| | Who Cites adjl-rep? |
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| adjl-rep | Def AdjListRep == mkgraphrep(AdjList,  A.adjl-graph(A), A.adjl-obj{\\\\v:l,i:l}(A)) |
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| adjl-obj | Def adjl-obj{\\\\v:l,i:l}(A) == mkgraphobj(x,y. x =A= y, ext{assert_eq_adjl}(A), f,s,x. adjl-edge-accum(A;s',x'.f(s',x');s;x), ext{adjl_DASH_edge_DASH_accum_DASH_properties}{i:l}(A), f,s. adjl-vertex-accum(A;s',x'.f(s',x');s), ext{adjl_DASH_vertex_DASH_accum_DASH_properties}{i:l}(A),  ) |
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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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| adjlist | Def AdjList == size:size  (size List) |
| | | Thm* AdjList  Type |
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| mkgraphrep | Def mkgraphrep(type, graph, obj) == < type,graph,obj > |
| | | Thm*  type:Type, graph:(typeGraph), obj:(r:typeGraphObject(graph(r))). mkgraphrep(type, graph, obj) Graph Representation |
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| adjl-vertex-accum | Def adjl-vertex-accum(A;s',x.f(s';x);s) == primrec(A.size;s;x,s'. f(s';x)) |
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| 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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| eq_adjl | Def x =A= y == x= y |
| | | Thm* A:AdjList, x,y:Vertices(adjl-graph(A)). x =A= 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:(VV), eqw:(x,y:V. eq(x,y)   x = y), eacc:( T:Type. (TVT)TVT), eaccw:(T:Type, s:T, x:V, f:(TVT).  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. (TVT)TT), vaccw:(T:Type, s:T, f:(TVT). 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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| adjl_size | Def t.size == 1of(t) |
| | | Thm* t:AdjList. t.size |
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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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| adjl_out | Def t.out == 2of(t) |
| | | Thm* t:AdjList. t.out t.size(t.size List) |
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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: . 0 n  n < ||l|| l[n] A |
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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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| int_seg | Def {i..j } == {k:| i k < j } |
| | | Thm* m,n:. {m..n} Type |
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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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| nat | Def == {i:| 0i } |
| | | Thm* Type |
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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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| 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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| eq_int | Def i=j == if i=j true ; false fi |
| | | Thm* i,j:. (i=j) |
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| nth_tl | Def nth_tl(n;as) == if n0 as else nth_tl(n-1;tl(as)) fi (recursive) |
| | | Thm* A:Type, as:A List, i:. nth_tl(i;as) A List |
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| hd | Def hd(l) == Case of l; nil "?" ; h.t h |
| | | Thm* A:Type, l:A List. ||l|| 1 hd(l) A |
| | | Thm* A:Type, l:A List . hd(l) A |
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| lelt | Def i j < k == ij & j < k |
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| le | Def AB ==  B < A |
| | | Thm* i,j:. (ij) Prop |
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| tl | Def tl(l) == Case of l; nil nil ; h.t t |
| | | Thm* A:Type, l:A List. tl(l) A List |
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| le_int | Def ij == j < i |
| | | Thm* i,j:. (ij) |
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| not | Def A == A False |
| | | Thm* A:Prop. (A) Prop |
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| lt_int | Def i < j == if i < j true ; false fi |
| | | Thm* i,j:. (i < j) |
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| bnot | Def b == if b false else true fi |
| | | Thm* b:. b |
