| Who Cites adjm-obj? |
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adjm-obj | Def adjm-obj{\\\\v:l,i:l}(M) == mkgraphobj( x,y. x =M= y, ext{assert_eq_adjm}(M), f,s,x. adjm-edge-accum(M;s',x'.f(s',x');s;x), ext{adjm_DASH_edge_DASH_accum_DASH_properties}{i:l}(M), f,s. adjm-vertex-accum(M;s',x'.f(s',x');s), ext{adjm_DASH_vertex_DASH_accum_DASH_properties}{i:l}(M), ) |
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adjm-vertex-accum | Def adjm-vertex-accum(M;s',x.f(s';x);s) == primrec(M.size;s; x,s'. f(s';x)) |
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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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eq_adjm | Def x =M= y == x= y |
| | Thm* M:AdjMatrix, x,y:Vertices(adjm-graph(M)). x =M= y  |
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mkgraphobj | Def mkgraphobj(eq, eqw, eacc, eaccw, vacc, vaccw, other) == < eq,eqw,eacc,eaccw,vacc,vaccw,other > |
| | Thm* For any graph
eq:(V V  ), eqw:( x,y:V. eq(x,y)  x = y), eacc:( T:Type. (T V T) T V T), eaccw:( T:Type, s:T, x:V, f:(T V T). L:V List. ( y:V. x-the_graph- > y  (y L)) & eacc(f,s,x) = list_accum(s',x'.f(s',x');s;L)), vacc:( T:Type. (T V T) T T), vaccw:( T:Type, s:T, f:(T V T). L:V List. no_repeats(V;L) & ( y:V. (y L)) & vacc(f,s) = list_accum(s',x'.f(s',x');s;L)), other:Top. mkgraphobj(eq, eqw, eacc, eaccw, vacc, vaccw, other) GraphObject(the_graph) |
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adjm_size | Def t.size == 1of(t) |
| | Thm* t:AdjMatrix. t.size  |
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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:( n T T). primrec(n;b;c) T |
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adjm_adj | Def t.adj == 2of(t) |
| | Thm* t:AdjMatrix. t.adj t.size  t.size   |
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eq_int | Def i= j == if i=j true ; false fi |
| | Thm* i,j: . (i= j)  |
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pi1 | Def 1of(t) == t.1 |
| | Thm* A:Type, B:(A Type), p:(a:A B(a)). 1of(p) A |
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pi2 | Def 2of(t) == t.2 |
| | Thm* A:Type, B:(A Type), p:(a:A B(a)). 2of(p) B(1of(p)) |