TPTP Problem File: NUM923+3.p
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%------------------------------------------------------------------------------
% File : NUM923+3 : TPTP v8.2.0. Released v5.3.0.
% Domain : Number Theory
% Problem : Sum of two squares line 23, 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_l23 [Bla11]
% Status : Theorem
% Rating : 1.00 v5.3.0
% Syntax : Number of formulae : 1211 ( 400 unt; 0 def)
% Number of atoms : 2667 ( 857 equ)
% Maximal formula atoms : 9 ( 2 avg)
% Number of connectives : 1702 ( 246 ~; 69 |; 116 &)
% ( 307 <=>; 964 =>; 0 <=; 0 <~>)
% Maximal formula depth : 16 ( 5 avg)
% Maximal term depth : 11 ( 2 avg)
% Number of predicates : 3 ( 2 usr; 0 prp; 1-2 aty)
% Number of functors : 75 ( 75 usr; 27 con; 0-8 aty)
% Number of variables : 2759 (2714 !; 45 ?)
% SPC : FOF_THM_RFO_SEQ
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2011-08-09 17:17:31
% : Encoded with monomorphized guards.
%------------------------------------------------------------------------------
%----Explicit typings (8)
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_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__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_c_hAPP_000tc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_000tc__HOL__Obool,axiom,
! [B_1_1,B_2_1] : is_bool(hAPP_P603027463t_bool(B_1_1,B_2_1)) ).
fof(gsy_c_hAPP_000tc__prod_Itc__Int__Oint_Mtc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_,axiom,
! [B_1_1,B_2_1] : is_bool(hAPP_P178748804t_bool(B_1_1,B_2_1)) ).
fof(gsy_c_member_000tc__RealDef__Oreal,axiom,
! [B_1_1,B_2_1] : is_bool(member_real(B_1_1,B_2_1)) ).
%----Relevant facts (1199)
fof(fact_0_xzgcda__linear__aux1,axiom,
! [A,R,B,M,C,D_1,N_1] : hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(hAPP_int_int(minus_minus_int(A),hAPP_int_int(times_times_int(R),B))),M)),hAPP_int_int(times_times_int(hAPP_int_int(minus_minus_int(C),hAPP_int_int(times_times_int(R),D_1))),N_1)) = hAPP_int_int(minus_minus_int(hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(A),M)),hAPP_int_int(times_times_int(C),N_1))),hAPP_int_int(times_times_int(R),hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(B),M)),hAPP_int_int(times_times_int(D_1),N_1)))) ).
fof(fact_1_mult__diff__mult,axiom,
! [X_46,Y_46,A_117,B_92] : hAPP_int_int(minus_minus_int(hAPP_int_int(times_times_int(X_46),Y_46)),hAPP_int_int(times_times_int(A_117),B_92)) = hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(X_46),hAPP_int_int(minus_minus_int(Y_46),B_92))),hAPP_int_int(times_times_int(hAPP_int_int(minus_minus_int(X_46),A_117)),B_92)) ).
fof(fact_2_mult__diff__mult,axiom,
! [X_46,Y_46,A_117,B_92] : hAPP_real_real(minus_minus_real(hAPP_real_real(times_times_real(X_46),Y_46)),hAPP_real_real(times_times_real(A_117),B_92)) = hAPP_real_real(plus_plus_real(hAPP_real_real(times_times_real(X_46),hAPP_real_real(minus_minus_real(Y_46),B_92))),hAPP_real_real(times_times_real(hAPP_real_real(minus_minus_real(X_46),A_117)),B_92)) ).
fof(fact_3_mult_Oprod__diff__prod,axiom,
! [X_1,Y_1,A_116,B_91] : hAPP_real_real(minus_minus_real(hAPP_real_real(times_times_real(X_1),Y_1)),hAPP_real_real(times_times_real(A_116),B_91)) = hAPP_real_real(plus_plus_real(hAPP_real_real(plus_plus_real(hAPP_real_real(times_times_real(hAPP_real_real(minus_minus_real(X_1),A_116)),hAPP_real_real(minus_minus_real(Y_1),B_91))),hAPP_real_real(times_times_real(hAPP_real_real(minus_minus_real(X_1),A_116)),B_91))),hAPP_real_real(times_times_real(A_116),hAPP_real_real(minus_minus_real(Y_1),B_91))) ).
fof(fact_4_eq__add__iff2,axiom,
! [Aa,E_1,C_1,Ba,D_2] :
( hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(Aa),E_1)),C_1) = hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(Ba),E_1)),D_2)
<=> C_1 = hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(hAPP_int_int(minus_minus_int(Ba),Aa)),E_1)),D_2) ) ).
fof(fact_5_eq__add__iff2,axiom,
! [Aa,E_1,C_1,Ba,D_2] :
( hAPP_real_real(plus_plus_real(hAPP_real_real(times_times_real(Aa),E_1)),C_1) = hAPP_real_real(plus_plus_real(hAPP_real_real(times_times_real(Ba),E_1)),D_2)
<=> C_1 = hAPP_real_real(plus_plus_real(hAPP_real_real(times_times_real(hAPP_real_real(minus_minus_real(Ba),Aa)),E_1)),D_2) ) ).
fof(fact_6_eq__add__iff1,axiom,
! [Aa,E_1,C_1,Ba,D_2] :
( hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(Aa),E_1)),C_1) = hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(Ba),E_1)),D_2)
<=> hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(hAPP_int_int(minus_minus_int(Aa),Ba)),E_1)),C_1) = D_2 ) ).
fof(fact_7_eq__add__iff1,axiom,
! [Aa,E_1,C_1,Ba,D_2] :
( hAPP_real_real(plus_plus_real(hAPP_real_real(times_times_real(Aa),E_1)),C_1) = hAPP_real_real(plus_plus_real(hAPP_real_real(times_times_real(Ba),E_1)),D_2)
<=> hAPP_real_real(plus_plus_real(hAPP_real_real(times_times_real(hAPP_real_real(minus_minus_real(Aa),Ba)),E_1)),C_1) = D_2 ) ).
fof(fact_8_is__sum2sq__def,axiom,
! [X_1] :
( hBOOL(hAPP_int_bool(twoSqu1154269391sum2sq,X_1))
<=> ? [A_49,B_65] : twoSqu1241645765sum2sq(product_Pair_int_int(A_49,B_65)) = X_1 ) ).
fof(fact_9_Int2_Oaux1,axiom,
! [A,B,C] :
( hAPP_int_int(minus_minus_int(A),B) = C
=> A = hAPP_int_int(plus_plus_int(C),B) ) ).
fof(fact_10_zdiff__zmult__distrib2,axiom,
! [W_1,Z1,Z2] : hAPP_int_int(times_times_int(W_1),hAPP_int_int(minus_minus_int(Z1),Z2)) = hAPP_int_int(minus_minus_int(hAPP_int_int(times_times_int(W_1),Z1)),hAPP_int_int(times_times_int(W_1),Z2)) ).
fof(fact_11_zdiff__zmult__distrib,axiom,
! [Z1,Z2,W_1] : hAPP_int_int(times_times_int(hAPP_int_int(minus_minus_int(Z1),Z2)),W_1) = hAPP_int_int(minus_minus_int(hAPP_int_int(times_times_int(Z1),W_1)),hAPP_int_int(times_times_int(Z2),W_1)) ).
fof(fact_12_zadd__zmult__distrib2,axiom,
! [W_1,Z1,Z2] : hAPP_int_int(times_times_int(W_1),hAPP_int_int(plus_plus_int(Z1),Z2)) = hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(W_1),Z1)),hAPP_int_int(times_times_int(W_1),Z2)) ).
fof(fact_13_zadd__zmult__distrib,axiom,
! [Z1,Z2,W_1] : hAPP_int_int(times_times_int(hAPP_int_int(plus_plus_int(Z1),Z2)),W_1) = hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(Z1),W_1)),hAPP_int_int(times_times_int(Z2),W_1)) ).
fof(fact_14_diff__add__cancel,axiom,
! [A_115,B_90] : hAPP_int_int(plus_plus_int(hAPP_int_int(minus_minus_int(A_115),B_90)),B_90) = A_115 ).
fof(fact_15_diff__add__cancel,axiom,
! [A_115,B_90] : hAPP_real_real(plus_plus_real(hAPP_real_real(minus_minus_real(A_115),B_90)),B_90) = A_115 ).
fof(fact_16_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A_114,B_89,C_54] : hAPP_int_int(times_times_int(hAPP_int_int(times_times_int(A_114),B_89)),C_54) = hAPP_int_int(times_times_int(A_114),hAPP_int_int(times_times_int(B_89),C_54)) ).
fof(fact_17_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A_114,B_89,C_54] : hAPP_real_real(times_times_real(hAPP_real_real(times_times_real(A_114),B_89)),C_54) = hAPP_real_real(times_times_real(A_114),hAPP_real_real(times_times_real(B_89),C_54)) ).
fof(fact_18_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A_114,B_89,C_54] : hAPP_nat_nat(times_times_nat(hAPP_nat_nat(times_times_nat(A_114),B_89)),C_54) = hAPP_nat_nat(times_times_nat(A_114),hAPP_nat_nat(times_times_nat(B_89),C_54)) ).
fof(fact_19_add__right__imp__eq,axiom,
! [B_88,A_113,C_53] :
( hAPP_int_int(plus_plus_int(B_88),A_113) = hAPP_int_int(plus_plus_int(C_53),A_113)
=> B_88 = C_53 ) ).
fof(fact_20_add__right__imp__eq,axiom,
! [B_88,A_113,C_53] :
( hAPP_real_real(plus_plus_real(B_88),A_113) = hAPP_real_real(plus_plus_real(C_53),A_113)
=> B_88 = C_53 ) ).
fof(fact_21_add__right__imp__eq,axiom,
! [B_88,A_113,C_53] :
( hAPP_nat_nat(plus_plus_nat(B_88),A_113) = hAPP_nat_nat(plus_plus_nat(C_53),A_113)
=> B_88 = C_53 ) ).
fof(fact_22_add__imp__eq,axiom,
! [A_112,B_87,C_52] :
( hAPP_int_int(plus_plus_int(A_112),B_87) = hAPP_int_int(plus_plus_int(A_112),C_52)
=> B_87 = C_52 ) ).
fof(fact_23_add__imp__eq,axiom,
! [A_112,B_87,C_52] :
( hAPP_real_real(plus_plus_real(A_112),B_87) = hAPP_real_real(plus_plus_real(A_112),C_52)
=> B_87 = C_52 ) ).
fof(fact_24_add__imp__eq,axiom,
! [A_112,B_87,C_52] :
( hAPP_nat_nat(plus_plus_nat(A_112),B_87) = hAPP_nat_nat(plus_plus_nat(A_112),C_52)
=> B_87 = C_52 ) ).
fof(fact_25_add__left__imp__eq,axiom,
! [A_111,B_86,C_51] :
( hAPP_int_int(plus_plus_int(A_111),B_86) = hAPP_int_int(plus_plus_int(A_111),C_51)
=> B_86 = C_51 ) ).
fof(fact_26_add__left__imp__eq,axiom,
! [A_111,B_86,C_51] :
( hAPP_real_real(plus_plus_real(A_111),B_86) = hAPP_real_real(plus_plus_real(A_111),C_51)
=> B_86 = C_51 ) ).
fof(fact_27_add__left__imp__eq,axiom,
! [A_111,B_86,C_51] :
( hAPP_nat_nat(plus_plus_nat(A_111),B_86) = hAPP_nat_nat(plus_plus_nat(A_111),C_51)
=> B_86 = C_51 ) ).
fof(fact_28_add__right__cancel,axiom,
! [Ba,Aa,C_1] :
( hAPP_int_int(plus_plus_int(Ba),Aa) = hAPP_int_int(plus_plus_int(C_1),Aa)
<=> Ba = C_1 ) ).
fof(fact_29_add__right__cancel,axiom,
! [Ba,Aa,C_1] :
( hAPP_real_real(plus_plus_real(Ba),Aa) = hAPP_real_real(plus_plus_real(C_1),Aa)
<=> Ba = C_1 ) ).
fof(fact_30_add__right__cancel,axiom,
! [Ba,Aa,C_1] :
( hAPP_nat_nat(plus_plus_nat(Ba),Aa) = hAPP_nat_nat(plus_plus_nat(C_1),Aa)
<=> Ba = C_1 ) ).
fof(fact_31_add__left__cancel,axiom,
! [Aa,Ba,C_1] :
( hAPP_int_int(plus_plus_int(Aa),Ba) = hAPP_int_int(plus_plus_int(Aa),C_1)
<=> Ba = C_1 ) ).
fof(fact_32_add__left__cancel,axiom,
! [Aa,Ba,C_1] :
( hAPP_real_real(plus_plus_real(Aa),Ba) = hAPP_real_real(plus_plus_real(Aa),C_1)
<=> Ba = C_1 ) ).
fof(fact_33_add__left__cancel,axiom,
! [Aa,Ba,C_1] :
( hAPP_nat_nat(plus_plus_nat(Aa),Ba) = hAPP_nat_nat(plus_plus_nat(Aa),C_1)
<=> Ba = C_1 ) ).
fof(fact_34_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A_110,B_85,C_50] : hAPP_int_int(plus_plus_int(hAPP_int_int(plus_plus_int(A_110),B_85)),C_50) = hAPP_int_int(plus_plus_int(A_110),hAPP_int_int(plus_plus_int(B_85),C_50)) ).
fof(fact_35_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A_110,B_85,C_50] : hAPP_real_real(plus_plus_real(hAPP_real_real(plus_plus_real(A_110),B_85)),C_50) = hAPP_real_real(plus_plus_real(A_110),hAPP_real_real(plus_plus_real(B_85),C_50)) ).
fof(fact_36_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A_110,B_85,C_50] : hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(plus_plus_nat(A_110),B_85)),C_50) = hAPP_nat_nat(plus_plus_nat(A_110),hAPP_nat_nat(plus_plus_nat(B_85),C_50)) ).
fof(fact_37_diff__eq__diff__eq,axiom,
! [Aa,Ba,C_1,D_2] :
( hAPP_int_int(minus_minus_int(Aa),Ba) = hAPP_int_int(minus_minus_int(C_1),D_2)
=> ( Aa = Ba
<=> C_1 = D_2 ) ) ).
fof(fact_38_diff__eq__diff__eq,axiom,
! [Aa,Ba,C_1,D_2] :
( hAPP_real_real(minus_minus_real(Aa),Ba) = hAPP_real_real(minus_minus_real(C_1),D_2)
=> ( Aa = Ba
<=> C_1 = D_2 ) ) ).
fof(fact_39_zmult__assoc,axiom,
! [Z1,Z2,Z3] : hAPP_int_int(times_times_int(hAPP_int_int(times_times_int(Z1),Z2)),Z3) = hAPP_int_int(times_times_int(Z1),hAPP_int_int(times_times_int(Z2),Z3)) ).
fof(fact_40_zmult__commute,axiom,
! [Z_1,W_1] : hAPP_int_int(times_times_int(Z_1),W_1) = hAPP_int_int(times_times_int(W_1),Z_1) ).
fof(fact_41_zadd__assoc,axiom,
! [Z1,Z2,Z3] : hAPP_int_int(plus_plus_int(hAPP_int_int(plus_plus_int(Z1),Z2)),Z3) = hAPP_int_int(plus_plus_int(Z1),hAPP_int_int(plus_plus_int(Z2),Z3)) ).
fof(fact_42_zadd__left__commute,axiom,
! [X,Y,Z_1] : hAPP_int_int(plus_plus_int(X),hAPP_int_int(plus_plus_int(Y),Z_1)) = hAPP_int_int(plus_plus_int(Y),hAPP_int_int(plus_plus_int(X),Z_1)) ).
fof(fact_43_zadd__commute,axiom,
! [Z_1,W_1] : hAPP_int_int(plus_plus_int(Z_1),W_1) = hAPP_int_int(plus_plus_int(W_1),Z_1) ).
fof(fact_44_combine__common__factor,axiom,
! [A_109,E_2,B_84,C_49] : hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(A_109),E_2)),hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(B_84),E_2)),C_49)) = hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(hAPP_int_int(plus_plus_int(A_109),B_84)),E_2)),C_49) ).
fof(fact_45_combine__common__factor,axiom,
! [A_109,E_2,B_84,C_49] : hAPP_real_real(plus_plus_real(hAPP_real_real(times_times_real(A_109),E_2)),hAPP_real_real(plus_plus_real(hAPP_real_real(times_times_real(B_84),E_2)),C_49)) = hAPP_real_real(plus_plus_real(hAPP_real_real(times_times_real(hAPP_real_real(plus_plus_real(A_109),B_84)),E_2)),C_49) ).
fof(fact_46_combine__common__factor,axiom,
! [A_109,E_2,B_84,C_49] : hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(times_times_nat(A_109),E_2)),hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(times_times_nat(B_84),E_2)),C_49)) = hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(times_times_nat(hAPP_nat_nat(plus_plus_nat(A_109),B_84)),E_2)),C_49) ).
fof(fact_47_mult__left_Oadd,axiom,
! [X_45,Y_45,Ya_1] : hAPP_real_real(times_times_real(hAPP_real_real(plus_plus_real(X_45),Y_45)),Ya_1) = hAPP_real_real(plus_plus_real(hAPP_real_real(times_times_real(X_45),Ya_1)),hAPP_real_real(times_times_real(Y_45),Ya_1)) ).
fof(fact_48_mult_Oadd__left,axiom,
! [A_108,A_107,B_83] : hAPP_real_real(times_times_real(hAPP_real_real(plus_plus_real(A_108),A_107)),B_83) = hAPP_real_real(plus_plus_real(hAPP_real_real(times_times_real(A_108),B_83)),hAPP_real_real(times_times_real(A_107),B_83)) ).
fof(fact_49_comm__semiring__class_Odistrib,axiom,
! [A_106,B_82,C_48] : hAPP_int_int(times_times_int(hAPP_int_int(plus_plus_int(A_106),B_82)),C_48) = hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(A_106),C_48)),hAPP_int_int(times_times_int(B_82),C_48)) ).
fof(fact_50_comm__semiring__class_Odistrib,axiom,
! [A_106,B_82,C_48] : hAPP_real_real(times_times_real(hAPP_real_real(plus_plus_real(A_106),B_82)),C_48) = hAPP_real_real(plus_plus_real(hAPP_real_real(times_times_real(A_106),C_48)),hAPP_real_real(times_times_real(B_82),C_48)) ).
fof(fact_51_comm__semiring__class_Odistrib,axiom,
! [A_106,B_82,C_48] : hAPP_nat_nat(times_times_nat(hAPP_nat_nat(plus_plus_nat(A_106),B_82)),C_48) = hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(times_times_nat(A_106),C_48)),hAPP_nat_nat(times_times_nat(B_82),C_48)) ).
fof(fact_52_mult__right_Oadd,axiom,
! [Xa_1,X_44,Y_44] : hAPP_real_real(times_times_real(Xa_1),hAPP_real_real(plus_plus_real(X_44),Y_44)) = hAPP_real_real(plus_plus_real(hAPP_real_real(times_times_real(Xa_1),X_44)),hAPP_real_real(times_times_real(Xa_1),Y_44)) ).
fof(fact_53_mult_Oadd__right,axiom,
! [A_105,B_81,B_80] : hAPP_real_real(times_times_real(A_105),hAPP_real_real(plus_plus_real(B_81),B_80)) = hAPP_real_real(plus_plus_real(hAPP_real_real(times_times_real(A_105),B_81)),hAPP_real_real(times_times_real(A_105),B_80)) ).
fof(fact_54_mult__left_Odiff,axiom,
! [X_43,Y_43,Ya] : hAPP_real_real(times_times_real(hAPP_real_real(minus_minus_real(X_43),Y_43)),Ya) = hAPP_real_real(minus_minus_real(hAPP_real_real(times_times_real(X_43),Ya)),hAPP_real_real(times_times_real(Y_43),Ya)) ).
fof(fact_55_mult_Odiff__left,axiom,
! [A_104,A_103,B_79] : hAPP_real_real(times_times_real(hAPP_real_real(minus_minus_real(A_104),A_103)),B_79) = hAPP_real_real(minus_minus_real(hAPP_real_real(times_times_real(A_104),B_79)),hAPP_real_real(times_times_real(A_103),B_79)) ).
fof(fact_56_mult__right_Odiff,axiom,
! [Xa,X_42,Y_42] : hAPP_real_real(times_times_real(Xa),hAPP_real_real(minus_minus_real(X_42),Y_42)) = hAPP_real_real(minus_minus_real(hAPP_real_real(times_times_real(Xa),X_42)),hAPP_real_real(times_times_real(Xa),Y_42)) ).
fof(fact_57_mult_Odiff__right,axiom,
! [A_102,B_78,B_77] : hAPP_real_real(times_times_real(A_102),hAPP_real_real(minus_minus_real(B_78),B_77)) = hAPP_real_real(minus_minus_real(hAPP_real_real(times_times_real(A_102),B_78)),hAPP_real_real(times_times_real(A_102),B_77)) ).
fof(fact_58_add__diff__add,axiom,
! [A_101,C_47,B_76,D_12] : hAPP_int_int(minus_minus_int(hAPP_int_int(plus_plus_int(A_101),C_47)),hAPP_int_int(plus_plus_int(B_76),D_12)) = hAPP_int_int(plus_plus_int(hAPP_int_int(minus_minus_int(A_101),B_76)),hAPP_int_int(minus_minus_int(C_47),D_12)) ).
fof(fact_59_add__diff__add,axiom,
! [A_101,C_47,B_76,D_12] : hAPP_real_real(minus_minus_real(hAPP_real_real(plus_plus_real(A_101),C_47)),hAPP_real_real(plus_plus_real(B_76),D_12)) = hAPP_real_real(plus_plus_real(hAPP_real_real(minus_minus_real(A_101),B_76)),hAPP_real_real(minus_minus_real(C_47),D_12)) ).
fof(fact_60_add__diff__cancel,axiom,
! [A_100,B_75] : hAPP_int_int(minus_minus_int(hAPP_int_int(plus_plus_int(A_100),B_75)),B_75) = A_100 ).
fof(fact_61_add__diff__cancel,axiom,
! [A_100,B_75] : hAPP_real_real(minus_minus_real(hAPP_real_real(plus_plus_real(A_100),B_75)),B_75) = A_100 ).
fof(fact_62_crossproduct__eq,axiom,
! [W,Y_1,X_1,Z] :
( hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(W),Y_1)),hAPP_int_int(times_times_int(X_1),Z)) = hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(W),Z)),hAPP_int_int(times_times_int(X_1),Y_1))
<=> ( W = X_1
| Y_1 = Z ) ) ).
fof(fact_63_crossproduct__eq,axiom,
! [W,Y_1,X_1,Z] :
( hAPP_real_real(plus_plus_real(hAPP_real_real(times_times_real(W),Y_1)),hAPP_real_real(times_times_real(X_1),Z)) = hAPP_real_real(plus_plus_real(hAPP_real_real(times_times_real(W),Z)),hAPP_real_real(times_times_real(X_1),Y_1))
<=> ( W = X_1
| Y_1 = Z ) ) ).
fof(fact_64_crossproduct__eq,axiom,
! [W,Y_1,X_1,Z] :
( hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(times_times_nat(W),Y_1)),hAPP_nat_nat(times_times_nat(X_1),Z)) = hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(times_times_nat(W),Z)),hAPP_nat_nat(times_times_nat(X_1),Y_1))
<=> ( W = X_1
| Y_1 = Z ) ) ).
fof(fact_65_comm__semiring__1__class_Onormalizing__semiring__rules_I1_J,axiom,
! [A_99,M_7,B_74] : hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(A_99),M_7)),hAPP_int_int(times_times_int(B_74),M_7)) = hAPP_int_int(times_times_int(hAPP_int_int(plus_plus_int(A_99),B_74)),M_7) ).
fof(fact_66_comm__semiring__1__class_Onormalizing__semiring__rules_I1_J,axiom,
! [A_99,M_7,B_74] : hAPP_real_real(plus_plus_real(hAPP_real_real(times_times_real(A_99),M_7)),hAPP_real_real(times_times_real(B_74),M_7)) = hAPP_real_real(times_times_real(hAPP_real_real(plus_plus_real(A_99),B_74)),M_7) ).
fof(fact_67_comm__semiring__1__class_Onormalizing__semiring__rules_I1_J,axiom,
! [A_99,M_7,B_74] : hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(times_times_nat(A_99),M_7)),hAPP_nat_nat(times_times_nat(B_74),M_7)) = hAPP_nat_nat(times_times_nat(hAPP_nat_nat(plus_plus_nat(A_99),B_74)),M_7) ).
fof(fact_68_comm__semiring__1__class_Onormalizing__semiring__rules_I8_J,axiom,
! [A_98,B_73,C_46] : hAPP_int_int(times_times_int(hAPP_int_int(plus_plus_int(A_98),B_73)),C_46) = hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(A_98),C_46)),hAPP_int_int(times_times_int(B_73),C_46)) ).
fof(fact_69_comm__semiring__1__class_Onormalizing__semiring__rules_I8_J,axiom,
! [A_98,B_73,C_46] : hAPP_real_real(times_times_real(hAPP_real_real(plus_plus_real(A_98),B_73)),C_46) = hAPP_real_real(plus_plus_real(hAPP_real_real(times_times_real(A_98),C_46)),hAPP_real_real(times_times_real(B_73),C_46)) ).
fof(fact_70_comm__semiring__1__class_Onormalizing__semiring__rules_I8_J,axiom,
! [A_98,B_73,C_46] : hAPP_nat_nat(times_times_nat(hAPP_nat_nat(plus_plus_nat(A_98),B_73)),C_46) = hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(times_times_nat(A_98),C_46)),hAPP_nat_nat(times_times_nat(B_73),C_46)) ).
fof(fact_71_crossproduct__noteq,axiom,
! [C_1,D_2,Aa,Ba] :
( ( Aa != Ba
& C_1 != D_2 )
<=> hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(Aa),C_1)),hAPP_int_int(times_times_int(Ba),D_2)) != hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(Aa),D_2)),hAPP_int_int(times_times_int(Ba),C_1)) ) ).
fof(fact_72_crossproduct__noteq,axiom,
! [C_1,D_2,Aa,Ba] :
( ( Aa != Ba
& C_1 != D_2 )
<=> hAPP_real_real(plus_plus_real(hAPP_real_real(times_times_real(Aa),C_1)),hAPP_real_real(times_times_real(Ba),D_2)) != hAPP_real_real(plus_plus_real(hAPP_real_real(times_times_real(Aa),D_2)),hAPP_real_real(times_times_real(Ba),C_1)) ) ).
fof(fact_73_crossproduct__noteq,axiom,
! [C_1,D_2,Aa,Ba] :
( ( Aa != Ba
& C_1 != D_2 )
<=> hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(times_times_nat(Aa),C_1)),hAPP_nat_nat(times_times_nat(Ba),D_2)) != hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(times_times_nat(Aa),D_2)),hAPP_nat_nat(times_times_nat(Ba),C_1)) ) ).
fof(fact_74_comm__semiring__1__class_Onormalizing__semiring__rules_I34_J,axiom,
! [X_41,Y_41,Z_17] : hAPP_int_int(times_times_int(X_41),hAPP_int_int(plus_plus_int(Y_41),Z_17)) = hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(X_41),Y_41)),hAPP_int_int(times_times_int(X_41),Z_17)) ).
fof(fact_75_comm__semiring__1__class_Onormalizing__semiring__rules_I34_J,axiom,
! [X_41,Y_41,Z_17] : hAPP_real_real(times_times_real(X_41),hAPP_real_real(plus_plus_real(Y_41),Z_17)) = hAPP_real_real(plus_plus_real(hAPP_real_real(times_times_real(X_41),Y_41)),hAPP_real_real(times_times_real(X_41),Z_17)) ).
fof(fact_76_comm__semiring__1__class_Onormalizing__semiring__rules_I34_J,axiom,
! [X_41,Y_41,Z_17] : hAPP_nat_nat(times_times_nat(X_41),hAPP_nat_nat(plus_plus_nat(Y_41),Z_17)) = hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(times_times_nat(X_41),Y_41)),hAPP_nat_nat(times_times_nat(X_41),Z_17)) ).
fof(fact_77_Pair__inject,axiom,
! [A_97,B_72,A_96,B_71] :
( product_Pair_int_int(A_97,B_72) = product_Pair_int_int(A_96,B_71)
=> ~ ( A_97 = A_96
=> B_72 != B_71 ) ) ).
fof(fact_78_Pair__inject,axiom,
! [A_97,B_72,A_96,B_71] :
( produc282740534nt_int(A_97,B_72) = produc282740534nt_int(A_96,B_71)
=> ~ ( A_97 = A_96
=> B_72 != B_71 ) ) ).
fof(fact_79_Pair__eq,axiom,
! [Aa,Ba,A_95,B_70] :
( product_Pair_int_int(Aa,Ba) = product_Pair_int_int(A_95,B_70)
<=> ( Aa = A_95
& Ba = B_70 ) ) ).
fof(fact_80_Pair__eq,axiom,
! [Aa,Ba,A_95,B_70] :
( produc282740534nt_int(Aa,Ba) = produc282740534nt_int(A_95,B_70)
<=> ( Aa = A_95
& Ba = B_70 ) ) ).
fof(fact_81_split__paired__All,axiom,
! [P_1] :
( ! [X1] : hBOOL(hAPP_P603027463t_bool(P_1,X1))
<=> ! [A_49,B_65] : hBOOL(hAPP_P603027463t_bool(P_1,product_Pair_int_int(A_49,B_65))) ) ).
fof(fact_82_split__paired__All,axiom,
! [P_1] :
( ! [X1] : hBOOL(hAPP_P178748804t_bool(P_1,X1))
<=> ! [A_49,B_65] : hBOOL(hAPP_P178748804t_bool(P_1,produc282740534nt_int(A_49,B_65))) ) ).
fof(fact_83_comm__semiring__1__class_Onormalizing__semiring__rules_I20_J,axiom,
! [A_94,B_69,C_45,D_11] : hAPP_int_int(plus_plus_int(hAPP_int_int(plus_plus_int(A_94),B_69)),hAPP_int_int(plus_plus_int(C_45),D_11)) = hAPP_int_int(plus_plus_int(hAPP_int_int(plus_plus_int(A_94),C_45)),hAPP_int_int(plus_plus_int(B_69),D_11)) ).
fof(fact_84_comm__semiring__1__class_Onormalizing__semiring__rules_I20_J,axiom,
! [A_94,B_69,C_45,D_11] : hAPP_real_real(plus_plus_real(hAPP_real_real(plus_plus_real(A_94),B_69)),hAPP_real_real(plus_plus_real(C_45),D_11)) = hAPP_real_real(plus_plus_real(hAPP_real_real(plus_plus_real(A_94),C_45)),hAPP_real_real(plus_plus_real(B_69),D_11)) ).
fof(fact_85_comm__semiring__1__class_Onormalizing__semiring__rules_I20_J,axiom,
! [A_94,B_69,C_45,D_11] : hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(plus_plus_nat(A_94),B_69)),hAPP_nat_nat(plus_plus_nat(C_45),D_11)) = hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(plus_plus_nat(A_94),C_45)),hAPP_nat_nat(plus_plus_nat(B_69),D_11)) ).
fof(fact_86_comm__semiring__1__class_Onormalizing__semiring__rules_I23_J,axiom,
! [A_93,B_68,C_44] : hAPP_int_int(plus_plus_int(hAPP_int_int(plus_plus_int(A_93),B_68)),C_44) = hAPP_int_int(plus_plus_int(hAPP_int_int(plus_plus_int(A_93),C_44)),B_68) ).
fof(fact_87_comm__semiring__1__class_Onormalizing__semiring__rules_I23_J,axiom,
! [A_93,B_68,C_44] : hAPP_real_real(plus_plus_real(hAPP_real_real(plus_plus_real(A_93),B_68)),C_44) = hAPP_real_real(plus_plus_real(hAPP_real_real(plus_plus_real(A_93),C_44)),B_68) ).
fof(fact_88_comm__semiring__1__class_Onormalizing__semiring__rules_I23_J,axiom,
! [A_93,B_68,C_44] : hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(plus_plus_nat(A_93),B_68)),C_44) = hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(plus_plus_nat(A_93),C_44)),B_68) ).
fof(fact_89_comm__semiring__1__class_Onormalizing__semiring__rules_I7_J,axiom,
! [A_92,B_67] : hAPP_int_int(times_times_int(A_92),B_67) = hAPP_int_int(times_times_int(B_67),A_92) ).
fof(fact_90_comm__semiring__1__class_Onormalizing__semiring__rules_I7_J,axiom,
! [A_92,B_67] : hAPP_real_real(times_times_real(A_92),B_67) = hAPP_real_real(times_times_real(B_67),A_92) ).
fof(fact_91_comm__semiring__1__class_Onormalizing__semiring__rules_I7_J,axiom,
! [A_92,B_67] : hAPP_nat_nat(times_times_nat(A_92),B_67) = hAPP_nat_nat(times_times_nat(B_67),A_92) ).
fof(fact_92_comm__semiring__1__class_Onormalizing__semiring__rules_I19_J,axiom,
! [Lx_6,Rx_6,Ry_4] : hAPP_int_int(times_times_int(Lx_6),hAPP_int_int(times_times_int(Rx_6),Ry_4)) = hAPP_int_int(times_times_int(Rx_6),hAPP_int_int(times_times_int(Lx_6),Ry_4)) ).
fof(fact_93_comm__semiring__1__class_Onormalizing__semiring__rules_I19_J,axiom,
! [Lx_6,Rx_6,Ry_4] : hAPP_real_real(times_times_real(Lx_6),hAPP_real_real(times_times_real(Rx_6),Ry_4)) = hAPP_real_real(times_times_real(Rx_6),hAPP_real_real(times_times_real(Lx_6),Ry_4)) ).
fof(fact_94_comm__semiring__1__class_Onormalizing__semiring__rules_I19_J,axiom,
! [Lx_6,Rx_6,Ry_4] : hAPP_nat_nat(times_times_nat(Lx_6),hAPP_nat_nat(times_times_nat(Rx_6),Ry_4)) = hAPP_nat_nat(times_times_nat(Rx_6),hAPP_nat_nat(times_times_nat(Lx_6),Ry_4)) ).
fof(fact_95_comm__semiring__1__class_Onormalizing__semiring__rules_I18_J,axiom,
! [Lx_5,Rx_5,Ry_3] : hAPP_int_int(times_times_int(Lx_5),hAPP_int_int(times_times_int(Rx_5),Ry_3)) = hAPP_int_int(times_times_int(hAPP_int_int(times_times_int(Lx_5),Rx_5)),Ry_3) ).
fof(fact_96_comm__semiring__1__class_Onormalizing__semiring__rules_I18_J,axiom,
! [Lx_5,Rx_5,Ry_3] : hAPP_real_real(times_times_real(Lx_5),hAPP_real_real(times_times_real(Rx_5),Ry_3)) = hAPP_real_real(times_times_real(hAPP_real_real(times_times_real(Lx_5),Rx_5)),Ry_3) ).
fof(fact_97_comm__semiring__1__class_Onormalizing__semiring__rules_I18_J,axiom,
! [Lx_5,Rx_5,Ry_3] : hAPP_nat_nat(times_times_nat(Lx_5),hAPP_nat_nat(times_times_nat(Rx_5),Ry_3)) = hAPP_nat_nat(times_times_nat(hAPP_nat_nat(times_times_nat(Lx_5),Rx_5)),Ry_3) ).
fof(fact_98_comm__semiring__1__class_Onormalizing__semiring__rules_I17_J,axiom,
! [Lx_4,Ly_4,Rx_4] : hAPP_int_int(times_times_int(hAPP_int_int(times_times_int(Lx_4),Ly_4)),Rx_4) = hAPP_int_int(times_times_int(Lx_4),hAPP_int_int(times_times_int(Ly_4),Rx_4)) ).
fof(fact_99_comm__semiring__1__class_Onormalizing__semiring__rules_I17_J,axiom,
! [Lx_4,Ly_4,Rx_4] : hAPP_real_real(times_times_real(hAPP_real_real(times_times_real(Lx_4),Ly_4)),Rx_4) = hAPP_real_real(times_times_real(Lx_4),hAPP_real_real(times_times_real(Ly_4),Rx_4)) ).
