TPTP Problem File: SWW470+3.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : SWW470+3 : TPTP v8.2.0. Released v5.3.0.
% Domain : Software Verification
% Problem : Hoare's Logic with Procedures line 200, 1000 axioms selected
% Version : Especial.
% English :
% Refs : [BN10] Boehme & Nipkow (2010), Sledgehammer: Judgement Day
% : [Bla11] Blanchette (2011), Email to Geoff Sutcliffe
% Source : [Bla11]
% Names : hoare_1000_fofmg_l200 [Bla11]
% Status : Theorem
% Rating : 0.56 v7.5.0, 0.69 v7.4.0, 0.63 v7.3.0, 0.62 v7.1.0, 0.52 v7.0.0, 0.63 v6.4.0, 0.65 v6.3.0, 0.58 v6.2.0, 0.72 v6.1.0, 0.83 v5.5.0, 0.93 v5.4.0, 0.96 v5.3.0
% Syntax : Number of formulae : 1399 ( 496 unt; 0 def)
% Number of atoms : 3162 (1143 equ)
% Maximal formula atoms : 9 ( 2 avg)
% Number of connectives : 2100 ( 337 ~; 66 |; 138 &)
% ( 203 <=>;1356 =>; 0 <=; 0 <~>)
% Maximal formula depth : 16 ( 5 avg)
% Maximal term depth : 11 ( 2 avg)
% Number of predicates : 4 ( 3 usr; 0 prp; 1-2 aty)
% Number of functors : 602 ( 602 usr; 231 con; 0-3 aty)
% Number of variables : 3856 (3776 !; 80 ?)
% SPC : FOF_THM_RFO_SEQ
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2011-08-09 17:22:57
% : Encoded with monomorphized guards.
%------------------------------------------------------------------------------
%----Explicit typings (35)
fof(gsy_c_Big__Operators_Osemilattice__big_000tc__Hoare____Mirabelle____uwgpyvfjxg__,axiom,
! [B_1_1,B_2_1] : is_bool(big_se1603268663iple_a(B_1_1,B_2_1)) ).
fof(gsy_c_Big__Operators_Osemilattice__big_000tc__Int__Oint,axiom,
! [B_1_1,B_2_1] : is_bool(big_se913005884ig_int(B_1_1,B_2_1)) ).
fof(gsy_c_Big__Operators_Osemilattice__big_000tc__Nat__Onat,axiom,
! [B_1_1,B_2_1] : is_bool(big_se275732192ig_nat(B_1_1,B_2_1)) ).
fof(gsy_c_Finite__Set_Ocomp__fun__commute_000tc__Hoare____Mirabelle____uwgpyvfjxg__O,axiom,
! [B_1_1] : is_bool(finite1428139793iple_a(B_1_1)) ).
fof(gsy_c_Finite__Set_Ocomp__fun__commute_000tc__Hoare____Mirabelle____uwgpyvfjxg__O_001,axiom,
! [B_1_1] : is_bool(finite2114894759a_bool(B_1_1)) ).
fof(gsy_c_Finite__Set_Ocomp__fun__commute_000tc__Int__Oint_000tc__Int__Oint,axiom,
! [B_1_1] : is_bool(finite1973466193nt_int(B_1_1)) ).
fof(gsy_c_Finite__Set_Ocomp__fun__commute_000tc__Int__Oint_000tc__fun_Itc__Int__Oint,axiom,
! [B_1_1] : is_bool(finite1321096241t_bool(B_1_1)) ).
fof(gsy_c_Finite__Set_Ocomp__fun__commute_000tc__Nat__Onat_000tc__Nat__Onat,axiom,
! [B_1_1] : is_bool(finite2130160977at_nat(B_1_1)) ).
fof(gsy_c_Finite__Set_Ocomp__fun__commute_000tc__Nat__Onat_000tc__fun_Itc__Nat__Onat,axiom,
! [B_1_1] : is_bool(finite1071749497t_bool(B_1_1)) ).
fof(gsy_c_Finite__Set_Ocomp__fun__idem_000tc__Hoare____Mirabelle____uwgpyvfjxg__Otri,axiom,
! [B_1_1] : is_bool(finite648855052iple_a(B_1_1)) ).
fof(gsy_c_Finite__Set_Ocomp__fun__idem_000tc__Hoare____Mirabelle____uwgpyvfjxg__Otri_002,axiom,
! [B_1_1] : is_bool(finite1803123436a_bool(B_1_1)) ).
fof(gsy_c_Finite__Set_Ocomp__fun__idem_000tc__Int__Oint_000tc__Int__Oint,axiom,
! [B_1_1] : is_bool(finite1704255308nt_int(B_1_1)) ).
fof(gsy_c_Finite__Set_Ocomp__fun__idem_000tc__Int__Oint_000tc__fun_Itc__Int__Oint_Mt,axiom,
! [B_1_1] : is_bool(finite58652534t_bool(B_1_1)) ).
fof(gsy_c_Finite__Set_Ocomp__fun__idem_000tc__Nat__Onat_000tc__Nat__Onat,axiom,
! [B_1_1] : is_bool(finite1860950092at_nat(B_1_1)) ).
fof(gsy_c_Finite__Set_Ocomp__fun__idem_000tc__Nat__Onat_000tc__fun_Itc__Nat__Onat_Mt,axiom,
! [B_1_1] : is_bool(finite1956789438t_bool(B_1_1)) ).
fof(gsy_c_Finite__Set_Ofolding__one_000tc__Hoare____Mirabelle____uwgpyvfjxg__Otriple,axiom,
! [B_1_1,B_2_1] : is_bool(finite1753440478iple_a(B_1_1,B_2_1)) ).
fof(gsy_c_Finite__Set_Ofolding__one_000tc__Int__Oint,axiom,
! [B_1_1,B_2_1] : is_bool(finite1626084323ne_int(B_1_1,B_2_1)) ).
fof(gsy_c_Finite__Set_Ofolding__one_000tc__Nat__Onat,axiom,
! [B_1_1,B_2_1] : is_bool(finite988810631ne_nat(B_1_1,B_2_1)) ).
fof(gsy_c_Finite__Set_Ofolding__one__idem_000tc__Hoare____Mirabelle____uwgpyvfjxg__O,axiom,
! [B_1_1,B_2_1] : is_bool(finite1033474011iple_a(B_1_1,B_2_1)) ).
fof(gsy_c_Finite__Set_Ofolding__one__idem_000tc__Int__Oint,axiom,
! [B_1_1,B_2_1] : is_bool(finite1432773856em_int(B_1_1,B_2_1)) ).
fof(gsy_c_Finite__Set_Ofolding__one__idem_000tc__Nat__Onat,axiom,
! [B_1_1,B_2_1] : is_bool(finite795500164em_nat(B_1_1,B_2_1)) ).
fof(gsy_c_Orderings_Obot__class_Obot_000tc__HOL__Obool,axiom,
is_bool(bot_bot_bool) ).
fof(gsy_c_fFalse,hypothesis,
is_bool(fFalse) ).
fof(gsy_c_fTrue,axiom,
is_bool(fTrue) ).
fof(gsy_c_hAPP_000tc__Com__Ostate_000tc__HOL__Obool,axiom,
! [B_1_1,B_2_1] : is_bool(hAPP_state_bool(B_1_1,B_2_1)) ).
fof(gsy_c_hAPP_000tc__HOL__Obool_000tc__HOL__Obool,axiom,
! [B_1_1,B_2_1] :
( is_bool(B_2_1)
=> is_bool(hAPP_bool_bool(B_1_1,B_2_1)) ) ).
fof(gsy_c_hAPP_000tc__Hoare____Mirabelle____uwgpyvfjxg__Otriple_It__a_J_000tc__HOL__,axiom,
! [B_1_1,B_2_1] : is_bool(hAPP_H1037229737a_bool(B_1_1,B_2_1)) ).
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__fun_Itc__Hoare____Mirabelle____uwgpyvfjxg__Otriple_It__a_J_Mtc,hypothesis,
! [B_1_1,B_2_1] : is_bool(hAPP_f1695230391l_bool(B_1_1,B_2_1)) ).
fof(gsy_c_hAPP_000tc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J_000tc__HOL__Obool,axiom,
! [B_1_1,B_2_1] : is_bool(hAPP_f448129468l_bool(B_1_1,B_2_1)) ).
fof(gsy_c_hAPP_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_000tc__HOL__Obool,axiom,
! [B_1_1,B_2_1] : is_bool(hAPP_f54304608l_bool(B_1_1,B_2_1)) ).
fof(gsy_c_hAPP_000tc__fun_Itc__fun_Itc__Hoare____Mirabelle____uwgpyvfjxg__Otriple_It,axiom,
! [B_1_1,B_2_1] : is_bool(hAPP_f2110825313l_bool(B_1_1,B_2_1)) ).
fof(gsy_c_hAPP_000tc__fun_Itc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J_Mtc__HOL__Obool_,axiom,
! [B_1_1,B_2_1] : is_bool(hAPP_f215623910l_bool(B_1_1,B_2_1)) ).
fof(gsy_c_hAPP_000tc__fun_Itc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_Mtc__HOL__Obool_,axiom,
! [B_1_1,B_2_1] : is_bool(hAPP_f1637334154l_bool(B_1_1,B_2_1)) ).
%----Relevant facts (1198)
fof(fact_0_empty,axiom,
! [Ga] : hBOOL(hAPP_f1695230391l_bool(hoare_2102800559rivs_a(Ga),bot_bo797238721a_bool)) ).
fof(fact_1_triple_Oinject,axiom,
! [Fun1_2,Com_2,Fun2_2,Fun1_1,Com_1,Fun2_1] :
( hoare_1916936827iple_a(Fun1_2,Com_2,Fun2_2) = hoare_1916936827iple_a(Fun1_1,Com_1,Fun2_1)
<=> ( Fun1_2 = Fun1_1
& Com_2 = Com_1
& Fun2_2 = Fun2_1 ) ) ).
fof(fact_2_cut,axiom,
! [Ga,G_1,Ts] :
( hBOOL(hAPP_f1695230391l_bool(hoare_2102800559rivs_a(G_1),Ts))
=> ( hBOOL(hAPP_f1695230391l_bool(hoare_2102800559rivs_a(Ga),G_1))
=> hBOOL(hAPP_f1695230391l_bool(hoare_2102800559rivs_a(Ga),Ts)) ) ) ).
fof(fact_3_hoare__derivs_Oinsert,axiom,
! [Ts,Ga,T_2] :
( hBOOL(hAPP_f1695230391l_bool(hoare_2102800559rivs_a(Ga),hAPP_f20753329a_bool(hAPP_H1743777351a_bool(insert956547291iple_a,T_2),bot_bo797238721a_bool)))
=> ( hBOOL(hAPP_f1695230391l_bool(hoare_2102800559rivs_a(Ga),Ts))
=> hBOOL(hAPP_f1695230391l_bool(hoare_2102800559rivs_a(Ga),hAPP_f20753329a_bool(hAPP_H1743777351a_bool(insert956547291iple_a,T_2),Ts))) ) ) ).
fof(fact_4_constant,axiom,
! [Ga,Pa,Ca,Q_1,C_5] :
( ( hBOOL(C_5)
=> hBOOL(hAPP_f1695230391l_bool(hoare_2102800559rivs_a(Ga),hAPP_f20753329a_bool(hAPP_H1743777351a_bool(insert956547291iple_a,hoare_1916936827iple_a(Pa,Ca,Q_1)),bot_bo797238721a_bool))) )
=> hBOOL(hAPP_f1695230391l_bool(hoare_2102800559rivs_a(Ga),hAPP_f20753329a_bool(hAPP_H1743777351a_bool(insert956547291iple_a,hoare_1916936827iple_a(hAPP_b540892988e_bool(hAPP_f1824947087e_bool(cOMBC_41962815e_bool,hAPP_f340725611e_bool(hAPP_f1006724181e_bool(cOMBB_1348041619bool_a,cOMBC_231445413l_bool),hAPP_f1509969235l_bool(hAPP_f1178339559l_bool(cOMBB_1355796797bool_a,hAPP_f1561913689l_bool(cOMBB_188601460_state,fconj)),Pa))),C_5),Ca,Q_1)),bot_bo797238721a_bool))) ) ).
fof(fact_5_escape,axiom,
! [Ga,Ca,Q_1,Pa] :
( ! [Z_11,S_2] :
( hBOOL(hAPP_state_bool(hAPP_a2036067514e_bool(Pa,Z_11),S_2))
=> hBOOL(hAPP_f1695230391l_bool(hoare_2102800559rivs_a(Ga),hAPP_f20753329a_bool(hAPP_H1743777351a_bool(insert956547291iple_a,hoare_1916936827iple_a(hAPP_f762886889e_bool(cOMBK_1458035955bool_a,hAPP_s1806633685e_bool(hAPP_f817621513e_bool(cOMBC_2027030106e_bool,fequal_state),S_2)),Ca,hAPP_f762886889e_bool(cOMBK_1458035955bool_a,hAPP_a2036067514e_bool(Q_1,Z_11)))),bot_bo797238721a_bool))) )
=> hBOOL(hAPP_f1695230391l_bool(hoare_2102800559rivs_a(Ga),hAPP_f20753329a_bool(hAPP_H1743777351a_bool(insert956547291iple_a,hoare_1916936827iple_a(Pa,Ca,Q_1)),bot_bo797238721a_bool))) ) ).
fof(fact_6_conseq2,axiom,
! [Q_1,Ga,Pa,Ca,Q_3] :
( hBOOL(hAPP_f1695230391l_bool(hoare_2102800559rivs_a(Ga),hAPP_f20753329a_bool(hAPP_H1743777351a_bool(insert956547291iple_a,hoare_1916936827iple_a(Pa,Ca,Q_3)),bot_bo797238721a_bool)))
=> ( ! [Z_11,S_2] :
( hBOOL(hAPP_state_bool(hAPP_a2036067514e_bool(Q_3,Z_11),S_2))
=> hBOOL(hAPP_state_bool(hAPP_a2036067514e_bool(Q_1,Z_11),S_2)) )
=> hBOOL(hAPP_f1695230391l_bool(hoare_2102800559rivs_a(Ga),hAPP_f20753329a_bool(hAPP_H1743777351a_bool(insert956547291iple_a,hoare_1916936827iple_a(Pa,Ca,Q_1)),bot_bo797238721a_bool))) ) ) ).
fof(fact_7_conseq1,axiom,
! [Pa,Ga,P_2,Ca,Q_1] :
( hBOOL(hAPP_f1695230391l_bool(hoare_2102800559rivs_a(Ga),hAPP_f20753329a_bool(hAPP_H1743777351a_bool(insert956547291iple_a,hoare_1916936827iple_a(P_2,Ca,Q_1)),bot_bo797238721a_bool)))
=> ( ! [Z_11,S_2] :
( hBOOL(hAPP_state_bool(hAPP_a2036067514e_bool(Pa,Z_11),S_2))
=> hBOOL(hAPP_state_bool(hAPP_a2036067514e_bool(P_2,Z_11),S_2)) )
=> hBOOL(hAPP_f1695230391l_bool(hoare_2102800559rivs_a(Ga),hAPP_f20753329a_bool(hAPP_H1743777351a_bool(insert956547291iple_a,hoare_1916936827iple_a(Pa,Ca,Q_1)),bot_bo797238721a_bool))) ) ) ).
fof(fact_8_conseq12,axiom,
! [Q_1,Pa,Ga,P_2,Ca,Q_3] :
( hBOOL(hAPP_f1695230391l_bool(hoare_2102800559rivs_a(Ga),hAPP_f20753329a_bool(hAPP_H1743777351a_bool(insert956547291iple_a,hoare_1916936827iple_a(P_2,Ca,Q_3)),bot_bo797238721a_bool)))
=> ( ! [Z_11,S_2] :
( hBOOL(hAPP_state_bool(hAPP_a2036067514e_bool(Pa,Z_11),S_2))
=> ! [S_3] :
( ! [Z_12] :
( hBOOL(hAPP_state_bool(hAPP_a2036067514e_bool(P_2,Z_12),S_2))
=> hBOOL(hAPP_state_bool(hAPP_a2036067514e_bool(Q_3,Z_12),S_3)) )
=> hBOOL(hAPP_state_bool(hAPP_a2036067514e_bool(Q_1,Z_11),S_3)) ) )
=> hBOOL(hAPP_f1695230391l_bool(hoare_2102800559rivs_a(Ga),hAPP_f20753329a_bool(hAPP_H1743777351a_bool(insert956547291iple_a,hoare_1916936827iple_a(Pa,Ca,Q_1)),bot_bo797238721a_bool))) ) ) ).
fof(fact_9_insertE,axiom,
! [A_13,Ba,A] :
( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_13),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,Ba),A)))
=> ( A_13 != Ba
=> hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_13),A)) ) ) ).
fof(fact_10_insertE,axiom,
! [A_13,Ba,A] :
( hBOOL(hAPP_f1695230391l_bool(hAPP_H622608077l_bool(member1797258804iple_a,A_13),hAPP_f20753329a_bool(hAPP_H1743777351a_bool(insert956547291iple_a,Ba),A)))
=> ( A_13 != Ba
=> hBOOL(hAPP_f1695230391l_bool(hAPP_H622608077l_bool(member1797258804iple_a,A_13),A)) ) ) ).
fof(fact_11_insertE,axiom,
! [A_13,Ba,A] :
( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,A_13),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,Ba),A)))
=> ( A_13 != Ba
=> hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,A_13),A)) ) ) ).
fof(fact_12_insertCI,axiom,
! [Ba,A_13,B] :
( ( ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_13),B))
=> A_13 = Ba )
=> hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_13),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,Ba),B))) ) ).
fof(fact_13_insertCI,axiom,
! [Ba,A_13,B] :
( ( ~ hBOOL(hAPP_f1695230391l_bool(hAPP_H622608077l_bool(member1797258804iple_a,A_13),B))
=> A_13 = Ba )
=> hBOOL(hAPP_f1695230391l_bool(hAPP_H622608077l_bool(member1797258804iple_a,A_13),hAPP_f20753329a_bool(hAPP_H1743777351a_bool(insert956547291iple_a,Ba),B))) ) ).
fof(fact_14_insertCI,axiom,
! [Ba,A_13,B] :
( ( ~ hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,A_13),B))
=> A_13 = Ba )
=> hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,A_13),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,Ba),B))) ) ).
fof(fact_15_emptyE,axiom,
! [A_13] : ~ hBOOL(hAPP_f1695230391l_bool(hAPP_H622608077l_bool(member1797258804iple_a,A_13),bot_bo797238721a_bool)) ).
fof(fact_16_emptyE,axiom,
! [A_13] : ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_13),bot_bot_fun_nat_bool)) ).
fof(fact_17_emptyE,axiom,
! [A_13] : ~ hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,A_13),bot_bot_fun_int_bool)) ).
fof(fact_18_singleton__conv2,axiom,
! [A_13] : hAPP_f800510211t_bool(collect_nat,hAPP_n1699378549t_bool(fequal_nat,A_13)) = hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_13),bot_bot_fun_nat_bool) ).
fof(fact_19_singleton__conv2,axiom,
! [A_13] : hAPP_f20753329a_bool(collec351493750iple_a,hAPP_H426895267a_bool(fequal963300192iple_a,A_13)) = hAPP_f20753329a_bool(hAPP_H1743777351a_bool(insert956547291iple_a,A_13),bot_bo797238721a_bool) ).
fof(fact_20_singleton__conv2,axiom,
! [A_13] : hAPP_f1805168059t_bool(collect_int,hAPP_i1948725293t_bool(fequal_int,A_13)) = hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,A_13),bot_bot_fun_int_bool) ).
fof(fact_21_singleton__conv,axiom,
! [A_13] : hAPP_f800510211t_bool(collect_nat,hAPP_n1699378549t_bool(hAPP_f229349961t_bool(cOMBC_nat_nat_bool,fequal_nat),A_13)) = hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_13),bot_bot_fun_nat_bool) ).
fof(fact_22_singleton__conv,axiom,
! [A_13] : hAPP_f20753329a_bool(collec351493750iple_a,hAPP_H426895267a_bool(hAPP_f920293029a_bool(cOMBC_1940896922a_bool,fequal963300192iple_a),A_13)) = hAPP_f20753329a_bool(hAPP_H1743777351a_bool(insert956547291iple_a,A_13),bot_bo797238721a_bool) ).
fof(fact_23_singleton__conv,axiom,
! [A_13] : hAPP_f1805168059t_bool(collect_int,hAPP_i1948725293t_bool(hAPP_f428220345t_bool(cOMBC_int_int_bool,fequal_int),A_13)) = hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,A_13),bot_bot_fun_int_bool) ).
fof(fact_24_Collect__conv__if2,axiom,
! [Pa,A_13] :
( ( hBOOL(hAPP_nat_bool(Pa,A_13))
=> hAPP_f800510211t_bool(collect_nat,hAPP_f800510211t_bool(hAPP_f561022312t_bool(cOMBS_nat_bool_bool,hAPP_f1146629647l_bool(hAPP_f1080886329l_bool(cOMBB_1015721476ol_nat,fconj),hAPP_n1699378549t_bool(fequal_nat,A_13))),Pa)) = hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_13),bot_bot_fun_nat_bool) )
& ( ~ hBOOL(hAPP_nat_bool(Pa,A_13))
=> hAPP_f800510211t_bool(collect_nat,hAPP_f800510211t_bool(hAPP_f561022312t_bool(cOMBS_nat_bool_bool,hAPP_f1146629647l_bool(hAPP_f1080886329l_bool(cOMBB_1015721476ol_nat,fconj),hAPP_n1699378549t_bool(fequal_nat,A_13))),Pa)) = bot_bot_fun_nat_bool ) ) ).
fof(fact_25_Collect__conv__if2,axiom,
! [Pa,A_13] :
( ( hBOOL(hAPP_H1037229737a_bool(Pa,A_13))
=> hAPP_f20753329a_bool(collec351493750iple_a,hAPP_f20753329a_bool(hAPP_f607497069a_bool(cOMBS_154988028l_bool,hAPP_f146754017l_bool(hAPP_f239971723l_bool(cOMBB_1866391387iple_a,fconj),hAPP_H426895267a_bool(fequal963300192iple_a,A_13))),Pa)) = hAPP_f20753329a_bool(hAPP_H1743777351a_bool(insert956547291iple_a,A_13),bot_bo797238721a_bool) )
& ( ~ hBOOL(hAPP_H1037229737a_bool(Pa,A_13))
=> hAPP_f20753329a_bool(collec351493750iple_a,hAPP_f20753329a_bool(hAPP_f607497069a_bool(cOMBS_154988028l_bool,hAPP_f146754017l_bool(hAPP_f239971723l_bool(cOMBB_1866391387iple_a,fconj),hAPP_H426895267a_bool(fequal963300192iple_a,A_13))),Pa)) = bot_bo797238721a_bool ) ) ).
fof(fact_26_Collect__conv__if2,axiom,
! [Pa,A_13] :
( ( hBOOL(hAPP_int_bool(Pa,A_13))
=> hAPP_f1805168059t_bool(collect_int,hAPP_f1805168059t_bool(hAPP_f727283836t_bool(cOMBS_int_bool_bool,hAPP_f2144054103l_bool(hAPP_f1734373249l_bool(cOMBB_1652995168ol_int,fconj),hAPP_i1948725293t_bool(fequal_int,A_13))),Pa)) = hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,A_13),bot_bot_fun_int_bool) )
& ( ~ hBOOL(hAPP_int_bool(Pa,A_13))
=> hAPP_f1805168059t_bool(collect_int,hAPP_f1805168059t_bool(hAPP_f727283836t_bool(cOMBS_int_bool_bool,hAPP_f2144054103l_bool(hAPP_f1734373249l_bool(cOMBB_1652995168ol_int,fconj),hAPP_i1948725293t_bool(fequal_int,A_13))),Pa)) = bot_bot_fun_int_bool ) ) ).
fof(fact_27_Collect__conv__if,axiom,
! [Pa,A_13] :
( ( hBOOL(hAPP_nat_bool(Pa,A_13))
=> hAPP_f800510211t_bool(collect_nat,hAPP_f800510211t_bool(hAPP_f561022312t_bool(cOMBS_nat_bool_bool,hAPP_f1146629647l_bool(hAPP_f1080886329l_bool(cOMBB_1015721476ol_nat,fconj),hAPP_n1699378549t_bool(hAPP_f229349961t_bool(cOMBC_nat_nat_bool,fequal_nat),A_13))),Pa)) = hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_13),bot_bot_fun_nat_bool) )
& ( ~ hBOOL(hAPP_nat_bool(Pa,A_13))
=> hAPP_f800510211t_bool(collect_nat,hAPP_f800510211t_bool(hAPP_f561022312t_bool(cOMBS_nat_bool_bool,hAPP_f1146629647l_bool(hAPP_f1080886329l_bool(cOMBB_1015721476ol_nat,fconj),hAPP_n1699378549t_bool(hAPP_f229349961t_bool(cOMBC_nat_nat_bool,fequal_nat),A_13))),Pa)) = bot_bot_fun_nat_bool ) ) ).
fof(fact_28_Collect__conv__if,axiom,
! [Pa,A_13] :
( ( hBOOL(hAPP_H1037229737a_bool(Pa,A_13))
=> hAPP_f20753329a_bool(collec351493750iple_a,hAPP_f20753329a_bool(hAPP_f607497069a_bool(cOMBS_154988028l_bool,hAPP_f146754017l_bool(hAPP_f239971723l_bool(cOMBB_1866391387iple_a,fconj),hAPP_H426895267a_bool(hAPP_f920293029a_bool(cOMBC_1940896922a_bool,fequal963300192iple_a),A_13))),Pa)) = hAPP_f20753329a_bool(hAPP_H1743777351a_bool(insert956547291iple_a,A_13),bot_bo797238721a_bool) )
& ( ~ hBOOL(hAPP_H1037229737a_bool(Pa,A_13))
=> hAPP_f20753329a_bool(collec351493750iple_a,hAPP_f20753329a_bool(hAPP_f607497069a_bool(cOMBS_154988028l_bool,hAPP_f146754017l_bool(hAPP_f239971723l_bool(cOMBB_1866391387iple_a,fconj),hAPP_H426895267a_bool(hAPP_f920293029a_bool(cOMBC_1940896922a_bool,fequal963300192iple_a),A_13))),Pa)) = bot_bo797238721a_bool ) ) ).
fof(fact_29_Collect__conv__if,axiom,
! [Pa,A_13] :
( ( hBOOL(hAPP_int_bool(Pa,A_13))
=> hAPP_f1805168059t_bool(collect_int,hAPP_f1805168059t_bool(hAPP_f727283836t_bool(cOMBS_int_bool_bool,hAPP_f2144054103l_bool(hAPP_f1734373249l_bool(cOMBB_1652995168ol_int,fconj),hAPP_i1948725293t_bool(hAPP_f428220345t_bool(cOMBC_int_int_bool,fequal_int),A_13))),Pa)) = hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,A_13),bot_bot_fun_int_bool) )
& ( ~ hBOOL(hAPP_int_bool(Pa,A_13))
=> hAPP_f1805168059t_bool(collect_int,hAPP_f1805168059t_bool(hAPP_f727283836t_bool(cOMBS_int_bool_bool,hAPP_f2144054103l_bool(hAPP_f1734373249l_bool(cOMBB_1652995168ol_int,fconj),hAPP_i1948725293t_bool(hAPP_f428220345t_bool(cOMBC_int_int_bool,fequal_int),A_13))),Pa)) = bot_bot_fun_int_bool ) ) ).
fof(fact_30_equals0D,axiom,
! [A_13,A] :
( A = bot_bo797238721a_bool
=> ~ hBOOL(hAPP_f1695230391l_bool(hAPP_H622608077l_bool(member1797258804iple_a,A_13),A)) ) ).
fof(fact_31_equals0D,axiom,
! [A_13,A] :
( A = bot_bot_fun_nat_bool
=> ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_13),A)) ) ).
fof(fact_32_equals0D,axiom,
! [A_13,A] :
( A = bot_bot_fun_int_bool
=> ~ hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,A_13),A)) ) ).
fof(fact_33_Collect__empty__eq,axiom,
! [Pa] :
( hAPP_f20753329a_bool(collec351493750iple_a,Pa) = bot_bo797238721a_bool
<=> ! [X_2] : ~ hBOOL(hAPP_H1037229737a_bool(Pa,X_2)) ) ).
fof(fact_34_Collect__empty__eq,axiom,
! [Pa] :
( hAPP_f1805168059t_bool(collect_int,Pa) = bot_bot_fun_int_bool
<=> ! [X_2] : ~ hBOOL(hAPP_int_bool(Pa,X_2)) ) ).
fof(fact_35_Collect__empty__eq,axiom,
! [Pa] :
( hAPP_f800510211t_bool(collect_nat,Pa) = bot_bot_fun_nat_bool
<=> ! [X_2] : ~ hBOOL(hAPP_nat_bool(Pa,X_2)) ) ).
fof(fact_36_empty__iff,axiom,
! [Ca] : ~ hBOOL(hAPP_f1695230391l_bool(hAPP_H622608077l_bool(member1797258804iple_a,Ca),bot_bo797238721a_bool)) ).
fof(fact_37_empty__iff,axiom,
! [Ca] : ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,Ca),bot_bot_fun_nat_bool)) ).
fof(fact_38_empty__iff,axiom,
! [Ca] : ~ hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Ca),bot_bot_fun_int_bool)) ).
fof(fact_39_empty__Collect__eq,axiom,
! [Pa] :
( bot_bo797238721a_bool = hAPP_f20753329a_bool(collec351493750iple_a,Pa)
<=> ! [X_2] : ~ hBOOL(hAPP_H1037229737a_bool(Pa,X_2)) ) ).
fof(fact_40_empty__Collect__eq,axiom,
! [Pa] :
( bot_bot_fun_int_bool = hAPP_f1805168059t_bool(collect_int,Pa)
<=> ! [X_2] : ~ hBOOL(hAPP_int_bool(Pa,X_2)) ) ).
fof(fact_41_empty__Collect__eq,axiom,
! [Pa] :
( bot_bot_fun_nat_bool = hAPP_f800510211t_bool(collect_nat,Pa)
<=> ! [X_2] : ~ hBOOL(hAPP_nat_bool(Pa,X_2)) ) ).
fof(fact_42_ex__in__conv,axiom,
! [A] :
( ? [X_2] : hBOOL(hAPP_f1695230391l_bool(hAPP_H622608077l_bool(member1797258804iple_a,X_2),A))
<=> A != bot_bo797238721a_bool ) ).
fof(fact_43_ex__in__conv,axiom,
! [A] :
( ? [X_2] : hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_2),A))
<=> A != bot_bot_fun_nat_bool ) ).
fof(fact_44_ex__in__conv,axiom,
! [A] :
( ? [X_2] : hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_2),A))
<=> A != bot_bot_fun_int_bool ) ).
fof(fact_45_all__not__in__conv,axiom,
! [A] :
( ! [X_2] : ~ hBOOL(hAPP_f1695230391l_bool(hAPP_H622608077l_bool(member1797258804iple_a,X_2),A))
<=> A = bot_bo797238721a_bool ) ).
fof(fact_46_all__not__in__conv,axiom,
! [A] :
( ! [X_2] : ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_2),A))
<=> A = bot_bot_fun_nat_bool ) ).
fof(fact_47_all__not__in__conv,axiom,
! [A] :
( ! [X_2] : ~ hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_2),A))
<=> A = bot_bot_fun_int_bool ) ).
fof(fact_48_empty__def,axiom,
bot_bo797238721a_bool = hAPP_f20753329a_bool(collec351493750iple_a,hAPP_b1882817719a_bool(cOMBK_235286536iple_a,fFalse)) ).
fof(fact_49_empty__def,axiom,
bot_bot_fun_int_bool = hAPP_f1805168059t_bool(collect_int,hAPP_b396694332t_bool(cOMBK_bool_int,fFalse)) ).
fof(fact_50_empty__def,axiom,
bot_bot_fun_nat_bool = hAPP_f800510211t_bool(collect_nat,hAPP_b1013836512t_bool(cOMBK_bool_nat,fFalse)) ).
fof(fact_51_insert__absorb,axiom,
! [A_13,A] :
( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_13),A))
=> hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_13),A) = A ) ).
fof(fact_52_insert__absorb,axiom,
! [A_13,A] :
( hBOOL(hAPP_f1695230391l_bool(hAPP_H622608077l_bool(member1797258804iple_a,A_13),A))
=> hAPP_f20753329a_bool(hAPP_H1743777351a_bool(insert956547291iple_a,A_13),A) = A ) ).
fof(fact_53_insert__absorb,axiom,
! [A_13,A] :
( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,A_13),A))
=> hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,A_13),A) = A ) ).
fof(fact_54_insertI2,axiom,
! [Ba,A_13,B] :
( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_13),B))
=> hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_13),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,Ba),B))) ) ).
fof(fact_55_insertI2,axiom,
! [Ba,A_13,B] :
( hBOOL(hAPP_f1695230391l_bool(hAPP_H622608077l_bool(member1797258804iple_a,A_13),B))
=> hBOOL(hAPP_f1695230391l_bool(hAPP_H622608077l_bool(member1797258804iple_a,A_13),hAPP_f20753329a_bool(hAPP_H1743777351a_bool(insert956547291iple_a,Ba),B))) ) ).
fof(fact_56_insertI2,axiom,
! [Ba,A_13,B] :
( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,A_13),B))
=> hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,A_13),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,Ba),B))) ) ).
fof(fact_57_insert__ident,axiom,
! [B,X_1,A] :
( ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A))
=> ( ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),B))
=> ( hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),A) = hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),B)
<=> A = B ) ) ) ).
fof(fact_58_insert__ident,axiom,
! [B,X_1,A] :
( ~ hBOOL(hAPP_f1695230391l_bool(hAPP_H622608077l_bool(member1797258804iple_a,X_1),A))
=> ( ~ hBOOL(hAPP_f1695230391l_bool(hAPP_H622608077l_bool(member1797258804iple_a,X_1),B))
=> ( hAPP_f20753329a_bool(hAPP_H1743777351a_bool(insert956547291iple_a,X_1),A) = hAPP_f20753329a_bool(hAPP_H1743777351a_bool(insert956547291iple_a,X_1),B)
<=> A = B ) ) ) ).
fof(fact_59_insert__ident,axiom,
! [B,X_1,A] :
( ~ hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),A))
=> ( ~ hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),B))
=> ( hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),A) = hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),B)
<=> A = B ) ) ) ).
fof(fact_60_insert__code,axiom,
! [Y_1,A,X_1] :
( hBOOL(hAPP_nat_bool(hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,Y_1),A),X_1))
<=> ( Y_1 = X_1
| hBOOL(hAPP_nat_bool(A,X_1)) ) ) ).
fof(fact_61_insert__code,axiom,
! [Y_1,A,X_1] :
( hBOOL(hAPP_H1037229737a_bool(hAPP_f20753329a_bool(hAPP_H1743777351a_bool(insert956547291iple_a,Y_1),A),X_1))
<=> ( Y_1 = X_1
| hBOOL(hAPP_H1037229737a_bool(A,X_1)) ) ) ).
fof(fact_62_insert__code,axiom,
! [Y_1,A,X_1] :
( hBOOL(hAPP_int_bool(hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,Y_1),A),X_1))
<=> ( Y_1 = X_1
| hBOOL(hAPP_int_bool(A,X_1)) ) ) ).
fof(fact_63_insert__iff,axiom,
! [A_13,Ba,A] :
( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_13),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,Ba),A)))
<=> ( A_13 = Ba
| hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_13),A)) ) ) ).
fof(fact_64_insert__iff,axiom,
! [A_13,Ba,A] :
( hBOOL(hAPP_f1695230391l_bool(hAPP_H622608077l_bool(member1797258804iple_a,A_13),hAPP_f20753329a_bool(hAPP_H1743777351a_bool(insert956547291iple_a,Ba),A)))
<=> ( A_13 = Ba
| hBOOL(hAPP_f1695230391l_bool(hAPP_H622608077l_bool(member1797258804iple_a,A_13),A)) ) ) ).
fof(fact_65_insert__iff,axiom,
! [A_13,Ba,A] :
( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,A_13),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,Ba),A)))
<=> ( A_13 = Ba
| hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,A_13),A)) ) ) ).
fof(fact_66_insert__commute,axiom,
! [X_1,Y_1,A] : hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,Y_1),A)) = hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,Y_1),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),A)) ).
fof(fact_67_insert__commute,axiom,
! [X_1,Y_1,A] : hAPP_f20753329a_bool(hAPP_H1743777351a_bool(insert956547291iple_a,X_1),hAPP_f20753329a_bool(hAPP_H1743777351a_bool(insert956547291iple_a,Y_1),A)) = hAPP_f20753329a_bool(hAPP_H1743777351a_bool(insert956547291iple_a,Y_1),hAPP_f20753329a_bool(hAPP_H1743777351a_bool(insert956547291iple_a,X_1),A)) ).
fof(fact_68_insert__commute,axiom,
! [X_1,Y_1,A] : hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,Y_1),A)) = hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,Y_1),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),A)) ).
fof(fact_69_insert__absorb2,axiom,
! [X_1,A] : hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),A)) = hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),A) ).
fof(fact_70_insert__absorb2,axiom,
! [X_1,A] : hAPP_f20753329a_bool(hAPP_H1743777351a_bool(insert956547291iple_a,X_1),hAPP_f20753329a_bool(hAPP_H1743777351a_bool(insert956547291iple_a,X_1),A)) = hAPP_f20753329a_bool(hAPP_H1743777351a_bool(insert956547291iple_a,X_1),A) ).
fof(fact_71_insert__absorb2,axiom,
! [X_1,A] : hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),A)) = hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),A) ).
fof(fact_72_insert__Collect,axiom,
! [A_13,Pa] : hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_13),hAPP_f800510211t_bool(collect_nat,Pa)) = hAPP_f800510211t_bool(collect_nat,hAPP_f800510211t_bool(hAPP_f561022312t_bool(cOMBS_nat_bool_bool,hAPP_f1146629647l_bool(hAPP_f1080886329l_bool(cOMBB_1015721476ol_nat,fimplies),hAPP_f800510211t_bool(hAPP_f894608603t_bool(cOMBB_bool_bool_nat,fNot),hAPP_n1699378549t_bool(hAPP_f229349961t_bool(cOMBC_nat_nat_bool,fequal_nat),A_13)))),Pa)) ).
fof(fact_73_insert__Collect,axiom,
! [A_13,Pa] : hAPP_f20753329a_bool(hAPP_H1743777351a_bool(insert956547291iple_a,A_13),hAPP_f20753329a_bool(collec351493750iple_a,Pa)) = hAPP_f20753329a_bool(collec351493750iple_a,hAPP_f20753329a_bool(hAPP_f607497069a_bool(cOMBS_154988028l_bool,hAPP_f146754017l_bool(hAPP_f239971723l_bool(cOMBB_1866391387iple_a,fimplies),hAPP_f20753329a_bool(hAPP_f604201481a_bool(cOMBB_1882975613iple_a,fNot),hAPP_H426895267a_bool(hAPP_f920293029a_bool(cOMBC_1940896922a_bool,fequal963300192iple_a),A_13)))),Pa)) ).
fof(fact_74_insert__Collect,axiom,
! [A_13,Pa] : hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,A_13),hAPP_f1805168059t_bool(collect_int,Pa)) = hAPP_f1805168059t_bool(collect_int,hAPP_f1805168059t_bool(hAPP_f727283836t_bool(cOMBS_int_bool_bool,hAPP_f2144054103l_bool(hAPP_f1734373249l_bool(cOMBB_1652995168ol_int,fimplies),hAPP_f1805168059t_bool(hAPP_f627970963t_bool(cOMBB_bool_bool_int,fNot),hAPP_i1948725293t_bool(hAPP_f428220345t_bool(cOMBC_int_int_bool,fequal_int),A_13)))),Pa)) ).
fof(fact_75_insert__compr,axiom,
! [A_13,B] : hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_13),B) = hAPP_f800510211t_bool(collect_nat,hAPP_f800510211t_bool(hAPP_f561022312t_bool(cOMBS_nat_bool_bool,hAPP_f1146629647l_bool(hAPP_f1080886329l_bool(cOMBB_1015721476ol_nat,fdisj),hAPP_n1699378549t_bool(hAPP_f229349961t_bool(cOMBC_nat_nat_bool,fequal_nat),A_13))),hAPP_f800510211t_bool(hAPP_f1722879237t_bool(cOMBC_226598744l_bool,member_nat),B))) ).
fof(fact_76_insert__compr,axiom,
! [A_13,B] : hAPP_f20753329a_bool(hAPP_H1743777351a_bool(insert956547291iple_a,A_13),B) = hAPP_f20753329a_bool(collec351493750iple_a,hAPP_f20753329a_bool(hAPP_f607497069a_bool(cOMBS_154988028l_bool,hAPP_f146754017l_bool(hAPP_f239971723l_bool(cOMBB_1866391387iple_a,fdisj),hAPP_H426895267a_bool(hAPP_f920293029a_bool(cOMBC_1940896922a_bool,fequal963300192iple_a),A_13))),hAPP_f20753329a_bool(hAPP_f247708275a_bool(cOMBC_1519416966l_bool,member1797258804iple_a),B))) ).
fof(fact_77_insert__compr,axiom,
! [A_13,B] : hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,A_13),B) = hAPP_f1805168059t_bool(collect_int,hAPP_f1805168059t_bool(hAPP_f727283836t_bool(cOMBS_int_bool_bool,hAPP_f2144054103l_bool(hAPP_f1734373249l_bool(cOMBB_1652995168ol_int,fdisj),hAPP_i1948725293t_bool(hAPP_f428220345t_bool(cOMBC_int_int_bool,fequal_int),A_13))),hAPP_f1805168059t_bool(hAPP_f202917053t_bool(cOMBC_94739984l_bool,member_int),B))) ).
fof(fact_78_insertI1,axiom,
! [A_13,B] : hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_13),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_13),B))) ).
fof(fact_79_insertI1,axiom,
! [A_13,B] : hBOOL(hAPP_f1695230391l_bool(hAPP_H622608077l_bool(member1797258804iple_a,A_13),hAPP_f20753329a_bool(hAPP_H1743777351a_bool(insert956547291iple_a,A_13),B))) ).
fof(fact_80_insertI1,axiom,
! [A_13,B] : hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,A_13),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,A_13),B))) ).
fof(fact_81_insert__compr__raw,axiom,
! [X_2,Xa] : hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_2),Xa) = hAPP_f800510211t_bool(collect_nat,hAPP_f800510211t_bool(hAPP_f561022312t_bool(cOMBS_nat_bool_bool,hAPP_f1146629647l_bool(hAPP_f1080886329l_bool(cOMBB_1015721476ol_nat,fdisj),hAPP_n1699378549t_bool(hAPP_f229349961t_bool(cOMBC_nat_nat_bool,fequal_nat),X_2))),hAPP_f800510211t_bool(hAPP_f1722879237t_bool(cOMBC_226598744l_bool,member_nat),Xa))) ).
fof(fact_82_insert__compr__raw,axiom,
! [X_2,Xa] : hAPP_f20753329a_bool(hAPP_H1743777351a_bool(insert956547291iple_a,X_2),Xa) = hAPP_f20753329a_bool(collec351493750iple_a,hAPP_f20753329a_bool(hAPP_f607497069a_bool(cOMBS_154988028l_bool,hAPP_f146754017l_bool(hAPP_f239971723l_bool(cOMBB_1866391387iple_a,fdisj),hAPP_H426895267a_bool(hAPP_f920293029a_bool(cOMBC_1940896922a_bool,fequal963300192iple_a),X_2))),hAPP_f20753329a_bool(hAPP_f247708275a_bool(cOMBC_1519416966l_bool,member1797258804iple_a),Xa))) ).
fof(fact_83_insert__compr__raw,axiom,
! [X_2,Xa] : hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_2),Xa) = hAPP_f1805168059t_bool(collect_int,hAPP_f1805168059t_bool(hAPP_f727283836t_bool(cOMBS_int_bool_bool,hAPP_f2144054103l_bool(hAPP_f1734373249l_bool(cOMBB_1652995168ol_int,fdisj),hAPP_i1948725293t_bool(hAPP_f428220345t_bool(cOMBC_int_int_bool,fequal_int),X_2))),hAPP_f1805168059t_bool(hAPP_f202917053t_bool(cOMBC_94739984l_bool,member_int),Xa))) ).
fof(fact_84_singleton__inject,axiom,
! [A_13,Ba] :
( hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_13),bot_bot_fun_nat_bool) = hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,Ba),bot_bot_fun_nat_bool)
=> A_13 = Ba ) ).
fof(fact_85_singleton__inject,axiom,
! [A_13,Ba] :
( hAPP_f20753329a_bool(hAPP_H1743777351a_bool(insert956547291iple_a,A_13),bot_bo797238721a_bool) = hAPP_f20753329a_bool(hAPP_H1743777351a_bool(insert956547291iple_a,Ba),bot_bo797238721a_bool)
=> A_13 = Ba ) ).
fof(fact_86_singleton__inject,axiom,
! [A_13,Ba] :
( hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,A_13),bot_bot_fun_int_bool) = hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,Ba),bot_bot_fun_int_bool)
=> A_13 = Ba ) ).
fof(fact_87_singletonE,axiom,
! [Ba,A_13] :
( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,Ba),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_13),bot_bot_fun_nat_bool)))
=> Ba = A_13 ) ).
fof(fact_88_singletonE,axiom,
! [Ba,A_13] :
( hBOOL(hAPP_f1695230391l_bool(hAPP_H622608077l_bool(member1797258804iple_a,Ba),hAPP_f20753329a_bool(hAPP_H1743777351a_bool(insert956547291iple_a,A_13),bot_bo797238721a_bool)))
=> Ba = A_13 ) ).
fof(fact_89_singletonE,axiom,
! [Ba,A_13] :
( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Ba),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,A_13),bot_bot_fun_int_bool)))
=> Ba = A_13 ) ).
fof(fact_90_doubleton__eq__iff,axiom,
! [A_13,Ba,Ca,D_1] :
( hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_13),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,Ba),bot_bot_fun_nat_bool)) = hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,Ca),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,D_1),bot_bot_fun_nat_bool))
<=> ( ( A_13 = Ca
& Ba = D_1 )
| ( A_13 = D_1
& Ba = Ca ) ) ) ).
fof(fact_91_doubleton__eq__iff,axiom,
! [A_13,Ba,Ca,D_1] :
( hAPP_f20753329a_bool(hAPP_H1743777351a_bool(insert956547291iple_a,A_13),hAPP_f20753329a_bool(hAPP_H1743777351a_bool(insert956547291iple_a,Ba),bot_bo797238721a_bool)) = hAPP_f20753329a_bool(hAPP_H1743777351a_bool(insert956547291iple_a,Ca),hAPP_f20753329a_bool(hAPP_H1743777351a_bool(insert956547291iple_a,D_1),bot_bo797238721a_bool))
<=> ( ( A_13 = Ca
& Ba = D_1 )
| ( A_13 = D_1
& Ba = Ca ) ) ) ).
fof(fact_92_doubleton__eq__iff,axiom,
! [A_13,Ba,Ca,D_1] :
( hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,A_13),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,Ba),bot_bot_fun_int_bool)) = hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,Ca),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,D_1),bot_bot_fun_int_bool))
<=> ( ( A_13 = Ca
& Ba = D_1 )
| ( A_13 = D_1
& Ba = Ca ) ) ) ).
fof(fact_93_singleton__iff,axiom,
! [Ba,A_13] :
( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,Ba),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_13),bot_bot_fun_nat_bool)))
<=> Ba = A_13 ) ).
fof(fact_94_singleton__iff,axiom,
! [Ba,A_13] :
( hBOOL(hAPP_f1695230391l_bool(hAPP_H622608077l_bool(member1797258804iple_a,Ba),hAPP_f20753329a_bool(hAPP_H1743777351a_bool(insert956547291iple_a,A_13),bot_bo797238721a_bool)))
<=> Ba = A_13 ) ).
fof(fact_95_singleton__iff,axiom,
! [Ba,A_13] :
( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Ba),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,A_13),bot_bot_fun_int_bool)))
<=> Ba = A_13 ) ).
fof(fact_96_insert__not__empty,axiom,
! [A_13,A] : hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_13),A) != bot_bot_fun_nat_bool ).
fof(fact_97_insert__not__empty,axiom,
! [A_13,A] : hAPP_f20753329a_bool(hAPP_H1743777351a_bool(insert956547291iple_a,A_13),A) != bot_bo797238721a_bool ).
fof(fact_98_insert__not__empty,axiom,
! [A_13,A] : hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,A_13),A) != bot_bot_fun_int_bool ).
fof(fact_99_empty__not__insert,axiom,
! [A_13,A] : bot_bot_fun_nat_bool != hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_13),A) ).
fof(fact_100_empty__not__insert,axiom,
! [A_13,A] : bot_bo797238721a_bool != hAPP_f20753329a_bool(hAPP_H1743777351a_bool(insert956547291iple_a,A_13),A) ).
fof(fact_101_empty__not__insert,axiom,
! [A_13,A] : bot_bot_fun_int_bool != hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,A_13),A) ).
fof(fact_102_the__elem__eq,axiom,
! [X_1] : hAPP_f22106695ol_nat(the_elem_nat,hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),bot_bot_fun_nat_bool)) = X_1 ).
fof(fact_103_the__elem__eq,axiom,
! [X_1] : hAPP_f1693662087iple_a(the_el1519802624iple_a,hAPP_f20753329a_bool(hAPP_H1743777351a_bool(insert956547291iple_a,X_1),bot_bo797238721a_bool)) = X_1 ).
fof(fact_104_the__elem__eq,axiom,
! [X_1] : hAPP_f1594865479ol_int(the_elem_int,hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),bot_bot_fun_int_bool)) = X_1 ).
fof(fact_105_bot__apply,axiom,
! [X_1] :
( hBOOL(hAPP_H1037229737a_bool(bot_bo797238721a_bool,X_1))
<=> hBOOL(bot_bot_bool) ) ).
fof(fact_106_bot__apply,axiom,
! [X_1] :
( hBOOL(hAPP_int_bool(bot_bot_fun_int_bool,X_1))
<=> hBOOL(bot_bot_bool) ) ).
fof(fact_107_bot__apply,axiom,
! [X_1] :
( hBOOL(hAPP_nat_bool(bot_bot_fun_nat_bool,X_1))
<=> hBOOL(bot_bot_bool) ) ).
fof(fact_108_bot__fun__def,axiom,
! [X_2] :
( hBOOL(hAPP_H1037229737a_bool(bot_bo797238721a_bool,X_2))
<=> hBOOL(bot_bot_bool) ) ).
fof(fact_109_bot__fun__def,axiom,
! [X_2] :
( hBOOL(hAPP_int_bool(bot_bot_fun_int_bool,X_2))
<=> hBOOL(bot_bot_bool) ) ).
fof(fact_110_bot__fun__def,axiom,
! [X_2] :
( hBOOL(hAPP_nat_bool(bot_bot_fun_nat_bool,X_2))
<=> hBOOL(bot_bot_bool) ) ).
fof(fact_111_hoare__derivs_OSkip,axiom,
! [Ga,Pa] : hBOOL(hAPP_f1695230391l_bool(hoare_2102800559rivs_a(Ga),hAPP_f20753329a_bool(hAPP_H1743777351a_bool(insert956547291iple_a,hoare_1916936827iple_a(Pa,skip,Pa)),bot_bo797238721a_bool))) ).
fof(fact_112_Comp,axiom,
! [D_1,R_1,Ga,Pa,Ca,Q_1] :
( hBOOL(hAPP_f1695230391l_bool(hoare_2102800559rivs_a(Ga),hAPP_f20753329a_bool(hAPP_H1743777351a_bool(insert956547291iple_a,hoare_1916936827iple_a(Pa,Ca,Q_1)),bot_bo797238721a_bool)))
=> ( hBOOL(hAPP_f1695230391l_bool(hoare_2102800559rivs_a(Ga),hAPP_f20753329a_bool(hAPP_H1743777351a_bool(insert956547291iple_a,hoare_1916936827iple_a(Q_1,D_1,R_1)),bot_bo797238721a_bool)))
=> hBOOL(hAPP_f1695230391l_bool(hoare_2102800559rivs_a(Ga),hAPP_f20753329a_bool(hAPP_H1743777351a_bool(insert956547291iple_a,hoare_1916936827iple_a(Pa,semi(Ca,D_1),R_1)),bot_bo797238721a_bool))) ) ) ).
fof(fact_113_triple_Oexhaust,axiom,
! [Y_1] :
~ ! [Fun1,Com,Fun2] : Y_1 != hoare_1916936827iple_a(Fun1,Com,Fun2) ).
fof(fact_114_Set_Oset__insert,axiom,
! [X_1,A] :
( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A))
=> ~ ! [B_35] :
( A = hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),B_35)
=> hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),B_35)) ) ) ).
fof(fact_115_Set_Oset__insert,axiom,
! [X_1,A] :
( hBOOL(hAPP_f1695230391l_bool(hAPP_H622608077l_bool(member1797258804iple_a,X_1),A))
=> ~ ! [B_35] :
( A = hAPP_f20753329a_bool(hAPP_H1743777351a_bool(insert956547291iple_a,X_1),B_35)
=> hBOOL(hAPP_f1695230391l_bool(hAPP_H622608077l_bool(member1797258804iple_a,X_1),B_35)) ) ) ).
fof(fact_116_Set_Oset__insert,axiom,
! [X_1,A] :
( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),A))
=> ~ ! [B_35] :
( A = hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),B_35)
=> hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),B_35)) ) ) ).
fof(fact_117_mk__disjoint__insert,axiom,
! [A_13,A] :
( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_13),A))
=> ? [B_35] :
( A = hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_13),B_35)
& ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_13),B_35)) ) ) ).
fof(fact_118_mk__disjoint__insert,axiom,
! [A_13,A] :
( hBOOL(hAPP_f1695230391l_bool(hAPP_H622608077l_bool(member1797258804iple_a,A_13),A))
=> ? [B_35] :
( A = hAPP_f20753329a_bool(hAPP_H1743777351a_bool(insert956547291iple_a,A_13),B_35)
& ~ hBOOL(hAPP_f1695230391l_bool(hAPP_H622608077l_bool(member1797258804iple_a,A_13),B_35)) ) ) ).
fof(fact_119_mk__disjoint__insert,axiom,
! [A_13,A] :
( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,A_13),A))
=> ? [B_35] :
( A = hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,A_13),B_35)
& ~ hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,A_13),B_35)) ) ) ).
fof(fact_120_equals0I,axiom,
! [A] :
( ! [Y_21] : ~ hBOOL(hAPP_f1695230391l_bool(hAPP_H622608077l_bool(member1797258804iple_a,Y_21),A))
=> A = bot_bo797238721a_bool ) ).
fof(fact_121_equals0I,axiom,
! [A] :
( ! [Y_21] : ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,Y_21),A))
=> A = bot_bot_fun_nat_bool ) ).
fof(fact_122_equals0I,axiom,
! [A] :
( ! [Y_21] : ~ hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Y_21),A))
=> A = bot_bot_fun_int_bool ) ).
fof(fact_123_conseq,axiom,
! [Q_1,Ga,Ca,Pa] :
( ! [Z_11,S_2] :
( hBOOL(hAPP_state_bool(hAPP_a2036067514e_bool(Pa,Z_11),S_2))
=> ? [P_1,Q_2] :
( hBOOL(hAPP_f1695230391l_bool(hoare_2102800559rivs_a(Ga),hAPP_f20753329a_bool(hAPP_H1743777351a_bool(insert956547291iple_a,hoare_1916936827iple_a(P_1,Ca,Q_2)),bot_bo797238721a_bool)))
& ! [S_3] :
( ! [Z_12] :
( hBOOL(hAPP_state_bool(hAPP_a2036067514e_bool(P_1,Z_12),S_2))
=> hBOOL(hAPP_state_bool(hAPP_a2036067514e_bool(Q_2,Z_12),S_3)) )
=> hBOOL(hAPP_state_bool(hAPP_a2036067514e_bool(Q_1,Z_11),S_3)) ) ) )
=> hBOOL(hAPP_f1695230391l_bool(hoare_2102800559rivs_a(Ga),hAPP_f20753329a_bool(hAPP_H1743777351a_bool(insert956547291iple_a,hoare_1916936827iple_a(Pa,Ca,Q_1)),bot_bo797238721a_bool))) ) ).
fof(fact_124_nonempty__iff,axiom,
! [A] :
( A != bot_bot_fun_nat_bool
<=> ? [X_2,B_35] :
( A = hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_2),B_35)
& ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_2),B_35)) ) ) ).
fof(fact_125_nonempty__iff,axiom,
! [A] :
( A != bot_bo797238721a_bool
<=> ? [X_2,B_35] :
( A = hAPP_f20753329a_bool(hAPP_H1743777351a_bool(insert956547291iple_a,X_2),B_35)
& ~ hBOOL(hAPP_f1695230391l_bool(hAPP_H622608077l_bool(member1797258804iple_a,X_2),B_35)) ) ) ).
fof(fact_126_nonempty__iff,axiom,
! [A] :
( A != bot_bot_fun_int_bool
<=> ? [X_2,B_35] :
( A = hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_2),B_35)
& ~ hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_2),B_35)) ) ) ).
fof(fact_127_com_Osimps_I13_J,axiom,
! [Com1_2,Com2_2] : semi(Com1_2,Com2_2) != skip ).
fof(fact_128_com_Osimps_I12_J,axiom,
! [Com1_2,Com2_2] : skip != semi(Com1_2,Com2_2) ).
fof(fact_129_the__elem__def,axiom,
! [X_28] : hAPP_f22106695ol_nat(the_elem_nat,X_28) = hAPP_f22106695ol_nat(the_nat,hAPP_f158894502t_bool(hAPP_f1777703707t_bool(cOMBB_955900739ol_nat,hAPP_f103356543l_bool(fequal_fun_nat_bool,X_28)),hAPP_f1570313510t_bool(hAPP_f688831301t_bool(cOMBC_178881787t_bool,insert_nat),bot_bot_fun_nat_bool))) ).
fof(fact_130_the__elem__def,axiom,
! [X_28] : hAPP_f1693662087iple_a(the_el1519802624iple_a,X_28) = hAPP_f1693662087iple_a(the_Ho830102290iple_a,hAPP_f1194757675a_bool(hAPP_f1697223433a_bool(cOMBB_1610959875iple_a,hAPP_f1767618879l_bool(fequal874423448a_bool,X_28)),hAPP_f1181346091a_bool(hAPP_f571476211a_bool(cOMBC_1570214144a_bool,insert956547291iple_a),bot_bo797238721a_bool))) ).
fof(fact_131_the__elem__def,axiom,
! [X_28] : hAPP_f1594865479ol_int(the_elem_int,X_28) = hAPP_f1594865479ol_int(the_int,hAPP_f2119767738t_bool(hAPP_f423804115t_bool(cOMBB_1418110531ol_int,hAPP_f284875647l_bool(fequal_fun_int_bool,X_28)),hAPP_f1048215610t_bool(hAPP_f472159229t_bool(cOMBC_1683390479t_bool,insert_int),bot_bot_fun_int_bool))) ).
fof(fact_132_com_Osimps_I3_J,axiom,
! [Com1_1,Com2_1,Com1,Com2] :
( semi(Com1_1,Com2_1) = semi(Com1,Com2)
<=> ( Com1_1 = Com1
& Com2_1 = Com2 ) ) ).
fof(fact_133_fold1Set__sing,axiom,
! [F,A_13,Ba] :
( hBOOL(hAPP_nat_bool(finite_fold1Set_nat(F,hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_13),bot_bot_fun_nat_bool)),Ba))
<=> A_13 = Ba ) ).
fof(fact_134_fold1Set__sing,axiom,
! [F,A_13,Ba] :
( hBOOL(hAPP_H1037229737a_bool(finite1621280049iple_a(F,hAPP_f20753329a_bool(hAPP_H1743777351a_bool(insert956547291iple_a,A_13),bot_bo797238721a_bool)),Ba))
<=> A_13 = Ba ) ).
fof(fact_135_fold1Set__sing,axiom,
! [F,A_13,Ba] :
( hBOOL(hAPP_int_bool(finite_fold1Set_int(F,hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,A_13),bot_bot_fun_int_bool)),Ba))
<=> A_13 = Ba ) ).
fof(fact_136_folding__one_Osingleton,axiom,
! [X_1,F,F_1] :
( hBOOL(finite988810631ne_nat(F,F_1))
=> hAPP_f22106695ol_nat(F_1,hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),bot_bot_fun_nat_bool)) = X_1 ) ).
fof(fact_137_folding__one_Osingleton,axiom,
! [X_1,F,F_1] :
( hBOOL(finite1753440478iple_a(F,F_1))
=> hAPP_f1693662087iple_a(F_1,hAPP_f20753329a_bool(hAPP_H1743777351a_bool(insert956547291iple_a,X_1),bot_bo797238721a_bool)) = X_1 ) ).
fof(fact_138_folding__one_Osingleton,axiom,
! [X_1,F,F_1] :
( hBOOL(finite1626084323ne_int(F,F_1))
=> hAPP_f1594865479ol_int(F_1,hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),bot_bot_fun_int_bool)) = X_1 ) ).
fof(fact_139_bot__empty__eq,axiom,
! [X_2] :
( hBOOL(hAPP_H1037229737a_bool(bot_bo797238721a_bool,X_2))
<=> hBOOL(hAPP_f1695230391l_bool(hAPP_H622608077l_bool(member1797258804iple_a,X_2),bot_bo797238721a_bool)) ) ).
fof(fact_140_bot__empty__eq,axiom,
! [X_2] :
( hBOOL(hAPP_nat_bool(bot_bot_fun_nat_bool,X_2))
<=> hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_2),bot_bot_fun_nat_bool)) ) ).
fof(fact_141_bot__empty__eq,axiom,
! [X_2] :
( hBOOL(hAPP_int_bool(bot_bot_fun_int_bool,X_2))
<=> hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_2),bot_bot_fun_int_bool)) ) ).
fof(fact_142_Ass,axiom,
! [Ga,Pa,X_28,A_13] : hBOOL(hAPP_f1695230391l_bool(hoare_2102800559rivs_a(Ga),hAPP_f20753329a_bool(hAPP_H1743777351a_bool(insert956547291iple_a,hoare_1916936827iple_a(hAPP_f100967412e_bool(hAPP_f1460451647e_bool(cOMBC_524597097e_bool,hAPP_f1406200875e_bool(hAPP_f635443597e_bool(cOMBB_2139825703bool_a,cOMBB_844853809_state),Pa)),hAPP_f871651461_state(hAPP_f851239890_state(cOMBS_777315357_state,hAPP_v365393659_state(hAPP_f1542232213_state(cOMBC_1193272608_state,update),X_28)),A_13)),ass(X_28,A_13),Pa)),bot_bo797238721a_bool))) ).
