%   ORIGINAL: h4/rat/RAT__DIV__CONG2
% 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/NOT__CLAUSES_c2: ~F <=> 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/integer/INT__MUL__SYM: !y x. h4/integer/int__mul x y = h4/integer/int__mul y x
% Assm: h4/integer/INT__MUL__ASSOC: !z y x. h4/integer/int__mul x (h4/integer/int__mul y z) = h4/integer/int__mul (h4/integer/int__mul x y) z
% Assm: h4/integer/INT__EQ__LMUL: !z y x. h4/integer/int__mul x y = h4/integer/int__mul x z <=> x = h4/integer/int__of__num h4/num/0 \/ y = z
% Assm: h4/intExtension/INT__MUL__POS__SIGN: !b a. h4/integer/int__lt (h4/integer/int__of__num h4/num/0) a ==> h4/integer/int__lt (h4/integer/int__of__num h4/num/0) b ==> h4/integer/int__lt (h4/integer/int__of__num h4/num/0) (h4/integer/int__mul a b)
% Assm: h4/frac/frac__mul__def: !f2 f1. h4/frac/frac__mul f1 f2 = h4/frac/abs__frac (h4/pair/_2C (h4/integer/int__mul (h4/frac/frac__nmr f1) (h4/frac/frac__nmr f2)) (h4/integer/int__mul (h4/frac/frac__dnm f1) (h4/frac/frac__dnm f2)))
% Assm: h4/frac/frac__div__def: !f2 f1. h4/frac/frac__div f1 f2 = h4/frac/frac__mul f1 (h4/frac/frac__minv f2)
% Assm: h4/frac/FRAC__DNMPOS: !f. h4/integer/int__lt (h4/integer/int__of__num h4/num/0) (h4/frac/frac__dnm f)
% Assm: h4/frac/NMR: !b a. h4/integer/int__lt (h4/integer/int__of__num h4/num/0) b ==> h4/frac/frac__nmr (h4/frac/abs__frac (h4/pair/_2C a b)) = a
% Assm: h4/frac/DNM: !b a. h4/integer/int__lt (h4/integer/int__of__num h4/num/0) b ==> h4/frac/frac__dnm (h4/frac/abs__frac (h4/pair/_2C a b)) = b
% Assm: h4/rat/rat__equiv__def: !f2 f1. h4/rat/rat__equiv f1 f2 <=> h4/integer/int__mul (h4/frac/frac__nmr f1) (h4/frac/frac__dnm f2) = h4/integer/int__mul (h4/frac/frac__nmr f2) (h4/frac/frac__dnm f1)
% Assm: h4/rat/RAT__ABS__EQUIV: !f2 f1. h4/rat/abs__rat f1 = h4/rat/abs__rat f2 <=> h4/rat/rat__equiv f1 f2
% Assm: h4/rat/RAT__MINV__CONG: !x. ~(h4/frac/frac__nmr x = h4/integer/int__of__num h4/num/0) ==> h4/rat/abs__rat (h4/frac/frac__minv (h4/rat/rep__rat (h4/rat/abs__rat x))) = h4/rat/abs__rat (h4/frac/frac__minv x)
% Goal: !y x. ~(h4/frac/frac__nmr y = h4/integer/int__of__num h4/num/0) ==> h4/rat/abs__rat (h4/frac/frac__div x (h4/rat/rep__rat (h4/rat/abs__rat y))) = h4/rat/abs__rat (h4/frac/frac__div x y)
%   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_NOTu_u_CLAUSESu_c2]: ~F <=> 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_integers_INTu_u_MULu_u_SYM]: !y x. h4/integer/int__mul x y = h4/integer/int__mul y x
% Assm [h4s_integers_INTu_u_MULu_u_ASSOC]: !z y x. h4/integer/int__mul x (h4/integer/int__mul y z) = h4/integer/int__mul (h4/integer/int__mul x y) z
