TPTP Problem File: NUM923+2.p
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%------------------------------------------------------------------------------
% File : NUM923+2 : TPTP v8.2.0. Released v5.3.0.
% Domain : Number Theory
% Problem : Sum of two squares line 23, 500 axioms selected
% Version : Especial.
% English :
% Refs : [BN10] Boehme & Nipkow (2010), Sledgehammer: Judgement Day
% : [Bla11] Blanchette (2011), Email to Geoff Sutcliffe
% Source : [Bla11]
% Names : s2s_500_fofmg_l23 [Bla11]
% Status : Theorem
% Rating : 1.00 v6.2.0, 0.96 v6.1.0, 0.97 v6.0.0, 1.00 v5.4.0, 0.96 v5.3.0
% Syntax : Number of formulae : 651 ( 195 unt; 0 def)
% Number of atoms : 1536 ( 426 equ)
% Maximal formula atoms : 8 ( 2 avg)
% Number of connectives : 1029 ( 144 ~; 41 |; 51 &)
% ( 130 <=>; 663 =>; 0 <=; 0 <~>)
% Maximal formula depth : 16 ( 5 avg)
% Maximal term depth : 8 ( 2 avg)
% Number of predicates : 3 ( 2 usr; 0 prp; 1-2 aty)
% Number of functors : 39 ( 39 usr; 13 con; 0-8 aty)
% Number of variables : 1599 (1575 !; 24 ?)
% SPC : FOF_THM_RFO_SEQ
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2011-08-09 14:10:46
% : Encoded with monomorphized guards.
%------------------------------------------------------------------------------
%----Explicit typings (4)
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__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)) ).
%----Relevant facts (646)
fof(fact_0_xzgcda__linear__aux1,axiom,
! [A_19,R_1,B,M,C_31,D,N] : plus_plus_int(times_times_int(minus_minus_int(A_19,times_times_int(R_1,B)),M),times_times_int(minus_minus_int(C_31,times_times_int(R_1,D)),N)) = minus_minus_int(plus_plus_int(times_times_int(A_19,M),times_times_int(C_31,N)),times_times_int(R_1,plus_plus_int(times_times_int(B,M),times_times_int(D,N)))) ).
fof(fact_1_mult__diff__mult,axiom,
! [X_44,Y_43,A_146,B_112] : minus_minus_int(times_times_int(X_44,Y_43),times_times_int(A_146,B_112)) = plus_plus_int(times_times_int(X_44,minus_minus_int(Y_43,B_112)),times_times_int(minus_minus_int(X_44,A_146),B_112)) ).
fof(fact_2_eq__add__iff2,axiom,
! [Aa,E,C_4,Ba,D_3] :
( plus_plus_int(times_times_int(Aa,E),C_4) = plus_plus_int(times_times_int(Ba,E),D_3)
<=> C_4 = plus_plus_int(times_times_int(minus_minus_int(Ba,Aa),E),D_3) ) ).
fof(fact_3_eq__add__iff1,axiom,
! [Aa,E,C_4,Ba,D_3] :
( plus_plus_int(times_times_int(Aa,E),C_4) = plus_plus_int(times_times_int(Ba,E),D_3)
<=> plus_plus_int(times_times_int(minus_minus_int(Aa,Ba),E),C_4) = D_3 ) ).
fof(fact_4_is__sum2sq__def,axiom,
! [X_1] :
( hBOOL(hAPP_int_bool(twoSqu142715416sum2sq,X_1))
<=> ? [A_88,B_92] : twoSqu140629262sum2sq(product_Pair_int_int(A_88,B_92)) = X_1 ) ).
fof(fact_5_Int2_Oaux1,axiom,
! [A_19,B,C_31] :
( minus_minus_int(A_19,B) = C_31
=> A_19 = plus_plus_int(C_31,B) ) ).
fof(fact_6_zdiff__zmult__distrib2,axiom,
! [W,Z1,Z2] : times_times_int(W,minus_minus_int(Z1,Z2)) = minus_minus_int(times_times_int(W,Z1),times_times_int(W,Z2)) ).
fof(fact_7_zdiff__zmult__distrib,axiom,
! [Z1,Z2,W] : times_times_int(minus_minus_int(Z1,Z2),W) = minus_minus_int(times_times_int(Z1,W),times_times_int(Z2,W)) ).
fof(fact_8_zadd__zmult__distrib2,axiom,
! [W,Z1,Z2] : times_times_int(W,plus_plus_int(Z1,Z2)) = plus_plus_int(times_times_int(W,Z1),times_times_int(W,Z2)) ).
fof(fact_9_zadd__zmult__distrib,axiom,
! [Z1,Z2,W] : times_times_int(plus_plus_int(Z1,Z2),W) = plus_plus_int(times_times_int(Z1,W),times_times_int(Z2,W)) ).
fof(fact_10_diff__add__cancel,axiom,
! [A_145,B_111] : plus_plus_int(minus_minus_int(A_145,B_111),B_111) = A_145 ).
fof(fact_11_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A_144,B_110,C_67] : times_times_int(times_times_int(A_144,B_110),C_67) = times_times_int(A_144,times_times_int(B_110,C_67)) ).
fof(fact_12_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A_144,B_110,C_67] : times_times_nat(times_times_nat(A_144,B_110),C_67) = times_times_nat(A_144,times_times_nat(B_110,C_67)) ).
fof(fact_13_add__right__imp__eq,axiom,
! [B_109,A_143,C_66] :
( plus_plus_int(B_109,A_143) = plus_plus_int(C_66,A_143)
=> B_109 = C_66 ) ).
fof(fact_14_add__right__imp__eq,axiom,
! [B_109,A_143,C_66] :
( plus_plus_nat(B_109,A_143) = plus_plus_nat(C_66,A_143)
=> B_109 = C_66 ) ).
fof(fact_15_add__imp__eq,axiom,
! [A_142,B_108,C_65] :
( plus_plus_int(A_142,B_108) = plus_plus_int(A_142,C_65)
=> B_108 = C_65 ) ).
fof(fact_16_add__imp__eq,axiom,
! [A_142,B_108,C_65] :
( plus_plus_nat(A_142,B_108) = plus_plus_nat(A_142,C_65)
=> B_108 = C_65 ) ).
fof(fact_17_add__left__imp__eq,axiom,
! [A_141,B_107,C_64] :
( plus_plus_int(A_141,B_107) = plus_plus_int(A_141,C_64)
=> B_107 = C_64 ) ).
fof(fact_18_add__left__imp__eq,axiom,
! [A_141,B_107,C_64] :
( plus_plus_nat(A_141,B_107) = plus_plus_nat(A_141,C_64)
=> B_107 = C_64 ) ).
fof(fact_19_add__right__cancel,axiom,
! [Ba,Aa,C_4] :
( plus_plus_int(Ba,Aa) = plus_plus_int(C_4,Aa)
<=> Ba = C_4 ) ).
fof(fact_20_add__right__cancel,axiom,
! [Ba,Aa,C_4] :
( plus_plus_nat(Ba,Aa) = plus_plus_nat(C_4,Aa)
<=> Ba = C_4 ) ).
fof(fact_21_add__left__cancel,axiom,
! [Aa,Ba,C_4] :
( plus_plus_int(Aa,Ba) = plus_plus_int(Aa,C_4)
<=> Ba = C_4 ) ).
fof(fact_22_add__left__cancel,axiom,
! [Aa,Ba,C_4] :
( plus_plus_nat(Aa,Ba) = plus_plus_nat(Aa,C_4)
<=> Ba = C_4 ) ).
fof(fact_23_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A_140,B_106,C_63] : plus_plus_int(plus_plus_int(A_140,B_106),C_63) = plus_plus_int(A_140,plus_plus_int(B_106,C_63)) ).
fof(fact_24_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A_140,B_106,C_63] : plus_plus_nat(plus_plus_nat(A_140,B_106),C_63) = plus_plus_nat(A_140,plus_plus_nat(B_106,C_63)) ).
fof(fact_25_diff__eq__diff__eq,axiom,
! [Aa,Ba,C_4,D_3] :
( minus_minus_int(Aa,Ba) = minus_minus_int(C_4,D_3)
=> ( Aa = Ba
<=> C_4 = D_3 ) ) ).
fof(fact_26_zmult__assoc,axiom,
! [Z1,Z2,Z3] : times_times_int(times_times_int(Z1,Z2),Z3) = times_times_int(Z1,times_times_int(Z2,Z3)) ).
fof(fact_27_zmult__commute,axiom,
! [Z,W] : times_times_int(Z,W) = times_times_int(W,Z) ).
fof(fact_28_zadd__assoc,axiom,
! [Z1,Z2,Z3] : plus_plus_int(plus_plus_int(Z1,Z2),Z3) = plus_plus_int(Z1,plus_plus_int(Z2,Z3)) ).
fof(fact_29_zadd__left__commute,axiom,
! [X,Y,Z] : plus_plus_int(X,plus_plus_int(Y,Z)) = plus_plus_int(Y,plus_plus_int(X,Z)) ).
fof(fact_30_zadd__commute,axiom,
! [Z,W] : plus_plus_int(Z,W) = plus_plus_int(W,Z) ).
fof(fact_31_combine__common__factor,axiom,
! [A_139,E_1,B_105,C_62] : plus_plus_int(times_times_int(A_139,E_1),plus_plus_int(times_times_int(B_105,E_1),C_62)) = plus_plus_int(times_times_int(plus_plus_int(A_139,B_105),E_1),C_62) ).
fof(fact_32_combine__common__factor,axiom,
! [A_139,E_1,B_105,C_62] : plus_plus_nat(times_times_nat(A_139,E_1),plus_plus_nat(times_times_nat(B_105,E_1),C_62)) = plus_plus_nat(times_times_nat(plus_plus_nat(A_139,B_105),E_1),C_62) ).
fof(fact_33_comm__semiring__class_Odistrib,axiom,
! [A_138,B_104,C_61] : times_times_int(plus_plus_int(A_138,B_104),C_61) = plus_plus_int(times_times_int(A_138,C_61),times_times_int(B_104,C_61)) ).
fof(fact_34_comm__semiring__class_Odistrib,axiom,
! [A_138,B_104,C_61] : times_times_nat(plus_plus_nat(A_138,B_104),C_61) = plus_plus_nat(times_times_nat(A_138,C_61),times_times_nat(B_104,C_61)) ).
fof(fact_35_add__diff__add,axiom,
! [A_137,C_60,B_103,D_18] : minus_minus_int(plus_plus_int(A_137,C_60),plus_plus_int(B_103,D_18)) = plus_plus_int(minus_minus_int(A_137,B_103),minus_minus_int(C_60,D_18)) ).
fof(fact_36_add__diff__cancel,axiom,
! [A_136,B_102] : minus_minus_int(plus_plus_int(A_136,B_102),B_102) = A_136 ).
fof(fact_37_crossproduct__eq,axiom,
! [W_1,Y_3,X_1,Z_1] :
( plus_plus_int(times_times_int(W_1,Y_3),times_times_int(X_1,Z_1)) = plus_plus_int(times_times_int(W_1,Z_1),times_times_int(X_1,Y_3))
<=> ( W_1 = X_1
| Y_3 = Z_1 ) ) ).
fof(fact_38_crossproduct__eq,axiom,
! [W_1,Y_3,X_1,Z_1] :
( plus_plus_nat(times_times_nat(W_1,Y_3),times_times_nat(X_1,Z_1)) = plus_plus_nat(times_times_nat(W_1,Z_1),times_times_nat(X_1,Y_3))
<=> ( W_1 = X_1
| Y_3 = Z_1 ) ) ).
fof(fact_39_comm__semiring__1__class_Onormalizing__semiring__rules_I1_J,axiom,
! [A_135,M_6,B_101] : plus_plus_int(times_times_int(A_135,M_6),times_times_int(B_101,M_6)) = times_times_int(plus_plus_int(A_135,B_101),M_6) ).
fof(fact_40_comm__semiring__1__class_Onormalizing__semiring__rules_I1_J,axiom,
! [A_135,M_6,B_101] : plus_plus_nat(times_times_nat(A_135,M_6),times_times_nat(B_101,M_6)) = times_times_nat(plus_plus_nat(A_135,B_101),M_6) ).
fof(fact_41_comm__semiring__1__class_Onormalizing__semiring__rules_I8_J,axiom,
! [A_134,B_100,C_59] : times_times_int(plus_plus_int(A_134,B_100),C_59) = plus_plus_int(times_times_int(A_134,C_59),times_times_int(B_100,C_59)) ).
fof(fact_42_comm__semiring__1__class_Onormalizing__semiring__rules_I8_J,axiom,
! [A_134,B_100,C_59] : times_times_nat(plus_plus_nat(A_134,B_100),C_59) = plus_plus_nat(times_times_nat(A_134,C_59),times_times_nat(B_100,C_59)) ).
fof(fact_43_crossproduct__noteq,axiom,
! [C_4,D_3,Aa,Ba] :
( ( Aa != Ba
& C_4 != D_3 )
<=> plus_plus_int(times_times_int(Aa,C_4),times_times_int(Ba,D_3)) != plus_plus_int(times_times_int(Aa,D_3),times_times_int(Ba,C_4)) ) ).
fof(fact_44_crossproduct__noteq,axiom,
! [C_4,D_3,Aa,Ba] :
( ( Aa != Ba
& C_4 != D_3 )
<=> plus_plus_nat(times_times_nat(Aa,C_4),times_times_nat(Ba,D_3)) != plus_plus_nat(times_times_nat(Aa,D_3),times_times_nat(Ba,C_4)) ) ).
fof(fact_45_comm__semiring__1__class_Onormalizing__semiring__rules_I34_J,axiom,
! [X_43,Y_42,Z_12] : times_times_int(X_43,plus_plus_int(Y_42,Z_12)) = plus_plus_int(times_times_int(X_43,Y_42),times_times_int(X_43,Z_12)) ).
fof(fact_46_comm__semiring__1__class_Onormalizing__semiring__rules_I34_J,axiom,
! [X_43,Y_42,Z_12] : times_times_nat(X_43,plus_plus_nat(Y_42,Z_12)) = plus_plus_nat(times_times_nat(X_43,Y_42),times_times_nat(X_43,Z_12)) ).
fof(fact_47_Pair__inject,axiom,
! [A_133,B_99,A_132,B_98] :
( product_Pair_int_int(A_133,B_99) = product_Pair_int_int(A_132,B_98)
=> ~ ( A_133 = A_132
=> B_99 != B_98 ) ) ).
fof(fact_48_Pair__inject,axiom,
! [A_133,B_99,A_132,B_98] :
( produc282740534nt_int(A_133,B_99) = produc282740534nt_int(A_132,B_98)
=> ~ ( A_133 = A_132
=> B_99 != B_98 ) ) ).
fof(fact_49_Pair__eq,axiom,
! [Aa,Ba,A_131,B_97] :
( product_Pair_int_int(Aa,Ba) = product_Pair_int_int(A_131,B_97)
<=> ( Aa = A_131
& Ba = B_97 ) ) ).
fof(fact_50_Pair__eq,axiom,
! [Aa,Ba,A_131,B_97] :
( produc282740534nt_int(Aa,Ba) = produc282740534nt_int(A_131,B_97)
<=> ( Aa = A_131
& Ba = B_97 ) ) ).
fof(fact_51_split__paired__All,axiom,
! [P] :
( ! [X1] : hBOOL(hAPP_P603027463t_bool(P,X1))
<=> ! [A_88,B_92] : hBOOL(hAPP_P603027463t_bool(P,product_Pair_int_int(A_88,B_92))) ) ).
fof(fact_52_split__paired__All,axiom,
! [P] :
( ! [X1] : hBOOL(hAPP_P178748804t_bool(P,X1))
<=> ! [A_88,B_92] : hBOOL(hAPP_P178748804t_bool(P,produc282740534nt_int(A_88,B_92))) ) ).
fof(fact_53_comm__semiring__1__class_Onormalizing__semiring__rules_I20_J,axiom,
! [A_130,B_96,C_58,D_17] : plus_plus_int(plus_plus_int(A_130,B_96),plus_plus_int(C_58,D_17)) = plus_plus_int(plus_plus_int(A_130,C_58),plus_plus_int(B_96,D_17)) ).
fof(fact_54_comm__semiring__1__class_Onormalizing__semiring__rules_I20_J,axiom,
! [A_130,B_96,C_58,D_17] : plus_plus_nat(plus_plus_nat(A_130,B_96),plus_plus_nat(C_58,D_17)) = plus_plus_nat(plus_plus_nat(A_130,C_58),plus_plus_nat(B_96,D_17)) ).
fof(fact_55_comm__semiring__1__class_Onormalizing__semiring__rules_I7_J,axiom,
! [A_129,B_95] : times_times_int(A_129,B_95) = times_times_int(B_95,A_129) ).
