%   ORIGINAL: h4/sptree/domain__foldi
% 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/arithmetic/ADD__0: !m. h4/arithmetic/_2B m h4/num/0 = m
% Assm: h4/arithmetic/ADD__SYM: !n m. h4/arithmetic/_2B m n = h4/arithmetic/_2B n m
% Assm: h4/arithmetic/MULT__LEFT__1: !m. h4/arithmetic/_2A (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) m = m
% Assm: h4/pred__set/UNION__EMPTY_c0: !s. h4/pred__set/UNION h4/pred__set/EMPTY s = s
% Assm: h4/pred__set/IMAGE__ID: !s. h4/pred__set/IMAGE (\x. x) s = s
% Assm: h4/sptree/lrnext__thm_c0: h4/sptree/lrnext h4/num/0 = h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)
% Assm: h4/sptree/set__foldi__keys: !t i a. h4/sptree/foldi (\k v a0. h4/pred__set/INSERT k a0) i a t = h4/pred__set/UNION a (h4/pred__set/IMAGE (\n. h4/arithmetic/_2B i (h4/arithmetic/_2A (h4/sptree/lrnext i) n)) (h4/sptree/domain t))
% Goal: !t. h4/sptree/domain t = h4/sptree/foldi (\k v a. h4/pred__set/INSERT k a) h4/num/0 h4/pred__set/EMPTY t
%   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_arithmetics_ADDu_u_0]: !m. h4/arithmetic/_2B m h4/num/0 = m
% Assm [h4s_arithmetics_ADDu_u_SYM]: !n m. h4/arithmetic/_2B m n = h4/arithmetic/_2B n m
% Assm [h4s_arithmetics_MULTu_u_LEFTu_u_1]: !m. h4/arithmetic/_2A (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) m = m
% Assm [h4s_predu_u_sets_UNIONu_u_EMPTYu_c0]: !s. h4/pred__set/UNION h4/pred__set/EMPTY s = s
% Assm [h4s_predu_u_sets_IMAGEu_u_ID]: !_0. (!x. happ _0 x = x) ==> (!s. h4/pred__set/IMAGE _0 s = s)
% Assm [h4s_sptrees_lrnextu_u_thmu_c0]: h4/sptree/lrnext h4/num/0 = h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)
% Assm [h4s_sptrees_setu_u_foldiu_u_keys]: !_3. (!i n. happ (happ _3 i) n = h4/arithmetic/_2B i (h4/arithmetic/_2A (h4/sptree/lrnext i) n)) ==> (!_2. (!k a0. happ (happ _2 k) a0 = h4/pred__set/INSERT k a0) ==> (!_1. (!k v. happ (happ _1 k) v = happ _2 k) ==> (!_0. (!k. happ _0 k = happ _1 k) ==> (!t i a. h4/sptree/foldi _0 i a t = h4/pred__set/UNION a (h4/pred__set/IMAGE (happ _3 i) (h4/sptree/domain t))))))
% Goal: !_2. (!k a. happ (happ _2 k) a = h4/pred__set/INSERT k a) ==> (!_1. (!k v. happ (happ _1 k) v = happ _2 k) ==> (!_0. (!k. happ _0 k = happ _1 k) ==> (!t. h4/sptree/domain t = h4/sptree/foldi _0 h4/num/0 h4/pred__set/EMPTY t)))
fof(aHLu_TRUTH, axiom, p(s(t_bool,t0))).
fof(aHLu_FALSITY, axiom, ~ (p(s(t_bool,f)))).
fof(aHLu_BOOLu_CASES, axiom, ![V_t]: (s(t_bool,V_t) = s(t_bool,t0) | s(t_bool,V_t) = s(t_bool,f))).
fof(aHLu_EXT, axiom, ![TV_Q82265,TV_Q82261]: ![V_f, V_g]: (![V_x]: s(TV_Q82261,happ(s(t_fun(TV_Q82265,TV_Q82261),V_f),s(TV_Q82265,V_x))) = s(TV_Q82261,happ(s(t_fun(TV_Q82265,TV_Q82261),V_g),s(TV_Q82265,V_x))) => s(t_fun(TV_Q82265,TV_Q82261),V_f) = s(t_fun(TV_Q82265,TV_Q82261),V_g))).
