%------------------------------------------------------------------------------
% File     : Leo-III---1.7.7
% Problem  : SWW470+3 : TPTP v8.1.2. Released v5.3.0.
% Transfm  : none
% Format   : tptp:raw
% Command  : run_Leo-III %s %d

% Computer : n009.cluster.edu
% Model    : x86_64 x86_64
% CPU      : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory   : 8042.1875MB
% OS       : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit  : 300s
% DateTime : Fri May 19 12:23:59 EDT 2023

% Result   : Theorem 93.85s 24.35s
% Output   : Refutation 94.89s
% Verified : 
% SZS Type : Refutation
%            Derivation depth      :    3
%            Number of leaves      : 1450
% Syntax   : Number of formulae    : 2581 ( 674 unt; 321 typ;   0 def)
%            Number of atoms       : 5824 (1990 equ;   0 cnn)
%            Maximal formula atoms :   14 (   2 avg)
%            Number of connectives : 39792 ( 636   ~; 141   |; 478   &;35592   @)
%                                         ( 183 <=>;2762  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   24 (   9 avg)
%            Number of types       :    2 (   0 usr)
%            Number of type conns  :  363 ( 363   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :  323 ( 321 usr; 125 con; 0-3 aty)
%            Number of variables   : 6362 (   0   ^;6177   !; 185   ?;6362   :)

% Comments : 
%------------------------------------------------------------------------------
thf(hBOOL_type,type,
    hBOOL: $i > $o ).

thf(hAPP_f1695230391l_bool_type,type,
    hAPP_f1695230391l_bool: $i > $i > $i ).

thf(hoare_2102800559rivs_a_type,type,
    hoare_2102800559rivs_a: $i > $i ).

thf(g_type,type,
    g: $i ).

thf(hAPP_f20753329a_bool_type,type,
    hAPP_f20753329a_bool: $i > $i > $i ).

thf(hAPP_H1743777351a_bool_type,type,
    hAPP_H1743777351a_bool: $i > $i > $i ).

thf(insert956547291iple_a_type,type,
    insert956547291iple_a: $i ).

thf(hoare_1916936827iple_a_type,type,
    hoare_1916936827iple_a: $i > $i > $i > $i ).

thf(hAPP_f762886889e_bool_type,type,
    hAPP_f762886889e_bool: $i > $i > $i ).

thf(cOMBK_1458035955bool_a_type,type,
    cOMBK_1458035955bool_a: $i ).

thf(hAPP_b2019457360e_bool_type,type,
    hAPP_b2019457360e_bool: $i > $i > $i ).

thf(cOMBK_bool_state_type,type,
    cOMBK_bool_state: $i ).

thf(fFalse_type,type,
    fFalse: $i ).

thf(c_type,type,
    c: $i ).

thf(hAPP_f1261923407e_bool_type,type,
    hAPP_f1261923407e_bool: $i > $i > $i ).

thf(cOMBC_892787026e_bool_type,type,
    cOMBC_892787026e_bool: $i ).

thf(hAPP_f963367678e_bool_type,type,
    hAPP_f963367678e_bool: $i > $i > $i ).

thf(hAPP_f375255701e_bool_type,type,
    hAPP_f375255701e_bool: $i > $i > $i ).

thf(cOMBB_145932198bool_a_type,type,
    cOMBB_145932198bool_a: $i ).

thf(cOMBS_1378840469l_bool_type,type,
    cOMBS_1378840469l_bool: $i ).

thf(hAPP_f1509969235l_bool_type,type,
    hAPP_f1509969235l_bool: $i > $i > $i ).

thf(hAPP_f1178339559l_bool_type,type,
    hAPP_f1178339559l_bool: $i > $i > $i ).

thf(cOMBB_1355796797bool_a_type,type,
    cOMBB_1355796797bool_a: $i ).

thf(hAPP_f1561913689l_bool_type,type,
    hAPP_f1561913689l_bool: $i > $i > $i ).

thf(cOMBB_188601460_state_type,type,
    cOMBB_188601460_state: $i ).

thf(fconj_type,type,
    fconj: $i ).

thf(p_type,type,
    p: $i ).

thf(hAPP_f1759915619e_bool_type,type,
    hAPP_f1759915619e_bool: $i > $i > $i ).

thf(hAPP_f2073279419e_bool_type,type,
    hAPP_f2073279419e_bool: $i > $i > $i ).

thf(cOMBB_160679318_state_type,type,
    cOMBB_160679318_state: $i ).

thf(fNot_type,type,
    fNot: $i ).

thf(b_type,type,
    b: $i ).

thf(bot_bo797238721a_bool_type,type,
    bot_bo797238721a_bool: $i ).

thf(hAPP_int_bool_type,type,
    hAPP_int_bool: $i > $i > $i ).

thf(hAPP_i1948725293t_bool_type,type,
    hAPP_i1948725293t_bool: $i > $i > $i ).

thf(ord_less_eq_int_type,type,
    ord_less_eq_int: $i ).

thf(hAPP_int_int_type,type,
    hAPP_int_int: $i > $i > $i ).

thf(uminus_uminus_int_type,type,
    uminus_uminus_int: $i ).

thf(hAPP_nat_int_type,type,
    hAPP_nat_int: $i > $i > $i ).

thf(semiri1621563631at_int_type,type,
    semiri1621563631at_int: $i ).

thf(hAPP_int_fun_int_int_type,type,
    hAPP_int_fun_int_int: $i > $i > $i ).

thf(plus_plus_int_type,type,
    plus_plus_int: $i ).

thf(ord_min_int_type,type,
    ord_min_int: $i ).

thf(one_one_int_type,type,
    one_one_int: $i ).

thf(times_times_int_type,type,
    times_times_int: $i ).

thf(minus_minus_int_type,type,
    minus_minus_int: $i ).

thf(is_bool_type,type,
    is_bool: $i > $o ).

thf(finite1704255308nt_int_type,type,
    finite1704255308nt_int: $i > $i ).

thf(hAPP_nat_bool_type,type,
    hAPP_nat_bool: $i > $i > $i ).

thf(hAPP_n1699378549t_bool_type,type,
    hAPP_n1699378549t_bool: $i > $i > $i ).

thf(ord_less_eq_nat_type,type,
    ord_less_eq_nat: $i ).

thf(hAPP_f22106695ol_nat_type,type,
    hAPP_f22106695ol_nat: $i > $i > $i ).

thf(the_nat_type,type,
    the_nat: $i ).

thf(hAPP_f448129468l_bool_type,type,
    hAPP_f448129468l_bool: $i > $i > $i ).

thf(finite_finite_int_type,type,
    finite_finite_int: $i ).

thf(hAPP_f1594865479ol_int_type,type,
    hAPP_f1594865479ol_int: $i > $i > $i ).

thf(finite_fold_int_int_type,type,
    finite_fold_int_int: $i > $i > $i ).

thf(hAPP_f1805168059t_bool_type,type,
    hAPP_f1805168059t_bool: $i > $i > $i ).

thf(hAPP_i1529485324t_bool_type,type,
    hAPP_i1529485324t_bool: $i > $i > $i ).

thf(insert_int_type,type,
    insert_int: $i ).

thf(ord_less_nat_type,type,
    ord_less_nat: $i ).

thf(fequal_int_type,type,
    fequal_int: $i ).

thf(finite772772422nt_int_type,type,
    finite772772422nt_int: $i > $i ).

thf(hAPP_i2112223885l_bool_type,type,
    hAPP_i2112223885l_bool: $i > $i > $i ).

thf(member_int_type,type,
    member_int: $i ).

thf(finite_fold1Set_int_type,type,
    finite_fold1Set_int: $i > $i > $i ).

thf(finite1973466193nt_int_type,type,
    finite1973466193nt_int: $i > $i ).

thf(hAPP_nat_nat_type,type,
    hAPP_nat_nat: $i > $i > $i ).

thf(hAPP_nat_fun_nat_nat_type,type,
    hAPP_nat_fun_nat_nat: $i > $i > $i ).

thf(ord_min_nat_type,type,
    ord_min_nat: $i ).

thf(finite_fold1_nat_type,type,
    finite_fold1_nat: $i > $i ).

thf(finite_fold1Set_nat_type,type,
    finite_fold1Set_nat: $i > $i > $i ).

thf(finite_fold1_int_type,type,
    finite_fold1_int: $i > $i ).

thf(hAPP_H1229305626_a_int_type,type,
    hAPP_H1229305626_a_int: $i > $i > $i ).

thf(hAPP_i1475897073_a_int_type,type,
    hAPP_i1475897073_a_int: $i > $i > $i ).

thf(cOMBK_1007576201iple_a_type,type,
    cOMBK_1007576201iple_a: $i ).

thf(bot_bot_fun_int_bool_type,type,
    bot_bot_fun_int_bool: $i ).

thf(hAPP_f54304608l_bool_type,type,
    hAPP_f54304608l_bool: $i > $i > $i ).

thf(hAPP_n215258509l_bool_type,type,
    hAPP_n215258509l_bool: $i > $i > $i ).

thf(member_nat_type,type,
    member_nat: $i ).

thf(hAPP_f782000547ol_nat_type,type,
    hAPP_f782000547ol_nat: $i > $i > $i ).

thf(big_co387207925at_nat_type,type,
    big_co387207925at_nat: $i ).

thf(hAPP_H1675210280t_bool_type,type,
    hAPP_H1675210280t_bool: $i > $i > $i ).

thf(hAPP_f2142913731t_bool_type,type,
    hAPP_f2142913731t_bool: $i > $i > $i ).

thf(hAPP_f1329415375t_bool_type,type,
    hAPP_f1329415375t_bool: $i > $i > $i ).

thf(cOMBB_1505691757iple_a_type,type,
    cOMBB_1505691757iple_a: $i ).

thf(fequal_nat_type,type,
    fequal_nat: $i ).

thf(finite1321096241t_bool_type,type,
    finite1321096241t_bool: $i > $i ).

thf(finite_finite_nat_type,type,
    finite_finite_nat: $i ).

thf(big_linorder_Min_nat_type,type,
    big_linorder_Min_nat: $i ).

thf(plus_plus_nat_type,type,
    plus_plus_nat: $i ).

thf(hAPP_f966999552at_nat_type,type,
    hAPP_f966999552at_nat: $i > $i > $i ).

thf(hAPP_f724467227at_nat_type,type,
    hAPP_f724467227at_nat: $i > $i > $i ).

thf(cOMBB_161916371at_nat_type,type,
    cOMBB_161916371at_nat: $i ).

thf(hAPP_f2097660464ol_nat_type,type,
    hAPP_f2097660464ol_nat: $i > $i > $i ).

thf(hAPP_n1919155532a_bool_type,type,
    hAPP_n1919155532a_bool: $i > $i > $i ).

thf(finite1626084323ne_int_type,type,
    finite1626084323ne_int: $i > $i > $i ).

thf(hAPP_f284875647l_bool_type,type,
    hAPP_f284875647l_bool: $i > $i > $i ).

thf(ord_le951220754t_bool_type,type,
    ord_le951220754t_bool: $i ).

thf(minus_minus_nat_type,type,
    minus_minus_nat: $i ).

thf(big_linorder_Min_int_type,type,
    big_linorder_Min_int: $i ).

thf(big_se275732192ig_nat_type,type,
    big_se275732192ig_nat: $i > $i > $i ).

thf(hAPP_f800510211t_bool_type,type,
    hAPP_f800510211t_bool: $i > $i > $i ).

thf(hAPP_f1730770594t_bool_type,type,
    hAPP_f1730770594t_bool: $i > $i > $i ).

thf(minus_2067140911t_bool_type,type,
    minus_2067140911t_bool: $i ).

thf(hAPP_n1512601776t_bool_type,type,
    hAPP_n1512601776t_bool: $i > $i > $i ).

thf(insert_nat_type,type,
    insert_nat: $i ).

thf(bot_bot_fun_nat_bool_type,type,
    bot_bot_fun_nat_bool: $i ).

thf(finite1071749497t_bool_type,type,
    finite1071749497t_bool: $i > $i ).

thf(finite1578363458t_bool_type,type,
    finite1578363458t_bool: $i > $i > $i ).

thf(one_one_nat_type,type,
    one_one_nat: $i ).

thf(ord_lessThan_nat_type,type,
    ord_lessThan_nat: $i ).

thf(hAPP_f103356543l_bool_type,type,
    hAPP_f103356543l_bool: $i > $i > $i ).

thf(ord_le1568362934t_bool_type,type,
    ord_le1568362934t_bool: $i ).

thf(image_nat_nat_type,type,
    image_nat_nat: $i > $i > $i ).

thf(finite1827710202t_bool_type,type,
    finite1827710202t_bool: $i > $i > $i ).

thf(hAPP_H622608077l_bool_type,type,
    hAPP_H622608077l_bool: $i > $i > $i ).

thf(member1797258804iple_a_type,type,
    member1797258804iple_a: $i ).

thf(image_685909450_a_int_type,type,
    image_685909450_a_int: $i > $i > $i ).

thf(hAPP_H592031934_a_nat_type,type,
    hAPP_H592031934_a_nat: $i > $i > $i ).

thf(hAPP_n1653940209_a_nat_type,type,
    hAPP_n1653940209_a_nat: $i > $i > $i ).

thf(cOMBK_1028443109iple_a_type,type,
    cOMBK_1028443109iple_a: $i ).

thf(finite2130160977at_nat_type,type,
    finite2130160977at_nat: $i > $i ).

thf(finite929467206at_nat_type,type,
    finite929467206at_nat: $i > $i ).

thf(hAPP_f1223193598t_bool_type,type,
    hAPP_f1223193598t_bool: $i > $i > $i ).

thf(minus_1449998731t_bool_type,type,
    minus_1449998731t_bool: $i ).

thf(image_nat_int_type,type,
    image_nat_int: $i > $i > $i ).

thf(image_int_int_type,type,
    image_int_int: $i > $i > $i ).

thf(cOMBK_int_int_type,type,
    cOMBK_int_int: $i ).

thf(hAPP_f1926459811ol_int_type,type,
    hAPP_f1926459811ol_int: $i > $i > $i ).

thf(big_co230513141nt_int_type,type,
    big_co230513141nt_int: $i ).

thf(finite_fold_nat_nat_type,type,
    finite_fold_nat_nat: $i > $i > $i ).

thf(times_times_nat_type,type,
    times_times_nat: $i ).

thf(finite1432773856em_int_type,type,
    finite1432773856em_int: $i > $i > $i ).

thf(finite988810631ne_nat_type,type,
    finite988810631ne_nat: $i > $i > $i ).

thf(hAPP_H1037229737a_bool_type,type,
    hAPP_H1037229737a_bool: $i > $i > $i ).

thf(hAPP_f228417655a_bool_type,type,
    hAPP_f228417655a_bool: $i > $i > $i ).

thf(cOMBC_364739t_bool_type,type,
    cOMBC_364739t_bool: $i ).

thf(hAPP_H144868812t_bool_type,type,
    hAPP_H144868812t_bool: $i > $i > $i ).

thf(big_co1705425894at_nat_type,type,
    big_co1705425894at_nat: $i > $i ).

thf(hAPP_f604201481a_bool_type,type,
    hAPP_f604201481a_bool: $i > $i > $i ).

thf(cOMBB_1882975613iple_a_type,type,
    cOMBB_1882975613iple_a: $i ).

thf(hAPP_bool_bool_type,type,
    hAPP_bool_bool: $i > $i > $i ).

thf(collect_int_type,type,
    collect_int: $i ).

thf(if_nat_type,type,
    if_nat: $i > $i ).

thf(hAPP_i240634960iple_a_type,type,
    hAPP_i240634960iple_a: $i > $i > $i ).

thf(image_1844722432iple_a_type,type,
    image_1844722432iple_a: $i > $i > $i ).

thf(hAPP_H426895267a_bool_type,type,
    hAPP_H426895267a_bool: $i > $i > $i ).

thf(hAPP_f960023481a_bool_type,type,
    hAPP_f960023481a_bool: $i > $i > $i ).

thf(hAPP_f1786170683a_bool_type,type,
    hAPP_f1786170683a_bool: $i > $i > $i ).

thf(cOMBB_683172471iple_a_type,type,
    cOMBB_683172471iple_a: $i ).

thf(hAPP_H905846293iple_a_type,type,
    hAPP_H905846293iple_a: $i > $i > $i ).

thf(finite795500164em_nat_type,type,
    finite795500164em_nat: $i > $i > $i ).

thf(hAPP_b589554111l_bool_type,type,
    hAPP_b589554111l_bool: $i > $i > $i ).

thf(minus_minus_bool_type,type,
    minus_minus_bool: $i ).

thf(big_co1548731110nt_int_type,type,
    big_co1548731110nt_int: $i > $i ).

thf(hAPP_f659380387ol_int_type,type,
    hAPP_f659380387ol_int: $i > $i > $i ).

thf(big_co195215938at_int_type,type,
    big_co195215938at_int: $i > $i ).

thf(the_elem_int_type,type,
    the_elem_int: $i ).

thf(finite1738664244iple_a_type,type,
    finite1738664244iple_a: $i ).

thf(hAPP_f1371399797ol_nat_type,type,
    hAPP_f1371399797ol_nat: $i > $i > $i ).

thf(big_co1340561246_a_nat_type,type,
    big_co1340561246_a_nat: $i ).

thf(hAPP_f2087582327a_bool_type,type,
    hAPP_f2087582327a_bool: $i > $i > $i ).

thf(hAPP_f1396758363a_bool_type,type,
    hAPP_f1396758363a_bool: $i > $i > $i ).

thf(cOMBB_391046962ol_nat_type,type,
    cOMBB_391046962ol_nat: $i ).

thf(collec351493750iple_a_type,type,
    collec351493750iple_a: $i ).

thf(hAPP_f607497069a_bool_type,type,
    hAPP_f607497069a_bool: $i > $i > $i ).

thf(cOMBS_154988028l_bool_type,type,
    cOMBS_154988028l_bool: $i ).

thf(hAPP_f146754017l_bool_type,type,
    hAPP_f146754017l_bool: $i > $i > $i ).

thf(hAPP_f239971723l_bool_type,type,
    hAPP_f239971723l_bool: $i > $i > $i ).

thf(cOMBB_1866391387iple_a_type,type,
    cOMBB_1866391387iple_a: $i ).

thf(hAPP_f247708275a_bool_type,type,
    hAPP_f247708275a_bool: $i > $i > $i ).

thf(cOMBC_1519416966l_bool_type,type,
    cOMBC_1519416966l_bool: $i ).

thf(hAPP_f1872860171t_bool_type,type,
    hAPP_f1872860171t_bool: $i > $i > $i ).

thf(hAPP_f1981448543t_bool_type,type,
    hAPP_f1981448543t_bool: $i > $i > $i ).

thf(cOMBB_1391527205iple_a_type,type,
    cOMBB_1391527205iple_a: $i ).

thf(image_48635758_a_nat_type,type,
    image_48635758_a_nat: $i > $i > $i ).

thf(ord_max_nat_type,type,
    ord_max_nat: $i ).

thf(hAPP_i611776424a_bool_type,type,
    hAPP_i611776424a_bool: $i > $i > $i ).

thf(hAPP_f920351535a_bool_type,type,
    hAPP_f920351535a_bool: $i > $i > $i ).

thf(cOMBC_1530706207t_bool_type,type,
    cOMBC_1530706207t_bool: $i ).

thf(cOMBK_nat_nat_type,type,
    cOMBK_nat_nat: $i ).

thf(finite1860950092at_nat_type,type,
    finite1860950092at_nat: $i > $i ).

thf(hAPP_state_bool_type,type,
    hAPP_state_bool: $i > $i > $i ).

thf(hAPP_f746301080e_bool_type,type,
    hAPP_f746301080e_bool: $i > $i > $i ).

thf(hAPP_f162060345e_bool_type,type,
    hAPP_f162060345e_bool: $i > $i > $i ).

thf(cOMBB_844853809_state_type,type,
    cOMBB_844853809_state: $i ).

thf(hAPP_state_state_type,type,
    hAPP_state_state: $i > $i > $i ).

thf(big_linorder_Max_nat_type,type,
    big_linorder_Max_nat: $i ).

thf(fequal_fun_int_bool_type,type,
    fequal_fun_int_bool: $i ).

thf(hAPP_f1693662087iple_a_type,type,
    hAPP_f1693662087iple_a: $i > $i > $i ).

thf(the_Ho830102290iple_a_type,type,
    the_Ho830102290iple_a: $i ).

thf(fimplies_type,type,
    fimplies: $i ).

thf(hAPP_f920293029a_bool_type,type,
    hAPP_f920293029a_bool: $i > $i > $i ).

thf(cOMBC_1940896922a_bool_type,type,
    cOMBC_1940896922a_bool: $i ).

thf(fequal963300192iple_a_type,type,
    fequal963300192iple_a: $i ).

thf(bot_bot_bool_type,type,
    bot_bot_bool: $i ).

thf(hAPP_n261501868iple_a_type,type,
    hAPP_n261501868iple_a: $i > $i > $i ).

thf(image_1865589340iple_a_type,type,
    image_1865589340iple_a: $i > $i > $i ).

thf(hAPP_f215623910l_bool_type,type,
    hAPP_f215623910l_bool: $i > $i > $i ).

thf(collect_nat_type,type,
    collect_nat: $i ).

thf(hAPP_a1224971408e_bool_type,type,
    hAPP_a1224971408e_bool: $i > $i > $i ).

thf(hAPP_f1406200875e_bool_type,type,
    hAPP_f1406200875e_bool: $i > $i > $i ).

thf(hAPP_f635443597e_bool_type,type,
    hAPP_f635443597e_bool: $i > $i > $i ).

thf(cOMBB_2139825703bool_a_type,type,
    cOMBB_2139825703bool_a: $i ).

thf(hAPP_a2036067514e_bool_type,type,
    hAPP_a2036067514e_bool: $i > $i > $i ).

thf(hAPP_f587450508ol_int_type,type,
    hAPP_f587450508ol_int: $i > $i > $i ).

thf(hAPP_f1544320301ol_int_type,type,
    hAPP_f1544320301ol_int: $i > $i > $i ).

thf(big_co1977834938_a_int_type,type,
    big_co1977834938_a_int: $i ).

thf(hAPP_f1706700729a_bool_type,type,
    hAPP_f1706700729a_bool: $i > $i > $i ).

thf(minus_85316870a_bool_type,type,
    minus_85316870a_bool: $i ).

thf(ord_less_eq_bool_type,type,
    ord_less_eq_bool: $i ).

thf(fdisj_type,type,
    fdisj: $i ).

thf(ord_at238088361st_nat_type,type,
    ord_at238088361st_nat: $i > $i ).

thf(semi_type,type,
    semi: $i > $i > $i ).

thf(ass_type,type,
    ass: $i > $i > $i ).

thf(finite58652534t_bool_type,type,
    finite58652534t_bool: $i > $i ).

thf(finite1753440478iple_a_type,type,
    finite1753440478iple_a: $i > $i > $i ).

thf(fTrue_type,type,
    fTrue: $i ).

thf(hAPP_b396694332t_bool_type,type,
    hAPP_b396694332t_bool: $i > $i > $i ).

thf(cOMBK_bool_int_type,type,
    cOMBK_bool_int: $i ).

thf(skip_type,type,
    skip: $i ).

thf(image_533441733iple_a_type,type,
    image_533441733iple_a: $i > $i > $i ).

thf(big_co1815942735_a_nat_type,type,
    big_co1815942735_a_nat: $i > $i ).

thf(hAPP_f1399552105l_bool_type,type,
    hAPP_f1399552105l_bool: $i > $i > $i ).

thf(cOMBC_605892544l_bool_type,type,
    cOMBC_605892544l_bool: $i ).

thf(hAPP_f1767618879l_bool_type,type,
    hAPP_f1767618879l_bool: $i > $i > $i ).

thf(ord_le1523781517a_bool_type,type,
    ord_le1523781517a_bool: $i ).

thf(finite1033474011iple_a_type,type,
    finite1033474011iple_a: $i > $i > $i ).

thf(hAPP_H2085992369iple_a_type,type,
    hAPP_H2085992369iple_a: $i > $i > $i ).

thf(finite1428139793iple_a_type,type,
    finite1428139793iple_a: $i > $i ).

thf(finite1205970312iple_a_type,type,
    finite1205970312iple_a: $i > $i > $i ).

thf(finite2114894759a_bool_type,type,
    finite2114894759a_bool: $i > $i ).

thf(finite1956789438t_bool_type,type,
    finite1956789438t_bool: $i > $i ).

thf(big_co305732779_a_int_type,type,
    big_co305732779_a_int: $i > $i ).

thf(fequal_fun_nat_bool_type,type,
    fequal_fun_nat_bool: $i ).

thf(the_elem_nat_type,type,
    the_elem_nat: $i ).

thf(hAPP_f1637334154l_bool_type,type,
    hAPP_f1637334154l_bool: $i > $i > $i ).

thf(hAPP_a723219176e_bool_type,type,
    hAPP_a723219176e_bool: $i > $i > $i ).

thf(hAPP_f340725611e_bool_type,type,
    hAPP_f340725611e_bool: $i > $i > $i ).

thf(hAPP_f1006724181e_bool_type,type,
    hAPP_f1006724181e_bool: $i > $i > $i ).

thf(cOMBB_1348041619bool_a_type,type,
    cOMBB_1348041619bool_a: $i ).

thf(hAPP_f167292325e_bool_type,type,
    hAPP_f167292325e_bool: $i > $i > $i ).

thf(hAPP_a849909144l_bool_type,type,
    hAPP_a849909144l_bool: $i > $i > $i ).

thf(hAPP_f2110825313l_bool_type,type,
    hAPP_f2110825313l_bool: $i > $i > $i ).

thf(hAPP_s1892499976_state_type,type,
    hAPP_s1892499976_state: $i > $i > $i ).

thf(hAPP_v365393659_state_type,type,
    hAPP_v365393659_state: $i > $i > $i ).

thf(hAPP_f1542232213_state_type,type,
    hAPP_f1542232213_state: $i > $i > $i ).

thf(cOMBC_1193272608_state_type,type,
    cOMBC_1193272608_state: $i ).

thf(hAPP_v594194232_state_type,type,
    hAPP_v594194232_state: $i > $i > $i ).

thf(hAPP_s712361723_state_type,type,
    hAPP_s712361723_state: $i > $i > $i ).

thf(is_vname_type,type,
    is_vname: $i > $o ).

thf(cOMBK_2052406350iple_a_type,type,
    cOMBK_2052406350iple_a: $i ).

thf(finite648855052iple_a_type,type,
    finite648855052iple_a: $i > $i ).

thf(finite1465856449iple_a_type,type,
    finite1465856449iple_a: $i > $i ).

thf(finite1621280049iple_a_type,type,
    finite1621280049iple_a: $i > $i > $i ).

thf(hAPP_H2130837971l_bool_type,type,
    hAPP_H2130837971l_bool: $i > $i > $i ).

thf(hAPP_b1882817719a_bool_type,type,
    hAPP_b1882817719a_bool: $i > $i > $i ).

thf(cOMBK_235286536iple_a_type,type,
    cOMBK_235286536iple_a: $i ).

thf(hAPP_s1806633685e_bool_type,type,
    hAPP_s1806633685e_bool: $i > $i > $i ).

thf(hAPP_f817621513e_bool_type,type,
    hAPP_f817621513e_bool: $i > $i > $i ).

thf(cOMBC_2027030106e_bool_type,type,
    cOMBC_2027030106e_bool: $i ).

thf(hAPP_b1013836512t_bool_type,type,
    hAPP_b1013836512t_bool: $i > $i > $i ).

thf(cOMBK_bool_nat_type,type,
    cOMBK_bool_nat: $i ).

thf(finite1803123436a_bool_type,type,
    finite1803123436a_bool: $i > $i ).

thf(hAPP_f100967412e_bool_type,type,
    hAPP_f100967412e_bool: $i > $i > $i ).

thf(hAPP_f1460451647e_bool_type,type,
    hAPP_f1460451647e_bool: $i > $i > $i ).

thf(cOMBC_524597097e_bool_type,type,
    cOMBC_524597097e_bool: $i ).

thf(hAPP_f643944041l_bool_type,type,
    hAPP_f643944041l_bool: $i > $i > $i ).

thf(cOMBC_1693257480l_bool_type,type,
    cOMBC_1693257480l_bool: $i ).

thf(hAPP_f871651461_state_type,type,
    hAPP_f871651461_state: $i > $i > $i ).

thf(hAPP_f851239890_state_type,type,
    hAPP_f851239890_state: $i > $i > $i ).

thf(cOMBS_777315357_state_type,type,
    cOMBS_777315357_state: $i ).

thf(hAPP_nat_state_type,type,
    hAPP_nat_state: $i > $i > $i ).

thf(hAPP_state_nat_type,type,
    hAPP_state_nat: $i > $i > $i ).

thf(big_se1603268663iple_a_type,type,
    big_se1603268663iple_a: $i > $i > $i ).

thf(hAPP_b540892988e_bool_type,type,
    hAPP_b540892988e_bool: $i > $i > $i ).

thf(hAPP_f1824947087e_bool_type,type,
    hAPP_f1824947087e_bool: $i > $i > $i ).

thf(cOMBC_41962815e_bool_type,type,
    cOMBC_41962815e_bool: $i ).

thf(cOMBC_231445413l_bool_type,type,
    cOMBC_231445413l_bool: $i ).

thf(update_type,type,
    update: $i ).

thf(hAPP_f1194757675a_bool_type,type,
    hAPP_f1194757675a_bool: $i > $i > $i ).

thf(hAPP_f1697223433a_bool_type,type,
    hAPP_f1697223433a_bool: $i > $i > $i ).

thf(cOMBB_1610959875iple_a_type,type,
    cOMBB_1610959875iple_a: $i ).

thf(hAPP_f1259673775l_bool_type,type,
    hAPP_f1259673775l_bool: $i > $i > $i ).

thf(finite1215486064a_bool_type,type,
    finite1215486064a_bool: $i > $i > $i ).

thf(hAPP_f644196280e_bool_type,type,
    hAPP_f644196280e_bool: $i > $i > $i ).

thf(hAPP_s58564346l_bool_type,type,
    hAPP_s58564346l_bool: $i > $i > $i ).

thf(hAPP_f1577095245l_bool_type,type,
    hAPP_f1577095245l_bool: $i > $i > $i ).

thf(finite1152845682a_bool_type,type,
    finite1152845682a_bool: $i > $i ).

thf(hAPP_a1200519163e_bool_type,type,
    hAPP_a1200519163e_bool: $i > $i > $i ).

thf(finite90892550iple_a_type,type,
    finite90892550iple_a: $i > $i ).

thf(hAPP_f550111323a_bool_type,type,
    hAPP_f550111323a_bool: $i > $i > $i ).

thf(hAPP_f615286271a_bool_type,type,
    hAPP_f615286271a_bool: $i > $i > $i ).

thf(cOMBB_1176122337iple_a_type,type,
    cOMBB_1176122337iple_a: $i ).

thf(hAPP_f957014749a_bool_type,type,
    hAPP_f957014749a_bool: $i > $i > $i ).

thf(cOMBC_288844080a_bool_type,type,
    cOMBC_288844080a_bool: $i ).

thf(hAPP_f1181346091a_bool_type,type,
    hAPP_f1181346091a_bool: $i > $i > $i ).

thf(hAPP_f571476211a_bool_type,type,
    hAPP_f571476211a_bool: $i > $i > $i ).

thf(cOMBC_1570214144a_bool_type,type,
    cOMBC_1570214144a_bool: $i ).

thf(finite2012431853t_bool_type,type,
    finite2012431853t_bool: $i ).

thf(hAPP_f1246832597l_bool_type,type,
    hAPP_f1246832597l_bool: $i > $i > $i ).

thf(collect_fun_nat_bool_type,type,
    collect_fun_nat_bool: $i ).

thf(fequal874423448a_bool_type,type,
    fequal874423448a_bool: $i ).

thf(fequal_state_type,type,
    fequal_state: $i ).

thf(the_el1519802624iple_a_type,type,
    the_el1519802624iple_a: $i ).

thf(ord_gr375877188st_nat_type,type,
    ord_gr375877188st_nat: $i > $i ).

thf(finite268574148a_bool_type,type,
    finite268574148a_bool: $i ).

thf(hAPP_f396832789l_bool_type,type,
    hAPP_f396832789l_bool: $i > $i > $i ).

thf(collec999179778a_bool_type,type,
    collec999179778a_bool: $i ).

thf(hAPP_f2100528361l_bool_type,type,
    hAPP_f2100528361l_bool: $i > $i > $i ).

thf(cOMBC_627599030l_bool_type,type,
    cOMBC_627599030l_bool: $i ).

thf(finite1395289673t_bool_type,type,
    finite1395289673t_bool: $i ).

thf(hAPP_f1948010709l_bool_type,type,
    hAPP_f1948010709l_bool: $i > $i > $i ).

thf(collect_fun_int_bool_type,type,
    collect_fun_int_bool: $i ).

thf(hAPP_f258822235a_bool_type,type,
    hAPP_f258822235a_bool: $i > $i > $i ).

thf(the_fu1555441254a_bool_type,type,
    the_fu1555441254a_bool: $i ).

thf(hAPP_f1718504751a_bool_type,type,
    hAPP_f1718504751a_bool: $i > $i > $i ).

thf(hAPP_f481410067a_bool_type,type,
    hAPP_f481410067a_bool: $i > $i > $i ).

thf(cOMBB_1028320654ol_int_type,type,
    cOMBB_1028320654ol_int: $i ).

thf(partia1949573335iple_a_type,type,
    partia1949573335iple_a: $i > $i ).

thf(293,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ A ) @ B )
        = ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ C ) @ B ) )
    <=> ( A = C ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_1047_nat__add__right__cancel) ).

thf(2199,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ A ) @ B )
          = ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ C ) @ B ) )
       => ( A = C ) )
      & ( ( A = C )
       => ( ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ A ) @ B )
          = ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ C ) @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[293]) ).

thf(757,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ A ) @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ B ) @ C ) ) )
     => ( ( A != B )
       => ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ A ) @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_9_insertE) ).

thf(3873,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ A ) @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ B ) @ C ) ) )
     => ( ( A != B )
       => ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ A ) @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[757]) ).

thf(1084,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_f1767618879l_bool @ ord_le1523781517a_bool @ C ) @ D ) )
     => ( ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_f1767618879l_bool @ ord_le1523781517a_bool @ A ) @ B ) )
       => ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_f1767618879l_bool @ ord_le1523781517a_bool @ ( hAPP_f20753329a_bool @ ( hAPP_f1706700729a_bool @ minus_85316870a_bool @ C ) @ B ) ) @ ( hAPP_f20753329a_bool @ ( hAPP_f1706700729a_bool @ minus_85316870a_bool @ D ) @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_884_Diff__mono) ).

thf(5116,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_f1767618879l_bool @ ord_le1523781517a_bool @ C ) @ D ) )
     => ( ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_f1767618879l_bool @ ord_le1523781517a_bool @ A ) @ B ) )
       => ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_f1767618879l_bool @ ord_le1523781517a_bool @ ( hAPP_f20753329a_bool @ ( hAPP_f1706700729a_bool @ minus_85316870a_bool @ C ) @ B ) ) @ ( hAPP_f20753329a_bool @ ( hAPP_f1706700729a_bool @ minus_85316870a_bool @ D ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1084]) ).

thf(94,axiom,
    ! [A: $i,B: $i] :
      ( ( A != B )
     => ( ~ ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_nat @ A ) @ B ) )
       => ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_nat @ B ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_1120_linorder__neqE__nat) ).

thf(1469,plain,
    ! [A: $i,B: $i] :
      ( ( A != B )
     => ( ~ ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_nat @ A ) @ B ) )
       => ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_nat @ B ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[94]) ).

thf(273,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ B ) @ C ) )
     => ( ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ minus_minus_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ C ) @ A ) ) @ B )
        = ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ minus_minus_nat @ C ) @ B ) ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_1037_diff__add__assoc2) ).

thf(2131,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ B ) @ C ) )
     => ( ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ minus_minus_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ C ) @ A ) ) @ B )
        = ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ minus_minus_nat @ C ) @ B ) ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[273]) ).

thf(670,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( finite1956789438t_bool @ D ) )
     => ( ( hBOOL @ ( hAPP_f54304608l_bool @ finite_finite_nat @ C ) )
       => ( ( hAPP_f800510211t_bool @ ( finite1578363458t_bool @ D @ A ) @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ B ) @ C ) )
          = ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ D @ B ) @ ( hAPP_f800510211t_bool @ ( finite1578363458t_bool @ D @ A ) @ C ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_576_comp__fun__idem_Ofold__insert__idem) ).

thf(3581,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( finite1956789438t_bool @ D ) )
     => ( ( hBOOL @ ( hAPP_f54304608l_bool @ finite_finite_nat @ C ) )
       => ( ( hAPP_f800510211t_bool @ ( finite1578363458t_bool @ D @ A ) @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ B ) @ C ) )
          = ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ D @ B ) @ ( hAPP_f800510211t_bool @ ( finite1578363458t_bool @ D @ A ) @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[670]) ).

thf(742,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ A ) @ B ) )
     => ( ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ A ) @ B )
        = B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_51_insert__absorb) ).

thf(3830,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ A ) @ B ) )
     => ( ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ A ) @ B )
        = B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[742]) ).

thf(785,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f54304608l_bool @ finite_finite_nat @ C ) )
     => ? [D: $i] : ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ ( finite929467206at_nat @ A ) @ B ) @ C ) @ D ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_330_finite__imp__fold__graph) ).

thf(3969,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f54304608l_bool @ finite_finite_nat @ C ) )
     => ? [D: $i] : ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ ( finite929467206at_nat @ A ) @ B ) @ C ) @ D ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[785]) ).

thf(1051,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP_f20753329a_bool @ ( hAPP_f1706700729a_bool @ minus_85316870a_bool @ A ) @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ B ) @ C ) )
      = ( 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 @ B ) @ bot_bo797238721a_bool ) ) ) @ C ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_415_Diff__insert2) ).

thf(4992,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP_f20753329a_bool @ ( hAPP_f1706700729a_bool @ minus_85316870a_bool @ A ) @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ B ) @ C ) )
      = ( 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 @ B ) @ bot_bo797238721a_bool ) ) ) @ C ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1051]) ).

thf(556,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ A ) @ B ) ) @ C )
      = ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ A ) @ C ) ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ B ) @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_703_comm__semiring__1__class_Onormalizing__semiring__rules_I8_J) ).

thf(3172,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ A ) @ B ) ) @ C )
      = ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ A ) @ C ) ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ B ) @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[556]) ).

thf(126,axiom,
    ! [A: $i,B: $i] :
      ( ( ( hAPP_n1699378549t_bool @ ord_lessThan_nat @ A )
        = ( hAPP_n1699378549t_bool @ ord_lessThan_nat @ B ) )
    <=> ( A = B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_817_lessThan__eq__iff) ).

thf(1586,plain,
    ! [A: $i,B: $i] :
      ( ( ( ( hAPP_n1699378549t_bool @ ord_lessThan_nat @ A )
          = ( hAPP_n1699378549t_bool @ ord_lessThan_nat @ B ) )
       => ( A = B ) )
      & ( ( A = B )
       => ( ( hAPP_n1699378549t_bool @ ord_lessThan_nat @ A )
          = ( hAPP_n1699378549t_bool @ ord_lessThan_nat @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[126]) ).

thf(842,axiom,
    ! [A: $i] :
      ( ( A != bot_bot_fun_nat_bool )
    <=> ? [B: $i,C: $i] :
          ( ( A
            = ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ B ) @ C ) )
          & ~ ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ B ) @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_124_nonempty__iff) ).

thf(4201,plain,
    ! [A: $i] :
      ( ( ( A != bot_bot_fun_nat_bool )
       => ? [B: $i,C: $i] :
            ( ( A
              = ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ B ) @ C ) )
            & ~ ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ B ) @ C ) ) ) )
      & ( ? [B: $i,C: $i] :
            ( ( A
              = ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ B ) @ C ) )
            & ~ ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ B ) @ C ) ) )
       => ( A != bot_bot_fun_nat_bool ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[842]) ).

thf(364,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ B ) @ C ) )
     => ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ ( hAPP_nat_int @ A @ B ) ) @ ( image_nat_int @ A @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_182_imageI) ).

thf(2452,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ B ) @ C ) )
     => ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ ( hAPP_nat_int @ A @ B ) ) @ ( image_nat_int @ A @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[364]) ).

thf(213,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_nat @ B ) @ C ) )
     => ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_nat @ B ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ A ) @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_1104_trans__less__add2) ).

thf(1879,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_nat @ B ) @ C ) )
     => ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_nat @ B ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ A ) @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[213]) ).

thf(165,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( finite1432773856em_int @ C @ D ) )
     => ( ( hBOOL @ ( hAPP_f448129468l_bool @ finite_finite_int @ B ) )
       => ( ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ A ) @ B ) )
         => ( ( hAPP_int_int @ ( hAPP_int_fun_int_int @ C @ A ) @ ( hAPP_f1594865479ol_int @ D @ B ) )
            = ( hAPP_f1594865479ol_int @ D @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_433_folding__one__idem_Oin__idem) ).

thf(1718,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( finite1432773856em_int @ C @ D ) )
     => ( ( hBOOL @ ( hAPP_f448129468l_bool @ finite_finite_int @ B ) )
       => ( ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ A ) @ B ) )
         => ( ( hAPP_int_int @ ( hAPP_int_fun_int_int @ C @ A ) @ ( hAPP_f1594865479ol_int @ D @ B ) )
            = ( hAPP_f1594865479ol_int @ D @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[165]) ).

thf(780,axiom,
    ! [A: $i,B: $i] :
      ( ( 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 ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_723_equalityI) ).

thf(3944,plain,
    ! [A: $i,B: $i] :
      ( ( 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 ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[780]) ).

thf(77,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ ord_min_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ ord_min_nat @ A ) @ B ) ) @ C )
      = ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ ord_min_nat @ A ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ ord_min_nat @ B ) @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_953_min__max_Oinf__assoc) ).

thf(1411,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ ord_min_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ ord_min_nat @ A ) @ B ) ) @ C )
      = ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ ord_min_nat @ A ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ ord_min_nat @ B ) @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[77]) ).

thf(1079,axiom,
    ! [A: $i,B: $i] :
      ( ( ( hBOOL @ ( hAPP_H1037229737a_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_H426895267a_bool @ fequal963300192iple_a @ B ) ) ) @ A ) )
          = ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ B ) @ bot_bo797238721a_bool ) ) )
      & ( ~ ( hBOOL @ ( hAPP_H1037229737a_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_H426895267a_bool @ fequal963300192iple_a @ B ) ) ) @ A ) )
          = bot_bo797238721a_bool ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_25_Collect__conv__if2) ).

thf(5088,plain,
    ! [A: $i,B: $i] :
      ( ( ( hBOOL @ ( hAPP_H1037229737a_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_H426895267a_bool @ fequal963300192iple_a @ B ) ) ) @ A ) )
          = ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ B ) @ bot_bo797238721a_bool ) ) )
      & ( ~ ( hBOOL @ ( hAPP_H1037229737a_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_H426895267a_bool @ fequal963300192iple_a @ B ) ) ) @ A ) )
          = bot_bo797238721a_bool ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1079]) ).

thf(442,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ A ) @ ( hAPP_f1805168059t_bool @ ( hAPP_f1223193598t_bool @ minus_1449998731t_bool @ B ) @ C ) ) )
     => ~ ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ A ) @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_390_DiffD2) ).

thf(2744,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ A ) @ ( hAPP_f1805168059t_bool @ ( hAPP_f1223193598t_bool @ minus_1449998731t_bool @ B ) @ C ) ) )
     => ~ ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ A ) @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[442]) ).

thf(321,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f54304608l_bool @ finite_finite_nat @ C ) )
     => ( ( hAPP_f22106695ol_nat @ ( hAPP_f782000547ol_nat @ big_co387207925at_nat @ A ) @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ B ) @ C ) )
        = ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ ( hAPP_nat_nat @ A @ B ) ) @ ( hAPP_f22106695ol_nat @ ( hAPP_f782000547ol_nat @ big_co387207925at_nat @ A ) @ ( hAPP_f800510211t_bool @ ( hAPP_f1730770594t_bool @ minus_2067140911t_bool @ C ) @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ B ) @ bot_bot_fun_nat_bool ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_620_setsum_Oinsert__remove) ).

thf(2295,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f54304608l_bool @ finite_finite_nat @ C ) )
     => ( ( hAPP_f22106695ol_nat @ ( hAPP_f782000547ol_nat @ big_co387207925at_nat @ A ) @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ B ) @ C ) )
        = ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ ( hAPP_nat_nat @ A @ B ) ) @ ( hAPP_f22106695ol_nat @ ( hAPP_f782000547ol_nat @ big_co387207925at_nat @ A ) @ ( hAPP_f800510211t_bool @ ( hAPP_f1730770594t_bool @ minus_2067140911t_bool @ C ) @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ B ) @ bot_bot_fun_nat_bool ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[321]) ).

thf(325,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_nat @ A ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ minus_minus_nat @ B ) @ C ) ) )
    <=> ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ A ) @ C ) ) @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_1131_less__diff__conv) ).

thf(2306,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_nat @ A ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ minus_minus_nat @ B ) @ C ) ) )
       => ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ A ) @ C ) ) @ B ) ) )
      & ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ A ) @ C ) ) @ B ) )
       => ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_nat @ A ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ minus_minus_nat @ B ) @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[325]) ).

thf(932,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP_a2036067514e_bool @ ( hAPP_b540892988e_bool @ ( hAPP_f1824947087e_bool @ cOMBC_41962815e_bool @ A ) @ B ) @ C )
      = ( hAPP_b2019457360e_bool @ ( hAPP_a723219176e_bool @ A @ C ) @ B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',help_COMBC_1_1_COMBC_000t__a_000tc__HOL__Obool_000tc__fun_Itc__Com__Ostate_Mtc__) ).

thf(4571,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP_a2036067514e_bool @ ( hAPP_b540892988e_bool @ ( hAPP_f1824947087e_bool @ cOMBC_41962815e_bool @ A ) @ B ) @ C )
      = ( hAPP_b2019457360e_bool @ ( hAPP_a723219176e_bool @ A @ C ) @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[932]) ).

thf(237,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ A ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ B ) @ C ) )
      = ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ B ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ A ) @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_673_comm__semiring__1__class_Onormalizing__semiring__rules_I22_J) ).

thf(1995,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ A ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ B ) @ C ) )
      = ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ B ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ A ) @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[237]) ).

thf(312,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ A ) @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ B ) @ C ) ) )
     => ( ( A != B )
       => ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ A ) @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_11_insertE) ).

thf(2263,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ A ) @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ B ) @ C ) ) )
     => ( ( A != B )
       => ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ A ) @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[312]) ).

thf(1020,axiom,
    ! [A: $i,B: $i] :
      ( ~ ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_f1767618879l_bool @ fequal874423448a_bool @ A ) @ B ) )
      | ( A = B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',help_fequal_1_1_fequal_000tc__fun_Itc__Hoare____Mirabelle____uwgpyvfjxg__Otriple) ).

thf(4899,plain,
    ! [A: $i,B: $i] :
      ( ~ ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_f1767618879l_bool @ fequal874423448a_bool @ A ) @ B ) )
      | ( A = B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1020]) ).

thf(732,axiom,
    ! [A: $i,B: $i] :
      ( ( is_bool @ B )
     => ( is_bool @ ( hAPP_bool_bool @ A @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',gsy_c_hAPP_000tc__HOL__Obool_000tc__HOL__Obool) ).

thf(3798,plain,
    ! [A: $i,B: $i] :
      ( ( is_bool @ B )
     => ( is_bool @ ( hAPP_bool_bool @ A @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[732]) ).

thf(168,axiom,
    ! [A: $i,B: $i] : ( is_bool @ ( finite988810631ne_nat @ A @ B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',gsy_c_Finite__Set_Ofolding__one_000tc__Nat__Onat) ).

thf(1729,plain,
    ! [A: $i,B: $i] : ( is_bool @ ( finite988810631ne_nat @ A @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[168]) ).

thf(696,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( C = D )
     => ( ! [E: $i] :
            ( ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ E ) @ D ) )
           => ( ( hAPP_H592031934_a_nat @ A @ E )
              = ( hAPP_H592031934_a_nat @ B @ E ) ) )
       => ( ( hAPP_f2097660464ol_nat @ ( hAPP_f1371399797ol_nat @ big_co1340561246_a_nat @ A ) @ C )
          = ( hAPP_f2097660464ol_nat @ ( hAPP_f1371399797ol_nat @ big_co1340561246_a_nat @ B ) @ D ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_708_setsum__cong) ).

thf(3666,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( C = D )
     => ( ! [E: $i] :
            ( ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ E ) @ D ) )
           => ( ( hAPP_H592031934_a_nat @ A @ E )
              = ( hAPP_H592031934_a_nat @ B @ E ) ) )
       => ( ( hAPP_f2097660464ol_nat @ ( hAPP_f1371399797ol_nat @ big_co1340561246_a_nat @ A ) @ C )
          = ( hAPP_f2097660464ol_nat @ ( hAPP_f1371399797ol_nat @ big_co1340561246_a_nat @ B ) @ D ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[696]) ).

thf(971,axiom,
    hBOOL @ ( hAPP_f1695230391l_bool @ finite1738664244iple_a @ bot_bo797238721a_bool ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_279_finite_OemptyI) ).

thf(4729,plain,
    hBOOL @ ( hAPP_f1695230391l_bool @ finite1738664244iple_a @ bot_bo797238721a_bool ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[971]) ).

thf(839,axiom,
    ! [A: $i] :
      ( ( hBOOL @ ( hAPP_f54304608l_bool @ finite_finite_nat @ A ) )
    <=> ( ( A = bot_bot_fun_nat_bool )
        | ? [B: $i,C: $i] :
            ( ( A
              = ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ C ) @ B ) )
            & ( hBOOL @ ( hAPP_f54304608l_bool @ finite_finite_nat @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_314_finite_Osimps) ).

thf(4183,plain,
    ! [A: $i] :
      ( ( ( hBOOL @ ( hAPP_f54304608l_bool @ finite_finite_nat @ A ) )
       => ( ( A = bot_bot_fun_nat_bool )
          | ? [B: $i,C: $i] :
              ( ( A
                = ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ C ) @ B ) )
              & ( hBOOL @ ( hAPP_f54304608l_bool @ finite_finite_nat @ B ) ) ) ) )
      & ( ( ( A = bot_bot_fun_nat_bool )
          | ? [B: $i,C: $i] :
              ( ( A
                = ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ C ) @ B ) )
              & ( hBOOL @ ( hAPP_f54304608l_bool @ finite_finite_nat @ B ) ) ) )
       => ( hBOOL @ ( hAPP_f54304608l_bool @ finite_finite_nat @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[839]) ).

thf(863,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ B ) @ C ) )
     => ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ A ) @ B ) ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ A ) @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_994_mult__le__mono2) ).

thf(4290,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ B ) @ C ) )
     => ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ A ) @ B ) ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ A ) @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[863]) ).

thf(394,axiom,
    ! [A: $i,B: $i] : ( is_bool @ ( finite1432773856em_int @ A @ B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',gsy_c_Finite__Set_Ofolding__one__idem_000tc__Int__Oint) ).

thf(2559,plain,
    ! [A: $i,B: $i] : ( is_bool @ ( finite1432773856em_int @ A @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[394]) ).

thf(1105,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( finite1753440478iple_a @ C @ D ) )
     => ( ( hBOOL @ ( hAPP_f1695230391l_bool @ finite1738664244iple_a @ B ) )
       => ( ~ ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ A ) @ B ) )
         => ( ( hAPP_f1693662087iple_a @ D @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ A ) @ B ) )
            = ( hAPP_f1693662087iple_a @ ( finite1205970312iple_a @ C @ A ) @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_582_folding__one_Oeq__fold_H) ).

thf(5205,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( finite1753440478iple_a @ C @ D ) )
     => ( ( hBOOL @ ( hAPP_f1695230391l_bool @ finite1738664244iple_a @ B ) )
       => ( ~ ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ A ) @ B ) )
         => ( ( hAPP_f1693662087iple_a @ D @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ A ) @ B ) )
            = ( hAPP_f1693662087iple_a @ ( finite1205970312iple_a @ C @ A ) @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1105]) ).

thf(632,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP_f54304608l_bool @ finite_finite_nat @ B ) )
     => ( hBOOL @ ( hAPP_f54304608l_bool @ finite_finite_nat @ ( image_nat_nat @ A @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_291_finite__imageI) ).

thf(3449,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP_f54304608l_bool @ finite_finite_nat @ B ) )
     => ( hBOOL @ ( hAPP_f54304608l_bool @ finite_finite_nat @ ( image_nat_nat @ A @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[632]) ).

thf(794,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( finite1432773856em_int @ C @ D ) )
     => ( ( hBOOL @ ( hAPP_f448129468l_bool @ finite_finite_int @ B ) )
       => ( ( A != bot_bot_fun_int_bool )
         => ( ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_f284875647l_bool @ ord_le951220754t_bool @ A ) @ B ) )
           => ( ( hAPP_int_int @ ( hAPP_int_fun_int_int @ C @ ( hAPP_f1594865479ol_int @ D @ A ) ) @ ( hAPP_f1594865479ol_int @ D @ B ) )
              = ( hAPP_f1594865479ol_int @ D @ B ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_911_folding__one__idem_Osubset__idem) ).

thf(4015,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( finite1432773856em_int @ C @ D ) )
     => ( ( hBOOL @ ( hAPP_f448129468l_bool @ finite_finite_int @ B ) )
       => ( ( A != bot_bot_fun_int_bool )
         => ( ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_f284875647l_bool @ ord_le951220754t_bool @ A ) @ B ) )
           => ( ( hAPP_int_int @ ( hAPP_int_fun_int_int @ C @ ( hAPP_f1594865479ol_int @ D @ A ) ) @ ( hAPP_f1594865479ol_int @ D @ B ) )
              = ( hAPP_f1594865479ol_int @ D @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[794]) ).

thf(29,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_int_bool @ B @ C ) )
     => ( ! [D: $i] :
            ( ( hBOOL @ ( hAPP_int_bool @ ( hAPP_i1948725293t_bool @ ord_less_eq_int @ C ) @ D ) )
           => ( ( hBOOL @ ( hAPP_int_bool @ B @ D ) )
             => ( hBOOL @ ( hAPP_int_bool @ B @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ D ) @ one_one_int ) ) ) ) )
       => ( ! [D: $i] :
              ( ( hBOOL @ ( hAPP_int_bool @ ( hAPP_i1948725293t_bool @ ord_less_eq_int @ D ) @ C ) )
             => ( ( hBOOL @ ( hAPP_int_bool @ B @ D ) )
               => ( hBOOL @ ( hAPP_int_bool @ B @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ minus_minus_int @ D ) @ one_one_int ) ) ) ) )
         => ( hBOOL @ ( hAPP_int_bool @ B @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_1174_int__induct) ).

thf(1211,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_int_bool @ B @ C ) )
     => ( ! [D: $i] :
            ( ( hBOOL @ ( hAPP_int_bool @ ( hAPP_i1948725293t_bool @ ord_less_eq_int @ C ) @ D ) )
           => ( ( hBOOL @ ( hAPP_int_bool @ B @ D ) )
             => ( hBOOL @ ( hAPP_int_bool @ B @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ D ) @ one_one_int ) ) ) ) )
       => ( ! [D: $i] :
              ( ( hBOOL @ ( hAPP_int_bool @ ( hAPP_i1948725293t_bool @ ord_less_eq_int @ D ) @ C ) )
             => ( ( hBOOL @ ( hAPP_int_bool @ B @ D ) )
               => ( hBOOL @ ( hAPP_int_bool @ B @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ minus_minus_int @ D ) @ one_one_int ) ) ) ) )
         => ( hBOOL @ ( hAPP_int_bool @ B @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[29]) ).

thf(965,axiom,
    is_bool @ fFalse,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',gsy_c_fFalse) ).

thf(4714,plain,
    is_bool @ fFalse,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[965]) ).

thf(529,axiom,
    ! [A: $i,B: $i] :
      ( ( hAPP_int_int @ ( hAPP_int_fun_int_int @ minus_minus_int @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ A ) @ B ) ) @ B )
      = A ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_646_add__diff__cancel) ).

thf(3072,plain,
    ! [A: $i,B: $i] :
      ( ( hAPP_int_int @ ( hAPP_int_fun_int_int @ minus_minus_int @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ A ) @ B ) ) @ B )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[529]) ).

thf(496,axiom,
    ! [A: $i,B: $i] :
      ( ~ ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ A ) @ B ) )
     => ( ( hAPP_f1805168059t_bool @ ( hAPP_f1223193598t_bool @ minus_1449998731t_bool @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ A ) @ B ) ) @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ A ) @ bot_bot_fun_int_bool ) )
        = B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_422_Diff__insert__absorb) ).

thf(2940,plain,
    ! [A: $i,B: $i] :
      ( ~ ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ A ) @ B ) )
     => ( ( hAPP_f1805168059t_bool @ ( hAPP_f1223193598t_bool @ minus_1449998731t_bool @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ A ) @ B ) ) @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ A ) @ bot_bot_fun_int_bool ) )
        = B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[496]) ).

thf(600,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_f103356543l_bool @ ord_le1568362934t_bool @ B ) @ C ) )
     => ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_f103356543l_bool @ ord_le1568362934t_bool @ C ) @ A ) )
       => ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_f103356543l_bool @ ord_le1568362934t_bool @ B ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_749_order__trans) ).

thf(3317,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_f103356543l_bool @ ord_le1568362934t_bool @ B ) @ C ) )
     => ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_f103356543l_bool @ ord_le1568362934t_bool @ C ) @ A ) )
       => ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_f103356543l_bool @ ord_le1568362934t_bool @ B ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[600]) ).

thf(48,axiom,
    ! [A: $i,B: $i] :
      ( ~ ( hBOOL @ ( hAPP_int_bool @ ( hAPP_i1948725293t_bool @ ord_less_eq_int @ A ) @ B ) )
     => ( hBOOL @ ( hAPP_int_bool @ ( hAPP_i1948725293t_bool @ ord_less_eq_int @ B ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_736_linorder__le__cases) ).

thf(1290,plain,
    ! [A: $i,B: $i] :
      ( ~ ( hBOOL @ ( hAPP_int_bool @ ( hAPP_i1948725293t_bool @ ord_less_eq_int @ A ) @ B ) )
     => ( hBOOL @ ( hAPP_int_bool @ ( hAPP_i1948725293t_bool @ ord_less_eq_int @ B ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[48]) ).

thf(752,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ A ) @ B ) ) @ C )
      = ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ A ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ B ) @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_685_comm__semiring__1__class_Onormalizing__semiring__rules_I17_J) ).

thf(3858,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ A ) @ B ) ) @ C )
      = ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ A ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ B ) @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[752]) ).

thf(106,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ ord_min_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ minus_minus_nat @ A ) @ B ) ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ minus_minus_nat @ C ) @ B ) )
      = ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ minus_minus_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ ord_min_nat @ A ) @ C ) ) @ B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_1014_min__diff) ).

thf(1522,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ ord_min_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ minus_minus_nat @ A ) @ B ) ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ minus_minus_nat @ C ) @ B ) )
      = ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ minus_minus_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ ord_min_nat @ A ) @ C ) ) @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[106]) ).

thf(735,axiom,
    ! [A: $i] : ( is_bool @ ( finite648855052iple_a @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',gsy_c_Finite__Set_Ocomp__fun__idem_000tc__Hoare____Mirabelle____uwgpyvfjxg__Otri) ).

thf(3813,plain,
    ! [A: $i] : ( is_bool @ ( finite648855052iple_a @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[735]) ).

thf(452,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ A ) @ B ) )
     => ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ B ) @ A ) )
       => ( A = B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_755_order__antisym) ).

thf(2776,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ A ) @ B ) )
     => ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ B ) @ A ) )
       => ( A = B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[452]) ).

thf(1099,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ! [E: $i,F: $i] :
          ( ( hBOOL @ ( hAPP_state_bool @ ( hAPP_a2036067514e_bool @ D @ E ) @ F ) )
         => ? [G: $i,H: $i] :
              ( ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hoare_2102800559rivs_a @ B ) @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ ( hoare_1916936827iple_a @ G @ C @ H ) ) @ bot_bo797238721a_bool ) ) )
              & ! [I: $i] :
                  ( ! [J: $i] :
                      ( ( hBOOL @ ( hAPP_state_bool @ ( hAPP_a2036067514e_bool @ G @ J ) @ F ) )
                     => ( hBOOL @ ( hAPP_state_bool @ ( hAPP_a2036067514e_bool @ H @ J ) @ I ) ) )
                 => ( hBOOL @ ( hAPP_state_bool @ ( hAPP_a2036067514e_bool @ A @ E ) @ I ) ) ) ) )
     => ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hoare_2102800559rivs_a @ B ) @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ ( hoare_1916936827iple_a @ D @ C @ A ) ) @ bot_bo797238721a_bool ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_123_conseq) ).

thf(5173,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ! [E: $i,F: $i] :
          ( ( hBOOL @ ( hAPP_state_bool @ ( hAPP_a2036067514e_bool @ D @ E ) @ F ) )
         => ? [G: $i,H: $i] :
              ( ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hoare_2102800559rivs_a @ B ) @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ ( hoare_1916936827iple_a @ G @ C @ H ) ) @ bot_bo797238721a_bool ) ) )
              & ! [I: $i] :
                  ( ! [J: $i] :
                      ( ( hBOOL @ ( hAPP_state_bool @ ( hAPP_a2036067514e_bool @ G @ J ) @ F ) )
                     => ( hBOOL @ ( hAPP_state_bool @ ( hAPP_a2036067514e_bool @ H @ J ) @ I ) ) )
                 => ( hBOOL @ ( hAPP_state_bool @ ( hAPP_a2036067514e_bool @ A @ E ) @ I ) ) ) ) )
     => ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hoare_2102800559rivs_a @ B ) @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ ( hoare_1916936827iple_a @ D @ C @ A ) ) @ bot_bo797238721a_bool ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1099]) ).

thf(1039,axiom,
    ! [A: $i,B: $i] :
      ( ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ ( if_nat @ fFalse ) @ A ) @ B )
      = B ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',help_If_2_1_If_000tc__Nat__Onat_T) ).

thf(4960,plain,
    ! [A: $i,B: $i] :
      ( ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ ( if_nat @ fFalse ) @ A ) @ B )
      = B ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1039]) ).

thf(648,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_f284875647l_bool @ ord_le951220754t_bool @ B ) @ C ) )
     => ( ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_f284875647l_bool @ ord_le951220754t_bool @ C ) @ A ) )
       => ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_f284875647l_bool @ ord_le951220754t_bool @ B ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_792_subset__trans) ).

thf(3498,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_f284875647l_bool @ ord_le951220754t_bool @ B ) @ C ) )
     => ( ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_f284875647l_bool @ ord_le951220754t_bool @ C ) @ A ) )
       => ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_f284875647l_bool @ ord_le951220754t_bool @ B ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[648]) ).

thf(890,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ A ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ B ) @ C ) )
      = ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ A ) @ B ) ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ A ) @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_1052_add__mult__distrib2) ).

thf(4408,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ A ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ B ) @ C ) )
      = ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ A ) @ B ) ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ A ) @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[890]) ).

thf(51,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP_int_bool @ ( hAPP_i1948725293t_bool @ ord_less_eq_int @ A ) @ B ) )
     => ( ( hBOOL @ ( hAPP_int_bool @ ( hAPP_i1948725293t_bool @ ord_less_eq_int @ B ) @ A ) )
       => ( A = B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_1085_zle__antisym) ).

thf(1305,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP_int_bool @ ( hAPP_i1948725293t_bool @ ord_less_eq_int @ A ) @ B ) )
     => ( ( hBOOL @ ( hAPP_int_bool @ ( hAPP_i1948725293t_bool @ ord_less_eq_int @ B ) @ A ) )
       => ( A = B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[51]) ).

thf(908,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f1695230391l_bool @ finite1738664244iple_a @ C ) )
     => ( ~ ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ B ) @ C ) )
       => ( ( hAPP_f2097660464ol_nat @ ( hAPP_f1371399797ol_nat @ big_co1340561246_a_nat @ A ) @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ B ) @ C ) )
          = ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ ( hAPP_H592031934_a_nat @ A @ B ) ) @ ( hAPP_f2097660464ol_nat @ ( hAPP_f1371399797ol_nat @ big_co1340561246_a_nat @ A ) @ C ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_636_setsum__insert) ).

thf(4487,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f1695230391l_bool @ finite1738664244iple_a @ C ) )
     => ( ~ ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ B ) @ C ) )
       => ( ( hAPP_f2097660464ol_nat @ ( hAPP_f1371399797ol_nat @ big_co1340561246_a_nat @ A ) @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ B ) @ C ) )
          = ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ ( hAPP_H592031934_a_nat @ A @ B ) ) @ ( hAPP_f2097660464ol_nat @ ( hAPP_f1371399797ol_nat @ big_co1340561246_a_nat @ A ) @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[908]) ).

thf(905,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_f1767618879l_bool @ ord_le1523781517a_bool @ B ) @ C ) )
     => ( ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ A ) @ B ) )
       => ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ A ) @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_794_set__mp) ).

thf(4479,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_f1767618879l_bool @ ord_le1523781517a_bool @ B ) @ C ) )
     => ( ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ A ) @ B ) )
       => ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ A ) @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[905]) ).

thf(98,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_int_bool @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ ( finite772772422nt_int @ A ) @ B ) @ bot_bot_fun_int_bool ) @ C ) )
     => ( C = B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_303_empty__fold__graphE) ).

thf(1486,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_int_bool @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ ( finite772772422nt_int @ A ) @ B ) @ bot_bot_fun_int_bool ) @ C ) )
     => ( C = B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[98]) ).

thf(206,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ D ) @ E ) )
     => ( ( ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ D ) @ A ) ) @ B )
          = ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ E ) @ A ) ) @ C ) )
      <=> ( B
          = ( 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 @ E ) @ D ) ) @ A ) ) @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_1040_nat__eq__add__iff2) ).

thf(1855,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ D ) @ E ) )
     => ( ( ( ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ D ) @ A ) ) @ B )
            = ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ E ) @ A ) ) @ C ) )
         => ( B
            = ( 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 @ E ) @ D ) ) @ A ) ) @ C ) ) )
        & ( ( B
            = ( 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 @ E ) @ D ) ) @ A ) ) @ C ) )
         => ( ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ D ) @ A ) ) @ B )
            = ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ E ) @ A ) ) @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[206]) ).

thf(264,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( finite1626084323ne_int @ C @ D ) )
     => ( ( hBOOL @ ( hAPP_f448129468l_bool @ finite_finite_int @ B ) )
       => ( ~ ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ A ) @ B ) )
         => ( ( B != bot_bot_fun_int_bool )
           => ( ( hAPP_f1594865479ol_int @ D @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ A ) @ B ) )
              = ( hAPP_int_int @ ( hAPP_int_fun_int_int @ C @ A ) @ ( hAPP_f1594865479ol_int @ D @ B ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_269_folding__one_Oinsert) ).

thf(2077,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( finite1626084323ne_int @ C @ D ) )
     => ( ( hBOOL @ ( hAPP_f448129468l_bool @ finite_finite_int @ B ) )
       => ( ~ ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ A ) @ B ) )
         => ( ( B != bot_bot_fun_int_bool )
           => ( ( hAPP_f1594865479ol_int @ D @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ A ) @ B ) )
              = ( hAPP_int_int @ ( hAPP_int_fun_int_int @ C @ A ) @ ( hAPP_f1594865479ol_int @ D @ B ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[264]) ).

thf(1030,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( finite1753440478iple_a @ C @ D ) )
     => ( ( hBOOL @ ( hAPP_f1695230391l_bool @ finite1738664244iple_a @ B ) )
       => ( ~ ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ A ) @ B ) )
         => ( ( B != bot_bo797238721a_bool )
           => ( ( hAPP_f1693662087iple_a @ D @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ A ) @ B ) )
              = ( hAPP_H905846293iple_a @ ( hAPP_H2085992369iple_a @ C @ A ) @ ( hAPP_f1693662087iple_a @ D @ B ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_268_folding__one_Oinsert) ).

thf(4931,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( finite1753440478iple_a @ C @ D ) )
     => ( ( hBOOL @ ( hAPP_f1695230391l_bool @ finite1738664244iple_a @ B ) )
       => ( ~ ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ A ) @ B ) )
         => ( ( B != bot_bo797238721a_bool )
           => ( ( hAPP_f1693662087iple_a @ D @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ A ) @ B ) )
              = ( hAPP_H905846293iple_a @ ( hAPP_H2085992369iple_a @ C @ A ) @ ( hAPP_f1693662087iple_a @ D @ B ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1030]) ).

thf(1075,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_H1037229737a_bool @ ( finite1621280049iple_a @ A @ B ) @ C ) )
    <=> ? [D: $i,E: $i,F: $i] :
          ( ( B
            = ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ D ) @ E ) )
          & ( C = F )
          & ( hBOOL @ ( hAPP_H1037229737a_bool @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ ( finite90892550iple_a @ A ) @ D ) @ E ) @ F ) )
          & ~ ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ D ) @ E ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_333_fold1Set_Osimps) ).

thf(5069,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( hBOOL @ ( hAPP_H1037229737a_bool @ ( finite1621280049iple_a @ A @ B ) @ C ) )
       => ? [D: $i,E: $i,F: $i] :
            ( ( B
              = ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ D ) @ E ) )
            & ( C = F )
            & ( hBOOL @ ( hAPP_H1037229737a_bool @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ ( finite90892550iple_a @ A ) @ D ) @ E ) @ F ) )
            & ~ ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ D ) @ E ) ) ) )
      & ( ? [D: $i,E: $i,F: $i] :
            ( ( B
              = ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ D ) @ E ) )
            & ( C = F )
            & ( hBOOL @ ( hAPP_H1037229737a_bool @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ ( finite90892550iple_a @ A ) @ D ) @ E ) @ F ) )
            & ~ ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ D ) @ E ) ) )
       => ( hBOOL @ ( hAPP_H1037229737a_bool @ ( finite1621280049iple_a @ A @ B ) @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1075]) ).

thf(489,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_f103356543l_bool @ ord_le1568362934t_bool @ A ) @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ B ) @ C ) ) )
    <=> ( ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ B ) @ 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 @ B ) @ bot_bot_fun_nat_bool ) ) ) @ C ) ) )
        & ( ~ ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ B ) @ A ) )
         => ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_f103356543l_bool @ ord_le1568362934t_bool @ A ) @ C ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_907_subset__insert__iff) ).

thf(2908,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_f103356543l_bool @ ord_le1568362934t_bool @ A ) @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ B ) @ C ) ) )
       => ( ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ B ) @ 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 @ B ) @ bot_bot_fun_nat_bool ) ) ) @ C ) ) )
          & ( ~ ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ B ) @ A ) )
           => ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_f103356543l_bool @ ord_le1568362934t_bool @ A ) @ C ) ) ) ) )
      & ( ( ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ B ) @ 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 @ B ) @ bot_bot_fun_nat_bool ) ) ) @ C ) ) )
          & ( ~ ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ B ) @ A ) )
           => ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_f103356543l_bool @ ord_le1568362934t_bool @ A ) @ C ) ) ) )
       => ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_f103356543l_bool @ ord_le1568362934t_bool @ A ) @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ B ) @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[489]) ).

thf(259,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( finite795500164em_nat @ C @ D ) )
     => ( ! [E: $i,F: $i] :
            ( ( hAPP_nat_nat @ B @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ C @ E ) @ F ) )
            = ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ C @ ( hAPP_nat_nat @ B @ E ) ) @ ( hAPP_nat_nat @ B @ F ) ) )
       => ( ( hBOOL @ ( hAPP_f54304608l_bool @ finite_finite_nat @ A ) )
         => ( ( A != bot_bot_fun_nat_bool )
           => ( ( hAPP_nat_nat @ B @ ( hAPP_f22106695ol_nat @ D @ A ) )
              = ( hAPP_f22106695ol_nat @ D @ ( image_nat_nat @ B @ A ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_438_folding__one__idem_Ohom__commute) ).

thf(2062,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( finite795500164em_nat @ C @ D ) )
     => ( ! [E: $i,F: $i] :
            ( ( hAPP_nat_nat @ B @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ C @ E ) @ F ) )
            = ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ C @ ( hAPP_nat_nat @ B @ E ) ) @ ( hAPP_nat_nat @ B @ F ) ) )
       => ( ( hBOOL @ ( hAPP_f54304608l_bool @ finite_finite_nat @ A ) )
         => ( ( A != bot_bot_fun_nat_bool )
           => ( ( hAPP_nat_nat @ B @ ( hAPP_f22106695ol_nat @ D @ A ) )
              = ( hAPP_f22106695ol_nat @ D @ ( image_nat_nat @ B @ A ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[259]) ).

thf(499,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ A ) @ B ) ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ A ) @ C ) ) )
    <=> ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ B ) @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_829_add__le__cancel__left) ).

thf(2953,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ A ) @ B ) ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ A ) @ C ) ) )
       => ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ B ) @ C ) ) )
      & ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ B ) @ C ) )
       => ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ A ) @ B ) ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ A ) @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[499]) ).

thf(399,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_nat @ C ) @ D ) )
     => ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_nat @ A ) @ B ) )
       => ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ C ) @ A ) ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ D ) @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_1102_add__less__mono) ).

thf(2590,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_nat @ C ) @ D ) )
     => ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_nat @ A ) @ B ) )
       => ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ C ) @ A ) ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ D ) @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[399]) ).

thf(1125,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP_f1695230391l_bool @ finite1738664244iple_a @ B ) )
     => ( hBOOL @ ( hAPP_f1695230391l_bool @ finite1738664244iple_a @ ( hAPP_f20753329a_bool @ ( hAPP_f1706700729a_bool @ minus_85316870a_bool @ B ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_370_finite__Diff) ).

thf(5297,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP_f1695230391l_bool @ finite1738664244iple_a @ B ) )
     => ( hBOOL @ ( hAPP_f1695230391l_bool @ finite1738664244iple_a @ ( hAPP_f20753329a_bool @ ( hAPP_f1706700729a_bool @ minus_85316870a_bool @ B ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1125]) ).

thf(635,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( finite58652534t_bool @ D ) )
     => ( ( hBOOL @ ( hAPP_f448129468l_bool @ finite_finite_int @ C ) )
       => ( ( hAPP_f1805168059t_bool @ ( finite1827710202t_bool @ D @ A ) @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ B ) @ C ) )
          = ( hAPP_f1805168059t_bool @ ( finite1827710202t_bool @ D @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ D @ B ) @ A ) ) @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_574_comp__fun__idem_Ofold__insert__idem2) ).

thf(3460,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( finite58652534t_bool @ D ) )
     => ( ( hBOOL @ ( hAPP_f448129468l_bool @ finite_finite_int @ C ) )
       => ( ( hAPP_f1805168059t_bool @ ( finite1827710202t_bool @ D @ A ) @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ B ) @ C ) )
          = ( hAPP_f1805168059t_bool @ ( finite1827710202t_bool @ D @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ D @ B ) @ A ) ) @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[635]) ).

thf(120,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ A ) @ B ) )
     => ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ B ) @ A ) )
       => ( A = B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_1002_le__antisym) ).

thf(1569,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ A ) @ B ) )
     => ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ B ) @ A ) )
       => ( A = B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[120]) ).

thf(941,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( finite2114894759a_bool @ C ) )
     => ( ( hBOOL @ ( hAPP_f1695230391l_bool @ finite1738664244iple_a @ B ) )
       => ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_f1767618879l_bool @ ( hAPP_f1577095245l_bool @ ( finite1152845682a_bool @ C ) @ A ) @ B ) @ ( hAPP_f20753329a_bool @ ( finite1215486064a_bool @ C @ A ) @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_518_comp__fun__commute_Ofold__graph__fold) ).

thf(4603,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( finite2114894759a_bool @ C ) )
     => ( ( hBOOL @ ( hAPP_f1695230391l_bool @ finite1738664244iple_a @ B ) )
       => ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_f1767618879l_bool @ ( hAPP_f1577095245l_bool @ ( finite1152845682a_bool @ C ) @ A ) @ B ) @ ( hAPP_f20753329a_bool @ ( finite1215486064a_bool @ C @ A ) @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[941]) ).

thf(26,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_int_bool @ ( hAPP_i1948725293t_bool @ ord_less_eq_int @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ A ) @ B ) ) @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ C ) @ B ) ) )
     => ( hBOOL @ ( hAPP_int_bool @ ( hAPP_i1948725293t_bool @ ord_less_eq_int @ A ) @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_820_add__le__imp__le__right) ).

thf(1205,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_int_bool @ ( hAPP_i1948725293t_bool @ ord_less_eq_int @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ A ) @ B ) ) @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ C ) @ B ) ) )
     => ( hBOOL @ ( hAPP_int_bool @ ( hAPP_i1948725293t_bool @ ord_less_eq_int @ A ) @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[26]) ).

thf(146,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f448129468l_bool @ finite_finite_int @ C ) )
     => ( ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ B ) @ C ) )
       => ( ( hAPP_f1594865479ol_int @ ( hAPP_f1926459811ol_int @ big_co230513141nt_int @ A ) @ C )
          = ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ ( hAPP_int_int @ A @ B ) ) @ ( hAPP_f1594865479ol_int @ ( hAPP_f1926459811ol_int @ big_co230513141nt_int @ A ) @ ( hAPP_f1805168059t_bool @ ( hAPP_f1223193598t_bool @ minus_1449998731t_bool @ C ) @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ B ) @ bot_bot_fun_int_bool ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_617_setsum__diff1_H) ).

thf(1655,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f448129468l_bool @ finite_finite_int @ C ) )
     => ( ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ B ) @ C ) )
       => ( ( hAPP_f1594865479ol_int @ ( hAPP_f1926459811ol_int @ big_co230513141nt_int @ A ) @ C )
          = ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ ( hAPP_int_int @ A @ B ) ) @ ( hAPP_f1594865479ol_int @ ( hAPP_f1926459811ol_int @ big_co230513141nt_int @ A ) @ ( hAPP_f1805168059t_bool @ ( hAPP_f1223193598t_bool @ minus_1449998731t_bool @ C ) @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ B ) @ bot_bot_fun_int_bool ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[146]) ).

thf(893,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hAPP_int_int @ ( hAPP_int_fun_int_int @ times_times_int @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ times_times_int @ A ) @ B ) ) @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ times_times_int @ C ) @ D ) )
      = ( hAPP_int_int @ ( hAPP_int_fun_int_int @ times_times_int @ A ) @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ times_times_int @ B ) @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ times_times_int @ C ) @ D ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_682_comm__semiring__1__class_Onormalizing__semiring__rules_I14_J) ).

thf(4418,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hAPP_int_int @ ( hAPP_int_fun_int_int @ times_times_int @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ times_times_int @ A ) @ B ) ) @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ times_times_int @ C ) @ D ) )
      = ( hAPP_int_int @ ( hAPP_int_fun_int_int @ times_times_int @ A ) @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ times_times_int @ B ) @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ times_times_int @ C ) @ D ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[893]) ).

thf(406,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( finite1432773856em_int @ C @ D ) )
     => ( ! [E: $i,F: $i] :
            ( ( hAPP_int_int @ B @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ C @ E ) @ F ) )
            = ( hAPP_int_int @ ( hAPP_int_fun_int_int @ C @ ( hAPP_int_int @ B @ E ) ) @ ( hAPP_int_int @ B @ F ) ) )
       => ( ( hBOOL @ ( hAPP_f448129468l_bool @ finite_finite_int @ A ) )
         => ( ( A != bot_bot_fun_int_bool )
           => ( ( hAPP_int_int @ B @ ( hAPP_f1594865479ol_int @ D @ A ) )
              = ( hAPP_f1594865479ol_int @ D @ ( image_int_int @ B @ A ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_437_folding__one__idem_Ohom__commute) ).

thf(2607,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( finite1432773856em_int @ C @ D ) )
     => ( ! [E: $i,F: $i] :
            ( ( hAPP_int_int @ B @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ C @ E ) @ F ) )
            = ( hAPP_int_int @ ( hAPP_int_fun_int_int @ C @ ( hAPP_int_int @ B @ E ) ) @ ( hAPP_int_int @ B @ F ) ) )
       => ( ( hBOOL @ ( hAPP_f448129468l_bool @ finite_finite_int @ A ) )
         => ( ( A != bot_bot_fun_int_bool )
           => ( ( hAPP_int_int @ B @ ( hAPP_f1594865479ol_int @ D @ A ) )
              = ( hAPP_f1594865479ol_int @ D @ ( image_int_int @ B @ A ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[406]) ).

thf(176,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ A ) @ B ) )
     => ( ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ minus_minus_nat @ B ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ minus_minus_nat @ B ) @ A ) )
        = A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_1029_diff__diff__cancel) ).

thf(1756,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ A ) @ B ) )
     => ( ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ minus_minus_nat @ B ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ minus_minus_nat @ B ) @ A ) )
        = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[176]) ).

thf(544,axiom,
    ! [A: $i] : ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_f103356543l_bool @ ord_le1568362934t_bool @ A ) @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_719_order__refl) ).

thf(3135,plain,
    ! [A: $i] : ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_f103356543l_bool @ ord_le1568362934t_bool @ A ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[544]) ).

thf(1119,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_H1037229737a_bool @ ( finite1621280049iple_a @ A @ B ) @ C ) )
     => ( B != bot_bo797238721a_bool ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_227_fold1Set__nonempty) ).

thf(5257,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_H1037229737a_bool @ ( finite1621280049iple_a @ A @ B ) @ C ) )
     => ( B != bot_bo797238721a_bool ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1119]) ).

thf(680,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_int_bool @ B @ C ) )
     => ( ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_f284875647l_bool @ ord_le951220754t_bool @ B ) @ A ) )
       => ( hBOOL @ ( hAPP_int_bool @ A @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_813_rev__predicate1D) ).

thf(3616,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_int_bool @ B @ C ) )
     => ( ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_f284875647l_bool @ ord_le951220754t_bool @ B ) @ A ) )
       => ( hBOOL @ ( hAPP_int_bool @ A @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[680]) ).

thf(897,axiom,
    ! [A: $i] :
      ( ? [B: $i] :
          ( ( hBOOL @ ( hAPP_H1037229737a_bool @ A @ B ) )
          & ! [C: $i] :
              ( ( hBOOL @ ( hAPP_H1037229737a_bool @ A @ C ) )
             => ( C = B ) ) )
     => ( hBOOL @ ( hAPP_H1037229737a_bool @ A @ ( hAPP_f1693662087iple_a @ the_Ho830102290iple_a @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_342_theI_H) ).

thf(4440,plain,
    ! [A: $i] :
      ( ? [B: $i] :
          ( ( hBOOL @ ( hAPP_H1037229737a_bool @ A @ B ) )
          & ! [C: $i] :
              ( ( hBOOL @ ( hAPP_H1037229737a_bool @ A @ C ) )
             => ( C = B ) ) )
     => ( hBOOL @ ( hAPP_H1037229737a_bool @ A @ ( hAPP_f1693662087iple_a @ the_Ho830102290iple_a @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[897]) ).

thf(703,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_f103356543l_bool @ ord_le1568362934t_bool @ C ) @ D ) )
     => ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_f103356543l_bool @ ord_le1568362934t_bool @ A ) @ B ) )
       => ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_f103356543l_bool @ ord_le1568362934t_bool @ ( hAPP_f800510211t_bool @ ( hAPP_f1730770594t_bool @ minus_2067140911t_bool @ C ) @ B ) ) @ ( hAPP_f800510211t_bool @ ( hAPP_f1730770594t_bool @ minus_2067140911t_bool @ D ) @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_885_Diff__mono) ).

thf(3693,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_f103356543l_bool @ ord_le1568362934t_bool @ C ) @ D ) )
     => ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_f103356543l_bool @ ord_le1568362934t_bool @ A ) @ B ) )
       => ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_f103356543l_bool @ ord_le1568362934t_bool @ ( hAPP_f800510211t_bool @ ( hAPP_f1730770594t_bool @ minus_2067140911t_bool @ C ) @ B ) ) @ ( hAPP_f800510211t_bool @ ( hAPP_f1730770594t_bool @ minus_2067140911t_bool @ D ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[703]) ).

thf(411,axiom,
    ! [A: $i,B: $i] :
      ( ( ( hAPP_nat_int @ semiri1621563631at_int @ A )
        = ( hAPP_nat_int @ semiri1621563631at_int @ B ) )
    <=> ( A = B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_1158_transfer__int__nat__relations_I1_J) ).

thf(2627,plain,
    ! [A: $i,B: $i] :
      ( ( ( ( hAPP_nat_int @ semiri1621563631at_int @ A )
          = ( hAPP_nat_int @ semiri1621563631at_int @ B ) )
       => ( A = B ) )
      & ( ( A = B )
       => ( ( hAPP_nat_int @ semiri1621563631at_int @ A )
          = ( hAPP_nat_int @ semiri1621563631at_int @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[411]) ).

thf(178,axiom,
    ! [A: $i,B: $i] :
      ( ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ A ) @ ( hAPP_f1805168059t_bool @ ( hAPP_f1223193598t_bool @ minus_1449998731t_bool @ B ) @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ A ) @ bot_bot_fun_int_bool ) ) )
      = ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ A ) @ B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_419_insert__Diff__single) ).

thf(1761,plain,
    ! [A: $i,B: $i] :
      ( ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ A ) @ ( hAPP_f1805168059t_bool @ ( hAPP_f1223193598t_bool @ minus_1449998731t_bool @ B ) @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ A ) @ bot_bot_fun_int_bool ) ) )
      = ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ A ) @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[178]) ).

thf(669,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( B = C )
     => ( ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_f284875647l_bool @ ord_le951220754t_bool @ A ) @ C ) )
       => ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_f284875647l_bool @ ord_le951220754t_bool @ A ) @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_764_xt1_I3_J) ).

thf(3577,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( B = C )
     => ( ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_f284875647l_bool @ ord_le951220754t_bool @ A ) @ C ) )
       => ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_f284875647l_bool @ ord_le951220754t_bool @ A ) @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[669]) ).

thf(983,axiom,
    ! [A: $i] :
      ( ( 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 ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_732_finite__Collect__subsets) ).

thf(4762,plain,
    ! [A: $i] :
      ( ( 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 ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[983]) ).

thf(749,axiom,
    ! [A: $i] :
      ( ( hAPP_f1805168059t_bool @ collect_int @ A )
      = A ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_198_Collect__def) ).

thf(3848,plain,
    ! [A: $i] :
      ( ( hAPP_f1805168059t_bool @ collect_int @ A )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[749]) ).

thf(517,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP_f448129468l_bool @ finite_finite_int @ B ) )
     => ( ( hBOOL @ ( hAPP_f448129468l_bool @ A @ B ) )
       => ( ! [C: $i,D: $i] :
              ( ( hBOOL @ ( hAPP_f448129468l_bool @ finite_finite_int @ D ) )
             => ( ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ C ) @ D ) )
               => ( ( hBOOL @ ( hAPP_f448129468l_bool @ A @ D ) )
                 => ( hBOOL @ ( hAPP_f448129468l_bool @ A @ ( hAPP_f1805168059t_bool @ ( hAPP_f1223193598t_bool @ minus_1449998731t_bool @ D ) @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ C ) @ bot_bot_fun_int_bool ) ) ) ) ) ) )
         => ( hBOOL @ ( hAPP_f448129468l_bool @ A @ bot_bot_fun_int_bool ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_441_finite__empty__induct) ).

thf(3017,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP_f448129468l_bool @ finite_finite_int @ B ) )
     => ( ( hBOOL @ ( hAPP_f448129468l_bool @ A @ B ) )
       => ( ! [C: $i,D: $i] :
              ( ( hBOOL @ ( hAPP_f448129468l_bool @ finite_finite_int @ D ) )
             => ( ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ C ) @ D ) )
               => ( ( hBOOL @ ( hAPP_f448129468l_bool @ A @ D ) )
                 => ( hBOOL @ ( hAPP_f448129468l_bool @ A @ ( hAPP_f1805168059t_bool @ ( hAPP_f1223193598t_bool @ minus_1449998731t_bool @ D ) @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ C ) @ bot_bot_fun_int_bool ) ) ) ) ) ) )
         => ( hBOOL @ ( hAPP_f448129468l_bool @ A @ bot_bot_fun_int_bool ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[517]) ).

thf(925,axiom,
    ! [A: $i] :
      ( ( hBOOL @ ( hAPP_H1037229737a_bool @ bot_bo797238721a_bool @ A ) )
    <=> ( hBOOL @ bot_bot_bool ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_108_bot__fun__def) ).

thf(4543,plain,
    ! [A: $i] :
      ( ( ( hBOOL @ ( hAPP_H1037229737a_bool @ bot_bo797238721a_bool @ A ) )
       => ( hBOOL @ bot_bot_bool ) )
      & ( ( hBOOL @ bot_bot_bool )
       => ( hBOOL @ ( hAPP_H1037229737a_bool @ bot_bo797238721a_bool @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[925]) ).

thf(188,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP_H1037229737a_bool @ ( hAPP_f20753329a_bool @ ( hAPP_f604201481a_bool @ cOMBB_1882975613iple_a @ A ) @ B ) @ C )
      = ( hAPP_bool_bool @ A @ ( hAPP_H1037229737a_bool @ B @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',help_COMBB_1_1_COMBB_000tc__HOL__Obool_000tc__HOL__Obool_000tc__Hoare____Mirabel) ).

thf(1788,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP_H1037229737a_bool @ ( hAPP_f20753329a_bool @ ( hAPP_f604201481a_bool @ cOMBB_1882975613iple_a @ A ) @ B ) @ C )
      = ( hAPP_bool_bool @ A @ ( hAPP_H1037229737a_bool @ B @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[188]) ).

thf(82,axiom,
    ! [A: $i] : ( is_bool @ ( finite1321096241t_bool @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',gsy_c_Finite__Set_Ocomp__fun__commute_000tc__Int__Oint_000tc__fun_Itc__Int__Oint) ).

thf(1426,plain,
    ! [A: $i] : ( is_bool @ ( finite1321096241t_bool @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[82]) ).

thf(514,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( C = D )
     => ( ! [E: $i] :
            ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ E ) @ D ) )
           => ( ( hAPP_nat_nat @ A @ E )
              = ( hAPP_nat_nat @ B @ E ) ) )
       => ( ( image_nat_nat @ A @ C )
          = ( image_nat_nat @ B @ D ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_259_image__cong) ).

thf(3004,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( C = D )
     => ( ! [E: $i] :
            ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ E ) @ D ) )
           => ( ( hAPP_nat_nat @ A @ E )
              = ( hAPP_nat_nat @ B @ E ) ) )
       => ( ( image_nat_nat @ A @ C )
          = ( image_nat_nat @ B @ D ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[514]) ).

thf(250,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_nat @ B ) @ C ) )
     => ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_nat @ B ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ C ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_1105_trans__less__add1) ).

thf(2035,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_nat @ B ) @ C ) )
     => ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_nat @ B ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ C ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[250]) ).

thf(821,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ A ) @ B ) ) @ C )
      = ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ A ) @ C ) ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ B ) @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_701_comm__semiring__class_Odistrib) ).

thf(4110,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ A ) @ B ) ) @ C )
      = ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ A ) @ C ) ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ B ) @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[821]) ).

thf(846,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ A ) @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ B ) @ bot_bot_fun_nat_bool ) )
        = ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ C ) @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ D ) @ bot_bot_fun_nat_bool ) ) )
    <=> ( ( ( A = C )
          & ( B = D ) )
        | ( ( A = D )
          & ( B = C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_90_doubleton__eq__iff) ).

thf(4220,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ A ) @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ B ) @ bot_bot_fun_nat_bool ) )
          = ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ C ) @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ D ) @ bot_bot_fun_nat_bool ) ) )
       => ( ( ( A = C )
            & ( B = D ) )
          | ( ( A = D )
            & ( B = C ) ) ) )
      & ( ( ( ( A = C )
            & ( B = D ) )
          | ( ( A = D )
            & ( B = C ) ) )
       => ( ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ A ) @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ B ) @ bot_bot_fun_nat_bool ) )
          = ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ C ) @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ D ) @ bot_bot_fun_nat_bool ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[846]) ).

thf(919,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f1695230391l_bool @ finite1738664244iple_a @ C ) )
     => ( ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ B ) @ C ) )
       => ( ( hAPP_f587450508ol_int @ ( big_co305732779_a_int @ A ) @ C )
          = ( hAPP_int_int @ ( hAPP_int_fun_int_int @ times_times_int @ ( hAPP_H1229305626_a_int @ A @ B ) ) @ ( hAPP_f587450508ol_int @ ( big_co305732779_a_int @ A ) @ ( hAPP_f20753329a_bool @ ( hAPP_f1706700729a_bool @ minus_85316870a_bool @ C ) @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ B ) @ bot_bo797238721a_bool ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_476_setprod_Oremove) ).

thf(4528,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f1695230391l_bool @ finite1738664244iple_a @ C ) )
     => ( ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ B ) @ C ) )
       => ( ( hAPP_f587450508ol_int @ ( big_co305732779_a_int @ A ) @ C )
          = ( hAPP_int_int @ ( hAPP_int_fun_int_int @ times_times_int @ ( hAPP_H1229305626_a_int @ A @ B ) ) @ ( hAPP_f587450508ol_int @ ( big_co305732779_a_int @ A ) @ ( hAPP_f20753329a_bool @ ( hAPP_f1706700729a_bool @ minus_85316870a_bool @ C ) @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ B ) @ bot_bo797238721a_bool ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[919]) ).

thf(557,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( finite58652534t_bool @ C ) )
     => ( ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ C @ A ) @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ C @ A ) @ B ) )
        = ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ C @ A ) @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_451_comp__fun__idem_Ofun__left__idem) ).

thf(3175,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( finite58652534t_bool @ C ) )
     => ( ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ C @ A ) @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ C @ A ) @ B ) )
        = ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ C @ A ) @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[557]) ).

thf(663,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( finite1860950092at_nat @ D ) )
     => ( ( hBOOL @ ( hAPP_f54304608l_bool @ finite_finite_nat @ C ) )
       => ( ( hAPP_f22106695ol_nat @ ( finite_fold_nat_nat @ D @ A ) @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ B ) @ C ) )
          = ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ D @ B ) @ ( hAPP_f22106695ol_nat @ ( finite_fold_nat_nat @ D @ A ) @ C ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_575_comp__fun__idem_Ofold__insert__idem) ).

thf(3562,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( finite1860950092at_nat @ D ) )
     => ( ( hBOOL @ ( hAPP_f54304608l_bool @ finite_finite_nat @ C ) )
       => ( ( hAPP_f22106695ol_nat @ ( finite_fold_nat_nat @ D @ A ) @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ B ) @ C ) )
          = ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ D @ B ) @ ( hAPP_f22106695ol_nat @ ( finite_fold_nat_nat @ D @ A ) @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[663]) ).

thf(851,axiom,
    ! [A: $i,B: $i] :
      ( ~ ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_f284875647l_bool @ fequal_fun_int_bool @ A ) @ B ) )
      | ( A = B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',help_fequal_1_1_fequal_000tc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J_T) ).

thf(4247,plain,
    ! [A: $i,B: $i] :
      ( ~ ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_f284875647l_bool @ fequal_fun_int_bool @ A ) @ B ) )
      | ( A = B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[851]) ).

thf(1055,axiom,
    ! [A: $i] :
      ( ( 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 ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_731_finite__Collect__subsets) ).

thf(5005,plain,
    ! [A: $i] :
      ( ( 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 ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1055]) ).

thf(873,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP_H1037229737a_bool @ A @ B ) )
     => ( ! [C: $i] :
            ( ( hBOOL @ ( hAPP_H1037229737a_bool @ A @ C ) )
           => ( C = B ) )
       => ( hBOOL @ ( hAPP_H1037229737a_bool @ A @ ( hAPP_f1693662087iple_a @ the_Ho830102290iple_a @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_336_theI) ).

thf(4343,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP_H1037229737a_bool @ A @ B ) )
     => ( ! [C: $i] :
            ( ( hBOOL @ ( hAPP_H1037229737a_bool @ A @ C ) )
           => ( C = B ) )
       => ( hBOOL @ ( hAPP_H1037229737a_bool @ A @ ( hAPP_f1693662087iple_a @ the_Ho830102290iple_a @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[873]) ).

thf(1113,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( finite2114894759a_bool @ D ) )
     => ( ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_f1767618879l_bool @ ( hAPP_f1577095245l_bool @ ( finite1152845682a_bool @ D ) @ A ) @ B ) @ C ) )
       => ( ( hAPP_f20753329a_bool @ ( finite1215486064a_bool @ D @ A ) @ B )
          = C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_497_comp__fun__commute_Ofold__equality) ).

thf(5227,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( finite2114894759a_bool @ D ) )
     => ( ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_f1767618879l_bool @ ( hAPP_f1577095245l_bool @ ( finite1152845682a_bool @ D ) @ A ) @ B ) @ C ) )
       => ( ( hAPP_f20753329a_bool @ ( finite1215486064a_bool @ D @ A ) @ B )
          = C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1113]) ).

thf(186,axiom,
    ! [A: $i] :
      ( ( image_nat_nat @ A @ bot_bot_fun_nat_bool )
      = bot_bot_fun_nat_bool ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_208_image__empty) ).

thf(1782,plain,
    ! [A: $i] :
      ( ( image_nat_nat @ A @ bot_bot_fun_nat_bool )
      = bot_bot_fun_nat_bool ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[186]) ).

thf(245,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ A ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ B ) @ C ) )
      = ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ A ) @ B ) ) @ C ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_671_comm__semiring__1__class_Onormalizing__semiring__rules_I25_J) ).

thf(2020,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ A ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ B ) @ C ) )
      = ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ A ) @ B ) ) @ C ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[245]) ).

thf(945,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ A ) @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ B ) @ bot_bo797238721a_bool ) ) )
    <=> ( A = B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_94_singleton__iff) ).

thf(4618,plain,
    ! [A: $i,B: $i] :
      ( ( ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ A ) @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ B ) @ bot_bo797238721a_bool ) ) )
       => ( A = B ) )
      & ( ( A = B )
       => ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ A ) @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ B ) @ bot_bo797238721a_bool ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[945]) ).

thf(135,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ C ) @ D ) )
     => ( ( A
          = ( hAPP_H1229305626_a_int @ B @ C ) )
       => ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ A ) @ ( image_685909450_a_int @ B @ D ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_172_rev__image__eqI) ).

thf(1613,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ C ) @ D ) )
     => ( ( A
          = ( hAPP_H1229305626_a_int @ B @ C ) )
       => ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ A ) @ ( image_685909450_a_int @ B @ D ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[135]) ).

thf(480,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ B ) @ C ) )
     => ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ B ) @ A ) ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ C ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_827_add__right__mono) ).

thf(2886,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ B ) @ C ) )
     => ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ B ) @ A ) ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ C ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[480]) ).

thf(266,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( B
        = ( hAPP_int_int @ C @ D ) )
     => ( ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ D ) @ A ) )
       => ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ B ) @ ( image_int_int @ C @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_159_image__eqI) ).

thf(2082,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( B
        = ( hAPP_int_int @ C @ D ) )
     => ( ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ D ) @ A ) )
       => ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ B ) @ ( image_int_int @ C @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[266]) ).

thf(1040,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hoare_2102800559rivs_a @ B ) @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ C ) @ bot_bo797238721a_bool ) ) )
     => ( ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hoare_2102800559rivs_a @ B ) @ A ) )
       => ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hoare_2102800559rivs_a @ B ) @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ C ) @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_3_hoare__derivs_Oinsert) ).

thf(4963,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hoare_2102800559rivs_a @ B ) @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ C ) @ bot_bo797238721a_bool ) ) )
     => ( ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hoare_2102800559rivs_a @ B ) @ A ) )
       => ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hoare_2102800559rivs_a @ B ) @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ C ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1040]) ).

thf(759,axiom,
    ! [A: $i,B: $i] :
      ( ( is_bool @ A )
     => ( ( hAPP_H1037229737a_bool @ ( hAPP_b1882817719a_bool @ cOMBK_235286536iple_a @ A ) @ B )
        = A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',help_COMBK_1_1_COMBK_000tc__HOL__Obool_000tc__Hoare____Mirabelle____uwgpyvfjxg__) ).

thf(3878,plain,
    ! [A: $i,B: $i] :
      ( ( is_bool @ A )
     => ( ( hAPP_H1037229737a_bool @ ( hAPP_b1882817719a_bool @ cOMBK_235286536iple_a @ A ) @ B )
        = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[759]) ).

thf(701,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_f284875647l_bool @ ord_le951220754t_bool @ B ) @ C ) )
     => ( ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_f284875647l_bool @ ord_le951220754t_bool @ C ) @ A ) )
       => ( ( hAPP_f1805168059t_bool @ ( hAPP_f1223193598t_bool @ minus_1449998731t_bool @ C ) @ ( hAPP_f1805168059t_bool @ ( hAPP_f1223193598t_bool @ minus_1449998731t_bool @ A ) @ B ) )
          = B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_880_double__diff) ).