fof(fact_100_comm__semiring__1__class_Onormalizing__semiring__rules_I17_J,axiom,
! [Lx_4,Ly_4,Rx_4] : hAPP_nat_nat(times_times_nat(hAPP_nat_nat(times_times_nat(Lx_4),Ly_4)),Rx_4) = hAPP_nat_nat(times_times_nat(Lx_4),hAPP_nat_nat(times_times_nat(Ly_4),Rx_4)) ).
fof(fact_101_comm__semiring__1__class_Onormalizing__semiring__rules_I16_J,axiom,
! [Lx_3,Ly_3,Rx_3] : hAPP_int_int(times_times_int(hAPP_int_int(times_times_int(Lx_3),Ly_3)),Rx_3) = hAPP_int_int(times_times_int(hAPP_int_int(times_times_int(Lx_3),Rx_3)),Ly_3) ).
fof(fact_102_comm__semiring__1__class_Onormalizing__semiring__rules_I16_J,axiom,
! [Lx_3,Ly_3,Rx_3] : hAPP_real_real(times_times_real(hAPP_real_real(times_times_real(Lx_3),Ly_3)),Rx_3) = hAPP_real_real(times_times_real(hAPP_real_real(times_times_real(Lx_3),Rx_3)),Ly_3) ).
fof(fact_103_comm__semiring__1__class_Onormalizing__semiring__rules_I16_J,axiom,
! [Lx_3,Ly_3,Rx_3] : hAPP_nat_nat(times_times_nat(hAPP_nat_nat(times_times_nat(Lx_3),Ly_3)),Rx_3) = hAPP_nat_nat(times_times_nat(hAPP_nat_nat(times_times_nat(Lx_3),Rx_3)),Ly_3) ).
fof(fact_104_comm__semiring__1__class_Onormalizing__semiring__rules_I14_J,axiom,
! [Lx_2,Ly_2,Rx_2,Ry_2] : hAPP_int_int(times_times_int(hAPP_int_int(times_times_int(Lx_2),Ly_2)),hAPP_int_int(times_times_int(Rx_2),Ry_2)) = hAPP_int_int(times_times_int(Lx_2),hAPP_int_int(times_times_int(Ly_2),hAPP_int_int(times_times_int(Rx_2),Ry_2))) ).
fof(fact_105_comm__semiring__1__class_Onormalizing__semiring__rules_I14_J,axiom,
! [Lx_2,Ly_2,Rx_2,Ry_2] : hAPP_real_real(times_times_real(hAPP_real_real(times_times_real(Lx_2),Ly_2)),hAPP_real_real(times_times_real(Rx_2),Ry_2)) = hAPP_real_real(times_times_real(Lx_2),hAPP_real_real(times_times_real(Ly_2),hAPP_real_real(times_times_real(Rx_2),Ry_2))) ).
fof(fact_106_comm__semiring__1__class_Onormalizing__semiring__rules_I14_J,axiom,
! [Lx_2,Ly_2,Rx_2,Ry_2] : hAPP_nat_nat(times_times_nat(hAPP_nat_nat(times_times_nat(Lx_2),Ly_2)),hAPP_nat_nat(times_times_nat(Rx_2),Ry_2)) = hAPP_nat_nat(times_times_nat(Lx_2),hAPP_nat_nat(times_times_nat(Ly_2),hAPP_nat_nat(times_times_nat(Rx_2),Ry_2))) ).
fof(fact_107_comm__semiring__1__class_Onormalizing__semiring__rules_I15_J,axiom,
! [Lx_1,Ly_1,Rx_1,Ry_1] : hAPP_int_int(times_times_int(hAPP_int_int(times_times_int(Lx_1),Ly_1)),hAPP_int_int(times_times_int(Rx_1),Ry_1)) = hAPP_int_int(times_times_int(Rx_1),hAPP_int_int(times_times_int(hAPP_int_int(times_times_int(Lx_1),Ly_1)),Ry_1)) ).
fof(fact_108_comm__semiring__1__class_Onormalizing__semiring__rules_I15_J,axiom,
! [Lx_1,Ly_1,Rx_1,Ry_1] : hAPP_real_real(times_times_real(hAPP_real_real(times_times_real(Lx_1),Ly_1)),hAPP_real_real(times_times_real(Rx_1),Ry_1)) = hAPP_real_real(times_times_real(Rx_1),hAPP_real_real(times_times_real(hAPP_real_real(times_times_real(Lx_1),Ly_1)),Ry_1)) ).
fof(fact_109_comm__semiring__1__class_Onormalizing__semiring__rules_I15_J,axiom,
! [Lx_1,Ly_1,Rx_1,Ry_1] : hAPP_nat_nat(times_times_nat(hAPP_nat_nat(times_times_nat(Lx_1),Ly_1)),hAPP_nat_nat(times_times_nat(Rx_1),Ry_1)) = hAPP_nat_nat(times_times_nat(Rx_1),hAPP_nat_nat(times_times_nat(hAPP_nat_nat(times_times_nat(Lx_1),Ly_1)),Ry_1)) ).
fof(fact_110_comm__semiring__1__class_Onormalizing__semiring__rules_I13_J,axiom,
! [Lx,Ly,Rx,Ry] : hAPP_int_int(times_times_int(hAPP_int_int(times_times_int(Lx),Ly)),hAPP_int_int(times_times_int(Rx),Ry)) = hAPP_int_int(times_times_int(hAPP_int_int(times_times_int(Lx),Rx)),hAPP_int_int(times_times_int(Ly),Ry)) ).
fof(fact_111_comm__semiring__1__class_Onormalizing__semiring__rules_I13_J,axiom,
! [Lx,Ly,Rx,Ry] : hAPP_real_real(times_times_real(hAPP_real_real(times_times_real(Lx),Ly)),hAPP_real_real(times_times_real(Rx),Ry)) = hAPP_real_real(times_times_real(hAPP_real_real(times_times_real(Lx),Rx)),hAPP_real_real(times_times_real(Ly),Ry)) ).
fof(fact_112_comm__semiring__1__class_Onormalizing__semiring__rules_I13_J,axiom,
! [Lx,Ly,Rx,Ry] : hAPP_nat_nat(times_times_nat(hAPP_nat_nat(times_times_nat(Lx),Ly)),hAPP_nat_nat(times_times_nat(Rx),Ry)) = hAPP_nat_nat(times_times_nat(hAPP_nat_nat(times_times_nat(Lx),Rx)),hAPP_nat_nat(times_times_nat(Ly),Ry)) ).
fof(fact_113_comm__semiring__1__class_Onormalizing__semiring__rules_I24_J,axiom,
! [A_91,C_43] : hAPP_int_int(plus_plus_int(A_91),C_43) = hAPP_int_int(plus_plus_int(C_43),A_91) ).
fof(fact_114_comm__semiring__1__class_Onormalizing__semiring__rules_I24_J,axiom,
! [A_91,C_43] : hAPP_real_real(plus_plus_real(A_91),C_43) = hAPP_real_real(plus_plus_real(C_43),A_91) ).
fof(fact_115_comm__semiring__1__class_Onormalizing__semiring__rules_I24_J,axiom,
! [A_91,C_43] : hAPP_nat_nat(plus_plus_nat(A_91),C_43) = hAPP_nat_nat(plus_plus_nat(C_43),A_91) ).
fof(fact_116_comm__semiring__1__class_Onormalizing__semiring__rules_I22_J,axiom,
! [A_90,C_42,D_10] : hAPP_int_int(plus_plus_int(A_90),hAPP_int_int(plus_plus_int(C_42),D_10)) = hAPP_int_int(plus_plus_int(C_42),hAPP_int_int(plus_plus_int(A_90),D_10)) ).
fof(fact_117_comm__semiring__1__class_Onormalizing__semiring__rules_I22_J,axiom,
! [A_90,C_42,D_10] : hAPP_real_real(plus_plus_real(A_90),hAPP_real_real(plus_plus_real(C_42),D_10)) = hAPP_real_real(plus_plus_real(C_42),hAPP_real_real(plus_plus_real(A_90),D_10)) ).
fof(fact_118_comm__semiring__1__class_Onormalizing__semiring__rules_I22_J,axiom,
! [A_90,C_42,D_10] : hAPP_nat_nat(plus_plus_nat(A_90),hAPP_nat_nat(plus_plus_nat(C_42),D_10)) = hAPP_nat_nat(plus_plus_nat(C_42),hAPP_nat_nat(plus_plus_nat(A_90),D_10)) ).
fof(fact_119_comm__semiring__1__class_Onormalizing__semiring__rules_I25_J,axiom,
! [A_89,C_41,D_9] : hAPP_int_int(plus_plus_int(A_89),hAPP_int_int(plus_plus_int(C_41),D_9)) = hAPP_int_int(plus_plus_int(hAPP_int_int(plus_plus_int(A_89),C_41)),D_9) ).
fof(fact_120_comm__semiring__1__class_Onormalizing__semiring__rules_I25_J,axiom,
! [A_89,C_41,D_9] : hAPP_real_real(plus_plus_real(A_89),hAPP_real_real(plus_plus_real(C_41),D_9)) = hAPP_real_real(plus_plus_real(hAPP_real_real(plus_plus_real(A_89),C_41)),D_9) ).
fof(fact_121_comm__semiring__1__class_Onormalizing__semiring__rules_I25_J,axiom,
! [A_89,C_41,D_9] : hAPP_nat_nat(plus_plus_nat(A_89),hAPP_nat_nat(plus_plus_nat(C_41),D_9)) = hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(plus_plus_nat(A_89),C_41)),D_9) ).
fof(fact_122_comm__semiring__1__class_Onormalizing__semiring__rules_I21_J,axiom,
! [A_88,B_66,C_40] : hAPP_int_int(plus_plus_int(hAPP_int_int(plus_plus_int(A_88),B_66)),C_40) = hAPP_int_int(plus_plus_int(A_88),hAPP_int_int(plus_plus_int(B_66),C_40)) ).
fof(fact_123_comm__semiring__1__class_Onormalizing__semiring__rules_I21_J,axiom,
! [A_88,B_66,C_40] : hAPP_real_real(plus_plus_real(hAPP_real_real(plus_plus_real(A_88),B_66)),C_40) = hAPP_real_real(plus_plus_real(A_88),hAPP_real_real(plus_plus_real(B_66),C_40)) ).
fof(fact_124_comm__semiring__1__class_Onormalizing__semiring__rules_I21_J,axiom,
! [A_88,B_66,C_40] : hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(plus_plus_nat(A_88),B_66)),C_40) = hAPP_nat_nat(plus_plus_nat(A_88),hAPP_nat_nat(plus_plus_nat(B_66),C_40)) ).
fof(fact_125_split__paired__Ex,axiom,
! [P_1] :
( ? [X1] : hBOOL(hAPP_P603027463t_bool(P_1,X1))
<=> ? [A_49,B_65] : hBOOL(hAPP_P603027463t_bool(P_1,product_Pair_int_int(A_49,B_65))) ) ).
fof(fact_126_split__paired__Ex,axiom,
! [P_1] :
( ? [X1] : hBOOL(hAPP_P178748804t_bool(P_1,X1))
<=> ? [A_49,B_65] : hBOOL(hAPP_P178748804t_bool(P_1,produc282740534nt_int(A_49,B_65))) ) ).
fof(fact_127_prod__induct3,axiom,
! [X_1,P_1] :
( ! [A_49,B_65,C_39] : hBOOL(hAPP_P178748804t_bool(P_1,produc282740534nt_int(A_49,product_Pair_int_int(B_65,C_39))))
=> hBOOL(hAPP_P178748804t_bool(P_1,X_1)) ) ).
fof(fact_128_prod__cases3,axiom,
! [Y_40] :
~ ! [A_49,B_65,C_39] : Y_40 != produc282740534nt_int(A_49,product_Pair_int_int(B_65,C_39)) ).
fof(fact_129_PairE,axiom,
! [P_3] :
~ ! [X_2,Y_2] : P_3 != product_Pair_int_int(X_2,Y_2) ).
fof(fact_130_PairE,axiom,
! [P_3] :
~ ! [X_2,Y_2] : P_3 != produc282740534nt_int(X_2,Y_2) ).
fof(fact_131_mem__def,axiom,
! [X_1,A_87] :
( hBOOL(member_real(X_1,A_87))
<=> hBOOL(hAPP_real_bool(A_87,X_1)) ) ).
fof(fact_132_prod_Oexhaust,axiom,
! [Y_39] :
~ ! [A_49,B_65] : Y_39 != product_Pair_int_int(A_49,B_65) ).
fof(fact_133_prod_Oexhaust,axiom,
! [Y_39] :
~ ! [A_49,B_65] : Y_39 != produc282740534nt_int(A_49,B_65) ).
fof(fact_134_curryI,axiom,
! [F,Aa,Ba] :
( hBOOL(hAPP_P603027463t_bool(F,product_Pair_int_int(Aa,Ba)))
=> hBOOL(hAPP_int_bool(produc262399358t_bool(F,Aa),Ba)) ) ).
fof(fact_135_curryI,axiom,
! [F,Aa,Ba] :
( hBOOL(hAPP_P178748804t_bool(F,produc282740534nt_int(Aa,Ba)))
=> hBOOL(hAPP_P603027463t_bool(produc97574839t_bool(F,Aa),Ba)) ) ).
fof(fact_136_le__add__iff1,axiom,
! [Aa,E_1,C_1,Ba,D_2] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(Aa),E_1)),C_1)),hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(Ba),E_1)),D_2)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(hAPP_int_int(minus_minus_int(Aa),Ba)),E_1)),C_1)),D_2)) ) ).
fof(fact_137_le__add__iff1,axiom,
! [Aa,E_1,C_1,Ba,D_2] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(plus_plus_real(hAPP_real_real(times_times_real(Aa),E_1)),C_1)),hAPP_real_real(plus_plus_real(hAPP_real_real(times_times_real(Ba),E_1)),D_2)))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(plus_plus_real(hAPP_real_real(times_times_real(hAPP_real_real(minus_minus_real(Aa),Ba)),E_1)),C_1)),D_2)) ) ).
fof(fact_138_le__add__iff2,axiom,
! [Aa,E_1,C_1,Ba,D_2] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(Aa),E_1)),C_1)),hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(Ba),E_1)),D_2)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,C_1),hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(hAPP_int_int(minus_minus_int(Ba),Aa)),E_1)),D_2))) ) ).
fof(fact_139_le__add__iff2,axiom,
! [Aa,E_1,C_1,Ba,D_2] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(plus_plus_real(hAPP_real_real(times_times_real(Aa),E_1)),C_1)),hAPP_real_real(plus_plus_real(hAPP_real_real(times_times_real(Ba),E_1)),D_2)))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,C_1),hAPP_real_real(plus_plus_real(hAPP_real_real(times_times_real(hAPP_real_real(minus_minus_real(Ba),Aa)),E_1)),D_2))) ) ).
fof(fact_140_real__squared__diff__one__factored,axiom,
! [X_40] : hAPP_int_int(minus_minus_int(hAPP_int_int(times_times_int(X_40),X_40)),one_one_int) = hAPP_int_int(times_times_int(hAPP_int_int(plus_plus_int(X_40),one_one_int)),hAPP_int_int(minus_minus_int(X_40),one_one_int)) ).
fof(fact_141_real__squared__diff__one__factored,axiom,
! [X_40] : hAPP_real_real(minus_minus_real(hAPP_real_real(times_times_real(X_40),X_40)),one_one_real) = hAPP_real_real(times_times_real(hAPP_real_real(plus_plus_real(X_40),one_one_real)),hAPP_real_real(minus_minus_real(X_40),one_one_real)) ).
fof(fact_142_less__add__iff1,axiom,
! [Aa,E_1,C_1,Ba,D_2] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(Aa),E_1)),C_1)),hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(Ba),E_1)),D_2)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(hAPP_int_int(minus_minus_int(Aa),Ba)),E_1)),C_1)),D_2)) ) ).
fof(fact_143_less__add__iff1,axiom,
! [Aa,E_1,C_1,Ba,D_2] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(plus_plus_real(hAPP_real_real(times_times_real(Aa),E_1)),C_1)),hAPP_real_real(plus_plus_real(hAPP_real_real(times_times_real(Ba),E_1)),D_2)))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(plus_plus_real(hAPP_real_real(times_times_real(hAPP_real_real(minus_minus_real(Aa),Ba)),E_1)),C_1)),D_2)) ) ).
fof(fact_144_less__add__iff2,axiom,
! [Aa,E_1,C_1,Ba,D_2] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(Aa),E_1)),C_1)),hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(Ba),E_1)),D_2)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,C_1),hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(hAPP_int_int(minus_minus_int(Ba),Aa)),E_1)),D_2))) ) ).
fof(fact_145_less__add__iff2,axiom,
! [Aa,E_1,C_1,Ba,D_2] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(plus_plus_real(hAPP_real_real(times_times_real(Aa),E_1)),C_1)),hAPP_real_real(plus_plus_real(hAPP_real_real(times_times_real(Ba),E_1)),D_2)))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,C_1),hAPP_real_real(plus_plus_real(hAPP_real_real(times_times_real(hAPP_real_real(minus_minus_real(Ba),Aa)),E_1)),D_2))) ) ).
fof(fact_146_add__number__of__diff1,axiom,
! [V_17,W_11,C_38] : hAPP_int_int(plus_plus_int(number_number_of_int(V_17)),hAPP_int_int(minus_minus_int(number_number_of_int(W_11)),C_38)) = hAPP_int_int(minus_minus_int(number_number_of_int(hAPP_int_int(plus_plus_int(V_17),W_11))),C_38) ).
fof(fact_147_add__number__of__diff1,axiom,
! [V_17,W_11,C_38] : hAPP_real_real(plus_plus_real(number267125858f_real(V_17)),hAPP_real_real(minus_minus_real(number267125858f_real(W_11)),C_38)) = hAPP_real_real(minus_minus_real(number267125858f_real(hAPP_int_int(plus_plus_int(V_17),W_11))),C_38) ).
fof(fact_148_curry__conv,axiom,
! [F,Aa,Ba] :
( hBOOL(hAPP_int_bool(produc262399358t_bool(F,Aa),Ba))
<=> hBOOL(hAPP_P603027463t_bool(F,product_Pair_int_int(Aa,Ba))) ) ).
fof(fact_149_curry__conv,axiom,
! [F,Aa,Ba] :
( hBOOL(hAPP_P603027463t_bool(produc97574839t_bool(F,Aa),Ba))
<=> hBOOL(hAPP_P178748804t_bool(F,produc282740534nt_int(Aa,Ba))) ) ).
fof(fact_150_less__1__mult,axiom,
! [N_3,M_6] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),M_6))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),N_3))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),hAPP_int_int(times_times_int(M_6),N_3))) ) ) ).
fof(fact_151_less__1__mult,axiom,
! [N_3,M_6] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),M_6))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),N_3))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),hAPP_real_real(times_times_real(M_6),N_3))) ) ) ).
fof(fact_152_less__1__mult,axiom,
! [N_3,M_6] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_nat),M_6))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_nat),N_3))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_nat),hAPP_nat_nat(times_times_nat(M_6),N_3))) ) ) ).
fof(fact_153_less__add__one,axiom,
! [A_86] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_86),hAPP_int_int(plus_plus_int(A_86),one_one_int))) ).
fof(fact_154_less__add__one,axiom,
! [A_86] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_86),hAPP_real_real(plus_plus_real(A_86),one_one_real))) ).
fof(fact_155_less__add__one,axiom,
! [A_86] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_86),hAPP_nat_nat(plus_plus_nat(A_86),one_one_nat))) ).
fof(fact_156_add__less__le__mono,axiom,
! [C_37,D_8,A_85,B_64] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_85),B_64))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,C_37),D_8))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(plus_plus_int(A_85),C_37)),hAPP_int_int(plus_plus_int(B_64),D_8))) ) ) ).
fof(fact_157_add__less__le__mono,axiom,
! [C_37,D_8,A_85,B_64] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_85),B_64))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,C_37),D_8))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(plus_plus_real(A_85),C_37)),hAPP_real_real(plus_plus_real(B_64),D_8))) ) ) ).
fof(fact_158_add__less__le__mono,axiom,
! [C_37,D_8,A_85,B_64] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_85),B_64))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,C_37),D_8))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(plus_plus_nat(A_85),C_37)),hAPP_nat_nat(plus_plus_nat(B_64),D_8))) ) ) ).
fof(fact_159_add__le__less__mono,axiom,
! [C_36,D_7,A_84,B_63] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_84),B_63))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,C_36),D_7))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(plus_plus_int(A_84),C_36)),hAPP_int_int(plus_plus_int(B_63),D_7))) ) ) ).
fof(fact_160_add__le__less__mono,axiom,
! [C_36,D_7,A_84,B_63] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_84),B_63))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,C_36),D_7))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(plus_plus_real(A_84),C_36)),hAPP_real_real(plus_plus_real(B_63),D_7))) ) ) ).
fof(fact_161_add__le__less__mono,axiom,
! [C_36,D_7,A_84,B_63] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_84),B_63))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,C_36),D_7))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(plus_plus_nat(A_84),C_36)),hAPP_nat_nat(plus_plus_nat(B_63),D_7))) ) ) ).
fof(fact_162_one__reorient,axiom,
! [X_1] :
( one_one_int = X_1
<=> X_1 = one_one_int ) ).
fof(fact_163_one__reorient,axiom,
! [X_1] :
( one_one_real = X_1
<=> X_1 = one_one_real ) ).
fof(fact_164_one__reorient,axiom,
! [X_1] :
( one_one_nat = X_1
<=> X_1 = one_one_nat ) ).
fof(fact_165_zle__add1__eq__le,axiom,
! [W,Z] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,W),hAPP_int_int(plus_plus_int(Z),one_one_int)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,W),Z)) ) ).
fof(fact_166_number__of__reorient,axiom,
! [W,X_1] :
( number_number_of_int(W) = X_1
<=> X_1 = number_number_of_int(W) ) ).
fof(fact_167_number__of__reorient,axiom,
! [W,X_1] :
( number267125858f_real(W) = X_1
<=> X_1 = number267125858f_real(W) ) ).
fof(fact_168_number__of__reorient,axiom,
! [W,X_1] :
( number_number_of_nat(W) = X_1
<=> X_1 = number_number_of_nat(W) ) ).
fof(fact_169_le__number__of__eq__not__less,axiom,
! [V_1,W] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,number_number_of_int(V_1)),number_number_of_int(W)))
<=> ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,number_number_of_int(W)),number_number_of_int(V_1))) ) ).
fof(fact_170_le__number__of__eq__not__less,axiom,
! [V_1,W] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,number267125858f_real(V_1)),number267125858f_real(W)))
<=> ~ hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,number267125858f_real(W)),number267125858f_real(V_1))) ) ).
fof(fact_171_le__number__of__eq__not__less,axiom,
! [V_1,W] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,number_number_of_nat(V_1)),number_number_of_nat(W)))
<=> ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,number_number_of_nat(W)),number_number_of_nat(V_1))) ) ).
fof(fact_172_eq__number__of,axiom,
! [X_1,Y_1] :
( number_number_of_int(X_1) = number_number_of_int(Y_1)
<=> X_1 = Y_1 ) ).
fof(fact_173_eq__number__of,axiom,
! [X_1,Y_1] :
( number267125858f_real(X_1) = number267125858f_real(Y_1)
<=> X_1 = Y_1 ) ).
fof(fact_174_less__number__of,axiom,
! [X_1,Y_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,number_number_of_int(X_1)),number_number_of_int(Y_1)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_1),Y_1)) ) ).
fof(fact_175_less__number__of,axiom,
! [X_1,Y_1] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,number267125858f_real(X_1)),number267125858f_real(Y_1)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_1),Y_1)) ) ).
fof(fact_176_le__number__of,axiom,
! [X_1,Y_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,number_number_of_int(X_1)),number_number_of_int(Y_1)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_1),Y_1)) ) ).
fof(fact_177_le__number__of,axiom,
! [X_1,Y_1] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,number267125858f_real(X_1)),number267125858f_real(Y_1)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_1),Y_1)) ) ).
fof(fact_178_add1__zle__eq,axiom,
! [W,Z] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(plus_plus_int(W),one_one_int)),Z))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,W),Z)) ) ).
fof(fact_179_linorder__neqE__linordered__idom,axiom,
! [X_39,Y_38] :
( X_39 != Y_38
=> ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_39),Y_38))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Y_38),X_39)) ) ) ).
fof(fact_180_linorder__neqE__linordered__idom,axiom,
! [X_39,Y_38] :
( X_39 != Y_38
=> ( ~ hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_39),Y_38))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Y_38),X_39)) ) ) ).
fof(fact_181_zless__imp__add1__zle,axiom,
! [W_1,Z_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,W_1),Z_1))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(plus_plus_int(W_1),one_one_int)),Z_1)) ) ).
fof(fact_182_order__le__neq__implies__less,axiom,
! [X_38,Y_37] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_38),Y_37))
=> ( X_38 != Y_37
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_38),Y_37)) ) ) ).
fof(fact_183_order__le__neq__implies__less,axiom,
! [X_38,Y_37] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_38),Y_37))
=> ( X_38 != Y_37
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_38),Y_37)) ) ) ).
fof(fact_184_order__le__neq__implies__less,axiom,
! [X_38,Y_37] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_38),Y_37))
=> ( X_38 != Y_37
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_38),Y_37)) ) ) ).
fof(fact_185_zle__diff1__eq,axiom,
! [W,Z] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,W),hAPP_int_int(minus_minus_int(Z),one_one_int)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,W),Z)) ) ).
fof(fact_186_zadd__zless__mono,axiom,
! [Z_16,Z_1,W_10,W_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,W_10),W_1))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Z_16),Z_1))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(plus_plus_int(W_10),Z_16)),hAPP_int_int(plus_plus_int(W_1),Z_1))) ) ) ).
fof(fact_187_zless__add1__eq,axiom,
! [W,Z] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,W),hAPP_int_int(plus_plus_int(Z),one_one_int)))
<=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,W),Z))
| W = Z ) ) ).
fof(fact_188_add__less__cancel__right,axiom,
! [Aa,C_1,Ba] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(plus_plus_int(Aa),C_1)),hAPP_int_int(plus_plus_int(Ba),C_1)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Aa),Ba)) ) ).
fof(fact_189_add__less__cancel__right,axiom,
! [Aa,C_1,Ba] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(plus_plus_real(Aa),C_1)),hAPP_real_real(plus_plus_real(Ba),C_1)))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Aa),Ba)) ) ).
fof(fact_190_add__less__cancel__right,axiom,
! [Aa,C_1,Ba] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(plus_plus_nat(Aa),C_1)),hAPP_nat_nat(plus_plus_nat(Ba),C_1)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Aa),Ba)) ) ).
fof(fact_191_add__less__cancel__left,axiom,
! [C_1,Aa,Ba] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(plus_plus_int(C_1),Aa)),hAPP_int_int(plus_plus_int(C_1),Ba)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Aa),Ba)) ) ).
fof(fact_192_add__less__cancel__left,axiom,
! [C_1,Aa,Ba] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(plus_plus_real(C_1),Aa)),hAPP_real_real(plus_plus_real(C_1),Ba)))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Aa),Ba)) ) ).
fof(fact_193_add__less__cancel__left,axiom,
! [C_1,Aa,Ba] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(plus_plus_nat(C_1),Aa)),hAPP_nat_nat(plus_plus_nat(C_1),Ba)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Aa),Ba)) ) ).
fof(fact_194_add__strict__right__mono,axiom,
! [C_35,A_83,B_62] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_83),B_62))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(plus_plus_int(A_83),C_35)),hAPP_int_int(plus_plus_int(B_62),C_35))) ) ).
fof(fact_195_add__strict__right__mono,axiom,
! [C_35,A_83,B_62] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_83),B_62))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(plus_plus_real(A_83),C_35)),hAPP_real_real(plus_plus_real(B_62),C_35))) ) ).
fof(fact_196_add__strict__right__mono,axiom,
! [C_35,A_83,B_62] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_83),B_62))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(plus_plus_nat(A_83),C_35)),hAPP_nat_nat(plus_plus_nat(B_62),C_35))) ) ).
fof(fact_197_add__strict__left__mono,axiom,
! [C_34,A_82,B_61] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_82),B_61))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(plus_plus_int(C_34),A_82)),hAPP_int_int(plus_plus_int(C_34),B_61))) ) ).
fof(fact_198_add__strict__left__mono,axiom,
! [C_34,A_82,B_61] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_82),B_61))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(plus_plus_real(C_34),A_82)),hAPP_real_real(plus_plus_real(C_34),B_61))) ) ).
fof(fact_199_add__strict__left__mono,axiom,
! [C_34,A_82,B_61] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_82),B_61))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(plus_plus_nat(C_34),A_82)),hAPP_nat_nat(plus_plus_nat(C_34),B_61))) ) ).
fof(fact_200_add__strict__mono,axiom,
! [C_33,D_6,A_81,B_60] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_81),B_60))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,C_33),D_6))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(plus_plus_int(A_81),C_33)),hAPP_int_int(plus_plus_int(B_60),D_6))) ) ) ).
fof(fact_201_add__strict__mono,axiom,
! [C_33,D_6,A_81,B_60] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_81),B_60))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,C_33),D_6))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(plus_plus_real(A_81),C_33)),hAPP_real_real(plus_plus_real(B_60),D_6))) ) ) ).
fof(fact_202_add__strict__mono,axiom,
! [C_33,D_6,A_81,B_60] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_81),B_60))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,C_33),D_6))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(plus_plus_nat(A_81),C_33)),hAPP_nat_nat(plus_plus_nat(B_60),D_6))) ) ) ).
fof(fact_203_add__less__imp__less__right,axiom,
! [A_80,C_32,B_59] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(plus_plus_int(A_80),C_32)),hAPP_int_int(plus_plus_int(B_59),C_32)))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_80),B_59)) ) ).
fof(fact_204_add__less__imp__less__right,axiom,
! [A_80,C_32,B_59] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(plus_plus_real(A_80),C_32)),hAPP_real_real(plus_plus_real(B_59),C_32)))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_80),B_59)) ) ).
fof(fact_205_add__less__imp__less__right,axiom,
! [A_80,C_32,B_59] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(plus_plus_nat(A_80),C_32)),hAPP_nat_nat(plus_plus_nat(B_59),C_32)))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_80),B_59)) ) ).
fof(fact_206_add__less__imp__less__left,axiom,
! [C_31,A_79,B_58] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(plus_plus_int(C_31),A_79)),hAPP_int_int(plus_plus_int(C_31),B_58)))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_79),B_58)) ) ).
fof(fact_207_add__less__imp__less__left,axiom,
! [C_31,A_79,B_58] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(plus_plus_real(C_31),A_79)),hAPP_real_real(plus_plus_real(C_31),B_58)))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_79),B_58)) ) ).
fof(fact_208_add__less__imp__less__left,axiom,
! [C_31,A_79,B_58] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(plus_plus_nat(C_31),A_79)),hAPP_nat_nat(plus_plus_nat(C_31),B_58)))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_79),B_58)) ) ).
fof(fact_209_comm__semiring__1__class_Onormalizing__semiring__rules_I11_J,axiom,
! [A_78] : hAPP_int_int(times_times_int(one_one_int),A_78) = A_78 ).
fof(fact_210_comm__semiring__1__class_Onormalizing__semiring__rules_I11_J,axiom,
! [A_78] : hAPP_real_real(times_times_real(one_one_real),A_78) = A_78 ).
fof(fact_211_comm__semiring__1__class_Onormalizing__semiring__rules_I11_J,axiom,
! [A_78] : hAPP_nat_nat(times_times_nat(one_one_nat),A_78) = A_78 ).
fof(fact_212_comm__semiring__1__class_Onormalizing__semiring__rules_I12_J,axiom,
! [A_77] : hAPP_int_int(times_times_int(A_77),one_one_int) = A_77 ).
fof(fact_213_comm__semiring__1__class_Onormalizing__semiring__rules_I12_J,axiom,
! [A_77] : hAPP_real_real(times_times_real(A_77),one_one_real) = A_77 ).
fof(fact_214_comm__semiring__1__class_Onormalizing__semiring__rules_I12_J,axiom,
! [A_77] : hAPP_nat_nat(times_times_nat(A_77),one_one_nat) = A_77 ).
fof(fact_215_mult__1__left,axiom,
! [A_76] : hAPP_int_int(times_times_int(one_one_int),A_76) = A_76 ).
fof(fact_216_mult__1__left,axiom,
! [A_76] : hAPP_real_real(times_times_real(one_one_real),A_76) = A_76 ).
fof(fact_217_mult__1__left,axiom,
! [A_76] : hAPP_nat_nat(times_times_nat(one_one_nat),A_76) = A_76 ).
fof(fact_218_mult__1,axiom,
! [A_75] : hAPP_int_int(times_times_int(one_one_int),A_75) = A_75 ).
fof(fact_219_mult__1,axiom,
! [A_75] : hAPP_real_real(times_times_real(one_one_real),A_75) = A_75 ).
fof(fact_220_mult__1,axiom,
! [A_75] : hAPP_nat_nat(times_times_nat(one_one_nat),A_75) = A_75 ).
fof(fact_221_mult__1__right,axiom,
! [A_74] : hAPP_int_int(times_times_int(A_74),one_one_int) = A_74 ).
fof(fact_222_mult__1__right,axiom,
! [A_74] : hAPP_real_real(times_times_real(A_74),one_one_real) = A_74 ).
fof(fact_223_mult__1__right,axiom,
! [A_74] : hAPP_nat_nat(times_times_nat(A_74),one_one_nat) = A_74 ).
fof(fact_224_mult_Ocomm__neutral,axiom,
! [A_73] : hAPP_int_int(times_times_int(A_73),one_one_int) = A_73 ).
fof(fact_225_mult_Ocomm__neutral,axiom,
! [A_73] : hAPP_real_real(times_times_real(A_73),one_one_real) = A_73 ).
fof(fact_226_mult_Ocomm__neutral,axiom,
! [A_73] : hAPP_nat_nat(times_times_nat(A_73),one_one_nat) = A_73 ).
fof(fact_227_diff__eq__diff__less,axiom,
! [Aa,Ba,C_1,D_2] :
( hAPP_int_int(minus_minus_int(Aa),Ba) = hAPP_int_int(minus_minus_int(C_1),D_2)
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Aa),Ba))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,C_1),D_2)) ) ) ).
fof(fact_228_diff__eq__diff__less,axiom,
! [Aa,Ba,C_1,D_2] :
( hAPP_real_real(minus_minus_real(Aa),Ba) = hAPP_real_real(minus_minus_real(C_1),D_2)
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Aa),Ba))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,C_1),D_2)) ) ) ).
fof(fact_229_add__le__cancel__right,axiom,
! [Aa,C_1,Ba] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(plus_plus_int(Aa),C_1)),hAPP_int_int(plus_plus_int(Ba),C_1)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Aa),Ba)) ) ).
fof(fact_230_add__le__cancel__right,axiom,
! [Aa,C_1,Ba] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(plus_plus_real(Aa),C_1)),hAPP_real_real(plus_plus_real(Ba),C_1)))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Aa),Ba)) ) ).
fof(fact_231_add__le__cancel__right,axiom,
! [Aa,C_1,Ba] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(plus_plus_nat(Aa),C_1)),hAPP_nat_nat(plus_plus_nat(Ba),C_1)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Aa),Ba)) ) ).
fof(fact_232_add__le__cancel__left,axiom,
! [C_1,Aa,Ba] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(plus_plus_int(C_1),Aa)),hAPP_int_int(plus_plus_int(C_1),Ba)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Aa),Ba)) ) ).
fof(fact_233_add__le__cancel__left,axiom,
! [C_1,Aa,Ba] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(plus_plus_real(C_1),Aa)),hAPP_real_real(plus_plus_real(C_1),Ba)))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Aa),Ba)) ) ).
fof(fact_234_add__le__cancel__left,axiom,
! [C_1,Aa,Ba] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(plus_plus_nat(C_1),Aa)),hAPP_nat_nat(plus_plus_nat(C_1),Ba)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Aa),Ba)) ) ).
