%   ORIGINAL: 'h4/thm/poly/POLY_ROOTS_FINITE_SET_'
% 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: !p. ~('h4/const/poly/poly' p = 'h4/const/poly/poly' 'h4/const/list/NIL') ==> 'h4/const/pred_set/FINITE' ('h4/const/pred_set/GSPEC' (\x. 'h4/const/pair/,' x ('h4/const/poly/poly' p x = 'h4/const/real/real_of_num' 'h4/const/num/0')))
%   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. (!p x. ?v. (v <=> happ ('h4/const/poly/poly' p) x = 'h4/const/real/real_of_num' 'h4/const/num/0') /\ happ (happ _0 p) x = 'h4/const/pair/,' x v) ==> (!p. ~('h4/const/poly/poly' p = 'h4/const/poly/poly' 'h4/const/list/NIL') ==> 'h4/const/pred_set/FINITE' ('h4/const/pred_set/GSPEC' (happ _0 p)))
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_3f48110,V_3f48106]: ![F, G]: (![X]: s(V_3f48106,happ(s(fun(V_3f48110,V_3f48106),F),s(V_3f48110,X))) = s(V_3f48106,happ(s(fun(V_3f48110,V_3f48106),G),s(V_3f48110,X))) => s(fun(V_3f48110,V_3f48106),F) = s(fun(V_3f48110,V_3f48106),G))).
fof('h4/thm/poly/POLY_ROOTS_FINITE_SET_', conjecture, ![V__0]: (![P, X]: ?[V]: ((p(s(bool,V)) <=> s('h4/type/realax/real',happ(s(fun('h4/type/realax/real','h4/type/realax/real'),'h4/const/poly/poly'(s('h4/type/list/list'('h4/type/realax/real'),P))),s('h4/type/realax/real',X))) = s('h4/type/realax/real','h4/const/real/real_of_num'(s('h4/type/num/num','h4/const/num/0')))) & s('h4/type/pair/prod'('h4/type/realax/real',bool),happ(s(fun('h4/type/realax/real','h4/type/pair/prod'('h4/type/realax/real',bool)),happ(s(fun('h4/type/list/list'('h4/type/realax/real'),fun('h4/type/realax/real','h4/type/pair/prod'('h4/type/realax/real',bool))),V__0),s('h4/type/list/list'('h4/type/realax/real'),P))),s('h4/type/realax/real',X))) = s('h4/type/pair/prod'('h4/type/realax/real',bool),'h4/const/pair/,'(s('h4/type/realax/real',X),s(bool,V)))) => ![P]: (~ (s(fun('h4/type/realax/real','h4/type/realax/real'),'h4/const/poly/poly'(s('h4/type/list/list'('h4/type/realax/real'),P))) = s(fun('h4/type/realax/real','h4/type/realax/real'),'h4/const/poly/poly'(s('h4/type/list/list'('h4/type/realax/real'),'h4/const/list/NIL')))) => p(s(bool,'h4/const/pred_set/FINITE'(s(fun('h4/type/realax/real',bool),'h4/const/pred_set/GSPEC'(s(fun('h4/type/realax/real','h4/type/pair/prod'('h4/type/realax/real',bool)),happ(s(fun('h4/type/list/list'('h4/type/realax/real'),fun('h4/type/realax/real','h4/type/pair/prod'('h4/type/realax/real',bool))),V__0),s('h4/type/list/list'('h4/type/realax/real'),P))))))))))).
