TPTP Problem File: NUM925+3.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : NUM925+3 : TPTP v8.2.0. Released v5.3.0.
% Domain : Number Theory
% Problem : Sum of two squares line 192, 1000 axioms selected
% Version : Especial.
% English :
% Refs : [BN10] Boehme & Nipkow (2010), Sledgehammer: Judgement Day
% : [Bla11] Blanchette (2011), Email to Geoff Sutcliffe
% Source : [Bla11]
% Names : s2s_1000_fofmg_l192 [Bla11]
% Status : Theorem
% Rating : 0.31 v8.2.0, 0.33 v7.5.0, 0.38 v7.4.0, 0.20 v7.3.0, 0.28 v7.1.0, 0.35 v7.0.0, 0.33 v6.4.0, 0.38 v6.3.0, 0.29 v6.2.0, 0.36 v6.1.0, 0.50 v6.0.0, 0.48 v5.5.0, 0.67 v5.4.0, 0.71 v5.3.0
% Syntax : Number of formulae : 1233 ( 454 unt; 0 def)
% Number of atoms : 2822 ( 890 equ)
% Maximal formula atoms : 9 ( 2 avg)
% Number of connectives : 1808 ( 219 ~; 66 |; 205 &)
% ( 335 <=>; 983 =>; 0 <=; 0 <~>)
% Maximal formula depth : 13 ( 4 avg)
% Maximal term depth : 10 ( 2 avg)
% Number of predicates : 4 ( 3 usr; 0 prp; 1-2 aty)
% Number of functors : 69 ( 69 usr; 33 con; 0-3 aty)
% Number of variables : 2527 (2507 !; 20 ?)
% SPC : FOF_THM_RFO_SEQ
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2011-08-09 17:41:47
% : Encoded with monomorphized guards.
%------------------------------------------------------------------------------
%----Explicit typings (28)
fof(gsy_c_Groups_Oabs__class_Oabs_000tc__Int__Oint,axiom,
! [B_1_1] :
( is_int(B_1_1)
=> is_int(abs_abs_int(B_1_1)) ) ).
fof(gsy_c_Groups_Ominus__class_Ominus_000tc__Int__Oint,axiom,
! [B_1_1,B_2_1] :
( ( is_int(B_1_1)
& is_int(B_2_1) )
=> is_int(minus_minus_int(B_1_1,B_2_1)) ) ).
fof(gsy_c_Groups_Oone__class_Oone_000tc__Int__Oint,hypothesis,
is_int(one_one_int) ).
fof(gsy_c_Groups_Oplus__class_Oplus_000tc__Int__Oint,hypothesis,
! [B_1_1,B_2_1] :
( ( is_int(B_1_1)
& is_int(B_2_1) )
=> is_int(plus_plus_int(B_1_1,B_2_1)) ) ).
fof(gsy_c_Groups_Otimes__class_Otimes_000tc__Int__Oint,axiom,
! [B_1_1,B_2_1] :
( ( is_int(B_1_1)
& is_int(B_2_1) )
=> is_int(times_times_int(B_1_1,B_2_1)) ) ).
fof(gsy_c_Groups_Ozero__class_Ozero_000tc__Int__Oint,hypothesis,
is_int(zero_zero_int) ).
fof(gsy_c_HOL_Oundefined_000tc__Int__Oint,axiom,
is_int(undefined_int(int)) ).
fof(gsy_c_If_000tc__Int__Oint,axiom,
! [B_1_1,B_2_1,B_3_1] :
( ( is_bool(B_1_1)
& is_int(B_2_1)
& is_int(B_3_1) )
=> is_int(if_int(B_1_1,B_2_1,B_3_1)) ) ).
fof(gsy_c_Int_OBit0,hypothesis,
! [B_1_1] :
( is_int(B_1_1)
=> is_int(bit0(B_1_1)) ) ).
fof(gsy_c_Int_OBit1,hypothesis,
! [B_1_1] :
( is_int(B_1_1)
=> is_int(bit1(B_1_1)) ) ).
fof(gsy_c_Int_OMin,axiom,
is_int(min) ).
fof(gsy_c_Int_OPls,hypothesis,
is_int(pls) ).
fof(gsy_c_Int_Onumber__class_Onumber__of_000tc__Int__Oint,axiom,
! [B_1_1] :
( is_int(B_1_1)
=> is_int(number_number_of_int(B_1_1)) ) ).
fof(gsy_c_Int_Osucc,axiom,
! [B_1_1] :
( is_int(B_1_1)
=> is_int(succ(B_1_1)) ) ).
fof(gsy_c_Residues_OLegendre,axiom,
! [B_1_1,B_2_1] :
( ( is_int(B_1_1)
& is_int(B_2_1) )
=> is_int(legendre(B_1_1,B_2_1)) ) ).
fof(gsy_c_fFalse,axiom,
is_bool(fFalse) ).
fof(gsy_c_fTrue,axiom,
is_bool(fTrue) ).
fof(gsy_c_hAPP_000tc__Int__Oint_000tc__HOL__Obool,axiom,
! [B_1_1,B_2_1] :
( is_int(B_2_1)
=> is_bool(hAPP_int_bool(B_1_1,B_2_1)) ) ).
fof(gsy_c_hAPP_000tc__Nat__Onat_000tc__HOL__Obool,axiom,
! [B_1_1,B_2_1] : is_bool(hAPP_nat_bool(B_1_1,B_2_1)) ).
fof(gsy_c_hAPP_000tc__Nat__Onat_000tc__Int__Oint,hypothesis,
! [B_1_1,B_2_1] : is_int(hAPP_nat_int(B_1_1,B_2_1)) ).
fof(gsy_c_hAPP_000tc__RealDef__Oreal_000tc__HOL__Obool,axiom,
! [B_1_1,B_2_1] : is_bool(hAPP_real_bool(B_1_1,B_2_1)) ).
fof(gsy_v_m,axiom,
is_int(m) ).
fof(gsy_v_m1____,axiom,
is_int(m1) ).
fof(gsy_v_s1____,axiom,
is_int(s1) ).
fof(gsy_v_s____,axiom,
is_int(s) ).
fof(gsy_v_t____,axiom,
is_int(t) ).
fof(gsy_v_x____,axiom,
is_int(x) ).
fof(gsy_v_y____,axiom,
is_int(y) ).
%----Relevant facts (1198)
fof(fact_0_n1pos,axiom,
hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),plus_plus_int(one_one_int,hAPP_nat_int(semiri1621563631at_int,n)))) ).
fof(fact_1_t1,axiom,
hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),t)) ).
fof(fact_2_sum__power2__eq__zero__iff,axiom,
! [Xa,Ya] :
( ( is_int(Xa)
& is_int(Ya) )
=> ( plus_plus_int(hAPP_nat_int(power_power_int(Xa),number_number_of_nat(bit0(bit1(pls)))),hAPP_nat_int(power_power_int(Ya),number_number_of_nat(bit0(bit1(pls))))) = zero_zero_int
<=> ( Xa = zero_zero_int
& Ya = zero_zero_int ) ) ) ).
fof(fact_3_sum__power2__eq__zero__iff,axiom,
! [Xa,Ya] :
( plus_plus_real(hAPP_nat_real(power_power_real(Xa),number_number_of_nat(bit0(bit1(pls)))),hAPP_nat_real(power_power_real(Ya),number_number_of_nat(bit0(bit1(pls))))) = zero_zero_real
<=> ( Xa = zero_zero_real
& Ya = zero_zero_real ) ) ).
fof(fact_4_one__power2,axiom,
hAPP_nat_int(power_power_int(one_one_int),number_number_of_nat(bit0(bit1(pls)))) = one_one_int ).
fof(fact_5_one__power2,axiom,
hAPP_nat_nat(power_power_nat(one_one_nat),number_number_of_nat(bit0(bit1(pls)))) = one_one_nat ).
fof(fact_6_one__power2,axiom,
hAPP_nat_real(power_power_real(one_one_real),number_number_of_nat(bit0(bit1(pls)))) = one_one_real ).
fof(fact_7_zero__power2,axiom,
hAPP_nat_int(power_power_int(zero_zero_int),number_number_of_nat(bit0(bit1(pls)))) = zero_zero_int ).
fof(fact_8_zero__power2,axiom,
hAPP_nat_nat(power_power_nat(zero_zero_nat),number_number_of_nat(bit0(bit1(pls)))) = zero_zero_nat ).
fof(fact_9_zero__power2,axiom,
hAPP_nat_real(power_power_real(zero_zero_real),number_number_of_nat(bit0(bit1(pls)))) = zero_zero_real ).
fof(fact_10_zero__eq__power2,axiom,
! [A_1] :
( is_int(A_1)
=> ( hAPP_nat_int(power_power_int(A_1),number_number_of_nat(bit0(bit1(pls)))) = zero_zero_int
<=> A_1 = zero_zero_int ) ) ).
fof(fact_11_zero__eq__power2,axiom,
! [A_1] :
( hAPP_nat_real(power_power_real(A_1),number_number_of_nat(bit0(bit1(pls)))) = zero_zero_real
<=> A_1 = zero_zero_real ) ).
fof(fact_12_add__special_I2_J,axiom,
! [W_13] : plus_plus_int(one_one_int,number_number_of_int(W_13)) = number_number_of_int(plus_plus_int(bit1(pls),W_13)) ).
fof(fact_13_add__special_I2_J,axiom,
! [W_13] : plus_plus_real(one_one_real,number267125858f_real(W_13)) = number267125858f_real(plus_plus_int(bit1(pls),W_13)) ).
fof(fact_14_add__special_I3_J,axiom,
! [V_17] : plus_plus_int(number_number_of_int(V_17),one_one_int) = number_number_of_int(plus_plus_int(V_17,bit1(pls))) ).
fof(fact_15_add__special_I3_J,axiom,
! [V_17] : plus_plus_real(number267125858f_real(V_17),one_one_real) = number267125858f_real(plus_plus_int(V_17,bit1(pls))) ).
fof(fact_16_one__add__one__is__two,axiom,
plus_plus_int(one_one_int,one_one_int) = number_number_of_int(bit0(bit1(pls))) ).
fof(fact_17_one__add__one__is__two,axiom,
plus_plus_real(one_one_real,one_one_real) = number267125858f_real(bit0(bit1(pls))) ).
fof(fact_18_semiring__one__add__one__is__two,axiom,
plus_plus_int(one_one_int,one_one_int) = number_number_of_int(bit0(bit1(pls))) ).
fof(fact_19_semiring__one__add__one__is__two,axiom,
plus_plus_nat(one_one_nat,one_one_nat) = number_number_of_nat(bit0(bit1(pls))) ).
fof(fact_20_semiring__one__add__one__is__two,axiom,
plus_plus_real(one_one_real,one_one_real) = number267125858f_real(bit0(bit1(pls))) ).
fof(fact_21_quartic__square__square,axiom,
! [X] : hAPP_nat_int(power_power_int(hAPP_nat_int(power_power_int(X),number_number_of_nat(bit0(bit1(pls))))),number_number_of_nat(bit0(bit1(pls)))) = hAPP_nat_int(power_power_int(X),number_number_of_nat(bit0(bit0(bit1(pls))))) ).
fof(fact_22_power__0__left__number__of,axiom,
! [W_12] :
( ( number_number_of_nat(W_12) = zero_zero_nat
=> hAPP_nat_int(power_power_int(zero_zero_int),number_number_of_nat(W_12)) = one_one_int )
& ( number_number_of_nat(W_12) != zero_zero_nat
=> hAPP_nat_int(power_power_int(zero_zero_int),number_number_of_nat(W_12)) = zero_zero_int ) ) ).
fof(fact_23_power__0__left__number__of,axiom,
! [W_12] :
( ( number_number_of_nat(W_12) = zero_zero_nat
=> hAPP_nat_nat(power_power_nat(zero_zero_nat),number_number_of_nat(W_12)) = one_one_nat )
& ( number_number_of_nat(W_12) != zero_zero_nat
=> hAPP_nat_nat(power_power_nat(zero_zero_nat),number_number_of_nat(W_12)) = zero_zero_nat ) ) ).
fof(fact_24_power__0__left__number__of,axiom,
! [W_12] :
( ( number_number_of_nat(W_12) = zero_zero_nat
=> hAPP_nat_real(power_power_real(zero_zero_real),number_number_of_nat(W_12)) = one_one_real )
& ( number_number_of_nat(W_12) != zero_zero_nat
=> hAPP_nat_real(power_power_real(zero_zero_real),number_number_of_nat(W_12)) = zero_zero_real ) ) ).
fof(fact_25_semiring__norm_I110_J,axiom,
one_one_int = number_number_of_int(bit1(pls)) ).
fof(fact_26_semiring__norm_I110_J,axiom,
one_one_real = number267125858f_real(bit1(pls)) ).
fof(fact_27_numeral__1__eq__1,axiom,
number_number_of_int(bit1(pls)) = one_one_int ).
fof(fact_28_numeral__1__eq__1,axiom,
number267125858f_real(bit1(pls)) = one_one_real ).
fof(fact_29_n0,axiom,
hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),n)) ).
fof(fact_30_zless__linear,axiom,
! [X,Y] :
( ( is_int(X)
& is_int(Y) )
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X),Y))
| X = Y
| hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Y),X)) ) ) ).
fof(fact_31_less__number__of__int__code,axiom,
! [K,L_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,number_number_of_int(K)),number_number_of_int(L_1)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K),L_1)) ) ).
fof(fact_32_plus__numeral__code_I9_J,axiom,
! [V_1,W] : plus_plus_int(number_number_of_int(V_1),number_number_of_int(W)) = number_number_of_int(plus_plus_int(V_1,W)) ).
fof(fact_33_less__number__of,axiom,
! [Xa,Ya] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,number_number_of_int(Xa)),number_number_of_int(Ya)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Xa),Ya)) ) ).
fof(fact_34_less__number__of,axiom,
! [Xa,Ya] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,number267125858f_real(Xa)),number267125858f_real(Ya)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Xa),Ya)) ) ).
fof(fact_35_zero__is__num__zero,axiom,
zero_zero_int = number_number_of_int(pls) ).
fof(fact_36_zpower__int,axiom,
! [M,N] : hAPP_nat_int(power_power_int(hAPP_nat_int(semiri1621563631at_int,M)),N) = hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(power_power_nat(M),N)) ).
fof(fact_37_int__power,axiom,
! [M,N] : hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(power_power_nat(M),N)) = hAPP_nat_int(power_power_int(hAPP_nat_int(semiri1621563631at_int,M)),N) ).
fof(fact_38_zadd__int__left,axiom,
! [M,N,Z] : plus_plus_int(hAPP_nat_int(semiri1621563631at_int,M),plus_plus_int(hAPP_nat_int(semiri1621563631at_int,N),Z)) = plus_plus_int(hAPP_nat_int(semiri1621563631at_int,plus_plus_nat(M,N)),Z) ).
fof(fact_39_zadd__int,axiom,
! [M,N] : plus_plus_int(hAPP_nat_int(semiri1621563631at_int,M),hAPP_nat_int(semiri1621563631at_int,N)) = hAPP_nat_int(semiri1621563631at_int,plus_plus_nat(M,N)) ).
fof(fact_40_int__1,axiom,
hAPP_nat_int(semiri1621563631at_int,one_one_nat) = one_one_int ).
fof(fact_41_nat__number__of__Pls,axiom,
number_number_of_nat(pls) = zero_zero_nat ).
fof(fact_42_semiring__norm_I113_J,axiom,
zero_zero_nat = number_number_of_nat(pls) ).
fof(fact_43_int__eq__0__conv,axiom,
! [Na] :
( hAPP_nat_int(semiri1621563631at_int,Na) = zero_zero_int
<=> Na = zero_zero_nat ) ).
fof(fact_44_int__0,axiom,
hAPP_nat_int(semiri1621563631at_int,zero_zero_nat) = zero_zero_int ).
fof(fact_45_nat__1__add__1,axiom,
plus_plus_nat(one_one_nat,one_one_nat) = number_number_of_nat(bit0(bit1(pls))) ).
fof(fact_46_less__int__code_I16_J,axiom,
! [K1,K2] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,bit1(K1)),bit1(K2)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K1),K2)) ) ).
fof(fact_47_rel__simps_I17_J,axiom,
! [K,L_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,bit1(K)),bit1(L_1)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K),L_1)) ) ).
fof(fact_48_rel__simps_I2_J,axiom,
~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,pls),pls)) ).
fof(fact_49_less__int__code_I13_J,axiom,
! [K1,K2] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,bit0(K1)),bit0(K2)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K1),K2)) ) ).
fof(fact_50_rel__simps_I14_J,axiom,
! [K,L_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,bit0(K)),bit0(L_1)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K),L_1)) ) ).
fof(fact_51_zadd__strict__right__mono,axiom,
! [K_1,I_2,J_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,I_2),J_1))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,plus_plus_int(I_2,K_1)),plus_plus_int(J_1,K_1))) ) ).
fof(fact_52_add__nat__number__of,axiom,
! [V_2,V_1] :
( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,V_1),pls))
=> plus_plus_nat(number_number_of_nat(V_1),number_number_of_nat(V_2)) = number_number_of_nat(V_2) )
& ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,V_1),pls))
=> ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,V_2),pls))
=> plus_plus_nat(number_number_of_nat(V_1),number_number_of_nat(V_2)) = number_number_of_nat(V_1) )
& ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,V_2),pls))
=> plus_plus_nat(number_number_of_nat(V_1),number_number_of_nat(V_2)) = number_number_of_nat(plus_plus_int(V_1,V_2)) ) ) ) ) ).
fof(fact_53_one__is__num__one,axiom,
one_one_int = number_number_of_int(bit1(pls)) ).
fof(fact_54_nat__numeral__1__eq__1,axiom,
number_number_of_nat(bit1(pls)) = one_one_nat ).
fof(fact_55_Numeral1__eq1__nat,axiom,
one_one_nat = number_number_of_nat(bit1(pls)) ).
fof(fact_56_eq__number__of,axiom,
! [Xa,Ya] :
( ( is_int(Xa)
& is_int(Ya) )
=> ( number_number_of_int(Xa) = number_number_of_int(Ya)
<=> Xa = Ya ) ) ).
fof(fact_57_eq__number__of,axiom,
! [Xa,Ya] :
( ( is_int(Xa)
& is_int(Ya) )
=> ( number267125858f_real(Xa) = number267125858f_real(Ya)
<=> Xa = Ya ) ) ).
fof(fact_58_number__of__reorient,axiom,
! [Wa,Xa] :
( number_number_of_nat(Wa) = Xa
<=> Xa = number_number_of_nat(Wa) ) ).
fof(fact_59_number__of__reorient,axiom,
! [Wa,Xa] :
( is_int(Xa)
=> ( number_number_of_int(Wa) = Xa
<=> Xa = number_number_of_int(Wa) ) ) ).
fof(fact_60_number__of__reorient,axiom,
! [Wa,Xa] :
( number267125858f_real(Wa) = Xa
<=> Xa = number267125858f_real(Wa) ) ).
fof(fact_61_rel__simps_I51_J,axiom,
! [K,L_1] :
( ( is_int(K)
& is_int(L_1) )
=> ( bit1(K) = bit1(L_1)
<=> K = L_1 ) ) ).
fof(fact_62_rel__simps_I48_J,axiom,
! [K,L_1] :
( ( is_int(K)
& is_int(L_1) )
=> ( bit0(K) = bit0(L_1)
<=> K = L_1 ) ) ).
fof(fact_63_even__less__0__iff,axiom,
! [A_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,plus_plus_int(A_1,A_1)),zero_zero_int))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_1),zero_zero_int)) ) ).
fof(fact_64_even__less__0__iff,axiom,
! [A_1] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,plus_plus_real(A_1,A_1)),zero_zero_real))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_1),zero_zero_real)) ) ).
fof(fact_65_zadd__assoc,axiom,
! [Z1,Z2,Z3] : plus_plus_int(plus_plus_int(Z1,Z2),Z3) = plus_plus_int(Z1,plus_plus_int(Z2,Z3)) ).
fof(fact_66_zadd__left__commute,axiom,
! [X,Y,Z] : plus_plus_int(X,plus_plus_int(Y,Z)) = plus_plus_int(Y,plus_plus_int(X,Z)) ).
fof(fact_67_zadd__commute,axiom,
! [Z,W] : plus_plus_int(Z,W) = plus_plus_int(W,Z) ).
fof(fact_68_int__int__eq,axiom,
! [Ma,Na] :
( hAPP_nat_int(semiri1621563631at_int,Ma) = hAPP_nat_int(semiri1621563631at_int,Na)
<=> Ma = Na ) ).
fof(fact_69_less__special_I3_J,axiom,
! [Xa] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,number_number_of_int(Xa)),zero_zero_int))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Xa),pls)) ) ).
fof(fact_70_less__special_I3_J,axiom,
! [Xa] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,number267125858f_real(Xa)),zero_zero_real))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Xa),pls)) ) ).
fof(fact_71_less__special_I1_J,axiom,
! [Ya] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),number_number_of_int(Ya)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,pls),Ya)) ) ).
fof(fact_72_less__special_I1_J,axiom,
! [Ya] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),number267125858f_real(Ya)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,pls),Ya)) ) ).
fof(fact_73_rel__simps_I12_J,axiom,
! [K] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,bit1(K)),pls))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K),pls)) ) ).
fof(fact_74_less__int__code_I15_J,axiom,
! [K1,K2] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,bit1(K1)),bit0(K2)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K1),K2)) ) ).
fof(fact_75_rel__simps_I16_J,axiom,
! [K,L_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,bit1(K)),bit0(L_1)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K),L_1)) ) ).
fof(fact_76_rel__simps_I10_J,axiom,
! [K] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,bit0(K)),pls))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K),pls)) ) ).
fof(fact_77_rel__simps_I4_J,axiom,
! [K] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,pls),bit0(K)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,pls),K)) ) ).
fof(fact_78_bin__less__0__simps_I4_J,axiom,
! [Wa] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,bit1(Wa)),zero_zero_int))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Wa),zero_zero_int)) ) ).
fof(fact_79_bin__less__0__simps_I1_J,axiom,
~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,pls),zero_zero_int)) ).
fof(fact_80_bin__less__0__simps_I3_J,axiom,
! [Wa] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,bit0(Wa)),zero_zero_int))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Wa),zero_zero_int)) ) ).
fof(fact_81_int__0__less__1,axiom,
hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),one_one_int)) ).
fof(fact_82_zless__add1__eq,axiom,
! [Wa,Z_1] :
( ( is_int(Wa)
& is_int(Z_1) )
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Wa),plus_plus_int(Z_1,one_one_int)))
<=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Wa),Z_1))
| Wa = Z_1 ) ) ) ).
fof(fact_83_int__less__0__conv,axiom,
! [K_1] : ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_nat_int(semiri1621563631at_int,K_1)),zero_zero_int)) ).
fof(fact_84_less__special_I4_J,axiom,
! [Xa] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,number_number_of_int(Xa)),one_one_int))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Xa),bit1(pls))) ) ).
fof(fact_85_less__special_I4_J,axiom,
! [Xa] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,number267125858f_real(Xa)),one_one_real))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Xa),bit1(pls))) ) ).
fof(fact_86_less__special_I2_J,axiom,
! [Ya] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),number_number_of_int(Ya)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,bit1(pls)),Ya)) ) ).
fof(fact_87_less__special_I2_J,axiom,
! [Ya] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),number267125858f_real(Ya)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,bit1(pls)),Ya)) ) ).
fof(fact_88_odd__less__0,axiom,
! [Z_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,plus_plus_int(plus_plus_int(one_one_int,Z_1),Z_1)),zero_zero_int))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Z_1),zero_zero_int)) ) ).
fof(fact_89_double__eq__0__iff,axiom,
! [A_1] :
( is_int(A_1)
=> ( plus_plus_int(A_1,A_1) = zero_zero_int
<=> A_1 = zero_zero_int ) ) ).
fof(fact_90_double__eq__0__iff,axiom,
! [A_1] :
( plus_plus_real(A_1,A_1) = zero_zero_real
<=> A_1 = zero_zero_real ) ).
fof(fact_91_rel__simps_I46_J,axiom,
! [K_1] : bit1(K_1) != pls ).
fof(fact_92_rel__simps_I39_J,axiom,
! [L] : pls != bit1(L) ).
fof(fact_93_rel__simps_I50_J,axiom,
! [K_1,L] : bit1(K_1) != bit0(L) ).
fof(fact_94_rel__simps_I49_J,axiom,
! [K_1,L] : bit0(K_1) != bit1(L) ).
fof(fact_95_rel__simps_I44_J,axiom,
! [K] :
( is_int(K)
=> ( bit0(K) = pls
<=> K = pls ) ) ).
fof(fact_96_rel__simps_I38_J,axiom,
! [L_1] :
( is_int(L_1)
=> ( pls = bit0(L_1)
<=> pls = L_1 ) ) ).
fof(fact_97_Bit0__Pls,axiom,
bit0(pls) = pls ).
fof(fact_98_Pls__def,axiom,
pls = zero_zero_int ).
fof(fact_99_int__0__neq__1,axiom,
zero_zero_int != one_one_int ).
fof(fact_100_add__Pls__right,axiom,
! [K_1] :
( is_int(K_1)
=> plus_plus_int(K_1,pls) = K_1 ) ).
fof(fact_101_add__Pls,axiom,
! [K_1] :
( is_int(K_1)
=> plus_plus_int(pls,K_1) = K_1 ) ).
fof(fact_102_add__Bit0__Bit0,axiom,
! [K_1,L] : plus_plus_int(bit0(K_1),bit0(L)) = bit0(plus_plus_int(K_1,L)) ).
fof(fact_103_Bit0__def,axiom,
! [K_1] : bit0(K_1) = plus_plus_int(K_1,K_1) ).
fof(fact_104_zadd__0__right,axiom,
! [Z] :
( is_int(Z)
=> plus_plus_int(Z,zero_zero_int) = Z ) ).
fof(fact_105_zadd__0,axiom,
! [Z] :
( is_int(Z)
=> plus_plus_int(zero_zero_int,Z) = Z ) ).
fof(fact_106_semiring__numeral__0__eq__0,axiom,
number_number_of_int(pls) = zero_zero_int ).
fof(fact_107_semiring__numeral__0__eq__0,axiom,
number_number_of_nat(pls) = zero_zero_nat ).
fof(fact_108_semiring__numeral__0__eq__0,axiom,
number267125858f_real(pls) = zero_zero_real ).
fof(fact_109_number__of__Pls,axiom,
number_number_of_int(pls) = zero_zero_int ).
fof(fact_110_number__of__Pls,axiom,
number267125858f_real(pls) = zero_zero_real ).
fof(fact_111_semiring__norm_I112_J,axiom,
zero_zero_int = number_number_of_int(pls) ).
fof(fact_112_semiring__norm_I112_J,axiom,
zero_zero_real = number267125858f_real(pls) ).
fof(fact_113_add__numeral__0,axiom,
! [A_100] :
( is_int(A_100)
=> plus_plus_int(number_number_of_int(pls),A_100) = A_100 ) ).
fof(fact_114_add__numeral__0,axiom,
! [A_100] : plus_plus_real(number267125858f_real(pls),A_100) = A_100 ).
fof(fact_115_add__numeral__0__right,axiom,
! [A_99] :
( is_int(A_99)
=> plus_plus_int(A_99,number_number_of_int(pls)) = A_99 ) ).
fof(fact_116_add__numeral__0__right,axiom,
! [A_99] : plus_plus_real(A_99,number267125858f_real(pls)) = A_99 ).
fof(fact_117_power__eq__0__iff__number__of,axiom,
! [A_1,Wa] :
( is_int(A_1)
=> ( hAPP_nat_int(power_power_int(A_1),number_number_of_nat(Wa)) = zero_zero_int
<=> ( A_1 = zero_zero_int
& number_number_of_nat(Wa) != zero_zero_nat ) ) ) ).
fof(fact_118_power__eq__0__iff__number__of,axiom,
! [A_1,Wa] :
( hAPP_nat_nat(power_power_nat(A_1),number_number_of_nat(Wa)) = zero_zero_nat
<=> ( A_1 = zero_zero_nat
& number_number_of_nat(Wa) != zero_zero_nat ) ) ).
fof(fact_119_power__eq__0__iff__number__of,axiom,
! [A_1,Wa] :
( hAPP_nat_real(power_power_real(A_1),number_number_of_nat(Wa)) = zero_zero_real
<=> ( A_1 = zero_zero_real
& number_number_of_nat(Wa) != zero_zero_nat ) ) ).
fof(fact_120_add__number__of__left,axiom,
! [V_16,W_11,Z_6] : plus_plus_int(number_number_of_int(V_16),plus_plus_int(number_number_of_int(W_11),Z_6)) = plus_plus_int(number_number_of_int(plus_plus_int(V_16,W_11)),Z_6) ).
fof(fact_121_add__number__of__left,axiom,
! [V_16,W_11,Z_6] : plus_plus_real(number267125858f_real(V_16),plus_plus_real(number267125858f_real(W_11),Z_6)) = plus_plus_real(number267125858f_real(plus_plus_int(V_16,W_11)),Z_6) ).
fof(fact_122_add__number__of__eq,axiom,
! [V_15,W_10] : plus_plus_int(number_number_of_int(V_15),number_number_of_int(W_10)) = number_number_of_int(plus_plus_int(V_15,W_10)) ).
fof(fact_123_add__number__of__eq,axiom,
! [V_15,W_10] : plus_plus_real(number267125858f_real(V_15),number267125858f_real(W_10)) = number267125858f_real(plus_plus_int(V_15,W_10)) ).
fof(fact_124_number__of__add,axiom,
! [V_14,W_9] : number_number_of_int(plus_plus_int(V_14,W_9)) = plus_plus_int(number_number_of_int(V_14),number_number_of_int(W_9)) ).
fof(fact_125_number__of__add,axiom,
! [V_14,W_9] : number267125858f_real(plus_plus_int(V_14,W_9)) = plus_plus_real(number267125858f_real(V_14),number267125858f_real(W_9)) ).
fof(fact_126_add__Bit1__Bit0,axiom,
! [K_1,L] : plus_plus_int(bit1(K_1),bit0(L)) = bit1(plus_plus_int(K_1,L)) ).
fof(fact_127_add__Bit0__Bit1,axiom,
! [K_1,L] : plus_plus_int(bit0(K_1),bit1(L)) = bit1(plus_plus_int(K_1,L)) ).
fof(fact_128_Bit1__def,axiom,
! [K_1] : bit1(K_1) = plus_plus_int(plus_plus_int(one_one_int,K_1),K_1) ).
fof(fact_129_odd__nonzero,axiom,
! [Z] : plus_plus_int(plus_plus_int(one_one_int,Z),Z) != zero_zero_int ).
fof(fact_130_number__of__int,axiom,
! [N_26] : number_number_of_nat(hAPP_nat_int(semiri1621563631at_int,N_26)) = hAPP_nat_nat(semiri984289939at_nat,N_26) ).
fof(fact_131_number__of__int,axiom,
! [N_26] : number_number_of_int(hAPP_nat_int(semiri1621563631at_int,N_26)) = hAPP_nat_int(semiri1621563631at_int,N_26) ).
fof(fact_132_number__of__int,axiom,
! [N_26] : number267125858f_real(hAPP_nat_int(semiri1621563631at_int,N_26)) = hAPP_nat_real(semiri132038758t_real,N_26) ).
fof(fact_133_zero__less__power2,axiom,
! [A_1] :
( is_int(A_1)
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_nat_int(power_power_int(A_1),number_number_of_nat(bit0(bit1(pls))))))
<=> A_1 != zero_zero_int ) ) ).
fof(fact_134_zero__less__power2,axiom,
! [A_1] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_nat_real(power_power_real(A_1),number_number_of_nat(bit0(bit1(pls))))))
<=> A_1 != zero_zero_real ) ).
fof(fact_135_power2__less__0,axiom,
! [A_98] : ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_nat_int(power_power_int(A_98),number_number_of_nat(bit0(bit1(pls))))),zero_zero_int)) ).
fof(fact_136_power2__less__0,axiom,
! [A_98] : ~ hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_nat_real(power_power_real(A_98),number_number_of_nat(bit0(bit1(pls))))),zero_zero_real)) ).
fof(fact_137_sum__power2__gt__zero__iff,axiom,
! [Xa,Ya] :
( ( is_int(Xa)
& is_int(Ya) )
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),plus_plus_int(hAPP_nat_int(power_power_int(Xa),number_number_of_nat(bit0(bit1(pls)))),hAPP_nat_int(power_power_int(Ya),number_number_of_nat(bit0(bit1(pls)))))))
<=> ( Xa != zero_zero_int
| Ya != zero_zero_int ) ) ) ).
fof(fact_138_sum__power2__gt__zero__iff,axiom,
! [Xa,Ya] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),plus_plus_real(hAPP_nat_real(power_power_real(Xa),number_number_of_nat(bit0(bit1(pls)))),hAPP_nat_real(power_power_real(Ya),number_number_of_nat(bit0(bit1(pls)))))))
<=> ( Xa != zero_zero_real
| Ya != zero_zero_real ) ) ).
fof(fact_139_not__sum__power2__lt__zero,axiom,
! [X_15,Y_11] : ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,plus_plus_int(hAPP_nat_int(power_power_int(X_15),number_number_of_nat(bit0(bit1(pls)))),hAPP_nat_int(power_power_int(Y_11),number_number_of_nat(bit0(bit1(pls)))))),zero_zero_int)) ).
fof(fact_140_not__sum__power2__lt__zero,axiom,
! [X_15,Y_11] : ~ hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,plus_plus_real(hAPP_nat_real(power_power_real(X_15),number_number_of_nat(bit0(bit1(pls)))),hAPP_nat_real(power_power_real(Y_11),number_number_of_nat(bit0(bit1(pls)))))),zero_zero_real)) ).
fof(fact_141_number__of__Bit0,axiom,
! [W_8] : number_number_of_int(bit0(W_8)) = plus_plus_int(plus_plus_int(zero_zero_int,number_number_of_int(W_8)),number_number_of_int(W_8)) ).
fof(fact_142_number__of__Bit0,axiom,
! [W_8] : number267125858f_real(bit0(W_8)) = plus_plus_real(plus_plus_real(zero_zero_real,number267125858f_real(W_8)),number267125858f_real(W_8)) ).
fof(fact_143_number__of__Bit1,axiom,
! [W_7] : number_number_of_int(bit1(W_7)) = plus_plus_int(plus_plus_int(one_one_int,number_number_of_int(W_7)),number_number_of_int(W_7)) ).
fof(fact_144_number__of__Bit1,axiom,
! [W_7] : number267125858f_real(bit1(W_7)) = plus_plus_real(plus_plus_real(one_one_real,number267125858f_real(W_7)),number267125858f_real(W_7)) ).
fof(fact_145_semiring__numeral__1__eq__1,axiom,
number_number_of_int(bit1(pls)) = one_one_int ).
fof(fact_146_semiring__numeral__1__eq__1,axiom,
number_number_of_nat(bit1(pls)) = one_one_nat ).
fof(fact_147_semiring__numeral__1__eq__1,axiom,
number267125858f_real(bit1(pls)) = one_one_real ).
fof(fact_148_mn,axiom,
hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,m1),plus_plus_int(one_one_int,hAPP_nat_int(semiri1621563631at_int,n)))) ).