thf(3688,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_f284875647l_bool @ ord_le951220754t_bool @ B ) @ C ) )
     => ( ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_f284875647l_bool @ ord_le951220754t_bool @ C ) @ A ) )
       => ( ( hAPP_f1805168059t_bool @ ( hAPP_f1223193598t_bool @ minus_1449998731t_bool @ C ) @ ( hAPP_f1805168059t_bool @ ( hAPP_f1223193598t_bool @ minus_1449998731t_bool @ A ) @ B ) )
          = B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[701]) ).

thf(1061,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( finite1753440478iple_a @ B @ C ) )
     => ( ( hAPP_f1693662087iple_a @ C @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ A ) @ bot_bo797238721a_bool ) )
        = A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_137_folding__one_Osingleton) ).

thf(5024,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( finite1753440478iple_a @ B @ C ) )
     => ( ( hAPP_f1693662087iple_a @ C @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ A ) @ bot_bo797238721a_bool ) )
        = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1061]) ).

thf(885,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_f103356543l_bool @ ord_le1568362934t_bool @ B ) @ C ) )
     => ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_f103356543l_bool @ ord_le1568362934t_bool @ C ) @ A ) )
       => ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_f103356543l_bool @ ord_le1568362934t_bool @ B ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_793_subset__trans) ).

thf(4386,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_f103356543l_bool @ ord_le1568362934t_bool @ B ) @ C ) )
     => ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_f103356543l_bool @ ord_le1568362934t_bool @ C ) @ A ) )
       => ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_f103356543l_bool @ ord_le1568362934t_bool @ B ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[885]) ).

thf(503,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ B ) @ C ) )
     => ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ ( hAPP_H1229305626_a_int @ A @ B ) ) @ ( image_685909450_a_int @ A @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_180_imageI) ).

thf(2972,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ B ) @ C ) )
     => ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ ( hAPP_H1229305626_a_int @ A @ B ) ) @ ( image_685909450_a_int @ A @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[503]) ).

thf(687,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ times_times_int @ A ) @ B ) ) @ C )
        = ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ times_times_int @ D ) @ B ) ) @ E ) )
    <=> ( ( 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 ) @ D ) ) @ B ) ) @ C )
        = E ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_644_eq__add__iff1) ).

thf(3634,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( ( ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ times_times_int @ A ) @ B ) ) @ C )
          = ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ times_times_int @ D ) @ B ) ) @ E ) )
       => ( ( 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 ) @ D ) ) @ B ) ) @ C )
          = E ) )
      & ( ( ( 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 ) @ D ) ) @ B ) ) @ C )
          = E )
       => ( ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ times_times_int @ A ) @ B ) ) @ C )
          = ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ times_times_int @ D ) @ B ) ) @ E ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[687]) ).

thf(358,axiom,
    ! [A: $i] :
      ~ ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ A ) @ bot_bot_fun_int_bool ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_17_emptyE) ).

thf(2432,plain,
    ! [A: $i] :
      ~ ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ A ) @ bot_bot_fun_int_bool ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[358]) ).

thf(218,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( C = D )
     => ( ! [E: $i] :
            ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ E ) @ D ) )
           => ( ( hAPP_nat_int @ A @ E )
              = ( hAPP_nat_int @ B @ E ) ) )
       => ( ( image_nat_int @ A @ C )
          = ( image_nat_int @ B @ D ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_260_image__cong) ).

thf(1914,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( C = D )
     => ( ! [E: $i] :
            ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ E ) @ D ) )
           => ( ( hAPP_nat_int @ A @ E )
              = ( hAPP_nat_int @ B @ E ) ) )
       => ( ( image_nat_int @ A @ C )
          = ( image_nat_int @ B @ D ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[218]) ).

thf(911,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( semi @ A @ B )
        = ( semi @ C @ D ) )
    <=> ( ( A = C )
        & ( B = D ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_132_com_Osimps_I3_J) ).

thf(4496,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( ( semi @ A @ B )
          = ( semi @ C @ D ) )
       => ( ( A = C )
          & ( B = D ) ) )
      & ( ( ( A = C )
          & ( B = D ) )
       => ( ( semi @ A @ B )
          = ( semi @ C @ D ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[911]) ).

thf(72,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_nat @ A ) @ B ) )
     => ( A != B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_1117_less__not__refl3) ).

thf(1391,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_nat @ A ) @ B ) )
     => ( A != B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[72]) ).

thf(1086,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f1695230391l_bool @ finite1738664244iple_a @ C ) )
     => ( ( hAPP_f587450508ol_int @ ( big_co305732779_a_int @ A ) @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ B ) @ C ) )
        = ( hAPP_int_int @ ( hAPP_int_fun_int_int @ times_times_int @ ( hAPP_H1229305626_a_int @ A @ B ) ) @ ( hAPP_f587450508ol_int @ ( big_co305732779_a_int @ A ) @ ( hAPP_f20753329a_bool @ ( hAPP_f1706700729a_bool @ minus_85316870a_bool @ C ) @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ B ) @ bot_bo797238721a_bool ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_470_setprod_Oinsert__remove) ).

thf(5123,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f1695230391l_bool @ finite1738664244iple_a @ C ) )
     => ( ( hAPP_f587450508ol_int @ ( big_co305732779_a_int @ A ) @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ B ) @ C ) )
        = ( hAPP_int_int @ ( hAPP_int_fun_int_int @ times_times_int @ ( hAPP_H1229305626_a_int @ A @ B ) ) @ ( hAPP_f587450508ol_int @ ( big_co305732779_a_int @ A ) @ ( hAPP_f20753329a_bool @ ( hAPP_f1706700729a_bool @ minus_85316870a_bool @ C ) @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ B ) @ bot_bo797238721a_bool ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1086]) ).

thf(579,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP_f448129468l_bool @ ( hAPP_f284875647l_bool @ ( hAPP_f1399552105l_bool @ cOMBC_605892544l_bool @ A ) @ B ) @ C )
      = ( hAPP_f448129468l_bool @ ( hAPP_f284875647l_bool @ A @ C ) @ B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',help_COMBC_1_1_COMBC_000tc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J_000tc__fun_Itc_) ).

thf(3249,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP_f448129468l_bool @ ( hAPP_f284875647l_bool @ ( hAPP_f1399552105l_bool @ cOMBC_605892544l_bool @ A ) @ B ) @ C )
      = ( hAPP_f448129468l_bool @ ( hAPP_f284875647l_bool @ A @ C ) @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[579]) ).

thf(814,axiom,
    ! [A: $i,B: $i] :
      ( bot_bot_fun_nat_bool
     != ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ A ) @ B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_99_empty__not__insert) ).

thf(4079,plain,
    ! [A: $i,B: $i] :
      ( bot_bot_fun_nat_bool
     != ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ A ) @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[814]) ).

thf(87,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ ord_min_nat @ A ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ ord_min_nat @ B ) @ C ) )
      = ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ ord_min_nat @ B ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ ord_min_nat @ A ) @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_955_min__max_Oinf__left__commute) ).

thf(1441,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ ord_min_nat @ A ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ ord_min_nat @ B ) @ C ) )
      = ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ ord_min_nat @ B ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ ord_min_nat @ A ) @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[87]) ).

thf(241,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_nat @ B ) @ C ) )
     => ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ A ) @ B ) )
       => ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ minus_minus_nat @ B ) @ A ) ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ minus_minus_nat @ C ) @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_1133_diff__less__mono) ).

thf(2010,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_nat @ B ) @ C ) )
     => ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ A ) @ B ) )
       => ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ minus_minus_nat @ B ) @ A ) ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ minus_minus_nat @ C ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[241]) ).

thf(222,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_nat_bool @ ( finite_fold1Set_nat @ A @ B ) @ C ) )
     => ( B != bot_bot_fun_nat_bool ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_229_fold1Set__nonempty) ).

thf(1935,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_nat_bool @ ( finite_fold1Set_nat @ A @ B ) @ C ) )
     => ( B != bot_bot_fun_nat_bool ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[222]) ).

thf(781,axiom,
    ! [A: $i,B: $i] :
      ( ( 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 ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_782_order__eq__iff) ).

thf(3947,plain,
    ! [A: $i,B: $i] :
      ( ( ( 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 ) ) ) )
      & ( ( ( 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 ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[781]) ).

thf(997,axiom,
    ! [A: $i,B: $i] : ( is_bool @ ( hAPP_f1695230391l_bool @ A @ B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',gsy_c_hAPP_000tc__fun_Itc__Hoare____Mirabelle____uwgpyvfjxg__Otriple_It__a_J_Mtc) ).

thf(4803,plain,
    ! [A: $i,B: $i] : ( is_bool @ ( hAPP_f1695230391l_bool @ A @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[997]) ).

thf(823,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ B ) @ C ) )
     => ( ( image_48635758_a_nat @ ( hAPP_n1653940209_a_nat @ cOMBK_1028443109iple_a @ A ) @ C )
        = ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ A ) @ bot_bot_fun_nat_bool ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_233_image__constant) ).

thf(4124,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ B ) @ C ) )
     => ( ( image_48635758_a_nat @ ( hAPP_n1653940209_a_nat @ cOMBK_1028443109iple_a @ A ) @ C )
        = ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ A ) @ bot_bot_fun_nat_bool ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[823]) ).

thf(144,axiom,
    ! [A: $i,B: $i] :
      ( ( ( B = bot_bot_fun_int_bool )
       => ( ( image_int_int @ ( hAPP_int_fun_int_int @ cOMBK_int_int @ A ) @ B )
          = bot_bot_fun_int_bool ) )
      & ( ( B != bot_bot_fun_int_bool )
       => ( ( image_int_int @ ( hAPP_int_fun_int_int @ cOMBK_int_int @ A ) @ B )
          = ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ A ) @ bot_bot_fun_int_bool ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_152_image__constant__conv) ).

thf(1645,plain,
    ! [A: $i,B: $i] :
      ( ( ( B = bot_bot_fun_int_bool )
       => ( ( image_int_int @ ( hAPP_int_fun_int_int @ cOMBK_int_int @ A ) @ B )
          = bot_bot_fun_int_bool ) )
      & ( ( B != bot_bot_fun_int_bool )
       => ( ( image_int_int @ ( hAPP_int_fun_int_int @ cOMBK_int_int @ A ) @ B )
          = ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ A ) @ bot_bot_fun_int_bool ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[144]) ).

thf(219,axiom,
    ! [A: $i,B: $i] :
      ( ( A != B )
      | ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ fequal_nat @ A ) @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',help_fequal_2_1_fequal_000tc__Nat__Onat_T) ).

thf(1921,plain,
    ! [A: $i,B: $i] :
      ( ( A != B )
      | ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ fequal_nat @ A ) @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[219]) ).

thf(71,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i] :
          ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ D ) @ C ) )
         => ( ( hAPP_nat_nat @ A @ D )
            = ( hAPP_nat_nat @ B @ D ) ) )
     => ( ( hAPP_f22106695ol_nat @ ( hAPP_f782000547ol_nat @ big_co387207925at_nat @ A ) @ C )
        = ( hAPP_f22106695ol_nat @ ( hAPP_f782000547ol_nat @ big_co387207925at_nat @ B ) @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_706_setsum__cong2) ).

thf(1386,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i] :
          ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ D ) @ C ) )
         => ( ( hAPP_nat_nat @ A @ D )
            = ( hAPP_nat_nat @ B @ D ) ) )
     => ( ( hAPP_f22106695ol_nat @ ( hAPP_f782000547ol_nat @ big_co387207925at_nat @ A ) @ C )
        = ( hAPP_f22106695ol_nat @ ( hAPP_f782000547ol_nat @ big_co387207925at_nat @ B ) @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[71]) ).

thf(589,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f1695230391l_bool @ finite1738664244iple_a @ C ) )
     => ( ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_f1767618879l_bool @ ord_le1523781517a_bool @ C ) @ B ) )
       => ( ( hAPP_f2097660464ol_nat @ ( hAPP_f1371399797ol_nat @ big_co1340561246_a_nat @ A ) @ ( hAPP_f20753329a_bool @ ( hAPP_f1706700729a_bool @ minus_85316870a_bool @ B ) @ C ) )
          = ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ minus_minus_nat @ ( hAPP_f2097660464ol_nat @ ( hAPP_f1371399797ol_nat @ big_co1340561246_a_nat @ A ) @ B ) ) @ ( hAPP_f2097660464ol_nat @ ( hAPP_f1371399797ol_nat @ big_co1340561246_a_nat @ A ) @ C ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_715_setsum__diff__nat) ).

thf(3281,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f1695230391l_bool @ finite1738664244iple_a @ C ) )
     => ( ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_f1767618879l_bool @ ord_le1523781517a_bool @ C ) @ B ) )
       => ( ( hAPP_f2097660464ol_nat @ ( hAPP_f1371399797ol_nat @ big_co1340561246_a_nat @ A ) @ ( hAPP_f20753329a_bool @ ( hAPP_f1706700729a_bool @ minus_85316870a_bool @ B ) @ C ) )
          = ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ minus_minus_nat @ ( hAPP_f2097660464ol_nat @ ( hAPP_f1371399797ol_nat @ big_co1340561246_a_nat @ A ) @ B ) ) @ ( hAPP_f2097660464ol_nat @ ( hAPP_f1371399797ol_nat @ big_co1340561246_a_nat @ A ) @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[589]) ).

thf(167,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ minus_minus_nat @ A ) @ B ) ) @ C ) )
    <=> ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ A ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ C ) @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_1025_le__diff__conv) ).

thf(1724,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ minus_minus_nat @ A ) @ B ) ) @ C ) )
       => ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ A ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ C ) @ B ) ) ) )
      & ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ A ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ C ) @ B ) ) )
       => ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ minus_minus_nat @ A ) @ B ) ) @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[167]) ).

thf(96,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ A ) @ B ) )
    <=> ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_nat @ A ) @ B ) )
        | ( A = B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_1110_le__eq__less__or__eq) ).

thf(1475,plain,
    ! [A: $i,B: $i] :
      ( ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ A ) @ B ) )
       => ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_nat @ A ) @ B ) )
          | ( A = B ) ) )
      & ( ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_nat @ A ) @ B ) )
          | ( A = B ) )
       => ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ A ) @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[96]) ).

thf(1121,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_f1767618879l_bool @ ord_le1523781517a_bool @ A ) @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ B ) @ bot_bo797238721a_bool ) ) )
     => ( ( A = bot_bo797238721a_bool )
        | ( A
          = ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ B ) @ bot_bo797238721a_bool ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_893_subset__singletonD) ).

thf(5269,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_f1767618879l_bool @ ord_le1523781517a_bool @ A ) @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ B ) @ bot_bo797238721a_bool ) ) )
     => ( ( A = bot_bo797238721a_bool )
        | ( A
          = ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ B ) @ bot_bo797238721a_bool ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1121]) ).

thf(192,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ B ) @ C ) )
     => ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ C ) @ A ) )
       => ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ B ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_751_order__trans) ).

thf(1803,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ B ) @ C ) )
     => ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ C ) @ A ) )
       => ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ B ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[192]) ).

thf(848,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f54304608l_bool @ finite_finite_nat @ C ) )
     => ( ( hAPP_f22106695ol_nat @ ( big_co1705425894at_nat @ A ) @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ B ) @ C ) )
        = ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ ( hAPP_nat_nat @ A @ B ) ) @ ( hAPP_f22106695ol_nat @ ( big_co1705425894at_nat @ A ) @ ( hAPP_f800510211t_bool @ ( hAPP_f1730770594t_bool @ minus_2067140911t_bool @ C ) @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ B ) @ bot_bot_fun_nat_bool ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_466_setprod_Oinsert__remove) ).

thf(4238,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f54304608l_bool @ finite_finite_nat @ C ) )
     => ( ( hAPP_f22106695ol_nat @ ( big_co1705425894at_nat @ A ) @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ B ) @ C ) )
        = ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ ( hAPP_nat_nat @ A @ B ) ) @ ( hAPP_f22106695ol_nat @ ( big_co1705425894at_nat @ A ) @ ( hAPP_f800510211t_bool @ ( hAPP_f1730770594t_bool @ minus_2067140911t_bool @ C ) @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ B ) @ bot_bot_fun_nat_bool ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[848]) ).

thf(689,axiom,
    ! [A: $i,B: $i] :
      ( ( hAPP_int_int @ ( hAPP_int_fun_int_int @ times_times_int @ ( hAPP_nat_int @ semiri1621563631at_int @ A ) ) @ ( hAPP_nat_int @ semiri1621563631at_int @ B ) )
      = ( hAPP_nat_int @ semiri1621563631at_int @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ A ) @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_1067_zmult__int) ).

thf(3644,plain,
    ! [A: $i,B: $i] :
      ( ( hAPP_int_int @ ( hAPP_int_fun_int_int @ times_times_int @ ( hAPP_nat_int @ semiri1621563631at_int @ A ) ) @ ( hAPP_nat_int @ semiri1621563631at_int @ B ) )
      = ( hAPP_nat_int @ semiri1621563631at_int @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ A ) @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[689]) ).

thf(960,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_f1767618879l_bool @ ord_le1523781517a_bool @ A ) @ ( image_533441733iple_a @ B @ C ) ) )
    <=> ? [D: $i] :
          ( ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_f1767618879l_bool @ ord_le1523781517a_bool @ D ) @ C ) )
          & ( A
            = ( image_533441733iple_a @ B @ D ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_877_subset__image__iff) ).

thf(4692,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_f1767618879l_bool @ ord_le1523781517a_bool @ A ) @ ( image_533441733iple_a @ B @ C ) ) )
       => ? [D: $i] :
            ( ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_f1767618879l_bool @ ord_le1523781517a_bool @ D ) @ C ) )
            & ( A
              = ( image_533441733iple_a @ B @ D ) ) ) )
      & ( ? [D: $i] :
            ( ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_f1767618879l_bool @ ord_le1523781517a_bool @ D ) @ C ) )
            & ( A
              = ( image_533441733iple_a @ B @ D ) ) )
       => ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_f1767618879l_bool @ ord_le1523781517a_bool @ A ) @ ( image_533441733iple_a @ B @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[960]) ).

thf(668,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP_a723219176e_bool @ ( hAPP_f340725611e_bool @ ( hAPP_f1006724181e_bool @ cOMBB_1348041619bool_a @ A ) @ B ) @ C )
      = ( hAPP_f167292325e_bool @ A @ ( hAPP_a849909144l_bool @ B @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',help_COMBB_1_1_COMBB_000tc__fun_Itc__Com__Ostate_Mtc__fun_Itc__HOL__Obool_Mtc__H) ).

thf(3574,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP_a723219176e_bool @ ( hAPP_f340725611e_bool @ ( hAPP_f1006724181e_bool @ cOMBB_1348041619bool_a @ A ) @ B ) @ C )
      = ( hAPP_f167292325e_bool @ A @ ( hAPP_a849909144l_bool @ B @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[668]) ).

thf(527,axiom,
    ! [A: $i,B: $i] :
      ( ( A != B )
      | ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_f284875647l_bool @ fequal_fun_int_bool @ A ) @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',help_fequal_2_1_fequal_000tc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J_T) ).

thf(3061,plain,
    ! [A: $i,B: $i] :
      ( ( A != B )
      | ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_f284875647l_bool @ fequal_fun_int_bool @ A ) @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[527]) ).

thf(56,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_nat @ B ) @ C ) )
       => ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ A @ C ) @ B ) ) )
     => ( ( ( B = C )
         => ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ A @ C ) @ B ) ) )
       => ( ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_nat @ C ) @ B ) )
           => ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ A @ C ) @ B ) ) )
         => ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ A @ C ) @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_1116_nat__less__cases) ).

thf(1330,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_nat @ B ) @ C ) )
       => ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ A @ C ) @ B ) ) )
     => ( ( ( B = C )
         => ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ A @ C ) @ B ) ) )
       => ( ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_nat @ C ) @ B ) )
           => ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ A @ C ) @ B ) ) )
         => ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ A @ C ) @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[56]) ).

thf(425,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( finite2130160977at_nat @ D ) )
     => ( ( hBOOL @ ( hAPP_f54304608l_bool @ finite_finite_nat @ C ) )
       => ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ B ) @ C ) )
         => ( ( hAPP_f22106695ol_nat @ ( finite_fold_nat_nat @ D @ A ) @ C )
            = ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ D @ B ) @ ( hAPP_f22106695ol_nat @ ( finite_fold_nat_nat @ D @ A ) @ ( hAPP_f800510211t_bool @ ( hAPP_f1730770594t_bool @ minus_2067140911t_bool @ C ) @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ B ) @ bot_bot_fun_nat_bool ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_540_comp__fun__commute_Ofold__rec) ).

thf(2682,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( finite2130160977at_nat @ D ) )
     => ( ( hBOOL @ ( hAPP_f54304608l_bool @ finite_finite_nat @ C ) )
       => ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ B ) @ C ) )
         => ( ( hAPP_f22106695ol_nat @ ( finite_fold_nat_nat @ D @ A ) @ C )
            = ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ D @ B ) @ ( hAPP_f22106695ol_nat @ ( finite_fold_nat_nat @ D @ A ) @ ( hAPP_f800510211t_bool @ ( hAPP_f1730770594t_bool @ minus_2067140911t_bool @ C ) @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ B ) @ bot_bot_fun_nat_bool ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[425]) ).

thf(21,axiom,
    ! [A: $i] : ( is_bool @ ( finite1704255308nt_int @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',gsy_c_Finite__Set_Ocomp__fun__idem_000tc__Int__Oint_000tc__Int__Oint) ).

thf(1191,plain,
    ! [A: $i] : ( is_bool @ ( finite1704255308nt_int @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[21]) ).

thf(931,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( finite2114894759a_bool @ D ) )
     => ( ( hBOOL @ ( hAPP_f1695230391l_bool @ finite1738664244iple_a @ C ) )
       => ( ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ D @ A ) @ ( hAPP_f20753329a_bool @ ( finite1215486064a_bool @ D @ B ) @ C ) )
          = ( hAPP_f20753329a_bool @ ( finite1215486064a_bool @ D @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ D @ A ) @ B ) ) @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_482_comp__fun__commute_Ofold__fun__comm) ).

thf(4568,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( finite2114894759a_bool @ D ) )
     => ( ( hBOOL @ ( hAPP_f1695230391l_bool @ finite1738664244iple_a @ C ) )
       => ( ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ D @ A ) @ ( hAPP_f20753329a_bool @ ( finite1215486064a_bool @ D @ B ) @ C ) )
          = ( hAPP_f20753329a_bool @ ( finite1215486064a_bool @ D @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ D @ A ) @ B ) ) @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[931]) ).

thf(3,axiom,
    ! [A: $i,B: $i] : ( hBOOL @ ( hAPP_int_bool @ ( hAPP_i1948725293t_bool @ ord_less_eq_int @ ( hAPP_int_int @ uminus_uminus_int @ ( hAPP_nat_int @ semiri1621563631at_int @ A ) ) ) @ ( hAPP_nat_int @ semiri1621563631at_int @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_1071_negative__zle) ).

thf(1132,plain,
    ! [A: $i,B: $i] : ( hBOOL @ ( hAPP_int_bool @ ( hAPP_i1948725293t_bool @ ord_less_eq_int @ ( hAPP_int_int @ uminus_uminus_int @ ( hAPP_nat_int @ semiri1621563631at_int @ A ) ) ) @ ( hAPP_nat_int @ semiri1621563631at_int @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[3]) ).

thf(710,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_f103356543l_bool @ ord_le1568362934t_bool @ A ) @ B ) )
     => ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_f103356543l_bool @ ord_le1568362934t_bool @ B ) @ A ) )
       => ( B = A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_745_xt1_I5_J) ).

thf(3713,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_f103356543l_bool @ ord_le1568362934t_bool @ A ) @ B ) )
     => ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_f103356543l_bool @ ord_le1568362934t_bool @ B ) @ A ) )
       => ( B = A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[710]) ).

thf(432,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i] :
          ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_nat @ D ) @ E ) )
         => ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_nat @ ( hAPP_nat_nat @ C @ D ) ) @ ( hAPP_nat_nat @ C @ E ) ) ) )
     => ( 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 @ C @ A ) ) @ B ) ) @ ( hAPP_nat_nat @ C @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ A ) @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_1170_mono__nat__linear__lb) ).

thf(2703,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i] :
          ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_nat @ D ) @ E ) )
         => ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_nat @ ( hAPP_nat_nat @ C @ D ) ) @ ( hAPP_nat_nat @ C @ E ) ) ) )
     => ( 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 @ C @ A ) ) @ B ) ) @ ( hAPP_nat_nat @ C @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ A ) @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[432]) ).

thf(738,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ B ) @ C ) )
     => ( ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_f1767618879l_bool @ ord_le1523781517a_bool @ C ) @ A ) )
       => ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ B ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_797_set__rev__mp) ).

thf(3820,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ B ) @ C ) )
     => ( ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_f1767618879l_bool @ ord_le1523781517a_bool @ C ) @ A ) )
       => ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ B ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[738]) ).

thf(139,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( finite2130160977at_nat @ E ) )
     => ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ ( finite929467206at_nat @ E ) @ B ) @ C ) @ D ) )
       => ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ A ) @ C ) )
         => ? [F: $i] :
              ( ( D
                = ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ E @ A ) @ F ) )
              & ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ ( finite929467206at_nat @ E ) @ B ) @ ( hAPP_f800510211t_bool @ ( hAPP_f1730770594t_bool @ minus_2067140911t_bool @ C ) @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ A ) @ bot_bot_fun_nat_bool ) ) ) @ F ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_457_comp__fun__commute_Ofold__graph__insertE__aux) ).

thf(1628,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( finite2130160977at_nat @ E ) )
     => ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ ( finite929467206at_nat @ E ) @ B ) @ C ) @ D ) )
       => ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ A ) @ C ) )
         => ? [F: $i] :
              ( ( D
                = ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ E @ A ) @ F ) )
              & ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ ( finite929467206at_nat @ E ) @ B ) @ ( hAPP_f800510211t_bool @ ( hAPP_f1730770594t_bool @ minus_2067140911t_bool @ C ) @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ A ) @ bot_bot_fun_nat_bool ) ) ) @ F ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[139]) ).

thf(778,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ A ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ B ) @ C ) )
      = ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ A ) @ B ) ) @ C ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_689_comm__semiring__1__class_Onormalizing__semiring__rules_I18_J) ).

thf(3938,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ A ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ B ) @ C ) )
      = ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ A ) @ B ) ) @ C ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[778]) ).

thf(563,axiom,
    ! [A: $i,B: $i] :
      ( ( is_bool @ A )
     => ( ( hAPP_int_bool @ ( hAPP_b396694332t_bool @ cOMBK_bool_int @ A ) @ B )
        = A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',help_COMBK_1_1_COMBK_000tc__HOL__Obool_000tc__Int__Oint_U) ).

thf(3195,plain,
    ! [A: $i,B: $i] :
      ( ( is_bool @ A )
     => ( ( hAPP_int_bool @ ( hAPP_b396694332t_bool @ cOMBK_bool_int @ A ) @ B )
        = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[563]) ).

thf(430,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ B ) @ C ) )
     => ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ ord_min_nat @ A ) @ B ) ) @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_976_min__max_Ole__infI2) ).

thf(2698,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ B ) @ C ) )
     => ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ ord_min_nat @ A ) @ B ) ) @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[430]) ).

thf(50,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP_int_bool @ ( hAPP_i1948725293t_bool @ ord_less_eq_int @ A ) @ B ) )
    <=> ? [C: $i] :
          ( B
          = ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ A ) @ ( hAPP_nat_int @ semiri1621563631at_int @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_1074_zle__iff__zadd) ).

thf(1298,plain,
    ! [A: $i,B: $i] :
      ( ( ( hBOOL @ ( hAPP_int_bool @ ( hAPP_i1948725293t_bool @ ord_less_eq_int @ A ) @ B ) )
       => ? [C: $i] :
            ( B
            = ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ A ) @ ( hAPP_nat_int @ semiri1621563631at_int @ C ) ) ) )
      & ( ? [C: $i] :
            ( B
            = ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ A ) @ ( hAPP_nat_int @ semiri1621563631at_int @ C ) ) )
       => ( hBOOL @ ( hAPP_int_bool @ ( hAPP_i1948725293t_bool @ ord_less_eq_int @ A ) @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[50]) ).

thf(811,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP_f54304608l_bool @ finite_finite_nat @ B ) )
     => ( hBOOL @ ( hAPP_f1695230391l_bool @ finite1738664244iple_a @ ( image_1865589340iple_a @ A @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_286_finite__imageI) ).

thf(4070,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP_f54304608l_bool @ finite_finite_nat @ B ) )
     => ( hBOOL @ ( hAPP_f1695230391l_bool @ finite1738664244iple_a @ ( image_1865589340iple_a @ A @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[811]) ).

thf(542,axiom,
    ! [A: $i] :
      ( ( hBOOL @ ( hAPP_bool_bool @ ( hAPP_b589554111l_bool @ ord_less_eq_bool @ A ) @ bot_bot_bool ) )
    <=> ( ( hBOOL @ A )
      <=> ( hBOOL @ bot_bot_bool ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_837_bot__unique) ).

thf(3116,plain,
    ! [A: $i] :
      ( ( ( hBOOL @ ( hAPP_bool_bool @ ( hAPP_b589554111l_bool @ ord_less_eq_bool @ A ) @ bot_bot_bool ) )
       => ( ( ( hBOOL @ A )
           => ( hBOOL @ bot_bot_bool ) )
          & ( ( hBOOL @ bot_bot_bool )
           => ( hBOOL @ A ) ) ) )
      & ( ( ( ( hBOOL @ A )
           => ( hBOOL @ bot_bot_bool ) )
          & ( ( hBOOL @ bot_bot_bool )
           => ( hBOOL @ A ) ) )
       => ( hBOOL @ ( hAPP_bool_bool @ ( hAPP_b589554111l_bool @ ord_less_eq_bool @ A ) @ bot_bot_bool ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[542]) ).

thf(830,axiom,
    ! [A: $i,B: $i] :
      ( ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ minus_minus_int @ A ) @ B ) ) @ B )
      = A ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_645_diff__add__cancel) ).

thf(4151,plain,
    ! [A: $i,B: $i] :
      ( ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ minus_minus_int @ A ) @ B ) ) @ B )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[830]) ).

thf(1073,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( finite1428139793iple_a @ D ) )
     => ( ( hBOOL @ ( hAPP_f1695230391l_bool @ finite1738664244iple_a @ C ) )
       => ( ( hAPP_f1693662087iple_a @ ( finite1205970312iple_a @ D @ A ) @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ B ) @ C ) )
          = ( hAPP_H905846293iple_a @ ( hAPP_H2085992369iple_a @ D @ B ) @ ( hAPP_f1693662087iple_a @ ( finite1205970312iple_a @ D @ A ) @ ( hAPP_f20753329a_bool @ ( hAPP_f1706700729a_bool @ minus_85316870a_bool @ C ) @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ B ) @ bot_bo797238721a_bool ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_548_comp__fun__commute_Ofold__insert__remove) ).

thf(5064,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( finite1428139793iple_a @ D ) )
     => ( ( hBOOL @ ( hAPP_f1695230391l_bool @ finite1738664244iple_a @ C ) )
       => ( ( hAPP_f1693662087iple_a @ ( finite1205970312iple_a @ D @ A ) @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ B ) @ C ) )
          = ( hAPP_H905846293iple_a @ ( hAPP_H2085992369iple_a @ D @ B ) @ ( hAPP_f1693662087iple_a @ ( finite1205970312iple_a @ D @ A ) @ ( hAPP_f20753329a_bool @ ( hAPP_f1706700729a_bool @ minus_85316870a_bool @ C ) @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ B ) @ bot_bo797238721a_bool ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1073]) ).

thf(377,axiom,
    ! [A: $i,B: $i] :
      ~ ( hBOOL @ ( hAPP_nat_bool @ ( finite_fold1Set_nat @ A @ bot_bot_fun_nat_bool ) @ B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_232_empty__fold1SetE) ).

thf(2501,plain,
    ! [A: $i,B: $i] :
      ~ ( hBOOL @ ( hAPP_nat_bool @ ( finite_fold1Set_nat @ A @ bot_bot_fun_nat_bool ) @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[377]) ).

thf(825,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP_int_int @ ( hAPP_int_fun_int_int @ times_times_int @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ A ) @ B ) ) @ C )
      = ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ times_times_int @ A ) @ C ) ) @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ times_times_int @ B ) @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_702_comm__semiring__1__class_Onormalizing__semiring__rules_I8_J) ).

thf(4130,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP_int_int @ ( hAPP_int_fun_int_int @ times_times_int @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ A ) @ B ) ) @ C )
      = ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ times_times_int @ A ) @ C ) ) @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ times_times_int @ B ) @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[825]) ).

thf(53,axiom,
    ! [A: $i] :
      ( ? [B: $i] :
          ( ( hBOOL @ ( hAPP_nat_bool @ A @ B ) )
          & ! [C: $i] :
              ( ( hBOOL @ ( hAPP_nat_bool @ A @ C ) )
             => ( C = B ) ) )
     => ( hBOOL @ ( hAPP_nat_bool @ A @ ( hAPP_f22106695ol_nat @ the_nat @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_343_theI_H) ).

thf(1313,plain,
    ! [A: $i] :
      ( ? [B: $i] :
          ( ( hBOOL @ ( hAPP_nat_bool @ A @ B ) )
          & ! [C: $i] :
              ( ( hBOOL @ ( hAPP_nat_bool @ A @ C ) )
             => ( C = B ) ) )
     => ( hBOOL @ ( hAPP_nat_bool @ A @ ( hAPP_f22106695ol_nat @ the_nat @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[53]) ).

thf(288,axiom,
    ! [A: $i,B: $i] :
      ( ( 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 ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_382_Diff__idemp) ).

thf(2185,plain,
    ! [A: $i,B: $i] :
      ( ( 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 ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[288]) ).

thf(91,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( finite1973466193nt_int @ C ) )
     => ( ( hBOOL @ ( hAPP_f448129468l_bool @ finite_finite_int @ B ) )
       => ( hBOOL @ ( hAPP_int_bool @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ ( finite772772422nt_int @ C ) @ A ) @ B ) @ ( hAPP_f1594865479ol_int @ ( finite_fold_int_int @ C @ A ) @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_521_comp__fun__commute_Ofold__graph__fold) ).

thf(1455,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( finite1973466193nt_int @ C ) )
     => ( ( hBOOL @ ( hAPP_f448129468l_bool @ finite_finite_int @ B ) )
       => ( hBOOL @ ( hAPP_int_bool @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ ( finite772772422nt_int @ C ) @ A ) @ B ) @ ( hAPP_f1594865479ol_int @ ( finite_fold_int_int @ C @ A ) @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[91]) ).

thf(718,axiom,
    ! [A: $i] :
      ( ( hAPP_int_int @ ( hAPP_int_fun_int_int @ times_times_int @ one_one_int ) @ A )
      = A ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_1099_zmult__1) ).

thf(3749,plain,
    ! [A: $i] :
      ( ( hAPP_int_int @ ( hAPP_int_fun_int_int @ times_times_int @ one_one_int ) @ A )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[718]) ).

thf(748,axiom,
    ! [A: $i,B: $i] :
      ( ( hAPP_f1693662087iple_a @ ( finite1465856449iple_a @ A ) @ B )
      = ( hAPP_f1693662087iple_a @ the_Ho830102290iple_a @ ( finite1621280049iple_a @ A @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_164_fold1__def) ).

thf(3845,plain,
    ! [A: $i,B: $i] :
      ( ( hAPP_f1693662087iple_a @ ( finite1465856449iple_a @ A ) @ B )
      = ( hAPP_f1693662087iple_a @ the_Ho830102290iple_a @ ( finite1621280049iple_a @ A @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[748]) ).

thf(797,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( B
        = ( hAPP_H592031934_a_nat @ C @ D ) )
     => ( ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ D ) @ A ) )
       => ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ B ) @ ( image_48635758_a_nat @ C @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_158_image__eqI) ).

thf(4023,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( B
        = ( hAPP_H592031934_a_nat @ C @ D ) )
     => ( ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ D ) @ A ) )
       => ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ B ) @ ( image_48635758_a_nat @ C @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[797]) ).

thf(113,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_nat @ A ) @ B ) )
     => ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ A ) @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_1111_less__imp__le__nat) ).

thf(1545,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_nat @ A ) @ B ) )
     => ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ A ) @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[113]) ).

thf(284,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( finite1973466193nt_int @ D ) )
     => ( ( hBOOL @ ( hAPP_f448129468l_bool @ finite_finite_int @ C ) )
       => ( ~ ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ B ) @ C ) )
         => ( ( hAPP_f1594865479ol_int @ ( finite_fold_int_int @ D @ A ) @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ B ) @ C ) )
            = ( hAPP_int_int @ ( hAPP_int_fun_int_int @ D @ B ) @ ( hAPP_f1594865479ol_int @ ( finite_fold_int_int @ D @ A ) @ C ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_515_comp__fun__commute_Ofold__insert) ).

thf(2169,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( finite1973466193nt_int @ D ) )
     => ( ( hBOOL @ ( hAPP_f448129468l_bool @ finite_finite_int @ C ) )
       => ( ~ ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ B ) @ C ) )
         => ( ( hAPP_f1594865479ol_int @ ( finite_fold_int_int @ D @ A ) @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ B ) @ C ) )
            = ( hAPP_int_int @ ( hAPP_int_fun_int_int @ D @ B ) @ ( hAPP_f1594865479ol_int @ ( finite_fold_int_int @ D @ A ) @ C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[284]) ).

thf(38,axiom,
    ! [A: $i,B: $i] :
      ( ( A = B )
     => ( hBOOL @ ( hAPP_int_bool @ ( hAPP_i1948725293t_bool @ ord_less_eq_int @ A ) @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_780_order__eq__refl) ).

thf(1247,plain,
    ! [A: $i,B: $i] :
      ( ( A = B )
     => ( hBOOL @ ( hAPP_int_bool @ ( hAPP_i1948725293t_bool @ ord_less_eq_int @ A ) @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[38]) ).

thf(956,axiom,
    ! [A: $i,B: $i,C: $i] : ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_f1767618879l_bool @ ord_le1523781517a_bool @ ( hAPP_f20753329a_bool @ ( hAPP_f1706700729a_bool @ minus_85316870a_bool @ ( image_533441733iple_a @ A @ B ) ) @ ( image_533441733iple_a @ A @ C ) ) ) @ ( image_533441733iple_a @ A @ ( hAPP_f20753329a_bool @ ( hAPP_f1706700729a_bool @ minus_85316870a_bool @ B ) @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_899_image__diff__subset) ).

thf(4664,plain,
    ! [A: $i,B: $i,C: $i] : ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_f1767618879l_bool @ ord_le1523781517a_bool @ ( hAPP_f20753329a_bool @ ( hAPP_f1706700729a_bool @ minus_85316870a_bool @ ( image_533441733iple_a @ A @ B ) ) @ ( image_533441733iple_a @ A @ C ) ) ) @ ( image_533441733iple_a @ A @ ( hAPP_f20753329a_bool @ ( hAPP_f1706700729a_bool @ minus_85316870a_bool @ B ) @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[956]) ).

thf(1070,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( finite1803123436a_bool @ C ) )
     => ( ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ C @ A ) @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ C @ A ) @ B ) )
        = ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ C @ A ) @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_452_comp__fun__idem_Ofun__left__idem) ).

thf(5052,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( finite1803123436a_bool @ C ) )
     => ( ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ C @ A ) @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ C @ A ) @ B ) )
        = ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ C @ A ) @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1070]) ).

thf(693,axiom,
    is_bool @ fTrue,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',gsy_c_fTrue) ).

thf(3657,plain,
    is_bool @ fTrue,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[693]) ).

thf(856,axiom,
    ! [A: $i,B: $i] :
      ( ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ A ) @ B )
     != bot_bot_fun_nat_bool ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_96_insert__not__empty) ).

thf(4265,plain,
    ! [A: $i,B: $i] :
      ( ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ A ) @ B )
     != bot_bot_fun_nat_bool ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[856]) ).

thf(746,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ A ) @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ B ) @ C ) )
      = ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ A ) @ B ) ) @ C ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_672_comm__semiring__1__class_Onormalizing__semiring__rules_I25_J) ).

thf(3840,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ A ) @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ B ) @ C ) )
      = ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ A ) @ B ) ) @ C ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[746]) ).

thf(619,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP_f54304608l_bool @ finite_finite_nat @ B ) )
     => ( ( B != bot_bot_fun_nat_bool )
       => ( ! [C: $i] : ( hBOOL @ ( hAPP_f54304608l_bool @ A @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ C ) @ bot_bot_fun_nat_bool ) ) )
         => ( ! [C: $i,D: $i] :
                ( ( hBOOL @ ( hAPP_f54304608l_bool @ finite_finite_nat @ D ) )
               => ( ( D != bot_bot_fun_nat_bool )
                 => ( ~ ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ C ) @ D ) )
                   => ( ( hBOOL @ ( hAPP_f54304608l_bool @ A @ D ) )
                     => ( hBOOL @ ( hAPP_f54304608l_bool @ A @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ C ) @ D ) ) ) ) ) ) )
           => ( hBOOL @ ( hAPP_f54304608l_bool @ A @ B ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_350_finite__ne__induct) ).

thf(3374,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP_f54304608l_bool @ finite_finite_nat @ B ) )
     => ( ( B != bot_bot_fun_nat_bool )
       => ( ! [C: $i] : ( hBOOL @ ( hAPP_f54304608l_bool @ A @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ C ) @ bot_bot_fun_nat_bool ) ) )
         => ( ! [C: $i,D: $i] :
                ( ( hBOOL @ ( hAPP_f54304608l_bool @ finite_finite_nat @ D ) )
               => ( ( D != bot_bot_fun_nat_bool )
                 => ( ~ ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ C ) @ D ) )
                   => ( ( hBOOL @ ( hAPP_f54304608l_bool @ A @ D ) )
                     => ( hBOOL @ ( hAPP_f54304608l_bool @ A @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ C ) @ D ) ) ) ) ) ) )
           => ( hBOOL @ ( hAPP_f54304608l_bool @ A @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[619]) ).

thf(620,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ A ) @ B )
        = ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ C ) @ B ) )
    <=> ( A = C ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_662_add__right__cancel) ).

thf(3385,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ A ) @ B )
          = ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ C ) @ B ) )
       => ( A = C ) )
      & ( ( A = C )
       => ( ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ A ) @ B )
          = ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ C ) @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[620]) ).

thf(636,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_f103356543l_bool @ ord_le1568362934t_bool @ B ) @ C ) )
     => ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_f103356543l_bool @ ord_le1568362934t_bool @ A ) @ B ) )
       => ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_f103356543l_bool @ ord_le1568362934t_bool @ A ) @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_741_xt1_I6_J) ).

thf(3463,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_f103356543l_bool @ ord_le1568362934t_bool @ B ) @ C ) )
     => ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_f103356543l_bool @ ord_le1568362934t_bool @ A ) @ B ) )
       => ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_f103356543l_bool @ ord_le1568362934t_bool @ A ) @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[636]) ).

thf(70,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( C = D )
     => ( ! [E: $i] :
            ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ E ) @ D ) )
           => ( ( hAPP_nat_nat @ A @ E )
              = ( hAPP_nat_nat @ B @ E ) ) )
       => ( ( hAPP_f22106695ol_nat @ ( hAPP_f782000547ol_nat @ big_co387207925at_nat @ A ) @ C )
          = ( hAPP_f22106695ol_nat @ ( hAPP_f782000547ol_nat @ big_co387207925at_nat @ B ) @ D ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_709_setsum__cong) ).

thf(1379,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( C = D )
     => ( ! [E: $i] :
            ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ E ) @ D ) )
           => ( ( hAPP_nat_nat @ A @ E )
              = ( hAPP_nat_nat @ B @ E ) ) )
       => ( ( hAPP_f22106695ol_nat @ ( hAPP_f782000547ol_nat @ big_co387207925at_nat @ A ) @ C )
          = ( hAPP_f22106695ol_nat @ ( hAPP_f782000547ol_nat @ big_co387207925at_nat @ B ) @ D ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[70]) ).

thf(1103,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP_f20753329a_bool @ ( finite1215486064a_bool @ A @ B ) @ C )
      = ( hAPP_f258822235a_bool @ the_fu1555441254a_bool @ ( hAPP_f1767618879l_bool @ ( hAPP_f1577095245l_bool @ ( finite1152845682a_bool @ A ) @ B ) @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_567_fold__def) ).

thf(5199,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP_f20753329a_bool @ ( finite1215486064a_bool @ A @ B ) @ C )
      = ( hAPP_f258822235a_bool @ the_fu1555441254a_bool @ ( hAPP_f1767618879l_bool @ ( hAPP_f1577095245l_bool @ ( finite1152845682a_bool @ A ) @ B ) @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1103]) ).

thf(9,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_int_bool @ ( hAPP_i1948725293t_bool @ ord_less_eq_int @ B ) @ C ) )
     => ( ( hBOOL @ ( hAPP_int_bool @ ( hAPP_i1948725293t_bool @ ord_less_eq_int @ C ) @ A ) )
       => ( hBOOL @ ( hAPP_int_bool @ ( hAPP_i1948725293t_bool @ ord_less_eq_int @ B ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_1084_zle__trans) ).

thf(1147,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_int_bool @ ( hAPP_i1948725293t_bool @ ord_less_eq_int @ B ) @ C ) )
     => ( ( hBOOL @ ( hAPP_int_bool @ ( hAPP_i1948725293t_bool @ ord_less_eq_int @ C ) @ A ) )
       => ( hBOOL @ ( hAPP_int_bool @ ( hAPP_i1948725293t_bool @ ord_less_eq_int @ B ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[9]) ).

thf(161,axiom,
    ! [A: $i] :
      ( ( hAPP_int_int @ uminus_uminus_int @ ( hAPP_int_int @ uminus_uminus_int @ A ) )
      = A ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_1139_zminus__zminus) ).

thf(1704,plain,
    ! [A: $i] :
      ( ( hAPP_int_int @ uminus_uminus_int @ ( hAPP_int_int @ uminus_uminus_int @ A ) )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[161]) ).

thf(215,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP_int_bool @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ ( finite772772422nt_int @ A ) @ B ) @ C ) @ D ) )
    <=> ( ( ( C = bot_bot_fun_int_bool )
          & ( D = B ) )
        | ? [E: $i,F: $i,G: $i] :
            ( ( C
              = ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ E ) @ F ) )
            & ( D
              = ( hAPP_int_int @ ( hAPP_int_fun_int_int @ A @ E ) @ G ) )
            & ~ ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ E ) @ F ) )
            & ( hBOOL @ ( hAPP_int_bool @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ ( finite772772422nt_int @ A ) @ B ) @ F ) @ G ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_346_fold__graph_Osimps) ).

thf(1890,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( hBOOL @ ( hAPP_int_bool @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ ( finite772772422nt_int @ A ) @ B ) @ C ) @ D ) )
       => ( ( ( C = bot_bot_fun_int_bool )
            & ( D = B ) )
          | ? [E: $i,F: $i,G: $i] :
              ( ( C
                = ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ E ) @ F ) )
              & ( D
                = ( hAPP_int_int @ ( hAPP_int_fun_int_int @ A @ E ) @ G ) )
              & ~ ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ E ) @ F ) )
              & ( hBOOL @ ( hAPP_int_bool @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ ( finite772772422nt_int @ A ) @ B ) @ F ) @ G ) ) ) ) )
      & ( ( ( ( C = bot_bot_fun_int_bool )
            & ( D = B ) )
          | ? [E: $i,F: $i,G: $i] :
              ( ( C
                = ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ E ) @ F ) )
              & ( D
                = ( hAPP_int_int @ ( hAPP_int_fun_int_int @ A @ E ) @ G ) )
              & ~ ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ E ) @ F ) )
              & ( hBOOL @ ( hAPP_int_bool @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ ( finite772772422nt_int @ A ) @ B ) @ F ) @ G ) ) ) )
       => ( hBOOL @ ( hAPP_int_bool @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ ( finite772772422nt_int @ A ) @ B ) @ C ) @ D ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[215]) ).

thf(59,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP_int_bool @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ ( finite772772422nt_int @ A ) @ B ) @ C ) @ D ) )
     => ( ~ ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ B ) @ C ) )
       => ( hBOOL @ ( hAPP_int_bool @ ( finite_fold1Set_int @ A @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ B ) @ C ) ) @ D ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_263_fold1Set_Ointros) ).

thf(1347,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP_int_bool @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ ( finite772772422nt_int @ A ) @ B ) @ C ) @ D ) )
     => ( ~ ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ B ) @ C ) )
       => ( hBOOL @ ( hAPP_int_bool @ ( finite_fold1Set_int @ A @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ B ) @ C ) ) @ D ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[59]) ).

thf(121,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f54304608l_bool @ finite_finite_nat @ C ) )
     => ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ B ) @ C ) )
       => ( ( hAPP_f22106695ol_nat @ ( hAPP_f782000547ol_nat @ big_co387207925at_nat @ A ) @ C )
          = ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ ( hAPP_nat_nat @ A @ B ) ) @ ( hAPP_f22106695ol_nat @ ( hAPP_f782000547ol_nat @ big_co387207925at_nat @ A ) @ ( hAPP_f800510211t_bool @ ( hAPP_f1730770594t_bool @ minus_2067140911t_bool @ C ) @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ B ) @ bot_bot_fun_nat_bool ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_614_setsum__diff1_H) ).

thf(1572,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f54304608l_bool @ finite_finite_nat @ C ) )
     => ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ B ) @ C ) )
       => ( ( hAPP_f22106695ol_nat @ ( hAPP_f782000547ol_nat @ big_co387207925at_nat @ A ) @ C )
          = ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ ( hAPP_nat_nat @ A @ B ) ) @ ( hAPP_f22106695ol_nat @ ( hAPP_f782000547ol_nat @ big_co387207925at_nat @ A ) @ ( hAPP_f800510211t_bool @ ( hAPP_f1730770594t_bool @ minus_2067140911t_bool @ C ) @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ B ) @ bot_bot_fun_nat_bool ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[121]) ).

thf(715,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP_f54304608l_bool @ finite_finite_nat @ B ) )
     => ( hBOOL @ ( hAPP_f54304608l_bool @ finite_finite_nat @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ A ) @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_282_finite_OinsertI) ).

thf(3734,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP_f54304608l_bool @ finite_finite_nat @ B ) )
     => ( hBOOL @ ( hAPP_f54304608l_bool @ finite_finite_nat @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ A ) @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[715]) ).

thf(382,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( finite1973466193nt_int @ D ) )
     => ( ( hBOOL @ ( hAPP_f448129468l_bool @ finite_finite_int @ C ) )
       => ( ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ B ) @ C ) )
         => ( ( hAPP_f1594865479ol_int @ ( finite_fold_int_int @ D @ A ) @ C )
            = ( hAPP_int_int @ ( hAPP_int_fun_int_int @ D @ B ) @ ( hAPP_f1594865479ol_int @ ( finite_fold_int_int @ D @ A ) @ ( hAPP_f1805168059t_bool @ ( hAPP_f1223193598t_bool @ minus_1449998731t_bool @ C ) @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ B ) @ bot_bot_fun_int_bool ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_544_comp__fun__commute_Ofold__rec) ).

thf(2520,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( finite1973466193nt_int @ D ) )
     => ( ( hBOOL @ ( hAPP_f448129468l_bool @ finite_finite_int @ C ) )
       => ( ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ B ) @ C ) )
         => ( ( hAPP_f1594865479ol_int @ ( finite_fold_int_int @ D @ A ) @ C )
            = ( hAPP_int_int @ ( hAPP_int_fun_int_int @ D @ B ) @ ( hAPP_f1594865479ol_int @ ( finite_fold_int_int @ D @ A ) @ ( hAPP_f1805168059t_bool @ ( hAPP_f1223193598t_bool @ minus_1449998731t_bool @ C ) @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ B ) @ bot_bot_fun_int_bool ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[382]) ).

thf(143,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ A ) @ ( image_nat_int @ B @ C ) ) )
     => ~ ! [D: $i] :
            ( ( A
              = ( hAPP_nat_int @ B @ D ) )
           => ~ ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ D ) @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_247_imageE) ).

thf(1641,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ A ) @ ( image_nat_int @ B @ C ) ) )
     => ~ ! [D: $i] :
            ( ( A
              = ( hAPP_nat_int @ B @ D ) )
           => ~ ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ D ) @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[143]) ).

thf(744,axiom,
    ! [A: $i,B: $i] :
      ( ~ ( hBOOL @ ( hAPP_bool_bool @ ( hAPP_b589554111l_bool @ fdisj @ A ) @ B ) )
      | ( hBOOL @ A )
      | ( hBOOL @ B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',help_fdisj_3_1_U) ).

thf(3835,plain,
    ! [A: $i,B: $i] :
      ( ~ ( hBOOL @ ( hAPP_bool_bool @ ( hAPP_b589554111l_bool @ fdisj @ A ) @ B ) )
      | ( hBOOL @ A )
      | ( hBOOL @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[744]) ).

thf(708,axiom,
    ! [A: $i] :
      ( ( hAPP_f1693662087iple_a @ the_Ho830102290iple_a @ ( hAPP_H426895267a_bool @ ( hAPP_f920293029a_bool @ cOMBC_1940896922a_bool @ fequal963300192iple_a ) @ A ) )
      = A ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_252_the__eq__trivial) ).

thf(3708,plain,
    ! [A: $i] :
      ( ( hAPP_f1693662087iple_a @ the_Ho830102290iple_a @ ( hAPP_H426895267a_bool @ ( hAPP_f920293029a_bool @ cOMBC_1940896922a_bool @ fequal963300192iple_a ) @ A ) )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[708]) ).

thf(884,axiom,
    ! [A: $i,B: $i] :
      ( ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ A ) @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ A ) @ B ) )
      = ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ A ) @ B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_69_insert__absorb2) ).

thf(4383,plain,
    ! [A: $i,B: $i] :
      ( ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ A ) @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ A ) @ B ) )
      = ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ A ) @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[884]) ).

thf(100,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_int_bool @ ( finite_fold1Set_int @ A @ B ) @ C ) )
    <=> ? [D: $i,E: $i,F: $i] :
          ( ( B
            = ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ D ) @ E ) )
          & ( C = F )
          & ( hBOOL @ ( hAPP_int_bool @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ ( finite772772422nt_int @ A ) @ D ) @ E ) @ F ) )
          & ~ ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ D ) @ E ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_334_fold1Set_Osimps) ).

thf(1494,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( hBOOL @ ( hAPP_int_bool @ ( finite_fold1Set_int @ A @ B ) @ C ) )
       => ? [D: $i,E: $i,F: $i] :
            ( ( B
              = ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ D ) @ E ) )
            & ( C = F )
            & ( hBOOL @ ( hAPP_int_bool @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ ( finite772772422nt_int @ A ) @ D ) @ E ) @ F ) )
            & ~ ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ D ) @ E ) ) ) )
      & ( ? [D: $i,E: $i,F: $i] :
            ( ( B
              = ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ D ) @ E ) )
            & ( C = F )
            & ( hBOOL @ ( hAPP_int_bool @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ ( finite772772422nt_int @ A ) @ D ) @ E ) @ F ) )
            & ~ ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ D ) @ E ) ) )
       => ( hBOOL @ ( hAPP_int_bool @ ( finite_fold1Set_int @ A @ B ) @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[100]) ).

thf(912,axiom,
    hBOOL @ ( hAPP_f448129468l_bool @ finite_finite_int @ bot_bot_fun_int_bool ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_281_finite_OemptyI) ).

thf(4507,plain,
    hBOOL @ ( hAPP_f448129468l_bool @ finite_finite_int @ bot_bot_fun_int_bool ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[912]) ).

thf(212,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ A ) @ B ) )
     => ~ ! [C: $i] :
            ( ( B
              = ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ A ) @ C ) )
           => ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ A ) @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_116_Set_Oset__insert) ).

thf(1875,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ A ) @ B ) )
     => ~ ! [C: $i] :
            ( ( B
              = ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ A ) @ C ) )
           => ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ A ) @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[212]) ).

thf(184,axiom,
    ! [A: $i,B: $i] : ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ minus_minus_nat @ A ) @ B ) ) @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_1023_diff__le__self) ).

thf(1778,plain,
    ! [A: $i,B: $i] : ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ minus_minus_nat @ A ) @ B ) ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[184]) ).

thf(986,axiom,
    ! [A: $i] :
      ( ( is_bool @ A )
     => ( ( A = fTrue )
        | ( A = fFalse ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',help_fFalse_1_1_T) ).

thf(4771,plain,
    ! [A: $i] :
      ( ( is_bool @ A )
     => ( ( A = fTrue )
        | ( A = fFalse ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[986]) ).

thf(573,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f1695230391l_bool @ finite1738664244iple_a @ C ) )
     => ( ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ B ) @ C ) )
       => ( ( hAPP_f2097660464ol_nat @ ( big_co1815942735_a_nat @ A ) @ C )
          = ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ ( hAPP_H592031934_a_nat @ A @ B ) ) @ ( hAPP_f2097660464ol_nat @ ( big_co1815942735_a_nat @ A ) @ ( hAPP_f20753329a_bool @ ( hAPP_f1706700729a_bool @ minus_85316870a_bool @ C ) @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ B ) @ bot_bo797238721a_bool ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_473_setprod_Oremove) ).

thf(3231,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f1695230391l_bool @ finite1738664244iple_a @ C ) )
     => ( ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ B ) @ C ) )
       => ( ( hAPP_f2097660464ol_nat @ ( big_co1815942735_a_nat @ A ) @ C )
          = ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ ( hAPP_H592031934_a_nat @ A @ B ) ) @ ( hAPP_f2097660464ol_nat @ ( big_co1815942735_a_nat @ A ) @ ( hAPP_f20753329a_bool @ ( hAPP_f1706700729a_bool @ minus_85316870a_bool @ C ) @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ B ) @ bot_bo797238721a_bool ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[573]) ).

thf(915,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_f103356543l_bool @ ord_le1568362934t_bool @ B ) @ C ) )
     => ( hBOOL @ ( hAPP_bool_bool @ ( hAPP_b589554111l_bool @ ord_less_eq_bool @ ( hAPP_nat_bool @ B @ A ) ) @ ( hAPP_nat_bool @ C @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_777_le__funD) ).

thf(4514,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_f103356543l_bool @ ord_le1568362934t_bool @ B ) @ C ) )
     => ( hBOOL @ ( hAPP_bool_bool @ ( hAPP_b589554111l_bool @ ord_less_eq_bool @ ( hAPP_nat_bool @ B @ A ) ) @ ( hAPP_nat_bool @ C @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[915]) ).

thf(346,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ B ) @ C ) )
     => ( ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ minus_minus_nat @ A ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ minus_minus_nat @ C ) @ B ) )
        = ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ minus_minus_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ A ) @ B ) ) @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_1024_diff__diff__right) ).

thf(2388,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ B ) @ C ) )
     => ( ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ minus_minus_nat @ A ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ minus_minus_nat @ C ) @ B ) )
        = ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ minus_minus_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ A ) @ B ) ) @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[346]) ).

thf(1042,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ A ) @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ B ) @ C ) )
      = ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ B ) @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ A ) @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_67_insert__commute) ).

thf(4967,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ A ) @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ B ) @ C ) )
      = ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ B ) @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ A ) @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1042]) ).

thf(584,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP_int_int @ ( hAPP_int_fun_int_int @ times_times_int @ A ) @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ times_times_int @ B ) @ C ) )
      = ( hAPP_int_int @ ( hAPP_int_fun_int_int @ times_times_int @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ times_times_int @ A ) @ B ) ) @ C ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_690_comm__semiring__1__class_Onormalizing__semiring__rules_I18_J) ).

thf(3268,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP_int_int @ ( hAPP_int_fun_int_int @ times_times_int @ A ) @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ times_times_int @ B ) @ C ) )
      = ( hAPP_int_int @ ( hAPP_int_fun_int_int @ times_times_int @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ times_times_int @ A ) @ B ) ) @ C ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[584]) ).

thf(981,axiom,
    ! [A: $i,B: $i] :
      ( ( hAPP_f20753329a_bool @ ( finite1215486064a_bool @ A @ B ) @ bot_bo797238721a_bool )
      = B ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_500_fold__empty) ).

thf(4756,plain,
    ! [A: $i,B: $i] :
      ( ( hAPP_f20753329a_bool @ ( finite1215486064a_bool @ A @ B ) @ bot_bo797238721a_bool )
      = B ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[981]) ).

thf(730,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f1695230391l_bool @ finite1738664244iple_a @ C ) )
     => ( ! [D: $i] :
            ( ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ D ) @ C ) )
           => ( ( hAPP_H592031934_a_nat @ A @ D )
              = ( hAPP_H592031934_a_nat @ B @ D ) ) )
       => ( ( hAPP_f2097660464ol_nat @ ( hAPP_f1371399797ol_nat @ big_co1340561246_a_nat @ A ) @ C )
          = ( hAPP_f2097660464ol_nat @ ( hAPP_f1371399797ol_nat @ big_co1340561246_a_nat @ B ) @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_598_setsum_Ocong) ).

thf(3788,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f1695230391l_bool @ finite1738664244iple_a @ C ) )
     => ( ! [D: $i] :
            ( ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ D ) @ C ) )
           => ( ( hAPP_H592031934_a_nat @ A @ D )
              = ( hAPP_H592031934_a_nat @ B @ D ) ) )
       => ( ( hAPP_f2097660464ol_nat @ ( hAPP_f1371399797ol_nat @ big_co1340561246_a_nat @ A ) @ C )
          = ( hAPP_f2097660464ol_nat @ ( hAPP_f1371399797ol_nat @ big_co1340561246_a_nat @ B ) @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[730]) ).

thf(602,axiom,
    ! [A: $i] : ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_f103356543l_bool @ ord_le1568362934t_bool @ bot_bot_fun_nat_bool ) @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_727_empty__subsetI) ).

thf(3322,plain,
    ! [A: $i] : ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_f103356543l_bool @ ord_le1568362934t_bool @ bot_bot_fun_nat_bool ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[602]) ).

thf(654,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_f103356543l_bool @ ord_le1568362934t_bool @ B ) @ C ) )
     => ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_f284875647l_bool @ ord_le951220754t_bool @ ( image_nat_int @ A @ B ) ) @ ( image_nat_int @ A @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_874_image__mono) ).

thf(3524,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_f103356543l_bool @ ord_le1568362934t_bool @ B ) @ C ) )
     => ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_f284875647l_bool @ ord_le951220754t_bool @ ( image_nat_int @ A @ B ) ) @ ( image_nat_int @ A @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[654]) ).

thf(564,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( B = C )
     => ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_f103356543l_bool @ ord_le1568362934t_bool @ A ) @ C ) )
       => ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_f103356543l_bool @ ord_le1568362934t_bool @ A ) @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_765_xt1_I3_J) ).

thf(3199,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( B = C )
     => ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_f103356543l_bool @ ord_le1568362934t_bool @ A ) @ C ) )
       => ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_f103356543l_bool @ ord_le1568362934t_bool @ A ) @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[564]) ).

thf(973,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hoare_2102800559rivs_a @ B ) @ C ) )
     => ( ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_f1767618879l_bool @ ord_le1523781517a_bool @ A ) @ C ) )
       => ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hoare_2102800559rivs_a @ B ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_889_weaken) ).

thf(4733,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hoare_2102800559rivs_a @ B ) @ C ) )
     => ( ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_f1767618879l_bool @ ord_le1523781517a_bool @ A ) @ C ) )
       => ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hoare_2102800559rivs_a @ B ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[973]) ).

thf(416,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ A ) @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ B ) @ bot_bot_fun_int_bool ) ) )
    <=> ( A = B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_95_singleton__iff) ).

thf(2650,plain,
    ! [A: $i,B: $i] :
      ( ( ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ A ) @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ B ) @ bot_bot_fun_int_bool ) ) )
       => ( A = B ) )
      & ( ( A = B )
       => ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ A ) @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ B ) @ bot_bot_fun_int_bool ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[416]) ).

thf(969,axiom,
    ! [A: $i,B: $i] : ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_f1767618879l_bool @ ord_le1523781517a_bool @ ( hAPP_f20753329a_bool @ ( hAPP_f1706700729a_bool @ minus_85316870a_bool @ A ) @ B ) ) @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_887_Diff__subset) ).

thf(4724,plain,
    ! [A: $i,B: $i] : ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_f1767618879l_bool @ ord_le1523781517a_bool @ ( hAPP_f20753329a_bool @ ( hAPP_f1706700729a_bool @ minus_85316870a_bool @ A ) @ B ) ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[969]) ).

thf(1064,axiom,
    ! [A: $i] :
      ( ! [B: $i] :
          ~ ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ B ) @ A ) )
    <=> ( A = bot_bo797238721a_bool ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_45_all__not__in__conv) ).

thf(5032,plain,
    ! [A: $i] :
      ( ( ! [B: $i] :
            ~ ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ B ) @ A ) )
       => ( A = bot_bo797238721a_bool ) )
      & ( ( A = bot_bo797238721a_bool )
       => ! [B: $i] :
            ~ ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ B ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1064]) ).

thf(319,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_nat @ B ) @ C ) )
     => ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_nat @ B ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ C ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_1125_termination__basic__simps_I1_J) ).

thf(2291,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_nat @ B ) @ C ) )
     => ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_nat @ B ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ C ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[319]) ).

thf(332,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ A ) @ B ) )
    <=> ? [C: $i] :
          ( B
          = ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ A ) @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_1011_le__iff__add) ).

thf(2335,plain,
    ! [A: $i,B: $i] :
      ( ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ A ) @ B ) )
       => ? [C: $i] :
            ( B
            = ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ A ) @ C ) ) )
      & ( ? [C: $i] :
            ( B
            = ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ A ) @ C ) )
       => ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ A ) @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[332]) ).

thf(990,axiom,
    ! [A: $i] : ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_f1767618879l_bool @ ord_le1523781517a_bool @ bot_bo797238721a_bool ) @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_729_empty__subsetI) ).

thf(4783,plain,
    ! [A: $i] : ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_f1767618879l_bool @ ord_le1523781517a_bool @ bot_bo797238721a_bool ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[990]) ).

thf(750,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP_f448129468l_bool @ finite_finite_int @ B ) )
     => ( hBOOL @ ( hAPP_f448129468l_bool @ finite_finite_int @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ A ) @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_284_finite_OinsertI) ).

thf(3851,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP_f448129468l_bool @ finite_finite_int @ B ) )
     => ( hBOOL @ ( hAPP_f448129468l_bool @ finite_finite_int @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ A ) @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[750]) ).

thf(244,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( finite1321096241t_bool @ D ) )
     => ( ( hBOOL @ ( hAPP_f448129468l_bool @ finite_finite_int @ C ) )
       => ( ( hAPP_f1805168059t_bool @ ( finite1827710202t_bool @ D @ A ) @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ B ) @ C ) )
          = ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ D @ B ) @ ( hAPP_f1805168059t_bool @ ( finite1827710202t_bool @ D @ A ) @ ( hAPP_f1805168059t_bool @ ( hAPP_f1223193598t_bool @ minus_1449998731t_bool @ C ) @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ B ) @ bot_bot_fun_int_bool ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_551_comp__fun__commute_Ofold__insert__remove) ).

thf(2017,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( finite1321096241t_bool @ D ) )
     => ( ( hBOOL @ ( hAPP_f448129468l_bool @ finite_finite_int @ C ) )
       => ( ( hAPP_f1805168059t_bool @ ( finite1827710202t_bool @ D @ A ) @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ B ) @ C ) )
          = ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ D @ B ) @ ( hAPP_f1805168059t_bool @ ( finite1827710202t_bool @ D @ A ) @ ( hAPP_f1805168059t_bool @ ( hAPP_f1223193598t_bool @ minus_1449998731t_bool @ C ) @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ B ) @ bot_bot_fun_int_bool ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[244]) ).

thf(300,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ A ) @ B ) )
    <=> ( ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ ord_min_nat @ A ) @ B )
        = A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_982_min__max_Ole__iff__inf) ).

thf(2222,plain,
    ! [A: $i,B: $i] :
      ( ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ A ) @ B ) )
       => ( ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ ord_min_nat @ A ) @ B )
          = A ) )
      & ( ( ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ ord_min_nat @ A ) @ B )
          = A )
       => ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ A ) @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[300]) ).

thf(891,axiom,
    ! [A: $i] : ( hBOOL @ ( hAPP_bool_bool @ ( hAPP_b589554111l_bool @ ord_less_eq_bool @ bot_bot_bool ) @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_842_bot__least) ).

thf(4411,plain,
    ! [A: $i] : ( hBOOL @ ( hAPP_bool_bool @ ( hAPP_b589554111l_bool @ ord_less_eq_bool @ bot_bot_bool ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[891]) ).

thf(99,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_int_bool @ ( finite_fold1Set_int @ A @ B ) @ C ) )
     => ( B != bot_bot_fun_int_bool ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_228_fold1Set__nonempty) ).

thf(1489,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_int_bool @ ( finite_fold1Set_int @ A @ B ) @ C ) )
     => ( B != bot_bot_fun_int_bool ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[99]) ).

thf(498,axiom,
    ! [A: $i] :
      ( ( hAPP_f1805168059t_bool @ ( hAPP_f1223193598t_bool @ minus_1449998731t_bool @ bot_bot_fun_int_bool ) @ A )
      = bot_bot_fun_int_bool ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_392_empty__Diff) ).

thf(2950,plain,
    ! [A: $i] :
      ( ( hAPP_f1805168059t_bool @ ( hAPP_f1223193598t_bool @ minus_1449998731t_bool @ bot_bot_fun_int_bool ) @ A )
      = bot_bot_fun_int_bool ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[498]) ).

thf(1049,axiom,
    hBOOL @ ( finite58652534t_bool @ insert_int ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_456_comp__fun__idem__insert) ).

thf(4988,plain,
    hBOOL @ ( finite58652534t_bool @ insert_int ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1049]) ).

thf(871,axiom,
    ! [A: $i] :
      ( ( hBOOL @ ( hAPP_f448129468l_bool @ finite_finite_int @ A ) )
    <=> ( ( A = bot_bot_fun_int_bool )
        | ? [B: $i,C: $i] :
            ( ( A
              = ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ C ) @ B ) )
            & ( hBOOL @ ( hAPP_f448129468l_bool @ finite_finite_int @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_316_finite_Osimps) ).

thf(4329,plain,
    ! [A: $i] :
      ( ( ( hBOOL @ ( hAPP_f448129468l_bool @ finite_finite_int @ A ) )
       => ( ( A = bot_bot_fun_int_bool )
          | ? [B: $i,C: $i] :
              ( ( A
                = ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ C ) @ B ) )
              & ( hBOOL @ ( hAPP_f448129468l_bool @ finite_finite_int @ B ) ) ) ) )
      & ( ( ( A = bot_bot_fun_int_bool )
          | ? [B: $i,C: $i] :
              ( ( A
                = ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ C ) @ B ) )
              & ( hBOOL @ ( hAPP_f448129468l_bool @ finite_finite_int @ B ) ) ) )
       => ( hBOOL @ ( hAPP_f448129468l_bool @ finite_finite_int @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[871]) ).

thf(329,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ~ ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ D ) @ E ) )
     => ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ ( finite929467206at_nat @ A ) @ B ) @ E ) @ C ) )
       => ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ ( finite929467206at_nat @ A ) @ B ) @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ D ) @ E ) ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ A @ D ) @ C ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_305_fold__graph_OinsertI) ).

thf(2324,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ~ ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ D ) @ E ) )
     => ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ ( finite929467206at_nat @ A ) @ B ) @ E ) @ C ) )
       => ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ ( finite929467206at_nat @ A ) @ B ) @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ D ) @ E ) ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ A @ D ) @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[329]) ).

thf(936,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_f1767618879l_bool @ ord_le1523781517a_bool @ B ) @ C ) )
     => ( ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_f1767618879l_bool @ ord_le1523781517a_bool @ C ) @ A ) )
       => ( ( hAPP_f20753329a_bool @ ( hAPP_f1706700729a_bool @ minus_85316870a_bool @ C ) @ ( hAPP_f20753329a_bool @ ( hAPP_f1706700729a_bool @ minus_85316870a_bool @ A ) @ B ) )
          = B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_881_double__diff) ).

thf(4583,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_f1767618879l_bool @ ord_le1523781517a_bool @ B ) @ C ) )
     => ( ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_f1767618879l_bool @ ord_le1523781517a_bool @ C ) @ A ) )
       => ( ( hAPP_f20753329a_bool @ ( hAPP_f1706700729a_bool @ minus_85316870a_bool @ C ) @ ( hAPP_f20753329a_bool @ ( hAPP_f1706700729a_bool @ minus_85316870a_bool @ A ) @ B ) )
          = B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[936]) ).

thf(934,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP_state_bool @ ( hAPP_f1759915619e_bool @ ( hAPP_f644196280e_bool @ cOMBS_1378840469l_bool @ A ) @ B ) @ C )
      = ( hAPP_bool_bool @ ( hAPP_s58564346l_bool @ A @ C ) @ ( hAPP_state_bool @ B @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',help_COMBS_1_1_COMBS_000tc__Com__Ostate_000tc__HOL__Obool_000tc__HOL__Obool_U) ).

thf(4578,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP_state_bool @ ( hAPP_f1759915619e_bool @ ( hAPP_f644196280e_bool @ cOMBS_1378840469l_bool @ A ) @ B ) @ C )
      = ( hAPP_bool_bool @ ( hAPP_s58564346l_bool @ A @ C ) @ ( hAPP_state_bool @ B @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[934]) ).

thf(530,axiom,
    is_bool @ bot_bot_bool,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',gsy_c_Orderings_Obot__class_Obot_000tc__HOL__Obool) ).

thf(3075,plain,
    is_bool @ bot_bot_bool,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[530]) ).

thf(812,axiom,
    ! [A: $i,B: $i] :
      ( ( A = B )
     => ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_f103356543l_bool @ ord_le1568362934t_bool @ A ) @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_779_order__eq__refl) ).

thf(4072,plain,
    ! [A: $i,B: $i] :
      ( ( A = B )
     => ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_f103356543l_bool @ ord_le1568362934t_bool @ A ) @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[812]) ).

thf(795,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP_f1695230391l_bool @ finite1738664244iple_a @ B ) )
     => ( hBOOL @ ( hAPP_f54304608l_bool @ finite_finite_nat @ ( image_48635758_a_nat @ A @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_288_finite__imageI) ).

thf(4018,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP_f1695230391l_bool @ finite1738664244iple_a @ B ) )
     => ( hBOOL @ ( hAPP_f54304608l_bool @ finite_finite_nat @ ( image_48635758_a_nat @ A @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[795]) ).

thf(1062,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_f1767618879l_bool @ ord_le1523781517a_bool @ A ) @ B ) )
     => ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hoare_2102800559rivs_a @ B ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_890_asm) ).

thf(5027,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_f1767618879l_bool @ ord_le1523781517a_bool @ A ) @ B ) )
     => ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hoare_2102800559rivs_a @ B ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1062]) ).

thf(1058,axiom,
    ! [A: $i] :
      ( ( hBOOL @ ( hAPP_H1037229737a_bool @ bot_bo797238721a_bool @ A ) )
    <=> ( hBOOL @ bot_bot_bool ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_105_bot__apply) ).

thf(5013,plain,
    ! [A: $i] :
      ( ( ( hBOOL @ ( hAPP_H1037229737a_bool @ bot_bo797238721a_bool @ A ) )
       => ( hBOOL @ bot_bot_bool ) )
      & ( ( hBOOL @ bot_bot_bool )
       => ( hBOOL @ ( hAPP_H1037229737a_bool @ bot_bo797238721a_bool @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1058]) ).

thf(68,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i] :
          ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_nat @ D ) @ E ) )
         => ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_nat @ ( hAPP_nat_nat @ C @ D ) ) @ ( hAPP_nat_nat @ C @ E ) ) ) )
     => ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ A ) @ B ) )
       => ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ ( hAPP_nat_nat @ C @ A ) ) @ ( hAPP_nat_nat @ C @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_1173_less__mono__imp__le__mono) ).

thf(1373,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i] :
          ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_nat @ D ) @ E ) )
         => ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_nat @ ( hAPP_nat_nat @ C @ D ) ) @ ( hAPP_nat_nat @ C @ E ) ) ) )
     => ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ A ) @ B ) )
       => ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ ( hAPP_nat_nat @ C @ A ) ) @ ( hAPP_nat_nat @ C @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[68]) ).

thf(511,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( finite988810631ne_nat @ C @ D ) )
     => ( ( hBOOL @ ( hAPP_f54304608l_bool @ finite_finite_nat @ B ) )
       => ( ~ ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ A ) @ B ) )
         => ( ( B != bot_bot_fun_nat_bool )
           => ( ( hAPP_f22106695ol_nat @ D @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ A ) @ B ) )
              = ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ C @ A ) @ ( hAPP_f22106695ol_nat @ D @ B ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_267_folding__one_Oinsert) ).

thf(2996,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( finite988810631ne_nat @ C @ D ) )
     => ( ( hBOOL @ ( hAPP_f54304608l_bool @ finite_finite_nat @ B ) )
       => ( ~ ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ A ) @ B ) )
         => ( ( B != bot_bot_fun_nat_bool )
           => ( ( hAPP_f22106695ol_nat @ D @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ A ) @ B ) )
              = ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ C @ A ) @ ( hAPP_f22106695ol_nat @ D @ B ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[511]) ).

thf(594,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ ord_min_int @ A ) @ B ) ) @ C )
      = ( hAPP_int_int @ ( hAPP_int_fun_int_int @ ord_min_int @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ A ) @ C ) ) @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ B ) @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_988_min__add__distrib__left) ).

thf(3295,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ ord_min_int @ A ) @ B ) ) @ C )
      = ( hAPP_int_int @ ( hAPP_int_fun_int_int @ ord_min_int @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ A ) @ C ) ) @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ B ) @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[594]) ).

thf(454,axiom,
    ! [A: $i] :
      ~ ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ A ) @ bot_bot_fun_int_bool ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_38_empty__iff) ).

thf(2782,plain,
    ! [A: $i] :
      ~ ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ A ) @ bot_bot_fun_int_bool ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[454]) ).

thf(483,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_f103356543l_bool @ ord_le1568362934t_bool @ B ) @ C ) )
     => ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ A ) @ B ) )
       => ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ A ) @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_726_subsetD) ).

thf(2893,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_f103356543l_bool @ ord_le1568362934t_bool @ B ) @ C ) )
     => ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ A ) @ B ) )
       => ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ A ) @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[483]) ).

thf(140,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP_f1805168059t_bool @ ( hAPP_f1223193598t_bool @ minus_1449998731t_bool @ A ) @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ B ) @ C ) )
      = ( 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 @ B ) @ bot_bot_fun_int_bool ) ) ) @ C ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_416_Diff__insert2) ).

thf(1632,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP_f1805168059t_bool @ ( hAPP_f1223193598t_bool @ minus_1449998731t_bool @ A ) @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ B ) @ C ) )
      = ( 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 @ B ) @ bot_bot_fun_int_bool ) ) ) @ C ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[140]) ).

thf(370,axiom,
    ! [A: $i,B: $i] :
      ( ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ A ) @ ( hAPP_int_int @ uminus_uminus_int @ B ) )
      = ( hAPP_int_int @ ( hAPP_int_fun_int_int @ minus_minus_int @ A ) @ B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_1079_diff__int__def__symmetric) ).

thf(2469,plain,
    ! [A: $i,B: $i] :
      ( ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ A ) @ ( hAPP_int_int @ uminus_uminus_int @ B ) )
      = ( hAPP_int_int @ ( hAPP_int_fun_int_int @ minus_minus_int @ A ) @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[370]) ).

thf(640,axiom,
    ! [A: $i,B: $i] :
      ( ( 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 ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_855_rev__finite__subset) ).

thf(3473,plain,
    ! [A: $i,B: $i] :
      ( ( 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 ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[640]) ).

thf(706,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ A ) @ B ) ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ C ) @ B ) ) @ D ) )
      = ( 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 ) @ C ) ) @ B ) ) @ D ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_1054_left__add__mult__distrib) ).

thf(3702,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ A ) @ B ) ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ C ) @ B ) ) @ D ) )
      = ( 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 ) @ C ) ) @ B ) ) @ D ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[706]) ).

thf(788,axiom,
    ! [A: $i,B: $i] :
      ( ( ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ A ) @ B )
        = one_one_nat )
    <=> ( ( A = one_one_nat )
        & ( B = one_one_nat ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_1060_nat__mult__eq__1__iff) ).

thf(3983,plain,
    ! [A: $i,B: $i] :
      ( ( ( ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ A ) @ B )
          = one_one_nat )
       => ( ( A = one_one_nat )
          & ( B = one_one_nat ) ) )
      & ( ( ( A = one_one_nat )
          & ( B = one_one_nat ) )
       => ( ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ A ) @ B )
          = one_one_nat ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[788]) ).

thf(154,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ B ) @ C ) )
     => ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ B ) @ A ) )
       => ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ B ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ ord_min_nat @ C ) @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_968_min__max_Oinf__greatest) ).

thf(1682,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ B ) @ C ) )
     => ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ B ) @ A ) )
       => ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ B ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ ord_min_nat @ C ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[154]) ).

thf(5,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP_int_bool @ ( hAPP_i1948725293t_bool @ ord_less_eq_int @ A ) @ B ) )
     => ( ( hAPP_int_int @ ( hAPP_int_fun_int_int @ ord_min_int @ A ) @ B )
        = A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_975_min__max_Oinf__absorb1) ).

thf(1136,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP_int_bool @ ( hAPP_i1948725293t_bool @ ord_less_eq_int @ A ) @ B ) )
     => ( ( hAPP_int_int @ ( hAPP_int_fun_int_int @ ord_min_int @ A ) @ B )
        = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[5]) ).

thf(630,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_H1037229737a_bool @ ( hAPP_f20753329a_bool @ ( hAPP_f1706700729a_bool @ minus_85316870a_bool @ A ) @ B ) @ C ) )
    <=> ( hBOOL @ ( hAPP_bool_bool @ ( hAPP_b589554111l_bool @ minus_minus_bool @ ( hAPP_H1037229737a_bool @ A @ C ) ) @ ( hAPP_H1037229737a_bool @ B @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_443_minus__apply) ).

thf(3438,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( hBOOL @ ( hAPP_H1037229737a_bool @ ( hAPP_f20753329a_bool @ ( hAPP_f1706700729a_bool @ minus_85316870a_bool @ A ) @ B ) @ C ) )
       => ( hBOOL @ ( hAPP_bool_bool @ ( hAPP_b589554111l_bool @ minus_minus_bool @ ( hAPP_H1037229737a_bool @ A @ C ) ) @ ( hAPP_H1037229737a_bool @ B @ C ) ) ) )
      & ( ( hBOOL @ ( hAPP_bool_bool @ ( hAPP_b589554111l_bool @ minus_minus_bool @ ( hAPP_H1037229737a_bool @ A @ C ) ) @ ( hAPP_H1037229737a_bool @ B @ C ) ) )
       => ( hBOOL @ ( hAPP_H1037229737a_bool @ ( hAPP_f20753329a_bool @ ( hAPP_f1706700729a_bool @ minus_85316870a_bool @ A ) @ B ) @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[630]) ).

thf(804,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_nat_bool @ ( finite_fold1Set_nat @ A @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ B ) @ bot_bot_fun_nat_bool ) ) @ C ) )
    <=> ( B = C ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_133_fold1Set__sing) ).

thf(4050,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( hBOOL @ ( hAPP_nat_bool @ ( finite_fold1Set_nat @ A @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ B ) @ bot_bot_fun_nat_bool ) ) @ C ) )
       => ( B = C ) )
      & ( ( B = C )
       => ( hBOOL @ ( hAPP_nat_bool @ ( finite_fold1Set_nat @ A @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ B ) @ bot_bot_fun_nat_bool ) ) @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[804]) ).

thf(389,axiom,
    ! [A: $i] :
      ( ( hBOOL @ ( hAPP_f54304608l_bool @ finite_finite_nat @ A ) )
    <=> ? [B: $i] :
        ! [C: $i] :
          ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ C ) @ A ) )
         => ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ C ) @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_949_finite__nat__set__iff__bounded__le) ).

thf(2540,plain,
    ! [A: $i] :
      ( ( ( hBOOL @ ( hAPP_f54304608l_bool @ finite_finite_nat @ A ) )
       => ? [B: $i] :
          ! [C: $i] :
            ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ C ) @ A ) )
           => ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ C ) @ B ) ) ) )
      & ( ? [B: $i] :
          ! [C: $i] :
            ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ C ) @ A ) )
           => ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ C ) @ B ) ) )
       => ( hBOOL @ ( hAPP_f54304608l_bool @ finite_finite_nat @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[389]) ).

thf(1013,axiom,
    ! [A: $i,B: $i] :
      ( ( B = bot_bo797238721a_bool )
     => ~ ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ A ) @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_30_equals0D) ).

thf(4871,plain,
    ! [A: $i,B: $i] :
      ( ( B = bot_bo797238721a_bool )
     => ~ ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ A ) @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1013]) ).

thf(773,axiom,
    ! [A: $i] : ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_f284875647l_bool @ ord_le951220754t_bool @ bot_bot_fun_int_bool ) @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_728_empty__subsetI) ).

thf(3923,plain,
    ! [A: $i] : ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_f284875647l_bool @ ord_le951220754t_bool @ bot_bot_fun_int_bool ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[773]) ).

thf(592,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( finite1033474011iple_a @ C @ D ) )
     => ( ( hBOOL @ ( hAPP_f1695230391l_bool @ finite1738664244iple_a @ B ) )
       => ( ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ A ) @ B ) )
         => ( ( hAPP_H905846293iple_a @ ( hAPP_H2085992369iple_a @ C @ A ) @ ( hAPP_f1693662087iple_a @ D @ B ) )
            = ( hAPP_f1693662087iple_a @ D @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_431_folding__one__idem_Oin__idem) ).

thf(3290,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( finite1033474011iple_a @ C @ D ) )
     => ( ( hBOOL @ ( hAPP_f1695230391l_bool @ finite1738664244iple_a @ B ) )
       => ( ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ A ) @ B ) )
         => ( ( hAPP_H905846293iple_a @ ( hAPP_H2085992369iple_a @ C @ A ) @ ( hAPP_f1693662087iple_a @ D @ B ) )
            = ( hAPP_f1693662087iple_a @ D @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[592]) ).

thf(17,axiom,
    ! [A: $i] : ( hBOOL @ ( hAPP_int_bool @ ( hAPP_i1948725293t_bool @ ord_less_eq_int @ A ) @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_720_order__refl) ).

thf(1183,plain,
    ! [A: $i] : ( hBOOL @ ( hAPP_int_bool @ ( hAPP_i1948725293t_bool @ ord_less_eq_int @ A ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[17]) ).

thf(262,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_f284875647l_bool @ ord_le951220754t_bool @ B ) @ C ) )
     => ( ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ A ) @ B ) )
       => ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ A ) @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_795_set__mp) ).

thf(2073,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_f284875647l_bool @ ord_le951220754t_bool @ B ) @ C ) )
     => ( ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ A ) @ B ) )
       => ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ A ) @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[262]) ).

thf(658,axiom,
    ! [A: $i,B: $i] :
      ( ( 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 ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_808_set__eq__subset) ).

thf(3537,plain,
    ! [A: $i,B: $i] :
      ( ( ( 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 ) ) ) )
      & ( ( ( 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 ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[658]) ).

thf(466,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ B ) @ C ) )
     => ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ B ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ C ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_1009_trans__le__add1) ).

thf(2833,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ B ) @ C ) )
     => ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ B ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ C ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[466]) ).

thf(1110,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP_H426895267a_bool @ ( hAPP_f1181346091a_bool @ ( hAPP_f571476211a_bool @ cOMBC_1570214144a_bool @ A ) @ B ) @ C )
      = ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ A @ C ) @ B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',help_COMBC_1_1_COMBC_000tc__Hoare____Mirabelle____uwgpyvfjxg__Otriple_It__a_J_00_053) ).

thf(5220,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP_H426895267a_bool @ ( hAPP_f1181346091a_bool @ ( hAPP_f571476211a_bool @ cOMBC_1570214144a_bool @ A ) @ B ) @ C )
      = ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ A @ C ) @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1110]) ).

thf(373,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ D ) @ E ) )
     => ( ( 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 @ D ) @ A ) ) @ B ) ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ E ) @ A ) ) @ C ) ) )
      <=> ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ B ) @ ( 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 @ E ) @ D ) ) @ A ) ) @ C ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_1042_nat__le__add__iff2) ).

thf(2483,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ D ) @ E ) )
     => ( ( ( 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 @ D ) @ A ) ) @ B ) ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ E ) @ A ) ) @ C ) ) )
         => ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ B ) @ ( 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 @ E ) @ D ) ) @ A ) ) @ C ) ) ) )
        & ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ B ) @ ( 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 @ E ) @ D ) ) @ A ) ) @ C ) ) )
         => ( 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 @ D ) @ A ) ) @ B ) ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ E ) @ A ) ) @ C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[373]) ).

thf(137,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ A ) @ B ) )
     => ( ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ A ) @ B )
        = B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_53_insert__absorb) ).

thf(1622,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ A ) @ B ) )
     => ( ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ A ) @ B )
        = B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[137]) ).

thf(1004,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP_a2036067514e_bool @ ( hAPP_f762886889e_bool @ ( hAPP_f1261923407e_bool @ cOMBC_892787026e_bool @ A ) @ B ) @ C )
      = ( hAPP_f1759915619e_bool @ ( hAPP_a1200519163e_bool @ A @ C ) @ B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',help_COMBC_1_1_COMBC_000t__a_000tc__fun_Itc__Com__Ostate_Mtc__HOL__Obool_J_000tc) ).

thf(4825,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP_a2036067514e_bool @ ( hAPP_f762886889e_bool @ ( hAPP_f1261923407e_bool @ cOMBC_892787026e_bool @ A ) @ B ) @ C )
      = ( hAPP_f1759915619e_bool @ ( hAPP_a1200519163e_bool @ A @ C ) @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1004]) ).

thf(922,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] : ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hoare_2102800559rivs_a @ A ) @ ( 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 ) @ B ) ) @ ( hAPP_f871651461_state @ ( hAPP_f851239890_state @ cOMBS_777315357_state @ ( hAPP_v365393659_state @ ( hAPP_f1542232213_state @ cOMBC_1193272608_state @ update ) @ C ) ) @ D ) ) @ ( ass @ C @ D ) @ B ) ) @ bot_bo797238721a_bool ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_142_Ass) ).

thf(4536,plain,
    ! [A: $i,B: $i,C: $i,D: $i] : ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hoare_2102800559rivs_a @ A ) @ ( 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 ) @ B ) ) @ ( hAPP_f871651461_state @ ( hAPP_f851239890_state @ cOMBS_777315357_state @ ( hAPP_v365393659_state @ ( hAPP_f1542232213_state @ cOMBC_1193272608_state @ update ) @ C ) ) @ D ) ) @ ( ass @ C @ D ) @ B ) ) @ bot_bo797238721a_bool ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[922]) ).

thf(75,axiom,
    ! [A: $i,B: $i] :
      ( ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ ord_min_nat @ A ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ ord_min_nat @ A ) @ B ) )
      = ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ ord_min_nat @ A ) @ B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_958_min__max_Oinf_Oleft__idem) ).

thf(1405,plain,
    ! [A: $i,B: $i] :
      ( ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ ord_min_nat @ A ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ ord_min_nat @ A ) @ B ) )
      = ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ ord_min_nat @ A ) @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[75]) ).

thf(756,axiom,
    ! [A: $i,B: $i] :
      ( ! [C: $i] :
          ( ( hBOOL @ ( hAPP_nat_bool @ B @ C ) )
         => ( hBOOL @ ( hAPP_nat_bool @ A @ C ) ) )
     => ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_f103356543l_bool @ ord_le1568362934t_bool @ ( hAPP_f800510211t_bool @ collect_nat @ B ) ) @ ( hAPP_f800510211t_bool @ collect_nat @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_942_Collect__mono) ).

thf(3870,plain,
    ! [A: $i,B: $i] :
      ( ! [C: $i] :
          ( ( hBOOL @ ( hAPP_nat_bool @ B @ C ) )
         => ( hBOOL @ ( hAPP_nat_bool @ A @ C ) ) )
     => ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_f103356543l_bool @ ord_le1568362934t_bool @ ( hAPP_f800510211t_bool @ collect_nat @ B ) ) @ ( hAPP_f800510211t_bool @ collect_nat @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[756]) ).

thf(766,axiom,
    ! [A: $i,B: $i] :
      ( ~ ( hBOOL @ B )
      | ( hBOOL @ ( hAPP_bool_bool @ ( hAPP_b589554111l_bool @ fimplies @ A ) @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',help_fimplies_2_1_U) ).

thf(3906,plain,
    ! [A: $i,B: $i] :
      ( ~ ( hBOOL @ B )
      | ( hBOOL @ ( hAPP_bool_bool @ ( hAPP_b589554111l_bool @ fimplies @ A ) @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[766]) ).

thf(947,axiom,
    ! [A: $i,B: $i] :
      ( ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ A ) @ ( hAPP_f20753329a_bool @ ( hAPP_f1706700729a_bool @ minus_85316870a_bool @ B ) @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ A ) @ bot_bo797238721a_bool ) ) )
      = ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ A ) @ B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_418_insert__Diff__single) ).

thf(4628,plain,
    ! [A: $i,B: $i] :
      ( ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ A ) @ ( hAPP_f20753329a_bool @ ( hAPP_f1706700729a_bool @ minus_85316870a_bool @ B ) @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ A ) @ bot_bo797238721a_bool ) ) )
      = ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ A ) @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[947]) ).

thf(436,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ B ) @ C ) )
     => ( ( C = A )
       => ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ B ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_763_ord__le__eq__trans) ).

thf(2718,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ B ) @ C ) )
     => ( ( C = A )
       => ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ B ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[436]) ).

thf(560,axiom,
    ! [A: $i,B: $i] : ( is_bool @ ( finite1753440478iple_a @ A @ B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',gsy_c_Finite__Set_Ofolding__one_000tc__Hoare____Mirabelle____uwgpyvfjxg__Otriple) ).

thf(3184,plain,
    ! [A: $i,B: $i] : ( is_bool @ ( finite1753440478iple_a @ A @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[560]) ).

thf(967,axiom,
    ! [A: $i,B: $i] :
      ( ( hAPP_a2036067514e_bool @ ( hAPP_f762886889e_bool @ cOMBK_1458035955bool_a @ A ) @ B )
      = A ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',help_COMBK_1_1_COMBK_000tc__fun_Itc__Com__Ostate_Mtc__HOL__Obool_J_000t__a_U) ).

thf(4718,plain,
    ! [A: $i,B: $i] :
      ( ( hAPP_a2036067514e_bool @ ( hAPP_f762886889e_bool @ cOMBK_1458035955bool_a @ A ) @ B )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[967]) ).

thf(979,axiom,
    ! [A: $i] :
      ( ( hBOOL @ A )
      | ( hBOOL @ ( hAPP_bool_bool @ fNot @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',help_fNot_2_1_U) ).

thf(4751,plain,
    ! [A: $i] :
      ( ( hBOOL @ A )
      | ( hBOOL @ ( hAPP_bool_bool @ fNot @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[979]) ).

thf(383,axiom,
    ! [A: $i,B: $i] :
      ( ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ minus_minus_nat @ A ) @ B ) ) @ B )
      = ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ ord_max_nat @ A ) @ B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_1058_nat__minus__add__max) ).

thf(2523,plain,
    ! [A: $i,B: $i] :
      ( ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ minus_minus_nat @ A ) @ B ) ) @ B )
      = ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ ord_max_nat @ A ) @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[383]) ).

thf(521,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ A ) @ B ) )
     => ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ B ) @ A ) )
      <=> ( B = A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_775_order__antisym__conv) ).

thf(3036,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ A ) @ B ) )
     => ( ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ B ) @ A ) )
         => ( B = A ) )
        & ( ( B = A )
         => ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ B ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[521]) ).

thf(386,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ B ) @ C ) )
     => ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ A ) @ B ) )
       => ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ A ) @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_743_xt1_I6_J) ).

thf(2530,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ B ) @ C ) )
     => ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ A ) @ B ) )
       => ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ A ) @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[386]) ).

thf(698,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( finite1860950092at_nat @ D ) )
     => ( ( hBOOL @ ( hAPP_f54304608l_bool @ finite_finite_nat @ C ) )
       => ( ( hAPP_f22106695ol_nat @ ( finite_fold_nat_nat @ D @ A ) @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ B ) @ C ) )
          = ( hAPP_f22106695ol_nat @ ( finite_fold_nat_nat @ D @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ D @ B ) @ A ) ) @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_569_comp__fun__idem_Ofold__insert__idem2) ).

thf(3676,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( finite1860950092at_nat @ D ) )
     => ( ( hBOOL @ ( hAPP_f54304608l_bool @ finite_finite_nat @ C ) )
       => ( ( hAPP_f22106695ol_nat @ ( finite_fold_nat_nat @ D @ A ) @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ B ) @ C ) )
          = ( hAPP_f22106695ol_nat @ ( finite_fold_nat_nat @ D @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ D @ B ) @ A ) ) @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[698]) ).

thf(275,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ A ) @ B )
        = ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ A ) @ C ) )
     => ( B = C ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_657_add__left__imp__eq) ).

thf(2137,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ A ) @ B )
        = ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ A ) @ C ) )
     => ( B = C ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[275]) ).

thf(950,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP_f1693662087iple_a @ ( finite1205970312iple_a @ A @ B ) @ C )
      = ( hAPP_f1693662087iple_a @ the_Ho830102290iple_a @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ ( finite90892550iple_a @ A ) @ B ) @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_565_fold__def) ).

thf(4643,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP_f1693662087iple_a @ ( finite1205970312iple_a @ A @ B ) @ C )
      = ( hAPP_f1693662087iple_a @ the_Ho830102290iple_a @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ ( finite90892550iple_a @ A ) @ B ) @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[950]) ).

thf(224,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( 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 @ B ) @ bot_bot_fun_int_bool ) ) ) @ C ) )
     => ( ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ B ) @ A ) )
       => ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_f284875647l_bool @ ord_le951220754t_bool @ A ) @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ B ) @ C ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_905_diff__single__insert) ).

thf(1943,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( 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 @ B ) @ bot_bot_fun_int_bool ) ) ) @ C ) )
     => ( ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ B ) @ A ) )
       => ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_f284875647l_bool @ ord_le951220754t_bool @ A ) @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ B ) @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[224]) ).

thf(36,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP_int_bool @ ( hAPP_i1948725293t_bool @ ord_less_eq_int @ A ) @ B ) )
     => ( ( hBOOL @ ( hAPP_int_bool @ ( hAPP_i1948725293t_bool @ ord_less_eq_int @ B ) @ A ) )
       => ( A = B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_754_order__antisym) ).

thf(1242,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP_int_bool @ ( hAPP_i1948725293t_bool @ ord_less_eq_int @ A ) @ B ) )
     => ( ( hBOOL @ ( hAPP_int_bool @ ( hAPP_i1948725293t_bool @ ord_less_eq_int @ B ) @ A ) )
       => ( A = B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[36]) ).

thf(861,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_f103356543l_bool @ ord_le1568362934t_bool @ B ) @ C ) )
     => ( hBOOL @ ( hAPP_bool_bool @ ( hAPP_b589554111l_bool @ ord_less_eq_bool @ ( hAPP_nat_bool @ B @ A ) ) @ ( hAPP_nat_bool @ C @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_739_le__funE) ).

thf(4285,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_f103356543l_bool @ ord_le1568362934t_bool @ B ) @ C ) )
     => ( hBOOL @ ( hAPP_bool_bool @ ( hAPP_b589554111l_bool @ ord_less_eq_bool @ ( hAPP_nat_bool @ B @ A ) ) @ ( hAPP_nat_bool @ C @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[861]) ).

thf(629,axiom,
    ! [A: $i,B: $i] :
      ( ( hAPP_int_int @ ( hAPP_int_fun_int_int @ cOMBK_int_int @ A ) @ B )
      = A ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',help_COMBK_1_1_COMBK_000tc__Int__Oint_000tc__Int__Oint_U) ).

thf(3435,plain,
    ! [A: $i,B: $i] :
      ( ( hAPP_int_int @ ( hAPP_int_fun_int_int @ cOMBK_int_int @ A ) @ B )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[629]) ).

thf(961,axiom,
    ! [A: $i,B: $i] :
      ( ! [C: $i] :
          ( ( hBOOL @ ( hAPP_H1037229737a_bool @ B @ C ) )
         => ( hBOOL @ ( hAPP_H1037229737a_bool @ A @ C ) ) )
     => ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_f1767618879l_bool @ ord_le1523781517a_bool @ ( hAPP_f20753329a_bool @ collec351493750iple_a @ B ) ) @ ( hAPP_f20753329a_bool @ collec351493750iple_a @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_941_Collect__mono) ).

thf(4700,plain,
    ! [A: $i,B: $i] :
      ( ! [C: $i] :
          ( ( hBOOL @ ( hAPP_H1037229737a_bool @ B @ C ) )
         => ( hBOOL @ ( hAPP_H1037229737a_bool @ A @ C ) ) )
     => ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_f1767618879l_bool @ ord_le1523781517a_bool @ ( hAPP_f20753329a_bool @ collec351493750iple_a @ B ) ) @ ( hAPP_f20753329a_bool @ collec351493750iple_a @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[961]) ).

thf(515,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ B ) @ C ) )
     => ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ B ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ A ) @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_1008_trans__le__add2) ).