fof(fact_56_comm__semiring__1__class_Onormalizing__semiring__rules_I7_J,axiom,
! [A_129,B_95] : times_times_nat(A_129,B_95) = times_times_nat(B_95,A_129) ).
fof(fact_57_comm__semiring__1__class_Onormalizing__semiring__rules_I19_J,axiom,
! [Lx_6,Rx_6,Ry_4] : times_times_int(Lx_6,times_times_int(Rx_6,Ry_4)) = times_times_int(Rx_6,times_times_int(Lx_6,Ry_4)) ).
fof(fact_58_comm__semiring__1__class_Onormalizing__semiring__rules_I19_J,axiom,
! [Lx_6,Rx_6,Ry_4] : times_times_nat(Lx_6,times_times_nat(Rx_6,Ry_4)) = times_times_nat(Rx_6,times_times_nat(Lx_6,Ry_4)) ).
fof(fact_59_comm__semiring__1__class_Onormalizing__semiring__rules_I18_J,axiom,
! [Lx_5,Rx_5,Ry_3] : times_times_int(Lx_5,times_times_int(Rx_5,Ry_3)) = times_times_int(times_times_int(Lx_5,Rx_5),Ry_3) ).
fof(fact_60_comm__semiring__1__class_Onormalizing__semiring__rules_I18_J,axiom,
! [Lx_5,Rx_5,Ry_3] : times_times_nat(Lx_5,times_times_nat(Rx_5,Ry_3)) = times_times_nat(times_times_nat(Lx_5,Rx_5),Ry_3) ).
fof(fact_61_comm__semiring__1__class_Onormalizing__semiring__rules_I17_J,axiom,
! [Lx_4,Ly_4,Rx_4] : times_times_int(times_times_int(Lx_4,Ly_4),Rx_4) = times_times_int(Lx_4,times_times_int(Ly_4,Rx_4)) ).
fof(fact_62_comm__semiring__1__class_Onormalizing__semiring__rules_I17_J,axiom,
! [Lx_4,Ly_4,Rx_4] : times_times_nat(times_times_nat(Lx_4,Ly_4),Rx_4) = times_times_nat(Lx_4,times_times_nat(Ly_4,Rx_4)) ).
fof(fact_63_comm__semiring__1__class_Onormalizing__semiring__rules_I16_J,axiom,
! [Lx_3,Ly_3,Rx_3] : times_times_int(times_times_int(Lx_3,Ly_3),Rx_3) = times_times_int(times_times_int(Lx_3,Rx_3),Ly_3) ).
fof(fact_64_comm__semiring__1__class_Onormalizing__semiring__rules_I16_J,axiom,
! [Lx_3,Ly_3,Rx_3] : times_times_nat(times_times_nat(Lx_3,Ly_3),Rx_3) = times_times_nat(times_times_nat(Lx_3,Rx_3),Ly_3) ).
fof(fact_65_comm__semiring__1__class_Onormalizing__semiring__rules_I14_J,axiom,
! [Lx_2,Ly_2,Rx_2,Ry_2] : times_times_int(times_times_int(Lx_2,Ly_2),times_times_int(Rx_2,Ry_2)) = times_times_int(Lx_2,times_times_int(Ly_2,times_times_int(Rx_2,Ry_2))) ).
fof(fact_66_comm__semiring__1__class_Onormalizing__semiring__rules_I14_J,axiom,
! [Lx_2,Ly_2,Rx_2,Ry_2] : times_times_nat(times_times_nat(Lx_2,Ly_2),times_times_nat(Rx_2,Ry_2)) = times_times_nat(Lx_2,times_times_nat(Ly_2,times_times_nat(Rx_2,Ry_2))) ).
fof(fact_67_comm__semiring__1__class_Onormalizing__semiring__rules_I15_J,axiom,
! [Lx_1,Ly_1,Rx_1,Ry_1] : times_times_int(times_times_int(Lx_1,Ly_1),times_times_int(Rx_1,Ry_1)) = times_times_int(Rx_1,times_times_int(times_times_int(Lx_1,Ly_1),Ry_1)) ).
fof(fact_68_comm__semiring__1__class_Onormalizing__semiring__rules_I15_J,axiom,
! [Lx_1,Ly_1,Rx_1,Ry_1] : times_times_nat(times_times_nat(Lx_1,Ly_1),times_times_nat(Rx_1,Ry_1)) = times_times_nat(Rx_1,times_times_nat(times_times_nat(Lx_1,Ly_1),Ry_1)) ).
fof(fact_69_comm__semiring__1__class_Onormalizing__semiring__rules_I13_J,axiom,
! [Lx,Ly,Rx,Ry] : times_times_int(times_times_int(Lx,Ly),times_times_int(Rx,Ry)) = times_times_int(times_times_int(Lx,Rx),times_times_int(Ly,Ry)) ).
fof(fact_70_comm__semiring__1__class_Onormalizing__semiring__rules_I13_J,axiom,
! [Lx,Ly,Rx,Ry] : times_times_nat(times_times_nat(Lx,Ly),times_times_nat(Rx,Ry)) = times_times_nat(times_times_nat(Lx,Rx),times_times_nat(Ly,Ry)) ).
fof(fact_71_comm__semiring__1__class_Onormalizing__semiring__rules_I24_J,axiom,
! [A_128,C_57] : plus_plus_int(A_128,C_57) = plus_plus_int(C_57,A_128) ).
fof(fact_72_comm__semiring__1__class_Onormalizing__semiring__rules_I24_J,axiom,
! [A_128,C_57] : plus_plus_nat(A_128,C_57) = plus_plus_nat(C_57,A_128) ).
fof(fact_73_comm__semiring__1__class_Onormalizing__semiring__rules_I22_J,axiom,
! [A_127,C_56,D_16] : plus_plus_int(A_127,plus_plus_int(C_56,D_16)) = plus_plus_int(C_56,plus_plus_int(A_127,D_16)) ).
fof(fact_74_comm__semiring__1__class_Onormalizing__semiring__rules_I22_J,axiom,
! [A_127,C_56,D_16] : plus_plus_nat(A_127,plus_plus_nat(C_56,D_16)) = plus_plus_nat(C_56,plus_plus_nat(A_127,D_16)) ).
fof(fact_75_comm__semiring__1__class_Onormalizing__semiring__rules_I25_J,axiom,
! [A_126,C_55,D_15] : plus_plus_int(A_126,plus_plus_int(C_55,D_15)) = plus_plus_int(plus_plus_int(A_126,C_55),D_15) ).
fof(fact_76_comm__semiring__1__class_Onormalizing__semiring__rules_I25_J,axiom,
! [A_126,C_55,D_15] : plus_plus_nat(A_126,plus_plus_nat(C_55,D_15)) = plus_plus_nat(plus_plus_nat(A_126,C_55),D_15) ).
fof(fact_77_comm__semiring__1__class_Onormalizing__semiring__rules_I21_J,axiom,
! [A_125,B_94,C_54] : plus_plus_int(plus_plus_int(A_125,B_94),C_54) = plus_plus_int(A_125,plus_plus_int(B_94,C_54)) ).
fof(fact_78_comm__semiring__1__class_Onormalizing__semiring__rules_I21_J,axiom,
! [A_125,B_94,C_54] : plus_plus_nat(plus_plus_nat(A_125,B_94),C_54) = plus_plus_nat(A_125,plus_plus_nat(B_94,C_54)) ).
fof(fact_79_comm__semiring__1__class_Onormalizing__semiring__rules_I23_J,axiom,
! [A_124,B_93,C_53] : plus_plus_int(plus_plus_int(A_124,B_93),C_53) = plus_plus_int(plus_plus_int(A_124,C_53),B_93) ).
fof(fact_80_comm__semiring__1__class_Onormalizing__semiring__rules_I23_J,axiom,
! [A_124,B_93,C_53] : plus_plus_nat(plus_plus_nat(A_124,B_93),C_53) = plus_plus_nat(plus_plus_nat(A_124,C_53),B_93) ).
fof(fact_81_split__paired__Ex,axiom,
! [P] :
( ? [X1] : hBOOL(hAPP_P603027463t_bool(P,X1))
<=> ? [A_88,B_92] : hBOOL(hAPP_P603027463t_bool(P,product_Pair_int_int(A_88,B_92))) ) ).
fof(fact_82_split__paired__Ex,axiom,
! [P] :
( ? [X1] : hBOOL(hAPP_P178748804t_bool(P,X1))
<=> ? [A_88,B_92] : hBOOL(hAPP_P178748804t_bool(P,produc282740534nt_int(A_88,B_92))) ) ).
fof(fact_83_prod__induct3,axiom,
! [X_1,P] :
( ! [A_88,B_92,C_52] : hBOOL(hAPP_P178748804t_bool(P,produc282740534nt_int(A_88,product_Pair_int_int(B_92,C_52))))
=> hBOOL(hAPP_P178748804t_bool(P,X_1)) ) ).
fof(fact_84_prod__cases3,axiom,
! [Y_41] :
~ ! [A_88,B_92,C_52] : Y_41 != produc282740534nt_int(A_88,product_Pair_int_int(B_92,C_52)) ).
fof(fact_85_prod_Oexhaust,axiom,
! [Y_40] :
~ ! [A_88,B_92] : Y_40 != product_Pair_int_int(A_88,B_92) ).
fof(fact_86_prod_Oexhaust,axiom,
! [Y_40] :
~ ! [A_88,B_92] : Y_40 != produc282740534nt_int(A_88,B_92) ).
fof(fact_87_PairE,axiom,
! [P_2] :
~ ! [X_3,Y_39] : P_2 != product_Pair_int_int(X_3,Y_39) ).
fof(fact_88_PairE,axiom,
! [P_2] :
~ ! [X_3,Y_39] : P_2 != produc282740534nt_int(X_3,Y_39) ).
fof(fact_89_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_90_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_91_le__add__iff1,axiom,
! [Aa,E,C_4,Ba,D_3] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,plus_plus_int(times_times_int(Aa,E),C_4)),plus_plus_int(times_times_int(Ba,E),D_3)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,plus_plus_int(times_times_int(minus_minus_int(Aa,Ba),E),C_4)),D_3)) ) ).
fof(fact_92_le__add__iff2,axiom,
! [Aa,E,C_4,Ba,D_3] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,plus_plus_int(times_times_int(Aa,E),C_4)),plus_plus_int(times_times_int(Ba,E),D_3)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,C_4),plus_plus_int(times_times_int(minus_minus_int(Ba,Aa),E),D_3))) ) ).
fof(fact_93_real__squared__diff__one__factored,axiom,
! [X_42] : minus_minus_int(times_times_int(X_42,X_42),one_one_int) = times_times_int(plus_plus_int(X_42,one_one_int),minus_minus_int(X_42,one_one_int)) ).
fof(fact_94_less__add__iff1,axiom,
! [Aa,E,C_4,Ba,D_3] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,plus_plus_int(times_times_int(Aa,E),C_4)),plus_plus_int(times_times_int(Ba,E),D_3)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,plus_plus_int(times_times_int(minus_minus_int(Aa,Ba),E),C_4)),D_3)) ) ).
fof(fact_95_less__add__iff2,axiom,
! [Aa,E,C_4,Ba,D_3] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,plus_plus_int(times_times_int(Aa,E),C_4)),plus_plus_int(times_times_int(Ba,E),D_3)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,C_4),plus_plus_int(times_times_int(minus_minus_int(Ba,Aa),E),D_3))) ) ).
fof(fact_96_one__reorient,axiom,
! [X_1] :
( one_one_int = X_1
<=> X_1 = one_one_int ) ).
fof(fact_97_one__reorient,axiom,
! [X_1] :
( one_one_nat = X_1
<=> X_1 = one_one_nat ) ).
fof(fact_98_zle__add1__eq__le,axiom,
! [W_1,Z_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,W_1),plus_plus_int(Z_1,one_one_int)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,W_1),Z_1)) ) ).
fof(fact_99_add1__zle__eq,axiom,
! [W_1,Z_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,plus_plus_int(W_1,one_one_int)),Z_1))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,W_1),Z_1)) ) ).
fof(fact_100_linorder__neqE__linordered__idom,axiom,
! [X_41,Y_38] :
( X_41 != Y_38
=> ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_41),Y_38))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Y_38),X_41)) ) ) ).
fof(fact_101_zless__imp__add1__zle,axiom,
! [W,Z] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,W),Z))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,plus_plus_int(W,one_one_int)),Z)) ) ).
fof(fact_102_order__le__neq__implies__less,axiom,
! [X_40,Y_37] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_40),Y_37))
=> ( X_40 != Y_37
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_40),Y_37)) ) ) ).
fof(fact_103_order__le__neq__implies__less,axiom,
! [X_40,Y_37] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_40),Y_37))
=> ( X_40 != Y_37
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_40),Y_37)) ) ) ).
fof(fact_104_zless__add1__eq,axiom,
! [W_1,Z_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,W_1),plus_plus_int(Z_1,one_one_int)))
<=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,W_1),Z_1))
| W_1 = Z_1 ) ) ).
fof(fact_105_zadd__zless__mono,axiom,
! [Z_11,Z,W_2,W] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,W_2),W))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Z_11),Z))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,plus_plus_int(W_2,Z_11)),plus_plus_int(W,Z))) ) ) ).
fof(fact_106_add__less__le__mono,axiom,
! [C_51,D_14,A_123,B_91] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_123),B_91))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,C_51),D_14))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,plus_plus_int(A_123,C_51)),plus_plus_int(B_91,D_14))) ) ) ).
fof(fact_107_add__less__le__mono,axiom,
! [C_51,D_14,A_123,B_91] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_123),B_91))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,C_51),D_14))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,plus_plus_nat(A_123,C_51)),plus_plus_nat(B_91,D_14))) ) ) ).
fof(fact_108_add__le__less__mono,axiom,
! [C_50,D_13,A_122,B_90] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_122),B_90))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,C_50),D_13))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,plus_plus_int(A_122,C_50)),plus_plus_int(B_90,D_13))) ) ) ).
fof(fact_109_add__le__less__mono,axiom,
! [C_50,D_13,A_122,B_90] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_122),B_90))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,C_50),D_13))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,plus_plus_nat(A_122,C_50)),plus_plus_nat(B_90,D_13))) ) ) ).
fof(fact_110_less__add__one,axiom,
! [A_121] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_121),plus_plus_int(A_121,one_one_int))) ).
fof(fact_111_less__add__one,axiom,
! [A_121] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_121),plus_plus_nat(A_121,one_one_nat))) ).
fof(fact_112_less__1__mult,axiom,
! [N_2,M_5] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),M_5))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),N_2))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),times_times_int(M_5,N_2))) ) ) ).
fof(fact_113_less__1__mult,axiom,
! [N_2,M_5] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_nat),M_5))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_nat),N_2))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_nat),times_times_nat(M_5,N_2))) ) ) ).
fof(fact_114_zle__diff1__eq,axiom,
! [W_1,Z_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,W_1),minus_minus_int(Z_1,one_one_int)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,W_1),Z_1)) ) ).
fof(fact_115_add__less__cancel__right,axiom,
! [Aa,C_4,Ba] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,plus_plus_int(Aa,C_4)),plus_plus_int(Ba,C_4)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Aa),Ba)) ) ).
fof(fact_116_add__less__cancel__right,axiom,
! [Aa,C_4,Ba] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,plus_plus_nat(Aa,C_4)),plus_plus_nat(Ba,C_4)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Aa),Ba)) ) ).
fof(fact_117_add__less__cancel__left,axiom,
! [C_4,Aa,Ba] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,plus_plus_int(C_4,Aa)),plus_plus_int(C_4,Ba)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Aa),Ba)) ) ).
fof(fact_118_add__less__cancel__left,axiom,
! [C_4,Aa,Ba] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,plus_plus_nat(C_4,Aa)),plus_plus_nat(C_4,Ba)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Aa),Ba)) ) ).
fof(fact_119_add__strict__right__mono,axiom,
! [C_49,A_120,B_89] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_120),B_89))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,plus_plus_int(A_120,C_49)),plus_plus_int(B_89,C_49))) ) ).
fof(fact_120_add__strict__right__mono,axiom,
! [C_49,A_120,B_89] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_120),B_89))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,plus_plus_nat(A_120,C_49)),plus_plus_nat(B_89,C_49))) ) ).
fof(fact_121_add__strict__left__mono,axiom,
! [C_48,A_119,B_88] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_119),B_88))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,plus_plus_int(C_48,A_119)),plus_plus_int(C_48,B_88))) ) ).
fof(fact_122_add__strict__left__mono,axiom,
! [C_48,A_119,B_88] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_119),B_88))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,plus_plus_nat(C_48,A_119)),plus_plus_nat(C_48,B_88))) ) ).
fof(fact_123_add__strict__mono,axiom,
! [C_47,D_12,A_118,B_87] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_118),B_87))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,C_47),D_12))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,plus_plus_int(A_118,C_47)),plus_plus_int(B_87,D_12))) ) ) ).
