(16steps total) PrintForm Definitions Lemmas graph 1 3 Sections Graphs Doc

At: dfs induction2
For any graph the_obj:GraphObject(the_graph), P:(VTraversalTraversalProp), s:Traversal, i:V. (s:Traversal, i:V. (inl(i) s) (inr(i) s) P(i,s,nil)) (s1,s2,s3:Traversal, i,j:V. i-the_graph- > j paren(V;s2) paren(V;s3) P(j,s1,s2) P(i,s2 @ s1,s3) P(i,s1,s3 @ s2)) (s1,s2:Traversal, i:V. (inl(i) s1) (inr(i) s1) P(i,[inr(i) / s1],s2) P(i,s1,[inl(i) / (s2 @ [inr(i)])])) (s':Traversal. P(i,s,s') & l_disjoint(V+V;s';s) & no_repeats(V+V;s') & paren(V;s') & dfs(the_obj;s;i) = (s' @ s))
By:
Auto
THEN
Inst Thm* graphobj-properties [the_graph;the_obj]
THEN
Analyze -1
THEN
Analyze -1
THEN
Inst Thm* dfs_induction [the_graph;the_obj;i,s1,s2. P(i,s1,s2) & l_disjoint(Vertices(the_graph)+Vertices(the_graph);s2;s1) & no_repeats(Vertices(the_graph)+Vertices(the_graph);s2) & paren(Vertices(the_graph);s2);s;i]
THEN
AllHyps (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)))
THEN
EasyHyp
Generated subgoals:

11. the_graph: Graph
2. the_obj: GraphObject(the_graph)
3. P: Vertices(the_graph)traversal(the_graph)traversal(the_graph)Prop
4. s: traversal(the_graph)
5. i: Vertices(the_graph)
6. s:traversal(the_graph), i:Vertices(the_graph). (inl(i) s) (inr(i) s) P(i,s,nil)
7. s1,s2,s3:traversal(the_graph), i,j:Vertices(the_graph). i-the_graph- > j paren(Vertices(the_graph);s2) paren(Vertices(the_graph);s3) P(j,s1,s2) P(i,s2 @ s1,s3) P(i,s1,s3 @ s2)
8. s1,s2:traversal(the_graph), i:Vertices(the_graph). (inl(i) s1) (inr(i) s1) P(i,[inr(i) / s1],s2) P(i,s1,[inl(i) / (s2 @ [inr(i)])])
9. x,y:Vertices(the_graph). the_obj.eq(x,y) x = y
10. 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)
11. 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)
12. s1: traversal(the_graph)
13. i1: Vertices(the_graph)
14. (inl(i1) s1) (inr(i1) s1)
l_disjoint(Vertices(the_graph)+Vertices(the_graph);nil;s1)		1 step
 
21. the_graph: Graph
2. the_obj: GraphObject(the_graph)
3. P: Vertices(the_graph)traversal(the_graph)traversal(the_graph)Prop
4. s: traversal(the_graph)
5. i: Vertices(the_graph)
6. s:traversal(the_graph), i:Vertices(the_graph). (inl(i) s) (inr(i) s) P(i,s,nil)
7. s1,s2,s3:traversal(the_graph), i,j:Vertices(the_graph). i-the_graph- > j paren(Vertices(the_graph);s2) paren(Vertices(the_graph);s3) P(j,s1,s2) P(i,s2 @ s1,s3) P(i,s1,s3 @ s2)
8. s1,s2:traversal(the_graph), i:Vertices(the_graph). (inl(i) s1) (inr(i) s1) P(i,[inr(i) / s1],s2) P(i,s1,[inl(i) / (s2 @ [inr(i)])])
9. x,y:Vertices(the_graph). the_obj.eq(x,y) x = y
10. 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)
11. 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)
12. s1: traversal(the_graph)
13. i1: Vertices(the_graph)
14. (inl(i1) s1) (inr(i1) s1)
no_repeats(Vertices(the_graph)+Vertices(the_graph);nil)		1 step
 