fof(fact_235_add__right__mono,axiom,
! [C_30,A_72,B_57] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_72),B_57))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(plus_plus_int(A_72),C_30)),hAPP_int_int(plus_plus_int(B_57),C_30))) ) ).
fof(fact_236_add__right__mono,axiom,
! [C_30,A_72,B_57] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_72),B_57))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(plus_plus_real(A_72),C_30)),hAPP_real_real(plus_plus_real(B_57),C_30))) ) ).
fof(fact_237_add__right__mono,axiom,
! [C_30,A_72,B_57] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_72),B_57))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(plus_plus_nat(A_72),C_30)),hAPP_nat_nat(plus_plus_nat(B_57),C_30))) ) ).
fof(fact_238_add__left__mono,axiom,
! [C_29,A_71,B_56] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_71),B_56))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(plus_plus_int(C_29),A_71)),hAPP_int_int(plus_plus_int(C_29),B_56))) ) ).
fof(fact_239_add__left__mono,axiom,
! [C_29,A_71,B_56] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_71),B_56))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(plus_plus_real(C_29),A_71)),hAPP_real_real(plus_plus_real(C_29),B_56))) ) ).
fof(fact_240_add__left__mono,axiom,
! [C_29,A_71,B_56] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_71),B_56))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(plus_plus_nat(C_29),A_71)),hAPP_nat_nat(plus_plus_nat(C_29),B_56))) ) ).
fof(fact_241_add__mono,axiom,
! [C_28,D_5,A_70,B_55] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_70),B_55))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,C_28),D_5))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(plus_plus_int(A_70),C_28)),hAPP_int_int(plus_plus_int(B_55),D_5))) ) ) ).
fof(fact_242_add__mono,axiom,
! [C_28,D_5,A_70,B_55] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_70),B_55))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,C_28),D_5))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(plus_plus_real(A_70),C_28)),hAPP_real_real(plus_plus_real(B_55),D_5))) ) ) ).
fof(fact_243_add__mono,axiom,
! [C_28,D_5,A_70,B_55] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_70),B_55))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,C_28),D_5))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(plus_plus_nat(A_70),C_28)),hAPP_nat_nat(plus_plus_nat(B_55),D_5))) ) ) ).
fof(fact_244_add__le__imp__le__right,axiom,
! [A_69,C_27,B_54] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(plus_plus_int(A_69),C_27)),hAPP_int_int(plus_plus_int(B_54),C_27)))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_69),B_54)) ) ).
fof(fact_245_add__le__imp__le__right,axiom,
! [A_69,C_27,B_54] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(plus_plus_real(A_69),C_27)),hAPP_real_real(plus_plus_real(B_54),C_27)))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_69),B_54)) ) ).
fof(fact_246_add__le__imp__le__right,axiom,
! [A_69,C_27,B_54] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(plus_plus_nat(A_69),C_27)),hAPP_nat_nat(plus_plus_nat(B_54),C_27)))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_69),B_54)) ) ).
fof(fact_247_add__le__imp__le__left,axiom,
! [C_26,A_68,B_53] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(plus_plus_int(C_26),A_68)),hAPP_int_int(plus_plus_int(C_26),B_53)))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_68),B_53)) ) ).
fof(fact_248_add__le__imp__le__left,axiom,
! [C_26,A_68,B_53] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(plus_plus_real(C_26),A_68)),hAPP_real_real(plus_plus_real(C_26),B_53)))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_68),B_53)) ) ).
fof(fact_249_add__le__imp__le__left,axiom,
! [C_26,A_68,B_53] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(plus_plus_nat(C_26),A_68)),hAPP_nat_nat(plus_plus_nat(C_26),B_53)))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_68),B_53)) ) ).
fof(fact_250_diff__eq__diff__less__eq,axiom,
! [Aa,Ba,C_1,D_2] :
( hAPP_int_int(minus_minus_int(Aa),Ba) = hAPP_int_int(minus_minus_int(C_1),D_2)
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Aa),Ba))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,C_1),D_2)) ) ) ).
fof(fact_251_diff__eq__diff__less__eq,axiom,
! [Aa,Ba,C_1,D_2] :
( hAPP_real_real(minus_minus_real(Aa),Ba) = hAPP_real_real(minus_minus_real(C_1),D_2)
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Aa),Ba))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,C_1),D_2)) ) ) ).
fof(fact_252_times__numeral__code_I5_J,axiom,
! [V,W_1] : hAPP_int_int(times_times_int(number_number_of_int(V)),number_number_of_int(W_1)) = number_number_of_int(hAPP_int_int(times_times_int(V),W_1)) ).
fof(fact_253_plus__numeral__code_I9_J,axiom,
! [V,W_1] : hAPP_int_int(plus_plus_int(number_number_of_int(V)),number_number_of_int(W_1)) = number_number_of_int(hAPP_int_int(plus_plus_int(V),W_1)) ).
fof(fact_254_zadd__strict__right__mono,axiom,
! [K,I,J] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,I),J))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(plus_plus_int(I),K)),hAPP_int_int(plus_plus_int(J),K))) ) ).
fof(fact_255_zadd__left__mono,axiom,
! [K,I,J] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,I),J))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(plus_plus_int(K),I)),hAPP_int_int(plus_plus_int(K),J))) ) ).
fof(fact_256_zmult__1,axiom,
! [Z_1] : hAPP_int_int(times_times_int(one_one_int),Z_1) = Z_1 ).
fof(fact_257_zmult__1__right,axiom,
! [Z_1] : hAPP_int_int(times_times_int(Z_1),one_one_int) = Z_1 ).
fof(fact_258_right__distrib__number__of,axiom,
! [V_16,B_52,C_25] : hAPP_int_int(times_times_int(number_number_of_int(V_16)),hAPP_int_int(plus_plus_int(B_52),C_25)) = hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(number_number_of_int(V_16)),B_52)),hAPP_int_int(times_times_int(number_number_of_int(V_16)),C_25)) ).
fof(fact_259_right__distrib__number__of,axiom,
! [V_16,B_52,C_25] : hAPP_real_real(times_times_real(number267125858f_real(V_16)),hAPP_real_real(plus_plus_real(B_52),C_25)) = hAPP_real_real(plus_plus_real(hAPP_real_real(times_times_real(number267125858f_real(V_16)),B_52)),hAPP_real_real(times_times_real(number267125858f_real(V_16)),C_25)) ).
fof(fact_260_right__distrib__number__of,axiom,
! [V_16,B_52,C_25] : hAPP_nat_nat(times_times_nat(number_number_of_nat(V_16)),hAPP_nat_nat(plus_plus_nat(B_52),C_25)) = hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(times_times_nat(number_number_of_nat(V_16)),B_52)),hAPP_nat_nat(times_times_nat(number_number_of_nat(V_16)),C_25)) ).
fof(fact_261_left__distrib__number__of,axiom,
! [A_67,B_51,V_15] : hAPP_int_int(times_times_int(hAPP_int_int(plus_plus_int(A_67),B_51)),number_number_of_int(V_15)) = hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(A_67),number_number_of_int(V_15))),hAPP_int_int(times_times_int(B_51),number_number_of_int(V_15))) ).
fof(fact_262_left__distrib__number__of,axiom,
! [A_67,B_51,V_15] : hAPP_real_real(times_times_real(hAPP_real_real(plus_plus_real(A_67),B_51)),number267125858f_real(V_15)) = hAPP_real_real(plus_plus_real(hAPP_real_real(times_times_real(A_67),number267125858f_real(V_15))),hAPP_real_real(times_times_real(B_51),number267125858f_real(V_15))) ).
fof(fact_263_left__distrib__number__of,axiom,
! [A_67,B_51,V_15] : hAPP_nat_nat(times_times_nat(hAPP_nat_nat(plus_plus_nat(A_67),B_51)),number_number_of_nat(V_15)) = hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(times_times_nat(A_67),number_number_of_nat(V_15))),hAPP_nat_nat(times_times_nat(B_51),number_number_of_nat(V_15))) ).
fof(fact_264_right__diff__distrib__number__of,axiom,
! [V_14,B_50,C_24] : hAPP_int_int(times_times_int(number_number_of_int(V_14)),hAPP_int_int(minus_minus_int(B_50),C_24)) = hAPP_int_int(minus_minus_int(hAPP_int_int(times_times_int(number_number_of_int(V_14)),B_50)),hAPP_int_int(times_times_int(number_number_of_int(V_14)),C_24)) ).
fof(fact_265_right__diff__distrib__number__of,axiom,
! [V_14,B_50,C_24] : hAPP_real_real(times_times_real(number267125858f_real(V_14)),hAPP_real_real(minus_minus_real(B_50),C_24)) = hAPP_real_real(minus_minus_real(hAPP_real_real(times_times_real(number267125858f_real(V_14)),B_50)),hAPP_real_real(times_times_real(number267125858f_real(V_14)),C_24)) ).
fof(fact_266_left__diff__distrib__number__of,axiom,
! [A_66,B_49,V_13] : hAPP_int_int(times_times_int(hAPP_int_int(minus_minus_int(A_66),B_49)),number_number_of_int(V_13)) = hAPP_int_int(minus_minus_int(hAPP_int_int(times_times_int(A_66),number_number_of_int(V_13))),hAPP_int_int(times_times_int(B_49),number_number_of_int(V_13))) ).
fof(fact_267_left__diff__distrib__number__of,axiom,
! [A_66,B_49,V_13] : hAPP_real_real(times_times_real(hAPP_real_real(minus_minus_real(A_66),B_49)),number267125858f_real(V_13)) = hAPP_real_real(minus_minus_real(hAPP_real_real(times_times_real(A_66),number267125858f_real(V_13))),hAPP_real_real(times_times_real(B_49),number267125858f_real(V_13))) ).
fof(fact_268_number__of__mult,axiom,
! [V_12,W_9] : number_number_of_int(hAPP_int_int(times_times_int(V_12),W_9)) = hAPP_int_int(times_times_int(number_number_of_int(V_12)),number_number_of_int(W_9)) ).
fof(fact_269_number__of__mult,axiom,
! [V_12,W_9] : number267125858f_real(hAPP_int_int(times_times_int(V_12),W_9)) = hAPP_real_real(times_times_real(number267125858f_real(V_12)),number267125858f_real(W_9)) ).
fof(fact_270_arith__simps_I32_J,axiom,
! [V_11,W_8] : hAPP_int_int(times_times_int(number_number_of_int(V_11)),number_number_of_int(W_8)) = number_number_of_int(hAPP_int_int(times_times_int(V_11),W_8)) ).
fof(fact_271_arith__simps_I32_J,axiom,
! [V_11,W_8] : hAPP_real_real(times_times_real(number267125858f_real(V_11)),number267125858f_real(W_8)) = number267125858f_real(hAPP_int_int(times_times_int(V_11),W_8)) ).
fof(fact_272_mult__number__of__left,axiom,
! [V_10,W_7,Z_15] : hAPP_int_int(times_times_int(number_number_of_int(V_10)),hAPP_int_int(times_times_int(number_number_of_int(W_7)),Z_15)) = hAPP_int_int(times_times_int(number_number_of_int(hAPP_int_int(times_times_int(V_10),W_7))),Z_15) ).
fof(fact_273_mult__number__of__left,axiom,
! [V_10,W_7,Z_15] : hAPP_real_real(times_times_real(number267125858f_real(V_10)),hAPP_real_real(times_times_real(number267125858f_real(W_7)),Z_15)) = hAPP_real_real(times_times_real(number267125858f_real(hAPP_int_int(times_times_int(V_10),W_7))),Z_15) ).
fof(fact_274_number__of__add,axiom,
! [V_9,W_6] : number_number_of_int(hAPP_int_int(plus_plus_int(V_9),W_6)) = hAPP_int_int(plus_plus_int(number_number_of_int(V_9)),number_number_of_int(W_6)) ).
fof(fact_275_number__of__add,axiom,
! [V_9,W_6] : number267125858f_real(hAPP_int_int(plus_plus_int(V_9),W_6)) = hAPP_real_real(plus_plus_real(number267125858f_real(V_9)),number267125858f_real(W_6)) ).
fof(fact_276_add__number__of__eq,axiom,
! [V_8,W_5] : hAPP_int_int(plus_plus_int(number_number_of_int(V_8)),number_number_of_int(W_5)) = number_number_of_int(hAPP_int_int(plus_plus_int(V_8),W_5)) ).
fof(fact_277_add__number__of__eq,axiom,
! [V_8,W_5] : hAPP_real_real(plus_plus_real(number267125858f_real(V_8)),number267125858f_real(W_5)) = number267125858f_real(hAPP_int_int(plus_plus_int(V_8),W_5)) ).
fof(fact_278_add__number__of__left,axiom,
! [V_7,W_4,Z_14] : hAPP_int_int(plus_plus_int(number_number_of_int(V_7)),hAPP_int_int(plus_plus_int(number_number_of_int(W_4)),Z_14)) = hAPP_int_int(plus_plus_int(number_number_of_int(hAPP_int_int(plus_plus_int(V_7),W_4))),Z_14) ).
fof(fact_279_add__number__of__left,axiom,
! [V_7,W_4,Z_14] : hAPP_real_real(plus_plus_real(number267125858f_real(V_7)),hAPP_real_real(plus_plus_real(number267125858f_real(W_4)),Z_14)) = hAPP_real_real(plus_plus_real(number267125858f_real(hAPP_int_int(plus_plus_int(V_7),W_4))),Z_14) ).
fof(fact_280_number__of__diff,axiom,
! [V_6,W_3] : number_number_of_int(hAPP_int_int(minus_minus_int(V_6),W_3)) = hAPP_int_int(minus_minus_int(number_number_of_int(V_6)),number_number_of_int(W_3)) ).
fof(fact_281_number__of__diff,axiom,
! [V_6,W_3] : number267125858f_real(hAPP_int_int(minus_minus_int(V_6),W_3)) = hAPP_real_real(minus_minus_real(number267125858f_real(V_6)),number267125858f_real(W_3)) ).
fof(fact_282_comm__semiring__1__class_Onormalizing__semiring__rules_I4_J,axiom,
! [M_5] : hAPP_int_int(plus_plus_int(M_5),M_5) = hAPP_int_int(times_times_int(hAPP_int_int(plus_plus_int(one_one_int),one_one_int)),M_5) ).
fof(fact_283_comm__semiring__1__class_Onormalizing__semiring__rules_I4_J,axiom,
! [M_5] : hAPP_real_real(plus_plus_real(M_5),M_5) = hAPP_real_real(times_times_real(hAPP_real_real(plus_plus_real(one_one_real),one_one_real)),M_5) ).
fof(fact_284_comm__semiring__1__class_Onormalizing__semiring__rules_I4_J,axiom,
! [M_5] : hAPP_nat_nat(plus_plus_nat(M_5),M_5) = hAPP_nat_nat(times_times_nat(hAPP_nat_nat(plus_plus_nat(one_one_nat),one_one_nat)),M_5) ).
fof(fact_285_comm__semiring__1__class_Onormalizing__semiring__rules_I3_J,axiom,
! [M_4,A_65] : hAPP_int_int(plus_plus_int(M_4),hAPP_int_int(times_times_int(A_65),M_4)) = hAPP_int_int(times_times_int(hAPP_int_int(plus_plus_int(A_65),one_one_int)),M_4) ).
fof(fact_286_comm__semiring__1__class_Onormalizing__semiring__rules_I3_J,axiom,
! [M_4,A_65] : hAPP_real_real(plus_plus_real(M_4),hAPP_real_real(times_times_real(A_65),M_4)) = hAPP_real_real(times_times_real(hAPP_real_real(plus_plus_real(A_65),one_one_real)),M_4) ).
fof(fact_287_comm__semiring__1__class_Onormalizing__semiring__rules_I3_J,axiom,
! [M_4,A_65] : hAPP_nat_nat(plus_plus_nat(M_4),hAPP_nat_nat(times_times_nat(A_65),M_4)) = hAPP_nat_nat(times_times_nat(hAPP_nat_nat(plus_plus_nat(A_65),one_one_nat)),M_4) ).
fof(fact_288_comm__semiring__1__class_Onormalizing__semiring__rules_I2_J,axiom,
! [A_64,M_3] : hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(A_64),M_3)),M_3) = hAPP_int_int(times_times_int(hAPP_int_int(plus_plus_int(A_64),one_one_int)),M_3) ).
fof(fact_289_comm__semiring__1__class_Onormalizing__semiring__rules_I2_J,axiom,
! [A_64,M_3] : hAPP_real_real(plus_plus_real(hAPP_real_real(times_times_real(A_64),M_3)),M_3) = hAPP_real_real(times_times_real(hAPP_real_real(plus_plus_real(A_64),one_one_real)),M_3) ).
fof(fact_290_comm__semiring__1__class_Onormalizing__semiring__rules_I2_J,axiom,
! [A_64,M_3] : hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(times_times_nat(A_64),M_3)),M_3) = hAPP_nat_nat(times_times_nat(hAPP_nat_nat(plus_plus_nat(A_64),one_one_nat)),M_3) ).
fof(fact_291_curryD,axiom,
! [F,Aa,Ba] :
( hBOOL(hAPP_int_bool(produc262399358t_bool(F,Aa),Ba))
=> hBOOL(hAPP_P603027463t_bool(F,product_Pair_int_int(Aa,Ba))) ) ).
fof(fact_292_curryD,axiom,
! [F,Aa,Ba] :
( hBOOL(hAPP_P603027463t_bool(produc97574839t_bool(F,Aa),Ba))
=> hBOOL(hAPP_P178748804t_bool(F,produc282740534nt_int(Aa,Ba))) ) ).
fof(fact_293_curryE,axiom,
! [F,Aa,Ba] :
( hBOOL(hAPP_int_bool(produc262399358t_bool(F,Aa),Ba))
=> hBOOL(hAPP_P603027463t_bool(F,product_Pair_int_int(Aa,Ba))) ) ).
fof(fact_294_curryE,axiom,
! [F,Aa,Ba] :
( hBOOL(hAPP_P603027463t_bool(produc97574839t_bool(F,Aa),Ba))
=> hBOOL(hAPP_P178748804t_bool(F,produc282740534nt_int(Aa,Ba))) ) ).
fof(fact_295_order__refl,axiom,
! [X_37] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_37),X_37)) ).
fof(fact_296_order__refl,axiom,
! [X_37] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_37),X_37)) ).
fof(fact_297_order__refl,axiom,
! [X_37] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_37),X_37)) ).
fof(fact_298_xt1_I8_J,axiom,
! [Z_13,Y_36,X_36] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y_36),X_36))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Z_13),Y_36))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Z_13),X_36)) ) ) ).
fof(fact_299_xt1_I8_J,axiom,
! [Z_13,Y_36,X_36] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y_36),X_36))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Z_13),Y_36))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Z_13),X_36)) ) ) ).
fof(fact_300_xt1_I8_J,axiom,
! [Z_13,Y_36,X_36] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Y_36),X_36))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Z_13),Y_36))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Z_13),X_36)) ) ) ).
fof(fact_301_order__le__less__trans,axiom,
! [Z_12,X_35,Y_35] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_35),Y_35))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Y_35),Z_12))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_35),Z_12)) ) ) ).
fof(fact_302_order__le__less__trans,axiom,
! [Z_12,X_35,Y_35] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_35),Y_35))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Y_35),Z_12))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_35),Z_12)) ) ) ).
fof(fact_303_order__le__less__trans,axiom,
! [Z_12,X_35,Y_35] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_35),Y_35))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Y_35),Z_12))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_35),Z_12)) ) ) ).
fof(fact_304_xt1_I7_J,axiom,
! [Z_11,Y_34,X_34] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Y_34),X_34))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Z_11),Y_34))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Z_11),X_34)) ) ) ).
fof(fact_305_xt1_I7_J,axiom,
! [Z_11,Y_34,X_34] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Y_34),X_34))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Z_11),Y_34))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Z_11),X_34)) ) ) ).
fof(fact_306_xt1_I7_J,axiom,
! [Z_11,Y_34,X_34] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Y_34),X_34))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Z_11),Y_34))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Z_11),X_34)) ) ) ).
fof(fact_307_order__less__le__trans,axiom,
! [Z_10,X_33,Y_33] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_33),Y_33))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y_33),Z_10))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_33),Z_10)) ) ) ).
fof(fact_308_order__less__le__trans,axiom,
! [Z_10,X_33,Y_33] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_33),Y_33))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y_33),Z_10))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_33),Z_10)) ) ) ).
fof(fact_309_order__less__le__trans,axiom,
! [Z_10,X_33,Y_33] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_33),Y_33))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Y_33),Z_10))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_33),Z_10)) ) ) ).
fof(fact_310_xt1_I11_J,axiom,
! [B_48,A_63] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_48),A_63))
=> ( A_63 != B_48
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_48),A_63)) ) ) ).
fof(fact_311_xt1_I11_J,axiom,
! [B_48,A_63] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B_48),A_63))
=> ( A_63 != B_48
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,B_48),A_63)) ) ) ).
fof(fact_312_xt1_I11_J,axiom,
! [B_48,A_63] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,B_48),A_63))
=> ( A_63 != B_48
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,B_48),A_63)) ) ) ).
fof(fact_313_order__le__neq__trans,axiom,
! [A_62,B_47] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_62),B_47))
=> ( A_62 != B_47
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_62),B_47)) ) ) ).
fof(fact_314_order__le__neq__trans,axiom,
! [A_62,B_47] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_62),B_47))
=> ( A_62 != B_47
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_62),B_47)) ) ) ).
fof(fact_315_order__le__neq__trans,axiom,
! [A_62,B_47] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_62),B_47))
=> ( A_62 != B_47
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_62),B_47)) ) ) ).
fof(fact_316_order__le__imp__less__or__eq,axiom,
! [X_32,Y_32] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_32),Y_32))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_32),Y_32))
| X_32 = Y_32 ) ) ).
fof(fact_317_order__le__imp__less__or__eq,axiom,
! [X_32,Y_32] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_32),Y_32))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_32),Y_32))
| X_32 = Y_32 ) ) ).
fof(fact_318_order__le__imp__less__or__eq,axiom,
! [X_32,Y_32] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_32),Y_32))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_32),Y_32))
| X_32 = Y_32 ) ) ).
fof(fact_319_linorder__antisym__conv2,axiom,
! [X_1,Y_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_1),Y_1))
=> ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_1),Y_1))
<=> X_1 = Y_1 ) ) ).
fof(fact_320_linorder__antisym__conv2,axiom,
! [X_1,Y_1] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_1),Y_1))
=> ( ~ hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_1),Y_1))
<=> X_1 = Y_1 ) ) ).
fof(fact_321_linorder__antisym__conv2,axiom,
! [X_1,Y_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_1),Y_1))
=> ( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_1),Y_1))
<=> X_1 = Y_1 ) ) ).
fof(fact_322_order__less__imp__le,axiom,
! [X_31,Y_31] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_31),Y_31))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_31),Y_31)) ) ).
fof(fact_323_order__less__imp__le,axiom,
! [X_31,Y_31] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_31),Y_31))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_31),Y_31)) ) ).
fof(fact_324_order__less__imp__le,axiom,
! [X_31,Y_31] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_31),Y_31))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_31),Y_31)) ) ).
fof(fact_325_zle__refl,axiom,
! [W_1] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,W_1),W_1)) ).
fof(fact_326_number__of__is__id,axiom,
! [K] : number_number_of_int(K) = K ).
fof(fact_327_zle__linear,axiom,
! [Z_1,W_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Z_1),W_1))
| hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,W_1),Z_1)) ) ).
fof(fact_328_zless__le,axiom,
! [Z,W] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Z),W))
<=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Z),W))
& Z != W ) ) ).
fof(fact_329_zless__linear,axiom,
! [X,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_330_less__number__of__int__code,axiom,
! [K_1,L] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,number_number_of_int(K_1)),number_number_of_int(L)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K_1),L)) ) ).
fof(fact_331_less__eq__number__of__int__code,axiom,
! [K_1,L] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,number_number_of_int(K_1)),number_number_of_int(L)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K_1),L)) ) ).
fof(fact_332_zle__trans,axiom,
! [K,I,J] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,I),J))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,J),K))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,I),K)) ) ) ).
fof(fact_333_zle__antisym,axiom,
! [Z_1,W_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Z_1),W_1))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,W_1),Z_1))
=> Z_1 = W_1 ) ) ).
fof(fact_334_linorder__linear,axiom,
! [X_30,Y_30] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_30),Y_30))
| hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y_30),X_30)) ) ).
fof(fact_335_linorder__linear,axiom,
! [X_30,Y_30] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_30),Y_30))
| hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y_30),X_30)) ) ).
fof(fact_336_linorder__linear,axiom,
! [X_30,Y_30] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_30),Y_30))
| hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Y_30),X_30)) ) ).
fof(fact_337_order__eq__iff,axiom,
! [X_1,Y_1] :
( X_1 = Y_1
<=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_1),Y_1))
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y_1),X_1)) ) ) ).
fof(fact_338_order__eq__iff,axiom,
! [X_1,Y_1] :
( X_1 = Y_1
<=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_1),Y_1))
& hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y_1),X_1)) ) ) ).
fof(fact_339_order__eq__iff,axiom,
! [X_1,Y_1] :
( X_1 = Y_1
<=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_1),Y_1))
& hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Y_1),X_1)) ) ) ).
fof(fact_340_order__eq__refl,axiom,
! [X_29,Y_29] :
( X_29 = Y_29
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_29),Y_29)) ) ).
fof(fact_341_order__eq__refl,axiom,
! [X_29,Y_29] :
( X_29 = Y_29
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_29),Y_29)) ) ).
fof(fact_342_order__eq__refl,axiom,
! [X_29,Y_29] :
( X_29 = Y_29
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_29),Y_29)) ) ).
fof(fact_343_order__antisym__conv,axiom,
! [Y_1,X_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y_1),X_1))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_1),Y_1))
<=> X_1 = Y_1 ) ) ).
fof(fact_344_order__antisym__conv,axiom,
! [Y_1,X_1] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y_1),X_1))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_1),Y_1))
<=> X_1 = Y_1 ) ) ).
fof(fact_345_order__antisym__conv,axiom,
! [Y_1,X_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Y_1),X_1))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_1),Y_1))
<=> X_1 = Y_1 ) ) ).
fof(fact_346_ord__eq__le__trans,axiom,
! [C_23,A_61,B_46] :
( A_61 = B_46
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_46),C_23))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_61),C_23)) ) ) ).
fof(fact_347_ord__eq__le__trans,axiom,
! [C_23,A_61,B_46] :
( A_61 = B_46
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B_46),C_23))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_61),C_23)) ) ) ).
fof(fact_348_ord__eq__le__trans,axiom,
! [C_23,A_61,B_46] :
( A_61 = B_46
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,B_46),C_23))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_61),C_23)) ) ) ).
fof(fact_349_xt1_I3_J,axiom,
! [C_22,A_60,B_45] :
( A_60 = B_45
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,C_22),B_45))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,C_22),A_60)) ) ) ).
fof(fact_350_xt1_I3_J,axiom,
! [C_22,A_60,B_45] :
( A_60 = B_45
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,C_22),B_45))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,C_22),A_60)) ) ) ).
fof(fact_351_xt1_I3_J,axiom,
! [C_22,A_60,B_45] :
( A_60 = B_45
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,C_22),B_45))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,C_22),A_60)) ) ) ).
fof(fact_352_ord__le__eq__trans,axiom,
! [C_21,A_59,B_44] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_59),B_44))
=> ( B_44 = C_21
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_59),C_21)) ) ) ).
fof(fact_353_ord__le__eq__trans,axiom,
! [C_21,A_59,B_44] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_59),B_44))
=> ( B_44 = C_21
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_59),C_21)) ) ) ).
fof(fact_354_ord__le__eq__trans,axiom,
! [C_21,A_59,B_44] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_59),B_44))
=> ( B_44 = C_21
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_59),C_21)) ) ) ).
fof(fact_355_xt1_I4_J,axiom,
! [C_20,B_43,A_58] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_43),A_58))
=> ( B_43 = C_20
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,C_20),A_58)) ) ) ).
fof(fact_356_xt1_I4_J,axiom,
! [C_20,B_43,A_58] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B_43),A_58))
=> ( B_43 = C_20
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,C_20),A_58)) ) ) ).
fof(fact_357_xt1_I4_J,axiom,
! [C_20,B_43,A_58] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,B_43),A_58))
=> ( B_43 = C_20
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,C_20),A_58)) ) ) ).
fof(fact_358_order__antisym,axiom,
! [X_28,Y_28] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_28),Y_28))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y_28),X_28))
=> X_28 = Y_28 ) ) ).
fof(fact_359_order__antisym,axiom,
! [X_28,Y_28] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_28),Y_28))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y_28),X_28))
=> X_28 = Y_28 ) ) ).
fof(fact_360_order__antisym,axiom,
! [X_28,Y_28] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_28),Y_28))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Y_28),X_28))
=> X_28 = Y_28 ) ) ).
fof(fact_361_order__trans,axiom,
! [Z_9,X_27,Y_27] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_27),Y_27))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y_27),Z_9))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_27),Z_9)) ) ) ).
fof(fact_362_order__trans,axiom,
! [Z_9,X_27,Y_27] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_27),Y_27))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y_27),Z_9))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_27),Z_9)) ) ) ).
fof(fact_363_order__trans,axiom,
! [Z_9,X_27,Y_27] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_27),Y_27))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Y_27),Z_9))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_27),Z_9)) ) ) ).
fof(fact_364_xt1_I5_J,axiom,
! [Y_26,X_26] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y_26),X_26))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_26),Y_26))
=> X_26 = Y_26 ) ) ).
fof(fact_365_xt1_I5_J,axiom,
! [Y_26,X_26] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y_26),X_26))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_26),Y_26))
=> X_26 = Y_26 ) ) ).
fof(fact_366_xt1_I5_J,axiom,
! [Y_26,X_26] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Y_26),X_26))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_26),Y_26))
=> X_26 = Y_26 ) ) ).
fof(fact_367_xt1_I6_J,axiom,
! [Z_8,Y_25,X_25] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y_25),X_25))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Z_8),Y_25))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Z_8),X_25)) ) ) ).
fof(fact_368_xt1_I6_J,axiom,
! [Z_8,Y_25,X_25] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y_25),X_25))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Z_8),Y_25))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Z_8),X_25)) ) ) ).
fof(fact_369_xt1_I6_J,axiom,
! [Z_8,Y_25,X_25] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Y_25),X_25))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Z_8),Y_25))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Z_8),X_25)) ) ) ).
fof(fact_370_linorder__le__cases,axiom,
! [X_24,Y_24] :
( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_24),Y_24))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y_24),X_24)) ) ).
fof(fact_371_linorder__le__cases,axiom,
! [X_24,Y_24] :
( ~ hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_24),Y_24))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y_24),X_24)) ) ).
fof(fact_372_linorder__le__cases,axiom,
! [X_24,Y_24] :
( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_24),Y_24))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Y_24),X_24)) ) ).
fof(fact_373_order__less__irrefl,axiom,
! [X_23] : ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_23),X_23)) ).
fof(fact_374_order__less__irrefl,axiom,
! [X_23] : ~ hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_23),X_23)) ).
fof(fact_375_order__less__irrefl,axiom,
! [X_23] : ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_23),X_23)) ).
fof(fact_376_linorder__neq__iff,axiom,
! [X_1,Y_1] :
( X_1 != Y_1
<=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_1),Y_1))
| hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Y_1),X_1)) ) ) ).
fof(fact_377_linorder__neq__iff,axiom,
! [X_1,Y_1] :
( X_1 != Y_1
<=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_1),Y_1))
| hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Y_1),X_1)) ) ) ).
fof(fact_378_linorder__neq__iff,axiom,
! [X_1,Y_1] :
( X_1 != Y_1
<=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_1),Y_1))
| hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Y_1),X_1)) ) ) ).
fof(fact_379_not__less__iff__gr__or__eq,axiom,
! [X_1,Y_1] :
( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_1),Y_1))
<=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Y_1),X_1))
| X_1 = Y_1 ) ) ).
fof(fact_380_not__less__iff__gr__or__eq,axiom,
! [X_1,Y_1] :
( ~ hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_1),Y_1))
<=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Y_1),X_1))
| X_1 = Y_1 ) ) ).
fof(fact_381_not__less__iff__gr__or__eq,axiom,
! [X_1,Y_1] :
( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_1),Y_1))
<=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Y_1),X_1))
| X_1 = Y_1 ) ) ).
fof(fact_382_linorder__less__linear,axiom,
! [X_22,Y_23] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_22),Y_23))
| X_22 = Y_23
| hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Y_23),X_22)) ) ).
fof(fact_383_linorder__less__linear,axiom,
! [X_22,Y_23] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_22),Y_23))
| X_22 = Y_23
| hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Y_23),X_22)) ) ).
fof(fact_384_linorder__less__linear,axiom,
! [X_22,Y_23] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_22),Y_23))
| X_22 = Y_23
| hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Y_23),X_22)) ) ).
fof(fact_385_linorder__antisym__conv3,axiom,
! [Y_1,X_1] :
( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Y_1),X_1))
=> ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_1),Y_1))
<=> X_1 = Y_1 ) ) ).
fof(fact_386_linorder__antisym__conv3,axiom,
! [Y_1,X_1] :
( ~ hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Y_1),X_1))
=> ( ~ hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_1),Y_1))
<=> X_1 = Y_1 ) ) ).
fof(fact_387_linorder__antisym__conv3,axiom,
! [Y_1,X_1] :
( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Y_1),X_1))
=> ( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_1),Y_1))
<=> X_1 = Y_1 ) ) ).
fof(fact_388_linorder__neqE,axiom,
! [X_21,Y_22] :
( X_21 != Y_22
=> ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_21),Y_22))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Y_22),X_21)) ) ) ).
fof(fact_389_linorder__neqE,axiom,
! [X_21,Y_22] :
( X_21 != Y_22
=> ( ~ hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_21),Y_22))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Y_22),X_21)) ) ) ).
fof(fact_390_linorder__neqE,axiom,
! [X_21,Y_22] :
( X_21 != Y_22
=> ( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_21),Y_22))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Y_22),X_21)) ) ) ).
fof(fact_391_less__imp__neq,axiom,
! [X_20,Y_21] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_20),Y_21))
=> X_20 != Y_21 ) ).
fof(fact_392_less__imp__neq,axiom,
! [X_20,Y_21] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_20),Y_21))
=> X_20 != Y_21 ) ).
fof(fact_393_less__imp__neq,axiom,
! [X_20,Y_21] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_20),Y_21))
=> X_20 != Y_21 ) ).
fof(fact_394_order__less__not__sym,axiom,
! [X_19,Y_20] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_19),Y_20))
=> ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Y_20),X_19)) ) ).
fof(fact_395_order__less__not__sym,axiom,
! [X_19,Y_20] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_19),Y_20))
=> ~ hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Y_20),X_19)) ) ).
fof(fact_396_order__less__not__sym,axiom,
! [X_19,Y_20] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_19),Y_20))
=> ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Y_20),X_19)) ) ).
fof(fact_397_order__less__imp__not__less,axiom,
! [X_18,Y_19] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_18),Y_19))
=> ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Y_19),X_18)) ) ).
fof(fact_398_order__less__imp__not__less,axiom,
! [X_18,Y_19] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_18),Y_19))
=> ~ hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Y_19),X_18)) ) ).
fof(fact_399_order__less__imp__not__less,axiom,
! [X_18,Y_19] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_18),Y_19))
=> ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Y_19),X_18)) ) ).
fof(fact_400_order__less__imp__not__eq,axiom,
! [X_17,Y_18] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_17),Y_18))
=> X_17 != Y_18 ) ).
fof(fact_401_order__less__imp__not__eq,axiom,
! [X_17,Y_18] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_17),Y_18))
=> X_17 != Y_18 ) ).
fof(fact_402_order__less__imp__not__eq,axiom,
! [X_17,Y_18] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_17),Y_18))
=> X_17 != Y_18 ) ).
fof(fact_403_order__less__imp__not__eq2,axiom,
! [X_16,Y_17] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_16),Y_17))
=> Y_17 != X_16 ) ).
fof(fact_404_order__less__imp__not__eq2,axiom,
! [X_16,Y_17] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_16),Y_17))
=> Y_17 != X_16 ) ).
