Nuprl - Some Definitions

biject	`Def Bij(A; B; f) == Inj(A; B; f) & Surj(A; B; f)` `Thm* A,B:Type, f:(AB). Bij(A; B; f) Prop`
inv_funs	`Def InvFuns(A; B; f; g) == g o f = Id & f o g = Id` `Thm* A,B:Type, f:(AB), g:(BA). InvFuns(A; B; f; g) Prop`
compose	`Def (f o g)(x) == f(g(x))` `Thm* A,B,C:Type, f:(BC), g:(AB). f o g AC`
exp	`Def (basepower) == if power=0 1 else base(basepower-1) fi (recursive) Thm* n,k:. (nk)` `Thm* n,k:. (nk)`
int_seg	`Def {i..j} == {k:\| i k < j }` `Thm* m,n:. {m..n} Type`
length	`Def \|\|as\|\| == Case of as; nil 0 ; a.as' \|\|as'\|\|+1 (recursive) Thm* A:Type, l:A. \|\|l\|\|` `Thm \|\|nil\|\|`
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*`
nat	`Def == {i:\| 0i }` `Thm* Type`
surject	`Def Surj(A; B; f) == b:B. a:A. f(a) = b` `Thm* A,B:Type, f:(AB). Surj(A; B; f) Prop`
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`
eq_int	`Def i=j == if i=j true ; false fi` `Thm* i,j:. i=j`
lelt	`Def i j < k == ij & j < k`
tidentity	`Def Id == Id` `Thm* A:Type. Id AA`
le_int	`Def ij == j < i` `Thm* i,j:. ij`
le	`Def AB == B < A` `Thm* i,j:. ij Prop`
identity	`Def Id(x) == x` `Thm* A:Type. Id AA`
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`

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