Nuprl - Some Definitions

action_set	`Def ActionSet(T) == car:TypeTcarcar` `Thm* T:Type{i}. ActionSet(T) Type{i'}`
aset_act	`Def a.act == 2of(a)` `Thm* T:Type, a:ActionSet(T). a.act Ta.cara.car`
aset_car	`Def a.car == 1of(a)` `Thm* T:Type, a:ActionSet(T). a.car Type`
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`
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*`
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`
rev_implies	`Def P Q == Q P` `Thm* A,B:Prop. (A B) Prop`
pi2	`Def 2of(t) == t.2` `Thm* A:Type, B:(AType), p:a:AB(a). 2of(p) B(1of(p))`
pi1	`Def 1of(t) == t.1` `Thm* A:Type, B:(AType), p:a:AB(a). 1of(p) A`
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`
lelt	`Def i j < k == ij & j < k`
le_int	`Def ij == j < i` `Thm* i,j:. ij`
le	`Def AB == B < A` `Thm* i,j:. ij Prop`
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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