%   ORIGINAL: h4/patricia/PERM__REMOVE
% 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/IMP__CLAUSES_c1: !t. t ==> 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/patricia/TRANSFORM__IS__PTREE: !t f. h4/patricia/IS__PTREE t ==> h4/patricia/IS__PTREE (h4/patricia/TRANSFORM f t)
% Assm: h4/patricia/REMOVE__TRANSFORM: !t k f. h4/patricia/TRANSFORM f (h4/patricia/REMOVE t k) = h4/patricia/REMOVE (h4/patricia/TRANSFORM f t) k
% Assm: h4/patricia/TRAVERSE__TRANSFORM: !t f. h4/patricia/TRAVERSE (h4/patricia/TRANSFORM f t) = h4/patricia/TRAVERSE t
% Assm: h4/patricia/PERM__DELETE__PTREE: !t k. h4/patricia/IS__PTREE t /\ h4/bool/IN k (h4/list/LIST__TO__SET (h4/patricia/TRAVERSE t)) ==> h4/sorting/PERM (h4/patricia/TRAVERSE (h4/patricia/REMOVE t k)) (h4/list/FILTER (\x. ~(x = k)) (h4/patricia/TRAVERSE t))
% Goal: !t k. h4/patricia/IS__PTREE t /\ h4/bool/IN k (h4/list/LIST__TO__SET (h4/patricia/TRAVERSE t)) ==> h4/sorting/PERM (h4/patricia/TRAVERSE (h4/patricia/REMOVE t k)) (h4/list/FILTER (\x. ~(x = k)) (h4/patricia/TRAVERSE 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_bools_TRUTH]: T
% Assm [h4s_bools_ANDu_u_CLAUSESu_c0]: !t. T /\ t <=> t
% Assm [h4s_bools_IMPu_u_CLAUSESu_c1]: !t. t ==> 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_patricias_TRANSFORMu_u_ISu_u_PTREE]: !t f. h4/patricia/IS__PTREE t ==> h4/patricia/IS__PTREE (h4/patricia/TRANSFORM f t)
% Assm [h4s_patricias_REMOVEu_u_TRANSFORM]: !t k f. h4/patricia/TRANSFORM f (h4/patricia/REMOVE t k) = h4/patricia/REMOVE (h4/patricia/TRANSFORM f t) k
% Assm [h4s_patricias_TRAVERSEu_u_TRANSFORM]: !t f. h4/patricia/TRAVERSE (h4/patricia/TRANSFORM f t) = h4/patricia/TRAVERSE t
% Assm [h4s_patricias_PERMu_u_DELETEu_u_PTREE]: !_0. (!k x. happ (happ _0 k) x <=> ~(x = k)) ==> (!t k. h4/patricia/IS__PTREE t /\ h4/bool/IN k (h4/list/LIST__TO__SET (h4/patricia/TRAVERSE t)) ==> h4/sorting/PERM (h4/patricia/TRAVERSE (h4/patricia/REMOVE t k)) (h4/list/FILTER (happ _0 k) (h4/patricia/TRAVERSE t)))
% Goal: !_0. (!k x. happ (happ _0 k) x <=> ~(x = k)) ==> (!t k. h4/patricia/IS__PTREE t /\ h4/bool/IN k (h4/list/LIST__TO__SET (h4/patricia/TRAVERSE t)) ==> h4/sorting/PERM (h4/patricia/TRAVERSE (h4/patricia/REMOVE t k)) (h4/list/FILTER (happ _0 k) (h4/patricia/TRAVERSE 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_Q287931,TV_Q287927]: ![V_f, V_g]: (![V_x]: s(TV_Q287927,happ(s(t_fun(TV_Q287931,TV_Q287927),V_f),s(TV_Q287931,V_x))) = s(TV_Q287927,happ(s(t_fun(TV_Q287931,TV_Q287927),V_g),s(TV_Q287931,V_x))) => s(t_fun(TV_Q287931,TV_Q287927),V_f) = s(t_fun(TV_Q287931,TV_Q287927),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_IMPu_u_CLAUSESu_c1, axiom, ![V_t]: ((p(s(t_bool,V_t)) => p(s(t_bool,t0))) <=> p(s(t_bool,t0)))).
