TPTP Problem File: SLH0602^1.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_01273_048621__5950290_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1386 ( 621 unt; 114 typ; 0 def)
% Number of atoms : 3272 (1133 equ; 0 cnn)
% Maximal formula atoms : 12 ( 2 avg)
% Number of connectives : 9199 ( 326 ~; 65 |; 143 &;7321 @)
% ( 0 <=>;1344 =>; 0 <=; 0 <~>)
% Maximal formula depth : 18 ( 6 avg)
% Number of types : 27 ( 26 usr)
% Number of type conns : 421 ( 421 >; 0 *; 0 +; 0 <<)
% Number of symbols : 91 ( 88 usr; 9 con; 0-3 aty)
% Number of variables : 3322 ( 200 ^;3028 !; 94 ?;3322 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 09:01:00.372
%------------------------------------------------------------------------------
% Could-be-implicit typings (26)
thf(ty_n_t__Set__Oset_It__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_J,type,
set_Pr4708930517165415495nt_int: $tType ).
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set_Pr5872125604998073543nt_int: $tType ).
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thf(ty_n_t__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(ty_n_t__Set__Oset_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_J,type,
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set_Pr7486745082216227783et_int: $tType ).
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produc212874708166070503et_int: $tType ).
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set_Pr2560585780119916871nt_int: $tType ).
thf(ty_n_t__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,
produc1219242969750017639nt_int: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Int__Oint_M_062_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J,type,
produc8551481072490612790e_term: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Set__Oset_It__Int__Oint_J_Mt__Set__Oset_It__Int__Oint_J_J_J,type,
set_Pr2522554150109002629et_int: $tType ).
thf(ty_n_t__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J,type,
option6357759511663192854e_term: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Set__Oset_It__Int__Oint_J_J_J,type,
set_Pr4810089274464741491et_int: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Set__Oset_It__Int__Oint_J_Mt__Set__Oset_It__Int__Oint_J_J,type,
produc2115011035271226405et_int: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
set_Pr958786334691620121nt_int: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__Set__Oset_It__Int__Oint_J_J,type,
produc9133624956312949779et_int: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
product_prod_nat_nat: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
product_prod_int_int: $tType ).
thf(ty_n_t__Set__Oset_It__Int__Oint_J,type,
set_int: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
thf(ty_n_t__Int__Oint,type,
int: $tType ).
% Explicit typings (88)
thf(sy_c_BNF__Cardinal__Order__Relation_OrelChain_001t__Int__Oint_001t__Int__Oint,type,
bNF_Ca1965613569405424510nt_int: set_Pr958786334691620121nt_int > ( int > int ) > $o ).
thf(sy_c_BNF__Cardinal__Order__Relation_OrelChain_001t__Int__Oint_001t__Nat__Onat,type,
bNF_Ca1968104039914474786nt_nat: set_Pr958786334691620121nt_int > ( int > nat ) > $o ).
thf(sy_c_BNF__Cardinal__Order__Relation_OrelChain_001t__Int__Oint_001t__Set__Oset_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
bNF_Ca8719598144974034247nt_int: set_Pr958786334691620121nt_int > ( int > set_Pr958786334691620121nt_int ) > $o ).
thf(sy_c_BNF__Cardinal__Order__Relation_OrelChain_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Int__Oint,type,
bNF_Ca1641342347952694721nt_int: set_Pr2560585780119916871nt_int > ( product_prod_int_int > int ) > $o ).
thf(sy_c_BNF__Cardinal__Order__Relation_OrelChain_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Nat__Onat,type,
bNF_Ca1643832818461744997nt_nat: set_Pr2560585780119916871nt_int > ( product_prod_int_int > nat ) > $o ).
thf(sy_c_BNF__Cardinal__Order__Relation_OrelChain_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Set__Oset_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
bNF_Ca5742924509254848324nt_int: set_Pr2560585780119916871nt_int > ( product_prod_int_int > set_Pr958786334691620121nt_int ) > $o ).
thf(sy_c_GCD_Obezw,type,
bezw: nat > nat > product_prod_int_int ).
thf(sy_c_Groups_Oabs__class_Oabs_001t__Int__Oint,type,
abs_abs_int: int > 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_Otimes__class_Otimes_001t__Int__Oint,type,
times_times_int: int > int > int ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Int__Oint,type,
uminus_uminus_int: int > int ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_It__Int__Oint_J,type,
uminus1532241313380277803et_int: set_int > set_int ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
uminus6221592323253981072nt_int: set_Pr958786334691620121nt_int > set_Pr958786334691620121nt_int ).
