(44steps total) PrintForm Definitions Lemmas graph 1 3 Sections Graphs Doc

At: dfs-df-traversal 2 2 2 1 2 1 1 3 1 1
1. the_graph: Graph
2. the_obj: GraphObject(the_graph)
3. s1: traversal(the_graph)
4. s2: traversal(the_graph)
5. i1: Vertices(the_graph)
6. (inr(i1) s1)
7. (inl(i1) s1)
8. paren(Vertices(the_graph);s2)
9. l_disjoint(Vertices(the_graph)+Vertices(the_graph);s2;s1)
10. no_repeats(Vertices(the_graph)+Vertices(the_graph);s2)
11. j:Vertices(the_graph). (inr(j) s2) i1-the_graph- > *j
12. j:Vertices(the_graph). i1-the_graph- > j j = i1 (inl(j) s2) (inl(j) s1) (inr(j) s1)
13. df-traversal(the_graph;s1)
14. j:Vertices(the_graph). (inr(j) s1) (inl(j) s1) j-the_graph- > *i1
15. df-traversal(the_graph;s2 @ [inr(i1) / s1])
16. j:Vertices(the_graph). i1-the_graph- > *j non-trivial-loop(the_graph;j)
17. j: Vertices(the_graph)
18. z: Vertices(the_graph)
19. z = i1
20. i1-the_graph- > z
21. z-the_graph- > *j
22. (inl(z) s2)
(inr(j) s2 @ [inr(i1) / s1])
By:
Inst Thm* s:T List, z:T. (z s) (s1,s2:T List. s = (s1 @ [z / s2])) [Vertices(the_graph)+Vertices(the_graph);s2;inl(z)]
THEN
ExRepD
THEN
let h 15 in (Unfold `df-traversal` h) THEN (Analyze h) THEN (InstHyp [z;s3;s2@0 @ [inr(i1) / s1];j] h+1)
Generated subgoals:

115. i:Vertices(the_graph), s1@0,s2@0:traversal(the_graph). (s2 @ [inr(i1) / s1]) = (s1@0 @ [inr(i)] @ s2@0) traversal(the_graph) (j:Vertices(the_graph). (inr(j) s2@0) (inl(j) s2@0) j-the_graph- > *i)
16. i:Vertices(the_graph), s1@0,s2@0:traversal(the_graph). (j:Vertices(the_graph). i-the_graph- > *j non-trivial-loop(the_graph;j)) (s2 @ [inr(i1) / s1]) = (s1@0 @ [inl(i)] @ s2@0) traversal(the_graph) (j:Vertices(the_graph). i-the_graph- > *j (inr(j) s2@0))
17. j:Vertices(the_graph). i1-the_graph- > *j non-trivial-loop(the_graph;j)
18. j: Vertices(the_graph)
19. z: Vertices(the_graph)
20. z = i1
21. i1-the_graph- > z
22. z-the_graph- > *j
23. (inl(z) s2)
24. s3: (Vertices(the_graph)+Vertices(the_graph)) List
25. s2@0: (Vertices(the_graph)+Vertices(the_graph)) List
26. s2 = (s3 @ [inl(z) / s2@0])
27. j1: Vertices(the_graph)
28. z-the_graph- > *j1
non-trivial-loop(the_graph;j1)		1 step
 
215. i:Vertices(the_graph), s1@0,s2@0:traversal(the_graph). (s2 @ [inr(i1) / s1]) = (s1@0 @ [inr(i)] @ s2@0) traversal(the_graph) (j:Vertices(the_graph). (inr(j) s2@0) (inl(j) s2@0) j-the_graph- > *i)
16. i:Vertices(the_graph), s1@0,s2@0:traversal(the_graph). (j:Vertices(the_graph). i-the_graph- > *j non-trivial-loop(the_graph;j)) (s2 @ [inr(i1) / s1]) = (s1@0 @ [inl(i)] @ s2@0) traversal(the_graph) (j:Vertices(the_graph). i-the_graph- > *j (inr(j) s2@0))
17. j:Vertices(the_graph). i1-the_graph- > *j non-trivial-loop(the_graph;j)
18. j: Vertices(the_graph)
19. z: Vertices(the_graph)
20. z = i1
21. i1-the_graph- > z
22. z-the_graph- > *j
23. (inl(z) s2)
24. s3: (Vertices(the_graph)+Vertices(the_graph)) List
25. s2@0: (Vertices(the_graph)+Vertices(the_graph)) List
26. s2 = (s3 @ [inl(z) / s2@0])
(s2 @ [inr(i1) / s1]) = (s3 @ [inl(z)] @ s2@0 @ [inr(i1) / s1]) traversal(the_graph)		1 step
 
315. i:Vertices(the_graph), s1@0,s2@0:traversal(the_graph). (s2 @ [inr(i1) / s1]) = (s1@0 @ [inr(i)] @ s2@0) traversal(the_graph) (j:Vertices(the_graph). (inr(j) s2@0) (inl(j) s2@0) j-the_graph- > *i)
16. i:Vertices(the_graph), s1@0,s2@0:traversal(the_graph). (j:Vertices(the_graph). i-the_graph- > *j non-trivial-loop(the_graph;j)) (s2 @ [inr(i1) / s1]) = (s1@0 @ [inl(i)] @ s2@0) traversal(the_graph) (j:Vertices(the_graph). i-the_graph- > *j (inr(j) s2@0))
17. j:Vertices(the_graph). i1-the_graph- > *j non-trivial-loop(the_graph;j)
18. j: Vertices(the_graph)
19. z: Vertices(the_graph)
20. z = i1
21. i1-the_graph- > z
22. z-the_graph- > *j
23. (inl(z) s2)
24. s3: (Vertices(the_graph)+Vertices(the_graph)) List
25. s2@0: (Vertices(the_graph)+Vertices(the_graph)) List
26. s2 = (s3 @ [inl(z) / s2@0])
27. (inr(j) s2@0 @ [inr(i1) / s1])
(inr(j) s2 @ [inr(i1) / s1])		3 steps

About:
listconsconsnilunioninlinr
universeequalimpliesorallexists

(44steps total) PrintForm Definitions Lemmas graph 1 3 Sections Graphs Doc