%   ORIGINAL: h4/bit/BIT__DIV2
% 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/REFL__CLAUSE: !x. x = x <=> T
% 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/arithmetic/EXP0_c1: !n m. h4/arithmetic/EXP m (h4/num/SUC n) = h4/arithmetic/_2A m (h4/arithmetic/EXP m n)
% Assm: h4/arithmetic/SUB__MONO__EQ: !n m. h4/arithmetic/_2D (h4/num/SUC n) (h4/num/SUC m) = h4/arithmetic/_2D n m
% Assm: h4/arithmetic/DIV__DIV__DIV__MULT: !n m. h4/prim__rec/_3C h4/num/0 m /\ h4/prim__rec/_3C h4/num/0 n ==> (!x. h4/arithmetic/DIV (h4/arithmetic/DIV x m) n = h4/arithmetic/DIV x (h4/arithmetic/_2A m n))
% Assm: h4/numeral/numeral__distrib_c21: !n. h4/prim__rec/_3C h4/num/0 (h4/arithmetic/NUMERAL n) <=> h4/prim__rec/_3C h4/arithmetic/ZERO n
% Assm: h4/numeral/numeral__lt_c1: !n. h4/prim__rec/_3C h4/arithmetic/ZERO (h4/arithmetic/BIT2 n) <=> T
% Assm: h4/bit/BIT__def: !n b. h4/bit/BIT b n <=> h4/bit/BITS b b n = h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)
% Assm: h4/bit/ZERO__LT__TWOEXP: !n. h4/prim__rec/_3C h4/num/0 (h4/arithmetic/EXP (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO)) n)
% Assm: h4/bit/BITS__THM: !n l h. h4/bit/BITS h l n = h4/arithmetic/MOD (h4/arithmetic/DIV n (h4/arithmetic/EXP (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO)) l)) (h4/arithmetic/EXP (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO)) (h4/arithmetic/_2D (h4/num/SUC h) l))
% Goal: !n i. h4/bit/BIT n (h4/arithmetic/DIV i (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO))) <=> h4/bit/BIT (h4/num/SUC n) i
%   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_REFLu_u_CLAUSE]: !x. x = x <=> T
% 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_arithmetics_EXP0u_c1]: !n m. h4/arithmetic/EXP m (h4/num/SUC n) = h4/arithmetic/_2A m (h4/arithmetic/EXP m n)
% Assm [h4s_arithmetics_SUBu_u_MONOu_u_EQ]: !n m. h4/arithmetic/_2D (h4/num/SUC n) (h4/num/SUC m) = h4/arithmetic/_2D n m
% Assm [h4s_arithmetics_DIVu_u_DIVu_u_DIVu_u_MULT]: !n m. h4/prim__rec/_3C h4/num/0 m /\ h4/prim__rec/_3C h4/num/0 n ==> (!x. h4/arithmetic/DIV (h4/arithmetic/DIV x m) n = h4/arithmetic/DIV x (h4/arithmetic/_2A m n))
% Assm [h4s_numerals_numeralu_u_distribu_c21]: !n. h4/prim__rec/_3C h4/num/0 (h4/arithmetic/NUMERAL n) <=> h4/prim__rec/_3C h4/arithmetic/ZERO n
% Assm [h4s_numerals_numeralu_u_ltu_c1]: !n. h4/prim__rec/_3C h4/arithmetic/ZERO (h4/arithmetic/BIT2 n) <=> T
% Assm [h4s_bits_BITu_u_def]: !n b. h4/bit/BIT b n <=> h4/bit/BITS b b n = h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)
% Assm [h4s_bits_ZEROu_u_LTu_u_TWOEXP]: !n. h4/prim__rec/_3C h4/num/0 (h4/arithmetic/EXP (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO)) n)
% Assm [h4s_bits_BITSu_u_THM]: !n l h. h4/bit/BITS h l n = h4/arithmetic/MOD (h4/arithmetic/DIV n (h4/arithmetic/EXP (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO)) l)) (h4/arithmetic/EXP (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO)) (h4/arithmetic/_2D (h4/num/SUC h) l))
% Goal: !n i. h4/bit/BIT n (h4/arithmetic/DIV i (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO))) <=> h4/bit/BIT (h4/num/SUC n) i
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_Q202770,TV_Q202766]: ![V_f, V_g]: (![V_x]: s(TV_Q202766,happ(s(t_fun(TV_Q202770,TV_Q202766),V_f),s(TV_Q202770,V_x))) = s(TV_Q202766,happ(s(t_fun(TV_Q202770,TV_Q202766),V_g),s(TV_Q202770,V_x))) => s(t_fun(TV_Q202770,TV_Q202766),V_f) = s(t_fun(TV_Q202770,TV_Q202766),V_g))).
