| | Who Cites non-trivial-loop-free? |
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| non-trivial-loop-free | Def non-trivial-loop-free(G) ==  i:Vertices(G).  non-trivial-loop(G;i) |
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| non-trivial-loop | Def non-trivial-loop(G;i) ==  j:Vertices(G). j = i & i-G- > *j & j-G- > *i |
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| connect | Def x-the_graph- > *y == p:Vertices(the_graph) List. path(the_graph;p) & p[0] = x & last(p) = y |
| | | Thm* For any graph
x,y:V. x-the_graph- > *y  Prop |
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| path | Def path(the_graph;p) == 0 < ||p|| & (i: (||p||-1). p[i]-the_graph- > p[(i+1)]) |
| | | Thm* For any graph
p:V List. path(the_graph;p) Prop |
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| int_seg | Def {i..j } == {k: | i  k < j } |
| | | Thm* m,n:. {m..n} Type |
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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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| not | Def A == A   False |
| | | Thm* A:Prop. (A) 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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| gr_v | Def Vertices(t) == 1of(t) |
| | | Thm* t:Graph. Vertices(t) Type |
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| gr_f | Def Incidence(t) == 1of(2of(2of(t))) |
| | | Thm* t:Graph. Incidence(t) Edges(t)  Vertices(t) Vertices(t) |
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| gr_e | Def Edges(t) == 1of(2of(t)) |
| | | Thm* t:Graph. Edges(t) 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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| last | Def last(L) == L[(||L||-1)] |
| | | Thm* T:Type, L:T List. null(L) last(L) T |
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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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| 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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| nth_tl | Def nth_tl(n;as) == if n 0 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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| 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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| 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 |
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| pi2 | Def 2of(t) == t.2 |
| | | Thm* A:Type, B:(AType), p:(a:AB(a)). 2of(p) B(1of(p)) |