fof(fact_149_of__nat__less__two__power,axiom,
! [N_25] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_nat_int(semiri1621563631at_int,N_25)),hAPP_nat_int(power_power_int(number_number_of_int(bit0(bit1(pls)))),N_25))) ).
fof(fact_150_of__nat__less__two__power,axiom,
! [N_25] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_nat_real(semiri132038758t_real,N_25)),hAPP_nat_real(power_power_real(number267125858f_real(bit0(bit1(pls)))),N_25))) ).
fof(fact_151_transfer__int__nat__numerals_I3_J,axiom,
number_number_of_int(bit0(bit1(pls))) = hAPP_nat_int(semiri1621563631at_int,number_number_of_nat(bit0(bit1(pls)))) ).
fof(fact_152_transfer__int__nat__numerals_I4_J,axiom,
number_number_of_int(bit1(bit1(pls))) = hAPP_nat_int(semiri1621563631at_int,number_number_of_nat(bit1(bit1(pls)))) ).
fof(fact_153_realpow__two__sum__zero__iff,axiom,
! [Xa,Ya] :
( plus_plus_real(hAPP_nat_real(power_power_real(Xa),number_number_of_nat(bit0(bit1(pls)))),hAPP_nat_real(power_power_real(Ya),number_number_of_nat(bit0(bit1(pls))))) = zero_zero_real
<=> ( Xa = zero_zero_real
& Ya = zero_zero_real ) ) ).
fof(fact_154_of__nat__0__less__iff,axiom,
! [Na] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_nat_int(semiri1621563631at_int,Na)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),Na)) ) ).
fof(fact_155_of__nat__0__less__iff,axiom,
! [Na] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_nat_nat(semiri984289939at_nat,Na)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),Na)) ) ).
fof(fact_156_of__nat__0__less__iff,axiom,
! [Na] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_nat_real(semiri132038758t_real,Na)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),Na)) ) ).
fof(fact_157_one__less__power,axiom,
! [N_24,A_97] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),A_97))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_24))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),hAPP_nat_int(power_power_int(A_97),N_24))) ) ) ).
fof(fact_158_one__less__power,axiom,
! [N_24,A_97] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_nat),A_97))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_24))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_nat),hAPP_nat_nat(power_power_nat(A_97),N_24))) ) ) ).
fof(fact_159_one__less__power,axiom,
! [N_24,A_97] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),A_97))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_24))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),hAPP_nat_real(power_power_real(A_97),N_24))) ) ) ).
fof(fact_160_power__0__left,axiom,
! [N_23] :
( ( N_23 = zero_zero_nat
=> hAPP_nat_int(power_power_int(zero_zero_int),N_23) = one_one_int )
& ( N_23 != zero_zero_nat
=> hAPP_nat_int(power_power_int(zero_zero_int),N_23) = zero_zero_int ) ) ).
fof(fact_161_power__0__left,axiom,
! [N_23] :
( ( N_23 = zero_zero_nat
=> hAPP_nat_nat(power_power_nat(zero_zero_nat),N_23) = one_one_nat )
& ( N_23 != zero_zero_nat
=> hAPP_nat_nat(power_power_nat(zero_zero_nat),N_23) = zero_zero_nat ) ) ).
fof(fact_162_power__0__left,axiom,
! [N_23] :
( ( N_23 = zero_zero_nat
=> hAPP_nat_real(power_power_real(zero_zero_real),N_23) = one_one_real )
& ( N_23 != zero_zero_nat
=> hAPP_nat_real(power_power_real(zero_zero_real),N_23) = zero_zero_real ) ) ).
fof(fact_163_power__strict__decreasing,axiom,
! [A_96,N_22,N_21] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_22),N_21))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_96))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_96),one_one_int))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_nat_int(power_power_int(A_96),N_21)),hAPP_nat_int(power_power_int(A_96),N_22))) ) ) ) ).
fof(fact_164_power__strict__decreasing,axiom,
! [A_96,N_22,N_21] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_22),N_21))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),A_96))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_96),one_one_nat))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(power_power_nat(A_96),N_21)),hAPP_nat_nat(power_power_nat(A_96),N_22))) ) ) ) ).
fof(fact_165_power__strict__decreasing,axiom,
! [A_96,N_22,N_21] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_22),N_21))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_96))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_96),one_one_real))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_nat_real(power_power_real(A_96),N_21)),hAPP_nat_real(power_power_real(A_96),N_22))) ) ) ) ).
fof(fact_166_zero__less__two,axiom,
hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),plus_plus_int(one_one_int,one_one_int))) ).
fof(fact_167_zero__less__two,axiom,
hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),plus_plus_nat(one_one_nat,one_one_nat))) ).
fof(fact_168_zero__less__two,axiom,
hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),plus_plus_real(one_one_real,one_one_real))) ).
fof(fact_169_int__gr__induct,axiom,
! [P_1,K,I_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K),I_1))
=> ( hBOOL(hAPP_int_bool(P_1,plus_plus_int(K,one_one_int)))
=> ( ! [I] :
( is_int(I)
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K),I))
=> ( hBOOL(hAPP_int_bool(P_1,I))
=> hBOOL(hAPP_int_bool(P_1,plus_plus_int(I,one_one_int))) ) ) )
=> hBOOL(hAPP_int_bool(P_1,I_1)) ) ) ) ).
fof(fact_170_transfer__int__nat__numerals_I1_J,axiom,
zero_zero_int = hAPP_nat_int(semiri1621563631at_int,zero_zero_nat) ).
fof(fact_171_tn0,axiom,
hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),tn)) ).
fof(fact_172_less__zeroE,axiom,
! [N] : ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N),zero_zero_nat)) ).
fof(fact_173_real__zero__not__eq__one,axiom,
zero_zero_real != one_one_real ).
fof(fact_174_less__not__refl,axiom,
! [N] : ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N),N)) ).
fof(fact_175_not__add__less1,axiom,
! [I_2,J_1] : ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,plus_plus_nat(I_2,J_1)),I_2)) ).
fof(fact_176_not__add__less2,axiom,
! [J_1,I_2] : ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,plus_plus_nat(J_1,I_2)),I_2)) ).
fof(fact_177_number__of__is__id,axiom,
! [K_1] :
( is_int(K_1)
=> number_number_of_int(K_1) = K_1 ) ).
fof(fact_178_nat__neq__iff,axiom,
! [Ma,Na] :
( Ma != Na
<=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),Na))
| hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Na),Ma)) ) ) ).
fof(fact_179_nat__add__commute,axiom,
! [M,N] : plus_plus_nat(M,N) = plus_plus_nat(N,M) ).
fof(fact_180_nat__add__left__commute,axiom,
! [X,Y,Z] : plus_plus_nat(X,plus_plus_nat(Y,Z)) = plus_plus_nat(Y,plus_plus_nat(X,Z)) ).
fof(fact_181_nat__add__assoc,axiom,
! [M,N,K_1] : plus_plus_nat(plus_plus_nat(M,N),K_1) = plus_plus_nat(M,plus_plus_nat(N,K_1)) ).
fof(fact_182_nat__add__left__cancel,axiom,
! [K,Ma,Na] :
( plus_plus_nat(K,Ma) = plus_plus_nat(K,Na)
<=> Ma = Na ) ).
fof(fact_183_nat__add__right__cancel,axiom,
! [Ma,K,Na] :
( plus_plus_nat(Ma,K) = plus_plus_nat(Na,K)
<=> Ma = Na ) ).
fof(fact_184_nat__add__left__cancel__less,axiom,
! [K,Ma,Na] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,plus_plus_nat(K,Ma)),plus_plus_nat(K,Na)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),Na)) ) ).
fof(fact_185_linorder__neqE__nat,axiom,
! [X,Y] :
( X != Y
=> ( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X),Y))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Y),X)) ) ) ).
fof(fact_186_less__irrefl__nat,axiom,
! [N] : ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N),N)) ).
fof(fact_187_less__not__refl2,axiom,
! [N,M] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N),M))
=> M != N ) ).
fof(fact_188_less__not__refl3,axiom,
! [S_1,T_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,S_1),T_1))
=> S_1 != T_1 ) ).
fof(fact_189_trans__less__add1,axiom,
! [M,I_2,J_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_2),J_1))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_2),plus_plus_nat(J_1,M))) ) ).
fof(fact_190_trans__less__add2,axiom,
! [M,I_2,J_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_2),J_1))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_2),plus_plus_nat(M,J_1))) ) ).
fof(fact_191_add__less__mono1,axiom,
! [K_1,I_2,J_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_2),J_1))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,plus_plus_nat(I_2,K_1)),plus_plus_nat(J_1,K_1))) ) ).
fof(fact_192_add__less__mono,axiom,
! [K_1,L,I_2,J_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_2),J_1))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,K_1),L))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,plus_plus_nat(I_2,K_1)),plus_plus_nat(J_1,L))) ) ) ).
fof(fact_193_less__add__eq__less,axiom,
! [M,N,K_1,L] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,K_1),L))
=> ( plus_plus_nat(M,L) = plus_plus_nat(K_1,N)
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N)) ) ) ).
fof(fact_194_add__lessD1,axiom,
! [I_2,J_1,K_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,plus_plus_nat(I_2,J_1)),K_1))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_2),K_1)) ) ).
fof(fact_195_nat__less__cases,axiom,
! [P_1,Ma,Na] :
( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),Na))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(P_1,Na),Ma)) )
=> ( ( Ma = Na
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(P_1,Na),Ma)) )
=> ( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Na),Ma))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(P_1,Na),Ma)) )
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(P_1,Na),Ma)) ) ) ) ).
fof(fact_196_gr0I,axiom,
! [N] :
( N != zero_zero_nat
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N)) ) ).
fof(fact_197_gr__implies__not0,axiom,
! [M,N] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N))
=> N != zero_zero_nat ) ).
fof(fact_198_nat__power__less__imp__less,axiom,
! [M,N,I_2] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),I_2))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(power_power_nat(I_2),M)),hAPP_nat_nat(power_power_nat(I_2),N)))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N)) ) ) ).
fof(fact_199_less__nat__zero__code,axiom,
! [N] : ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N),zero_zero_nat)) ).
fof(fact_200_nat__zero__less__power__iff,axiom,
! [Xa,Na] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_nat_nat(power_power_nat(Xa),Na)))
<=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),Xa))
| Na = zero_zero_nat ) ) ).
fof(fact_201_add__gr__0,axiom,
! [Ma,Na] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),plus_plus_nat(Ma,Na)))
<=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),Ma))
| hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),Na)) ) ) ).
fof(fact_202_neq0__conv,axiom,
! [Na] :
( Na != zero_zero_nat
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),Na)) ) ).
fof(fact_203_not__less0,axiom,
! [N] : ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N),zero_zero_nat)) ).
fof(fact_204_zero__less__power__nat__eq,axiom,
! [Xa,Na] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_nat_nat(power_power_nat(Xa),Na)))
<=> ( Na = zero_zero_nat
| hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),Xa)) ) ) ).
fof(fact_205_Nat__Transfer_Otransfer__int__nat__relations_I2_J,axiom,
! [Xa,Ya] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_nat_int(semiri1621563631at_int,Xa)),hAPP_nat_int(semiri1621563631at_int,Ya)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Xa),Ya)) ) ).
fof(fact_206_zero__less__power__nat__eq__number__of,axiom,
! [Xa,Wa] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_nat_nat(power_power_nat(Xa),number_number_of_nat(Wa))))
<=> ( number_number_of_nat(Wa) = zero_zero_nat
| hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),Xa)) ) ) ).
fof(fact_207_zless__int,axiom,
! [Ma,Na] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_nat_int(semiri1621563631at_int,Ma)),hAPP_nat_int(semiri1621563631at_int,Na)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),Na)) ) ).
fof(fact_208_less__nat__number__of,axiom,
! [Va,V_3] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,number_number_of_nat(Va)),number_number_of_nat(V_3)))
<=> ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Va),V_3))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,pls),V_3)) )
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Va),V_3)) ) ) ).
fof(fact_209_linorder__neqE__linordered__idom,axiom,
! [X_14,Y_10] :
( ( is_int(X_14)
& is_int(Y_10) )
=> ( X_14 != Y_10
=> ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_14),Y_10))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Y_10),X_14)) ) ) ) ).
fof(fact_210_linorder__neqE__linordered__idom,axiom,
! [X_14,Y_10] :
( X_14 != Y_10
=> ( ~ hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_14),Y_10))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Y_10),X_14)) ) ) ).
fof(fact_211_add__eq__self__zero,axiom,
! [M,N] :
( plus_plus_nat(M,N) = M
=> N = zero_zero_nat ) ).
fof(fact_212_add__is__0,axiom,
! [Ma,Na] :
( plus_plus_nat(Ma,Na) = zero_zero_nat
<=> ( Ma = zero_zero_nat
& Na = zero_zero_nat ) ) ).
fof(fact_213_Nat_Oadd__0__right,axiom,
! [M] : plus_plus_nat(M,zero_zero_nat) = M ).
fof(fact_214_plus__nat_Oadd__0,axiom,
! [N] : plus_plus_nat(zero_zero_nat,N) = N ).
fof(fact_215_power__one__right,axiom,
! [A_95] :
( is_int(A_95)
=> hAPP_nat_int(power_power_int(A_95),one_one_nat) = A_95 ) ).
fof(fact_216_power__one__right,axiom,
! [A_95] : hAPP_nat_nat(power_power_nat(A_95),one_one_nat) = A_95 ).
fof(fact_217_power__one__right,axiom,
! [A_95] : hAPP_nat_real(power_power_real(A_95),one_one_nat) = A_95 ).
fof(fact_218_of__nat__eq__iff,axiom,
! [Ma,Na] :
( hAPP_nat_real(semiri132038758t_real,Ma) = hAPP_nat_real(semiri132038758t_real,Na)
<=> Ma = Na ) ).
fof(fact_219_of__nat__eq__iff,axiom,
! [Ma,Na] :
( hAPP_nat_nat(semiri984289939at_nat,Ma) = hAPP_nat_nat(semiri984289939at_nat,Na)
<=> Ma = Na ) ).
fof(fact_220_of__nat__eq__iff,axiom,
! [Ma,Na] :
( hAPP_nat_int(semiri1621563631at_int,Ma) = hAPP_nat_int(semiri1621563631at_int,Na)
<=> Ma = Na ) ).
fof(fact_221_Nat__Transfer_Otransfer__int__nat__relations_I1_J,axiom,
! [Xa,Ya] :
( hAPP_nat_int(semiri1621563631at_int,Xa) = hAPP_nat_int(semiri1621563631at_int,Ya)
<=> Xa = Ya ) ).
fof(fact_222_int__if__cong,axiom,
! [Xa,Ya,P_1] :
( ( hBOOL(P_1)
=> hAPP_nat_int(semiri1621563631at_int,Xa) = hAPP_nat_int(semiri1621563631at_int,if_nat(P_1,Xa,Ya)) )
& ( ~ hBOOL(P_1)
=> hAPP_nat_int(semiri1621563631at_int,Ya) = hAPP_nat_int(semiri1621563631at_int,if_nat(P_1,Xa,Ya)) ) ) ).
fof(fact_223_less__0__number__of,axiom,
! [Va] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),number_number_of_nat(Va)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,pls),Va)) ) ).
fof(fact_224_zero__less__int__conv,axiom,
! [Na] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_nat_int(semiri1621563631at_int,Na)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),Na)) ) ).
fof(fact_225_one__neq__zero,axiom,
one_one_int != zero_zero_int ).
fof(fact_226_one__neq__zero,axiom,
one_one_nat != zero_zero_nat ).
fof(fact_227_one__neq__zero,axiom,
one_one_real != zero_zero_real ).
fof(fact_228_zero__neq__one,axiom,
zero_zero_int != one_one_int ).
fof(fact_229_zero__neq__one,axiom,
zero_zero_nat != one_one_nat ).
fof(fact_230_zero__neq__one,axiom,
zero_zero_real != one_one_real ).
fof(fact_231_field__power__not__zero,axiom,
! [N_20,A_94] :
( is_int(A_94)
=> ( A_94 != zero_zero_int
=> hAPP_nat_int(power_power_int(A_94),N_20) != zero_zero_int ) ) ).
fof(fact_232_field__power__not__zero,axiom,
! [N_20,A_94] :
( A_94 != zero_zero_real
=> hAPP_nat_real(power_power_real(A_94),N_20) != zero_zero_real ) ).
fof(fact_233_power__one,axiom,
! [N_19] : hAPP_nat_int(power_power_int(one_one_int),N_19) = one_one_int ).
fof(fact_234_power__one,axiom,
! [N_19] : hAPP_nat_nat(power_power_nat(one_one_nat),N_19) = one_one_nat ).
fof(fact_235_power__one,axiom,
! [N_19] : hAPP_nat_real(power_power_real(one_one_real),N_19) = one_one_real ).
fof(fact_236_of__nat__less__iff,axiom,
! [Ma,Na] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_nat_int(semiri1621563631at_int,Ma)),hAPP_nat_int(semiri1621563631at_int,Na)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),Na)) ) ).
fof(fact_237_of__nat__less__iff,axiom,
! [Ma,Na] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(semiri984289939at_nat,Ma)),hAPP_nat_nat(semiri984289939at_nat,Na)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),Na)) ) ).
fof(fact_238_of__nat__less__iff,axiom,
! [Ma,Na] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_nat_real(semiri132038758t_real,Ma)),hAPP_nat_real(semiri132038758t_real,Na)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),Na)) ) ).
fof(fact_239_less__imp__of__nat__less,axiom,
! [M_10,N_18] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_10),N_18))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_nat_int(semiri1621563631at_int,M_10)),hAPP_nat_int(semiri1621563631at_int,N_18))) ) ).
fof(fact_240_less__imp__of__nat__less,axiom,
! [M_10,N_18] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_10),N_18))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(semiri984289939at_nat,M_10)),hAPP_nat_nat(semiri984289939at_nat,N_18))) ) ).
fof(fact_241_less__imp__of__nat__less,axiom,
! [M_10,N_18] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_10),N_18))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_nat_real(semiri132038758t_real,M_10)),hAPP_nat_real(semiri132038758t_real,N_18))) ) ).
fof(fact_242_of__nat__less__imp__less,axiom,
! [M_9,N_17] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_nat_int(semiri1621563631at_int,M_9)),hAPP_nat_int(semiri1621563631at_int,N_17)))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_9),N_17)) ) ).
fof(fact_243_of__nat__less__imp__less,axiom,
! [M_9,N_17] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(semiri984289939at_nat,M_9)),hAPP_nat_nat(semiri984289939at_nat,N_17)))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_9),N_17)) ) ).
fof(fact_244_of__nat__less__imp__less,axiom,
! [M_9,N_17] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_nat_real(semiri132038758t_real,M_9)),hAPP_nat_real(semiri132038758t_real,N_17)))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_9),N_17)) ) ).
fof(fact_245_of__nat__add,axiom,
! [M_8,N_16] : hAPP_nat_int(semiri1621563631at_int,plus_plus_nat(M_8,N_16)) = plus_plus_int(hAPP_nat_int(semiri1621563631at_int,M_8),hAPP_nat_int(semiri1621563631at_int,N_16)) ).
fof(fact_246_of__nat__add,axiom,
! [M_8,N_16] : hAPP_nat_nat(semiri984289939at_nat,plus_plus_nat(M_8,N_16)) = plus_plus_nat(hAPP_nat_nat(semiri984289939at_nat,M_8),hAPP_nat_nat(semiri984289939at_nat,N_16)) ).
fof(fact_247_of__nat__add,axiom,
! [M_8,N_16] : hAPP_nat_real(semiri132038758t_real,plus_plus_nat(M_8,N_16)) = plus_plus_real(hAPP_nat_real(semiri132038758t_real,M_8),hAPP_nat_real(semiri132038758t_real,N_16)) ).
fof(fact_248_of__nat__1,axiom,
hAPP_nat_int(semiri1621563631at_int,one_one_nat) = one_one_int ).
fof(fact_249_of__nat__1,axiom,
hAPP_nat_nat(semiri984289939at_nat,one_one_nat) = one_one_nat ).
fof(fact_250_of__nat__1,axiom,
hAPP_nat_real(semiri132038758t_real,one_one_nat) = one_one_real ).
fof(fact_251_of__nat__power,axiom,
! [M_7,N_15] : hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(power_power_nat(M_7),N_15)) = hAPP_nat_int(power_power_int(hAPP_nat_int(semiri1621563631at_int,M_7)),N_15) ).
fof(fact_252_of__nat__power,axiom,
! [M_7,N_15] : hAPP_nat_nat(semiri984289939at_nat,hAPP_nat_nat(power_power_nat(M_7),N_15)) = hAPP_nat_nat(power_power_nat(hAPP_nat_nat(semiri984289939at_nat,M_7)),N_15) ).
fof(fact_253_of__nat__power,axiom,
! [M_7,N_15] : hAPP_nat_real(semiri132038758t_real,hAPP_nat_nat(power_power_nat(M_7),N_15)) = hAPP_nat_real(power_power_real(hAPP_nat_real(semiri132038758t_real,M_7)),N_15) ).
fof(fact_254_transfer__int__nat__numerals_I2_J,axiom,
one_one_int = hAPP_nat_int(semiri1621563631at_int,one_one_nat) ).
fof(fact_255_Nat__Transfer_Otransfer__int__nat__functions_I1_J,axiom,
! [X,Y] : plus_plus_int(hAPP_nat_int(semiri1621563631at_int,X),hAPP_nat_int(semiri1621563631at_int,Y)) = hAPP_nat_int(semiri1621563631at_int,plus_plus_nat(X,Y)) ).
fof(fact_256_Nat__Transfer_Otransfer__int__nat__functions_I4_J,axiom,
! [X,N] : hAPP_nat_int(power_power_int(hAPP_nat_int(semiri1621563631at_int,X)),N) = hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(power_power_nat(X),N)) ).
fof(fact_257_pos__add__strict,axiom,
! [B_61,C_45,A_93] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_93))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_61),C_45))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_61),plus_plus_int(A_93,C_45))) ) ) ).
fof(fact_258_pos__add__strict,axiom,
! [B_61,C_45,A_93] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),A_93))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,B_61),C_45))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,B_61),plus_plus_nat(A_93,C_45))) ) ) ).
fof(fact_259_pos__add__strict,axiom,
! [B_61,C_45,A_93] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_93))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,B_61),C_45))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,B_61),plus_plus_real(A_93,C_45))) ) ) ).
fof(fact_260_not__one__less__zero,axiom,
~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),zero_zero_int)) ).
fof(fact_261_not__one__less__zero,axiom,
~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_nat),zero_zero_nat)) ).
fof(fact_262_not__one__less__zero,axiom,
~ hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),zero_zero_real)) ).
fof(fact_263_zero__less__one,axiom,
hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),one_one_int)) ).
fof(fact_264_zero__less__one,axiom,
hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),one_one_nat)) ).
fof(fact_265_zero__less__one,axiom,
hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),one_one_real)) ).
fof(fact_266_zero__less__power,axiom,
! [N_14,A_92] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_92))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_nat_int(power_power_int(A_92),N_14))) ) ).
fof(fact_267_zero__less__power,axiom,
! [N_14,A_92] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),A_92))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_nat_nat(power_power_nat(A_92),N_14))) ) ).
fof(fact_268_zero__less__power,axiom,
! [N_14,A_92] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_92))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_nat_real(power_power_real(A_92),N_14))) ) ).
fof(fact_269_less__add__one,axiom,
! [A_91] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_91),plus_plus_int(A_91,one_one_int))) ).
fof(fact_270_less__add__one,axiom,
! [A_91] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_91),plus_plus_nat(A_91,one_one_nat))) ).
fof(fact_271_less__add__one,axiom,
! [A_91] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_91),plus_plus_real(A_91,one_one_real))) ).
fof(fact_272_power__inject__exp,axiom,
! [Ma,Na,A_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),A_1))
=> ( hAPP_nat_int(power_power_int(A_1),Ma) = hAPP_nat_int(power_power_int(A_1),Na)
<=> Ma = Na ) ) ).
fof(fact_273_power__inject__exp,axiom,
! [Ma,Na,A_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_nat),A_1))
=> ( hAPP_nat_nat(power_power_nat(A_1),Ma) = hAPP_nat_nat(power_power_nat(A_1),Na)
<=> Ma = Na ) ) ).
fof(fact_274_power__inject__exp,axiom,
! [Ma,Na,A_1] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),A_1))
=> ( hAPP_nat_real(power_power_real(A_1),Ma) = hAPP_nat_real(power_power_real(A_1),Na)
<=> Ma = Na ) ) ).
fof(fact_275_power__strict__increasing__iff,axiom,
! [Xa,Ya,B] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),B))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_nat_int(power_power_int(B),Xa)),hAPP_nat_int(power_power_int(B),Ya)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Xa),Ya)) ) ) ).
fof(fact_276_power__strict__increasing__iff,axiom,
! [Xa,Ya,B] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_nat),B))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(power_power_nat(B),Xa)),hAPP_nat_nat(power_power_nat(B),Ya)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Xa),Ya)) ) ) ).
fof(fact_277_power__strict__increasing__iff,axiom,
! [Xa,Ya,B] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),B))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_nat_real(power_power_real(B),Xa)),hAPP_nat_real(power_power_real(B),Ya)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Xa),Ya)) ) ) ).
fof(fact_278_power__less__imp__less__exp,axiom,
! [M_6,N_13,A_90] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),A_90))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_nat_int(power_power_int(A_90),M_6)),hAPP_nat_int(power_power_int(A_90),N_13)))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_6),N_13)) ) ) ).
fof(fact_279_power__less__imp__less__exp,axiom,
! [M_6,N_13,A_90] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_nat),A_90))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(power_power_nat(A_90),M_6)),hAPP_nat_nat(power_power_nat(A_90),N_13)))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_6),N_13)) ) ) ).
fof(fact_280_power__less__imp__less__exp,axiom,
! [M_6,N_13,A_90] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),A_90))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_nat_real(power_power_real(A_90),M_6)),hAPP_nat_real(power_power_real(A_90),N_13)))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_6),N_13)) ) ) ).
fof(fact_281_power__strict__increasing,axiom,
! [A_89,N_12,N_11] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_12),N_11))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),A_89))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_nat_int(power_power_int(A_89),N_12)),hAPP_nat_int(power_power_int(A_89),N_11))) ) ) ).
fof(fact_282_power__strict__increasing,axiom,
! [A_89,N_12,N_11] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_12),N_11))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_nat),A_89))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(power_power_nat(A_89),N_12)),hAPP_nat_nat(power_power_nat(A_89),N_11))) ) ) ).
fof(fact_283_power__strict__increasing,axiom,
! [A_89,N_12,N_11] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_12),N_11))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),A_89))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_nat_real(power_power_real(A_89),N_12)),hAPP_nat_real(power_power_real(A_89),N_11))) ) ) ).
fof(fact_284_power__eq__0__iff,axiom,
! [A_1,Na] :
( is_int(A_1)
=> ( hAPP_nat_int(power_power_int(A_1),Na) = zero_zero_int
<=> ( A_1 = zero_zero_int
& Na != zero_zero_nat ) ) ) ).
fof(fact_285_power__eq__0__iff,axiom,
! [A_1,Na] :
( hAPP_nat_nat(power_power_nat(A_1),Na) = zero_zero_nat
<=> ( A_1 = zero_zero_nat
& Na != zero_zero_nat ) ) ).
fof(fact_286_power__eq__0__iff,axiom,
! [A_1,Na] :
( hAPP_nat_real(power_power_real(A_1),Na) = zero_zero_real
<=> ( A_1 = zero_zero_real
& Na != zero_zero_nat ) ) ).
fof(fact_287_of__nat__less__0__iff,axiom,
! [M_5] : ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_nat_int(semiri1621563631at_int,M_5)),zero_zero_int)) ).
fof(fact_288_of__nat__less__0__iff,axiom,
! [M_5] : ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(semiri984289939at_nat,M_5)),zero_zero_nat)) ).
fof(fact_289_of__nat__less__0__iff,axiom,
! [M_5] : ~ hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_nat_real(semiri132038758t_real,M_5)),zero_zero_real)) ).
fof(fact_290_power__0,axiom,
! [A_88] : hAPP_nat_int(power_power_int(A_88),zero_zero_nat) = one_one_int ).
fof(fact_291_power__0,axiom,
! [A_88] : hAPP_nat_nat(power_power_nat(A_88),zero_zero_nat) = one_one_nat ).
fof(fact_292_power__0,axiom,
! [A_88] : hAPP_nat_real(power_power_real(A_88),zero_zero_nat) = one_one_real ).
fof(fact_293_of__nat__0,axiom,
hAPP_nat_int(semiri1621563631at_int,zero_zero_nat) = zero_zero_int ).
fof(fact_294_of__nat__0,axiom,
hAPP_nat_nat(semiri984289939at_nat,zero_zero_nat) = zero_zero_nat ).
fof(fact_295_of__nat__0,axiom,
hAPP_nat_real(semiri132038758t_real,zero_zero_nat) = zero_zero_real ).
fof(fact_296_pos2,axiom,
hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),number_number_of_nat(bit0(bit1(pls))))) ).
fof(fact_297_zero__less__imp__eq__int,axiom,
! [K_1] :
( is_int(K_1)
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),K_1))
=> ? [N_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1))
& K_1 = hAPP_nat_int(semiri1621563631at_int,N_1) ) ) ) ).
fof(fact_298_less__imp__add__positive,axiom,
! [I_2,J_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_2),J_1))
=> ? [K_2] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),K_2))
& plus_plus_nat(I_2,K_2) = J_1 ) ) ).
fof(fact_299_exp__eq__1,axiom,
! [Xa,Na] :
( hAPP_nat_nat(power_power_nat(Xa),Na) = one_one_nat
<=> ( Xa = one_one_nat
| Na = zero_zero_nat ) ) ).
fof(fact_300_comm__semiring__1__class_Onormalizing__semiring__rules_I32_J,axiom,
! [X_13] : hAPP_nat_int(power_power_int(X_13),zero_zero_nat) = one_one_int ).
fof(fact_301_comm__semiring__1__class_Onormalizing__semiring__rules_I32_J,axiom,
! [X_13] : hAPP_nat_nat(power_power_nat(X_13),zero_zero_nat) = one_one_nat ).
fof(fact_302_comm__semiring__1__class_Onormalizing__semiring__rules_I32_J,axiom,
! [X_13] : hAPP_nat_real(power_power_real(X_13),zero_zero_nat) = one_one_real ).
fof(fact_303_zero__less__double__add__iff__zero__less__single__add,axiom,
! [A_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),plus_plus_int(A_1,A_1)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_1)) ) ).
fof(fact_304_zero__less__double__add__iff__zero__less__single__add,axiom,
! [A_1] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),plus_plus_real(A_1,A_1)))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_1)) ) ).
fof(fact_305_double__add__less__zero__iff__single__add__less__zero,axiom,
! [A_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,plus_plus_int(A_1,A_1)),zero_zero_int))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_1),zero_zero_int)) ) ).
fof(fact_306_double__add__less__zero__iff__single__add__less__zero,axiom,
! [A_1] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,plus_plus_real(A_1,A_1)),zero_zero_real))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_1),zero_zero_real)) ) ).
fof(fact_307_add__pos__pos,axiom,
! [B_60,A_87] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_87))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_60))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),plus_plus_int(A_87,B_60))) ) ) ).
fof(fact_308_add__pos__pos,axiom,
! [B_60,A_87] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),A_87))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),B_60))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),plus_plus_nat(A_87,B_60))) ) ) ).
fof(fact_309_add__pos__pos,axiom,
! [B_60,A_87] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_87))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),B_60))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),plus_plus_real(A_87,B_60))) ) ) ).
fof(fact_310_add__neg__neg,axiom,
! [B_59,A_86] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_86),zero_zero_int))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_59),zero_zero_int))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,plus_plus_int(A_86,B_59)),zero_zero_int)) ) ) ).
fof(fact_311_add__neg__neg,axiom,
! [B_59,A_86] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_86),zero_zero_nat))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,B_59),zero_zero_nat))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,plus_plus_nat(A_86,B_59)),zero_zero_nat)) ) ) ).
fof(fact_312_add__neg__neg,axiom,
! [B_59,A_86] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_86),zero_zero_real))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,B_59),zero_zero_real))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,plus_plus_real(A_86,B_59)),zero_zero_real)) ) ) ).
fof(fact_313_zero__reorient,axiom,
! [Xa] :
( is_int(Xa)
=> ( zero_zero_int = Xa
<=> Xa = zero_zero_int ) ) ).
fof(fact_314_zero__reorient,axiom,
! [Xa] :
( zero_zero_nat = Xa
<=> Xa = zero_zero_nat ) ).
fof(fact_315_zero__reorient,axiom,
! [Xa] :
( zero_zero_real = Xa
<=> Xa = zero_zero_real ) ).
fof(fact_316_add__right__imp__eq,axiom,
! [B_58,A_85,C_44] :
( ( is_int(B_58)
& is_int(C_44) )
=> ( plus_plus_int(B_58,A_85) = plus_plus_int(C_44,A_85)
=> B_58 = C_44 ) ) ).
fof(fact_317_add__right__imp__eq,axiom,
! [B_58,A_85,C_44] :
( plus_plus_nat(B_58,A_85) = plus_plus_nat(C_44,A_85)
=> B_58 = C_44 ) ).
fof(fact_318_add__right__imp__eq,axiom,
! [B_58,A_85,C_44] :
( plus_plus_real(B_58,A_85) = plus_plus_real(C_44,A_85)
=> B_58 = C_44 ) ).
fof(fact_319_add__imp__eq,axiom,
! [A_84,B_57,C_43] :
( ( is_int(B_57)
& is_int(C_43) )
=> ( plus_plus_int(A_84,B_57) = plus_plus_int(A_84,C_43)
=> B_57 = C_43 ) ) ).
fof(fact_320_add__imp__eq,axiom,
! [A_84,B_57,C_43] :
( plus_plus_nat(A_84,B_57) = plus_plus_nat(A_84,C_43)
=> B_57 = C_43 ) ).
fof(fact_321_add__imp__eq,axiom,
! [A_84,B_57,C_43] :
( plus_plus_real(A_84,B_57) = plus_plus_real(A_84,C_43)
=> B_57 = C_43 ) ).
fof(fact_322_add__left__imp__eq,axiom,
! [A_83,B_56,C_42] :
( ( is_int(B_56)
& is_int(C_42) )
=> ( plus_plus_int(A_83,B_56) = plus_plus_int(A_83,C_42)
=> B_56 = C_42 ) ) ).
fof(fact_323_add__left__imp__eq,axiom,
! [A_83,B_56,C_42] :
( plus_plus_nat(A_83,B_56) = plus_plus_nat(A_83,C_42)
=> B_56 = C_42 ) ).
fof(fact_324_add__left__imp__eq,axiom,
! [A_83,B_56,C_42] :
( plus_plus_real(A_83,B_56) = plus_plus_real(A_83,C_42)
=> B_56 = C_42 ) ).
fof(fact_325_comm__semiring__1__class_Onormalizing__semiring__rules_I20_J,axiom,
! [A_82,B_55,C_41,D_14] : plus_plus_int(plus_plus_int(A_82,B_55),plus_plus_int(C_41,D_14)) = plus_plus_int(plus_plus_int(A_82,C_41),plus_plus_int(B_55,D_14)) ).
