Thm* r,k: , L,R: List. 2 k  ||R|| = ||L||  ( i: ||L||. 0 < L[i]  R[i]- > L[i--]^k)  r- > R^k-1  r+1- > L^k | [Ramsey-recursion] |
Thm* k: , L: List, r1,r2: . r1 r2  r1- > L^k  r2- > L^k | [arrows-monotone1] |
Thm* n,m: , f:( n  m). Inj( n; m; f)  n m | [pigeon-hole] |
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] |