TPTP Problem File: SLH0467^1.p
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%------------------------------------------------------------------------------
% File : SLH0000^1 : TPTP v8.2.0. Released v8.2.0.
% Domain : Archive of Formal Proofs
% Problem :
% Version : Especial.
% English :
% Refs : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source : [Des23]
% Names : Knights_Tour/0000_KnightsTour/prob_01501_058523__6010510_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1576 ( 573 unt; 308 typ; 0 def)
% Number of atoms : 3393 (1519 equ; 0 cnn)
% Maximal formula atoms : 13 ( 2 avg)
% Number of connectives : 11599 ( 472 ~; 108 |; 229 &;9367 @)
% ( 0 <=>;1423 =>; 0 <=; 0 <~>)
% Maximal formula depth : 23 ( 7 avg)
% Number of types : 68 ( 67 usr)
% Number of type conns : 751 ( 751 >; 0 *; 0 +; 0 <<)
% Number of symbols : 244 ( 241 usr; 26 con; 0-3 aty)
% Number of variables : 3884 ( 138 ^;3583 !; 163 ?;3884 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 09:01:09.619
%------------------------------------------------------------------------------
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% Explicit typings (241)
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bNF_Gr9175161896313899224nt_int: set_li3855193629254082847nt_int > product_prod_int_int > set_li3855193629254082847nt_int ).
thf(sy_c_BNF__Greatest__Fixpoint_OSucc_001t__Int__Oint,type,
bNF_Gr6350390219475566417cc_int: set_list_int > list_int > set_int ).
thf(sy_c_BNF__Greatest__Fixpoint_OSucc_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
bNF_Gr2185362013987207252nt_int: set_li3855193629254082847nt_int > list_P5707943133018811711nt_int > set_Pr958786334691620121nt_int ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Int__Oint,type,
minus_minus_int: int > int > int ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat,type,
minus_minus_nat: nat > nat > nat ).
thf(sy_c_Groups_Oone__class_Oone_001t__Int__Oint,type,
one_one_int: int ).
thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat,type,
one_one_nat: nat ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Int__Oint,type,
plus_plus_int: int > int > int ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat,type,
plus_plus_nat: nat > nat > nat ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Int__Oint,type,
zero_zero_int: int ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
zero_zero_nat: nat ).
thf(sy_c_If_001t__Int__Oint,type,
if_int: $o > int > int > int ).
thf(sy_c_If_001t__Nat__Onat,type,
if_nat: $o > nat > nat > nat ).
thf(sy_c_KnightsTour_Oboard__exec,type,
board_exec: nat > nat > set_Pr958786334691620121nt_int ).
thf(sy_c_KnightsTour_Oboard__exec__aux,type,
board_exec_aux: nat > set_int > set_Pr958786334691620121nt_int ).
thf(sy_c_KnightsTour_Oknights__circuit,type,
knights_circuit: set_Pr958786334691620121nt_int > list_P5707943133018811711nt_int > $o ).
thf(sy_c_KnightsTour_Ostep__in,type,
step_in: list_P5707943133018811711nt_int > product_prod_int_int > product_prod_int_int > $o ).
thf(sy_c_KnightsTour_Otrans__path,type,
trans_path: product_prod_int_int > list_P5707943133018811711nt_int > list_P5707943133018811711nt_int ).
thf(sy_c_KnightsTour_Otrans__path__rel,type,
trans_path_rel: produc4058024888802116461nt_int > produc4058024888802116461nt_int > $o ).
thf(sy_c_KnightsTour_Ovalid__step,type,
valid_step: product_prod_int_int > product_prod_int_int > $o ).
thf(sy_c_List_Oappend_001t__Int__Oint,type,
append_int: list_int > list_int > list_int ).
thf(sy_c_List_Oappend_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
append7030698103840186580nt_int: list_P5707943133018811711nt_int > list_P5707943133018811711nt_int > list_P5707943133018811711nt_int ).
thf(sy_c_List_Oappend_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
append3046253387908738819nt_int: list_P314425023053850222nt_int > list_P314425023053850222nt_int > list_P314425023053850222nt_int ).
thf(sy_c_List_Oappend_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
append9111546825139864322nt_int: list_P2336717926344734829nt_int > list_P2336717926344734829nt_int > list_P2336717926344734829nt_int ).
thf(sy_c_List_Obind_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
bind_P8056059132564497095nt_int: list_P5707943133018811711nt_int > ( product_prod_int_int > list_P5707943133018811711nt_int ) > list_P5707943133018811711nt_int ).
thf(sy_c_List_Obutlast_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
butlas2390654596765537291nt_int: list_P5707943133018811711nt_int > list_P5707943133018811711nt_int ).
thf(sy_c_List_Odistinct__adj_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
distin7999467515711547031nt_int: list_P5707943133018811711nt_int > $o ).
thf(sy_c_List_Odrop_001t__Int__Oint,type,
drop_int: nat > list_int > list_int ).
thf(sy_c_List_Odrop_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
drop_P5690361596310759935nt_int: nat > list_P5707943133018811711nt_int > list_P5707943133018811711nt_int ).
thf(sy_c_List_Odrop_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
drop_P5227224544534396845nt_int: nat > list_P2336717926344734829nt_int > list_P2336717926344734829nt_int ).
thf(sy_c_List_Oenumerate_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
enumer5233495138702836510nt_int: nat > list_P5707943133018811711nt_int > list_P314425023053850222nt_int ).
thf(sy_c_List_Oenumerate_001t__Set__Oset_It__Int__Oint_J,type,
enumerate_set_int: nat > list_set_int > list_P7388728034488748185et_int ).
thf(sy_c_List_Olast_001_062_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J,type,
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thf(sy_c_List_Olast_001_062_It__Product____Type__Oprod_It__Int__Oint_M_062_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J,type,
last_P8974135382477003148e_term: list_P1743416141875011707e_term > produc8551481072490612790e_term > option6357759511663192854e_term ).
thf(sy_c_List_Olast_001t__Int__Oint,type,
last_int: list_int > int ).
thf(sy_c_List_Olast_001t__List__Olist_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
last_l5818330359162608606nt_int: list_l1670014477004246597nt_int > list_P5707943133018811711nt_int ).
thf(sy_c_List_Olast_001t__Nat__Onat,type,
last_nat: list_nat > nat ).
thf(sy_c_List_Olast_001t__Product____Type__Oprod_I_062_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
last_P858631638320919470nt_int: list_P8915022641806594461nt_int > produc7773217078559923341nt_int ).
thf(sy_c_List_Olast_001t__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Int__Oint_M_062_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
last_P7958387789370520228nt_int: list_P651320350408439699nt_int > produc2285326912895808259nt_int ).
thf(sy_c_List_Olast_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
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thf(sy_c_List_Olast_001t__Product____Type__Oprod_It__Int__Oint_Mt__List__Olist_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_J,type,
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thf(sy_c_List_Olast_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Set__Oset_It__Int__Oint_J_J,type,
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thf(sy_c_List_Olast_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
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thf(sy_c_List_Olast_001t__Set__Oset_It__Int__Oint_J,type,
last_set_int: list_set_int > set_int ).
thf(sy_c_List_Olenlex_001t__Int__Oint,type,
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thf(sy_c_List_Olenlex_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
lenlex6370358691973319492nt_int: set_Pr2560585780119916871nt_int > set_Pr6304946757569631943nt_int ).
thf(sy_c_List_Olex_001t__Int__Oint,type,
lex_int: set_Pr958786334691620121nt_int > set_Pr765067013931698361st_int ).
thf(sy_c_List_Olex_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
lex_Pr5393148144989827363nt_int: set_Pr2560585780119916871nt_int > set_Pr6304946757569631943nt_int ).
thf(sy_c_List_Olexn_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
lexn_P6893940911914197855nt_int: set_Pr2560585780119916871nt_int > nat > set_Pr6304946757569631943nt_int ).
thf(sy_c_List_Olexord_001t__Int__Oint,type,
lexord_int: set_Pr958786334691620121nt_int > set_Pr765067013931698361st_int ).
thf(sy_c_List_Olexord_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
lexord8886728381914977324nt_int: set_Pr2560585780119916871nt_int > set_Pr6304946757569631943nt_int ).
thf(sy_c_List_Olist_OCons_001_062_It__Int__Oint_Mt__Nat__Onat_J,type,
cons_int_nat: ( int > nat ) > list_int_nat > list_int_nat ).
thf(sy_c_List_Olist_OCons_001_062_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J,type,
cons_i7166360444231718571e_term: ( int > option6357759511663192854e_term ) > list_i8448526496819171953e_term > list_i8448526496819171953e_term ).
thf(sy_c_List_Olist_OCons_001_062_It__Product____Type__Oprod_It__Int__Oint_M_062_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J,type,
cons_P2630085844062958645e_term: ( produc8551481072490612790e_term > option6357759511663192854e_term ) > list_P1743416141875011707e_term > list_P1743416141875011707e_term ).
thf(sy_c_List_Olist_OCons_001t__Int__Oint,type,
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thf(sy_c_List_Olist_OCons_001t__List__Olist_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
cons_l7309679040211256053nt_int: list_P5707943133018811711nt_int > list_l1670014477004246597nt_int > list_l1670014477004246597nt_int ).
thf(sy_c_List_Olist_OCons_001t__Nat__Onat,type,
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thf(sy_c_List_Olist_OCons_001t__Product____Type__Oprod_I_062_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
cons_P2743708091642732631nt_int: produc7773217078559923341nt_int > list_P8915022641806594461nt_int > list_P8915022641806594461nt_int ).
thf(sy_c_List_Olist_OCons_001t__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Int__Oint_M_062_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
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thf(sy_c_List_Olist_OCons_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
cons_P3334398858971670639nt_int: product_prod_int_int > list_P5707943133018811711nt_int > list_P5707943133018811711nt_int ).
thf(sy_c_List_Olist_OCons_001t__Product____Type__Oprod_It__Int__Oint_Mt__List__Olist_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_J,type,
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thf(sy_c_List_Olist_OCons_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
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thf(sy_c_List_Olist_OCons_001t__Set__Oset_It__Int__Oint_J,type,
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thf(sy_c_List_Olist_ONil_001_062_It__Int__Oint_Mt__Nat__Onat_J,type,
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thf(sy_c_List_Olist_ONil_001t__Int__Oint,type,
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member4298863958545693205nt_int: produc6531732081377312382nt_int > set_Pr6887017113042850356nt_int > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__Set__Oset_It__Int__Oint_J_J_J,type,
member4311088582351766364et_int: produc9097395316738159027et_int > set_Pr2163364003801676051et_int > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_Mt__List__Olist_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_J,type,
member1390679175989562640nt_int: produc1089560213143673063nt_int > set_Pr6304946757569631943nt_int > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
member8440522571783428010at_nat: product_prod_nat_nat > set_Pr1261947904930325089at_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Set__Oset_It__Int__Oint_J_J,type,
member1292241183792264892et_int: produc9133624956312949779et_int > set_Pr4810089274464741491et_int > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_I_062_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_Mt__Product____Type__Oprod_I_062_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_J,type,
member4085533954029916580nt_int: produc2501202720802129403nt_int > set_Pr8634505666381077339nt_int > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Int__Oint_M_062_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_Mt__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Int__Oint_M_062_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_J,type,
member6582457606847315088nt_int: produc2432570611225516007nt_int > set_Pr4708930517165415495nt_int > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
member8566619992076573584nt_int: produc1219242969750017639nt_int > set_Pr2560585780119916871nt_int > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Int__Oint_Mt__List__Olist_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__List__Olist_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_J_J,type,
member211307870107589146nt_int: produc2223382658075010929nt_int > set_Pr5463530407906606289nt_int > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Set__Oset_It__Int__Oint_J_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Set__Oset_It__Int__Oint_J_J_J,type,
member5126324565730479632et_int: produc212874708166070503et_int > set_Pr7486745082216227783et_int > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Set__Oset_It__Int__Oint_J_Mt__Set__Oset_It__Int__Oint_J_J,type,
member2572552093476627150et_int: produc2115011035271226405et_int > set_Pr2522554150109002629et_int > $o ).
thf(sy_v_i_092_060_094sub_062h,type,
i_h: int ).
thf(sy_v_i_092_060_094sub_062l,type,
i_l: int ).
thf(sy_v_j_092_060_094sub_062h,type,
j_h: int ).
thf(sy_v_j_092_060_094sub_062l,type,
j_l: int ).
thf(sy_v_m_092_060_094sub_0621,type,
m_1: nat ).
thf(sy_v_ps_092_060_094sub_0621,type,
ps_1: list_P5707943133018811711nt_int ).
thf(sy_v_ps_092_060_094sub_0622,type,
ps_2: list_P5707943133018811711nt_int ).
thf(sy_v_s_092_060_094sub_062i,type,
s_i: product_prod_int_int ).
thf(sy_v_s_092_060_094sub_062j,type,
s_j: product_prod_int_int ).
thf(sy_v_thesis____,type,
thesis: $o ).
% Relevant facts (1261)
thf(fact_0__092_060open_0621_A_060_Alength_Aps_092_060_094sub_0622_092_060close_062,axiom,
ord_less_nat @ one_one_nat @ ( size_s5157815400016825771nt_int @ ps_2 ) ).
% \<open>1 < length ps\<^sub>2\<close>
thf(fact_1__092_060open_062length_Aps_092_060_094sub_0622_A_061_A1_A_092_060or_062_A1_A_060_Alength_Aps_092_060_094sub_0622_092_060close_062,axiom,
( ( ( size_s5157815400016825771nt_int @ ps_2 )
= one_one_nat )
| ( ord_less_nat @ one_one_nat @ ( size_s5157815400016825771nt_int @ ps_2 ) ) ) ).
% \<open>length ps\<^sub>2 = 1 \<or> 1 < length ps\<^sub>2\<close>
thf(fact_2_list_Oinject,axiom,
! [X21: product_prod_int_int,X22: list_P5707943133018811711nt_int,Y21: product_prod_int_int,Y22: list_P5707943133018811711nt_int] :
( ( ( cons_P3334398858971670639nt_int @ X21 @ X22 )
= ( cons_P3334398858971670639nt_int @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% list.inject
thf(fact_3_prod_Oinject,axiom,
! [X1: nat,X2: set_int,Y1: nat,Y2: set_int] :
( ( ( produc29655638201817675et_int @ X1 @ X2 )
= ( produc29655638201817675et_int @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X2 = Y2 ) ) ) ).
% prod.inject
thf(fact_4_prod_Oinject,axiom,
! [X1: int,X2: list_P5707943133018811711nt_int,Y1: int,Y2: list_P5707943133018811711nt_int] :
( ( ( produc8814303788642274490nt_int @ X1 @ X2 )
= ( produc8814303788642274490nt_int @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X2 = Y2 ) ) ) ).
% prod.inject
thf(fact_5_prod_Oinject,axiom,
! [X1: int,X2: int,Y1: int,Y2: int] :
( ( ( product_Pair_int_int @ X1 @ X2 )
= ( product_Pair_int_int @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X2 = Y2 ) ) ) ).
% prod.inject
thf(fact_6_prod_Oinject,axiom,
! [X1: produc8551481072490612790e_term > option6357759511663192854e_term,X2: product_prod_int_int,Y1: produc8551481072490612790e_term > option6357759511663192854e_term,Y2: product_prod_int_int] :
( ( ( produc5700946648718959541nt_int @ X1 @ X2 )
= ( produc5700946648718959541nt_int @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X2 = Y2 ) ) ) ).
% prod.inject
thf(fact_7_prod_Oinject,axiom,
! [X1: int > option6357759511663192854e_term,X2: product_prod_int_int,Y1: int > option6357759511663192854e_term,Y2: product_prod_int_int] :
( ( ( produc4305682042979456191nt_int @ X1 @ X2 )
= ( produc4305682042979456191nt_int @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X2 = Y2 ) ) ) ).
% prod.inject
thf(fact_8_old_Oprod_Oinject,axiom,
! [A: nat,B: set_int,A2: nat,B2: set_int] :
( ( ( produc29655638201817675et_int @ A @ B )
= ( produc29655638201817675et_int @ A2 @ B2 ) )
= ( ( A = A2 )
& ( B = B2 ) ) ) ).
% old.prod.inject
thf(fact_9_old_Oprod_Oinject,axiom,
! [A: int,B: list_P5707943133018811711nt_int,A2: int,B2: list_P5707943133018811711nt_int] :
( ( ( produc8814303788642274490nt_int @ A @ B )
= ( produc8814303788642274490nt_int @ A2 @ B2 ) )
= ( ( A = A2 )
& ( B = B2 ) ) ) ).
% old.prod.inject
thf(fact_10_old_Oprod_Oinject,axiom,
! [A: int,B: int,A2: int,B2: int] :
( ( ( product_Pair_int_int @ A @ B )
= ( product_Pair_int_int @ A2 @ B2 ) )
= ( ( A = A2 )
& ( B = B2 ) ) ) ).
% old.prod.inject
thf(fact_11_old_Oprod_Oinject,axiom,
! [A: produc8551481072490612790e_term > option6357759511663192854e_term,B: product_prod_int_int,A2: produc8551481072490612790e_term > option6357759511663192854e_term,B2: product_prod_int_int] :
( ( ( produc5700946648718959541nt_int @ A @ B )
= ( produc5700946648718959541nt_int @ A2 @ B2 ) )
= ( ( A = A2 )
& ( B = B2 ) ) ) ).
% old.prod.inject
thf(fact_12_old_Oprod_Oinject,axiom,
! [A: int > option6357759511663192854e_term,B: product_prod_int_int,A2: int > option6357759511663192854e_term,B2: product_prod_int_int] :
( ( ( produc4305682042979456191nt_int @ A @ B )
= ( produc4305682042979456191nt_int @ A2 @ B2 ) )
= ( ( A = A2 )
& ( B = B2 ) ) ) ).
% old.prod.inject
thf(fact_13_assms_I5_J,axiom,
( ( last_P3305686521732843992nt_int @ ps_2 )
= ( product_Pair_int_int @ i_l @ j_l ) ) ).
% assms(5)
thf(fact_14_assms_I4_J,axiom,
( ( hd_Pro282112905867057956nt_int @ ps_2 )
= ( product_Pair_int_int @ i_h @ j_h ) ) ).
% assms(4)
thf(fact_15_small__lazy_H_Ocases,axiom,
! [X: product_prod_int_int] :
~ ! [D: int,I: int] :
( X
!= ( product_Pair_int_int @ D @ I ) ) ).
% small_lazy'.cases
thf(fact_16_not__Cons__self2,axiom,
! [X: product_prod_int_int,Xs: list_P5707943133018811711nt_int] :
( ( cons_P3334398858971670639nt_int @ X @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_17_old_Oprod_Oexhaust,axiom,
! [Y: produc9133624956312949779et_int] :
~ ! [A3: nat,B3: set_int] :
( Y
!= ( produc29655638201817675et_int @ A3 @ B3 ) ) ).
% old.prod.exhaust
thf(fact_18_old_Oprod_Oexhaust,axiom,
! [Y: produc661532565036771336nt_int] :
~ ! [A3: int,B3: list_P5707943133018811711nt_int] :
( Y
!= ( produc8814303788642274490nt_int @ A3 @ B3 ) ) ).
% old.prod.exhaust
thf(fact_19_old_Oprod_Oexhaust,axiom,
! [Y: product_prod_int_int] :
~ ! [A3: int,B3: int] :
( Y
!= ( product_Pair_int_int @ A3 @ B3 ) ) ).
% old.prod.exhaust
thf(fact_20_old_Oprod_Oexhaust,axiom,
! [Y: produc2285326912895808259nt_int] :
~ ! [A3: produc8551481072490612790e_term > option6357759511663192854e_term,B3: product_prod_int_int] :
( Y
!= ( produc5700946648718959541nt_int @ A3 @ B3 ) ) ).
% old.prod.exhaust
thf(fact_21_old_Oprod_Oexhaust,axiom,
! [Y: produc7773217078559923341nt_int] :
~ ! [A3: int > option6357759511663192854e_term,B3: product_prod_int_int] :
( Y
!= ( produc4305682042979456191nt_int @ A3 @ B3 ) ) ).
% old.prod.exhaust
thf(fact_22_surj__pair,axiom,
! [P: produc9133624956312949779et_int] :
? [X3: nat,Y3: set_int] :
( P
= ( produc29655638201817675et_int @ X3 @ Y3 ) ) ).
% surj_pair
thf(fact_23_surj__pair,axiom,
! [P: produc661532565036771336nt_int] :
? [X3: int,Y3: list_P5707943133018811711nt_int] :
( P
= ( produc8814303788642274490nt_int @ X3 @ Y3 ) ) ).
% surj_pair
thf(fact_24_surj__pair,axiom,
! [P: product_prod_int_int] :
? [X3: int,Y3: int] :
( P
= ( product_Pair_int_int @ X3 @ Y3 ) ) ).
% surj_pair
thf(fact_25_surj__pair,axiom,
! [P: produc2285326912895808259nt_int] :
? [X3: produc8551481072490612790e_term > option6357759511663192854e_term,Y3: product_prod_int_int] :
( P
= ( produc5700946648718959541nt_int @ X3 @ Y3 ) ) ).
% surj_pair
thf(fact_26_surj__pair,axiom,
! [P: produc7773217078559923341nt_int] :
? [X3: int > option6357759511663192854e_term,Y3: product_prod_int_int] :
( P
= ( produc4305682042979456191nt_int @ X3 @ Y3 ) ) ).
% surj_pair
thf(fact_27_prod__cases,axiom,
! [P2: produc9133624956312949779et_int > $o,P: produc9133624956312949779et_int] :
( ! [A3: nat,B3: set_int] : ( P2 @ ( produc29655638201817675et_int @ A3 @ B3 ) )
=> ( P2 @ P ) ) ).
% prod_cases
thf(fact_28_prod__cases,axiom,
! [P2: produc661532565036771336nt_int > $o,P: produc661532565036771336nt_int] :
( ! [A3: int,B3: list_P5707943133018811711nt_int] : ( P2 @ ( produc8814303788642274490nt_int @ A3 @ B3 ) )
=> ( P2 @ P ) ) ).
% prod_cases
thf(fact_29_prod__cases,axiom,
! [P2: product_prod_int_int > $o,P: product_prod_int_int] :
( ! [A3: int,B3: int] : ( P2 @ ( product_Pair_int_int @ A3 @ B3 ) )
=> ( P2 @ P ) ) ).
% prod_cases
thf(fact_30_prod__cases,axiom,
! [P2: produc2285326912895808259nt_int > $o,P: produc2285326912895808259nt_int] :
( ! [A3: produc8551481072490612790e_term > option6357759511663192854e_term,B3: product_prod_int_int] : ( P2 @ ( produc5700946648718959541nt_int @ A3 @ B3 ) )
=> ( P2 @ P ) ) ).
% prod_cases
thf(fact_31_prod__cases,axiom,
! [P2: produc7773217078559923341nt_int > $o,P: produc7773217078559923341nt_int] :
( ! [A3: int > option6357759511663192854e_term,B3: product_prod_int_int] : ( P2 @ ( produc4305682042979456191nt_int @ A3 @ B3 ) )
=> ( P2 @ P ) ) ).
% prod_cases
thf(fact_32_length__induct,axiom,
! [P2: list_P5707943133018811711nt_int > $o,Xs: list_P5707943133018811711nt_int] :
( ! [Xs2: list_P5707943133018811711nt_int] :
( ! [Ys: list_P5707943133018811711nt_int] :
( ( ord_less_nat @ ( size_s5157815400016825771nt_int @ Ys ) @ ( size_s5157815400016825771nt_int @ Xs2 ) )
=> ( P2 @ Ys ) )
=> ( P2 @ Xs2 ) )
=> ( P2 @ Xs ) ) ).
% length_induct
thf(fact_33_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs2: list_P5707943133018811711nt_int] :
( ( size_s5157815400016825771nt_int @ Xs2 )
= N ) ).
% Ex_list_of_length
thf(fact_34_neq__if__length__neq,axiom,
! [Xs: list_P5707943133018811711nt_int,Ys2: list_P5707943133018811711nt_int] :
( ( ( size_s5157815400016825771nt_int @ Xs )
!= ( size_s5157815400016825771nt_int @ Ys2 ) )
=> ( Xs != Ys2 ) ) ).
% neq_if_length_neq
thf(fact_35_len1__hd__last,axiom,
! [Xs: list_P5707943133018811711nt_int] :
( ( ( size_s5157815400016825771nt_int @ Xs )
= one_one_nat )
=> ( ( hd_Pro282112905867057956nt_int @ Xs )
= ( last_P3305686521732843992nt_int @ Xs ) ) ) ).
% len1_hd_last
thf(fact_36_list_Osel_I1_J,axiom,
! [X21: product_prod_int_int,X22: list_P5707943133018811711nt_int] :
( ( hd_Pro282112905867057956nt_int @ ( cons_P3334398858971670639nt_int @ X21 @ X22 ) )
= X21 ) ).
% list.sel(1)
thf(fact_37_step__checker_Ocases,axiom,
! [X: produc1219242969750017639nt_int] :
~ ! [I: int,J: int,I2: int,J2: int] :
( X
!= ( produc3646306378393792727nt_int @ ( product_Pair_int_int @ I @ J ) @ ( product_Pair_int_int @ I2 @ J2 ) ) ) ).
% step_checker.cases
thf(fact_38_list__len__g__1__split,axiom,
! [Xs: list_P5707943133018811711nt_int] :
( ( ord_less_nat @ one_one_nat @ ( size_s5157815400016825771nt_int @ Xs ) )
=> ? [X_1: product_prod_int_int,X_2: product_prod_int_int,Xs3: list_P5707943133018811711nt_int] :
( Xs
= ( cons_P3334398858971670639nt_int @ X_1 @ ( cons_P3334398858971670639nt_int @ X_2 @ Xs3 ) ) ) ) ).
% list_len_g_1_split
thf(fact_39_prod__induct3,axiom,
! [P2: produc2285326912895808259nt_int > $o,X: produc2285326912895808259nt_int] :
( ! [A3: produc8551481072490612790e_term > option6357759511663192854e_term,B3: int,C: int] : ( P2 @ ( produc5700946648718959541nt_int @ A3 @ ( product_Pair_int_int @ B3 @ C ) ) )
=> ( P2 @ X ) ) ).
% prod_induct3
thf(fact_40_prod__induct3,axiom,
! [P2: produc7773217078559923341nt_int > $o,X: produc7773217078559923341nt_int] :
( ! [A3: int > option6357759511663192854e_term,B3: int,C: int] : ( P2 @ ( produc4305682042979456191nt_int @ A3 @ ( product_Pair_int_int @ B3 @ C ) ) )
=> ( P2 @ X ) ) ).
% prod_induct3
thf(fact_41_prod__cases3,axiom,
! [Y: produc2285326912895808259nt_int] :
~ ! [A3: produc8551481072490612790e_term > option6357759511663192854e_term,B3: int,C: int] :
( Y
!= ( produc5700946648718959541nt_int @ A3 @ ( product_Pair_int_int @ B3 @ C ) ) ) ).
% prod_cases3
thf(fact_42_prod__cases3,axiom,
! [Y: produc7773217078559923341nt_int] :
~ ! [A3: int > option6357759511663192854e_term,B3: int,C: int] :
( Y
!= ( produc4305682042979456191nt_int @ A3 @ ( product_Pair_int_int @ B3 @ C ) ) ) ).
% prod_cases3
thf(fact_43_Pair__inject,axiom,
! [A: nat,B: set_int,A2: nat,B2: set_int] :
( ( ( produc29655638201817675et_int @ A @ B )
= ( produc29655638201817675et_int @ A2 @ B2 ) )
=> ~ ( ( A = A2 )
=> ( B != B2 ) ) ) ).
% Pair_inject
thf(fact_44_Pair__inject,axiom,
! [A: int,B: list_P5707943133018811711nt_int,A2: int,B2: list_P5707943133018811711nt_int] :
( ( ( produc8814303788642274490nt_int @ A @ B )
= ( produc8814303788642274490nt_int @ A2 @ B2 ) )
=> ~ ( ( A = A2 )
=> ( B != B2 ) ) ) ).
% Pair_inject
thf(fact_45_Pair__inject,axiom,
! [A: int,B: int,A2: int,B2: int] :
( ( ( product_Pair_int_int @ A @ B )
= ( product_Pair_int_int @ A2 @ B2 ) )
=> ~ ( ( A = A2 )
=> ( B != B2 ) ) ) ).
% Pair_inject
thf(fact_46_Pair__inject,axiom,
! [A: produc8551481072490612790e_term > option6357759511663192854e_term,B: product_prod_int_int,A2: produc8551481072490612790e_term > option6357759511663192854e_term,B2: product_prod_int_int] :
( ( ( produc5700946648718959541nt_int @ A @ B )
= ( produc5700946648718959541nt_int @ A2 @ B2 ) )
=> ~ ( ( A = A2 )
=> ( B != B2 ) ) ) ).
% Pair_inject
thf(fact_47_Pair__inject,axiom,
! [A: int > option6357759511663192854e_term,B: product_prod_int_int,A2: int > option6357759511663192854e_term,B2: product_prod_int_int] :
( ( ( produc4305682042979456191nt_int @ A @ B )
= ( produc4305682042979456191nt_int @ A2 @ B2 ) )
=> ~ ( ( A = A2 )
=> ( B != B2 ) ) ) ).
% Pair_inject
thf(fact_48_less__numeral__extra_I4_J,axiom,
~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).
% less_numeral_extra(4)
thf(fact_49_less__numeral__extra_I4_J,axiom,
~ ( ord_less_int @ one_one_int @ one_one_int ) ).
% less_numeral_extra(4)
thf(fact_50_in__measures_I2_J,axiom,
! [X: int,Y: int,F: int > nat,Fs: list_int_nat] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ ( measures_int @ ( cons_int_nat @ F @ Fs ) ) )
= ( ( ord_less_nat @ ( F @ X ) @ ( F @ Y ) )
| ( ( ( F @ X )
= ( F @ Y ) )
& ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ ( measures_int @ Fs ) ) ) ) ) ).
% in_measures(2)
thf(fact_51_Cons__lenlex__iff,axiom,
! [M: int,Ms: list_int,N: int,Ns: list_int,R: set_Pr958786334691620121nt_int] :
( ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ ( cons_int @ M @ Ms ) @ ( cons_int @ N @ Ns ) ) @ ( lenlex_int @ R ) )
= ( ( ord_less_nat @ ( size_size_list_int @ Ms ) @ ( size_size_list_int @ Ns ) )
| ( ( ( size_size_list_int @ Ms )
= ( size_size_list_int @ Ns ) )
& ( member5262025264175285858nt_int @ ( product_Pair_int_int @ M @ N ) @ R ) )
| ( ( M = N )
& ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Ms @ Ns ) @ ( lenlex_int @ R ) ) ) ) ) ).
% Cons_lenlex_iff
thf(fact_52_Cons__lenlex__iff,axiom,
! [M: product_prod_int_int,Ms: list_P5707943133018811711nt_int,N: product_prod_int_int,Ns: list_P5707943133018811711nt_int,R: set_Pr2560585780119916871nt_int] :
( ( member1390679175989562640nt_int @ ( produc1932183703851549015nt_int @ ( cons_P3334398858971670639nt_int @ M @ Ms ) @ ( cons_P3334398858971670639nt_int @ N @ Ns ) ) @ ( lenlex6370358691973319492nt_int @ R ) )
= ( ( ord_less_nat @ ( size_s5157815400016825771nt_int @ Ms ) @ ( size_s5157815400016825771nt_int @ Ns ) )
| ( ( ( size_s5157815400016825771nt_int @ Ms )
= ( size_s5157815400016825771nt_int @ Ns ) )
& ( member8566619992076573584nt_int @ ( produc3646306378393792727nt_int @ M @ N ) @ R ) )
| ( ( M = N )
& ( member1390679175989562640nt_int @ ( produc1932183703851549015nt_int @ Ms @ Ns ) @ ( lenlex6370358691973319492nt_int @ R ) ) ) ) ) ).
% Cons_lenlex_iff
thf(fact_53_Cons__in__lex,axiom,
! [X: int,Xs: list_int,Y: int,Ys2: list_int,R: set_Pr958786334691620121nt_int] :
( ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ ( cons_int @ X @ Xs ) @ ( cons_int @ Y @ Ys2 ) ) @ ( lex_int @ R ) )
= ( ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ R )
& ( ( size_size_list_int @ Xs )
= ( size_size_list_int @ Ys2 ) ) )
| ( ( X = Y )
& ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Xs @ Ys2 ) @ ( lex_int @ R ) ) ) ) ) ).
% Cons_in_lex
thf(fact_54_Cons__in__lex,axiom,
! [X: product_prod_int_int,Xs: list_P5707943133018811711nt_int,Y: product_prod_int_int,Ys2: list_P5707943133018811711nt_int,R: set_Pr2560585780119916871nt_int] :
( ( member1390679175989562640nt_int @ ( produc1932183703851549015nt_int @ ( cons_P3334398858971670639nt_int @ X @ Xs ) @ ( cons_P3334398858971670639nt_int @ Y @ Ys2 ) ) @ ( lex_Pr5393148144989827363nt_int @ R ) )
= ( ( ( member8566619992076573584nt_int @ ( produc3646306378393792727nt_int @ X @ Y ) @ R )
& ( ( size_s5157815400016825771nt_int @ Xs )
= ( size_s5157815400016825771nt_int @ Ys2 ) ) )
| ( ( X = Y )
& ( member1390679175989562640nt_int @ ( produc1932183703851549015nt_int @ Xs @ Ys2 ) @ ( lex_Pr5393148144989827363nt_int @ R ) ) ) ) ) ).
% Cons_in_lex
thf(fact_55_in__measure,axiom,
! [X: int,Y: int,F: int > nat] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ ( measure_int @ F ) )
= ( ord_less_nat @ ( F @ X ) @ ( F @ Y ) ) ) ).
% in_measure
thf(fact_56_measures__less,axiom,
! [F: int > nat,X: int,Y: int,Fs: list_int_nat] :
( ( ord_less_nat @ ( F @ X ) @ ( F @ Y ) )
=> ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ ( measures_int @ ( cons_int_nat @ F @ Fs ) ) ) ) ).
% measures_less
thf(fact_57_List_Olast__drop,axiom,
! [N: nat,Xs: list_P5707943133018811711nt_int] :
( ( ord_less_nat @ N @ ( size_s5157815400016825771nt_int @ Xs ) )
=> ( ( last_P3305686521732843992nt_int @ ( drop_P5690361596310759935nt_int @ N @ Xs ) )
= ( last_P3305686521732843992nt_int @ Xs ) ) ) ).
% List.last_drop
thf(fact_58_full__exhaustive__int_H_Ocases,axiom,
! [X: produc2285326912895808259nt_int] :
~ ! [F2: produc8551481072490612790e_term > option6357759511663192854e_term,D: int,I: int] :
( X
!= ( produc5700946648718959541nt_int @ F2 @ ( product_Pair_int_int @ D @ I ) ) ) ).
% full_exhaustive_int'.cases
thf(fact_59_exhaustive__int_H_Ocases,axiom,
! [X: produc7773217078559923341nt_int] :
~ ! [F2: int > option6357759511663192854e_term,D: int,I: int] :
( X
!= ( produc4305682042979456191nt_int @ F2 @ ( product_Pair_int_int @ D @ I ) ) ) ).
% exhaustive_int'.cases
thf(fact_60_lenlex__irreflexive,axiom,
! [R: set_Pr958786334691620121nt_int,Xs: list_int] :
( ! [X3: int] :
~ ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X3 @ X3 ) @ R )
=> ~ ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Xs @ Xs ) @ ( lenlex_int @ R ) ) ) ).
% lenlex_irreflexive
thf(fact_61_mem__Collect__eq,axiom,
! [A: product_prod_int_int,P2: product_prod_int_int > $o] :
( ( member5262025264175285858nt_int @ A @ ( collec213857154873943460nt_int @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_62_mem__Collect__eq,axiom,
! [A: int,P2: int > $o] :
( ( member_int @ A @ ( collect_int @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_63_Collect__mem__eq,axiom,
! [A4: set_Pr958786334691620121nt_int] :
( ( collec213857154873943460nt_int
@ ^ [X4: product_prod_int_int] : ( member5262025264175285858nt_int @ X4 @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_64_Collect__mem__eq,axiom,
! [A4: set_int] :
( ( collect_int
@ ^ [X4: int] : ( member_int @ X4 @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_65_KnightsTour_Olast__drop,axiom,
! [K: nat,Xs: list_P5707943133018811711nt_int] :
( ( ord_less_nat @ K @ ( size_s5157815400016825771nt_int @ Xs ) )
=> ( ( last_P3305686521732843992nt_int @ Xs )
= ( last_P3305686521732843992nt_int @ ( drop_P5690361596310759935nt_int @ K @ Xs ) ) ) ) ).
% KnightsTour.last_drop
thf(fact_66_one__reorient,axiom,
! [X: nat] :
( ( one_one_nat = X )
= ( X = one_one_nat ) ) ).
% one_reorient
thf(fact_67_one__reorient,axiom,
! [X: int] :
( ( one_one_int = X )
= ( X = one_one_int ) ) ).
% one_reorient
thf(fact_68_knights__circuit__rotate__to,axiom,
! [B: set_Pr958786334691620121nt_int,Ps: list_P5707943133018811711nt_int,K: nat,S_i: product_prod_int_int] :
( ( knights_circuit @ B @ Ps )
=> ( ( ( hd_Pro282112905867057956nt_int @ ( drop_P5690361596310759935nt_int @ K @ Ps ) )
= S_i )
=> ( ( ord_less_nat @ K @ ( size_s5157815400016825771nt_int @ Ps ) )
=> ? [Ps2: list_P5707943133018811711nt_int] :
( ( knights_circuit @ B @ Ps2 )
& ( ( hd_Pro282112905867057956nt_int @ Ps2 )
= S_i ) ) ) ) ) ).
% knights_circuit_rotate_to
thf(fact_69_in__lex__prod,axiom,
! [A: nat,B: set_int,A2: nat,B2: set_int,R: set_Pr1261947904930325089at_nat,S: set_Pr2522554150109002629et_int] :
( ( member5126324565730479632et_int @ ( produc985091676681408599et_int @ ( produc29655638201817675et_int @ A @ B ) @ ( produc29655638201817675et_int @ A2 @ B2 ) ) @ ( lex_prod_nat_set_int @ R @ S ) )
= ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ A2 ) @ R )
| ( ( A = A2 )
& ( member2572552093476627150et_int @ ( produc6363374080413544029et_int @ B @ B2 ) @ S ) ) ) ) ).
% in_lex_prod
thf(fact_70_in__lex__prod,axiom,
! [A: int,B: list_P5707943133018811711nt_int,A2: int,B2: list_P5707943133018811711nt_int,R: set_Pr958786334691620121nt_int,S: set_Pr6304946757569631943nt_int] :
( ( member211307870107589146nt_int @ ( produc1418755309762820265nt_int @ ( produc8814303788642274490nt_int @ A @ B ) @ ( produc8814303788642274490nt_int @ A2 @ B2 ) ) @ ( lex_pr1845011153532943010nt_int @ R @ S ) )
= ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A @ A2 ) @ R )
| ( ( A = A2 )
& ( member1390679175989562640nt_int @ ( produc1932183703851549015nt_int @ B @ B2 ) @ S ) ) ) ) ).
% in_lex_prod
thf(fact_71_in__lex__prod,axiom,
! [A: int,B: int,A2: int,B2: int,R: set_Pr958786334691620121nt_int,S: set_Pr958786334691620121nt_int] :
( ( member8566619992076573584nt_int @ ( produc3646306378393792727nt_int @ ( product_Pair_int_int @ A @ B ) @ ( product_Pair_int_int @ A2 @ B2 ) ) @ ( lex_prod_int_int @ R @ S ) )
= ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A @ A2 ) @ R )
| ( ( A = A2 )
& ( member5262025264175285858nt_int @ ( product_Pair_int_int @ B @ B2 ) @ S ) ) ) ) ).
% in_lex_prod
thf(fact_72_in__lex__prod,axiom,
! [A: produc8551481072490612790e_term > option6357759511663192854e_term,B: product_prod_int_int,A2: produc8551481072490612790e_term > option6357759511663192854e_term,B2: product_prod_int_int,R: set_Pr4675205556198584599e_term,S: set_Pr2560585780119916871nt_int] :
( ( member6582457606847315088nt_int @ ( produc7601053194514725023nt_int @ ( produc5700946648718959541nt_int @ A @ B ) @ ( produc5700946648718959541nt_int @ A2 @ B2 ) ) @ ( lex_pr4548132154224949661nt_int @ R @ S ) )
= ( ( member8796044733138646368e_term @ ( produc8124735558134971759e_term @ A @ A2 ) @ R )
| ( ( A = A2 )
& ( member8566619992076573584nt_int @ ( produc3646306378393792727nt_int @ B @ B2 ) @ S ) ) ) ) ).
% in_lex_prod
thf(fact_73_in__lex__prod,axiom,
! [A: int > option6357759511663192854e_term,B: product_prod_int_int,A2: int > option6357759511663192854e_term,B2: product_prod_int_int,R: set_Pr7140365069852192003e_term,S: set_Pr2560585780119916871nt_int] :
( ( member4085533954029916580nt_int @ ( produc8406175334058502835nt_int @ ( produc4305682042979456191nt_int @ A @ B ) @ ( produc4305682042979456191nt_int @ A2 @ B2 ) ) @ ( lex_pr7234932505866811047nt_int @ R @ S ) )
= ( ( member6538663773838357324e_term @ ( produc4510790414113558107e_term @ A @ A2 ) @ R )
| ( ( A = A2 )
& ( member8566619992076573584nt_int @ ( produc3646306378393792727nt_int @ B @ B2 ) @ S ) ) ) ) ).
% in_lex_prod
thf(fact_74_measures__lesseq,axiom,
! [F: int > nat,X: int,Y: int,Fs: list_int_nat] :
( ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) )
=> ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ ( measures_int @ Fs ) )
=> ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ ( measures_int @ ( cons_int_nat @ F @ Fs ) ) ) ) ) ).
% measures_lesseq
thf(fact_75_mlex__iff,axiom,
! [X: int,Y: int,F: int > nat,R2: set_Pr958786334691620121nt_int] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ ( mlex_prod_int @ F @ R2 ) )
= ( ( ord_less_nat @ ( F @ X ) @ ( F @ Y ) )
| ( ( ( F @ X )
= ( F @ Y ) )
& ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ R2 ) ) ) ) ).
% mlex_iff
thf(fact_76_mlex__less,axiom,
! [F: int > nat,X: int,Y: int,R2: set_Pr958786334691620121nt_int] :
( ( ord_less_nat @ ( F @ X ) @ ( F @ Y ) )
=> ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ ( mlex_prod_int @ F @ R2 ) ) ) ).
% mlex_less
thf(fact_77_same__fstI,axiom,
! [P2: nat > $o,X: nat,Y4: set_int,Y: set_int,R2: nat > set_Pr2522554150109002629et_int] :
( ( P2 @ X )
=> ( ( member2572552093476627150et_int @ ( produc6363374080413544029et_int @ Y4 @ Y ) @ ( R2 @ X ) )
=> ( member5126324565730479632et_int @ ( produc985091676681408599et_int @ ( produc29655638201817675et_int @ X @ Y4 ) @ ( produc29655638201817675et_int @ X @ Y ) ) @ ( same_fst_nat_set_int @ P2 @ R2 ) ) ) ) ).
% same_fstI
thf(fact_78_same__fstI,axiom,
! [P2: int > $o,X: int,Y4: list_P5707943133018811711nt_int,Y: list_P5707943133018811711nt_int,R2: int > set_Pr6304946757569631943nt_int] :
( ( P2 @ X )
=> ( ( member1390679175989562640nt_int @ ( produc1932183703851549015nt_int @ Y4 @ Y ) @ ( R2 @ X ) )
=> ( member211307870107589146nt_int @ ( produc1418755309762820265nt_int @ ( produc8814303788642274490nt_int @ X @ Y4 ) @ ( produc8814303788642274490nt_int @ X @ Y ) ) @ ( same_f3003577036519843749nt_int @ P2 @ R2 ) ) ) ) ).
% same_fstI
thf(fact_79_same__fstI,axiom,
! [P2: ( produc8551481072490612790e_term > option6357759511663192854e_term ) > $o,X: produc8551481072490612790e_term > option6357759511663192854e_term,Y4: product_prod_int_int,Y: product_prod_int_int,R2: ( produc8551481072490612790e_term > option6357759511663192854e_term ) > set_Pr2560585780119916871nt_int] :
( ( P2 @ X )
=> ( ( member8566619992076573584nt_int @ ( produc3646306378393792727nt_int @ Y4 @ Y ) @ ( R2 @ X ) )
=> ( member6582457606847315088nt_int @ ( produc7601053194514725023nt_int @ ( produc5700946648718959541nt_int @ X @ Y4 ) @ ( produc5700946648718959541nt_int @ X @ Y ) ) @ ( same_f5472592420709775776nt_int @ P2 @ R2 ) ) ) ) ).
% same_fstI
thf(fact_80_same__fstI,axiom,
! [P2: ( int > option6357759511663192854e_term ) > $o,X: int > option6357759511663192854e_term,Y4: product_prod_int_int,Y: product_prod_int_int,R2: ( int > option6357759511663192854e_term ) > set_Pr2560585780119916871nt_int] :
( ( P2 @ X )
=> ( ( member8566619992076573584nt_int @ ( produc3646306378393792727nt_int @ Y4 @ Y ) @ ( R2 @ X ) )
=> ( member4085533954029916580nt_int @ ( produc8406175334058502835nt_int @ ( produc4305682042979456191nt_int @ X @ Y4 ) @ ( produc4305682042979456191nt_int @ X @ Y ) ) @ ( same_f6662725367016992042nt_int @ P2 @ R2 ) ) ) ) ).
% same_fstI
thf(fact_81_same__fstI,axiom,
! [P2: int > $o,X: int,Y4: int,Y: int,R2: int > set_Pr958786334691620121nt_int] :
( ( P2 @ X )
=> ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ Y4 @ Y ) @ ( R2 @ X ) )
=> ( member8566619992076573584nt_int @ ( produc3646306378393792727nt_int @ ( product_Pair_int_int @ X @ Y4 ) @ ( product_Pair_int_int @ X @ Y ) ) @ ( same_fst_int_int @ P2 @ R2 ) ) ) ) ).
% same_fstI
thf(fact_82_lexord__cons__cons,axiom,
! [A: product_prod_int_int,X: list_P5707943133018811711nt_int,B: product_prod_int_int,Y: list_P5707943133018811711nt_int,R: set_Pr2560585780119916871nt_int] :
( ( member1390679175989562640nt_int @ ( produc1932183703851549015nt_int @ ( cons_P3334398858971670639nt_int @ A @ X ) @ ( cons_P3334398858971670639nt_int @ B @ Y ) ) @ ( lexord8886728381914977324nt_int @ R ) )
= ( ( member8566619992076573584nt_int @ ( produc3646306378393792727nt_int @ A @ B ) @ R )
| ( ( A = B )
& ( member1390679175989562640nt_int @ ( produc1932183703851549015nt_int @ X @ Y ) @ ( lexord8886728381914977324nt_int @ R ) ) ) ) ) ).
% lexord_cons_cons
thf(fact_83_lexord__cons__cons,axiom,
! [A: int,X: list_int,B: int,Y: list_int,R: set_Pr958786334691620121nt_int] :
( ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ ( cons_int @ A @ X ) @ ( cons_int @ B @ Y ) ) @ ( lexord_int @ R ) )
= ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A @ B ) @ R )
| ( ( A = B )
& ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ X @ Y ) @ ( lexord_int @ R ) ) ) ) ) ).
% lexord_cons_cons
thf(fact_84_lexn__length,axiom,
! [Xs: list_P5707943133018811711nt_int,Ys2: list_P5707943133018811711nt_int,R: set_Pr2560585780119916871nt_int,N: nat] :
( ( member1390679175989562640nt_int @ ( produc1932183703851549015nt_int @ Xs @ Ys2 ) @ ( lexn_P6893940911914197855nt_int @ R @ N ) )
=> ( ( ( size_s5157815400016825771nt_int @ Xs )
= N )
& ( ( size_s5157815400016825771nt_int @ Ys2 )
= N ) ) ) ).
% lexn_length
thf(fact_85_Cons__listrel1__Cons,axiom,
! [X: product_prod_int_int,Xs: list_P5707943133018811711nt_int,Y: product_prod_int_int,Ys2: list_P5707943133018811711nt_int,R: set_Pr2560585780119916871nt_int] :
( ( member1390679175989562640nt_int @ ( produc1932183703851549015nt_int @ ( cons_P3334398858971670639nt_int @ X @ Xs ) @ ( cons_P3334398858971670639nt_int @ Y @ Ys2 ) ) @ ( listre1649617614542993432nt_int @ R ) )
= ( ( ( member8566619992076573584nt_int @ ( produc3646306378393792727nt_int @ X @ Y ) @ R )
& ( Xs = Ys2 ) )
| ( ( X = Y )
& ( member1390679175989562640nt_int @ ( produc1932183703851549015nt_int @ Xs @ Ys2 ) @ ( listre1649617614542993432nt_int @ R ) ) ) ) ) ).
% Cons_listrel1_Cons
thf(fact_86_Cons__listrel1__Cons,axiom,
! [X: int,Xs: list_int,Y: int,Ys2: list_int,R: set_Pr958786334691620121nt_int] :
( ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ ( cons_int @ X @ Xs ) @ ( cons_int @ Y @ Ys2 ) ) @ ( listrel1_int @ R ) )
= ( ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ R )
& ( Xs = Ys2 ) )
| ( ( X = Y )
& ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Xs @ Ys2 ) @ ( listrel1_int @ R ) ) ) ) ) ).
% Cons_listrel1_Cons
thf(fact_87_hd__drop__conv__nth,axiom,
! [N: nat,Xs: list_P5707943133018811711nt_int] :
( ( ord_less_nat @ N @ ( size_s5157815400016825771nt_int @ Xs ) )
=> ( ( hd_Pro282112905867057956nt_int @ ( drop_P5690361596310759935nt_int @ N @ Xs ) )
= ( nth_Pr4439495888332055232nt_int @ Xs @ N ) ) ) ).
% hd_drop_conv_nth
thf(fact_88_lexord__lex,axiom,
! [X: list_P5707943133018811711nt_int,Y: list_P5707943133018811711nt_int,R: set_Pr2560585780119916871nt_int] :
( ( member1390679175989562640nt_int @ ( produc1932183703851549015nt_int @ X @ Y ) @ ( lex_Pr5393148144989827363nt_int @ R ) )
= ( ( member1390679175989562640nt_int @ ( produc1932183703851549015nt_int @ X @ Y ) @ ( lexord8886728381914977324nt_int @ R ) )
& ( ( size_s5157815400016825771nt_int @ X )
= ( size_s5157815400016825771nt_int @ Y ) ) ) ) ).
% lexord_lex
thf(fact_89_mlex__leq,axiom,
! [F: int > nat,X: int,Y: int,R2: set_Pr958786334691620121nt_int] :
( ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) )
=> ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ R2 )
=> ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ ( mlex_prod_int @ F @ R2 ) ) ) ) ).
% mlex_leq
thf(fact_90_le__numeral__extra_I4_J,axiom,
ord_less_eq_nat @ one_one_nat @ one_one_nat ).
% le_numeral_extra(4)
thf(fact_91_le__numeral__extra_I4_J,axiom,
ord_less_eq_int @ one_one_int @ one_one_int ).
% le_numeral_extra(4)
thf(fact_92_nth__equalityI,axiom,
! [Xs: list_P5707943133018811711nt_int,Ys2: list_P5707943133018811711nt_int] :
( ( ( size_s5157815400016825771nt_int @ Xs )
= ( size_s5157815400016825771nt_int @ Ys2 ) )
=> ( ! [I: nat] :
( ( ord_less_nat @ I @ ( size_s5157815400016825771nt_int @ Xs ) )
=> ( ( nth_Pr4439495888332055232nt_int @ Xs @ I )
= ( nth_Pr4439495888332055232nt_int @ Ys2 @ I ) ) )
=> ( Xs = Ys2 ) ) ) ).
% nth_equalityI
thf(fact_93_Skolem__list__nth,axiom,
! [K: nat,P2: nat > product_prod_int_int > $o] :
( ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ K )
=> ? [X5: product_prod_int_int] : ( P2 @ I3 @ X5 ) ) )
= ( ? [Xs4: list_P5707943133018811711nt_int] :
( ( ( size_s5157815400016825771nt_int @ Xs4 )
= K )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ K )
=> ( P2 @ I3 @ ( nth_Pr4439495888332055232nt_int @ Xs4 @ I3 ) ) ) ) ) ) ).
% Skolem_list_nth
thf(fact_94_list__eq__iff__nth__eq,axiom,
( ( ^ [Y5: list_P5707943133018811711nt_int,Z: list_P5707943133018811711nt_int] : ( Y5 = Z ) )
= ( ^ [Xs4: list_P5707943133018811711nt_int,Ys3: list_P5707943133018811711nt_int] :
( ( ( size_s5157815400016825771nt_int @ Xs4 )
= ( size_s5157815400016825771nt_int @ Ys3 ) )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s5157815400016825771nt_int @ Xs4 ) )
=> ( ( nth_Pr4439495888332055232nt_int @ Xs4 @ I3 )
= ( nth_Pr4439495888332055232nt_int @ Ys3 @ I3 ) ) ) ) ) ) ).
% list_eq_iff_nth_eq
thf(fact_95_nth__via__drop,axiom,
! [N: nat,Xs: list_P5707943133018811711nt_int,Y: product_prod_int_int,Ys2: list_P5707943133018811711nt_int] :
( ( ( drop_P5690361596310759935nt_int @ N @ Xs )
= ( cons_P3334398858971670639nt_int @ Y @ Ys2 ) )
=> ( ( nth_Pr4439495888332055232nt_int @ Xs @ N )
= Y ) ) ).
% nth_via_drop
thf(fact_96_impossible__Cons,axiom,
! [Xs: list_P5707943133018811711nt_int,Ys2: list_P5707943133018811711nt_int,X: product_prod_int_int] :
( ( ord_less_eq_nat @ ( size_s5157815400016825771nt_int @ Xs ) @ ( size_s5157815400016825771nt_int @ Ys2 ) )
=> ( Xs
!= ( cons_P3334398858971670639nt_int @ X @ Ys2 ) ) ) ).
% impossible_Cons
thf(fact_97_listrel1I2,axiom,
! [Xs: list_P5707943133018811711nt_int,Ys2: list_P5707943133018811711nt_int,R: set_Pr2560585780119916871nt_int,X: product_prod_int_int] :
( ( member1390679175989562640nt_int @ ( produc1932183703851549015nt_int @ Xs @ Ys2 ) @ ( listre1649617614542993432nt_int @ R ) )
=> ( member1390679175989562640nt_int @ ( produc1932183703851549015nt_int @ ( cons_P3334398858971670639nt_int @ X @ Xs ) @ ( cons_P3334398858971670639nt_int @ X @ Ys2 ) ) @ ( listre1649617614542993432nt_int @ R ) ) ) ).
% listrel1I2
thf(fact_98_listrel1__eq__len,axiom,
! [Xs: list_P5707943133018811711nt_int,Ys2: list_P5707943133018811711nt_int,R: set_Pr2560585780119916871nt_int] :
( ( member1390679175989562640nt_int @ ( produc1932183703851549015nt_int @ Xs @ Ys2 ) @ ( listre1649617614542993432nt_int @ R ) )
=> ( ( size_s5157815400016825771nt_int @ Xs )
= ( size_s5157815400016825771nt_int @ Ys2 ) ) ) ).
% listrel1_eq_len
thf(fact_99_lexord__irreflexive,axiom,
! [R: set_Pr958786334691620121nt_int,Xs: list_int] :
( ! [X3: int] :
~ ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X3 @ X3 ) @ R )
=> ~ ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Xs @ Xs ) @ ( lexord_int @ R ) ) ) ).