fof(fact_143_fold1__singleton,axiom,
! [F,A_13] : hAPP_f22106695ol_nat(finite_fold1_nat(F),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_13),bot_bot_fun_nat_bool)) = A_13 ).
fof(fact_144_fold1__singleton,axiom,
! [F,A_13] : hAPP_f1693662087iple_a(finite1465856449iple_a(F),hAPP_f20753329a_bool(hAPP_H1743777351a_bool(insert956547291iple_a,A_13),bot_bo797238721a_bool)) = A_13 ).
fof(fact_145_fold1__singleton,axiom,
! [F,A_13] : hAPP_f1594865479ol_int(finite_fold1_int(F),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,A_13),bot_bot_fun_int_bool)) = A_13 ).
fof(fact_146_fold1__singleton__def,axiom,
! [A_13,G,F] :
( G = finite_fold1_nat(F)
=> hAPP_f22106695ol_nat(G,hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_13),bot_bot_fun_nat_bool)) = A_13 ) ).
fof(fact_147_fold1__singleton__def,axiom,
! [A_13,G,F] :
( G = finite1465856449iple_a(F)
=> hAPP_f1693662087iple_a(G,hAPP_f20753329a_bool(hAPP_H1743777351a_bool(insert956547291iple_a,A_13),bot_bo797238721a_bool)) = A_13 ) ).
fof(fact_148_fold1__singleton__def,axiom,
! [A_13,G,F] :
( G = finite_fold1_int(F)
=> hAPP_f1594865479ol_int(G,hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,A_13),bot_bot_fun_int_bool)) = A_13 ) ).
fof(fact_149_image__constant__conv,axiom,
! [Ca,A] :
( ( A = bot_bo797238721a_bool
=> image_48635758_a_nat(hAPP_n1653940209_a_nat(cOMBK_1028443109iple_a,Ca),A) = bot_bot_fun_nat_bool )
& ( A != bot_bo797238721a_bool
=> image_48635758_a_nat(hAPP_n1653940209_a_nat(cOMBK_1028443109iple_a,Ca),A) = hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,Ca),bot_bot_fun_nat_bool) ) ) ).
fof(fact_150_image__constant__conv,axiom,
! [Ca,A] :
( ( A = bot_bot_fun_int_bool
=> image_int_nat(hAPP_nat_fun_int_nat(cOMBK_nat_int,Ca),A) = bot_bot_fun_nat_bool )
& ( A != bot_bot_fun_int_bool
=> image_int_nat(hAPP_nat_fun_int_nat(cOMBK_nat_int,Ca),A) = hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,Ca),bot_bot_fun_nat_bool) ) ) ).
fof(fact_151_image__constant__conv,axiom,
! [Ca,A] :
( ( A = bot_bo797238721a_bool
=> image_533441733iple_a(hAPP_H2085992369iple_a(cOMBK_2052406350iple_a,Ca),A) = bot_bo797238721a_bool )
& ( A != bot_bo797238721a_bool
=> image_533441733iple_a(hAPP_H2085992369iple_a(cOMBK_2052406350iple_a,Ca),A) = hAPP_f20753329a_bool(hAPP_H1743777351a_bool(insert956547291iple_a,Ca),bot_bo797238721a_bool) ) ) ).
fof(fact_152_image__constant__conv,axiom,
! [Ca,A] :
( ( A = bot_bot_fun_int_bool
=> image_int_int(hAPP_int_fun_int_int(cOMBK_int_int,Ca),A) = bot_bot_fun_int_bool )
& ( A != bot_bot_fun_int_bool
=> image_int_int(hAPP_int_fun_int_int(cOMBK_int_int,Ca),A) = hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,Ca),bot_bot_fun_int_bool) ) ) ).
fof(fact_153_image__constant__conv,axiom,
! [Ca,A] :
( ( A = bot_bot_fun_nat_bool
=> image_nat_nat(hAPP_nat_fun_nat_nat(cOMBK_nat_nat,Ca),A) = bot_bot_fun_nat_bool )
& ( A != bot_bot_fun_nat_bool
=> image_nat_nat(hAPP_nat_fun_nat_nat(cOMBK_nat_nat,Ca),A) = hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,Ca),bot_bot_fun_nat_bool) ) ) ).
fof(fact_154_image__constant__conv,axiom,
! [Ca,A] :
( ( A = bot_bot_fun_nat_bool
=> image_nat_int(hAPP_int_fun_nat_int(cOMBK_int_nat,Ca),A) = bot_bot_fun_int_bool )
& ( A != bot_bot_fun_nat_bool
=> image_nat_int(hAPP_int_fun_nat_int(cOMBK_int_nat,Ca),A) = hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,Ca),bot_bot_fun_int_bool) ) ) ).
fof(fact_155_image__eqI,axiom,
! [A,Ba,F,X_1] :
( Ba = hAPP_H905846293iple_a(F,X_1)
=> ( hBOOL(hAPP_f1695230391l_bool(hAPP_H622608077l_bool(member1797258804iple_a,X_1),A))
=> hBOOL(hAPP_f1695230391l_bool(hAPP_H622608077l_bool(member1797258804iple_a,Ba),image_533441733iple_a(F,A))) ) ) ).
fof(fact_156_image__eqI,axiom,
! [A,Ba,F,X_1] :
( Ba = hAPP_n261501868iple_a(F,X_1)
=> ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A))
=> hBOOL(hAPP_f1695230391l_bool(hAPP_H622608077l_bool(member1797258804iple_a,Ba),image_1865589340iple_a(F,A))) ) ) ).
fof(fact_157_image__eqI,axiom,
! [A,Ba,F,X_1] :
( Ba = hAPP_i240634960iple_a(F,X_1)
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),A))
=> hBOOL(hAPP_f1695230391l_bool(hAPP_H622608077l_bool(member1797258804iple_a,Ba),image_1844722432iple_a(F,A))) ) ) ).
fof(fact_158_image__eqI,axiom,
! [A,Ba,F,X_1] :
( Ba = hAPP_H592031934_a_nat(F,X_1)
=> ( hBOOL(hAPP_f1695230391l_bool(hAPP_H622608077l_bool(member1797258804iple_a,X_1),A))
=> hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,Ba),image_48635758_a_nat(F,A))) ) ) ).
fof(fact_159_image__eqI,axiom,
! [A,Ba,F,X_1] :
( Ba = hAPP_int_int(F,X_1)
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),A))
=> hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Ba),image_int_int(F,A))) ) ) ).
fof(fact_160_image__eqI,axiom,
! [A,Ba,F,X_1] :
( Ba = hAPP_H1229305626_a_int(F,X_1)
=> ( hBOOL(hAPP_f1695230391l_bool(hAPP_H622608077l_bool(member1797258804iple_a,X_1),A))
=> hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Ba),image_685909450_a_int(F,A))) ) ) ).
fof(fact_161_image__eqI,axiom,
! [A,Ba,F,X_1] :
( Ba = hAPP_nat_nat(F,X_1)
=> ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A))
=> hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,Ba),image_nat_nat(F,A))) ) ) ).
fof(fact_162_image__eqI,axiom,
! [A,Ba,F,X_1] :
( Ba = hAPP_nat_int(F,X_1)
=> ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A))
=> hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Ba),image_nat_int(F,A))) ) ) ).
fof(fact_163_fold1__def,axiom,
! [F,A] : hAPP_f1594865479ol_int(finite_fold1_int(F),A) = hAPP_f1594865479ol_int(the_int,finite_fold1Set_int(F,A)) ).
fof(fact_164_fold1__def,axiom,
! [F,A] : hAPP_f1693662087iple_a(finite1465856449iple_a(F),A) = hAPP_f1693662087iple_a(the_Ho830102290iple_a,finite1621280049iple_a(F,A)) ).
fof(fact_165_fold1__def,axiom,
! [F,A] : hAPP_f22106695ol_nat(finite_fold1_nat(F),A) = hAPP_f22106695ol_nat(the_nat,finite_fold1Set_nat(F,A)) ).
fof(fact_166_com_Osimps_I1_J,axiom,
! [Vname_1,Fun_1,Vname,Fun] :
( ( is_vname(Vname_1)
& is_vname(Vname) )
=> ( ass(Vname_1,Fun_1) = ass(Vname,Fun)
<=> ( Vname_1 = Vname
& Fun_1 = Fun ) ) ) ).
fof(fact_167_rev__image__eqI,axiom,
! [Ba,F,X_1,A] :
( hBOOL(hAPP_f1695230391l_bool(hAPP_H622608077l_bool(member1797258804iple_a,X_1),A))
=> ( Ba = hAPP_H905846293iple_a(F,X_1)
=> hBOOL(hAPP_f1695230391l_bool(hAPP_H622608077l_bool(member1797258804iple_a,Ba),image_533441733iple_a(F,A))) ) ) ).
fof(fact_168_rev__image__eqI,axiom,
! [Ba,F,X_1,A] :
( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A))
=> ( Ba = hAPP_n261501868iple_a(F,X_1)
=> hBOOL(hAPP_f1695230391l_bool(hAPP_H622608077l_bool(member1797258804iple_a,Ba),image_1865589340iple_a(F,A))) ) ) ).
fof(fact_169_rev__image__eqI,axiom,
! [Ba,F,X_1,A] :
( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),A))
=> ( Ba = hAPP_i240634960iple_a(F,X_1)
=> hBOOL(hAPP_f1695230391l_bool(hAPP_H622608077l_bool(member1797258804iple_a,Ba),image_1844722432iple_a(F,A))) ) ) ).
fof(fact_170_rev__image__eqI,axiom,
! [Ba,F,X_1,A] :
( hBOOL(hAPP_f1695230391l_bool(hAPP_H622608077l_bool(member1797258804iple_a,X_1),A))
=> ( Ba = hAPP_H592031934_a_nat(F,X_1)
=> hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,Ba),image_48635758_a_nat(F,A))) ) ) ).
fof(fact_171_rev__image__eqI,axiom,
! [Ba,F,X_1,A] :
( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),A))
=> ( Ba = hAPP_int_int(F,X_1)
=> hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Ba),image_int_int(F,A))) ) ) ).
fof(fact_172_rev__image__eqI,axiom,
! [Ba,F,X_1,A] :
( hBOOL(hAPP_f1695230391l_bool(hAPP_H622608077l_bool(member1797258804iple_a,X_1),A))
=> ( Ba = hAPP_H1229305626_a_int(F,X_1)
=> hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Ba),image_685909450_a_int(F,A))) ) ) ).
fof(fact_173_rev__image__eqI,axiom,
! [Ba,F,X_1,A] :
( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A))
=> ( Ba = hAPP_nat_nat(F,X_1)
=> hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,Ba),image_nat_nat(F,A))) ) ) ).
fof(fact_174_rev__image__eqI,axiom,
! [Ba,F,X_1,A] :
( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A))
=> ( Ba = hAPP_nat_int(F,X_1)
=> hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Ba),image_nat_int(F,A))) ) ) ).
fof(fact_175_imageI,axiom,
! [F,X_1,A] :
( hBOOL(hAPP_f1695230391l_bool(hAPP_H622608077l_bool(member1797258804iple_a,X_1),A))
=> hBOOL(hAPP_f1695230391l_bool(hAPP_H622608077l_bool(member1797258804iple_a,hAPP_H905846293iple_a(F,X_1)),image_533441733iple_a(F,A))) ) ).
fof(fact_176_imageI,axiom,
! [F,X_1,A] :
( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A))
=> hBOOL(hAPP_f1695230391l_bool(hAPP_H622608077l_bool(member1797258804iple_a,hAPP_n261501868iple_a(F,X_1)),image_1865589340iple_a(F,A))) ) ).
fof(fact_177_imageI,axiom,
! [F,X_1,A] :
( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),A))
=> hBOOL(hAPP_f1695230391l_bool(hAPP_H622608077l_bool(member1797258804iple_a,hAPP_i240634960iple_a(F,X_1)),image_1844722432iple_a(F,A))) ) ).
fof(fact_178_imageI,axiom,
! [F,X_1,A] :
( hBOOL(hAPP_f1695230391l_bool(hAPP_H622608077l_bool(member1797258804iple_a,X_1),A))
=> hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,hAPP_H592031934_a_nat(F,X_1)),image_48635758_a_nat(F,A))) ) ).
fof(fact_179_imageI,axiom,
! [F,X_1,A] :
( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),A))
=> hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hAPP_int_int(F,X_1)),image_int_int(F,A))) ) ).
fof(fact_180_imageI,axiom,
! [F,X_1,A] :
( hBOOL(hAPP_f1695230391l_bool(hAPP_H622608077l_bool(member1797258804iple_a,X_1),A))
=> hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hAPP_H1229305626_a_int(F,X_1)),image_685909450_a_int(F,A))) ) ).
fof(fact_181_imageI,axiom,
! [F,X_1,A] :
( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A))
=> hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,hAPP_nat_nat(F,X_1)),image_nat_nat(F,A))) ) ).
fof(fact_182_imageI,axiom,
! [F,X_1,A] :
( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A))
=> hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hAPP_nat_int(F,X_1)),image_nat_int(F,A))) ) ).
fof(fact_183_image__iff,axiom,
! [Z_1,F,A] :
( hBOOL(hAPP_f1695230391l_bool(hAPP_H622608077l_bool(member1797258804iple_a,Z_1),image_533441733iple_a(F,A)))
<=> ? [X_2] :
( hBOOL(hAPP_f1695230391l_bool(hAPP_H622608077l_bool(member1797258804iple_a,X_2),A))
& Z_1 = hAPP_H905846293iple_a(F,X_2) ) ) ).
fof(fact_184_image__iff,axiom,
! [Z_1,F,A] :
( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Z_1),image_int_int(F,A)))
<=> ? [X_2] :
( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_2),A))
& Z_1 = hAPP_int_int(F,X_2) ) ) ).
fof(fact_185_image__iff,axiom,
! [Z_1,F,A] :
( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,Z_1),image_nat_nat(F,A)))
<=> ? [X_2] :
( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_2),A))
& Z_1 = hAPP_nat_nat(F,X_2) ) ) ).
fof(fact_186_image__iff,axiom,
! [Z_1,F,A] :
( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Z_1),image_nat_int(F,A)))
<=> ? [X_2] :
( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_2),A))
& Z_1 = hAPP_nat_int(F,X_2) ) ) ).
fof(fact_187_image__image,axiom,
! [F,G,A] : image_nat_int(F,image_int_nat(G,A)) = image_int_int(hAPP_f147134065nt_int(hAPP_f1673907925nt_int(cOMBB_nat_int_int,F),G),A) ).
fof(fact_188_image__image,axiom,
! [F,G,A] : image_int_int(F,image_nat_int(G,A)) = image_nat_int(hAPP_f1139079189at_int(hAPP_f1431025877at_int(cOMBB_int_int_nat,F),G),A) ).
fof(fact_189_image__ident,axiom,
! [Y_23] : image_int_int(cOMBI_int,Y_23) = Y_23 ).
fof(fact_190_image__ident,axiom,
! [Y_23] : image_533441733iple_a(cOMBI_1301683934iple_a,Y_23) = Y_23 ).
fof(fact_191_image__ident,axiom,
! [Y_23] : image_nat_nat(cOMBI_nat,Y_23) = Y_23 ).
fof(fact_192_com_Osimps_I24_J,axiom,
! [Vname_1,Fun_1,Com1,Com2] : ass(Vname_1,Fun_1) != semi(Com1,Com2) ).
fof(fact_193_com_Osimps_I25_J,axiom,
! [Com1,Com2,Vname_1,Fun_1] : semi(Com1,Com2) != ass(Vname_1,Fun_1) ).
fof(fact_194_com_Osimps_I8_J,axiom,
! [Vname,Fun] : skip != ass(Vname,Fun) ).
fof(fact_195_mem__def,axiom,
! [X_1,A] :
( hBOOL(hAPP_f1695230391l_bool(hAPP_H622608077l_bool(member1797258804iple_a,X_1),A))
<=> hBOOL(hAPP_H1037229737a_bool(A,X_1)) ) ).
fof(fact_196_mem__def,axiom,
! [X_1,A] :
( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A))
<=> hBOOL(hAPP_nat_bool(A,X_1)) ) ).
fof(fact_197_mem__def,axiom,
! [X_1,A] :
( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),A))
<=> hBOOL(hAPP_int_bool(A,X_1)) ) ).
fof(fact_198_Collect__def,axiom,
! [Pa] : hAPP_f1805168059t_bool(collect_int,Pa) = Pa ).
fof(fact_199_Collect__def,axiom,
! [Pa] : hAPP_f20753329a_bool(collec351493750iple_a,Pa) = Pa ).
fof(fact_200_Collect__def,axiom,
! [Pa] : hAPP_f800510211t_bool(collect_nat,Pa) = Pa ).
fof(fact_201_com_Osimps_I9_J,axiom,
! [Vname,Fun] : ass(Vname,Fun) != skip ).
fof(fact_202_image__is__empty,axiom,
! [F,A] :
( image_533441733iple_a(F,A) = bot_bo797238721a_bool
<=> A = bot_bo797238721a_bool ) ).
fof(fact_203_image__is__empty,axiom,
! [F,A] :
( image_int_int(F,A) = bot_bot_fun_int_bool
<=> A = bot_bot_fun_int_bool ) ).
fof(fact_204_image__is__empty,axiom,
! [F,A] :
( image_nat_nat(F,A) = bot_bot_fun_nat_bool
<=> A = bot_bot_fun_nat_bool ) ).
fof(fact_205_image__is__empty,axiom,
! [F,A] :
( image_nat_int(F,A) = bot_bot_fun_int_bool
<=> A = bot_bot_fun_nat_bool ) ).
fof(fact_206_image__empty,axiom,
! [F] : image_533441733iple_a(F,bot_bo797238721a_bool) = bot_bo797238721a_bool ).
fof(fact_207_image__empty,axiom,
! [F] : image_int_int(F,bot_bot_fun_int_bool) = bot_bot_fun_int_bool ).
fof(fact_208_image__empty,axiom,
! [F] : image_nat_nat(F,bot_bot_fun_nat_bool) = bot_bot_fun_nat_bool ).
fof(fact_209_image__empty,axiom,
! [F] : image_nat_int(F,bot_bot_fun_nat_bool) = bot_bot_fun_int_bool ).
fof(fact_210_empty__is__image,axiom,
! [F,A] :
( bot_bo797238721a_bool = image_533441733iple_a(F,A)
<=> A = bot_bo797238721a_bool ) ).
fof(fact_211_empty__is__image,axiom,
! [F,A] :
( bot_bot_fun_int_bool = image_int_int(F,A)
<=> A = bot_bot_fun_int_bool ) ).
fof(fact_212_empty__is__image,axiom,
! [F,A] :
( bot_bot_fun_nat_bool = image_nat_nat(F,A)
<=> A = bot_bot_fun_nat_bool ) ).
fof(fact_213_empty__is__image,axiom,
! [F,A] :
( bot_bot_fun_int_bool = image_nat_int(F,A)
<=> A = bot_bot_fun_nat_bool ) ).
fof(fact_214_insert__image,axiom,
! [F,X_1,A] :
( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),A))
=> hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,hAPP_int_nat(F,X_1)),image_int_nat(F,A)) = image_int_nat(F,A) ) ).
fof(fact_215_insert__image,axiom,
! [F,X_1,A] :
( hBOOL(hAPP_f1695230391l_bool(hAPP_H622608077l_bool(member1797258804iple_a,X_1),A))
=> hAPP_f20753329a_bool(hAPP_H1743777351a_bool(insert956547291iple_a,hAPP_H905846293iple_a(F,X_1)),image_533441733iple_a(F,A)) = image_533441733iple_a(F,A) ) ).
fof(fact_216_insert__image,axiom,
! [F,X_1,A] :
( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),A))
=> hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,hAPP_int_int(F,X_1)),image_int_int(F,A)) = image_int_int(F,A) ) ).
fof(fact_217_insert__image,axiom,
! [F,X_1,A] :
( hBOOL(hAPP_f1695230391l_bool(hAPP_H622608077l_bool(member1797258804iple_a,X_1),A))
=> hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,hAPP_H1229305626_a_int(F,X_1)),image_685909450_a_int(F,A)) = image_685909450_a_int(F,A) ) ).
fof(fact_218_insert__image,axiom,
! [F,X_1,A] :
( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A))
=> hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,hAPP_nat_nat(F,X_1)),image_nat_nat(F,A)) = image_nat_nat(F,A) ) ).
fof(fact_219_insert__image,axiom,
! [F,X_1,A] :
( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A))
=> hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,hAPP_nat_int(F,X_1)),image_nat_int(F,A)) = image_nat_int(F,A) ) ).
fof(fact_220_image__insert,axiom,
! [F,A_13,B] : image_48635758_a_nat(F,hAPP_f20753329a_bool(hAPP_H1743777351a_bool(insert956547291iple_a,A_13),B)) = hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,hAPP_H592031934_a_nat(F,A_13)),image_48635758_a_nat(F,B)) ).
fof(fact_221_image__insert,axiom,
! [F,A_13,B] : image_int_nat(F,hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,A_13),B)) = hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,hAPP_int_nat(F,A_13)),image_int_nat(F,B)) ).
fof(fact_222_image__insert,axiom,
! [F,A_13,B] : image_533441733iple_a(F,hAPP_f20753329a_bool(hAPP_H1743777351a_bool(insert956547291iple_a,A_13),B)) = hAPP_f20753329a_bool(hAPP_H1743777351a_bool(insert956547291iple_a,hAPP_H905846293iple_a(F,A_13)),image_533441733iple_a(F,B)) ).
fof(fact_223_image__insert,axiom,
! [F,A_13,B] : image_1865589340iple_a(F,hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_13),B)) = hAPP_f20753329a_bool(hAPP_H1743777351a_bool(insert956547291iple_a,hAPP_n261501868iple_a(F,A_13)),image_1865589340iple_a(F,B)) ).
fof(fact_224_image__insert,axiom,
! [F,A_13,B] : image_int_int(F,hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,A_13),B)) = hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,hAPP_int_int(F,A_13)),image_int_int(F,B)) ).
fof(fact_225_image__insert,axiom,
! [F,A_13,B] : image_nat_nat(F,hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_13),B)) = hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,hAPP_nat_nat(F,A_13)),image_nat_nat(F,B)) ).
fof(fact_226_image__insert,axiom,
! [F,A_13,B] : image_nat_int(F,hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_13),B)) = hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,hAPP_nat_int(F,A_13)),image_nat_int(F,B)) ).
fof(fact_227_fold1Set__nonempty,axiom,
! [F,A,X_1] :
( hBOOL(hAPP_H1037229737a_bool(finite1621280049iple_a(F,A),X_1))
=> A != bot_bo797238721a_bool ) ).
fof(fact_228_fold1Set__nonempty,axiom,
! [F,A,X_1] :
( hBOOL(hAPP_int_bool(finite_fold1Set_int(F,A),X_1))
=> A != bot_bot_fun_int_bool ) ).
fof(fact_229_fold1Set__nonempty,axiom,
! [F,A,X_1] :
( hBOOL(hAPP_nat_bool(finite_fold1Set_nat(F,A),X_1))
=> A != bot_bot_fun_nat_bool ) ).
fof(fact_230_empty__fold1SetE,axiom,
! [F,X_1] : ~ hBOOL(hAPP_H1037229737a_bool(finite1621280049iple_a(F,bot_bo797238721a_bool),X_1)) ).
fof(fact_231_empty__fold1SetE,axiom,
! [F,X_1] : ~ hBOOL(hAPP_int_bool(finite_fold1Set_int(F,bot_bot_fun_int_bool),X_1)) ).
fof(fact_232_empty__fold1SetE,axiom,
! [F,X_1] : ~ hBOOL(hAPP_nat_bool(finite_fold1Set_nat(F,bot_bot_fun_nat_bool),X_1)) ).
fof(fact_233_image__constant,axiom,
! [Ca,X_1,A] :
( hBOOL(hAPP_f1695230391l_bool(hAPP_H622608077l_bool(member1797258804iple_a,X_1),A))
=> image_48635758_a_nat(hAPP_n1653940209_a_nat(cOMBK_1028443109iple_a,Ca),A) = hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,Ca),bot_bot_fun_nat_bool) ) ).
fof(fact_234_image__constant,axiom,
! [Ca,X_1,A] :
( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),A))
=> image_int_nat(hAPP_nat_fun_int_nat(cOMBK_nat_int,Ca),A) = hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,Ca),bot_bot_fun_nat_bool) ) ).
fof(fact_235_image__constant,axiom,
! [Ca,X_1,A] :
( hBOOL(hAPP_f1695230391l_bool(hAPP_H622608077l_bool(member1797258804iple_a,X_1),A))
=> image_533441733iple_a(hAPP_H2085992369iple_a(cOMBK_2052406350iple_a,Ca),A) = hAPP_f20753329a_bool(hAPP_H1743777351a_bool(insert956547291iple_a,Ca),bot_bo797238721a_bool) ) ).
fof(fact_236_image__constant,axiom,
! [Ca,X_1,A] :
( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),A))
=> image_int_int(hAPP_int_fun_int_int(cOMBK_int_int,Ca),A) = hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,Ca),bot_bot_fun_int_bool) ) ).
fof(fact_237_image__constant,axiom,
! [Ca,X_1,A] :
( hBOOL(hAPP_f1695230391l_bool(hAPP_H622608077l_bool(member1797258804iple_a,X_1),A))
=> image_685909450_a_int(hAPP_i1475897073_a_int(cOMBK_1007576201iple_a,Ca),A) = hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,Ca),bot_bot_fun_int_bool) ) ).
fof(fact_238_image__constant,axiom,
! [Ca,X_1,A] :
( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A))
=> image_nat_nat(hAPP_nat_fun_nat_nat(cOMBK_nat_nat,Ca),A) = hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,Ca),bot_bot_fun_nat_bool) ) ).
fof(fact_239_image__constant,axiom,
! [Ca,X_1,A] :
( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A))
=> image_nat_int(hAPP_int_fun_nat_int(cOMBK_int_nat,Ca),A) = hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,Ca),bot_bot_fun_int_bool) ) ).
fof(fact_240_imageE,axiom,
! [Ba,F,A] :
( hBOOL(hAPP_f1695230391l_bool(hAPP_H622608077l_bool(member1797258804iple_a,Ba),image_533441733iple_a(F,A)))
=> ~ ! [X_2] :
( Ba = hAPP_H905846293iple_a(F,X_2)
=> ~ hBOOL(hAPP_f1695230391l_bool(hAPP_H622608077l_bool(member1797258804iple_a,X_2),A)) ) ) ).
fof(fact_241_imageE,axiom,
! [Ba,F,A] :
( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,Ba),image_48635758_a_nat(F,A)))
=> ~ ! [X_2] :
( Ba = hAPP_H592031934_a_nat(F,X_2)
=> ~ hBOOL(hAPP_f1695230391l_bool(hAPP_H622608077l_bool(member1797258804iple_a,X_2),A)) ) ) ).
fof(fact_242_imageE,axiom,
! [Ba,F,A] :
( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Ba),image_685909450_a_int(F,A)))
=> ~ ! [X_2] :
( Ba = hAPP_H1229305626_a_int(F,X_2)
=> ~ hBOOL(hAPP_f1695230391l_bool(hAPP_H622608077l_bool(member1797258804iple_a,X_2),A)) ) ) ).
fof(fact_243_imageE,axiom,
! [Ba,F,A] :
( hBOOL(hAPP_f1695230391l_bool(hAPP_H622608077l_bool(member1797258804iple_a,Ba),image_1865589340iple_a(F,A)))
=> ~ ! [X_2] :
( Ba = hAPP_n261501868iple_a(F,X_2)
=> ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_2),A)) ) ) ).
fof(fact_244_imageE,axiom,
! [Ba,F,A] :
( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Ba),image_int_int(F,A)))
=> ~ ! [X_2] :
( Ba = hAPP_int_int(F,X_2)
=> ~ hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_2),A)) ) ) ).
fof(fact_245_imageE,axiom,
! [Ba,F,A] :
( hBOOL(hAPP_f1695230391l_bool(hAPP_H622608077l_bool(member1797258804iple_a,Ba),image_1844722432iple_a(F,A)))
=> ~ ! [X_2] :
( Ba = hAPP_i240634960iple_a(F,X_2)
=> ~ hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_2),A)) ) ) ).
fof(fact_246_imageE,axiom,
! [Ba,F,A] :
( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,Ba),image_nat_nat(F,A)))
=> ~ ! [X_2] :
( Ba = hAPP_nat_nat(F,X_2)
=> ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_2),A)) ) ) ).
fof(fact_247_imageE,axiom,
! [Ba,F,A] :
( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Ba),image_nat_int(F,A)))
=> ~ ! [X_2] :
( Ba = hAPP_nat_int(F,X_2)
=> ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_2),A)) ) ) ).
fof(fact_248_the__sym__eq__trivial,axiom,
! [X_1] : hAPP_f1594865479ol_int(the_int,hAPP_i1948725293t_bool(fequal_int,X_1)) = X_1 ).
fof(fact_249_the__sym__eq__trivial,axiom,
! [X_1] : hAPP_f1693662087iple_a(the_Ho830102290iple_a,hAPP_H426895267a_bool(fequal963300192iple_a,X_1)) = X_1 ).
fof(fact_250_the__sym__eq__trivial,axiom,
! [X_1] : hAPP_f22106695ol_nat(the_nat,hAPP_n1699378549t_bool(fequal_nat,X_1)) = X_1 ).
fof(fact_251_the__eq__trivial,axiom,
! [A_13] : hAPP_f1594865479ol_int(the_int,hAPP_i1948725293t_bool(hAPP_f428220345t_bool(cOMBC_int_int_bool,fequal_int),A_13)) = A_13 ).
fof(fact_252_the__eq__trivial,axiom,
! [A_13] : hAPP_f1693662087iple_a(the_Ho830102290iple_a,hAPP_H426895267a_bool(hAPP_f920293029a_bool(cOMBC_1940896922a_bool,fequal963300192iple_a),A_13)) = A_13 ).
fof(fact_253_the__eq__trivial,axiom,
! [A_13] : hAPP_f22106695ol_nat(the_nat,hAPP_n1699378549t_bool(hAPP_f229349961t_bool(cOMBC_nat_nat_bool,fequal_nat),A_13)) = A_13 ).
fof(fact_254_If__def,axiom,
! [X_1,Y_1,Pa] :
( ( hBOOL(Pa)
=> X_1 = hAPP_f1594865479ol_int(the_int,hAPP_f1805168059t_bool(hAPP_f727283836t_bool(cOMBS_int_bool_bool,hAPP_f2144054103l_bool(hAPP_f1734373249l_bool(cOMBB_1652995168ol_int,fconj),hAPP_f1805168059t_bool(hAPP_f627970963t_bool(cOMBB_bool_bool_int,hAPP_b589554111l_bool(fimplies,Pa)),hAPP_i1948725293t_bool(hAPP_f428220345t_bool(cOMBC_int_int_bool,fequal_int),X_1)))),hAPP_f1805168059t_bool(hAPP_f627970963t_bool(cOMBB_bool_bool_int,hAPP_b589554111l_bool(fimplies,hAPP_bool_bool(fNot,Pa))),hAPP_i1948725293t_bool(hAPP_f428220345t_bool(cOMBC_int_int_bool,fequal_int),Y_1)))) )
& ( ~ hBOOL(Pa)
=> Y_1 = hAPP_f1594865479ol_int(the_int,hAPP_f1805168059t_bool(hAPP_f727283836t_bool(cOMBS_int_bool_bool,hAPP_f2144054103l_bool(hAPP_f1734373249l_bool(cOMBB_1652995168ol_int,fconj),hAPP_f1805168059t_bool(hAPP_f627970963t_bool(cOMBB_bool_bool_int,hAPP_b589554111l_bool(fimplies,Pa)),hAPP_i1948725293t_bool(hAPP_f428220345t_bool(cOMBC_int_int_bool,fequal_int),X_1)))),hAPP_f1805168059t_bool(hAPP_f627970963t_bool(cOMBB_bool_bool_int,hAPP_b589554111l_bool(fimplies,hAPP_bool_bool(fNot,Pa))),hAPP_i1948725293t_bool(hAPP_f428220345t_bool(cOMBC_int_int_bool,fequal_int),Y_1)))) ) ) ).
fof(fact_255_If__def,axiom,
! [X_1,Y_1,Pa] :
( ( hBOOL(Pa)
=> X_1 = hAPP_f1693662087iple_a(the_Ho830102290iple_a,hAPP_f20753329a_bool(hAPP_f607497069a_bool(cOMBS_154988028l_bool,hAPP_f146754017l_bool(hAPP_f239971723l_bool(cOMBB_1866391387iple_a,fconj),hAPP_f20753329a_bool(hAPP_f604201481a_bool(cOMBB_1882975613iple_a,hAPP_b589554111l_bool(fimplies,Pa)),hAPP_H426895267a_bool(hAPP_f920293029a_bool(cOMBC_1940896922a_bool,fequal963300192iple_a),X_1)))),hAPP_f20753329a_bool(hAPP_f604201481a_bool(cOMBB_1882975613iple_a,hAPP_b589554111l_bool(fimplies,hAPP_bool_bool(fNot,Pa))),hAPP_H426895267a_bool(hAPP_f920293029a_bool(cOMBC_1940896922a_bool,fequal963300192iple_a),Y_1)))) )
& ( ~ hBOOL(Pa)
=> Y_1 = hAPP_f1693662087iple_a(the_Ho830102290iple_a,hAPP_f20753329a_bool(hAPP_f607497069a_bool(cOMBS_154988028l_bool,hAPP_f146754017l_bool(hAPP_f239971723l_bool(cOMBB_1866391387iple_a,fconj),hAPP_f20753329a_bool(hAPP_f604201481a_bool(cOMBB_1882975613iple_a,hAPP_b589554111l_bool(fimplies,Pa)),hAPP_H426895267a_bool(hAPP_f920293029a_bool(cOMBC_1940896922a_bool,fequal963300192iple_a),X_1)))),hAPP_f20753329a_bool(hAPP_f604201481a_bool(cOMBB_1882975613iple_a,hAPP_b589554111l_bool(fimplies,hAPP_bool_bool(fNot,Pa))),hAPP_H426895267a_bool(hAPP_f920293029a_bool(cOMBC_1940896922a_bool,fequal963300192iple_a),Y_1)))) ) ) ).
fof(fact_256_If__def,axiom,
! [X_1,Y_1,Pa] :
( ( hBOOL(Pa)
=> X_1 = hAPP_f22106695ol_nat(the_nat,hAPP_f800510211t_bool(hAPP_f561022312t_bool(cOMBS_nat_bool_bool,hAPP_f1146629647l_bool(hAPP_f1080886329l_bool(cOMBB_1015721476ol_nat,fconj),hAPP_f800510211t_bool(hAPP_f894608603t_bool(cOMBB_bool_bool_nat,hAPP_b589554111l_bool(fimplies,Pa)),hAPP_n1699378549t_bool(hAPP_f229349961t_bool(cOMBC_nat_nat_bool,fequal_nat),X_1)))),hAPP_f800510211t_bool(hAPP_f894608603t_bool(cOMBB_bool_bool_nat,hAPP_b589554111l_bool(fimplies,hAPP_bool_bool(fNot,Pa))),hAPP_n1699378549t_bool(hAPP_f229349961t_bool(cOMBC_nat_nat_bool,fequal_nat),Y_1)))) )
& ( ~ hBOOL(Pa)
=> Y_1 = hAPP_f22106695ol_nat(the_nat,hAPP_f800510211t_bool(hAPP_f561022312t_bool(cOMBS_nat_bool_bool,hAPP_f1146629647l_bool(hAPP_f1080886329l_bool(cOMBB_1015721476ol_nat,fconj),hAPP_f800510211t_bool(hAPP_f894608603t_bool(cOMBB_bool_bool_nat,hAPP_b589554111l_bool(fimplies,Pa)),hAPP_n1699378549t_bool(hAPP_f229349961t_bool(cOMBC_nat_nat_bool,fequal_nat),X_1)))),hAPP_f800510211t_bool(hAPP_f894608603t_bool(cOMBB_bool_bool_nat,hAPP_b589554111l_bool(fimplies,hAPP_bool_bool(fNot,Pa))),hAPP_n1699378549t_bool(hAPP_f229349961t_bool(cOMBC_nat_nat_bool,fequal_nat),Y_1)))) ) ) ).
fof(fact_257_image__cong,axiom,
! [F,G,M_3,N_3] :
( M_3 = N_3
=> ( ! [X_2] :
( hBOOL(hAPP_f1695230391l_bool(hAPP_H622608077l_bool(member1797258804iple_a,X_2),N_3))
=> hAPP_H905846293iple_a(F,X_2) = hAPP_H905846293iple_a(G,X_2) )
=> image_533441733iple_a(F,M_3) = image_533441733iple_a(G,N_3) ) ) ).
fof(fact_258_image__cong,axiom,
! [F,G,M_3,N_3] :
( M_3 = N_3
=> ( ! [X_2] :
( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_2),N_3))
=> hAPP_int_int(F,X_2) = hAPP_int_int(G,X_2) )
=> image_int_int(F,M_3) = image_int_int(G,N_3) ) ) ).
fof(fact_259_image__cong,axiom,
! [F,G,M_3,N_3] :
( M_3 = N_3
=> ( ! [X_2] :
( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_2),N_3))
=> hAPP_nat_nat(F,X_2) = hAPP_nat_nat(G,X_2) )
=> image_nat_nat(F,M_3) = image_nat_nat(G,N_3) ) ) ).
fof(fact_260_image__cong,axiom,
! [F,G,M_3,N_3] :
( M_3 = N_3
=> ( ! [X_2] :
( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_2),N_3))
=> hAPP_nat_int(F,X_2) = hAPP_nat_int(G,X_2) )
=> image_nat_int(F,M_3) = image_nat_int(G,N_3) ) ) ).
fof(fact_261_fold1Set_Ointros,axiom,
! [F,A_13,A,X_1] :
( hBOOL(hAPP_nat_bool(hAPP_f800510211t_bool(hAPP_n1512601776t_bool(finite929467206at_nat(F),A_13),A),X_1))
=> ( ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_13),A))
=> hBOOL(hAPP_nat_bool(finite_fold1Set_nat(F,hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_13),A)),X_1)) ) ) ).
fof(fact_262_fold1Set_Ointros,axiom,
! [F,A_13,A,X_1] :
( hBOOL(hAPP_H1037229737a_bool(hAPP_f20753329a_bool(hAPP_H1743777351a_bool(finite90892550iple_a(F),A_13),A),X_1))
=> ( ~ hBOOL(hAPP_f1695230391l_bool(hAPP_H622608077l_bool(member1797258804iple_a,A_13),A))
=> hBOOL(hAPP_H1037229737a_bool(finite1621280049iple_a(F,hAPP_f20753329a_bool(hAPP_H1743777351a_bool(insert956547291iple_a,A_13),A)),X_1)) ) ) ).
fof(fact_263_fold1Set_Ointros,axiom,
! [F,A_13,A,X_1] :
( hBOOL(hAPP_int_bool(hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(finite772772422nt_int(F),A_13),A),X_1))
=> ( ~ hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,A_13),A))
=> hBOOL(hAPP_int_bool(finite_fold1Set_int(F,hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,A_13),A)),X_1)) ) ) ).
fof(fact_264_the__equality,axiom,
! [Pa,A_13] :
( hBOOL(hAPP_int_bool(Pa,A_13))
=> ( ! [X_2] :
( hBOOL(hAPP_int_bool(Pa,X_2))
=> X_2 = A_13 )
=> hAPP_f1594865479ol_int(the_int,Pa) = A_13 ) ) ).
fof(fact_265_the__equality,axiom,
! [Pa,A_13] :
( hBOOL(hAPP_H1037229737a_bool(Pa,A_13))
=> ( ! [X_2] :
( hBOOL(hAPP_H1037229737a_bool(Pa,X_2))
=> X_2 = A_13 )
=> hAPP_f1693662087iple_a(the_Ho830102290iple_a,Pa) = A_13 ) ) ).
fof(fact_266_the__equality,axiom,
! [Pa,A_13] :
( hBOOL(hAPP_nat_bool(Pa,A_13))
=> ( ! [X_2] :
( hBOOL(hAPP_nat_bool(Pa,X_2))
=> X_2 = A_13 )
=> hAPP_f22106695ol_nat(the_nat,Pa) = A_13 ) ) ).
fof(fact_267_folding__one_Oinsert,axiom,
! [X_1,A,F,F_1] :
( hBOOL(finite988810631ne_nat(F,F_1))
=> ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
=> ( ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A))
=> ( A != bot_bot_fun_nat_bool
=> hAPP_f22106695ol_nat(F_1,hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),A)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(F,X_1),hAPP_f22106695ol_nat(F_1,A)) ) ) ) ) ).
fof(fact_268_folding__one_Oinsert,axiom,
! [X_1,A,F,F_1] :
( hBOOL(finite1753440478iple_a(F,F_1))
=> ( hBOOL(hAPP_f1695230391l_bool(finite1738664244iple_a,A))
=> ( ~ hBOOL(hAPP_f1695230391l_bool(hAPP_H622608077l_bool(member1797258804iple_a,X_1),A))
=> ( A != bot_bo797238721a_bool
=> hAPP_f1693662087iple_a(F_1,hAPP_f20753329a_bool(hAPP_H1743777351a_bool(insert956547291iple_a,X_1),A)) = hAPP_H905846293iple_a(hAPP_H2085992369iple_a(F,X_1),hAPP_f1693662087iple_a(F_1,A)) ) ) ) ) ).
fof(fact_269_folding__one_Oinsert,axiom,
! [X_1,A,F,F_1] :
( hBOOL(finite1626084323ne_int(F,F_1))
=> ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,A))
=> ( ~ hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),A))
=> ( A != bot_bot_fun_int_bool
=> hAPP_f1594865479ol_int(F_1,hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),A)) = hAPP_int_int(hAPP_int_fun_int_int(F,X_1),hAPP_f1594865479ol_int(F_1,A)) ) ) ) ) ).
fof(fact_270_folding__one_Oeq__fold,axiom,
! [A,F,F_1] :
( hBOOL(finite1753440478iple_a(F,F_1))
=> ( hBOOL(hAPP_f1695230391l_bool(finite1738664244iple_a,A))
=> hAPP_f1693662087iple_a(F_1,A) = hAPP_f1693662087iple_a(finite1465856449iple_a(F),A) ) ) ).
fof(fact_271_folding__one_Oeq__fold,axiom,
! [A,F,F_1] :
( hBOOL(finite988810631ne_nat(F,F_1))
=> ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
=> hAPP_f22106695ol_nat(F_1,A) = hAPP_f22106695ol_nat(finite_fold1_nat(F),A) ) ) ).
fof(fact_272_folding__one_Oeq__fold,axiom,
! [A,F,F_1] :
( hBOOL(finite1626084323ne_int(F,F_1))
=> ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,A))
=> hAPP_f1594865479ol_int(F_1,A) = hAPP_f1594865479ol_int(finite_fold1_int(F),A) ) ) ).
fof(fact_273_folding__one_Oclosed,axiom,
! [A,F,F_1] :
( hBOOL(finite988810631ne_nat(F,F_1))
=> ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
=> ( A != bot_bot_fun_nat_bool
=> ( ! [X_2,Y_21] : hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(F,X_2),Y_21)),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_2),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,Y_21),bot_bot_fun_nat_bool))))
=> hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,hAPP_f22106695ol_nat(F_1,A)),A)) ) ) ) ) ).
fof(fact_274_folding__one_Oclosed,axiom,
! [A,F,F_1] :
( hBOOL(finite1753440478iple_a(F,F_1))
=> ( hBOOL(hAPP_f1695230391l_bool(finite1738664244iple_a,A))
=> ( A != bot_bo797238721a_bool
=> ( ! [X_2,Y_21] : hBOOL(hAPP_f1695230391l_bool(hAPP_H622608077l_bool(member1797258804iple_a,hAPP_H905846293iple_a(hAPP_H2085992369iple_a(F,X_2),Y_21)),hAPP_f20753329a_bool(hAPP_H1743777351a_bool(insert956547291iple_a,X_2),hAPP_f20753329a_bool(hAPP_H1743777351a_bool(insert956547291iple_a,Y_21),bot_bo797238721a_bool))))
=> hBOOL(hAPP_f1695230391l_bool(hAPP_H622608077l_bool(member1797258804iple_a,hAPP_f1693662087iple_a(F_1,A)),A)) ) ) ) ) ).
fof(fact_275_folding__one_Oclosed,axiom,
! [A,F,F_1] :
( hBOOL(finite1626084323ne_int(F,F_1))
=> ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,A))
=> ( A != bot_bot_fun_int_bool
=> ( ! [X_2,Y_21] : hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hAPP_int_int(hAPP_int_fun_int_int(F,X_2),Y_21)),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_2),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,Y_21),bot_bot_fun_int_bool))))
=> hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hAPP_f1594865479ol_int(F_1,A)),A)) ) ) ) ) ).
fof(fact_276_finite__Collect__conjI,axiom,
! [Q_1,Pa] :
( ( hBOOL(hAPP_f1695230391l_bool(finite1738664244iple_a,hAPP_f20753329a_bool(collec351493750iple_a,Pa)))
| hBOOL(hAPP_f1695230391l_bool(finite1738664244iple_a,hAPP_f20753329a_bool(collec351493750iple_a,Q_1))) )
=> hBOOL(hAPP_f1695230391l_bool(finite1738664244iple_a,hAPP_f20753329a_bool(collec351493750iple_a,hAPP_f20753329a_bool(hAPP_f607497069a_bool(cOMBS_154988028l_bool,hAPP_f146754017l_bool(hAPP_f239971723l_bool(cOMBB_1866391387iple_a,fconj),Pa)),Q_1)))) ) ).
fof(fact_277_finite__Collect__conjI,axiom,
! [Q_1,Pa] :
( ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,hAPP_f1805168059t_bool(collect_int,Pa)))
| hBOOL(hAPP_f448129468l_bool(finite_finite_int,hAPP_f1805168059t_bool(collect_int,Q_1))) )
=> hBOOL(hAPP_f448129468l_bool(finite_finite_int,hAPP_f1805168059t_bool(collect_int,hAPP_f1805168059t_bool(hAPP_f727283836t_bool(cOMBS_int_bool_bool,hAPP_f2144054103l_bool(hAPP_f1734373249l_bool(cOMBB_1652995168ol_int,fconj),Pa)),Q_1)))) ) ).
fof(fact_278_finite__Collect__conjI,axiom,
! [Q_1,Pa] :
( ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,hAPP_f800510211t_bool(collect_nat,Pa)))
| hBOOL(hAPP_f54304608l_bool(finite_finite_nat,hAPP_f800510211t_bool(collect_nat,Q_1))) )
=> hBOOL(hAPP_f54304608l_bool(finite_finite_nat,hAPP_f800510211t_bool(collect_nat,hAPP_f800510211t_bool(hAPP_f561022312t_bool(cOMBS_nat_bool_bool,hAPP_f1146629647l_bool(hAPP_f1080886329l_bool(cOMBB_1015721476ol_nat,fconj),Pa)),Q_1)))) ) ).
fof(fact_279_finite_OemptyI,axiom,
hBOOL(hAPP_f1695230391l_bool(finite1738664244iple_a,bot_bo797238721a_bool)) ).
fof(fact_280_finite_OemptyI,axiom,
hBOOL(hAPP_f54304608l_bool(finite_finite_nat,bot_bot_fun_nat_bool)) ).
fof(fact_281_finite_OemptyI,axiom,
hBOOL(hAPP_f448129468l_bool(finite_finite_int,bot_bot_fun_int_bool)) ).
fof(fact_282_finite_OinsertI,axiom,
! [A_13,A] :
( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
=> hBOOL(hAPP_f54304608l_bool(finite_finite_nat,hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_13),A))) ) ).
fof(fact_283_finite_OinsertI,axiom,
! [A_13,A] :
( hBOOL(hAPP_f1695230391l_bool(finite1738664244iple_a,A))
=> hBOOL(hAPP_f1695230391l_bool(finite1738664244iple_a,hAPP_f20753329a_bool(hAPP_H1743777351a_bool(insert956547291iple_a,A_13),A))) ) ).
fof(fact_284_finite_OinsertI,axiom,
! [A_13,A] :
( hBOOL(hAPP_f448129468l_bool(finite_finite_int,A))
=> hBOOL(hAPP_f448129468l_bool(finite_finite_int,hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,A_13),A))) ) ).
fof(fact_285_finite__imageI,axiom,
! [H,F_1] :
( hBOOL(hAPP_f1695230391l_bool(finite1738664244iple_a,F_1))
=> hBOOL(hAPP_f1695230391l_bool(finite1738664244iple_a,image_533441733iple_a(H,F_1))) ) ).
fof(fact_286_finite__imageI,axiom,
! [H,F_1] :
( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,F_1))
=> hBOOL(hAPP_f1695230391l_bool(finite1738664244iple_a,image_1865589340iple_a(H,F_1))) ) ).
fof(fact_287_finite__imageI,axiom,
! [H,F_1] :
( hBOOL(hAPP_f448129468l_bool(finite_finite_int,F_1))
=> hBOOL(hAPP_f1695230391l_bool(finite1738664244iple_a,image_1844722432iple_a(H,F_1))) ) ).
fof(fact_288_finite__imageI,axiom,
! [H,F_1] :
( hBOOL(hAPP_f1695230391l_bool(finite1738664244iple_a,F_1))
=> hBOOL(hAPP_f54304608l_bool(finite_finite_nat,image_48635758_a_nat(H,F_1))) ) ).
fof(fact_289_finite__imageI,axiom,
! [H,F_1] :
( hBOOL(hAPP_f448129468l_bool(finite_finite_int,F_1))
=> hBOOL(hAPP_f448129468l_bool(finite_finite_int,image_int_int(H,F_1))) ) ).
fof(fact_290_finite__imageI,axiom,
! [H,F_1] :
( hBOOL(hAPP_f1695230391l_bool(finite1738664244iple_a,F_1))
=> hBOOL(hAPP_f448129468l_bool(finite_finite_int,image_685909450_a_int(H,F_1))) ) ).
fof(fact_291_finite__imageI,axiom,
! [H,F_1] :
( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,F_1))
=> hBOOL(hAPP_f54304608l_bool(finite_finite_nat,image_nat_nat(H,F_1))) ) ).
fof(fact_292_finite__imageI,axiom,
! [H,F_1] :
( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,F_1))
=> hBOOL(hAPP_f448129468l_bool(finite_finite_int,image_nat_int(H,F_1))) ) ).
fof(fact_293_finite__Collect__disjI,axiom,
! [Pa,Q_1] :
( hBOOL(hAPP_f1695230391l_bool(finite1738664244iple_a,hAPP_f20753329a_bool(collec351493750iple_a,hAPP_f20753329a_bool(hAPP_f607497069a_bool(cOMBS_154988028l_bool,hAPP_f146754017l_bool(hAPP_f239971723l_bool(cOMBB_1866391387iple_a,fdisj),Pa)),Q_1))))
<=> ( hBOOL(hAPP_f1695230391l_bool(finite1738664244iple_a,hAPP_f20753329a_bool(collec351493750iple_a,Pa)))
& hBOOL(hAPP_f1695230391l_bool(finite1738664244iple_a,hAPP_f20753329a_bool(collec351493750iple_a,Q_1))) ) ) ).
fof(fact_294_finite__Collect__disjI,axiom,
! [Pa,Q_1] :
( hBOOL(hAPP_f448129468l_bool(finite_finite_int,hAPP_f1805168059t_bool(collect_int,hAPP_f1805168059t_bool(hAPP_f727283836t_bool(cOMBS_int_bool_bool,hAPP_f2144054103l_bool(hAPP_f1734373249l_bool(cOMBB_1652995168ol_int,fdisj),Pa)),Q_1))))
<=> ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,hAPP_f1805168059t_bool(collect_int,Pa)))
& hBOOL(hAPP_f448129468l_bool(finite_finite_int,hAPP_f1805168059t_bool(collect_int,Q_1))) ) ) ).
fof(fact_295_finite__Collect__disjI,axiom,
! [Pa,Q_1] :
( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,hAPP_f800510211t_bool(collect_nat,hAPP_f800510211t_bool(hAPP_f561022312t_bool(cOMBS_nat_bool_bool,hAPP_f1146629647l_bool(hAPP_f1080886329l_bool(cOMBB_1015721476ol_nat,fdisj),Pa)),Q_1))))
<=> ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,hAPP_f800510211t_bool(collect_nat,Pa)))
& hBOOL(hAPP_f54304608l_bool(finite_finite_nat,hAPP_f800510211t_bool(collect_nat,Q_1))) ) ) ).
fof(fact_296_finite__insert,axiom,
! [A_13,A] :
( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_13),A)))
<=> hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A)) ) ).
fof(fact_297_finite__insert,axiom,
! [A_13,A] :
( hBOOL(hAPP_f1695230391l_bool(finite1738664244iple_a,hAPP_f20753329a_bool(hAPP_H1743777351a_bool(insert956547291iple_a,A_13),A)))
<=> hBOOL(hAPP_f1695230391l_bool(finite1738664244iple_a,A)) ) ).
fof(fact_298_finite__insert,axiom,
! [A_13,A] :
( hBOOL(hAPP_f448129468l_bool(finite_finite_int,hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,A_13),A)))
<=> hBOOL(hAPP_f448129468l_bool(finite_finite_int,A)) ) ).
fof(fact_299_fold__graph_OemptyI,axiom,
! [F,Z_1] : hBOOL(hAPP_H1037229737a_bool(hAPP_f20753329a_bool(hAPP_H1743777351a_bool(finite90892550iple_a(F),Z_1),bot_bo797238721a_bool),Z_1)) ).
fof(fact_300_fold__graph_OemptyI,axiom,
! [F,Z_1] : hBOOL(hAPP_int_bool(hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(finite772772422nt_int(F),Z_1),bot_bot_fun_int_bool),Z_1)) ).
fof(fact_301_fold__graph_OemptyI,axiom,
! [F,Z_1] : hBOOL(hAPP_nat_bool(hAPP_f800510211t_bool(hAPP_n1512601776t_bool(finite929467206at_nat(F),Z_1),bot_bot_fun_nat_bool),Z_1)) ).
fof(fact_302_empty__fold__graphE,axiom,
! [F,Z_1,X_1] :
( hBOOL(hAPP_H1037229737a_bool(hAPP_f20753329a_bool(hAPP_H1743777351a_bool(finite90892550iple_a(F),Z_1),bot_bo797238721a_bool),X_1))
=> X_1 = Z_1 ) ).
fof(fact_303_empty__fold__graphE,axiom,
! [F,Z_1,X_1] :
( hBOOL(hAPP_int_bool(hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(finite772772422nt_int(F),Z_1),bot_bot_fun_int_bool),X_1))
=> X_1 = Z_1 ) ).
fof(fact_304_empty__fold__graphE,axiom,
! [F,Z_1,X_1] :
( hBOOL(hAPP_nat_bool(hAPP_f800510211t_bool(hAPP_n1512601776t_bool(finite929467206at_nat(F),Z_1),bot_bot_fun_nat_bool),X_1))
=> X_1 = Z_1 ) ).
fof(fact_305_fold__graph_OinsertI,axiom,
! [F,Z_1,Y_1,X_1,A] :
( ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A))
=> ( hBOOL(hAPP_nat_bool(hAPP_f800510211t_bool(hAPP_n1512601776t_bool(finite929467206at_nat(F),Z_1),A),Y_1))
=> hBOOL(hAPP_nat_bool(hAPP_f800510211t_bool(hAPP_n1512601776t_bool(finite929467206at_nat(F),Z_1),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),A)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(F,X_1),Y_1))) ) ) ).
fof(fact_306_fold__graph_OinsertI,axiom,
! [F,Z_1,Y_1,X_1,A] :
( ~ hBOOL(hAPP_f1695230391l_bool(hAPP_H622608077l_bool(member1797258804iple_a,X_1),A))
=> ( hBOOL(hAPP_H1037229737a_bool(hAPP_f20753329a_bool(hAPP_H1743777351a_bool(finite90892550iple_a(F),Z_1),A),Y_1))
=> hBOOL(hAPP_H1037229737a_bool(hAPP_f20753329a_bool(hAPP_H1743777351a_bool(finite90892550iple_a(F),Z_1),hAPP_f20753329a_bool(hAPP_H1743777351a_bool(insert956547291iple_a,X_1),A)),hAPP_H905846293iple_a(hAPP_H2085992369iple_a(F,X_1),Y_1))) ) ) ).
fof(fact_307_fold__graph_OinsertI,axiom,
! [F,Z_1,Y_1,X_1,A] :
( ~ hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),A))
=> ( hBOOL(hAPP_int_bool(hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(finite772772422nt_int(F),Z_1),A),Y_1))
=> hBOOL(hAPP_int_bool(hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(finite772772422nt_int(F),Z_1),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),A)),hAPP_int_int(hAPP_int_fun_int_int(F,X_1),Y_1))) ) ) ).
fof(fact_308_insert__fold1SetE,axiom,
! [F,A_13,X_28,X_1] :
( hBOOL(hAPP_nat_bool(finite_fold1Set_nat(F,hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_13),X_28)),X_1))
=> ~ ! [A_12,A_43] :
( hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_13),X_28) = hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_12),A_43)
=> ( hBOOL(hAPP_nat_bool(hAPP_f800510211t_bool(hAPP_n1512601776t_bool(finite929467206at_nat(F),A_12),A_43),X_1))
=> hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_12),A_43)) ) ) ) ).
fof(fact_309_insert__fold1SetE,axiom,
! [F,A_13,X_28,X_1] :
( hBOOL(hAPP_H1037229737a_bool(finite1621280049iple_a(F,hAPP_f20753329a_bool(hAPP_H1743777351a_bool(insert956547291iple_a,A_13),X_28)),X_1))
=> ~ ! [A_12,A_43] :
( hAPP_f20753329a_bool(hAPP_H1743777351a_bool(insert956547291iple_a,A_13),X_28) = hAPP_f20753329a_bool(hAPP_H1743777351a_bool(insert956547291iple_a,A_12),A_43)
=> ( hBOOL(hAPP_H1037229737a_bool(hAPP_f20753329a_bool(hAPP_H1743777351a_bool(finite90892550iple_a(F),A_12),A_43),X_1))
=> hBOOL(hAPP_f1695230391l_bool(hAPP_H622608077l_bool(member1797258804iple_a,A_12),A_43)) ) ) ) ).
fof(fact_310_insert__fold1SetE,axiom,
! [F,A_13,X_28,X_1] :
( hBOOL(hAPP_int_bool(finite_fold1Set_int(F,hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,A_13),X_28)),X_1))
=> ~ ! [A_12,A_43] :
( hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,A_13),X_28) = hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,A_12),A_43)
=> ( hBOOL(hAPP_int_bool(hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(finite772772422nt_int(F),A_12),A_43),X_1))
=> hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,A_12),A_43)) ) ) ) ).
fof(fact_311_finite__nonempty__imp__fold1Set,axiom,
! [F,A] :
( hBOOL(hAPP_f1695230391l_bool(finite1738664244iple_a,A))
=> ( A != bot_bo797238721a_bool
=> ? [X1] : hBOOL(hAPP_H1037229737a_bool(finite1621280049iple_a(F,A),X1)) ) ) ).
fof(fact_312_finite__nonempty__imp__fold1Set,axiom,
! [F,A] :
( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
=> ( A != bot_bot_fun_nat_bool
=> ? [X1] : hBOOL(hAPP_nat_bool(finite_fold1Set_nat(F,A),X1)) ) ) ).
fof(fact_313_finite__nonempty__imp__fold1Set,axiom,
! [F,A] :
( hBOOL(hAPP_f448129468l_bool(finite_finite_int,A))
=> ( A != bot_bot_fun_int_bool
=> ? [X1] : hBOOL(hAPP_int_bool(finite_fold1Set_int(F,A),X1)) ) ) ).
fof(fact_314_finite_Osimps,axiom,
! [A_13] :
( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A_13))
<=> ( A_13 = bot_bot_fun_nat_bool
| ? [A_43,A_12] :
( A_13 = hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_12),A_43)
& hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A_43)) ) ) ) ).
fof(fact_315_finite_Osimps,axiom,
! [A_13] :
( hBOOL(hAPP_f1695230391l_bool(finite1738664244iple_a,A_13))
<=> ( A_13 = bot_bo797238721a_bool
| ? [A_43,A_12] :
( A_13 = hAPP_f20753329a_bool(hAPP_H1743777351a_bool(insert956547291iple_a,A_12),A_43)
& hBOOL(hAPP_f1695230391l_bool(finite1738664244iple_a,A_43)) ) ) ) ).
fof(fact_316_finite_Osimps,axiom,
! [A_13] :
( hBOOL(hAPP_f448129468l_bool(finite_finite_int,A_13))
<=> ( A_13 = bot_bot_fun_int_bool
| ? [A_43,A_12] :
( A_13 = hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,A_12),A_43)
& hBOOL(hAPP_f448129468l_bool(finite_finite_int,A_43)) ) ) ) ).
fof(fact_317_finite__induct,axiom,
! [Pa,F_1] :
( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,F_1))
=> ( hBOOL(hAPP_f54304608l_bool(Pa,bot_bot_fun_nat_bool))
=> ( ! [X_2,F_2] :
( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,F_2))
=> ( ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_2),F_2))
=> ( hBOOL(hAPP_f54304608l_bool(Pa,F_2))
=> hBOOL(hAPP_f54304608l_bool(Pa,hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_2),F_2))) ) ) )
=> hBOOL(hAPP_f54304608l_bool(Pa,F_1)) ) ) ) ).
fof(fact_318_finite__induct,axiom,
! [Pa,F_1] :
( hBOOL(hAPP_f1695230391l_bool(finite1738664244iple_a,F_1))
=> ( hBOOL(hAPP_f1695230391l_bool(Pa,bot_bo797238721a_bool))
=> ( ! [X_2,F_2] :
( hBOOL(hAPP_f1695230391l_bool(finite1738664244iple_a,F_2))
=> ( ~ hBOOL(hAPP_f1695230391l_bool(hAPP_H622608077l_bool(member1797258804iple_a,X_2),F_2))
=> ( hBOOL(hAPP_f1695230391l_bool(Pa,F_2))
=> hBOOL(hAPP_f1695230391l_bool(Pa,hAPP_f20753329a_bool(hAPP_H1743777351a_bool(insert956547291iple_a,X_2),F_2))) ) ) )
=> hBOOL(hAPP_f1695230391l_bool(Pa,F_1)) ) ) ) ).