thf(3011,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ B ) @ C ) )
     => ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ B ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ A ) @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[515]) ).

thf(841,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_f1767618879l_bool @ ord_le1523781517a_bool @ B ) @ C ) )
     => ( ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ A ) @ B ) )
       => ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ A ) @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_724_subsetD) ).

thf(4199,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_f1767618879l_bool @ ord_le1523781517a_bool @ B ) @ C ) )
     => ( ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ A ) @ B ) )
       => ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ A ) @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[841]) ).

thf(980,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ A ) @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ B ) @ bot_bo797238721a_bool ) ) )
     => ( A = B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_88_singletonE) ).

thf(4753,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ A ) @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ B ) @ bot_bo797238721a_bool ) ) )
     => ( A = B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[980]) ).

thf(158,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ A ) @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ B ) @ C ) )
      = ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ B ) @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ A ) @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_68_insert__commute) ).

thf(1691,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ A ) @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ B ) @ C ) )
      = ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ B ) @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ A ) @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[158]) ).

thf(555,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ A ) @ B ) ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ C ) @ B ) ) @ D ) )
      = ( 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 ) @ C ) ) @ B ) ) @ D ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_649_combine__common__factor) ).

thf(3169,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ A ) @ B ) ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ C ) @ B ) ) @ D ) )
      = ( 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 ) @ C ) ) @ B ) ) @ D ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[555]) ).

thf(920,axiom,
    ~ ( hBOOL @ fFalse ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',help_fFalse_1_1_U) ).

thf(4531,plain,
    ~ ( hBOOL @ fFalse ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[920]) ).

thf(561,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f448129468l_bool @ finite_finite_int @ C ) )
     => ( ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_f284875647l_bool @ ord_le951220754t_bool @ C ) @ B ) )
       => ( ( hBOOL @ ( hAPP_f448129468l_bool @ A @ bot_bot_fun_int_bool ) )
         => ( ! [D: $i,E: $i] :
                ( ( hBOOL @ ( hAPP_f448129468l_bool @ finite_finite_int @ E ) )
               => ( ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ D ) @ B ) )
                 => ( ~ ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ D ) @ E ) )
                   => ( ( hBOOL @ ( hAPP_f448129468l_bool @ A @ E ) )
                     => ( hBOOL @ ( hAPP_f448129468l_bool @ A @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ D ) @ E ) ) ) ) ) ) )
           => ( hBOOL @ ( hAPP_f448129468l_bool @ A @ C ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_914_finite__subset__induct) ).

thf(3186,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f448129468l_bool @ finite_finite_int @ C ) )
     => ( ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_f284875647l_bool @ ord_le951220754t_bool @ C ) @ B ) )
       => ( ( hBOOL @ ( hAPP_f448129468l_bool @ A @ bot_bot_fun_int_bool ) )
         => ( ! [D: $i,E: $i] :
                ( ( hBOOL @ ( hAPP_f448129468l_bool @ finite_finite_int @ E ) )
               => ( ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ D ) @ B ) )
                 => ( ~ ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ D ) @ E ) )
                   => ( ( hBOOL @ ( hAPP_f448129468l_bool @ A @ E ) )
                     => ( hBOOL @ ( hAPP_f448129468l_bool @ A @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ D ) @ E ) ) ) ) ) ) )
           => ( hBOOL @ ( hAPP_f448129468l_bool @ A @ C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[561]) ).

thf(1097,axiom,
    ! [A: $i,B: $i] :
      ( ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ A ) @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ A ) @ B ) )
      = ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ A ) @ B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_70_insert__absorb2) ).

thf(5167,plain,
    ! [A: $i,B: $i] :
      ( ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ A ) @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ A ) @ B ) )
      = ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ A ) @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1097]) ).

thf(253,axiom,
    ! [A: $i,B: $i] :
      ( ( B = bot_bot_fun_nat_bool )
     => ~ ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ A ) @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_31_equals0D) ).

thf(2047,plain,
    ! [A: $i,B: $i] :
      ( ( B = bot_bot_fun_nat_bool )
     => ~ ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ A ) @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[253]) ).

thf(451,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ minus_minus_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ A ) @ B ) ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ A ) @ C ) )
      = ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ minus_minus_nat @ B ) @ C ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_1021_diff__cancel) ).

thf(2773,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ minus_minus_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ A ) @ B ) ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ A ) @ C ) )
      = ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ minus_minus_nat @ B ) @ C ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[451]) ).

thf(699,axiom,
    ! [A: $i] :
      ( ( ( hAPP_f1805168059t_bool @ collect_int @ A )
        = bot_bot_fun_int_bool )
    <=> ! [B: $i] :
          ~ ( hBOOL @ ( hAPP_int_bool @ A @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_34_Collect__empty__eq) ).

thf(3679,plain,
    ! [A: $i] :
      ( ( ( ( hAPP_f1805168059t_bool @ collect_int @ A )
          = bot_bot_fun_int_bool )
       => ! [B: $i] :
            ~ ( hBOOL @ ( hAPP_int_bool @ A @ B ) ) )
      & ( ! [B: $i] :
            ~ ( hBOOL @ ( hAPP_int_bool @ A @ B ) )
       => ( ( hAPP_f1805168059t_bool @ collect_int @ A )
          = bot_bot_fun_int_bool ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[699]) ).

thf(549,axiom,
    ! [A: $i,B: $i] :
      ( ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ 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 @ fdisj ) @ ( hAPP_H426895267a_bool @ ( hAPP_f920293029a_bool @ cOMBC_1940896922a_bool @ fequal963300192iple_a ) @ A ) ) ) @ ( hAPP_f20753329a_bool @ ( hAPP_f247708275a_bool @ cOMBC_1519416966l_bool @ member1797258804iple_a ) @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_82_insert__compr__raw) ).

thf(3151,plain,
    ! [A: $i,B: $i] :
      ( ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ 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 @ fdisj ) @ ( hAPP_H426895267a_bool @ ( hAPP_f920293029a_bool @ cOMBC_1940896922a_bool @ fequal963300192iple_a ) @ A ) ) ) @ ( hAPP_f20753329a_bool @ ( hAPP_f247708275a_bool @ cOMBC_1519416966l_bool @ member1797258804iple_a ) @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[549]) ).

thf(538,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( C != D )
        & ( A != B ) )
    <=> ( ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ C ) @ A ) ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ D ) @ B ) )
       != ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ C ) @ B ) ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ D ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_698_crossproduct__noteq) ).

thf(3097,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( ( C != D )
          & ( A != B ) )
       => ( ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ C ) @ A ) ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ D ) @ B ) )
         != ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ C ) @ B ) ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ D ) @ A ) ) ) )
      & ( ( ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ C ) @ A ) ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ D ) @ B ) )
         != ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ C ) @ B ) ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ D ) @ A ) ) )
       => ( ( C != D )
          & ( A != B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[538]) ).

thf(1018,axiom,
    ! [A: $i] :
      ( ( A != bot_bo797238721a_bool )
    <=> ? [B: $i,C: $i] :
          ( ( A
            = ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ B ) @ C ) )
          & ~ ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ B ) @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_125_nonempty__iff) ).

thf(4888,plain,
    ! [A: $i] :
      ( ( ( A != bot_bo797238721a_bool )
       => ? [B: $i,C: $i] :
            ( ( A
              = ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ B ) @ C ) )
            & ~ ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ B ) @ C ) ) ) )
      & ( ? [B: $i,C: $i] :
            ( ( A
              = ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ B ) @ C ) )
            & ~ ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ B ) @ C ) ) )
       => ( A != bot_bo797238721a_bool ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1018]) ).

thf(86,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ ord_min_nat @ A ) @ B ) ) @ C )
      = ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ ord_min_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ A ) @ C ) ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ B ) @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_989_min__add__distrib__left) ).

thf(1438,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ ord_min_nat @ A ) @ B ) ) @ C )
      = ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ ord_min_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ A ) @ C ) ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ B ) @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[86]) ).

thf(712,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP_f448129468l_bool @ finite_finite_int @ B ) )
     => ( hBOOL @ ( hAPP_f448129468l_bool @ finite_finite_int @ ( image_int_int @ A @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_289_finite__imageI) ).

thf(3720,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP_f448129468l_bool @ finite_finite_int @ B ) )
     => ( hBOOL @ ( hAPP_f448129468l_bool @ finite_finite_int @ ( image_int_int @ A @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[712]) ).

thf(25,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_int_bool @ ( hAPP_i1948725293t_bool @ ord_less_eq_int @ A ) @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ ord_min_int @ B ) @ C ) ) )
    <=> ( ( hBOOL @ ( hAPP_int_bool @ ( hAPP_i1948725293t_bool @ ord_less_eq_int @ A ) @ B ) )
        & ( hBOOL @ ( hAPP_int_bool @ ( hAPP_i1948725293t_bool @ ord_less_eq_int @ A ) @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_981_min__max_Ole__inf__iff) ).

thf(1199,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( hBOOL @ ( hAPP_int_bool @ ( hAPP_i1948725293t_bool @ ord_less_eq_int @ A ) @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ ord_min_int @ B ) @ C ) ) )
       => ( ( hBOOL @ ( hAPP_int_bool @ ( hAPP_i1948725293t_bool @ ord_less_eq_int @ A ) @ B ) )
          & ( hBOOL @ ( hAPP_int_bool @ ( hAPP_i1948725293t_bool @ ord_less_eq_int @ A ) @ C ) ) ) )
      & ( ( ( hBOOL @ ( hAPP_int_bool @ ( hAPP_i1948725293t_bool @ ord_less_eq_int @ A ) @ B ) )
          & ( hBOOL @ ( hAPP_int_bool @ ( hAPP_i1948725293t_bool @ ord_less_eq_int @ A ) @ C ) ) )
       => ( hBOOL @ ( hAPP_int_bool @ ( hAPP_i1948725293t_bool @ ord_less_eq_int @ A ) @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ ord_min_int @ B ) @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[25]) ).

thf(339,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ B ) @ C ) )
     => ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ B ) @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ A ) @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_56_insertI2) ).

thf(2364,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ B ) @ C ) )
     => ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ B ) @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ A ) @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[339]) ).

thf(282,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_nat @ B ) @ C ) )
     => ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_nat @ B ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ A ) @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_1124_termination__basic__simps_I2_J) ).

thf(2164,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_nat @ B ) @ C ) )
     => ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_nat @ B ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ A ) @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[282]) ).

thf(813,axiom,
    ! [A: $i,B: $i] :
      ( ( ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ A ) @ bot_bot_fun_int_bool )
        = ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ B ) @ bot_bot_fun_int_bool ) )
     => ( A = B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_86_singleton__inject) ).

thf(4076,plain,
    ! [A: $i,B: $i] :
      ( ( ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ A ) @ bot_bot_fun_int_bool )
        = ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ B ) @ bot_bot_fun_int_bool ) )
     => ( A = B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[813]) ).

thf(880,axiom,
    ! [A: $i] : ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_f284875647l_bool @ ord_le951220754t_bool @ A ) @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_809_subset__refl) ).

thf(4370,plain,
    ! [A: $i] : ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_f284875647l_bool @ ord_le951220754t_bool @ A ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[880]) ).

thf(723,axiom,
    ! [A: $i,B: $i] : ( is_bool @ ( finite1033474011iple_a @ A @ B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',gsy_c_Finite__Set_Ofolding__one__idem_000tc__Hoare____Mirabelle____uwgpyvfjxg__O) ).

thf(3767,plain,
    ! [A: $i,B: $i] : ( is_bool @ ( finite1033474011iple_a @ A @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[723]) ).

thf(302,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f54304608l_bool @ finite_finite_nat @ C ) )
     => ( ~ ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ B ) @ C ) )
       => ( ( hAPP_f659380387ol_int @ ( big_co195215938at_int @ A ) @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ B ) @ C ) )
          = ( hAPP_int_int @ ( hAPP_int_fun_int_int @ times_times_int @ ( hAPP_nat_int @ A @ B ) ) @ ( hAPP_f659380387ol_int @ ( big_co195215938at_int @ A ) @ C ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_558_setprod_Oinsert) ).

thf(2232,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f54304608l_bool @ finite_finite_nat @ C ) )
     => ( ~ ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ B ) @ C ) )
       => ( ( hAPP_f659380387ol_int @ ( big_co195215938at_int @ A ) @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ B ) @ C ) )
          = ( hAPP_int_int @ ( hAPP_int_fun_int_int @ times_times_int @ ( hAPP_nat_int @ A @ B ) ) @ ( hAPP_f659380387ol_int @ ( big_co195215938at_int @ A ) @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[302]) ).

thf(171,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP_f800510211t_bool @ ( hAPP_f1730770594t_bool @ minus_2067140911t_bool @ A ) @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ B ) @ C ) )
      = ( hAPP_f800510211t_bool @ ( hAPP_f1730770594t_bool @ minus_2067140911t_bool @ ( hAPP_f800510211t_bool @ ( hAPP_f1730770594t_bool @ minus_2067140911t_bool @ A ) @ C ) ) @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ B ) @ bot_bot_fun_nat_bool ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_411_Diff__insert) ).

thf(1736,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP_f800510211t_bool @ ( hAPP_f1730770594t_bool @ minus_2067140911t_bool @ A ) @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ B ) @ C ) )
      = ( hAPP_f800510211t_bool @ ( hAPP_f1730770594t_bool @ minus_2067140911t_bool @ ( hAPP_f800510211t_bool @ ( hAPP_f1730770594t_bool @ minus_2067140911t_bool @ A ) @ C ) ) @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ B ) @ bot_bot_fun_nat_bool ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[171]) ).

thf(868,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP_a2036067514e_bool @ ( hAPP_f100967412e_bool @ ( hAPP_f1460451647e_bool @ cOMBC_524597097e_bool @ A ) @ B ) @ C )
      = ( hAPP_f746301080e_bool @ ( hAPP_a1224971408e_bool @ A @ C ) @ B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',help_COMBC_1_1_COMBC_000t__a_000tc__fun_Itc__Com__Ostate_Mtc__Com__Ostate_J_000t) ).

thf(4312,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP_a2036067514e_bool @ ( hAPP_f100967412e_bool @ ( hAPP_f1460451647e_bool @ cOMBC_524597097e_bool @ A ) @ B ) @ C )
      = ( hAPP_f746301080e_bool @ ( hAPP_a1224971408e_bool @ A @ C ) @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[868]) ).

thf(806,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_f284875647l_bool @ ord_le951220754t_bool @ B ) @ C ) )
     => ( ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_f284875647l_bool @ ord_le951220754t_bool @ A ) @ B ) )
       => ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_f284875647l_bool @ ord_le951220754t_bool @ A ) @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_740_xt1_I6_J) ).

thf(4059,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_f284875647l_bool @ ord_le951220754t_bool @ B ) @ C ) )
     => ( ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_f284875647l_bool @ ord_le951220754t_bool @ A ) @ B ) )
       => ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_f284875647l_bool @ ord_le951220754t_bool @ A ) @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[806]) ).

thf(488,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP_state_bool @ ( hAPP_f746301080e_bool @ ( hAPP_f162060345e_bool @ cOMBB_844853809_state @ A ) @ B ) @ C )
      = ( hAPP_state_bool @ A @ ( hAPP_state_state @ B @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',help_COMBB_1_1_COMBB_000tc__Com__Ostate_000tc__HOL__Obool_000tc__Com__Ostate_U) ).

thf(2905,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP_state_bool @ ( hAPP_f746301080e_bool @ ( hAPP_f162060345e_bool @ cOMBB_844853809_state @ A ) @ B ) @ C )
      = ( hAPP_state_bool @ A @ ( hAPP_state_state @ B @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[488]) ).

thf(209,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ A ) @ B ) ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ C ) @ B ) ) )
     => ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ A ) @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_821_add__le__imp__le__right) ).

thf(1868,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ A ) @ B ) ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ C ) @ B ) ) )
     => ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ A ) @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[209]) ).

thf(361,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ~ ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ B ) @ C ) )
     => ( ~ ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ B ) @ A ) )
       => ( ( ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ B ) @ C )
            = ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ B ) @ A ) )
        <=> ( C = A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_59_insert__ident) ).

thf(2441,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ~ ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ B ) @ C ) )
     => ( ~ ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ B ) @ A ) )
       => ( ( ( ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ B ) @ C )
              = ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ B ) @ A ) )
           => ( C = A ) )
          & ( ( C = A )
           => ( ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ B ) @ C )
              = ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ B ) @ A ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[361]) ).

thf(1069,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( finite1428139793iple_a @ E ) )
     => ( ( hBOOL @ ( hAPP_H1037229737a_bool @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ ( finite90892550iple_a @ E ) @ B ) @ C ) @ D ) )
       => ( ( hBOOL @ ( hAPP_H1037229737a_bool @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ ( finite90892550iple_a @ E ) @ B ) @ C ) @ A ) )
         => ( A = D ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_537_comp__fun__commute_Ofold__graph__determ) ).

thf(5049,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( finite1428139793iple_a @ E ) )
     => ( ( hBOOL @ ( hAPP_H1037229737a_bool @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ ( finite90892550iple_a @ E ) @ B ) @ C ) @ D ) )
       => ( ( hBOOL @ ( hAPP_H1037229737a_bool @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ ( finite90892550iple_a @ E ) @ B ) @ C ) @ A ) )
         => ( A = D ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1069]) ).

thf(964,axiom,
    ! [A: $i,B: $i] : ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_f1767618879l_bool @ ord_le1523781517a_bool @ A ) @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ B ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_871_subset__insertI) ).

thf(4712,plain,
    ! [A: $i,B: $i] : ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_f1767618879l_bool @ ord_le1523781517a_bool @ A ) @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ B ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[964]) ).

thf(902,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ A ) @ B ) ) @ C )
      = ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ A ) @ C ) ) @ B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_666_comm__semiring__1__class_Onormalizing__semiring__rules_I23_J) ).

thf(4466,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ A ) @ B ) ) @ C )
      = ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ A ) @ C ) ) @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[902]) ).

thf(882,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f448129468l_bool @ finite_finite_int @ C ) )
     => ( ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_f284875647l_bool @ ord_le951220754t_bool @ C ) @ ( image_int_int @ A @ B ) ) )
       => ? [D: $i] :
            ( ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_f284875647l_bool @ ord_le951220754t_bool @ D ) @ B ) )
            & ( hBOOL @ ( hAPP_f448129468l_bool @ finite_finite_int @ D ) )
            & ( C
              = ( image_int_int @ A @ D ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_919_finite__subset__image) ).

thf(4375,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f448129468l_bool @ finite_finite_int @ C ) )
     => ( ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_f284875647l_bool @ ord_le951220754t_bool @ C ) @ ( image_int_int @ A @ B ) ) )
       => ? [D: $i] :
            ( ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_f284875647l_bool @ ord_le951220754t_bool @ D ) @ B ) )
            & ( hBOOL @ ( hAPP_f448129468l_bool @ finite_finite_int @ D ) )
            & ( C
              = ( image_int_int @ A @ D ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[882]) ).

thf(1015,axiom,
    ! [A: $i,B: $i] :
      ( ( A != B )
      | ( hBOOL @ ( hAPP_state_bool @ ( hAPP_s1806633685e_bool @ fequal_state @ A ) @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',help_fequal_2_1_fequal_000tc__Com__Ostate_T) ).

thf(4878,plain,
    ! [A: $i,B: $i] :
      ( ( A != B )
      | ( hBOOL @ ( hAPP_state_bool @ ( hAPP_s1806633685e_bool @ fequal_state @ A ) @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1015]) ).

thf(991,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ A ) @ B ) )
     => ~ ! [C: $i] :
            ( ( B
              = ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ A ) @ C ) )
           => ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ A ) @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_115_Set_Oset__insert) ).

thf(4785,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ A ) @ B ) )
     => ~ ! [C: $i] :
            ( ( B
              = ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ A ) @ C ) )
           => ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ A ) @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[991]) ).

thf(67,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i] :
          ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ D ) @ C ) )
         => ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ ( hAPP_nat_nat @ A @ D ) ) @ ( hAPP_nat_nat @ B @ D ) ) ) )
     => ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ ( hAPP_f22106695ol_nat @ ( hAPP_f782000547ol_nat @ big_co387207925at_nat @ A ) @ C ) ) @ ( hAPP_f22106695ol_nat @ ( hAPP_f782000547ol_nat @ big_co387207925at_nat @ B ) @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_925_setsum__mono) ).

thf(1370,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i] :
          ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ D ) @ C ) )
         => ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ ( hAPP_nat_nat @ A @ D ) ) @ ( hAPP_nat_nat @ B @ D ) ) ) )
     => ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ ( hAPP_f22106695ol_nat @ ( hAPP_f782000547ol_nat @ big_co387207925at_nat @ A ) @ C ) ) @ ( hAPP_f22106695ol_nat @ ( hAPP_f782000547ol_nat @ big_co387207925at_nat @ B ) @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[67]) ).

thf(249,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( finite988810631ne_nat @ C @ D ) )
     => ( ( hBOOL @ ( hAPP_f54304608l_bool @ finite_finite_nat @ B ) )
       => ( ~ ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ A ) @ B ) )
         => ( ( hAPP_f22106695ol_nat @ D @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ A ) @ B ) )
            = ( hAPP_f22106695ol_nat @ ( finite_fold_nat_nat @ C @ A ) @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_581_folding__one_Oeq__fold_H) ).

thf(2032,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( finite988810631ne_nat @ C @ D ) )
     => ( ( hBOOL @ ( hAPP_f54304608l_bool @ finite_finite_nat @ B ) )
       => ( ~ ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ A ) @ B ) )
         => ( ( hAPP_f22106695ol_nat @ D @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ A ) @ B ) )
            = ( hAPP_f22106695ol_nat @ ( finite_fold_nat_nat @ C @ A ) @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[249]) ).

thf(404,axiom,
    ! [A: $i,B: $i] :
      ( ( 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 ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_384_Diff__idemp) ).

thf(2601,plain,
    ! [A: $i,B: $i] :
      ( ( 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 ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[404]) ).

thf(741,axiom,
    ! [A: $i,B: $i] :
      ( ( 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 ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_752_order__antisym) ).

thf(3827,plain,
    ! [A: $i,B: $i] :
      ( ( 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 ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[741]) ).

thf(709,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_f103356543l_bool @ ord_le1568362934t_bool @ B ) @ C ) )
     => ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_f103356543l_bool @ ord_le1568362934t_bool @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ A ) @ B ) ) @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ A ) @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_857_insert__mono) ).

thf(3711,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_f103356543l_bool @ ord_le1568362934t_bool @ B ) @ C ) )
     => ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_f103356543l_bool @ ord_le1568362934t_bool @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ A ) @ B ) ) @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ A ) @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[709]) ).

thf(440,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ B ) @ C ) )
     => ( ( image_nat_nat @ ( hAPP_nat_fun_nat_nat @ cOMBK_nat_nat @ A ) @ C )
        = ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ A ) @ bot_bot_fun_nat_bool ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_238_image__constant) ).

thf(2733,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ B ) @ C ) )
     => ( ( image_nat_nat @ ( hAPP_nat_fun_nat_nat @ cOMBK_nat_nat @ A ) @ C )
        = ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ A ) @ bot_bot_fun_nat_bool ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[440]) ).

thf(459,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ A ) @ B ) )
    <=> ( hBOOL @ ( hAPP_nat_bool @ B @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_196_mem__def) ).

thf(2805,plain,
    ! [A: $i,B: $i] :
      ( ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ A ) @ B ) )
       => ( hBOOL @ ( hAPP_nat_bool @ B @ A ) ) )
      & ( ( hBOOL @ ( hAPP_nat_bool @ B @ A ) )
       => ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ A ) @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[459]) ).

thf(308,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ B ) @ C ) )
     => ( ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ A ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ minus_minus_nat @ C ) @ B ) )
        = ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ minus_minus_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ A ) @ C ) ) @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_1031_add__diff__assoc) ).

thf(2249,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ B ) @ C ) )
     => ( ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ A ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ minus_minus_nat @ C ) @ B ) )
        = ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ minus_minus_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ A ) @ C ) ) @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[308]) ).

thf(676,axiom,
    hBOOL @ ( hAPP_f54304608l_bool @ finite_finite_nat @ bot_bot_fun_nat_bool ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_280_finite_OemptyI) ).

thf(3603,plain,
    hBOOL @ ( hAPP_f54304608l_bool @ finite_finite_nat @ bot_bot_fun_nat_bool ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[676]) ).

thf(105,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( finite1626084323ne_int @ C @ D ) )
     => ( ( hBOOL @ ( hAPP_f448129468l_bool @ finite_finite_int @ B ) )
       => ( ~ ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ A ) @ B ) )
         => ( ( hAPP_f1594865479ol_int @ D @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ A ) @ B ) )
            = ( hAPP_f1594865479ol_int @ ( finite_fold_int_int @ C @ A ) @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_583_folding__one_Oeq__fold_H) ).

thf(1519,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( finite1626084323ne_int @ C @ D ) )
     => ( ( hBOOL @ ( hAPP_f448129468l_bool @ finite_finite_int @ B ) )
       => ( ~ ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ A ) @ B ) )
         => ( ( hAPP_f1594865479ol_int @ D @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ A ) @ B ) )
            = ( hAPP_f1594865479ol_int @ ( finite_fold_int_int @ C @ A ) @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[105]) ).

thf(402,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( image_nat_int @ A @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ B ) @ C ) )
      = ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ ( hAPP_nat_int @ A @ B ) ) @ ( image_nat_int @ A @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_226_image__insert) ).

thf(2596,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( image_nat_int @ A @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ B ) @ C ) )
      = ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ ( hAPP_nat_int @ A @ B ) ) @ ( image_nat_int @ A @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[402]) ).

thf(755,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP_H2130837971l_bool @ ( hAPP_f146754017l_bool @ ( hAPP_f239971723l_bool @ cOMBB_1866391387iple_a @ A ) @ B ) @ C )
      = ( hAPP_b589554111l_bool @ A @ ( hAPP_H1037229737a_bool @ B @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',help_COMBB_1_1_COMBB_000tc__HOL__Obool_000tc__fun_Itc__HOL__Obool_Mtc__HOL__Oboo_027) ).

thf(3867,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP_H2130837971l_bool @ ( hAPP_f146754017l_bool @ ( hAPP_f239971723l_bool @ cOMBB_1866391387iple_a @ A ) @ B ) @ C )
      = ( hAPP_b589554111l_bool @ A @ ( hAPP_H1037229737a_bool @ B @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[755]) ).

thf(62,axiom,
    ! [A: $i,B: $i] :
      ( ( hAPP_f22106695ol_nat @ ( finite_fold1_nat @ A ) @ B )
      = ( hAPP_f22106695ol_nat @ the_nat @ ( finite_fold1Set_nat @ A @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_165_fold1__def) ).

thf(1355,plain,
    ! [A: $i,B: $i] :
      ( ( hAPP_f22106695ol_nat @ ( finite_fold1_nat @ A ) @ B )
      = ( hAPP_f22106695ol_nat @ the_nat @ ( finite_fold1Set_nat @ A @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[62]) ).

thf(124,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ A ) @ B ) ) @ C )
      = ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ A ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ B ) @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_669_comm__semiring__1__class_Onormalizing__semiring__rules_I21_J) ).

thf(1580,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ A ) @ B ) ) @ C )
      = ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ A ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ B ) @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[124]) ).

thf(1050,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP_state_bool @ ( hAPP_b2019457360e_bool @ ( hAPP_f167292325e_bool @ cOMBC_231445413l_bool @ A ) @ B ) @ C )
      = ( hAPP_bool_bool @ ( hAPP_s58564346l_bool @ A @ C ) @ B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',help_COMBC_1_1_COMBC_000tc__Com__Ostate_000tc__HOL__Obool_000tc__HOL__Obool_U) ).

thf(4989,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP_state_bool @ ( hAPP_b2019457360e_bool @ ( hAPP_f167292325e_bool @ cOMBC_231445413l_bool @ A ) @ B ) @ C )
      = ( hAPP_bool_bool @ ( hAPP_s58564346l_bool @ A @ C ) @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1050]) ).

thf(1067,axiom,
    ! [A: $i,B: $i] :
      ~ ( hBOOL @ ( hAPP_H1037229737a_bool @ ( finite1621280049iple_a @ A @ bot_bo797238721a_bool ) @ B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_230_empty__fold1SetE) ).

thf(5044,plain,
    ! [A: $i,B: $i] :
      ~ ( hBOOL @ ( hAPP_H1037229737a_bool @ ( finite1621280049iple_a @ A @ bot_bo797238721a_bool ) @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1067]) ).

thf(518,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( B
        = ( hAPP_H1229305626_a_int @ C @ D ) )
     => ( ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ D ) @ A ) )
       => ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ B ) @ ( image_685909450_a_int @ C @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_160_image__eqI) ).

thf(3022,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( B
        = ( hAPP_H1229305626_a_int @ C @ D ) )
     => ( ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ D ) @ A ) )
       => ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ B ) @ ( image_685909450_a_int @ C @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[518]) ).

thf(853,axiom,
    ! [A: $i] :
      ( ( hBOOL @ ( hAPP_nat_bool @ bot_bot_fun_nat_bool @ A ) )
    <=> ( hBOOL @ bot_bot_bool ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_107_bot__apply) ).

thf(4252,plain,
    ! [A: $i] :
      ( ( ( hBOOL @ ( hAPP_nat_bool @ bot_bot_fun_nat_bool @ A ) )
       => ( hBOOL @ bot_bot_bool ) )
      & ( ( hBOOL @ bot_bot_bool )
       => ( hBOOL @ ( hAPP_nat_bool @ bot_bot_fun_nat_bool @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[853]) ).

thf(719,axiom,
    ! [A: $i,B: $i] :
      ( ( hAPP_H905846293iple_a @ ( hAPP_H2085992369iple_a @ cOMBK_2052406350iple_a @ A ) @ B )
      = A ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',help_COMBK_1_1_COMBK_000tc__Hoare____Mirabelle____uwgpyvfjxg__Otriple_It__a_J_00) ).

thf(3752,plain,
    ! [A: $i,B: $i] :
      ( ( hAPP_H905846293iple_a @ ( hAPP_H2085992369iple_a @ cOMBK_2052406350iple_a @ A ) @ B )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[719]) ).

thf(576,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP_f1695230391l_bool @ finite1738664244iple_a @ B ) )
     => ( hBOOL @ ( hAPP_f448129468l_bool @ finite_finite_int @ ( image_685909450_a_int @ A @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_290_finite__imageI) ).

thf(3239,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP_f1695230391l_bool @ finite1738664244iple_a @ B ) )
     => ( hBOOL @ ( hAPP_f448129468l_bool @ finite_finite_int @ ( image_685909450_a_int @ A @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[576]) ).

thf(618,axiom,
    ! [A: $i,B: $i] :
      ( ( hAPP_int_int @ ( hAPP_int_fun_int_int @ times_times_int @ ( hAPP_int_int @ uminus_uminus_int @ A ) ) @ B )
      = ( hAPP_int_int @ uminus_uminus_int @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ times_times_int @ A ) @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_1078_zmult__zminus) ).

thf(3371,plain,
    ! [A: $i,B: $i] :
      ( ( hAPP_int_int @ ( hAPP_int_fun_int_int @ times_times_int @ ( hAPP_int_int @ uminus_uminus_int @ A ) ) @ B )
      = ( hAPP_int_int @ uminus_uminus_int @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ times_times_int @ A ) @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[618]) ).

thf(761,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ B ) @ C ) )
     => ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ ( hAPP_n261501868iple_a @ A @ B ) ) @ ( image_1865589340iple_a @ A @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_176_imageI) ).

thf(3884,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ B ) @ C ) )
     => ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ ( hAPP_n261501868iple_a @ A @ B ) ) @ ( image_1865589340iple_a @ A @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[761]) ).

thf(495,axiom,
    ! [A: $i,B: $i] :
      ( ( hAPP_int_int @ ( hAPP_int_fun_int_int @ minus_minus_int @ A ) @ B )
      = ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ A ) @ ( hAPP_int_int @ uminus_uminus_int @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_1080_diff__int__def) ).

thf(2937,plain,
    ! [A: $i,B: $i] :
      ( ( hAPP_int_int @ ( hAPP_int_fun_int_int @ minus_minus_int @ A ) @ B )
      = ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ A ) @ ( hAPP_int_int @ uminus_uminus_int @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[495]) ).

thf(550,axiom,
    ! [A: $i,B: $i] : ( hBOOL @ ( hAPP_f54304608l_bool @ finite_finite_nat @ ( hAPP_n1699378549t_bool @ ( ord_at238088361st_nat @ A ) @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_1072_finite__atLeastAtMost) ).

thf(3154,plain,
    ! [A: $i,B: $i] : ( hBOOL @ ( hAPP_f54304608l_bool @ finite_finite_nat @ ( hAPP_n1699378549t_bool @ ( ord_at238088361st_nat @ A ) @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[550]) ).

thf(910,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( finite1956789438t_bool @ D ) )
     => ( ( hBOOL @ ( hAPP_f54304608l_bool @ finite_finite_nat @ C ) )
       => ( ( hAPP_f800510211t_bool @ ( finite1578363458t_bool @ D @ A ) @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ B ) @ C ) )
          = ( hAPP_f800510211t_bool @ ( finite1578363458t_bool @ D @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ D @ B ) @ A ) ) @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_570_comp__fun__idem_Ofold__insert__idem2) ).

thf(4493,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( finite1956789438t_bool @ D ) )
     => ( ( hBOOL @ ( hAPP_f54304608l_bool @ finite_finite_nat @ C ) )
       => ( ( hAPP_f800510211t_bool @ ( finite1578363458t_bool @ D @ A ) @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ B ) @ C ) )
          = ( hAPP_f800510211t_bool @ ( finite1578363458t_bool @ D @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ D @ B ) @ A ) ) @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[910]) ).

thf(1078,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f1695230391l_bool @ finite1738664244iple_a @ C ) )
     => ( ( hAPP_f2097660464ol_nat @ ( big_co1815942735_a_nat @ A ) @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ B ) @ C ) )
        = ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ ( hAPP_H592031934_a_nat @ A @ B ) ) @ ( hAPP_f2097660464ol_nat @ ( big_co1815942735_a_nat @ A ) @ ( hAPP_f20753329a_bool @ ( hAPP_f1706700729a_bool @ minus_85316870a_bool @ C ) @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ B ) @ bot_bo797238721a_bool ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_467_setprod_Oinsert__remove) ).

thf(5085,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f1695230391l_bool @ finite1738664244iple_a @ C ) )
     => ( ( hAPP_f2097660464ol_nat @ ( big_co1815942735_a_nat @ A ) @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ B ) @ C ) )
        = ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ ( hAPP_H592031934_a_nat @ A @ B ) ) @ ( hAPP_f2097660464ol_nat @ ( big_co1815942735_a_nat @ A ) @ ( hAPP_f20753329a_bool @ ( hAPP_f1706700729a_bool @ minus_85316870a_bool @ C ) @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ B ) @ bot_bo797238721a_bool ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1078]) ).

thf(482,axiom,
    ! [A: $i,B: $i] :
      ( ! [C: $i] :
          ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ C ) @ B ) )
         => ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ C ) @ A ) ) )
     => ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_f103356543l_bool @ ord_le1568362934t_bool @ B ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_918_subsetI) ).

thf(2890,plain,
    ! [A: $i,B: $i] :
      ( ! [C: $i] :
          ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ C ) @ B ) )
         => ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ C ) @ A ) ) )
     => ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_f103356543l_bool @ ord_le1568362934t_bool @ B ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[482]) ).

thf(1006,axiom,
    ! [A: $i,B: $i] :
      ( ( ( image_533441733iple_a @ A @ B )
        = bot_bo797238721a_bool )
    <=> ( B = bot_bo797238721a_bool ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_202_image__is__empty) ).

thf(4836,plain,
    ! [A: $i,B: $i] :
      ( ( ( ( image_533441733iple_a @ A @ B )
          = bot_bo797238721a_bool )
       => ( B = bot_bo797238721a_bool ) )
      & ( ( B = bot_bo797238721a_bool )
       => ( ( image_533441733iple_a @ A @ B )
          = bot_bo797238721a_bool ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1006]) ).

thf(955,axiom,
    ! [A: $i,B: $i] :
      ( ~ ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ A ) @ B ) )
     => ( ( hAPP_f20753329a_bool @ ( hAPP_f1706700729a_bool @ minus_85316870a_bool @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ A ) @ B ) ) @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ A ) @ bot_bo797238721a_bool ) )
        = B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_421_Diff__insert__absorb) ).

thf(4661,plain,
    ! [A: $i,B: $i] :
      ( ~ ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ A ) @ B ) )
     => ( ( hAPP_f20753329a_bool @ ( hAPP_f1706700729a_bool @ minus_85316870a_bool @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ A ) @ B ) ) @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ A ) @ bot_bo797238721a_bool ) )
        = B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[955]) ).

thf(385,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ B ) @ C ) )
     => ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ C ) @ A ) )
       => ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ B ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_1001_le__trans) ).

thf(2528,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ B ) @ C ) )
     => ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ C ) @ A ) )
       => ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ B ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[385]) ).

thf(727,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ass @ A @ B )
     != ( semi @ C @ D ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_192_com_Osimps_I24_J) ).

thf(3778,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ass @ A @ B )
     != ( semi @ C @ D ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[727]) ).

thf(35,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_int_bool @ ( hAPP_i1948725293t_bool @ ord_less_eq_int @ B ) @ C ) )
     => ( ( hBOOL @ ( hAPP_int_bool @ A @ C ) )
       => ( ! [D: $i] :
              ( ( hBOOL @ ( hAPP_int_bool @ ( hAPP_i1948725293t_bool @ ord_less_eq_int @ D ) @ C ) )
             => ( ( hBOOL @ ( hAPP_int_bool @ A @ D ) )
               => ( hBOOL @ ( hAPP_int_bool @ A @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ minus_minus_int @ D ) @ one_one_int ) ) ) ) )
         => ( hBOOL @ ( hAPP_int_bool @ A @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_1148_int__le__induct) ).

thf(1238,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_int_bool @ ( hAPP_i1948725293t_bool @ ord_less_eq_int @ B ) @ C ) )
     => ( ( hBOOL @ ( hAPP_int_bool @ A @ C ) )
       => ( ! [D: $i] :
              ( ( hBOOL @ ( hAPP_int_bool @ ( hAPP_i1948725293t_bool @ ord_less_eq_int @ D ) @ C ) )
             => ( ( hBOOL @ ( hAPP_int_bool @ A @ D ) )
               => ( hBOOL @ ( hAPP_int_bool @ A @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ minus_minus_int @ D ) @ one_one_int ) ) ) ) )
         => ( hBOOL @ ( hAPP_int_bool @ A @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[35]) ).

thf(243,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_nat @ B ) @ C ) )
     => ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_nat @ B ) @ A ) )
       => ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ minus_minus_nat @ A ) @ C ) ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ minus_minus_nat @ A ) @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_1115_diff__less__mono2) ).

thf(2015,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_nat @ B ) @ C ) )
     => ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_nat @ B ) @ A ) )
       => ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ minus_minus_nat @ A ) @ C ) ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ minus_minus_nat @ A ) @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[243]) ).

thf(763,axiom,
    ! [A: $i] :
      ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_f103356543l_bool @ ord_le1568362934t_bool @ A ) @ bot_bot_fun_nat_bool ) )
    <=> ( A = bot_bot_fun_nat_bool ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_838_bot__unique) ).

thf(3889,plain,
    ! [A: $i] :
      ( ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_f103356543l_bool @ ord_le1568362934t_bool @ A ) @ bot_bot_fun_nat_bool ) )
       => ( A = bot_bot_fun_nat_bool ) )
      & ( ( A = bot_bot_fun_nat_bool )
       => ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_f103356543l_bool @ ord_le1568362934t_bool @ A ) @ bot_bot_fun_nat_bool ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[763]) ).

thf(639,axiom,
    ! [A: $i] : ( hBOOL @ ( hAPP_f54304608l_bool @ finite_finite_nat @ ( hAPP_n1699378549t_bool @ ord_lessThan_nat @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_947_finite__lessThan) ).

thf(3471,plain,
    ! [A: $i] : ( hBOOL @ ( hAPP_f54304608l_bool @ finite_finite_nat @ ( hAPP_n1699378549t_bool @ ord_lessThan_nat @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[639]) ).

thf(118,axiom,
    ! [A: $i,B: $i] : ( is_bool @ ( big_se275732192ig_nat @ A @ B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',gsy_c_Big__Operators_Osemilattice__big_000tc__Nat__Onat) ).

thf(1560,plain,
    ! [A: $i,B: $i] : ( is_bool @ ( big_se275732192ig_nat @ A @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[118]) ).

thf(734,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_f103356543l_bool @ ord_le1568362934t_bool @ A ) @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ B ) @ bot_bot_fun_nat_bool ) ) )
     => ( ( A = bot_bot_fun_nat_bool )
        | ( A
          = ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ B ) @ bot_bot_fun_nat_bool ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_891_subset__singletonD) ).

thf(3810,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_f103356543l_bool @ ord_le1568362934t_bool @ A ) @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ B ) @ bot_bot_fun_nat_bool ) ) )
     => ( ( A = bot_bot_fun_nat_bool )
        | ( A
          = ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ B ) @ bot_bot_fun_nat_bool ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[734]) ).

thf(943,axiom,
    ! [A: $i] :
      ( ( ( hAPP_f20753329a_bool @ collec351493750iple_a @ A )
        = bot_bo797238721a_bool )
    <=> ! [B: $i] :
          ~ ( hBOOL @ ( hAPP_H1037229737a_bool @ A @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_33_Collect__empty__eq) ).

thf(4608,plain,
    ! [A: $i] :
      ( ( ( ( hAPP_f20753329a_bool @ collec351493750iple_a @ A )
          = bot_bo797238721a_bool )
       => ! [B: $i] :
            ~ ( hBOOL @ ( hAPP_H1037229737a_bool @ A @ B ) ) )
      & ( ! [B: $i] :
            ~ ( hBOOL @ ( hAPP_H1037229737a_bool @ A @ B ) )
       => ( ( hAPP_f20753329a_bool @ collec351493750iple_a @ A )
          = bot_bo797238721a_bool ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[943]) ).

thf(536,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_f103356543l_bool @ ord_le1568362934t_bool @ B ) @ C ) )
     => ( ( C = A )
       => ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_f103356543l_bool @ ord_le1568362934t_bool @ B ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_761_ord__le__eq__trans) ).

thf(3090,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_f103356543l_bool @ ord_le1568362934t_bool @ B ) @ C ) )
     => ( ( C = A )
       => ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_f103356543l_bool @ ord_le1568362934t_bool @ B ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[536]) ).

thf(599,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( finite1428139793iple_a @ D ) )
     => ( ( hBOOL @ ( hAPP_f1695230391l_bool @ finite1738664244iple_a @ C ) )
       => ( ( hAPP_H905846293iple_a @ ( hAPP_H2085992369iple_a @ D @ A ) @ ( hAPP_f1693662087iple_a @ ( finite1205970312iple_a @ D @ B ) @ C ) )
          = ( hAPP_f1693662087iple_a @ ( finite1205970312iple_a @ D @ ( hAPP_H905846293iple_a @ ( hAPP_H2085992369iple_a @ D @ A ) @ B ) ) @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_481_comp__fun__commute_Ofold__fun__comm) ).

thf(3314,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( finite1428139793iple_a @ D ) )
     => ( ( hBOOL @ ( hAPP_f1695230391l_bool @ finite1738664244iple_a @ C ) )
       => ( ( hAPP_H905846293iple_a @ ( hAPP_H2085992369iple_a @ D @ A ) @ ( hAPP_f1693662087iple_a @ ( finite1205970312iple_a @ D @ B ) @ C ) )
          = ( hAPP_f1693662087iple_a @ ( finite1205970312iple_a @ D @ ( hAPP_H905846293iple_a @ ( hAPP_H2085992369iple_a @ D @ A ) @ B ) ) @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[599]) ).

thf(16,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( 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 ) @ B ) ) @ C ) ) @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ times_times_int @ D ) @ B ) ) @ E ) ) )
    <=> ( hBOOL @ ( hAPP_int_bool @ ( hAPP_i1948725293t_bool @ ord_less_eq_int @ C ) @ ( 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 @ D ) @ A ) ) @ B ) ) @ E ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_903_le__add__iff2) ).

thf(1178,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( ( 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 ) @ B ) ) @ C ) ) @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ times_times_int @ D ) @ B ) ) @ E ) ) )
       => ( hBOOL @ ( hAPP_int_bool @ ( hAPP_i1948725293t_bool @ ord_less_eq_int @ C ) @ ( 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 @ D ) @ A ) ) @ B ) ) @ E ) ) ) )
      & ( ( hBOOL @ ( hAPP_int_bool @ ( hAPP_i1948725293t_bool @ ord_less_eq_int @ C ) @ ( 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 @ D ) @ A ) ) @ B ) ) @ E ) ) )
       => ( 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 ) @ B ) ) @ C ) ) @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ times_times_int @ D ) @ B ) ) @ E ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[16]) ).

thf(1060,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f1695230391l_bool @ finite1738664244iple_a @ C ) )
     => ( ( hAPP_f587450508ol_int @ ( hAPP_f1544320301ol_int @ big_co1977834938_a_int @ A ) @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ B ) @ C ) )
        = ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ ( hAPP_H1229305626_a_int @ A @ B ) ) @ ( hAPP_f587450508ol_int @ ( hAPP_f1544320301ol_int @ big_co1977834938_a_int @ A ) @ ( hAPP_f20753329a_bool @ ( hAPP_f1706700729a_bool @ minus_85316870a_bool @ C ) @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ B ) @ bot_bo797238721a_bool ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_622_setsum_Oinsert__remove) ).

thf(5021,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f1695230391l_bool @ finite1738664244iple_a @ C ) )
     => ( ( hAPP_f587450508ol_int @ ( hAPP_f1544320301ol_int @ big_co1977834938_a_int @ A ) @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ B ) @ C ) )
        = ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ ( hAPP_H1229305626_a_int @ A @ B ) ) @ ( hAPP_f587450508ol_int @ ( hAPP_f1544320301ol_int @ big_co1977834938_a_int @ A ) @ ( hAPP_f20753329a_bool @ ( hAPP_f1706700729a_bool @ minus_85316870a_bool @ C ) @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ B ) @ bot_bo797238721a_bool ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1060]) ).

thf(927,axiom,
    ! [A: $i] : ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hoare_2102800559rivs_a @ A ) @ bot_bo797238721a_bool ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_0_empty) ).

thf(4553,plain,
    ! [A: $i] : ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hoare_2102800559rivs_a @ A ) @ bot_bo797238721a_bool ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[927]) ).

thf(286,axiom,
    ! [A: $i] :
      ~ ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ A ) @ bot_bot_fun_nat_bool ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_37_empty__iff) ).

thf(2179,plain,
    ! [A: $i] :
      ~ ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ A ) @ bot_bot_fun_nat_bool ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[286]) ).

thf(296,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ C ) @ D ) )
     => ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ A ) @ B ) )
       => ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ C ) @ A ) ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ D ) @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_1006_add__le__mono) ).

thf(2213,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ C ) @ D ) )
     => ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ A ) @ B ) )
       => ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ C ) @ A ) ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ D ) @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[296]) ).

thf(523,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ B ) @ C ) )
     => ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ ( hAPP_int_int @ A @ B ) ) @ ( image_int_int @ A @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_179_imageI) ).

thf(3050,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ B ) @ C ) )
     => ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ ( hAPP_int_int @ A @ B ) ) @ ( image_int_int @ A @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[523]) ).

thf(23,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_int_bool @ ( hAPP_i1948725293t_bool @ ord_less_eq_int @ B ) @ C ) )
     => ( hBOOL @ ( hAPP_int_bool @ ( hAPP_i1948725293t_bool @ ord_less_eq_int @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ B ) @ A ) ) @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ C ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_826_add__right__mono) ).

thf(1195,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_int_bool @ ( hAPP_i1948725293t_bool @ ord_less_eq_int @ B ) @ C ) )
     => ( hBOOL @ ( hAPP_int_bool @ ( hAPP_i1948725293t_bool @ ord_less_eq_int @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ B ) @ A ) ) @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ C ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[23]) ).

thf(46,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_int_bool @ ( hAPP_i1948725293t_bool @ ord_less_eq_int @ B ) @ C ) )
     => ( ( B = A )
       => ( hBOOL @ ( hAPP_int_bool @ ( hAPP_i1948725293t_bool @ ord_less_eq_int @ A ) @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_758_xt1_I4_J) ).

thf(1281,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_int_bool @ ( hAPP_i1948725293t_bool @ ord_less_eq_int @ B ) @ C ) )
     => ( ( B = A )
       => ( hBOOL @ ( hAPP_int_bool @ ( hAPP_i1948725293t_bool @ ord_less_eq_int @ A ) @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[46]) ).

thf(690,axiom,
    ! [A: $i,B: $i] :
      ( ( A != B )
      | ( hBOOL @ ( hAPP_H1037229737a_bool @ ( hAPP_H426895267a_bool @ fequal963300192iple_a @ A ) @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',help_fequal_2_1_fequal_000tc__Hoare____Mirabelle____uwgpyvfjxg__Otriple_It__a_J_) ).

thf(3647,plain,
    ! [A: $i,B: $i] :
      ( ( A != B )
      | ( hBOOL @ ( hAPP_H1037229737a_bool @ ( hAPP_H426895267a_bool @ fequal963300192iple_a @ A ) @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[690]) ).

thf(798,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( 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 @ B ) @ C ) ) ) )
    <=> ( hBOOL @ ( hAPP_f54304608l_bool @ finite_finite_nat @ ( hAPP_f800510211t_bool @ ( hAPP_f1730770594t_bool @ minus_2067140911t_bool @ A ) @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_426_finite__Diff__insert) ).

thf(4027,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( 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 @ B ) @ C ) ) ) )
       => ( hBOOL @ ( hAPP_f54304608l_bool @ finite_finite_nat @ ( hAPP_f800510211t_bool @ ( hAPP_f1730770594t_bool @ minus_2067140911t_bool @ A ) @ C ) ) ) )
      & ( ( hBOOL @ ( hAPP_f54304608l_bool @ finite_finite_nat @ ( hAPP_f800510211t_bool @ ( hAPP_f1730770594t_bool @ minus_2067140911t_bool @ A ) @ C ) ) )
       => ( 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 @ B ) @ C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[798]) ).

thf(500,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f448129468l_bool @ finite_finite_int @ C ) )
     => ( ( hAPP_f1594865479ol_int @ ( hAPP_f1926459811ol_int @ big_co230513141nt_int @ A ) @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ B ) @ C ) )
        = ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ ( hAPP_int_int @ A @ B ) ) @ ( hAPP_f1594865479ol_int @ ( hAPP_f1926459811ol_int @ big_co230513141nt_int @ A ) @ ( hAPP_f1805168059t_bool @ ( hAPP_f1223193598t_bool @ minus_1449998731t_bool @ C ) @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ B ) @ bot_bot_fun_int_bool ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_623_setsum_Oinsert__remove) ).

thf(2958,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f448129468l_bool @ finite_finite_int @ C ) )
     => ( ( hAPP_f1594865479ol_int @ ( hAPP_f1926459811ol_int @ big_co230513141nt_int @ A ) @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ B ) @ C ) )
        = ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ ( hAPP_int_int @ A @ B ) ) @ ( hAPP_f1594865479ol_int @ ( hAPP_f1926459811ol_int @ big_co230513141nt_int @ A ) @ ( hAPP_f1805168059t_bool @ ( hAPP_f1223193598t_bool @ minus_1449998731t_bool @ C ) @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ B ) @ bot_bot_fun_int_bool ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[500]) ).

thf(81,axiom,
    ! [A: $i] :
      ( ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ ord_min_nat @ A ) @ A )
      = A ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_952_Min_Oidem) ).

thf(1423,plain,
    ! [A: $i] :
      ( ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ ord_min_nat @ A ) @ A )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[81]) ).

thf(598,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ times_times_int @ A ) @ B ) ) @ C )
        = ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ times_times_int @ D ) @ B ) ) @ E ) )
    <=> ( C
        = ( 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 @ D ) @ A ) ) @ B ) ) @ E ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_643_eq__add__iff2) ).

thf(3307,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( ( ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ times_times_int @ A ) @ B ) ) @ C )
          = ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ times_times_int @ D ) @ B ) ) @ E ) )
       => ( C
          = ( 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 @ D ) @ A ) ) @ B ) ) @ E ) ) )
      & ( ( C
          = ( 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 @ D ) @ A ) ) @ B ) ) @ E ) )
       => ( ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ times_times_int @ A ) @ B ) ) @ C )
          = ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ times_times_int @ D ) @ B ) ) @ E ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[598]) ).

thf(585,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ A ) @ B ) ) @ C )
      = ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ A ) @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ B ) @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_670_comm__semiring__1__class_Onormalizing__semiring__rules_I21_J) ).

thf(3271,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ A ) @ B ) ) @ C )
      = ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ A ) @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ B ) @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[585]) ).

thf(554,axiom,
    ! [A: $i,B: $i] :
      ( ( ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ A ) @ bot_bot_fun_nat_bool )
        = ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ B ) @ bot_bot_fun_nat_bool ) )
     => ( A = B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_84_singleton__inject) ).

thf(3166,plain,
    ! [A: $i,B: $i] :
      ( ( ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ A ) @ bot_bot_fun_nat_bool )
        = ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ B ) @ bot_bot_fun_nat_bool ) )
     => ( A = B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[554]) ).

thf(782,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i] :
          ( ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ D ) @ C ) )
         => ( ( hAPP_H592031934_a_nat @ A @ D )
            = ( hAPP_H592031934_a_nat @ B @ D ) ) )
     => ( ( hAPP_f2097660464ol_nat @ ( hAPP_f1371399797ol_nat @ big_co1340561246_a_nat @ A ) @ C )
        = ( hAPP_f2097660464ol_nat @ ( hAPP_f1371399797ol_nat @ big_co1340561246_a_nat @ B ) @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_705_setsum__cong2) ).

thf(3958,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i] :
          ( ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ D ) @ C ) )
         => ( ( hAPP_H592031934_a_nat @ A @ D )
            = ( hAPP_H592031934_a_nat @ B @ D ) ) )
     => ( ( hAPP_f2097660464ol_nat @ ( hAPP_f1371399797ol_nat @ big_co1340561246_a_nat @ A ) @ C )
        = ( hAPP_f2097660464ol_nat @ ( hAPP_f1371399797ol_nat @ big_co1340561246_a_nat @ B ) @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[782]) ).

thf(865,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP_int_int @ ( hAPP_int_fun_int_int @ times_times_int @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ times_times_int @ A ) @ B ) ) @ C )
      = ( hAPP_int_int @ ( hAPP_int_fun_int_int @ times_times_int @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ times_times_int @ A ) @ C ) ) @ B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_684_comm__semiring__1__class_Onormalizing__semiring__rules_I16_J) ).

thf(4295,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP_int_int @ ( hAPP_int_fun_int_int @ times_times_int @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ times_times_int @ A ) @ B ) ) @ C )
      = ( hAPP_int_int @ ( hAPP_int_fun_int_int @ times_times_int @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ times_times_int @ A ) @ C ) ) @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[865]) ).

thf(368,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( finite795500164em_nat @ C @ D ) )
     => ( ( hBOOL @ ( hAPP_f54304608l_bool @ finite_finite_nat @ B ) )
       => ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ A ) @ B ) )
         => ( ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ C @ A ) @ ( hAPP_f22106695ol_nat @ D @ B ) )
            = ( hAPP_f22106695ol_nat @ D @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_432_folding__one__idem_Oin__idem) ).

thf(2464,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( finite795500164em_nat @ C @ D ) )
     => ( ( hBOOL @ ( hAPP_f54304608l_bool @ finite_finite_nat @ B ) )
       => ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ A ) @ B ) )
         => ( ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ C @ A ) @ ( hAPP_f22106695ol_nat @ D @ B ) )
            = ( hAPP_f22106695ol_nat @ D @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[368]) ).

thf(860,axiom,
    ! [A: $i,B: $i] :
      ( ( 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 ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_1154_transfer__nat__int__set__relations_I5_J) ).

thf(4280,plain,
    ! [A: $i,B: $i] :
      ( ( ( 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 ) ) ) )
      & ( ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_f284875647l_bool @ ord_le951220754t_bool @ ( image_nat_int @ semiri1621563631at_int @ A ) ) @ ( image_nat_int @ semiri1621563631at_int @ B ) ) )
       => ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_f103356543l_bool @ ord_le1568362934t_bool @ A ) @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[860]) ).

thf(1127,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( finite1753440478iple_a @ B @ C ) )
     => ( ( hBOOL @ ( hAPP_f1695230391l_bool @ finite1738664244iple_a @ A ) )
       => ( ( hAPP_f1693662087iple_a @ C @ A )
          = ( hAPP_f1693662087iple_a @ ( finite1465856449iple_a @ B ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_270_folding__one_Oeq__fold) ).

thf(5302,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( finite1753440478iple_a @ B @ C ) )
     => ( ( hBOOL @ ( hAPP_f1695230391l_bool @ finite1738664244iple_a @ A ) )
       => ( ( hAPP_f1693662087iple_a @ C @ A )
          = ( hAPP_f1693662087iple_a @ ( finite1465856449iple_a @ B ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1127]) ).

thf(131,axiom,
    ! [A: $i,B: $i] :
      ( ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ A ) @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ A ) @ B ) )
      = ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ A ) @ B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_71_insert__absorb2) ).

thf(1602,plain,
    ! [A: $i,B: $i] :
      ( ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ A ) @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ A ) @ B ) )
      = ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ A ) @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[131]) ).

thf(681,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_f103356543l_bool @ ord_le1568362934t_bool @ B ) @ C ) )
     => ( ( hBOOL @ ( hAPP_nat_bool @ B @ A ) )
       => ( hBOOL @ ( hAPP_nat_bool @ C @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_812_predicate1D) ).

thf(3618,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_f103356543l_bool @ ord_le1568362934t_bool @ B ) @ C ) )
     => ( ( hBOOL @ ( hAPP_nat_bool @ B @ A ) )
       => ( hBOOL @ ( hAPP_nat_bool @ C @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[681]) ).

thf(545,axiom,
    ! [A: $i,B: $i] :
      ( ~ ( hBOOL @ ( hAPP_bool_bool @ ( hAPP_b589554111l_bool @ fimplies @ A ) @ B ) )
      | ~ ( hBOOL @ A )
      | ( hBOOL @ B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',help_fimplies_3_1_U) ).

thf(3137,plain,
    ! [A: $i,B: $i] :
      ( ~ ( hBOOL @ ( hAPP_bool_bool @ ( hAPP_b589554111l_bool @ fimplies @ A ) @ B ) )
      | ~ ( hBOOL @ A )
      | ( hBOOL @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[545]) ).

thf(528,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( hBOOL @ C )
       => ( A
          = ( 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 @ C ) ) @ ( hAPP_H426895267a_bool @ ( hAPP_f920293029a_bool @ cOMBC_1940896922a_bool @ fequal963300192iple_a ) @ A ) ) ) ) @ ( hAPP_f20753329a_bool @ ( hAPP_f604201481a_bool @ cOMBB_1882975613iple_a @ ( hAPP_b589554111l_bool @ fimplies @ ( hAPP_bool_bool @ fNot @ C ) ) ) @ ( hAPP_H426895267a_bool @ ( hAPP_f920293029a_bool @ cOMBC_1940896922a_bool @ fequal963300192iple_a ) @ B ) ) ) ) ) )
      & ( ~ ( hBOOL @ C )
       => ( B
          = ( 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 @ C ) ) @ ( hAPP_H426895267a_bool @ ( hAPP_f920293029a_bool @ cOMBC_1940896922a_bool @ fequal963300192iple_a ) @ A ) ) ) ) @ ( hAPP_f20753329a_bool @ ( hAPP_f604201481a_bool @ cOMBB_1882975613iple_a @ ( hAPP_b589554111l_bool @ fimplies @ ( hAPP_bool_bool @ fNot @ C ) ) ) @ ( hAPP_H426895267a_bool @ ( hAPP_f920293029a_bool @ cOMBC_1940896922a_bool @ fequal963300192iple_a ) @ B ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_255_If__def) ).

thf(3065,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( hBOOL @ C )
       => ( A
          = ( 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 @ C ) ) @ ( hAPP_H426895267a_bool @ ( hAPP_f920293029a_bool @ cOMBC_1940896922a_bool @ fequal963300192iple_a ) @ A ) ) ) ) @ ( hAPP_f20753329a_bool @ ( hAPP_f604201481a_bool @ cOMBB_1882975613iple_a @ ( hAPP_b589554111l_bool @ fimplies @ ( hAPP_bool_bool @ fNot @ C ) ) ) @ ( hAPP_H426895267a_bool @ ( hAPP_f920293029a_bool @ cOMBC_1940896922a_bool @ fequal963300192iple_a ) @ B ) ) ) ) ) )
      & ( ~ ( hBOOL @ C )
       => ( B
          = ( 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 @ C ) ) @ ( hAPP_H426895267a_bool @ ( hAPP_f920293029a_bool @ cOMBC_1940896922a_bool @ fequal963300192iple_a ) @ A ) ) ) ) @ ( hAPP_f20753329a_bool @ ( hAPP_f604201481a_bool @ cOMBB_1882975613iple_a @ ( hAPP_b589554111l_bool @ fimplies @ ( hAPP_bool_bool @ fNot @ C ) ) ) @ ( hAPP_H426895267a_bool @ ( hAPP_f920293029a_bool @ cOMBC_1940896922a_bool @ fequal963300192iple_a ) @ B ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[528]) ).

thf(1100,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ B ) @ C ) )
       => ( ( hAPP_f20753329a_bool @ ( hAPP_f1706700729a_bool @ minus_85316870a_bool @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ B ) @ A ) ) @ C )
          = ( hAPP_f20753329a_bool @ ( hAPP_f1706700729a_bool @ minus_85316870a_bool @ A ) @ C ) ) )
      & ( ~ ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ B ) @ C ) )
       => ( ( hAPP_f20753329a_bool @ ( hAPP_f1706700729a_bool @ minus_85316870a_bool @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ B ) @ A ) ) @ C )
          = ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ B ) @ ( hAPP_f20753329a_bool @ ( hAPP_f1706700729a_bool @ minus_85316870a_bool @ A ) @ C ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_404_insert__Diff__if) ).

thf(5177,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ B ) @ C ) )
       => ( ( hAPP_f20753329a_bool @ ( hAPP_f1706700729a_bool @ minus_85316870a_bool @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ B ) @ A ) ) @ C )
          = ( hAPP_f20753329a_bool @ ( hAPP_f1706700729a_bool @ minus_85316870a_bool @ A ) @ C ) ) )
      & ( ~ ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ B ) @ C ) )
       => ( ( hAPP_f20753329a_bool @ ( hAPP_f1706700729a_bool @ minus_85316870a_bool @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ B ) @ A ) ) @ C )
          = ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ B ) @ ( hAPP_f20753329a_bool @ ( hAPP_f1706700729a_bool @ minus_85316870a_bool @ A ) @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1100]) ).

thf(611,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_f284875647l_bool @ ord_le951220754t_bool @ B ) @ C ) )
     => ( hBOOL @ ( hAPP_bool_bool @ ( hAPP_b589554111l_bool @ ord_less_eq_bool @ ( hAPP_int_bool @ B @ A ) ) @ ( hAPP_int_bool @ C @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_738_le__funE) ).

thf(3349,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_f284875647l_bool @ ord_le951220754t_bool @ B ) @ C ) )
     => ( hBOOL @ ( hAPP_bool_bool @ ( hAPP_b589554111l_bool @ ord_less_eq_bool @ ( hAPP_int_bool @ B @ A ) ) @ ( hAPP_int_bool @ C @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[611]) ).

thf(675,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP_int_int @ ( hAPP_int_fun_int_int @ times_times_int @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ times_times_int @ A ) @ B ) ) @ C )
      = ( hAPP_int_int @ ( hAPP_int_fun_int_int @ times_times_int @ A ) @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ times_times_int @ B ) @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_686_comm__semiring__1__class_Onormalizing__semiring__rules_I17_J) ).

thf(3600,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP_int_int @ ( hAPP_int_fun_int_int @ times_times_int @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ times_times_int @ A ) @ B ) ) @ C )
      = ( hAPP_int_int @ ( hAPP_int_fun_int_int @ times_times_int @ A ) @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ times_times_int @ B ) @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[675]) ).

thf(92,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP_nat_nat @ ( hAPP_f966999552at_nat @ ( hAPP_f724467227at_nat @ cOMBB_161916371at_nat @ A ) @ B ) @ C )
      = ( hAPP_f2097660464ol_nat @ A @ ( hAPP_n1919155532a_bool @ B @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',help_COMBB_1_1_COMBB_000tc__fun_Itc__Hoare____Mirabelle____uwgpyvfjxg__Otriple_I) ).

thf(1457,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP_nat_nat @ ( hAPP_f966999552at_nat @ ( hAPP_f724467227at_nat @ cOMBB_161916371at_nat @ A ) @ B ) @ C )
      = ( hAPP_f2097660464ol_nat @ A @ ( hAPP_n1919155532a_bool @ B @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[92]) ).

thf(547,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP_H1037229737a_bool @ A @ B ) )
     => ( ! [C: $i] :
            ( ( hBOOL @ ( hAPP_H1037229737a_bool @ A @ C ) )
           => ( C = B ) )
       => ( ( hAPP_f1693662087iple_a @ the_Ho830102290iple_a @ A )
          = B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_265_the__equality) ).

thf(3144,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP_H1037229737a_bool @ A @ B ) )
     => ( ! [C: $i] :
            ( ( hBOOL @ ( hAPP_H1037229737a_bool @ A @ C ) )
           => ( C = B ) )
       => ( ( hAPP_f1693662087iple_a @ the_Ho830102290iple_a @ A )
          = B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[547]) ).

thf(12,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_int_bool @ ( hAPP_i1948725293t_bool @ ord_less_eq_int @ B ) @ C ) )
     => ( hBOOL @ ( hAPP_int_bool @ ( hAPP_i1948725293t_bool @ ord_less_eq_int @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ ord_min_int @ B ) @ A ) ) @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_979_min__max_Ole__infI1) ).

thf(1159,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_int_bool @ ( hAPP_i1948725293t_bool @ ord_less_eq_int @ B ) @ C ) )
     => ( hBOOL @ ( hAPP_int_bool @ ( hAPP_i1948725293t_bool @ ord_less_eq_int @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ ord_min_int @ B ) @ A ) ) @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[12]) ).

thf(201,axiom,
    ! [A: $i,B: $i] :
      ~ ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ A ) @ B ) ) @ B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_1107_not__add__less2) ).

thf(1836,plain,
    ! [A: $i,B: $i] :
      ~ ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ A ) @ B ) ) @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[201]) ).

thf(1034,axiom,
    ! [A: $i] :
      ( ( image_533441733iple_a @ A @ bot_bo797238721a_bool )
      = bot_bo797238721a_bool ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_206_image__empty) ).

thf(4942,plain,
    ! [A: $i] :
      ( ( image_533441733iple_a @ A @ bot_bo797238721a_bool )
      = bot_bo797238721a_bool ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1034]) ).

thf(390,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ B ) @ C ) )
     => ( ( hAPP_f800510211t_bool @ ( hAPP_f1730770594t_bool @ minus_2067140911t_bool @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ B ) @ A ) ) @ C )
        = ( hAPP_f800510211t_bool @ ( hAPP_f1730770594t_bool @ minus_2067140911t_bool @ A ) @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_406_insert__Diff1) ).

thf(2547,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ B ) @ C ) )
     => ( ( hAPP_f800510211t_bool @ ( hAPP_f1730770594t_bool @ minus_2067140911t_bool @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ B ) @ A ) ) @ C )
        = ( hAPP_f800510211t_bool @ ( hAPP_f1730770594t_bool @ minus_2067140911t_bool @ A ) @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[390]) ).

thf(376,axiom,
    ! [A: $i] :
      ( ( A != bot_bot_fun_int_bool )
    <=> ? [B: $i,C: $i] :
          ( ( A
            = ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ B ) @ C ) )
          & ~ ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ B ) @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_126_nonempty__iff) ).

thf(2492,plain,
    ! [A: $i] :
      ( ( ( A != bot_bot_fun_int_bool )
       => ? [B: $i,C: $i] :
            ( ( A
              = ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ B ) @ C ) )
            & ~ ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ B ) @ C ) ) ) )
      & ( ? [B: $i,C: $i] :
            ( ( A
              = ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ B ) @ C ) )
            & ~ ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ B ) @ C ) ) )
       => ( A != bot_bot_fun_int_bool ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[376]) ).

thf(520,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP_f54304608l_bool @ finite_finite_nat @ B ) )
     => ( ( B != bot_bot_fun_nat_bool )
       => ? [C: $i] : ( hBOOL @ ( hAPP_nat_bool @ ( finite_fold1Set_nat @ A @ B ) @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_312_finite__nonempty__imp__fold1Set) ).

thf(3033,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP_f54304608l_bool @ finite_finite_nat @ B ) )
     => ( ( B != bot_bot_fun_nat_bool )
       => ? [C: $i] : ( hBOOL @ ( hAPP_nat_bool @ ( finite_fold1Set_nat @ A @ B ) @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[520]) ).

thf(625,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ A ) @ ( image_533441733iple_a @ B @ C ) ) )
     => ~ ! [D: $i] :
            ( ( A
              = ( hAPP_H905846293iple_a @ B @ D ) )
           => ~ ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ D ) @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_240_imageE) ).

thf(3408,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ A ) @ ( image_533441733iple_a @ B @ C ) ) )
     => ~ ! [D: $i] :
            ( ( A
              = ( hAPP_H905846293iple_a @ B @ D ) )
           => ~ ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ D ) @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[625]) ).

thf(714,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP_int_int @ ( hAPP_int_fun_int_int @ times_times_int @ A ) @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ B ) @ C ) )
      = ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ times_times_int @ A ) @ B ) ) @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ times_times_int @ A ) @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_697_comm__semiring__1__class_Onormalizing__semiring__rules_I34_J) ).

thf(3731,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP_int_int @ ( hAPP_int_fun_int_int @ times_times_int @ A ) @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ B ) @ C ) )
      = ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ times_times_int @ A ) @ B ) ) @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ times_times_int @ A ) @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[714]) ).

thf(728,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_f103356543l_bool @ ord_le1568362934t_bool @ B ) @ C ) )
     => ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_f103356543l_bool @ ord_le1568362934t_bool @ C ) @ A ) )
       => ( ( hAPP_f800510211t_bool @ ( hAPP_f1730770594t_bool @ minus_2067140911t_bool @ C ) @ ( hAPP_f800510211t_bool @ ( hAPP_f1730770594t_bool @ minus_2067140911t_bool @ A ) @ B ) )
          = B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_882_double__diff) ).

thf(3782,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_f103356543l_bool @ ord_le1568362934t_bool @ B ) @ C ) )
     => ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_f103356543l_bool @ ord_le1568362934t_bool @ C ) @ A ) )
       => ( ( hAPP_f800510211t_bool @ ( hAPP_f1730770594t_bool @ minus_2067140911t_bool @ C ) @ ( hAPP_f800510211t_bool @ ( hAPP_f1730770594t_bool @ minus_2067140911t_bool @ A ) @ B ) )
          = B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[728]) ).

thf(1038,axiom,
    ! [A: $i,B: $i] :
      ( ~ ( hBOOL @ ( hAPP_state_bool @ ( hAPP_s1806633685e_bool @ fequal_state @ A ) @ B ) )
      | ( A = B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',help_fequal_1_1_fequal_000tc__Com__Ostate_T) ).

thf(4957,plain,
    ! [A: $i,B: $i] :
      ( ~ ( hBOOL @ ( hAPP_state_bool @ ( hAPP_s1806633685e_bool @ fequal_state @ A ) @ B ) )
      | ( A = B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1038]) ).

thf(643,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( B = C )
     => ( ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_f284875647l_bool @ ord_le951220754t_bool @ C ) @ A ) )
       => ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_f284875647l_bool @ ord_le951220754t_bool @ B ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_768_ord__eq__le__trans) ).

thf(3482,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( B = C )
     => ( ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_f284875647l_bool @ ord_le951220754t_bool @ C ) @ A ) )
       => ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_f284875647l_bool @ ord_le951220754t_bool @ B ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[643]) ).

thf(343,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ A ) @ ( hAPP_f1805168059t_bool @ ( hAPP_f1223193598t_bool @ minus_1449998731t_bool @ B ) @ C ) ) )
    <=> ( ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ A ) @ B ) )
        & ~ ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ A ) @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_381_Diff__iff) ).

thf(2377,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ A ) @ ( hAPP_f1805168059t_bool @ ( hAPP_f1223193598t_bool @ minus_1449998731t_bool @ B ) @ C ) ) )
       => ( ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ A ) @ B ) )
          & ~ ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ A ) @ C ) ) ) )
      & ( ( ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ A ) @ B ) )
          & ~ ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ A ) @ C ) ) )
       => ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ A ) @ ( hAPP_f1805168059t_bool @ ( hAPP_f1223193598t_bool @ minus_1449998731t_bool @ B ) @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[343]) ).

thf(1044,axiom,
    ! [A: $i,B: $i] :
      ( ( 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 ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_854_rev__finite__subset) ).

thf(4974,plain,
    ! [A: $i,B: $i] :
      ( ( 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 ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1044]) ).

thf(484,axiom,
    ! [A: $i,B: $i] :
      ( ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ minus_minus_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ A ) @ B ) ) @ B )
      = A ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_1016_diff__add__inverse2) ).

thf(2895,plain,
    ! [A: $i,B: $i] :
      ( ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ minus_minus_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ A ) @ B ) ) @ B )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[484]) ).

thf(469,axiom,
    ! [A: $i] :
      ( ( hBOOL @ ( hAPP_f54304608l_bool @ finite_finite_nat @ A ) )
    <=> ? [B: $i] :
        ! [C: $i] :
          ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ C ) @ A ) )
         => ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_nat @ C ) @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_1127_finite__nat__set__iff__bounded) ).

thf(2841,plain,
    ! [A: $i] :
      ( ( ( hBOOL @ ( hAPP_f54304608l_bool @ finite_finite_nat @ A ) )
       => ? [B: $i] :
          ! [C: $i] :
            ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ C ) @ A ) )
           => ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_nat @ C ) @ B ) ) ) )
      & ( ? [B: $i] :
          ! [C: $i] :
            ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ C ) @ A ) )
           => ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_nat @ C ) @ B ) ) )
       => ( hBOOL @ ( hAPP_f54304608l_bool @ finite_finite_nat @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[469]) ).

thf(231,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( B
        = ( hAPP_i240634960iple_a @ C @ D ) )
     => ( ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ D ) @ A ) )
       => ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ B ) @ ( image_1844722432iple_a @ C @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_157_image__eqI) ).

thf(1970,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( B
        = ( hAPP_i240634960iple_a @ C @ D ) )
     => ( ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ D ) @ A ) )
       => ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ B ) @ ( image_1844722432iple_a @ C @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[231]) ).

thf(918,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( finite1973466193nt_int @ D ) )
     => ( ( hAPP_int_int @ ( hAPP_int_fun_int_int @ D @ A ) @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ D @ B ) @ C ) )
        = ( hAPP_int_int @ ( hAPP_int_fun_int_int @ D @ B ) @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ D @ A ) @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_487_comp__fun__commute_Ofun__left__comm) ).

thf(4525,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( finite1973466193nt_int @ D ) )
     => ( ( hAPP_int_int @ ( hAPP_int_fun_int_int @ D @ A ) @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ D @ B ) @ C ) )
        = ( hAPP_int_int @ ( hAPP_int_fun_int_int @ D @ B ) @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ D @ A ) @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[918]) ).

thf(388,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ B ) @ C ) )
     => ( ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ minus_minus_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ A ) @ C ) ) @ B )
        = ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ A ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ minus_minus_nat @ C ) @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_1035_diff__add__assoc) ).

thf(2537,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ B ) @ C ) )
     => ( ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ minus_minus_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ A ) @ C ) ) @ B )
        = ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ A ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ minus_minus_nat @ C ) @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[388]) ).

thf(793,axiom,
    ! [A: $i,B: $i] :
      ( ~ ( hBOOL @ ( hAPP_f1695230391l_bool @ finite1738664244iple_a @ B ) )
     => ( ( hBOOL @ ( hAPP_f448129468l_bool @ finite_finite_int @ ( image_685909450_a_int @ A @ B ) ) )
       => ? [C: $i] :
            ( ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ C ) @ B ) )
            & ~ ( 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 ) @ B ) ) ) @ ( hAPP_i611776424a_bool @ ( hAPP_f920351535a_bool @ cOMBC_1530706207t_bool @ ( hAPP_f2142913731t_bool @ ( hAPP_f1329415375t_bool @ cOMBB_1505691757iple_a @ fequal_int ) @ A ) ) @ ( hAPP_H1229305626_a_int @ A @ C ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_323_pigeonhole__infinite) ).

thf(4012,plain,
    ! [A: $i,B: $i] :
      ( ~ ( hBOOL @ ( hAPP_f1695230391l_bool @ finite1738664244iple_a @ B ) )
     => ( ( hBOOL @ ( hAPP_f448129468l_bool @ finite_finite_int @ ( image_685909450_a_int @ A @ B ) ) )
       => ? [C: $i] :
            ( ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ C ) @ B ) )
            & ~ ( 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 ) @ B ) ) ) @ ( hAPP_i611776424a_bool @ ( hAPP_f920351535a_bool @ cOMBC_1530706207t_bool @ ( hAPP_f2142913731t_bool @ ( hAPP_f1329415375t_bool @ cOMBB_1505691757iple_a @ fequal_int ) @ A ) ) @ ( hAPP_H1229305626_a_int @ A @ C ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[793]) ).

thf(1115,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( finite1428139793iple_a @ D ) )
     => ( ( hBOOL @ ( hAPP_f1695230391l_bool @ finite1738664244iple_a @ C ) )
       => ( ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ B ) @ C ) )
         => ( ( hAPP_f1693662087iple_a @ ( finite1205970312iple_a @ D @ A ) @ C )
            = ( hAPP_H905846293iple_a @ ( hAPP_H2085992369iple_a @ D @ B ) @ ( hAPP_f1693662087iple_a @ ( finite1205970312iple_a @ D @ A ) @ ( hAPP_f20753329a_bool @ ( hAPP_f1706700729a_bool @ minus_85316870a_bool @ C ) @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ B ) @ bot_bo797238721a_bool ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_542_comp__fun__commute_Ofold__rec) ).

thf(5233,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( finite1428139793iple_a @ D ) )
     => ( ( hBOOL @ ( hAPP_f1695230391l_bool @ finite1738664244iple_a @ C ) )
       => ( ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ B ) @ C ) )
         => ( ( hAPP_f1693662087iple_a @ ( finite1205970312iple_a @ D @ A ) @ C )
            = ( hAPP_H905846293iple_a @ ( hAPP_H2085992369iple_a @ D @ B ) @ ( hAPP_f1693662087iple_a @ ( finite1205970312iple_a @ D @ A ) @ ( hAPP_f20753329a_bool @ ( hAPP_f1706700729a_bool @ minus_85316870a_bool @ C ) @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ B ) @ bot_bo797238721a_bool ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1115]) ).

thf(566,axiom,
    ! [A: $i,B: $i] :
      ( ( ass @ A @ B )
     != skip ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_201_com_Osimps_I9_J) ).

thf(3205,plain,
    ! [A: $i,B: $i] :
      ( ( ass @ A @ B )
     != skip ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[566]) ).

thf(999,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( finite2114894759a_bool @ D ) )
     => ( ( hBOOL @ ( hAPP_f1695230391l_bool @ finite1738664244iple_a @ C ) )
       => ( ~ ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ B ) @ C ) )
         => ( ( hAPP_f20753329a_bool @ ( finite1215486064a_bool @ D @ A ) @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ B ) @ C ) )
            = ( hAPP_f20753329a_bool @ ( finite1215486064a_bool @ D @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ D @ B ) @ A ) ) @ C ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_508_comp__fun__commute_Ofold__insert2) ).

thf(4808,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( finite2114894759a_bool @ D ) )
     => ( ( hBOOL @ ( hAPP_f1695230391l_bool @ finite1738664244iple_a @ C ) )
       => ( ~ ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ B ) @ C ) )
         => ( ( hAPP_f20753329a_bool @ ( finite1215486064a_bool @ D @ A ) @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ B ) @ C ) )
            = ( hAPP_f20753329a_bool @ ( finite1215486064a_bool @ D @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ D @ B ) @ A ) ) @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[999]) ).

thf(1130,axiom,
    ! [A: $i,B: $i] :
      ( ( ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_f1767618879l_bool @ ord_le1523781517a_bool @ A ) @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ B ) @ bot_bo797238721a_bool ) ) )
       => ( ( hAPP_f1693662087iple_a @ ( partia1949573335iple_a @ B ) @ A )
          = B ) )
      & ( ~ ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_f1767618879l_bool @ ord_le1523781517a_bool @ A ) @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ B ) @ bot_bo797238721a_bool ) ) )
       => ( ( hAPP_f1693662087iple_a @ ( partia1949573335iple_a @ B ) @ 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 @ B ) @ bot_bo797238721a_bool ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_933_flat__lub__def) ).

thf(5313,plain,
    ! [A: $i,B: $i] :
      ( ( ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_f1767618879l_bool @ ord_le1523781517a_bool @ A ) @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ B ) @ bot_bo797238721a_bool ) ) )
       => ( ( hAPP_f1693662087iple_a @ ( partia1949573335iple_a @ B ) @ A )
          = B ) )
      & ( ~ ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_f1767618879l_bool @ ord_le1523781517a_bool @ A ) @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ B ) @ bot_bo797238721a_bool ) ) )
       => ( ( hAPP_f1693662087iple_a @ ( partia1949573335iple_a @ B ) @ 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 @ B ) @ bot_bo797238721a_bool ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1130]) ).

thf(705,axiom,
    ! [A: $i,B: $i] :
      ( ( 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 ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_722_equalityI) ).

thf(3699,plain,
    ! [A: $i,B: $i] :
      ( ( 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 ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[705]) ).

thf(323,axiom,
    ! [A: $i] : ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ A ) @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_721_order__refl) ).

thf(2300,plain,
    ! [A: $i] : ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ A ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[323]) ).

thf(44,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP_int_bool @ ( hAPP_i1948725293t_bool @ ord_less_eq_int @ A ) @ B ) )
    <=> ( ( hAPP_int_int @ ( hAPP_int_fun_int_int @ ord_min_int @ A ) @ B )
        = A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_983_min__max_Ole__iff__inf) ).

thf(1270,plain,
    ! [A: $i,B: $i] :
      ( ( ( hBOOL @ ( hAPP_int_bool @ ( hAPP_i1948725293t_bool @ ord_less_eq_int @ A ) @ B ) )
       => ( ( hAPP_int_int @ ( hAPP_int_fun_int_int @ ord_min_int @ A ) @ B )
          = A ) )
      & ( ( ( hAPP_int_int @ ( hAPP_int_fun_int_int @ ord_min_int @ A ) @ B )
          = A )
       => ( hBOOL @ ( hAPP_int_bool @ ( hAPP_i1948725293t_bool @ ord_less_eq_int @ A ) @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[44]) ).

thf(182,axiom,
    ! [A: $i] :
      ( ( hAPP_f1805168059t_bool @ ( hAPP_f1223193598t_bool @ minus_1449998731t_bool @ A ) @ A )
      = bot_bot_fun_int_bool ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_398_Diff__cancel) ).

thf(1772,plain,
    ! [A: $i] :
      ( ( hAPP_f1805168059t_bool @ ( hAPP_f1223193598t_bool @ minus_1449998731t_bool @ A ) @ A )
      = bot_bot_fun_int_bool ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[182]) ).

thf(41,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP_int_bool @ ( hAPP_i1948725293t_bool @ ord_less_eq_int @ A ) @ B ) )
      | ( hBOOL @ ( hAPP_int_bool @ ( hAPP_i1948725293t_bool @ ord_less_eq_int @ B ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_1083_zle__linear) ).

thf(1259,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP_int_bool @ ( hAPP_i1948725293t_bool @ ord_less_eq_int @ A ) @ B ) )
      | ( hBOOL @ ( hAPP_int_bool @ ( hAPP_i1948725293t_bool @ ord_less_eq_int @ B ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[41]) ).

thf(110,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ ord_min_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ ord_min_nat @ A ) @ B ) ) @ C )
      = ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ ord_min_nat @ A ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ ord_min_nat @ B ) @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_954_min__max_Oinf_Oassoc) ).

thf(1534,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ ord_min_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ ord_min_nat @ A ) @ B ) ) @ C )
      = ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ ord_min_nat @ A ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ ord_min_nat @ B ) @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[110]) ).

thf(190,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ A ) @ B )
        = ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ C ) @ B ) )
     => ( A = C ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_653_add__right__imp__eq) ).

thf(1794,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ A ) @ B )
        = ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ C ) @ B ) )
     => ( A = C ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[190]) ).

thf(914,axiom,
    ! [A: $i] : ( is_bool @ ( finite1428139793iple_a @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',gsy_c_Finite__Set_Ocomp__fun__commute_000tc__Hoare____Mirabelle____uwgpyvfjxg__O) ).

thf(4512,plain,
    ! [A: $i] : ( is_bool @ ( finite1428139793iple_a @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[914]) ).

thf(426,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ A ) @ B ) ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ C ) @ B ) ) )
    <=> ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ A ) @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_831_add__le__cancel__right) ).

thf(2685,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ A ) @ B ) ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ C ) @ B ) ) )
       => ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ A ) @ C ) ) )
      & ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ A ) @ C ) )
       => ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ A ) @ B ) ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ C ) @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[426]) ).

thf(992,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP_H1037229737a_bool @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ ( finite90892550iple_a @ A ) @ B ) @ C ) @ D ) )
     => ( ~ ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ B ) @ C ) )
       => ( hBOOL @ ( hAPP_H1037229737a_bool @ ( finite1621280049iple_a @ A @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ B ) @ C ) ) @ D ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_262_fold1Set_Ointros) ).

thf(4789,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP_H1037229737a_bool @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ ( finite90892550iple_a @ A ) @ B ) @ C ) @ D ) )
     => ( ~ ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ B ) @ C ) )
       => ( hBOOL @ ( hAPP_H1037229737a_bool @ ( finite1621280049iple_a @ A @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ B ) @ C ) ) @ D ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[992]) ).

thf(803,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ B )
      | ( hBOOL @ ( hAPP_bool_bool @ ( hAPP_b589554111l_bool @ fimplies @ B ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',help_fimplies_1_1_U) ).

thf(4048,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ B )
      | ( hBOOL @ ( hAPP_bool_bool @ ( hAPP_b589554111l_bool @ fimplies @ B ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[803]) ).

thf(112,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP_nat_bool @ A @ B ) )
     => ( ! [C: $i] :
            ( ( hBOOL @ ( hAPP_nat_bool @ A @ C ) )
           => ( C = B ) )
       => ( ( hAPP_f22106695ol_nat @ the_nat @ A )
          = B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_266_the__equality) ).

thf(1540,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP_nat_bool @ A @ B ) )
     => ( ! [C: $i] :
            ( ( hBOOL @ ( hAPP_nat_bool @ A @ C ) )
           => ( C = B ) )
       => ( ( hAPP_f22106695ol_nat @ the_nat @ A )
          = B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[112]) ).

thf(783,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_f103356543l_bool @ ord_le1568362934t_bool @ B ) @ C ) )
     => ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_f103356543l_bool @ ord_le1568362934t_bool @ B ) @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ A ) @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_860_subset__insertI2) ).

thf(3963,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_f103356543l_bool @ ord_le1568362934t_bool @ B ) @ C ) )
     => ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_f103356543l_bool @ ord_le1568362934t_bool @ B ) @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ A ) @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[783]) ).

thf(770,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_f284875647l_bool @ ord_le951220754t_bool @ B ) @ C ) )
     => ( hBOOL @ ( hAPP_bool_bool @ ( hAPP_b589554111l_bool @ ord_less_eq_bool @ ( hAPP_int_bool @ B @ A ) ) @ ( hAPP_int_bool @ C @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_776_le__funD) ).

thf(3916,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_f284875647l_bool @ ord_le951220754t_bool @ B ) @ C ) )
     => ( hBOOL @ ( hAPP_bool_bool @ ( hAPP_b589554111l_bool @ ord_less_eq_bool @ ( hAPP_int_bool @ B @ A ) ) @ ( hAPP_int_bool @ C @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[770]) ).

thf(575,axiom,
    ! [A: $i] : ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_f103356543l_bool @ ord_le1568362934t_bool @ bot_bot_fun_nat_bool ) @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_843_bot__least) ).

thf(3237,plain,
    ! [A: $i] : ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_f103356543l_bool @ ord_le1568362934t_bool @ bot_bot_fun_nat_bool ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[575]) ).

thf(779,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( finite988810631ne_nat @ B @ C ) )
     => ( ( hAPP_f22106695ol_nat @ C @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ A ) @ bot_bot_fun_nat_bool ) )
        = A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_136_folding__one_Osingleton) ).

thf(3941,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( finite988810631ne_nat @ B @ C ) )
     => ( ( hAPP_f22106695ol_nat @ C @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ A ) @ bot_bot_fun_nat_bool ) )
        = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[779]) ).

thf(535,axiom,
    ! [A: $i] :
      ( ( hAPP_f800510211t_bool @ collect_nat @ ( hAPP_n1699378549t_bool @ fequal_nat @ A ) )
      = ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ A ) @ bot_bot_fun_nat_bool ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_18_singleton__conv2) ).

thf(3087,plain,
    ! [A: $i] :
      ( ( hAPP_f800510211t_bool @ collect_nat @ ( hAPP_n1699378549t_bool @ fequal_nat @ A ) )
      = ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ A ) @ bot_bot_fun_nat_bool ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[535]) ).

thf(855,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP_f448129468l_bool @ finite_finite_int @ B ) )
     => ( hBOOL @ ( hAPP_f1695230391l_bool @ finite1738664244iple_a @ ( image_1844722432iple_a @ A @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_287_finite__imageI) ).

thf(4263,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP_f448129468l_bool @ finite_finite_int @ B ) )
     => ( hBOOL @ ( hAPP_f1695230391l_bool @ finite1738664244iple_a @ ( image_1844722432iple_a @ A @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[855]) ).

thf(367,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ C ) @ D ) )
     => ( ( A
          = ( hAPP_i240634960iple_a @ B @ C ) )
       => ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ A ) @ ( image_1844722432iple_a @ B @ D ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_169_rev__image__eqI) ).

thf(2460,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ C ) @ D ) )
     => ( ( A
          = ( hAPP_i240634960iple_a @ B @ C ) )
       => ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ A ) @ ( image_1844722432iple_a @ B @ D ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[367]) ).

thf(64,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP_f448129468l_bool @ finite_finite_int @ B ) )
     => ( ~ ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ A ) @ B ) )
       => ( ( hAPP_f1594865479ol_int @ ( finite_fold1_int @ times_times_int ) @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ A ) @ B ) )
          = ( hAPP_f1594865479ol_int @ ( finite_fold_int_int @ times_times_int @ A ) @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_588_fold1__eq__fold) ).

thf(1361,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP_f448129468l_bool @ finite_finite_int @ B ) )
     => ( ~ ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ A ) @ B ) )
       => ( ( hAPP_f1594865479ol_int @ ( finite_fold1_int @ times_times_int ) @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ A ) @ B ) )
          = ( hAPP_f1594865479ol_int @ ( finite_fold_int_int @ times_times_int @ A ) @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[64]) ).

thf(1047,axiom,
    ! [A: $i] :
      ( ( 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 ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_730_finite__Collect__subsets) ).

thf(4983,plain,
    ! [A: $i] :
      ( ( 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 ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1047]) ).

thf(395,axiom,
    ! [A: $i] :
      ( ? [B: $i] : ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ B ) @ A ) )
    <=> ( A != bot_bot_fun_int_bool ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_44_ex__in__conv) ).

thf(2561,plain,
    ! [A: $i] :
      ( ( ? [B: $i] : ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ B ) @ A ) )
       => ( A != bot_bot_fun_int_bool ) )
      & ( ( A != bot_bot_fun_int_bool )
       => ? [B: $i] : ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ B ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[395]) ).

thf(475,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( finite1071749497t_bool @ D ) )
     => ( ( hBOOL @ ( hAPP_f54304608l_bool @ finite_finite_nat @ C ) )
       => ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ B ) @ C ) )
         => ( ( hAPP_f800510211t_bool @ ( finite1578363458t_bool @ D @ A ) @ C )
            = ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ D @ B ) @ ( hAPP_f800510211t_bool @ ( finite1578363458t_bool @ D @ A ) @ ( hAPP_f800510211t_bool @ ( hAPP_f1730770594t_bool @ minus_2067140911t_bool @ C ) @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ B ) @ bot_bot_fun_nat_bool ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_541_comp__fun__commute_Ofold__rec) ).

thf(2868,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( finite1071749497t_bool @ D ) )
     => ( ( hBOOL @ ( hAPP_f54304608l_bool @ finite_finite_nat @ C ) )
       => ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ B ) @ C ) )
         => ( ( hAPP_f800510211t_bool @ ( finite1578363458t_bool @ D @ A ) @ C )
            = ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ D @ B ) @ ( hAPP_f800510211t_bool @ ( finite1578363458t_bool @ D @ A ) @ ( hAPP_f800510211t_bool @ ( hAPP_f1730770594t_bool @ minus_2067140911t_bool @ C ) @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ B ) @ bot_bot_fun_nat_bool ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[475]) ).

thf(265,axiom,
    ( one_one_int
    = ( hAPP_nat_int @ semiri1621563631at_int @ one_one_nat ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_1166_transfer__int__nat__numerals_I2_J) ).

thf(2080,plain,
    ( one_one_int
    = ( hAPP_nat_int @ semiri1621563631at_int @ one_one_nat ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[265]) ).

thf(977,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ~ ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ B ) @ C ) )
       => ( B = A ) )
     => ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ B ) @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ A ) @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_13_insertCI) ).

thf(4744,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ~ ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ B ) @ C ) )
       => ( B = A ) )
     => ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ B ) @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ A ) @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[977]) ).

thf(334,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( finite988810631ne_nat @ B @ C ) )
     => ( ( hBOOL @ ( hAPP_f54304608l_bool @ finite_finite_nat @ A ) )
       => ( ( hAPP_f22106695ol_nat @ C @ A )
          = ( hAPP_f22106695ol_nat @ ( finite_fold1_nat @ B ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_271_folding__one_Oeq__fold) ).

thf(2349,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( finite988810631ne_nat @ B @ C ) )
     => ( ( hBOOL @ ( hAPP_f54304608l_bool @ finite_finite_nat @ A ) )
       => ( ( hAPP_f22106695ol_nat @ C @ A )
          = ( hAPP_f22106695ol_nat @ ( finite_fold1_nat @ B ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[334]) ).

thf(27,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_int_bool @ ( hAPP_i1948725293t_bool @ ord_less_eq_int @ B ) @ C ) )
     => ( ( hBOOL @ ( hAPP_int_bool @ ( hAPP_i1948725293t_bool @ ord_less_eq_int @ B ) @ A ) )
       => ( hBOOL @ ( hAPP_int_bool @ ( hAPP_i1948725293t_bool @ ord_less_eq_int @ B ) @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ ord_min_int @ C ) @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_969_min__max_Oinf__greatest) ).

thf(1207,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_int_bool @ ( hAPP_i1948725293t_bool @ ord_less_eq_int @ B ) @ C ) )
     => ( ( hBOOL @ ( hAPP_int_bool @ ( hAPP_i1948725293t_bool @ ord_less_eq_int @ B ) @ A ) )
       => ( hBOOL @ ( hAPP_int_bool @ ( hAPP_i1948725293t_bool @ ord_less_eq_int @ B ) @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ ord_min_int @ C ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[27]) ).

thf(661,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f1695230391l_bool @ finite1738664244iple_a @ C ) )
     => ( ( ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ B ) @ C ) )
         => ( ( hAPP_f587450508ol_int @ ( hAPP_f1544320301ol_int @ big_co1977834938_a_int @ A ) @ ( hAPP_f20753329a_bool @ ( hAPP_f1706700729a_bool @ minus_85316870a_bool @ C ) @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ B ) @ bot_bo797238721a_bool ) ) )
            = ( hAPP_int_int @ ( hAPP_int_fun_int_int @ minus_minus_int @ ( hAPP_f587450508ol_int @ ( hAPP_f1544320301ol_int @ big_co1977834938_a_int @ A ) @ C ) ) @ ( hAPP_H1229305626_a_int @ A @ B ) ) ) )
        & ( ~ ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ B ) @ C ) )
         => ( ( hAPP_f587450508ol_int @ ( hAPP_f1544320301ol_int @ big_co1977834938_a_int @ A ) @ ( hAPP_f20753329a_bool @ ( hAPP_f1706700729a_bool @ minus_85316870a_bool @ C ) @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ B ) @ bot_bo797238721a_bool ) ) )
            = ( hAPP_f587450508ol_int @ ( hAPP_f1544320301ol_int @ big_co1977834938_a_int @ A ) @ C ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_461_setsum__diff1) ).

thf(3555,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f1695230391l_bool @ finite1738664244iple_a @ C ) )
     => ( ( ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ B ) @ C ) )
         => ( ( hAPP_f587450508ol_int @ ( hAPP_f1544320301ol_int @ big_co1977834938_a_int @ A ) @ ( hAPP_f20753329a_bool @ ( hAPP_f1706700729a_bool @ minus_85316870a_bool @ C ) @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ B ) @ bot_bo797238721a_bool ) ) )
            = ( hAPP_int_int @ ( hAPP_int_fun_int_int @ minus_minus_int @ ( hAPP_f587450508ol_int @ ( hAPP_f1544320301ol_int @ big_co1977834938_a_int @ A ) @ C ) ) @ ( hAPP_H1229305626_a_int @ A @ B ) ) ) )
        & ( ~ ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ B ) @ C ) )
         => ( ( hAPP_f587450508ol_int @ ( hAPP_f1544320301ol_int @ big_co1977834938_a_int @ A ) @ ( hAPP_f20753329a_bool @ ( hAPP_f1706700729a_bool @ minus_85316870a_bool @ C ) @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ B ) @ bot_bo797238721a_bool ) ) )
            = ( hAPP_f587450508ol_int @ ( hAPP_f1544320301ol_int @ big_co1977834938_a_int @ A ) @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[661]) ).

thf(726,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ A ) @ B ) ) @ C )
      = ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ A ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ B ) @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_687_ab__semigroup__mult__class_Omult__ac_I1_J) ).

thf(3775,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ A ) @ B ) ) @ C )
      = ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ A ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ B ) @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[726]) ).

thf(1036,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( finite2114894759a_bool @ D ) )
     => ( ( hBOOL @ ( hAPP_f1695230391l_bool @ finite1738664244iple_a @ C ) )
       => ( ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ B ) @ C ) )
         => ( ( hAPP_f20753329a_bool @ ( finite1215486064a_bool @ D @ A ) @ C )
            = ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ D @ B ) @ ( hAPP_f20753329a_bool @ ( finite1215486064a_bool @ D @ A ) @ ( hAPP_f20753329a_bool @ ( hAPP_f1706700729a_bool @ minus_85316870a_bool @ C ) @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ B ) @ bot_bo797238721a_bool ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_543_comp__fun__commute_Ofold__rec) ).

thf(4951,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( finite2114894759a_bool @ D ) )
     => ( ( hBOOL @ ( hAPP_f1695230391l_bool @ finite1738664244iple_a @ C ) )
       => ( ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ B ) @ C ) )
         => ( ( hAPP_f20753329a_bool @ ( finite1215486064a_bool @ D @ A ) @ C )
            = ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ D @ B ) @ ( hAPP_f20753329a_bool @ ( finite1215486064a_bool @ D @ A ) @ ( hAPP_f20753329a_bool @ ( hAPP_f1706700729a_bool @ minus_85316870a_bool @ C ) @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ B ) @ bot_bo797238721a_bool ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1036]) ).

thf(246,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ~ ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ B ) @ C ) )
     => ( ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_f284875647l_bool @ ord_le951220754t_bool @ C ) @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ B ) @ A ) ) )
      <=> ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_f284875647l_bool @ ord_le951220754t_bool @ C ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_864_subset__insert) ).

thf(2023,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ~ ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ B ) @ C ) )
     => ( ( ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_f284875647l_bool @ ord_le951220754t_bool @ C ) @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ B ) @ A ) ) )
         => ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_f284875647l_bool @ ord_le951220754t_bool @ C ) @ A ) ) )
        & ( ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_f284875647l_bool @ ord_le951220754t_bool @ C ) @ A ) )
         => ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_f284875647l_bool @ ord_le951220754t_bool @ C ) @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ B ) @ A ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[246]) ).

thf(604,axiom,
    ! [A: $i] :
      ( ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ A ) @ one_one_nat )
      = A ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_1061_nat__mult__1__right) ).

thf(3326,plain,
    ! [A: $i] :
      ( ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ A ) @ one_one_nat )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[604]) ).

thf(1076,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( finite648855052iple_a @ D ) )
     => ( ( hBOOL @ ( hAPP_f1695230391l_bool @ finite1738664244iple_a @ C ) )
       => ( ( hAPP_f1693662087iple_a @ ( finite1205970312iple_a @ D @ A ) @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ B ) @ C ) )
          = ( hAPP_H905846293iple_a @ ( hAPP_H2085992369iple_a @ D @ B ) @ ( hAPP_f1693662087iple_a @ ( finite1205970312iple_a @ D @ A ) @ C ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_577_comp__fun__idem_Ofold__insert__idem) ).