% lexord_irreflexive
thf(fact_100_lexord__linear,axiom,
! [R: set_Pr958786334691620121nt_int,X: list_int,Y: list_int] :
( ! [A3: int,B3: int] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A3 @ B3 ) @ R )
| ( A3 = B3 )
| ( member5262025264175285858nt_int @ ( product_Pair_int_int @ B3 @ A3 ) @ R ) )
=> ( ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ X @ Y ) @ ( lexord_int @ R ) )
| ( X = Y )
| ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Y @ X ) @ ( lexord_int @ R ) ) ) ) ).
% lexord_linear
thf(fact_101_Cons__listrel1E2,axiom,
! [Xs: list_P5707943133018811711nt_int,Y: product_prod_int_int,Ys2: list_P5707943133018811711nt_int,R: set_Pr2560585780119916871nt_int] :
( ( member1390679175989562640nt_int @ ( produc1932183703851549015nt_int @ Xs @ ( cons_P3334398858971670639nt_int @ Y @ Ys2 ) ) @ ( listre1649617614542993432nt_int @ R ) )
=> ( ! [X3: product_prod_int_int] :
( ( Xs
= ( cons_P3334398858971670639nt_int @ X3 @ Ys2 ) )
=> ~ ( member8566619992076573584nt_int @ ( produc3646306378393792727nt_int @ X3 @ Y ) @ R ) )
=> ~ ! [Zs: list_P5707943133018811711nt_int] :
( ( Xs
= ( cons_P3334398858971670639nt_int @ Y @ Zs ) )
=> ~ ( member1390679175989562640nt_int @ ( produc1932183703851549015nt_int @ Zs @ Ys2 ) @ ( listre1649617614542993432nt_int @ R ) ) ) ) ) ).
% Cons_listrel1E2
thf(fact_102_Cons__listrel1E2,axiom,
! [Xs: list_int,Y: int,Ys2: list_int,R: set_Pr958786334691620121nt_int] :
( ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Xs @ ( cons_int @ Y @ Ys2 ) ) @ ( listrel1_int @ R ) )
=> ( ! [X3: int] :
( ( Xs
= ( cons_int @ X3 @ Ys2 ) )
=> ~ ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X3 @ Y ) @ R ) )
=> ~ ! [Zs: list_int] :
( ( Xs
= ( cons_int @ Y @ Zs ) )
=> ~ ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Zs @ Ys2 ) @ ( listrel1_int @ R ) ) ) ) ) ).
% Cons_listrel1E2
thf(fact_103_Cons__listrel1E1,axiom,
! [X: product_prod_int_int,Xs: list_P5707943133018811711nt_int,Ys2: list_P5707943133018811711nt_int,R: set_Pr2560585780119916871nt_int] :
( ( member1390679175989562640nt_int @ ( produc1932183703851549015nt_int @ ( cons_P3334398858971670639nt_int @ X @ Xs ) @ Ys2 ) @ ( listre1649617614542993432nt_int @ R ) )
=> ( ! [Y3: product_prod_int_int] :
( ( Ys2
= ( cons_P3334398858971670639nt_int @ Y3 @ Xs ) )
=> ~ ( member8566619992076573584nt_int @ ( produc3646306378393792727nt_int @ X @ Y3 ) @ R ) )
=> ~ ! [Zs: list_P5707943133018811711nt_int] :
( ( Ys2
= ( cons_P3334398858971670639nt_int @ X @ Zs ) )
=> ~ ( member1390679175989562640nt_int @ ( produc1932183703851549015nt_int @ Xs @ Zs ) @ ( listre1649617614542993432nt_int @ R ) ) ) ) ) ).
% Cons_listrel1E1
thf(fact_104_Cons__listrel1E1,axiom,
! [X: int,Xs: list_int,Ys2: list_int,R: set_Pr958786334691620121nt_int] :
( ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ ( cons_int @ X @ Xs ) @ Ys2 ) @ ( listrel1_int @ R ) )
=> ( ! [Y3: int] :
( ( Ys2
= ( cons_int @ Y3 @ Xs ) )
=> ~ ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y3 ) @ R ) )
=> ~ ! [Zs: list_int] :
( ( Ys2
= ( cons_int @ X @ Zs ) )
=> ~ ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Xs @ Zs ) @ ( listrel1_int @ R ) ) ) ) ) ).
% Cons_listrel1E1
thf(fact_105_listrel1I1,axiom,
! [X: product_prod_int_int,Y: product_prod_int_int,R: set_Pr2560585780119916871nt_int,Xs: list_P5707943133018811711nt_int] :
( ( member8566619992076573584nt_int @ ( produc3646306378393792727nt_int @ X @ Y ) @ R )
=> ( member1390679175989562640nt_int @ ( produc1932183703851549015nt_int @ ( cons_P3334398858971670639nt_int @ X @ Xs ) @ ( cons_P3334398858971670639nt_int @ Y @ Xs ) ) @ ( listre1649617614542993432nt_int @ R ) ) ) ).
% listrel1I1
thf(fact_106_listrel1I1,axiom,
! [X: int,Y: int,R: set_Pr958786334691620121nt_int,Xs: list_int] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ R )
=> ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ ( cons_int @ X @ Xs ) @ ( cons_int @ Y @ Xs ) ) @ ( listrel1_int @ R ) ) ) ).
% listrel1I1
thf(fact_107_lenlex__length,axiom,
! [Ms: list_P5707943133018811711nt_int,Ns: list_P5707943133018811711nt_int,R: set_Pr2560585780119916871nt_int] :
( ( member1390679175989562640nt_int @ ( produc1932183703851549015nt_int @ Ms @ Ns ) @ ( lenlex6370358691973319492nt_int @ R ) )
=> ( ord_less_eq_nat @ ( size_s5157815400016825771nt_int @ Ms ) @ ( size_s5157815400016825771nt_int @ Ns ) ) ) ).
% lenlex_length
thf(fact_108_order__refl,axiom,
! [X: nat] : ( ord_less_eq_nat @ X @ X ) ).
% order_refl
thf(fact_109_order__refl,axiom,
! [X: int] : ( ord_less_eq_int @ X @ X ) ).
% order_refl
thf(fact_110_dual__order_Orefl,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% dual_order.refl
thf(fact_111_dual__order_Orefl,axiom,
! [A: int] : ( ord_less_eq_int @ A @ A ) ).
% dual_order.refl
thf(fact_112_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
& ( M2 != N2 ) ) ) ) ).
% nat_less_le
thf(fact_113_less__imp__le__nat,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% less_imp_le_nat
thf(fact_114_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ N2 )
| ( M2 = N2 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_115_less__or__eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( ( ord_less_nat @ M @ N )
| ( M = N ) )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% less_or_eq_imp_le
thf(fact_116_le__neq__implies__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( M != N )
=> ( ord_less_nat @ M @ N ) ) ) ).
% le_neq_implies_less
thf(fact_117_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I4: nat,J3: nat] :
( ! [I: nat,J: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ ( F @ I ) @ ( F @ J ) ) )
=> ( ( ord_less_eq_nat @ I4 @ J3 )
=> ( ord_less_eq_nat @ ( F @ I4 ) @ ( F @ J3 ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_118_nat__descend__induct,axiom,
! [N: nat,P2: nat > $o,M: nat] :
( ! [K2: nat] :
( ( ord_less_nat @ N @ K2 )
=> ( P2 @ K2 ) )
=> ( ! [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N )
=> ( ! [I5: nat] :
( ( ord_less_nat @ K2 @ I5 )
=> ( P2 @ I5 ) )
=> ( P2 @ K2 ) ) )
=> ( P2 @ M ) ) ) ).
% nat_descend_induct
thf(fact_119_minf_I8_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z2 )
=> ~ ( ord_less_eq_nat @ T @ X6 ) ) ).
% minf(8)
thf(fact_120_minf_I8_J,axiom,
! [T: int] :
? [Z2: int] :
! [X6: int] :
( ( ord_less_int @ X6 @ Z2 )
=> ~ ( ord_less_eq_int @ T @ X6 ) ) ).
% minf(8)
thf(fact_121_order__antisym__conv,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ( ( ord_less_eq_nat @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_122_order__antisym__conv,axiom,
! [Y: int,X: int] :
( ( ord_less_eq_int @ Y @ X )
=> ( ( ord_less_eq_int @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_123_linorder__le__cases,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_eq_nat @ X @ Y )
=> ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_le_cases
thf(fact_124_linorder__le__cases,axiom,
! [X: int,Y: int] :
( ~ ( ord_less_eq_int @ X @ Y )
=> ( ord_less_eq_int @ Y @ X ) ) ).
% linorder_le_cases
thf(fact_125_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_126_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > int,C2: int] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_127_ord__le__eq__subst,axiom,
! [A: int,B: int,F: int > nat,C2: nat] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_128_ord__le__eq__subst,axiom,
! [A: int,B: int,F: int > int,C2: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_129_ord__eq__le__subst,axiom,
! [A: nat,F: nat > nat,B: nat,C2: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_130_ord__eq__le__subst,axiom,
! [A: int,F: nat > int,B: nat,C2: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_131_ord__eq__le__subst,axiom,
! [A: nat,F: int > nat,B: int,C2: int] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C2 )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_132_ord__eq__le__subst,axiom,
! [A: int,F: int > int,B: int,C2: int] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C2 )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_133_linorder__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
| ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_linear
thf(fact_134_linorder__linear,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
| ( ord_less_eq_int @ Y @ X ) ) ).
% linorder_linear
thf(fact_135_order__eq__refl,axiom,
! [X: nat,Y: nat] :
( ( X = Y )
=> ( ord_less_eq_nat @ X @ Y ) ) ).
% order_eq_refl
thf(fact_136_order__eq__refl,axiom,
! [X: int,Y: int] :
( ( X = Y )
=> ( ord_less_eq_int @ X @ Y ) ) ).
% order_eq_refl
thf(fact_137_order__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C2 )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_138_order__subst2,axiom,
! [A: nat,B: nat,F: nat > int,C2: int] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C2 )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_139_order__subst2,axiom,
! [A: int,B: int,F: int > nat,C2: nat] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C2 )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_140_order__subst2,axiom,
! [A: int,B: int,F: int > int,C2: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C2 )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_141_order__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_142_order__subst1,axiom,
! [A: nat,F: int > nat,B: int,C2: int] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C2 )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_143_order__subst1,axiom,
! [A: int,F: nat > int,B: nat,C2: nat] :
( ( ord_less_eq_int @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_144_order__subst1,axiom,
! [A: int,F: int > int,B: int,C2: int] :
( ( ord_less_eq_int @ A @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C2 )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_145_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: nat,Z: nat] : ( Y5 = Z ) )
= ( ^ [A5: nat,B4: nat] :
( ( ord_less_eq_nat @ A5 @ B4 )
& ( ord_less_eq_nat @ B4 @ A5 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_146_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: int,Z: int] : ( Y5 = Z ) )
= ( ^ [A5: int,B4: int] :
( ( ord_less_eq_int @ A5 @ B4 )
& ( ord_less_eq_int @ B4 @ A5 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_147_antisym,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_148_antisym,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_149_dual__order_Otrans,axiom,
! [B: nat,A: nat,C2: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_eq_nat @ C2 @ B )
=> ( ord_less_eq_nat @ C2 @ A ) ) ) ).
% dual_order.trans
thf(fact_150_dual__order_Otrans,axiom,
! [B: int,A: int,C2: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( ord_less_eq_int @ C2 @ B )
=> ( ord_less_eq_int @ C2 @ A ) ) ) ).
% dual_order.trans
thf(fact_151_dual__order_Oantisym,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_152_dual__order_Oantisym,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( ord_less_eq_int @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_153_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: nat,Z: nat] : ( Y5 = Z ) )
= ( ^ [A5: nat,B4: nat] :
( ( ord_less_eq_nat @ B4 @ A5 )
& ( ord_less_eq_nat @ A5 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_154_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: int,Z: int] : ( Y5 = Z ) )
= ( ^ [A5: int,B4: int] :
( ( ord_less_eq_int @ B4 @ A5 )
& ( ord_less_eq_int @ A5 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_155_linorder__wlog,axiom,
! [P2: nat > nat > $o,A: nat,B: nat] :
( ! [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
=> ( P2 @ A3 @ B3 ) )
=> ( ! [A3: nat,B3: nat] :
( ( P2 @ B3 @ A3 )
=> ( P2 @ A3 @ B3 ) )
=> ( P2 @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_156_linorder__wlog,axiom,
! [P2: int > int > $o,A: int,B: int] :
( ! [A3: int,B3: int] :
( ( ord_less_eq_int @ A3 @ B3 )
=> ( P2 @ A3 @ B3 ) )
=> ( ! [A3: int,B3: int] :
( ( P2 @ B3 @ A3 )
=> ( P2 @ A3 @ B3 ) )
=> ( P2 @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_157_order__trans,axiom,
! [X: nat,Y: nat,Z3: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z3 )
=> ( ord_less_eq_nat @ X @ Z3 ) ) ) ).
% order_trans
thf(fact_158_order__trans,axiom,
! [X: int,Y: int,Z3: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ord_less_eq_int @ Y @ Z3 )
=> ( ord_less_eq_int @ X @ Z3 ) ) ) ).
% order_trans
thf(fact_159_order_Otrans,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ord_less_eq_nat @ A @ C2 ) ) ) ).
% order.trans
thf(fact_160_order_Otrans,axiom,
! [A: int,B: int,C2: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ B @ C2 )
=> ( ord_less_eq_int @ A @ C2 ) ) ) ).
% order.trans
thf(fact_161_order__antisym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_162_order__antisym,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ord_less_eq_int @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_163_ord__le__eq__trans,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( B = C2 )
=> ( ord_less_eq_nat @ A @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_164_ord__le__eq__trans,axiom,
! [A: int,B: int,C2: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( B = C2 )
=> ( ord_less_eq_int @ A @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_165_ord__eq__le__trans,axiom,
! [A: nat,B: nat,C2: nat] :
( ( A = B )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ord_less_eq_nat @ A @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_166_ord__eq__le__trans,axiom,
! [A: int,B: int,C2: int] :
( ( A = B )
=> ( ( ord_less_eq_int @ B @ C2 )
=> ( ord_less_eq_int @ A @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_167_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: nat,Z: nat] : ( Y5 = Z ) )
= ( ^ [X4: nat,Y6: nat] :
( ( ord_less_eq_nat @ X4 @ Y6 )
& ( ord_less_eq_nat @ Y6 @ X4 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_168_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: int,Z: int] : ( Y5 = Z ) )
= ( ^ [X4: int,Y6: int] :
( ( ord_less_eq_int @ X4 @ Y6 )
& ( ord_less_eq_int @ Y6 @ X4 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_169_le__cases3,axiom,
! [X: nat,Y: nat,Z3: nat] :
( ( ( ord_less_eq_nat @ X @ Y )
=> ~ ( ord_less_eq_nat @ Y @ Z3 ) )
=> ( ( ( ord_less_eq_nat @ Y @ X )
=> ~ ( ord_less_eq_nat @ X @ Z3 ) )
=> ( ( ( ord_less_eq_nat @ X @ Z3 )
=> ~ ( ord_less_eq_nat @ Z3 @ Y ) )
=> ( ( ( ord_less_eq_nat @ Z3 @ Y )
=> ~ ( ord_less_eq_nat @ Y @ X ) )
=> ( ( ( ord_less_eq_nat @ Y @ Z3 )
=> ~ ( ord_less_eq_nat @ Z3 @ X ) )
=> ~ ( ( ord_less_eq_nat @ Z3 @ X )
=> ~ ( ord_less_eq_nat @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_170_le__cases3,axiom,
! [X: int,Y: int,Z3: int] :
( ( ( ord_less_eq_int @ X @ Y )
=> ~ ( ord_less_eq_int @ Y @ Z3 ) )
=> ( ( ( ord_less_eq_int @ Y @ X )
=> ~ ( ord_less_eq_int @ X @ Z3 ) )
=> ( ( ( ord_less_eq_int @ X @ Z3 )
=> ~ ( ord_less_eq_int @ Z3 @ Y ) )
=> ( ( ( ord_less_eq_int @ Z3 @ Y )
=> ~ ( ord_less_eq_int @ Y @ X ) )
=> ( ( ( ord_less_eq_int @ Y @ Z3 )
=> ~ ( ord_less_eq_int @ Z3 @ X ) )
=> ~ ( ( ord_less_eq_int @ Z3 @ X )
=> ~ ( ord_less_eq_int @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_171_nle__le,axiom,
! [A: nat,B: nat] :
( ( ~ ( ord_less_eq_nat @ A @ B ) )
= ( ( ord_less_eq_nat @ B @ A )
& ( B != A ) ) ) ).
% nle_le
thf(fact_172_nle__le,axiom,
! [A: int,B: int] :
( ( ~ ( ord_less_eq_int @ A @ B ) )
= ( ( ord_less_eq_int @ B @ A )
& ( B != A ) ) ) ).
% nle_le
thf(fact_173_order__less__imp__not__less,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_imp_not_less
thf(fact_174_order__less__imp__not__less,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ~ ( ord_less_int @ Y @ X ) ) ).
% order_less_imp_not_less
thf(fact_175_order__less__imp__not__eq2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_176_order__less__imp__not__eq2,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_177_order__less__imp__not__eq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_178_order__less__imp__not__eq,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_179_linorder__less__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
| ( X = Y )
| ( ord_less_nat @ Y @ X ) ) ).
% linorder_less_linear
thf(fact_180_linorder__less__linear,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
| ( X = Y )
| ( ord_less_int @ Y @ X ) ) ).
% linorder_less_linear
thf(fact_181_order__less__imp__triv,axiom,
! [X: nat,Y: nat,P2: $o] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ X )
=> P2 ) ) ).
% order_less_imp_triv
thf(fact_182_order__less__imp__triv,axiom,
! [X: int,Y: int,P2: $o] :
( ( ord_less_int @ X @ Y )
=> ( ( ord_less_int @ Y @ X )
=> P2 ) ) ).
% order_less_imp_triv
thf(fact_183_order__less__not__sym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_not_sym
thf(fact_184_order__less__not__sym,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ~ ( ord_less_int @ Y @ X ) ) ).
% order_less_not_sym
thf(fact_185_order__less__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C2 )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% order_less_subst2
thf(fact_186_order__less__subst2,axiom,
! [A: nat,B: nat,F: nat > int,C2: int] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C2 )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C2 ) ) ) ) ).
% order_less_subst2
thf(fact_187_order__less__subst2,axiom,
! [A: int,B: int,F: int > nat,C2: nat] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C2 )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% order_less_subst2
thf(fact_188_order__less__subst2,axiom,
! [A: int,B: int,F: int > int,C2: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C2 )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C2 ) ) ) ) ).
% order_less_subst2
thf(fact_189_order__less__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C2: nat] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% order_less_subst1
thf(fact_190_order__less__subst1,axiom,
! [A: nat,F: int > nat,B: int,C2: int] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C2 )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% order_less_subst1
thf(fact_191_order__less__subst1,axiom,
! [A: int,F: nat > int,B: nat,C2: nat] :
( ( ord_less_int @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A @ ( F @ C2 ) ) ) ) ) ).
% order_less_subst1
thf(fact_192_order__less__subst1,axiom,
! [A: int,F: int > int,B: int,C2: int] :
( ( ord_less_int @ A @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C2 )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A @ ( F @ C2 ) ) ) ) ) ).
% order_less_subst1
thf(fact_193_order__less__irrefl,axiom,
! [X: nat] :
~ ( ord_less_nat @ X @ X ) ).
% order_less_irrefl
thf(fact_194_order__less__irrefl,axiom,
! [X: int] :
~ ( ord_less_int @ X @ X ) ).
% order_less_irrefl
thf(fact_195_ord__less__eq__subst,axiom,
! [A: nat,B: nat,F: nat > nat,C2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_less_eq_subst
thf(fact_196_ord__less__eq__subst,axiom,
! [A: nat,B: nat,F: nat > int,C2: int] :
( ( ord_less_nat @ A @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_less_eq_subst
thf(fact_197_ord__less__eq__subst,axiom,
! [A: int,B: int,F: int > nat,C2: nat] :
( ( ord_less_int @ A @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_less_eq_subst
thf(fact_198_ord__less__eq__subst,axiom,
! [A: int,B: int,F: int > int,C2: int] :
( ( ord_less_int @ A @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_less_eq_subst
thf(fact_199_ord__eq__less__subst,axiom,
! [A: nat,F: nat > nat,B: nat,C2: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_200_ord__eq__less__subst,axiom,
! [A: int,F: nat > int,B: nat,C2: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_201_ord__eq__less__subst,axiom,
! [A: nat,F: int > nat,B: int,C2: int] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_int @ B @ C2 )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_202_ord__eq__less__subst,axiom,
! [A: int,F: int > int,B: int,C2: int] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_int @ B @ C2 )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_203_order__less__trans,axiom,
! [X: nat,Y: nat,Z3: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ Z3 )
=> ( ord_less_nat @ X @ Z3 ) ) ) ).
% order_less_trans
thf(fact_204_order__less__trans,axiom,
! [X: int,Y: int,Z3: int] :
( ( ord_less_int @ X @ Y )
=> ( ( ord_less_int @ Y @ Z3 )
=> ( ord_less_int @ X @ Z3 ) ) ) ).
% order_less_trans
thf(fact_205_order__less__asym_H,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% order_less_asym'
thf(fact_206_order__less__asym_H,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ~ ( ord_less_int @ B @ A ) ) ).
% order_less_asym'
thf(fact_207_linorder__neq__iff,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
= ( ( ord_less_nat @ X @ Y )
| ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neq_iff
thf(fact_208_linorder__neq__iff,axiom,
! [X: int,Y: int] :
( ( X != Y )
= ( ( ord_less_int @ X @ Y )
| ( ord_less_int @ Y @ X ) ) ) ).
% linorder_neq_iff
thf(fact_209_order__less__asym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_asym
thf(fact_210_order__less__asym,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ~ ( ord_less_int @ Y @ X ) ) ).
% order_less_asym
thf(fact_211_linorder__neqE,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neqE
thf(fact_212_linorder__neqE,axiom,
! [X: int,Y: int] :
( ( X != Y )
=> ( ~ ( ord_less_int @ X @ Y )
=> ( ord_less_int @ Y @ X ) ) ) ).
% linorder_neqE
thf(fact_213_dual__order_Ostrict__implies__not__eq,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_214_dual__order_Ostrict__implies__not__eq,axiom,
! [B: int,A: int] :
( ( ord_less_int @ B @ A )
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_215_order_Ostrict__implies__not__eq,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_216_order_Ostrict__implies__not__eq,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_217_dual__order_Ostrict__trans,axiom,
! [B: nat,A: nat,C2: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( ord_less_nat @ C2 @ B )
=> ( ord_less_nat @ C2 @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_218_dual__order_Ostrict__trans,axiom,
! [B: int,A: int,C2: int] :
( ( ord_less_int @ B @ A )
=> ( ( ord_less_int @ C2 @ B )
=> ( ord_less_int @ C2 @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_219_not__less__iff__gr__or__eq,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X @ Y ) )
= ( ( ord_less_nat @ Y @ X )
| ( X = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_220_not__less__iff__gr__or__eq,axiom,
! [X: int,Y: int] :
( ( ~ ( ord_less_int @ X @ Y ) )
= ( ( ord_less_int @ Y @ X )
| ( X = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_221_order_Ostrict__trans,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ord_less_nat @ A @ C2 ) ) ) ).
% order.strict_trans
thf(fact_222_order_Ostrict__trans,axiom,
! [A: int,B: int,C2: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ B @ C2 )
=> ( ord_less_int @ A @ C2 ) ) ) ).
% order.strict_trans
thf(fact_223_linorder__less__wlog,axiom,
! [P2: nat > nat > $o,A: nat,B: nat] :
( ! [A3: nat,B3: nat] :
( ( ord_less_nat @ A3 @ B3 )
=> ( P2 @ A3 @ B3 ) )
=> ( ! [A3: nat] : ( P2 @ A3 @ A3 )
=> ( ! [A3: nat,B3: nat] :
( ( P2 @ B3 @ A3 )
=> ( P2 @ A3 @ B3 ) )
=> ( P2 @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_224_linorder__less__wlog,axiom,
! [P2: int > int > $o,A: int,B: int] :
( ! [A3: int,B3: int] :
( ( ord_less_int @ A3 @ B3 )
=> ( P2 @ A3 @ B3 ) )
=> ( ! [A3: int] : ( P2 @ A3 @ A3 )
=> ( ! [A3: int,B3: int] :
( ( P2 @ B3 @ A3 )
=> ( P2 @ A3 @ B3 ) )
=> ( P2 @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_225_exists__least__iff,axiom,
( ( ^ [P3: nat > $o] :
? [X7: nat] : ( P3 @ X7 ) )
= ( ^ [P4: nat > $o] :
? [N2: nat] :
( ( P4 @ N2 )
& ! [M2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ~ ( P4 @ M2 ) ) ) ) ) ).
% exists_least_iff
thf(fact_226_dual__order_Oirrefl,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% dual_order.irrefl
thf(fact_227_dual__order_Oirrefl,axiom,
! [A: int] :
~ ( ord_less_int @ A @ A ) ).
% dual_order.irrefl
thf(fact_228_dual__order_Oasym,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ~ ( ord_less_nat @ A @ B ) ) ).
% dual_order.asym
thf(fact_229_dual__order_Oasym,axiom,
! [B: int,A: int] :
( ( ord_less_int @ B @ A )
=> ~ ( ord_less_int @ A @ B ) ) ).
% dual_order.asym
thf(fact_230_linorder__cases,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ( X != Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_231_linorder__cases,axiom,
! [X: int,Y: int] :
( ~ ( ord_less_int @ X @ Y )
=> ( ( X != Y )
=> ( ord_less_int @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_232_antisym__conv3,axiom,
! [Y: nat,X: nat] :
( ~ ( ord_less_nat @ Y @ X )
=> ( ( ~ ( ord_less_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_233_antisym__conv3,axiom,
! [Y: int,X: int] :
( ~ ( ord_less_int @ Y @ X )
=> ( ( ~ ( ord_less_int @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_234_less__induct,axiom,
! [P2: nat > $o,A: nat] :
( ! [X3: nat] :
( ! [Y7: nat] :
( ( ord_less_nat @ Y7 @ X3 )
=> ( P2 @ Y7 ) )
=> ( P2 @ X3 ) )
=> ( P2 @ A ) ) ).
% less_induct
thf(fact_235_ord__less__eq__trans,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( B = C2 )
=> ( ord_less_nat @ A @ C2 ) ) ) ).
% ord_less_eq_trans
thf(fact_236_ord__less__eq__trans,axiom,
! [A: int,B: int,C2: int] :
( ( ord_less_int @ A @ B )
=> ( ( B = C2 )
=> ( ord_less_int @ A @ C2 ) ) ) ).
% ord_less_eq_trans
thf(fact_237_ord__eq__less__trans,axiom,
! [A: nat,B: nat,C2: nat] :
( ( A = B )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ord_less_nat @ A @ C2 ) ) ) ).
% ord_eq_less_trans
thf(fact_238_ord__eq__less__trans,axiom,
! [A: int,B: int,C2: int] :
( ( A = B )
=> ( ( ord_less_int @ B @ C2 )
=> ( ord_less_int @ A @ C2 ) ) ) ).
% ord_eq_less_trans
thf(fact_239_order_Oasym,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% order.asym
thf(fact_240_order_Oasym,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ~ ( ord_less_int @ B @ A ) ) ).
% order.asym
thf(fact_241_less__imp__neq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_242_less__imp__neq,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_243_gt__ex,axiom,
! [X: nat] :
? [X_1: nat] : ( ord_less_nat @ X @ X_1 ) ).
% gt_ex
thf(fact_244_gt__ex,axiom,
! [X: int] :
? [X_1: int] : ( ord_less_int @ X @ X_1 ) ).
% gt_ex
thf(fact_245_lt__ex,axiom,
! [X: int] :
? [Y3: int] : ( ord_less_int @ Y3 @ X ) ).
% lt_ex
thf(fact_246_pinf_I1_J,axiom,
! [P2: nat > $o,P5: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z4: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z4 @ X3 )
=> ( ( P2 @ X3 )
= ( P5 @ X3 ) ) )
=> ( ? [Z4: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z4 @ X3 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z2: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z2 @ X6 )
=> ( ( ( P2 @ X6 )
& ( Q @ X6 ) )
= ( ( P5 @ X6 )
& ( Q2 @ X6 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_247_pinf_I1_J,axiom,
! [P2: int > $o,P5: int > $o,Q: int > $o,Q2: int > $o] :
( ? [Z4: int] :
! [X3: int] :
( ( ord_less_int @ Z4 @ X3 )
=> ( ( P2 @ X3 )
= ( P5 @ X3 ) ) )
=> ( ? [Z4: int] :
! [X3: int] :
( ( ord_less_int @ Z4 @ X3 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z2: int] :
! [X6: int] :
( ( ord_less_int @ Z2 @ X6 )
=> ( ( ( P2 @ X6 )
& ( Q @ X6 ) )
= ( ( P5 @ X6 )
& ( Q2 @ X6 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_248_pinf_I2_J,axiom,
! [P2: nat > $o,P5: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z4: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z4 @ X3 )
=> ( ( P2 @ X3 )
= ( P5 @ X3 ) ) )
=> ( ? [Z4: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z4 @ X3 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z2: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z2 @ X6 )
=> ( ( ( P2 @ X6 )
| ( Q @ X6 ) )
= ( ( P5 @ X6 )
| ( Q2 @ X6 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_249_pinf_I2_J,axiom,
! [P2: int > $o,P5: int > $o,Q: int > $o,Q2: int > $o] :
( ? [Z4: int] :
! [X3: int] :
( ( ord_less_int @ Z4 @ X3 )
=> ( ( P2 @ X3 )
= ( P5 @ X3 ) ) )
=> ( ? [Z4: int] :
! [X3: int] :
( ( ord_less_int @ Z4 @ X3 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z2: int] :
! [X6: int] :
( ( ord_less_int @ Z2 @ X6 )
=> ( ( ( P2 @ X6 )
| ( Q @ X6 ) )
= ( ( P5 @ X6 )
| ( Q2 @ X6 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_250_pinf_I3_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z2 @ X6 )
=> ( X6 != T ) ) ).
% pinf(3)
thf(fact_251_pinf_I3_J,axiom,
! [T: int] :
? [Z2: int] :
! [X6: int] :
( ( ord_less_int @ Z2 @ X6 )
=> ( X6 != T ) ) ).
% pinf(3)
thf(fact_252_pinf_I4_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z2 @ X6 )
=> ( X6 != T ) ) ).
% pinf(4)
thf(fact_253_pinf_I4_J,axiom,
! [T: int] :
? [Z2: int] :
! [X6: int] :
( ( ord_less_int @ Z2 @ X6 )
=> ( X6 != T ) ) ).
% pinf(4)
thf(fact_254_pinf_I5_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z2 @ X6 )
=> ~ ( ord_less_nat @ X6 @ T ) ) ).
% pinf(5)
thf(fact_255_pinf_I5_J,axiom,
! [T: int] :
? [Z2: int] :
! [X6: int] :
( ( ord_less_int @ Z2 @ X6 )
=> ~ ( ord_less_int @ X6 @ T ) ) ).
% pinf(5)
thf(fact_256_pinf_I7_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z2 @ X6 )
=> ( ord_less_nat @ T @ X6 ) ) ).
% pinf(7)
thf(fact_257_pinf_I7_J,axiom,
! [T: int] :
? [Z2: int] :
! [X6: int] :
( ( ord_less_int @ Z2 @ X6 )
=> ( ord_less_int @ T @ X6 ) ) ).
% pinf(7)
thf(fact_258_minf_I1_J,axiom,
! [P2: nat > $o,P5: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z4: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z4 )
=> ( ( P2 @ X3 )
= ( P5 @ X3 ) ) )
=> ( ? [Z4: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z4 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z2: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z2 )
=> ( ( ( P2 @ X6 )
& ( Q @ X6 ) )
= ( ( P5 @ X6 )
& ( Q2 @ X6 ) ) ) ) ) ) ).
% minf(1)
thf(fact_259_minf_I1_J,axiom,
! [P2: int > $o,P5: int > $o,Q: int > $o,Q2: int > $o] :
( ? [Z4: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z4 )
=> ( ( P2 @ X3 )
= ( P5 @ X3 ) ) )
=> ( ? [Z4: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z4 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z2: int] :
! [X6: int] :
( ( ord_less_int @ X6 @ Z2 )
=> ( ( ( P2 @ X6 )
& ( Q @ X6 ) )
= ( ( P5 @ X6 )
& ( Q2 @ X6 ) ) ) ) ) ) ).
% minf(1)
thf(fact_260_minf_I2_J,axiom,
! [P2: nat > $o,P5: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z4: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z4 )
=> ( ( P2 @ X3 )
= ( P5 @ X3 ) ) )
=> ( ? [Z4: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z4 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z2: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z2 )
=> ( ( ( P2 @ X6 )
| ( Q @ X6 ) )
= ( ( P5 @ X6 )
| ( Q2 @ X6 ) ) ) ) ) ) ).
% minf(2)
thf(fact_261_minf_I2_J,axiom,
! [P2: int > $o,P5: int > $o,Q: int > $o,Q2: int > $o] :
( ? [Z4: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z4 )
=> ( ( P2 @ X3 )
= ( P5 @ X3 ) ) )
=> ( ? [Z4: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z4 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z2: int] :
! [X6: int] :
( ( ord_less_int @ X6 @ Z2 )
=> ( ( ( P2 @ X6 )
| ( Q @ X6 ) )
= ( ( P5 @ X6 )
| ( Q2 @ X6 ) ) ) ) ) ) ).
% minf(2)
thf(fact_262_minf_I3_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z2 )
=> ( X6 != T ) ) ).
% minf(3)
thf(fact_263_minf_I3_J,axiom,
! [T: int] :
? [Z2: int] :
! [X6: int] :
( ( ord_less_int @ X6 @ Z2 )
=> ( X6 != T ) ) ).
% minf(3)
thf(fact_264_minf_I4_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z2 )
=> ( X6 != T ) ) ).
% minf(4)
thf(fact_265_minf_I4_J,axiom,
! [T: int] :
? [Z2: int] :
! [X6: int] :
( ( ord_less_int @ X6 @ Z2 )
=> ( X6 != T ) ) ).
% minf(4)
thf(fact_266_minf_I5_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z2 )
=> ( ord_less_nat @ X6 @ T ) ) ).
% minf(5)
thf(fact_267_minf_I5_J,axiom,
! [T: int] :
? [Z2: int] :
! [X6: int] :
( ( ord_less_int @ X6 @ Z2 )
=> ( ord_less_int @ X6 @ T ) ) ).
% minf(5)
thf(fact_268_minf_I7_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z2 )
=> ~ ( ord_less_nat @ T @ X6 ) ) ).
% minf(7)
thf(fact_269_minf_I7_J,axiom,
! [T: int] :
? [Z2: int] :
! [X6: int] :
( ( ord_less_int @ X6 @ Z2 )
=> ~ ( ord_less_int @ T @ X6 ) ) ).
% minf(7)
thf(fact_270_linorder__neqE__nat,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neqE_nat
thf(fact_271_infinite__descent,axiom,
! [P2: nat > $o,N: nat] :
( ! [N3: nat] :
( ~ ( P2 @ N3 )
=> ? [M3: nat] :
( ( ord_less_nat @ M3 @ N3 )
& ~ ( P2 @ M3 ) ) )
=> ( P2 @ N ) ) ).
% infinite_descent
thf(fact_272_nat__less__induct,axiom,
! [P2: nat > $o,N: nat] :
( ! [N3: nat] :
( ! [M3: nat] :
( ( ord_less_nat @ M3 @ N3 )
=> ( P2 @ M3 ) )
=> ( P2 @ N3 ) )
=> ( P2 @ N ) ) ).
% nat_less_induct
thf(fact_273_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_274_less__not__refl3,axiom,
! [S: nat,T: nat] :
( ( ord_less_nat @ S @ T )
=> ( S != T ) ) ).
% less_not_refl3
thf(fact_275_less__not__refl2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
=> ( M != N ) ) ).
% less_not_refl2
thf(fact_276_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_not_refl
thf(fact_277_nat__neq__iff,axiom,
! [M: nat,N: nat] :
( ( M != N )
= ( ( ord_less_nat @ M @ N )
| ( ord_less_nat @ N @ M ) ) ) ).
% nat_neq_iff
thf(fact_278_size__neq__size__imp__neq,axiom,
! [X: list_P5707943133018811711nt_int,Y: list_P5707943133018811711nt_int] :
( ( ( size_s5157815400016825771nt_int @ X )
!= ( size_s5157815400016825771nt_int @ Y ) )
=> ( X != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_279_Nat_Oex__has__greatest__nat,axiom,
! [P2: nat > $o,K: nat,B: nat] :
( ( P2 @ K )
=> ( ! [Y3: nat] :
( ( P2 @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ B ) )
=> ? [X3: nat] :
( ( P2 @ X3 )
& ! [Y7: nat] :
( ( P2 @ Y7 )
=> ( ord_less_eq_nat @ Y7 @ X3 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_280_nat__le__linear,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
| ( ord_less_eq_nat @ N @ M ) ) ).
% nat_le_linear
thf(fact_281_le__antisym,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( M = N ) ) ) ).
% le_antisym
thf(fact_282_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% eq_imp_le
thf(fact_283_le__trans,axiom,
! [I4: nat,J3: nat,K: nat] :
( ( ord_less_eq_nat @ I4 @ J3 )
=> ( ( ord_less_eq_nat @ J3 @ K )
=> ( ord_less_eq_nat @ I4 @ K ) ) ) ).
% le_trans
thf(fact_284_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_285_order__le__imp__less__or__eq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_nat @ X @ Y )
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_286_order__le__imp__less__or__eq,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ord_less_int @ X @ Y )
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_287_linorder__le__less__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
| ( ord_less_nat @ Y @ X ) ) ).
% linorder_le_less_linear
thf(fact_288_linorder__le__less__linear,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
| ( ord_less_int @ Y @ X ) ) ).
% linorder_le_less_linear
thf(fact_289_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C2 )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% order_less_le_subst2
thf(fact_290_order__less__le__subst2,axiom,
! [A: int,B: int,F: int > nat,C2: nat] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C2 )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% order_less_le_subst2
thf(fact_291_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > int,C2: int] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C2 )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C2 ) ) ) ) ).
% order_less_le_subst2
thf(fact_292_order__less__le__subst2,axiom,
! [A: int,B: int,F: int > int,C2: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C2 )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C2 ) ) ) ) ).
% order_less_le_subst2
thf(fact_293_order__less__le__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C2: nat] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_294_order__less__le__subst1,axiom,
! [A: int,F: nat > int,B: nat,C2: nat] :
( ( ord_less_int @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A @ ( F @ C2 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_295_order__less__le__subst1,axiom,
! [A: nat,F: int > nat,B: int,C2: int] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C2 )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_296_order__less__le__subst1,axiom,
! [A: int,F: int > int,B: int,C2: int] :
( ( ord_less_int @ A @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C2 )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A @ ( F @ C2 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_297_order__le__less__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C2 )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_298_order__le__less__subst2,axiom,
! [A: nat,B: nat,F: nat > int,C2: int] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C2 )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_299_order__le__less__subst2,axiom,
! [A: int,B: int,F: int > nat,C2: nat] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C2 )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_300_order__le__less__subst2,axiom,
! [A: int,B: int,F: int > int,C2: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C2 )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_301_order__le__less__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_302_order__le__less__subst1,axiom,
! [A: nat,F: int > nat,B: int,C2: int] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C2 )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_303_order__le__less__subst1,axiom,
! [A: int,F: nat > int,B: nat,C2: nat] :
( ( ord_less_eq_int @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A @ ( F @ C2 ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_304_order__le__less__subst1,axiom,
! [A: int,F: int > int,B: int,C2: int] :
( ( ord_less_eq_int @ A @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C2 )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A @ ( F @ C2 ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_305_order__less__le__trans,axiom,
! [X: nat,Y: nat,Z3: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z3 )
=> ( ord_less_nat @ X @ Z3 ) ) ) ).
% order_less_le_trans
thf(fact_306_order__less__le__trans,axiom,
! [X: int,Y: int,Z3: int] :
( ( ord_less_int @ X @ Y )
=> ( ( ord_less_eq_int @ Y @ Z3 )
=> ( ord_less_int @ X @ Z3 ) ) ) ).
% order_less_le_trans
thf(fact_307_order__le__less__trans,axiom,
! [X: nat,Y: nat,Z3: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ Z3 )
=> ( ord_less_nat @ X @ Z3 ) ) ) ).
% order_le_less_trans
thf(fact_308_order__le__less__trans,axiom,
! [X: int,Y: int,Z3: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ord_less_int @ Y @ Z3 )
=> ( ord_less_int @ X @ Z3 ) ) ) ).
% order_le_less_trans
thf(fact_309_order__neq__le__trans,axiom,
! [A: nat,B: nat] :
( ( A != B )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_nat @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_310_order__neq__le__trans,axiom,
! [A: int,B: int] :
( ( A != B )
=> ( ( ord_less_eq_int @ A @ B )
=> ( ord_less_int @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_311_order__le__neq__trans,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( A != B )
=> ( ord_less_nat @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_312_order__le__neq__trans,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( A != B )
=> ( ord_less_int @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_313_order__less__imp__le,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ord_less_eq_nat @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_314_order__less__imp__le,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ( ord_less_eq_int @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_315_linorder__not__less,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X @ Y ) )
= ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_not_less
thf(fact_316_linorder__not__less,axiom,
! [X: int,Y: int] :
( ( ~ ( ord_less_int @ X @ Y ) )
= ( ord_less_eq_int @ Y @ X ) ) ).
% linorder_not_less
thf(fact_317_linorder__not__le,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_eq_nat @ X @ Y ) )
= ( ord_less_nat @ Y @ X ) ) ).
% linorder_not_le
thf(fact_318_linorder__not__le,axiom,
! [X: int,Y: int] :
( ( ~ ( ord_less_eq_int @ X @ Y ) )
= ( ord_less_int @ Y @ X ) ) ).
% linorder_not_le
thf(fact_319_order__less__le,axiom,
( ord_less_nat
= ( ^ [X4: nat,Y6: nat] :
( ( ord_less_eq_nat @ X4 @ Y6 )
& ( X4 != Y6 ) ) ) ) ).
% order_less_le
thf(fact_320_order__less__le,axiom,
( ord_less_int
= ( ^ [X4: int,Y6: int] :
( ( ord_less_eq_int @ X4 @ Y6 )
& ( X4 != Y6 ) ) ) ) ).
% order_less_le
thf(fact_321_order__le__less,axiom,
( ord_less_eq_nat
= ( ^ [X4: nat,Y6: nat] :
( ( ord_less_nat @ X4 @ Y6 )
| ( X4 = Y6 ) ) ) ) ).
% order_le_less
thf(fact_322_order__le__less,axiom,
( ord_less_eq_int
= ( ^ [X4: int,Y6: int] :
( ( ord_less_int @ X4 @ Y6 )
| ( X4 = Y6 ) ) ) ) ).
% order_le_less
thf(fact_323_dual__order_Ostrict__implies__order,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( ord_less_eq_nat @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_324_dual__order_Ostrict__implies__order,axiom,
! [B: int,A: int] :
( ( ord_less_int @ B @ A )
=> ( ord_less_eq_int @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_325_order_Ostrict__implies__order,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_326_order_Ostrict__implies__order,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_eq_int @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_327_dual__order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [B4: nat,A5: nat] :
( ( ord_less_eq_nat @ B4 @ A5 )
& ~ ( ord_less_eq_nat @ A5 @ B4 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_328_dual__order_Ostrict__iff__not,axiom,
( ord_less_int
= ( ^ [B4: int,A5: int] :
( ( ord_less_eq_int @ B4 @ A5 )
& ~ ( ord_less_eq_int @ A5 @ B4 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_329_dual__order_Ostrict__trans2,axiom,
! [B: nat,A: nat,C2: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( ord_less_eq_nat @ C2 @ B )
=> ( ord_less_nat @ C2 @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_330_dual__order_Ostrict__trans2,axiom,
! [B: int,A: int,C2: int] :
( ( ord_less_int @ B @ A )
=> ( ( ord_less_eq_int @ C2 @ B )
=> ( ord_less_int @ C2 @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_331_dual__order_Ostrict__trans1,axiom,
! [B: nat,A: nat,C2: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_nat @ C2 @ B )
=> ( ord_less_nat @ C2 @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_332_dual__order_Ostrict__trans1,axiom,
! [B: int,A: int,C2: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( ord_less_int @ C2 @ B )
=> ( ord_less_int @ C2 @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_333_dual__order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [B4: nat,A5: nat] :
( ( ord_less_eq_nat @ B4 @ A5 )
& ( A5 != B4 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_334_dual__order_Ostrict__iff__order,axiom,
( ord_less_int
= ( ^ [B4: int,A5: int] :
( ( ord_less_eq_int @ B4 @ A5 )
& ( A5 != B4 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_335_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [B4: nat,A5: nat] :
( ( ord_less_nat @ B4 @ A5 )
| ( A5 = B4 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_336_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_int
= ( ^ [B4: int,A5: int] :
( ( ord_less_int @ B4 @ A5 )
| ( A5 = B4 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_337_order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [A5: nat,B4: nat] :
( ( ord_less_eq_nat @ A5 @ B4 )
& ~ ( ord_less_eq_nat @ B4 @ A5 ) ) ) ) ).
% order.strict_iff_not
thf(fact_338_order_Ostrict__iff__not,axiom,
( ord_less_int
= ( ^ [A5: int,B4: int] :
( ( ord_less_eq_int @ A5 @ B4 )
& ~ ( ord_less_eq_int @ B4 @ A5 ) ) ) ) ).
% order.strict_iff_not
thf(fact_339_order_Ostrict__trans2,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ord_less_nat @ A @ C2 ) ) ) ).
% order.strict_trans2
thf(fact_340_order_Ostrict__trans2,axiom,
! [A: int,B: int,C2: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_int @ B @ C2 )
=> ( ord_less_int @ A @ C2 ) ) ) ).
% order.strict_trans2
thf(fact_341_order_Ostrict__trans1,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ord_less_nat @ A @ C2 ) ) ) ).
% order.strict_trans1
thf(fact_342_order_Ostrict__trans1,axiom,
! [A: int,B: int,C2: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_int @ B @ C2 )
=> ( ord_less_int @ A @ C2 ) ) ) ).
% order.strict_trans1
thf(fact_343_order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [A5: nat,B4: nat] :
( ( ord_less_eq_nat @ A5 @ B4 )
& ( A5 != B4 ) ) ) ) ).
% order.strict_iff_order
thf(fact_344_order_Ostrict__iff__order,axiom,
( ord_less_int
= ( ^ [A5: int,B4: int] :
( ( ord_less_eq_int @ A5 @ B4 )
& ( A5 != B4 ) ) ) ) ).
% order.strict_iff_order
thf(fact_345_order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [A5: nat,B4: nat] :
( ( ord_less_nat @ A5 @ B4 )
| ( A5 = B4 ) ) ) ) ).
% order.order_iff_strict
thf(fact_346_order_Oorder__iff__strict,axiom,
( ord_less_eq_int
= ( ^ [A5: int,B4: int] :
( ( ord_less_int @ A5 @ B4 )
| ( A5 = B4 ) ) ) ) ).
% order.order_iff_strict
thf(fact_347_not__le__imp__less,axiom,
! [Y: nat,X: nat] :
( ~ ( ord_less_eq_nat @ Y @ X )
=> ( ord_less_nat @ X @ Y ) ) ).
% not_le_imp_less
thf(fact_348_not__le__imp__less,axiom,
! [Y: int,X: int] :
( ~ ( ord_less_eq_int @ Y @ X )
=> ( ord_less_int @ X @ Y ) ) ).
% not_le_imp_less
thf(fact_349_less__le__not__le,axiom,
( ord_less_nat
= ( ^ [X4: nat,Y6: nat] :
( ( ord_less_eq_nat @ X4 @ Y6 )
& ~ ( ord_less_eq_nat @ Y6 @ X4 ) ) ) ) ).
% less_le_not_le
thf(fact_350_less__le__not__le,axiom,
( ord_less_int
= ( ^ [X4: int,Y6: int] :
( ( ord_less_eq_int @ X4 @ Y6 )
& ~ ( ord_less_eq_int @ Y6 @ X4 ) ) ) ) ).
% less_le_not_le
thf(fact_351_antisym__conv2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ~ ( ord_less_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_352_antisym__conv2,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ~ ( ord_less_int @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_353_antisym__conv1,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_354_antisym__conv1,axiom,
! [X: int,Y: int] :
( ~ ( ord_less_int @ X @ Y )
=> ( ( ord_less_eq_int @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_355_nless__le,axiom,
! [A: nat,B: nat] :
( ( ~ ( ord_less_nat @ A @ B ) )
= ( ~ ( ord_less_eq_nat @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_356_nless__le,axiom,
! [A: int,B: int] :
( ( ~ ( ord_less_int @ A @ B ) )
= ( ~ ( ord_less_eq_int @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_357_leI,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_eq_nat @ Y @ X ) ) ).
% leI
thf(fact_358_leI,axiom,
! [X: int,Y: int] :
( ~ ( ord_less_int @ X @ Y )
=> ( ord_less_eq_int @ Y @ X ) ) ).
% leI
thf(fact_359_leD,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ~ ( ord_less_nat @ X @ Y ) ) ).
% leD
thf(fact_360_leD,axiom,
! [Y: int,X: int] :
( ( ord_less_eq_int @ Y @ X )
=> ~ ( ord_less_int @ X @ Y ) ) ).
% leD
thf(fact_361_pinf_I6_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z2 @ X6 )
=> ~ ( ord_less_eq_nat @ X6 @ T ) ) ).
% pinf(6)
thf(fact_362_pinf_I6_J,axiom,
! [T: int] :
? [Z2: int] :
! [X6: int] :
( ( ord_less_int @ Z2 @ X6 )
=> ~ ( ord_less_eq_int @ X6 @ T ) ) ).
% pinf(6)
thf(fact_363_pinf_I8_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z2 @ X6 )
=> ( ord_less_eq_nat @ T @ X6 ) ) ).
% pinf(8)
thf(fact_364_pinf_I8_J,axiom,
! [T: int] :
? [Z2: int] :
! [X6: int] :
( ( ord_less_int @ Z2 @ X6 )
=> ( ord_less_eq_int @ T @ X6 ) ) ).
% pinf(8)
thf(fact_365_minf_I6_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z2 )
=> ( ord_less_eq_nat @ X6 @ T ) ) ).
% minf(6)
thf(fact_366_minf_I6_J,axiom,
! [T: int] :
? [Z2: int] :
! [X6: int] :
( ( ord_less_int @ X6 @ Z2 )
=> ( ord_less_eq_int @ X6 @ T ) ) ).
% minf(6)
thf(fact_367_complete__interval,axiom,
! [A: nat,B: nat,P2: nat > $o] :
( ( ord_less_nat @ A @ B )
=> ( ( P2 @ A )
=> ( ~ ( P2 @ B )
=> ? [C: nat] :
( ( ord_less_eq_nat @ A @ C )
& ( ord_less_eq_nat @ C @ B )
& ! [X6: nat] :
( ( ( ord_less_eq_nat @ A @ X6 )
& ( ord_less_nat @ X6 @ C ) )
=> ( P2 @ X6 ) )
& ! [D2: nat] :
( ! [X3: nat] :
( ( ( ord_less_eq_nat @ A @ X3 )
& ( ord_less_nat @ X3 @ D2 ) )
=> ( P2 @ X3 ) )
=> ( ord_less_eq_nat @ D2 @ C ) ) ) ) ) ) ).
% complete_interval
thf(fact_368_complete__interval,axiom,
! [A: int,B: int,P2: int > $o] :
( ( ord_less_int @ A @ B )
=> ( ( P2 @ A )
=> ( ~ ( P2 @ B )
=> ? [C: int] :
( ( ord_less_eq_int @ A @ C )
& ( ord_less_eq_int @ C @ B )
& ! [X6: int] :
( ( ( ord_less_eq_int @ A @ X6 )
& ( ord_less_int @ X6 @ C ) )
=> ( P2 @ X6 ) )
& ! [D2: int] :
( ! [X3: int] :
( ( ( ord_less_eq_int @ A @ X3 )
& ( ord_less_int @ X3 @ D2 ) )
=> ( P2 @ X3 ) )
=> ( ord_less_eq_int @ D2 @ C ) ) ) ) ) ) ).
% complete_interval
thf(fact_369_verit__comp__simplify1_I3_J,axiom,
! [B2: nat,A2: nat] :
( ( ~ ( ord_less_eq_nat @ B2 @ A2 ) )
= ( ord_less_nat @ A2 @ B2 ) ) ).
% verit_comp_simplify1(3)
thf(fact_370_verit__comp__simplify1_I3_J,axiom,
! [B2: int,A2: int] :
( ( ~ ( ord_less_eq_int @ B2 @ A2 ) )
= ( ord_less_int @ A2 @ B2 ) ) ).
% verit_comp_simplify1(3)
thf(fact_371_listrel1__iff__update,axiom,
! [Xs: list_int,Ys2: list_int,R: set_Pr958786334691620121nt_int] :
( ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Xs @ Ys2 ) @ ( listrel1_int @ R ) )
= ( ? [Y6: int,N2: nat] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ ( nth_int @ Xs @ N2 ) @ Y6 ) @ R )
& ( ord_less_nat @ N2 @ ( size_size_list_int @ Xs ) )
& ( Ys2
= ( list_update_int @ Xs @ N2 @ Y6 ) ) ) ) ) ).
% listrel1_iff_update
thf(fact_372_listrel1__iff__update,axiom,
! [Xs: list_P5707943133018811711nt_int,Ys2: list_P5707943133018811711nt_int,R: set_Pr2560585780119916871nt_int] :
( ( member1390679175989562640nt_int @ ( produc1932183703851549015nt_int @ Xs @ Ys2 ) @ ( listre1649617614542993432nt_int @ R ) )
= ( ? [Y6: product_prod_int_int,N2: nat] :
( ( member8566619992076573584nt_int @ ( produc3646306378393792727nt_int @ ( nth_Pr4439495888332055232nt_int @ Xs @ N2 ) @ Y6 ) @ R )
& ( ord_less_nat @ N2 @ ( size_s5157815400016825771nt_int @ Xs ) )
& ( Ys2
= ( list_u3002344382305578791nt_int @ Xs @ N2 @ Y6 ) ) ) ) ) ).
% listrel1_iff_update
thf(fact_373_listrel__iff__nth,axiom,
! [Xs: list_nat,Ys2: list_set_int,R: set_Pr4810089274464741491et_int] :
( ( member4311088582351766364et_int @ ( produc6467668960084403947et_int @ Xs @ Ys2 ) @ ( listrel_nat_set_int @ R ) )
= ( ( ( size_size_list_nat @ Xs )
= ( size_s763970886550748294et_int @ Ys2 ) )
& ! [N2: nat] :
( ( ord_less_nat @ N2 @ ( size_size_list_nat @ Xs ) )
=> ( member1292241183792264892et_int @ ( produc29655638201817675et_int @ ( nth_nat @ Xs @ N2 ) @ ( nth_set_int @ Ys2 @ N2 ) ) @ R ) ) ) ) ).
% listrel_iff_nth
thf(fact_374_listrel__iff__nth,axiom,
! [Xs: list_int,Ys2: list_l1670014477004246597nt_int,R: set_Pr1843331182504696254nt_int] :
( ( member4298863958545693205nt_int @ ( produc1403559775789145904nt_int @ Xs @ Ys2 ) @ ( listre1889173885445251040nt_int @ R ) )
= ( ( ( size_size_list_int @ Xs )
= ( size_s2969076144586574001nt_int @ Ys2 ) )
& ! [N2: nat] :
( ( ord_less_nat @ N2 @ ( size_size_list_int @ Xs ) )
=> ( member3263328415308337823nt_int @ ( produc8814303788642274490nt_int @ ( nth_int @ Xs @ N2 ) @ ( nth_li549691060785553606nt_int @ Ys2 @ N2 ) ) @ R ) ) ) ) ).
