Nuprl - Proof: dfs-df-traversal

(44steps total) PrintForm Definitions Lemmas graph 1 3 Sections Graphs Doc

For any graph the_obj:GraphObject(the_graph), s:Traversal, i:V. df-traversal(the_graph;s) (j:V. (inr(j) s) (inl(j) s) j-the_graph- > *i) df-traversal(the_graph;dfs(the_obj;s;i))

By: Auto THEN Inst Thm* dfs_induction4 [the_graph;the_obj;i,s1,s2. df-traversal(the_graph;s1) (j:Vertices(the_graph). (inr(j) s1) (inl(j) s1) j-the_graph- > *i) df-traversal(the_graph;s2 @ s1);s;i] THEN All Reduce THEN ExRepD THEN Repeat ThinTrivial THEN Try (RWO Thm* a,b,c:Top List. ((a @ b) @ c) ~ (a @ b @ c) 0) THEN AllHyps (h.(HypSubst h 0) THEN (Complete Auto))

Generated subgoals:

1 1. the_graph: Graph
2. the_obj: GraphObject(the_graph)
3. s: traversal(the_graph)
4. i: Vertices(the_graph)
5. df-traversal(the_graph;s)
6. j:Vertices(the_graph). (inr(j) s) (inl(j) s) j-the_graph- > *i
7. s1: traversal(the_graph)
8. s2: traversal(the_graph)
9. s3: traversal(the_graph)
10. i1: Vertices(the_graph)
11. j: Vertices(the_graph)
12. i1-the_graph- > j
13. paren(Vertices(the_graph);s2)
14. k:Vertices(the_graph). (inr(k) s2) j-the_graph- > *k
15. paren(Vertices(the_graph);s3)
16. df-traversal(the_graph;s2 @ s1) (j:Vertices(the_graph). (inr(j) s2 @ s1) (inl(j) s2 @ s1) j-the_graph- > *i1) df-traversal(the_graph;s3 @ s2 @ s1)
17. df-traversal(the_graph;s1)
18. j:Vertices(the_graph). (inr(j) s1) (inl(j) s1) j-the_graph- > *i1
19. (j@0:Vertices(the_graph). (inr(j@0) s1) (inl(j@0) s1) j@0-the_graph- > *j) df-traversal(the_graph;s2 @ s1)
df-traversal(the_graph;s3 @ s2 @ s1) 8 steps

2 1. the_graph: Graph
2. the_obj: GraphObject(the_graph)
3. s: traversal(the_graph)
4. i: Vertices(the_graph)
5. df-traversal(the_graph;s)
6. j:Vertices(the_graph). (inr(j) s) (inl(j) s) j-the_graph- > *i
7. s1: traversal(the_graph)
8. s2: traversal(the_graph)
9. i1: Vertices(the_graph)
10. (inr(i1) s1)
11. (inl(i1) s1)
12. paren(Vertices(the_graph);s2)
13. l_disjoint(Vertices(the_graph)+Vertices(the_graph);s2;s1)
14. no_repeats(Vertices(the_graph)+Vertices(the_graph);s2)
15. j:Vertices(the_graph). (inr(j) s2) i1-the_graph- > *j
16. j:Vertices(the_graph). i1-the_graph- > j j = i1 (inl(j) s2) (inl(j) s1) (inr(j) s1)
17. df-traversal(the_graph;[inr(i1) / s1]) (j:Vertices(the_graph). (inr(j) [inr(i1) / s1]) (inl(j) [inr(i1) / s1]) j-the_graph- > *i1) df-traversal(the_graph;s2 @ [inr(i1) / s1])
18. df-traversal(the_graph;s1)
19. j:Vertices(the_graph). (inr(j) s1) (inl(j) s1) j-the_graph- > *i1
df-traversal(the_graph;[inl(i1) / (s2 @ [inr(i1)] @ s1)]) 35 steps

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(44steps total) PrintForm Definitions Lemmas graph 1 3 Sections Graphs Doc

1	1. `the_graph:` `Graph` 2. `the_obj:` `GraphObject(the_graph)` 3. `s:` `traversal(the_graph)` 4. `i:` `Vertices(the_graph)` 5. `df-traversal(the_graph;s)` 6. `j:Vertices(the_graph). (inr(j) s) (inl(j) s) j-the_graph- > i` 7. `s1:` `traversal(the_graph)` 8. `s2:` `traversal(the_graph)` 9. `s3:` `traversal(the_graph)` 10. `i1:` `Vertices(the_graph)` 11. `j:` `Vertices(the_graph)` 12. `i1-the_graph- > j` 13. `paren(Vertices(the_graph);s2)` 14. `k:Vertices(the_graph). (inr(k) s2) j-the_graph- > k` 15. `paren(Vertices(the_graph);s3)` 16. `df-traversal(the_graph;s2 @ s1) (j:Vertices(the_graph). (inr(j) s2 @ s1) (inl(j) s2 @ s1) j-the_graph- > i1) df-traversal(the_graph;s3 @ s2 @ s1)` 17. `df-traversal(the_graph;s1)` 18. `j:Vertices(the_graph). (inr(j) s1) (inl(j) s1) j-the_graph- > i1` 19. `(j@0:Vertices(the_graph). (inr(j@0) s1) (inl(j@0) s1) j@0-the_graph- > *j) df-traversal(the_graph;s2 @ s1)` `df-traversal(the_graph;s3 @ s2 @ s1)`	8 steps

2	1. `the_graph:` `Graph` 2. `the_obj:` `GraphObject(the_graph)` 3. `s:` `traversal(the_graph)` 4. `i:` `Vertices(the_graph)` 5. `df-traversal(the_graph;s)` 6. `j:Vertices(the_graph). (inr(j) s) (inl(j) s) j-the_graph- > i` 7. `s1:` `traversal(the_graph)` 8. `s2:` `traversal(the_graph)` 9. `i1:` `Vertices(the_graph)` 10. `(inr(i1) s1)` 11. `(inl(i1) s1)` 12. `paren(Vertices(the_graph);s2)` 13. `l_disjoint(Vertices(the_graph)+Vertices(the_graph);s2;s1)` 14. `no_repeats(Vertices(the_graph)+Vertices(the_graph);s2)` 15. `j:Vertices(the_graph). (inr(j) s2) i1-the_graph- > j` 16. `j:Vertices(the_graph). i1-the_graph- > j j = i1 (inl(j) s2) (inl(j) s1) (inr(j) s1)` 17. `df-traversal(the_graph;[inr(i1) / s1]) (j:Vertices(the_graph). (inr(j) [inr(i1) / s1]) (inl(j) [inr(i1) / s1]) j-the_graph- > i1) df-traversal(the_graph;s2 @ [inr(i1) / s1])` 18. `df-traversal(the_graph;s1)` 19. `j:Vertices(the_graph). (inr(j) s1) (inl(j) s1) j-the_graph- > i1` `df-traversal(the_graph;[inl(i1) / (s2 @ [inr(i1)] @ s1)])`	35 steps