thf(5080,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( finite648855052iple_a @ D ) )
     => ( ( hBOOL @ ( hAPP_f1695230391l_bool @ finite1738664244iple_a @ C ) )
       => ( ( hAPP_f1693662087iple_a @ ( finite1205970312iple_a @ D @ A ) @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ B ) @ C ) )
          = ( hAPP_H905846293iple_a @ ( hAPP_H2085992369iple_a @ D @ B ) @ ( hAPP_f1693662087iple_a @ ( finite1205970312iple_a @ D @ A ) @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1076]) ).

thf(49,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP_int_bool @ ( hAPP_i1948725293t_bool @ ord_less_eq_int @ A ) @ B ) )
     => ( ( hBOOL @ ( hAPP_int_bool @ ( hAPP_i1948725293t_bool @ ord_less_eq_int @ B ) @ A ) )
      <=> ( B = A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_774_order__antisym__conv) ).

thf(1292,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP_int_bool @ ( hAPP_i1948725293t_bool @ ord_less_eq_int @ A ) @ B ) )
     => ( ( ( hBOOL @ ( hAPP_int_bool @ ( hAPP_i1948725293t_bool @ ord_less_eq_int @ B ) @ A ) )
         => ( B = A ) )
        & ( ( B = A )
         => ( hBOOL @ ( hAPP_int_bool @ ( hAPP_i1948725293t_bool @ ord_less_eq_int @ B ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[49]) ).

thf(537,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP_a1224971408e_bool @ ( hAPP_f1406200875e_bool @ ( hAPP_f635443597e_bool @ cOMBB_2139825703bool_a @ A ) @ B ) @ C )
      = ( hAPP_f162060345e_bool @ A @ ( hAPP_a2036067514e_bool @ B @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',help_COMBB_1_1_COMBB_000tc__fun_Itc__Com__Ostate_Mtc__HOL__Obool_J_000tc__fun_It_037) ).

thf(3094,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP_a1224971408e_bool @ ( hAPP_f1406200875e_bool @ ( hAPP_f635443597e_bool @ cOMBB_2139825703bool_a @ A ) @ B ) @ C )
      = ( hAPP_f162060345e_bool @ A @ ( hAPP_a2036067514e_bool @ B @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[537]) ).

thf(968,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( big_se1603268663iple_a @ B @ C ) )
     => ( ( hBOOL @ ( hAPP_f1695230391l_bool @ finite1738664244iple_a @ A ) )
       => ( ( hAPP_f1693662087iple_a @ C @ A )
          = ( hAPP_f1693662087iple_a @ ( finite1465856449iple_a @ B ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_353_semilattice__big_OF__eq) ).

thf(4721,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( big_se1603268663iple_a @ B @ C ) )
     => ( ( hBOOL @ ( hAPP_f1695230391l_bool @ finite1738664244iple_a @ A ) )
       => ( ( hAPP_f1693662087iple_a @ C @ A )
          = ( hAPP_f1693662087iple_a @ ( finite1465856449iple_a @ B ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[968]) ).

thf(172,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f54304608l_bool @ finite_finite_nat @ C ) )
     => ( ~ ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ B ) @ C ) )
       => ( ( hAPP_f22106695ol_nat @ ( hAPP_f782000547ol_nat @ big_co387207925at_nat @ A ) @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ B ) @ C ) )
          = ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ ( hAPP_nat_nat @ A @ B ) ) @ ( hAPP_f22106695ol_nat @ ( hAPP_f782000547ol_nat @ big_co387207925at_nat @ A ) @ C ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_633_setsum__insert) ).

thf(1739,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f54304608l_bool @ finite_finite_nat @ C ) )
     => ( ~ ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ B ) @ C ) )
       => ( ( hAPP_f22106695ol_nat @ ( hAPP_f782000547ol_nat @ big_co387207925at_nat @ A ) @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ B ) @ C ) )
          = ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ ( hAPP_nat_nat @ A @ B ) ) @ ( hAPP_f22106695ol_nat @ ( hAPP_f782000547ol_nat @ big_co387207925at_nat @ A ) @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[172]) ).

thf(894,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ A ) @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ B ) @ C ) ) )
    <=> ( ( A = B )
        | ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ A ) @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_63_insert__iff) ).

thf(4421,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ A ) @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ B ) @ C ) ) )
       => ( ( A = B )
          | ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ A ) @ C ) ) ) )
      & ( ( ( A = B )
          | ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ A ) @ C ) ) )
       => ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ A ) @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ B ) @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[894]) ).

thf(869,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_f284875647l_bool @ ord_le951220754t_bool @ B ) @ C ) )
     => ( ( B = A )
       => ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_f284875647l_bool @ ord_le951220754t_bool @ A ) @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_756_xt1_I4_J) ).

thf(4315,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_f284875647l_bool @ ord_le951220754t_bool @ B ) @ C ) )
     => ( ( B = A )
       => ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_f284875647l_bool @ ord_le951220754t_bool @ A ) @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[869]) ).

thf(78,axiom,
    ! [A: $i] :
      ( ( hAPP_f22106695ol_nat @ the_nat @ ( hAPP_n1699378549t_bool @ fequal_nat @ A ) )
      = A ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_250_the__sym__eq__trivial) ).

thf(1414,plain,
    ! [A: $i] :
      ( ( hAPP_f22106695ol_nat @ the_nat @ ( hAPP_n1699378549t_bool @ fequal_nat @ A ) )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[78]) ).

thf(203,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ B ) @ C ) )
     => ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_f103356543l_bool @ ord_le1568362934t_bool @ C ) @ A ) )
       => ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ B ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_799_set__rev__mp) ).

thf(1842,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ B ) @ C ) )
     => ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_f103356543l_bool @ ord_le1568362934t_bool @ C ) @ A ) )
       => ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ B ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[203]) ).

thf(189,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f54304608l_bool @ finite_finite_nat @ C ) )
     => ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ B ) @ C ) )
       => ( ( hAPP_f22106695ol_nat @ ( big_co1705425894at_nat @ A ) @ C )
          = ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ ( hAPP_nat_nat @ A @ B ) ) @ ( hAPP_f22106695ol_nat @ ( big_co1705425894at_nat @ A ) @ ( hAPP_f800510211t_bool @ ( hAPP_f1730770594t_bool @ minus_2067140911t_bool @ C ) @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ B ) @ bot_bot_fun_nat_bool ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_472_setprod_Oremove) ).

thf(1791,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f54304608l_bool @ finite_finite_nat @ C ) )
     => ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ B ) @ C ) )
       => ( ( hAPP_f22106695ol_nat @ ( big_co1705425894at_nat @ A ) @ C )
          = ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ ( hAPP_nat_nat @ A @ B ) ) @ ( hAPP_f22106695ol_nat @ ( big_co1705425894at_nat @ A ) @ ( hAPP_f800510211t_bool @ ( hAPP_f1730770594t_bool @ minus_2067140911t_bool @ C ) @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ B ) @ bot_bot_fun_nat_bool ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[189]) ).

thf(285,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ A ) @ B )
        = ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ A ) @ C ) )
    <=> ( B = C ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_1048_nat__add__left__cancel) ).

thf(2172,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ A ) @ B )
          = ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ A ) @ C ) )
       => ( B = C ) )
      & ( ( B = C )
       => ( ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ A ) @ B )
          = ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ A ) @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[285]) ).

thf(1011,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ A ) @ ( hAPP_f20753329a_bool @ ( hAPP_f1706700729a_bool @ minus_85316870a_bool @ B ) @ C ) ) )
     => ~ ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ A ) @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_388_DiffD2) ).

thf(4867,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ A ) @ ( hAPP_f20753329a_bool @ ( hAPP_f1706700729a_bool @ minus_85316870a_bool @ B ) @ C ) ) )
     => ~ ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ A ) @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1011]) ).

thf(898,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_f284875647l_bool @ ord_le951220754t_bool @ A ) @ ( image_int_int @ B @ C ) ) )
    <=> ? [D: $i] :
          ( ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_f284875647l_bool @ ord_le951220754t_bool @ D ) @ C ) )
          & ( A
            = ( image_int_int @ B @ D ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_876_subset__image__iff) ).

thf(4444,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_f284875647l_bool @ ord_le951220754t_bool @ A ) @ ( image_int_int @ B @ C ) ) )
       => ? [D: $i] :
            ( ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_f284875647l_bool @ ord_le951220754t_bool @ D ) @ C ) )
            & ( A
              = ( image_int_int @ B @ D ) ) ) )
      & ( ? [D: $i] :
            ( ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_f284875647l_bool @ ord_le951220754t_bool @ D ) @ C ) )
            & ( A
              = ( image_int_int @ B @ D ) ) )
       => ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_f284875647l_bool @ ord_le951220754t_bool @ A ) @ ( image_int_int @ B @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[898]) ).

thf(419,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f448129468l_bool @ finite_finite_int @ C ) )
     => ( ( hAPP_f1594865479ol_int @ ( big_co1548731110nt_int @ A ) @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ B ) @ C ) )
        = ( hAPP_int_int @ ( hAPP_int_fun_int_int @ times_times_int @ ( hAPP_int_int @ A @ B ) ) @ ( hAPP_f1594865479ol_int @ ( big_co1548731110nt_int @ A ) @ ( hAPP_f1805168059t_bool @ ( hAPP_f1223193598t_bool @ minus_1449998731t_bool @ C ) @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ B ) @ bot_bot_fun_int_bool ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_471_setprod_Oinsert__remove) ).

thf(2666,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f448129468l_bool @ finite_finite_int @ C ) )
     => ( ( hAPP_f1594865479ol_int @ ( big_co1548731110nt_int @ A ) @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ B ) @ C ) )
        = ( hAPP_int_int @ ( hAPP_int_fun_int_int @ times_times_int @ ( hAPP_int_int @ A @ B ) ) @ ( hAPP_f1594865479ol_int @ ( big_co1548731110nt_int @ A ) @ ( hAPP_f1805168059t_bool @ ( hAPP_f1223193598t_bool @ minus_1449998731t_bool @ C ) @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ B ) @ bot_bot_fun_int_bool ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[419]) ).

thf(371,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_f103356543l_bool @ ord_le1568362934t_bool @ B ) @ C ) )
     => ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ A ) @ B ) )
       => ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ A ) @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_796_set__mp) ).

thf(2472,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_f103356543l_bool @ ord_le1568362934t_bool @ B ) @ C ) )
     => ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ A ) @ B ) )
       => ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ A ) @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[371]) ).

thf(838,axiom,
    ! [A: $i,B: $i] :
      ( ( hAPP_int_int @ ( hAPP_int_fun_int_int @ times_times_int @ A ) @ B )
      = ( hAPP_int_int @ ( hAPP_int_fun_int_int @ times_times_int @ B ) @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_1088_zmult__commute) ).

thf(4180,plain,
    ! [A: $i,B: $i] :
      ( ( hAPP_int_int @ ( hAPP_int_fun_int_int @ times_times_int @ A ) @ B )
      = ( hAPP_int_int @ ( hAPP_int_fun_int_int @ times_times_int @ B ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[838]) ).

thf(90,axiom,
    ! [A: $i,B: $i] :
      ( ( A != B )
      | ( hBOOL @ ( hAPP_int_bool @ ( hAPP_i1948725293t_bool @ fequal_int @ A ) @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',help_fequal_2_1_fequal_000tc__Int__Oint_T) ).

thf(1451,plain,
    ! [A: $i,B: $i] :
      ( ( A != B )
      | ( hBOOL @ ( hAPP_int_bool @ ( hAPP_i1948725293t_bool @ fequal_int @ A ) @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[90]) ).

thf(913,axiom,
    ! [A: $i,B: $i] :
      ( ( A = B )
     => ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_f103356543l_bool @ ord_le1568362934t_bool @ A ) @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_806_equalityD1) ).

thf(4508,plain,
    ! [A: $i,B: $i] :
      ( ( A = B )
     => ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_f103356543l_bool @ ord_le1568362934t_bool @ A ) @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[913]) ).

thf(234,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ~ ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ D ) @ E ) )
     => ( ( hBOOL @ ( hAPP_int_bool @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ ( finite772772422nt_int @ A ) @ B ) @ E ) @ C ) )
       => ( hBOOL @ ( hAPP_int_bool @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ ( finite772772422nt_int @ A ) @ B ) @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ D ) @ E ) ) @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ A @ D ) @ C ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_307_fold__graph_OinsertI) ).

thf(1984,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ~ ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ D ) @ E ) )
     => ( ( hBOOL @ ( hAPP_int_bool @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ ( finite772772422nt_int @ A ) @ B ) @ E ) @ C ) )
       => ( hBOOL @ ( hAPP_int_bool @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ ( finite772772422nt_int @ A ) @ B ) @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ D ) @ E ) ) @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ A @ D ) @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[234]) ).

thf(1128,axiom,
    ! [A: $i] :
      ( ~ ( hBOOL @ ( hAPP_bool_bool @ fNot @ A ) )
      | ~ ( hBOOL @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',help_fNot_1_1_U) ).

thf(5305,plain,
    ! [A: $i] :
      ( ~ ( hBOOL @ ( hAPP_bool_bool @ fNot @ A ) )
      | ~ ( hBOOL @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1128]) ).

thf(1109,axiom,
    ! [A: $i] :
      ( ( is_bool @ A )
     => ( ( A = fTrue )
        | ( A = fFalse ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',help_If_3_1_If_000tc__Nat__Onat_T) ).

thf(5217,plain,
    ! [A: $i] :
      ( ( is_bool @ A )
     => ( ( A = fTrue )
        | ( A = fFalse ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1109]) ).

thf(164,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP_int_int @ ( hAPP_int_fun_int_int @ minus_minus_int @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ ord_min_int @ A ) @ B ) ) @ C )
      = ( hAPP_int_int @ ( hAPP_int_fun_int_int @ ord_min_int @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ minus_minus_int @ A ) @ C ) ) @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ minus_minus_int @ B ) @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_990_min__diff__distrib__left) ).

thf(1715,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP_int_int @ ( hAPP_int_fun_int_int @ minus_minus_int @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ ord_min_int @ A ) @ B ) ) @ C )
      = ( hAPP_int_int @ ( hAPP_int_fun_int_int @ ord_min_int @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ minus_minus_int @ A ) @ C ) ) @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ minus_minus_int @ B ) @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[164]) ).

thf(615,axiom,
    ! [A: $i,B: $i] :
      ( ~ ( hBOOL @ B )
      | ( hBOOL @ ( hAPP_bool_bool @ ( hAPP_b589554111l_bool @ fdisj @ A ) @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',help_fdisj_2_1_U) ).

thf(3364,plain,
    ! [A: $i,B: $i] :
      ( ~ ( hBOOL @ B )
      | ( hBOOL @ ( hAPP_bool_bool @ ( hAPP_b589554111l_bool @ fdisj @ A ) @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[615]) ).

thf(502,axiom,
    ! [A: $i,B: $i] :
      ( ( A = B )
     => ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ A ) @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_1000_eq__imp__le) ).

thf(2968,plain,
    ! [A: $i,B: $i] :
      ( ( A = B )
     => ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ A ) @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[502]) ).

thf(378,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f1695230391l_bool @ finite1738664244iple_a @ C ) )
     => ( ( hAPP_f2097660464ol_nat @ ( hAPP_f1371399797ol_nat @ big_co1340561246_a_nat @ A ) @ C )
        = ( 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 @ A ) ) @ ( 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 ) @ C ) ) ) ) @ ( hAPP_f228417655a_bool @ cOMBC_364739t_bool @ ( hAPP_f1872860171t_bool @ ( hAPP_f1981448543t_bool @ cOMBB_1391527205iple_a @ fequal_nat ) @ B ) ) ) ) ) ) @ ( image_48635758_a_nat @ B @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_592_setsum__image__gen) ).

thf(2504,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f1695230391l_bool @ finite1738664244iple_a @ C ) )
     => ( ( hAPP_f2097660464ol_nat @ ( hAPP_f1371399797ol_nat @ big_co1340561246_a_nat @ A ) @ C )
        = ( 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 @ A ) ) @ ( 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 ) @ C ) ) ) ) @ ( hAPP_f228417655a_bool @ cOMBC_364739t_bool @ ( hAPP_f1872860171t_bool @ ( hAPP_f1981448543t_bool @ cOMBB_1391527205iple_a @ fequal_nat ) @ B ) ) ) ) ) ) @ ( image_48635758_a_nat @ B @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[378]) ).

thf(644,axiom,
    ! [A: $i,B: $i] :
      ( ( 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 ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_383_Diff__idemp) ).

thf(3486,plain,
    ! [A: $i,B: $i] :
      ( ( 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 ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[644]) ).

thf(692,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ A ) @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ B ) @ C ) )
      = ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ B ) @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ A ) @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_674_comm__semiring__1__class_Onormalizing__semiring__rules_I22_J) ).

thf(3654,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ A ) @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ B ) @ C ) )
      = ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ B ) @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ A ) @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[692]) ).

thf(613,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_f284875647l_bool @ ord_le951220754t_bool @ A ) @ ( image_nat_int @ B @ C ) ) )
    <=> ? [D: $i] :
          ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_f103356543l_bool @ ord_le1568362934t_bool @ D ) @ C ) )
          & ( A
            = ( image_nat_int @ B @ D ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_878_subset__image__iff) ).

thf(3354,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_f284875647l_bool @ ord_le951220754t_bool @ A ) @ ( image_nat_int @ B @ C ) ) )
       => ? [D: $i] :
            ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_f103356543l_bool @ ord_le1568362934t_bool @ D ) @ C ) )
            & ( A
              = ( image_nat_int @ B @ D ) ) ) )
      & ( ? [D: $i] :
            ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_f103356543l_bool @ ord_le1568362934t_bool @ D ) @ C ) )
            & ( A
              = ( image_nat_int @ B @ D ) ) )
       => ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_f284875647l_bool @ ord_le951220754t_bool @ A ) @ ( image_nat_int @ B @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[613]) ).

thf(810,axiom,
    ! [A: $i,B: $i] : ( is_bool @ ( hAPP_H1037229737a_bool @ A @ B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',gsy_c_hAPP_000tc__Hoare____Mirabelle____uwgpyvfjxg__Otriple_It__a_J_000tc__HOL__) ).

thf(4068,plain,
    ! [A: $i,B: $i] : ( is_bool @ ( hAPP_H1037229737a_bool @ A @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[810]) ).

thf(486,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ A ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ ord_min_nat @ B ) @ C ) ) )
     => ~ ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ A ) @ B ) )
         => ~ ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ A ) @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_964_min__max_Ole__infE) ).

thf(2900,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ A ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ ord_min_nat @ B ) @ C ) ) )
     => ~ ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ A ) @ B ) )
         => ~ ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ A ) @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[486]) ).

thf(1043,axiom,
    ! [A: $i,B: $i] :
      ( ( is_bool @ A )
     => ( ( hAPP_state_bool @ ( hAPP_b2019457360e_bool @ cOMBK_bool_state @ A ) @ B )
        = A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',help_COMBK_1_1_COMBK_000tc__HOL__Obool_000tc__Com__Ostate_U) ).

thf(4970,plain,
    ! [A: $i,B: $i] :
      ( ( is_bool @ A )
     => ( ( hAPP_state_bool @ ( hAPP_b2019457360e_bool @ cOMBK_bool_state @ A ) @ B )
        = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1043]) ).

thf(1,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 ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',conj_0) ).

thf(2,negated_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 ) ) ),
    inference(neg_conjecture,[status(cth)],[1]) ).

thf(1131,plain,
    ~ ( 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 ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2]) ).

thf(150,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ A ) @ B )
        = ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ A ) @ C ) )
    <=> ( B = C ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_663_add__left__cancel) ).

thf(1668,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ A ) @ B )
          = ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ A ) @ C ) )
       => ( B = C ) )
      & ( ( B = C )
       => ( ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ A ) @ B )
          = ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ A ) @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[150]) ).

thf(232,axiom,
    ! [A: $i,B: $i] :
      ( ~ ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ fequal_nat @ A ) @ B ) )
      | ( A = B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',help_fequal_1_1_fequal_000tc__Nat__Onat_T) ).

thf(1974,plain,
    ! [A: $i,B: $i] :
      ( ~ ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ fequal_nat @ A ) @ B ) )
      | ( A = B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[232]) ).

thf(116,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f54304608l_bool @ finite_finite_nat @ C ) )
     => ( ! [D: $i] :
            ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ D ) @ C ) )
           => ( ( hAPP_nat_nat @ A @ D )
              = ( hAPP_nat_nat @ B @ D ) ) )
       => ( ( hAPP_f22106695ol_nat @ ( hAPP_f782000547ol_nat @ big_co387207925at_nat @ A ) @ C )
          = ( hAPP_f22106695ol_nat @ ( hAPP_f782000547ol_nat @ big_co387207925at_nat @ B ) @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_599_setsum_Ocong) ).

thf(1552,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f54304608l_bool @ finite_finite_nat @ C ) )
     => ( ! [D: $i] :
            ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ D ) @ C ) )
           => ( ( hAPP_nat_nat @ A @ D )
              = ( hAPP_nat_nat @ B @ D ) ) )
       => ( ( hAPP_f22106695ol_nat @ ( hAPP_f782000547ol_nat @ big_co387207925at_nat @ A ) @ C )
          = ( hAPP_f22106695ol_nat @ ( hAPP_f782000547ol_nat @ big_co387207925at_nat @ B ) @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[116]) ).

thf(471,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP_nat_bool @ ( finite_fold1Set_nat @ A @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ B ) @ C ) ) @ D ) )
     => ~ ! [E: $i,F: $i] :
            ( ( ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ B ) @ C )
              = ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ E ) @ F ) )
           => ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ ( finite929467206at_nat @ A ) @ E ) @ F ) @ D ) )
             => ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ E ) @ F ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_308_insert__fold1SetE) ).

thf(2851,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP_nat_bool @ ( finite_fold1Set_nat @ A @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ B ) @ C ) ) @ D ) )
     => ~ ! [E: $i,F: $i] :
            ( ( ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ B ) @ C )
              = ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ E ) @ F ) )
           => ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ ( finite929467206at_nat @ A ) @ E ) @ F ) @ D ) )
             => ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ E ) @ F ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[471]) ).

thf(336,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ A ) @ B )
        = ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ A ) @ C ) )
     => ( B = C ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_655_add__imp__eq) ).

thf(2355,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ A ) @ B )
        = ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ A ) @ C ) )
     => ( B = C ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[336]) ).

thf(221,axiom,
    ! [A: $i] :
      ( ( hBOOL @ ( hAPP_f448129468l_bool @ finite_finite_int @ A ) )
     => ( ( A != bot_bot_fun_int_bool )
       => ( ! [B: $i,C: $i] : ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ times_times_int @ B ) @ C ) ) @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ B ) @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ C ) @ 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 ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_435_fold1__in) ).

thf(1932,plain,
    ! [A: $i] :
      ( ( hBOOL @ ( hAPP_f448129468l_bool @ finite_finite_int @ A ) )
     => ( ( A != bot_bot_fun_int_bool )
       => ( ! [B: $i,C: $i] : ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ times_times_int @ B ) @ C ) ) @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ B ) @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ C ) @ 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 ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[221]) ).

thf(850,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f1695230391l_bool @ finite1738664244iple_a @ C ) )
     => ( ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ B ) @ C ) )
       => ( ( hAPP_f587450508ol_int @ ( hAPP_f1544320301ol_int @ big_co1977834938_a_int @ A ) @ C )
          = ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ ( hAPP_H1229305626_a_int @ A @ B ) ) @ ( hAPP_f587450508ol_int @ ( hAPP_f1544320301ol_int @ big_co1977834938_a_int @ A ) @ ( hAPP_f20753329a_bool @ ( hAPP_f1706700729a_bool @ minus_85316870a_bool @ C ) @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ B ) @ bot_bo797238721a_bool ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_616_setsum__diff1_H) ).

thf(4244,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f1695230391l_bool @ finite1738664244iple_a @ C ) )
     => ( ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ B ) @ C ) )
       => ( ( hAPP_f587450508ol_int @ ( hAPP_f1544320301ol_int @ big_co1977834938_a_int @ A ) @ C )
          = ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ ( hAPP_H1229305626_a_int @ A @ B ) ) @ ( hAPP_f587450508ol_int @ ( hAPP_f1544320301ol_int @ big_co1977834938_a_int @ A ) @ ( hAPP_f20753329a_bool @ ( hAPP_f1706700729a_bool @ minus_85316870a_bool @ C ) @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ B ) @ bot_bo797238721a_bool ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[850]) ).

thf(571,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ A ) @ B ) ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ C ) @ D ) )
      = ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ A ) @ C ) ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ B ) @ D ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_677_comm__semiring__1__class_Onormalizing__semiring__rules_I13_J) ).

thf(3222,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ A ) @ B ) ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ C ) @ D ) )
      = ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ A ) @ C ) ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ B ) @ D ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[571]) ).

thf(1031,axiom,
    ! [A: $i,B: $i] :
      ( ( 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 ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_851_finite__subset) ).

thf(4934,plain,
    ! [A: $i,B: $i] :
      ( ( 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 ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1031]) ).

thf(924,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP_H1037229737a_bool @ ( hAPP_f1194757675a_bool @ ( hAPP_f1697223433a_bool @ cOMBB_1610959875iple_a @ A ) @ B ) @ C )
      = ( hAPP_f1695230391l_bool @ A @ ( hAPP_H426895267a_bool @ B @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',help_COMBB_1_1_COMBB_000tc__fun_Itc__Hoare____Mirabelle____uwgpyvfjxg__Otriple_I_041) ).

thf(4540,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP_H1037229737a_bool @ ( hAPP_f1194757675a_bool @ ( hAPP_f1697223433a_bool @ cOMBB_1610959875iple_a @ A ) @ B ) @ C )
      = ( hAPP_f1695230391l_bool @ A @ ( hAPP_H426895267a_bool @ B @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[924]) ).

thf(461,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( finite988810631ne_nat @ C @ D ) )
     => ( ( hBOOL @ ( hAPP_f54304608l_bool @ finite_finite_nat @ B ) )
       => ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ A ) @ B ) )
         => ( ( ( ( hAPP_f800510211t_bool @ ( hAPP_f1730770594t_bool @ minus_2067140911t_bool @ B ) @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ A ) @ bot_bot_fun_nat_bool ) )
                = bot_bot_fun_nat_bool )
             => ( ( hAPP_f22106695ol_nat @ D @ B )
                = A ) )
            & ( ( ( hAPP_f800510211t_bool @ ( hAPP_f1730770594t_bool @ minus_2067140911t_bool @ B ) @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ A ) @ bot_bot_fun_nat_bool ) )
               != bot_bot_fun_nat_bool )
             => ( ( hAPP_f22106695ol_nat @ D @ B )
                = ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ C @ A ) @ ( hAPP_f22106695ol_nat @ D @ ( hAPP_f800510211t_bool @ ( hAPP_f1730770594t_bool @ minus_2067140911t_bool @ B ) @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ A ) @ bot_bot_fun_nat_bool ) ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_356_folding__one_Oremove) ).

thf(2813,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( finite988810631ne_nat @ C @ D ) )
     => ( ( hBOOL @ ( hAPP_f54304608l_bool @ finite_finite_nat @ B ) )
       => ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ A ) @ B ) )
         => ( ( ( ( hAPP_f800510211t_bool @ ( hAPP_f1730770594t_bool @ minus_2067140911t_bool @ B ) @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ A ) @ bot_bot_fun_nat_bool ) )
                = bot_bot_fun_nat_bool )
             => ( ( hAPP_f22106695ol_nat @ D @ B )
                = A ) )
            & ( ( ( hAPP_f800510211t_bool @ ( hAPP_f1730770594t_bool @ minus_2067140911t_bool @ B ) @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ A ) @ bot_bot_fun_nat_bool ) )
               != bot_bot_fun_nat_bool )
             => ( ( hAPP_f22106695ol_nat @ D @ B )
                = ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ C @ A ) @ ( hAPP_f22106695ol_nat @ D @ ( hAPP_f800510211t_bool @ ( hAPP_f1730770594t_bool @ minus_2067140911t_bool @ B ) @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ A ) @ bot_bot_fun_nat_bool ) ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[461]) ).

thf(874,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f1695230391l_bool @ finite1738664244iple_a @ C ) )
     => ( ~ ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ B ) @ C ) )
       => ( ( hAPP_f2097660464ol_nat @ ( big_co1815942735_a_nat @ A ) @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ B ) @ C ) )
          = ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ ( hAPP_H592031934_a_nat @ A @ B ) ) @ ( hAPP_f2097660464ol_nat @ ( big_co1815942735_a_nat @ A ) @ C ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_557_setprod_Oinsert) ).

thf(4347,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f1695230391l_bool @ finite1738664244iple_a @ C ) )
     => ( ~ ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ B ) @ C ) )
       => ( ( hAPP_f2097660464ol_nat @ ( big_co1815942735_a_nat @ A ) @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ B ) @ C ) )
          = ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ ( hAPP_H592031934_a_nat @ A @ B ) ) @ ( hAPP_f2097660464ol_nat @ ( big_co1815942735_a_nat @ A ) @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[874]) ).

thf(255,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ A ) @ B ) ) @ C )
      = ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ A ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ B ) @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_667_ab__semigroup__add__class_Oadd__ac_I1_J) ).

thf(2052,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ A ) @ B ) ) @ C )
      = ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ A ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ B ) @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[255]) ).

thf(130,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ A ) @ B ) )
     => ( ( hAPP_int_int @ ( hAPP_int_fun_int_int @ minus_minus_int @ ( hAPP_nat_int @ semiri1621563631at_int @ B ) ) @ ( hAPP_nat_int @ semiri1621563631at_int @ A ) )
        = ( hAPP_nat_int @ semiri1621563631at_int @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ minus_minus_nat @ B ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_1070_zdiff__int) ).

thf(1599,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ A ) @ B ) )
     => ( ( hAPP_int_int @ ( hAPP_int_fun_int_int @ minus_minus_int @ ( hAPP_nat_int @ semiri1621563631at_int @ B ) ) @ ( hAPP_nat_int @ semiri1621563631at_int @ A ) )
        = ( hAPP_nat_int @ semiri1621563631at_int @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ minus_minus_nat @ B ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[130]) ).

thf(1021,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP_H1743777351a_bool @ ( hAPP_f550111323a_bool @ ( hAPP_f615286271a_bool @ cOMBB_1176122337iple_a @ A ) @ B ) @ C )
      = ( hAPP_f1706700729a_bool @ A @ ( hAPP_H426895267a_bool @ B @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',help_COMBB_1_1_COMBB_000tc__fun_Itc__Hoare____Mirabelle____uwgpyvfjxg__Otriple_I_056) ).

thf(4902,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP_H1743777351a_bool @ ( hAPP_f550111323a_bool @ ( hAPP_f615286271a_bool @ cOMBB_1176122337iple_a @ A ) @ B ) @ C )
      = ( hAPP_f1706700729a_bool @ A @ ( hAPP_H426895267a_bool @ B @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1021]) ).

thf(276,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i] :
          ( ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ D ) @ C ) )
         => ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ ( hAPP_int_int @ A @ D ) ) @ B ) ) )
     => ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_f284875647l_bool @ ord_le951220754t_bool @ ( image_int_int @ A @ C ) ) @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_926_image__subsetI) ).

thf(2140,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i] :
          ( ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ D ) @ C ) )
         => ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ ( hAPP_int_int @ A @ D ) ) @ B ) ) )
     => ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_f284875647l_bool @ ord_le951220754t_bool @ ( image_int_int @ A @ C ) ) @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[276]) ).

thf(447,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( finite1626084323ne_int @ C @ D ) )
     => ( ( hBOOL @ ( hAPP_f448129468l_bool @ finite_finite_int @ B ) )
       => ( ( ( ( hAPP_f1805168059t_bool @ ( hAPP_f1223193598t_bool @ minus_1449998731t_bool @ B ) @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ A ) @ bot_bot_fun_int_bool ) )
              = bot_bot_fun_int_bool )
           => ( ( hAPP_f1594865479ol_int @ D @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ A ) @ B ) )
              = A ) )
          & ( ( ( hAPP_f1805168059t_bool @ ( hAPP_f1223193598t_bool @ minus_1449998731t_bool @ B ) @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ A ) @ bot_bot_fun_int_bool ) )
             != bot_bot_fun_int_bool )
           => ( ( hAPP_f1594865479ol_int @ D @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ A ) @ B ) )
              = ( hAPP_int_int @ ( hAPP_int_fun_int_int @ C @ A ) @ ( hAPP_f1594865479ol_int @ D @ ( hAPP_f1805168059t_bool @ ( hAPP_f1223193598t_bool @ minus_1449998731t_bool @ B ) @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ A ) @ bot_bot_fun_int_bool ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_361_folding__one_Oinsert__remove) ).

thf(2756,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( finite1626084323ne_int @ C @ D ) )
     => ( ( hBOOL @ ( hAPP_f448129468l_bool @ finite_finite_int @ B ) )
       => ( ( ( ( hAPP_f1805168059t_bool @ ( hAPP_f1223193598t_bool @ minus_1449998731t_bool @ B ) @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ A ) @ bot_bot_fun_int_bool ) )
              = bot_bot_fun_int_bool )
           => ( ( hAPP_f1594865479ol_int @ D @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ A ) @ B ) )
              = A ) )
          & ( ( ( hAPP_f1805168059t_bool @ ( hAPP_f1223193598t_bool @ minus_1449998731t_bool @ B ) @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ A ) @ bot_bot_fun_int_bool ) )
             != bot_bot_fun_int_bool )
           => ( ( hAPP_f1594865479ol_int @ D @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ A ) @ B ) )
              = ( hAPP_int_int @ ( hAPP_int_fun_int_int @ C @ A ) @ ( hAPP_f1594865479ol_int @ D @ ( hAPP_f1805168059t_bool @ ( hAPP_f1223193598t_bool @ minus_1449998731t_bool @ B ) @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ A ) @ bot_bot_fun_int_bool ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[447]) ).

thf(533,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ B ) @ C ) )
     => ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ B ) @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ A ) @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_54_insertI2) ).

thf(3083,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ B ) @ C ) )
     => ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ B ) @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ A ) @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[533]) ).

thf(553,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( semi @ A @ B )
     != ( ass @ C @ D ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_193_com_Osimps_I25_J) ).

thf(3162,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( semi @ A @ B )
     != ( ass @ C @ D ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[553]) ).

thf(177,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ A ) @ B ) ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ A ) @ C ) ) )
     => ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ B ) @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_819_add__le__imp__le__left) ).

thf(1759,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ A ) @ B ) ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ A ) @ C ) ) )
     => ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ B ) @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[177]) ).

thf(569,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_f284875647l_bool @ ord_le951220754t_bool @ B ) @ C ) )
     => ( ( C = A )
       => ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_f284875647l_bool @ ord_le951220754t_bool @ B ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_760_ord__le__eq__trans) ).

thf(3215,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_f284875647l_bool @ ord_le951220754t_bool @ B ) @ C ) )
     => ( ( C = A )
       => ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_f284875647l_bool @ ord_le951220754t_bool @ B ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[569]) ).

thf(107,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( finite1704255308nt_int @ D ) )
     => ( ( hBOOL @ ( hAPP_f448129468l_bool @ finite_finite_int @ C ) )
       => ( ( hAPP_f1594865479ol_int @ ( finite_fold_int_int @ D @ A ) @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ B ) @ C ) )
          = ( hAPP_f1594865479ol_int @ ( finite_fold_int_int @ D @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ D @ B ) @ A ) ) @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_573_comp__fun__idem_Ofold__insert__idem2) ).

thf(1525,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( finite1704255308nt_int @ D ) )
     => ( ( hBOOL @ ( hAPP_f448129468l_bool @ finite_finite_int @ C ) )
       => ( ( hAPP_f1594865479ol_int @ ( finite_fold_int_int @ D @ A ) @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ B ) @ C ) )
          = ( hAPP_f1594865479ol_int @ ( finite_fold_int_int @ D @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ D @ B ) @ A ) ) @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[107]) ).

thf(720,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ~ ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ B ) @ C ) )
     => ( ~ ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ B ) @ A ) )
       => ( ( ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ B ) @ C )
            = ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ B ) @ A ) )
        <=> ( C = A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_57_insert__ident) ).

thf(3755,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ~ ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ B ) @ C ) )
     => ( ~ ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ B ) @ A ) )
       => ( ( ( ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ B ) @ C )
              = ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ B ) @ A ) )
           => ( C = A ) )
          & ( ( C = A )
           => ( ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ B ) @ C )
              = ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ B ) @ A ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[720]) ).

thf(492,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_f800510211t_bool @ ( hAPP_f1730770594t_bool @ minus_2067140911t_bool @ A ) @ B ) @ C ) )
    <=> ( hBOOL @ ( hAPP_bool_bool @ ( hAPP_b589554111l_bool @ minus_minus_bool @ ( hAPP_nat_bool @ A @ C ) ) @ ( hAPP_nat_bool @ B @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_444_minus__apply) ).

thf(2926,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_f800510211t_bool @ ( hAPP_f1730770594t_bool @ minus_2067140911t_bool @ A ) @ B ) @ C ) )
       => ( hBOOL @ ( hAPP_bool_bool @ ( hAPP_b589554111l_bool @ minus_minus_bool @ ( hAPP_nat_bool @ A @ C ) ) @ ( hAPP_nat_bool @ B @ C ) ) ) )
      & ( ( hBOOL @ ( hAPP_bool_bool @ ( hAPP_b589554111l_bool @ minus_minus_bool @ ( hAPP_nat_bool @ A @ C ) ) @ ( hAPP_nat_bool @ B @ C ) ) )
       => ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_f800510211t_bool @ ( hAPP_f1730770594t_bool @ minus_2067140911t_bool @ A ) @ B ) @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[492]) ).

thf(876,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ times_times_int @ A ) @ B ) ) @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ times_times_int @ C ) @ B ) ) @ D ) )
      = ( 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 ) @ C ) ) @ B ) ) @ D ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_650_combine__common__factor) ).

thf(4352,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ times_times_int @ A ) @ B ) ) @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ times_times_int @ C ) @ B ) ) @ D ) )
      = ( 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 ) @ C ) ) @ B ) ) @ D ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[876]) ).

thf(946,axiom,
    ! [A: $i] :
      ( ( hAPP_f20753329a_bool @ ( hAPP_f1706700729a_bool @ minus_85316870a_bool @ A ) @ A )
      = bot_bo797238721a_bool ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_397_Diff__cancel) ).

thf(4625,plain,
    ! [A: $i] :
      ( ( hAPP_f20753329a_bool @ ( hAPP_f1706700729a_bool @ minus_85316870a_bool @ A ) @ A )
      = bot_bo797238721a_bool ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[946]) ).

thf(295,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( B
        = ( finite_fold1_nat @ C ) )
     => ( ( hAPP_f22106695ol_nat @ B @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ A ) @ bot_bot_fun_nat_bool ) )
        = A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_146_fold1__singleton__def) ).

thf(2208,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( B
        = ( finite_fold1_nat @ C ) )
     => ( ( hAPP_f22106695ol_nat @ B @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ A ) @ bot_bot_fun_nat_bool ) )
        = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[295]) ).

thf(479,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( finite2130160977at_nat @ D ) )
     => ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ ( finite929467206at_nat @ D ) @ A ) @ B ) @ C ) )
       => ( ( hAPP_f22106695ol_nat @ ( finite_fold_nat_nat @ D @ A ) @ B )
          = C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_494_comp__fun__commute_Ofold__equality) ).

thf(2883,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( finite2130160977at_nat @ D ) )
     => ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ ( finite929467206at_nat @ D ) @ A ) @ B ) @ C ) )
       => ( ( hAPP_f22106695ol_nat @ ( finite_fold_nat_nat @ D @ A ) @ B )
          = C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[479]) ).

thf(444,axiom,
    ! [A: $i] :
      ( ( hAPP_f800510211t_bool @ ( hAPP_f1730770594t_bool @ minus_2067140911t_bool @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ A ) @ bot_bot_fun_nat_bool ) ) @ ( hAPP_n1699378549t_bool @ ord_lessThan_nat @ A ) )
      = ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ A ) @ bot_bot_fun_nat_bool ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_713_single__Diff__lessThan) ).

thf(2748,plain,
    ! [A: $i] :
      ( ( hAPP_f800510211t_bool @ ( hAPP_f1730770594t_bool @ minus_2067140911t_bool @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ A ) @ bot_bot_fun_nat_bool ) ) @ ( hAPP_n1699378549t_bool @ ord_lessThan_nat @ A ) )
      = ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ A ) @ bot_bot_fun_nat_bool ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[444]) ).

thf(790,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ A ) @ B )
        = ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ A ) @ C ) )
    <=> ( B = C ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_664_add__left__cancel) ).

thf(3999,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ A ) @ B )
          = ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ A ) @ C ) )
       => ( B = C ) )
      & ( ( B = C )
       => ( ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ A ) @ B )
          = ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ A ) @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[790]) ).

thf(19,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP_int_bool @ ( hAPP_i1948725293t_bool @ ord_less_eq_int @ C ) @ D ) )
     => ( ( hBOOL @ ( hAPP_int_bool @ ( hAPP_i1948725293t_bool @ ord_less_eq_int @ A ) @ B ) )
       => ( hBOOL @ ( hAPP_int_bool @ ( hAPP_i1948725293t_bool @ ord_less_eq_int @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ ord_min_int @ C ) @ A ) ) @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ ord_min_int @ D ) @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_967_min__max_Oinf__mono) ).

thf(1187,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP_int_bool @ ( hAPP_i1948725293t_bool @ ord_less_eq_int @ C ) @ D ) )
     => ( ( hBOOL @ ( hAPP_int_bool @ ( hAPP_i1948725293t_bool @ ord_less_eq_int @ A ) @ B ) )
       => ( hBOOL @ ( hAPP_int_bool @ ( hAPP_i1948725293t_bool @ ord_less_eq_int @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ ord_min_int @ C ) @ A ) ) @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ ord_min_int @ D ) @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[19]) ).

thf(1052,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP_f1695230391l_bool @ finite1738664244iple_a @ B ) )
     => ( ( B != bot_bo797238721a_bool )
       => ? [C: $i] : ( hBOOL @ ( hAPP_H1037229737a_bool @ ( finite1621280049iple_a @ A @ B ) @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_311_finite__nonempty__imp__fold1Set) ).

thf(4995,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP_f1695230391l_bool @ finite1738664244iple_a @ B ) )
     => ( ( B != bot_bo797238721a_bool )
       => ? [C: $i] : ( hBOOL @ ( hAPP_H1037229737a_bool @ ( finite1621280049iple_a @ A @ B ) @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1052]) ).

thf(1098,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP_f1695230391l_bool @ ( hAPP_f1767618879l_bool @ ( hAPP_f2100528361l_bool @ cOMBC_627599030l_bool @ A ) @ B ) @ C )
      = ( hAPP_f1695230391l_bool @ ( hAPP_f1767618879l_bool @ A @ C ) @ B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',help_COMBC_1_1_COMBC_000tc__fun_Itc__Hoare____Mirabelle____uwgpyvfjxg__Otriple_I) ).

thf(5170,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP_f1695230391l_bool @ ( hAPP_f1767618879l_bool @ ( hAPP_f2100528361l_bool @ cOMBC_627599030l_bool @ A ) @ B ) @ C )
      = ( hAPP_f1695230391l_bool @ ( hAPP_f1767618879l_bool @ A @ C ) @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1098]) ).

thf(834,axiom,
    ! [A: $i] :
      ( ( hBOOL @ ( hAPP_nat_bool @ bot_bot_fun_nat_bool @ A ) )
    <=> ( hBOOL @ bot_bot_bool ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_110_bot__fun__def) ).

thf(4163,plain,
    ! [A: $i] :
      ( ( ( hBOOL @ ( hAPP_nat_bool @ bot_bot_fun_nat_bool @ A ) )
       => ( hBOOL @ bot_bot_bool ) )
      & ( ( hBOOL @ bot_bot_bool )
       => ( hBOOL @ ( hAPP_nat_bool @ bot_bot_fun_nat_bool @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[834]) ).

thf(740,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f1695230391l_bool @ finite1738664244iple_a @ C ) )
     => ( ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ B ) @ C ) )
       => ( ( hAPP_f2097660464ol_nat @ ( hAPP_f1371399797ol_nat @ big_co1340561246_a_nat @ A ) @ C )
          = ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ ( hAPP_H592031934_a_nat @ A @ B ) ) @ ( hAPP_f2097660464ol_nat @ ( hAPP_f1371399797ol_nat @ big_co1340561246_a_nat @ A ) @ ( hAPP_f20753329a_bool @ ( hAPP_f1706700729a_bool @ minus_85316870a_bool @ C ) @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ B ) @ bot_bo797238721a_bool ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_612_setsum_Oremove) ).

thf(3824,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f1695230391l_bool @ finite1738664244iple_a @ C ) )
     => ( ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ B ) @ C ) )
       => ( ( hAPP_f2097660464ol_nat @ ( hAPP_f1371399797ol_nat @ big_co1340561246_a_nat @ A ) @ C )
          = ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ ( hAPP_H592031934_a_nat @ A @ B ) ) @ ( hAPP_f2097660464ol_nat @ ( hAPP_f1371399797ol_nat @ big_co1340561246_a_nat @ A ) @ ( hAPP_f20753329a_bool @ ( hAPP_f1706700729a_bool @ minus_85316870a_bool @ C ) @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ B ) @ bot_bo797238721a_bool ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[740]) ).

thf(829,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( image_nat_nat @ A @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ B ) @ C ) )
      = ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ ( hAPP_nat_nat @ A @ B ) ) @ ( image_nat_nat @ A @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_225_image__insert) ).

thf(4148,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( image_nat_nat @ A @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ B ) @ C ) )
      = ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ ( hAPP_nat_nat @ A @ B ) ) @ ( image_nat_nat @ A @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[829]) ).

thf(412,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ A ) @ B ) )
     => ( ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ A ) @ ( hAPP_f1805168059t_bool @ ( hAPP_f1223193598t_bool @ minus_1449998731t_bool @ B ) @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ A ) @ bot_bot_fun_int_bool ) ) )
        = B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_425_insert__Diff) ).

thf(2634,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ A ) @ B ) )
     => ( ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ A ) @ ( hAPP_f1805168059t_bool @ ( hAPP_f1223193598t_bool @ minus_1449998731t_bool @ B ) @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ A ) @ bot_bot_fun_int_bool ) ) )
        = B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[412]) ).

thf(85,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP_nat_bool @ A @ B ) )
     => ( ! [C: $i] :
            ( ( hBOOL @ ( hAPP_nat_bool @ A @ C ) )
           => ( C = B ) )
       => ( hBOOL @ ( hAPP_nat_bool @ A @ ( hAPP_f22106695ol_nat @ the_nat @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_337_theI) ).

thf(1434,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP_nat_bool @ A @ B ) )
     => ( ! [C: $i] :
            ( ( hBOOL @ ( hAPP_nat_bool @ A @ C ) )
           => ( C = B ) )
       => ( hBOOL @ ( hAPP_nat_bool @ A @ ( hAPP_f22106695ol_nat @ the_nat @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[85]) ).

thf(765,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP_int_int @ ( hAPP_int_fun_int_int @ times_times_int @ A ) @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ B ) @ C ) )
      = ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ times_times_int @ A ) @ B ) ) @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ times_times_int @ A ) @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_1095_zadd__zmult__distrib2) ).

thf(3903,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP_int_int @ ( hAPP_int_fun_int_int @ times_times_int @ A ) @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ B ) @ C ) )
      = ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ times_times_int @ A ) @ B ) ) @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ times_times_int @ A ) @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[765]) ).

thf(195,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( big_se275732192ig_nat @ B @ C ) )
     => ( ( hBOOL @ ( hAPP_f54304608l_bool @ finite_finite_nat @ A ) )
       => ( ( hAPP_f22106695ol_nat @ C @ A )
          = ( hAPP_f22106695ol_nat @ ( finite_fold1_nat @ B ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_354_semilattice__big_OF__eq) ).

thf(1814,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( big_se275732192ig_nat @ B @ C ) )
     => ( ( hBOOL @ ( hAPP_f54304608l_bool @ finite_finite_nat @ A ) )
       => ( ( hAPP_f22106695ol_nat @ C @ A )
          = ( hAPP_f22106695ol_nat @ ( finite_fold1_nat @ B ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[195]) ).

thf(572,axiom,
    ! [A: $i,B: $i] :
      ( ( 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 ) @ A ) ) @ B ) ) ) )
    <=> ( ( hBOOL @ ( hAPP_f1695230391l_bool @ finite1738664244iple_a @ ( hAPP_f20753329a_bool @ collec351493750iple_a @ A ) ) )
        & ( hBOOL @ ( hAPP_f1695230391l_bool @ finite1738664244iple_a @ ( hAPP_f20753329a_bool @ collec351493750iple_a @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_293_finite__Collect__disjI) ).

thf(3225,plain,
    ! [A: $i,B: $i] :
      ( ( ( 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 ) @ A ) ) @ B ) ) ) )
       => ( ( hBOOL @ ( hAPP_f1695230391l_bool @ finite1738664244iple_a @ ( hAPP_f20753329a_bool @ collec351493750iple_a @ A ) ) )
          & ( hBOOL @ ( hAPP_f1695230391l_bool @ finite1738664244iple_a @ ( hAPP_f20753329a_bool @ collec351493750iple_a @ B ) ) ) ) )
      & ( ( ( hBOOL @ ( hAPP_f1695230391l_bool @ finite1738664244iple_a @ ( hAPP_f20753329a_bool @ collec351493750iple_a @ A ) ) )
          & ( hBOOL @ ( hAPP_f1695230391l_bool @ finite1738664244iple_a @ ( hAPP_f20753329a_bool @ collec351493750iple_a @ B ) ) ) )
       => ( 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 ) @ A ) ) @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[572]) ).

thf(448,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ A ) @ B ) ) @ C ) )
     => ~ ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ A ) @ C ) )
         => ~ ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ B ) @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_1003_add__leE) ).

thf(2761,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ A ) @ B ) ) @ C ) )
     => ~ ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ A ) @ C ) )
         => ~ ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ B ) @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[448]) ).

thf(840,axiom,
    ! [A: $i,B: $i] :
      ( ( A = B )
     => ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_f284875647l_bool @ ord_le951220754t_bool @ A ) @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_778_order__eq__refl) ).

thf(4195,plain,
    ! [A: $i,B: $i] :
      ( ( A = B )
     => ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_f284875647l_bool @ ord_le951220754t_bool @ A ) @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[840]) ).

thf(588,axiom,
    ! [A: $i] : ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ A ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ A ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ A ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_996_le__cube) ).

thf(3279,plain,
    ! [A: $i] : ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ A ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ A ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ A ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[588]) ).

thf(672,axiom,
    ! [A: $i] : ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_f103356543l_bool @ ord_le1568362934t_bool @ A ) @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_810_subset__refl) ).

thf(3587,plain,
    ! [A: $i] : ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_f103356543l_bool @ ord_le1568362934t_bool @ A ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[672]) ).

thf(257,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ A ) @ B ) ) @ C ) )
     => ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ A ) @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_1004_add__leD1) ).

thf(2057,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ A ) @ B ) ) @ C ) )
     => ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ A ) @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[257]) ).

thf(163,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP_f448129468l_bool @ finite_finite_int @ B ) )
     => ( ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ A ) @ B ) )
       => ( ( hAPP_int_int @ ( hAPP_int_fun_int_int @ ord_min_int @ A ) @ ( hAPP_f1594865479ol_int @ big_linorder_Min_int @ B ) )
          = ( hAPP_f1594865479ol_int @ big_linorder_Min_int @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_950_Min_Oin__idem) ).

thf(1712,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP_f448129468l_bool @ finite_finite_int @ B ) )
     => ( ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ A ) @ B ) )
       => ( ( hAPP_int_int @ ( hAPP_int_fun_int_int @ ord_min_int @ A ) @ ( hAPP_f1594865479ol_int @ big_linorder_Min_int @ B ) )
          = ( hAPP_f1594865479ol_int @ big_linorder_Min_int @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[163]) ).

thf(350,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f54304608l_bool @ finite_finite_nat @ C ) )
     => ( ~ ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ B ) @ C ) )
       => ( ( hAPP_f22106695ol_nat @ ( hAPP_f782000547ol_nat @ big_co387207925at_nat @ A ) @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ B ) @ C ) )
          = ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ ( hAPP_nat_nat @ A @ B ) ) @ ( hAPP_f22106695ol_nat @ ( hAPP_f782000547ol_nat @ big_co387207925at_nat @ A ) @ C ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_638_setsum_Oinsert) ).

thf(2408,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f54304608l_bool @ finite_finite_nat @ C ) )
     => ( ~ ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ B ) @ C ) )
       => ( ( hAPP_f22106695ol_nat @ ( hAPP_f782000547ol_nat @ big_co387207925at_nat @ A ) @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ B ) @ C ) )
          = ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ ( hAPP_nat_nat @ A @ B ) ) @ ( hAPP_f22106695ol_nat @ ( hAPP_f782000547ol_nat @ big_co387207925at_nat @ A ) @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[350]) ).

thf(149,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ A ) @ B ) ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ A ) @ C ) ) )
    <=> ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ B ) @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_1010_nat__add__left__cancel__le) ).

thf(1663,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ A ) @ B ) ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ A ) @ C ) ) )
       => ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ B ) @ C ) ) )
      & ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ B ) @ C ) )
       => ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ A ) @ B ) ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ A ) @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[149]) ).

thf(268,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_f284875647l_bool @ ord_le951220754t_bool @ A ) @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ B ) @ C ) ) )
    <=> ( ( ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ B ) @ 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 @ B ) @ bot_bot_fun_int_bool ) ) ) @ C ) ) )
        & ( ~ ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ B ) @ A ) )
         => ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_f284875647l_bool @ ord_le951220754t_bool @ A ) @ C ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_908_subset__insert__iff) ).

thf(2089,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_f284875647l_bool @ ord_le951220754t_bool @ A ) @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ B ) @ C ) ) )
       => ( ( ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ B ) @ 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 @ B ) @ bot_bot_fun_int_bool ) ) ) @ C ) ) )
          & ( ~ ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ B ) @ A ) )
           => ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_f284875647l_bool @ ord_le951220754t_bool @ A ) @ C ) ) ) ) )
      & ( ( ( ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ B ) @ 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 @ B ) @ bot_bot_fun_int_bool ) ) ) @ C ) ) )
          & ( ~ ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ B ) @ A ) )
           => ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_f284875647l_bool @ ord_le951220754t_bool @ A ) @ C ) ) ) )
       => ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_f284875647l_bool @ ord_le951220754t_bool @ A ) @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ B ) @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[268]) ).

thf(1124,axiom,
    ! [A: $i,B: $i] : ( hBOOL @ ( hAPP_H1037229737a_bool @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ ( finite90892550iple_a @ A ) @ B ) @ bot_bo797238721a_bool ) @ B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_299_fold__graph_OemptyI) ).

thf(5295,plain,
    ! [A: $i,B: $i] : ( hBOOL @ ( hAPP_H1037229737a_bool @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ ( finite90892550iple_a @ A ) @ B ) @ bot_bo797238721a_bool ) @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1124]) ).

thf(409,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ D ) @ E ) )
     => ( ( 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 @ E ) @ A ) ) @ B ) ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ D ) @ A ) ) @ C ) ) )
      <=> ( 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 @ E ) @ D ) ) @ A ) ) @ B ) ) @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_1055_nat__le__add__iff1) ).

thf(2617,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ D ) @ E ) )
     => ( ( ( 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 @ E ) @ A ) ) @ B ) ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ D ) @ A ) ) @ C ) ) )
         => ( 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 @ E ) @ D ) ) @ A ) ) @ B ) ) @ C ) ) )
        & ( ( 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 @ E ) @ D ) ) @ A ) ) @ B ) ) @ C ) )
         => ( 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 @ E ) @ A ) ) @ B ) ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ D ) @ A ) ) @ C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[409]) ).

thf(1087,axiom,
    ! [A: $i,B: $i] :
      ( ~ ( hBOOL @ B )
      | ~ ( hBOOL @ A )
      | ( hBOOL @ ( hAPP_bool_bool @ ( hAPP_b589554111l_bool @ fconj @ B ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',help_fconj_1_1_U) ).

thf(5126,plain,
    ! [A: $i,B: $i] :
      ( ~ ( hBOOL @ B )
      | ~ ( hBOOL @ A )
      | ( hBOOL @ ( hAPP_bool_bool @ ( hAPP_b589554111l_bool @ fconj @ B ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1087]) ).

thf(418,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ A ) @ ( image_685909450_a_int @ B @ C ) ) )
     => ~ ! [D: $i] :
            ( ( A
              = ( hAPP_H1229305626_a_int @ B @ D ) )
           => ~ ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ D ) @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_242_imageE) ).

thf(2662,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ A ) @ ( image_685909450_a_int @ B @ C ) ) )
     => ~ ! [D: $i] :
            ( ( A
              = ( hAPP_H1229305626_a_int @ B @ D ) )
           => ~ ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ D ) @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[418]) ).

thf(18,axiom,
    ! [A: $i,B: $i] : ( hBOOL @ ( hAPP_int_bool @ ( hAPP_i1948725293t_bool @ ord_less_eq_int @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ ord_min_int @ A ) @ B ) ) @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_987_min__max_Oinf__le1) ).

thf(1185,plain,
    ! [A: $i,B: $i] : ( hBOOL @ ( hAPP_int_bool @ ( hAPP_i1948725293t_bool @ ord_less_eq_int @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ ord_min_int @ A ) @ B ) ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[18]) ).

thf(303,axiom,
    ! [A: $i,B: $i] :
      ( ( B != bot_bot_fun_int_bool )
     => ( ( hBOOL @ ( hAPP_f448129468l_bool @ finite_finite_int @ B ) )
       => ( ~ ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ A ) @ B ) )
         => ( ( hAPP_f1594865479ol_int @ ( finite_fold1_int @ times_times_int ) @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ A ) @ B ) )
            = ( hAPP_int_int @ ( hAPP_int_fun_int_int @ times_times_int @ A ) @ ( hAPP_f1594865479ol_int @ ( finite_fold1_int @ times_times_int ) @ B ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_363_fold1__insert) ).

thf(2235,plain,
    ! [A: $i,B: $i] :
      ( ( B != bot_bot_fun_int_bool )
     => ( ( hBOOL @ ( hAPP_f448129468l_bool @ finite_finite_int @ B ) )
       => ( ~ ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ A ) @ B ) )
         => ( ( hAPP_f1594865479ol_int @ ( finite_fold1_int @ times_times_int ) @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ A ) @ B ) )
            = ( hAPP_int_int @ ( hAPP_int_fun_int_int @ times_times_int @ A ) @ ( hAPP_f1594865479ol_int @ ( finite_fold1_int @ times_times_int ) @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[303]) ).

thf(239,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( B
        = ( hAPP_nat_int @ C @ D ) )
     => ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ D ) @ A ) )
       => ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ B ) @ ( image_nat_int @ C @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_162_image__eqI) ).

thf(2001,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( B
        = ( hAPP_nat_int @ C @ D ) )
     => ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ D ) @ A ) )
       => ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ B ) @ ( image_nat_int @ C @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[239]) ).

thf(423,axiom,
    ! [A: $i,B: $i] :
      ( ~ ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ A ) @ B ) )
     => ( ( hAPP_f800510211t_bool @ ( hAPP_f1730770594t_bool @ minus_2067140911t_bool @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ A ) @ B ) ) @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ A ) @ bot_bot_fun_nat_bool ) )
        = B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_420_Diff__insert__absorb) ).

thf(2677,plain,
    ! [A: $i,B: $i] :
      ( ~ ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ A ) @ B ) )
     => ( ( hAPP_f800510211t_bool @ ( hAPP_f1730770594t_bool @ minus_2067140911t_bool @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ A ) @ B ) ) @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ A ) @ bot_bot_fun_nat_bool ) )
        = B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[423]) ).

thf(883,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ A ) @ B )
        = ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ A ) @ C ) )
     => ( B = C ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_658_add__left__imp__eq) ).

thf(4380,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ A ) @ B )
        = ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ A ) @ C ) )
     => ( B = C ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[883]) ).

thf(228,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( finite988810631ne_nat @ B @ C ) )
     => ( ( hBOOL @ ( hAPP_f54304608l_bool @ finite_finite_nat @ A ) )
       => ( ( A != bot_bot_fun_nat_bool )
         => ( ! [D: $i,E: $i] : ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ B @ D ) @ E ) ) @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ D ) @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ E ) @ bot_bot_fun_nat_bool ) ) ) )
           => ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ ( hAPP_f22106695ol_nat @ C @ A ) ) @ A ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_273_folding__one_Oclosed) ).

thf(1959,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( finite988810631ne_nat @ B @ C ) )
     => ( ( hBOOL @ ( hAPP_f54304608l_bool @ finite_finite_nat @ A ) )
       => ( ( A != bot_bot_fun_nat_bool )
         => ( ! [D: $i,E: $i] : ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ B @ D ) @ E ) ) @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ D ) @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ E ) @ bot_bot_fun_nat_bool ) ) ) )
           => ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ ( hAPP_f22106695ol_nat @ C @ A ) ) @ A ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[228]) ).

thf(522,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ A ) @ ( image_nat_nat @ B @ C ) ) )
    <=> ? [D: $i] :
          ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ D ) @ C ) )
          & ( A
            = ( hAPP_nat_nat @ B @ D ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_185_image__iff) ).

thf(3042,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ A ) @ ( image_nat_nat @ B @ C ) ) )
       => ? [D: $i] :
            ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ D ) @ C ) )
            & ( A
              = ( hAPP_nat_nat @ B @ D ) ) ) )
      & ( ? [D: $i] :
            ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ D ) @ C ) )
            & ( A
              = ( hAPP_nat_nat @ B @ D ) ) )
       => ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ A ) @ ( image_nat_nat @ B @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[522]) ).

thf(678,axiom,
    ! [A: $i,B: $i] : ( is_bool @ ( hAPP_f2110825313l_bool @ A @ B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',gsy_c_hAPP_000tc__fun_Itc__fun_Itc__Hoare____Mirabelle____uwgpyvfjxg__Otriple_It) ).

thf(3607,plain,
    ! [A: $i,B: $i] : ( is_bool @ ( hAPP_f2110825313l_bool @ A @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[678]) ).

thf(646,axiom,
    ! [A: $i] :
      ( ( hAPP_f22106695ol_nat @ the_elem_nat @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ A ) @ bot_bot_fun_nat_bool ) )
      = A ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_102_the__elem__eq) ).

thf(3491,plain,
    ! [A: $i] :
      ( ( hAPP_f22106695ol_nat @ the_elem_nat @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ A ) @ bot_bot_fun_nat_bool ) )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[646]) ).

thf(1114,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP_i611776424a_bool @ ( hAPP_f1718504751a_bool @ ( hAPP_f481410067a_bool @ cOMBB_1028320654ol_int @ A ) @ B ) @ C )
      = ( hAPP_f20753329a_bool @ A @ ( hAPP_i611776424a_bool @ B @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',help_COMBB_1_1_COMBB_000tc__fun_Itc__Hoare____Mirabelle____uwgpyvfjxg__Otriple_I_047) ).

thf(5230,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP_i611776424a_bool @ ( hAPP_f1718504751a_bool @ ( hAPP_f481410067a_bool @ cOMBB_1028320654ol_int @ A ) @ B ) @ C )
      = ( hAPP_f20753329a_bool @ A @ ( hAPP_i611776424a_bool @ B @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1114]) ).

thf(930,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ~ ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ B ) @ C ) )
     => ( ~ ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ B ) @ A ) )
       => ( ( ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ B ) @ C )
            = ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ B ) @ A ) )
        <=> ( C = A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_58_insert__ident) ).

thf(4562,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ~ ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ B ) @ C ) )
     => ( ~ ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ B ) @ A ) )
       => ( ( ( ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ B ) @ C )
              = ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ B ) @ A ) )
           => ( C = A ) )
          & ( ( C = A )
           => ( ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ B ) @ C )
              = ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ B ) @ A ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[930]) ).

thf(995,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ A ) @ ( hAPP_f20753329a_bool @ ( hAPP_f1706700729a_bool @ minus_85316870a_bool @ B ) @ C ) ) )
     => ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ A ) @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_385_DiffD1) ).

thf(4797,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ A ) @ ( hAPP_f20753329a_bool @ ( hAPP_f1706700729a_bool @ minus_85316870a_bool @ B ) @ C ) ) )
     => ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ A ) @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[995]) ).

thf(128,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ A ) @ B ) ) @ C ) )
     => ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ B ) @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_1005_add__leD2) ).

thf(1595,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ A ) @ B ) ) @ C ) )
     => ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ B ) @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[128]) ).

thf(1118,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ A ) @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ B ) @ bot_bo797238721a_bool ) )
        = ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ C ) @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ D ) @ bot_bo797238721a_bool ) ) )
    <=> ( ( ( A = C )
          & ( B = D ) )
        | ( ( A = D )
          & ( B = C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_91_doubleton__eq__iff) ).

thf(5241,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ A ) @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ B ) @ bot_bo797238721a_bool ) )
          = ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ C ) @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ D ) @ bot_bo797238721a_bool ) ) )
       => ( ( ( A = C )
            & ( B = D ) )
          | ( ( A = D )
            & ( B = C ) ) ) )
      & ( ( ( ( A = C )
            & ( B = D ) )
          | ( ( A = D )
            & ( B = C ) ) )
       => ( ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ A ) @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ B ) @ bot_bo797238721a_bool ) )
          = ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ C ) @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ D ) @ bot_bo797238721a_bool ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1118]) ).

thf(39,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP_int_bool @ ( hAPP_i1948725293t_bool @ ord_less_eq_int @ ( hAPP_nat_int @ semiri1621563631at_int @ A ) ) @ ( hAPP_nat_int @ semiri1621563631at_int @ B ) ) )
    <=> ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ A ) @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_1066_zle__int) ).

thf(1251,plain,
    ! [A: $i,B: $i] :
      ( ( ( hBOOL @ ( hAPP_int_bool @ ( hAPP_i1948725293t_bool @ ord_less_eq_int @ ( hAPP_nat_int @ semiri1621563631at_int @ A ) ) @ ( hAPP_nat_int @ semiri1621563631at_int @ B ) ) )
       => ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ A ) @ B ) ) )
      & ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ A ) @ B ) )
       => ( hBOOL @ ( hAPP_int_bool @ ( hAPP_i1948725293t_bool @ ord_less_eq_int @ ( hAPP_nat_int @ semiri1621563631at_int @ A ) ) @ ( hAPP_nat_int @ semiri1621563631at_int @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[39]) ).

thf(179,axiom,
    ! [A: $i,B: $i] :
      ( ~ ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ A ) @ B ) )
     => ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ B ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_737_linorder__le__cases) ).

thf(1764,plain,
    ! [A: $i,B: $i] :
      ( ~ ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ A ) @ B ) )
     => ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ B ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[179]) ).

thf(352,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ A ) @ B ) )
     => ( ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ ord_min_nat @ A ) @ B )
        = A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_974_min__max_Oinf__absorb1) ).

thf(2413,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ A ) @ B ) )
     => ( ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ ord_min_nat @ A ) @ B )
        = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[352]) ).

thf(347,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_f103356543l_bool @ ord_le1568362934t_bool @ A ) @ B ) )
     => ( ( A != bot_bot_fun_nat_bool )
       => ( ( hBOOL @ ( hAPP_f54304608l_bool @ finite_finite_nat @ B ) )
         => ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ ( hAPP_f22106695ol_nat @ big_linorder_Min_nat @ B ) ) @ ( hAPP_f22106695ol_nat @ big_linorder_Min_nat @ A ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_944_Min__antimono) ).

thf(2391,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_f103356543l_bool @ ord_le1568362934t_bool @ A ) @ B ) )
     => ( ( A != bot_bot_fun_nat_bool )
       => ( ( hBOOL @ ( hAPP_f54304608l_bool @ finite_finite_nat @ B ) )
         => ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ ( hAPP_f22106695ol_nat @ big_linorder_Min_nat @ B ) ) @ ( hAPP_f22106695ol_nat @ big_linorder_Min_nat @ A ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[347]) ).

thf(153,axiom,
    ! [A: $i,B: $i] : ( is_bool @ ( hAPP_int_bool @ A @ B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',gsy_c_hAPP_000tc__Int__Oint_000tc__HOL__Obool) ).

thf(1680,plain,
    ! [A: $i,B: $i] : ( is_bool @ ( hAPP_int_bool @ A @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[153]) ).

thf(1080,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP_f1695230391l_bool @ finite1738664244iple_a @ B ) )
     => ( ( B != bot_bo797238721a_bool )
       => ( ! [C: $i] : ( hBOOL @ ( hAPP_f1695230391l_bool @ A @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ C ) @ bot_bo797238721a_bool ) ) )
         => ( ! [C: $i,D: $i] :
                ( ( hBOOL @ ( hAPP_f1695230391l_bool @ finite1738664244iple_a @ D ) )
               => ( ( D != bot_bo797238721a_bool )
                 => ( ~ ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ C ) @ D ) )
                   => ( ( hBOOL @ ( hAPP_f1695230391l_bool @ A @ D ) )
                     => ( hBOOL @ ( hAPP_f1695230391l_bool @ A @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ C ) @ D ) ) ) ) ) ) )
           => ( hBOOL @ ( hAPP_f1695230391l_bool @ A @ B ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_351_finite__ne__induct) ).

thf(5095,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP_f1695230391l_bool @ finite1738664244iple_a @ B ) )
     => ( ( B != bot_bo797238721a_bool )
       => ( ! [C: $i] : ( hBOOL @ ( hAPP_f1695230391l_bool @ A @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ C ) @ bot_bo797238721a_bool ) ) )
         => ( ! [C: $i,D: $i] :
                ( ( hBOOL @ ( hAPP_f1695230391l_bool @ finite1738664244iple_a @ D ) )
               => ( ( D != bot_bo797238721a_bool )
                 => ( ~ ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ C ) @ D ) )
                   => ( ( hBOOL @ ( hAPP_f1695230391l_bool @ A @ D ) )
                     => ( hBOOL @ ( hAPP_f1695230391l_bool @ A @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ C ) @ D ) ) ) ) ) ) )
           => ( hBOOL @ ( hAPP_f1695230391l_bool @ A @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1080]) ).

thf(878,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( finite1432773856em_int @ B @ C ) )
     => ( ( hAPP_int_int @ ( hAPP_int_fun_int_int @ B @ A ) @ A )
        = A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_373_folding__one__idem_Oidem) ).

thf(4359,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( finite1432773856em_int @ B @ C ) )
     => ( ( hAPP_int_int @ ( hAPP_int_fun_int_int @ B @ A ) @ A )
        = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[878]) ).

thf(586,axiom,
    ! [A: $i,B: $i] :
      ( ! [C: $i] :
          ( ( hBOOL @ ( hAPP_int_bool @ B @ C ) )
         => ( hBOOL @ ( hAPP_int_bool @ A @ C ) ) )
     => ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_f284875647l_bool @ ord_le951220754t_bool @ B ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_938_predicate1I) ).

thf(3274,plain,
    ! [A: $i,B: $i] :
      ( ! [C: $i] :
          ( ( hBOOL @ ( hAPP_int_bool @ B @ C ) )
         => ( hBOOL @ ( hAPP_int_bool @ A @ C ) ) )
     => ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_f284875647l_bool @ ord_le951220754t_bool @ B ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[586]) ).

thf(857,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f54304608l_bool @ finite_finite_nat @ C ) )
     => ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_f103356543l_bool @ ord_le1568362934t_bool @ C ) @ ( image_nat_nat @ A @ B ) ) )
       => ? [D: $i] :
            ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_f103356543l_bool @ ord_le1568362934t_bool @ D ) @ B ) )
            & ( hBOOL @ ( hAPP_f54304608l_bool @ finite_finite_nat @ D ) )
            & ( C
              = ( image_nat_nat @ A @ D ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_922_finite__subset__image) ).

thf(4269,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f54304608l_bool @ finite_finite_nat @ C ) )
     => ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_f103356543l_bool @ ord_le1568362934t_bool @ C ) @ ( image_nat_nat @ A @ B ) ) )
       => ? [D: $i] :
            ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_f103356543l_bool @ ord_le1568362934t_bool @ D ) @ B ) )
            & ( hBOOL @ ( hAPP_f54304608l_bool @ finite_finite_nat @ D ) )
            & ( C
              = ( image_nat_nat @ A @ D ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[857]) ).

thf(700,axiom,
    ! [A: $i,B: $i] :
      ( ! [C: $i] : ( hBOOL @ ( hAPP_bool_bool @ ( hAPP_b589554111l_bool @ ord_less_eq_bool @ ( hAPP_nat_bool @ A @ C ) ) @ ( hAPP_nat_bool @ B @ C ) ) )
     => ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_f103356543l_bool @ ord_le1568362934t_bool @ A ) @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_931_le__funI) ).

thf(3686,plain,
    ! [A: $i,B: $i] :
      ( ! [C: $i] : ( hBOOL @ ( hAPP_bool_bool @ ( hAPP_b589554111l_bool @ ord_less_eq_bool @ ( hAPP_nat_bool @ A @ C ) ) @ ( hAPP_nat_bool @ B @ C ) ) )
     => ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_f103356543l_bool @ ord_le1568362934t_bool @ A ) @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[700]) ).

thf(11,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_int_bool @ ( hAPP_i1948725293t_bool @ ord_less_eq_int @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ A ) @ B ) ) @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ C ) @ B ) ) )
    <=> ( hBOOL @ ( hAPP_int_bool @ ( hAPP_i1948725293t_bool @ ord_less_eq_int @ A ) @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_830_add__le__cancel__right) ).

thf(1154,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( hBOOL @ ( hAPP_int_bool @ ( hAPP_i1948725293t_bool @ ord_less_eq_int @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ A ) @ B ) ) @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ C ) @ B ) ) )
       => ( hBOOL @ ( hAPP_int_bool @ ( hAPP_i1948725293t_bool @ ord_less_eq_int @ A ) @ C ) ) )
      & ( ( hBOOL @ ( hAPP_int_bool @ ( hAPP_i1948725293t_bool @ ord_less_eq_int @ A ) @ C ) )
       => ( hBOOL @ ( hAPP_int_bool @ ( hAPP_i1948725293t_bool @ ord_less_eq_int @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ A ) @ B ) ) @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ C ) @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[11]) ).

thf(1053,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_H1037229737a_bool @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ ( finite90892550iple_a @ A ) @ B ) @ bot_bo797238721a_bool ) @ C ) )
     => ( C = B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_302_empty__fold__graphE) ).

thf(4998,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_H1037229737a_bool @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ ( finite90892550iple_a @ A ) @ B ) @ bot_bo797238721a_bool ) @ C ) )
     => ( C = B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1053]) ).

thf(200,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP_f22106695ol_nat @ ( finite_fold_nat_nat @ A @ B ) @ C )
      = ( hAPP_f22106695ol_nat @ the_nat @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ ( finite929467206at_nat @ A ) @ B ) @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_564_fold__def) ).

thf(1833,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP_f22106695ol_nat @ ( finite_fold_nat_nat @ A @ B ) @ C )
      = ( hAPP_f22106695ol_nat @ the_nat @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ ( finite929467206at_nat @ A ) @ B ) @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[200]) ).

thf(872,axiom,
    ! [A: $i,B: $i] :
      ( ( 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 ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_856_rev__finite__subset) ).

thf(4341,plain,
    ! [A: $i,B: $i] :
      ( ( 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 ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[872]) ).

thf(601,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( finite1432773856em_int @ C @ D ) )
     => ( ( hBOOL @ ( hAPP_f448129468l_bool @ finite_finite_int @ B ) )
       => ( ( B != bot_bot_fun_int_bool )
         => ( ( hAPP_f1594865479ol_int @ D @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ A ) @ B ) )
            = ( hAPP_int_int @ ( hAPP_int_fun_int_int @ C @ A ) @ ( hAPP_f1594865479ol_int @ D @ B ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_349_folding__one__idem_Oinsert__idem) ).

thf(3319,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( finite1432773856em_int @ C @ D ) )
     => ( ( hBOOL @ ( hAPP_f448129468l_bool @ finite_finite_int @ B ) )
       => ( ( B != bot_bot_fun_int_bool )
         => ( ( hAPP_f1594865479ol_int @ D @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ A ) @ B ) )
            = ( hAPP_int_int @ ( hAPP_int_fun_int_int @ C @ A ) @ ( hAPP_f1594865479ol_int @ D @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[601]) ).

thf(650,axiom,
    ! [A: $i,B: $i] :
      ( ( 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 ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_376_set__diff__eq) ).

thf(3504,plain,
    ! [A: $i,B: $i] :
      ( ( 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 ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[650]) ).

thf(581,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP_f54304608l_bool @ finite_finite_nat @ B ) )
     => ( ( hBOOL @ ( hAPP_f54304608l_bool @ A @ bot_bot_fun_nat_bool ) )
       => ( ! [C: $i,D: $i] :
              ( ( hBOOL @ ( hAPP_f54304608l_bool @ finite_finite_nat @ D ) )
             => ( ~ ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ C ) @ D ) )
               => ( ( hBOOL @ ( hAPP_f54304608l_bool @ A @ D ) )
                 => ( hBOOL @ ( hAPP_f54304608l_bool @ A @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ C ) @ D ) ) ) ) ) )
         => ( hBOOL @ ( hAPP_f54304608l_bool @ A @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_317_finite__induct) ).

thf(3255,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP_f54304608l_bool @ finite_finite_nat @ B ) )
     => ( ( hBOOL @ ( hAPP_f54304608l_bool @ A @ bot_bot_fun_nat_bool ) )
       => ( ! [C: $i,D: $i] :
              ( ( hBOOL @ ( hAPP_f54304608l_bool @ finite_finite_nat @ D ) )
             => ( ~ ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ C ) @ D ) )
               => ( ( hBOOL @ ( hAPP_f54304608l_bool @ A @ D ) )
                 => ( hBOOL @ ( hAPP_f54304608l_bool @ A @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ C ) @ D ) ) ) ) ) )
         => ( hBOOL @ ( hAPP_f54304608l_bool @ A @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[581]) ).

thf(375,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f54304608l_bool @ finite_finite_nat @ C ) )
     => ( ( hAPP_f659380387ol_int @ ( big_co195215938at_int @ A ) @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ B ) @ C ) )
        = ( hAPP_int_int @ ( hAPP_int_fun_int_int @ times_times_int @ ( hAPP_nat_int @ A @ B ) ) @ ( hAPP_f659380387ol_int @ ( big_co195215938at_int @ A ) @ ( hAPP_f800510211t_bool @ ( hAPP_f1730770594t_bool @ minus_2067140911t_bool @ C ) @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ B ) @ bot_bot_fun_nat_bool ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_469_setprod_Oinsert__remove) ).

thf(2489,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f54304608l_bool @ finite_finite_nat @ C ) )
     => ( ( hAPP_f659380387ol_int @ ( big_co195215938at_int @ A ) @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ B ) @ C ) )
        = ( hAPP_int_int @ ( hAPP_int_fun_int_int @ times_times_int @ ( hAPP_nat_int @ A @ B ) ) @ ( hAPP_f659380387ol_int @ ( big_co195215938at_int @ A ) @ ( hAPP_f800510211t_bool @ ( hAPP_f1730770594t_bool @ minus_2067140911t_bool @ C ) @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ B ) @ bot_bot_fun_nat_bool ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[375]) ).

thf(716,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( B
        = ( hAPP_H905846293iple_a @ C @ D ) )
     => ( ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ D ) @ A ) )
       => ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ B ) @ ( image_533441733iple_a @ C @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_155_image__eqI) ).

thf(3736,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( B
        = ( hAPP_H905846293iple_a @ C @ D ) )
     => ( ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ D ) @ A ) )
       => ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ B ) @ ( image_533441733iple_a @ C @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[716]) ).

thf(799,axiom,
    ! [A: $i,B: $i] : ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_f284875647l_bool @ ord_le951220754t_bool @ A ) @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ B ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_870_subset__insertI) ).

thf(4032,plain,
    ! [A: $i,B: $i] : ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_f284875647l_bool @ ord_le951220754t_bool @ A ) @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ B ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[799]) ).

thf(677,axiom,
    ! [A: $i,B: $i] :
      ( ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ A ) @ B )
      = ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ B ) @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_1093_zadd__commute) ).

thf(3604,plain,
    ! [A: $i,B: $i] :
      ( ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ A ) @ B )
      = ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ B ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[677]) ).

thf(274,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ minus_minus_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ A ) @ B ) ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ C ) @ B ) )
      = ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ minus_minus_nat @ A ) @ C ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_1022_diff__cancel2) ).

thf(2134,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ minus_minus_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ A ) @ B ) ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ C ) @ B ) )
      = ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ minus_minus_nat @ A ) @ C ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[274]) ).

thf(574,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP_int_int @ ( hAPP_int_fun_int_int @ times_times_int @ A ) @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ times_times_int @ B ) @ C ) )
      = ( hAPP_int_int @ ( hAPP_int_fun_int_int @ times_times_int @ B ) @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ times_times_int @ A ) @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_692_comm__semiring__1__class_Onormalizing__semiring__rules_I19_J) ).

thf(3234,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP_int_int @ ( hAPP_int_fun_int_int @ times_times_int @ A ) @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ times_times_int @ B ) @ C ) )
      = ( hAPP_int_int @ ( hAPP_int_fun_int_int @ times_times_int @ B ) @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ times_times_int @ A ) @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[574]) ).

thf(774,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ A ) @ B ) ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ C ) @ D ) )
      = ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ C ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ A ) @ B ) ) @ D ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_679_comm__semiring__1__class_Onormalizing__semiring__rules_I15_J) ).

thf(3925,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ A ) @ B ) ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ C ) @ D ) )
      = ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ C ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ A ) @ B ) ) @ D ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[774]) ).

thf(473,axiom,
    ! [A: $i,B: $i] :
      ( ( B = bot_bot_fun_int_bool )
     => ~ ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ A ) @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_32_equals0D) ).

thf(2859,plain,
    ! [A: $i,B: $i] :
      ( ( B = bot_bot_fun_int_bool )
     => ~ ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ A ) @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[473]) ).

thf(456,axiom,
    ! [A: $i,B: $i] :
      ( ~ ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_nat @ A ) @ B ) )
     => ( ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ B ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ minus_minus_nat @ A ) @ B ) )
        = A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_1130_add__diff__inverse) ).

thf(2789,plain,
    ! [A: $i,B: $i] :
      ( ~ ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_nat @ A ) @ B ) )
     => ( ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ B ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ minus_minus_nat @ A ) @ B ) )
        = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[456]) ).

thf(653,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_f284875647l_bool @ ord_le951220754t_bool @ A ) @ B ) )
    <=> ! [C: $i] : ( hBOOL @ ( hAPP_bool_bool @ ( hAPP_b589554111l_bool @ ord_less_eq_bool @ ( hAPP_int_bool @ A @ C ) ) @ ( hAPP_int_bool @ B @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_788_le__fun__def) ).

thf(3519,plain,
    ! [A: $i,B: $i] :
      ( ( ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_f284875647l_bool @ ord_le951220754t_bool @ A ) @ B ) )
       => ! [C: $i] : ( hBOOL @ ( hAPP_bool_bool @ ( hAPP_b589554111l_bool @ ord_less_eq_bool @ ( hAPP_int_bool @ A @ C ) ) @ ( hAPP_int_bool @ B @ C ) ) ) )
      & ( ! [C: $i] : ( hBOOL @ ( hAPP_bool_bool @ ( hAPP_b589554111l_bool @ ord_less_eq_bool @ ( hAPP_int_bool @ A @ C ) ) @ ( hAPP_int_bool @ B @ C ) ) )
       => ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_f284875647l_bool @ ord_le951220754t_bool @ A ) @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[653]) ).

thf(679,axiom,
    ! [A: $i] :
      ( ( bot_bot_fun_int_bool
        = ( hAPP_f1805168059t_bool @ collect_int @ A ) )
    <=> ! [B: $i] :
          ~ ( hBOOL @ ( hAPP_int_bool @ A @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_40_empty__Collect__eq) ).

thf(3609,plain,
    ! [A: $i] :
      ( ( ( bot_bot_fun_int_bool
          = ( hAPP_f1805168059t_bool @ collect_int @ A ) )
       => ! [B: $i] :
            ~ ( hBOOL @ ( hAPP_int_bool @ A @ B ) ) )
      & ( ! [B: $i] :
            ~ ( hBOOL @ ( hAPP_int_bool @ A @ B ) )
       => ( bot_bot_fun_int_bool
          = ( hAPP_f1805168059t_bool @ collect_int @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[679]) ).

thf(417,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP_f54304608l_bool @ finite_finite_nat @ B ) )
     => ( ( hBOOL @ ( hAPP_f54304608l_bool @ A @ B ) )
       => ( ! [C: $i,D: $i] :
              ( ( hBOOL @ ( hAPP_f54304608l_bool @ finite_finite_nat @ D ) )
             => ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ C ) @ D ) )
               => ( ( hBOOL @ ( hAPP_f54304608l_bool @ A @ D ) )
                 => ( hBOOL @ ( hAPP_f54304608l_bool @ A @ ( hAPP_f800510211t_bool @ ( hAPP_f1730770594t_bool @ minus_2067140911t_bool @ D ) @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ C ) @ bot_bot_fun_nat_bool ) ) ) ) ) ) )
         => ( hBOOL @ ( hAPP_f54304608l_bool @ A @ bot_bot_fun_nat_bool ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_439_finite__empty__induct) ).

thf(2657,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP_f54304608l_bool @ finite_finite_nat @ B ) )
     => ( ( hBOOL @ ( hAPP_f54304608l_bool @ A @ B ) )
       => ( ! [C: $i,D: $i] :
              ( ( hBOOL @ ( hAPP_f54304608l_bool @ finite_finite_nat @ D ) )
             => ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ C ) @ D ) )
               => ( ( hBOOL @ ( hAPP_f54304608l_bool @ A @ D ) )
                 => ( hBOOL @ ( hAPP_f54304608l_bool @ A @ ( hAPP_f800510211t_bool @ ( hAPP_f1730770594t_bool @ minus_2067140911t_bool @ D ) @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ C ) @ bot_bot_fun_nat_bool ) ) ) ) ) ) )
         => ( hBOOL @ ( hAPP_f54304608l_bool @ A @ bot_bot_fun_nat_bool ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[417]) ).

thf(289,axiom,
    ! [A: $i,B: $i] :
      ( ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ ( hAPP_nat_int @ semiri1621563631at_int @ A ) ) @ ( hAPP_nat_int @ semiri1621563631at_int @ B ) )
      = ( hAPP_nat_int @ semiri1621563631at_int @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ A ) @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_1065_zadd__int) ).

thf(2188,plain,
    ! [A: $i,B: $i] :
      ( ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ ( hAPP_nat_int @ semiri1621563631at_int @ A ) ) @ ( hAPP_nat_int @ semiri1621563631at_int @ B ) )
      = ( hAPP_nat_int @ semiri1621563631at_int @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ A ) @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[289]) ).

thf(1074,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ B ) @ C ) )
     => ( ~ ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ B ) @ A ) )
       => ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ B ) @ ( hAPP_f20753329a_bool @ ( hAPP_f1706700729a_bool @ minus_85316870a_bool @ C ) @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_364_DiffI) ).

thf(5067,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ B ) @ C ) )
     => ( ~ ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ B ) @ A ) )
       => ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ B ) @ ( hAPP_f20753329a_bool @ ( hAPP_f1706700729a_bool @ minus_85316870a_bool @ C ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1074]) ).

thf(1029,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP_f1695230391l_bool @ finite1738664244iple_a @ B ) )
     => ( ( hBOOL @ ( hAPP_f1695230391l_bool @ A @ B ) )
       => ( ! [C: $i,D: $i] :
              ( ( hBOOL @ ( hAPP_f1695230391l_bool @ finite1738664244iple_a @ D ) )
             => ( ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ C ) @ D ) )
               => ( ( hBOOL @ ( hAPP_f1695230391l_bool @ A @ D ) )
                 => ( hBOOL @ ( hAPP_f1695230391l_bool @ A @ ( hAPP_f20753329a_bool @ ( hAPP_f1706700729a_bool @ minus_85316870a_bool @ D ) @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ C ) @ bot_bo797238721a_bool ) ) ) ) ) ) )
         => ( hBOOL @ ( hAPP_f1695230391l_bool @ A @ bot_bo797238721a_bool ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_440_finite__empty__induct) ).

thf(4926,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP_f1695230391l_bool @ finite1738664244iple_a @ B ) )
     => ( ( hBOOL @ ( hAPP_f1695230391l_bool @ A @ B ) )
       => ( ! [C: $i,D: $i] :
              ( ( hBOOL @ ( hAPP_f1695230391l_bool @ finite1738664244iple_a @ D ) )
             => ( ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ C ) @ D ) )
               => ( ( hBOOL @ ( hAPP_f1695230391l_bool @ A @ D ) )
                 => ( hBOOL @ ( hAPP_f1695230391l_bool @ A @ ( hAPP_f20753329a_bool @ ( hAPP_f1706700729a_bool @ minus_85316870a_bool @ D ) @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ C ) @ bot_bo797238721a_bool ) ) ) ) ) ) )
         => ( hBOOL @ ( hAPP_f1695230391l_bool @ A @ bot_bo797238721a_bool ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1029]) ).

thf(422,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP_H1037229737a_bool @ ( hAPP_i611776424a_bool @ ( hAPP_f920351535a_bool @ cOMBC_1530706207t_bool @ A ) @ B ) @ C )
      = ( hAPP_int_bool @ ( hAPP_H1675210280t_bool @ A @ C ) @ B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',help_COMBC_1_1_COMBC_000tc__Hoare____Mirabelle____uwgpyvfjxg__Otriple_It__a_J_00_011) ).

thf(2674,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP_H1037229737a_bool @ ( hAPP_i611776424a_bool @ ( hAPP_f920351535a_bool @ cOMBC_1530706207t_bool @ A ) @ B ) @ C )
      = ( hAPP_int_bool @ ( hAPP_H1675210280t_bool @ A @ C ) @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[422]) ).

thf(942,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP_a1200519163e_bool @ ( hAPP_f963367678e_bool @ ( hAPP_f375255701e_bool @ cOMBB_145932198bool_a @ A ) @ B ) @ C )
      = ( hAPP_f644196280e_bool @ A @ ( hAPP_a849909144l_bool @ B @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',help_COMBB_1_1_COMBB_000tc__fun_Itc__Com__Ostate_Mtc__fun_Itc__HOL__Obool_Mtc__H_045) ).

thf(4605,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP_a1200519163e_bool @ ( hAPP_f963367678e_bool @ ( hAPP_f375255701e_bool @ cOMBB_145932198bool_a @ A ) @ B ) @ C )
      = ( hAPP_f644196280e_bool @ A @ ( hAPP_a849909144l_bool @ B @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[942]) ).

thf(509,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ A ) @ ( hAPP_f800510211t_bool @ ( hAPP_f1730770594t_bool @ minus_2067140911t_bool @ B ) @ C ) ) )
     => ~ ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ A ) @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_389_DiffD2) ).

thf(2991,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ A ) @ ( hAPP_f800510211t_bool @ ( hAPP_f1730770594t_bool @ minus_2067140911t_bool @ B ) @ C ) ) )
     => ~ ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ A ) @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[509]) ).

thf(1117,axiom,
    ! [A: $i,B: $i] :
      ( ~ ( hBOOL @ ( hAPP_bool_bool @ ( hAPP_b589554111l_bool @ fconj @ A ) @ B ) )
      | ( hBOOL @ B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',help_fconj_3_1_U) ).

thf(5239,plain,
    ! [A: $i,B: $i] :
      ( ~ ( hBOOL @ ( hAPP_bool_bool @ ( hAPP_b589554111l_bool @ fconj @ A ) @ B ) )
      | ( hBOOL @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1117]) ).

thf(230,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP_f448129468l_bool @ finite_finite_int @ B ) )
     => ( ( hBOOL @ ( hAPP_f448129468l_bool @ A @ bot_bot_fun_int_bool ) )
       => ( ! [C: $i,D: $i] :
              ( ( hBOOL @ ( hAPP_f448129468l_bool @ finite_finite_int @ D ) )
             => ( ~ ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ C ) @ D ) )
               => ( ( hBOOL @ ( hAPP_f448129468l_bool @ A @ D ) )
                 => ( hBOOL @ ( hAPP_f448129468l_bool @ A @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ C ) @ D ) ) ) ) ) )
         => ( hBOOL @ ( hAPP_f448129468l_bool @ A @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_319_finite__induct) ).

thf(1965,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP_f448129468l_bool @ finite_finite_int @ B ) )
     => ( ( hBOOL @ ( hAPP_f448129468l_bool @ A @ bot_bot_fun_int_bool ) )
       => ( ! [C: $i,D: $i] :
              ( ( hBOOL @ ( hAPP_f448129468l_bool @ finite_finite_int @ D ) )
             => ( ~ ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ C ) @ D ) )
               => ( ( hBOOL @ ( hAPP_f448129468l_bool @ A @ D ) )
                 => ( hBOOL @ ( hAPP_f448129468l_bool @ A @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ C ) @ D ) ) ) ) ) )
         => ( hBOOL @ ( hAPP_f448129468l_bool @ A @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[230]) ).

thf(32,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( B = C )
     => ( ( hBOOL @ ( hAPP_int_bool @ ( hAPP_i1948725293t_bool @ ord_less_eq_int @ A ) @ C ) )
       => ( hBOOL @ ( hAPP_int_bool @ ( hAPP_i1948725293t_bool @ ord_less_eq_int @ A ) @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_766_xt1_I3_J) ).

thf(1226,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( B = C )
     => ( ( hBOOL @ ( hAPP_int_bool @ ( hAPP_i1948725293t_bool @ ord_less_eq_int @ A ) @ C ) )
       => ( hBOOL @ ( hAPP_int_bool @ ( hAPP_i1948725293t_bool @ ord_less_eq_int @ A ) @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[32]) ).

thf(429,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ B ) @ C ) )
     => ( ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ ( hAPP_int_int @ A @ B ) ) @ ( image_int_int @ A @ C ) )
        = ( image_int_int @ A @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_216_insert__image) ).

thf(2695,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ B ) @ C ) )
     => ( ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ ( hAPP_int_int @ A @ B ) ) @ ( image_int_int @ A @ C ) )
        = ( image_int_int @ A @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[429]) ).

thf(40,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_int_bool @ ( hAPP_i1948725293t_bool @ ord_less_eq_int @ A ) @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ ord_min_int @ B ) @ C ) ) )
     => ~ ( ( hBOOL @ ( hAPP_int_bool @ ( hAPP_i1948725293t_bool @ ord_less_eq_int @ A ) @ B ) )
         => ~ ( hBOOL @ ( hAPP_int_bool @ ( hAPP_i1948725293t_bool @ ord_less_eq_int @ A ) @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_965_min__max_Ole__infE) ).

thf(1256,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_int_bool @ ( hAPP_i1948725293t_bool @ ord_less_eq_int @ A ) @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ ord_min_int @ B ) @ C ) ) )
     => ~ ( ( hBOOL @ ( hAPP_int_bool @ ( hAPP_i1948725293t_bool @ ord_less_eq_int @ A ) @ B ) )
         => ~ ( hBOOL @ ( hAPP_int_bool @ ( hAPP_i1948725293t_bool @ ord_less_eq_int @ A ) @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[40]) ).

thf(6,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_int_bool @ ( hAPP_i1948725293t_bool @ ord_less_eq_int @ B ) @ C ) )
     => ( ( hBOOL @ ( hAPP_int_bool @ ( hAPP_i1948725293t_bool @ ord_less_eq_int @ C ) @ A ) )
       => ( hBOOL @ ( hAPP_int_bool @ ( hAPP_i1948725293t_bool @ ord_less_eq_int @ B ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_750_order__trans) ).

thf(1139,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_int_bool @ ( hAPP_i1948725293t_bool @ ord_less_eq_int @ B ) @ C ) )
     => ( ( hBOOL @ ( hAPP_int_bool @ ( hAPP_i1948725293t_bool @ ord_less_eq_int @ C ) @ A ) )
       => ( hBOOL @ ( hAPP_int_bool @ ( hAPP_i1948725293t_bool @ ord_less_eq_int @ B ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[6]) ).

thf(1065,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ B ) @ C ) )
     => ( ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ ( hAPP_H905846293iple_a @ A @ B ) ) @ ( image_533441733iple_a @ A @ C ) )
        = ( image_533441733iple_a @ A @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_215_insert__image) ).

thf(5039,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ B ) @ C ) )
     => ( ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ ( hAPP_H905846293iple_a @ A @ B ) ) @ ( image_533441733iple_a @ A @ C ) )
        = ( image_533441733iple_a @ A @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1065]) ).

thf(892,axiom,
    ! [A: $i] :
      ( ( hBOOL @ ( hAPP_f54304608l_bool @ finite_finite_nat @ A ) )
    <=> ( hBOOL @ ( hAPP_f448129468l_bool @ finite_finite_int @ ( image_nat_int @ semiri1621563631at_int @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_1153_transfer__nat__int__set__relations_I1_J) ).

thf(4413,plain,
    ! [A: $i] :
      ( ( ( hBOOL @ ( hAPP_f54304608l_bool @ finite_finite_nat @ A ) )
       => ( hBOOL @ ( hAPP_f448129468l_bool @ finite_finite_int @ ( image_nat_int @ semiri1621563631at_int @ A ) ) ) )
      & ( ( hBOOL @ ( hAPP_f448129468l_bool @ finite_finite_int @ ( image_nat_int @ semiri1621563631at_int @ A ) ) )
       => ( hBOOL @ ( hAPP_f54304608l_bool @ finite_finite_nat @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[892]) ).

thf(984,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( image_533441733iple_a @ A @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ B ) @ C ) )
      = ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ ( hAPP_H905846293iple_a @ A @ B ) ) @ ( image_533441733iple_a @ A @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_222_image__insert) ).

thf(4764,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( image_533441733iple_a @ A @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ B ) @ C ) )
      = ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ ( hAPP_H905846293iple_a @ A @ B ) ) @ ( image_533441733iple_a @ A @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[984]) ).

thf(446,axiom,
    ! [A: $i,B: $i] :
      ( ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ A ) @ B )
      = ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ B ) @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_675_comm__semiring__1__class_Onormalizing__semiring__rules_I24_J) ).

thf(2753,plain,
    ! [A: $i,B: $i] :
      ( ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ A ) @ B )
      = ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ B ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[446]) ).

thf(256,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ B ) @ C ) )
     => ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ ( hAPP_nat_nat @ A @ B ) ) @ ( image_nat_nat @ A @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_181_imageI) ).

thf(2055,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ B ) @ C ) )
     => ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ ( hAPP_nat_nat @ A @ B ) ) @ ( image_nat_nat @ A @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[256]) ).

thf(433,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( B = C )
     => ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ A ) @ C ) )
       => ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ A ) @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_767_xt1_I3_J) ).

thf(2706,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( B = C )
     => ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ A ) @ C ) )
       => ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ A ) @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[433]) ).

thf(283,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ D ) @ E ) )
     => ( ( 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 @ D ) @ A ) ) @ B ) ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ E ) @ A ) ) @ C ) )
        = ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ minus_minus_nat @ B ) @ ( 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 @ E ) @ D ) ) @ A ) ) @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_1041_nat__diff__add__eq2) ).

thf(2166,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ D ) @ E ) )
     => ( ( 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 @ D ) @ A ) ) @ B ) ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ E ) @ A ) ) @ C ) )
        = ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ minus_minus_nat @ B ) @ ( 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 @ E ) @ D ) ) @ A ) ) @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[283]) ).

thf(1093,axiom,
    hBOOL @ ( finite1803123436a_bool @ insert956547291iple_a ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_455_comp__fun__idem__insert) ).

thf(5158,plain,
    hBOOL @ ( finite1803123436a_bool @ insert956547291iple_a ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1093]) ).

thf(372,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ A ) @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ B ) @ C ) ) )
    <=> ( ( A = B )
        | ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ A ) @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_65_insert__iff) ).

thf(2474,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ A ) @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ B ) @ C ) ) )
       => ( ( A = B )
          | ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ A ) @ C ) ) ) )
      & ( ( ( A = B )
          | ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ A ) @ C ) ) )
       => ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ A ) @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ B ) @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[372]) ).

thf(151,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ minus_minus_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ minus_minus_nat @ A ) @ B ) ) @ C )
      = ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ minus_minus_nat @ A ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ B ) @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_1018_diff__diff__left) ).

thf(1675,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ minus_minus_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ minus_minus_nat @ A ) @ B ) ) @ C )
      = ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ minus_minus_nat @ A ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ B ) @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[151]) ).

thf(657,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f448129468l_bool @ finite_finite_int @ C ) )
     => ( ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_f284875647l_bool @ ord_le951220754t_bool @ C ) @ ( image_nat_int @ A @ B ) ) )
       => ? [D: $i] :
            ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_f103356543l_bool @ ord_le1568362934t_bool @ D ) @ B ) )
            & ( hBOOL @ ( hAPP_f54304608l_bool @ finite_finite_nat @ D ) )
            & ( C
              = ( image_nat_int @ A @ D ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_921_finite__subset__image) ).