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_Int_Onat,type,
nat2: int > nat ).
thf(sy_c_Int_Oring__1__class_Oof__int_001t__Int__Oint,type,
ring_1_of_int_int: int > int ).
thf(sy_c_KnightsTour_Oboard,type,
board: nat > nat > set_Pr958786334691620121nt_int ).
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_Omirror1__board,type,
mirror1_board: int > set_Pr958786334691620121nt_int > set_Pr958786334691620121nt_int ).
thf(sy_c_KnightsTour_Omirror2__board,type,
mirror2_board: int > set_Pr958786334691620121nt_int > set_Pr958786334691620121nt_int ).
thf(sy_c_KnightsTour_Otrans__board,type,
trans_board: product_prod_int_int > set_Pr958786334691620121nt_int > set_Pr958786334691620121nt_int ).
thf(sy_c_KnightsTour_Otranspose__board,type,
transpose_board: set_Pr958786334691620121nt_int > set_Pr958786334691620121nt_int ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Int__Oint,type,
sup_sup_int: int > int > int ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Nat__Onat,type,
sup_sup_nat: nat > nat > nat ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Int__Oint_J,type,
sup_sup_set_int: set_int > set_int > set_int ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
sup_su6024340866399070445nt_int: set_Pr958786334691620121nt_int > set_Pr958786334691620121nt_int > set_Pr958786334691620121nt_int ).
thf(sy_c_Nat_OSuc,type,
suc: nat > nat ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Int__Oint,type,
semiri1314217659103216013at_int: nat > int ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Nat__Onat,type,
semiri1316708129612266289at_nat: nat > nat ).
thf(sy_c_Num_Oneg__numeral__class_Odbl__dec_001t__Int__Oint,type,
neg_nu3811975205180677377ec_int: int > int ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Int__Oint,type,
ord_less_int: int > int > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
ord_less_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Int__Oint_J,type,
ord_less_set_int: set_int > set_int > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
ord_le7563427860532173253nt_int: set_Pr958786334691620121nt_int > set_Pr958786334691620121nt_int > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Int__Oint,type,
ord_less_eq_int: int > int > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
ord_less_eq_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Int__Oint_J,type,
ord_less_eq_set_int: set_int > set_int > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_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_J,type,
ord_le135402666524580259nt_int: set_Pr1872883991513573699nt_int > set_Pr1872883991513573699nt_int > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_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_J,type,
ord_le8725513860283290265nt_int: set_Pr9222295170931077689nt_int > set_Pr9222295170931077689nt_int > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
ord_le2843351958646193337nt_int: set_Pr958786334691620121nt_int > set_Pr958786334691620121nt_int > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Set__Oset_It__Int__Oint_J_J_J,type,
ord_le8255767777184198675et_int: set_Pr4810089274464741491et_int > set_Pr4810089274464741491et_int > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__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_J,type,
ord_le6090609446090860775nt_int: set_Pr2560585780119916871nt_int > set_Pr2560585780119916871nt_int > $o ).
thf(sy_c_Orderings_Oorder__class_OGreatest_001t__Int__Oint,type,
order_Greatest_int: ( int > $o ) > int ).
thf(sy_c_Orderings_Oorder__class_OGreatest_001t__Nat__Onat,type,
order_Greatest_nat: ( nat > $o ) > nat ).
thf(sy_c_Orderings_Oorder__class_OGreatest_001t__Set__Oset_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
order_3894005715824938610nt_int: ( set_Pr958786334691620121nt_int > $o ) > set_Pr958786334691620121nt_int ).
thf(sy_c_Parity_Ounique__euclidean__semiring__with__nat__division__class_Odivides__aux_001t__Int__Oint,type,
unique5329631941980267465ux_int: product_prod_int_int > $o ).
thf(sy_c_Parity_Ounique__euclidean__semiring__with__nat__division__class_Odivides__aux_001t__Nat__Onat,type,
unique5332122412489317741ux_nat: product_prod_nat_nat > $o ).
