%   ORIGINAL: h4/rat/RAT__LES__RMUL__NEG
% 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/AND__CLAUSES_c0: !t. T /\ t <=> 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/numeral/numeral__distrib_c17: !n. h4/arithmetic/NUMERAL n = h4/num/0 <=> n = h4/arithmetic/ZERO
% Assm: h4/numeral/numeral__eq_c1: !n. h4/arithmetic/BIT1 n = h4/arithmetic/ZERO <=> F
% Assm: h4/integer/INT__MUL__LID: !x. h4/integer/int__mul (h4/integer/int__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) x = x
% Assm: h4/integer/INT__NEG__EQ: !y x. h4/integer/int__neg x = y <=> x = h4/integer/int__neg y
% Assm: h4/integer/INT__INJ: !n m. h4/integer/int__of__num m = h4/integer/int__of__num n <=> m = n
% Assm: h4/integer/INT__EQ__NEG: !y x. h4/integer/int__neg x = h4/integer/int__neg y <=> x = y
% Assm: h4/integer/INT__EQ__CALCULATE_c1: !y x. h4/integer/int__neg x = h4/integer/int__neg y <=> x = y
% Assm: h4/integer/INT__EQ__CALCULATE_c2: !n m. h4/integer/int__of__num n = h4/integer/int__neg (h4/integer/int__of__num m) <=> n = h4/num/0 /\ m = h4/num/0
% Assm: h4/rat/rat__les__def: !r2 r1. h4/rat/rat__les r1 r2 <=> h4/rat/rat__sgn (h4/rat/rat__sub r2 r1) = h4/integer/int__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))
% Assm: h4/rat/RAT__MUL__ASSOC: !c b a. h4/rat/rat__mul a (h4/rat/rat__mul b c) = h4/rat/rat__mul (h4/rat/rat__mul a b) c
% Assm: h4/rat/RAT__ADD__COMM: !b a. h4/rat/rat__add a b = h4/rat/rat__add b a
% Assm: h4/rat/RAT__ADD__LID: !a. h4/rat/rat__add (h4/rat/rat__of__num h4/num/0) a = a
% Assm: h4/rat/RAT__MUL__RID: !a. h4/rat/rat__mul a (h4/rat/rat__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) = a
% Assm: h4/rat/RAT__MUL__RINV: !a. ~(a = h4/rat/rat__of__num h4/num/0) ==> h4/rat/rat__mul a (h4/rat/rat__minv a) = h4/rat/rat__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))
% Assm: h4/rat/RAT__RDISTRIB: !c b a. h4/rat/rat__mul (h4/rat/rat__add a b) c = h4/rat/rat__add (h4/rat/rat__mul a c) (h4/rat/rat__mul b c)
% Assm: h4/rat/RAT__SUB__ADDAINV: !r2 r1. h4/rat/rat__sub r1 r2 = h4/rat/rat__add r1 (h4/rat/rat__ainv r2)
% Assm: h4/rat/RAT__AINV__ADD: !r2 r1. h4/rat/rat__ainv (h4/rat/rat__add r1 r2) = h4/rat/rat__add (h4/rat/rat__ainv r1) (h4/rat/rat__ainv r2)
% Assm: h4/rat/RAT__AINV__LMUL: !r2 r1. h4/rat/rat__ainv (h4/rat/rat__mul r1 r2) = h4/rat/rat__mul (h4/rat/rat__ainv r1) r2
% Assm: h4/rat/RAT__SGN__CLAUSES_c0: !r1. h4/rat/rat__sgn r1 = h4/integer/int__neg (h4/integer/int__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) <=> h4/rat/rat__les r1 (h4/rat/rat__of__num h4/num/0)
% Assm: h4/rat/RAT__SGN__CLAUSES_c1: !r1. h4/rat/rat__sgn r1 = h4/integer/int__of__num h4/num/0 <=> r1 = h4/rat/rat__of__num h4/num/0
% Assm: h4/rat/RAT__SGN__CLAUSES_c2: !r1. h4/rat/rat__sgn r1 = h4/integer/int__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) <=> h4/rat/rat__gre r1 (h4/rat/rat__of__num h4/num/0)
