Nuprl - Proof: df-traversal-cons2 2

(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. j:Vertices(the_graph). i@0-the_graph- > *j non-trivial-loop(the_graph;j)
11. [inr(i) / s] = (s1 @ [inl(i@0)] @ s2) traversal(the_graph)
12. j: Vertices(the_graph)
13. i@0-the_graph- > *j

(inr(j) s2)

By: Analyze -6 THEN All Reduce THEN SplitCons -3

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. j:Vertices(the_graph). i@0-the_graph- > *j non-trivial-loop(the_graph;j)
10. j: Vertices(the_graph)
11. i@0-the_graph- > *j
12. inr(i) = inl(i@0) Vertices(the_graph)+Vertices(the_graph)
13. s = s2
(inr(j) s2) 1 step

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. j:Vertices(the_graph). i@0-the_graph- > *j non-trivial-loop(the_graph;j)
12. j: Vertices(the_graph)
13. i@0-the_graph- > *j
14. inr(i) = u
15. s = (v @ [inl(i@0) / s2]) traversal(the_graph)
(inr(j) s2) 1 step

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. `j:Vertices(the_graph). i@0-the_graph- > j non-trivial-loop(the_graph;j)` 10. `j:` `Vertices(the_graph)` 11. `i@0-the_graph- > *j` 12. `inr(i) = inl(i@0) Vertices(the_graph)+Vertices(the_graph)` 13. `s = s2` `(inr(j) s2)`	1 step

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. `j:Vertices(the_graph). i@0-the_graph- > j non-trivial-loop(the_graph;j)` 12. `j:` `Vertices(the_graph)` 13. `i@0-the_graph- > *j` 14. `inr(i) = u` 15. `s = (v @ [inl(i@0) / s2]) traversal(the_graph)` `(inr(j) s2)`	1 step