thf(sy_c_Product__Type_OPair_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_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
produc4305682042979456191nt_int: ( int > option6357759511663192854e_term ) > product_prod_int_int > produc7773217078559923341nt_int ).
thf(sy_c_Product__Type_OPair_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_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
produc5700946648718959541nt_int: ( produc8551481072490612790e_term > option6357759511663192854e_term ) > product_prod_int_int > produc2285326912895808259nt_int ).
thf(sy_c_Product__Type_OPair_001t__Int__Oint_001t__Int__Oint,type,
product_Pair_int_int: int > int > product_prod_int_int ).
thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Nat__Onat,type,
product_Pair_nat_nat: nat > nat > product_prod_nat_nat ).
thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Set__Oset_It__Int__Oint_J,type,
produc29655638201817675et_int: nat > set_int > produc9133624956312949779et_int ).
thf(sy_c_Product__Type_OPair_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_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,
produc8406175334058502835nt_int: produc7773217078559923341nt_int > produc7773217078559923341nt_int > produc2501202720802129403nt_int ).
thf(sy_c_Product__Type_OPair_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_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,
produc7601053194514725023nt_int: produc2285326912895808259nt_int > produc2285326912895808259nt_int > produc2432570611225516007nt_int ).
thf(sy_c_Product__Type_OPair_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
produc3646306378393792727nt_int: product_prod_int_int > product_prod_int_int > produc1219242969750017639nt_int ).
thf(sy_c_Product__Type_OPair_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Set__Oset_It__Int__Oint_J_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Set__Oset_It__Int__Oint_J_J,type,
produc985091676681408599et_int: produc9133624956312949779et_int > produc9133624956312949779et_int > produc212874708166070503et_int ).
thf(sy_c_Product__Type_OPair_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_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,
produc1923601798490594135nt_int: produc1219242969750017639nt_int > produc1219242969750017639nt_int > produc340838079399958759nt_int ).
thf(sy_c_Product__Type_OPair_001t__Set__Oset_It__Int__Oint_J_001t__Set__Oset_It__Int__Oint_J,type,
produc6363374080413544029et_int: set_int > set_int > produc2115011035271226405et_int ).
thf(sy_c_Set_OCollect_001t__Int__Oint,type,
collect_int: ( int > $o ) > set_int ).
thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
collec213857154873943460nt_int: ( product_prod_int_int > $o ) > set_Pr958786334691620121nt_int ).
thf(sy_c_Wfrec_Osame__fst_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_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
same_f6662725367016992042nt_int: ( ( int > option6357759511663192854e_term ) > $o ) > ( ( int > option6357759511663192854e_term ) > set_Pr2560585780119916871nt_int ) > set_Pr8634505666381077339nt_int ).
thf(sy_c_Wfrec_Osame__fst_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_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
same_f5472592420709775776nt_int: ( ( produc8551481072490612790e_term > option6357759511663192854e_term ) > $o ) > ( ( produc8551481072490612790e_term > option6357759511663192854e_term ) > set_Pr2560585780119916871nt_int ) > set_Pr4708930517165415495nt_int ).
thf(sy_c_Wfrec_Osame__fst_001t__Int__Oint_001t__Int__Oint,type,
same_fst_int_int: ( int > $o ) > ( int > set_Pr958786334691620121nt_int ) > set_Pr2560585780119916871nt_int ).
thf(sy_c_Wfrec_Osame__fst_001t__Nat__Onat_001t__Set__Oset_It__Int__Oint_J,type,
same_fst_nat_set_int: ( nat > $o ) > ( nat > set_Pr2522554150109002629et_int ) > set_Pr7486745082216227783et_int ).
thf(sy_c_Wfrec_Osame__fst_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
same_f2440470920016040620nt_int: ( product_prod_int_int > $o ) > ( product_prod_int_int > set_Pr2560585780119916871nt_int ) > set_Pr5872125604998073543nt_int ).
thf(sy_c_member_001t__Int__Oint,type,
member_int: int > set_int > $o ).
thf(sy_c_member_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,
member7034335876925520548nt_int: produc7773217078559923341nt_int > set_Pr1872883991513573699nt_int > $o ).
