Nuprl - Proof: dfl-traversal-consr 1

(11steps total) PrintForm Definitions Lemmas graph 1 2 Sections Graphs Doc

1. the_graph: Graph
2. L: Vertices(the_graph) List
3. i: Vertices(the_graph)
4. s: traversal(the_graph)
5. j:Vertices(the_graph). (inr(j) s) (inl(j) s) j-the_graph- > *i
6. L-the_graph- > *i
7. i:Vertices(the_graph), s1,s2:traversal(the_graph). s = (s1 @ [inr(i)] @ s2) traversal(the_graph) (j:Vertices(the_graph). (inr(j) s2) (inl(j) s2) j-the_graph- > *i)
8. j:Vertices(the_graph). (inr(j) s) L-the_graph- > *j
9. i:Vertices(the_graph), s1,s2:traversal(the_graph). (j:Vertices(the_graph). i-the_graph- > *j non-trivial-loop(the_graph;j)) s = (s1 @ [inl(i)] @ s2) traversal(the_graph) L-the_graph- > *i
10. i@0: Vertices(the_graph)
11. s1: traversal(the_graph)
12. s2: traversal(the_graph)
13. [inr(i) / s] = (s1 @ [inr(i@0)] @ s2) traversal(the_graph)
14. j: Vertices(the_graph)
15. (inr(j) s2)
16. (inl(j) s2)

j-the_graph- > *i@0

By: Analyze -6 THEN All Reduce

Generated subgoals:

1 7. i:Vertices(the_graph), s1,s2:traversal(the_graph). s = (s1 @ [inr(i) / s2]) traversal(the_graph) (j:Vertices(the_graph). (inr(j) s2) (inl(j) s2) j-the_graph- > *i)
8. j:Vertices(the_graph). (inr(j) s) L-the_graph- > *j
9. i:Vertices(the_graph), s1,s2:traversal(the_graph). (j:Vertices(the_graph). i-the_graph- > *j non-trivial-loop(the_graph;j)) s = (s1 @ [inl(i) / s2]) traversal(the_graph) L-the_graph- > *i
10. i@0: Vertices(the_graph)
11. s2: traversal(the_graph)
12. [inr(i) / s] = [inr(i@0) / s2]
13. j: Vertices(the_graph)
14. (inr(j) s2)
15. (inl(j) s2)
j-the_graph- > *i@0 2 steps

2 7. i:Vertices(the_graph), s1,s2:traversal(the_graph). s = (s1 @ [inr(i) / s2]) traversal(the_graph) (j:Vertices(the_graph). (inr(j) s2) (inl(j) s2) j-the_graph- > *i)
8. j:Vertices(the_graph). (inr(j) s) L-the_graph- > *j
9. i:Vertices(the_graph), s1,s2:traversal(the_graph). (j:Vertices(the_graph). i-the_graph- > *j non-trivial-loop(the_graph;j)) s = (s1 @ [inl(i) / s2]) traversal(the_graph) L-the_graph- > *i
10. i@0: Vertices(the_graph)
11. u: Vertices(the_graph)+Vertices(the_graph)
12. v: (Vertices(the_graph)+Vertices(the_graph)) List
13. s2: traversal(the_graph)
14. [inr(i) / s] = [u / (v @ [inr(i@0) / s2])] traversal(the_graph)
15. j: Vertices(the_graph)
16. (inr(j) s2)
17. (inl(j) s2)
j-the_graph- > *i@0 2 steps

About:

(11steps total) PrintForm Definitions Lemmas graph 1 2 Sections Graphs Doc

1	7. `i:Vertices(the_graph), s1,s2:traversal(the_graph). s = (s1 @ [inr(i) / s2]) traversal(the_graph) (j:Vertices(the_graph). (inr(j) s2) (inl(j) s2) j-the_graph- > i)` 8. `j:Vertices(the_graph). (inr(j) s) L-the_graph- > j` 9. `i:Vertices(the_graph), s1,s2:traversal(the_graph). (j:Vertices(the_graph). i-the_graph- > j non-trivial-loop(the_graph;j)) s = (s1 @ [inl(i) / s2]) traversal(the_graph) L-the_graph- > i` 10. `i@0:` `Vertices(the_graph)` 11. `s2:` `traversal(the_graph)` 12. `[inr(i) / s] = [inr(i@0) / s2]` 13. `j:` `Vertices(the_graph)` 14. `(inr(j) s2)` 15. `(inl(j) s2)` `j-the_graph- > *i@0`	2 steps

2	7. `i:Vertices(the_graph), s1,s2:traversal(the_graph). s = (s1 @ [inr(i) / s2]) traversal(the_graph) (j:Vertices(the_graph). (inr(j) s2) (inl(j) s2) j-the_graph- > i)` 8. `j:Vertices(the_graph). (inr(j) s) L-the_graph- > j` 9. `i:Vertices(the_graph), s1,s2:traversal(the_graph). (j:Vertices(the_graph). i-the_graph- > j non-trivial-loop(the_graph;j)) s = (s1 @ [inl(i) / s2]) traversal(the_graph) L-the_graph- > i` 10. `i@0:` `Vertices(the_graph)` 11. `u:` `Vertices(the_graph)+Vertices(the_graph)` 12. `v:` `(Vertices(the_graph)+Vertices(the_graph)) List` 13. `s2:` `traversal(the_graph)` 14. `[inr(i) / s] = [u / (v @ [inr(i@0) / s2])] traversal(the_graph)` 15. `j:` `Vertices(the_graph)` 16. `(inr(j) s2)` 17. `(inl(j) s2)` `j-the_graph- > *i@0`	2 steps