%   ORIGINAL: h4/patricia/PTREE__OF__NUMSET__NUMSET__OF__PTREE
% 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/IMP__ANTISYM__AX: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm: h4/bool/FALSITY: !t. F ==> t
% Assm: h4/bool/AND__CLAUSES_c0: !t. T /\ t <=> t
% Assm: h4/bool/OR__CLAUSES_c2: !t. F \/ t <=> t
% Assm: h4/bool/REFL__CLAUSE: !x. x = x <=> 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/patricia/EMPTY__IS__PTREE: h4/patricia/IS__PTREE h4/patricia/Empty
% Assm: h4/patricia/IN__NUMSET__OF__PTREE: !t n. h4/patricia/IS__PTREE t ==> (h4/bool/IN n (h4/patricia/NUMSET__OF__PTREE t) <=> h4/patricia/IN__PTREE n t)
% 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/PTREE__OF__NUMSET__IS__PTREE__EMPTY: !s. h4/patricia/IS__PTREE (h4/patricia/PTREE__OF__NUMSET h4/patricia/Empty s)
% Assm: h4/patricia/FINITE__NUMSET__OF__PTREE: !t. h4/pred__set/FINITE (h4/patricia/NUMSET__OF__PTREE t)
% Assm: h4/patricia/IN__PTREE__OF__NUMSET: !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)
% Assm: h4/patricia/IN__PTREE__EMPTY: !n. ~h4/patricia/IN__PTREE n h4/patricia/Empty
% Assm: h4/patricia/PTREE__EXTENSION: !t2 t1. h4/patricia/IS__PTREE t1 /\ h4/patricia/IS__PTREE t2 ==> (t1 = t2 <=> (!x. h4/patricia/IN__PTREE x t1 <=> h4/patricia/IN__PTREE x t2))
% Goal: !t s. h4/patricia/IS__PTREE t /\ h4/pred__set/FINITE s ==> h4/patricia/PTREE__OF__NUMSET h4/patricia/Empty (h4/pred__set/UNION (h4/patricia/NUMSET__OF__PTREE t) s) = h4/patricia/PTREE__OF__NUMSET t 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_IMPu_u_ANTISYMu_u_AX]: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm [h4s_bools_FALSITY]: !t. F ==> t
% Assm [h4s_bools_ANDu_u_CLAUSESu_c0]: !t. T /\ t <=> t
% Assm [h4s_bools_ORu_u_CLAUSESu_c2]: !t. F \/ t <=> t
% Assm [h4s_bools_REFLu_u_CLAUSE]: !x. x = x <=> 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_patricias_EMPTYu_u_ISu_u_PTREE]: h4/patricia/IS__PTREE h4/patricia/Empty
% Assm [h4s_patricias_INu_u_NUMSETu_u_OFu_u_PTREE]: !t n. h4/patricia/IS__PTREE t ==> (h4/bool/IN n (h4/patricia/NUMSET__OF__PTREE t) <=> h4/patricia/IN__PTREE n t)
% 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_PTREEu_u_OFu_u_NUMSETu_u_ISu_u_PTREEu_u_EMPTY]: !s. h4/patricia/IS__PTREE (h4/patricia/PTREE__OF__NUMSET h4/patricia/Empty s)
% Assm [h4s_patricias_FINITEu_u_NUMSETu_u_OFu_u_PTREE]: !t. h4/pred__set/FINITE (h4/patricia/NUMSET__OF__PTREE t)
% Assm [h4s_patricias_INu_u_PTREEu_u_OFu_u_NUMSET]: !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)
% Assm [h4s_patricias_INu_u_PTREEu_u_EMPTY]: !n. ~h4/patricia/IN__PTREE n h4/patricia/Empty
% Assm [h4s_patricias_PTREEu_u_EXTENSION]: !t2 t1. h4/patricia/IS__PTREE t1 /\ h4/patricia/IS__PTREE t2 ==> (t1 = t2 <=> (!x. h4/patricia/IN__PTREE x t1 <=> h4/patricia/IN__PTREE x t2))
% Goal: !t s. h4/patricia/IS__PTREE t /\ h4/pred__set/FINITE s ==> h4/patricia/PTREE__OF__NUMSET h4/patricia/Empty (h4/pred__set/UNION (h4/patricia/NUMSET__OF__PTREE t) s) = h4/patricia/PTREE__OF__NUMSET t 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_Q287226,TV_Q287222]: ![V_f, V_g]: (![V_x]: s(TV_Q287222,happ(s(t_fun(TV_Q287226,TV_Q287222),V_f),s(TV_Q287226,V_x))) = s(TV_Q287222,happ(s(t_fun(TV_Q287226,TV_Q287222),V_g),s(TV_Q287226,V_x))) => s(t_fun(TV_Q287226,TV_Q287222),V_f) = s(t_fun(TV_Q287226,TV_Q287222),V_g))).
