%   ORIGINAL: 'h4/thm/pred_set/KoenigsLemma_'
% Assm: HL_TRUTH: T
% Assm: HL_FALSITY: ~F
% Assm: HL_BOOL_CASES: !t. (t <=> T) \/ (t <=> F)
% Assm: HL_EXT: !f g. (!x. f x = g x) ==> f = g
% Goal: !R. (!x. 'h4/const/pred_set/FINITE' ('h4/const/pred_set/GSPEC' (\y. 'h4/const/pair/,' y (R x y)))) ==> (!x. ~'h4/const/pred_set/FINITE' ('h4/const/pred_set/GSPEC' (\y. 'h4/const/pair/,' y ('h4/const/relation/RTC' R x y))) ==> (?f. f 'h4/const/num/0' = x /\ (!n. R (f n) (f ('h4/const/num/SUC' n)))))
%   PROCESSED
% Assm ['HL_TRUTH']: T
% Assm ['HL_FALSITY']: ~F
% Assm ['HL_BOOL_CASES']: !t. (t <=> T) \/ (t <=> F)
% Assm ['HL_EXT']: !f g. (!x. happ f x = happ g x) ==> f = g
% Goal: !_0. (!R x y. happ (happ (happ _0 R) x) y = 'h4/const/pair/,' y (happ (happ R x) y)) ==> (!R. (!x. 'h4/const/pred_set/FINITE' ('h4/const/pred_set/GSPEC' (happ (happ _0 R) x))) ==> (!x. ~'h4/const/pred_set/FINITE' ('h4/const/pred_set/GSPEC' (happ (happ _0 ('h4/const/relation/RTC' R)) x)) ==> (?f. happ f 'h4/const/num/0' = x /\ (!n. happ (happ R (happ f n)) (happ f ('h4/const/num/SUC' n))))))
fof('HL_TRUTH', axiom, p(s(bool,'T'))).
fof('HL_FALSITY', axiom, ~ (p(s(bool,'F')))).
fof('HL_BOOL_CASES', axiom, ![T]: (s(bool,T) = s(bool,'T') | s(bool,T) = s(bool,'F'))).
fof('HL_EXT', axiom, ![V_3f23927,V_3f23923]: ![F, G]: (![X]: s(V_3f23923,happ(s(fun(V_3f23927,V_3f23923),F),s(V_3f23927,X))) = s(V_3f23923,happ(s(fun(V_3f23927,V_3f23923),G),s(V_3f23927,X))) => s(fun(V_3f23927,V_3f23923),F) = s(fun(V_3f23927,V_3f23923),G))).
fof('h4/thm/pred_set/KoenigsLemma_', conjecture, ![A]: ![V__0]: (![R, X, Y]: s('h4/type/pair/prod'(A,bool),happ(s(fun(A,'h4/type/pair/prod'(A,bool)),happ(s(fun(A,fun(A,'h4/type/pair/prod'(A,bool))),happ(s(fun(fun(A,fun(A,bool)),fun(A,fun(A,'h4/type/pair/prod'(A,bool)))),V__0),s(fun(A,fun(A,bool)),R))),s(A,X))),s(A,Y))) = s('h4/type/pair/prod'(A,bool),'h4/const/pair/,'(s(A,Y),s(bool,happ(s(fun(A,bool),happ(s(fun(A,fun(A,bool)),R),s(A,X))),s(A,Y))))) => ![R]: (![X]: p(s(bool,'h4/const/pred_set/FINITE'(s(fun(A,bool),'h4/const/pred_set/GSPEC'(s(fun(A,'h4/type/pair/prod'(A,bool)),happ(s(fun(A,fun(A,'h4/type/pair/prod'(A,bool))),happ(s(fun(fun(A,fun(A,bool)),fun(A,fun(A,'h4/type/pair/prod'(A,bool)))),V__0),s(fun(A,fun(A,bool)),R))),s(A,X)))))))) => ![X]: (~ (p(s(bool,'h4/const/pred_set/FINITE'(s(fun(A,bool),'h4/const/pred_set/GSPEC'(s(fun(A,'h4/type/pair/prod'(A,bool)),happ(s(fun(A,fun(A,'h4/type/pair/prod'(A,bool))),happ(s(fun(fun(A,fun(A,bool)),fun(A,fun(A,'h4/type/pair/prod'(A,bool)))),V__0),s(fun(A,fun(A,bool)),'h4/const/relation/RTC'(s(fun(A,fun(A,bool)),R))))),s(A,X))))))))) => ?[F]: (s(A,happ(s(fun('h4/type/num/num',A),F),s('h4/type/num/num','h4/const/num/0'))) = s(A,X) & ![N]: p(s(bool,happ(s(fun(A,bool),happ(s(fun(A,fun(A,bool)),R),s(A,happ(s(fun('h4/type/num/num',A),F),s('h4/type/num/num',N))))),s(A,happ(s(fun('h4/type/num/num',A),F),s('h4/type/num/num','h4/const/num/SUC'(s('h4/type/num/num',N))))))))))))).