fof(fact_405_order__less__imp__not__eq2,axiom,
! [X_16,Y_17] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_16),Y_17))
=> Y_17 != X_16 ) ).
fof(fact_406_order__less__imp__triv,axiom,
! [P_1,X_1,Y_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_1),Y_1))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Y_1),X_1))
=> hBOOL(P_1) ) ) ).
fof(fact_407_order__less__imp__triv,axiom,
! [P_1,X_1,Y_1] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_1),Y_1))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Y_1),X_1))
=> hBOOL(P_1) ) ) ).
fof(fact_408_order__less__imp__triv,axiom,
! [P_1,X_1,Y_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_1),Y_1))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Y_1),X_1))
=> hBOOL(P_1) ) ) ).
fof(fact_409_order__less__asym_H,axiom,
! [A_57,B_42] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_57),B_42))
=> ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_42),A_57)) ) ).
fof(fact_410_order__less__asym_H,axiom,
! [A_57,B_42] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_57),B_42))
=> ~ hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,B_42),A_57)) ) ).
fof(fact_411_order__less__asym_H,axiom,
! [A_57,B_42] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_57),B_42))
=> ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,B_42),A_57)) ) ).
fof(fact_412_xt1_I9_J,axiom,
! [B_41,A_56] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_41),A_56))
=> ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_56),B_41)) ) ).
fof(fact_413_xt1_I9_J,axiom,
! [B_41,A_56] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,B_41),A_56))
=> ~ hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_56),B_41)) ) ).
fof(fact_414_xt1_I9_J,axiom,
! [B_41,A_56] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,B_41),A_56))
=> ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_56),B_41)) ) ).
fof(fact_415_ord__eq__less__trans,axiom,
! [C_19,A_55,B_40] :
( A_55 = B_40
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_40),C_19))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_55),C_19)) ) ) ).
fof(fact_416_ord__eq__less__trans,axiom,
! [C_19,A_55,B_40] :
( A_55 = B_40
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,B_40),C_19))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_55),C_19)) ) ) ).
fof(fact_417_ord__eq__less__trans,axiom,
! [C_19,A_55,B_40] :
( A_55 = B_40
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,B_40),C_19))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_55),C_19)) ) ) ).
fof(fact_418_xt1_I1_J,axiom,
! [C_18,A_54,B_39] :
( A_54 = B_39
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,C_18),B_39))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,C_18),A_54)) ) ) ).
fof(fact_419_xt1_I1_J,axiom,
! [C_18,A_54,B_39] :
( A_54 = B_39
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,C_18),B_39))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,C_18),A_54)) ) ) ).
fof(fact_420_xt1_I1_J,axiom,
! [C_18,A_54,B_39] :
( A_54 = B_39
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,C_18),B_39))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,C_18),A_54)) ) ) ).
fof(fact_421_ord__less__eq__trans,axiom,
! [C_17,A_53,B_38] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_53),B_38))
=> ( B_38 = C_17
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_53),C_17)) ) ) ).
fof(fact_422_ord__less__eq__trans,axiom,
! [C_17,A_53,B_38] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_53),B_38))
=> ( B_38 = C_17
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_53),C_17)) ) ) ).
fof(fact_423_ord__less__eq__trans,axiom,
! [C_17,A_53,B_38] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_53),B_38))
=> ( B_38 = C_17
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_53),C_17)) ) ) ).
fof(fact_424_xt1_I2_J,axiom,
! [C_16,B_37,A_52] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_37),A_52))
=> ( B_37 = C_16
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,C_16),A_52)) ) ) ).
fof(fact_425_xt1_I2_J,axiom,
! [C_16,B_37,A_52] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,B_37),A_52))
=> ( B_37 = C_16
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,C_16),A_52)) ) ) ).
fof(fact_426_xt1_I2_J,axiom,
! [C_16,B_37,A_52] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,B_37),A_52))
=> ( B_37 = C_16
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,C_16),A_52)) ) ) ).
fof(fact_427_order__less__trans,axiom,
! [Z_7,X_15,Y_16] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_15),Y_16))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Y_16),Z_7))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_15),Z_7)) ) ) ).
fof(fact_428_order__less__trans,axiom,
! [Z_7,X_15,Y_16] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_15),Y_16))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Y_16),Z_7))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_15),Z_7)) ) ) ).
fof(fact_429_order__less__trans,axiom,
! [Z_7,X_15,Y_16] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_15),Y_16))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Y_16),Z_7))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_15),Z_7)) ) ) ).
fof(fact_430_xt1_I10_J,axiom,
! [Z_6,Y_15,X_14] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Y_15),X_14))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Z_6),Y_15))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Z_6),X_14)) ) ) ).
fof(fact_431_xt1_I10_J,axiom,
! [Z_6,Y_15,X_14] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Y_15),X_14))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Z_6),Y_15))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Z_6),X_14)) ) ) ).
fof(fact_432_xt1_I10_J,axiom,
! [Z_6,Y_15,X_14] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Y_15),X_14))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Z_6),Y_15))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Z_6),X_14)) ) ) ).
fof(fact_433_order__less__asym,axiom,
! [X_13,Y_14] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_13),Y_14))
=> ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Y_14),X_13)) ) ).
fof(fact_434_order__less__asym,axiom,
! [X_13,Y_14] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_13),Y_14))
=> ~ hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Y_14),X_13)) ) ).
fof(fact_435_order__less__asym,axiom,
! [X_13,Y_14] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_13),Y_14))
=> ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Y_14),X_13)) ) ).
fof(fact_436_linorder__cases,axiom,
! [X_12,Y_13] :
( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_12),Y_13))
=> ( X_12 != Y_13
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Y_13),X_12)) ) ) ).
fof(fact_437_linorder__cases,axiom,
! [X_12,Y_13] :
( ~ hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_12),Y_13))
=> ( X_12 != Y_13
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Y_13),X_12)) ) ) ).
fof(fact_438_linorder__cases,axiom,
! [X_12,Y_13] :
( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_12),Y_13))
=> ( X_12 != Y_13
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Y_13),X_12)) ) ) ).
fof(fact_439_linorder__not__less,axiom,
! [X_1,Y_1] :
( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_1),Y_1))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y_1),X_1)) ) ).
fof(fact_440_linorder__not__less,axiom,
! [X_1,Y_1] :
( ~ hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_1),Y_1))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y_1),X_1)) ) ).
fof(fact_441_linorder__not__less,axiom,
! [X_1,Y_1] :
( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_1),Y_1))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Y_1),X_1)) ) ).
fof(fact_442_linorder__not__le,axiom,
! [X_1,Y_1] :
( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_1),Y_1))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Y_1),X_1)) ) ).
fof(fact_443_linorder__not__le,axiom,
! [X_1,Y_1] :
( ~ hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_1),Y_1))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Y_1),X_1)) ) ).
fof(fact_444_linorder__not__le,axiom,
! [X_1,Y_1] :
( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_1),Y_1))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Y_1),X_1)) ) ).
fof(fact_445_linorder__le__less__linear,axiom,
! [X_11,Y_12] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_11),Y_12))
| hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Y_12),X_11)) ) ).
fof(fact_446_linorder__le__less__linear,axiom,
! [X_11,Y_12] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_11),Y_12))
| hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Y_12),X_11)) ) ).
fof(fact_447_linorder__le__less__linear,axiom,
! [X_11,Y_12] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_11),Y_12))
| hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Y_12),X_11)) ) ).
fof(fact_448_order__less__le,axiom,
! [X_1,Y_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_1),Y_1))
<=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_1),Y_1))
& X_1 != Y_1 ) ) ).
fof(fact_449_order__less__le,axiom,
! [X_1,Y_1] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_1),Y_1))
<=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_1),Y_1))
& X_1 != Y_1 ) ) ).
fof(fact_450_order__less__le,axiom,
! [X_1,Y_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_1),Y_1))
<=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_1),Y_1))
& X_1 != Y_1 ) ) ).
fof(fact_451_less__le__not__le,axiom,
! [X_1,Y_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_1),Y_1))
<=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_1),Y_1))
& ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y_1),X_1)) ) ) ).
fof(fact_452_less__le__not__le,axiom,
! [X_1,Y_1] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_1),Y_1))
<=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_1),Y_1))
& ~ hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y_1),X_1)) ) ) ).
fof(fact_453_less__le__not__le,axiom,
! [X_1,Y_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_1),Y_1))
<=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_1),Y_1))
& ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Y_1),X_1)) ) ) ).
fof(fact_454_order__le__less,axiom,
! [X_1,Y_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_1),Y_1))
<=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_1),Y_1))
| X_1 = Y_1 ) ) ).
fof(fact_455_order__le__less,axiom,
! [X_1,Y_1] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_1),Y_1))
<=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_1),Y_1))
| X_1 = Y_1 ) ) ).
fof(fact_456_order__le__less,axiom,
! [X_1,Y_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_1),Y_1))
<=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_1),Y_1))
| X_1 = Y_1 ) ) ).
fof(fact_457_leI,axiom,
! [X_10,Y_11] :
( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_10),Y_11))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y_11),X_10)) ) ).
fof(fact_458_leI,axiom,
! [X_10,Y_11] :
( ~ hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_10),Y_11))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y_11),X_10)) ) ).
fof(fact_459_leI,axiom,
! [X_10,Y_11] :
( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_10),Y_11))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Y_11),X_10)) ) ).
fof(fact_460_not__leE,axiom,
! [Y_10,X_9] :
( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y_10),X_9))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_9),Y_10)) ) ).
fof(fact_461_not__leE,axiom,
! [Y_10,X_9] :
( ~ hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y_10),X_9))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_9),Y_10)) ) ).
fof(fact_462_not__leE,axiom,
! [Y_10,X_9] :
( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Y_10),X_9))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_9),Y_10)) ) ).
fof(fact_463_linorder__antisym__conv1,axiom,
! [X_1,Y_1] :
( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_1),Y_1))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_1),Y_1))
<=> X_1 = Y_1 ) ) ).
fof(fact_464_linorder__antisym__conv1,axiom,
! [X_1,Y_1] :
( ~ hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_1),Y_1))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_1),Y_1))
<=> X_1 = Y_1 ) ) ).
fof(fact_465_linorder__antisym__conv1,axiom,
! [X_1,Y_1] :
( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_1),Y_1))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_1),Y_1))
<=> X_1 = Y_1 ) ) ).
fof(fact_466_order__neq__le__trans,axiom,
! [A_51,B_36] :
( A_51 != B_36
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_51),B_36))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_51),B_36)) ) ) ).
fof(fact_467_order__neq__le__trans,axiom,
! [A_51,B_36] :
( A_51 != B_36
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_51),B_36))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_51),B_36)) ) ) ).
fof(fact_468_order__neq__le__trans,axiom,
! [A_51,B_36] :
( A_51 != B_36
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_51),B_36))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_51),B_36)) ) ) ).
fof(fact_469_xt1_I12_J,axiom,
! [A_50,B_35] :
( A_50 != B_35
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_35),A_50))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_35),A_50)) ) ) ).
fof(fact_470_xt1_I12_J,axiom,
! [A_50,B_35] :
( A_50 != B_35
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B_35),A_50))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,B_35),A_50)) ) ) ).
fof(fact_471_xt1_I12_J,axiom,
! [A_50,B_35] :
( A_50 != B_35
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,B_35),A_50))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,B_35),A_50)) ) ) ).
fof(fact_472_leD,axiom,
! [Y_9,X_8] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y_9),X_8))
=> ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_8),Y_9)) ) ).
fof(fact_473_leD,axiom,
! [Y_9,X_8] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y_9),X_8))
=> ~ hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_8),Y_9)) ) ).
fof(fact_474_leD,axiom,
! [Y_9,X_8] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Y_9),X_8))
=> ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_8),Y_9)) ) ).
fof(fact_475_norR__mem__unique__aux,axiom,
! [A,B] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A),hAPP_int_int(minus_minus_int(B),one_one_int)))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A),B)) ) ).
fof(fact_476_int__le__induct,axiom,
! [P_1,I_1,K_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,I_1),K_1))
=> ( hBOOL(hAPP_int_bool(P_1,K_1))
=> ( ! [I_2] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,I_2),K_1))
=> ( hBOOL(hAPP_int_bool(P_1,I_2))
=> hBOOL(hAPP_int_bool(P_1,hAPP_int_int(minus_minus_int(I_2),one_one_int))) ) )
=> hBOOL(hAPP_int_bool(P_1,I_1)) ) ) ) ).
fof(fact_477_int__less__induct,axiom,
! [P_1,I_1,K_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,I_1),K_1))
=> ( hBOOL(hAPP_int_bool(P_1,hAPP_int_int(minus_minus_int(K_1),one_one_int)))
=> ( ! [I_2] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,I_2),K_1))
=> ( hBOOL(hAPP_int_bool(P_1,I_2))
=> hBOOL(hAPP_int_bool(P_1,hAPP_int_int(minus_minus_int(I_2),one_one_int))) ) )
=> hBOOL(hAPP_int_bool(P_1,I_1)) ) ) ) ).
fof(fact_478_int__ge__induct,axiom,
! [P_1,K_1,I_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K_1),I_1))
=> ( hBOOL(hAPP_int_bool(P_1,K_1))
=> ( ! [I_2] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K_1),I_2))
=> ( hBOOL(hAPP_int_bool(P_1,I_2))
=> hBOOL(hAPP_int_bool(P_1,hAPP_int_int(plus_plus_int(I_2),one_one_int))) ) )
=> hBOOL(hAPP_int_bool(P_1,I_1)) ) ) ) ).
fof(fact_479_int__gr__induct,axiom,
! [P_1,K_1,I_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K_1),I_1))
=> ( hBOOL(hAPP_int_bool(P_1,hAPP_int_int(plus_plus_int(K_1),one_one_int)))
=> ( ! [I_2] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K_1),I_2))
=> ( hBOOL(hAPP_int_bool(P_1,I_2))
=> hBOOL(hAPP_int_bool(P_1,hAPP_int_int(plus_plus_int(I_2),one_one_int))) ) )
=> hBOOL(hAPP_int_bool(P_1,I_1)) ) ) ) ).
fof(fact_480_dense__le,axiom,
! [Z_5,Y_8] :
( ! [X_2] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_2),Y_8))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_2),Z_5)) )
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y_8),Z_5)) ) ).
fof(fact_481_dense__le__bounded,axiom,
! [Z_4,X_7,Y_7] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_7),Y_7))
=> ( ! [W_2] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_7),W_2))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,W_2),Y_7))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,W_2),Z_4)) ) )
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y_7),Z_4)) ) ) ).
fof(fact_482_int__induct,axiom,
! [I_1,P_1,K_1] :
( hBOOL(hAPP_int_bool(P_1,K_1))
=> ( ! [I_2] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K_1),I_2))
=> ( hBOOL(hAPP_int_bool(P_1,I_2))
=> hBOOL(hAPP_int_bool(P_1,hAPP_int_int(plus_plus_int(I_2),one_one_int))) ) )
=> ( ! [I_2] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,I_2),K_1))
=> ( hBOOL(hAPP_int_bool(P_1,I_2))
=> hBOOL(hAPP_int_bool(P_1,hAPP_int_int(minus_minus_int(I_2),one_one_int))) ) )
=> hBOOL(hAPP_int_bool(P_1,I_1)) ) ) ) ).
fof(fact_483_Euler_Oaux2,axiom,
! [B,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),C))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A),B))
| hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B),A)) ) ) ) ).
fof(fact_484_d22set__induct__old,axiom,
! [X_1,P_1] :
( ! [A_49] :
( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),A_49))
=> hBOOL(hAPP_int_bool(P_1,hAPP_int_int(minus_minus_int(A_49),one_one_int))) )
=> hBOOL(hAPP_int_bool(P_1,A_49)) )
=> hBOOL(hAPP_int_bool(P_1,X_1)) ) ).
fof(fact_485_convex__bound__lt,axiom,
! [V_5,U_2,Y_6,X_6,A_48] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_6),A_48))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Y_6),A_48))
=> ( 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_5))
=> ( hAPP_int_int(plus_plus_int(U_2),V_5) = one_one_int
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(U_2),X_6)),hAPP_int_int(times_times_int(V_5),Y_6))),A_48)) ) ) ) ) ) ).
fof(fact_486_convex__bound__lt,axiom,
! [V_5,U_2,Y_6,X_6,A_48] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_6),A_48))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Y_6),A_48))
=> ( 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_5))
=> ( hAPP_real_real(plus_plus_real(U_2),V_5) = one_one_real
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(plus_plus_real(hAPP_real_real(times_times_real(U_2),X_6)),hAPP_real_real(times_times_real(V_5),Y_6))),A_48)) ) ) ) ) ) ).
fof(fact_487_number__of__le__ceiling,axiom,
! [V_1,X_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,number_number_of_int(V_1)),archim856651990g_real(X_1)))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(minus_minus_real(number267125858f_real(V_1)),one_one_real)),X_1)) ) ).
fof(fact_488_ceiling__less__number__of,axiom,
! [X_1,V_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,archim856651990g_real(X_1)),number_number_of_int(V_1)))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_1),hAPP_real_real(minus_minus_real(number267125858f_real(V_1)),one_one_real))) ) ).
fof(fact_489_zero__less__ceiling,axiom,
! [X_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),archim856651990g_real(X_1)))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_1)) ) ).
fof(fact_490_zero__reorient,axiom,
! [X_1] :
( zero_zero_int = X_1
<=> X_1 = zero_zero_int ) ).
fof(fact_491_zero__reorient,axiom,
! [X_1] :
( zero_zero_real = X_1
<=> X_1 = zero_zero_real ) ).
fof(fact_492_zero__reorient,axiom,
! [X_1] :
( zero_zero_nat = X_1
<=> X_1 = zero_zero_nat ) ).
fof(fact_493_ceiling__zero,axiom,
archim856651990g_real(zero_zero_real) = zero_zero_int ).
fof(fact_494_ceiling__le__zero,axiom,
! [X_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,archim856651990g_real(X_1)),zero_zero_int))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_1),zero_zero_real)) ) ).
fof(fact_495_ceiling__less__one,axiom,
! [X_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,archim856651990g_real(X_1)),one_one_int))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_1),zero_zero_real)) ) ).
fof(fact_496_one__le__ceiling,axiom,
! [X_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,one_one_int),archim856651990g_real(X_1)))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_1)) ) ).
fof(fact_497_comm__semiring__1__class_Onormalizing__semiring__rules_I9_J,axiom,
! [A_47] : hAPP_nat_nat(times_times_nat(zero_zero_nat),A_47) = zero_zero_nat ).
fof(fact_498_comm__semiring__1__class_Onormalizing__semiring__rules_I9_J,axiom,
! [A_47] : hAPP_real_real(times_times_real(zero_zero_real),A_47) = zero_zero_real ).
fof(fact_499_comm__semiring__1__class_Onormalizing__semiring__rules_I9_J,axiom,
! [A_47] : hAPP_int_int(times_times_int(zero_zero_int),A_47) = zero_zero_int ).
fof(fact_500_comm__semiring__1__class_Onormalizing__semiring__rules_I10_J,axiom,
! [A_46] : hAPP_nat_nat(times_times_nat(A_46),zero_zero_nat) = zero_zero_nat ).
fof(fact_501_comm__semiring__1__class_Onormalizing__semiring__rules_I10_J,axiom,
! [A_46] : hAPP_real_real(times_times_real(A_46),zero_zero_real) = zero_zero_real ).
fof(fact_502_comm__semiring__1__class_Onormalizing__semiring__rules_I10_J,axiom,
! [A_46] : hAPP_int_int(times_times_int(A_46),zero_zero_int) = zero_zero_int ).
fof(fact_503_mult__zero__left,axiom,
! [A_45] : hAPP_nat_nat(times_times_nat(zero_zero_nat),A_45) = zero_zero_nat ).
fof(fact_504_mult__zero__left,axiom,
! [A_45] : hAPP_real_real(times_times_real(zero_zero_real),A_45) = zero_zero_real ).
fof(fact_505_mult__zero__left,axiom,
! [A_45] : hAPP_int_int(times_times_int(zero_zero_int),A_45) = zero_zero_int ).
fof(fact_506_mult_Ozero__left,axiom,
! [B_34] : hAPP_real_real(times_times_real(zero_zero_real),B_34) = zero_zero_real ).
fof(fact_507_mult__left_Ozero,axiom,
! [Y_5] : hAPP_real_real(times_times_real(zero_zero_real),Y_5) = zero_zero_real ).
fof(fact_508_mult__zero__right,axiom,
! [A_44] : hAPP_nat_nat(times_times_nat(A_44),zero_zero_nat) = zero_zero_nat ).
fof(fact_509_mult__zero__right,axiom,
! [A_44] : hAPP_real_real(times_times_real(A_44),zero_zero_real) = zero_zero_real ).
fof(fact_510_mult__zero__right,axiom,
! [A_44] : hAPP_int_int(times_times_int(A_44),zero_zero_int) = zero_zero_int ).
fof(fact_511_mult_Ozero__right,axiom,
! [A_43] : hAPP_real_real(times_times_real(A_43),zero_zero_real) = zero_zero_real ).
fof(fact_512_mult__right_Ozero,axiom,
! [X_5] : hAPP_real_real(times_times_real(X_5),zero_zero_real) = zero_zero_real ).
fof(fact_513_mult__eq__0__iff,axiom,
! [Aa,Ba] :
( hAPP_real_real(times_times_real(Aa),Ba) = zero_zero_real
<=> ( Aa = zero_zero_real
| Ba = zero_zero_real ) ) ).
fof(fact_514_mult__eq__0__iff,axiom,
! [Aa,Ba] :
( hAPP_int_int(times_times_int(Aa),Ba) = zero_zero_int
<=> ( Aa = zero_zero_int
| Ba = zero_zero_int ) ) ).
fof(fact_515_no__zero__divisors,axiom,
! [B_33,A_42] :
( A_42 != zero_zero_nat
=> ( B_33 != zero_zero_nat
=> hAPP_nat_nat(times_times_nat(A_42),B_33) != zero_zero_nat ) ) ).
fof(fact_516_no__zero__divisors,axiom,
! [B_33,A_42] :
( A_42 != zero_zero_real
=> ( B_33 != zero_zero_real
=> hAPP_real_real(times_times_real(A_42),B_33) != zero_zero_real ) ) ).
fof(fact_517_no__zero__divisors,axiom,
! [B_33,A_42] :
( A_42 != zero_zero_int
=> ( B_33 != zero_zero_int
=> hAPP_int_int(times_times_int(A_42),B_33) != zero_zero_int ) ) ).
fof(fact_518_divisors__zero,axiom,
! [A_41,B_32] :
( hAPP_nat_nat(times_times_nat(A_41),B_32) = zero_zero_nat
=> ( A_41 = zero_zero_nat
| B_32 = zero_zero_nat ) ) ).
fof(fact_519_divisors__zero,axiom,
! [A_41,B_32] :
( hAPP_real_real(times_times_real(A_41),B_32) = zero_zero_real
=> ( A_41 = zero_zero_real
| B_32 = zero_zero_real ) ) ).
fof(fact_520_divisors__zero,axiom,
! [A_41,B_32] :
( hAPP_int_int(times_times_int(A_41),B_32) = zero_zero_int
=> ( A_41 = zero_zero_int
| B_32 = zero_zero_int ) ) ).
fof(fact_521_comm__semiring__1__class_Onormalizing__semiring__rules_I5_J,axiom,
! [A_40] : hAPP_nat_nat(plus_plus_nat(zero_zero_nat),A_40) = A_40 ).
fof(fact_522_comm__semiring__1__class_Onormalizing__semiring__rules_I5_J,axiom,
! [A_40] : hAPP_real_real(plus_plus_real(zero_zero_real),A_40) = A_40 ).
fof(fact_523_comm__semiring__1__class_Onormalizing__semiring__rules_I5_J,axiom,
! [A_40] : hAPP_int_int(plus_plus_int(zero_zero_int),A_40) = A_40 ).
fof(fact_524_comm__semiring__1__class_Onormalizing__semiring__rules_I6_J,axiom,
! [A_39] : hAPP_nat_nat(plus_plus_nat(A_39),zero_zero_nat) = A_39 ).
fof(fact_525_comm__semiring__1__class_Onormalizing__semiring__rules_I6_J,axiom,
! [A_39] : hAPP_real_real(plus_plus_real(A_39),zero_zero_real) = A_39 ).
fof(fact_526_comm__semiring__1__class_Onormalizing__semiring__rules_I6_J,axiom,
! [A_39] : hAPP_int_int(plus_plus_int(A_39),zero_zero_int) = A_39 ).
fof(fact_527_add__0__iff,axiom,
! [Ba,Aa] :
( Ba = hAPP_nat_nat(plus_plus_nat(Ba),Aa)
<=> Aa = zero_zero_nat ) ).
fof(fact_528_add__0__iff,axiom,
! [Ba,Aa] :
( Ba = hAPP_real_real(plus_plus_real(Ba),Aa)
<=> Aa = zero_zero_real ) ).
fof(fact_529_add__0__iff,axiom,
! [Ba,Aa] :
( Ba = hAPP_int_int(plus_plus_int(Ba),Aa)
<=> Aa = zero_zero_int ) ).
fof(fact_530_add__0__left,axiom,
! [A_38] : hAPP_nat_nat(plus_plus_nat(zero_zero_nat),A_38) = A_38 ).
fof(fact_531_add__0__left,axiom,
! [A_38] : hAPP_real_real(plus_plus_real(zero_zero_real),A_38) = A_38 ).
fof(fact_532_add__0__left,axiom,
! [A_38] : hAPP_int_int(plus_plus_int(zero_zero_int),A_38) = A_38 ).
fof(fact_533_add__0,axiom,
! [A_37] : hAPP_nat_nat(plus_plus_nat(zero_zero_nat),A_37) = A_37 ).
fof(fact_534_add__0,axiom,
! [A_37] : hAPP_real_real(plus_plus_real(zero_zero_real),A_37) = A_37 ).
fof(fact_535_add__0,axiom,
! [A_37] : hAPP_int_int(plus_plus_int(zero_zero_int),A_37) = A_37 ).
fof(fact_536_double__zero__sym,axiom,
! [Aa] :
( zero_zero_real = hAPP_real_real(plus_plus_real(Aa),Aa)
<=> Aa = zero_zero_real ) ).
fof(fact_537_double__zero__sym,axiom,
! [Aa] :
( zero_zero_int = hAPP_int_int(plus_plus_int(Aa),Aa)
<=> Aa = zero_zero_int ) ).
fof(fact_538_add__0__right,axiom,
! [A_36] : hAPP_nat_nat(plus_plus_nat(A_36),zero_zero_nat) = A_36 ).
fof(fact_539_add__0__right,axiom,
! [A_36] : hAPP_real_real(plus_plus_real(A_36),zero_zero_real) = A_36 ).
fof(fact_540_add__0__right,axiom,
! [A_36] : hAPP_int_int(plus_plus_int(A_36),zero_zero_int) = A_36 ).
fof(fact_541_add_Ocomm__neutral,axiom,
! [A_35] : hAPP_nat_nat(plus_plus_nat(A_35),zero_zero_nat) = A_35 ).
fof(fact_542_add_Ocomm__neutral,axiom,
! [A_35] : hAPP_real_real(plus_plus_real(A_35),zero_zero_real) = A_35 ).
fof(fact_543_add_Ocomm__neutral,axiom,
! [A_35] : hAPP_int_int(plus_plus_int(A_35),zero_zero_int) = A_35 ).
fof(fact_544_double__eq__0__iff,axiom,
! [Aa] :
( hAPP_real_real(plus_plus_real(Aa),Aa) = zero_zero_real
<=> Aa = zero_zero_real ) ).
fof(fact_545_double__eq__0__iff,axiom,
! [Aa] :
( hAPP_int_int(plus_plus_int(Aa),Aa) = zero_zero_int
<=> Aa = zero_zero_int ) ).
fof(fact_546_one__neq__zero,axiom,
one_one_nat != zero_zero_nat ).
fof(fact_547_one__neq__zero,axiom,
one_one_real != zero_zero_real ).
fof(fact_548_one__neq__zero,axiom,
one_one_int != zero_zero_int ).
fof(fact_549_zero__neq__one,axiom,
zero_zero_nat != one_one_nat ).
fof(fact_550_zero__neq__one,axiom,
zero_zero_real != one_one_real ).
fof(fact_551_zero__neq__one,axiom,
zero_zero_int != one_one_int ).
fof(fact_552_diff__0__right,axiom,
! [A_34] : hAPP_real_real(minus_minus_real(A_34),zero_zero_real) = A_34 ).
fof(fact_553_diff__0__right,axiom,
! [A_34] : hAPP_int_int(minus_minus_int(A_34),zero_zero_int) = A_34 ).
fof(fact_554_diff__self,axiom,
! [A_33] : hAPP_real_real(minus_minus_real(A_33),A_33) = zero_zero_real ).
fof(fact_555_diff__self,axiom,
! [A_33] : hAPP_int_int(minus_minus_int(A_33),A_33) = zero_zero_int ).
fof(fact_556_eq__iff__diff__eq__0,axiom,
! [Aa,Ba] :
( Aa = Ba
<=> hAPP_real_real(minus_minus_real(Aa),Ba) = zero_zero_real ) ).
fof(fact_557_eq__iff__diff__eq__0,axiom,
! [Aa,Ba] :
( Aa = Ba
<=> hAPP_int_int(minus_minus_int(Aa),Ba) = zero_zero_int ) ).
fof(fact_558_right__minus__eq,axiom,
! [Aa,Ba] :
( hAPP_real_real(minus_minus_real(Aa),Ba) = zero_zero_real
<=> Aa = Ba ) ).
fof(fact_559_right__minus__eq,axiom,
! [Aa,Ba] :
( hAPP_int_int(minus_minus_int(Aa),Ba) = zero_zero_int
<=> Aa = Ba ) ).
fof(fact_560_int__0__neq__1,axiom,
zero_zero_int != one_one_int ).
fof(fact_561_zadd__0,axiom,
! [Z_1] : hAPP_int_int(plus_plus_int(zero_zero_int),Z_1) = Z_1 ).
fof(fact_562_zadd__0__right,axiom,
! [Z_1] : hAPP_int_int(plus_plus_int(Z_1),zero_zero_int) = Z_1 ).
fof(fact_563_ceiling__less__cancel,axiom,
! [X_4,Y_4] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,archim856651990g_real(X_4)),archim856651990g_real(Y_4)))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_4),Y_4)) ) ).
fof(fact_564_ceiling__mono,axiom,
! [Y_3,X_3] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y_3),X_3))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,archim856651990g_real(Y_3)),archim856651990g_real(X_3))) ) ).
fof(fact_565_ceiling__one,axiom,
archim856651990g_real(one_one_real) = one_one_int ).
fof(fact_566_ceiling__number__of,axiom,
! [V_4] : archim856651990g_real(number267125858f_real(V_4)) = number_number_of_int(V_4) ).
fof(fact_567_zero__le__square,axiom,
! [A_32] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(times_times_real(A_32),A_32))) ).
fof(fact_568_zero__le__square,axiom,
! [A_32] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(times_times_int(A_32),A_32))) ).
fof(fact_569_zero__le__mult__iff,axiom,
! [Aa,Ba] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(times_times_real(Aa),Ba)))
<=> ( ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),Aa))
& hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),Ba)) )
| ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Aa),zero_zero_real))
& hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Ba),zero_zero_real)) ) ) ) ).
fof(fact_570_zero__le__mult__iff,axiom,
! [Aa,Ba] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(times_times_int(Aa),Ba)))
<=> ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Aa))
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Ba)) )
| ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Aa),zero_zero_int))
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Ba),zero_zero_int)) ) ) ) ).
fof(fact_571_mult__le__0__iff,axiom,
! [Aa,Ba] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(times_times_real(Aa),Ba)),zero_zero_real))
<=> ( ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),Aa))
& hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Ba),zero_zero_real)) )
| ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Aa),zero_zero_real))
& hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),Ba)) ) ) ) ).
fof(fact_572_mult__le__0__iff,axiom,
! [Aa,Ba] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(times_times_int(Aa),Ba)),zero_zero_int))
<=> ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Aa))
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Ba),zero_zero_int)) )
| ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Aa),zero_zero_int))
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Ba)) ) ) ) ).
fof(fact_573_mult__nonneg__nonneg,axiom,
! [B_31,A_31] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),A_31))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),B_31))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),hAPP_nat_nat(times_times_nat(A_31),B_31))) ) ) ).
fof(fact_574_mult__nonneg__nonneg,axiom,
! [B_31,A_31] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_31))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),B_31))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(times_times_real(A_31),B_31))) ) ) ).
fof(fact_575_mult__nonneg__nonneg,axiom,
! [B_31,A_31] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_31))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),B_31))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(times_times_int(A_31),B_31))) ) ) ).
fof(fact_576_mult__nonneg__nonpos,axiom,
! [B_30,A_30] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),A_30))
=> ( 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,hAPP_nat_nat(times_times_nat(A_30),B_30)),zero_zero_nat)) ) ) ).
fof(fact_577_mult__nonneg__nonpos,axiom,
! [B_30,A_30] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_30))
=> ( 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,hAPP_real_real(times_times_real(A_30),B_30)),zero_zero_real)) ) ) ).
fof(fact_578_mult__nonneg__nonpos,axiom,
! [B_30,A_30] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_30))
=> ( 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,hAPP_int_int(times_times_int(A_30),B_30)),zero_zero_int)) ) ) ).
fof(fact_579_mult__nonneg__nonpos2,axiom,
! [B_29,A_29] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),A_29))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,B_29),zero_zero_nat))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(times_times_nat(B_29),A_29)),zero_zero_nat)) ) ) ).
fof(fact_580_mult__nonneg__nonpos2,axiom,
! [B_29,A_29] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_29))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B_29),zero_zero_real))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(times_times_real(B_29),A_29)),zero_zero_real)) ) ) ).
fof(fact_581_mult__nonneg__nonpos2,axiom,
! [B_29,A_29] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_29))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_29),zero_zero_int))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(times_times_int(B_29),A_29)),zero_zero_int)) ) ) ).
fof(fact_582_mult__nonpos__nonneg,axiom,
! [B_28,A_28] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_28),zero_zero_nat))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),B_28))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(times_times_nat(A_28),B_28)),zero_zero_nat)) ) ) ).
fof(fact_583_mult__nonpos__nonneg,axiom,
! [B_28,A_28] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_28),zero_zero_real))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),B_28))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(times_times_real(A_28),B_28)),zero_zero_real)) ) ) ).
fof(fact_584_mult__nonpos__nonneg,axiom,
! [B_28,A_28] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_28),zero_zero_int))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),B_28))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(times_times_int(A_28),B_28)),zero_zero_int)) ) ) ).
fof(fact_585_mult__nonpos__nonpos,axiom,
! [B_27,A_27] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_27),zero_zero_real))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B_27),zero_zero_real))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(times_times_real(A_27),B_27))) ) ) ).
fof(fact_586_mult__nonpos__nonpos,axiom,
! [B_27,A_27] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_27),zero_zero_int))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_27),zero_zero_int))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(times_times_int(A_27),B_27))) ) ) ).
fof(fact_587_mult__right__mono,axiom,
! [C_15,A_26,B_26] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_26),B_26))
=> ( 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_eq_nat,hAPP_nat_nat(times_times_nat(A_26),C_15)),hAPP_nat_nat(times_times_nat(B_26),C_15))) ) ) ).
fof(fact_588_mult__right__mono,axiom,
! [C_15,A_26,B_26] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_26),B_26))
=> ( 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_eq_real,hAPP_real_real(times_times_real(A_26),C_15)),hAPP_real_real(times_times_real(B_26),C_15))) ) ) ).
fof(fact_589_mult__right__mono,axiom,
! [C_15,A_26,B_26] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_26),B_26))
=> ( 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_eq_int,hAPP_int_int(times_times_int(A_26),C_15)),hAPP_int_int(times_times_int(B_26),C_15))) ) ) ).
fof(fact_590_mult__left__mono,axiom,
! [C_14,A_25,B_25] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_25),B_25))
=> ( 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_eq_nat,hAPP_nat_nat(times_times_nat(C_14),A_25)),hAPP_nat_nat(times_times_nat(C_14),B_25))) ) ) ).
