Nuprl - Some Definitions

Thms automata 5 Sections AutomataTheory Doc

lang_rel	`Def L-induced Equiv(x,y) == z:A. L(z @ x) L(z @ y)` `Thm A:Type, L:LangOver(A). L-induced Equiv AAProp`
append	`Def as @ bs == Case of as; nil bs ; a.as' a.(as' @ bs) (recursive)` `Thm* T:Type, as,bs:T. (as @ bs) T`
auto_lang	`Def LangOf(DA)(l) == DA(l)` `Thm* Alph,St:Type, A:Automata(Alph;St). LangOf(A) LangOver(Alph)`
assert	`Def b == if b True else False fi` `Thm* b:. b Prop`
automata	`Def Automata(Alph;States) == (StatesAlphStates)States(States)` `Thm* Alph,States:Type{i}. Automata(Alph;States) Type{i'}`
decidable	`Def Dec(P) == P P` `Thm* A:Prop. Dec(A) Prop`
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`
int_seg	`Def {i..j} == {k:\| i k < j }` `Thm* m,n:. {m..n} Type`
nat	`Def == {i:\| 0i }` `Thm* Type`
lelt	`Def i j < k == ij & j < k`
le	`Def AB == B < A` `Thm* i,j:. ij Prop`
one_one_corr	`Def A ~ B == f:(AB), g:(BA). InvFuns(A; B; f; g)` `Thm* A,B:Type. (A ~ B) Prop`
accept_list	`Def DA(l) == FinalState(DA)(Result(DA)l)` `Thm* Alph,St:Type, A:Automata(Alph;St), l:Alph*. A(l)`
not	`Def A == A False` `Thm* A:Prop. (A) 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`
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`
rev_implies	`Def P Q == Q P` `Thm* A,B:Prop. (A B) 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`
compute_list	`Def Result(DA)l == if null(l) InitialState(DA) else DA((Result(DA)tl(l)),hd(l)) fi (recursive)` `Thm* Alph,St:Type, A:Automata(Alph;St), l:Alph*. (Result(A)l) St`
DA_fin	`Def FinalState(a) == 2of(2of(a))` `Thm* Alph,States:Type, a:Automata(Alph;States). FinalState(a) States`
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`
tidentity	`Def Id == Id` `Thm* A:Type. Id AA`
compose	`Def (f o g)(x) == f(g(x))` `Thm* A,B,C:Type, f:(BC), g:(AB). f o g AC`
hd	`Def hd(l) == Case of l; nil "?" ; h.t h` `Thm* A:Type, l:A*. \|\|l\|\|1 hd(l) A`
tl	`Def tl(l) == Case of l; nil nil ; h.t t` `Thm* A:Type, l:A. tl(l) A`
DA_act	`Def a == 1of(a)` `Thm* Alph,States:Type, a:Automata(Alph;States). a StatesAlphStates`
DA_init	`Def InitialState(a) == 1of(2of(a))` `Thm* Alph,States:Type, a:Automata(Alph;States). InitialState(a) States`
null	`Def null(as) == Case of as; nil true ; a.as' false Thm* T:Type, as:T. null(as)` `Thm null(nil)`
pi2	`Def 2of(t) == t.2` `Thm* A:Type, B:(AType), p:a:AB(a). 2of(p) B(1of(p))`
identity	`Def Id(x) == x` `Thm* A:Type. Id AA`
pi1	`Def 1of(t) == t.1` `Thm* A:Type, B:(AType), p:a:AB(a). 1of(p) A`

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