Nuprl - Proof: dfsl-properties 2 1 3 2 1

(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. (inl(i) s)
11. s2: traversal(the_graph)
12. l_disjoint(Vertices(the_graph)+Vertices(the_graph);s2;s)
13. paren(Vertices(the_graph);s2)
14. no_repeats(Vertices(the_graph)+Vertices(the_graph);s2) & (inr(i) s2)
15. j:Vertices(the_graph). (inr(j) s2) i-the_graph- > *j
16. df-traversal(the_graph;([inl(i)] @ s2 @ [inr(i)]) @ s)

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

By: AllHyps (h.ParallelOp h) THEN ExRepD THEN Analyze 0 THEN Analyze 0 THEN Try Trivial

Generated subgoals:

1 9. df-traversal(the_graph;s)
10. i:Vertices(the_graph). (inl(i) s) L1-the_graph- > *i
11. (i:Vertices(the_graph). L1-the_graph- > *i non-trivial-loop(the_graph;i)) (L1@0,L2:Vertices(the_graph) List. L1 = (L1@0 @ L2) (s1,s2:traversal(the_graph). s = (s2 @ s1) traversal(the_graph) & paren(Vertices(the_graph);s1) & paren(Vertices(the_graph);s2) & (j:Vertices(the_graph). ((inl(j) s1) L1@0-the_graph- > *j) & ((inl(j) s2) L2-the_graph- > *j & L1@0-the_graph- > *j))))
12. (inl(i) s)
13. s2: traversal(the_graph)
14. l_disjoint(Vertices(the_graph)+Vertices(the_graph);s2;s)
15. paren(Vertices(the_graph);s2)
16. no_repeats(Vertices(the_graph)+Vertices(the_graph);s2)
17. (inr(i) s2)
18. j:Vertices(the_graph). (inr(j) s2) i-the_graph- > *j
19. df-traversal(the_graph;([inl(i)] @ s2 @ [inr(i)]) @ s)
i@0:Vertices(the_graph). (inl(i@0) ([inl(i)] @ s2 @ [inr(i)]) @ s) L1 @ [i]-the_graph- > *i@0 11 steps

2 9. df-traversal(the_graph;s)
10. i:Vertices(the_graph). (inl(i) s) L1-the_graph- > *i
11. (i:Vertices(the_graph). L1-the_graph- > *i non-trivial-loop(the_graph;i)) (L1@0,L2:Vertices(the_graph) List. L1 = (L1@0 @ L2) (s1,s2:traversal(the_graph). s = (s2 @ s1) traversal(the_graph) & paren(Vertices(the_graph);s1) & paren(Vertices(the_graph);s2) & (j:Vertices(the_graph). ((inl(j) s1) L1@0-the_graph- > *j) & ((inl(j) s2) L2-the_graph- > *j & L1@0-the_graph- > *j))))
12. (inl(i) s)
13. s2: traversal(the_graph)
14. l_disjoint(Vertices(the_graph)+Vertices(the_graph);s2;s)
15. paren(Vertices(the_graph);s2)
16. no_repeats(Vertices(the_graph)+Vertices(the_graph);s2)
17. (inr(i) s2)
18. j:Vertices(the_graph). (inr(j) s2) i-the_graph- > *j
19. df-traversal(the_graph;([inl(i)] @ s2 @ [inr(i)]) @ s)
20. i@0:Vertices(the_graph). L1 @ [i]-the_graph- > *i@0 non-trivial-loop(the_graph;i@0)
L1@0,L2:Vertices(the_graph) List. (L1 @ [i]) = (L1@0 @ L2) (s1,s2@0:traversal(the_graph). (([inl(i)] @ s2 @ [inr(i)]) @ s) = (s2@0 @ s1) traversal(the_graph) & paren(Vertices(the_graph);s1) & paren(Vertices(the_graph);s2@0) & (j:Vertices(the_graph). ((inl(j) s1) L1@0-the_graph- > *j) & ((inl(j) s2@0) L2-the_graph- > *j & L1@0-the_graph- > *j))) 135 steps

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

1	9. `df-traversal(the_graph;s)` 10. `i:Vertices(the_graph). (inl(i) s) L1-the_graph- > i` 11. `(i:Vertices(the_graph). L1-the_graph- > i non-trivial-loop(the_graph;i)) (L1@0,L2:Vertices(the_graph) List. L1 = (L1@0 @ L2) (s1,s2:traversal(the_graph). s = (s2 @ s1) traversal(the_graph) & paren(Vertices(the_graph);s1) & paren(Vertices(the_graph);s2) & (j:Vertices(the_graph). ((inl(j) s1) L1@0-the_graph- > j) & ((inl(j) s2) L2-the_graph- > j & L1@0-the_graph- > j))))` 12. `(inl(i) s)` 13. `s2:` `traversal(the_graph)` 14. `l_disjoint(Vertices(the_graph)+Vertices(the_graph);s2;s)` 15. `paren(Vertices(the_graph);s2)` 16. `no_repeats(Vertices(the_graph)+Vertices(the_graph);s2)` 17. `(inr(i) s2)` 18. `j:Vertices(the_graph). (inr(j) s2) i-the_graph- > j` 19. `df-traversal(the_graph;([inl(i)] @ s2 @ [inr(i)]) @ s)` `i@0:Vertices(the_graph). (inl(i@0) ([inl(i)] @ s2 @ [inr(i)]) @ s) L1 @ [i]-the_graph- > *i@0`	11 steps

2	9. `df-traversal(the_graph;s)` 10. `i:Vertices(the_graph). (inl(i) s) L1-the_graph- > i` 11. `(i:Vertices(the_graph). L1-the_graph- > i non-trivial-loop(the_graph;i)) (L1@0,L2:Vertices(the_graph) List. L1 = (L1@0 @ L2) (s1,s2:traversal(the_graph). s = (s2 @ s1) traversal(the_graph) & paren(Vertices(the_graph);s1) & paren(Vertices(the_graph);s2) & (j:Vertices(the_graph). ((inl(j) s1) L1@0-the_graph- > j) & ((inl(j) s2) L2-the_graph- > j & L1@0-the_graph- > j))))` 12. `(inl(i) s)` 13. `s2:` `traversal(the_graph)` 14. `l_disjoint(Vertices(the_graph)+Vertices(the_graph);s2;s)` 15. `paren(Vertices(the_graph);s2)` 16. `no_repeats(Vertices(the_graph)+Vertices(the_graph);s2)` 17. `(inr(i) s2)` 18. `j:Vertices(the_graph). (inr(j) s2) i-the_graph- > j` 19. `df-traversal(the_graph;([inl(i)] @ s2 @ [inr(i)]) @ s)` 20. `i@0:Vertices(the_graph). L1 @ [i]-the_graph- > i@0 non-trivial-loop(the_graph;i@0)` `L1@0,L2:Vertices(the_graph) List. (L1 @ [i]) = (L1@0 @ L2) (s1,s2@0:traversal(the_graph). (([inl(i)] @ s2 @ [inr(i)]) @ s) = (s2@0 @ s1) traversal(the_graph) & paren(Vertices(the_graph);s1) & paren(Vertices(the_graph);s2@0) & (j:Vertices(the_graph). ((inl(j) s1) L1@0-the_graph- > j) & ((inl(j) s2@0) L2-the_graph- > j & L1@0-the_graph- > j)))`	135 steps