fof(fact_591_mult__left__mono,axiom,
! [C_14,A_25,B_25] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_25),B_25))
=> ( 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_eq_real,hAPP_real_real(times_times_real(C_14),A_25)),hAPP_real_real(times_times_real(C_14),B_25))) ) ) ).
fof(fact_592_mult__left__mono,axiom,
! [C_14,A_25,B_25] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_25),B_25))
=> ( 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_eq_int,hAPP_int_int(times_times_int(C_14),A_25)),hAPP_int_int(times_times_int(C_14),B_25))) ) ) ).
fof(fact_593_comm__mult__left__mono,axiom,
! [C_13,A_24,B_24] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_24),B_24))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),C_13))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(times_times_nat(C_13),A_24)),hAPP_nat_nat(times_times_nat(C_13),B_24))) ) ) ).
fof(fact_594_comm__mult__left__mono,axiom,
! [C_13,A_24,B_24] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_24),B_24))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),C_13))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(times_times_real(C_13),A_24)),hAPP_real_real(times_times_real(C_13),B_24))) ) ) ).
fof(fact_595_comm__mult__left__mono,axiom,
! [C_13,A_24,B_24] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_24),B_24))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),C_13))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(times_times_int(C_13),A_24)),hAPP_int_int(times_times_int(C_13),B_24))) ) ) ).
fof(fact_596_mult__right__mono__neg,axiom,
! [C_12,B_23,A_23] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B_23),A_23))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,C_12),zero_zero_real))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(times_times_real(A_23),C_12)),hAPP_real_real(times_times_real(B_23),C_12))) ) ) ).
fof(fact_597_mult__right__mono__neg,axiom,
! [C_12,B_23,A_23] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_23),A_23))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,C_12),zero_zero_int))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(times_times_int(A_23),C_12)),hAPP_int_int(times_times_int(B_23),C_12))) ) ) ).
fof(fact_598_mult__left__mono__neg,axiom,
! [C_11,B_22,A_22] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B_22),A_22))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,C_11),zero_zero_real))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(times_times_real(C_11),A_22)),hAPP_real_real(times_times_real(C_11),B_22))) ) ) ).
fof(fact_599_mult__left__mono__neg,axiom,
! [C_11,B_22,A_22] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_22),A_22))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,C_11),zero_zero_int))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(times_times_int(C_11),A_22)),hAPP_int_int(times_times_int(C_11),B_22))) ) ) ).
fof(fact_600_mult__mono_H,axiom,
! [C_10,D_4,A_21,B_21] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_21),B_21))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,C_10),D_4))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),A_21))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),C_10))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(times_times_nat(A_21),C_10)),hAPP_nat_nat(times_times_nat(B_21),D_4))) ) ) ) ) ).
fof(fact_601_mult__mono_H,axiom,
! [C_10,D_4,A_21,B_21] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_21),B_21))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,C_10),D_4))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_21))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),C_10))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(times_times_real(A_21),C_10)),hAPP_real_real(times_times_real(B_21),D_4))) ) ) ) ) ).
fof(fact_602_mult__mono_H,axiom,
! [C_10,D_4,A_21,B_21] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_21),B_21))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,C_10),D_4))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_21))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),C_10))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(times_times_int(A_21),C_10)),hAPP_int_int(times_times_int(B_21),D_4))) ) ) ) ) ).
fof(fact_603_mult__mono,axiom,
! [C_9,D_3,A_20,B_20] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_20),B_20))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,C_9),D_3))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),B_20))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),C_9))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(times_times_nat(A_20),C_9)),hAPP_nat_nat(times_times_nat(B_20),D_3))) ) ) ) ) ).
fof(fact_604_mult__mono,axiom,
! [C_9,D_3,A_20,B_20] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_20),B_20))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,C_9),D_3))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),B_20))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),C_9))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(times_times_real(A_20),C_9)),hAPP_real_real(times_times_real(B_20),D_3))) ) ) ) ) ).
fof(fact_605_mult__mono,axiom,
! [C_9,D_3,A_20,B_20] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_20),B_20))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,C_9),D_3))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),B_20))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),C_9))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(times_times_int(A_20),C_9)),hAPP_int_int(times_times_int(B_20),D_3))) ) ) ) ) ).
fof(fact_606_split__mult__pos__le,axiom,
! [B_19,A_19] :
( ( ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_19))
& hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),B_19)) )
| ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_19),zero_zero_real))
& hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B_19),zero_zero_real)) ) )
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(times_times_real(A_19),B_19))) ) ).
fof(fact_607_split__mult__pos__le,axiom,
! [B_19,A_19] :
( ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_19))
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),B_19)) )
| ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_19),zero_zero_int))
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_19),zero_zero_int)) ) )
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(times_times_int(A_19),B_19))) ) ).
fof(fact_608_split__mult__neg__le,axiom,
! [B_18,A_18] :
( ( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),A_18))
& hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,B_18),zero_zero_nat)) )
| ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_18),zero_zero_nat))
& hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),B_18)) ) )
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(times_times_nat(A_18),B_18)),zero_zero_nat)) ) ).
fof(fact_609_split__mult__neg__le,axiom,
! [B_18,A_18] :
( ( ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_18))
& hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B_18),zero_zero_real)) )
| ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_18),zero_zero_real))
& hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),B_18)) ) )
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(times_times_real(A_18),B_18)),zero_zero_real)) ) ).
fof(fact_610_split__mult__neg__le,axiom,
! [B_18,A_18] :
( ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_18))
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_18),zero_zero_int)) )
| ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_18),zero_zero_int))
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),B_18)) ) )
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(times_times_int(A_18),B_18)),zero_zero_int)) ) ).
fof(fact_611_not__square__less__zero,axiom,
! [A_17] : ~ hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(times_times_real(A_17),A_17)),zero_zero_real)) ).
fof(fact_612_not__square__less__zero,axiom,
! [A_17] : ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(times_times_int(A_17),A_17)),zero_zero_int)) ).
fof(fact_613_mult__less__cancel__right__disj,axiom,
! [Aa,C_1,Ba] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(times_times_real(Aa),C_1)),hAPP_real_real(times_times_real(Ba),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,Aa),Ba)) )
| ( 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,Ba),Aa)) ) ) ) ).
fof(fact_614_mult__less__cancel__right__disj,axiom,
! [Aa,C_1,Ba] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(times_times_int(Aa),C_1)),hAPP_int_int(times_times_int(Ba),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,Aa),Ba)) )
| ( 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,Ba),Aa)) ) ) ) ).
fof(fact_615_mult__less__cancel__left__disj,axiom,
! [C_1,Aa,Ba] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(times_times_real(C_1),Aa)),hAPP_real_real(times_times_real(C_1),Ba)))
<=> ( ( 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,Aa),Ba)) )
| ( 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,Ba),Aa)) ) ) ) ).
fof(fact_616_mult__less__cancel__left__disj,axiom,
! [C_1,Aa,Ba] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(times_times_int(C_1),Aa)),hAPP_int_int(times_times_int(C_1),Ba)))
<=> ( ( 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,Aa),Ba)) )
| ( 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,Ba),Aa)) ) ) ) ).
fof(fact_617_mult__less__cancel__left__pos,axiom,
! [Aa,Ba,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,hAPP_real_real(times_times_real(C_1),Aa)),hAPP_real_real(times_times_real(C_1),Ba)))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Aa),Ba)) ) ) ).
fof(fact_618_mult__less__cancel__left__pos,axiom,
! [Aa,Ba,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,hAPP_int_int(times_times_int(C_1),Aa)),hAPP_int_int(times_times_int(C_1),Ba)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Aa),Ba)) ) ) ).
fof(fact_619_mult__pos__pos,axiom,
! [B_17,A_16] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),A_16))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),B_17))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_nat_nat(times_times_nat(A_16),B_17))) ) ) ).
fof(fact_620_mult__pos__pos,axiom,
! [B_17,A_16] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_16))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),B_17))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(times_times_real(A_16),B_17))) ) ) ).
fof(fact_621_mult__pos__pos,axiom,
! [B_17,A_16] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_16))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_17))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_int_int(times_times_int(A_16),B_17))) ) ) ).
fof(fact_622_mult__pos__neg,axiom,
! [B_16,A_15] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),A_15))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,B_16),zero_zero_nat))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(times_times_nat(A_15),B_16)),zero_zero_nat)) ) ) ).
fof(fact_623_mult__pos__neg,axiom,
! [B_16,A_15] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_15))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,B_16),zero_zero_real))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(times_times_real(A_15),B_16)),zero_zero_real)) ) ) ).
fof(fact_624_mult__pos__neg,axiom,
! [B_16,A_15] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_15))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_16),zero_zero_int))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(times_times_int(A_15),B_16)),zero_zero_int)) ) ) ).
fof(fact_625_mult__pos__neg2,axiom,
! [B_15,A_14] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),A_14))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,B_15),zero_zero_nat))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(times_times_nat(B_15),A_14)),zero_zero_nat)) ) ) ).
fof(fact_626_mult__pos__neg2,axiom,
! [B_15,A_14] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_14))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,B_15),zero_zero_real))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(times_times_real(B_15),A_14)),zero_zero_real)) ) ) ).
fof(fact_627_mult__pos__neg2,axiom,
! [B_15,A_14] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_14))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_15),zero_zero_int))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(times_times_int(B_15),A_14)),zero_zero_int)) ) ) ).
fof(fact_628_zero__less__mult__pos,axiom,
! [A_13,B_14] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_nat_nat(times_times_nat(A_13),B_14)))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),A_13))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),B_14)) ) ) ).
fof(fact_629_zero__less__mult__pos,axiom,
! [A_13,B_14] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(times_times_real(A_13),B_14)))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_13))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),B_14)) ) ) ).
fof(fact_630_zero__less__mult__pos,axiom,
! [A_13,B_14] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_int_int(times_times_int(A_13),B_14)))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_13))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_14)) ) ) ).
fof(fact_631_zero__less__mult__pos2,axiom,
! [B_13,A_12] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_nat_nat(times_times_nat(B_13),A_12)))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),A_12))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),B_13)) ) ) ).
fof(fact_632_zero__less__mult__pos2,axiom,
! [B_13,A_12] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(times_times_real(B_13),A_12)))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_12))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),B_13)) ) ) ).
fof(fact_633_zero__less__mult__pos2,axiom,
! [B_13,A_12] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_int_int(times_times_int(B_13),A_12)))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_12))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_13)) ) ) ).
fof(fact_634_mult__less__cancel__left__neg,axiom,
! [Aa,Ba,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_real,hAPP_real_real(times_times_real(C_1),Aa)),hAPP_real_real(times_times_real(C_1),Ba)))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Ba),Aa)) ) ) ).
fof(fact_635_mult__less__cancel__left__neg,axiom,
! [Aa,Ba,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_int,hAPP_int_int(times_times_int(C_1),Aa)),hAPP_int_int(times_times_int(C_1),Ba)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Ba),Aa)) ) ) ).
fof(fact_636_mult__neg__pos,axiom,
! [B_12,A_11] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_11),zero_zero_nat))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),B_12))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(times_times_nat(A_11),B_12)),zero_zero_nat)) ) ) ).
fof(fact_637_mult__neg__pos,axiom,
! [B_12,A_11] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_11),zero_zero_real))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),B_12))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(times_times_real(A_11),B_12)),zero_zero_real)) ) ) ).
fof(fact_638_mult__neg__pos,axiom,
! [B_12,A_11] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_11),zero_zero_int))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_12))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(times_times_int(A_11),B_12)),zero_zero_int)) ) ) ).
fof(fact_639_mult__neg__neg,axiom,
! [B_11,A_10] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_10),zero_zero_real))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,B_11),zero_zero_real))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(times_times_real(A_10),B_11))) ) ) ).
fof(fact_640_mult__neg__neg,axiom,
! [B_11,A_10] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_10),zero_zero_int))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_11),zero_zero_int))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_int_int(times_times_int(A_10),B_11))) ) ) ).
fof(fact_641_mult__strict__right__mono,axiom,
! [C_8,A_9,B_10] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_9),B_10))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),C_8))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(times_times_nat(A_9),C_8)),hAPP_nat_nat(times_times_nat(B_10),C_8))) ) ) ).
fof(fact_642_mult__strict__right__mono,axiom,
! [C_8,A_9,B_10] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_9),B_10))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),C_8))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(times_times_real(A_9),C_8)),hAPP_real_real(times_times_real(B_10),C_8))) ) ) ).
fof(fact_643_mult__strict__right__mono,axiom,
! [C_8,A_9,B_10] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_9),B_10))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),C_8))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(times_times_int(A_9),C_8)),hAPP_int_int(times_times_int(B_10),C_8))) ) ) ).
fof(fact_644_mult__strict__left__mono,axiom,
! [C_7,A_8,B_9] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_8),B_9))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),C_7))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(times_times_nat(C_7),A_8)),hAPP_nat_nat(times_times_nat(C_7),B_9))) ) ) ).
fof(fact_645_mult__strict__left__mono,axiom,
! [C_7,A_8,B_9] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_8),B_9))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),C_7))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(times_times_real(C_7),A_8)),hAPP_real_real(times_times_real(C_7),B_9))) ) ) ).
fof(fact_646_mult__strict__left__mono,axiom,
! [C_7,A_8,B_9] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_8),B_9))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),C_7))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(times_times_int(C_7),A_8)),hAPP_int_int(times_times_int(C_7),B_9))) ) ) ).
fof(fact_647_comm__mult__strict__left__mono,axiom,
! [C_6,A_7,B_8] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_7),B_8))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),C_6))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(times_times_nat(C_6),A_7)),hAPP_nat_nat(times_times_nat(C_6),B_8))) ) ) ).
fof(fact_648_comm__mult__strict__left__mono,axiom,
! [C_6,A_7,B_8] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_7),B_8))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),C_6))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(times_times_real(C_6),A_7)),hAPP_real_real(times_times_real(C_6),B_8))) ) ) ).
fof(fact_649_comm__mult__strict__left__mono,axiom,
! [C_6,A_7,B_8] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_7),B_8))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),C_6))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(times_times_int(C_6),A_7)),hAPP_int_int(times_times_int(C_6),B_8))) ) ) ).
fof(fact_650_mult__strict__right__mono__neg,axiom,
! [C_5,B_7,A_6] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,B_7),A_6))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,C_5),zero_zero_real))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(times_times_real(A_6),C_5)),hAPP_real_real(times_times_real(B_7),C_5))) ) ) ).
fof(fact_651_mult__strict__right__mono__neg,axiom,
! [C_5,B_7,A_6] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_7),A_6))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,C_5),zero_zero_int))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(times_times_int(A_6),C_5)),hAPP_int_int(times_times_int(B_7),C_5))) ) ) ).
fof(fact_652_mult__strict__left__mono__neg,axiom,
! [C_4,B_6,A_5] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,B_6),A_5))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,C_4),zero_zero_real))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(times_times_real(C_4),A_5)),hAPP_real_real(times_times_real(C_4),B_6))) ) ) ).
fof(fact_653_mult__strict__left__mono__neg,axiom,
! [C_4,B_6,A_5] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_6),A_5))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,C_4),zero_zero_int))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(times_times_int(C_4),A_5)),hAPP_int_int(times_times_int(C_4),B_6))) ) ) ).
fof(fact_654_zero__le__double__add__iff__zero__le__single__add,axiom,
! [Aa] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(plus_plus_real(Aa),Aa)))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),Aa)) ) ).
fof(fact_655_zero__le__double__add__iff__zero__le__single__add,axiom,
! [Aa] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(plus_plus_int(Aa),Aa)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Aa)) ) ).
fof(fact_656_double__add__le__zero__iff__single__add__le__zero,axiom,
! [Aa] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(plus_plus_real(Aa),Aa)),zero_zero_real))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Aa),zero_zero_real)) ) ).
fof(fact_657_double__add__le__zero__iff__single__add__le__zero,axiom,
! [Aa] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(plus_plus_int(Aa),Aa)),zero_zero_int))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Aa),zero_zero_int)) ) ).
fof(fact_658_add__nonneg__nonneg,axiom,
! [B_5,A_4] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),A_4))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),B_5))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),hAPP_nat_nat(plus_plus_nat(A_4),B_5))) ) ) ).
fof(fact_659_add__nonneg__nonneg,axiom,
! [B_5,A_4] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_4))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),B_5))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(plus_plus_real(A_4),B_5))) ) ) ).
fof(fact_660_add__nonneg__nonneg,axiom,
! [B_5,A_4] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_4))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),B_5))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(plus_plus_int(A_4),B_5))) ) ) ).
fof(fact_661_add__nonneg__eq__0__iff,axiom,
! [Y_1,X_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),X_1))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),Y_1))
=> ( hAPP_nat_nat(plus_plus_nat(X_1),Y_1) = zero_zero_nat
<=> ( X_1 = zero_zero_nat
& Y_1 = zero_zero_nat ) ) ) ) ).
fof(fact_662_add__nonneg__eq__0__iff,axiom,
! [Y_1,X_1] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X_1))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),Y_1))
=> ( hAPP_real_real(plus_plus_real(X_1),Y_1) = zero_zero_real
<=> ( X_1 = zero_zero_real
& Y_1 = zero_zero_real ) ) ) ) ).
fof(fact_663_add__nonneg__eq__0__iff,axiom,
! [Y_1,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_eq_int,zero_zero_int),Y_1))
=> ( hAPP_int_int(plus_plus_int(X_1),Y_1) = zero_zero_int
<=> ( X_1 = zero_zero_int
& Y_1 = zero_zero_int ) ) ) ) ).
fof(fact_664_add__increasing,axiom,
! [B_4,C_3,A_3] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),A_3))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,B_4),C_3))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,B_4),hAPP_nat_nat(plus_plus_nat(A_3),C_3))) ) ) ).
fof(fact_665_add__increasing,axiom,
! [B_4,C_3,A_3] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_3))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B_4),C_3))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B_4),hAPP_real_real(plus_plus_real(A_3),C_3))) ) ) ).
fof(fact_666_add__increasing,axiom,
! [B_4,C_3,A_3] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_3))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_4),C_3))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_4),hAPP_int_int(plus_plus_int(A_3),C_3))) ) ) ).
fof(fact_667_add__increasing2,axiom,
! [B_3,A_2,C_2] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),C_2))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,B_3),A_2))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,B_3),hAPP_nat_nat(plus_plus_nat(A_2),C_2))) ) ) ).
fof(fact_668_add__increasing2,axiom,
! [B_3,A_2,C_2] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),C_2))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B_3),A_2))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B_3),hAPP_real_real(plus_plus_real(A_2),C_2))) ) ) ).
fof(fact_669_add__increasing2,axiom,
! [B_3,A_2,C_2] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),C_2))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_3),A_2))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_3),hAPP_int_int(plus_plus_int(A_2),C_2))) ) ) ).
fof(fact_670_add__nonpos__nonpos,axiom,
! [B_2,A_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_1),zero_zero_nat))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,B_2),zero_zero_nat))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(plus_plus_nat(A_1),B_2)),zero_zero_nat)) ) ) ).
fof(fact_671_add__nonpos__nonpos,axiom,
! [B_2,A_1] :
( 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_2),zero_zero_real))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(plus_plus_real(A_1),B_2)),zero_zero_real)) ) ) ).
fof(fact_672_add__nonpos__nonpos,axiom,
! [B_2,A_1] :
( 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_2),zero_zero_int))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(plus_plus_int(A_1),B_2)),zero_zero_int)) ) ) ).
fof(fact_673_zero__less__double__add__iff__zero__less__single__add,axiom,
! [Aa] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(plus_plus_real(Aa),Aa)))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Aa)) ) ).
fof(fact_674_zero__less__double__add__iff__zero__less__single__add,axiom,
! [Aa] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_int_int(plus_plus_int(Aa),Aa)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),Aa)) ) ).
fof(fact_675_Euler_Oaux1,axiom,
! [A,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 != hAPP_int_int(minus_minus_int(A),one_one_int)
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X),hAPP_int_int(minus_minus_int(A),one_one_int))) ) ) ) ).
fof(fact_676_int__0__less__1,axiom,
hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),one_one_int)) ).
fof(fact_677_zmult__zless__mono2,axiom,
! [K,I,J] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,I),J))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),K))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(times_times_int(K),I)),hAPP_int_int(times_times_int(K),J))) ) ) ).
fof(fact_678_odd__nonzero,axiom,
! [Z_1] : hAPP_int_int(plus_plus_int(hAPP_int_int(plus_plus_int(one_one_int),Z_1)),Z_1) != zero_zero_int ).
fof(fact_679_less__bin__lemma,axiom,
! [K_1,L] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K_1),L))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(minus_minus_int(K_1),L)),zero_zero_int)) ) ).
fof(fact_680_int__one__le__iff__zero__less,axiom,
! [Z] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,one_one_int),Z))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),Z)) ) ).
fof(fact_681_pos__zmult__eq__1__iff,axiom,
! [N_2,M_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),M_1))
=> ( hAPP_int_int(times_times_int(M_1),N_2) = one_one_int
<=> ( M_1 = one_one_int
& N_2 = one_one_int ) ) ) ).
fof(fact_682_odd__less__0,axiom,
! [Z] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(plus_plus_int(hAPP_int_int(plus_plus_int(one_one_int),Z)),Z)),zero_zero_int))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Z),zero_zero_int)) ) ).
fof(fact_683_le__imp__0__less,axiom,
! [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_int,zero_zero_int),hAPP_int_int(plus_plus_int(one_one_int),Z_1))) ) ).
fof(fact_684_self__quotient__aux2,axiom,
! [R,Q,A] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A))
=> ( A = hAPP_int_int(plus_plus_int(R),hAPP_int_int(times_times_int(A),Q))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),R))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Q),one_one_int)) ) ) ) ).
fof(fact_685_self__quotient__aux1,axiom,
! [R,Q,A] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A))
=> ( A = hAPP_int_int(plus_plus_int(R),hAPP_int_int(times_times_int(A),Q))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,R),A))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,one_one_int),Q)) ) ) ) ).
fof(fact_686_zdiv__mono2__neg__lemma,axiom,
! [B,Q,R,B_1,Q_1,R_1] :
( hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(B),Q)),R) = hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(B_1),Q_1)),R_1)
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(B_1),Q_1)),R_1)),zero_zero_int))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,R),B))
=> ( 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_1))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_1),B))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Q_1),Q)) ) ) ) ) ) ) ).
fof(fact_687_unique__quotient__lemma__neg,axiom,
! [B,Q_1,R_1,Q,R] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(B),Q_1)),R_1)),hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(B),Q)),R)))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,R),zero_zero_int))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B),R))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B),R_1))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Q),Q_1)) ) ) ) ) ).
fof(fact_688_zdiv__mono2__lemma,axiom,
! [B,Q,R,B_1,Q_1,R_1] :
( hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(B),Q)),R) = hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(B_1),Q_1)),R_1)
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(B_1),Q_1)),R_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,zero_zero_int),R))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_1))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_1),B))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Q),Q_1)) ) ) ) ) ) ) ).
fof(fact_689_unique__quotient__lemma,axiom,
! [B,Q_1,R_1,Q,R] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(B),Q_1)),R_1)),hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(B),Q)),R)))
=> ( 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,R_1),B))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,R),B))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Q_1),Q)) ) ) ) ) ).
fof(fact_690_q__neg__lemma,axiom,
! [B_1,Q_1,R_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(B_1),Q_1)),R_1)),zero_zero_int))
=> ( 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_1))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Q_1),zero_zero_int)) ) ) ) ).
fof(fact_691_q__pos__lemma,axiom,
! [B_1,Q_1,R_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(B_1),Q_1)),R_1)))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,R_1),B_1))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_1))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Q_1)) ) ) ) ).
fof(fact_692_xzgcd__linear,axiom,
! [M,R,S,T,N_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),N_1))
=> ( xzgcd(M,N_1) = produc282740534nt_int(R,product_Pair_int_int(S,T))
=> R = hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(S),M)),hAPP_int_int(times_times_int(T),N_1)) ) ) ).
fof(fact_693_xzgcda__linear,axiom,
! [M,N_1,R_1,S_1,S,T_1,T,Rn,Sn,Tn,R] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),R))
=> ( xzgcda(M,N_1,R_1,R,S_1,S,T_1,T) = produc282740534nt_int(Rn,product_Pair_int_int(Sn,Tn))
=> ( R_1 = hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(S_1),M)),hAPP_int_int(times_times_int(T_1),N_1))
=> ( R = hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(S),M)),hAPP_int_int(times_times_int(T),N_1))
=> Rn = hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(Sn),M)),hAPP_int_int(times_times_int(Tn),N_1)) ) ) ) ) ).
fof(fact_694_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),hAPP_int_int(plus_plus_int(X),Y))) ) ) ).
fof(fact_695_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),hAPP_int_int(times_times_int(X),Y))) ) ) ).
fof(fact_696_pos__zmult__pos,axiom,
! [B,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),hAPP_int_int(times_times_int(A),B)))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B)) ) ) ).
fof(fact_697_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_698_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_699_xzgcd__def,axiom,
! [M,N_1] : xzgcd(M,N_1) = xzgcda(M,N_1,M,N_1,one_one_int,zero_zero_int,zero_zero_int,one_one_int) ).
fof(fact_700_decr__mult__lemma,axiom,
! [K_1,P_1,D_2] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),D_2))
=> ( ! [X_2] :
( hBOOL(hAPP_int_bool(P_1,X_2))
=> hBOOL(hAPP_int_bool(P_1,hAPP_int_int(minus_minus_int(X_2),D_2))) )
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),K_1))
=> ! [X_2] :
( hBOOL(hAPP_int_bool(P_1,X_2))
=> hBOOL(hAPP_int_bool(P_1,hAPP_int_int(minus_minus_int(X_2),hAPP_int_int(times_times_int(K_1),D_2)))) ) ) ) ) ).
fof(fact_701_incr__mult__lemma,axiom,
! [K_1,P_1,D_2] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),D_2))
=> ( ! [X_2] :
( hBOOL(hAPP_int_bool(P_1,X_2))
=> hBOOL(hAPP_int_bool(P_1,hAPP_int_int(plus_plus_int(X_2),D_2))) )
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),K_1))
=> ! [X_2] :
( hBOOL(hAPP_int_bool(P_1,X_2))
=> hBOOL(hAPP_int_bool(P_1,hAPP_int_int(plus_plus_int(X_2),hAPP_int_int(times_times_int(K_1),D_2)))) ) ) ) ) ).
fof(fact_702_divmod__int__relI,axiom,
! [A,B,Q,R] :
( A = hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(B),Q)),R)
=> ( ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),R))
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,R),B)) ) )
& ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B),R))
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,R),zero_zero_int)) ) ) )
=> hBOOL(hAPP_P603027463t_bool(divmod_int_rel(A,B),product_Pair_int_int(Q,R))) ) ) ).
fof(fact_703_tsub__def,axiom,
! [Y,X] :
( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y),X))
=> nat_tsub(X,Y) = hAPP_int_int(minus_minus_int(X),Y) )
& ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y),X))
=> nat_tsub(X,Y) = zero_zero_int ) ) ).
fof(fact_704_divmod__int__rel__0,axiom,
! [B] :
( B != zero_zero_int
=> hBOOL(hAPP_P603027463t_bool(divmod_int_rel(zero_zero_int,B),product_Pair_int_int(zero_zero_int,zero_zero_int))) ) ).
fof(fact_705_self__remainder,axiom,
! [A,Q,R] :
( hBOOL(hAPP_P603027463t_bool(divmod_int_rel(A,A),product_Pair_int_int(Q,R)))
=> ( A != zero_zero_int
=> R = zero_zero_int ) ) ).
fof(fact_706_unique__quotient,axiom,
! [Q_1,R_1,A,B,Q,R] :
( hBOOL(hAPP_P603027463t_bool(divmod_int_rel(A,B),product_Pair_int_int(Q,R)))
=> ( hBOOL(hAPP_P603027463t_bool(divmod_int_rel(A,B),product_Pair_int_int(Q_1,R_1)))
=> ( B != zero_zero_int
=> Q = Q_1 ) ) ) ).
fof(fact_707_unique__remainder,axiom,
! [Q_1,R_1,A,B,Q,R] :
( hBOOL(hAPP_P603027463t_bool(divmod_int_rel(A,B),product_Pair_int_int(Q,R)))
=> ( hBOOL(hAPP_P603027463t_bool(divmod_int_rel(A,B),product_Pair_int_int(Q_1,R_1)))
=> ( B != zero_zero_int
=> R = R_1 ) ) ) ).
fof(fact_708_self__quotient,axiom,
! [A,Q,R] :
( hBOOL(hAPP_P603027463t_bool(divmod_int_rel(A,A),product_Pair_int_int(Q,R)))
=> ( A != zero_zero_int
=> Q = one_one_int ) ) ).
fof(fact_709_Nat__Transfer_Otransfer__nat__int__function__closures_I3_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),nat_tsub(X,Y))) ) ) ).
fof(fact_710_tsub__eq,axiom,
! [Y,X] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y),X))
=> nat_tsub(X,Y) = hAPP_int_int(minus_minus_int(X),Y) ) ).
fof(fact_711_conj__le__cong,axiom,
! [P_2,P_1,X_1] :
( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_1))
=> ( hBOOL(P_1)
<=> hBOOL(P_2) ) )
=> ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_1))
& hBOOL(P_1) )
<=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_1))
& hBOOL(P_2) ) ) ) ).
fof(fact_712_imp__le__cong,axiom,
! [P_2,P_1,X_1] :
( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_1))
=> ( hBOOL(P_1)
<=> hBOOL(P_2) ) )
=> ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_1))
=> hBOOL(P_1) )
<=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_1))
=> hBOOL(P_2) ) ) ) ).
fof(fact_713_minusinfinity,axiom,
! [P_1,P1,D_2] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),D_2))
=> ( ! [X_2,K_2] :
( hBOOL(hAPP_int_bool(P1,X_2))
<=> hBOOL(hAPP_int_bool(P1,hAPP_int_int(minus_minus_int(X_2),hAPP_int_int(times_times_int(K_2),D_2)))) )
=> ( ? [Z_3] :
! [X_2] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_2),Z_3))
=> ( hBOOL(hAPP_int_bool(P_1,X_2))
<=> hBOOL(hAPP_int_bool(P1,X_2)) ) )
=> ( ? [X1] : hBOOL(hAPP_int_bool(P1,X1))
=> ? [X1] : hBOOL(hAPP_int_bool(P_1,X1)) ) ) ) ) ).
fof(fact_714_plusinfinity,axiom,
! [P_1,P_2,D_2] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),D_2))
=> ( ! [X_2,K_2] :
( hBOOL(hAPP_int_bool(P_2,X_2))
<=> hBOOL(hAPP_int_bool(P_2,hAPP_int_int(minus_minus_int(X_2),hAPP_int_int(times_times_int(K_2),D_2)))) )
=> ( ? [Z_3] :
! [X_2] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Z_3),X_2))
=> ( hBOOL(hAPP_int_bool(P_1,X_2))
<=> hBOOL(hAPP_int_bool(P_2,X_2)) ) )
=> ( ? [X1] : hBOOL(hAPP_int_bool(P_2,X1))
=> ? [X1] : hBOOL(hAPP_int_bool(P_1,X1)) ) ) ) ) ).
fof(fact_715_zfact_Osimps,axiom,
! [N_1] :
( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,N_1),zero_zero_int))
=> zfact(N_1) = one_one_int )
& ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,N_1),zero_zero_int))
=> zfact(N_1) = hAPP_int_int(times_times_int(N_1),zfact(hAPP_int_int(minus_minus_int(N_1),one_one_int))) ) ) ).
fof(fact_716_le__mult__floor,axiom,
! [B,A] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),B))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(times_times_int(archim1246769320r_real(A)),archim1246769320r_real(B))),archim1246769320r_real(hAPP_real_real(times_times_real(A),B)))) ) ) ).
fof(fact_717_posDivAlg__correct,axiom,
! [B,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,zero_zero_int),B))
=> hBOOL(hAPP_P603027463t_bool(divmod_int_rel(A,B),posDivAlg(A,B))) ) ) ).
fof(fact_718_incr__lemma,axiom,
! [Z_1,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_1),hAPP_int_int(plus_plus_int(X),hAPP_int_int(times_times_int(hAPP_int_int(plus_plus_int(abs_abs_int(hAPP_int_int(minus_minus_int(X),Z_1))),one_one_int)),D_1)))) ) ).
fof(fact_719_decr__lemma,axiom,
! [X,Z_1,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,hAPP_int_int(minus_minus_int(X),hAPP_int_int(times_times_int(hAPP_int_int(plus_plus_int(abs_abs_int(hAPP_int_int(minus_minus_int(X),Z_1))),one_one_int)),D_1))),Z_1)) ) ).
fof(fact_720_real__le__refl,axiom,
! [W_1] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,W_1),W_1)) ).
fof(fact_721_real__le__linear,axiom,
! [Z_1,W_1] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Z_1),W_1))
| hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,W_1),Z_1)) ) ).
fof(fact_722_real__add__left__mono,axiom,
! [Z_1,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,hAPP_real_real(plus_plus_real(Z_1),X)),hAPP_real_real(plus_plus_real(Z_1),Y))) ) ).
fof(fact_723_real__le__trans,axiom,
! [K,I,J] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,I),J))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,J),K))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,I),K)) ) ) ).
fof(fact_724_real__le__antisym,axiom,
! [Z_1,W_1] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Z_1),W_1))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,W_1),Z_1))
=> Z_1 = W_1 ) ) ).
fof(fact_725_real__add__mult__distrib,axiom,
! [Z1,Z2,W_1] : hAPP_real_real(times_times_real(hAPP_real_real(plus_plus_real(Z1),Z2)),W_1) = hAPP_real_real(plus_plus_real(hAPP_real_real(times_times_real(Z1),W_1)),hAPP_real_real(times_times_real(Z2),W_1)) ).
fof(fact_726_real__mult__assoc,axiom,
! [Z1,Z2,Z3] : hAPP_real_real(times_times_real(hAPP_real_real(times_times_real(Z1),Z2)),Z3) = hAPP_real_real(times_times_real(Z1),hAPP_real_real(times_times_real(Z2),Z3)) ).
fof(fact_727_real__mult__commute,axiom,
! [Z_1,W_1] : hAPP_real_real(times_times_real(Z_1),W_1) = hAPP_real_real(times_times_real(W_1),Z_1) ).
fof(fact_728_real__mult__1,axiom,
! [Z_1] : hAPP_real_real(times_times_real(one_one_real),Z_1) = Z_1 ).
fof(fact_729_less__eq__real__def,axiom,
! [X_1,Y_1] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_1),Y_1))
<=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_1),Y_1))
| X_1 = Y_1 ) ) ).
fof(fact_730_real__less__def,axiom,
! [X_1,Y_1] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_1),Y_1))
<=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_1),Y_1))
& X_1 != Y_1 ) ) ).
fof(fact_731_real__le__eq__diff,axiom,
! [X_1,Y_1] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_1),Y_1))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(minus_minus_real(X_1),Y_1)),zero_zero_real)) ) ).
fof(fact_732_real__mult__left__cancel,axiom,
! [Aa,Ba,C_1] :
( C_1 != zero_zero_real
=> ( hAPP_real_real(times_times_real(C_1),Aa) = hAPP_real_real(times_times_real(C_1),Ba)
<=> Aa = Ba ) ) ).
fof(fact_733_real__mult__right__cancel,axiom,
! [Aa,Ba,C_1] :
( C_1 != zero_zero_real
=> ( hAPP_real_real(times_times_real(Aa),C_1) = hAPP_real_real(times_times_real(Ba),C_1)
<=> Aa = Ba ) ) ).
fof(fact_734_real__two__squares__add__zero__iff,axiom,
! [X_1,Y_1] :
( hAPP_real_real(plus_plus_real(hAPP_real_real(times_times_real(X_1),X_1)),hAPP_real_real(times_times_real(Y_1),Y_1)) = zero_zero_real
<=> ( X_1 = zero_zero_real
& Y_1 = zero_zero_real ) ) ).
fof(fact_735_real__zero__not__eq__one,axiom,
zero_zero_real != one_one_real ).
