| | Some definitions of interest. |
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| append | Def as @ bs == Case of as; nil  bs ; a.as' [a / (as' @ bs)] (recursive) |
| | | Thm*  T:Type, as,bs:T List. (as @ bs)  T List |
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| 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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| assert | Def b == if b True else False fi |
| | | Thm* b:. b Prop |
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| dfs | Def dfs(the_obj;s;i) == if member-paren(x,y.the_obj.eq(x,y);i;s) s else [inl(i) / (the_obj.eacc(( s',j. dfs(the_obj;s';j)),[inr(i) / s],i))] fi (recursive) |
| | | Thm* For any graph
the_obj:GraphObject(the_graph), s:Traversal, i:V. dfs(the_obj;s;i) Traversal |
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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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| traversal | Def traversal(G) == (Vertices(G)+Vertices(G)) List |
| | | Thm* For any graph
Traversal Type |
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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_eq | Def t.eq == 1of(t) |
| | | Thm* For any graph
t:GraphObject(the_graph). t.eq VV |
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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 |
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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' |
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| member-paren | Def member-paren(x,y.E(x;y);i;s) == (xs.InjCase(x; a. E(a;i), E(a;i))) |
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| no_repeats | Def no_repeats(T;l) == i,j:. i < ||l|| j < ||l||  i = j l[i] = l[j] T |
| | | Thm* T:Type, l:T List. no_repeats(T;l) Prop |
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| not | Def A == A False |
| | | Thm* A:Prop. (A) Prop |