% Assm [h4s_integers_INTu_u_EQu_u_LMUL]: !z y x. h4/integer/int__mul x y = h4/integer/int__mul x z <=> x = h4/integer/int__of__num h4/num/0 \/ y = z
% Assm [h4s_intExtensions_INTu_u_MULu_u_POSu_u_SIGN]: !b a. h4/integer/int__lt (h4/integer/int__of__num h4/num/0) a ==> h4/integer/int__lt (h4/integer/int__of__num h4/num/0) b ==> h4/integer/int__lt (h4/integer/int__of__num h4/num/0) (h4/integer/int__mul a b)
% Assm [h4s_fracs_fracu_u_mulu_u_def]: !f2 f1. h4/frac/frac__mul f1 f2 = h4/frac/abs__frac (h4/pair/_2C (h4/integer/int__mul (h4/frac/frac__nmr f1) (h4/frac/frac__nmr f2)) (h4/integer/int__mul (h4/frac/frac__dnm f1) (h4/frac/frac__dnm f2)))
% Assm [h4s_fracs_fracu_u_divu_u_def]: !f2 f1. h4/frac/frac__div f1 f2 = h4/frac/frac__mul f1 (h4/frac/frac__minv f2)
% Assm [h4s_fracs_FRACu_u_DNMPOS]: !f. h4/integer/int__lt (h4/integer/int__of__num h4/num/0) (h4/frac/frac__dnm f)
% Assm [h4s_fracs_NMR]: !b a. h4/integer/int__lt (h4/integer/int__of__num h4/num/0) b ==> h4/frac/frac__nmr (h4/frac/abs__frac (h4/pair/_2C a b)) = a
% Assm [h4s_fracs_DNM]: !b a. h4/integer/int__lt (h4/integer/int__of__num h4/num/0) b ==> h4/frac/frac__dnm (h4/frac/abs__frac (h4/pair/_2C a b)) = b
% Assm [h4s_rats_ratu_u_equivu_u_def]: !f2 f1. h4/rat/rat__equiv f1 f2 <=> h4/integer/int__mul (h4/frac/frac__nmr f1) (h4/frac/frac__dnm f2) = h4/integer/int__mul (h4/frac/frac__nmr f2) (h4/frac/frac__dnm f1)
% Assm [h4s_rats_RATu_u_ABSu_u_EQUIV]: !f2 f1. h4/rat/abs__rat f1 = h4/rat/abs__rat f2 <=> h4/rat/rat__equiv f1 f2
% Assm [h4s_rats_RATu_u_MINVu_u_CONG]: !x. ~(h4/frac/frac__nmr x = h4/integer/int__of__num h4/num/0) ==> h4/rat/abs__rat (h4/frac/frac__minv (h4/rat/rep__rat (h4/rat/abs__rat x))) = h4/rat/abs__rat (h4/frac/frac__minv x)
% Goal: !y x. ~(h4/frac/frac__nmr y = h4/integer/int__of__num h4/num/0) ==> h4/rat/abs__rat (h4/frac/frac__div x (h4/rat/rep__rat (h4/rat/abs__rat y))) = h4/rat/abs__rat (h4/frac/frac__div x y)
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_Q92038,TV_Q92034]: ![V_f, V_g]: (![V_x]: s(TV_Q92034,happ(s(t_fun(TV_Q92038,TV_Q92034),V_f),s(TV_Q92038,V_x))) = s(TV_Q92034,happ(s(t_fun(TV_Q92038,TV_Q92034),V_g),s(TV_Q92038,V_x))) => s(t_fun(TV_Q92038,TV_Q92034),V_f) = s(t_fun(TV_Q92038,TV_Q92034),V_g))).
fof(ah4s_bools_TRUTH, axiom, p(s(t_bool,t))).
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_NOTu_u_CLAUSESu_c2, axiom, (~ (p(s(t_bool,f))) <=> p(s(t_bool,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_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_integers_INTu_u_MULu_u_SYM, axiom, ![V_y, V_x]: s(t_h4s_integers_int,h4s_integers_intu_u_mul(s(t_h4s_integers_int,V_x),s(t_h4s_integers_int,V_y))) = s(t_h4s_integers_int,h4s_integers_intu_u_mul(s(t_h4s_integers_int,V_y),s(t_h4s_integers_int,V_x)))).