fof(fact_736_real__mult__less__iff1,axiom,
! [X_1,Y_1,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,hAPP_real_real(times_times_real(X_1),Z)),hAPP_real_real(times_times_real(Y_1),Z)))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_1),Y_1)) ) ) ).
fof(fact_737_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),hAPP_real_real(times_times_real(X),Y))) ) ) ).
fof(fact_738_real__mult__less__mono2,axiom,
! [X,Y,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,X),Y))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(times_times_real(Z_1),X)),hAPP_real_real(times_times_real(Z_1),Y))) ) ) ).
fof(fact_739_real__mult__le__cancel__iff2,axiom,
! [X_1,Y_1,Z] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Z))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(times_times_real(Z),X_1)),hAPP_real_real(times_times_real(Z),Y_1)))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_1),Y_1)) ) ) ).
fof(fact_740_real__mult__le__cancel__iff1,axiom,
! [X_1,Y_1,Z] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Z))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(times_times_real(X_1),Z)),hAPP_real_real(times_times_real(Y_1),Z)))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_1),Y_1)) ) ) ).
fof(fact_741_abs__zmult__eq__1,axiom,
! [M,N_1] :
( abs_abs_int(hAPP_int_int(times_times_int(M),N_1)) = one_one_int
=> abs_abs_int(M) = one_one_int ) ).
fof(fact_742_zabs__less__one__iff,axiom,
! [Z] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,abs_abs_int(Z)),one_one_int))
<=> Z = zero_zero_int ) ).
fof(fact_743_posDivAlg__0,axiom,
! [B] : posDivAlg(zero_zero_int,B) = product_Pair_int_int(zero_zero_int,zero_zero_int) ).
fof(fact_744_pdivmod__posDivAlg,axiom,
! [K,L_1] :
( ( L_1 = zero_zero_int
=> pdivmod(K,L_1) = product_Pair_int_int(zero_zero_int,abs_abs_int(K)) )
& ( L_1 != zero_zero_int
=> pdivmod(K,L_1) = posDivAlg(abs_abs_int(K),abs_abs_int(L_1)) ) ) ).
fof(fact_745_nat__intermed__int__val,axiom,
! [K_1,F,N_2,M_1] :
( ! [I_2] :
( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_1),I_2))
& hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_2),N_2)) )
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,abs_abs_int(hAPP_int_int(minus_minus_int(hAPP_nat_int(F,hAPP_nat_nat(plus_plus_nat(I_2),one_one_nat))),hAPP_nat_int(F,I_2)))),one_one_int)) )
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),N_2))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_nat_int(F,M_1)),K_1))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K_1),hAPP_nat_int(F,N_2)))
=> ? [I_2] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_1),I_2))
& hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_2),N_2))
& hAPP_nat_int(F,I_2) = K_1 ) ) ) ) ) ).
fof(fact_746_nat0__intermed__int__val,axiom,
! [K_1,F,N_2] :
( ! [I_2] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_2),N_2))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,abs_abs_int(hAPP_int_int(minus_minus_int(hAPP_nat_int(F,hAPP_nat_nat(plus_plus_nat(I_2),one_one_nat))),hAPP_nat_int(F,I_2)))),one_one_int)) )
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_nat_int(F,zero_zero_nat)),K_1))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K_1),hAPP_nat_int(F,N_2)))
=> ? [I_2] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_2),N_2))
& hAPP_nat_int(F,I_2) = K_1 ) ) ) ) ).
fof(fact_747_int__val__lemma,axiom,
! [K_1,F,N_2] :
( ! [I_2] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_2),N_2))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,abs_abs_int(hAPP_int_int(minus_minus_int(hAPP_nat_int(F,hAPP_nat_nat(plus_plus_nat(I_2),one_one_nat))),hAPP_nat_int(F,I_2)))),one_one_int)) )
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_nat_int(F,zero_zero_nat)),K_1))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K_1),hAPP_nat_int(F,N_2)))
=> ? [I_2] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_2),N_2))
& hAPP_nat_int(F,I_2) = K_1 ) ) ) ) ).
fof(fact_748_mult__eq__if,axiom,
! [N_1,M] :
( ( M = zero_zero_nat
=> hAPP_nat_nat(times_times_nat(M),N_1) = zero_zero_nat )
& ( M != zero_zero_nat
=> hAPP_nat_nat(times_times_nat(M),N_1) = hAPP_nat_nat(plus_plus_nat(N_1),hAPP_nat_nat(times_times_nat(hAPP_nat_nat(minus_minus_nat(M),one_one_nat)),N_1)) ) ) ).
fof(fact_749_abs__add__one__not__less__self,axiom,
! [X] : ~ hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(plus_plus_real(abs_abs_real(X)),one_one_real)),X)) ).
fof(fact_750_abs__add__one__gt__zero,axiom,
! [X] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(plus_plus_real(one_one_real),abs_abs_real(X)))) ).
fof(fact_751_le0,axiom,
! [N_1] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),N_1)) ).
fof(fact_752_less__zeroE,axiom,
! [N_1] : ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_1),zero_zero_nat)) ).
fof(fact_753_sin__bound__lemma,axiom,
! [U_1,V,X,Y] :
( X = Y
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,abs_abs_real(U_1)),V))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,abs_abs_real(hAPP_real_real(minus_minus_real(hAPP_real_real(plus_plus_real(X),U_1)),Y))),V)) ) ) ).
fof(fact_754_nat__less__add__iff2,axiom,
! [U,M_1,N_2,I_1,J_2] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_1),J_2))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(times_times_nat(I_1),U)),M_1)),hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(times_times_nat(J_2),U)),N_2)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(times_times_nat(hAPP_nat_nat(minus_minus_nat(J_2),I_1)),U)),N_2))) ) ) ).
fof(fact_755_diff__mult__distrib2,axiom,
! [K,M,N_1] : hAPP_nat_nat(times_times_nat(K),hAPP_nat_nat(minus_minus_nat(M),N_1)) = hAPP_nat_nat(minus_minus_nat(hAPP_nat_nat(times_times_nat(K),M)),hAPP_nat_nat(times_times_nat(K),N_1)) ).
fof(fact_756_diff__mult__distrib,axiom,
! [M,N_1,K] : hAPP_nat_nat(times_times_nat(hAPP_nat_nat(minus_minus_nat(M),N_1)),K) = hAPP_nat_nat(minus_minus_nat(hAPP_nat_nat(times_times_nat(M),K)),hAPP_nat_nat(times_times_nat(N_1),K)) ).
fof(fact_757_diff__commute,axiom,
! [I,J,K] : hAPP_nat_nat(minus_minus_nat(hAPP_nat_nat(minus_minus_nat(I),J)),K) = hAPP_nat_nat(minus_minus_nat(hAPP_nat_nat(minus_minus_nat(I),K)),J) ).
fof(fact_758_nat__mult__commute,axiom,
! [M,N_1] : hAPP_nat_nat(times_times_nat(M),N_1) = hAPP_nat_nat(times_times_nat(N_1),M) ).
fof(fact_759_nat__mult__assoc,axiom,
! [M,N_1,K] : hAPP_nat_nat(times_times_nat(hAPP_nat_nat(times_times_nat(M),N_1)),K) = hAPP_nat_nat(times_times_nat(M),hAPP_nat_nat(times_times_nat(N_1),K)) ).
fof(fact_760_mult__0,axiom,
! [N_1] : hAPP_nat_nat(times_times_nat(zero_zero_nat),N_1) = zero_zero_nat ).
fof(fact_761_mult__0__right,axiom,
! [M] : hAPP_nat_nat(times_times_nat(M),zero_zero_nat) = zero_zero_nat ).
fof(fact_762_mult__is__0,axiom,
! [M_1,N_2] :
( hAPP_nat_nat(times_times_nat(M_1),N_2) = zero_zero_nat
<=> ( M_1 = zero_zero_nat
| N_2 = zero_zero_nat ) ) ).
fof(fact_763_nat__mult__eq__cancel__disj,axiom,
! [K_1,M_1,N_2] :
( hAPP_nat_nat(times_times_nat(K_1),M_1) = hAPP_nat_nat(times_times_nat(K_1),N_2)
<=> ( K_1 = zero_zero_nat
| M_1 = N_2 ) ) ).
fof(fact_764_mult__cancel1,axiom,
! [K_1,M_1,N_2] :
( hAPP_nat_nat(times_times_nat(K_1),M_1) = hAPP_nat_nat(times_times_nat(K_1),N_2)
<=> ( M_1 = N_2
| K_1 = zero_zero_nat ) ) ).
fof(fact_765_mult__cancel2,axiom,
! [M_1,K_1,N_2] :
( hAPP_nat_nat(times_times_nat(M_1),K_1) = hAPP_nat_nat(times_times_nat(N_2),K_1)
<=> ( M_1 = N_2
| K_1 = zero_zero_nat ) ) ).
fof(fact_766_diffs0__imp__equal,axiom,
! [M,N_1] :
( hAPP_nat_nat(minus_minus_nat(M),N_1) = zero_zero_nat
=> ( hAPP_nat_nat(minus_minus_nat(N_1),M) = zero_zero_nat
=> M = N_1 ) ) ).
fof(fact_767_diff__self__eq__0,axiom,
! [M] : hAPP_nat_nat(minus_minus_nat(M),M) = zero_zero_nat ).
fof(fact_768_minus__nat_Odiff__0,axiom,
! [M] : hAPP_nat_nat(minus_minus_nat(M),zero_zero_nat) = M ).
fof(fact_769_diff__0__eq__0,axiom,
! [N_1] : hAPP_nat_nat(minus_minus_nat(zero_zero_nat),N_1) = zero_zero_nat ).
fof(fact_770_nat__less__cases,axiom,
! [P_1,M_1,N_2] :
( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),N_2))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(P_1,N_2),M_1)) )
=> ( ( M_1 = N_2
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(P_1,N_2),M_1)) )
=> ( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_2),M_1))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(P_1,N_2),M_1)) )
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(P_1,N_2),M_1)) ) ) ) ).
fof(fact_771_less__not__refl3,axiom,
! [S,T] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,S),T))
=> S != T ) ).
fof(fact_772_less__not__refl2,axiom,
! [N_1,M] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_1),M))
=> M != N_1 ) ).
fof(fact_773_less__irrefl__nat,axiom,
! [N_1] : ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_1),N_1)) ).
fof(fact_774_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_775_nat__neq__iff,axiom,
! [M_1,N_2] :
( M_1 != N_2
<=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),N_2))
| hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_2),M_1)) ) ) ).
fof(fact_776_less__not__refl,axiom,
! [N_1] : ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_1),N_1)) ).
fof(fact_777_diff__less__mono2,axiom,
! [L_1,M,N_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N_1))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),L_1))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(minus_minus_nat(L_1),N_1)),hAPP_nat_nat(minus_minus_nat(L_1),M))) ) ) ).
fof(fact_778_less__imp__diff__less,axiom,
! [N_1,J,K] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,J),K))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(minus_minus_nat(J),N_1)),K)) ) ).
fof(fact_779_diff__add__inverse2,axiom,
! [M,N_1] : hAPP_nat_nat(minus_minus_nat(hAPP_nat_nat(plus_plus_nat(M),N_1)),N_1) = M ).
fof(fact_780_diff__add__inverse,axiom,
! [N_1,M] : hAPP_nat_nat(minus_minus_nat(hAPP_nat_nat(plus_plus_nat(N_1),M)),N_1) = M ).
fof(fact_781_diff__diff__left,axiom,
! [I,J,K] : hAPP_nat_nat(minus_minus_nat(hAPP_nat_nat(minus_minus_nat(I),J)),K) = hAPP_nat_nat(minus_minus_nat(I),hAPP_nat_nat(plus_plus_nat(J),K)) ).
fof(fact_782_Nat_Odiff__cancel,axiom,
! [K,M,N_1] : hAPP_nat_nat(minus_minus_nat(hAPP_nat_nat(plus_plus_nat(K),M)),hAPP_nat_nat(plus_plus_nat(K),N_1)) = hAPP_nat_nat(minus_minus_nat(M),N_1) ).
fof(fact_783_diff__cancel2,axiom,
! [M,K,N_1] : hAPP_nat_nat(minus_minus_nat(hAPP_nat_nat(plus_plus_nat(M),K)),hAPP_nat_nat(plus_plus_nat(N_1),K)) = hAPP_nat_nat(minus_minus_nat(M),N_1) ).
fof(fact_784_nat__add__right__cancel,axiom,
! [M_1,K_1,N_2] :
( hAPP_nat_nat(plus_plus_nat(M_1),K_1) = hAPP_nat_nat(plus_plus_nat(N_2),K_1)
<=> M_1 = N_2 ) ).
fof(fact_785_nat__add__left__cancel,axiom,
! [K_1,M_1,N_2] :
( hAPP_nat_nat(plus_plus_nat(K_1),M_1) = hAPP_nat_nat(plus_plus_nat(K_1),N_2)
<=> M_1 = N_2 ) ).
fof(fact_786_nat__add__assoc,axiom,
! [M,N_1,K] : hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(plus_plus_nat(M),N_1)),K) = hAPP_nat_nat(plus_plus_nat(M),hAPP_nat_nat(plus_plus_nat(N_1),K)) ).
fof(fact_787_nat__add__left__commute,axiom,
! [X,Y,Z_1] : hAPP_nat_nat(plus_plus_nat(X),hAPP_nat_nat(plus_plus_nat(Y),Z_1)) = hAPP_nat_nat(plus_plus_nat(Y),hAPP_nat_nat(plus_plus_nat(X),Z_1)) ).
fof(fact_788_nat__add__commute,axiom,
! [M,N_1] : hAPP_nat_nat(plus_plus_nat(M),N_1) = hAPP_nat_nat(plus_plus_nat(N_1),M) ).
fof(fact_789_add__mult__distrib2,axiom,
! [K,M,N_1] : hAPP_nat_nat(times_times_nat(K),hAPP_nat_nat(plus_plus_nat(M),N_1)) = hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(times_times_nat(K),M)),hAPP_nat_nat(times_times_nat(K),N_1)) ).
fof(fact_790_add__mult__distrib,axiom,
! [M,N_1,K] : hAPP_nat_nat(times_times_nat(hAPP_nat_nat(plus_plus_nat(M),N_1)),K) = hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(times_times_nat(M),K)),hAPP_nat_nat(times_times_nat(N_1),K)) ).
fof(fact_791_left__add__mult__distrib,axiom,
! [I,U_1,J,K] : hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(times_times_nat(I),U_1)),hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(times_times_nat(J),U_1)),K)) = hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(times_times_nat(hAPP_nat_nat(plus_plus_nat(I),J)),U_1)),K) ).
fof(fact_792_Nat_Odiff__le__self,axiom,
! [M,N_1] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(minus_minus_nat(M),N_1)),M)) ).
fof(fact_793_diff__le__mono2,axiom,
! [L_1,M,N_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N_1))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(minus_minus_nat(L_1),N_1)),hAPP_nat_nat(minus_minus_nat(L_1),M))) ) ).
fof(fact_794_diff__le__mono,axiom,
! [L_1,M,N_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N_1))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(minus_minus_nat(M),L_1)),hAPP_nat_nat(minus_minus_nat(N_1),L_1))) ) ).
fof(fact_795_diff__diff__cancel,axiom,
! [I,N_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I),N_1))
=> hAPP_nat_nat(minus_minus_nat(N_1),hAPP_nat_nat(minus_minus_nat(N_1),I)) = I ) ).
fof(fact_796_eq__diff__iff,axiom,
! [N_2,K_1,M_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_1),M_1))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_1),N_2))
=> ( hAPP_nat_nat(minus_minus_nat(M_1),K_1) = hAPP_nat_nat(minus_minus_nat(N_2),K_1)
<=> M_1 = N_2 ) ) ) ).
fof(fact_797_Nat_Odiff__diff__eq,axiom,
! [N_1,K,M] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K),M))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K),N_1))
=> hAPP_nat_nat(minus_minus_nat(hAPP_nat_nat(minus_minus_nat(M),K)),hAPP_nat_nat(minus_minus_nat(N_1),K)) = hAPP_nat_nat(minus_minus_nat(M),N_1) ) ) ).
fof(fact_798_le__diff__iff,axiom,
! [N_2,K_1,M_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_1),M_1))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_1),N_2))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(minus_minus_nat(M_1),K_1)),hAPP_nat_nat(minus_minus_nat(N_2),K_1)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_1),N_2)) ) ) ) ).
fof(fact_799_le__refl,axiom,
! [N_1] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_1),N_1)) ).
fof(fact_800_nat__le__linear,axiom,
! [M,N_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N_1))
| hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_1),M)) ) ).
fof(fact_801_eq__imp__le,axiom,
! [M,N_1] :
( M = N_1
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N_1)) ) ).
fof(fact_802_le__trans,axiom,
! [K,I,J] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I),J))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,J),K))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I),K)) ) ) ).
fof(fact_803_le__antisym,axiom,
! [M,N_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N_1))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_1),M))
=> M = N_1 ) ) ).
fof(fact_804_mult__le__mono,axiom,
! [K,L_1,I,J] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I),J))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K),L_1))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(times_times_nat(I),K)),hAPP_nat_nat(times_times_nat(J),L_1))) ) ) ).
fof(fact_805_mult__le__mono2,axiom,
! [K,I,J] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I),J))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(times_times_nat(K),I)),hAPP_nat_nat(times_times_nat(K),J))) ) ).
fof(fact_806_mult__le__mono1,axiom,
! [K,I,J] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I),J))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(times_times_nat(I),K)),hAPP_nat_nat(times_times_nat(J),K))) ) ).
fof(fact_807_le__cube,axiom,
! [M] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),hAPP_nat_nat(times_times_nat(M),hAPP_nat_nat(times_times_nat(M),M)))) ).
fof(fact_808_le__square,axiom,
! [M] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),hAPP_nat_nat(times_times_nat(M),M))) ).
fof(fact_809_nat__mult__1,axiom,
! [N_1] : hAPP_nat_nat(times_times_nat(one_one_nat),N_1) = N_1 ).
fof(fact_810_nat__1__eq__mult__iff,axiom,
! [M_1,N_2] :
( one_one_nat = hAPP_nat_nat(times_times_nat(M_1),N_2)
<=> ( M_1 = one_one_nat
& N_2 = one_one_nat ) ) ).
fof(fact_811_nat__mult__1__right,axiom,
! [N_1] : hAPP_nat_nat(times_times_nat(N_1),one_one_nat) = N_1 ).
fof(fact_812_nat__mult__eq__1__iff,axiom,
! [M_1,N_2] :
( hAPP_nat_nat(times_times_nat(M_1),N_2) = one_one_nat
<=> ( M_1 = one_one_nat
& N_2 = one_one_nat ) ) ).
fof(fact_813_zero__less__diff,axiom,
! [N_2,M_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_nat_nat(minus_minus_nat(N_2),M_1)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),N_2)) ) ).
fof(fact_814_diff__less,axiom,
! [M,N_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),M))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(minus_minus_nat(M),N_1)),M)) ) ) ).
fof(fact_815_not__less0,axiom,
! [N_1] : ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_1),zero_zero_nat)) ).
fof(fact_816_neq0__conv,axiom,
! [N_2] :
( N_2 != zero_zero_nat
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_2)) ) ).
fof(fact_817_less__nat__zero__code,axiom,
! [N_1] : ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_1),zero_zero_nat)) ).
fof(fact_818_gr__implies__not0,axiom,
! [M,N_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N_1))
=> N_1 != zero_zero_nat ) ).
fof(fact_819_gr0I,axiom,
! [N_1] :
( N_1 != zero_zero_nat
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1)) ) ).
fof(fact_820_mult__less__mono2,axiom,
! [K,I,J] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I),J))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),K))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(times_times_nat(K),I)),hAPP_nat_nat(times_times_nat(K),J))) ) ) ).
fof(fact_821_mult__less__mono1,axiom,
! [K,I,J] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I),J))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),K))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(times_times_nat(I),K)),hAPP_nat_nat(times_times_nat(J),K))) ) ) ).
fof(fact_822_nat__mult__less__cancel1,axiom,
! [M_1,N_2,K_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,hAPP_nat_nat(times_times_nat(K_1),M_1)),hAPP_nat_nat(times_times_nat(K_1),N_2)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),N_2)) ) ) ).
fof(fact_823_nat__mult__eq__cancel1,axiom,
! [M_1,N_2,K_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),K_1))
=> ( hAPP_nat_nat(times_times_nat(K_1),M_1) = hAPP_nat_nat(times_times_nat(K_1),N_2)
<=> M_1 = N_2 ) ) ).
fof(fact_824_mult__less__cancel2,axiom,
! [M_1,K_1,N_2] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(times_times_nat(M_1),K_1)),hAPP_nat_nat(times_times_nat(N_2),K_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,M_1),N_2)) ) ) ).
fof(fact_825_mult__less__cancel1,axiom,
! [K_1,M_1,N_2] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(times_times_nat(K_1),M_1)),hAPP_nat_nat(times_times_nat(K_1),N_2)))
<=> ( 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,M_1),N_2)) ) ) ).
fof(fact_826_nat__0__less__mult__iff,axiom,
! [M_1,N_2] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_nat_nat(times_times_nat(M_1),N_2)))
<=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),M_1))
& hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_2)) ) ) ).
fof(fact_827_plus__nat_Oadd__0,axiom,
! [N_1] : hAPP_nat_nat(plus_plus_nat(zero_zero_nat),N_1) = N_1 ).
fof(fact_828_Nat_Oadd__0__right,axiom,
! [M] : hAPP_nat_nat(plus_plus_nat(M),zero_zero_nat) = M ).
fof(fact_829_add__is__0,axiom,
! [M_1,N_2] :
( hAPP_nat_nat(plus_plus_nat(M_1),N_2) = zero_zero_nat
<=> ( M_1 = zero_zero_nat
& N_2 = zero_zero_nat ) ) ).
fof(fact_830_add__eq__self__zero,axiom,
! [M,N_1] :
( hAPP_nat_nat(plus_plus_nat(M),N_1) = M
=> N_1 = zero_zero_nat ) ).
fof(fact_831_diff__add__0,axiom,
! [N_1,M] : hAPP_nat_nat(minus_minus_nat(N_1),hAPP_nat_nat(plus_plus_nat(N_1),M)) = zero_zero_nat ).
fof(fact_832_diff__is__0__eq,axiom,
! [M_1,N_2] :
( hAPP_nat_nat(minus_minus_nat(M_1),N_2) = zero_zero_nat
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_1),N_2)) ) ).
fof(fact_833_diff__is__0__eq_H,axiom,
! [M,N_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N_1))
=> hAPP_nat_nat(minus_minus_nat(M),N_1) = zero_zero_nat ) ).
fof(fact_834_less__eq__nat_Osimps_I1_J,axiom,
! [N_1] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),N_1)) ).
fof(fact_835_le__0__eq,axiom,
! [N_2] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_2),zero_zero_nat))
<=> N_2 = zero_zero_nat ) ).
fof(fact_836_not__add__less1,axiom,
! [I,J] : ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(plus_plus_nat(I),J)),I)) ).
fof(fact_837_not__add__less2,axiom,
! [J,I] : ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(plus_plus_nat(J),I)),I)) ).
fof(fact_838_nat__add__left__cancel__less,axiom,
! [K_1,M_1,N_2] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(plus_plus_nat(K_1),M_1)),hAPP_nat_nat(plus_plus_nat(K_1),N_2)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),N_2)) ) ).
fof(fact_839_trans__less__add1,axiom,
! [M,I,J] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I),J))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I),hAPP_nat_nat(plus_plus_nat(J),M))) ) ).
fof(fact_840_trans__less__add2,axiom,
! [M,I,J] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I),J))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I),hAPP_nat_nat(plus_plus_nat(M),J))) ) ).
fof(fact_841_add__less__mono1,axiom,
! [K,I,J] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I),J))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(plus_plus_nat(I),K)),hAPP_nat_nat(plus_plus_nat(J),K))) ) ).
fof(fact_842_add__less__mono,axiom,
! [K,L_1,I,J] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I),J))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,K),L_1))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(plus_plus_nat(I),K)),hAPP_nat_nat(plus_plus_nat(J),L_1))) ) ) ).
fof(fact_843_less__add__eq__less,axiom,
! [M,N_1,K,L_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,K),L_1))
=> ( hAPP_nat_nat(plus_plus_nat(M),L_1) = hAPP_nat_nat(plus_plus_nat(K),N_1)
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N_1)) ) ) ).
fof(fact_844_add__lessD1,axiom,
! [I,J,K] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(plus_plus_nat(I),J)),K))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I),K)) ) ).
fof(fact_845_add__diff__inverse,axiom,
! [M,N_1] :
( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N_1))
=> hAPP_nat_nat(plus_plus_nat(N_1),hAPP_nat_nat(minus_minus_nat(M),N_1)) = M ) ).
fof(fact_846_less__diff__conv,axiom,
! [I_1,J_2,K_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_1),hAPP_nat_nat(minus_minus_nat(J_2),K_1)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(plus_plus_nat(I_1),K_1)),J_2)) ) ).
fof(fact_847_mult__eq__self__implies__10,axiom,
! [M,N_1] :
( M = hAPP_nat_nat(times_times_nat(M),N_1)
=> ( N_1 = one_one_nat
| M = zero_zero_nat ) ) ).
fof(fact_848_less__or__eq__imp__le,axiom,
! [M,N_1] :
( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N_1))
| M = N_1 )
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N_1)) ) ).
fof(fact_849_le__neq__implies__less,axiom,
! [M,N_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N_1))
=> ( M != N_1
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N_1)) ) ) ).
fof(fact_850_less__imp__le__nat,axiom,
! [M,N_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N_1))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N_1)) ) ).
fof(fact_851_le__eq__less__or__eq,axiom,
! [M_1,N_2] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_1),N_2))
<=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),N_2))
| M_1 = N_2 ) ) ).
fof(fact_852_nat__less__le,axiom,
! [M_1,N_2] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),N_2))
<=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_1),N_2))
& M_1 != N_2 ) ) ).
fof(fact_853_less__diff__iff,axiom,
! [N_2,K_1,M_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_1),M_1))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_1),N_2))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(minus_minus_nat(M_1),K_1)),hAPP_nat_nat(minus_minus_nat(N_2),K_1)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),N_2)) ) ) ) ).
fof(fact_854_diff__less__mono,axiom,
! [C,A,B] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A),B))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,C),A))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(minus_minus_nat(A),C)),hAPP_nat_nat(minus_minus_nat(B),C))) ) ) ).
fof(fact_855_diff__diff__right,axiom,
! [I,K,J] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K),J))
=> hAPP_nat_nat(minus_minus_nat(I),hAPP_nat_nat(minus_minus_nat(J),K)) = hAPP_nat_nat(minus_minus_nat(hAPP_nat_nat(plus_plus_nat(I),K)),J) ) ).
fof(fact_856_le__diff__conv,axiom,
! [J_2,K_1,I_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(minus_minus_nat(J_2),K_1)),I_1))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,J_2),hAPP_nat_nat(plus_plus_nat(I_1),K_1))) ) ).
fof(fact_857_le__add__diff,axiom,
! [M,K,N_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K),N_1))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),hAPP_nat_nat(minus_minus_nat(hAPP_nat_nat(plus_plus_nat(N_1),M)),K))) ) ).
fof(fact_858_le__add__diff__inverse,axiom,
! [N_1,M] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_1),M))
=> hAPP_nat_nat(plus_plus_nat(N_1),hAPP_nat_nat(minus_minus_nat(M),N_1)) = M ) ).
fof(fact_859_add__diff__assoc,axiom,
! [I,K,J] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K),J))
=> hAPP_nat_nat(plus_plus_nat(I),hAPP_nat_nat(minus_minus_nat(J),K)) = hAPP_nat_nat(minus_minus_nat(hAPP_nat_nat(plus_plus_nat(I),J)),K) ) ).
fof(fact_860_le__diff__conv2,axiom,
! [I_1,K_1,J_2] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_1),J_2))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_1),hAPP_nat_nat(minus_minus_nat(J_2),K_1)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(plus_plus_nat(I_1),K_1)),J_2)) ) ) ).
fof(fact_861_le__add__diff__inverse2,axiom,
! [N_1,M] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_1),M))
=> hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(minus_minus_nat(M),N_1)),N_1) = M ) ).
fof(fact_862_le__imp__diff__is__add,axiom,
! [K_1,I_1,J_2] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_1),J_2))
=> ( hAPP_nat_nat(minus_minus_nat(J_2),I_1) = K_1
<=> J_2 = hAPP_nat_nat(plus_plus_nat(K_1),I_1) ) ) ).
fof(fact_863_diff__add__assoc,axiom,
! [I,K,J] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K),J))
=> hAPP_nat_nat(minus_minus_nat(hAPP_nat_nat(plus_plus_nat(I),J)),K) = hAPP_nat_nat(plus_plus_nat(I),hAPP_nat_nat(minus_minus_nat(J),K)) ) ).
fof(fact_864_add__diff__assoc2,axiom,
! [I,K,J] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K),J))
=> hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(minus_minus_nat(J),K)),I) = hAPP_nat_nat(minus_minus_nat(hAPP_nat_nat(plus_plus_nat(J),I)),K) ) ).
fof(fact_865_diff__add__assoc2,axiom,
! [I,K,J] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K),J))
=> hAPP_nat_nat(minus_minus_nat(hAPP_nat_nat(plus_plus_nat(J),I)),K) = hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(minus_minus_nat(J),K)),I) ) ).
fof(fact_866_nat__eq__add__iff2,axiom,
! [U,M_1,N_2,I_1,J_2] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_1),J_2))
=> ( hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(times_times_nat(I_1),U)),M_1) = hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(times_times_nat(J_2),U)),N_2)
<=> M_1 = hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(times_times_nat(hAPP_nat_nat(minus_minus_nat(J_2),I_1)),U)),N_2) ) ) ).
fof(fact_867_nat__diff__add__eq2,axiom,
! [U_1,M,N_1,I,J] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I),J))
=> hAPP_nat_nat(minus_minus_nat(hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(times_times_nat(I),U_1)),M)),hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(times_times_nat(J),U_1)),N_1)) = hAPP_nat_nat(minus_minus_nat(M),hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(times_times_nat(hAPP_nat_nat(minus_minus_nat(J),I)),U_1)),N_1)) ) ).
fof(fact_868_nat__le__add__iff2,axiom,
! [U,M_1,N_2,I_1,J_2] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_1),J_2))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(times_times_nat(I_1),U)),M_1)),hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(times_times_nat(J_2),U)),N_2)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_1),hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(times_times_nat(hAPP_nat_nat(minus_minus_nat(J_2),I_1)),U)),N_2))) ) ) ).
fof(fact_869_nat__eq__add__iff1,axiom,
! [U,M_1,N_2,J_2,I_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,J_2),I_1))
=> ( hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(times_times_nat(I_1),U)),M_1) = hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(times_times_nat(J_2),U)),N_2)
<=> hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(times_times_nat(hAPP_nat_nat(minus_minus_nat(I_1),J_2)),U)),M_1) = N_2 ) ) ).
fof(fact_870_nat__diff__add__eq1,axiom,
! [U_1,M,N_1,J,I] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,J),I))
=> hAPP_nat_nat(minus_minus_nat(hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(times_times_nat(I),U_1)),M)),hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(times_times_nat(J),U_1)),N_1)) = hAPP_nat_nat(minus_minus_nat(hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(times_times_nat(hAPP_nat_nat(minus_minus_nat(I),J)),U_1)),M)),N_1) ) ).
fof(fact_871_nat__le__add__iff1,axiom,
! [U,M_1,N_2,J_2,I_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,J_2),I_1))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(times_times_nat(I_1),U)),M_1)),hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(times_times_nat(J_2),U)),N_2)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(times_times_nat(hAPP_nat_nat(minus_minus_nat(I_1),J_2)),U)),M_1)),N_2)) ) ) ).
fof(fact_872_add__leE,axiom,
! [M,K,N_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(plus_plus_nat(M),K)),N_1))
=> ~ ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N_1))
=> ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K),N_1)) ) ) ).
fof(fact_873_add__leD1,axiom,
! [M,K,N_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(plus_plus_nat(M),K)),N_1))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N_1)) ) ).
fof(fact_874_add__leD2,axiom,
! [M,K,N_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(plus_plus_nat(M),K)),N_1))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K),N_1)) ) ).
fof(fact_875_add__le__mono,axiom,
! [K,L_1,I,J] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I),J))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K),L_1))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(plus_plus_nat(I),K)),hAPP_nat_nat(plus_plus_nat(J),L_1))) ) ) ).
fof(fact_876_add__le__mono1,axiom,
! [K,I,J] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I),J))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(plus_plus_nat(I),K)),hAPP_nat_nat(plus_plus_nat(J),K))) ) ).
fof(fact_877_trans__le__add2,axiom,
! [M,I,J] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I),J))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I),hAPP_nat_nat(plus_plus_nat(M),J))) ) ).
fof(fact_878_trans__le__add1,axiom,
! [M,I,J] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I),J))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I),hAPP_nat_nat(plus_plus_nat(J),M))) ) ).
fof(fact_879_nat__add__left__cancel__le,axiom,
! [K_1,M_1,N_2] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(plus_plus_nat(K_1),M_1)),hAPP_nat_nat(plus_plus_nat(K_1),N_2)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_1),N_2)) ) ).
fof(fact_880_le__iff__add,axiom,
! [M_1,N_2] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_1),N_2))
<=> ? [K_2] : N_2 = hAPP_nat_nat(plus_plus_nat(M_1),K_2) ) ).
fof(fact_881_le__add1,axiom,
! [N_1,M] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_1),hAPP_nat_nat(plus_plus_nat(N_1),M))) ).
fof(fact_882_le__add2,axiom,
! [N_1,M] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_1),hAPP_nat_nat(plus_plus_nat(M),N_1))) ).
fof(fact_883_eq__diff__eq_H,axiom,
! [X_1,Y_1,Z] :
( X_1 = hAPP_real_real(minus_minus_real(Y_1),Z)
<=> Y_1 = hAPP_real_real(plus_plus_real(X_1),Z) ) ).
fof(fact_884_nat__diff__split__asm,axiom,
! [P_1,Aa,Ba] :
( hBOOL(hAPP_nat_bool(P_1,hAPP_nat_nat(minus_minus_nat(Aa),Ba)))
<=> ~ ( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Aa),Ba))
& ~ hBOOL(hAPP_nat_bool(P_1,zero_zero_nat)) )
| ? [D] :
( Aa = hAPP_nat_nat(plus_plus_nat(Ba),D)
& ~ hBOOL(hAPP_nat_bool(P_1,D)) ) ) ) ).
fof(fact_885_nat__diff__split,axiom,
! [P_1,Aa,Ba] :
( hBOOL(hAPP_nat_bool(P_1,hAPP_nat_nat(minus_minus_nat(Aa),Ba)))
<=> ( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Aa),Ba))
=> hBOOL(hAPP_nat_bool(P_1,zero_zero_nat)) )
& ! [D] :
( Aa = hAPP_nat_nat(plus_plus_nat(Ba),D)
=> hBOOL(hAPP_nat_bool(P_1,D)) ) ) ) ).
fof(fact_886_add__gr__0,axiom,
! [M_1,N_2] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_nat_nat(plus_plus_nat(M_1),N_2)))
<=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),M_1))
| hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_2)) ) ) ).
fof(fact_887_mult__le__cancel1,axiom,
! [K_1,M_1,N_2] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(times_times_nat(K_1),M_1)),hAPP_nat_nat(times_times_nat(K_1),N_2)))
<=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),K_1))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_1),N_2)) ) ) ).
fof(fact_888_mult__le__cancel2,axiom,
! [M_1,K_1,N_2] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(times_times_nat(M_1),K_1)),hAPP_nat_nat(times_times_nat(N_2),K_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_eq_nat,M_1),N_2)) ) ) ).
fof(fact_889_nat__mult__le__cancel1,axiom,
! [M_1,N_2,K_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_eq_nat,hAPP_nat_nat(times_times_nat(K_1),M_1)),hAPP_nat_nat(times_times_nat(K_1),N_2)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_1),N_2)) ) ) ).
