%   ORIGINAL: 'h4/thm/list/EL_LUPDATE_'
% 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: !ys x i k. 'h4/const/list/EL' i ('h4/const/list/LUPDATE' x k ys) = 'h4/const/bool/COND' (i = k /\ 'h4/const/prim_rec/<' k ('h4/const/list/LENGTH' ys)) x ('h4/const/list/EL' i ys)
%   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: !ys x i k. ?v. (v <=> i = k /\ 'h4/const/prim_rec/<' k ('h4/const/list/LENGTH' ys)) /\ 'h4/const/list/EL' i ('h4/const/list/LUPDATE' x k ys) = 'h4/const/bool/COND' v x ('h4/const/list/EL' i ys)
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_3f10342,V_3f10338]: ![F, G]: (![X]: s(V_3f10338,happ(s(fun(V_3f10342,V_3f10338),F),s(V_3f10342,X))) = s(V_3f10338,happ(s(fun(V_3f10342,V_3f10338),G),s(V_3f10342,X))) => s(fun(V_3f10342,V_3f10338),F) = s(fun(V_3f10342,V_3f10338),G))).
fof('h4/thm/list/EL_LUPDATE_', conjecture, ![A]: ![Ys, X, I, K]: ?[V]: ((p(s(bool,V)) <=> (s('h4/type/num/num',I) = s('h4/type/num/num',K) & p(s(bool,'h4/const/prim_rec/<'(s('h4/type/num/num',K),s('h4/type/num/num','h4/const/list/LENGTH'(s('h4/type/list/list'(A),Ys)))))))) & s(A,'h4/const/list/EL'(s('h4/type/num/num',I),s('h4/type/list/list'(A),'h4/const/list/LUPDATE'(s(A,X),s('h4/type/num/num',K),s('h4/type/list/list'(A),Ys))))) = s(A,'h4/const/bool/COND'(s(bool,V),s(A,X),s(A,'h4/const/list/EL'(s('h4/type/num/num',I),s('h4/type/list/list'(A),Ys))))))).