% Assm: h4/rat/RAT__SGN__AINV: !r1. h4/integer/int__neg (h4/rat/rat__sgn (h4/rat/rat__ainv r1)) = h4/rat/rat__sgn r1
% Assm: h4/rat/RAT__SGN__MUL: !r2 r1. h4/rat/rat__sgn (h4/rat/rat__mul r1 r2) = h4/integer/int__mul (h4/rat/rat__sgn r1) (h4/rat/rat__sgn r2)
% Assm: h4/rat/RAT__SGN__MINV: !r1. ~(r1 = h4/rat/rat__of__num h4/num/0) ==> h4/rat/rat__sgn (h4/rat/rat__minv r1) = h4/rat/rat__sgn r1
% Assm: h4/rat/RAT__AINV__EQ: !r2 r1. h4/rat/rat__ainv r1 = r2 <=> r1 = h4/rat/rat__ainv r2
% Goal: !r3 r2 r1. h4/rat/rat__les r3 (h4/rat/rat__of__num h4/num/0) ==> (h4/rat/rat__les (h4/rat/rat__mul r2 r3) (h4/rat/rat__mul r1 r3) <=> h4/rat/rat__les r1 r2)
%   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_ANDu_u_CLAUSESu_c0]: !t. T /\ t <=> 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_numerals_numeralu_u_distribu_c17]: !n. h4/arithmetic/NUMERAL n = h4/num/0 <=> n = h4/arithmetic/ZERO
% Assm [h4s_numerals_numeralu_u_equ_c1]: !n. h4/arithmetic/BIT1 n = h4/arithmetic/ZERO <=> F
% Assm [h4s_integers_INTu_u_MULu_u_LID]: !x. h4/integer/int__mul (h4/integer/int__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) x = x
% Assm [h4s_integers_INTu_u_NEGu_u_EQ]: !y x. h4/integer/int__neg x = y <=> x = h4/integer/int__neg y
% Assm [h4s_integers_INTu_u_INJ]: !n m. h4/integer/int__of__num m = h4/integer/int__of__num n <=> m = n
% Assm [h4s_integers_INTu_u_EQu_u_NEG]: !y x. h4/integer/int__neg x = h4/integer/int__neg y <=> x = y
% Assm [h4s_integers_INTu_u_EQu_u_CALCULATEu_c1]: !y x. h4/integer/int__neg x = h4/integer/int__neg y <=> x = y
% Assm [h4s_integers_INTu_u_EQu_u_CALCULATEu_c2]: !n m. h4/integer/int__of__num n = h4/integer/int__neg (h4/integer/int__of__num m) <=> n = h4/num/0 /\ m = h4/num/0
% Assm [h4s_rats_ratu_u_lesu_u_def]: !r2 r1. h4/rat/rat__les r1 r2 <=> h4/rat/rat__sgn (h4/rat/rat__sub r2 r1) = h4/integer/int__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))
% Assm [h4s_rats_RATu_u_MULu_u_ASSOC]: !c b a. h4/rat/rat__mul a (h4/rat/rat__mul b c) = h4/rat/rat__mul (h4/rat/rat__mul a b) c
% Assm [h4s_rats_RATu_u_ADDu_u_COMM]: !b a. h4/rat/rat__add a b = h4/rat/rat__add b a
% Assm [h4s_rats_RATu_u_ADDu_u_LID]: !a. h4/rat/rat__add (h4/rat/rat__of__num h4/num/0) a = a
% Assm [h4s_rats_RATu_u_MULu_u_RID]: !a. h4/rat/rat__mul a (h4/rat/rat__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) = a
% Assm [h4s_rats_RATu_u_MULu_u_RINV]: !a. ~(a = h4/rat/rat__of__num h4/num/0) ==> h4/rat/rat__mul a (h4/rat/rat__minv a) = h4/rat/rat__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))
% Assm [h4s_rats_RATu_u_RDISTRIB]: !c b a. h4/rat/rat__mul (h4/rat/rat__add a b) c = h4/rat/rat__add (h4/rat/rat__mul a c) (h4/rat/rat__mul b c)
% Assm [h4s_rats_RATu_u_SUBu_u_ADDAINV]: !r2 r1. h4/rat/rat__sub r1 r2 = h4/rat/rat__add r1 (h4/rat/rat__ainv r2)