fof(fact_319_finite__induct,axiom,
! [Pa,F_1] :
( hBOOL(hAPP_f448129468l_bool(finite_finite_int,F_1))
=> ( hBOOL(hAPP_f448129468l_bool(Pa,bot_bot_fun_int_bool))
=> ( ! [X_2,F_2] :
( hBOOL(hAPP_f448129468l_bool(finite_finite_int,F_2))
=> ( ~ hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_2),F_2))
=> ( hBOOL(hAPP_f448129468l_bool(Pa,F_2))
=> hBOOL(hAPP_f448129468l_bool(Pa,hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_2),F_2))) ) ) )
=> hBOOL(hAPP_f448129468l_bool(Pa,F_1)) ) ) ) ).
fof(fact_320_pigeonhole__infinite,axiom,
! [F,A] :
( ~ hBOOL(hAPP_f1695230391l_bool(finite1738664244iple_a,A))
=> ( hBOOL(hAPP_f1695230391l_bool(finite1738664244iple_a,image_533441733iple_a(F,A)))
=> ? [X_2] :
( hBOOL(hAPP_f1695230391l_bool(hAPP_H622608077l_bool(member1797258804iple_a,X_2),A))
& ~ hBOOL(hAPP_f1695230391l_bool(finite1738664244iple_a,hAPP_f20753329a_bool(collec351493750iple_a,hAPP_f20753329a_bool(hAPP_f607497069a_bool(cOMBS_154988028l_bool,hAPP_f146754017l_bool(hAPP_f239971723l_bool(cOMBB_1866391387iple_a,fconj),hAPP_f20753329a_bool(hAPP_f247708275a_bool(cOMBC_1519416966l_bool,member1797258804iple_a),A))),hAPP_H426895267a_bool(hAPP_f920293029a_bool(cOMBC_1940896922a_bool,hAPP_f960023481a_bool(hAPP_f1786170683a_bool(cOMBB_683172471iple_a,fequal963300192iple_a),F)),hAPP_H905846293iple_a(F,X_2)))))) ) ) ) ).
fof(fact_321_pigeonhole__infinite,axiom,
! [F,A] :
( ~ hBOOL(hAPP_f1695230391l_bool(finite1738664244iple_a,A))
=> ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,image_48635758_a_nat(F,A)))
=> ? [X_2] :
( hBOOL(hAPP_f1695230391l_bool(hAPP_H622608077l_bool(member1797258804iple_a,X_2),A))
& ~ hBOOL(hAPP_f1695230391l_bool(finite1738664244iple_a,hAPP_f20753329a_bool(collec351493750iple_a,hAPP_f20753329a_bool(hAPP_f607497069a_bool(cOMBS_154988028l_bool,hAPP_f146754017l_bool(hAPP_f239971723l_bool(cOMBB_1866391387iple_a,fconj),hAPP_f20753329a_bool(hAPP_f247708275a_bool(cOMBC_1519416966l_bool,member1797258804iple_a),A))),hAPP_n1919155532a_bool(hAPP_f228417655a_bool(cOMBC_364739t_bool,hAPP_f1872860171t_bool(hAPP_f1981448543t_bool(cOMBB_1391527205iple_a,fequal_nat),F)),hAPP_H592031934_a_nat(F,X_2)))))) ) ) ) ).
fof(fact_322_pigeonhole__infinite,axiom,
! [F,A] :
( ~ hBOOL(hAPP_f448129468l_bool(finite_finite_int,A))
=> ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,image_int_nat(F,A)))
=> ? [X_2] :
( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_2),A))
& ~ hBOOL(hAPP_f448129468l_bool(finite_finite_int,hAPP_f1805168059t_bool(collect_int,hAPP_f1805168059t_bool(hAPP_f727283836t_bool(cOMBS_int_bool_bool,hAPP_f2144054103l_bool(hAPP_f1734373249l_bool(cOMBB_1652995168ol_int,fconj),hAPP_f1805168059t_bool(hAPP_f202917053t_bool(cOMBC_94739984l_bool,member_int),A))),hAPP_n1082236369t_bool(hAPP_f175561985t_bool(cOMBC_int_nat_bool,hAPP_f876579787t_bool(hAPP_f2026117279t_bool(cOMBB_1437810218ol_int,fequal_nat),F)),hAPP_int_nat(F,X_2)))))) ) ) ) ).
fof(fact_323_pigeonhole__infinite,axiom,
! [F,A] :
( ~ hBOOL(hAPP_f1695230391l_bool(finite1738664244iple_a,A))
=> ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,image_685909450_a_int(F,A)))
=> ? [X_2] :
( hBOOL(hAPP_f1695230391l_bool(hAPP_H622608077l_bool(member1797258804iple_a,X_2),A))
& ~ hBOOL(hAPP_f1695230391l_bool(finite1738664244iple_a,hAPP_f20753329a_bool(collec351493750iple_a,hAPP_f20753329a_bool(hAPP_f607497069a_bool(cOMBS_154988028l_bool,hAPP_f146754017l_bool(hAPP_f239971723l_bool(cOMBB_1866391387iple_a,fconj),hAPP_f20753329a_bool(hAPP_f247708275a_bool(cOMBC_1519416966l_bool,member1797258804iple_a),A))),hAPP_i611776424a_bool(hAPP_f920351535a_bool(cOMBC_1530706207t_bool,hAPP_f2142913731t_bool(hAPP_f1329415375t_bool(cOMBB_1505691757iple_a,fequal_int),F)),hAPP_H1229305626_a_int(F,X_2)))))) ) ) ) ).
fof(fact_324_pigeonhole__infinite,axiom,
! [F,A] :
( ~ hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
=> ( hBOOL(hAPP_f1695230391l_bool(finite1738664244iple_a,image_1865589340iple_a(F,A)))
=> ? [X_2] :
( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_2),A))
& ~ hBOOL(hAPP_f54304608l_bool(finite_finite_nat,hAPP_f800510211t_bool(collect_nat,hAPP_f800510211t_bool(hAPP_f561022312t_bool(cOMBS_nat_bool_bool,hAPP_f1146629647l_bool(hAPP_f1080886329l_bool(cOMBB_1015721476ol_nat,fconj),hAPP_f800510211t_bool(hAPP_f1722879237t_bool(cOMBC_226598744l_bool,member_nat),A))),hAPP_H144868812t_bool(hAPP_f1886543991t_bool(cOMBC_533513393a_bool,hAPP_f260875897a_bool(hAPP_f1930382715a_bool(cOMBB_1063143712ol_nat,fequal963300192iple_a),F)),hAPP_n261501868iple_a(F,X_2)))))) ) ) ) ).
fof(fact_325_pigeonhole__infinite,axiom,
! [F,A] :
( ~ hBOOL(hAPP_f448129468l_bool(finite_finite_int,A))
=> ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,image_int_int(F,A)))
=> ? [X_2] :
( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_2),A))
& ~ hBOOL(hAPP_f448129468l_bool(finite_finite_int,hAPP_f1805168059t_bool(collect_int,hAPP_f1805168059t_bool(hAPP_f727283836t_bool(cOMBS_int_bool_bool,hAPP_f2144054103l_bool(hAPP_f1734373249l_bool(cOMBB_1652995168ol_int,fconj),hAPP_f1805168059t_bool(hAPP_f202917053t_bool(cOMBC_94739984l_bool,member_int),A))),hAPP_i1948725293t_bool(hAPP_f428220345t_bool(cOMBC_int_int_bool,hAPP_f1791153283t_bool(hAPP_f1399575567t_bool(cOMBB_118231410ol_int,fequal_int),F)),hAPP_int_int(F,X_2)))))) ) ) ) ).
fof(fact_326_pigeonhole__infinite,axiom,
! [F,A] :
( ~ hBOOL(hAPP_f448129468l_bool(finite_finite_int,A))
=> ( hBOOL(hAPP_f1695230391l_bool(finite1738664244iple_a,image_1844722432iple_a(F,A)))
=> ? [X_2] :
( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_2),A))
& ~ hBOOL(hAPP_f448129468l_bool(finite_finite_int,hAPP_f1805168059t_bool(collect_int,hAPP_f1805168059t_bool(hAPP_f727283836t_bool(cOMBS_int_bool_bool,hAPP_f2144054103l_bool(hAPP_f1734373249l_bool(cOMBB_1652995168ol_int,fconj),hAPP_f1805168059t_bool(hAPP_f202917053t_bool(cOMBC_94739984l_bool,member_int),A))),hAPP_H1675210280t_bool(hAPP_f936407855t_bool(cOMBC_1719122773a_bool,hAPP_f298605433a_bool(hAPP_f1421841531a_bool(cOMBB_1700417404ol_int,fequal963300192iple_a),F)),hAPP_i240634960iple_a(F,X_2)))))) ) ) ) ).
fof(fact_327_pigeonhole__infinite,axiom,
! [F,A] :
( ~ hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
=> ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,image_nat_nat(F,A)))
=> ? [X_2] :
( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_2),A))
& ~ hBOOL(hAPP_f54304608l_bool(finite_finite_nat,hAPP_f800510211t_bool(collect_nat,hAPP_f800510211t_bool(hAPP_f561022312t_bool(cOMBS_nat_bool_bool,hAPP_f1146629647l_bool(hAPP_f1080886329l_bool(cOMBB_1015721476ol_nat,fconj),hAPP_f800510211t_bool(hAPP_f1722879237t_bool(cOMBC_226598744l_bool,member_nat),A))),hAPP_n1699378549t_bool(hAPP_f229349961t_bool(cOMBC_nat_nat_bool,hAPP_f618557131t_bool(hAPP_f1505651103t_bool(cOMBB_800536526ol_nat,fequal_nat),F)),hAPP_nat_nat(F,X_2)))))) ) ) ) ).
fof(fact_328_pigeonhole__infinite,axiom,
! [F,A] :
( ~ hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
=> ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,image_nat_int(F,A)))
=> ? [X_2] :
( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_2),A))
& ~ hBOOL(hAPP_f54304608l_bool(finite_finite_nat,hAPP_f800510211t_bool(collect_nat,hAPP_f800510211t_bool(hAPP_f561022312t_bool(cOMBS_nat_bool_bool,hAPP_f1146629647l_bool(hAPP_f1080886329l_bool(cOMBB_1015721476ol_nat,fconj),hAPP_f800510211t_bool(hAPP_f1722879237t_bool(cOMBC_226598744l_bool,member_nat),A))),hAPP_i418383825t_bool(hAPP_f482008321t_bool(cOMBC_nat_int_bool,hAPP_f1533130627t_bool(hAPP_f879109391t_bool(cOMBB_1628441366ol_nat,fequal_int),F)),hAPP_nat_int(F,X_2)))))) ) ) ) ).
fof(fact_329_finite__imp__fold__graph,axiom,
! [F,Z_1,A] :
( hBOOL(hAPP_f1695230391l_bool(finite1738664244iple_a,A))
=> ? [X1] : hBOOL(hAPP_H1037229737a_bool(hAPP_f20753329a_bool(hAPP_H1743777351a_bool(finite90892550iple_a(F),Z_1),A),X1)) ) ).
fof(fact_330_finite__imp__fold__graph,axiom,
! [F,Z_1,A] :
( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
=> ? [X1] : hBOOL(hAPP_nat_bool(hAPP_f800510211t_bool(hAPP_n1512601776t_bool(finite929467206at_nat(F),Z_1),A),X1)) ) ).
fof(fact_331_finite__imp__fold__graph,axiom,
! [F,Z_1,A] :
( hBOOL(hAPP_f448129468l_bool(finite_finite_int,A))
=> ? [X1] : hBOOL(hAPP_int_bool(hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(finite772772422nt_int(F),Z_1),A),X1)) ) ).
fof(fact_332_fold1Set_Osimps,axiom,
! [F,A1,A2] :
( hBOOL(hAPP_nat_bool(finite_fold1Set_nat(F,A1),A2))
<=> ? [A_12,A_43,X_2] :
( A1 = hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_12),A_43)
& A2 = X_2
& hBOOL(hAPP_nat_bool(hAPP_f800510211t_bool(hAPP_n1512601776t_bool(finite929467206at_nat(F),A_12),A_43),X_2))
& ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_12),A_43)) ) ) ).
fof(fact_333_fold1Set_Osimps,axiom,
! [F,A1,A2] :
( hBOOL(hAPP_H1037229737a_bool(finite1621280049iple_a(F,A1),A2))
<=> ? [A_12,A_43,X_2] :
( A1 = hAPP_f20753329a_bool(hAPP_H1743777351a_bool(insert956547291iple_a,A_12),A_43)
& A2 = X_2
& hBOOL(hAPP_H1037229737a_bool(hAPP_f20753329a_bool(hAPP_H1743777351a_bool(finite90892550iple_a(F),A_12),A_43),X_2))
& ~ hBOOL(hAPP_f1695230391l_bool(hAPP_H622608077l_bool(member1797258804iple_a,A_12),A_43)) ) ) ).
fof(fact_334_fold1Set_Osimps,axiom,
! [F,A1,A2] :
( hBOOL(hAPP_int_bool(finite_fold1Set_int(F,A1),A2))
<=> ? [A_12,A_43,X_2] :
( A1 = hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,A_12),A_43)
& A2 = X_2
& hBOOL(hAPP_int_bool(hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(finite772772422nt_int(F),A_12),A_43),X_2))
& ~ hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,A_12),A_43)) ) ) ).
fof(fact_335_theI,axiom,
! [Pa,A_13] :
( hBOOL(hAPP_int_bool(Pa,A_13))
=> ( ! [X_2] :
( hBOOL(hAPP_int_bool(Pa,X_2))
=> X_2 = A_13 )
=> hBOOL(hAPP_int_bool(Pa,hAPP_f1594865479ol_int(the_int,Pa))) ) ) ).
fof(fact_336_theI,axiom,
! [Pa,A_13] :
( hBOOL(hAPP_H1037229737a_bool(Pa,A_13))
=> ( ! [X_2] :
( hBOOL(hAPP_H1037229737a_bool(Pa,X_2))
=> X_2 = A_13 )
=> hBOOL(hAPP_H1037229737a_bool(Pa,hAPP_f1693662087iple_a(the_Ho830102290iple_a,Pa))) ) ) ).
fof(fact_337_theI,axiom,
! [Pa,A_13] :
( hBOOL(hAPP_nat_bool(Pa,A_13))
=> ( ! [X_2] :
( hBOOL(hAPP_nat_bool(Pa,X_2))
=> X_2 = A_13 )
=> hBOOL(hAPP_nat_bool(Pa,hAPP_f22106695ol_nat(the_nat,Pa))) ) ) ).
fof(fact_338_the1__equality,axiom,
! [A_13,Pa] :
( ? [X_2] :
( hBOOL(hAPP_int_bool(Pa,X_2))
& ! [Y_21] :
( hBOOL(hAPP_int_bool(Pa,Y_21))
=> Y_21 = X_2 ) )
=> ( hBOOL(hAPP_int_bool(Pa,A_13))
=> hAPP_f1594865479ol_int(the_int,Pa) = A_13 ) ) ).
fof(fact_339_the1__equality,axiom,
! [A_13,Pa] :
( ? [X_2] :
( hBOOL(hAPP_H1037229737a_bool(Pa,X_2))
& ! [Y_21] :
( hBOOL(hAPP_H1037229737a_bool(Pa,Y_21))
=> Y_21 = X_2 ) )
=> ( hBOOL(hAPP_H1037229737a_bool(Pa,A_13))
=> hAPP_f1693662087iple_a(the_Ho830102290iple_a,Pa) = A_13 ) ) ).
fof(fact_340_the1__equality,axiom,
! [A_13,Pa] :
( ? [X_2] :
( hBOOL(hAPP_nat_bool(Pa,X_2))
& ! [Y_21] :
( hBOOL(hAPP_nat_bool(Pa,Y_21))
=> Y_21 = X_2 ) )
=> ( hBOOL(hAPP_nat_bool(Pa,A_13))
=> hAPP_f22106695ol_nat(the_nat,Pa) = A_13 ) ) ).
fof(fact_341_theI_H,axiom,
! [Pa] :
( ? [X_2] :
( hBOOL(hAPP_int_bool(Pa,X_2))
& ! [Y_21] :
( hBOOL(hAPP_int_bool(Pa,Y_21))
=> Y_21 = X_2 ) )
=> hBOOL(hAPP_int_bool(Pa,hAPP_f1594865479ol_int(the_int,Pa))) ) ).
fof(fact_342_theI_H,axiom,
! [Pa] :
( ? [X_2] :
( hBOOL(hAPP_H1037229737a_bool(Pa,X_2))
& ! [Y_21] :
( hBOOL(hAPP_H1037229737a_bool(Pa,Y_21))
=> Y_21 = X_2 ) )
=> hBOOL(hAPP_H1037229737a_bool(Pa,hAPP_f1693662087iple_a(the_Ho830102290iple_a,Pa))) ) ).
fof(fact_343_theI_H,axiom,
! [Pa] :
( ? [X_2] :
( hBOOL(hAPP_nat_bool(Pa,X_2))
& ! [Y_21] :
( hBOOL(hAPP_nat_bool(Pa,Y_21))
=> Y_21 = X_2 ) )
=> hBOOL(hAPP_nat_bool(Pa,hAPP_f22106695ol_nat(the_nat,Pa))) ) ).
fof(fact_344_fold__graph_Osimps,axiom,
! [F,Z_1,A1,A2] :
( hBOOL(hAPP_nat_bool(hAPP_f800510211t_bool(hAPP_n1512601776t_bool(finite929467206at_nat(F),Z_1),A1),A2))
<=> ( ( A1 = bot_bot_fun_nat_bool
& A2 = Z_1 )
| ? [X_2,A_43,Y_21] :
( A1 = hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_2),A_43)
& A2 = hAPP_nat_nat(hAPP_nat_fun_nat_nat(F,X_2),Y_21)
& ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_2),A_43))
& hBOOL(hAPP_nat_bool(hAPP_f800510211t_bool(hAPP_n1512601776t_bool(finite929467206at_nat(F),Z_1),A_43),Y_21)) ) ) ) ).
fof(fact_345_fold__graph_Osimps,axiom,
! [F,Z_1,A1,A2] :
( hBOOL(hAPP_H1037229737a_bool(hAPP_f20753329a_bool(hAPP_H1743777351a_bool(finite90892550iple_a(F),Z_1),A1),A2))
<=> ( ( A1 = bot_bo797238721a_bool
& A2 = Z_1 )
| ? [X_2,A_43,Y_21] :
( A1 = hAPP_f20753329a_bool(hAPP_H1743777351a_bool(insert956547291iple_a,X_2),A_43)
& A2 = hAPP_H905846293iple_a(hAPP_H2085992369iple_a(F,X_2),Y_21)
& ~ hBOOL(hAPP_f1695230391l_bool(hAPP_H622608077l_bool(member1797258804iple_a,X_2),A_43))
& hBOOL(hAPP_H1037229737a_bool(hAPP_f20753329a_bool(hAPP_H1743777351a_bool(finite90892550iple_a(F),Z_1),A_43),Y_21)) ) ) ) ).
fof(fact_346_fold__graph_Osimps,axiom,
! [F,Z_1,A1,A2] :
( hBOOL(hAPP_int_bool(hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(finite772772422nt_int(F),Z_1),A1),A2))
<=> ( ( A1 = bot_bot_fun_int_bool
& A2 = Z_1 )
| ? [X_2,A_43,Y_21] :
( A1 = hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_2),A_43)
& A2 = hAPP_int_int(hAPP_int_fun_int_int(F,X_2),Y_21)
& ~ hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_2),A_43))
& hBOOL(hAPP_int_bool(hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(finite772772422nt_int(F),Z_1),A_43),Y_21)) ) ) ) ).
fof(fact_347_folding__one__idem_Oinsert__idem,axiom,
! [X_1,A,F,F_1] :
( hBOOL(finite795500164em_nat(F,F_1))
=> ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
=> ( A != bot_bot_fun_nat_bool
=> hAPP_f22106695ol_nat(F_1,hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),A)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(F,X_1),hAPP_f22106695ol_nat(F_1,A)) ) ) ) ).
fof(fact_348_folding__one__idem_Oinsert__idem,axiom,
! [X_1,A,F,F_1] :
( hBOOL(finite1033474011iple_a(F,F_1))
=> ( hBOOL(hAPP_f1695230391l_bool(finite1738664244iple_a,A))
=> ( A != bot_bo797238721a_bool
=> hAPP_f1693662087iple_a(F_1,hAPP_f20753329a_bool(hAPP_H1743777351a_bool(insert956547291iple_a,X_1),A)) = hAPP_H905846293iple_a(hAPP_H2085992369iple_a(F,X_1),hAPP_f1693662087iple_a(F_1,A)) ) ) ) ).
fof(fact_349_folding__one__idem_Oinsert__idem,axiom,
! [X_1,A,F,F_1] :
( hBOOL(finite1432773856em_int(F,F_1))
=> ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,A))
=> ( A != bot_bot_fun_int_bool
=> hAPP_f1594865479ol_int(F_1,hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),A)) = hAPP_int_int(hAPP_int_fun_int_int(F,X_1),hAPP_f1594865479ol_int(F_1,A)) ) ) ) ).
fof(fact_350_finite__ne__induct,axiom,
! [Pa,F_1] :
( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,F_1))
=> ( F_1 != bot_bot_fun_nat_bool
=> ( ! [X_2] : hBOOL(hAPP_f54304608l_bool(Pa,hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_2),bot_bot_fun_nat_bool)))
=> ( ! [X_2,F_2] :
( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,F_2))
=> ( F_2 != bot_bot_fun_nat_bool
=> ( ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_2),F_2))
=> ( hBOOL(hAPP_f54304608l_bool(Pa,F_2))
=> hBOOL(hAPP_f54304608l_bool(Pa,hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_2),F_2))) ) ) ) )
=> hBOOL(hAPP_f54304608l_bool(Pa,F_1)) ) ) ) ) ).
fof(fact_351_finite__ne__induct,axiom,
! [Pa,F_1] :
( hBOOL(hAPP_f1695230391l_bool(finite1738664244iple_a,F_1))
=> ( F_1 != bot_bo797238721a_bool
=> ( ! [X_2] : hBOOL(hAPP_f1695230391l_bool(Pa,hAPP_f20753329a_bool(hAPP_H1743777351a_bool(insert956547291iple_a,X_2),bot_bo797238721a_bool)))
=> ( ! [X_2,F_2] :
( hBOOL(hAPP_f1695230391l_bool(finite1738664244iple_a,F_2))
=> ( F_2 != bot_bo797238721a_bool
=> ( ~ hBOOL(hAPP_f1695230391l_bool(hAPP_H622608077l_bool(member1797258804iple_a,X_2),F_2))
=> ( hBOOL(hAPP_f1695230391l_bool(Pa,F_2))
=> hBOOL(hAPP_f1695230391l_bool(Pa,hAPP_f20753329a_bool(hAPP_H1743777351a_bool(insert956547291iple_a,X_2),F_2))) ) ) ) )
=> hBOOL(hAPP_f1695230391l_bool(Pa,F_1)) ) ) ) ) ).
fof(fact_352_finite__ne__induct,axiom,
! [Pa,F_1] :
( hBOOL(hAPP_f448129468l_bool(finite_finite_int,F_1))
=> ( F_1 != bot_bot_fun_int_bool
=> ( ! [X_2] : hBOOL(hAPP_f448129468l_bool(Pa,hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_2),bot_bot_fun_int_bool)))
=> ( ! [X_2,F_2] :
( hBOOL(hAPP_f448129468l_bool(finite_finite_int,F_2))
=> ( F_2 != bot_bot_fun_int_bool
=> ( ~ hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_2),F_2))
=> ( hBOOL(hAPP_f448129468l_bool(Pa,F_2))
=> hBOOL(hAPP_f448129468l_bool(Pa,hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_2),F_2))) ) ) ) )
=> hBOOL(hAPP_f448129468l_bool(Pa,F_1)) ) ) ) ) ).
fof(fact_353_semilattice__big_OF__eq,axiom,
! [A,F,F_1] :
( hBOOL(big_se1603268663iple_a(F,F_1))
=> ( hBOOL(hAPP_f1695230391l_bool(finite1738664244iple_a,A))
=> hAPP_f1693662087iple_a(F_1,A) = hAPP_f1693662087iple_a(finite1465856449iple_a(F),A) ) ) ).
fof(fact_354_semilattice__big_OF__eq,axiom,
! [A,F,F_1] :
( hBOOL(big_se275732192ig_nat(F,F_1))
=> ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
=> hAPP_f22106695ol_nat(F_1,A) = hAPP_f22106695ol_nat(finite_fold1_nat(F),A) ) ) ).
fof(fact_355_semilattice__big_OF__eq,axiom,
! [A,F,F_1] :
( hBOOL(big_se913005884ig_int(F,F_1))
=> ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,A))
=> hAPP_f1594865479ol_int(F_1,A) = hAPP_f1594865479ol_int(finite_fold1_int(F),A) ) ) ).
fof(fact_356_folding__one_Oremove,axiom,
! [X_1,A,F,F_1] :
( hBOOL(finite988810631ne_nat(F,F_1))
=> ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
=> ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A))
=> ( ( hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),bot_bot_fun_nat_bool)) = bot_bot_fun_nat_bool
=> hAPP_f22106695ol_nat(F_1,A) = X_1 )
& ( hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),bot_bot_fun_nat_bool)) != bot_bot_fun_nat_bool
=> hAPP_f22106695ol_nat(F_1,A) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(F,X_1),hAPP_f22106695ol_nat(F_1,hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),bot_bot_fun_nat_bool)))) ) ) ) ) ) ).
fof(fact_357_folding__one_Oremove,axiom,
! [X_1,A,F,F_1] :
( hBOOL(finite1753440478iple_a(F,F_1))
=> ( hBOOL(hAPP_f1695230391l_bool(finite1738664244iple_a,A))
=> ( hBOOL(hAPP_f1695230391l_bool(hAPP_H622608077l_bool(member1797258804iple_a,X_1),A))
=> ( ( hAPP_f20753329a_bool(hAPP_f1706700729a_bool(minus_85316870a_bool,A),hAPP_f20753329a_bool(hAPP_H1743777351a_bool(insert956547291iple_a,X_1),bot_bo797238721a_bool)) = bot_bo797238721a_bool
=> hAPP_f1693662087iple_a(F_1,A) = X_1 )
& ( hAPP_f20753329a_bool(hAPP_f1706700729a_bool(minus_85316870a_bool,A),hAPP_f20753329a_bool(hAPP_H1743777351a_bool(insert956547291iple_a,X_1),bot_bo797238721a_bool)) != bot_bo797238721a_bool
=> hAPP_f1693662087iple_a(F_1,A) = hAPP_H905846293iple_a(hAPP_H2085992369iple_a(F,X_1),hAPP_f1693662087iple_a(F_1,hAPP_f20753329a_bool(hAPP_f1706700729a_bool(minus_85316870a_bool,A),hAPP_f20753329a_bool(hAPP_H1743777351a_bool(insert956547291iple_a,X_1),bot_bo797238721a_bool)))) ) ) ) ) ) ).
fof(fact_358_folding__one_Oremove,axiom,
! [X_1,A,F,F_1] :
( hBOOL(finite1626084323ne_int(F,F_1))
=> ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,A))
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),A))
=> ( ( hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),bot_bot_fun_int_bool)) = bot_bot_fun_int_bool
=> hAPP_f1594865479ol_int(F_1,A) = X_1 )
& ( hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),bot_bot_fun_int_bool)) != bot_bot_fun_int_bool
=> hAPP_f1594865479ol_int(F_1,A) = hAPP_int_int(hAPP_int_fun_int_int(F,X_1),hAPP_f1594865479ol_int(F_1,hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),bot_bot_fun_int_bool)))) ) ) ) ) ) ).
fof(fact_359_folding__one_Oinsert__remove,axiom,
! [X_1,A,F,F_1] :
( hBOOL(finite988810631ne_nat(F,F_1))
=> ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
=> ( ( hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),bot_bot_fun_nat_bool)) = bot_bot_fun_nat_bool
=> hAPP_f22106695ol_nat(F_1,hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),A)) = X_1 )
& ( hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),bot_bot_fun_nat_bool)) != bot_bot_fun_nat_bool
=> hAPP_f22106695ol_nat(F_1,hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),A)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(F,X_1),hAPP_f22106695ol_nat(F_1,hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),bot_bot_fun_nat_bool)))) ) ) ) ) ).
fof(fact_360_folding__one_Oinsert__remove,axiom,
! [X_1,A,F,F_1] :
( hBOOL(finite1753440478iple_a(F,F_1))
=> ( hBOOL(hAPP_f1695230391l_bool(finite1738664244iple_a,A))
=> ( ( hAPP_f20753329a_bool(hAPP_f1706700729a_bool(minus_85316870a_bool,A),hAPP_f20753329a_bool(hAPP_H1743777351a_bool(insert956547291iple_a,X_1),bot_bo797238721a_bool)) = bot_bo797238721a_bool
=> hAPP_f1693662087iple_a(F_1,hAPP_f20753329a_bool(hAPP_H1743777351a_bool(insert956547291iple_a,X_1),A)) = X_1 )
& ( hAPP_f20753329a_bool(hAPP_f1706700729a_bool(minus_85316870a_bool,A),hAPP_f20753329a_bool(hAPP_H1743777351a_bool(insert956547291iple_a,X_1),bot_bo797238721a_bool)) != bot_bo797238721a_bool
=> hAPP_f1693662087iple_a(F_1,hAPP_f20753329a_bool(hAPP_H1743777351a_bool(insert956547291iple_a,X_1),A)) = hAPP_H905846293iple_a(hAPP_H2085992369iple_a(F,X_1),hAPP_f1693662087iple_a(F_1,hAPP_f20753329a_bool(hAPP_f1706700729a_bool(minus_85316870a_bool,A),hAPP_f20753329a_bool(hAPP_H1743777351a_bool(insert956547291iple_a,X_1),bot_bo797238721a_bool)))) ) ) ) ) ).
fof(fact_361_folding__one_Oinsert__remove,axiom,
! [X_1,A,F,F_1] :
( hBOOL(finite1626084323ne_int(F,F_1))
=> ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,A))
=> ( ( hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),bot_bot_fun_int_bool)) = bot_bot_fun_int_bool
=> hAPP_f1594865479ol_int(F_1,hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),A)) = X_1 )
& ( hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),bot_bot_fun_int_bool)) != bot_bot_fun_int_bool
=> hAPP_f1594865479ol_int(F_1,hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),A)) = hAPP_int_int(hAPP_int_fun_int_int(F,X_1),hAPP_f1594865479ol_int(F_1,hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),bot_bot_fun_int_bool)))) ) ) ) ) ).
fof(fact_362_fold1__insert,axiom,
! [X_1,A] :
( A != bot_bot_fun_nat_bool
=> ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
=> ( ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A))
=> hAPP_f22106695ol_nat(finite_fold1_nat(times_times_nat),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),A)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,X_1),hAPP_f22106695ol_nat(finite_fold1_nat(times_times_nat),A)) ) ) ) ).
fof(fact_363_fold1__insert,axiom,
! [X_1,A] :
( A != bot_bot_fun_int_bool
=> ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,A))
=> ( ~ hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),A))
=> hAPP_f1594865479ol_int(finite_fold1_int(times_times_int),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),A)) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X_1),hAPP_f1594865479ol_int(finite_fold1_int(times_times_int),A)) ) ) ) ).
fof(fact_364_DiffI,axiom,
! [B,Ca,A] :
( hBOOL(hAPP_f1695230391l_bool(hAPP_H622608077l_bool(member1797258804iple_a,Ca),A))
=> ( ~ hBOOL(hAPP_f1695230391l_bool(hAPP_H622608077l_bool(member1797258804iple_a,Ca),B))
=> hBOOL(hAPP_f1695230391l_bool(hAPP_H622608077l_bool(member1797258804iple_a,Ca),hAPP_f20753329a_bool(hAPP_f1706700729a_bool(minus_85316870a_bool,A),B))) ) ) ).
fof(fact_365_DiffI,axiom,
! [B,Ca,A] :
( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,Ca),A))
=> ( ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,Ca),B))
=> hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,Ca),hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),B))) ) ) ).
fof(fact_366_DiffI,axiom,
! [B,Ca,A] :
( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Ca),A))
=> ( ~ hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Ca),B))
=> hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Ca),hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A),B))) ) ) ).
fof(fact_367_DiffE,axiom,
! [Ca,A,B] :
( hBOOL(hAPP_f1695230391l_bool(hAPP_H622608077l_bool(member1797258804iple_a,Ca),hAPP_f20753329a_bool(hAPP_f1706700729a_bool(minus_85316870a_bool,A),B)))
=> ~ ( hBOOL(hAPP_f1695230391l_bool(hAPP_H622608077l_bool(member1797258804iple_a,Ca),A))
=> hBOOL(hAPP_f1695230391l_bool(hAPP_H622608077l_bool(member1797258804iple_a,Ca),B)) ) ) ).
fof(fact_368_DiffE,axiom,
! [Ca,A,B] :
( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,Ca),hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),B)))
=> ~ ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,Ca),A))
=> hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,Ca),B)) ) ) ).
fof(fact_369_DiffE,axiom,
! [Ca,A,B] :
( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Ca),hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A),B)))
=> ~ ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Ca),A))
=> hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Ca),B)) ) ) ).
fof(fact_370_finite__Diff,axiom,
! [B,A] :
( hBOOL(hAPP_f1695230391l_bool(finite1738664244iple_a,A))
=> hBOOL(hAPP_f1695230391l_bool(finite1738664244iple_a,hAPP_f20753329a_bool(hAPP_f1706700729a_bool(minus_85316870a_bool,A),B))) ) ).
fof(fact_371_finite__Diff,axiom,
! [B,A] :
( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
=> hBOOL(hAPP_f54304608l_bool(finite_finite_nat,hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),B))) ) ).
fof(fact_372_finite__Diff,axiom,
! [B,A] :
( hBOOL(hAPP_f448129468l_bool(finite_finite_int,A))
=> hBOOL(hAPP_f448129468l_bool(finite_finite_int,hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A),B))) ) ).
fof(fact_373_folding__one__idem_Oidem,axiom,
! [X_1,F,F_1] :
( hBOOL(finite1432773856em_int(F,F_1))
=> hAPP_int_int(hAPP_int_fun_int_int(F,X_1),X_1) = X_1 ) ).
fof(fact_374_folding__one__idem_Oidem,axiom,
! [X_1,F,F_1] :
( hBOOL(finite1033474011iple_a(F,F_1))
=> hAPP_H905846293iple_a(hAPP_H2085992369iple_a(F,X_1),X_1) = X_1 ) ).
fof(fact_375_folding__one__idem_Oidem,axiom,
! [X_1,F,F_1] :
( hBOOL(finite795500164em_nat(F,F_1))
=> hAPP_nat_nat(hAPP_nat_fun_nat_nat(F,X_1),X_1) = X_1 ) ).
fof(fact_376_set__diff__eq,axiom,
! [A,B] : hAPP_f20753329a_bool(hAPP_f1706700729a_bool(minus_85316870a_bool,A),B) = hAPP_f20753329a_bool(collec351493750iple_a,hAPP_f20753329a_bool(hAPP_f607497069a_bool(cOMBS_154988028l_bool,hAPP_f146754017l_bool(hAPP_f239971723l_bool(cOMBB_1866391387iple_a,fconj),hAPP_f20753329a_bool(hAPP_f247708275a_bool(cOMBC_1519416966l_bool,member1797258804iple_a),A))),hAPP_f20753329a_bool(hAPP_f604201481a_bool(cOMBB_1882975613iple_a,fNot),hAPP_f20753329a_bool(hAPP_f247708275a_bool(cOMBC_1519416966l_bool,member1797258804iple_a),B)))) ).
fof(fact_377_set__diff__eq,axiom,
! [A,B] : hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),B) = hAPP_f800510211t_bool(collect_nat,hAPP_f800510211t_bool(hAPP_f561022312t_bool(cOMBS_nat_bool_bool,hAPP_f1146629647l_bool(hAPP_f1080886329l_bool(cOMBB_1015721476ol_nat,fconj),hAPP_f800510211t_bool(hAPP_f1722879237t_bool(cOMBC_226598744l_bool,member_nat),A))),hAPP_f800510211t_bool(hAPP_f894608603t_bool(cOMBB_bool_bool_nat,fNot),hAPP_f800510211t_bool(hAPP_f1722879237t_bool(cOMBC_226598744l_bool,member_nat),B)))) ).
fof(fact_378_set__diff__eq,axiom,
! [A,B] : hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A),B) = hAPP_f1805168059t_bool(collect_int,hAPP_f1805168059t_bool(hAPP_f727283836t_bool(cOMBS_int_bool_bool,hAPP_f2144054103l_bool(hAPP_f1734373249l_bool(cOMBB_1652995168ol_int,fconj),hAPP_f1805168059t_bool(hAPP_f202917053t_bool(cOMBC_94739984l_bool,member_int),A))),hAPP_f1805168059t_bool(hAPP_f627970963t_bool(cOMBB_bool_bool_int,fNot),hAPP_f1805168059t_bool(hAPP_f202917053t_bool(cOMBC_94739984l_bool,member_int),B)))) ).
fof(fact_379_Diff__iff,axiom,
! [Ca,A,B] :
( hBOOL(hAPP_f1695230391l_bool(hAPP_H622608077l_bool(member1797258804iple_a,Ca),hAPP_f20753329a_bool(hAPP_f1706700729a_bool(minus_85316870a_bool,A),B)))
<=> ( hBOOL(hAPP_f1695230391l_bool(hAPP_H622608077l_bool(member1797258804iple_a,Ca),A))
& ~ hBOOL(hAPP_f1695230391l_bool(hAPP_H622608077l_bool(member1797258804iple_a,Ca),B)) ) ) ).
fof(fact_380_Diff__iff,axiom,
! [Ca,A,B] :
( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,Ca),hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),B)))
<=> ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,Ca),A))
& ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,Ca),B)) ) ) ).
fof(fact_381_Diff__iff,axiom,
! [Ca,A,B] :
( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Ca),hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A),B)))
<=> ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Ca),A))
& ~ hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Ca),B)) ) ) ).
fof(fact_382_Diff__idemp,axiom,
! [A,B] : hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A),B)),B) = hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A),B) ).
fof(fact_383_Diff__idemp,axiom,
! [A,B] : hAPP_f20753329a_bool(hAPP_f1706700729a_bool(minus_85316870a_bool,hAPP_f20753329a_bool(hAPP_f1706700729a_bool(minus_85316870a_bool,A),B)),B) = hAPP_f20753329a_bool(hAPP_f1706700729a_bool(minus_85316870a_bool,A),B) ).
fof(fact_384_Diff__idemp,axiom,
! [A,B] : hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),B)),B) = hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),B) ).
fof(fact_385_DiffD1,axiom,
! [Ca,A,B] :
( hBOOL(hAPP_f1695230391l_bool(hAPP_H622608077l_bool(member1797258804iple_a,Ca),hAPP_f20753329a_bool(hAPP_f1706700729a_bool(minus_85316870a_bool,A),B)))
=> hBOOL(hAPP_f1695230391l_bool(hAPP_H622608077l_bool(member1797258804iple_a,Ca),A)) ) ).
fof(fact_386_DiffD1,axiom,
! [Ca,A,B] :
( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,Ca),hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),B)))
=> hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,Ca),A)) ) ).
fof(fact_387_DiffD1,axiom,
! [Ca,A,B] :
( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Ca),hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A),B)))
=> hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Ca),A)) ) ).
fof(fact_388_DiffD2,axiom,
! [Ca,A,B] :
( hBOOL(hAPP_f1695230391l_bool(hAPP_H622608077l_bool(member1797258804iple_a,Ca),hAPP_f20753329a_bool(hAPP_f1706700729a_bool(minus_85316870a_bool,A),B)))
=> ~ hBOOL(hAPP_f1695230391l_bool(hAPP_H622608077l_bool(member1797258804iple_a,Ca),B)) ) ).
fof(fact_389_DiffD2,axiom,
! [Ca,A,B] :
( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,Ca),hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),B)))
=> ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,Ca),B)) ) ).
fof(fact_390_DiffD2,axiom,
! [Ca,A,B] :
( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Ca),hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A),B)))
=> ~ hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Ca),B)) ) ).
fof(fact_391_empty__Diff,axiom,
! [A] : hAPP_f20753329a_bool(hAPP_f1706700729a_bool(minus_85316870a_bool,bot_bo797238721a_bool),A) = bot_bo797238721a_bool ).
fof(fact_392_empty__Diff,axiom,
! [A] : hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,bot_bot_fun_int_bool),A) = bot_bot_fun_int_bool ).
fof(fact_393_empty__Diff,axiom,
! [A] : hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,bot_bot_fun_nat_bool),A) = bot_bot_fun_nat_bool ).
fof(fact_394_Diff__empty,axiom,
! [A] : hAPP_f20753329a_bool(hAPP_f1706700729a_bool(minus_85316870a_bool,A),bot_bo797238721a_bool) = A ).
fof(fact_395_Diff__empty,axiom,
! [A] : hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A),bot_bot_fun_int_bool) = A ).
fof(fact_396_Diff__empty,axiom,
! [A] : hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),bot_bot_fun_nat_bool) = A ).
fof(fact_397_Diff__cancel,axiom,
! [A] : hAPP_f20753329a_bool(hAPP_f1706700729a_bool(minus_85316870a_bool,A),A) = bot_bo797238721a_bool ).
fof(fact_398_Diff__cancel,axiom,
! [A] : hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A),A) = bot_bot_fun_int_bool ).
fof(fact_399_Diff__cancel,axiom,
! [A] : hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),A) = bot_bot_fun_nat_bool ).
fof(fact_400_finite__Diff2,axiom,
! [A,B] :
( hBOOL(hAPP_f1695230391l_bool(finite1738664244iple_a,B))
=> ( hBOOL(hAPP_f1695230391l_bool(finite1738664244iple_a,hAPP_f20753329a_bool(hAPP_f1706700729a_bool(minus_85316870a_bool,A),B)))
<=> hBOOL(hAPP_f1695230391l_bool(finite1738664244iple_a,A)) ) ) ).
fof(fact_401_finite__Diff2,axiom,
! [A,B] :
( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,B))
=> ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),B)))
<=> hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A)) ) ) ).
fof(fact_402_finite__Diff2,axiom,
! [A,B] :
( hBOOL(hAPP_f448129468l_bool(finite_finite_int,B))
=> ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A),B)))
<=> hBOOL(hAPP_f448129468l_bool(finite_finite_int,A)) ) ) ).
fof(fact_403_insert__Diff__if,axiom,
! [A,X_1,B] :
( ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),B))
=> hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),A)),B) = hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),B) )
& ( ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),B))
=> hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),A)),B) = hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),B)) ) ) ).
fof(fact_404_insert__Diff__if,axiom,
! [A,X_1,B] :
( ( hBOOL(hAPP_f1695230391l_bool(hAPP_H622608077l_bool(member1797258804iple_a,X_1),B))
=> hAPP_f20753329a_bool(hAPP_f1706700729a_bool(minus_85316870a_bool,hAPP_f20753329a_bool(hAPP_H1743777351a_bool(insert956547291iple_a,X_1),A)),B) = hAPP_f20753329a_bool(hAPP_f1706700729a_bool(minus_85316870a_bool,A),B) )
& ( ~ hBOOL(hAPP_f1695230391l_bool(hAPP_H622608077l_bool(member1797258804iple_a,X_1),B))
=> hAPP_f20753329a_bool(hAPP_f1706700729a_bool(minus_85316870a_bool,hAPP_f20753329a_bool(hAPP_H1743777351a_bool(insert956547291iple_a,X_1),A)),B) = hAPP_f20753329a_bool(hAPP_H1743777351a_bool(insert956547291iple_a,X_1),hAPP_f20753329a_bool(hAPP_f1706700729a_bool(minus_85316870a_bool,A),B)) ) ) ).
fof(fact_405_insert__Diff__if,axiom,
! [A,X_1,B] :
( ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),B))
=> hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),A)),B) = hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A),B) )
& ( ~ hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),B))
=> hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),A)),B) = hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A),B)) ) ) ).
fof(fact_406_insert__Diff1,axiom,
! [A,X_1,B] :
( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),B))
=> hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),A)),B) = hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),B) ) ).
fof(fact_407_insert__Diff1,axiom,
! [A,X_1,B] :
( hBOOL(hAPP_f1695230391l_bool(hAPP_H622608077l_bool(member1797258804iple_a,X_1),B))
=> hAPP_f20753329a_bool(hAPP_f1706700729a_bool(minus_85316870a_bool,hAPP_f20753329a_bool(hAPP_H1743777351a_bool(insert956547291iple_a,X_1),A)),B) = hAPP_f20753329a_bool(hAPP_f1706700729a_bool(minus_85316870a_bool,A),B) ) ).
fof(fact_408_insert__Diff1,axiom,
! [A,X_1,B] :
( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),B))
=> hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),A)),B) = hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A),B) ) ).
fof(fact_409_fold__graph__permute__diff,axiom,
! [A_13,Ba,A,X_1] :
( hBOOL(hAPP_nat_bool(hAPP_f800510211t_bool(hAPP_n1512601776t_bool(finite929467206at_nat(times_times_nat),Ba),A),X_1))
=> ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_13),A))
=> ( ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,Ba),A))
=> hBOOL(hAPP_nat_bool(hAPP_f800510211t_bool(hAPP_n1512601776t_bool(finite929467206at_nat(times_times_nat),A_13),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,Ba),hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_13),bot_bot_fun_nat_bool)))),X_1)) ) ) ) ).
fof(fact_410_fold__graph__permute__diff,axiom,
! [A_13,Ba,A,X_1] :
( hBOOL(hAPP_int_bool(hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(finite772772422nt_int(times_times_int),Ba),A),X_1))
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,A_13),A))
=> ( ~ hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Ba),A))
=> hBOOL(hAPP_int_bool(hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(finite772772422nt_int(times_times_int),A_13),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,Ba),hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,A_13),bot_bot_fun_int_bool)))),X_1)) ) ) ) ).
fof(fact_411_Diff__insert,axiom,
! [A,A_13,B] : hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_13),B)) = hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),B)),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_13),bot_bot_fun_nat_bool)) ).
fof(fact_412_Diff__insert,axiom,
! [A,A_13,B] : hAPP_f20753329a_bool(hAPP_f1706700729a_bool(minus_85316870a_bool,A),hAPP_f20753329a_bool(hAPP_H1743777351a_bool(insert956547291iple_a,A_13),B)) = hAPP_f20753329a_bool(hAPP_f1706700729a_bool(minus_85316870a_bool,hAPP_f20753329a_bool(hAPP_f1706700729a_bool(minus_85316870a_bool,A),B)),hAPP_f20753329a_bool(hAPP_H1743777351a_bool(insert956547291iple_a,A_13),bot_bo797238721a_bool)) ).
fof(fact_413_Diff__insert,axiom,
! [A,A_13,B] : hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,A_13),B)) = hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A),B)),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,A_13),bot_bot_fun_int_bool)) ).
fof(fact_414_Diff__insert2,axiom,
! [A,A_13,B] : hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_13),B)) = hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_13),bot_bot_fun_nat_bool))),B) ).
fof(fact_415_Diff__insert2,axiom,
! [A,A_13,B] : hAPP_f20753329a_bool(hAPP_f1706700729a_bool(minus_85316870a_bool,A),hAPP_f20753329a_bool(hAPP_H1743777351a_bool(insert956547291iple_a,A_13),B)) = hAPP_f20753329a_bool(hAPP_f1706700729a_bool(minus_85316870a_bool,hAPP_f20753329a_bool(hAPP_f1706700729a_bool(minus_85316870a_bool,A),hAPP_f20753329a_bool(hAPP_H1743777351a_bool(insert956547291iple_a,A_13),bot_bo797238721a_bool))),B) ).
fof(fact_416_Diff__insert2,axiom,
! [A,A_13,B] : hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,A_13),B)) = hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,A_13),bot_bot_fun_int_bool))),B) ).
fof(fact_417_insert__Diff__single,axiom,
! [A_13,A] : hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_13),hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_13),bot_bot_fun_nat_bool))) = hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_13),A) ).
fof(fact_418_insert__Diff__single,axiom,
! [A_13,A] : hAPP_f20753329a_bool(hAPP_H1743777351a_bool(insert956547291iple_a,A_13),hAPP_f20753329a_bool(hAPP_f1706700729a_bool(minus_85316870a_bool,A),hAPP_f20753329a_bool(hAPP_H1743777351a_bool(insert956547291iple_a,A_13),bot_bo797238721a_bool))) = hAPP_f20753329a_bool(hAPP_H1743777351a_bool(insert956547291iple_a,A_13),A) ).
fof(fact_419_insert__Diff__single,axiom,
! [A_13,A] : hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,A_13),hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,A_13),bot_bot_fun_int_bool))) = hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,A_13),A) ).
fof(fact_420_Diff__insert__absorb,axiom,
! [X_1,A] :
( ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A))
=> hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),A)),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),bot_bot_fun_nat_bool)) = A ) ).
fof(fact_421_Diff__insert__absorb,axiom,
! [X_1,A] :
( ~ hBOOL(hAPP_f1695230391l_bool(hAPP_H622608077l_bool(member1797258804iple_a,X_1),A))
=> hAPP_f20753329a_bool(hAPP_f1706700729a_bool(minus_85316870a_bool,hAPP_f20753329a_bool(hAPP_H1743777351a_bool(insert956547291iple_a,X_1),A)),hAPP_f20753329a_bool(hAPP_H1743777351a_bool(insert956547291iple_a,X_1),bot_bo797238721a_bool)) = A ) ).
fof(fact_422_Diff__insert__absorb,axiom,
! [X_1,A] :
( ~ hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),A))
=> hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),A)),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),bot_bot_fun_int_bool)) = A ) ).
fof(fact_423_insert__Diff,axiom,
! [A_13,A] :
( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_13),A))
=> hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_13),hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_13),bot_bot_fun_nat_bool))) = A ) ).
fof(fact_424_insert__Diff,axiom,
! [A_13,A] :
( hBOOL(hAPP_f1695230391l_bool(hAPP_H622608077l_bool(member1797258804iple_a,A_13),A))
=> hAPP_f20753329a_bool(hAPP_H1743777351a_bool(insert956547291iple_a,A_13),hAPP_f20753329a_bool(hAPP_f1706700729a_bool(minus_85316870a_bool,A),hAPP_f20753329a_bool(hAPP_H1743777351a_bool(insert956547291iple_a,A_13),bot_bo797238721a_bool))) = A ) ).
fof(fact_425_insert__Diff,axiom,
! [A_13,A] :
( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,A_13),A))
=> hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,A_13),hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,A_13),bot_bot_fun_int_bool))) = A ) ).
fof(fact_426_finite__Diff__insert,axiom,
! [A,A_13,B] :
( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_13),B))))
<=> hBOOL(hAPP_f54304608l_bool(finite_finite_nat,hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),B))) ) ).
fof(fact_427_finite__Diff__insert,axiom,
! [A,A_13,B] :
( hBOOL(hAPP_f1695230391l_bool(finite1738664244iple_a,hAPP_f20753329a_bool(hAPP_f1706700729a_bool(minus_85316870a_bool,A),hAPP_f20753329a_bool(hAPP_H1743777351a_bool(insert956547291iple_a,A_13),B))))
<=> hBOOL(hAPP_f1695230391l_bool(finite1738664244iple_a,hAPP_f20753329a_bool(hAPP_f1706700729a_bool(minus_85316870a_bool,A),B))) ) ).
fof(fact_428_finite__Diff__insert,axiom,
! [A,A_13,B] :
( hBOOL(hAPP_f448129468l_bool(finite_finite_int,hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,A_13),B))))
<=> hBOOL(hAPP_f448129468l_bool(finite_finite_int,hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A),B))) ) ).
fof(fact_429_fold__graph__insert__swap,axiom,
! [Z_1,Ba,A,Y_1] :
( hBOOL(hAPP_nat_bool(hAPP_f800510211t_bool(hAPP_n1512601776t_bool(finite929467206at_nat(times_times_nat),Ba),A),Y_1))
=> ( ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,Ba),A))
=> hBOOL(hAPP_nat_bool(hAPP_f800510211t_bool(hAPP_n1512601776t_bool(finite929467206at_nat(times_times_nat),Z_1),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,Ba),A)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Z_1),Y_1))) ) ) ).
fof(fact_430_fold__graph__insert__swap,axiom,
! [Z_1,Ba,A,Y_1] :
( hBOOL(hAPP_int_bool(hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(finite772772422nt_int(times_times_int),Ba),A),Y_1))
=> ( ~ hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Ba),A))
=> hBOOL(hAPP_int_bool(hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(finite772772422nt_int(times_times_int),Z_1),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,Ba),A)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Z_1),Y_1))) ) ) ).
fof(fact_431_folding__one__idem_Oin__idem,axiom,
! [X_1,A,F,F_1] :
( hBOOL(finite1033474011iple_a(F,F_1))
=> ( hBOOL(hAPP_f1695230391l_bool(finite1738664244iple_a,A))
=> ( hBOOL(hAPP_f1695230391l_bool(hAPP_H622608077l_bool(member1797258804iple_a,X_1),A))
=> hAPP_H905846293iple_a(hAPP_H2085992369iple_a(F,X_1),hAPP_f1693662087iple_a(F_1,A)) = hAPP_f1693662087iple_a(F_1,A) ) ) ) ).
fof(fact_432_folding__one__idem_Oin__idem,axiom,
! [X_1,A,F,F_1] :
( hBOOL(finite795500164em_nat(F,F_1))
=> ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
=> ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A))
=> hAPP_nat_nat(hAPP_nat_fun_nat_nat(F,X_1),hAPP_f22106695ol_nat(F_1,A)) = hAPP_f22106695ol_nat(F_1,A) ) ) ) ).
fof(fact_433_folding__one__idem_Oin__idem,axiom,
! [X_1,A,F,F_1] :
( hBOOL(finite1432773856em_int(F,F_1))
=> ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,A))
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),A))
=> hAPP_int_int(hAPP_int_fun_int_int(F,X_1),hAPP_f1594865479ol_int(F_1,A)) = hAPP_f1594865479ol_int(F_1,A) ) ) ) ).
fof(fact_434_fold1__in,axiom,
! [A] :
( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
=> ( A != bot_bot_fun_nat_bool
=> ( ! [X_2,Y_21] : hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,X_2),Y_21)),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_2),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,Y_21),bot_bot_fun_nat_bool))))
=> hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,hAPP_f22106695ol_nat(finite_fold1_nat(times_times_nat),A)),A)) ) ) ) ).
fof(fact_435_fold1__in,axiom,
! [A] :
( hBOOL(hAPP_f448129468l_bool(finite_finite_int,A))
=> ( A != bot_bot_fun_int_bool
=> ( ! [X_2,Y_21] : hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X_2),Y_21)),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_2),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,Y_21),bot_bot_fun_int_bool))))
=> hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hAPP_f1594865479ol_int(finite_fold1_int(times_times_int),A)),A)) ) ) ) ).
fof(fact_436_folding__one__idem_Ohom__commute,axiom,
! [N_3,H,F,F_1] :
( hBOOL(finite1033474011iple_a(F,F_1))
=> ( ! [X_2,Y_21] : hAPP_H905846293iple_a(H,hAPP_H905846293iple_a(hAPP_H2085992369iple_a(F,X_2),Y_21)) = hAPP_H905846293iple_a(hAPP_H2085992369iple_a(F,hAPP_H905846293iple_a(H,X_2)),hAPP_H905846293iple_a(H,Y_21))
=> ( hBOOL(hAPP_f1695230391l_bool(finite1738664244iple_a,N_3))
=> ( N_3 != bot_bo797238721a_bool
=> hAPP_H905846293iple_a(H,hAPP_f1693662087iple_a(F_1,N_3)) = hAPP_f1693662087iple_a(F_1,image_533441733iple_a(H,N_3)) ) ) ) ) ).
fof(fact_437_folding__one__idem_Ohom__commute,axiom,
! [N_3,H,F,F_1] :
( hBOOL(finite1432773856em_int(F,F_1))
=> ( ! [X_2,Y_21] : hAPP_int_int(H,hAPP_int_int(hAPP_int_fun_int_int(F,X_2),Y_21)) = hAPP_int_int(hAPP_int_fun_int_int(F,hAPP_int_int(H,X_2)),hAPP_int_int(H,Y_21))
=> ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,N_3))
=> ( N_3 != bot_bot_fun_int_bool
=> hAPP_int_int(H,hAPP_f1594865479ol_int(F_1,N_3)) = hAPP_f1594865479ol_int(F_1,image_int_int(H,N_3)) ) ) ) ) ).
fof(fact_438_folding__one__idem_Ohom__commute,axiom,
! [N_3,H,F,F_1] :
( hBOOL(finite795500164em_nat(F,F_1))
=> ( ! [X_2,Y_21] : hAPP_nat_nat(H,hAPP_nat_nat(hAPP_nat_fun_nat_nat(F,X_2),Y_21)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(F,hAPP_nat_nat(H,X_2)),hAPP_nat_nat(H,Y_21))
=> ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,N_3))
=> ( N_3 != bot_bot_fun_nat_bool
=> hAPP_nat_nat(H,hAPP_f22106695ol_nat(F_1,N_3)) = hAPP_f22106695ol_nat(F_1,image_nat_nat(H,N_3)) ) ) ) ) ).
fof(fact_439_finite__empty__induct,axiom,
! [Pa,A] :
( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
=> ( hBOOL(hAPP_f54304608l_bool(Pa,A))
=> ( ! [A_12,A_43] :
( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A_43))
=> ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_12),A_43))
=> ( hBOOL(hAPP_f54304608l_bool(Pa,A_43))
=> hBOOL(hAPP_f54304608l_bool(Pa,hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A_43),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_12),bot_bot_fun_nat_bool)))) ) ) )
=> hBOOL(hAPP_f54304608l_bool(Pa,bot_bot_fun_nat_bool)) ) ) ) ).
fof(fact_440_finite__empty__induct,axiom,
! [Pa,A] :
( hBOOL(hAPP_f1695230391l_bool(finite1738664244iple_a,A))
=> ( hBOOL(hAPP_f1695230391l_bool(Pa,A))
=> ( ! [A_12,A_43] :
( hBOOL(hAPP_f1695230391l_bool(finite1738664244iple_a,A_43))
=> ( hBOOL(hAPP_f1695230391l_bool(hAPP_H622608077l_bool(member1797258804iple_a,A_12),A_43))
=> ( hBOOL(hAPP_f1695230391l_bool(Pa,A_43))
=> hBOOL(hAPP_f1695230391l_bool(Pa,hAPP_f20753329a_bool(hAPP_f1706700729a_bool(minus_85316870a_bool,A_43),hAPP_f20753329a_bool(hAPP_H1743777351a_bool(insert956547291iple_a,A_12),bot_bo797238721a_bool)))) ) ) )
=> hBOOL(hAPP_f1695230391l_bool(Pa,bot_bo797238721a_bool)) ) ) ) ).
fof(fact_441_finite__empty__induct,axiom,
! [Pa,A] :
( hBOOL(hAPP_f448129468l_bool(finite_finite_int,A))
=> ( hBOOL(hAPP_f448129468l_bool(Pa,A))
=> ( ! [A_12,A_43] :
( hBOOL(hAPP_f448129468l_bool(finite_finite_int,A_43))
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,A_12),A_43))
=> ( hBOOL(hAPP_f448129468l_bool(Pa,A_43))
=> hBOOL(hAPP_f448129468l_bool(Pa,hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A_43),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,A_12),bot_bot_fun_int_bool)))) ) ) )
=> hBOOL(hAPP_f448129468l_bool(Pa,bot_bot_fun_int_bool)) ) ) ) ).
fof(fact_442_minus__apply,axiom,
! [A,B,X_1] :
( hBOOL(hAPP_int_bool(hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A),B),X_1))
<=> hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(minus_minus_bool,hAPP_int_bool(A,X_1)),hAPP_int_bool(B,X_1))) ) ).
fof(fact_443_minus__apply,axiom,
! [A,B,X_1] :
( hBOOL(hAPP_H1037229737a_bool(hAPP_f20753329a_bool(hAPP_f1706700729a_bool(minus_85316870a_bool,A),B),X_1))
<=> hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(minus_minus_bool,hAPP_H1037229737a_bool(A,X_1)),hAPP_H1037229737a_bool(B,X_1))) ) ).
fof(fact_444_minus__apply,axiom,
! [A,B,X_1] :
( hBOOL(hAPP_nat_bool(hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),B),X_1))
<=> hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(minus_minus_bool,hAPP_nat_bool(A,X_1)),hAPP_nat_bool(B,X_1))) ) ).
fof(fact_445_fun__diff__def,axiom,
! [A,B,X_2] :
( hBOOL(hAPP_int_bool(hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A),B),X_2))
<=> hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(minus_minus_bool,hAPP_int_bool(A,X_2)),hAPP_int_bool(B,X_2))) ) ).
fof(fact_446_fun__diff__def,axiom,
! [A,B,X_2] :
( hBOOL(hAPP_H1037229737a_bool(hAPP_f20753329a_bool(hAPP_f1706700729a_bool(minus_85316870a_bool,A),B),X_2))
<=> hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(minus_minus_bool,hAPP_H1037229737a_bool(A,X_2)),hAPP_H1037229737a_bool(B,X_2))) ) ).
fof(fact_447_fun__diff__def,axiom,
! [A,B,X_2] :
( hBOOL(hAPP_nat_bool(hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),B),X_2))
<=> hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(minus_minus_bool,hAPP_nat_bool(A,X_2)),hAPP_nat_bool(B,X_2))) ) ).
fof(fact_448_comp__fun__idem__remove,axiom,
hBOOL(finite1956789438t_bool(hAPP_f10625010t_bool(hAPP_f1115950719t_bool(cOMBB_1654519265ol_nat,hAPP_f1311642927t_bool(cOMBC_646315179t_bool,minus_2067140911t_bool)),hAPP_f1570313510t_bool(hAPP_f688831301t_bool(cOMBC_178881787t_bool,insert_nat),bot_bot_fun_nat_bool)))) ).
fof(fact_449_comp__fun__idem__remove,axiom,
hBOOL(finite1803123436a_bool(hAPP_f550111323a_bool(hAPP_f615286271a_bool(cOMBB_1176122337iple_a,hAPP_f957014749a_bool(cOMBC_288844080a_bool,minus_85316870a_bool)),hAPP_f1181346091a_bool(hAPP_f571476211a_bool(cOMBC_1570214144a_bool,insert956547291iple_a),bot_bo797238721a_bool)))) ).
