%   ORIGINAL: h4/sptree/delete__compute_c8
% 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/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/COND__CLAUSES_c0: !t2 t1. h4/bool/COND T t1 t2 = t1
% Assm: h4/bool/AND__IMP__INTRO: !t3 t2 t1. t1 ==> t2 ==> t3 <=> t1 /\ t2 ==> t3
% Assm: h4/bool/COND__CONG: !y_27 y x_27 x Q P. (P <=> Q) /\ (Q ==> x = x_27) /\ (~Q ==> y = y_27) ==> h4/bool/COND P x y = h4/bool/COND Q x_27 y_27
% Assm: h4/arithmetic/ALT__ZERO: h4/arithmetic/ZERO = h4/num/0
% Assm: h4/numeral/numeral__evenodd_c0: h4/arithmetic/EVEN h4/arithmetic/ZERO
% Assm: h4/sptree/delete__def_c3: !t2 t1 k a. h4/sptree/delete k (h4/sptree/BS t1 a t2) = h4/bool/COND (k = h4/num/0) (h4/sptree/BN t1 t2) (h4/bool/COND (h4/arithmetic/EVEN k) (h4/sptree/mk__BS (h4/sptree/delete (h4/arithmetic/DIV (h4/arithmetic/_2D k (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO))) t1) a t2) (h4/sptree/mk__BS t1 a (h4/sptree/delete (h4/arithmetic/DIV (h4/arithmetic/_2D k (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO))) t2)))
% Goal: !t2 t1 a. h4/sptree/delete h4/arithmetic/ZERO (h4/sptree/BS t1 a t2) = h4/sptree/BN t1 t2
%   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_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_CONDu_u_CLAUSESu_c0]: !t2 t1. h4/bool/COND T t1 t2 = t1
% Assm [h4s_bools_ANDu_u_IMPu_u_INTRO]: !t3 t2 t1. t1 ==> t2 ==> t3 <=> t1 /\ t2 ==> t3
% Assm [h4s_bools_CONDu_u_CONG]: !y_27 y x_27 x Q P. (P <=> Q) /\ (Q ==> x = x_27) /\ (~Q ==> y = y_27) ==> h4/bool/COND P x y = h4/bool/COND Q x_27 y_27
% Assm [h4s_arithmetics_ALTu_u_ZERO]: h4/arithmetic/ZERO = h4/num/0
% Assm [h4s_numerals_numeralu_u_evenoddu_c0]: h4/arithmetic/EVEN h4/arithmetic/ZERO
% Assm [h4s_sptrees_deleteu_u_defu_c3]: !t2 t1 k a. ?v. (v <=> k = h4/num/0) /\ h4/sptree/delete k (h4/sptree/BS t1 a t2) = h4/bool/COND v (h4/sptree/BN t1 t2) (h4/bool/COND (h4/arithmetic/EVEN k) (h4/sptree/mk__BS (h4/sptree/delete (h4/arithmetic/DIV (h4/arithmetic/_2D k (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO))) t1) a t2) (h4/sptree/mk__BS t1 a (h4/sptree/delete (h4/arithmetic/DIV (h4/arithmetic/_2D k (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO))) t2)))
% Goal: !t2 t1 a. h4/sptree/delete h4/arithmetic/ZERO (h4/sptree/BS t1 a t2) = h4/sptree/BN t1 t2
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_Q84089,TV_Q84085]: ![V_f, V_g]: (![V_x]: s(TV_Q84085,happ(s(t_fun(TV_Q84089,TV_Q84085),V_f),s(TV_Q84089,V_x))) = s(TV_Q84085,happ(s(t_fun(TV_Q84089,TV_Q84085),V_g),s(TV_Q84089,V_x))) => s(t_fun(TV_Q84089,TV_Q84085),V_f) = s(t_fun(TV_Q84089,TV_Q84085),V_g))).
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,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,t) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_CONDu_u_CLAUSESu_c0, axiom, ![TV_u_27a]: ![V_t2, V_t1]: s(TV_u_27a,h4s_bools_cond(s(t_bool,t),s(TV_u_27a,V_t1),s(TV_u_27a,V_t2))) = s(TV_u_27a,V_t1)).