thf(3532,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f448129468l_bool @ finite_finite_int @ C ) )
     => ( ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_f284875647l_bool @ ord_le951220754t_bool @ C ) @ ( image_nat_int @ A @ B ) ) )
       => ? [D: $i] :
            ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_f103356543l_bool @ ord_le1568362934t_bool @ D ) @ B ) )
            & ( hBOOL @ ( hAPP_f54304608l_bool @ finite_finite_nat @ D ) )
            & ( C
              = ( image_nat_int @ A @ D ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[657]) ).

thf(83,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP_f54304608l_bool @ finite_finite_nat @ B ) )
     => ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ A ) @ B ) )
       => ( ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ ord_min_nat @ A ) @ ( hAPP_f22106695ol_nat @ big_linorder_Min_nat @ B ) )
          = ( hAPP_f22106695ol_nat @ big_linorder_Min_nat @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_951_Min_Oin__idem) ).

thf(1428,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP_f54304608l_bool @ finite_finite_nat @ B ) )
     => ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ A ) @ B ) )
       => ( ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ ord_min_nat @ A ) @ ( hAPP_f22106695ol_nat @ big_linorder_Min_nat @ B ) )
          = ( hAPP_f22106695ol_nat @ big_linorder_Min_nat @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[83]) ).

thf(281,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f448129468l_bool @ finite_finite_int @ C ) )
     => ( ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ B ) @ C ) )
       => ( ( hAPP_f1594865479ol_int @ ( big_co1548731110nt_int @ A ) @ C )
          = ( hAPP_int_int @ ( hAPP_int_fun_int_int @ times_times_int @ ( hAPP_int_int @ A @ B ) ) @ ( hAPP_f1594865479ol_int @ ( big_co1548731110nt_int @ A ) @ ( hAPP_f1805168059t_bool @ ( hAPP_f1223193598t_bool @ minus_1449998731t_bool @ C ) @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ B ) @ bot_bot_fun_int_bool ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_477_setprod_Oremove) ).

thf(2161,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f448129468l_bool @ finite_finite_int @ C ) )
     => ( ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ B ) @ C ) )
       => ( ( hAPP_f1594865479ol_int @ ( big_co1548731110nt_int @ A ) @ C )
          = ( hAPP_int_int @ ( hAPP_int_fun_int_int @ times_times_int @ ( hAPP_int_int @ A @ B ) ) @ ( hAPP_f1594865479ol_int @ ( big_co1548731110nt_int @ A ) @ ( hAPP_f1805168059t_bool @ ( hAPP_f1223193598t_bool @ minus_1449998731t_bool @ C ) @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ B ) @ bot_bot_fun_int_bool ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[281]) ).

thf(824,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ minus_minus_nat @ A ) @ B ) ) @ C )
      = ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ minus_minus_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ A ) @ C ) ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ B ) @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_1020_diff__mult__distrib) ).

thf(4127,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ minus_minus_nat @ A ) @ B ) ) @ C )
      = ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ minus_minus_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ A ) @ C ) ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ B ) @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[824]) ).

thf(299,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( finite2130160977at_nat @ C ) )
     => ( ( hBOOL @ ( hAPP_f54304608l_bool @ finite_finite_nat @ B ) )
       => ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ ( finite929467206at_nat @ C ) @ A ) @ B ) @ ( hAPP_f22106695ol_nat @ ( finite_fold_nat_nat @ C @ A ) @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_519_comp__fun__commute_Ofold__graph__fold) ).

thf(2220,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( finite2130160977at_nat @ C ) )
     => ( ( hBOOL @ ( hAPP_f54304608l_bool @ finite_finite_nat @ B ) )
       => ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ ( finite929467206at_nat @ C ) @ A ) @ B ) @ ( hAPP_f22106695ol_nat @ ( finite_fold_nat_nat @ C @ A ) @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[299]) ).

thf(248,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ B ) @ C ) )
     => ( ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ minus_minus_nat @ C ) @ B ) ) @ A )
        = ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ minus_minus_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ C ) @ A ) ) @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_1036_add__diff__assoc2) ).

thf(2029,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ B ) @ C ) )
     => ( ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ minus_minus_nat @ C ) @ B ) ) @ A )
        = ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ minus_minus_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ C ) @ A ) ) @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[248]) ).

thf(819,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i] :
          ( ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ D ) @ C ) )
         => ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ ( hAPP_H905846293iple_a @ A @ D ) ) @ B ) ) )
     => ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_f1767618879l_bool @ ord_le1523781517a_bool @ ( image_533441733iple_a @ A @ C ) ) @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_927_image__subsetI) ).

thf(4096,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i] :
          ( ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ D ) @ C ) )
         => ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ ( hAPP_H905846293iple_a @ A @ D ) ) @ B ) ) )
     => ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_f1767618879l_bool @ ord_le1523781517a_bool @ ( image_533441733iple_a @ A @ C ) ) @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[819]) ).

thf(330,axiom,
    ! [A: $i,B: $i] :
      ( ( ( image_nat_nat @ A @ B )
        = bot_bot_fun_nat_bool )
    <=> ( B = bot_bot_fun_nat_bool ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_204_image__is__empty) ).

thf(2326,plain,
    ! [A: $i,B: $i] :
      ( ( ( ( image_nat_nat @ A @ B )
          = bot_bot_fun_nat_bool )
       => ( B = bot_bot_fun_nat_bool ) )
      & ( ( B = bot_bot_fun_nat_bool )
       => ( ( image_nat_nat @ A @ B )
          = bot_bot_fun_nat_bool ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[330]) ).

thf(162,axiom,
    ! [A: $i] :
      ( ( hBOOL @ ( hAPP_int_bool @ bot_bot_fun_int_bool @ A ) )
    <=> ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ A ) @ bot_bot_fun_int_bool ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_141_bot__empty__eq) ).

thf(1707,plain,
    ! [A: $i] :
      ( ( ( hBOOL @ ( hAPP_int_bool @ bot_bot_fun_int_bool @ A ) )
       => ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ A ) @ bot_bot_fun_int_bool ) ) )
      & ( ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ A ) @ bot_bot_fun_int_bool ) )
       => ( hBOOL @ ( hAPP_int_bool @ bot_bot_fun_int_bool @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[162]) ).

thf(1085,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( finite1753440478iple_a @ C @ D ) )
     => ( ( hBOOL @ ( hAPP_f1695230391l_bool @ finite1738664244iple_a @ B ) )
       => ( ( ( ( hAPP_f20753329a_bool @ ( hAPP_f1706700729a_bool @ minus_85316870a_bool @ B ) @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ A ) @ bot_bo797238721a_bool ) )
              = bot_bo797238721a_bool )
           => ( ( hAPP_f1693662087iple_a @ D @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ A ) @ B ) )
              = A ) )
          & ( ( ( hAPP_f20753329a_bool @ ( hAPP_f1706700729a_bool @ minus_85316870a_bool @ B ) @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ A ) @ bot_bo797238721a_bool ) )
             != bot_bo797238721a_bool )
           => ( ( hAPP_f1693662087iple_a @ D @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ A ) @ B ) )
              = ( hAPP_H905846293iple_a @ ( hAPP_H2085992369iple_a @ C @ A ) @ ( hAPP_f1693662087iple_a @ D @ ( hAPP_f20753329a_bool @ ( hAPP_f1706700729a_bool @ minus_85316870a_bool @ B ) @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ A ) @ bot_bo797238721a_bool ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_360_folding__one_Oinsert__remove) ).

thf(5118,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( finite1753440478iple_a @ C @ D ) )
     => ( ( hBOOL @ ( hAPP_f1695230391l_bool @ finite1738664244iple_a @ B ) )
       => ( ( ( ( hAPP_f20753329a_bool @ ( hAPP_f1706700729a_bool @ minus_85316870a_bool @ B ) @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ A ) @ bot_bo797238721a_bool ) )
              = bot_bo797238721a_bool )
           => ( ( hAPP_f1693662087iple_a @ D @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ A ) @ B ) )
              = A ) )
          & ( ( ( hAPP_f20753329a_bool @ ( hAPP_f1706700729a_bool @ minus_85316870a_bool @ B ) @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ A ) @ bot_bo797238721a_bool ) )
             != bot_bo797238721a_bool )
           => ( ( hAPP_f1693662087iple_a @ D @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ A ) @ B ) )
              = ( hAPP_H905846293iple_a @ ( hAPP_H2085992369iple_a @ C @ A ) @ ( hAPP_f1693662087iple_a @ D @ ( hAPP_f20753329a_bool @ ( hAPP_f1706700729a_bool @ minus_85316870a_bool @ B ) @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ A ) @ bot_bo797238721a_bool ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1085]) ).

thf(320,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_nat @ B ) @ C ) )
     => ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ minus_minus_nat @ B ) @ A ) ) @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_1114_less__imp__diff__less) ).

thf(2293,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_nat @ B ) @ C ) )
     => ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ minus_minus_nat @ B ) @ A ) ) @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[320]) ).

thf(691,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP_s1892499976_state @ ( hAPP_v365393659_state @ ( hAPP_f1542232213_state @ cOMBC_1193272608_state @ A ) @ B ) @ C )
      = ( hAPP_v594194232_state @ ( hAPP_s712361723_state @ A @ C ) @ B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',help_COMBC_1_1_COMBC_000tc__Com__Ostate_000tc__Com__Ovname_000tc__fun_Itc__Nat__) ).

thf(3651,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP_s1892499976_state @ ( hAPP_v365393659_state @ ( hAPP_f1542232213_state @ cOMBC_1193272608_state @ A ) @ B ) @ C )
      = ( hAPP_v594194232_state @ ( hAPP_s712361723_state @ A @ C ) @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[691]) ).

thf(704,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( finite1033474011iple_a @ B @ C ) )
     => ( ( hAPP_H905846293iple_a @ ( hAPP_H2085992369iple_a @ B @ A ) @ A )
        = A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_374_folding__one__idem_Oidem) ).

thf(3695,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( finite1033474011iple_a @ B @ C ) )
     => ( ( hAPP_H905846293iple_a @ ( hAPP_H2085992369iple_a @ B @ A ) @ A )
        = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[704]) ).

thf(820,axiom,
    ! [A: $i,B: $i] :
      ( ( 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 ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_807_set__eq__subset) ).

thf(4099,plain,
    ! [A: $i,B: $i] :
      ( ( ( 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 ) ) ) )
      & ( ( ( 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 ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[820]) ).

thf(836,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_f284875647l_bool @ ord_le951220754t_bool @ A ) @ B ) )
     => ( ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_f284875647l_bool @ ord_le951220754t_bool @ B ) @ A ) )
      <=> ( B = A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_772_order__antisym__conv) ).

thf(4171,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_f284875647l_bool @ ord_le951220754t_bool @ A ) @ B ) )
     => ( ( ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_f284875647l_bool @ ord_le951220754t_bool @ B ) @ A ) )
         => ( B = A ) )
        & ( ( B = A )
         => ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_f284875647l_bool @ ord_le951220754t_bool @ B ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[836]) ).

thf(847,axiom,
    ! [A: $i,B: $i] : ( is_bool @ ( hAPP_f448129468l_bool @ A @ B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',gsy_c_hAPP_000tc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J_000tc__HOL__Obool) ).

thf(4236,plain,
    ! [A: $i,B: $i] : ( is_bool @ ( hAPP_f448129468l_bool @ A @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[847]) ).

thf(695,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f1695230391l_bool @ finite1738664244iple_a @ C ) )
     => ( ~ ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ B ) @ C ) )
       => ( ( hAPP_f587450508ol_int @ ( big_co305732779_a_int @ A ) @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ B ) @ C ) )
          = ( hAPP_int_int @ ( hAPP_int_fun_int_int @ times_times_int @ ( hAPP_H1229305626_a_int @ A @ B ) ) @ ( hAPP_f587450508ol_int @ ( big_co305732779_a_int @ A ) @ C ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_555_setprod__insert) ).

thf(3663,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f1695230391l_bool @ finite1738664244iple_a @ C ) )
     => ( ~ ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ B ) @ C ) )
       => ( ( hAPP_f587450508ol_int @ ( big_co305732779_a_int @ A ) @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ B ) @ C ) )
          = ( hAPP_int_int @ ( hAPP_int_fun_int_int @ times_times_int @ ( hAPP_H1229305626_a_int @ A @ B ) ) @ ( hAPP_f587450508ol_int @ ( big_co305732779_a_int @ A ) @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[695]) ).

thf(208,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ A ) @ B ) ) @ C )
      = ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ A ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ B ) @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_1049_nat__add__assoc) ).

thf(1865,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ A ) @ B ) ) @ C )
      = ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ A ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ B ) @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[208]) ).

thf(141,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i] :
          ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ D ) @ C ) )
         => ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ ( hAPP_nat_nat @ A @ D ) ) @ B ) ) )
     => ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_f103356543l_bool @ ord_le1568362934t_bool @ ( image_nat_nat @ A @ C ) ) @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_929_image__subsetI) ).

thf(1635,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i] :
          ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ D ) @ C ) )
         => ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ ( hAPP_nat_nat @ A @ D ) ) @ B ) ) )
     => ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_f103356543l_bool @ ord_le1568362934t_bool @ ( image_nat_nat @ A @ C ) ) @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[141]) ).

thf(513,axiom,
    ! [A: $i,B: $i] : ( is_bool @ ( finite795500164em_nat @ A @ B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',gsy_c_Finite__Set_Ofolding__one__idem_000tc__Nat__Onat) ).

thf(3002,plain,
    ! [A: $i,B: $i] : ( is_bool @ ( finite795500164em_nat @ A @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[513]) ).

thf(170,axiom,
    ! [A: $i,B: $i] :
      ( ! [C: $i] :
          ( ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ C ) @ B ) )
         => ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ C ) @ A ) ) )
     => ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_f284875647l_bool @ ord_le951220754t_bool @ B ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_917_subsetI) ).

thf(1733,plain,
    ! [A: $i,B: $i] :
      ( ! [C: $i] :
          ( ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ C ) @ B ) )
         => ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ C ) @ A ) ) )
     => ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_f284875647l_bool @ ord_le951220754t_bool @ B ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[170]) ).

thf(366,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f54304608l_bool @ finite_finite_nat @ C ) )
     => ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ B ) @ C ) )
       => ( ( hAPP_f659380387ol_int @ ( big_co195215938at_int @ A ) @ C )
          = ( hAPP_int_int @ ( hAPP_int_fun_int_int @ times_times_int @ ( hAPP_nat_int @ A @ B ) ) @ ( hAPP_f659380387ol_int @ ( big_co195215938at_int @ A ) @ ( hAPP_f800510211t_bool @ ( hAPP_f1730770594t_bool @ minus_2067140911t_bool @ C ) @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ B ) @ bot_bot_fun_nat_bool ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_475_setprod_Oremove) ).

thf(2457,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f54304608l_bool @ finite_finite_nat @ C ) )
     => ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ B ) @ C ) )
       => ( ( hAPP_f659380387ol_int @ ( big_co195215938at_int @ A ) @ C )
          = ( hAPP_int_int @ ( hAPP_int_fun_int_int @ times_times_int @ ( hAPP_nat_int @ A @ B ) ) @ ( hAPP_f659380387ol_int @ ( big_co195215938at_int @ A ) @ ( hAPP_f800510211t_bool @ ( hAPP_f1730770594t_bool @ minus_2067140911t_bool @ C ) @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ B ) @ bot_bot_fun_nat_bool ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[366]) ).

thf(393,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( 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 @ B ) @ bot_bot_fun_nat_bool ) ) ) @ C ) )
     => ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ B ) @ A ) )
       => ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_f103356543l_bool @ ord_le1568362934t_bool @ A ) @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ B ) @ C ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_904_diff__single__insert) ).

thf(2557,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( 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 @ B ) @ bot_bot_fun_nat_bool ) ) ) @ C ) )
     => ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ B ) @ A ) )
       => ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_f103356543l_bool @ ord_le1568362934t_bool @ A ) @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ B ) @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[393]) ).

thf(949,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( finite1033474011iple_a @ C @ D ) )
     => ( ( hBOOL @ ( hAPP_f1695230391l_bool @ finite1738664244iple_a @ B ) )
       => ( ( B != bot_bo797238721a_bool )
         => ( ( hAPP_f1693662087iple_a @ D @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ A ) @ B ) )
            = ( hAPP_H905846293iple_a @ ( hAPP_H2085992369iple_a @ C @ A ) @ ( hAPP_f1693662087iple_a @ D @ B ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_348_folding__one__idem_Oinsert__idem) ).

thf(4640,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( finite1033474011iple_a @ C @ D ) )
     => ( ( hBOOL @ ( hAPP_f1695230391l_bool @ finite1738664244iple_a @ B ) )
       => ( ( B != bot_bo797238721a_bool )
         => ( ( hAPP_f1693662087iple_a @ D @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ A ) @ B ) )
            = ( hAPP_H905846293iple_a @ ( hAPP_H2085992369iple_a @ C @ A ) @ ( hAPP_f1693662087iple_a @ D @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[949]) ).

thf(688,axiom,
    ! [A: $i] :
      ( ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ one_one_nat ) @ A )
      = A ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_1063_nat__mult__1) ).

thf(3641,plain,
    ! [A: $i] :
      ( ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ one_one_nat ) @ A )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[688]) ).

thf(356,axiom,
    ! [A: $i] :
      ~ ! [B: $i,C: $i] :
          ( A
         != ( hAPP_int_int @ ( hAPP_int_fun_int_int @ minus_minus_int @ ( hAPP_nat_int @ semiri1621563631at_int @ B ) ) @ ( hAPP_nat_int @ semiri1621563631at_int @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_1172_int__diff__cases) ).

thf(2425,plain,
    ! [A: $i] :
      ~ ! [B: $i,C: $i] :
          ( A
         != ( hAPP_int_int @ ( hAPP_int_fun_int_int @ minus_minus_int @ ( hAPP_nat_int @ semiri1621563631at_int @ B ) ) @ ( hAPP_nat_int @ semiri1621563631at_int @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[356]) ).

thf(627,axiom,
    ! [A: $i,B: $i] :
      ( ( 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 ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_753_order__antisym) ).

thf(3416,plain,
    ! [A: $i,B: $i] :
      ( ( 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 ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[627]) ).

thf(596,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP_int_int @ ( hAPP_int_fun_int_int @ times_times_int @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ A ) @ B ) ) @ C )
      = ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ times_times_int @ A ) @ C ) ) @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ times_times_int @ B ) @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_700_comm__semiring__class_Odistrib) ).

thf(3302,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP_int_int @ ( hAPP_int_fun_int_int @ times_times_int @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ A ) @ B ) ) @ C )
      = ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ times_times_int @ A ) @ C ) ) @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ times_times_int @ B ) @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[596]) ).

thf(665,axiom,
    ! [A: $i] : ( is_bool @ ( finite1956789438t_bool @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',gsy_c_Finite__Set_Ocomp__fun__idem_000tc__Nat__Onat_000tc__fun_Itc__Nat__Onat_Mt) ).

thf(3568,plain,
    ! [A: $i] : ( is_bool @ ( finite1956789438t_bool @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[665]) ).

thf(142,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i] :
          ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ D ) @ C ) )
         => ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ ( hAPP_nat_int @ A @ D ) ) @ B ) ) )
     => ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_f284875647l_bool @ ord_le951220754t_bool @ ( image_nat_int @ A @ C ) ) @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_928_image__subsetI) ).

thf(1638,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i] :
          ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ D ) @ C ) )
         => ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ ( hAPP_nat_int @ A @ D ) ) @ B ) ) )
     => ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_f284875647l_bool @ ord_le951220754t_bool @ ( image_nat_int @ A @ C ) ) @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[142]) ).

thf(31,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_int_bool @ ( hAPP_i1948725293t_bool @ ord_less_eq_int @ B ) @ C ) )
     => ( ( hBOOL @ ( hAPP_int_bool @ ( hAPP_i1948725293t_bool @ ord_less_eq_int @ A ) @ B ) )
       => ( hBOOL @ ( hAPP_int_bool @ ( hAPP_i1948725293t_bool @ ord_less_eq_int @ A ) @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_742_xt1_I6_J) ).

thf(1224,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_int_bool @ ( hAPP_i1948725293t_bool @ ord_less_eq_int @ B ) @ C ) )
     => ( ( hBOOL @ ( hAPP_int_bool @ ( hAPP_i1948725293t_bool @ ord_less_eq_int @ A ) @ B ) )
       => ( hBOOL @ ( hAPP_int_bool @ ( hAPP_i1948725293t_bool @ ord_less_eq_int @ A ) @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[31]) ).

thf(337,axiom,
    ! [A: $i,B: $i] :
      ( ( B != bot_bot_fun_nat_bool )
     => ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_f103356543l_bool @ ord_le1568362934t_bool @ B ) @ A ) )
       => ( ( hBOOL @ ( hAPP_f54304608l_bool @ finite_finite_nat @ A ) )
         => ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ ( hAPP_f22106695ol_nat @ ( finite_fold1_nat @ ord_min_nat ) @ A ) ) @ ( hAPP_f22106695ol_nat @ ( finite_fold1_nat @ ord_min_nat ) @ B ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_937_fold1__antimono) ).

thf(2358,plain,
    ! [A: $i,B: $i] :
      ( ( B != bot_bot_fun_nat_bool )
     => ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_f103356543l_bool @ ord_le1568362934t_bool @ B ) @ A ) )
       => ( ( hBOOL @ ( hAPP_f54304608l_bool @ finite_finite_nat @ A ) )
         => ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ ( hAPP_f22106695ol_nat @ ( finite_fold1_nat @ ord_min_nat ) @ A ) ) @ ( hAPP_f22106695ol_nat @ ( finite_fold1_nat @ ord_min_nat ) @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[337]) ).

thf(327,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( finite1432773856em_int @ C @ D ) )
     => ( ( hBOOL @ ( hAPP_f448129468l_bool @ finite_finite_int @ B ) )
       => ( ( hAPP_f1594865479ol_int @ D @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ A ) @ B ) )
          = ( hAPP_f1594865479ol_int @ ( finite_fold_int_int @ C @ A ) @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_586_folding__one__idem_Oeq__fold__idem_H) ).

thf(2319,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( finite1432773856em_int @ C @ D ) )
     => ( ( hBOOL @ ( hAPP_f448129468l_bool @ finite_finite_int @ B ) )
       => ( ( hAPP_f1594865479ol_int @ D @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ A ) @ B ) )
          = ( hAPP_f1594865479ol_int @ ( finite_fold_int_int @ C @ A ) @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[327]) ).

thf(896,axiom,
    ! [A: $i] :
      ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_f103356543l_bool @ ord_le1568362934t_bool @ A ) @ bot_bot_fun_nat_bool ) )
    <=> ( A = bot_bot_fun_nat_bool ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_848_subset__empty) ).

thf(4433,plain,
    ! [A: $i] :
      ( ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_f103356543l_bool @ ord_le1568362934t_bool @ A ) @ bot_bot_fun_nat_bool ) )
       => ( A = bot_bot_fun_nat_bool ) )
      & ( ( A = bot_bot_fun_nat_bool )
       => ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_f103356543l_bool @ ord_le1568362934t_bool @ A ) @ bot_bot_fun_nat_bool ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[896]) ).

thf(397,axiom,
    ! [A: $i,B: $i] :
      ( ( A = B )
    <=> ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ A ) @ B ) )
        & ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ B ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_785_order__eq__iff) ).

thf(2573,plain,
    ! [A: $i,B: $i] :
      ( ( ( A = B )
       => ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ A ) @ B ) )
          & ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ B ) @ A ) ) ) )
      & ( ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ A ) @ B ) )
          & ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ B ) @ A ) ) )
       => ( A = B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[397]) ).

thf(278,axiom,
    ! [A: $i] :
      ( ( image_int_int @ A @ bot_bot_fun_int_bool )
      = bot_bot_fun_int_bool ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_207_image__empty) ).

thf(2148,plain,
    ! [A: $i] :
      ( ( image_int_int @ A @ bot_bot_fun_int_bool )
      = bot_bot_fun_int_bool ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[278]) ).

thf(380,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ A ) @ B ) )
     => ( ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ minus_minus_nat @ B ) @ A ) ) @ A )
        = B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_1033_le__add__diff__inverse2) ).

thf(2510,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ A ) @ B ) )
     => ( ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ minus_minus_nat @ B ) @ A ) ) @ A )
        = B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[380]) ).

thf(28,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_int_bool @ ( hAPP_i1948725293t_bool @ ord_less_eq_int @ B ) @ C ) )
     => ( hBOOL @ ( hAPP_int_bool @ ( hAPP_i1948725293t_bool @ ord_less_eq_int @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ A ) @ B ) ) @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ A ) @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_824_add__left__mono) ).

thf(1209,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_int_bool @ ( hAPP_i1948725293t_bool @ ord_less_eq_int @ B ) @ C ) )
     => ( hBOOL @ ( hAPP_int_bool @ ( hAPP_i1948725293t_bool @ ord_less_eq_int @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ A ) @ B ) ) @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ A ) @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[28]) ).

thf(1090,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ A ) @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ B ) @ C ) ) )
    <=> ( ( A = B )
        | ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ A ) @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_64_insert__iff) ).

thf(5143,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ A ) @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ B ) @ C ) ) )
       => ( ( A = B )
          | ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ A ) @ C ) ) ) )
      & ( ( ( A = B )
          | ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ A ) @ C ) ) )
       => ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ A ) @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ B ) @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1090]) ).

thf(76,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ A ) @ B ) )
     => ( ( A != B )
       => ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_nat @ A ) @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_1112_le__neq__implies__less) ).

thf(1408,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ A ) @ B ) )
     => ( ( A != B )
       => ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_nat @ A ) @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[76]) ).

thf(344,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ B ) @ C ) )
     => ( ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ ( hAPP_nat_nat @ A @ B ) ) @ ( image_nat_nat @ A @ C ) )
        = ( image_nat_nat @ A @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_218_insert__image) ).

thf(2383,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ B ) @ C ) )
     => ( ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ ( hAPP_nat_nat @ A @ B ) ) @ ( image_nat_nat @ A @ C ) )
        = ( image_nat_nat @ A @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[344]) ).

thf(722,axiom,
    ! [A: $i,B: $i] :
      ( ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ cOMBK_nat_nat @ A ) @ B )
      = A ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',help_COMBK_1_1_COMBK_000tc__Nat__Onat_000tc__Nat__Onat_U) ).

thf(3764,plain,
    ! [A: $i,B: $i] :
      ( ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ cOMBK_nat_nat @ A ) @ B )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[722]) ).

thf(628,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ A ) @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ B ) @ bot_bot_fun_int_bool ) )
        = ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ C ) @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ D ) @ bot_bot_fun_int_bool ) ) )
    <=> ( ( ( A = C )
          & ( B = D ) )
        | ( ( A = D )
          & ( B = C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_92_doubleton__eq__iff) ).

thf(3419,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ A ) @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ B ) @ bot_bot_fun_int_bool ) )
          = ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ C ) @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ D ) @ bot_bot_fun_int_bool ) ) )
       => ( ( ( A = C )
            & ( B = D ) )
          | ( ( A = D )
            & ( B = C ) ) ) )
      & ( ( ( ( A = C )
            & ( B = D ) )
          | ( ( A = D )
            & ( B = C ) ) )
       => ( ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ A ) @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ B ) @ bot_bot_fun_int_bool ) )
          = ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ C ) @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ D ) @ bot_bot_fun_int_bool ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[628]) ).

thf(307,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( finite795500164em_nat @ C @ D ) )
     => ( ( hBOOL @ ( hAPP_f54304608l_bool @ finite_finite_nat @ B ) )
       => ( ( hAPP_f22106695ol_nat @ D @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ A ) @ B ) )
          = ( hAPP_f22106695ol_nat @ ( finite_fold_nat_nat @ C @ A ) @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_584_folding__one__idem_Oeq__fold__idem_H) ).

thf(2246,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( finite795500164em_nat @ C @ D ) )
     => ( ( hBOOL @ ( hAPP_f54304608l_bool @ finite_finite_nat @ B ) )
       => ( ( hAPP_f22106695ol_nat @ D @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ A ) @ B ) )
          = ( hAPP_f22106695ol_nat @ ( finite_fold_nat_nat @ C @ A ) @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[307]) ).

thf(937,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP_f1695230391l_bool @ finite1738664244iple_a @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ A ) @ B ) ) )
    <=> ( hBOOL @ ( hAPP_f1695230391l_bool @ finite1738664244iple_a @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_297_finite__insert) ).

thf(4586,plain,
    ! [A: $i,B: $i] :
      ( ( ( hBOOL @ ( hAPP_f1695230391l_bool @ finite1738664244iple_a @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ A ) @ B ) ) )
       => ( hBOOL @ ( hAPP_f1695230391l_bool @ finite1738664244iple_a @ B ) ) )
      & ( ( hBOOL @ ( hAPP_f1695230391l_bool @ finite1738664244iple_a @ B ) )
       => ( hBOOL @ ( hAPP_f1695230391l_bool @ finite1738664244iple_a @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ A ) @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[937]) ).

thf(497,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ B ) @ C ) )
       => ( ( hAPP_f22106695ol_nat @ ( hAPP_f782000547ol_nat @ big_co387207925at_nat @ A ) @ ( hAPP_f800510211t_bool @ ( hAPP_f1730770594t_bool @ minus_2067140911t_bool @ C ) @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ B ) @ 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 @ A ) @ C ) ) @ ( hAPP_nat_nat @ A @ B ) ) ) )
      & ( ~ ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ B ) @ C ) )
       => ( ( hAPP_f22106695ol_nat @ ( hAPP_f782000547ol_nat @ big_co387207925at_nat @ A ) @ ( hAPP_f800510211t_bool @ ( hAPP_f1730770594t_bool @ minus_2067140911t_bool @ C ) @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ B ) @ bot_bot_fun_nat_bool ) ) )
          = ( hAPP_f22106695ol_nat @ ( hAPP_f782000547ol_nat @ big_co387207925at_nat @ A ) @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_560_setsum__diff1__nat) ).

thf(2943,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ B ) @ C ) )
       => ( ( hAPP_f22106695ol_nat @ ( hAPP_f782000547ol_nat @ big_co387207925at_nat @ A ) @ ( hAPP_f800510211t_bool @ ( hAPP_f1730770594t_bool @ minus_2067140911t_bool @ C ) @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ B ) @ 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 @ A ) @ C ) ) @ ( hAPP_nat_nat @ A @ B ) ) ) )
      & ( ~ ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ B ) @ C ) )
       => ( ( hAPP_f22106695ol_nat @ ( hAPP_f782000547ol_nat @ big_co387207925at_nat @ A ) @ ( hAPP_f800510211t_bool @ ( hAPP_f1730770594t_bool @ minus_2067140911t_bool @ C ) @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ B ) @ bot_bot_fun_nat_bool ) ) )
          = ( hAPP_f22106695ol_nat @ ( hAPP_f782000547ol_nat @ big_co387207925at_nat @ A ) @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[497]) ).

thf(292,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ B ) @ C ) )
     => ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ A ) @ B ) ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ A ) @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_825_add__left__mono) ).

thf(2197,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ B ) @ C ) )
     => ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ A ) @ B ) ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ A ) @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[292]) ).

thf(338,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_nat @ C ) @ D ) )
     => ( ( ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ A ) @ D )
          = ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ C ) @ B ) )
       => ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_nat @ A ) @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_1101_less__add__eq__less) ).

thf(2361,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_nat @ C ) @ D ) )
     => ( ( ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ A ) @ D )
          = ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ C ) @ B ) )
       => ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_nat @ A ) @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[338]) ).

thf(326,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ A ) @ ( image_int_int @ B @ C ) ) )
    <=> ? [D: $i] :
          ( ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ D ) @ C ) )
          & ( A
            = ( hAPP_int_int @ B @ D ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_184_image__iff) ).

thf(2311,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ A ) @ ( image_int_int @ B @ C ) ) )
       => ? [D: $i] :
            ( ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ D ) @ C ) )
            & ( A
              = ( hAPP_int_int @ B @ D ) ) ) )
      & ( ? [D: $i] :
            ( ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ D ) @ C ) )
            & ( A
              = ( hAPP_int_int @ B @ D ) ) )
       => ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ A ) @ ( image_int_int @ B @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[326]) ).

thf(279,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( finite1973466193nt_int @ E ) )
     => ( ( hBOOL @ ( hAPP_int_bool @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ ( finite772772422nt_int @ E ) @ A ) @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ B ) @ C ) ) @ D ) )
       => ( ~ ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ B ) @ C ) )
         => ~ ! [F: $i] :
                ( ( D
                  = ( hAPP_int_int @ ( hAPP_int_fun_int_int @ E @ B ) @ F ) )
               => ~ ( hBOOL @ ( hAPP_int_bool @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ ( finite772772422nt_int @ E ) @ A ) @ C ) @ F ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_591_comp__fun__commute_Ofold__graph__insertE) ).

thf(2151,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( finite1973466193nt_int @ E ) )
     => ( ( hBOOL @ ( hAPP_int_bool @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ ( finite772772422nt_int @ E ) @ A ) @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ B ) @ C ) ) @ D ) )
       => ( ~ ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ B ) @ C ) )
         => ~ ! [F: $i] :
                ( ( D
                  = ( hAPP_int_int @ ( hAPP_int_fun_int_int @ E @ B ) @ F ) )
               => ~ ( hBOOL @ ( hAPP_int_bool @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ ( finite772772422nt_int @ E ) @ A ) @ C ) @ F ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[279]) ).

thf(301,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ D ) @ E ) )
     => ( ( 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 @ D ) @ A ) ) @ B ) ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ E ) @ A ) ) @ C ) ) )
      <=> ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_nat @ B ) @ ( 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 @ E ) @ D ) ) @ A ) ) @ C ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_1135_nat__less__add__iff2) ).

thf(2229,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ D ) @ E ) )
     => ( ( ( 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 @ D ) @ A ) ) @ B ) ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ E ) @ A ) ) @ C ) ) )
         => ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_nat @ B ) @ ( 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 @ E ) @ D ) ) @ A ) ) @ C ) ) ) )
        & ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_nat @ B ) @ ( 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 @ E ) @ D ) ) @ A ) ) @ C ) ) )
         => ( 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 @ D ) @ A ) ) @ B ) ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ E ) @ A ) ) @ C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[301]) ).

thf(736,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_f284875647l_bool @ ord_le951220754t_bool @ A ) @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ B ) @ bot_bot_fun_int_bool ) ) )
     => ( ( A = bot_bot_fun_int_bool )
        | ( A
          = ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ B ) @ bot_bot_fun_int_bool ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_892_subset__singletonD) ).

thf(3815,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_f284875647l_bool @ ord_le951220754t_bool @ A ) @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ B ) @ bot_bot_fun_int_bool ) ) )
     => ( ( A = bot_bot_fun_int_bool )
        | ( A
          = ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ B ) @ bot_bot_fun_int_bool ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[736]) ).

thf(413,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f54304608l_bool @ finite_finite_nat @ C ) )
     => ( ~ ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ B ) @ C ) )
       => ( ( hAPP_f659380387ol_int @ ( big_co195215938at_int @ A ) @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ B ) @ C ) )
          = ( hAPP_int_int @ ( hAPP_int_fun_int_int @ times_times_int @ ( hAPP_nat_int @ A @ B ) ) @ ( hAPP_f659380387ol_int @ ( big_co195215938at_int @ A ) @ C ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_554_setprod__insert) ).

thf(2637,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f54304608l_bool @ finite_finite_nat @ C ) )
     => ( ~ ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ B ) @ C ) )
       => ( ( hAPP_f659380387ol_int @ ( big_co195215938at_int @ A ) @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ B ) @ C ) )
          = ( hAPP_int_int @ ( hAPP_int_fun_int_int @ times_times_int @ ( hAPP_nat_int @ A @ B ) ) @ ( hAPP_f659380387ol_int @ ( big_co195215938at_int @ A ) @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[413]) ).

thf(660,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ A ) @ ( image_48635758_a_nat @ B @ C ) ) )
     => ~ ! [D: $i] :
            ( ( A
              = ( hAPP_H592031934_a_nat @ B @ D ) )
           => ~ ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ D ) @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_241_imageE) ).

thf(3551,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ A ) @ ( image_48635758_a_nat @ B @ C ) ) )
     => ~ ! [D: $i] :
            ( ( A
              = ( hAPP_H592031934_a_nat @ B @ D ) )
           => ~ ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ D ) @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[660]) ).

thf(355,axiom,
    ! [A: $i,B: $i] :
      ( ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ A ) @ ( hAPP_f800510211t_bool @ ( hAPP_f1730770594t_bool @ minus_2067140911t_bool @ B ) @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ A ) @ bot_bot_fun_nat_bool ) ) )
      = ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ A ) @ B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_417_insert__Diff__single) ).

thf(2422,plain,
    ! [A: $i,B: $i] :
      ( ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ A ) @ ( hAPP_f800510211t_bool @ ( hAPP_f1730770594t_bool @ minus_2067140911t_bool @ B ) @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ A ) @ bot_bot_fun_nat_bool ) ) )
      = ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ A ) @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[355]) ).

thf(591,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hAPP_int_int @ ( hAPP_int_fun_int_int @ times_times_int @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ times_times_int @ A ) @ B ) ) @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ times_times_int @ C ) @ D ) )
      = ( hAPP_int_int @ ( hAPP_int_fun_int_int @ times_times_int @ C ) @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ times_times_int @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ times_times_int @ A ) @ B ) ) @ D ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_680_comm__semiring__1__class_Onormalizing__semiring__rules_I15_J) ).

thf(3287,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hAPP_int_int @ ( hAPP_int_fun_int_int @ times_times_int @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ times_times_int @ A ) @ B ) ) @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ times_times_int @ C ) @ D ) )
      = ( hAPP_int_int @ ( hAPP_int_fun_int_int @ times_times_int @ C ) @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ times_times_int @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ times_times_int @ A ) @ B ) ) @ D ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[591]) ).

thf(1005,axiom,
    ! [A: $i,B: $i] :
      ( ( ( B = bot_bo797238721a_bool )
       => ( ( image_533441733iple_a @ ( hAPP_H2085992369iple_a @ cOMBK_2052406350iple_a @ A ) @ B )
          = bot_bo797238721a_bool ) )
      & ( ( B != bot_bo797238721a_bool )
       => ( ( image_533441733iple_a @ ( hAPP_H2085992369iple_a @ cOMBK_2052406350iple_a @ A ) @ B )
          = ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ A ) @ bot_bo797238721a_bool ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_151_image__constant__conv) ).

thf(4828,plain,
    ! [A: $i,B: $i] :
      ( ( ( B = bot_bo797238721a_bool )
       => ( ( image_533441733iple_a @ ( hAPP_H2085992369iple_a @ cOMBK_2052406350iple_a @ A ) @ B )
          = bot_bo797238721a_bool ) )
      & ( ( B != bot_bo797238721a_bool )
       => ( ( image_533441733iple_a @ ( hAPP_H2085992369iple_a @ cOMBK_2052406350iple_a @ A ) @ B )
          = ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ A ) @ bot_bo797238721a_bool ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1005]) ).

thf(652,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( B = C )
     => ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_f103356543l_bool @ ord_le1568362934t_bool @ C ) @ A ) )
       => ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_f103356543l_bool @ ord_le1568362934t_bool @ B ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_769_ord__eq__le__trans) ).

thf(3515,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( B = C )
     => ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_f103356543l_bool @ ord_le1568362934t_bool @ C ) @ A ) )
       => ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_f103356543l_bool @ ord_le1568362934t_bool @ B ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[652]) ).

thf(1025,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( finite648855052iple_a @ D ) )
     => ( ( hBOOL @ ( hAPP_f1695230391l_bool @ finite1738664244iple_a @ C ) )
       => ( ( hAPP_f1693662087iple_a @ ( finite1205970312iple_a @ D @ A ) @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ B ) @ C ) )
          = ( hAPP_f1693662087iple_a @ ( finite1205970312iple_a @ D @ ( hAPP_H905846293iple_a @ ( hAPP_H2085992369iple_a @ D @ B ) @ A ) ) @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_571_comp__fun__idem_Ofold__insert__idem2) ).

thf(4914,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( finite648855052iple_a @ D ) )
     => ( ( hBOOL @ ( hAPP_f1695230391l_bool @ finite1738664244iple_a @ C ) )
       => ( ( hAPP_f1693662087iple_a @ ( finite1205970312iple_a @ D @ A ) @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ B ) @ C ) )
          = ( hAPP_f1693662087iple_a @ ( finite1205970312iple_a @ D @ ( hAPP_H905846293iple_a @ ( hAPP_H2085992369iple_a @ D @ B ) @ A ) ) @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1025]) ).

thf(60,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( finite1973466193nt_int @ E ) )
     => ( ( hBOOL @ ( hAPP_int_bool @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ ( finite772772422nt_int @ E ) @ B ) @ C ) @ D ) )
       => ( ( hBOOL @ ( hAPP_int_bool @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ ( finite772772422nt_int @ E ) @ B ) @ C ) @ A ) )
         => ( A = D ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_535_comp__fun__commute_Ofold__graph__determ) ).

thf(1349,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( finite1973466193nt_int @ E ) )
     => ( ( hBOOL @ ( hAPP_int_bool @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ ( finite772772422nt_int @ E ) @ B ) @ C ) @ D ) )
       => ( ( hBOOL @ ( hAPP_int_bool @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ ( finite772772422nt_int @ E ) @ B ) @ C ) @ A ) )
         => ( A = D ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[60]) ).

thf(940,axiom,
    ( bot_bo797238721a_bool
    = ( hAPP_f20753329a_bool @ collec351493750iple_a @ ( hAPP_b1882817719a_bool @ cOMBK_235286536iple_a @ fFalse ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_48_empty__def) ).

thf(4601,plain,
    ( bot_bo797238721a_bool
    = ( hAPP_f20753329a_bool @ collec351493750iple_a @ ( hAPP_b1882817719a_bool @ cOMBK_235286536iple_a @ fFalse ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[940]) ).

thf(638,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( finite1033474011iple_a @ C @ D ) )
     => ( ! [E: $i,F: $i] :
            ( ( hAPP_H905846293iple_a @ B @ ( hAPP_H905846293iple_a @ ( hAPP_H2085992369iple_a @ C @ E ) @ F ) )
            = ( hAPP_H905846293iple_a @ ( hAPP_H2085992369iple_a @ C @ ( hAPP_H905846293iple_a @ B @ E ) ) @ ( hAPP_H905846293iple_a @ B @ F ) ) )
       => ( ( hBOOL @ ( hAPP_f1695230391l_bool @ finite1738664244iple_a @ A ) )
         => ( ( A != bot_bo797238721a_bool )
           => ( ( hAPP_H905846293iple_a @ B @ ( hAPP_f1693662087iple_a @ D @ A ) )
              = ( hAPP_f1693662087iple_a @ D @ ( image_533441733iple_a @ B @ A ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_436_folding__one__idem_Ohom__commute) ).

thf(3468,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( finite1033474011iple_a @ C @ D ) )
     => ( ! [E: $i,F: $i] :
            ( ( hAPP_H905846293iple_a @ B @ ( hAPP_H905846293iple_a @ ( hAPP_H2085992369iple_a @ C @ E ) @ F ) )
            = ( hAPP_H905846293iple_a @ ( hAPP_H2085992369iple_a @ C @ ( hAPP_H905846293iple_a @ B @ E ) ) @ ( hAPP_H905846293iple_a @ B @ F ) ) )
       => ( ( hBOOL @ ( hAPP_f1695230391l_bool @ finite1738664244iple_a @ A ) )
         => ( ( A != bot_bo797238721a_bool )
           => ( ( hAPP_H905846293iple_a @ B @ ( hAPP_f1693662087iple_a @ D @ A ) )
              = ( hAPP_f1693662087iple_a @ D @ ( image_533441733iple_a @ B @ A ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[638]) ).

thf(998,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP_n1919155532a_bool @ ( hAPP_f2087582327a_bool @ ( hAPP_f1396758363a_bool @ cOMBB_391046962ol_nat @ A ) @ B ) @ C )
      = ( hAPP_f20753329a_bool @ A @ ( hAPP_n1919155532a_bool @ B @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',help_COMBB_1_1_COMBB_000tc__fun_Itc__Hoare____Mirabelle____uwgpyvfjxg__Otriple_I_048) ).

thf(4805,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP_n1919155532a_bool @ ( hAPP_f2087582327a_bool @ ( hAPP_f1396758363a_bool @ cOMBB_391046962ol_nat @ A ) @ B ) @ C )
      = ( hAPP_f20753329a_bool @ A @ ( hAPP_n1919155532a_bool @ B @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[998]) ).

thf(420,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( image_int_int @ A @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ B ) @ C ) )
      = ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ ( hAPP_int_int @ A @ B ) ) @ ( image_int_int @ A @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_224_image__insert) ).

thf(2669,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( image_int_int @ A @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ B ) @ C ) )
      = ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ ( hAPP_int_int @ A @ B ) ) @ ( image_int_int @ A @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[420]) ).

thf(844,axiom,
    ! [A: $i,B: $i] :
      ( ( semi @ A @ B )
     != skip ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_127_com_Osimps_I13_J) ).

thf(4213,plain,
    ! [A: $i,B: $i] :
      ( ( semi @ A @ B )
     != skip ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[844]) ).

thf(831,axiom,
    ! [A: $i,B: $i] :
      ( ( A = B )
     => ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_f284875647l_bool @ ord_le951220754t_bool @ A ) @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_805_equalityD1) ).

thf(4154,plain,
    ! [A: $i,B: $i] :
      ( ( A = B )
     => ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_f284875647l_bool @ ord_le951220754t_bool @ A ) @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[831]) ).

thf(472,axiom,
    ! [A: $i,B: $i] :
      ( ( 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 ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_402_finite__Diff2) ).

thf(2856,plain,
    ! [A: $i,B: $i] :
      ( ( 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 ) ) )
        & ( ( 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 ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[472]) ).

thf(649,axiom,
    ! [A: $i,B: $i] :
      ( skip
     != ( semi @ A @ B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_128_com_Osimps_I12_J) ).

thf(3500,plain,
    ! [A: $i,B: $i] :
      ( skip
     != ( semi @ A @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[649]) ).

thf(476,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ B ) @ C ) )
     => ( ( B = A )
       => ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ A ) @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_759_xt1_I4_J) ).

thf(2871,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ B ) @ C ) )
     => ( ( B = A )
       => ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ A ) @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[476]) ).

thf(69,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP_H1675210280t_bool @ ( hAPP_f2142913731t_bool @ ( hAPP_f1329415375t_bool @ cOMBB_1505691757iple_a @ A ) @ B ) @ C )
      = ( hAPP_i1948725293t_bool @ A @ ( hAPP_H1229305626_a_int @ B @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',help_COMBB_1_1_COMBB_000tc__Int__Oint_000tc__fun_Itc__Int__Oint_Mtc__HOL__Obool__020) ).

thf(1376,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP_H1675210280t_bool @ ( hAPP_f2142913731t_bool @ ( hAPP_f1329415375t_bool @ cOMBB_1505691757iple_a @ A ) @ B ) @ C )
      = ( hAPP_i1948725293t_bool @ A @ ( hAPP_H1229305626_a_int @ B @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[69]) ).

thf(260,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_int_bool @ ( hAPP_f1805168059t_bool @ ( hAPP_f1223193598t_bool @ minus_1449998731t_bool @ A ) @ B ) @ C ) )
    <=> ( hBOOL @ ( hAPP_bool_bool @ ( hAPP_b589554111l_bool @ minus_minus_bool @ ( hAPP_int_bool @ A @ C ) ) @ ( hAPP_int_bool @ B @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_442_minus__apply) ).

thf(2065,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( hBOOL @ ( hAPP_int_bool @ ( hAPP_f1805168059t_bool @ ( hAPP_f1223193598t_bool @ minus_1449998731t_bool @ A ) @ B ) @ C ) )
       => ( hBOOL @ ( hAPP_bool_bool @ ( hAPP_b589554111l_bool @ minus_minus_bool @ ( hAPP_int_bool @ A @ C ) ) @ ( hAPP_int_bool @ B @ C ) ) ) )
      & ( ( hBOOL @ ( hAPP_bool_bool @ ( hAPP_b589554111l_bool @ minus_minus_bool @ ( hAPP_int_bool @ A @ C ) ) @ ( hAPP_int_bool @ B @ C ) ) )
       => ( hBOOL @ ( hAPP_int_bool @ ( hAPP_f1805168059t_bool @ ( hAPP_f1223193598t_bool @ minus_1449998731t_bool @ A ) @ B ) @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[260]) ).

thf(886,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( C != D )
        & ( A != B ) )
    <=> ( ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ times_times_int @ C ) @ A ) ) @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ times_times_int @ D ) @ B ) )
       != ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ times_times_int @ C ) @ B ) ) @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ times_times_int @ D ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_699_crossproduct__noteq) ).

thf(4388,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( ( C != D )
          & ( A != B ) )
       => ( ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ times_times_int @ C ) @ A ) ) @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ times_times_int @ D ) @ B ) )
         != ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ times_times_int @ C ) @ B ) ) @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ times_times_int @ D ) @ A ) ) ) )
      & ( ( ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ times_times_int @ C ) @ A ) ) @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ times_times_int @ D ) @ B ) )
         != ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ times_times_int @ C ) @ B ) ) @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ times_times_int @ D ) @ A ) ) )
       => ( ( C != D )
          & ( A != B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[886]) ).

thf(1104,axiom,
    ! [A: $i] :
      ~ ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ A ) @ bot_bo797238721a_bool ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_36_empty__iff) ).

thf(5202,plain,
    ! [A: $i] :
      ~ ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ A ) @ bot_bo797238721a_bool ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1104]) ).

thf(407,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ~ ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ B ) @ C ) )
       => ( B = A ) )
     => ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ B ) @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ A ) @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_14_insertCI) ).

thf(2610,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ~ ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ B ) @ C ) )
       => ( B = A ) )
     => ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ B ) @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ A ) @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[407]) ).

thf(859,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( finite1033474011iple_a @ C @ D ) )
     => ( ( hBOOL @ ( hAPP_f1695230391l_bool @ finite1738664244iple_a @ B ) )
       => ( ( A != bot_bo797238721a_bool )
         => ( ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_f1767618879l_bool @ ord_le1523781517a_bool @ A ) @ B ) )
           => ( ( hAPP_H905846293iple_a @ ( hAPP_H2085992369iple_a @ C @ ( hAPP_f1693662087iple_a @ D @ A ) ) @ ( hAPP_f1693662087iple_a @ D @ B ) )
              = ( hAPP_f1693662087iple_a @ D @ B ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_910_folding__one__idem_Osubset__idem) ).

thf(4277,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( finite1033474011iple_a @ C @ D ) )
     => ( ( hBOOL @ ( hAPP_f1695230391l_bool @ finite1738664244iple_a @ B ) )
       => ( ( A != bot_bo797238721a_bool )
         => ( ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_f1767618879l_bool @ ord_le1523781517a_bool @ A ) @ B ) )
           => ( ( hAPP_H905846293iple_a @ ( hAPP_H2085992369iple_a @ C @ ( hAPP_f1693662087iple_a @ D @ A ) ) @ ( hAPP_f1693662087iple_a @ D @ B ) )
              = ( hAPP_f1693662087iple_a @ D @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[859]) ).

thf(562,axiom,
    ! [A: $i,B: $i] :
      ( ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ ( if_nat @ fTrue ) @ A ) @ B )
      = A ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',help_If_1_1_If_000tc__Nat__Onat_T) ).

thf(3192,plain,
    ! [A: $i,B: $i] :
      ( ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ ( if_nat @ fTrue ) @ A ) @ B )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[562]) ).

thf(491,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i] :
          ( ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ D ) @ C ) )
         => ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ ( hAPP_H592031934_a_nat @ A @ D ) ) @ ( hAPP_H592031934_a_nat @ B @ D ) ) ) )
     => ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ ( hAPP_f2097660464ol_nat @ ( hAPP_f1371399797ol_nat @ big_co1340561246_a_nat @ A ) @ C ) ) @ ( hAPP_f2097660464ol_nat @ ( hAPP_f1371399797ol_nat @ big_co1340561246_a_nat @ B ) @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_924_setsum__mono) ).

thf(2923,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i] :
          ( ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ D ) @ C ) )
         => ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ ( hAPP_H592031934_a_nat @ A @ D ) ) @ ( hAPP_H592031934_a_nat @ B @ D ) ) ) )
     => ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ ( hAPP_f2097660464ol_nat @ ( hAPP_f1371399797ol_nat @ big_co1340561246_a_nat @ A ) @ C ) ) @ ( hAPP_f2097660464ol_nat @ ( hAPP_f1371399797ol_nat @ big_co1340561246_a_nat @ B ) @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[491]) ).

thf(504,axiom,
    ! [A: $i] :
      ( ( hAPP_f1805168059t_bool @ ( hAPP_f1223193598t_bool @ minus_1449998731t_bool @ A ) @ bot_bot_fun_int_bool )
      = A ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_395_Diff__empty) ).

thf(2974,plain,
    ! [A: $i] :
      ( ( hAPP_f1805168059t_bool @ ( hAPP_f1223193598t_bool @ minus_1449998731t_bool @ A ) @ bot_bot_fun_int_bool )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[504]) ).

thf(227,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( hBOOL @ C )
       => ( ( hAPP_nat_int @ semiri1621563631at_int @ A )
          = ( hAPP_nat_int @ semiri1621563631at_int @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ ( if_nat @ C ) @ A ) @ B ) ) ) )
      & ( ~ ( hBOOL @ C )
       => ( ( hAPP_nat_int @ semiri1621563631at_int @ B )
          = ( hAPP_nat_int @ semiri1621563631at_int @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ ( if_nat @ C ) @ A ) @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_1157_int__if__cong) ).

thf(1952,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( hBOOL @ C )
       => ( ( hAPP_nat_int @ semiri1621563631at_int @ A )
          = ( hAPP_nat_int @ semiri1621563631at_int @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ ( if_nat @ C ) @ A ) @ B ) ) ) )
      & ( ~ ( hBOOL @ C )
       => ( ( hAPP_nat_int @ semiri1621563631at_int @ B )
          = ( hAPP_nat_int @ semiri1621563631at_int @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ ( if_nat @ C ) @ A ) @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[227]) ).

thf(157,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( finite1071749497t_bool @ D ) )
     => ( ( hBOOL @ ( hAPP_f54304608l_bool @ finite_finite_nat @ C ) )
       => ( ~ ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ B ) @ C ) )
         => ( ( hAPP_f800510211t_bool @ ( finite1578363458t_bool @ D @ A ) @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ B ) @ C ) )
            = ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ D @ B ) @ ( hAPP_f800510211t_bool @ ( finite1578363458t_bool @ D @ A ) @ C ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_512_comp__fun__commute_Ofold__insert) ).

thf(1688,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( finite1071749497t_bool @ D ) )
     => ( ( hBOOL @ ( hAPP_f54304608l_bool @ finite_finite_nat @ C ) )
       => ( ~ ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ B ) @ C ) )
         => ( ( hAPP_f800510211t_bool @ ( finite1578363458t_bool @ D @ A ) @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ B ) @ C ) )
            = ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ D @ B ) @ ( hAPP_f800510211t_bool @ ( finite1578363458t_bool @ D @ A ) @ C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[157]) ).

thf(570,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ B ) @ C ) )
     => ( ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ ( hAPP_H1229305626_a_int @ A @ B ) ) @ ( image_685909450_a_int @ A @ C ) )
        = ( image_685909450_a_int @ A @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_217_insert__image) ).

thf(3219,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ B ) @ C ) )
     => ( ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ ( hAPP_H1229305626_a_int @ A @ B ) ) @ ( image_685909450_a_int @ A @ C ) )
        = ( image_685909450_a_int @ A @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[570]) ).

thf(464,axiom,
    ! [A: $i,B: $i] :
      ( ( ( image_nat_int @ A @ B )
        = bot_bot_fun_int_bool )
    <=> ( B = bot_bot_fun_nat_bool ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_205_image__is__empty) ).

thf(2824,plain,
    ! [A: $i,B: $i] :
      ( ( ( ( image_nat_int @ A @ B )
          = bot_bot_fun_int_bool )
       => ( B = bot_bot_fun_nat_bool ) )
      & ( ( B = bot_bot_fun_nat_bool )
       => ( ( image_nat_int @ A @ B )
          = bot_bot_fun_int_bool ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[464]) ).

thf(1083,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ~ ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ D ) @ E ) )
     => ( ( hBOOL @ ( hAPP_H1037229737a_bool @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ ( finite90892550iple_a @ A ) @ B ) @ E ) @ C ) )
       => ( hBOOL @ ( hAPP_H1037229737a_bool @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ ( finite90892550iple_a @ A ) @ B ) @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ D ) @ E ) ) @ ( hAPP_H905846293iple_a @ ( hAPP_H2085992369iple_a @ A @ D ) @ C ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_306_fold__graph_OinsertI) ).

thf(5114,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ~ ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ D ) @ E ) )
     => ( ( hBOOL @ ( hAPP_H1037229737a_bool @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ ( finite90892550iple_a @ A ) @ B ) @ E ) @ C ) )
       => ( hBOOL @ ( hAPP_H1037229737a_bool @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ ( finite90892550iple_a @ A ) @ B ) @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ D ) @ E ) ) @ ( hAPP_H905846293iple_a @ ( hAPP_H2085992369iple_a @ A @ D ) @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1083]) ).

thf(287,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ ( finite929467206at_nat @ A ) @ B ) @ bot_bot_fun_nat_bool ) @ C ) )
     => ( C = B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_304_empty__fold__graphE) ).

thf(2182,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ ( finite929467206at_nat @ A ) @ B ) @ bot_bot_fun_nat_bool ) @ C ) )
     => ( C = B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[287]) ).

thf(888,axiom,
    ! [A: $i,B: $i] :
      ( ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ A ) @ B )
      = ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ B ) @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_693_comm__semiring__1__class_Onormalizing__semiring__rules_I7_J) ).

thf(4401,plain,
    ! [A: $i,B: $i] :
      ( ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ A ) @ B )
      = ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ B ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[888]) ).

thf(895,axiom,
    ! [A: $i,B: $i] :
      ( ( hAPP_int_int @ uminus_uminus_int @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ A ) @ B ) )
      = ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ ( hAPP_int_int @ uminus_uminus_int @ A ) ) @ ( hAPP_int_int @ uminus_uminus_int @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_1081_zminus__zadd__distrib) ).

thf(4430,plain,
    ! [A: $i,B: $i] :
      ( ( hAPP_int_int @ uminus_uminus_int @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ A ) @ B ) )
      = ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ ( hAPP_int_int @ uminus_uminus_int @ A ) ) @ ( hAPP_int_int @ uminus_uminus_int @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[895]) ).

thf(1016,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ B ) @ C ) )
     => ( ( hAPP_f20753329a_bool @ ( hAPP_f1706700729a_bool @ minus_85316870a_bool @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ B ) @ A ) ) @ C )
        = ( hAPP_f20753329a_bool @ ( hAPP_f1706700729a_bool @ minus_85316870a_bool @ A ) @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_407_insert__Diff1) ).

thf(4882,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ B ) @ C ) )
     => ( ( hAPP_f20753329a_bool @ ( hAPP_f1706700729a_bool @ minus_85316870a_bool @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ B ) @ A ) ) @ C )
        = ( hAPP_f20753329a_bool @ ( hAPP_f1706700729a_bool @ minus_85316870a_bool @ A ) @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1016]) ).

thf(792,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ A ) @ B ) ) @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ C ) @ D ) )
      = ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ A ) @ C ) ) @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ B ) @ D ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_660_comm__semiring__1__class_Onormalizing__semiring__rules_I20_J) ).

thf(4009,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ A ) @ B ) ) @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ C ) @ D ) )
      = ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ A ) @ C ) ) @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ B ) @ D ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[792]) ).

thf(525,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ B ) @ C ) )
     => ( ( hAPP_f1805168059t_bool @ ( hAPP_f1223193598t_bool @ minus_1449998731t_bool @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ B ) @ A ) ) @ C )
        = ( hAPP_f1805168059t_bool @ ( hAPP_f1223193598t_bool @ minus_1449998731t_bool @ A ) @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_408_insert__Diff1) ).

thf(3055,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ B ) @ C ) )
     => ( ( hAPP_f1805168059t_bool @ ( hAPP_f1223193598t_bool @ minus_1449998731t_bool @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ B ) @ A ) ) @ C )
        = ( hAPP_f1805168059t_bool @ ( hAPP_f1223193598t_bool @ minus_1449998731t_bool @ A ) @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[525]) ).

thf(117,axiom,
    ! [A: $i] :
      ~ ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_nat @ A ) @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_1119_less__irrefl__nat) ).

thf(1557,plain,
    ! [A: $i] :
      ~ ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_nat @ A ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[117]) ).

thf(1024,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ A ) @ B ) )
     => ( ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ A ) @ ( hAPP_f20753329a_bool @ ( hAPP_f1706700729a_bool @ minus_85316870a_bool @ B ) @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ A ) @ bot_bo797238721a_bool ) ) )
        = B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_424_insert__Diff) ).

thf(4911,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ A ) @ B ) )
     => ( ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ A ) @ ( hAPP_f20753329a_bool @ ( hAPP_f1706700729a_bool @ minus_85316870a_bool @ B ) @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ A ) @ bot_bo797238721a_bool ) ) )
        = B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1024]) ).

thf(271,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ C ) @ D ) )
     => ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ A ) @ B ) )
       => ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ ord_min_nat @ C ) @ A ) ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ ord_min_nat @ D ) @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_966_min__max_Oinf__mono) ).

thf(2111,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ C ) @ D ) )
     => ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ A ) @ B ) )
       => ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ ord_min_nat @ C ) @ A ) ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ ord_min_nat @ D ) @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[271]) ).

thf(808,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_f284875647l_bool @ ord_le951220754t_bool @ B ) @ C ) )
     => ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_f284875647l_bool @ ord_le951220754t_bool @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ A ) @ B ) ) @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ A ) @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_858_insert__mono) ).

thf(4064,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_f284875647l_bool @ ord_le951220754t_bool @ B ) @ C ) )
     => ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_f284875647l_bool @ ord_le951220754t_bool @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ A ) @ B ) ) @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ A ) @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[808]) ).

thf(236,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ B ) @ C ) )
     => ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ ord_min_nat @ B ) @ A ) ) @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_978_min__max_Ole__infI1) ).

thf(1993,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ B ) @ C ) )
     => ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ ord_min_nat @ B ) @ A ) ) @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[236]) ).

thf(169,axiom,
    ! [A: $i,B: $i] : ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ ord_min_nat @ A ) @ B ) ) @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_986_min__max_Oinf__le1) ).

thf(1731,plain,
    ! [A: $i,B: $i] : ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ ord_min_nat @ A ) @ B ) ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[169]) ).

thf(733,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ A ) @ B ) ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ C ) @ D ) )
        = ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ A ) @ D ) ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ C ) @ B ) ) )
    <=> ( ( A = C )
        | ( B = D ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_647_crossproduct__eq) ).

thf(3800,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ A ) @ B ) ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ C ) @ D ) )
          = ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ A ) @ D ) ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ C ) @ B ) ) )
       => ( ( A = C )
          | ( B = D ) ) )
      & ( ( ( A = C )
          | ( B = D ) )
       => ( ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ A ) @ B ) ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ C ) @ D ) )
          = ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ A ) @ D ) ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ C ) @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[733]) ).

thf(667,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_f284875647l_bool @ ord_le951220754t_bool @ B ) @ C ) )
     => ( ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_f284875647l_bool @ ord_le951220754t_bool @ C ) @ A ) )
       => ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_f284875647l_bool @ ord_le951220754t_bool @ B ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_748_order__trans) ).

thf(3572,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_f284875647l_bool @ ord_le951220754t_bool @ B ) @ C ) )
     => ( ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_f284875647l_bool @ ord_le951220754t_bool @ C ) @ A ) )
       => ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_f284875647l_bool @ ord_le951220754t_bool @ B ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[667]) ).

thf(1019,axiom,
    ! [A: $i,B: $i] : ( hBOOL @ ( hAPP_f54304608l_bool @ finite_finite_nat @ ( hAPP_n1699378549t_bool @ ( ord_gr375877188st_nat @ A ) @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_1177_finite__greaterThanAtMost) ).

thf(4897,plain,
    ! [A: $i,B: $i] : ( hBOOL @ ( hAPP_f54304608l_bool @ finite_finite_nat @ ( hAPP_n1699378549t_bool @ ( ord_gr375877188st_nat @ A ) @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1019]) ).

thf(1048,axiom,
    ! [A: $i,B: $i] :
      ( ( hAPP_f1693662087iple_a @ ( finite1205970312iple_a @ A @ B ) @ bot_bo797238721a_bool )
      = B ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_499_fold__empty) ).

thf(4985,plain,
    ! [A: $i,B: $i] :
      ( ( hAPP_f1693662087iple_a @ ( finite1205970312iple_a @ A @ B ) @ bot_bo797238721a_bool )
      = B ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1048]) ).

thf(899,axiom,
    ! [A: $i] :
      ( ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_f284875647l_bool @ ord_le951220754t_bool @ A ) @ bot_bot_fun_int_bool ) )
    <=> ( A = bot_bot_fun_int_bool ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_849_subset__empty) ).

thf(4452,plain,
    ! [A: $i] :
      ( ( ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_f284875647l_bool @ ord_le951220754t_bool @ A ) @ bot_bot_fun_int_bool ) )
       => ( A = bot_bot_fun_int_bool ) )
      & ( ( A = bot_bot_fun_int_bool )
       => ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_f284875647l_bool @ ord_le951220754t_bool @ A ) @ bot_bot_fun_int_bool ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[899]) ).

thf(455,axiom,
    ! [A: $i,B: $i] :
      ( ( A = B )
     => ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ A ) @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_781_order__eq__refl) ).

thf(2785,plain,
    ! [A: $i,B: $i] :
      ( ( A = B )
     => ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ A ) @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[455]) ).

thf(633,axiom,
    ! [A: $i,B: $i] :
      ( ( A != B )
      | ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_f103356543l_bool @ fequal_fun_nat_bool @ A ) @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',help_fequal_2_1_fequal_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_T) ).

thf(3451,plain,
    ! [A: $i,B: $i] :
      ( ( A != B )
      | ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_f103356543l_bool @ fequal_fun_nat_bool @ A ) @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[633]) ).

thf(10,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( 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 ) @ B ) ) @ C ) ) @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ times_times_int @ D ) @ B ) ) @ E ) ) )
    <=> ( 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 ) @ D ) ) @ B ) ) @ C ) ) @ E ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_902_le__add__iff1) ).

thf(1149,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( ( 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 ) @ B ) ) @ C ) ) @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ times_times_int @ D ) @ B ) ) @ E ) ) )
       => ( 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 ) @ D ) ) @ B ) ) @ C ) ) @ E ) ) )
      & ( ( 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 ) @ D ) ) @ B ) ) @ C ) ) @ E ) )
       => ( 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 ) @ B ) ) @ C ) ) @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ times_times_int @ D ) @ B ) ) @ E ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[10]) ).

thf(595,axiom,
    ! [A: $i,B: $i] :
      ( ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ A ) @ B )
     != bot_bot_fun_int_bool ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_98_insert__not__empty) ).

thf(3298,plain,
    ! [A: $i,B: $i] :
      ( ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ A ) @ B )
     != bot_bot_fun_int_bool ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[595]) ).

thf(786,axiom,
    ! [A: $i] :
      ( ( hBOOL @ ( hAPP_f54304608l_bool @ finite_finite_nat @ A ) )
     => ( ( A != bot_bot_fun_nat_bool )
       => ( ! [B: $i,C: $i] : ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ B ) @ C ) ) @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ B ) @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ C ) @ 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 ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_434_fold1__in) ).

thf(3971,plain,
    ! [A: $i] :
      ( ( hBOOL @ ( hAPP_f54304608l_bool @ finite_finite_nat @ A ) )
     => ( ( A != bot_bot_fun_nat_bool )
       => ( ! [B: $i,C: $i] : ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ B ) @ C ) ) @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ B ) @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ C ) @ 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 ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[786]) ).

thf(58,axiom,
    ! [A: $i,B: $i] :
      ( ~ ( hBOOL @ ( hAPP_int_bool @ ( hAPP_i1948725293t_bool @ fequal_int @ A ) @ B ) )
      | ( A = B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',help_fequal_1_1_fequal_000tc__Int__Oint_T) ).

thf(1344,plain,
    ! [A: $i,B: $i] :
      ( ~ ( hBOOL @ ( hAPP_int_bool @ ( hAPP_i1948725293t_bool @ fequal_int @ A ) @ B ) )
      | ( A = B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[58]) ).

thf(93,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_nat @ A ) @ B ) )
    <=> ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ A ) @ B ) )
        & ( A != B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_1109_nat__less__le) ).

thf(1460,plain,
    ! [A: $i,B: $i] :
      ( ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_nat @ A ) @ B ) )
       => ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ A ) @ B ) )
          & ( A != B ) ) )
      & ( ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ A ) @ B ) )
          & ( A != B ) )
       => ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_nat @ A ) @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[93]) ).

thf(318,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ ( finite929467206at_nat @ times_times_nat ) @ B ) @ C ) @ D ) )
     => ( ~ ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ B ) @ C ) )
       => ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ ( finite929467206at_nat @ times_times_nat ) @ A ) @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ B ) @ C ) ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ A ) @ D ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_429_fold__graph__insert__swap) ).

thf(2289,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ ( finite929467206at_nat @ times_times_nat ) @ B ) @ C ) @ D ) )
     => ( ~ ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ B ) @ C ) )
       => ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ ( finite929467206at_nat @ times_times_nat ) @ A ) @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ B ) @ C ) ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ A ) @ D ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[318]) ).

thf(415,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP_H144868812t_bool @ ( hAPP_f1872860171t_bool @ ( hAPP_f1981448543t_bool @ cOMBB_1391527205iple_a @ A ) @ B ) @ C )
      = ( hAPP_n1699378549t_bool @ A @ ( hAPP_H592031934_a_nat @ B @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',help_COMBB_1_1_COMBB_000tc__Nat__Onat_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool__021) ).

thf(2647,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP_H144868812t_bool @ ( hAPP_f1872860171t_bool @ ( hAPP_f1981448543t_bool @ cOMBB_1391527205iple_a @ A ) @ B ) @ C )
      = ( hAPP_n1699378549t_bool @ A @ ( hAPP_H592031934_a_nat @ B @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[415]) ).

thf(970,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( finite1803123436a_bool @ D ) )
     => ( ( hBOOL @ ( hAPP_f1695230391l_bool @ finite1738664244iple_a @ C ) )
       => ( ( hAPP_f20753329a_bool @ ( finite1215486064a_bool @ D @ A ) @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ B ) @ C ) )
          = ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ D @ B ) @ ( hAPP_f20753329a_bool @ ( finite1215486064a_bool @ D @ A ) @ C ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_578_comp__fun__idem_Ofold__insert__idem) ).

thf(4726,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( finite1803123436a_bool @ D ) )
     => ( ( hBOOL @ ( hAPP_f1695230391l_bool @ finite1738664244iple_a @ C ) )
       => ( ( hAPP_f20753329a_bool @ ( finite1215486064a_bool @ D @ A ) @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ B ) @ C ) )
          = ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ D @ B ) @ ( hAPP_f20753329a_bool @ ( finite1215486064a_bool @ D @ A ) @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[970]) ).

thf(309,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( finite1973466193nt_int @ D ) )
     => ( ( hBOOL @ ( hAPP_f448129468l_bool @ finite_finite_int @ C ) )
       => ( ( hAPP_int_int @ ( hAPP_int_fun_int_int @ D @ A ) @ ( hAPP_f1594865479ol_int @ ( finite_fold_int_int @ D @ B ) @ C ) )
          = ( hAPP_f1594865479ol_int @ ( finite_fold_int_int @ D @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ D @ A ) @ B ) ) @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_485_comp__fun__commute_Ofold__fun__comm) ).

thf(2252,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( finite1973466193nt_int @ D ) )
     => ( ( hBOOL @ ( hAPP_f448129468l_bool @ finite_finite_int @ C ) )
       => ( ( hAPP_int_int @ ( hAPP_int_fun_int_int @ D @ A ) @ ( hAPP_f1594865479ol_int @ ( finite_fold_int_int @ D @ B ) @ C ) )
          = ( hAPP_f1594865479ol_int @ ( finite_fold_int_int @ D @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ D @ A ) @ B ) ) @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[309]) ).

thf(753,axiom,
    ! [A: $i,B: $i] :
      ( ! [C: $i] :
          ( ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ C ) @ B ) )
         => ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ C ) @ A ) ) )
     => ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_f1767618879l_bool @ ord_le1523781517a_bool @ B ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_916_subsetI) ).

thf(3861,plain,
    ! [A: $i,B: $i] :
      ( ! [C: $i] :
          ( ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ C ) @ B ) )
         => ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ C ) @ A ) ) )
     => ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_f1767618879l_bool @ ord_le1523781517a_bool @ B ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[753]) ).

thf(392,axiom,
    ! [A: $i] :
      ( ! [B: $i] :
          ~ ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ B ) @ A ) )
     => ( A = bot_bot_fun_int_bool ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_122_equals0I) ).

thf(2553,plain,
    ! [A: $i] :
      ( ! [B: $i] :
          ~ ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ B ) @ A ) )
     => ( A = bot_bot_fun_int_bool ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[392]) ).

thf(1056,axiom,
    ! [A: $i] :
      ( ( hAPP_f20753329a_bool @ ( hAPP_f1706700729a_bool @ minus_85316870a_bool @ bot_bo797238721a_bool ) @ A )
      = bot_bo797238721a_bool ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_391_empty__Diff) ).

thf(5007,plain,
    ! [A: $i] :
      ( ( hAPP_f20753329a_bool @ ( hAPP_f1706700729a_bool @ minus_85316870a_bool @ bot_bo797238721a_bool ) @ A )
      = bot_bo797238721a_bool ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1056]) ).

thf(434,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( finite2130160977at_nat @ E ) )
     => ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ ( finite929467206at_nat @ E ) @ A ) @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ B ) @ C ) ) @ D ) )
       => ( ~ ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ B ) @ C ) )
         => ~ ! [F: $i] :
                ( ( D
                  = ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ E @ B ) @ F ) )
               => ~ ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ ( finite929467206at_nat @ E ) @ A ) @ C ) @ F ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_589_comp__fun__commute_Ofold__graph__insertE) ).

thf(2710,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( finite2130160977at_nat @ E ) )
     => ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ ( finite929467206at_nat @ E ) @ A ) @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ B ) @ C ) ) @ D ) )
       => ( ~ ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ B ) @ C ) )
         => ~ ! [F: $i] :
                ( ( D
                  = ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ E @ B ) @ F ) )
               => ~ ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ ( finite929467206at_nat @ E ) @ A ) @ C ) @ F ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[434]) ).

thf(197,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( hAPP_int_int @ ( hAPP_int_fun_int_int @ minus_minus_int @ A ) @ B )
        = ( hAPP_int_int @ ( hAPP_int_fun_int_int @ minus_minus_int @ C ) @ D ) )
     => ( ( A = B )
      <=> ( C = D ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_695_diff__eq__diff__eq) ).

thf(1820,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( hAPP_int_int @ ( hAPP_int_fun_int_int @ minus_minus_int @ A ) @ B )
        = ( hAPP_int_int @ ( hAPP_int_fun_int_int @ minus_minus_int @ C ) @ D ) )
     => ( ( ( A = B )
         => ( C = D ) )
        & ( ( C = D )
         => ( A = B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[197]) ).

thf(438,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ A ) @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ B ) @ bot_bot_fun_int_bool ) ) )
     => ( A = B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_89_singletonE) ).

thf(2725,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ A ) @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ B ) @ bot_bot_fun_int_bool ) ) )
     => ( A = B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[438]) ).

thf(685,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ A ) @ B )
        = ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ C ) @ B ) )
     => ( A = C ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_654_add__right__imp__eq) ).

thf(3627,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ A ) @ B )
        = ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ C ) @ B ) )
     => ( A = C ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[685]) ).

thf(467,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ D ) @ E ) )
     => ( ( 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 @ E ) @ A ) ) @ B ) ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ D ) @ A ) ) @ C ) )
        = ( 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 @ E ) @ D ) ) @ A ) ) @ B ) ) @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_1044_nat__diff__add__eq1) ).

thf(2835,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ D ) @ E ) )
     => ( ( 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 @ E ) @ A ) ) @ B ) ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ D ) @ A ) ) @ C ) )
        = ( 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 @ E ) @ D ) ) @ A ) ) @ B ) ) @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[467]) ).

thf(15,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_int_bool @ ( hAPP_i1948725293t_bool @ ord_less_eq_int @ B ) @ C ) )
     => ( hBOOL @ ( hAPP_int_bool @ ( hAPP_i1948725293t_bool @ ord_less_eq_int @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ ord_min_int @ A ) @ B ) ) @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_977_min__max_Ole__infI2) ).

thf(1176,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_int_bool @ ( hAPP_i1948725293t_bool @ ord_less_eq_int @ B ) @ C ) )
     => ( hBOOL @ ( hAPP_int_bool @ ( hAPP_i1948725293t_bool @ ord_less_eq_int @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ ord_min_int @ A ) @ B ) ) @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[15]) ).

thf(577,axiom,
    ! [A: $i,B: $i] :
      ( ( B != bot_bot_fun_nat_bool )
     => ( ( hBOOL @ ( hAPP_f54304608l_bool @ finite_finite_nat @ B ) )
       => ( ~ ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ A ) @ B ) )
         => ( ( hAPP_f22106695ol_nat @ ( finite_fold1_nat @ times_times_nat ) @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ A ) @ B ) )
            = ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ A ) @ ( hAPP_f22106695ol_nat @ ( finite_fold1_nat @ times_times_nat ) @ B ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_362_fold1__insert) ).

thf(3241,plain,
    ! [A: $i,B: $i] :
      ( ( B != bot_bot_fun_nat_bool )
     => ( ( hBOOL @ ( hAPP_f54304608l_bool @ finite_finite_nat @ B ) )
       => ( ~ ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ A ) @ B ) )
         => ( ( hAPP_f22106695ol_nat @ ( finite_fold1_nat @ times_times_nat ) @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ A ) @ B ) )
            = ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ A ) @ ( hAPP_f22106695ol_nat @ ( finite_fold1_nat @ times_times_nat ) @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[577]) ).

thf(1126,axiom,
    ! [A: $i,B: $i] :
      ( ( ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ A ) @ bot_bo797238721a_bool )
        = ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ B ) @ bot_bo797238721a_bool ) )
     => ( A = B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_85_singleton__inject) ).

thf(5299,plain,
    ! [A: $i,B: $i] :
      ( ( ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ A ) @ bot_bo797238721a_bool )
        = ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ B ) @ bot_bo797238721a_bool ) )
     => ( A = B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1126]) ).

thf(558,axiom,
    ! [A: $i,B: $i] :
      ( ( hAPP_int_int @ ( hAPP_int_fun_int_int @ times_times_int @ A ) @ B )
      = ( hAPP_int_int @ ( hAPP_int_fun_int_int @ times_times_int @ B ) @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_694_comm__semiring__1__class_Onormalizing__semiring__rules_I7_J) ).

thf(3178,plain,
    ! [A: $i,B: $i] :
      ( ( hAPP_int_int @ ( hAPP_int_fun_int_int @ times_times_int @ A ) @ B )
      = ( hAPP_int_int @ ( hAPP_int_fun_int_int @ times_times_int @ B ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[558]) ).

thf(512,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( finite2130160977at_nat @ D ) )
     => ( ( hBOOL @ ( hAPP_f54304608l_bool @ finite_finite_nat @ C ) )
       => ( ( hAPP_f22106695ol_nat @ ( finite_fold_nat_nat @ D @ A ) @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ B ) @ C ) )
          = ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ D @ B ) @ ( hAPP_f22106695ol_nat @ ( finite_fold_nat_nat @ D @ A ) @ ( hAPP_f800510211t_bool @ ( hAPP_f1730770594t_bool @ minus_2067140911t_bool @ C ) @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ B ) @ bot_bot_fun_nat_bool ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_546_comp__fun__commute_Ofold__insert__remove) ).

thf(2999,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( finite2130160977at_nat @ D ) )
     => ( ( hBOOL @ ( hAPP_f54304608l_bool @ finite_finite_nat @ C ) )
       => ( ( hAPP_f22106695ol_nat @ ( finite_fold_nat_nat @ D @ A ) @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ B ) @ C ) )
          = ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ D @ B ) @ ( hAPP_f22106695ol_nat @ ( finite_fold_nat_nat @ D @ A ) @ ( hAPP_f800510211t_bool @ ( hAPP_f1730770594t_bool @ minus_2067140911t_bool @ C ) @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ B ) @ bot_bot_fun_nat_bool ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[512]) ).

thf(119,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( C = D )
     => ( ! [E: $i] :
            ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ E ) @ D ) )
           => ( ( hAPP_nat_nat @ A @ E )
              = ( hAPP_nat_nat @ B @ E ) ) )
       => ( ( hAPP_f22106695ol_nat @ ( hAPP_f782000547ol_nat @ big_co387207925at_nat @ A ) @ C )
          = ( hAPP_f22106695ol_nat @ ( hAPP_f782000547ol_nat @ big_co387207925at_nat @ B ) @ D ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_712_setsum_OF__cong) ).

thf(1562,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( C = D )
     => ( ! [E: $i] :
            ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ E ) @ D ) )
           => ( ( hAPP_nat_nat @ A @ E )
              = ( hAPP_nat_nat @ B @ E ) ) )
       => ( ( hAPP_f22106695ol_nat @ ( hAPP_f782000547ol_nat @ big_co387207925at_nat @ A ) @ C )
          = ( hAPP_f22106695ol_nat @ ( hAPP_f782000547ol_nat @ big_co387207925at_nat @ B ) @ D ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[119]) ).

thf(263,axiom,
    ! [A: $i,B: $i] : ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ A ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ B ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_1013_le__add2) ).

thf(2075,plain,
    ! [A: $i,B: $i] : ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ A ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ B ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[263]) ).

thf(354,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( finite1973466193nt_int @ E ) )
     => ( ( hBOOL @ ( hAPP_int_bool @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ ( finite772772422nt_int @ E ) @ B ) @ C ) @ D ) )
       => ( ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ A ) @ C ) )
         => ? [F: $i] :
              ( ( D
                = ( hAPP_int_int @ ( hAPP_int_fun_int_int @ E @ A ) @ F ) )
              & ( hBOOL @ ( hAPP_int_bool @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ ( finite772772422nt_int @ E ) @ B ) @ ( hAPP_f1805168059t_bool @ ( hAPP_f1223193598t_bool @ minus_1449998731t_bool @ C ) @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ A ) @ bot_bot_fun_int_bool ) ) ) @ F ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_459_comp__fun__commute_Ofold__graph__insertE__aux) ).

thf(2418,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( finite1973466193nt_int @ E ) )
     => ( ( hBOOL @ ( hAPP_int_bool @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ ( finite772772422nt_int @ E ) @ B ) @ C ) @ D ) )
       => ( ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ A ) @ C ) )
         => ? [F: $i] :
              ( ( D
                = ( hAPP_int_int @ ( hAPP_int_fun_int_int @ E @ A ) @ F ) )
              & ( hBOOL @ ( hAPP_int_bool @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ ( finite772772422nt_int @ E ) @ B ) @ ( hAPP_f1805168059t_bool @ ( hAPP_f1223193598t_bool @ minus_1449998731t_bool @ C ) @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ A ) @ bot_bot_fun_int_bool ) ) ) @ F ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[354]) ).

thf(254,axiom,
    hBOOL @ ( finite1973466193nt_int @ times_times_int ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_534_comp__fun__commute) ).

thf(2051,plain,
    hBOOL @ ( finite1973466193nt_int @ times_times_int ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[254]) ).

thf(391,axiom,
    ! [A: $i,B: $i] :
      ( ( hAPP_f22106695ol_nat @ ( finite_fold1_nat @ A ) @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ B ) @ bot_bot_fun_nat_bool ) )
      = B ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_143_fold1__singleton) ).

thf(2550,plain,
    ! [A: $i,B: $i] :
      ( ( hAPP_f22106695ol_nat @ ( finite_fold1_nat @ A ) @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ B ) @ bot_bot_fun_nat_bool ) )
      = B ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[391]) ).

thf(37,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_int_bool @ ( hAPP_i1948725293t_bool @ ord_less_eq_int @ B ) @ C ) )
     => ( hBOOL @ ( hAPP_int_bool @ ( hAPP_i1948725293t_bool @ ord_less_eq_int @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ A ) @ B ) ) @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ A ) @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_1097_zadd__left__mono) ).

thf(1245,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_int_bool @ ( hAPP_i1948725293t_bool @ ord_less_eq_int @ B ) @ C ) )
     => ( hBOOL @ ( hAPP_int_bool @ ( hAPP_i1948725293t_bool @ ord_less_eq_int @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ A ) @ B ) ) @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ A ) @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[37]) ).

thf(108,axiom,
    ! [A: $i,B: $i] :
      ( ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ ord_min_nat @ A ) @ B )
      = ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ ord_min_nat @ B ) @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_960_min__max_Oinf_Ocommute) ).

thf(1528,plain,
    ! [A: $i,B: $i] :
      ( ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ ord_min_nat @ A ) @ B )
      = ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ ord_min_nat @ B ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[108]) ).

thf(410,axiom,
    ! [A: $i] :
      ( ! [B: $i] :
          ~ ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ B ) @ A ) )
    <=> ( A = bot_bot_fun_nat_bool ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_46_all__not__in__conv) ).

thf(2620,plain,
    ! [A: $i] :
      ( ( ! [B: $i] :
            ~ ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ B ) @ A ) )
       => ( A = bot_bot_fun_nat_bool ) )
      & ( ( A = bot_bot_fun_nat_bool )
       => ! [B: $i] :
            ~ ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ B ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[410]) ).

thf(540,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ A ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ B ) @ C ) )
      = ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ A ) @ B ) ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ A ) @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_696_comm__semiring__1__class_Onormalizing__semiring__rules_I34_J) ).

thf(3110,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ A ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ B ) @ C ) )
      = ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ A ) @ B ) ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ A ) @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[540]) ).

thf(267,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ B ) @ C ) )
     => ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ B ) @ A ) )
       => ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ minus_minus_nat @ C ) @ B ) ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ minus_minus_nat @ A ) @ B ) ) )
        <=> ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ C ) @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_992_le__diff__iff) ).

thf(2086,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ B ) @ C ) )
     => ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ B ) @ A ) )
       => ( ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ minus_minus_nat @ C ) @ B ) ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ minus_minus_nat @ A ) @ B ) ) )
           => ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ C ) @ A ) ) )
          & ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ C ) @ A ) )
           => ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ minus_minus_nat @ C ) @ B ) ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ minus_minus_nat @ A ) @ B ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[267]) ).

thf(816,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ A ) @ B ) ) @ C )
      = ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ A ) @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ B ) @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_668_ab__semigroup__add__class_Oadd__ac_I1_J) ).

thf(4085,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ A ) @ B ) ) @ C )
      = ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ A ) @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ B ) @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[816]) ).

thf(552,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_f284875647l_bool @ ord_le951220754t_bool @ A ) @ B ) )
     => ( ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_f284875647l_bool @ ord_le951220754t_bool @ B ) @ A ) )
       => ( B = A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_744_xt1_I5_J) ).

thf(3159,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_f284875647l_bool @ ord_le951220754t_bool @ A ) @ B ) )
     => ( ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_f284875647l_bool @ ord_le951220754t_bool @ B ) @ A ) )
       => ( B = A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[552]) ).

thf(341,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_f800510211t_bool @ ( hAPP_f1730770594t_bool @ minus_2067140911t_bool @ A ) @ B ) @ C ) )
    <=> ( hBOOL @ ( hAPP_bool_bool @ ( hAPP_b589554111l_bool @ minus_minus_bool @ ( hAPP_nat_bool @ A @ C ) ) @ ( hAPP_nat_bool @ B @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_447_fun__diff__def) ).

thf(2369,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_f800510211t_bool @ ( hAPP_f1730770594t_bool @ minus_2067140911t_bool @ A ) @ B ) @ C ) )
       => ( hBOOL @ ( hAPP_bool_bool @ ( hAPP_b589554111l_bool @ minus_minus_bool @ ( hAPP_nat_bool @ A @ C ) ) @ ( hAPP_nat_bool @ B @ C ) ) ) )
      & ( ( hBOOL @ ( hAPP_bool_bool @ ( hAPP_b589554111l_bool @ minus_minus_bool @ ( hAPP_nat_bool @ A @ C ) ) @ ( hAPP_nat_bool @ B @ C ) ) )
       => ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_f800510211t_bool @ ( hAPP_f1730770594t_bool @ minus_2067140911t_bool @ A ) @ B ) @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[341]) ).

thf(493,axiom,
    ! [A: $i] :
      ( ( hAPP_f800510211t_bool @ ( hAPP_f1730770594t_bool @ minus_2067140911t_bool @ A ) @ bot_bot_fun_nat_bool )
      = A ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_396_Diff__empty) ).

thf(2931,plain,
    ! [A: $i] :
      ( ( hAPP_f800510211t_bool @ ( hAPP_f1730770594t_bool @ minus_2067140911t_bool @ A ) @ bot_bot_fun_nat_bool )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[493]) ).

thf(1066,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ B ) @ C ) )
     => ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ B ) @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ A ) @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_55_insertI2) ).

thf(5042,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ B ) @ C ) )
     => ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ B ) @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ A ) @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1066]) ).

thf(306,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ A ) @ B ) ) @ C ) )
     => ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_nat @ A ) @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_1100_add__lessD1) ).

thf(2244,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ A ) @ B ) ) @ C ) )
     => ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_nat @ A ) @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[306]) ).

thf(183,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f54304608l_bool @ finite_finite_nat @ C ) )
     => ( ~ ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ B ) @ C ) )
       => ( ( hAPP_f22106695ol_nat @ ( big_co1705425894at_nat @ A ) @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ B ) @ C ) )
          = ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ ( hAPP_nat_nat @ A @ B ) ) @ ( hAPP_f22106695ol_nat @ ( big_co1705425894at_nat @ A ) @ C ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_552_setprod__insert) ).

thf(1775,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f54304608l_bool @ finite_finite_nat @ C ) )
     => ( ~ ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ B ) @ C ) )
       => ( ( hAPP_f22106695ol_nat @ ( big_co1705425894at_nat @ A ) @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ B ) @ C ) )
          = ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ ( hAPP_nat_nat @ A @ B ) ) @ ( hAPP_f22106695ol_nat @ ( big_co1705425894at_nat @ A ) @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[183]) ).

thf(400,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ B ) @ C ) )
     => ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ ( hAPP_i240634960iple_a @ A @ B ) ) @ ( image_1844722432iple_a @ A @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_177_imageI) ).

thf(2592,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ B ) @ C ) )
     => ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ ( hAPP_i240634960iple_a @ A @ B ) ) @ ( image_1844722432iple_a @ A @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[400]) ).

thf(962,axiom,
    ! [A: $i] :
      ( ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_f1767618879l_bool @ ord_le1523781517a_bool @ A ) @ bot_bo797238721a_bool ) )
    <=> ( A = bot_bo797238721a_bool ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_840_bot__unique) ).

thf(4703,plain,
    ! [A: $i] :
      ( ( ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_f1767618879l_bool @ ord_le1523781517a_bool @ A ) @ bot_bo797238721a_bool ) )
       => ( A = bot_bo797238721a_bool ) )
      & ( ( A = bot_bo797238721a_bool )
       => ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_f1767618879l_bool @ ord_le1523781517a_bool @ A ) @ bot_bo797238721a_bool ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[962]) ).

thf(269,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ B ) @ C ) )
     => ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ B ) @ A ) )
       => ( ( ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ minus_minus_nat @ C ) @ B )
            = ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ minus_minus_nat @ A ) @ B ) )
        <=> ( C = A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_1038_eq__diff__iff) ).

thf(2101,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ B ) @ C ) )
     => ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ B ) @ A ) )
       => ( ( ( ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ minus_minus_nat @ C ) @ B )
              = ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ minus_minus_nat @ A ) @ B ) )
           => ( C = A ) )
          & ( ( C = A )
           => ( ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ minus_minus_nat @ C ) @ B )
              = ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ minus_minus_nat @ A ) @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[269]) ).

thf(65,axiom,
    ! [A: $i,B: $i] :
      ( ( hAPP_H1229305626_a_int @ ( hAPP_i1475897073_a_int @ cOMBK_1007576201iple_a @ A ) @ B )
      = A ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',help_COMBK_1_1_COMBK_000tc__Int__Oint_000tc__Hoare____Mirabelle____uwgpyvfjxg__O) ).

thf(1364,plain,
    ! [A: $i,B: $i] :
      ( ( hAPP_H1229305626_a_int @ ( hAPP_i1475897073_a_int @ cOMBK_1007576201iple_a @ A ) @ B )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[65]) ).

thf(1037,axiom,
    ! [A: $i,B: $i] :
      ( ( 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 ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_400_finite__Diff2) ).

thf(4954,plain,
    ! [A: $i,B: $i] :
      ( ( 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 ) ) )
        & ( ( hBOOL @ ( hAPP_f1695230391l_bool @ finite1738664244iple_a @ A ) )
         => ( hBOOL @ ( hAPP_f1695230391l_bool @ finite1738664244iple_a @ ( hAPP_f20753329a_bool @ ( hAPP_f1706700729a_bool @ minus_85316870a_bool @ A ) @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1037]) ).

thf(767,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP_int_int @ ( hAPP_int_fun_int_int @ times_times_int @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ minus_minus_int @ A ) @ B ) ) @ C )
      = ( hAPP_int_int @ ( hAPP_int_fun_int_int @ minus_minus_int @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ times_times_int @ A ) @ C ) ) @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ times_times_int @ B ) @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_1089_zdiff__zmult__distrib) ).

thf(3908,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP_int_int @ ( hAPP_int_fun_int_int @ times_times_int @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ minus_minus_int @ A ) @ B ) ) @ C )
      = ( hAPP_int_int @ ( hAPP_int_fun_int_int @ minus_minus_int @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ times_times_int @ A ) @ C ) ) @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ times_times_int @ B ) @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[767]) ).

thf(758,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_f284875647l_bool @ ord_le951220754t_bool @ B ) @ C ) )
     => ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_f284875647l_bool @ ord_le951220754t_bool @ ( image_int_int @ A @ B ) ) @ ( image_int_int @ A @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_872_image__mono) ).

thf(3876,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_f284875647l_bool @ ord_le951220754t_bool @ B ) @ C ) )
     => ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_f284875647l_bool @ ord_le951220754t_bool @ ( image_int_int @ A @ B ) ) @ ( image_int_int @ A @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[758]) ).

thf(159,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ A ) @ ( image_nat_int @ B @ C ) ) )
    <=> ? [D: $i] :
          ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ D ) @ C ) )
          & ( A
            = ( hAPP_nat_int @ B @ D ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_186_image__iff) ).

thf(1694,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ A ) @ ( image_nat_int @ B @ C ) ) )
       => ? [D: $i] :
            ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ D ) @ C ) )
            & ( A
              = ( hAPP_nat_int @ B @ D ) ) ) )
      & ( ? [D: $i] :
            ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ D ) @ C ) )
            & ( A
              = ( hAPP_nat_int @ B @ D ) ) )
       => ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ A ) @ ( image_nat_int @ B @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[159]) ).

thf(428,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ B ) @ C ) )
     => ( ~ ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ B ) @ A ) )
       => ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ B ) @ ( hAPP_f800510211t_bool @ ( hAPP_f1730770594t_bool @ minus_2067140911t_bool @ C ) @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_365_DiffI) ).

thf(2693,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ B ) @ C ) )
     => ( ~ ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ B ) @ A ) )
       => ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ B ) @ ( hAPP_f800510211t_bool @ ( hAPP_f1730770594t_bool @ minus_2067140911t_bool @ C ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[428]) ).

thf(1041,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP_f1695230391l_bool @ finite1738664244iple_a @ B ) )
     => ( hBOOL @ ( hAPP_f1695230391l_bool @ finite1738664244iple_a @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ A ) @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_283_finite_OinsertI) ).

thf(4965,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP_f1695230391l_bool @ finite1738664244iple_a @ B ) )
     => ( hBOOL @ ( hAPP_f1695230391l_bool @ finite1738664244iple_a @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ A ) @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1041]) ).

thf(906,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP_int_int @ ( hAPP_int_fun_int_int @ times_times_int @ A ) @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ minus_minus_int @ B ) @ C ) )
      = ( hAPP_int_int @ ( hAPP_int_fun_int_int @ minus_minus_int @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ times_times_int @ A ) @ B ) ) @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ times_times_int @ A ) @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_1090_zdiff__zmult__distrib2) ).

thf(4481,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP_int_int @ ( hAPP_int_fun_int_int @ times_times_int @ A ) @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ minus_minus_int @ B ) @ C ) )
      = ( hAPP_int_int @ ( hAPP_int_fun_int_int @ minus_minus_int @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ times_times_int @ A ) @ B ) ) @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ times_times_int @ A ) @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[906]) ).

thf(832,axiom,
    ! [A: $i,B: $i] :
      ( ~ ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_f103356543l_bool @ fequal_fun_nat_bool @ A ) @ B ) )
      | ( A = B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',help_fequal_1_1_fequal_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_T) ).

thf(4158,plain,
    ! [A: $i,B: $i] :
      ( ~ ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_f103356543l_bool @ fequal_fun_nat_bool @ A ) @ B ) )
      | ( A = B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[832]) ).

thf(193,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP_int_bool @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ ( finite772772422nt_int @ times_times_int ) @ B ) @ C ) @ D ) )
     => ( ~ ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ B ) @ C ) )
       => ( hBOOL @ ( hAPP_int_bool @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ ( finite772772422nt_int @ times_times_int ) @ A ) @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ B ) @ C ) ) @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ times_times_int @ A ) @ D ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_430_fold__graph__insert__swap) ).

thf(1805,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP_int_bool @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ ( finite772772422nt_int @ times_times_int ) @ B ) @ C ) @ D ) )
     => ( ~ ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ B ) @ C ) )
       => ( hBOOL @ ( hAPP_int_bool @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ ( finite772772422nt_int @ times_times_int ) @ A ) @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ B ) @ C ) ) @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ times_times_int @ A ) @ D ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[193]) ).

thf(240,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( finite1626084323ne_int @ C @ D ) )
     => ( ( hBOOL @ ( hAPP_f448129468l_bool @ finite_finite_int @ B ) )
       => ( ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ A ) @ B ) )
         => ( ( ( ( hAPP_f1805168059t_bool @ ( hAPP_f1223193598t_bool @ minus_1449998731t_bool @ B ) @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ A ) @ bot_bot_fun_int_bool ) )
                = bot_bot_fun_int_bool )
             => ( ( hAPP_f1594865479ol_int @ D @ B )
                = A ) )
            & ( ( ( hAPP_f1805168059t_bool @ ( hAPP_f1223193598t_bool @ minus_1449998731t_bool @ B ) @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ A ) @ bot_bot_fun_int_bool ) )
               != bot_bot_fun_int_bool )
             => ( ( hAPP_f1594865479ol_int @ D @ B )
                = ( hAPP_int_int @ ( hAPP_int_fun_int_int @ C @ A ) @ ( hAPP_f1594865479ol_int @ D @ ( hAPP_f1805168059t_bool @ ( hAPP_f1223193598t_bool @ minus_1449998731t_bool @ B ) @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ A ) @ bot_bot_fun_int_bool ) ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_358_folding__one_Oremove) ).

thf(2005,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( finite1626084323ne_int @ C @ D ) )
     => ( ( hBOOL @ ( hAPP_f448129468l_bool @ finite_finite_int @ B ) )
       => ( ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ A ) @ B ) )
         => ( ( ( ( hAPP_f1805168059t_bool @ ( hAPP_f1223193598t_bool @ minus_1449998731t_bool @ B ) @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ A ) @ bot_bot_fun_int_bool ) )
                = bot_bot_fun_int_bool )
             => ( ( hAPP_f1594865479ol_int @ D @ B )
                = A ) )
            & ( ( ( hAPP_f1805168059t_bool @ ( hAPP_f1223193598t_bool @ minus_1449998731t_bool @ B ) @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ A ) @ bot_bot_fun_int_bool ) )
               != bot_bot_fun_int_bool )
             => ( ( hAPP_f1594865479ol_int @ D @ B )
                = ( hAPP_int_int @ ( hAPP_int_fun_int_int @ C @ A ) @ ( hAPP_f1594865479ol_int @ D @ ( hAPP_f1805168059t_bool @ ( hAPP_f1223193598t_bool @ minus_1449998731t_bool @ B ) @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ A ) @ bot_bot_fun_int_bool ) ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[240]) ).

thf(976,axiom,
    ! [A: $i] :
      ( ( hAPP_f20753329a_bool @ collec351493750iple_a @ ( hAPP_H426895267a_bool @ ( hAPP_f920293029a_bool @ cOMBC_1940896922a_bool @ fequal963300192iple_a ) @ A ) )
      = ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ A ) @ bot_bo797238721a_bool ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_22_singleton__conv) ).

thf(4741,plain,
    ! [A: $i] :
      ( ( hAPP_f20753329a_bool @ collec351493750iple_a @ ( hAPP_H426895267a_bool @ ( hAPP_f920293029a_bool @ cOMBC_1940896922a_bool @ fequal963300192iple_a ) @ A ) )
      = ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ A ) @ bot_bo797238721a_bool ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[976]) ).