fof(fact_124_add__strict__mono,axiom,
! [C_47,D_12,A_118,B_87] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_118),B_87))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,C_47),D_12))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,plus_plus_nat(A_118,C_47)),plus_plus_nat(B_87,D_12))) ) ) ).
fof(fact_125_add__less__imp__less__right,axiom,
! [A_117,C_46,B_86] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,plus_plus_int(A_117,C_46)),plus_plus_int(B_86,C_46)))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_117),B_86)) ) ).
fof(fact_126_add__less__imp__less__right,axiom,
! [A_117,C_46,B_86] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,plus_plus_nat(A_117,C_46)),plus_plus_nat(B_86,C_46)))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_117),B_86)) ) ).
fof(fact_127_add__less__imp__less__left,axiom,
! [C_45,A_116,B_85] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,plus_plus_int(C_45,A_116)),plus_plus_int(C_45,B_85)))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_116),B_85)) ) ).
fof(fact_128_add__less__imp__less__left,axiom,
! [C_45,A_116,B_85] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,plus_plus_nat(C_45,A_116)),plus_plus_nat(C_45,B_85)))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_116),B_85)) ) ).
fof(fact_129_comm__semiring__1__class_Onormalizing__semiring__rules_I11_J,axiom,
! [A_115] : times_times_int(one_one_int,A_115) = A_115 ).
fof(fact_130_comm__semiring__1__class_Onormalizing__semiring__rules_I11_J,axiom,
! [A_115] : times_times_nat(one_one_nat,A_115) = A_115 ).
fof(fact_131_comm__semiring__1__class_Onormalizing__semiring__rules_I12_J,axiom,
! [A_114] : times_times_int(A_114,one_one_int) = A_114 ).
fof(fact_132_comm__semiring__1__class_Onormalizing__semiring__rules_I12_J,axiom,
! [A_114] : times_times_nat(A_114,one_one_nat) = A_114 ).
fof(fact_133_mult__1__left,axiom,
! [A_113] : times_times_int(one_one_int,A_113) = A_113 ).
fof(fact_134_mult__1__left,axiom,
! [A_113] : times_times_nat(one_one_nat,A_113) = A_113 ).
fof(fact_135_mult__1,axiom,
! [A_112] : times_times_int(one_one_int,A_112) = A_112 ).
fof(fact_136_mult__1,axiom,
! [A_112] : times_times_nat(one_one_nat,A_112) = A_112 ).
fof(fact_137_mult__1__right,axiom,
! [A_111] : times_times_int(A_111,one_one_int) = A_111 ).
fof(fact_138_mult__1__right,axiom,
! [A_111] : times_times_nat(A_111,one_one_nat) = A_111 ).
fof(fact_139_mult_Ocomm__neutral,axiom,
! [A_110] : times_times_int(A_110,one_one_int) = A_110 ).
fof(fact_140_mult_Ocomm__neutral,axiom,
! [A_110] : times_times_nat(A_110,one_one_nat) = A_110 ).
fof(fact_141_diff__eq__diff__less,axiom,
! [Aa,Ba,C_4,D_3] :
( minus_minus_int(Aa,Ba) = minus_minus_int(C_4,D_3)
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Aa),Ba))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,C_4),D_3)) ) ) ).
fof(fact_142_add__le__cancel__right,axiom,
! [Aa,C_4,Ba] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,plus_plus_int(Aa,C_4)),plus_plus_int(Ba,C_4)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Aa),Ba)) ) ).
fof(fact_143_add__le__cancel__right,axiom,
! [Aa,C_4,Ba] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,plus_plus_nat(Aa,C_4)),plus_plus_nat(Ba,C_4)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Aa),Ba)) ) ).
fof(fact_144_add__le__cancel__left,axiom,
! [C_4,Aa,Ba] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,plus_plus_int(C_4,Aa)),plus_plus_int(C_4,Ba)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Aa),Ba)) ) ).
fof(fact_145_add__le__cancel__left,axiom,
! [C_4,Aa,Ba] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,plus_plus_nat(C_4,Aa)),plus_plus_nat(C_4,Ba)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Aa),Ba)) ) ).
fof(fact_146_add__right__mono,axiom,
! [C_44,A_109,B_84] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_109),B_84))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,plus_plus_int(A_109,C_44)),plus_plus_int(B_84,C_44))) ) ).
fof(fact_147_add__right__mono,axiom,
! [C_44,A_109,B_84] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_109),B_84))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,plus_plus_nat(A_109,C_44)),plus_plus_nat(B_84,C_44))) ) ).
fof(fact_148_add__left__mono,axiom,
! [C_43,A_108,B_83] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_108),B_83))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,plus_plus_int(C_43,A_108)),plus_plus_int(C_43,B_83))) ) ).
fof(fact_149_add__left__mono,axiom,
! [C_43,A_108,B_83] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_108),B_83))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,plus_plus_nat(C_43,A_108)),plus_plus_nat(C_43,B_83))) ) ).
fof(fact_150_add__mono,axiom,
! [C_42,D_11,A_107,B_82] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_107),B_82))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,C_42),D_11))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,plus_plus_int(A_107,C_42)),plus_plus_int(B_82,D_11))) ) ) ).
fof(fact_151_add__mono,axiom,
! [C_42,D_11,A_107,B_82] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_107),B_82))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,C_42),D_11))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,plus_plus_nat(A_107,C_42)),plus_plus_nat(B_82,D_11))) ) ) ).
fof(fact_152_add__le__imp__le__right,axiom,
! [A_106,C_41,B_81] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,plus_plus_int(A_106,C_41)),plus_plus_int(B_81,C_41)))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_106),B_81)) ) ).
fof(fact_153_add__le__imp__le__right,axiom,
! [A_106,C_41,B_81] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,plus_plus_nat(A_106,C_41)),plus_plus_nat(B_81,C_41)))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_106),B_81)) ) ).
fof(fact_154_add__le__imp__le__left,axiom,
! [C_40,A_105,B_80] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,plus_plus_int(C_40,A_105)),plus_plus_int(C_40,B_80)))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_105),B_80)) ) ).
fof(fact_155_add__le__imp__le__left,axiom,
! [C_40,A_105,B_80] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,plus_plus_nat(C_40,A_105)),plus_plus_nat(C_40,B_80)))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_105),B_80)) ) ).
fof(fact_156_diff__eq__diff__less__eq,axiom,
! [Aa,Ba,C_4,D_3] :
( minus_minus_int(Aa,Ba) = minus_minus_int(C_4,D_3)
=> ( 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_4),D_3)) ) ) ).
fof(fact_157_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,plus_plus_int(I,K)),plus_plus_int(J,K))) ) ).
fof(fact_158_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,plus_plus_int(K,I)),plus_plus_int(K,J))) ) ).
fof(fact_159_zmult__1,axiom,
! [Z] : times_times_int(one_one_int,Z) = Z ).
fof(fact_160_zmult__1__right,axiom,
! [Z] : times_times_int(Z,one_one_int) = Z ).
fof(fact_161_comm__semiring__1__class_Onormalizing__semiring__rules_I4_J,axiom,
! [M_4] : plus_plus_int(M_4,M_4) = times_times_int(plus_plus_int(one_one_int,one_one_int),M_4) ).
fof(fact_162_comm__semiring__1__class_Onormalizing__semiring__rules_I4_J,axiom,
! [M_4] : plus_plus_nat(M_4,M_4) = times_times_nat(plus_plus_nat(one_one_nat,one_one_nat),M_4) ).
fof(fact_163_comm__semiring__1__class_Onormalizing__semiring__rules_I3_J,axiom,
! [M_3,A_104] : plus_plus_nat(M_3,times_times_nat(A_104,M_3)) = times_times_nat(plus_plus_nat(A_104,one_one_nat),M_3) ).
fof(fact_164_comm__semiring__1__class_Onormalizing__semiring__rules_I3_J,axiom,
! [M_3,A_104] : plus_plus_int(M_3,times_times_int(A_104,M_3)) = times_times_int(plus_plus_int(A_104,one_one_int),M_3) ).
fof(fact_165_comm__semiring__1__class_Onormalizing__semiring__rules_I2_J,axiom,
! [A_103,M_2] : plus_plus_nat(times_times_nat(A_103,M_2),M_2) = times_times_nat(plus_plus_nat(A_103,one_one_nat),M_2) ).
fof(fact_166_comm__semiring__1__class_Onormalizing__semiring__rules_I2_J,axiom,
! [A_103,M_2] : plus_plus_int(times_times_int(A_103,M_2),M_2) = times_times_int(plus_plus_int(A_103,one_one_int),M_2) ).
fof(fact_167_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_168_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_169_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_170_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_171_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_172_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_173_order__refl,axiom,
! [X_39] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_39),X_39)) ).
fof(fact_174_order__refl,axiom,
! [X_39] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_39),X_39)) ).
fof(fact_175_xt1_I8_J,axiom,
! [Z_10,Y_36,X_38] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Y_36),X_38))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Z_10),Y_36))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Z_10),X_38)) ) ) ).
fof(fact_176_xt1_I8_J,axiom,
! [Z_10,Y_36,X_38] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y_36),X_38))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Z_10),Y_36))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Z_10),X_38)) ) ) ).
fof(fact_177_order__le__less__trans,axiom,
! [Z_9,X_37,Y_35] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_37),Y_35))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Y_35),Z_9))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_37),Z_9)) ) ) ).
fof(fact_178_order__le__less__trans,axiom,
! [Z_9,X_37,Y_35] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_37),Y_35))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Y_35),Z_9))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_37),Z_9)) ) ) ).
fof(fact_179_xt1_I7_J,axiom,
! [Z_8,Y_34,X_36] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Y_34),X_36))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Z_8),Y_34))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Z_8),X_36)) ) ) ).
fof(fact_180_xt1_I7_J,axiom,
! [Z_8,Y_34,X_36] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Y_34),X_36))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Z_8),Y_34))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Z_8),X_36)) ) ) ).
fof(fact_181_order__less__le__trans,axiom,
! [Z_7,X_35,Y_33] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_35),Y_33))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Y_33),Z_7))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_35),Z_7)) ) ) ).
fof(fact_182_order__less__le__trans,axiom,
! [Z_7,X_35,Y_33] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_35),Y_33))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y_33),Z_7))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_35),Z_7)) ) ) ).
fof(fact_183_xt1_I11_J,axiom,
! [B_79,A_102] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,B_79),A_102))
=> ( A_102 != B_79
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,B_79),A_102)) ) ) ).
fof(fact_184_xt1_I11_J,axiom,
! [B_79,A_102] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_79),A_102))
=> ( A_102 != B_79
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_79),A_102)) ) ) ).
fof(fact_185_order__le__neq__trans,axiom,
! [A_101,B_78] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_101),B_78))
=> ( A_101 != B_78
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_101),B_78)) ) ) ).
fof(fact_186_order__le__neq__trans,axiom,
! [A_101,B_78] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_101),B_78))
=> ( A_101 != B_78
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_101),B_78)) ) ) ).
fof(fact_187_order__le__imp__less__or__eq,axiom,
! [X_34,Y_32] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_34),Y_32))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_34),Y_32))
| X_34 = Y_32 ) ) ).
fof(fact_188_order__le__imp__less__or__eq,axiom,
! [X_34,Y_32] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_34),Y_32))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_34),Y_32))
| X_34 = Y_32 ) ) ).
fof(fact_189_zle__refl,axiom,
! [W] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,W),W)) ).
fof(fact_190_zle__linear,axiom,
! [Z,W] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Z),W))
| hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,W),Z)) ) ).
fof(fact_191_zless__le,axiom,
! [Z_1,W_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Z_1),W_1))
<=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Z_1),W_1))
& Z_1 != W_1 ) ) ).
fof(fact_192_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_193_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_194_zle__antisym,axiom,
! [Z,W] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Z),W))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,W),Z))
=> Z = W ) ) ).
fof(fact_195_linorder__linear,axiom,
! [X_33,Y_31] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_33),Y_31))
| hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Y_31),X_33)) ) ).
fof(fact_196_linorder__linear,axiom,
! [X_33,Y_31] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_33),Y_31))
| hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y_31),X_33)) ) ).
fof(fact_197_order__eq__iff,axiom,
! [X_1,Y_3] :
( X_1 = Y_3
<=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_1),Y_3))
& hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Y_3),X_1)) ) ) ).
fof(fact_198_order__eq__iff,axiom,
! [X_1,Y_3] :
( X_1 = Y_3
<=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_1),Y_3))
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y_3),X_1)) ) ) ).
fof(fact_199_order__eq__refl,axiom,
! [X_32,Y_30] :
( X_32 = Y_30
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_32),Y_30)) ) ).
fof(fact_200_order__eq__refl,axiom,
! [X_32,Y_30] :
( X_32 = Y_30
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_32),Y_30)) ) ).
fof(fact_201_order__antisym__conv,axiom,
! [Y_3,X_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Y_3),X_1))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_1),Y_3))
<=> X_1 = Y_3 ) ) ).
fof(fact_202_order__antisym__conv,axiom,
! [Y_3,X_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y_3),X_1))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_1),Y_3))
<=> X_1 = Y_3 ) ) ).
fof(fact_203_ord__eq__le__trans,axiom,
! [C_39,A_100,B_77] :
( A_100 = B_77
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,B_77),C_39))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_100),C_39)) ) ) ).
fof(fact_204_ord__eq__le__trans,axiom,
! [C_39,A_100,B_77] :
( A_100 = B_77
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_77),C_39))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_100),C_39)) ) ) ).
fof(fact_205_xt1_I3_J,axiom,
! [C_38,A_99,B_76] :
( A_99 = B_76
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,C_38),B_76))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,C_38),A_99)) ) ) ).
fof(fact_206_xt1_I3_J,axiom,
! [C_38,A_99,B_76] :
( A_99 = B_76
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,C_38),B_76))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,C_38),A_99)) ) ) ).
fof(fact_207_ord__le__eq__trans,axiom,
! [C_37,A_98,B_75] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_98),B_75))
=> ( B_75 = C_37
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_98),C_37)) ) ) ).
fof(fact_208_ord__le__eq__trans,axiom,
! [C_37,A_98,B_75] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_98),B_75))
=> ( B_75 = C_37
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_98),C_37)) ) ) ).
fof(fact_209_xt1_I4_J,axiom,
! [C_36,B_74,A_97] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,B_74),A_97))
=> ( B_74 = C_36
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,C_36),A_97)) ) ) ).
fof(fact_210_xt1_I4_J,axiom,
! [C_36,B_74,A_97] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_74),A_97))
=> ( B_74 = C_36
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,C_36),A_97)) ) ) ).
fof(fact_211_order__antisym,axiom,
! [X_31,Y_29] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_31),Y_29))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Y_29),X_31))
=> X_31 = Y_29 ) ) ).
fof(fact_212_order__antisym,axiom,
! [X_31,Y_29] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_31),Y_29))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y_29),X_31))
=> X_31 = Y_29 ) ) ).
fof(fact_213_order__trans,axiom,
! [Z_6,X_30,Y_28] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_30),Y_28))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Y_28),Z_6))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_30),Z_6)) ) ) ).
fof(fact_214_order__trans,axiom,
! [Z_6,X_30,Y_28] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_30),Y_28))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y_28),Z_6))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_30),Z_6)) ) ) ).
fof(fact_215_xt1_I5_J,axiom,
! [Y_27,X_29] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Y_27),X_29))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_29),Y_27))
=> X_29 = Y_27 ) ) ).
fof(fact_216_xt1_I5_J,axiom,
! [Y_27,X_29] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y_27),X_29))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_29),Y_27))
=> X_29 = Y_27 ) ) ).
fof(fact_217_xt1_I6_J,axiom,
! [Z_5,Y_26,X_28] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Y_26),X_28))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Z_5),Y_26))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Z_5),X_28)) ) ) ).
fof(fact_218_xt1_I6_J,axiom,
! [Z_5,Y_26,X_28] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y_26),X_28))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Z_5),Y_26))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Z_5),X_28)) ) ) ).
fof(fact_219_linorder__le__cases,axiom,
! [X_27,Y_25] :
( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_27),Y_25))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Y_25),X_27)) ) ).
fof(fact_220_linorder__le__cases,axiom,
! [X_27,Y_25] :
( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_27),Y_25))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y_25),X_27)) ) ).
fof(fact_221_order__less__irrefl,axiom,
! [X_26] : ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_26),X_26)) ).
fof(fact_222_order__less__irrefl,axiom,
! [X_26] : ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_26),X_26)) ).
fof(fact_223_linorder__neq__iff,axiom,
! [X_1,Y_3] :
( X_1 != Y_3
<=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_1),Y_3))
| hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Y_3),X_1)) ) ) ).
