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

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

At: dfsl-properties 2 1 3 2 1 2 1 2 2 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@0 = (L1 @ e)
26. [i] = (e @ L2)
27. i:Vertices(the_graph). (inl(i) s) L1-the_graph- > *i

L2 = nil e = nil

By: Lemmaize [-2] THEN Analyze -2 THEN Analyze 3 THEN All Reduce

Generated subgoals:

1 2. i: Vertices(the_graph)
3. [i] = nil
nil = nil Vertices(the_graph) List nil = nil Vertices(the_graph) List 1 step

2 2. i: Vertices(the_graph)
3. u: Vertices(the_graph)
4. v: Vertices(the_graph) List
5. [i] = [u / v]
[u / v] = nil nil = nil Vertices(the_graph) List 1 step

3 2. i: Vertices(the_graph)
3. u: Vertices(the_graph)
4. v: Vertices(the_graph) List
5. [i] = [u / (v @ nil)]
nil = nil Vertices(the_graph) List [u / v] = nil 1 step

4 2. i: Vertices(the_graph)
3. u1: Vertices(the_graph)
4. v1: Vertices(the_graph) List
5. u: Vertices(the_graph)
6. v: Vertices(the_graph) List
7. [i] = [u / (v @ [u1 / v1])]
[u1 / v1] = nil [u / v] = nil 1 step

About:

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

1	2. `i:` `Vertices(the_graph)` 3. `[i] = nil` `nil = nil Vertices(the_graph) List nil = nil Vertices(the_graph) List`	1 step

2	2. `i:` `Vertices(the_graph)` 3. `u:` `Vertices(the_graph)` 4. `v:` `Vertices(the_graph) List` 5. `[i] = [u / v]` `[u / v] = nil nil = nil Vertices(the_graph) List`	1 step

3	2. `i:` `Vertices(the_graph)` 3. `u:` `Vertices(the_graph)` 4. `v:` `Vertices(the_graph) List` 5. `[i] = [u / (v @ nil)]` `nil = nil Vertices(the_graph) List [u / v] = nil`	1 step

4	2. `i:` `Vertices(the_graph)` 3. `u1:` `Vertices(the_graph)` 4. `v1:` `Vertices(the_graph) List` 5. `u:` `Vertices(the_graph)` 6. `v:` `Vertices(the_graph) List` 7. `[i] = [u / (v @ [u1 / v1])]` `[u1 / v1] = nil [u / v] = nil`	1 step