fof(fact_450_comp__fun__idem__remove,axiom,
hBOOL(finite58652534t_bool(hAPP_f1468280982t_bool(hAPP_f1596157055t_bool(cOMBB_765314529ol_int,hAPP_f194616807t_bool(cOMBC_922884543t_bool,minus_1449998731t_bool)),hAPP_f1048215610t_bool(hAPP_f472159229t_bool(cOMBC_1683390479t_bool,insert_int),bot_bot_fun_int_bool)))) ).
fof(fact_451_comp__fun__idem_Ofun__left__idem,axiom,
! [X_1,Z_1,F] :
( hBOOL(finite58652534t_bool(F))
=> hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(F,X_1),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(F,X_1),Z_1)) = hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(F,X_1),Z_1) ) ).
fof(fact_452_comp__fun__idem_Ofun__left__idem,axiom,
! [X_1,Z_1,F] :
( hBOOL(finite1803123436a_bool(F))
=> hAPP_f20753329a_bool(hAPP_H1743777351a_bool(F,X_1),hAPP_f20753329a_bool(hAPP_H1743777351a_bool(F,X_1),Z_1)) = hAPP_f20753329a_bool(hAPP_H1743777351a_bool(F,X_1),Z_1) ) ).
fof(fact_453_comp__fun__idem_Ofun__left__idem,axiom,
! [X_1,Z_1,F] :
( hBOOL(finite1956789438t_bool(F))
=> hAPP_f800510211t_bool(hAPP_n1512601776t_bool(F,X_1),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(F,X_1),Z_1)) = hAPP_f800510211t_bool(hAPP_n1512601776t_bool(F,X_1),Z_1) ) ).
fof(fact_454_comp__fun__idem__insert,axiom,
hBOOL(finite1956789438t_bool(insert_nat)) ).
fof(fact_455_comp__fun__idem__insert,axiom,
hBOOL(finite1803123436a_bool(insert956547291iple_a)) ).
fof(fact_456_comp__fun__idem__insert,axiom,
hBOOL(finite58652534t_bool(insert_int)) ).
fof(fact_457_comp__fun__commute_Ofold__graph__insertE__aux,axiom,
! [A_13,Z_1,A,Y_1,F] :
( hBOOL(finite2130160977at_nat(F))
=> ( hBOOL(hAPP_nat_bool(hAPP_f800510211t_bool(hAPP_n1512601776t_bool(finite929467206at_nat(F),Z_1),A),Y_1))
=> ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_13),A))
=> ? [Y_22] :
( Y_1 = hAPP_nat_nat(hAPP_nat_fun_nat_nat(F,A_13),Y_22)
& hBOOL(hAPP_nat_bool(hAPP_f800510211t_bool(hAPP_n1512601776t_bool(finite929467206at_nat(F),Z_1),hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_13),bot_bot_fun_nat_bool))),Y_22)) ) ) ) ) ).
fof(fact_458_comp__fun__commute_Ofold__graph__insertE__aux,axiom,
! [A_13,Z_1,A,Y_1,F] :
( hBOOL(finite1428139793iple_a(F))
=> ( hBOOL(hAPP_H1037229737a_bool(hAPP_f20753329a_bool(hAPP_H1743777351a_bool(finite90892550iple_a(F),Z_1),A),Y_1))
=> ( hBOOL(hAPP_f1695230391l_bool(hAPP_H622608077l_bool(member1797258804iple_a,A_13),A))
=> ? [Y_22] :
( Y_1 = hAPP_H905846293iple_a(hAPP_H2085992369iple_a(F,A_13),Y_22)
& hBOOL(hAPP_H1037229737a_bool(hAPP_f20753329a_bool(hAPP_H1743777351a_bool(finite90892550iple_a(F),Z_1),hAPP_f20753329a_bool(hAPP_f1706700729a_bool(minus_85316870a_bool,A),hAPP_f20753329a_bool(hAPP_H1743777351a_bool(insert956547291iple_a,A_13),bot_bo797238721a_bool))),Y_22)) ) ) ) ) ).
fof(fact_459_comp__fun__commute_Ofold__graph__insertE__aux,axiom,
! [A_13,Z_1,A,Y_1,F] :
( hBOOL(finite1973466193nt_int(F))
=> ( hBOOL(hAPP_int_bool(hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(finite772772422nt_int(F),Z_1),A),Y_1))
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,A_13),A))
=> ? [Y_22] :
( Y_1 = hAPP_int_int(hAPP_int_fun_int_int(F,A_13),Y_22)
& hBOOL(hAPP_int_bool(hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(finite772772422nt_int(F),Z_1),hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,A_13),bot_bot_fun_int_bool))),Y_22)) ) ) ) ) ).
fof(fact_460_setsum__diff1,axiom,
! [F,A_13,A] :
( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
=> ( ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_13),A))
=> hAPP_f659380387ol_int(hAPP_f1599440987ol_int(big_co1024481617at_int,F),hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_13),bot_bot_fun_nat_bool))) = hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_f659380387ol_int(hAPP_f1599440987ol_int(big_co1024481617at_int,F),A)),hAPP_nat_int(F,A_13)) )
& ( ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_13),A))
=> hAPP_f659380387ol_int(hAPP_f1599440987ol_int(big_co1024481617at_int,F),hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_13),bot_bot_fun_nat_bool))) = hAPP_f659380387ol_int(hAPP_f1599440987ol_int(big_co1024481617at_int,F),A) ) ) ) ).
fof(fact_461_setsum__diff1,axiom,
! [F,A_13,A] :
( hBOOL(hAPP_f1695230391l_bool(finite1738664244iple_a,A))
=> ( ( hBOOL(hAPP_f1695230391l_bool(hAPP_H622608077l_bool(member1797258804iple_a,A_13),A))
=> hAPP_f587450508ol_int(hAPP_f1544320301ol_int(big_co1977834938_a_int,F),hAPP_f20753329a_bool(hAPP_f1706700729a_bool(minus_85316870a_bool,A),hAPP_f20753329a_bool(hAPP_H1743777351a_bool(insert956547291iple_a,A_13),bot_bo797238721a_bool))) = hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_f587450508ol_int(hAPP_f1544320301ol_int(big_co1977834938_a_int,F),A)),hAPP_H1229305626_a_int(F,A_13)) )
& ( ~ hBOOL(hAPP_f1695230391l_bool(hAPP_H622608077l_bool(member1797258804iple_a,A_13),A))
=> hAPP_f587450508ol_int(hAPP_f1544320301ol_int(big_co1977834938_a_int,F),hAPP_f20753329a_bool(hAPP_f1706700729a_bool(minus_85316870a_bool,A),hAPP_f20753329a_bool(hAPP_H1743777351a_bool(insert956547291iple_a,A_13),bot_bo797238721a_bool))) = hAPP_f587450508ol_int(hAPP_f1544320301ol_int(big_co1977834938_a_int,F),A) ) ) ) ).
fof(fact_462_setsum__diff1,axiom,
! [F,A_13,A] :
( hBOOL(hAPP_f448129468l_bool(finite_finite_int,A))
=> ( ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,A_13),A))
=> hAPP_f1594865479ol_int(hAPP_f1926459811ol_int(big_co230513141nt_int,F),hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,A_13),bot_bot_fun_int_bool))) = hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_f1594865479ol_int(hAPP_f1926459811ol_int(big_co230513141nt_int,F),A)),hAPP_int_int(F,A_13)) )
& ( ~ hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,A_13),A))
=> hAPP_f1594865479ol_int(hAPP_f1926459811ol_int(big_co230513141nt_int,F),hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,A_13),bot_bot_fun_int_bool))) = hAPP_f1594865479ol_int(hAPP_f1926459811ol_int(big_co230513141nt_int,F),A) ) ) ) ).
fof(fact_463_setsum__diff1__ring,axiom,
! [F,A_13,A] :
( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
=> ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_13),A))
=> hAPP_f659380387ol_int(hAPP_f1599440987ol_int(big_co1024481617at_int,F),hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_13),bot_bot_fun_nat_bool))) = hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_f659380387ol_int(hAPP_f1599440987ol_int(big_co1024481617at_int,F),A)),hAPP_nat_int(F,A_13)) ) ) ).
fof(fact_464_setsum__diff1__ring,axiom,
! [F,A_13,A] :
( hBOOL(hAPP_f1695230391l_bool(finite1738664244iple_a,A))
=> ( hBOOL(hAPP_f1695230391l_bool(hAPP_H622608077l_bool(member1797258804iple_a,A_13),A))
=> hAPP_f587450508ol_int(hAPP_f1544320301ol_int(big_co1977834938_a_int,F),hAPP_f20753329a_bool(hAPP_f1706700729a_bool(minus_85316870a_bool,A),hAPP_f20753329a_bool(hAPP_H1743777351a_bool(insert956547291iple_a,A_13),bot_bo797238721a_bool))) = hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_f587450508ol_int(hAPP_f1544320301ol_int(big_co1977834938_a_int,F),A)),hAPP_H1229305626_a_int(F,A_13)) ) ) ).
fof(fact_465_setsum__diff1__ring,axiom,
! [F,A_13,A] :
( hBOOL(hAPP_f448129468l_bool(finite_finite_int,A))
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,A_13),A))
=> hAPP_f1594865479ol_int(hAPP_f1926459811ol_int(big_co230513141nt_int,F),hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,A_13),bot_bot_fun_int_bool))) = hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_f1594865479ol_int(hAPP_f1926459811ol_int(big_co230513141nt_int,F),A)),hAPP_int_int(F,A_13)) ) ) ).
fof(fact_466_setprod_Oinsert__remove,axiom,
! [G,X_1,A] :
( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
=> hAPP_f22106695ol_nat(big_co1705425894at_nat(G),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),A)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(G,X_1)),hAPP_f22106695ol_nat(big_co1705425894at_nat(G),hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),bot_bot_fun_nat_bool)))) ) ).
fof(fact_467_setprod_Oinsert__remove,axiom,
! [G,X_1,A] :
( hBOOL(hAPP_f1695230391l_bool(finite1738664244iple_a,A))
=> hAPP_f2097660464ol_nat(big_co1815942735_a_nat(G),hAPP_f20753329a_bool(hAPP_H1743777351a_bool(insert956547291iple_a,X_1),A)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_H592031934_a_nat(G,X_1)),hAPP_f2097660464ol_nat(big_co1815942735_a_nat(G),hAPP_f20753329a_bool(hAPP_f1706700729a_bool(minus_85316870a_bool,A),hAPP_f20753329a_bool(hAPP_H1743777351a_bool(insert956547291iple_a,X_1),bot_bo797238721a_bool)))) ) ).
fof(fact_468_setprod_Oinsert__remove,axiom,
! [G,X_1,A] :
( hBOOL(hAPP_f448129468l_bool(finite_finite_int,A))
=> hAPP_f957591787ol_nat(big_co911457418nt_nat(G),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),A)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(G,X_1)),hAPP_f957591787ol_nat(big_co911457418nt_nat(G),hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),bot_bot_fun_int_bool)))) ) ).
fof(fact_469_setprod_Oinsert__remove,axiom,
! [G,X_1,A] :
( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
=> hAPP_f659380387ol_int(big_co195215938at_int(G),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),A)) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_nat_int(G,X_1)),hAPP_f659380387ol_int(big_co195215938at_int(G),hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),bot_bot_fun_nat_bool)))) ) ).
fof(fact_470_setprod_Oinsert__remove,axiom,
! [G,X_1,A] :
( hBOOL(hAPP_f1695230391l_bool(finite1738664244iple_a,A))
=> hAPP_f587450508ol_int(big_co305732779_a_int(G),hAPP_f20753329a_bool(hAPP_H1743777351a_bool(insert956547291iple_a,X_1),A)) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_H1229305626_a_int(G,X_1)),hAPP_f587450508ol_int(big_co305732779_a_int(G),hAPP_f20753329a_bool(hAPP_f1706700729a_bool(minus_85316870a_bool,A),hAPP_f20753329a_bool(hAPP_H1743777351a_bool(insert956547291iple_a,X_1),bot_bo797238721a_bool)))) ) ).
fof(fact_471_setprod_Oinsert__remove,axiom,
! [G,X_1,A] :
( hBOOL(hAPP_f448129468l_bool(finite_finite_int,A))
=> hAPP_f1594865479ol_int(big_co1548731110nt_int(G),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),A)) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(G,X_1)),hAPP_f1594865479ol_int(big_co1548731110nt_int(G),hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),bot_bot_fun_int_bool)))) ) ).
fof(fact_472_setprod_Oremove,axiom,
! [G,X_1,A] :
( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
=> ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A))
=> hAPP_f22106695ol_nat(big_co1705425894at_nat(G),A) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(G,X_1)),hAPP_f22106695ol_nat(big_co1705425894at_nat(G),hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),bot_bot_fun_nat_bool)))) ) ) ).
fof(fact_473_setprod_Oremove,axiom,
! [G,X_1,A] :
( hBOOL(hAPP_f1695230391l_bool(finite1738664244iple_a,A))
=> ( hBOOL(hAPP_f1695230391l_bool(hAPP_H622608077l_bool(member1797258804iple_a,X_1),A))
=> hAPP_f2097660464ol_nat(big_co1815942735_a_nat(G),A) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_H592031934_a_nat(G,X_1)),hAPP_f2097660464ol_nat(big_co1815942735_a_nat(G),hAPP_f20753329a_bool(hAPP_f1706700729a_bool(minus_85316870a_bool,A),hAPP_f20753329a_bool(hAPP_H1743777351a_bool(insert956547291iple_a,X_1),bot_bo797238721a_bool)))) ) ) ).
fof(fact_474_setprod_Oremove,axiom,
! [G,X_1,A] :
( hBOOL(hAPP_f448129468l_bool(finite_finite_int,A))
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),A))
=> hAPP_f957591787ol_nat(big_co911457418nt_nat(G),A) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(G,X_1)),hAPP_f957591787ol_nat(big_co911457418nt_nat(G),hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),bot_bot_fun_int_bool)))) ) ) ).
fof(fact_475_setprod_Oremove,axiom,
! [G,X_1,A] :
( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
=> ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A))
=> hAPP_f659380387ol_int(big_co195215938at_int(G),A) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_nat_int(G,X_1)),hAPP_f659380387ol_int(big_co195215938at_int(G),hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),bot_bot_fun_nat_bool)))) ) ) ).
fof(fact_476_setprod_Oremove,axiom,
! [G,X_1,A] :
( hBOOL(hAPP_f1695230391l_bool(finite1738664244iple_a,A))
=> ( hBOOL(hAPP_f1695230391l_bool(hAPP_H622608077l_bool(member1797258804iple_a,X_1),A))
=> hAPP_f587450508ol_int(big_co305732779_a_int(G),A) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_H1229305626_a_int(G,X_1)),hAPP_f587450508ol_int(big_co305732779_a_int(G),hAPP_f20753329a_bool(hAPP_f1706700729a_bool(minus_85316870a_bool,A),hAPP_f20753329a_bool(hAPP_H1743777351a_bool(insert956547291iple_a,X_1),bot_bo797238721a_bool)))) ) ) ).
fof(fact_477_setprod_Oremove,axiom,
! [G,X_1,A] :
( hBOOL(hAPP_f448129468l_bool(finite_finite_int,A))
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),A))
=> hAPP_f1594865479ol_int(big_co1548731110nt_int(G),A) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(G,X_1)),hAPP_f1594865479ol_int(big_co1548731110nt_int(G),hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),bot_bot_fun_int_bool)))) ) ) ).
fof(fact_478_minus__fold__remove,axiom,
! [B,A] :
( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
=> hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,B),A) = hAPP_f800510211t_bool(finite1578363458t_bool(hAPP_f10625010t_bool(hAPP_f1115950719t_bool(cOMBB_1654519265ol_nat,hAPP_f1311642927t_bool(cOMBC_646315179t_bool,minus_2067140911t_bool)),hAPP_f1570313510t_bool(hAPP_f688831301t_bool(cOMBC_178881787t_bool,insert_nat),bot_bot_fun_nat_bool)),B),A) ) ).
fof(fact_479_minus__fold__remove,axiom,
! [B,A] :
( hBOOL(hAPP_f1695230391l_bool(finite1738664244iple_a,A))
=> hAPP_f20753329a_bool(hAPP_f1706700729a_bool(minus_85316870a_bool,B),A) = hAPP_f20753329a_bool(finite1215486064a_bool(hAPP_f550111323a_bool(hAPP_f615286271a_bool(cOMBB_1176122337iple_a,hAPP_f957014749a_bool(cOMBC_288844080a_bool,minus_85316870a_bool)),hAPP_f1181346091a_bool(hAPP_f571476211a_bool(cOMBC_1570214144a_bool,insert956547291iple_a),bot_bo797238721a_bool)),B),A) ) ).
fof(fact_480_minus__fold__remove,axiom,
! [B,A] :
( hBOOL(hAPP_f448129468l_bool(finite_finite_int,A))
=> hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,B),A) = hAPP_f1805168059t_bool(finite1827710202t_bool(hAPP_f1468280982t_bool(hAPP_f1596157055t_bool(cOMBB_765314529ol_int,hAPP_f194616807t_bool(cOMBC_922884543t_bool,minus_1449998731t_bool)),hAPP_f1048215610t_bool(hAPP_f472159229t_bool(cOMBC_1683390479t_bool,insert_int),bot_bot_fun_int_bool)),B),A) ) ).
fof(fact_481_comp__fun__commute_Ofold__fun__comm,axiom,
! [X_1,Z_1,A,F] :
( hBOOL(finite1428139793iple_a(F))
=> ( hBOOL(hAPP_f1695230391l_bool(finite1738664244iple_a,A))
=> hAPP_H905846293iple_a(hAPP_H2085992369iple_a(F,X_1),hAPP_f1693662087iple_a(finite1205970312iple_a(F,Z_1),A)) = hAPP_f1693662087iple_a(finite1205970312iple_a(F,hAPP_H905846293iple_a(hAPP_H2085992369iple_a(F,X_1),Z_1)),A) ) ) ).
fof(fact_482_comp__fun__commute_Ofold__fun__comm,axiom,
! [X_1,Z_1,A,F] :
( hBOOL(finite2114894759a_bool(F))
=> ( hBOOL(hAPP_f1695230391l_bool(finite1738664244iple_a,A))
=> hAPP_f20753329a_bool(hAPP_H1743777351a_bool(F,X_1),hAPP_f20753329a_bool(finite1215486064a_bool(F,Z_1),A)) = hAPP_f20753329a_bool(finite1215486064a_bool(F,hAPP_f20753329a_bool(hAPP_H1743777351a_bool(F,X_1),Z_1)),A) ) ) ).
fof(fact_483_comp__fun__commute_Ofold__fun__comm,axiom,
! [X_1,Z_1,A,F] :
( hBOOL(finite2130160977at_nat(F))
=> ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
=> hAPP_nat_nat(hAPP_nat_fun_nat_nat(F,X_1),hAPP_f22106695ol_nat(finite_fold_nat_nat(F,Z_1),A)) = hAPP_f22106695ol_nat(finite_fold_nat_nat(F,hAPP_nat_nat(hAPP_nat_fun_nat_nat(F,X_1),Z_1)),A) ) ) ).
fof(fact_484_comp__fun__commute_Ofold__fun__comm,axiom,
! [X_1,Z_1,A,F] :
( hBOOL(finite1071749497t_bool(F))
=> ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
=> hAPP_f800510211t_bool(hAPP_n1512601776t_bool(F,X_1),hAPP_f800510211t_bool(finite1578363458t_bool(F,Z_1),A)) = hAPP_f800510211t_bool(finite1578363458t_bool(F,hAPP_f800510211t_bool(hAPP_n1512601776t_bool(F,X_1),Z_1)),A) ) ) ).
fof(fact_485_comp__fun__commute_Ofold__fun__comm,axiom,
! [X_1,Z_1,A,F] :
( hBOOL(finite1973466193nt_int(F))
=> ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,A))
=> hAPP_int_int(hAPP_int_fun_int_int(F,X_1),hAPP_f1594865479ol_int(finite_fold_int_int(F,Z_1),A)) = hAPP_f1594865479ol_int(finite_fold_int_int(F,hAPP_int_int(hAPP_int_fun_int_int(F,X_1),Z_1)),A) ) ) ).
fof(fact_486_comp__fun__commute_Ofold__fun__comm,axiom,
! [X_1,Z_1,A,F] :
( hBOOL(finite1321096241t_bool(F))
=> ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,A))
=> hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(F,X_1),hAPP_f1805168059t_bool(finite1827710202t_bool(F,Z_1),A)) = hAPP_f1805168059t_bool(finite1827710202t_bool(F,hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(F,X_1),Z_1)),A) ) ) ).
fof(fact_487_comp__fun__commute_Ofun__left__comm,axiom,
! [X_1,Y_1,Z_1,F] :
( hBOOL(finite1973466193nt_int(F))
=> hAPP_int_int(hAPP_int_fun_int_int(F,X_1),hAPP_int_int(hAPP_int_fun_int_int(F,Y_1),Z_1)) = hAPP_int_int(hAPP_int_fun_int_int(F,Y_1),hAPP_int_int(hAPP_int_fun_int_int(F,X_1),Z_1)) ) ).
fof(fact_488_comp__fun__commute_Ofun__left__comm,axiom,
! [X_1,Y_1,Z_1,F] :
( hBOOL(finite2130160977at_nat(F))
=> hAPP_nat_nat(hAPP_nat_fun_nat_nat(F,X_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(F,Y_1),Z_1)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(F,Y_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(F,X_1),Z_1)) ) ).
fof(fact_489_setsum__commute,axiom,
! [F,B,A] : hAPP_f22106695ol_nat(hAPP_f782000547ol_nat(big_co387207925at_nat,hAPP_f1473602334at_nat(hAPP_f630508183at_nat(cOMBC_1505178355ol_nat,hAPP_f299305025ol_nat(hAPP_f1632710027ol_nat(cOMBB_1453893592at_nat,big_co1740723097nt_nat),F)),B)),A) = hAPP_f957591787ol_nat(hAPP_f1109019371ol_nat(big_co1740723097nt_nat,hAPP_f787214110nt_nat(hAPP_f261731407nt_nat(cOMBC_462113011ol_nat,hAPP_f1501159417ol_nat(hAPP_f1029311995ol_nat(cOMBB_1562322300at_int,big_co387207925at_nat),hAPP_f446977493at_nat(cOMBC_nat_int_nat,F))),A)),B) ).
fof(fact_490_setsum__commute,axiom,
! [F,B,A] : hAPP_f22106695ol_nat(hAPP_f782000547ol_nat(big_co387207925at_nat,hAPP_f2122367971at_nat(hAPP_f870149665at_nat(cOMBC_2036134968ol_nat,hAPP_f883871819ol_nat(hAPP_f424754463ol_nat(cOMBB_1716606670at_nat,big_co1340561246_a_nat),F)),B)),A) = hAPP_f2097660464ol_nat(hAPP_f1371399797ol_nat(big_co1340561246_a_nat,hAPP_f1464798435_a_nat(hAPP_f1012303887_a_nat(cOMBC_1159891896ol_nat,hAPP_f1669765561ol_nat(hAPP_f1073041467ol_nat(cOMBB_10565751iple_a,big_co387207925at_nat),hAPP_f2146713109at_nat(cOMBC_441305270_a_nat,F))),A)),B) ).
fof(fact_491_setsum__commute,axiom,
! [F,B,A] : hAPP_f957591787ol_nat(hAPP_f1109019371ol_nat(big_co1740723097nt_nat,hAPP_f787214110nt_nat(hAPP_f261731407nt_nat(cOMBC_462113011ol_nat,hAPP_f1501159417ol_nat(hAPP_f1029311995ol_nat(cOMBB_1562322300at_int,big_co387207925at_nat),F)),B)),A) = hAPP_f22106695ol_nat(hAPP_f782000547ol_nat(big_co387207925at_nat,hAPP_f1473602334at_nat(hAPP_f630508183at_nat(cOMBC_1505178355ol_nat,hAPP_f299305025ol_nat(hAPP_f1632710027ol_nat(cOMBB_1453893592at_nat,big_co1740723097nt_nat),hAPP_f879494613nt_nat(cOMBC_int_nat_nat,F))),A)),B) ).
fof(fact_492_setsum__commute,axiom,
! [F,B,A] : hAPP_f2097660464ol_nat(hAPP_f1371399797ol_nat(big_co1340561246_a_nat,hAPP_f1464798435_a_nat(hAPP_f1012303887_a_nat(cOMBC_1159891896ol_nat,hAPP_f1669765561ol_nat(hAPP_f1073041467ol_nat(cOMBB_10565751iple_a,big_co387207925at_nat),F)),B)),A) = hAPP_f22106695ol_nat(hAPP_f782000547ol_nat(big_co387207925at_nat,hAPP_f2122367971at_nat(hAPP_f870149665at_nat(cOMBC_2036134968ol_nat,hAPP_f883871819ol_nat(hAPP_f424754463ol_nat(cOMBB_1716606670at_nat,big_co1340561246_a_nat),hAPP_f1903462933_a_nat(cOMBC_667342884at_nat,F))),A)),B) ).
fof(fact_493_comp__fun__commute_Ofold__equality,axiom,
! [Z_1,A,Y_1,F] :
( hBOOL(finite1973466193nt_int(F))
=> ( hBOOL(hAPP_int_bool(hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(finite772772422nt_int(F),Z_1),A),Y_1))
=> hAPP_f1594865479ol_int(finite_fold_int_int(F,Z_1),A) = Y_1 ) ) ).
fof(fact_494_comp__fun__commute_Ofold__equality,axiom,
! [Z_1,A,Y_1,F] :
( hBOOL(finite2130160977at_nat(F))
=> ( hBOOL(hAPP_nat_bool(hAPP_f800510211t_bool(hAPP_n1512601776t_bool(finite929467206at_nat(F),Z_1),A),Y_1))
=> hAPP_f22106695ol_nat(finite_fold_nat_nat(F,Z_1),A) = Y_1 ) ) ).
fof(fact_495_comp__fun__commute_Ofold__equality,axiom,
! [Z_1,A,Y_1,F] :
( hBOOL(finite1428139793iple_a(F))
=> ( hBOOL(hAPP_H1037229737a_bool(hAPP_f20753329a_bool(hAPP_H1743777351a_bool(finite90892550iple_a(F),Z_1),A),Y_1))
=> hAPP_f1693662087iple_a(finite1205970312iple_a(F,Z_1),A) = Y_1 ) ) ).
fof(fact_496_comp__fun__commute_Ofold__equality,axiom,
! [Z_1,A,Y_1,F] :
( hBOOL(finite1321096241t_bool(F))
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(hAPP_f1634429330l_bool(finite1008073724t_bool(F),Z_1),A),Y_1))
=> hAPP_f1805168059t_bool(finite1827710202t_bool(F,Z_1),A) = Y_1 ) ) ).
fof(fact_497_comp__fun__commute_Ofold__equality,axiom,
! [Z_1,A,Y_1,F] :
( hBOOL(finite2114894759a_bool(F))
=> ( hBOOL(hAPP_f1695230391l_bool(hAPP_f1767618879l_bool(hAPP_f1577095245l_bool(finite1152845682a_bool(F),Z_1),A),Y_1))
=> hAPP_f20753329a_bool(finite1215486064a_bool(F,Z_1),A) = Y_1 ) ) ).
fof(fact_498_comp__fun__commute_Ofold__equality,axiom,
! [Z_1,A,Y_1,F] :
( hBOOL(finite1071749497t_bool(F))
=> ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(hAPP_f1266913334l_bool(finite758726980t_bool(F),Z_1),A),Y_1))
=> hAPP_f800510211t_bool(finite1578363458t_bool(F,Z_1),A) = Y_1 ) ) ).
fof(fact_499_fold__empty,axiom,
! [F,Z_1] : hAPP_f1693662087iple_a(finite1205970312iple_a(F,Z_1),bot_bo797238721a_bool) = Z_1 ).
fof(fact_500_fold__empty,axiom,
! [F,Z_1] : hAPP_f20753329a_bool(finite1215486064a_bool(F,Z_1),bot_bo797238721a_bool) = Z_1 ).
fof(fact_501_fold__empty,axiom,
! [F,Z_1] : hAPP_f1594865479ol_int(finite_fold_int_int(F,Z_1),bot_bot_fun_int_bool) = Z_1 ).
fof(fact_502_fold__empty,axiom,
! [F,Z_1] : hAPP_f1805168059t_bool(finite1827710202t_bool(F,Z_1),bot_bot_fun_int_bool) = Z_1 ).
fof(fact_503_fold__empty,axiom,
! [F,Z_1] : hAPP_f22106695ol_nat(finite_fold_nat_nat(F,Z_1),bot_bot_fun_nat_bool) = Z_1 ).
fof(fact_504_fold__empty,axiom,
! [F,Z_1] : hAPP_f800510211t_bool(finite1578363458t_bool(F,Z_1),bot_bot_fun_nat_bool) = Z_1 ).
fof(fact_505_comp__fun__commute_Ofold__insert2,axiom,
! [Z_1,X_1,A,F] :
( hBOOL(finite2130160977at_nat(F))
=> ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
=> ( ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A))
=> hAPP_f22106695ol_nat(finite_fold_nat_nat(F,Z_1),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),A)) = hAPP_f22106695ol_nat(finite_fold_nat_nat(F,hAPP_nat_nat(hAPP_nat_fun_nat_nat(F,X_1),Z_1)),A) ) ) ) ).
fof(fact_506_comp__fun__commute_Ofold__insert2,axiom,
! [Z_1,X_1,A,F] :
( hBOOL(finite1071749497t_bool(F))
=> ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
=> ( ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A))
=> hAPP_f800510211t_bool(finite1578363458t_bool(F,Z_1),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),A)) = hAPP_f800510211t_bool(finite1578363458t_bool(F,hAPP_f800510211t_bool(hAPP_n1512601776t_bool(F,X_1),Z_1)),A) ) ) ) ).
fof(fact_507_comp__fun__commute_Ofold__insert2,axiom,
! [Z_1,X_1,A,F] :
( hBOOL(finite1428139793iple_a(F))
=> ( hBOOL(hAPP_f1695230391l_bool(finite1738664244iple_a,A))
=> ( ~ hBOOL(hAPP_f1695230391l_bool(hAPP_H622608077l_bool(member1797258804iple_a,X_1),A))
=> hAPP_f1693662087iple_a(finite1205970312iple_a(F,Z_1),hAPP_f20753329a_bool(hAPP_H1743777351a_bool(insert956547291iple_a,X_1),A)) = hAPP_f1693662087iple_a(finite1205970312iple_a(F,hAPP_H905846293iple_a(hAPP_H2085992369iple_a(F,X_1),Z_1)),A) ) ) ) ).
fof(fact_508_comp__fun__commute_Ofold__insert2,axiom,
! [Z_1,X_1,A,F] :
( hBOOL(finite2114894759a_bool(F))
=> ( hBOOL(hAPP_f1695230391l_bool(finite1738664244iple_a,A))
=> ( ~ hBOOL(hAPP_f1695230391l_bool(hAPP_H622608077l_bool(member1797258804iple_a,X_1),A))
=> hAPP_f20753329a_bool(finite1215486064a_bool(F,Z_1),hAPP_f20753329a_bool(hAPP_H1743777351a_bool(insert956547291iple_a,X_1),A)) = hAPP_f20753329a_bool(finite1215486064a_bool(F,hAPP_f20753329a_bool(hAPP_H1743777351a_bool(F,X_1),Z_1)),A) ) ) ) ).
fof(fact_509_comp__fun__commute_Ofold__insert2,axiom,
! [Z_1,X_1,A,F] :
( hBOOL(finite1973466193nt_int(F))
=> ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,A))
=> ( ~ hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),A))
=> hAPP_f1594865479ol_int(finite_fold_int_int(F,Z_1),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),A)) = hAPP_f1594865479ol_int(finite_fold_int_int(F,hAPP_int_int(hAPP_int_fun_int_int(F,X_1),Z_1)),A) ) ) ) ).
fof(fact_510_comp__fun__commute_Ofold__insert2,axiom,
! [Z_1,X_1,A,F] :
( hBOOL(finite1321096241t_bool(F))
=> ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,A))
=> ( ~ hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),A))
=> hAPP_f1805168059t_bool(finite1827710202t_bool(F,Z_1),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),A)) = hAPP_f1805168059t_bool(finite1827710202t_bool(F,hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(F,X_1),Z_1)),A) ) ) ) ).
fof(fact_511_comp__fun__commute_Ofold__insert,axiom,
! [Z_1,X_1,A,F] :
( hBOOL(finite2130160977at_nat(F))
=> ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
=> ( ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A))
=> hAPP_f22106695ol_nat(finite_fold_nat_nat(F,Z_1),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),A)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(F,X_1),hAPP_f22106695ol_nat(finite_fold_nat_nat(F,Z_1),A)) ) ) ) ).
fof(fact_512_comp__fun__commute_Ofold__insert,axiom,
! [Z_1,X_1,A,F] :
( hBOOL(finite1071749497t_bool(F))
=> ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
=> ( ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A))
=> hAPP_f800510211t_bool(finite1578363458t_bool(F,Z_1),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),A)) = hAPP_f800510211t_bool(hAPP_n1512601776t_bool(F,X_1),hAPP_f800510211t_bool(finite1578363458t_bool(F,Z_1),A)) ) ) ) ).
fof(fact_513_comp__fun__commute_Ofold__insert,axiom,
! [Z_1,X_1,A,F] :
( hBOOL(finite1428139793iple_a(F))
=> ( hBOOL(hAPP_f1695230391l_bool(finite1738664244iple_a,A))
=> ( ~ hBOOL(hAPP_f1695230391l_bool(hAPP_H622608077l_bool(member1797258804iple_a,X_1),A))
=> hAPP_f1693662087iple_a(finite1205970312iple_a(F,Z_1),hAPP_f20753329a_bool(hAPP_H1743777351a_bool(insert956547291iple_a,X_1),A)) = hAPP_H905846293iple_a(hAPP_H2085992369iple_a(F,X_1),hAPP_f1693662087iple_a(finite1205970312iple_a(F,Z_1),A)) ) ) ) ).
fof(fact_514_comp__fun__commute_Ofold__insert,axiom,
! [Z_1,X_1,A,F] :
( hBOOL(finite2114894759a_bool(F))
=> ( hBOOL(hAPP_f1695230391l_bool(finite1738664244iple_a,A))
=> ( ~ hBOOL(hAPP_f1695230391l_bool(hAPP_H622608077l_bool(member1797258804iple_a,X_1),A))
=> hAPP_f20753329a_bool(finite1215486064a_bool(F,Z_1),hAPP_f20753329a_bool(hAPP_H1743777351a_bool(insert956547291iple_a,X_1),A)) = hAPP_f20753329a_bool(hAPP_H1743777351a_bool(F,X_1),hAPP_f20753329a_bool(finite1215486064a_bool(F,Z_1),A)) ) ) ) ).
fof(fact_515_comp__fun__commute_Ofold__insert,axiom,
! [Z_1,X_1,A,F] :
( hBOOL(finite1973466193nt_int(F))
=> ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,A))
=> ( ~ hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),A))
=> hAPP_f1594865479ol_int(finite_fold_int_int(F,Z_1),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),A)) = hAPP_int_int(hAPP_int_fun_int_int(F,X_1),hAPP_f1594865479ol_int(finite_fold_int_int(F,Z_1),A)) ) ) ) ).
fof(fact_516_comp__fun__commute_Ofold__insert,axiom,
! [Z_1,X_1,A,F] :
( hBOOL(finite1321096241t_bool(F))
=> ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,A))
=> ( ~ hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),A))
=> hAPP_f1805168059t_bool(finite1827710202t_bool(F,Z_1),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),A)) = hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(F,X_1),hAPP_f1805168059t_bool(finite1827710202t_bool(F,Z_1),A)) ) ) ) ).
fof(fact_517_comp__fun__commute_Ofold__graph__fold,axiom,
! [Z_1,A,F] :
( hBOOL(finite1428139793iple_a(F))
=> ( hBOOL(hAPP_f1695230391l_bool(finite1738664244iple_a,A))
=> hBOOL(hAPP_H1037229737a_bool(hAPP_f20753329a_bool(hAPP_H1743777351a_bool(finite90892550iple_a(F),Z_1),A),hAPP_f1693662087iple_a(finite1205970312iple_a(F,Z_1),A))) ) ) ).
fof(fact_518_comp__fun__commute_Ofold__graph__fold,axiom,
! [Z_1,A,F] :
( hBOOL(finite2114894759a_bool(F))
=> ( hBOOL(hAPP_f1695230391l_bool(finite1738664244iple_a,A))
=> hBOOL(hAPP_f1695230391l_bool(hAPP_f1767618879l_bool(hAPP_f1577095245l_bool(finite1152845682a_bool(F),Z_1),A),hAPP_f20753329a_bool(finite1215486064a_bool(F,Z_1),A))) ) ) ).
fof(fact_519_comp__fun__commute_Ofold__graph__fold,axiom,
! [Z_1,A,F] :
( hBOOL(finite2130160977at_nat(F))
=> ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
=> hBOOL(hAPP_nat_bool(hAPP_f800510211t_bool(hAPP_n1512601776t_bool(finite929467206at_nat(F),Z_1),A),hAPP_f22106695ol_nat(finite_fold_nat_nat(F,Z_1),A))) ) ) ).
fof(fact_520_comp__fun__commute_Ofold__graph__fold,axiom,
! [Z_1,A,F] :
( hBOOL(finite1071749497t_bool(F))
=> ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
=> hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(hAPP_f1266913334l_bool(finite758726980t_bool(F),Z_1),A),hAPP_f800510211t_bool(finite1578363458t_bool(F,Z_1),A))) ) ) ).
fof(fact_521_comp__fun__commute_Ofold__graph__fold,axiom,
! [Z_1,A,F] :
( hBOOL(finite1973466193nt_int(F))
=> ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,A))
=> hBOOL(hAPP_int_bool(hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(finite772772422nt_int(F),Z_1),A),hAPP_f1594865479ol_int(finite_fold_int_int(F,Z_1),A))) ) ) ).
fof(fact_522_comp__fun__commute_Ofold__graph__fold,axiom,
! [Z_1,A,F] :
( hBOOL(finite1321096241t_bool(F))
=> ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,A))
=> hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(hAPP_f1634429330l_bool(finite1008073724t_bool(F),Z_1),A),hAPP_f1805168059t_bool(finite1827710202t_bool(F,Z_1),A))) ) ) ).
fof(fact_523_setsum__product,axiom,
! [F,A,G,B] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_f22106695ol_nat(hAPP_f782000547ol_nat(big_co387207925at_nat,F),A)),hAPP_f957591787ol_nat(hAPP_f1109019371ol_nat(big_co1740723097nt_nat,G),B)) = hAPP_f22106695ol_nat(hAPP_f782000547ol_nat(big_co387207925at_nat,hAPP_f1473602334at_nat(hAPP_f630508183at_nat(cOMBC_1505178355ol_nat,hAPP_f299305025ol_nat(hAPP_f1632710027ol_nat(cOMBB_1453893592at_nat,big_co1740723097nt_nat),hAPP_f237327688nt_nat(hAPP_f1633513941nt_nat(cOMBC_1218031117nt_nat,hAPP_f2080483477nt_nat(hAPP_f1701814485nt_nat(cOMBB_737513486at_nat,cOMBB_nat_nat_int),hAPP_f1639111240at_nat(hAPP_f1731313045at_nat(cOMBB_963856155at_nat,times_times_nat),F))),G))),B)),A) ).
fof(fact_524_setsum__product,axiom,
! [F,A,G,B] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_f22106695ol_nat(hAPP_f782000547ol_nat(big_co387207925at_nat,F),A)),hAPP_f2097660464ol_nat(hAPP_f1371399797ol_nat(big_co1340561246_a_nat,G),B)) = hAPP_f22106695ol_nat(hAPP_f782000547ol_nat(big_co387207925at_nat,hAPP_f2122367971at_nat(hAPP_f870149665at_nat(cOMBC_2036134968ol_nat,hAPP_f883871819ol_nat(hAPP_f424754463ol_nat(cOMBB_1716606670at_nat,big_co1340561246_a_nat),hAPP_f1393576712_a_nat(hAPP_f145529813_a_nat(cOMBC_309106125_a_nat,hAPP_f1485673429_a_nat(hAPP_f2006236373_a_nat(cOMBB_2005430478at_nat,cOMBB_1250632933iple_a),hAPP_f1639111240at_nat(hAPP_f1731313045at_nat(cOMBB_963856155at_nat,times_times_nat),F))),G))),B)),A) ).
fof(fact_525_setsum__product,axiom,
! [F,A,G,B] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_f957591787ol_nat(hAPP_f1109019371ol_nat(big_co1740723097nt_nat,F),A)),hAPP_f22106695ol_nat(hAPP_f782000547ol_nat(big_co387207925at_nat,G),B)) = hAPP_f957591787ol_nat(hAPP_f1109019371ol_nat(big_co1740723097nt_nat,hAPP_f787214110nt_nat(hAPP_f261731407nt_nat(cOMBC_462113011ol_nat,hAPP_f1501159417ol_nat(hAPP_f1029311995ol_nat(cOMBB_1562322300at_int,big_co387207925at_nat),hAPP_f1169617132at_nat(hAPP_f701299925at_nat(cOMBC_1294079849at_nat,hAPP_f1033905301at_nat(hAPP_f1315855317at_nat(cOMBB_891709290at_int,cOMBB_nat_nat_nat),hAPP_f1463450952at_nat(hAPP_f1087393429at_nat(cOMBB_1601129847at_int,times_times_nat),F))),G))),B)),A) ).
fof(fact_526_setsum__product,axiom,
! [F,A,G,B] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_f2097660464ol_nat(hAPP_f1371399797ol_nat(big_co1340561246_a_nat,F),A)),hAPP_f22106695ol_nat(hAPP_f782000547ol_nat(big_co387207925at_nat,G),B)) = hAPP_f2097660464ol_nat(hAPP_f1371399797ol_nat(big_co1340561246_a_nat,hAPP_f1464798435_a_nat(hAPP_f1012303887_a_nat(cOMBC_1159891896ol_nat,hAPP_f1669765561ol_nat(hAPP_f1073041467ol_nat(cOMBB_10565751iple_a,big_co387207925at_nat),hAPP_f221335601at_nat(hAPP_f388255893at_nat(cOMBC_1038836708at_nat,hAPP_f1879857877at_nat(hAPP_f481016213at_nat(cOMBB_1672982117iple_a,cOMBB_nat_nat_nat),hAPP_f429024520at_nat(hAPP_f703420885at_nat(cOMBB_789959282iple_a,times_times_nat),F))),G))),B)),A) ).
fof(fact_527_setsum__left__distrib,axiom,
! [F,A,R_2] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_f22106695ol_nat(hAPP_f782000547ol_nat(big_co387207925at_nat,F),A)),R_2) = hAPP_f22106695ol_nat(hAPP_f782000547ol_nat(big_co387207925at_nat,hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,hAPP_f1639111240at_nat(hAPP_f1731313045at_nat(cOMBB_963856155at_nat,times_times_nat),F)),R_2)),A) ).
fof(fact_528_setsum__left__distrib,axiom,
! [F,A,R_2] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_f957591787ol_nat(hAPP_f1109019371ol_nat(big_co1740723097nt_nat,F),A)),R_2) = hAPP_f957591787ol_nat(hAPP_f1109019371ol_nat(big_co1740723097nt_nat,hAPP_nat_fun_int_nat(hAPP_f879494613nt_nat(cOMBC_int_nat_nat,hAPP_f1463450952at_nat(hAPP_f1087393429at_nat(cOMBB_1601129847at_int,times_times_nat),F)),R_2)),A) ).
fof(fact_529_setsum__left__distrib,axiom,
! [F,A,R_2] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_f2097660464ol_nat(hAPP_f1371399797ol_nat(big_co1340561246_a_nat,F),A)),R_2) = hAPP_f2097660464ol_nat(hAPP_f1371399797ol_nat(big_co1340561246_a_nat,hAPP_n1653940209_a_nat(hAPP_f1903462933_a_nat(cOMBC_667342884at_nat,hAPP_f429024520at_nat(hAPP_f703420885at_nat(cOMBB_789959282iple_a,times_times_nat),F)),R_2)),A) ).
fof(fact_530_setsum__right__distrib,axiom,
! [R_2,F,A] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,R_2),hAPP_f22106695ol_nat(hAPP_f782000547ol_nat(big_co387207925at_nat,F),A)) = hAPP_f22106695ol_nat(hAPP_f782000547ol_nat(big_co387207925at_nat,hAPP_f1914919701at_nat(hAPP_f1585078997at_nat(cOMBB_nat_nat_nat,hAPP_nat_fun_nat_nat(times_times_nat,R_2)),F)),A) ).
fof(fact_531_setsum__right__distrib,axiom,
! [R_2,F,A] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,R_2),hAPP_f957591787ol_nat(hAPP_f1109019371ol_nat(big_co1740723097nt_nat,F),A)) = hAPP_f957591787ol_nat(hAPP_f1109019371ol_nat(big_co1740723097nt_nat,hAPP_f2132704789nt_nat(hAPP_f901215189nt_nat(cOMBB_nat_nat_int,hAPP_nat_fun_nat_nat(times_times_nat,R_2)),F)),A) ).
fof(fact_532_setsum__right__distrib,axiom,
! [R_2,F,A] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,R_2),hAPP_f2097660464ol_nat(hAPP_f1371399797ol_nat(big_co1340561246_a_nat,F),A)) = hAPP_f2097660464ol_nat(hAPP_f1371399797ol_nat(big_co1340561246_a_nat,hAPP_f106743253_a_nat(hAPP_f500900629_a_nat(cOMBB_1250632933iple_a,hAPP_nat_fun_nat_nat(times_times_nat,R_2)),F)),A) ).
fof(fact_533_comp__fun__commute,axiom,
hBOOL(finite2130160977at_nat(times_times_nat)) ).
fof(fact_534_comp__fun__commute,axiom,
hBOOL(finite1973466193nt_int(times_times_int)) ).
fof(fact_535_comp__fun__commute_Ofold__graph__determ,axiom,
! [Y_1,Z_1,A,X_1,F] :
( hBOOL(finite1973466193nt_int(F))
=> ( hBOOL(hAPP_int_bool(hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(finite772772422nt_int(F),Z_1),A),X_1))
=> ( hBOOL(hAPP_int_bool(hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(finite772772422nt_int(F),Z_1),A),Y_1))
=> Y_1 = X_1 ) ) ) ).
fof(fact_536_comp__fun__commute_Ofold__graph__determ,axiom,
! [Y_1,Z_1,A,X_1,F] :
( hBOOL(finite2130160977at_nat(F))
=> ( hBOOL(hAPP_nat_bool(hAPP_f800510211t_bool(hAPP_n1512601776t_bool(finite929467206at_nat(F),Z_1),A),X_1))
=> ( hBOOL(hAPP_nat_bool(hAPP_f800510211t_bool(hAPP_n1512601776t_bool(finite929467206at_nat(F),Z_1),A),Y_1))
=> Y_1 = X_1 ) ) ) ).
fof(fact_537_comp__fun__commute_Ofold__graph__determ,axiom,
! [Y_1,Z_1,A,X_1,F] :
( hBOOL(finite1428139793iple_a(F))
=> ( hBOOL(hAPP_H1037229737a_bool(hAPP_f20753329a_bool(hAPP_H1743777351a_bool(finite90892550iple_a(F),Z_1),A),X_1))
=> ( hBOOL(hAPP_H1037229737a_bool(hAPP_f20753329a_bool(hAPP_H1743777351a_bool(finite90892550iple_a(F),Z_1),A),Y_1))
=> Y_1 = X_1 ) ) ) ).
fof(fact_538_setprod_Odistrib,axiom,
! [G,H,A] :
( hBOOL(hAPP_f1695230391l_bool(finite1738664244iple_a,A))
=> hAPP_f2097660464ol_nat(big_co1815942735_a_nat(hAPP_f106743253_a_nat(hAPP_f1418102009_a_nat(cOMBS_1756972052at_nat,hAPP_f429024520at_nat(hAPP_f703420885at_nat(cOMBB_789959282iple_a,times_times_nat),G)),H)),A) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_f2097660464ol_nat(big_co1815942735_a_nat(G),A)),hAPP_f2097660464ol_nat(big_co1815942735_a_nat(H),A)) ) ).
fof(fact_539_setprod_Odistrib,axiom,
! [G,H,A] :
( hBOOL(hAPP_f1695230391l_bool(finite1738664244iple_a,A))
=> hAPP_f587450508ol_int(big_co305732779_a_int(hAPP_f1561000661_a_int(hAPP_f789902073_a_int(cOMBS_1600277268nt_int,hAPP_f231878828nt_int(hAPP_f1621780181nt_int(cOMBB_1885471694iple_a,times_times_int),G)),H)),A) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_f587450508ol_int(big_co305732779_a_int(G),A)),hAPP_f587450508ol_int(big_co305732779_a_int(H),A)) ) ).
fof(fact_540_comp__fun__commute_Ofold__rec,axiom,
! [Z_1,X_1,A,F] :
( hBOOL(finite2130160977at_nat(F))
=> ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
=> ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A))
=> hAPP_f22106695ol_nat(finite_fold_nat_nat(F,Z_1),A) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(F,X_1),hAPP_f22106695ol_nat(finite_fold_nat_nat(F,Z_1),hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),bot_bot_fun_nat_bool)))) ) ) ) ).
fof(fact_541_comp__fun__commute_Ofold__rec,axiom,
! [Z_1,X_1,A,F] :
( hBOOL(finite1071749497t_bool(F))
=> ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
=> ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A))
=> hAPP_f800510211t_bool(finite1578363458t_bool(F,Z_1),A) = hAPP_f800510211t_bool(hAPP_n1512601776t_bool(F,X_1),hAPP_f800510211t_bool(finite1578363458t_bool(F,Z_1),hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),bot_bot_fun_nat_bool)))) ) ) ) ).
fof(fact_542_comp__fun__commute_Ofold__rec,axiom,
! [Z_1,X_1,A,F] :
( hBOOL(finite1428139793iple_a(F))
=> ( hBOOL(hAPP_f1695230391l_bool(finite1738664244iple_a,A))
=> ( hBOOL(hAPP_f1695230391l_bool(hAPP_H622608077l_bool(member1797258804iple_a,X_1),A))
=> hAPP_f1693662087iple_a(finite1205970312iple_a(F,Z_1),A) = hAPP_H905846293iple_a(hAPP_H2085992369iple_a(F,X_1),hAPP_f1693662087iple_a(finite1205970312iple_a(F,Z_1),hAPP_f20753329a_bool(hAPP_f1706700729a_bool(minus_85316870a_bool,A),hAPP_f20753329a_bool(hAPP_H1743777351a_bool(insert956547291iple_a,X_1),bot_bo797238721a_bool)))) ) ) ) ).
fof(fact_543_comp__fun__commute_Ofold__rec,axiom,
! [Z_1,X_1,A,F] :
( hBOOL(finite2114894759a_bool(F))
=> ( hBOOL(hAPP_f1695230391l_bool(finite1738664244iple_a,A))
=> ( hBOOL(hAPP_f1695230391l_bool(hAPP_H622608077l_bool(member1797258804iple_a,X_1),A))
=> hAPP_f20753329a_bool(finite1215486064a_bool(F,Z_1),A) = hAPP_f20753329a_bool(hAPP_H1743777351a_bool(F,X_1),hAPP_f20753329a_bool(finite1215486064a_bool(F,Z_1),hAPP_f20753329a_bool(hAPP_f1706700729a_bool(minus_85316870a_bool,A),hAPP_f20753329a_bool(hAPP_H1743777351a_bool(insert956547291iple_a,X_1),bot_bo797238721a_bool)))) ) ) ) ).
fof(fact_544_comp__fun__commute_Ofold__rec,axiom,
! [Z_1,X_1,A,F] :
( hBOOL(finite1973466193nt_int(F))
=> ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,A))
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),A))
=> hAPP_f1594865479ol_int(finite_fold_int_int(F,Z_1),A) = hAPP_int_int(hAPP_int_fun_int_int(F,X_1),hAPP_f1594865479ol_int(finite_fold_int_int(F,Z_1),hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),bot_bot_fun_int_bool)))) ) ) ) ).
fof(fact_545_comp__fun__commute_Ofold__rec,axiom,
! [Z_1,X_1,A,F] :
( hBOOL(finite1321096241t_bool(F))
=> ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,A))
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),A))
=> hAPP_f1805168059t_bool(finite1827710202t_bool(F,Z_1),A) = hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(F,X_1),hAPP_f1805168059t_bool(finite1827710202t_bool(F,Z_1),hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),bot_bot_fun_int_bool)))) ) ) ) ).
fof(fact_546_comp__fun__commute_Ofold__insert__remove,axiom,
! [Z_1,X_1,A,F] :
( hBOOL(finite2130160977at_nat(F))
=> ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
=> hAPP_f22106695ol_nat(finite_fold_nat_nat(F,Z_1),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),A)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(F,X_1),hAPP_f22106695ol_nat(finite_fold_nat_nat(F,Z_1),hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),bot_bot_fun_nat_bool)))) ) ) ).
fof(fact_547_comp__fun__commute_Ofold__insert__remove,axiom,
! [Z_1,X_1,A,F] :
( hBOOL(finite1071749497t_bool(F))
=> ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
=> hAPP_f800510211t_bool(finite1578363458t_bool(F,Z_1),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),A)) = hAPP_f800510211t_bool(hAPP_n1512601776t_bool(F,X_1),hAPP_f800510211t_bool(finite1578363458t_bool(F,Z_1),hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),bot_bot_fun_nat_bool)))) ) ) ).
fof(fact_548_comp__fun__commute_Ofold__insert__remove,axiom,
! [Z_1,X_1,A,F] :
( hBOOL(finite1428139793iple_a(F))
=> ( hBOOL(hAPP_f1695230391l_bool(finite1738664244iple_a,A))
=> hAPP_f1693662087iple_a(finite1205970312iple_a(F,Z_1),hAPP_f20753329a_bool(hAPP_H1743777351a_bool(insert956547291iple_a,X_1),A)) = hAPP_H905846293iple_a(hAPP_H2085992369iple_a(F,X_1),hAPP_f1693662087iple_a(finite1205970312iple_a(F,Z_1),hAPP_f20753329a_bool(hAPP_f1706700729a_bool(minus_85316870a_bool,A),hAPP_f20753329a_bool(hAPP_H1743777351a_bool(insert956547291iple_a,X_1),bot_bo797238721a_bool)))) ) ) ).
fof(fact_549_comp__fun__commute_Ofold__insert__remove,axiom,
! [Z_1,X_1,A,F] :
( hBOOL(finite2114894759a_bool(F))
=> ( hBOOL(hAPP_f1695230391l_bool(finite1738664244iple_a,A))
=> hAPP_f20753329a_bool(finite1215486064a_bool(F,Z_1),hAPP_f20753329a_bool(hAPP_H1743777351a_bool(insert956547291iple_a,X_1),A)) = hAPP_f20753329a_bool(hAPP_H1743777351a_bool(F,X_1),hAPP_f20753329a_bool(finite1215486064a_bool(F,Z_1),hAPP_f20753329a_bool(hAPP_f1706700729a_bool(minus_85316870a_bool,A),hAPP_f20753329a_bool(hAPP_H1743777351a_bool(insert956547291iple_a,X_1),bot_bo797238721a_bool)))) ) ) ).
fof(fact_550_comp__fun__commute_Ofold__insert__remove,axiom,
! [Z_1,X_1,A,F] :
( hBOOL(finite1973466193nt_int(F))
=> ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,A))
=> hAPP_f1594865479ol_int(finite_fold_int_int(F,Z_1),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),A)) = hAPP_int_int(hAPP_int_fun_int_int(F,X_1),hAPP_f1594865479ol_int(finite_fold_int_int(F,Z_1),hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),bot_bot_fun_int_bool)))) ) ) ).
fof(fact_551_comp__fun__commute_Ofold__insert__remove,axiom,
! [Z_1,X_1,A,F] :
( hBOOL(finite1321096241t_bool(F))
=> ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,A))
=> hAPP_f1805168059t_bool(finite1827710202t_bool(F,Z_1),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),A)) = hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(F,X_1),hAPP_f1805168059t_bool(finite1827710202t_bool(F,Z_1),hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),bot_bot_fun_int_bool)))) ) ) ).
fof(fact_552_setprod__insert,axiom,
! [F,A_13,A] :
( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
=> ( ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_13),A))
=> hAPP_f22106695ol_nat(big_co1705425894at_nat(F),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_13),A)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(F,A_13)),hAPP_f22106695ol_nat(big_co1705425894at_nat(F),A)) ) ) ).
fof(fact_553_setprod__insert,axiom,
! [F,A_13,A] :
( hBOOL(hAPP_f1695230391l_bool(finite1738664244iple_a,A))
=> ( ~ hBOOL(hAPP_f1695230391l_bool(hAPP_H622608077l_bool(member1797258804iple_a,A_13),A))
=> hAPP_f2097660464ol_nat(big_co1815942735_a_nat(F),hAPP_f20753329a_bool(hAPP_H1743777351a_bool(insert956547291iple_a,A_13),A)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_H592031934_a_nat(F,A_13)),hAPP_f2097660464ol_nat(big_co1815942735_a_nat(F),A)) ) ) ).
fof(fact_554_setprod__insert,axiom,
! [F,A_13,A] :
( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
=> ( ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_13),A))
=> hAPP_f659380387ol_int(big_co195215938at_int(F),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_13),A)) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_nat_int(F,A_13)),hAPP_f659380387ol_int(big_co195215938at_int(F),A)) ) ) ).
fof(fact_555_setprod__insert,axiom,
! [F,A_13,A] :
( hBOOL(hAPP_f1695230391l_bool(finite1738664244iple_a,A))
=> ( ~ hBOOL(hAPP_f1695230391l_bool(hAPP_H622608077l_bool(member1797258804iple_a,A_13),A))
=> hAPP_f587450508ol_int(big_co305732779_a_int(F),hAPP_f20753329a_bool(hAPP_H1743777351a_bool(insert956547291iple_a,A_13),A)) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_H1229305626_a_int(F,A_13)),hAPP_f587450508ol_int(big_co305732779_a_int(F),A)) ) ) ).
fof(fact_556_setprod_Oinsert,axiom,
! [G,X_1,A] :
( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
=> ( ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A))
=> hAPP_f22106695ol_nat(big_co1705425894at_nat(G),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),A)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(G,X_1)),hAPP_f22106695ol_nat(big_co1705425894at_nat(G),A)) ) ) ).
fof(fact_557_setprod_Oinsert,axiom,
! [G,X_1,A] :
( hBOOL(hAPP_f1695230391l_bool(finite1738664244iple_a,A))
=> ( ~ hBOOL(hAPP_f1695230391l_bool(hAPP_H622608077l_bool(member1797258804iple_a,X_1),A))
=> hAPP_f2097660464ol_nat(big_co1815942735_a_nat(G),hAPP_f20753329a_bool(hAPP_H1743777351a_bool(insert956547291iple_a,X_1),A)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_H592031934_a_nat(G,X_1)),hAPP_f2097660464ol_nat(big_co1815942735_a_nat(G),A)) ) ) ).
fof(fact_558_setprod_Oinsert,axiom,
! [G,X_1,A] :
( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
=> ( ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A))
=> hAPP_f659380387ol_int(big_co195215938at_int(G),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),A)) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_nat_int(G,X_1)),hAPP_f659380387ol_int(big_co195215938at_int(G),A)) ) ) ).
fof(fact_559_setprod_Oinsert,axiom,
! [G,X_1,A] :
( hBOOL(hAPP_f1695230391l_bool(finite1738664244iple_a,A))
=> ( ~ hBOOL(hAPP_f1695230391l_bool(hAPP_H622608077l_bool(member1797258804iple_a,X_1),A))
=> hAPP_f587450508ol_int(big_co305732779_a_int(G),hAPP_f20753329a_bool(hAPP_H1743777351a_bool(insert956547291iple_a,X_1),A)) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_H1229305626_a_int(G,X_1)),hAPP_f587450508ol_int(big_co305732779_a_int(G),A)) ) ) ).
fof(fact_560_setsum__diff1__nat,axiom,
! [F,A_13,A] :
( ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_13),A))
=> hAPP_f22106695ol_nat(hAPP_f782000547ol_nat(big_co387207925at_nat,F),hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_13),bot_bot_fun_nat_bool))) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_f22106695ol_nat(hAPP_f782000547ol_nat(big_co387207925at_nat,F),A)),hAPP_nat_nat(F,A_13)) )
& ( ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_13),A))
=> hAPP_f22106695ol_nat(hAPP_f782000547ol_nat(big_co387207925at_nat,F),hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_13),bot_bot_fun_nat_bool))) = hAPP_f22106695ol_nat(hAPP_f782000547ol_nat(big_co387207925at_nat,F),A) ) ) ).
fof(fact_561_setsum__diff1__nat,axiom,
! [F,A_13,A] :
( ( hBOOL(hAPP_f1695230391l_bool(hAPP_H622608077l_bool(member1797258804iple_a,A_13),A))
=> hAPP_f2097660464ol_nat(hAPP_f1371399797ol_nat(big_co1340561246_a_nat,F),hAPP_f20753329a_bool(hAPP_f1706700729a_bool(minus_85316870a_bool,A),hAPP_f20753329a_bool(hAPP_H1743777351a_bool(insert956547291iple_a,A_13),bot_bo797238721a_bool))) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_f2097660464ol_nat(hAPP_f1371399797ol_nat(big_co1340561246_a_nat,F),A)),hAPP_H592031934_a_nat(F,A_13)) )
& ( ~ hBOOL(hAPP_f1695230391l_bool(hAPP_H622608077l_bool(member1797258804iple_a,A_13),A))
=> hAPP_f2097660464ol_nat(hAPP_f1371399797ol_nat(big_co1340561246_a_nat,F),hAPP_f20753329a_bool(hAPP_f1706700729a_bool(minus_85316870a_bool,A),hAPP_f20753329a_bool(hAPP_H1743777351a_bool(insert956547291iple_a,A_13),bot_bo797238721a_bool))) = hAPP_f2097660464ol_nat(hAPP_f1371399797ol_nat(big_co1340561246_a_nat,F),A) ) ) ).