31. the_graph: Graph
2. the_obj: GraphObject(the_graph)
3. P: Vertices(the_graph)traversal(the_graph)traversal(the_graph)Prop
4. s: traversal(the_graph)
5. i: Vertices(the_graph)
6. s:traversal(the_graph), i:Vertices(the_graph). (inl(i) s) (inr(i) s) P(i,s,nil)
7. s1,s2,s3:traversal(the_graph), i,j:Vertices(the_graph). i-the_graph- > j paren(Vertices(the_graph);s2) paren(Vertices(the_graph);s3) P(j,s1,s2) P(i,s2 @ s1,s3) P(i,s1,s3 @ s2)
8. s1,s2:traversal(the_graph), i:Vertices(the_graph). (inl(i) s1) (inr(i) s1) P(i,[inr(i) / s1],s2) P(i,s1,[inl(i) / (s2 @ [inr(i)])])
9. x,y:Vertices(the_graph). the_obj.eq(x,y) x = y
10. 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)
11. 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)
12. s1: traversal(the_graph)
13. i1: Vertices(the_graph)
14. (inl(i1) s1) (inr(i1) s1)
paren(Vertices(the_graph);nil)		1 step
 
41. the_graph: Graph
2. the_obj: GraphObject(the_graph)
3. P: Vertices(the_graph)traversal(the_graph)traversal(the_graph)Prop
4. s: traversal(the_graph)
5. i: Vertices(the_graph)
6. s:traversal(the_graph), i:Vertices(the_graph). (inl(i) s) (inr(i) s) P(i,s,nil)
7. s1,s2,s3:traversal(the_graph), i,j:Vertices(the_graph). i-the_graph- > j paren(Vertices(the_graph);s2) paren(Vertices(the_graph);s3) P(j,s1,s2) P(i,s2 @ s1,s3) P(i,s1,s3 @ s2)
8. s1,s2:traversal(the_graph), i:Vertices(the_graph). (inl(i) s1) (inr(i) s1) P(i,[inr(i) / s1],s2) P(i,s1,[inl(i) / (s2 @ [inr(i)])])
9. x,y:Vertices(the_graph). the_obj.eq(x,y) x = y
10. 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)
11. 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)
12. s1: traversal(the_graph)
13. s2: traversal(the_graph)
14. s3: traversal(the_graph)
15. i1: Vertices(the_graph)
16. j: Vertices(the_graph)
17. i1-the_graph- > j
18. P(j,s1,s2)
19. l_disjoint(Vertices(the_graph)+Vertices(the_graph);s2;s1)
20. no_repeats(Vertices(the_graph)+Vertices(the_graph);s2)
21. paren(Vertices(the_graph);s2)
22. P(i1,s2 @ s1,s3)
23. l_disjoint(Vertices(the_graph)+Vertices(the_graph);s3;s2 @ s1)
24. no_repeats(Vertices(the_graph)+Vertices(the_graph);s3)
25. paren(Vertices(the_graph);s3)
l_disjoint(Vertices(the_graph)+Vertices(the_graph);s3 @ s2;s1)		1 step
 
51. the_graph: Graph
2. the_obj: GraphObject(the_graph)
3. P: Vertices(the_graph)traversal(the_graph)traversal(the_graph)Prop
4. s: traversal(the_graph)
5. i: Vertices(the_graph)
6. s:traversal(the_graph), i:Vertices(the_graph). (inl(i) s) (inr(i) s) P(i,s,nil)
7. s1,s2,s3:traversal(the_graph), i,j:Vertices(the_graph). i-the_graph- > j paren(Vertices(the_graph);s2) paren(Vertices(the_graph);s3) P(j,s1,s2) P(i,s2 @ s1,s3) P(i,s1,s3 @ s2)
8. s1,s2:traversal(the_graph), i:Vertices(the_graph). (inl(i) s1) (inr(i) s1) P(i,[inr(i) / s1],s2) P(i,s1,[inl(i) / (s2 @ [inr(i)])])
9. x,y:Vertices(the_graph). the_obj.eq(x,y) x = y
10. 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)
11. 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)
12. s1: traversal(the_graph)
13. s2: traversal(the_graph)
14. s3: traversal(the_graph)
15. i1: Vertices(the_graph)
16. j: Vertices(the_graph)
17. i1-the_graph- > j
18. P(j,s1,s2)
19. l_disjoint(Vertices(the_graph)+Vertices(the_graph);s2;s1)
20. no_repeats(Vertices(the_graph)+Vertices(the_graph);s2)
21. paren(Vertices(the_graph);s2)
22. P(i1,s2 @ s1,s3)
23. l_disjoint(Vertices(the_graph)+Vertices(the_graph);s3;s2 @ s1)
24. no_repeats(Vertices(the_graph)+Vertices(the_graph);s3)
25. paren(Vertices(the_graph);s3)
no_repeats(Vertices(the_graph)+Vertices(the_graph);s3 @ s2)		1 step
 