fof(ah4s_bools_TRUTH, axiom, p(s(t_bool,t))).
fof(ah4s_bools_ANDu_u_CLAUSESu_c0, axiom, ![V_t]: ((p(s(t_bool,t)) & 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,t)))).
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_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_arithmetics_EXP0u_c1, axiom, ![V_n, V_m]: s(t_h4s_nums_num,h4s_arithmetics_exp(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n))))) = s(t_h4s_nums_num,h4s_arithmetics_u_2a(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,h4s_arithmetics_exp(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n)))))).
fof(ah4s_arithmetics_SUBu_u_MONOu_u_EQ, axiom, ![V_n, V_m]: s(t_h4s_nums_num,h4s_arithmetics_u_2d(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_m))))) = s(t_h4s_nums_num,h4s_arithmetics_u_2d(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_m)))).
fof(ah4s_arithmetics_DIVu_u_DIVu_u_DIVu_u_MULT, axiom, ![V_n, V_m]: ((p(s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,h4s_nums_0),s(t_h4s_nums_num,V_m)))) & p(s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,h4s_nums_0),s(t_h4s_nums_num,V_n))))) => ![V_x]: s(t_h4s_nums_num,h4s_arithmetics_div(s(t_h4s_nums_num,h4s_arithmetics_div(s(t_h4s_nums_num,V_x),s(t_h4s_nums_num,V_m))),s(t_h4s_nums_num,V_n))) = s(t_h4s_nums_num,h4s_arithmetics_div(s(t_h4s_nums_num,V_x),s(t_h4s_nums_num,h4s_arithmetics_u_2a(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n))))))).
fof(ah4s_numerals_numeralu_u_distribu_c21, axiom, ![V_n]: s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,h4s_nums_0),s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,V_n))))) = s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,h4s_arithmetics_zero),s(t_h4s_nums_num,V_n)))).
fof(ah4s_numerals_numeralu_u_ltu_c1, axiom, ![V_n]: s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,h4s_arithmetics_zero),s(t_h4s_nums_num,h4s_arithmetics_bit2(s(t_h4s_nums_num,V_n))))) = s(t_bool,t)).
fof(ah4s_bits_BITu_u_def, axiom, ![V_n, V_b]: (p(s(t_bool,h4s_bits_bit(s(t_h4s_nums_num,V_b),s(t_h4s_nums_num,V_n)))) <=> s(t_h4s_nums_num,h4s_bits_bits(s(t_h4s_nums_num,V_b),s(t_h4s_nums_num,V_b),s(t_h4s_nums_num,V_n))) = 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_bits_ZEROu_u_LTu_u_TWOEXP, axiom, ![V_n]: p(s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,h4s_nums_0),s(t_h4s_nums_num,h4s_arithmetics_exp(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_nums_num,V_n))))))).
fof(ah4s_bits_BITSu_u_THM, axiom, ![V_n, V_l, V_h]: s(t_h4s_nums_num,h4s_bits_bits(s(t_h4s_nums_num,V_h),s(t_h4s_nums_num,V_l),s(t_h4s_nums_num,V_n))) = s(t_h4s_nums_num,h4s_arithmetics_mod(s(t_h4s_nums_num,h4s_arithmetics_div(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,h4s_arithmetics_exp(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_nums_num,V_l))))),s(t_h4s_nums_num,h4s_arithmetics_exp(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_nums_num,h4s_arithmetics_u_2d(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_h))),s(t_h4s_nums_num,V_l)))))))).
fof(ch4s_bits_BITu_u_DIV2, conjecture, ![V_n, V_i]: s(t_bool,h4s_bits_bit(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,h4s_arithmetics_div(s(t_h4s_nums_num,V_i),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_bool,h4s_bits_bit(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,V_i)))).
