%   ORIGINAL: 'h4/thm/list/EXISTS_GENLIST_'
% 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: !f P n. 'h4/const/list/EXISTS' P ('h4/const/list/GENLIST' f n) <=> (?i. 'h4/const/prim_rec/<' i n /\ P (f i))
%   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: !f P n. 'h4/const/list/EXISTS' P ('h4/const/list/GENLIST' f n) <=> (?i. 'h4/const/prim_rec/<' i n /\ happ P (happ f i))
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_3f9391,V_3f9387]: ![F, G]: (![X]: s(V_3f9387,happ(s(fun(V_3f9391,V_3f9387),F),s(V_3f9391,X))) = s(V_3f9387,happ(s(fun(V_3f9391,V_3f9387),G),s(V_3f9391,X))) => s(fun(V_3f9391,V_3f9387),F) = s(fun(V_3f9391,V_3f9387),G))).
fof('h4/thm/list/EXISTS_GENLIST_', conjecture, ![A]: ![F, P, N]: (p(s(bool,'h4/const/list/EXISTS'(s(fun(A,bool),P),s('h4/type/list/list'(A),'h4/const/list/GENLIST'(s(fun('h4/type/num/num',A),F),s('h4/type/num/num',N)))))) <=> ?[I]: (p(s(bool,'h4/const/prim_rec/<'(s('h4/type/num/num',I),s('h4/type/num/num',N)))) & p(s(bool,happ(s(fun(A,bool),P),s(A,happ(s(fun('h4/type/num/num',A),F),s('h4/type/num/num',I))))))))).