fof(fact_326_comm__semiring__1__class_Onormalizing__semiring__rules_I20_J,axiom,
! [A_82,B_55,C_41,D_14] : plus_plus_nat(plus_plus_nat(A_82,B_55),plus_plus_nat(C_41,D_14)) = plus_plus_nat(plus_plus_nat(A_82,C_41),plus_plus_nat(B_55,D_14)) ).
fof(fact_327_comm__semiring__1__class_Onormalizing__semiring__rules_I20_J,axiom,
! [A_82,B_55,C_41,D_14] : plus_plus_real(plus_plus_real(A_82,B_55),plus_plus_real(C_41,D_14)) = plus_plus_real(plus_plus_real(A_82,C_41),plus_plus_real(B_55,D_14)) ).
fof(fact_328_add__right__cancel,axiom,
! [B,A_1,C_1] :
( ( is_int(B)
& is_int(C_1) )
=> ( plus_plus_int(B,A_1) = plus_plus_int(C_1,A_1)
<=> B = C_1 ) ) ).
fof(fact_329_add__right__cancel,axiom,
! [B,A_1,C_1] :
( plus_plus_nat(B,A_1) = plus_plus_nat(C_1,A_1)
<=> B = C_1 ) ).
fof(fact_330_add__right__cancel,axiom,
! [B,A_1,C_1] :
( plus_plus_real(B,A_1) = plus_plus_real(C_1,A_1)
<=> B = C_1 ) ).
fof(fact_331_add__left__cancel,axiom,
! [A_1,B,C_1] :
( ( is_int(B)
& is_int(C_1) )
=> ( plus_plus_int(A_1,B) = plus_plus_int(A_1,C_1)
<=> B = C_1 ) ) ).
fof(fact_332_add__left__cancel,axiom,
! [A_1,B,C_1] :
( plus_plus_nat(A_1,B) = plus_plus_nat(A_1,C_1)
<=> B = C_1 ) ).
fof(fact_333_add__left__cancel,axiom,
! [A_1,B,C_1] :
( plus_plus_real(A_1,B) = plus_plus_real(A_1,C_1)
<=> B = C_1 ) ).
fof(fact_334_comm__semiring__1__class_Onormalizing__semiring__rules_I23_J,axiom,
! [A_81,B_54,C_40] : plus_plus_int(plus_plus_int(A_81,B_54),C_40) = plus_plus_int(plus_plus_int(A_81,C_40),B_54) ).
fof(fact_335_comm__semiring__1__class_Onormalizing__semiring__rules_I23_J,axiom,
! [A_81,B_54,C_40] : plus_plus_nat(plus_plus_nat(A_81,B_54),C_40) = plus_plus_nat(plus_plus_nat(A_81,C_40),B_54) ).
fof(fact_336_comm__semiring__1__class_Onormalizing__semiring__rules_I23_J,axiom,
! [A_81,B_54,C_40] : plus_plus_real(plus_plus_real(A_81,B_54),C_40) = plus_plus_real(plus_plus_real(A_81,C_40),B_54) ).
fof(fact_337_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A_80,B_53,C_39] : plus_plus_int(plus_plus_int(A_80,B_53),C_39) = plus_plus_int(A_80,plus_plus_int(B_53,C_39)) ).
fof(fact_338_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A_80,B_53,C_39] : plus_plus_nat(plus_plus_nat(A_80,B_53),C_39) = plus_plus_nat(A_80,plus_plus_nat(B_53,C_39)) ).
fof(fact_339_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A_80,B_53,C_39] : plus_plus_real(plus_plus_real(A_80,B_53),C_39) = plus_plus_real(A_80,plus_plus_real(B_53,C_39)) ).
fof(fact_340_comm__semiring__1__class_Onormalizing__semiring__rules_I21_J,axiom,
! [A_79,B_52,C_38] : plus_plus_int(plus_plus_int(A_79,B_52),C_38) = plus_plus_int(A_79,plus_plus_int(B_52,C_38)) ).
fof(fact_341_comm__semiring__1__class_Onormalizing__semiring__rules_I21_J,axiom,
! [A_79,B_52,C_38] : plus_plus_nat(plus_plus_nat(A_79,B_52),C_38) = plus_plus_nat(A_79,plus_plus_nat(B_52,C_38)) ).
fof(fact_342_comm__semiring__1__class_Onormalizing__semiring__rules_I21_J,axiom,
! [A_79,B_52,C_38] : plus_plus_real(plus_plus_real(A_79,B_52),C_38) = plus_plus_real(A_79,plus_plus_real(B_52,C_38)) ).
fof(fact_343_comm__semiring__1__class_Onormalizing__semiring__rules_I25_J,axiom,
! [A_78,C_37,D_13] : plus_plus_int(A_78,plus_plus_int(C_37,D_13)) = plus_plus_int(plus_plus_int(A_78,C_37),D_13) ).
fof(fact_344_comm__semiring__1__class_Onormalizing__semiring__rules_I25_J,axiom,
! [A_78,C_37,D_13] : plus_plus_nat(A_78,plus_plus_nat(C_37,D_13)) = plus_plus_nat(plus_plus_nat(A_78,C_37),D_13) ).
fof(fact_345_comm__semiring__1__class_Onormalizing__semiring__rules_I25_J,axiom,
! [A_78,C_37,D_13] : plus_plus_real(A_78,plus_plus_real(C_37,D_13)) = plus_plus_real(plus_plus_real(A_78,C_37),D_13) ).
fof(fact_346_comm__semiring__1__class_Onormalizing__semiring__rules_I22_J,axiom,
! [A_77,C_36,D_12] : plus_plus_int(A_77,plus_plus_int(C_36,D_12)) = plus_plus_int(C_36,plus_plus_int(A_77,D_12)) ).
fof(fact_347_comm__semiring__1__class_Onormalizing__semiring__rules_I22_J,axiom,
! [A_77,C_36,D_12] : plus_plus_nat(A_77,plus_plus_nat(C_36,D_12)) = plus_plus_nat(C_36,plus_plus_nat(A_77,D_12)) ).
fof(fact_348_comm__semiring__1__class_Onormalizing__semiring__rules_I22_J,axiom,
! [A_77,C_36,D_12] : plus_plus_real(A_77,plus_plus_real(C_36,D_12)) = plus_plus_real(C_36,plus_plus_real(A_77,D_12)) ).
fof(fact_349_comm__semiring__1__class_Onormalizing__semiring__rules_I24_J,axiom,
! [A_76,C_35] : plus_plus_int(A_76,C_35) = plus_plus_int(C_35,A_76) ).
fof(fact_350_comm__semiring__1__class_Onormalizing__semiring__rules_I24_J,axiom,
! [A_76,C_35] : plus_plus_nat(A_76,C_35) = plus_plus_nat(C_35,A_76) ).
fof(fact_351_comm__semiring__1__class_Onormalizing__semiring__rules_I24_J,axiom,
! [A_76,C_35] : plus_plus_real(A_76,C_35) = plus_plus_real(C_35,A_76) ).
fof(fact_352_one__reorient,axiom,
! [Xa] :
( is_int(Xa)
=> ( one_one_int = Xa
<=> Xa = one_one_int ) ) ).
fof(fact_353_one__reorient,axiom,
! [Xa] :
( one_one_nat = Xa
<=> Xa = one_one_nat ) ).
fof(fact_354_one__reorient,axiom,
! [Xa] :
( one_one_real = Xa
<=> Xa = one_one_real ) ).
fof(fact_355_add__0__iff,axiom,
! [B,A_1] :
( ( is_int(B)
& is_int(A_1) )
=> ( B = plus_plus_int(B,A_1)
<=> A_1 = zero_zero_int ) ) ).
fof(fact_356_add__0__iff,axiom,
! [B,A_1] :
( B = plus_plus_nat(B,A_1)
<=> A_1 = zero_zero_nat ) ).
fof(fact_357_add__0__iff,axiom,
! [B,A_1] :
( B = plus_plus_real(B,A_1)
<=> A_1 = zero_zero_real ) ).
fof(fact_358_add_Ocomm__neutral,axiom,
! [A_75] :
( is_int(A_75)
=> plus_plus_int(A_75,zero_zero_int) = A_75 ) ).
fof(fact_359_add_Ocomm__neutral,axiom,
! [A_75] : plus_plus_nat(A_75,zero_zero_nat) = A_75 ).
fof(fact_360_add_Ocomm__neutral,axiom,
! [A_75] : plus_plus_real(A_75,zero_zero_real) = A_75 ).
fof(fact_361_comm__semiring__1__class_Onormalizing__semiring__rules_I6_J,axiom,
! [A_74] :
( is_int(A_74)
=> plus_plus_int(A_74,zero_zero_int) = A_74 ) ).
fof(fact_362_comm__semiring__1__class_Onormalizing__semiring__rules_I6_J,axiom,
! [A_74] : plus_plus_nat(A_74,zero_zero_nat) = A_74 ).
fof(fact_363_comm__semiring__1__class_Onormalizing__semiring__rules_I6_J,axiom,
! [A_74] : plus_plus_real(A_74,zero_zero_real) = A_74 ).
fof(fact_364_add__0__right,axiom,
! [A_73] :
( is_int(A_73)
=> plus_plus_int(A_73,zero_zero_int) = A_73 ) ).
fof(fact_365_add__0__right,axiom,
! [A_73] : plus_plus_nat(A_73,zero_zero_nat) = A_73 ).
fof(fact_366_add__0__right,axiom,
! [A_73] : plus_plus_real(A_73,zero_zero_real) = A_73 ).
fof(fact_367_double__zero__sym,axiom,
! [A_1] :
( is_int(A_1)
=> ( zero_zero_int = plus_plus_int(A_1,A_1)
<=> A_1 = zero_zero_int ) ) ).
fof(fact_368_double__zero__sym,axiom,
! [A_1] :
( zero_zero_real = plus_plus_real(A_1,A_1)
<=> A_1 = zero_zero_real ) ).
fof(fact_369_add__0,axiom,
! [A_72] :
( is_int(A_72)
=> plus_plus_int(zero_zero_int,A_72) = A_72 ) ).
fof(fact_370_add__0,axiom,
! [A_72] : plus_plus_nat(zero_zero_nat,A_72) = A_72 ).
fof(fact_371_add__0,axiom,
! [A_72] : plus_plus_real(zero_zero_real,A_72) = A_72 ).
fof(fact_372_comm__semiring__1__class_Onormalizing__semiring__rules_I5_J,axiom,
! [A_71] :
( is_int(A_71)
=> plus_plus_int(zero_zero_int,A_71) = A_71 ) ).
fof(fact_373_comm__semiring__1__class_Onormalizing__semiring__rules_I5_J,axiom,
! [A_71] : plus_plus_nat(zero_zero_nat,A_71) = A_71 ).
fof(fact_374_comm__semiring__1__class_Onormalizing__semiring__rules_I5_J,axiom,
! [A_71] : plus_plus_real(zero_zero_real,A_71) = A_71 ).
fof(fact_375_add__0__left,axiom,
! [A_70] :
( is_int(A_70)
=> plus_plus_int(zero_zero_int,A_70) = A_70 ) ).
fof(fact_376_add__0__left,axiom,
! [A_70] : plus_plus_nat(zero_zero_nat,A_70) = A_70 ).
fof(fact_377_add__0__left,axiom,
! [A_70] : plus_plus_real(zero_zero_real,A_70) = A_70 ).
fof(fact_378_add__less__imp__less__left,axiom,
! [C_34,A_69,B_51] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,plus_plus_int(C_34,A_69)),plus_plus_int(C_34,B_51)))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_69),B_51)) ) ).
fof(fact_379_add__less__imp__less__left,axiom,
! [C_34,A_69,B_51] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,plus_plus_nat(C_34,A_69)),plus_plus_nat(C_34,B_51)))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_69),B_51)) ) ).
fof(fact_380_add__less__imp__less__left,axiom,
! [C_34,A_69,B_51] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,plus_plus_real(C_34,A_69)),plus_plus_real(C_34,B_51)))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_69),B_51)) ) ).
fof(fact_381_add__less__imp__less__right,axiom,
! [A_68,C_33,B_50] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,plus_plus_int(A_68,C_33)),plus_plus_int(B_50,C_33)))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_68),B_50)) ) ).
fof(fact_382_add__less__imp__less__right,axiom,
! [A_68,C_33,B_50] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,plus_plus_nat(A_68,C_33)),plus_plus_nat(B_50,C_33)))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_68),B_50)) ) ).
fof(fact_383_add__less__imp__less__right,axiom,
! [A_68,C_33,B_50] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,plus_plus_real(A_68,C_33)),plus_plus_real(B_50,C_33)))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_68),B_50)) ) ).
fof(fact_384_add__strict__mono,axiom,
! [C_32,D_11,A_67,B_49] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_67),B_49))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,C_32),D_11))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,plus_plus_int(A_67,C_32)),plus_plus_int(B_49,D_11))) ) ) ).
fof(fact_385_add__strict__mono,axiom,
! [C_32,D_11,A_67,B_49] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_67),B_49))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,C_32),D_11))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,plus_plus_nat(A_67,C_32)),plus_plus_nat(B_49,D_11))) ) ) ).
fof(fact_386_add__strict__mono,axiom,
! [C_32,D_11,A_67,B_49] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_67),B_49))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,C_32),D_11))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,plus_plus_real(A_67,C_32)),plus_plus_real(B_49,D_11))) ) ) ).
fof(fact_387_add__strict__left__mono,axiom,
! [C_31,A_66,B_48] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_66),B_48))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,plus_plus_int(C_31,A_66)),plus_plus_int(C_31,B_48))) ) ).
fof(fact_388_add__strict__left__mono,axiom,
! [C_31,A_66,B_48] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_66),B_48))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,plus_plus_nat(C_31,A_66)),plus_plus_nat(C_31,B_48))) ) ).
fof(fact_389_add__strict__left__mono,axiom,
! [C_31,A_66,B_48] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_66),B_48))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,plus_plus_real(C_31,A_66)),plus_plus_real(C_31,B_48))) ) ).
fof(fact_390_add__strict__right__mono,axiom,
! [C_30,A_65,B_47] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_65),B_47))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,plus_plus_int(A_65,C_30)),plus_plus_int(B_47,C_30))) ) ).
fof(fact_391_add__strict__right__mono,axiom,
! [C_30,A_65,B_47] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_65),B_47))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,plus_plus_nat(A_65,C_30)),plus_plus_nat(B_47,C_30))) ) ).
fof(fact_392_add__strict__right__mono,axiom,
! [C_30,A_65,B_47] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_65),B_47))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,plus_plus_real(A_65,C_30)),plus_plus_real(B_47,C_30))) ) ).
fof(fact_393_add__less__cancel__left,axiom,
! [C_1,A_1,B] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,plus_plus_int(C_1,A_1)),plus_plus_int(C_1,B)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_1),B)) ) ).
fof(fact_394_add__less__cancel__left,axiom,
! [C_1,A_1,B] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,plus_plus_nat(C_1,A_1)),plus_plus_nat(C_1,B)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_1),B)) ) ).
fof(fact_395_add__less__cancel__left,axiom,
! [C_1,A_1,B] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,plus_plus_real(C_1,A_1)),plus_plus_real(C_1,B)))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_1),B)) ) ).
fof(fact_396_add__less__cancel__right,axiom,
! [A_1,C_1,B] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,plus_plus_int(A_1,C_1)),plus_plus_int(B,C_1)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_1),B)) ) ).
fof(fact_397_add__less__cancel__right,axiom,
! [A_1,C_1,B] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,plus_plus_nat(A_1,C_1)),plus_plus_nat(B,C_1)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_1),B)) ) ).
fof(fact_398_add__less__cancel__right,axiom,
! [A_1,C_1,B] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,plus_plus_real(A_1,C_1)),plus_plus_real(B,C_1)))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_1),B)) ) ).
fof(fact_399_comm__semiring__1__class_Onormalizing__semiring__rules_I33_J,axiom,
! [X_12] :
( is_int(X_12)
=> hAPP_nat_int(power_power_int(X_12),one_one_nat) = X_12 ) ).
fof(fact_400_comm__semiring__1__class_Onormalizing__semiring__rules_I33_J,axiom,
! [X_12] : hAPP_nat_nat(power_power_nat(X_12),one_one_nat) = X_12 ).
fof(fact_401_comm__semiring__1__class_Onormalizing__semiring__rules_I33_J,axiom,
! [X_12] : hAPP_nat_real(power_power_real(X_12),one_one_nat) = X_12 ).
fof(fact_402_nat__power__eq__0__iff,axiom,
! [Ma,Na] :
( hAPP_nat_nat(power_power_nat(Ma),Na) = zero_zero_nat
<=> ( Na != zero_zero_nat
& Ma = zero_zero_nat ) ) ).
fof(fact_403_realpow__pos__nth__unique,axiom,
! [A,N] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A))
=> ? [X_1] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_1))
& hAPP_nat_real(power_power_real(X_1),N) = A
& ! [Y_1] :
( ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Y_1))
& hAPP_nat_real(power_power_real(Y_1),N) = A )
=> Y_1 = X_1 ) ) ) ) ).
fof(fact_404_realpow__pos__nth,axiom,
! [A,N] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A))
=> ? [R] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),R))
& hAPP_nat_real(power_power_real(R),N) = A ) ) ) ).
fof(fact_405_tpos,axiom,
hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,one_one_int),t)) ).
fof(fact_406_nat__number__of__add__1,axiom,
! [V_1] :
( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,V_1),pls))
=> plus_plus_nat(number_number_of_nat(V_1),one_one_nat) = one_one_nat )
& ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,V_1),pls))
=> plus_plus_nat(number_number_of_nat(V_1),one_one_nat) = number_number_of_nat(succ(V_1)) ) ) ).
fof(fact_407_nat__1__add__number__of,axiom,
! [V_1] :
( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,V_1),pls))
=> plus_plus_nat(one_one_nat,number_number_of_nat(V_1)) = one_one_nat )
& ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,V_1),pls))
=> plus_plus_nat(one_one_nat,number_number_of_nat(V_1)) = number_number_of_nat(succ(V_1)) ) ) ).
fof(fact_408_zadd__power3,axiom,
! [A,B_1] : hAPP_nat_int(power_power_int(plus_plus_int(A,B_1)),number_number_of_nat(bit1(bit1(pls)))) = plus_plus_int(plus_plus_int(plus_plus_int(hAPP_nat_int(power_power_int(A),number_number_of_nat(bit1(bit1(pls)))),times_times_int(times_times_int(number_number_of_int(bit1(bit1(pls))),hAPP_nat_int(power_power_int(A),number_number_of_nat(bit0(bit1(pls))))),B_1)),times_times_int(times_times_int(number_number_of_int(bit1(bit1(pls))),A),hAPP_nat_int(power_power_int(B_1),number_number_of_nat(bit0(bit1(pls)))))),hAPP_nat_int(power_power_int(B_1),number_number_of_nat(bit1(bit1(pls))))) ).
fof(fact_409_zadd__power2,axiom,
! [A,B_1] : hAPP_nat_int(power_power_int(plus_plus_int(A,B_1)),number_number_of_nat(bit0(bit1(pls)))) = plus_plus_int(plus_plus_int(hAPP_nat_int(power_power_int(A),number_number_of_nat(bit0(bit1(pls)))),times_times_int(times_times_int(number_number_of_int(bit0(bit1(pls))),A),B_1)),hAPP_nat_int(power_power_int(B_1),number_number_of_nat(bit0(bit1(pls))))) ).
fof(fact_410_tn,axiom,
tn = minus_minus_nat(nat(t),one_one_nat) ).
fof(fact_411__096_B_Bthesis_O_A_I_B_Btn_O_A_091_124_Atn_A_061_Anat_At_A_N_A1_059_A0_,axiom,
hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),minus_minus_nat(nat(t),one_one_nat))) ).
fof(fact_412_le__iff__diff__le__0,axiom,
! [A_1,B] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_1),B))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,minus_minus_int(A_1,B)),zero_zero_int)) ) ).
fof(fact_413_le__iff__diff__le__0,axiom,
! [A_1,B] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_1),B))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,minus_minus_real(A_1,B)),zero_zero_real)) ) ).
fof(fact_414_comm__semiring__1__class_Onormalizing__semiring__rules_I7_J,axiom,
! [A_64,B_46] : times_times_int(A_64,B_46) = times_times_int(B_46,A_64) ).
fof(fact_415_comm__semiring__1__class_Onormalizing__semiring__rules_I7_J,axiom,
! [A_64,B_46] : times_times_nat(A_64,B_46) = times_times_nat(B_46,A_64) ).
fof(fact_416_comm__semiring__1__class_Onormalizing__semiring__rules_I7_J,axiom,
! [A_64,B_46] : times_times_real(A_64,B_46) = times_times_real(B_46,A_64) ).
fof(fact_417_comm__semiring__1__class_Onormalizing__semiring__rules_I19_J,axiom,
! [Lx_6,Rx_6,Ry_4] : times_times_int(Lx_6,times_times_int(Rx_6,Ry_4)) = times_times_int(Rx_6,times_times_int(Lx_6,Ry_4)) ).
fof(fact_418_comm__semiring__1__class_Onormalizing__semiring__rules_I19_J,axiom,
! [Lx_6,Rx_6,Ry_4] : times_times_nat(Lx_6,times_times_nat(Rx_6,Ry_4)) = times_times_nat(Rx_6,times_times_nat(Lx_6,Ry_4)) ).
fof(fact_419_comm__semiring__1__class_Onormalizing__semiring__rules_I19_J,axiom,
! [Lx_6,Rx_6,Ry_4] : times_times_real(Lx_6,times_times_real(Rx_6,Ry_4)) = times_times_real(Rx_6,times_times_real(Lx_6,Ry_4)) ).
fof(fact_420_comm__semiring__1__class_Onormalizing__semiring__rules_I18_J,axiom,
! [Lx_5,Rx_5,Ry_3] : times_times_int(Lx_5,times_times_int(Rx_5,Ry_3)) = times_times_int(times_times_int(Lx_5,Rx_5),Ry_3) ).
fof(fact_421_comm__semiring__1__class_Onormalizing__semiring__rules_I18_J,axiom,
! [Lx_5,Rx_5,Ry_3] : times_times_nat(Lx_5,times_times_nat(Rx_5,Ry_3)) = times_times_nat(times_times_nat(Lx_5,Rx_5),Ry_3) ).
fof(fact_422_comm__semiring__1__class_Onormalizing__semiring__rules_I18_J,axiom,
! [Lx_5,Rx_5,Ry_3] : times_times_real(Lx_5,times_times_real(Rx_5,Ry_3)) = times_times_real(times_times_real(Lx_5,Rx_5),Ry_3) ).
fof(fact_423_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A_63,B_45,C_29] : times_times_int(times_times_int(A_63,B_45),C_29) = times_times_int(A_63,times_times_int(B_45,C_29)) ).
fof(fact_424_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A_63,B_45,C_29] : times_times_nat(times_times_nat(A_63,B_45),C_29) = times_times_nat(A_63,times_times_nat(B_45,C_29)) ).
fof(fact_425_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A_63,B_45,C_29] : times_times_real(times_times_real(A_63,B_45),C_29) = times_times_real(A_63,times_times_real(B_45,C_29)) ).
fof(fact_426_comm__semiring__1__class_Onormalizing__semiring__rules_I17_J,axiom,
! [Lx_4,Ly_4,Rx_4] : times_times_int(times_times_int(Lx_4,Ly_4),Rx_4) = times_times_int(Lx_4,times_times_int(Ly_4,Rx_4)) ).
fof(fact_427_comm__semiring__1__class_Onormalizing__semiring__rules_I17_J,axiom,
! [Lx_4,Ly_4,Rx_4] : times_times_nat(times_times_nat(Lx_4,Ly_4),Rx_4) = times_times_nat(Lx_4,times_times_nat(Ly_4,Rx_4)) ).
fof(fact_428_comm__semiring__1__class_Onormalizing__semiring__rules_I17_J,axiom,
! [Lx_4,Ly_4,Rx_4] : times_times_real(times_times_real(Lx_4,Ly_4),Rx_4) = times_times_real(Lx_4,times_times_real(Ly_4,Rx_4)) ).
fof(fact_429_comm__semiring__1__class_Onormalizing__semiring__rules_I16_J,axiom,
! [Lx_3,Ly_3,Rx_3] : times_times_int(times_times_int(Lx_3,Ly_3),Rx_3) = times_times_int(times_times_int(Lx_3,Rx_3),Ly_3) ).
fof(fact_430_comm__semiring__1__class_Onormalizing__semiring__rules_I16_J,axiom,
! [Lx_3,Ly_3,Rx_3] : times_times_nat(times_times_nat(Lx_3,Ly_3),Rx_3) = times_times_nat(times_times_nat(Lx_3,Rx_3),Ly_3) ).
fof(fact_431_comm__semiring__1__class_Onormalizing__semiring__rules_I16_J,axiom,
! [Lx_3,Ly_3,Rx_3] : times_times_real(times_times_real(Lx_3,Ly_3),Rx_3) = times_times_real(times_times_real(Lx_3,Rx_3),Ly_3) ).
fof(fact_432_comm__semiring__1__class_Onormalizing__semiring__rules_I14_J,axiom,
! [Lx_2,Ly_2,Rx_2,Ry_2] : times_times_int(times_times_int(Lx_2,Ly_2),times_times_int(Rx_2,Ry_2)) = times_times_int(Lx_2,times_times_int(Ly_2,times_times_int(Rx_2,Ry_2))) ).
fof(fact_433_comm__semiring__1__class_Onormalizing__semiring__rules_I14_J,axiom,
! [Lx_2,Ly_2,Rx_2,Ry_2] : times_times_nat(times_times_nat(Lx_2,Ly_2),times_times_nat(Rx_2,Ry_2)) = times_times_nat(Lx_2,times_times_nat(Ly_2,times_times_nat(Rx_2,Ry_2))) ).
fof(fact_434_comm__semiring__1__class_Onormalizing__semiring__rules_I14_J,axiom,
! [Lx_2,Ly_2,Rx_2,Ry_2] : times_times_real(times_times_real(Lx_2,Ly_2),times_times_real(Rx_2,Ry_2)) = times_times_real(Lx_2,times_times_real(Ly_2,times_times_real(Rx_2,Ry_2))) ).
fof(fact_435_comm__semiring__1__class_Onormalizing__semiring__rules_I15_J,axiom,
! [Lx_1,Ly_1,Rx_1,Ry_1] : times_times_real(times_times_real(Lx_1,Ly_1),times_times_real(Rx_1,Ry_1)) = times_times_real(Rx_1,times_times_real(times_times_real(Lx_1,Ly_1),Ry_1)) ).
fof(fact_436_comm__semiring__1__class_Onormalizing__semiring__rules_I15_J,axiom,
! [Lx_1,Ly_1,Rx_1,Ry_1] : times_times_nat(times_times_nat(Lx_1,Ly_1),times_times_nat(Rx_1,Ry_1)) = times_times_nat(Rx_1,times_times_nat(times_times_nat(Lx_1,Ly_1),Ry_1)) ).
fof(fact_437_comm__semiring__1__class_Onormalizing__semiring__rules_I15_J,axiom,
! [Lx_1,Ly_1,Rx_1,Ry_1] : times_times_int(times_times_int(Lx_1,Ly_1),times_times_int(Rx_1,Ry_1)) = times_times_int(Rx_1,times_times_int(times_times_int(Lx_1,Ly_1),Ry_1)) ).
fof(fact_438_comm__semiring__1__class_Onormalizing__semiring__rules_I13_J,axiom,
! [Lx,Ly,Rx,Ry] : times_times_real(times_times_real(Lx,Ly),times_times_real(Rx,Ry)) = times_times_real(times_times_real(Lx,Rx),times_times_real(Ly,Ry)) ).
fof(fact_439_comm__semiring__1__class_Onormalizing__semiring__rules_I13_J,axiom,
! [Lx,Ly,Rx,Ry] : times_times_nat(times_times_nat(Lx,Ly),times_times_nat(Rx,Ry)) = times_times_nat(times_times_nat(Lx,Rx),times_times_nat(Ly,Ry)) ).
fof(fact_440_comm__semiring__1__class_Onormalizing__semiring__rules_I13_J,axiom,
! [Lx,Ly,Rx,Ry] : times_times_int(times_times_int(Lx,Ly),times_times_int(Rx,Ry)) = times_times_int(times_times_int(Lx,Rx),times_times_int(Ly,Ry)) ).
fof(fact_441_diff__eq__diff__eq,axiom,
! [A_1,B,C_1,D] :
( ( is_int(A_1)
& is_int(B)
& is_int(C_1)
& is_int(D) )
=> ( minus_minus_int(A_1,B) = minus_minus_int(C_1,D)
=> ( A_1 = B
<=> C_1 = D ) ) ) ).
fof(fact_442_diff__eq__diff__eq,axiom,
! [A_1,B,C_1,D] :
( minus_minus_real(A_1,B) = minus_minus_real(C_1,D)
=> ( A_1 = B
<=> C_1 = D ) ) ).
fof(fact_443_diff__eq__diff__less__eq,axiom,
! [A_1,B,C_1,D] :
( minus_minus_int(A_1,B) = minus_minus_int(C_1,D)
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_1),B))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,C_1),D)) ) ) ).
fof(fact_444_diff__eq__diff__less__eq,axiom,
! [A_1,B,C_1,D] :
( minus_minus_real(A_1,B) = minus_minus_real(C_1,D)
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_1),B))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,C_1),D)) ) ) ).
fof(fact_445_semiring__mult__number__of,axiom,
! [V_13,V_12] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),V_12))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),V_13))
=> times_times_real(number267125858f_real(V_12),number267125858f_real(V_13)) = number267125858f_real(times_times_int(V_12,V_13)) ) ) ).
fof(fact_446_semiring__mult__number__of,axiom,
! [V_13,V_12] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),V_12))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),V_13))
=> times_times_nat(number_number_of_nat(V_12),number_number_of_nat(V_13)) = number_number_of_nat(times_times_int(V_12,V_13)) ) ) ).
fof(fact_447_semiring__mult__number__of,axiom,
! [V_13,V_12] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),V_12))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),V_13))
=> times_times_int(number_number_of_int(V_12),number_number_of_int(V_13)) = number_number_of_int(times_times_int(V_12,V_13)) ) ) ).
fof(fact_448_zle__refl,axiom,
! [W] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,W),W)) ).
fof(fact_449_zmult__commute,axiom,
! [Z,W] : times_times_int(Z,W) = times_times_int(W,Z) ).
fof(fact_450_zle__linear,axiom,
! [Z,W] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Z),W))
| hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,W),Z)) ) ).
fof(fact_451_zmult__assoc,axiom,
! [Z1,Z2,Z3] : times_times_int(times_times_int(Z1,Z2),Z3) = times_times_int(Z1,times_times_int(Z2,Z3)) ).
fof(fact_452_zle__trans,axiom,
! [K_1,I_2,J_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,I_2),J_1))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,J_1),K_1))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,I_2),K_1)) ) ) ).
fof(fact_453_zle__antisym,axiom,
! [Z,W] :
( ( is_int(Z)
& is_int(W) )
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Z),W))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,W),Z))
=> Z = W ) ) ) ).
fof(fact_454_le__number__of,axiom,
! [Xa,Ya] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,number267125858f_real(Xa)),number267125858f_real(Ya)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Xa),Ya)) ) ).
fof(fact_455_le__number__of,axiom,
! [Xa,Ya] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,number_number_of_int(Xa)),number_number_of_int(Ya)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Xa),Ya)) ) ).
fof(fact_456_number__of__mult,axiom,
! [V_11,W_6] : number267125858f_real(times_times_int(V_11,W_6)) = times_times_real(number267125858f_real(V_11),number267125858f_real(W_6)) ).
fof(fact_457_number__of__mult,axiom,
! [V_11,W_6] : number_number_of_int(times_times_int(V_11,W_6)) = times_times_int(number_number_of_int(V_11),number_number_of_int(W_6)) ).
fof(fact_458_arith__simps_I32_J,axiom,
! [V_10,W_5] : times_times_real(number267125858f_real(V_10),number267125858f_real(W_5)) = number267125858f_real(times_times_int(V_10,W_5)) ).
fof(fact_459_arith__simps_I32_J,axiom,
! [V_10,W_5] : times_times_int(number_number_of_int(V_10),number_number_of_int(W_5)) = number_number_of_int(times_times_int(V_10,W_5)) ).
fof(fact_460_mult__number__of__left,axiom,
! [V_9,W_4,Z_5] : times_times_real(number267125858f_real(V_9),times_times_real(number267125858f_real(W_4),Z_5)) = times_times_real(number267125858f_real(times_times_int(V_9,W_4)),Z_5) ).
fof(fact_461_mult__number__of__left,axiom,
! [V_9,W_4,Z_5] : times_times_int(number_number_of_int(V_9),times_times_int(number_number_of_int(W_4),Z_5)) = times_times_int(number_number_of_int(times_times_int(V_9,W_4)),Z_5) ).
fof(fact_462_eq__nat__nat__iff,axiom,
! [Z_4,Z_1] :
( ( is_int(Z_4)
& is_int(Z_1) )
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Z_1))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Z_4))
=> ( nat(Z_1) = nat(Z_4)
<=> Z_1 = Z_4 ) ) ) ) ).
fof(fact_463_right__diff__distrib__number__of,axiom,
! [V_8,B_44,C_28] : times_times_int(number_number_of_int(V_8),minus_minus_int(B_44,C_28)) = minus_minus_int(times_times_int(number_number_of_int(V_8),B_44),times_times_int(number_number_of_int(V_8),C_28)) ).
fof(fact_464_right__diff__distrib__number__of,axiom,
! [V_8,B_44,C_28] : times_times_real(number267125858f_real(V_8),minus_minus_real(B_44,C_28)) = minus_minus_real(times_times_real(number267125858f_real(V_8),B_44),times_times_real(number267125858f_real(V_8),C_28)) ).
fof(fact_465_left__diff__distrib__number__of,axiom,
! [A_62,B_43,V_7] : times_times_int(minus_minus_int(A_62,B_43),number_number_of_int(V_7)) = minus_minus_int(times_times_int(A_62,number_number_of_int(V_7)),times_times_int(B_43,number_number_of_int(V_7))) ).
fof(fact_466_left__diff__distrib__number__of,axiom,
! [A_62,B_43,V_7] : times_times_real(minus_minus_real(A_62,B_43),number267125858f_real(V_7)) = minus_minus_real(times_times_real(A_62,number267125858f_real(V_7)),times_times_real(B_43,number267125858f_real(V_7))) ).
fof(fact_467_number__of__diff,axiom,
! [V_6,W_3] : number_number_of_int(minus_minus_int(V_6,W_3)) = minus_minus_int(number_number_of_int(V_6),number_number_of_int(W_3)) ).
fof(fact_468_number__of__diff,axiom,
! [V_6,W_3] : number267125858f_real(minus_minus_int(V_6,W_3)) = minus_minus_real(number267125858f_real(V_6),number267125858f_real(W_3)) ).
fof(fact_469_of__nat__mult,axiom,
! [M_4,N_10] : hAPP_nat_real(semiri132038758t_real,times_times_nat(M_4,N_10)) = times_times_real(hAPP_nat_real(semiri132038758t_real,M_4),hAPP_nat_real(semiri132038758t_real,N_10)) ).
