%   ORIGINAL: h4/update/LIST__UPDATE__THMS_c6
% 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/combin/o__THM: !x g f. h4/combin/o f g x = f (g x)
% Assm: h4/combin/I__THM: !x. h4/combin/I x = x
% Assm: h4/list/APPEND0_c0: !l. h4/list/APPEND h4/list/NIL l = l
% Assm: h4/list/APPEND0_c1: !l2 l1 h. h4/list/APPEND (h4/list/CONS h l1) l2 = h4/list/CONS h (h4/list/APPEND l1 l2)
% 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/update/LIST__UPDATE__def_c0: h4/update/LIST__UPDATE h4/list/NIL = h4/combin/I
% Assm: h4/update/LIST__UPDATE__def_c1: !t h. h4/update/LIST__UPDATE (h4/list/CONS h t) = h4/combin/o (h4/combin/UPDATE (h4/pair/FST h) (h4/pair/SND h)) (h4/update/LIST__UPDATE t)
% Goal: !l2 l1 f. h4/update/LIST__UPDATE l1 (h4/update/LIST__UPDATE l2 f) = h4/update/LIST__UPDATE (h4/list/APPEND l1 l2) f
%   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_combins_ou_u_THM]: !x g f. happ (h4/combin/o f g) x = happ f (happ g x)
% Assm [h4s_combins_Iu_u_THM]: !x. happ h4/combin/I x = x
% Assm [h4s_lists_APPEND0u_c0]: !l. h4/list/APPEND h4/list/NIL l = l
% Assm [h4s_lists_APPEND0u_c1]: !l2 l1 h. h4/list/APPEND (h4/list/CONS h l1) l2 = h4/list/CONS h (h4/list/APPEND l1 l2)
% 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_updates_LISTu_u_UPDATEu_u_defu_c0]: h4/update/LIST__UPDATE h4/list/NIL = h4/combin/I
% Assm [h4s_updates_LISTu_u_UPDATEu_u_defu_c1]: !t h. h4/update/LIST__UPDATE (h4/list/CONS h t) = h4/combin/o (h4/combin/UPDATE (h4/pair/FST h) (h4/pair/SND h)) (h4/update/LIST__UPDATE t)
% Goal: !l2 l1 f. happ (h4/update/LIST__UPDATE l1) (happ (h4/update/LIST__UPDATE l2) f) = happ (h4/update/LIST__UPDATE (h4/list/APPEND l1 l2)) f
fof(aHLu_TRUTH, axiom, p(s(t_bool,t))).
fof(aHLu_FALSITY, axiom, ~ (p(s(t_bool,f0)))).
fof(aHLu_BOOLu_CASES, axiom, ![V_t]: (s(t_bool,V_t) = s(t_bool,t) | s(t_bool,V_t) = s(t_bool,f0))).
fof(aHLu_EXT, axiom, ![TV_Q213450,TV_Q213446]: ![V_f, V_g]: (![V_x]: s(TV_Q213446,happ(s(t_fun(TV_Q213450,TV_Q213446),V_f),s(TV_Q213450,V_x))) = s(TV_Q213446,happ(s(t_fun(TV_Q213450,TV_Q213446),V_g),s(TV_Q213450,V_x))) => s(t_fun(TV_Q213450,TV_Q213446),V_f) = s(t_fun(TV_Q213450,TV_Q213446),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_combins_ou_u_THM, axiom, ![TV_u_27b,TV_u_27a,TV_u_27c]: ![V_x, V_g, V_f]: s(TV_u_27b,happ(s(t_fun(TV_u_27c,TV_u_27b),h4s_combins_o(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_fun(TV_u_27c,TV_u_27a),V_g))),s(TV_u_27c,V_x))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,happ(s(t_fun(TV_u_27c,TV_u_27a),V_g),s(TV_u_27c,V_x)))))).
fof(ah4s_combins_Iu_u_THM, axiom, ![TV_u_27a]: ![V_x]: s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),h4s_combins_i),s(TV_u_27a,V_x))) = s(TV_u_27a,V_x)).
fof(ah4s_lists_APPEND0u_c0, axiom, ![TV_u_27a]: ![V_l]: s(t_h4s_lists_list(TV_u_27a),h4s_lists_append(s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil),s(t_h4s_lists_list(TV_u_27a),V_l))) = s(t_h4s_lists_list(TV_u_27a),V_l)).
fof(ah4s_lists_APPEND0u_c1, axiom, ![TV_u_27a]: ![V_l2, V_l1, V_h]: s(t_h4s_lists_list(TV_u_27a),h4s_lists_append(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_l1))),s(t_h4s_lists_list(TV_u_27a),V_l2))) = 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),h4s_lists_append(s(t_h4s_lists_list(TV_u_27a),V_l1),s(t_h4s_lists_list(TV_u_27a),V_l2)))))).
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_updates_LISTu_u_UPDATEu_u_defu_c0, axiom, ![TV_u_27a,TV_u_27b]: s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(TV_u_27a,TV_u_27b)),h4s_updates_listu_u_update(s(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),h4s_lists_nil))) = s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(TV_u_27a,TV_u_27b)),h4s_combins_i)).
fof(ah4s_updates_LISTu_u_UPDATEu_u_defu_c1, axiom, ![TV_u_27a,TV_u_27b]: ![V_t, V_h]: s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(TV_u_27a,TV_u_27b)),h4s_updates_listu_u_update(s(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),h4s_lists_cons(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_h),s(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),V_t))))) = s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(TV_u_27a,TV_u_27b)),h4s_combins_o(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(TV_u_27a,TV_u_27b)),h4s_combins_update(s(TV_u_27a,h4s_pairs_fst(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_h))),s(TV_u_27b,h4s_pairs_snd(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_h))))),s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(TV_u_27a,TV_u_27b)),h4s_updates_listu_u_update(s(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),V_t)))))).
fof(ch4s_updates_LISTu_u_UPDATEu_u_THMSu_c6, conjecture, ![TV_u_27e,TV_u_27f]: ![V_l2, V_l1, V_f]: s(t_fun(TV_u_27e,TV_u_27f),happ(s(t_fun(t_fun(TV_u_27e,TV_u_27f),t_fun(TV_u_27e,TV_u_27f)),h4s_updates_listu_u_update(s(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27e,TV_u_27f)),V_l1))),s(t_fun(TV_u_27e,TV_u_27f),happ(s(t_fun(t_fun(TV_u_27e,TV_u_27f),t_fun(TV_u_27e,TV_u_27f)),h4s_updates_listu_u_update(s(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27e,TV_u_27f)),V_l2))),s(t_fun(TV_u_27e,TV_u_27f),V_f))))) = s(t_fun(TV_u_27e,TV_u_27f),happ(s(t_fun(t_fun(TV_u_27e,TV_u_27f),t_fun(TV_u_27e,TV_u_27f)),h4s_updates_listu_u_update(s(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27e,TV_u_27f)),h4s_lists_append(s(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27e,TV_u_27f)),V_l1),s(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27e,TV_u_27f)),V_l2))))),s(t_fun(TV_u_27e,TV_u_27f),V_f)))).