% listrel_iff_nth
thf(fact_375_listrel__iff__nth,axiom,
! [Xs: list_int,Ys2: list_int,R: set_Pr958786334691620121nt_int] :
( ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Xs @ Ys2 ) @ ( listrel_int_int @ R ) )
= ( ( ( size_size_list_int @ Xs )
= ( size_size_list_int @ Ys2 ) )
& ! [N2: nat] :
( ( ord_less_nat @ N2 @ ( size_size_list_int @ Xs ) )
=> ( member5262025264175285858nt_int @ ( product_Pair_int_int @ ( nth_int @ Xs @ N2 ) @ ( nth_int @ Ys2 @ N2 ) ) @ R ) ) ) ) ).
% listrel_iff_nth
thf(fact_376_listrel__iff__nth,axiom,
! [Xs: list_P1743416141875011707e_term,Ys2: list_P5707943133018811711nt_int,R: set_Pr9222295170931077689nt_int] :
( ( member1869596255364795664nt_int @ ( produc8723725229082728235nt_int @ Xs @ Ys2 ) @ ( listre7929554186369383643nt_int @ R ) )
= ( ( ( size_s3817825649112772879e_term @ Xs )
= ( size_s5157815400016825771nt_int @ Ys2 ) )
& ! [N2: nat] :
( ( ord_less_nat @ N2 @ ( size_s3817825649112772879e_term @ Xs ) )
=> ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ ( nth_Pr6123904574653836964e_term @ Xs @ N2 ) @ ( nth_Pr4439495888332055232nt_int @ Ys2 @ N2 ) ) @ R ) ) ) ) ).
% listrel_iff_nth
thf(fact_377_listrel__iff__nth,axiom,
! [Xs: list_i8448526496819171953e_term,Ys2: list_P5707943133018811711nt_int,R: set_Pr1872883991513573699nt_int] :
( ( member6768764179952475034nt_int @ ( produc2259991554960072885nt_int @ Xs @ Ys2 ) @ ( listre7622244038138809061nt_int @ R ) )
= ( ( ( size_s8881356780637927685e_term @ Xs )
= ( size_s5157815400016825771nt_int @ Ys2 ) )
& ! [N2: nat] :
( ( ord_less_nat @ N2 @ ( size_s8881356780637927685e_term @ Xs ) )
=> ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ ( nth_in8840386337573990810e_term @ Xs @ N2 ) @ ( nth_Pr4439495888332055232nt_int @ Ys2 @ N2 ) ) @ R ) ) ) ) ).
% listrel_iff_nth
thf(fact_378_listrel__iff__nth,axiom,
! [Xs: list_P5707943133018811711nt_int,Ys2: list_P5707943133018811711nt_int,R: set_Pr2560585780119916871nt_int] :
( ( member1390679175989562640nt_int @ ( produc1932183703851549015nt_int @ Xs @ Ys2 ) @ ( listre7525836092462517041nt_int @ R ) )
= ( ( ( size_s5157815400016825771nt_int @ Xs )
= ( size_s5157815400016825771nt_int @ Ys2 ) )
& ! [N2: nat] :
( ( ord_less_nat @ N2 @ ( size_s5157815400016825771nt_int @ Xs ) )
=> ( member8566619992076573584nt_int @ ( produc3646306378393792727nt_int @ ( nth_Pr4439495888332055232nt_int @ Xs @ N2 ) @ ( nth_Pr4439495888332055232nt_int @ Ys2 @ N2 ) ) @ R ) ) ) ) ).
% listrel_iff_nth
thf(fact_379_lex__take__index,axiom,
! [Xs: list_int,Ys2: list_int,R: set_Pr958786334691620121nt_int] :
( ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Xs @ Ys2 ) @ ( lex_int @ R ) )
=> ~ ! [I: nat] :
( ( ord_less_nat @ I @ ( size_size_list_int @ Xs ) )
=> ( ( ord_less_nat @ I @ ( size_size_list_int @ Ys2 ) )
=> ( ( ( take_int @ I @ Xs )
= ( take_int @ I @ Ys2 ) )
=> ~ ( member5262025264175285858nt_int @ ( product_Pair_int_int @ ( nth_int @ Xs @ I ) @ ( nth_int @ Ys2 @ I ) ) @ R ) ) ) ) ) ).
% lex_take_index
thf(fact_380_lex__take__index,axiom,
! [Xs: list_P5707943133018811711nt_int,Ys2: list_P5707943133018811711nt_int,R: set_Pr2560585780119916871nt_int] :
( ( member1390679175989562640nt_int @ ( produc1932183703851549015nt_int @ Xs @ Ys2 ) @ ( lex_Pr5393148144989827363nt_int @ R ) )
=> ~ ! [I: nat] :
( ( ord_less_nat @ I @ ( size_s5157815400016825771nt_int @ Xs ) )
=> ( ( ord_less_nat @ I @ ( size_s5157815400016825771nt_int @ Ys2 ) )
=> ( ( ( take_P8218740963776755879nt_int @ I @ Xs )
= ( take_P8218740963776755879nt_int @ I @ Ys2 ) )
=> ~ ( member8566619992076573584nt_int @ ( produc3646306378393792727nt_int @ ( nth_Pr4439495888332055232nt_int @ Xs @ I ) @ ( nth_Pr4439495888332055232nt_int @ Ys2 @ I ) ) @ R ) ) ) ) ) ).
% lex_take_index
thf(fact_381_rotate1__length01,axiom,
! [Xs: list_P5707943133018811711nt_int] :
( ( ord_less_eq_nat @ ( size_s5157815400016825771nt_int @ Xs ) @ one_one_nat )
=> ( ( rotate9137591727117064716nt_int @ Xs )
= Xs ) ) ).
% rotate1_length01
thf(fact_382_Cons__nth__drop__Suc,axiom,
! [I4: nat,Xs: list_P5707943133018811711nt_int] :
( ( ord_less_nat @ I4 @ ( size_s5157815400016825771nt_int @ Xs ) )
=> ( ( cons_P3334398858971670639nt_int @ ( nth_Pr4439495888332055232nt_int @ Xs @ I4 ) @ ( drop_P5690361596310759935nt_int @ ( suc @ I4 ) @ Xs ) )
= ( drop_P5690361596310759935nt_int @ I4 @ Xs ) ) ) ).
% Cons_nth_drop_Suc
thf(fact_383_nth__drop,axiom,
! [N: nat,Xs: list_P5707943133018811711nt_int,I4: nat] :
( ( ord_less_eq_nat @ N @ ( size_s5157815400016825771nt_int @ Xs ) )
=> ( ( nth_Pr4439495888332055232nt_int @ ( drop_P5690361596310759935nt_int @ N @ Xs ) @ I4 )
= ( nth_Pr4439495888332055232nt_int @ Xs @ ( plus_plus_nat @ N @ I4 ) ) ) ) ).
% nth_drop
thf(fact_384_rotate__length01,axiom,
! [Xs: list_P5707943133018811711nt_int,N: nat] :
( ( ord_less_eq_nat @ ( size_s5157815400016825771nt_int @ Xs ) @ one_one_nat )
=> ( ( rotate3281235723284208371nt_int @ N @ Xs )
= Xs ) ) ).
% rotate_length01
thf(fact_385_list__ex__length,axiom,
( list_e4511028299437703933nt_int
= ( ^ [P4: product_prod_int_int > $o,Xs4: list_P5707943133018811711nt_int] :
? [N2: nat] :
( ( ord_less_nat @ N2 @ ( size_s5157815400016825771nt_int @ Xs4 ) )
& ( P4 @ ( nth_Pr4439495888332055232nt_int @ Xs4 @ N2 ) ) ) ) ) ).
% list_ex_length
thf(fact_386_add__right__cancel,axiom,
! [B: nat,A: nat,C2: nat] :
( ( ( plus_plus_nat @ B @ A )
= ( plus_plus_nat @ C2 @ A ) )
= ( B = C2 ) ) ).
% add_right_cancel
thf(fact_387_add__right__cancel,axiom,
! [B: int,A: int,C2: int] :
( ( ( plus_plus_int @ B @ A )
= ( plus_plus_int @ C2 @ A ) )
= ( B = C2 ) ) ).
% add_right_cancel
thf(fact_388_add__left__cancel,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ( plus_plus_nat @ A @ B )
= ( plus_plus_nat @ A @ C2 ) )
= ( B = C2 ) ) ).
% add_left_cancel
thf(fact_389_add__left__cancel,axiom,
! [A: int,B: int,C2: int] :
( ( ( plus_plus_int @ A @ B )
= ( plus_plus_int @ A @ C2 ) )
= ( B = C2 ) ) ).
% add_left_cancel
thf(fact_390_old_Onat_Oinject,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( suc @ Nat )
= ( suc @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% old.nat.inject
thf(fact_391_nat_Oinject,axiom,
! [X2: nat,Y2: nat] :
( ( ( suc @ X2 )
= ( suc @ Y2 ) )
= ( X2 = Y2 ) ) ).
% nat.inject
thf(fact_392_add__le__cancel__right,axiom,
! [A: nat,C2: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ C2 ) )
= ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_cancel_right
thf(fact_393_add__le__cancel__right,axiom,
! [A: int,C2: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A @ C2 ) @ ( plus_plus_int @ B @ C2 ) )
= ( ord_less_eq_int @ A @ B ) ) ).
% add_le_cancel_right
thf(fact_394_add__le__cancel__left,axiom,
! [C2: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C2 @ A ) @ ( plus_plus_nat @ C2 @ B ) )
= ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_cancel_left
thf(fact_395_add__le__cancel__left,axiom,
! [C2: int,A: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ C2 @ A ) @ ( plus_plus_int @ C2 @ B ) )
= ( ord_less_eq_int @ A @ B ) ) ).
% add_le_cancel_left
thf(fact_396_add__less__cancel__left,axiom,
! [C2: nat,A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C2 @ A ) @ ( plus_plus_nat @ C2 @ B ) )
= ( ord_less_nat @ A @ B ) ) ).
% add_less_cancel_left
thf(fact_397_add__less__cancel__left,axiom,
! [C2: int,A: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ C2 @ A ) @ ( plus_plus_int @ C2 @ B ) )
= ( ord_less_int @ A @ B ) ) ).
% add_less_cancel_left
thf(fact_398_add__less__cancel__right,axiom,
! [A: nat,C2: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ C2 ) )
= ( ord_less_nat @ A @ B ) ) ).
% add_less_cancel_right
thf(fact_399_add__less__cancel__right,axiom,
! [A: int,C2: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ A @ C2 ) @ ( plus_plus_int @ B @ C2 ) )
= ( ord_less_int @ A @ B ) ) ).
% add_less_cancel_right
thf(fact_400_lessI,axiom,
! [N: nat] : ( ord_less_nat @ N @ ( suc @ N ) ) ).
% lessI
thf(fact_401_Suc__mono,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) ) ) ).
% Suc_mono
thf(fact_402_Suc__less__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% Suc_less_eq
thf(fact_403_Suc__le__mono,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( suc @ M ) )
= ( ord_less_eq_nat @ N @ M ) ) ).
% Suc_le_mono
thf(fact_404_add__Suc__right,axiom,
! [M: nat,N: nat] :
( ( plus_plus_nat @ M @ ( suc @ N ) )
= ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).
% add_Suc_right
thf(fact_405_nat__add__left__cancel__less,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% nat_add_left_cancel_less
thf(fact_406_nat__add__left__cancel__le,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% nat_add_left_cancel_le
thf(fact_407_drop__drop,axiom,
! [N: nat,M: nat,Xs: list_P5707943133018811711nt_int] :
( ( drop_P5690361596310759935nt_int @ N @ ( drop_P5690361596310759935nt_int @ M @ Xs ) )
= ( drop_P5690361596310759935nt_int @ ( plus_plus_nat @ N @ M ) @ Xs ) ) ).
% drop_drop
thf(fact_408_length__list__update,axiom,
! [Xs: list_P5707943133018811711nt_int,I4: nat,X: product_prod_int_int] :
( ( size_s5157815400016825771nt_int @ ( list_u3002344382305578791nt_int @ Xs @ I4 @ X ) )
= ( size_s5157815400016825771nt_int @ Xs ) ) ).
% length_list_update
thf(fact_409_length__rotate,axiom,
! [N: nat,Xs: list_P5707943133018811711nt_int] :
( ( size_s5157815400016825771nt_int @ ( rotate3281235723284208371nt_int @ N @ Xs ) )
= ( size_s5157815400016825771nt_int @ Xs ) ) ).
% length_rotate
thf(fact_410_length__rotate1,axiom,
! [Xs: list_P5707943133018811711nt_int] :
( ( size_s5157815400016825771nt_int @ ( rotate9137591727117064716nt_int @ Xs ) )
= ( size_s5157815400016825771nt_int @ Xs ) ) ).
% length_rotate1
thf(fact_411_list__ex__simps_I1_J,axiom,
! [P2: product_prod_int_int > $o,X: product_prod_int_int,Xs: list_P5707943133018811711nt_int] :
( ( list_e4511028299437703933nt_int @ P2 @ ( cons_P3334398858971670639nt_int @ X @ Xs ) )
= ( ( P2 @ X )
| ( list_e4511028299437703933nt_int @ P2 @ Xs ) ) ) ).
% list_ex_simps(1)
thf(fact_412_nth__Cons__Suc,axiom,
! [X: product_prod_int_int,Xs: list_P5707943133018811711nt_int,N: nat] :
( ( nth_Pr4439495888332055232nt_int @ ( cons_P3334398858971670639nt_int @ X @ Xs ) @ ( suc @ N ) )
= ( nth_Pr4439495888332055232nt_int @ Xs @ N ) ) ).
% nth_Cons_Suc
thf(fact_413_take__Suc__Cons,axiom,
! [N: nat,X: product_prod_int_int,Xs: list_P5707943133018811711nt_int] :
( ( take_P8218740963776755879nt_int @ ( suc @ N ) @ ( cons_P3334398858971670639nt_int @ X @ Xs ) )
= ( cons_P3334398858971670639nt_int @ X @ ( take_P8218740963776755879nt_int @ N @ Xs ) ) ) ).
% take_Suc_Cons
thf(fact_414_drop__Suc__Cons,axiom,
! [N: nat,X: product_prod_int_int,Xs: list_P5707943133018811711nt_int] :
( ( drop_P5690361596310759935nt_int @ ( suc @ N ) @ ( cons_P3334398858971670639nt_int @ X @ Xs ) )
= ( drop_P5690361596310759935nt_int @ N @ Xs ) ) ).
% drop_Suc_Cons
thf(fact_415_take__all,axiom,
! [Xs: list_P5707943133018811711nt_int,N: nat] :
( ( ord_less_eq_nat @ ( size_s5157815400016825771nt_int @ Xs ) @ N )
=> ( ( take_P8218740963776755879nt_int @ N @ Xs )
= Xs ) ) ).
% take_all
thf(fact_416_take__all__iff,axiom,
! [N: nat,Xs: list_P5707943133018811711nt_int] :
( ( ( take_P8218740963776755879nt_int @ N @ Xs )
= Xs )
= ( ord_less_eq_nat @ ( size_s5157815400016825771nt_int @ Xs ) @ N ) ) ).
% take_all_iff
thf(fact_417_nth__take,axiom,
! [I4: nat,N: nat,Xs: list_P5707943133018811711nt_int] :
( ( ord_less_nat @ I4 @ N )
=> ( ( nth_Pr4439495888332055232nt_int @ ( take_P8218740963776755879nt_int @ N @ Xs ) @ I4 )
= ( nth_Pr4439495888332055232nt_int @ Xs @ I4 ) ) ) ).
% nth_take
thf(fact_418_list__update__beyond,axiom,
! [Xs: list_P5707943133018811711nt_int,I4: nat,X: product_prod_int_int] :
( ( ord_less_eq_nat @ ( size_s5157815400016825771nt_int @ Xs ) @ I4 )
=> ( ( list_u3002344382305578791nt_int @ Xs @ I4 @ X )
= Xs ) ) ).
% list_update_beyond
thf(fact_419_take__update__cancel,axiom,
! [N: nat,M: nat,Xs: list_P5707943133018811711nt_int,Y: product_prod_int_int] :
( ( ord_less_eq_nat @ N @ M )
=> ( ( take_P8218740963776755879nt_int @ N @ ( list_u3002344382305578791nt_int @ Xs @ M @ Y ) )
= ( take_P8218740963776755879nt_int @ N @ Xs ) ) ) ).
% take_update_cancel
thf(fact_420_drop__update__cancel,axiom,
! [N: nat,M: nat,Xs: list_P5707943133018811711nt_int,X: product_prod_int_int] :
( ( ord_less_nat @ N @ M )
=> ( ( drop_P5690361596310759935nt_int @ M @ ( list_u3002344382305578791nt_int @ Xs @ N @ X ) )
= ( drop_P5690361596310759935nt_int @ M @ Xs ) ) ) ).
% drop_update_cancel
thf(fact_421_nth__list__update__eq,axiom,
! [I4: nat,Xs: list_P5707943133018811711nt_int,X: product_prod_int_int] :
( ( ord_less_nat @ I4 @ ( size_s5157815400016825771nt_int @ Xs ) )
=> ( ( nth_Pr4439495888332055232nt_int @ ( list_u3002344382305578791nt_int @ Xs @ I4 @ X ) @ I4 )
= X ) ) ).
% nth_list_update_eq
thf(fact_422_add__Suc__shift,axiom,
! [M: nat,N: nat] :
( ( plus_plus_nat @ ( suc @ M ) @ N )
= ( plus_plus_nat @ M @ ( suc @ N ) ) ) ).
% add_Suc_shift
thf(fact_423_n__not__Suc__n,axiom,
! [N: nat] :
( N
!= ( suc @ N ) ) ).
% n_not_Suc_n
thf(fact_424_Suc__inject,axiom,
! [X: nat,Y: nat] :
( ( ( suc @ X )
= ( suc @ Y ) )
=> ( X = Y ) ) ).
% Suc_inject
thf(fact_425_add__Suc,axiom,
! [M: nat,N: nat] :
( ( plus_plus_nat @ ( suc @ M ) @ N )
= ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).
% add_Suc
thf(fact_426_nat__arith_Osuc1,axiom,
! [A4: nat,K: nat,A: nat] :
( ( A4
= ( plus_plus_nat @ K @ A ) )
=> ( ( suc @ A4 )
= ( plus_plus_nat @ K @ ( suc @ A ) ) ) ) ).
% nat_arith.suc1
thf(fact_427_less__imp__Suc__add,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ? [K2: nat] :
( N
= ( suc @ ( plus_plus_nat @ M @ K2 ) ) ) ) ).
% less_imp_Suc_add
thf(fact_428_less__iff__Suc__add,axiom,
( ord_less_nat
= ( ^ [M2: nat,N2: nat] :
? [K3: nat] :
( N2
= ( suc @ ( plus_plus_nat @ M2 @ K3 ) ) ) ) ) ).
% less_iff_Suc_add
thf(fact_429_less__add__Suc2,axiom,
! [I4: nat,M: nat] : ( ord_less_nat @ I4 @ ( suc @ ( plus_plus_nat @ M @ I4 ) ) ) ).
% less_add_Suc2
thf(fact_430_less__add__Suc1,axiom,
! [I4: nat,M: nat] : ( ord_less_nat @ I4 @ ( suc @ ( plus_plus_nat @ I4 @ M ) ) ) ).
% less_add_Suc1
thf(fact_431_less__natE,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ~ ! [Q3: nat] :
( N
!= ( suc @ ( plus_plus_nat @ M @ Q3 ) ) ) ) ).
% less_natE
thf(fact_432_list__update__code_I3_J,axiom,
! [X: product_prod_int_int,Xs: list_P5707943133018811711nt_int,I4: nat,Y: product_prod_int_int] :
( ( list_u3002344382305578791nt_int @ ( cons_P3334398858971670639nt_int @ X @ Xs ) @ ( suc @ I4 ) @ Y )
= ( cons_P3334398858971670639nt_int @ X @ ( list_u3002344382305578791nt_int @ Xs @ I4 @ Y ) ) ) ).
% list_update_code(3)
thf(fact_433_add__right__imp__eq,axiom,
! [B: nat,A: nat,C2: nat] :
( ( ( plus_plus_nat @ B @ A )
= ( plus_plus_nat @ C2 @ A ) )
=> ( B = C2 ) ) ).
% add_right_imp_eq
thf(fact_434_add__right__imp__eq,axiom,
! [B: int,A: int,C2: int] :
( ( ( plus_plus_int @ B @ A )
= ( plus_plus_int @ C2 @ A ) )
=> ( B = C2 ) ) ).
% add_right_imp_eq
thf(fact_435_add__left__imp__eq,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ( plus_plus_nat @ A @ B )
= ( plus_plus_nat @ A @ C2 ) )
=> ( B = C2 ) ) ).
% add_left_imp_eq
thf(fact_436_add__left__imp__eq,axiom,
! [A: int,B: int,C2: int] :
( ( ( plus_plus_int @ A @ B )
= ( plus_plus_int @ A @ C2 ) )
=> ( B = C2 ) ) ).
% add_left_imp_eq
thf(fact_437_add_Oleft__commute,axiom,
! [B: nat,A: nat,C2: nat] :
( ( plus_plus_nat @ B @ ( plus_plus_nat @ A @ C2 ) )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C2 ) ) ) ).
% add.left_commute
thf(fact_438_add_Oleft__commute,axiom,
! [B: int,A: int,C2: int] :
( ( plus_plus_int @ B @ ( plus_plus_int @ A @ C2 ) )
= ( plus_plus_int @ A @ ( plus_plus_int @ B @ C2 ) ) ) ).
% add.left_commute
thf(fact_439_add_Ocommute,axiom,
( plus_plus_nat
= ( ^ [A5: nat,B4: nat] : ( plus_plus_nat @ B4 @ A5 ) ) ) ).
% add.commute
thf(fact_440_add_Ocommute,axiom,
( plus_plus_int
= ( ^ [A5: int,B4: int] : ( plus_plus_int @ B4 @ A5 ) ) ) ).
% add.commute
thf(fact_441_add_Oright__cancel,axiom,
! [B: int,A: int,C2: int] :
( ( ( plus_plus_int @ B @ A )
= ( plus_plus_int @ C2 @ A ) )
= ( B = C2 ) ) ).
% add.right_cancel
thf(fact_442_add_Oleft__cancel,axiom,
! [A: int,B: int,C2: int] :
( ( ( plus_plus_int @ A @ B )
= ( plus_plus_int @ A @ C2 ) )
= ( B = C2 ) ) ).
% add.left_cancel
thf(fact_443_add_Oassoc,axiom,
! [A: nat,B: nat,C2: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C2 )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C2 ) ) ) ).
% add.assoc
thf(fact_444_add_Oassoc,axiom,
! [A: int,B: int,C2: int] :
( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C2 )
= ( plus_plus_int @ A @ ( plus_plus_int @ B @ C2 ) ) ) ).
% add.assoc
thf(fact_445_group__cancel_Oadd2,axiom,
! [B5: nat,K: nat,B: nat,A: nat] :
( ( B5
= ( plus_plus_nat @ K @ B ) )
=> ( ( plus_plus_nat @ A @ B5 )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_446_group__cancel_Oadd2,axiom,
! [B5: int,K: int,B: int,A: int] :
( ( B5
= ( plus_plus_int @ K @ B ) )
=> ( ( plus_plus_int @ A @ B5 )
= ( plus_plus_int @ K @ ( plus_plus_int @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_447_group__cancel_Oadd1,axiom,
! [A4: nat,K: nat,A: nat,B: nat] :
( ( A4
= ( plus_plus_nat @ K @ A ) )
=> ( ( plus_plus_nat @ A4 @ B )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_448_group__cancel_Oadd1,axiom,
! [A4: int,K: int,A: int,B: int] :
( ( A4
= ( plus_plus_int @ K @ A ) )
=> ( ( plus_plus_int @ A4 @ B )
= ( plus_plus_int @ K @ ( plus_plus_int @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_449_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I4: nat,J3: nat,K: nat,L: nat] :
( ( ( I4 = J3 )
& ( K = L ) )
=> ( ( plus_plus_nat @ I4 @ K )
= ( plus_plus_nat @ J3 @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_450_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I4: int,J3: int,K: int,L: int] :
( ( ( I4 = J3 )
& ( K = L ) )
=> ( ( plus_plus_int @ I4 @ K )
= ( plus_plus_int @ J3 @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_451_is__num__normalize_I1_J,axiom,
! [A: int,B: int,C2: int] :
( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C2 )
= ( plus_plus_int @ A @ ( plus_plus_int @ B @ C2 ) ) ) ).
% is_num_normalize(1)
thf(fact_452_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: nat,B: nat,C2: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C2 )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C2 ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_453_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: int,B: int,C2: int] :
( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C2 )
= ( plus_plus_int @ A @ ( plus_plus_int @ B @ C2 ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_454_take__update__swap,axiom,
! [M: nat,Xs: list_P5707943133018811711nt_int,N: nat,X: product_prod_int_int] :
( ( take_P8218740963776755879nt_int @ M @ ( list_u3002344382305578791nt_int @ Xs @ N @ X ) )
= ( list_u3002344382305578791nt_int @ ( take_P8218740963776755879nt_int @ M @ Xs ) @ N @ X ) ) ).
% take_update_swap
thf(fact_455_take__equalityI,axiom,
! [Xs: list_P5707943133018811711nt_int,Ys2: list_P5707943133018811711nt_int] :
( ! [I: nat] :
( ( take_P8218740963776755879nt_int @ I @ Xs )
= ( take_P8218740963776755879nt_int @ I @ Ys2 ) )
=> ( Xs = Ys2 ) ) ).
% take_equalityI
thf(fact_456_take__drop,axiom,
! [N: nat,M: nat,Xs: list_P5707943133018811711nt_int] :
( ( take_P8218740963776755879nt_int @ N @ ( drop_P5690361596310759935nt_int @ M @ Xs ) )
= ( drop_P5690361596310759935nt_int @ M @ ( take_P8218740963776755879nt_int @ ( plus_plus_nat @ N @ M ) @ Xs ) ) ) ).
% take_drop
thf(fact_457_Suc__eq__plus1,axiom,
( suc
= ( ^ [N2: nat] : ( plus_plus_nat @ N2 @ one_one_nat ) ) ) ).
% Suc_eq_plus1
thf(fact_458_plus__1__eq__Suc,axiom,
( ( plus_plus_nat @ one_one_nat )
= suc ) ).
% plus_1_eq_Suc
thf(fact_459_Suc__eq__plus1__left,axiom,
( suc
= ( plus_plus_nat @ one_one_nat ) ) ).
% Suc_eq_plus1_left
thf(fact_460_add__le__imp__le__right,axiom,
! [A: nat,C2: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ C2 ) )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_imp_le_right
thf(fact_461_add__le__imp__le__right,axiom,
! [A: int,C2: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A @ C2 ) @ ( plus_plus_int @ B @ C2 ) )
=> ( ord_less_eq_int @ A @ B ) ) ).
% add_le_imp_le_right
thf(fact_462_add__le__imp__le__left,axiom,
! [C2: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C2 @ A ) @ ( plus_plus_nat @ C2 @ B ) )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_imp_le_left
thf(fact_463_add__le__imp__le__left,axiom,
! [C2: int,A: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ C2 @ A ) @ ( plus_plus_int @ C2 @ B ) )
=> ( ord_less_eq_int @ A @ B ) ) ).
% add_le_imp_le_left
thf(fact_464_le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [A5: nat,B4: nat] :
? [C3: nat] :
( B4
= ( plus_plus_nat @ A5 @ C3 ) ) ) ) ).
% le_iff_add
thf(fact_465_add__right__mono,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ C2 ) ) ) ).
% add_right_mono
thf(fact_466_add__right__mono,axiom,
! [A: int,B: int,C2: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ C2 ) @ ( plus_plus_int @ B @ C2 ) ) ) ).
% add_right_mono
thf(fact_467_less__eqE,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ~ ! [C: nat] :
( B
!= ( plus_plus_nat @ A @ C ) ) ) ).
% less_eqE
thf(fact_468_add__left__mono,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ C2 @ A ) @ ( plus_plus_nat @ C2 @ B ) ) ) ).
% add_left_mono
thf(fact_469_add__left__mono,axiom,
! [A: int,B: int,C2: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ord_less_eq_int @ ( plus_plus_int @ C2 @ A ) @ ( plus_plus_int @ C2 @ B ) ) ) ).
% add_left_mono
thf(fact_470_add__mono,axiom,
! [A: nat,B: nat,C2: nat,D3: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C2 @ D3 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ D3 ) ) ) ) ).
% add_mono
thf(fact_471_add__mono,axiom,
! [A: int,B: int,C2: int,D3: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ C2 @ D3 )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ C2 ) @ ( plus_plus_int @ B @ D3 ) ) ) ) ).
% add_mono
thf(fact_472_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I4: nat,J3: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I4 @ J3 )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I4 @ K ) @ ( plus_plus_nat @ J3 @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_473_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I4: int,J3: int,K: int,L: int] :
( ( ( ord_less_eq_int @ I4 @ J3 )
& ( ord_less_eq_int @ K @ L ) )
=> ( ord_less_eq_int @ ( plus_plus_int @ I4 @ K ) @ ( plus_plus_int @ J3 @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_474_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I4: nat,J3: nat,K: nat,L: nat] :
( ( ( I4 = J3 )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I4 @ K ) @ ( plus_plus_nat @ J3 @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_475_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I4: int,J3: int,K: int,L: int] :
( ( ( I4 = J3 )
& ( ord_less_eq_int @ K @ L ) )
=> ( ord_less_eq_int @ ( plus_plus_int @ I4 @ K ) @ ( plus_plus_int @ J3 @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_476_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I4: nat,J3: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I4 @ J3 )
& ( K = L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I4 @ K ) @ ( plus_plus_nat @ J3 @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_477_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I4: int,J3: int,K: int,L: int] :
( ( ( ord_less_eq_int @ I4 @ J3 )
& ( K = L ) )
=> ( ord_less_eq_int @ ( plus_plus_int @ I4 @ K ) @ ( plus_plus_int @ J3 @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_478_add__mono__thms__linordered__field_I5_J,axiom,
! [I4: nat,J3: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I4 @ J3 )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I4 @ K ) @ ( plus_plus_nat @ J3 @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_479_add__mono__thms__linordered__field_I5_J,axiom,
! [I4: int,J3: int,K: int,L: int] :
( ( ( ord_less_int @ I4 @ J3 )
& ( ord_less_int @ K @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I4 @ K ) @ ( plus_plus_int @ J3 @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_480_add__mono__thms__linordered__field_I2_J,axiom,
! [I4: nat,J3: nat,K: nat,L: nat] :
( ( ( I4 = J3 )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I4 @ K ) @ ( plus_plus_nat @ J3 @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_481_add__mono__thms__linordered__field_I2_J,axiom,
! [I4: int,J3: int,K: int,L: int] :
( ( ( I4 = J3 )
& ( ord_less_int @ K @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I4 @ K ) @ ( plus_plus_int @ J3 @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_482_add__mono__thms__linordered__field_I1_J,axiom,
! [I4: nat,J3: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I4 @ J3 )
& ( K = L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I4 @ K ) @ ( plus_plus_nat @ J3 @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_483_add__mono__thms__linordered__field_I1_J,axiom,
! [I4: int,J3: int,K: int,L: int] :
( ( ( ord_less_int @ I4 @ J3 )
& ( K = L ) )
=> ( ord_less_int @ ( plus_plus_int @ I4 @ K ) @ ( plus_plus_int @ J3 @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_484_add__strict__mono,axiom,
! [A: nat,B: nat,C2: nat,D3: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C2 @ D3 )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ D3 ) ) ) ) ).
% add_strict_mono
thf(fact_485_add__strict__mono,axiom,
! [A: int,B: int,C2: int,D3: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ C2 @ D3 )
=> ( ord_less_int @ ( plus_plus_int @ A @ C2 ) @ ( plus_plus_int @ B @ D3 ) ) ) ) ).
% add_strict_mono
thf(fact_486_add__strict__left__mono,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ C2 @ A ) @ ( plus_plus_nat @ C2 @ B ) ) ) ).
% add_strict_left_mono
thf(fact_487_add__strict__left__mono,axiom,
! [A: int,B: int,C2: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_int @ ( plus_plus_int @ C2 @ A ) @ ( plus_plus_int @ C2 @ B ) ) ) ).
% add_strict_left_mono
thf(fact_488_add__strict__right__mono,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ C2 ) ) ) ).
% add_strict_right_mono
thf(fact_489_add__strict__right__mono,axiom,
! [A: int,B: int,C2: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_int @ ( plus_plus_int @ A @ C2 ) @ ( plus_plus_int @ B @ C2 ) ) ) ).
% add_strict_right_mono
thf(fact_490_add__less__imp__less__left,axiom,
! [C2: nat,A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C2 @ A ) @ ( plus_plus_nat @ C2 @ B ) )
=> ( ord_less_nat @ A @ B ) ) ).
% add_less_imp_less_left
thf(fact_491_add__less__imp__less__left,axiom,
! [C2: int,A: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ C2 @ A ) @ ( plus_plus_int @ C2 @ B ) )
=> ( ord_less_int @ A @ B ) ) ).
% add_less_imp_less_left
thf(fact_492_add__less__imp__less__right,axiom,
! [A: nat,C2: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ C2 ) )
=> ( ord_less_nat @ A @ B ) ) ).
% add_less_imp_less_right
thf(fact_493_add__less__imp__less__right,axiom,
! [A: int,C2: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ A @ C2 ) @ ( plus_plus_int @ B @ C2 ) )
=> ( ord_less_int @ A @ B ) ) ).
% add_less_imp_less_right
thf(fact_494_Nat_OlessE,axiom,
! [I4: nat,K: nat] :
( ( ord_less_nat @ I4 @ K )
=> ( ( K
!= ( suc @ I4 ) )
=> ~ ! [J: nat] :
( ( ord_less_nat @ I4 @ J )
=> ( K
!= ( suc @ J ) ) ) ) ) ).
% Nat.lessE
thf(fact_495_Suc__lessD,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ N )
=> ( ord_less_nat @ M @ N ) ) ).
% Suc_lessD
thf(fact_496_Suc__lessE,axiom,
! [I4: nat,K: nat] :
( ( ord_less_nat @ ( suc @ I4 ) @ K )
=> ~ ! [J: nat] :
( ( ord_less_nat @ I4 @ J )
=> ( K
!= ( suc @ J ) ) ) ) ).
% Suc_lessE
thf(fact_497_Suc__lessI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ( ( suc @ M )
!= N )
=> ( ord_less_nat @ ( suc @ M ) @ N ) ) ) ).
% Suc_lessI
thf(fact_498_less__SucE,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
=> ( ~ ( ord_less_nat @ M @ N )
=> ( M = N ) ) ) ).
% less_SucE
thf(fact_499_less__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ M @ ( suc @ N ) ) ) ).
% less_SucI
thf(fact_500_Ex__less__Suc,axiom,
! [N: nat,P2: nat > $o] :
( ( ? [I3: nat] :
( ( ord_less_nat @ I3 @ ( suc @ N ) )
& ( P2 @ I3 ) ) )
= ( ( P2 @ N )
| ? [I3: nat] :
( ( ord_less_nat @ I3 @ N )
& ( P2 @ I3 ) ) ) ) ).
% Ex_less_Suc
thf(fact_501_less__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
= ( ( ord_less_nat @ M @ N )
| ( M = N ) ) ) ).
% less_Suc_eq
thf(fact_502_not__less__eq,axiom,
! [M: nat,N: nat] :
( ( ~ ( ord_less_nat @ M @ N ) )
= ( ord_less_nat @ N @ ( suc @ M ) ) ) ).
% not_less_eq
thf(fact_503_All__less__Suc,axiom,
! [N: nat,P2: nat > $o] :
( ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( suc @ N ) )
=> ( P2 @ I3 ) ) )
= ( ( P2 @ N )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ N )
=> ( P2 @ I3 ) ) ) ) ).
% All_less_Suc
thf(fact_504_Suc__less__eq2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ ( suc @ N ) @ M )
= ( ? [M4: nat] :
( ( M
= ( suc @ M4 ) )
& ( ord_less_nat @ N @ M4 ) ) ) ) ).
% Suc_less_eq2
thf(fact_505_less__antisym,axiom,
! [N: nat,M: nat] :
( ~ ( ord_less_nat @ N @ M )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
=> ( M = N ) ) ) ).
% less_antisym
thf(fact_506_Suc__less__SucD,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
=> ( ord_less_nat @ M @ N ) ) ).
% Suc_less_SucD
thf(fact_507_less__trans__Suc,axiom,
! [I4: nat,J3: nat,K: nat] :
( ( ord_less_nat @ I4 @ J3 )
=> ( ( ord_less_nat @ J3 @ K )
=> ( ord_less_nat @ ( suc @ I4 ) @ K ) ) ) ).
% less_trans_Suc
thf(fact_508_less__Suc__induct,axiom,
! [I4: nat,J3: nat,P2: nat > nat > $o] :
( ( ord_less_nat @ I4 @ J3 )
=> ( ! [I: nat] : ( P2 @ I @ ( suc @ I ) )
=> ( ! [I: nat,J: nat,K2: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ J @ K2 )
=> ( ( P2 @ I @ J )
=> ( ( P2 @ J @ K2 )
=> ( P2 @ I @ K2 ) ) ) ) )
=> ( P2 @ I4 @ J3 ) ) ) ) ).
% less_Suc_induct
thf(fact_509_strict__inc__induct,axiom,
! [I4: nat,J3: nat,P2: nat > $o] :
( ( ord_less_nat @ I4 @ J3 )
=> ( ! [I: nat] :
( ( J3
= ( suc @ I ) )
=> ( P2 @ I ) )
=> ( ! [I: nat] :
( ( ord_less_nat @ I @ J3 )
=> ( ( P2 @ ( suc @ I ) )
=> ( P2 @ I ) ) )
=> ( P2 @ I4 ) ) ) ) ).
% strict_inc_induct
thf(fact_510_not__less__less__Suc__eq,axiom,
! [N: nat,M: nat] :
( ~ ( ord_less_nat @ N @ M )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
= ( N = M ) ) ) ).
% not_less_less_Suc_eq
thf(fact_511_Suc__leD,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% Suc_leD
thf(fact_512_le__SucE,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ ( suc @ N ) )
=> ( ~ ( ord_less_eq_nat @ M @ N )
=> ( M
= ( suc @ N ) ) ) ) ).
% le_SucE
thf(fact_513_le__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ M @ ( suc @ N ) ) ) ).
% le_SucI
thf(fact_514_Suc__le__D,axiom,
! [N: nat,M5: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ M5 )
=> ? [M6: nat] :
( M5
= ( suc @ M6 ) ) ) ).
% Suc_le_D
thf(fact_515_le__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ ( suc @ N ) )
= ( ( ord_less_eq_nat @ M @ N )
| ( M
= ( suc @ N ) ) ) ) ).
% le_Suc_eq
thf(fact_516_Suc__n__not__le__n,axiom,
! [N: nat] :
~ ( ord_less_eq_nat @ ( suc @ N ) @ N ) ).
% Suc_n_not_le_n
thf(fact_517_not__less__eq__eq,axiom,
! [M: nat,N: nat] :
( ( ~ ( ord_less_eq_nat @ M @ N ) )
= ( ord_less_eq_nat @ ( suc @ N ) @ M ) ) ).
% not_less_eq_eq
thf(fact_518_full__nat__induct,axiom,
! [P2: nat > $o,N: nat] :
( ! [N3: nat] :
( ! [M3: nat] :
( ( ord_less_eq_nat @ ( suc @ M3 ) @ N3 )
=> ( P2 @ M3 ) )
=> ( P2 @ N3 ) )
=> ( P2 @ N ) ) ).
% full_nat_induct
thf(fact_519_nat__induct__at__least,axiom,
! [M: nat,N: nat,P2: nat > $o] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( P2 @ M )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ M @ N3 )
=> ( ( P2 @ N3 )
=> ( P2 @ ( suc @ N3 ) ) ) )
=> ( P2 @ N ) ) ) ) ).
% nat_induct_at_least
thf(fact_520_transitive__stepwise__le,axiom,
! [M: nat,N: nat,R2: nat > nat > $o] :
( ( ord_less_eq_nat @ M @ N )
=> ( ! [X3: nat] : ( R2 @ X3 @ X3 )
=> ( ! [X3: nat,Y3: nat,Z2: nat] :
( ( R2 @ X3 @ Y3 )
=> ( ( R2 @ Y3 @ Z2 )
=> ( R2 @ X3 @ Z2 ) ) )
=> ( ! [N3: nat] : ( R2 @ N3 @ ( suc @ N3 ) )
=> ( R2 @ M @ N ) ) ) ) ) ).
% transitive_stepwise_le
thf(fact_521_add__lessD1,axiom,
! [I4: nat,J3: nat,K: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ I4 @ J3 ) @ K )
=> ( ord_less_nat @ I4 @ K ) ) ).
% add_lessD1
thf(fact_522_add__less__mono,axiom,
! [I4: nat,J3: nat,K: nat,L: nat] :
( ( ord_less_nat @ I4 @ J3 )
=> ( ( ord_less_nat @ K @ L )
=> ( ord_less_nat @ ( plus_plus_nat @ I4 @ K ) @ ( plus_plus_nat @ J3 @ L ) ) ) ) ).
% add_less_mono
thf(fact_523_not__add__less1,axiom,
! [I4: nat,J3: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ I4 @ J3 ) @ I4 ) ).
% not_add_less1
thf(fact_524_not__add__less2,axiom,
! [J3: nat,I4: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ J3 @ I4 ) @ I4 ) ).
% not_add_less2
thf(fact_525_add__less__mono1,axiom,
! [I4: nat,J3: nat,K: nat] :
( ( ord_less_nat @ I4 @ J3 )
=> ( ord_less_nat @ ( plus_plus_nat @ I4 @ K ) @ ( plus_plus_nat @ J3 @ K ) ) ) ).
% add_less_mono1
thf(fact_526_trans__less__add1,axiom,
! [I4: nat,J3: nat,M: nat] :
( ( ord_less_nat @ I4 @ J3 )
=> ( ord_less_nat @ I4 @ ( plus_plus_nat @ J3 @ M ) ) ) ).
% trans_less_add1
thf(fact_527_trans__less__add2,axiom,
! [I4: nat,J3: nat,M: nat] :
( ( ord_less_nat @ I4 @ J3 )
=> ( ord_less_nat @ I4 @ ( plus_plus_nat @ M @ J3 ) ) ) ).
% trans_less_add2
thf(fact_528_less__add__eq__less,axiom,
! [K: nat,L: nat,M: nat,N: nat] :
( ( ord_less_nat @ K @ L )
=> ( ( ( plus_plus_nat @ M @ L )
= ( plus_plus_nat @ K @ N ) )
=> ( ord_less_nat @ M @ N ) ) ) ).
% less_add_eq_less
thf(fact_529_add__leE,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ~ ( ( ord_less_eq_nat @ M @ N )
=> ~ ( ord_less_eq_nat @ K @ N ) ) ) ).
% add_leE
thf(fact_530_le__add1,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M ) ) ).
% le_add1
thf(fact_531_le__add2,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M @ N ) ) ).
% le_add2
thf(fact_532_add__leD1,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% add_leD1
thf(fact_533_add__leD2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ( ord_less_eq_nat @ K @ N ) ) ).
% add_leD2
thf(fact_534_le__Suc__ex,axiom,
! [K: nat,L: nat] :
( ( ord_less_eq_nat @ K @ L )
=> ? [N3: nat] :
( L
= ( plus_plus_nat @ K @ N3 ) ) ) ).
% le_Suc_ex
thf(fact_535_add__le__mono,axiom,
! [I4: nat,J3: nat,K: nat,L: nat] :
( ( ord_less_eq_nat @ I4 @ J3 )
=> ( ( ord_less_eq_nat @ K @ L )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I4 @ K ) @ ( plus_plus_nat @ J3 @ L ) ) ) ) ).
% add_le_mono
thf(fact_536_add__le__mono1,axiom,
! [I4: nat,J3: nat,K: nat] :
( ( ord_less_eq_nat @ I4 @ J3 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I4 @ K ) @ ( plus_plus_nat @ J3 @ K ) ) ) ).
% add_le_mono1
thf(fact_537_trans__le__add1,axiom,
! [I4: nat,J3: nat,M: nat] :
( ( ord_less_eq_nat @ I4 @ J3 )
=> ( ord_less_eq_nat @ I4 @ ( plus_plus_nat @ J3 @ M ) ) ) ).
% trans_le_add1
thf(fact_538_trans__le__add2,axiom,
! [I4: nat,J3: nat,M: nat] :
( ( ord_less_eq_nat @ I4 @ J3 )
=> ( ord_less_eq_nat @ I4 @ ( plus_plus_nat @ M @ J3 ) ) ) ).
% trans_le_add2
thf(fact_539_nat__le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [M2: nat,N2: nat] :
? [K3: nat] :
( N2
= ( plus_plus_nat @ M2 @ K3 ) ) ) ) ).
% nat_le_iff_add
thf(fact_540_add__less__le__mono,axiom,
! [A: nat,B: nat,C2: nat,D3: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C2 @ D3 )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ D3 ) ) ) ) ).
% add_less_le_mono
thf(fact_541_add__less__le__mono,axiom,
! [A: int,B: int,C2: int,D3: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_int @ C2 @ D3 )
=> ( ord_less_int @ ( plus_plus_int @ A @ C2 ) @ ( plus_plus_int @ B @ D3 ) ) ) ) ).
% add_less_le_mono
thf(fact_542_add__le__less__mono,axiom,
! [A: nat,B: nat,C2: nat,D3: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ C2 @ D3 )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ D3 ) ) ) ) ).
% add_le_less_mono
thf(fact_543_add__le__less__mono,axiom,
! [A: int,B: int,C2: int,D3: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_int @ C2 @ D3 )
=> ( ord_less_int @ ( plus_plus_int @ A @ C2 ) @ ( plus_plus_int @ B @ D3 ) ) ) ) ).
% add_le_less_mono
thf(fact_544_add__mono__thms__linordered__field_I3_J,axiom,
! [I4: nat,J3: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I4 @ J3 )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I4 @ K ) @ ( plus_plus_nat @ J3 @ L ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_545_add__mono__thms__linordered__field_I3_J,axiom,
! [I4: int,J3: int,K: int,L: int] :
( ( ( ord_less_int @ I4 @ J3 )
& ( ord_less_eq_int @ K @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I4 @ K ) @ ( plus_plus_int @ J3 @ L ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_546_add__mono__thms__linordered__field_I4_J,axiom,
! [I4: nat,J3: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I4 @ J3 )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I4 @ K ) @ ( plus_plus_nat @ J3 @ L ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_547_add__mono__thms__linordered__field_I4_J,axiom,
! [I4: int,J3: int,K: int,L: int] :
( ( ( ord_less_eq_int @ I4 @ J3 )
& ( ord_less_int @ K @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I4 @ K ) @ ( plus_plus_int @ J3 @ L ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_548_lift__Suc__mono__le,axiom,
! [F: nat > nat,N: nat,N4: nat] :
( ! [N3: nat] : ( ord_less_eq_nat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_eq_nat @ N @ N4 )
=> ( ord_less_eq_nat @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_549_lift__Suc__mono__le,axiom,
! [F: nat > int,N: nat,N4: nat] :
( ! [N3: nat] : ( ord_less_eq_int @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_eq_nat @ N @ N4 )
=> ( ord_less_eq_int @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_550_lift__Suc__antimono__le,axiom,
! [F: nat > nat,N: nat,N4: nat] :
( ! [N3: nat] : ( ord_less_eq_nat @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
=> ( ( ord_less_eq_nat @ N @ N4 )
=> ( ord_less_eq_nat @ ( F @ N4 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_551_lift__Suc__antimono__le,axiom,
! [F: nat > int,N: nat,N4: nat] :
( ! [N3: nat] : ( ord_less_eq_int @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
=> ( ( ord_less_eq_nat @ N @ N4 )
=> ( ord_less_eq_int @ ( F @ N4 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_552_lift__Suc__mono__less,axiom,
! [F: nat > nat,N: nat,N4: nat] :
( ! [N3: nat] : ( ord_less_nat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_nat @ N @ N4 )
=> ( ord_less_nat @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_553_lift__Suc__mono__less,axiom,
! [F: nat > int,N: nat,N4: nat] :
( ! [N3: nat] : ( ord_less_int @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_nat @ N @ N4 )
=> ( ord_less_int @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_554_lift__Suc__mono__less__iff,axiom,
! [F: nat > nat,N: nat,M: nat] :
( ! [N3: nat] : ( ord_less_nat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_nat @ ( F @ N ) @ ( F @ M ) )
= ( ord_less_nat @ N @ M ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_555_lift__Suc__mono__less__iff,axiom,
! [F: nat > int,N: nat,M: nat] :
( ! [N3: nat] : ( ord_less_int @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_int @ ( F @ N ) @ ( F @ M ) )
= ( ord_less_nat @ N @ M ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_556_length__Suc__conv,axiom,
! [Xs: list_P5707943133018811711nt_int,N: nat] :
( ( ( size_s5157815400016825771nt_int @ Xs )
= ( suc @ N ) )
= ( ? [Y6: product_prod_int_int,Ys3: list_P5707943133018811711nt_int] :
( ( Xs
= ( cons_P3334398858971670639nt_int @ Y6 @ Ys3 ) )
& ( ( size_s5157815400016825771nt_int @ Ys3 )
= N ) ) ) ) ).
% length_Suc_conv
thf(fact_557_Suc__length__conv,axiom,
! [N: nat,Xs: list_P5707943133018811711nt_int] :
( ( ( suc @ N )
= ( size_s5157815400016825771nt_int @ Xs ) )
= ( ? [Y6: product_prod_int_int,Ys3: list_P5707943133018811711nt_int] :
( ( Xs
= ( cons_P3334398858971670639nt_int @ Y6 @ Ys3 ) )
& ( ( size_s5157815400016825771nt_int @ Ys3 )
= N ) ) ) ) ).
% Suc_length_conv
thf(fact_558_Suc__leI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_eq_nat @ ( suc @ M ) @ N ) ) ).
% Suc_leI
thf(fact_559_Suc__le__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
= ( ord_less_nat @ M @ N ) ) ).
% Suc_le_eq
thf(fact_560_dec__induct,axiom,
! [I4: nat,J3: nat,P2: nat > $o] :
( ( ord_less_eq_nat @ I4 @ J3 )
=> ( ( P2 @ I4 )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ I4 @ N3 )
=> ( ( ord_less_nat @ N3 @ J3 )
=> ( ( P2 @ N3 )
=> ( P2 @ ( suc @ N3 ) ) ) ) )
=> ( P2 @ J3 ) ) ) ) ).
% dec_induct
thf(fact_561_inc__induct,axiom,
! [I4: nat,J3: nat,P2: nat > $o] :
( ( ord_less_eq_nat @ I4 @ J3 )
=> ( ( P2 @ J3 )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ I4 @ N3 )
=> ( ( ord_less_nat @ N3 @ J3 )
=> ( ( P2 @ ( suc @ N3 ) )
=> ( P2 @ N3 ) ) ) )
=> ( P2 @ I4 ) ) ) ) ).
% inc_induct
thf(fact_562_Suc__le__lessD,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
=> ( ord_less_nat @ M @ N ) ) ).
% Suc_le_lessD
thf(fact_563_le__less__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
= ( N = M ) ) ) ).
% le_less_Suc_eq
thf(fact_564_less__Suc__eq__le,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% less_Suc_eq_le
thf(fact_565_less__eq__Suc__le,axiom,
( ord_less_nat
= ( ^ [N2: nat] : ( ord_less_eq_nat @ ( suc @ N2 ) ) ) ) ).
% less_eq_Suc_le
thf(fact_566_le__imp__less__Suc,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_nat @ M @ ( suc @ N ) ) ) ).
% le_imp_less_Suc
thf(fact_567_mono__nat__linear__lb,axiom,
! [F: nat > nat,M: nat,K: nat] :
( ! [M6: nat,N3: nat] :
( ( ord_less_nat @ M6 @ N3 )
=> ( ord_less_nat @ ( F @ M6 ) @ ( F @ N3 ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M ) @ K ) @ ( F @ ( plus_plus_nat @ M @ K ) ) ) ) ).
% mono_nat_linear_lb
thf(fact_568_listrel__eq__len,axiom,
! [Xs: list_P5707943133018811711nt_int,Ys2: list_P5707943133018811711nt_int,R: set_Pr2560585780119916871nt_int] :
( ( member1390679175989562640nt_int @ ( produc1932183703851549015nt_int @ Xs @ Ys2 ) @ ( listre7525836092462517041nt_int @ R ) )
=> ( ( size_s5157815400016825771nt_int @ Xs )
= ( size_s5157815400016825771nt_int @ Ys2 ) ) ) ).
% listrel_eq_len
thf(fact_569_Suc__le__length__iff,axiom,
! [N: nat,Xs: list_P5707943133018811711nt_int] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( size_s5157815400016825771nt_int @ Xs ) )
= ( ? [X4: product_prod_int_int,Ys3: list_P5707943133018811711nt_int] :
( ( Xs
= ( cons_P3334398858971670639nt_int @ X4 @ Ys3 ) )
& ( ord_less_eq_nat @ N @ ( size_s5157815400016825771nt_int @ Ys3 ) ) ) ) ) ).
% Suc_le_length_iff
thf(fact_570_verit__comp__simplify1_I2_J,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_571_verit__comp__simplify1_I2_J,axiom,
! [A: int] : ( ord_less_eq_int @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_572_verit__la__disequality,axiom,
! [A: nat,B: nat] :
( ( A = B )
| ~ ( ord_less_eq_nat @ A @ B )
| ~ ( ord_less_eq_nat @ B @ A ) ) ).
% verit_la_disequality
thf(fact_573_verit__la__disequality,axiom,
! [A: int,B: int] :
( ( A = B )
| ~ ( ord_less_eq_int @ A @ B )
| ~ ( ord_less_eq_int @ B @ A ) ) ).
% verit_la_disequality
thf(fact_574_nth__list__update,axiom,
! [I4: nat,Xs: list_P5707943133018811711nt_int,J3: nat,X: product_prod_int_int] :
( ( ord_less_nat @ I4 @ ( size_s5157815400016825771nt_int @ Xs ) )
=> ( ( ( I4 = J3 )
=> ( ( nth_Pr4439495888332055232nt_int @ ( list_u3002344382305578791nt_int @ Xs @ I4 @ X ) @ J3 )
= X ) )
& ( ( I4 != J3 )
=> ( ( nth_Pr4439495888332055232nt_int @ ( list_u3002344382305578791nt_int @ Xs @ I4 @ X ) @ J3 )
= ( nth_Pr4439495888332055232nt_int @ Xs @ J3 ) ) ) ) ) ).
% nth_list_update
thf(fact_575_list__update__same__conv,axiom,
! [I4: nat,Xs: list_P5707943133018811711nt_int,X: product_prod_int_int] :
( ( ord_less_nat @ I4 @ ( size_s5157815400016825771nt_int @ Xs ) )
=> ( ( ( list_u3002344382305578791nt_int @ Xs @ I4 @ X )
= Xs )
= ( ( nth_Pr4439495888332055232nt_int @ Xs @ I4 )
= X ) ) ) ).