fof(fact_470_of__nat__mult,axiom,
! [M_4,N_10] : hAPP_nat_nat(semiri984289939at_nat,times_times_nat(M_4,N_10)) = times_times_nat(hAPP_nat_nat(semiri984289939at_nat,M_4),hAPP_nat_nat(semiri984289939at_nat,N_10)) ).
fof(fact_471_of__nat__mult,axiom,
! [M_4,N_10] : hAPP_nat_int(semiri1621563631at_int,times_times_nat(M_4,N_10)) = times_times_int(hAPP_nat_int(semiri1621563631at_int,M_4),hAPP_nat_int(semiri1621563631at_int,N_10)) ).
fof(fact_472_of__nat__le__iff,axiom,
! [Ma,Na] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(semiri132038758t_real,Ma)),hAPP_nat_real(semiri132038758t_real,Na)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Ma),Na)) ) ).
fof(fact_473_of__nat__le__iff,axiom,
! [Ma,Na] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(semiri984289939at_nat,Ma)),hAPP_nat_nat(semiri984289939at_nat,Na)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Ma),Na)) ) ).
fof(fact_474_of__nat__le__iff,axiom,
! [Ma,Na] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_nat_int(semiri1621563631at_int,Ma)),hAPP_nat_int(semiri1621563631at_int,Na)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Ma),Na)) ) ).
fof(fact_475_diff__commute,axiom,
! [I_2,J_1,K_1] : minus_minus_nat(minus_minus_nat(I_2,J_1),K_1) = minus_minus_nat(minus_minus_nat(I_2,K_1),J_1) ).
fof(fact_476_nat__if__cong,axiom,
! [Xa,Ya,P_1] :
( ( hBOOL(P_1)
=> nat(Xa) = nat(if_int(P_1,Xa,Ya)) )
& ( ~ hBOOL(P_1)
=> nat(Ya) = nat(if_int(P_1,Xa,Ya)) ) ) ).
fof(fact_477_split__mult__neg__le,axiom,
! [B_42,A_61] :
( ( ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_61))
& hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B_42),zero_zero_real)) )
| ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_61),zero_zero_real))
& hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),B_42)) ) )
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,times_times_real(A_61,B_42)),zero_zero_real)) ) ).
fof(fact_478_split__mult__neg__le,axiom,
! [B_42,A_61] :
( ( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),A_61))
& hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,B_42),zero_zero_nat)) )
| ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_61),zero_zero_nat))
& hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),B_42)) ) )
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,times_times_nat(A_61,B_42)),zero_zero_nat)) ) ).
fof(fact_479_split__mult__neg__le,axiom,
! [B_42,A_61] :
( ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_61))
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_42),zero_zero_int)) )
| ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_61),zero_zero_int))
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),B_42)) ) )
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,times_times_int(A_61,B_42)),zero_zero_int)) ) ).
fof(fact_480_split__mult__pos__le,axiom,
! [B_41,A_60] :
( ( ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_60))
& hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),B_41)) )
| ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_60),zero_zero_real))
& hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B_41),zero_zero_real)) ) )
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),times_times_real(A_60,B_41))) ) ).
fof(fact_481_split__mult__pos__le,axiom,
! [B_41,A_60] :
( ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_60))
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),B_41)) )
| ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_60),zero_zero_int))
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_41),zero_zero_int)) ) )
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),times_times_int(A_60,B_41))) ) ).
fof(fact_482_mult__mono,axiom,
! [C_27,D_10,A_59,B_40] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_59),B_40))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,C_27),D_10))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),B_40))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),C_27))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,times_times_real(A_59,C_27)),times_times_real(B_40,D_10))) ) ) ) ) ).
fof(fact_483_mult__mono,axiom,
! [C_27,D_10,A_59,B_40] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_59),B_40))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,C_27),D_10))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),B_40))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),C_27))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,times_times_nat(A_59,C_27)),times_times_nat(B_40,D_10))) ) ) ) ) ).
fof(fact_484_mult__mono,axiom,
! [C_27,D_10,A_59,B_40] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_59),B_40))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,C_27),D_10))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),B_40))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),C_27))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,times_times_int(A_59,C_27)),times_times_int(B_40,D_10))) ) ) ) ) ).
fof(fact_485_mult__mono_H,axiom,
! [C_26,D_9,A_58,B_39] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_58),B_39))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,C_26),D_9))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_58))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),C_26))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,times_times_real(A_58,C_26)),times_times_real(B_39,D_9))) ) ) ) ) ).
fof(fact_486_mult__mono_H,axiom,
! [C_26,D_9,A_58,B_39] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_58),B_39))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,C_26),D_9))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),A_58))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),C_26))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,times_times_nat(A_58,C_26)),times_times_nat(B_39,D_9))) ) ) ) ) ).
fof(fact_487_mult__mono_H,axiom,
! [C_26,D_9,A_58,B_39] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_58),B_39))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,C_26),D_9))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_58))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),C_26))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,times_times_int(A_58,C_26)),times_times_int(B_39,D_9))) ) ) ) ) ).
fof(fact_488_mult__left__mono__neg,axiom,
! [C_25,B_38,A_57] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B_38),A_57))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,C_25),zero_zero_real))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,times_times_real(C_25,A_57)),times_times_real(C_25,B_38))) ) ) ).
fof(fact_489_mult__left__mono__neg,axiom,
! [C_25,B_38,A_57] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_38),A_57))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,C_25),zero_zero_int))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,times_times_int(C_25,A_57)),times_times_int(C_25,B_38))) ) ) ).
fof(fact_490_mult__right__mono__neg,axiom,
! [C_24,B_37,A_56] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B_37),A_56))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,C_24),zero_zero_real))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,times_times_real(A_56,C_24)),times_times_real(B_37,C_24))) ) ) ).
fof(fact_491_mult__right__mono__neg,axiom,
! [C_24,B_37,A_56] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_37),A_56))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,C_24),zero_zero_int))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,times_times_int(A_56,C_24)),times_times_int(B_37,C_24))) ) ) ).
fof(fact_492_comm__mult__left__mono,axiom,
! [C_23,A_55,B_36] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_55),B_36))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),C_23))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,times_times_real(C_23,A_55)),times_times_real(C_23,B_36))) ) ) ).
fof(fact_493_comm__mult__left__mono,axiom,
! [C_23,A_55,B_36] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_55),B_36))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),C_23))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,times_times_nat(C_23,A_55)),times_times_nat(C_23,B_36))) ) ) ).
fof(fact_494_comm__mult__left__mono,axiom,
! [C_23,A_55,B_36] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_55),B_36))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),C_23))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,times_times_int(C_23,A_55)),times_times_int(C_23,B_36))) ) ) ).
fof(fact_495_mult__left__mono,axiom,
! [C_22,A_54,B_35] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_54),B_35))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),C_22))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,times_times_real(C_22,A_54)),times_times_real(C_22,B_35))) ) ) ).
fof(fact_496_mult__left__mono,axiom,
! [C_22,A_54,B_35] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_54),B_35))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),C_22))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,times_times_nat(C_22,A_54)),times_times_nat(C_22,B_35))) ) ) ).
fof(fact_497_mult__left__mono,axiom,
! [C_22,A_54,B_35] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_54),B_35))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),C_22))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,times_times_int(C_22,A_54)),times_times_int(C_22,B_35))) ) ) ).
fof(fact_498_mult__right__mono,axiom,
! [C_21,A_53,B_34] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_53),B_34))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),C_21))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,times_times_real(A_53,C_21)),times_times_real(B_34,C_21))) ) ) ).
fof(fact_499_mult__right__mono,axiom,
! [C_21,A_53,B_34] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_53),B_34))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),C_21))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,times_times_nat(A_53,C_21)),times_times_nat(B_34,C_21))) ) ) ).
fof(fact_500_mult__right__mono,axiom,
! [C_21,A_53,B_34] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_53),B_34))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),C_21))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,times_times_int(A_53,C_21)),times_times_int(B_34,C_21))) ) ) ).
fof(fact_501_mult__nonpos__nonpos,axiom,
! [B_33,A_52] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_52),zero_zero_real))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B_33),zero_zero_real))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),times_times_real(A_52,B_33))) ) ) ).
fof(fact_502_mult__nonpos__nonpos,axiom,
! [B_33,A_52] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_52),zero_zero_int))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_33),zero_zero_int))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),times_times_int(A_52,B_33))) ) ) ).
fof(fact_503_mult__nonpos__nonneg,axiom,
! [B_32,A_51] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_51),zero_zero_real))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),B_32))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,times_times_real(A_51,B_32)),zero_zero_real)) ) ) ).
fof(fact_504_mult__nonpos__nonneg,axiom,
! [B_32,A_51] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_51),zero_zero_nat))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),B_32))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,times_times_nat(A_51,B_32)),zero_zero_nat)) ) ) ).
fof(fact_505_mult__nonpos__nonneg,axiom,
! [B_32,A_51] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_51),zero_zero_int))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),B_32))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,times_times_int(A_51,B_32)),zero_zero_int)) ) ) ).
fof(fact_506_mult__nonneg__nonpos2,axiom,
! [B_31,A_50] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_50))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B_31),zero_zero_real))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,times_times_real(B_31,A_50)),zero_zero_real)) ) ) ).
fof(fact_507_mult__nonneg__nonpos2,axiom,
! [B_31,A_50] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),A_50))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,B_31),zero_zero_nat))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,times_times_nat(B_31,A_50)),zero_zero_nat)) ) ) ).
fof(fact_508_mult__nonneg__nonpos2,axiom,
! [B_31,A_50] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_50))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_31),zero_zero_int))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,times_times_int(B_31,A_50)),zero_zero_int)) ) ) ).
fof(fact_509_mult__nonneg__nonpos,axiom,
! [B_30,A_49] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_49))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B_30),zero_zero_real))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,times_times_real(A_49,B_30)),zero_zero_real)) ) ) ).
fof(fact_510_mult__nonneg__nonpos,axiom,
! [B_30,A_49] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),A_49))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,B_30),zero_zero_nat))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,times_times_nat(A_49,B_30)),zero_zero_nat)) ) ) ).
fof(fact_511_mult__nonneg__nonpos,axiom,
! [B_30,A_49] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_49))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_30),zero_zero_int))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,times_times_int(A_49,B_30)),zero_zero_int)) ) ) ).
fof(fact_512_transfer__nat__int__relations_I1_J,axiom,
! [Ya,Xa] :
( ( is_int(Ya)
& is_int(Xa) )
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Xa))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Ya))
=> ( nat(Xa) = nat(Ya)
<=> Xa = Ya ) ) ) ) ).
fof(fact_513_Nat__Transfer_Otransfer__nat__int__function__closures_I2_J,axiom,
! [Y,X] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Y))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),times_times_int(X,Y))) ) ) ).
fof(fact_514_mult__nonneg__nonneg,axiom,
! [B_29,A_48] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_48))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),B_29))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),times_times_real(A_48,B_29))) ) ) ).
fof(fact_515_mult__nonneg__nonneg,axiom,
! [B_29,A_48] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),A_48))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),B_29))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),times_times_nat(A_48,B_29))) ) ) ).
fof(fact_516_mult__nonneg__nonneg,axiom,
! [B_29,A_48] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_48))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),B_29))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),times_times_int(A_48,B_29))) ) ) ).
fof(fact_517_le__add__iff1,axiom,
! [A_1,E_1,C_1,B,D] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,plus_plus_real(times_times_real(A_1,E_1),C_1)),plus_plus_real(times_times_real(B,E_1),D)))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,plus_plus_real(times_times_real(minus_minus_real(A_1,B),E_1),C_1)),D)) ) ).
fof(fact_518_le__add__iff1,axiom,
! [A_1,E_1,C_1,B,D] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,plus_plus_int(times_times_int(A_1,E_1),C_1)),plus_plus_int(times_times_int(B,E_1),D)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,plus_plus_int(times_times_int(minus_minus_int(A_1,B),E_1),C_1)),D)) ) ).
fof(fact_519_eq__add__iff1,axiom,
! [A_1,E_1,C_1,B,D] :
( plus_plus_real(times_times_real(A_1,E_1),C_1) = plus_plus_real(times_times_real(B,E_1),D)
<=> plus_plus_real(times_times_real(minus_minus_real(A_1,B),E_1),C_1) = D ) ).
fof(fact_520_eq__add__iff1,axiom,
! [A_1,E_1,C_1,B,D] :
( is_int(D)
=> ( plus_plus_int(times_times_int(A_1,E_1),C_1) = plus_plus_int(times_times_int(B,E_1),D)
<=> plus_plus_int(times_times_int(minus_minus_int(A_1,B),E_1),C_1) = D ) ) ).
fof(fact_521_le__add__iff2,axiom,
! [A_1,E_1,C_1,B,D] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,plus_plus_real(times_times_real(A_1,E_1),C_1)),plus_plus_real(times_times_real(B,E_1),D)))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,C_1),plus_plus_real(times_times_real(minus_minus_real(B,A_1),E_1),D))) ) ).
fof(fact_522_le__add__iff2,axiom,
! [A_1,E_1,C_1,B,D] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,plus_plus_int(times_times_int(A_1,E_1),C_1)),plus_plus_int(times_times_int(B,E_1),D)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,C_1),plus_plus_int(times_times_int(minus_minus_int(B,A_1),E_1),D))) ) ).
fof(fact_523_eq__add__iff2,axiom,
! [A_1,E_1,C_1,B,D] :
( plus_plus_real(times_times_real(A_1,E_1),C_1) = plus_plus_real(times_times_real(B,E_1),D)
<=> C_1 = plus_plus_real(times_times_real(minus_minus_real(B,A_1),E_1),D) ) ).
fof(fact_524_eq__add__iff2,axiom,
! [A_1,E_1,C_1,B,D] :
( is_int(C_1)
=> ( plus_plus_int(times_times_int(A_1,E_1),C_1) = plus_plus_int(times_times_int(B,E_1),D)
<=> C_1 = plus_plus_int(times_times_int(minus_minus_int(B,A_1),E_1),D) ) ) ).
fof(fact_525_mult__diff__mult,axiom,
! [X_11,Y_9,A_47,B_28] : minus_minus_real(times_times_real(X_11,Y_9),times_times_real(A_47,B_28)) = plus_plus_real(times_times_real(X_11,minus_minus_real(Y_9,B_28)),times_times_real(minus_minus_real(X_11,A_47),B_28)) ).
fof(fact_526_mult__diff__mult,axiom,
! [X_11,Y_9,A_47,B_28] : minus_minus_int(times_times_int(X_11,Y_9),times_times_int(A_47,B_28)) = plus_plus_int(times_times_int(X_11,minus_minus_int(Y_9,B_28)),times_times_int(minus_minus_int(X_11,A_47),B_28)) ).
fof(fact_527_mult__le__0__iff,axiom,
! [A_1,B] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,times_times_real(A_1,B)),zero_zero_real))
<=> ( ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_1))
& hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B),zero_zero_real)) )
| ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_1),zero_zero_real))
& hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),B)) ) ) ) ).
fof(fact_528_mult__le__0__iff,axiom,
! [A_1,B] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,times_times_int(A_1,B)),zero_zero_int))
<=> ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_1))
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B),zero_zero_int)) )
| ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_1),zero_zero_int))
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),B)) ) ) ) ).
fof(fact_529_zero__le__mult__iff,axiom,
! [A_1,B] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),times_times_real(A_1,B)))
<=> ( ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_1))
& hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),B)) )
| ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_1),zero_zero_real))
& hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B),zero_zero_real)) ) ) ) ).
fof(fact_530_zero__le__mult__iff,axiom,
! [A_1,B] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),times_times_int(A_1,B)))
<=> ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_1))
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),B)) )
| ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_1),zero_zero_int))
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B),zero_zero_int)) ) ) ) ).
fof(fact_531_all__nat,axiom,
! [P_1] :
( ! [X1] : hBOOL(hAPP_nat_bool(P_1,X1))
<=> ! [X_1] :
( is_int(X_1)
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_1))
=> hBOOL(hAPP_nat_bool(P_1,nat(X_1))) ) ) ) ).
fof(fact_532_ex__nat,axiom,
! [P_1] :
( ? [X1] : hBOOL(hAPP_nat_bool(P_1,X1))
<=> ? [X_1] :
( is_int(X_1)
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_1))
& hBOOL(hAPP_nat_bool(P_1,nat(X_1))) ) ) ).
fof(fact_533_zero__le__square,axiom,
! [A_46] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),times_times_real(A_46,A_46))) ).
fof(fact_534_zero__le__square,axiom,
! [A_46] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),times_times_int(A_46,A_46))) ).
fof(fact_535_less__add__iff1,axiom,
! [A_1,E_1,C_1,B,D] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,plus_plus_real(times_times_real(A_1,E_1),C_1)),plus_plus_real(times_times_real(B,E_1),D)))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,plus_plus_real(times_times_real(minus_minus_real(A_1,B),E_1),C_1)),D)) ) ).
fof(fact_536_less__add__iff1,axiom,
! [A_1,E_1,C_1,B,D] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,plus_plus_int(times_times_int(A_1,E_1),C_1)),plus_plus_int(times_times_int(B,E_1),D)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,plus_plus_int(times_times_int(minus_minus_int(A_1,B),E_1),C_1)),D)) ) ).
fof(fact_537_less__add__iff2,axiom,
! [A_1,E_1,C_1,B,D] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,plus_plus_real(times_times_real(A_1,E_1),C_1)),plus_plus_real(times_times_real(B,E_1),D)))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,C_1),plus_plus_real(times_times_real(minus_minus_real(B,A_1),E_1),D))) ) ).
fof(fact_538_less__add__iff2,axiom,
! [A_1,E_1,C_1,B,D] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,plus_plus_int(times_times_int(A_1,E_1),C_1)),plus_plus_int(times_times_int(B,E_1),D)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,C_1),plus_plus_int(times_times_int(minus_minus_int(B,A_1),E_1),D))) ) ).
fof(fact_539_real__squared__diff__one__factored,axiom,
! [X_10] : minus_minus_real(times_times_real(X_10,X_10),one_one_real) = times_times_real(plus_plus_real(X_10,one_one_real),minus_minus_real(X_10,one_one_real)) ).
fof(fact_540_real__squared__diff__one__factored,axiom,
! [X_10] : minus_minus_int(times_times_int(X_10,X_10),one_one_int) = times_times_int(plus_plus_int(X_10,one_one_int),minus_minus_int(X_10,one_one_int)) ).
fof(fact_541_mult__left__le__imp__le,axiom,
! [C_20,A_45,B_27] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,times_times_real(C_20,A_45)),times_times_real(C_20,B_27)))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),C_20))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_45),B_27)) ) ) ).
fof(fact_542_mult__left__le__imp__le,axiom,
! [C_20,A_45,B_27] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,times_times_nat(C_20,A_45)),times_times_nat(C_20,B_27)))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),C_20))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_45),B_27)) ) ) ).
fof(fact_543_mult__left__le__imp__le,axiom,
! [C_20,A_45,B_27] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,times_times_int(C_20,A_45)),times_times_int(C_20,B_27)))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),C_20))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_45),B_27)) ) ) ).
fof(fact_544_mult__right__le__imp__le,axiom,
! [A_44,C_19,B_26] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,times_times_real(A_44,C_19)),times_times_real(B_26,C_19)))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),C_19))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_44),B_26)) ) ) ).
fof(fact_545_mult__right__le__imp__le,axiom,
! [A_44,C_19,B_26] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,times_times_nat(A_44,C_19)),times_times_nat(B_26,C_19)))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),C_19))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_44),B_26)) ) ) ).
fof(fact_546_mult__right__le__imp__le,axiom,
! [A_44,C_19,B_26] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,times_times_int(A_44,C_19)),times_times_int(B_26,C_19)))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),C_19))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_44),B_26)) ) ) ).
fof(fact_547_mult__less__imp__less__left,axiom,
! [C_18,A_43,B_25] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,times_times_real(C_18,A_43)),times_times_real(C_18,B_25)))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),C_18))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_43),B_25)) ) ) ).
fof(fact_548_mult__less__imp__less__left,axiom,
! [C_18,A_43,B_25] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,times_times_nat(C_18,A_43)),times_times_nat(C_18,B_25)))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),C_18))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_43),B_25)) ) ) ).
fof(fact_549_mult__less__imp__less__left,axiom,
! [C_18,A_43,B_25] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,times_times_int(C_18,A_43)),times_times_int(C_18,B_25)))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),C_18))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_43),B_25)) ) ) ).
fof(fact_550_mult__left__less__imp__less,axiom,
! [C_17,A_42,B_24] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,times_times_real(C_17,A_42)),times_times_real(C_17,B_24)))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),C_17))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_42),B_24)) ) ) ).
fof(fact_551_mult__left__less__imp__less,axiom,
! [C_17,A_42,B_24] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,times_times_nat(C_17,A_42)),times_times_nat(C_17,B_24)))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),C_17))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_42),B_24)) ) ) ).
fof(fact_552_mult__left__less__imp__less,axiom,
! [C_17,A_42,B_24] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,times_times_int(C_17,A_42)),times_times_int(C_17,B_24)))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),C_17))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_42),B_24)) ) ) ).
fof(fact_553_mult__less__imp__less__right,axiom,
! [A_41,C_16,B_23] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,times_times_real(A_41,C_16)),times_times_real(B_23,C_16)))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),C_16))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_41),B_23)) ) ) ).
fof(fact_554_mult__less__imp__less__right,axiom,
! [A_41,C_16,B_23] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,times_times_nat(A_41,C_16)),times_times_nat(B_23,C_16)))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),C_16))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_41),B_23)) ) ) ).
fof(fact_555_mult__less__imp__less__right,axiom,
! [A_41,C_16,B_23] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,times_times_int(A_41,C_16)),times_times_int(B_23,C_16)))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),C_16))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_41),B_23)) ) ) ).
fof(fact_556_mult__right__less__imp__less,axiom,
! [A_40,C_15,B_22] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,times_times_real(A_40,C_15)),times_times_real(B_22,C_15)))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),C_15))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_40),B_22)) ) ) ).
fof(fact_557_mult__right__less__imp__less,axiom,
! [A_40,C_15,B_22] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,times_times_nat(A_40,C_15)),times_times_nat(B_22,C_15)))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),C_15))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_40),B_22)) ) ) ).
fof(fact_558_mult__right__less__imp__less,axiom,
! [A_40,C_15,B_22] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,times_times_int(A_40,C_15)),times_times_int(B_22,C_15)))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),C_15))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_40),B_22)) ) ) ).
fof(fact_559_mult__le__less__imp__less,axiom,
! [C_14,D_8,A_39,B_21] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_39),B_21))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,C_14),D_8))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_39))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),C_14))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,times_times_real(A_39,C_14)),times_times_real(B_21,D_8))) ) ) ) ) ).
fof(fact_560_mult__le__less__imp__less,axiom,
! [C_14,D_8,A_39,B_21] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_39),B_21))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,C_14),D_8))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),A_39))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),C_14))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,times_times_nat(A_39,C_14)),times_times_nat(B_21,D_8))) ) ) ) ) ).
fof(fact_561_mult__le__less__imp__less,axiom,
! [C_14,D_8,A_39,B_21] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_39),B_21))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,C_14),D_8))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_39))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),C_14))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,times_times_int(A_39,C_14)),times_times_int(B_21,D_8))) ) ) ) ) ).
fof(fact_562_mult__less__le__imp__less,axiom,
! [C_13,D_7,A_38,B_20] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_38),B_20))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,C_13),D_7))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_38))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),C_13))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,times_times_real(A_38,C_13)),times_times_real(B_20,D_7))) ) ) ) ) ).
fof(fact_563_mult__less__le__imp__less,axiom,
! [C_13,D_7,A_38,B_20] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_38),B_20))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,C_13),D_7))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),A_38))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),C_13))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,times_times_nat(A_38,C_13)),times_times_nat(B_20,D_7))) ) ) ) ) ).
fof(fact_564_mult__less__le__imp__less,axiom,
! [C_13,D_7,A_38,B_20] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_38),B_20))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,C_13),D_7))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_38))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),C_13))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,times_times_int(A_38,C_13)),times_times_int(B_20,D_7))) ) ) ) ) ).
fof(fact_565_mult__strict__mono_H,axiom,
! [C_12,D_6,A_37,B_19] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_37),B_19))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,C_12),D_6))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_37))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),C_12))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,times_times_real(A_37,C_12)),times_times_real(B_19,D_6))) ) ) ) ) ).
fof(fact_566_mult__strict__mono_H,axiom,
! [C_12,D_6,A_37,B_19] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_37),B_19))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,C_12),D_6))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),A_37))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),C_12))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,times_times_nat(A_37,C_12)),times_times_nat(B_19,D_6))) ) ) ) ) ).
fof(fact_567_mult__strict__mono_H,axiom,
! [C_12,D_6,A_37,B_19] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_37),B_19))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,C_12),D_6))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_37))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),C_12))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,times_times_int(A_37,C_12)),times_times_int(B_19,D_6))) ) ) ) ) ).
fof(fact_568_mult__strict__mono,axiom,
! [C_11,D_5,A_36,B_18] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_36),B_18))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,C_11),D_5))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),B_18))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),C_11))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,times_times_real(A_36,C_11)),times_times_real(B_18,D_5))) ) ) ) ) ).
fof(fact_569_mult__strict__mono,axiom,
! [C_11,D_5,A_36,B_18] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_36),B_18))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,C_11),D_5))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),B_18))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),C_11))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,times_times_nat(A_36,C_11)),times_times_nat(B_18,D_5))) ) ) ) ) ).
fof(fact_570_mult__strict__mono,axiom,
! [C_11,D_5,A_36,B_18] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_36),B_18))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,C_11),D_5))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_18))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),C_11))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,times_times_int(A_36,C_11)),times_times_int(B_18,D_5))) ) ) ) ) ).
fof(fact_571_mult__le__cancel__left__neg,axiom,
! [A_1,B,C_1] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,C_1),zero_zero_real))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,times_times_real(C_1,A_1)),times_times_real(C_1,B)))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B),A_1)) ) ) ).
fof(fact_572_mult__le__cancel__left__neg,axiom,
! [A_1,B,C_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,C_1),zero_zero_int))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,times_times_int(C_1,A_1)),times_times_int(C_1,B)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B),A_1)) ) ) ).
fof(fact_573_mult__le__cancel__left__pos,axiom,
! [A_1,B,C_1] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),C_1))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,times_times_real(C_1,A_1)),times_times_real(C_1,B)))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_1),B)) ) ) ).
fof(fact_574_mult__le__cancel__left__pos,axiom,
! [A_1,B,C_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),C_1))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,times_times_int(C_1,A_1)),times_times_int(C_1,B)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_1),B)) ) ) ).
fof(fact_575_sum__squares__ge__zero,axiom,
! [X_9,Y_8] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),plus_plus_real(times_times_real(X_9,X_9),times_times_real(Y_8,Y_8)))) ).
fof(fact_576_sum__squares__ge__zero,axiom,
! [X_9,Y_8] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),plus_plus_int(times_times_int(X_9,X_9),times_times_int(Y_8,Y_8)))) ).
fof(fact_577_sum__squares__le__zero__iff,axiom,
! [Xa,Ya] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,plus_plus_real(times_times_real(Xa,Xa),times_times_real(Ya,Ya))),zero_zero_real))
<=> ( Xa = zero_zero_real
& Ya = zero_zero_real ) ) ).
fof(fact_578_sum__squares__le__zero__iff,axiom,
! [Xa,Ya] :
( ( is_int(Xa)
& is_int(Ya) )
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,plus_plus_int(times_times_int(Xa,Xa),times_times_int(Ya,Ya))),zero_zero_int))
<=> ( Xa = zero_zero_int
& Ya = zero_zero_int ) ) ) ).
fof(fact_579_mult__right__le__one__le,axiom,
! [Y_7,X_8] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X_8))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),Y_7))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y_7),one_one_real))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,times_times_real(X_8,Y_7)),X_8)) ) ) ) ).
fof(fact_580_mult__right__le__one__le,axiom,
! [Y_7,X_8] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_8))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Y_7))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y_7),one_one_int))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,times_times_int(X_8,Y_7)),X_8)) ) ) ) ).
fof(fact_581_mult__left__le__one__le,axiom,
! [Y_6,X_7] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X_7))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),Y_6))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y_6),one_one_real))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,times_times_real(Y_6,X_7)),X_7)) ) ) ) ).
fof(fact_582_mult__left__le__one__le,axiom,
! [Y_6,X_7] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_7))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Y_6))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y_6),one_one_int))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,times_times_int(Y_6,X_7)),X_7)) ) ) ) ).
fof(fact_583_nat__le__0,axiom,
! [Z] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Z),zero_zero_int))
=> nat(Z) = zero_zero_nat ) ).
fof(fact_584_nat__0__iff,axiom,
! [I_1] :
( nat(I_1) = zero_zero_nat
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,I_1),zero_zero_int)) ) ).
fof(fact_585_power__increasing,axiom,
! [A_35,N_9,N_8] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_9),N_8))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,one_one_real),A_35))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(power_power_real(A_35),N_9)),hAPP_nat_real(power_power_real(A_35),N_8))) ) ) ).
fof(fact_586_power__increasing,axiom,
! [A_35,N_9,N_8] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_9),N_8))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,one_one_nat),A_35))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(power_power_nat(A_35),N_9)),hAPP_nat_nat(power_power_nat(A_35),N_8))) ) ) ).
fof(fact_587_power__increasing,axiom,
! [A_35,N_9,N_8] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_9),N_8))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,one_one_int),A_35))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_nat_int(power_power_int(A_35),N_9)),hAPP_nat_int(power_power_int(A_35),N_8))) ) ) ).
fof(fact_588_int__nat__eq,axiom,
! [Z] :
( is_int(Z)
=> ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Z))
=> hAPP_nat_int(semiri1621563631at_int,nat(Z)) = Z )
& ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Z))
=> hAPP_nat_int(semiri1621563631at_int,nat(Z)) = zero_zero_int ) ) ) ).
fof(fact_589_int__eq__iff,axiom,
! [Ma,Z_1] :
( is_int(Z_1)
=> ( hAPP_nat_int(semiri1621563631at_int,Ma) = Z_1
<=> ( Ma = nat(Z_1)
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Z_1)) ) ) ) ).
fof(fact_590_nat__0__le,axiom,
! [Z] :
( is_int(Z)
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Z))
=> hAPP_nat_int(semiri1621563631at_int,nat(Z)) = Z ) ) ).
fof(fact_591_right__minus__eq,axiom,
! [A_1,B] :
( minus_minus_real(A_1,B) = zero_zero_real
<=> A_1 = B ) ).
fof(fact_592_right__minus__eq,axiom,
! [A_1,B] :
( ( is_int(A_1)
& is_int(B) )
=> ( minus_minus_int(A_1,B) = zero_zero_int
<=> A_1 = B ) ) ).
fof(fact_593_eq__iff__diff__eq__0,axiom,
! [A_1,B] :
( A_1 = B
<=> minus_minus_real(A_1,B) = zero_zero_real ) ).
fof(fact_594_eq__iff__diff__eq__0,axiom,
! [A_1,B] :
( ( is_int(A_1)
& is_int(B) )
=> ( A_1 = B
<=> minus_minus_int(A_1,B) = zero_zero_int ) ) ).
fof(fact_595_diff__self,axiom,
! [A_34] : minus_minus_real(A_34,A_34) = zero_zero_real ).
fof(fact_596_diff__self,axiom,
! [A_34] : minus_minus_int(A_34,A_34) = zero_zero_int ).
fof(fact_597_diff__0__right,axiom,
! [A_33] : minus_minus_real(A_33,zero_zero_real) = A_33 ).
fof(fact_598_diff__0__right,axiom,
! [A_33] :
( is_int(A_33)
=> minus_minus_int(A_33,zero_zero_int) = A_33 ) ).
fof(fact_599_diff__eq__diff__less,axiom,
! [A_1,B,C_1,D] :
( minus_minus_int(A_1,B) = minus_minus_int(C_1,D)
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_1),B))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,C_1),D)) ) ) ).
fof(fact_600_diff__eq__diff__less,axiom,
! [A_1,B,C_1,D] :
( minus_minus_real(A_1,B) = minus_minus_real(C_1,D)
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_1),B))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,C_1),D)) ) ) ).
fof(fact_601_add__diff__add,axiom,
! [A_32,C_10,B_17,D_4] : minus_minus_real(plus_plus_real(A_32,C_10),plus_plus_real(B_17,D_4)) = plus_plus_real(minus_minus_real(A_32,B_17),minus_minus_real(C_10,D_4)) ).
fof(fact_602_add__diff__add,axiom,
! [A_32,C_10,B_17,D_4] : minus_minus_int(plus_plus_int(A_32,C_10),plus_plus_int(B_17,D_4)) = plus_plus_int(minus_minus_int(A_32,B_17),minus_minus_int(C_10,D_4)) ).
fof(fact_603_add__diff__cancel,axiom,
! [A_31,B_16] : minus_minus_real(plus_plus_real(A_31,B_16),B_16) = A_31 ).
fof(fact_604_add__diff__cancel,axiom,
! [A_31,B_16] :
( is_int(A_31)
=> minus_minus_int(plus_plus_int(A_31,B_16),B_16) = A_31 ) ).
fof(fact_605_diff__add__cancel,axiom,
! [A_30,B_15] : plus_plus_real(minus_minus_real(A_30,B_15),B_15) = A_30 ).
fof(fact_606_diff__add__cancel,axiom,
! [A_30,B_15] :
( is_int(A_30)
=> plus_plus_int(minus_minus_int(A_30,B_15),B_15) = A_30 ) ).
fof(fact_607_mult__zero__left,axiom,
! [A_29] : times_times_real(zero_zero_real,A_29) = zero_zero_real ).
fof(fact_608_mult__zero__left,axiom,
! [A_29] : times_times_nat(zero_zero_nat,A_29) = zero_zero_nat ).
fof(fact_609_mult__zero__left,axiom,
! [A_29] : times_times_int(zero_zero_int,A_29) = zero_zero_int ).
fof(fact_610_mult__zero__right,axiom,
! [A_28] : times_times_real(A_28,zero_zero_real) = zero_zero_real ).
fof(fact_611_mult__zero__right,axiom,
! [A_28] : times_times_nat(A_28,zero_zero_nat) = zero_zero_nat ).
fof(fact_612_mult__zero__right,axiom,
! [A_28] : times_times_int(A_28,zero_zero_int) = zero_zero_int ).
fof(fact_613_mult__eq__0__iff,axiom,
! [A_1,B] :
( times_times_real(A_1,B) = zero_zero_real
<=> ( A_1 = zero_zero_real
| B = zero_zero_real ) ) ).
fof(fact_614_mult__eq__0__iff,axiom,
! [A_1,B] :
( ( is_int(A_1)
& is_int(B) )
=> ( times_times_int(A_1,B) = zero_zero_int
<=> ( A_1 = zero_zero_int
| B = zero_zero_int ) ) ) ).
fof(fact_615_no__zero__divisors,axiom,
! [B_14,A_27] :
( A_27 != zero_zero_real
=> ( B_14 != zero_zero_real
=> times_times_real(A_27,B_14) != zero_zero_real ) ) ).
fof(fact_616_no__zero__divisors,axiom,
! [B_14,A_27] :
( A_27 != zero_zero_nat
=> ( B_14 != zero_zero_nat
=> times_times_nat(A_27,B_14) != zero_zero_nat ) ) ).
