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

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

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

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. i@0:Vertices(the_graph), s1,s2@0:traversal(the_graph). (([inl(i)] @ s2 @ [inr(i)]) @ s) = (s1 @ [inr(i@0)] @ s2@0) traversal(the_graph) (j:Vertices(the_graph). (inr(j) s2@0) (inl(j) s2@0) j-the_graph- > *i@0)
19. i@0:Vertices(the_graph), s1,s2@0:traversal(the_graph). (j:Vertices(the_graph). i@0-the_graph- > *j non-trivial-loop(the_graph;j)) (([inl(i)] @ s2 @ [inr(i)]) @ s) = (s1 @ [inl(i@0)] @ s2@0) traversal(the_graph) (j:Vertices(the_graph). i@0-the_graph- > *j (inr(j) s2@0))
20. i@0:Vertices(the_graph). L1 @ [i]-the_graph- > *i@0 non-trivial-loop(the_graph;i@0)
21. 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)))
22. L1@0: Vertices(the_graph) List
23. L2: Vertices(the_graph) List
24. (L1 @ [i]) = (L1@0 @ L2)
25. e: Vertices(the_graph) List
26. L1 = (L1@0 @ e)
27. L2 = (e @ [i])
28. s1: traversal(the_graph)
29. s3: traversal(the_graph)
30. s = (s3 @ s1) traversal(the_graph)
31. paren(Vertices(the_graph);s1)
32. paren(Vertices(the_graph);s3)
33. j:Vertices(the_graph). ((inl(j) s1) L1@0-the_graph- > *j) & ((inl(j) s3) e-the_graph- > *j & L1@0-the_graph- > *j)
34. j: Vertices(the_graph)
35. (inl(j) s3) e-the_graph- > *j & L1@0-the_graph- > *j
36. (inl(j) s3) (e-the_graph- > *j & L1@0-the_graph- > *j)
37. i-the_graph- > *j
38. L1@0-the_graph- > *j
39. (inr(j) (s2 @ [inr(i)]) @ s)

(inl(j) s2) (inl(j) s) inl(j) = inl(i) Vertices(the_graph)+Vertices(the_graph)

By: RWW Thm* x:T, l1,l2:T List. (x l1 @ l2) (x l1) (x l2) -1 THEN RWO Thm* a,x:T. (x [a]) x = a -1 THEN Analyze -1

Generated subgoals:

1 39. (inr(j) s2) inr(j) = inr(i) Vertices(the_graph)+Vertices(the_graph)
(inl(j) s2) (inl(j) s) inl(j) = inl(i) Vertices(the_graph)+Vertices(the_graph) 4 steps

2 39. (inr(j) s)
(inl(j) s2) (inl(j) s) inl(j) = inl(i) Vertices(the_graph)+Vertices(the_graph) 2 steps

About:

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

1	39. `(inr(j) s2) inr(j) = inr(i) Vertices(the_graph)+Vertices(the_graph)` `(inl(j) s2) (inl(j) s) inl(j) = inl(i) Vertices(the_graph)+Vertices(the_graph)`	4 steps

2	39. `(inr(j) s)` `(inl(j) s2) (inl(j) s) inl(j) = inl(i) Vertices(the_graph)+Vertices(the_graph)`	2 steps