% list_update_same_conv
thf(fact_576_verit__comp__simplify1_I1_J,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_577_verit__comp__simplify1_I1_J,axiom,
! [A: int] :
~ ( ord_less_int @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_578_listrel__Cons2,axiom,
! [Xs: list_P5707943133018811711nt_int,Y: product_prod_int_int,Ys2: list_P5707943133018811711nt_int,R: set_Pr2560585780119916871nt_int] :
( ( member1390679175989562640nt_int @ ( produc1932183703851549015nt_int @ Xs @ ( cons_P3334398858971670639nt_int @ Y @ Ys2 ) ) @ ( listre7525836092462517041nt_int @ R ) )
=> ~ ! [X3: product_prod_int_int,Xs2: list_P5707943133018811711nt_int] :
( ( Xs
= ( cons_P3334398858971670639nt_int @ X3 @ Xs2 ) )
=> ( ( member8566619992076573584nt_int @ ( produc3646306378393792727nt_int @ X3 @ Y ) @ R )
=> ~ ( member1390679175989562640nt_int @ ( produc1932183703851549015nt_int @ Xs2 @ Ys2 ) @ ( listre7525836092462517041nt_int @ R ) ) ) ) ) ).
% listrel_Cons2
thf(fact_579_listrel__Cons2,axiom,
! [Xs: list_nat,Y: set_int,Ys2: list_set_int,R: set_Pr4810089274464741491et_int] :
( ( member4311088582351766364et_int @ ( produc6467668960084403947et_int @ Xs @ ( cons_set_int @ Y @ Ys2 ) ) @ ( listrel_nat_set_int @ R ) )
=> ~ ! [X3: nat,Xs2: list_nat] :
( ( Xs
= ( cons_nat @ X3 @ Xs2 ) )
=> ( ( member1292241183792264892et_int @ ( produc29655638201817675et_int @ X3 @ Y ) @ R )
=> ~ ( member4311088582351766364et_int @ ( produc6467668960084403947et_int @ Xs2 @ Ys2 ) @ ( listrel_nat_set_int @ R ) ) ) ) ) ).
% listrel_Cons2
thf(fact_580_listrel__Cons2,axiom,
! [Xs: list_int,Y: list_P5707943133018811711nt_int,Ys2: list_l1670014477004246597nt_int,R: set_Pr1843331182504696254nt_int] :
( ( member4298863958545693205nt_int @ ( produc1403559775789145904nt_int @ Xs @ ( cons_l7309679040211256053nt_int @ Y @ Ys2 ) ) @ ( listre1889173885445251040nt_int @ R ) )
=> ~ ! [X3: int,Xs2: list_int] :
( ( Xs
= ( cons_int @ X3 @ Xs2 ) )
=> ( ( member3263328415308337823nt_int @ ( produc8814303788642274490nt_int @ X3 @ Y ) @ R )
=> ~ ( member4298863958545693205nt_int @ ( produc1403559775789145904nt_int @ Xs2 @ Ys2 ) @ ( listre1889173885445251040nt_int @ R ) ) ) ) ) ).
% listrel_Cons2
thf(fact_581_listrel__Cons2,axiom,
! [Xs: list_int,Y: int,Ys2: list_int,R: set_Pr958786334691620121nt_int] :
( ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Xs @ ( cons_int @ Y @ Ys2 ) ) @ ( listrel_int_int @ R ) )
=> ~ ! [X3: int,Xs2: list_int] :
( ( Xs
= ( cons_int @ X3 @ Xs2 ) )
=> ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X3 @ Y ) @ R )
=> ~ ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Xs2 @ Ys2 ) @ ( listrel_int_int @ R ) ) ) ) ) ).
% listrel_Cons2
thf(fact_582_listrel__Cons2,axiom,
! [Xs: list_P1743416141875011707e_term,Y: product_prod_int_int,Ys2: list_P5707943133018811711nt_int,R: set_Pr9222295170931077689nt_int] :
( ( member1869596255364795664nt_int @ ( produc8723725229082728235nt_int @ Xs @ ( cons_P3334398858971670639nt_int @ Y @ Ys2 ) ) @ ( listre7929554186369383643nt_int @ R ) )
=> ~ ! [X3: produc8551481072490612790e_term > option6357759511663192854e_term,Xs2: list_P1743416141875011707e_term] :
( ( Xs
= ( cons_P2630085844062958645e_term @ X3 @ Xs2 ) )
=> ( ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ X3 @ Y ) @ R )
=> ~ ( member1869596255364795664nt_int @ ( produc8723725229082728235nt_int @ Xs2 @ Ys2 ) @ ( listre7929554186369383643nt_int @ R ) ) ) ) ) ).
% listrel_Cons2
thf(fact_583_listrel__Cons2,axiom,
! [Xs: list_i8448526496819171953e_term,Y: product_prod_int_int,Ys2: list_P5707943133018811711nt_int,R: set_Pr1872883991513573699nt_int] :
( ( member6768764179952475034nt_int @ ( produc2259991554960072885nt_int @ Xs @ ( cons_P3334398858971670639nt_int @ Y @ Ys2 ) ) @ ( listre7622244038138809061nt_int @ R ) )
=> ~ ! [X3: int > option6357759511663192854e_term,Xs2: list_i8448526496819171953e_term] :
( ( Xs
= ( cons_i7166360444231718571e_term @ X3 @ Xs2 ) )
=> ( ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ X3 @ Y ) @ R )
=> ~ ( member6768764179952475034nt_int @ ( produc2259991554960072885nt_int @ Xs2 @ Ys2 ) @ ( listre7622244038138809061nt_int @ R ) ) ) ) ) ).
% listrel_Cons2
thf(fact_584_listrel__Cons1,axiom,
! [Y: product_prod_int_int,Ys2: list_P5707943133018811711nt_int,Xs: list_P5707943133018811711nt_int,R: set_Pr2560585780119916871nt_int] :
( ( member1390679175989562640nt_int @ ( produc1932183703851549015nt_int @ ( cons_P3334398858971670639nt_int @ Y @ Ys2 ) @ Xs ) @ ( listre7525836092462517041nt_int @ R ) )
=> ~ ! [Y3: product_prod_int_int,Ys4: list_P5707943133018811711nt_int] :
( ( Xs
= ( cons_P3334398858971670639nt_int @ Y3 @ Ys4 ) )
=> ( ( member8566619992076573584nt_int @ ( produc3646306378393792727nt_int @ Y @ Y3 ) @ R )
=> ~ ( member1390679175989562640nt_int @ ( produc1932183703851549015nt_int @ Ys2 @ Ys4 ) @ ( listre7525836092462517041nt_int @ R ) ) ) ) ) ).
% listrel_Cons1
thf(fact_585_listrel__Cons1,axiom,
! [Y: nat,Ys2: list_nat,Xs: list_set_int,R: set_Pr4810089274464741491et_int] :
( ( member4311088582351766364et_int @ ( produc6467668960084403947et_int @ ( cons_nat @ Y @ Ys2 ) @ Xs ) @ ( listrel_nat_set_int @ R ) )
=> ~ ! [Y3: set_int,Ys4: list_set_int] :
( ( Xs
= ( cons_set_int @ Y3 @ Ys4 ) )
=> ( ( member1292241183792264892et_int @ ( produc29655638201817675et_int @ Y @ Y3 ) @ R )
=> ~ ( member4311088582351766364et_int @ ( produc6467668960084403947et_int @ Ys2 @ Ys4 ) @ ( listrel_nat_set_int @ R ) ) ) ) ) ).
% listrel_Cons1
thf(fact_586_listrel__Cons1,axiom,
! [Y: int,Ys2: list_int,Xs: list_l1670014477004246597nt_int,R: set_Pr1843331182504696254nt_int] :
( ( member4298863958545693205nt_int @ ( produc1403559775789145904nt_int @ ( cons_int @ Y @ Ys2 ) @ Xs ) @ ( listre1889173885445251040nt_int @ R ) )
=> ~ ! [Y3: list_P5707943133018811711nt_int,Ys4: list_l1670014477004246597nt_int] :
( ( Xs
= ( cons_l7309679040211256053nt_int @ Y3 @ Ys4 ) )
=> ( ( member3263328415308337823nt_int @ ( produc8814303788642274490nt_int @ Y @ Y3 ) @ R )
=> ~ ( member4298863958545693205nt_int @ ( produc1403559775789145904nt_int @ Ys2 @ Ys4 ) @ ( listre1889173885445251040nt_int @ R ) ) ) ) ) ).
% listrel_Cons1
thf(fact_587_listrel__Cons1,axiom,
! [Y: int,Ys2: list_int,Xs: list_int,R: set_Pr958786334691620121nt_int] :
( ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ ( cons_int @ Y @ Ys2 ) @ Xs ) @ ( listrel_int_int @ R ) )
=> ~ ! [Y3: int,Ys4: list_int] :
( ( Xs
= ( cons_int @ Y3 @ Ys4 ) )
=> ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ Y @ Y3 ) @ R )
=> ~ ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Ys2 @ Ys4 ) @ ( listrel_int_int @ R ) ) ) ) ) ).
% listrel_Cons1
thf(fact_588_listrel__Cons1,axiom,
! [Y: produc8551481072490612790e_term > option6357759511663192854e_term,Ys2: list_P1743416141875011707e_term,Xs: list_P5707943133018811711nt_int,R: set_Pr9222295170931077689nt_int] :
( ( member1869596255364795664nt_int @ ( produc8723725229082728235nt_int @ ( cons_P2630085844062958645e_term @ Y @ Ys2 ) @ Xs ) @ ( listre7929554186369383643nt_int @ R ) )
=> ~ ! [Y3: product_prod_int_int,Ys4: list_P5707943133018811711nt_int] :
( ( Xs
= ( cons_P3334398858971670639nt_int @ Y3 @ Ys4 ) )
=> ( ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ Y @ Y3 ) @ R )
=> ~ ( member1869596255364795664nt_int @ ( produc8723725229082728235nt_int @ Ys2 @ Ys4 ) @ ( listre7929554186369383643nt_int @ R ) ) ) ) ) ).
% listrel_Cons1
thf(fact_589_listrel__Cons1,axiom,
! [Y: int > option6357759511663192854e_term,Ys2: list_i8448526496819171953e_term,Xs: list_P5707943133018811711nt_int,R: set_Pr1872883991513573699nt_int] :
( ( member6768764179952475034nt_int @ ( produc2259991554960072885nt_int @ ( cons_i7166360444231718571e_term @ Y @ Ys2 ) @ Xs ) @ ( listre7622244038138809061nt_int @ R ) )
=> ~ ! [Y3: product_prod_int_int,Ys4: list_P5707943133018811711nt_int] :
( ( Xs
= ( cons_P3334398858971670639nt_int @ Y3 @ Ys4 ) )
=> ( ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ Y @ Y3 ) @ R )
=> ~ ( member6768764179952475034nt_int @ ( produc2259991554960072885nt_int @ Ys2 @ Ys4 ) @ ( listre7622244038138809061nt_int @ R ) ) ) ) ) ).
% listrel_Cons1
thf(fact_590_listrel_OCons,axiom,
! [X: product_prod_int_int,Y: product_prod_int_int,R: set_Pr2560585780119916871nt_int,Xs: list_P5707943133018811711nt_int,Ys2: list_P5707943133018811711nt_int] :
( ( member8566619992076573584nt_int @ ( produc3646306378393792727nt_int @ X @ Y ) @ R )
=> ( ( member1390679175989562640nt_int @ ( produc1932183703851549015nt_int @ Xs @ Ys2 ) @ ( listre7525836092462517041nt_int @ R ) )
=> ( member1390679175989562640nt_int @ ( produc1932183703851549015nt_int @ ( cons_P3334398858971670639nt_int @ X @ Xs ) @ ( cons_P3334398858971670639nt_int @ Y @ Ys2 ) ) @ ( listre7525836092462517041nt_int @ R ) ) ) ) ).
% listrel.Cons
thf(fact_591_listrel_OCons,axiom,
! [X: nat,Y: set_int,R: set_Pr4810089274464741491et_int,Xs: list_nat,Ys2: list_set_int] :
( ( member1292241183792264892et_int @ ( produc29655638201817675et_int @ X @ Y ) @ R )
=> ( ( member4311088582351766364et_int @ ( produc6467668960084403947et_int @ Xs @ Ys2 ) @ ( listrel_nat_set_int @ R ) )
=> ( member4311088582351766364et_int @ ( produc6467668960084403947et_int @ ( cons_nat @ X @ Xs ) @ ( cons_set_int @ Y @ Ys2 ) ) @ ( listrel_nat_set_int @ R ) ) ) ) ).
% listrel.Cons
thf(fact_592_listrel_OCons,axiom,
! [X: int,Y: list_P5707943133018811711nt_int,R: set_Pr1843331182504696254nt_int,Xs: list_int,Ys2: list_l1670014477004246597nt_int] :
( ( member3263328415308337823nt_int @ ( produc8814303788642274490nt_int @ X @ Y ) @ R )
=> ( ( member4298863958545693205nt_int @ ( produc1403559775789145904nt_int @ Xs @ Ys2 ) @ ( listre1889173885445251040nt_int @ R ) )
=> ( member4298863958545693205nt_int @ ( produc1403559775789145904nt_int @ ( cons_int @ X @ Xs ) @ ( cons_l7309679040211256053nt_int @ Y @ Ys2 ) ) @ ( listre1889173885445251040nt_int @ R ) ) ) ) ).
% listrel.Cons
thf(fact_593_listrel_OCons,axiom,
! [X: int,Y: int,R: set_Pr958786334691620121nt_int,Xs: list_int,Ys2: list_int] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ R )
=> ( ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Xs @ Ys2 ) @ ( listrel_int_int @ R ) )
=> ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ ( cons_int @ X @ Xs ) @ ( cons_int @ Y @ Ys2 ) ) @ ( listrel_int_int @ R ) ) ) ) ).
% listrel.Cons
thf(fact_594_listrel_OCons,axiom,
! [X: produc8551481072490612790e_term > option6357759511663192854e_term,Y: product_prod_int_int,R: set_Pr9222295170931077689nt_int,Xs: list_P1743416141875011707e_term,Ys2: list_P5707943133018811711nt_int] :
( ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ X @ Y ) @ R )
=> ( ( member1869596255364795664nt_int @ ( produc8723725229082728235nt_int @ Xs @ Ys2 ) @ ( listre7929554186369383643nt_int @ R ) )
=> ( member1869596255364795664nt_int @ ( produc8723725229082728235nt_int @ ( cons_P2630085844062958645e_term @ X @ Xs ) @ ( cons_P3334398858971670639nt_int @ Y @ Ys2 ) ) @ ( listre7929554186369383643nt_int @ R ) ) ) ) ).
% listrel.Cons
thf(fact_595_listrel_OCons,axiom,
! [X: int > option6357759511663192854e_term,Y: product_prod_int_int,R: set_Pr1872883991513573699nt_int,Xs: list_i8448526496819171953e_term,Ys2: list_P5707943133018811711nt_int] :
( ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ X @ Y ) @ R )
=> ( ( member6768764179952475034nt_int @ ( produc2259991554960072885nt_int @ Xs @ Ys2 ) @ ( listre7622244038138809061nt_int @ R ) )
=> ( member6768764179952475034nt_int @ ( produc2259991554960072885nt_int @ ( cons_i7166360444231718571e_term @ X @ Xs ) @ ( cons_P3334398858971670639nt_int @ Y @ Ys2 ) ) @ ( listre7622244038138809061nt_int @ R ) ) ) ) ).
% listrel.Cons
thf(fact_596_nth__take__lemma,axiom,
! [K: nat,Xs: list_P5707943133018811711nt_int,Ys2: list_P5707943133018811711nt_int] :
( ( ord_less_eq_nat @ K @ ( size_s5157815400016825771nt_int @ Xs ) )
=> ( ( ord_less_eq_nat @ K @ ( size_s5157815400016825771nt_int @ Ys2 ) )
=> ( ! [I: nat] :
( ( ord_less_nat @ I @ K )
=> ( ( nth_Pr4439495888332055232nt_int @ Xs @ I )
= ( nth_Pr4439495888332055232nt_int @ Ys2 @ I ) ) )
=> ( ( take_P8218740963776755879nt_int @ K @ Xs )
= ( take_P8218740963776755879nt_int @ K @ Ys2 ) ) ) ) ) ).
% nth_take_lemma
thf(fact_597_length__Cons,axiom,
! [X: product_prod_int_int,Xs: list_P5707943133018811711nt_int] :
( ( size_s5157815400016825771nt_int @ ( cons_P3334398858971670639nt_int @ X @ Xs ) )
= ( suc @ ( size_s5157815400016825771nt_int @ Xs ) ) ) ).
% length_Cons
thf(fact_598_less__add__one,axiom,
! [A: nat] : ( ord_less_nat @ A @ ( plus_plus_nat @ A @ one_one_nat ) ) ).
% less_add_one
thf(fact_599_less__add__one,axiom,
! [A: int] : ( ord_less_int @ A @ ( plus_plus_int @ A @ one_one_int ) ) ).
% less_add_one
thf(fact_600_add__mono1,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ one_one_nat ) @ ( plus_plus_nat @ B @ one_one_nat ) ) ) ).
% add_mono1
thf(fact_601_add__mono1,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_int @ ( plus_plus_int @ A @ one_one_int ) @ ( plus_plus_int @ B @ one_one_int ) ) ) ).
% add_mono1
thf(fact_602_subsetI,axiom,
! [A4: set_Pr958786334691620121nt_int,B5: set_Pr958786334691620121nt_int] :
( ! [X3: product_prod_int_int] :
( ( member5262025264175285858nt_int @ X3 @ A4 )
=> ( member5262025264175285858nt_int @ X3 @ B5 ) )
=> ( ord_le2843351958646193337nt_int @ A4 @ B5 ) ) ).
% subsetI
thf(fact_603_subsetI,axiom,
! [A4: set_int,B5: set_int] :
( ! [X3: int] :
( ( member_int @ X3 @ A4 )
=> ( member_int @ X3 @ B5 ) )
=> ( ord_less_eq_set_int @ A4 @ B5 ) ) ).
% subsetI
thf(fact_604_linorder__neqE__linordered__idom,axiom,
! [X: int,Y: int] :
( ( X != Y )
=> ( ~ ( ord_less_int @ X @ Y )
=> ( ord_less_int @ Y @ X ) ) ) ).
% linorder_neqE_linordered_idom
thf(fact_605_subset__iff,axiom,
( ord_le2843351958646193337nt_int
= ( ^ [A6: set_Pr958786334691620121nt_int,B6: set_Pr958786334691620121nt_int] :
! [T2: product_prod_int_int] :
( ( member5262025264175285858nt_int @ T2 @ A6 )
=> ( member5262025264175285858nt_int @ T2 @ B6 ) ) ) ) ).
% subset_iff
thf(fact_606_subset__iff,axiom,
( ord_less_eq_set_int
= ( ^ [A6: set_int,B6: set_int] :
! [T2: int] :
( ( member_int @ T2 @ A6 )
=> ( member_int @ T2 @ B6 ) ) ) ) ).
% subset_iff
thf(fact_607_subset__eq,axiom,
( ord_le2843351958646193337nt_int
= ( ^ [A6: set_Pr958786334691620121nt_int,B6: set_Pr958786334691620121nt_int] :
! [X4: product_prod_int_int] :
( ( member5262025264175285858nt_int @ X4 @ A6 )
=> ( member5262025264175285858nt_int @ X4 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_608_subset__eq,axiom,
( ord_less_eq_set_int
= ( ^ [A6: set_int,B6: set_int] :
! [X4: int] :
( ( member_int @ X4 @ A6 )
=> ( member_int @ X4 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_609_subsetD,axiom,
! [A4: set_Pr958786334691620121nt_int,B5: set_Pr958786334691620121nt_int,C2: product_prod_int_int] :
( ( ord_le2843351958646193337nt_int @ A4 @ B5 )
=> ( ( member5262025264175285858nt_int @ C2 @ A4 )
=> ( member5262025264175285858nt_int @ C2 @ B5 ) ) ) ).
% subsetD
thf(fact_610_subsetD,axiom,
! [A4: set_int,B5: set_int,C2: int] :
( ( ord_less_eq_set_int @ A4 @ B5 )
=> ( ( member_int @ C2 @ A4 )
=> ( member_int @ C2 @ B5 ) ) ) ).
% subsetD
thf(fact_611_in__mono,axiom,
! [A4: set_Pr958786334691620121nt_int,B5: set_Pr958786334691620121nt_int,X: product_prod_int_int] :
( ( ord_le2843351958646193337nt_int @ A4 @ B5 )
=> ( ( member5262025264175285858nt_int @ X @ A4 )
=> ( member5262025264175285858nt_int @ X @ B5 ) ) ) ).
% in_mono
thf(fact_612_in__mono,axiom,
! [A4: set_int,B5: set_int,X: int] :
( ( ord_less_eq_set_int @ A4 @ B5 )
=> ( ( member_int @ X @ A4 )
=> ( member_int @ X @ B5 ) ) ) ).
% in_mono
thf(fact_613_bounded__Max__nat,axiom,
! [P2: nat > $o,X: nat,M7: nat] :
( ( P2 @ X )
=> ( ! [X3: nat] :
( ( P2 @ X3 )
=> ( ord_less_eq_nat @ X3 @ M7 ) )
=> ~ ! [M6: nat] :
( ( P2 @ M6 )
=> ~ ! [X6: nat] :
( ( P2 @ X6 )
=> ( ord_less_eq_nat @ X6 @ M6 ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_614_upd__conv__take__nth__drop,axiom,
! [I4: nat,Xs: list_P5707943133018811711nt_int,A: product_prod_int_int] :
( ( ord_less_nat @ I4 @ ( size_s5157815400016825771nt_int @ Xs ) )
=> ( ( list_u3002344382305578791nt_int @ Xs @ I4 @ A )
= ( append7030698103840186580nt_int @ ( take_P8218740963776755879nt_int @ I4 @ Xs ) @ ( cons_P3334398858971670639nt_int @ A @ ( drop_P5690361596310759935nt_int @ ( suc @ I4 ) @ Xs ) ) ) ) ) ).
% upd_conv_take_nth_drop
thf(fact_615_subrelI,axiom,
! [R: set_Pr4810089274464741491et_int,S: set_Pr4810089274464741491et_int] :
( ! [X3: nat,Y3: set_int] :
( ( member1292241183792264892et_int @ ( produc29655638201817675et_int @ X3 @ Y3 ) @ R )
=> ( member1292241183792264892et_int @ ( produc29655638201817675et_int @ X3 @ Y3 ) @ S ) )
=> ( ord_le8255767777184198675et_int @ R @ S ) ) ).
% subrelI
thf(fact_616_subrelI,axiom,
! [R: set_Pr1843331182504696254nt_int,S: set_Pr1843331182504696254nt_int] :
( ! [X3: int,Y3: list_P5707943133018811711nt_int] :
( ( member3263328415308337823nt_int @ ( produc8814303788642274490nt_int @ X3 @ Y3 ) @ R )
=> ( member3263328415308337823nt_int @ ( produc8814303788642274490nt_int @ X3 @ Y3 ) @ S ) )
=> ( ord_le5985196928369334814nt_int @ R @ S ) ) ).
% subrelI
thf(fact_617_subrelI,axiom,
! [R: set_Pr958786334691620121nt_int,S: set_Pr958786334691620121nt_int] :
( ! [X3: int,Y3: int] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X3 @ Y3 ) @ R )
=> ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X3 @ Y3 ) @ S ) )
=> ( ord_le2843351958646193337nt_int @ R @ S ) ) ).
% subrelI
thf(fact_618_subrelI,axiom,
! [R: set_Pr9222295170931077689nt_int,S: set_Pr9222295170931077689nt_int] :
( ! [X3: produc8551481072490612790e_term > option6357759511663192854e_term,Y3: product_prod_int_int] :
( ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ X3 @ Y3 ) @ R )
=> ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ X3 @ Y3 ) @ S ) )
=> ( ord_le8725513860283290265nt_int @ R @ S ) ) ).
% subrelI
thf(fact_619_subrelI,axiom,
! [R: set_Pr1872883991513573699nt_int,S: set_Pr1872883991513573699nt_int] :
( ! [X3: int > option6357759511663192854e_term,Y3: product_prod_int_int] :
( ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ X3 @ Y3 ) @ R )
=> ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ X3 @ Y3 ) @ S ) )
=> ( ord_le135402666524580259nt_int @ R @ S ) ) ).
% subrelI
thf(fact_620_id__take__nth__drop,axiom,
! [I4: nat,Xs: list_P5707943133018811711nt_int] :
( ( ord_less_nat @ I4 @ ( size_s5157815400016825771nt_int @ Xs ) )
=> ( Xs
= ( append7030698103840186580nt_int @ ( take_P8218740963776755879nt_int @ I4 @ Xs ) @ ( cons_P3334398858971670639nt_int @ ( nth_Pr4439495888332055232nt_int @ Xs @ I4 ) @ ( drop_P5690361596310759935nt_int @ ( suc @ I4 ) @ Xs ) ) ) ) ) ).
% id_take_nth_drop
thf(fact_621_lexord__take__index__conv,axiom,
! [X: list_int,Y: list_int,R: set_Pr958786334691620121nt_int] :
( ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ X @ Y ) @ ( lexord_int @ R ) )
= ( ( ( ord_less_nat @ ( size_size_list_int @ X ) @ ( size_size_list_int @ Y ) )
& ( ( take_int @ ( size_size_list_int @ X ) @ Y )
= X ) )
| ? [I3: nat] :
( ( ord_less_nat @ I3 @ ( ord_min_nat @ ( size_size_list_int @ X ) @ ( size_size_list_int @ Y ) ) )
& ( ( take_int @ I3 @ X )
= ( take_int @ I3 @ Y ) )
& ( member5262025264175285858nt_int @ ( product_Pair_int_int @ ( nth_int @ X @ I3 ) @ ( nth_int @ Y @ I3 ) ) @ R ) ) ) ) ).
% lexord_take_index_conv
thf(fact_622_lexord__take__index__conv,axiom,
! [X: list_P5707943133018811711nt_int,Y: list_P5707943133018811711nt_int,R: set_Pr2560585780119916871nt_int] :
( ( member1390679175989562640nt_int @ ( produc1932183703851549015nt_int @ X @ Y ) @ ( lexord8886728381914977324nt_int @ R ) )
= ( ( ( ord_less_nat @ ( size_s5157815400016825771nt_int @ X ) @ ( size_s5157815400016825771nt_int @ Y ) )
& ( ( take_P8218740963776755879nt_int @ ( size_s5157815400016825771nt_int @ X ) @ Y )
= X ) )
| ? [I3: nat] :
( ( ord_less_nat @ I3 @ ( ord_min_nat @ ( size_s5157815400016825771nt_int @ X ) @ ( size_s5157815400016825771nt_int @ Y ) ) )
& ( ( take_P8218740963776755879nt_int @ I3 @ X )
= ( take_P8218740963776755879nt_int @ I3 @ Y ) )
& ( member8566619992076573584nt_int @ ( produc3646306378393792727nt_int @ ( nth_Pr4439495888332055232nt_int @ X @ I3 ) @ ( nth_Pr4439495888332055232nt_int @ Y @ I3 ) ) @ R ) ) ) ) ).
% lexord_take_index_conv
thf(fact_623_same__append__eq,axiom,
! [Xs: list_P5707943133018811711nt_int,Ys2: list_P5707943133018811711nt_int,Zs2: list_P5707943133018811711nt_int] :
( ( ( append7030698103840186580nt_int @ Xs @ Ys2 )
= ( append7030698103840186580nt_int @ Xs @ Zs2 ) )
= ( Ys2 = Zs2 ) ) ).
% same_append_eq
thf(fact_624_append__same__eq,axiom,
! [Ys2: list_P5707943133018811711nt_int,Xs: list_P5707943133018811711nt_int,Zs2: list_P5707943133018811711nt_int] :
( ( ( append7030698103840186580nt_int @ Ys2 @ Xs )
= ( append7030698103840186580nt_int @ Zs2 @ Xs ) )
= ( Ys2 = Zs2 ) ) ).
% append_same_eq
thf(fact_625_append__assoc,axiom,
! [Xs: list_P5707943133018811711nt_int,Ys2: list_P5707943133018811711nt_int,Zs2: list_P5707943133018811711nt_int] :
( ( append7030698103840186580nt_int @ ( append7030698103840186580nt_int @ Xs @ Ys2 ) @ Zs2 )
= ( append7030698103840186580nt_int @ Xs @ ( append7030698103840186580nt_int @ Ys2 @ Zs2 ) ) ) ).
% append_assoc
thf(fact_626_append_Oassoc,axiom,
! [A: list_P5707943133018811711nt_int,B: list_P5707943133018811711nt_int,C2: list_P5707943133018811711nt_int] :
( ( append7030698103840186580nt_int @ ( append7030698103840186580nt_int @ A @ B ) @ C2 )
= ( append7030698103840186580nt_int @ A @ ( append7030698103840186580nt_int @ B @ C2 ) ) ) ).
% append.assoc
thf(fact_627_append__eq__append__conv,axiom,
! [Xs: list_P5707943133018811711nt_int,Ys2: list_P5707943133018811711nt_int,Us: list_P5707943133018811711nt_int,Vs: list_P5707943133018811711nt_int] :
( ( ( ( size_s5157815400016825771nt_int @ Xs )
= ( size_s5157815400016825771nt_int @ Ys2 ) )
| ( ( size_s5157815400016825771nt_int @ Us )
= ( size_s5157815400016825771nt_int @ Vs ) ) )
=> ( ( ( append7030698103840186580nt_int @ Xs @ Us )
= ( append7030698103840186580nt_int @ Ys2 @ Vs ) )
= ( ( Xs = Ys2 )
& ( Us = Vs ) ) ) ) ).
% append_eq_append_conv
thf(fact_628_min__Suc__Suc,axiom,
! [M: nat,N: nat] :
( ( ord_min_nat @ ( suc @ M ) @ ( suc @ N ) )
= ( suc @ ( ord_min_nat @ M @ N ) ) ) ).
% min_Suc_Suc
thf(fact_629_take__take,axiom,
! [N: nat,M: nat,Xs: list_P5707943133018811711nt_int] :
( ( take_P8218740963776755879nt_int @ N @ ( take_P8218740963776755879nt_int @ M @ Xs ) )
= ( take_P8218740963776755879nt_int @ ( ord_min_nat @ N @ M ) @ Xs ) ) ).
% take_take
thf(fact_630_list__ex__append,axiom,
! [P2: product_prod_int_int > $o,Xs: list_P5707943133018811711nt_int,Ys2: list_P5707943133018811711nt_int] :
( ( list_e4511028299437703933nt_int @ P2 @ ( append7030698103840186580nt_int @ Xs @ Ys2 ) )
= ( ( list_e4511028299437703933nt_int @ P2 @ Xs )
| ( list_e4511028299437703933nt_int @ P2 @ Ys2 ) ) ) ).
% list_ex_append
thf(fact_631_length__append,axiom,
! [Xs: list_P5707943133018811711nt_int,Ys2: list_P5707943133018811711nt_int] :
( ( size_s5157815400016825771nt_int @ ( append7030698103840186580nt_int @ Xs @ Ys2 ) )
= ( plus_plus_nat @ ( size_s5157815400016825771nt_int @ Xs ) @ ( size_s5157815400016825771nt_int @ Ys2 ) ) ) ).
% length_append
thf(fact_632_append__take__drop__id,axiom,
! [N: nat,Xs: list_P5707943133018811711nt_int] :
( ( append7030698103840186580nt_int @ ( take_P8218740963776755879nt_int @ N @ Xs ) @ ( drop_P5690361596310759935nt_int @ N @ Xs ) )
= Xs ) ).
% append_take_drop_id
thf(fact_633_length__take,axiom,
! [N: nat,Xs: list_P5707943133018811711nt_int] :
( ( size_s5157815400016825771nt_int @ ( take_P8218740963776755879nt_int @ N @ Xs ) )
= ( ord_min_nat @ ( size_s5157815400016825771nt_int @ Xs ) @ N ) ) ).
% length_take
thf(fact_634_nth__append__length,axiom,
! [Xs: list_P5707943133018811711nt_int,X: product_prod_int_int,Ys2: list_P5707943133018811711nt_int] :
( ( nth_Pr4439495888332055232nt_int @ ( append7030698103840186580nt_int @ Xs @ ( cons_P3334398858971670639nt_int @ X @ Ys2 ) ) @ ( size_s5157815400016825771nt_int @ Xs ) )
= X ) ).
% nth_append_length
thf(fact_635_nth__append__length__plus,axiom,
! [Xs: list_P5707943133018811711nt_int,Ys2: list_P5707943133018811711nt_int,N: nat] :
( ( nth_Pr4439495888332055232nt_int @ ( append7030698103840186580nt_int @ Xs @ Ys2 ) @ ( plus_plus_nat @ ( size_s5157815400016825771nt_int @ Xs ) @ N ) )
= ( nth_Pr4439495888332055232nt_int @ Ys2 @ N ) ) ).
% nth_append_length_plus
thf(fact_636_list__update__length,axiom,
! [Xs: list_P5707943133018811711nt_int,X: product_prod_int_int,Ys2: list_P5707943133018811711nt_int,Y: product_prod_int_int] :
( ( list_u3002344382305578791nt_int @ ( append7030698103840186580nt_int @ Xs @ ( cons_P3334398858971670639nt_int @ X @ Ys2 ) ) @ ( size_s5157815400016825771nt_int @ Xs ) @ Y )
= ( append7030698103840186580nt_int @ Xs @ ( cons_P3334398858971670639nt_int @ Y @ Ys2 ) ) ) ).
% list_update_length
thf(fact_637_min__absorb2,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ( ( ord_min_nat @ X @ Y )
= Y ) ) ).
% min_absorb2
thf(fact_638_min__absorb2,axiom,
! [Y: int,X: int] :
( ( ord_less_eq_int @ Y @ X )
=> ( ( ord_min_int @ X @ Y )
= Y ) ) ).
% min_absorb2
thf(fact_639_min__absorb1,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_min_nat @ X @ Y )
= X ) ) ).
% min_absorb1
thf(fact_640_min__absorb1,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ord_min_int @ X @ Y )
= X ) ) ).
% min_absorb1
thf(fact_641_min__def,axiom,
( ord_min_nat
= ( ^ [A5: nat,B4: nat] : ( if_nat @ ( ord_less_eq_nat @ A5 @ B4 ) @ A5 @ B4 ) ) ) ).
% min_def
thf(fact_642_min__def,axiom,
( ord_min_int
= ( ^ [A5: int,B4: int] : ( if_int @ ( ord_less_eq_int @ A5 @ B4 ) @ A5 @ B4 ) ) ) ).
% min_def
thf(fact_643_min__add__distrib__left,axiom,
! [X: int,Y: int,Z3: int] :
( ( plus_plus_int @ ( ord_min_int @ X @ Y ) @ Z3 )
= ( ord_min_int @ ( plus_plus_int @ X @ Z3 ) @ ( plus_plus_int @ Y @ Z3 ) ) ) ).
% min_add_distrib_left
thf(fact_644_min__add__distrib__left,axiom,
! [X: nat,Y: nat,Z3: nat] :
( ( plus_plus_nat @ ( ord_min_nat @ X @ Y ) @ Z3 )
= ( ord_min_nat @ ( plus_plus_nat @ X @ Z3 ) @ ( plus_plus_nat @ Y @ Z3 ) ) ) ).
% min_add_distrib_left
thf(fact_645_min__add__distrib__right,axiom,
! [X: int,Y: int,Z3: int] :
( ( plus_plus_int @ X @ ( ord_min_int @ Y @ Z3 ) )
= ( ord_min_int @ ( plus_plus_int @ X @ Y ) @ ( plus_plus_int @ X @ Z3 ) ) ) ).
% min_add_distrib_right
thf(fact_646_min__add__distrib__right,axiom,
! [X: nat,Y: nat,Z3: nat] :
( ( plus_plus_nat @ X @ ( ord_min_nat @ Y @ Z3 ) )
= ( ord_min_nat @ ( plus_plus_nat @ X @ Y ) @ ( plus_plus_nat @ X @ Z3 ) ) ) ).
% min_add_distrib_right
thf(fact_647_append__Cons,axiom,
! [X: product_prod_int_int,Xs: list_P5707943133018811711nt_int,Ys2: list_P5707943133018811711nt_int] :
( ( append7030698103840186580nt_int @ ( cons_P3334398858971670639nt_int @ X @ Xs ) @ Ys2 )
= ( cons_P3334398858971670639nt_int @ X @ ( append7030698103840186580nt_int @ Xs @ Ys2 ) ) ) ).
% append_Cons
thf(fact_648_Cons__eq__appendI,axiom,
! [X: product_prod_int_int,Xs1: list_P5707943133018811711nt_int,Ys2: list_P5707943133018811711nt_int,Xs: list_P5707943133018811711nt_int,Zs2: list_P5707943133018811711nt_int] :
( ( ( cons_P3334398858971670639nt_int @ X @ Xs1 )
= Ys2 )
=> ( ( Xs
= ( append7030698103840186580nt_int @ Xs1 @ Zs2 ) )
=> ( ( cons_P3334398858971670639nt_int @ X @ Xs )
= ( append7030698103840186580nt_int @ Ys2 @ Zs2 ) ) ) ) ).
% Cons_eq_appendI
thf(fact_649_append__eq__append__conv2,axiom,
! [Xs: list_P5707943133018811711nt_int,Ys2: list_P5707943133018811711nt_int,Zs2: list_P5707943133018811711nt_int,Ts: list_P5707943133018811711nt_int] :
( ( ( append7030698103840186580nt_int @ Xs @ Ys2 )
= ( append7030698103840186580nt_int @ Zs2 @ Ts ) )
= ( ? [Us2: list_P5707943133018811711nt_int] :
( ( ( Xs
= ( append7030698103840186580nt_int @ Zs2 @ Us2 ) )
& ( ( append7030698103840186580nt_int @ Us2 @ Ys2 )
= Ts ) )
| ( ( ( append7030698103840186580nt_int @ Xs @ Us2 )
= Zs2 )
& ( Ys2
= ( append7030698103840186580nt_int @ Us2 @ Ts ) ) ) ) ) ) ).
% append_eq_append_conv2
thf(fact_650_append__eq__appendI,axiom,
! [Xs: list_P5707943133018811711nt_int,Xs1: list_P5707943133018811711nt_int,Zs2: list_P5707943133018811711nt_int,Ys2: list_P5707943133018811711nt_int,Us: list_P5707943133018811711nt_int] :
( ( ( append7030698103840186580nt_int @ Xs @ Xs1 )
= Zs2 )
=> ( ( Ys2
= ( append7030698103840186580nt_int @ Xs1 @ Us ) )
=> ( ( append7030698103840186580nt_int @ Xs @ Ys2 )
= ( append7030698103840186580nt_int @ Zs2 @ Us ) ) ) ) ).
% append_eq_appendI
thf(fact_651_rotate__append,axiom,
! [L: list_P5707943133018811711nt_int,Q4: list_P5707943133018811711nt_int] :
( ( rotate3281235723284208371nt_int @ ( size_s5157815400016825771nt_int @ L ) @ ( append7030698103840186580nt_int @ L @ Q4 ) )
= ( append7030698103840186580nt_int @ Q4 @ L ) ) ).
% rotate_append
thf(fact_652_append__listrel1I,axiom,
! [Xs: list_P5707943133018811711nt_int,Ys2: list_P5707943133018811711nt_int,R: set_Pr2560585780119916871nt_int,Us: list_P5707943133018811711nt_int,Vs: list_P5707943133018811711nt_int] :
( ( ( ( member1390679175989562640nt_int @ ( produc1932183703851549015nt_int @ Xs @ Ys2 ) @ ( listre1649617614542993432nt_int @ R ) )
& ( Us = Vs ) )
| ( ( Xs = Ys2 )
& ( member1390679175989562640nt_int @ ( produc1932183703851549015nt_int @ Us @ Vs ) @ ( listre1649617614542993432nt_int @ R ) ) ) )
=> ( member1390679175989562640nt_int @ ( produc1932183703851549015nt_int @ ( append7030698103840186580nt_int @ Xs @ Us ) @ ( append7030698103840186580nt_int @ Ys2 @ Vs ) ) @ ( listre1649617614542993432nt_int @ R ) ) ) ).
% append_listrel1I
thf(fact_653_lexord__append__leftI,axiom,
! [U: list_P5707943133018811711nt_int,V: list_P5707943133018811711nt_int,R: set_Pr2560585780119916871nt_int,X: list_P5707943133018811711nt_int] :
( ( member1390679175989562640nt_int @ ( produc1932183703851549015nt_int @ U @ V ) @ ( lexord8886728381914977324nt_int @ R ) )
=> ( member1390679175989562640nt_int @ ( produc1932183703851549015nt_int @ ( append7030698103840186580nt_int @ X @ U ) @ ( append7030698103840186580nt_int @ X @ V ) ) @ ( lexord8886728381914977324nt_int @ R ) ) ) ).
% lexord_append_leftI
thf(fact_654_lex__append__leftI,axiom,
! [Ys2: list_P5707943133018811711nt_int,Zs2: list_P5707943133018811711nt_int,R: set_Pr2560585780119916871nt_int,Xs: list_P5707943133018811711nt_int] :
( ( member1390679175989562640nt_int @ ( produc1932183703851549015nt_int @ Ys2 @ Zs2 ) @ ( lex_Pr5393148144989827363nt_int @ R ) )
=> ( member1390679175989562640nt_int @ ( produc1932183703851549015nt_int @ ( append7030698103840186580nt_int @ Xs @ Ys2 ) @ ( append7030698103840186580nt_int @ Xs @ Zs2 ) ) @ ( lex_Pr5393148144989827363nt_int @ R ) ) ) ).
% lex_append_leftI
thf(fact_655_list__update__append1,axiom,
! [I4: nat,Xs: list_P5707943133018811711nt_int,Ys2: list_P5707943133018811711nt_int,X: product_prod_int_int] :
( ( ord_less_nat @ I4 @ ( size_s5157815400016825771nt_int @ Xs ) )
=> ( ( list_u3002344382305578791nt_int @ ( append7030698103840186580nt_int @ Xs @ Ys2 ) @ I4 @ X )
= ( append7030698103840186580nt_int @ ( list_u3002344382305578791nt_int @ Xs @ I4 @ X ) @ Ys2 ) ) ) ).
% list_update_append1
thf(fact_656_append__eq__conv__conj,axiom,
! [Xs: list_P5707943133018811711nt_int,Ys2: list_P5707943133018811711nt_int,Zs2: list_P5707943133018811711nt_int] :
( ( ( append7030698103840186580nt_int @ Xs @ Ys2 )
= Zs2 )
= ( ( Xs
= ( take_P8218740963776755879nt_int @ ( size_s5157815400016825771nt_int @ Xs ) @ Zs2 ) )
& ( Ys2
= ( drop_P5690361596310759935nt_int @ ( size_s5157815400016825771nt_int @ Xs ) @ Zs2 ) ) ) ) ).
% append_eq_conv_conj
thf(fact_657_take__add,axiom,
! [I4: nat,J3: nat,Xs: list_P5707943133018811711nt_int] :
( ( take_P8218740963776755879nt_int @ ( plus_plus_nat @ I4 @ J3 ) @ Xs )
= ( append7030698103840186580nt_int @ ( take_P8218740963776755879nt_int @ I4 @ Xs ) @ ( take_P8218740963776755879nt_int @ J3 @ ( drop_P5690361596310759935nt_int @ I4 @ Xs ) ) ) ) ).
% take_add
thf(fact_658_lexord__append__leftD,axiom,
! [X: list_P5707943133018811711nt_int,U: list_P5707943133018811711nt_int,V: list_P5707943133018811711nt_int,R: set_Pr2560585780119916871nt_int] :
( ( member1390679175989562640nt_int @ ( produc1932183703851549015nt_int @ ( append7030698103840186580nt_int @ X @ U ) @ ( append7030698103840186580nt_int @ X @ V ) ) @ ( lexord8886728381914977324nt_int @ R ) )
=> ( ! [A3: product_prod_int_int] :
~ ( member8566619992076573584nt_int @ ( produc3646306378393792727nt_int @ A3 @ A3 ) @ R )
=> ( member1390679175989562640nt_int @ ( produc1932183703851549015nt_int @ U @ V ) @ ( lexord8886728381914977324nt_int @ R ) ) ) ) ).
% lexord_append_leftD
thf(fact_659_lexord__append__leftD,axiom,
! [X: list_int,U: list_int,V: list_int,R: set_Pr958786334691620121nt_int] :
( ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ ( append_int @ X @ U ) @ ( append_int @ X @ V ) ) @ ( lexord_int @ R ) )
=> ( ! [A3: int] :
~ ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A3 @ A3 ) @ R )
=> ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ U @ V ) @ ( lexord_int @ R ) ) ) ) ).
% lexord_append_leftD
thf(fact_660_lexord__append__rightI,axiom,
! [Y: list_P5707943133018811711nt_int,X: list_P5707943133018811711nt_int,R: set_Pr2560585780119916871nt_int] :
( ? [B7: product_prod_int_int,Z4: list_P5707943133018811711nt_int] :
( Y
= ( cons_P3334398858971670639nt_int @ B7 @ Z4 ) )
=> ( member1390679175989562640nt_int @ ( produc1932183703851549015nt_int @ X @ ( append7030698103840186580nt_int @ X @ Y ) ) @ ( lexord8886728381914977324nt_int @ R ) ) ) ).
% lexord_append_rightI
thf(fact_661_lexord__sufE,axiom,
! [Xs: list_P5707943133018811711nt_int,Zs2: list_P5707943133018811711nt_int,Ys2: list_P5707943133018811711nt_int,Qs: list_P5707943133018811711nt_int,R: set_Pr2560585780119916871nt_int] :
( ( member1390679175989562640nt_int @ ( produc1932183703851549015nt_int @ ( append7030698103840186580nt_int @ Xs @ Zs2 ) @ ( append7030698103840186580nt_int @ Ys2 @ Qs ) ) @ ( lexord8886728381914977324nt_int @ R ) )
=> ( ( Xs != Ys2 )
=> ( ( ( size_s5157815400016825771nt_int @ Xs )
= ( size_s5157815400016825771nt_int @ Ys2 ) )
=> ( ( ( size_s5157815400016825771nt_int @ Zs2 )
= ( size_s5157815400016825771nt_int @ Qs ) )
=> ( member1390679175989562640nt_int @ ( produc1932183703851549015nt_int @ Xs @ Ys2 ) @ ( lexord8886728381914977324nt_int @ R ) ) ) ) ) ) ).
% lexord_sufE
thf(fact_662_lex__append__leftD,axiom,
! [R: set_Pr2560585780119916871nt_int,Xs: list_P5707943133018811711nt_int,Ys2: list_P5707943133018811711nt_int,Zs2: list_P5707943133018811711nt_int] :
( ! [X3: product_prod_int_int] :
~ ( member8566619992076573584nt_int @ ( produc3646306378393792727nt_int @ X3 @ X3 ) @ R )
=> ( ( member1390679175989562640nt_int @ ( produc1932183703851549015nt_int @ ( append7030698103840186580nt_int @ Xs @ Ys2 ) @ ( append7030698103840186580nt_int @ Xs @ Zs2 ) ) @ ( lex_Pr5393148144989827363nt_int @ R ) )
=> ( member1390679175989562640nt_int @ ( produc1932183703851549015nt_int @ Ys2 @ Zs2 ) @ ( lex_Pr5393148144989827363nt_int @ R ) ) ) ) ).
% lex_append_leftD
thf(fact_663_lex__append__leftD,axiom,
! [R: set_Pr958786334691620121nt_int,Xs: list_int,Ys2: list_int,Zs2: list_int] :
( ! [X3: int] :
~ ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X3 @ X3 ) @ R )
=> ( ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ ( append_int @ Xs @ Ys2 ) @ ( append_int @ Xs @ Zs2 ) ) @ ( lex_int @ R ) )
=> ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Ys2 @ Zs2 ) @ ( lex_int @ R ) ) ) ) ).
% lex_append_leftD
thf(fact_664_lex__append__left__iff,axiom,
! [R: set_Pr2560585780119916871nt_int,Xs: list_P5707943133018811711nt_int,Ys2: list_P5707943133018811711nt_int,Zs2: list_P5707943133018811711nt_int] :
( ! [X3: product_prod_int_int] :
~ ( member8566619992076573584nt_int @ ( produc3646306378393792727nt_int @ X3 @ X3 ) @ R )
=> ( ( member1390679175989562640nt_int @ ( produc1932183703851549015nt_int @ ( append7030698103840186580nt_int @ Xs @ Ys2 ) @ ( append7030698103840186580nt_int @ Xs @ Zs2 ) ) @ ( lex_Pr5393148144989827363nt_int @ R ) )
= ( member1390679175989562640nt_int @ ( produc1932183703851549015nt_int @ Ys2 @ Zs2 ) @ ( lex_Pr5393148144989827363nt_int @ R ) ) ) ) ).
% lex_append_left_iff
thf(fact_665_lex__append__left__iff,axiom,
! [R: set_Pr958786334691620121nt_int,Xs: list_int,Ys2: list_int,Zs2: list_int] :
( ! [X3: int] :
~ ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X3 @ X3 ) @ R )
=> ( ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ ( append_int @ Xs @ Ys2 ) @ ( append_int @ Xs @ Zs2 ) ) @ ( lex_int @ R ) )
= ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Ys2 @ Zs2 ) @ ( lex_int @ R ) ) ) ) ).
% lex_append_left_iff
thf(fact_666_lex__append__rightI,axiom,
! [Xs: list_P5707943133018811711nt_int,Ys2: list_P5707943133018811711nt_int,R: set_Pr2560585780119916871nt_int,Vs: list_P5707943133018811711nt_int,Us: list_P5707943133018811711nt_int] :
( ( member1390679175989562640nt_int @ ( produc1932183703851549015nt_int @ Xs @ Ys2 ) @ ( lex_Pr5393148144989827363nt_int @ R ) )
=> ( ( ( size_s5157815400016825771nt_int @ Vs )
= ( size_s5157815400016825771nt_int @ Us ) )
=> ( member1390679175989562640nt_int @ ( produc1932183703851549015nt_int @ ( append7030698103840186580nt_int @ Xs @ Us ) @ ( append7030698103840186580nt_int @ Ys2 @ Vs ) ) @ ( lex_Pr5393148144989827363nt_int @ R ) ) ) ) ).
% lex_append_rightI
thf(fact_667_lenlex__append1,axiom,
! [Us: list_P5707943133018811711nt_int,Xs: list_P5707943133018811711nt_int,R2: set_Pr2560585780119916871nt_int,Vs: list_P5707943133018811711nt_int,Ys2: list_P5707943133018811711nt_int] :
( ( member1390679175989562640nt_int @ ( produc1932183703851549015nt_int @ Us @ Xs ) @ ( lenlex6370358691973319492nt_int @ R2 ) )
=> ( ( ( size_s5157815400016825771nt_int @ Vs )
= ( size_s5157815400016825771nt_int @ Ys2 ) )
=> ( member1390679175989562640nt_int @ ( produc1932183703851549015nt_int @ ( append7030698103840186580nt_int @ Us @ Vs ) @ ( append7030698103840186580nt_int @ Xs @ Ys2 ) ) @ ( lenlex6370358691973319492nt_int @ R2 ) ) ) ) ).
% lenlex_append1
thf(fact_668_listrel1I,axiom,
! [X: product_prod_int_int,Y: product_prod_int_int,R: set_Pr2560585780119916871nt_int,Xs: list_P5707943133018811711nt_int,Us: list_P5707943133018811711nt_int,Vs: list_P5707943133018811711nt_int,Ys2: list_P5707943133018811711nt_int] :
( ( member8566619992076573584nt_int @ ( produc3646306378393792727nt_int @ X @ Y ) @ R )
=> ( ( Xs
= ( append7030698103840186580nt_int @ Us @ ( cons_P3334398858971670639nt_int @ X @ Vs ) ) )
=> ( ( Ys2
= ( append7030698103840186580nt_int @ Us @ ( cons_P3334398858971670639nt_int @ Y @ Vs ) ) )
=> ( member1390679175989562640nt_int @ ( produc1932183703851549015nt_int @ Xs @ Ys2 ) @ ( listre1649617614542993432nt_int @ R ) ) ) ) ) ).
% listrel1I
thf(fact_669_listrel1I,axiom,
! [X: int,Y: int,R: set_Pr958786334691620121nt_int,Xs: list_int,Us: list_int,Vs: list_int,Ys2: list_int] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ R )
=> ( ( Xs
= ( append_int @ Us @ ( cons_int @ X @ Vs ) ) )
=> ( ( Ys2
= ( append_int @ Us @ ( cons_int @ Y @ Vs ) ) )
=> ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Xs @ Ys2 ) @ ( listrel1_int @ R ) ) ) ) ) ).
% listrel1I
thf(fact_670_listrel1E,axiom,
! [Xs: list_P5707943133018811711nt_int,Ys2: list_P5707943133018811711nt_int,R: set_Pr2560585780119916871nt_int] :
( ( member1390679175989562640nt_int @ ( produc1932183703851549015nt_int @ Xs @ Ys2 ) @ ( listre1649617614542993432nt_int @ R ) )
=> ~ ! [X3: product_prod_int_int,Y3: product_prod_int_int] :
( ( member8566619992076573584nt_int @ ( produc3646306378393792727nt_int @ X3 @ Y3 ) @ R )
=> ! [Us3: list_P5707943133018811711nt_int,Vs2: list_P5707943133018811711nt_int] :
( ( Xs
= ( append7030698103840186580nt_int @ Us3 @ ( cons_P3334398858971670639nt_int @ X3 @ Vs2 ) ) )
=> ( Ys2
!= ( append7030698103840186580nt_int @ Us3 @ ( cons_P3334398858971670639nt_int @ Y3 @ Vs2 ) ) ) ) ) ) ).
% listrel1E
thf(fact_671_listrel1E,axiom,
! [Xs: list_int,Ys2: list_int,R: set_Pr958786334691620121nt_int] :
( ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Xs @ Ys2 ) @ ( listrel1_int @ R ) )
=> ~ ! [X3: int,Y3: int] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X3 @ Y3 ) @ R )
=> ! [Us3: list_int,Vs2: list_int] :
( ( Xs
= ( append_int @ Us3 @ ( cons_int @ X3 @ Vs2 ) ) )
=> ( Ys2
!= ( append_int @ Us3 @ ( cons_int @ Y3 @ Vs2 ) ) ) ) ) ) ).
% listrel1E
thf(fact_672_lexord__append__left__rightI,axiom,
! [A: product_prod_int_int,B: product_prod_int_int,R: set_Pr2560585780119916871nt_int,U: list_P5707943133018811711nt_int,X: list_P5707943133018811711nt_int,Y: list_P5707943133018811711nt_int] :
( ( member8566619992076573584nt_int @ ( produc3646306378393792727nt_int @ A @ B ) @ R )
=> ( member1390679175989562640nt_int @ ( produc1932183703851549015nt_int @ ( append7030698103840186580nt_int @ U @ ( cons_P3334398858971670639nt_int @ A @ X ) ) @ ( append7030698103840186580nt_int @ U @ ( cons_P3334398858971670639nt_int @ B @ Y ) ) ) @ ( lexord8886728381914977324nt_int @ R ) ) ) ).
% lexord_append_left_rightI
thf(fact_673_lexord__append__left__rightI,axiom,
! [A: int,B: int,R: set_Pr958786334691620121nt_int,U: list_int,X: list_int,Y: list_int] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A @ B ) @ R )
=> ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ ( append_int @ U @ ( cons_int @ A @ X ) ) @ ( append_int @ U @ ( cons_int @ B @ Y ) ) ) @ ( lexord_int @ R ) ) ) ).
