%   ORIGINAL: h4/EncodeVar/of__length__exists__zero
% 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/IMP__ANTISYM__AX: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm: h4/bool/FORALL__SIMP: !t. (!x. t) <=> t
% Assm: h4/bool/REFL__CLAUSE: !x. x = x <=> T
% Assm: h4/bool/UNWIND__THM2: !a P. (?x. x = a /\ P x) <=> P a
% Assm: h4/bool/RES__EXISTS__THM: !f P. h4/bool/RES__EXISTS P f <=> (?x. h4/bool/IN x P /\ f x)
% Assm: h4/list/LENGTH__NIL: !l. h4/list/LENGTH l = h4/num/0 <=> l = h4/list/NIL
% Assm: h4/EncodeVar/of__length__def: !n l. h4/bool/IN l (h4/EncodeVar/of__length n) <=> h4/list/LENGTH l = n
% Goal: !phi. h4/bool/RES__EXISTS (h4/EncodeVar/of__length h4/num/0) (\w. phi w) <=> phi h4/list/NIL
%   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_IMPu_u_ANTISYMu_u_AX]: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm [h4s_bools_FORALLu_u_SIMP]: !t. (!x. t) <=> t
% Assm [h4s_bools_REFLu_u_CLAUSE]: !x. x = x <=> T
% Assm [h4s_bools_UNWINDu_u_THM2]: !a P. (?x. x = a /\ happ P x) <=> happ P a
% Assm [h4s_bools_RESu_u_EXISTSu_u_THM]: !f P. h4/bool/RES__EXISTS P f <=> (?x. h4/bool/IN x P /\ happ f x)
% Assm [h4s_lists_LENGTHu_u_NIL]: !l. h4/list/LENGTH l = h4/num/0 <=> l = h4/list/NIL
% Assm [h4s_EncodeVars_ofu_u_lengthu_u_def]: !n l. h4/bool/IN l (h4/EncodeVar/of__length n) <=> h4/list/LENGTH l = n
% Goal: !_0. (!phi w. happ (happ _0 phi) w <=> happ phi w) ==> (!phi. h4/bool/RES__EXISTS (h4/EncodeVar/of__length h4/num/0) (happ _0 phi) <=> happ phi h4/list/NIL)
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_Q71920,TV_Q71916]: ![V_f, V_g]: (![V_x]: s(TV_Q71916,happ(s(t_fun(TV_Q71920,TV_Q71916),V_f),s(TV_Q71920,V_x))) = s(TV_Q71916,happ(s(t_fun(TV_Q71920,TV_Q71916),V_g),s(TV_Q71920,V_x))) => s(t_fun(TV_Q71920,TV_Q71916),V_f) = s(t_fun(TV_Q71920,TV_Q71916),V_g))).
fof(ah4s_bools_TRUTH, axiom, p(s(t_bool,t))).
fof(ah4s_bools_IMPu_u_ANTISYMu_u_AX, axiom, ![V_t2, V_t1]: ((p(s(t_bool,V_t1)) => p(s(t_bool,V_t2))) => ((p(s(t_bool,V_t2)) => p(s(t_bool,V_t1))) => s(t_bool,V_t1) = s(t_bool,V_t2)))).
fof(ah4s_bools_FORALLu_u_SIMP, axiom, ![TV_u_27a]: ![V_t]: (![V_x]: p(s(t_bool,V_t)) <=> p(s(t_bool,V_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_bools_UNWINDu_u_THM2, axiom, ![TV_u_27a]: ![V_a, V_P]: (?[V_x]: (s(TV_u_27a,V_x) = s(TV_u_27a,V_a) & p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x))))) <=> p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_a)))))).
fof(ah4s_bools_RESu_u_EXISTSu_u_THM, axiom, ![TV_u_27a]: ![V_f, V_P]: (p(s(t_bool,h4s_bools_resu_u_exists(s(t_fun(TV_u_27a,t_bool),V_P),s(t_fun(TV_u_27a,t_bool),V_f)))) <=> ?[V_x]: (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_P)))) & p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_f),s(TV_u_27a,V_x))))))).
fof(ah4s_lists_LENGTHu_u_NIL, axiom, ![TV_u_27a]: ![V_l]: (s(t_h4s_nums_num,h4s_lists_length(s(t_h4s_lists_list(TV_u_27a),V_l))) = s(t_h4s_nums_num,h4s_nums_0) <=> s(t_h4s_lists_list(TV_u_27a),V_l) = s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil))).
fof(ah4s_EncodeVars_ofu_u_lengthu_u_def, axiom, ![TV_u_27a]: ![V_n, V_l]: (p(s(t_bool,h4s_bools_in(s(t_h4s_lists_list(TV_u_27a),V_l),s(t_fun(t_h4s_lists_list(TV_u_27a),t_bool),h4s_encodevars_ofu_u_length(s(t_h4s_nums_num,V_n)))))) <=> s(t_h4s_nums_num,h4s_lists_length(s(t_h4s_lists_list(TV_u_27a),V_l))) = s(t_h4s_nums_num,V_n))).
fof(ch4s_EncodeVars_ofu_u_lengthu_u_existsu_u_zero, conjecture, ![TV_u_27a]: ![V_uu_0]: (![V_phi, V_w]: s(t_bool,happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_bool),happ(s(t_fun(t_fun(t_h4s_lists_list(TV_u_27a),t_bool),t_fun(t_h4s_lists_list(TV_u_27a),t_bool)),V_uu_0),s(t_fun(t_h4s_lists_list(TV_u_27a),t_bool),V_phi))),s(t_h4s_lists_list(TV_u_27a),V_w))) = s(t_bool,happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_bool),V_phi),s(t_h4s_lists_list(TV_u_27a),V_w))) => ![V_phi]: s(t_bool,h4s_bools_resu_u_exists(s(t_fun(t_h4s_lists_list(TV_u_27a),t_bool),h4s_encodevars_ofu_u_length(s(t_h4s_nums_num,h4s_nums_0))),s(t_fun(t_h4s_lists_list(TV_u_27a),t_bool),happ(s(t_fun(t_fun(t_h4s_lists_list(TV_u_27a),t_bool),t_fun(t_h4s_lists_list(TV_u_27a),t_bool)),V_uu_0),s(t_fun(t_h4s_lists_list(TV_u_27a),t_bool),V_phi))))) = s(t_bool,happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_bool),V_phi),s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil))))).
