Nuprl - Proof: dfs induction3 1 1 1

(26steps total) PrintForm Definitions Lemmas graph 1 3 Sections Graphs Doc

1. the_graph: Graph
2. the_obj: GraphObject(the_graph)
3. P: Vertices(the_graph)traversal(the_graph)traversal(the_graph)Prop
4. s1,s2:traversal(the_graph), i:Vertices(the_graph). P(i,s1,s2) l_disjoint(Vertices(the_graph)+Vertices(the_graph);s2;s1)
5. s1,s2:traversal(the_graph), i:Vertices(the_graph). P(i,s1,s2) no_repeats(Vertices(the_graph)+Vertices(the_graph);s2)
6. s:traversal(the_graph), i:Vertices(the_graph). member-paren(x,y.the_obj.eq(x,y);i;s) P(i,s,nil)
7. s1,s2,s3:traversal(the_graph), i,j:Vertices(the_graph). i-the_graph- > j P(j,s1,s2) P(i,s2 @ s1,s3) P(i,s1,s3 @ s2)
8. s1,s2:traversal(the_graph), i:Vertices(the_graph). member-paren(x,y.the_obj.eq(x,y);i;s1) (j:Vertices(the_graph). i-the_graph- > j j = i (inl(j) s2) member-paren(x,y.the_obj.eq(x,y);j;s1)) P(i,[inr(i) / s1],s2) P(i,s1,[inl(i) / (s2 @ [inr(i)])])
9. M: traversal(the_graph)
10. i:Vertices(the_graph), s:traversal(the_graph). M([inl(i) / s])M(s)
11. i:Vertices(the_graph), s:traversal(the_graph). member-paren(x,y.the_obj.eq(x,y);i;s) M([inr(i) / s]) < M(s)
12. d:
13. d1:. d1 < d (s:traversal(the_graph), i:Vertices(the_graph). M(s)d1 (s':traversal(the_graph). M(s' @ s)M(s) & P(i,s,s') & dfs(the_obj;s;i) = (s' @ s)))
14. s: traversal(the_graph)
15. i: Vertices(the_graph)
16. M(s)d
17. member-paren(x,y.the_obj.eq(x,y);i;s)
18. x,y:Vertices(the_graph). the_obj.eq(x,y) x = y
19. T:Type, s:T, x:Vertices(the_graph), f:(TVertices(the_graph)T). L:Vertices(the_graph) List. (y:Vertices(the_graph). x-the_graph- > y (y L)) & the_obj.eacc(f,s,x) = list_accum(s',x'.f(s',x');s;L)
20. T:Type, s:T, f:(TVertices(the_graph)T). L:Vertices(the_graph) List. no_repeats(Vertices(the_graph);L) & (y:Vertices(the_graph). (y L)) & the_obj.vacc(f,s) = list_accum(s',x'.f(s',x');s;L)

L:Vertices(the_graph) List. (y:Vertices(the_graph). (y L) i-the_graph- > y) (s2:traversal(the_graph). M(s2) < d member-paren(x,y.the_obj.eq(x,y);i;s2) (s':traversal(the_graph). M(s' @ s2)M(s2) & P(i,s2,s') & list_accum(s',x'.dfs(the_obj;s';x');s2;L) = (s' @ s2)))

By: InductionOnList THEN Reduce 0

Generated subgoals:

1 21. L: Vertices(the_graph) List
(y:Vertices(the_graph). (y nil) i-the_graph- > y) (s2:traversal(the_graph). M(s2) < d member-paren(x,y.the_obj.eq(x,y);i;s2) (s':traversal(the_graph). M(s' @ s2)M(s2) & P(i,s2,s') & s2 = (s' @ s2) traversal(the_graph))) 1 step

2 21. L: Vertices(the_graph) List
22. u: Vertices(the_graph)
23. v: Vertices(the_graph) List
24. (y:Vertices(the_graph). (y v) i-the_graph- > y) (s2:traversal(the_graph). M(s2) < d member-paren(x,y.the_obj.eq(x,y);i;s2) (s':traversal(the_graph). M(s' @ s2)M(s2) & P(i,s2,s') & list_accum(s',x'.dfs(the_obj;s';x');s2;v) = (s' @ s2)))
(y:Vertices(the_graph). (y [u / v]) i-the_graph- > y) (s2:traversal(the_graph). M(s2) < d member-paren(x,y.the_obj.eq(x,y);i;s2) (s':traversal(the_graph). M(s' @ s2)M(s2) & P(i,s2,s') & list_accum(s',x'.dfs(the_obj;s';x');dfs(the_obj;s2;u);v) = (s' @ s2))) 9 steps

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(26steps total) PrintForm Definitions Lemmas graph 1 3 Sections Graphs Doc

1	21. `L:` `Vertices(the_graph) List` `(y:Vertices(the_graph). (y nil) i-the_graph- > y) (s2:traversal(the_graph). M(s2) < d member-paren(x,y.the_obj.eq(x,y);i;s2) (s':traversal(the_graph). M(s' @ s2)M(s2) & P(i,s2,s') & s2 = (s' @ s2) traversal(the_graph)))`	1 step

2	21. `L:` `Vertices(the_graph) List` 22. `u:` `Vertices(the_graph)` 23. `v:` `Vertices(the_graph) List` 24. `(y:Vertices(the_graph). (y v) i-the_graph- > y) (s2:traversal(the_graph). M(s2) < d member-paren(x,y.the_obj.eq(x,y);i;s2) (s':traversal(the_graph). M(s' @ s2)M(s2) & P(i,s2,s') & list_accum(s',x'.dfs(the_obj;s';x');s2;v) = (s' @ s2)))` `(y:Vertices(the_graph). (y [u / v]) i-the_graph- > y) (s2:traversal(the_graph). M(s2) < d member-paren(x,y.the_obj.eq(x,y);i;s2) (s':traversal(the_graph). M(s' @ s2)M(s2) & P(i,s2,s') & list_accum(s',x'.dfs(the_obj;s';x');dfs(the_obj;s2;u);v) = (s' @ s2)))`	9 steps