% Assm [h4s_rats_RATu_u_AINVu_u_ADD]: !r2 r1. h4/rat/rat__ainv (h4/rat/rat__add r1 r2) = h4/rat/rat__add (h4/rat/rat__ainv r1) (h4/rat/rat__ainv r2)
% Assm [h4s_rats_RATu_u_AINVu_u_LMUL]: !r2 r1. h4/rat/rat__ainv (h4/rat/rat__mul r1 r2) = h4/rat/rat__mul (h4/rat/rat__ainv r1) r2
% Assm [h4s_rats_RATu_u_SGNu_u_CLAUSESu_c0]: !r1. h4/rat/rat__sgn r1 = h4/integer/int__neg (h4/integer/int__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) <=> h4/rat/rat__les r1 (h4/rat/rat__of__num h4/num/0)
% Assm [h4s_rats_RATu_u_SGNu_u_CLAUSESu_c1]: !r1. h4/rat/rat__sgn r1 = h4/integer/int__of__num h4/num/0 <=> r1 = h4/rat/rat__of__num h4/num/0
% Assm [h4s_rats_RATu_u_SGNu_u_CLAUSESu_c2]: !r1. h4/rat/rat__sgn r1 = h4/integer/int__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) <=> h4/rat/rat__gre r1 (h4/rat/rat__of__num h4/num/0)
% Assm [h4s_rats_RATu_u_SGNu_u_AINV]: !r1. h4/integer/int__neg (h4/rat/rat__sgn (h4/rat/rat__ainv r1)) = h4/rat/rat__sgn r1
% Assm [h4s_rats_RATu_u_SGNu_u_MUL]: !r2 r1. h4/rat/rat__sgn (h4/rat/rat__mul r1 r2) = h4/integer/int__mul (h4/rat/rat__sgn r1) (h4/rat/rat__sgn r2)
% Assm [h4s_rats_RATu_u_SGNu_u_MINV]: !r1. ~(r1 = h4/rat/rat__of__num h4/num/0) ==> h4/rat/rat__sgn (h4/rat/rat__minv r1) = h4/rat/rat__sgn r1
% Assm [h4s_rats_RATu_u_AINVu_u_EQ]: !r2 r1. h4/rat/rat__ainv r1 = r2 <=> r1 = h4/rat/rat__ainv r2
% Goal: !r3 r2 r1. h4/rat/rat__les r3 (h4/rat/rat__of__num h4/num/0) ==> (h4/rat/rat__les (h4/rat/rat__mul r2 r3) (h4/rat/rat__mul r1 r3) <=> h4/rat/rat__les r1 r2)
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_Q94604,TV_Q94600]: ![V_f, V_g]: (![V_x]: s(TV_Q94600,happ(s(t_fun(TV_Q94604,TV_Q94600),V_f),s(TV_Q94604,V_x))) = s(TV_Q94600,happ(s(t_fun(TV_Q94604,TV_Q94600),V_g),s(TV_Q94604,V_x))) => s(t_fun(TV_Q94604,TV_Q94600),V_f) = s(t_fun(TV_Q94604,TV_Q94600),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_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_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_numerals_numeralu_u_distribu_c17, axiom, ![V_n]: (s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,V_n))) = s(t_h4s_nums_num,h4s_nums_0) <=> s(t_h4s_nums_num,V_n) = s(t_h4s_nums_num,h4s_arithmetics_zero))).
fof(ah4s_numerals_numeralu_u_equ_c1, axiom, ![V_n]: (s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,V_n))) = s(t_h4s_nums_num,h4s_arithmetics_zero) <=> p(s(t_bool,f)))).
fof(ah4s_integers_INTu_u_MULu_u_LID, axiom, ![V_x]: s(t_h4s_integers_int,h4s_integers_intu_u_mul(s(t_h4s_integers_int,h4s_integers_intu_u_ofu_u_num(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_integers_int,V_x))) = s(t_h4s_integers_int,V_x)).
fof(ah4s_integers_INTu_u_NEGu_u_EQ, axiom, ![V_y, V_x]: (s(t_h4s_integers_int,h4s_integers_intu_u_neg(s(t_h4s_integers_int,V_x))) = s(t_h4s_integers_int,V_y) <=> s(t_h4s_integers_int,V_x) = s(t_h4s_integers_int,h4s_integers_intu_u_neg(s(t_h4s_integers_int,V_y))))).
