| | Some definitions of interest. |
|
| adjl-edge-accum | Def adjl-edge-accum(A;s',x'.f(s';x');s;x) == list_accum(s',x'.f(s';x');s;A.out(x)) |
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| adjl-graph | Def adjl-graph(G) == < vertices =  G.size, edges = x:G.size ||G.out(x)||, incidence =  e. < 1of(e),(G.out(1of(e)))[2of(e)] > > |
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| adjl_out | Def t.out == 2of(t) |
| | | Thm*  t:AdjList. t.out  t.size  (t.size List) |
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| adjl_size | Def t.size == 1of(t) |
| | | Thm* t:AdjList. t.size |
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| adjlist | Def AdjList == size:size(size List) |
| | | Thm* AdjList Type |
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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 |
|
| gr_v | Def Vertices(t) == 1of(t) |
| | | Thm* t:Graph. Vertices(t) Type |
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| iff | Def P   Q == (P  Q) & (P Q) |
| | | Thm* A,B:Prop. (A B) Prop |
|
| int_seg | Def {i..j } == {k: | i  k < j } |
| | | Thm* m,n:. {m..n} Type |
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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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| length | Def ||as|| == Case of as; nil  0 ; a.as' ||as'||+1 (recursive) |
| | | Thm* A:Type, l:A List. ||l|| |
| | | Thm* ||nil|| |
|
| 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' |
|
| nat | Def == {i:| 0i } |
| | | Thm* Type |
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| pi1 | Def 1of(t) == t.1 |
| | | Thm* A:Type, B:(AType), p:(a:AB(a)). 1of(p) A |
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| pi2 | Def 2of(t) == t.2 |
| | | Thm* A:Type, B:(AType), p:(a:AB(a)). 2of(p) B(1of(p)) |
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| select | Def l[i] == hd(nth_tl(i;l)) |
| | | Thm* A:Type, l:A List, n:. 0n n < ||l|| l[n] A |