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_CONDu_u_CONG, axiom, ![TV_u_27a]: ![V_yu_27, V_y, V_xu_27, V_x, V_Q, V_P]: ((s(t_bool,V_P) = s(t_bool,V_Q) & ((p(s(t_bool,V_Q)) => s(TV_u_27a,V_x) = s(TV_u_27a,V_xu_27)) & (~ (p(s(t_bool,V_Q))) => s(TV_u_27a,V_y) = s(TV_u_27a,V_yu_27)))) => s(TV_u_27a,h4s_bools_cond(s(t_bool,V_P),s(TV_u_27a,V_x),s(TV_u_27a,V_y))) = s(TV_u_27a,h4s_bools_cond(s(t_bool,V_Q),s(TV_u_27a,V_xu_27),s(TV_u_27a,V_yu_27))))).
fof(ah4s_arithmetics_ALTu_u_ZERO, axiom, s(t_h4s_nums_num,h4s_arithmetics_zero) = s(t_h4s_nums_num,h4s_nums_0)).
fof(ah4s_numerals_numeralu_u_evenoddu_c0, axiom, p(s(t_bool,h4s_arithmetics_even(s(t_h4s_nums_num,h4s_arithmetics_zero))))).
fof(ah4s_sptrees_deleteu_u_defu_c3, axiom, ![TV_u_27a]: ![V_t2, V_t1, V_k, V_a]: ?[V_v]: ((p(s(t_bool,V_v)) <=> s(t_h4s_nums_num,V_k) = s(t_h4s_nums_num,h4s_nums_0)) & s(t_h4s_sptrees_spt(TV_u_27a),h4s_sptrees_delete(s(t_h4s_nums_num,V_k),s(t_h4s_sptrees_spt(TV_u_27a),h4s_sptrees_bs(s(t_h4s_sptrees_spt(TV_u_27a),V_t1),s(TV_u_27a,V_a),s(t_h4s_sptrees_spt(TV_u_27a),V_t2))))) = s(t_h4s_sptrees_spt(TV_u_27a),h4s_bools_cond(s(t_bool,V_v),s(t_h4s_sptrees_spt(TV_u_27a),h4s_sptrees_bn(s(t_h4s_sptrees_spt(TV_u_27a),V_t1),s(t_h4s_sptrees_spt(TV_u_27a),V_t2))),s(t_h4s_sptrees_spt(TV_u_27a),h4s_bools_cond(s(t_bool,h4s_arithmetics_even(s(t_h4s_nums_num,V_k))),s(t_h4s_sptrees_spt(TV_u_27a),h4s_sptrees_mku_u_bs(s(t_h4s_sptrees_spt(TV_u_27a),h4s_sptrees_delete(s(t_h4s_nums_num,h4s_arithmetics_div(s(t_h4s_nums_num,h4s_arithmetics_u_2d(s(t_h4s_nums_num,V_k),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,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit2(s(t_h4s_nums_num,h4s_arithmetics_zero))))))),s(t_h4s_sptrees_spt(TV_u_27a),V_t1))),s(TV_u_27a,V_a),s(t_h4s_sptrees_spt(TV_u_27a),V_t2))),s(t_h4s_sptrees_spt(TV_u_27a),h4s_sptrees_mku_u_bs(s(t_h4s_sptrees_spt(TV_u_27a),V_t1),s(TV_u_27a,V_a),s(t_h4s_sptrees_spt(TV_u_27a),h4s_sptrees_delete(s(t_h4s_nums_num,h4s_arithmetics_div(s(t_h4s_nums_num,h4s_arithmetics_u_2d(s(t_h4s_nums_num,V_k),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,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit2(s(t_h4s_nums_num,h4s_arithmetics_zero))))))),s(t_h4s_sptrees_spt(TV_u_27a),V_t2))))))))))).
fof(ch4s_sptrees_deleteu_u_computeu_c8, conjecture, ![TV_u_27a]: ![V_t2, V_t1, V_a]: s(t_h4s_sptrees_spt(TV_u_27a),h4s_sptrees_delete(s(t_h4s_nums_num,h4s_arithmetics_zero),s(t_h4s_sptrees_spt(TV_u_27a),h4s_sptrees_bs(s(t_h4s_sptrees_spt(TV_u_27a),V_t1),s(TV_u_27a,V_a),s(t_h4s_sptrees_spt(TV_u_27a),V_t2))))) = s(t_h4s_sptrees_spt(TV_u_27a),h4s_sptrees_bn(s(t_h4s_sptrees_spt(TV_u_27a),V_t1),s(t_h4s_sptrees_spt(TV_u_27a),V_t2)))).
