%   ORIGINAL: h4/update/LIST__UPDATE__THMS_c2
% 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/combin/o__THM: !x g f. h4/combin/o f g x = f (g x)
% Assm: h4/combin/I__o__ID_c1: !f. h4/combin/o f h4/combin/I = f
% Assm: h4/pair/FST0: !y x. h4/pair/FST (h4/pair/_2C x y) = x
% Assm: h4/pair/SND0: !y x. h4/pair/SND (h4/pair/_2C x y) = y
% 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: !r2 r1 l2 l1 f. h4/combin/UPDATE l1 r1 (h4/combin/UPDATE l2 r2 f) = h4/update/LIST__UPDATE (h4/list/CONS (h4/pair/_2C l1 r1) (h4/list/CONS (h4/pair/_2C l2 r2) h4/list/NIL)) 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_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_ou_u_IDu_c1]: !f. h4/combin/o f h4/combin/I = f
% Assm [h4s_pairs_FST0]: !y x. h4/pair/FST (h4/pair/_2C x y) = x
% Assm [h4s_pairs_SND0]: !y x. h4/pair/SND (h4/pair/_2C x y) = y
% 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: !r2 r1 l2 l1 f. happ (h4/combin/UPDATE l1 r1) (happ (h4/combin/UPDATE l2 r2) f) = happ (h4/update/LIST__UPDATE (h4/list/CONS (h4/pair/_2C l1 r1) (h4/list/CONS (h4/pair/_2C l2 r2) h4/list/NIL))) 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_Q213350,TV_Q213346]: ![V_f, V_g]: (![V_x]: s(TV_Q213346,happ(s(t_fun(TV_Q213350,TV_Q213346),V_f),s(TV_Q213350,V_x))) = s(TV_Q213346,happ(s(t_fun(TV_Q213350,TV_Q213346),V_g),s(TV_Q213350,V_x))) => s(t_fun(TV_Q213350,TV_Q213346),V_f) = s(t_fun(TV_Q213350,TV_Q213346),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_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_ou_u_IDu_c1, axiom, ![TV_u_27a,TV_u_27b]: ![V_f]: s(t_fun(TV_u_27a,TV_u_27b),h4s_combins_o(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_fun(TV_u_27a,TV_u_27a),h4s_combins_i))) = s(t_fun(TV_u_27a,TV_u_27b),V_f)).
fof(ah4s_pairs_FST0, axiom, ![TV_u_27b,TV_u_27a]: ![V_y, V_x]: s(TV_u_27a,h4s_pairs_fst(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(TV_u_27b,V_y))))) = s(TV_u_27a,V_x)).
fof(ah4s_pairs_SND0, axiom, ![TV_u_27a,TV_u_27b]: ![V_y, V_x]: s(TV_u_27b,h4s_pairs_snd(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(TV_u_27b,V_y))))) = s(TV_u_27b,V_y)).
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_c2, conjecture, ![TV_u_27e,TV_u_27f]: ![V_r2, V_r1, 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_combins_update(s(TV_u_27e,V_l1),s(TV_u_27f,V_r1))),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_combins_update(s(TV_u_27e,V_l2),s(TV_u_27f,V_r2))),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_cons(s(t_h4s_pairs_prod(TV_u_27e,TV_u_27f),h4s_pairs_u_2c(s(TV_u_27e,V_l1),s(TV_u_27f,V_r1))),s(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27e,TV_u_27f)),h4s_lists_cons(s(t_h4s_pairs_prod(TV_u_27e,TV_u_27f),h4s_pairs_u_2c(s(TV_u_27e,V_l2),s(TV_u_27f,V_r2))),s(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27e,TV_u_27f)),h4s_lists_nil))))))),s(t_fun(TV_u_27e,TV_u_27f),V_f)))).
