%   ORIGINAL: 'h4/thm/poly/POLY_ROOTS_INDEX_LEMMA_'
% 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: !n p. ~('h4/const/poly/poly' p = 'h4/const/poly/poly' 'h4/const/list/NIL') /\ 'h4/const/list/LENGTH' p = n ==> (?i. !x. 'h4/const/poly/poly' p x = 'h4/const/real/real_of_num' 'h4/const/num/0' ==> (?m. 'h4/const/arithmetic/<=' m n /\ x = i m))
%   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: !n p. ~('h4/const/poly/poly' p = 'h4/const/poly/poly' 'h4/const/list/NIL') /\ 'h4/const/list/LENGTH' p = n ==> (?i. !x. happ ('h4/const/poly/poly' p) x = 'h4/const/real/real_of_num' 'h4/const/num/0' ==> (?m. 'h4/const/arithmetic/<=' m n /\ x = happ i m))
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_3f47034,V_3f47030]: ![F, G]: (![X]: s(V_3f47030,happ(s(fun(V_3f47034,V_3f47030),F),s(V_3f47034,X))) = s(V_3f47030,happ(s(fun(V_3f47034,V_3f47030),G),s(V_3f47034,X))) => s(fun(V_3f47034,V_3f47030),F) = s(fun(V_3f47034,V_3f47030),G))).
fof('h4/thm/poly/POLY_ROOTS_INDEX_LEMMA_', conjecture, ![N, 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')))) & s('h4/type/num/num','h4/const/list/LENGTH'(s('h4/type/list/list'('h4/type/realax/real'),P))) = s('h4/type/num/num',N)) => ?[I]: ![X]: (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'))) => ?[M]: (p(s(bool,'h4/const/arithmetic/<='(s('h4/type/num/num',M),s('h4/type/num/num',N)))) & s('h4/type/realax/real',X) = s('h4/type/realax/real',happ(s(fun('h4/type/num/num','h4/type/realax/real'),I),s('h4/type/num/num',M))))))).
