%   ORIGINAL: h4/bag/BAG__MERGE__IDEM
% 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/FORALL__SIMP: !t. (!x. t) <=> t
% Assm: h4/bool/REFL__CLAUSE: !x. x = x <=> T
% Assm: h4/bool/FUN__EQ__THM: !g f. f = g <=> (!x. f x = g x)
% Assm: h4/bool/COND__ID: !t b. h4/bool/COND b t t = t
% Assm: h4/bag/BAG__MERGE0: !b2 b1. h4/bag/BAG__MERGE b1 b2 = (\x. h4/bool/COND (h4/prim__rec/_3C (b1 x) (b2 x)) (b2 x) (b1 x))
% Goal: !b. h4/bag/BAG__MERGE b b = b
%   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_FORALLu_u_SIMP]: !t. (!x. t) <=> t
% Assm [h4s_bools_REFLu_u_CLAUSE]: !x. x = x <=> T
% Assm [h4s_bools_FUNu_u_EQu_u_THM]: !g f. f = g <=> (!x. happ f x = happ g x)
% Assm [h4s_bools_CONDu_u_ID]: !t b. h4/bool/COND b t t = t
% Assm [h4s_bags_BAGu_u_MERGE0]: !b2 b1 x. happ (h4/bag/BAG__MERGE b1 b2) x = h4/bool/COND (h4/prim__rec/_3C (happ b1 x) (happ b2 x)) (happ b2 x) (happ b1 x)
% Goal: !b. h4/bag/BAG__MERGE b b = b
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_Q226759,TV_Q226755]: ![V_f, V_g]: (![V_x]: s(TV_Q226755,happ(s(t_fun(TV_Q226759,TV_Q226755),V_f),s(TV_Q226759,V_x))) = s(TV_Q226755,happ(s(t_fun(TV_Q226759,TV_Q226755),V_g),s(TV_Q226759,V_x))) => s(t_fun(TV_Q226759,TV_Q226755),V_f) = s(t_fun(TV_Q226759,TV_Q226755),V_g))).
fof(ah4s_bools_TRUTH, axiom, p(s(t_bool,t))).
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_FUNu_u_EQu_u_THM, axiom, ![TV_u_27b,TV_u_27a]: ![V_g, V_f]: (s(t_fun(TV_u_27a,TV_u_27b),V_f) = s(t_fun(TV_u_27a,TV_u_27b),V_g) <=> ![V_x]: s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_x))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_g),s(TV_u_27a,V_x))))).
fof(ah4s_bools_CONDu_u_ID, axiom, ![TV_u_27a]: ![V_t, V_b]: s(TV_u_27a,h4s_bools_cond(s(t_bool,V_b),s(TV_u_27a,V_t),s(TV_u_27a,V_t))) = s(TV_u_27a,V_t)).
fof(ah4s_bags_BAGu_u_MERGE0, axiom, ![TV_u_27a]: ![V_b2, V_b1, V_x]: s(t_h4s_nums_num,happ(s(t_fun(TV_u_27a,t_h4s_nums_num),h4s_bags_bagu_u_merge(s(t_fun(TV_u_27a,t_h4s_nums_num),V_b1),s(t_fun(TV_u_27a,t_h4s_nums_num),V_b2))),s(TV_u_27a,V_x))) = s(t_h4s_nums_num,h4s_bools_cond(s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,happ(s(t_fun(TV_u_27a,t_h4s_nums_num),V_b1),s(TV_u_27a,V_x))),s(t_h4s_nums_num,happ(s(t_fun(TV_u_27a,t_h4s_nums_num),V_b2),s(TV_u_27a,V_x))))),s(t_h4s_nums_num,happ(s(t_fun(TV_u_27a,t_h4s_nums_num),V_b2),s(TV_u_27a,V_x))),s(t_h4s_nums_num,happ(s(t_fun(TV_u_27a,t_h4s_nums_num),V_b1),s(TV_u_27a,V_x)))))).
fof(ch4s_bags_BAGu_u_MERGEu_u_IDEM, conjecture, ![TV_u_27a]: ![V_b]: s(t_fun(TV_u_27a,t_h4s_nums_num),h4s_bags_bagu_u_merge(s(t_fun(TV_u_27a,t_h4s_nums_num),V_b),s(t_fun(TV_u_27a,t_h4s_nums_num),V_b))) = s(t_fun(TV_u_27a,t_h4s_nums_num),V_b)).
