| | Some definitions of interest. |
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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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| adjm-graph | Def adjm-graph(A) == < vertices = A.size, edges = {p:(A.sizeA.size)|  (A.adj(1of(p),2of(p))) }, incidence = e.e > |
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| adjm_to_adjl | Def adjm_to_adjl(m) == Case(m) Case mk_adjmatrix(n,f) = > mk_adjlist(n, i.filter(j.f(i,j);upto(0;n))) |
| | | Thm*  m:AdjMatrix. adjm_to_adjl(m)  AdjList |
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| adjmatrix | Def AdjMatrix == size:size  size |
| | | Thm* AdjMatrix Type |
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| assert | Def b == if b True else False fi |
| | | Thm* b:. b Prop |
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| graph-isomorphic | Def G  H ==  vmap:(Vertices(G)Vertices(H)), emap:(Edges(G)Edges(H)). Bij(Vertices(G); Vertices(H); vmap) & Bij(Edges(G); Edges(H); emap) & (vmap,vmap) o Incidence(G) = Incidence(H) o emap |
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| biject | Def Bij(A; B; f) == Inj(A; B; f) & Surj(A; B; f) |
| | | Thm* A,B:Type, f:(AB). Bij(A; B; f) Prop |
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| compose | Def (f o g)(x) == f(g(x)) |
| | | Thm* A,B,C:Type, f:(BC), g:(AB). f o g AC |
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| compose2 | Def (f1,f2) o g(x) == g(x)/x,y. < f1(x),f2(y) > |
| | | Thm* A,B,C,B',C':Type, g:(ABC), f1:(BB'), f2:(CC'). (f1,f2) o g AB'C' |
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| filter | Def filter(P;l) == reduce(a,v. if P(a) [a / v] else v fi;nil;l) |
| | | Thm* T:Type, P:(T), l:T List. filter(P;l) T List |
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| identity | Def Id(x) == x |
| | | Thm* A:Type. Id AA |
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| int_seg | Def {i..j } == {k: | i  k < j } |
| | | Thm* m,n:. {m..n} Type |
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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|| |
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| 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 |
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| upto | Def upto(i;j) == if i <  j [i / upto(i+1;j)] else nil fi (recursive) |
| | | Thm* i,j:. upto(i;j) {i..j} List |
