Thm* DivGraph_2 Graph | [divides-graph2_wf] |
Thm* EquivRel G,H:Graph. G H | [graph-isomorphic-equiv] |
Thm* G,H:Graph. G H  H G | [graph-isomorphic_inversion] |
Thm* G,H,K:Graph. G H  H K  G K | [graph-isomorphic_transitivity] |
Thm* G,H:Graph. G = H  G H | [graph-isomorphic_weakening] |
Thm* For any graph
L1,L2:V List, s:Traversal. L1-- > *L2  dfsl-traversal(the_graph;L1;s)  dfsl-traversal(the_graph;L1 @ L2;s) | [dfsl-traversal-append] |
Thm* For any graph
dfsl-traversal(the_graph;nil;nil) | [dfsl-traversal-nil] |
Thm* For any graph
L:V List, i:V, s:Traversal. (inr(i) s)  dfl-traversal(the_graph;L;s)  dfl-traversal(the_graph;L;[inl(i) / s]) | [dfl-traversal-consl] |
Thm* For any graph
L:V List, i:V, s:Traversal. ( j:V. (inr(j) s)  (inl(j) s)  j-the_graph- > *i)  L-- > *i  dfl-traversal(the_graph;L;s)  dfl-traversal(the_graph;L;[inr(i) / s]) | [dfl-traversal-consr] |
Thm* For any graph
L1,L2:V List, s:Traversal. L1-- > *L2  dfl-traversal(the_graph;L2;s)  dfl-traversal(the_graph;L1;s) | [dfl-traversal-connect] |
Thm* G:Graph. ( x,y:Vertices(G). Dec(x = y))  ( s:traversal(G). paren(Vertices(G);s)  no_repeats(Vertices(G)+Vertices(G);s)  df-traversal(G;s)  depthfirst-traversal(G;s)) | [paren-df-traversal] |
Thm* For any graph
df-traversal(the_graph;nil) | [df-traversal-nil] |
Thm* For any graph
i:V, s:Traversal. ( j:V. (inr(j) s)  (inl(j) s)  j-the_graph- > *i)  df-traversal(the_graph;s)  df-traversal(the_graph;[inr(i) / s]) | [df-traversal-cons2] |
Thm* For any graph
A,B,C:V List. A-- > *C B-- > *C  A @ B-- > *C | [list-list-connect-append2] |
Thm* For any graph
A,B,C:V List. A-- > *B @ C  A-- > *B & A-- > *C | [list-list-connect-append] |
Thm* For any graph
A,B:V List. A = B  A-- > *B | [list-list-connect_weakening] |
Thm* For any graph
A,B:V List. B A  A-- > *B | [list-list-connect-iseg] |
Thm* For any graph
A:V List, x:V. A-- > *[x]  A-- > *x | [list-list-connect-singleton] |
Thm* For any graph
A,B,C:V List. A-- > *B  B-- > *C  A-- > *C | [list-list-connect_transitivity] |
Thm* For any graph
A,B:V List, i:V. A @ B-- > *i  A-- > *i B-- > *i | [list-connect-append] |
Thm* For any graph
i,j:V. [i]-- > *j  i-the_graph- > *j | [list-connect-singleton] |
Thm* For any graph
( x,y:V. Dec(x = y))  ( x,y:V. x-the_graph- > *y  x = y ( z:V. z = x & x-the_graph- > z & z-the_graph- > *y)) | [connect-iff] |
Thm* For any graph
p:V List. 1 < ||p||  path(the_graph;p)  path(the_graph;tl(p)) | [path-tl] |
Thm* For any graph
x,y:V. x-the_graph- > y  x-the_graph- > *y | [edge-connect] |
Thm* For any graph
x,y:V. x = y  x-the_graph- > *y | [connect_weakening] |
Thm* For any graph
x,y,z:V. x-the_graph- > *y  y-the_graph- > *z  x-the_graph- > *z | [connect_transitivity] |
Thm* For any graph
x,y:V. x-the_graph- > *y Prop | [connect_wf] |
Thm* t:Graph. t.o Top | [gr_o_wf] |
Thm* t:Graph. Incidence(t) Edges(t) Vertices(t) Vertices(t) | [gr_f_wf] |
Thm* t:Graph. Edges(t) Type | [gr_e_wf] |
Thm* t:Graph. Vertices(t) Type | [gr_v_wf] |