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

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

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

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@0 = (L1 @ e)
26. [i] = (e @ L2)
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-the_graph- > *j
33. j:Vertices(the_graph). (inl(j) s3) nil-the_graph- > *j & L1-the_graph- > *j
34. j:Vertices(the_graph). (inl(j) s3) False
35. j:Vertices(the_graph). (inr(j) s3) False

s3 = nil

By: Analyze -8 THEN Analyze -9

Generated subgoals:

1 28. x: Vertices(the_graph)
29. v: (Vertices(the_graph)+Vertices(the_graph)) List
30. s = ([inl(x) / v] @ s1) traversal(the_graph)
31. paren(Vertices(the_graph);s1)
32. paren(Vertices(the_graph);[inl(x) / v])
33. j:Vertices(the_graph). (inl(j) s1) L1-the_graph- > *j
34. j:Vertices(the_graph). (inl(j) [inl(x) / v]) nil-the_graph- > *j & L1-the_graph- > *j
35. j:Vertices(the_graph). (inl(j) [inl(x) / v]) False
36. j:Vertices(the_graph). (inr(j) [inl(x) / v]) False
[inl(x) / v] = nil traversal(the_graph) 2 steps

2 28. y: Vertices(the_graph)
29. v: (Vertices(the_graph)+Vertices(the_graph)) List
30. s = ([inr(y) / v] @ s1) traversal(the_graph)
31. paren(Vertices(the_graph);s1)
32. paren(Vertices(the_graph);[inr(y) / v])
33. j:Vertices(the_graph). (inl(j) s1) L1-the_graph- > *j
34. j:Vertices(the_graph). (inl(j) [inr(y) / v]) nil-the_graph- > *j & L1-the_graph- > *j
35. j:Vertices(the_graph). (inl(j) [inr(y) / v]) False
36. j:Vertices(the_graph). (inr(j) [inr(y) / v]) False
[inr(y) / v] = nil traversal(the_graph) 2 steps

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

1	28. `x:` `Vertices(the_graph)` 29. `v:` `(Vertices(the_graph)+Vertices(the_graph)) List` 30. `s = ([inl(x) / v] @ s1) traversal(the_graph)` 31. `paren(Vertices(the_graph);s1)` 32. `paren(Vertices(the_graph);[inl(x) / v])` 33. `j:Vertices(the_graph). (inl(j) s1) L1-the_graph- > j` 34. `j:Vertices(the_graph). (inl(j) [inl(x) / v]) nil-the_graph- > j & L1-the_graph- > *j` 35. `j:Vertices(the_graph). (inl(j) [inl(x) / v]) False` 36. `j:Vertices(the_graph). (inr(j) [inl(x) / v]) False` `[inl(x) / v] = nil traversal(the_graph)`	2 steps

2	28. `y:` `Vertices(the_graph)` 29. `v:` `(Vertices(the_graph)+Vertices(the_graph)) List` 30. `s = ([inr(y) / v] @ s1) traversal(the_graph)` 31. `paren(Vertices(the_graph);s1)` 32. `paren(Vertices(the_graph);[inr(y) / v])` 33. `j:Vertices(the_graph). (inl(j) s1) L1-the_graph- > j` 34. `j:Vertices(the_graph). (inl(j) [inr(y) / v]) nil-the_graph- > j & L1-the_graph- > *j` 35. `j:Vertices(the_graph). (inl(j) [inr(y) / v]) False` 36. `j:Vertices(the_graph). (inr(j) [inr(y) / v]) False` `[inr(y) / v] = nil traversal(the_graph)`	2 steps