fof(fact_617_no__zero__divisors,axiom,
! [B_14,A_27] :
( ( is_int(B_14)
& is_int(A_27) )
=> ( A_27 != zero_zero_int
=> ( B_14 != zero_zero_int
=> times_times_int(A_27,B_14) != zero_zero_int ) ) ) ).
fof(fact_618_divisors__zero,axiom,
! [A_26,B_13] :
( times_times_real(A_26,B_13) = zero_zero_real
=> ( A_26 = zero_zero_real
| B_13 = zero_zero_real ) ) ).
fof(fact_619_divisors__zero,axiom,
! [A_26,B_13] :
( times_times_nat(A_26,B_13) = zero_zero_nat
=> ( A_26 = zero_zero_nat
| B_13 = zero_zero_nat ) ) ).
fof(fact_620_divisors__zero,axiom,
! [A_26,B_13] :
( ( is_int(A_26)
& is_int(B_13) )
=> ( times_times_int(A_26,B_13) = zero_zero_int
=> ( A_26 = zero_zero_int
| B_13 = zero_zero_int ) ) ) ).
fof(fact_621_comm__semiring__1__class_Onormalizing__semiring__rules_I10_J,axiom,
! [A_25] : times_times_real(A_25,zero_zero_real) = zero_zero_real ).
fof(fact_622_comm__semiring__1__class_Onormalizing__semiring__rules_I10_J,axiom,
! [A_25] : times_times_nat(A_25,zero_zero_nat) = zero_zero_nat ).
fof(fact_623_comm__semiring__1__class_Onormalizing__semiring__rules_I10_J,axiom,
! [A_25] : times_times_int(A_25,zero_zero_int) = zero_zero_int ).
fof(fact_624_comm__semiring__1__class_Onormalizing__semiring__rules_I9_J,axiom,
! [A_24] : times_times_real(zero_zero_real,A_24) = zero_zero_real ).
fof(fact_625_comm__semiring__1__class_Onormalizing__semiring__rules_I9_J,axiom,
! [A_24] : times_times_nat(zero_zero_nat,A_24) = zero_zero_nat ).
fof(fact_626_comm__semiring__1__class_Onormalizing__semiring__rules_I9_J,axiom,
! [A_24] : times_times_int(zero_zero_int,A_24) = zero_zero_int ).
fof(fact_627_diffs0__imp__equal,axiom,
! [M,N] :
( minus_minus_nat(M,N) = zero_zero_nat
=> ( minus_minus_nat(N,M) = zero_zero_nat
=> M = N ) ) ).
fof(fact_628_diff__self__eq__0,axiom,
! [M] : minus_minus_nat(M,M) = zero_zero_nat ).
fof(fact_629_minus__nat_Odiff__0,axiom,
! [M] : minus_minus_nat(M,zero_zero_nat) = M ).
fof(fact_630_diff__0__eq__0,axiom,
! [N] : minus_minus_nat(zero_zero_nat,N) = zero_zero_nat ).
fof(fact_631_comm__semiring__class_Odistrib,axiom,
! [A_23,B_12,C_9] : times_times_real(plus_plus_real(A_23,B_12),C_9) = plus_plus_real(times_times_real(A_23,C_9),times_times_real(B_12,C_9)) ).
fof(fact_632_comm__semiring__class_Odistrib,axiom,
! [A_23,B_12,C_9] : times_times_nat(plus_plus_nat(A_23,B_12),C_9) = plus_plus_nat(times_times_nat(A_23,C_9),times_times_nat(B_12,C_9)) ).
fof(fact_633_comm__semiring__class_Odistrib,axiom,
! [A_23,B_12,C_9] : times_times_int(plus_plus_int(A_23,B_12),C_9) = plus_plus_int(times_times_int(A_23,C_9),times_times_int(B_12,C_9)) ).
fof(fact_634_combine__common__factor,axiom,
! [A_22,E,B_11,C_8] : plus_plus_real(times_times_real(A_22,E),plus_plus_real(times_times_real(B_11,E),C_8)) = plus_plus_real(times_times_real(plus_plus_real(A_22,B_11),E),C_8) ).
fof(fact_635_combine__common__factor,axiom,
! [A_22,E,B_11,C_8] : plus_plus_nat(times_times_nat(A_22,E),plus_plus_nat(times_times_nat(B_11,E),C_8)) = plus_plus_nat(times_times_nat(plus_plus_nat(A_22,B_11),E),C_8) ).
fof(fact_636_combine__common__factor,axiom,
! [A_22,E,B_11,C_8] : plus_plus_int(times_times_int(A_22,E),plus_plus_int(times_times_int(B_11,E),C_8)) = plus_plus_int(times_times_int(plus_plus_int(A_22,B_11),E),C_8) ).
fof(fact_637_crossproduct__eq,axiom,
! [Wa,Ya,Xa,Z_1] :
( plus_plus_real(times_times_real(Wa,Ya),times_times_real(Xa,Z_1)) = plus_plus_real(times_times_real(Wa,Z_1),times_times_real(Xa,Ya))
<=> ( Wa = Xa
| Ya = Z_1 ) ) ).
fof(fact_638_crossproduct__eq,axiom,
! [Wa,Ya,Xa,Z_1] :
( plus_plus_nat(times_times_nat(Wa,Ya),times_times_nat(Xa,Z_1)) = plus_plus_nat(times_times_nat(Wa,Z_1),times_times_nat(Xa,Ya))
<=> ( Wa = Xa
| Ya = Z_1 ) ) ).
fof(fact_639_crossproduct__eq,axiom,
! [Wa,Ya,Xa,Z_1] :
( ( is_int(Wa)
& is_int(Ya)
& is_int(Xa)
& is_int(Z_1) )
=> ( plus_plus_int(times_times_int(Wa,Ya),times_times_int(Xa,Z_1)) = plus_plus_int(times_times_int(Wa,Z_1),times_times_int(Xa,Ya))
<=> ( Wa = Xa
| Ya = Z_1 ) ) ) ).
fof(fact_640_comm__semiring__1__class_Onormalizing__semiring__rules_I1_J,axiom,
! [A_21,M_3,B_10] : plus_plus_real(times_times_real(A_21,M_3),times_times_real(B_10,M_3)) = times_times_real(plus_plus_real(A_21,B_10),M_3) ).
fof(fact_641_comm__semiring__1__class_Onormalizing__semiring__rules_I1_J,axiom,
! [A_21,M_3,B_10] : plus_plus_nat(times_times_nat(A_21,M_3),times_times_nat(B_10,M_3)) = times_times_nat(plus_plus_nat(A_21,B_10),M_3) ).
fof(fact_642_comm__semiring__1__class_Onormalizing__semiring__rules_I1_J,axiom,
! [A_21,M_3,B_10] : plus_plus_int(times_times_int(A_21,M_3),times_times_int(B_10,M_3)) = times_times_int(plus_plus_int(A_21,B_10),M_3) ).
fof(fact_643_comm__semiring__1__class_Onormalizing__semiring__rules_I8_J,axiom,
! [A_20,B_9,C_7] : times_times_real(plus_plus_real(A_20,B_9),C_7) = plus_plus_real(times_times_real(A_20,C_7),times_times_real(B_9,C_7)) ).
fof(fact_644_comm__semiring__1__class_Onormalizing__semiring__rules_I8_J,axiom,
! [A_20,B_9,C_7] : times_times_nat(plus_plus_nat(A_20,B_9),C_7) = plus_plus_nat(times_times_nat(A_20,C_7),times_times_nat(B_9,C_7)) ).
fof(fact_645_comm__semiring__1__class_Onormalizing__semiring__rules_I8_J,axiom,
! [A_20,B_9,C_7] : times_times_int(plus_plus_int(A_20,B_9),C_7) = plus_plus_int(times_times_int(A_20,C_7),times_times_int(B_9,C_7)) ).
fof(fact_646_crossproduct__noteq,axiom,
! [C_1,D,A_1,B] :
( ( A_1 != B
& C_1 != D )
<=> plus_plus_real(times_times_real(A_1,C_1),times_times_real(B,D)) != plus_plus_real(times_times_real(A_1,D),times_times_real(B,C_1)) ) ).
fof(fact_647_crossproduct__noteq,axiom,
! [C_1,D,A_1,B] :
( ( A_1 != B
& C_1 != D )
<=> plus_plus_nat(times_times_nat(A_1,C_1),times_times_nat(B,D)) != plus_plus_nat(times_times_nat(A_1,D),times_times_nat(B,C_1)) ) ).
fof(fact_648_crossproduct__noteq,axiom,
! [C_1,D,A_1,B] :
( ( is_int(C_1)
& is_int(D)
& is_int(A_1)
& is_int(B) )
=> ( ( A_1 != B
& C_1 != D )
<=> plus_plus_int(times_times_int(A_1,C_1),times_times_int(B,D)) != plus_plus_int(times_times_int(A_1,D),times_times_int(B,C_1)) ) ) ).
fof(fact_649_comm__semiring__1__class_Onormalizing__semiring__rules_I34_J,axiom,
! [X_6,Y_5,Z_3] : times_times_real(X_6,plus_plus_real(Y_5,Z_3)) = plus_plus_real(times_times_real(X_6,Y_5),times_times_real(X_6,Z_3)) ).
fof(fact_650_comm__semiring__1__class_Onormalizing__semiring__rules_I34_J,axiom,
! [X_6,Y_5,Z_3] : times_times_nat(X_6,plus_plus_nat(Y_5,Z_3)) = plus_plus_nat(times_times_nat(X_6,Y_5),times_times_nat(X_6,Z_3)) ).
fof(fact_651_comm__semiring__1__class_Onormalizing__semiring__rules_I34_J,axiom,
! [X_6,Y_5,Z_3] : times_times_int(X_6,plus_plus_int(Y_5,Z_3)) = plus_plus_int(times_times_int(X_6,Y_5),times_times_int(X_6,Z_3)) ).
fof(fact_652_nat__int,axiom,
! [N] : nat(hAPP_nat_int(semiri1621563631at_int,N)) = N ).
fof(fact_653_mult_Ocomm__neutral,axiom,
! [A_19] : times_times_real(A_19,one_one_real) = A_19 ).
fof(fact_654_mult_Ocomm__neutral,axiom,
! [A_19] : times_times_nat(A_19,one_one_nat) = A_19 ).
fof(fact_655_mult_Ocomm__neutral,axiom,
! [A_19] :
( is_int(A_19)
=> times_times_int(A_19,one_one_int) = A_19 ) ).
fof(fact_656_comm__semiring__1__class_Onormalizing__semiring__rules_I12_J,axiom,
! [A_18] : times_times_real(A_18,one_one_real) = A_18 ).
fof(fact_657_comm__semiring__1__class_Onormalizing__semiring__rules_I12_J,axiom,
! [A_18] : times_times_nat(A_18,one_one_nat) = A_18 ).
fof(fact_658_comm__semiring__1__class_Onormalizing__semiring__rules_I12_J,axiom,
! [A_18] :
( is_int(A_18)
=> times_times_int(A_18,one_one_int) = A_18 ) ).
fof(fact_659_mult__1__right,axiom,
! [A_17] : times_times_real(A_17,one_one_real) = A_17 ).
fof(fact_660_mult__1__right,axiom,
! [A_17] : times_times_nat(A_17,one_one_nat) = A_17 ).
fof(fact_661_mult__1__right,axiom,
! [A_17] :
( is_int(A_17)
=> times_times_int(A_17,one_one_int) = A_17 ) ).
fof(fact_662_mult__1,axiom,
! [A_16] : times_times_real(one_one_real,A_16) = A_16 ).
fof(fact_663_mult__1,axiom,
! [A_16] : times_times_nat(one_one_nat,A_16) = A_16 ).
fof(fact_664_mult__1,axiom,
! [A_16] :
( is_int(A_16)
=> times_times_int(one_one_int,A_16) = A_16 ) ).
fof(fact_665_comm__semiring__1__class_Onormalizing__semiring__rules_I11_J,axiom,
! [A_15] : times_times_real(one_one_real,A_15) = A_15 ).
fof(fact_666_comm__semiring__1__class_Onormalizing__semiring__rules_I11_J,axiom,
! [A_15] : times_times_nat(one_one_nat,A_15) = A_15 ).
fof(fact_667_comm__semiring__1__class_Onormalizing__semiring__rules_I11_J,axiom,
! [A_15] :
( is_int(A_15)
=> times_times_int(one_one_int,A_15) = A_15 ) ).
fof(fact_668_mult__1__left,axiom,
! [A_14] : times_times_real(one_one_real,A_14) = A_14 ).
fof(fact_669_mult__1__left,axiom,
! [A_14] : times_times_nat(one_one_nat,A_14) = A_14 ).
fof(fact_670_mult__1__left,axiom,
! [A_14] :
( is_int(A_14)
=> times_times_int(one_one_int,A_14) = A_14 ) ).
fof(fact_671_add__le__imp__le__left,axiom,
! [C_6,A_13,B_8] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,plus_plus_real(C_6,A_13)),plus_plus_real(C_6,B_8)))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_13),B_8)) ) ).
fof(fact_672_add__le__imp__le__left,axiom,
! [C_6,A_13,B_8] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,plus_plus_nat(C_6,A_13)),plus_plus_nat(C_6,B_8)))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_13),B_8)) ) ).
fof(fact_673_add__le__imp__le__left,axiom,
! [C_6,A_13,B_8] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,plus_plus_int(C_6,A_13)),plus_plus_int(C_6,B_8)))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_13),B_8)) ) ).
fof(fact_674_add__le__imp__le__right,axiom,
! [A_12,C_5,B_7] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,plus_plus_real(A_12,C_5)),plus_plus_real(B_7,C_5)))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_12),B_7)) ) ).
fof(fact_675_add__le__imp__le__right,axiom,
! [A_12,C_5,B_7] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,plus_plus_nat(A_12,C_5)),plus_plus_nat(B_7,C_5)))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_12),B_7)) ) ).
fof(fact_676_add__le__imp__le__right,axiom,
! [A_12,C_5,B_7] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,plus_plus_int(A_12,C_5)),plus_plus_int(B_7,C_5)))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_12),B_7)) ) ).
fof(fact_677_add__mono,axiom,
! [C_4,D_3,A_11,B_6] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_11),B_6))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,C_4),D_3))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,plus_plus_real(A_11,C_4)),plus_plus_real(B_6,D_3))) ) ) ).
fof(fact_678_add__mono,axiom,
! [C_4,D_3,A_11,B_6] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_11),B_6))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,C_4),D_3))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,plus_plus_nat(A_11,C_4)),plus_plus_nat(B_6,D_3))) ) ) ).
fof(fact_679_add__mono,axiom,
! [C_4,D_3,A_11,B_6] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_11),B_6))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,C_4),D_3))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,plus_plus_int(A_11,C_4)),plus_plus_int(B_6,D_3))) ) ) ).
fof(fact_680_add__left__mono,axiom,
! [C_3,A_10,B_5] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_10),B_5))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,plus_plus_real(C_3,A_10)),plus_plus_real(C_3,B_5))) ) ).
fof(fact_681_add__left__mono,axiom,
! [C_3,A_10,B_5] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_10),B_5))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,plus_plus_nat(C_3,A_10)),plus_plus_nat(C_3,B_5))) ) ).
fof(fact_682_add__left__mono,axiom,
! [C_3,A_10,B_5] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_10),B_5))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,plus_plus_int(C_3,A_10)),plus_plus_int(C_3,B_5))) ) ).
fof(fact_683_add__right__mono,axiom,
! [C_2,A_9,B_4] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_9),B_4))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,plus_plus_real(A_9,C_2)),plus_plus_real(B_4,C_2))) ) ).
fof(fact_684_add__right__mono,axiom,
! [C_2,A_9,B_4] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_9),B_4))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,plus_plus_nat(A_9,C_2)),plus_plus_nat(B_4,C_2))) ) ).
fof(fact_685_add__right__mono,axiom,
! [C_2,A_9,B_4] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_9),B_4))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,plus_plus_int(A_9,C_2)),plus_plus_int(B_4,C_2))) ) ).
fof(fact_686_add__le__cancel__left,axiom,
! [C_1,A_1,B] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,plus_plus_real(C_1,A_1)),plus_plus_real(C_1,B)))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_1),B)) ) ).
fof(fact_687_add__le__cancel__left,axiom,
! [C_1,A_1,B] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,plus_plus_nat(C_1,A_1)),plus_plus_nat(C_1,B)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_1),B)) ) ).
fof(fact_688_add__le__cancel__left,axiom,
! [C_1,A_1,B] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,plus_plus_int(C_1,A_1)),plus_plus_int(C_1,B)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_1),B)) ) ).
fof(fact_689_add__le__cancel__right,axiom,
! [A_1,C_1,B] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,plus_plus_real(A_1,C_1)),plus_plus_real(B,C_1)))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_1),B)) ) ).
fof(fact_690_add__le__cancel__right,axiom,
! [A_1,C_1,B] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,plus_plus_nat(A_1,C_1)),plus_plus_nat(B,C_1)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_1),B)) ) ).
fof(fact_691_add__le__cancel__right,axiom,
! [A_1,C_1,B] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,plus_plus_int(A_1,C_1)),plus_plus_int(B,C_1)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_1),B)) ) ).
fof(fact_692_less__imp__diff__less,axiom,
! [N,J_1,K_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,J_1),K_1))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,minus_minus_nat(J_1,N)),K_1)) ) ).
fof(fact_693_diff__less__mono2,axiom,
! [L,M,N] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),L))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,minus_minus_nat(L,N)),minus_minus_nat(L,M))) ) ) ).
fof(fact_694_power__commutes,axiom,
! [A_8,N_7] : times_times_real(hAPP_nat_real(power_power_real(A_8),N_7),A_8) = times_times_real(A_8,hAPP_nat_real(power_power_real(A_8),N_7)) ).
fof(fact_695_power__commutes,axiom,
! [A_8,N_7] : times_times_nat(hAPP_nat_nat(power_power_nat(A_8),N_7),A_8) = times_times_nat(A_8,hAPP_nat_nat(power_power_nat(A_8),N_7)) ).
fof(fact_696_power__commutes,axiom,
! [A_8,N_7] : times_times_int(hAPP_nat_int(power_power_int(A_8),N_7),A_8) = times_times_int(A_8,hAPP_nat_int(power_power_int(A_8),N_7)) ).
fof(fact_697_power__mult__distrib,axiom,
! [A_7,B_3,N_6] : hAPP_nat_real(power_power_real(times_times_real(A_7,B_3)),N_6) = times_times_real(hAPP_nat_real(power_power_real(A_7),N_6),hAPP_nat_real(power_power_real(B_3),N_6)) ).
fof(fact_698_power__mult__distrib,axiom,
! [A_7,B_3,N_6] : hAPP_nat_nat(power_power_nat(times_times_nat(A_7,B_3)),N_6) = times_times_nat(hAPP_nat_nat(power_power_nat(A_7),N_6),hAPP_nat_nat(power_power_nat(B_3),N_6)) ).
fof(fact_699_power__mult__distrib,axiom,
! [A_7,B_3,N_6] : hAPP_nat_int(power_power_int(times_times_int(A_7,B_3)),N_6) = times_times_int(hAPP_nat_int(power_power_int(A_7),N_6),hAPP_nat_int(power_power_int(B_3),N_6)) ).
fof(fact_700_comm__semiring__1__class_Onormalizing__semiring__rules_I30_J,axiom,
! [X_5,Y_4,Q_2] : hAPP_nat_real(power_power_real(times_times_real(X_5,Y_4)),Q_2) = times_times_real(hAPP_nat_real(power_power_real(X_5),Q_2),hAPP_nat_real(power_power_real(Y_4),Q_2)) ).
fof(fact_701_comm__semiring__1__class_Onormalizing__semiring__rules_I30_J,axiom,
! [X_5,Y_4,Q_2] : hAPP_nat_nat(power_power_nat(times_times_nat(X_5,Y_4)),Q_2) = times_times_nat(hAPP_nat_nat(power_power_nat(X_5),Q_2),hAPP_nat_nat(power_power_nat(Y_4),Q_2)) ).
fof(fact_702_comm__semiring__1__class_Onormalizing__semiring__rules_I30_J,axiom,
! [X_5,Y_4,Q_2] : hAPP_nat_int(power_power_int(times_times_int(X_5,Y_4)),Q_2) = times_times_int(hAPP_nat_int(power_power_int(X_5),Q_2),hAPP_nat_int(power_power_int(Y_4),Q_2)) ).
fof(fact_703_diff__add__inverse2,axiom,
! [M,N] : minus_minus_nat(plus_plus_nat(M,N),N) = M ).
fof(fact_704_diff__add__inverse,axiom,
! [N,M] : minus_minus_nat(plus_plus_nat(N,M),N) = M ).
fof(fact_705_diff__diff__left,axiom,
! [I_2,J_1,K_1] : minus_minus_nat(minus_minus_nat(I_2,J_1),K_1) = minus_minus_nat(I_2,plus_plus_nat(J_1,K_1)) ).
fof(fact_706_Nat_Odiff__cancel,axiom,
! [K_1,M,N] : minus_minus_nat(plus_plus_nat(K_1,M),plus_plus_nat(K_1,N)) = minus_minus_nat(M,N) ).
fof(fact_707_diff__cancel2,axiom,
! [M,K_1,N] : minus_minus_nat(plus_plus_nat(M,K_1),plus_plus_nat(N,K_1)) = minus_minus_nat(M,N) ).
fof(fact_708_mult__Pls,axiom,
! [W] : times_times_int(pls,W) = pls ).
fof(fact_709_mult__Bit0,axiom,
! [K_1,L] : times_times_int(bit0(K_1),L) = bit0(times_times_int(K_1,L)) ).
fof(fact_710_rel__simps_I34_J,axiom,
! [K,L_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,bit1(K)),bit1(L_1)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K),L_1)) ) ).
fof(fact_711_less__eq__int__code_I16_J,axiom,
! [K1,K2] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,bit1(K1)),bit1(K2)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K1),K2)) ) ).
fof(fact_712_rel__simps_I19_J,axiom,
hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),pls)) ).
fof(fact_713_rel__simps_I31_J,axiom,
! [K,L_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,bit0(K)),bit0(L_1)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K),L_1)) ) ).
fof(fact_714_less__eq__int__code_I13_J,axiom,
! [K1,K2] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,bit0(K1)),bit0(K2)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K1),K2)) ) ).
fof(fact_715_Nat__Transfer_Otransfer__nat__int__function__closures_I5_J,axiom,
hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),zero_zero_int)) ).
fof(fact_716_zmult__1,axiom,
! [Z] :
( is_int(Z)
=> times_times_int(one_one_int,Z) = Z ) ).
fof(fact_717_zmult__1__right,axiom,
! [Z] :
( is_int(Z)
=> times_times_int(Z,one_one_int) = Z ) ).
fof(fact_718_zless__le,axiom,
! [Z_1,Wa] :
( ( is_int(Z_1)
& is_int(Wa) )
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Z_1),Wa))
<=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Z_1),Wa))
& Z_1 != Wa ) ) ) ).
fof(fact_719_zadd__zmult__distrib2,axiom,
! [W,Z1,Z2] : times_times_int(W,plus_plus_int(Z1,Z2)) = plus_plus_int(times_times_int(W,Z1),times_times_int(W,Z2)) ).
fof(fact_720_zadd__zmult__distrib,axiom,
! [Z1,Z2,W] : times_times_int(plus_plus_int(Z1,Z2),W) = plus_plus_int(times_times_int(Z1,W),times_times_int(Z2,W)) ).
fof(fact_721_zadd__left__mono,axiom,
! [K_1,I_2,J_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,I_2),J_1))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,plus_plus_int(K_1,I_2)),plus_plus_int(K_1,J_1))) ) ).
fof(fact_722_times__numeral__code_I5_J,axiom,
! [V_1,W] : times_times_int(number_number_of_int(V_1),number_number_of_int(W)) = number_number_of_int(times_times_int(V_1,W)) ).
fof(fact_723_less__eq__number__of__int__code,axiom,
! [K,L_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,number_number_of_int(K)),number_number_of_int(L_1)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K),L_1)) ) ).
fof(fact_724_convex__bound__le,axiom,
! [V_5,U_3,Y_3,X_4,A_6] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_4),A_6))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y_3),A_6))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),U_3))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),V_5))
=> ( plus_plus_real(U_3,V_5) = one_one_real
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,plus_plus_real(times_times_real(U_3,X_4),times_times_real(V_5,Y_3))),A_6)) ) ) ) ) ) ).
fof(fact_725_convex__bound__le,axiom,
! [V_5,U_3,Y_3,X_4,A_6] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_4),A_6))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y_3),A_6))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),U_3))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),V_5))
=> ( plus_plus_int(U_3,V_5) = one_one_int
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,plus_plus_int(times_times_int(U_3,X_4),times_times_int(V_5,Y_3))),A_6)) ) ) ) ) ) ).
fof(fact_726_transfer__nat__int__relations_I2_J,axiom,
! [Ya,Xa] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Xa))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Ya))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,nat(Xa)),nat(Ya)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Xa),Ya)) ) ) ) ).
fof(fact_727_nat__less__eq__zless,axiom,
! [Z_1,Wa] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Wa))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,nat(Wa)),nat(Z_1)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Wa),Z_1)) ) ) ).
fof(fact_728_le__special_I1_J,axiom,
! [Ya] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),number267125858f_real(Ya)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),Ya)) ) ).
fof(fact_729_le__special_I1_J,axiom,
! [Ya] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),number_number_of_int(Ya)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),Ya)) ) ).
fof(fact_730_le__special_I3_J,axiom,
! [Xa] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,number267125858f_real(Xa)),zero_zero_real))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Xa),pls)) ) ).
fof(fact_731_le__special_I3_J,axiom,
! [Xa] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,number_number_of_int(Xa)),zero_zero_int))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Xa),pls)) ) ).
fof(fact_732_power__decreasing,axiom,
! [A_5,N_5,N_4] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_5),N_4))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_5))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_5),one_one_real))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(power_power_real(A_5),N_4)),hAPP_nat_real(power_power_real(A_5),N_5))) ) ) ) ).
fof(fact_733_power__decreasing,axiom,
! [A_5,N_5,N_4] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_5),N_4))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),A_5))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_5),one_one_nat))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(power_power_nat(A_5),N_4)),hAPP_nat_nat(power_power_nat(A_5),N_5))) ) ) ) ).
fof(fact_734_power__decreasing,axiom,
! [A_5,N_5,N_4] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_5),N_4))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_5))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_5),one_one_int))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_nat_int(power_power_int(A_5),N_4)),hAPP_nat_int(power_power_int(A_5),N_5))) ) ) ) ).
fof(fact_735_power__increasing__iff,axiom,
! [Xa,Ya,B] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),B))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(power_power_real(B),Xa)),hAPP_nat_real(power_power_real(B),Ya)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Xa),Ya)) ) ) ).
fof(fact_736_power__increasing__iff,axiom,
! [Xa,Ya,B] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_nat),B))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(power_power_nat(B),Xa)),hAPP_nat_nat(power_power_nat(B),Ya)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Xa),Ya)) ) ) ).
fof(fact_737_power__increasing__iff,axiom,
! [Xa,Ya,B] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),B))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_nat_int(power_power_int(B),Xa)),hAPP_nat_int(power_power_int(B),Ya)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Xa),Ya)) ) ) ).
fof(fact_738_power__le__imp__le__exp,axiom,
! [M_2,N_3,A_4] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),A_4))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(power_power_real(A_4),M_2)),hAPP_nat_real(power_power_real(A_4),N_3)))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_2),N_3)) ) ) ).
fof(fact_739_power__le__imp__le__exp,axiom,
! [M_2,N_3,A_4] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_nat),A_4))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(power_power_nat(A_4),M_2)),hAPP_nat_nat(power_power_nat(A_4),N_3)))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_2),N_3)) ) ) ).
fof(fact_740_power__le__imp__le__exp,axiom,
! [M_2,N_3,A_4] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),A_4))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_nat_int(power_power_int(A_4),M_2)),hAPP_nat_int(power_power_int(A_4),N_3)))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_2),N_3)) ) ) ).
fof(fact_741_nat__eq__iff2,axiom,
! [Ma,Wa] :
( is_int(Wa)
=> ( Ma = nat(Wa)
<=> ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Wa))
=> Wa = hAPP_nat_int(semiri1621563631at_int,Ma) )
& ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Wa))
=> Ma = zero_zero_nat ) ) ) ) ).
fof(fact_742_nat__eq__iff,axiom,
! [Wa,Ma] :
( is_int(Wa)
=> ( nat(Wa) = Ma
<=> ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Wa))
=> Wa = hAPP_nat_int(semiri1621563631at_int,Ma) )
& ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Wa))
=> Ma = zero_zero_nat ) ) ) ) ).
fof(fact_743_Nat__Transfer_Otransfer__nat__int__functions_I1_J,axiom,
! [Y,X] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Y))
=> plus_plus_nat(nat(X),nat(Y)) = nat(plus_plus_int(X,Y)) ) ) ).
fof(fact_744_nat__add__distrib,axiom,
! [Z_2,Z] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Z))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Z_2))
=> nat(plus_plus_int(Z,Z_2)) = plus_plus_nat(nat(Z),nat(Z_2)) ) ) ).
fof(fact_745_int__eq__iff__number__of,axiom,
! [Ma,Va] :
( hAPP_nat_int(semiri1621563631at_int,Ma) = number_number_of_int(Va)
<=> ( Ma = nat(number_number_of_int(Va))
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),number_number_of_int(Va))) ) ) ).
fof(fact_746_Nat__Transfer_Otransfer__nat__int__functions_I4_J,axiom,
! [N,X] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X))
=> hAPP_nat_nat(power_power_nat(nat(X)),N) = nat(hAPP_nat_int(power_power_int(X),N)) ) ).
fof(fact_747_nat__power__eq,axiom,
! [N,Z] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Z))
=> nat(hAPP_nat_int(power_power_int(Z),N)) = hAPP_nat_nat(power_power_nat(nat(Z)),N) ) ).
fof(fact_748_convex__bound__lt,axiom,
! [V_4,U_2,Y_2,X_3,A_3] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_3),A_3))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Y_2),A_3))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),U_2))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),V_4))
=> ( plus_plus_real(U_2,V_4) = one_one_real
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,plus_plus_real(times_times_real(U_2,X_3),times_times_real(V_4,Y_2))),A_3)) ) ) ) ) ) ).
fof(fact_749_convex__bound__lt,axiom,
! [V_4,U_2,Y_2,X_3,A_3] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_3),A_3))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Y_2),A_3))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),U_2))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),V_4))
=> ( plus_plus_int(U_2,V_4) = one_one_int
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,plus_plus_int(times_times_int(U_2,X_3),times_times_int(V_4,Y_2))),A_3)) ) ) ) ) ) ).
fof(fact_750_nat__less__iff,axiom,
! [Ma,Wa] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Wa))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,nat(Wa)),Ma))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Wa),hAPP_nat_int(semiri1621563631at_int,Ma))) ) ) ).
fof(fact_751_le__special_I2_J,axiom,
! [Ya] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,one_one_real),number267125858f_real(Ya)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,bit1(pls)),Ya)) ) ).
fof(fact_752_le__special_I2_J,axiom,
! [Ya] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,one_one_int),number_number_of_int(Ya)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,bit1(pls)),Ya)) ) ).
fof(fact_753_le__special_I4_J,axiom,
! [Xa] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,number267125858f_real(Xa)),one_one_real))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Xa),bit1(pls))) ) ).
fof(fact_754_le__special_I4_J,axiom,
! [Xa] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,number_number_of_int(Xa)),one_one_int))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Xa),bit1(pls))) ) ).
fof(fact_755_realpow__minus__mult,axiom,
! [X_2,N_2] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_2))
=> times_times_real(hAPP_nat_real(power_power_real(X_2),minus_minus_nat(N_2,one_one_nat)),X_2) = hAPP_nat_real(power_power_real(X_2),N_2) ) ).
fof(fact_756_realpow__minus__mult,axiom,
! [X_2,N_2] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_2))
=> times_times_nat(hAPP_nat_nat(power_power_nat(X_2),minus_minus_nat(N_2,one_one_nat)),X_2) = hAPP_nat_nat(power_power_nat(X_2),N_2) ) ).
fof(fact_757_realpow__minus__mult,axiom,
! [X_2,N_2] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_2))
=> times_times_int(hAPP_nat_int(power_power_int(X_2),minus_minus_nat(N_2,one_one_nat)),X_2) = hAPP_nat_int(power_power_int(X_2),N_2) ) ).
fof(fact_758_less__iff__diff__less__0,axiom,
! [A_1,B] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_1),B))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,minus_minus_real(A_1,B)),zero_zero_real)) ) ).
fof(fact_759_less__iff__diff__less__0,axiom,
! [A_1,B] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_1),B))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,minus_minus_int(A_1,B)),zero_zero_int)) ) ).
fof(fact_760_transfer__nat__int__numerals_I1_J,axiom,
zero_zero_nat = nat(zero_zero_int) ).
fof(fact_761_nat__0,axiom,
nat(zero_zero_int) = zero_zero_nat ).
fof(fact_762_not__square__less__zero,axiom,
! [A_2] : ~ hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,times_times_real(A_2,A_2)),zero_zero_real)) ).
fof(fact_763_not__square__less__zero,axiom,
! [A_2] : ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,times_times_int(A_2,A_2)),zero_zero_int)) ).
fof(fact_764_mult__less__cancel__right__disj,axiom,
! [A_1,C_1,B] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,times_times_real(A_1,C_1)),times_times_real(B,C_1)))
<=> ( ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),C_1))
& hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_1),B)) )
| ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,C_1),zero_zero_real))
& hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,B),A_1)) ) ) ) ).
fof(fact_765_mult__less__cancel__right__disj,axiom,
! [A_1,C_1,B] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,times_times_int(A_1,C_1)),times_times_int(B,C_1)))
<=> ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),C_1))
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_1),B)) )
| ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,C_1),zero_zero_int))
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B),A_1)) ) ) ) ).
fof(fact_766_nat__number__of,axiom,
! [W] : nat(number_number_of_int(W)) = number_number_of_nat(W) ).
fof(fact_767_nat__number__of__def,axiom,
! [V_1] : number_number_of_nat(V_1) = nat(number_number_of_int(V_1)) ).
fof(fact_768_transfer__nat__int__numerals_I2_J,axiom,
one_one_nat = nat(one_one_int) ).
fof(fact_769_zero__less__diff,axiom,
! [Na,Ma] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),minus_minus_nat(Na,Ma)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),Na)) ) ).
fof(fact_770_diff__less,axiom,
! [M,N] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),M))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,minus_minus_nat(M,N)),M)) ) ) ).
fof(fact_771_diff__add__0,axiom,
! [N,M] : minus_minus_nat(N,plus_plus_nat(N,M)) = zero_zero_nat ).
fof(fact_772_less__diff__conv,axiom,
! [I_1,J,K] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_1),minus_minus_nat(J,K)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,plus_plus_nat(I_1,K)),J)) ) ).
fof(fact_773_add__diff__inverse,axiom,
! [M,N] :
( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N))
=> plus_plus_nat(N,minus_minus_nat(M,N)) = M ) ).
fof(fact_774_rel__simps_I22_J,axiom,
! [K] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),bit1(K)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),K)) ) ).