fof(ah4s_integers_INTu_u_INJ, axiom, ![V_n, V_m]: (s(t_h4s_integers_int,h4s_integers_intu_u_ofu_u_num(s(t_h4s_nums_num,V_m))) = s(t_h4s_integers_int,h4s_integers_intu_u_ofu_u_num(s(t_h4s_nums_num,V_n))) <=> s(t_h4s_nums_num,V_m) = s(t_h4s_nums_num,V_n))).
fof(ah4s_integers_INTu_u_EQu_u_NEG, axiom, ![V_y, V_x]: (s(t_h4s_integers_int,h4s_integers_intu_u_neg(s(t_h4s_integers_int,V_x))) = s(t_h4s_integers_int,h4s_integers_intu_u_neg(s(t_h4s_integers_int,V_y))) <=> s(t_h4s_integers_int,V_x) = s(t_h4s_integers_int,V_y))).
fof(ah4s_integers_INTu_u_EQu_u_CALCULATEu_c1, axiom, ![V_y, V_x]: (s(t_h4s_integers_int,h4s_integers_intu_u_neg(s(t_h4s_integers_int,V_x))) = s(t_h4s_integers_int,h4s_integers_intu_u_neg(s(t_h4s_integers_int,V_y))) <=> s(t_h4s_integers_int,V_x) = s(t_h4s_integers_int,V_y))).
fof(ah4s_integers_INTu_u_EQu_u_CALCULATEu_c2, axiom, ![V_n, V_m]: (s(t_h4s_integers_int,h4s_integers_intu_u_ofu_u_num(s(t_h4s_nums_num,V_n))) = s(t_h4s_integers_int,h4s_integers_intu_u_neg(s(t_h4s_integers_int,h4s_integers_intu_u_ofu_u_num(s(t_h4s_nums_num,V_m))))) <=> (s(t_h4s_nums_num,V_n) = s(t_h4s_nums_num,h4s_nums_0) & s(t_h4s_nums_num,V_m) = s(t_h4s_nums_num,h4s_nums_0)))).
fof(ah4s_rats_ratu_u_lesu_u_def, axiom, ![V_r2, V_r1]: (p(s(t_bool,h4s_rats_ratu_u_les(s(t_h4s_rats_rat,V_r1),s(t_h4s_rats_rat,V_r2)))) <=> s(t_h4s_integers_int,h4s_rats_ratu_u_sgn(s(t_h4s_rats_rat,h4s_rats_ratu_u_sub(s(t_h4s_rats_rat,V_r2),s(t_h4s_rats_rat,V_r1))))) = s(t_h4s_integers_int,h4s_integers_intu_u_ofu_u_num(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_rats_RATu_u_MULu_u_ASSOC, axiom, ![V_c, V_b, V_a]: s(t_h4s_rats_rat,h4s_rats_ratu_u_mul(s(t_h4s_rats_rat,V_a),s(t_h4s_rats_rat,h4s_rats_ratu_u_mul(s(t_h4s_rats_rat,V_b),s(t_h4s_rats_rat,V_c))))) = s(t_h4s_rats_rat,h4s_rats_ratu_u_mul(s(t_h4s_rats_rat,h4s_rats_ratu_u_mul(s(t_h4s_rats_rat,V_a),s(t_h4s_rats_rat,V_b))),s(t_h4s_rats_rat,V_c)))).
fof(ah4s_rats_RATu_u_ADDu_u_COMM, axiom, ![V_b, V_a]: s(t_h4s_rats_rat,h4s_rats_ratu_u_add(s(t_h4s_rats_rat,V_a),s(t_h4s_rats_rat,V_b))) = s(t_h4s_rats_rat,h4s_rats_ratu_u_add(s(t_h4s_rats_rat,V_b),s(t_h4s_rats_rat,V_a)))).
fof(ah4s_rats_RATu_u_ADDu_u_LID, axiom, ![V_a]: s(t_h4s_rats_rat,h4s_rats_ratu_u_add(s(t_h4s_rats_rat,h4s_rats_ratu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))),s(t_h4s_rats_rat,V_a))) = s(t_h4s_rats_rat,V_a)).
