%   ORIGINAL: h4/patricia/IN__PTREE__OF__NUMSET
% 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/AND__CLAUSES_c0: !t. T /\ t <=> t
% Assm: h4/bool/EQ__SYM__EQ: !y x. x = y <=> y = x
% Assm: h4/bool/EQ__CLAUSES_c1: !t. (t <=> T) <=> t
% Assm: h4/bool/AND__IMP__INTRO: !t3 t2 t1. t1 ==> t2 ==> t3 <=> t1 /\ t2 ==> t3
% Assm: h4/bool/IMP__CONG: !y_27 y x_27 x. (x <=> x_27) /\ (x_27 ==> (y <=> y_27)) ==> (x ==> y <=> x_27 ==> y_27)
% Assm: h4/pred__set/IN__UNION: !x t s. h4/bool/IN x (h4/pred__set/UNION s t) <=> h4/bool/IN x s \/ h4/bool/IN x t
% Assm: h4/pred__set/FINITE__UNION: !t s. h4/pred__set/FINITE (h4/pred__set/UNION s t) <=> h4/pred__set/FINITE s /\ h4/pred__set/FINITE t
% Assm: h4/list/SET__TO__LIST__IN__MEM: !s. h4/pred__set/FINITE s ==> (!x. h4/bool/IN x s <=> h4/bool/IN x (h4/list/LIST__TO__SET (h4/list/SET__TO__LIST s)))
% Assm: h4/sorting/PERM__MEM__EQ: !l2 l1. h4/sorting/PERM l1 l2 ==> (!x. h4/bool/IN x (h4/list/LIST__TO__SET l1) <=> h4/bool/IN x (h4/list/LIST__TO__SET l2))
% Assm: h4/patricia/IN__PTREE__def: !t n. h4/patricia/IN__PTREE n t <=> h4/option/IS__SOME (h4/patricia/PEEK t n)
% Assm: h4/patricia/PTREE__OF__NUMSET__def: !t s. h4/patricia/PTREE__OF__NUMSET t s = h4/list/FOLDL (h4/combin/C h4/patricia/INSERT__PTREE) t (h4/list/SET__TO__LIST s)
% Assm: h4/patricia/MEM__TRAVERSE__PEEK: !t k. h4/patricia/IS__PTREE t ==> (h4/bool/IN k (h4/list/LIST__TO__SET (h4/patricia/TRAVERSE t)) <=> h4/option/IS__SOME (h4/patricia/PEEK t k))
% Assm: h4/patricia/PTREE__OF__NUMSET__IS__PTREE: !t s. h4/patricia/IS__PTREE t ==> h4/patricia/IS__PTREE (h4/patricia/PTREE__OF__NUMSET t s)
% Assm: h4/patricia/MEM__TRAVERSE: !t k. h4/patricia/IS__PTREE t ==> (h4/bool/IN k (h4/list/LIST__TO__SET (h4/patricia/TRAVERSE t)) <=> h4/bool/IN k (h4/patricia/NUMSET__OF__PTREE t))
% Assm: h4/patricia/FINITE__NUMSET__OF__PTREE: !t. h4/pred__set/FINITE (h4/patricia/NUMSET__OF__PTREE t)
% Assm: h4/patricia/PERM__INSERT__PTREE: !t s. h4/pred__set/FINITE s ==> h4/patricia/IS__PTREE t ==> h4/sorting/PERM (h4/patricia/TRAVERSE (h4/list/FOLDL (h4/combin/C h4/patricia/INSERT__PTREE) t (h4/list/SET__TO__LIST s))) (h4/list/SET__TO__LIST (h4/pred__set/UNION (h4/patricia/NUMSET__OF__PTREE t) s))
% Goal: !t s n. h4/patricia/IS__PTREE t /\ h4/pred__set/FINITE s ==> (h4/patricia/IN__PTREE n (h4/patricia/PTREE__OF__NUMSET t s) <=> h4/patricia/IN__PTREE n t \/ h4/bool/IN n s)
%   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_ANDu_u_CLAUSESu_c0]: !t. T /\ t <=> t
% Assm [h4s_bools_EQu_u_SYMu_u_EQ]: !y x. x = y <=> y = x
% Assm [h4s_bools_EQu_u_CLAUSESu_c1]: !t. (t <=> T) <=> t
