Nuprl - Some Definitions

inject	`Def Inj(A; B; f) == a1,a2:A. f(a1) = f(a2) B a1 = a2` `Thm* A,B:Type, f:(AB). Inj(A; B; f) Prop`
int_seg	`Def {i..j} == {k:\| i k < j }` `Thm* m,n:. {m..n} Type`
listify	`Def listify(f; m; n) == if nm nil else f(m).listify(f; m+1; n) fi (recursive)` `Thm* T:Type, m,n:, f:({m..n}T). listify(f; m; n) T*`
mem_f	`Def mem_f(T;a;bs) == Case of bs; nil False ; b.bs' b = a T mem_f(T;a;bs') (recursive)` `Thm* T:Type, a:T, bs:T*. mem_f(T;a;bs) Prop`
nat	`Def == {i:\| 0i }` `Thm* Type`
select	`Def l[i] == hd(nth_tl(i;l))` `Thm* A:Type, l:A*, n:. 0n n < \|\|l\|\| l[n] A`
lelt	`Def i j < k == ij & j < k`
nth_tl	`Def nth_tl(n;as) == if n0 as else nth_tl(n-1;tl(as)) fi (recursive)` `Thm* A:Type, as:A, i:. nth_tl(i;as) A`
le_int	`Def ij == j < i` `Thm* i,j:. ij`
le	`Def AB == B < A` `Thm* i,j:. ij Prop`
hd	`Def hd(l) == Case of l; nil "?" ; h.t h` `Thm* A:Type, l:A*. \|\|l\|\|1 hd(l) A`
lt_int	`Def i < j == if i < j true ; false fi` `Thm* i,j:. i < j`
bnot	`Def b == if b false else true fi` `Thm* b:. b`
not	`Def A == A False` `Thm* A:Prop. (A) Prop`
tl	`Def tl(l) == Case of l; nil nil ; h.t t` `Thm* A:Type, l:A. tl(l) A`

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