fof(ah4s_arithmetics_ADDu_u_0, axiom, ![V_m]: s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,h4s_nums_0))) = s(t_h4s_nums_num,V_m)).
fof(ah4s_arithmetics_ADDu_u_SYM, axiom, ![V_n, V_m]: s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n))) = s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_m)))).
fof(ah4s_arithmetics_MULTu_u_LEFTu_u_1, axiom, ![V_m]: s(t_h4s_nums_num,h4s_arithmetics_u_2a(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero))))),s(t_h4s_nums_num,V_m))) = s(t_h4s_nums_num,V_m)).
fof(ah4s_predu_u_sets_UNIONu_u_EMPTYu_c0, axiom, ![TV_u_27a]: ![V_s]: s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_union(s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_empty),s(t_fun(TV_u_27a,t_bool),V_s))) = s(t_fun(TV_u_27a,t_bool),V_s)).
fof(ah4s_predu_u_sets_IMAGEu_u_ID, axiom, ![TV_u_27a]: ![V_uu_0]: (![V_x]: s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),V_uu_0),s(TV_u_27a,V_x))) = s(TV_u_27a,V_x) => ![V_s]: s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_image(s(t_fun(TV_u_27a,TV_u_27a),V_uu_0),s(t_fun(TV_u_27a,t_bool),V_s))) = s(t_fun(TV_u_27a,t_bool),V_s))).
fof(ah4s_sptrees_lrnextu_u_thmu_c0, axiom, s(t_h4s_nums_num,h4s_sptrees_lrnext(s(t_h4s_nums_num,h4s_nums_0))) = s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero)))))).
fof(ah4s_sptrees_setu_u_foldiu_u_keys, axiom, ![TV_u_27a]: ![V_uu_3]: (![V_i, V_n]: s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_nums_num)),V_uu_3),s(t_h4s_nums_num,V_i))),s(t_h4s_nums_num,V_n))) = s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_i),s(t_h4s_nums_num,h4s_arithmetics_u_2a(s(t_h4s_nums_num,h4s_sptrees_lrnext(s(t_h4s_nums_num,V_i))),s(t_h4s_nums_num,V_n))))) => ![V_uu_2]: (![V_k, V_a0]: s(t_fun(t_h4s_nums_num,t_bool),happ(s(t_fun(t_fun(t_h4s_nums_num,t_bool),t_fun(t_h4s_nums_num,t_bool)),happ(s(t_fun(t_h4s_nums_num,t_fun(t_fun(t_h4s_nums_num,t_bool),t_fun(t_h4s_nums_num,t_bool))),V_uu_2),s(t_h4s_nums_num,V_k))),s(t_fun(t_h4s_nums_num,t_bool),V_a0))) = 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),V_a0))) => ![V_uu_1]: (![V_k, V_v]: s(t_fun(t_fun(t_h4s_nums_num,t_bool),t_fun(t_h4s_nums_num,t_bool)),happ(s(t_fun(TV_u_27a,t_fun(t_fun(t_h4s_nums_num,t_bool),t_fun(t_h4s_nums_num,t_bool))),happ(s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_fun(t_fun(t_h4s_nums_num,t_bool),t_fun(t_h4s_nums_num,t_bool)))),V_uu_1),s(t_h4s_nums_num,V_k))),s(TV_u_27a,V_v))) = s(t_fun(t_fun(t_h4s_nums_num,t_bool),t_fun(t_h4s_nums_num,t_bool)),happ(s(t_fun(t_h4s_nums_num,t_fun(t_fun(t_h4s_nums_num,t_bool),t_fun(t_h4s_nums_num,t_bool))),V_uu_2),s(t_h4s_nums_num,V_k))) => ![V_uu_0]: (![V_k]: