| | Some definitions of interest. |
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| adjm-edge-accum | Def adjm-edge-accum(M;s',x'.f(s';x');s;x) == primrec(M.size;s; y,s'. if M.adj(x,y) f(s';y) else s' fi) |
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| adjm-graph | Def adjm-graph(A) == < vertices =  A.size, edges = {p:(A.size A.size)|  (A.adj(1of(p),2of(p))) }, incidence = e.e > |
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| adjm_adj | Def t.adj == 2of(t) |
| | | Thm*  t:AdjMatrix. t.adj  t.size  t.size |
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| adjm_size | Def t.size == 1of(t) |
| | | Thm* t:AdjMatrix. t.size |
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| adjmatrix | Def AdjMatrix == size:sizesize |
| | | Thm* AdjMatrix Type |
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| assert | Def b == if b True else False fi |
| | | Thm* b:. b Prop |
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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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| 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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| 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 |
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| 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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| 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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| mkgraph | Def < vertices = v, edges = e, incidence = f > == < v,e,f,o > |
| | | Thm* v,e:Type, f:(evv), o:Top. < vertices = v, edges = e, incidence = f > Graph |
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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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| primrec | Def primrec(n;b;c) == if n= 0 b else c(n-1,primrec(n-1;b;c)) fi (recursive) |
| | | Thm* T:Type, n:, b:T, c:(nTT). primrec(n;b;c) T |
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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 |