% lexord_append_left_rightI
thf(fact_674_append__eq__append__conv__if,axiom,
! [Xs_1: list_P5707943133018811711nt_int,Xs_2: list_P5707943133018811711nt_int,Ys_1: list_P5707943133018811711nt_int,Ys_2: list_P5707943133018811711nt_int] :
( ( ( append7030698103840186580nt_int @ Xs_1 @ Xs_2 )
= ( append7030698103840186580nt_int @ Ys_1 @ Ys_2 ) )
= ( ( ( ord_less_eq_nat @ ( size_s5157815400016825771nt_int @ Xs_1 ) @ ( size_s5157815400016825771nt_int @ Ys_1 ) )
=> ( ( Xs_1
= ( take_P8218740963776755879nt_int @ ( size_s5157815400016825771nt_int @ Xs_1 ) @ Ys_1 ) )
& ( Xs_2
= ( append7030698103840186580nt_int @ ( drop_P5690361596310759935nt_int @ ( size_s5157815400016825771nt_int @ Xs_1 ) @ Ys_1 ) @ Ys_2 ) ) ) )
& ( ~ ( ord_less_eq_nat @ ( size_s5157815400016825771nt_int @ Xs_1 ) @ ( size_s5157815400016825771nt_int @ Ys_1 ) )
=> ( ( ( take_P8218740963776755879nt_int @ ( size_s5157815400016825771nt_int @ Ys_1 ) @ Xs_1 )
= Ys_1 )
& ( ( append7030698103840186580nt_int @ ( drop_P5690361596310759935nt_int @ ( size_s5157815400016825771nt_int @ Ys_1 ) @ Xs_1 ) @ Xs_2 )
= Ys_2 ) ) ) ) ) ).
% append_eq_append_conv_if
thf(fact_675_lexord__sufI,axiom,
! [U: list_P5707943133018811711nt_int,W: list_P5707943133018811711nt_int,R: set_Pr2560585780119916871nt_int,V: list_P5707943133018811711nt_int,Z3: list_P5707943133018811711nt_int] :
( ( member1390679175989562640nt_int @ ( produc1932183703851549015nt_int @ U @ W ) @ ( lexord8886728381914977324nt_int @ R ) )
=> ( ( ord_less_eq_nat @ ( size_s5157815400016825771nt_int @ W ) @ ( size_s5157815400016825771nt_int @ U ) )
=> ( member1390679175989562640nt_int @ ( produc1932183703851549015nt_int @ ( append7030698103840186580nt_int @ U @ V ) @ ( append7030698103840186580nt_int @ W @ Z3 ) ) @ ( lexord8886728381914977324nt_int @ R ) ) ) ) ).
% lexord_sufI
thf(fact_676_min_Oabsorb3,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_min_nat @ A @ B )
= A ) ) ).
% min.absorb3
thf(fact_677_min_Oabsorb3,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_min_int @ A @ B )
= A ) ) ).
% min.absorb3
thf(fact_678_min_Oabsorb4,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( ord_min_nat @ A @ B )
= B ) ) ).
% min.absorb4
thf(fact_679_min_Oabsorb4,axiom,
! [B: int,A: int] :
( ( ord_less_int @ B @ A )
=> ( ( ord_min_int @ A @ B )
= B ) ) ).
% min.absorb4
thf(fact_680_min__less__iff__conj,axiom,
! [Z3: nat,X: nat,Y: nat] :
( ( ord_less_nat @ Z3 @ ( ord_min_nat @ X @ Y ) )
= ( ( ord_less_nat @ Z3 @ X )
& ( ord_less_nat @ Z3 @ Y ) ) ) ).
% min_less_iff_conj
thf(fact_681_min__less__iff__conj,axiom,
! [Z3: int,X: int,Y: int] :
( ( ord_less_int @ Z3 @ ( ord_min_int @ X @ Y ) )
= ( ( ord_less_int @ Z3 @ X )
& ( ord_less_int @ Z3 @ Y ) ) ) ).
% min_less_iff_conj
thf(fact_682_min_Oabsorb1,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_min_nat @ A @ B )
= A ) ) ).
% min.absorb1
thf(fact_683_min_Oabsorb1,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_min_int @ A @ B )
= A ) ) ).
% min.absorb1
thf(fact_684_min_Oabsorb2,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_min_nat @ A @ B )
= B ) ) ).
% min.absorb2
thf(fact_685_min_Oabsorb2,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( ord_min_int @ A @ B )
= B ) ) ).
% min.absorb2
thf(fact_686_min_Obounded__iff,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ ( ord_min_nat @ B @ C2 ) )
= ( ( ord_less_eq_nat @ A @ B )
& ( ord_less_eq_nat @ A @ C2 ) ) ) ).
% min.bounded_iff
thf(fact_687_min_Obounded__iff,axiom,
! [A: int,B: int,C2: int] :
( ( ord_less_eq_int @ A @ ( ord_min_int @ B @ C2 ) )
= ( ( ord_less_eq_int @ A @ B )
& ( ord_less_eq_int @ A @ C2 ) ) ) ).
% min.bounded_iff
thf(fact_688_min_Omono,axiom,
! [A: nat,C2: nat,B: nat,D3: nat] :
( ( ord_less_eq_nat @ A @ C2 )
=> ( ( ord_less_eq_nat @ B @ D3 )
=> ( ord_less_eq_nat @ ( ord_min_nat @ A @ B ) @ ( ord_min_nat @ C2 @ D3 ) ) ) ) ).
% min.mono
thf(fact_689_min_Omono,axiom,
! [A: int,C2: int,B: int,D3: int] :
( ( ord_less_eq_int @ A @ C2 )
=> ( ( ord_less_eq_int @ B @ D3 )
=> ( ord_less_eq_int @ ( ord_min_int @ A @ B ) @ ( ord_min_int @ C2 @ D3 ) ) ) ) ).
% min.mono
thf(fact_690_min_OorderE,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( A
= ( ord_min_nat @ A @ B ) ) ) ).
% min.orderE
thf(fact_691_min_OorderE,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ B )
=> ( A
= ( ord_min_int @ A @ B ) ) ) ).
% min.orderE
thf(fact_692_min_OorderI,axiom,
! [A: nat,B: nat] :
( ( A
= ( ord_min_nat @ A @ B ) )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% min.orderI
thf(fact_693_min_OorderI,axiom,
! [A: int,B: int] :
( ( A
= ( ord_min_int @ A @ B ) )
=> ( ord_less_eq_int @ A @ B ) ) ).
% min.orderI
thf(fact_694_min_OboundedE,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ ( ord_min_nat @ B @ C2 ) )
=> ~ ( ( ord_less_eq_nat @ A @ B )
=> ~ ( ord_less_eq_nat @ A @ C2 ) ) ) ).
% min.boundedE
thf(fact_695_min_OboundedE,axiom,
! [A: int,B: int,C2: int] :
( ( ord_less_eq_int @ A @ ( ord_min_int @ B @ C2 ) )
=> ~ ( ( ord_less_eq_int @ A @ B )
=> ~ ( ord_less_eq_int @ A @ C2 ) ) ) ).
% min.boundedE
thf(fact_696_min_OboundedI,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ A @ C2 )
=> ( ord_less_eq_nat @ A @ ( ord_min_nat @ B @ C2 ) ) ) ) ).
% min.boundedI
thf(fact_697_min_OboundedI,axiom,
! [A: int,B: int,C2: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ A @ C2 )
=> ( ord_less_eq_int @ A @ ( ord_min_int @ B @ C2 ) ) ) ) ).
% min.boundedI
thf(fact_698_min_Oorder__iff,axiom,
( ord_less_eq_nat
= ( ^ [A5: nat,B4: nat] :
( A5
= ( ord_min_nat @ A5 @ B4 ) ) ) ) ).
% min.order_iff
thf(fact_699_min_Oorder__iff,axiom,
( ord_less_eq_int
= ( ^ [A5: int,B4: int] :
( A5
= ( ord_min_int @ A5 @ B4 ) ) ) ) ).
% min.order_iff
thf(fact_700_min_Ocobounded1,axiom,
! [A: nat,B: nat] : ( ord_less_eq_nat @ ( ord_min_nat @ A @ B ) @ A ) ).
% min.cobounded1
thf(fact_701_min_Ocobounded1,axiom,
! [A: int,B: int] : ( ord_less_eq_int @ ( ord_min_int @ A @ B ) @ A ) ).
% min.cobounded1
thf(fact_702_min_Ocobounded2,axiom,
! [A: nat,B: nat] : ( ord_less_eq_nat @ ( ord_min_nat @ A @ B ) @ B ) ).
% min.cobounded2
thf(fact_703_min_Ocobounded2,axiom,
! [A: int,B: int] : ( ord_less_eq_int @ ( ord_min_int @ A @ B ) @ B ) ).
% min.cobounded2
thf(fact_704_min_Oabsorb__iff1,axiom,
( ord_less_eq_nat
= ( ^ [A5: nat,B4: nat] :
( ( ord_min_nat @ A5 @ B4 )
= A5 ) ) ) ).
% min.absorb_iff1
thf(fact_705_min_Oabsorb__iff1,axiom,
( ord_less_eq_int
= ( ^ [A5: int,B4: int] :
( ( ord_min_int @ A5 @ B4 )
= A5 ) ) ) ).
% min.absorb_iff1
thf(fact_706_min_Oabsorb__iff2,axiom,
( ord_less_eq_nat
= ( ^ [B4: nat,A5: nat] :
( ( ord_min_nat @ A5 @ B4 )
= B4 ) ) ) ).
% min.absorb_iff2
thf(fact_707_min_Oabsorb__iff2,axiom,
( ord_less_eq_int
= ( ^ [B4: int,A5: int] :
( ( ord_min_int @ A5 @ B4 )
= B4 ) ) ) ).
% min.absorb_iff2
thf(fact_708_min_OcoboundedI1,axiom,
! [A: nat,C2: nat,B: nat] :
( ( ord_less_eq_nat @ A @ C2 )
=> ( ord_less_eq_nat @ ( ord_min_nat @ A @ B ) @ C2 ) ) ).
% min.coboundedI1
thf(fact_709_min_OcoboundedI1,axiom,
! [A: int,C2: int,B: int] :
( ( ord_less_eq_int @ A @ C2 )
=> ( ord_less_eq_int @ ( ord_min_int @ A @ B ) @ C2 ) ) ).
% min.coboundedI1
thf(fact_710_min_OcoboundedI2,axiom,
! [B: nat,C2: nat,A: nat] :
( ( ord_less_eq_nat @ B @ C2 )
=> ( ord_less_eq_nat @ ( ord_min_nat @ A @ B ) @ C2 ) ) ).
% min.coboundedI2
thf(fact_711_min_OcoboundedI2,axiom,
! [B: int,C2: int,A: int] :
( ( ord_less_eq_int @ B @ C2 )
=> ( ord_less_eq_int @ ( ord_min_int @ A @ B ) @ C2 ) ) ).
% min.coboundedI2
thf(fact_712_min__le__iff__disj,axiom,
! [X: nat,Y: nat,Z3: nat] :
( ( ord_less_eq_nat @ ( ord_min_nat @ X @ Y ) @ Z3 )
= ( ( ord_less_eq_nat @ X @ Z3 )
| ( ord_less_eq_nat @ Y @ Z3 ) ) ) ).
% min_le_iff_disj
thf(fact_713_min__le__iff__disj,axiom,
! [X: int,Y: int,Z3: int] :
( ( ord_less_eq_int @ ( ord_min_int @ X @ Y ) @ Z3 )
= ( ( ord_less_eq_int @ X @ Z3 )
| ( ord_less_eq_int @ Y @ Z3 ) ) ) ).
% min_le_iff_disj
thf(fact_714_min_Ostrict__coboundedI2,axiom,
! [B: nat,C2: nat,A: nat] :
( ( ord_less_nat @ B @ C2 )
=> ( ord_less_nat @ ( ord_min_nat @ A @ B ) @ C2 ) ) ).
% min.strict_coboundedI2
thf(fact_715_min_Ostrict__coboundedI2,axiom,
! [B: int,C2: int,A: int] :
( ( ord_less_int @ B @ C2 )
=> ( ord_less_int @ ( ord_min_int @ A @ B ) @ C2 ) ) ).
% min.strict_coboundedI2
thf(fact_716_min_Ostrict__coboundedI1,axiom,
! [A: nat,C2: nat,B: nat] :
( ( ord_less_nat @ A @ C2 )
=> ( ord_less_nat @ ( ord_min_nat @ A @ B ) @ C2 ) ) ).
% min.strict_coboundedI1
thf(fact_717_min_Ostrict__coboundedI1,axiom,
! [A: int,C2: int,B: int] :
( ( ord_less_int @ A @ C2 )
=> ( ord_less_int @ ( ord_min_int @ A @ B ) @ C2 ) ) ).
% min.strict_coboundedI1
thf(fact_718_min_Ostrict__order__iff,axiom,
( ord_less_nat
= ( ^ [A5: nat,B4: nat] :
( ( A5
= ( ord_min_nat @ A5 @ B4 ) )
& ( A5 != B4 ) ) ) ) ).
% min.strict_order_iff
thf(fact_719_min_Ostrict__order__iff,axiom,
( ord_less_int
= ( ^ [A5: int,B4: int] :
( ( A5
= ( ord_min_int @ A5 @ B4 ) )
& ( A5 != B4 ) ) ) ) ).
% min.strict_order_iff
thf(fact_720_min_Ostrict__boundedE,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_nat @ A @ ( ord_min_nat @ B @ C2 ) )
=> ~ ( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ A @ C2 ) ) ) ).
% min.strict_boundedE
thf(fact_721_min_Ostrict__boundedE,axiom,
! [A: int,B: int,C2: int] :
( ( ord_less_int @ A @ ( ord_min_int @ B @ C2 ) )
=> ~ ( ( ord_less_int @ A @ B )
=> ~ ( ord_less_int @ A @ C2 ) ) ) ).
% min.strict_boundedE
thf(fact_722_min__less__iff__disj,axiom,
! [X: nat,Y: nat,Z3: nat] :
( ( ord_less_nat @ ( ord_min_nat @ X @ Y ) @ Z3 )
= ( ( ord_less_nat @ X @ Z3 )
| ( ord_less_nat @ Y @ Z3 ) ) ) ).
% min_less_iff_disj
thf(fact_723_min__less__iff__disj,axiom,
! [X: int,Y: int,Z3: int] :
( ( ord_less_int @ ( ord_min_int @ X @ Y ) @ Z3 )
= ( ( ord_less_int @ X @ Z3 )
| ( ord_less_int @ Y @ Z3 ) ) ) ).
% min_less_iff_disj
thf(fact_724_take__hd__drop,axiom,
! [N: nat,Xs: list_P5707943133018811711nt_int] :
( ( ord_less_nat @ N @ ( size_s5157815400016825771nt_int @ Xs ) )
=> ( ( append7030698103840186580nt_int @ ( take_P8218740963776755879nt_int @ N @ Xs ) @ ( cons_P3334398858971670639nt_int @ ( hd_Pro282112905867057956nt_int @ ( drop_P5690361596310759935nt_int @ N @ Xs ) ) @ nil_Pr2300489316682597567nt_int ) )
= ( take_P8218740963776755879nt_int @ ( suc @ N ) @ Xs ) ) ) ).
% take_hd_drop
thf(fact_725_take__Suc__conv__app__nth,axiom,
! [I4: nat,Xs: list_P5707943133018811711nt_int] :
( ( ord_less_nat @ I4 @ ( size_s5157815400016825771nt_int @ Xs ) )
=> ( ( take_P8218740963776755879nt_int @ ( suc @ I4 ) @ Xs )
= ( append7030698103840186580nt_int @ ( take_P8218740963776755879nt_int @ I4 @ Xs ) @ ( cons_P3334398858971670639nt_int @ ( nth_Pr4439495888332055232nt_int @ Xs @ I4 ) @ nil_Pr2300489316682597567nt_int ) ) ) ) ).
% take_Suc_conv_app_nth
thf(fact_726_bind__simps_I2_J,axiom,
! [X: product_prod_int_int,Xs: list_P5707943133018811711nt_int,F: product_prod_int_int > list_P5707943133018811711nt_int] :
( ( bind_P8056059132564497095nt_int @ ( cons_P3334398858971670639nt_int @ X @ Xs ) @ F )
= ( append7030698103840186580nt_int @ ( F @ X ) @ ( bind_P8056059132564497095nt_int @ Xs @ F ) ) ) ).
% bind_simps(2)
thf(fact_727_nth__enumerate__eq,axiom,
! [M: nat,Xs: list_set_int,N: nat] :
( ( ord_less_nat @ M @ ( size_s763970886550748294et_int @ Xs ) )
=> ( ( nth_Pr4328007639516273434et_int @ ( enumerate_set_int @ N @ Xs ) @ M )
= ( produc29655638201817675et_int @ ( plus_plus_nat @ N @ M ) @ ( nth_set_int @ Xs @ M ) ) ) ) ).
% nth_enumerate_eq
thf(fact_728_nth__enumerate__eq,axiom,
! [M: nat,Xs: list_P5707943133018811711nt_int,N: nat] :
( ( ord_less_nat @ M @ ( size_s5157815400016825771nt_int @ Xs ) )
=> ( ( nth_Pr7109764839600559639nt_int @ ( enumer5233495138702836510nt_int @ N @ Xs ) @ M )
= ( produc6532261156004778512nt_int @ ( plus_plus_nat @ N @ M ) @ ( nth_Pr4439495888332055232nt_int @ Xs @ M ) ) ) ) ).
% nth_enumerate_eq
thf(fact_729_append__is__Nil__conv,axiom,
! [Xs: list_P5707943133018811711nt_int,Ys2: list_P5707943133018811711nt_int] :
( ( ( append7030698103840186580nt_int @ Xs @ Ys2 )
= nil_Pr2300489316682597567nt_int )
= ( ( Xs = nil_Pr2300489316682597567nt_int )
& ( Ys2 = nil_Pr2300489316682597567nt_int ) ) ) ).
% append_is_Nil_conv
thf(fact_730_Nil__is__append__conv,axiom,
! [Xs: list_P5707943133018811711nt_int,Ys2: list_P5707943133018811711nt_int] :
( ( nil_Pr2300489316682597567nt_int
= ( append7030698103840186580nt_int @ Xs @ Ys2 ) )
= ( ( Xs = nil_Pr2300489316682597567nt_int )
& ( Ys2 = nil_Pr2300489316682597567nt_int ) ) ) ).
% Nil_is_append_conv
thf(fact_731_self__append__conv2,axiom,
! [Y: list_P5707943133018811711nt_int,Xs: list_P5707943133018811711nt_int] :
( ( Y
= ( append7030698103840186580nt_int @ Xs @ Y ) )
= ( Xs = nil_Pr2300489316682597567nt_int ) ) ).
% self_append_conv2
thf(fact_732_append__self__conv2,axiom,
! [Xs: list_P5707943133018811711nt_int,Ys2: list_P5707943133018811711nt_int] :
( ( ( append7030698103840186580nt_int @ Xs @ Ys2 )
= Ys2 )
= ( Xs = nil_Pr2300489316682597567nt_int ) ) ).
% append_self_conv2
thf(fact_733_self__append__conv,axiom,
! [Y: list_P5707943133018811711nt_int,Ys2: list_P5707943133018811711nt_int] :
( ( Y
= ( append7030698103840186580nt_int @ Y @ Ys2 ) )
= ( Ys2 = nil_Pr2300489316682597567nt_int ) ) ).
% self_append_conv
thf(fact_734_append__self__conv,axiom,
! [Xs: list_P5707943133018811711nt_int,Ys2: list_P5707943133018811711nt_int] :
( ( ( append7030698103840186580nt_int @ Xs @ Ys2 )
= Xs )
= ( Ys2 = nil_Pr2300489316682597567nt_int ) ) ).
% append_self_conv
thf(fact_735_append__Nil2,axiom,
! [Xs: list_P5707943133018811711nt_int] :
( ( append7030698103840186580nt_int @ Xs @ nil_Pr2300489316682597567nt_int )
= Xs ) ).
% append_Nil2
thf(fact_736_append_Oright__neutral,axiom,
! [A: list_P5707943133018811711nt_int] :
( ( append7030698103840186580nt_int @ A @ nil_Pr2300489316682597567nt_int )
= A ) ).
% append.right_neutral
thf(fact_737_list__update__nonempty,axiom,
! [Xs: list_P5707943133018811711nt_int,K: nat,X: product_prod_int_int] :
( ( ( list_u3002344382305578791nt_int @ Xs @ K @ X )
= nil_Pr2300489316682597567nt_int )
= ( Xs = nil_Pr2300489316682597567nt_int ) ) ).
% list_update_nonempty
thf(fact_738_rotate__is__Nil__conv,axiom,
! [N: nat,Xs: list_P5707943133018811711nt_int] :
( ( ( rotate3281235723284208371nt_int @ N @ Xs )
= nil_Pr2300489316682597567nt_int )
= ( Xs = nil_Pr2300489316682597567nt_int ) ) ).
% rotate_is_Nil_conv
thf(fact_739_rotate1__is__Nil__conv,axiom,
! [Xs: list_P5707943133018811711nt_int] :
( ( ( rotate9137591727117064716nt_int @ Xs )
= nil_Pr2300489316682597567nt_int )
= ( Xs = nil_Pr2300489316682597567nt_int ) ) ).
% rotate1_is_Nil_conv
thf(fact_740_enumerate__simps_I1_J,axiom,
! [N: nat] :
( ( enumer5233495138702836510nt_int @ N @ nil_Pr2300489316682597567nt_int )
= nil_Pr5165738950135863640nt_int ) ).
% enumerate_simps(1)
thf(fact_741_length__enumerate,axiom,
! [N: nat,Xs: list_P5707943133018811711nt_int] :
( ( size_s2001693051472072450nt_int @ ( enumer5233495138702836510nt_int @ N @ Xs ) )
= ( size_s5157815400016825771nt_int @ Xs ) ) ).
% length_enumerate
thf(fact_742_list__ex__simps_I2_J,axiom,
! [P2: product_prod_int_int > $o] :
~ ( list_e4511028299437703933nt_int @ P2 @ nil_Pr2300489316682597567nt_int ) ).
% list_ex_simps(2)
thf(fact_743_in__measures_I1_J,axiom,
! [X: int,Y: int] :
~ ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ ( measures_int @ nil_int_nat ) ) ).
% in_measures(1)
thf(fact_744_bind__simps_I1_J,axiom,
! [F: product_prod_int_int > list_P5707943133018811711nt_int] :
( ( bind_P8056059132564497095nt_int @ nil_Pr2300489316682597567nt_int @ F )
= nil_Pr2300489316682597567nt_int ) ).
% bind_simps(1)
thf(fact_745_append1__eq__conv,axiom,
! [Xs: list_P5707943133018811711nt_int,X: product_prod_int_int,Ys2: list_P5707943133018811711nt_int,Y: product_prod_int_int] :
( ( ( append7030698103840186580nt_int @ Xs @ ( cons_P3334398858971670639nt_int @ X @ nil_Pr2300489316682597567nt_int ) )
= ( append7030698103840186580nt_int @ Ys2 @ ( cons_P3334398858971670639nt_int @ Y @ nil_Pr2300489316682597567nt_int ) ) )
= ( ( Xs = Ys2 )
& ( X = Y ) ) ) ).
% append1_eq_conv
thf(fact_746_hd__append2,axiom,
! [Xs: list_P5707943133018811711nt_int,Ys2: list_P5707943133018811711nt_int] :
( ( Xs != nil_Pr2300489316682597567nt_int )
=> ( ( hd_Pro282112905867057956nt_int @ ( append7030698103840186580nt_int @ Xs @ Ys2 ) )
= ( hd_Pro282112905867057956nt_int @ Xs ) ) ) ).
% hd_append2
thf(fact_747_last__appendR,axiom,
! [Ys2: list_P5707943133018811711nt_int,Xs: list_P5707943133018811711nt_int] :
( ( Ys2 != nil_Pr2300489316682597567nt_int )
=> ( ( last_P3305686521732843992nt_int @ ( append7030698103840186580nt_int @ Xs @ Ys2 ) )
= ( last_P3305686521732843992nt_int @ Ys2 ) ) ) ).
% last_appendR
thf(fact_748_last__appendL,axiom,
! [Ys2: list_P5707943133018811711nt_int,Xs: list_P5707943133018811711nt_int] :
( ( Ys2 = nil_Pr2300489316682597567nt_int )
=> ( ( last_P3305686521732843992nt_int @ ( append7030698103840186580nt_int @ Xs @ Ys2 ) )
= ( last_P3305686521732843992nt_int @ Xs ) ) ) ).
% last_appendL
thf(fact_749_Nil__lenlex__iff1,axiom,
! [Ns: list_P5707943133018811711nt_int,R: set_Pr2560585780119916871nt_int] :
( ( member1390679175989562640nt_int @ ( produc1932183703851549015nt_int @ nil_Pr2300489316682597567nt_int @ Ns ) @ ( lenlex6370358691973319492nt_int @ R ) )
= ( Ns != nil_Pr2300489316682597567nt_int ) ) ).
% Nil_lenlex_iff1
thf(fact_750_drop__eq__Nil2,axiom,
! [N: nat,Xs: list_P5707943133018811711nt_int] :
( ( nil_Pr2300489316682597567nt_int
= ( drop_P5690361596310759935nt_int @ N @ Xs ) )
= ( ord_less_eq_nat @ ( size_s5157815400016825771nt_int @ Xs ) @ N ) ) ).
% drop_eq_Nil2
thf(fact_751_drop__eq__Nil,axiom,
! [N: nat,Xs: list_P5707943133018811711nt_int] :
( ( ( drop_P5690361596310759935nt_int @ N @ Xs )
= nil_Pr2300489316682597567nt_int )
= ( ord_less_eq_nat @ ( size_s5157815400016825771nt_int @ Xs ) @ N ) ) ).
% drop_eq_Nil
thf(fact_752_drop__all,axiom,
! [Xs: list_P5707943133018811711nt_int,N: nat] :
( ( ord_less_eq_nat @ ( size_s5157815400016825771nt_int @ Xs ) @ N )
=> ( ( drop_P5690361596310759935nt_int @ N @ Xs )
= nil_Pr2300489316682597567nt_int ) ) ).
% drop_all
thf(fact_753_last__snoc,axiom,
! [Xs: list_P5707943133018811711nt_int,X: product_prod_int_int] :
( ( last_P3305686521732843992nt_int @ ( append7030698103840186580nt_int @ Xs @ ( cons_P3334398858971670639nt_int @ X @ nil_Pr2300489316682597567nt_int ) ) )
= X ) ).
% last_snoc
thf(fact_754_lexord__Nil__left,axiom,
! [Y: list_P5707943133018811711nt_int,R: set_Pr2560585780119916871nt_int] :
( ( member1390679175989562640nt_int @ ( produc1932183703851549015nt_int @ nil_Pr2300489316682597567nt_int @ Y ) @ ( lexord8886728381914977324nt_int @ R ) )
= ( ? [A5: product_prod_int_int,X4: list_P5707943133018811711nt_int] :
( Y
= ( cons_P3334398858971670639nt_int @ A5 @ X4 ) ) ) ) ).
% lexord_Nil_left
thf(fact_755_enumerate__simps_I2_J,axiom,
! [N: nat,X: product_prod_int_int,Xs: list_P5707943133018811711nt_int] :
( ( enumer5233495138702836510nt_int @ N @ ( cons_P3334398858971670639nt_int @ X @ Xs ) )
= ( cons_P4639984832015440296nt_int @ ( produc6532261156004778512nt_int @ N @ X ) @ ( enumer5233495138702836510nt_int @ ( suc @ N ) @ Xs ) ) ) ).
% enumerate_simps(2)
thf(fact_756_enumerate__simps_I2_J,axiom,
! [N: nat,X: set_int,Xs: list_set_int] :
( ( enumerate_set_int @ N @ ( cons_set_int @ X @ Xs ) )
= ( cons_P5529800768674571465et_int @ ( produc29655638201817675et_int @ N @ X ) @ ( enumerate_set_int @ ( suc @ N ) @ Xs ) ) ) ).
% enumerate_simps(2)
thf(fact_757_append__Nil,axiom,
! [Ys2: list_P5707943133018811711nt_int] :
( ( append7030698103840186580nt_int @ nil_Pr2300489316682597567nt_int @ Ys2 )
= Ys2 ) ).
% append_Nil
thf(fact_758_append_Oleft__neutral,axiom,
! [A: list_P5707943133018811711nt_int] :
( ( append7030698103840186580nt_int @ nil_Pr2300489316682597567nt_int @ A )
= A ) ).
% append.left_neutral
thf(fact_759_eq__Nil__appendI,axiom,
! [Xs: list_P5707943133018811711nt_int,Ys2: list_P5707943133018811711nt_int] :
( ( Xs = Ys2 )
=> ( Xs
= ( append7030698103840186580nt_int @ nil_Pr2300489316682597567nt_int @ Ys2 ) ) ) ).
% eq_Nil_appendI
thf(fact_760_take__Nil,axiom,
! [N: nat] :
( ( take_P8218740963776755879nt_int @ N @ nil_Pr2300489316682597567nt_int )
= nil_Pr2300489316682597567nt_int ) ).
% take_Nil
thf(fact_761_drop__Nil,axiom,
! [N: nat] :
( ( drop_P5690361596310759935nt_int @ N @ nil_Pr2300489316682597567nt_int )
= nil_Pr2300489316682597567nt_int ) ).
% drop_Nil
thf(fact_762_KnightsTour_Otranspose_Ocases,axiom,
! [X: list_P5707943133018811711nt_int] :
( ( X != nil_Pr2300489316682597567nt_int )
=> ~ ! [S_i2: product_prod_int_int,Ps3: list_P5707943133018811711nt_int] :
( X
!= ( cons_P3334398858971670639nt_int @ S_i2 @ Ps3 ) ) ) ).
% KnightsTour.transpose.cases
thf(fact_763_path__checker_Ocases,axiom,
! [X: produc2007852851243229709nt_int] :
( ! [B3: set_Pr958786334691620121nt_int] :
( X
!= ( produc2261658324281137661nt_int @ B3 @ nil_Pr2300489316682597567nt_int ) )
=> ( ! [B3: set_Pr958786334691620121nt_int,S_i2: product_prod_int_int] :
( X
!= ( produc2261658324281137661nt_int @ B3 @ ( cons_P3334398858971670639nt_int @ S_i2 @ nil_Pr2300489316682597567nt_int ) ) )
=> ~ ! [B3: set_Pr958786334691620121nt_int,S_i2: product_prod_int_int,S_j: product_prod_int_int,Ps3: list_P5707943133018811711nt_int] :
( X
!= ( produc2261658324281137661nt_int @ B3 @ ( cons_P3334398858971670639nt_int @ S_i2 @ ( cons_P3334398858971670639nt_int @ S_j @ Ps3 ) ) ) ) ) ) ).
% path_checker.cases
thf(fact_764_mirror2__aux_Ocases,axiom,
! [X: produc661532565036771336nt_int] :
( ! [M6: int] :
( X
!= ( produc8814303788642274490nt_int @ M6 @ nil_Pr2300489316682597567nt_int ) )
=> ~ ! [M6: int,S_i2: product_prod_int_int,Ps3: list_P5707943133018811711nt_int] :
( X
!= ( produc8814303788642274490nt_int @ M6 @ ( cons_P3334398858971670639nt_int @ S_i2 @ Ps3 ) ) ) ) ).
% mirror2_aux.cases
thf(fact_765_list__update__code_I1_J,axiom,
! [I4: nat,Y: product_prod_int_int] :
( ( list_u3002344382305578791nt_int @ nil_Pr2300489316682597567nt_int @ I4 @ Y )
= nil_Pr2300489316682597567nt_int ) ).
% list_update_code(1)
thf(fact_766_list__update_Osimps_I1_J,axiom,
! [I4: nat,V: product_prod_int_int] :
( ( list_u3002344382305578791nt_int @ nil_Pr2300489316682597567nt_int @ I4 @ V )
= nil_Pr2300489316682597567nt_int ) ).
% list_update.simps(1)
thf(fact_767_List_Otranspose_Ocases,axiom,
! [X: list_l1670014477004246597nt_int] :
( ( X != nil_li8670148097206105925nt_int )
=> ( ! [Xss: list_l1670014477004246597nt_int] :
( X
!= ( cons_l7309679040211256053nt_int @ nil_Pr2300489316682597567nt_int @ Xss ) )
=> ~ ! [X3: product_prod_int_int,Xs2: list_P5707943133018811711nt_int,Xss: list_l1670014477004246597nt_int] :
( X
!= ( cons_l7309679040211256053nt_int @ ( cons_P3334398858971670639nt_int @ X3 @ Xs2 ) @ Xss ) ) ) ) ).
% List.transpose.cases
thf(fact_768_list_Odistinct_I1_J,axiom,
! [X21: product_prod_int_int,X22: list_P5707943133018811711nt_int] :
( nil_Pr2300489316682597567nt_int
!= ( cons_P3334398858971670639nt_int @ X21 @ X22 ) ) ).
% list.distinct(1)
thf(fact_769_list_OdiscI,axiom,
! [List: list_P5707943133018811711nt_int,X21: product_prod_int_int,X22: list_P5707943133018811711nt_int] :
( ( List
= ( cons_P3334398858971670639nt_int @ X21 @ X22 ) )
=> ( List != nil_Pr2300489316682597567nt_int ) ) ).
% list.discI
thf(fact_770_list_Oexhaust,axiom,
! [Y: list_P5707943133018811711nt_int] :
( ( Y != nil_Pr2300489316682597567nt_int )
=> ~ ! [X212: product_prod_int_int,X222: list_P5707943133018811711nt_int] :
( Y
!= ( cons_P3334398858971670639nt_int @ X212 @ X222 ) ) ) ).
% list.exhaust
thf(fact_771_remdups__adj_Ocases,axiom,
! [X: list_P5707943133018811711nt_int] :
( ( X != nil_Pr2300489316682597567nt_int )
=> ( ! [X3: product_prod_int_int] :
( X
!= ( cons_P3334398858971670639nt_int @ X3 @ nil_Pr2300489316682597567nt_int ) )
=> ~ ! [X3: product_prod_int_int,Y3: product_prod_int_int,Xs2: list_P5707943133018811711nt_int] :
( X
!= ( cons_P3334398858971670639nt_int @ X3 @ ( cons_P3334398858971670639nt_int @ Y3 @ Xs2 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_772_neq__Nil__conv,axiom,
! [Xs: list_P5707943133018811711nt_int] :
( ( Xs != nil_Pr2300489316682597567nt_int )
= ( ? [Y6: product_prod_int_int,Ys3: list_P5707943133018811711nt_int] :
( Xs
= ( cons_P3334398858971670639nt_int @ Y6 @ Ys3 ) ) ) ) ).
% neq_Nil_conv
thf(fact_773_list__induct2_H,axiom,
! [P2: list_P5707943133018811711nt_int > list_P5707943133018811711nt_int > $o,Xs: list_P5707943133018811711nt_int,Ys2: list_P5707943133018811711nt_int] :
( ( P2 @ nil_Pr2300489316682597567nt_int @ nil_Pr2300489316682597567nt_int )
=> ( ! [X3: product_prod_int_int,Xs2: list_P5707943133018811711nt_int] : ( P2 @ ( cons_P3334398858971670639nt_int @ X3 @ Xs2 ) @ nil_Pr2300489316682597567nt_int )
=> ( ! [Y3: product_prod_int_int,Ys4: list_P5707943133018811711nt_int] : ( P2 @ nil_Pr2300489316682597567nt_int @ ( cons_P3334398858971670639nt_int @ Y3 @ Ys4 ) )
=> ( ! [X3: product_prod_int_int,Xs2: list_P5707943133018811711nt_int,Y3: product_prod_int_int,Ys4: list_P5707943133018811711nt_int] :
( ( P2 @ Xs2 @ Ys4 )
=> ( P2 @ ( cons_P3334398858971670639nt_int @ X3 @ Xs2 ) @ ( cons_P3334398858971670639nt_int @ Y3 @ Ys4 ) ) )
=> ( P2 @ Xs @ Ys2 ) ) ) ) ) ).
% list_induct2'
thf(fact_774_list__nonempty__induct,axiom,
! [Xs: list_P5707943133018811711nt_int,P2: list_P5707943133018811711nt_int > $o] :
( ( Xs != nil_Pr2300489316682597567nt_int )
=> ( ! [X3: product_prod_int_int] : ( P2 @ ( cons_P3334398858971670639nt_int @ X3 @ nil_Pr2300489316682597567nt_int ) )
=> ( ! [X3: product_prod_int_int,Xs2: list_P5707943133018811711nt_int] :
( ( Xs2 != nil_Pr2300489316682597567nt_int )
=> ( ( P2 @ Xs2 )
=> ( P2 @ ( cons_P3334398858971670639nt_int @ X3 @ Xs2 ) ) ) )
=> ( P2 @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_775_successively_Ocases,axiom,
! [X: produc1050408459402128056nt_int] :
( ! [P6: product_prod_int_int > product_prod_int_int > $o] :
( X
!= ( produc3328129369365053992nt_int @ P6 @ nil_Pr2300489316682597567nt_int ) )
=> ( ! [P6: product_prod_int_int > product_prod_int_int > $o,X3: product_prod_int_int] :
( X
!= ( produc3328129369365053992nt_int @ P6 @ ( cons_P3334398858971670639nt_int @ X3 @ nil_Pr2300489316682597567nt_int ) ) )
=> ~ ! [P6: product_prod_int_int > product_prod_int_int > $o,X3: product_prod_int_int,Y3: product_prod_int_int,Xs2: list_P5707943133018811711nt_int] :
( X
!= ( produc3328129369365053992nt_int @ P6 @ ( cons_P3334398858971670639nt_int @ X3 @ ( cons_P3334398858971670639nt_int @ Y3 @ Xs2 ) ) ) ) ) ) ).
% successively.cases
thf(fact_776_sorted__wrt_Ocases,axiom,
! [X: produc1050408459402128056nt_int] :
( ! [P6: product_prod_int_int > product_prod_int_int > $o] :
( X
!= ( produc3328129369365053992nt_int @ P6 @ nil_Pr2300489316682597567nt_int ) )
=> ~ ! [P6: product_prod_int_int > product_prod_int_int > $o,X3: product_prod_int_int,Ys4: list_P5707943133018811711nt_int] :
( X
!= ( produc3328129369365053992nt_int @ P6 @ ( cons_P3334398858971670639nt_int @ X3 @ Ys4 ) ) ) ) ).
% sorted_wrt.cases
thf(fact_777_rotate1_Osimps_I1_J,axiom,
( ( rotate9137591727117064716nt_int @ nil_Pr2300489316682597567nt_int )
= nil_Pr2300489316682597567nt_int ) ).
% rotate1.simps(1)
thf(fact_778_list__induct3,axiom,
! [Xs: list_P5707943133018811711nt_int,Ys2: list_P5707943133018811711nt_int,Zs2: list_P5707943133018811711nt_int,P2: list_P5707943133018811711nt_int > list_P5707943133018811711nt_int > list_P5707943133018811711nt_int > $o] :
( ( ( size_s5157815400016825771nt_int @ Xs )
= ( size_s5157815400016825771nt_int @ Ys2 ) )
=> ( ( ( size_s5157815400016825771nt_int @ Ys2 )
= ( size_s5157815400016825771nt_int @ Zs2 ) )
=> ( ( P2 @ nil_Pr2300489316682597567nt_int @ nil_Pr2300489316682597567nt_int @ nil_Pr2300489316682597567nt_int )
=> ( ! [X3: product_prod_int_int,Xs2: list_P5707943133018811711nt_int,Y3: product_prod_int_int,Ys4: list_P5707943133018811711nt_int,Z2: product_prod_int_int,Zs: list_P5707943133018811711nt_int] :
( ( ( size_s5157815400016825771nt_int @ Xs2 )
= ( size_s5157815400016825771nt_int @ Ys4 ) )
=> ( ( ( size_s5157815400016825771nt_int @ Ys4 )
= ( size_s5157815400016825771nt_int @ Zs ) )
=> ( ( P2 @ Xs2 @ Ys4 @ Zs )
=> ( P2 @ ( cons_P3334398858971670639nt_int @ X3 @ Xs2 ) @ ( cons_P3334398858971670639nt_int @ Y3 @ Ys4 ) @ ( cons_P3334398858971670639nt_int @ Z2 @ Zs ) ) ) ) )
=> ( P2 @ Xs @ Ys2 @ Zs2 ) ) ) ) ) ).
% list_induct3
thf(fact_779_list__induct2,axiom,
! [Xs: list_P5707943133018811711nt_int,Ys2: list_P5707943133018811711nt_int,P2: list_P5707943133018811711nt_int > list_P5707943133018811711nt_int > $o] :
( ( ( size_s5157815400016825771nt_int @ Xs )
= ( size_s5157815400016825771nt_int @ Ys2 ) )
=> ( ( P2 @ nil_Pr2300489316682597567nt_int @ nil_Pr2300489316682597567nt_int )
=> ( ! [X3: product_prod_int_int,Xs2: list_P5707943133018811711nt_int,Y3: product_prod_int_int,Ys4: list_P5707943133018811711nt_int] :
( ( ( size_s5157815400016825771nt_int @ Xs2 )
= ( size_s5157815400016825771nt_int @ Ys4 ) )
=> ( ( P2 @ Xs2 @ Ys4 )
=> ( P2 @ ( cons_P3334398858971670639nt_int @ X3 @ Xs2 ) @ ( cons_P3334398858971670639nt_int @ Y3 @ Ys4 ) ) ) )
=> ( P2 @ Xs @ Ys2 ) ) ) ) ).
% list_induct2
thf(fact_780_rev__nonempty__induct,axiom,
! [Xs: list_P5707943133018811711nt_int,P2: list_P5707943133018811711nt_int > $o] :
( ( Xs != nil_Pr2300489316682597567nt_int )
=> ( ! [X3: product_prod_int_int] : ( P2 @ ( cons_P3334398858971670639nt_int @ X3 @ nil_Pr2300489316682597567nt_int ) )
=> ( ! [X3: product_prod_int_int,Xs2: list_P5707943133018811711nt_int] :
( ( Xs2 != nil_Pr2300489316682597567nt_int )
=> ( ( P2 @ Xs2 )
=> ( P2 @ ( append7030698103840186580nt_int @ Xs2 @ ( cons_P3334398858971670639nt_int @ X3 @ nil_Pr2300489316682597567nt_int ) ) ) ) )
=> ( P2 @ Xs ) ) ) ) ).
% rev_nonempty_induct
thf(fact_781_append__eq__Cons__conv,axiom,
! [Ys2: list_P5707943133018811711nt_int,Zs2: list_P5707943133018811711nt_int,X: product_prod_int_int,Xs: list_P5707943133018811711nt_int] :
( ( ( append7030698103840186580nt_int @ Ys2 @ Zs2 )
= ( cons_P3334398858971670639nt_int @ X @ Xs ) )
= ( ( ( Ys2 = nil_Pr2300489316682597567nt_int )
& ( Zs2
= ( cons_P3334398858971670639nt_int @ X @ Xs ) ) )
| ? [Ys5: list_P5707943133018811711nt_int] :
( ( Ys2
= ( cons_P3334398858971670639nt_int @ X @ Ys5 ) )
& ( ( append7030698103840186580nt_int @ Ys5 @ Zs2 )
= Xs ) ) ) ) ).
% append_eq_Cons_conv
thf(fact_782_Cons__eq__append__conv,axiom,
! [X: product_prod_int_int,Xs: list_P5707943133018811711nt_int,Ys2: list_P5707943133018811711nt_int,Zs2: list_P5707943133018811711nt_int] :
( ( ( cons_P3334398858971670639nt_int @ X @ Xs )
= ( append7030698103840186580nt_int @ Ys2 @ Zs2 ) )
= ( ( ( Ys2 = nil_Pr2300489316682597567nt_int )
& ( ( cons_P3334398858971670639nt_int @ X @ Xs )
= Zs2 ) )
| ? [Ys5: list_P5707943133018811711nt_int] :
( ( ( cons_P3334398858971670639nt_int @ X @ Ys5 )
= Ys2 )
& ( Xs
= ( append7030698103840186580nt_int @ Ys5 @ Zs2 ) ) ) ) ) ).
% Cons_eq_append_conv
thf(fact_783_rev__exhaust,axiom,
! [Xs: list_P5707943133018811711nt_int] :
( ( Xs != nil_Pr2300489316682597567nt_int )
=> ~ ! [Ys4: list_P5707943133018811711nt_int,Y3: product_prod_int_int] :
( Xs
!= ( append7030698103840186580nt_int @ Ys4 @ ( cons_P3334398858971670639nt_int @ Y3 @ nil_Pr2300489316682597567nt_int ) ) ) ) ).
% rev_exhaust
thf(fact_784_rev__induct,axiom,
! [P2: list_P5707943133018811711nt_int > $o,Xs: list_P5707943133018811711nt_int] :
( ( P2 @ nil_Pr2300489316682597567nt_int )
=> ( ! [X3: product_prod_int_int,Xs2: list_P5707943133018811711nt_int] :
( ( P2 @ Xs2 )
=> ( P2 @ ( append7030698103840186580nt_int @ Xs2 @ ( cons_P3334398858971670639nt_int @ X3 @ nil_Pr2300489316682597567nt_int ) ) ) )
=> ( P2 @ Xs ) ) ) ).
% rev_induct
thf(fact_785_splice_Ocases,axiom,
! [X: produc1089560213143673063nt_int] :
( ! [Ys4: list_P5707943133018811711nt_int] :
( X
!= ( produc1932183703851549015nt_int @ nil_Pr2300489316682597567nt_int @ Ys4 ) )
=> ~ ! [X3: product_prod_int_int,Xs2: list_P5707943133018811711nt_int,Ys4: list_P5707943133018811711nt_int] :
( X
!= ( produc1932183703851549015nt_int @ ( cons_P3334398858971670639nt_int @ X3 @ Xs2 ) @ Ys4 ) ) ) ).
% splice.cases
thf(fact_786_shuffles_Ocases,axiom,
! [X: produc1089560213143673063nt_int] :
( ! [Ys4: list_P5707943133018811711nt_int] :
( X
!= ( produc1932183703851549015nt_int @ nil_Pr2300489316682597567nt_int @ Ys4 ) )
=> ( ! [Xs2: list_P5707943133018811711nt_int] :
( X
!= ( produc1932183703851549015nt_int @ Xs2 @ nil_Pr2300489316682597567nt_int ) )
=> ~ ! [X3: product_prod_int_int,Xs2: list_P5707943133018811711nt_int,Y3: product_prod_int_int,Ys4: list_P5707943133018811711nt_int] :
( X
!= ( produc1932183703851549015nt_int @ ( cons_P3334398858971670639nt_int @ X3 @ Xs2 ) @ ( cons_P3334398858971670639nt_int @ Y3 @ Ys4 ) ) ) ) ) ).
% shuffles.cases
thf(fact_787_last_Osimps,axiom,
! [Xs: list_P5707943133018811711nt_int,X: product_prod_int_int] :
( ( ( Xs = nil_Pr2300489316682597567nt_int )
=> ( ( last_P3305686521732843992nt_int @ ( cons_P3334398858971670639nt_int @ X @ Xs ) )
= X ) )
& ( ( Xs != nil_Pr2300489316682597567nt_int )
=> ( ( last_P3305686521732843992nt_int @ ( cons_P3334398858971670639nt_int @ X @ Xs ) )
= ( last_P3305686521732843992nt_int @ Xs ) ) ) ) ).
% last.simps
thf(fact_788_last__ConsL,axiom,
! [Xs: list_P5707943133018811711nt_int,X: product_prod_int_int] :
( ( Xs = nil_Pr2300489316682597567nt_int )
=> ( ( last_P3305686521732843992nt_int @ ( cons_P3334398858971670639nt_int @ X @ Xs ) )
= X ) ) ).
% last_ConsL
thf(fact_789_last__ConsR,axiom,
! [Xs: list_P5707943133018811711nt_int,X: product_prod_int_int] :
( ( Xs != nil_Pr2300489316682597567nt_int )
=> ( ( last_P3305686521732843992nt_int @ ( cons_P3334398858971670639nt_int @ X @ Xs ) )
= ( last_P3305686521732843992nt_int @ Xs ) ) ) ).
% last_ConsR
thf(fact_790_longest__common__prefix,axiom,
! [Xs: list_P5707943133018811711nt_int,Ys2: list_P5707943133018811711nt_int] :
? [Ps3: list_P5707943133018811711nt_int,Xs3: list_P5707943133018811711nt_int,Ys6: list_P5707943133018811711nt_int] :
( ( Xs
= ( append7030698103840186580nt_int @ Ps3 @ Xs3 ) )
& ( Ys2
= ( append7030698103840186580nt_int @ Ps3 @ Ys6 ) )
& ( ( Xs3 = nil_Pr2300489316682597567nt_int )
| ( Ys6 = nil_Pr2300489316682597567nt_int )
| ( ( hd_Pro282112905867057956nt_int @ Xs3 )
!= ( hd_Pro282112905867057956nt_int @ Ys6 ) ) ) ) ).
% longest_common_prefix
thf(fact_791_hd__append,axiom,
! [Xs: list_P5707943133018811711nt_int,Ys2: list_P5707943133018811711nt_int] :
( ( ( Xs = nil_Pr2300489316682597567nt_int )
=> ( ( hd_Pro282112905867057956nt_int @ ( append7030698103840186580nt_int @ Xs @ Ys2 ) )
= ( hd_Pro282112905867057956nt_int @ Ys2 ) ) )
& ( ( Xs != nil_Pr2300489316682597567nt_int )
=> ( ( hd_Pro282112905867057956nt_int @ ( append7030698103840186580nt_int @ Xs @ Ys2 ) )
= ( hd_Pro282112905867057956nt_int @ Xs ) ) ) ) ).
% hd_append
thf(fact_792_longest__common__suffix,axiom,
! [Xs: list_P5707943133018811711nt_int,Ys2: list_P5707943133018811711nt_int] :
? [Ss: list_P5707943133018811711nt_int,Xs3: list_P5707943133018811711nt_int,Ys6: list_P5707943133018811711nt_int] :
( ( Xs
= ( append7030698103840186580nt_int @ Xs3 @ Ss ) )
& ( Ys2
= ( append7030698103840186580nt_int @ Ys6 @ Ss ) )
& ( ( Xs3 = nil_Pr2300489316682597567nt_int )
| ( Ys6 = nil_Pr2300489316682597567nt_int )
| ( ( last_P3305686521732843992nt_int @ Xs3 )
!= ( last_P3305686521732843992nt_int @ Ys6 ) ) ) ) ).
% longest_common_suffix
thf(fact_793_last__append,axiom,
! [Ys2: list_P5707943133018811711nt_int,Xs: list_P5707943133018811711nt_int] :
( ( ( Ys2 = nil_Pr2300489316682597567nt_int )
=> ( ( last_P3305686521732843992nt_int @ ( append7030698103840186580nt_int @ Xs @ Ys2 ) )
= ( last_P3305686521732843992nt_int @ Xs ) ) )
& ( ( Ys2 != nil_Pr2300489316682597567nt_int )
=> ( ( last_P3305686521732843992nt_int @ ( append7030698103840186580nt_int @ Xs @ Ys2 ) )
= ( last_P3305686521732843992nt_int @ Ys2 ) ) ) ) ).
% last_append
thf(fact_794_trans__path_Ocases,axiom,
! [X: produc4058024888802116461nt_int] :
( ! [K_1: int,K_2: int] :
( X
!= ( produc8677244595851196253nt_int @ ( product_Pair_int_int @ K_1 @ K_2 ) @ nil_Pr2300489316682597567nt_int ) )
=> ~ ! [K_1: int,K_2: int,I: int,J: int,Xs2: list_P5707943133018811711nt_int] :
( X
!= ( produc8677244595851196253nt_int @ ( product_Pair_int_int @ K_1 @ K_2 ) @ ( cons_P3334398858971670639nt_int @ ( product_Pair_int_int @ I @ J ) @ Xs2 ) ) ) ) ).
% trans_path.cases
thf(fact_795_hd__Nil__eq__last,axiom,
( ( hd_Pro282112905867057956nt_int @ nil_Pr2300489316682597567nt_int )
= ( last_P3305686521732843992nt_int @ nil_Pr2300489316682597567nt_int ) ) ).
% hd_Nil_eq_last
thf(fact_796_not__listrel1__Nil,axiom,
! [Xs: list_P5707943133018811711nt_int,R: set_Pr2560585780119916871nt_int] :
~ ( member1390679175989562640nt_int @ ( produc1932183703851549015nt_int @ Xs @ nil_Pr2300489316682597567nt_int ) @ ( listre1649617614542993432nt_int @ R ) ) ).
% not_listrel1_Nil
thf(fact_797_not__Nil__listrel1,axiom,
! [Xs: list_P5707943133018811711nt_int,R: set_Pr2560585780119916871nt_int] :
~ ( member1390679175989562640nt_int @ ( produc1932183703851549015nt_int @ nil_Pr2300489316682597567nt_int @ Xs ) @ ( listre1649617614542993432nt_int @ R ) ) ).
% not_Nil_listrel1
thf(fact_798_listrel__Nil2,axiom,
! [Xs: list_P5707943133018811711nt_int,R: set_Pr2560585780119916871nt_int] :
( ( member1390679175989562640nt_int @ ( produc1932183703851549015nt_int @ Xs @ nil_Pr2300489316682597567nt_int ) @ ( listre7525836092462517041nt_int @ R ) )
=> ( Xs = nil_Pr2300489316682597567nt_int ) ) ).
% listrel_Nil2
thf(fact_799_listrel__Nil1,axiom,
! [Xs: list_P5707943133018811711nt_int,R: set_Pr2560585780119916871nt_int] :
( ( member1390679175989562640nt_int @ ( produc1932183703851549015nt_int @ nil_Pr2300489316682597567nt_int @ Xs ) @ ( listre7525836092462517041nt_int @ R ) )
=> ( Xs = nil_Pr2300489316682597567nt_int ) ) ).
% listrel_Nil1
thf(fact_800_listrel_ONil,axiom,
! [R: set_Pr2560585780119916871nt_int] : ( member1390679175989562640nt_int @ ( produc1932183703851549015nt_int @ nil_Pr2300489316682597567nt_int @ nil_Pr2300489316682597567nt_int ) @ ( listre7525836092462517041nt_int @ R ) ) ).
% listrel.Nil
thf(fact_801_lexord__Nil__right,axiom,
! [X: list_P5707943133018811711nt_int,R: set_Pr2560585780119916871nt_int] :
~ ( member1390679175989562640nt_int @ ( produc1932183703851549015nt_int @ X @ nil_Pr2300489316682597567nt_int ) @ ( lexord8886728381914977324nt_int @ R ) ) ).
% lexord_Nil_right
thf(fact_802_Nil2__notin__lex,axiom,
! [Xs: list_P5707943133018811711nt_int,R: set_Pr2560585780119916871nt_int] :
~ ( member1390679175989562640nt_int @ ( produc1932183703851549015nt_int @ Xs @ nil_Pr2300489316682597567nt_int ) @ ( lex_Pr5393148144989827363nt_int @ R ) ) ).
% Nil2_notin_lex
thf(fact_803_Nil__notin__lex,axiom,
! [Ys2: list_P5707943133018811711nt_int,R: set_Pr2560585780119916871nt_int] :
~ ( member1390679175989562640nt_int @ ( produc1932183703851549015nt_int @ nil_Pr2300489316682597567nt_int @ Ys2 ) @ ( lex_Pr5393148144989827363nt_int @ R ) ) ).
% Nil_notin_lex
thf(fact_804_Nil__lenlex__iff2,axiom,
! [Ns: list_P5707943133018811711nt_int,R: set_Pr2560585780119916871nt_int] :
~ ( member1390679175989562640nt_int @ ( produc1932183703851549015nt_int @ Ns @ nil_Pr2300489316682597567nt_int ) @ ( lenlex6370358691973319492nt_int @ R ) ) ).
% Nil_lenlex_iff2
thf(fact_805_knights__circuit__rotate1,axiom,
! [B: set_Pr958786334691620121nt_int,S_i: product_prod_int_int,Ps: list_P5707943133018811711nt_int] :
( ( knights_circuit @ B @ ( cons_P3334398858971670639nt_int @ S_i @ Ps ) )
=> ( knights_circuit @ B @ ( append7030698103840186580nt_int @ Ps @ ( cons_P3334398858971670639nt_int @ S_i @ nil_Pr2300489316682597567nt_int ) ) ) ) ).