fof(fact_890_nat__less__add__iff1,axiom,
! [U,M_1,N_2,J_2,I_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,J_2),I_1))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(times_times_nat(I_1),U)),M_1)),hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(times_times_nat(J_2),U)),N_2)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(times_times_nat(hAPP_nat_nat(minus_minus_nat(I_1),J_2)),U)),M_1)),N_2)) ) ) ).
fof(fact_891_ex__least__nat__less,axiom,
! [N_2,P_1] :
( ~ hBOOL(hAPP_nat_bool(P_1,zero_zero_nat))
=> ( hBOOL(hAPP_nat_bool(P_1,N_2))
=> ? [K_2] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,K_2),N_2))
& ! [I_2] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_2),K_2))
=> ~ hBOOL(hAPP_nat_bool(P_1,I_2)) )
& hBOOL(hAPP_nat_bool(P_1,hAPP_nat_nat(plus_plus_nat(K_2),one_one_nat))) ) ) ) ).
fof(fact_892_rabs__ratiotest__lemma,axiom,
! [X,Y,C] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,C),zero_zero_real))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,abs_abs_real(X)),hAPP_real_real(times_times_real(C),abs_abs_real(Y))))
=> X = zero_zero_real ) ) ).
fof(fact_893_mono__nat__linear__lb,axiom,
! [M_1,K_1,F] :
( ! [M_2,N] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_2),N))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(F,M_2)),hAPP_nat_nat(F,N))) )
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(F,M_1)),K_1)),hAPP_nat_nat(F,hAPP_nat_nat(plus_plus_nat(M_1),K_1)))) ) ).
fof(fact_894_ex__least__nat__le,axiom,
! [N_2,P_1] :
( ~ hBOOL(hAPP_nat_bool(P_1,zero_zero_nat))
=> ( hBOOL(hAPP_nat_bool(P_1,N_2))
=> ? [K_2] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_2),N_2))
& ! [I_2] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_2),K_2))
=> ~ hBOOL(hAPP_nat_bool(P_1,I_2)) )
& hBOOL(hAPP_nat_bool(P_1,K_2)) ) ) ) ).
fof(fact_895_le__Suc__ex__iff,axiom,
! [K_1,L] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_1),L))
<=> ? [N] : L = hAPP_nat_nat(plus_plus_nat(K_1),N) ) ).
fof(fact_896_less__imp__add__positive,axiom,
! [I,J] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I),J))
=> ? [K_2] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),K_2))
& hAPP_nat_nat(plus_plus_nat(I),K_2) = J ) ) ).
fof(fact_897_not__real__square__gt__zero,axiom,
! [X_1] :
( ~ hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(times_times_real(X_1),X_1)))
<=> X_1 = zero_zero_real ) ).
fof(fact_898_mult__left__cancel,axiom,
! [N_1,M,K] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),K))
=> ( hAPP_nat_nat(times_times_nat(K),N_1) = hAPP_nat_nat(times_times_nat(K),M)
=> N_1 = M ) ) ).
fof(fact_899_lemma__interval,axiom,
! [B,A,X] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A),X))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X),B))
=> ? [D] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),D))
& ! [Y_2] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,abs_abs_real(hAPP_real_real(minus_minus_real(X),Y_2))),D))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A),Y_2))
& hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y_2),B)) ) ) ) ) ) ).
fof(fact_900_le__Suc__ex,axiom,
! [K,L_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K),L_1))
=> ? [N] : L_1 = hAPP_nat_nat(plus_plus_nat(K),N) ) ).
fof(fact_901_less__mono__imp__le__mono,axiom,
! [I_1,J_2,F] :
( ! [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,hAPP_nat_nat(F,I_2)),hAPP_nat_nat(F,J_1))) )
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_1),J_2))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(F,I_1)),hAPP_nat_nat(F,J_2))) ) ) ).
fof(fact_902_lemma__interval__lt,axiom,
! [B,A,X] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A),X))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X),B))
=> ? [D] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),D))
& ! [Y_2] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,abs_abs_real(hAPP_real_real(minus_minus_real(X),Y_2))),D))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A),Y_2))
& hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Y_2),B)) ) ) ) ) ) ).
fof(fact_903_real__lbound__gt__zero,axiom,
! [D2,D1] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),D1))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),D2))
=> ? [E] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),E))
& hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,E),D1))
& hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,E),D2)) ) ) ) ).
fof(fact_904_natceiling__add__one,axiom,
! [X] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X))
=> natceiling(hAPP_real_real(plus_plus_real(X),one_one_real)) = hAPP_nat_nat(plus_plus_nat(natceiling(X)),one_one_nat) ) ).
fof(fact_905_natceiling__number__of__eq,axiom,
! [N_1] : natceiling(number267125858f_real(N_1)) = number_number_of_nat(N_1) ).
fof(fact_906_natceiling__zero,axiom,
natceiling(zero_zero_real) = zero_zero_nat ).
fof(fact_907_zero__le__natceiling,axiom,
! [X] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),natceiling(X))) ).
fof(fact_908_natceiling__mono,axiom,
! [X,Y] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X),Y))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,natceiling(X)),natceiling(Y))) ) ).
fof(fact_909_natceiling__one,axiom,
natceiling(one_one_real) = one_one_nat ).
fof(fact_910_natceiling__neg,axiom,
! [X] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X),zero_zero_real))
=> natceiling(X) = zero_zero_nat ) ).
fof(fact_911_natceiling__le__eq__one,axiom,
! [X_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,natceiling(X_1)),one_one_nat))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_1),one_one_real)) ) ).
fof(fact_912_natfloor__add__one,axiom,
! [X] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X))
=> natfloor(hAPP_real_real(plus_plus_real(X),one_one_real)) = hAPP_nat_nat(plus_plus_nat(natfloor(X)),one_one_nat) ) ).
fof(fact_913_nat__mult__eq__one,axiom,
! [N_2,M_1] :
( hAPP_nat_nat(times_times_nat(N_2),M_1) = one_one_nat
<=> ( N_2 = one_one_nat
& M_1 = one_one_nat ) ) ).
fof(fact_914_natceiling__eq,axiom,
! [N_1,X] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_nat_real(real_nat,N_1)),X))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X),hAPP_real_real(plus_plus_real(hAPP_nat_real(real_nat,N_1)),one_one_real)))
=> natceiling(X) = hAPP_nat_nat(plus_plus_nat(N_1),one_one_nat) ) ) ).
fof(fact_915_real__of__nat__ge__zero,axiom,
! [N_1] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_nat_real(real_nat,N_1))) ).
fof(fact_916_natceiling__real__of__nat,axiom,
! [N_1] : natceiling(hAPP_nat_real(real_nat,N_1)) = N_1 ).
fof(fact_917_real__of__nat__less__iff,axiom,
! [N_2,M_1] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_nat_real(real_nat,N_2)),hAPP_nat_real(real_nat,M_1)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_2),M_1)) ) ).
fof(fact_918_real__of__nat__zero,axiom,
hAPP_nat_real(real_nat,zero_zero_nat) = zero_zero_real ).
fof(fact_919_real__of__nat__zero__iff,axiom,
! [N_2] :
( hAPP_nat_real(real_nat,N_2) = zero_zero_real
<=> N_2 = zero_zero_nat ) ).
fof(fact_920_abs__real__of__nat__cancel,axiom,
! [X] : abs_abs_real(hAPP_nat_real(real_nat,X)) = hAPP_nat_real(real_nat,X) ).
fof(fact_921_natfloor__real__of__nat,axiom,
! [N_1] : natfloor(hAPP_nat_real(real_nat,N_1)) = N_1 ).
fof(fact_922_Ints__real__of__nat,axiom,
! [N_2] : hBOOL(member_real(hAPP_nat_real(real_nat,N_2),ring_1_Ints_real)) ).
fof(fact_923_natfloor__number__of__eq,axiom,
! [N_1] : natfloor(number267125858f_real(N_1)) = number_number_of_nat(N_1) ).
fof(fact_924_real__of__nat__inject,axiom,
! [N_2,M_1] :
( hAPP_nat_real(real_nat,N_2) = hAPP_nat_real(real_nat,M_1)
<=> N_2 = M_1 ) ).
fof(fact_925_real__natfloor__le,axiom,
! [X] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(real_nat,natfloor(X))),X)) ) ).
fof(fact_926_le__natfloor,axiom,
! [X,A] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(real_nat,X)),A))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X),natfloor(A))) ) ).
fof(fact_927_real__natfloor__gt__diff__one,axiom,
! [X] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(minus_minus_real(X),one_one_real)),hAPP_nat_real(real_nat,natfloor(X)))) ).
fof(fact_928_natfloor__subtract,axiom,
! [A,X] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(real_nat,A)),X))
=> natfloor(hAPP_real_real(minus_minus_real(X),hAPP_nat_real(real_nat,A))) = hAPP_nat_nat(minus_minus_nat(natfloor(X)),A) ) ).
fof(fact_929_real__natfloor__add__one__gt,axiom,
! [X] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X),hAPP_real_real(plus_plus_real(hAPP_nat_real(real_nat,natfloor(X))),one_one_real))) ).
fof(fact_930_le__natfloor__eq,axiom,
! [Aa,X_1] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X_1))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Aa),natfloor(X_1)))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(real_nat,Aa)),X_1)) ) ) ).
fof(fact_931_not__real__of__nat__less__zero,axiom,
! [N_1] : ~ hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_nat_real(real_nat,N_1)),zero_zero_real)) ).
fof(fact_932_real__of__nat__le__iff,axiom,
! [N_2,M_1] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(real_nat,N_2)),hAPP_nat_real(real_nat,M_1)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_2),M_1)) ) ).
fof(fact_933_less__natfloor,axiom,
! [N_1,X] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X),hAPP_nat_real(real_nat,N_1)))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,natfloor(X)),N_1)) ) ) ).
fof(fact_934_real__of__nat__mult,axiom,
! [M,N_1] : hAPP_nat_real(real_nat,hAPP_nat_nat(times_times_nat(M),N_1)) = hAPP_real_real(times_times_real(hAPP_nat_real(real_nat,M)),hAPP_nat_real(real_nat,N_1)) ).
fof(fact_935_natfloor__add,axiom,
! [A,X] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X))
=> natfloor(hAPP_real_real(plus_plus_real(X),hAPP_nat_real(real_nat,A))) = hAPP_nat_nat(plus_plus_nat(natfloor(X)),A) ) ).
fof(fact_936_real__of__nat__add,axiom,
! [M,N_1] : hAPP_nat_real(real_nat,hAPP_nat_nat(plus_plus_nat(M),N_1)) = hAPP_real_real(plus_plus_real(hAPP_nat_real(real_nat,M)),hAPP_nat_real(real_nat,N_1)) ).
fof(fact_937_ge__natfloor__plus__one__imp__gt,axiom,
! [Z_1,N_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(plus_plus_nat(natfloor(Z_1)),one_one_nat)),N_1))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Z_1),hAPP_nat_real(real_nat,N_1))) ) ).
fof(fact_938_natfloor__eq,axiom,
! [N_1,X] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(real_nat,N_1)),X))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X),hAPP_real_real(plus_plus_real(hAPP_nat_real(real_nat,N_1)),one_one_real)))
=> natfloor(X) = N_1 ) ) ).
fof(fact_939_real__of__nat__1,axiom,
hAPP_nat_real(real_nat,one_one_nat) = one_one_real ).
fof(fact_940_real__natceiling__ge,axiom,
! [X] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X),hAPP_nat_real(real_nat,natceiling(X)))) ).
fof(fact_941_natfloor__zero,axiom,
natfloor(zero_zero_real) = zero_zero_nat ).
fof(fact_942_zero__le__natfloor,axiom,
! [X] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),natfloor(X))) ).
fof(fact_943_natfloor__mono,axiom,
! [X,Y] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X),Y))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,natfloor(X)),natfloor(Y))) ) ).
fof(fact_944_natfloor__one,axiom,
natfloor(one_one_real) = one_one_nat ).
fof(fact_945_real__of__nat__le__zero__cancel__iff,axiom,
! [N_2] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(real_nat,N_2)),zero_zero_real))
<=> N_2 = zero_zero_nat ) ).
fof(fact_946_real__of__nat__diff,axiom,
! [N_1,M] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_1),M))
=> hAPP_nat_real(real_nat,hAPP_nat_nat(minus_minus_nat(M),N_1)) = hAPP_real_real(minus_minus_real(hAPP_nat_real(real_nat,M)),hAPP_nat_real(real_nat,N_1)) ) ).
fof(fact_947_natceiling__le,axiom,
! [X,A] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X),hAPP_nat_real(real_nat,A)))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,natceiling(X)),A)) ) ).
fof(fact_948_natfloor__neg,axiom,
! [X] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X),zero_zero_real))
=> natfloor(X) = zero_zero_nat ) ).
fof(fact_949_real__of__nat__gt__zero__cancel__iff,axiom,
! [N_2] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_nat_real(real_nat,N_2)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_2)) ) ).
fof(fact_950_nat__less__real__le,axiom,
! [N_2,M_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_2),M_1))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(plus_plus_real(hAPP_nat_real(real_nat,N_2)),one_one_real)),hAPP_nat_real(real_nat,M_1))) ) ).
fof(fact_951_nat__le__real__less,axiom,
! [N_2,M_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_2),M_1))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_nat_real(real_nat,N_2)),hAPP_real_real(plus_plus_real(hAPP_nat_real(real_nat,M_1)),one_one_real))) ) ).
fof(fact_952_natceiling__le__eq,axiom,
! [Aa,X_1] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X_1))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,natceiling(X_1)),Aa))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_1),hAPP_nat_real(real_nat,Aa))) ) ) ).
fof(fact_953_natceiling__subtract,axiom,
! [A,X] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(real_nat,A)),X))
=> natceiling(hAPP_real_real(minus_minus_real(X),hAPP_nat_real(real_nat,A))) = hAPP_nat_nat(minus_minus_nat(natceiling(X)),A) ) ).
fof(fact_954_le__natfloor__eq__one,axiom,
! [X_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,one_one_nat),natfloor(X_1)))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,one_one_real),X_1)) ) ).
fof(fact_955_natceiling__add,axiom,
! [A,X] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X))
=> natceiling(hAPP_real_real(plus_plus_real(X),hAPP_nat_real(real_nat,A))) = hAPP_nat_nat(plus_plus_nat(natceiling(X)),A) ) ).
fof(fact_956_le__mult__natfloor,axiom,
! [B,A] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),B))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(times_times_nat(natfloor(A)),natfloor(B))),natfloor(hAPP_real_real(times_times_real(A),B)))) ) ) ).
fof(fact_957_reals__Archimedean6,axiom,
! [R] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),R))
=> ? [N] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(real_nat,hAPP_nat_nat(minus_minus_nat(N),one_one_nat))),R))
& hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,R),hAPP_nat_real(real_nat,N))) ) ) ).
fof(fact_958_reals__Archimedean3,axiom,
! [X] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X))
=> ! [Y_2] :
? [N] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Y_2),hAPP_real_real(times_times_real(hAPP_nat_real(real_nat,N)),X))) ) ).
fof(fact_959_natceiling__add__number__of,axiom,
! [X,N_1] :
( ~ hBOOL(hAPP_int_bool(nat_neg,number_number_of_int(N_1)))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X))
=> natceiling(hAPP_real_real(plus_plus_real(X),number267125858f_real(N_1))) = hAPP_nat_nat(plus_plus_nat(natceiling(X)),number_number_of_nat(N_1)) ) ) ).
fof(fact_960_not__neg__0,axiom,
~ hBOOL(hAPP_int_bool(nat_neg,zero_zero_int)) ).
fof(fact_961_not__neg__1,axiom,
~ hBOOL(hAPP_int_bool(nat_neg,one_one_int)) ).
fof(fact_962_neg__def,axiom,
! [Z_2] :
( hBOOL(hAPP_int_bool(nat_neg,Z_2))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Z_2),zero_zero_int)) ) ).
fof(fact_963_not__neg__eq__ge__0,axiom,
! [X_1] :
( ~ hBOOL(hAPP_int_bool(nat_neg,X_1))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_1)) ) ).
fof(fact_964_neg__imp__number__of__eq__0,axiom,
! [V] :
( hBOOL(hAPP_int_bool(nat_neg,number_number_of_int(V)))
=> number_number_of_nat(V) = zero_zero_nat ) ).
fof(fact_965_eq__nat__number__of,axiom,
! [V_1,V_3] :
( number_number_of_nat(V_1) = number_number_of_nat(V_3)
<=> ( ( hBOOL(hAPP_int_bool(nat_neg,number_number_of_int(V_1)))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,number_number_of_int(V_3)),zero_zero_int)) )
& ( ~ hBOOL(hAPP_int_bool(nat_neg,number_number_of_int(V_1)))
=> ( ( hBOOL(hAPP_int_bool(nat_neg,number_number_of_int(V_3)))
=> number_number_of_int(V_1) = zero_zero_int )
& ( ~ hBOOL(hAPP_int_bool(nat_neg,number_number_of_int(V_3)))
=> V_1 = V_3 ) ) ) ) ) ).
fof(fact_966_nat__number__of__add__left,axiom,
! [V_2,K,V] :
( ( hBOOL(hAPP_int_bool(nat_neg,number_number_of_int(V)))
=> hAPP_nat_nat(plus_plus_nat(number_number_of_nat(V)),hAPP_nat_nat(plus_plus_nat(number_number_of_nat(V_2)),K)) = hAPP_nat_nat(plus_plus_nat(number_number_of_nat(V_2)),K) )
& ( ~ hBOOL(hAPP_int_bool(nat_neg,number_number_of_int(V)))
=> ( ( hBOOL(hAPP_int_bool(nat_neg,number_number_of_int(V_2)))
=> hAPP_nat_nat(plus_plus_nat(number_number_of_nat(V)),hAPP_nat_nat(plus_plus_nat(number_number_of_nat(V_2)),K)) = hAPP_nat_nat(plus_plus_nat(number_number_of_nat(V)),K) )
& ( ~ hBOOL(hAPP_int_bool(nat_neg,number_number_of_int(V_2)))
=> hAPP_nat_nat(plus_plus_nat(number_number_of_nat(V)),hAPP_nat_nat(plus_plus_nat(number_number_of_nat(V_2)),K)) = hAPP_nat_nat(plus_plus_nat(number_number_of_nat(hAPP_int_int(plus_plus_int(V),V_2))),K) ) ) ) ) ).
fof(fact_967_real__of__nat__number__of,axiom,
! [V] :
( ( hBOOL(hAPP_int_bool(nat_neg,number_number_of_int(V)))
=> hAPP_nat_real(real_nat,number_number_of_nat(V)) = zero_zero_real )
& ( ~ hBOOL(hAPP_int_bool(nat_neg,number_number_of_int(V)))
=> hAPP_nat_real(real_nat,number_number_of_nat(V)) = number267125858f_real(V) ) ) ).
fof(fact_968_le__natfloor__eq__number__of,axiom,
! [X_1,N_2] :
( ~ hBOOL(hAPP_int_bool(nat_neg,number_number_of_int(N_2)))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X_1))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,number_number_of_nat(N_2)),natfloor(X_1)))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,number267125858f_real(N_2)),X_1)) ) ) ) ).
fof(fact_969_natceiling__le__eq__number__of,axiom,
! [X_1,N_2] :
( ~ hBOOL(hAPP_int_bool(nat_neg,number_number_of_int(N_2)))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X_1))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,natceiling(X_1)),number_number_of_nat(N_2)))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_1),number267125858f_real(N_2))) ) ) ) ).
fof(fact_970_natfloor__add__number__of,axiom,
! [X,N_1] :
( ~ hBOOL(hAPP_int_bool(nat_neg,number_number_of_int(N_1)))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X))
=> natfloor(hAPP_real_real(plus_plus_real(X),number267125858f_real(N_1))) = hAPP_nat_nat(plus_plus_nat(natfloor(X)),number_number_of_nat(N_1)) ) ) ).
fof(fact_971_reals__Archimedean2,axiom,
! [X] :
? [N] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X),hAPP_nat_real(real_nat,N))) ).
fof(fact_972_termination__basic__simps_I4_J,axiom,
! [Y,X,Z_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X),Z_1))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X),hAPP_nat_nat(plus_plus_nat(Y),Z_1))) ) ).
fof(fact_973_termination__basic__simps_I3_J,axiom,
! [Z_1,X,Y] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X),Y))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X),hAPP_nat_nat(plus_plus_nat(Y),Z_1))) ) ).
fof(fact_974_termination__basic__simps_I2_J,axiom,
! [Y,X,Z_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X),Z_1))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X),hAPP_nat_nat(plus_plus_nat(Y),Z_1))) ) ).
fof(fact_975_termination__basic__simps_I1_J,axiom,
! [Z_1,X,Y] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X),Y))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X),hAPP_nat_nat(plus_plus_nat(Y),Z_1))) ) ).
fof(fact_976_termination__basic__simps_I5_J,axiom,
! [X,Y] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X),Y))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X),Y)) ) ).
fof(fact_977_ceiling__eq2,axiom,
! [N_1,X] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_int_real(real_int,N_1)),X))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X),hAPP_real_real(plus_plus_real(hAPP_int_real(real_int,N_1)),one_one_real)))
=> archim856651990g_real(X) = hAPP_int_int(plus_plus_int(N_1),one_one_int) ) ) ).
fof(fact_978_real__of__int__add,axiom,
! [X,Y] : hAPP_int_real(real_int,hAPP_int_int(plus_plus_int(X),Y)) = hAPP_real_real(plus_plus_real(hAPP_int_real(real_int,X)),hAPP_int_real(real_int,Y)) ).
fof(fact_979_ceiling__add,axiom,
! [X,A] : archim856651990g_real(hAPP_real_real(plus_plus_real(X),hAPP_int_real(real_int,A))) = hAPP_int_int(plus_plus_int(archim856651990g_real(X)),A) ).
fof(fact_980_real__of__int__floor__cancel,axiom,
! [X_1] :
( hAPP_int_real(real_int,archim1246769320r_real(X_1)) = X_1
<=> ? [N] : X_1 = hAPP_int_real(real_int,N) ) ).
fof(fact_981_floor__real__of__int,axiom,
! [N_1] : archim1246769320r_real(hAPP_int_real(real_int,N_1)) = N_1 ).
fof(fact_982_real__of__int__ceiling__cancel,axiom,
! [X_1] :
( hAPP_int_real(real_int,archim856651990g_real(X_1)) = X_1
<=> ? [N] : X_1 = hAPP_int_real(real_int,N) ) ).
fof(fact_983_ceiling__real__of__int,axiom,
! [N_1] : archim856651990g_real(hAPP_int_real(real_int,N_1)) = N_1 ).
fof(fact_984_real__number__of,axiom,
! [V] : hAPP_int_real(real_int,number_number_of_int(V)) = number267125858f_real(V) ).
fof(fact_985_real__of__int__inject,axiom,
! [X_1,Y_1] :
( hAPP_int_real(real_int,X_1) = hAPP_int_real(real_int,Y_1)
<=> X_1 = Y_1 ) ).
fof(fact_986_real__of__int__ceiling__ge,axiom,
! [R] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,R),hAPP_int_real(real_int,archim856651990g_real(R)))) ).
fof(fact_987_real__of__one,axiom,
hAPP_int_real(real_int,one_one_int) = one_one_real ).
fof(fact_988_real__of__int__mult,axiom,
! [X,Y] : hAPP_int_real(real_int,hAPP_int_int(times_times_int(X),Y)) = hAPP_real_real(times_times_real(hAPP_int_real(real_int,X)),hAPP_int_real(real_int,Y)) ).
fof(fact_989_ceiling__le__eq,axiom,
! [X_1,Aa] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,archim856651990g_real(X_1)),Aa))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_1),hAPP_int_real(real_int,Aa))) ) ).
fof(fact_990_ceiling__le,axiom,
! [X,A] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X),hAPP_int_real(real_int,A)))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,archim856651990g_real(X)),A)) ) ).
fof(fact_991_ceiling__le__real,axiom,
! [X,A] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,archim856651990g_real(X)),A))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X),hAPP_int_real(real_int,A))) ) ).
fof(fact_992_real__of__int__le__iff,axiom,
! [X_1,Y_1] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(real_int,X_1)),hAPP_int_real(real_int,Y_1)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_1),Y_1)) ) ).
fof(fact_993_real__of__int__less__iff,axiom,
! [X_1,Y_1] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_int_real(real_int,X_1)),hAPP_int_real(real_int,Y_1)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_1),Y_1)) ) ).
fof(fact_994_less__ceiling__eq,axiom,
! [Aa,X_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Aa),archim856651990g_real(X_1)))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_int_real(real_int,Aa)),X_1)) ) ).
fof(fact_995_real__of__int__zero__cancel,axiom,
! [X_1] :
( hAPP_int_real(real_int,X_1) = zero_zero_real
<=> X_1 = zero_zero_int ) ).
fof(fact_996_real__of__int__zero,axiom,
hAPP_int_real(real_int,zero_zero_int) = zero_zero_real ).
fof(fact_997_ceiling__subtract,axiom,
! [X,A] : archim856651990g_real(hAPP_real_real(minus_minus_real(X),hAPP_int_real(real_int,A))) = hAPP_int_int(minus_minus_int(archim856651990g_real(X)),A) ).
fof(fact_998_real__of__int__diff,axiom,
! [X,Y] : hAPP_int_real(real_int,hAPP_int_int(minus_minus_int(X),Y)) = hAPP_real_real(minus_minus_real(hAPP_int_real(real_int,X)),hAPP_int_real(real_int,Y)) ).
fof(fact_999_real__of__int__floor__le,axiom,
! [R] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(real_int,archim1246769320r_real(R))),R)) ).
fof(fact_1000_real__of__int__abs,axiom,
! [X] : hAPP_int_real(real_int,abs_abs_int(X)) = abs_abs_real(hAPP_int_real(real_int,X)) ).
fof(fact_1001_Ints__real__of__int,axiom,
! [X_1] : hBOOL(member_real(hAPP_int_real(real_int,X_1),ring_1_Ints_real)) ).
fof(fact_1002_real__of__int__ceiling__le__add__one,axiom,
! [R] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(real_int,archim856651990g_real(R))),hAPP_real_real(plus_plus_real(R),one_one_real))) ).
fof(fact_1003_real__of__int__ceiling__diff__one__le,axiom,
! [R] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(minus_minus_real(hAPP_int_real(real_int,archim856651990g_real(R))),one_one_real)),R)) ).
fof(fact_1004_floor__less__eq,axiom,
! [X_1,Aa] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,archim1246769320r_real(X_1)),Aa))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_1),hAPP_int_real(real_int,Aa))) ) ).
fof(fact_1005_le__floor,axiom,
! [A,X] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(real_int,A)),X))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A),archim1246769320r_real(X))) ) ).
fof(fact_1006_le__floor__eq,axiom,
! [Aa,X_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Aa),archim1246769320r_real(X_1)))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(real_int,Aa)),X_1)) ) ).
fof(fact_1007_real__le__floor,axiom,
! [A,X] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A),archim1246769320r_real(X)))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(real_int,A)),X)) ) ).
fof(fact_1008_floor__add,axiom,
! [X,A] : archim1246769320r_real(hAPP_real_real(plus_plus_real(X),hAPP_int_real(real_int,A))) = hAPP_int_int(plus_plus_int(archim1246769320r_real(X)),A) ).
fof(fact_1009_floor__subtract,axiom,
! [X,A] : archim1246769320r_real(hAPP_real_real(minus_minus_real(X),hAPP_int_real(real_int,A))) = hAPP_int_int(minus_minus_int(archim1246769320r_real(X)),A) ).
fof(fact_1010_real__of__int__gt__zero__cancel__iff,axiom,
! [N_2] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_int_real(real_int,N_2)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),N_2)) ) ).
fof(fact_1011_real__of__int__lt__zero__cancel__iff,axiom,
! [N_2] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_int_real(real_int,N_2)),zero_zero_real))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,N_2),zero_zero_int)) ) ).
fof(fact_1012_real__of__int__ge__zero__cancel__iff,axiom,
! [N_2] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_int_real(real_int,N_2)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),N_2)) ) ).
fof(fact_1013_real__of__int__le__zero__cancel__iff,axiom,
! [N_2] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(real_int,N_2)),zero_zero_real))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,N_2),zero_zero_int)) ) ).
fof(fact_1014_number__of__less__real__of__int__iff,axiom,
! [N_2,M_1] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,number267125858f_real(N_2)),hAPP_int_real(real_int,M_1)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,number_number_of_int(N_2)),M_1)) ) ).
fof(fact_1015_number__of__less__real__of__int__iff2,axiom,
! [M_1,N_2] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_int_real(real_int,M_1)),number267125858f_real(N_2)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,M_1),number_number_of_int(N_2))) ) ).
fof(fact_1016_number__of__le__real__of__int__iff,axiom,
! [N_2,M_1] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,number267125858f_real(N_2)),hAPP_int_real(real_int,M_1)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,number_number_of_int(N_2)),M_1)) ) ).
fof(fact_1017_number__of__le__real__of__int__iff2,axiom,
! [M_1,N_2] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(real_int,M_1)),number267125858f_real(N_2)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,M_1),number_number_of_int(N_2))) ) ).
fof(fact_1018_int__less__real__le,axiom,
! [N_2,M_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,N_2),M_1))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(plus_plus_real(hAPP_int_real(real_int,N_2)),one_one_real)),hAPP_int_real(real_int,M_1))) ) ).
fof(fact_1019_int__le__real__less,axiom,
! [N_2,M_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,N_2),M_1))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_int_real(real_int,N_2)),hAPP_real_real(plus_plus_real(hAPP_int_real(real_int,M_1)),one_one_real))) ) ).
fof(fact_1020_lemma__floor2,axiom,
! [N_1,X] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_int_real(real_int,N_1)),hAPP_real_real(plus_plus_real(hAPP_int_real(real_int,X)),one_one_real)))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,N_1),X)) ) ).
fof(fact_1021_ceiling__eq3,axiom,
! [N_1,X] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(minus_minus_real(hAPP_int_real(real_int,N_1)),one_one_real)),X))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X),hAPP_int_real(real_int,N_1)))
=> archim856651990g_real(X) = N_1 ) ) ).
fof(fact_1022_ceiling__less__eq,axiom,
! [X_1,Aa] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,archim856651990g_real(X_1)),Aa))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_1),hAPP_real_real(minus_minus_real(hAPP_int_real(real_int,Aa)),one_one_real))) ) ).
fof(fact_1023_le__ceiling__eq,axiom,
! [Aa,X_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Aa),archim856651990g_real(X_1)))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(minus_minus_real(hAPP_int_real(real_int,Aa)),one_one_real)),X_1)) ) ).
fof(fact_1024_real__of__int__floor__add__one__ge,axiom,
! [R] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,R),hAPP_real_real(plus_plus_real(hAPP_int_real(real_int,archim1246769320r_real(R))),one_one_real))) ).
fof(fact_1025_real__of__int__floor__add__one__gt,axiom,
! [R] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,R),hAPP_real_real(plus_plus_real(hAPP_int_real(real_int,archim1246769320r_real(R))),one_one_real))) ).
fof(fact_1026_floor__eq,axiom,
! [N_1,X] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_int_real(real_int,N_1)),X))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X),hAPP_real_real(plus_plus_real(hAPP_int_real(real_int,N_1)),one_one_real)))
=> archim1246769320r_real(X) = N_1 ) ) ).
fof(fact_1027_real__of__int__floor__ge__diff__one,axiom,
! [R] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(minus_minus_real(R),one_one_real)),hAPP_int_real(real_int,archim1246769320r_real(R)))) ).
fof(fact_1028_real__of__int__floor__gt__diff__one,axiom,
! [R] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(minus_minus_real(R),one_one_real)),hAPP_int_real(real_int,archim1246769320r_real(R)))) ).
fof(fact_1029_lemma__floor,axiom,
! [N_1,M,R] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(real_int,M)),R))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,R),hAPP_real_real(plus_plus_real(hAPP_int_real(real_int,N_1)),one_one_real)))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,M),N_1)) ) ) ).
fof(fact_1030_ceiling__eq,axiom,
! [N_1,X] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_int_real(real_int,N_1)),X))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X),hAPP_real_real(plus_plus_real(hAPP_int_real(real_int,N_1)),one_one_real)))
=> archim856651990g_real(X) = hAPP_int_int(plus_plus_int(N_1),one_one_int) ) ) ).
fof(fact_1031_floor__eq2,axiom,
! [N_1,X] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(real_int,N_1)),X))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X),hAPP_real_real(plus_plus_real(hAPP_int_real(real_int,N_1)),one_one_real)))
=> archim1246769320r_real(X) = N_1 ) ) ).
fof(fact_1032_less__floor__eq,axiom,
! [Aa,X_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Aa),archim1246769320r_real(X_1)))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(plus_plus_real(hAPP_int_real(real_int,Aa)),one_one_real)),X_1)) ) ).
fof(fact_1033_floor__le__eq,axiom,
! [X_1,Aa] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,archim1246769320r_real(X_1)),Aa))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_1),hAPP_real_real(plus_plus_real(hAPP_int_real(real_int,Aa)),one_one_real))) ) ).
fof(fact_1034_reals__Archimedean__6b__int,axiom,
! [R] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),R))
=> ? [N] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(real_int,N)),R))
& hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,R),hAPP_int_real(real_int,hAPP_int_int(plus_plus_int(N),one_one_int)))) ) ) ).
fof(fact_1035_reals__Archimedean__6c__int,axiom,
! [R] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,R),zero_zero_real))
=> ? [N] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(real_int,N)),R))
& hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,R),hAPP_int_real(real_int,hAPP_int_int(plus_plus_int(N),one_one_int)))) ) ) ).
fof(fact_1036_real__lb__ub__int,axiom,
! [R] :
? [N] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(real_int,N)),R))
& hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,R),hAPP_int_real(real_int,hAPP_int_int(plus_plus_int(N),one_one_int)))) ) ).
fof(fact_1037_ceiling__real__of__nat,axiom,
! [N_1] : archim856651990g_real(hAPP_nat_real(real_nat,N_1)) = hAPP_nat_int(semiri1621563631at_int,N_1) ).
fof(fact_1038_int__less__0__conv,axiom,
! [K] : ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_nat_int(semiri1621563631at_int,K)),zero_zero_int)) ).
fof(fact_1039_zero__zle__int,axiom,
! [N_1] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_nat_int(semiri1621563631at_int,N_1))) ).
fof(fact_1040_transfer__int__nat__quantifiers_I1_J,axiom,
! [P_1] :
( ! [X_2] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_2))
=> hBOOL(hAPP_int_bool(P_1,X_2)) )
<=> ! [X_2] : hBOOL(hAPP_int_bool(P_1,hAPP_nat_int(semiri1621563631at_int,X_2))) ) ).
fof(fact_1041_transfer__int__nat__quantifiers_I2_J,axiom,
! [P_1] :
( ? [X_2] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_2))
& hBOOL(hAPP_int_bool(P_1,X_2)) )
<=> ? [X_2] : hBOOL(hAPP_int_bool(P_1,hAPP_nat_int(semiri1621563631at_int,X_2))) ) ).
fof(fact_1042_Nat__Transfer_Otransfer__nat__int__function__closures_I9_J,axiom,
! [Z_1] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_nat_int(semiri1621563631at_int,Z_1))) ).
fof(fact_1043_Nat__Transfer_Otransfer__int__nat__relations_I2_J,axiom,
! [X_1,Y_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_nat_int(semiri1621563631at_int,X_1)),hAPP_nat_int(semiri1621563631at_int,Y_1)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_1),Y_1)) ) ).
fof(fact_1044_zless__int,axiom,
! [M_1,N_2] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_nat_int(semiri1621563631at_int,M_1)),hAPP_nat_int(semiri1621563631at_int,N_2)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),N_2)) ) ).
fof(fact_1045_abs__int__eq,axiom,
! [M] : abs_abs_int(hAPP_nat_int(semiri1621563631at_int,M)) = hAPP_nat_int(semiri1621563631at_int,M) ).
fof(fact_1046_int__eq__0__conv,axiom,
! [N_2] :
( hAPP_nat_int(semiri1621563631at_int,N_2) = zero_zero_int
<=> N_2 = zero_zero_nat ) ).
fof(fact_1047_int__0,axiom,
hAPP_nat_int(semiri1621563631at_int,zero_zero_nat) = zero_zero_int ).
