Nuprl - Some Definitions

nfa 1 Sections AutomataTheory Doc

nd_accept_list	`Def NDA(l) == q:St. NDA(l)q & (F(NDA)(q))` `Thm* Alph,St:Type, NDA:NDA(Alph;St), l:Alph*. NDA(l) Prop`
nd_compute_list	`Def NDA(l)q == if null(l) q = I(NDA) St else t:St. NDA(tl(l))t & NDA(t,hd(l),q) fi (recursive)` `Thm* Alph,St:Type, NDA:NDA(Alph;St), l:Alph*, q:St. NDA(l)q Prop`
NDA_act	`Def n == 1of(n)` `Thm* Alph,States:Type, n:NDA(Alph;States). n StatesAlphStatesProp`
NDA_fin	`Def F(n) == 2of(2of(n))` `Thm* Alph,States:Type, n:NDA(Alph;States). F(n) States`
NDA_init	`Def I(n) == 1of(2of(n))` `Thm* Alph,States:Type, n:NDA(Alph;States). I(n) States`
accept_list	`Def DA(l) == FinalState(DA)(Result(DA)l)` `Thm* Alph,St:Type, A:Automata(Alph;St), l:Alph*. A(l)`
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'}`
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`
decidable	`Def Dec(P) == P P` `Thm* A:Prop. Dec(A) 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`
nd_automata	`Def NDA(Alph;States) == (StatesAlphStatesProp)States(States)` `Thm* Alph,States:Type{i}. nd_automata{i}(Alph;States) Type{i'}`
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`
pi1	`Def 1of(t) == t.1` `Thm* A:Type, B:(AType), p:a:AB(a). 1of(p) A`
DA_fin	`Def FinalState(a) == 2of(2of(a))` `Thm* Alph,States:Type, a:Automata(Alph;States). FinalState(a) States`
pi2	`Def 2of(t) == t.2` `Thm* A:Type, B:(AType), p:a:AB(a). 2of(p) B(1of(p))`
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`
null	`Def null(as) == Case of as; nil true ; a.as' false Thm* T:Type, as:T. null(as)` `Thm null(nil)`
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`
not	`Def A == A False` `Thm* A:Prop. (A) 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`
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`

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