thf(sy_c_member_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,
member7618704894036264090nt_int: produc2285326912895808259nt_int > set_Pr9222295170931077689nt_int > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
member5262025264175285858nt_int: product_prod_int_int > set_Pr958786334691620121nt_int > $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__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__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_Mt__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_J,type,
member8053542592415931152nt_int: produc340838079399958759nt_int > set_Pr5872125604998073543nt_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_m,type,
m: nat ).
thf(sy_v_n_092_060_094sub_0621,type,
n_1: nat ).
thf(sy_v_n_092_060_094sub_0622,type,
n_2: nat ).
% Relevant facts (1266)
thf(fact_0_board__concat,axiom,
! [N: nat,M_1: nat,M_2: nat] :
( ( sup_su6024340866399070445nt_int @ ( board @ N @ M_1 ) @ ( trans_board @ ( product_Pair_int_int @ zero_zero_int @ ( semiri1314217659103216013at_int @ M_1 ) ) @ ( board @ N @ M_2 ) ) )
= ( board @ N @ ( plus_plus_nat @ M_1 @ M_2 ) ) ) ).
% board_concat
thf(fact_1_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_2_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_3_of__nat__0,axiom,
( ( semiri1316708129612266289at_nat @ zero_zero_nat )
= zero_zero_nat ) ).
% of_nat_0
thf(fact_4_of__nat__0,axiom,
( ( semiri1314217659103216013at_int @ zero_zero_nat )
= zero_zero_int ) ).
% of_nat_0
thf(fact_5_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_6_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_7_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_8_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_9_add_Oright__neutral,axiom,
! [A: int] :
( ( plus_plus_int @ A @ zero_zero_int )
= A ) ).
% add.right_neutral
thf(fact_10_add_Oright__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.right_neutral
thf(fact_11_double__zero__sym,axiom,
! [A: int] :
( ( zero_zero_int
= ( plus_plus_int @ A @ A ) )
= ( A = zero_zero_int ) ) ).
% double_zero_sym
thf(fact_12_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_13_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_14_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_15_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_16_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_17_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_18_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_19_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_20_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_21_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_22_add__right__cancel,axiom,
! [B: nat,A: nat,C: nat] :
( ( ( plus_plus_nat @ B @ A )
= ( plus_plus_nat @ C @ A ) )
= ( B = C ) ) ).
% add_right_cancel
thf(fact_23_add__right__cancel,axiom,
! [B: int,A: int,C: int] :
( ( ( plus_plus_int @ B @ A )
= ( plus_plus_int @ C @ A ) )
= ( B = C ) ) ).
% add_right_cancel
thf(fact_24_add__left__cancel,axiom,
! [A: nat,B: nat,C: nat] :
( ( ( plus_plus_nat @ A @ B )
= ( plus_plus_nat @ A @ C ) )
= ( B = C ) ) ).
% add_left_cancel
thf(fact_25_add__left__cancel,axiom,
! [A: int,B: int,C: int] :
( ( ( plus_plus_int @ A @ B )
= ( plus_plus_int @ A @ C ) )
= ( B = C ) ) ).
% add_left_cancel
thf(fact_26_of__nat__eq__iff,axiom,
! [M: nat,N: nat] :
( ( ( semiri1314217659103216013at_int @ M )
= ( semiri1314217659103216013at_int @ N ) )
= ( M = N ) ) ).
% of_nat_eq_iff
thf(fact_27_of__nat__eq__iff,axiom,
! [M: nat,N: nat] :
( ( ( semiri1316708129612266289at_nat @ M )
= ( semiri1316708129612266289at_nat @ N ) )
= ( M = N ) ) ).
% of_nat_eq_iff
thf(fact_28_add__0,axiom,
! [A: int] :
( ( plus_plus_int @ zero_zero_int @ A )
= A ) ).
% add_0
thf(fact_29_add__0,axiom,
! [A: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A )
= A ) ).
% add_0
thf(fact_30_Nat_Oadd__0__right,axiom,
! [M: nat] :
( ( plus_plus_nat @ M @ zero_zero_nat )
= M ) ).
% Nat.add_0_right
thf(fact_31_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_32_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_33_plus__nat_Oadd__0,axiom,
! [N: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N )
= N ) ).
