Nuprl - Proof: dfs induction 1 1 2

(22steps 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) 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. x,y:Vertices(the_graph). the_obj.eq(x,y) x = y
13. 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)
14. 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)
15. d:
16. 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)))
17. s: traversal(the_graph)
18. i: Vertices(the_graph)
19. M(s)d
20. member-paren(x,y.the_obj.eq(x,y);i;s)
21. 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)))

s':traversal(the_graph). M(s' @ s)M(s) & P(i,s,s') & [inl(i) / (the_obj.eacc((s',j. dfs(the_obj;s';j)),[inr(i) / s],i))] = (s' @ s) traversal(the_graph)

By: ((InstHyp [traversal(the_graph);[inr(i) / s];i;s',j. dfs(the_obj;s';j)] 13) THEN ExRepD THEN (HypSubst -1 0) THEN (Thin -1) THEN (InstHyp [L;[inr(i) / s]] -3)) THENA (Auto THEN EasyHyp THEN (Try (RWO Thm* E:(TT). (x,y:T. E(x,y) x = y) (i:T, s:(T+T) List. member-paren(x,y.E(x,y);i;s) (inl(i) s) (inr(i) s)) 0)) THEN (Try (AutoInst [i;s])))

Generated subgoals:

1 22. L: Vertices(the_graph) List
23. y:Vertices(the_graph). i-the_graph- > y (y L)
(inl(i) [inr(i) / s]) (inr(i) [inr(i) / s]) 1 step

2 22. L: Vertices(the_graph) List
23. y:Vertices(the_graph). i-the_graph- > y (y L)
24. s':traversal(the_graph). M(s' @ [inr(i) / s])M([inr(i) / s]) & P(i,[inr(i) / s],s') & list_accum(s',x'.dfs(the_obj;s';x');[inr(i) / s];L) = (s' @ [inr(i) / s])
s':traversal(the_graph). M(s' @ s)M(s) & P(i,s,s') & [inl(i) / list_accum(s',x'.(s',j. dfs(the_obj;s';j))(s',x');[inr(i) / s];L)] = (s' @ s) traversal(the_graph) 2 steps

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(22steps total) PrintForm Definitions Lemmas graph 1 3 Sections Graphs Doc

1	22. `L:` `Vertices(the_graph) List` 23. `y:Vertices(the_graph). i-the_graph- > y (y L)` `(inl(i) [inr(i) / s]) (inr(i) [inr(i) / s])`	1 step

2	22. `L:` `Vertices(the_graph) List` 23. `y:Vertices(the_graph). i-the_graph- > y (y L)` 24. `s':traversal(the_graph). M(s' @ [inr(i) / s])M([inr(i) / s]) & P(i,[inr(i) / s],s') & list_accum(s',x'.dfs(the_obj;s';x');[inr(i) / s];L) = (s' @ [inr(i) / s])` `s':traversal(the_graph). M(s' @ s)M(s) & P(i,s,s') & [inl(i) / list_accum(s',x'.(s',j. dfs(the_obj;s';j))(s',x');[inr(i) / s];L)] = (s' @ s) traversal(the_graph)`	2 steps