Nuprl - Pf fin_dec_fin 2 1 2 2 4 1 1 5 1 2 1

PrintForm Definitions finite sets Sections AutomataTheory Doc

1. n:
2. 0 < n
3. T:Type, B:(TProp). (f:((n-1)T), g:(T(n-1)). InvFuns((n-1); T; f; g)) & (t:T. Dec(B(t))) (m:, f:(m{t:T| B(t) }), g:({t:T| B(t) }m). InvFuns(m; {t:T| B(t) }; f; g))
4. T: Type
5. B: TProp
6. f: nT
7. g: Tn
8. g o f = Id
9. f o g = Id
10. t:T. Dec(B(t))
11. f (n-1){t:T| g(t) < n-1 }
12. g {t:T| g(t) < n-1 }(n-1)
13. m:
14. f1: m{t:{t:T| g(t) < n-1 }| B(t) }
15. g@0: {t:{t:T| g(t) < n-1 }| B(t) }m
16. g@0 o f1 = Id
17. f1 o g@0 = Id
18. B(f(n-1))
19. f2: (m+1){t:T| B(t) }
20. f2 = (x.if x=m f(n-1) else f1(x) fi)
21. g2: {t:T| B(t) }(m+1)
22. g2 = (x.if g(x)=n-1 m else g@0(x) fi)
23. x: {t:T| B(t) }

if if g(x)=n-1 m else g@0(x) fi=m f(n-1) else f1(if g(x)=n-1 m else g@0(x) fi) fi = x {t:T| B(t) }

By: SplitOnNthConclITE 2 THENA Try (Complete Auto)

Generated subgoals:

1 24. g(x) = n-1
if m=m f(n-1) else f1(m) fi = x {t:T| B(t) }

2 24. g(x) = n-1
if g@0(x)=m f(n-1) else f1(g@0(x)) fi = x {t:T| B(t) }

About:

1	24. `g(x) = n-1` `if m=m f(n-1) else f1(m) fi = x {t:T\| B(t) }`
2	24. `g(x) = n-1` `if g@0(x)=m f(n-1) else f1(g@0(x)) fi = x {t:T\| B(t) }`