fof(fact_562_setsum__diff1__nat,axiom,
! [F,A_13,A] :
( ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,A_13),A))
=> hAPP_f957591787ol_nat(hAPP_f1109019371ol_nat(big_co1740723097nt_nat,F),hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,A_13),bot_bot_fun_int_bool))) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_f957591787ol_nat(hAPP_f1109019371ol_nat(big_co1740723097nt_nat,F),A)),hAPP_int_nat(F,A_13)) )
& ( ~ hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,A_13),A))
=> hAPP_f957591787ol_nat(hAPP_f1109019371ol_nat(big_co1740723097nt_nat,F),hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,A_13),bot_bot_fun_int_bool))) = hAPP_f957591787ol_nat(hAPP_f1109019371ol_nat(big_co1740723097nt_nat,F),A) ) ) ).
fof(fact_563_fold__def,axiom,
! [F,Z_1,A] : hAPP_f1594865479ol_int(finite_fold_int_int(F,Z_1),A) = hAPP_f1594865479ol_int(the_int,hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(finite772772422nt_int(F),Z_1),A)) ).
fof(fact_564_fold__def,axiom,
! [F,Z_1,A] : hAPP_f22106695ol_nat(finite_fold_nat_nat(F,Z_1),A) = hAPP_f22106695ol_nat(the_nat,hAPP_f800510211t_bool(hAPP_n1512601776t_bool(finite929467206at_nat(F),Z_1),A)) ).
fof(fact_565_fold__def,axiom,
! [F,Z_1,A] : hAPP_f1693662087iple_a(finite1205970312iple_a(F,Z_1),A) = hAPP_f1693662087iple_a(the_Ho830102290iple_a,hAPP_f20753329a_bool(hAPP_H1743777351a_bool(finite90892550iple_a(F),Z_1),A)) ).
fof(fact_566_fold__def,axiom,
! [F,Z_1,A] : hAPP_f1805168059t_bool(finite1827710202t_bool(F,Z_1),A) = hAPP_f631471077t_bool(the_fun_int_bool,hAPP_f284875647l_bool(hAPP_f1634429330l_bool(finite1008073724t_bool(F),Z_1),A)) ).
fof(fact_567_fold__def,axiom,
! [F,Z_1,A] : hAPP_f20753329a_bool(finite1215486064a_bool(F,Z_1),A) = hAPP_f258822235a_bool(the_fu1555441254a_bool,hAPP_f1767618879l_bool(hAPP_f1577095245l_bool(finite1152845682a_bool(F),Z_1),A)) ).
fof(fact_568_fold__def,axiom,
! [F,Z_1,A] : hAPP_f800510211t_bool(finite1578363458t_bool(F,Z_1),A) = hAPP_f582319405t_bool(the_fun_nat_bool,hAPP_f103356543l_bool(hAPP_f1266913334l_bool(finite758726980t_bool(F),Z_1),A)) ).
fof(fact_569_comp__fun__idem_Ofold__insert__idem2,axiom,
! [Z_1,X_1,A,F] :
( hBOOL(finite1860950092at_nat(F))
=> ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
=> hAPP_f22106695ol_nat(finite_fold_nat_nat(F,Z_1),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),A)) = hAPP_f22106695ol_nat(finite_fold_nat_nat(F,hAPP_nat_nat(hAPP_nat_fun_nat_nat(F,X_1),Z_1)),A) ) ) ).
fof(fact_570_comp__fun__idem_Ofold__insert__idem2,axiom,
! [Z_1,X_1,A,F] :
( hBOOL(finite1956789438t_bool(F))
=> ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
=> hAPP_f800510211t_bool(finite1578363458t_bool(F,Z_1),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),A)) = hAPP_f800510211t_bool(finite1578363458t_bool(F,hAPP_f800510211t_bool(hAPP_n1512601776t_bool(F,X_1),Z_1)),A) ) ) ).
fof(fact_571_comp__fun__idem_Ofold__insert__idem2,axiom,
! [Z_1,X_1,A,F] :
( hBOOL(finite648855052iple_a(F))
=> ( hBOOL(hAPP_f1695230391l_bool(finite1738664244iple_a,A))
=> hAPP_f1693662087iple_a(finite1205970312iple_a(F,Z_1),hAPP_f20753329a_bool(hAPP_H1743777351a_bool(insert956547291iple_a,X_1),A)) = hAPP_f1693662087iple_a(finite1205970312iple_a(F,hAPP_H905846293iple_a(hAPP_H2085992369iple_a(F,X_1),Z_1)),A) ) ) ).
fof(fact_572_comp__fun__idem_Ofold__insert__idem2,axiom,
! [Z_1,X_1,A,F] :
( hBOOL(finite1803123436a_bool(F))
=> ( hBOOL(hAPP_f1695230391l_bool(finite1738664244iple_a,A))
=> hAPP_f20753329a_bool(finite1215486064a_bool(F,Z_1),hAPP_f20753329a_bool(hAPP_H1743777351a_bool(insert956547291iple_a,X_1),A)) = hAPP_f20753329a_bool(finite1215486064a_bool(F,hAPP_f20753329a_bool(hAPP_H1743777351a_bool(F,X_1),Z_1)),A) ) ) ).
fof(fact_573_comp__fun__idem_Ofold__insert__idem2,axiom,
! [Z_1,X_1,A,F] :
( hBOOL(finite1704255308nt_int(F))
=> ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,A))
=> hAPP_f1594865479ol_int(finite_fold_int_int(F,Z_1),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),A)) = hAPP_f1594865479ol_int(finite_fold_int_int(F,hAPP_int_int(hAPP_int_fun_int_int(F,X_1),Z_1)),A) ) ) ).
fof(fact_574_comp__fun__idem_Ofold__insert__idem2,axiom,
! [Z_1,X_1,A,F] :
( hBOOL(finite58652534t_bool(F))
=> ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,A))
=> hAPP_f1805168059t_bool(finite1827710202t_bool(F,Z_1),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),A)) = hAPP_f1805168059t_bool(finite1827710202t_bool(F,hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(F,X_1),Z_1)),A) ) ) ).
fof(fact_575_comp__fun__idem_Ofold__insert__idem,axiom,
! [Z_1,X_1,A,F] :
( hBOOL(finite1860950092at_nat(F))
=> ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
=> hAPP_f22106695ol_nat(finite_fold_nat_nat(F,Z_1),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),A)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(F,X_1),hAPP_f22106695ol_nat(finite_fold_nat_nat(F,Z_1),A)) ) ) ).
fof(fact_576_comp__fun__idem_Ofold__insert__idem,axiom,
! [Z_1,X_1,A,F] :
( hBOOL(finite1956789438t_bool(F))
=> ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
=> hAPP_f800510211t_bool(finite1578363458t_bool(F,Z_1),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),A)) = hAPP_f800510211t_bool(hAPP_n1512601776t_bool(F,X_1),hAPP_f800510211t_bool(finite1578363458t_bool(F,Z_1),A)) ) ) ).
fof(fact_577_comp__fun__idem_Ofold__insert__idem,axiom,
! [Z_1,X_1,A,F] :
( hBOOL(finite648855052iple_a(F))
=> ( hBOOL(hAPP_f1695230391l_bool(finite1738664244iple_a,A))
=> hAPP_f1693662087iple_a(finite1205970312iple_a(F,Z_1),hAPP_f20753329a_bool(hAPP_H1743777351a_bool(insert956547291iple_a,X_1),A)) = hAPP_H905846293iple_a(hAPP_H2085992369iple_a(F,X_1),hAPP_f1693662087iple_a(finite1205970312iple_a(F,Z_1),A)) ) ) ).
fof(fact_578_comp__fun__idem_Ofold__insert__idem,axiom,
! [Z_1,X_1,A,F] :
( hBOOL(finite1803123436a_bool(F))
=> ( hBOOL(hAPP_f1695230391l_bool(finite1738664244iple_a,A))
=> hAPP_f20753329a_bool(finite1215486064a_bool(F,Z_1),hAPP_f20753329a_bool(hAPP_H1743777351a_bool(insert956547291iple_a,X_1),A)) = hAPP_f20753329a_bool(hAPP_H1743777351a_bool(F,X_1),hAPP_f20753329a_bool(finite1215486064a_bool(F,Z_1),A)) ) ) ).
fof(fact_579_comp__fun__idem_Ofold__insert__idem,axiom,
! [Z_1,X_1,A,F] :
( hBOOL(finite1704255308nt_int(F))
=> ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,A))
=> hAPP_f1594865479ol_int(finite_fold_int_int(F,Z_1),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),A)) = hAPP_int_int(hAPP_int_fun_int_int(F,X_1),hAPP_f1594865479ol_int(finite_fold_int_int(F,Z_1),A)) ) ) ).
fof(fact_580_comp__fun__idem_Ofold__insert__idem,axiom,
! [Z_1,X_1,A,F] :
( hBOOL(finite58652534t_bool(F))
=> ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,A))
=> hAPP_f1805168059t_bool(finite1827710202t_bool(F,Z_1),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),A)) = hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(F,X_1),hAPP_f1805168059t_bool(finite1827710202t_bool(F,Z_1),A)) ) ) ).
fof(fact_581_folding__one_Oeq__fold_H,axiom,
! [X_1,A,F,F_1] :
( hBOOL(finite988810631ne_nat(F,F_1))
=> ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
=> ( ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A))
=> hAPP_f22106695ol_nat(F_1,hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),A)) = hAPP_f22106695ol_nat(finite_fold_nat_nat(F,X_1),A) ) ) ) ).
fof(fact_582_folding__one_Oeq__fold_H,axiom,
! [X_1,A,F,F_1] :
( hBOOL(finite1753440478iple_a(F,F_1))
=> ( hBOOL(hAPP_f1695230391l_bool(finite1738664244iple_a,A))
=> ( ~ hBOOL(hAPP_f1695230391l_bool(hAPP_H622608077l_bool(member1797258804iple_a,X_1),A))
=> hAPP_f1693662087iple_a(F_1,hAPP_f20753329a_bool(hAPP_H1743777351a_bool(insert956547291iple_a,X_1),A)) = hAPP_f1693662087iple_a(finite1205970312iple_a(F,X_1),A) ) ) ) ).
fof(fact_583_folding__one_Oeq__fold_H,axiom,
! [X_1,A,F,F_1] :
( hBOOL(finite1626084323ne_int(F,F_1))
=> ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,A))
=> ( ~ hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),A))
=> hAPP_f1594865479ol_int(F_1,hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),A)) = hAPP_f1594865479ol_int(finite_fold_int_int(F,X_1),A) ) ) ) ).
fof(fact_584_folding__one__idem_Oeq__fold__idem_H,axiom,
! [A_13,A,F,F_1] :
( hBOOL(finite795500164em_nat(F,F_1))
=> ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
=> hAPP_f22106695ol_nat(F_1,hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_13),A)) = hAPP_f22106695ol_nat(finite_fold_nat_nat(F,A_13),A) ) ) ).
fof(fact_585_folding__one__idem_Oeq__fold__idem_H,axiom,
! [A_13,A,F,F_1] :
( hBOOL(finite1033474011iple_a(F,F_1))
=> ( hBOOL(hAPP_f1695230391l_bool(finite1738664244iple_a,A))
=> hAPP_f1693662087iple_a(F_1,hAPP_f20753329a_bool(hAPP_H1743777351a_bool(insert956547291iple_a,A_13),A)) = hAPP_f1693662087iple_a(finite1205970312iple_a(F,A_13),A) ) ) ).
fof(fact_586_folding__one__idem_Oeq__fold__idem_H,axiom,
! [A_13,A,F,F_1] :
( hBOOL(finite1432773856em_int(F,F_1))
=> ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,A))
=> hAPP_f1594865479ol_int(F_1,hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,A_13),A)) = hAPP_f1594865479ol_int(finite_fold_int_int(F,A_13),A) ) ) ).
fof(fact_587_fold1__eq__fold,axiom,
! [A_13,A] :
( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
=> ( ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_13),A))
=> hAPP_f22106695ol_nat(finite_fold1_nat(times_times_nat),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_13),A)) = hAPP_f22106695ol_nat(finite_fold_nat_nat(times_times_nat,A_13),A) ) ) ).
fof(fact_588_fold1__eq__fold,axiom,
! [A_13,A] :
( hBOOL(hAPP_f448129468l_bool(finite_finite_int,A))
=> ( ~ hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,A_13),A))
=> hAPP_f1594865479ol_int(finite_fold1_int(times_times_int),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,A_13),A)) = hAPP_f1594865479ol_int(finite_fold_int_int(times_times_int,A_13),A) ) ) ).
fof(fact_589_comp__fun__commute_Ofold__graph__insertE,axiom,
! [Z_1,X_1,A,V,F] :
( hBOOL(finite2130160977at_nat(F))
=> ( hBOOL(hAPP_nat_bool(hAPP_f800510211t_bool(hAPP_n1512601776t_bool(finite929467206at_nat(F),Z_1),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),A)),V))
=> ( ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A))
=> ~ ! [Y_21] :
( V = hAPP_nat_nat(hAPP_nat_fun_nat_nat(F,X_1),Y_21)
=> ~ hBOOL(hAPP_nat_bool(hAPP_f800510211t_bool(hAPP_n1512601776t_bool(finite929467206at_nat(F),Z_1),A),Y_21)) ) ) ) ) ).
fof(fact_590_comp__fun__commute_Ofold__graph__insertE,axiom,
! [Z_1,X_1,A,V,F] :
( hBOOL(finite1428139793iple_a(F))
=> ( hBOOL(hAPP_H1037229737a_bool(hAPP_f20753329a_bool(hAPP_H1743777351a_bool(finite90892550iple_a(F),Z_1),hAPP_f20753329a_bool(hAPP_H1743777351a_bool(insert956547291iple_a,X_1),A)),V))
=> ( ~ hBOOL(hAPP_f1695230391l_bool(hAPP_H622608077l_bool(member1797258804iple_a,X_1),A))
=> ~ ! [Y_21] :
( V = hAPP_H905846293iple_a(hAPP_H2085992369iple_a(F,X_1),Y_21)
=> ~ hBOOL(hAPP_H1037229737a_bool(hAPP_f20753329a_bool(hAPP_H1743777351a_bool(finite90892550iple_a(F),Z_1),A),Y_21)) ) ) ) ) ).
fof(fact_591_comp__fun__commute_Ofold__graph__insertE,axiom,
! [Z_1,X_1,A,V,F] :
( hBOOL(finite1973466193nt_int(F))
=> ( hBOOL(hAPP_int_bool(hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(finite772772422nt_int(F),Z_1),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),A)),V))
=> ( ~ hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),A))
=> ~ ! [Y_21] :
( V = hAPP_int_int(hAPP_int_fun_int_int(F,X_1),Y_21)
=> ~ hBOOL(hAPP_int_bool(hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(finite772772422nt_int(F),Z_1),A),Y_21)) ) ) ) ) ).
fof(fact_592_setsum__image__gen,axiom,
! [G,F,S_1] :
( hBOOL(hAPP_f1695230391l_bool(finite1738664244iple_a,S_1))
=> hAPP_f2097660464ol_nat(hAPP_f1371399797ol_nat(big_co1340561246_a_nat,G),S_1) = hAPP_f22106695ol_nat(hAPP_f782000547ol_nat(big_co387207925at_nat,hAPP_f966999552at_nat(hAPP_f724467227at_nat(cOMBB_161916371at_nat,hAPP_f1371399797ol_nat(big_co1340561246_a_nat,G)),hAPP_f2087582327a_bool(hAPP_f1396758363a_bool(cOMBB_391046962ol_nat,collec351493750iple_a),hAPP_f2087582327a_bool(hAPP_f1396758363a_bool(cOMBB_391046962ol_nat,hAPP_f607497069a_bool(cOMBS_154988028l_bool,hAPP_f146754017l_bool(hAPP_f239971723l_bool(cOMBB_1866391387iple_a,fconj),hAPP_f20753329a_bool(hAPP_f247708275a_bool(cOMBC_1519416966l_bool,member1797258804iple_a),S_1)))),hAPP_f228417655a_bool(cOMBC_364739t_bool,hAPP_f1872860171t_bool(hAPP_f1981448543t_bool(cOMBB_1391527205iple_a,fequal_nat),F)))))),image_48635758_a_nat(F,S_1)) ) ).
fof(fact_593_setsum__image__gen,axiom,
! [G,F,S_1] :
( hBOOL(hAPP_f448129468l_bool(finite_finite_int,S_1))
=> hAPP_f957591787ol_nat(hAPP_f1109019371ol_nat(big_co1740723097nt_nat,G),S_1) = hAPP_f22106695ol_nat(hAPP_f782000547ol_nat(big_co387207925at_nat,hAPP_f389300155at_nat(hAPP_f109563153at_nat(cOMBB_1381347800at_nat,hAPP_f1109019371ol_nat(big_co1740723097nt_nat,G)),hAPP_f2022049025t_bool(hAPP_f1986088027t_bool(cOMBB_2101530536ol_nat,collect_int),hAPP_f2022049025t_bool(hAPP_f1986088027t_bool(cOMBB_2101530536ol_nat,hAPP_f727283836t_bool(cOMBS_int_bool_bool,hAPP_f2144054103l_bool(hAPP_f1734373249l_bool(cOMBB_1652995168ol_int,fconj),hAPP_f1805168059t_bool(hAPP_f202917053t_bool(cOMBC_94739984l_bool,member_int),S_1)))),hAPP_f175561985t_bool(cOMBC_int_nat_bool,hAPP_f876579787t_bool(hAPP_f2026117279t_bool(cOMBB_1437810218ol_int,fequal_nat),F)))))),image_int_nat(F,S_1)) ) ).
fof(fact_594_setsum__image__gen,axiom,
! [G,F,S_1] :
( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,S_1))
=> hAPP_f22106695ol_nat(hAPP_f782000547ol_nat(big_co387207925at_nat,G),S_1) = hAPP_f2097660464ol_nat(hAPP_f1371399797ol_nat(big_co1340561246_a_nat,hAPP_f2085403241_a_nat(hAPP_f734830747_a_nat(cOMBB_1107515603iple_a,hAPP_f782000547ol_nat(big_co387207925at_nat,G)),hAPP_f27379319t_bool(hAPP_f196067081t_bool(cOMBB_437532599iple_a,collect_nat),hAPP_f27379319t_bool(hAPP_f196067081t_bool(cOMBB_437532599iple_a,hAPP_f561022312t_bool(cOMBS_nat_bool_bool,hAPP_f1146629647l_bool(hAPP_f1080886329l_bool(cOMBB_1015721476ol_nat,fconj),hAPP_f800510211t_bool(hAPP_f1722879237t_bool(cOMBC_226598744l_bool,member_nat),S_1)))),hAPP_f1886543991t_bool(cOMBC_533513393a_bool,hAPP_f260875897a_bool(hAPP_f1930382715a_bool(cOMBB_1063143712ol_nat,fequal963300192iple_a),F)))))),image_1865589340iple_a(F,S_1)) ) ).
fof(fact_595_setsum__image__gen,axiom,
! [G,F,S_1] :
( hBOOL(hAPP_f448129468l_bool(finite_finite_int,S_1))
=> hAPP_f957591787ol_nat(hAPP_f1109019371ol_nat(big_co1740723097nt_nat,G),S_1) = hAPP_f957591787ol_nat(hAPP_f1109019371ol_nat(big_co1740723097nt_nat,hAPP_f1134349059nt_nat(hAPP_f2100446809nt_nat(cOMBB_2018621492at_int,hAPP_f1109019371ol_nat(big_co1740723097nt_nat,G)),hAPP_f428220345t_bool(hAPP_f654702867t_bool(cOMBB_591320580ol_int,collect_int),hAPP_f428220345t_bool(hAPP_f654702867t_bool(cOMBB_591320580ol_int,hAPP_f727283836t_bool(cOMBS_int_bool_bool,hAPP_f2144054103l_bool(hAPP_f1734373249l_bool(cOMBB_1652995168ol_int,fconj),hAPP_f1805168059t_bool(hAPP_f202917053t_bool(cOMBC_94739984l_bool,member_int),S_1)))),hAPP_f428220345t_bool(cOMBC_int_int_bool,hAPP_f1791153283t_bool(hAPP_f1399575567t_bool(cOMBB_118231410ol_int,fequal_int),F)))))),image_int_int(F,S_1)) ) ).
fof(fact_596_setsum__image__gen,axiom,
! [G,F,S_1] :
( hBOOL(hAPP_f448129468l_bool(finite_finite_int,S_1))
=> hAPP_f957591787ol_nat(hAPP_f1109019371ol_nat(big_co1740723097nt_nat,G),S_1) = hAPP_f2097660464ol_nat(hAPP_f1371399797ol_nat(big_co1340561246_a_nat,hAPP_f342603021_a_nat(hAPP_f575660003_a_nat(cOMBB_1357028143iple_a,hAPP_f1109019371ol_nat(big_co1740723097nt_nat,G)),hAPP_f138254639t_bool(hAPP_f1998609161t_bool(cOMBB_1870764287iple_a,collect_int),hAPP_f138254639t_bool(hAPP_f1998609161t_bool(cOMBB_1870764287iple_a,hAPP_f727283836t_bool(cOMBS_int_bool_bool,hAPP_f2144054103l_bool(hAPP_f1734373249l_bool(cOMBB_1652995168ol_int,fconj),hAPP_f1805168059t_bool(hAPP_f202917053t_bool(cOMBC_94739984l_bool,member_int),S_1)))),hAPP_f936407855t_bool(cOMBC_1719122773a_bool,hAPP_f298605433a_bool(hAPP_f1421841531a_bool(cOMBB_1700417404ol_int,fequal963300192iple_a),F)))))),image_1844722432iple_a(F,S_1)) ) ).
fof(fact_597_setsum__image__gen,axiom,
! [G,F,S_1] :
( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,S_1))
=> hAPP_f22106695ol_nat(hAPP_f782000547ol_nat(big_co387207925at_nat,G),S_1) = hAPP_f957591787ol_nat(hAPP_f1109019371ol_nat(big_co1740723097nt_nat,hAPP_f1161717855nt_nat(hAPP_f1964560145nt_nat(cOMBB_1125113304at_int,hAPP_f782000547ol_nat(big_co387207925at_nat,G)),hAPP_f783004929t_bool(hAPP_f1410409747t_bool(cOMBB_1746576572ol_int,collect_nat),hAPP_f783004929t_bool(hAPP_f1410409747t_bool(cOMBB_1746576572ol_int,hAPP_f561022312t_bool(cOMBS_nat_bool_bool,hAPP_f1146629647l_bool(hAPP_f1080886329l_bool(cOMBB_1015721476ol_nat,fconj),hAPP_f800510211t_bool(hAPP_f1722879237t_bool(cOMBC_226598744l_bool,member_nat),S_1)))),hAPP_f482008321t_bool(cOMBC_nat_int_bool,hAPP_f1533130627t_bool(hAPP_f879109391t_bool(cOMBB_1628441366ol_nat,fequal_int),F)))))),image_nat_int(F,S_1)) ) ).
fof(fact_598_setsum_Ocong,axiom,
! [G,H,A] :
( hBOOL(hAPP_f1695230391l_bool(finite1738664244iple_a,A))
=> ( ! [X_2] :
( hBOOL(hAPP_f1695230391l_bool(hAPP_H622608077l_bool(member1797258804iple_a,X_2),A))
=> hAPP_H592031934_a_nat(G,X_2) = hAPP_H592031934_a_nat(H,X_2) )
=> hAPP_f2097660464ol_nat(hAPP_f1371399797ol_nat(big_co1340561246_a_nat,G),A) = hAPP_f2097660464ol_nat(hAPP_f1371399797ol_nat(big_co1340561246_a_nat,H),A) ) ) ).
fof(fact_599_setsum_Ocong,axiom,
! [G,H,A] :
( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
=> ( ! [X_2] :
( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_2),A))
=> hAPP_nat_nat(G,X_2) = hAPP_nat_nat(H,X_2) )
=> hAPP_f22106695ol_nat(hAPP_f782000547ol_nat(big_co387207925at_nat,G),A) = hAPP_f22106695ol_nat(hAPP_f782000547ol_nat(big_co387207925at_nat,H),A) ) ) ).
fof(fact_600_setsum_Ocong,axiom,
! [G,H,A] :
( hBOOL(hAPP_f448129468l_bool(finite_finite_int,A))
=> ( ! [X_2] :
( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_2),A))
=> hAPP_int_nat(G,X_2) = hAPP_int_nat(H,X_2) )
=> hAPP_f957591787ol_nat(hAPP_f1109019371ol_nat(big_co1740723097nt_nat,G),A) = hAPP_f957591787ol_nat(hAPP_f1109019371ol_nat(big_co1740723097nt_nat,H),A) ) ) ).
fof(fact_601_setsum__setsum__restrict,axiom,
! [F,R_1,T_1,S_1] :
( hBOOL(hAPP_f448129468l_bool(finite_finite_int,S_1))
=> ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,T_1))
=> hAPP_f957591787ol_nat(hAPP_f1109019371ol_nat(big_co1740723097nt_nat,hAPP_f1161717855nt_nat(hAPP_f369809936nt_nat(cOMBS_1063588099ol_nat,hAPP_f1501159417ol_nat(hAPP_f1029311995ol_nat(cOMBB_1562322300at_int,big_co387207925at_nat),F)),hAPP_f783004929t_bool(hAPP_f1410409747t_bool(cOMBB_1746576572ol_int,collect_nat),hAPP_f783004929t_bool(hAPP_f1410409747t_bool(cOMBB_1746576572ol_int,hAPP_f561022312t_bool(cOMBS_nat_bool_bool,hAPP_f1146629647l_bool(hAPP_f1080886329l_bool(cOMBB_1015721476ol_nat,fconj),hAPP_f800510211t_bool(hAPP_f1722879237t_bool(cOMBC_226598744l_bool,member_nat),T_1)))),R_1)))),S_1) = hAPP_f22106695ol_nat(hAPP_f782000547ol_nat(big_co387207925at_nat,hAPP_f389300155at_nat(hAPP_f1750007732at_nat(cOMBS_2106653443ol_nat,hAPP_f299305025ol_nat(hAPP_f1632710027ol_nat(cOMBB_1453893592at_nat,big_co1740723097nt_nat),hAPP_f879494613nt_nat(cOMBC_int_nat_nat,F))),hAPP_f2022049025t_bool(hAPP_f1986088027t_bool(cOMBB_2101530536ol_nat,collect_int),hAPP_f2022049025t_bool(hAPP_f1986088027t_bool(cOMBB_2101530536ol_nat,hAPP_f727283836t_bool(cOMBS_int_bool_bool,hAPP_f2144054103l_bool(hAPP_f1734373249l_bool(cOMBB_1652995168ol_int,fconj),hAPP_f1805168059t_bool(hAPP_f202917053t_bool(cOMBC_94739984l_bool,member_int),S_1)))),hAPP_f175561985t_bool(cOMBC_int_nat_bool,R_1))))),T_1) ) ) ).
fof(fact_602_setsum__setsum__restrict,axiom,
! [F,R_1,T_1,S_1] :
( hBOOL(hAPP_f1695230391l_bool(finite1738664244iple_a,S_1))
=> ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,T_1))
=> hAPP_f2097660464ol_nat(hAPP_f1371399797ol_nat(big_co1340561246_a_nat,hAPP_f2085403241_a_nat(hAPP_f1052116629_a_nat(cOMBS_397915080ol_nat,hAPP_f1669765561ol_nat(hAPP_f1073041467ol_nat(cOMBB_10565751iple_a,big_co387207925at_nat),F)),hAPP_f27379319t_bool(hAPP_f196067081t_bool(cOMBB_437532599iple_a,collect_nat),hAPP_f27379319t_bool(hAPP_f196067081t_bool(cOMBB_437532599iple_a,hAPP_f561022312t_bool(cOMBS_nat_bool_bool,hAPP_f1146629647l_bool(hAPP_f1080886329l_bool(cOMBB_1015721476ol_nat,fconj),hAPP_f800510211t_bool(hAPP_f1722879237t_bool(cOMBC_226598744l_bool,member_nat),T_1)))),R_1)))),S_1) = hAPP_f22106695ol_nat(hAPP_f782000547ol_nat(big_co387207925at_nat,hAPP_f966999552at_nat(hAPP_f264818750at_nat(cOMBS_1274158152ol_nat,hAPP_f883871819ol_nat(hAPP_f424754463ol_nat(cOMBB_1716606670at_nat,big_co1340561246_a_nat),hAPP_f1903462933_a_nat(cOMBC_667342884at_nat,F))),hAPP_f2087582327a_bool(hAPP_f1396758363a_bool(cOMBB_391046962ol_nat,collec351493750iple_a),hAPP_f2087582327a_bool(hAPP_f1396758363a_bool(cOMBB_391046962ol_nat,hAPP_f607497069a_bool(cOMBS_154988028l_bool,hAPP_f146754017l_bool(hAPP_f239971723l_bool(cOMBB_1866391387iple_a,fconj),hAPP_f20753329a_bool(hAPP_f247708275a_bool(cOMBC_1519416966l_bool,member1797258804iple_a),S_1)))),hAPP_f228417655a_bool(cOMBC_364739t_bool,R_1))))),T_1) ) ) ).
fof(fact_603_setsum__setsum__restrict,axiom,
! [F,R_1,T_1,S_1] :
( hBOOL(hAPP_f1695230391l_bool(finite1738664244iple_a,S_1))
=> ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,T_1))
=> hAPP_f2097660464ol_nat(hAPP_f1371399797ol_nat(big_co1340561246_a_nat,hAPP_f342603021_a_nat(hAPP_f270087133_a_nat(cOMBS_1333400172ol_nat,hAPP_f224922881ol_nat(hAPP_f543861963ol_nat(cOMBB_1075417391iple_a,big_co1740723097nt_nat),F)),hAPP_f138254639t_bool(hAPP_f1998609161t_bool(cOMBB_1870764287iple_a,collect_int),hAPP_f138254639t_bool(hAPP_f1998609161t_bool(cOMBB_1870764287iple_a,hAPP_f727283836t_bool(cOMBS_int_bool_bool,hAPP_f2144054103l_bool(hAPP_f1734373249l_bool(cOMBB_1652995168ol_int,fconj),hAPP_f1805168059t_bool(hAPP_f202917053t_bool(cOMBC_94739984l_bool,member_int),T_1)))),R_1)))),S_1) = hAPP_f957591787ol_nat(hAPP_f1109019371ol_nat(big_co1740723097nt_nat,hAPP_f2050777416nt_nat(hAPP_f1895921634nt_nat(cOMBS_202713324ol_nat,hAPP_f1442126667ol_nat(hAPP_f1072896543ol_nat(cOMBB_206396714at_int,big_co1340561246_a_nat),hAPP_f183615253_a_nat(cOMBC_2020858056nt_nat,F))),hAPP_f1718504751a_bool(hAPP_f481410067a_bool(cOMBB_1028320654ol_int,collec351493750iple_a),hAPP_f1718504751a_bool(hAPP_f481410067a_bool(cOMBB_1028320654ol_int,hAPP_f607497069a_bool(cOMBS_154988028l_bool,hAPP_f146754017l_bool(hAPP_f239971723l_bool(cOMBB_1866391387iple_a,fconj),hAPP_f20753329a_bool(hAPP_f247708275a_bool(cOMBC_1519416966l_bool,member1797258804iple_a),S_1)))),hAPP_f920351535a_bool(cOMBC_1530706207t_bool,R_1))))),T_1) ) ) ).
fof(fact_604_setsum__setsum__restrict,axiom,
! [F,R_1,T_1,S_1] :
( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,S_1))
=> ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,T_1))
=> hAPP_f22106695ol_nat(hAPP_f782000547ol_nat(big_co387207925at_nat,hAPP_f389300155at_nat(hAPP_f1750007732at_nat(cOMBS_2106653443ol_nat,hAPP_f299305025ol_nat(hAPP_f1632710027ol_nat(cOMBB_1453893592at_nat,big_co1740723097nt_nat),F)),hAPP_f2022049025t_bool(hAPP_f1986088027t_bool(cOMBB_2101530536ol_nat,collect_int),hAPP_f2022049025t_bool(hAPP_f1986088027t_bool(cOMBB_2101530536ol_nat,hAPP_f727283836t_bool(cOMBS_int_bool_bool,hAPP_f2144054103l_bool(hAPP_f1734373249l_bool(cOMBB_1652995168ol_int,fconj),hAPP_f1805168059t_bool(hAPP_f202917053t_bool(cOMBC_94739984l_bool,member_int),T_1)))),R_1)))),S_1) = hAPP_f957591787ol_nat(hAPP_f1109019371ol_nat(big_co1740723097nt_nat,hAPP_f1161717855nt_nat(hAPP_f369809936nt_nat(cOMBS_1063588099ol_nat,hAPP_f1501159417ol_nat(hAPP_f1029311995ol_nat(cOMBB_1562322300at_int,big_co387207925at_nat),hAPP_f446977493at_nat(cOMBC_nat_int_nat,F))),hAPP_f783004929t_bool(hAPP_f1410409747t_bool(cOMBB_1746576572ol_int,collect_nat),hAPP_f783004929t_bool(hAPP_f1410409747t_bool(cOMBB_1746576572ol_int,hAPP_f561022312t_bool(cOMBS_nat_bool_bool,hAPP_f1146629647l_bool(hAPP_f1080886329l_bool(cOMBB_1015721476ol_nat,fconj),hAPP_f800510211t_bool(hAPP_f1722879237t_bool(cOMBC_226598744l_bool,member_nat),S_1)))),hAPP_f482008321t_bool(cOMBC_nat_int_bool,R_1))))),T_1) ) ) ).
fof(fact_605_setsum__setsum__restrict,axiom,
! [F,R_1,T_1,S_1] :
( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,S_1))
=> ( hBOOL(hAPP_f1695230391l_bool(finite1738664244iple_a,T_1))
=> hAPP_f22106695ol_nat(hAPP_f782000547ol_nat(big_co387207925at_nat,hAPP_f966999552at_nat(hAPP_f264818750at_nat(cOMBS_1274158152ol_nat,hAPP_f883871819ol_nat(hAPP_f424754463ol_nat(cOMBB_1716606670at_nat,big_co1340561246_a_nat),F)),hAPP_f2087582327a_bool(hAPP_f1396758363a_bool(cOMBB_391046962ol_nat,collec351493750iple_a),hAPP_f2087582327a_bool(hAPP_f1396758363a_bool(cOMBB_391046962ol_nat,hAPP_f607497069a_bool(cOMBS_154988028l_bool,hAPP_f146754017l_bool(hAPP_f239971723l_bool(cOMBB_1866391387iple_a,fconj),hAPP_f20753329a_bool(hAPP_f247708275a_bool(cOMBC_1519416966l_bool,member1797258804iple_a),T_1)))),R_1)))),S_1) = hAPP_f2097660464ol_nat(hAPP_f1371399797ol_nat(big_co1340561246_a_nat,hAPP_f2085403241_a_nat(hAPP_f1052116629_a_nat(cOMBS_397915080ol_nat,hAPP_f1669765561ol_nat(hAPP_f1073041467ol_nat(cOMBB_10565751iple_a,big_co387207925at_nat),hAPP_f2146713109at_nat(cOMBC_441305270_a_nat,F))),hAPP_f27379319t_bool(hAPP_f196067081t_bool(cOMBB_437532599iple_a,collect_nat),hAPP_f27379319t_bool(hAPP_f196067081t_bool(cOMBB_437532599iple_a,hAPP_f561022312t_bool(cOMBS_nat_bool_bool,hAPP_f1146629647l_bool(hAPP_f1080886329l_bool(cOMBB_1015721476ol_nat,fconj),hAPP_f800510211t_bool(hAPP_f1722879237t_bool(cOMBC_226598744l_bool,member_nat),S_1)))),hAPP_f1886543991t_bool(cOMBC_533513393a_bool,R_1))))),T_1) ) ) ).
fof(fact_606_setsum__setsum__restrict,axiom,
! [F,R_1,T_1,S_1] :
( hBOOL(hAPP_f448129468l_bool(finite_finite_int,S_1))
=> ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,T_1))
=> hAPP_f957591787ol_nat(hAPP_f1109019371ol_nat(big_co1740723097nt_nat,hAPP_f1134349059nt_nat(hAPP_f446447448nt_nat(cOMBS_1999073191ol_nat,hAPP_f1548925761ol_nat(hAPP_f1331458699ol_nat(cOMBB_2091167284at_int,big_co1740723097nt_nat),F)),hAPP_f428220345t_bool(hAPP_f654702867t_bool(cOMBB_591320580ol_int,collect_int),hAPP_f428220345t_bool(hAPP_f654702867t_bool(cOMBB_591320580ol_int,hAPP_f727283836t_bool(cOMBS_int_bool_bool,hAPP_f2144054103l_bool(hAPP_f1734373249l_bool(cOMBB_1652995168ol_int,fconj),hAPP_f1805168059t_bool(hAPP_f202917053t_bool(cOMBC_94739984l_bool,member_int),T_1)))),R_1)))),S_1) = hAPP_f957591787ol_nat(hAPP_f1109019371ol_nat(big_co1740723097nt_nat,hAPP_f1134349059nt_nat(hAPP_f446447448nt_nat(cOMBS_1999073191ol_nat,hAPP_f1548925761ol_nat(hAPP_f1331458699ol_nat(cOMBB_2091167284at_int,big_co1740723097nt_nat),hAPP_f909851349nt_nat(cOMBC_int_int_nat,F))),hAPP_f428220345t_bool(hAPP_f654702867t_bool(cOMBB_591320580ol_int,collect_int),hAPP_f428220345t_bool(hAPP_f654702867t_bool(cOMBB_591320580ol_int,hAPP_f727283836t_bool(cOMBS_int_bool_bool,hAPP_f2144054103l_bool(hAPP_f1734373249l_bool(cOMBB_1652995168ol_int,fconj),hAPP_f1805168059t_bool(hAPP_f202917053t_bool(cOMBC_94739984l_bool,member_int),S_1)))),hAPP_f428220345t_bool(cOMBC_int_int_bool,R_1))))),T_1) ) ) ).
fof(fact_607_setsum__setsum__restrict,axiom,
! [F,R_1,T_1,S_1] :
( hBOOL(hAPP_f448129468l_bool(finite_finite_int,S_1))
=> ( hBOOL(hAPP_f1695230391l_bool(finite1738664244iple_a,T_1))
=> hAPP_f957591787ol_nat(hAPP_f1109019371ol_nat(big_co1740723097nt_nat,hAPP_f2050777416nt_nat(hAPP_f1895921634nt_nat(cOMBS_202713324ol_nat,hAPP_f1442126667ol_nat(hAPP_f1072896543ol_nat(cOMBB_206396714at_int,big_co1340561246_a_nat),F)),hAPP_f1718504751a_bool(hAPP_f481410067a_bool(cOMBB_1028320654ol_int,collec351493750iple_a),hAPP_f1718504751a_bool(hAPP_f481410067a_bool(cOMBB_1028320654ol_int,hAPP_f607497069a_bool(cOMBS_154988028l_bool,hAPP_f146754017l_bool(hAPP_f239971723l_bool(cOMBB_1866391387iple_a,fconj),hAPP_f20753329a_bool(hAPP_f247708275a_bool(cOMBC_1519416966l_bool,member1797258804iple_a),T_1)))),R_1)))),S_1) = hAPP_f2097660464ol_nat(hAPP_f1371399797ol_nat(big_co1340561246_a_nat,hAPP_f342603021_a_nat(hAPP_f270087133_a_nat(cOMBS_1333400172ol_nat,hAPP_f224922881ol_nat(hAPP_f543861963ol_nat(cOMBB_1075417391iple_a,big_co1740723097nt_nat),hAPP_f1451619093nt_nat(cOMBC_1917249810_a_nat,F))),hAPP_f138254639t_bool(hAPP_f1998609161t_bool(cOMBB_1870764287iple_a,collect_int),hAPP_f138254639t_bool(hAPP_f1998609161t_bool(cOMBB_1870764287iple_a,hAPP_f727283836t_bool(cOMBS_int_bool_bool,hAPP_f2144054103l_bool(hAPP_f1734373249l_bool(cOMBB_1652995168ol_int,fconj),hAPP_f1805168059t_bool(hAPP_f202917053t_bool(cOMBC_94739984l_bool,member_int),S_1)))),hAPP_f936407855t_bool(cOMBC_1719122773a_bool,R_1))))),T_1) ) ) ).
fof(fact_608_setsum_Oremove,axiom,
! [G,X_1,A] :
( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
=> ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A))
=> hAPP_f22106695ol_nat(hAPP_f782000547ol_nat(big_co387207925at_nat,G),A) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(G,X_1)),hAPP_f22106695ol_nat(hAPP_f782000547ol_nat(big_co387207925at_nat,G),hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),bot_bot_fun_nat_bool)))) ) ) ).
fof(fact_609_setsum_Oremove,axiom,
! [G,X_1,A] :
( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
=> ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A))
=> hAPP_f659380387ol_int(hAPP_f1599440987ol_int(big_co1024481617at_int,G),A) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_nat_int(G,X_1)),hAPP_f659380387ol_int(hAPP_f1599440987ol_int(big_co1024481617at_int,G),hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),bot_bot_fun_nat_bool)))) ) ) ).
fof(fact_610_setsum_Oremove,axiom,
! [G,X_1,A] :
( hBOOL(hAPP_f1695230391l_bool(finite1738664244iple_a,A))
=> ( hBOOL(hAPP_f1695230391l_bool(hAPP_H622608077l_bool(member1797258804iple_a,X_1),A))
=> hAPP_f587450508ol_int(hAPP_f1544320301ol_int(big_co1977834938_a_int,G),A) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_H1229305626_a_int(G,X_1)),hAPP_f587450508ol_int(hAPP_f1544320301ol_int(big_co1977834938_a_int,G),hAPP_f20753329a_bool(hAPP_f1706700729a_bool(minus_85316870a_bool,A),hAPP_f20753329a_bool(hAPP_H1743777351a_bool(insert956547291iple_a,X_1),bot_bo797238721a_bool)))) ) ) ).
fof(fact_611_setsum_Oremove,axiom,
! [G,X_1,A] :
( hBOOL(hAPP_f448129468l_bool(finite_finite_int,A))
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),A))
=> hAPP_f1594865479ol_int(hAPP_f1926459811ol_int(big_co230513141nt_int,G),A) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(G,X_1)),hAPP_f1594865479ol_int(hAPP_f1926459811ol_int(big_co230513141nt_int,G),hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),bot_bot_fun_int_bool)))) ) ) ).
fof(fact_612_setsum_Oremove,axiom,
! [G,X_1,A] :
( hBOOL(hAPP_f1695230391l_bool(finite1738664244iple_a,A))
=> ( hBOOL(hAPP_f1695230391l_bool(hAPP_H622608077l_bool(member1797258804iple_a,X_1),A))
=> hAPP_f2097660464ol_nat(hAPP_f1371399797ol_nat(big_co1340561246_a_nat,G),A) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_H592031934_a_nat(G,X_1)),hAPP_f2097660464ol_nat(hAPP_f1371399797ol_nat(big_co1340561246_a_nat,G),hAPP_f20753329a_bool(hAPP_f1706700729a_bool(minus_85316870a_bool,A),hAPP_f20753329a_bool(hAPP_H1743777351a_bool(insert956547291iple_a,X_1),bot_bo797238721a_bool)))) ) ) ).
fof(fact_613_setsum_Oremove,axiom,
! [G,X_1,A] :
( hBOOL(hAPP_f448129468l_bool(finite_finite_int,A))
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),A))
=> hAPP_f957591787ol_nat(hAPP_f1109019371ol_nat(big_co1740723097nt_nat,G),A) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_int_nat(G,X_1)),hAPP_f957591787ol_nat(hAPP_f1109019371ol_nat(big_co1740723097nt_nat,G),hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),bot_bot_fun_int_bool)))) ) ) ).
fof(fact_614_setsum__diff1_H,axiom,
! [F,A_13,A] :
( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
=> ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_13),A))
=> hAPP_f22106695ol_nat(hAPP_f782000547ol_nat(big_co387207925at_nat,F),A) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(F,A_13)),hAPP_f22106695ol_nat(hAPP_f782000547ol_nat(big_co387207925at_nat,F),hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_13),bot_bot_fun_nat_bool)))) ) ) ).
fof(fact_615_setsum__diff1_H,axiom,
! [F,A_13,A] :
( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
=> ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_13),A))
=> hAPP_f659380387ol_int(hAPP_f1599440987ol_int(big_co1024481617at_int,F),A) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_nat_int(F,A_13)),hAPP_f659380387ol_int(hAPP_f1599440987ol_int(big_co1024481617at_int,F),hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_13),bot_bot_fun_nat_bool)))) ) ) ).
fof(fact_616_setsum__diff1_H,axiom,
! [F,A_13,A] :
( hBOOL(hAPP_f1695230391l_bool(finite1738664244iple_a,A))
=> ( hBOOL(hAPP_f1695230391l_bool(hAPP_H622608077l_bool(member1797258804iple_a,A_13),A))
=> hAPP_f587450508ol_int(hAPP_f1544320301ol_int(big_co1977834938_a_int,F),A) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_H1229305626_a_int(F,A_13)),hAPP_f587450508ol_int(hAPP_f1544320301ol_int(big_co1977834938_a_int,F),hAPP_f20753329a_bool(hAPP_f1706700729a_bool(minus_85316870a_bool,A),hAPP_f20753329a_bool(hAPP_H1743777351a_bool(insert956547291iple_a,A_13),bot_bo797238721a_bool)))) ) ) ).
fof(fact_617_setsum__diff1_H,axiom,
! [F,A_13,A] :
( hBOOL(hAPP_f448129468l_bool(finite_finite_int,A))
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,A_13),A))
=> hAPP_f1594865479ol_int(hAPP_f1926459811ol_int(big_co230513141nt_int,F),A) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(F,A_13)),hAPP_f1594865479ol_int(hAPP_f1926459811ol_int(big_co230513141nt_int,F),hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,A_13),bot_bot_fun_int_bool)))) ) ) ).
fof(fact_618_setsum__diff1_H,axiom,
! [F,A_13,A] :
( hBOOL(hAPP_f1695230391l_bool(finite1738664244iple_a,A))
=> ( hBOOL(hAPP_f1695230391l_bool(hAPP_H622608077l_bool(member1797258804iple_a,A_13),A))
=> hAPP_f2097660464ol_nat(hAPP_f1371399797ol_nat(big_co1340561246_a_nat,F),A) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_H592031934_a_nat(F,A_13)),hAPP_f2097660464ol_nat(hAPP_f1371399797ol_nat(big_co1340561246_a_nat,F),hAPP_f20753329a_bool(hAPP_f1706700729a_bool(minus_85316870a_bool,A),hAPP_f20753329a_bool(hAPP_H1743777351a_bool(insert956547291iple_a,A_13),bot_bo797238721a_bool)))) ) ) ).
fof(fact_619_setsum__diff1_H,axiom,
! [F,A_13,A] :
( hBOOL(hAPP_f448129468l_bool(finite_finite_int,A))
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,A_13),A))
=> hAPP_f957591787ol_nat(hAPP_f1109019371ol_nat(big_co1740723097nt_nat,F),A) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_int_nat(F,A_13)),hAPP_f957591787ol_nat(hAPP_f1109019371ol_nat(big_co1740723097nt_nat,F),hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,A_13),bot_bot_fun_int_bool)))) ) ) ).
fof(fact_620_setsum_Oinsert__remove,axiom,
! [G,X_1,A] :
( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
=> hAPP_f22106695ol_nat(hAPP_f782000547ol_nat(big_co387207925at_nat,G),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),A)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(G,X_1)),hAPP_f22106695ol_nat(hAPP_f782000547ol_nat(big_co387207925at_nat,G),hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),bot_bot_fun_nat_bool)))) ) ).
fof(fact_621_setsum_Oinsert__remove,axiom,
! [G,X_1,A] :
( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
=> hAPP_f659380387ol_int(hAPP_f1599440987ol_int(big_co1024481617at_int,G),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),A)) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_nat_int(G,X_1)),hAPP_f659380387ol_int(hAPP_f1599440987ol_int(big_co1024481617at_int,G),hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),bot_bot_fun_nat_bool)))) ) ).
fof(fact_622_setsum_Oinsert__remove,axiom,
! [G,X_1,A] :
( hBOOL(hAPP_f1695230391l_bool(finite1738664244iple_a,A))
=> hAPP_f587450508ol_int(hAPP_f1544320301ol_int(big_co1977834938_a_int,G),hAPP_f20753329a_bool(hAPP_H1743777351a_bool(insert956547291iple_a,X_1),A)) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_H1229305626_a_int(G,X_1)),hAPP_f587450508ol_int(hAPP_f1544320301ol_int(big_co1977834938_a_int,G),hAPP_f20753329a_bool(hAPP_f1706700729a_bool(minus_85316870a_bool,A),hAPP_f20753329a_bool(hAPP_H1743777351a_bool(insert956547291iple_a,X_1),bot_bo797238721a_bool)))) ) ).
fof(fact_623_setsum_Oinsert__remove,axiom,
! [G,X_1,A] :
( hBOOL(hAPP_f448129468l_bool(finite_finite_int,A))
=> hAPP_f1594865479ol_int(hAPP_f1926459811ol_int(big_co230513141nt_int,G),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),A)) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(G,X_1)),hAPP_f1594865479ol_int(hAPP_f1926459811ol_int(big_co230513141nt_int,G),hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),bot_bot_fun_int_bool)))) ) ).
fof(fact_624_setsum_Oinsert__remove,axiom,
! [G,X_1,A] :
( hBOOL(hAPP_f1695230391l_bool(finite1738664244iple_a,A))
=> hAPP_f2097660464ol_nat(hAPP_f1371399797ol_nat(big_co1340561246_a_nat,G),hAPP_f20753329a_bool(hAPP_H1743777351a_bool(insert956547291iple_a,X_1),A)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_H592031934_a_nat(G,X_1)),hAPP_f2097660464ol_nat(hAPP_f1371399797ol_nat(big_co1340561246_a_nat,G),hAPP_f20753329a_bool(hAPP_f1706700729a_bool(minus_85316870a_bool,A),hAPP_f20753329a_bool(hAPP_H1743777351a_bool(insert956547291iple_a,X_1),bot_bo797238721a_bool)))) ) ).
fof(fact_625_setsum_Oinsert__remove,axiom,
! [G,X_1,A] :
( hBOOL(hAPP_f448129468l_bool(finite_finite_int,A))
=> hAPP_f957591787ol_nat(hAPP_f1109019371ol_nat(big_co1740723097nt_nat,G),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),A)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_int_nat(G,X_1)),hAPP_f957591787ol_nat(hAPP_f1109019371ol_nat(big_co1740723097nt_nat,G),hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),bot_bot_fun_int_bool)))) ) ).
fof(fact_626_setsum__addf,axiom,
! [F,G,A] : hAPP_f22106695ol_nat(hAPP_f782000547ol_nat(big_co387207925at_nat,hAPP_f1914919701at_nat(hAPP_f1408247010at_nat(cOMBS_nat_nat_nat,hAPP_f1639111240at_nat(hAPP_f1731313045at_nat(cOMBB_963856155at_nat,plus_plus_nat),F)),G)),A) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_f22106695ol_nat(hAPP_f782000547ol_nat(big_co387207925at_nat,F),A)),hAPP_f22106695ol_nat(hAPP_f782000547ol_nat(big_co387207925at_nat,G),A)) ).
fof(fact_627_setsum__addf,axiom,
! [F,G,A] : hAPP_f957591787ol_nat(hAPP_f1109019371ol_nat(big_co1740723097nt_nat,hAPP_f2132704789nt_nat(hAPP_f1399363134nt_nat(cOMBS_int_nat_nat,hAPP_f1463450952at_nat(hAPP_f1087393429at_nat(cOMBB_1601129847at_int,plus_plus_nat),F)),G)),A) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_f957591787ol_nat(hAPP_f1109019371ol_nat(big_co1740723097nt_nat,F),A)),hAPP_f957591787ol_nat(hAPP_f1109019371ol_nat(big_co1740723097nt_nat,G),A)) ).
fof(fact_628_setsum__addf,axiom,
! [F,G,A] : hAPP_f2097660464ol_nat(hAPP_f1371399797ol_nat(big_co1340561246_a_nat,hAPP_f106743253_a_nat(hAPP_f1418102009_a_nat(cOMBS_1756972052at_nat,hAPP_f429024520at_nat(hAPP_f703420885at_nat(cOMBB_789959282iple_a,plus_plus_nat),F)),G)),A) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_f2097660464ol_nat(hAPP_f1371399797ol_nat(big_co1340561246_a_nat,F),A)),hAPP_f2097660464ol_nat(hAPP_f1371399797ol_nat(big_co1340561246_a_nat,G),A)) ).
fof(fact_629_setsum_Odistrib,axiom,
! [G,H,A] :
( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
=> hAPP_f22106695ol_nat(hAPP_f782000547ol_nat(big_co387207925at_nat,hAPP_f1914919701at_nat(hAPP_f1408247010at_nat(cOMBS_nat_nat_nat,hAPP_f1639111240at_nat(hAPP_f1731313045at_nat(cOMBB_963856155at_nat,plus_plus_nat),G)),H)),A) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_f22106695ol_nat(hAPP_f782000547ol_nat(big_co387207925at_nat,G),A)),hAPP_f22106695ol_nat(hAPP_f782000547ol_nat(big_co387207925at_nat,H),A)) ) ).
fof(fact_630_setsum_Odistrib,axiom,
! [G,H,A] :
( hBOOL(hAPP_f1695230391l_bool(finite1738664244iple_a,A))
=> hAPP_f2097660464ol_nat(hAPP_f1371399797ol_nat(big_co1340561246_a_nat,hAPP_f106743253_a_nat(hAPP_f1418102009_a_nat(cOMBS_1756972052at_nat,hAPP_f429024520at_nat(hAPP_f703420885at_nat(cOMBB_789959282iple_a,plus_plus_nat),G)),H)),A) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_f2097660464ol_nat(hAPP_f1371399797ol_nat(big_co1340561246_a_nat,G),A)),hAPP_f2097660464ol_nat(hAPP_f1371399797ol_nat(big_co1340561246_a_nat,H),A)) ) ).
fof(fact_631_setsum_Odistrib,axiom,
! [G,H,A] :
( hBOOL(hAPP_f1695230391l_bool(finite1738664244iple_a,A))
=> hAPP_f587450508ol_int(hAPP_f1544320301ol_int(big_co1977834938_a_int,hAPP_f1561000661_a_int(hAPP_f789902073_a_int(cOMBS_1600277268nt_int,hAPP_f231878828nt_int(hAPP_f1621780181nt_int(cOMBB_1885471694iple_a,plus_plus_int),G)),H)),A) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_f587450508ol_int(hAPP_f1544320301ol_int(big_co1977834938_a_int,G),A)),hAPP_f587450508ol_int(hAPP_f1544320301ol_int(big_co1977834938_a_int,H),A)) ) ).
fof(fact_632_setsum_Odistrib,axiom,
! [G,H,A] :
( hBOOL(hAPP_f448129468l_bool(finite_finite_int,A))
=> hAPP_f957591787ol_nat(hAPP_f1109019371ol_nat(big_co1740723097nt_nat,hAPP_f2132704789nt_nat(hAPP_f1399363134nt_nat(cOMBS_int_nat_nat,hAPP_f1463450952at_nat(hAPP_f1087393429at_nat(cOMBB_1601129847at_int,plus_plus_nat),G)),H)),A) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_f957591787ol_nat(hAPP_f1109019371ol_nat(big_co1740723097nt_nat,G),A)),hAPP_f957591787ol_nat(hAPP_f1109019371ol_nat(big_co1740723097nt_nat,H),A)) ) ).
fof(fact_633_setsum__insert,axiom,
! [F,A_13,F_1] :
( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,F_1))
=> ( ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_13),F_1))
=> hAPP_f22106695ol_nat(hAPP_f782000547ol_nat(big_co387207925at_nat,F),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_13),F_1)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(F,A_13)),hAPP_f22106695ol_nat(hAPP_f782000547ol_nat(big_co387207925at_nat,F),F_1)) ) ) ).
fof(fact_634_setsum__insert,axiom,
! [F,A_13,F_1] :
( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,F_1))
=> ( ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_13),F_1))
=> hAPP_f659380387ol_int(hAPP_f1599440987ol_int(big_co1024481617at_int,F),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_13),F_1)) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_nat_int(F,A_13)),hAPP_f659380387ol_int(hAPP_f1599440987ol_int(big_co1024481617at_int,F),F_1)) ) ) ).
fof(fact_635_setsum__insert,axiom,
! [F,A_13,F_1] :
( hBOOL(hAPP_f1695230391l_bool(finite1738664244iple_a,F_1))
=> ( ~ hBOOL(hAPP_f1695230391l_bool(hAPP_H622608077l_bool(member1797258804iple_a,A_13),F_1))
=> hAPP_f587450508ol_int(hAPP_f1544320301ol_int(big_co1977834938_a_int,F),hAPP_f20753329a_bool(hAPP_H1743777351a_bool(insert956547291iple_a,A_13),F_1)) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_H1229305626_a_int(F,A_13)),hAPP_f587450508ol_int(hAPP_f1544320301ol_int(big_co1977834938_a_int,F),F_1)) ) ) ).
fof(fact_636_setsum__insert,axiom,
! [F,A_13,F_1] :
( hBOOL(hAPP_f1695230391l_bool(finite1738664244iple_a,F_1))
=> ( ~ hBOOL(hAPP_f1695230391l_bool(hAPP_H622608077l_bool(member1797258804iple_a,A_13),F_1))
=> hAPP_f2097660464ol_nat(hAPP_f1371399797ol_nat(big_co1340561246_a_nat,F),hAPP_f20753329a_bool(hAPP_H1743777351a_bool(insert956547291iple_a,A_13),F_1)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_H592031934_a_nat(F,A_13)),hAPP_f2097660464ol_nat(hAPP_f1371399797ol_nat(big_co1340561246_a_nat,F),F_1)) ) ) ).
fof(fact_637_setsum__insert,axiom,
! [F,A_13,F_1] :
( hBOOL(hAPP_f448129468l_bool(finite_finite_int,F_1))
=> ( ~ hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,A_13),F_1))
=> hAPP_f957591787ol_nat(hAPP_f1109019371ol_nat(big_co1740723097nt_nat,F),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,A_13),F_1)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_int_nat(F,A_13)),hAPP_f957591787ol_nat(hAPP_f1109019371ol_nat(big_co1740723097nt_nat,F),F_1)) ) ) ).
fof(fact_638_setsum_Oinsert,axiom,
! [G,X_1,A] :
( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
=> ( ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A))
=> hAPP_f22106695ol_nat(hAPP_f782000547ol_nat(big_co387207925at_nat,G),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),A)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(G,X_1)),hAPP_f22106695ol_nat(hAPP_f782000547ol_nat(big_co387207925at_nat,G),A)) ) ) ).
fof(fact_639_setsum_Oinsert,axiom,
! [G,X_1,A] :
( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
=> ( ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A))
=> hAPP_f659380387ol_int(hAPP_f1599440987ol_int(big_co1024481617at_int,G),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),A)) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_nat_int(G,X_1)),hAPP_f659380387ol_int(hAPP_f1599440987ol_int(big_co1024481617at_int,G),A)) ) ) ).
fof(fact_640_setsum_Oinsert,axiom,
! [G,X_1,A] :
( hBOOL(hAPP_f1695230391l_bool(finite1738664244iple_a,A))
=> ( ~ hBOOL(hAPP_f1695230391l_bool(hAPP_H622608077l_bool(member1797258804iple_a,X_1),A))
=> hAPP_f587450508ol_int(hAPP_f1544320301ol_int(big_co1977834938_a_int,G),hAPP_f20753329a_bool(hAPP_H1743777351a_bool(insert956547291iple_a,X_1),A)) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_H1229305626_a_int(G,X_1)),hAPP_f587450508ol_int(hAPP_f1544320301ol_int(big_co1977834938_a_int,G),A)) ) ) ).
fof(fact_641_setsum_Oinsert,axiom,
! [G,X_1,A] :
( hBOOL(hAPP_f1695230391l_bool(finite1738664244iple_a,A))
=> ( ~ hBOOL(hAPP_f1695230391l_bool(hAPP_H622608077l_bool(member1797258804iple_a,X_1),A))
=> hAPP_f2097660464ol_nat(hAPP_f1371399797ol_nat(big_co1340561246_a_nat,G),hAPP_f20753329a_bool(hAPP_H1743777351a_bool(insert956547291iple_a,X_1),A)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_H592031934_a_nat(G,X_1)),hAPP_f2097660464ol_nat(hAPP_f1371399797ol_nat(big_co1340561246_a_nat,G),A)) ) ) ).
fof(fact_642_setsum_Oinsert,axiom,
! [G,X_1,A] :
( hBOOL(hAPP_f448129468l_bool(finite_finite_int,A))
=> ( ~ hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),A))
=> hAPP_f957591787ol_nat(hAPP_f1109019371ol_nat(big_co1740723097nt_nat,G),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),A)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_int_nat(G,X_1)),hAPP_f957591787ol_nat(hAPP_f1109019371ol_nat(big_co1740723097nt_nat,G),A)) ) ) ).
fof(fact_643_eq__add__iff2,axiom,
! [A_13,E,Ca,Ba,D_1] :
( hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_13),E)),Ca) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Ba),E)),D_1)
<=> Ca = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Ba),A_13)),E)),D_1) ) ).
fof(fact_644_eq__add__iff1,axiom,
! [A_13,E,Ca,Ba,D_1] :
( hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_13),E)),Ca) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Ba),E)),D_1)
<=> hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,A_13),Ba)),E)),Ca) = D_1 ) ).
fof(fact_645_diff__add__cancel,axiom,
! [A_42,B_34] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,A_42),B_34)),B_34) = A_42 ).
fof(fact_646_add__diff__cancel,axiom,
! [A_41,B_33] : hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_41),B_33)),B_33) = A_41 ).
fof(fact_647_crossproduct__eq,axiom,
! [W,Y_1,X_1,Z_1] :
( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,W),Y_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,X_1),Z_1)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,W),Z_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,X_1),Y_1))
<=> ( W = X_1
| Y_1 = Z_1 ) ) ).
fof(fact_648_crossproduct__eq,axiom,
! [W,Y_1,X_1,Z_1] :
( hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,W),Y_1)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X_1),Z_1)) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,W),Z_1)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X_1),Y_1))
<=> ( W = X_1
| Y_1 = Z_1 ) ) ).
fof(fact_649_combine__common__factor,axiom,
! [A_40,E_1,B_32,C_28] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_40),E_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_32),E_1)),C_28)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_40),B_32)),E_1)),C_28) ).
fof(fact_650_combine__common__factor,axiom,
! [A_40,E_1,B_32,C_28] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_40),E_1)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_32),E_1)),C_28)) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_40),B_32)),E_1)),C_28) ).