s(t_fun(TV_u_27a,t_fun(t_fun(t_h4s_nums_num,t_bool),t_fun(t_h4s_nums_num,t_bool))),happ(s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_fun(t_fun(t_h4s_nums_num,t_bool),t_fun(t_h4s_nums_num,t_bool)))),V_uu_0),s(t_h4s_nums_num,V_k))) = s(t_fun(TV_u_27a,t_fun(t_fun(t_h4s_nums_num,t_bool),t_fun(t_h4s_nums_num,t_bool))),happ(s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_fun(t_fun(t_h4s_nums_num,t_bool),t_fun(t_h4s_nums_num,t_bool)))),V_uu_1),s(t_h4s_nums_num,V_k))) => ![V_t, V_i, V_a]: s(t_fun(t_h4s_nums_num,t_bool),h4s_sptrees_foldi(s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_fun(t_fun(t_h4s_nums_num,t_bool),t_fun(t_h4s_nums_num,t_bool)))),V_uu_0),s(t_h4s_nums_num,V_i),s(t_fun(t_h4s_nums_num,t_bool),V_a),s(t_h4s_sptrees_spt(TV_u_27a),V_t))) = s(t_fun(t_h4s_nums_num,t_bool),h4s_predu_u_sets_union(s(t_fun(t_h4s_nums_num,t_bool),V_a),s(t_fun(t_h4s_nums_num,t_bool),h4s_predu_u_sets_image(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_nums_num)),V_uu_3),s(t_h4s_nums_num,V_i))),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_foldi, conjecture, ![TV_u_27a]: ![V_uu_2]: (![V_k, V_a]: s(t_fun(t_h4s_nums_num,t_bool),happ(s(t_fun(t_fun(t_h4s_nums_num,t_bool),t_fun(t_h4s_nums_num,t_bool)),happ(s(t_fun(t_h4s_nums_num,t_fun(t_fun(t_h4s_nums_num,t_bool),t_fun(t_h4s_nums_num,t_bool))),V_uu_2),s(t_h4s_nums_num,V_k))),s(t_fun(t_h4s_nums_num,t_bool),V_a))) = 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),V_a))) => ![V_uu_1]: (![V_k, V_v]: s(t_fun(t_fun(t_h4s_nums_num,t_bool),t_fun(t_h4s_nums_num,t_bool)),happ(s(t_fun(TV_u_27a,t_fun(t_fun(t_h4s_nums_num,t_bool),t_fun(t_h4s_nums_num,t_bool))),happ(s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_fun(t_fun(t_h4s_nums_num,t_bool),t_fun(t_h4s_nums_num,t_bool)))),V_uu_1),s(t_h4s_nums_num,V_k))),s(TV_u_27a,V_v))) = s(t_fun(t_fun(t_h4s_nums_num,t_bool),t_fun(t_h4s_nums_num,t_bool)),happ(s(t_fun(t_h4s_nums_num,t_fun(t_fun(t_h4s_nums_num,t_bool),t_fun(t_h4s_nums_num,t_bool))),V_uu_2),s(t_h4s_nums_num,V_k))) => ![V_uu_0]: (![V_k]: s(t_fun(TV_u_27a,t_fun(t_fun(t_h4s_nums_num,t_bool),t_fun(t_h4s_nums_num,t_bool))),happ(s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_fun(t_fun(t_h4s_nums_num,t_bool),t_fun(t_h4s_nums_num,t_bool)))),V_uu_0),s(t_h4s_nums_num,V_k))) = s(t_fun(TV_u_27a,t_fun(t_fun(t_h4s_nums_num,t_bool),t_fun(t_h4s_nums_num,t_bool))),happ(s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_fun(t_fun(t_h4s_nums_num,t_bool),t_fun(t_h4s_nums_num,t_bool)))),V_uu_1),s(t_h4s_nums_num,V_k))) => ![V_t]: s(t_fun(t_h4s_nums_num,t_bool),h4s_sptrees_domain(s(t_h4s_sptrees_spt(TV_u_27a),V_t))) = s(t_fun(t_h4s_nums_num,t_bool),h4s_sptrees_foldi(s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_fun(t_fun(t_h4s_nums_num,t_bool),t_fun(t_h4s_nums_num,t_bool)))),V_uu_0),s(t_h4s_nums_num,h4s_nums_0),s(t_fun(t_h4s_nums_num,t_bool),h4s_predu_u_sets_empty),s(t_h4s_sptrees_spt(TV_u_27a),V_t))))))).
