Nuprl - Proof: df-traversal-cons2 1

(10steps total) PrintForm Definitions graph 1 2 Sections Graphs Doc

1. the_graph: Graph
2. i: Vertices(the_graph)
3. s: traversal(the_graph)
4. j:Vertices(the_graph). (inr(j) s) (inl(j) s) j-the_graph- > *i
5. 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)
6. 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) (j:Vertices(the_graph). i-the_graph- > *j (inr(j) s2))
7. i@0: Vertices(the_graph)
8. s1: traversal(the_graph)
9. s2: traversal(the_graph)
10. [inr(i) / s] = (s1 @ [inr(i@0)] @ s2) traversal(the_graph)
11. j: Vertices(the_graph)
12. (inr(j) s2)
13. (inl(j) s2)

j-the_graph- > *i@0

By: Analyze -6 THEN All Reduce

Generated subgoals:

1 5. 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)
6. 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) (j:Vertices(the_graph). i-the_graph- > *j (inr(j) s2))
7. i@0: Vertices(the_graph)
8. s2: traversal(the_graph)
9. [inr(i) / s] = [inr(i@0) / s2]
10. j: Vertices(the_graph)
11. (inr(j) s2)
12. (inl(j) s2)
j-the_graph- > *i@0 3 steps

2 5. 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)
6. 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) (j:Vertices(the_graph). i-the_graph- > *j (inr(j) s2))
7. i@0: Vertices(the_graph)
8. u: Vertices(the_graph)+Vertices(the_graph)
9. v: (Vertices(the_graph)+Vertices(the_graph)) List
10. s2: traversal(the_graph)
11. [inr(i) / s] = [u / (v @ [inr(i@0) / s2])] traversal(the_graph)
12. j: Vertices(the_graph)
13. (inr(j) s2)
14. (inl(j) s2)
j-the_graph- > *i@0 2 steps

About:

(10steps total) PrintForm Definitions graph 1 2 Sections Graphs Doc

1	5. `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)` 6. `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) (j:Vertices(the_graph). i-the_graph- > j (inr(j) s2))` 7. `i@0:` `Vertices(the_graph)` 8. `s2:` `traversal(the_graph)` 9. `[inr(i) / s] = [inr(i@0) / s2]` 10. `j:` `Vertices(the_graph)` 11. `(inr(j) s2)` 12. `(inl(j) s2)` `j-the_graph- > i@0`	3 steps

2	5. `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)` 6. `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) (j:Vertices(the_graph). i-the_graph- > j (inr(j) s2))` 7. `i@0:` `Vertices(the_graph)` 8. `u:` `Vertices(the_graph)+Vertices(the_graph)` 9. `v:` `(Vertices(the_graph)+Vertices(the_graph)) List` 10. `s2:` `traversal(the_graph)` 11. `[inr(i) / s] = [u / (v @ [inr(i@0) / s2])] traversal(the_graph)` 12. `j:` `Vertices(the_graph)` 13. `(inr(j) s2)` 14. `(inl(j) s2)` `j-the_graph- > i@0`	2 steps