| Rank | Theorem | Name |
| 15 | Thm* For any graph
 the_obj:GraphObject(the_graph), P:(V  TraversalTraversalProp), 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) (k:V. (inr(k) s2) j-the_graph- > *k) paren(V;s3) P(j,s1,s2) P(i,s2 @ s1,s3) P(i,s1,s3 @ s2)) (s1,s2:Traversal, i:V.  (inr(i) s1) (inl(i) s1) paren(V;s2) l_disjoint(V+V;s2;s1) no_repeats(V+V;s2) (j:V. (inr(j) s2) i-the_graph- > *j) (j:V. i-the_graph- > j j = i (inl(j) s2) (inl(j) s1) (inr(j) 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') & (j:V. (inr(j) s') i-the_graph- > *j) & dfs(the_obj;s;i) = (s' @ s)) | [dfs_induction4] |
| cites |
| 14 | Thm* For any graph
the_obj:GraphObject(the_graph), P:(VTraversalTraversalProp), s:Traversal, i:V. (s1,s2:Traversal, i:V. P(i,s1,s2) l_disjoint(V+V;s2;s1) & no_repeats(V+V;s2)) (s:Traversal, i:V. member-paren(x,y.the_obj.eq(x,y);i;s) P(i,s,nil)) (s1,s2,s3:Traversal, i,j:V. i-the_graph- > j P(j,s1,s2) P(i,s2 @ s1,s3) P(i,s1,s3 @ s2)) (s1,s2:Traversal, i:V. member-paren(x,y.the_obj.eq(x,y);i;s1) (j:V. 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)])])) (s':Traversal. P(i,s,s') & dfs(the_obj;s;i) = (s' @ s)) | [dfs_induction3] |
| 3 | Thm* a,b,c:T List. l_disjoint(T;c;a @ b)  l_disjoint(T;c;a) & l_disjoint(T;c;b) | [l_disjoint_append2] |
| 1 | Thm* s',s3:(T+T) List. paren(T;s') paren(T;s3) paren(T;s3 @ s') | [append_paren] |
| 3 | Thm* a,b,c:T List. l_disjoint(T;a @ b;c) l_disjoint(T;a;c) & l_disjoint(T;b;c) | [l_disjoint_append] |
| 2 | Thm* a,b:T List, t:T. l_disjoint(T;[t / a];b) (t b) & l_disjoint(T;a;b) | [l_disjoint_cons] |
| 2 | 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)) | [assert-member-paren] |
| 0 | Thm* For any graph
the_obj:GraphObject(the_graph). (x,y:V. the_obj.eq(x,y) x = y) & (T:Type, s:T, x:V, f:(TVT). L:V List. (y:V. x-the_graph- > y (y L)) & the_obj.eacc(f,s,x) = list_accum(s',x'.f(s',x');s;L)) & (T:Type, s:T, f:(TVT). L:V List. no_repeats(V;L) & (y:V. (y L)) & the_obj.vacc(f,s) = list_accum(s',x'.f(s',x');s;L)) | [graphobj-properties] |
| 2 | Thm* l1,l2:T List. no_repeats(T;l1 @ l2) l_disjoint(T;l1;l2) & no_repeats(T;l1) & no_repeats(T;l2) | [no_repeats_append_iff] |
| 1 | Thm* l:T List, x:T. no_repeats(T;[x / l]) no_repeats(T;l) & (x l) | [no_repeats_cons] |
| 2 | Thm* a,b:T List, t:T. l_disjoint(T;b;[t / a]) (t b) & l_disjoint(T;b;a) | [l_disjoint_cons2] |
| 3 | Thm* s':(T+T) List. paren(T;s') (i:T. (inl(i) s') (inr(i) s')) | [paren_balance1] |
| 0 | Thm* a,x:T. (x [a]) x = a | [member_singleton] |
| 2 | Thm* x:T, l1,l2:T List. (x l1 @ l2) (x l1) (x l2) | [member_append] |
| 1 | Thm* l:T List, a,x:T. (x [a / l]) x = a (x l) | [cons_member] |