Nuprl - Proof: dfsl-properties 2 1 3

(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. dfsl-traversal(the_graph;L1;s)
10. s': traversal(the_graph)
11. (inr(i) s) (inl(i) s) s' = nil
12. l_disjoint(Vertices(the_graph)+Vertices(the_graph);s';s)
13. no_repeats(Vertices(the_graph)+Vertices(the_graph);s')
14. paren(Vertices(the_graph);s')
15. j:Vertices(the_graph). (inr(j) s') i-the_graph- > *j
16. dfs(the_obj;s;i) = (s' @ s)
17. df-traversal(the_graph;s' @ s)
18. (inl(i) s)
19. s2: traversal(the_graph)
20. s' = ([inl(i)] @ s2 @ [inr(i)]) traversal(the_graph)

dfsl-traversal(the_graph;L1 @ [i];([inl(i)] @ s2 @ [inr(i)]) @ s)

By: let h 11 in (Thin h) THEN (AssertBY l_disjoint(Vertices(the_graph)+Vertices(the_graph);s2;s) ((MoveToHyp 0 h) THEN (SubstFor s' -1) THEN (RWW Thm* a,b,c:T List. l_disjoint(T;a @ b;c) l_disjoint(T;a;c) & l_disjoint(T;b;c) -1))) THEN (Thin h) THEN (AssertBY paren(Vertices(the_graph);s2) ((Inst Thm* paren_interval [Vertices(the_graph);s';nil;s2;nil;i]) THEN (All Reduce))) THEN (AssertBY (no_repeats(Vertices(the_graph)+Vertices(the_graph);s2) & (inr(i) s2)) (((MoveToHyp 0 h) THEN (SubstFor s' -1) THEN (RWW Thm* l1,l2:T List. no_repeats(T;l1 @ l2) l_disjoint(T;l1;l2) & no_repeats(T;l1) & no_repeats(T;l2) -1)) THEN (RWO Thm* a,b:T List, t:T. l_disjoint(T;b;[t / a]) (t b) & l_disjoint(T;b;a) -1))) THEN (Thin h) THEN (Thin h) THEN (AssertBY (j:Vertices(the_graph). (inr(j) s2) i-the_graph- > *j) (Auto THEN BackThruSomeHyp THEN (SubstFor s' 0)))

Generated subgoals:

1 11. j:Vertices(the_graph). (inr(j) s') i-the_graph- > *j
12. dfs(the_obj;s;i) = (s' @ s)
13. df-traversal(the_graph;s' @ s)
14. (inl(i) s)
15. s2: traversal(the_graph)
16. s' = ([inl(i)] @ s2 @ [inr(i)]) traversal(the_graph)
17. l_disjoint(Vertices(the_graph)+Vertices(the_graph);s2;s)
18. paren(Vertices(the_graph);s2)
19. no_repeats(Vertices(the_graph)+Vertices(the_graph);s2)
20. (inr(i) s2)
21. j: Vertices(the_graph)
22. (inr(j) s2)
(inr(j) [inl(i)] @ s2 @ [inr(i)]) 1 step

2 11. j:Vertices(the_graph). (inr(j) s') i-the_graph- > *j
12. dfs(the_obj;s;i) = (s' @ s)
13. df-traversal(the_graph;s' @ s)
14. (inl(i) s)
15. s2: traversal(the_graph)
16. s' = ([inl(i)] @ s2 @ [inr(i)]) traversal(the_graph)
17. l_disjoint(Vertices(the_graph)+Vertices(the_graph);s2;s)
18. paren(Vertices(the_graph);s2)
19. no_repeats(Vertices(the_graph)+Vertices(the_graph);s2) & (inr(i) s2)
20. j:Vertices(the_graph). (inr(j) s2) i-the_graph- > *j
dfsl-traversal(the_graph;L1 @ [i];([inl(i)] @ s2 @ [inr(i)]) @ s) 148 steps

About:

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

1	11. `j:Vertices(the_graph). (inr(j) s') i-the_graph- > *j` 12. `dfs(the_obj;s;i) = (s' @ s)` 13. `df-traversal(the_graph;s' @ s)` 14. `(inl(i) s)` 15. `s2:` `traversal(the_graph)` 16. `s' = ([inl(i)] @ s2 @ [inr(i)]) traversal(the_graph)` 17. `l_disjoint(Vertices(the_graph)+Vertices(the_graph);s2;s)` 18. `paren(Vertices(the_graph);s2)` 19. `no_repeats(Vertices(the_graph)+Vertices(the_graph);s2)` 20. `(inr(i) s2)` 21. `j:` `Vertices(the_graph)` 22. `(inr(j) s2)` `(inr(j) [inl(i)] @ s2 @ [inr(i)])`	1 step

2	11. `j:Vertices(the_graph). (inr(j) s') i-the_graph- > j` 12. `dfs(the_obj;s;i) = (s' @ s)` 13. `df-traversal(the_graph;s' @ s)` 14. `(inl(i) s)` 15. `s2:` `traversal(the_graph)` 16. `s' = ([inl(i)] @ s2 @ [inr(i)]) traversal(the_graph)` 17. `l_disjoint(Vertices(the_graph)+Vertices(the_graph);s2;s)` 18. `paren(Vertices(the_graph);s2)` 19. `no_repeats(Vertices(the_graph)+Vertices(the_graph);s2) & (inr(i) s2)` 20. `j:Vertices(the_graph). (inr(j) s2) i-the_graph- > j` `dfsl-traversal(the_graph;L1 @ [i];([inl(i)] @ s2 @ [inr(i)]) @ s)`	148 steps