fof(fact_775_rel__simps_I32_J,axiom,
! [K,L_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,bit0(K)),bit1(L_1)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K),L_1)) ) ).
fof(fact_776_less__eq__int__code_I14_J,axiom,
! [K1,K2] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,bit0(K1)),bit1(K2)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K1),K2)) ) ).
fof(fact_777_rel__simps_I21_J,axiom,
! [K] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),bit0(K)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),K)) ) ).
fof(fact_778_rel__simps_I27_J,axiom,
! [K] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,bit0(K)),pls))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K),pls)) ) ).
fof(fact_779_pos__zmult__pos,axiom,
! [B_1,A] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),times_times_int(A,B_1)))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_1)) ) ) ).
fof(fact_780_zmult__zless__mono2,axiom,
! [K_1,I_2,J_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,I_2),J_1))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),K_1))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,times_times_int(K_1,I_2)),times_times_int(K_1,J_1))) ) ) ).
fof(fact_781_Nat__Transfer_Otransfer__nat__int__function__closures_I6_J,axiom,
hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),one_one_int)) ).
fof(fact_782_Nat__Transfer_Otransfer__nat__int__function__closures_I1_J,axiom,
! [Y,X] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Y))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),plus_plus_int(X,Y))) ) ) ).
fof(fact_783_zadd__zless__mono,axiom,
! [Z_2,Z,W_2,W] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,W_2),W))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Z_2),Z))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,plus_plus_int(W_2,Z_2)),plus_plus_int(W,Z))) ) ) ).
fof(fact_784_Nat__Transfer_Otransfer__nat__int__function__closures_I9_J,axiom,
! [Z] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_nat_int(semiri1621563631at_int,Z))) ).
fof(fact_785_transfer__int__nat__quantifiers_I2_J,axiom,
! [P_1] :
( ? [X_1] :
( is_int(X_1)
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_1))
& hBOOL(hAPP_int_bool(P_1,X_1)) )
<=> ? [X_1] : hBOOL(hAPP_int_bool(P_1,hAPP_nat_int(semiri1621563631at_int,X_1))) ) ).
fof(fact_786_transfer__int__nat__quantifiers_I1_J,axiom,
! [P_1] :
( ! [X_1] :
( is_int(X_1)
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_1))
=> hBOOL(hAPP_int_bool(P_1,X_1)) ) )
<=> ! [X_1] : hBOOL(hAPP_int_bool(P_1,hAPP_nat_int(semiri1621563631at_int,X_1))) ) ).
fof(fact_787_zero__zle__int,axiom,
! [N] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_nat_int(semiri1621563631at_int,N))) ).
fof(fact_788_Nat__Transfer_Otransfer__nat__int__function__closures_I4_J,axiom,
! [N,X] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_nat_int(power_power_int(X),N))) ) ).
fof(fact_789_zle__iff__zadd,axiom,
! [Wa,Z_1] :
( is_int(Z_1)
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Wa),Z_1))
<=> ? [N_1] : Z_1 = plus_plus_int(Wa,hAPP_nat_int(semiri1621563631at_int,N_1)) ) ) ).
fof(fact_790_zpower__zadd__distrib,axiom,
! [X,Y,Z] : hAPP_nat_int(power_power_int(X),plus_plus_nat(Y,Z)) = times_times_int(hAPP_nat_int(power_power_int(X),Y),hAPP_nat_int(power_power_int(X),Z)) ).
fof(fact_791_nat__mono__iff,axiom,
! [Wa,Z_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),Z_1))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,nat(Wa)),nat(Z_1)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Wa),Z_1)) ) ) ).
fof(fact_792_zless__nat__conj,axiom,
! [Wa,Z_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,nat(Wa)),nat(Z_1)))
<=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),Z_1))
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Wa),Z_1)) ) ) ).
fof(fact_793_zless__nat__eq__int__zless,axiom,
! [Ma,Z_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),nat(Z_1)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_nat_int(semiri1621563631at_int,Ma)),Z_1)) ) ).
fof(fact_794_nat__diff__split__asm,axiom,
! [P_1,A_1,B] :
( hBOOL(hAPP_nat_bool(P_1,minus_minus_nat(A_1,B)))
<=> ~ ( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_1),B))
& ~ hBOOL(hAPP_nat_bool(P_1,zero_zero_nat)) )
| ? [D_2] :
( A_1 = plus_plus_nat(B,D_2)
& ~ hBOOL(hAPP_nat_bool(P_1,D_2)) ) ) ) ).
fof(fact_795_nat__diff__split,axiom,
! [P_1,A_1,B] :
( hBOOL(hAPP_nat_bool(P_1,minus_minus_nat(A_1,B)))
<=> ( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_1),B))
=> hBOOL(hAPP_nat_bool(P_1,zero_zero_nat)) )
& ! [D_2] :
( A_1 = plus_plus_nat(B,D_2)
=> hBOOL(hAPP_nat_bool(P_1,D_2)) ) ) ) ).
fof(fact_796_eq__0__number__of,axiom,
! [Va] :
( zero_zero_nat = number_number_of_nat(Va)
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Va),pls)) ) ).
fof(fact_797_eq__number__of__0,axiom,
! [Va] :
( number_number_of_nat(Va) = zero_zero_nat
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Va),pls)) ) ).
fof(fact_798_rel__simps_I5_J,axiom,
! [K] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,pls),bit1(K)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),K)) ) ).
fof(fact_799_rel__simps_I29_J,axiom,
! [K] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,bit1(K)),pls))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K),pls)) ) ).
fof(fact_800_rel__simps_I15_J,axiom,
! [K,L_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,bit0(K)),bit1(L_1)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K),L_1)) ) ).
fof(fact_801_less__int__code_I14_J,axiom,
! [K1,K2] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,bit0(K1)),bit1(K2)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K1),K2)) ) ).
fof(fact_802_rel__simps_I33_J,axiom,
! [K,L_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,bit1(K)),bit0(L_1)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K),L_1)) ) ).
fof(fact_803_less__eq__int__code_I15_J,axiom,
! [K1,K2] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,bit1(K1)),bit0(K2)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K1),K2)) ) ).
fof(fact_804_mult__Bit1,axiom,
! [K_1,L] : times_times_int(bit1(K_1),L) = plus_plus_int(bit0(times_times_int(K_1,L)),L) ).
fof(fact_805_pos__zmult__eq__1__iff,axiom,
! [Na,Ma] :
( ( is_int(Na)
& is_int(Ma) )
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),Ma))
=> ( times_times_int(Ma,Na) = one_one_int
<=> ( Ma = one_one_int
& Na = one_one_int ) ) ) ) ).
fof(fact_806_int__one__le__iff__zero__less,axiom,
! [Z_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,one_one_int),Z_1))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),Z_1)) ) ).
fof(fact_807_int__le__0__conv,axiom,
! [Na] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_nat_int(semiri1621563631at_int,Na)),zero_zero_int))
<=> Na = zero_zero_nat ) ).
fof(fact_808_succ__Pls,axiom,
succ(pls) = bit1(pls) ).
fof(fact_809_succ__Bit0,axiom,
! [K_1] : succ(bit0(K_1)) = bit1(K_1) ).
fof(fact_810_succ__Bit1,axiom,
! [K_1] : succ(bit1(K_1)) = bit0(succ(K_1)) ).
fof(fact_811_zle__add1__eq__le,axiom,
! [Wa,Z_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Wa),plus_plus_int(Z_1,one_one_int)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Wa),Z_1)) ) ).
fof(fact_812_add1__zle__eq,axiom,
! [Wa,Z_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,plus_plus_int(Wa,one_one_int)),Z_1))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Wa),Z_1)) ) ).
fof(fact_813_zless__imp__add1__zle,axiom,
! [W,Z] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,W),Z))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,plus_plus_int(W,one_one_int)),Z)) ) ).
fof(fact_814_succ__def,axiom,
! [K_1] : succ(K_1) = plus_plus_int(K_1,one_one_int) ).
fof(fact_815_zero__less__nat__eq,axiom,
! [Z_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),nat(Z_1)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),Z_1)) ) ).
fof(fact_816_transfer__nat__int__numerals_I4_J,axiom,
number_number_of_nat(bit1(bit1(pls))) = nat(number_number_of_int(bit1(bit1(pls)))) ).
fof(fact_817_split__nat,axiom,
! [P_1,I_1] :
( is_int(I_1)
=> ( hBOOL(hAPP_nat_bool(P_1,nat(I_1)))
<=> ( ! [N_1] :
( I_1 = hAPP_nat_int(semiri1621563631at_int,N_1)
=> hBOOL(hAPP_nat_bool(P_1,N_1)) )
& ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,I_1),zero_zero_int))
=> hBOOL(hAPP_nat_bool(P_1,zero_zero_nat)) ) ) ) ) ).
fof(fact_818_Nat__Transfer_Otransfer__nat__int__function__closures_I8_J,axiom,
hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),number_number_of_int(bit1(bit1(pls))))) ).
fof(fact_819_le__imp__0__less,axiom,
! [Z] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Z))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),plus_plus_int(one_one_int,Z))) ) ).
fof(fact_820_zmult__zless__mono2__lemma,axiom,
! [K_1,I_2,J_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,I_2),J_1))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),K_1))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,times_times_int(hAPP_nat_int(semiri1621563631at_int,K_1),I_2)),times_times_int(hAPP_nat_int(semiri1621563631at_int,K_1),J_1))) ) ) ).
fof(fact_821_add__Bit1__Bit1,axiom,
! [K_1,L] : plus_plus_int(bit1(K_1),bit1(L)) = bit0(plus_plus_int(K_1,succ(L))) ).
fof(fact_822_transfer__nat__int__numerals_I3_J,axiom,
number_number_of_nat(bit0(bit1(pls))) = nat(number_number_of_int(bit0(bit1(pls)))) ).
fof(fact_823_Nat__Transfer_Otransfer__nat__int__function__closures_I7_J,axiom,
hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),number_number_of_int(bit0(bit1(pls))))) ).
fof(fact_824_cube__square,axiom,
! [A] : times_times_int(A,hAPP_nat_int(power_power_int(A),number_number_of_nat(bit0(bit1(pls))))) = hAPP_nat_int(power_power_int(A),number_number_of_nat(bit1(bit1(pls)))) ).
fof(fact_825_power2__ge__self,axiom,
! [X] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X),hAPP_nat_int(power_power_int(X),number_number_of_nat(bit0(bit1(pls)))))) ).
fof(fact_826_int__pos__lt__two__imp__zero__or__one,axiom,
! [X] :
( is_int(X)
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X),number_number_of_int(bit0(bit1(pls)))))
=> ( X = zero_zero_int
| X = one_one_int ) ) ) ) ).
fof(fact_827_Euler_Oaux__1,axiom,
! [A,P_2] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),P_2))
=> hAPP_nat_int(power_power_int(A),nat(P_2)) = times_times_int(A,hAPP_nat_int(power_power_int(A),minus_minus_nat(nat(P_2),one_one_nat))) ) ).
fof(fact_828_self__quotient__aux2,axiom,
! [R_1,Q,A] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A))
=> ( A = plus_plus_int(R_1,times_times_int(A,Q))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),R_1))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Q),one_one_int)) ) ) ) ).
fof(fact_829_self__quotient__aux1,axiom,
! [R_1,Q,A] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A))
=> ( A = plus_plus_int(R_1,times_times_int(A,Q))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,R_1),A))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,one_one_int),Q)) ) ) ) ).
fof(fact_830_smaller_I2_J,axiom,
~ ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,plus_plus_int(one_one_int,hAPP_nat_int(semiri1621563631at_int,n))),plus_plus_int(times_times_int(number_number_of_int(bit0(bit0(bit1(pls)))),m),one_one_int)))
=> ~ hBOOL(hAPP_int_bool(twoSqu1154269391sum2sq,times_times_int(plus_plus_int(times_times_int(number_number_of_int(bit0(bit0(bit1(pls)))),m),one_one_int),plus_plus_int(one_one_int,hAPP_nat_int(semiri1621563631at_int,n))))) ) ).
fof(fact_831_q__pos__lemma,axiom,
! [B_2,Q_1,R_2] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),plus_plus_int(times_times_int(B_2,Q_1),R_2)))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,R_2),B_2))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_2))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Q_1)) ) ) ) ).
fof(fact_832_q__neg__lemma,axiom,
! [B_2,Q_1,R_2] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,plus_plus_int(times_times_int(B_2,Q_1),R_2)),zero_zero_int))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),R_2))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_2))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Q_1),zero_zero_int)) ) ) ) ).
fof(fact_833_nQ1,axiom,
~ hBOOL(hAPP_int_bool(twoSqu1154269391sum2sq,times_times_int(plus_plus_int(times_times_int(number_number_of_int(bit0(bit0(bit1(pls)))),m),one_one_int),plus_plus_int(one_one_int,hAPP_nat_int(semiri1621563631at_int,zero_zero_nat))))) ).
fof(fact_834_le0,axiom,
! [N] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),N)) ).
fof(fact_835_p0,axiom,
hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),plus_plus_int(times_times_int(number_number_of_int(bit0(bit0(bit1(pls)))),m),one_one_int))) ).
fof(fact_836_t__l__p,axiom,
hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,t),plus_plus_int(times_times_int(number_number_of_int(bit0(bit0(bit1(pls)))),m),one_one_int))) ).
fof(fact_837_qf1pt,axiom,
hBOOL(hAPP_int_bool(twoSqu1154269391sum2sq,times_times_int(plus_plus_int(times_times_int(number_number_of_int(bit0(bit0(bit1(pls)))),m),one_one_int),t))) ).
fof(fact_838_IH,axiom,
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,plus_plus_int(one_one_int,hAPP_nat_int(semiri1621563631at_int,n))),plus_plus_int(times_times_int(number_number_of_int(bit0(bit0(bit1(pls)))),m),one_one_int)))
& hBOOL(hAPP_int_bool(twoSqu1154269391sum2sq,times_times_int(plus_plus_int(times_times_int(number_number_of_int(bit0(bit0(bit1(pls)))),m),one_one_int),plus_plus_int(one_one_int,hAPP_nat_int(semiri1621563631at_int,n))))) ) ).
fof(fact_839__096_B_Bthesis_O_A_I_B_Bx_Ay_O_Ax_A_094_A2_A_L_Ay_A_094_A2_A_061_A_I4_A,axiom,
~ ! [X_1,Y_1] :
( ( is_int(X_1)
& is_int(Y_1) )
=> plus_plus_int(hAPP_nat_int(power_power_int(X_1),number_number_of_nat(bit0(bit1(pls)))),hAPP_nat_int(power_power_int(Y_1),number_number_of_nat(bit0(bit1(pls))))) != times_times_int(plus_plus_int(times_times_int(number_number_of_int(bit0(bit0(bit1(pls)))),m),one_one_int),plus_plus_int(one_one_int,hAPP_nat_int(semiri1621563631at_int,n))) ) ).
fof(fact_840_t,axiom,
plus_plus_int(hAPP_nat_int(power_power_int(s),number_number_of_nat(bit0(bit1(pls)))),one_one_int) = times_times_int(plus_plus_int(times_times_int(number_number_of_int(bit0(bit0(bit1(pls)))),m),one_one_int),t) ).
fof(fact_841_p,axiom,
hBOOL(hAPP_int_bool(zprime,plus_plus_int(times_times_int(number_number_of_int(bit0(bit0(bit1(pls)))),m),one_one_int))) ).
fof(fact_842__096t_A_061_A1_A_061_061_062_AEX_Ax_Ay_O_Ax_A_094_A2_A_L_Ay_A_094_A2_A_,axiom,
( t = one_one_int
=> ? [X_1,Y_1] :
( is_int(X_1)
& is_int(Y_1)
& plus_plus_int(hAPP_nat_int(power_power_int(X_1),number_number_of_nat(bit0(bit1(pls)))),hAPP_nat_int(power_power_int(Y_1),number_number_of_nat(bit0(bit1(pls))))) = plus_plus_int(times_times_int(number_number_of_int(bit0(bit0(bit1(pls)))),m),one_one_int) ) ) ).
fof(fact_843_le__refl,axiom,
! [N] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N),N)) ).
fof(fact_844_le__square,axiom,
! [M] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),times_times_nat(M,M))) ).
fof(fact_845_le__cube,axiom,
! [M] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),times_times_nat(M,times_times_nat(M,M)))) ).
fof(fact_846_nat__mult__commute,axiom,
! [M,N] : times_times_nat(M,N) = times_times_nat(N,M) ).
fof(fact_847_nat__le__linear,axiom,
! [M,N] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N))
| hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N),M)) ) ).
fof(fact_848_nat__mult__assoc,axiom,
! [M,N,K_1] : times_times_nat(times_times_nat(M,N),K_1) = times_times_nat(M,times_times_nat(N,K_1)) ).
fof(fact_849_eq__imp__le,axiom,
! [M,N] :
( M = N
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N)) ) ).
fof(fact_850_mult__le__mono1,axiom,
! [K_1,I_2,J_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_2),J_1))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,times_times_nat(I_2,K_1)),times_times_nat(J_1,K_1))) ) ).
fof(fact_851_mult__le__mono2,axiom,
! [K_1,I_2,J_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_2),J_1))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,times_times_nat(K_1,I_2)),times_times_nat(K_1,J_1))) ) ).
fof(fact_852_le__trans,axiom,
! [K_1,I_2,J_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_2),J_1))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,J_1),K_1))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_2),K_1)) ) ) ).
fof(fact_853_le__antisym,axiom,
! [M,N] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N),M))
=> M = N ) ) ).
fof(fact_854_mult__le__mono,axiom,
! [K_1,L,I_2,J_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_2),J_1))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_1),L))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,times_times_nat(I_2,K_1)),times_times_nat(J_1,L))) ) ) ).
fof(fact_855_real__le__eq__diff,axiom,
! [Xa,Ya] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Xa),Ya))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,minus_minus_real(Xa,Ya)),zero_zero_real)) ) ).
fof(fact_856_mult__le__cancel2,axiom,
! [Ma,K,Na] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,times_times_nat(Ma,K)),times_times_nat(Na,K)))
<=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),K))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Ma),Na)) ) ) ).
fof(fact_857_mult__le__cancel1,axiom,
! [K,Ma,Na] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,times_times_nat(K,Ma)),times_times_nat(K,Na)))
<=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),K))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Ma),Na)) ) ) ).
fof(fact_858_real__mult__le__cancel__iff1,axiom,
! [Xa,Ya,Z_1] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Z_1))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,times_times_real(Xa,Z_1)),times_times_real(Ya,Z_1)))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Xa),Ya)) ) ) ).
fof(fact_859_real__mult__le__cancel__iff2,axiom,
! [Xa,Ya,Z_1] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Z_1))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,times_times_real(Z_1,Xa)),times_times_real(Z_1,Ya)))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Xa),Ya)) ) ) ).
fof(fact_860_zdiff__int,axiom,
! [N,M] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N),M))
=> minus_minus_int(hAPP_nat_int(semiri1621563631at_int,M),hAPP_nat_int(semiri1621563631at_int,N)) = hAPP_nat_int(semiri1621563631at_int,minus_minus_nat(M,N)) ) ).
fof(fact_861_diff__bin__simps_I1_J,axiom,
! [K_1] :
( is_int(K_1)
=> minus_minus_int(K_1,pls) = K_1 ) ).
fof(fact_862_diff__bin__simps_I7_J,axiom,
! [K_1,L] : minus_minus_int(bit0(K_1),bit0(L)) = bit0(minus_minus_int(K_1,L)) ).
fof(fact_863_zdiff__zmult__distrib,axiom,
! [Z1,Z2,W] : times_times_int(minus_minus_int(Z1,Z2),W) = minus_minus_int(times_times_int(Z1,W),times_times_int(Z2,W)) ).
fof(fact_864_zdiff__zmult__distrib2,axiom,
! [W,Z1,Z2] : times_times_int(W,minus_minus_int(Z1,Z2)) = minus_minus_int(times_times_int(W,Z1),times_times_int(W,Z2)) ).
fof(fact_865_mult__0,axiom,
! [N] : times_times_nat(zero_zero_nat,N) = zero_zero_nat ).
fof(fact_866_mult__0__right,axiom,
! [M] : times_times_nat(M,zero_zero_nat) = zero_zero_nat ).
fof(fact_867_mult__is__0,axiom,
! [Ma,Na] :
( times_times_nat(Ma,Na) = zero_zero_nat
<=> ( Ma = zero_zero_nat
| Na = zero_zero_nat ) ) ).
fof(fact_868_mult__cancel1,axiom,
! [K,Ma,Na] :
( times_times_nat(K,Ma) = times_times_nat(K,Na)
<=> ( Ma = Na
| K = zero_zero_nat ) ) ).
fof(fact_869_mult__cancel2,axiom,
! [Ma,K,Na] :
( times_times_nat(Ma,K) = times_times_nat(Na,K)
<=> ( Ma = Na
| K = zero_zero_nat ) ) ).
fof(fact_870_less__eq__nat_Osimps_I1_J,axiom,
! [N] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),N)) ).
fof(fact_871_le__0__eq,axiom,
! [Na] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Na),zero_zero_nat))
<=> Na = zero_zero_nat ) ).
fof(fact_872_real__mult__left__cancel,axiom,
! [A_1,B,C_1] :
( C_1 != zero_zero_real
=> ( times_times_real(C_1,A_1) = times_times_real(C_1,B)
<=> A_1 = B ) ) ).
fof(fact_873_real__mult__right__cancel,axiom,
! [A_1,B,C_1] :
( C_1 != zero_zero_real
=> ( times_times_real(A_1,C_1) = times_times_real(B,C_1)
<=> A_1 = B ) ) ).
fof(fact_874_less__or__eq__imp__le,axiom,
! [M,N] :
( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N))
| M = N )
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N)) ) ).
fof(fact_875_le__neq__implies__less,axiom,
! [M,N] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N))
=> ( M != N
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N)) ) ) ).
fof(fact_876_less__imp__le__nat,axiom,
! [M,N] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N)) ) ).
fof(fact_877_le__eq__less__or__eq,axiom,
! [Ma,Na] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Ma),Na))
<=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),Na))
| Ma = Na ) ) ).
fof(fact_878_nat__less__le,axiom,
! [Ma,Na] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),Na))
<=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Ma),Na))
& Ma != Na ) ) ).
fof(fact_879_real__less__def,axiom,
! [Xa,Ya] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Xa),Ya))
<=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Xa),Ya))
& Xa != Ya ) ) ).
fof(fact_880_less__eq__real__def,axiom,
! [Xa,Ya] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Xa),Ya))
<=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Xa),Ya))
| Xa = Ya ) ) ).
fof(fact_881_add__mult__distrib,axiom,
! [M,N,K_1] : times_times_nat(plus_plus_nat(M,N),K_1) = plus_plus_nat(times_times_nat(M,K_1),times_times_nat(N,K_1)) ).
fof(fact_882_add__mult__distrib2,axiom,
! [K_1,M,N] : times_times_nat(K_1,plus_plus_nat(M,N)) = plus_plus_nat(times_times_nat(K_1,M),times_times_nat(K_1,N)) ).
fof(fact_883_add__leE,axiom,
! [M,K_1,N] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,plus_plus_nat(M,K_1)),N))
=> ~ ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N))
=> ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_1),N)) ) ) ).
fof(fact_884_add__leD1,axiom,
! [M,K_1,N] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,plus_plus_nat(M,K_1)),N))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N)) ) ).
fof(fact_885_add__leD2,axiom,
! [M,K_1,N] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,plus_plus_nat(M,K_1)),N))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_1),N)) ) ).
fof(fact_886_add__le__mono,axiom,
! [K_1,L,I_2,J_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_2),J_1))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_1),L))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,plus_plus_nat(I_2,K_1)),plus_plus_nat(J_1,L))) ) ) ).
fof(fact_887_add__le__mono1,axiom,
! [K_1,I_2,J_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_2),J_1))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,plus_plus_nat(I_2,K_1)),plus_plus_nat(J_1,K_1))) ) ).
fof(fact_888_trans__le__add2,axiom,
! [M,I_2,J_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_2),J_1))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_2),plus_plus_nat(M,J_1))) ) ).
fof(fact_889_trans__le__add1,axiom,
! [M,I_2,J_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_2),J_1))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_2),plus_plus_nat(J_1,M))) ) ).
fof(fact_890_nat__add__left__cancel__le,axiom,
! [K,Ma,Na] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,plus_plus_nat(K,Ma)),plus_plus_nat(K,Na)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Ma),Na)) ) ).
fof(fact_891_le__iff__add,axiom,
! [Ma,Na] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Ma),Na))
<=> ? [K_2] : Na = plus_plus_nat(Ma,K_2) ) ).
fof(fact_892_le__add1,axiom,
! [N,M] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N),plus_plus_nat(N,M))) ).
fof(fact_893_le__add2,axiom,
! [N,M] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N),plus_plus_nat(M,N))) ).
fof(fact_894_nat__mult__eq__one,axiom,
! [Na,Ma] :
( times_times_nat(Na,Ma) = one_one_nat
<=> ( Na = one_one_nat
& Ma = one_one_nat ) ) ).
fof(fact_895_nat__mult__eq__1__iff,axiom,
! [Ma,Na] :
( times_times_nat(Ma,Na) = one_one_nat
<=> ( Ma = one_one_nat
& Na = one_one_nat ) ) ).
fof(fact_896_nat__mult__1__right,axiom,
! [N] : times_times_nat(N,one_one_nat) = N ).
fof(fact_897_nat__1__eq__mult__iff,axiom,
! [Ma,Na] :
( one_one_nat = times_times_nat(Ma,Na)
<=> ( Ma = one_one_nat
& Na = one_one_nat ) ) ).
fof(fact_898_nat__mult__1,axiom,
! [N] : times_times_nat(one_one_nat,N) = N ).
fof(fact_899_diff__mult__distrib,axiom,
! [M,N,K_1] : times_times_nat(minus_minus_nat(M,N),K_1) = minus_minus_nat(times_times_nat(M,K_1),times_times_nat(N,K_1)) ).
fof(fact_900_diff__mult__distrib2,axiom,
! [K_1,M,N] : times_times_nat(K_1,minus_minus_nat(M,N)) = minus_minus_nat(times_times_nat(K_1,M),times_times_nat(K_1,N)) ).
fof(fact_901_real__mult__1,axiom,
! [Z] : times_times_real(one_one_real,Z) = Z ).
fof(fact_902_le__diff__iff,axiom,
! [Na,K,Ma] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K),Ma))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K),Na))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,minus_minus_nat(Ma,K)),minus_minus_nat(Na,K)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Ma),Na)) ) ) ) ).
fof(fact_903_Nat_Odiff__diff__eq,axiom,
! [N,K_1,M] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_1),M))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_1),N))
=> minus_minus_nat(minus_minus_nat(M,K_1),minus_minus_nat(N,K_1)) = minus_minus_nat(M,N) ) ) ).
fof(fact_904_eq__diff__iff,axiom,
! [Na,K,Ma] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K),Ma))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K),Na))
=> ( minus_minus_nat(Ma,K) = minus_minus_nat(Na,K)
<=> Ma = Na ) ) ) ).
fof(fact_905_diff__diff__cancel,axiom,
! [I_2,N] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_2),N))
=> minus_minus_nat(N,minus_minus_nat(N,I_2)) = I_2 ) ).
fof(fact_906_diff__le__mono,axiom,
! [L,M,N] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,minus_minus_nat(M,L)),minus_minus_nat(N,L))) ) ).
fof(fact_907_diff__le__mono2,axiom,
! [L,M,N] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,minus_minus_nat(L,N)),minus_minus_nat(L,M))) ) ).
fof(fact_908_Nat_Odiff__le__self,axiom,
! [M,N] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,minus_minus_nat(M,N)),M)) ).
fof(fact_909_real__add__mult__distrib,axiom,
! [Z1,Z2,W] : times_times_real(plus_plus_real(Z1,Z2),W) = plus_plus_real(times_times_real(Z1,W),times_times_real(Z2,W)) ).
fof(fact_910_zpower__zpower,axiom,
! [X,Y,Z] : hAPP_nat_int(power_power_int(hAPP_nat_int(power_power_int(X),Y)),Z) = hAPP_nat_int(power_power_int(X),times_times_nat(Y,Z)) ).
fof(fact_911_real__add__left__mono,axiom,
! [Z,X,Y] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X),Y))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,plus_plus_real(Z,X)),plus_plus_real(Z,Y))) ) ).
fof(fact_912_diff__bin__simps_I10_J,axiom,
! [K_1,L] : minus_minus_int(bit1(K_1),bit1(L)) = bit0(minus_minus_int(K_1,L)) ).
fof(fact_913_diff__bin__simps_I9_J,axiom,
! [K_1,L] : minus_minus_int(bit1(K_1),bit0(L)) = bit1(minus_minus_int(K_1,L)) ).
fof(fact_914_diff__bin__simps_I3_J,axiom,
! [L] : minus_minus_int(pls,bit0(L)) = bit0(minus_minus_int(pls,L)) ).
fof(fact_915_Euler_Oaux1,axiom,
! [A,X] :
( is_int(X)
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),X))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X),A))
=> ( X != minus_minus_int(A,one_one_int)
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X),minus_minus_int(A,one_one_int))) ) ) ) ) ).
fof(fact_916_less__bin__lemma,axiom,
! [K,L_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K),L_1))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,minus_minus_int(K,L_1)),zero_zero_int)) ) ).
fof(fact_917_mult__less__mono2,axiom,
! [K_1,I_2,J_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_2),J_1))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),K_1))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,times_times_nat(K_1,I_2)),times_times_nat(K_1,J_1))) ) ) ).
fof(fact_918_mult__less__mono1,axiom,
! [K_1,I_2,J_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_2),J_1))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),K_1))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,times_times_nat(I_2,K_1)),times_times_nat(J_1,K_1))) ) ) ).
fof(fact_919_mult__less__cancel2,axiom,
! [Ma,K,Na] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,times_times_nat(Ma,K)),times_times_nat(Na,K)))
<=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),K))
& hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),Na)) ) ) ).
fof(fact_920_mult__less__cancel1,axiom,
! [K,Ma,Na] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,times_times_nat(K,Ma)),times_times_nat(K,Na)))
<=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),K))
& hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),Na)) ) ) ).
fof(fact_921_nat__0__less__mult__iff,axiom,
! [Ma,Na] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),times_times_nat(Ma,Na)))
<=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),Ma))
& hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),Na)) ) ) ).
fof(fact_922_not__real__square__gt__zero,axiom,
! [Xa] :
( ~ hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),times_times_real(Xa,Xa)))
<=> Xa = zero_zero_real ) ).
fof(fact_923_real__mult__less__mono2,axiom,
! [X,Y,Z] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Z))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X),Y))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,times_times_real(Z,X)),times_times_real(Z,Y))) ) ) ).
fof(fact_924_real__mult__order,axiom,
! [Y,X] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Y))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),times_times_real(X,Y))) ) ) ).
fof(fact_925_real__mult__less__iff1,axiom,
! [Xa,Ya,Z_1] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Z_1))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,times_times_real(Xa,Z_1)),times_times_real(Ya,Z_1)))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Xa),Ya)) ) ) ).
fof(fact_926_mult__eq__self__implies__10,axiom,
! [M,N] :
( M = times_times_nat(M,N)
=> ( N = one_one_nat
| M = zero_zero_nat ) ) ).
fof(fact_927_diff__is__0__eq,axiom,
! [Ma,Na] :
( minus_minus_nat(Ma,Na) = zero_zero_nat
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Ma),Na)) ) ).
fof(fact_928_diff__is__0__eq_H,axiom,
! [M,N] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N))
=> minus_minus_nat(M,N) = zero_zero_nat ) ).
fof(fact_929_real__two__squares__add__zero__iff,axiom,
! [Xa,Ya] :
( plus_plus_real(times_times_real(Xa,Xa),times_times_real(Ya,Ya)) = zero_zero_real
<=> ( Xa = zero_zero_real
& Ya = zero_zero_real ) ) ).
fof(fact_930_diff__less__mono,axiom,
! [C,A,B_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A),B_1))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,C),A))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,minus_minus_nat(A,C)),minus_minus_nat(B_1,C))) ) ) ).
fof(fact_931_less__diff__iff,axiom,
! [Na,K,Ma] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K),Ma))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K),Na))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,minus_minus_nat(Ma,K)),minus_minus_nat(Na,K)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),Na)) ) ) ) ).
fof(fact_932_Nat__Transfer_Otransfer__int__nat__functions_I2_J,axiom,
! [X,Y] : times_times_int(hAPP_nat_int(semiri1621563631at_int,X),hAPP_nat_int(semiri1621563631at_int,Y)) = hAPP_nat_int(semiri1621563631at_int,times_times_nat(X,Y)) ).
fof(fact_933_zmult__int,axiom,
! [M,N] : times_times_int(hAPP_nat_int(semiri1621563631at_int,M),hAPP_nat_int(semiri1621563631at_int,N)) = hAPP_nat_int(semiri1621563631at_int,times_times_nat(M,N)) ).
fof(fact_934_int__mult,axiom,
! [M,N] : hAPP_nat_int(semiri1621563631at_int,times_times_nat(M,N)) = times_times_int(hAPP_nat_int(semiri1621563631at_int,M),hAPP_nat_int(semiri1621563631at_int,N)) ).
fof(fact_935_diff__diff__right,axiom,
! [I_2,K_1,J_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_1),J_1))
=> minus_minus_nat(I_2,minus_minus_nat(J_1,K_1)) = minus_minus_nat(plus_plus_nat(I_2,K_1),J_1) ) ).
fof(fact_936_le__diff__conv,axiom,
! [J,K,I_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,minus_minus_nat(J,K)),I_1))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,J),plus_plus_nat(I_1,K))) ) ).
fof(fact_937_le__add__diff,axiom,
! [M,K_1,N] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_1),N))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),minus_minus_nat(plus_plus_nat(N,M),K_1))) ) ).
fof(fact_938_le__add__diff__inverse,axiom,
! [N,M] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N),M))
=> plus_plus_nat(N,minus_minus_nat(M,N)) = M ) ).
fof(fact_939_add__diff__assoc,axiom,
! [I_2,K_1,J_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_1),J_1))
=> plus_plus_nat(I_2,minus_minus_nat(J_1,K_1)) = minus_minus_nat(plus_plus_nat(I_2,J_1),K_1) ) ).
fof(fact_940_le__diff__conv2,axiom,
! [I_1,K,J] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K),J))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_1),minus_minus_nat(J,K)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,plus_plus_nat(I_1,K)),J)) ) ) ).
fof(fact_941_le__add__diff__inverse2,axiom,
! [N,M] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N),M))
=> plus_plus_nat(minus_minus_nat(M,N),N) = M ) ).
fof(fact_942_le__imp__diff__is__add,axiom,
! [K,I_1,J] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_1),J))
=> ( minus_minus_nat(J,I_1) = K
<=> J = plus_plus_nat(K,I_1) ) ) ).
fof(fact_943_diff__add__assoc,axiom,
! [I_2,K_1,J_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_1),J_1))
=> minus_minus_nat(plus_plus_nat(I_2,J_1),K_1) = plus_plus_nat(I_2,minus_minus_nat(J_1,K_1)) ) ).
fof(fact_944_add__diff__assoc2,axiom,
! [I_2,K_1,J_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_1),J_1))
=> plus_plus_nat(minus_minus_nat(J_1,K_1),I_2) = minus_minus_nat(plus_plus_nat(J_1,I_2),K_1) ) ).
