%   ORIGINAL: h4/sptree/domain__sing
% 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/sptree/domain__def_c0: h4/sptree/domain h4/sptree/LN = h4/pred__set/EMPTY
% Assm: h4/sptree/domain__insert: !v t k. h4/sptree/domain (h4/sptree/insert k v t) = h4/pred__set/INSERT k (h4/sptree/domain t)
% Goal: !v k. h4/sptree/domain (h4/sptree/insert k v h4/sptree/LN) = h4/pred__set/INSERT k h4/pred__set/EMPTY
%   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_sptrees_domainu_u_defu_c0]: h4/sptree/domain h4/sptree/LN = h4/pred__set/EMPTY
% Assm [h4s_sptrees_domainu_u_insert]: !v t k. h4/sptree/domain (h4/sptree/insert k v t) = h4/pred__set/INSERT k (h4/sptree/domain t)
% Goal: !v k. h4/sptree/domain (h4/sptree/insert k v h4/sptree/LN) = h4/pred__set/INSERT k h4/pred__set/EMPTY
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_Q82130,TV_Q82126]: ![V_f, V_g]: (![V_x]: s(TV_Q82126,happ(s(t_fun(TV_Q82130,TV_Q82126),V_f),s(TV_Q82130,V_x))) = s(TV_Q82126,happ(s(t_fun(TV_Q82130,TV_Q82126),V_g),s(TV_Q82130,V_x))) => s(t_fun(TV_Q82130,TV_Q82126),V_f) = s(t_fun(TV_Q82130,TV_Q82126),V_g))).
fof(ah4s_sptrees_domainu_u_defu_c0, axiom, ![TV_u_27a]: s(t_fun(t_h4s_nums_num,t_bool),h4s_sptrees_domain(s(t_h4s_sptrees_spt(TV_u_27a),h4s_sptrees_ln))) = s(t_fun(t_h4s_nums_num,t_bool),h4s_predu_u_sets_empty)).
fof(ah4s_sptrees_domainu_u_insert, axiom, ![TV_u_27a]: ![V_v, V_t, V_k]: s(t_fun(t_h4s_nums_num,t_bool),h4s_sptrees_domain(s(t_h4s_sptrees_spt(TV_u_27a),h4s_sptrees_insert(s(t_h4s_nums_num,V_k),s(TV_u_27a,V_v),s(t_h4s_sptrees_spt(TV_u_27a),V_t))))) = s(t_fun(t_h4s_nums_num,t_bool),h4s_predu_u_sets_insert(s(t_h4s_nums_num,V_k),s(t_fun(t_h4s_nums_num,t_bool),h4s_sptrees_domain(s(t_h4s_sptrees_spt(TV_u_27a),V_t)))))).
fof(ch4s_sptrees_domainu_u_sing, conjecture, ![TV_u_27a]: ![V_v, V_k]: s(t_fun(t_h4s_nums_num,t_bool),h4s_sptrees_domain(s(t_h4s_sptrees_spt(TV_u_27a),h4s_sptrees_insert(s(t_h4s_nums_num,V_k),s(TV_u_27a,V_v),s(t_h4s_sptrees_spt(TV_u_27a),h4s_sptrees_ln))))) = s(t_fun(t_h4s_nums_num,t_bool),h4s_predu_u_sets_insert(s(t_h4s_nums_num,V_k),s(t_fun(t_h4s_nums_num,t_bool),h4s_predu_u_sets_empty)))).