thf(827,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( C = D )
     => ( ! [E: $i] :
            ( ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ E ) @ D ) )
           => ( ( hAPP_H905846293iple_a @ A @ E )
              = ( hAPP_H905846293iple_a @ B @ E ) ) )
       => ( ( image_533441733iple_a @ A @ C )
          = ( image_533441733iple_a @ B @ D ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_257_image__cong) ).

thf(4136,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( C = D )
     => ( ! [E: $i] :
            ( ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ E ) @ D ) )
           => ( ( hAPP_H905846293iple_a @ A @ E )
              = ( hAPP_H905846293iple_a @ B @ E ) ) )
       => ( ( image_533441733iple_a @ A @ C )
          = ( image_533441733iple_a @ B @ D ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[827]) ).

thf(211,axiom,
    ! [A: $i,B: $i] : ( is_bool @ ( hAPP_nat_bool @ A @ B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',gsy_c_hAPP_000tc__Nat__Onat_000tc__HOL__Obool) ).

thf(1873,plain,
    ! [A: $i,B: $i] : ( is_bool @ ( hAPP_nat_bool @ A @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[211]) ).

thf(314,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ A ) @ B ) )
      | ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ B ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_787_linorder__linear) ).

thf(2271,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ A ) @ B ) )
      | ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ B ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[314]) ).

thf(251,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ A ) @ ( hAPP_f1805168059t_bool @ ( hAPP_f1223193598t_bool @ minus_1449998731t_bool @ B ) @ C ) ) )
     => ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ A ) @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_387_DiffD1) ).

thf(2037,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ A ) @ ( hAPP_f1805168059t_bool @ ( hAPP_f1223193598t_bool @ minus_1449998731t_bool @ B ) @ C ) ) )
     => ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ A ) @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[251]) ).

thf(4,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_int_bool @ ( hAPP_i1948725293t_bool @ ord_less_eq_int @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ A ) @ B ) ) @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ A ) @ C ) ) )
     => ( hBOOL @ ( hAPP_int_bool @ ( hAPP_i1948725293t_bool @ ord_less_eq_int @ B ) @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_818_add__le__imp__le__left) ).

thf(1134,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_int_bool @ ( hAPP_i1948725293t_bool @ ord_less_eq_int @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ A ) @ B ) ) @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ A ) @ C ) ) )
     => ( hBOOL @ ( hAPP_int_bool @ ( hAPP_i1948725293t_bool @ ord_less_eq_int @ B ) @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[4]) ).

thf(837,axiom,
    ! [A: $i,B: $i] :
      ( ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ A ) @ B )
      = ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ B ) @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_1046_nat__mult__commute) ).

thf(4177,plain,
    ! [A: $i,B: $i] :
      ( ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ A ) @ B )
      = ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ B ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[837]) ).

thf(935,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hoare_2102800559rivs_a @ C ) @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ ( hoare_1916936827iple_a @ D @ E @ F ) ) @ bot_bo797238721a_bool ) ) )
     => ( ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hoare_2102800559rivs_a @ C ) @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ ( hoare_1916936827iple_a @ F @ A @ B ) ) @ bot_bo797238721a_bool ) ) )
       => ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hoare_2102800559rivs_a @ C ) @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ ( hoare_1916936827iple_a @ D @ ( semi @ E @ A ) @ B ) ) @ bot_bo797238721a_bool ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_112_Comp) ).

thf(4581,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hoare_2102800559rivs_a @ C ) @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ ( hoare_1916936827iple_a @ D @ E @ F ) ) @ bot_bo797238721a_bool ) ) )
     => ( ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hoare_2102800559rivs_a @ C ) @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ ( hoare_1916936827iple_a @ F @ A @ B ) ) @ bot_bo797238721a_bool ) ) )
       => ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hoare_2102800559rivs_a @ C ) @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ ( hoare_1916936827iple_a @ D @ ( semi @ E @ A ) @ B ) ) @ bot_bo797238721a_bool ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[935]) ).

thf(877,axiom,
    ! [A: $i,B: $i] :
      ( bot_bot_fun_int_bool
     != ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ A ) @ B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_101_empty__not__insert) ).

thf(4355,plain,
    ! [A: $i,B: $i] :
      ( bot_bot_fun_int_bool
     != ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ A ) @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[877]) ).

thf(835,axiom,
    ! [A: $i] :
      ( ( hAPP_f1693662087iple_a @ the_Ho830102290iple_a @ ( hAPP_H426895267a_bool @ fequal963300192iple_a @ A ) )
      = A ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_249_the__sym__eq__trivial) ).

thf(4168,plain,
    ! [A: $i] :
      ( ( hAPP_f1693662087iple_a @ the_Ho830102290iple_a @ ( hAPP_H426895267a_bool @ fequal963300192iple_a @ A ) )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[835]) ).

thf(1023,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( finite1428139793iple_a @ E ) )
     => ( ( hBOOL @ ( hAPP_H1037229737a_bool @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ ( finite90892550iple_a @ E ) @ B ) @ C ) @ D ) )
       => ( ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ A ) @ C ) )
         => ? [F: $i] :
              ( ( D
                = ( hAPP_H905846293iple_a @ ( hAPP_H2085992369iple_a @ E @ A ) @ F ) )
              & ( hBOOL @ ( hAPP_H1037229737a_bool @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ ( finite90892550iple_a @ E ) @ B ) @ ( hAPP_f20753329a_bool @ ( hAPP_f1706700729a_bool @ minus_85316870a_bool @ C ) @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ A ) @ bot_bo797238721a_bool ) ) ) @ F ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_458_comp__fun__commute_Ofold__graph__insertE__aux) ).

thf(4907,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( finite1428139793iple_a @ E ) )
     => ( ( hBOOL @ ( hAPP_H1037229737a_bool @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ ( finite90892550iple_a @ E ) @ B ) @ C ) @ D ) )
       => ( ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ A ) @ C ) )
         => ? [F: $i] :
              ( ( D
                = ( hAPP_H905846293iple_a @ ( hAPP_H2085992369iple_a @ E @ A ) @ F ) )
              & ( hBOOL @ ( hAPP_H1037229737a_bool @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ ( finite90892550iple_a @ E ) @ B ) @ ( hAPP_f20753329a_bool @ ( hAPP_f1706700729a_bool @ minus_85316870a_bool @ C ) @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ A ) @ bot_bo797238721a_bool ) ) ) @ F ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1023]) ).

thf(125,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( finite1071749497t_bool @ D ) )
     => ( ( hBOOL @ ( hAPP_f54304608l_bool @ finite_finite_nat @ C ) )
       => ( ~ ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ B ) @ C ) )
         => ( ( hAPP_f800510211t_bool @ ( finite1578363458t_bool @ D @ A ) @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ B ) @ C ) )
            = ( hAPP_f800510211t_bool @ ( finite1578363458t_bool @ D @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ D @ B ) @ A ) ) @ C ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_506_comp__fun__commute_Ofold__insert2) ).

thf(1583,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( finite1071749497t_bool @ D ) )
     => ( ( hBOOL @ ( hAPP_f54304608l_bool @ finite_finite_nat @ C ) )
       => ( ~ ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ B ) @ C ) )
         => ( ( hAPP_f800510211t_bool @ ( finite1578363458t_bool @ D @ A ) @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ B ) @ C ) )
            = ( hAPP_f800510211t_bool @ ( finite1578363458t_bool @ D @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ D @ B ) @ A ) ) @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[125]) ).

thf(777,axiom,
    ! [A: $i,B: $i] : ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_f103356543l_bool @ ord_le1568362934t_bool @ A ) @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ B ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_869_subset__insertI) ).

thf(3936,plain,
    ! [A: $i,B: $i] : ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_f103356543l_bool @ ord_le1568362934t_bool @ A ) @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ B ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[777]) ).

thf(270,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ A ) @ ( image_int_int @ B @ C ) ) )
     => ~ ! [D: $i] :
            ( ( A
              = ( hAPP_int_int @ B @ D ) )
           => ~ ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ D ) @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_244_imageE) ).

thf(2107,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ A ) @ ( image_int_int @ B @ C ) ) )
     => ~ ! [D: $i] :
            ( ( A
              = ( hAPP_int_int @ B @ D ) )
           => ~ ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ D ) @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[270]) ).

thf(593,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f448129468l_bool @ finite_finite_int @ C ) )
     => ( ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_f284875647l_bool @ ord_le951220754t_bool @ A ) @ ( image_int_int @ B @ C ) ) )
       => ( hBOOL @ ( hAPP_f448129468l_bool @ finite_finite_int @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_894_finite__surj) ).

thf(3293,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f448129468l_bool @ finite_finite_int @ C ) )
     => ( ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_f284875647l_bool @ ord_le951220754t_bool @ A ) @ ( image_int_int @ B @ C ) ) )
       => ( hBOOL @ ( hAPP_f448129468l_bool @ finite_finite_int @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[593]) ).

thf(939,axiom,
    ! [A: $i] :
      ( ( hAPP_f20753329a_bool @ ( hAPP_f1706700729a_bool @ minus_85316870a_bool @ A ) @ bot_bo797238721a_bool )
      = A ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_394_Diff__empty) ).

thf(4598,plain,
    ! [A: $i] :
      ( ( hAPP_f20753329a_bool @ ( hAPP_f1706700729a_bool @ minus_85316870a_bool @ A ) @ bot_bo797238721a_bool )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[939]) ).

thf(974,axiom,
    ! [A: $i,B: $i] :
      ( ( ( hBOOL @ ( hAPP_f1695230391l_bool @ finite1738664244iple_a @ ( hAPP_f20753329a_bool @ collec351493750iple_a @ B ) ) )
        | ( hBOOL @ ( hAPP_f1695230391l_bool @ finite1738664244iple_a @ ( hAPP_f20753329a_bool @ collec351493750iple_a @ 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 ) @ B ) ) @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_276_finite__Collect__conjI) ).

thf(4735,plain,
    ! [A: $i,B: $i] :
      ( ( ( hBOOL @ ( hAPP_f1695230391l_bool @ finite1738664244iple_a @ ( hAPP_f20753329a_bool @ collec351493750iple_a @ B ) ) )
        | ( hBOOL @ ( hAPP_f1695230391l_bool @ finite1738664244iple_a @ ( hAPP_f20753329a_bool @ collec351493750iple_a @ 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 ) @ B ) ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[974]) ).

thf(80,axiom,
    ! [A: $i,B: $i] :
      ( ( hAPP_f1594865479ol_int @ ( finite_fold1_int @ A ) @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ B ) @ bot_bot_fun_int_bool ) )
      = B ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_145_fold1__singleton) ).

thf(1420,plain,
    ! [A: $i,B: $i] :
      ( ( hAPP_f1594865479ol_int @ ( finite_fold1_int @ A ) @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ B ) @ bot_bot_fun_int_bool ) )
      = B ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[80]) ).

thf(478,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ A ) @ B ) )
    <=> ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ ( hAPP_nat_int @ semiri1621563631at_int @ A ) ) @ ( image_nat_int @ semiri1621563631at_int @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_1155_transfer__nat__int__set__relations_I2_J) ).

thf(2878,plain,
    ! [A: $i,B: $i] :
      ( ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ A ) @ B ) )
       => ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ ( hAPP_nat_int @ semiri1621563631at_int @ A ) ) @ ( image_nat_int @ semiri1621563631at_int @ B ) ) ) )
      & ( ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ ( hAPP_nat_int @ semiri1621563631at_int @ A ) ) @ ( image_nat_int @ semiri1621563631at_int @ B ) ) )
       => ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ A ) @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[478]) ).

thf(155,axiom,
    hBOOL @ ( finite2130160977at_nat @ times_times_nat ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_533_comp__fun__commute) ).

thf(1684,plain,
    hBOOL @ ( finite2130160977at_nat @ times_times_nat ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[155]) ).

thf(45,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_int_bool @ ( hAPP_i1948725293t_bool @ ord_less_eq_int @ B ) @ C ) )
     => ( ( C = A )
       => ( hBOOL @ ( hAPP_int_bool @ ( hAPP_i1948725293t_bool @ ord_less_eq_int @ B ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_762_ord__le__eq__trans) ).

thf(1277,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_int_bool @ ( hAPP_i1948725293t_bool @ ord_less_eq_int @ B ) @ C ) )
     => ( ( C = A )
       => ( hBOOL @ ( hAPP_int_bool @ ( hAPP_i1948725293t_bool @ ord_less_eq_int @ B ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[45]) ).

thf(870,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ times_times_int @ A ) @ B ) ) @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ times_times_int @ C ) @ D ) )
        = ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ times_times_int @ A ) @ D ) ) @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ times_times_int @ C ) @ B ) ) )
    <=> ( ( A = C )
        | ( B = D ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_648_crossproduct__eq) ).

thf(4319,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ times_times_int @ A ) @ B ) ) @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ times_times_int @ C ) @ D ) )
          = ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ times_times_int @ A ) @ D ) ) @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ times_times_int @ C ) @ B ) ) )
       => ( ( A = C )
          | ( B = D ) ) )
      & ( ( ( A = C )
          | ( B = D ) )
       => ( ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ times_times_int @ A ) @ B ) ) @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ times_times_int @ C ) @ D ) )
          = ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ times_times_int @ A ) @ D ) ) @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ times_times_int @ C ) @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[870]) ).

thf(864,axiom,
    ! [A: $i,B: $i] :
      ( ~ ( hBOOL @ ( hAPP_H1037229737a_bool @ ( hAPP_H426895267a_bool @ fequal963300192iple_a @ A ) @ B ) )
      | ( A = B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',help_fequal_1_1_fequal_000tc__Hoare____Mirabelle____uwgpyvfjxg__Otriple_It__a_J_) ).

thf(4292,plain,
    ! [A: $i,B: $i] :
      ( ~ ( hBOOL @ ( hAPP_H1037229737a_bool @ ( hAPP_H426895267a_bool @ fequal963300192iple_a @ A ) @ B ) )
      | ( A = B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[864]) ).

thf(363,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f448129468l_bool @ finite_finite_int @ C ) )
     => ( ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ B ) @ C ) )
       => ( ( hAPP_f1594865479ol_int @ ( hAPP_f1926459811ol_int @ big_co230513141nt_int @ A ) @ C )
          = ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ ( hAPP_int_int @ A @ B ) ) @ ( hAPP_f1594865479ol_int @ ( hAPP_f1926459811ol_int @ big_co230513141nt_int @ A ) @ ( hAPP_f1805168059t_bool @ ( hAPP_f1223193598t_bool @ minus_1449998731t_bool @ C ) @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ B ) @ bot_bot_fun_int_bool ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_611_setsum_Oremove) ).

thf(2449,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f448129468l_bool @ finite_finite_int @ C ) )
     => ( ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ B ) @ C ) )
       => ( ( hAPP_f1594865479ol_int @ ( hAPP_f1926459811ol_int @ big_co230513141nt_int @ A ) @ C )
          = ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ ( hAPP_int_int @ A @ B ) ) @ ( hAPP_f1594865479ol_int @ ( hAPP_f1926459811ol_int @ big_co230513141nt_int @ A ) @ ( hAPP_f1805168059t_bool @ ( hAPP_f1223193598t_bool @ minus_1449998731t_bool @ C ) @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ B ) @ bot_bot_fun_int_bool ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[363]) ).

thf(772,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_f103356543l_bool @ ord_le1568362934t_bool @ B ) @ C ) )
     => ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_f103356543l_bool @ ord_le1568362934t_bool @ ( image_nat_nat @ A @ B ) ) @ ( image_nat_nat @ A @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_875_image__mono) ).

thf(3921,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_f103356543l_bool @ ord_le1568362934t_bool @ B ) @ C ) )
     => ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_f103356543l_bool @ ord_le1568362934t_bool @ ( image_nat_nat @ A @ B ) ) @ ( image_nat_nat @ A @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[772]) ).

thf(57,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_nat @ A ) @ B ) )
     => ( B != A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_1118_less__not__refl2) ).

thf(1340,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_nat @ A ) @ B ) )
     => ( B != A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[57]) ).

thf(634,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_f103356543l_bool @ ord_le1568362934t_bool @ A ) @ B ) )
    <=> ! [C: $i] : ( hBOOL @ ( hAPP_bool_bool @ ( hAPP_b589554111l_bool @ ord_less_eq_bool @ ( hAPP_nat_bool @ A @ C ) ) @ ( hAPP_nat_bool @ B @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_789_le__fun__def) ).

thf(3455,plain,
    ! [A: $i,B: $i] :
      ( ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_f103356543l_bool @ ord_le1568362934t_bool @ A ) @ B ) )
       => ! [C: $i] : ( hBOOL @ ( hAPP_bool_bool @ ( hAPP_b589554111l_bool @ ord_less_eq_bool @ ( hAPP_nat_bool @ A @ C ) ) @ ( hAPP_nat_bool @ B @ C ) ) ) )
      & ( ! [C: $i] : ( hBOOL @ ( hAPP_bool_bool @ ( hAPP_b589554111l_bool @ ord_less_eq_bool @ ( hAPP_nat_bool @ A @ C ) ) @ ( hAPP_nat_bool @ B @ C ) ) )
       => ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_f103356543l_bool @ ord_le1568362934t_bool @ A ) @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[634]) ).

thf(95,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( finite1626084323ne_int @ B @ C ) )
     => ( ( hBOOL @ ( hAPP_f448129468l_bool @ finite_finite_int @ A ) )
       => ( ( hAPP_f1594865479ol_int @ C @ A )
          = ( hAPP_f1594865479ol_int @ ( finite_fold1_int @ B ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_272_folding__one_Oeq__fold) ).

thf(1472,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( finite1626084323ne_int @ B @ C ) )
     => ( ( hBOOL @ ( hAPP_f448129468l_bool @ finite_finite_int @ A ) )
       => ( ( hAPP_f1594865479ol_int @ C @ A )
          = ( hAPP_f1594865479ol_int @ ( finite_fold1_int @ B ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[95]) ).

thf(331,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ B ) @ C ) )
     => ( ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_f284875647l_bool @ ord_le951220754t_bool @ C ) @ A ) )
       => ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ B ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_798_set__rev__mp) ).

thf(2333,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ B ) @ C ) )
     => ( ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_f284875647l_bool @ ord_le951220754t_bool @ C ) @ A ) )
       => ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ B ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[331]) ).

thf(8,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP_int_bool @ ( hAPP_i1948725293t_bool @ ord_less_eq_int @ C ) @ D ) )
     => ( ( hBOOL @ ( hAPP_int_bool @ ( hAPP_i1948725293t_bool @ ord_less_eq_int @ A ) @ B ) )
       => ( hBOOL @ ( hAPP_int_bool @ ( hAPP_i1948725293t_bool @ ord_less_eq_int @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ C ) @ A ) ) @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ D ) @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_822_add__mono) ).

thf(1145,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP_int_bool @ ( hAPP_i1948725293t_bool @ ord_less_eq_int @ C ) @ D ) )
     => ( ( hBOOL @ ( hAPP_int_bool @ ( hAPP_i1948725293t_bool @ ord_less_eq_int @ A ) @ B ) )
       => ( hBOOL @ ( hAPP_int_bool @ ( hAPP_i1948725293t_bool @ ord_less_eq_int @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ C ) @ A ) ) @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ D ) @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[8]) ).

thf(822,axiom,
    ! [A: $i,B: $i] :
      ( ( 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 ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_783_order__eq__iff) ).

thf(4113,plain,
    ! [A: $i,B: $i] :
      ( ( ( 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 ) ) ) )
      & ( ( ( 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 ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[822]) ).

thf(322,axiom,
    ( bot_bot_fun_int_bool
    = ( image_nat_int @ semiri1621563631at_int @ bot_bot_fun_nat_bool ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_1161_Nat__Transfer_Otransfer__int__nat__set__functions_I2_J) ).

thf(2298,plain,
    ( bot_bot_fun_int_bool
    = ( image_nat_int @ semiri1621563631at_int @ bot_bot_fun_nat_bool ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[322]) ).

thf(769,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP_state_bool @ ( hAPP_s1806633685e_bool @ ( hAPP_f817621513e_bool @ cOMBC_2027030106e_bool @ A ) @ B ) @ C )
      = ( hAPP_state_bool @ ( hAPP_s1806633685e_bool @ A @ C ) @ B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',help_COMBC_1_1_COMBC_000tc__Com__Ostate_000tc__Com__Ostate_000tc__HOL__Obool_U) ).

thf(3913,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP_state_bool @ ( hAPP_s1806633685e_bool @ ( hAPP_f817621513e_bool @ cOMBC_2027030106e_bool @ A ) @ B ) @ C )
      = ( hAPP_state_bool @ ( hAPP_s1806633685e_bool @ A @ C ) @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[769]) ).

thf(909,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ A ) @ B ) ) @ C )
      = ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ A ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ B ) @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_1045_nat__mult__assoc) ).

thf(4490,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ A ) @ B ) ) @ C )
      = ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ A ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ B ) @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[909]) ).

thf(958,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP_f1695230391l_bool @ finite1738664244iple_a @ B ) )
     => ( ( hBOOL @ ( hAPP_f1695230391l_bool @ A @ bot_bo797238721a_bool ) )
       => ( ! [C: $i,D: $i] :
              ( ( hBOOL @ ( hAPP_f1695230391l_bool @ finite1738664244iple_a @ D ) )
             => ( ~ ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ C ) @ D ) )
               => ( ( hBOOL @ ( hAPP_f1695230391l_bool @ A @ D ) )
                 => ( hBOOL @ ( hAPP_f1695230391l_bool @ A @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ C ) @ D ) ) ) ) ) )
         => ( hBOOL @ ( hAPP_f1695230391l_bool @ A @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_318_finite__induct) ).

thf(4673,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP_f1695230391l_bool @ finite1738664244iple_a @ B ) )
     => ( ( hBOOL @ ( hAPP_f1695230391l_bool @ A @ bot_bo797238721a_bool ) )
       => ( ! [C: $i,D: $i] :
              ( ( hBOOL @ ( hAPP_f1695230391l_bool @ finite1738664244iple_a @ D ) )
             => ( ~ ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ C ) @ D ) )
               => ( ( hBOOL @ ( hAPP_f1695230391l_bool @ A @ D ) )
                 => ( hBOOL @ ( hAPP_f1695230391l_bool @ A @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ C ) @ D ) ) ) ) ) )
         => ( hBOOL @ ( hAPP_f1695230391l_bool @ A @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[958]) ).

thf(1027,axiom,
    hBOOL @ ( finite1956789438t_bool @ insert_nat ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_454_comp__fun__idem__insert) ).

thf(4922,plain,
    hBOOL @ ( finite1956789438t_bool @ insert_nat ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1027]) ).

thf(743,axiom,
    ! [A: $i,B: $i] : ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_f103356543l_bool @ ord_le1568362934t_bool @ ( hAPP_f800510211t_bool @ ( hAPP_f1730770594t_bool @ minus_2067140911t_bool @ A ) @ B ) ) @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_888_Diff__subset) ).

thf(3833,plain,
    ! [A: $i,B: $i] : ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_f103356543l_bool @ ord_le1568362934t_bool @ ( hAPP_f800510211t_bool @ ( hAPP_f1730770594t_bool @ minus_2067140911t_bool @ A ) @ B ) ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[743]) ).

thf(109,axiom,
    ! [A: $i,B: $i] :
      ( ( hAPP_f1594865479ol_int @ ( finite_fold_int_int @ A @ B ) @ bot_bot_fun_int_bool )
      = B ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_501_fold__empty) ).

thf(1531,plain,
    ! [A: $i,B: $i] :
      ( ( hAPP_f1594865479ol_int @ ( finite_fold_int_int @ A @ B ) @ bot_bot_fun_int_bool )
      = B ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[109]) ).

thf(481,axiom,
    ! [A: $i,B: $i] : ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ A ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ A ) @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_1012_le__add1) ).

thf(2888,plain,
    ! [A: $i,B: $i] : ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ A ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ A ) @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[481]) ).

thf(1059,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP_state_bool @ ( hAPP_f1759915619e_bool @ ( hAPP_f2073279419e_bool @ cOMBB_160679318_state @ A ) @ B ) @ C )
      = ( hAPP_bool_bool @ A @ ( hAPP_state_bool @ B @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',help_COMBB_1_1_COMBB_000tc__HOL__Obool_000tc__HOL__Obool_000tc__Com__Ostate_U) ).

thf(5018,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP_state_bool @ ( hAPP_f1759915619e_bool @ ( hAPP_f2073279419e_bool @ cOMBB_160679318_state @ A ) @ B ) @ C )
      = ( hAPP_bool_bool @ A @ ( hAPP_state_bool @ B @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1059]) ).

thf(465,axiom,
    ! [A: $i] : ( is_bool @ ( finite1071749497t_bool @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',gsy_c_Finite__Set_Ocomp__fun__commute_000tc__Nat__Onat_000tc__fun_Itc__Nat__Onat) ).

thf(2831,plain,
    ! [A: $i] : ( is_bool @ ( finite1071749497t_bool @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[465]) ).

thf(802,axiom,
    ! [A: $i,B: $i] :
      ( ! [C: $i] :
          ( ( hBOOL @ ( hAPP_int_bool @ B @ C ) )
         => ( hBOOL @ ( hAPP_int_bool @ A @ C ) ) )
     => ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_f284875647l_bool @ ord_le951220754t_bool @ ( hAPP_f1805168059t_bool @ collect_int @ B ) ) @ ( hAPP_f1805168059t_bool @ collect_int @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_940_Collect__mono) ).

thf(4045,plain,
    ! [A: $i,B: $i] :
      ( ! [C: $i] :
          ( ( hBOOL @ ( hAPP_int_bool @ B @ C ) )
         => ( hBOOL @ ( hAPP_int_bool @ A @ C ) ) )
     => ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_f284875647l_bool @ ord_le951220754t_bool @ ( hAPP_f1805168059t_bool @ collect_int @ B ) ) @ ( hAPP_f1805168059t_bool @ collect_int @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[802]) ).

thf(954,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP_f1695230391l_bool @ finite1738664244iple_a @ B ) )
     => ( ( hAPP_f20753329a_bool @ ( hAPP_f1706700729a_bool @ minus_85316870a_bool @ A ) @ B )
        = ( 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 ) ) @ A ) @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_479_minus__fold__remove) ).

thf(4658,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP_f1695230391l_bool @ finite1738664244iple_a @ B ) )
     => ( ( hAPP_f20753329a_bool @ ( hAPP_f1706700729a_bool @ minus_85316870a_bool @ A ) @ B )
        = ( 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 ) ) @ A ) @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[954]) ).

thf(381,axiom,
    ! [A: $i,B: $i] :
      ( ( bot_bot_fun_int_bool
        = ( image_int_int @ A @ B ) )
    <=> ( B = bot_bot_fun_int_bool ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_211_empty__is__image) ).

thf(2513,plain,
    ! [A: $i,B: $i] :
      ( ( ( bot_bot_fun_int_bool
          = ( image_int_int @ A @ B ) )
       => ( B = bot_bot_fun_int_bool ) )
      & ( ( B = bot_bot_fun_int_bool )
       => ( bot_bot_fun_int_bool
          = ( image_int_int @ A @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[381]) ).

thf(978,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( 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 @ B ) @ bot_bo797238721a_bool ) ) ) @ C ) )
     => ( ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ B ) @ A ) )
       => ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_f1767618879l_bool @ ord_le1523781517a_bool @ A ) @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ B ) @ C ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_906_diff__single__insert) ).

thf(4749,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( 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 @ B ) @ bot_bo797238721a_bool ) ) ) @ C ) )
     => ( ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ B ) @ A ) )
       => ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_f1767618879l_bool @ ord_le1523781517a_bool @ A ) @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ B ) @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[978]) ).

thf(114,axiom,
    ! [A: $i] :
      ( ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ ord_min_nat @ A ) @ A )
      = A ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_961_min__max_Oinf_Oidem) ).

thf(1547,plain,
    ! [A: $i] :
      ( ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ ord_min_nat @ A ) @ A )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[114]) ).

thf(1068,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_f1767618879l_bool @ ord_le1523781517a_bool @ B ) @ C ) )
     => ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_f1767618879l_bool @ ord_le1523781517a_bool @ ( image_533441733iple_a @ A @ B ) ) @ ( image_533441733iple_a @ A @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_873_image__mono) ).

thf(5047,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_f1767618879l_bool @ ord_le1523781517a_bool @ B ) @ C ) )
     => ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_f1767618879l_bool @ ord_le1523781517a_bool @ ( image_533441733iple_a @ A @ B ) ) @ ( image_533441733iple_a @ A @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1068]) ).

thf(104,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP_f448129468l_bool @ finite_finite_int @ B ) )
     => ( ( B != bot_bot_fun_int_bool )
       => ? [C: $i] : ( hBOOL @ ( hAPP_int_bool @ ( finite_fold1Set_int @ A @ B ) @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_313_finite__nonempty__imp__fold1Set) ).

thf(1516,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP_f448129468l_bool @ finite_finite_int @ B ) )
     => ( ( B != bot_bot_fun_int_bool )
       => ? [C: $i] : ( hBOOL @ ( hAPP_int_bool @ ( finite_fold1Set_int @ A @ B ) @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[104]) ).

thf(474,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ A ) @ B ) ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ A ) @ C ) ) )
    <=> ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_nat @ B ) @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_1106_nat__add__left__cancel__less) ).

thf(2863,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ A ) @ B ) ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ A ) @ C ) ) )
       => ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_nat @ B ) @ C ) ) )
      & ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_nat @ B ) @ C ) )
       => ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ A ) @ B ) ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ A ) @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[474]) ).

thf(349,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_nat_bool @ ( finite_fold1Set_nat @ A @ B ) @ C ) )
    <=> ? [D: $i,E: $i,F: $i] :
          ( ( B
            = ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ D ) @ E ) )
          & ( C = F )
          & ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ ( finite929467206at_nat @ A ) @ D ) @ E ) @ F ) )
          & ~ ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ D ) @ E ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_332_fold1Set_Osimps) ).

thf(2397,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( hBOOL @ ( hAPP_nat_bool @ ( finite_fold1Set_nat @ A @ B ) @ C ) )
       => ? [D: $i,E: $i,F: $i] :
            ( ( B
              = ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ D ) @ E ) )
            & ( C = F )
            & ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ ( finite929467206at_nat @ A ) @ D ) @ E ) @ F ) )
            & ~ ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ D ) @ E ) ) ) )
      & ( ? [D: $i,E: $i,F: $i] :
            ( ( B
              = ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ D ) @ E ) )
            & ( C = F )
            & ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ ( finite929467206at_nat @ A ) @ D ) @ E ) @ F ) )
            & ~ ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ D ) @ E ) ) )
       => ( hBOOL @ ( hAPP_nat_bool @ ( finite_fold1Set_nat @ A @ B ) @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[349]) ).

thf(651,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ A ) @ ( image_533441733iple_a @ B @ C ) ) )
    <=> ? [D: $i] :
          ( ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ D ) @ C ) )
          & ( A
            = ( hAPP_H905846293iple_a @ B @ D ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_183_image__iff) ).

thf(3507,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ A ) @ ( image_533441733iple_a @ B @ C ) ) )
       => ? [D: $i] :
            ( ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ D ) @ C ) )
            & ( A
              = ( hAPP_H905846293iple_a @ B @ D ) ) ) )
      & ( ? [D: $i] :
            ( ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ D ) @ C ) )
            & ( A
              = ( hAPP_H905846293iple_a @ B @ D ) ) )
       => ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ A ) @ ( image_533441733iple_a @ B @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[651]) ).

thf(505,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_f103356543l_bool @ ord_le1568362934t_bool @ A ) @ B ) )
     => ( ( A != bot_bot_fun_nat_bool )
       => ( ( hBOOL @ ( hAPP_f54304608l_bool @ finite_finite_nat @ B ) )
         => ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ ( hAPP_f22106695ol_nat @ big_linorder_Max_nat @ A ) ) @ ( hAPP_f22106695ol_nat @ big_linorder_Max_nat @ B ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_946_Max__mono) ).

thf(2977,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_f103356543l_bool @ ord_le1568362934t_bool @ A ) @ B ) )
     => ( ( A != bot_bot_fun_nat_bool )
       => ( ( hBOOL @ ( hAPP_f54304608l_bool @ finite_finite_nat @ B ) )
         => ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ ( hAPP_f22106695ol_nat @ big_linorder_Max_nat @ A ) ) @ ( hAPP_f22106695ol_nat @ big_linorder_Max_nat @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[505]) ).

thf(1082,axiom,
    ! [A: $i] :
      ( ( hAPP_f1693662087iple_a @ the_el1519802624iple_a @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ A ) @ bot_bo797238721a_bool ) )
      = A ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_103_the__elem__eq) ).

thf(5111,plain,
    ! [A: $i] :
      ( ( hAPP_f1693662087iple_a @ the_el1519802624iple_a @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ A ) @ bot_bo797238721a_bool ) )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1082]) ).

thf(1033,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( finite1428139793iple_a @ D ) )
     => ( ( hBOOL @ ( hAPP_f1695230391l_bool @ finite1738664244iple_a @ C ) )
       => ( ~ ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ B ) @ C ) )
         => ( ( hAPP_f1693662087iple_a @ ( finite1205970312iple_a @ D @ A ) @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ B ) @ C ) )
            = ( hAPP_f1693662087iple_a @ ( finite1205970312iple_a @ D @ ( hAPP_H905846293iple_a @ ( hAPP_H2085992369iple_a @ D @ B ) @ A ) ) @ C ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_507_comp__fun__commute_Ofold__insert2) ).

thf(4939,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( finite1428139793iple_a @ D ) )
     => ( ( hBOOL @ ( hAPP_f1695230391l_bool @ finite1738664244iple_a @ C ) )
       => ( ~ ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ B ) @ C ) )
         => ( ( hAPP_f1693662087iple_a @ ( finite1205970312iple_a @ D @ A ) @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ B ) @ C ) )
            = ( hAPP_f1693662087iple_a @ ( finite1205970312iple_a @ D @ ( hAPP_H905846293iple_a @ ( hAPP_H2085992369iple_a @ D @ B ) @ A ) ) @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1033]) ).

thf(74,axiom,
    ! [A: $i,B: $i] :
      ( ? [C: $i] :
          ( ( hBOOL @ ( hAPP_nat_bool @ B @ C ) )
          & ! [D: $i] :
              ( ( hBOOL @ ( hAPP_nat_bool @ B @ D ) )
             => ( D = C ) ) )
     => ( ( hBOOL @ ( hAPP_nat_bool @ B @ A ) )
       => ( ( hAPP_f22106695ol_nat @ the_nat @ B )
          = A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_340_the1__equality) ).

thf(1400,plain,
    ! [A: $i,B: $i] :
      ( ? [C: $i] :
          ( ( hBOOL @ ( hAPP_nat_bool @ B @ C ) )
          & ! [D: $i] :
              ( ( hBOOL @ ( hAPP_nat_bool @ B @ D ) )
             => ( D = C ) ) )
     => ( ( hBOOL @ ( hAPP_nat_bool @ B @ A ) )
       => ( ( hAPP_f22106695ol_nat @ the_nat @ B )
          = A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[74]) ).

thf(277,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( finite988810631ne_nat @ C @ D ) )
     => ( ( hBOOL @ ( hAPP_f54304608l_bool @ finite_finite_nat @ B ) )
       => ( ( ( ( hAPP_f800510211t_bool @ ( hAPP_f1730770594t_bool @ minus_2067140911t_bool @ B ) @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ A ) @ bot_bot_fun_nat_bool ) )
              = bot_bot_fun_nat_bool )
           => ( ( hAPP_f22106695ol_nat @ D @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ A ) @ B ) )
              = A ) )
          & ( ( ( hAPP_f800510211t_bool @ ( hAPP_f1730770594t_bool @ minus_2067140911t_bool @ B ) @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ A ) @ bot_bot_fun_nat_bool ) )
             != bot_bot_fun_nat_bool )
           => ( ( hAPP_f22106695ol_nat @ D @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ A ) @ B ) )
              = ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ C @ A ) @ ( hAPP_f22106695ol_nat @ D @ ( hAPP_f800510211t_bool @ ( hAPP_f1730770594t_bool @ minus_2067140911t_bool @ B ) @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ A ) @ bot_bot_fun_nat_bool ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_359_folding__one_Oinsert__remove) ).

thf(2143,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( finite988810631ne_nat @ C @ D ) )
     => ( ( hBOOL @ ( hAPP_f54304608l_bool @ finite_finite_nat @ B ) )
       => ( ( ( ( hAPP_f800510211t_bool @ ( hAPP_f1730770594t_bool @ minus_2067140911t_bool @ B ) @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ A ) @ bot_bot_fun_nat_bool ) )
              = bot_bot_fun_nat_bool )
           => ( ( hAPP_f22106695ol_nat @ D @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ A ) @ B ) )
              = A ) )
          & ( ( ( hAPP_f800510211t_bool @ ( hAPP_f1730770594t_bool @ minus_2067140911t_bool @ B ) @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ A ) @ bot_bot_fun_nat_bool ) )
             != bot_bot_fun_nat_bool )
           => ( ( hAPP_f22106695ol_nat @ D @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ A ) @ B ) )
              = ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ C @ A ) @ ( hAPP_f22106695ol_nat @ D @ ( hAPP_f800510211t_bool @ ( hAPP_f1730770594t_bool @ minus_2067140911t_bool @ B ) @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ A ) @ bot_bot_fun_nat_bool ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[277]) ).

thf(242,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP_H426895267a_bool @ ( hAPP_f960023481a_bool @ ( hAPP_f1786170683a_bool @ cOMBB_683172471iple_a @ A ) @ B ) @ C )
      = ( hAPP_H426895267a_bool @ A @ ( hAPP_H905846293iple_a @ B @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',help_COMBB_1_1_COMBB_000tc__Hoare____Mirabelle____uwgpyvfjxg__Otriple_It__a_J_00_050) ).

thf(2012,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP_H426895267a_bool @ ( hAPP_f960023481a_bool @ ( hAPP_f1786170683a_bool @ cOMBB_683172471iple_a @ A ) @ B ) @ C )
      = ( hAPP_H426895267a_bool @ A @ ( hAPP_H905846293iple_a @ B @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[242]) ).

thf(132,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_f284875647l_bool @ ord_le951220754t_bool @ B ) @ C ) )
     => ( ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ A ) @ B ) )
       => ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ A ) @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_801_in__mono) ).

thf(1605,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_f284875647l_bool @ ord_le951220754t_bool @ B ) @ C ) )
     => ( ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ A ) @ B ) )
       => ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ A ) @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[132]) ).

thf(582,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ A ) @ ( image_1865589340iple_a @ B @ C ) ) )
     => ~ ! [D: $i] :
            ( ( A
              = ( hAPP_n261501868iple_a @ B @ D ) )
           => ~ ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ D ) @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_243_imageE) ).

thf(3260,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ A ) @ ( image_1865589340iple_a @ B @ C ) ) )
     => ~ ! [D: $i] :
            ( ( A
              = ( hAPP_n261501868iple_a @ B @ D ) )
           => ~ ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ D ) @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[582]) ).

thf(862,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP_H1037229737a_bool @ ( hAPP_f20753329a_bool @ ( hAPP_f607497069a_bool @ cOMBS_154988028l_bool @ A ) @ B ) @ C )
      = ( hAPP_bool_bool @ ( hAPP_H2130837971l_bool @ A @ C ) @ ( hAPP_H1037229737a_bool @ B @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',help_COMBS_1_1_COMBS_000tc__Hoare____Mirabelle____uwgpyvfjxg__Otriple_It__a_J_00_015) ).

thf(4287,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP_H1037229737a_bool @ ( hAPP_f20753329a_bool @ ( hAPP_f607497069a_bool @ cOMBS_154988028l_bool @ A ) @ B ) @ C )
      = ( hAPP_bool_bool @ ( hAPP_H2130837971l_bool @ A @ C ) @ ( hAPP_H1037229737a_bool @ B @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[862]) ).

thf(903,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP_state_state @ ( hAPP_f871651461_state @ ( hAPP_f851239890_state @ cOMBS_777315357_state @ A ) @ B ) @ C )
      = ( hAPP_nat_state @ ( hAPP_s1892499976_state @ A @ C ) @ ( hAPP_state_nat @ B @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',help_COMBS_1_1_COMBS_000tc__Com__Ostate_000tc__Nat__Onat_000tc__Com__Ostate_U) ).

thf(4469,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP_state_state @ ( hAPP_f871651461_state @ ( hAPP_f851239890_state @ cOMBS_777315357_state @ A ) @ B ) @ C )
      = ( hAPP_nat_state @ ( hAPP_s1892499976_state @ A @ C ) @ ( hAPP_state_nat @ B @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[903]) ).

thf(185,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ A ) @ B ) )
      | ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ B ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_999_nat__le__linear) ).

thf(1780,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ A ) @ B ) )
      | ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ B ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[185]) ).

thf(463,axiom,
    ! [A: $i,B: $i] :
      ( ! [C: $i] :
          ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ C ) @ B ) )
         => ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_nat @ C ) @ A ) ) )
     => ( hBOOL @ ( hAPP_f54304608l_bool @ finite_finite_nat @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_1171_bounded__nat__set__is__finite) ).

thf(2821,plain,
    ! [A: $i,B: $i] :
      ( ! [C: $i] :
          ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ C ) @ B ) )
         => ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_nat @ C ) @ A ) ) )
     => ( hBOOL @ ( hAPP_f54304608l_bool @ finite_finite_nat @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[463]) ).

thf(567,axiom,
    ! [A: $i,B: $i] :
      ( ~ ( hBOOL @ ( hAPP_f1695230391l_bool @ finite1738664244iple_a @ B ) )
     => ( ( hBOOL @ ( hAPP_f1695230391l_bool @ finite1738664244iple_a @ ( image_533441733iple_a @ A @ B ) ) )
       => ? [C: $i] :
            ( ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ C ) @ B ) )
            & ~ ( 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 ) @ B ) ) ) @ ( hAPP_H426895267a_bool @ ( hAPP_f920293029a_bool @ cOMBC_1940896922a_bool @ ( hAPP_f960023481a_bool @ ( hAPP_f1786170683a_bool @ cOMBB_683172471iple_a @ fequal963300192iple_a ) @ A ) ) @ ( hAPP_H905846293iple_a @ A @ C ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_320_pigeonhole__infinite) ).

thf(3209,plain,
    ! [A: $i,B: $i] :
      ( ~ ( hBOOL @ ( hAPP_f1695230391l_bool @ finite1738664244iple_a @ B ) )
     => ( ( hBOOL @ ( hAPP_f1695230391l_bool @ finite1738664244iple_a @ ( image_533441733iple_a @ A @ B ) ) )
       => ? [C: $i] :
            ( ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ C ) @ B ) )
            & ~ ( 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 ) @ B ) ) ) @ ( hAPP_H426895267a_bool @ ( hAPP_f920293029a_bool @ cOMBC_1940896922a_bool @ ( hAPP_f960023481a_bool @ ( hAPP_f1786170683a_bool @ cOMBB_683172471iple_a @ fequal963300192iple_a ) @ A ) ) @ ( hAPP_H905846293iple_a @ A @ C ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[567]) ).

thf(175,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( finite1973466193nt_int @ D ) )
     => ( ( hBOOL @ ( hAPP_f448129468l_bool @ finite_finite_int @ C ) )
       => ( ~ ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ B ) @ C ) )
         => ( ( hAPP_f1594865479ol_int @ ( finite_fold_int_int @ D @ A ) @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ B ) @ C ) )
            = ( hAPP_f1594865479ol_int @ ( finite_fold_int_int @ D @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ D @ B ) @ A ) ) @ C ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_509_comp__fun__commute_Ofold__insert2) ).

thf(1753,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( finite1973466193nt_int @ D ) )
     => ( ( hBOOL @ ( hAPP_f448129468l_bool @ finite_finite_int @ C ) )
       => ( ~ ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ B ) @ C ) )
         => ( ( hAPP_f1594865479ol_int @ ( finite_fold_int_int @ D @ A ) @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ B ) @ C ) )
            = ( hAPP_f1594865479ol_int @ ( finite_fold_int_int @ D @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ D @ B ) @ A ) ) @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[175]) ).

thf(526,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ A ) @ B ) ) @ C )
      = ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ A ) @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ B ) @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_1091_zadd__assoc) ).

thf(3058,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ A ) @ B ) ) @ C )
      = ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ A ) @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ B ) @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[526]) ).

thf(345,axiom,
    ! [A: $i] : ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ A ) @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_998_le__refl) ).

thf(2386,plain,
    ! [A: $i] : ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ A ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[345]) ).

thf(33,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( hAPP_int_int @ ( hAPP_int_fun_int_int @ minus_minus_int @ A ) @ B )
        = ( hAPP_int_int @ ( hAPP_int_fun_int_int @ minus_minus_int @ C ) @ D ) )
     => ( ( hBOOL @ ( hAPP_int_bool @ ( hAPP_i1948725293t_bool @ ord_less_eq_int @ A ) @ B ) )
      <=> ( hBOOL @ ( hAPP_int_bool @ ( hAPP_i1948725293t_bool @ ord_less_eq_int @ C ) @ D ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_847_diff__eq__diff__less__eq) ).

thf(1230,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( hAPP_int_int @ ( hAPP_int_fun_int_int @ minus_minus_int @ A ) @ B )
        = ( hAPP_int_int @ ( hAPP_int_fun_int_int @ minus_minus_int @ C ) @ D ) )
     => ( ( ( hBOOL @ ( hAPP_int_bool @ ( hAPP_i1948725293t_bool @ ord_less_eq_int @ A ) @ B ) )
         => ( hBOOL @ ( hAPP_int_bool @ ( hAPP_i1948725293t_bool @ ord_less_eq_int @ C ) @ D ) ) )
        & ( ( hBOOL @ ( hAPP_int_bool @ ( hAPP_i1948725293t_bool @ ord_less_eq_int @ C ) @ D ) )
         => ( hBOOL @ ( hAPP_int_bool @ ( hAPP_i1948725293t_bool @ ord_less_eq_int @ A ) @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[33]) ).

thf(97,axiom,
    ! [A: $i,B: $i] : ( is_bool @ ( finite1626084323ne_int @ A @ B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',gsy_c_Finite__Set_Ofolding__one_000tc__Int__Oint) ).

thf(1484,plain,
    ! [A: $i,B: $i] : ( is_bool @ ( finite1626084323ne_int @ A @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[97]) ).

thf(532,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ A ) @ B ) ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ C ) @ D ) )
      = ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ A ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ B ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ C ) @ D ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_681_comm__semiring__1__class_Onormalizing__semiring__rules_I14_J) ).

thf(3080,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ A ) @ B ) ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ C ) @ D ) )
      = ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ A ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ B ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ C ) @ D ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[532]) ).

thf(396,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( B = C )
     => ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ C ) @ A ) )
       => ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ B ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_771_ord__eq__le__trans) ).

thf(2569,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( B = C )
     => ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ C ) @ A ) )
       => ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ B ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[396]) ).

thf(1120,axiom,
    ! [A: $i,B: $i] :
      ( ( bot_bo797238721a_bool
        = ( image_533441733iple_a @ A @ B ) )
    <=> ( B = bot_bo797238721a_bool ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_210_empty__is__image) ).

thf(5262,plain,
    ! [A: $i,B: $i] :
      ( ( ( bot_bo797238721a_bool
          = ( image_533441733iple_a @ A @ B ) )
       => ( B = bot_bo797238721a_bool ) )
      & ( ( B = bot_bo797238721a_bool )
       => ( bot_bo797238721a_bool
          = ( image_533441733iple_a @ A @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1120]) ).

thf(365,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ B ) @ C ) )
     => ( ( image_685909450_a_int @ ( hAPP_i1475897073_a_int @ cOMBK_1007576201iple_a @ A ) @ C )
        = ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ A ) @ bot_bot_fun_int_bool ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_237_image__constant) ).

thf(2454,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ B ) @ C ) )
     => ( ( image_685909450_a_int @ ( hAPP_i1475897073_a_int @ cOMBK_1007576201iple_a @ A ) @ C )
        = ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ A ) @ bot_bot_fun_int_bool ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[365]) ).

thf(490,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ A ) @ B ) )
     => ( ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ A ) @ ( hAPP_f800510211t_bool @ ( hAPP_f1730770594t_bool @ minus_2067140911t_bool @ B ) @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ A ) @ bot_bot_fun_nat_bool ) ) )
        = B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_423_insert__Diff) ).

thf(2920,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ A ) @ B ) )
     => ( ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ A ) @ ( hAPP_f800510211t_bool @ ( hAPP_f1730770594t_bool @ minus_2067140911t_bool @ B ) @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ A ) @ bot_bot_fun_nat_bool ) ) )
        = B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[490]) ).

thf(431,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ B ) @ C ) )
     => ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ B ) @ A ) )
       => ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ minus_minus_nat @ C ) @ B ) ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ minus_minus_nat @ A ) @ B ) ) )
        <=> ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_nat @ C ) @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_1132_less__diff__iff) ).

thf(2700,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ B ) @ C ) )
     => ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ B ) @ A ) )
       => ( ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ minus_minus_nat @ C ) @ B ) ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ minus_minus_nat @ A ) @ B ) ) )
           => ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_nat @ C ) @ A ) ) )
          & ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_nat @ C ) @ A ) )
           => ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ minus_minus_nat @ C ) @ B ) ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ minus_minus_nat @ A ) @ B ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[431]) ).

thf(609,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP_H1037229737a_bool @ ( hAPP_H426895267a_bool @ ( hAPP_f920293029a_bool @ cOMBC_1940896922a_bool @ A ) @ B ) @ C )
      = ( hAPP_H1037229737a_bool @ ( hAPP_H426895267a_bool @ A @ C ) @ B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',help_COMBC_1_1_COMBC_000tc__Hoare____Mirabelle____uwgpyvfjxg__Otriple_It__a_J_00_029) ).

thf(3344,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP_H1037229737a_bool @ ( hAPP_H426895267a_bool @ ( hAPP_f920293029a_bool @ cOMBC_1940896922a_bool @ A ) @ B ) @ C )
      = ( hAPP_H1037229737a_bool @ ( hAPP_H426895267a_bool @ A @ C ) @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[609]) ).

thf(187,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( finite1973466193nt_int @ D ) )
     => ( ( hBOOL @ ( hAPP_f448129468l_bool @ finite_finite_int @ C ) )
       => ( ( hAPP_f1594865479ol_int @ ( finite_fold_int_int @ D @ A ) @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ B ) @ C ) )
          = ( hAPP_int_int @ ( hAPP_int_fun_int_int @ D @ B ) @ ( hAPP_f1594865479ol_int @ ( finite_fold_int_int @ D @ A ) @ ( hAPP_f1805168059t_bool @ ( hAPP_f1223193598t_bool @ minus_1449998731t_bool @ C ) @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ B ) @ bot_bot_fun_int_bool ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_550_comp__fun__commute_Ofold__insert__remove) ).

thf(1785,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( finite1973466193nt_int @ D ) )
     => ( ( hBOOL @ ( hAPP_f448129468l_bool @ finite_finite_int @ C ) )
       => ( ( hAPP_f1594865479ol_int @ ( finite_fold_int_int @ D @ A ) @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ B ) @ C ) )
          = ( hAPP_int_int @ ( hAPP_int_fun_int_int @ D @ B ) @ ( hAPP_f1594865479ol_int @ ( finite_fold_int_int @ D @ A ) @ ( hAPP_f1805168059t_bool @ ( hAPP_f1223193598t_bool @ minus_1449998731t_bool @ C ) @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ B ) @ bot_bot_fun_int_bool ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[187]) ).

thf(34,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP_int_bool @ ( hAPP_i1948725293t_bool @ ord_less_eq_int @ A ) @ B ) )
     => ( ( hBOOL @ ( hAPP_int_bool @ ( hAPP_i1948725293t_bool @ ord_less_eq_int @ B ) @ A ) )
       => ( B = A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_746_xt1_I5_J) ).

thf(1235,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP_int_bool @ ( hAPP_i1948725293t_bool @ ord_less_eq_int @ A ) @ B ) )
     => ( ( hBOOL @ ( hAPP_int_bool @ ( hAPP_i1948725293t_bool @ ord_less_eq_int @ B ) @ A ) )
       => ( B = A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[34]) ).

thf(316,axiom,
    ! [A: $i,B: $i] :
      ( ( bot_bot_fun_nat_bool
        = ( image_nat_nat @ A @ B ) )
    <=> ( B = bot_bot_fun_nat_bool ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_212_empty__is__image) ).

thf(2276,plain,
    ! [A: $i,B: $i] :
      ( ( ( bot_bot_fun_nat_bool
          = ( image_nat_nat @ A @ B ) )
       => ( B = bot_bot_fun_nat_bool ) )
      & ( ( B = bot_bot_fun_nat_bool )
       => ( bot_bot_fun_nat_bool
          = ( image_nat_nat @ A @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[316]) ).

thf(54,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( finite1704255308nt_int @ D ) )
     => ( ( hBOOL @ ( hAPP_f448129468l_bool @ finite_finite_int @ C ) )
       => ( ( hAPP_f1594865479ol_int @ ( finite_fold_int_int @ D @ A ) @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ B ) @ C ) )
          = ( hAPP_int_int @ ( hAPP_int_fun_int_int @ D @ B ) @ ( hAPP_f1594865479ol_int @ ( finite_fold_int_int @ D @ A ) @ C ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_579_comp__fun__idem_Ofold__insert__idem) ).

thf(1317,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( finite1704255308nt_int @ D ) )
     => ( ( hBOOL @ ( hAPP_f448129468l_bool @ finite_finite_int @ C ) )
       => ( ( hAPP_f1594865479ol_int @ ( finite_fold_int_int @ D @ A ) @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ B ) @ C ) )
          = ( hAPP_int_int @ ( hAPP_int_fun_int_int @ D @ B ) @ ( hAPP_f1594865479ol_int @ ( finite_fold_int_int @ D @ A ) @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[54]) ).

thf(605,axiom,
    ! [A: $i] : ( is_bool @ ( finite2114894759a_bool @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',gsy_c_Finite__Set_Ocomp__fun__commute_000tc__Hoare____Mirabelle____uwgpyvfjxg__O_001) ).

thf(3329,plain,
    ! [A: $i] : ( is_bool @ ( finite2114894759a_bool @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[605]) ).

thf(887,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP_f54304608l_bool @ ( hAPP_f103356543l_bool @ ( hAPP_f643944041l_bool @ cOMBC_1693257480l_bool @ A ) @ B ) @ C )
      = ( hAPP_f54304608l_bool @ ( hAPP_f103356543l_bool @ A @ C ) @ B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',help_COMBC_1_1_COMBC_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_000tc__fun_Itc_) ).

thf(4398,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP_f54304608l_bool @ ( hAPP_f103356543l_bool @ ( hAPP_f643944041l_bool @ cOMBC_1693257480l_bool @ A ) @ B ) @ C )
      = ( hAPP_f54304608l_bool @ ( hAPP_f103356543l_bool @ A @ C ) @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[887]) ).

thf(1022,axiom,
    ! [A: $i,B: $i] : ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hoare_2102800559rivs_a @ A ) @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ ( hoare_1916936827iple_a @ B @ skip @ B ) ) @ bot_bo797238721a_bool ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_111_hoare__derivs_OSkip) ).

thf(4905,plain,
    ! [A: $i,B: $i] : ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hoare_2102800559rivs_a @ A ) @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ ( hoare_1916936827iple_a @ B @ skip @ B ) ) @ bot_bo797238721a_bool ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1022]) ).

thf(597,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP_f54304608l_bool @ finite_finite_nat @ B ) )
     => ( hBOOL @ ( hAPP_f448129468l_bool @ finite_finite_int @ ( image_nat_int @ A @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_292_finite__imageI) ).

thf(3305,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP_f54304608l_bool @ finite_finite_nat @ B ) )
     => ( hBOOL @ ( hAPP_f448129468l_bool @ finite_finite_int @ ( image_nat_int @ A @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[597]) ).

thf(737,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f1695230391l_bool @ finite1738664244iple_a @ C ) )
     => ( ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_f1767618879l_bool @ ord_le1523781517a_bool @ A ) @ ( image_533441733iple_a @ B @ C ) ) )
       => ( hBOOL @ ( hAPP_f1695230391l_bool @ finite1738664244iple_a @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_895_finite__surj) ).

thf(3818,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f1695230391l_bool @ finite1738664244iple_a @ C ) )
     => ( ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_f1767618879l_bool @ ord_le1523781517a_bool @ A ) @ ( image_533441733iple_a @ B @ C ) ) )
       => ( hBOOL @ ( hAPP_f1695230391l_bool @ finite1738664244iple_a @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[737]) ).

thf(702,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_f1767618879l_bool @ ord_le1523781517a_bool @ B ) @ C ) )
     => ( ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ A ) @ B ) )
       => ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ A ) @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_800_in__mono) ).

thf(3691,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_f1767618879l_bool @ ord_le1523781517a_bool @ B ) @ C ) )
     => ( ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ A ) @ B ) )
       => ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ A ) @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[702]) ).

thf(607,axiom,
    ! [A: $i] : ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ A ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ A ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_997_le__square) ).

thf(3334,plain,
    ! [A: $i] : ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ A ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ A ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[607]) ).

thf(713,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( is_vname @ A )
        & ( is_vname @ C ) )
     => ( ( ( ass @ A @ B )
          = ( ass @ C @ D ) )
      <=> ( ( A = C )
          & ( B = D ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_166_com_Osimps_I1_J) ).

thf(3722,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( is_vname @ A )
        & ( is_vname @ C ) )
     => ( ( ( ( ass @ A @ B )
            = ( ass @ C @ D ) )
         => ( ( A = C )
            & ( B = D ) ) )
        & ( ( ( A = C )
            & ( B = D ) )
         => ( ( ass @ A @ B )
            = ( ass @ C @ D ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[713]) ).

thf(796,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( finite795500164em_nat @ C @ D ) )
     => ( ( hBOOL @ ( hAPP_f54304608l_bool @ finite_finite_nat @ B ) )
       => ( ( A != bot_bot_fun_nat_bool )
         => ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_f103356543l_bool @ ord_le1568362934t_bool @ A ) @ B ) )
           => ( ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ C @ ( hAPP_f22106695ol_nat @ D @ A ) ) @ ( hAPP_f22106695ol_nat @ D @ B ) )
              = ( hAPP_f22106695ol_nat @ D @ B ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_912_folding__one__idem_Osubset__idem) ).

thf(4020,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( finite795500164em_nat @ C @ D ) )
     => ( ( hBOOL @ ( hAPP_f54304608l_bool @ finite_finite_nat @ B ) )
       => ( ( A != bot_bot_fun_nat_bool )
         => ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_f103356543l_bool @ ord_le1568362934t_bool @ A ) @ B ) )
           => ( ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ C @ ( hAPP_f22106695ol_nat @ D @ A ) ) @ ( hAPP_f22106695ol_nat @ D @ B ) )
              = ( hAPP_f22106695ol_nat @ D @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[796]) ).

thf(641,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_f103356543l_bool @ ord_le1568362934t_bool @ B ) @ C ) )
     => ( ( B = A )
       => ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_f103356543l_bool @ ord_le1568362934t_bool @ A ) @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_757_xt1_I4_J) ).

thf(3475,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_f103356543l_bool @ ord_le1568362934t_bool @ B ) @ C ) )
     => ( ( B = A )
       => ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_f103356543l_bool @ ord_le1568362934t_bool @ A ) @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[641]) ).

thf(519,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ A ) @ B )
        = ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ C ) @ B ) )
    <=> ( A = C ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_661_add__right__cancel) ).

thf(3026,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ A ) @ B )
          = ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ C ) @ B ) )
       => ( A = C ) )
      & ( ( A = C )
       => ( ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ A ) @ B )
          = ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ C ) @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[519]) ).

thf(357,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ A ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ B ) @ C ) )
      = ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ B ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ A ) @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_1050_nat__add__left__commute) ).

thf(2429,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ A ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ B ) @ C ) )
      = ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ B ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ A ) @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[357]) ).

thf(1091,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP_f20753329a_bool @ ( hAPP_f1706700729a_bool @ minus_85316870a_bool @ A ) @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ B ) @ C ) )
      = ( hAPP_f20753329a_bool @ ( hAPP_f1706700729a_bool @ minus_85316870a_bool @ ( hAPP_f20753329a_bool @ ( hAPP_f1706700729a_bool @ minus_85316870a_bool @ A ) @ C ) ) @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ B ) @ bot_bo797238721a_bool ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_412_Diff__insert) ).

thf(5152,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP_f20753329a_bool @ ( hAPP_f1706700729a_bool @ minus_85316870a_bool @ A ) @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ B ) @ C ) )
      = ( hAPP_f20753329a_bool @ ( hAPP_f1706700729a_bool @ minus_85316870a_bool @ ( hAPP_f20753329a_bool @ ( hAPP_f1706700729a_bool @ minus_85316870a_bool @ A ) @ C ) ) @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ B ) @ bot_bo797238721a_bool ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1091]) ).

thf(147,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( finite2130160977at_nat @ D ) )
     => ( ( hBOOL @ ( hAPP_f54304608l_bool @ finite_finite_nat @ C ) )
       => ( ~ ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ B ) @ C ) )
         => ( ( hAPP_f22106695ol_nat @ ( finite_fold_nat_nat @ D @ A ) @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ B ) @ C ) )
            = ( hAPP_f22106695ol_nat @ ( finite_fold_nat_nat @ D @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ D @ B ) @ A ) ) @ C ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_505_comp__fun__commute_Ofold__insert2) ).

thf(1658,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( finite2130160977at_nat @ D ) )
     => ( ( hBOOL @ ( hAPP_f54304608l_bool @ finite_finite_nat @ C ) )
       => ( ~ ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ B ) @ C ) )
         => ( ( hAPP_f22106695ol_nat @ ( finite_fold_nat_nat @ D @ A ) @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ B ) @ C ) )
            = ( hAPP_f22106695ol_nat @ ( finite_fold_nat_nat @ D @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ D @ B ) @ A ) ) @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[147]) ).

thf(328,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_f284875647l_bool @ ord_le951220754t_bool @ B ) @ C ) )
     => ( ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ A ) @ B ) )
       => ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ A ) @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_725_subsetD) ).

thf(2322,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_f284875647l_bool @ ord_le951220754t_bool @ B ) @ C ) )
     => ( ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ A ) @ B ) )
       => ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ A ) @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[328]) ).

thf(1054,axiom,
    ! [A: $i] :
      ~ ! [B: $i,C: $i,D: $i] :
          ( A
         != ( hoare_1916936827iple_a @ B @ C @ D ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_113_triple_Oexhaust) ).

thf(5001,plain,
    ! [A: $i] :
      ~ ! [B: $i,C: $i,D: $i] :
          ( A
         != ( hoare_1916936827iple_a @ B @ C @ D ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1054]) ).

thf(374,axiom,
    ! [A: $i] :
      ( ( hAPP_f1594865479ol_int @ the_elem_int @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ A ) @ bot_bot_fun_int_bool ) )
      = A ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_104_the__elem__eq) ).

thf(2486,plain,
    ! [A: $i] :
      ( ( hAPP_f1594865479ol_int @ the_elem_int @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ A ) @ bot_bot_fun_int_bool ) )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[374]) ).

thf(849,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ A ) @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ B ) @ bot_bot_fun_nat_bool ) ) )
     => ( A = B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_87_singletonE) ).

thf(4241,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ A ) @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ B ) @ bot_bot_fun_nat_bool ) ) )
     => ( A = B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[849]) ).

thf(403,axiom,
    ! [A: $i] : ( is_bool @ ( finite1973466193nt_int @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',gsy_c_Finite__Set_Ocomp__fun__commute_000tc__Int__Oint_000tc__Int__Oint) ).

thf(2599,plain,
    ! [A: $i] : ( is_bool @ ( finite1973466193nt_int @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[403]) ).

thf(487,axiom,
    ! [A: $i,B: $i] : ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ ( finite929467206at_nat @ A ) @ B ) @ bot_bot_fun_nat_bool ) @ B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_301_fold__graph_OemptyI) ).

thf(2903,plain,
    ! [A: $i,B: $i] : ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ ( finite929467206at_nat @ A ) @ B ) @ bot_bot_fun_nat_bool ) @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[487]) ).

thf(996,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ! [E: $i,F: $i] :
          ( ( hBOOL @ ( hAPP_state_bool @ ( hAPP_a2036067514e_bool @ D @ E ) @ F ) )
         => ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hoare_2102800559rivs_a @ A ) @ ( 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 ) @ F ) ) @ B @ ( hAPP_f762886889e_bool @ cOMBK_1458035955bool_a @ ( hAPP_a2036067514e_bool @ C @ E ) ) ) ) @ bot_bo797238721a_bool ) ) ) )
     => ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hoare_2102800559rivs_a @ A ) @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ ( hoare_1916936827iple_a @ D @ B @ C ) ) @ bot_bo797238721a_bool ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_5_escape) ).

thf(4800,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ! [E: $i,F: $i] :
          ( ( hBOOL @ ( hAPP_state_bool @ ( hAPP_a2036067514e_bool @ D @ E ) @ F ) )
         => ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hoare_2102800559rivs_a @ A ) @ ( 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 ) @ F ) ) @ B @ ( hAPP_f762886889e_bool @ cOMBK_1458035955bool_a @ ( hAPP_a2036067514e_bool @ C @ E ) ) ) ) @ bot_bo797238721a_bool ) ) ) )
     => ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hoare_2102800559rivs_a @ A ) @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ ( hoare_1916936827iple_a @ D @ B @ C ) ) @ bot_bo797238721a_bool ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[996]) ).

thf(683,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ C ) @ D ) )
     => ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ A ) @ B ) )
       => ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ C ) @ A ) ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ D ) @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_993_mult__le__mono) ).

thf(3623,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ C ) @ D ) )
     => ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ A ) @ B ) )
       => ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ C ) @ A ) ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ D ) @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[683]) ).

thf(626,axiom,
    ! [A: $i,B: $i] :
      ( ( A = B )
     => ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_f284875647l_bool @ ord_le951220754t_bool @ B ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_803_equalityD2) ).

thf(3412,plain,
    ! [A: $i,B: $i] :
      ( ( A = B )
     => ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_f284875647l_bool @ ord_le951220754t_bool @ B ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[626]) ).

thf(47,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP_int_bool @ ( hAPP_i1948725293t_bool @ ord_less_eq_int @ ( hAPP_nat_int @ semiri1621563631at_int @ A ) ) @ ( hAPP_nat_int @ semiri1621563631at_int @ B ) ) )
    <=> ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ A ) @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_1163_transfer__int__nat__relations_I3_J) ).

thf(1285,plain,
    ! [A: $i,B: $i] :
      ( ( ( hBOOL @ ( hAPP_int_bool @ ( hAPP_i1948725293t_bool @ ord_less_eq_int @ ( hAPP_nat_int @ semiri1621563631at_int @ A ) ) @ ( hAPP_nat_int @ semiri1621563631at_int @ B ) ) )
       => ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ A ) @ B ) ) )
      & ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ A ) @ B ) )
       => ( hBOOL @ ( hAPP_int_bool @ ( hAPP_i1948725293t_bool @ ord_less_eq_int @ ( hAPP_nat_int @ semiri1621563631at_int @ A ) ) @ ( hAPP_nat_int @ semiri1621563631at_int @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[47]) ).

thf(751,axiom,
    ! [A: $i,B: $i] :
      ( ? [C: $i] :
          ( ( hBOOL @ ( hAPP_H1037229737a_bool @ B @ C ) )
          & ! [D: $i] :
              ( ( hBOOL @ ( hAPP_H1037229737a_bool @ B @ D ) )
             => ( D = C ) ) )
     => ( ( hBOOL @ ( hAPP_H1037229737a_bool @ B @ A ) )
       => ( ( hAPP_f1693662087iple_a @ the_Ho830102290iple_a @ B )
          = A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_339_the1__equality) ).

thf(3853,plain,
    ! [A: $i,B: $i] :
      ( ? [C: $i] :
          ( ( hBOOL @ ( hAPP_H1037229737a_bool @ B @ C ) )
          & ! [D: $i] :
              ( ( hBOOL @ ( hAPP_H1037229737a_bool @ B @ D ) )
             => ( D = C ) ) )
     => ( ( hBOOL @ ( hAPP_H1037229737a_bool @ B @ A ) )
       => ( ( hAPP_f1693662087iple_a @ the_Ho830102290iple_a @ B )
          = A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[751]) ).

thf(1081,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( B
        = ( finite1465856449iple_a @ C ) )
     => ( ( hAPP_f1693662087iple_a @ B @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ A ) @ bot_bo797238721a_bool ) )
        = A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_147_fold1__singleton__def) ).

thf(5106,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( B
        = ( finite1465856449iple_a @ C ) )
     => ( ( hAPP_f1693662087iple_a @ B @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ A ) @ bot_bo797238721a_bool ) )
        = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1081]) ).

thf(724,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP_int_int @ ( hAPP_int_fun_int_int @ times_times_int @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ times_times_int @ A ) @ B ) ) @ C )
      = ( hAPP_int_int @ ( hAPP_int_fun_int_int @ times_times_int @ A ) @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ times_times_int @ B ) @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_688_ab__semigroup__mult__class_Omult__ac_I1_J) ).

thf(3769,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP_int_int @ ( hAPP_int_fun_int_int @ times_times_int @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ times_times_int @ A ) @ B ) ) @ C )
      = ( hAPP_int_int @ ( hAPP_int_fun_int_int @ times_times_int @ A ) @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ times_times_int @ B ) @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[724]) ).

thf(1014,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( finite1428139793iple_a @ D ) )
     => ( ( hBOOL @ ( hAPP_f1695230391l_bool @ finite1738664244iple_a @ C ) )
       => ( ~ ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ B ) @ C ) )
         => ( ( hAPP_f1693662087iple_a @ ( finite1205970312iple_a @ D @ A ) @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ B ) @ C ) )
            = ( hAPP_H905846293iple_a @ ( hAPP_H2085992369iple_a @ D @ B ) @ ( hAPP_f1693662087iple_a @ ( finite1205970312iple_a @ D @ A ) @ C ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_513_comp__fun__commute_Ofold__insert) ).

thf(4875,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( finite1428139793iple_a @ D ) )
     => ( ( hBOOL @ ( hAPP_f1695230391l_bool @ finite1738664244iple_a @ C ) )
       => ( ~ ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ B ) @ C ) )
         => ( ( hAPP_f1693662087iple_a @ ( finite1205970312iple_a @ D @ A ) @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ B ) @ C ) )
            = ( hAPP_H905846293iple_a @ ( hAPP_H2085992369iple_a @ D @ B ) @ ( hAPP_f1693662087iple_a @ ( finite1205970312iple_a @ D @ A ) @ C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1014]) ).

thf(63,axiom,
    ! [A: $i,B: $i] :
      ( ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ ord_min_nat @ A ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ ord_min_nat @ A ) @ B ) )
      = ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ ord_min_nat @ A ) @ B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_957_min__max_Oinf__left__idem) ).

thf(1358,plain,
    ! [A: $i,B: $i] :
      ( ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ ord_min_nat @ A ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ ord_min_nat @ A ) @ B ) )
      = ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ ord_min_nat @ A ) @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[63]) ).

thf(1106,axiom,
    ! [A: $i] :
      ( ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_f1767618879l_bool @ ord_le1523781517a_bool @ A ) @ bot_bo797238721a_bool ) )
     => ( A = bot_bo797238721a_bool ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_835_le__bot) ).

thf(5208,plain,
    ! [A: $i] :
      ( ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_f1767618879l_bool @ ord_le1523781517a_bool @ A ) @ bot_bo797238721a_bool ) )
     => ( A = bot_bo797238721a_bool ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1106]) ).

thf(351,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f448129468l_bool @ finite_finite_int @ C ) )
     => ? [D: $i] : ( hBOOL @ ( hAPP_int_bool @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ ( finite772772422nt_int @ A ) @ B ) @ C ) @ D ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_331_finite__imp__fold__graph) ).

thf(2411,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f448129468l_bool @ finite_finite_int @ C ) )
     => ? [D: $i] : ( hBOOL @ ( hAPP_int_bool @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ ( finite772772422nt_int @ A ) @ B ) @ C ) @ D ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[351]) ).

thf(805,axiom,
    ! [A: $i,B: $i] :
      ( ( 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 ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_852_finite__subset) ).

thf(4057,plain,
    ! [A: $i,B: $i] :
      ( ( 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 ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[805]) ).

thf(160,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP_f448129468l_bool @ finite_finite_int @ B ) )
     => ( hBOOL @ ( hAPP_f448129468l_bool @ finite_finite_int @ ( hAPP_f1805168059t_bool @ ( hAPP_f1223193598t_bool @ minus_1449998731t_bool @ B ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_372_finite__Diff) ).

thf(1702,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP_f448129468l_bool @ finite_finite_int @ B ) )
     => ( hBOOL @ ( hAPP_f448129468l_bool @ finite_finite_int @ ( hAPP_f1805168059t_bool @ ( hAPP_f1223193598t_bool @ minus_1449998731t_bool @ B ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[160]) ).

thf(1045,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ A ) @ B ) )
    <=> ( hBOOL @ ( hAPP_H1037229737a_bool @ B @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_195_mem__def) ).

thf(4976,plain,
    ! [A: $i,B: $i] :
      ( ( ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ A ) @ B ) )
       => ( hBOOL @ ( hAPP_H1037229737a_bool @ B @ A ) ) )
      & ( ( hBOOL @ ( hAPP_H1037229737a_bool @ B @ A ) )
       => ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ A ) @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1045]) ).

thf(30,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP_int_bool @ ( hAPP_i1948725293t_bool @ ord_less_eq_int @ A ) @ B ) )
     => ( ( hAPP_int_int @ ( hAPP_int_fun_int_int @ ord_min_int @ B ) @ A )
        = A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_973_min__max_Oinf__absorb2) ).

thf(1221,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP_int_bool @ ( hAPP_i1948725293t_bool @ ord_less_eq_int @ A ) @ B ) )
     => ( ( hAPP_int_int @ ( hAPP_int_fun_int_int @ ord_min_int @ B ) @ A )
        = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[30]) ).

thf(1112,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 ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_449_comp__fun__idem__remove) ).

thf(5226,plain,
    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 ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1112]) ).

thf(191,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ D ) @ E ) )
     => ( ( ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ E ) @ A ) ) @ B )
          = ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ D ) @ A ) ) @ C ) )
      <=> ( ( 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 @ E ) @ D ) ) @ A ) ) @ B )
          = C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_1043_nat__eq__add__iff1) ).

thf(1797,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ D ) @ E ) )
     => ( ( ( ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ E ) @ A ) ) @ B )
            = ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ D ) @ A ) ) @ C ) )
         => ( ( 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 @ E ) @ D ) ) @ A ) ) @ B )
            = C ) )
        & ( ( ( 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 @ E ) @ D ) ) @ A ) ) @ B )
            = C )
         => ( ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ E ) @ A ) ) @ B )
            = ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ D ) @ A ) ) @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[191]) ).

thf(439,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ A ) @ B ) )
    <=> ( hBOOL @ ( hAPP_int_bool @ B @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_197_mem__def) ).

thf(2728,plain,
    ! [A: $i,B: $i] :
      ( ( ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ A ) @ B ) )
       => ( hBOOL @ ( hAPP_int_bool @ B @ A ) ) )
      & ( ( hBOOL @ ( hAPP_int_bool @ B @ A ) )
       => ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ A ) @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[439]) ).

thf(1077,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( finite1428139793iple_a @ C ) )
     => ( ( hBOOL @ ( hAPP_f1695230391l_bool @ finite1738664244iple_a @ B ) )
       => ( hBOOL @ ( hAPP_H1037229737a_bool @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ ( finite90892550iple_a @ C ) @ A ) @ B ) @ ( hAPP_f1693662087iple_a @ ( finite1205970312iple_a @ C @ A ) @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_517_comp__fun__commute_Ofold__graph__fold) ).

thf(5083,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( finite1428139793iple_a @ C ) )
     => ( ( hBOOL @ ( hAPP_f1695230391l_bool @ finite1738664244iple_a @ B ) )
       => ( hBOOL @ ( hAPP_H1037229737a_bool @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ ( finite90892550iple_a @ C ) @ A ) @ B ) @ ( hAPP_f1693662087iple_a @ ( finite1205970312iple_a @ C @ A ) @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1077]) ).

thf(1002,axiom,
    ! [A: $i,B: $i] :
      ( bot_bo797238721a_bool
     != ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ A ) @ B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_100_empty__not__insert) ).

thf(4819,plain,
    ! [A: $i,B: $i] :
      ( bot_bo797238721a_bool
     != ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ A ) @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1002]) ).

thf(180,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP_H1037229737a_bool @ ( hAPP_n1919155532a_bool @ ( hAPP_f228417655a_bool @ cOMBC_364739t_bool @ A ) @ B ) @ C )
      = ( hAPP_nat_bool @ ( hAPP_H144868812t_bool @ A @ C ) @ B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',help_COMBC_1_1_COMBC_000tc__Hoare____Mirabelle____uwgpyvfjxg__Otriple_It__a_J_00_012) ).

thf(1766,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP_H1037229737a_bool @ ( hAPP_n1919155532a_bool @ ( hAPP_f228417655a_bool @ cOMBC_364739t_bool @ A ) @ B ) @ C )
      = ( hAPP_nat_bool @ ( hAPP_H144868812t_bool @ A @ C ) @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[180]) ).

thf(539,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ A ) @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ B ) @ C ) )
      = ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ B ) @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ A ) @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_1092_zadd__left__commute) ).

thf(3107,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ A ) @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ B ) @ C ) )
      = ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ B ) @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ A ) @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[539]) ).

thf(468,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( finite1321096241t_bool @ D ) )
     => ( ( hBOOL @ ( hAPP_f448129468l_bool @ finite_finite_int @ C ) )
       => ( ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ B ) @ C ) )
         => ( ( hAPP_f1805168059t_bool @ ( finite1827710202t_bool @ D @ A ) @ C )
            = ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ D @ B ) @ ( hAPP_f1805168059t_bool @ ( finite1827710202t_bool @ D @ A ) @ ( hAPP_f1805168059t_bool @ ( hAPP_f1223193598t_bool @ minus_1449998731t_bool @ C ) @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ B ) @ bot_bot_fun_int_bool ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_545_comp__fun__commute_Ofold__rec) ).

thf(2838,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( finite1321096241t_bool @ D ) )
     => ( ( hBOOL @ ( hAPP_f448129468l_bool @ finite_finite_int @ C ) )
       => ( ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ B ) @ C ) )
         => ( ( hAPP_f1805168059t_bool @ ( finite1827710202t_bool @ D @ A ) @ C )
            = ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ D @ B ) @ ( hAPP_f1805168059t_bool @ ( finite1827710202t_bool @ D @ A ) @ ( hAPP_f1805168059t_bool @ ( hAPP_f1223193598t_bool @ minus_1449998731t_bool @ C ) @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ B ) @ bot_bot_fun_int_bool ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[468]) ).

thf(272,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ ( finite929467206at_nat @ A ) @ B ) @ C ) @ D ) )
    <=> ( ( ( C = bot_bot_fun_nat_bool )
          & ( D = B ) )
        | ? [E: $i,F: $i,G: $i] :
            ( ( C
              = ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ E ) @ F ) )
            & ( D
              = ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ A @ E ) @ G ) )
            & ~ ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ E ) @ F ) )
            & ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ ( finite929467206at_nat @ A ) @ B ) @ F ) @ G ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_344_fold__graph_Osimps) ).

thf(2113,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ ( finite929467206at_nat @ A ) @ B ) @ C ) @ D ) )
       => ( ( ( C = bot_bot_fun_nat_bool )
            & ( D = B ) )
          | ? [E: $i,F: $i,G: $i] :
              ( ( C
                = ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ E ) @ F ) )
              & ( D
                = ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ A @ E ) @ G ) )
              & ~ ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ E ) @ F ) )
              & ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ ( finite929467206at_nat @ A ) @ B ) @ F ) @ G ) ) ) ) )
      & ( ( ( ( C = bot_bot_fun_nat_bool )
            & ( D = B ) )
          | ? [E: $i,F: $i,G: $i] :
              ( ( C
                = ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ E ) @ F ) )
              & ( D
                = ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ A @ E ) @ G ) )
              & ~ ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ E ) @ F ) )
              & ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ ( finite929467206at_nat @ A ) @ B ) @ F ) @ G ) ) ) )
       => ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ ( finite929467206at_nat @ A ) @ B ) @ C ) @ D ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[272]) ).

thf(1111,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ B ) @ C ) )
     => ( ( image_533441733iple_a @ ( hAPP_H2085992369iple_a @ cOMBK_2052406350iple_a @ A ) @ C )
        = ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ A ) @ bot_bo797238721a_bool ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_235_image__constant) ).

thf(5223,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ B ) @ C ) )
     => ( ( image_533441733iple_a @ ( hAPP_H2085992369iple_a @ cOMBK_2052406350iple_a @ A ) @ C )
        = ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ A ) @ bot_bo797238721a_bool ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1111]) ).

thf(7,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_int_bool @ ( hAPP_i1948725293t_bool @ ord_less_eq_int @ B ) @ C ) )
     => ( ( hBOOL @ ( hAPP_int_bool @ A @ B ) )
       => ( ! [D: $i] :
              ( ( hBOOL @ ( hAPP_int_bool @ ( hAPP_i1948725293t_bool @ ord_less_eq_int @ B ) @ D ) )
             => ( ( hBOOL @ ( hAPP_int_bool @ A @ D ) )
               => ( hBOOL @ ( hAPP_int_bool @ A @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ D ) @ one_one_int ) ) ) ) )
         => ( hBOOL @ ( hAPP_int_bool @ A @ C ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_1169_int__ge__induct) ).

thf(1141,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_int_bool @ ( hAPP_i1948725293t_bool @ ord_less_eq_int @ B ) @ C ) )
     => ( ( hBOOL @ ( hAPP_int_bool @ A @ B ) )
       => ( ! [D: $i] :
              ( ( hBOOL @ ( hAPP_int_bool @ ( hAPP_i1948725293t_bool @ ord_less_eq_int @ B ) @ D ) )
             => ( ( hBOOL @ ( hAPP_int_bool @ A @ D ) )
               => ( hBOOL @ ( hAPP_int_bool @ A @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ D ) @ one_one_int ) ) ) ) )
         => ( hBOOL @ ( hAPP_int_bool @ A @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[7]) ).

thf(826,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP_H1037229737a_bool @ ( hAPP_f20753329a_bool @ ( hAPP_f247708275a_bool @ cOMBC_1519416966l_bool @ A ) @ B ) @ C )
      = ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ A @ C ) @ B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',help_COMBC_1_1_COMBC_000tc__Hoare____Mirabelle____uwgpyvfjxg__Otriple_It__a_J_00_042) ).

thf(4133,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP_H1037229737a_bool @ ( hAPP_f20753329a_bool @ ( hAPP_f247708275a_bool @ cOMBC_1519416966l_bool @ A ) @ B ) @ C )
      = ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ A @ C ) @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[826]) ).

thf(828,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP_f448129468l_bool @ finite_finite_int @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ A ) @ B ) ) )
    <=> ( hBOOL @ ( hAPP_f448129468l_bool @ finite_finite_int @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_298_finite__insert) ).

thf(4143,plain,
    ! [A: $i,B: $i] :
      ( ( ( hBOOL @ ( hAPP_f448129468l_bool @ finite_finite_int @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ A ) @ B ) ) )
       => ( hBOOL @ ( hAPP_f448129468l_bool @ finite_finite_int @ B ) ) )
      & ( ( hBOOL @ ( hAPP_f448129468l_bool @ finite_finite_int @ B ) )
       => ( hBOOL @ ( hAPP_f448129468l_bool @ finite_finite_int @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ A ) @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[828]) ).

thf(923,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_f1767618879l_bool @ ord_le1523781517a_bool @ B ) @ C ) )
     => ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_f1767618879l_bool @ ord_le1523781517a_bool @ B ) @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ A ) @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_862_subset__insertI2) ).

thf(4538,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_f1767618879l_bool @ ord_le1523781517a_bool @ B ) @ C ) )
     => ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_f1767618879l_bool @ ord_le1523781517a_bool @ B ) @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ A ) @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[923]) ).

thf(818,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ~ ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ B ) @ C ) )
     => ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_f103356543l_bool @ ord_le1568362934t_bool @ C ) @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ B ) @ A ) ) )
      <=> ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_f103356543l_bool @ ord_le1568362934t_bool @ C ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_863_subset__insert) ).

thf(4093,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ~ ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ B ) @ C ) )
     => ( ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_f103356543l_bool @ ord_le1568362934t_bool @ C ) @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ B ) @ A ) ) )
         => ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_f103356543l_bool @ ord_le1568362934t_bool @ C ) @ A ) ) )
        & ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_f103356543l_bool @ ord_le1568362934t_bool @ C ) @ A ) )
         => ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_f103356543l_bool @ ord_le1568362934t_bool @ C ) @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ B ) @ A ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[818]) ).

thf(437,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ B ) @ C ) )
     => ( ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ ( hAPP_nat_int @ A @ B ) ) @ ( image_nat_int @ A @ C ) )
        = ( image_nat_int @ A @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_219_insert__image) ).

thf(2722,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ B ) @ C ) )
     => ( ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ ( hAPP_nat_int @ A @ B ) ) @ ( image_nat_int @ A @ C ) )
        = ( image_nat_int @ A @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[437]) ).

thf(214,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_int_bool @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ A ) @ B ) @ C ) )
    <=> ( ( A = C )
        | ( hBOOL @ ( hAPP_int_bool @ B @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_62_insert__code) ).

thf(1881,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( hBOOL @ ( hAPP_int_bool @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ A ) @ B ) @ C ) )
       => ( ( A = C )
          | ( hBOOL @ ( hAPP_int_bool @ B @ C ) ) ) )
      & ( ( ( A = C )
          | ( hBOOL @ ( hAPP_int_bool @ B @ C ) ) )
       => ( hBOOL @ ( hAPP_int_bool @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ A ) @ B ) @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[214]) ).

thf(551,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( finite1071749497t_bool @ D ) )
     => ( ( hBOOL @ ( hAPP_f54304608l_bool @ finite_finite_nat @ C ) )
       => ( ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ D @ A ) @ ( hAPP_f800510211t_bool @ ( finite1578363458t_bool @ D @ B ) @ C ) )
          = ( hAPP_f800510211t_bool @ ( finite1578363458t_bool @ D @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ D @ A ) @ B ) ) @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_484_comp__fun__commute_Ofold__fun__comm) ).

thf(3156,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( finite1071749497t_bool @ D ) )
     => ( ( hBOOL @ ( hAPP_f54304608l_bool @ finite_finite_nat @ C ) )
       => ( ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ D @ A ) @ ( hAPP_f800510211t_bool @ ( finite1578363458t_bool @ D @ B ) @ C ) )
          = ( hAPP_f800510211t_bool @ ( finite1578363458t_bool @ D @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ D @ A ) @ B ) ) @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[551]) ).

thf(994,axiom,
    ! [A: $i,B: $i] : ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ A ) @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ A ) @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_79_insertI1) ).

thf(4795,plain,
    ! [A: $i,B: $i] : ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ A ) @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ A ) @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[994]) ).

thf(470,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( finite2130160977at_nat @ D ) )
     => ( ( hBOOL @ ( hAPP_f54304608l_bool @ finite_finite_nat @ C ) )
       => ( ~ ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ B ) @ C ) )
         => ( ( hAPP_f22106695ol_nat @ ( finite_fold_nat_nat @ D @ A ) @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ B ) @ C ) )
            = ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ D @ B ) @ ( hAPP_f22106695ol_nat @ ( finite_fold_nat_nat @ D @ A ) @ C ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_511_comp__fun__commute_Ofold__insert) ).

thf(2848,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( finite2130160977at_nat @ D ) )
     => ( ( hBOOL @ ( hAPP_f54304608l_bool @ finite_finite_nat @ C ) )
       => ( ~ ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ B ) @ C ) )
         => ( ( hAPP_f22106695ol_nat @ ( finite_fold_nat_nat @ D @ A ) @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ B ) @ C ) )
            = ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ D @ B ) @ ( hAPP_f22106695ol_nat @ ( finite_fold_nat_nat @ D @ A ) @ C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[470]) ).

thf(616,axiom,
    ! [A: $i,B: $i] :
      ( ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ A ) @ B )
      = ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ B ) @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_676_comm__semiring__1__class_Onormalizing__semiring__rules_I24_J) ).

thf(3366,plain,
    ! [A: $i,B: $i] :
      ( ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ A ) @ B )
      = ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ B ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[616]) ).

thf(313,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_int_bool @ ( hAPP_f1805168059t_bool @ ( hAPP_f1223193598t_bool @ minus_1449998731t_bool @ A ) @ B ) @ C ) )
    <=> ( hBOOL @ ( hAPP_bool_bool @ ( hAPP_b589554111l_bool @ minus_minus_bool @ ( hAPP_int_bool @ A @ C ) ) @ ( hAPP_int_bool @ B @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_445_fun__diff__def) ).

thf(2266,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( hBOOL @ ( hAPP_int_bool @ ( hAPP_f1805168059t_bool @ ( hAPP_f1223193598t_bool @ minus_1449998731t_bool @ A ) @ B ) @ C ) )
       => ( hBOOL @ ( hAPP_bool_bool @ ( hAPP_b589554111l_bool @ minus_minus_bool @ ( hAPP_int_bool @ A @ C ) ) @ ( hAPP_int_bool @ B @ C ) ) ) )
      & ( ( hBOOL @ ( hAPP_bool_bool @ ( hAPP_b589554111l_bool @ minus_minus_bool @ ( hAPP_int_bool @ A @ C ) ) @ ( hAPP_int_bool @ B @ C ) ) )
       => ( hBOOL @ ( hAPP_int_bool @ ( hAPP_f1805168059t_bool @ ( hAPP_f1223193598t_bool @ minus_1449998731t_bool @ A ) @ B ) @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[313]) ).

thf(916,axiom,
    ! [A: $i] :
      ( ( ( hAPP_f800510211t_bool @ collect_nat @ A )
        = bot_bot_fun_nat_bool )
    <=> ! [B: $i] :
          ~ ( hBOOL @ ( hAPP_nat_bool @ A @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_35_Collect__empty__eq) ).

thf(4516,plain,
    ! [A: $i] :
      ( ( ( ( hAPP_f800510211t_bool @ collect_nat @ A )
          = bot_bot_fun_nat_bool )
       => ! [B: $i] :
            ~ ( hBOOL @ ( hAPP_nat_bool @ A @ B ) ) )
      & ( ! [B: $i] :
            ~ ( hBOOL @ ( hAPP_nat_bool @ A @ B ) )
       => ( ( hAPP_f800510211t_bool @ collect_nat @ A )
          = bot_bot_fun_nat_bool ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[916]) ).

thf(398,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ A ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ ord_min_nat @ B ) @ C ) ) )
    <=> ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ A ) @ B ) )
        & ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ A ) @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_980_min__max_Ole__inf__iff) ).

thf(2584,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ A ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ ord_min_nat @ B ) @ C ) ) )
       => ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ A ) @ B ) )
          & ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ A ) @ C ) ) ) )
      & ( ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ A ) @ B ) )
          & ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ A ) @ C ) ) )
       => ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ A ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ ord_min_nat @ B ) @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[398]) ).

thf(854,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_f103356543l_bool @ ord_le1568362934t_bool @ A ) @ B ) )
     => ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_f103356543l_bool @ ord_le1568362934t_bool @ B ) @ A ) )
      <=> ( B = A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_773_order__antisym__conv) ).

thf(4257,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_f103356543l_bool @ ord_le1568362934t_bool @ A ) @ B ) )
     => ( ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_f103356543l_bool @ ord_le1568362934t_bool @ B ) @ A ) )
         => ( B = A ) )
        & ( ( B = A )
         => ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_f103356543l_bool @ ord_le1568362934t_bool @ B ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[854]) ).

thf(845,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( finite58652534t_bool @ D ) )
     => ( ( hBOOL @ ( hAPP_f448129468l_bool @ finite_finite_int @ C ) )
       => ( ( hAPP_f1805168059t_bool @ ( finite1827710202t_bool @ D @ A ) @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ B ) @ C ) )
          = ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ D @ B ) @ ( hAPP_f1805168059t_bool @ ( finite1827710202t_bool @ D @ A ) @ C ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_580_comp__fun__idem_Ofold__insert__idem) ).

thf(4217,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( finite58652534t_bool @ D ) )
     => ( ( hBOOL @ ( hAPP_f448129468l_bool @ finite_finite_int @ C ) )
       => ( ( hAPP_f1805168059t_bool @ ( finite1827710202t_bool @ D @ A ) @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ B ) @ C ) )
          = ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ D @ B ) @ ( hAPP_f1805168059t_bool @ ( finite1827710202t_bool @ D @ A ) @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[845]) ).

thf(136,axiom,
    ! [A: $i] :
      ( ( hBOOL @ ( hAPP_nat_bool @ bot_bot_fun_nat_bool @ A ) )
    <=> ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ A ) @ bot_bot_fun_nat_bool ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_140_bot__empty__eq) ).

thf(1617,plain,
    ! [A: $i] :
      ( ( ( hBOOL @ ( hAPP_nat_bool @ bot_bot_fun_nat_bool @ A ) )
       => ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ A ) @ bot_bot_fun_nat_bool ) ) )
      & ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ A ) @ bot_bot_fun_nat_bool ) )
       => ( hBOOL @ ( hAPP_nat_bool @ bot_bot_fun_nat_bool @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[136]) ).

thf(926,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( finite1753440478iple_a @ C @ D ) )
     => ( ( hBOOL @ ( hAPP_f1695230391l_bool @ finite1738664244iple_a @ B ) )
       => ( ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ A ) @ B ) )
         => ( ( ( ( hAPP_f20753329a_bool @ ( hAPP_f1706700729a_bool @ minus_85316870a_bool @ B ) @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ A ) @ bot_bo797238721a_bool ) )
                = bot_bo797238721a_bool )
             => ( ( hAPP_f1693662087iple_a @ D @ B )
                = A ) )
            & ( ( ( hAPP_f20753329a_bool @ ( hAPP_f1706700729a_bool @ minus_85316870a_bool @ B ) @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ A ) @ bot_bo797238721a_bool ) )
               != bot_bo797238721a_bool )
             => ( ( hAPP_f1693662087iple_a @ D @ B )
                = ( hAPP_H905846293iple_a @ ( hAPP_H2085992369iple_a @ C @ A ) @ ( hAPP_f1693662087iple_a @ D @ ( hAPP_f20753329a_bool @ ( hAPP_f1706700729a_bool @ minus_85316870a_bool @ B ) @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ A ) @ bot_bo797238721a_bool ) ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_357_folding__one_Oremove) ).

thf(4548,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( finite1753440478iple_a @ C @ D ) )
     => ( ( hBOOL @ ( hAPP_f1695230391l_bool @ finite1738664244iple_a @ B ) )
       => ( ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ A ) @ B ) )
         => ( ( ( ( hAPP_f20753329a_bool @ ( hAPP_f1706700729a_bool @ minus_85316870a_bool @ B ) @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ A ) @ bot_bo797238721a_bool ) )
                = bot_bo797238721a_bool )
             => ( ( hAPP_f1693662087iple_a @ D @ B )
                = A ) )
            & ( ( ( hAPP_f20753329a_bool @ ( hAPP_f1706700729a_bool @ minus_85316870a_bool @ B ) @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ A ) @ bot_bo797238721a_bool ) )
               != bot_bo797238721a_bool )
             => ( ( hAPP_f1693662087iple_a @ D @ B )
                = ( hAPP_H905846293iple_a @ ( hAPP_H2085992369iple_a @ C @ A ) @ ( hAPP_f1693662087iple_a @ D @ ( hAPP_f20753329a_bool @ ( hAPP_f1706700729a_bool @ minus_85316870a_bool @ B ) @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ A ) @ bot_bo797238721a_bool ) ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[926]) ).

thf(335,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( finite1626084323ne_int @ B @ C ) )
     => ( ( hAPP_f1594865479ol_int @ C @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ A ) @ bot_bot_fun_int_bool ) )
        = A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_138_folding__one_Osingleton) ).

thf(2352,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( finite1626084323ne_int @ B @ C ) )
     => ( ( hAPP_f1594865479ol_int @ C @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ A ) @ bot_bot_fun_int_bool ) )
        = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[335]) ).

thf(673,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( finite795500164em_nat @ B @ C ) )
     => ( ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ B @ A ) @ A )
        = A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_375_folding__one__idem_Oidem) ).

thf(3589,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( finite795500164em_nat @ B @ C ) )
     => ( ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ B @ A ) @ A )
        = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[673]) ).

thf(156,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ A ) @ B ) ) @ C )
      = ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ A ) @ C ) ) @ B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_665_comm__semiring__1__class_Onormalizing__semiring__rules_I23_J) ).

thf(1685,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ A ) @ B ) ) @ C )
      = ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ A ) @ C ) ) @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[156]) ).

thf(226,axiom,
    ! [A: $i] :
      ( ( image_nat_int @ A @ bot_bot_fun_nat_bool )
      = bot_bot_fun_int_bool ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_209_image__empty) ).

thf(1949,plain,
    ! [A: $i] :
      ( ( image_nat_int @ A @ bot_bot_fun_nat_bool )
      = bot_bot_fun_int_bool ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[226]) ).

thf(196,axiom,
    ! [A: $i] :
      ( ( hAPP_f1805168059t_bool @ collect_int @ ( hAPP_i1948725293t_bool @ fequal_int @ A ) )
      = ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ A ) @ bot_bot_fun_int_bool ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_20_singleton__conv2) ).

thf(1817,plain,
    ! [A: $i] :
      ( ( hAPP_f1805168059t_bool @ collect_int @ ( hAPP_i1948725293t_bool @ fequal_int @ A ) )
      = ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ A ) @ bot_bot_fun_int_bool ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[196]) ).

thf(953,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f1695230391l_bool @ finite1738664244iple_a @ C ) )
     => ? [D: $i] : ( hBOOL @ ( hAPP_H1037229737a_bool @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ ( finite90892550iple_a @ A ) @ B ) @ C ) @ D ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_329_finite__imp__fold__graph) ).

thf(4656,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f1695230391l_bool @ finite1738664244iple_a @ C ) )
     => ? [D: $i] : ( hBOOL @ ( hAPP_H1037229737a_bool @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ ( finite90892550iple_a @ A ) @ B ) @ C ) @ D ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[953]) ).

thf(204,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ C ) @ D ) )
     => ( ( A
          = ( hAPP_nat_int @ B @ C ) )
       => ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ A ) @ ( image_nat_int @ B @ D ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_174_rev__image__eqI) ).

thf(1844,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ C ) @ D ) )
     => ( ( A
          = ( hAPP_nat_int @ B @ C ) )
       => ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ A ) @ ( image_nat_int @ B @ D ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[204]) ).

thf(775,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ~ ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ B ) @ C ) )
       => ( B = A ) )
     => ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ B ) @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ A ) @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_12_insertCI) ).

thf(3928,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ~ ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ B ) @ C ) )
       => ( B = A ) )
     => ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ B ) @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ A ) @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[775]) ).

thf(477,axiom,
    ! [A: $i,B: $i] :
      ( ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ ( hAPP_nat_int @ semiri1621563631at_int @ A ) ) @ ( hAPP_nat_int @ semiri1621563631at_int @ B ) )
      = ( hAPP_nat_int @ semiri1621563631at_int @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ A ) @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_1164_Nat__Transfer_Otransfer__int__nat__functions_I1_J) ).

thf(2875,plain,
    ! [A: $i,B: $i] :
      ( ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ ( hAPP_nat_int @ semiri1621563631at_int @ A ) ) @ ( hAPP_nat_int @ semiri1621563631at_int @ B ) )
      = ( hAPP_nat_int @ semiri1621563631at_int @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ A ) @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[477]) ).

thf(223,axiom,
    ! [A: $i,B: $i] :
      ( ( hAPP_f800510211t_bool @ ( finite1578363458t_bool @ A @ B ) @ bot_bot_fun_nat_bool )
      = B ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_504_fold__empty) ).

thf(1940,plain,
    ! [A: $i,B: $i] :
      ( ( hAPP_f800510211t_bool @ ( finite1578363458t_bool @ A @ B ) @ bot_bot_fun_nat_bool )
      = B ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[223]) ).

thf(359,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ A ) @ B ) )
     => ( ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ ord_min_nat @ B ) @ A )
        = A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_972_min__max_Oinf__absorb2) ).

thf(2435,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ A ) @ B ) )
     => ( ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ ord_min_nat @ B ) @ A )
        = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[359]) ).

thf(590,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f1695230391l_bool @ finite1738664244iple_a @ C ) )
     => ( ~ ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ B ) @ C ) )
       => ( ( hAPP_f2097660464ol_nat @ ( hAPP_f1371399797ol_nat @ big_co1340561246_a_nat @ A ) @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ B ) @ C ) )
          = ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ ( hAPP_H592031934_a_nat @ A @ B ) ) @ ( hAPP_f2097660464ol_nat @ ( hAPP_f1371399797ol_nat @ big_co1340561246_a_nat @ A ) @ C ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_641_setsum_Oinsert) ).