fof(fact_945_diff__add__assoc2,axiom,
! [I_2,K_1,J_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_1),J_1))
=> minus_minus_nat(plus_plus_nat(J_1,I_2),K_1) = plus_plus_nat(minus_minus_nat(J_1,K_1),I_2) ) ).
fof(fact_946_Nat__Transfer_Otransfer__int__nat__relations_I3_J,axiom,
! [Xa,Ya] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_nat_int(semiri1621563631at_int,Xa)),hAPP_nat_int(semiri1621563631at_int,Ya)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Xa),Ya)) ) ).
fof(fact_947_zle__int,axiom,
! [Ma,Na] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_nat_int(semiri1621563631at_int,Ma)),hAPP_nat_int(semiri1621563631at_int,Na)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Ma),Na)) ) ).
fof(fact_948_is__mult__sum2sq,axiom,
! [Y,X] :
( hBOOL(hAPP_int_bool(twoSqu1154269391sum2sq,X))
=> ( hBOOL(hAPP_int_bool(twoSqu1154269391sum2sq,Y))
=> hBOOL(hAPP_int_bool(twoSqu1154269391sum2sq,times_times_int(X,Y))) ) ) ).
fof(fact_949_zle__diff1__eq,axiom,
! [Wa,Z_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Wa),minus_minus_int(Z_1,one_one_int)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Wa),Z_1)) ) ).
fof(fact_950_le__nat__number__of,axiom,
! [Va,V_3] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,number_number_of_nat(Va)),number_number_of_nat(V_3)))
<=> ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Va),V_3))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Va),pls)) ) ) ).
fof(fact_951_transfer__nat__int__relations_I3_J,axiom,
! [Ya,Xa] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Xa))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Ya))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,nat(Xa)),nat(Ya)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Xa),Ya)) ) ) ) ).
fof(fact_952_nat__diff__distrib,axiom,
! [Z,Z_2] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Z_2))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Z_2),Z))
=> nat(minus_minus_int(Z,Z_2)) = minus_minus_nat(nat(Z),nat(Z_2)) ) ) ).
fof(fact_953_nat__mult__distrib,axiom,
! [Z_2,Z] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Z))
=> nat(times_times_int(Z,Z_2)) = times_times_nat(nat(Z),nat(Z_2)) ) ).
fof(fact_954_Nat__Transfer_Otransfer__nat__int__functions_I2_J,axiom,
! [Y,X] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Y))
=> times_times_nat(nat(X),nat(Y)) = nat(times_times_int(X,Y)) ) ) ).
fof(fact_955_mult__eq__if,axiom,
! [N,M] :
( ( M = zero_zero_nat
=> times_times_nat(M,N) = zero_zero_nat )
& ( M != zero_zero_nat
=> times_times_nat(M,N) = plus_plus_nat(N,times_times_nat(minus_minus_nat(M,one_one_nat),N)) ) ) ).
fof(fact_956_nat__le__eq__zle,axiom,
! [Z_1,Wa] :
( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),Wa))
| hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Z_1)) )
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,nat(Wa)),nat(Z_1)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Wa),Z_1)) ) ) ).
fof(fact_957_power__eq__if,axiom,
! [P_2,M] :
( ( M = zero_zero_nat
=> hAPP_nat_nat(power_power_nat(P_2),M) = one_one_nat )
& ( M != zero_zero_nat
=> hAPP_nat_nat(power_power_nat(P_2),M) = times_times_nat(P_2,hAPP_nat_nat(power_power_nat(P_2),minus_minus_nat(M,one_one_nat))) ) ) ).
fof(fact_958_nat__mult__2,axiom,
! [Z] : times_times_nat(number_number_of_nat(bit0(bit1(pls))),Z) = plus_plus_nat(Z,Z) ).
fof(fact_959_nat__mult__2__right,axiom,
! [Z] : times_times_nat(Z,number_number_of_nat(bit0(bit1(pls)))) = plus_plus_nat(Z,Z) ).
fof(fact_960_nat__number__of__mult__left,axiom,
! [V_2,K_1,V_1] :
( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,V_1),pls))
=> times_times_nat(number_number_of_nat(V_1),times_times_nat(number_number_of_nat(V_2),K_1)) = zero_zero_nat )
& ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,V_1),pls))
=> times_times_nat(number_number_of_nat(V_1),times_times_nat(number_number_of_nat(V_2),K_1)) = times_times_nat(number_number_of_nat(times_times_int(V_1,V_2)),K_1) ) ) ).
fof(fact_961_mult__nat__number__of,axiom,
! [V_2,V_1] :
( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,V_1),pls))
=> times_times_nat(number_number_of_nat(V_1),number_number_of_nat(V_2)) = zero_zero_nat )
& ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,V_1),pls))
=> times_times_nat(number_number_of_nat(V_1),number_number_of_nat(V_2)) = number_number_of_nat(times_times_int(V_1,V_2)) ) ) ).
fof(fact_962_four__x__squared,axiom,
! [X] : times_times_real(number267125858f_real(bit0(bit0(bit1(pls)))),hAPP_nat_real(power_power_real(X),number_number_of_nat(bit0(bit1(pls))))) = hAPP_nat_real(power_power_real(times_times_real(number267125858f_real(bit0(bit1(pls))),X)),number_number_of_nat(bit0(bit1(pls)))) ).
fof(fact_963_two__realpow__ge__one,axiom,
! [N] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,one_one_real),hAPP_nat_real(power_power_real(number267125858f_real(bit0(bit1(pls)))),N))) ).
fof(fact_964_Euler_Oaux2,axiom,
! [B_1,A,C] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A),C))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_1),C))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A),B_1))
| hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_1),A)) ) ) ) ).
fof(fact_965_zspecial__product,axiom,
! [A,B_1] : times_times_int(plus_plus_int(A,B_1),minus_minus_int(A,B_1)) = minus_minus_int(hAPP_nat_int(power_power_int(A),number_number_of_nat(bit0(bit1(pls)))),hAPP_nat_int(power_power_int(B_1),number_number_of_nat(bit0(bit1(pls))))) ).
fof(fact_966_diff__square,axiom,
! [X,Y] : minus_minus_nat(hAPP_nat_nat(power_power_nat(X),number_number_of_nat(bit0(bit1(pls)))),hAPP_nat_nat(power_power_nat(Y),number_number_of_nat(bit0(bit1(pls))))) = times_times_nat(plus_plus_nat(X,Y),minus_minus_nat(X,Y)) ).
fof(fact_967_zdiff__power3,axiom,
! [A,B_1] : hAPP_nat_int(power_power_int(minus_minus_int(A,B_1)),number_number_of_nat(bit1(bit1(pls)))) = minus_minus_int(plus_plus_int(minus_minus_int(hAPP_nat_int(power_power_int(A),number_number_of_nat(bit1(bit1(pls)))),times_times_int(times_times_int(number_number_of_int(bit1(bit1(pls))),hAPP_nat_int(power_power_int(A),number_number_of_nat(bit0(bit1(pls))))),B_1)),times_times_int(times_times_int(number_number_of_int(bit1(bit1(pls))),A),hAPP_nat_int(power_power_int(B_1),number_number_of_nat(bit0(bit1(pls)))))),hAPP_nat_int(power_power_int(B_1),number_number_of_nat(bit1(bit1(pls))))) ).
fof(fact_968_zdiff__power2,axiom,
! [A,B_1] : hAPP_nat_int(power_power_int(minus_minus_int(A,B_1)),number_number_of_nat(bit0(bit1(pls)))) = plus_plus_int(minus_minus_int(hAPP_nat_int(power_power_int(A),number_number_of_nat(bit0(bit1(pls)))),times_times_int(times_times_int(number_number_of_int(bit0(bit1(pls))),A),B_1)),hAPP_nat_int(power_power_int(B_1),number_number_of_nat(bit0(bit1(pls))))) ).
fof(fact_969_zdiv__mono2__neg__lemma,axiom,
! [B_1,Q,R_1,B_2,Q_1,R_2] :
( plus_plus_int(times_times_int(B_1,Q),R_1) = plus_plus_int(times_times_int(B_2,Q_1),R_2)
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,plus_plus_int(times_times_int(B_2,Q_1),R_2)),zero_zero_int))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,R_1),B_1))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),R_2))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_2))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_2),B_1))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Q_1),Q)) ) ) ) ) ) ) ).
fof(fact_970_unique__quotient__lemma__neg,axiom,
! [B_1,Q_1,R_2,Q,R_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,plus_plus_int(times_times_int(B_1,Q_1),R_2)),plus_plus_int(times_times_int(B_1,Q),R_1)))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,R_1),zero_zero_int))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_1),R_1))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_1),R_2))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Q),Q_1)) ) ) ) ) ).
fof(fact_971_zdiv__mono2__lemma,axiom,
! [B_1,Q,R_1,B_2,Q_1,R_2] :
( plus_plus_int(times_times_int(B_1,Q),R_1) = plus_plus_int(times_times_int(B_2,Q_1),R_2)
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),plus_plus_int(times_times_int(B_2,Q_1),R_2)))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,R_2),B_2))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),R_1))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_2))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_2),B_1))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Q),Q_1)) ) ) ) ) ) ) ).
fof(fact_972_unique__quotient__lemma,axiom,
! [B_1,Q_1,R_2,Q,R_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,plus_plus_int(times_times_int(B_1,Q_1),R_2)),plus_plus_int(times_times_int(B_1,Q),R_1)))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),R_2))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,R_2),B_1))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,R_1),B_1))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Q_1),Q)) ) ) ) ) ).
fof(fact_973_xy,axiom,
plus_plus_int(hAPP_nat_int(power_power_int(x),number_number_of_nat(bit0(bit1(pls)))),hAPP_nat_int(power_power_int(y),number_number_of_nat(bit0(bit1(pls))))) = times_times_int(plus_plus_int(times_times_int(number_number_of_int(bit0(bit0(bit1(pls)))),m),one_one_int),plus_plus_int(one_one_int,hAPP_nat_int(semiri1621563631at_int,n))) ).
fof(fact_974_Int2_Oaux__1,axiom,
! [P_2] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,number_number_of_int(bit0(bit1(pls)))),P_2))
=> minus_minus_nat(nat(P_2),number_number_of_nat(bit0(bit1(pls)))) = nat(minus_minus_int(P_2,number_number_of_int(bit0(bit1(pls))))) ) ).
fof(fact_975_Int2_Oaux__2,axiom,
! [P_2] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,number_number_of_int(bit0(bit1(pls)))),P_2))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),nat(minus_minus_int(P_2,number_number_of_int(bit0(bit1(pls))))))) ) ).
fof(fact_976__096_B_Bthesis_O_A_I_B_Bt_O_As_____A_094_A2_A_L_A1_A_061_A_I4_A_K_Am_A_,axiom,
~ ! [T] :
( is_int(T)
=> plus_plus_int(hAPP_nat_int(power_power_int(s),number_number_of_nat(bit0(bit1(pls)))),one_one_int) != times_times_int(plus_plus_int(times_times_int(number_number_of_int(bit0(bit0(bit1(pls)))),m),one_one_int),T) ) ).
fof(fact_977_real__le__refl,axiom,
! [W] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,W),W)) ).
fof(fact_978_real__mult__commute,axiom,
! [Z,W] : times_times_real(Z,W) = times_times_real(W,Z) ).
fof(fact_979_real__le__linear,axiom,
! [Z,W] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Z),W))
| hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,W),Z)) ) ).
fof(fact_980_real__mult__assoc,axiom,
! [Z1,Z2,Z3] : times_times_real(times_times_real(Z1,Z2),Z3) = times_times_real(Z1,times_times_real(Z2,Z3)) ).
fof(fact_981_real__le__trans,axiom,
! [K_1,I_2,J_1] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,I_2),J_1))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,J_1),K_1))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,I_2),K_1)) ) ) ).
fof(fact_982_real__le__antisym,axiom,
! [Z,W] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Z),W))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,W),Z))
=> Z = W ) ) ).
fof(fact_983_zprime__2,axiom,
hBOOL(hAPP_int_bool(zprime,number_number_of_int(bit0(bit1(pls))))) ).
fof(fact_984_Int2_Oaux1,axiom,
! [A,B_1,C] :
( is_int(A)
=> ( minus_minus_int(A,B_1) = C
=> A = plus_plus_int(C,B_1) ) ) ).
fof(fact_985__0964_A_K_Am_A_L_A1_Advd_As_____A_094_A2_A_L_A1_096,axiom,
hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,plus_plus_int(times_times_int(number_number_of_int(bit0(bit0(bit1(pls)))),m),one_one_int)),plus_plus_int(hAPP_nat_int(power_power_int(s),number_number_of_nat(bit0(bit1(pls)))),one_one_int))) ).
fof(fact_986_real__sum__squared__expand,axiom,
! [X,Y] : hAPP_nat_real(power_power_real(plus_plus_real(X,Y)),number_number_of_nat(bit0(bit1(pls)))) = plus_plus_real(plus_plus_real(hAPP_nat_real(power_power_real(X),number_number_of_nat(bit0(bit1(pls)))),hAPP_nat_real(power_power_real(Y),number_number_of_nat(bit0(bit1(pls))))),times_times_real(times_times_real(number267125858f_real(bit0(bit1(pls))),X),Y)) ).
fof(fact_987__096s_____A_094_A2_A_N_A_N1_A_061_As_____A_094_A2_A_L_A1_096,axiom,
minus_minus_int(hAPP_nat_int(power_power_int(s),number_number_of_nat(bit0(bit1(pls)))),number_number_of_int(min)) = plus_plus_int(hAPP_nat_int(power_power_int(s),number_number_of_nat(bit0(bit1(pls)))),one_one_int) ).
fof(fact_988__0964_A_K_Am_A_L_A1_Advd_As_____A_094_A2_A_N_A_N1_096,axiom,
hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,plus_plus_int(times_times_int(number_number_of_int(bit0(bit0(bit1(pls)))),m),one_one_int)),minus_minus_int(hAPP_nat_int(power_power_int(s),number_number_of_nat(bit0(bit1(pls)))),number_number_of_int(min)))) ).
fof(fact_989__096Legendre_A_N1_A_I4_A_K_Am_A_L_A1_J_A_061_A1_096,axiom,
legendre(number_number_of_int(min),plus_plus_int(times_times_int(number_number_of_int(bit0(bit0(bit1(pls)))),m),one_one_int)) = one_one_int ).
fof(fact_990_zdvd__zdiffD,axiom,
! [K_1,M,N] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K_1),minus_minus_int(M,N)))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K_1),N))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K_1),M)) ) ) ).
fof(fact_991_zprime__zdvd__zmult__better,axiom,
! [M,N,P_2] :
( hBOOL(hAPP_int_bool(zprime,P_2))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,P_2),times_times_int(M,N)))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,P_2),M))
| hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,P_2),N)) ) ) ) ).
fof(fact_992_rel__simps_I24_J,axiom,
hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,min),min)) ).
fof(fact_993_zdvd__bounds,axiom,
! [N,M] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,N),M))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,M),zero_zero_int))
| hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,N),M)) ) ) ).
fof(fact_994_zprime__zdvd__power,axiom,
! [A,N,P_2] :
( hBOOL(hAPP_int_bool(zprime,P_2))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,P_2),hAPP_nat_int(power_power_int(A),N)))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,P_2),A)) ) ) ).
fof(fact_995_zdvd__not__zless,axiom,
! [N,M] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),M))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,M),N))
=> ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,N),M)) ) ) ).
fof(fact_996_zdvd__mult__cancel,axiom,
! [K_1,M,N] :
( is_int(K_1)
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,times_times_int(K_1,M)),times_times_int(K_1,N)))
=> ( K_1 != zero_zero_int
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,M),N)) ) ) ) ).
fof(fact_997_zdvd__antisym__nonneg,axiom,
! [N,M] :
( ( is_int(N)
& is_int(M) )
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),M))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),N))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,M),N))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,N),M))
=> M = N ) ) ) ) ) ).
fof(fact_998_zdvd__reduce,axiom,
! [K,Na,Ma] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K),plus_plus_int(Na,times_times_int(K,Ma))))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K),Na)) ) ).
fof(fact_999_zdvd__period,axiom,
! [C_1,Xa,Ta,A_1,D] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A_1),D))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A_1),plus_plus_int(Xa,Ta)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A_1),plus_plus_int(plus_plus_int(Xa,times_times_int(C_1,D)),Ta))) ) ) ).
fof(fact_1000_rel__simps_I9_J,axiom,
! [K] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,min),bit1(K)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,min),K)) ) ).
fof(fact_1001_rel__simps_I13_J,axiom,
! [K] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,bit1(K)),min))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K),min)) ) ).
fof(fact_1002_rel__simps_I6_J,axiom,
hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,min),pls)) ).
fof(fact_1003_rel__simps_I3_J,axiom,
~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,pls),min)) ).
fof(fact_1004_rel__simps_I8_J,axiom,
! [K] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,min),bit0(K)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,min),K)) ) ).
fof(fact_1005_rel__simps_I26_J,axiom,
! [K] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,min),bit1(K)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,min),K)) ) ).
fof(fact_1006_rel__simps_I30_J,axiom,
! [K] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,bit1(K)),min))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K),min)) ) ).
fof(fact_1007_bin__less__0__simps_I2_J,axiom,
hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,min),zero_zero_int)) ).
fof(fact_1008_rel__simps_I23_J,axiom,
hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,min),pls)) ).
fof(fact_1009_rel__simps_I20_J,axiom,
~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),min)) ).
fof(fact_1010_rel__simps_I28_J,axiom,
! [K] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,bit0(K)),min))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K),min)) ) ).
fof(fact_1011_eq__number__of__Pls__Min,axiom,
number_number_of_int(pls) != number_number_of_int(min) ).
fof(fact_1012_rel__simps_I7_J,axiom,
~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,min),min)) ).
fof(fact_1013_rel__simps_I42_J,axiom,
! [L] : min != bit0(L) ).
fof(fact_1014_rel__simps_I45_J,axiom,
! [K_1] : bit0(K_1) != min ).
fof(fact_1015_rel__simps_I40_J,axiom,
min != pls ).
fof(fact_1016_rel__simps_I37_J,axiom,
pls != min ).
fof(fact_1017_Bit1__Min,axiom,
bit1(min) = min ).
fof(fact_1018_rel__simps_I43_J,axiom,
! [L_1] :
( is_int(L_1)
=> ( min = bit1(L_1)
<=> min = L_1 ) ) ).
fof(fact_1019_rel__simps_I47_J,axiom,
! [K] :
( is_int(K)
=> ( bit1(K) = min
<=> K = min ) ) ).
fof(fact_1020_succ__Min,axiom,
succ(min) = pls ).
fof(fact_1021_diff__bin__simps_I2_J,axiom,
! [K_1] : minus_minus_int(K_1,min) = succ(K_1) ).
fof(fact_1022_zdvd__imp__le,axiom,
! [Z,N] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,Z),N))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),N))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Z),N)) ) ) ).
fof(fact_1023_zpower__zdvd__prop1,axiom,
! [P_2,Y,N] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,P_2),Y))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,P_2),hAPP_nat_int(power_power_int(Y),N))) ) ) ).
fof(fact_1024_rel__simps_I25_J,axiom,
! [K] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,min),bit0(K)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,min),K)) ) ).
fof(fact_1025_rel__simps_I11_J,axiom,
! [K] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,bit0(K)),min))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K),min)) ) ).
fof(fact_1026_diff__bin__simps_I4_J,axiom,
! [L] : minus_minus_int(pls,bit1(L)) = bit1(minus_minus_int(min,L)) ).
fof(fact_1027_diff__bin__simps_I6_J,axiom,
! [L] : minus_minus_int(min,bit1(L)) = bit0(minus_minus_int(min,L)) ).
fof(fact_1028_diff__bin__simps_I5_J,axiom,
! [L] : minus_minus_int(min,bit0(L)) = bit1(minus_minus_int(min,L)) ).
fof(fact_1029_zmult__eq__1__iff,axiom,
! [Ma,Na] :
( ( is_int(Ma)
& is_int(Na) )
=> ( times_times_int(Ma,Na) = one_one_int
<=> ( ( Ma = one_one_int
& Na = one_one_int )
| ( Ma = number_number_of_int(min)
& Na = number_number_of_int(min) ) ) ) ) ).
fof(fact_1030_pos__zmult__eq__1__iff__lemma,axiom,
! [M,N] :
( is_int(M)
=> ( times_times_int(M,N) = one_one_int
=> ( M = one_one_int
| M = number_number_of_int(min) ) ) ) ).
fof(fact_1031_zprime__power__zdvd__cancel__right,axiom,
! [N,A,B_1,P_2] :
( hBOOL(hAPP_int_bool(zprime,P_2))
=> ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,P_2),B_1))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_nat_int(power_power_int(P_2),N)),times_times_int(A,B_1)))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_nat_int(power_power_int(P_2),N)),A)) ) ) ) ).
fof(fact_1032_zprime__power__zdvd__cancel__left,axiom,
! [N,B_1,A,P_2] :
( hBOOL(hAPP_int_bool(zprime,P_2))
=> ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,P_2),A))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_nat_int(power_power_int(P_2),N)),times_times_int(A,B_1)))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_nat_int(power_power_int(P_2),N)),B_1)) ) ) ) ).
fof(fact_1033_zpower__zdvd__prop2,axiom,
! [Y,N,P_2] :
( hBOOL(hAPP_int_bool(zprime,P_2))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,P_2),hAPP_nat_int(power_power_int(Y),N)))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,P_2),Y)) ) ) ) ).
fof(fact_1034__096QuadRes_A_I4_A_K_Am_A_L_A1_J_A_N1_096,axiom,
hBOOL(hAPP_int_bool(quadRes(plus_plus_int(times_times_int(number_number_of_int(bit0(bit0(bit1(pls)))),m),one_one_int)),number_number_of_int(min))) ).
fof(fact_1035_s,axiom,
hBOOL(hAPP_int_bool(zcong(hAPP_nat_int(power_power_int(s),number_number_of_nat(bit0(bit1(pls)))),number_number_of_int(min)),plus_plus_int(times_times_int(number_number_of_int(bit0(bit0(bit1(pls)))),m),one_one_int))) ).
fof(fact_1036__096_126_AQuadRes_A_I4_A_K_Am_A_L_A1_J_A_N1_A_061_061_062_ALegendre_A_,axiom,
( ~ hBOOL(hAPP_int_bool(quadRes(plus_plus_int(times_times_int(number_number_of_int(bit0(bit0(bit1(pls)))),m),one_one_int)),number_number_of_int(min)))
=> legendre(number_number_of_int(min),plus_plus_int(times_times_int(number_number_of_int(bit0(bit0(bit1(pls)))),m),one_one_int)) != one_one_int ) ).
fof(fact_1037_s1,axiom,
hBOOL(hAPP_int_bool(zcong(hAPP_nat_int(power_power_int(s1),number_number_of_nat(bit0(bit1(pls)))),number_number_of_int(min)),plus_plus_int(times_times_int(number_number_of_int(bit0(bit0(bit1(pls)))),m),one_one_int))) ).
fof(fact_1038_s0p,axiom,
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),s))
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,s),plus_plus_int(times_times_int(number_number_of_int(bit0(bit0(bit1(pls)))),m),one_one_int)))
& hBOOL(hAPP_int_bool(zcong(s1,s),plus_plus_int(times_times_int(number_number_of_int(bit0(bit0(bit1(pls)))),m),one_one_int))) ) ).
fof(fact_1039__096_B_Bthesis_O_A_I_B_Bs1_O_A_091s1_A_094_A2_A_061_A_N1_093_A_Imod_A4,axiom,
~ ! [S1] :
( is_int(S1)
=> ~ hBOOL(hAPP_int_bool(zcong(hAPP_nat_int(power_power_int(S1),number_number_of_nat(bit0(bit1(pls)))),number_number_of_int(min)),plus_plus_int(times_times_int(number_number_of_int(bit0(bit0(bit1(pls)))),m),one_one_int))) ) ).
fof(fact_1040_divides__mul__r,axiom,
! [C,A,B_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A),B_1))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,times_times_nat(A,C)),times_times_nat(B_1,C))) ) ).
fof(fact_1041_divides__mul__l,axiom,
! [C,A,B_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A),B_1))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,times_times_nat(C,A)),times_times_nat(C,B_1))) ) ).
fof(fact_1042_dvd__diff__nat,axiom,
! [N,K_1,M] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K_1),M))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K_1),N))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K_1),minus_minus_nat(M,N))) ) ) ).
fof(fact_1043_zcong__eq__zdvd__prop,axiom,
! [Xa,P_3] :
( hBOOL(hAPP_int_bool(zcong(Xa,zero_zero_int),P_3))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,P_3),Xa)) ) ).
fof(fact_1044_zcong__zero__equiv__div,axiom,
! [A_1,Ma] :
( hBOOL(hAPP_int_bool(zcong(A_1,zero_zero_int),Ma))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,Ma),A_1)) ) ).
fof(fact_1045_nat__dvd__not__less,axiom,
! [N,M] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),M))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N))
=> ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,N),M)) ) ) ).
fof(fact_1046_divides__ge,axiom,
! [A,B_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A),B_1))
=> ( B_1 = zero_zero_nat
| hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A),B_1)) ) ) ).
fof(fact_1047_nat__mult__dvd__cancel__disj_H,axiom,
! [Ma,K,Na] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,times_times_nat(Ma,K)),times_times_nat(Na,K)))
<=> ( K = zero_zero_nat
| hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Ma),Na)) ) ) ).
fof(fact_1048_dvd__diffD,axiom,
! [K_1,M,N] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K_1),minus_minus_nat(M,N)))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K_1),N))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N),M))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K_1),M)) ) ) ) ).
fof(fact_1049_dvd__diffD1,axiom,
! [K_1,M,N] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K_1),minus_minus_nat(M,N)))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K_1),M))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N),M))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K_1),N)) ) ) ) ).
fof(fact_1050_divides__rev,axiom,
! [A,N,B_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(power_power_nat(A),N)),hAPP_nat_nat(power_power_nat(B_1),N)))
=> ( N != zero_zero_nat
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A),B_1)) ) ) ).
fof(fact_1051_divides__exp2,axiom,
! [X,Y,N] :
( N != zero_zero_nat
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(power_power_nat(X),N)),Y))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),Y)) ) ) ).
fof(fact_1052_Nat__Transfer_Otransfer__int__nat__relations_I4_J,axiom,
! [Xa,Ya] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_nat_int(semiri1621563631at_int,Xa)),hAPP_nat_int(semiri1621563631at_int,Ya)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Xa),Ya)) ) ).
fof(fact_1053_zdvd__int,axiom,
! [Xa,Ya] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Xa),Ya))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_nat_int(semiri1621563631at_int,Xa)),hAPP_nat_int(semiri1621563631at_int,Ya))) ) ).
fof(fact_1054_divides__exp,axiom,
! [N,X,Y] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),Y))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(power_power_nat(X),N)),hAPP_nat_nat(power_power_nat(Y),N))) ) ).
fof(fact_1055_nat__dvd__1__iff__1,axiom,
! [Ma] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Ma),one_one_nat))
<=> Ma = one_one_nat ) ).
fof(fact_1056_dvd__reduce,axiom,
! [K,Na] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K),plus_plus_nat(Na,K)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K),Na)) ) ).
fof(fact_1057_divides__add__revr,axiom,
! [B_1,D_1,A] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,D_1),A))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,D_1),plus_plus_nat(A,B_1)))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,D_1),B_1)) ) ) ).
fof(fact_1058_zcong__id,axiom,
! [M] : hBOOL(hAPP_int_bool(zcong(M,zero_zero_int),M)) ).
fof(fact_1059_zcong__shift,axiom,
! [C,A,B_1,M] :
( hBOOL(hAPP_int_bool(zcong(A,B_1),M))
=> hBOOL(hAPP_int_bool(zcong(plus_plus_int(A,C),plus_plus_int(B_1,C)),M)) ) ).
fof(fact_1060_zcong__eq__trans,axiom,
! [D_1,C,A,B_1,M] :
( hBOOL(hAPP_int_bool(zcong(A,B_1),M))
=> ( B_1 = C
=> ( hBOOL(hAPP_int_bool(zcong(C,D_1),M))
=> hBOOL(hAPP_int_bool(zcong(A,D_1),M)) ) ) ) ).
fof(fact_1061_zcong__zpower,axiom,
! [Z,X,Y,M] :
( hBOOL(hAPP_int_bool(zcong(X,Y),M))
=> hBOOL(hAPP_int_bool(zcong(hAPP_nat_int(power_power_int(X),Z),hAPP_nat_int(power_power_int(Y),Z)),M)) ) ).
fof(fact_1062_zcong__less__eq,axiom,
! [M,Y,X] :
( ( is_int(Y)
& is_int(X) )
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),X))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),Y))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),M))
=> ( hBOOL(hAPP_int_bool(zcong(X,Y),M))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X),M))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Y),M))
=> X = Y ) ) ) ) ) ) ) ).
fof(fact_1063_zcong__not__zero,axiom,
! [M,X] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),X))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X),M))
=> ~ hBOOL(hAPP_int_bool(zcong(X,zero_zero_int),M)) ) ) ).
fof(fact_1064_zcong__zmult__prop1,axiom,
! [C_1,D,A_1,B,Ma] :
( hBOOL(hAPP_int_bool(zcong(A_1,B),Ma))
=> ( hBOOL(hAPP_int_bool(zcong(C_1,times_times_int(A_1,D)),Ma))
<=> hBOOL(hAPP_int_bool(zcong(C_1,times_times_int(B,D)),Ma)) ) ) ).
fof(fact_1065_zcong__zmult__prop2,axiom,
! [C_1,D,A_1,B,Ma] :
( hBOOL(hAPP_int_bool(zcong(A_1,B),Ma))
=> ( hBOOL(hAPP_int_bool(zcong(C_1,times_times_int(D,A_1)),Ma))
<=> hBOOL(hAPP_int_bool(zcong(C_1,times_times_int(D,B)),Ma)) ) ) ).
fof(fact_1066_Int2_Ozcong__zero,axiom,
! [M,X] :
( is_int(X)
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X),M))
=> ( hBOOL(hAPP_int_bool(zcong(X,zero_zero_int),M))
=> X = zero_zero_int ) ) ) ) ).
fof(fact_1067_zcong__zmult__prop3,axiom,
! [Y,X,P_2] :
( hBOOL(hAPP_int_bool(zprime,P_2))
=> ( ~ hBOOL(hAPP_int_bool(zcong(X,zero_zero_int),P_2))
=> ( ~ hBOOL(hAPP_int_bool(zcong(Y,zero_zero_int),P_2))
=> ~ hBOOL(hAPP_int_bool(zcong(times_times_int(X,Y),zero_zero_int),P_2)) ) ) ) ).
fof(fact_1068_dvd__imp__le,axiom,
! [K_1,N] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K_1),N))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_1),N)) ) ) ).
fof(fact_1069_dvd__mult__cancel,axiom,
! [K_1,M,N] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,times_times_nat(K_1,M)),times_times_nat(K_1,N)))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),K_1))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,M),N)) ) ) ).
fof(fact_1070_zcong__zprime__prod__zero,axiom,
! [B_1,A,P_2] :
( hBOOL(hAPP_int_bool(zprime,P_2))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A))
=> ( hBOOL(hAPP_int_bool(zcong(times_times_int(A,B_1),zero_zero_int),P_2))
=> ( hBOOL(hAPP_int_bool(zcong(A,zero_zero_int),P_2))
| hBOOL(hAPP_int_bool(zcong(B_1,zero_zero_int),P_2)) ) ) ) ) ).
fof(fact_1071_zcong__zprime__prod__zero__contra,axiom,
! [B_1,A,P_2] :
( hBOOL(hAPP_int_bool(zprime,P_2))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A))
=> ( ( ~ hBOOL(hAPP_int_bool(zcong(A,zero_zero_int),P_2))
& ~ hBOOL(hAPP_int_bool(zcong(B_1,zero_zero_int),P_2)) )
=> ~ hBOOL(hAPP_int_bool(zcong(times_times_int(A,B_1),zero_zero_int),P_2)) ) ) ) ).
fof(fact_1072_divides__div__not,axiom,
! [X,Q,N,R_1] :
( X = plus_plus_nat(times_times_nat(Q,N),R_1)
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),R_1))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,R_1),N))
=> ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,N),X)) ) ) ) ).
fof(fact_1073_dvd__mult__cancel1,axiom,
! [Na,Ma] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),Ma))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,times_times_nat(Ma,Na)),Ma))
<=> Na = one_one_nat ) ) ).
fof(fact_1074_dvd__mult__cancel2,axiom,
! [Na,Ma] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),Ma))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,times_times_nat(Na,Ma)),Ma))
<=> Na = one_one_nat ) ) ).
fof(fact_1075_transfer__nat__int__relations_I4_J,axiom,
! [Ya,Xa] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Xa))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Ya))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,nat(Xa)),nat(Ya)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,Xa),Ya)) ) ) ) ).
fof(fact_1076_power__dvd__imp__le,axiom,
! [I_2,M,N] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(power_power_nat(I_2),M)),hAPP_nat_nat(power_power_nat(I_2),N)))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_nat),I_2))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N)) ) ) ).
fof(fact_1077_nat__dvd__iff,axiom,
! [Z_1,Ma] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,nat(Z_1)),Ma))
<=> ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Z_1))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,Z_1),hAPP_nat_int(semiri1621563631at_int,Ma))) )
& ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Z_1))
=> Ma = zero_zero_nat ) ) ) ).
fof(fact_1078_zcong__neg__1__impl__ne__1,axiom,
! [X,P_2] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,number_number_of_int(bit0(bit1(pls)))),P_2))
=> ( hBOOL(hAPP_int_bool(zcong(X,number_number_of_int(min)),P_2))
=> ~ hBOOL(hAPP_int_bool(zcong(X,one_one_int),P_2)) ) ) ).
fof(fact_1079_Euler_Oaux____1,axiom,
! [Y,X,P_2] :
( ~ hBOOL(hAPP_int_bool(zcong(X,zero_zero_int),P_2))
=> ( hBOOL(hAPP_int_bool(zcong(hAPP_nat_int(power_power_int(Y),number_number_of_nat(bit0(bit1(pls)))),X),P_2))
=> ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,P_2),Y)) ) ) ).
fof(fact_1080_divides__cases,axiom,
! [N,M] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,N),M))
=> ( M = zero_zero_nat
| M = N
| hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,times_times_nat(number_number_of_nat(bit0(bit1(pls))),N)),M)) ) ) ).
fof(fact_1081_Legendre__1mod4,axiom,
! [M] :
( hBOOL(hAPP_int_bool(zprime,plus_plus_int(times_times_int(number_number_of_int(bit0(bit0(bit1(pls)))),M),one_one_int)))
=> legendre(number_number_of_int(min),plus_plus_int(times_times_int(number_number_of_int(bit0(bit0(bit1(pls)))),M),one_one_int)) = one_one_int ) ).
fof(fact_1082__096_091s_____A_094_A2_A_061_As1_A_094_A2_093_A_Imod_A4_A_K_Am_A_L_A1_,axiom,
hBOOL(hAPP_int_bool(zcong(hAPP_nat_int(power_power_int(s),number_number_of_nat(bit0(bit1(pls)))),hAPP_nat_int(power_power_int(s1),number_number_of_nat(bit0(bit1(pls))))),plus_plus_int(times_times_int(number_number_of_int(bit0(bit0(bit1(pls)))),m),one_one_int))) ).
