Nuprl - Proof: topsortl-properties 1 2 2 2 2 1 2 1

(47steps 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. i:Vertices(the_graph). (i L)
5. paren(Vertices(the_graph);dfsl(the_obj;L))
6. no_repeats(Vertices(the_graph)+Vertices(the_graph);dfsl(the_obj;L))
7. dfsl-traversal(the_graph;L;dfsl(the_obj;L))
8. i:Vertices(the_graph). (i L) (inr(i) dfsl(the_obj;L)) & (inl(i) dfsl(the_obj;L))
9. non-trivial-loop-free(the_graph)
10. L1,L2:Vertices(the_graph) List. L = (L1 @ L2) (s1,s2:Vertices(the_graph) List. topsortl(the_obj;L) = (s2 @ s1) & (j:Vertices(the_graph). ((j s1) L1-the_graph- > *j) & ((j s2) L2-the_graph- > *j & L1-the_graph- > *j)))
11. i:Vertices(the_graph), s1,s2:Vertices(the_graph) List. topsortl(the_obj;L) = (s1 @ [i] @ s2) (j:Vertices(the_graph). j = i i-the_graph- > *j (j s2))
12. i:Vertices(the_graph). (i topsortl(the_obj;L))
13. i,j:Vertices(the_graph). j = i i-the_graph- > *j i before j topsortl(the_obj;L)
14. i: Vertices(the_graph)
15. j: Vertices(the_graph)
16. A: Vertices(the_graph) List
17. B: Vertices(the_graph) List
18. L = (A @ B)
19. (j A)
20. (i B)
21. j-the_graph- > *i
22. k:Vertices(the_graph). k before j L k-the_graph- > *i
23. s1: Vertices(the_graph) List
24. s2: Vertices(the_graph) List
25. A = (s1 @ [j / s2])
26. (j s1)
27. s1@0: Vertices(the_graph) List
28. s2@0: Vertices(the_graph) List
29. topsortl(the_obj;L) = (s2@0 @ s1@0)
30. j@0:Vertices(the_graph). ((j@0 s1@0) s1 @ [j]-the_graph- > *j@0) & ((j@0 s2@0) s2 @ B-the_graph- > *j@0 & s1 @ [j]-the_graph- > *j@0)

(i s2@0)

By: InstHyp [i] -1 THEN Analyze -1 THEN Analyze -1 THEN Analyze -1 THEN RWO Thm* For any graph A,B:V List, i:V. A @ B-- > *i A-- > *i B-- > *i 0 THEN RWW Thm* For any graph i,j:V. [i]-- > *j i-the_graph- > *j 0

Generated subgoals:

1 31. (i s1@0) s1 @ [j]-the_graph- > *i
32. (i s1@0) s1 @ [j]-the_graph- > *i
33. (i s2@0) s2 @ B-the_graph- > *i & s1 @ [j]-the_graph- > *i
s2-the_graph- > *i B-the_graph- > *i 1 step

2 31. (i s1@0) s1 @ [j]-the_graph- > *i
32. (i s1@0) s1 @ [j]-the_graph- > *i
33. (i s2@0) s2 @ B-the_graph- > *i & s1 @ [j]-the_graph- > *i
(s1-the_graph- > *i j-the_graph- > *i) 6 steps

About:

(47steps total) PrintForm Definitions Lemmas graph 1 3 Sections Graphs Doc

1	31. `(i s1@0) s1 @ [j]-the_graph- > i` 32. `(i s1@0) s1 @ [j]-the_graph- > i` 33. `(i s2@0) s2 @ B-the_graph- > i & s1 @ [j]-the_graph- > i` `s2-the_graph- > i B-the_graph- > i`	1 step

2	31. `(i s1@0) s1 @ [j]-the_graph- > i` 32. `(i s1@0) s1 @ [j]-the_graph- > i` 33. `(i s2@0) s2 @ B-the_graph- > i & s1 @ [j]-the_graph- > i` `(s1-the_graph- > i j-the_graph- > i)`	6 steps