%   ORIGINAL: h4/container/EVERY__LIST__TO__BAG
% 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/ETA__AX: !t. (\x. t x) = t
% Assm: h4/bool/TRUTH: T
% Assm: h4/bool/REFL__CLAUSE: !x. x = x <=> T
% Assm: h4/bool/EQ__CLAUSES_c0: !t. (T <=> t) <=> t
% Assm: h4/bool/EQ__CLAUSES_c1: !t. (t <=> T) <=> t
% Assm: h4/list/EVERY__DEF_c0: !P. h4/list/EVERY P h4/list/NIL <=> T
% Assm: h4/list/EVERY__DEF_c1: !t h P. h4/list/EVERY P (h4/list/CONS h t) <=> P h /\ h4/list/EVERY P t
% Assm: h4/list/list__induction: !P. P h4/list/NIL /\ (!t. P t ==> (!h. P (h4/list/CONS h t))) ==> (!l. P l)
% Assm: h4/bag/BAG__EVERY__THM_c0: !P. h4/bag/BAG__EVERY P h4/bag/EMPTY__BAG
% Assm: h4/bag/BAG__EVERY__THM_c1: !e b P. h4/bag/BAG__EVERY P (h4/bag/BAG__INSERT e b) <=> P e /\ h4/bag/BAG__EVERY P b
% Assm: h4/container/LIST__TO__BAG__def_c0: h4/container/LIST__TO__BAG h4/list/NIL = h4/bag/EMPTY__BAG
% Assm: h4/container/LIST__TO__BAG__def_c1: !t h. h4/container/LIST__TO__BAG (h4/list/CONS h t) = h4/bag/BAG__INSERT h (h4/container/LIST__TO__BAG t)
% Goal: !ls P. h4/bag/BAG__EVERY P (h4/container/LIST__TO__BAG ls) <=> h4/list/EVERY P ls
%   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_ETAu_u_AX]: !t x. happ t x = happ t x
% Assm [h4s_bools_TRUTH]: T
% Assm [h4s_bools_REFLu_u_CLAUSE]: !x. x = x <=> T
% Assm [h4s_bools_EQu_u_CLAUSESu_c0]: !t. (T <=> t) <=> t
% Assm [h4s_bools_EQu_u_CLAUSESu_c1]: !t. (t <=> T) <=> t
% Assm [h4s_lists_EVERYu_u_DEFu_c0]: !P. h4/list/EVERY P h4/list/NIL <=> T
% Assm [h4s_lists_EVERYu_u_DEFu_c1]: !t h P. h4/list/EVERY P (h4/list/CONS h t) <=> happ P h /\ h4/list/EVERY P t
% Assm [h4s_lists_listu_u_induction]: !P. happ P h4/list/NIL /\ (!t. happ P t ==> (!h. happ P (h4/list/CONS h t))) ==> (!l. happ P l)
% Assm [h4s_bags_BAGu_u_EVERYu_u_THMu_c0]: !P. h4/bag/BAG__EVERY P h4/bag/EMPTY__BAG
% Assm [h4s_bags_BAGu_u_EVERYu_u_THMu_c1]: !e b P. h4/bag/BAG__EVERY P (h4/bag/BAG__INSERT e b) <=> happ P e /\ h4/bag/BAG__EVERY P b
% Assm [h4s_containers_LISTu_u_TOu_u_BAGu_u_defu_c0]: h4/container/LIST__TO__BAG h4/list/NIL = h4/bag/EMPTY__BAG
% Assm [h4s_containers_LISTu_u_TOu_u_BAGu_u_defu_c1]: !t h. h4/container/LIST__TO__BAG (h4/list/CONS h t) = h4/bag/BAG__INSERT h (h4/container/LIST__TO__BAG t)
% Goal: !ls P. h4/bag/BAG__EVERY P (h4/container/LIST__TO__BAG ls) <=> h4/list/EVERY P ls
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_Q300011,TV_Q300007]: ![V_f, V_g]: (![V_x]: s(TV_Q300007,happ(s(t_fun(TV_Q300011,TV_Q300007),V_f),s(TV_Q300011,V_x))) = s(TV_Q300007,happ(s(t_fun(TV_Q300011,TV_Q300007),V_g),s(TV_Q300011,V_x))) => s(t_fun(TV_Q300011,TV_Q300007),V_f) = s(t_fun(TV_Q300011,TV_Q300007),V_g))).
fof(ah4s_bools_ETAu_u_AX, axiom, ![TV_u_27b,TV_u_27a]: ![V_t, V_x]: s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_t),s(TV_u_27a,V_x))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_t),s(TV_u_27a,V_x)))).
