)
10. i:Vertices(the_graph). (inl(i) s) L1-the_graph- > *i
11. (inl(i) s)
12. s2: traversal(the_graph)
13. l_disjoint(Vertices(the_graph)+Vertices(the_graph);s2;s)
14. paren(Vertices(the_graph);s2)
15. no_repeats(Vertices(the_graph)+Vertices(the_graph);s2)
16. (inr(i) s2)
17. j:Vertices(the_graph). (inr(j) s2) i-the_graph- > *j
18. df-traversal(the_graph;([inl(i)] @ s2 @ [inr(i)]) @ s)
19. i@0:Vertices(the_graph). L1 @ [i]-the_graph- > *i@0 non-trivial-loop(the_graph;i@0)
20. L1@0,L2:Vertices(the_graph) List. L1 = (L1@0 @ L2) (s1,s2:traversal(the_graph). s = (s2 @ s1) traversal(the_graph) & paren(Vertices(the_graph);s1) & paren(Vertices(the_graph);s2) & (j:Vertices(the_graph). ((inl(j) s1) L1@0-the_graph- > *j) & ((inl(j) s2) L2-the_graph- > *j & L1@0-the_graph- > *j)))
21. L1@0: Vertices(the_graph) List
22. L2: Vertices(the_graph) List
23. (L1 @ [i]) = (L1@0 @ L2)
24. e: Vertices(the_graph) List
25. L1 = (L1@0 @ e)
26. L2 = (e @ [i])

s1,s2@0:traversal(the_graph). (([inl(i)] @ s2 @ [inr(i)]) @ s) = (s2@0 @ s1) traversal(the_graph) & paren(Vertices(the_graph);s1) & paren(Vertices(the_graph);s2@0) & (j:Vertices(the_graph). ((inl(j) s1) L1@0-the_graph- > *j) & ((inl(j) s2@0) L2-the_graph- > *j & L1@0-the_graph- > *j))

By: InstHyp [L1@0;e] -7 THEN ExRepD THEN InstConcl [s1;([inl(i)] @ s2 @ [inr(i)]) @ s3] THEN Analyze 0

Generated subgoals:

1 27. s1: traversal(the_graph)
28. s3: traversal(the_graph)
29. s = (s3 @ s1) traversal(the_graph)
30. paren(Vertices(the_graph);s1)
31. paren(Vertices(the_graph);s3)
32. j:Vertices(the_graph). ((inl(j) s1) L1@0-the_graph- > *j) & ((inl(j) s3) e-the_graph- > *j & L1@0-the_graph- > *j)
(([inl(i)] @ s2 @ [inr(i)]) @ s) = ((([inl(i)] @ s2 @ [inr(i)]) @ s3) @ s1) traversal(the_graph) 1 step

2 27. s1: traversal(the_graph)
28. s3: traversal(the_graph)
29. s = (s3 @ s1) traversal(the_graph)
30. paren(Vertices(the_graph);s1)
31. paren(Vertices(the_graph);s3)
32. j:Vertices(the_graph). ((inl(j) s1) L1@0-the_graph- > *j) & ((inl(j) s3) e-the_graph- > *j & L1@0-the_graph- > *j)
paren(Vertices(the_graph);s1) & paren(Vertices(the_graph);([inl(i)] @ s2 @ [inr(i)]) @ s3) & (j:Vertices(the_graph). ((inl(j) s1) L1@0-the_graph- > *j) & ((inl(j) ([inl(i)] @ s2 @ [inr(i)]) @ s3) L2-the_graph- > *j & L1@0-the_graph- > *j)) 47 steps

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(158steps total) PrintForm Definitions Lemmas graph 1 3 Sections Graphs Doc

1	27. `s1:` `traversal(the_graph)` 28. `s3:` `traversal(the_graph)` 29. `s = (s3 @ s1) traversal(the_graph)` 30. `paren(Vertices(the_graph);s1)` 31. `paren(Vertices(the_graph);s3)` 32. `j:Vertices(the_graph). ((inl(j) s1) L1@0-the_graph- > j) & ((inl(j) s3) e-the_graph- > j & L1@0-the_graph- > *j)` `(([inl(i)] @ s2 @ [inr(i)]) @ s) = ((([inl(i)] @ s2 @ [inr(i)]) @ s3) @ s1) traversal(the_graph)`	1 step

2	27. `s1:` `traversal(the_graph)` 28. `s3:` `traversal(the_graph)` 29. `s = (s3 @ s1) traversal(the_graph)` 30. `paren(Vertices(the_graph);s1)` 31. `paren(Vertices(the_graph);s3)` 32. `j:Vertices(the_graph). ((inl(j) s1) L1@0-the_graph- > j) & ((inl(j) s3) e-the_graph- > j & L1@0-the_graph- > j)` `paren(Vertices(the_graph);s1) & paren(Vertices(the_graph);([inl(i)] @ s2 @ [inr(i)]) @ s3) & (j:Vertices(the_graph). ((inl(j) s1) L1@0-the_graph- > j) & ((inl(j) ([inl(i)] @ s2 @ [inr(i)]) @ s3) L2-the_graph- > j & L1@0-the_graph- > j))`	47 steps