thf(3284,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f1695230391l_bool @ finite1738664244iple_a @ C ) )
     => ( ~ ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ B ) @ C ) )
       => ( ( hAPP_f2097660464ol_nat @ ( hAPP_f1371399797ol_nat @ big_co1340561246_a_nat @ A ) @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ B ) @ C ) )
          = ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ ( hAPP_H592031934_a_nat @ A @ B ) ) @ ( hAPP_f2097660464ol_nat @ ( hAPP_f1371399797ol_nat @ big_co1340561246_a_nat @ A ) @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[590]) ).

thf(546,axiom,
    ! [A: $i] :
      ( ( hBOOL @ ( hAPP_int_bool @ bot_bot_fun_int_bool @ A ) )
    <=> ( hBOOL @ bot_bot_bool ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_109_bot__fun__def) ).

thf(3139,plain,
    ! [A: $i] :
      ( ( ( hBOOL @ ( hAPP_int_bool @ bot_bot_fun_int_bool @ A ) )
       => ( hBOOL @ bot_bot_bool ) )
      & ( ( hBOOL @ bot_bot_bool )
       => ( hBOOL @ ( hAPP_int_bool @ bot_bot_fun_int_bool @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[546]) ).

thf(682,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP_int_int @ ( hAPP_int_fun_int_int @ times_times_int @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ A ) @ B ) ) @ C )
      = ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ times_times_int @ A ) @ C ) ) @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ times_times_int @ B ) @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_1096_zadd__zmult__distrib) ).

thf(3620,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP_int_int @ ( hAPP_int_fun_int_int @ times_times_int @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ A ) @ B ) ) @ C )
      = ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ times_times_int @ A ) @ C ) ) @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ times_times_int @ B ) @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[682]) ).

thf(202,axiom,
    ! [A: $i,B: $i] :
      ( ( hAPP_f1805168059t_bool @ ( finite1827710202t_bool @ A @ B ) @ bot_bot_fun_int_bool )
      = B ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_502_fold__empty) ).

thf(1839,plain,
    ! [A: $i,B: $i] :
      ( ( hAPP_f1805168059t_bool @ ( finite1827710202t_bool @ A @ B ) @ bot_bot_fun_int_bool )
      = B ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[202]) ).

thf(671,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hAPP_int_int @ ( hAPP_int_fun_int_int @ times_times_int @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ times_times_int @ A ) @ B ) ) @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ times_times_int @ C ) @ D ) )
      = ( hAPP_int_int @ ( hAPP_int_fun_int_int @ times_times_int @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ times_times_int @ A ) @ C ) ) @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ times_times_int @ B ) @ D ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_678_comm__semiring__1__class_Onormalizing__semiring__rules_I13_J) ).

thf(3584,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hAPP_int_int @ ( hAPP_int_fun_int_int @ times_times_int @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ times_times_int @ A ) @ B ) ) @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ times_times_int @ C ) @ D ) )
      = ( hAPP_int_int @ ( hAPP_int_fun_int_int @ times_times_int @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ times_times_int @ A ) @ C ) ) @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ times_times_int @ B ) @ D ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[671]) ).

thf(664,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ times_times_int @ A ) @ B ) ) @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ times_times_int @ C ) @ B ) )
      = ( hAPP_int_int @ ( hAPP_int_fun_int_int @ times_times_int @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ A ) @ C ) ) @ B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_652_comm__semiring__1__class_Onormalizing__semiring__rules_I1_J) ).

thf(3565,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ times_times_int @ A ) @ B ) ) @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ times_times_int @ C ) @ B ) )
      = ( hAPP_int_int @ ( hAPP_int_fun_int_int @ times_times_int @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ A ) @ C ) ) @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[664]) ).

thf(623,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( B
        = ( hAPP_n261501868iple_a @ C @ D ) )
     => ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ D ) @ A ) )
       => ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ B ) @ ( image_1865589340iple_a @ C @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_156_image__eqI) ).

thf(3401,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( B
        = ( hAPP_n261501868iple_a @ C @ D ) )
     => ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ D ) @ A ) )
       => ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ B ) @ ( image_1865589340iple_a @ C @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[623]) ).

thf(1035,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_f1767618879l_bool @ ord_le1523781517a_bool @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ A ) @ B ) ) @ C ) )
    <=> ( ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ A ) @ C ) )
        & ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_f1767618879l_bool @ ord_le1523781517a_bool @ B ) @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_868_insert__subset) ).

thf(4945,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_f1767618879l_bool @ ord_le1523781517a_bool @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ A ) @ B ) ) @ C ) )
       => ( ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ A ) @ C ) )
          & ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_f1767618879l_bool @ ord_le1523781517a_bool @ B ) @ C ) ) ) )
      & ( ( ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ A ) @ C ) )
          & ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_f1767618879l_bool @ ord_le1523781517a_bool @ B ) @ C ) ) )
       => ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_f1767618879l_bool @ ord_le1523781517a_bool @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ A ) @ B ) ) @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1035]) ).

thf(462,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ A ) @ ( hAPP_f1805168059t_bool @ ( hAPP_f1223193598t_bool @ minus_1449998731t_bool @ B ) @ C ) ) )
     => ~ ( ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ A ) @ B ) )
         => ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ A ) @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_369_DiffE) ).

thf(2818,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ A ) @ ( hAPP_f1805168059t_bool @ ( hAPP_f1223193598t_bool @ minus_1449998731t_bool @ B ) @ C ) ) )
     => ~ ( ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ A ) @ B ) )
         => ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ A ) @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[462]) ).

thf(659,axiom,
    ! [A: $i] :
      ( ( hAPP_f800510211t_bool @ collect_nat @ A )
      = A ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_200_Collect__def) ).

thf(3548,plain,
    ! [A: $i] :
      ( ( hAPP_f800510211t_bool @ collect_nat @ A )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[659]) ).

thf(291,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( finite1321096241t_bool @ D ) )
     => ( ( hBOOL @ ( hAPP_f448129468l_bool @ finite_finite_int @ C ) )
       => ( ~ ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ B ) @ C ) )
         => ( ( hAPP_f1805168059t_bool @ ( finite1827710202t_bool @ D @ A ) @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ B ) @ C ) )
            = ( hAPP_f1805168059t_bool @ ( finite1827710202t_bool @ D @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ D @ B ) @ A ) ) @ C ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_510_comp__fun__commute_Ofold__insert2) ).

thf(2194,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( finite1321096241t_bool @ D ) )
     => ( ( hBOOL @ ( hAPP_f448129468l_bool @ finite_finite_int @ C ) )
       => ( ~ ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ B ) @ C ) )
         => ( ( hAPP_f1805168059t_bool @ ( finite1827710202t_bool @ D @ A ) @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ B ) @ C ) )
            = ( hAPP_f1805168059t_bool @ ( finite1827710202t_bool @ D @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ D @ B ) @ A ) ) @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[291]) ).

thf(297,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP_f1805168059t_bool @ ( hAPP_f1223193598t_bool @ minus_1449998731t_bool @ A ) @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ B ) @ C ) )
      = ( hAPP_f1805168059t_bool @ ( hAPP_f1223193598t_bool @ minus_1449998731t_bool @ ( hAPP_f1805168059t_bool @ ( hAPP_f1223193598t_bool @ minus_1449998731t_bool @ A ) @ C ) ) @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ B ) @ bot_bot_fun_int_bool ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_413_Diff__insert) ).

thf(2215,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP_f1805168059t_bool @ ( hAPP_f1223193598t_bool @ minus_1449998731t_bool @ A ) @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ B ) @ C ) )
      = ( hAPP_f1805168059t_bool @ ( hAPP_f1223193598t_bool @ minus_1449998731t_bool @ ( hAPP_f1805168059t_bool @ ( hAPP_f1223193598t_bool @ minus_1449998731t_bool @ A ) @ C ) ) @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ B ) @ bot_bot_fun_int_bool ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[297]) ).

thf(929,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP_a849909144l_bool @ ( hAPP_f1509969235l_bool @ ( hAPP_f1178339559l_bool @ cOMBB_1355796797bool_a @ A ) @ B ) @ C )
      = ( hAPP_f1259673775l_bool @ A @ ( hAPP_a2036067514e_bool @ B @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',help_COMBB_1_1_COMBB_000tc__fun_Itc__Com__Ostate_Mtc__HOL__Obool_J_000tc__fun_It) ).

thf(4559,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP_a849909144l_bool @ ( hAPP_f1509969235l_bool @ ( hAPP_f1178339559l_bool @ cOMBB_1355796797bool_a @ A ) @ B ) @ C )
      = ( hAPP_f1259673775l_bool @ A @ ( hAPP_a2036067514e_bool @ B @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[929]) ).

thf(975,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( finite1033474011iple_a @ C @ D ) )
     => ( ( hBOOL @ ( hAPP_f1695230391l_bool @ finite1738664244iple_a @ B ) )
       => ( ( hAPP_f1693662087iple_a @ D @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ A ) @ B ) )
          = ( hAPP_f1693662087iple_a @ ( finite1205970312iple_a @ C @ A ) @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_585_folding__one__idem_Oeq__fold__idem_H) ).

thf(4738,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( finite1033474011iple_a @ C @ D ) )
     => ( ( hBOOL @ ( hAPP_f1695230391l_bool @ finite1738664244iple_a @ B ) )
       => ( ( hAPP_f1693662087iple_a @ D @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ A ) @ B ) )
          = ( hAPP_f1693662087iple_a @ ( finite1205970312iple_a @ C @ A ) @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[975]) ).

thf(989,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( image_48635758_a_nat @ A @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ B ) @ C ) )
      = ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ ( hAPP_H592031934_a_nat @ A @ B ) ) @ ( image_48635758_a_nat @ A @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_220_image__insert) ).

thf(4780,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( image_48635758_a_nat @ A @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ B ) @ C ) )
      = ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ ( hAPP_H592031934_a_nat @ A @ B ) ) @ ( image_48635758_a_nat @ A @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[989]) ).

thf(684,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_f284875647l_bool @ ord_le951220754t_bool @ C ) @ D ) )
     => ( ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_f284875647l_bool @ ord_le951220754t_bool @ A ) @ B ) )
       => ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_f284875647l_bool @ ord_le951220754t_bool @ ( hAPP_f1805168059t_bool @ ( hAPP_f1223193598t_bool @ minus_1449998731t_bool @ C ) @ B ) ) @ ( hAPP_f1805168059t_bool @ ( hAPP_f1223193598t_bool @ minus_1449998731t_bool @ D ) @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_883_Diff__mono) ).

thf(3625,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_f284875647l_bool @ ord_le951220754t_bool @ C ) @ D ) )
     => ( ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_f284875647l_bool @ ord_le951220754t_bool @ A ) @ B ) )
       => ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_f284875647l_bool @ ord_le951220754t_bool @ ( hAPP_f1805168059t_bool @ ( hAPP_f1223193598t_bool @ minus_1449998731t_bool @ C ) @ B ) ) @ ( hAPP_f1805168059t_bool @ ( hAPP_f1223193598t_bool @ minus_1449998731t_bool @ D ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[684]) ).

thf(951,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hoare_2102800559rivs_a @ B ) @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ ( hoare_1916936827iple_a @ C @ D @ E ) ) @ bot_bo797238721a_bool ) ) )
     => ( ! [F: $i,G: $i] :
            ( ( hBOOL @ ( hAPP_state_bool @ ( hAPP_a2036067514e_bool @ E @ F ) @ G ) )
           => ( hBOOL @ ( hAPP_state_bool @ ( hAPP_a2036067514e_bool @ A @ F ) @ G ) ) )
       => ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hoare_2102800559rivs_a @ B ) @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ ( hoare_1916936827iple_a @ C @ D @ A ) ) @ bot_bo797238721a_bool ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_6_conseq2) ).

thf(4646,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hoare_2102800559rivs_a @ B ) @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ ( hoare_1916936827iple_a @ C @ D @ E ) ) @ bot_bo797238721a_bool ) ) )
     => ( ! [F: $i,G: $i] :
            ( ( hBOOL @ ( hAPP_state_bool @ ( hAPP_a2036067514e_bool @ E @ F ) @ G ) )
           => ( hBOOL @ ( hAPP_state_bool @ ( hAPP_a2036067514e_bool @ A @ F ) @ G ) ) )
       => ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hoare_2102800559rivs_a @ B ) @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ ( hoare_1916936827iple_a @ C @ D @ A ) ) @ bot_bo797238721a_bool ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[951]) ).

thf(20,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_int_bool @ ( hAPP_i1948725293t_bool @ ord_less_eq_int @ B ) @ C ) )
     => ( ( hBOOL @ ( hAPP_int_bool @ ( hAPP_i1948725293t_bool @ ord_less_eq_int @ B ) @ A ) )
       => ( hBOOL @ ( hAPP_int_bool @ ( hAPP_i1948725293t_bool @ ord_less_eq_int @ B ) @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ ord_min_int @ C ) @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_971_min__max_Ole__infI) ).

thf(1189,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_int_bool @ ( hAPP_i1948725293t_bool @ ord_less_eq_int @ B ) @ C ) )
     => ( ( hBOOL @ ( hAPP_int_bool @ ( hAPP_i1948725293t_bool @ ord_less_eq_int @ B ) @ A ) )
       => ( hBOOL @ ( hAPP_int_bool @ ( hAPP_i1948725293t_bool @ ord_less_eq_int @ B ) @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ ord_min_int @ C ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[20]) ).

thf(791,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ A ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ minus_minus_nat @ B ) @ C ) )
      = ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ minus_minus_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ A ) @ B ) ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ A ) @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_1015_diff__mult__distrib2) ).

thf(4006,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ A ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ minus_minus_nat @ B ) @ C ) )
      = ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ minus_minus_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ A ) @ B ) ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ A ) @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[791]) ).

thf(333,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ B ) @ C ) )
       => ( ( hAPP_f800510211t_bool @ ( hAPP_f1730770594t_bool @ minus_2067140911t_bool @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ B ) @ A ) ) @ C )
          = ( hAPP_f800510211t_bool @ ( hAPP_f1730770594t_bool @ minus_2067140911t_bool @ A ) @ C ) ) )
      & ( ~ ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ B ) @ C ) )
       => ( ( hAPP_f800510211t_bool @ ( hAPP_f1730770594t_bool @ minus_2067140911t_bool @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ B ) @ A ) ) @ C )
          = ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ B ) @ ( hAPP_f800510211t_bool @ ( hAPP_f1730770594t_bool @ minus_2067140911t_bool @ A ) @ C ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_403_insert__Diff__if) ).

thf(2342,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ B ) @ C ) )
       => ( ( hAPP_f800510211t_bool @ ( hAPP_f1730770594t_bool @ minus_2067140911t_bool @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ B ) @ A ) ) @ C )
          = ( hAPP_f800510211t_bool @ ( hAPP_f1730770594t_bool @ minus_2067140911t_bool @ A ) @ C ) ) )
      & ( ~ ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ B ) @ C ) )
       => ( ( hAPP_f800510211t_bool @ ( hAPP_f1730770594t_bool @ minus_2067140911t_bool @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ B ) @ A ) ) @ C )
          = ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ B ) @ ( hAPP_f800510211t_bool @ ( hAPP_f1730770594t_bool @ minus_2067140911t_bool @ A ) @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[333]) ).

thf(122,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( finite1071749497t_bool @ D ) )
     => ( ( hBOOL @ ( hAPP_f54304608l_bool @ finite_finite_nat @ C ) )
       => ( ( hAPP_f800510211t_bool @ ( finite1578363458t_bool @ D @ A ) @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ B ) @ C ) )
          = ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ D @ B ) @ ( hAPP_f800510211t_bool @ ( finite1578363458t_bool @ D @ A ) @ ( hAPP_f800510211t_bool @ ( hAPP_f1730770594t_bool @ minus_2067140911t_bool @ C ) @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ B ) @ bot_bot_fun_nat_bool ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_547_comp__fun__commute_Ofold__insert__remove) ).

thf(1575,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( finite1071749497t_bool @ D ) )
     => ( ( hBOOL @ ( hAPP_f54304608l_bool @ finite_finite_nat @ C ) )
       => ( ( hAPP_f800510211t_bool @ ( finite1578363458t_bool @ D @ A ) @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ B ) @ C ) )
          = ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ D @ B ) @ ( hAPP_f800510211t_bool @ ( finite1578363458t_bool @ D @ A ) @ ( hAPP_f800510211t_bool @ ( hAPP_f1730770594t_bool @ minus_2067140911t_bool @ C ) @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ B ) @ bot_bot_fun_nat_bool ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[122]) ).

thf(901,axiom,
    ! [A: $i,B: $i] :
      ( ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ 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 @ fdisj ) @ ( hAPP_H426895267a_bool @ ( hAPP_f920293029a_bool @ cOMBC_1940896922a_bool @ fequal963300192iple_a ) @ A ) ) ) @ ( hAPP_f20753329a_bool @ ( hAPP_f247708275a_bool @ cOMBC_1519416966l_bool @ member1797258804iple_a ) @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_76_insert__compr) ).

thf(4463,plain,
    ! [A: $i,B: $i] :
      ( ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ 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 @ fdisj ) @ ( hAPP_H426895267a_bool @ ( hAPP_f920293029a_bool @ cOMBC_1940896922a_bool @ fequal963300192iple_a ) @ A ) ) ) @ ( hAPP_f20753329a_bool @ ( hAPP_f247708275a_bool @ cOMBC_1519416966l_bool @ member1797258804iple_a ) @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[901]) ).

thf(457,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP_f448129468l_bool @ finite_finite_int @ B ) )
     => ( ( B != bot_bot_fun_int_bool )
       => ( ! [C: $i] : ( hBOOL @ ( hAPP_f448129468l_bool @ A @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ C ) @ bot_bot_fun_int_bool ) ) )
         => ( ! [C: $i,D: $i] :
                ( ( hBOOL @ ( hAPP_f448129468l_bool @ finite_finite_int @ D ) )
               => ( ( D != bot_bot_fun_int_bool )
                 => ( ~ ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ C ) @ D ) )
                   => ( ( hBOOL @ ( hAPP_f448129468l_bool @ A @ D ) )
                     => ( hBOOL @ ( hAPP_f448129468l_bool @ A @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ C ) @ D ) ) ) ) ) ) )
           => ( hBOOL @ ( hAPP_f448129468l_bool @ A @ B ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_352_finite__ne__induct) ).

thf(2792,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP_f448129468l_bool @ finite_finite_int @ B ) )
     => ( ( B != bot_bot_fun_int_bool )
       => ( ! [C: $i] : ( hBOOL @ ( hAPP_f448129468l_bool @ A @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ C ) @ bot_bot_fun_int_bool ) ) )
         => ( ! [C: $i,D: $i] :
                ( ( hBOOL @ ( hAPP_f448129468l_bool @ finite_finite_int @ D ) )
               => ( ( D != bot_bot_fun_int_bool )
                 => ( ~ ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ C ) @ D ) )
                   => ( ( hBOOL @ ( hAPP_f448129468l_bool @ A @ D ) )
                     => ( hBOOL @ ( hAPP_f448129468l_bool @ A @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ C ) @ D ) ) ) ) ) ) )
           => ( hBOOL @ ( hAPP_f448129468l_bool @ A @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[457]) ).

thf(194,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( C = D )
     => ( ! [E: $i] :
            ( ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ E ) @ D ) )
           => ( ( hAPP_int_int @ A @ E )
              = ( hAPP_int_int @ B @ E ) ) )
       => ( ( image_int_int @ A @ C )
          = ( image_int_int @ B @ D ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_258_image__cong) ).

thf(1807,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( C = D )
     => ( ! [E: $i] :
            ( ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ E ) @ D ) )
           => ( ( hAPP_int_int @ A @ E )
              = ( hAPP_int_int @ B @ E ) ) )
       => ( ( image_int_int @ A @ C )
          = ( image_int_int @ B @ D ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[194]) ).

thf(807,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f1695230391l_bool @ finite1738664244iple_a @ C ) )
     => ( ~ ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ B ) @ C ) )
       => ( ( hAPP_f587450508ol_int @ ( hAPP_f1544320301ol_int @ big_co1977834938_a_int @ A ) @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ B ) @ C ) )
          = ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ ( hAPP_H1229305626_a_int @ A @ B ) ) @ ( hAPP_f587450508ol_int @ ( hAPP_f1544320301ol_int @ big_co1977834938_a_int @ A ) @ C ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_640_setsum_Oinsert) ).

thf(4061,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f1695230391l_bool @ finite1738664244iple_a @ C ) )
     => ( ~ ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ B ) @ C ) )
       => ( ( hAPP_f587450508ol_int @ ( hAPP_f1544320301ol_int @ big_co1977834938_a_int @ A ) @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ B ) @ C ) )
          = ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ ( hAPP_H1229305626_a_int @ A @ B ) ) @ ( hAPP_f587450508ol_int @ ( hAPP_f1544320301ol_int @ big_co1977834938_a_int @ A ) @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[807]) ).

thf(1101,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( finite1753440478iple_a @ B @ C ) )
     => ( ( hBOOL @ ( hAPP_f1695230391l_bool @ finite1738664244iple_a @ A ) )
       => ( ( A != bot_bo797238721a_bool )
         => ( ! [D: $i,E: $i] : ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ ( hAPP_H905846293iple_a @ ( hAPP_H2085992369iple_a @ B @ D ) @ E ) ) @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ D ) @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ E ) @ bot_bo797238721a_bool ) ) ) )
           => ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ ( hAPP_f1693662087iple_a @ C @ A ) ) @ A ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_274_folding__one_Oclosed) ).

thf(5184,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( finite1753440478iple_a @ B @ C ) )
     => ( ( hBOOL @ ( hAPP_f1695230391l_bool @ finite1738664244iple_a @ A ) )
       => ( ( A != bot_bo797238721a_bool )
         => ( ! [D: $i,E: $i] : ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ ( hAPP_H905846293iple_a @ ( hAPP_H2085992369iple_a @ B @ D ) @ E ) ) @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ D ) @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ E ) @ bot_bo797238721a_bool ) ) ) )
           => ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ ( hAPP_f1693662087iple_a @ C @ A ) ) @ A ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1101]) ).

thf(972,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ A ) @ ( hAPP_f20753329a_bool @ ( hAPP_f1706700729a_bool @ minus_85316870a_bool @ B ) @ C ) ) )
     => ~ ( ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ A ) @ B ) )
         => ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ A ) @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_367_DiffE) ).

thf(4730,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ A ) @ ( hAPP_f20753329a_bool @ ( hAPP_f1706700729a_bool @ minus_85316870a_bool @ B ) @ C ) ) )
     => ~ ( ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ A ) @ B ) )
         => ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ A ) @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[972]) ).

thf(959,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( ( hoare_1916936827iple_a @ A @ B @ C )
        = ( hoare_1916936827iple_a @ D @ E @ F ) )
    <=> ( ( A = D )
        & ( B = E )
        & ( C = F ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_1_triple_Oinject) ).

thf(4678,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( ( ( hoare_1916936827iple_a @ A @ B @ C )
          = ( hoare_1916936827iple_a @ D @ E @ F ) )
       => ( ( A = D )
          & ( B = E )
          & ( C = F ) ) )
      & ( ( ( A = D )
          & ( B = E )
          & ( C = F ) )
       => ( ( hoare_1916936827iple_a @ A @ B @ C )
          = ( hoare_1916936827iple_a @ D @ E @ F ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[959]) ).

thf(238,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ A ) @ ( hAPP_f800510211t_bool @ ( hAPP_f1730770594t_bool @ minus_2067140911t_bool @ B ) @ C ) ) )
     => ~ ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ A ) @ B ) )
         => ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ A ) @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_368_DiffE) ).

thf(1998,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ A ) @ ( hAPP_f800510211t_bool @ ( hAPP_f1730770594t_bool @ minus_2067140911t_bool @ B ) @ C ) ) )
     => ~ ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ A ) @ B ) )
         => ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ A ) @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[238]) ).

thf(933,axiom,
    ! [A: $i] :
      ( ! [B: $i] :
          ~ ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ B ) @ A ) )
     => ( A = bot_bo797238721a_bool ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_120_equals0I) ).

thf(4574,plain,
    ! [A: $i] :
      ( ! [B: $i] :
          ~ ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ B ) @ A ) )
     => ( A = bot_bo797238721a_bool ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[933]) ).

thf(674,axiom,
    ! [A: $i] :
      ( ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_f284875647l_bool @ ord_le951220754t_bool @ A ) @ bot_bot_fun_int_bool ) )
    <=> ( A = bot_bot_fun_int_bool ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_839_bot__unique) ).

thf(3593,plain,
    ! [A: $i] :
      ( ( ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_f284875647l_bool @ ord_le951220754t_bool @ A ) @ bot_bot_fun_int_bool ) )
       => ( A = bot_bot_fun_int_bool ) )
      & ( ( A = bot_bot_fun_int_bool )
       => ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_f284875647l_bool @ ord_le951220754t_bool @ A ) @ bot_bot_fun_int_bool ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[674]) ).

thf(768,axiom,
    ! [A: $i,B: $i] :
      ( ( 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 ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_853_finite__subset) ).

thf(3911,plain,
    ! [A: $i,B: $i] :
      ( ( 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 ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[768]) ).

thf(315,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ B ) @ C ) )
     => ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ B ) @ A ) )
       => ( ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ minus_minus_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ minus_minus_nat @ C ) @ B ) ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ minus_minus_nat @ A ) @ B ) )
          = ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ minus_minus_nat @ C ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_1039_Nat_Odiff__diff__eq) ).

thf(2273,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ B ) @ C ) )
     => ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ B ) @ A ) )
       => ( ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ minus_minus_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ minus_minus_nat @ C ) @ B ) ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ minus_minus_nat @ A ) @ B ) )
          = ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ minus_minus_nat @ C ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[315]) ).

thf(317,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ A ) @ ( hAPP_f800510211t_bool @ ( hAPP_f1730770594t_bool @ minus_2067140911t_bool @ B ) @ C ) ) )
    <=> ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ A ) @ B ) )
        & ~ ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ A ) @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_380_Diff__iff) ).

thf(2283,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ A ) @ ( hAPP_f800510211t_bool @ ( hAPP_f1730770594t_bool @ minus_2067140911t_bool @ B ) @ C ) ) )
       => ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ A ) @ B ) )
          & ~ ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ A ) @ C ) ) ) )
      & ( ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ A ) @ B ) )
          & ~ ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ A ) @ C ) ) )
       => ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ A ) @ ( hAPP_f800510211t_bool @ ( hAPP_f1730770594t_bool @ minus_2067140911t_bool @ B ) @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[317]) ).

thf(957,axiom,
    ! [A: $i] :
      ( ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_f1767618879l_bool @ ord_le1523781517a_bool @ A ) @ bot_bo797238721a_bool ) )
    <=> ( A = bot_bo797238721a_bool ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_850_subset__empty) ).

thf(4666,plain,
    ! [A: $i] :
      ( ( ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_f1767618879l_bool @ ord_le1523781517a_bool @ A ) @ bot_bo797238721a_bool ) )
       => ( A = bot_bo797238721a_bool ) )
      & ( ( A = bot_bo797238721a_bool )
       => ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_f1767618879l_bool @ ord_le1523781517a_bool @ A ) @ bot_bo797238721a_bool ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[957]) ).

thf(852,axiom,
    ! [A: $i] : ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_f284875647l_bool @ ord_le951220754t_bool @ bot_bot_fun_int_bool ) @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_844_bot__least) ).

thf(4250,plain,
    ! [A: $i] : ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_f284875647l_bool @ ord_le951220754t_bool @ bot_bot_fun_int_bool ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[852]) ).

thf(608,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_f103356543l_bool @ ord_le1568362934t_bool @ A ) @ ( image_nat_nat @ B @ C ) ) )
    <=> ? [D: $i] :
          ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_f103356543l_bool @ ord_le1568362934t_bool @ D ) @ C ) )
          & ( A
            = ( image_nat_nat @ B @ D ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_879_subset__image__iff) ).

thf(3336,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_f103356543l_bool @ ord_le1568362934t_bool @ A ) @ ( image_nat_nat @ B @ C ) ) )
       => ? [D: $i] :
            ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_f103356543l_bool @ ord_le1568362934t_bool @ D ) @ C ) )
            & ( A
              = ( image_nat_nat @ B @ D ) ) ) )
      & ( ? [D: $i] :
            ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_f103356543l_bool @ ord_le1568362934t_bool @ D ) @ C ) )
            & ( A
              = ( image_nat_nat @ B @ D ) ) )
       => ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_f103356543l_bool @ ord_le1568362934t_bool @ A ) @ ( image_nat_nat @ B @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[608]) ).

thf(174,axiom,
    ! [A: $i] :
      ( ? [B: $i] : ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ B ) @ A ) )
    <=> ( A != bot_bot_fun_nat_bool ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_43_ex__in__conv) ).

thf(1745,plain,
    ! [A: $i] :
      ( ( ? [B: $i] : ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ B ) @ A ) )
       => ( A != bot_bot_fun_nat_bool ) )
      & ( ( A != bot_bot_fun_nat_bool )
       => ? [B: $i] : ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ B ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[174]) ).

thf(606,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f1695230391l_bool @ finite1738664244iple_a @ C ) )
     => ( ~ ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ B ) @ C ) )
       => ( ( hAPP_f2097660464ol_nat @ ( big_co1815942735_a_nat @ A ) @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ B ) @ C ) )
          = ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ ( hAPP_H592031934_a_nat @ A @ B ) ) @ ( hAPP_f2097660464ol_nat @ ( big_co1815942735_a_nat @ A ) @ C ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_553_setprod__insert) ).

thf(3331,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f1695230391l_bool @ finite1738664244iple_a @ C ) )
     => ( ~ ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ B ) @ C ) )
       => ( ( hAPP_f2097660464ol_nat @ ( big_co1815942735_a_nat @ A ) @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ B ) @ C ) )
          = ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ ( hAPP_H592031934_a_nat @ A @ B ) ) @ ( hAPP_f2097660464ol_nat @ ( big_co1815942735_a_nat @ A ) @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[606]) ).

thf(494,axiom,
    ! [A: $i,B: $i] :
      ~ ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ A ) @ B ) ) @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_1108_not__add__less1) ).

thf(2934,plain,
    ! [A: $i,B: $i] :
      ~ ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ A ) @ B ) ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[494]) ).

thf(637,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ A ) @ B )
        = ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ A ) @ C ) )
     => ( B = C ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_656_add__imp__eq) ).

thf(3465,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ A ) @ B )
        = ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ A ) @ C ) )
     => ( B = C ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[637]) ).

thf(1071,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ A ) @ ( hAPP_f20753329a_bool @ ( hAPP_f1706700729a_bool @ minus_85316870a_bool @ B ) @ C ) ) )
    <=> ( ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ A ) @ B ) )
        & ~ ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ A ) @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_379_Diff__iff) ).

thf(5055,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ A ) @ ( hAPP_f20753329a_bool @ ( hAPP_f1706700729a_bool @ minus_85316870a_bool @ B ) @ C ) ) )
       => ( ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ A ) @ B ) )
          & ~ ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ A ) @ C ) ) ) )
      & ( ( ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ A ) @ B ) )
          & ~ ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ A ) @ C ) ) )
       => ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ A ) @ ( hAPP_f20753329a_bool @ ( hAPP_f1706700729a_bool @ minus_85316870a_bool @ B ) @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1071]) ).

thf(963,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hoare_2102800559rivs_a @ B ) @ C ) )
     => ( ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hoare_2102800559rivs_a @ A ) @ B ) )
       => ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hoare_2102800559rivs_a @ A ) @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_2_cut) ).

thf(4710,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hoare_2102800559rivs_a @ B ) @ C ) )
     => ( ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hoare_2102800559rivs_a @ A ) @ B ) )
       => ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hoare_2102800559rivs_a @ A ) @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[963]) ).

thf(508,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( finite1626084323ne_int @ B @ C ) )
     => ( ( hBOOL @ ( hAPP_f448129468l_bool @ finite_finite_int @ A ) )
       => ( ( A != bot_bot_fun_int_bool )
         => ( ! [D: $i,E: $i] : ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ B @ D ) @ E ) ) @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ D ) @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ E ) @ bot_bot_fun_int_bool ) ) ) )
           => ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ ( hAPP_f1594865479ol_int @ C @ A ) ) @ A ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_275_folding__one_Oclosed) ).

thf(2988,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( finite1626084323ne_int @ B @ C ) )
     => ( ( hBOOL @ ( hAPP_f448129468l_bool @ finite_finite_int @ A ) )
       => ( ( A != bot_bot_fun_int_bool )
         => ( ! [D: $i,E: $i] : ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ B @ D ) @ E ) ) @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ D ) @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ E ) @ bot_bot_fun_int_bool ) ) ) )
           => ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ ( hAPP_f1594865479ol_int @ C @ A ) ) @ A ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[508]) ).

thf(754,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ A ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ B ) @ C ) )
      = ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ B ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ A ) @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_691_comm__semiring__1__class_Onormalizing__semiring__rules_I19_J) ).

thf(3864,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ A ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ B ) @ C ) )
      = ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ B ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ A ) @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[754]) ).

thf(401,axiom,
    ! [A: $i,B: $i] : ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ A ) @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ A ) @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_80_insertI1) ).

thf(2594,plain,
    ! [A: $i,B: $i] : ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ A ) @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ A ) @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[401]) ).

thf(166,axiom,
    ! [A: $i,B: $i] :
      ( ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ A ) @ B )
      = ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ B ) @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_1051_nat__add__commute) ).

thf(1721,plain,
    ! [A: $i,B: $i] :
      ( ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ A ) @ B )
      = ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ B ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[166]) ).

thf(1122,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP_H1037229737a_bool @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ ( finite90892550iple_a @ A ) @ B ) @ C ) @ D ) )
    <=> ( ( ( C = bot_bo797238721a_bool )
          & ( D = B ) )
        | ? [E: $i,F: $i,G: $i] :
            ( ( C
              = ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ E ) @ F ) )
            & ( D
              = ( hAPP_H905846293iple_a @ ( hAPP_H2085992369iple_a @ A @ E ) @ G ) )
            & ~ ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ E ) @ F ) )
            & ( hBOOL @ ( hAPP_H1037229737a_bool @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ ( finite90892550iple_a @ A ) @ B ) @ F ) @ G ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_345_fold__graph_Osimps) ).

thf(5272,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( hBOOL @ ( hAPP_H1037229737a_bool @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ ( finite90892550iple_a @ A ) @ B ) @ C ) @ D ) )
       => ( ( ( C = bot_bo797238721a_bool )
            & ( D = B ) )
          | ? [E: $i,F: $i,G: $i] :
              ( ( C
                = ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ E ) @ F ) )
              & ( D
                = ( hAPP_H905846293iple_a @ ( hAPP_H2085992369iple_a @ A @ E ) @ G ) )
              & ~ ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ E ) @ F ) )
              & ( hBOOL @ ( hAPP_H1037229737a_bool @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ ( finite90892550iple_a @ A ) @ B ) @ F ) @ G ) ) ) ) )
      & ( ( ( ( C = bot_bo797238721a_bool )
            & ( D = B ) )
          | ? [E: $i,F: $i,G: $i] :
              ( ( C
                = ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ E ) @ F ) )
              & ( D
                = ( hAPP_H905846293iple_a @ ( hAPP_H2085992369iple_a @ A @ E ) @ G ) )
              & ~ ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ E ) @ F ) )
              & ( hBOOL @ ( hAPP_H1037229737a_bool @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ ( finite90892550iple_a @ A ) @ B ) @ F ) @ G ) ) ) )
       => ( hBOOL @ ( hAPP_H1037229737a_bool @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ ( finite90892550iple_a @ A ) @ B ) @ C ) @ D ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1122]) ).

thf(944,axiom,
    ! [A: $i] :
      ~ ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ A ) @ bot_bo797238721a_bool ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_15_emptyE) ).

thf(4615,plain,
    ! [A: $i] :
      ~ ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ A ) @ bot_bo797238721a_bool ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[944]) ).

thf(387,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f448129468l_bool @ finite_finite_int @ C ) )
     => ( ( ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ B ) @ C ) )
         => ( ( hAPP_f1594865479ol_int @ ( hAPP_f1926459811ol_int @ big_co230513141nt_int @ A ) @ ( hAPP_f1805168059t_bool @ ( hAPP_f1223193598t_bool @ minus_1449998731t_bool @ C ) @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ B ) @ 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 @ A ) @ C ) ) @ ( hAPP_int_int @ A @ B ) ) ) )
        & ( ~ ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ B ) @ C ) )
         => ( ( hAPP_f1594865479ol_int @ ( hAPP_f1926459811ol_int @ big_co230513141nt_int @ A ) @ ( hAPP_f1805168059t_bool @ ( hAPP_f1223193598t_bool @ minus_1449998731t_bool @ C ) @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ B ) @ bot_bot_fun_int_bool ) ) )
            = ( hAPP_f1594865479ol_int @ ( hAPP_f1926459811ol_int @ big_co230513141nt_int @ A ) @ C ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_462_setsum__diff1) ).

thf(2532,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f448129468l_bool @ finite_finite_int @ C ) )
     => ( ( ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ B ) @ C ) )
         => ( ( hAPP_f1594865479ol_int @ ( hAPP_f1926459811ol_int @ big_co230513141nt_int @ A ) @ ( hAPP_f1805168059t_bool @ ( hAPP_f1223193598t_bool @ minus_1449998731t_bool @ C ) @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ B ) @ 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 @ A ) @ C ) ) @ ( hAPP_int_int @ A @ B ) ) ) )
        & ( ~ ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ B ) @ C ) )
         => ( ( hAPP_f1594865479ol_int @ ( hAPP_f1926459811ol_int @ big_co230513141nt_int @ A ) @ ( hAPP_f1805168059t_bool @ ( hAPP_f1223193598t_bool @ minus_1449998731t_bool @ C ) @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ B ) @ bot_bot_fun_int_bool ) ) )
            = ( hAPP_f1594865479ol_int @ ( hAPP_f1926459811ol_int @ big_co230513141nt_int @ A ) @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[387]) ).

thf(102,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( B
        = ( finite_fold1_int @ C ) )
     => ( ( hAPP_f1594865479ol_int @ B @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ A ) @ bot_bot_fun_int_bool ) )
        = A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_148_fold1__singleton__def) ).

thf(1508,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( B
        = ( finite_fold1_int @ C ) )
     => ( ( hAPP_f1594865479ol_int @ B @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ A ) @ bot_bot_fun_int_bool ) )
        = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[102]) ).

thf(617,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_f284875647l_bool @ ord_le951220754t_bool @ B ) @ C ) )
     => ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_f284875647l_bool @ ord_le951220754t_bool @ B ) @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ A ) @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_861_subset__insertI2) ).

thf(3369,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_f284875647l_bool @ ord_le951220754t_bool @ B ) @ C ) )
     => ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_f284875647l_bool @ ord_le951220754t_bool @ B ) @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ A ) @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[617]) ).

thf(801,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ A ) @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ B ) @ C ) )
      = ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ B ) @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ A ) @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_66_insert__commute) ).

thf(4042,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ A ) @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ B ) @ C ) )
      = ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ B ) @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ A ) @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[801]) ).

thf(966,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ~ ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ B ) @ C ) )
     => ( ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_f1767618879l_bool @ ord_le1523781517a_bool @ C ) @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ B ) @ A ) ) )
      <=> ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_f1767618879l_bool @ ord_le1523781517a_bool @ C ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_865_subset__insert) ).

thf(4715,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ~ ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ B ) @ C ) )
     => ( ( ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_f1767618879l_bool @ ord_le1523781517a_bool @ C ) @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ B ) @ A ) ) )
         => ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_f1767618879l_bool @ ord_le1523781517a_bool @ C ) @ A ) ) )
        & ( ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_f1767618879l_bool @ ord_le1523781517a_bool @ C ) @ A ) )
         => ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_f1767618879l_bool @ ord_le1523781517a_bool @ C ) @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ B ) @ A ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[966]) ).

thf(993,axiom,
    ! [A: $i,B: $i] :
      ( ( A != B )
      | ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_f1767618879l_bool @ fequal874423448a_bool @ A ) @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',help_fequal_2_1_fequal_000tc__fun_Itc__Hoare____Mirabelle____uwgpyvfjxg__Otriple) ).

thf(4791,plain,
    ! [A: $i,B: $i] :
      ( ( A != B )
      | ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_f1767618879l_bool @ fequal874423448a_bool @ A ) @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[993]) ).

thf(247,axiom,
    ! [A: $i,B: $i] :
      ( ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ minus_minus_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ A ) @ B ) ) @ A )
      = B ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_1017_diff__add__inverse) ).

thf(2026,plain,
    ! [A: $i,B: $i] :
      ( ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ minus_minus_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ A ) @ B ) ) @ A )
      = B ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[247]) ).

thf(686,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ A ) @ B ) )
     => ~ ! [C: $i] :
            ( ( B
              = ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ A ) @ C ) )
           => ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ A ) @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_114_Set_Oset__insert) ).

thf(3630,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ A ) @ B ) )
     => ~ ! [C: $i] :
            ( ( B
              = ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ A ) @ C ) )
           => ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ A ) @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[686]) ).

thf(1057,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( finite2114894759a_bool @ D ) )
     => ( ( hBOOL @ ( hAPP_f1695230391l_bool @ finite1738664244iple_a @ C ) )
       => ( ~ ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ B ) @ C ) )
         => ( ( hAPP_f20753329a_bool @ ( finite1215486064a_bool @ D @ A ) @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ B ) @ C ) )
            = ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ D @ B ) @ ( hAPP_f20753329a_bool @ ( finite1215486064a_bool @ D @ A ) @ C ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_514_comp__fun__commute_Ofold__insert) ).

thf(5010,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( finite2114894759a_bool @ D ) )
     => ( ( hBOOL @ ( hAPP_f1695230391l_bool @ finite1738664244iple_a @ C ) )
       => ( ~ ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ B ) @ C ) )
         => ( ( hAPP_f20753329a_bool @ ( finite1215486064a_bool @ D @ A ) @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ B ) @ C ) )
            = ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ D @ B ) @ ( hAPP_f20753329a_bool @ ( finite1215486064a_bool @ D @ A ) @ C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1057]) ).

thf(449,axiom,
    ! [A: $i,B: $i] :
      ( ( bot_bot_fun_int_bool
        = ( image_nat_int @ A @ B ) )
    <=> ( B = bot_bot_fun_nat_bool ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_213_empty__is__image) ).

thf(2764,plain,
    ! [A: $i,B: $i] :
      ( ( ( bot_bot_fun_int_bool
          = ( image_nat_int @ A @ B ) )
       => ( B = bot_bot_fun_nat_bool ) )
      & ( ( B = bot_bot_fun_nat_bool )
       => ( bot_bot_fun_int_bool
          = ( image_nat_int @ A @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[449]) ).

thf(580,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ A ) @ B ) ) @ C )
      = ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ A ) @ C ) ) @ B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_683_comm__semiring__1__class_Onormalizing__semiring__rules_I16_J) ).

thf(3252,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ A ) @ B ) ) @ C )
      = ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ A ) @ C ) ) @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[580]) ).

thf(1003,axiom,
    ! [A: $i] : ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_f1767618879l_bool @ ord_le1523781517a_bool @ bot_bo797238721a_bool ) @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_845_bot__least) ).

thf(4823,plain,
    ! [A: $i] : ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_f1767618879l_bool @ ord_le1523781517a_bool @ bot_bo797238721a_bool ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1003]) ).

thf(384,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ C ) @ D ) )
     => ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ A ) @ B ) )
       => ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ C ) @ A ) ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ D ) @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_823_add__mono) ).

thf(2526,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ C ) @ D ) )
     => ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ A ) @ B ) )
       => ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ C ) @ A ) ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ D ) @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[384]) ).

thf(353,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ B ) @ C ) )
     => ( ~ ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ B ) @ A ) )
       => ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ B ) @ ( hAPP_f1805168059t_bool @ ( hAPP_f1223193598t_bool @ minus_1449998731t_bool @ C ) @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_366_DiffI) ).

thf(2416,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ B ) @ C ) )
     => ( ~ ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ B ) @ A ) )
       => ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ B ) @ ( hAPP_f1805168059t_bool @ ( hAPP_f1223193598t_bool @ minus_1449998731t_bool @ C ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[353]) ).

thf(348,axiom,
    ! [A: $i] :
      ( ( hAPP_f800510211t_bool @ ( hAPP_f1730770594t_bool @ minus_2067140911t_bool @ bot_bot_fun_nat_bool ) @ A )
      = bot_bot_fun_nat_bool ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_393_empty__Diff) ).

thf(2394,plain,
    ! [A: $i] :
      ( ( hAPP_f800510211t_bool @ ( hAPP_f1730770594t_bool @ minus_2067140911t_bool @ bot_bot_fun_nat_bool ) @ A )
      = bot_bot_fun_nat_bool ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[348]) ).

thf(534,axiom,
    ! [A: $i,B: $i] : ( is_bool @ ( hAPP_f215623910l_bool @ A @ B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',gsy_c_hAPP_000tc__fun_Itc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J_Mtc__HOL__Obool_) ).

thf(3085,plain,
    ! [A: $i,B: $i] : ( is_bool @ ( hAPP_f215623910l_bool @ A @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[534]) ).

thf(362,axiom,
    ! [A: $i,B: $i] : ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ ord_min_nat @ A ) @ B ) ) @ B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_984_min__max_Oinf__le2) ).

thf(2447,plain,
    ! [A: $i,B: $i] : ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ ord_min_nat @ A ) @ B ) ) @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[362]) ).

thf(207,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( B
        = ( hAPP_nat_nat @ C @ D ) )
     => ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ D ) @ A ) )
       => ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ B ) @ ( image_nat_nat @ C @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_161_image__eqI) ).

thf(1861,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( B
        = ( hAPP_nat_nat @ C @ D ) )
     => ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ D ) @ A ) )
       => ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ B ) @ ( image_nat_nat @ C @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[207]) ).

thf(938,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_H1037229737a_bool @ ( finite1621280049iple_a @ A @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ B ) @ bot_bo797238721a_bool ) ) @ C ) )
    <=> ( B = C ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_134_fold1Set__sing) ).

thf(4591,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( hBOOL @ ( hAPP_H1037229737a_bool @ ( finite1621280049iple_a @ A @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ B ) @ bot_bo797238721a_bool ) ) @ C ) )
       => ( B = C ) )
      & ( ( B = C )
       => ( hBOOL @ ( hAPP_H1037229737a_bool @ ( finite1621280049iple_a @ A @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ B ) @ bot_bo797238721a_bool ) ) @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[938]) ).

thf(982,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP_s58564346l_bool @ ( hAPP_f1259673775l_bool @ ( hAPP_f1561913689l_bool @ cOMBB_188601460_state @ A ) @ B ) @ C )
      = ( hAPP_b589554111l_bool @ A @ ( hAPP_state_bool @ B @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',help_COMBB_1_1_COMBB_000tc__HOL__Obool_000tc__fun_Itc__HOL__Obool_Mtc__HOL__Oboo_008) ).

thf(4759,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP_s58564346l_bool @ ( hAPP_f1259673775l_bool @ ( hAPP_f1561913689l_bool @ cOMBB_188601460_state @ A ) @ B ) @ C )
      = ( hAPP_b589554111l_bool @ A @ ( hAPP_state_bool @ B @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[982]) ).

thf(987,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( finite1428139793iple_a @ D ) )
     => ( ( hBOOL @ ( hAPP_H1037229737a_bool @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ ( finite90892550iple_a @ D ) @ A ) @ B ) @ C ) )
       => ( ( hAPP_f1693662087iple_a @ ( finite1205970312iple_a @ D @ A ) @ B )
          = C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_495_comp__fun__commute_Ofold__equality) ).

thf(4774,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( finite1428139793iple_a @ D ) )
     => ( ( hBOOL @ ( hAPP_H1037229737a_bool @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ ( finite90892550iple_a @ D ) @ A ) @ B ) @ C ) )
       => ( ( hAPP_f1693662087iple_a @ ( finite1205970312iple_a @ D @ A ) @ B )
          = C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[987]) ).

thf(548,axiom,
    ! [A: $i,B: $i] :
      ( ! [C: $i] : ( hBOOL @ ( hAPP_bool_bool @ ( hAPP_b589554111l_bool @ ord_less_eq_bool @ ( hAPP_int_bool @ A @ C ) ) @ ( hAPP_int_bool @ B @ C ) ) )
     => ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_f284875647l_bool @ ord_le951220754t_bool @ A ) @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_930_le__funI) ).

thf(3149,plain,
    ! [A: $i,B: $i] :
      ( ! [C: $i] : ( hBOOL @ ( hAPP_bool_bool @ ( hAPP_b589554111l_bool @ ord_less_eq_bool @ ( hAPP_int_bool @ A @ C ) ) @ ( hAPP_int_bool @ B @ C ) ) )
     => ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_f284875647l_bool @ ord_le951220754t_bool @ A ) @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[548]) ).

thf(148,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_f103356543l_bool @ ord_le1568362934t_bool @ B ) @ C ) )
     => ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ A ) @ B ) )
       => ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ A ) @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_802_in__mono) ).

thf(1661,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_f103356543l_bool @ ord_le1568362934t_bool @ B ) @ C ) )
     => ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ A ) @ B ) )
       => ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ A ) @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[148]) ).

thf(1107,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hoare_2102800559rivs_a @ C ) @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ ( hoare_1916936827iple_a @ D @ E @ F ) ) @ bot_bo797238721a_bool ) ) )
     => ( ! [G: $i,H: $i] :
            ( ( hBOOL @ ( hAPP_state_bool @ ( hAPP_a2036067514e_bool @ B @ G ) @ H ) )
           => ! [I: $i] :
                ( ! [J: $i] :
                    ( ( hBOOL @ ( hAPP_state_bool @ ( hAPP_a2036067514e_bool @ D @ J ) @ H ) )
                   => ( hBOOL @ ( hAPP_state_bool @ ( hAPP_a2036067514e_bool @ F @ J ) @ I ) ) )
               => ( hBOOL @ ( hAPP_state_bool @ ( hAPP_a2036067514e_bool @ A @ G ) @ I ) ) ) )
       => ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hoare_2102800559rivs_a @ C ) @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ ( hoare_1916936827iple_a @ B @ E @ A ) ) @ bot_bo797238721a_bool ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_8_conseq12) ).

thf(5211,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hoare_2102800559rivs_a @ C ) @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ ( hoare_1916936827iple_a @ D @ E @ F ) ) @ bot_bo797238721a_bool ) ) )
     => ( ! [G: $i,H: $i] :
            ( ( hBOOL @ ( hAPP_state_bool @ ( hAPP_a2036067514e_bool @ B @ G ) @ H ) )
           => ! [I: $i] :
                ( ! [J: $i] :
                    ( ( hBOOL @ ( hAPP_state_bool @ ( hAPP_a2036067514e_bool @ D @ J ) @ H ) )
                   => ( hBOOL @ ( hAPP_state_bool @ ( hAPP_a2036067514e_bool @ F @ J ) @ I ) ) )
               => ( hBOOL @ ( hAPP_state_bool @ ( hAPP_a2036067514e_bool @ A @ G ) @ I ) ) ) )
       => ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hoare_2102800559rivs_a @ C ) @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ ( hoare_1916936827iple_a @ B @ E @ A ) ) @ bot_bo797238721a_bool ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1107]) ).

thf(1088,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_f1767618879l_bool @ ord_le1523781517a_bool @ A ) @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ B ) @ C ) ) )
    <=> ( ( ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ B ) @ 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 @ B ) @ bot_bo797238721a_bool ) ) ) @ C ) ) )
        & ( ~ ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ B ) @ A ) )
         => ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_f1767618879l_bool @ ord_le1523781517a_bool @ A ) @ C ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_909_subset__insert__iff) ).

thf(5128,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_f1767618879l_bool @ ord_le1523781517a_bool @ A ) @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ B ) @ C ) ) )
       => ( ( ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ B ) @ 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 @ B ) @ bot_bo797238721a_bool ) ) ) @ C ) ) )
          & ( ~ ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ B ) @ A ) )
           => ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_f1767618879l_bool @ ord_le1523781517a_bool @ A ) @ C ) ) ) ) )
      & ( ( ( ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ B ) @ 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 @ B ) @ bot_bo797238721a_bool ) ) ) @ C ) ) )
          & ( ~ ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ B ) @ A ) )
           => ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_f1767618879l_bool @ ord_le1523781517a_bool @ A ) @ C ) ) ) )
       => ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_f1767618879l_bool @ ord_le1523781517a_bool @ A ) @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ B ) @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1088]) ).

thf(88,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP_int_bool @ ( finite_fold1Set_int @ A @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ B ) @ C ) ) @ D ) )
     => ~ ! [E: $i,F: $i] :
            ( ( ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ B ) @ C )
              = ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ E ) @ F ) )
           => ( ( hBOOL @ ( hAPP_int_bool @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ ( finite772772422nt_int @ A ) @ E ) @ F ) @ D ) )
             => ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ E ) @ F ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_310_insert__fold1SetE) ).

thf(1444,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP_int_bool @ ( finite_fold1Set_int @ A @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ B ) @ C ) ) @ D ) )
     => ~ ! [E: $i,F: $i] :
            ( ( ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ B ) @ C )
              = ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ E ) @ F ) )
           => ( ( hBOOL @ ( hAPP_int_bool @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ ( finite772772422nt_int @ A ) @ E ) @ F ) @ D ) )
             => ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ E ) @ F ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[88]) ).

thf(867,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_f103356543l_bool @ ord_le1568362934t_bool @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ A ) @ B ) ) @ C ) )
    <=> ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ A ) @ C ) )
        & ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_f103356543l_bool @ ord_le1568362934t_bool @ B ) @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_866_insert__subset) ).

thf(4306,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_f103356543l_bool @ ord_le1568362934t_bool @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ A ) @ B ) ) @ C ) )
       => ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ A ) @ C ) )
          & ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_f103356543l_bool @ ord_le1568362934t_bool @ B ) @ C ) ) ) )
      & ( ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ A ) @ C ) )
          & ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_f103356543l_bool @ ord_le1568362934t_bool @ B ) @ C ) ) )
       => ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_f103356543l_bool @ ord_le1568362934t_bool @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ A ) @ B ) ) @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[867]) ).

thf(453,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f54304608l_bool @ finite_finite_nat @ C ) )
     => ( ~ ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ B ) @ C ) )
       => ( ( hAPP_f22106695ol_nat @ ( big_co1705425894at_nat @ A ) @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ B ) @ C ) )
          = ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ ( hAPP_nat_nat @ A @ B ) ) @ ( hAPP_f22106695ol_nat @ ( big_co1705425894at_nat @ A ) @ C ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_556_setprod_Oinsert) ).

thf(2779,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f54304608l_bool @ finite_finite_nat @ C ) )
     => ( ~ ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ B ) @ C ) )
       => ( ( hAPP_f22106695ol_nat @ ( big_co1705425894at_nat @ A ) @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ B ) @ C ) )
          = ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ ( hAPP_nat_nat @ A @ B ) ) @ ( hAPP_f22106695ol_nat @ ( big_co1705425894at_nat @ A ) @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[453]) ).

thf(13,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( B = C )
     => ( ( hBOOL @ ( hAPP_int_bool @ ( hAPP_i1948725293t_bool @ ord_less_eq_int @ C ) @ A ) )
       => ( hBOOL @ ( hAPP_int_bool @ ( hAPP_i1948725293t_bool @ ord_less_eq_int @ B ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_770_ord__eq__le__trans) ).

thf(1161,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( B = C )
     => ( ( hBOOL @ ( hAPP_int_bool @ ( hAPP_i1948725293t_bool @ ord_less_eq_int @ C ) @ A ) )
       => ( hBOOL @ ( hAPP_int_bool @ ( hAPP_i1948725293t_bool @ ord_less_eq_int @ B ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[13]) ).

thf(559,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f1695230391l_bool @ finite1738664244iple_a @ C ) )
     => ( ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ B ) @ C ) )
       => ( ( hAPP_f2097660464ol_nat @ ( hAPP_f1371399797ol_nat @ big_co1340561246_a_nat @ A ) @ C )
          = ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ ( hAPP_H592031934_a_nat @ A @ B ) ) @ ( hAPP_f2097660464ol_nat @ ( hAPP_f1371399797ol_nat @ big_co1340561246_a_nat @ A ) @ ( hAPP_f20753329a_bool @ ( hAPP_f1706700729a_bool @ minus_85316870a_bool @ C ) @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ B ) @ bot_bo797238721a_bool ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_618_setsum__diff1_H) ).

thf(3181,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f1695230391l_bool @ finite1738664244iple_a @ C ) )
     => ( ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ B ) @ C ) )
       => ( ( hAPP_f2097660464ol_nat @ ( hAPP_f1371399797ol_nat @ big_co1340561246_a_nat @ A ) @ C )
          = ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ ( hAPP_H592031934_a_nat @ A @ B ) ) @ ( hAPP_f2097660464ol_nat @ ( hAPP_f1371399797ol_nat @ big_co1340561246_a_nat @ A ) @ ( hAPP_f20753329a_bool @ ( hAPP_f1706700729a_bool @ minus_85316870a_bool @ C ) @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ B ) @ bot_bo797238721a_bool ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[559]) ).

thf(1009,axiom,
    ! [A: $i] :
      ( ? [B: $i] : ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ B ) @ A ) )
    <=> ( A != bot_bo797238721a_bool ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_42_ex__in__conv) ).

thf(4855,plain,
    ! [A: $i] :
      ( ( ? [B: $i] : ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ B ) @ A ) )
       => ( A != bot_bo797238721a_bool ) )
      & ( ( A != bot_bo797238721a_bool )
       => ? [B: $i] : ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ B ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1009]) ).

thf(603,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ B ) @ C ) )
     => ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ ( hAPP_H592031934_a_nat @ A @ B ) ) @ ( image_48635758_a_nat @ A @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_178_imageI) ).

thf(3324,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ B ) @ C ) )
     => ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ ( hAPP_H592031934_a_nat @ A @ B ) ) @ ( image_48635758_a_nat @ A @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[603]) ).

thf(952,axiom,
    ! [A: $i] :
      ( ( bot_bo797238721a_bool
        = ( hAPP_f20753329a_bool @ collec351493750iple_a @ A ) )
    <=> ! [B: $i] :
          ~ ( hBOOL @ ( hAPP_H1037229737a_bool @ A @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_39_empty__Collect__eq) ).

thf(4649,plain,
    ! [A: $i] :
      ( ( ( bot_bo797238721a_bool
          = ( hAPP_f20753329a_bool @ collec351493750iple_a @ A ) )
       => ! [B: $i] :
            ~ ( hBOOL @ ( hAPP_H1037229737a_bool @ A @ B ) ) )
      & ( ! [B: $i] :
            ~ ( hBOOL @ ( hAPP_H1037229737a_bool @ A @ B ) )
       => ( bot_bo797238721a_bool
          = ( hAPP_f20753329a_bool @ collec351493750iple_a @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[952]) ).

thf(1096,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP_f20753329a_bool @ ( hAPP_f1706700729a_bool @ ( hAPP_f957014749a_bool @ cOMBC_288844080a_bool @ A ) @ B ) @ C )
      = ( hAPP_f20753329a_bool @ ( hAPP_f1706700729a_bool @ A @ C ) @ B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',help_COMBC_1_1_COMBC_000tc__fun_Itc__Hoare____Mirabelle____uwgpyvfjxg__Otriple_I_055) ).

thf(5164,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP_f20753329a_bool @ ( hAPP_f1706700729a_bool @ ( hAPP_f957014749a_bool @ cOMBC_288844080a_bool @ A ) @ B ) @ C )
      = ( hAPP_f20753329a_bool @ ( hAPP_f1706700729a_bool @ A @ C ) @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1096]) ).

thf(1095,axiom,
    ( bot_bot_fun_nat_bool
    = ( hAPP_f800510211t_bool @ collect_nat @ ( hAPP_b1013836512t_bool @ cOMBK_bool_nat @ fFalse ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_50_empty__def) ).

thf(5162,plain,
    ( bot_bot_fun_nat_bool
    = ( hAPP_f800510211t_bool @ collect_nat @ ( hAPP_b1013836512t_bool @ cOMBK_bool_nat @ fFalse ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1095]) ).

thf(52,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_int_bool @ ( hAPP_i1948725293t_bool @ ord_less_eq_int @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ A ) @ B ) ) @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ A ) @ C ) ) )
    <=> ( hBOOL @ ( hAPP_int_bool @ ( hAPP_i1948725293t_bool @ ord_less_eq_int @ B ) @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_828_add__le__cancel__left) ).

thf(1308,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( hBOOL @ ( hAPP_int_bool @ ( hAPP_i1948725293t_bool @ ord_less_eq_int @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ A ) @ B ) ) @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ A ) @ C ) ) )
       => ( hBOOL @ ( hAPP_int_bool @ ( hAPP_i1948725293t_bool @ ord_less_eq_int @ B ) @ C ) ) )
      & ( ( hBOOL @ ( hAPP_int_bool @ ( hAPP_i1948725293t_bool @ ord_less_eq_int @ B ) @ C ) )
       => ( hBOOL @ ( hAPP_int_bool @ ( hAPP_i1948725293t_bool @ ord_less_eq_int @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ A ) @ B ) ) @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ A ) @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[52]) ).

thf(764,axiom,
    ! [A: $i,B: $i] :
      ( ( 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 ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_790_equalityE) ).

thf(3896,plain,
    ! [A: $i,B: $i] :
      ( ( 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 ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[764]) ).

thf(656,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f1695230391l_bool @ finite1738664244iple_a @ C ) )
     => ( ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ B ) @ C ) )
       => ( ( hAPP_f587450508ol_int @ ( hAPP_f1544320301ol_int @ big_co1977834938_a_int @ A ) @ C )
          = ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ ( hAPP_H1229305626_a_int @ A @ B ) ) @ ( hAPP_f587450508ol_int @ ( hAPP_f1544320301ol_int @ big_co1977834938_a_int @ A ) @ ( hAPP_f20753329a_bool @ ( hAPP_f1706700729a_bool @ minus_85316870a_bool @ C ) @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ B ) @ bot_bo797238721a_bool ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_610_setsum_Oremove) ).

thf(3529,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f1695230391l_bool @ finite1738664244iple_a @ C ) )
     => ( ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ B ) @ C ) )
       => ( ( hAPP_f587450508ol_int @ ( hAPP_f1544320301ol_int @ big_co1977834938_a_int @ A ) @ C )
          = ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ ( hAPP_H1229305626_a_int @ A @ B ) ) @ ( hAPP_f587450508ol_int @ ( hAPP_f1544320301ol_int @ big_co1977834938_a_int @ A ) @ ( hAPP_f20753329a_bool @ ( hAPP_f1706700729a_bool @ minus_85316870a_bool @ C ) @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ B ) @ bot_bo797238721a_bool ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[656]) ).

thf(22,axiom,
    ! [A: $i,B: $i] : ( hBOOL @ ( hAPP_int_bool @ ( hAPP_i1948725293t_bool @ ord_less_eq_int @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ ord_min_int @ A ) @ B ) ) @ B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_985_min__max_Oinf__le2) ).

thf(1193,plain,
    ! [A: $i,B: $i] : ( hBOOL @ ( hAPP_int_bool @ ( hAPP_i1948725293t_bool @ ord_less_eq_int @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ ord_min_int @ A ) @ B ) ) @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[22]) ).

thf(731,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f1695230391l_bool @ finite1738664244iple_a @ C ) )
     => ( ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_f1767618879l_bool @ ord_le1523781517a_bool @ C ) @ ( image_533441733iple_a @ A @ B ) ) )
       => ? [D: $i] :
            ( ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_f1767618879l_bool @ ord_le1523781517a_bool @ D ) @ B ) )
            & ( hBOOL @ ( hAPP_f1695230391l_bool @ finite1738664244iple_a @ D ) )
            & ( C
              = ( image_533441733iple_a @ A @ D ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_920_finite__subset__image) ).

thf(3793,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f1695230391l_bool @ finite1738664244iple_a @ C ) )
     => ( ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_f1767618879l_bool @ ord_le1523781517a_bool @ C ) @ ( image_533441733iple_a @ A @ B ) ) )
       => ? [D: $i] :
            ( ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_f1767618879l_bool @ ord_le1523781517a_bool @ D ) @ B ) )
            & ( hBOOL @ ( hAPP_f1695230391l_bool @ finite1738664244iple_a @ D ) )
            & ( C
              = ( image_533441733iple_a @ A @ D ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[731]) ).

thf(424,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ B ) @ C ) )
     => ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ B ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ C ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_1056_termination__basic__simps_I3_J) ).

thf(2680,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ B ) @ C ) )
     => ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ B ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ C ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[424]) ).

thf(133,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP_f800510211t_bool @ ( hAPP_f1730770594t_bool @ minus_2067140911t_bool @ A ) @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ B ) @ C ) )
      = ( 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 @ B ) @ bot_bot_fun_nat_bool ) ) ) @ C ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_414_Diff__insert2) ).

thf(1607,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP_f800510211t_bool @ ( hAPP_f1730770594t_bool @ minus_2067140911t_bool @ A ) @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ B ) @ C ) )
      = ( 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 @ B ) @ bot_bot_fun_nat_bool ) ) ) @ C ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[133]) ).

thf(889,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ C ) @ D ) )
     => ( ( A
          = ( hAPP_H592031934_a_nat @ B @ C ) )
       => ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ A ) @ ( image_48635758_a_nat @ B @ D ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_170_rev__image__eqI) ).

thf(4404,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ C ) @ D ) )
     => ( ( A
          = ( hAPP_H592031934_a_nat @ B @ C ) )
       => ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ A ) @ ( image_48635758_a_nat @ B @ D ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[889]) ).

thf(612,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( finite1956789438t_bool @ C ) )
     => ( ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ C @ A ) @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ C @ A ) @ B ) )
        = ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ C @ A ) @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_453_comp__fun__idem_Ofun__left__idem) ).

thf(3351,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( finite1956789438t_bool @ C ) )
     => ( ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ C @ A ) @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ C @ A ) @ B ) )
        = ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ C @ A ) @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[612]) ).

thf(103,axiom,
    ! [A: $i,B: $i] :
      ( ( B != bot_bot_fun_int_bool )
     => ( ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_f284875647l_bool @ ord_le951220754t_bool @ B ) @ A ) )
       => ( ( hBOOL @ ( hAPP_f448129468l_bool @ finite_finite_int @ A ) )
         => ( hBOOL @ ( hAPP_int_bool @ ( hAPP_i1948725293t_bool @ ord_less_eq_int @ ( hAPP_f1594865479ol_int @ ( finite_fold1_int @ ord_min_int ) @ A ) ) @ ( hAPP_f1594865479ol_int @ ( finite_fold1_int @ ord_min_int ) @ B ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_936_fold1__antimono) ).

thf(1513,plain,
    ! [A: $i,B: $i] :
      ( ( B != bot_bot_fun_int_bool )
     => ( ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_f284875647l_bool @ ord_le951220754t_bool @ B ) @ A ) )
       => ( ( hBOOL @ ( hAPP_f448129468l_bool @ finite_finite_int @ A ) )
         => ( hBOOL @ ( hAPP_int_bool @ ( hAPP_i1948725293t_bool @ ord_less_eq_int @ ( hAPP_f1594865479ol_int @ ( finite_fold1_int @ ord_min_int ) @ A ) ) @ ( hAPP_f1594865479ol_int @ ( finite_fold1_int @ ord_min_int ) @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[103]) ).

thf(1012,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_f1767618879l_bool @ ord_le1523781517a_bool @ B ) @ C ) )
     => ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_f1767618879l_bool @ ord_le1523781517a_bool @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ A ) @ B ) ) @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ A ) @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_859_insert__mono) ).

thf(4869,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_f1767618879l_bool @ ord_le1523781517a_bool @ B ) @ C ) )
     => ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_f1767618879l_bool @ ord_le1523781517a_bool @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ A ) @ B ) ) @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ A ) @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1012]) ).

thf(881,axiom,
    ! [A: $i] :
      ( ( hBOOL @ ( hAPP_bool_bool @ ( hAPP_b589554111l_bool @ ord_less_eq_bool @ A ) @ bot_bot_bool ) )
     => ( ( hBOOL @ A )
      <=> ( hBOOL @ bot_bot_bool ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_832_le__bot) ).

thf(4372,plain,
    ! [A: $i] :
      ( ( hBOOL @ ( hAPP_bool_bool @ ( hAPP_b589554111l_bool @ ord_less_eq_bool @ A ) @ bot_bot_bool ) )
     => ( ( ( hBOOL @ A )
         => ( hBOOL @ bot_bot_bool ) )
        & ( ( hBOOL @ bot_bot_bool )
         => ( hBOOL @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[881]) ).

thf(721,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f1695230391l_bool @ finite1738664244iple_a @ C ) )
     => ( ~ ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ B ) @ C ) )
       => ( ( hAPP_f587450508ol_int @ ( hAPP_f1544320301ol_int @ big_co1977834938_a_int @ A ) @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ B ) @ C ) )
          = ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ ( hAPP_H1229305626_a_int @ A @ B ) ) @ ( hAPP_f587450508ol_int @ ( hAPP_f1544320301ol_int @ big_co1977834938_a_int @ A ) @ C ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_635_setsum__insert) ).

thf(3761,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f1695230391l_bool @ finite1738664244iple_a @ C ) )
     => ( ~ ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ B ) @ C ) )
       => ( ( hAPP_f587450508ol_int @ ( hAPP_f1544320301ol_int @ big_co1977834938_a_int @ A ) @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ B ) @ C ) )
          = ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ ( hAPP_H1229305626_a_int @ A @ B ) ) @ ( hAPP_f587450508ol_int @ ( hAPP_f1544320301ol_int @ big_co1977834938_a_int @ A ) @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[721]) ).

thf(342,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ A ) @ B ) ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ C ) @ D ) )
      = ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ A ) @ C ) ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ B ) @ D ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_659_comm__semiring__1__class_Onormalizing__semiring__rules_I20_J) ).

thf(2374,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ A ) @ B ) ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ C ) @ D ) )
      = ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ A ) @ C ) ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ B ) @ D ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[342]) ).

thf(324,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ A ) @ ( image_1844722432iple_a @ B @ C ) ) )
     => ~ ! [D: $i] :
            ( ( A
              = ( hAPP_i240634960iple_a @ B @ D ) )
           => ~ ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ D ) @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_245_imageE) ).

thf(2302,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ A ) @ ( image_1844722432iple_a @ B @ C ) ) )
     => ~ ! [D: $i] :
            ( ( A
              = ( hAPP_i240634960iple_a @ B @ D ) )
           => ~ ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ D ) @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[324]) ).

thf(707,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ A ) @ B ) ) @ C )
      = ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ A ) @ C ) ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ B ) @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_1053_add__mult__distrib) ).

thf(3705,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ A ) @ B ) ) @ C )
      = ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ A ) @ C ) ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ B ) @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[707]) ).

thf(507,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( 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 @ B ) @ C ) ) ) )
    <=> ( hBOOL @ ( hAPP_f448129468l_bool @ finite_finite_int @ ( hAPP_f1805168059t_bool @ ( hAPP_f1223193598t_bool @ minus_1449998731t_bool @ A ) @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_428_finite__Diff__insert) ).

thf(2983,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( 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 @ B ) @ C ) ) ) )
       => ( hBOOL @ ( hAPP_f448129468l_bool @ finite_finite_int @ ( hAPP_f1805168059t_bool @ ( hAPP_f1223193598t_bool @ minus_1449998731t_bool @ A ) @ C ) ) ) )
      & ( ( hBOOL @ ( hAPP_f448129468l_bool @ finite_finite_int @ ( hAPP_f1805168059t_bool @ ( hAPP_f1223193598t_bool @ minus_1449998731t_bool @ A ) @ C ) ) )
       => ( 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 @ B ) @ C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[507]) ).

thf(1000,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_f1767618879l_bool @ ord_le1523781517a_bool @ ( hAPP_f20753329a_bool @ ( hAPP_f247708275a_bool @ cOMBC_1519416966l_bool @ member1797258804iple_a ) @ A ) ) @ ( hAPP_f20753329a_bool @ ( hAPP_f247708275a_bool @ cOMBC_1519416966l_bool @ member1797258804iple_a ) @ B ) ) )
    <=> ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_f1767618879l_bool @ ord_le1523781517a_bool @ A ) @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_733_pred__subset__eq) ).

thf(4811,plain,
    ! [A: $i,B: $i] :
      ( ( ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_f1767618879l_bool @ ord_le1523781517a_bool @ ( hAPP_f20753329a_bool @ ( hAPP_f247708275a_bool @ cOMBC_1519416966l_bool @ member1797258804iple_a ) @ A ) ) @ ( hAPP_f20753329a_bool @ ( hAPP_f247708275a_bool @ cOMBC_1519416966l_bool @ member1797258804iple_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 ) @ B ) )
       => ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_f1767618879l_bool @ ord_le1523781517a_bool @ ( hAPP_f20753329a_bool @ ( hAPP_f247708275a_bool @ cOMBC_1519416966l_bool @ member1797258804iple_a ) @ A ) ) @ ( hAPP_f20753329a_bool @ ( hAPP_f247708275a_bool @ cOMBC_1519416966l_bool @ member1797258804iple_a ) @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1000]) ).

thf(583,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ A ) @ B ) )
     => ? [C: $i] :
          ( ( B
            = ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ A ) @ C ) )
          & ~ ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ A ) @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_117_mk__disjoint__insert) ).

thf(3264,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ A ) @ B ) )
     => ? [C: $i] :
          ( ( B
            = ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ A ) @ C ) )
          & ~ ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ A ) @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[583]) ).

thf(917,axiom,
    ! [A: $i,B: $i] : ( is_bool @ ( big_se1603268663iple_a @ A @ B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',gsy_c_Big__Operators_Osemilattice__big_000tc__Hoare____Mirabelle____uwgpyvfjxg__) ).

thf(4523,plain,
    ! [A: $i,B: $i] : ( is_bool @ ( big_se1603268663iple_a @ A @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[917]) ).

thf(1007,axiom,
    ! [A: $i] :
      ( ( hBOOL @ ( hAPP_H1037229737a_bool @ bot_bo797238721a_bool @ A ) )
    <=> ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ A ) @ bot_bo797238721a_bool ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_139_bot__empty__eq) ).

thf(4843,plain,
    ! [A: $i] :
      ( ( ( hBOOL @ ( hAPP_H1037229737a_bool @ bot_bo797238721a_bool @ A ) )
       => ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ A ) @ bot_bo797238721a_bool ) ) )
      & ( ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ A ) @ bot_bo797238721a_bool ) )
       => ( hBOOL @ ( hAPP_H1037229737a_bool @ bot_bo797238721a_bool @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1007]) ).

thf(1017,axiom,
    ! [A: $i,B: $i] :
      ( ( hAPP_f1693662087iple_a @ ( finite1465856449iple_a @ A ) @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ B ) @ bot_bo797238721a_bool ) )
      = B ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_144_fold1__singleton) ).

thf(4885,plain,
    ! [A: $i,B: $i] :
      ( ( hAPP_f1693662087iple_a @ ( finite1465856449iple_a @ A ) @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ B ) @ bot_bo797238721a_bool ) )
      = B ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1017]) ).

thf(565,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_f284875647l_bool @ ord_le951220754t_bool @ B ) @ C ) )
     => ( ( hBOOL @ ( hAPP_int_bool @ B @ A ) )
       => ( hBOOL @ ( hAPP_int_bool @ C @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_811_predicate1D) ).

thf(3203,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_f284875647l_bool @ ord_le951220754t_bool @ B ) @ C ) )
     => ( ( hBOOL @ ( hAPP_int_bool @ B @ A ) )
       => ( hBOOL @ ( hAPP_int_bool @ C @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[565]) ).

thf(789,axiom,
    ! [A: $i] :
      ( ( bot_bot_fun_nat_bool
        = ( hAPP_f800510211t_bool @ collect_nat @ A ) )
    <=> ! [B: $i] :
          ~ ( hBOOL @ ( hAPP_nat_bool @ A @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_41_empty__Collect__eq) ).

thf(3992,plain,
    ! [A: $i] :
      ( ( ( bot_bot_fun_nat_bool
          = ( hAPP_f800510211t_bool @ collect_nat @ A ) )
       => ! [B: $i] :
            ~ ( hBOOL @ ( hAPP_nat_bool @ A @ B ) ) )
      & ( ! [B: $i] :
            ~ ( hBOOL @ ( hAPP_nat_bool @ A @ B ) )
       => ( bot_bot_fun_nat_bool
          = ( hAPP_f800510211t_bool @ collect_nat @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[789]) ).

thf(216,axiom,
    ! [A: $i] :
      ( ! [B: $i] :
          ~ ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ B ) @ A ) )
     => ( A = bot_bot_fun_nat_bool ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_121_equals0I) ).

thf(1908,plain,
    ! [A: $i] :
      ( ! [B: $i] :
          ~ ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ B ) @ A ) )
     => ( A = bot_bot_fun_nat_bool ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[216]) ).

thf(694,axiom,
    ! [A: $i] :
      ( ( hBOOL @ ( hAPP_int_bool @ bot_bot_fun_int_bool @ A ) )
    <=> ( hBOOL @ bot_bot_bool ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_106_bot__apply) ).

thf(3658,plain,
    ! [A: $i] :
      ( ( ( hBOOL @ ( hAPP_int_bool @ bot_bot_fun_int_bool @ A ) )
       => ( hBOOL @ bot_bot_bool ) )
      & ( ( hBOOL @ bot_bot_bool )
       => ( hBOOL @ ( hAPP_int_bool @ bot_bot_fun_int_bool @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[694]) ).

thf(1102,axiom,
    ! [A: $i] :
      ( ( hBOOL @ ( hAPP_f1695230391l_bool @ finite1738664244iple_a @ A ) )
    <=> ( ( A = bot_bo797238721a_bool )
        | ? [B: $i,C: $i] :
            ( ( A
              = ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ C ) @ B ) )
            & ( hBOOL @ ( hAPP_f1695230391l_bool @ finite1738664244iple_a @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_315_finite_Osimps) ).

thf(5187,plain,
    ! [A: $i] :
      ( ( ( hBOOL @ ( hAPP_f1695230391l_bool @ finite1738664244iple_a @ A ) )
       => ( ( A = bot_bo797238721a_bool )
          | ? [B: $i,C: $i] :
              ( ( A
                = ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ C ) @ B ) )
              & ( hBOOL @ ( hAPP_f1695230391l_bool @ finite1738664244iple_a @ B ) ) ) ) )
      & ( ( ( A = bot_bo797238721a_bool )
          | ? [B: $i,C: $i] :
              ( ( A
                = ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ C ) @ B ) )
              & ( hBOOL @ ( hAPP_f1695230391l_bool @ finite1738664244iple_a @ B ) ) ) )
       => ( hBOOL @ ( hAPP_f1695230391l_bool @ finite1738664244iple_a @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1102]) ).

thf(298,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_nat @ B ) @ C ) )
     => ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ B ) @ A ) ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ C ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_1103_add__less__mono1) ).

thf(2218,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_nat @ B ) @ C ) )
     => ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ B ) @ A ) ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ C ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[298]) ).

thf(311,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_f103356543l_bool @ ord_le1568362934t_bool @ ( hAPP_n1699378549t_bool @ ord_lessThan_nat @ A ) ) @ ( hAPP_n1699378549t_bool @ ord_lessThan_nat @ B ) ) )
    <=> ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ A ) @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_816_lessThan__subset__iff) ).

thf(2258,plain,
    ! [A: $i,B: $i] :
      ( ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_f103356543l_bool @ ord_le1568362934t_bool @ ( hAPP_n1699378549t_bool @ ord_lessThan_nat @ A ) ) @ ( hAPP_n1699378549t_bool @ ord_lessThan_nat @ B ) ) )
       => ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ A ) @ B ) ) )
      & ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ A ) @ B ) )
       => ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_f103356543l_bool @ ord_le1568362934t_bool @ ( hAPP_n1699378549t_bool @ ord_lessThan_nat @ A ) ) @ ( hAPP_n1699378549t_bool @ ord_lessThan_nat @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[311]) ).

thf(1001,axiom,
    ! [A: $i] :
      ( ( hAPP_f1693662087iple_a @ the_el1519802624iple_a @ A )
      = ( hAPP_f1693662087iple_a @ the_Ho830102290iple_a @ ( hAPP_f1194757675a_bool @ ( hAPP_f1697223433a_bool @ cOMBB_1610959875iple_a @ ( hAPP_f1767618879l_bool @ fequal874423448a_bool @ A ) ) @ ( hAPP_f1181346091a_bool @ ( hAPP_f571476211a_bool @ cOMBC_1570214144a_bool @ insert956547291iple_a ) @ bot_bo797238721a_bool ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_130_the__elem__def) ).

thf(4816,plain,
    ! [A: $i] :
      ( ( hAPP_f1693662087iple_a @ the_el1519802624iple_a @ A )
      = ( hAPP_f1693662087iple_a @ the_Ho830102290iple_a @ ( hAPP_f1194757675a_bool @ ( hAPP_f1697223433a_bool @ cOMBB_1610959875iple_a @ ( hAPP_f1767618879l_bool @ fequal874423448a_bool @ A ) ) @ ( hAPP_f1181346091a_bool @ ( hAPP_f571476211a_bool @ cOMBC_1570214144a_bool @ insert956547291iple_a ) @ bot_bo797238721a_bool ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1001]) ).

thf(1026,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP_H1037229737a_bool @ ( finite1621280049iple_a @ A @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ B ) @ C ) ) @ D ) )
     => ~ ! [E: $i,F: $i] :
            ( ( ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ B ) @ C )
              = ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ E ) @ F ) )
           => ( ( hBOOL @ ( hAPP_H1037229737a_bool @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ ( finite90892550iple_a @ A ) @ E ) @ F ) @ D ) )
             => ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ E ) @ F ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_309_insert__fold1SetE) ).

thf(4917,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP_H1037229737a_bool @ ( finite1621280049iple_a @ A @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ B ) @ C ) ) @ D ) )
     => ~ ! [E: $i,F: $i] :
            ( ( ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ B ) @ C )
              = ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ E ) @ F ) )
           => ( ( hBOOL @ ( hAPP_H1037229737a_bool @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ ( finite90892550iple_a @ A ) @ E ) @ F ) @ D ) )
             => ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ E ) @ F ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1026]) ).

thf(123,axiom,
    ( ( hAPP_nat_int @ semiri1621563631at_int @ one_one_nat )
    = one_one_int ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_1069_int__1) ).

thf(1578,plain,
    ( ( hAPP_nat_int @ semiri1621563631at_int @ one_one_nat )
    = one_one_int ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[123]) ).

thf(904,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( C = D )
     => ( ! [E: $i] :
            ( ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ E ) @ D ) )
           => ( ( hAPP_H592031934_a_nat @ A @ E )
              = ( hAPP_H592031934_a_nat @ B @ E ) ) )
       => ( ( hAPP_f2097660464ol_nat @ ( hAPP_f1371399797ol_nat @ big_co1340561246_a_nat @ A ) @ C )
          = ( hAPP_f2097660464ol_nat @ ( hAPP_f1371399797ol_nat @ big_co1340561246_a_nat @ B ) @ D ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_711_setsum_OF__cong) ).

thf(4472,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( C = D )
     => ( ! [E: $i] :
            ( ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ E ) @ D ) )
           => ( ( hAPP_H592031934_a_nat @ A @ E )
              = ( hAPP_H592031934_a_nat @ B @ E ) ) )
       => ( ( hAPP_f2097660464ol_nat @ ( hAPP_f1371399797ol_nat @ big_co1340561246_a_nat @ A ) @ C )
          = ( hAPP_f2097660464ol_nat @ ( hAPP_f1371399797ol_nat @ big_co1340561246_a_nat @ B ) @ D ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[904]) ).

thf(152,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ B ) @ C ) )
     => ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ minus_minus_nat @ A ) @ C ) ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ minus_minus_nat @ A ) @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_1027_diff__le__mono2) ).

thf(1678,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ B ) @ C ) )
     => ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ minus_minus_nat @ A ) @ C ) ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ minus_minus_nat @ A ) @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[152]) ).

thf(729,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP_int_int @ ( hAPP_int_fun_int_int @ times_times_int @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ times_times_int @ A ) @ B ) ) @ C )
      = ( hAPP_int_int @ ( hAPP_int_fun_int_int @ times_times_int @ A ) @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ times_times_int @ B ) @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_1086_zmult__assoc) ).

thf(3785,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP_int_int @ ( hAPP_int_fun_int_int @ times_times_int @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ times_times_int @ A ) @ B ) ) @ C )
      = ( hAPP_int_int @ ( hAPP_int_fun_int_int @ times_times_int @ A ) @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ times_times_int @ B ) @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[729]) ).

thf(280,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_f284875647l_bool @ ord_le951220754t_bool @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ A ) @ B ) ) @ C ) )
    <=> ( ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ A ) @ C ) )
        & ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_f284875647l_bool @ ord_le951220754t_bool @ B ) @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_867_insert__subset) ).

thf(2155,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_f284875647l_bool @ ord_le951220754t_bool @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ A ) @ B ) ) @ C ) )
       => ( ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ A ) @ C ) )
          & ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_f284875647l_bool @ ord_le951220754t_bool @ B ) @ C ) ) ) )
      & ( ( ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ A ) @ C ) )
          & ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_f284875647l_bool @ ord_le951220754t_bool @ B ) @ C ) ) )
       => ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_f284875647l_bool @ ord_le951220754t_bool @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ A ) @ B ) ) @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[280]) ).

thf(310,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ B ) @ C ) )
     => ( ( image_int_int @ ( hAPP_int_fun_int_int @ cOMBK_int_int @ A ) @ C )
        = ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ A ) @ bot_bot_fun_int_bool ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_236_image__constant) ).

thf(2255,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ B ) @ C ) )
     => ( ( image_int_int @ ( hAPP_int_fun_int_int @ cOMBK_int_int @ A ) @ C )
        = ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ A ) @ bot_bot_fun_int_bool ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[310]) ).

thf(760,axiom,
    ! [A: $i,B: $i] :
      ( ~ ( hBOOL @ B )
      | ( hBOOL @ ( hAPP_bool_bool @ ( hAPP_b589554111l_bool @ fdisj @ B ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',help_fdisj_1_1_U) ).

thf(3882,plain,
    ! [A: $i,B: $i] :
      ( ~ ( hBOOL @ B )
      | ( hBOOL @ ( hAPP_bool_bool @ ( hAPP_b589554111l_bool @ fdisj @ B ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[760]) ).

thf(73,axiom,
    ! [A: $i,B: $i] :
      ( ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_nat @ A ) @ B ) )
        | ( A = B ) )
     => ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ A ) @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_1113_less__or__eq__imp__le) ).

thf(1395,plain,
    ! [A: $i,B: $i] :
      ( ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_nat @ A ) @ B ) )
        | ( A = B ) )
     => ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ A ) @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[73]) ).

thf(138,axiom,
    ! [A: $i,B: $i] :
      ( ( hAPP_H592031934_a_nat @ ( hAPP_n1653940209_a_nat @ cOMBK_1028443109iple_a @ A ) @ B )
      = A ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',help_COMBK_1_1_COMBK_000tc__Nat__Onat_000tc__Hoare____Mirabelle____uwgpyvfjxg__O) ).

thf(1625,plain,
    ! [A: $i,B: $i] :
      ( ( hAPP_H592031934_a_nat @ ( hAPP_n1653940209_a_nat @ cOMBK_1028443109iple_a @ A ) @ B )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[138]) ).

thf(127,axiom,
    ! [A: $i,B: $i,C: $i] : ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_f103356543l_bool @ ord_le1568362934t_bool @ ( hAPP_f800510211t_bool @ ( hAPP_f1730770594t_bool @ minus_2067140911t_bool @ ( image_nat_nat @ A @ B ) ) @ ( image_nat_nat @ A @ C ) ) ) @ ( image_nat_nat @ A @ ( hAPP_f800510211t_bool @ ( hAPP_f1730770594t_bool @ minus_2067140911t_bool @ B ) @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_901_image__diff__subset) ).

thf(1593,plain,
    ! [A: $i,B: $i,C: $i] : ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_f103356543l_bool @ ord_le1568362934t_bool @ ( hAPP_f800510211t_bool @ ( hAPP_f1730770594t_bool @ minus_2067140911t_bool @ ( image_nat_nat @ A @ B ) ) @ ( image_nat_nat @ A @ C ) ) ) @ ( image_nat_nat @ A @ ( hAPP_f800510211t_bool @ ( hAPP_f1730770594t_bool @ minus_2067140911t_bool @ B ) @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[127]) ).

thf(745,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( finite795500164em_nat @ C @ D ) )
     => ( ( hBOOL @ ( hAPP_f54304608l_bool @ finite_finite_nat @ B ) )
       => ( ( B != bot_bot_fun_nat_bool )
         => ( ( hAPP_f22106695ol_nat @ D @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ A ) @ B ) )
            = ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ C @ A ) @ ( hAPP_f22106695ol_nat @ D @ B ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_347_folding__one__idem_Oinsert__idem) ).

thf(3837,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( finite795500164em_nat @ C @ D ) )
     => ( ( hBOOL @ ( hAPP_f54304608l_bool @ finite_finite_nat @ B ) )
       => ( ( B != bot_bot_fun_nat_bool )
         => ( ( hAPP_f22106695ol_nat @ D @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ A ) @ B ) )
            = ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ C @ A ) @ ( hAPP_f22106695ol_nat @ D @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[745]) ).

thf(405,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f54304608l_bool @ finite_finite_nat @ C ) )
     => ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_f103356543l_bool @ ord_le1568362934t_bool @ C ) @ B ) )
       => ( ( hAPP_f22106695ol_nat @ ( hAPP_f782000547ol_nat @ big_co387207925at_nat @ A ) @ ( hAPP_f800510211t_bool @ ( hAPP_f1730770594t_bool @ minus_2067140911t_bool @ B ) @ C ) )
          = ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ minus_minus_nat @ ( hAPP_f22106695ol_nat @ ( hAPP_f782000547ol_nat @ big_co387207925at_nat @ A ) @ B ) ) @ ( hAPP_f22106695ol_nat @ ( hAPP_f782000547ol_nat @ big_co387207925at_nat @ A ) @ C ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_717_setsum__diff__nat) ).

thf(2604,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f54304608l_bool @ finite_finite_nat @ C ) )
     => ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_f103356543l_bool @ ord_le1568362934t_bool @ C ) @ B ) )
       => ( ( hAPP_f22106695ol_nat @ ( hAPP_f782000547ol_nat @ big_co387207925at_nat @ A ) @ ( hAPP_f800510211t_bool @ ( hAPP_f1730770594t_bool @ minus_2067140911t_bool @ B ) @ C ) )
          = ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ minus_minus_nat @ ( hAPP_f22106695ol_nat @ ( hAPP_f782000547ol_nat @ big_co387207925at_nat @ A ) @ B ) ) @ ( hAPP_f22106695ol_nat @ ( hAPP_f782000547ol_nat @ big_co387207925at_nat @ A ) @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[405]) ).

thf(441,axiom,
    ! [A: $i,B: $i] :
      ( ( A = B )
    <=> ( ( image_nat_int @ semiri1621563631at_int @ A )
        = ( image_nat_int @ semiri1621563631at_int @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_1156_transfer__nat__int__set__relations_I3_J) ).

thf(2736,plain,
    ! [A: $i,B: $i] :
      ( ( ( A = B )
       => ( ( image_nat_int @ semiri1621563631at_int @ A )
          = ( image_nat_int @ semiri1621563631at_int @ B ) ) )
      & ( ( ( image_nat_int @ semiri1621563631at_int @ A )
          = ( image_nat_int @ semiri1621563631at_int @ B ) )
       => ( A = B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[441]) ).

thf(235,axiom,
    ! [A: $i,B: $i] :
      ( ( ( hAPP_nat_int @ semiri1621563631at_int @ A )
        = ( hAPP_nat_int @ semiri1621563631at_int @ B ) )
    <=> ( A = B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_1077_int__int__eq) ).

thf(1986,plain,
    ! [A: $i,B: $i] :
      ( ( ( ( hAPP_nat_int @ semiri1621563631at_int @ A )
          = ( hAPP_nat_int @ semiri1621563631at_int @ B ) )
       => ( A = B ) )
      & ( ( A = B )
       => ( ( hAPP_nat_int @ semiri1621563631at_int @ A )
          = ( hAPP_nat_int @ semiri1621563631at_int @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[235]) ).

thf(739,axiom,
    ! [A: $i,B: $i] : ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ A ) @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ A ) @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_78_insertI1) ).

thf(3822,plain,
    ! [A: $i,B: $i] : ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ A ) @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ A ) @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[739]) ).

thf(445,axiom,
    ! [A: $i,B: $i,C: $i] : ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_f284875647l_bool @ ord_le951220754t_bool @ ( hAPP_f1805168059t_bool @ ( hAPP_f1223193598t_bool @ minus_1449998731t_bool @ ( image_int_int @ A @ B ) ) @ ( image_int_int @ A @ C ) ) ) @ ( image_int_int @ A @ ( hAPP_f1805168059t_bool @ ( hAPP_f1223193598t_bool @ minus_1449998731t_bool @ B ) @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_898_image__diff__subset) ).

thf(2751,plain,
    ! [A: $i,B: $i,C: $i] : ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_f284875647l_bool @ ord_le951220754t_bool @ ( hAPP_f1805168059t_bool @ ( hAPP_f1223193598t_bool @ minus_1449998731t_bool @ ( image_int_int @ A @ B ) ) @ ( image_int_int @ A @ C ) ) ) @ ( image_int_int @ A @ ( hAPP_f1805168059t_bool @ ( hAPP_f1223193598t_bool @ minus_1449998731t_bool @ B ) @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[445]) ).

thf(988,axiom,
    ! [A: $i,B: $i] :
      ( ~ ( hBOOL @ ( hAPP_bool_bool @ ( hAPP_b589554111l_bool @ fconj @ A ) @ B ) )
      | ( hBOOL @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',help_fconj_2_1_U) ).

thf(4777,plain,
    ! [A: $i,B: $i] :
      ( ~ ( hBOOL @ ( hAPP_bool_bool @ ( hAPP_b589554111l_bool @ fconj @ A ) @ B ) )
      | ( hBOOL @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[988]) ).

thf(948,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_H1037229737a_bool @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ A ) @ B ) @ C ) )
    <=> ( ( A = C )
        | ( hBOOL @ ( hAPP_H1037229737a_bool @ B @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_61_insert__code) ).

thf(4631,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( hBOOL @ ( hAPP_H1037229737a_bool @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ A ) @ B ) @ C ) )
       => ( ( A = C )
          | ( hBOOL @ ( hAPP_H1037229737a_bool @ B @ C ) ) ) )
      & ( ( ( A = C )
          | ( hBOOL @ ( hAPP_H1037229737a_bool @ B @ C ) ) )
       => ( hBOOL @ ( hAPP_H1037229737a_bool @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ A ) @ B ) @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[948]) ).

thf(443,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ B ) @ C ) )
     => ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ A ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ minus_minus_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ C ) @ A ) ) @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_1026_le__add__diff) ).

thf(2746,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ B ) @ C ) )
     => ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ A ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ minus_minus_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ C ) @ A ) ) @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[443]) ).

thf(642,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( finite2130160977at_nat @ D ) )
     => ( ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ D @ A ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ D @ B ) @ C ) )
        = ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ D @ B ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ D @ A ) @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_488_comp__fun__commute_Ofun__left__comm) ).

thf(3479,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( finite2130160977at_nat @ D ) )
     => ( ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ D @ A ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ D @ B ) @ C ) )
        = ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ D @ B ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ D @ A ) @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[642]) ).

thf(340,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ B ) @ C ) )
     => ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ A ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ minus_minus_nat @ C ) @ B ) ) )
      <=> ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ A ) @ B ) ) @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_1032_le__diff__conv2) ).

thf(2366,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ B ) @ C ) )
     => ( ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ A ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ minus_minus_nat @ C ) @ B ) ) )
         => ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ A ) @ B ) ) @ C ) ) )
        & ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ A ) @ B ) ) @ C ) )
         => ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ A ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ minus_minus_nat @ C ) @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[340]) ).

thf(435,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ C ) @ D ) )
     => ( ( A
          = ( hAPP_int_int @ B @ C ) )
       => ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ A ) @ ( image_int_int @ B @ D ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_171_rev__image__eqI) ).

thf(2714,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ C ) @ D ) )
     => ( ( A
          = ( hAPP_int_int @ B @ C ) )
       => ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ A ) @ ( image_int_int @ B @ D ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[435]) ).

thf(800,axiom,
    ! [A: $i,B: $i] :
      ( ( ( B = bot_bo797238721a_bool )
       => ( ( image_48635758_a_nat @ ( hAPP_n1653940209_a_nat @ cOMBK_1028443109iple_a @ A ) @ B )
          = bot_bot_fun_nat_bool ) )
      & ( ( B != bot_bo797238721a_bool )
       => ( ( image_48635758_a_nat @ ( hAPP_n1653940209_a_nat @ cOMBK_1028443109iple_a @ A ) @ B )
          = ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ A ) @ bot_bot_fun_nat_bool ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_149_image__constant__conv) ).

thf(4034,plain,
    ! [A: $i,B: $i] :
      ( ( ( B = bot_bo797238721a_bool )
       => ( ( image_48635758_a_nat @ ( hAPP_n1653940209_a_nat @ cOMBK_1028443109iple_a @ A ) @ B )
          = bot_bot_fun_nat_bool ) )
      & ( ( B != bot_bo797238721a_bool )
       => ( ( image_48635758_a_nat @ ( hAPP_n1653940209_a_nat @ cOMBK_1028443109iple_a @ A ) @ B )
          = ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ A ) @ bot_bot_fun_nat_bool ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[800]) ).

thf(1108,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP_f1695230391l_bool @ finite1738664244iple_a @ B ) )
     => ( hBOOL @ ( hAPP_f1695230391l_bool @ finite1738664244iple_a @ ( image_533441733iple_a @ A @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_285_finite__imageI) ).

thf(5215,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP_f1695230391l_bool @ finite1738664244iple_a @ B ) )
     => ( hBOOL @ ( hAPP_f1695230391l_bool @ finite1738664244iple_a @ ( image_533441733iple_a @ A @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1108]) ).

thf(252,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ B ) @ C ) )
       => ( ( hAPP_f1805168059t_bool @ ( hAPP_f1223193598t_bool @ minus_1449998731t_bool @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ B ) @ A ) ) @ C )
          = ( hAPP_f1805168059t_bool @ ( hAPP_f1223193598t_bool @ minus_1449998731t_bool @ A ) @ C ) ) )
      & ( ~ ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ B ) @ C ) )
       => ( ( hAPP_f1805168059t_bool @ ( hAPP_f1223193598t_bool @ minus_1449998731t_bool @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ B ) @ A ) ) @ C )
          = ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ B ) @ ( hAPP_f1805168059t_bool @ ( hAPP_f1223193598t_bool @ minus_1449998731t_bool @ A ) @ C ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_405_insert__Diff__if) ).

thf(2040,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ B ) @ C ) )
       => ( ( hAPP_f1805168059t_bool @ ( hAPP_f1223193598t_bool @ minus_1449998731t_bool @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ B ) @ A ) ) @ C )
          = ( hAPP_f1805168059t_bool @ ( hAPP_f1223193598t_bool @ minus_1449998731t_bool @ A ) @ C ) ) )
      & ( ~ ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ B ) @ C ) )
       => ( ( hAPP_f1805168059t_bool @ ( hAPP_f1223193598t_bool @ minus_1449998731t_bool @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ B ) @ A ) ) @ C )
          = ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ B ) @ ( hAPP_f1805168059t_bool @ ( hAPP_f1223193598t_bool @ minus_1449998731t_bool @ A ) @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[252]) ).

thf(14,axiom,
    ! [A: $i,B: $i] :
      ( ( A = B )
    <=> ( ( hBOOL @ ( hAPP_int_bool @ ( hAPP_i1948725293t_bool @ ord_less_eq_int @ A ) @ B ) )
        & ( hBOOL @ ( hAPP_int_bool @ ( hAPP_i1948725293t_bool @ ord_less_eq_int @ B ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_784_order__eq__iff) ).

thf(1165,plain,
    ! [A: $i,B: $i] :
      ( ( ( A = B )
       => ( ( hBOOL @ ( hAPP_int_bool @ ( hAPP_i1948725293t_bool @ ord_less_eq_int @ A ) @ B ) )
          & ( hBOOL @ ( hAPP_int_bool @ ( hAPP_i1948725293t_bool @ ord_less_eq_int @ B ) @ A ) ) ) )
      & ( ( ( hBOOL @ ( hAPP_int_bool @ ( hAPP_i1948725293t_bool @ ord_less_eq_int @ A ) @ B ) )
          & ( hBOOL @ ( hAPP_int_bool @ ( hAPP_i1948725293t_bool @ ord_less_eq_int @ B ) @ A ) ) )
       => ( A = B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[14]) ).

thf(747,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ B ) @ C ) )
     => ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ B ) @ A ) ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ C ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_995_mult__le__mono1) ).

thf(3843,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ B ) @ C ) )
     => ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ B ) @ A ) ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ C ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[747]) ).

thf(173,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ A ) @ ( hAPP_f800510211t_bool @ ( hAPP_f1730770594t_bool @ minus_2067140911t_bool @ B ) @ C ) ) )
     => ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ A ) @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_386_DiffD1) ).

thf(1742,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ A ) @ ( hAPP_f800510211t_bool @ ( hAPP_f1730770594t_bool @ minus_2067140911t_bool @ B ) @ C ) ) )
     => ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ A ) @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[173]) ).