fof(fact_224_linorder__neq__iff,axiom,
! [X_1,Y_3] :
( X_1 != Y_3
<=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_1),Y_3))
| hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Y_3),X_1)) ) ) ).
fof(fact_225_not__less__iff__gr__or__eq,axiom,
! [X_1,Y_3] :
( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_1),Y_3))
<=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Y_3),X_1))
| X_1 = Y_3 ) ) ).
fof(fact_226_not__less__iff__gr__or__eq,axiom,
! [X_1,Y_3] :
( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_1),Y_3))
<=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Y_3),X_1))
| X_1 = Y_3 ) ) ).
fof(fact_227_linorder__less__linear,axiom,
! [X_25,Y_24] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_25),Y_24))
| X_25 = Y_24
| hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Y_24),X_25)) ) ).
fof(fact_228_linorder__less__linear,axiom,
! [X_25,Y_24] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_25),Y_24))
| X_25 = Y_24
| hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Y_24),X_25)) ) ).
fof(fact_229_linorder__antisym__conv3,axiom,
! [Y_3,X_1] :
( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Y_3),X_1))
=> ( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_1),Y_3))
<=> X_1 = Y_3 ) ) ).
fof(fact_230_linorder__antisym__conv3,axiom,
! [Y_3,X_1] :
( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Y_3),X_1))
=> ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_1),Y_3))
<=> X_1 = Y_3 ) ) ).
fof(fact_231_linorder__neqE,axiom,
! [X_24,Y_23] :
( X_24 != Y_23
=> ( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_24),Y_23))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Y_23),X_24)) ) ) ).
fof(fact_232_linorder__neqE,axiom,
! [X_24,Y_23] :
( X_24 != Y_23
=> ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_24),Y_23))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Y_23),X_24)) ) ) ).
fof(fact_233_less__imp__neq,axiom,
! [X_23,Y_22] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_23),Y_22))
=> X_23 != Y_22 ) ).
fof(fact_234_less__imp__neq,axiom,
! [X_23,Y_22] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_23),Y_22))
=> X_23 != Y_22 ) ).
fof(fact_235_order__less__not__sym,axiom,
! [X_22,Y_21] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_22),Y_21))
=> ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Y_21),X_22)) ) ).
fof(fact_236_order__less__not__sym,axiom,
! [X_22,Y_21] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_22),Y_21))
=> ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Y_21),X_22)) ) ).
fof(fact_237_order__less__imp__not__less,axiom,
! [X_21,Y_20] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_21),Y_20))
=> ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Y_20),X_21)) ) ).
fof(fact_238_order__less__imp__not__less,axiom,
! [X_21,Y_20] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_21),Y_20))
=> ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Y_20),X_21)) ) ).
fof(fact_239_order__less__imp__not__eq,axiom,
! [X_20,Y_19] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_20),Y_19))
=> X_20 != Y_19 ) ).
fof(fact_240_order__less__imp__not__eq,axiom,
! [X_20,Y_19] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_20),Y_19))
=> X_20 != Y_19 ) ).
fof(fact_241_order__less__imp__not__eq2,axiom,
! [X_19,Y_18] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_19),Y_18))
=> Y_18 != X_19 ) ).
fof(fact_242_order__less__imp__not__eq2,axiom,
! [X_19,Y_18] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_19),Y_18))
=> Y_18 != X_19 ) ).
fof(fact_243_order__less__imp__triv,axiom,
! [P,X_1,Y_3] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_1),Y_3))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Y_3),X_1))
=> hBOOL(P) ) ) ).
fof(fact_244_order__less__imp__triv,axiom,
! [P,X_1,Y_3] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_1),Y_3))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Y_3),X_1))
=> hBOOL(P) ) ) ).
fof(fact_245_order__less__asym_H,axiom,
! [A_96,B_73] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_96),B_73))
=> ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,B_73),A_96)) ) ).
fof(fact_246_order__less__asym_H,axiom,
! [A_96,B_73] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_96),B_73))
=> ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_73),A_96)) ) ).
fof(fact_247_xt1_I9_J,axiom,
! [B_72,A_95] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,B_72),A_95))
=> ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_95),B_72)) ) ).
fof(fact_248_xt1_I9_J,axiom,
! [B_72,A_95] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_72),A_95))
=> ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_95),B_72)) ) ).
fof(fact_249_ord__eq__less__trans,axiom,
! [C_35,A_94,B_71] :
( A_94 = B_71
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,B_71),C_35))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_94),C_35)) ) ) ).
fof(fact_250_ord__eq__less__trans,axiom,
! [C_35,A_94,B_71] :
( A_94 = B_71
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_71),C_35))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_94),C_35)) ) ) ).
fof(fact_251_xt1_I1_J,axiom,
! [C_34,A_93,B_70] :
( A_93 = B_70
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,C_34),B_70))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,C_34),A_93)) ) ) ).
fof(fact_252_xt1_I1_J,axiom,
! [C_34,A_93,B_70] :
( A_93 = B_70
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,C_34),B_70))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,C_34),A_93)) ) ) ).
fof(fact_253_ord__less__eq__trans,axiom,
! [C_33,A_92,B_69] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_92),B_69))
=> ( B_69 = C_33
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_92),C_33)) ) ) ).
fof(fact_254_ord__less__eq__trans,axiom,
! [C_33,A_92,B_69] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_92),B_69))
=> ( B_69 = C_33
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_92),C_33)) ) ) ).
fof(fact_255_xt1_I2_J,axiom,
! [C_32,B_68,A_91] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,B_68),A_91))
=> ( B_68 = C_32
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,C_32),A_91)) ) ) ).
fof(fact_256_xt1_I2_J,axiom,
! [C_32,B_68,A_91] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_68),A_91))
=> ( B_68 = C_32
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,C_32),A_91)) ) ) ).
fof(fact_257_order__less__trans,axiom,
! [Z_4,X_18,Y_17] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_18),Y_17))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Y_17),Z_4))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_18),Z_4)) ) ) ).
fof(fact_258_order__less__trans,axiom,
! [Z_4,X_18,Y_17] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_18),Y_17))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Y_17),Z_4))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_18),Z_4)) ) ) ).
fof(fact_259_xt1_I10_J,axiom,
! [Z_3,Y_16,X_17] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Y_16),X_17))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Z_3),Y_16))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Z_3),X_17)) ) ) ).
fof(fact_260_xt1_I10_J,axiom,
! [Z_3,Y_16,X_17] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Y_16),X_17))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Z_3),Y_16))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Z_3),X_17)) ) ) ).
fof(fact_261_order__less__asym,axiom,
! [X_16,Y_15] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_16),Y_15))
=> ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Y_15),X_16)) ) ).
fof(fact_262_order__less__asym,axiom,
! [X_16,Y_15] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_16),Y_15))
=> ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Y_15),X_16)) ) ).
fof(fact_263_linorder__cases,axiom,
! [X_15,Y_14] :
( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_15),Y_14))
=> ( X_15 != Y_14
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Y_14),X_15)) ) ) ).
fof(fact_264_linorder__cases,axiom,
! [X_15,Y_14] :
( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_15),Y_14))
=> ( X_15 != Y_14
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Y_14),X_15)) ) ) ).
fof(fact_265_linorder__not__less,axiom,
! [X_1,Y_3] :
( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_1),Y_3))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Y_3),X_1)) ) ).
fof(fact_266_linorder__not__less,axiom,
! [X_1,Y_3] :
( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_1),Y_3))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y_3),X_1)) ) ).
fof(fact_267_linorder__not__le,axiom,
! [X_1,Y_3] :
( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_1),Y_3))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Y_3),X_1)) ) ).
fof(fact_268_linorder__not__le,axiom,
! [X_1,Y_3] :
( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_1),Y_3))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Y_3),X_1)) ) ).
fof(fact_269_linorder__le__less__linear,axiom,
! [X_14,Y_13] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_14),Y_13))
| hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Y_13),X_14)) ) ).
fof(fact_270_linorder__le__less__linear,axiom,
! [X_14,Y_13] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_14),Y_13))
| hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Y_13),X_14)) ) ).
fof(fact_271_order__less__le,axiom,
! [X_1,Y_3] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_1),Y_3))
<=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_1),Y_3))
& X_1 != Y_3 ) ) ).
fof(fact_272_order__less__le,axiom,
! [X_1,Y_3] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_1),Y_3))
<=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_1),Y_3))
& X_1 != Y_3 ) ) ).
fof(fact_273_less__le__not__le,axiom,
! [X_1,Y_3] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_1),Y_3))
<=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_1),Y_3))
& ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Y_3),X_1)) ) ) ).
fof(fact_274_less__le__not__le,axiom,
! [X_1,Y_3] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_1),Y_3))
<=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_1),Y_3))
& ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y_3),X_1)) ) ) ).
fof(fact_275_order__le__less,axiom,
! [X_1,Y_3] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_1),Y_3))
<=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_1),Y_3))
| X_1 = Y_3 ) ) ).
fof(fact_276_order__le__less,axiom,
! [X_1,Y_3] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_1),Y_3))
<=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_1),Y_3))
| X_1 = Y_3 ) ) ).
fof(fact_277_leI,axiom,
! [X_13,Y_12] :
( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_13),Y_12))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Y_12),X_13)) ) ).
fof(fact_278_leI,axiom,
! [X_13,Y_12] :
( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_13),Y_12))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y_12),X_13)) ) ).
fof(fact_279_not__leE,axiom,
! [Y_11,X_12] :
( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Y_11),X_12))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_12),Y_11)) ) ).
fof(fact_280_not__leE,axiom,
! [Y_11,X_12] :
( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y_11),X_12))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_12),Y_11)) ) ).
fof(fact_281_linorder__antisym__conv1,axiom,
! [X_1,Y_3] :
( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_1),Y_3))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_1),Y_3))
<=> X_1 = Y_3 ) ) ).
fof(fact_282_linorder__antisym__conv1,axiom,
! [X_1,Y_3] :
( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_1),Y_3))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_1),Y_3))
<=> X_1 = Y_3 ) ) ).
fof(fact_283_order__neq__le__trans,axiom,
! [A_90,B_67] :
( A_90 != B_67
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_90),B_67))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_90),B_67)) ) ) ).
fof(fact_284_order__neq__le__trans,axiom,
! [A_90,B_67] :
( A_90 != B_67
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_90),B_67))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_90),B_67)) ) ) ).
fof(fact_285_xt1_I12_J,axiom,
! [A_89,B_66] :
( A_89 != B_66
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,B_66),A_89))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,B_66),A_89)) ) ) ).
fof(fact_286_xt1_I12_J,axiom,
! [A_89,B_66] :
( A_89 != B_66
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_66),A_89))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_66),A_89)) ) ) ).
fof(fact_287_leD,axiom,
! [Y_10,X_11] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Y_10),X_11))
=> ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_11),Y_10)) ) ).
fof(fact_288_leD,axiom,
! [Y_10,X_11] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y_10),X_11))
=> ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_11),Y_10)) ) ).
fof(fact_289_order__less__imp__le,axiom,
! [X_10,Y_9] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_10),Y_9))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_10),Y_9)) ) ).
fof(fact_290_order__less__imp__le,axiom,
! [X_10,Y_9] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_10),Y_9))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_10),Y_9)) ) ).
fof(fact_291_linorder__antisym__conv2,axiom,
! [X_1,Y_3] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_1),Y_3))
=> ( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_1),Y_3))
<=> X_1 = Y_3 ) ) ).
fof(fact_292_linorder__antisym__conv2,axiom,
! [X_1,Y_3] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_1),Y_3))
=> ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_1),Y_3))
<=> X_1 = Y_3 ) ) ).
fof(fact_293_norR__mem__unique__aux,axiom,
! [A_19,B] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_19),minus_minus_int(B,one_one_int)))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_19),B)) ) ).
fof(fact_294_int__le__induct,axiom,
! [P,I_1,K_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,I_1),K_1))
=> ( hBOOL(hAPP_int_bool(P,K_1))
=> ( ! [I_2] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,I_2),K_1))
=> ( hBOOL(hAPP_int_bool(P,I_2))
=> hBOOL(hAPP_int_bool(P,minus_minus_int(I_2,one_one_int))) ) )
=> hBOOL(hAPP_int_bool(P,I_1)) ) ) ) ).
fof(fact_295_int__less__induct,axiom,
! [P,I_1,K_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,I_1),K_1))
=> ( hBOOL(hAPP_int_bool(P,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,I_2))
=> hBOOL(hAPP_int_bool(P,minus_minus_int(I_2,one_one_int))) ) )
=> hBOOL(hAPP_int_bool(P,I_1)) ) ) ) ).
fof(fact_296_int__ge__induct,axiom,
! [P,K_1,I_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K_1),I_1))
=> ( hBOOL(hAPP_int_bool(P,K_1))
=> ( ! [I_2] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K_1),I_2))
=> ( hBOOL(hAPP_int_bool(P,I_2))
=> hBOOL(hAPP_int_bool(P,plus_plus_int(I_2,one_one_int))) ) )
=> hBOOL(hAPP_int_bool(P,I_1)) ) ) ) ).
fof(fact_297_int__gr__induct,axiom,
! [P,K_1,I_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K_1),I_1))
=> ( hBOOL(hAPP_int_bool(P,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,I_2))
=> hBOOL(hAPP_int_bool(P,plus_plus_int(I_2,one_one_int))) ) )
=> hBOOL(hAPP_int_bool(P,I_1)) ) ) ) ).
fof(fact_298_int__induct,axiom,
! [I_1,P,K_1] :
( hBOOL(hAPP_int_bool(P,K_1))
=> ( ! [I_2] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K_1),I_2))
=> ( hBOOL(hAPP_int_bool(P,I_2))
=> hBOOL(hAPP_int_bool(P,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,I_2))
=> hBOOL(hAPP_int_bool(P,minus_minus_int(I_2,one_one_int))) ) )
=> hBOOL(hAPP_int_bool(P,I_1)) ) ) ) ).
fof(fact_299_Euler_Oaux2,axiom,
! [B,A_19,C_31] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_19),C_31))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B),C_31))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_19),B))
| hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B),A_19)) ) ) ) ).
fof(fact_300_d22set__induct__old,axiom,
! [X_1,P] :
( ! [A_88] :
( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),A_88))
=> hBOOL(hAPP_int_bool(P,minus_minus_int(A_88,one_one_int))) )
=> hBOOL(hAPP_int_bool(P,A_88)) )
=> hBOOL(hAPP_int_bool(P,X_1)) ) ).
fof(fact_301_convex__bound__lt,axiom,
! [V_1,U_1,Y_8,X_9,A_87] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_9),A_87))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Y_8),A_87))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),U_1))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),V_1))
=> ( plus_plus_int(U_1,V_1) = one_one_int
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,plus_plus_int(times_times_int(U_1,X_9),times_times_int(V_1,Y_8))),A_87)) ) ) ) ) ) ).
fof(fact_302_comm__semiring__1__class_Onormalizing__semiring__rules_I9_J,axiom,
! [A_86] : times_times_nat(zero_zero_nat,A_86) = zero_zero_nat ).
fof(fact_303_comm__semiring__1__class_Onormalizing__semiring__rules_I9_J,axiom,
! [A_86] : times_times_int(zero_zero_int,A_86) = zero_zero_int ).
fof(fact_304_comm__semiring__1__class_Onormalizing__semiring__rules_I10_J,axiom,
! [A_85] : times_times_nat(A_85,zero_zero_nat) = zero_zero_nat ).
fof(fact_305_comm__semiring__1__class_Onormalizing__semiring__rules_I10_J,axiom,
! [A_85] : times_times_int(A_85,zero_zero_int) = zero_zero_int ).
fof(fact_306_mult__zero__left,axiom,
! [A_84] : times_times_nat(zero_zero_nat,A_84) = zero_zero_nat ).
fof(fact_307_mult__zero__left,axiom,
! [A_84] : times_times_int(zero_zero_int,A_84) = zero_zero_int ).
fof(fact_308_mult__zero__right,axiom,
! [A_83] : times_times_nat(A_83,zero_zero_nat) = zero_zero_nat ).
fof(fact_309_mult__zero__right,axiom,
! [A_83] : times_times_int(A_83,zero_zero_int) = zero_zero_int ).
fof(fact_310_mult__eq__0__iff,axiom,
! [Aa,Ba] :
( times_times_int(Aa,Ba) = zero_zero_int
<=> ( Aa = zero_zero_int
| Ba = zero_zero_int ) ) ).
fof(fact_311_no__zero__divisors,axiom,
! [B_65,A_82] :
( A_82 != zero_zero_nat
=> ( B_65 != zero_zero_nat
=> times_times_nat(A_82,B_65) != zero_zero_nat ) ) ).
fof(fact_312_no__zero__divisors,axiom,
! [B_65,A_82] :
( A_82 != zero_zero_int
=> ( B_65 != zero_zero_int
=> times_times_int(A_82,B_65) != zero_zero_int ) ) ).
