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

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

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)))

L1@0,L2:Vertices(the_graph) List. (L1 @ [i]) = (L1@0 @ L2) (s1,s2@0:traversal(the_graph). (([inl(i)] @ s2 @ [inr(i)]) @ s) = (s2@0 @ s1) traversal(the_graph) & paren(Vertices(the_graph);s1) & paren(Vertices(the_graph);s2@0) & (j:Vertices(the_graph). ((inl(j) s1) L1@0-the_graph- > *j) & ((inl(j) s2@0) L2-the_graph- > *j & L1@0-the_graph- > *j)))

By: Auto THEN EqualAppendsD -1 THEN Analyze -1

Generated subgoals:

1 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])
s1,s2@0:traversal(the_graph). (([inl(i)] @ s2 @ [inr(i)]) @ s) = (s2@0 @ s1) traversal(the_graph) & paren(Vertices(the_graph);s1) & paren(Vertices(the_graph);s2@0) & (j:Vertices(the_graph). ((inl(j) s1) L1@0-the_graph- > *j) & ((inl(j) s2@0) L2-the_graph- > *j & L1@0-the_graph- > *j)) 49 steps

2 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)
s1,s2@0:traversal(the_graph). (([inl(i)] @ s2 @ [inr(i)]) @ s) = (s2@0 @ s1) traversal(the_graph) & paren(Vertices(the_graph);s1) & paren(Vertices(the_graph);s2@0) & (j:Vertices(the_graph). ((inl(j) s1) L1@0-the_graph- > *j) & ((inl(j) s2@0) L2-the_graph- > *j & L1@0-the_graph- > *j)) 84 steps

About:

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

1	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])` `s1,s2@0:traversal(the_graph). (([inl(i)] @ s2 @ [inr(i)]) @ s) = (s2@0 @ s1) traversal(the_graph) & paren(Vertices(the_graph);s1) & paren(Vertices(the_graph);s2@0) & (j:Vertices(the_graph). ((inl(j) s1) L1@0-the_graph- > j) & ((inl(j) s2@0) L2-the_graph- > j & L1@0-the_graph- > *j))`	49 steps

2	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)` `s1,s2@0:traversal(the_graph). (([inl(i)] @ s2 @ [inr(i)]) @ s) = (s2@0 @ s1) traversal(the_graph) & paren(Vertices(the_graph);s1) & paren(Vertices(the_graph);s2@0) & (j:Vertices(the_graph). ((inl(j) s1) L1@0-the_graph- > j) & ((inl(j) s2@0) L2-the_graph- > j & L1@0-the_graph- > *j))`	84 steps