% knights_circuit_rotate1
thf(fact_806_same__length__different,axiom,
! [Xs: list_P5707943133018811711nt_int,Ys2: list_P5707943133018811711nt_int] :
( ( Xs != Ys2 )
=> ( ( ( size_s5157815400016825771nt_int @ Xs )
= ( size_s5157815400016825771nt_int @ Ys2 ) )
=> ? [Pre: list_P5707943133018811711nt_int,X3: product_prod_int_int,Xs3: list_P5707943133018811711nt_int,Y3: product_prod_int_int,Ys6: list_P5707943133018811711nt_int] :
( ( X3 != Y3 )
& ( Xs
= ( append7030698103840186580nt_int @ Pre @ ( append7030698103840186580nt_int @ ( cons_P3334398858971670639nt_int @ X3 @ nil_Pr2300489316682597567nt_int ) @ Xs3 ) ) )
& ( Ys2
= ( append7030698103840186580nt_int @ Pre @ ( append7030698103840186580nt_int @ ( cons_P3334398858971670639nt_int @ Y3 @ nil_Pr2300489316682597567nt_int ) @ Ys6 ) ) ) ) ) ) ).
% same_length_different
thf(fact_807_rotate1_Osimps_I2_J,axiom,
! [X: product_prod_int_int,Xs: list_P5707943133018811711nt_int] :
( ( rotate9137591727117064716nt_int @ ( cons_P3334398858971670639nt_int @ X @ Xs ) )
= ( append7030698103840186580nt_int @ Xs @ ( cons_P3334398858971670639nt_int @ X @ nil_Pr2300489316682597567nt_int ) ) ) ).
% rotate1.simps(2)
thf(fact_808_length__append__singleton,axiom,
! [Xs: list_P5707943133018811711nt_int,X: product_prod_int_int] :
( ( size_s5157815400016825771nt_int @ ( append7030698103840186580nt_int @ Xs @ ( cons_P3334398858971670639nt_int @ X @ nil_Pr2300489316682597567nt_int ) ) )
= ( suc @ ( size_s5157815400016825771nt_int @ Xs ) ) ) ).
% length_append_singleton
thf(fact_809_length__Suc__conv__rev,axiom,
! [Xs: list_P5707943133018811711nt_int,N: nat] :
( ( ( size_s5157815400016825771nt_int @ Xs )
= ( suc @ N ) )
= ( ? [Y6: product_prod_int_int,Ys3: list_P5707943133018811711nt_int] :
( ( Xs
= ( append7030698103840186580nt_int @ Ys3 @ ( cons_P3334398858971670639nt_int @ Y6 @ nil_Pr2300489316682597567nt_int ) ) )
& ( ( size_s5157815400016825771nt_int @ Ys3 )
= N ) ) ) ) ).
% length_Suc_conv_rev
thf(fact_810_listrel_Osimps,axiom,
! [A1: list_P5707943133018811711nt_int,A22: list_P5707943133018811711nt_int,R: set_Pr2560585780119916871nt_int] :
( ( member1390679175989562640nt_int @ ( produc1932183703851549015nt_int @ A1 @ A22 ) @ ( listre7525836092462517041nt_int @ R ) )
= ( ( ( A1 = nil_Pr2300489316682597567nt_int )
& ( A22 = nil_Pr2300489316682597567nt_int ) )
| ? [X4: product_prod_int_int,Y6: product_prod_int_int,Xs4: list_P5707943133018811711nt_int,Ys3: list_P5707943133018811711nt_int] :
( ( A1
= ( cons_P3334398858971670639nt_int @ X4 @ Xs4 ) )
& ( A22
= ( cons_P3334398858971670639nt_int @ Y6 @ Ys3 ) )
& ( member8566619992076573584nt_int @ ( produc3646306378393792727nt_int @ X4 @ Y6 ) @ R )
& ( member1390679175989562640nt_int @ ( produc1932183703851549015nt_int @ Xs4 @ Ys3 ) @ ( listre7525836092462517041nt_int @ R ) ) ) ) ) ).
% listrel.simps
thf(fact_811_listrel_Osimps,axiom,
! [A1: list_nat,A22: list_set_int,R: set_Pr4810089274464741491et_int] :
( ( member4311088582351766364et_int @ ( produc6467668960084403947et_int @ A1 @ A22 ) @ ( listrel_nat_set_int @ R ) )
= ( ( ( A1 = nil_nat )
& ( A22 = nil_set_int ) )
| ? [X4: nat,Y6: set_int,Xs4: list_nat,Ys3: list_set_int] :
( ( A1
= ( cons_nat @ X4 @ Xs4 ) )
& ( A22
= ( cons_set_int @ Y6 @ Ys3 ) )
& ( member1292241183792264892et_int @ ( produc29655638201817675et_int @ X4 @ Y6 ) @ R )
& ( member4311088582351766364et_int @ ( produc6467668960084403947et_int @ Xs4 @ Ys3 ) @ ( listrel_nat_set_int @ R ) ) ) ) ) ).
% listrel.simps
thf(fact_812_listrel_Osimps,axiom,
! [A1: list_int,A22: list_l1670014477004246597nt_int,R: set_Pr1843331182504696254nt_int] :
( ( member4298863958545693205nt_int @ ( produc1403559775789145904nt_int @ A1 @ A22 ) @ ( listre1889173885445251040nt_int @ R ) )
= ( ( ( A1 = nil_int )
& ( A22 = nil_li8670148097206105925nt_int ) )
| ? [X4: int,Y6: list_P5707943133018811711nt_int,Xs4: list_int,Ys3: list_l1670014477004246597nt_int] :
( ( A1
= ( cons_int @ X4 @ Xs4 ) )
& ( A22
= ( cons_l7309679040211256053nt_int @ Y6 @ Ys3 ) )
& ( member3263328415308337823nt_int @ ( produc8814303788642274490nt_int @ X4 @ Y6 ) @ R )
& ( member4298863958545693205nt_int @ ( produc1403559775789145904nt_int @ Xs4 @ Ys3 ) @ ( listre1889173885445251040nt_int @ R ) ) ) ) ) ).
% listrel.simps
thf(fact_813_listrel_Osimps,axiom,
! [A1: list_int,A22: list_int,R: set_Pr958786334691620121nt_int] :
( ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ A1 @ A22 ) @ ( listrel_int_int @ R ) )
= ( ( ( A1 = nil_int )
& ( A22 = nil_int ) )
| ? [X4: int,Y6: int,Xs4: list_int,Ys3: list_int] :
( ( A1
= ( cons_int @ X4 @ Xs4 ) )
& ( A22
= ( cons_int @ Y6 @ Ys3 ) )
& ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X4 @ Y6 ) @ R )
& ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Xs4 @ Ys3 ) @ ( listrel_int_int @ R ) ) ) ) ) ).
% listrel.simps
thf(fact_814_listrel_Osimps,axiom,
! [A1: list_P1743416141875011707e_term,A22: list_P5707943133018811711nt_int,R: set_Pr9222295170931077689nt_int] :
( ( member1869596255364795664nt_int @ ( produc8723725229082728235nt_int @ A1 @ A22 ) @ ( listre7929554186369383643nt_int @ R ) )
= ( ( ( A1 = nil_Pr2193041437141079013e_term )
& ( A22 = nil_Pr2300489316682597567nt_int ) )
| ? [X4: produc8551481072490612790e_term > option6357759511663192854e_term,Y6: product_prod_int_int,Xs4: list_P1743416141875011707e_term,Ys3: list_P5707943133018811711nt_int] :
( ( A1
= ( cons_P2630085844062958645e_term @ X4 @ Xs4 ) )
& ( A22
= ( cons_P3334398858971670639nt_int @ Y6 @ Ys3 ) )
& ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ X4 @ Y6 ) @ R )
& ( member1869596255364795664nt_int @ ( produc8723725229082728235nt_int @ Xs4 @ Ys3 ) @ ( listre7929554186369383643nt_int @ R ) ) ) ) ) ).
% listrel.simps
thf(fact_815_listrel_Osimps,axiom,
! [A1: list_i8448526496819171953e_term,A22: list_P5707943133018811711nt_int,R: set_Pr1872883991513573699nt_int] :
( ( member6768764179952475034nt_int @ ( produc2259991554960072885nt_int @ A1 @ A22 ) @ ( listre7622244038138809061nt_int @ R ) )
= ( ( ( A1 = nil_in3447728264653889115e_term )
& ( A22 = nil_Pr2300489316682597567nt_int ) )
| ? [X4: int > option6357759511663192854e_term,Y6: product_prod_int_int,Xs4: list_i8448526496819171953e_term,Ys3: list_P5707943133018811711nt_int] :
( ( A1
= ( cons_i7166360444231718571e_term @ X4 @ Xs4 ) )
& ( A22
= ( cons_P3334398858971670639nt_int @ Y6 @ Ys3 ) )
& ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ X4 @ Y6 ) @ R )
& ( member6768764179952475034nt_int @ ( produc2259991554960072885nt_int @ Xs4 @ Ys3 ) @ ( listre7622244038138809061nt_int @ R ) ) ) ) ) ).
% listrel.simps
thf(fact_816_listrel_Ocases,axiom,
! [A1: list_P5707943133018811711nt_int,A22: list_P5707943133018811711nt_int,R: set_Pr2560585780119916871nt_int] :
( ( member1390679175989562640nt_int @ ( produc1932183703851549015nt_int @ A1 @ A22 ) @ ( listre7525836092462517041nt_int @ R ) )
=> ( ( ( A1 = nil_Pr2300489316682597567nt_int )
=> ( A22 != nil_Pr2300489316682597567nt_int ) )
=> ~ ! [X3: product_prod_int_int,Y3: product_prod_int_int,Xs2: list_P5707943133018811711nt_int] :
( ( A1
= ( cons_P3334398858971670639nt_int @ X3 @ Xs2 ) )
=> ! [Ys4: list_P5707943133018811711nt_int] :
( ( A22
= ( cons_P3334398858971670639nt_int @ Y3 @ Ys4 ) )
=> ( ( member8566619992076573584nt_int @ ( produc3646306378393792727nt_int @ X3 @ Y3 ) @ R )
=> ~ ( member1390679175989562640nt_int @ ( produc1932183703851549015nt_int @ Xs2 @ Ys4 ) @ ( listre7525836092462517041nt_int @ R ) ) ) ) ) ) ) ).
% listrel.cases
thf(fact_817_listrel_Ocases,axiom,
! [A1: list_nat,A22: list_set_int,R: set_Pr4810089274464741491et_int] :
( ( member4311088582351766364et_int @ ( produc6467668960084403947et_int @ A1 @ A22 ) @ ( listrel_nat_set_int @ R ) )
=> ( ( ( A1 = nil_nat )
=> ( A22 != nil_set_int ) )
=> ~ ! [X3: nat,Y3: set_int,Xs2: list_nat] :
( ( A1
= ( cons_nat @ X3 @ Xs2 ) )
=> ! [Ys4: list_set_int] :
( ( A22
= ( cons_set_int @ Y3 @ Ys4 ) )
=> ( ( member1292241183792264892et_int @ ( produc29655638201817675et_int @ X3 @ Y3 ) @ R )
=> ~ ( member4311088582351766364et_int @ ( produc6467668960084403947et_int @ Xs2 @ Ys4 ) @ ( listrel_nat_set_int @ R ) ) ) ) ) ) ) ).
% listrel.cases
thf(fact_818_listrel_Ocases,axiom,
! [A1: list_int,A22: list_l1670014477004246597nt_int,R: set_Pr1843331182504696254nt_int] :
( ( member4298863958545693205nt_int @ ( produc1403559775789145904nt_int @ A1 @ A22 ) @ ( listre1889173885445251040nt_int @ R ) )
=> ( ( ( A1 = nil_int )
=> ( A22 != nil_li8670148097206105925nt_int ) )
=> ~ ! [X3: int,Y3: list_P5707943133018811711nt_int,Xs2: list_int] :
( ( A1
= ( cons_int @ X3 @ Xs2 ) )
=> ! [Ys4: list_l1670014477004246597nt_int] :
( ( A22
= ( cons_l7309679040211256053nt_int @ Y3 @ Ys4 ) )
=> ( ( member3263328415308337823nt_int @ ( produc8814303788642274490nt_int @ X3 @ Y3 ) @ R )
=> ~ ( member4298863958545693205nt_int @ ( produc1403559775789145904nt_int @ Xs2 @ Ys4 ) @ ( listre1889173885445251040nt_int @ R ) ) ) ) ) ) ) ).
% listrel.cases
thf(fact_819_listrel_Ocases,axiom,
! [A1: list_int,A22: list_int,R: set_Pr958786334691620121nt_int] :
( ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ A1 @ A22 ) @ ( listrel_int_int @ R ) )
=> ( ( ( A1 = nil_int )
=> ( A22 != nil_int ) )
=> ~ ! [X3: int,Y3: int,Xs2: list_int] :
( ( A1
= ( cons_int @ X3 @ Xs2 ) )
=> ! [Ys4: list_int] :
( ( A22
= ( cons_int @ Y3 @ Ys4 ) )
=> ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X3 @ Y3 ) @ R )
=> ~ ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Xs2 @ Ys4 ) @ ( listrel_int_int @ R ) ) ) ) ) ) ) ).
% listrel.cases
thf(fact_820_listrel_Ocases,axiom,
! [A1: list_P1743416141875011707e_term,A22: list_P5707943133018811711nt_int,R: set_Pr9222295170931077689nt_int] :
( ( member1869596255364795664nt_int @ ( produc8723725229082728235nt_int @ A1 @ A22 ) @ ( listre7929554186369383643nt_int @ R ) )
=> ( ( ( A1 = nil_Pr2193041437141079013e_term )
=> ( A22 != nil_Pr2300489316682597567nt_int ) )
=> ~ ! [X3: produc8551481072490612790e_term > option6357759511663192854e_term,Y3: product_prod_int_int,Xs2: list_P1743416141875011707e_term] :
( ( A1
= ( cons_P2630085844062958645e_term @ X3 @ Xs2 ) )
=> ! [Ys4: list_P5707943133018811711nt_int] :
( ( A22
= ( cons_P3334398858971670639nt_int @ Y3 @ Ys4 ) )
=> ( ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ X3 @ Y3 ) @ R )
=> ~ ( member1869596255364795664nt_int @ ( produc8723725229082728235nt_int @ Xs2 @ Ys4 ) @ ( listre7929554186369383643nt_int @ R ) ) ) ) ) ) ) ).
% listrel.cases
thf(fact_821_listrel_Ocases,axiom,
! [A1: list_i8448526496819171953e_term,A22: list_P5707943133018811711nt_int,R: set_Pr1872883991513573699nt_int] :
( ( member6768764179952475034nt_int @ ( produc2259991554960072885nt_int @ A1 @ A22 ) @ ( listre7622244038138809061nt_int @ R ) )
=> ( ( ( A1 = nil_in3447728264653889115e_term )
=> ( A22 != nil_Pr2300489316682597567nt_int ) )
=> ~ ! [X3: int > option6357759511663192854e_term,Y3: product_prod_int_int,Xs2: list_i8448526496819171953e_term] :
( ( A1
= ( cons_i7166360444231718571e_term @ X3 @ Xs2 ) )
=> ! [Ys4: list_P5707943133018811711nt_int] :
( ( A22
= ( cons_P3334398858971670639nt_int @ Y3 @ Ys4 ) )
=> ( ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ X3 @ Y3 ) @ R )
=> ~ ( member6768764179952475034nt_int @ ( produc2259991554960072885nt_int @ Xs2 @ Ys4 ) @ ( listre7622244038138809061nt_int @ R ) ) ) ) ) ) ) ).
% listrel.cases
thf(fact_822_enumerate__append__eq,axiom,
! [N: nat,Xs: list_P5707943133018811711nt_int,Ys2: list_P5707943133018811711nt_int] :
( ( enumer5233495138702836510nt_int @ N @ ( append7030698103840186580nt_int @ Xs @ Ys2 ) )
= ( append3046253387908738819nt_int @ ( enumer5233495138702836510nt_int @ N @ Xs ) @ ( enumer5233495138702836510nt_int @ ( plus_plus_nat @ N @ ( size_s5157815400016825771nt_int @ Xs ) ) @ Ys2 ) ) ) ).
% enumerate_append_eq
thf(fact_823_snoc__listrel1__snoc__iff,axiom,
! [Xs: list_P5707943133018811711nt_int,X: product_prod_int_int,Ys2: list_P5707943133018811711nt_int,Y: product_prod_int_int,R: set_Pr2560585780119916871nt_int] :
( ( member1390679175989562640nt_int @ ( produc1932183703851549015nt_int @ ( append7030698103840186580nt_int @ Xs @ ( cons_P3334398858971670639nt_int @ X @ nil_Pr2300489316682597567nt_int ) ) @ ( append7030698103840186580nt_int @ Ys2 @ ( cons_P3334398858971670639nt_int @ Y @ nil_Pr2300489316682597567nt_int ) ) ) @ ( listre1649617614542993432nt_int @ R ) )
= ( ( ( member1390679175989562640nt_int @ ( produc1932183703851549015nt_int @ Xs @ Ys2 ) @ ( listre1649617614542993432nt_int @ R ) )
& ( X = Y ) )
| ( ( Xs = Ys2 )
& ( member8566619992076573584nt_int @ ( produc3646306378393792727nt_int @ X @ Y ) @ R ) ) ) ) ).
% snoc_listrel1_snoc_iff
thf(fact_824_snoc__listrel1__snoc__iff,axiom,
! [Xs: list_int,X: int,Ys2: list_int,Y: int,R: set_Pr958786334691620121nt_int] :
( ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ ( append_int @ Xs @ ( cons_int @ X @ nil_int ) ) @ ( append_int @ Ys2 @ ( cons_int @ Y @ nil_int ) ) ) @ ( listrel1_int @ R ) )
= ( ( ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Xs @ Ys2 ) @ ( listrel1_int @ R ) )
& ( X = Y ) )
| ( ( Xs = Ys2 )
& ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ R ) ) ) ) ).
% snoc_listrel1_snoc_iff
thf(fact_825_SuccI,axiom,
! [Kl: list_int,K: int,Kl2: set_list_int] :
( ( member_list_int @ ( append_int @ Kl @ ( cons_int @ K @ nil_int ) ) @ Kl2 )
=> ( member_int @ K @ ( bNF_Gr6350390219475566417cc_int @ Kl2 @ Kl ) ) ) ).
% SuccI
thf(fact_826_SuccI,axiom,
! [Kl: list_P5707943133018811711nt_int,K: product_prod_int_int,Kl2: set_li3855193629254082847nt_int] :
( ( member2764346250752101224nt_int @ ( append7030698103840186580nt_int @ Kl @ ( cons_P3334398858971670639nt_int @ K @ nil_Pr2300489316682597567nt_int ) ) @ Kl2 )
=> ( member5262025264175285858nt_int @ K @ ( bNF_Gr2185362013987207252nt_int @ Kl2 @ Kl ) ) ) ).
% SuccI
thf(fact_827_SuccD,axiom,
! [K: int,Kl2: set_list_int,Kl: list_int] :
( ( member_int @ K @ ( bNF_Gr6350390219475566417cc_int @ Kl2 @ Kl ) )
=> ( member_list_int @ ( append_int @ Kl @ ( cons_int @ K @ nil_int ) ) @ Kl2 ) ) ).
% SuccD
thf(fact_828_SuccD,axiom,
! [K: product_prod_int_int,Kl2: set_li3855193629254082847nt_int,Kl: list_P5707943133018811711nt_int] :
( ( member5262025264175285858nt_int @ K @ ( bNF_Gr2185362013987207252nt_int @ Kl2 @ Kl ) )
=> ( member2764346250752101224nt_int @ ( append7030698103840186580nt_int @ Kl @ ( cons_P3334398858971670639nt_int @ K @ nil_Pr2300489316682597567nt_int ) ) @ Kl2 ) ) ).
% SuccD
thf(fact_829_nth__zip,axiom,
! [I4: nat,Xs: list_nat,Ys2: list_set_int] :
( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Xs ) )
=> ( ( ord_less_nat @ I4 @ ( size_s763970886550748294et_int @ Ys2 ) )
=> ( ( nth_Pr4328007639516273434et_int @ ( zip_nat_set_int @ Xs @ Ys2 ) @ I4 )
= ( produc29655638201817675et_int @ ( nth_nat @ Xs @ I4 ) @ ( nth_set_int @ Ys2 @ I4 ) ) ) ) ) ).
% nth_zip
thf(fact_830_nth__zip,axiom,
! [I4: nat,Xs: list_int,Ys2: list_l1670014477004246597nt_int] :
( ( ord_less_nat @ I4 @ ( size_size_list_int @ Xs ) )
=> ( ( ord_less_nat @ I4 @ ( size_s2969076144586574001nt_int @ Ys2 ) )
=> ( ( nth_Pr4686562112285265857nt_int @ ( zip_in8018307065301790106nt_int @ Xs @ Ys2 ) @ I4 )
= ( produc8814303788642274490nt_int @ ( nth_int @ Xs @ I4 ) @ ( nth_li549691060785553606nt_int @ Ys2 @ I4 ) ) ) ) ) ).
% nth_zip
thf(fact_831_nth__zip,axiom,
! [I4: nat,Xs: list_int,Ys2: list_int] :
( ( ord_less_nat @ I4 @ ( size_size_list_int @ Xs ) )
=> ( ( ord_less_nat @ I4 @ ( size_size_list_int @ Ys2 ) )
=> ( ( nth_Pr4439495888332055232nt_int @ ( zip_int_int @ Xs @ Ys2 ) @ I4 )
= ( product_Pair_int_int @ ( nth_int @ Xs @ I4 ) @ ( nth_int @ Ys2 @ I4 ) ) ) ) ) ).
% nth_zip
thf(fact_832_nth__zip,axiom,
! [I4: nat,Xs: list_P1743416141875011707e_term,Ys2: list_P5707943133018811711nt_int] :
( ( ord_less_nat @ I4 @ ( size_s3817825649112772879e_term @ Xs ) )
=> ( ( ord_less_nat @ I4 @ ( size_s5157815400016825771nt_int @ Ys2 ) )
=> ( ( nth_Pr5768845505533887420nt_int @ ( zip_Pr4168994715204986005nt_int @ Xs @ Ys2 ) @ I4 )
= ( produc5700946648718959541nt_int @ ( nth_Pr6123904574653836964e_term @ Xs @ I4 ) @ ( nth_Pr4439495888332055232nt_int @ Ys2 @ I4 ) ) ) ) ) ).
% nth_zip
thf(fact_833_nth__zip,axiom,
! [I4: nat,Xs: list_i8448526496819171953e_term,Ys2: list_P5707943133018811711nt_int] :
( ( ord_less_nat @ I4 @ ( size_s8881356780637927685e_term @ Xs ) )
=> ( ( ord_less_nat @ I4 @ ( size_s5157815400016825771nt_int @ Ys2 ) )
=> ( ( nth_Pr7284623736940738758nt_int @ ( zip_in8766932505889695135nt_int @ Xs @ Ys2 ) @ I4 )
= ( produc4305682042979456191nt_int @ ( nth_in8840386337573990810e_term @ Xs @ I4 ) @ ( nth_Pr4439495888332055232nt_int @ Ys2 @ I4 ) ) ) ) ) ).
% nth_zip
thf(fact_834_nth__zip,axiom,
! [I4: nat,Xs: list_P5707943133018811711nt_int,Ys2: list_P5707943133018811711nt_int] :
( ( ord_less_nat @ I4 @ ( size_s5157815400016825771nt_int @ Xs ) )
=> ( ( ord_less_nat @ I4 @ ( size_s5157815400016825771nt_int @ Ys2 ) )
=> ( ( nth_Pr7104135640663552750nt_int @ ( zip_Pr2148635498163357687nt_int @ Xs @ Ys2 ) @ I4 )
= ( produc3646306378393792727nt_int @ ( nth_Pr4439495888332055232nt_int @ Xs @ I4 ) @ ( nth_Pr4439495888332055232nt_int @ Ys2 @ I4 ) ) ) ) ) ).
% nth_zip
thf(fact_835_append__butlast__last__id,axiom,
! [Xs: list_P5707943133018811711nt_int] :
( ( Xs != nil_Pr2300489316682597567nt_int )
=> ( ( append7030698103840186580nt_int @ ( butlas2390654596765537291nt_int @ Xs ) @ ( cons_P3334398858971670639nt_int @ ( last_P3305686521732843992nt_int @ Xs ) @ nil_Pr2300489316682597567nt_int ) )
= Xs ) ) ).
% append_butlast_last_id
thf(fact_836_zip__eq__Nil__iff,axiom,
! [Xs: list_P5707943133018811711nt_int,Ys2: list_P5707943133018811711nt_int] :
( ( ( zip_Pr2148635498163357687nt_int @ Xs @ Ys2 )
= nil_Pr2941907410405800557nt_int )
= ( ( Xs = nil_Pr2300489316682597567nt_int )
| ( Ys2 = nil_Pr2300489316682597567nt_int ) ) ) ).
% zip_eq_Nil_iff
thf(fact_837_zip__eq__Nil__iff,axiom,
! [Xs: list_int,Ys2: list_int] :
( ( ( zip_int_int @ Xs @ Ys2 )
= nil_Pr2300489316682597567nt_int )
= ( ( Xs = nil_int )
| ( Ys2 = nil_int ) ) ) ).
% zip_eq_Nil_iff
thf(fact_838_Nil__eq__zip__iff,axiom,
! [Xs: list_P5707943133018811711nt_int,Ys2: list_P5707943133018811711nt_int] :
( ( nil_Pr2941907410405800557nt_int
= ( zip_Pr2148635498163357687nt_int @ Xs @ Ys2 ) )
= ( ( Xs = nil_Pr2300489316682597567nt_int )
| ( Ys2 = nil_Pr2300489316682597567nt_int ) ) ) ).
% Nil_eq_zip_iff
thf(fact_839_Nil__eq__zip__iff,axiom,
! [Xs: list_int,Ys2: list_int] :
( ( nil_Pr2300489316682597567nt_int
= ( zip_int_int @ Xs @ Ys2 ) )
= ( ( Xs = nil_int )
| ( Ys2 = nil_int ) ) ) ).
% Nil_eq_zip_iff
thf(fact_840_zip__Nil,axiom,
! [Ys2: list_int] :
( ( zip_int_int @ nil_int @ Ys2 )
= nil_Pr2300489316682597567nt_int ) ).
% zip_Nil
thf(fact_841_zip__Cons__Cons,axiom,
! [X: product_prod_int_int,Xs: list_P5707943133018811711nt_int,Y: product_prod_int_int,Ys2: list_P5707943133018811711nt_int] :
( ( zip_Pr2148635498163357687nt_int @ ( cons_P3334398858971670639nt_int @ X @ Xs ) @ ( cons_P3334398858971670639nt_int @ Y @ Ys2 ) )
= ( cons_P9091998269994194461nt_int @ ( produc3646306378393792727nt_int @ X @ Y ) @ ( zip_Pr2148635498163357687nt_int @ Xs @ Ys2 ) ) ) ).
% zip_Cons_Cons
thf(fact_842_zip__Cons__Cons,axiom,
! [X: nat,Xs: list_nat,Y: set_int,Ys2: list_set_int] :
( ( zip_nat_set_int @ ( cons_nat @ X @ Xs ) @ ( cons_set_int @ Y @ Ys2 ) )
= ( cons_P5529800768674571465et_int @ ( produc29655638201817675et_int @ X @ Y ) @ ( zip_nat_set_int @ Xs @ Ys2 ) ) ) ).
% zip_Cons_Cons
thf(fact_843_zip__Cons__Cons,axiom,
! [X: int,Xs: list_int,Y: list_P5707943133018811711nt_int,Ys2: list_l1670014477004246597nt_int] :
( ( zip_in8018307065301790106nt_int @ ( cons_int @ X @ Xs ) @ ( cons_l7309679040211256053nt_int @ Y @ Ys2 ) )
= ( cons_P9095479431110121170nt_int @ ( produc8814303788642274490nt_int @ X @ Y ) @ ( zip_in8018307065301790106nt_int @ Xs @ Ys2 ) ) ) ).
% zip_Cons_Cons
thf(fact_844_zip__Cons__Cons,axiom,
! [X: int,Xs: list_int,Y: int,Ys2: list_int] :
( ( zip_int_int @ ( cons_int @ X @ Xs ) @ ( cons_int @ Y @ Ys2 ) )
= ( cons_P3334398858971670639nt_int @ ( product_Pair_int_int @ X @ Y ) @ ( zip_int_int @ Xs @ Ys2 ) ) ) ).
% zip_Cons_Cons
thf(fact_845_zip__Cons__Cons,axiom,
! [X: produc8551481072490612790e_term > option6357759511663192854e_term,Xs: list_P1743416141875011707e_term,Y: product_prod_int_int,Ys2: list_P5707943133018811711nt_int] :
( ( zip_Pr4168994715204986005nt_int @ ( cons_P2630085844062958645e_term @ X @ Xs ) @ ( cons_P3334398858971670639nt_int @ Y @ Ys2 ) )
= ( cons_P6018425551955479501nt_int @ ( produc5700946648718959541nt_int @ X @ Y ) @ ( zip_Pr4168994715204986005nt_int @ Xs @ Ys2 ) ) ) ).
% zip_Cons_Cons
thf(fact_846_zip__Cons__Cons,axiom,
! [X: int > option6357759511663192854e_term,Xs: list_i8448526496819171953e_term,Y: product_prod_int_int,Ys2: list_P5707943133018811711nt_int] :
( ( zip_in8766932505889695135nt_int @ ( cons_i7166360444231718571e_term @ X @ Xs ) @ ( cons_P3334398858971670639nt_int @ Y @ Ys2 ) )
= ( cons_P2743708091642732631nt_int @ ( produc4305682042979456191nt_int @ X @ Y ) @ ( zip_in8766932505889695135nt_int @ Xs @ Ys2 ) ) ) ).
% zip_Cons_Cons
thf(fact_847_butlast__snoc,axiom,
! [Xs: list_P5707943133018811711nt_int,X: product_prod_int_int] :
( ( butlas2390654596765537291nt_int @ ( append7030698103840186580nt_int @ Xs @ ( cons_P3334398858971670639nt_int @ X @ nil_Pr2300489316682597567nt_int ) ) )
= Xs ) ).
% butlast_snoc
thf(fact_848_length__zip,axiom,
! [Xs: list_P5707943133018811711nt_int,Ys2: list_P5707943133018811711nt_int] :
( ( size_s6770063216428074713nt_int @ ( zip_Pr2148635498163357687nt_int @ Xs @ Ys2 ) )
= ( ord_min_nat @ ( size_s5157815400016825771nt_int @ Xs ) @ ( size_s5157815400016825771nt_int @ Ys2 ) ) ) ).
% length_zip
thf(fact_849_length__zip,axiom,
! [Xs: list_int,Ys2: list_int] :
( ( size_s5157815400016825771nt_int @ ( zip_int_int @ Xs @ Ys2 ) )
= ( ord_min_nat @ ( size_size_list_int @ Xs ) @ ( size_size_list_int @ Ys2 ) ) ) ).
% length_zip
thf(fact_850_zip__append,axiom,
! [Xs: list_int,Us: list_int,Ys2: list_int,Vs: list_int] :
( ( ( size_size_list_int @ Xs )
= ( size_size_list_int @ Us ) )
=> ( ( zip_int_int @ ( append_int @ Xs @ Ys2 ) @ ( append_int @ Us @ Vs ) )
= ( append7030698103840186580nt_int @ ( zip_int_int @ Xs @ Us ) @ ( zip_int_int @ Ys2 @ Vs ) ) ) ) ).
% zip_append
thf(fact_851_zip__append,axiom,
! [Xs: list_P5707943133018811711nt_int,Us: list_P5707943133018811711nt_int,Ys2: list_P5707943133018811711nt_int,Vs: list_P5707943133018811711nt_int] :
( ( ( size_s5157815400016825771nt_int @ Xs )
= ( size_s5157815400016825771nt_int @ Us ) )
=> ( ( zip_Pr2148635498163357687nt_int @ ( append7030698103840186580nt_int @ Xs @ Ys2 ) @ ( append7030698103840186580nt_int @ Us @ Vs ) )
= ( append9111546825139864322nt_int @ ( zip_Pr2148635498163357687nt_int @ Xs @ Us ) @ ( zip_Pr2148635498163357687nt_int @ Ys2 @ Vs ) ) ) ) ).
% zip_append
thf(fact_852_butlast_Osimps_I1_J,axiom,
( ( butlas2390654596765537291nt_int @ nil_Pr2300489316682597567nt_int )
= nil_Pr2300489316682597567nt_int ) ).
% butlast.simps(1)
thf(fact_853_zip_Osimps_I1_J,axiom,
! [Xs: list_int] :
( ( zip_int_int @ Xs @ nil_int )
= nil_Pr2300489316682597567nt_int ) ).
% zip.simps(1)
thf(fact_854_take__zip,axiom,
! [N: nat,Xs: list_P5707943133018811711nt_int,Ys2: list_P5707943133018811711nt_int] :
( ( take_P5614214687868072533nt_int @ N @ ( zip_Pr2148635498163357687nt_int @ Xs @ Ys2 ) )
= ( zip_Pr2148635498163357687nt_int @ ( take_P8218740963776755879nt_int @ N @ Xs ) @ ( take_P8218740963776755879nt_int @ N @ Ys2 ) ) ) ).
% take_zip
thf(fact_855_take__zip,axiom,
! [N: nat,Xs: list_int,Ys2: list_int] :
( ( take_P8218740963776755879nt_int @ N @ ( zip_int_int @ Xs @ Ys2 ) )
= ( zip_int_int @ ( take_int @ N @ Xs ) @ ( take_int @ N @ Ys2 ) ) ) ).
% take_zip
thf(fact_856_drop__butlast,axiom,
! [N: nat,Xs: list_P5707943133018811711nt_int] :
( ( drop_P5690361596310759935nt_int @ N @ ( butlas2390654596765537291nt_int @ Xs ) )
= ( butlas2390654596765537291nt_int @ ( drop_P5690361596310759935nt_int @ N @ Xs ) ) ) ).
% drop_butlast
thf(fact_857_zip__update,axiom,
! [Xs: list_nat,I4: nat,X: nat,Ys2: list_set_int,Y: set_int] :
( ( zip_nat_set_int @ ( list_update_nat @ Xs @ I4 @ X ) @ ( list_update_set_int @ Ys2 @ I4 @ Y ) )
= ( list_u7092962573381667713et_int @ ( zip_nat_set_int @ Xs @ Ys2 ) @ I4 @ ( produc29655638201817675et_int @ X @ Y ) ) ) ).
% zip_update
thf(fact_858_zip__update,axiom,
! [Xs: list_int,I4: nat,X: int,Ys2: list_l1670014477004246597nt_int,Y: list_P5707943133018811711nt_int] :
( ( zip_in8018307065301790106nt_int @ ( list_update_int @ Xs @ I4 @ X ) @ ( list_u1995434435150319021nt_int @ Ys2 @ I4 @ Y ) )
= ( list_u6063430246987531802nt_int @ ( zip_in8018307065301790106nt_int @ Xs @ Ys2 ) @ I4 @ ( produc8814303788642274490nt_int @ X @ Y ) ) ) ).
% zip_update
thf(fact_859_zip__update,axiom,
! [Xs: list_int,I4: nat,X: int,Ys2: list_int,Y: int] :
( ( zip_int_int @ ( list_update_int @ Xs @ I4 @ X ) @ ( list_update_int @ Ys2 @ I4 @ Y ) )
= ( list_u3002344382305578791nt_int @ ( zip_int_int @ Xs @ Ys2 ) @ I4 @ ( product_Pair_int_int @ X @ Y ) ) ) ).
% zip_update
thf(fact_860_zip__update,axiom,
! [Xs: list_P1743416141875011707e_term,I4: nat,X: produc8551481072490612790e_term > option6357759511663192854e_term,Ys2: list_P5707943133018811711nt_int,Y: product_prod_int_int] :
( ( zip_Pr4168994715204986005nt_int @ ( list_u3533491785856317309e_term @ Xs @ I4 @ X ) @ ( list_u3002344382305578791nt_int @ Ys2 @ I4 @ Y ) )
= ( list_u7736365598306452245nt_int @ ( zip_Pr4168994715204986005nt_int @ Xs @ Ys2 ) @ I4 @ ( produc5700946648718959541nt_int @ X @ Y ) ) ) ).
% zip_update
thf(fact_861_zip__update,axiom,
! [Xs: list_i8448526496819171953e_term,I4: nat,X: int > option6357759511663192854e_term,Ys2: list_P5707943133018811711nt_int,Y: product_prod_int_int] :
( ( zip_in8766932505889695135nt_int @ ( list_u8946639151299769843e_term @ Xs @ I4 @ X ) @ ( list_u3002344382305578791nt_int @ Ys2 @ I4 @ Y ) )
= ( list_u4780935413889332127nt_int @ ( zip_in8766932505889695135nt_int @ Xs @ Ys2 ) @ I4 @ ( produc4305682042979456191nt_int @ X @ Y ) ) ) ).
% zip_update
thf(fact_862_drop__zip,axiom,
! [N: nat,Xs: list_P5707943133018811711nt_int,Ys2: list_P5707943133018811711nt_int] :
( ( drop_P5227224544534396845nt_int @ N @ ( zip_Pr2148635498163357687nt_int @ Xs @ Ys2 ) )
= ( zip_Pr2148635498163357687nt_int @ ( drop_P5690361596310759935nt_int @ N @ Xs ) @ ( drop_P5690361596310759935nt_int @ N @ Ys2 ) ) ) ).
% drop_zip
thf(fact_863_drop__zip,axiom,
! [N: nat,Xs: list_int,Ys2: list_int] :
( ( drop_P5690361596310759935nt_int @ N @ ( zip_int_int @ Xs @ Ys2 ) )
= ( zip_int_int @ ( drop_int @ N @ Xs ) @ ( drop_int @ N @ Ys2 ) ) ) ).
% drop_zip
thf(fact_864_butlast_Osimps_I2_J,axiom,
! [Xs: list_P5707943133018811711nt_int,X: product_prod_int_int] :
( ( ( Xs = nil_Pr2300489316682597567nt_int )
=> ( ( butlas2390654596765537291nt_int @ ( cons_P3334398858971670639nt_int @ X @ Xs ) )
= nil_Pr2300489316682597567nt_int ) )
& ( ( Xs != nil_Pr2300489316682597567nt_int )
=> ( ( butlas2390654596765537291nt_int @ ( cons_P3334398858971670639nt_int @ X @ Xs ) )
= ( cons_P3334398858971670639nt_int @ X @ ( butlas2390654596765537291nt_int @ Xs ) ) ) ) ) ).
% butlast.simps(2)
thf(fact_865_butlast__append,axiom,
! [Ys2: list_P5707943133018811711nt_int,Xs: list_P5707943133018811711nt_int] :
( ( ( Ys2 = nil_Pr2300489316682597567nt_int )
=> ( ( butlas2390654596765537291nt_int @ ( append7030698103840186580nt_int @ Xs @ Ys2 ) )
= ( butlas2390654596765537291nt_int @ Xs ) ) )
& ( ( Ys2 != nil_Pr2300489316682597567nt_int )
=> ( ( butlas2390654596765537291nt_int @ ( append7030698103840186580nt_int @ Xs @ Ys2 ) )
= ( append7030698103840186580nt_int @ Xs @ ( butlas2390654596765537291nt_int @ Ys2 ) ) ) ) ) ).
% butlast_append
thf(fact_866_zip__eq__ConsE,axiom,
! [Xs: list_P5707943133018811711nt_int,Ys2: list_P5707943133018811711nt_int,Xy: produc1219242969750017639nt_int,Xys: list_P2336717926344734829nt_int] :
( ( ( zip_Pr2148635498163357687nt_int @ Xs @ Ys2 )
= ( cons_P9091998269994194461nt_int @ Xy @ Xys ) )
=> ~ ! [X3: product_prod_int_int,Xs3: list_P5707943133018811711nt_int] :
( ( Xs
= ( cons_P3334398858971670639nt_int @ X3 @ Xs3 ) )
=> ! [Y3: product_prod_int_int,Ys6: list_P5707943133018811711nt_int] :
( ( Ys2
= ( cons_P3334398858971670639nt_int @ Y3 @ Ys6 ) )
=> ( ( Xy
= ( produc3646306378393792727nt_int @ X3 @ Y3 ) )
=> ( Xys
!= ( zip_Pr2148635498163357687nt_int @ Xs3 @ Ys6 ) ) ) ) ) ) ).
% zip_eq_ConsE
thf(fact_867_zip__eq__ConsE,axiom,
! [Xs: list_nat,Ys2: list_set_int,Xy: produc9133624956312949779et_int,Xys: list_P7388728034488748185et_int] :
( ( ( zip_nat_set_int @ Xs @ Ys2 )
= ( cons_P5529800768674571465et_int @ Xy @ Xys ) )
=> ~ ! [X3: nat,Xs3: list_nat] :
( ( Xs
= ( cons_nat @ X3 @ Xs3 ) )
=> ! [Y3: set_int,Ys6: list_set_int] :
( ( Ys2
= ( cons_set_int @ Y3 @ Ys6 ) )
=> ( ( Xy
= ( produc29655638201817675et_int @ X3 @ Y3 ) )
=> ( Xys
!= ( zip_nat_set_int @ Xs3 @ Ys6 ) ) ) ) ) ) ).
% zip_eq_ConsE
thf(fact_868_zip__eq__ConsE,axiom,
! [Xs: list_int,Ys2: list_l1670014477004246597nt_int,Xy: produc661532565036771336nt_int,Xys: list_P6957225279449687320nt_int] :
( ( ( zip_in8018307065301790106nt_int @ Xs @ Ys2 )
= ( cons_P9095479431110121170nt_int @ Xy @ Xys ) )
=> ~ ! [X3: int,Xs3: list_int] :
( ( Xs
= ( cons_int @ X3 @ Xs3 ) )
=> ! [Y3: list_P5707943133018811711nt_int,Ys6: list_l1670014477004246597nt_int] :
( ( Ys2
= ( cons_l7309679040211256053nt_int @ Y3 @ Ys6 ) )
=> ( ( Xy
= ( produc8814303788642274490nt_int @ X3 @ Y3 ) )
=> ( Xys
!= ( zip_in8018307065301790106nt_int @ Xs3 @ Ys6 ) ) ) ) ) ) ).
% zip_eq_ConsE
thf(fact_869_zip__eq__ConsE,axiom,
! [Xs: list_int,Ys2: list_int,Xy: product_prod_int_int,Xys: list_P5707943133018811711nt_int] :
( ( ( zip_int_int @ Xs @ Ys2 )
= ( cons_P3334398858971670639nt_int @ Xy @ Xys ) )
=> ~ ! [X3: int,Xs3: list_int] :
( ( Xs
= ( cons_int @ X3 @ Xs3 ) )
=> ! [Y3: int,Ys6: list_int] :
( ( Ys2
= ( cons_int @ Y3 @ Ys6 ) )
=> ( ( Xy
= ( product_Pair_int_int @ X3 @ Y3 ) )
=> ( Xys
!= ( zip_int_int @ Xs3 @ Ys6 ) ) ) ) ) ) ).
% zip_eq_ConsE
thf(fact_870_zip__eq__ConsE,axiom,
! [Xs: list_P1743416141875011707e_term,Ys2: list_P5707943133018811711nt_int,Xy: produc2285326912895808259nt_int,Xys: list_P651320350408439699nt_int] :
( ( ( zip_Pr4168994715204986005nt_int @ Xs @ Ys2 )
= ( cons_P6018425551955479501nt_int @ Xy @ Xys ) )
=> ~ ! [X3: produc8551481072490612790e_term > option6357759511663192854e_term,Xs3: list_P1743416141875011707e_term] :
( ( Xs
= ( cons_P2630085844062958645e_term @ X3 @ Xs3 ) )
=> ! [Y3: product_prod_int_int,Ys6: list_P5707943133018811711nt_int] :
( ( Ys2
= ( cons_P3334398858971670639nt_int @ Y3 @ Ys6 ) )
=> ( ( Xy
= ( produc5700946648718959541nt_int @ X3 @ Y3 ) )
=> ( Xys
!= ( zip_Pr4168994715204986005nt_int @ Xs3 @ Ys6 ) ) ) ) ) ) ).
% zip_eq_ConsE
thf(fact_871_zip__eq__ConsE,axiom,
! [Xs: list_i8448526496819171953e_term,Ys2: list_P5707943133018811711nt_int,Xy: produc7773217078559923341nt_int,Xys: list_P8915022641806594461nt_int] :
( ( ( zip_in8766932505889695135nt_int @ Xs @ Ys2 )
= ( cons_P2743708091642732631nt_int @ Xy @ Xys ) )
=> ~ ! [X3: int > option6357759511663192854e_term,Xs3: list_i8448526496819171953e_term] :
( ( Xs
= ( cons_i7166360444231718571e_term @ X3 @ Xs3 ) )
=> ! [Y3: product_prod_int_int,Ys6: list_P5707943133018811711nt_int] :
( ( Ys2
= ( cons_P3334398858971670639nt_int @ Y3 @ Ys6 ) )
=> ( ( Xy
= ( produc4305682042979456191nt_int @ X3 @ Y3 ) )
=> ( Xys
!= ( zip_in8766932505889695135nt_int @ Xs3 @ Ys6 ) ) ) ) ) ) ).
% zip_eq_ConsE
thf(fact_872_zip__obtain__same__length,axiom,
! [Xs: list_P5707943133018811711nt_int,Ys2: list_P5707943133018811711nt_int,P2: list_P2336717926344734829nt_int > $o] :
( ! [Zs: list_P5707943133018811711nt_int,Ws: list_P5707943133018811711nt_int,N3: nat] :
( ( ( size_s5157815400016825771nt_int @ Zs )
= ( size_s5157815400016825771nt_int @ Ws ) )
=> ( ( N3
= ( ord_min_nat @ ( size_s5157815400016825771nt_int @ Xs ) @ ( size_s5157815400016825771nt_int @ Ys2 ) ) )
=> ( ( Zs
= ( take_P8218740963776755879nt_int @ N3 @ Xs ) )
=> ( ( Ws
= ( take_P8218740963776755879nt_int @ N3 @ Ys2 ) )
=> ( P2 @ ( zip_Pr2148635498163357687nt_int @ Zs @ Ws ) ) ) ) ) )
=> ( P2 @ ( zip_Pr2148635498163357687nt_int @ Xs @ Ys2 ) ) ) ).
% zip_obtain_same_length
thf(fact_873_nth__butlast,axiom,
! [N: nat,Xs: list_P5707943133018811711nt_int] :
( ( ord_less_nat @ N @ ( size_s5157815400016825771nt_int @ ( butlas2390654596765537291nt_int @ Xs ) ) )
=> ( ( nth_Pr4439495888332055232nt_int @ ( butlas2390654596765537291nt_int @ Xs ) @ N )
= ( nth_Pr4439495888332055232nt_int @ Xs @ N ) ) ) ).
% nth_butlast
thf(fact_874_take__butlast,axiom,
! [N: nat,Xs: list_P5707943133018811711nt_int] :
( ( ord_less_nat @ N @ ( size_s5157815400016825771nt_int @ Xs ) )
=> ( ( take_P8218740963776755879nt_int @ N @ ( butlas2390654596765537291nt_int @ Xs ) )
= ( take_P8218740963776755879nt_int @ N @ Xs ) ) ) ).
% take_butlast
thf(fact_875_zip__append2,axiom,
! [Xs: list_int,Ys2: list_int,Zs2: list_int] :
( ( zip_int_int @ Xs @ ( append_int @ Ys2 @ Zs2 ) )
= ( append7030698103840186580nt_int @ ( zip_int_int @ ( take_int @ ( size_size_list_int @ Ys2 ) @ Xs ) @ Ys2 ) @ ( zip_int_int @ ( drop_int @ ( size_size_list_int @ Ys2 ) @ Xs ) @ Zs2 ) ) ) ).
% zip_append2
thf(fact_876_zip__append2,axiom,
! [Xs: list_P5707943133018811711nt_int,Ys2: list_P5707943133018811711nt_int,Zs2: list_P5707943133018811711nt_int] :
( ( zip_Pr2148635498163357687nt_int @ Xs @ ( append7030698103840186580nt_int @ Ys2 @ Zs2 ) )
= ( append9111546825139864322nt_int @ ( zip_Pr2148635498163357687nt_int @ ( take_P8218740963776755879nt_int @ ( size_s5157815400016825771nt_int @ Ys2 ) @ Xs ) @ Ys2 ) @ ( zip_Pr2148635498163357687nt_int @ ( drop_P5690361596310759935nt_int @ ( size_s5157815400016825771nt_int @ Ys2 ) @ Xs ) @ Zs2 ) ) ) ).
% zip_append2
thf(fact_877_zip__append1,axiom,
! [Xs: list_int,Ys2: list_int,Zs2: list_int] :
( ( zip_int_int @ ( append_int @ Xs @ Ys2 ) @ Zs2 )
= ( append7030698103840186580nt_int @ ( zip_int_int @ Xs @ ( take_int @ ( size_size_list_int @ Xs ) @ Zs2 ) ) @ ( zip_int_int @ Ys2 @ ( drop_int @ ( size_size_list_int @ Xs ) @ Zs2 ) ) ) ) ).
% zip_append1
thf(fact_878_zip__append1,axiom,
! [Xs: list_P5707943133018811711nt_int,Ys2: list_P5707943133018811711nt_int,Zs2: list_P5707943133018811711nt_int] :
( ( zip_Pr2148635498163357687nt_int @ ( append7030698103840186580nt_int @ Xs @ Ys2 ) @ Zs2 )
= ( append9111546825139864322nt_int @ ( zip_Pr2148635498163357687nt_int @ Xs @ ( take_P8218740963776755879nt_int @ ( size_s5157815400016825771nt_int @ Xs ) @ Zs2 ) ) @ ( zip_Pr2148635498163357687nt_int @ Ys2 @ ( drop_P5690361596310759935nt_int @ ( size_s5157815400016825771nt_int @ Xs ) @ Zs2 ) ) ) ) ).
% zip_append1
thf(fact_879_hd__zip,axiom,
! [Xs: list_P5707943133018811711nt_int,Ys2: list_P5707943133018811711nt_int] :
( ( Xs != nil_Pr2300489316682597567nt_int )
=> ( ( Ys2 != nil_Pr2300489316682597567nt_int )
=> ( ( hd_Pro8821803587750610258nt_int @ ( zip_Pr2148635498163357687nt_int @ Xs @ Ys2 ) )
= ( produc3646306378393792727nt_int @ ( hd_Pro282112905867057956nt_int @ Xs ) @ ( hd_Pro282112905867057956nt_int @ Ys2 ) ) ) ) ) ).
% hd_zip
thf(fact_880_hd__zip,axiom,
! [Xs: list_nat,Ys2: list_set_int] :
( ( Xs != nil_nat )
=> ( ( Ys2 != nil_set_int )
=> ( ( hd_Pro4409541185786561406et_int @ ( zip_nat_set_int @ Xs @ Ys2 ) )
= ( produc29655638201817675et_int @ ( hd_nat @ Xs ) @ ( hd_set_int @ Ys2 ) ) ) ) ) ).
% hd_zip
thf(fact_881_hd__zip,axiom,
! [Xs: list_int,Ys2: list_l1670014477004246597nt_int] :
( ( Xs != nil_int )
=> ( ( Ys2 != nil_li8670148097206105925nt_int )
=> ( ( hd_Pro5866865383777520733nt_int @ ( zip_in8018307065301790106nt_int @ Xs @ Ys2 ) )
= ( produc8814303788642274490nt_int @ ( hd_int @ Xs ) @ ( hd_lis8550591025403828010nt_int @ Ys2 ) ) ) ) ) ).
% hd_zip
thf(fact_882_hd__zip,axiom,
! [Xs: list_int,Ys2: list_int] :
( ( Xs != nil_int )
=> ( ( Ys2 != nil_int )
=> ( ( hd_Pro282112905867057956nt_int @ ( zip_int_int @ Xs @ Ys2 ) )
= ( product_Pair_int_int @ ( hd_int @ Xs ) @ ( hd_int @ Ys2 ) ) ) ) ) ).
% hd_zip
thf(fact_883_hd__zip,axiom,
! [Xs: list_P1743416141875011707e_term,Ys2: list_P5707943133018811711nt_int] :
( ( Xs != nil_Pr2193041437141079013e_term )
=> ( ( Ys2 != nil_Pr2300489316682597567nt_int )
=> ( ( hd_Pro8246561224800902232nt_int @ ( zip_Pr4168994715204986005nt_int @ Xs @ Ys2 ) )
= ( produc5700946648718959541nt_int @ ( hd_Pro8443970089900138816e_term @ Xs ) @ ( hd_Pro282112905867057956nt_int @ Ys2 ) ) ) ) ) ).
% hd_zip
thf(fact_884_hd__zip,axiom,
! [Xs: list_i8448526496819171953e_term,Ys2: list_P5707943133018811711nt_int] :
( ( Xs != nil_in3447728264653889115e_term )
=> ( ( Ys2 != nil_Pr2300489316682597567nt_int )
=> ( ( hd_Pro2201378251908260706nt_int @ ( zip_in8766932505889695135nt_int @ Xs @ Ys2 ) )
= ( produc4305682042979456191nt_int @ ( hd_int7405675249218303542e_term @ Xs ) @ ( hd_Pro282112905867057956nt_int @ Ys2 ) ) ) ) ) ).
% hd_zip
thf(fact_885_last__zip,axiom,
! [Xs: list_nat,Ys2: list_set_int] :
( ( Xs != nil_nat )
=> ( ( Ys2 != nil_set_int )
=> ( ( ( size_size_list_nat @ Xs )
= ( size_s763970886550748294et_int @ Ys2 ) )
=> ( ( last_P2058727373477175858et_int @ ( zip_nat_set_int @ Xs @ Ys2 ) )
= ( produc29655638201817675et_int @ ( last_nat @ Xs ) @ ( last_set_int @ Ys2 ) ) ) ) ) ) ).
% last_zip
thf(fact_886_last__zip,axiom,
! [Xs: list_int,Ys2: list_l1670014477004246597nt_int] :
( ( Xs != nil_int )
=> ( ( Ys2 != nil_li8670148097206105925nt_int )
=> ( ( ( size_size_list_int @ Xs )
= ( size_s2969076144586574001nt_int @ Ys2 ) )
=> ( ( last_P2520469823115007657nt_int @ ( zip_in8018307065301790106nt_int @ Xs @ Ys2 ) )
= ( produc8814303788642274490nt_int @ ( last_int @ Xs ) @ ( last_l5818330359162608606nt_int @ Ys2 ) ) ) ) ) ) ).
% last_zip
thf(fact_887_last__zip,axiom,
! [Xs: list_int,Ys2: list_int] :
( ( Xs != nil_int )
=> ( ( Ys2 != nil_int )
=> ( ( ( size_size_list_int @ Xs )
= ( size_size_list_int @ Ys2 ) )
=> ( ( last_P3305686521732843992nt_int @ ( zip_int_int @ Xs @ Ys2 ) )
= ( product_Pair_int_int @ ( last_int @ Xs ) @ ( last_int @ Ys2 ) ) ) ) ) ) ).
% last_zip
thf(fact_888_last__zip,axiom,
! [Xs: list_P1743416141875011707e_term,Ys2: list_P5707943133018811711nt_int] :
( ( Xs != nil_Pr2193041437141079013e_term )
=> ( ( Ys2 != nil_Pr2300489316682597567nt_int )
=> ( ( ( size_s3817825649112772879e_term @ Xs )
= ( size_s5157815400016825771nt_int @ Ys2 ) )
=> ( ( last_P7958387789370520228nt_int @ ( zip_Pr4168994715204986005nt_int @ Xs @ Ys2 ) )
= ( produc5700946648718959541nt_int @ ( last_P8974135382477003148e_term @ Xs ) @ ( last_P3305686521732843992nt_int @ Ys2 ) ) ) ) ) ) ).