fof(fact_313_divisors__zero,axiom,
! [A_81,B_64] :
( times_times_nat(A_81,B_64) = zero_zero_nat
=> ( A_81 = zero_zero_nat
| B_64 = zero_zero_nat ) ) ).
fof(fact_314_divisors__zero,axiom,
! [A_81,B_64] :
( times_times_int(A_81,B_64) = zero_zero_int
=> ( A_81 = zero_zero_int
| B_64 = zero_zero_int ) ) ).
fof(fact_315_comm__semiring__1__class_Onormalizing__semiring__rules_I5_J,axiom,
! [A_80] : plus_plus_nat(zero_zero_nat,A_80) = A_80 ).
fof(fact_316_comm__semiring__1__class_Onormalizing__semiring__rules_I5_J,axiom,
! [A_80] : plus_plus_int(zero_zero_int,A_80) = A_80 ).
fof(fact_317_comm__semiring__1__class_Onormalizing__semiring__rules_I6_J,axiom,
! [A_79] : plus_plus_nat(A_79,zero_zero_nat) = A_79 ).
fof(fact_318_comm__semiring__1__class_Onormalizing__semiring__rules_I6_J,axiom,
! [A_79] : plus_plus_int(A_79,zero_zero_int) = A_79 ).
fof(fact_319_add__0__iff,axiom,
! [Ba,Aa] :
( Ba = plus_plus_nat(Ba,Aa)
<=> Aa = zero_zero_nat ) ).
fof(fact_320_add__0__iff,axiom,
! [Ba,Aa] :
( Ba = plus_plus_int(Ba,Aa)
<=> Aa = zero_zero_int ) ).
fof(fact_321_add__0__left,axiom,
! [A_78] : plus_plus_nat(zero_zero_nat,A_78) = A_78 ).
fof(fact_322_add__0__left,axiom,
! [A_78] : plus_plus_int(zero_zero_int,A_78) = A_78 ).
fof(fact_323_add__0,axiom,
! [A_77] : plus_plus_nat(zero_zero_nat,A_77) = A_77 ).
fof(fact_324_add__0,axiom,
! [A_77] : plus_plus_int(zero_zero_int,A_77) = A_77 ).
fof(fact_325_double__zero__sym,axiom,
! [Aa] :
( zero_zero_int = plus_plus_int(Aa,Aa)
<=> Aa = zero_zero_int ) ).
fof(fact_326_add__0__right,axiom,
! [A_76] : plus_plus_nat(A_76,zero_zero_nat) = A_76 ).
fof(fact_327_add__0__right,axiom,
! [A_76] : plus_plus_int(A_76,zero_zero_int) = A_76 ).
fof(fact_328_add_Ocomm__neutral,axiom,
! [A_75] : plus_plus_nat(A_75,zero_zero_nat) = A_75 ).
fof(fact_329_add_Ocomm__neutral,axiom,
! [A_75] : plus_plus_int(A_75,zero_zero_int) = A_75 ).
fof(fact_330_double__eq__0__iff,axiom,
! [Aa] :
( plus_plus_int(Aa,Aa) = zero_zero_int
<=> Aa = zero_zero_int ) ).
fof(fact_331_zero__reorient,axiom,
! [X_1] :
( zero_zero_nat = X_1
<=> X_1 = zero_zero_nat ) ).
fof(fact_332_zero__reorient,axiom,
! [X_1] :
( zero_zero_int = X_1
<=> X_1 = zero_zero_int ) ).
fof(fact_333_diff__0__right,axiom,
! [A_74] : minus_minus_int(A_74,zero_zero_int) = A_74 ).
fof(fact_334_diff__self,axiom,
! [A_73] : minus_minus_int(A_73,A_73) = zero_zero_int ).
fof(fact_335_eq__iff__diff__eq__0,axiom,
! [Aa,Ba] :
( Aa = Ba
<=> minus_minus_int(Aa,Ba) = zero_zero_int ) ).
fof(fact_336_right__minus__eq,axiom,
! [Aa,Ba] :
( minus_minus_int(Aa,Ba) = zero_zero_int
<=> Aa = Ba ) ).
fof(fact_337_one__neq__zero,axiom,
one_one_nat != zero_zero_nat ).
fof(fact_338_one__neq__zero,axiom,
one_one_int != zero_zero_int ).
fof(fact_339_zero__neq__one,axiom,
zero_zero_nat != one_one_nat ).
fof(fact_340_zero__neq__one,axiom,
zero_zero_int != one_one_int ).
fof(fact_341_int__0__neq__1,axiom,
zero_zero_int != one_one_int ).
fof(fact_342_zadd__0__right,axiom,
! [Z] : plus_plus_int(Z,zero_zero_int) = Z ).
fof(fact_343_zadd__0,axiom,
! [Z] : plus_plus_int(zero_zero_int,Z) = Z ).
fof(fact_344_split__mult__neg__le,axiom,
! [B_63,A_72] :
( ( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),A_72))
& hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,B_63),zero_zero_nat)) )
| ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_72),zero_zero_nat))
& hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),B_63)) ) )
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,times_times_nat(A_72,B_63)),zero_zero_nat)) ) ).
fof(fact_345_split__mult__neg__le,axiom,
! [B_63,A_72] :
( ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_72))
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_63),zero_zero_int)) )
| ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_72),zero_zero_int))
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),B_63)) ) )
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,times_times_int(A_72,B_63)),zero_zero_int)) ) ).
fof(fact_346_split__mult__pos__le,axiom,
! [B_62,A_71] :
( ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_71))
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),B_62)) )
| ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_71),zero_zero_int))
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_62),zero_zero_int)) ) )
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),times_times_int(A_71,B_62))) ) ).
fof(fact_347_mult__mono,axiom,
! [C_30,D_10,A_70,B_61] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_70),B_61))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,C_30),D_10))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),B_61))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),C_30))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,times_times_nat(A_70,C_30)),times_times_nat(B_61,D_10))) ) ) ) ) ).
fof(fact_348_mult__mono,axiom,
! [C_30,D_10,A_70,B_61] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_70),B_61))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,C_30),D_10))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),B_61))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),C_30))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,times_times_int(A_70,C_30)),times_times_int(B_61,D_10))) ) ) ) ) ).
fof(fact_349_mult__mono_H,axiom,
! [C_29,D_9,A_69,B_60] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_69),B_60))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,C_29),D_9))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),A_69))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),C_29))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,times_times_nat(A_69,C_29)),times_times_nat(B_60,D_9))) ) ) ) ) ).
fof(fact_350_mult__mono_H,axiom,
! [C_29,D_9,A_69,B_60] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_69),B_60))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,C_29),D_9))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_69))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),C_29))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,times_times_int(A_69,C_29)),times_times_int(B_60,D_9))) ) ) ) ) ).
fof(fact_351_mult__left__mono__neg,axiom,
! [C_28,B_59,A_68] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_59),A_68))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,C_28),zero_zero_int))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,times_times_int(C_28,A_68)),times_times_int(C_28,B_59))) ) ) ).
fof(fact_352_mult__right__mono__neg,axiom,
! [C_27,B_58,A_67] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_58),A_67))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,C_27),zero_zero_int))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,times_times_int(A_67,C_27)),times_times_int(B_58,C_27))) ) ) ).
fof(fact_353_comm__mult__left__mono,axiom,
! [C_26,A_66,B_57] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_66),B_57))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),C_26))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,times_times_nat(C_26,A_66)),times_times_nat(C_26,B_57))) ) ) ).
fof(fact_354_comm__mult__left__mono,axiom,
! [C_26,A_66,B_57] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_66),B_57))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),C_26))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,times_times_int(C_26,A_66)),times_times_int(C_26,B_57))) ) ) ).
fof(fact_355_mult__left__mono,axiom,
! [C_25,A_65,B_56] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_65),B_56))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),C_25))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,times_times_nat(C_25,A_65)),times_times_nat(C_25,B_56))) ) ) ).
fof(fact_356_mult__left__mono,axiom,
! [C_25,A_65,B_56] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_65),B_56))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),C_25))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,times_times_int(C_25,A_65)),times_times_int(C_25,B_56))) ) ) ).
fof(fact_357_mult__right__mono,axiom,
! [C_24,A_64,B_55] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_64),B_55))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),C_24))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,times_times_nat(A_64,C_24)),times_times_nat(B_55,C_24))) ) ) ).
fof(fact_358_mult__right__mono,axiom,
! [C_24,A_64,B_55] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_64),B_55))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),C_24))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,times_times_int(A_64,C_24)),times_times_int(B_55,C_24))) ) ) ).
fof(fact_359_mult__nonpos__nonpos,axiom,
! [B_54,A_63] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_63),zero_zero_int))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_54),zero_zero_int))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),times_times_int(A_63,B_54))) ) ) ).
fof(fact_360_mult__nonpos__nonneg,axiom,
! [B_53,A_62] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_62),zero_zero_nat))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),B_53))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,times_times_nat(A_62,B_53)),zero_zero_nat)) ) ) ).
fof(fact_361_mult__nonpos__nonneg,axiom,
! [B_53,A_62] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_62),zero_zero_int))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),B_53))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,times_times_int(A_62,B_53)),zero_zero_int)) ) ) ).
fof(fact_362_mult__nonneg__nonpos2,axiom,
! [B_52,A_61] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),A_61))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,B_52),zero_zero_nat))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,times_times_nat(B_52,A_61)),zero_zero_nat)) ) ) ).
fof(fact_363_mult__nonneg__nonpos2,axiom,
! [B_52,A_61] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_61))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_52),zero_zero_int))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,times_times_int(B_52,A_61)),zero_zero_int)) ) ) ).
fof(fact_364_mult__nonneg__nonpos,axiom,
! [B_51,A_60] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),A_60))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,B_51),zero_zero_nat))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,times_times_nat(A_60,B_51)),zero_zero_nat)) ) ) ).
fof(fact_365_mult__nonneg__nonpos,axiom,
! [B_51,A_60] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_60))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_51),zero_zero_int))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,times_times_int(A_60,B_51)),zero_zero_int)) ) ) ).
fof(fact_366_mult__nonneg__nonneg,axiom,
! [B_50,A_59] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),A_59))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),B_50))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),times_times_nat(A_59,B_50))) ) ) ).
fof(fact_367_mult__nonneg__nonneg,axiom,
! [B_50,A_59] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_59))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),B_50))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),times_times_int(A_59,B_50))) ) ) ).
fof(fact_368_mult__le__0__iff,axiom,
! [Aa,Ba] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_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_369_zero__le__mult__iff,axiom,
! [Aa,Ba] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_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_370_zero__le__square,axiom,
! [A_58] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),times_times_int(A_58,A_58))) ).
fof(fact_371_mult__strict__left__mono__neg,axiom,
! [C_23,B_49,A_57] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_49),A_57))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,C_23),zero_zero_int))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,times_times_int(C_23,A_57)),times_times_int(C_23,B_49))) ) ) ).
fof(fact_372_mult__strict__right__mono__neg,axiom,
! [C_22,B_48,A_56] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_48),A_56))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,C_22),zero_zero_int))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,times_times_int(A_56,C_22)),times_times_int(B_48,C_22))) ) ) ).
fof(fact_373_comm__mult__strict__left__mono,axiom,
! [C_21,A_55,B_47] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_55),B_47))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),C_21))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,times_times_nat(C_21,A_55)),times_times_nat(C_21,B_47))) ) ) ).
fof(fact_374_comm__mult__strict__left__mono,axiom,
! [C_21,A_55,B_47] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_55),B_47))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),C_21))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,times_times_int(C_21,A_55)),times_times_int(C_21,B_47))) ) ) ).
fof(fact_375_mult__strict__left__mono,axiom,
! [C_20,A_54,B_46] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_54),B_46))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),C_20))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,times_times_nat(C_20,A_54)),times_times_nat(C_20,B_46))) ) ) ).
fof(fact_376_mult__strict__left__mono,axiom,
! [C_20,A_54,B_46] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_54),B_46))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),C_20))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,times_times_int(C_20,A_54)),times_times_int(C_20,B_46))) ) ) ).
fof(fact_377_mult__strict__right__mono,axiom,
! [C_19,A_53,B_45] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_53),B_45))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),C_19))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,times_times_nat(A_53,C_19)),times_times_nat(B_45,C_19))) ) ) ).
fof(fact_378_mult__strict__right__mono,axiom,
! [C_19,A_53,B_45] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_53),B_45))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),C_19))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,times_times_int(A_53,C_19)),times_times_int(B_45,C_19))) ) ) ).
fof(fact_379_mult__neg__neg,axiom,
! [B_44,A_52] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_52),zero_zero_int))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_44),zero_zero_int))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),times_times_int(A_52,B_44))) ) ) ).
fof(fact_380_mult__neg__pos,axiom,
! [B_43,A_51] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_51),zero_zero_nat))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),B_43))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,times_times_nat(A_51,B_43)),zero_zero_nat)) ) ) ).
fof(fact_381_mult__neg__pos,axiom,
! [B_43,A_51] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_51),zero_zero_int))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_43))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,times_times_int(A_51,B_43)),zero_zero_int)) ) ) ).
fof(fact_382_mult__less__cancel__left__neg,axiom,
! [Aa,Ba,C_4] :
( 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,times_times_int(C_4,Aa)),times_times_int(C_4,Ba)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Ba),Aa)) ) ) ).
fof(fact_383_zero__less__mult__pos2,axiom,
! [B_42,A_50] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),times_times_nat(B_42,A_50)))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),A_50))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),B_42)) ) ) ).
fof(fact_384_zero__less__mult__pos2,axiom,
! [B_42,A_50] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),times_times_int(B_42,A_50)))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_50))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_42)) ) ) ).
fof(fact_385_zero__less__mult__pos,axiom,
! [A_49,B_41] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),times_times_nat(A_49,B_41)))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),A_49))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),B_41)) ) ) ).
fof(fact_386_zero__less__mult__pos,axiom,
! [A_49,B_41] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),times_times_int(A_49,B_41)))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_49))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_41)) ) ) ).
fof(fact_387_mult__pos__neg2,axiom,
! [B_40,A_48] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),A_48))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,B_40),zero_zero_nat))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,times_times_nat(B_40,A_48)),zero_zero_nat)) ) ) ).
fof(fact_388_mult__pos__neg2,axiom,
! [B_40,A_48] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_48))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_40),zero_zero_int))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,times_times_int(B_40,A_48)),zero_zero_int)) ) ) ).
fof(fact_389_mult__pos__neg,axiom,
! [B_39,A_47] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),A_47))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,B_39),zero_zero_nat))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,times_times_nat(A_47,B_39)),zero_zero_nat)) ) ) ).
fof(fact_390_mult__pos__neg,axiom,
! [B_39,A_47] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_47))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_39),zero_zero_int))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,times_times_int(A_47,B_39)),zero_zero_int)) ) ) ).
fof(fact_391_mult__pos__pos,axiom,
! [B_38,A_46] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),A_46))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),B_38))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),times_times_nat(A_46,B_38))) ) ) ).
fof(fact_392_mult__pos__pos,axiom,
! [B_38,A_46] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_46))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_38))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),times_times_int(A_46,B_38))) ) ) ).
fof(fact_393_mult__less__cancel__left__pos,axiom,
! [Aa,Ba,C_4] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),C_4))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,times_times_int(C_4,Aa)),times_times_int(C_4,Ba)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Aa),Ba)) ) ) ).
fof(fact_394_mult__less__cancel__left__disj,axiom,
! [C_4,Aa,Ba] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,times_times_int(C_4,Aa)),times_times_int(C_4,Ba)))
<=> ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),C_4))
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Aa),Ba)) )
| ( 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,Ba),Aa)) ) ) ) ).
fof(fact_395_mult__less__cancel__right__disj,axiom,
! [Aa,C_4,Ba] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,times_times_int(Aa,C_4)),times_times_int(Ba,C_4)))
<=> ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),C_4))
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Aa),Ba)) )
| ( 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,Ba),Aa)) ) ) ) ).
fof(fact_396_not__square__less__zero,axiom,
! [A_45] : ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,times_times_int(A_45,A_45)),zero_zero_int)) ).
fof(fact_397_add__nonpos__nonpos,axiom,
! [B_37,A_44] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_44),zero_zero_nat))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,B_37),zero_zero_nat))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,plus_plus_nat(A_44,B_37)),zero_zero_nat)) ) ) ).
fof(fact_398_add__nonpos__nonpos,axiom,
! [B_37,A_44] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_44),zero_zero_int))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_37),zero_zero_int))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,plus_plus_int(A_44,B_37)),zero_zero_int)) ) ) ).
fof(fact_399_add__increasing2,axiom,
! [B_36,A_43,C_18] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),C_18))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,B_36),A_43))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,B_36),plus_plus_nat(A_43,C_18))) ) ) ).
