%   ORIGINAL: h4/fcp/FCP__EXISTS0
% 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
% Assm: h4/bool/TRUTH: T
% Assm: h4/bool/REFL__CLAUSE: !x. x = x <=> T
% Assm: h4/list/EXISTS__GENLIST: !n f P. h4/list/EXISTS P (h4/list/GENLIST f n) <=> (?i. h4/prim__rec/_3C i n /\ P (f i))
% Assm: h4/fcp/FCP__EXISTS__def: !v P. h4/fcp/FCP__EXISTS P v <=> (?i. h4/prim__rec/_3C i (h4/fcp/dimindex h4/bool/the__value) /\ P (h4/fcp/fcp__index v i))
% Assm: h4/fcp/V2L__def: !v. h4/fcp/V2L v = h4/list/GENLIST (h4/fcp/fcp__index v) (h4/fcp/dimindex h4/bool/the__value)
% Goal: !v P. h4/fcp/FCP__EXISTS P v <=> h4/list/EXISTS P (h4/fcp/V2L v)
%   PROCESSED
% Assm [HLu_TRUTH]: T
% Assm [HLu_FALSITY]: ~F
% Assm [HLu_BOOLu_CASES]: !t. (t <=> T) \/ (t <=> F)
% Assm [HLu_EXT]: !f g. (!x. happ f x = happ g x) ==> f = g
% Assm [h4s_bools_TRUTH]: T
% Assm [h4s_bools_REFLu_u_CLAUSE]: !x. x = x <=> T
% Assm [h4s_lists_EXISTSu_u_GENLIST]: !n f P. h4/list/EXISTS P (h4/list/GENLIST f n) <=> (?i. h4/prim__rec/_3C i n /\ happ P (happ f i))
% Assm [h4s_fcps_FCPu_u_EXISTSu_u_def]: !v P. h4/fcp/FCP__EXISTS P v <=> (?i. h4/prim__rec/_3C i (h4/fcp/dimindex h4/bool/the__value) /\ happ P (happ (h4/fcp/fcp__index v) i))
% Assm [h4s_fcps_V2Lu_u_def]: !v. h4/fcp/V2L v = h4/list/GENLIST (h4/fcp/fcp__index v) (h4/fcp/dimindex h4/bool/the__value)
% Goal: !v P. h4/fcp/FCP__EXISTS P v <=> h4/list/EXISTS P (h4/fcp/V2L v)
fof(aHLu_TRUTH, axiom, p(s(t_bool,t))).
fof(aHLu_FALSITY, axiom, ~ (p(s(t_bool,f)))).
fof(aHLu_BOOLu_CASES, axiom, ![V_t]: (s(t_bool,V_t) = s(t_bool,t) | s(t_bool,V_t) = s(t_bool,f))).
fof(aHLu_EXT, axiom, ![TV_Q191089,TV_Q191085]: ![V_f, V_g]: (![V_x]: s(TV_Q191085,happ(s(t_fun(TV_Q191089,TV_Q191085),V_f),s(TV_Q191089,V_x))) = s(TV_Q191085,happ(s(t_fun(TV_Q191089,TV_Q191085),V_g),s(TV_Q191089,V_x))) => s(t_fun(TV_Q191089,TV_Q191085),V_f) = s(t_fun(TV_Q191089,TV_Q191085),V_g))).
fof(ah4s_bools_TRUTH, axiom, p(s(t_bool,t))).
fof(ah4s_bools_REFLu_u_CLAUSE, axiom, ![TV_u_27a]: ![V_x]: (s(TV_u_27a,V_x) = s(TV_u_27a,V_x) <=> p(s(t_bool,t)))).
fof(ah4s_lists_EXISTSu_u_GENLIST, axiom, ![TV_u_27a]: ![V_n, V_f, V_P]: (p(s(t_bool,h4s_lists_exists(s(t_fun(TV_u_27a,t_bool),V_P),s(t_h4s_lists_list(TV_u_27a),h4s_lists_genlist(s(t_fun(t_h4s_nums_num,TV_u_27a),V_f),s(t_h4s_nums_num,V_n)))))) <=> ?[V_i]: (p(s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,V_i),s(t_h4s_nums_num,V_n)))) & p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,happ(s(t_fun(t_h4s_nums_num,TV_u_27a),V_f),s(t_h4s_nums_num,V_i))))))))).
fof(ah4s_fcps_FCPu_u_EXISTSu_u_def, axiom, ![TV_u_27b,TV_u_27a]: ![V_v, V_P]: (p(s(t_bool,h4s_fcps_fcpu_u_exists(s(t_fun(TV_u_27b,t_bool),V_P),s(t_h4s_fcps_cart(TV_u_27b,TV_u_27a),V_v)))) <=> ?[V_i]: (p(s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,V_i),s(t_h4s_nums_num,h4s_fcps_dimindex(s(t_h4s_bools_itself(TV_u_27a),h4s_bools_theu_u_value)))))) & p(s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),V_P),s(TV_u_27b,happ(s(t_fun(t_h4s_nums_num,TV_u_27b),h4s_fcps_fcpu_u_index(s(t_h4s_fcps_cart(TV_u_27b,TV_u_27a),V_v))),s(t_h4s_nums_num,V_i))))))))).
fof(ah4s_fcps_V2Lu_u_def, axiom, ![TV_u_27a,TV_u_27b]: ![V_v]: s(t_h4s_lists_list(TV_u_27a),h4s_fcps_v2l(s(t_h4s_fcps_cart(TV_u_27a,TV_u_27b),V_v))) = s(t_h4s_lists_list(TV_u_27a),h4s_lists_genlist(s(t_fun(t_h4s_nums_num,TV_u_27a),h4s_fcps_fcpu_u_index(s(t_h4s_fcps_cart(TV_u_27a,TV_u_27b),V_v))),s(t_h4s_nums_num,h4s_fcps_dimindex(s(t_h4s_bools_itself(TV_u_27b),h4s_bools_theu_u_value)))))).
fof(ch4s_fcps_FCPu_u_EXISTS0, conjecture, ![TV_u_27a,TV_u_27b]: ![V_v, V_P]: s(t_bool,h4s_fcps_fcpu_u_exists(s(t_fun(TV_u_27a,t_bool),V_P),s(t_h4s_fcps_cart(TV_u_27a,TV_u_27b),V_v))) = s(t_bool,h4s_lists_exists(s(t_fun(TV_u_27a,t_bool),V_P),s(t_h4s_lists_list(TV_u_27a),h4s_fcps_v2l(s(t_h4s_fcps_cart(TV_u_27a,TV_u_27b),V_v)))))).