61. the_graph: Graph
2. the_obj: GraphObject(the_graph)
3. P: Vertices(the_graph)traversal(the_graph)traversal(the_graph)Prop
4. s: traversal(the_graph)
5. i: Vertices(the_graph)
6. s:traversal(the_graph), i:Vertices(the_graph). (inl(i) s) (inr(i) s) P(i,s,nil)
7. s1,s2,s3:traversal(the_graph), i,j:Vertices(the_graph). i-the_graph- > j paren(Vertices(the_graph);s2) paren(Vertices(the_graph);s3) P(j,s1,s2) P(i,s2 @ s1,s3) P(i,s1,s3 @ s2)
8. s1,s2:traversal(the_graph), i:Vertices(the_graph). (inl(i) s1) (inr(i) s1) P(i,[inr(i) / s1],s2) P(i,s1,[inl(i) / (s2 @ [inr(i)])])
9. x,y:Vertices(the_graph). the_obj.eq(x,y) x = y
10. 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)
11. 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)
12. s1: traversal(the_graph)
13. s2: traversal(the_graph)
14. s3: traversal(the_graph)
15. i1: Vertices(the_graph)
16. j: Vertices(the_graph)
17. i1-the_graph- > j
18. P(j,s1,s2)
19. l_disjoint(Vertices(the_graph)+Vertices(the_graph);s2;s1)
20. no_repeats(Vertices(the_graph)+Vertices(the_graph);s2)
21. paren(Vertices(the_graph);s2)
22. P(i1,s2 @ s1,s3)
23. l_disjoint(Vertices(the_graph)+Vertices(the_graph);s3;s2 @ s1)
24. no_repeats(Vertices(the_graph)+Vertices(the_graph);s3)
25. paren(Vertices(the_graph);s3)
paren(Vertices(the_graph);s3 @ s2)		1 step
 
71. the_graph: Graph
2. the_obj: GraphObject(the_graph)
3. P: Vertices(the_graph)traversal(the_graph)traversal(the_graph)Prop
4. s: traversal(the_graph)
5. i: Vertices(the_graph)
6. s:traversal(the_graph), i:Vertices(the_graph). (inl(i) s) (inr(i) s) P(i,s,nil)
7. s1,s2,s3:traversal(the_graph), i,j:Vertices(the_graph). i-the_graph- > j paren(Vertices(the_graph);s2) paren(Vertices(the_graph);s3) P(j,s1,s2) P(i,s2 @ s1,s3) P(i,s1,s3 @ s2)
8. s1,s2:traversal(the_graph), i:Vertices(the_graph). (inl(i) s1) (inr(i) s1) P(i,[inr(i) / s1],s2) P(i,s1,[inl(i) / (s2 @ [inr(i)])])
9. x,y:Vertices(the_graph). the_obj.eq(x,y) x = y
10. 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)
11. 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)
12. s1: traversal(the_graph)
13. s2: traversal(the_graph)
14. i1: Vertices(the_graph)
15. ((inl(i1) s1) (inr(i1) s1))
16. P(i1,[inr(i1) / s1],s2)
17. l_disjoint(Vertices(the_graph)+Vertices(the_graph);s2;[inr(i1) / s1])
18. no_repeats(Vertices(the_graph)+Vertices(the_graph);s2)
19. paren(Vertices(the_graph);s2)
P(i1,s1,[inl(i1) / (s2 @ [inr(i1)])])		1 step
 