fof(fact_1048_transfer__int__nat__numerals_I1_J,axiom,
zero_zero_int = hAPP_nat_int(semiri1621563631at_int,zero_zero_nat) ).
fof(fact_1049_transfer__int__nat__numerals_I2_J,axiom,
one_one_int = hAPP_nat_int(semiri1621563631at_int,one_one_nat) ).
fof(fact_1050_int__1,axiom,
hAPP_nat_int(semiri1621563631at_int,one_one_nat) = one_one_int ).
fof(fact_1051_Nat__Transfer_Otransfer__int__nat__functions_I1_J,axiom,
! [X,Y] : hAPP_int_int(plus_plus_int(hAPP_nat_int(semiri1621563631at_int,X)),hAPP_nat_int(semiri1621563631at_int,Y)) = hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(plus_plus_nat(X),Y)) ).
fof(fact_1052_zadd__int,axiom,
! [M,N_1] : hAPP_int_int(plus_plus_int(hAPP_nat_int(semiri1621563631at_int,M)),hAPP_nat_int(semiri1621563631at_int,N_1)) = hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(plus_plus_nat(M),N_1)) ).
fof(fact_1053_zadd__int__left,axiom,
! [M,N_1,Z_1] : hAPP_int_int(plus_plus_int(hAPP_nat_int(semiri1621563631at_int,M)),hAPP_int_int(plus_plus_int(hAPP_nat_int(semiri1621563631at_int,N_1)),Z_1)) = hAPP_int_int(plus_plus_int(hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(plus_plus_nat(M),N_1))),Z_1) ).
fof(fact_1054_zle__iff__zadd,axiom,
! [W,Z] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,W),Z))
<=> ? [N] : Z = hAPP_int_int(plus_plus_int(W),hAPP_nat_int(semiri1621563631at_int,N)) ) ).
fof(fact_1055_zmult__int,axiom,
! [M,N_1] : hAPP_int_int(times_times_int(hAPP_nat_int(semiri1621563631at_int,M)),hAPP_nat_int(semiri1621563631at_int,N_1)) = hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(times_times_nat(M),N_1)) ).
fof(fact_1056_int__mult,axiom,
! [M,N_1] : hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(times_times_nat(M),N_1)) = hAPP_int_int(times_times_int(hAPP_nat_int(semiri1621563631at_int,M)),hAPP_nat_int(semiri1621563631at_int,N_1)) ).
fof(fact_1057_Nat__Transfer_Otransfer__int__nat__functions_I2_J,axiom,
! [X,Y] : hAPP_int_int(times_times_int(hAPP_nat_int(semiri1621563631at_int,X)),hAPP_nat_int(semiri1621563631at_int,Y)) = hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(times_times_nat(X),Y)) ).
fof(fact_1058_zle__int,axiom,
! [M_1,N_2] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_nat_int(semiri1621563631at_int,M_1)),hAPP_nat_int(semiri1621563631at_int,N_2)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_1),N_2)) ) ).
fof(fact_1059_Nat__Transfer_Otransfer__int__nat__relations_I3_J,axiom,
! [X_1,Y_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_nat_int(semiri1621563631at_int,X_1)),hAPP_nat_int(semiri1621563631at_int,Y_1)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_1),Y_1)) ) ).
fof(fact_1060_not__neg__int,axiom,
! [N_1] : ~ hBOOL(hAPP_int_bool(nat_neg,hAPP_nat_int(semiri1621563631at_int,N_1))) ).
fof(fact_1061_real__of__nat__def,axiom,
real_nat = semiri132038758t_real ).
fof(fact_1062_real__of__int__of__nat__eq,axiom,
! [N_1] : hAPP_int_real(real_int,hAPP_nat_int(semiri1621563631at_int,N_1)) = hAPP_nat_real(real_nat,N_1) ).
fof(fact_1063_real__eq__of__nat,axiom,
real_nat = semiri132038758t_real ).
fof(fact_1064_floor__real__of__nat,axiom,
! [N_1] : archim1246769320r_real(hAPP_nat_real(real_nat,N_1)) = hAPP_nat_int(semiri1621563631at_int,N_1) ).
fof(fact_1065_Nat__Transfer_Otransfer__int__nat__functions_I3_J,axiom,
! [X,Y] : nat_tsub(hAPP_nat_int(semiri1621563631at_int,X),hAPP_nat_int(semiri1621563631at_int,Y)) = hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(minus_minus_nat(X),Y)) ).
fof(fact_1066_int__le__0__conv,axiom,
! [N_2] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_nat_int(semiri1621563631at_int,N_2)),zero_zero_int))
<=> N_2 = zero_zero_nat ) ).
fof(fact_1067_zdiff__int,axiom,
! [N_1,M] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_1),M))
=> hAPP_int_int(minus_minus_int(hAPP_nat_int(semiri1621563631at_int,M)),hAPP_nat_int(semiri1621563631at_int,N_1)) = hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(minus_minus_nat(M),N_1)) ) ).
fof(fact_1068_zero__less__int__conv,axiom,
! [N_2] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_nat_int(semiri1621563631at_int,N_2)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_2)) ) ).
fof(fact_1069_zmult__zless__mono2__lemma,axiom,
! [K,I,J] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,I),J))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),K))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(times_times_int(hAPP_nat_int(semiri1621563631at_int,K)),I)),hAPP_int_int(times_times_int(hAPP_nat_int(semiri1621563631at_int,K)),J))) ) ) ).
fof(fact_1070_int__nat__number__of,axiom,
! [V] :
( ( hBOOL(hAPP_int_bool(nat_neg,number_number_of_int(V)))
=> hAPP_nat_int(semiri1621563631at_int,number_number_of_nat(V)) = zero_zero_int )
& ( ~ hBOOL(hAPP_int_bool(nat_neg,number_number_of_int(V)))
=> hAPP_nat_int(semiri1621563631at_int,number_number_of_nat(V)) = number_number_of_int(V) ) ) ).
fof(fact_1071_zdiff__int__split,axiom,
! [P_1,X_1,Y_1] :
( hBOOL(hAPP_int_bool(P_1,hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(minus_minus_nat(X_1),Y_1))))
<=> ( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Y_1),X_1))
=> hBOOL(hAPP_int_bool(P_1,hAPP_int_int(minus_minus_int(hAPP_nat_int(semiri1621563631at_int,X_1)),hAPP_nat_int(semiri1621563631at_int,Y_1)))) )
& ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_1),Y_1))
=> hBOOL(hAPP_int_bool(P_1,zero_zero_int)) ) ) ) ).
fof(fact_1072_zero__less__imp__eq__int,axiom,
! [K] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),K))
=> ? [N] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N))
& K = hAPP_nat_int(semiri1621563631at_int,N) ) ) ).
fof(fact_1073_int__int__eq,axiom,
! [M_1,N_2] :
( hAPP_nat_int(semiri1621563631at_int,M_1) = hAPP_nat_int(semiri1621563631at_int,N_2)
<=> M_1 = N_2 ) ).
fof(fact_1074_int__if__cong,axiom,
! [X_1,Y_1,P_1] :
( ( hBOOL(P_1)
=> hAPP_nat_int(semiri1621563631at_int,X_1) = hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(if_nat(P_1,X_1),Y_1)) )
& ( ~ hBOOL(P_1)
=> hAPP_nat_int(semiri1621563631at_int,Y_1) = hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(if_nat(P_1,X_1),Y_1)) ) ) ).
fof(fact_1075_Nat__Transfer_Otransfer__int__nat__relations_I1_J,axiom,
! [X_1,Y_1] :
( hAPP_nat_int(semiri1621563631at_int,X_1) = hAPP_nat_int(semiri1621563631at_int,Y_1)
<=> X_1 = Y_1 ) ).
fof(fact_1076_zero__le__imp__eq__int,axiom,
! [K] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),K))
=> ? [N] : K = hAPP_nat_int(semiri1621563631at_int,N) ) ).
fof(fact_1077_nonneg__eq__int,axiom,
! [Z_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Z_1))
=> ~ ! [M_2] : Z_1 != hAPP_nat_int(semiri1621563631at_int,M_2) ) ).
fof(fact_1078_nonneg__int__cases,axiom,
! [K] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),K))
=> ~ ! [N] : K != hAPP_nat_int(semiri1621563631at_int,N) ) ).
fof(fact_1079_int__diff__cases,axiom,
! [Z_1] :
~ ! [M_2,N] : Z_1 != hAPP_int_int(minus_minus_int(hAPP_nat_int(semiri1621563631at_int,M_2)),hAPP_nat_int(semiri1621563631at_int,N)) ).
fof(fact_1080_mult__nat__number__of,axiom,
! [V_2,V] :
( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,V),pls))
=> hAPP_nat_nat(times_times_nat(number_number_of_nat(V)),number_number_of_nat(V_2)) = zero_zero_nat )
& ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,V),pls))
=> hAPP_nat_nat(times_times_nat(number_number_of_nat(V)),number_number_of_nat(V_2)) = number_number_of_nat(hAPP_int_int(times_times_int(V),V_2)) ) ) ).
fof(fact_1081_nat__number__of__mult__left,axiom,
! [V_2,K,V] :
( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,V),pls))
=> hAPP_nat_nat(times_times_nat(number_number_of_nat(V)),hAPP_nat_nat(times_times_nat(number_number_of_nat(V_2)),K)) = zero_zero_nat )
& ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,V),pls))
=> hAPP_nat_nat(times_times_nat(number_number_of_nat(V)),hAPP_nat_nat(times_times_nat(number_number_of_nat(V_2)),K)) = hAPP_nat_nat(times_times_nat(number_number_of_nat(hAPP_int_int(times_times_int(V),V_2))),K) ) ) ).
fof(fact_1082_diff__bin__simps_I1_J,axiom,
! [K] : hAPP_int_int(minus_minus_int(K),pls) = K ).
fof(fact_1083_mult__Pls,axiom,
! [W_1] : hAPP_int_int(times_times_int(pls),W_1) = pls ).
fof(fact_1084_rel__simps_I2_J,axiom,
~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,pls),pls)) ).
fof(fact_1085_add__Pls,axiom,
! [K] : hAPP_int_int(plus_plus_int(pls),K) = K ).
fof(fact_1086_add__Pls__right,axiom,
! [K] : hAPP_int_int(plus_plus_int(K),pls) = K ).
fof(fact_1087_Pls__def,axiom,
pls = zero_zero_int ).
fof(fact_1088_rel__simps_I19_J,axiom,
hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),pls)) ).
fof(fact_1089_zero__is__num__zero,axiom,
zero_zero_int = number_number_of_int(pls) ).
fof(fact_1090_bin__less__0__simps_I1_J,axiom,
~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,pls),zero_zero_int)) ).
fof(fact_1091_nat__number__of__Pls,axiom,
number_number_of_nat(pls) = zero_zero_nat ).
fof(fact_1092_semiring__norm_I113_J,axiom,
zero_zero_nat = number_number_of_nat(pls) ).
fof(fact_1093_not__neg__number__of__Pls,axiom,
~ hBOOL(hAPP_int_bool(nat_neg,number_number_of_int(pls))) ).
fof(fact_1094_eq__0__number__of,axiom,
! [V_1] :
( zero_zero_nat = number_number_of_nat(V_1)
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,V_1),pls)) ) ).
fof(fact_1095_eq__number__of__0,axiom,
! [V_1] :
( number_number_of_nat(V_1) = zero_zero_nat
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,V_1),pls)) ) ).
fof(fact_1096_less__nat__number__of,axiom,
! [V_1,V_3] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,number_number_of_nat(V_1)),number_number_of_nat(V_3)))
<=> ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,V_1),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,V_1),V_3)) ) ) ).
fof(fact_1097_number__of2,axiom,
hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),number_number_of_int(pls))) ).
fof(fact_1098_le__nat__number__of,axiom,
! [V_1,V_3] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,number_number_of_nat(V_1)),number_number_of_nat(V_3)))
<=> ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,V_1),V_3))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,V_1),pls)) ) ) ).
fof(fact_1099_less__0__number__of,axiom,
! [V_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),number_number_of_nat(V_1)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,pls),V_1)) ) ).
fof(fact_1100_add__nat__number__of,axiom,
! [V_2,V] :
( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,V),pls))
=> hAPP_nat_nat(plus_plus_nat(number_number_of_nat(V)),number_number_of_nat(V_2)) = number_number_of_nat(V_2) )
& ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,V),pls))
=> ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,V_2),pls))
=> hAPP_nat_nat(plus_plus_nat(number_number_of_nat(V)),number_number_of_nat(V_2)) = number_number_of_nat(V) )
& ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,V_2),pls))
=> hAPP_nat_nat(plus_plus_nat(number_number_of_nat(V)),number_number_of_nat(V_2)) = number_number_of_nat(hAPP_int_int(plus_plus_int(V),V_2)) ) ) ) ) ).
fof(fact_1101_nat__1__add__number__of,axiom,
! [V] :
( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,V),pls))
=> hAPP_nat_nat(plus_plus_nat(one_one_nat),number_number_of_nat(V)) = one_one_nat )
& ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,V),pls))
=> hAPP_nat_nat(plus_plus_nat(one_one_nat),number_number_of_nat(V)) = number_number_of_nat(succ(V)) ) ) ).
fof(fact_1102_nat__number__of__add__1,axiom,
! [V] :
( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,V),pls))
=> hAPP_nat_nat(plus_plus_nat(number_number_of_nat(V)),one_one_nat) = one_one_nat )
& ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,V),pls))
=> hAPP_nat_nat(plus_plus_nat(number_number_of_nat(V)),one_one_nat) = number_number_of_nat(succ(V)) ) ) ).
fof(fact_1103_succ__def,axiom,
! [K] : succ(K) = hAPP_int_int(plus_plus_int(K),one_one_int) ).
fof(fact_1104_nat__number__of__diff__1,axiom,
! [V] :
( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,V),pls))
=> hAPP_nat_nat(minus_minus_nat(number_number_of_nat(V)),one_one_nat) = zero_zero_nat )
& ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,V),pls))
=> hAPP_nat_nat(minus_minus_nat(number_number_of_nat(V)),one_one_nat) = number_number_of_nat(pred(V)) ) ) ).
fof(fact_1105_Suc__nat__number__of__add,axiom,
! [N_1,V] :
( ( hBOOL(hAPP_int_bool(nat_neg,number_number_of_int(V)))
=> hAPP_nat_nat(suc,hAPP_nat_nat(plus_plus_nat(number_number_of_nat(V)),N_1)) = hAPP_nat_nat(plus_plus_nat(one_one_nat),N_1) )
& ( ~ hBOOL(hAPP_int_bool(nat_neg,number_number_of_int(V)))
=> hAPP_nat_nat(suc,hAPP_nat_nat(plus_plus_nat(number_number_of_nat(V)),N_1)) = hAPP_nat_nat(plus_plus_nat(number_number_of_nat(succ(V))),N_1) ) ) ).
fof(fact_1106_lessI,axiom,
! [N_1] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_1),hAPP_nat_nat(suc,N_1))) ).
fof(fact_1107_Suc__mono,axiom,
! [M,N_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N_1))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(suc,M)),hAPP_nat_nat(suc,N_1))) ) ).
fof(fact_1108_zero__less__Suc,axiom,
! [N_1] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_nat_nat(suc,N_1))) ).
fof(fact_1109_succ__pred,axiom,
! [X] : succ(pred(X)) = X ).
fof(fact_1110_less__Suc__eq__0__disj,axiom,
! [M_1,N_2] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),hAPP_nat_nat(suc,N_2)))
<=> ( M_1 = zero_zero_nat
| ? [J_1] :
( M_1 = hAPP_nat_nat(suc,J_1)
& hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,J_1),N_2)) ) ) ) ).
fof(fact_1111_less__Suc0,axiom,
! [N_2] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_2),hAPP_nat_nat(suc,zero_zero_nat)))
<=> N_2 = zero_zero_nat ) ).
fof(fact_1112_gr0__conv__Suc,axiom,
! [N_2] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_2))
<=> ? [M_2] : N_2 = hAPP_nat_nat(suc,M_2) ) ).
fof(fact_1113_add__is__1,axiom,
! [M_1,N_2] :
( hAPP_nat_nat(plus_plus_nat(M_1),N_2) = hAPP_nat_nat(suc,zero_zero_nat)
<=> ( ( M_1 = hAPP_nat_nat(suc,zero_zero_nat)
& N_2 = zero_zero_nat )
| ( M_1 = zero_zero_nat
& N_2 = hAPP_nat_nat(suc,zero_zero_nat) ) ) ) ).
fof(fact_1114_one__is__add,axiom,
! [M_1,N_2] :
( hAPP_nat_nat(suc,zero_zero_nat) = hAPP_nat_nat(plus_plus_nat(M_1),N_2)
<=> ( ( M_1 = hAPP_nat_nat(suc,zero_zero_nat)
& N_2 = zero_zero_nat )
| ( M_1 = zero_zero_nat
& N_2 = hAPP_nat_nat(suc,zero_zero_nat) ) ) ) ).
fof(fact_1115_mult__eq__1__iff,axiom,
! [M_1,N_2] :
( hAPP_nat_nat(times_times_nat(M_1),N_2) = hAPP_nat_nat(suc,zero_zero_nat)
<=> ( M_1 = hAPP_nat_nat(suc,zero_zero_nat)
& N_2 = hAPP_nat_nat(suc,zero_zero_nat) ) ) ).
fof(fact_1116_less__iff__Suc__add,axiom,
! [M_1,N_2] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),N_2))
<=> ? [K_2] : N_2 = hAPP_nat_nat(suc,hAPP_nat_nat(plus_plus_nat(M_1),K_2)) ) ).
fof(fact_1117_less__add__Suc2,axiom,
! [I,M] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I),hAPP_nat_nat(suc,hAPP_nat_nat(plus_plus_nat(M),I)))) ).
fof(fact_1118_less__add__Suc1,axiom,
! [I,M] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I),hAPP_nat_nat(suc,hAPP_nat_nat(plus_plus_nat(I),M)))) ).
fof(fact_1119_One__nat__def,axiom,
one_one_nat = hAPP_nat_nat(suc,zero_zero_nat) ).
fof(fact_1120_less__eq__Suc__le,axiom,
! [N_2,M_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_2),M_1))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(suc,N_2)),M_1)) ) ).
fof(fact_1121_less__Suc__eq__le,axiom,
! [M_1,N_2] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),hAPP_nat_nat(suc,N_2)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_1),N_2)) ) ).
fof(fact_1122_Suc__le__eq,axiom,
! [M_1,N_2] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(suc,M_1)),N_2))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),N_2)) ) ).
fof(fact_1123_le__imp__less__Suc,axiom,
! [M,N_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N_1))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),hAPP_nat_nat(suc,N_1))) ) ).
fof(fact_1124_Suc__leI,axiom,
! [M,N_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N_1))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(suc,M)),N_1)) ) ).
fof(fact_1125_le__less__Suc__eq,axiom,
! [M_1,N_2] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_1),N_2))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_2),hAPP_nat_nat(suc,M_1)))
<=> N_2 = M_1 ) ) ).
fof(fact_1126_Suc__le__lessD,axiom,
! [M,N_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(suc,M)),N_1))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N_1)) ) ).
fof(fact_1127_Suc__mult__less__cancel1,axiom,
! [K_1,M_1,N_2] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(times_times_nat(hAPP_nat_nat(suc,K_1)),M_1)),hAPP_nat_nat(times_times_nat(hAPP_nat_nat(suc,K_1)),N_2)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),N_2)) ) ).
fof(fact_1128_diff__less__Suc,axiom,
! [M,N_1] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(minus_minus_nat(M),N_1)),hAPP_nat_nat(suc,M))) ).
fof(fact_1129_mult__Suc,axiom,
! [M,N_1] : hAPP_nat_nat(times_times_nat(hAPP_nat_nat(suc,M)),N_1) = hAPP_nat_nat(plus_plus_nat(N_1),hAPP_nat_nat(times_times_nat(M),N_1)) ).
fof(fact_1130_mult__Suc__right,axiom,
! [M,N_1] : hAPP_nat_nat(times_times_nat(M),hAPP_nat_nat(suc,N_1)) = hAPP_nat_nat(plus_plus_nat(M),hAPP_nat_nat(times_times_nat(M),N_1)) ).
fof(fact_1131_Suc__mult__le__cancel1,axiom,
! [K_1,M_1,N_2] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(times_times_nat(hAPP_nat_nat(suc,K_1)),M_1)),hAPP_nat_nat(times_times_nat(hAPP_nat_nat(suc,K_1)),N_2)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_1),N_2)) ) ).
fof(fact_1132_Suc__eq__plus1__left,axiom,
! [N_1] : hAPP_nat_nat(suc,N_1) = hAPP_nat_nat(plus_plus_nat(one_one_nat),N_1) ).
fof(fact_1133_Suc__eq__plus1,axiom,
! [N_1] : hAPP_nat_nat(suc,N_1) = hAPP_nat_nat(plus_plus_nat(N_1),one_one_nat) ).
fof(fact_1134_Suc__diff__le,axiom,
! [N_1,M] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_1),M))
=> hAPP_nat_nat(minus_minus_nat(hAPP_nat_nat(suc,M)),N_1) = hAPP_nat_nat(suc,hAPP_nat_nat(minus_minus_nat(M),N_1)) ) ).
fof(fact_1135_diff__Suc__1,axiom,
! [N_1] : hAPP_nat_nat(minus_minus_nat(hAPP_nat_nat(suc,N_1)),one_one_nat) = N_1 ).
fof(fact_1136_diff__Suc__eq__diff__pred,axiom,
! [M,N_1] : hAPP_nat_nat(minus_minus_nat(M),hAPP_nat_nat(suc,N_1)) = hAPP_nat_nat(minus_minus_nat(hAPP_nat_nat(minus_minus_nat(M),one_one_nat)),N_1) ).
fof(fact_1137_Suc__leD,axiom,
! [M,N_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(suc,M)),N_1))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N_1)) ) ).
fof(fact_1138_le__SucE,axiom,
! [M,N_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),hAPP_nat_nat(suc,N_1)))
=> ( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N_1))
=> M = hAPP_nat_nat(suc,N_1) ) ) ).
fof(fact_1139_le__SucI,axiom,
! [M,N_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N_1))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),hAPP_nat_nat(suc,N_1))) ) ).
fof(fact_1140_Suc__le__mono,axiom,
! [N_2,M_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(suc,N_2)),hAPP_nat_nat(suc,M_1)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_2),M_1)) ) ).
fof(fact_1141_le__Suc__eq,axiom,
! [M_1,N_2] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_1),hAPP_nat_nat(suc,N_2)))
<=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_1),N_2))
| M_1 = hAPP_nat_nat(suc,N_2) ) ) ).
fof(fact_1142_not__less__eq__eq,axiom,
! [M_1,N_2] :
( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_1),N_2))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(suc,N_2)),M_1)) ) ).
fof(fact_1143_Suc__n__not__le__n,axiom,
! [N_1] : ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(suc,N_1)),N_1)) ).
fof(fact_1144_Zero__not__Suc,axiom,
! [M] : zero_zero_nat != hAPP_nat_nat(suc,M) ).
fof(fact_1145_nat_Osimps_I2_J,axiom,
! [Nat_3] : zero_zero_nat != hAPP_nat_nat(suc,Nat_3) ).
fof(fact_1146_Suc__not__Zero,axiom,
! [M] : hAPP_nat_nat(suc,M) != zero_zero_nat ).
fof(fact_1147_nat_Osimps_I3_J,axiom,
! [Nat_2] : hAPP_nat_nat(suc,Nat_2) != zero_zero_nat ).
fof(fact_1148_Zero__neq__Suc,axiom,
! [M] : zero_zero_nat != hAPP_nat_nat(suc,M) ).
fof(fact_1149_Suc__neq__Zero,axiom,
! [M] : hAPP_nat_nat(suc,M) != zero_zero_nat ).
fof(fact_1150_not__less__eq,axiom,
! [M_1,N_2] :
( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),N_2))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_2),hAPP_nat_nat(suc,M_1))) ) ).
fof(fact_1151_less__Suc__eq,axiom,
! [M_1,N_2] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),hAPP_nat_nat(suc,N_2)))
<=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),N_2))
| M_1 = N_2 ) ) ).
fof(fact_1152_Suc__less__eq,axiom,
! [M_1,N_2] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(suc,M_1)),hAPP_nat_nat(suc,N_2)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),N_2)) ) ).
fof(fact_1153_not__less__less__Suc__eq,axiom,
! [N_2,M_1] :
( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_2),M_1))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_2),hAPP_nat_nat(suc,M_1)))
<=> N_2 = M_1 ) ) ).
fof(fact_1154_less__antisym,axiom,
! [N_1,M] :
( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_1),M))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_1),hAPP_nat_nat(suc,M)))
=> M = N_1 ) ) ).
fof(fact_1155_less__SucI,axiom,
! [M,N_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N_1))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),hAPP_nat_nat(suc,N_1))) ) ).
fof(fact_1156_Suc__lessI,axiom,
! [M,N_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N_1))
=> ( hAPP_nat_nat(suc,M) != N_1
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(suc,M)),N_1)) ) ) ).
fof(fact_1157_less__trans__Suc,axiom,
! [K,I,J] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I),J))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,J),K))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(suc,I)),K)) ) ) ).
fof(fact_1158_less__SucE,axiom,
! [M,N_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),hAPP_nat_nat(suc,N_1)))
=> ( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N_1))
=> M = N_1 ) ) ).
fof(fact_1159_Suc__lessD,axiom,
! [M,N_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(suc,M)),N_1))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N_1)) ) ).
fof(fact_1160_Suc__less__SucD,axiom,
! [M,N_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(suc,M)),hAPP_nat_nat(suc,N_1)))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N_1)) ) ).
fof(fact_1161_add__Suc__right,axiom,
! [M,N_1] : hAPP_nat_nat(plus_plus_nat(M),hAPP_nat_nat(suc,N_1)) = hAPP_nat_nat(suc,hAPP_nat_nat(plus_plus_nat(M),N_1)) ).
fof(fact_1162_add__Suc,axiom,
! [M,N_1] : hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(suc,M)),N_1) = hAPP_nat_nat(suc,hAPP_nat_nat(plus_plus_nat(M),N_1)) ).
fof(fact_1163_add__Suc__shift,axiom,
! [M,N_1] : hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(suc,M)),N_1) = hAPP_nat_nat(plus_plus_nat(M),hAPP_nat_nat(suc,N_1)) ).
fof(fact_1164_Suc__diff__diff,axiom,
! [M,N_1,K] : hAPP_nat_nat(minus_minus_nat(hAPP_nat_nat(minus_minus_nat(hAPP_nat_nat(suc,M)),N_1)),hAPP_nat_nat(suc,K)) = hAPP_nat_nat(minus_minus_nat(hAPP_nat_nat(minus_minus_nat(M),N_1)),K) ).
fof(fact_1165_diff__Suc__Suc,axiom,
! [M,N_1] : hAPP_nat_nat(minus_minus_nat(hAPP_nat_nat(suc,M)),hAPP_nat_nat(suc,N_1)) = hAPP_nat_nat(minus_minus_nat(M),N_1) ).
fof(fact_1166_n__not__Suc__n,axiom,
! [N_1] : N_1 != hAPP_nat_nat(suc,N_1) ).
fof(fact_1167_Suc__n__not__n,axiom,
! [N_1] : hAPP_nat_nat(suc,N_1) != N_1 ).
fof(fact_1168_nat_Oinject,axiom,
! [Nat_1,Nat] :
( hAPP_nat_nat(suc,Nat_1) = hAPP_nat_nat(suc,Nat)
<=> Nat_1 = Nat ) ).
fof(fact_1169_Suc__inject,axiom,
! [X,Y] :
( hAPP_nat_nat(suc,X) = hAPP_nat_nat(suc,Y)
=> X = Y ) ).
fof(fact_1170_Suc__mult__cancel1,axiom,
! [K_1,M_1,N_2] :
( hAPP_nat_nat(times_times_nat(hAPP_nat_nat(suc,K_1)),M_1) = hAPP_nat_nat(times_times_nat(hAPP_nat_nat(suc,K_1)),N_2)
<=> M_1 = N_2 ) ).
fof(fact_1171_pred__def,axiom,
! [K] : pred(K) = hAPP_int_int(minus_minus_int(K),one_one_int) ).
fof(fact_1172_le__iff__pred__less,axiom,
! [K_1,L] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K_1),L))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,pred(K_1)),L)) ) ).
fof(fact_1173_Suc__diff__number__of,axiom,
! [M,V] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,pls),V))
=> hAPP_nat_nat(minus_minus_nat(hAPP_nat_nat(suc,M)),number_number_of_nat(V)) = hAPP_nat_nat(minus_minus_nat(M),number_number_of_nat(pred(V))) ) ).
fof(fact_1174_one__less__mult,axiom,
! [M,N_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(suc,zero_zero_nat)),N_1))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(suc,zero_zero_nat)),M))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(suc,zero_zero_nat)),hAPP_nat_nat(times_times_nat(M),N_1))) ) ) ).
fof(fact_1175_n__less__n__mult__m,axiom,
! [M,N_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(suc,zero_zero_nat)),N_1))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(suc,zero_zero_nat)),M))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_1),hAPP_nat_nat(times_times_nat(N_1),M))) ) ) ).
fof(fact_1176_n__less__m__mult__n,axiom,
! [M,N_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(suc,zero_zero_nat)),N_1))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(suc,zero_zero_nat)),M))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_1),hAPP_nat_nat(times_times_nat(M),N_1))) ) ) ).
fof(fact_1177_mn__eq__m__one,axiom,
! [N_1,M] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),M))
=> ( hAPP_nat_nat(times_times_nat(M),N_1) = M
=> N_1 = hAPP_nat_nat(suc,zero_zero_nat) ) ) ).
fof(fact_1178_prod__mn__less__k,axiom,
! [M,K,N_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),K))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(suc,zero_zero_nat)),M))
=> ( hAPP_nat_nat(times_times_nat(M),N_1) = K
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_1),K)) ) ) ) ) ).
fof(fact_1179_one__less__k,axiom,
! [M,K] :
( M != hAPP_nat_nat(times_times_nat(M),K)
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(suc,zero_zero_nat)),hAPP_nat_nat(times_times_nat(M),K)))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(suc,zero_zero_nat)),K)) ) ) ).
fof(fact_1180_one__less__m,axiom,
! [M,K] :
( M != hAPP_nat_nat(times_times_nat(M),K)
=> ( M != hAPP_nat_nat(suc,zero_zero_nat)
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(suc,zero_zero_nat)),M)) ) ) ).
fof(fact_1181_Suc__pred,axiom,
! [N_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1))
=> hAPP_nat_nat(suc,hAPP_nat_nat(minus_minus_nat(N_1),hAPP_nat_nat(suc,zero_zero_nat))) = N_1 ) ).
fof(fact_1182_diff__Suc__less,axiom,
! [I,N_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(minus_minus_nat(N_1),hAPP_nat_nat(suc,I))),N_1)) ) ).
fof(fact_1183_one__le__mult__iff,axiom,
! [M_1,N_2] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(suc,zero_zero_nat)),hAPP_nat_nat(times_times_nat(M_1),N_2)))
<=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(suc,zero_zero_nat)),M_1))
& hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(suc,zero_zero_nat)),N_2)) ) ) ).
fof(fact_1184_real__of__nat__Suc__gt__zero,axiom,
! [N_1] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_nat_real(real_nat,hAPP_nat_nat(suc,N_1)))) ).
fof(fact_1185_int__Suc0__eq__1,axiom,
hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(suc,zero_zero_nat)) = one_one_int ).
fof(fact_1186_real__of__nat__one,axiom,
hAPP_nat_real(real_nat,hAPP_nat_nat(suc,zero_zero_nat)) = one_one_real ).
fof(fact_1187_diff__Suc__diff__eq1,axiom,
! [M,K,J] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K),J))
=> hAPP_nat_nat(minus_minus_nat(M),hAPP_nat_nat(suc,hAPP_nat_nat(minus_minus_nat(J),K))) = hAPP_nat_nat(minus_minus_nat(hAPP_nat_nat(plus_plus_nat(M),K)),hAPP_nat_nat(suc,J)) ) ).
fof(fact_1188_diff__Suc__diff__eq2,axiom,
! [M,K,J] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K),J))
=> hAPP_nat_nat(minus_minus_nat(hAPP_nat_nat(suc,hAPP_nat_nat(minus_minus_nat(J),K))),M) = hAPP_nat_nat(minus_minus_nat(hAPP_nat_nat(suc,J)),hAPP_nat_nat(plus_plus_nat(K),M)) ) ).
fof(fact_1189_zless__iff__Suc__zadd,axiom,
! [W,Z] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,W),Z))
<=> ? [N] : Z = hAPP_int_int(plus_plus_int(W),hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(suc,N))) ) ).
fof(fact_1190_int__Suc,axiom,
! [M] : hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(suc,M)) = hAPP_int_int(plus_plus_int(one_one_int),hAPP_nat_int(semiri1621563631at_int,M)) ).
fof(fact_1191_real__of__nat__Suc,axiom,
! [N_1] : hAPP_nat_real(real_nat,hAPP_nat_nat(suc,N_1)) = hAPP_real_real(plus_plus_real(hAPP_nat_real(real_nat,N_1)),one_one_real) ).
fof(fact_1192_Suc__pred_H,axiom,
! [N_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1))
=> N_1 = hAPP_nat_nat(suc,hAPP_nat_nat(minus_minus_nat(N_1),one_one_nat)) ) ).
fof(fact_1193_Suc__diff__1,axiom,
! [N_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1))
=> hAPP_nat_nat(suc,hAPP_nat_nat(minus_minus_nat(N_1),one_one_nat)) = N_1 ) ).
fof(fact_1194_add__eq__if,axiom,
! [N_1,M] :
( ( M = zero_zero_nat
=> hAPP_nat_nat(plus_plus_nat(M),N_1) = N_1 )
& ( M != zero_zero_nat
=> hAPP_nat_nat(plus_plus_nat(M),N_1) = hAPP_nat_nat(suc,hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(minus_minus_nat(M),one_one_nat)),N_1)) ) ) ).
fof(fact_1195_expand__Suc,axiom,
! [V] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),number_number_of_nat(V)))
=> number_number_of_nat(V) = hAPP_nat_nat(suc,hAPP_nat_nat(minus_minus_nat(number_number_of_nat(V)),one_one_nat)) ) ).
fof(fact_1196_neg__number__of__pred__iff__0,axiom,
! [V_1] :
( hBOOL(hAPP_int_bool(nat_neg,number_number_of_int(pred(V_1))))
<=> number_number_of_nat(V_1) = zero_zero_nat ) ).
fof(fact_1197_Suc__nat__number__of,axiom,
! [V] :
( ( hBOOL(hAPP_int_bool(nat_neg,number_number_of_int(V)))
=> hAPP_nat_nat(suc,number_number_of_nat(V)) = one_one_nat )
& ( ~ hBOOL(hAPP_int_bool(nat_neg,number_number_of_int(V)))
=> hAPP_nat_nat(suc,number_number_of_nat(V)) = number_number_of_nat(succ(V)) ) ) ).
fof(fact_1198_reals__Archimedean6a,axiom,
! [R] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),R))
=> ? [N] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(real_nat,N)),R))
& hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,R),hAPP_nat_real(real_nat,hAPP_nat_nat(suc,N)))) ) ) ).
%----Helper facts (3)
fof(help_If_1_1_If_000tc__Nat__Onat_T,axiom,
! [X,Y] : hAPP_nat_nat(if_nat(fTrue,X),Y) = X ).
fof(help_If_2_1_If_000tc__Nat__Onat_T,axiom,
! [X,Y] : hAPP_nat_nat(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_int_int(times_times_int(twoSqu1241645765sum2sq(product_Pair_int_int(a,b))),twoSqu1241645765sum2sq(product_Pair_int_int(p,q))) = twoSqu1241645765sum2sq(product_Pair_int_int(hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(a),p)),hAPP_int_int(times_times_int(b),q)),hAPP_int_int(minus_minus_int(hAPP_int_int(times_times_int(a),q)),hAPP_int_int(times_times_int(b),p)))) ).
%------------------------------------------------------------------------------