% last_zip
thf(fact_889_last__zip,axiom,
! [Xs: list_i8448526496819171953e_term,Ys2: list_P5707943133018811711nt_int] :
( ( Xs != nil_in3447728264653889115e_term )
=> ( ( Ys2 != nil_Pr2300489316682597567nt_int )
=> ( ( ( size_s8881356780637927685e_term @ Xs )
= ( size_s5157815400016825771nt_int @ Ys2 ) )
=> ( ( last_P858631638320919470nt_int @ ( zip_in8766932505889695135nt_int @ Xs @ Ys2 ) )
= ( produc4305682042979456191nt_int @ ( last_i813233280922151042e_term @ Xs ) @ ( last_P3305686521732843992nt_int @ Ys2 ) ) ) ) ) ) ).
% last_zip
thf(fact_890_last__zip,axiom,
! [Xs: list_P5707943133018811711nt_int,Ys2: list_P5707943133018811711nt_int] :
( ( Xs != nil_Pr2300489316682597567nt_int )
=> ( ( Ys2 != nil_Pr2300489316682597567nt_int )
=> ( ( ( size_s5157815400016825771nt_int @ Xs )
= ( size_s5157815400016825771nt_int @ Ys2 ) )
=> ( ( last_P6195391842956826118nt_int @ ( zip_Pr2148635498163357687nt_int @ Xs @ Ys2 ) )
= ( produc3646306378393792727nt_int @ ( last_P3305686521732843992nt_int @ Xs ) @ ( last_P3305686521732843992nt_int @ Ys2 ) ) ) ) ) ) ).
% last_zip
thf(fact_891_snoc__eq__iff__butlast,axiom,
! [Xs: list_P5707943133018811711nt_int,X: product_prod_int_int,Ys2: list_P5707943133018811711nt_int] :
( ( ( append7030698103840186580nt_int @ Xs @ ( cons_P3334398858971670639nt_int @ X @ nil_Pr2300489316682597567nt_int ) )
= Ys2 )
= ( ( Ys2 != nil_Pr2300489316682597567nt_int )
& ( ( butlas2390654596765537291nt_int @ Ys2 )
= Xs )
& ( ( last_P3305686521732843992nt_int @ Ys2 )
= X ) ) ) ).
% snoc_eq_iff_butlast
thf(fact_892_Succ__Shift,axiom,
! [Kl2: set_li3855193629254082847nt_int,K: product_prod_int_int,Kl: list_P5707943133018811711nt_int] :
( ( bNF_Gr2185362013987207252nt_int @ ( bNF_Gr9175161896313899224nt_int @ Kl2 @ K ) @ Kl )
= ( bNF_Gr2185362013987207252nt_int @ Kl2 @ ( cons_P3334398858971670639nt_int @ K @ Kl ) ) ) ).
% Succ_Shift
thf(fact_893_product__lists_Osimps_I1_J,axiom,
( ( produc5568053154996169768nt_int @ nil_li8670148097206105925nt_int )
= ( cons_l7309679040211256053nt_int @ nil_Pr2300489316682597567nt_int @ nil_li8670148097206105925nt_int ) ) ).
% product_lists.simps(1)
thf(fact_894_trans__path__non__nil__last,axiom,
! [Ps: list_P5707943133018811711nt_int,K_12: int,K_22: int,I4: int,J3: int] :
( ( Ps != nil_Pr2300489316682597567nt_int )
=> ( ( last_P3305686521732843992nt_int @ ( trans_path @ ( product_Pair_int_int @ K_12 @ K_22 ) @ Ps ) )
= ( last_P3305686521732843992nt_int @ ( trans_path @ ( product_Pair_int_int @ K_12 @ K_22 ) @ ( cons_P3334398858971670639nt_int @ ( product_Pair_int_int @ I4 @ J3 ) @ Ps ) ) ) ) ) ).
% trans_path_non_nil_last
thf(fact_895_take__trans,axiom,
! [K: nat,K_12: int,K_22: int,Ps: list_P5707943133018811711nt_int] :
( ( take_P8218740963776755879nt_int @ K @ ( trans_path @ ( product_Pair_int_int @ K_12 @ K_22 ) @ Ps ) )
= ( trans_path @ ( product_Pair_int_int @ K_12 @ K_22 ) @ ( take_P8218740963776755879nt_int @ K @ Ps ) ) ) ).
% take_trans
thf(fact_896_trans__path_Osimps_I2_J,axiom,
! [K_12: int,K_22: int,I4: int,J3: int,Xs: list_P5707943133018811711nt_int] :
( ( trans_path @ ( product_Pair_int_int @ K_12 @ K_22 ) @ ( cons_P3334398858971670639nt_int @ ( product_Pair_int_int @ I4 @ J3 ) @ Xs ) )
= ( cons_P3334398858971670639nt_int @ ( product_Pair_int_int @ ( plus_plus_int @ I4 @ K_12 ) @ ( plus_plus_int @ J3 @ K_22 ) ) @ ( trans_path @ ( product_Pair_int_int @ K_12 @ K_22 ) @ Xs ) ) ) ).
% trans_path.simps(2)
thf(fact_897_trans__path_Osimps_I1_J,axiom,
! [K_12: int,K_22: int] :
( ( trans_path @ ( product_Pair_int_int @ K_12 @ K_22 ) @ nil_Pr2300489316682597567nt_int )
= nil_Pr2300489316682597567nt_int ) ).
% trans_path.simps(1)
thf(fact_898_trans__path__non__nil,axiom,
! [Ps: list_P5707943133018811711nt_int,K_12: int,K_22: int] :
( ( Ps != nil_Pr2300489316682597567nt_int )
=> ( ( trans_path @ ( product_Pair_int_int @ K_12 @ K_22 ) @ Ps )
!= nil_Pr2300489316682597567nt_int ) ) ).
% trans_path_non_nil
thf(fact_899_ShiftD,axiom,
! [Kl: list_P5707943133018811711nt_int,Kl2: set_li3855193629254082847nt_int,K: product_prod_int_int] :
( ( member2764346250752101224nt_int @ Kl @ ( bNF_Gr9175161896313899224nt_int @ Kl2 @ K ) )
=> ( member2764346250752101224nt_int @ ( cons_P3334398858971670639nt_int @ K @ Kl ) @ Kl2 ) ) ).
% ShiftD
thf(fact_900_trans__path__length,axiom,
! [K_12: int,K_22: int] :
( size_s5157815400016825771nt_int
= ( ^ [Ps4: list_P5707943133018811711nt_int] : ( size_s5157815400016825771nt_int @ ( trans_path @ ( product_Pair_int_int @ K_12 @ K_22 ) @ Ps4 ) ) ) ) ).
% trans_path_length
thf(fact_901_drop__trans,axiom,
! [K: nat,K_12: int,K_22: int,Ps: list_P5707943133018811711nt_int] :
( ( drop_P5690361596310759935nt_int @ K @ ( trans_path @ ( product_Pair_int_int @ K_12 @ K_22 ) @ Ps ) )
= ( trans_path @ ( product_Pair_int_int @ K_12 @ K_22 ) @ ( drop_P5690361596310759935nt_int @ K @ Ps ) ) ) ).
% drop_trans
thf(fact_902_trans__path_Oelims,axiom,
! [X: product_prod_int_int,Xa: list_P5707943133018811711nt_int,Y: list_P5707943133018811711nt_int] :
( ( ( trans_path @ X @ Xa )
= Y )
=> ( ( ? [K_1: int,K_2: int] :
( X
= ( product_Pair_int_int @ K_1 @ K_2 ) )
=> ( ( Xa = nil_Pr2300489316682597567nt_int )
=> ( Y != nil_Pr2300489316682597567nt_int ) ) )
=> ~ ! [K_1: int,K_2: int] :
( ( X
= ( product_Pair_int_int @ K_1 @ K_2 ) )
=> ! [I: int,J: int,Xs2: list_P5707943133018811711nt_int] :
( ( Xa
= ( cons_P3334398858971670639nt_int @ ( product_Pair_int_int @ I @ J ) @ Xs2 ) )
=> ( Y
!= ( cons_P3334398858971670639nt_int @ ( product_Pair_int_int @ ( plus_plus_int @ I @ K_1 ) @ ( plus_plus_int @ J @ K_2 ) ) @ ( trans_path @ ( product_Pair_int_int @ K_1 @ K_2 ) @ Xs2 ) ) ) ) ) ) ) ).
% trans_path.elims
thf(fact_903_last__trans__path,axiom,
! [Ps: list_P5707943133018811711nt_int,I4: int,J3: int,K_12: int,K_22: int] :
( ( Ps != nil_Pr2300489316682597567nt_int )
=> ( ( ( last_P3305686521732843992nt_int @ Ps )
= ( product_Pair_int_int @ I4 @ J3 ) )
=> ( ( last_P3305686521732843992nt_int @ ( trans_path @ ( product_Pair_int_int @ K_12 @ K_22 ) @ Ps ) )
= ( product_Pair_int_int @ ( plus_plus_int @ I4 @ K_12 ) @ ( plus_plus_int @ J3 @ K_22 ) ) ) ) ) ).
% last_trans_path
thf(fact_904_hd__trans__path,axiom,
! [Ps: list_P5707943133018811711nt_int,I4: int,J3: int,K_12: int,K_22: int] :
( ( Ps != nil_Pr2300489316682597567nt_int )
=> ( ( ( hd_Pro282112905867057956nt_int @ Ps )
= ( product_Pair_int_int @ I4 @ J3 ) )
=> ( ( hd_Pro282112905867057956nt_int @ ( trans_path @ ( product_Pair_int_int @ K_12 @ K_22 ) @ Ps ) )
= ( product_Pair_int_int @ ( plus_plus_int @ I4 @ K_12 ) @ ( plus_plus_int @ J3 @ K_22 ) ) ) ) ) ).
% hd_trans_path
thf(fact_905_subseqs_Osimps_I1_J,axiom,
( ( subseq1357044202310323342nt_int @ nil_Pr2300489316682597567nt_int )
= ( cons_l7309679040211256053nt_int @ nil_Pr2300489316682597567nt_int @ nil_li8670148097206105925nt_int ) ) ).
% subseqs.simps(1)
thf(fact_906_rotate1__hd__tl,axiom,
! [Xs: list_P5707943133018811711nt_int] :
( ( Xs != nil_Pr2300489316682597567nt_int )
=> ( ( rotate9137591727117064716nt_int @ Xs )
= ( append7030698103840186580nt_int @ ( tl_Pro1049539609081552168nt_int @ Xs ) @ ( cons_P3334398858971670639nt_int @ ( hd_Pro282112905867057956nt_int @ Xs ) @ nil_Pr2300489316682597567nt_int ) ) ) ) ).
% rotate1_hd_tl
thf(fact_907_distinct__adj__append__iff,axiom,
! [Xs: list_P5707943133018811711nt_int,Ys2: list_P5707943133018811711nt_int] :
( ( distin7999467515711547031nt_int @ ( append7030698103840186580nt_int @ Xs @ Ys2 ) )
= ( ( distin7999467515711547031nt_int @ Xs )
& ( distin7999467515711547031nt_int @ Ys2 )
& ( ( Xs = nil_Pr2300489316682597567nt_int )
| ( Ys2 = nil_Pr2300489316682597567nt_int )
| ( ( last_P3305686521732843992nt_int @ Xs )
!= ( hd_Pro282112905867057956nt_int @ Ys2 ) ) ) ) ) ).
% distinct_adj_append_iff
thf(fact_908_distinct__adj__Cons__Cons,axiom,
! [X: product_prod_int_int,Y: product_prod_int_int,Xs: list_P5707943133018811711nt_int] :
( ( distin7999467515711547031nt_int @ ( cons_P3334398858971670639nt_int @ X @ ( cons_P3334398858971670639nt_int @ Y @ Xs ) ) )
= ( ( X != Y )
& ( distin7999467515711547031nt_int @ ( cons_P3334398858971670639nt_int @ Y @ Xs ) ) ) ) ).
% distinct_adj_Cons_Cons
thf(fact_909_tl__append2,axiom,
! [Xs: list_P5707943133018811711nt_int,Ys2: list_P5707943133018811711nt_int] :
( ( Xs != nil_Pr2300489316682597567nt_int )
=> ( ( tl_Pro1049539609081552168nt_int @ ( append7030698103840186580nt_int @ Xs @ Ys2 ) )
= ( append7030698103840186580nt_int @ ( tl_Pro1049539609081552168nt_int @ Xs ) @ Ys2 ) ) ) ).
% tl_append2
thf(fact_910_list_Ocollapse,axiom,
! [List: list_P5707943133018811711nt_int] :
( ( List != nil_Pr2300489316682597567nt_int )
=> ( ( cons_P3334398858971670639nt_int @ ( hd_Pro282112905867057956nt_int @ List ) @ ( tl_Pro1049539609081552168nt_int @ List ) )
= List ) ) ).
% list.collapse
thf(fact_911_hd__Cons__tl,axiom,
! [Xs: list_P5707943133018811711nt_int] :
( ( Xs != nil_Pr2300489316682597567nt_int )
=> ( ( cons_P3334398858971670639nt_int @ ( hd_Pro282112905867057956nt_int @ Xs ) @ ( tl_Pro1049539609081552168nt_int @ Xs ) )
= Xs ) ) ).
% hd_Cons_tl
thf(fact_912_list_Osel_I2_J,axiom,
( ( tl_Pro1049539609081552168nt_int @ nil_Pr2300489316682597567nt_int )
= nil_Pr2300489316682597567nt_int ) ).
% list.sel(2)
thf(fact_913_list_Osel_I3_J,axiom,
! [X21: product_prod_int_int,X22: list_P5707943133018811711nt_int] :
( ( tl_Pro1049539609081552168nt_int @ ( cons_P3334398858971670639nt_int @ X21 @ X22 ) )
= X22 ) ).
% list.sel(3)
thf(fact_914_distinct__adj__ConsD,axiom,
! [X: product_prod_int_int,Xs: list_P5707943133018811711nt_int] :
( ( distin7999467515711547031nt_int @ ( cons_P3334398858971670639nt_int @ X @ Xs ) )
=> ( distin7999467515711547031nt_int @ Xs ) ) ).
% distinct_adj_ConsD
thf(fact_915_distinct__adj__Nil,axiom,
distin7999467515711547031nt_int @ nil_Pr2300489316682597567nt_int ).
% distinct_adj_Nil
thf(fact_916_distinct__adj__appendD1,axiom,
! [Xs: list_P5707943133018811711nt_int,Ys2: list_P5707943133018811711nt_int] :
( ( distin7999467515711547031nt_int @ ( append7030698103840186580nt_int @ Xs @ Ys2 ) )
=> ( distin7999467515711547031nt_int @ Xs ) ) ).
% distinct_adj_appendD1
thf(fact_917_distinct__adj__appendD2,axiom,
! [Xs: list_P5707943133018811711nt_int,Ys2: list_P5707943133018811711nt_int] :
( ( distin7999467515711547031nt_int @ ( append7030698103840186580nt_int @ Xs @ Ys2 ) )
=> ( distin7999467515711547031nt_int @ Ys2 ) ) ).
% distinct_adj_appendD2
thf(fact_918_tl__drop,axiom,
! [N: nat,Xs: list_P5707943133018811711nt_int] :
( ( tl_Pro1049539609081552168nt_int @ ( drop_P5690361596310759935nt_int @ N @ Xs ) )
= ( drop_P5690361596310759935nt_int @ N @ ( tl_Pro1049539609081552168nt_int @ Xs ) ) ) ).
% tl_drop
thf(fact_919_Nil__tl,axiom,
! [Xs: list_P5707943133018811711nt_int] :
( ( nil_Pr2300489316682597567nt_int
= ( tl_Pro1049539609081552168nt_int @ Xs ) )
= ( ( Xs = nil_Pr2300489316682597567nt_int )
| ? [X4: product_prod_int_int] :
( Xs
= ( cons_P3334398858971670639nt_int @ X4 @ nil_Pr2300489316682597567nt_int ) ) ) ) ).
% Nil_tl
thf(fact_920_tl__Nil,axiom,
! [Xs: list_P5707943133018811711nt_int] :
( ( ( tl_Pro1049539609081552168nt_int @ Xs )
= nil_Pr2300489316682597567nt_int )
= ( ( Xs = nil_Pr2300489316682597567nt_int )
| ? [X4: product_prod_int_int] :
( Xs
= ( cons_P3334398858971670639nt_int @ X4 @ nil_Pr2300489316682597567nt_int ) ) ) ) ).
% tl_Nil
thf(fact_921_tl__append__if,axiom,
! [Xs: list_P5707943133018811711nt_int,Ys2: list_P5707943133018811711nt_int] :
( ( ( Xs = nil_Pr2300489316682597567nt_int )
=> ( ( tl_Pro1049539609081552168nt_int @ ( append7030698103840186580nt_int @ Xs @ Ys2 ) )
= ( tl_Pro1049539609081552168nt_int @ Ys2 ) ) )
& ( ( Xs != nil_Pr2300489316682597567nt_int )
=> ( ( tl_Pro1049539609081552168nt_int @ ( append7030698103840186580nt_int @ Xs @ Ys2 ) )
= ( append7030698103840186580nt_int @ ( tl_Pro1049539609081552168nt_int @ Xs ) @ Ys2 ) ) ) ) ).
% tl_append_if
thf(fact_922_take__tl,axiom,
! [N: nat,Xs: list_P5707943133018811711nt_int] :
( ( take_P8218740963776755879nt_int @ N @ ( tl_Pro1049539609081552168nt_int @ Xs ) )
= ( tl_Pro1049539609081552168nt_int @ ( take_P8218740963776755879nt_int @ ( suc @ N ) @ Xs ) ) ) ).
% take_tl
thf(fact_923_drop__Suc,axiom,
! [N: nat,Xs: list_P5707943133018811711nt_int] :
( ( drop_P5690361596310759935nt_int @ ( suc @ N ) @ Xs )
= ( drop_P5690361596310759935nt_int @ N @ ( tl_Pro1049539609081552168nt_int @ Xs ) ) ) ).
% drop_Suc
thf(fact_924_list_Oexpand,axiom,
! [List: list_P5707943133018811711nt_int,List2: list_P5707943133018811711nt_int] :
( ( ( List = nil_Pr2300489316682597567nt_int )
= ( List2 = nil_Pr2300489316682597567nt_int ) )
=> ( ( ( List != nil_Pr2300489316682597567nt_int )
=> ( ( List2 != nil_Pr2300489316682597567nt_int )
=> ( ( ( hd_Pro282112905867057956nt_int @ List )
= ( hd_Pro282112905867057956nt_int @ List2 ) )
& ( ( tl_Pro1049539609081552168nt_int @ List )
= ( tl_Pro1049539609081552168nt_int @ List2 ) ) ) ) )
=> ( List = List2 ) ) ) ).
% list.expand
thf(fact_925_last__tl,axiom,
! [Xs: list_P5707943133018811711nt_int] :
( ( ( Xs = nil_Pr2300489316682597567nt_int )
| ( ( tl_Pro1049539609081552168nt_int @ Xs )
!= nil_Pr2300489316682597567nt_int ) )
=> ( ( last_P3305686521732843992nt_int @ ( tl_Pro1049539609081552168nt_int @ Xs ) )
= ( last_P3305686521732843992nt_int @ Xs ) ) ) ).
% last_tl
thf(fact_926_distinct__adj__singleton,axiom,
! [X: product_prod_int_int] : ( distin7999467515711547031nt_int @ ( cons_P3334398858971670639nt_int @ X @ nil_Pr2300489316682597567nt_int ) ) ).
% distinct_adj_singleton
thf(fact_927_list_Oexhaust__sel,axiom,
! [List: list_P5707943133018811711nt_int] :
( ( List != nil_Pr2300489316682597567nt_int )
=> ( List
= ( cons_P3334398858971670639nt_int @ ( hd_Pro282112905867057956nt_int @ List ) @ ( tl_Pro1049539609081552168nt_int @ List ) ) ) ) ).
% list.exhaust_sel
thf(fact_928_distinct__adj__Cons,axiom,
! [X: product_prod_int_int,Xs: list_P5707943133018811711nt_int] :
( ( distin7999467515711547031nt_int @ ( cons_P3334398858971670639nt_int @ X @ Xs ) )
= ( ( Xs = nil_Pr2300489316682597567nt_int )
| ( ( X
!= ( hd_Pro282112905867057956nt_int @ Xs ) )
& ( distin7999467515711547031nt_int @ Xs ) ) ) ) ).
% distinct_adj_Cons
thf(fact_929_nth__tl,axiom,
! [N: nat,Xs: list_P5707943133018811711nt_int] :
( ( ord_less_nat @ N @ ( size_s5157815400016825771nt_int @ ( tl_Pro1049539609081552168nt_int @ Xs ) ) )
=> ( ( nth_Pr4439495888332055232nt_int @ ( tl_Pro1049539609081552168nt_int @ Xs ) @ N )
= ( nth_Pr4439495888332055232nt_int @ Xs @ ( suc @ N ) ) ) ) ).
% nth_tl
thf(fact_930_take__Suc,axiom,
! [Xs: list_P5707943133018811711nt_int,N: nat] :
( ( Xs != nil_Pr2300489316682597567nt_int )
=> ( ( take_P8218740963776755879nt_int @ ( suc @ N ) @ Xs )
= ( cons_P3334398858971670639nt_int @ ( hd_Pro282112905867057956nt_int @ Xs ) @ ( take_P8218740963776755879nt_int @ N @ ( tl_Pro1049539609081552168nt_int @ Xs ) ) ) ) ) ).
% take_Suc
thf(fact_931_zle__add1__eq__le,axiom,
! [W: int,Z3: int] :
( ( ord_less_int @ W @ ( plus_plus_int @ Z3 @ one_one_int ) )
= ( ord_less_eq_int @ W @ Z3 ) ) ).
% zle_add1_eq_le
thf(fact_932_verit__la__generic,axiom,
! [A: int,X: int] :
( ( ord_less_eq_int @ A @ X )
| ( A = X )
| ( ord_less_eq_int @ X @ A ) ) ).
% verit_la_generic
thf(fact_933_add1__zle__eq,axiom,
! [W: int,Z3: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ W @ one_one_int ) @ Z3 )
= ( ord_less_int @ W @ Z3 ) ) ).
% add1_zle_eq
thf(fact_934_int__ge__induct,axiom,
! [K: int,I4: int,P2: int > $o] :
( ( ord_less_eq_int @ K @ I4 )
=> ( ( P2 @ K )
=> ( ! [I: int] :
( ( ord_less_eq_int @ K @ I )
=> ( ( P2 @ I )
=> ( P2 @ ( plus_plus_int @ I @ one_one_int ) ) ) )
=> ( P2 @ I4 ) ) ) ) ).
% int_ge_induct
thf(fact_935_zless__imp__add1__zle,axiom,
! [W: int,Z3: int] :
( ( ord_less_int @ W @ Z3 )
=> ( ord_less_eq_int @ ( plus_plus_int @ W @ one_one_int ) @ Z3 ) ) ).
% zless_imp_add1_zle
thf(fact_936_int__gr__induct,axiom,
! [K: int,I4: int,P2: int > $o] :
( ( ord_less_int @ K @ I4 )
=> ( ( P2 @ ( plus_plus_int @ K @ one_one_int ) )
=> ( ! [I: int] :
( ( ord_less_int @ K @ I )
=> ( ( P2 @ I )
=> ( P2 @ ( plus_plus_int @ I @ one_one_int ) ) ) )
=> ( P2 @ I4 ) ) ) ) ).
% int_gr_induct
thf(fact_937_zless__add1__eq,axiom,
! [W: int,Z3: int] :
( ( ord_less_int @ W @ ( plus_plus_int @ Z3 @ one_one_int ) )
= ( ( ord_less_int @ W @ Z3 )
| ( W = Z3 ) ) ) ).
% zless_add1_eq
thf(fact_938_trans__path_Opelims,axiom,
! [X: product_prod_int_int,Xa: list_P5707943133018811711nt_int,Y: list_P5707943133018811711nt_int] :
( ( ( trans_path @ X @ Xa )
= Y )
=> ( ( accp_P6701151336149146870nt_int @ trans_path_rel @ ( produc8677244595851196253nt_int @ X @ Xa ) )
=> ( ! [K_1: int,K_2: int] :
( ( X
= ( product_Pair_int_int @ K_1 @ K_2 ) )
=> ( ( Xa = nil_Pr2300489316682597567nt_int )
=> ( ( Y = nil_Pr2300489316682597567nt_int )
=> ~ ( accp_P6701151336149146870nt_int @ trans_path_rel @ ( produc8677244595851196253nt_int @ ( product_Pair_int_int @ K_1 @ K_2 ) @ nil_Pr2300489316682597567nt_int ) ) ) ) )
=> ~ ! [K_1: int,K_2: int] :
( ( X
= ( product_Pair_int_int @ K_1 @ K_2 ) )
=> ! [I: int,J: int,Xs2: list_P5707943133018811711nt_int] :
( ( Xa
= ( cons_P3334398858971670639nt_int @ ( product_Pair_int_int @ I @ J ) @ Xs2 ) )
=> ( ( Y
= ( cons_P3334398858971670639nt_int @ ( product_Pair_int_int @ ( plus_plus_int @ I @ K_1 ) @ ( plus_plus_int @ J @ K_2 ) ) @ ( trans_path @ ( product_Pair_int_int @ K_1 @ K_2 ) @ Xs2 ) ) )
=> ~ ( accp_P6701151336149146870nt_int @ trans_path_rel @ ( produc8677244595851196253nt_int @ ( product_Pair_int_int @ K_1 @ K_2 ) @ ( cons_P3334398858971670639nt_int @ ( product_Pair_int_int @ I @ J ) @ Xs2 ) ) ) ) ) ) ) ) ) ).
% trans_path.pelims
thf(fact_939_assms_I7_J,axiom,
valid_step @ ( product_Pair_int_int @ i_l @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ m_1 ) @ j_l ) ) @ s_j ).
% assms(7)
thf(fact_940_assms_I6_J,axiom,
valid_step @ s_i @ ( product_Pair_int_int @ i_h @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ m_1 ) @ j_h ) ) ).
% assms(6)
thf(fact_941_of__nat__eq__iff,axiom,
! [M: nat,N: nat] :
( ( ( semiri1314217659103216013at_int @ M )
= ( semiri1314217659103216013at_int @ N ) )
= ( M = N ) ) ).
% of_nat_eq_iff
thf(fact_942_of__nat__le__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% of_nat_le_iff
thf(fact_943_of__nat__le__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% of_nat_le_iff
thf(fact_944_of__nat__less__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% of_nat_less_iff
thf(fact_945_of__nat__less__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% of_nat_less_iff
thf(fact_946_of__nat__add,axiom,
! [M: nat,N: nat] :
( ( semiri1316708129612266289at_nat @ ( plus_plus_nat @ M @ N ) )
= ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% of_nat_add
thf(fact_947_of__nat__add,axiom,
! [M: nat,N: nat] :
( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ M @ N ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% of_nat_add
thf(fact_948_of__nat__eq__1__iff,axiom,
! [N: nat] :
( ( ( semiri1316708129612266289at_nat @ N )
= one_one_nat )
= ( N = one_one_nat ) ) ).
% of_nat_eq_1_iff
thf(fact_949_of__nat__eq__1__iff,axiom,
! [N: nat] :
( ( ( semiri1314217659103216013at_int @ N )
= one_one_int )
= ( N = one_one_nat ) ) ).
% of_nat_eq_1_iff
thf(fact_950_of__nat__1__eq__iff,axiom,
! [N: nat] :
( ( one_one_nat
= ( semiri1316708129612266289at_nat @ N ) )
= ( N = one_one_nat ) ) ).
% of_nat_1_eq_iff
thf(fact_951_of__nat__1__eq__iff,axiom,
! [N: nat] :
( ( one_one_int
= ( semiri1314217659103216013at_int @ N ) )
= ( N = one_one_nat ) ) ).
% of_nat_1_eq_iff
thf(fact_952_of__nat__1,axiom,
( ( semiri1316708129612266289at_nat @ one_one_nat )
= one_one_nat ) ).
% of_nat_1
thf(fact_953_of__nat__1,axiom,
( ( semiri1314217659103216013at_int @ one_one_nat )
= one_one_int ) ).
% of_nat_1
thf(fact_954_of__nat__Suc,axiom,
! [M: nat] :
( ( semiri1316708129612266289at_nat @ ( suc @ M ) )
= ( plus_plus_nat @ one_one_nat @ ( semiri1316708129612266289at_nat @ M ) ) ) ).
% of_nat_Suc
thf(fact_955_of__nat__Suc,axiom,
! [M: nat] :
( ( semiri1314217659103216013at_int @ ( suc @ M ) )
= ( plus_plus_int @ one_one_int @ ( semiri1314217659103216013at_int @ M ) ) ) ).
% of_nat_Suc
thf(fact_956_assms_I3_J,axiom,
step_in @ ps_1 @ s_i @ s_j ).
% assms(3)
thf(fact_957_accp__subset__induct,axiom,
! [D4: produc4058024888802116461nt_int > $o,R2: produc4058024888802116461nt_int > produc4058024888802116461nt_int > $o,X: produc4058024888802116461nt_int,P2: produc4058024888802116461nt_int > $o] :
( ( ord_le348346290460294320_int_o @ D4 @ ( accp_P6701151336149146870nt_int @ R2 ) )
=> ( ! [X3: produc4058024888802116461nt_int,Z2: produc4058024888802116461nt_int] :
( ( D4 @ X3 )
=> ( ( R2 @ Z2 @ X3 )
=> ( D4 @ Z2 ) ) )
=> ( ( D4 @ X )
=> ( ! [X3: produc4058024888802116461nt_int] :
( ( D4 @ X3 )
=> ( ! [Z4: produc4058024888802116461nt_int] :
( ( R2 @ Z4 @ X3 )
=> ( P2 @ Z4 ) )
=> ( P2 @ X3 ) ) )
=> ( P2 @ X ) ) ) ) ) ).
% accp_subset_induct
thf(fact_958_accp__subset__induct,axiom,
! [D4: product_prod_int_int > $o,R2: product_prod_int_int > product_prod_int_int > $o,X: product_prod_int_int,P2: product_prod_int_int > $o] :
( ( ord_le8369615600986905444_int_o @ D4 @ ( accp_P1096762738010456898nt_int @ R2 ) )
=> ( ! [X3: product_prod_int_int,Z2: product_prod_int_int] :
( ( D4 @ X3 )
=> ( ( R2 @ Z2 @ X3 )
=> ( D4 @ Z2 ) ) )
=> ( ( D4 @ X )
=> ( ! [X3: product_prod_int_int] :
( ( D4 @ X3 )
=> ( ! [Z4: product_prod_int_int] :
( ( R2 @ Z4 @ X3 )
=> ( P2 @ Z4 ) )
=> ( P2 @ X3 ) ) )
=> ( P2 @ X ) ) ) ) ) ).
% accp_subset_induct
thf(fact_959_accp__subset,axiom,
! [R1: produc4058024888802116461nt_int > produc4058024888802116461nt_int > $o,R22: produc4058024888802116461nt_int > produc4058024888802116461nt_int > $o] :
( ( ord_le9187342105224860622_int_o @ R1 @ R22 )
=> ( ord_le348346290460294320_int_o @ ( accp_P6701151336149146870nt_int @ R22 ) @ ( accp_P6701151336149146870nt_int @ R1 ) ) ) ).
% accp_subset
thf(fact_960_accp__subset,axiom,
! [R1: product_prod_int_int > product_prod_int_int > $o,R22: product_prod_int_int > product_prod_int_int > $o] :
( ( ord_le1598226405681992910_int_o @ R1 @ R22 )
=> ( ord_le8369615600986905444_int_o @ ( accp_P1096762738010456898nt_int @ R22 ) @ ( accp_P1096762738010456898nt_int @ R1 ) ) ) ).
% accp_subset
thf(fact_961_nat__int__comparison_I1_J,axiom,
( ( ^ [Y5: nat,Z: nat] : ( Y5 = Z ) )
= ( ^ [A5: nat,B4: nat] :
( ( semiri1314217659103216013at_int @ A5 )
= ( semiri1314217659103216013at_int @ B4 ) ) ) ) ).
% nat_int_comparison(1)
thf(fact_962_int__if,axiom,
! [P2: $o,A: nat,B: nat] :
( ( P2
=> ( ( semiri1314217659103216013at_int @ ( if_nat @ P2 @ A @ B ) )
= ( semiri1314217659103216013at_int @ A ) ) )
& ( ~ P2
=> ( ( semiri1314217659103216013at_int @ ( if_nat @ P2 @ A @ B ) )
= ( semiri1314217659103216013at_int @ B ) ) ) ) ).
% int_if
thf(fact_963_valid__step__non__transitive,axiom,
! [S_i: product_prod_int_int,S_j2: product_prod_int_int,S_k: product_prod_int_int] :
( ( valid_step @ S_i @ S_j2 )
=> ( ( valid_step @ S_j2 @ S_k )
=> ~ ( valid_step @ S_i @ S_k ) ) ) ).
% valid_step_non_transitive
thf(fact_964_valid__step__rev,axiom,
! [S_i: product_prod_int_int,S_j2: product_prod_int_int] :
( ( valid_step @ S_i @ S_j2 )
=> ( valid_step @ S_j2 @ S_i ) ) ).
% valid_step_rev
thf(fact_965_valid__step__neq,axiom,
! [S_i: product_prod_int_int,S_j2: product_prod_int_int] :
( ( valid_step @ S_i @ S_j2 )
=> ( S_i != S_j2 ) ) ).
% valid_step_neq
thf(fact_966_accp_Ocases,axiom,
! [R: produc4058024888802116461nt_int > produc4058024888802116461nt_int > $o,A: produc4058024888802116461nt_int] :
( ( accp_P6701151336149146870nt_int @ R @ A )
=> ! [Y7: produc4058024888802116461nt_int] :
( ( R @ Y7 @ A )
=> ( accp_P6701151336149146870nt_int @ R @ Y7 ) ) ) ).
% accp.cases
thf(fact_967_accp_Ocases,axiom,
! [R: product_prod_int_int > product_prod_int_int > $o,A: product_prod_int_int] :
( ( accp_P1096762738010456898nt_int @ R @ A )
=> ! [Y7: product_prod_int_int] :
( ( R @ Y7 @ A )
=> ( accp_P1096762738010456898nt_int @ R @ Y7 ) ) ) ).
% accp.cases
thf(fact_968_accp_Osimps,axiom,
( accp_P6701151336149146870nt_int
= ( ^ [R3: produc4058024888802116461nt_int > produc4058024888802116461nt_int > $o,A5: produc4058024888802116461nt_int] :
? [X4: produc4058024888802116461nt_int] :
( ( A5 = X4 )
& ! [Y6: produc4058024888802116461nt_int] :
( ( R3 @ Y6 @ X4 )
=> ( accp_P6701151336149146870nt_int @ R3 @ Y6 ) ) ) ) ) ).
% accp.simps
thf(fact_969_accp_Osimps,axiom,
( accp_P1096762738010456898nt_int
= ( ^ [R3: product_prod_int_int > product_prod_int_int > $o,A5: product_prod_int_int] :
? [X4: product_prod_int_int] :
( ( A5 = X4 )
& ! [Y6: product_prod_int_int] :
( ( R3 @ Y6 @ X4 )
=> ( accp_P1096762738010456898nt_int @ R3 @ Y6 ) ) ) ) ) ).
% accp.simps
thf(fact_970_accpI,axiom,
! [R: produc4058024888802116461nt_int > produc4058024888802116461nt_int > $o,X: produc4058024888802116461nt_int] :
( ! [Y3: produc4058024888802116461nt_int] :
( ( R @ Y3 @ X )
=> ( accp_P6701151336149146870nt_int @ R @ Y3 ) )
=> ( accp_P6701151336149146870nt_int @ R @ X ) ) ).
% accpI
thf(fact_971_accpI,axiom,
! [R: product_prod_int_int > product_prod_int_int > $o,X: product_prod_int_int] :
( ! [Y3: product_prod_int_int] :
( ( R @ Y3 @ X )
=> ( accp_P1096762738010456898nt_int @ R @ Y3 ) )
=> ( accp_P1096762738010456898nt_int @ R @ X ) ) ).
% accpI
thf(fact_972_accp__induct,axiom,
! [R: produc4058024888802116461nt_int > produc4058024888802116461nt_int > $o,A: produc4058024888802116461nt_int,P2: produc4058024888802116461nt_int > $o] :
( ( accp_P6701151336149146870nt_int @ R @ A )
=> ( ! [X3: produc4058024888802116461nt_int] :
( ( accp_P6701151336149146870nt_int @ R @ X3 )
=> ( ! [Y7: produc4058024888802116461nt_int] :
( ( R @ Y7 @ X3 )
=> ( P2 @ Y7 ) )
=> ( P2 @ X3 ) ) )
=> ( P2 @ A ) ) ) ).
% accp_induct
thf(fact_973_accp__induct,axiom,
! [R: product_prod_int_int > product_prod_int_int > $o,A: product_prod_int_int,P2: product_prod_int_int > $o] :
( ( accp_P1096762738010456898nt_int @ R @ A )
=> ( ! [X3: product_prod_int_int] :
( ( accp_P1096762738010456898nt_int @ R @ X3 )
=> ( ! [Y7: product_prod_int_int] :
( ( R @ Y7 @ X3 )
=> ( P2 @ Y7 ) )
=> ( P2 @ X3 ) ) )
=> ( P2 @ A ) ) ) ).
% accp_induct
thf(fact_974_accp__downward,axiom,
! [R: produc4058024888802116461nt_int > produc4058024888802116461nt_int > $o,B: produc4058024888802116461nt_int,A: produc4058024888802116461nt_int] :
( ( accp_P6701151336149146870nt_int @ R @ B )
=> ( ( R @ A @ B )
=> ( accp_P6701151336149146870nt_int @ R @ A ) ) ) ).
% accp_downward
thf(fact_975_accp__downward,axiom,
! [R: product_prod_int_int > product_prod_int_int > $o,B: product_prod_int_int,A: product_prod_int_int] :
( ( accp_P1096762738010456898nt_int @ R @ B )
=> ( ( R @ A @ B )
=> ( accp_P1096762738010456898nt_int @ R @ A ) ) ) ).
% accp_downward
thf(fact_976_not__accp__down,axiom,
! [R2: produc4058024888802116461nt_int > produc4058024888802116461nt_int > $o,X: produc4058024888802116461nt_int] :
( ~ ( accp_P6701151336149146870nt_int @ R2 @ X )
=> ~ ! [Z2: produc4058024888802116461nt_int] :
( ( R2 @ Z2 @ X )
=> ( accp_P6701151336149146870nt_int @ R2 @ Z2 ) ) ) ).
% not_accp_down
thf(fact_977_not__accp__down,axiom,
! [R2: product_prod_int_int > product_prod_int_int > $o,X: product_prod_int_int] :
( ~ ( accp_P1096762738010456898nt_int @ R2 @ X )
=> ~ ! [Z2: product_prod_int_int] :
( ( R2 @ Z2 @ X )
=> ( accp_P1096762738010456898nt_int @ R2 @ Z2 ) ) ) ).
% not_accp_down
thf(fact_978_accp__induct__rule,axiom,
! [R: produc4058024888802116461nt_int > produc4058024888802116461nt_int > $o,A: produc4058024888802116461nt_int,P2: produc4058024888802116461nt_int > $o] :
( ( accp_P6701151336149146870nt_int @ R @ A )
=> ( ! [X3: produc4058024888802116461nt_int] :
( ( accp_P6701151336149146870nt_int @ R @ X3 )
=> ( ! [Y7: produc4058024888802116461nt_int] :
( ( R @ Y7 @ X3 )
=> ( P2 @ Y7 ) )
=> ( P2 @ X3 ) ) )
=> ( P2 @ A ) ) ) ).
% accp_induct_rule
thf(fact_979_accp__induct__rule,axiom,
! [R: product_prod_int_int > product_prod_int_int > $o,A: product_prod_int_int,P2: product_prod_int_int > $o] :
( ( accp_P1096762738010456898nt_int @ R @ A )
=> ( ! [X3: product_prod_int_int] :
( ( accp_P1096762738010456898nt_int @ R @ X3 )
=> ( ! [Y7: product_prod_int_int] :
( ( R @ Y7 @ X3 )
=> ( P2 @ Y7 ) )
=> ( P2 @ X3 ) ) )
=> ( P2 @ A ) ) ) ).
% accp_induct_rule
thf(fact_980_of__nat__mono,axiom,
! [I4: nat,J3: nat] :
( ( ord_less_eq_nat @ I4 @ J3 )
=> ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ I4 ) @ ( semiri1316708129612266289at_nat @ J3 ) ) ) ).
% of_nat_mono
thf(fact_981_of__nat__mono,axiom,
! [I4: nat,J3: nat] :
( ( ord_less_eq_nat @ I4 @ J3 )
=> ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ I4 ) @ ( semiri1314217659103216013at_int @ J3 ) ) ) ).
% of_nat_mono
thf(fact_982_of__nat__less__imp__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) )
=> ( ord_less_nat @ M @ N ) ) ).
% of_nat_less_imp_less
thf(fact_983_of__nat__less__imp__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
=> ( ord_less_nat @ M @ N ) ) ).
% of_nat_less_imp_less
thf(fact_984_less__imp__of__nat__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% less_imp_of_nat_less
thf(fact_985_less__imp__of__nat__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% less_imp_of_nat_less
thf(fact_986_nat__int__comparison_I3_J,axiom,
( ord_less_eq_nat
= ( ^ [A5: nat,B4: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A5 ) @ ( semiri1314217659103216013at_int @ B4 ) ) ) ) ).
% nat_int_comparison(3)
thf(fact_987_zle__int,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% zle_int
thf(fact_988_nat__int__comparison_I2_J,axiom,
( ord_less_nat
= ( ^ [A5: nat,B4: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A5 ) @ ( semiri1314217659103216013at_int @ B4 ) ) ) ) ).
% nat_int_comparison(2)
thf(fact_989_int__plus,axiom,
! [N: nat,M: nat] :
( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ N @ M ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ ( semiri1314217659103216013at_int @ M ) ) ) ).
% int_plus
thf(fact_990_int__ops_I5_J,axiom,
! [A: nat,B: nat] :
( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ A @ B ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ).
% int_ops(5)
thf(fact_991_zadd__int__left,axiom,
! [M: nat,N: nat,Z3: int] :
( ( plus_plus_int @ ( semiri1314217659103216013at_int @ M ) @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ Z3 ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ ( plus_plus_nat @ M @ N ) ) @ Z3 ) ) ).
% zadd_int_left
thf(fact_992_zle__iff__zadd,axiom,
( ord_less_eq_int
= ( ^ [W2: int,Z5: int] :
? [N2: nat] :
( Z5
= ( plus_plus_int @ W2 @ ( semiri1314217659103216013at_int @ N2 ) ) ) ) ) ).
% zle_iff_zadd
thf(fact_993_int__ops_I2_J,axiom,
( ( semiri1314217659103216013at_int @ one_one_nat )
= one_one_int ) ).
% int_ops(2)
thf(fact_994_of__nat__min,axiom,
! [X: nat,Y: nat] :
( ( semiri1316708129612266289at_nat @ ( ord_min_nat @ X @ Y ) )
= ( ord_min_nat @ ( semiri1316708129612266289at_nat @ X ) @ ( semiri1316708129612266289at_nat @ Y ) ) ) ).
% of_nat_min
thf(fact_995_of__nat__min,axiom,
! [X: nat,Y: nat] :
( ( semiri1314217659103216013at_int @ ( ord_min_nat @ X @ Y ) )
= ( ord_min_int @ ( semiri1314217659103216013at_int @ X ) @ ( semiri1314217659103216013at_int @ Y ) ) ) ).
% of_nat_min
thf(fact_996_trans__valid__step,axiom,
! [I4: int,J3: int,I6: int,J4: int,K_12: int,K_22: int] :
( ( valid_step @ ( product_Pair_int_int @ I4 @ J3 ) @ ( product_Pair_int_int @ I6 @ J4 ) )
=> ( valid_step @ ( product_Pair_int_int @ ( plus_plus_int @ I4 @ K_12 ) @ ( plus_plus_int @ J3 @ K_22 ) ) @ ( product_Pair_int_int @ ( plus_plus_int @ I6 @ K_12 ) @ ( plus_plus_int @ J4 @ K_22 ) ) ) ) ).
% trans_valid_step
thf(fact_997_int__Suc,axiom,
! [N: nat] :
( ( semiri1314217659103216013at_int @ ( suc @ N ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) ) ).
% int_Suc
thf(fact_998_int__ops_I4_J,axiom,
! [A: nat] :
( ( semiri1314217659103216013at_int @ ( suc @ A ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ A ) @ one_one_int ) ) ).
% int_ops(4)
thf(fact_999_zless__iff__Suc__zadd,axiom,
( ord_less_int
= ( ^ [W2: int,Z5: int] :
? [N2: nat] :
( Z5
= ( plus_plus_int @ W2 @ ( semiri1314217659103216013at_int @ ( suc @ N2 ) ) ) ) ) ) ).
% zless_iff_Suc_zadd
thf(fact_1000_board__exec__leq,axiom,
! [I4: int,J3: int,N: nat,M: nat] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ I4 @ J3 ) @ ( board_exec @ N @ M ) )
= ( ( ord_less_eq_int @ one_one_int @ I4 )
& ( ord_less_eq_int @ I4 @ ( semiri1314217659103216013at_int @ N ) )
& ( ord_less_eq_int @ one_one_int @ J3 )
& ( ord_less_eq_int @ J3 @ ( semiri1314217659103216013at_int @ M ) ) ) ) ).
% board_exec_leq
thf(fact_1001_board__exec__aux__leq__mem,axiom,
! [I4: int,J3: int,K: nat,M7: set_int] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ I4 @ J3 ) @ ( board_exec_aux @ K @ M7 ) )
= ( ( ord_less_eq_int @ one_one_int @ I4 )
& ( ord_less_eq_int @ I4 @ ( semiri1314217659103216013at_int @ K ) )
& ( member_int @ J3 @ M7 ) ) ) ).
% board_exec_aux_leq_mem
thf(fact_1002_butlast__take,axiom,
! [N: nat,Xs: list_P5707943133018811711nt_int] :
( ( ord_less_eq_nat @ N @ ( size_s5157815400016825771nt_int @ Xs ) )
=> ( ( butlas2390654596765537291nt_int @ ( take_P8218740963776755879nt_int @ N @ Xs ) )
= ( take_P8218740963776755879nt_int @ ( minus_minus_nat @ N @ one_one_nat ) @ Xs ) ) ) ).
% butlast_take
thf(fact_1003_add__diff__cancel__right_H,axiom,
! [A: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ B )
= A ) ).
% add_diff_cancel_right'
thf(fact_1004_add__diff__cancel__right_H,axiom,
! [A: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ B )
= A ) ).
% add_diff_cancel_right'
thf(fact_1005_add__diff__cancel__right,axiom,
! [A: nat,C2: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ C2 ) )
= ( minus_minus_nat @ A @ B ) ) ).
% add_diff_cancel_right
thf(fact_1006_add__diff__cancel__right,axiom,
! [A: int,C2: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ A @ C2 ) @ ( plus_plus_int @ B @ C2 ) )
= ( minus_minus_int @ A @ B ) ) ).
% add_diff_cancel_right
thf(fact_1007_add__diff__cancel__left_H,axiom,
! [A: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ A )
= B ) ).
% add_diff_cancel_left'
thf(fact_1008_add__diff__cancel__left_H,axiom,
! [A: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ A )
= B ) ).
% add_diff_cancel_left'
thf(fact_1009_add__diff__cancel__left,axiom,
! [C2: nat,A: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ C2 @ A ) @ ( plus_plus_nat @ C2 @ B ) )
= ( minus_minus_nat @ A @ B ) ) ).
% add_diff_cancel_left
thf(fact_1010_add__diff__cancel__left,axiom,
! [C2: int,A: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ C2 @ A ) @ ( plus_plus_int @ C2 @ B ) )
= ( minus_minus_int @ A @ B ) ) ).
% add_diff_cancel_left
thf(fact_1011_diff__add__cancel,axiom,
! [A: int,B: int] :
( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ B )
= A ) ).
% diff_add_cancel
thf(fact_1012_add__diff__cancel,axiom,
! [A: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ B )
= A ) ).
% add_diff_cancel
thf(fact_1013_diff__Suc__Suc,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ ( suc @ M ) @ ( suc @ N ) )
= ( minus_minus_nat @ M @ N ) ) ).
% diff_Suc_Suc
thf(fact_1014_Suc__diff__diff,axiom,
! [M: nat,N: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ ( suc @ M ) @ N ) @ ( suc @ K ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M @ N ) @ K ) ) ).
% Suc_diff_diff
thf(fact_1015_diff__diff__cancel,axiom,
! [I4: nat,N: nat] :
( ( ord_less_eq_nat @ I4 @ N )
=> ( ( minus_minus_nat @ N @ ( minus_minus_nat @ N @ I4 ) )
= I4 ) ) ).
% diff_diff_cancel
thf(fact_1016_diff__diff__left,axiom,
! [I4: nat,J3: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I4 @ J3 ) @ K )
= ( minus_minus_nat @ I4 @ ( plus_plus_nat @ J3 @ K ) ) ) ).
% diff_diff_left
thf(fact_1017_le__add__diff__inverse,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( plus_plus_nat @ B @ ( minus_minus_nat @ A @ B ) )
= A ) ) ).
% le_add_diff_inverse
thf(fact_1018_le__add__diff__inverse,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( plus_plus_int @ B @ ( minus_minus_int @ A @ B ) )
= A ) ) ).
% le_add_diff_inverse
thf(fact_1019_le__add__diff__inverse2,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ A @ B ) @ B )
= A ) ) ).
% le_add_diff_inverse2
thf(fact_1020_le__add__diff__inverse2,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ B )
= A ) ) ).
% le_add_diff_inverse2
thf(fact_1021_Nat_Odiff__diff__right,axiom,
! [K: nat,J3: nat,I4: nat] :
( ( ord_less_eq_nat @ K @ J3 )
=> ( ( minus_minus_nat @ I4 @ ( minus_minus_nat @ J3 @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I4 @ K ) @ J3 ) ) ) ).
% Nat.diff_diff_right
thf(fact_1022_Nat_Oadd__diff__assoc2,axiom,
! [K: nat,J3: nat,I4: nat] :
( ( ord_less_eq_nat @ K @ J3 )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ J3 @ K ) @ I4 )
= ( minus_minus_nat @ ( plus_plus_nat @ J3 @ I4 ) @ K ) ) ) ).
% Nat.add_diff_assoc2
thf(fact_1023_Nat_Oadd__diff__assoc,axiom,
! [K: nat,J3: nat,I4: nat] :
( ( ord_less_eq_nat @ K @ J3 )
=> ( ( plus_plus_nat @ I4 @ ( minus_minus_nat @ J3 @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I4 @ J3 ) @ K ) ) ) ).
% Nat.add_diff_assoc
thf(fact_1024_diff__Suc__1,axiom,
! [N: nat] :
( ( minus_minus_nat @ ( suc @ N ) @ one_one_nat )
= N ) ).
% diff_Suc_1
thf(fact_1025_length__drop,axiom,
! [N: nat,Xs: list_P5707943133018811711nt_int] :
( ( size_s5157815400016825771nt_int @ ( drop_P5690361596310759935nt_int @ N @ Xs ) )
= ( minus_minus_nat @ ( size_s5157815400016825771nt_int @ Xs ) @ N ) ) ).
% length_drop
thf(fact_1026_diff__Suc__diff__eq1,axiom,
! [K: nat,J3: nat,I4: nat] :
( ( ord_less_eq_nat @ K @ J3 )
=> ( ( minus_minus_nat @ I4 @ ( suc @ ( minus_minus_nat @ J3 @ K ) ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I4 @ K ) @ ( suc @ J3 ) ) ) ) ).
% diff_Suc_diff_eq1
thf(fact_1027_diff__Suc__diff__eq2,axiom,
! [K: nat,J3: nat,I4: nat] :
( ( ord_less_eq_nat @ K @ J3 )
=> ( ( minus_minus_nat @ ( suc @ ( minus_minus_nat @ J3 @ K ) ) @ I4 )
= ( minus_minus_nat @ ( suc @ J3 ) @ ( plus_plus_nat @ K @ I4 ) ) ) ) ).
% diff_Suc_diff_eq2
thf(fact_1028_take__append,axiom,
! [N: nat,Xs: list_P5707943133018811711nt_int,Ys2: list_P5707943133018811711nt_int] :
( ( take_P8218740963776755879nt_int @ N @ ( append7030698103840186580nt_int @ Xs @ Ys2 ) )
= ( append7030698103840186580nt_int @ ( take_P8218740963776755879nt_int @ N @ Xs ) @ ( take_P8218740963776755879nt_int @ ( minus_minus_nat @ N @ ( size_s5157815400016825771nt_int @ Xs ) ) @ Ys2 ) ) ) ).
% take_append
thf(fact_1029_drop__append,axiom,
! [N: nat,Xs: list_P5707943133018811711nt_int,Ys2: list_P5707943133018811711nt_int] :
( ( drop_P5690361596310759935nt_int @ N @ ( append7030698103840186580nt_int @ Xs @ Ys2 ) )
= ( append7030698103840186580nt_int @ ( drop_P5690361596310759935nt_int @ N @ Xs ) @ ( drop_P5690361596310759935nt_int @ ( minus_minus_nat @ N @ ( size_s5157815400016825771nt_int @ Xs ) ) @ Ys2 ) ) ) ).
% drop_append
thf(fact_1030_length__tl,axiom,
! [Xs: list_P5707943133018811711nt_int] :
( ( size_s5157815400016825771nt_int @ ( tl_Pro1049539609081552168nt_int @ Xs ) )
= ( minus_minus_nat @ ( size_s5157815400016825771nt_int @ Xs ) @ one_one_nat ) ) ).
% length_tl
thf(fact_1031_length__butlast,axiom,
! [Xs: list_P5707943133018811711nt_int] :
( ( size_s5157815400016825771nt_int @ ( butlas2390654596765537291nt_int @ Xs ) )
= ( minus_minus_nat @ ( size_s5157815400016825771nt_int @ Xs ) @ one_one_nat ) ) ).
% length_butlast
thf(fact_1032_of__nat__diff,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ N @ M )
=> ( ( semiri1316708129612266289at_nat @ ( minus_minus_nat @ M @ N ) )
= ( minus_minus_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ) ).
% of_nat_diff
thf(fact_1033_of__nat__diff,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ N @ M )
=> ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ M @ N ) )
= ( minus_minus_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ) ).
% of_nat_diff
thf(fact_1034_add__le__add__imp__diff__le,axiom,
! [I4: nat,K: nat,N: nat,J3: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ I4 @ K ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( plus_plus_nat @ J3 @ K ) )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ I4 @ K ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( plus_plus_nat @ J3 @ K ) )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ N @ K ) @ J3 ) ) ) ) ) ).
% add_le_add_imp_diff_le
thf(fact_1035_add__le__add__imp__diff__le,axiom,
! [I4: int,K: int,N: int,J3: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ I4 @ K ) @ N )
=> ( ( ord_less_eq_int @ N @ ( plus_plus_int @ J3 @ K ) )
=> ( ( ord_less_eq_int @ ( plus_plus_int @ I4 @ K ) @ N )
=> ( ( ord_less_eq_int @ N @ ( plus_plus_int @ J3 @ K ) )
=> ( ord_less_eq_int @ ( minus_minus_int @ N @ K ) @ J3 ) ) ) ) ) ).
% add_le_add_imp_diff_le
thf(fact_1036_add__le__imp__le__diff,axiom,
! [I4: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ I4 @ K ) @ N )
=> ( ord_less_eq_nat @ I4 @ ( minus_minus_nat @ N @ K ) ) ) ).