fof(fact_651_comm__semiring__1__class_Onormalizing__semiring__rules_I1_J,axiom,
! [A_39,M_4,B_31] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_39),M_4)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_31),M_4)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_39),B_31)),M_4) ).
fof(fact_652_comm__semiring__1__class_Onormalizing__semiring__rules_I1_J,axiom,
! [A_39,M_4,B_31] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_39),M_4)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_31),M_4)) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_39),B_31)),M_4) ).
fof(fact_653_add__right__imp__eq,axiom,
! [B_30,A_38,C_27] :
( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,B_30),A_38) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,C_27),A_38)
=> B_30 = C_27 ) ).
fof(fact_654_add__right__imp__eq,axiom,
! [B_30,A_38,C_27] :
( hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,B_30),A_38) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,C_27),A_38)
=> B_30 = C_27 ) ).
fof(fact_655_add__imp__eq,axiom,
! [A_37,B_29,C_26] :
( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_37),B_29) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_37),C_26)
=> B_29 = C_26 ) ).
fof(fact_656_add__imp__eq,axiom,
! [A_37,B_29,C_26] :
( hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_37),B_29) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_37),C_26)
=> B_29 = C_26 ) ).
fof(fact_657_add__left__imp__eq,axiom,
! [A_36,B_28,C_25] :
( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_36),B_28) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_36),C_25)
=> B_28 = C_25 ) ).
fof(fact_658_add__left__imp__eq,axiom,
! [A_36,B_28,C_25] :
( hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_36),B_28) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_36),C_25)
=> B_28 = C_25 ) ).
fof(fact_659_comm__semiring__1__class_Onormalizing__semiring__rules_I20_J,axiom,
! [A_35,B_27,C_24,D_6] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_35),B_27)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,C_24),D_6)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_35),C_24)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,B_27),D_6)) ).
fof(fact_660_comm__semiring__1__class_Onormalizing__semiring__rules_I20_J,axiom,
! [A_35,B_27,C_24,D_6] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_35),B_27)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,C_24),D_6)) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_35),C_24)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,B_27),D_6)) ).
fof(fact_661_add__right__cancel,axiom,
! [Ba,A_13,Ca] :
( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Ba),A_13) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Ca),A_13)
<=> Ba = Ca ) ).
fof(fact_662_add__right__cancel,axiom,
! [Ba,A_13,Ca] :
( hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Ba),A_13) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Ca),A_13)
<=> Ba = Ca ) ).
fof(fact_663_add__left__cancel,axiom,
! [A_13,Ba,Ca] :
( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_13),Ba) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_13),Ca)
<=> Ba = Ca ) ).
fof(fact_664_add__left__cancel,axiom,
! [A_13,Ba,Ca] :
( hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_13),Ba) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_13),Ca)
<=> Ba = Ca ) ).
fof(fact_665_comm__semiring__1__class_Onormalizing__semiring__rules_I23_J,axiom,
! [A_34,B_26,C_23] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_34),B_26)),C_23) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_34),C_23)),B_26) ).
fof(fact_666_comm__semiring__1__class_Onormalizing__semiring__rules_I23_J,axiom,
! [A_34,B_26,C_23] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_34),B_26)),C_23) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_34),C_23)),B_26) ).
fof(fact_667_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A_33,B_25,C_22] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_33),B_25)),C_22) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_33),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,B_25),C_22)) ).
fof(fact_668_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A_33,B_25,C_22] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_33),B_25)),C_22) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_33),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,B_25),C_22)) ).
fof(fact_669_comm__semiring__1__class_Onormalizing__semiring__rules_I21_J,axiom,
! [A_32,B_24,C_21] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_32),B_24)),C_21) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_32),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,B_24),C_21)) ).
fof(fact_670_comm__semiring__1__class_Onormalizing__semiring__rules_I21_J,axiom,
! [A_32,B_24,C_21] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_32),B_24)),C_21) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_32),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,B_24),C_21)) ).
fof(fact_671_comm__semiring__1__class_Onormalizing__semiring__rules_I25_J,axiom,
! [A_31,C_20,D_5] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_31),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,C_20),D_5)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_31),C_20)),D_5) ).
fof(fact_672_comm__semiring__1__class_Onormalizing__semiring__rules_I25_J,axiom,
! [A_31,C_20,D_5] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_31),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,C_20),D_5)) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_31),C_20)),D_5) ).
fof(fact_673_comm__semiring__1__class_Onormalizing__semiring__rules_I22_J,axiom,
! [A_30,C_19,D_4] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_30),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,C_19),D_4)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,C_19),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_30),D_4)) ).
fof(fact_674_comm__semiring__1__class_Onormalizing__semiring__rules_I22_J,axiom,
! [A_30,C_19,D_4] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_30),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,C_19),D_4)) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,C_19),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_30),D_4)) ).
fof(fact_675_comm__semiring__1__class_Onormalizing__semiring__rules_I24_J,axiom,
! [A_29,C_18] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_29),C_18) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,C_18),A_29) ).
fof(fact_676_comm__semiring__1__class_Onormalizing__semiring__rules_I24_J,axiom,
! [A_29,C_18] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_29),C_18) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,C_18),A_29) ).
fof(fact_677_comm__semiring__1__class_Onormalizing__semiring__rules_I13_J,axiom,
! [Lx_6,Ly_4,Rx_6,Ry_4] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Lx_6),Ly_4)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Rx_6),Ry_4)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Lx_6),Rx_6)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Ly_4),Ry_4)) ).
fof(fact_678_comm__semiring__1__class_Onormalizing__semiring__rules_I13_J,axiom,
! [Lx_6,Ly_4,Rx_6,Ry_4] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Lx_6),Ly_4)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Rx_6),Ry_4)) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Lx_6),Rx_6)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Ly_4),Ry_4)) ).
fof(fact_679_comm__semiring__1__class_Onormalizing__semiring__rules_I15_J,axiom,
! [Lx_5,Ly_3,Rx_5,Ry_3] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Lx_5),Ly_3)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Rx_5),Ry_3)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Rx_5),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Lx_5),Ly_3)),Ry_3)) ).
fof(fact_680_comm__semiring__1__class_Onormalizing__semiring__rules_I15_J,axiom,
! [Lx_5,Ly_3,Rx_5,Ry_3] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Lx_5),Ly_3)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Rx_5),Ry_3)) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Rx_5),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Lx_5),Ly_3)),Ry_3)) ).
fof(fact_681_comm__semiring__1__class_Onormalizing__semiring__rules_I14_J,axiom,
! [Lx_4,Ly_2,Rx_4,Ry_2] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Lx_4),Ly_2)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Rx_4),Ry_2)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Lx_4),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Ly_2),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Rx_4),Ry_2))) ).
fof(fact_682_comm__semiring__1__class_Onormalizing__semiring__rules_I14_J,axiom,
! [Lx_4,Ly_2,Rx_4,Ry_2] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Lx_4),Ly_2)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Rx_4),Ry_2)) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Lx_4),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Ly_2),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Rx_4),Ry_2))) ).
fof(fact_683_comm__semiring__1__class_Onormalizing__semiring__rules_I16_J,axiom,
! [Lx_3,Ly_1,Rx_3] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Lx_3),Ly_1)),Rx_3) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Lx_3),Rx_3)),Ly_1) ).
fof(fact_684_comm__semiring__1__class_Onormalizing__semiring__rules_I16_J,axiom,
! [Lx_3,Ly_1,Rx_3] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Lx_3),Ly_1)),Rx_3) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Lx_3),Rx_3)),Ly_1) ).
fof(fact_685_comm__semiring__1__class_Onormalizing__semiring__rules_I17_J,axiom,
! [Lx_2,Ly,Rx_2] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Lx_2),Ly)),Rx_2) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Lx_2),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Ly),Rx_2)) ).
fof(fact_686_comm__semiring__1__class_Onormalizing__semiring__rules_I17_J,axiom,
! [Lx_2,Ly,Rx_2] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Lx_2),Ly)),Rx_2) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Lx_2),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Ly),Rx_2)) ).
fof(fact_687_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A_28,B_23,C_17] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_28),B_23)),C_17) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_28),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_23),C_17)) ).
fof(fact_688_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A_28,B_23,C_17] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_28),B_23)),C_17) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_28),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_23),C_17)) ).
fof(fact_689_comm__semiring__1__class_Onormalizing__semiring__rules_I18_J,axiom,
! [Lx_1,Rx_1,Ry_1] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Lx_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Rx_1),Ry_1)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Lx_1),Rx_1)),Ry_1) ).
fof(fact_690_comm__semiring__1__class_Onormalizing__semiring__rules_I18_J,axiom,
! [Lx_1,Rx_1,Ry_1] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Lx_1),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Rx_1),Ry_1)) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Lx_1),Rx_1)),Ry_1) ).
fof(fact_691_comm__semiring__1__class_Onormalizing__semiring__rules_I19_J,axiom,
! [Lx,Rx,Ry] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Lx),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Rx),Ry)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Rx),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Lx),Ry)) ).
fof(fact_692_comm__semiring__1__class_Onormalizing__semiring__rules_I19_J,axiom,
! [Lx,Rx,Ry] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Lx),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Rx),Ry)) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Rx),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Lx),Ry)) ).
fof(fact_693_comm__semiring__1__class_Onormalizing__semiring__rules_I7_J,axiom,
! [A_27,B_22] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_27),B_22) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_22),A_27) ).
fof(fact_694_comm__semiring__1__class_Onormalizing__semiring__rules_I7_J,axiom,
! [A_27,B_22] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_27),B_22) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_22),A_27) ).
fof(fact_695_diff__eq__diff__eq,axiom,
! [A_13,Ba,Ca,D_1] :
( hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,A_13),Ba) = hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Ca),D_1)
=> ( A_13 = Ba
<=> Ca = D_1 ) ) ).
fof(fact_696_comm__semiring__1__class_Onormalizing__semiring__rules_I34_J,axiom,
! [X_27,Y_20,Z_10] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,X_27),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Y_20),Z_10)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,X_27),Y_20)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,X_27),Z_10)) ).
fof(fact_697_comm__semiring__1__class_Onormalizing__semiring__rules_I34_J,axiom,
! [X_27,Y_20,Z_10] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X_27),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Y_20),Z_10)) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X_27),Y_20)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X_27),Z_10)) ).
fof(fact_698_crossproduct__noteq,axiom,
! [Ca,D_1,A_13,Ba] :
( ( A_13 != Ba
& Ca != D_1 )
<=> hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_13),Ca)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Ba),D_1)) != hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_13),D_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Ba),Ca)) ) ).
fof(fact_699_crossproduct__noteq,axiom,
! [Ca,D_1,A_13,Ba] :
( ( A_13 != Ba
& Ca != D_1 )
<=> hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_13),Ca)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Ba),D_1)) != hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_13),D_1)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Ba),Ca)) ) ).
fof(fact_700_comm__semiring__class_Odistrib,axiom,
! [A_26,B_21,C_16] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_26),B_21)),C_16) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_26),C_16)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_21),C_16)) ).
fof(fact_701_comm__semiring__class_Odistrib,axiom,
! [A_26,B_21,C_16] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_26),B_21)),C_16) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_26),C_16)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_21),C_16)) ).
fof(fact_702_comm__semiring__1__class_Onormalizing__semiring__rules_I8_J,axiom,
! [A_25,B_20,C_15] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_25),B_20)),C_15) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_25),C_15)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_20),C_15)) ).
fof(fact_703_comm__semiring__1__class_Onormalizing__semiring__rules_I8_J,axiom,
! [A_25,B_20,C_15] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_25),B_20)),C_15) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_25),C_15)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_20),C_15)) ).
fof(fact_704_setsum__cong2,axiom,
! [F,G,A] :
( ! [X_2] :
( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_2),A))
=> hAPP_int_nat(F,X_2) = hAPP_int_nat(G,X_2) )
=> hAPP_f957591787ol_nat(hAPP_f1109019371ol_nat(big_co1740723097nt_nat,F),A) = hAPP_f957591787ol_nat(hAPP_f1109019371ol_nat(big_co1740723097nt_nat,G),A) ) ).
fof(fact_705_setsum__cong2,axiom,
! [F,G,A] :
( ! [X_2] :
( hBOOL(hAPP_f1695230391l_bool(hAPP_H622608077l_bool(member1797258804iple_a,X_2),A))
=> hAPP_H592031934_a_nat(F,X_2) = hAPP_H592031934_a_nat(G,X_2) )
=> hAPP_f2097660464ol_nat(hAPP_f1371399797ol_nat(big_co1340561246_a_nat,F),A) = hAPP_f2097660464ol_nat(hAPP_f1371399797ol_nat(big_co1340561246_a_nat,G),A) ) ).
fof(fact_706_setsum__cong2,axiom,
! [F,G,A] :
( ! [X_2] :
( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_2),A))
=> hAPP_nat_nat(F,X_2) = hAPP_nat_nat(G,X_2) )
=> hAPP_f22106695ol_nat(hAPP_f782000547ol_nat(big_co387207925at_nat,F),A) = hAPP_f22106695ol_nat(hAPP_f782000547ol_nat(big_co387207925at_nat,G),A) ) ).
fof(fact_707_setsum__cong,axiom,
! [F,G,A,B] :
( A = B
=> ( ! [X_2] :
( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_2),B))
=> hAPP_int_nat(F,X_2) = hAPP_int_nat(G,X_2) )
=> hAPP_f957591787ol_nat(hAPP_f1109019371ol_nat(big_co1740723097nt_nat,F),A) = hAPP_f957591787ol_nat(hAPP_f1109019371ol_nat(big_co1740723097nt_nat,G),B) ) ) ).
fof(fact_708_setsum__cong,axiom,
! [F,G,A,B] :
( A = B
=> ( ! [X_2] :
( hBOOL(hAPP_f1695230391l_bool(hAPP_H622608077l_bool(member1797258804iple_a,X_2),B))
=> hAPP_H592031934_a_nat(F,X_2) = hAPP_H592031934_a_nat(G,X_2) )
=> hAPP_f2097660464ol_nat(hAPP_f1371399797ol_nat(big_co1340561246_a_nat,F),A) = hAPP_f2097660464ol_nat(hAPP_f1371399797ol_nat(big_co1340561246_a_nat,G),B) ) ) ).
fof(fact_709_setsum__cong,axiom,
! [F,G,A,B] :
( A = B
=> ( ! [X_2] :
( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_2),B))
=> hAPP_nat_nat(F,X_2) = hAPP_nat_nat(G,X_2) )
=> hAPP_f22106695ol_nat(hAPP_f782000547ol_nat(big_co387207925at_nat,F),A) = hAPP_f22106695ol_nat(hAPP_f782000547ol_nat(big_co387207925at_nat,G),B) ) ) ).
fof(fact_710_setsum_OF__cong,axiom,
! [H,G,A,B] :
( A = B
=> ( ! [X_2] :
( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_2),B))
=> hAPP_int_nat(H,X_2) = hAPP_int_nat(G,X_2) )
=> hAPP_f957591787ol_nat(hAPP_f1109019371ol_nat(big_co1740723097nt_nat,H),A) = hAPP_f957591787ol_nat(hAPP_f1109019371ol_nat(big_co1740723097nt_nat,G),B) ) ) ).
fof(fact_711_setsum_OF__cong,axiom,
! [H,G,A,B] :
( A = B
=> ( ! [X_2] :
( hBOOL(hAPP_f1695230391l_bool(hAPP_H622608077l_bool(member1797258804iple_a,X_2),B))
=> hAPP_H592031934_a_nat(H,X_2) = hAPP_H592031934_a_nat(G,X_2) )
=> hAPP_f2097660464ol_nat(hAPP_f1371399797ol_nat(big_co1340561246_a_nat,H),A) = hAPP_f2097660464ol_nat(hAPP_f1371399797ol_nat(big_co1340561246_a_nat,G),B) ) ) ).
fof(fact_712_setsum_OF__cong,axiom,
! [H,G,A,B] :
( A = B
=> ( ! [X_2] :
( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_2),B))
=> hAPP_nat_nat(H,X_2) = hAPP_nat_nat(G,X_2) )
=> hAPP_f22106695ol_nat(hAPP_f782000547ol_nat(big_co387207925at_nat,H),A) = hAPP_f22106695ol_nat(hAPP_f782000547ol_nat(big_co387207925at_nat,G),B) ) ) ).
fof(fact_713_single__Diff__lessThan,axiom,
! [K_1] : hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,K_1),bot_bot_fun_nat_bool)),hAPP_n1699378549t_bool(ord_lessThan_nat,K_1)) = hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,K_1),bot_bot_fun_nat_bool) ).
fof(fact_714_single__Diff__lessThan,axiom,
! [K_1] : hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,K_1),bot_bot_fun_int_bool)),hAPP_i1948725293t_bool(ord_lessThan_int,K_1)) = hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,K_1),bot_bot_fun_int_bool) ).
fof(fact_715_setsum__diff__nat,axiom,
! [F,A,B] :
( hBOOL(hAPP_f1695230391l_bool(finite1738664244iple_a,B))
=> ( hBOOL(hAPP_f1695230391l_bool(hAPP_f1767618879l_bool(ord_le1523781517a_bool,B),A))
=> hAPP_f2097660464ol_nat(hAPP_f1371399797ol_nat(big_co1340561246_a_nat,F),hAPP_f20753329a_bool(hAPP_f1706700729a_bool(minus_85316870a_bool,A),B)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_f2097660464ol_nat(hAPP_f1371399797ol_nat(big_co1340561246_a_nat,F),A)),hAPP_f2097660464ol_nat(hAPP_f1371399797ol_nat(big_co1340561246_a_nat,F),B)) ) ) ).
fof(fact_716_setsum__diff__nat,axiom,
! [F,A,B] :
( hBOOL(hAPP_f448129468l_bool(finite_finite_int,B))
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,B),A))
=> hAPP_f957591787ol_nat(hAPP_f1109019371ol_nat(big_co1740723097nt_nat,F),hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A),B)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_f957591787ol_nat(hAPP_f1109019371ol_nat(big_co1740723097nt_nat,F),A)),hAPP_f957591787ol_nat(hAPP_f1109019371ol_nat(big_co1740723097nt_nat,F),B)) ) ) ).
fof(fact_717_setsum__diff__nat,axiom,
! [F,A,B] :
( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,B))
=> ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,B),A))
=> hAPP_f22106695ol_nat(hAPP_f782000547ol_nat(big_co387207925at_nat,F),hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),B)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_f22106695ol_nat(hAPP_f782000547ol_nat(big_co387207925at_nat,F),A)),hAPP_f22106695ol_nat(hAPP_f782000547ol_nat(big_co387207925at_nat,F),B)) ) ) ).
fof(fact_718_order__refl,axiom,
! [X_1] : hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,X_1),X_1)) ).
fof(fact_719_order__refl,axiom,
! [X_1] : hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,X_1),X_1)) ).
fof(fact_720_order__refl,axiom,
! [X_26] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_26),X_26)) ).
fof(fact_721_order__refl,axiom,
! [X_26] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_26),X_26)) ).
fof(fact_722_equalityI,axiom,
! [A,B] :
( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,A),B))
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,B),A))
=> A = B ) ) ).
fof(fact_723_equalityI,axiom,
! [A,B] :
( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A),B))
=> ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,B),A))
=> A = B ) ) ).
fof(fact_724_subsetD,axiom,
! [Ca,A,B] :
( hBOOL(hAPP_f1695230391l_bool(hAPP_f1767618879l_bool(ord_le1523781517a_bool,A),B))
=> ( hBOOL(hAPP_f1695230391l_bool(hAPP_H622608077l_bool(member1797258804iple_a,Ca),A))
=> hBOOL(hAPP_f1695230391l_bool(hAPP_H622608077l_bool(member1797258804iple_a,Ca),B)) ) ) ).
fof(fact_725_subsetD,axiom,
! [Ca,A,B] :
( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,A),B))
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Ca),A))
=> hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Ca),B)) ) ) ).
fof(fact_726_subsetD,axiom,
! [Ca,A,B] :
( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A),B))
=> ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,Ca),A))
=> hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,Ca),B)) ) ) ).
fof(fact_727_empty__subsetI,axiom,
! [A] : hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,bot_bot_fun_nat_bool),A)) ).
fof(fact_728_empty__subsetI,axiom,
! [A] : hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,bot_bot_fun_int_bool),A)) ).
fof(fact_729_empty__subsetI,axiom,
! [A] : hBOOL(hAPP_f1695230391l_bool(hAPP_f1767618879l_bool(ord_le1523781517a_bool,bot_bo797238721a_bool),A)) ).
fof(fact_730_finite__Collect__subsets,axiom,
! [A] :
( hBOOL(hAPP_f1695230391l_bool(finite1738664244iple_a,A))
=> hBOOL(hAPP_f2110825313l_bool(finite268574148a_bool,hAPP_f396832789l_bool(collec999179778a_bool,hAPP_f1767618879l_bool(hAPP_f2100528361l_bool(cOMBC_627599030l_bool,ord_le1523781517a_bool),A)))) ) ).
fof(fact_731_finite__Collect__subsets,axiom,
! [A] :
( hBOOL(hAPP_f448129468l_bool(finite_finite_int,A))
=> hBOOL(hAPP_f215623910l_bool(finite1395289673t_bool,hAPP_f1948010709l_bool(collect_fun_int_bool,hAPP_f284875647l_bool(hAPP_f1399552105l_bool(cOMBC_605892544l_bool,ord_le951220754t_bool),A)))) ) ).
fof(fact_732_finite__Collect__subsets,axiom,
! [A] :
( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
=> hBOOL(hAPP_f1637334154l_bool(finite2012431853t_bool,hAPP_f1246832597l_bool(collect_fun_nat_bool,hAPP_f103356543l_bool(hAPP_f643944041l_bool(cOMBC_1693257480l_bool,ord_le1568362934t_bool),A)))) ) ).
fof(fact_733_pred__subset__eq,axiom,
! [R_1,S_1] :
( hBOOL(hAPP_f1695230391l_bool(hAPP_f1767618879l_bool(ord_le1523781517a_bool,hAPP_f20753329a_bool(hAPP_f247708275a_bool(cOMBC_1519416966l_bool,member1797258804iple_a),R_1)),hAPP_f20753329a_bool(hAPP_f247708275a_bool(cOMBC_1519416966l_bool,member1797258804iple_a),S_1)))
<=> hBOOL(hAPP_f1695230391l_bool(hAPP_f1767618879l_bool(ord_le1523781517a_bool,R_1),S_1)) ) ).
fof(fact_734_pred__subset__eq,axiom,
! [R_1,S_1] :
( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,hAPP_f1805168059t_bool(hAPP_f202917053t_bool(cOMBC_94739984l_bool,member_int),R_1)),hAPP_f1805168059t_bool(hAPP_f202917053t_bool(cOMBC_94739984l_bool,member_int),S_1)))
<=> hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,R_1),S_1)) ) ).
fof(fact_735_pred__subset__eq,axiom,
! [R_1,S_1] :
( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,hAPP_f800510211t_bool(hAPP_f1722879237t_bool(cOMBC_226598744l_bool,member_nat),R_1)),hAPP_f800510211t_bool(hAPP_f1722879237t_bool(cOMBC_226598744l_bool,member_nat),S_1)))
<=> hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,R_1),S_1)) ) ).
fof(fact_736_linorder__le__cases,axiom,
! [X_25,Y_19] :
( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_25),Y_19))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y_19),X_25)) ) ).
fof(fact_737_linorder__le__cases,axiom,
! [X_25,Y_19] :
( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_25),Y_19))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Y_19),X_25)) ) ).
fof(fact_738_le__funE,axiom,
! [X_1,F,G] :
( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,F),G))
=> hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(ord_less_eq_bool,hAPP_int_bool(F,X_1)),hAPP_int_bool(G,X_1))) ) ).
fof(fact_739_le__funE,axiom,
! [X_1,F,G] :
( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,F),G))
=> hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(ord_less_eq_bool,hAPP_nat_bool(F,X_1)),hAPP_nat_bool(G,X_1))) ) ).
fof(fact_740_xt1_I6_J,axiom,
! [Z_1,Y_1,X_1] :
( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,Y_1),X_1))
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,Z_1),Y_1))
=> hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,Z_1),X_1)) ) ) ).
fof(fact_741_xt1_I6_J,axiom,
! [Z_1,Y_1,X_1] :
( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,Y_1),X_1))
=> ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,Z_1),Y_1))
=> hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,Z_1),X_1)) ) ) ).
fof(fact_742_xt1_I6_J,axiom,
! [Z_9,Y_18,X_24] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y_18),X_24))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Z_9),Y_18))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Z_9),X_24)) ) ) ).
fof(fact_743_xt1_I6_J,axiom,
! [Z_9,Y_18,X_24] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Y_18),X_24))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Z_9),Y_18))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Z_9),X_24)) ) ) ).
fof(fact_744_xt1_I5_J,axiom,
! [Y_1,X_1] :
( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,Y_1),X_1))
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,X_1),Y_1))
=> X_1 = Y_1 ) ) ).
fof(fact_745_xt1_I5_J,axiom,
! [Y_1,X_1] :
( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,Y_1),X_1))
=> ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,X_1),Y_1))
=> X_1 = Y_1 ) ) ).
fof(fact_746_xt1_I5_J,axiom,
! [Y_17,X_23] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y_17),X_23))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_23),Y_17))
=> X_23 = Y_17 ) ) ).
fof(fact_747_xt1_I5_J,axiom,
! [Y_17,X_23] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Y_17),X_23))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_23),Y_17))
=> X_23 = Y_17 ) ) ).
fof(fact_748_order__trans,axiom,
! [Z_1,X_1,Y_1] :
( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,X_1),Y_1))
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,Y_1),Z_1))
=> hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,X_1),Z_1)) ) ) ).
fof(fact_749_order__trans,axiom,
! [Z_1,X_1,Y_1] :
( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,X_1),Y_1))
=> ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,Y_1),Z_1))
=> hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,X_1),Z_1)) ) ) ).
fof(fact_750_order__trans,axiom,
! [Z_8,X_22,Y_16] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_22),Y_16))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y_16),Z_8))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_22),Z_8)) ) ) ).
fof(fact_751_order__trans,axiom,
! [Z_8,X_22,Y_16] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_22),Y_16))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Y_16),Z_8))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_22),Z_8)) ) ) ).
fof(fact_752_order__antisym,axiom,
! [X_1,Y_1] :
( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,X_1),Y_1))
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,Y_1),X_1))
=> X_1 = Y_1 ) ) ).
fof(fact_753_order__antisym,axiom,
! [X_1,Y_1] :
( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,X_1),Y_1))
=> ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,Y_1),X_1))
=> X_1 = Y_1 ) ) ).
fof(fact_754_order__antisym,axiom,
! [X_21,Y_15] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_21),Y_15))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y_15),X_21))
=> X_21 = Y_15 ) ) ).
fof(fact_755_order__antisym,axiom,
! [X_21,Y_15] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_21),Y_15))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Y_15),X_21))
=> X_21 = Y_15 ) ) ).
fof(fact_756_xt1_I4_J,axiom,
! [Ca,Ba,A_13] :
( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,Ba),A_13))
=> ( Ba = Ca
=> hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,Ca),A_13)) ) ) ).
fof(fact_757_xt1_I4_J,axiom,
! [Ca,Ba,A_13] :
( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,Ba),A_13))
=> ( Ba = Ca
=> hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,Ca),A_13)) ) ) ).
fof(fact_758_xt1_I4_J,axiom,
! [C_14,B_19,A_24] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_19),A_24))
=> ( B_19 = C_14
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,C_14),A_24)) ) ) ).
fof(fact_759_xt1_I4_J,axiom,
! [C_14,B_19,A_24] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,B_19),A_24))
=> ( B_19 = C_14
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,C_14),A_24)) ) ) ).
fof(fact_760_ord__le__eq__trans,axiom,
! [Ca,A_13,Ba] :
( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,A_13),Ba))
=> ( Ba = Ca
=> hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,A_13),Ca)) ) ) ).
fof(fact_761_ord__le__eq__trans,axiom,
! [Ca,A_13,Ba] :
( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A_13),Ba))
=> ( Ba = Ca
=> hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A_13),Ca)) ) ) ).
fof(fact_762_ord__le__eq__trans,axiom,
! [C_13,A_23,B_18] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_23),B_18))
=> ( B_18 = C_13
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_23),C_13)) ) ) ).
fof(fact_763_ord__le__eq__trans,axiom,
! [C_13,A_23,B_18] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_23),B_18))
=> ( B_18 = C_13
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_23),C_13)) ) ) ).
fof(fact_764_xt1_I3_J,axiom,
! [Ca,A_13,Ba] :
( A_13 = Ba
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,Ca),Ba))
=> hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,Ca),A_13)) ) ) ).
fof(fact_765_xt1_I3_J,axiom,
! [Ca,A_13,Ba] :
( A_13 = Ba
=> ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,Ca),Ba))
=> hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,Ca),A_13)) ) ) ).
fof(fact_766_xt1_I3_J,axiom,
! [C_12,A_22,B_17] :
( A_22 = B_17
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,C_12),B_17))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,C_12),A_22)) ) ) ).
fof(fact_767_xt1_I3_J,axiom,
! [C_12,A_22,B_17] :
( A_22 = B_17
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,C_12),B_17))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,C_12),A_22)) ) ) ).
fof(fact_768_ord__eq__le__trans,axiom,
! [Ca,A_13,Ba] :
( A_13 = Ba
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,Ba),Ca))
=> hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,A_13),Ca)) ) ) ).
fof(fact_769_ord__eq__le__trans,axiom,
! [Ca,A_13,Ba] :
( A_13 = Ba
=> ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,Ba),Ca))
=> hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A_13),Ca)) ) ) ).
fof(fact_770_ord__eq__le__trans,axiom,
! [C_11,A_21,B_16] :
( A_21 = B_16
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_16),C_11))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_21),C_11)) ) ) ).
fof(fact_771_ord__eq__le__trans,axiom,
! [C_11,A_21,B_16] :
( A_21 = B_16
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,B_16),C_11))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_21),C_11)) ) ) ).
fof(fact_772_order__antisym__conv,axiom,
! [Y_1,X_1] :
( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,Y_1),X_1))
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,X_1),Y_1))
<=> X_1 = Y_1 ) ) ).
fof(fact_773_order__antisym__conv,axiom,
! [Y_1,X_1] :
( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,Y_1),X_1))
=> ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,X_1),Y_1))
<=> X_1 = Y_1 ) ) ).
fof(fact_774_order__antisym__conv,axiom,
! [Y_1,X_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y_1),X_1))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_1),Y_1))
<=> X_1 = Y_1 ) ) ).
fof(fact_775_order__antisym__conv,axiom,
! [Y_1,X_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Y_1),X_1))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_1),Y_1))
<=> X_1 = Y_1 ) ) ).
fof(fact_776_le__funD,axiom,
! [X_1,F,G] :
( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,F),G))
=> hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(ord_less_eq_bool,hAPP_int_bool(F,X_1)),hAPP_int_bool(G,X_1))) ) ).
fof(fact_777_le__funD,axiom,
! [X_1,F,G] :
( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,F),G))
=> hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(ord_less_eq_bool,hAPP_nat_bool(F,X_1)),hAPP_nat_bool(G,X_1))) ) ).
fof(fact_778_order__eq__refl,axiom,
! [X_1,Y_1] :
( X_1 = Y_1
=> hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,X_1),Y_1)) ) ).
fof(fact_779_order__eq__refl,axiom,
! [X_1,Y_1] :
( X_1 = Y_1
=> hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,X_1),Y_1)) ) ).
fof(fact_780_order__eq__refl,axiom,
! [X_20,Y_14] :
( X_20 = Y_14
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_20),Y_14)) ) ).
fof(fact_781_order__eq__refl,axiom,
! [X_20,Y_14] :
( X_20 = Y_14
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_20),Y_14)) ) ).
fof(fact_782_order__eq__iff,axiom,
! [X_1,Y_1] :
( X_1 = Y_1
<=> ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,X_1),Y_1))
& hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,Y_1),X_1)) ) ) ).
fof(fact_783_order__eq__iff,axiom,
! [X_1,Y_1] :
( X_1 = Y_1
<=> ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,X_1),Y_1))
& hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,Y_1),X_1)) ) ) ).
fof(fact_784_order__eq__iff,axiom,
! [X_1,Y_1] :
( X_1 = Y_1
<=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_1),Y_1))
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y_1),X_1)) ) ) ).
fof(fact_785_order__eq__iff,axiom,
! [X_1,Y_1] :
( X_1 = Y_1
<=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_1),Y_1))
& hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Y_1),X_1)) ) ) ).
fof(fact_786_linorder__linear,axiom,
! [X_19,Y_13] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_19),Y_13))
| hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y_13),X_19)) ) ).
fof(fact_787_linorder__linear,axiom,
! [X_19,Y_13] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_19),Y_13))
| hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Y_13),X_19)) ) ).
fof(fact_788_le__fun__def,axiom,
! [F,G] :
( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,F),G))
<=> ! [X_2] : hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(ord_less_eq_bool,hAPP_int_bool(F,X_2)),hAPP_int_bool(G,X_2))) ) ).
fof(fact_789_le__fun__def,axiom,
! [F,G] :
( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,F),G))
<=> ! [X_2] : hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(ord_less_eq_bool,hAPP_nat_bool(F,X_2)),hAPP_nat_bool(G,X_2))) ) ).
fof(fact_790_equalityE,axiom,
! [A,B] :
( A = B
=> ~ ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,A),B))
=> ~ hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,B),A)) ) ) ).
fof(fact_791_equalityE,axiom,
! [A,B] :
( A = B
=> ~ ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A),B))
=> ~ hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,B),A)) ) ) ).
fof(fact_792_subset__trans,axiom,
! [C_5,A,B] :
( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,A),B))
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,B),C_5))
=> hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,A),C_5)) ) ) ).
fof(fact_793_subset__trans,axiom,
! [C_5,A,B] :
( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A),B))
=> ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,B),C_5))
=> hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A),C_5)) ) ) ).
fof(fact_794_set__mp,axiom,
! [X_1,A,B] :
( hBOOL(hAPP_f1695230391l_bool(hAPP_f1767618879l_bool(ord_le1523781517a_bool,A),B))
=> ( hBOOL(hAPP_f1695230391l_bool(hAPP_H622608077l_bool(member1797258804iple_a,X_1),A))
=> hBOOL(hAPP_f1695230391l_bool(hAPP_H622608077l_bool(member1797258804iple_a,X_1),B)) ) ) ).
fof(fact_795_set__mp,axiom,
! [X_1,A,B] :
( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,A),B))
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),A))
=> hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),B)) ) ) ).
fof(fact_796_set__mp,axiom,
! [X_1,A,B] :
( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A),B))
=> ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A))
=> hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),B)) ) ) ).
fof(fact_797_set__rev__mp,axiom,
! [B,X_1,A] :
( hBOOL(hAPP_f1695230391l_bool(hAPP_H622608077l_bool(member1797258804iple_a,X_1),A))
=> ( hBOOL(hAPP_f1695230391l_bool(hAPP_f1767618879l_bool(ord_le1523781517a_bool,A),B))
=> hBOOL(hAPP_f1695230391l_bool(hAPP_H622608077l_bool(member1797258804iple_a,X_1),B)) ) ) ).
fof(fact_798_set__rev__mp,axiom,
! [B,X_1,A] :
( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),A))
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,A),B))
=> hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),B)) ) ) ).
fof(fact_799_set__rev__mp,axiom,
! [B,X_1,A] :
( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A))
=> ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A),B))
=> hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),B)) ) ) ).
fof(fact_800_in__mono,axiom,
! [X_1,A,B] :
( hBOOL(hAPP_f1695230391l_bool(hAPP_f1767618879l_bool(ord_le1523781517a_bool,A),B))
=> ( hBOOL(hAPP_f1695230391l_bool(hAPP_H622608077l_bool(member1797258804iple_a,X_1),A))
=> hBOOL(hAPP_f1695230391l_bool(hAPP_H622608077l_bool(member1797258804iple_a,X_1),B)) ) ) ).
fof(fact_801_in__mono,axiom,
! [X_1,A,B] :
( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,A),B))
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),A))
=> hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),B)) ) ) ).
fof(fact_802_in__mono,axiom,
! [X_1,A,B] :
( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A),B))
=> ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A))
=> hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),B)) ) ) ).
fof(fact_803_equalityD2,axiom,
! [A,B] :
( A = B
=> hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,B),A)) ) ).
fof(fact_804_equalityD2,axiom,
! [A,B] :
( A = B
=> hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,B),A)) ) ).
fof(fact_805_equalityD1,axiom,
! [A,B] :
( A = B
=> hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,A),B)) ) ).
fof(fact_806_equalityD1,axiom,
! [A,B] :
( A = B
=> hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A),B)) ) ).
fof(fact_807_set__eq__subset,axiom,
! [A,B] :
( A = B
<=> ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,A),B))
& hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,B),A)) ) ) ).
fof(fact_808_set__eq__subset,axiom,
! [A,B] :
( A = B
<=> ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A),B))
& hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,B),A)) ) ) ).
fof(fact_809_subset__refl,axiom,
! [A] : hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,A),A)) ).
fof(fact_810_subset__refl,axiom,
! [A] : hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A),A)) ).
fof(fact_811_predicate1D,axiom,
! [X_1,Pa,Q_1] :
( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,Pa),Q_1))
=> ( hBOOL(hAPP_int_bool(Pa,X_1))
=> hBOOL(hAPP_int_bool(Q_1,X_1)) ) ) ).
fof(fact_812_predicate1D,axiom,
! [X_1,Pa,Q_1] :
( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,Pa),Q_1))
=> ( hBOOL(hAPP_nat_bool(Pa,X_1))
=> hBOOL(hAPP_nat_bool(Q_1,X_1)) ) ) ).
fof(fact_813_rev__predicate1D,axiom,
! [Q_1,Pa,X_1] :
( hBOOL(hAPP_int_bool(Pa,X_1))
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,Pa),Q_1))
=> hBOOL(hAPP_int_bool(Q_1,X_1)) ) ) ).
fof(fact_814_rev__predicate1D,axiom,
! [Q_1,Pa,X_1] :
( hBOOL(hAPP_nat_bool(Pa,X_1))
=> ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,Pa),Q_1))
=> hBOOL(hAPP_nat_bool(Q_1,X_1)) ) ) ).
fof(fact_815_lessThan__subset__iff,axiom,
! [X_1,Y_1] :
( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,hAPP_i1948725293t_bool(ord_lessThan_int,X_1)),hAPP_i1948725293t_bool(ord_lessThan_int,Y_1)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_1),Y_1)) ) ).
fof(fact_816_lessThan__subset__iff,axiom,
! [X_1,Y_1] :
( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,hAPP_n1699378549t_bool(ord_lessThan_nat,X_1)),hAPP_n1699378549t_bool(ord_lessThan_nat,Y_1)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_1),Y_1)) ) ).
fof(fact_817_lessThan__eq__iff,axiom,
! [X_1,Y_1] :
( hAPP_n1699378549t_bool(ord_lessThan_nat,X_1) = hAPP_n1699378549t_bool(ord_lessThan_nat,Y_1)
<=> X_1 = Y_1 ) ).
fof(fact_818_add__le__imp__le__left,axiom,
! [C_10,A_20,B_15] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,C_10),A_20)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,C_10),B_15)))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_20),B_15)) ) ).
fof(fact_819_add__le__imp__le__left,axiom,
! [C_10,A_20,B_15] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,C_10),A_20)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,C_10),B_15)))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_20),B_15)) ) ).
fof(fact_820_add__le__imp__le__right,axiom,
! [A_19,C_9,B_14] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_19),C_9)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,B_14),C_9)))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_19),B_14)) ) ).
fof(fact_821_add__le__imp__le__right,axiom,
! [A_19,C_9,B_14] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_19),C_9)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,B_14),C_9)))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_19),B_14)) ) ).
fof(fact_822_add__mono,axiom,
! [C_8,D_3,A_18,B_13] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_18),B_13))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,C_8),D_3))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_18),C_8)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,B_13),D_3))) ) ) ).
fof(fact_823_add__mono,axiom,
! [C_8,D_3,A_18,B_13] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_18),B_13))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,C_8),D_3))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_18),C_8)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,B_13),D_3))) ) ) ).
fof(fact_824_add__left__mono,axiom,
! [C_7,A_17,B_12] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_17),B_12))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,C_7),A_17)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,C_7),B_12))) ) ).
fof(fact_825_add__left__mono,axiom,
! [C_7,A_17,B_12] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_17),B_12))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,C_7),A_17)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,C_7),B_12))) ) ).
fof(fact_826_add__right__mono,axiom,
! [C_6,A_16,B_11] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_16),B_11))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_16),C_6)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,B_11),C_6))) ) ).
fof(fact_827_add__right__mono,axiom,
! [C_6,A_16,B_11] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_16),B_11))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_16),C_6)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,B_11),C_6))) ) ).
fof(fact_828_add__le__cancel__left,axiom,
! [Ca,A_13,Ba] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Ca),A_13)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Ca),Ba)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_13),Ba)) ) ).
fof(fact_829_add__le__cancel__left,axiom,
! [Ca,A_13,Ba] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Ca),A_13)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Ca),Ba)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_13),Ba)) ) ).
fof(fact_830_add__le__cancel__right,axiom,
! [A_13,Ca,Ba] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_13),Ca)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Ba),Ca)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_13),Ba)) ) ).
fof(fact_831_add__le__cancel__right,axiom,
! [A_13,Ca,Ba] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_13),Ca)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Ba),Ca)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_13),Ba)) ) ).
fof(fact_832_le__bot,axiom,
! [A_13] :
( hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(ord_less_eq_bool,A_13),bot_bot_bool))
=> ( hBOOL(A_13)
<=> hBOOL(bot_bot_bool) ) ) ).
fof(fact_833_le__bot,axiom,
! [A_13] :
( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A_13),bot_bot_fun_nat_bool))
=> A_13 = bot_bot_fun_nat_bool ) ).
fof(fact_834_le__bot,axiom,
! [A_13] :
( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,A_13),bot_bot_fun_int_bool))
=> A_13 = bot_bot_fun_int_bool ) ).
fof(fact_835_le__bot,axiom,
! [A_13] :
( hBOOL(hAPP_f1695230391l_bool(hAPP_f1767618879l_bool(ord_le1523781517a_bool,A_13),bot_bo797238721a_bool))
=> A_13 = bot_bo797238721a_bool ) ).
fof(fact_836_le__bot,axiom,
! [A_15] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_15),bot_bot_nat))
=> A_15 = bot_bot_nat ) ).
fof(fact_837_bot__unique,axiom,
! [A_13] :
( hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(ord_less_eq_bool,A_13),bot_bot_bool))
<=> ( hBOOL(A_13)
<=> hBOOL(bot_bot_bool) ) ) ).
fof(fact_838_bot__unique,axiom,
! [A_13] :
( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A_13),bot_bot_fun_nat_bool))
<=> A_13 = bot_bot_fun_nat_bool ) ).
fof(fact_839_bot__unique,axiom,
! [A_13] :
( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,A_13),bot_bot_fun_int_bool))
<=> A_13 = bot_bot_fun_int_bool ) ).
fof(fact_840_bot__unique,axiom,
! [A_13] :
( hBOOL(hAPP_f1695230391l_bool(hAPP_f1767618879l_bool(ord_le1523781517a_bool,A_13),bot_bo797238721a_bool))
<=> A_13 = bot_bo797238721a_bool ) ).
fof(fact_841_bot__unique,axiom,
! [A_13] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_13),bot_bot_nat))
<=> A_13 = bot_bot_nat ) ).
fof(fact_842_bot__least,axiom,
! [A_13] : hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(ord_less_eq_bool,bot_bot_bool),A_13)) ).
fof(fact_843_bot__least,axiom,
! [A_13] : hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,bot_bot_fun_nat_bool),A_13)) ).
fof(fact_844_bot__least,axiom,
! [A_13] : hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,bot_bot_fun_int_bool),A_13)) ).
fof(fact_845_bot__least,axiom,
! [A_13] : hBOOL(hAPP_f1695230391l_bool(hAPP_f1767618879l_bool(ord_le1523781517a_bool,bot_bo797238721a_bool),A_13)) ).
fof(fact_846_bot__least,axiom,
! [A_14] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,bot_bot_nat),A_14)) ).
fof(fact_847_diff__eq__diff__less__eq,axiom,
! [A_13,Ba,Ca,D_1] :
( hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,A_13),Ba) = hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Ca),D_1)
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_13),Ba))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Ca),D_1)) ) ) ).
fof(fact_848_subset__empty,axiom,
! [A] :
( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A),bot_bot_fun_nat_bool))
<=> A = bot_bot_fun_nat_bool ) ).
fof(fact_849_subset__empty,axiom,
! [A] :
( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,A),bot_bot_fun_int_bool))
<=> A = bot_bot_fun_int_bool ) ).
fof(fact_850_subset__empty,axiom,
! [A] :
( hBOOL(hAPP_f1695230391l_bool(hAPP_f1767618879l_bool(ord_le1523781517a_bool,A),bot_bo797238721a_bool))
<=> A = bot_bo797238721a_bool ) ).
fof(fact_851_finite__subset,axiom,
! [A,B] :
( hBOOL(hAPP_f1695230391l_bool(hAPP_f1767618879l_bool(ord_le1523781517a_bool,A),B))
=> ( hBOOL(hAPP_f1695230391l_bool(finite1738664244iple_a,B))
=> hBOOL(hAPP_f1695230391l_bool(finite1738664244iple_a,A)) ) ) ).
fof(fact_852_finite__subset,axiom,
! [A,B] :
( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,A),B))
=> ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,B))
=> hBOOL(hAPP_f448129468l_bool(finite_finite_int,A)) ) ) ).
fof(fact_853_finite__subset,axiom,
! [A,B] :
( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A),B))
=> ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,B))
=> hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A)) ) ) ).
fof(fact_854_rev__finite__subset,axiom,
! [A,B] :
( hBOOL(hAPP_f1695230391l_bool(finite1738664244iple_a,B))
=> ( hBOOL(hAPP_f1695230391l_bool(hAPP_f1767618879l_bool(ord_le1523781517a_bool,A),B))
=> hBOOL(hAPP_f1695230391l_bool(finite1738664244iple_a,A)) ) ) ).
fof(fact_855_rev__finite__subset,axiom,
! [A,B] :
( hBOOL(hAPP_f448129468l_bool(finite_finite_int,B))
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,A),B))
=> hBOOL(hAPP_f448129468l_bool(finite_finite_int,A)) ) ) ).
fof(fact_856_rev__finite__subset,axiom,
! [A,B] :
( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,B))
=> ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A),B))
=> hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A)) ) ) ).
fof(fact_857_insert__mono,axiom,
! [A_13,C_5,D_2] :
( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,C_5),D_2))
=> hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_13),C_5)),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_13),D_2))) ) ).
fof(fact_858_insert__mono,axiom,
! [A_13,C_5,D_2] :
( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,C_5),D_2))
=> hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,A_13),C_5)),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,A_13),D_2))) ) ).
fof(fact_859_insert__mono,axiom,
! [A_13,C_5,D_2] :
( hBOOL(hAPP_f1695230391l_bool(hAPP_f1767618879l_bool(ord_le1523781517a_bool,C_5),D_2))
=> hBOOL(hAPP_f1695230391l_bool(hAPP_f1767618879l_bool(ord_le1523781517a_bool,hAPP_f20753329a_bool(hAPP_H1743777351a_bool(insert956547291iple_a,A_13),C_5)),hAPP_f20753329a_bool(hAPP_H1743777351a_bool(insert956547291iple_a,A_13),D_2))) ) ).
fof(fact_860_subset__insertI2,axiom,
! [Ba,A,B] :
( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A),B))
=> hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,Ba),B))) ) ).
fof(fact_861_subset__insertI2,axiom,
! [Ba,A,B] :
( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,A),B))
=> hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,A),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,Ba),B))) ) ).
fof(fact_862_subset__insertI2,axiom,
! [Ba,A,B] :
( hBOOL(hAPP_f1695230391l_bool(hAPP_f1767618879l_bool(ord_le1523781517a_bool,A),B))
=> hBOOL(hAPP_f1695230391l_bool(hAPP_f1767618879l_bool(ord_le1523781517a_bool,A),hAPP_f20753329a_bool(hAPP_H1743777351a_bool(insert956547291iple_a,Ba),B))) ) ).
fof(fact_863_subset__insert,axiom,
! [B,X_1,A] :
( ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A))
=> ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),B)))
<=> hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A),B)) ) ) ).
fof(fact_864_subset__insert,axiom,
! [B,X_1,A] :
( ~ hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),A))
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,A),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),B)))
<=> hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,A),B)) ) ) ).
fof(fact_865_subset__insert,axiom,
! [B,X_1,A] :
( ~ hBOOL(hAPP_f1695230391l_bool(hAPP_H622608077l_bool(member1797258804iple_a,X_1),A))
=> ( hBOOL(hAPP_f1695230391l_bool(hAPP_f1767618879l_bool(ord_le1523781517a_bool,A),hAPP_f20753329a_bool(hAPP_H1743777351a_bool(insert956547291iple_a,X_1),B)))
<=> hBOOL(hAPP_f1695230391l_bool(hAPP_f1767618879l_bool(ord_le1523781517a_bool,A),B)) ) ) ).
fof(fact_866_insert__subset,axiom,
! [X_1,A,B] :
( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),A)),B))
<=> ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),B))
& hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A),B)) ) ) ).
fof(fact_867_insert__subset,axiom,
! [X_1,A,B] :
( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),A)),B))
<=> ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),B))
& hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,A),B)) ) ) ).
fof(fact_868_insert__subset,axiom,
! [X_1,A,B] :
( hBOOL(hAPP_f1695230391l_bool(hAPP_f1767618879l_bool(ord_le1523781517a_bool,hAPP_f20753329a_bool(hAPP_H1743777351a_bool(insert956547291iple_a,X_1),A)),B))
<=> ( hBOOL(hAPP_f1695230391l_bool(hAPP_H622608077l_bool(member1797258804iple_a,X_1),B))
& hBOOL(hAPP_f1695230391l_bool(hAPP_f1767618879l_bool(ord_le1523781517a_bool,A),B)) ) ) ).
fof(fact_869_subset__insertI,axiom,
! [B,A_13] : hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,B),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_13),B))) ).
fof(fact_870_subset__insertI,axiom,
! [B,A_13] : hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,B),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,A_13),B))) ).
fof(fact_871_subset__insertI,axiom,
! [B,A_13] : hBOOL(hAPP_f1695230391l_bool(hAPP_f1767618879l_bool(ord_le1523781517a_bool,B),hAPP_f20753329a_bool(hAPP_H1743777351a_bool(insert956547291iple_a,A_13),B))) ).
fof(fact_872_image__mono,axiom,
! [F,A,B] :
( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,A),B))
=> hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,image_int_int(F,A)),image_int_int(F,B))) ) ).
fof(fact_873_image__mono,axiom,
! [F,A,B] :
( hBOOL(hAPP_f1695230391l_bool(hAPP_f1767618879l_bool(ord_le1523781517a_bool,A),B))
=> hBOOL(hAPP_f1695230391l_bool(hAPP_f1767618879l_bool(ord_le1523781517a_bool,image_533441733iple_a(F,A)),image_533441733iple_a(F,B))) ) ).
fof(fact_874_image__mono,axiom,
! [F,A,B] :
( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A),B))
=> hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,image_nat_int(F,A)),image_nat_int(F,B))) ) ).
fof(fact_875_image__mono,axiom,
! [F,A,B] :
( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A),B))
=> hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,image_nat_nat(F,A)),image_nat_nat(F,B))) ) ).
fof(fact_876_subset__image__iff,axiom,
! [B,F,A] :
( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,B),image_int_int(F,A)))
<=> ? [AA] :
( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,AA),A))
& B = image_int_int(F,AA) ) ) ).
fof(fact_877_subset__image__iff,axiom,
! [B,F,A] :
( hBOOL(hAPP_f1695230391l_bool(hAPP_f1767618879l_bool(ord_le1523781517a_bool,B),image_533441733iple_a(F,A)))
<=> ? [AA] :
( hBOOL(hAPP_f1695230391l_bool(hAPP_f1767618879l_bool(ord_le1523781517a_bool,AA),A))
& B = image_533441733iple_a(F,AA) ) ) ).
fof(fact_878_subset__image__iff,axiom,
! [B,F,A] :
( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,B),image_nat_int(F,A)))
<=> ? [AA] :
( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,AA),A))
& B = image_nat_int(F,AA) ) ) ).
fof(fact_879_subset__image__iff,axiom,
! [B,F,A] :
( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,B),image_nat_nat(F,A)))
<=> ? [AA] :
( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,AA),A))
& B = image_nat_nat(F,AA) ) ) ).
fof(fact_880_double__diff,axiom,
! [C_5,A,B] :
( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,A),B))
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,B),C_5))
=> hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,B),hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,C_5),A)) = A ) ) ).
fof(fact_881_double__diff,axiom,
! [C_5,A,B] :
( hBOOL(hAPP_f1695230391l_bool(hAPP_f1767618879l_bool(ord_le1523781517a_bool,A),B))
=> ( hBOOL(hAPP_f1695230391l_bool(hAPP_f1767618879l_bool(ord_le1523781517a_bool,B),C_5))
=> hAPP_f20753329a_bool(hAPP_f1706700729a_bool(minus_85316870a_bool,B),hAPP_f20753329a_bool(hAPP_f1706700729a_bool(minus_85316870a_bool,C_5),A)) = A ) ) ).
fof(fact_882_double__diff,axiom,
! [C_5,A,B] :
( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A),B))
=> ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,B),C_5))
=> hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,B),hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,C_5),A)) = A ) ) ).
fof(fact_883_Diff__mono,axiom,
! [D_2,B,A,C_5] :
( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,A),C_5))
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,D_2),B))
=> hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A),B)),hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,C_5),D_2))) ) ) ).
fof(fact_884_Diff__mono,axiom,
! [D_2,B,A,C_5] :
( hBOOL(hAPP_f1695230391l_bool(hAPP_f1767618879l_bool(ord_le1523781517a_bool,A),C_5))
=> ( hBOOL(hAPP_f1695230391l_bool(hAPP_f1767618879l_bool(ord_le1523781517a_bool,D_2),B))
=> hBOOL(hAPP_f1695230391l_bool(hAPP_f1767618879l_bool(ord_le1523781517a_bool,hAPP_f20753329a_bool(hAPP_f1706700729a_bool(minus_85316870a_bool,A),B)),hAPP_f20753329a_bool(hAPP_f1706700729a_bool(minus_85316870a_bool,C_5),D_2))) ) ) ).
fof(fact_885_Diff__mono,axiom,
! [D_2,B,A,C_5] :
( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A),C_5))
=> ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,D_2),B))
=> hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),B)),hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,C_5),D_2))) ) ) ).
fof(fact_886_Diff__subset,axiom,
! [A,B] : hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A),B)),A)) ).
fof(fact_887_Diff__subset,axiom,
! [A,B] : hBOOL(hAPP_f1695230391l_bool(hAPP_f1767618879l_bool(ord_le1523781517a_bool,hAPP_f20753329a_bool(hAPP_f1706700729a_bool(minus_85316870a_bool,A),B)),A)) ).
fof(fact_888_Diff__subset,axiom,
! [A,B] : hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),B)),A)) ).
fof(fact_889_weaken,axiom,
! [Ts,Ga,Ts_1] :
( hBOOL(hAPP_f1695230391l_bool(hoare_2102800559rivs_a(Ga),Ts_1))
=> ( hBOOL(hAPP_f1695230391l_bool(hAPP_f1767618879l_bool(ord_le1523781517a_bool,Ts),Ts_1))
=> hBOOL(hAPP_f1695230391l_bool(hoare_2102800559rivs_a(Ga),Ts)) ) ) ).
fof(fact_890_asm,axiom,
! [Ts,Ga] :
( hBOOL(hAPP_f1695230391l_bool(hAPP_f1767618879l_bool(ord_le1523781517a_bool,Ts),Ga))
=> hBOOL(hAPP_f1695230391l_bool(hoare_2102800559rivs_a(Ga),Ts)) ) ).
fof(fact_891_subset__singletonD,axiom,
! [A,X_1] :
( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),bot_bot_fun_nat_bool)))
=> ( A = bot_bot_fun_nat_bool
| A = hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),bot_bot_fun_nat_bool) ) ) ).
fof(fact_892_subset__singletonD,axiom,
! [A,X_1] :
( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,A),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),bot_bot_fun_int_bool)))
=> ( A = bot_bot_fun_int_bool
| A = hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),bot_bot_fun_int_bool) ) ) ).
fof(fact_893_subset__singletonD,axiom,
! [A,X_1] :
( hBOOL(hAPP_f1695230391l_bool(hAPP_f1767618879l_bool(ord_le1523781517a_bool,A),hAPP_f20753329a_bool(hAPP_H1743777351a_bool(insert956547291iple_a,X_1),bot_bo797238721a_bool)))
=> ( A = bot_bo797238721a_bool
| A = hAPP_f20753329a_bool(hAPP_H1743777351a_bool(insert956547291iple_a,X_1),bot_bo797238721a_bool) ) ) ).
fof(fact_894_finite__surj,axiom,
! [B,F,A] :
( hBOOL(hAPP_f448129468l_bool(finite_finite_int,A))
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,B),image_int_int(F,A)))
=> hBOOL(hAPP_f448129468l_bool(finite_finite_int,B)) ) ) ).
fof(fact_895_finite__surj,axiom,
! [B,F,A] :
( hBOOL(hAPP_f1695230391l_bool(finite1738664244iple_a,A))
=> ( hBOOL(hAPP_f1695230391l_bool(hAPP_f1767618879l_bool(ord_le1523781517a_bool,B),image_533441733iple_a(F,A)))
=> hBOOL(hAPP_f1695230391l_bool(finite1738664244iple_a,B)) ) ) ).
fof(fact_896_finite__surj,axiom,
! [B,F,A] :
( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,B),image_nat_int(F,A)))
=> hBOOL(hAPP_f448129468l_bool(finite_finite_int,B)) ) ) ).
fof(fact_897_finite__surj,axiom,
! [B,F,A] :
( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
=> ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,B),image_nat_nat(F,A)))
=> hBOOL(hAPP_f54304608l_bool(finite_finite_nat,B)) ) ) ).
fof(fact_898_image__diff__subset,axiom,
! [F,A,B] : hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,image_int_int(F,A)),image_int_int(F,B))),image_int_int(F,hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A),B)))) ).
fof(fact_899_image__diff__subset,axiom,
! [F,A,B] : hBOOL(hAPP_f1695230391l_bool(hAPP_f1767618879l_bool(ord_le1523781517a_bool,hAPP_f20753329a_bool(hAPP_f1706700729a_bool(minus_85316870a_bool,image_533441733iple_a(F,A)),image_533441733iple_a(F,B))),image_533441733iple_a(F,hAPP_f20753329a_bool(hAPP_f1706700729a_bool(minus_85316870a_bool,A),B)))) ).
fof(fact_900_image__diff__subset,axiom,
! [F,A,B] : hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,image_nat_int(F,A)),image_nat_int(F,B))),image_nat_int(F,hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),B)))) ).
fof(fact_901_image__diff__subset,axiom,
! [F,A,B] : hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,image_nat_nat(F,A)),image_nat_nat(F,B))),image_nat_nat(F,hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),B)))) ).
fof(fact_902_le__add__iff1,axiom,
! [A_13,E,Ca,Ba,D_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_13),E)),Ca)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Ba),E)),D_1)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,A_13),Ba)),E)),Ca)),D_1)) ) ).
fof(fact_903_le__add__iff2,axiom,
! [A_13,E,Ca,Ba,D_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_13),E)),Ca)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Ba),E)),D_1)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Ca),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Ba),A_13)),E)),D_1))) ) ).
fof(fact_904_diff__single__insert,axiom,
! [A,X_1,B] :
( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),bot_bot_fun_nat_bool))),B))
=> ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A))
=> hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),B))) ) ) ).
fof(fact_905_diff__single__insert,axiom,
! [A,X_1,B] :
( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),bot_bot_fun_int_bool))),B))
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),A))
=> hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,A),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),B))) ) ) ).
fof(fact_906_diff__single__insert,axiom,
! [A,X_1,B] :
( hBOOL(hAPP_f1695230391l_bool(hAPP_f1767618879l_bool(ord_le1523781517a_bool,hAPP_f20753329a_bool(hAPP_f1706700729a_bool(minus_85316870a_bool,A),hAPP_f20753329a_bool(hAPP_H1743777351a_bool(insert956547291iple_a,X_1),bot_bo797238721a_bool))),B))
=> ( hBOOL(hAPP_f1695230391l_bool(hAPP_H622608077l_bool(member1797258804iple_a,X_1),A))
=> hBOOL(hAPP_f1695230391l_bool(hAPP_f1767618879l_bool(ord_le1523781517a_bool,A),hAPP_f20753329a_bool(hAPP_H1743777351a_bool(insert956547291iple_a,X_1),B))) ) ) ).
fof(fact_907_subset__insert__iff,axiom,
! [A,X_1,B] :
( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),B)))
<=> ( ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A))
=> hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),bot_bot_fun_nat_bool))),B)) )
& ( ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A))
=> hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A),B)) ) ) ) ).
fof(fact_908_subset__insert__iff,axiom,
! [A,X_1,B] :
( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,A),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),B)))
<=> ( ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),A))
=> hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),bot_bot_fun_int_bool))),B)) )
& ( ~ hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),A))
=> hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,A),B)) ) ) ) ).
fof(fact_909_subset__insert__iff,axiom,
! [A,X_1,B] :
( hBOOL(hAPP_f1695230391l_bool(hAPP_f1767618879l_bool(ord_le1523781517a_bool,A),hAPP_f20753329a_bool(hAPP_H1743777351a_bool(insert956547291iple_a,X_1),B)))
<=> ( ( hBOOL(hAPP_f1695230391l_bool(hAPP_H622608077l_bool(member1797258804iple_a,X_1),A))
=> hBOOL(hAPP_f1695230391l_bool(hAPP_f1767618879l_bool(ord_le1523781517a_bool,hAPP_f20753329a_bool(hAPP_f1706700729a_bool(minus_85316870a_bool,A),hAPP_f20753329a_bool(hAPP_H1743777351a_bool(insert956547291iple_a,X_1),bot_bo797238721a_bool))),B)) )
& ( ~ hBOOL(hAPP_f1695230391l_bool(hAPP_H622608077l_bool(member1797258804iple_a,X_1),A))
=> hBOOL(hAPP_f1695230391l_bool(hAPP_f1767618879l_bool(ord_le1523781517a_bool,A),B)) ) ) ) ).
