Nuprl - Some Definitions

Thms myhill nerode Sections AutomataTheory Doc

append	`Def as @ bs == Case of as; nil bs ; a.as' a.(as' @ bs) (recursive)` `Thm* T:Type, as,bs:T. (as @ bs) T`
assert	`Def b == if b True else False fi` `Thm* b:. b Prop`
beq	`Def p = q == if p q else q fi` `Thm* p,q:. p = q`
bnot	`Def b == if b false else true fi` `Thm* b:. b`
equiv_rel	`Def EquivRel x,y:T. E(x;y) == Refl(T;x,y.E(x;y)) & Sym x,y:T. E(x;y) & Trans x,y:T. E(x;y)` `Thm* T:Type, E:(TTProp). (EquivRel x,y:T. E(x,y)) Prop`
finite	`Def Fin(s) == n:, f:(ns). Bij(n; s; f)` `Thm* T:Type. Fin(T) Prop`
iff	`Def P Q == (P Q) & (P Q)` `Thm* A,B:Prop. (A B) Prop`
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`
trans	`Def Trans x,y:T. E(x;y) == a,b,c:T. E(a;b) E(b;c) E(a;c)` `Thm* T:Type, E:(TTProp). Trans x,y:T. E(x,y) Prop`
sym	`Def Sym x,y:T. E(x;y) == a,b:T. E(a;b) E(b;a)` `Thm* T:Type, E:(TTProp). Sym x,y:T. E(x,y) Prop`
refl	`Def Refl(T;x,y.E(x;y)) == a:T. E(a;a)` `Thm* T:Type, E:(TTProp). Refl(T;x,y.E(x,y)) Prop`
int_seg	`Def {i..j} == {k:\| i k < j }` `Thm* m,n:. {m..n} 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`
nat	`Def == {i:\| 0i }` `Thm* Type`
rev_implies	`Def P Q == Q P` `Thm* A,B:Prop. (A B) Prop`
lelt	`Def i j < k == ij & j < k`
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`
le	`Def AB == B < A` `Thm* i,j:. ij Prop`
not	`Def A == A False` `Thm* A:Prop. (A) Prop`

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