fof(ah4s_rats_RATu_u_MULu_u_RID, axiom, ![V_a]: s(t_h4s_rats_rat,h4s_rats_ratu_u_mul(s(t_h4s_rats_rat,V_a),s(t_h4s_rats_rat,h4s_rats_ratu_u_ofu_u_num(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_rats_rat,V_a)).
fof(ah4s_rats_RATu_u_MULu_u_RINV, axiom, ![V_a]: (~ (s(t_h4s_rats_rat,V_a) = s(t_h4s_rats_rat,h4s_rats_ratu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0)))) => s(t_h4s_rats_rat,h4s_rats_ratu_u_mul(s(t_h4s_rats_rat,V_a),s(t_h4s_rats_rat,h4s_rats_ratu_u_minv(s(t_h4s_rats_rat,V_a))))) = s(t_h4s_rats_rat,h4s_rats_ratu_u_ofu_u_num(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_rats_RATu_u_RDISTRIB, axiom, ![V_c, V_b, V_a]: s(t_h4s_rats_rat,h4s_rats_ratu_u_mul(s(t_h4s_rats_rat,h4s_rats_ratu_u_add(s(t_h4s_rats_rat,V_a),s(t_h4s_rats_rat,V_b))),s(t_h4s_rats_rat,V_c))) = s(t_h4s_rats_rat,h4s_rats_ratu_u_add(s(t_h4s_rats_rat,h4s_rats_ratu_u_mul(s(t_h4s_rats_rat,V_a),s(t_h4s_rats_rat,V_c))),s(t_h4s_rats_rat,h4s_rats_ratu_u_mul(s(t_h4s_rats_rat,V_b),s(t_h4s_rats_rat,V_c)))))).
fof(ah4s_rats_RATu_u_SUBu_u_ADDAINV, axiom, ![V_r2, V_r1]: s(t_h4s_rats_rat,h4s_rats_ratu_u_sub(s(t_h4s_rats_rat,V_r1),s(t_h4s_rats_rat,V_r2))) = s(t_h4s_rats_rat,h4s_rats_ratu_u_add(s(t_h4s_rats_rat,V_r1),s(t_h4s_rats_rat,h4s_rats_ratu_u_ainv(s(t_h4s_rats_rat,V_r2)))))).
fof(ah4s_rats_RATu_u_AINVu_u_ADD, axiom, ![V_r2, V_r1]: s(t_h4s_rats_rat,h4s_rats_ratu_u_ainv(s(t_h4s_rats_rat,h4s_rats_ratu_u_add(s(t_h4s_rats_rat,V_r1),s(t_h4s_rats_rat,V_r2))))) = s(t_h4s_rats_rat,h4s_rats_ratu_u_add(s(t_h4s_rats_rat,h4s_rats_ratu_u_ainv(s(t_h4s_rats_rat,V_r1))),s(t_h4s_rats_rat,h4s_rats_ratu_u_ainv(s(t_h4s_rats_rat,V_r2)))))).
fof(ah4s_rats_RATu_u_AINVu_u_LMUL, axiom, ![V_r2, V_r1]: s(t_h4s_rats_rat,h4s_rats_ratu_u_ainv(s(t_h4s_rats_rat,h4s_rats_ratu_u_mul(s(t_h4s_rats_rat,V_r1),s(t_h4s_rats_rat,V_r2))))) = s(t_h4s_rats_rat,h4s_rats_ratu_u_mul(s(t_h4s_rats_rat,h4s_rats_ratu_u_ainv(s(t_h4s_rats_rat,V_r1))),s(t_h4s_rats_rat,V_r2)))).
fof(ah4s_rats_RATu_u_SGNu_u_CLAUSESu_c0, axiom, ![V_r1]: (s(t_h4s_integers_int,h4s_rats_ratu_u_sgn(s(t_h4s_rats_rat,V_r1))) = s(t_h4s_integers_int,h4s_integers_intu_u_neg(s(t_h4s_integers_int,h4s_integers_intu_u_ofu_u_num(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero))))))))) <=> p(s(t_bool,h4s_rats_ratu_u_les(s(t_h4s_rats_rat,V_r1),s(t_h4s_rats_rat,h4s_rats_ratu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0)))))))).