% Assm [h4s_bools_ANDu_u_IMPu_u_INTRO]: !t3 t2 t1. t1 ==> t2 ==> t3 <=> t1 /\ t2 ==> t3
% Assm [h4s_bools_IMPu_u_CONG]: !y_27 y x_27 x. (x <=> x_27) /\ (x_27 ==> (y <=> y_27)) ==> (x ==> y <=> x_27 ==> y_27)
% Assm [h4s_predu_u_sets_INu_u_UNION]: !x t s. h4/bool/IN x (h4/pred__set/UNION s t) <=> h4/bool/IN x s \/ h4/bool/IN x t
% Assm [h4s_predu_u_sets_FINITEu_u_UNION]: !t s. h4/pred__set/FINITE (h4/pred__set/UNION s t) <=> h4/pred__set/FINITE s /\ h4/pred__set/FINITE t
% Assm [h4s_lists_SETu_u_TOu_u_LISTu_u_INu_u_MEM]: !s. h4/pred__set/FINITE s ==> (!x. h4/bool/IN x s <=> h4/bool/IN x (h4/list/LIST__TO__SET (h4/list/SET__TO__LIST s)))
% Assm [h4s_sortings_PERMu_u_MEMu_u_EQ]: !l2 l1. h4/sorting/PERM l1 l2 ==> (!x. h4/bool/IN x (h4/list/LIST__TO__SET l1) <=> h4/bool/IN x (h4/list/LIST__TO__SET l2))
% Assm [h4s_patricias_INu_u_PTREEu_u_def]: !t n. h4/patricia/IN__PTREE n t <=> h4/option/IS__SOME (h4/patricia/PEEK t n)
% Assm [h4s_patricias_PTREEu_u_OFu_u_NUMSETu_u_def]: !t s. h4/patricia/PTREE__OF__NUMSET t s = h4/list/FOLDL (h4/combin/C h4/patricia/INSERT__PTREE) t (h4/list/SET__TO__LIST s)
% Assm [h4s_patricias_MEMu_u_TRAVERSEu_u_PEEK]: !t k. h4/patricia/IS__PTREE t ==> (h4/bool/IN k (h4/list/LIST__TO__SET (h4/patricia/TRAVERSE t)) <=> h4/option/IS__SOME (h4/patricia/PEEK t k))
% Assm [h4s_patricias_PTREEu_u_OFu_u_NUMSETu_u_ISu_u_PTREE]: !t s. h4/patricia/IS__PTREE t ==> h4/patricia/IS__PTREE (h4/patricia/PTREE__OF__NUMSET t s)
% Assm [h4s_patricias_MEMu_u_TRAVERSE]: !t k. h4/patricia/IS__PTREE t ==> (h4/bool/IN k (h4/list/LIST__TO__SET (h4/patricia/TRAVERSE t)) <=> h4/bool/IN k (h4/patricia/NUMSET__OF__PTREE t))
% Assm [h4s_patricias_FINITEu_u_NUMSETu_u_OFu_u_PTREE]: !t. h4/pred__set/FINITE (h4/patricia/NUMSET__OF__PTREE t)
% Assm [h4s_patricias_PERMu_u_INSERTu_u_PTREE]: !t s. h4/pred__set/FINITE s ==> h4/patricia/IS__PTREE t ==> h4/sorting/PERM (h4/patricia/TRAVERSE (h4/list/FOLDL (h4/combin/C h4/patricia/INSERT__PTREE) t (h4/list/SET__TO__LIST s))) (h4/list/SET__TO__LIST (h4/pred__set/UNION (h4/patricia/NUMSET__OF__PTREE t) s))
% Goal: !t s n. h4/patricia/IS__PTREE t /\ h4/pred__set/FINITE s ==> (h4/patricia/IN__PTREE n (h4/patricia/PTREE__OF__NUMSET t s) <=> h4/patricia/IN__PTREE n t \/ h4/bool/IN n s)
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_Q287125,TV_Q287121]: ![V_f, V_g]: (![V_x]: s(TV_Q287121,happ(s(t_fun(TV_Q287125,TV_Q287121),V_f),s(TV_Q287125,V_x))) = s(TV_Q287121,happ(s(t_fun(TV_Q287125,TV_Q287121),V_g),s(TV_Q287125,V_x))) => s(t_fun(TV_Q287125,TV_Q287121),V_f) = s(t_fun(TV_Q287125,TV_Q287121),V_g))).
