Nuprl - Proof: Ramsey-recursion 1 2 1 1 2 1 1 1 1 3 2 2 1 2 2 1 1 2

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At: Ramsey-recursion 1 2 1 1 2 1 1 1 1 3 2 2 1 2 2 1 1 2

1. r:
2. k:
3. L: List
4. R: List
5. 2k
6. ||R|| = ||L||
7. i:||L||. 0 < L[i] R[i]- > L[i--]^k
8. r- > R^k-1
9. n:
10. r+1n
11. G: {s:(n List)| ||s|| = k & (x,y:||s||. x < y s[x] < s[y]) }||L||
12. c: ||R||
13. f: R[c](n-1)
14. increasing(f;R[c])
15. s:R[c] List. ||s|| = k-1 (x,y:||s||. x < y s[x] < s[y]) G(map(f;s) @ [(n-1)]) = c
16. 0 < L[c]
17. R[c]- > L[c--]^k
18. (s. G(map(f;s))) {s:(R[c] List)| ||s|| = k & (x,y:||s||. x < y s[x] < s[y]) }||L[c--]||
19. c@0: ||L[c--]||
20. f@0: (L[c]-1)R[c]
21. increasing(f@0;L[c]-1)
22. s:(L[c]-1) List. ||s|| = k (x,y:||s||. x < y s[x] < s[y]) G(map(f;map(f@0;s))) = c
23. increasing(f o f@0[L[c]-1:=n-1];L[c])
24. a:
25. L[c] = a
26. s: a List
27. u: a
28. v: a List
29. (i:||v||. v[i] = a-1) (map(f o f@0[a-1:=n-1];v) ~ map(f;map(f@0;v)))
30. (i:(||v||+1). [u / v][i] = a-1)