fof(fact_910_folding__one__idem_Osubset__idem,axiom,
! [B,A,F,F_1] :
( hBOOL(finite1033474011iple_a(F,F_1))
=> ( hBOOL(hAPP_f1695230391l_bool(finite1738664244iple_a,A))
=> ( B != bot_bo797238721a_bool
=> ( hBOOL(hAPP_f1695230391l_bool(hAPP_f1767618879l_bool(ord_le1523781517a_bool,B),A))
=> hAPP_H905846293iple_a(hAPP_H2085992369iple_a(F,hAPP_f1693662087iple_a(F_1,B)),hAPP_f1693662087iple_a(F_1,A)) = hAPP_f1693662087iple_a(F_1,A) ) ) ) ) ).
fof(fact_911_folding__one__idem_Osubset__idem,axiom,
! [B,A,F,F_1] :
( hBOOL(finite1432773856em_int(F,F_1))
=> ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,A))
=> ( B != bot_bot_fun_int_bool
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,B),A))
=> hAPP_int_int(hAPP_int_fun_int_int(F,hAPP_f1594865479ol_int(F_1,B)),hAPP_f1594865479ol_int(F_1,A)) = hAPP_f1594865479ol_int(F_1,A) ) ) ) ) ).
fof(fact_912_folding__one__idem_Osubset__idem,axiom,
! [B,A,F,F_1] :
( hBOOL(finite795500164em_nat(F,F_1))
=> ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
=> ( B != bot_bot_fun_nat_bool
=> ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,B),A))
=> hAPP_nat_nat(hAPP_nat_fun_nat_nat(F,hAPP_f22106695ol_nat(F_1,B)),hAPP_f22106695ol_nat(F_1,A)) = hAPP_f22106695ol_nat(F_1,A) ) ) ) ) ).
fof(fact_913_finite__subset__induct,axiom,
! [Pa,A,F_1] :
( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,F_1))
=> ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,F_1),A))
=> ( hBOOL(hAPP_f54304608l_bool(Pa,bot_bot_fun_nat_bool))
=> ( ! [A_12,F_2] :
( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,F_2))
=> ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_12),A))
=> ( ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_12),F_2))
=> ( hBOOL(hAPP_f54304608l_bool(Pa,F_2))
=> hBOOL(hAPP_f54304608l_bool(Pa,hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_12),F_2))) ) ) ) )
=> hBOOL(hAPP_f54304608l_bool(Pa,F_1)) ) ) ) ) ).
fof(fact_914_finite__subset__induct,axiom,
! [Pa,A,F_1] :
( hBOOL(hAPP_f448129468l_bool(finite_finite_int,F_1))
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,F_1),A))
=> ( hBOOL(hAPP_f448129468l_bool(Pa,bot_bot_fun_int_bool))
=> ( ! [A_12,F_2] :
( hBOOL(hAPP_f448129468l_bool(finite_finite_int,F_2))
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,A_12),A))
=> ( ~ hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,A_12),F_2))
=> ( hBOOL(hAPP_f448129468l_bool(Pa,F_2))
=> hBOOL(hAPP_f448129468l_bool(Pa,hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,A_12),F_2))) ) ) ) )
=> hBOOL(hAPP_f448129468l_bool(Pa,F_1)) ) ) ) ) ).
fof(fact_915_finite__subset__induct,axiom,
! [Pa,A,F_1] :
( hBOOL(hAPP_f1695230391l_bool(finite1738664244iple_a,F_1))
=> ( hBOOL(hAPP_f1695230391l_bool(hAPP_f1767618879l_bool(ord_le1523781517a_bool,F_1),A))
=> ( hBOOL(hAPP_f1695230391l_bool(Pa,bot_bo797238721a_bool))
=> ( ! [A_12,F_2] :
( hBOOL(hAPP_f1695230391l_bool(finite1738664244iple_a,F_2))
=> ( hBOOL(hAPP_f1695230391l_bool(hAPP_H622608077l_bool(member1797258804iple_a,A_12),A))
=> ( ~ hBOOL(hAPP_f1695230391l_bool(hAPP_H622608077l_bool(member1797258804iple_a,A_12),F_2))
=> ( hBOOL(hAPP_f1695230391l_bool(Pa,F_2))
=> hBOOL(hAPP_f1695230391l_bool(Pa,hAPP_f20753329a_bool(hAPP_H1743777351a_bool(insert956547291iple_a,A_12),F_2))) ) ) ) )
=> hBOOL(hAPP_f1695230391l_bool(Pa,F_1)) ) ) ) ) ).
fof(fact_916_subsetI,axiom,
! [B,A] :
( ! [X_2] :
( hBOOL(hAPP_f1695230391l_bool(hAPP_H622608077l_bool(member1797258804iple_a,X_2),A))
=> hBOOL(hAPP_f1695230391l_bool(hAPP_H622608077l_bool(member1797258804iple_a,X_2),B)) )
=> hBOOL(hAPP_f1695230391l_bool(hAPP_f1767618879l_bool(ord_le1523781517a_bool,A),B)) ) ).
fof(fact_917_subsetI,axiom,
! [B,A] :
( ! [X_2] :
( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_2),A))
=> hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_2),B)) )
=> hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,A),B)) ) ).
fof(fact_918_subsetI,axiom,
! [B,A] :
( ! [X_2] :
( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_2),A))
=> hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_2),B)) )
=> hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A),B)) ) ).
fof(fact_919_finite__subset__image,axiom,
! [F,A,B] :
( hBOOL(hAPP_f448129468l_bool(finite_finite_int,B))
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,B),image_int_int(F,A)))
=> ? [C_4] :
( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,C_4),A))
& hBOOL(hAPP_f448129468l_bool(finite_finite_int,C_4))
& B = image_int_int(F,C_4) ) ) ) ).
fof(fact_920_finite__subset__image,axiom,
! [F,A,B] :
( hBOOL(hAPP_f1695230391l_bool(finite1738664244iple_a,B))
=> ( hBOOL(hAPP_f1695230391l_bool(hAPP_f1767618879l_bool(ord_le1523781517a_bool,B),image_533441733iple_a(F,A)))
=> ? [C_4] :
( hBOOL(hAPP_f1695230391l_bool(hAPP_f1767618879l_bool(ord_le1523781517a_bool,C_4),A))
& hBOOL(hAPP_f1695230391l_bool(finite1738664244iple_a,C_4))
& B = image_533441733iple_a(F,C_4) ) ) ) ).
fof(fact_921_finite__subset__image,axiom,
! [F,A,B] :
( hBOOL(hAPP_f448129468l_bool(finite_finite_int,B))
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,B),image_nat_int(F,A)))
=> ? [C_4] :
( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,C_4),A))
& hBOOL(hAPP_f54304608l_bool(finite_finite_nat,C_4))
& B = image_nat_int(F,C_4) ) ) ) ).
fof(fact_922_finite__subset__image,axiom,
! [F,A,B] :
( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,B))
=> ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,B),image_nat_nat(F,A)))
=> ? [C_4] :
( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,C_4),A))
& hBOOL(hAPP_f54304608l_bool(finite_finite_nat,C_4))
& B = image_nat_nat(F,C_4) ) ) ) ).
fof(fact_923_setsum__mono,axiom,
! [F,G,K_3] :
( ! [I_2] :
( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,I_2),K_3))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_int_nat(F,I_2)),hAPP_int_nat(G,I_2))) )
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_f957591787ol_nat(hAPP_f1109019371ol_nat(big_co1740723097nt_nat,F),K_3)),hAPP_f957591787ol_nat(hAPP_f1109019371ol_nat(big_co1740723097nt_nat,G),K_3))) ) ).
fof(fact_924_setsum__mono,axiom,
! [F,G,K_3] :
( ! [I_2] :
( hBOOL(hAPP_f1695230391l_bool(hAPP_H622608077l_bool(member1797258804iple_a,I_2),K_3))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_H592031934_a_nat(F,I_2)),hAPP_H592031934_a_nat(G,I_2))) )
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_f2097660464ol_nat(hAPP_f1371399797ol_nat(big_co1340561246_a_nat,F),K_3)),hAPP_f2097660464ol_nat(hAPP_f1371399797ol_nat(big_co1340561246_a_nat,G),K_3))) ) ).
fof(fact_925_setsum__mono,axiom,
! [F,G,K_3] :
( ! [I_2] :
( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,I_2),K_3))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(F,I_2)),hAPP_nat_nat(G,I_2))) )
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_f22106695ol_nat(hAPP_f782000547ol_nat(big_co387207925at_nat,F),K_3)),hAPP_f22106695ol_nat(hAPP_f782000547ol_nat(big_co387207925at_nat,G),K_3))) ) ).
fof(fact_926_image__subsetI,axiom,
! [F,B,A] :
( ! [X_2] :
( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_2),A))
=> hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hAPP_int_int(F,X_2)),B)) )
=> hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,image_int_int(F,A)),B)) ) ).
fof(fact_927_image__subsetI,axiom,
! [F,B,A] :
( ! [X_2] :
( hBOOL(hAPP_f1695230391l_bool(hAPP_H622608077l_bool(member1797258804iple_a,X_2),A))
=> hBOOL(hAPP_f1695230391l_bool(hAPP_H622608077l_bool(member1797258804iple_a,hAPP_H905846293iple_a(F,X_2)),B)) )
=> hBOOL(hAPP_f1695230391l_bool(hAPP_f1767618879l_bool(ord_le1523781517a_bool,image_533441733iple_a(F,A)),B)) ) ).
fof(fact_928_image__subsetI,axiom,
! [F,B,A] :
( ! [X_2] :
( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_2),A))
=> hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hAPP_nat_int(F,X_2)),B)) )
=> hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,image_nat_int(F,A)),B)) ) ).
fof(fact_929_image__subsetI,axiom,
! [F,B,A] :
( ! [X_2] :
( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_2),A))
=> hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,hAPP_nat_nat(F,X_2)),B)) )
=> hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,image_nat_nat(F,A)),B)) ) ).
fof(fact_930_le__funI,axiom,
! [F,G] :
( ! [X_2] : hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(ord_less_eq_bool,hAPP_int_bool(F,X_2)),hAPP_int_bool(G,X_2)))
=> hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,F),G)) ) ).
fof(fact_931_le__funI,axiom,
! [F,G] :
( ! [X_2] : hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(ord_less_eq_bool,hAPP_nat_bool(F,X_2)),hAPP_nat_bool(G,X_2)))
=> hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,F),G)) ) ).
fof(fact_932_flat__lub__def,axiom,
! [A,Ba] :
( ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,A),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,Ba),bot_bot_fun_int_bool)))
=> hAPP_f1594865479ol_int(partial_flat_lub_int(Ba),A) = Ba )
& ( ~ hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,A),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,Ba),bot_bot_fun_int_bool)))
=> hAPP_f1594865479ol_int(partial_flat_lub_int(Ba),A) = hAPP_f1594865479ol_int(the_int,hAPP_f1805168059t_bool(hAPP_f202917053t_bool(cOMBC_94739984l_bool,member_int),hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,Ba),bot_bot_fun_int_bool)))) ) ) ).
fof(fact_933_flat__lub__def,axiom,
! [A,Ba] :
( ( hBOOL(hAPP_f1695230391l_bool(hAPP_f1767618879l_bool(ord_le1523781517a_bool,A),hAPP_f20753329a_bool(hAPP_H1743777351a_bool(insert956547291iple_a,Ba),bot_bo797238721a_bool)))
=> hAPP_f1693662087iple_a(partia1949573335iple_a(Ba),A) = Ba )
& ( ~ hBOOL(hAPP_f1695230391l_bool(hAPP_f1767618879l_bool(ord_le1523781517a_bool,A),hAPP_f20753329a_bool(hAPP_H1743777351a_bool(insert956547291iple_a,Ba),bot_bo797238721a_bool)))
=> hAPP_f1693662087iple_a(partia1949573335iple_a(Ba),A) = hAPP_f1693662087iple_a(the_Ho830102290iple_a,hAPP_f20753329a_bool(hAPP_f247708275a_bool(cOMBC_1519416966l_bool,member1797258804iple_a),hAPP_f20753329a_bool(hAPP_f1706700729a_bool(minus_85316870a_bool,A),hAPP_f20753329a_bool(hAPP_H1743777351a_bool(insert956547291iple_a,Ba),bot_bo797238721a_bool)))) ) ) ).
fof(fact_934_flat__lub__def,axiom,
! [A,Ba] :
( ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,Ba),bot_bot_fun_nat_bool)))
=> hAPP_f22106695ol_nat(partial_flat_lub_nat(Ba),A) = Ba )
& ( ~ hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,Ba),bot_bot_fun_nat_bool)))
=> hAPP_f22106695ol_nat(partial_flat_lub_nat(Ba),A) = hAPP_f22106695ol_nat(the_nat,hAPP_f800510211t_bool(hAPP_f1722879237t_bool(cOMBC_226598744l_bool,member_nat),hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,Ba),bot_bot_fun_nat_bool)))) ) ) ).
fof(fact_935_sum__diff__distrib,axiom,
! [N_1,Q_1,Pa] :
( ! [X_2] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(Q_1,X_2)),hAPP_nat_nat(Pa,X_2)))
=> hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_f22106695ol_nat(hAPP_f782000547ol_nat(big_co387207925at_nat,Pa),hAPP_n1699378549t_bool(ord_lessThan_nat,N_1))),hAPP_f22106695ol_nat(hAPP_f782000547ol_nat(big_co387207925at_nat,Q_1),hAPP_n1699378549t_bool(ord_lessThan_nat,N_1))) = hAPP_f22106695ol_nat(hAPP_f782000547ol_nat(big_co387207925at_nat,hAPP_f1914919701at_nat(hAPP_f1408247010at_nat(cOMBS_nat_nat_nat,hAPP_f1639111240at_nat(hAPP_f1731313045at_nat(cOMBB_963856155at_nat,minus_minus_nat),Pa)),Q_1)),hAPP_n1699378549t_bool(ord_lessThan_nat,N_1)) ) ).
fof(fact_936_fold1__antimono,axiom,
! [B,A] :
( A != bot_bot_fun_int_bool
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,A),B))
=> ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,B))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_f1594865479ol_int(finite_fold1_int(ord_min_int),B)),hAPP_f1594865479ol_int(finite_fold1_int(ord_min_int),A))) ) ) ) ).
fof(fact_937_fold1__antimono,axiom,
! [B,A] :
( A != bot_bot_fun_nat_bool
=> ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A),B))
=> ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,B))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_f22106695ol_nat(finite_fold1_nat(ord_min_nat),B)),hAPP_f22106695ol_nat(finite_fold1_nat(ord_min_nat),A))) ) ) ) ).
fof(fact_938_predicate1I,axiom,
! [Q_1,Pa] :
( ! [X_2] :
( hBOOL(hAPP_int_bool(Pa,X_2))
=> hBOOL(hAPP_int_bool(Q_1,X_2)) )
=> hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,Pa),Q_1)) ) ).
fof(fact_939_predicate1I,axiom,
! [Q_1,Pa] :
( ! [X_2] :
( hBOOL(hAPP_nat_bool(Pa,X_2))
=> hBOOL(hAPP_nat_bool(Q_1,X_2)) )
=> hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,Pa),Q_1)) ) ).
fof(fact_940_Collect__mono,axiom,
! [Q_1,Pa] :
( ! [X_2] :
( hBOOL(hAPP_int_bool(Pa,X_2))
=> hBOOL(hAPP_int_bool(Q_1,X_2)) )
=> hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,hAPP_f1805168059t_bool(collect_int,Pa)),hAPP_f1805168059t_bool(collect_int,Q_1))) ) ).
fof(fact_941_Collect__mono,axiom,
! [Q_1,Pa] :
( ! [X_2] :
( hBOOL(hAPP_H1037229737a_bool(Pa,X_2))
=> hBOOL(hAPP_H1037229737a_bool(Q_1,X_2)) )
=> hBOOL(hAPP_f1695230391l_bool(hAPP_f1767618879l_bool(ord_le1523781517a_bool,hAPP_f20753329a_bool(collec351493750iple_a,Pa)),hAPP_f20753329a_bool(collec351493750iple_a,Q_1))) ) ).
fof(fact_942_Collect__mono,axiom,
! [Q_1,Pa] :
( ! [X_2] :
( hBOOL(hAPP_nat_bool(Pa,X_2))
=> hBOOL(hAPP_nat_bool(Q_1,X_2)) )
=> hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,hAPP_f800510211t_bool(collect_nat,Pa)),hAPP_f800510211t_bool(collect_nat,Q_1))) ) ).
fof(fact_943_Min__antimono,axiom,
! [M_3,N_3] :
( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,M_3),N_3))
=> ( M_3 != bot_bot_fun_int_bool
=> ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,N_3))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_f1594865479ol_int(big_linorder_Min_int,N_3)),hAPP_f1594865479ol_int(big_linorder_Min_int,M_3))) ) ) ) ).
fof(fact_944_Min__antimono,axiom,
! [M_3,N_3] :
( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,M_3),N_3))
=> ( M_3 != bot_bot_fun_nat_bool
=> ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,N_3))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_f22106695ol_nat(big_linorder_Min_nat,N_3)),hAPP_f22106695ol_nat(big_linorder_Min_nat,M_3))) ) ) ) ).
fof(fact_945_Max__mono,axiom,
! [M_3,N_3] :
( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,M_3),N_3))
=> ( M_3 != bot_bot_fun_int_bool
=> ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,N_3))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_f1594865479ol_int(big_linorder_Max_int,M_3)),hAPP_f1594865479ol_int(big_linorder_Max_int,N_3))) ) ) ) ).
fof(fact_946_Max__mono,axiom,
! [M_3,N_3] :
( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,M_3),N_3))
=> ( M_3 != bot_bot_fun_nat_bool
=> ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,N_3))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_f22106695ol_nat(big_linorder_Max_nat,M_3)),hAPP_f22106695ol_nat(big_linorder_Max_nat,N_3))) ) ) ) ).
fof(fact_947_finite__lessThan,axiom,
! [K_1] : hBOOL(hAPP_f54304608l_bool(finite_finite_nat,hAPP_n1699378549t_bool(ord_lessThan_nat,K_1))) ).
fof(fact_948_finite__Collect__le__nat,axiom,
! [K_1] : hBOOL(hAPP_f54304608l_bool(finite_finite_nat,hAPP_f800510211t_bool(collect_nat,hAPP_n1699378549t_bool(hAPP_f229349961t_bool(cOMBC_nat_nat_bool,ord_less_eq_nat),K_1)))) ).
fof(fact_949_finite__nat__set__iff__bounded__le,axiom,
! [N_3] :
( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,N_3))
<=> ? [M_1] :
! [X_2] :
( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_2),N_3))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_2),M_1)) ) ) ).
fof(fact_950_Min_Oin__idem,axiom,
! [X_1,A] :
( hBOOL(hAPP_f448129468l_bool(finite_finite_int,A))
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),A))
=> hAPP_int_int(hAPP_int_fun_int_int(ord_min_int,X_1),hAPP_f1594865479ol_int(big_linorder_Min_int,A)) = hAPP_f1594865479ol_int(big_linorder_Min_int,A) ) ) ).
fof(fact_951_Min_Oin__idem,axiom,
! [X_1,A] :
( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
=> ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A))
=> hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,X_1),hAPP_f22106695ol_nat(big_linorder_Min_nat,A)) = hAPP_f22106695ol_nat(big_linorder_Min_nat,A) ) ) ).
fof(fact_952_Min_Oidem,axiom,
! [X_18] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,X_18),X_18) = X_18 ).
fof(fact_953_min__max_Oinf__assoc,axiom,
! [X_17,Y_12,Z_7] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,X_17),Y_12)),Z_7) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,X_17),hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,Y_12),Z_7)) ).
fof(fact_954_min__max_Oinf_Oassoc,axiom,
! [A_11,B_10,C_3] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,A_11),B_10)),C_3) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,A_11),hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,B_10),C_3)) ).
fof(fact_955_min__max_Oinf__left__commute,axiom,
! [X_16,Y_11,Z_6] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,X_16),hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,Y_11),Z_6)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,Y_11),hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,X_16),Z_6)) ).
fof(fact_956_min__max_Oinf_Oleft__commute,axiom,
! [B_9,A_10,C_2] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,B_9),hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,A_10),C_2)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,A_10),hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,B_9),C_2)) ).
fof(fact_957_min__max_Oinf__left__idem,axiom,
! [X_15,Y_10] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,X_15),hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,X_15),Y_10)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,X_15),Y_10) ).
fof(fact_958_min__max_Oinf_Oleft__idem,axiom,
! [A_9,B_8] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,A_9),hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,A_9),B_8)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,A_9),B_8) ).
fof(fact_959_min__max_Oinf__commute,axiom,
! [X_14,Y_9] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,X_14),Y_9) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,Y_9),X_14) ).
fof(fact_960_min__max_Oinf_Ocommute,axiom,
! [A_8,B_7] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,A_8),B_7) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,B_7),A_8) ).
fof(fact_961_min__max_Oinf_Oidem,axiom,
! [A_7] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,A_7),A_7) = A_7 ).
fof(fact_962_min__le__iff__disj,axiom,
! [X_1,Y_1,Z_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,X_1),Y_1)),Z_1))
<=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_1),Z_1))
| hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Y_1),Z_1)) ) ) ).
fof(fact_963_min__le__iff__disj,axiom,
! [X_1,Y_1,Z_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(ord_min_int,X_1),Y_1)),Z_1))
<=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_1),Z_1))
| hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y_1),Z_1)) ) ) ).
fof(fact_964_min__max_Ole__infE,axiom,
! [X_13,A_6,B_6] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_13),hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,A_6),B_6)))
=> ~ ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_13),A_6))
=> ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_13),B_6)) ) ) ).
fof(fact_965_min__max_Ole__infE,axiom,
! [X_13,A_6,B_6] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_13),hAPP_int_int(hAPP_int_fun_int_int(ord_min_int,A_6),B_6)))
=> ~ ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_13),A_6))
=> ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_13),B_6)) ) ) ).
fof(fact_966_min__max_Oinf__mono,axiom,
! [B_5,D,A_5,C_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_5),C_1))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,B_5),D))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,A_5),B_5)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,C_1),D))) ) ) ).
fof(fact_967_min__max_Oinf__mono,axiom,
! [B_5,D,A_5,C_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_5),C_1))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_5),D))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(ord_min_int,A_5),B_5)),hAPP_int_int(hAPP_int_fun_int_int(ord_min_int,C_1),D))) ) ) ).
fof(fact_968_min__max_Oinf__greatest,axiom,
! [Z_5,X_12,Y_8] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_12),Y_8))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_12),Z_5))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_12),hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,Y_8),Z_5))) ) ) ).
fof(fact_969_min__max_Oinf__greatest,axiom,
! [Z_5,X_12,Y_8] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_12),Y_8))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_12),Z_5))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_12),hAPP_int_int(hAPP_int_fun_int_int(ord_min_int,Y_8),Z_5))) ) ) ).
fof(fact_970_min__max_Ole__infI,axiom,
! [B_4,X_11,A_4] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_11),A_4))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_11),B_4))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_11),hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,A_4),B_4))) ) ) ).
fof(fact_971_min__max_Ole__infI,axiom,
! [B_4,X_11,A_4] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_11),A_4))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_11),B_4))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_11),hAPP_int_int(hAPP_int_fun_int_int(ord_min_int,A_4),B_4))) ) ) ).
fof(fact_972_min__max_Oinf__absorb2,axiom,
! [Y_7,X_10] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Y_7),X_10))
=> hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,X_10),Y_7) = Y_7 ) ).
fof(fact_973_min__max_Oinf__absorb2,axiom,
! [Y_7,X_10] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y_7),X_10))
=> hAPP_int_int(hAPP_int_fun_int_int(ord_min_int,X_10),Y_7) = Y_7 ) ).
fof(fact_974_min__max_Oinf__absorb1,axiom,
! [X_9,Y_6] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_9),Y_6))
=> hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,X_9),Y_6) = X_9 ) ).
fof(fact_975_min__max_Oinf__absorb1,axiom,
! [X_9,Y_6] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_9),Y_6))
=> hAPP_int_int(hAPP_int_fun_int_int(ord_min_int,X_9),Y_6) = X_9 ) ).
fof(fact_976_min__max_Ole__infI2,axiom,
! [A_3,B_3,X_8] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,B_3),X_8))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,A_3),B_3)),X_8)) ) ).
fof(fact_977_min__max_Ole__infI2,axiom,
! [A_3,B_3,X_8] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_3),X_8))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(ord_min_int,A_3),B_3)),X_8)) ) ).
fof(fact_978_min__max_Ole__infI1,axiom,
! [B_2,A_2,X_7] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_2),X_7))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,A_2),B_2)),X_7)) ) ).
fof(fact_979_min__max_Ole__infI1,axiom,
! [B_2,A_2,X_7] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_2),X_7))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(ord_min_int,A_2),B_2)),X_7)) ) ).
fof(fact_980_min__max_Ole__inf__iff,axiom,
! [X_1,Y_1,Z_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,Y_1),Z_1)))
<=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_1),Y_1))
& hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_1),Z_1)) ) ) ).
fof(fact_981_min__max_Ole__inf__iff,axiom,
! [X_1,Y_1,Z_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_1),hAPP_int_int(hAPP_int_fun_int_int(ord_min_int,Y_1),Z_1)))
<=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_1),Y_1))
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_1),Z_1)) ) ) ).
fof(fact_982_min__max_Ole__iff__inf,axiom,
! [X_1,Y_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_1),Y_1))
<=> hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,X_1),Y_1) = X_1 ) ).
fof(fact_983_min__max_Ole__iff__inf,axiom,
! [X_1,Y_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_1),Y_1))
<=> hAPP_int_int(hAPP_int_fun_int_int(ord_min_int,X_1),Y_1) = X_1 ) ).
fof(fact_984_min__max_Oinf__le2,axiom,
! [X_6,Y_5] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,X_6),Y_5)),Y_5)) ).
fof(fact_985_min__max_Oinf__le2,axiom,
! [X_6,Y_5] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(ord_min_int,X_6),Y_5)),Y_5)) ).
fof(fact_986_min__max_Oinf__le1,axiom,
! [X_5,Y_4] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,X_5),Y_4)),X_5)) ).
fof(fact_987_min__max_Oinf__le1,axiom,
! [X_5,Y_4] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(ord_min_int,X_5),Y_4)),X_5)) ).
fof(fact_988_min__add__distrib__left,axiom,
! [X_4,Y_3,Z_4] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(ord_min_int,X_4),Y_3)),Z_4) = hAPP_int_int(hAPP_int_fun_int_int(ord_min_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X_4),Z_4)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Y_3),Z_4)) ).
fof(fact_989_min__add__distrib__left,axiom,
! [X_4,Y_3,Z_4] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,X_4),Y_3)),Z_4) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,X_4),Z_4)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Y_3),Z_4)) ).
fof(fact_990_min__diff__distrib__left,axiom,
! [X_3,Y_2,Z_3] : hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_int_int(hAPP_int_fun_int_int(ord_min_int,X_3),Y_2)),Z_3) = hAPP_int_int(hAPP_int_fun_int_int(ord_min_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X_3),Z_3)),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Y_2),Z_3)) ).
fof(fact_991_finite__less__ub,axiom,
! [U,F] :
( ! [N_2] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_2),hAPP_nat_nat(F,N_2)))
=> hBOOL(hAPP_f54304608l_bool(finite_finite_nat,hAPP_f800510211t_bool(collect_nat,hAPP_n1699378549t_bool(hAPP_f229349961t_bool(cOMBC_nat_nat_bool,hAPP_f618557131t_bool(hAPP_f1505651103t_bool(cOMBB_800536526ol_nat,ord_less_eq_nat),F)),U)))) ) ).
fof(fact_992_le__diff__iff,axiom,
! [N_1,K_1,M_2] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_1),M_2))
=> ( 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,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M_2),K_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_1),K_1)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_2),N_1)) ) ) ) ).
fof(fact_993_mult__le__mono,axiom,
! [K,L_1,I,J] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I),J))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K),L_1))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,I),K)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,J),L_1))) ) ) ).
fof(fact_994_mult__le__mono2,axiom,
! [K,I,J] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I),J))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K),I)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K),J))) ) ).
fof(fact_995_mult__le__mono1,axiom,
! [K,I,J] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I),J))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,I),K)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,J),K))) ) ).
fof(fact_996_le__cube,axiom,
! [M] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M),M)))) ).
fof(fact_997_le__square,axiom,
! [M] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M),M))) ).
fof(fact_998_le__refl,axiom,
! [N] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N),N)) ).
fof(fact_999_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_1000_eq__imp__le,axiom,
! [M,N] :
( M = N
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N)) ) ).
fof(fact_1001_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_1002_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_1003_add__leE,axiom,
! [M,K,N] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),K)),N))
=> ~ ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N))
=> ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K),N)) ) ) ).
fof(fact_1004_add__leD1,axiom,
! [M,K,N] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),K)),N))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N)) ) ).
fof(fact_1005_add__leD2,axiom,
! [M,K,N] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),K)),N))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K),N)) ) ).
fof(fact_1006_add__le__mono,axiom,
! [K,L_1,I,J] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I),J))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K),L_1))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I),K)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,J),L_1))) ) ) ).
fof(fact_1007_add__le__mono1,axiom,
! [K,I,J] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I),J))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I),K)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,J),K))) ) ).
fof(fact_1008_trans__le__add2,axiom,
! [M,I,J] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I),J))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),J))) ) ).
fof(fact_1009_trans__le__add1,axiom,
! [M,I,J] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I),J))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,J),M))) ) ).
fof(fact_1010_nat__add__left__cancel__le,axiom,
! [K_1,M_2,N_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,K_1),M_2)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,K_1),N_1)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_2),N_1)) ) ).
fof(fact_1011_le__iff__add,axiom,
! [M_2,N_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_2),N_1))
<=> ? [K_2] : N_1 = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_2),K_2) ) ).
fof(fact_1012_le__add1,axiom,
! [N,M] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N),M))) ).
fof(fact_1013_le__add2,axiom,
! [N,M] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),N))) ).
fof(fact_1014_min__diff,axiom,
! [M,I,N] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),I)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N),I)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,M),N)),I) ).
fof(fact_1015_diff__mult__distrib2,axiom,
! [K,M,N] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),N)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K),M)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K),N)) ).
fof(fact_1016_diff__add__inverse2,axiom,
! [M,N] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),N)),N) = M ).
fof(fact_1017_diff__add__inverse,axiom,
! [N,M] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N),M)),N) = M ).
fof(fact_1018_diff__diff__left,axiom,
! [I,J,K] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,I),J)),K) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,I),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,J),K)) ).
fof(fact_1019_diff__commute,axiom,
! [I,J,K] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,I),J)),K) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,I),K)),J) ).
fof(fact_1020_diff__mult__distrib,axiom,
! [M,N,K] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),N)),K) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M),K)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,N),K)) ).
fof(fact_1021_diff__cancel,axiom,
! [K,M,N] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,K),M)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,K),N)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),N) ).
fof(fact_1022_diff__cancel2,axiom,
! [M,K,N] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),K)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N),K)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),N) ).
fof(fact_1023_diff__le__self,axiom,
! [M,N] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),N)),M)) ).
fof(fact_1024_diff__diff__right,axiom,
! [I,K,J] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K),J))
=> hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,I),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,J),K)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I),K)),J) ) ).
fof(fact_1025_le__diff__conv,axiom,
! [J_1,K_1,I_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,J_1),K_1)),I_1))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,J_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I_1),K_1))) ) ).
fof(fact_1026_le__add__diff,axiom,
! [M,K,N] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K),N))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N),M)),K))) ) ).
fof(fact_1027_diff__le__mono2,axiom,
! [L_1,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,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,L_1),N)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,L_1),M))) ) ).
fof(fact_1028_diff__le__mono,axiom,
! [L_1,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,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),L_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N),L_1))) ) ).
fof(fact_1029_diff__diff__cancel,axiom,
! [I,N] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I),N))
=> hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N),I)) = I ) ).
fof(fact_1030_le__add__diff__inverse,axiom,
! [N,M] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N),M))
=> hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),N)) = M ) ).
fof(fact_1031_add__diff__assoc,axiom,
! [I,K,J] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K),J))
=> hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,J),K)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I),J)),K) ) ).
fof(fact_1032_le__diff__conv2,axiom,
! [I_1,K_1,J_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_1),J_1))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,J_1),K_1)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I_1),K_1)),J_1)) ) ) ).
fof(fact_1033_le__add__diff__inverse2,axiom,
! [N,M] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N),M))
=> hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),N)),N) = M ) ).
fof(fact_1034_le__imp__diff__is__add,axiom,
! [K_1,I_1,J_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_1),J_1))
=> ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,J_1),I_1) = K_1
<=> J_1 = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,K_1),I_1) ) ) ).
fof(fact_1035_diff__add__assoc,axiom,
! [I,K,J] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K),J))
=> hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I),J)),K) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,J),K)) ) ).
fof(fact_1036_add__diff__assoc2,axiom,
! [I,K,J] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K),J))
=> hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,J),K)),I) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,J),I)),K) ) ).
fof(fact_1037_diff__add__assoc2,axiom,
! [I,K,J] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K),J))
=> hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,J),I)),K) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,J),K)),I) ) ).
fof(fact_1038_eq__diff__iff,axiom,
! [N_1,K_1,M_2] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_1),M_2))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_1),N_1))
=> ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M_2),K_1) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_1),K_1)
<=> M_2 = N_1 ) ) ) ).
fof(fact_1039_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))
=> hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),K)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N),K)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),N) ) ) ).
fof(fact_1040_nat__eq__add__iff2,axiom,
! [U,M_2,N_1,I_1,J_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_1),J_1))
=> ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,I_1),U)),M_2) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,J_1),U)),N_1)
<=> M_2 = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,J_1),I_1)),U)),N_1) ) ) ).
fof(fact_1041_nat__diff__add__eq2,axiom,
! [U_1,M,N,I,J] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I),J))
=> hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,I),U_1)),M)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,J),U_1)),N)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,J),I)),U_1)),N)) ) ).
fof(fact_1042_nat__le__add__iff2,axiom,
! [U,M_2,N_1,I_1,J_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_1),J_1))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,I_1),U)),M_2)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,J_1),U)),N_1)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_2),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,J_1),I_1)),U)),N_1))) ) ) ).
fof(fact_1043_nat__eq__add__iff1,axiom,
! [U,M_2,N_1,J_1,I_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,J_1),I_1))
=> ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,I_1),U)),M_2) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,J_1),U)),N_1)
<=> hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,I_1),J_1)),U)),M_2) = N_1 ) ) ).
fof(fact_1044_nat__diff__add__eq1,axiom,
! [U_1,M,N,J,I] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,J),I))
=> hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,I),U_1)),M)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,J),U_1)),N)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,I),J)),U_1)),M)),N) ) ).
fof(fact_1045_nat__mult__assoc,axiom,
! [M,N,K] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M),N)),K) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,N),K)) ).
fof(fact_1046_nat__mult__commute,axiom,
! [M,N] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M),N) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,N),M) ).
fof(fact_1047_nat__add__right__cancel,axiom,
! [M_2,K_1,N_1] :
( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_2),K_1) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N_1),K_1)
<=> M_2 = N_1 ) ).
fof(fact_1048_nat__add__left__cancel,axiom,
! [K_1,M_2,N_1] :
( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,K_1),M_2) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,K_1),N_1)
<=> M_2 = N_1 ) ).
fof(fact_1049_nat__add__assoc,axiom,
! [M,N,K] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),N)),K) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N),K)) ).
fof(fact_1050_nat__add__left__commute,axiom,
! [X,Y,Z] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,X),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Y),Z)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Y),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,X),Z)) ).
fof(fact_1051_nat__add__commute,axiom,
! [M,N] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),N) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N),M) ).
fof(fact_1052_add__mult__distrib2,axiom,
! [K,M,N] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),N)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K),M)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K),N)) ).
fof(fact_1053_add__mult__distrib,axiom,
! [M,N,K] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),N)),K) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M),K)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,N),K)) ).
fof(fact_1054_left__add__mult__distrib,axiom,
! [I,U_1,J,K] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,I),U_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,J),U_1)),K)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I),J)),U_1)),K) ).
fof(fact_1055_nat__le__add__iff1,axiom,
! [U,M_2,N_1,J_1,I_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,J_1),I_1))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,I_1),U)),M_2)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,J_1),U)),N_1)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,I_1),J_1)),U)),M_2)),N_1)) ) ) ).
fof(fact_1056_termination__basic__simps_I3_J,axiom,
! [Z,X,Y] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X),Y))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Y),Z))) ) ).
fof(fact_1057_termination__basic__simps_I4_J,axiom,
! [Y,X,Z] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X),Z))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Y),Z))) ) ).
fof(fact_1058_nat__minus__add__max,axiom,
! [N,M] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N),M)),M) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_max_nat,N),M) ).
fof(fact_1059_card__lessThan,axiom,
! [U] : hAPP_f22106695ol_nat(finite_card_nat,hAPP_n1699378549t_bool(ord_lessThan_nat,U)) = U ).
fof(fact_1060_nat__mult__eq__1__iff,axiom,
! [M_2,N_1] :
( hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M_2),N_1) = one_one_nat
<=> ( M_2 = one_one_nat
& N_1 = one_one_nat ) ) ).
fof(fact_1061_nat__mult__1__right,axiom,
! [N] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,N),one_one_nat) = N ).
fof(fact_1062_nat__1__eq__mult__iff,axiom,
! [M_2,N_1] :
( one_one_nat = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M_2),N_1)
<=> ( M_2 = one_one_nat
& N_1 = one_one_nat ) ) ).
fof(fact_1063_nat__mult__1,axiom,
! [N] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,one_one_nat),N) = N ).
fof(fact_1064_zadd__int__left,axiom,
! [M,N,Z] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_nat_int(semiri1621563631at_int,M)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_nat_int(semiri1621563631at_int,N)),Z)) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),N))),Z) ).
fof(fact_1065_zadd__int,axiom,
! [M,N] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_nat_int(semiri1621563631at_int,M)),hAPP_nat_int(semiri1621563631at_int,N)) = hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),N)) ).
fof(fact_1066_zle__int,axiom,
! [M_2,N_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_nat_int(semiri1621563631at_int,M_2)),hAPP_nat_int(semiri1621563631at_int,N_1)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_2),N_1)) ) ).
fof(fact_1067_zmult__int,axiom,
! [M,N] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_nat_int(semiri1621563631at_int,M)),hAPP_nat_int(semiri1621563631at_int,N)) = hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M),N)) ).
fof(fact_1068_int__mult,axiom,
! [M,N] : hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M),N)) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_nat_int(semiri1621563631at_int,M)),hAPP_nat_int(semiri1621563631at_int,N)) ).
fof(fact_1069_int__1,axiom,
hAPP_nat_int(semiri1621563631at_int,one_one_nat) = one_one_int ).
fof(fact_1070_zdiff__int,axiom,
! [N,M] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N),M))
=> hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_nat_int(semiri1621563631at_int,M)),hAPP_nat_int(semiri1621563631at_int,N)) = hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),N)) ) ).
fof(fact_1071_negative__zle,axiom,
! [N,M] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(uminus_uminus_int,hAPP_nat_int(semiri1621563631at_int,N))),hAPP_nat_int(semiri1621563631at_int,M))) ).
fof(fact_1072_finite__atLeastAtMost,axiom,
! [L,U] : hBOOL(hAPP_f54304608l_bool(finite_finite_nat,hAPP_n1699378549t_bool(ord_at238088361st_nat(L),U))) ).
fof(fact_1073_finite__Collect__less__nat,axiom,
! [K_1] : hBOOL(hAPP_f54304608l_bool(finite_finite_nat,hAPP_f800510211t_bool(collect_nat,hAPP_n1699378549t_bool(hAPP_f229349961t_bool(cOMBC_nat_nat_bool,ord_less_nat),K_1)))) ).
fof(fact_1074_zle__iff__zadd,axiom,
! [W,Z_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,W),Z_1))
<=> ? [N_2] : Z_1 = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,W),hAPP_nat_int(semiri1621563631at_int,N_2)) ) ).
fof(fact_1075_int__power,axiom,
! [M,N] : hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,M),N)) = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_nat_int(semiri1621563631at_int,M)),N) ).
fof(fact_1076_zpower__int,axiom,
! [M,N] : hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_nat_int(semiri1621563631at_int,M)),N) = hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,M),N)) ).
fof(fact_1077_int__int__eq,axiom,
! [M_2,N_1] :
( hAPP_nat_int(semiri1621563631at_int,M_2) = hAPP_nat_int(semiri1621563631at_int,N_1)
<=> M_2 = N_1 ) ).
fof(fact_1078_zmult__zminus,axiom,
! [Z,W_1] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(uminus_uminus_int,Z)),W_1) = hAPP_int_int(uminus_uminus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Z),W_1)) ).
fof(fact_1079_diff__int__def__symmetric,axiom,
! [Z,W_1] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Z),hAPP_int_int(uminus_uminus_int,W_1)) = hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Z),W_1) ).
fof(fact_1080_diff__int__def,axiom,
! [Z,W_1] : hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Z),W_1) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Z),hAPP_int_int(uminus_uminus_int,W_1)) ).
fof(fact_1081_zminus__zadd__distrib,axiom,
! [Z,W_1] : hAPP_int_int(uminus_uminus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Z),W_1)) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(uminus_uminus_int,Z)),hAPP_int_int(uminus_uminus_int,W_1)) ).
fof(fact_1082_zle__refl,axiom,
! [W_1] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,W_1),W_1)) ).
fof(fact_1083_zle__linear,axiom,
! [Z,W_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Z),W_1))
| hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,W_1),Z)) ) ).
fof(fact_1084_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_1085_zle__antisym,axiom,
! [Z,W_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Z),W_1))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,W_1),Z))
=> Z = W_1 ) ) ).
fof(fact_1086_zmult__assoc,axiom,
! [Z1,Z2,Z3] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Z1),Z2)),Z3) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Z1),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Z2),Z3)) ).
fof(fact_1087_zpower__zpower,axiom,
! [X,Y,Z] : hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X),Y)),Z) = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Y),Z)) ).
fof(fact_1088_zmult__commute,axiom,
! [Z,W_1] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Z),W_1) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,W_1),Z) ).
fof(fact_1089_zdiff__zmult__distrib,axiom,
! [Z1,Z2,W_1] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Z1),Z2)),W_1) = hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Z1),W_1)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Z2),W_1)) ).
fof(fact_1090_zdiff__zmult__distrib2,axiom,
! [W_1,Z1,Z2] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,W_1),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Z1),Z2)) = hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,W_1),Z1)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,W_1),Z2)) ).
fof(fact_1091_zadd__assoc,axiom,
! [Z1,Z2,Z3] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Z1),Z2)),Z3) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Z1),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Z2),Z3)) ).
fof(fact_1092_zadd__left__commute,axiom,
! [X,Y,Z] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Y),Z)) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Y),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X),Z)) ).
fof(fact_1093_zadd__commute,axiom,
! [Z,W_1] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Z),W_1) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,W_1),Z) ).
fof(fact_1094_zpower__zadd__distrib,axiom,
! [X,Y,Z] : hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Y),Z)) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X),Y)),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X),Z)) ).
fof(fact_1095_zadd__zmult__distrib2,axiom,
! [W_1,Z1,Z2] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,W_1),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Z1),Z2)) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,W_1),Z1)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,W_1),Z2)) ).
fof(fact_1096_zadd__zmult__distrib,axiom,
! [Z1,Z2,W_1] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Z1),Z2)),W_1) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Z1),W_1)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Z2),W_1)) ).
fof(fact_1097_zadd__left__mono,axiom,
! [K,I,J] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,I),J))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,K),I)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,K),J))) ) ).
fof(fact_1098_zmult__1__right,axiom,
! [Z] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Z),one_one_int) = Z ).
fof(fact_1099_zmult__1,axiom,
! [Z] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,one_one_int),Z) = Z ).
fof(fact_1100_add__lessD1,axiom,
! [I,J,K] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I),J)),K))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I),K)) ) ).
fof(fact_1101_less__add__eq__less,axiom,
! [M,N,K,L_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,K),L_1))
=> ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),L_1) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,K),N)
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N)) ) ) ).
fof(fact_1102_add__less__mono,axiom,
! [K,L_1,I,J] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I),J))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,K),L_1))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I),K)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,J),L_1))) ) ) ).
fof(fact_1103_add__less__mono1,axiom,
! [K,I,J] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I),J))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I),K)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,J),K))) ) ).
fof(fact_1104_trans__less__add2,axiom,
! [M,I,J] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I),J))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),J))) ) ).
fof(fact_1105_trans__less__add1,axiom,
! [M,I,J] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I),J))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,J),M))) ) ).
fof(fact_1106_nat__add__left__cancel__less,axiom,
! [K_1,M_2,N_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,K_1),M_2)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,K_1),N_1)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_2),N_1)) ) ).
fof(fact_1107_not__add__less2,axiom,
! [J,I] : ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,J),I)),I)) ).
fof(fact_1108_not__add__less1,axiom,
! [I,J] : ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I),J)),I)) ).
fof(fact_1109_nat__less__le,axiom,
! [M_2,N_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_2),N_1))
<=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_2),N_1))
& M_2 != N_1 ) ) ).
fof(fact_1110_le__eq__less__or__eq,axiom,
! [M_2,N_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_2),N_1))
<=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_2),N_1))
| M_2 = N_1 ) ) ).
fof(fact_1111_less__imp__le__nat,axiom,
! [M,N] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N)) ) ).
fof(fact_1112_le__neq__implies__less,axiom,
! [M,N] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N))
=> ( M != N
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N)) ) ) ).
fof(fact_1113_less__or__eq__imp__le,axiom,
! [M,N] :
( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N))
| M = N )
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N)) ) ).
fof(fact_1114_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,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,J),N)),K)) ) ).
fof(fact_1115_diff__less__mono2,axiom,
! [L_1,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_1))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,L_1),N)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,L_1),M))) ) ) ).
fof(fact_1116_nat__less__cases,axiom,
! [Pa,M_2,N_1] :
( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_2),N_1))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(Pa,N_1),M_2)) )
=> ( ( M_2 = N_1
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(Pa,N_1),M_2)) )
=> ( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_1),M_2))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(Pa,N_1),M_2)) )
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(Pa,N_1),M_2)) ) ) ) ).
fof(fact_1117_less__not__refl3,axiom,
! [S,T] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,S),T))
=> S != T ) ).
fof(fact_1118_less__not__refl2,axiom,
! [N,M] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N),M))
=> M != N ) ).
fof(fact_1119_less__irrefl__nat,axiom,
! [N] : ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N),N)) ).
fof(fact_1120_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_1121_nat__neq__iff,axiom,
! [M_2,N_1] :
( M_2 != N_1
<=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_2),N_1))
| hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_1),M_2)) ) ) ).
fof(fact_1122_less__not__refl,axiom,
! [N] : ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N),N)) ).
fof(fact_1123_termination__basic__simps_I5_J,axiom,
! [X,Y] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X),Y))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X),Y)) ) ).
fof(fact_1124_termination__basic__simps_I2_J,axiom,
! [Y,X,Z] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X),Z))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Y),Z))) ) ).
fof(fact_1125_termination__basic__simps_I1_J,axiom,
! [Z,X,Y] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X),Y))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Y),Z))) ) ).
fof(fact_1126_image__add__atLeastAtMost,axiom,
! [K_1,I_1,J_1] : image_nat_nat(hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,plus_plus_nat),K_1),hAPP_n1699378549t_bool(ord_at238088361st_nat(I_1),J_1)) = hAPP_n1699378549t_bool(ord_at238088361st_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I_1),K_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,J_1),K_1)) ).
fof(fact_1127_finite__nat__set__iff__bounded,axiom,
! [N_3] :
( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,N_3))
<=> ? [M_1] :
! [X_2] :
( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_2),N_3))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_2),M_1)) ) ) ).
fof(fact_1128_finite__M__bounded__by__nat,axiom,
! [Pa,I_1] : hBOOL(hAPP_f54304608l_bool(finite_finite_nat,hAPP_f800510211t_bool(collect_nat,hAPP_f800510211t_bool(hAPP_f561022312t_bool(cOMBS_nat_bool_bool,hAPP_f1146629647l_bool(hAPP_f1080886329l_bool(cOMBB_1015721476ol_nat,fconj),Pa)),hAPP_n1699378549t_bool(hAPP_f229349961t_bool(cOMBC_nat_nat_bool,ord_less_nat),I_1))))) ).
fof(fact_1129_card__Collect__less__nat,axiom,
! [N_1] : hAPP_f22106695ol_nat(finite_card_nat,hAPP_f800510211t_bool(collect_nat,hAPP_n1699378549t_bool(hAPP_f229349961t_bool(cOMBC_nat_nat_bool,ord_less_nat),N_1))) = N_1 ).
fof(fact_1130_add__diff__inverse,axiom,
! [M,N] :
( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N))
=> hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),N)) = M ) ).
fof(fact_1131_less__diff__conv,axiom,
! [I_1,J_1,K_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,J_1),K_1)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I_1),K_1)),J_1)) ) ).
fof(fact_1132_less__diff__iff,axiom,
! [N_1,K_1,M_2] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_1),M_2))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_1),N_1))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M_2),K_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_1),K_1)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_2),N_1)) ) ) ) ).
fof(fact_1133_diff__less__mono,axiom,
! [C,A_1,B_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_1),B_1))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,C),A_1))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,A_1),C)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,B_1),C))) ) ) ).
fof(fact_1134_nat__less__add__iff1,axiom,
! [U,M_2,N_1,J_1,I_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,J_1),I_1))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,I_1),U)),M_2)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,J_1),U)),N_1)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,I_1),J_1)),U)),M_2)),N_1)) ) ) ).
fof(fact_1135_nat__less__add__iff2,axiom,
! [U,M_2,N_1,I_1,J_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_1),J_1))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,I_1),U)),M_2)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,J_1),U)),N_1)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_2),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,J_1),I_1)),U)),N_1))) ) ) ).
fof(fact_1136_not__int__zless__negative,axiom,
! [N,M] : ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_nat_int(semiri1621563631at_int,N)),hAPP_int_int(uminus_uminus_int,hAPP_nat_int(semiri1621563631at_int,M)))) ).
fof(fact_1137_zless__le,axiom,
! [Z_1,W] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Z_1),W))
<=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Z_1),W))
& Z_1 != W ) ) ).
fof(fact_1138_zadd__strict__right__mono,axiom,
! [K,I,J] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,I),J))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,I),K)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,J),K))) ) ).
fof(fact_1139_zminus__zminus,axiom,
! [Z] : hAPP_int_int(uminus_uminus_int,hAPP_int_int(uminus_uminus_int,Z)) = Z ).
fof(fact_1140_zadd__zless__mono,axiom,
! [Z_2,Z,W_2,W_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,W_2),W_1))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Z_2),Z))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,W_2),Z_2)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,W_1),Z))) ) ) ).
fof(fact_1141_zless__int,axiom,
! [M_2,N_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_nat_int(semiri1621563631at_int,M_2)),hAPP_nat_int(semiri1621563631at_int,N_1)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_2),N_1)) ) ).
fof(fact_1142_zless__add1__eq,axiom,
! [W,Z_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,W),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Z_1),one_one_int)))
<=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,W),Z_1))
| W = Z_1 ) ) ).
fof(fact_1143_atLeastAtMostPlus1__int__conv,axiom,
! [M_2,N_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,M_2),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,one_one_int),N_1)))
=> hAPP_i1948725293t_bool(ord_at875362053st_int(M_2),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,one_one_int),N_1)) = hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,one_one_int),N_1)),hAPP_i1948725293t_bool(ord_at875362053st_int(M_2),N_1)) ) ).
fof(fact_1144_zless__imp__add1__zle,axiom,
! [W_1,Z] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,W_1),Z))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,W_1),one_one_int)),Z)) ) ).
fof(fact_1145_add1__zle__eq,axiom,
! [W,Z_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,W),one_one_int)),Z_1))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,W),Z_1)) ) ).
fof(fact_1146_zle__add1__eq__le,axiom,
! [W,Z_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,W),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Z_1),one_one_int)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,W),Z_1)) ) ).
fof(fact_1147_zle__diff1__eq,axiom,
! [W,Z_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,W),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Z_1),one_one_int)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,W),Z_1)) ) ).
fof(fact_1148_int__le__induct,axiom,
! [Pa,I_1,K_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,I_1),K_1))
=> ( hBOOL(hAPP_int_bool(Pa,K_1))
=> ( ! [I_2] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,I_2),K_1))
=> ( hBOOL(hAPP_int_bool(Pa,I_2))
=> hBOOL(hAPP_int_bool(Pa,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,I_2),one_one_int))) ) )
=> hBOOL(hAPP_int_bool(Pa,I_1)) ) ) ) ).
fof(fact_1149_finite__atLeastAtMost__int,axiom,
! [L,U] : hBOOL(hAPP_f448129468l_bool(finite_finite_int,hAPP_i1948725293t_bool(ord_at875362053st_int(L),U))) ).
fof(fact_1150_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_1151_simp__from__to,axiom,
! [J_1,I_1] :
( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,J_1),I_1))
=> hAPP_i1948725293t_bool(ord_at875362053st_int(I_1),J_1) = bot_bot_fun_int_bool )
& ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,J_1),I_1))
=> hAPP_i1948725293t_bool(ord_at875362053st_int(I_1),J_1) = hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,I_1),hAPP_i1948725293t_bool(ord_at875362053st_int(hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,I_1),one_one_int)),J_1)) ) ) ).
fof(fact_1152_int__less__induct,axiom,
! [Pa,I_1,K_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,I_1),K_1))
=> ( hBOOL(hAPP_int_bool(Pa,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,K_1),one_one_int)))
=> ( ! [I_2] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,I_2),K_1))
=> ( hBOOL(hAPP_int_bool(Pa,I_2))
=> hBOOL(hAPP_int_bool(Pa,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,I_2),one_one_int))) ) )
=> hBOOL(hAPP_int_bool(Pa,I_1)) ) ) ) ).
fof(fact_1153_transfer__nat__int__set__relations_I1_J,axiom,
! [A] :
( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
<=> hBOOL(hAPP_f448129468l_bool(finite_finite_int,image_nat_int(semiri1621563631at_int,A))) ) ).
fof(fact_1154_transfer__nat__int__set__relations_I5_J,axiom,
! [A,B] :
( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A),B))
<=> hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,image_nat_int(semiri1621563631at_int,A)),image_nat_int(semiri1621563631at_int,B))) ) ).
fof(fact_1155_transfer__nat__int__set__relations_I2_J,axiom,
! [X_1,A] :
( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A))
<=> hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hAPP_nat_int(semiri1621563631at_int,X_1)),image_nat_int(semiri1621563631at_int,A))) ) ).
fof(fact_1156_transfer__nat__int__set__relations_I3_J,axiom,
! [A,B] :
( A = B
<=> image_nat_int(semiri1621563631at_int,A) = image_nat_int(semiri1621563631at_int,B) ) ).
fof(fact_1157_int__if__cong,axiom,
! [X_1,Y_1,Pa] :
( ( hBOOL(Pa)
=> hAPP_nat_int(semiri1621563631at_int,X_1) = hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(hAPP_nat_fun_nat_nat(if_nat(Pa),X_1),Y_1)) )
& ( ~ hBOOL(Pa)
=> hAPP_nat_int(semiri1621563631at_int,Y_1) = hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(hAPP_nat_fun_nat_nat(if_nat(Pa),X_1),Y_1)) ) ) ).
fof(fact_1158_transfer__int__nat__relations_I1_J,axiom,
! [X_1,Y_1] :
( hAPP_nat_int(semiri1621563631at_int,X_1) = hAPP_nat_int(semiri1621563631at_int,Y_1)
<=> X_1 = Y_1 ) ).
fof(fact_1159_transfer__nat__int__set__relations_I4_J,axiom,
! [A,B] :
( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le382113706t_bool,A),B))
<=> hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le1912455174t_bool,image_nat_int(semiri1621563631at_int,A)),image_nat_int(semiri1621563631at_int,B))) ) ).
fof(fact_1160_Nat__Transfer_Otransfer__nat__int__set__functions_I1_J,axiom,
! [A] : hAPP_f22106695ol_nat(finite_card_nat,A) = hAPP_f957591787ol_nat(finite_card_int,image_nat_int(semiri1621563631at_int,A)) ).
fof(fact_1161_Nat__Transfer_Otransfer__int__nat__set__functions_I2_J,axiom,
bot_bot_fun_int_bool = image_nat_int(semiri1621563631at_int,bot_bot_fun_nat_bool) ).
fof(fact_1162_transfer__int__nat__relations_I2_J,axiom,
! [X_1,Y_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_nat_int(semiri1621563631at_int,X_1)),hAPP_nat_int(semiri1621563631at_int,Y_1)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_1),Y_1)) ) ).
fof(fact_1163_transfer__int__nat__relations_I3_J,axiom,
! [X_1,Y_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_nat_int(semiri1621563631at_int,X_1)),hAPP_nat_int(semiri1621563631at_int,Y_1)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_1),Y_1)) ) ).
fof(fact_1164_Nat__Transfer_Otransfer__int__nat__functions_I1_J,axiom,
! [X,Y] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_nat_int(semiri1621563631at_int,X)),hAPP_nat_int(semiri1621563631at_int,Y)) = hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,X),Y)) ).
fof(fact_1165_Nat__Transfer_Otransfer__int__nat__functions_I2_J,axiom,
! [X,Y] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_nat_int(semiri1621563631at_int,X)),hAPP_nat_int(semiri1621563631at_int,Y)) = hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,X),Y)) ).
fof(fact_1166_transfer__int__nat__numerals_I2_J,axiom,
one_one_int = hAPP_nat_int(semiri1621563631at_int,one_one_nat) ).
fof(fact_1167_Nat__Transfer_Otransfer__int__nat__functions_I4_J,axiom,
! [X,N] : hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_nat_int(semiri1621563631at_int,X)),N) = hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X),N)) ).
fof(fact_1168_int__gr__induct,axiom,
! [Pa,K_1,I_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K_1),I_1))
=> ( hBOOL(hAPP_int_bool(Pa,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,K_1),one_one_int)))
=> ( ! [I_2] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K_1),I_2))
=> ( hBOOL(hAPP_int_bool(Pa,I_2))
=> hBOOL(hAPP_int_bool(Pa,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,I_2),one_one_int))) ) )
=> hBOOL(hAPP_int_bool(Pa,I_1)) ) ) ) ).
fof(fact_1169_int__ge__induct,axiom,
! [Pa,K_1,I_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K_1),I_1))
=> ( hBOOL(hAPP_int_bool(Pa,K_1))
=> ( ! [I_2] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K_1),I_2))
=> ( hBOOL(hAPP_int_bool(Pa,I_2))
=> hBOOL(hAPP_int_bool(Pa,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,I_2),one_one_int))) ) )
=> hBOOL(hAPP_int_bool(Pa,I_1)) ) ) ) ).
fof(fact_1170_mono__nat__linear__lb,axiom,
! [M_2,K_1,F] :
( ! [M_1,N_2] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),N_2))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(F,M_1)),hAPP_nat_nat(F,N_2))) )
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(F,M_2)),K_1)),hAPP_nat_nat(F,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_2),K_1)))) ) ).
fof(fact_1171_bounded__nat__set__is__finite,axiom,
! [N_1,N_3] :
( ! [X_2] :
( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_2),N_3))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_2),N_1)) )
=> hBOOL(hAPP_f54304608l_bool(finite_finite_nat,N_3)) ) ).
fof(fact_1172_int__diff__cases,axiom,
! [Z] :
~ ! [M_1,N_2] : Z != hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_nat_int(semiri1621563631at_int,M_1)),hAPP_nat_int(semiri1621563631at_int,N_2)) ).
fof(fact_1173_less__mono__imp__le__mono,axiom,
! [I_1,J_1,F] :
( ! [I_2,J_2] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_2),J_2))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(F,I_2)),hAPP_nat_nat(F,J_2))) )
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_1),J_1))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(F,I_1)),hAPP_nat_nat(F,J_1))) ) ) ).
fof(fact_1174_int__induct,axiom,
! [I_1,Pa,K_1] :
( hBOOL(hAPP_int_bool(Pa,K_1))
=> ( ! [I_2] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K_1),I_2))
=> ( hBOOL(hAPP_int_bool(Pa,I_2))
=> hBOOL(hAPP_int_bool(Pa,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,I_2),one_one_int))) ) )
=> ( ! [I_2] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,I_2),K_1))
=> ( hBOOL(hAPP_int_bool(Pa,I_2))
=> hBOOL(hAPP_int_bool(Pa,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,I_2),one_one_int))) ) )
=> hBOOL(hAPP_int_bool(Pa,I_1)) ) ) ) ).
fof(fact_1175_less__zeroE,axiom,
! [N] : ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N),zero_zero_nat)) ).
fof(fact_1176_le0,axiom,
! [N] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),N)) ).
fof(fact_1177_finite__greaterThanAtMost,axiom,
! [L,U] : hBOOL(hAPP_f54304608l_bool(finite_finite_nat,hAPP_n1699378549t_bool(ord_gr375877188st_nat(L),U))) ).
fof(fact_1178_Nat__Transfer_Otransfer__nat__int__function__closures_I4_J,axiom,
! [N,X] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X),N))) ) ).
fof(fact_1179_transfer__int__nat__quantifiers_I1_J,axiom,
! [Pa] :
( ! [X_2] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_2))
=> hBOOL(hAPP_int_bool(Pa,X_2)) )
<=> ! [X_2] : hBOOL(hAPP_int_bool(Pa,hAPP_nat_int(semiri1621563631at_int,X_2))) ) ).
fof(fact_1180_transfer__int__nat__quantifiers_I2_J,axiom,
! [Pa] :
( ? [X_2] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_2))
& hBOOL(hAPP_int_bool(Pa,X_2)) )
<=> ? [X_2] : hBOOL(hAPP_int_bool(Pa,hAPP_nat_int(semiri1621563631at_int,X_2))) ) ).
fof(fact_1181_Nat__Transfer_Otransfer__nat__int__function__closures_I9_J,axiom,
! [Z] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_nat_int(semiri1621563631at_int,Z))) ).
fof(fact_1182_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_1183_Nat__Transfer_Otransfer__nat__int__function__closures_I1_J,axiom,
! [Y,X] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Y))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X),Y))) ) ) ).
fof(fact_1184_Nat__Transfer_Otransfer__nat__int__function__closures_I2_J,axiom,
! [Y,X] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Y))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X),Y))) ) ) ).
fof(fact_1185_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_1186_int__less__0__conv,axiom,
! [K] : ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_nat_int(semiri1621563631at_int,K)),zero_zero_int)) ).
fof(fact_1187_zero__less__int__conv,axiom,
! [N_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_nat_int(semiri1621563631at_int,N_1)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1)) ) ).
fof(fact_1188_less__bin__lemma,axiom,
! [K_1,L] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K_1),L))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,K_1),L)),zero_zero_int)) ) ).
fof(fact_1189_zmult__zless__mono2,axiom,
! [K,I,J] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,I),J))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),K))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K),I)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K),J))) ) ) ).
fof(fact_1190_int__0__less__1,axiom,
hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),one_one_int)) ).
fof(fact_1191_nat__zero__less__power__iff,axiom,
! [X_1,N_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X_1),N_1)))
<=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),X_1))
| N_1 = zero_zero_nat ) ) ).
fof(fact_1192_nat__power__less__imp__less,axiom,
! [M,N,I] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),I))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,I),M)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,I),N)))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N)) ) ) ).
fof(fact_1193_gr0I,axiom,
! [N] :
( N != zero_zero_nat
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N)) ) ).
fof(fact_1194_gr__implies__not0,axiom,
! [M,N] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N))
=> N != zero_zero_nat ) ).
fof(fact_1195_less__nat__zero__code,axiom,
! [N] : ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N),zero_zero_nat)) ).
fof(fact_1196_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_1197_not__less0,axiom,
! [N] : ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N),zero_zero_nat)) ).
