(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
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])
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)
33. j: Vertices(the_graph)
34. (inl(j) s3) e-the_graph- > *j & L1@0-the_graph- > *j
35. i-the_graph- > *j
36. L1@0-the_graph- > *j
(inl(j) s2) (inl(j) s) inl(j) = inl(i) Vertices(the_graph)+Vertices(the_graph)
By: let h 18 in (Unfold `df-traversal` h) THEN (Analyze h) THEN (InstHyp [i;nil;(s2 @ [inr(i)]) @ s;j] h+1)
Generated subgoals:

118. 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. j1: Vertices(the_graph)
40. i-the_graph- > *j1
non-trivial-loop(the_graph;j1)		2 steps
 
218. 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
(([inl(i)] @ s2 @ [inr(i)]) @ s) = (nil @ [inl(i)] @ (s2 @ [inr(i)]) @ s) traversal(the_graph)		1 step
 
318. 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)		7 steps

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