% plus_nat.add_0
thf(fact_34_trans__board__correct,axiom,
! [I: int,J: int,B: set_Pr958786334691620121nt_int,K_1: int,K_2: int] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ I @ J ) @ B )
= ( member5262025264175285858nt_int @ ( product_Pair_int_int @ ( plus_plus_int @ I @ K_1 ) @ ( plus_plus_int @ J @ K_2 ) ) @ ( trans_board @ ( product_Pair_int_int @ K_1 @ K_2 ) @ B ) ) ) ).
% trans_board_correct
thf(fact_35_zero__reorient,axiom,
! [X: int] :
( ( zero_zero_int = X )
= ( X = zero_zero_int ) ) ).
% zero_reorient
thf(fact_36_zero__reorient,axiom,
! [X: nat] :
( ( zero_zero_nat = X )
= ( X = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_37_add__right__imp__eq,axiom,
! [B: nat,A: nat,C: nat] :
( ( ( plus_plus_nat @ B @ A )
= ( plus_plus_nat @ C @ A ) )
=> ( B = C ) ) ).
% add_right_imp_eq
thf(fact_38_add__right__imp__eq,axiom,
! [B: int,A: int,C: int] :
( ( ( plus_plus_int @ B @ A )
= ( plus_plus_int @ C @ A ) )
=> ( B = C ) ) ).
% add_right_imp_eq
thf(fact_39_add__left__imp__eq,axiom,
! [A: nat,B: nat,C: nat] :
( ( ( plus_plus_nat @ A @ B )
= ( plus_plus_nat @ A @ C ) )
=> ( B = C ) ) ).
% add_left_imp_eq
thf(fact_40_add__left__imp__eq,axiom,
! [A: int,B: int,C: int] :
( ( ( plus_plus_int @ A @ B )
= ( plus_plus_int @ A @ C ) )
=> ( B = C ) ) ).
% add_left_imp_eq
thf(fact_41_add_Oleft__commute,axiom,
! [B: nat,A: nat,C: nat] :
( ( plus_plus_nat @ B @ ( plus_plus_nat @ A @ C ) )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% add.left_commute
thf(fact_42_add_Oleft__commute,axiom,
! [B: int,A: int,C: int] :
( ( plus_plus_int @ B @ ( plus_plus_int @ A @ C ) )
= ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% add.left_commute
thf(fact_43_add_Ocommute,axiom,
( plus_plus_nat
= ( ^ [A2: nat,B2: nat] : ( plus_plus_nat @ B2 @ A2 ) ) ) ).
% add.commute
thf(fact_44_add_Ocommute,axiom,
( plus_plus_int
= ( ^ [A2: int,B2: int] : ( plus_plus_int @ B2 @ A2 ) ) ) ).
% add.commute
thf(fact_45_add_Oright__cancel,axiom,
! [B: int,A: int,C: int] :
( ( ( plus_plus_int @ B @ A )
= ( plus_plus_int @ C @ A ) )
= ( B = C ) ) ).
% add.right_cancel
thf(fact_46_add_Oleft__cancel,axiom,
! [A: int,B: int,C: int] :
( ( ( plus_plus_int @ A @ B )
= ( plus_plus_int @ A @ C ) )
= ( B = C ) ) ).
% add.left_cancel
thf(fact_47_add_Oassoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% add.assoc
thf(fact_48_add_Oassoc,axiom,
! [A: int,B: int,C: int] :
( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C )
= ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% add.assoc
thf(fact_49_group__cancel_Oadd2,axiom,
! [B3: nat,K: nat,B: nat,A: nat] :
( ( B3
= ( plus_plus_nat @ K @ B ) )
=> ( ( plus_plus_nat @ A @ B3 )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_50_group__cancel_Oadd2,axiom,
! [B3: int,K: int,B: int,A: int] :
( ( B3
= ( plus_plus_int @ K @ B ) )
=> ( ( plus_plus_int @ A @ B3 )
= ( plus_plus_int @ K @ ( plus_plus_int @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_51_group__cancel_Oadd1,axiom,
! [A3: nat,K: nat,A: nat,B: nat] :
( ( A3
= ( plus_plus_nat @ K @ A ) )
=> ( ( plus_plus_nat @ A3 @ B )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_52_group__cancel_Oadd1,axiom,
! [A3: int,K: int,A: int,B: int] :
( ( A3
= ( plus_plus_int @ K @ A ) )
=> ( ( plus_plus_int @ A3 @ B )
= ( plus_plus_int @ K @ ( plus_plus_int @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_53_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( I = J )
& ( K = L ) )
=> ( ( plus_plus_nat @ I @ K )
= ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_54_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( I = J )
& ( K = L ) )
=> ( ( plus_plus_int @ I @ K )
= ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_55_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: nat,B: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_56_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: int,B: int,C: int] :
( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C )
= ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_57_add_Ogroup__left__neutral,axiom,
! [A: int] :
( ( plus_plus_int @ zero_zero_int @ A )
= A ) ).