% add_le_imp_le_diff
thf(fact_1037_add__le__imp__le__diff,axiom,
! [I4: int,K: int,N: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ I4 @ K ) @ N )
=> ( ord_less_eq_int @ I4 @ ( minus_minus_int @ N @ K ) ) ) ).
% add_le_imp_le_diff
thf(fact_1038_diff__le__eq,axiom,
! [A: int,B: int,C2: int] :
( ( ord_less_eq_int @ ( minus_minus_int @ A @ B ) @ C2 )
= ( ord_less_eq_int @ A @ ( plus_plus_int @ C2 @ B ) ) ) ).
% diff_le_eq
thf(fact_1039_le__diff__eq,axiom,
! [A: int,C2: int,B: int] :
( ( ord_less_eq_int @ A @ ( minus_minus_int @ C2 @ B ) )
= ( ord_less_eq_int @ ( plus_plus_int @ A @ B ) @ C2 ) ) ).
% le_diff_eq
thf(fact_1040_diff__add,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ A )
= B ) ) ).
% diff_add
thf(fact_1041_le__add__diff,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ C2 @ ( minus_minus_nat @ ( plus_plus_nat @ B @ C2 ) @ A ) ) ) ).
% le_add_diff
thf(fact_1042_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C2 @ ( minus_minus_nat @ B @ A ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ C2 @ A ) @ B ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_diff_conv2
thf(fact_1043_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ C2 @ ( minus_minus_nat @ B @ A ) )
= ( minus_minus_nat @ ( plus_plus_nat @ C2 @ B ) @ A ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc
thf(fact_1044_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ C2 @ B ) @ A )
= ( plus_plus_nat @ C2 @ ( minus_minus_nat @ B @ A ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc
thf(fact_1045_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ C2 )
= ( minus_minus_nat @ ( plus_plus_nat @ B @ C2 ) @ A ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
thf(fact_1046_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ B @ C2 ) @ A )
= ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ C2 ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
thf(fact_1047_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( minus_minus_nat @ C2 @ ( minus_minus_nat @ B @ A ) )
= ( minus_minus_nat @ ( plus_plus_nat @ C2 @ A ) @ B ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_diff_right
thf(fact_1048_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ A @ ( minus_minus_nat @ B @ A ) )
= B ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_inverse
thf(fact_1049_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( ( ( minus_minus_nat @ B @ A )
= C2 )
= ( B
= ( plus_plus_nat @ C2 @ A ) ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add
thf(fact_1050_linordered__semidom__class_Oadd__diff__inverse,axiom,
! [A: nat,B: nat] :
( ~ ( ord_less_nat @ A @ B )
=> ( ( plus_plus_nat @ B @ ( minus_minus_nat @ A @ B ) )
= A ) ) ).
% linordered_semidom_class.add_diff_inverse
thf(fact_1051_linordered__semidom__class_Oadd__diff__inverse,axiom,
! [A: int,B: int] :
( ~ ( ord_less_int @ A @ B )
=> ( ( plus_plus_int @ B @ ( minus_minus_int @ A @ B ) )
= A ) ) ).
% linordered_semidom_class.add_diff_inverse
thf(fact_1052_less__diff__eq,axiom,
! [A: int,C2: int,B: int] :
( ( ord_less_int @ A @ ( minus_minus_int @ C2 @ B ) )
= ( ord_less_int @ ( plus_plus_int @ A @ B ) @ C2 ) ) ).
% less_diff_eq
thf(fact_1053_diff__less__eq,axiom,
! [A: int,B: int,C2: int] :
( ( ord_less_int @ ( minus_minus_int @ A @ B ) @ C2 )
= ( ord_less_int @ A @ ( plus_plus_int @ C2 @ B ) ) ) ).
% diff_less_eq
thf(fact_1054_diff__add__inverse2,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ N )
= M ) ).
% diff_add_inverse2
thf(fact_1055_diff__add__inverse,axiom,
! [N: nat,M: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ N @ M ) @ N )
= M ) ).
% diff_add_inverse
thf(fact_1056_diff__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N @ K ) )
= ( minus_minus_nat @ M @ N ) ) ).
% diff_cancel2
thf(fact_1057_Nat_Odiff__cancel,axiom,
! [K: nat,M: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
= ( minus_minus_nat @ M @ N ) ) ).
% Nat.diff_cancel
thf(fact_1058_diff__diff__eq,axiom,
! [A: nat,B: nat,C2: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C2 )
= ( minus_minus_nat @ A @ ( plus_plus_nat @ B @ C2 ) ) ) ).
% diff_diff_eq
thf(fact_1059_diff__diff__eq,axiom,
! [A: int,B: int,C2: int] :
( ( minus_minus_int @ ( minus_minus_int @ A @ B ) @ C2 )
= ( minus_minus_int @ A @ ( plus_plus_int @ B @ C2 ) ) ) ).
% diff_diff_eq
thf(fact_1060_add__implies__diff,axiom,
! [C2: nat,B: nat,A: nat] :
( ( ( plus_plus_nat @ C2 @ B )
= A )
=> ( C2
= ( minus_minus_nat @ A @ B ) ) ) ).
% add_implies_diff
thf(fact_1061_add__implies__diff,axiom,
! [C2: int,B: int,A: int] :
( ( ( plus_plus_int @ C2 @ B )
= A )
=> ( C2
= ( minus_minus_int @ A @ B ) ) ) ).
% add_implies_diff
thf(fact_1062_diff__add__eq__diff__diff__swap,axiom,
! [A: int,B: int,C2: int] :
( ( minus_minus_int @ A @ ( plus_plus_int @ B @ C2 ) )
= ( minus_minus_int @ ( minus_minus_int @ A @ C2 ) @ B ) ) ).
% diff_add_eq_diff_diff_swap
thf(fact_1063_diff__add__eq,axiom,
! [A: int,B: int,C2: int] :
( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ C2 )
= ( minus_minus_int @ ( plus_plus_int @ A @ C2 ) @ B ) ) ).
% diff_add_eq
thf(fact_1064_diff__diff__eq2,axiom,
! [A: int,B: int,C2: int] :
( ( minus_minus_int @ A @ ( minus_minus_int @ B @ C2 ) )
= ( minus_minus_int @ ( plus_plus_int @ A @ C2 ) @ B ) ) ).
% diff_diff_eq2
thf(fact_1065_add__diff__eq,axiom,
! [A: int,B: int,C2: int] :
( ( plus_plus_int @ A @ ( minus_minus_int @ B @ C2 ) )
= ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ C2 ) ) ).
% add_diff_eq
thf(fact_1066_eq__diff__eq,axiom,
! [A: int,C2: int,B: int] :
( ( A
= ( minus_minus_int @ C2 @ B ) )
= ( ( plus_plus_int @ A @ B )
= C2 ) ) ).
% eq_diff_eq
thf(fact_1067_diff__eq__eq,axiom,
! [A: int,B: int,C2: int] :
( ( ( minus_minus_int @ A @ B )
= C2 )
= ( A
= ( plus_plus_int @ C2 @ B ) ) ) ).
% diff_eq_eq
thf(fact_1068_group__cancel_Osub1,axiom,
! [A4: int,K: int,A: int,B: int] :
( ( A4
= ( plus_plus_int @ K @ A ) )
=> ( ( minus_minus_int @ A4 @ B )
= ( plus_plus_int @ K @ ( minus_minus_int @ A @ B ) ) ) ) ).
% group_cancel.sub1
thf(fact_1069_min__diff__distrib__left,axiom,
! [X: int,Y: int,Z3: int] :
( ( minus_minus_int @ ( ord_min_int @ X @ Y ) @ Z3 )
= ( ord_min_int @ ( minus_minus_int @ X @ Z3 ) @ ( minus_minus_int @ Y @ Z3 ) ) ) ).
% min_diff_distrib_left
thf(fact_1070_min__diff,axiom,
! [M: nat,I4: nat,N: nat] :
( ( ord_min_nat @ ( minus_minus_nat @ M @ I4 ) @ ( minus_minus_nat @ N @ I4 ) )
= ( minus_minus_nat @ ( ord_min_nat @ M @ N ) @ I4 ) ) ).
% min_diff
thf(fact_1071_zero__induct__lemma,axiom,
! [P2: nat > $o,K: nat,I4: nat] :
( ( P2 @ K )
=> ( ! [N3: nat] :
( ( P2 @ ( suc @ N3 ) )
=> ( P2 @ N3 ) )
=> ( P2 @ ( minus_minus_nat @ K @ I4 ) ) ) ) ).
% zero_induct_lemma
thf(fact_1072_diff__commute,axiom,
! [I4: nat,J3: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I4 @ J3 ) @ K )
= ( minus_minus_nat @ ( minus_minus_nat @ I4 @ K ) @ J3 ) ) ).
% diff_commute
thf(fact_1073_diff__right__commute,axiom,
! [A: nat,C2: nat,B: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A @ C2 ) @ B )
= ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C2 ) ) ).
% diff_right_commute
thf(fact_1074_diff__right__commute,axiom,
! [A: int,C2: int,B: int] :
( ( minus_minus_int @ ( minus_minus_int @ A @ C2 ) @ B )
= ( minus_minus_int @ ( minus_minus_int @ A @ B ) @ C2 ) ) ).
% diff_right_commute
thf(fact_1075_diff__eq__diff__eq,axiom,
! [A: int,B: int,C2: int,D3: int] :
( ( ( minus_minus_int @ A @ B )
= ( minus_minus_int @ C2 @ D3 ) )
=> ( ( A = B )
= ( C2 = D3 ) ) ) ).
% diff_eq_diff_eq
thf(fact_1076_step__in__Cons,axiom,
! [Ps: list_P5707943133018811711nt_int,S_i: product_prod_int_int,S_j2: product_prod_int_int,S_k: product_prod_int_int] :
( ( step_in @ Ps @ S_i @ S_j2 )
=> ( step_in @ ( cons_P3334398858971670639nt_int @ S_k @ Ps ) @ S_i @ S_j2 ) ) ).
% step_in_Cons
thf(fact_1077_eq__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ( minus_minus_nat @ M @ K )
= ( minus_minus_nat @ N @ K ) )
= ( M = N ) ) ) ) ).
% eq_diff_iff
thf(fact_1078_le__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( ord_less_eq_nat @ M @ N ) ) ) ) ).
% le_diff_iff
thf(fact_1079_Nat_Odiff__diff__eq,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( minus_minus_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( minus_minus_nat @ M @ N ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_1080_diff__le__mono,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ M @ L ) @ ( minus_minus_nat @ N @ L ) ) ) ).
% diff_le_mono
thf(fact_1081_diff__le__self,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ M ) ).
% diff_le_self
thf(fact_1082_le__diff__iff_H,axiom,
! [A: nat,C2: nat,B: nat] :
( ( ord_less_eq_nat @ A @ C2 )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ C2 @ A ) @ ( minus_minus_nat @ C2 @ B ) )
= ( ord_less_eq_nat @ B @ A ) ) ) ) ).
% le_diff_iff'
thf(fact_1083_diff__le__mono2,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ).
% diff_le_mono2
thf(fact_1084_diff__less__mono2,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_nat @ M @ N )
=> ( ( ord_less_nat @ M @ L )
=> ( ord_less_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ) ).
% diff_less_mono2
thf(fact_1085_less__imp__diff__less,axiom,
! [J3: nat,K: nat,N: nat] :
( ( ord_less_nat @ J3 @ K )
=> ( ord_less_nat @ ( minus_minus_nat @ J3 @ N ) @ K ) ) ).
% less_imp_diff_less
thf(fact_1086_diff__strict__mono,axiom,
! [A: int,B: int,D3: int,C2: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ D3 @ C2 )
=> ( ord_less_int @ ( minus_minus_int @ A @ C2 ) @ ( minus_minus_int @ B @ D3 ) ) ) ) ).
% diff_strict_mono
thf(fact_1087_diff__eq__diff__less,axiom,
! [A: int,B: int,C2: int,D3: int] :
( ( ( minus_minus_int @ A @ B )
= ( minus_minus_int @ C2 @ D3 ) )
=> ( ( ord_less_int @ A @ B )
= ( ord_less_int @ C2 @ D3 ) ) ) ).
% diff_eq_diff_less
thf(fact_1088_diff__strict__left__mono,axiom,
! [B: int,A: int,C2: int] :
( ( ord_less_int @ B @ A )
=> ( ord_less_int @ ( minus_minus_int @ C2 @ A ) @ ( minus_minus_int @ C2 @ B ) ) ) ).
% diff_strict_left_mono
thf(fact_1089_diff__strict__right__mono,axiom,
! [A: int,B: int,C2: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_int @ ( minus_minus_int @ A @ C2 ) @ ( minus_minus_int @ B @ C2 ) ) ) ).
% diff_strict_right_mono
thf(fact_1090_diff__eq__diff__less__eq,axiom,
! [A: int,B: int,C2: int,D3: int] :
( ( ( minus_minus_int @ A @ B )
= ( minus_minus_int @ C2 @ D3 ) )
=> ( ( ord_less_eq_int @ A @ B )
= ( ord_less_eq_int @ C2 @ D3 ) ) ) ).
% diff_eq_diff_less_eq
thf(fact_1091_diff__right__mono,axiom,
! [A: int,B: int,C2: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ord_less_eq_int @ ( minus_minus_int @ A @ C2 ) @ ( minus_minus_int @ B @ C2 ) ) ) ).
% diff_right_mono
thf(fact_1092_diff__left__mono,axiom,
! [B: int,A: int,C2: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ord_less_eq_int @ ( minus_minus_int @ C2 @ A ) @ ( minus_minus_int @ C2 @ B ) ) ) ).
% diff_left_mono
thf(fact_1093_diff__mono,axiom,
! [A: int,B: int,D3: int,C2: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ D3 @ C2 )
=> ( ord_less_eq_int @ ( minus_minus_int @ A @ C2 ) @ ( minus_minus_int @ B @ D3 ) ) ) ) ).
% diff_mono
thf(fact_1094_diff__less__Suc,axiom,
! [M: nat,N: nat] : ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ ( suc @ M ) ) ).
% diff_less_Suc
thf(fact_1095_Suc__diff__Suc,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
=> ( ( suc @ ( minus_minus_nat @ M @ ( suc @ N ) ) )
= ( minus_minus_nat @ M @ N ) ) ) ).
% Suc_diff_Suc
thf(fact_1096_Suc__diff__le,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ N @ M )
=> ( ( minus_minus_nat @ ( suc @ M ) @ N )
= ( suc @ ( minus_minus_nat @ M @ N ) ) ) ) ).
% Suc_diff_le
thf(fact_1097_less__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( ord_less_nat @ M @ N ) ) ) ) ).
% less_diff_iff
thf(fact_1098_diff__less__mono,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C2 @ A )
=> ( ord_less_nat @ ( minus_minus_nat @ A @ C2 ) @ ( minus_minus_nat @ B @ C2 ) ) ) ) ).
% diff_less_mono
thf(fact_1099_add__diff__inverse__nat,axiom,
! [M: nat,N: nat] :
( ~ ( ord_less_nat @ M @ N )
=> ( ( plus_plus_nat @ N @ ( minus_minus_nat @ M @ N ) )
= M ) ) ).
% add_diff_inverse_nat
thf(fact_1100_less__diff__conv,axiom,
! [I4: nat,J3: nat,K: nat] :
( ( ord_less_nat @ I4 @ ( minus_minus_nat @ J3 @ K ) )
= ( ord_less_nat @ ( plus_plus_nat @ I4 @ K ) @ J3 ) ) ).
% less_diff_conv
thf(fact_1101_le__diff__conv,axiom,
! [J3: nat,K: nat,I4: nat] :
( ( ord_less_eq_nat @ ( minus_minus_nat @ J3 @ K ) @ I4 )
= ( ord_less_eq_nat @ J3 @ ( plus_plus_nat @ I4 @ K ) ) ) ).
% le_diff_conv
thf(fact_1102_Nat_Ole__diff__conv2,axiom,
! [K: nat,J3: nat,I4: nat] :
( ( ord_less_eq_nat @ K @ J3 )
=> ( ( ord_less_eq_nat @ I4 @ ( minus_minus_nat @ J3 @ K ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ I4 @ K ) @ J3 ) ) ) ).
% Nat.le_diff_conv2
thf(fact_1103_Nat_Odiff__add__assoc,axiom,
! [K: nat,J3: nat,I4: nat] :
( ( ord_less_eq_nat @ K @ J3 )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ I4 @ J3 ) @ K )
= ( plus_plus_nat @ I4 @ ( minus_minus_nat @ J3 @ K ) ) ) ) ).
% Nat.diff_add_assoc
thf(fact_1104_Nat_Odiff__add__assoc2,axiom,
! [K: nat,J3: nat,I4: nat] :
( ( ord_less_eq_nat @ K @ J3 )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ J3 @ I4 ) @ K )
= ( plus_plus_nat @ ( minus_minus_nat @ J3 @ K ) @ I4 ) ) ) ).
% Nat.diff_add_assoc2
thf(fact_1105_Nat_Ole__imp__diff__is__add,axiom,
! [I4: nat,J3: nat,K: nat] :
( ( ord_less_eq_nat @ I4 @ J3 )
=> ( ( ( minus_minus_nat @ J3 @ I4 )
= K )
= ( J3
= ( plus_plus_nat @ K @ I4 ) ) ) ) ).
% Nat.le_imp_diff_is_add
thf(fact_1106_diff__Suc__eq__diff__pred,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ M @ ( suc @ N ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M @ one_one_nat ) @ N ) ) ).
% diff_Suc_eq_diff_pred
thf(fact_1107_drop__take,axiom,
! [N: nat,M: nat,Xs: list_P5707943133018811711nt_int] :
( ( drop_P5690361596310759935nt_int @ N @ ( take_P8218740963776755879nt_int @ M @ Xs ) )
= ( take_P8218740963776755879nt_int @ ( minus_minus_nat @ M @ N ) @ ( drop_P5690361596310759935nt_int @ N @ Xs ) ) ) ).
% drop_take
thf(fact_1108_step__in__prepend,axiom,
! [Ps: list_P5707943133018811711nt_int,S_i: product_prod_int_int,S_j2: product_prod_int_int,Ps5: list_P5707943133018811711nt_int] :
( ( step_in @ Ps @ S_i @ S_j2 )
=> ( step_in @ ( append7030698103840186580nt_int @ Ps5 @ Ps ) @ S_i @ S_j2 ) ) ).
% step_in_prepend
thf(fact_1109_step__in__append,axiom,
! [Ps: list_P5707943133018811711nt_int,S_i: product_prod_int_int,S_j2: product_prod_int_int,Ps5: list_P5707943133018811711nt_int] :
( ( step_in @ Ps @ S_i @ S_j2 )
=> ( step_in @ ( append7030698103840186580nt_int @ Ps @ Ps5 ) @ S_i @ S_j2 ) ) ).
% step_in_append
thf(fact_1110_less__diff__conv2,axiom,
! [K: nat,J3: nat,I4: nat] :
( ( ord_less_eq_nat @ K @ J3 )
=> ( ( ord_less_nat @ ( minus_minus_nat @ J3 @ K ) @ I4 )
= ( ord_less_nat @ J3 @ ( plus_plus_nat @ I4 @ K ) ) ) ) ).
% less_diff_conv2
thf(fact_1111_drop__update__swap,axiom,
! [M: nat,N: nat,Xs: list_P5707943133018811711nt_int,X: product_prod_int_int] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( drop_P5690361596310759935nt_int @ M @ ( list_u3002344382305578791nt_int @ Xs @ N @ X ) )
= ( list_u3002344382305578791nt_int @ ( drop_P5690361596310759935nt_int @ M @ Xs ) @ ( minus_minus_nat @ N @ M ) @ X ) ) ) ).
% drop_update_swap
thf(fact_1112_tl__take,axiom,
! [N: nat,Xs: list_P5707943133018811711nt_int] :
( ( tl_Pro1049539609081552168nt_int @ ( take_P8218740963776755879nt_int @ N @ Xs ) )
= ( take_P8218740963776755879nt_int @ ( minus_minus_nat @ N @ one_one_nat ) @ ( tl_Pro1049539609081552168nt_int @ Xs ) ) ) ).
% tl_take
thf(fact_1113_nth__append,axiom,
! [N: nat,Xs: list_P5707943133018811711nt_int,Ys2: list_P5707943133018811711nt_int] :
( ( ( ord_less_nat @ N @ ( size_s5157815400016825771nt_int @ Xs ) )
=> ( ( nth_Pr4439495888332055232nt_int @ ( append7030698103840186580nt_int @ Xs @ Ys2 ) @ N )
= ( nth_Pr4439495888332055232nt_int @ Xs @ N ) ) )
& ( ~ ( ord_less_nat @ N @ ( size_s5157815400016825771nt_int @ Xs ) )
=> ( ( nth_Pr4439495888332055232nt_int @ ( append7030698103840186580nt_int @ Xs @ Ys2 ) @ N )
= ( nth_Pr4439495888332055232nt_int @ Ys2 @ ( minus_minus_nat @ N @ ( size_s5157815400016825771nt_int @ Xs ) ) ) ) ) ) ).
% nth_append
thf(fact_1114_list__update__append,axiom,
! [N: nat,Xs: list_P5707943133018811711nt_int,Ys2: list_P5707943133018811711nt_int,X: product_prod_int_int] :
( ( ( ord_less_nat @ N @ ( size_s5157815400016825771nt_int @ Xs ) )
=> ( ( list_u3002344382305578791nt_int @ ( append7030698103840186580nt_int @ Xs @ Ys2 ) @ N @ X )
= ( append7030698103840186580nt_int @ ( list_u3002344382305578791nt_int @ Xs @ N @ X ) @ Ys2 ) ) )
& ( ~ ( ord_less_nat @ N @ ( size_s5157815400016825771nt_int @ Xs ) )
=> ( ( list_u3002344382305578791nt_int @ ( append7030698103840186580nt_int @ Xs @ Ys2 ) @ N @ X )
= ( append7030698103840186580nt_int @ Xs @ ( list_u3002344382305578791nt_int @ Ys2 @ ( minus_minus_nat @ N @ ( size_s5157815400016825771nt_int @ Xs ) ) @ X ) ) ) ) ) ).
% list_update_append
thf(fact_1115_butlast__conv__take,axiom,
( butlas2390654596765537291nt_int
= ( ^ [Xs4: list_P5707943133018811711nt_int] : ( take_P8218740963776755879nt_int @ ( minus_minus_nat @ ( size_s5157815400016825771nt_int @ Xs4 ) @ one_one_nat ) @ Xs4 ) ) ) ).
% butlast_conv_take
thf(fact_1116_butlast__list__update,axiom,
! [K: nat,Xs: list_P5707943133018811711nt_int,X: product_prod_int_int] :
( ( ( K
= ( minus_minus_nat @ ( size_s5157815400016825771nt_int @ Xs ) @ one_one_nat ) )
=> ( ( butlas2390654596765537291nt_int @ ( list_u3002344382305578791nt_int @ Xs @ K @ X ) )
= ( butlas2390654596765537291nt_int @ Xs ) ) )
& ( ( K
!= ( minus_minus_nat @ ( size_s5157815400016825771nt_int @ Xs ) @ one_one_nat ) )
=> ( ( butlas2390654596765537291nt_int @ ( list_u3002344382305578791nt_int @ Xs @ K @ X ) )
= ( list_u3002344382305578791nt_int @ ( butlas2390654596765537291nt_int @ Xs ) @ K @ X ) ) ) ) ).
% butlast_list_update
thf(fact_1117_trans__step__in,axiom,
! [Ps: list_P5707943133018811711nt_int,I4: int,J3: int,I6: int,J4: int,K_12: int,K_22: int] :
( ( step_in @ Ps @ ( product_Pair_int_int @ I4 @ J3 ) @ ( product_Pair_int_int @ I6 @ J4 ) )
=> ( step_in @ ( trans_path @ ( product_Pair_int_int @ K_12 @ K_22 ) @ Ps ) @ ( product_Pair_int_int @ ( plus_plus_int @ I4 @ K_12 ) @ ( plus_plus_int @ J3 @ K_22 ) ) @ ( product_Pair_int_int @ ( plus_plus_int @ I6 @ K_12 ) @ ( plus_plus_int @ J4 @ K_22 ) ) ) ) ).
% trans_step_in
thf(fact_1118_last__conv__nth,axiom,
! [Xs: list_P5707943133018811711nt_int] :
( ( Xs != nil_Pr2300489316682597567nt_int )
=> ( ( last_P3305686521732843992nt_int @ Xs )
= ( nth_Pr4439495888332055232nt_int @ Xs @ ( minus_minus_nat @ ( size_s5157815400016825771nt_int @ Xs ) @ one_one_nat ) ) ) ) ).
% last_conv_nth
thf(fact_1119_last__list__update,axiom,
! [Xs: list_P5707943133018811711nt_int,K: nat,X: product_prod_int_int] :
( ( Xs != nil_Pr2300489316682597567nt_int )
=> ( ( ( K
= ( minus_minus_nat @ ( size_s5157815400016825771nt_int @ Xs ) @ one_one_nat ) )
=> ( ( last_P3305686521732843992nt_int @ ( list_u3002344382305578791nt_int @ Xs @ K @ X ) )
= X ) )
& ( ( K
!= ( minus_minus_nat @ ( size_s5157815400016825771nt_int @ Xs ) @ one_one_nat ) )
=> ( ( last_P3305686521732843992nt_int @ ( list_u3002344382305578791nt_int @ Xs @ K @ X ) )
= ( last_P3305686521732843992nt_int @ Xs ) ) ) ) ) ).
% last_list_update
thf(fact_1120_upto_Opinduct,axiom,
! [A0: int,A1: int,P2: int > int > $o] :
( ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ A0 @ A1 ) )
=> ( ! [I: int,J: int] :
( ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ I @ J ) )
=> ( ( ( ord_less_eq_int @ I @ J )
=> ( P2 @ ( plus_plus_int @ I @ one_one_int ) @ J ) )
=> ( P2 @ I @ J ) ) )
=> ( P2 @ A0 @ A1 ) ) ) ).
% upto.pinduct
thf(fact_1121_fold__atLeastAtMost__nat_Opinduct,axiom,
! [A0: nat > set_int > set_int,A1: nat,A22: nat,A32: set_int,P2: ( nat > set_int > set_int ) > nat > nat > set_int > $o] :
( ( accp_P6808974393373575412et_int @ set_fo6235691086258096971et_int @ ( produc4506484520578140579et_int @ A0 @ ( produc4111907023490026090et_int @ A1 @ ( produc29655638201817675et_int @ A22 @ A32 ) ) ) )
=> ( ! [F2: nat > set_int > set_int,A3: nat,B3: nat,Acc: set_int] :
( ( accp_P6808974393373575412et_int @ set_fo6235691086258096971et_int @ ( produc4506484520578140579et_int @ F2 @ ( produc4111907023490026090et_int @ A3 @ ( produc29655638201817675et_int @ B3 @ Acc ) ) ) )
=> ( ( ~ ( ord_less_nat @ B3 @ A3 )
=> ( P2 @ F2 @ ( plus_plus_nat @ A3 @ one_one_nat ) @ B3 @ ( F2 @ A3 @ Acc ) ) )
=> ( P2 @ F2 @ A3 @ B3 @ Acc ) ) )
=> ( P2 @ A0 @ A1 @ A22 @ A32 ) ) ) ).
% fold_atLeastAtMost_nat.pinduct
thf(fact_1122_step__in__def,axiom,
( step_in
= ( ^ [Ps4: list_P5707943133018811711nt_int,S_i3: product_prod_int_int,S_j3: product_prod_int_int] :
? [K3: nat] :
( ( ord_less_nat @ zero_zero_nat @ K3 )
& ( ord_less_nat @ K3 @ ( size_s5157815400016825771nt_int @ Ps4 ) )
& ( ( last_P3305686521732843992nt_int @ ( take_P8218740963776755879nt_int @ K3 @ Ps4 ) )
= S_i3 )
& ( ( hd_Pro282112905867057956nt_int @ ( drop_P5690361596310759935nt_int @ K3 @ Ps4 ) )
= S_j3 ) ) ) ) ).
% step_in_def
thf(fact_1123_le__zero__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_1124_not__gr__zero,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_1125_double__eq__0__iff,axiom,
! [A: int] :
( ( ( plus_plus_int @ A @ A )
= zero_zero_int )
= ( A = zero_zero_int ) ) ).
% double_eq_0_iff
thf(fact_1126_add_Oright__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.right_neutral
thf(fact_1127_add_Oright__neutral,axiom,
! [A: int] :
( ( plus_plus_int @ A @ zero_zero_int )
= A ) ).
% add.right_neutral
thf(fact_1128_double__zero__sym,axiom,
! [A: int] :
( ( zero_zero_int
= ( plus_plus_int @ A @ A ) )
= ( A = zero_zero_int ) ) ).
% double_zero_sym
thf(fact_1129_add__cancel__left__left,axiom,
! [B: nat,A: nat] :
( ( ( plus_plus_nat @ B @ A )
= A )
= ( B = zero_zero_nat ) ) ).
% add_cancel_left_left
thf(fact_1130_add__cancel__left__left,axiom,
! [B: int,A: int] :
( ( ( plus_plus_int @ B @ A )
= A )
= ( B = zero_zero_int ) ) ).
% add_cancel_left_left
thf(fact_1131_add__cancel__left__right,axiom,
! [A: nat,B: nat] :
( ( ( plus_plus_nat @ A @ B )
= A )
= ( B = zero_zero_nat ) ) ).
% add_cancel_left_right
thf(fact_1132_add__cancel__left__right,axiom,
! [A: int,B: int] :
( ( ( plus_plus_int @ A @ B )
= A )
= ( B = zero_zero_int ) ) ).
% add_cancel_left_right
thf(fact_1133_add__cancel__right__left,axiom,
! [A: nat,B: nat] :
( ( A
= ( plus_plus_nat @ B @ A ) )
= ( B = zero_zero_nat ) ) ).
% add_cancel_right_left
thf(fact_1134_add__cancel__right__left,axiom,
! [A: int,B: int] :
( ( A
= ( plus_plus_int @ B @ A ) )
= ( B = zero_zero_int ) ) ).
% add_cancel_right_left
thf(fact_1135_add__cancel__right__right,axiom,
! [A: nat,B: nat] :
( ( A
= ( plus_plus_nat @ A @ B ) )
= ( B = zero_zero_nat ) ) ).
% add_cancel_right_right
thf(fact_1136_add__cancel__right__right,axiom,
! [A: int,B: int] :
( ( A
= ( plus_plus_int @ A @ B ) )
= ( B = zero_zero_int ) ) ).
% add_cancel_right_right
thf(fact_1137_add__eq__0__iff__both__eq__0,axiom,
! [X: nat,Y: nat] :
( ( ( plus_plus_nat @ X @ Y )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ).
% add_eq_0_iff_both_eq_0
thf(fact_1138_zero__eq__add__iff__both__eq__0,axiom,
! [X: nat,Y: nat] :
( ( zero_zero_nat
= ( plus_plus_nat @ X @ Y ) )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ).
% zero_eq_add_iff_both_eq_0
thf(fact_1139_add__0,axiom,
! [A: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A )
= A ) ).
% add_0
thf(fact_1140_add__0,axiom,
! [A: int] :
( ( plus_plus_int @ zero_zero_int @ A )
= A ) ).
% add_0
thf(fact_1141_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: nat] :
( ( minus_minus_nat @ A @ A )
= zero_zero_nat ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_1142_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: int] :
( ( minus_minus_int @ A @ A )
= zero_zero_int ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_1143_diff__zero,axiom,
! [A: nat] :
( ( minus_minus_nat @ A @ zero_zero_nat )
= A ) ).
% diff_zero
thf(fact_1144_diff__zero,axiom,
! [A: int] :
( ( minus_minus_int @ A @ zero_zero_int )
= A ) ).
% diff_zero
thf(fact_1145_zero__diff,axiom,
! [A: nat] :
( ( minus_minus_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% zero_diff
thf(fact_1146_diff__0__right,axiom,
! [A: int] :
( ( minus_minus_int @ A @ zero_zero_int )
= A ) ).
% diff_0_right
thf(fact_1147_diff__self,axiom,
! [A: int] :
( ( minus_minus_int @ A @ A )
= zero_zero_int ) ).
% diff_self
thf(fact_1148_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_1149_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% neq0_conv
thf(fact_1150_bot__nat__0_Onot__eq__extremum,axiom,
! [A: nat] :
( ( A != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ A ) ) ).
% bot_nat_0.not_eq_extremum
thf(fact_1151_bot__nat__0_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% bot_nat_0.extremum
thf(fact_1152_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_1153_add__is__0,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus_nat @ M @ N )
= zero_zero_nat )
= ( ( M = zero_zero_nat )
& ( N = zero_zero_nat ) ) ) ).
% add_is_0
thf(fact_1154_Nat_Oadd__0__right,axiom,
! [M: nat] :
( ( plus_plus_nat @ M @ zero_zero_nat )
= M ) ).
% Nat.add_0_right
thf(fact_1155_diff__self__eq__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ M )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_1156_diff__0__eq__0,axiom,
! [N: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_1157_drop0,axiom,
( ( drop_P5690361596310759935nt_int @ zero_zero_nat )
= ( ^ [X4: list_P5707943133018811711nt_int] : X4 ) ) ).
% drop0
thf(fact_1158_min__0R,axiom,
! [N: nat] :
( ( ord_min_nat @ N @ zero_zero_nat )
= zero_zero_nat ) ).
% min_0R
thf(fact_1159_min__0L,axiom,
! [N: nat] :
( ( ord_min_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% min_0L
thf(fact_1160_add__le__same__cancel1,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ B @ A ) @ B )
= ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).
% add_le_same_cancel1
thf(fact_1161_add__le__same__cancel1,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ B @ A ) @ B )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% add_le_same_cancel1
thf(fact_1162_add__le__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ B )
= ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).
% add_le_same_cancel2
thf(fact_1163_add__le__same__cancel2,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A @ B ) @ B )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% add_le_same_cancel2
thf(fact_1164_le__add__same__cancel1,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ A @ B ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% le_add_same_cancel1
thf(fact_1165_le__add__same__cancel1,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ ( plus_plus_int @ A @ B ) )
= ( ord_less_eq_int @ zero_zero_int @ B ) ) ).
% le_add_same_cancel1
thf(fact_1166_le__add__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ B @ A ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% le_add_same_cancel2
thf(fact_1167_le__add__same__cancel2,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ ( plus_plus_int @ B @ A ) )
= ( ord_less_eq_int @ zero_zero_int @ B ) ) ).
% le_add_same_cancel2
thf(fact_1168_double__add__le__zero__iff__single__add__le__zero,axiom,
! [A: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A @ A ) @ zero_zero_int )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% double_add_le_zero_iff_single_add_le_zero
thf(fact_1169_zero__le__double__add__iff__zero__le__single__add,axiom,
! [A: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A @ A ) )
= ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% zero_le_double_add_iff_zero_le_single_add
thf(fact_1170_diff__ge__0__iff__ge,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( minus_minus_int @ A @ B ) )
= ( ord_less_eq_int @ B @ A ) ) ).
% diff_ge_0_iff_ge
thf(fact_1171_add__less__same__cancel1,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ B @ A ) @ B )
= ( ord_less_nat @ A @ zero_zero_nat ) ) ).
% add_less_same_cancel1
thf(fact_1172_add__less__same__cancel1,axiom,
! [B: int,A: int] :
( ( ord_less_int @ ( plus_plus_int @ B @ A ) @ B )
= ( ord_less_int @ A @ zero_zero_int ) ) ).
% add_less_same_cancel1
thf(fact_1173_add__less__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ B )
= ( ord_less_nat @ A @ zero_zero_nat ) ) ).
% add_less_same_cancel2
thf(fact_1174_add__less__same__cancel2,axiom,
! [A: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ A @ B ) @ B )
= ( ord_less_int @ A @ zero_zero_int ) ) ).
% add_less_same_cancel2
thf(fact_1175_less__add__same__cancel1,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ ( plus_plus_nat @ A @ B ) )
= ( ord_less_nat @ zero_zero_nat @ B ) ) ).
% less_add_same_cancel1
thf(fact_1176_less__add__same__cancel1,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ ( plus_plus_int @ A @ B ) )
= ( ord_less_int @ zero_zero_int @ B ) ) ).
% less_add_same_cancel1
thf(fact_1177_less__add__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ ( plus_plus_nat @ B @ A ) )
= ( ord_less_nat @ zero_zero_nat @ B ) ) ).
% less_add_same_cancel2
thf(fact_1178_less__add__same__cancel2,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ ( plus_plus_int @ B @ A ) )
= ( ord_less_int @ zero_zero_int @ B ) ) ).
% less_add_same_cancel2
thf(fact_1179_double__add__less__zero__iff__single__add__less__zero,axiom,
! [A: int] :
( ( ord_less_int @ ( plus_plus_int @ A @ A ) @ zero_zero_int )
= ( ord_less_int @ A @ zero_zero_int ) ) ).
% double_add_less_zero_iff_single_add_less_zero
thf(fact_1180_zero__less__double__add__iff__zero__less__single__add,axiom,
! [A: int] :
( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ A ) )
= ( ord_less_int @ zero_zero_int @ A ) ) ).
% zero_less_double_add_iff_zero_less_single_add
thf(fact_1181_diff__gt__0__iff__gt,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ ( minus_minus_int @ A @ B ) )
= ( ord_less_int @ B @ A ) ) ).
% diff_gt_0_iff_gt
thf(fact_1182_diff__add__zero,axiom,
! [A: nat,B: nat] :
( ( minus_minus_nat @ A @ ( plus_plus_nat @ A @ B ) )
= zero_zero_nat ) ).
% diff_add_zero
thf(fact_1183_diff__numeral__special_I9_J,axiom,
( ( minus_minus_int @ one_one_int @ one_one_int )
= zero_zero_int ) ).
% diff_numeral_special(9)
thf(fact_1184_of__nat__0,axiom,
( ( semiri1316708129612266289at_nat @ zero_zero_nat )
= zero_zero_nat ) ).
% of_nat_0
thf(fact_1185_of__nat__0,axiom,
( ( semiri1314217659103216013at_int @ zero_zero_nat )
= zero_zero_int ) ).
% of_nat_0
thf(fact_1186_of__nat__0__eq__iff,axiom,
! [N: nat] :
( ( zero_zero_nat
= ( semiri1316708129612266289at_nat @ N ) )
= ( zero_zero_nat = N ) ) ).
% of_nat_0_eq_iff
thf(fact_1187_of__nat__0__eq__iff,axiom,
! [N: nat] :
( ( zero_zero_int
= ( semiri1314217659103216013at_int @ N ) )
= ( zero_zero_nat = N ) ) ).
% of_nat_0_eq_iff
thf(fact_1188_of__nat__eq__0__iff,axiom,
! [M: nat] :
( ( ( semiri1316708129612266289at_nat @ M )
= zero_zero_nat )
= ( M = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
thf(fact_1189_of__nat__eq__0__iff,axiom,
! [M: nat] :
( ( ( semiri1314217659103216013at_int @ M )
= zero_zero_int )
= ( M = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
thf(fact_1190_zero__less__Suc,axiom,
! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N ) ) ).
% zero_less_Suc
thf(fact_1191_less__Suc0,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ ( suc @ zero_zero_nat ) )
= ( N = zero_zero_nat ) ) ).
% less_Suc0
thf(fact_1192_length__0__conv,axiom,
! [Xs: list_P5707943133018811711nt_int] :
( ( ( size_s5157815400016825771nt_int @ Xs )
= zero_zero_nat )
= ( Xs = nil_Pr2300489316682597567nt_int ) ) ).
% length_0_conv
thf(fact_1193_add__gr__0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ M )
| ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% add_gr_0
thf(fact_1194_zero__less__diff,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N @ M ) )
= ( ord_less_nat @ M @ N ) ) ).
% zero_less_diff
thf(fact_1195_diff__is__0__eq_H,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( minus_minus_nat @ M @ N )
= zero_zero_nat ) ) ).
% diff_is_0_eq'
thf(fact_1196_diff__is__0__eq,axiom,
! [M: nat,N: nat] :
( ( ( minus_minus_nat @ M @ N )
= zero_zero_nat )
= ( ord_less_eq_nat @ M @ N ) ) ).
% diff_is_0_eq
thf(fact_1197_less__one,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ one_one_nat )
= ( N = zero_zero_nat ) ) ).
% less_one
thf(fact_1198_zle__diff1__eq,axiom,
! [W: int,Z3: int] :
( ( ord_less_eq_int @ W @ ( minus_minus_int @ Z3 @ one_one_int ) )
= ( ord_less_int @ W @ Z3 ) ) ).
% zle_diff1_eq
thf(fact_1199_Suc__pred,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( suc @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) )
= N ) ) ).
% Suc_pred
thf(fact_1200_Suc__diff__1,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( suc @ ( minus_minus_nat @ N @ one_one_nat ) )
= N ) ) ).
% Suc_diff_1
thf(fact_1201_diffs0__imp__equal,axiom,
! [M: nat,N: nat] :
( ( ( minus_minus_nat @ M @ N )
= zero_zero_nat )
=> ( ( ( minus_minus_nat @ N @ M )
= zero_zero_nat )
=> ( M = N ) ) ) ).
% diffs0_imp_equal
thf(fact_1202_minus__nat_Odiff__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ zero_zero_nat )
= M ) ).
% minus_nat.diff_0
thf(fact_1203_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_1204_bot__nat__0_Oextremum__unique,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
= ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_unique
thf(fact_1205_bot__nat__0_Oextremum__uniqueI,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_uniqueI
thf(fact_1206_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_1207_int__le__induct,axiom,
! [I4: int,K: int,P2: int > $o] :
( ( ord_less_eq_int @ I4 @ K )
=> ( ( P2 @ K )
=> ( ! [I: int] :
( ( ord_less_eq_int @ I @ K )
=> ( ( P2 @ I )
=> ( P2 @ ( minus_minus_int @ I @ one_one_int ) ) ) )
=> ( P2 @ I4 ) ) ) ) ).
% int_le_induct
thf(fact_1208_infinite__descent0,axiom,
! [P2: nat > $o,N: nat] :
( ( P2 @ zero_zero_nat )
=> ( ! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ~ ( P2 @ N3 )
=> ? [M3: nat] :
( ( ord_less_nat @ M3 @ N3 )
& ~ ( P2 @ M3 ) ) ) )
=> ( P2 @ N ) ) ) ).
% infinite_descent0
thf(fact_1209_gr__implies__not0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_1210_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_zeroE
thf(fact_1211_not__less0,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less0
thf(fact_1212_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr0
thf(fact_1213_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr0I
thf(fact_1214_bot__nat__0_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_1215_not0__implies__Suc,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ? [M6: nat] :
( N
= ( suc @ M6 ) ) ) ).
% not0_implies_Suc
thf(fact_1216_Zero__not__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
% Zero_not_Suc
thf(fact_1217_Zero__neq__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
% Zero_neq_Suc
thf(fact_1218_Suc__neq__Zero,axiom,
! [M: nat] :
( ( suc @ M )
!= zero_zero_nat ) ).
% Suc_neq_Zero
thf(fact_1219_zero__induct,axiom,
! [P2: nat > $o,K: nat] :
( ( P2 @ K )
=> ( ! [N3: nat] :
( ( P2 @ ( suc @ N3 ) )
=> ( P2 @ N3 ) )
=> ( P2 @ zero_zero_nat ) ) ) ).
% zero_induct
thf(fact_1220_diff__induct,axiom,
! [P2: nat > nat > $o,M: nat,N: nat] :
( ! [X3: nat] : ( P2 @ X3 @ zero_zero_nat )
=> ( ! [Y3: nat] : ( P2 @ zero_zero_nat @ ( suc @ Y3 ) )
=> ( ! [X3: nat,Y3: nat] :
( ( P2 @ X3 @ Y3 )
=> ( P2 @ ( suc @ X3 ) @ ( suc @ Y3 ) ) )
=> ( P2 @ M @ N ) ) ) ) ).
% diff_induct
thf(fact_1221_nat__induct,axiom,
! [P2: nat > $o,N: nat] :
( ( P2 @ zero_zero_nat )
=> ( ! [N3: nat] :
( ( P2 @ N3 )
=> ( P2 @ ( suc @ N3 ) ) )
=> ( P2 @ N ) ) ) ).
% nat_induct
thf(fact_1222_old_Onat_Oexhaust,axiom,
! [Y: nat] :
( ( Y != zero_zero_nat )
=> ~ ! [Nat3: nat] :
( Y
!= ( suc @ Nat3 ) ) ) ).
% old.nat.exhaust
thf(fact_1223_nat_OdiscI,axiom,
! [Nat: nat,X2: nat] :
( ( Nat
= ( suc @ X2 ) )
=> ( Nat != zero_zero_nat ) ) ).
% nat.discI
thf(fact_1224_old_Onat_Odistinct_I1_J,axiom,
! [Nat2: nat] :
( zero_zero_nat
!= ( suc @ Nat2 ) ) ).
% old.nat.distinct(1)
thf(fact_1225_old_Onat_Odistinct_I2_J,axiom,
! [Nat2: nat] :
( ( suc @ Nat2 )
!= zero_zero_nat ) ).
% old.nat.distinct(2)
thf(fact_1226_nat_Odistinct_I1_J,axiom,
! [X2: nat] :
( zero_zero_nat
!= ( suc @ X2 ) ) ).
% nat.distinct(1)
thf(fact_1227_row__exec_Ocases,axiom,
! [X: nat] :
( ( X != zero_zero_nat )
=> ~ ! [V2: nat] :
( X
!= ( suc @ V2 ) ) ) ).
% row_exec.cases
thf(fact_1228_board__exec__aux_Ocases,axiom,
! [X: produc9133624956312949779et_int] :
( ! [M8: set_int] :
( X
!= ( produc29655638201817675et_int @ zero_zero_nat @ M8 ) )
=> ~ ! [V2: nat,M8: set_int] :
( X
!= ( produc29655638201817675et_int @ ( suc @ V2 ) @ M8 ) ) ) ).
% board_exec_aux.cases
thf(fact_1229_plus__nat_Oadd__0,axiom,
! [N: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N )
= N ) ).
% plus_nat.add_0
thf(fact_1230_add__eq__self__zero,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus_nat @ M @ N )
= M )
=> ( N = zero_zero_nat ) ) ).
% add_eq_self_zero
thf(fact_1231_less__Suc__eq__0__disj,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
= ( ( M = zero_zero_nat )
| ? [J5: nat] :
( ( M
= ( suc @ J5 ) )
& ( ord_less_nat @ J5 @ N ) ) ) ) ).
% less_Suc_eq_0_disj
thf(fact_1232_gr0__implies__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ? [M6: nat] :
( N
= ( suc @ M6 ) ) ) ).
% gr0_implies_Suc
thf(fact_1233_All__less__Suc2,axiom,
! [N: nat,P2: nat > $o] :
( ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( suc @ N ) )
=> ( P2 @ I3 ) ) )
= ( ( P2 @ zero_zero_nat )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ N )
=> ( P2 @ ( suc @ I3 ) ) ) ) ) ).
% All_less_Suc2
thf(fact_1234_gr0__conv__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
= ( ? [M2: nat] :
( N
= ( suc @ M2 ) ) ) ) ).
% gr0_conv_Suc
thf(fact_1235_Ex__less__Suc2,axiom,
! [N: nat,P2: nat > $o] :
( ( ? [I3: nat] :
( ( ord_less_nat @ I3 @ ( suc @ N ) )
& ( P2 @ I3 ) ) )
= ( ( P2 @ zero_zero_nat )
| ? [I3: nat] :
( ( ord_less_nat @ I3 @ N )
& ( P2 @ ( suc @ I3 ) ) ) ) ) ).
% Ex_less_Suc2
thf(fact_1236_add__is__1,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus_nat @ M @ N )
= ( suc @ zero_zero_nat ) )
= ( ( ( M
= ( suc @ zero_zero_nat ) )
& ( N = zero_zero_nat ) )
| ( ( M = zero_zero_nat )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ) ).
% add_is_1
thf(fact_1237_one__is__add,axiom,
! [M: nat,N: nat] :
( ( ( suc @ zero_zero_nat )
= ( plus_plus_nat @ M @ N ) )
= ( ( ( M
= ( suc @ zero_zero_nat ) )
& ( N = zero_zero_nat ) )
| ( ( M = zero_zero_nat )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ) ).
% one_is_add
thf(fact_1238_ex__least__nat__le,axiom,
! [P2: nat > $o,N: nat] :
( ( P2 @ N )
=> ( ~ ( P2 @ zero_zero_nat )
=> ? [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N )
& ! [I5: nat] :
( ( ord_less_nat @ I5 @ K2 )
=> ~ ( P2 @ I5 ) )
& ( P2 @ K2 ) ) ) ) ).
% ex_least_nat_le
thf(fact_1239_less__imp__add__positive,axiom,
! [I4: nat,J3: nat] :
( ( ord_less_nat @ I4 @ J3 )
=> ? [K2: nat] :
( ( ord_less_nat @ zero_zero_nat @ K2 )
& ( ( plus_plus_nat @ I4 @ K2 )
= J3 ) ) ) ).
% less_imp_add_positive
thf(fact_1240_diff__less,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ M ) ) ) ).
% diff_less
thf(fact_1241_One__nat__def,axiom,
( one_one_nat
= ( suc @ zero_zero_nat ) ) ).
% One_nat_def
thf(fact_1242_diff__add__0,axiom,
! [N: nat,M: nat] :
( ( minus_minus_nat @ N @ ( plus_plus_nat @ N @ M ) )
= zero_zero_nat ) ).
% diff_add_0
thf(fact_1243_int__induct,axiom,
! [P2: int > $o,K: int,I4: int] :
( ( P2 @ K )
=> ( ! [I: int] :
( ( ord_less_eq_int @ K @ I )
=> ( ( P2 @ I )
=> ( P2 @ ( plus_plus_int @ I @ one_one_int ) ) ) )
=> ( ! [I: int] :
( ( ord_less_eq_int @ I @ K )
=> ( ( P2 @ I )
=> ( P2 @ ( minus_minus_int @ I @ one_one_int ) ) ) )
=> ( P2 @ I4 ) ) ) ) ).
% int_induct
thf(fact_1244_ex__least__nat__less,axiom,
! [P2: nat > $o,N: nat] :
( ( P2 @ N )
=> ( ~ ( P2 @ zero_zero_nat )
=> ? [K2: nat] :
( ( ord_less_nat @ K2 @ N )
& ! [I5: nat] :
( ( ord_less_eq_nat @ I5 @ K2 )
=> ~ ( P2 @ I5 ) )
& ( P2 @ ( suc @ K2 ) ) ) ) ) ).
% ex_least_nat_less
thf(fact_1245_diff__Suc__less,axiom,
! [N: nat,I4: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_nat @ ( minus_minus_nat @ N @ ( suc @ I4 ) ) @ N ) ) ).
% diff_Suc_less
thf(fact_1246_nat__induct__non__zero,axiom,
! [N: nat,P2: nat > $o] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( P2 @ one_one_nat )
=> ( ! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ( P2 @ N3 )
=> ( P2 @ ( suc @ N3 ) ) ) )
=> ( P2 @ N ) ) ) ) ).
% nat_induct_non_zero
thf(fact_1247_nat__diff__split__asm,axiom,
! [P2: nat > $o,A: nat,B: nat] :
( ( P2 @ ( minus_minus_nat @ A @ B ) )
= ( ~ ( ( ( ord_less_nat @ A @ B )
& ~ ( P2 @ zero_zero_nat ) )
| ? [D5: nat] :
( ( A
= ( plus_plus_nat @ B @ D5 ) )
& ~ ( P2 @ D5 ) ) ) ) ) ).
% nat_diff_split_asm
thf(fact_1248_nat__diff__split,axiom,
! [P2: nat > $o,A: nat,B: nat] :
( ( P2 @ ( minus_minus_nat @ A @ B ) )
= ( ( ( ord_less_nat @ A @ B )
=> ( P2 @ zero_zero_nat ) )
& ! [D5: nat] :
( ( A
= ( plus_plus_nat @ B @ D5 ) )
=> ( P2 @ D5 ) ) ) ) ).
% nat_diff_split
thf(fact_1249_Suc__pred_H,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( N
= ( suc @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% Suc_pred'
thf(fact_1250_Suc__diff__eq__diff__pred,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( minus_minus_nat @ ( suc @ M ) @ N )
= ( minus_minus_nat @ M @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% Suc_diff_eq_diff_pred
thf(fact_1251_add__eq__if,axiom,
( plus_plus_nat
= ( ^ [M2: nat,N2: nat] : ( if_nat @ ( M2 = zero_zero_nat ) @ N2 @ ( suc @ ( plus_plus_nat @ ( minus_minus_nat @ M2 @ one_one_nat ) @ N2 ) ) ) ) ) ).
% add_eq_if
thf(fact_1252_int__ops_I1_J,axiom,
( ( semiri1314217659103216013at_int @ zero_zero_nat )
= zero_zero_int ) ).
% int_ops(1)
thf(fact_1253_conj__le__cong,axiom,
! [X: int,X8: int,P2: $o,P5: $o] :
( ( X = X8 )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X8 )
=> ( P2 = P5 ) )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X )
& P2 )
= ( ( ord_less_eq_int @ zero_zero_int @ X8 )
& P5 ) ) ) ) ).
% conj_le_cong
thf(fact_1254_imp__le__cong,axiom,
! [X: int,X8: int,P2: $o,P5: $o] :
( ( X = X8 )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X8 )
=> ( P2 = P5 ) )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X )
=> P2 )
= ( ( ord_less_eq_int @ zero_zero_int @ X8 )
=> P5 ) ) ) ) ).
% imp_le_cong
thf(fact_1255_less__eq__int__code_I1_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% less_eq_int_code(1)
thf(fact_1256_plus__int__code_I1_J,axiom,
! [K: int] :
( ( plus_plus_int @ K @ zero_zero_int )
= K ) ).
% plus_int_code(1)
thf(fact_1257_plus__int__code_I2_J,axiom,
! [L: int] :
( ( plus_plus_int @ zero_zero_int @ L )
= L ) ).
% plus_int_code(2)
thf(fact_1258_nonneg__int__cases,axiom,
! [K: int] :
( ( ord_less_eq_int @ zero_zero_int @ K )
=> ~ ! [N3: nat] :
( K
!= ( semiri1314217659103216013at_int @ N3 ) ) ) ).
% nonneg_int_cases
thf(fact_1259_zero__le__imp__eq__int,axiom,
! [K: int] :
( ( ord_less_eq_int @ zero_zero_int @ K )
=> ? [N3: nat] :
( K
= ( semiri1314217659103216013at_int @ N3 ) ) ) ).
% zero_le_imp_eq_int
thf(fact_1260_odd__nonzero,axiom,
! [Z3: int] :
( ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z3 ) @ Z3 )
!= zero_zero_int ) ).
% odd_nonzero
% Helper facts (5)
thf(help_If_2_1_If_001t__Int__Oint_T,axiom,
! [X: int,Y: int] :
( ( if_int @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Int__Oint_T,axiom,
! [X: int,Y: int] :
( ( if_int @ $true @ X @ Y )
= X ) ).
thf(help_If_3_1_If_001t__Nat__Onat_T,axiom,
! [P2: $o] :
( ( P2 = $true )
| ( P2 = $false ) ) ).
thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y: nat] :
( ( if_nat @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y: nat] :
( ( if_nat @ $true @ X @ Y )
= X ) ).
% Conjectures (2)
thf(conj_0,hypothesis,
! [I_1: int,J_1: int,I_2: int,J_2: int,Ps_2: list_P5707943133018811711nt_int] :
( ( ps_2
= ( cons_P3334398858971670639nt_int @ ( product_Pair_int_int @ I_1 @ J_1 ) @ ( cons_P3334398858971670639nt_int @ ( product_Pair_int_int @ I_2 @ J_2 ) @ Ps_2 ) ) )
=> thesis ) ).
thf(conj_1,conjecture,
thesis ).
%------------------------------------------------------------------------------