fof(ah4s_rats_RATu_u_SGNu_u_CLAUSESu_c1, axiom, ![V_r1]: (s(t_h4s_integers_int,h4s_rats_ratu_u_sgn(s(t_h4s_rats_rat,V_r1))) = 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,V_r1) = s(t_h4s_rats_rat,h4s_rats_ratu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))))).
fof(ah4s_rats_RATu_u_SGNu_u_CLAUSESu_c2, axiom, ![V_r1]: (s(t_h4s_integers_int,h4s_rats_ratu_u_sgn(s(t_h4s_rats_rat,V_r1))) = s(t_h4s_integers_int,h4s_integers_intu_u_ofu_u_num(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero))))))) <=> p(s(t_bool,h4s_rats_ratu_u_gre(s(t_h4s_rats_rat,V_r1),s(t_h4s_rats_rat,h4s_rats_ratu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0)))))))).
fof(ah4s_rats_RATu_u_SGNu_u_AINV, axiom, ![V_r1]: s(t_h4s_integers_int,h4s_integers_intu_u_neg(s(t_h4s_integers_int,h4s_rats_ratu_u_sgn(s(t_h4s_rats_rat,h4s_rats_ratu_u_ainv(s(t_h4s_rats_rat,V_r1))))))) = s(t_h4s_integers_int,h4s_rats_ratu_u_sgn(s(t_h4s_rats_rat,V_r1)))).
fof(ah4s_rats_RATu_u_SGNu_u_MUL, axiom, ![V_r2, V_r1]: s(t_h4s_integers_int,h4s_rats_ratu_u_sgn(s(t_h4s_rats_rat,h4s_rats_ratu_u_mul(s(t_h4s_rats_rat,V_r1),s(t_h4s_rats_rat,V_r2))))) = s(t_h4s_integers_int,h4s_integers_intu_u_mul(s(t_h4s_integers_int,h4s_rats_ratu_u_sgn(s(t_h4s_rats_rat,V_r1))),s(t_h4s_integers_int,h4s_rats_ratu_u_sgn(s(t_h4s_rats_rat,V_r2)))))).
fof(ah4s_rats_RATu_u_SGNu_u_MINV, axiom, ![V_r1]: (~ (s(t_h4s_rats_rat,V_r1) = s(t_h4s_rats_rat,h4s_rats_ratu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0)))) => s(t_h4s_integers_int,h4s_rats_ratu_u_sgn(s(t_h4s_rats_rat,h4s_rats_ratu_u_minv(s(t_h4s_rats_rat,V_r1))))) = s(t_h4s_integers_int,h4s_rats_ratu_u_sgn(s(t_h4s_rats_rat,V_r1))))).
fof(ah4s_rats_RATu_u_AINVu_u_EQ, axiom, ![V_r2, V_r1]: (s(t_h4s_rats_rat,h4s_rats_ratu_u_ainv(s(t_h4s_rats_rat,V_r1))) = s(t_h4s_rats_rat,V_r2) <=> s(t_h4s_rats_rat,V_r1) = s(t_h4s_rats_rat,h4s_rats_ratu_u_ainv(s(t_h4s_rats_rat,V_r2))))).
fof(ch4s_rats_RATu_u_LESu_u_RMULu_u_NEG, conjecture, ![V_r3, V_r2, V_r1]: (p(s(t_bool,h4s_rats_ratu_u_les(s(t_h4s_rats_rat,V_r3),s(t_h4s_rats_rat,h4s_rats_ratu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0)))))) => s(t_bool,h4s_rats_ratu_u_les(s(t_h4s_rats_rat,h4s_rats_ratu_u_mul(s(t_h4s_rats_rat,V_r2),s(t_h4s_rats_rat,V_r3))),s(t_h4s_rats_rat,h4s_rats_ratu_u_mul(s(t_h4s_rats_rat,V_r1),s(t_h4s_rats_rat,V_r3))))) = s(t_bool,h4s_rats_ratu_u_les(s(t_h4s_rats_rat,V_r1),s(t_h4s_rats_rat,V_r2))))).