fof(ah4s_bools_TRUTH, axiom, p(s(t_bool,t0))).
fof(ah4s_bools_ANDu_u_CLAUSESu_c0, axiom, ![V_t]: ((p(s(t_bool,t0)) & p(s(t_bool,V_t))) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_EQu_u_SYMu_u_EQ, axiom, ![TV_u_27a]: ![V_y, V_x]: (s(TV_u_27a,V_x) = s(TV_u_27a,V_y) <=> s(TV_u_27a,V_y) = s(TV_u_27a,V_x))).
fof(ah4s_bools_EQu_u_CLAUSESu_c1, axiom, ![V_t]: (s(t_bool,V_t) = s(t_bool,t0) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_ANDu_u_IMPu_u_INTRO, axiom, ![V_t3, V_t2, V_t1]: ((p(s(t_bool,V_t1)) => (p(s(t_bool,V_t2)) => p(s(t_bool,V_t3)))) <=> ((p(s(t_bool,V_t1)) & p(s(t_bool,V_t2))) => p(s(t_bool,V_t3))))).
fof(ah4s_bools_IMPu_u_CONG, axiom, ![V_yu_27, V_y, V_xu_27, V_x]: ((s(t_bool,V_x) = s(t_bool,V_xu_27) & (p(s(t_bool,V_xu_27)) => s(t_bool,V_y) = s(t_bool,V_yu_27))) => ((p(s(t_bool,V_x)) => p(s(t_bool,V_y))) <=> (p(s(t_bool,V_xu_27)) => p(s(t_bool,V_yu_27)))))).
fof(ah4s_predu_u_sets_INu_u_UNION, axiom, ![TV_u_27a]: ![V_x, V_t, V_s]: (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_union(s(t_fun(TV_u_27a,t_bool),V_s),s(t_fun(TV_u_27a,t_bool),V_t)))))) <=> (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_s)))) | p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_t))))))).
fof(ah4s_predu_u_sets_FINITEu_u_UNION, axiom, ![TV_u_27a]: ![V_t, V_s]: (p(s(t_bool,h4s_predu_u_sets_finite(s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_union(s(t_fun(TV_u_27a,t_bool),V_s),s(t_fun(TV_u_27a,t_bool),V_t)))))) <=> (p(s(t_bool,h4s_predu_u_sets_finite(s(t_fun(TV_u_27a,t_bool),V_s)))) & p(s(t_bool,h4s_predu_u_sets_finite(s(t_fun(TV_u_27a,t_bool),V_t))))))).
fof(ah4s_lists_SETu_u_TOu_u_LISTu_u_INu_u_MEM, axiom, ![TV_u_27a]: ![V_s]: (p(s(t_bool,h4s_predu_u_sets_finite(s(t_fun(TV_u_27a,t_bool),V_s)))) => ![V_x]: s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_s))) = s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),h4s_lists_listu_u_tou_u_set(s(t_h4s_lists_list(TV_u_27a),h4s_lists_setu_u_tou_u_list(s(t_fun(TV_u_27a,t_bool),V_s))))))))).