fof(fact_400_add__increasing2,axiom,
! [B_36,A_43,C_18] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),C_18))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_36),A_43))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_36),plus_plus_int(A_43,C_18))) ) ) ).
fof(fact_401_add__increasing,axiom,
! [B_35,C_17,A_42] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),A_42))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,B_35),C_17))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,B_35),plus_plus_nat(A_42,C_17))) ) ) ).
fof(fact_402_add__increasing,axiom,
! [B_35,C_17,A_42] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_42))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_35),C_17))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_35),plus_plus_int(A_42,C_17))) ) ) ).
fof(fact_403_add__nonneg__eq__0__iff,axiom,
! [Y_3,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_3))
=> ( plus_plus_nat(X_1,Y_3) = zero_zero_nat
<=> ( X_1 = zero_zero_nat
& Y_3 = zero_zero_nat ) ) ) ) ).
fof(fact_404_add__nonneg__eq__0__iff,axiom,
! [Y_3,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_3))
=> ( plus_plus_int(X_1,Y_3) = zero_zero_int
<=> ( X_1 = zero_zero_int
& Y_3 = zero_zero_int ) ) ) ) ).
fof(fact_405_add__nonneg__nonneg,axiom,
! [B_34,A_41] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),A_41))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),B_34))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),plus_plus_nat(A_41,B_34))) ) ) ).
fof(fact_406_add__nonneg__nonneg,axiom,
! [B_34,A_41] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_41))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),B_34))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),plus_plus_int(A_41,B_34))) ) ) ).
fof(fact_407_double__add__le__zero__iff__single__add__le__zero,axiom,
! [Aa] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_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_408_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),plus_plus_int(Aa,Aa)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Aa)) ) ).
fof(fact_409_add__neg__neg,axiom,
! [B_33,A_40] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_40),zero_zero_nat))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,B_33),zero_zero_nat))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,plus_plus_nat(A_40,B_33)),zero_zero_nat)) ) ) ).
fof(fact_410_add__neg__neg,axiom,
! [B_33,A_40] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_40),zero_zero_int))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_33),zero_zero_int))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,plus_plus_int(A_40,B_33)),zero_zero_int)) ) ) ).
fof(fact_411_pos__add__strict,axiom,
! [B_32,C_16,A_39] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),A_39))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,B_32),C_16))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,B_32),plus_plus_nat(A_39,C_16))) ) ) ).
fof(fact_412_pos__add__strict,axiom,
! [B_32,C_16,A_39] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_39))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_32),C_16))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_32),plus_plus_int(A_39,C_16))) ) ) ).
fof(fact_413_add__pos__pos,axiom,
! [B_31,A_38] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),A_38))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),B_31))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),plus_plus_nat(A_38,B_31))) ) ) ).
fof(fact_414_add__pos__pos,axiom,
! [B_31,A_38] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_38))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_31))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),plus_plus_int(A_38,B_31))) ) ) ).
fof(fact_415_double__add__less__zero__iff__single__add__less__zero,axiom,
! [Aa] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,plus_plus_int(Aa,Aa)),zero_zero_int))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Aa),zero_zero_int)) ) ).
fof(fact_416_even__less__0__iff,axiom,
! [Aa] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,plus_plus_int(Aa,Aa)),zero_zero_int))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Aa),zero_zero_int)) ) ).
fof(fact_417_zero__less__double__add__iff__zero__less__single__add,axiom,
! [Aa] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),plus_plus_int(Aa,Aa)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),Aa)) ) ).
fof(fact_418_zero__le__one,axiom,
hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),one_one_nat)) ).
fof(fact_419_zero__le__one,axiom,
hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),one_one_int)) ).
fof(fact_420_not__one__le__zero,axiom,
~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,one_one_nat),zero_zero_nat)) ).
fof(fact_421_not__one__le__zero,axiom,
~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,one_one_int),zero_zero_int)) ).
fof(fact_422_zero__less__one,axiom,
hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),one_one_nat)) ).
fof(fact_423_zero__less__one,axiom,
hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),one_one_int)) ).
fof(fact_424_not__one__less__zero,axiom,
~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_nat),zero_zero_nat)) ).
fof(fact_425_not__one__less__zero,axiom,
~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),zero_zero_int)) ).
fof(fact_426_le__iff__diff__le__0,axiom,
! [Aa,Ba] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Aa),Ba))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,minus_minus_int(Aa,Ba)),zero_zero_int)) ) ).
fof(fact_427_add__scale__eq__noteq,axiom,
! [C_15,D_8,A_37,B_30,R_3] :
( R_3 != zero_zero_nat
=> ( ( A_37 = B_30
& C_15 != D_8 )
=> plus_plus_nat(A_37,times_times_nat(R_3,C_15)) != plus_plus_nat(B_30,times_times_nat(R_3,D_8)) ) ) ).
fof(fact_428_add__scale__eq__noteq,axiom,
! [C_15,D_8,A_37,B_30,R_3] :
( R_3 != zero_zero_int
=> ( ( A_37 = B_30
& C_15 != D_8 )
=> plus_plus_int(A_37,times_times_int(R_3,C_15)) != plus_plus_int(B_30,times_times_int(R_3,D_8)) ) ) ).
fof(fact_429_less__iff__diff__less__0,axiom,
! [Aa,Ba] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Aa),Ba))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,minus_minus_int(Aa,Ba)),zero_zero_int)) ) ).
fof(fact_430_Euler_Oaux1,axiom,
! [A_19,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_19))
=> ( X != minus_minus_int(A_19,one_one_int)
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X),minus_minus_int(A_19,one_one_int))) ) ) ) ).
fof(fact_431_int__0__less__1,axiom,
hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),one_one_int)) ).
fof(fact_432_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,times_times_int(K,I)),times_times_int(K,J))) ) ) ).
fof(fact_433_odd__nonzero,axiom,
! [Z] : plus_plus_int(plus_plus_int(one_one_int,Z),Z) != zero_zero_int ).
fof(fact_434_less__bin__lemma,axiom,
! [K_1,L_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K_1),L_1))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,minus_minus_int(K_1,L_1)),zero_zero_int)) ) ).
fof(fact_435_mult__left__le__imp__le,axiom,
! [C_14,A_36,B_29] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,times_times_nat(C_14,A_36)),times_times_nat(C_14,B_29)))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),C_14))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_36),B_29)) ) ) ).
fof(fact_436_mult__left__le__imp__le,axiom,
! [C_14,A_36,B_29] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,times_times_int(C_14,A_36)),times_times_int(C_14,B_29)))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),C_14))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_36),B_29)) ) ) ).
fof(fact_437_mult__right__le__imp__le,axiom,
! [A_35,C_13,B_28] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,times_times_nat(A_35,C_13)),times_times_nat(B_28,C_13)))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),C_13))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_35),B_28)) ) ) ).
fof(fact_438_mult__right__le__imp__le,axiom,
! [A_35,C_13,B_28] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,times_times_int(A_35,C_13)),times_times_int(B_28,C_13)))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),C_13))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_35),B_28)) ) ) ).
fof(fact_439_mult__less__imp__less__left,axiom,
! [C_12,A_34,B_27] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,times_times_nat(C_12,A_34)),times_times_nat(C_12,B_27)))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),C_12))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_34),B_27)) ) ) ).
fof(fact_440_mult__less__imp__less__left,axiom,
! [C_12,A_34,B_27] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,times_times_int(C_12,A_34)),times_times_int(C_12,B_27)))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),C_12))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_34),B_27)) ) ) ).
fof(fact_441_mult__left__less__imp__less,axiom,
! [C_11,A_33,B_26] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,times_times_nat(C_11,A_33)),times_times_nat(C_11,B_26)))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),C_11))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_33),B_26)) ) ) ).
fof(fact_442_mult__left__less__imp__less,axiom,
! [C_11,A_33,B_26] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,times_times_int(C_11,A_33)),times_times_int(C_11,B_26)))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),C_11))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_33),B_26)) ) ) ).
fof(fact_443_mult__less__imp__less__right,axiom,
! [A_32,C_10,B_25] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,times_times_nat(A_32,C_10)),times_times_nat(B_25,C_10)))
=> ( 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_nat,A_32),B_25)) ) ) ).
fof(fact_444_mult__less__imp__less__right,axiom,
! [A_32,C_10,B_25] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,times_times_int(A_32,C_10)),times_times_int(B_25,C_10)))
=> ( 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_int,A_32),B_25)) ) ) ).
fof(fact_445_mult__right__less__imp__less,axiom,
! [A_31,C_9,B_24] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,times_times_nat(A_31,C_9)),times_times_nat(B_24,C_9)))
=> ( 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_nat,A_31),B_24)) ) ) ).
fof(fact_446_mult__right__less__imp__less,axiom,
! [A_31,C_9,B_24] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,times_times_int(A_31,C_9)),times_times_int(B_24,C_9)))
=> ( 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_int,A_31),B_24)) ) ) ).
fof(fact_447_mult__le__less__imp__less,axiom,
! [C_8,D_7,A_30,B_23] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_30),B_23))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,C_8),D_7))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),A_30))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),C_8))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,times_times_nat(A_30,C_8)),times_times_nat(B_23,D_7))) ) ) ) ) ).
fof(fact_448_mult__le__less__imp__less,axiom,
! [C_8,D_7,A_30,B_23] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_30),B_23))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,C_8),D_7))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_30))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),C_8))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,times_times_int(A_30,C_8)),times_times_int(B_23,D_7))) ) ) ) ) ).
fof(fact_449_mult__less__le__imp__less,axiom,
! [C_7,D_6,A_29,B_22] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_29),B_22))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,C_7),D_6))
=> ( 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_nat,zero_zero_nat),C_7))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,times_times_nat(A_29,C_7)),times_times_nat(B_22,D_6))) ) ) ) ) ).
fof(fact_450_mult__less__le__imp__less,axiom,
! [C_7,D_6,A_29,B_22] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_29),B_22))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,C_7),D_6))
=> ( 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_int,zero_zero_int),C_7))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,times_times_int(A_29,C_7)),times_times_int(B_22,D_6))) ) ) ) ) ).
fof(fact_451_mult__strict__mono_H,axiom,
! [C_6,D_5,A_28,B_21] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_28),B_21))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,C_6),D_5))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),A_28))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),C_6))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,times_times_nat(A_28,C_6)),times_times_nat(B_21,D_5))) ) ) ) ) ).
fof(fact_452_mult__strict__mono_H,axiom,
! [C_6,D_5,A_28,B_21] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_28),B_21))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,C_6),D_5))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_28))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),C_6))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,times_times_int(A_28,C_6)),times_times_int(B_21,D_5))) ) ) ) ) ).
fof(fact_453_mult__strict__mono,axiom,
! [C_5,D_4,A_27,B_20] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_27),B_20))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,C_5),D_4))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),B_20))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),C_5))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,times_times_nat(A_27,C_5)),times_times_nat(B_20,D_4))) ) ) ) ) ).
fof(fact_454_mult__strict__mono,axiom,
! [C_5,D_4,A_27,B_20] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_27),B_20))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,C_5),D_4))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_20))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),C_5))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,times_times_int(A_27,C_5)),times_times_int(B_20,D_4))) ) ) ) ) ).
fof(fact_455_mult__le__cancel__left__neg,axiom,
! [Aa,Ba,C_4] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,C_4),zero_zero_int))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,times_times_int(C_4,Aa)),times_times_int(C_4,Ba)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Ba),Aa)) ) ) ).
fof(fact_456_mult__le__cancel__left__pos,axiom,
! [Aa,Ba,C_4] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),C_4))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,times_times_int(C_4,Aa)),times_times_int(C_4,Ba)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Aa),Ba)) ) ) ).
fof(fact_457_add__nonpos__neg,axiom,
! [B_19,A_26] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_26),zero_zero_nat))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,B_19),zero_zero_nat))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,plus_plus_nat(A_26,B_19)),zero_zero_nat)) ) ) ).
fof(fact_458_add__nonpos__neg,axiom,
! [B_19,A_26] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_26),zero_zero_int))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_19),zero_zero_int))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,plus_plus_int(A_26,B_19)),zero_zero_int)) ) ) ).
fof(fact_459_add__neg__nonpos,axiom,
! [B_18,A_25] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_25),zero_zero_nat))
=> ( 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_nat,plus_plus_nat(A_25,B_18)),zero_zero_nat)) ) ) ).
fof(fact_460_add__neg__nonpos,axiom,
! [B_18,A_25] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_25),zero_zero_int))
=> ( 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_int,plus_plus_int(A_25,B_18)),zero_zero_int)) ) ) ).
fof(fact_461_add__strict__increasing2,axiom,
! [B_17,C_3,A_24] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),A_24))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,B_17),C_3))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,B_17),plus_plus_nat(A_24,C_3))) ) ) ).
fof(fact_462_add__strict__increasing2,axiom,
! [B_17,C_3,A_24] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_24))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_17),C_3))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_17),plus_plus_int(A_24,C_3))) ) ) ).
fof(fact_463_add__strict__increasing,axiom,
! [B_16,C_2,A_23] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),A_23))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,B_16),C_2))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,B_16),plus_plus_nat(A_23,C_2))) ) ) ).
fof(fact_464_add__strict__increasing,axiom,
! [B_16,C_2,A_23] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_23))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_16),C_2))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_16),plus_plus_int(A_23,C_2))) ) ) ).
fof(fact_465_add__nonneg__pos,axiom,
! [B_15,A_22] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),A_22))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),B_15))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),plus_plus_nat(A_22,B_15))) ) ) ).
fof(fact_466_add__nonneg__pos,axiom,
! [B_15,A_22] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_22))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_15))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),plus_plus_int(A_22,B_15))) ) ) ).
fof(fact_467_add__pos__nonneg,axiom,
! [B_14,A_21] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),A_21))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),B_14))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),plus_plus_nat(A_21,B_14))) ) ) ).
fof(fact_468_add__pos__nonneg,axiom,
! [B_14,A_21] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_21))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),B_14))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),plus_plus_int(A_21,B_14))) ) ) ).
fof(fact_469_mult__left__le__one__le,axiom,
! [Y_7,X_8] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_8))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Y_7))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y_7),one_one_int))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,times_times_int(Y_7,X_8)),X_8)) ) ) ) ).
fof(fact_470_mult__right__le__one__le,axiom,
! [Y_6,X_7] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_7))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Y_6))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y_6),one_one_int))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,times_times_int(X_7,Y_6)),X_7)) ) ) ) ).
fof(fact_471_zero__less__two,axiom,
hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),plus_plus_nat(one_one_nat,one_one_nat))) ).
fof(fact_472_zero__less__two,axiom,
hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),plus_plus_int(one_one_int,one_one_int))) ).
fof(fact_473_int__one__le__iff__zero__less,axiom,
! [Z_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,one_one_int),Z_1))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),Z_1)) ) ).
fof(fact_474_pos__zmult__eq__1__iff,axiom,
! [N_1,M_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),M_1))
=> ( times_times_int(M_1,N_1) = one_one_int
<=> ( M_1 = one_one_int
& N_1 = one_one_int ) ) ) ).
fof(fact_475_odd__less__0,axiom,
! [Z_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,plus_plus_int(plus_plus_int(one_one_int,Z_1),Z_1)),zero_zero_int))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Z_1),zero_zero_int)) ) ).
fof(fact_476_convex__bound__le,axiom,
! [V,U,Y_5,X_6,A_20] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_6),A_20))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y_5),A_20))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),U))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),V))
=> ( plus_plus_int(U,V) = one_one_int
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,plus_plus_int(times_times_int(U,X_6),times_times_int(V,Y_5))),A_20)) ) ) ) ) ) ).
fof(fact_477_le__imp__0__less,axiom,
! [Z] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Z))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),plus_plus_int(one_one_int,Z))) ) ).
fof(fact_478_self__quotient__aux1,axiom,
! [R_1,Q,A_19] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_19))
=> ( A_19 = plus_plus_int(R_1,times_times_int(A_19,Q))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,R_1),A_19))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,one_one_int),Q)) ) ) ) ).
fof(fact_479_self__quotient__aux2,axiom,
! [R_1,Q,A_19] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_19))
=> ( A_19 = plus_plus_int(R_1,times_times_int(A_19,Q))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),R_1))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Q),one_one_int)) ) ) ) ).
fof(fact_480_q__pos__lemma,axiom,
! [B_13,Q_1,R_2] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),plus_plus_int(times_times_int(B_13,Q_1),R_2)))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,R_2),B_13))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_13))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Q_1)) ) ) ) ).
fof(fact_481_q__neg__lemma,axiom,
! [B_13,Q_1,R_2] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,plus_plus_int(times_times_int(B_13,Q_1),R_2)),zero_zero_int))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),R_2))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_13))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Q_1),zero_zero_int)) ) ) ) ).