fof(ah4s_integers_INTu_u_MULu_u_ASSOC, axiom, ![V_z, V_y, V_x]: s(t_h4s_integers_int,h4s_integers_intu_u_mul(s(t_h4s_integers_int,V_x),s(t_h4s_integers_int,h4s_integers_intu_u_mul(s(t_h4s_integers_int,V_y),s(t_h4s_integers_int,V_z))))) = s(t_h4s_integers_int,h4s_integers_intu_u_mul(s(t_h4s_integers_int,h4s_integers_intu_u_mul(s(t_h4s_integers_int,V_x),s(t_h4s_integers_int,V_y))),s(t_h4s_integers_int,V_z)))).
fof(ah4s_integers_INTu_u_EQu_u_LMUL, axiom, ![V_z, V_y, V_x]: (s(t_h4s_integers_int,h4s_integers_intu_u_mul(s(t_h4s_integers_int,V_x),s(t_h4s_integers_int,V_y))) = s(t_h4s_integers_int,h4s_integers_intu_u_mul(s(t_h4s_integers_int,V_x),s(t_h4s_integers_int,V_z))) <=> (s(t_h4s_integers_int,V_x) = s(t_h4s_integers_int,h4s_integers_intu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))) | s(t_h4s_integers_int,V_y) = s(t_h4s_integers_int,V_z)))).
fof(ah4s_intExtensions_INTu_u_MULu_u_POSu_u_SIGN, axiom, ![V_b, V_a]: (p(s(t_bool,h4s_integers_intu_u_lt(s(t_h4s_integers_int,h4s_integers_intu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))),s(t_h4s_integers_int,V_a)))) => (p(s(t_bool,h4s_integers_intu_u_lt(s(t_h4s_integers_int,h4s_integers_intu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))),s(t_h4s_integers_int,V_b)))) => p(s(t_bool,h4s_integers_intu_u_lt(s(t_h4s_integers_int,h4s_integers_intu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))),s(t_h4s_integers_int,h4s_integers_intu_u_mul(s(t_h4s_integers_int,V_a),s(t_h4s_integers_int,V_b))))))))).
fof(ah4s_fracs_fracu_u_mulu_u_def, axiom, ![V_f2, V_f1]: s(t_h4s_fracs_frac,h4s_fracs_fracu_u_mul(s(t_h4s_fracs_frac,V_f1),s(t_h4s_fracs_frac,V_f2))) = s(t_h4s_fracs_frac,h4s_fracs_absu_u_frac(s(t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int),h4s_pairs_u_2c(s(t_h4s_integers_int,h4s_integers_intu_u_mul(s(t_h4s_integers_int,h4s_fracs_fracu_u_nmr(s(t_h4s_fracs_frac,V_f1))),s(t_h4s_integers_int,h4s_fracs_fracu_u_nmr(s(t_h4s_fracs_frac,V_f2))))),s(t_h4s_integers_int,h4s_integers_intu_u_mul(s(t_h4s_integers_int,h4s_fracs_fracu_u_dnm(s(t_h4s_fracs_frac,V_f1))),s(t_h4s_integers_int,h4s_fracs_fracu_u_dnm(s(t_h4s_fracs_frac,V_f2)))))))))).
fof(ah4s_fracs_fracu_u_divu_u_def, axiom, ![V_f2, V_f1]: s(t_h4s_fracs_frac,h4s_fracs_fracu_u_div(s(t_h4s_fracs_frac,V_f1),s(t_h4s_fracs_frac,V_f2))) = s(t_h4s_fracs_frac,h4s_fracs_fracu_u_mul(s(t_h4s_fracs_frac,V_f1),s(t_h4s_fracs_frac,h4s_fracs_fracu_u_minv(s(t_h4s_fracs_frac,V_f2)))))).
fof(ah4s_fracs_FRACu_u_DNMPOS, axiom, ![V_f]: p(s(t_bool,h4s_integers_intu_u_lt(s(t_h4s_integers_int,h4s_integers_intu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))),s(t_h4s_integers_int,h4s_fracs_fracu_u_dnm(s(t_h4s_fracs_frac,V_f))))))).
fof(ah4s_fracs_NMR, axiom, ![V_b, V_a]: (p(s(t_bool,h4s_integers_intu_u_lt(s(t_h4s_integers_int,h4s_integers_intu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))),s(t_h4s_integers_int,V_b)))) => s(t_h4s_integers_int,h4s_fracs_fracu_u_nmr(s(t_h4s_fracs_frac,h4s_fracs_absu_u_frac(s(t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int),h4s_pairs_u_2c(s(t_h4s_integers_int,V_a),s(t_h4s_integers_int,V_b))))))) = s(t_h4s_integers_int,V_a))).