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_bools_EQu_u_CLAUSESu_c0, axiom, ![V_t]: (s(t_bool,t) = s(t_bool,V_t) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_EQu_u_CLAUSESu_c1, axiom, ![V_t]: (s(t_bool,V_t) = s(t_bool,t) <=> p(s(t_bool,V_t)))).
fof(ah4s_lists_EVERYu_u_DEFu_c0, axiom, ![TV_u_27a]: ![V_P]: s(t_bool,h4s_lists_every(s(t_fun(TV_u_27a,t_bool),V_P),s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil))) = s(t_bool,t)).
fof(ah4s_lists_EVERYu_u_DEFu_c1, axiom, ![TV_u_27a]: ![V_t, V_h, V_P]: (p(s(t_bool,h4s_lists_every(s(t_fun(TV_u_27a,t_bool),V_P),s(t_h4s_lists_list(TV_u_27a),h4s_lists_cons(s(TV_u_27a,V_h),s(t_h4s_lists_list(TV_u_27a),V_t)))))) <=> (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_h)))) & p(s(t_bool,h4s_lists_every(s(t_fun(TV_u_27a,t_bool),V_P),s(t_h4s_lists_list(TV_u_27a),V_t))))))).
fof(ah4s_lists_listu_u_induction, axiom, ![TV_u_27a]: ![V_P]: ((p(s(t_bool,happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_bool),V_P),s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil)))) & ![V_t]: (p(s(t_bool,happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_bool),V_P),s(t_h4s_lists_list(TV_u_27a),V_t)))) => ![V_h]: p(s(t_bool,happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_bool),V_P),s(t_h4s_lists_list(TV_u_27a),h4s_lists_cons(s(TV_u_27a,V_h),s(t_h4s_lists_list(TV_u_27a),V_t)))))))) => ![V_l]: p(s(t_bool,happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_bool),V_P),s(t_h4s_lists_list(TV_u_27a),V_l)))))).
fof(ah4s_bags_BAGu_u_EVERYu_u_THMu_c0, axiom, ![TV_u_27a]: ![V_P]: p(s(t_bool,h4s_bags_bagu_u_every(s(t_fun(TV_u_27a,t_bool),V_P),s(t_fun(TV_u_27a,t_h4s_nums_num),h4s_bags_emptyu_u_bag))))).
fof(ah4s_bags_BAGu_u_EVERYu_u_THMu_c1, axiom, ![TV_u_27b]: ![V_e, V_b, V_P]: (p(s(t_bool,h4s_bags_bagu_u_every(s(t_fun(TV_u_27b,t_bool),V_P),s(t_fun(TV_u_27b,t_h4s_nums_num),h4s_bags_bagu_u_insert(s(TV_u_27b,V_e),s(t_fun(TV_u_27b,t_h4s_nums_num),V_b)))))) <=> (p(s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),V_P),s(TV_u_27b,V_e)))) & p(s(t_bool,h4s_bags_bagu_u_every(s(t_fun(TV_u_27b,t_bool),V_P),s(t_fun(TV_u_27b,t_h4s_nums_num),V_b))))))).
fof(ah4s_containers_LISTu_u_TOu_u_BAGu_u_defu_c0, axiom, ![TV_u_27a]: s(t_fun(TV_u_27a,t_h4s_nums_num),h4s_containers_listu_u_tou_u_bag(s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil))) = s(t_fun(TV_u_27a,t_h4s_nums_num),h4s_bags_emptyu_u_bag)).
fof(ah4s_containers_LISTu_u_TOu_u_BAGu_u_defu_c1, axiom, ![TV_u_27a]: ![V_t, V_h]: s(t_fun(TV_u_27a,t_h4s_nums_num),h4s_containers_listu_u_tou_u_bag(s(t_h4s_lists_list(TV_u_27a),h4s_lists_cons(s(TV_u_27a,V_h),s(t_h4s_lists_list(TV_u_27a),V_t))))) = s(t_fun(TV_u_27a,t_h4s_nums_num),h4s_bags_bagu_u_insert(s(TV_u_27a,V_h),s(t_fun(TV_u_27a,t_h4s_nums_num),h4s_containers_listu_u_tou_u_bag(s(t_h4s_lists_list(TV_u_27a),V_t)))))).
fof(ch4s_containers_EVERYu_u_LISTu_u_TOu_u_BAG, conjecture, ![TV_u_27a]: ![V_ls, V_P]: s(t_bool,h4s_bags_bagu_u_every(s(t_fun(TV_u_27a,t_bool),V_P),s(t_fun(TV_u_27a,t_h4s_nums_num),h4s_containers_listu_u_tou_u_bag(s(t_h4s_lists_list(TV_u_27a),V_ls))))) = s(t_bool,h4s_lists_every(s(t_fun(TV_u_27a,t_bool),V_P),s(t_h4s_lists_list(TV_u_27a),V_ls)))).
