Selected Objects

def	automata	Automata(Alph;States) == (StatesAlphStates)States(States)
def	DA_act	a == 1of(a)
def	DA_init	InitialState(a) == 1of(2of(a))
def	DA_fin	FinalState(a) == 2of(2of(a))
def	compute_list	(rec) Result(DA)l == if null(l) InitialState(DA) else DA((Result(DA)tl(l)),hd(l)) fi
THM	compute_list_qwf	Auto:Automata(Alph;St), l:x,y:Alph*//((Result(Auto)x) = (Result(Auto)y)). (Result(Auto)l) St
THM	compute_l_inv	Auto:Automata(Alph;St), x,y,z:Alph*. (Result(Auto)x) = (Result(Auto)y) (Result(Auto)z @ x) = (Result(Auto)z @ y)
def	accept_list	DA(l) == FinalState(DA)(Result(DA)l)
THM	accept_list_qwf	Auto:Automata(Alph;St), l:x,y:Alph*//((Result(Auto)x) = (Result(Auto)y)). Auto(l)
def	auto_lang	LangOf(DA)(l) == DA(l)
THM	action_auto	Auto:Automata(Alph;St), l:Alph*. ( < St,l,s. Auto(s,l) > :lInitialState(Auto)) = (Result(Auto)l)
THM	pos_states	Auto:Automata(Alph;St). Fin(St) (n:. #(St)=n )
THM	reach_lemma	S:ActionSet(Alph), si:S.car, nn:, f:(nnAlph), g:(Alphnn). Fin(S.car) InvFuns(nn; Alph; f; g) (n:. RL:{y:{x:(S.car*)| 0 < ||x|| & ||x||n+1 }| y[(||y||-1)] = si }. (s:S.car. (w:Alph*. (S:wsi) = s) mem_f(S.car;s;RL)) ||RL|| = n+1 & (i:||RL||, j:i. RL[i] = RL[j]) & (s:S.car. mem_f(S.car;s;RL) (w:Alph*. (S:wsi) = s)) & (k:. knn (RLa:S.car*. (i:{1..||RL||}, a:Alph. mem_f(S.car;S.act(a,RL[i]);RL) mem_f(S.car;S.act(a,RL[i]);RLa)) & (a:Alph. g(a) < k mem_f(S.car;S.act(a,hd(RL));RL) mem_f(S.car;S.act(a,hd(RL));RLa)) & (s:S.car. mem_f(S.car;s;RLa) (w:Alph*. (S:wsi) = s)))))
THM	reach_aux	S:ActionSet(Alph), si:S.car. Fin(S.car) Fin(Alph) (RL:S.car*. s:S.car. (w:Alph*. (S:wsi) = s) mem_f(S.car;s;RL))
THM	reach_list	Auto:Automata(Alph;St). Fin(St) & Fin(Alph) (RL:St*. s:St. (w:Alph*. (Result(Auto)w) = s) mem_f(St;s;RL))
THM	reach_dec	Auto:Automata(Alph;St). Fin(Alph) Fin(St) (s:St. Dec(w:Alph*. (Result(Auto)w) = s))