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

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

At: dfsl-properties 2 1 3 2 1 2 1 2 2 2 2 1 2 1 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
26. [i] = L2
27. i:Vertices(the_graph). (inl(i) s) L1-the_graph- > *i
28. j: Vertices(the_graph)
29. (inl(j) s) L1-the_graph- > *j
30. [i]-the_graph- > *j & (inl(j) s)

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

By: Unfold `df-traversal` 18 THEN Analyze 18 THEN InstHyp [i;nil;(s2 @ [inr(i)]) @ s;j] 19

Generated subgoals:

1 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@0 = L1
27. [i] = L2
28. i:Vertices(the_graph). (inl(i) s) L1-the_graph- > *i
29. j: Vertices(the_graph)
30. (inl(j) s) L1-the_graph- > *j
31. (inl(j) s) L1-the_graph- > *j
32. [i]-the_graph- > *j
33. (inl(j) s)
34. j1: Vertices(the_graph)
35. i-the_graph- > *j1
non-trivial-loop(the_graph;j1) 2 steps

2 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@0 = L1
27. [i] = L2
28. i:Vertices(the_graph). (inl(i) s) L1-the_graph- > *i
29. j: Vertices(the_graph)
30. (inl(j) s) L1-the_graph- > *j
31. (inl(j) s) L1-the_graph- > *j
32. [i]-the_graph- > *j
33. (inl(j) s)
(([inl(i)] @ s2 @ [inr(i)]) @ s) = (nil @ [inl(i)] @ (s2 @ [inr(i)]) @ s) traversal(the_graph) 1 step

3 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@0 = L1
27. [i] = L2
28. i:Vertices(the_graph). (inl(i) s) L1-the_graph- > *i
29. j: Vertices(the_graph)
30. (inl(j) s) L1-the_graph- > *j
31. (inl(j) s) L1-the_graph- > *j
32. [i]-the_graph- > *j
33. (inl(j) s)
i-the_graph- > *j 1 step

4 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@0 = L1
27. [i] = L2
28. i:Vertices(the_graph). (inl(i) s) L1-the_graph- > *i
29. j: Vertices(the_graph)
30. (inl(j) s) L1-the_graph- > *j
31. [i]-the_graph- > *j & (inl(j) s)
32. (inr(j) (s2 @ [inr(i)]) @ s)
inl(j) = inl(i) Vertices(the_graph)+Vertices(the_graph) (inl(j) s2) inl(j) = inr(i) Vertices(the_graph)+Vertices(the_graph) 6 steps

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

1	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@0 = L1` 27. `[i] = L2` 28. `i:Vertices(the_graph). (inl(i) s) L1-the_graph- > i` 29. `j:` `Vertices(the_graph)` 30. `(inl(j) s) L1-the_graph- > j` 31. `(inl(j) s) L1-the_graph- > j` 32. `[i]-the_graph- > j` 33. `(inl(j) s)` 34. `j1:` `Vertices(the_graph)` 35. `i-the_graph- > j1` `non-trivial-loop(the_graph;j1)`	2 steps

2	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@0 = L1` 27. `[i] = L2` 28. `i:Vertices(the_graph). (inl(i) s) L1-the_graph- > i` 29. `j:` `Vertices(the_graph)` 30. `(inl(j) s) L1-the_graph- > j` 31. `(inl(j) s) L1-the_graph- > j` 32. `[i]-the_graph- > *j` 33. `(inl(j) s)` `(([inl(i)] @ s2 @ [inr(i)]) @ s) = (nil @ [inl(i)] @ (s2 @ [inr(i)]) @ s) traversal(the_graph)`	1 step

3	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@0 = L1` 27. `[i] = L2` 28. `i:Vertices(the_graph). (inl(i) s) L1-the_graph- > i` 29. `j:` `Vertices(the_graph)` 30. `(inl(j) s) L1-the_graph- > j` 31. `(inl(j) s) L1-the_graph- > j` 32. `[i]-the_graph- > j` 33. `(inl(j) s)` `i-the_graph- > j`	1 step

4	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@0 = L1` 27. `[i] = L2` 28. `i:Vertices(the_graph). (inl(i) s) L1-the_graph- > i` 29. `j:` `Vertices(the_graph)` 30. `(inl(j) s) L1-the_graph- > j` 31. `[i]-the_graph- > j & (inl(j) s)` 32. `(inr(j) (s2 @ [inr(i)]) @ s)` `inl(j) = inl(i) Vertices(the_graph)+Vertices(the_graph) (inl(j) s2) inl(j) = inr(i) Vertices(the_graph)+Vertices(the_graph)`	6 steps