fof(ah4s_sortings_PERMu_u_MEMu_u_EQ, axiom, ![TV_u_27a]: ![V_l2, V_l1]: (p(s(t_bool,h4s_sortings_perm(s(t_h4s_lists_list(TV_u_27a),V_l1),s(t_h4s_lists_list(TV_u_27a),V_l2)))) => ![V_x]: s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),h4s_lists_listu_u_tou_u_set(s(t_h4s_lists_list(TV_u_27a),V_l1))))) = s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),h4s_lists_listu_u_tou_u_set(s(t_h4s_lists_list(TV_u_27a),V_l2))))))).
fof(ah4s_patricias_INu_u_PTREEu_u_def, axiom, ![V_t, V_n]: s(t_bool,h4s_patricias_inu_u_ptree(s(t_h4s_nums_num,V_n),s(t_h4s_patricias_ptree(t_h4s_ones_one),V_t))) = s(t_bool,h4s_options_isu_u_some(s(t_h4s_options_option(t_h4s_ones_one),h4s_patricias_peek(s(t_h4s_patricias_ptree(t_h4s_ones_one),V_t),s(t_h4s_nums_num,V_n)))))).
fof(ah4s_patricias_PTREEu_u_OFu_u_NUMSETu_u_def, axiom, ![V_t, V_s]: s(t_h4s_patricias_ptree(t_h4s_ones_one),h4s_patricias_ptreeu_u_ofu_u_numset(s(t_h4s_patricias_ptree(t_h4s_ones_one),V_t),s(t_fun(t_h4s_nums_num,t_bool),V_s))) = s(t_h4s_patricias_ptree(t_h4s_ones_one),h4s_lists_foldl(s(t_fun(t_h4s_patricias_ptree(t_h4s_ones_one),t_fun(t_h4s_nums_num,t_h4s_patricias_ptree(t_h4s_ones_one))),h4s_combins_c(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_patricias_ptree(t_h4s_ones_one),t_h4s_patricias_ptree(t_h4s_ones_one))),h4s_patricias_insertu_u_ptree))),s(t_h4s_patricias_ptree(t_h4s_ones_one),V_t),s(t_h4s_lists_list(t_h4s_nums_num),h4s_lists_setu_u_tou_u_list(s(t_fun(t_h4s_nums_num,t_bool),V_s)))))).
fof(ah4s_patricias_MEMu_u_TRAVERSEu_u_PEEK, axiom, ![TV_u_27a]: ![V_t, V_k]: (p(s(t_bool,h4s_patricias_isu_u_ptree(s(t_h4s_patricias_ptree(TV_u_27a),V_t)))) => s(t_bool,h4s_bools_in(s(t_h4s_nums_num,V_k),s(t_fun(t_h4s_nums_num,t_bool),h4s_lists_listu_u_tou_u_set(s(t_h4s_lists_list(t_h4s_nums_num),h4s_patricias_traverse(s(t_h4s_patricias_ptree(TV_u_27a),V_t))))))) = s(t_bool,h4s_options_isu_u_some(s(t_h4s_options_option(TV_u_27a),h4s_patricias_peek(s(t_h4s_patricias_ptree(TV_u_27a),V_t),s(t_h4s_nums_num,V_k))))))).
fof(ah4s_patricias_PTREEu_u_OFu_u_NUMSETu_u_ISu_u_PTREE, axiom, ![V_t, V_s]: (p(s(t_bool,h4s_patricias_isu_u_ptree(s(t_h4s_patricias_ptree(t_h4s_ones_one),V_t)))) => p(s(t_bool,h4s_patricias_isu_u_ptree(s(t_h4s_patricias_ptree(t_h4s_ones_one),h4s_patricias_ptreeu_u_ofu_u_numset(s(t_h4s_patricias_ptree(t_h4s_ones_one),V_t),s(t_fun(t_h4s_nums_num,t_bool),V_s)))))))).
