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

(99steps total) PrintForm Definitions Lemmas graph 1 2 Sections Graphs Doc

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

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]). increasing(f@0;L[c]-1) & (s:(L[c]-1) List. ||s|| = k (x,y:||s||. x < y s[x] < s[y]) G(map(f;map(f@0;s))) = c)

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)

By: ExRepD THEN InstConcl [f o f@0[L[c]-1:=n-1]]

Generated subgoals:

1 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
f o f@0[L[c]-1:=n-1] L[c]n 3 steps

2 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
increasing(f o f@0[L[c]-1:=n-1];L[c]) & (s:L[c] List. ||s|| = k (x,y:||s||. x < y s[x] < s[y]) G(map(f o f@0[L[c]-1:=n-1];s)) = c) 50 steps

3 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. f1: L[c]n
(increasing(f1;L[c]) & (s:L[c] List. ||s|| = k (x,y:||s||. x < y s[x] < s[y]) G(map(f1;s)) = c)) Prop 2 steps

About:

(99steps total) PrintForm Definitions Lemmas graph 1 2 Sections Graphs Doc

1	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` `f o f@0[L[c]-1:=n-1] L[c]n`	3 steps

2	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` `increasing(f o f@0[L[c]-1:=n-1];L[c]) & (s:L[c] List. \|\|s\|\| = k (x,y:\|\|s\|\|. x < y s[x] < s[y]) G(map(f o f@0[L[c]-1:=n-1];s)) = c)`	50 steps

3	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. `f1:` `L[c]n` `(increasing(f1;L[c]) & (s:L[c] List. \|\|s\|\| = k (x,y:\|\|s\|\|. x < y s[x] < s[y]) G(map(f1;s)) = c)) Prop`	2 steps