| | Some definitions of interest. |
|
| graphobj | Def GraphObject(the_graph) == eq:Vertices(the_graph)  Vertices(the_graph)  ( x,y:Vertices(the_graph).  (eq(x,y))   x = y)(eacc:( T:Type. (TVertices(the_graph)T)TVertices(the_graph)T)(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)) & eacc(f,s,x) = list_accum(s',x'.f(s',x');s;L))(vacc:(T:Type. (TVertices(the_graph)T)TT)(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)) & vacc(f,s) = list_accum(s',x'.f(s',x');s;L))Top)) |
| | | Thm* the_graph:Graph. GraphObject(the_graph) Type{i'} |
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| edge | Def x-the_graph- > y == e:Edges(the_graph). Incidence(the_graph)(e) = < x,y > |
| | | Thm* For any graph
x,y:V. x-the_graph- > y Prop |
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| gr_v | Def Vertices(t) == 1of(t) |
| | | Thm* t:Graph. Vertices(t) Type |
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| graph | Def Graph == v:Typee:Type(evv)Top |
| | | Thm* Graph Type{i'} |
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| gro_eacc | Def t.eacc == 1of(2of(2of(t))) |
| | | Thm* For any graph
t:GraphObject(the_graph), T:Type. t.eacc (TVT)TVT |
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| gro_eaccw | Def t.eaccw == 1of(2of(2of(2of(t)))) |
| | | Thm* For any graph
t:GraphObject(the_graph). t.eaccw (T:Type, s:T, x:V, f:(TVT). L:V List. (y:V. x-the_graph- > y (y L)) & t.eacc(f,s,x) = list_accum(s',x'.f(s',x');s;L)) |
|
| gro_eq | Def t.eq == 1of(t) |
| | | Thm* For any graph
t:GraphObject(the_graph). t.eq VV |
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| gro_eqw | Def t.eqw == 1of(2of(t)) |
| | | Thm* For any graph
t:GraphObject(the_graph). t.eqw (x,y:V. t.eq(x,y) x = y) |
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| gro_other | Def t.other == 2of(2of(2of(2of(2of(2of(t)))))) |
| | | Thm* For any graph
t:GraphObject(the_graph). t.other Top |
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| gro_vacc | Def t.vacc == 1of(2of(2of(2of(2of(t))))) |
| | | Thm* For any graph
t:GraphObject(the_graph), T:Type. t.vacc (TVT)TT |
|
| gro_vaccw | Def t.vaccw == 1of(2of(2of(2of(2of(2of(t)))))) |
| | | Thm* For any graph
t:GraphObject(the_graph). t.vaccw (T:Type, s:T, f:(TVT). L:V List. no_repeats(V;L) & (y:V. (y L)) & t.vacc(f,s) = list_accum(s',x'.f(s',x');s;L)) |
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| iff | Def P Q == (P  Q) & (P Q) |
| | | Thm* A,B:Prop. (A B) Prop |
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| l_member | Def (x l) == i: . i < ||l|| & x = l[i] T |
| | | Thm* T:Type, x:T, l:T List. (x l) Prop |
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| list_accum | Def list_accum(x,a.f(x;a);y;l) == Case of l; nil  y ; b.l' list_accum(x,a.f(x;a);f(y;b);l') (recursive) |
| | | Thm* T,T':Type, l:T List, y:T', f:(T'TT'). list_accum(x,a.f(x,a);y;l) T' |