fof(ah4s_patricias_MEMu_u_TRAVERSE, axiom, ![V_t, V_k]: (p(s(t_bool,h4s_patricias_isu_u_ptree(s(t_h4s_patricias_ptree(t_h4s_ones_one),V_t)))) => s(t_bool,h4s_bools_in(s(t_h4s_nums_num,V_k),s(t_fun(t_h4s_nums_num,t_bool),h4s_lists_listu_u_tou_u_set(s(t_h4s_lists_list(t_h4s_nums_num),h4s_patricias_traverse(s(t_h4s_patricias_ptree(t_h4s_ones_one),V_t))))))) = s(t_bool,h4s_bools_in(s(t_h4s_nums_num,V_k),s(t_fun(t_h4s_nums_num,t_bool),h4s_patricias_numsetu_u_ofu_u_ptree(s(t_h4s_patricias_ptree(t_h4s_ones_one),V_t))))))).
fof(ah4s_patricias_FINITEu_u_NUMSETu_u_OFu_u_PTREE, axiom, ![V_t]: p(s(t_bool,h4s_predu_u_sets_finite(s(t_fun(t_h4s_nums_num,t_bool),h4s_patricias_numsetu_u_ofu_u_ptree(s(t_h4s_patricias_ptree(t_h4s_ones_one),V_t))))))).
fof(ah4s_patricias_PERMu_u_INSERTu_u_PTREE, axiom, ![V_t, V_s]: (p(s(t_bool,h4s_predu_u_sets_finite(s(t_fun(t_h4s_nums_num,t_bool),V_s)))) => (p(s(t_bool,h4s_patricias_isu_u_ptree(s(t_h4s_patricias_ptree(t_h4s_ones_one),V_t)))) => p(s(t_bool,h4s_sortings_perm(s(t_h4s_lists_list(t_h4s_nums_num),h4s_patricias_traverse(s(t_h4s_patricias_ptree(t_h4s_ones_one),h4s_lists_foldl(s(t_fun(t_h4s_patricias_ptree(t_h4s_ones_one),t_fun(t_h4s_nums_num,t_h4s_patricias_ptree(t_h4s_ones_one))),h4s_combins_c(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_patricias_ptree(t_h4s_ones_one),t_h4s_patricias_ptree(t_h4s_ones_one))),h4s_patricias_insertu_u_ptree))),s(t_h4s_patricias_ptree(t_h4s_ones_one),V_t),s(t_h4s_lists_list(t_h4s_nums_num),h4s_lists_setu_u_tou_u_list(s(t_fun(t_h4s_nums_num,t_bool),V_s))))))),s(t_h4s_lists_list(t_h4s_nums_num),h4s_lists_setu_u_tou_u_list(s(t_fun(t_h4s_nums_num,t_bool),h4s_predu_u_sets_union(s(t_fun(t_h4s_nums_num,t_bool),h4s_patricias_numsetu_u_ofu_u_ptree(s(t_h4s_patricias_ptree(t_h4s_ones_one),V_t))),s(t_fun(t_h4s_nums_num,t_bool),V_s))))))))))).
fof(ch4s_patricias_INu_u_PTREEu_u_OFu_u_NUMSET, conjecture, ![V_t, V_s, V_n]: ((p(s(t_bool,h4s_patricias_isu_u_ptree(s(t_h4s_patricias_ptree(t_h4s_ones_one),V_t)))) & p(s(t_bool,h4s_predu_u_sets_finite(s(t_fun(t_h4s_nums_num,t_bool),V_s))))) => (p(s(t_bool,h4s_patricias_inu_u_ptree(s(t_h4s_nums_num,V_n),s(t_h4s_patricias_ptree(t_h4s_ones_one),h4s_patricias_ptreeu_u_ofu_u_numset(s(t_h4s_patricias_ptree(t_h4s_ones_one),V_t),s(t_fun(t_h4s_nums_num,t_bool),V_s)))))) <=> (p(s(t_bool,h4s_patricias_inu_u_ptree(s(t_h4s_nums_num,V_n),s(t_h4s_patricias_ptree(t_h4s_ones_one),V_t)))) | p(s(t_bool,h4s_bools_in(s(t_h4s_nums_num,V_n),s(t_fun(t_h4s_nums_num,t_bool),V_s)))))))).