%----Helper facts (165)
fof(help_fNot_1_1_U,axiom,
! [P] :
( ~ hBOOL(hAPP_bool_bool(fNot,P))
| ~ hBOOL(P) ) ).
fof(help_fNot_2_1_U,axiom,
! [P] :
( hBOOL(P)
| hBOOL(hAPP_bool_bool(fNot,P)) ) ).
fof(help_fconj_1_1_U,axiom,
! [Q,P] :
( ~ hBOOL(P)
| ~ hBOOL(Q)
| hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(fconj,P),Q)) ) ).
fof(help_fconj_2_1_U,axiom,
! [P,Q] :
( ~ hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(fconj,P),Q))
| hBOOL(P) ) ).
fof(help_fconj_3_1_U,axiom,
! [P,Q] :
( ~ hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(fconj,P),Q))
| hBOOL(Q) ) ).
fof(help_fdisj_1_1_U,axiom,
! [Q,P] :
( ~ hBOOL(P)
| hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(fdisj,P),Q)) ) ).
fof(help_fdisj_2_1_U,axiom,
! [P,Q] :
( ~ hBOOL(Q)
| hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(fdisj,P),Q)) ) ).
fof(help_fdisj_3_1_U,axiom,
! [P,Q] :
( ~ hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(fdisj,P),Q))
| hBOOL(P)
| hBOOL(Q) ) ).
fof(help_fFalse_1_1_U,axiom,
~ hBOOL(fFalse) ).
fof(help_fFalse_1_1_T,axiom,
! [P] :
( is_bool(P)
=> ( P = fTrue
| P = fFalse ) ) ).
fof(help_fimplies_1_1_U,axiom,
! [Q,P] :
( hBOOL(P)
| hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(fimplies,P),Q)) ) ).
fof(help_fimplies_2_1_U,axiom,
! [P,Q] :
( ~ hBOOL(Q)
| hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(fimplies,P),Q)) ) ).
fof(help_fimplies_3_1_U,axiom,
! [P,Q] :
( ~ hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(fimplies,P),Q))
| ~ hBOOL(P)
| hBOOL(Q) ) ).
fof(help_If_1_1_If_000tc__Nat__Onat_T,axiom,
! [X,Y] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(if_nat(fTrue),X),Y) = X ).
fof(help_If_2_1_If_000tc__Nat__Onat_T,axiom,
! [X,Y] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(if_nat(fFalse),X),Y) = Y ).
fof(help_If_3_1_If_000tc__Nat__Onat_T,axiom,
! [P] :
( is_bool(P)
=> ( P = fTrue
| P = fFalse ) ) ).
fof(help_COMBI_1_1_COMBI_000tc__Int__Oint_U,axiom,
! [P] : hAPP_int_int(cOMBI_int,P) = P ).
fof(help_COMBI_1_1_COMBI_000tc__Nat__Onat_U,axiom,
! [P] : hAPP_nat_nat(cOMBI_nat,P) = P ).
fof(help_fequal_1_1_fequal_000tc__Int__Oint_T,axiom,
! [X,Y] :
( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(fequal_int,X),Y))
| X = Y ) ).
fof(help_fequal_2_1_fequal_000tc__Int__Oint_T,axiom,
! [X,Y] :
( X != Y
| hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(fequal_int,X),Y)) ) ).
fof(help_fequal_1_1_fequal_000tc__Nat__Onat_T,axiom,
! [X,Y] :
( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(fequal_nat,X),Y))
| X = Y ) ).
fof(help_fequal_2_1_fequal_000tc__Nat__Onat_T,axiom,
! [X,Y] :
( X != Y
| hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(fequal_nat,X),Y)) ) ).
fof(help_fequal_1_1_fequal_000tc__Com__Ostate_T,axiom,
! [X,Y] :
( ~ hBOOL(hAPP_state_bool(hAPP_s1806633685e_bool(fequal_state,X),Y))
| X = Y ) ).
fof(help_fequal_2_1_fequal_000tc__Com__Ostate_T,axiom,
! [X,Y] :
( X != Y
| hBOOL(hAPP_state_bool(hAPP_s1806633685e_bool(fequal_state,X),Y)) ) ).
fof(help_COMBK_1_1_COMBK_000tc__Int__Oint_000tc__Int__Oint_U,axiom,
! [P,Q] : hAPP_int_int(hAPP_int_fun_int_int(cOMBK_int_int,P),Q) = P ).
fof(help_COMBK_1_1_COMBK_000tc__Int__Oint_000tc__Nat__Onat_U,axiom,
! [P,Q] : hAPP_nat_int(hAPP_int_fun_nat_int(cOMBK_int_nat,P),Q) = P ).
fof(help_COMBK_1_1_COMBK_000tc__Nat__Onat_000tc__Int__Oint_U,axiom,
! [P,Q] : hAPP_int_nat(hAPP_nat_fun_int_nat(cOMBK_nat_int,P),Q) = P ).
fof(help_COMBK_1_1_COMBK_000tc__Nat__Onat_000tc__Nat__Onat_U,axiom,
! [P,Q] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(cOMBK_nat_nat,P),Q) = P ).
fof(help_COMBK_1_1_COMBK_000tc__HOL__Obool_000tc__Int__Oint_U,axiom,
! [P,Q] :
( is_bool(P)
=> hAPP_int_bool(hAPP_b396694332t_bool(cOMBK_bool_int,P),Q) = P ) ).
fof(help_COMBK_1_1_COMBK_000tc__HOL__Obool_000tc__Nat__Onat_U,axiom,
! [P,Q] :
( is_bool(P)
=> hAPP_nat_bool(hAPP_b1013836512t_bool(cOMBK_bool_nat,P),Q) = P ) ).
fof(help_COMBK_1_1_COMBK_000tc__HOL__Obool_000tc__Com__Ostate_U,axiom,
! [P,Q] :
( is_bool(P)
=> hAPP_state_bool(hAPP_b2019457360e_bool(cOMBK_bool_state,P),Q) = P ) ).
fof(help_fequal_1_1_fequal_000tc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J_T,axiom,
! [X,Y] :
( ~ hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(fequal_fun_int_bool,X),Y))
| X = Y ) ).
fof(help_fequal_2_1_fequal_000tc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J_T,axiom,
! [X,Y] :
( X != Y
| hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(fequal_fun_int_bool,X),Y)) ) ).
fof(help_fequal_1_1_fequal_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_T,axiom,
! [X,Y] :
( ~ hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(fequal_fun_nat_bool,X),Y))
| X = Y ) ).
fof(help_fequal_2_1_fequal_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_T,axiom,
! [X,Y] :
( X != Y
| hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(fequal_fun_nat_bool,X),Y)) ) ).
fof(help_COMBB_1_1_COMBB_000tc__Int__Oint_000tc__Int__Oint_000tc__Nat__Onat_U,axiom,
! [P,Q,R] : hAPP_nat_int(hAPP_f1139079189at_int(hAPP_f1431025877at_int(cOMBB_int_int_nat,P),Q),R) = hAPP_int_int(P,hAPP_nat_int(Q,R)) ).
fof(help_COMBB_1_1_COMBB_000tc__Nat__Onat_000tc__Int__Oint_000tc__Int__Oint_U,axiom,
! [P,Q,R] : hAPP_int_int(hAPP_f147134065nt_int(hAPP_f1673907925nt_int(cOMBB_nat_int_int,P),Q),R) = hAPP_nat_int(P,hAPP_int_nat(Q,R)) ).
fof(help_COMBB_1_1_COMBB_000tc__Nat__Onat_000tc__Nat__Onat_000tc__Int__Oint_U,axiom,
! [P,Q,R] : hAPP_int_nat(hAPP_f2132704789nt_nat(hAPP_f901215189nt_nat(cOMBB_nat_nat_int,P),Q),R) = hAPP_nat_nat(P,hAPP_int_nat(Q,R)) ).
fof(help_COMBB_1_1_COMBB_000tc__Nat__Onat_000tc__Nat__Onat_000tc__Nat__Onat_U,axiom,
! [P,Q,R] : hAPP_nat_nat(hAPP_f1914919701at_nat(hAPP_f1585078997at_nat(cOMBB_nat_nat_nat,P),Q),R) = hAPP_nat_nat(P,hAPP_nat_nat(Q,R)) ).
fof(help_COMBC_1_1_COMBC_000tc__Int__Oint_000tc__Int__Oint_000tc__Nat__Onat_U,axiom,
! [P,Q,R] : hAPP_int_nat(hAPP_int_fun_int_nat(hAPP_f909851349nt_nat(cOMBC_int_int_nat,P),Q),R) = hAPP_int_nat(hAPP_int_fun_int_nat(P,R),Q) ).
fof(help_COMBC_1_1_COMBC_000tc__Int__Oint_000tc__Nat__Onat_000tc__Nat__Onat_U,axiom,
! [P,Q,R] : hAPP_int_nat(hAPP_nat_fun_int_nat(hAPP_f879494613nt_nat(cOMBC_int_nat_nat,P),Q),R) = hAPP_nat_nat(hAPP_int_fun_nat_nat(P,R),Q) ).
fof(help_COMBC_1_1_COMBC_000tc__Nat__Onat_000tc__Int__Oint_000tc__Nat__Onat_U,axiom,
! [P,Q,R] : hAPP_nat_nat(hAPP_int_fun_nat_nat(hAPP_f446977493at_nat(cOMBC_nat_int_nat,P),Q),R) = hAPP_int_nat(hAPP_nat_fun_int_nat(P,R),Q) ).
fof(help_COMBC_1_1_COMBC_000tc__Nat__Onat_000tc__Nat__Onat_000tc__Nat__Onat_U,axiom,
! [P,Q,R] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,P),Q),R) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(P,R),Q) ).
fof(help_COMBS_1_1_COMBS_000tc__Int__Oint_000tc__Nat__Onat_000tc__Nat__Onat_U,axiom,
! [P,Q,R] : hAPP_int_nat(hAPP_f2132704789nt_nat(hAPP_f1399363134nt_nat(cOMBS_int_nat_nat,P),Q),R) = hAPP_nat_nat(hAPP_int_fun_nat_nat(P,R),hAPP_int_nat(Q,R)) ).
fof(help_COMBS_1_1_COMBS_000tc__Nat__Onat_000tc__Nat__Onat_000tc__Nat__Onat_U,axiom,
! [P,Q,R] : hAPP_nat_nat(hAPP_f1914919701at_nat(hAPP_f1408247010at_nat(cOMBS_nat_nat_nat,P),Q),R) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(P,R),hAPP_nat_nat(Q,R)) ).
fof(help_COMBC_1_1_COMBC_000tc__Int__Oint_000tc__Int__Oint_000tc__HOL__Obool_U,axiom,
! [P,Q,R] : hAPP_int_bool(hAPP_i1948725293t_bool(hAPP_f428220345t_bool(cOMBC_int_int_bool,P),Q),R) = hAPP_int_bool(hAPP_i1948725293t_bool(P,R),Q) ).
fof(help_COMBC_1_1_COMBC_000tc__Int__Oint_000tc__Nat__Onat_000tc__HOL__Obool_U,axiom,
! [P,Q,R] : hAPP_int_bool(hAPP_n1082236369t_bool(hAPP_f175561985t_bool(cOMBC_int_nat_bool,P),Q),R) = hAPP_nat_bool(hAPP_i418383825t_bool(P,R),Q) ).
fof(help_COMBC_1_1_COMBC_000tc__Nat__Onat_000tc__Int__Oint_000tc__HOL__Obool_U,axiom,
! [P,Q,R] : hAPP_nat_bool(hAPP_i418383825t_bool(hAPP_f482008321t_bool(cOMBC_nat_int_bool,P),Q),R) = hAPP_int_bool(hAPP_n1082236369t_bool(P,R),Q) ).
fof(help_COMBC_1_1_COMBC_000tc__Nat__Onat_000tc__Nat__Onat_000tc__HOL__Obool_U,axiom,
! [P,Q,R] : hAPP_nat_bool(hAPP_n1699378549t_bool(hAPP_f229349961t_bool(cOMBC_nat_nat_bool,P),Q),R) = hAPP_nat_bool(hAPP_n1699378549t_bool(P,R),Q) ).
fof(help_COMBB_1_1_COMBB_000tc__HOL__Obool_000tc__HOL__Obool_000tc__Int__Oint_U,axiom,
! [P,Q,R] : hAPP_int_bool(hAPP_f1805168059t_bool(hAPP_f627970963t_bool(cOMBB_bool_bool_int,P),Q),R) = hAPP_bool_bool(P,hAPP_int_bool(Q,R)) ).
fof(help_COMBB_1_1_COMBB_000tc__HOL__Obool_000tc__HOL__Obool_000tc__Nat__Onat_U,axiom,
! [P,Q,R] : hAPP_nat_bool(hAPP_f800510211t_bool(hAPP_f894608603t_bool(cOMBB_bool_bool_nat,P),Q),R) = hAPP_bool_bool(P,hAPP_nat_bool(Q,R)) ).
fof(help_COMBS_1_1_COMBS_000tc__Int__Oint_000tc__HOL__Obool_000tc__HOL__Obool_U,axiom,
! [P,Q,R] : hAPP_int_bool(hAPP_f1805168059t_bool(hAPP_f727283836t_bool(cOMBS_int_bool_bool,P),Q),R) = hAPP_bool_bool(hAPP_i68813070l_bool(P,R),hAPP_int_bool(Q,R)) ).
fof(help_COMBS_1_1_COMBS_000tc__Nat__Onat_000tc__HOL__Obool_000tc__HOL__Obool_U,axiom,
! [P,Q,R] : hAPP_nat_bool(hAPP_f800510211t_bool(hAPP_f561022312t_bool(cOMBS_nat_bool_bool,P),Q),R) = hAPP_bool_bool(hAPP_n1006566506l_bool(P,R),hAPP_nat_bool(Q,R)) ).
fof(help_COMBK_1_1_COMBK_000tc__fun_Itc__Com__Ostate_Mtc__HOL__Obool_J_000t__a_U,axiom,
! [P,Q] : hAPP_a2036067514e_bool(hAPP_f762886889e_bool(cOMBK_1458035955bool_a,P),Q) = P ).
fof(help_COMBB_1_1_COMBB_000tc__HOL__Obool_000tc__HOL__Obool_000tc__Com__Ostate_U,axiom,
! [P,Q,R] : hAPP_state_bool(hAPP_f1759915619e_bool(hAPP_f2073279419e_bool(cOMBB_160679318_state,P),Q),R) = hAPP_bool_bool(P,hAPP_state_bool(Q,R)) ).
fof(help_COMBC_1_1_COMBC_000tc__Com__Ostate_000tc__HOL__Obool_000tc__HOL__Obool_U,axiom,
! [P,Q,R] : hAPP_state_bool(hAPP_b2019457360e_bool(hAPP_f167292325e_bool(cOMBC_231445413l_bool,P),Q),R) = hAPP_bool_bool(hAPP_s58564346l_bool(P,R),Q) ).
fof(help_COMBS_1_1_COMBS_000tc__Com__Ostate_000tc__HOL__Obool_000tc__HOL__Obool_U,axiom,
! [P,Q,R] : hAPP_state_bool(hAPP_f1759915619e_bool(hAPP_f644196280e_bool(cOMBS_1378840469l_bool,P),Q),R) = hAPP_bool_bool(hAPP_s58564346l_bool(P,R),hAPP_state_bool(Q,R)) ).
fof(help_COMBS_1_1_COMBS_000tc__Com__Ostate_000tc__Nat__Onat_000tc__Com__Ostate_U,axiom,
! [P,Q,R] : hAPP_state_state(hAPP_f871651461_state(hAPP_f851239890_state(cOMBS_777315357_state,P),Q),R) = hAPP_nat_state(hAPP_s1892499976_state(P,R),hAPP_state_nat(Q,R)) ).
fof(help_COMBB_1_1_COMBB_000tc__Com__Ostate_000tc__HOL__Obool_000tc__Com__Ostate_U,axiom,
! [P,Q,R] : hAPP_state_bool(hAPP_f746301080e_bool(hAPP_f162060345e_bool(cOMBB_844853809_state,P),Q),R) = hAPP_state_bool(P,hAPP_state_state(Q,R)) ).
fof(help_COMBC_1_1_COMBC_000tc__Com__Ostate_000tc__Com__Ostate_000tc__HOL__Obool_U,axiom,
! [P,Q,R] : hAPP_state_bool(hAPP_s1806633685e_bool(hAPP_f817621513e_bool(cOMBC_2027030106e_bool,P),Q),R) = hAPP_state_bool(hAPP_s1806633685e_bool(P,R),Q) ).
fof(help_COMBI_1_1_COMBI_000tc__Hoare____Mirabelle____uwgpyvfjxg__Otriple_It__a_J_U,axiom,
! [P] : hAPP_H905846293iple_a(cOMBI_1301683934iple_a,P) = P ).
fof(help_fequal_1_1_fequal_000tc__Hoare____Mirabelle____uwgpyvfjxg__Otriple_It__a_J_,axiom,
! [X,Y] :
( ~ hBOOL(hAPP_H1037229737a_bool(hAPP_H426895267a_bool(fequal963300192iple_a,X),Y))
| X = Y ) ).
fof(help_fequal_2_1_fequal_000tc__Hoare____Mirabelle____uwgpyvfjxg__Otriple_It__a_J_,axiom,
! [X,Y] :
( X != Y
| hBOOL(hAPP_H1037229737a_bool(hAPP_H426895267a_bool(fequal963300192iple_a,X),Y)) ) ).
fof(help_COMBC_1_1_COMBC_000t__a_000tc__HOL__Obool_000tc__fun_Itc__Com__Ostate_Mtc__,axiom,
! [P,Q,R] : hAPP_a2036067514e_bool(hAPP_b540892988e_bool(hAPP_f1824947087e_bool(cOMBC_41962815e_bool,P),Q),R) = hAPP_b2019457360e_bool(hAPP_a723219176e_bool(P,R),Q) ).
fof(help_COMBK_1_1_COMBK_000tc__Int__Oint_000tc__Hoare____Mirabelle____uwgpyvfjxg__O,axiom,
! [P,Q] : hAPP_H1229305626_a_int(hAPP_i1475897073_a_int(cOMBK_1007576201iple_a,P),Q) = P ).
fof(help_COMBK_1_1_COMBK_000tc__Nat__Onat_000tc__Hoare____Mirabelle____uwgpyvfjxg__O,axiom,
! [P,Q] : hAPP_H592031934_a_nat(hAPP_n1653940209_a_nat(cOMBK_1028443109iple_a,P),Q) = P ).
fof(help_COMBK_1_1_COMBK_000tc__HOL__Obool_000tc__Hoare____Mirabelle____uwgpyvfjxg__,axiom,
! [P,Q] :
( is_bool(P)
=> hAPP_H1037229737a_bool(hAPP_b1882817719a_bool(cOMBK_235286536iple_a,P),Q) = P ) ).
fof(help_COMBB_1_1_COMBB_000tc__Nat__Onat_000tc__fun_Itc__Nat__Onat_Mtc__Nat__Onat_J,axiom,
! [P,Q,R] : hAPP_int_fun_nat_nat(hAPP_f1463450952at_nat(hAPP_f1087393429at_nat(cOMBB_1601129847at_int,P),Q),R) = hAPP_nat_fun_nat_nat(P,hAPP_int_nat(Q,R)) ).
fof(help_COMBB_1_1_COMBB_000tc__Nat__Onat_000tc__fun_Itc__Nat__Onat_Mtc__Nat__Onat_J_003,axiom,
! [P,Q,R] : hAPP_nat_fun_nat_nat(hAPP_f1639111240at_nat(hAPP_f1731313045at_nat(cOMBB_963856155at_nat,P),Q),R) = hAPP_nat_fun_nat_nat(P,hAPP_nat_nat(Q,R)) ).
fof(help_COMBB_1_1_COMBB_000tc__Int__Oint_000tc__fun_Itc__Int__Oint_Mtc__HOL__Obool_,axiom,
! [P,Q,R] : hAPP_i1948725293t_bool(hAPP_f1791153283t_bool(hAPP_f1399575567t_bool(cOMBB_118231410ol_int,P),Q),R) = hAPP_i1948725293t_bool(P,hAPP_int_int(Q,R)) ).
fof(help_COMBB_1_1_COMBB_000tc__Int__Oint_000tc__fun_Itc__Int__Oint_Mtc__HOL__Obool__004,axiom,
! [P,Q,R] : hAPP_n1082236369t_bool(hAPP_f1533130627t_bool(hAPP_f879109391t_bool(cOMBB_1628441366ol_nat,P),Q),R) = hAPP_i1948725293t_bool(P,hAPP_nat_int(Q,R)) ).
fof(help_COMBB_1_1_COMBB_000tc__Nat__Onat_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_,axiom,
! [P,Q,R] : hAPP_i418383825t_bool(hAPP_f876579787t_bool(hAPP_f2026117279t_bool(cOMBB_1437810218ol_int,P),Q),R) = hAPP_n1699378549t_bool(P,hAPP_int_nat(Q,R)) ).
fof(help_COMBB_1_1_COMBB_000tc__Nat__Onat_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool__005,axiom,
! [P,Q,R] : hAPP_n1699378549t_bool(hAPP_f618557131t_bool(hAPP_f1505651103t_bool(cOMBB_800536526ol_nat,P),Q),R) = hAPP_n1699378549t_bool(P,hAPP_nat_nat(Q,R)) ).
fof(help_COMBB_1_1_COMBB_000tc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J_000tc__Nat__Ona,axiom,
! [P,Q,R] : hAPP_int_nat(hAPP_f1134349059nt_nat(hAPP_f2100446809nt_nat(cOMBB_2018621492at_int,P),Q),R) = hAPP_f957591787ol_nat(P,hAPP_i1948725293t_bool(Q,R)) ).
fof(help_COMBB_1_1_COMBB_000tc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J_000tc__Nat__Ona_006,axiom,
! [P,Q,R] : hAPP_nat_nat(hAPP_f389300155at_nat(hAPP_f109563153at_nat(cOMBB_1381347800at_nat,P),Q),R) = hAPP_f957591787ol_nat(P,hAPP_n1082236369t_bool(Q,R)) ).
fof(help_COMBB_1_1_COMBB_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_000tc__Nat__Ona,axiom,
! [P,Q,R] : hAPP_int_nat(hAPP_f1161717855nt_nat(hAPP_f1964560145nt_nat(cOMBB_1125113304at_int,P),Q),R) = hAPP_f22106695ol_nat(P,hAPP_i418383825t_bool(Q,R)) ).
fof(help_COMBC_1_1_COMBC_000tc__Int__Oint_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_,axiom,
! [P,Q,R] : hAPP_int_nat(hAPP_f787214110nt_nat(hAPP_f261731407nt_nat(cOMBC_462113011ol_nat,P),Q),R) = hAPP_f22106695ol_nat(hAPP_i1530422220ol_nat(P,R),Q) ).
fof(help_COMBC_1_1_COMBC_000tc__Nat__Onat_000tc__fun_Itc__Int__Oint_Mtc__HOL__Obool_,axiom,
! [P,Q,R] : hAPP_nat_nat(hAPP_f1473602334at_nat(hAPP_f630508183at_nat(cOMBC_1505178355ol_nat,P),Q),R) = hAPP_f957591787ol_nat(hAPP_n259507732ol_nat(P,R),Q) ).
fof(help_COMBS_1_1_COMBS_000tc__Int__Oint_000tc__fun_Itc__Int__Oint_Mtc__HOL__Obool_,axiom,
! [P,Q,R] : hAPP_int_nat(hAPP_f1134349059nt_nat(hAPP_f446447448nt_nat(cOMBS_1999073191ol_nat,P),Q),R) = hAPP_f957591787ol_nat(hAPP_i318423664ol_nat(P,R),hAPP_i1948725293t_bool(Q,R)) ).
fof(help_COMBS_1_1_COMBS_000tc__Int__Oint_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_,axiom,
! [P,Q,R] : hAPP_int_nat(hAPP_f1161717855nt_nat(hAPP_f369809936nt_nat(cOMBS_1063588099ol_nat,P),Q),R) = hAPP_f22106695ol_nat(hAPP_i1530422220ol_nat(P,R),hAPP_i418383825t_bool(Q,R)) ).
fof(help_COMBS_1_1_COMBS_000tc__Nat__Onat_000tc__fun_Itc__Int__Oint_Mtc__HOL__Obool_,axiom,
! [P,Q,R] : hAPP_nat_nat(hAPP_f389300155at_nat(hAPP_f1750007732at_nat(cOMBS_2106653443ol_nat,P),Q),R) = hAPP_f957591787ol_nat(hAPP_n259507732ol_nat(P,R),hAPP_n1082236369t_bool(Q,R)) ).
fof(help_COMBB_1_1_COMBB_000tc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J_000tc__HOL__Obo,axiom,
! [P,Q,R] : hAPP_int_bool(hAPP_f2119767738t_bool(hAPP_f423804115t_bool(cOMBB_1418110531ol_int,P),Q),R) = hAPP_f448129468l_bool(P,hAPP_i1948725293t_bool(Q,R)) ).
fof(help_COMBB_1_1_COMBB_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_000tc__HOL__Obo,axiom,
! [P,Q,R] : hAPP_nat_bool(hAPP_f158894502t_bool(hAPP_f1777703707t_bool(cOMBB_955900739ol_nat,P),Q),R) = hAPP_f54304608l_bool(P,hAPP_n1699378549t_bool(Q,R)) ).
fof(help_COMBC_1_1_COMBC_000tc__Int__Oint_000tc__fun_Itc__Int__Oint_Mtc__HOL__Obool_,axiom,
! [P,Q,R] : hAPP_int_bool(hAPP_f1805168059t_bool(hAPP_f202917053t_bool(cOMBC_94739984l_bool,P),Q),R) = hAPP_f448129468l_bool(hAPP_i2112223885l_bool(P,R),Q) ).
fof(help_COMBC_1_1_COMBC_000tc__Nat__Onat_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_,axiom,
! [P,Q,R] : hAPP_nat_bool(hAPP_f800510211t_bool(hAPP_f1722879237t_bool(cOMBC_226598744l_bool,P),Q),R) = hAPP_f54304608l_bool(hAPP_n215258509l_bool(P,R),Q) ).
fof(help_COMBB_1_1_COMBB_000tc__HOL__Obool_000tc__fun_Itc__HOL__Obool_Mtc__HOL__Oboo,axiom,
! [P,Q,R] : hAPP_i68813070l_bool(hAPP_f2144054103l_bool(hAPP_f1734373249l_bool(cOMBB_1652995168ol_int,P),Q),R) = hAPP_b589554111l_bool(P,hAPP_int_bool(Q,R)) ).
fof(help_COMBB_1_1_COMBB_000tc__HOL__Obool_000tc__fun_Itc__HOL__Obool_Mtc__HOL__Oboo_007,axiom,
! [P,Q,R] : hAPP_n1006566506l_bool(hAPP_f1146629647l_bool(hAPP_f1080886329l_bool(cOMBB_1015721476ol_nat,P),Q),R) = hAPP_b589554111l_bool(P,hAPP_nat_bool(Q,R)) ).
fof(help_COMBB_1_1_COMBB_000tc__HOL__Obool_000tc__fun_Itc__HOL__Obool_Mtc__HOL__Oboo_008,axiom,
! [P,Q,R] : hAPP_s58564346l_bool(hAPP_f1259673775l_bool(hAPP_f1561913689l_bool(cOMBB_188601460_state,P),Q),R) = hAPP_b589554111l_bool(P,hAPP_state_bool(Q,R)) ).
fof(help_COMBC_1_1_COMBC_000tc__Com__Ostate_000tc__Com__Ovname_000tc__fun_Itc__Nat__,axiom,
! [P,Q,R] : hAPP_s1892499976_state(hAPP_v365393659_state(hAPP_f1542232213_state(cOMBC_1193272608_state,P),Q),R) = hAPP_v594194232_state(hAPP_s712361723_state(P,R),Q) ).
fof(help_fequal_1_1_fequal_000tc__fun_Itc__Hoare____Mirabelle____uwgpyvfjxg__Otriple,axiom,
! [X,Y] :
( ~ hBOOL(hAPP_f1695230391l_bool(hAPP_f1767618879l_bool(fequal874423448a_bool,X),Y))
| X = Y ) ).
fof(help_fequal_2_1_fequal_000tc__fun_Itc__Hoare____Mirabelle____uwgpyvfjxg__Otriple,axiom,
! [X,Y] :
( X != Y
| hBOOL(hAPP_f1695230391l_bool(hAPP_f1767618879l_bool(fequal874423448a_bool,X),Y)) ) ).
fof(help_COMBB_1_1_COMBB_000tc__Nat__Onat_000tc__Nat__Onat_000tc__Hoare____Mirabelle,axiom,
! [P,Q,R] : hAPP_H592031934_a_nat(hAPP_f106743253_a_nat(hAPP_f500900629_a_nat(cOMBB_1250632933iple_a,P),Q),R) = hAPP_nat_nat(P,hAPP_H592031934_a_nat(Q,R)) ).
fof(help_COMBC_1_1_COMBC_000tc__Hoare____Mirabelle____uwgpyvfjxg__Otriple_It__a_J_00,axiom,
! [P,Q,R] : hAPP_H592031934_a_nat(hAPP_i1313984149_a_nat(hAPP_f183615253_a_nat(cOMBC_2020858056nt_nat,P),Q),R) = hAPP_int_nat(hAPP_H978941973nt_nat(P,R),Q) ).
fof(help_COMBC_1_1_COMBC_000tc__Hoare____Mirabelle____uwgpyvfjxg__Otriple_It__a_J_00_009,axiom,
! [P,Q,R] : hAPP_H592031934_a_nat(hAPP_n1653940209_a_nat(hAPP_f1903462933_a_nat(cOMBC_667342884at_nat,P),Q),R) = hAPP_nat_nat(hAPP_H1772910449at_nat(P,R),Q) ).
fof(help_COMBC_1_1_COMBC_000tc__Int__Oint_000tc__Hoare____Mirabelle____uwgpyvfjxg__O,axiom,
! [P,Q,R] : hAPP_int_nat(hAPP_H978941973nt_nat(hAPP_f1451619093nt_nat(cOMBC_1917249810_a_nat,P),Q),R) = hAPP_H592031934_a_nat(hAPP_i1313984149_a_nat(P,R),Q) ).
fof(help_COMBC_1_1_COMBC_000tc__Nat__Onat_000tc__Hoare____Mirabelle____uwgpyvfjxg__O,axiom,
! [P,Q,R] : hAPP_nat_nat(hAPP_H1772910449at_nat(hAPP_f2146713109at_nat(cOMBC_441305270_a_nat,P),Q),R) = hAPP_H592031934_a_nat(hAPP_n1653940209_a_nat(P,R),Q) ).
fof(help_COMBS_1_1_COMBS_000tc__Hoare____Mirabelle____uwgpyvfjxg__Otriple_It__a_J_00,axiom,
! [P,Q,R] : hAPP_H1229305626_a_int(hAPP_f1561000661_a_int(hAPP_f789902073_a_int(cOMBS_1600277268nt_int,P),Q),R) = hAPP_int_int(hAPP_H1140854897nt_int(P,R),hAPP_H1229305626_a_int(Q,R)) ).
fof(help_COMBS_1_1_COMBS_000tc__Hoare____Mirabelle____uwgpyvfjxg__Otriple_It__a_J_00_010,axiom,
! [P,Q,R] : hAPP_H592031934_a_nat(hAPP_f106743253_a_nat(hAPP_f1418102009_a_nat(cOMBS_1756972052at_nat,P),Q),R) = hAPP_nat_nat(hAPP_H1772910449at_nat(P,R),hAPP_H592031934_a_nat(Q,R)) ).
fof(help_COMBC_1_1_COMBC_000tc__Hoare____Mirabelle____uwgpyvfjxg__Otriple_It__a_J_00_011,axiom,
! [P,Q,R] : hAPP_H1037229737a_bool(hAPP_i611776424a_bool(hAPP_f920351535a_bool(cOMBC_1530706207t_bool,P),Q),R) = hAPP_int_bool(hAPP_H1675210280t_bool(P,R),Q) ).
fof(help_COMBC_1_1_COMBC_000tc__Hoare____Mirabelle____uwgpyvfjxg__Otriple_It__a_J_00_012,axiom,
! [P,Q,R] : hAPP_H1037229737a_bool(hAPP_n1919155532a_bool(hAPP_f228417655a_bool(cOMBC_364739t_bool,P),Q),R) = hAPP_nat_bool(hAPP_H144868812t_bool(P,R),Q) ).
fof(help_COMBC_1_1_COMBC_000tc__Int__Oint_000tc__Hoare____Mirabelle____uwgpyvfjxg__O_013,axiom,
! [P,Q,R] : hAPP_int_bool(hAPP_H1675210280t_bool(hAPP_f936407855t_bool(cOMBC_1719122773a_bool,P),Q),R) = hAPP_H1037229737a_bool(hAPP_i611776424a_bool(P,R),Q) ).
fof(help_COMBC_1_1_COMBC_000tc__Nat__Onat_000tc__Hoare____Mirabelle____uwgpyvfjxg__O_014,axiom,
! [P,Q,R] : hAPP_nat_bool(hAPP_H144868812t_bool(hAPP_f1886543991t_bool(cOMBC_533513393a_bool,P),Q),R) = hAPP_H1037229737a_bool(hAPP_n1919155532a_bool(P,R),Q) ).
fof(help_COMBB_1_1_COMBB_000tc__HOL__Obool_000tc__HOL__Obool_000tc__Hoare____Mirabel,axiom,
! [P,Q,R] : hAPP_H1037229737a_bool(hAPP_f20753329a_bool(hAPP_f604201481a_bool(cOMBB_1882975613iple_a,P),Q),R) = hAPP_bool_bool(P,hAPP_H1037229737a_bool(Q,R)) ).
fof(help_COMBS_1_1_COMBS_000tc__Hoare____Mirabelle____uwgpyvfjxg__Otriple_It__a_J_00_015,axiom,
! [P,Q,R] : hAPP_H1037229737a_bool(hAPP_f20753329a_bool(hAPP_f607497069a_bool(cOMBS_154988028l_bool,P),Q),R) = hAPP_bool_bool(hAPP_H2130837971l_bool(P,R),hAPP_H1037229737a_bool(Q,R)) ).
fof(help_COMBC_1_1_COMBC_000t__a_000tc__fun_Itc__Com__Ostate_Mtc__HOL__Obool_J_000tc,axiom,
! [P,Q,R] : hAPP_a2036067514e_bool(hAPP_f762886889e_bool(hAPP_f1261923407e_bool(cOMBC_892787026e_bool,P),Q),R) = hAPP_f1759915619e_bool(hAPP_a1200519163e_bool(P,R),Q) ).
fof(help_COMBC_1_1_COMBC_000t__a_000tc__fun_Itc__Com__Ostate_Mtc__Com__Ostate_J_000t,axiom,
! [P,Q,R] : hAPP_a2036067514e_bool(hAPP_f100967412e_bool(hAPP_f1460451647e_bool(cOMBC_524597097e_bool,P),Q),R) = hAPP_f746301080e_bool(hAPP_a1224971408e_bool(P,R),Q) ).
fof(help_COMBC_1_1_COMBC_000tc__Int__Oint_000tc__fun_Itc__Nat__Onat_Mtc__Nat__Onat_J,axiom,
! [P,Q,R] : hAPP_int_fun_nat_nat(hAPP_f1169617132at_nat(hAPP_f701299925at_nat(cOMBC_1294079849at_nat,P),Q),R) = hAPP_f1914919701at_nat(hAPP_i1876697324at_nat(P,R),Q) ).
fof(help_COMBC_1_1_COMBC_000tc__Nat__Onat_000tc__fun_Itc__Int__Oint_Mtc__Nat__Onat_J,axiom,
! [P,Q,R] : hAPP_nat_fun_int_nat(hAPP_f237327688nt_nat(hAPP_f1633513941nt_nat(cOMBC_1218031117nt_nat,P),Q),R) = hAPP_f2132704789nt_nat(hAPP_n74706760nt_nat(P,R),Q) ).
fof(help_COMBB_1_1_COMBB_000tc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J_000tc__fun_Itc_,axiom,
! [P,Q,R] : hAPP_i1948725293t_bool(hAPP_f428220345t_bool(hAPP_f654702867t_bool(cOMBB_591320580ol_int,P),Q),R) = hAPP_f1805168059t_bool(P,hAPP_i1948725293t_bool(Q,R)) ).
fof(help_COMBB_1_1_COMBB_000tc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J_000tc__fun_Itc__016,axiom,
! [P,Q,R] : hAPP_n1082236369t_bool(hAPP_f2022049025t_bool(hAPP_f1986088027t_bool(cOMBB_2101530536ol_nat,P),Q),R) = hAPP_f1805168059t_bool(P,hAPP_n1082236369t_bool(Q,R)) ).
fof(help_COMBB_1_1_COMBB_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_000tc__fun_Itc_,axiom,
! [P,Q,R] : hAPP_i418383825t_bool(hAPP_f783004929t_bool(hAPP_f1410409747t_bool(cOMBB_1746576572ol_int,P),Q),R) = hAPP_f800510211t_bool(P,hAPP_i418383825t_bool(Q,R)) ).
fof(help_COMBC_1_1_COMBC_000tc__Int__Oint_000tc__fun_Itc__Int__Oint_Mtc__HOL__Obool__017,axiom,
! [P,Q,R] : hAPP_i1948725293t_bool(hAPP_f1048215610t_bool(hAPP_f472159229t_bool(cOMBC_1683390479t_bool,P),Q),R) = hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(P,R),Q) ).
fof(help_COMBC_1_1_COMBC_000tc__Nat__Onat_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool__018,axiom,
! [P,Q,R] : hAPP_n1699378549t_bool(hAPP_f1570313510t_bool(hAPP_f688831301t_bool(cOMBC_178881787t_bool,P),Q),R) = hAPP_f800510211t_bool(hAPP_n1512601776t_bool(P,R),Q) ).
fof(help_COMBC_1_1_COMBC_000tc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J_000tc__fun_Itc_,axiom,
! [P,Q,R] : hAPP_f448129468l_bool(hAPP_f284875647l_bool(hAPP_f1399552105l_bool(cOMBC_605892544l_bool,P),Q),R) = hAPP_f448129468l_bool(hAPP_f284875647l_bool(P,R),Q) ).
fof(help_COMBC_1_1_COMBC_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_000tc__fun_Itc_,axiom,
! [P,Q,R] : hAPP_f54304608l_bool(hAPP_f103356543l_bool(hAPP_f643944041l_bool(cOMBC_1693257480l_bool,P),Q),R) = hAPP_f54304608l_bool(hAPP_f103356543l_bool(P,R),Q) ).
fof(help_COMBK_1_1_COMBK_000tc__Hoare____Mirabelle____uwgpyvfjxg__Otriple_It__a_J_00,axiom,
! [P,Q] : hAPP_H905846293iple_a(hAPP_H2085992369iple_a(cOMBK_2052406350iple_a,P),Q) = P ).
fof(help_COMBB_1_1_COMBB_000tc__Int__Oint_000tc__fun_Itc__Int__Oint_Mtc__Int__Oint_J,axiom,
! [P,Q,R] : hAPP_H1140854897nt_int(hAPP_f231878828nt_int(hAPP_f1621780181nt_int(cOMBB_1885471694iple_a,P),Q),R) = hAPP_int_fun_int_int(P,hAPP_H1229305626_a_int(Q,R)) ).
fof(help_COMBB_1_1_COMBB_000tc__Nat__Onat_000tc__fun_Itc__Nat__Onat_Mtc__Nat__Onat_J_019,axiom,
! [P,Q,R] : hAPP_H1772910449at_nat(hAPP_f429024520at_nat(hAPP_f703420885at_nat(cOMBB_789959282iple_a,P),Q),R) = hAPP_nat_fun_nat_nat(P,hAPP_H592031934_a_nat(Q,R)) ).
fof(help_COMBB_1_1_COMBB_000tc__Int__Oint_000tc__fun_Itc__Int__Oint_Mtc__HOL__Obool__020,axiom,
! [P,Q,R] : hAPP_H1675210280t_bool(hAPP_f2142913731t_bool(hAPP_f1329415375t_bool(cOMBB_1505691757iple_a,P),Q),R) = hAPP_i1948725293t_bool(P,hAPP_H1229305626_a_int(Q,R)) ).
fof(help_COMBB_1_1_COMBB_000tc__Nat__Onat_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool__021,axiom,
! [P,Q,R] : hAPP_H144868812t_bool(hAPP_f1872860171t_bool(hAPP_f1981448543t_bool(cOMBB_1391527205iple_a,P),Q),R) = hAPP_n1699378549t_bool(P,hAPP_H592031934_a_nat(Q,R)) ).
fof(help_COMBB_1_1_COMBB_000tc__fun_Itc__Hoare____Mirabelle____uwgpyvfjxg__Otriple_I,axiom,
! [P,Q,R] : hAPP_nat_nat(hAPP_f966999552at_nat(hAPP_f724467227at_nat(cOMBB_161916371at_nat,P),Q),R) = hAPP_f2097660464ol_nat(P,hAPP_n1919155532a_bool(Q,R)) ).
fof(help_COMBB_1_1_COMBB_000tc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J_000tc__Nat__Ona_022,axiom,
! [P,Q,R] : hAPP_H592031934_a_nat(hAPP_f342603021_a_nat(hAPP_f575660003_a_nat(cOMBB_1357028143iple_a,P),Q),R) = hAPP_f957591787ol_nat(P,hAPP_H1675210280t_bool(Q,R)) ).
fof(help_COMBB_1_1_COMBB_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_000tc__Nat__Ona_023,axiom,
! [P,Q,R] : hAPP_H592031934_a_nat(hAPP_f2085403241_a_nat(hAPP_f734830747_a_nat(cOMBB_1107515603iple_a,P),Q),R) = hAPP_f22106695ol_nat(P,hAPP_H144868812t_bool(Q,R)) ).
fof(help_COMBC_1_1_COMBC_000tc__Hoare____Mirabelle____uwgpyvfjxg__Otriple_It__a_J_00_024,axiom,
! [P,Q,R] : hAPP_H592031934_a_nat(hAPP_f1464798435_a_nat(hAPP_f1012303887_a_nat(cOMBC_1159891896ol_nat,P),Q),R) = hAPP_f22106695ol_nat(hAPP_H1993634375ol_nat(P,R),Q) ).
fof(help_COMBC_1_1_COMBC_000tc__Nat__Onat_000tc__fun_Itc__Hoare____Mirabelle____uwgp,axiom,
! [P,Q,R] : hAPP_nat_nat(hAPP_f2122367971at_nat(hAPP_f870149665at_nat(cOMBC_2036134968ol_nat,P),Q),R) = hAPP_f2097660464ol_nat(hAPP_n130255193ol_nat(P,R),Q) ).
fof(help_COMBS_1_1_COMBS_000tc__Hoare____Mirabelle____uwgpyvfjxg__Otriple_It__a_J_00_025,axiom,
! [P,Q,R] : hAPP_H592031934_a_nat(hAPP_f342603021_a_nat(hAPP_f270087133_a_nat(cOMBS_1333400172ol_nat,P),Q),R) = hAPP_f957591787ol_nat(hAPP_H781635819ol_nat(P,R),hAPP_H1675210280t_bool(Q,R)) ).
fof(help_COMBS_1_1_COMBS_000tc__Hoare____Mirabelle____uwgpyvfjxg__Otriple_It__a_J_00_026,axiom,
! [P,Q,R] : hAPP_H592031934_a_nat(hAPP_f2085403241_a_nat(hAPP_f1052116629_a_nat(cOMBS_397915080ol_nat,P),Q),R) = hAPP_f22106695ol_nat(hAPP_H1993634375ol_nat(P,R),hAPP_H144868812t_bool(Q,R)) ).
fof(help_COMBS_1_1_COMBS_000tc__Int__Oint_000tc__fun_Itc__Hoare____Mirabelle____uwgp,axiom,
! [P,Q,R] : hAPP_int_nat(hAPP_f2050777416nt_nat(hAPP_f1895921634nt_nat(cOMBS_202713324ol_nat,P),Q),R) = hAPP_f2097660464ol_nat(hAPP_i51375541ol_nat(P,R),hAPP_i611776424a_bool(Q,R)) ).
fof(help_COMBS_1_1_COMBS_000tc__Nat__Onat_000tc__fun_Itc__Hoare____Mirabelle____uwgp,axiom,
! [P,Q,R] : hAPP_nat_nat(hAPP_f966999552at_nat(hAPP_f264818750at_nat(cOMBS_1274158152ol_nat,P),Q),R) = hAPP_f2097660464ol_nat(hAPP_n130255193ol_nat(P,R),hAPP_n1919155532a_bool(Q,R)) ).
fof(help_COMBB_1_1_COMBB_000tc__HOL__Obool_000tc__fun_Itc__HOL__Obool_Mtc__HOL__Oboo_027,axiom,
! [P,Q,R] : hAPP_H2130837971l_bool(hAPP_f146754017l_bool(hAPP_f239971723l_bool(cOMBB_1866391387iple_a,P),Q),R) = hAPP_b589554111l_bool(P,hAPP_H1037229737a_bool(Q,R)) ).
fof(help_COMBB_1_1_COMBB_000tc__fun_Itc__Com__Ostate_Mtc__HOL__Obool_J_000tc__fun_It,axiom,
! [P,Q,R] : hAPP_a849909144l_bool(hAPP_f1509969235l_bool(hAPP_f1178339559l_bool(cOMBB_1355796797bool_a,P),Q),R) = hAPP_f1259673775l_bool(P,hAPP_a2036067514e_bool(Q,R)) ).
fof(help_COMBB_1_1_COMBB_000tc__fun_Itc__Int__Oint_Mtc__Nat__Onat_J_000tc__fun_Itc__,axiom,
! [P,Q,R] : hAPP_i318423664ol_nat(hAPP_f1548925761ol_nat(hAPP_f1331458699ol_nat(cOMBB_2091167284at_int,P),Q),R) = hAPP_f1109019371ol_nat(P,hAPP_int_fun_int_nat(Q,R)) ).
fof(help_COMBB_1_1_COMBB_000tc__fun_Itc__Int__Oint_Mtc__Nat__Onat_J_000tc__fun_Itc___028,axiom,
! [P,Q,R] : hAPP_n259507732ol_nat(hAPP_f299305025ol_nat(hAPP_f1632710027ol_nat(cOMBB_1453893592at_nat,P),Q),R) = hAPP_f1109019371ol_nat(P,hAPP_nat_fun_int_nat(Q,R)) ).
fof(help_COMBB_1_1_COMBB_000tc__fun_Itc__Nat__Onat_Mtc__Nat__Onat_J_000tc__fun_Itc__,axiom,
! [P,Q,R] : hAPP_i1530422220ol_nat(hAPP_f1501159417ol_nat(hAPP_f1029311995ol_nat(cOMBB_1562322300at_int,P),Q),R) = hAPP_f782000547ol_nat(P,hAPP_int_fun_nat_nat(Q,R)) ).
fof(help_COMBC_1_1_COMBC_000tc__Hoare____Mirabelle____uwgpyvfjxg__Otriple_It__a_J_00_029,axiom,
! [P,Q,R] : hAPP_H1037229737a_bool(hAPP_H426895267a_bool(hAPP_f920293029a_bool(cOMBC_1940896922a_bool,P),Q),R) = hAPP_H1037229737a_bool(hAPP_H426895267a_bool(P,R),Q) ).
fof(help_COMBC_1_1_COMBC_000tc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J_000tc__fun_Itc__030,axiom,
! [P,Q,R] : hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(hAPP_f194616807t_bool(cOMBC_922884543t_bool,P),Q),R) = hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(P,R),Q) ).
fof(help_COMBC_1_1_COMBC_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_000tc__fun_Itc__031,axiom,
! [P,Q,R] : hAPP_f800510211t_bool(hAPP_f1730770594t_bool(hAPP_f1311642927t_bool(cOMBC_646315179t_bool,P),Q),R) = hAPP_f800510211t_bool(hAPP_f1730770594t_bool(P,R),Q) ).
fof(help_COMBC_1_1_COMBC_000tc__Hoare____Mirabelle____uwgpyvfjxg__Otriple_It__a_J_00_032,axiom,
! [P,Q,R] : hAPP_H1772910449at_nat(hAPP_f221335601at_nat(hAPP_f388255893at_nat(cOMBC_1038836708at_nat,P),Q),R) = hAPP_f1914919701at_nat(hAPP_H1651200817at_nat(P,R),Q) ).
fof(help_COMBB_1_1_COMBB_000tc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J_000tc__fun_Itc__033,axiom,
! [P,Q,R] : hAPP_H1675210280t_bool(hAPP_f138254639t_bool(hAPP_f1998609161t_bool(cOMBB_1870764287iple_a,P),Q),R) = hAPP_f1805168059t_bool(P,hAPP_H1675210280t_bool(Q,R)) ).
fof(help_COMBB_1_1_COMBB_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_000tc__fun_Itc__034,axiom,
! [P,Q,R] : hAPP_H144868812t_bool(hAPP_f27379319t_bool(hAPP_f196067081t_bool(cOMBB_437532599iple_a,P),Q),R) = hAPP_f800510211t_bool(P,hAPP_H144868812t_bool(Q,R)) ).
fof(help_COMBB_1_1_COMBB_000tc__fun_Itc__Com__Ostate_Mtc__fun_Itc__HOL__Obool_Mtc__H,axiom,
! [P,Q,R] : hAPP_a723219176e_bool(hAPP_f340725611e_bool(hAPP_f1006724181e_bool(cOMBB_1348041619bool_a,P),Q),R) = hAPP_f167292325e_bool(P,hAPP_a849909144l_bool(Q,R)) ).
fof(help_COMBB_1_1_COMBB_000tc__fun_Itc__Nat__Onat_Mtc__Nat__Onat_J_000tc__fun_Itc___035,axiom,
! [P,Q,R] : hAPP_n74706760nt_nat(hAPP_f2080483477nt_nat(hAPP_f1701814485nt_nat(cOMBB_737513486at_nat,P),Q),R) = hAPP_f901215189nt_nat(P,hAPP_nat_fun_nat_nat(Q,R)) ).
fof(help_COMBB_1_1_COMBB_000tc__fun_Itc__Nat__Onat_Mtc__Nat__Onat_J_000tc__fun_Itc___036,axiom,
! [P,Q,R] : hAPP_i1876697324at_nat(hAPP_f1033905301at_nat(hAPP_f1315855317at_nat(cOMBB_891709290at_int,P),Q),R) = hAPP_f1585078997at_nat(P,hAPP_int_fun_nat_nat(Q,R)) ).
fof(help_COMBB_1_1_COMBB_000tc__fun_Itc__Com__Ostate_Mtc__HOL__Obool_J_000tc__fun_It_037,axiom,
! [P,Q,R] : hAPP_a1224971408e_bool(hAPP_f1406200875e_bool(hAPP_f635443597e_bool(cOMBB_2139825703bool_a,P),Q),R) = hAPP_f162060345e_bool(P,hAPP_a2036067514e_bool(Q,R)) ).
fof(help_COMBB_1_1_COMBB_000tc__Hoare____Mirabelle____uwgpyvfjxg__Otriple_It__a_J_00,axiom,
! [P,Q,R] : hAPP_i611776424a_bool(hAPP_f298605433a_bool(hAPP_f1421841531a_bool(cOMBB_1700417404ol_int,P),Q),R) = hAPP_H426895267a_bool(P,hAPP_i240634960iple_a(Q,R)) ).
fof(help_COMBB_1_1_COMBB_000tc__Hoare____Mirabelle____uwgpyvfjxg__Otriple_It__a_J_00_038,axiom,
! [P,Q,R] : hAPP_n1919155532a_bool(hAPP_f260875897a_bool(hAPP_f1930382715a_bool(cOMBB_1063143712ol_nat,P),Q),R) = hAPP_H426895267a_bool(P,hAPP_n261501868iple_a(Q,R)) ).
fof(help_COMBB_1_1_COMBB_000tc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J_000tc__fun_Itc__039,axiom,
! [P,Q,R] : hAPP_i1529485324t_bool(hAPP_f1468280982t_bool(hAPP_f1596157055t_bool(cOMBB_765314529ol_int,P),Q),R) = hAPP_f1223193598t_bool(P,hAPP_i1948725293t_bool(Q,R)) ).
fof(help_COMBB_1_1_COMBB_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_000tc__fun_Itc__040,axiom,
! [P,Q,R] : hAPP_n1512601776t_bool(hAPP_f10625010t_bool(hAPP_f1115950719t_bool(cOMBB_1654519265ol_nat,P),Q),R) = hAPP_f1730770594t_bool(P,hAPP_n1699378549t_bool(Q,R)) ).
fof(help_COMBB_1_1_COMBB_000tc__fun_Itc__Hoare____Mirabelle____uwgpyvfjxg__Otriple_I_041,axiom,
! [P,Q,R] : hAPP_H1037229737a_bool(hAPP_f1194757675a_bool(hAPP_f1697223433a_bool(cOMBB_1610959875iple_a,P),Q),R) = hAPP_f1695230391l_bool(P,hAPP_H426895267a_bool(Q,R)) ).
fof(help_COMBC_1_1_COMBC_000tc__Hoare____Mirabelle____uwgpyvfjxg__Otriple_It__a_J_00_042,axiom,
! [P,Q,R] : hAPP_H1037229737a_bool(hAPP_f20753329a_bool(hAPP_f247708275a_bool(cOMBC_1519416966l_bool,P),Q),R) = hAPP_f1695230391l_bool(hAPP_H622608077l_bool(P,R),Q) ).
fof(help_COMBB_1_1_COMBB_000tc__fun_Itc__Int__Oint_Mtc__Nat__Onat_J_000tc__fun_Itc___043,axiom,
! [P,Q,R] : hAPP_H781635819ol_nat(hAPP_f224922881ol_nat(hAPP_f543861963ol_nat(cOMBB_1075417391iple_a,P),Q),R) = hAPP_f1109019371ol_nat(P,hAPP_H978941973nt_nat(Q,R)) ).
fof(help_COMBB_1_1_COMBB_000tc__fun_Itc__Nat__Onat_Mtc__Nat__Onat_J_000tc__fun_Itc___044,axiom,
! [P,Q,R] : hAPP_H1993634375ol_nat(hAPP_f1669765561ol_nat(hAPP_f1073041467ol_nat(cOMBB_10565751iple_a,P),Q),R) = hAPP_f782000547ol_nat(P,hAPP_H1772910449at_nat(Q,R)) ).
fof(help_COMBB_1_1_COMBB_000tc__fun_Itc__Com__Ostate_Mtc__fun_Itc__HOL__Obool_Mtc__H_045,axiom,
! [P,Q,R] : hAPP_a1200519163e_bool(hAPP_f963367678e_bool(hAPP_f375255701e_bool(cOMBB_145932198bool_a,P),Q),R) = hAPP_f644196280e_bool(P,hAPP_a849909144l_bool(Q,R)) ).
fof(help_COMBC_1_1_COMBC_000tc__Nat__Onat_000tc__fun_Itc__Hoare____Mirabelle____uwgp_046,axiom,
! [P,Q,R] : hAPP_n1653940209_a_nat(hAPP_f1393576712_a_nat(hAPP_f145529813_a_nat(cOMBC_309106125_a_nat,P),Q),R) = hAPP_f106743253_a_nat(hAPP_n352002696_a_nat(P,R),Q) ).
fof(help_COMBB_1_1_COMBB_000tc__fun_Itc__Hoare____Mirabelle____uwgpyvfjxg__Otriple_I_047,axiom,
! [P,Q,R] : hAPP_i611776424a_bool(hAPP_f1718504751a_bool(hAPP_f481410067a_bool(cOMBB_1028320654ol_int,P),Q),R) = hAPP_f20753329a_bool(P,hAPP_i611776424a_bool(Q,R)) ).
fof(help_COMBB_1_1_COMBB_000tc__fun_Itc__Hoare____Mirabelle____uwgpyvfjxg__Otriple_I_048,axiom,
! [P,Q,R] : hAPP_n1919155532a_bool(hAPP_f2087582327a_bool(hAPP_f1396758363a_bool(cOMBB_391046962ol_nat,P),Q),R) = hAPP_f20753329a_bool(P,hAPP_n1919155532a_bool(Q,R)) ).
fof(help_COMBC_1_1_COMBC_000tc__fun_Itc__Hoare____Mirabelle____uwgpyvfjxg__Otriple_I,axiom,
! [P,Q,R] : hAPP_f1695230391l_bool(hAPP_f1767618879l_bool(hAPP_f2100528361l_bool(cOMBC_627599030l_bool,P),Q),R) = hAPP_f1695230391l_bool(hAPP_f1767618879l_bool(P,R),Q) ).
fof(help_COMBB_1_1_COMBB_000tc__fun_Itc__Nat__Onat_Mtc__Nat__Onat_J_000tc__fun_Itc___049,axiom,
! [P,Q,R] : hAPP_H1651200817at_nat(hAPP_f1879857877at_nat(hAPP_f481016213at_nat(cOMBB_1672982117iple_a,P),Q),R) = hAPP_f1585078997at_nat(P,hAPP_H1772910449at_nat(Q,R)) ).
fof(help_COMBB_1_1_COMBB_000tc__Hoare____Mirabelle____uwgpyvfjxg__Otriple_It__a_J_00_050,axiom,
! [P,Q,R] : hAPP_H426895267a_bool(hAPP_f960023481a_bool(hAPP_f1786170683a_bool(cOMBB_683172471iple_a,P),Q),R) = hAPP_H426895267a_bool(P,hAPP_H905846293iple_a(Q,R)) ).
fof(help_COMBB_1_1_COMBB_000tc__fun_Itc__Hoare____Mirabelle____uwgpyvfjxg__Otriple_I_051,axiom,
! [P,Q,R] : hAPP_i51375541ol_nat(hAPP_f1442126667ol_nat(hAPP_f1072896543ol_nat(cOMBB_206396714at_int,P),Q),R) = hAPP_f1371399797ol_nat(P,hAPP_i1313984149_a_nat(Q,R)) ).
fof(help_COMBB_1_1_COMBB_000tc__fun_Itc__Hoare____Mirabelle____uwgpyvfjxg__Otriple_I_052,axiom,
! [P,Q,R] : hAPP_n130255193ol_nat(hAPP_f883871819ol_nat(hAPP_f424754463ol_nat(cOMBB_1716606670at_nat,P),Q),R) = hAPP_f1371399797ol_nat(P,hAPP_n1653940209_a_nat(Q,R)) ).
fof(help_COMBC_1_1_COMBC_000tc__Hoare____Mirabelle____uwgpyvfjxg__Otriple_It__a_J_00_053,axiom,
! [P,Q,R] : hAPP_H426895267a_bool(hAPP_f1181346091a_bool(hAPP_f571476211a_bool(cOMBC_1570214144a_bool,P),Q),R) = hAPP_f20753329a_bool(hAPP_H1743777351a_bool(P,R),Q) ).
fof(help_COMBB_1_1_COMBB_000tc__fun_Itc__Nat__Onat_Mtc__Nat__Onat_J_000tc__fun_Itc___054,axiom,
! [P,Q,R] : hAPP_n352002696_a_nat(hAPP_f1485673429_a_nat(hAPP_f2006236373_a_nat(cOMBB_2005430478at_nat,P),Q),R) = hAPP_f500900629_a_nat(P,hAPP_nat_fun_nat_nat(Q,R)) ).
fof(help_COMBC_1_1_COMBC_000tc__fun_Itc__Hoare____Mirabelle____uwgpyvfjxg__Otriple_I_055,axiom,
! [P,Q,R] : hAPP_f20753329a_bool(hAPP_f1706700729a_bool(hAPP_f957014749a_bool(cOMBC_288844080a_bool,P),Q),R) = hAPP_f20753329a_bool(hAPP_f1706700729a_bool(P,R),Q) ).
fof(help_COMBB_1_1_COMBB_000tc__fun_Itc__Hoare____Mirabelle____uwgpyvfjxg__Otriple_I_056,axiom,
! [P,Q,R] : hAPP_H1743777351a_bool(hAPP_f550111323a_bool(hAPP_f615286271a_bool(cOMBB_1176122337iple_a,P),Q),R) = hAPP_f1706700729a_bool(P,hAPP_H426895267a_bool(Q,R)) ).
%----Conjectures (1)
fof(conj_0,conjecture,
hBOOL(hAPP_f1695230391l_bool(hoare_2102800559rivs_a(g),hAPP_f20753329a_bool(hAPP_H1743777351a_bool(insert956547291iple_a,hoare_1916936827iple_a(hAPP_f762886889e_bool(cOMBK_1458035955bool_a,hAPP_b2019457360e_bool(cOMBK_bool_state,fFalse)),c,hAPP_f762886889e_bool(hAPP_f1261923407e_bool(cOMBC_892787026e_bool,hAPP_f963367678e_bool(hAPP_f375255701e_bool(cOMBB_145932198bool_a,cOMBS_1378840469l_bool),hAPP_f1509969235l_bool(hAPP_f1178339559l_bool(cOMBB_1355796797bool_a,hAPP_f1561913689l_bool(cOMBB_188601460_state,fconj)),p))),hAPP_f1759915619e_bool(hAPP_f2073279419e_bool(cOMBB_160679318_state,fNot),b)))),bot_bo797238721a_bool))) ).
%------------------------------------------------------------------------------