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_patricias_TRANSFORMu_u_ISu_u_PTREE, axiom, ![TV_u_27b,TV_u_27a]: ![V_t, V_f]: (p(s(t_bool,h4s_patricias_isu_u_ptree(s(t_h4s_patricias_ptree(TV_u_27a),V_t)))) => p(s(t_bool,h4s_patricias_isu_u_ptree(s(t_h4s_patricias_ptree(TV_u_27b),h4s_patricias_transform(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_h4s_patricias_ptree(TV_u_27a),V_t)))))))).
fof(ah4s_patricias_REMOVEu_u_TRANSFORM, axiom, ![TV_u_27b,TV_u_27a]: ![V_t, V_k, V_f]: s(t_h4s_patricias_ptree(TV_u_27b),h4s_patricias_transform(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_h4s_patricias_ptree(TV_u_27a),h4s_patricias_remove(s(t_h4s_patricias_ptree(TV_u_27a),V_t),s(t_h4s_nums_num,V_k))))) = s(t_h4s_patricias_ptree(TV_u_27b),h4s_patricias_remove(s(t_h4s_patricias_ptree(TV_u_27b),h4s_patricias_transform(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_h4s_patricias_ptree(TV_u_27a),V_t))),s(t_h4s_nums_num,V_k)))).
fof(ah4s_patricias_TRAVERSEu_u_TRANSFORM, axiom, ![TV_u_27b,TV_u_27a]: ![V_t, V_f]: s(t_h4s_lists_list(t_h4s_nums_num),h4s_patricias_traverse(s(t_h4s_patricias_ptree(TV_u_27b),h4s_patricias_transform(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_h4s_patricias_ptree(TV_u_27a),V_t))))) = s(t_h4s_lists_list(t_h4s_nums_num),h4s_patricias_traverse(s(t_h4s_patricias_ptree(TV_u_27a),V_t)))).
fof(ah4s_patricias_PERMu_u_DELETEu_u_PTREE, axiom, ![V_uu_0]: (![V_k, V_x]: (p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_bool)),V_uu_0),s(t_h4s_nums_num,V_k))),s(t_h4s_nums_num,V_x)))) <=> ~ (s(t_h4s_nums_num,V_x) = s(t_h4s_nums_num,V_k))) => ![V_t, V_k]: ((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_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))))))))) => 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_patricias_remove(s(t_h4s_patricias_ptree(t_h4s_ones_one),V_t),s(t_h4s_nums_num,V_k))))),s(t_h4s_lists_list(t_h4s_nums_num),h4s_lists_filter(s(t_fun(t_h4s_nums_num,t_bool),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_bool)),V_uu_0),s(t_h4s_nums_num,V_k))),s(t_h4s_lists_list(t_h4s_nums_num),h4s_patricias_traverse(s(t_h4s_patricias_ptree(t_h4s_ones_one),V_t))))))))))).
fof(ch4s_patricias_PERMu_u_REMOVE, conjecture, ![TV_u_27a]: ![V_uu_0]: (![V_k, V_x]: (p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_bool)),V_uu_0),s(t_h4s_nums_num,V_k))),s(t_h4s_nums_num,V_x)))) <=> ~ (s(t_h4s_nums_num,V_x) = s(t_h4s_nums_num,V_k))) => ![V_t, V_k]: ((p(s(t_bool,h4s_patricias_isu_u_ptree(s(t_h4s_patricias_ptree(TV_u_27a),V_t)))) & p(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))))))))) => p(s(t_bool,h4s_sortings_perm(s(t_h4s_lists_list(t_h4s_nums_num),h4s_patricias_traverse(s(t_h4s_patricias_ptree(TV_u_27a),h4s_patricias_remove(s(t_h4s_patricias_ptree(TV_u_27a),V_t),s(t_h4s_nums_num,V_k))))),s(t_h4s_lists_list(t_h4s_nums_num),h4s_lists_filter(s(t_fun(t_h4s_nums_num,t_bool),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_bool)),V_uu_0),s(t_h4s_nums_num,V_k))),s(t_h4s_lists_list(t_h4s_nums_num),h4s_patricias_traverse(s(t_h4s_patricias_ptree(TV_u_27a),V_t))))))))))).