81. the_graph: Graph
2. the_obj: GraphObject(the_graph)
3. P: Vertices(the_graph)traversal(the_graph)traversal(the_graph)Prop
4. s: traversal(the_graph)
5. i: Vertices(the_graph)
6. s:traversal(the_graph), i:Vertices(the_graph). (inl(i) s) (inr(i) s) P(i,s,nil)
7. s1,s2,s3:traversal(the_graph), i,j:Vertices(the_graph). i-the_graph- > j paren(Vertices(the_graph);s2) paren(Vertices(the_graph);s3) P(j,s1,s2) P(i,s2 @ s1,s3) P(i,s1,s3 @ s2)
8. s1,s2:traversal(the_graph), i:Vertices(the_graph). (inl(i) s1) (inr(i) s1) P(i,[inr(i) / s1],s2) P(i,s1,[inl(i) / (s2 @ [inr(i)])])
9. x,y:Vertices(the_graph). the_obj.eq(x,y) x = y
10. 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)
11. 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)
12. s1: traversal(the_graph)
13. s2: traversal(the_graph)
14. i1: Vertices(the_graph)
15. ((inl(i1) s1) (inr(i1) s1))
16. P(i1,[inr(i1) / s1],s2)
17. l_disjoint(Vertices(the_graph)+Vertices(the_graph);s2;[inr(i1) / s1])
18. no_repeats(Vertices(the_graph)+Vertices(the_graph);s2)
19. paren(Vertices(the_graph);s2)
l_disjoint(Vertices(the_graph)+Vertices(the_graph);[inl(i1) / (s2 @ [inr(i1)])];s1)		1 step
 
91. the_graph: Graph
2. the_obj: GraphObject(the_graph)
3. P: Vertices(the_graph)traversal(the_graph)traversal(the_graph)Prop
4. s: traversal(the_graph)
5. i: Vertices(the_graph)
6. s:traversal(the_graph), i:Vertices(the_graph). (inl(i) s) (inr(i) s) P(i,s,nil)
7. s1,s2,s3:traversal(the_graph), i,j:Vertices(the_graph). i-the_graph- > j paren(Vertices(the_graph);s2) paren(Vertices(the_graph);s3) P(j,s1,s2) P(i,s2 @ s1,s3) P(i,s1,s3 @ s2)
8. s1,s2:traversal(the_graph), i:Vertices(the_graph). (inl(i) s1) (inr(i) s1) P(i,[inr(i) / s1],s2) P(i,s1,[inl(i) / (s2 @ [inr(i)])])
9. x,y:Vertices(the_graph). the_obj.eq(x,y) x = y
10. 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)
11. 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)
12. s1: traversal(the_graph)
13. s2: traversal(the_graph)
14. i1: Vertices(the_graph)
15. ((inl(i1) s1) (inr(i1) s1))
16. P(i1,[inr(i1) / s1],s2)
17. l_disjoint(Vertices(the_graph)+Vertices(the_graph);s2;[inr(i1) / s1])
18. no_repeats(Vertices(the_graph)+Vertices(the_graph);s2)
19. paren(Vertices(the_graph);s2)
no_repeats(Vertices(the_graph)+Vertices(the_graph);[inl(i1) / (s2 @ [inr(i1)])])		6 steps
 
101. the_graph: Graph
2. the_obj: GraphObject(the_graph)
3. P: Vertices(the_graph)traversal(the_graph)traversal(the_graph)Prop
4. s: traversal(the_graph)
5. i: Vertices(the_graph)
6. s:traversal(the_graph), i:Vertices(the_graph). (inl(i) s) (inr(i) s) P(i,s,nil)
7. s1,s2,s3:traversal(the_graph), i,j:Vertices(the_graph). i-the_graph- > j paren(Vertices(the_graph);s2) paren(Vertices(the_graph);s3) P(j,s1,s2) P(i,s2 @ s1,s3) P(i,s1,s3 @ s2)
8. s1,s2:traversal(the_graph), i:Vertices(the_graph). (inl(i) s1) (inr(i) s1) P(i,[inr(i) / s1],s2) P(i,s1,[inl(i) / (s2 @ [inr(i)])])
9. x,y:Vertices(the_graph). the_obj.eq(x,y) x = y
10. 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)
11. 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)
12. s1: traversal(the_graph)
13. s2: traversal(the_graph)
14. i1: Vertices(the_graph)
15. ((inl(i1) s1) (inr(i1) s1))
16. P(i1,[inr(i1) / s1],s2)
17. l_disjoint(Vertices(the_graph)+Vertices(the_graph);s2;[inr(i1) / s1])
18. no_repeats(Vertices(the_graph)+Vertices(the_graph);s2)
19. paren(Vertices(the_graph);s2)
paren(Vertices(the_graph);[inl(i1) / (s2 @ [inr(i1)])])		1 step

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(16steps total) PrintForm Definitions Lemmas graph 1 3 Sections Graphs Doc