fof(fact_482_zdiv__mono2__neg__lemma,axiom,
! [B,Q,R_1,B_13,Q_1,R_2] :
( plus_plus_int(times_times_int(B,Q),R_1) = plus_plus_int(times_times_int(B_13,Q_1),R_2)
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,plus_plus_int(times_times_int(B_13,Q_1),R_2)),zero_zero_int))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,R_1),B))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),R_2))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_13))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_13),B))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Q_1),Q)) ) ) ) ) ) ) ).
fof(fact_483_unique__quotient__lemma__neg,axiom,
! [B,Q_1,R_2,Q,R_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,plus_plus_int(times_times_int(B,Q_1),R_2)),plus_plus_int(times_times_int(B,Q),R_1)))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,R_1),zero_zero_int))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B),R_1))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B),R_2))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Q),Q_1)) ) ) ) ) ).
fof(fact_484_zdiv__mono2__lemma,axiom,
! [B,Q,R_1,B_13,Q_1,R_2] :
( plus_plus_int(times_times_int(B,Q),R_1) = plus_plus_int(times_times_int(B_13,Q_1),R_2)
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),plus_plus_int(times_times_int(B_13,Q_1),R_2)))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,R_2),B_13))
=> ( 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_13))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_13),B))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Q),Q_1)) ) ) ) ) ) ) ).
fof(fact_485_unique__quotient__lemma,axiom,
! [B,Q_1,R_2,Q,R_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,plus_plus_int(times_times_int(B,Q_1),R_2)),plus_plus_int(times_times_int(B,Q),R_1)))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),R_2))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,R_2),B))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,R_1),B))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Q_1),Q)) ) ) ) ) ).
fof(fact_486_xzgcd__linear,axiom,
! [M,R_1,S,T,N] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),N))
=> ( xzgcd(M,N) = produc282740534nt_int(R_1,product_Pair_int_int(S,T))
=> R_1 = plus_plus_int(times_times_int(S,M),times_times_int(T,N)) ) ) ).
fof(fact_487_not__sum__squares__lt__zero,axiom,
! [X_5,Y_4] : ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,plus_plus_int(times_times_int(X_5,X_5),times_times_int(Y_4,Y_4))),zero_zero_int)) ).
fof(fact_488_sum__squares__gt__zero__iff,axiom,
! [X_1,Y_3] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),plus_plus_int(times_times_int(X_1,X_1),times_times_int(Y_3,Y_3))))
<=> ( X_1 != zero_zero_int
| Y_3 != zero_zero_int ) ) ).
fof(fact_489_sum__squares__eq__zero__iff,axiom,
! [X_1,Y_3] :
( plus_plus_int(times_times_int(X_1,X_1),times_times_int(Y_3,Y_3)) = zero_zero_int
<=> ( X_1 = zero_zero_int
& Y_3 = zero_zero_int ) ) ).
fof(fact_490_sum__squares__le__zero__iff,axiom,
! [X_1,Y_3] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,plus_plus_int(times_times_int(X_1,X_1),times_times_int(Y_3,Y_3))),zero_zero_int))
<=> ( X_1 = zero_zero_int
& Y_3 = zero_zero_int ) ) ).
fof(fact_491_sum__squares__ge__zero,axiom,
! [X_4,Y_2] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),plus_plus_int(times_times_int(X_4,X_4),times_times_int(Y_2,Y_2)))) ).
fof(fact_492_xzgcda__linear,axiom,
! [M,N,R_2,S_1,S,T_5,T,Rn,Sn,Tn,R_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),R_1))
=> ( xzgcda(M,N,R_2,R_1,S_1,S,T_5,T) = produc282740534nt_int(Rn,product_Pair_int_int(Sn,Tn))
=> ( R_2 = plus_plus_int(times_times_int(S_1,M),times_times_int(T_5,N))
=> ( R_1 = plus_plus_int(times_times_int(S,M),times_times_int(T,N))
=> Rn = plus_plus_int(times_times_int(Sn,M),times_times_int(Tn,N)) ) ) ) ) ).
fof(fact_493_Nat__Transfer_Otransfer__nat__int__function__closures_I1_J,axiom,
! [Y,X] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Y))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),plus_plus_int(X,Y))) ) ) ).
fof(fact_494_Nat__Transfer_Otransfer__nat__int__function__closures_I2_J,axiom,
! [Y,X] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Y))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),times_times_int(X,Y))) ) ) ).
fof(fact_495_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_496_xzgcd__def,axiom,
! [M,N] : xzgcd(M,N) = xzgcda(M,N,M,N,one_one_int,zero_zero_int,zero_zero_int,one_one_int) ).
fof(fact_497_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_498_pos__zmult__pos,axiom,
! [B,A_19] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_19))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),times_times_int(A_19,B)))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B)) ) ) ).
fof(fact_499_decr__mult__lemma,axiom,
! [K_1,P,D_3] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),D_3))
=> ( ! [X_3] :
( hBOOL(hAPP_int_bool(P,X_3))
=> hBOOL(hAPP_int_bool(P,minus_minus_int(X_3,D_3))) )
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),K_1))
=> ! [X_3] :
( hBOOL(hAPP_int_bool(P,X_3))
=> hBOOL(hAPP_int_bool(P,minus_minus_int(X_3,times_times_int(K_1,D_3)))) ) ) ) ) ).
fof(fact_500_incr__mult__lemma,axiom,
! [K_1,P,D_3] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),D_3))
=> ( ! [X_3] :
( hBOOL(hAPP_int_bool(P,X_3))
=> hBOOL(hAPP_int_bool(P,plus_plus_int(X_3,D_3))) )
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),K_1))
=> ! [X_3] :
( hBOOL(hAPP_int_bool(P,X_3))
=> hBOOL(hAPP_int_bool(P,plus_plus_int(X_3,times_times_int(K_1,D_3)))) ) ) ) ) ).
fof(fact_501_imp__le__cong,axiom,
! [P_1,P,X_1] :
( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_1))
=> ( hBOOL(P)
<=> hBOOL(P_1) ) )
=> ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_1))
=> hBOOL(P) )
<=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_1))
=> hBOOL(P_1) ) ) ) ).
fof(fact_502_conj__le__cong,axiom,
! [P_1,P,X_1] :
( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_1))
=> ( hBOOL(P)
<=> hBOOL(P_1) ) )
=> ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_1))
& hBOOL(P) )
<=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_1))
& hBOOL(P_1) ) ) ) ).
fof(fact_503_minf_I8_J,axiom,
! [T_4] :
? [Z_2] :
! [X_3] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_3),Z_2))
=> ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,T_4),X_3)) ) ).
fof(fact_504_minf_I8_J,axiom,
! [T_4] :
? [Z_2] :
! [X_3] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_3),Z_2))
=> ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,T_4),X_3)) ) ).
fof(fact_505_pinf_I8_J,axiom,
! [T_3] :
? [Z_2] :
! [X_3] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Z_2),X_3))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,T_3),X_3)) ) ).
fof(fact_506_pinf_I8_J,axiom,
! [T_3] :
? [Z_2] :
! [X_3] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Z_2),X_3))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,T_3),X_3)) ) ).
fof(fact_507_pinf_I6_J,axiom,
! [T_2] :
? [Z_2] :
! [X_3] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Z_2),X_3))
=> ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_3),T_2)) ) ).
fof(fact_508_pinf_I6_J,axiom,
! [T_2] :
? [Z_2] :
! [X_3] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Z_2),X_3))
=> ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_3),T_2)) ) ).
fof(fact_509_minf_I6_J,axiom,
! [T_1] :
? [Z_2] :
! [X_3] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_3),Z_2))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_3),T_1)) ) ).
fof(fact_510_minf_I6_J,axiom,
! [T_1] :
? [Z_2] :
! [X_3] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_3),Z_2))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_3),T_1)) ) ).
fof(fact_511_divmod__int__relI,axiom,
! [A_19,B,Q,R_1] :
( A_19 = plus_plus_int(times_times_int(B,Q),R_1)
=> ( ( ( 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_1))
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,R_1),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_1))
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,R_1),zero_zero_int)) ) ) )
=> hBOOL(hAPP_P603027463t_bool(divmod_int_rel(A_19,B),product_Pair_int_int(Q,R_1))) ) ) ).
fof(fact_512_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_513_self__remainder,axiom,
! [A_19,Q,R_1] :
( hBOOL(hAPP_P603027463t_bool(divmod_int_rel(A_19,A_19),product_Pair_int_int(Q,R_1)))
=> ( A_19 != zero_zero_int
=> R_1 = zero_zero_int ) ) ).
fof(fact_514_unique__quotient,axiom,
! [Q_1,R_2,A_19,B,Q,R_1] :
( hBOOL(hAPP_P603027463t_bool(divmod_int_rel(A_19,B),product_Pair_int_int(Q,R_1)))
=> ( hBOOL(hAPP_P603027463t_bool(divmod_int_rel(A_19,B),product_Pair_int_int(Q_1,R_2)))
=> ( B != zero_zero_int
=> Q = Q_1 ) ) ) ).
fof(fact_515_unique__remainder,axiom,
! [Q_1,R_2,A_19,B,Q,R_1] :
( hBOOL(hAPP_P603027463t_bool(divmod_int_rel(A_19,B),product_Pair_int_int(Q,R_1)))
=> ( hBOOL(hAPP_P603027463t_bool(divmod_int_rel(A_19,B),product_Pair_int_int(Q_1,R_2)))
=> ( B != zero_zero_int
=> R_1 = R_2 ) ) ) ).
fof(fact_516_self__quotient,axiom,
! [A_19,Q,R_1] :
( hBOOL(hAPP_P603027463t_bool(divmod_int_rel(A_19,A_19),product_Pair_int_int(Q,R_1)))
=> ( A_19 != zero_zero_int
=> Q = one_one_int ) ) ).
fof(fact_517_minusinfinity,axiom,
! [P,P1,D_3] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),D_3))
=> ( ! [X_3,K_2] :
( hBOOL(hAPP_int_bool(P1,X_3))
<=> hBOOL(hAPP_int_bool(P1,minus_minus_int(X_3,times_times_int(K_2,D_3)))) )
=> ( ? [Z_2] :
! [X_3] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_3),Z_2))
=> ( hBOOL(hAPP_int_bool(P,X_3))
<=> hBOOL(hAPP_int_bool(P1,X_3)) ) )
=> ( ? [X1] : hBOOL(hAPP_int_bool(P1,X1))
=> ? [X1] : hBOOL(hAPP_int_bool(P,X1)) ) ) ) ) ).
fof(fact_518_plusinfinity,axiom,
! [P,P_1,D_3] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),D_3))
=> ( ! [X_3,K_2] :
( hBOOL(hAPP_int_bool(P_1,X_3))
<=> hBOOL(hAPP_int_bool(P_1,minus_minus_int(X_3,times_times_int(K_2,D_3)))) )
=> ( ? [Z_2] :
! [X_3] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Z_2),X_3))
=> ( hBOOL(hAPP_int_bool(P,X_3))
<=> hBOOL(hAPP_int_bool(P_1,X_3)) ) )
=> ( ? [X1] : hBOOL(hAPP_int_bool(P_1,X1))
=> ? [X1] : hBOOL(hAPP_int_bool(P,X1)) ) ) ) ) ).
fof(fact_519_tsub__def,axiom,
! [Y,X] :
( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y),X))
=> nat_tsub(X,Y) = 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_520_zfact_Osimps,axiom,
! [N] :
( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,N),zero_zero_int))
=> zfact(N) = one_one_int )
& ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,N),zero_zero_int))
=> zfact(N) = times_times_int(N,zfact(minus_minus_int(N,one_one_int))) ) ) ).
fof(fact_521_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_522_tsub__eq,axiom,
! [Y,X] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y),X))
=> nat_tsub(X,Y) = minus_minus_int(X,Y) ) ).
fof(fact_523_posDivAlg__correct,axiom,
! [B,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_int,zero_zero_int),B))
=> hBOOL(hAPP_P603027463t_bool(divmod_int_rel(A_19,B),posDivAlg(A_19,B))) ) ) ).
fof(fact_524_decr__lemma,axiom,
! [X,Z,D] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),D))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,minus_minus_int(X,times_times_int(plus_plus_int(abs_abs_int(minus_minus_int(X,Z)),one_one_int),D))),Z)) ) ).
fof(fact_525_abs__eq__0,axiom,
! [Aa] :
( abs_abs_int(Aa) = zero_zero_int
<=> Aa = zero_zero_int ) ).
fof(fact_526_abs__zero,axiom,
abs_abs_int(zero_zero_int) = zero_zero_int ).
fof(fact_527_abs__of__nonneg,axiom,
! [A_18] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_18))
=> abs_abs_int(A_18) = A_18 ) ).
fof(fact_528_abs__le__zero__iff,axiom,
! [Aa] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,abs_abs_int(Aa)),zero_zero_int))
<=> Aa = zero_zero_int ) ).
fof(fact_529_abs__ge__zero,axiom,
! [A_17] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),abs_abs_int(A_17))) ).
fof(fact_530_abs__of__pos,axiom,
! [A_16] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_16))
=> abs_abs_int(A_16) = A_16 ) ).
fof(fact_531_zero__less__abs__iff,axiom,
! [Aa] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),abs_abs_int(Aa)))
<=> Aa != zero_zero_int ) ).
fof(fact_532_abs__not__less__zero,axiom,
! [A_15] : ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,abs_abs_int(A_15)),zero_zero_int)) ).
fof(fact_533_abs__mult__less,axiom,
! [B_12,D_2,A_14,C_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,abs_abs_int(A_14)),C_1))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,abs_abs_int(B_12)),D_2))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,times_times_int(abs_abs_int(A_14),abs_abs_int(B_12))),times_times_int(C_1,D_2))) ) ) ).
fof(fact_534_abs__triangle__ineq,axiom,
! [A_13,B_11] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,abs_abs_int(plus_plus_int(A_13,B_11))),plus_plus_int(abs_abs_int(A_13),abs_abs_int(B_11)))) ).
fof(fact_535_abs__triangle__ineq3,axiom,
! [A_12,B_10] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,abs_abs_int(minus_minus_int(abs_abs_int(A_12),abs_abs_int(B_10)))),abs_abs_int(minus_minus_int(A_12,B_10)))) ).
fof(fact_536_abs__triangle__ineq2,axiom,
! [A_11,B_9] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,minus_minus_int(abs_abs_int(A_11),abs_abs_int(B_9))),abs_abs_int(minus_minus_int(A_11,B_9)))) ).
fof(fact_537_abs__triangle__ineq2__sym,axiom,
! [A_10,B_8] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,minus_minus_int(abs_abs_int(A_10),abs_abs_int(B_8))),abs_abs_int(minus_minus_int(B_8,A_10)))) ).
fof(fact_538_abs__zmult__eq__1,axiom,
! [M,N] :
( abs_abs_int(times_times_int(M,N)) = one_one_int
=> abs_abs_int(M) = one_one_int ) ).
fof(fact_539_abs__ge__self,axiom,
! [A_9] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_9),abs_abs_int(A_9))) ).
fof(fact_540_abs__le__D1,axiom,
! [A_8,B_7] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,abs_abs_int(A_8)),B_7))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_8),B_7)) ) ).
fof(fact_541_abs__mult,axiom,
! [A_7,B_6] : abs_abs_int(times_times_int(A_7,B_6)) = times_times_int(abs_abs_int(A_7),abs_abs_int(B_6)) ).
fof(fact_542_abs__mult__self,axiom,
! [A_6] : times_times_int(abs_abs_int(A_6),abs_abs_int(A_6)) = times_times_int(A_6,A_6) ).
fof(fact_543_abs__add__abs,axiom,
! [A_5,B_5] : abs_abs_int(plus_plus_int(abs_abs_int(A_5),abs_abs_int(B_5))) = plus_plus_int(abs_abs_int(A_5),abs_abs_int(B_5)) ).
fof(fact_544_abs__one,axiom,
abs_abs_int(one_one_int) = one_one_int ).
fof(fact_545_abs__minus__commute,axiom,
! [A_4,B_4] : abs_abs_int(minus_minus_int(A_4,B_4)) = abs_abs_int(minus_minus_int(B_4,A_4)) ).
fof(fact_546_abs__idempotent,axiom,
! [A_3] : abs_abs_int(abs_abs_int(A_3)) = abs_abs_int(A_3) ).
fof(fact_547_abs__mult__pos,axiom,
! [Y_1,X_2] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_2))
=> times_times_int(abs_abs_int(Y_1),X_2) = abs_abs_int(times_times_int(Y_1,X_2)) ) ).
fof(fact_548_abs__eq__mult,axiom,
! [B_3,A_2] :
( ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_2))
| hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_2),zero_zero_int)) )
& ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),B_3))
| hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_3),zero_zero_int)) ) )
=> abs_abs_int(times_times_int(A_2,B_3)) = times_times_int(abs_abs_int(A_2),abs_abs_int(B_3)) ) ).
