Thm* n,k: , c:( n  k). p:( k ( List)). sum(||p(j)|| | j < k) = n & ( j: k, x,y: ||p(j)||. x < y  (p(j))[x] > (p(j))[y]) & ( j: k, x: ||p(j)||. (p(j))[x] < n & c((p(j))[x]) = j) | [finite-partition] |
Def x-the_graph- > *y == p:Vertices(the_graph) List. path(the_graph;p) & p[0] = x & last(p) = y | [connect] |
Def paren(T;s) == s = nil (T+T) List ( t:T, s':(T+T) List. s = ([inl(t)] @ s' @ [inr(t)]) & paren(T;s')) ( s',s'':(T+T) List. ||s'|| < ||s|| & ||s''|| < ||s|| & s = (s' @ s'') & paren(T;s') & paren(T;s'')) (recursive) | [paren] |
Def r- > L^k == n: . r n  ( G:({s:( n List)| ||s|| = k & ( x,y: ||s||. x < y  s[x] < s[y]) }  ||L||). c: ||L||, f:( L[c]  n). increasing(f;L[c]) & ( s: L[c] List. ||s|| = k  ( x,y: ||s||. x < y  s[x] < s[y])  G(map(f;s)) = c)) | [arrows] |