thf(1094,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ A ) @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ B ) @ C ) ) )
     => ( ( A != B )
       => ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ A ) @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_10_insertE) ).

thf(5159,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ A ) @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ B ) @ C ) ) )
     => ( ( A != B )
       => ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ A ) @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1094]) ).

thf(875,axiom,
    ! [A: $i,B: $i] : ( is_bool @ ( hAPP_state_bool @ A @ B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',gsy_c_hAPP_000tc__Com__Ostate_000tc__HOL__Obool) ).

thf(4350,plain,
    ! [A: $i,B: $i] : ( is_bool @ ( hAPP_state_bool @ A @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[875]) ).

thf(1046,axiom,
    ( bot_bot_fun_int_bool
    = ( hAPP_f1805168059t_bool @ collect_int @ ( hAPP_b396694332t_bool @ cOMBK_bool_int @ fFalse ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_49_empty__def) ).

thf(4981,plain,
    ( bot_bot_fun_int_bool
    = ( hAPP_f1805168059t_bool @ collect_int @ ( hAPP_b396694332t_bool @ cOMBK_bool_int @ fFalse ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1046]) ).

thf(662,axiom,
    ! [A: $i] : ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_f284875647l_bool @ ord_le951220754t_bool @ A ) @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_718_order__refl) ).

thf(3560,plain,
    ! [A: $i] : ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_f284875647l_bool @ ord_le951220754t_bool @ A ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[662]) ).

thf(610,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f54304608l_bool @ finite_finite_nat @ C ) )
     => ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_f103356543l_bool @ ord_le1568362934t_bool @ A ) @ ( image_nat_nat @ B @ C ) ) )
       => ( hBOOL @ ( hAPP_f54304608l_bool @ finite_finite_nat @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_897_finite__surj) ).

thf(3347,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f54304608l_bool @ finite_finite_nat @ C ) )
     => ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_f103356543l_bool @ ord_le1568362934t_bool @ A ) @ ( image_nat_nat @ B @ C ) ) )
       => ( hBOOL @ ( hAPP_f54304608l_bool @ finite_finite_nat @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[610]) ).

thf(587,axiom,
    ! [A: $i,B: $i] : ( is_bool @ ( hAPP_f54304608l_bool @ A @ B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',gsy_c_hAPP_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_000tc__HOL__Obool) ).

thf(3277,plain,
    ! [A: $i,B: $i] : ( is_bool @ ( hAPP_f54304608l_bool @ A @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[587]) ).

thf(843,axiom,
    ! [A: $i,B: $i] :
      ( ( hAPP_nat_int @ semiri1621563631at_int @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ A ) @ B ) )
      = ( hAPP_int_int @ ( hAPP_int_fun_int_int @ times_times_int @ ( hAPP_nat_int @ semiri1621563631at_int @ A ) ) @ ( hAPP_nat_int @ semiri1621563631at_int @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_1068_int__mult) ).

thf(4210,plain,
    ! [A: $i,B: $i] :
      ( ( hAPP_nat_int @ semiri1621563631at_int @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ A ) @ B ) )
      = ( hAPP_int_int @ ( hAPP_int_fun_int_int @ times_times_int @ ( hAPP_nat_int @ semiri1621563631at_int @ A ) ) @ ( hAPP_nat_int @ semiri1621563631at_int @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[843]) ).

thf(1123,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f1695230391l_bool @ finite1738664244iple_a @ ( hAPP_f20753329a_bool @ ( hAPP_f1706700729a_bool @ minus_85316870a_bool @ A ) @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ B ) @ C ) ) ) )
    <=> ( hBOOL @ ( hAPP_f1695230391l_bool @ finite1738664244iple_a @ ( hAPP_f20753329a_bool @ ( hAPP_f1706700729a_bool @ minus_85316870a_bool @ A ) @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_427_finite__Diff__insert) ).

thf(5290,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( hBOOL @ ( hAPP_f1695230391l_bool @ finite1738664244iple_a @ ( hAPP_f20753329a_bool @ ( hAPP_f1706700729a_bool @ minus_85316870a_bool @ A ) @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ B ) @ C ) ) ) )
       => ( hBOOL @ ( hAPP_f1695230391l_bool @ finite1738664244iple_a @ ( hAPP_f20753329a_bool @ ( hAPP_f1706700729a_bool @ minus_85316870a_bool @ A ) @ C ) ) ) )
      & ( ( hBOOL @ ( hAPP_f1695230391l_bool @ finite1738664244iple_a @ ( hAPP_f20753329a_bool @ ( hAPP_f1706700729a_bool @ minus_85316870a_bool @ A ) @ C ) ) )
       => ( hBOOL @ ( hAPP_f1695230391l_bool @ finite1738664244iple_a @ ( hAPP_f20753329a_bool @ ( hAPP_f1706700729a_bool @ minus_85316870a_bool @ A ) @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ B ) @ C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1123]) ).

thf(815,axiom,
    ! [A: $i] : ( is_bool @ ( finite1803123436a_bool @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',gsy_c_Finite__Set_Ocomp__fun__idem_000tc__Hoare____Mirabelle____uwgpyvfjxg__Otri_002) ).

thf(4083,plain,
    ! [A: $i] : ( is_bool @ ( finite1803123436a_bool @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[815]) ).

thf(261,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( finite2130160977at_nat @ E ) )
     => ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ ( finite929467206at_nat @ E ) @ B ) @ C ) @ D ) )
       => ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ ( finite929467206at_nat @ E ) @ B ) @ C ) @ A ) )
         => ( A = D ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_536_comp__fun__commute_Ofold__graph__determ) ).

thf(2070,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( finite2130160977at_nat @ E ) )
     => ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ ( finite929467206at_nat @ E ) @ B ) @ C ) @ D ) )
       => ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ ( finite929467206at_nat @ E ) @ B ) @ C ) @ A ) )
         => ( A = D ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[261]) ).

thf(711,axiom,
    ! [A: $i,B: $i] :
      ( ( A = B )
     => ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_f103356543l_bool @ ord_le1568362934t_bool @ B ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_804_equalityD2) ).

thf(3716,plain,
    ! [A: $i,B: $i] :
      ( ( A = B )
     => ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_f103356543l_bool @ ord_le1568362934t_bool @ B ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[711]) ).

thf(501,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_int_bool @ ( finite_fold1Set_int @ A @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ B ) @ bot_bot_fun_int_bool ) ) @ C ) )
    <=> ( B = C ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_135_fold1Set__sing) ).

thf(2961,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( hBOOL @ ( hAPP_int_bool @ ( finite_fold1Set_int @ A @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ B ) @ bot_bot_fun_int_bool ) ) @ C ) )
       => ( B = C ) )
      & ( ( B = C )
       => ( hBOOL @ ( hAPP_int_bool @ ( finite_fold1Set_int @ A @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ B ) @ bot_bot_fun_int_bool ) ) @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[501]) ).

thf(24,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP_int_bool @ ( hAPP_i1948725293t_bool @ ord_less_eq_int @ A ) @ B ) )
      | ( hBOOL @ ( hAPP_int_bool @ ( hAPP_i1948725293t_bool @ ord_less_eq_int @ B ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_786_linorder__linear) ).

thf(1197,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP_int_bool @ ( hAPP_i1948725293t_bool @ ord_less_eq_int @ A ) @ B ) )
      | ( hBOOL @ ( hAPP_int_bool @ ( hAPP_i1948725293t_bool @ ord_less_eq_int @ B ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[24]) ).

thf(697,axiom,
    ! [A: $i,B: $i] :
      ( ( hAPP_int_int @ ( hAPP_int_fun_int_int @ times_times_int @ ( hAPP_nat_int @ semiri1621563631at_int @ A ) ) @ ( hAPP_nat_int @ semiri1621563631at_int @ B ) )
      = ( hAPP_nat_int @ semiri1621563631at_int @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ A ) @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_1165_Nat__Transfer_Otransfer__int__nat__functions_I2_J) ).

thf(3673,plain,
    ! [A: $i,B: $i] :
      ( ( hAPP_int_int @ ( hAPP_int_fun_int_int @ times_times_int @ ( hAPP_nat_int @ semiri1621563631at_int @ A ) ) @ ( hAPP_nat_int @ semiri1621563631at_int @ B ) )
      = ( hAPP_nat_int @ semiri1621563631at_int @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ A ) @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[697]) ).

thf(134,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( finite1321096241t_bool @ D ) )
     => ( ( hBOOL @ ( hAPP_f448129468l_bool @ finite_finite_int @ C ) )
       => ( ~ ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ B ) @ C ) )
         => ( ( hAPP_f1805168059t_bool @ ( finite1827710202t_bool @ D @ A ) @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ B ) @ C ) )
            = ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ D @ B ) @ ( hAPP_f1805168059t_bool @ ( finite1827710202t_bool @ D @ A ) @ C ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_516_comp__fun__commute_Ofold__insert) ).

thf(1610,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( finite1321096241t_bool @ D ) )
     => ( ( hBOOL @ ( hAPP_f448129468l_bool @ finite_finite_int @ C ) )
       => ( ~ ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ B ) @ C ) )
         => ( ( hAPP_f1805168059t_bool @ ( finite1827710202t_bool @ D @ A ) @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ B ) @ C ) )
            = ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ D @ B ) @ ( hAPP_f1805168059t_bool @ ( finite1827710202t_bool @ D @ A ) @ C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[134]) ).

thf(43,axiom,
    ! [A: $i] : ( hBOOL @ ( hAPP_int_bool @ ( hAPP_i1948725293t_bool @ ord_less_eq_int @ A ) @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_1082_zle__refl) ).

thf(1268,plain,
    ! [A: $i] : ( hBOOL @ ( hAPP_int_bool @ ( hAPP_i1948725293t_bool @ ord_less_eq_int @ A ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[43]) ).

thf(458,axiom,
    ! [A: $i] : ( is_bool @ ( finite1860950092at_nat @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',gsy_c_Finite__Set_Ocomp__fun__idem_000tc__Nat__Onat_000tc__Nat__Onat) ).

thf(2803,plain,
    ! [A: $i] : ( is_bool @ ( finite1860950092at_nat @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[458]) ).

thf(1089,axiom,
    ! [A: $i] :
      ( ( hAPP_f20753329a_bool @ collec351493750iple_a @ A )
      = A ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_199_Collect__def) ).

thf(5140,plain,
    ! [A: $i] :
      ( ( hAPP_f20753329a_bool @ collec351493750iple_a @ A )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1089]) ).

thf(369,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ B ) @ C ) )
     => ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ B ) @ A ) ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ C ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_1007_add__le__mono1) ).

thf(2467,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ B ) @ C ) )
     => ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ B ) @ A ) ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ C ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[369]) ).

thf(809,axiom,
    ! [A: $i] : ( is_bool @ ( finite58652534t_bool @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',gsy_c_Finite__Set_Ocomp__fun__idem_000tc__Int__Oint_000tc__fun_Itc__Int__Oint_Mt) ).

thf(4066,plain,
    ! [A: $i] : ( is_bool @ ( finite58652534t_bool @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[809]) ).

thf(784,axiom,
    ! [A: $i,B: $i] :
      ( ( is_bool @ A )
     => ( ( hAPP_nat_bool @ ( hAPP_b1013836512t_bool @ cOMBK_bool_nat @ A ) @ B )
        = A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',help_COMBK_1_1_COMBK_000tc__HOL__Obool_000tc__Nat__Onat_U) ).

thf(3965,plain,
    ! [A: $i,B: $i] :
      ( ( is_bool @ A )
     => ( ( hAPP_nat_bool @ ( hAPP_b1013836512t_bool @ cOMBK_bool_nat @ A ) @ B )
        = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[784]) ).

thf(379,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( finite1321096241t_bool @ D ) )
     => ( ( hBOOL @ ( hAPP_f448129468l_bool @ finite_finite_int @ C ) )
       => ( ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ D @ A ) @ ( hAPP_f1805168059t_bool @ ( finite1827710202t_bool @ D @ B ) @ C ) )
          = ( hAPP_f1805168059t_bool @ ( finite1827710202t_bool @ D @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ D @ A ) @ B ) ) @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_486_comp__fun__commute_Ofold__fun__comm) ).

thf(2507,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( finite1321096241t_bool @ D ) )
     => ( ( hBOOL @ ( hAPP_f448129468l_bool @ finite_finite_int @ C ) )
       => ( ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ D @ A ) @ ( hAPP_f1805168059t_bool @ ( finite1827710202t_bool @ D @ B ) @ C ) )
          = ( hAPP_f1805168059t_bool @ ( finite1827710202t_bool @ D @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ D @ A ) @ B ) ) @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[379]) ).

thf(621,axiom,
    ! [A: $i,B: $i] : ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_f284875647l_bool @ ord_le951220754t_bool @ ( hAPP_f1805168059t_bool @ ( hAPP_f1223193598t_bool @ minus_1449998731t_bool @ A ) @ B ) ) @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_886_Diff__subset) ).

thf(3392,plain,
    ! [A: $i,B: $i] : ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_f284875647l_bool @ ord_le951220754t_bool @ ( hAPP_f1805168059t_bool @ ( hAPP_f1223193598t_bool @ minus_1449998731t_bool @ A ) @ B ) ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[621]) ).

thf(220,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ B ) @ C ) )
     => ( ( ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ minus_minus_nat @ C ) @ B )
          = A )
      <=> ( C
          = ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ A ) @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_1034_le__imp__diff__is__add) ).

thf(1925,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ B ) @ C ) )
     => ( ( ( ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ minus_minus_nat @ C ) @ B )
            = A )
         => ( C
            = ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ A ) @ B ) ) )
        & ( ( C
            = ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ A ) @ B ) )
         => ( ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ minus_minus_nat @ C ) @ B )
            = A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[220]) ).

thf(1032,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ A ) @ B ) )
     => ( ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ A ) @ B )
        = B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_52_insert__absorb) ).

thf(4936,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ A ) @ B ) )
     => ( ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ A ) @ B )
        = B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1032]) ).

thf(647,axiom,
    ! [A: $i,B: $i] :
      ( skip
     != ( ass @ A @ B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_194_com_Osimps_I8_J) ).

thf(3494,plain,
    ! [A: $i,B: $i] :
      ( skip
     != ( ass @ A @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[647]) ).

thf(879,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ A ) @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ B ) @ bot_bot_fun_nat_bool ) ) )
    <=> ( A = B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_93_singleton__iff) ).

thf(4363,plain,
    ! [A: $i,B: $i] :
      ( ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ A ) @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ B ) @ bot_bot_fun_nat_bool ) ) )
       => ( A = B ) )
      & ( ( A = B )
       => ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ A ) @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ B ) @ bot_bot_fun_nat_bool ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[879]) ).

thf(1028,axiom,
    ! [A: $i] :
      ( ( hAPP_f20753329a_bool @ collec351493750iple_a @ ( hAPP_H426895267a_bool @ fequal963300192iple_a @ A ) )
      = ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ A ) @ bot_bo797238721a_bool ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_19_singleton__conv2) ).

thf(4923,plain,
    ! [A: $i] :
      ( ( hAPP_f20753329a_bool @ collec351493750iple_a @ ( hAPP_H426895267a_bool @ fequal963300192iple_a @ A ) )
      = ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ A ) @ bot_bo797238721a_bool ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1028]) ).

thf(578,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_H1037229737a_bool @ ( hAPP_f20753329a_bool @ ( hAPP_f1706700729a_bool @ minus_85316870a_bool @ A ) @ B ) @ C ) )
    <=> ( hBOOL @ ( hAPP_bool_bool @ ( hAPP_b589554111l_bool @ minus_minus_bool @ ( hAPP_H1037229737a_bool @ A @ C ) ) @ ( hAPP_H1037229737a_bool @ B @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_446_fun__diff__def) ).

thf(3244,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( hBOOL @ ( hAPP_H1037229737a_bool @ ( hAPP_f20753329a_bool @ ( hAPP_f1706700729a_bool @ minus_85316870a_bool @ A ) @ B ) @ C ) )
       => ( hBOOL @ ( hAPP_bool_bool @ ( hAPP_b589554111l_bool @ minus_minus_bool @ ( hAPP_H1037229737a_bool @ A @ C ) ) @ ( hAPP_H1037229737a_bool @ B @ C ) ) ) )
      & ( ( hBOOL @ ( hAPP_bool_bool @ ( hAPP_b589554111l_bool @ minus_minus_bool @ ( hAPP_H1037229737a_bool @ A @ C ) ) @ ( hAPP_H1037229737a_bool @ B @ C ) ) )
       => ( hBOOL @ ( hAPP_H1037229737a_bool @ ( hAPP_f20753329a_bool @ ( hAPP_f1706700729a_bool @ minus_85316870a_bool @ A ) @ B ) @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[578]) ).

thf(305,axiom,
    ! [A: $i,B: $i] :
      ( ( hAPP_f22106695ol_nat @ ( finite_fold_nat_nat @ A @ B ) @ bot_bot_fun_nat_bool )
      = B ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_503_fold__empty) ).

thf(2241,plain,
    ! [A: $i,B: $i] :
      ( ( hAPP_f22106695ol_nat @ ( finite_fold_nat_nat @ A @ B ) @ bot_bot_fun_nat_bool )
      = B ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[305]) ).

thf(1008,axiom,
    ! [A: $i,B: $i] :
      ( ( ( hBOOL @ ( hAPP_H1037229737a_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_H426895267a_bool @ ( hAPP_f920293029a_bool @ cOMBC_1940896922a_bool @ fequal963300192iple_a ) @ B ) ) ) @ A ) )
          = ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ B ) @ bot_bo797238721a_bool ) ) )
      & ( ~ ( hBOOL @ ( hAPP_H1037229737a_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_H426895267a_bool @ ( hAPP_f920293029a_bool @ cOMBC_1940896922a_bool @ fequal963300192iple_a ) @ B ) ) ) @ A ) )
          = bot_bo797238721a_bool ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_28_Collect__conv__if) ).

thf(4848,plain,
    ! [A: $i,B: $i] :
      ( ( ( hBOOL @ ( hAPP_H1037229737a_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_H426895267a_bool @ ( hAPP_f920293029a_bool @ cOMBC_1940896922a_bool @ fequal963300192iple_a ) @ B ) ) ) @ A ) )
          = ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ B ) @ bot_bo797238721a_bool ) ) )
      & ( ~ ( hBOOL @ ( hAPP_H1037229737a_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_H426895267a_bool @ ( hAPP_f920293029a_bool @ cOMBC_1940896922a_bool @ fequal963300192iple_a ) @ B ) ) ) @ A ) )
          = bot_bo797238721a_bool ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1008]) ).

thf(614,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_nat_bool @ B @ C ) )
     => ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_f103356543l_bool @ ord_le1568362934t_bool @ B ) @ A ) )
       => ( hBOOL @ ( hAPP_nat_bool @ A @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_814_rev__predicate1D) ).

thf(3362,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_nat_bool @ B @ C ) )
     => ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_f103356543l_bool @ ord_le1568362934t_bool @ B ) @ A ) )
       => ( hBOOL @ ( hAPP_nat_bool @ A @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[614]) ).

thf(460,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ minus_minus_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ minus_minus_nat @ A ) @ B ) ) @ C )
      = ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ minus_minus_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ minus_minus_nat @ A ) @ C ) ) @ B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_1019_diff__commute) ).

thf(2810,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ minus_minus_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ minus_minus_nat @ A ) @ B ) ) @ C )
      = ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ minus_minus_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ minus_minus_nat @ A ) @ C ) ) @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[460]) ).

thf(61,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ ord_min_nat @ A ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ ord_min_nat @ B ) @ C ) )
      = ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ ord_min_nat @ B ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ ord_min_nat @ A ) @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_956_min__max_Oinf_Oleft__commute) ).

thf(1352,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ ord_min_nat @ A ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ ord_min_nat @ B ) @ C ) )
      = ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ ord_min_nat @ B ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ ord_min_nat @ A ) @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[61]) ).

thf(543,axiom,
    ! [A: $i,B: $i] :
      ( ( 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 ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_791_equalityE) ).

thf(3128,plain,
    ! [A: $i,B: $i] :
      ( ( 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 ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[543]) ).

thf(199,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ A ) @ B ) )
     => ? [C: $i] :
          ( ( B
            = ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ A ) @ C ) )
          & ~ ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ A ) @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_119_mk__disjoint__insert) ).

thf(1829,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ A ) @ B ) )
     => ? [C: $i] :
          ( ( B
            = ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ A ) @ C ) )
          & ~ ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ A ) @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[199]) ).

thf(233,axiom,
    ! [A: $i] :
      ( ! [B: $i] :
          ~ ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ B ) @ A ) )
    <=> ( A = bot_bot_fun_int_bool ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_47_all__not__in__conv) ).

thf(1977,plain,
    ! [A: $i] :
      ( ( ! [B: $i] :
            ~ ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ B ) @ A ) )
       => ( A = bot_bot_fun_int_bool ) )
      & ( ( A = bot_bot_fun_int_bool )
       => ! [B: $i] :
            ~ ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ B ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[233]) ).

thf(787,axiom,
    ! [A: $i,B: $i] :
      ( ( one_one_nat
        = ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ A ) @ B ) )
    <=> ( ( A = one_one_nat )
        & ( B = one_one_nat ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_1062_nat__1__eq__mult__iff) ).

thf(3974,plain,
    ! [A: $i,B: $i] :
      ( ( ( one_one_nat
          = ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ A ) @ B ) )
       => ( ( A = one_one_nat )
          & ( B = one_one_nat ) ) )
      & ( ( ( A = one_one_nat )
          & ( B = one_one_nat ) )
       => ( one_one_nat
          = ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ A ) @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[787]) ).

thf(198,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ B ) @ C ) )
     => ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ B ) @ A ) )
       => ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ B ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ ord_min_nat @ C ) @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_970_min__max_Ole__infI) ).

thf(1827,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ B ) @ C ) )
     => ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ B ) @ A ) )
       => ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ B ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ ord_min_nat @ C ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[198]) ).

thf(42,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_int_bool @ ( hAPP_i1948725293t_bool @ ord_less_eq_int @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ ord_min_int @ A ) @ B ) ) @ C ) )
    <=> ( ( hBOOL @ ( hAPP_int_bool @ ( hAPP_i1948725293t_bool @ ord_less_eq_int @ A ) @ C ) )
        | ( hBOOL @ ( hAPP_int_bool @ ( hAPP_i1948725293t_bool @ ord_less_eq_int @ B ) @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_963_min__le__iff__disj) ).

thf(1261,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( hBOOL @ ( hAPP_int_bool @ ( hAPP_i1948725293t_bool @ ord_less_eq_int @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ ord_min_int @ A ) @ B ) ) @ C ) )
       => ( ( hBOOL @ ( hAPP_int_bool @ ( hAPP_i1948725293t_bool @ ord_less_eq_int @ A ) @ C ) )
          | ( hBOOL @ ( hAPP_int_bool @ ( hAPP_i1948725293t_bool @ ord_less_eq_int @ B ) @ C ) ) ) )
      & ( ( ( hBOOL @ ( hAPP_int_bool @ ( hAPP_i1948725293t_bool @ ord_less_eq_int @ A ) @ C ) )
          | ( hBOOL @ ( hAPP_int_bool @ ( hAPP_i1948725293t_bool @ ord_less_eq_int @ B ) @ C ) ) )
       => ( hBOOL @ ( hAPP_int_bool @ ( hAPP_i1948725293t_bool @ ord_less_eq_int @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ ord_min_int @ A ) @ B ) ) @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[42]) ).

thf(485,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ B ) @ C ) )
     => ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ B ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ A ) @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_1057_termination__basic__simps_I4_J) ).

thf(2898,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ B ) @ C ) )
     => ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ B ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ A ) @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[485]) ).

thf(666,axiom,
    ! [A: $i,B: $i] : ( is_bool @ ( hAPP_f1637334154l_bool @ A @ B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',gsy_c_hAPP_000tc__fun_Itc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_Mtc__HOL__Obool_) ).

thf(3570,plain,
    ! [A: $i,B: $i] : ( is_bool @ ( hAPP_f1637334154l_bool @ A @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[666]) ).

thf(111,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_f284875647l_bool @ ord_le951220754t_bool @ A ) @ B ) )
     => ( ( A != bot_bot_fun_int_bool )
       => ( ( hBOOL @ ( hAPP_f448129468l_bool @ finite_finite_int @ B ) )
         => ( hBOOL @ ( hAPP_int_bool @ ( hAPP_i1948725293t_bool @ ord_less_eq_int @ ( hAPP_f1594865479ol_int @ big_linorder_Min_int @ B ) ) @ ( hAPP_f1594865479ol_int @ big_linorder_Min_int @ A ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_943_Min__antimono) ).

thf(1537,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_f284875647l_bool @ ord_le951220754t_bool @ A ) @ B ) )
     => ( ( A != bot_bot_fun_int_bool )
       => ( ( hBOOL @ ( hAPP_f448129468l_bool @ finite_finite_int @ B ) )
         => ( hBOOL @ ( hAPP_int_bool @ ( hAPP_i1948725293t_bool @ ord_less_eq_int @ ( hAPP_f1594865479ol_int @ big_linorder_Min_int @ B ) ) @ ( hAPP_f1594865479ol_int @ big_linorder_Min_int @ A ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[111]) ).

thf(725,axiom,
    ! [A: $i] :
      ( ( hAPP_int_int @ ( hAPP_int_fun_int_int @ times_times_int @ A ) @ one_one_int )
      = A ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_1098_zmult__1__right) ).

thf(3772,plain,
    ! [A: $i] :
      ( ( hAPP_int_int @ ( hAPP_int_fun_int_int @ times_times_int @ A ) @ one_one_int )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[725]) ).

thf(1072,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hoare_2102800559rivs_a @ B ) @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ ( hoare_1916936827iple_a @ C @ D @ E ) ) @ bot_bo797238721a_bool ) ) )
     => ( ! [F: $i,G: $i] :
            ( ( hBOOL @ ( hAPP_state_bool @ ( hAPP_a2036067514e_bool @ A @ F ) @ G ) )
           => ( hBOOL @ ( hAPP_state_bool @ ( hAPP_a2036067514e_bool @ C @ F ) @ G ) ) )
       => ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hoare_2102800559rivs_a @ B ) @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ ( hoare_1916936827iple_a @ A @ D @ E ) ) @ bot_bo797238721a_bool ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_7_conseq1) ).

thf(5061,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hoare_2102800559rivs_a @ B ) @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ ( hoare_1916936827iple_a @ C @ D @ E ) ) @ bot_bo797238721a_bool ) ) )
     => ( ! [F: $i,G: $i] :
            ( ( hBOOL @ ( hAPP_state_bool @ ( hAPP_a2036067514e_bool @ A @ F ) @ G ) )
           => ( hBOOL @ ( hAPP_state_bool @ ( hAPP_a2036067514e_bool @ C @ F ) @ G ) ) )
       => ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hoare_2102800559rivs_a @ B ) @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ ( hoare_1916936827iple_a @ A @ D @ E ) ) @ bot_bo797238721a_bool ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1072]) ).

thf(115,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_nat @ A ) @ B ) )
     => ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ A ) @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_1123_termination__basic__simps_I5_J) ).

thf(1550,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_nat @ A ) @ B ) )
     => ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ A ) @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[115]) ).

thf(907,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f1695230391l_bool @ finite1738664244iple_a @ C ) )
     => ( ( hAPP_f2097660464ol_nat @ ( hAPP_f1371399797ol_nat @ big_co1340561246_a_nat @ A ) @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ B ) @ C ) )
        = ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ ( hAPP_H592031934_a_nat @ A @ B ) ) @ ( hAPP_f2097660464ol_nat @ ( hAPP_f1371399797ol_nat @ big_co1340561246_a_nat @ A ) @ ( hAPP_f20753329a_bool @ ( hAPP_f1706700729a_bool @ minus_85316870a_bool @ C ) @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ B ) @ bot_bo797238721a_bool ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_624_setsum_Oinsert__remove) ).

thf(4484,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f1695230391l_bool @ finite1738664244iple_a @ C ) )
     => ( ( hAPP_f2097660464ol_nat @ ( hAPP_f1371399797ol_nat @ big_co1340561246_a_nat @ A ) @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ B ) @ C ) )
        = ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ ( hAPP_H592031934_a_nat @ A @ B ) ) @ ( hAPP_f2097660464ol_nat @ ( hAPP_f1371399797ol_nat @ big_co1340561246_a_nat @ A ) @ ( hAPP_f20753329a_bool @ ( hAPP_f1706700729a_bool @ minus_85316870a_bool @ C ) @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ B ) @ bot_bo797238721a_bool ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[907]) ).

thf(645,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ B ) @ C ) )
     => ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ ( hAPP_H905846293iple_a @ A @ B ) ) @ ( image_533441733iple_a @ A @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_175_imageI) ).

thf(3489,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ B ) @ C ) )
     => ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ ( hAPP_H905846293iple_a @ A @ B ) ) @ ( image_533441733iple_a @ A @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[645]) ).

thf(762,axiom,
    ! [A: $i] :
      ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_f103356543l_bool @ ord_le1568362934t_bool @ A ) @ bot_bot_fun_nat_bool ) )
     => ( A = bot_bot_fun_nat_bool ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_833_le__bot) ).

thf(3886,plain,
    ! [A: $i] :
      ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_f103356543l_bool @ ord_le1568362934t_bool @ A ) @ bot_bot_fun_nat_bool ) )
     => ( A = bot_bot_fun_nat_bool ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[762]) ).

thf(1116,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( finite1803123436a_bool @ D ) )
     => ( ( hBOOL @ ( hAPP_f1695230391l_bool @ finite1738664244iple_a @ C ) )
       => ( ( hAPP_f20753329a_bool @ ( finite1215486064a_bool @ D @ A ) @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ B ) @ C ) )
          = ( hAPP_f20753329a_bool @ ( finite1215486064a_bool @ D @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ D @ B ) @ A ) ) @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_572_comp__fun__idem_Ofold__insert__idem2) ).

thf(5236,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( finite1803123436a_bool @ D ) )
     => ( ( hBOOL @ ( hAPP_f1695230391l_bool @ finite1738664244iple_a @ C ) )
       => ( ( hAPP_f20753329a_bool @ ( finite1215486064a_bool @ D @ A ) @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ B ) @ C ) )
          = ( hAPP_f20753329a_bool @ ( finite1215486064a_bool @ D @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ D @ B ) @ A ) ) @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1116]) ).

thf(129,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ B ) @ C ) )
     => ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ minus_minus_nat @ B ) @ A ) ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ minus_minus_nat @ C ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_1028_diff__le__mono) ).

thf(1597,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ B ) @ C ) )
     => ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ minus_minus_nat @ B ) @ A ) ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ minus_minus_nat @ C ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[129]) ).

thf(717,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ A ) @ B ) @ C ) )
    <=> ( ( A = C )
        | ( hBOOL @ ( hAPP_nat_bool @ B @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_60_insert__code) ).

thf(3740,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ A ) @ B ) @ C ) )
       => ( ( A = C )
          | ( hBOOL @ ( hAPP_nat_bool @ B @ C ) ) ) )
      & ( ( ( A = C )
          | ( hBOOL @ ( hAPP_nat_bool @ B @ C ) ) )
       => ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ A ) @ B ) @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[717]) ).

thf(84,axiom,
    ! [A: $i] :
      ~ ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_nat @ A ) @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_1122_less__not__refl) ).

thf(1431,plain,
    ! [A: $i] :
      ~ ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_nat @ A ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[84]) ).

thf(531,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ C ) @ D ) )
     => ( ( A
          = ( hAPP_n261501868iple_a @ B @ C ) )
       => ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ A ) @ ( image_1865589340iple_a @ B @ D ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_168_rev__image__eqI) ).

thf(3076,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ C ) @ D ) )
     => ( ( A
          = ( hAPP_n261501868iple_a @ B @ C ) )
       => ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ A ) @ ( image_1865589340iple_a @ B @ D ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[531]) ).

thf(541,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f1695230391l_bool @ finite1738664244iple_a @ C ) )
     => ( ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ B ) @ C ) )
       => ( ( hAPP_f587450508ol_int @ ( hAPP_f1544320301ol_int @ big_co1977834938_a_int @ A ) @ ( hAPP_f20753329a_bool @ ( hAPP_f1706700729a_bool @ minus_85316870a_bool @ C ) @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ B ) @ bot_bo797238721a_bool ) ) )
          = ( hAPP_int_int @ ( hAPP_int_fun_int_int @ minus_minus_int @ ( hAPP_f587450508ol_int @ ( hAPP_f1544320301ol_int @ big_co1977834938_a_int @ A ) @ C ) ) @ ( hAPP_H1229305626_a_int @ A @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_464_setsum__diff1__ring) ).

thf(3113,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f1695230391l_bool @ finite1738664244iple_a @ C ) )
     => ( ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ B ) @ C ) )
       => ( ( hAPP_f587450508ol_int @ ( hAPP_f1544320301ol_int @ big_co1977834938_a_int @ A ) @ ( hAPP_f20753329a_bool @ ( hAPP_f1706700729a_bool @ minus_85316870a_bool @ C ) @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ B ) @ bot_bo797238721a_bool ) ) )
          = ( hAPP_int_int @ ( hAPP_int_fun_int_int @ minus_minus_int @ ( hAPP_f587450508ol_int @ ( hAPP_f1544320301ol_int @ big_co1977834938_a_int @ A ) @ C ) ) @ ( hAPP_H1229305626_a_int @ A @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[541]) ).

thf(622,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ B ) @ C ) )
       => ( ( hAPP_f2097660464ol_nat @ ( hAPP_f1371399797ol_nat @ big_co1340561246_a_nat @ A ) @ ( hAPP_f20753329a_bool @ ( hAPP_f1706700729a_bool @ minus_85316870a_bool @ C ) @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ B ) @ bot_bo797238721a_bool ) ) )
          = ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ minus_minus_nat @ ( hAPP_f2097660464ol_nat @ ( hAPP_f1371399797ol_nat @ big_co1340561246_a_nat @ A ) @ C ) ) @ ( hAPP_H592031934_a_nat @ A @ B ) ) ) )
      & ( ~ ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ B ) @ C ) )
       => ( ( hAPP_f2097660464ol_nat @ ( hAPP_f1371399797ol_nat @ big_co1340561246_a_nat @ A ) @ ( hAPP_f20753329a_bool @ ( hAPP_f1706700729a_bool @ minus_85316870a_bool @ C ) @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ B ) @ bot_bo797238721a_bool ) ) )
          = ( hAPP_f2097660464ol_nat @ ( hAPP_f1371399797ol_nat @ big_co1340561246_a_nat @ A ) @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_561_setsum__diff1__nat) ).

thf(3394,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ B ) @ C ) )
       => ( ( hAPP_f2097660464ol_nat @ ( hAPP_f1371399797ol_nat @ big_co1340561246_a_nat @ A ) @ ( hAPP_f20753329a_bool @ ( hAPP_f1706700729a_bool @ minus_85316870a_bool @ C ) @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ B ) @ bot_bo797238721a_bool ) ) )
          = ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ minus_minus_nat @ ( hAPP_f2097660464ol_nat @ ( hAPP_f1371399797ol_nat @ big_co1340561246_a_nat @ A ) @ C ) ) @ ( hAPP_H592031934_a_nat @ A @ B ) ) ) )
      & ( ~ ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ B ) @ C ) )
       => ( ( hAPP_f2097660464ol_nat @ ( hAPP_f1371399797ol_nat @ big_co1340561246_a_nat @ A ) @ ( hAPP_f20753329a_bool @ ( hAPP_f1706700729a_bool @ minus_85316870a_bool @ C ) @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ B ) @ bot_bo797238721a_bool ) ) )
          = ( hAPP_f2097660464ol_nat @ ( hAPP_f1371399797ol_nat @ big_co1340561246_a_nat @ A ) @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[622]) ).

thf(101,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( finite1973466193nt_int @ D ) )
     => ( ( hBOOL @ ( hAPP_int_bool @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ ( finite772772422nt_int @ D ) @ A ) @ B ) @ C ) )
       => ( ( hAPP_f1594865479ol_int @ ( finite_fold_int_int @ D @ A ) @ B )
          = C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_493_comp__fun__commute_Ofold__equality) ).

thf(1505,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( finite1973466193nt_int @ D ) )
     => ( ( hBOOL @ ( hAPP_int_bool @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ ( finite772772422nt_int @ D ) @ A ) @ B ) @ C ) )
       => ( ( hAPP_f1594865479ol_int @ ( finite_fold_int_int @ D @ A ) @ B )
          = C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[101]) ).

thf(771,axiom,
    ! [A: $i] :
      ( ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_f284875647l_bool @ ord_le951220754t_bool @ A ) @ bot_bot_fun_int_bool ) )
     => ( A = bot_bot_fun_int_bool ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_834_le__bot) ).

thf(3918,plain,
    ! [A: $i] :
      ( ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_f284875647l_bool @ ord_le951220754t_bool @ A ) @ bot_bot_fun_int_bool ) )
     => ( A = bot_bot_fun_int_bool ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[771]) ).

thf(817,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP_f54304608l_bool @ finite_finite_nat @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ A ) @ B ) ) )
    <=> ( hBOOL @ ( hAPP_f54304608l_bool @ finite_finite_nat @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_296_finite__insert) ).

thf(4088,plain,
    ! [A: $i,B: $i] :
      ( ( ( hBOOL @ ( hAPP_f54304608l_bool @ finite_finite_nat @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ A ) @ B ) ) )
       => ( hBOOL @ ( hAPP_f54304608l_bool @ finite_finite_nat @ B ) ) )
      & ( ( hBOOL @ ( hAPP_f54304608l_bool @ finite_finite_nat @ B ) )
       => ( hBOOL @ ( hAPP_f54304608l_bool @ finite_finite_nat @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ A ) @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[817]) ).

thf(55,axiom,
    ! [A: $i,B: $i] :
      ( ( A != B )
    <=> ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_nat @ A ) @ B ) )
        | ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_nat @ B ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_1121_nat__neq__iff) ).

thf(1320,plain,
    ! [A: $i,B: $i] :
      ( ( ( A != B )
       => ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_nat @ A ) @ B ) )
          | ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_nat @ B ) @ A ) ) ) )
      & ( ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_nat @ A ) @ B ) )
          | ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_nat @ B ) @ A ) ) )
       => ( A != B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[55]) ).

thf(858,axiom,
    ! [A: $i,B: $i] :
      ( ! [C: $i] :
          ( ( hBOOL @ ( hAPP_nat_bool @ B @ C ) )
         => ( hBOOL @ ( hAPP_nat_bool @ A @ C ) ) )
     => ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_f103356543l_bool @ ord_le1568362934t_bool @ B ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_939_predicate1I) ).

thf(4274,plain,
    ! [A: $i,B: $i] :
      ( ! [C: $i] :
          ( ( hBOOL @ ( hAPP_nat_bool @ B @ C ) )
         => ( hBOOL @ ( hAPP_nat_bool @ A @ C ) ) )
     => ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_f103356543l_bool @ ord_le1568362934t_bool @ B ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[858]) ).

thf(229,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ ( hAPP_nat_int @ semiri1621563631at_int @ A ) ) @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ ( hAPP_nat_int @ semiri1621563631at_int @ B ) ) @ C ) )
      = ( 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 @ A ) @ B ) ) ) @ C ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_1064_zadd__int__left) ).

thf(1962,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ ( hAPP_nat_int @ semiri1621563631at_int @ A ) ) @ ( hAPP_int_int @ ( hAPP_int_fun_int_int @ plus_plus_int @ ( hAPP_nat_int @ semiri1621563631at_int @ B ) ) @ C ) )
      = ( 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 @ A ) @ B ) ) ) @ C ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[229]) ).

thf(1129,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f1695230391l_bool @ finite1738664244iple_a @ C ) )
     => ( ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_f1767618879l_bool @ ord_le1523781517a_bool @ C ) @ B ) )
       => ( ( hBOOL @ ( hAPP_f1695230391l_bool @ A @ bot_bo797238721a_bool ) )
         => ( ! [D: $i,E: $i] :
                ( ( hBOOL @ ( hAPP_f1695230391l_bool @ finite1738664244iple_a @ E ) )
               => ( ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ D ) @ B ) )
                 => ( ~ ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ D ) @ E ) )
                   => ( ( hBOOL @ ( hAPP_f1695230391l_bool @ A @ E ) )
                     => ( hBOOL @ ( hAPP_f1695230391l_bool @ A @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ D ) @ E ) ) ) ) ) ) )
           => ( hBOOL @ ( hAPP_f1695230391l_bool @ A @ C ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_915_finite__subset__induct) ).

thf(5307,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f1695230391l_bool @ finite1738664244iple_a @ C ) )
     => ( ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_f1767618879l_bool @ ord_le1523781517a_bool @ C ) @ B ) )
       => ( ( hBOOL @ ( hAPP_f1695230391l_bool @ A @ bot_bo797238721a_bool ) )
         => ( ! [D: $i,E: $i] :
                ( ( hBOOL @ ( hAPP_f1695230391l_bool @ finite1738664244iple_a @ E ) )
               => ( ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ D ) @ B ) )
                 => ( ~ ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ D ) @ E ) )
                   => ( ( hBOOL @ ( hAPP_f1695230391l_bool @ A @ E ) )
                     => ( hBOOL @ ( hAPP_f1695230391l_bool @ A @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ D ) @ E ) ) ) ) ) ) )
           => ( hBOOL @ ( hAPP_f1695230391l_bool @ A @ C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1129]) ).

thf(427,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ A ) @ B ) )
     => ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ B ) @ A ) )
       => ( B = A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_747_xt1_I5_J) ).

thf(2690,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ A ) @ B ) )
     => ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ B ) @ A ) )
       => ( B = A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[427]) ).

thf(833,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f54304608l_bool @ finite_finite_nat @ C ) )
     => ( ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_f284875647l_bool @ ord_le951220754t_bool @ A ) @ ( image_nat_int @ B @ C ) ) )
       => ( hBOOL @ ( hAPP_f448129468l_bool @ finite_finite_int @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_896_finite__surj) ).

thf(4161,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f54304608l_bool @ finite_finite_nat @ C ) )
     => ( ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_f284875647l_bool @ ord_le951220754t_bool @ A ) @ ( image_nat_int @ B @ C ) ) )
       => ( hBOOL @ ( hAPP_f448129468l_bool @ finite_finite_int @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[833]) ).

thf(866,axiom,
    ! [A: $i,B: $i] :
      ( ( ( B = bot_bot_fun_nat_bool )
       => ( ( image_nat_nat @ ( hAPP_nat_fun_nat_nat @ cOMBK_nat_nat @ A ) @ B )
          = bot_bot_fun_nat_bool ) )
      & ( ( B != bot_bot_fun_nat_bool )
       => ( ( image_nat_nat @ ( hAPP_nat_fun_nat_nat @ cOMBK_nat_nat @ A ) @ B )
          = ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ A ) @ bot_bot_fun_nat_bool ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_153_image__constant__conv) ).

thf(4298,plain,
    ! [A: $i,B: $i] :
      ( ( ( B = bot_bot_fun_nat_bool )
       => ( ( image_nat_nat @ ( hAPP_nat_fun_nat_nat @ cOMBK_nat_nat @ A ) @ B )
          = bot_bot_fun_nat_bool ) )
      & ( ( B != bot_bot_fun_nat_bool )
       => ( ( image_nat_nat @ ( hAPP_nat_fun_nat_nat @ cOMBK_nat_nat @ A ) @ B )
          = ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ A ) @ bot_bot_fun_nat_bool ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[866]) ).

thf(921,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( ( hBOOL @ E )
       => ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hoare_2102800559rivs_a @ A ) @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ ( hoare_1916936827iple_a @ B @ C @ D ) ) @ bot_bo797238721a_bool ) ) ) )
     => ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hoare_2102800559rivs_a @ A ) @ ( 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 ) ) @ B ) ) ) @ E ) @ C @ D ) ) @ bot_bo797238721a_bool ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_4_constant) ).

thf(4533,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( ( hBOOL @ E )
       => ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hoare_2102800559rivs_a @ A ) @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ ( hoare_1916936827iple_a @ B @ C @ D ) ) @ bot_bo797238721a_bool ) ) ) )
     => ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hoare_2102800559rivs_a @ A ) @ ( 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 ) ) @ B ) ) ) @ E ) @ C @ D ) ) @ bot_bo797238721a_bool ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[921]) ).

thf(516,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ C ) @ D ) )
     => ( ( A
          = ( hAPP_nat_nat @ B @ C ) )
       => ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ A ) @ ( image_nat_nat @ B @ D ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_173_rev__image__eqI) ).

thf(3013,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ C ) @ D ) )
     => ( ( A
          = ( hAPP_nat_nat @ B @ C ) )
       => ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ A ) @ ( image_nat_nat @ B @ D ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[516]) ).

thf(655,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ A ) @ B ) ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ C ) @ B ) )
      = ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ A ) @ C ) ) @ B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_651_comm__semiring__1__class_Onormalizing__semiring__rules_I1_J) ).

thf(3526,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ A ) @ B ) ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ C ) @ B ) )
      = ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ A ) @ C ) ) @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[655]) ).

thf(421,axiom,
    ! [A: $i] : ( is_bool @ ( finite2130160977at_nat @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',gsy_c_Finite__Set_Ocomp__fun__commute_000tc__Nat__Onat_000tc__Nat__Onat) ).

thf(2672,plain,
    ! [A: $i] : ( is_bool @ ( finite2130160977at_nat @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[421]) ).

thf(414,axiom,
    ! [A: $i,B: $i] :
      ( ( ( image_int_int @ A @ B )
        = bot_bot_fun_int_bool )
    <=> ( B = bot_bot_fun_int_bool ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_203_image__is__empty) ).

thf(2640,plain,
    ! [A: $i,B: $i] :
      ( ( ( ( image_int_int @ A @ B )
          = bot_bot_fun_int_bool )
       => ( B = bot_bot_fun_int_bool ) )
      & ( ( B = bot_bot_fun_int_bool )
       => ( ( image_int_int @ A @ B )
          = bot_bot_fun_int_bool ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[414]) ).

thf(1092,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( finite2114894759a_bool @ D ) )
     => ( ( hBOOL @ ( hAPP_f1695230391l_bool @ finite1738664244iple_a @ C ) )
       => ( ( hAPP_f20753329a_bool @ ( finite1215486064a_bool @ D @ A ) @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ B ) @ C ) )
          = ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ D @ B ) @ ( hAPP_f20753329a_bool @ ( finite1215486064a_bool @ D @ A ) @ ( hAPP_f20753329a_bool @ ( hAPP_f1706700729a_bool @ minus_85316870a_bool @ C ) @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ B ) @ bot_bo797238721a_bool ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_549_comp__fun__commute_Ofold__insert__remove) ).

thf(5155,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( finite2114894759a_bool @ D ) )
     => ( ( hBOOL @ ( hAPP_f1695230391l_bool @ finite1738664244iple_a @ C ) )
       => ( ( hAPP_f20753329a_bool @ ( finite1215486064a_bool @ D @ A ) @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ B ) @ C ) )
          = ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ D @ B ) @ ( hAPP_f20753329a_bool @ ( finite1215486064a_bool @ D @ A ) @ ( hAPP_f20753329a_bool @ ( hAPP_f1706700729a_bool @ minus_85316870a_bool @ C ) @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ B ) @ bot_bo797238721a_bool ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1092]) ).

thf(1063,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( image_1865589340iple_a @ A @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ B ) @ C ) )
      = ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ ( hAPP_n261501868iple_a @ A @ B ) ) @ ( image_1865589340iple_a @ A @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_223_image__insert) ).

thf(5029,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( image_1865589340iple_a @ A @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ B ) @ C ) )
      = ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ ( hAPP_n261501868iple_a @ A @ B ) ) @ ( image_1865589340iple_a @ A @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1063]) ).

thf(145,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP_int_bool @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ ( finite772772422nt_int @ times_times_int ) @ B ) @ C ) @ D ) )
     => ( ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ A ) @ C ) )
       => ( ~ ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ B ) @ C ) )
         => ( hBOOL @ ( hAPP_int_bool @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ ( finite772772422nt_int @ times_times_int ) @ A ) @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ B ) @ ( hAPP_f1805168059t_bool @ ( hAPP_f1223193598t_bool @ minus_1449998731t_bool @ C ) @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ A ) @ bot_bot_fun_int_bool ) ) ) ) @ D ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_410_fold__graph__permute__diff) ).

thf(1653,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP_int_bool @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ ( finite772772422nt_int @ times_times_int ) @ B ) @ C ) @ D ) )
     => ( ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ A ) @ C ) )
       => ( ~ ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ B ) @ C ) )
         => ( hBOOL @ ( hAPP_int_bool @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ ( finite772772422nt_int @ times_times_int ) @ A ) @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ B ) @ ( hAPP_f1805168059t_bool @ ( hAPP_f1223193598t_bool @ minus_1449998731t_bool @ C ) @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ A ) @ bot_bot_fun_int_bool ) ) ) ) @ D ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[145]) ).

thf(66,axiom,
    ! [A: $i,B: $i] :
      ~ ( hBOOL @ ( hAPP_int_bool @ ( finite_fold1Set_int @ A @ bot_bot_fun_int_bool ) @ B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_231_empty__fold1SetE) ).

thf(1367,plain,
    ! [A: $i,B: $i] :
      ~ ( hBOOL @ ( hAPP_int_bool @ ( finite_fold1Set_int @ A @ bot_bot_fun_int_bool ) @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[66]) ).

thf(225,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ A ) @ ( image_nat_nat @ B @ C ) ) )
     => ~ ! [D: $i] :
            ( ( A
              = ( hAPP_nat_nat @ B @ D ) )
           => ~ ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ D ) @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_246_imageE) ).

thf(1945,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ A ) @ ( image_nat_nat @ B @ C ) ) )
     => ~ ! [D: $i] :
            ( ( A
              = ( hAPP_nat_nat @ B @ D ) )
           => ~ ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ D ) @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[225]) ).

thf(568,axiom,
    ! [A: $i,B: $i] :
      ( ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ A ) @ ( hAPP_f20753329a_bool @ collec351493750iple_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 @ fimplies ) @ ( hAPP_f20753329a_bool @ ( hAPP_f604201481a_bool @ cOMBB_1882975613iple_a @ fNot ) @ ( hAPP_H426895267a_bool @ ( hAPP_f920293029a_bool @ cOMBC_1940896922a_bool @ fequal963300192iple_a ) @ A ) ) ) ) @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_73_insert__Collect) ).

thf(3212,plain,
    ! [A: $i,B: $i] :
      ( ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ A ) @ ( hAPP_f20753329a_bool @ collec351493750iple_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 @ fimplies ) @ ( hAPP_f20753329a_bool @ ( hAPP_f604201481a_bool @ cOMBB_1882975613iple_a @ fNot ) @ ( hAPP_H426895267a_bool @ ( hAPP_f920293029a_bool @ cOMBC_1940896922a_bool @ fequal963300192iple_a ) @ A ) ) ) ) @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[568]) ).

thf(928,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ A ) @ B ) )
     => ? [C: $i] :
          ( ( B
            = ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ A ) @ C ) )
          & ~ ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ A ) @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_118_mk__disjoint__insert) ).

thf(4555,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ A ) @ B ) )
     => ? [C: $i] :
          ( ( B
            = ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ A ) @ C ) )
          & ~ ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ A ) @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[928]) ).

thf(294,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ ( finite929467206at_nat @ times_times_nat ) @ B ) @ C ) @ D ) )
     => ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ A ) @ C ) )
       => ( ~ ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ B ) @ C ) )
         => ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ ( finite929467206at_nat @ times_times_nat ) @ A ) @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ B ) @ ( hAPP_f800510211t_bool @ ( hAPP_f1730770594t_bool @ minus_2067140911t_bool @ C ) @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ A ) @ bot_bot_fun_nat_bool ) ) ) ) @ D ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_409_fold__graph__permute__diff) ).

thf(2206,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ ( finite929467206at_nat @ times_times_nat ) @ B ) @ C ) @ D ) )
     => ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ A ) @ C ) )
       => ( ~ ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ B ) @ C ) )
         => ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ ( finite929467206at_nat @ times_times_nat ) @ A ) @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ B ) @ ( hAPP_f800510211t_bool @ ( hAPP_f1730770594t_bool @ minus_2067140911t_bool @ C ) @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ A ) @ bot_bot_fun_nat_bool ) ) ) ) @ D ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[294]) ).

thf(985,axiom,
    ! [A: $i,B: $i] :
      ( ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ A ) @ B )
     != bot_bo797238721a_bool ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_97_insert__not__empty) ).

thf(4767,plain,
    ! [A: $i,B: $i] :
      ( ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ A ) @ B )
     != bot_bo797238721a_bool ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[985]) ).

thf(624,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f1695230391l_bool @ finite1738664244iple_a @ C ) )
     => ( ~ ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ B ) @ C ) )
       => ( ( hAPP_f587450508ol_int @ ( big_co305732779_a_int @ A ) @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ B ) @ C ) )
          = ( hAPP_int_int @ ( hAPP_int_fun_int_int @ times_times_int @ ( hAPP_H1229305626_a_int @ A @ B ) ) @ ( hAPP_f587450508ol_int @ ( big_co305732779_a_int @ A ) @ C ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_559_setprod_Oinsert) ).

thf(3405,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f1695230391l_bool @ finite1738664244iple_a @ C ) )
     => ( ~ ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ B ) @ C ) )
       => ( ( hAPP_f587450508ol_int @ ( big_co305732779_a_int @ A ) @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ B ) @ C ) )
          = ( hAPP_int_int @ ( hAPP_int_fun_int_int @ times_times_int @ ( hAPP_H1229305626_a_int @ A @ B ) ) @ ( hAPP_f587450508ol_int @ ( big_co305732779_a_int @ A ) @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[624]) ).

thf(205,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ ord_min_nat @ A ) @ B ) ) @ C ) )
    <=> ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ A ) @ C ) )
        | ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ B ) @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_962_min__le__iff__disj) ).

thf(1848,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ ord_min_nat @ A ) @ B ) ) @ C ) )
       => ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ A ) @ C ) )
          | ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ B ) @ C ) ) ) )
      & ( ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ A ) @ C ) )
          | ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ B ) @ C ) ) )
       => ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ ord_min_nat @ A ) @ B ) ) @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[205]) ).

thf(258,axiom,
    ! [A: $i] :
      ( ( hAPP_f800510211t_bool @ ( hAPP_f1730770594t_bool @ minus_2067140911t_bool @ A ) @ A )
      = bot_bot_fun_nat_bool ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_399_Diff__cancel) ).

thf(2059,plain,
    ! [A: $i] :
      ( ( hAPP_f800510211t_bool @ ( hAPP_f1730770594t_bool @ minus_2067140911t_bool @ A ) @ A )
      = bot_bot_fun_nat_bool ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[258]) ).

thf(631,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f54304608l_bool @ finite_finite_nat @ C ) )
     => ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_f103356543l_bool @ ord_le1568362934t_bool @ C ) @ B ) )
       => ( ( hBOOL @ ( hAPP_f54304608l_bool @ A @ bot_bot_fun_nat_bool ) )
         => ( ! [D: $i,E: $i] :
                ( ( hBOOL @ ( hAPP_f54304608l_bool @ finite_finite_nat @ E ) )
               => ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ D ) @ B ) )
                 => ( ~ ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ D ) @ E ) )
                   => ( ( hBOOL @ ( hAPP_f54304608l_bool @ A @ E ) )
                     => ( hBOOL @ ( hAPP_f54304608l_bool @ A @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ D ) @ E ) ) ) ) ) ) )
           => ( hBOOL @ ( hAPP_f54304608l_bool @ A @ C ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_913_finite__subset__induct) ).

thf(3443,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f54304608l_bool @ finite_finite_nat @ C ) )
     => ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_f103356543l_bool @ ord_le1568362934t_bool @ C ) @ B ) )
       => ( ( hBOOL @ ( hAPP_f54304608l_bool @ A @ bot_bot_fun_nat_bool ) )
         => ( ! [D: $i,E: $i] :
                ( ( hBOOL @ ( hAPP_f54304608l_bool @ finite_finite_nat @ E ) )
               => ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ D ) @ B ) )
                 => ( ~ ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ D ) @ E ) )
                   => ( ( hBOOL @ ( hAPP_f54304608l_bool @ A @ E ) )
                     => ( hBOOL @ ( hAPP_f54304608l_bool @ A @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ D ) @ E ) ) ) ) ) ) )
           => ( hBOOL @ ( hAPP_f54304608l_bool @ A @ C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[631]) ).

thf(360,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f448129468l_bool @ finite_finite_int @ C ) )
     => ( ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ B ) @ C ) )
       => ( ( hAPP_f1594865479ol_int @ ( hAPP_f1926459811ol_int @ big_co230513141nt_int @ A ) @ ( hAPP_f1805168059t_bool @ ( hAPP_f1223193598t_bool @ minus_1449998731t_bool @ C ) @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ B ) @ 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 @ A ) @ C ) ) @ ( hAPP_int_int @ A @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_465_setsum__diff1__ring) ).

thf(2438,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f448129468l_bool @ finite_finite_int @ C ) )
     => ( ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_i2112223885l_bool @ member_int @ B ) @ C ) )
       => ( ( hAPP_f1594865479ol_int @ ( hAPP_f1926459811ol_int @ big_co230513141nt_int @ A ) @ ( hAPP_f1805168059t_bool @ ( hAPP_f1223193598t_bool @ minus_1449998731t_bool @ C ) @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ insert_int @ B ) @ 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 @ A ) @ C ) ) @ ( hAPP_int_int @ A @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[360]) ).

thf(304,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP_f54304608l_bool @ finite_finite_nat @ B ) )
     => ( ~ ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ A ) @ B ) )
       => ( ( hAPP_f22106695ol_nat @ ( finite_fold1_nat @ times_times_nat ) @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ A ) @ B ) )
          = ( hAPP_f22106695ol_nat @ ( finite_fold_nat_nat @ times_times_nat @ A ) @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_587_fold1__eq__fold) ).

thf(2238,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP_f54304608l_bool @ finite_finite_nat @ B ) )
     => ( ~ ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ A ) @ B ) )
       => ( ( hAPP_f22106695ol_nat @ ( finite_fold1_nat @ times_times_nat ) @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ A ) @ B ) )
          = ( hAPP_f22106695ol_nat @ ( finite_fold_nat_nat @ times_times_nat @ A ) @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[304]) ).

thf(408,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ ( finite929467206at_nat @ A ) @ B ) @ C ) @ D ) )
     => ( ~ ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ B ) @ C ) )
       => ( hBOOL @ ( hAPP_nat_bool @ ( finite_fold1Set_nat @ A @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ B ) @ C ) ) @ D ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_261_fold1Set_Ointros) ).

thf(2615,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ ( finite929467206at_nat @ A ) @ B ) @ C ) @ D ) )
     => ( ~ ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ B ) @ C ) )
       => ( hBOOL @ ( hAPP_nat_bool @ ( finite_fold1Set_nat @ A @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ B ) @ C ) ) @ D ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[408]) ).

thf(181,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ D ) @ E ) )
     => ( ( 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 @ E ) @ A ) ) @ B ) ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ D ) @ A ) ) @ C ) ) )
      <=> ( 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 @ E ) @ D ) ) @ A ) ) @ B ) ) @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_1134_nat__less__add__iff1) ).

thf(1769,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ D ) @ E ) )
     => ( ( ( 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 @ E ) @ A ) ) @ B ) ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ D ) @ A ) ) @ C ) ) )
         => ( 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 @ E ) @ D ) ) @ A ) ) @ B ) ) @ C ) ) )
        & ( ( 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 @ E ) @ D ) ) @ A ) ) @ B ) ) @ C ) )
         => ( 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 @ E ) @ A ) ) @ B ) ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ times_times_nat @ D ) @ A ) ) @ C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[181]) ).

thf(524,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f54304608l_bool @ finite_finite_nat @ C ) )
     => ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ B ) @ C ) )
       => ( ( hAPP_f22106695ol_nat @ ( hAPP_f782000547ol_nat @ big_co387207925at_nat @ A ) @ C )
          = ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ ( hAPP_nat_nat @ A @ B ) ) @ ( hAPP_f22106695ol_nat @ ( hAPP_f782000547ol_nat @ big_co387207925at_nat @ A ) @ ( hAPP_f800510211t_bool @ ( hAPP_f1730770594t_bool @ minus_2067140911t_bool @ C ) @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ B ) @ bot_bot_fun_nat_bool ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_608_setsum_Oremove) ).

thf(3052,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP_f54304608l_bool @ finite_finite_nat @ C ) )
     => ( ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ B ) @ C ) )
       => ( ( hAPP_f22106695ol_nat @ ( hAPP_f782000547ol_nat @ big_co387207925at_nat @ A ) @ C )
          = ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ ( hAPP_nat_nat @ A @ B ) ) @ ( hAPP_f22106695ol_nat @ ( hAPP_f782000547ol_nat @ big_co387207925at_nat @ A ) @ ( hAPP_f800510211t_bool @ ( hAPP_f1730770594t_bool @ minus_2067140911t_bool @ C ) @ ( hAPP_f800510211t_bool @ ( hAPP_n1512601776t_bool @ insert_nat @ B ) @ bot_bot_fun_nat_bool ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[524]) ).

thf(1010,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( finite1428139793iple_a @ E ) )
     => ( ( hBOOL @ ( hAPP_H1037229737a_bool @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ ( finite90892550iple_a @ E ) @ A ) @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ B ) @ C ) ) @ D ) )
       => ( ~ ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ B ) @ C ) )
         => ~ ! [F: $i] :
                ( ( D
                  = ( hAPP_H905846293iple_a @ ( hAPP_H2085992369iple_a @ E @ B ) @ F ) )
               => ~ ( hBOOL @ ( hAPP_H1037229737a_bool @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ ( finite90892550iple_a @ E ) @ A ) @ C ) @ F ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_590_comp__fun__commute_Ofold__graph__insertE) ).

thf(4863,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( finite1428139793iple_a @ E ) )
     => ( ( hBOOL @ ( hAPP_H1037229737a_bool @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ ( finite90892550iple_a @ E ) @ A ) @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ insert956547291iple_a @ B ) @ C ) ) @ D ) )
       => ( ~ ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ B ) @ C ) )
         => ~ ! [F: $i] :
                ( ( D
                  = ( hAPP_H905846293iple_a @ ( hAPP_H2085992369iple_a @ E @ B ) @ F ) )
               => ~ ( hBOOL @ ( hAPP_H1037229737a_bool @ ( hAPP_f20753329a_bool @ ( hAPP_H1743777351a_bool @ ( finite90892550iple_a @ E ) @ A ) @ C ) @ F ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1010]) ).

thf(510,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ A ) @ B ) )
     => ( ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ A ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ minus_minus_nat @ B ) @ A ) )
        = B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_1030_le__add__diff__inverse) ).

thf(2993,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP_nat_bool @ ( hAPP_n1699378549t_bool @ ord_less_eq_nat @ A ) @ B ) )
     => ( ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ plus_plus_nat @ A ) @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ minus_minus_nat @ B ) @ A ) )
        = B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[510]) ).

thf(290,axiom,
    ! [A: $i,B: $i] :
      ( ( 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 ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_401_finite__Diff2) ).

thf(2191,plain,
    ! [A: $i,B: $i] :
      ( ( 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 ) ) )
        & ( ( 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 ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[290]) ).

thf(89,axiom,
    ! [A: $i,B: $i] : ( hBOOL @ ( hAPP_int_bool @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ ( finite772772422nt_int @ A ) @ B ) @ bot_bot_fun_int_bool ) @ B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_300_fold__graph_OemptyI) ).

thf(1449,plain,
    ! [A: $i,B: $i] : ( hBOOL @ ( hAPP_int_bool @ ( hAPP_f1805168059t_bool @ ( hAPP_i1529485324t_bool @ ( finite772772422nt_int @ A ) @ B ) @ bot_bot_fun_int_bool ) @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[89]) ).

thf(776,axiom,
    ! [A: $i,B: $i] :
      ( ~ ( hBOOL @ ( hAPP_f1695230391l_bool @ finite1738664244iple_a @ B ) )
     => ( ( hBOOL @ ( hAPP_f54304608l_bool @ finite_finite_nat @ ( image_48635758_a_nat @ A @ B ) ) )
       => ? [C: $i] :
            ( ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ C ) @ B ) )
            & ~ ( 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 ) @ B ) ) ) @ ( hAPP_n1919155532a_bool @ ( hAPP_f228417655a_bool @ cOMBC_364739t_bool @ ( hAPP_f1872860171t_bool @ ( hAPP_f1981448543t_bool @ cOMBB_1391527205iple_a @ fequal_nat ) @ A ) ) @ ( hAPP_H592031934_a_nat @ A @ C ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_321_pigeonhole__infinite) ).

thf(3933,plain,
    ! [A: $i,B: $i] :
      ( ~ ( hBOOL @ ( hAPP_f1695230391l_bool @ finite1738664244iple_a @ B ) )
     => ( ( hBOOL @ ( hAPP_f54304608l_bool @ finite_finite_nat @ ( image_48635758_a_nat @ A @ B ) ) )
       => ? [C: $i] :
            ( ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ C ) @ B ) )
            & ~ ( 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 ) @ B ) ) ) @ ( hAPP_n1919155532a_bool @ ( hAPP_f228417655a_bool @ cOMBC_364739t_bool @ ( hAPP_f1872860171t_bool @ ( hAPP_f1981448543t_bool @ cOMBB_1391527205iple_a @ fequal_nat ) @ A ) ) @ ( hAPP_H592031934_a_nat @ A @ C ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[776]) ).

thf(900,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ C ) @ D ) )
     => ( ( A
          = ( hAPP_H905846293iple_a @ B @ C ) )
       => ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ A ) @ ( image_533441733iple_a @ B @ D ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_167_rev__image__eqI) ).

thf(4459,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ C ) @ D ) )
     => ( ( A
          = ( hAPP_H905846293iple_a @ B @ C ) )
       => ( hBOOL @ ( hAPP_f1695230391l_bool @ ( hAPP_H622608077l_bool @ member1797258804iple_a @ A ) @ ( image_533441733iple_a @ B @ D ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[900]) ).

thf(217,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP_f54304608l_bool @ finite_finite_nat @ B ) )
     => ( hBOOL @ ( hAPP_f54304608l_bool @ finite_finite_nat @ ( hAPP_f800510211t_bool @ ( hAPP_f1730770594t_bool @ minus_2067140911t_bool @ B ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_371_finite__Diff) ).

thf(1912,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP_f54304608l_bool @ finite_finite_nat @ B ) )
     => ( hBOOL @ ( hAPP_f54304608l_bool @ finite_finite_nat @ ( hAPP_f800510211t_bool @ ( hAPP_f1730770594t_bool @ minus_2067140911t_bool @ B ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[217]) ).

thf(506,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( finite2130160977at_nat @ D ) )
     => ( ( hBOOL @ ( hAPP_f54304608l_bool @ finite_finite_nat @ C ) )
       => ( ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ D @ A ) @ ( hAPP_f22106695ol_nat @ ( finite_fold_nat_nat @ D @ B ) @ C ) )
          = ( hAPP_f22106695ol_nat @ ( finite_fold_nat_nat @ D @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ D @ A ) @ B ) ) @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_483_comp__fun__commute_Ofold__fun__comm) ).

thf(2980,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( finite2130160977at_nat @ D ) )
     => ( ( hBOOL @ ( hAPP_f54304608l_bool @ finite_finite_nat @ C ) )
       => ( ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ D @ A ) @ ( hAPP_f22106695ol_nat @ ( finite_fold_nat_nat @ D @ B ) @ C ) )
          = ( hAPP_f22106695ol_nat @ ( finite_fold_nat_nat @ D @ ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ D @ A ) @ B ) ) @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[506]) ).

thf(79,axiom,
    ! [A: $i,B: $i] :
      ( ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ ord_min_nat @ A ) @ B )
      = ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ ord_min_nat @ B ) @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_959_min__max_Oinf__commute) ).

thf(1417,plain,
    ! [A: $i,B: $i] :
      ( ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ ord_min_nat @ A ) @ B )
      = ( hAPP_nat_nat @ ( hAPP_nat_fun_nat_nat @ ord_min_nat @ B ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[79]) ).

thf(210,axiom,
    ! [A: $i] :
      ~ ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ A ) @ bot_bot_fun_nat_bool ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_16_emptyE) ).

thf(1870,plain,
    ! [A: $i] :
      ~ ( hBOOL @ ( hAPP_f54304608l_bool @ ( hAPP_n215258509l_bool @ member_nat @ A ) @ bot_bot_fun_nat_bool ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[210]) ).

thf(450,axiom,
    ! [A: $i,B: $i,C: $i] : ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_f284875647l_bool @ ord_le951220754t_bool @ ( hAPP_f1805168059t_bool @ ( hAPP_f1223193598t_bool @ minus_1449998731t_bool @ ( image_nat_int @ A @ B ) ) @ ( image_nat_int @ A @ C ) ) ) @ ( image_nat_int @ A @ ( hAPP_f800510211t_bool @ ( hAPP_f1730770594t_bool @ minus_2067140911t_bool @ B ) @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_900_image__diff__subset) ).

thf(2771,plain,
    ! [A: $i,B: $i,C: $i] : ( hBOOL @ ( hAPP_f448129468l_bool @ ( hAPP_f284875647l_bool @ ord_le951220754t_bool @ ( hAPP_f1805168059t_bool @ ( hAPP_f1223193598t_bool @ minus_1449998731t_bool @ ( image_nat_int @ A @ B ) ) @ ( image_nat_int @ A @ C ) ) ) @ ( image_nat_int @ A @ ( hAPP_f800510211t_bool @ ( hAPP_f1730770594t_bool @ minus_2067140911t_bool @ B ) @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[450]) ).

thf(21243,plain,
    $false,
    inference(cvc4,[status(thm)],[2199,3873,5116,1469,2131,3581,3830,3969,4992,3172,1586,4201,2452,1879,1718,3944,1411,5088,2744,2295,2306,4571,1995,2263,4899,3798,1729,3666,4729,4183,4290,2559,5205,3449,4015,1211,4714,3072,2940,3317,1290,3858,1522,3813,2776,5173,4960,3498,4408,1305,4487,4479,1486,1855,2077,4931,5069,2908,2062,2953,2590,5297,3460,1569,4603,1205,1655,4418,2607,1756,3135,5257,3616,4440,3693,2627,1761,3577,4762,3848,3017,4543,1788,1426,3004,2035,4110,4220,4528,3175,3562,4247,5005,4343,5227,1782,2020,4618,1613,2886,2082,4963,3878,3688,5024,4386,2972,3634,2432,1914,4496,1391,5123,3249,4079,1441,2010,1935,3947,4803,4124,1645,1921,1386,3281,1724,1475,5269,1803,4238,3644,4692,3574,3061,1330,2682,1191,4568,1132,3713,2703,3820,1628,3938,3195,2698,1298,4070,3116,4151,5064,2501,4130,1313,2185,1455,3749,3845,4023,1545,2169,1247,4664,5052,3657,4265,3840,3374,3385,3463,1379,5199,1147,1704,1890,1347,1572,3734,2520,1641,3835,3708,4383,1494,4507,1875,1778,4771,3231,4514,2388,4967,3268,4756,3788,3322,3524,3199,4733,2650,4724,5032,2291,2335,4783,3851,2017,2222,4411,1489,2950,4988,4329,2324,4583,4578,3075,4072,4018,5027,5013,1373,2996,3295,2782,2893,1632,2469,3473,3702,3983,1682,1136,3438,4050,2540,4871,3923,3290,1183,2073,3537,2833,5220,2483,1622,4825,4536,1405,3870,3906,4628,2718,3184,4718,4751,2523,3036,2530,3676,2137,4643,1943,1242,4285,3435,4700,3011,4199,4753,1691,3169,4531,3186,5167,2047,2773,3679,3151,3097,4888,1438,3720,1199,2364,2164,4076,4370,3767,2232,1736,4312,4059,2905,1868,2441,5049,4712,4466,4375,4878,4785,1370,2032,2601,3827,3711,2733,2805,2249,3603,1519,2596,3867,1355,1580,4989,5044,3022,4252,3752,3239,3371,3884,2937,3154,4493,5085,2890,4836,4661,2528,3778,1238,2015,3889,3471,1560,3810,4608,3090,3314,1178,5021,4553,2179,2213,3050,1195,1281,3647,4027,2958,1423,3307,3271,3166,3958,4295,2464,4280,5302,1602,3618,3137,3065,5177,3349,3600,1457,3144,1159,1836,4942,2547,2492,3033,3408,3731,3782,4957,3482,2377,4974,2895,2841,1970,4525,2537,4012,5233,3205,4808,5313,3699,2300,1270,1772,1259,1534,1794,4512,2685,4789,4048,1540,3963,3916,3237,3941,3087,4263,2460,1361,4983,2561,2868,2080,4744,2349,1207,3555,3775,4951,2023,3326,5080,1292,3094,4721,1739,4421,4315,1414,1842,1791,2172,4867,4444,2666,2472,4180,1451,4508,1984,5305,5217,1715,3364,2968,2504,3486,3654,3354,4068,2900,4970,1131,1668,1974,1552,2851,2355,1932,4244,3222,4934,4540,2813,4347,2052,1599,4902,2140,2756,3083,3162,1759,3215,1525,3755,2926,4352,4625,2208,2883,2748,3999,1187,4995,5170,4163,3824,4148,2634,1434,3903,1814,3225,2761,4195,3279,3587,2057,1712,2408,1663,2089,5295,2617,5126,2662,1185,2235,2001,2677,4380,1959,3042,3607,3491,5230,4562,4797,1595,5241,1251,1764,2413,2391,1680,5095,4359,3274,4269,3686,1154,4998,1833,4341,3319,3504,3255,2489,3736,4032,3604,2134,3234,3925,2859,2789,3519,3609,2657,2188,5067,4926,2674,4605,2991,5239,1965,1226,2695,1256,1139,5039,4413,4764,2753,2055,2706,2166,5158,2474,1675,3532,1428,2161,4127,2220,2029,4096,2326,1707,5118,2293,3651,3695,4099,4171,4236,3663,1865,1635,3002,1733,2457,2557,4640,3641,2425,3416,3302,3568,1638,1224,2358,2319,4433,2573,2148,2510,1209,5143,1408,2383,3764,3419,2246,4586,2943,2197,2361,2311,2151,2229,3815,2637,3551,2422,3287,4828,3515,4914,1349,4601,3468,4805,2669,4213,4154,2856,3500,2871,1376,2065,4388,5202,2610,4277,3192,2923,2974,1952,1688,3219,2824,5114,2182,4401,4430,4882,4009,3055,1557,4911,2111,4064,1993,1731,3800,3572,4897,4985,4452,2785,3451,1149,3298,3971,1344,1460,2289,2647,4726,2252,3861,2553,5007,2710,1820,2725,3627,2835,1176,3241,5299,3178,2999,1562,2075,2418,2051,2550,1245,1528,2620,3110,2086,4085,3159,2369,2931,5042,2244,1775,2592,4703,2101,1364,4954,3908,3876,1694,2693,4965,4481,4158,1805,2005,4741,4136,1873,2271,2037,1134,4177,4581,4355,4168,4907,1583,3936,2107,3293,4598,4735,1420,2878,1684,1277,4319,4292,2449,3921,1340,3455,1472,2333,1145,4113,2298,3913,4490,4673,4922,3833,1531,2888,5018,2831,4045,4658,2513,4749,1547,5047,1516,2863,2397,3507,2977,5111,4939,1400,2143,2012,1605,3260,4287,4469,1780,2821,3209,1753,3058,2386,1230,1484,3080,2569,5262,2454,2920,2700,3344,1785,1235,2276,1317,3329,4398,4905,3305,3818,3691,3334,3722,4020,3475,3026,2429,5152,1658,2322,5001,2486,4241,2599,2903,4800,3623,3412,1285,3853,5106,3769,4875,1358,5208,2411,4057,1702,4976,1221,5226,1797,2728,5083,4819,1766,3107,2838,2113,5223,1141,4133,4143,4538,4093,2722,1881,3156,4795,2848,3366,2266,4516,2584,4257,4217,1617,4548,2352,3589,1685,1949,1817,4656,1844,3928,2875,1940,2435,3284,3139,3620,1839,3584,3565,3401,4945,2818,3548,2194,2215,4559,4738,4780,3625,4646,1189,4006,2342,1575,4463,2792,1807,4061,5184,4730,4678,1998,4574,3593,3911,2273,2283,4666,4250,3336,1745,3331,2934,3465,5055,4710,2988,3864,2594,1721,5272,4615,2532,1508,3369,4042,4715,4791,2026,3630,5010,2764,3252,4823,2526,2416,2394,3085,2447,1861,4591,4759,4774,3149,1661,5211,5128,1444,4306,2779,1161,3181,4855,3324,4649,5164,5162,1308,3896,3529,1193,3793,2680,1607,4404,3351,1513,4869,4372,3761,2374,2302,3705,2983,4811,3264,4523,4843,4885,3203,3992,1908,3658,5187,2218,2258,4816,4917,1578,4472,1678,3785,2155,2255,3882,1395,1625,1593,3837,2604,2736,1986,3822,2751,4777,4631,2746,3479,2366,2714,4034,5215,2040,1165,3843,1742,5159,4350,4981,3560,3347,3277,4210,5290,4083,2070,3716,2961,1197,3673,1610,1268,2803,5140,2467,4066,3965,2507,3392,1925,4936,3494,4363,4923,3244,2241,4848,3362,2810,1352,3128,1829,1977,3974,1827,1261,2898,3570,1537,3772,5061,1550,4484,3489,3886,5236,1597,3740,1431,3076,3113,3394,1505,3918,4088,1320,4274,1962,5307,2690,4161,4298,4533,3013,3526,2672,2640,5155,5029,1653,1367,1945,3212,4555,2206,4767,3405,1848,2059,3443,2438,2238,2615,1769,3052,4863,2993,2191,1449,3933,4459,1912,2980,1417,1870,2771]) ).

%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.07/0.13  % Problem  : SWW470+3 : TPTP v8.1.2. Released v5.3.0.
% 0.07/0.16  % Command  : run_Leo-III %s %d
% 0.16/0.37  % Computer : n009.cluster.edu
% 0.16/0.37  % Model    : x86_64 x86_64
% 0.16/0.37  % CPU      : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.16/0.37  % Memory   : 8042.1875MB
% 0.16/0.37  % OS       : Linux 3.10.0-693.el7.x86_64
% 0.16/0.37  % CPULimit : 300
% 0.16/0.37  % WCLimit  : 300
% 0.16/0.37  % DateTime : Thu May 18 22:14:18 EDT 2023
% 0.16/0.37  % CPUTime  : 
% 0.99/0.91  % [INFO] 	 Parsing problem /export/starexec/sandbox2/benchmark/theBenchmark.p ... 
% 2.06/1.22  % [INFO] 	 Parsing done (309ms). 
% 2.06/1.23  % [INFO] 	 Running in sequential loop mode. 
% 2.68/1.43  % [INFO] 	 eprover registered as external prover. 
% 2.68/1.44  % [INFO] 	 cvc4 registered as external prover. 
% 2.68/1.44  % [INFO] 	 Scanning for conjecture ... 
% 3.51/1.63  % [INFO] 	 Found a conjecture and 1398 axioms. Running axiom selection ... 
% 5.09/2.09  % [INFO] 	 Axiom selection finished. Selected 1128 axioms (removed 270 axioms). 
% 6.20/2.52  % [INFO] 	 Problem is first-order (TPTP FOF). 
% 6.37/2.55  % [INFO] 	 Type checking passed. 
% 6.37/2.55  % [CONFIG] 	 Using configuration: timeout(300) with strategy<name(default),share(1.0),primSubst(3),sos(false),unifierCount(4),uniDepth(8),boolExt(true),choice(true),renaming(true),funcspec(false), domConstr(0),specialInstances(39),restrictUniAttempts(true),termOrdering(CPO)>.  Searching for refutation ... 
% 93.85/24.34  % External prover 'cvc4' found a proof!
% 93.85/24.34  % [INFO] 	 Killing All external provers ... 
% 93.85/24.34  % Time passed: 23810ms (effective reasoning time: 23109ms)
% 93.85/24.34  % Solved by strategy<name(default),share(1.0),primSubst(3),sos(false),unifierCount(4),uniDepth(8),boolExt(true),choice(true),renaming(true),funcspec(false), domConstr(0),specialInstances(39),restrictUniAttempts(true),termOrdering(CPO)>
% 93.85/24.35  % Axioms used in derivation (1128): fact_271_folding__one_Oeq__fold, fact_686_comm__semiring__1__class_Onormalizing__semiring__rules_I17_J, fact_67_insert__commute, fact_235_image__constant, gsy_c_Finite__Set_Ocomp__fun__commute_000tc__Int__Oint_000tc__Int__Oint, fact_1040_nat__eq__add__iff2, fact_681_comm__semiring__1__class_Onormalizing__semiring__rules_I14_J, fact_419_insert__Diff__single, fact_318_finite__induct, fact_728_empty__subsetI, fact_497_comp__fun__commute_Ofold__equality, fact_43_ex__in__conv, fact_931_le__funI, fact_442_minus__apply, fact_206_image__empty, fact_291_finite__imageI, fact_424_insert__Diff, help_fimplies_2_1_U, fact_567_fold__def, fact_76_insert__compr, fact_1121_nat__neq__iff, fact_68_insert__commute, fact_4_constant, fact_92_doubleton__eq__iff, fact_779_order__eq__refl, fact_1122_less__not__refl, fact_766_xt1_I3_J, fact_107_bot__apply, fact_929_image__subsetI, fact_674_comm__semiring__1__class_Onormalizing__semiring__rules_I22_J, help_If_3_1_If_000tc__Nat__Onat_T, fact_243_imageE, fact_1123_termination__basic__simps_I5_J, gsy_c_Big__Operators_Osemilattice__big_000tc__Nat__Onat, fact_172_rev__image__eqI, fact_263_fold1Set_Ointros, fact_223_image__insert, fact_86_singleton__inject, fact_1003_add__leE, fact_365_DiffI, help_COMBK_1_1_COMBK_000tc__Int__Oint_000tc__Hoare____Mirabelle____uwgpyvfjxg__O, fact_59_insert__ident, fact_1013_le__add2, gsy_c_hAPP_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_000tc__HOL__Obool, fact_502_fold__empty, fact_576_comp__fun__idem_Ofold__insert__idem, fact_963_min__le__iff__disj, gsy_c_Finite__Set_Ocomp__fun__commute_000tc__Int__Oint_000tc__fun_Itc__Int__Oint, fact_778_order__eq__refl, fact_797_set__rev__mp, fact_876_subset__image__iff, fact_702_comm__semiring__1__class_Onormalizing__semiring__rules_I8_J, fact_1062_nat__1__eq__mult__iff, fact_614_setsum__diff1_H, fact_296_finite__insert, fact_920_finite__subset__image, fact_1088_zmult__commute, fact_19_singleton__conv2, fact_207_image__empty, fact_661_add__right__cancel, fact_35_Collect__empty__eq, fact_507_comp__fun__commute_Ofold__insert2, fact_201_com_Osimps_I9_J, fact_360_folding__one_Oinsert__remove, fact_421_Diff__insert__absorb, fact_36_empty__iff, fact_763_ord__le__eq__trans, fact_64_insert__iff, fact_691_comm__semiring__1__class_Onormalizing__semiring__rules_I19_J, fact_1100_add__lessD1, fact_89_singletonE, fact_793_subset__trans, fact_300_fold__graph_OemptyI, fact_749_order__trans, fact_397_Diff__cancel, fact_742_xt1_I6_J, fact_157_image__eqI, fact_352_finite__ne__induct, fact_901_image__diff__subset, fact_697_comm__semiring__1__class_Onormalizing__semiring__rules_I34_J, fact_82_insert__compr__raw, fact_499_fold__empty, help_COMBC_1_1_COMBC_000tc__Com__Ostate_000tc__HOL__Obool_000tc__HOL__Obool_U, fact_1169_int__ge__induct, fact_118_mk__disjoint__insert, fact_988_min__add__distrib__left, fact_6_conseq2, fact_477_setprod_Oremove, fact_1173_less__mono__imp__le__mono, fact_860_subset__insertI2, fact_762_ord__le__eq__trans, fact_956_min__max_Oinf_Oleft__commute, fact_88_singletonE, fact_1085_zle__antisym, fact_106_bot__apply, fact_556_setprod_Oinsert, fact_57_insert__ident, fact_839_bot__unique, help_fconj_3_1_U, fact_1104_trans__less__add2, fact_546_comp__fun__commute_Ofold__insert__remove, fact_259_image__cong, fact_38_empty__iff, fact_1050_nat__add__left__commute, fact_307_fold__graph_OinsertI, fact_370_finite__Diff, fact_220_image__insert, fact_165_fold1__def, gsy_c_Finite__Set_Ocomp__fun__idem_000tc__Int__Oint_000tc__fun_Itc__Int__Oint_Mt, fact_521_comp__fun__commute_Ofold__graph__fold, help_COMBK_1_1_COMBK_000tc__HOL__Obool_000tc__Nat__Onat_U, fact_1008_trans__le__add2, fact_957_min__max_Oinf__left__idem, help_fequal_1_1_fequal_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_T, help_fequal_2_1_fequal_000tc__Nat__Onat_T, fact_1177_finite__greaterThanAtMost, fact_394_Diff__empty, fact_1051_nat__add__commute, fact_588_fold1__eq__fold, gsy_c_Finite__Set_Ocomp__fun__idem_000tc__Nat__Onat_000tc__Nat__Onat, fact_193_com_Osimps_I25_J, fact_78_insertI1, fact_554_setprod__insert, fact_623_setsum_Oinsert__remove, fact_465_setsum__diff1__ring, fact_133_fold1Set__sing, fact_367_DiffE, fact_347_folding__one__idem_Oinsert__idem, fact_164_fold1__def, fact_1089_zdiff__zmult__distrib, fact_310_insert__fold1SetE, fact_1148_int__le__induct, fact_47_all__not__in__conv, fact_740_xt1_I6_J, fact_455_comp__fun__idem__insert, fact_939_predicate1I, fact_897_finite__surj, fact_413_Diff__insert, fact_808_set__eq__subset, fact_1109_nat__less__le, fact_313_finite__nonempty__imp__fold1Set, fact_407_insert__Diff1, fact_1082_zle__refl, fact_246_imageE, fact_1012_le__add1, fact_795_set__mp, help_fequal_1_1_fequal_000tc__Com__Ostate_T, fact_244_imageE, fact_782_order__eq__iff, fact_371_finite__Diff, fact_659_comm__semiring__1__class_Onormalizing__semiring__rules_I20_J, fact_336_theI, fact_845_bot__least, fact_409_fold__graph__permute__diff, fact_708_setsum__cong, fact_976_min__max_Ole__infI2, fact_485_comp__fun__commute_Ofold__fun__comm, fact_752_order__antisym, fact_722_equalityI, fact_62_insert__code, fact_784_order__eq__iff, fact_534_comp__fun__commute, gsy_c_Big__Operators_Osemilattice__big_000tc__Hoare____Mirabelle____uwgpyvfjxg__, fact_868_insert__subset, fact_683_comm__semiring__1__class_Onormalizing__semiring__rules_I16_J, fact_375_folding__one__idem_Oidem, gsy_c_Finite__Set_Ocomp__fun__idem_000tc__Hoare____Mirabelle____uwgpyvfjxg__Otri_002, fact_303_empty__fold__graphE, fact_94_singleton__iff, fact_551_comp__fun__commute_Ofold__insert__remove, fact_1064_zadd__int__left, fact_756_xt1_I4_J, help_COMBB_1_1_COMBB_000tc__fun_Itc__Com__Ostate_Mtc__fun_Itc__HOL__Obool_Mtc__H_045, fact_309_insert__fold1SetE, fact_372_finite__Diff, help_fequal_2_1_fequal_000tc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J_T, fact_1061_nat__mult__1__right, fact_3_hoare__derivs_Oinsert, fact_441_finite__empty__induct, fact_916_subsetI, help_fequal_2_1_fequal_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_T, fact_146_fold1__singleton__def, fact_692_comm__semiring__1__class_Onormalizing__semiring__rules_I19_J, fact_433_folding__one__idem_Oin__idem, fact_1105_trans__less__add1, fact_1046_nat__mult__commute, fact_331_finite__imp__fold__graph, fact_755_order__antisym, gsy_c_Finite__Set_Ocomp__fun__idem_000tc__Nat__Onat_000tc__fun_Itc__Nat__Onat_Mt, help_COMBC_1_1_COMBC_000tc__Com__Ostate_000tc__Com__Ovname_000tc__fun_Itc__Nat__, fact_995_mult__le__mono1, fact_732_finite__Collect__subsets, fact_186_image__iff, fact_581_folding__one_Oeq__fold_H, fact_5_escape, fact_1154_transfer__nat__int__set__relations_I5_J, fact_985_min__max_Oinf__le2, fact_476_setprod_Oremove, fact_168_rev__image__eqI, fact_1093_zadd__commute, fact_282_finite_OinsertI, fact_488_comp__fun__commute_Ofun__left__comm, fact_212_empty__is__image, fact_937_fold1__antimono, fact_438_folding__one__idem_Ohom__commute, fact_705_setsum__cong2, fact_1103_add__less__mono1, fact_827_add__right__mono, fact_1053_add__mult__distrib, fact_878_subset__image__iff, help_COMBC_1_1_COMBC_000tc__fun_Itc__Hoare____Mirabelle____uwgpyvfjxg__Otriple_I, fact_649_combine__common__factor, fact_1058_nat__minus__add__max, fact_464_setsum__diff1__ring, fact_218_insert__image, fact_553_setprod__insert, fact_738_le__funE, fact_0_empty, fact_640_setsum_Oinsert, fact_694_comm__semiring__1__class_Onormalizing__semiring__rules_I7_J, fact_208_image__empty, fact_947_finite__lessThan, fact_851_finite__subset, fact_481_comp__fun__commute_Ofold__fun__comm, help_fequal_2_1_fequal_000tc__Hoare____Mirabelle____uwgpyvfjxg__Otriple_It__a_J_, fact_1084_zle__trans, fact_1108_not__add__less1, gsy_c_hAPP_000tc__Hoare____Mirabelle____uwgpyvfjxg__Otriple_It__a_J_000tc__HOL__, fact_73_insert__Collect, fact_279_finite_OemptyI, fact_289_finite__imageI, fact_709_setsum__cong, help_COMBB_1_1_COMBB_000tc__Hoare____Mirabelle____uwgpyvfjxg__Otriple_It__a_J_00_050, gsy_c_Finite__Set_Ofolding__one__idem_000tc__Hoare____Mirabelle____uwgpyvfjxg__O, fact_1171_bounded__nat__set__is__finite, fact_733_pred__subset__eq, fact_863_subset__insert, fact_53_insert__absorb, fact_1027_diff__le__mono2, fact_651_comm__semiring__1__class_Onormalizing__semiring__rules_I1_J, fact_101_empty__not__insert, fact_726_subsetD, fact_849_subset__empty, fact_504_fold__empty, fact_1006_add__le__mono, fact_343_theI_H, fact_891_subset__singletonD, fact_435_fold1__in, fact_388_DiffD2, fact_706_setsum__cong2, fact_1101_less__add__eq__less, gsy_c_Finite__Set_Ocomp__fun__commute_000tc__Nat__Onat_000tc__fun_Itc__Nat__Onat, fact_385_DiffD1, fact_848_subset__empty, fact_340_the1__equality, fact_850_subset__empty, fact_1031_add__diff__assoc, help_COMBC_1_1_COMBC_000tc__Hoare____Mirabelle____uwgpyvfjxg__Otriple_It__a_J_00_042, fact_646_add__diff__cancel, fact_857_insert__mono, fact_200_Collect__def, fact_110_bot__fun__def, fact_648_crossproduct__eq, fact_203_image__is__empty, fact_1110_le__eq__less__or__eq, fact_1068_int__mult, fact_431_folding__one__idem_Oin__idem, fact_334_fold1Set_Osimps, fact_281_finite_OemptyI, fact_270_folding__one_Oeq__fold, fact_975_min__max_Oinf__absorb1, help_COMBC_1_1_COMBC_000tc__Hoare____Mirabelle____uwgpyvfjxg__Otriple_It__a_J_00_053, fact_1090_zdiff__zmult__distrib2, fact_962_min__le__iff__disj, fact_494_comp__fun__commute_Ofold__equality, fact_731_finite__Collect__subsets, fact_209_image__empty, help_COMBS_1_1_COMBS_000tc__Com__Ostate_000tc__Nat__Onat_000tc__Com__Ostate_U, fact_821_add__le__imp__le__right, fact_345_fold__graph_Osimps, fact_776_le__funD, fact_979_min__max_Ole__infI1, gsy_c_Finite__Set_Ofolding__one_000tc__Int__Oint, gsy_c_fFalse, fact_864_subset__insert, fact_700_comm__semiring__class_Odistrib, fact_573_comp__fun__idem_Ofold__insert__idem2, fact_996_le__cube, fact_20_singleton__conv2, fact_610_setsum_Oremove, fact_351_finite__ne__induct, fact_274_folding__one_Oclosed, fact_267_folding__one_Oinsert, fact_978_min__max_Ole__infI1, fact_953_min__max_Oinf__assoc, fact_980_min__max_Ole__inf__iff, fact_1119_less__irrefl__nat, fact_1096_zadd__zmult__distrib, fact_182_imageI, help_fNot_2_1_U, fact_830_add__le__cancel__right, fact_399_Diff__cancel, fact_31_equals0D, fact_443_minus__apply, fact_1011_le__iff__add, fact_112_Comp, fact_161_image__eqI, fact_564_fold__def, fact_229_fold1Set__nonempty, fact_199_Collect__def, gsy_c_fTrue, fact_311_finite__nonempty__imp__fold1Set, fact_561_setsum__diff1__nat, fact_558_setprod_Oinsert, fact_1041_nat__diff__add__eq2, gsy_c_Finite__Set_Ofolding__one__idem_000tc__Nat__Onat, fact_1161_Nat__Transfer_Otransfer__int__nat__set__functions_I2_J, fact_666_comm__semiring__1__class_Onormalizing__semiring__rules_I23_J, help_COMBK_1_1_COMBK_000tc__HOL__Obool_000tc__Com__Ostate_U, help_COMBC_1_1_COMBC_000t__a_000tc__fun_Itc__Com__Ostate_Mtc__HOL__Obool_J_000tc, fact_1038_eq__diff__iff, fact_872_image__mono, help_fNot_1_1_U, fact_512_comp__fun__commute_Ofold__insert, fact_292_finite__imageI, fact_99_empty__not__insert, fact_1014_min__diff, fact_727_empty__subsetI, fact_537_comp__fun__commute_Ofold__graph__determ, fact_288_finite__imageI, fact_1032_le__diff__conv2, fact_162_image__eqI, fact_1078_zmult__zminus, fact_215_insert__image, fact_147_fold1__singleton__def, help_COMBC_1_1_COMBC_000tc__fun_Itc__Hoare____Mirabelle____uwgpyvfjxg__Otriple_I_055, fact_312_finite__nonempty__imp__fold1Set, fact_467_setprod_Oinsert__remove, fact_111_hoare__derivs_OSkip, fact_1048_nat__add__left__cancel, fact_216_insert__image, fact_454_comp__fun__idem__insert, fact_757_xt1_I4_J, fact_719_order__refl, fact_366_DiffI, fact_1070_zdiff__int, fact_204_image__is__empty, fact_180_imageI, fact_810_subset__refl, fact_418_insert__Diff__single, fact_65_insert__iff, fact_986_min__max_Oinf__le1, fact_981_min__max_Ole__inf__iff, help_fFalse_1_1_T, fact_737_linorder__le__cases, gsy_c_Finite__Set_Ocomp__fun__idem_000tc__Hoare____Mirabelle____uwgpyvfjxg__Otri, fact_833_le__bot, fact_381_Diff__iff, fact_127_com_Osimps_I13_J, fact_1015_diff__mult__distrib2, fact_969_min__max_Oinf__greatest, fact_838_bot__unique, fact_816_lessThan__subset__iff, fact_820_add__le__imp__le__right, fact_786_linorder__linear, fact_1025_le__diff__conv, help_COMBC_1_1_COMBC_000tc__Hoare____Mirabelle____uwgpyvfjxg__Otriple_It__a_J_00_011, fact_226_image__insert, fact_134_fold1Set__sing, fact_395_Diff__empty, fact_415_Diff__insert2, fact_684_comm__semiring__1__class_Onormalizing__semiring__rules_I16_J, fact_1091_zadd__assoc, fact_675_comm__semiring__1__class_Onormalizing__semiring__rules_I24_J, fact_616_setsum__diff1_H, fact_983_min__max_Ole__iff__inf, fact_695_diff__eq__diff__eq, fact_844_bot__least, fact_34_Collect__empty__eq, fact_723_equalityI, fact_8_conseq12, fact_2_cut, fact_1044_nat__diff__add__eq1, fact_773_order__antisym__conv, fact_15_emptyE, fact_1060_nat__mult__eq__1__iff, fact_545_comp__fun__commute_Ofold__rec, fact_970_min__max_Ole__infI, fact_374_folding__one__idem_Oidem, fact_232_empty__fold1SetE, fact_888_Diff__subset, fact_508_comp__fun__commute_Ofold__insert2, fact_333_fold1Set_Osimps, fact_1114_less__imp__diff__less, fact_308_insert__fold1SetE, help_COMBB_1_1_COMBB_000tc__HOL__Obool_000tc__fun_Itc__HOL__Obool_Mtc__HOL__Oboo_008, help_COMBC_1_1_COMBC_000tc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J_000tc__fun_Itc_, fact_936_fold1__antimono, help_COMBB_1_1_COMBB_000tc__fun_Itc__Com__Ostate_Mtc__HOL__Obool_J_000tc__fun_It_037, fact_320_pigeonhole__infinite, fact_924_setsum__mono, fact_624_setsum_Oinsert__remove, fact_451_comp__fun__idem_Ofun__left__idem, fact_696_comm__semiring__1__class_Onormalizing__semiring__rules_I34_J, fact_87_singletonE, fact_46_all__not__in__conv, fact_941_Collect__mono, fact_687_ab__semigroup__mult__class_Omult__ac_I1_J, fact_1057_termination__basic__simps_I4_J, fact_171_rev__image__eqI, fact_1106_nat__add__left__cancel__less, fact_825_add__left__mono, fact_1172_int__diff__cases, fact_273_folding__one_Oclosed, fact_862_subset__insertI2, fact_557_setprod_Oinsert, gsy_c_Finite__Set_Ocomp__fun__commute_000tc__Hoare____Mirabelle____uwgpyvfjxg__O, fact_456_comp__fun__idem__insert, fact_1004_add__leD1, fact_1111_less__imp__le__nat, fact_85_singleton__inject, fact_1155_transfer__nat__int__set__relations_I2_J, fact_1163_transfer__int__nat__relations_I3_J, fact_1157_int__if__cong, fact_362_fold1__insert, fact_302_empty__fold__graphE, gsy_c_hAPP_000tc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J_000tc__HOL__Obool, fact_1095_zadd__zmult__distrib2, fact_116_Set_Oset__insert, fact_907_subset__insert__iff, fact_578_comp__fun__idem_Ofold__insert__idem, fact_874_image__mono, fact_63_insert__iff, fact_673_comm__semiring__1__class_Onormalizing__semiring__rules_I22_J, fact_173_rev__image__eqI, fact_1098_zmult__1__right, fact_52_insert__absorb, fact_913_finite__subset__induct, fact_260_image__cong, fact_1000_eq__imp__le, fact_181_imageI, fact_314_finite_Osimps, fact_440_finite__empty__induct, fact_636_setsum__insert, fact_194_com_Osimps_I8_J, fact_30_equals0D, fact_93_singleton__iff, fact_396_Diff__empty, fact_870_subset__insertI, help_COMBB_1_1_COMBB_000tc__fun_Itc__Hoare____Mirabelle____uwgpyvfjxg__Otriple_I_047, help_COMBB_1_1_COMBB_000tc__fun_Itc__Hoare____Mirabelle____uwgpyvfjxg__Otriple_I_041, fact_192_com_Osimps_I24_J, fact_997_le__square, fact_641_setsum_Oinsert, fact_824_add__left__mono, fact_1131_less__diff__conv, fact_222_image__insert, fact_652_comm__semiring__1__class_Onormalizing__semiring__rules_I1_J, fact_1107_not__add__less2, fact_856_rev__finite__subset, fact_548_comp__fun__commute_Ofold__insert__remove, fact_445_fun__diff__def, fact_444_minus__apply, gsy_c_hAPP_000tc__fun_Itc__Hoare____Mirabelle____uwgpyvfjxg__Otriple_It__a_J_Mtc, gsy_c_hAPP_000tc__Com__Ostate_000tc__HOL__Obool, fact_955_min__max_Oinf__left__commute, fact_583_folding__one_Oeq__fold_H, 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% 93.85/24.35  % No. of inferences in proof: 2260
% 93.85/24.35  % SZS status Theorem for /export/starexec/sandbox2/benchmark/theBenchmark.p : 23810 ms resp. 23109 ms w/o parsing
% 94.65/24.79  % SZS output start Refutation for /export/starexec/sandbox2/benchmark/theBenchmark.p
% See solution above
% 94.89/24.80  % [INFO] 	 Killing All external provers ... 
%------------------------------------------------------------------------------