Thm* Auto:Automata(Alph;St). Fin(Alph) Fin(St) (s:St. Dec(w:Alph*. (Result(Auto)w) = s)) reach_dec

Thm* Auto:Automata(Alph;St). Fin(St) & Fin(Alph) (RL:St*. s:St. (w:Alph*. (Result(Auto)w) = s) mem_f(St;s;RL)) reach_list

Thm* 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)) reach_aux

Thm* 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))))) reach_lemma

Thm* Auto:Automata(Alph;St). Fin(St) (n:. #(St)=n ) pos_states

In prior sections: finite sets list 3 autom