fof(fact_549_abs__diff__triangle__ineq,axiom,
! [A_1,B_2,C,D_1] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,abs_abs_int(minus_minus_int(plus_plus_int(A_1,B_2),plus_plus_int(C,D_1)))),plus_plus_int(abs_abs_int(minus_minus_int(A_1,C)),abs_abs_int(minus_minus_int(B_2,D_1))))) ).
fof(fact_550_abs__triangle__ineq4,axiom,
! [A,B_1] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,abs_abs_int(minus_minus_int(A,B_1))),plus_plus_int(abs_abs_int(A),abs_abs_int(B_1)))) ).
fof(fact_551_zabs__less__one__iff,axiom,
! [Z_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,abs_abs_int(Z_1)),one_one_int))
<=> Z_1 = zero_zero_int ) ).
fof(fact_552_posDivAlg__0,axiom,
! [B] : posDivAlg(zero_zero_int,B) = product_Pair_int_int(zero_zero_int,zero_zero_int) ).
fof(fact_553_incr__lemma,axiom,
! [Z,X,D] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),D))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Z),plus_plus_int(X,times_times_int(plus_plus_int(abs_abs_int(minus_minus_int(X,Z)),one_one_int),D)))) ) ).
fof(fact_554_abs__diff__less__iff,axiom,
! [X_1,Aa,R] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,abs_abs_int(minus_minus_int(X_1,Aa))),R))
<=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,minus_minus_int(Aa,R)),X_1))
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_1),plus_plus_int(Aa,R))) ) ) ).
fof(fact_555_pdivmod__posDivAlg,axiom,
! [K,L] :
( ( L = zero_zero_int
=> pdivmod(K,L) = product_Pair_int_int(zero_zero_int,abs_abs_int(K)) )
& ( L != zero_zero_int
=> pdivmod(K,L) = posDivAlg(abs_abs_int(K),abs_abs_int(L)) ) ) ).
fof(fact_556_nat0__intermed__int__val,axiom,
! [K_1,F,N_1] :
( ! [I_2] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_2),N_1))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,abs_abs_int(minus_minus_int(hAPP_nat_int(F,plus_plus_nat(I_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_1)))
=> ? [I_2] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_2),N_1))
& hAPP_nat_int(F,I_2) = K_1 ) ) ) ) ).
fof(fact_557_int__val__lemma,axiom,
! [K_1,F,N_1] :
( ! [I_2] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_2),N_1))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,abs_abs_int(minus_minus_int(hAPP_nat_int(F,plus_plus_nat(I_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_1)))
=> ? [I_2] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_2),N_1))
& hAPP_nat_int(F,I_2) = K_1 ) ) ) ) ).
fof(fact_558_mult__eq__if,axiom,
! [N,M] :
( ( M = zero_zero_nat
=> times_times_nat(M,N) = zero_zero_nat )
& ( M != zero_zero_nat
=> times_times_nat(M,N) = plus_plus_nat(N,times_times_nat(minus_minus_nat(M,one_one_nat),N)) ) ) ).
fof(fact_559_le0,axiom,
! [N] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),N)) ).
fof(fact_560_less__zeroE,axiom,
! [N] : ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N),zero_zero_nat)) ).
fof(fact_561_nat__mult__commute,axiom,
! [M,N] : times_times_nat(M,N) = times_times_nat(N,M) ).
fof(fact_562_nat__mult__assoc,axiom,
! [M,N,K] : times_times_nat(times_times_nat(M,N),K) = times_times_nat(M,times_times_nat(N,K)) ).
fof(fact_563_diff__mult__distrib,axiom,
! [M,N,K] : times_times_nat(minus_minus_nat(M,N),K) = minus_minus_nat(times_times_nat(M,K),times_times_nat(N,K)) ).
fof(fact_564_diff__commute,axiom,
! [I,J,K] : minus_minus_nat(minus_minus_nat(I,J),K) = minus_minus_nat(minus_minus_nat(I,K),J) ).
fof(fact_565_diff__mult__distrib2,axiom,
! [K,M,N] : times_times_nat(K,minus_minus_nat(M,N)) = minus_minus_nat(times_times_nat(K,M),times_times_nat(K,N)) ).
fof(fact_566_mult__0,axiom,
! [N] : times_times_nat(zero_zero_nat,N) = zero_zero_nat ).
fof(fact_567_mult__0__right,axiom,
! [M] : times_times_nat(M,zero_zero_nat) = zero_zero_nat ).
fof(fact_568_mult__is__0,axiom,
! [M_1,N_1] :
( times_times_nat(M_1,N_1) = zero_zero_nat
<=> ( M_1 = zero_zero_nat
| N_1 = zero_zero_nat ) ) ).
fof(fact_569_mult__cancel1,axiom,
! [K_1,M_1,N_1] :
( times_times_nat(K_1,M_1) = times_times_nat(K_1,N_1)
<=> ( M_1 = N_1
| K_1 = zero_zero_nat ) ) ).
fof(fact_570_mult__cancel2,axiom,
! [M_1,K_1,N_1] :
( times_times_nat(M_1,K_1) = times_times_nat(N_1,K_1)
<=> ( M_1 = N_1
| K_1 = zero_zero_nat ) ) ).
fof(fact_571_diffs0__imp__equal,axiom,
! [M,N] :
( minus_minus_nat(M,N) = zero_zero_nat
=> ( minus_minus_nat(N,M) = zero_zero_nat
=> M = N ) ) ).
fof(fact_572_diff__self__eq__0,axiom,
! [M] : minus_minus_nat(M,M) = zero_zero_nat ).
fof(fact_573_minus__nat_Odiff__0,axiom,
! [M] : minus_minus_nat(M,zero_zero_nat) = M ).
fof(fact_574_diff__0__eq__0,axiom,
! [N] : minus_minus_nat(zero_zero_nat,N) = zero_zero_nat ).
fof(fact_575_less__imp__diff__less,axiom,
! [N,J,K] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,J),K))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,minus_minus_nat(J,N)),K)) ) ).
fof(fact_576_diff__less__mono2,axiom,
! [L,M,N] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),L))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,minus_minus_nat(L,N)),minus_minus_nat(L,M))) ) ) ).
fof(fact_577_less__not__refl,axiom,
! [N] : ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N),N)) ).
fof(fact_578_nat__neq__iff,axiom,
! [M_1,N_1] :
( M_1 != N_1
<=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),N_1))
| hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_1),M_1)) ) ) ).
fof(fact_579_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_580_less__irrefl__nat,axiom,
! [N] : ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N),N)) ).
fof(fact_581_less__not__refl2,axiom,
! [N,M] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N),M))
=> M != N ) ).
fof(fact_582_less__not__refl3,axiom,
! [S,T] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,S),T))
=> S != T ) ).
fof(fact_583_nat__less__cases,axiom,
! [P,M_1,N_1] :
( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),N_1))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(P,N_1),M_1)) )
=> ( ( M_1 = N_1
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(P,N_1),M_1)) )
=> ( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_1),M_1))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(P,N_1),M_1)) )
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(P,N_1),M_1)) ) ) ) ).
fof(fact_584_nat__add__commute,axiom,
! [M,N] : plus_plus_nat(M,N) = plus_plus_nat(N,M) ).
fof(fact_585_nat__add__left__commute,axiom,
! [X,Y,Z] : plus_plus_nat(X,plus_plus_nat(Y,Z)) = plus_plus_nat(Y,plus_plus_nat(X,Z)) ).
fof(fact_586_add__mult__distrib2,axiom,
! [K,M,N] : times_times_nat(K,plus_plus_nat(M,N)) = plus_plus_nat(times_times_nat(K,M),times_times_nat(K,N)) ).
fof(fact_587_nat__add__assoc,axiom,
! [M,N,K] : plus_plus_nat(plus_plus_nat(M,N),K) = plus_plus_nat(M,plus_plus_nat(N,K)) ).
fof(fact_588_add__mult__distrib,axiom,
! [M,N,K] : times_times_nat(plus_plus_nat(M,N),K) = plus_plus_nat(times_times_nat(M,K),times_times_nat(N,K)) ).
fof(fact_589_nat__add__left__cancel,axiom,
! [K_1,M_1,N_1] :
( plus_plus_nat(K_1,M_1) = plus_plus_nat(K_1,N_1)
<=> M_1 = N_1 ) ).
fof(fact_590_nat__add__right__cancel,axiom,
! [M_1,K_1,N_1] :
( plus_plus_nat(M_1,K_1) = plus_plus_nat(N_1,K_1)
<=> M_1 = N_1 ) ).
fof(fact_591_diff__cancel2,axiom,
! [M,K,N] : minus_minus_nat(plus_plus_nat(M,K),plus_plus_nat(N,K)) = minus_minus_nat(M,N) ).
fof(fact_592_Nat_Odiff__cancel,axiom,
! [K,M,N] : minus_minus_nat(plus_plus_nat(K,M),plus_plus_nat(K,N)) = minus_minus_nat(M,N) ).
fof(fact_593_diff__diff__left,axiom,
! [I,J,K] : minus_minus_nat(minus_minus_nat(I,J),K) = minus_minus_nat(I,plus_plus_nat(J,K)) ).
fof(fact_594_diff__add__inverse,axiom,
! [N,M] : minus_minus_nat(plus_plus_nat(N,M),N) = M ).
fof(fact_595_diff__add__inverse2,axiom,
! [M,N] : minus_minus_nat(plus_plus_nat(M,N),N) = M ).
fof(fact_596_Nat_Odiff__le__self,axiom,
! [M,N] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,minus_minus_nat(M,N)),M)) ).
fof(fact_597_diff__le__mono2,axiom,
! [L,M,N] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,minus_minus_nat(L,N)),minus_minus_nat(L,M))) ) ).
fof(fact_598_diff__le__mono,axiom,
! [L,M,N] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,minus_minus_nat(M,L)),minus_minus_nat(N,L))) ) ).
fof(fact_599_diff__diff__cancel,axiom,
! [I,N] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I),N))
=> minus_minus_nat(N,minus_minus_nat(N,I)) = I ) ).
fof(fact_600_eq__diff__iff,axiom,
! [N_1,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_1))
=> ( minus_minus_nat(M_1,K_1) = minus_minus_nat(N_1,K_1)
<=> M_1 = N_1 ) ) ) ).
fof(fact_601_Nat_Odiff__diff__eq,axiom,
! [N,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))
=> minus_minus_nat(minus_minus_nat(M,K),minus_minus_nat(N,K)) = minus_minus_nat(M,N) ) ) ).
fof(fact_602_le__diff__iff,axiom,
! [N_1,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_1))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,minus_minus_nat(M_1,K_1)),minus_minus_nat(N_1,K_1)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_1),N_1)) ) ) ) ).
fof(fact_603_le__refl,axiom,
! [N] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N),N)) ).
fof(fact_604_nat__le__linear,axiom,
! [M,N] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N))
| hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N),M)) ) ).
fof(fact_605_eq__imp__le,axiom,
! [M,N] :
( M = N
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N)) ) ).
fof(fact_606_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_607_le__antisym,axiom,
! [M,N] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N),M))
=> M = N ) ) ).
fof(fact_608_mult__le__mono,axiom,
! [K,L,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))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,times_times_nat(I,K)),times_times_nat(J,L))) ) ) ).
fof(fact_609_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,times_times_nat(K,I)),times_times_nat(K,J))) ) ).
fof(fact_610_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,times_times_nat(I,K)),times_times_nat(J,K))) ) ).
fof(fact_611_le__cube,axiom,
! [M] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),times_times_nat(M,times_times_nat(M,M)))) ).
fof(fact_612_le__square,axiom,
! [M] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),times_times_nat(M,M))) ).
fof(fact_613_nat__mult__1,axiom,
! [N] : times_times_nat(one_one_nat,N) = N ).
fof(fact_614_nat__1__eq__mult__iff,axiom,
! [M_1,N_1] :
( one_one_nat = times_times_nat(M_1,N_1)
<=> ( M_1 = one_one_nat
& N_1 = one_one_nat ) ) ).
fof(fact_615_nat__mult__1__right,axiom,
! [N] : times_times_nat(N,one_one_nat) = N ).
fof(fact_616_nat__mult__eq__1__iff,axiom,
! [M_1,N_1] :
( times_times_nat(M_1,N_1) = one_one_nat
<=> ( M_1 = one_one_nat
& N_1 = one_one_nat ) ) ).
fof(fact_617_zero__less__diff,axiom,
! [N_1,M_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),minus_minus_nat(N_1,M_1)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),N_1)) ) ).
fof(fact_618_diff__less,axiom,
! [M,N] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),M))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,minus_minus_nat(M,N)),M)) ) ) ).
fof(fact_619_not__less0,axiom,
! [N] : ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N),zero_zero_nat)) ).
fof(fact_620_neq0__conv,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_621_less__nat__zero__code,axiom,
! [N] : ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N),zero_zero_nat)) ).
fof(fact_622_gr__implies__not0,axiom,
! [M,N] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N))
=> N != zero_zero_nat ) ).
fof(fact_623_gr0I,axiom,
! [N] :
( N != zero_zero_nat
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N)) ) ).
fof(fact_624_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,times_times_nat(K,I)),times_times_nat(K,J))) ) ) ).
fof(fact_625_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,times_times_nat(I,K)),times_times_nat(J,K))) ) ) ).
fof(fact_626_mult__less__cancel2,axiom,
! [M_1,K_1,N_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,times_times_nat(M_1,K_1)),times_times_nat(N_1,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_1)) ) ) ).
fof(fact_627_mult__less__cancel1,axiom,
! [K_1,M_1,N_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,times_times_nat(K_1,M_1)),times_times_nat(K_1,N_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_1)) ) ) ).
fof(fact_628_nat__0__less__mult__iff,axiom,
! [M_1,N_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),times_times_nat(M_1,N_1)))
<=> ( 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_1)) ) ) ).
fof(fact_629_plus__nat_Oadd__0,axiom,
! [N] : plus_plus_nat(zero_zero_nat,N) = N ).
fof(fact_630_Nat_Oadd__0__right,axiom,
! [M] : plus_plus_nat(M,zero_zero_nat) = M ).
fof(fact_631_add__is__0,axiom,
! [M_1,N_1] :
( plus_plus_nat(M_1,N_1) = zero_zero_nat
<=> ( M_1 = zero_zero_nat
& N_1 = zero_zero_nat ) ) ).
fof(fact_632_add__eq__self__zero,axiom,
! [M,N] :
( plus_plus_nat(M,N) = M
=> N = zero_zero_nat ) ).
fof(fact_633_diff__add__0,axiom,
! [N,M] : minus_minus_nat(N,plus_plus_nat(N,M)) = zero_zero_nat ).
fof(fact_634_diff__is__0__eq,axiom,
! [M_1,N_1] :
( minus_minus_nat(M_1,N_1) = zero_zero_nat
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_1),N_1)) ) ).
fof(fact_635_diff__is__0__eq_H,axiom,
! [M,N] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N))
=> minus_minus_nat(M,N) = zero_zero_nat ) ).
fof(fact_636_less__eq__nat_Osimps_I1_J,axiom,
! [N] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),N)) ).
fof(fact_637_le__0__eq,axiom,
! [N_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_1),zero_zero_nat))
<=> N_1 = zero_zero_nat ) ).
fof(fact_638_less__diff__conv,axiom,
! [I_1,J_1,K_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_1),minus_minus_nat(J_1,K_1)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,plus_plus_nat(I_1,K_1)),J_1)) ) ).
fof(fact_639_add__diff__inverse,axiom,
! [M,N] :
( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N))
=> plus_plus_nat(N,minus_minus_nat(M,N)) = M ) ).
fof(fact_640_add__lessD1,axiom,
! [I,J,K] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,plus_plus_nat(I,J)),K))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I),K)) ) ).
fof(fact_641_less__add__eq__less,axiom,
! [M,N,K,L] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,K),L))
=> ( plus_plus_nat(M,L) = plus_plus_nat(K,N)
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N)) ) ) ).
fof(fact_642_add__less__mono,axiom,
! [K,L,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))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,plus_plus_nat(I,K)),plus_plus_nat(J,L))) ) ) ).
fof(fact_643_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,plus_plus_nat(I,K)),plus_plus_nat(J,K))) ) ).
fof(fact_644_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),plus_plus_nat(M,J))) ) ).
fof(fact_645_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),plus_plus_nat(J,M))) ) ).
%----Conjectures (1)
fof(conj_0,conjecture,
times_times_int(twoSqu140629262sum2sq(product_Pair_int_int(a,b)),twoSqu140629262sum2sq(product_Pair_int_int(p,q))) = twoSqu140629262sum2sq(product_Pair_int_int(plus_plus_int(times_times_int(a,p),times_times_int(b,q)),minus_minus_int(times_times_int(a,q),times_times_int(b,p)))) ).
%------------------------------------------------------------------------------