fof(ah4s_fracs_DNM, axiom, ![V_b, V_a]: (p(s(t_bool,h4s_integers_intu_u_lt(s(t_h4s_integers_int,h4s_integers_intu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))),s(t_h4s_integers_int,V_b)))) => s(t_h4s_integers_int,h4s_fracs_fracu_u_dnm(s(t_h4s_fracs_frac,h4s_fracs_absu_u_frac(s(t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int),h4s_pairs_u_2c(s(t_h4s_integers_int,V_a),s(t_h4s_integers_int,V_b))))))) = s(t_h4s_integers_int,V_b))).
fof(ah4s_rats_ratu_u_equivu_u_def, axiom, ![V_f2, V_f1]: (p(s(t_bool,h4s_rats_ratu_u_equiv(s(t_h4s_fracs_frac,V_f1),s(t_h4s_fracs_frac,V_f2)))) <=> s(t_h4s_integers_int,h4s_integers_intu_u_mul(s(t_h4s_integers_int,h4s_fracs_fracu_u_nmr(s(t_h4s_fracs_frac,V_f1))),s(t_h4s_integers_int,h4s_fracs_fracu_u_dnm(s(t_h4s_fracs_frac,V_f2))))) = s(t_h4s_integers_int,h4s_integers_intu_u_mul(s(t_h4s_integers_int,h4s_fracs_fracu_u_nmr(s(t_h4s_fracs_frac,V_f2))),s(t_h4s_integers_int,h4s_fracs_fracu_u_dnm(s(t_h4s_fracs_frac,V_f1))))))).
fof(ah4s_rats_RATu_u_ABSu_u_EQUIV, axiom, ![V_f2, V_f1]: (s(t_h4s_rats_rat,h4s_rats_absu_u_rat(s(t_h4s_fracs_frac,V_f1))) = s(t_h4s_rats_rat,h4s_rats_absu_u_rat(s(t_h4s_fracs_frac,V_f2))) <=> p(s(t_bool,h4s_rats_ratu_u_equiv(s(t_h4s_fracs_frac,V_f1),s(t_h4s_fracs_frac,V_f2)))))).
fof(ah4s_rats_RATu_u_MINVu_u_CONG, axiom, ![V_x]: (~ (s(t_h4s_integers_int,h4s_fracs_fracu_u_nmr(s(t_h4s_fracs_frac,V_x))) = s(t_h4s_integers_int,h4s_integers_intu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0)))) => s(t_h4s_rats_rat,h4s_rats_absu_u_rat(s(t_h4s_fracs_frac,h4s_fracs_fracu_u_minv(s(t_h4s_fracs_frac,h4s_rats_repu_u_rat(s(t_h4s_rats_rat,h4s_rats_absu_u_rat(s(t_h4s_fracs_frac,V_x))))))))) = s(t_h4s_rats_rat,h4s_rats_absu_u_rat(s(t_h4s_fracs_frac,h4s_fracs_fracu_u_minv(s(t_h4s_fracs_frac,V_x))))))).
fof(ch4s_rats_RATu_u_DIVu_u_CONG2, conjecture, ![V_y, V_x]: (~ (s(t_h4s_integers_int,h4s_fracs_fracu_u_nmr(s(t_h4s_fracs_frac,V_y))) = s(t_h4s_integers_int,h4s_integers_intu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0)))) => s(t_h4s_rats_rat,h4s_rats_absu_u_rat(s(t_h4s_fracs_frac,h4s_fracs_fracu_u_div(s(t_h4s_fracs_frac,V_x),s(t_h4s_fracs_frac,h4s_rats_repu_u_rat(s(t_h4s_rats_rat,h4s_rats_absu_u_rat(s(t_h4s_fracs_frac,V_y))))))))) = s(t_h4s_rats_rat,h4s_rats_absu_u_rat(s(t_h4s_fracs_frac,h4s_fracs_fracu_u_div(s(t_h4s_fracs_frac,V_x),s(t_h4s_fracs_frac,V_y))))))).