fof(fact_1083__096EX_B_As_O_A0_A_060_061_As_A_G_As_A_060_A4_A_K_Am_A_L_A1_A_G_A_091s,axiom,
? [X_1] :
( is_int(X_1)
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_1))
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_1),plus_plus_int(times_times_int(number_number_of_int(bit0(bit0(bit1(pls)))),m),one_one_int)))
& hBOOL(hAPP_int_bool(zcong(s1,X_1),plus_plus_int(times_times_int(number_number_of_int(bit0(bit0(bit1(pls)))),m),one_one_int)))
& ! [Y_1] :
( is_int(Y_1)
=> ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Y_1))
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Y_1),plus_plus_int(times_times_int(number_number_of_int(bit0(bit0(bit1(pls)))),m),one_one_int)))
& hBOOL(hAPP_int_bool(zcong(s1,Y_1),plus_plus_int(times_times_int(number_number_of_int(bit0(bit0(bit1(pls)))),m),one_one_int))) )
=> Y_1 = X_1 ) ) ) ).
fof(fact_1084_dvd_Oorder__refl,axiom,
! [X] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),X)) ).
fof(fact_1085__096_B_Bthesis_O_A_I_B_Bs_O_A0_A_060_061_As_A_G_As_A_060_A4_A_K_Am_A_L,axiom,
~ ! [S] :
( is_int(S)
=> ~ ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),S))
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,S),plus_plus_int(times_times_int(number_number_of_int(bit0(bit0(bit1(pls)))),m),one_one_int)))
& hBOOL(hAPP_int_bool(zcong(s1,S),plus_plus_int(times_times_int(number_number_of_int(bit0(bit0(bit1(pls)))),m),one_one_int))) ) ) ).
fof(fact_1086_dvd_Oless__asym,axiom,
! [X,Y] :
( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),Y))
& ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y),X)) )
=> ~ ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y),X))
& ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),Y)) ) ) ).
fof(fact_1087_dvd_Oless__trans,axiom,
! [Z,X,Y] :
( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),Y))
& ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y),X)) )
=> ( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y),Z))
& ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Z),Y)) )
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),Z))
& ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Z),X)) ) ) ) ).
fof(fact_1088_dvd_Oless__asym_H,axiom,
! [A,B_1] :
( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A),B_1))
& ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,B_1),A)) )
=> ~ ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,B_1),A))
& ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A),B_1)) ) ) ).
fof(fact_1089_dvd_Oless__le__trans,axiom,
! [Z,X,Y] :
( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),Y))
& ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y),X)) )
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y),Z))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),Z))
& ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Z),X)) ) ) ) ).
fof(fact_1090_dvd_Oord__less__eq__trans,axiom,
! [C,A,B_1] :
( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A),B_1))
& ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,B_1),A)) )
=> ( B_1 = C
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A),C))
& ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,C),A)) ) ) ) ).
fof(fact_1091_dvd_Oless__imp__triv,axiom,
! [P_1,Xa,Ya] :
( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Xa),Ya))
& ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Ya),Xa)) )
=> ( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Ya),Xa))
& ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Xa),Ya)) )
=> hBOOL(P_1) ) ) ).
fof(fact_1092_dvd_Oless__imp__not__eq2,axiom,
! [X,Y] :
( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),Y))
& ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y),X)) )
=> Y != X ) ).
fof(fact_1093_dvd_Oless__imp__not__eq,axiom,
! [X,Y] :
( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),Y))
& ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y),X)) )
=> X != Y ) ).
fof(fact_1094_dvd_Oless__imp__not__less,axiom,
! [X,Y] :
( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),Y))
& ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y),X)) )
=> ~ ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y),X))
& ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),Y)) ) ) ).
fof(fact_1095_dvd_Oless__imp__le,axiom,
! [X,Y] :
( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),Y))
& ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y),X)) )
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),Y)) ) ).
fof(fact_1096_dvd_Oless__not__sym,axiom,
! [X,Y] :
( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),Y))
& ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y),X)) )
=> ~ ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y),X))
& ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),Y)) ) ) ).
fof(fact_1097_dvd_Oless__imp__neq,axiom,
! [X,Y] :
( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),Y))
& ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y),X)) )
=> X != Y ) ).
fof(fact_1098_dvd_Ole__less__trans,axiom,
! [Z,X,Y] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),Y))
=> ( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y),Z))
& ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Z),Y)) )
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),Z))
& ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Z),X)) ) ) ) ).
fof(fact_1099_dvd_Oord__eq__less__trans,axiom,
! [C,A,B_1] :
( A = B_1
=> ( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,B_1),C))
& ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,C),B_1)) )
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A),C))
& ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,C),A)) ) ) ) ).
fof(fact_1100_dvd_Oorder__trans,axiom,
! [Z,X,Y] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),Y))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y),Z))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),Z)) ) ) ).
fof(fact_1101_dvd_Oantisym,axiom,
! [X,Y] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),Y))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y),X))
=> X = Y ) ) ).
fof(fact_1102_dvd__antisym,axiom,
! [M,N] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,M),N))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,N),M))
=> M = N ) ) ).
fof(fact_1103_dvd_Oord__le__eq__trans,axiom,
! [C,A,B_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A),B_1))
=> ( B_1 = C
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A),C)) ) ) ).
fof(fact_1104_dvd_Oord__eq__le__trans,axiom,
! [C,A,B_1] :
( A = B_1
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,B_1),C))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A),C)) ) ) ).
fof(fact_1105_dvd_Ole__neq__trans,axiom,
! [A,B_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A),B_1))
=> ( A != B_1
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A),B_1))
& ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,B_1),A)) ) ) ) ).
fof(fact_1106_dvd_Ole__imp__less__or__eq,axiom,
! [X,Y] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),Y))
=> ( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),Y))
& ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y),X)) )
| X = Y ) ) ).
fof(fact_1107_dvd_Oantisym__conv,axiom,
! [Ya,Xa] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Ya),Xa))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Xa),Ya))
<=> Xa = Ya ) ) ).
fof(fact_1108_dvd_Oeq__refl,axiom,
! [X,Y] :
( X = Y
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),Y)) ) ).
fof(fact_1109_dvd_Oneq__le__trans,axiom,
! [A,B_1] :
( A != B_1
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A),B_1))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A),B_1))
& ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,B_1),A)) ) ) ) ).
fof(fact_1110_dvd_Oless__le__not__le,axiom,
! [Xa,Ya] :
( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Xa),Ya))
& ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Ya),Xa)) )
<=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Xa),Ya))
& ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Ya),Xa)) ) ) ).
fof(fact_1111_dvd_Oless__le,axiom,
! [Xa,Ya] :
( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Xa),Ya))
& ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Ya),Xa)) )
<=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Xa),Ya))
& Xa != Ya ) ) ).
fof(fact_1112_dvd_Ole__less,axiom,
! [Xa,Ya] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Xa),Ya))
<=> ( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Xa),Ya))
& ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Ya),Xa)) )
| Xa = Ya ) ) ).
fof(fact_1113_dvd_Oeq__iff,axiom,
! [Xa,Ya] :
( Xa = Ya
<=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Xa),Ya))
& hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Ya),Xa)) ) ) ).
fof(fact_1114_dvd_Oless__irrefl,axiom,
! [X] :
~ ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),X))
& ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),X)) ) ).
fof(fact_1115_divides__antisym,axiom,
! [Xa,Ya] :
( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Xa),Ya))
& hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Ya),Xa)) )
<=> Xa = Ya ) ).
fof(fact_1116_neg__one__power__eq__mod__m,axiom,
! [J_1,K_1,M] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,number_number_of_int(bit0(bit1(pls)))),M))
=> ( hBOOL(hAPP_int_bool(zcong(hAPP_nat_int(power_power_int(number_number_of_int(min)),J_1),hAPP_nat_int(power_power_int(number_number_of_int(min)),K_1)),M))
=> hAPP_nat_int(power_power_int(number_number_of_int(min)),J_1) = hAPP_nat_int(power_power_int(number_number_of_int(min)),K_1) ) ) ).
fof(fact_1117_one__not__neg__one__mod__m,axiom,
! [M] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,number_number_of_int(bit0(bit1(pls)))),M))
=> ~ hBOOL(hAPP_int_bool(zcong(one_one_int,number_number_of_int(min)),M)) ) ).
fof(fact_1118_neg__one__power,axiom,
! [N] :
( hAPP_nat_int(power_power_int(number_number_of_int(min)),N) = one_one_int
| hAPP_nat_int(power_power_int(number_number_of_int(min)),N) = number_number_of_int(min) ) ).
fof(fact_1119_Legendre__def,axiom,
! [A,P_2] :
( ( hBOOL(hAPP_int_bool(zcong(A,zero_zero_int),P_2))
=> legendre(A,P_2) = zero_zero_int )
& ( ~ hBOOL(hAPP_int_bool(zcong(A,zero_zero_int),P_2))
=> ( ( hBOOL(hAPP_int_bool(quadRes(P_2),A))
=> legendre(A,P_2) = one_one_int )
& ( ~ hBOOL(hAPP_int_bool(quadRes(P_2),A))
=> legendre(A,P_2) = number_number_of_int(min) ) ) ) ) ).
fof(fact_1120_QuadRes__def,axiom,
! [Ma,Xa] :
( hBOOL(hAPP_int_bool(quadRes(Ma),Xa))
<=> ? [Y_1] :
( is_int(Y_1)
& hBOOL(hAPP_int_bool(zcong(hAPP_nat_int(power_power_int(Y_1),number_number_of_nat(bit0(bit1(pls)))),Xa),Ma)) ) ) ).
fof(fact_1121_Little__Fermat,axiom,
! [X,P_2] :
( hBOOL(hAPP_int_bool(zprime,P_2))
=> ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,P_2),X))
=> hBOOL(hAPP_int_bool(zcong(hAPP_nat_int(power_power_int(X),nat(minus_minus_int(P_2,one_one_int))),one_one_int),P_2)) ) ) ).
fof(fact_1122_zcong__square__zless,axiom,
! [A,P_2] :
( is_int(A)
=> ( hBOOL(hAPP_int_bool(zprime,P_2))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A),P_2))
=> ( hBOOL(hAPP_int_bool(zcong(times_times_int(A,A),one_one_int),P_2))
=> ( A = one_one_int
| A = minus_minus_int(P_2,one_one_int) ) ) ) ) ) ) ).
fof(fact_1123_zcong__trans,axiom,
! [C,A,B_1,M] :
( hBOOL(hAPP_int_bool(zcong(A,B_1),M))
=> ( hBOOL(hAPP_int_bool(zcong(B_1,C),M))
=> hBOOL(hAPP_int_bool(zcong(A,C),M)) ) ) ).
fof(fact_1124_zcong__refl,axiom,
! [K_1,M] : hBOOL(hAPP_int_bool(zcong(K_1,K_1),M)) ).
fof(fact_1125_zcong__sym,axiom,
! [A_1,B,Ma] :
( hBOOL(hAPP_int_bool(zcong(A_1,B),Ma))
<=> hBOOL(hAPP_int_bool(zcong(B,A_1),Ma)) ) ).
fof(fact_1126_IntPrimes_Ozcong__zero,axiom,
! [A_1,B] :
( ( is_int(A_1)
& is_int(B) )
=> ( hBOOL(hAPP_int_bool(zcong(A_1,B),zero_zero_int))
<=> A_1 = B ) ) ).
fof(fact_1127_zcong__1,axiom,
! [A,B_1] : hBOOL(hAPP_int_bool(zcong(A,B_1),one_one_int)) ).
fof(fact_1128_zcong__zmult__self,axiom,
! [A,M,B_1] : hBOOL(hAPP_int_bool(zcong(times_times_int(A,M),times_times_int(B_1,M)),M)) ).
fof(fact_1129_zcong__scalar,axiom,
! [K_1,A,B_1,M] :
( hBOOL(hAPP_int_bool(zcong(A,B_1),M))
=> hBOOL(hAPP_int_bool(zcong(times_times_int(A,K_1),times_times_int(B_1,K_1)),M)) ) ).
fof(fact_1130_zcong__scalar2,axiom,
! [K_1,A,B_1,M] :
( hBOOL(hAPP_int_bool(zcong(A,B_1),M))
=> hBOOL(hAPP_int_bool(zcong(times_times_int(K_1,A),times_times_int(K_1,B_1)),M)) ) ).
fof(fact_1131_zcong__zmult,axiom,
! [C,D_1,A,B_1,M] :
( hBOOL(hAPP_int_bool(zcong(A,B_1),M))
=> ( hBOOL(hAPP_int_bool(zcong(C,D_1),M))
=> hBOOL(hAPP_int_bool(zcong(times_times_int(A,C),times_times_int(B_1,D_1)),M)) ) ) ).
fof(fact_1132_zcong__zadd,axiom,
! [C,D_1,A,B_1,M] :
( hBOOL(hAPP_int_bool(zcong(A,B_1),M))
=> ( hBOOL(hAPP_int_bool(zcong(C,D_1),M))
=> hBOOL(hAPP_int_bool(zcong(plus_plus_int(A,C),plus_plus_int(B_1,D_1)),M)) ) ) ).
fof(fact_1133_zcong__zdiff,axiom,
! [C,D_1,A,B_1,M] :
( hBOOL(hAPP_int_bool(zcong(A,B_1),M))
=> ( hBOOL(hAPP_int_bool(zcong(C,D_1),M))
=> hBOOL(hAPP_int_bool(zcong(minus_minus_int(A,C),minus_minus_int(B_1,D_1)),M)) ) ) ).
fof(fact_1134_zcong__not,axiom,
! [B_1,M,A] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A),M))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_1))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_1),A))
=> ~ hBOOL(hAPP_int_bool(zcong(A,B_1),M)) ) ) ) ) ).
fof(fact_1135_xzgcda__linear__aux1,axiom,
! [A,R_1,B_1,M,C,D_1,N] : plus_plus_int(times_times_int(minus_minus_int(A,times_times_int(R_1,B_1)),M),times_times_int(minus_minus_int(C,times_times_int(R_1,D_1)),N)) = minus_minus_int(plus_plus_int(times_times_int(A,M),times_times_int(C,N)),times_times_int(R_1,plus_plus_int(times_times_int(B_1,M),times_times_int(D_1,N)))) ).
fof(fact_1136_zcong__iff__lin,axiom,
! [A_1,B,Ma] :
( is_int(B)
=> ( hBOOL(hAPP_int_bool(zcong(A_1,B),Ma))
<=> ? [K_2] :
( is_int(K_2)
& B = plus_plus_int(A_1,times_times_int(Ma,K_2)) ) ) ) ).
fof(fact_1137_zcong__def,axiom,
! [A_1,B,Ma] :
( hBOOL(hAPP_int_bool(zcong(A_1,B),Ma))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,Ma),minus_minus_int(A_1,B))) ) ).
fof(fact_1138_norR__mem__unique__aux,axiom,
! [A,B_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A),minus_minus_int(B_1,one_one_int)))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A),B_1)) ) ).
fof(fact_1139_zcong__zless__0,axiom,
! [M,A] :
( is_int(A)
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A),M))
=> ( hBOOL(hAPP_int_bool(zcong(A,zero_zero_int),M))
=> A = zero_zero_int ) ) ) ) ).
fof(fact_1140_zcong__zless__imp__eq,axiom,
! [B_1,M,A] :
( ( is_int(B_1)
& is_int(A) )
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A),M))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),B_1))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_1),M))
=> ( hBOOL(hAPP_int_bool(zcong(A,B_1),M))
=> A = B_1 ) ) ) ) ) ) ).
fof(fact_1141_zprime__zdvd__zmult,axiom,
! [N,P_2,M] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),M))
=> ( hBOOL(hAPP_int_bool(zprime,P_2))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,P_2),times_times_int(M,N)))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,P_2),M))
| hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,P_2),N)) ) ) ) ) ).
fof(fact_1142_zprime__def,axiom,
! [P_3] :
( is_int(P_3)
=> ( hBOOL(hAPP_int_bool(zprime,P_3))
<=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),P_3))
& ! [M_1] :
( is_int(M_1)
=> ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),M_1))
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,M_1),P_3)) )
=> ( M_1 = one_one_int
| M_1 = P_3 ) ) ) ) ) ) ).
fof(fact_1143_zcong__square,axiom,
! [A,P_2] :
( hBOOL(hAPP_int_bool(zprime,P_2))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A))
=> ( hBOOL(hAPP_int_bool(zcong(times_times_int(A,A),one_one_int),P_2))
=> ( hBOOL(hAPP_int_bool(zcong(A,one_one_int),P_2))
| hBOOL(hAPP_int_bool(zcong(A,minus_minus_int(P_2,one_one_int)),P_2)) ) ) ) ) ).
fof(fact_1144_zdiff__int__split,axiom,
! [P_1,Xa,Ya] :
( hBOOL(hAPP_int_bool(P_1,hAPP_nat_int(semiri1621563631at_int,minus_minus_nat(Xa,Ya))))
<=> ( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Ya),Xa))
=> hBOOL(hAPP_int_bool(P_1,minus_minus_int(hAPP_nat_int(semiri1621563631at_int,Xa),hAPP_nat_int(semiri1621563631at_int,Ya)))) )
& ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Xa),Ya))
=> hBOOL(hAPP_int_bool(P_1,zero_zero_int)) ) ) ) ).
fof(fact_1145_prime__g__5,axiom,
! [P_2] :
( is_int(P_2)
=> ( hBOOL(hAPP_int_bool(zprime,P_2))
=> ( P_2 != number_number_of_int(bit0(bit1(pls)))
=> ( P_2 != number_number_of_int(bit1(bit1(pls)))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,number_number_of_int(bit1(bit0(bit1(pls))))),P_2)) ) ) ) ) ).
fof(fact_1146_conj__le__cong,axiom,
! [P_4,P_1,Xa] :
( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Xa))
=> ( hBOOL(P_1)
<=> hBOOL(P_4) ) )
=> ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Xa))
& hBOOL(P_1) )
<=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Xa))
& hBOOL(P_4) ) ) ) ).
fof(fact_1147_imp__le__cong,axiom,
! [P_4,P_1,Xa] :
( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Xa))
=> ( hBOOL(P_1)
<=> hBOOL(P_4) ) )
=> ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Xa))
=> hBOOL(P_1) )
<=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Xa))
=> hBOOL(P_4) ) ) ) ).
fof(fact_1148_zdvd__mono,axiom,
! [Ma,Ta,K] :
( is_int(K)
=> ( K != zero_zero_int
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,Ma),Ta))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,times_times_int(K,Ma)),times_times_int(K,Ta))) ) ) ) ).
fof(fact_1149_number__of2,axiom,
hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),number_number_of_int(pls))) ).
fof(fact_1150_inv__not__p__minus__1__aux,axiom,
! [A_1,P_3] :
( hBOOL(hAPP_int_bool(zcong(times_times_int(A_1,minus_minus_int(P_3,one_one_int)),one_one_int),P_3))
<=> hBOOL(hAPP_int_bool(zcong(A_1,minus_minus_int(P_3,one_one_int)),P_3)) ) ).
fof(fact_1151_zcong__zpower__zmult,axiom,
! [Z,X,Y,P_2] :
( hBOOL(hAPP_int_bool(zcong(hAPP_nat_int(power_power_int(X),Y),one_one_int),P_2))
=> hBOOL(hAPP_int_bool(zcong(hAPP_nat_int(power_power_int(X),times_times_nat(Y,Z)),one_one_int),P_2)) ) ).
fof(fact_1152_number__of1,axiom,
! [N] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),number_number_of_int(N)))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),number_number_of_int(bit0(N))))
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),number_number_of_int(bit1(N)))) ) ) ).
fof(fact_1153__096_B_Bthesis_O_A_I_B_Bs_Aw_O_Aw_A_061_Ay_A_N_As_A_K_A_I1_A_L_Aint_An,axiom,
~ ! [S,W_1] :
( ( is_int(S)
& is_int(W_1) )
=> ~ ( W_1 = minus_minus_int(y,times_times_int(S,plus_plus_int(one_one_int,hAPP_nat_int(semiri1621563631at_int,n))))
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,times_times_int(number_number_of_int(bit0(bit1(pls))),abs_abs_int(W_1))),plus_plus_int(one_one_int,hAPP_nat_int(semiri1621563631at_int,n)))) ) ) ).
fof(fact_1154__096_B_Bthesis_O_A_I_B_Br_Av_O_Av_A_061_Ax_A_N_Ar_A_K_A_I1_A_L_Aint_An,axiom,
~ ! [R,V] :
( ( is_int(R)
& is_int(V) )
=> ~ ( V = minus_minus_int(x,times_times_int(R,plus_plus_int(one_one_int,hAPP_nat_int(semiri1621563631at_int,n))))
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,times_times_int(number_number_of_int(bit0(bit1(pls))),abs_abs_int(V))),plus_plus_int(one_one_int,hAPP_nat_int(semiri1621563631at_int,n)))) ) ) ).
fof(fact_1155_dvd__imp__le__int,axiom,
! [D_1,I_2] :
( is_int(I_2)
=> ( I_2 != zero_zero_int
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,D_1),I_2))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,abs_abs_int(D_1)),abs_abs_int(I_2))) ) ) ) ).
fof(fact_1156_nat__abs__mult__distrib,axiom,
! [W,Z] : nat(abs_abs_int(times_times_int(W,Z))) = times_times_nat(nat(abs_abs_int(W)),nat(abs_abs_int(Z))) ).
fof(fact_1157_zdvd__antisym__abs,axiom,
! [A,B_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A),B_1))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,B_1),A))
=> abs_abs_int(A) = abs_abs_int(B_1) ) ) ).
fof(fact_1158_zdvd1__eq,axiom,
! [Xa] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,Xa),one_one_int))
<=> abs_abs_int(Xa) = one_one_int ) ).
fof(fact_1159_abs__int__eq,axiom,
! [M] : abs_abs_int(hAPP_nat_int(semiri1621563631at_int,M)) = hAPP_nat_int(semiri1621563631at_int,M) ).
fof(fact_1160_abs__zmult__eq__1,axiom,
! [M,N] :
( abs_abs_int(times_times_int(M,N)) = one_one_int
=> abs_abs_int(M) = one_one_int ) ).
fof(fact_1161_zero__le__zpower__abs,axiom,
! [X,N] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_nat_int(power_power_int(abs_abs_int(X)),N))) ).
fof(fact_1162_zabs__less__one__iff,axiom,
! [Z_1] :
( is_int(Z_1)
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,abs_abs_int(Z_1)),one_one_int))
<=> Z_1 = zero_zero_int ) ) ).
fof(fact_1163_abs__eq__1__iff,axiom,
! [Z_1] :
( is_int(Z_1)
=> ( abs_abs_int(Z_1) = one_one_int
<=> ( Z_1 = one_one_int
| Z_1 = number_number_of_int(min) ) ) ) ).
fof(fact_1164_abs__power3__distrib,axiom,
! [X] : abs_abs_int(hAPP_nat_int(power_power_int(X),number_number_of_nat(bit1(bit1(pls))))) = hAPP_nat_int(power_power_int(abs_abs_int(X)),number_number_of_nat(bit1(bit1(pls)))) ).
fof(fact_1165_zero__less__zpower__abs__iff,axiom,
! [Xa,Na] :
( is_int(Xa)
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_nat_int(power_power_int(abs_abs_int(Xa)),Na)))
<=> ( Xa != zero_zero_int
| Na = zero_zero_nat ) ) ) ).
fof(fact_1166_zdvd__mult__cancel1,axiom,
! [Na,Ma] :
( is_int(Ma)
=> ( Ma != zero_zero_int
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,times_times_int(Ma,Na)),Ma))
<=> abs_abs_int(Na) = one_one_int ) ) ) ).
fof(fact_1167_int__dvd__iff,axiom,
! [Ma,Z_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_nat_int(semiri1621563631at_int,Ma)),Z_1))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Ma),nat(abs_abs_int(Z_1)))) ) ).
fof(fact_1168_dvd__int__iff,axiom,
! [Z_1,Ma] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,Z_1),hAPP_nat_int(semiri1621563631at_int,Ma)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,nat(abs_abs_int(Z_1))),Ma)) ) ).
fof(fact_1169_abs__power2__distrib,axiom,
! [A] : abs_abs_int(hAPP_nat_int(power_power_int(A),number_number_of_nat(bit0(bit1(pls))))) = hAPP_nat_int(power_power_int(abs_abs_int(A)),number_number_of_nat(bit0(bit1(pls)))) ).
fof(fact_1170_power2__eq__iff__abs__eq,axiom,
! [A_1,B] :
( hAPP_nat_int(power_power_int(A_1),number_number_of_nat(bit0(bit1(pls)))) = hAPP_nat_int(power_power_int(B),number_number_of_nat(bit0(bit1(pls))))
<=> abs_abs_int(A_1) = abs_abs_int(B) ) ).
fof(fact_1171_power2__eq1__iff,axiom,
! [A] :
( hAPP_nat_int(power_power_int(A),number_number_of_nat(bit0(bit1(pls)))) = one_one_int
=> abs_abs_int(A) = one_one_int ) ).
fof(fact_1172_incr__lemma,axiom,
! [Z,X,D_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),D_1))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Z),plus_plus_int(X,times_times_int(plus_plus_int(abs_abs_int(minus_minus_int(X,Z)),one_one_int),D_1)))) ) ).
fof(fact_1173_decr__lemma,axiom,
! [X,Z,D_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),D_1))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,minus_minus_int(X,times_times_int(plus_plus_int(abs_abs_int(minus_minus_int(X,Z)),one_one_int),D_1))),Z)) ) ).
fof(fact_1174_best__division__abs,axiom,
! [Y,X] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),X))
=> ? [N_1] :
( is_int(N_1)
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,times_times_int(number_number_of_int(bit0(bit1(pls))),abs_abs_int(minus_minus_int(Y,times_times_int(N_1,X))))),X)) ) ) ).
fof(fact_1175_decr__mult__lemma,axiom,
! [K,P_1,D] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),D))
=> ( ! [X_1] :
( is_int(X_1)
=> ( hBOOL(hAPP_int_bool(P_1,X_1))
=> hBOOL(hAPP_int_bool(P_1,minus_minus_int(X_1,D))) ) )
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),K))
=> ! [X_1] :
( hBOOL(hAPP_int_bool(P_1,X_1))
=> hBOOL(hAPP_int_bool(P_1,minus_minus_int(X_1,times_times_int(K,D)))) ) ) ) ) ).
fof(fact_1176_abs__add__one__not__less__self,axiom,
! [X] : ~ hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,plus_plus_real(abs_abs_real(X),one_one_real)),X)) ).
fof(fact_1177_abs__add__one__gt__zero,axiom,
! [X] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),plus_plus_real(one_one_real,abs_abs_real(X)))) ).
fof(fact_1178_less__one__imp__sqr__less__one,axiom,
! [X] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,abs_abs_real(X)),one_one_real))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_nat_real(power_power_real(X),number_number_of_nat(bit0(bit1(pls))))),one_one_real)) ) ).
fof(fact_1179_ex__least__nat__less,axiom,
! [Na,P_1] :
( ~ hBOOL(hAPP_nat_bool(P_1,zero_zero_nat))
=> ( hBOOL(hAPP_nat_bool(P_1,Na))
=> ? [K_2] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,K_2),Na))
& ! [I] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I),K_2))
=> ~ hBOOL(hAPP_nat_bool(P_1,I)) )
& hBOOL(hAPP_nat_bool(P_1,plus_plus_nat(K_2,one_one_nat))) ) ) ) ).
fof(fact_1180_incr__mult__lemma,axiom,
! [K,P_1,D] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),D))
=> ( ! [X_1] :
( is_int(X_1)
=> ( hBOOL(hAPP_int_bool(P_1,X_1))
=> hBOOL(hAPP_int_bool(P_1,plus_plus_int(X_1,D))) ) )
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),K))
=> ! [X_1] :
( hBOOL(hAPP_int_bool(P_1,X_1))
=> hBOOL(hAPP_int_bool(P_1,plus_plus_int(X_1,times_times_int(K,D)))) ) ) ) ) ).
fof(fact_1181_nat__less__add__iff1,axiom,
! [U,Ma,Na,J,I_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,J),I_1))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,plus_plus_nat(times_times_nat(I_1,U),Ma)),plus_plus_nat(times_times_nat(J,U),Na)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,plus_plus_nat(times_times_nat(minus_minus_nat(I_1,J),U),Ma)),Na)) ) ) ).
fof(fact_1182_nat__less__add__iff2,axiom,
! [U,Ma,Na,I_1,J] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_1),J))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,plus_plus_nat(times_times_nat(I_1,U),Ma)),plus_plus_nat(times_times_nat(J,U),Na)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),plus_plus_nat(times_times_nat(minus_minus_nat(J,I_1),U),Na))) ) ) ).
fof(fact_1183_nat__mult__eq__cancel__disj,axiom,
! [K,Ma,Na] :
( times_times_nat(K,Ma) = times_times_nat(K,Na)
<=> ( K = zero_zero_nat
| Ma = Na ) ) ).
fof(fact_1184_left__add__mult__distrib,axiom,
! [I_2,U_1,J_1,K_1] : plus_plus_nat(times_times_nat(I_2,U_1),plus_plus_nat(times_times_nat(J_1,U_1),K_1)) = plus_plus_nat(times_times_nat(plus_plus_nat(I_2,J_1),U_1),K_1) ).
fof(fact_1185_nat__mult__less__cancel1,axiom,
! [Ma,Na,K] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),K))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,times_times_nat(K,Ma)),times_times_nat(K,Na)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),Na)) ) ) ).
fof(fact_1186_nat__mult__eq__cancel1,axiom,
! [Ma,Na,K] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),K))
=> ( times_times_nat(K,Ma) = times_times_nat(K,Na)
<=> Ma = Na ) ) ).
fof(fact_1187_nat__mult__dvd__cancel__disj,axiom,
! [K,Ma,Na] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,times_times_nat(K,Ma)),times_times_nat(K,Na)))
<=> ( K = zero_zero_nat
| hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Ma),Na)) ) ) ).
fof(fact_1188_nat__mult__dvd__cancel1,axiom,
! [Ma,Na,K] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),K))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,times_times_nat(K,Ma)),times_times_nat(K,Na)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Ma),Na)) ) ) ).
fof(fact_1189_nat__mult__le__cancel1,axiom,
! [Ma,Na,K] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),K))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,times_times_nat(K,Ma)),times_times_nat(K,Na)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Ma),Na)) ) ) ).
fof(fact_1190_nat__le__add__iff1,axiom,
! [U,Ma,Na,J,I_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,J),I_1))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,plus_plus_nat(times_times_nat(I_1,U),Ma)),plus_plus_nat(times_times_nat(J,U),Na)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,plus_plus_nat(times_times_nat(minus_minus_nat(I_1,J),U),Ma)),Na)) ) ) ).
fof(fact_1191_nat__diff__add__eq1,axiom,
! [U_1,M,N,J_1,I_2] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,J_1),I_2))
=> minus_minus_nat(plus_plus_nat(times_times_nat(I_2,U_1),M),plus_plus_nat(times_times_nat(J_1,U_1),N)) = minus_minus_nat(plus_plus_nat(times_times_nat(minus_minus_nat(I_2,J_1),U_1),M),N) ) ).
fof(fact_1192_nat__eq__add__iff1,axiom,
! [U,Ma,Na,J,I_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,J),I_1))
=> ( plus_plus_nat(times_times_nat(I_1,U),Ma) = plus_plus_nat(times_times_nat(J,U),Na)
<=> plus_plus_nat(times_times_nat(minus_minus_nat(I_1,J),U),Ma) = Na ) ) ).
fof(fact_1193_nat__le__add__iff2,axiom,
! [U,Ma,Na,I_1,J] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_1),J))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,plus_plus_nat(times_times_nat(I_1,U),Ma)),plus_plus_nat(times_times_nat(J,U),Na)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Ma),plus_plus_nat(times_times_nat(minus_minus_nat(J,I_1),U),Na))) ) ) ).
fof(fact_1194_nat__diff__add__eq2,axiom,
! [U_1,M,N,I_2,J_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_2),J_1))
=> minus_minus_nat(plus_plus_nat(times_times_nat(I_2,U_1),M),plus_plus_nat(times_times_nat(J_1,U_1),N)) = minus_minus_nat(M,plus_plus_nat(times_times_nat(minus_minus_nat(J_1,I_2),U_1),N)) ) ).
fof(fact_1195_nat__eq__add__iff2,axiom,
! [U,Ma,Na,I_1,J] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_1),J))
=> ( plus_plus_nat(times_times_nat(I_1,U),Ma) = plus_plus_nat(times_times_nat(J,U),Na)
<=> Ma = plus_plus_nat(times_times_nat(minus_minus_nat(J,I_1),U),Na) ) ) ).
fof(fact_1196_nat0__intermed__int__val,axiom,
! [K,F,Na] :
( is_int(K)
=> ( ! [I] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I),Na))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,abs_abs_int(minus_minus_int(hAPP_nat_int(F,plus_plus_nat(I,one_one_nat)),hAPP_nat_int(F,I)))),one_one_int)) )
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_nat_int(F,zero_zero_nat)),K))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K),hAPP_nat_int(F,Na)))
=> ? [I] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I),Na))
& hAPP_nat_int(F,I) = K ) ) ) ) ) ).
fof(fact_1197_int__val__lemma,axiom,
! [K,F,Na] :
( is_int(K)
=> ( ! [I] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I),Na))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,abs_abs_int(minus_minus_int(hAPP_nat_int(F,plus_plus_nat(I,one_one_nat)),hAPP_nat_int(F,I)))),one_one_int)) )
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_nat_int(F,zero_zero_nat)),K))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K),hAPP_nat_int(F,Na)))
=> ? [I] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I),Na))
& hAPP_nat_int(F,I) = K ) ) ) ) ) ).
%----Helper facts (6)
fof(help_If_1_1_If_000tc__Int__Oint_T,axiom,
! [X,Y] :
( is_int(X)
=> if_int(fTrue,X,Y) = X ) ).
fof(help_If_2_1_If_000tc__Int__Oint_T,axiom,
! [X,Y] :
( is_int(Y)
=> if_int(fFalse,X,Y) = Y ) ).
fof(help_If_3_1_If_000tc__Int__Oint_T,axiom,
! [P] :
( is_bool(P)
=> ( P = fTrue
| P = fFalse ) ) ).
fof(help_If_1_1_If_000tc__Nat__Onat_T,axiom,
! [X,Y] : if_nat(fTrue,X,Y) = X ).
fof(help_If_2_1_If_000tc__Nat__Onat_T,axiom,
! [X,Y] : if_nat(fFalse,X,Y) = Y ).
fof(help_If_3_1_If_000tc__Nat__Onat_T,axiom,
! [P] :
( is_bool(P)
=> ( P = fTrue
| P = fFalse ) ) ).
%----Conjectures (1)
fof(conj_0,conjecture,
hAPP_nat_int(power_power_int(plus_plus_int(one_one_int,hAPP_nat_int(semiri1621563631at_int,n))),number_number_of_nat(bit0(bit1(pls)))) != zero_zero_int ).
%------------------------------------------------------------------------------