fof(ah4s_bools_TRUTH, axiom, p(s(t_bool,t0))).
fof(ah4s_bools_IMPu_u_ANTISYMu_u_AX, axiom, ![V_t2, V_t1]: ((p(s(t_bool,V_t1)) => p(s(t_bool,V_t2))) => ((p(s(t_bool,V_t2)) => p(s(t_bool,V_t1))) => s(t_bool,V_t1) = s(t_bool,V_t2)))).
fof(ah4s_bools_FALSITY, axiom, ![V_t]: (p(s(t_bool,f)) => p(s(t_bool,V_t)))).
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_ORu_u_CLAUSESu_c2, axiom, ![V_t]: ((p(s(t_bool,f)) | 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,t0)))).
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_patricias_EMPTYu_u_ISu_u_PTREE, axiom, ![TV_u_27a]: p(s(t_bool,h4s_patricias_isu_u_ptree(s(t_h4s_patricias_ptree(TV_u_27a),h4s_patricias_empty))))).
fof(ah4s_patricias_INu_u_NUMSETu_u_OFu_u_PTREE, axiom, ![V_t, V_n]: (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_n),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_bool,h4s_patricias_inu_u_ptree(s(t_h4s_nums_num,V_n),s(t_h4s_patricias_ptree(t_h4s_ones_one),V_t))))).
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_PTREEu_u_OFu_u_NUMSETu_u_ISu_u_PTREEu_u_EMPTY, axiom, ![V_s]: 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),h4s_patricias_empty),s(t_fun(t_h4s_nums_num,t_bool),V_s))))))).
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_INu_u_PTREEu_u_OFu_u_NUMSET, axiom, ![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)))))))).
fof(ah4s_patricias_INu_u_PTREEu_u_EMPTY, axiom, ![V_n]: ~ (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_empty)))))).
fof(ah4s_patricias_PTREEu_u_EXTENSION, axiom, ![V_t2, V_t1]: ((p(s(t_bool,h4s_patricias_isu_u_ptree(s(t_h4s_patricias_ptree(t_h4s_ones_one),V_t1)))) & p(s(t_bool,h4s_patricias_isu_u_ptree(s(t_h4s_patricias_ptree(t_h4s_ones_one),V_t2))))) => (s(t_h4s_patricias_ptree(t_h4s_ones_one),V_t1) = s(t_h4s_patricias_ptree(t_h4s_ones_one),V_t2) <=> ![V_x]: s(t_bool,h4s_patricias_inu_u_ptree(s(t_h4s_nums_num,V_x),s(t_h4s_patricias_ptree(t_h4s_ones_one),V_t1))) = s(t_bool,h4s_patricias_inu_u_ptree(s(t_h4s_nums_num,V_x),s(t_h4s_patricias_ptree(t_h4s_ones_one),V_t2)))))).
fof(ch4s_patricias_PTREEu_u_OFu_u_NUMSETu_u_NUMSETu_u_OFu_u_PTREE, conjecture, ![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_predu_u_sets_finite(s(t_fun(t_h4s_nums_num,t_bool),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),h4s_patricias_empty),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))))) = 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))))).