% add.group_left_neutral
thf(fact_58_add_Ocomm__neutral,axiom,
! [A: int] :
( ( plus_plus_int @ A @ zero_zero_int )
= A ) ).
% add.comm_neutral
thf(fact_59_add_Ocomm__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.comm_neutral
thf(fact_60_comm__monoid__add__class_Oadd__0,axiom,
! [A: int] :
( ( plus_plus_int @ zero_zero_int @ A )
= A ) ).
% comm_monoid_add_class.add_0
thf(fact_61_comm__monoid__add__class_Oadd__0,axiom,
! [A: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A )
= A ) ).
% comm_monoid_add_class.add_0
thf(fact_62_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_63_int__ops_I1_J,axiom,
( ( semiri1314217659103216013at_int @ zero_zero_nat )
= zero_zero_int ) ).
% int_ops(1)
thf(fact_64_Un__iff,axiom,
! [C: int,A3: set_int,B3: set_int] :
( ( member_int @ C @ ( sup_sup_set_int @ A3 @ B3 ) )
= ( ( member_int @ C @ A3 )
| ( member_int @ C @ B3 ) ) ) ).
% Un_iff
thf(fact_65_Un__iff,axiom,
! [C: product_prod_int_int,A3: set_Pr958786334691620121nt_int,B3: set_Pr958786334691620121nt_int] :
( ( member5262025264175285858nt_int @ C @ ( sup_su6024340866399070445nt_int @ A3 @ B3 ) )
= ( ( member5262025264175285858nt_int @ C @ A3 )
| ( member5262025264175285858nt_int @ C @ B3 ) ) ) ).
% Un_iff
thf(fact_66_UnCI,axiom,
! [C: int,B3: set_int,A3: set_int] :
( ( ~ ( member_int @ C @ B3 )
=> ( member_int @ C @ A3 ) )
=> ( member_int @ C @ ( sup_sup_set_int @ A3 @ B3 ) ) ) ).
% UnCI
thf(fact_67_UnCI,axiom,
! [C: product_prod_int_int,B3: set_Pr958786334691620121nt_int,A3: set_Pr958786334691620121nt_int] :
( ( ~ ( member5262025264175285858nt_int @ C @ B3 )
=> ( member5262025264175285858nt_int @ C @ A3 ) )
=> ( member5262025264175285858nt_int @ C @ ( sup_su6024340866399070445nt_int @ A3 @ B3 ) ) ) ).
% UnCI
thf(fact_68_mem__Collect__eq,axiom,
! [A: product_prod_int_int,P: product_prod_int_int > $o] :
( ( member5262025264175285858nt_int @ A @ ( collec213857154873943460nt_int @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_69_mem__Collect__eq,axiom,
! [A: int,P: int > $o] :
( ( member_int @ A @ ( collect_int @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_70_Collect__mem__eq,axiom,
! [A3: set_Pr958786334691620121nt_int] :
( ( collec213857154873943460nt_int
@ ^ [X2: product_prod_int_int] : ( member5262025264175285858nt_int @ X2 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_71_Collect__mem__eq,axiom,
! [A3: set_int] :
( ( collect_int
@ ^ [X2: int] : ( member_int @ X2 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_72_sup_Oright__idem,axiom,
! [A: set_Pr958786334691620121nt_int,B: set_Pr958786334691620121nt_int] :
( ( sup_su6024340866399070445nt_int @ ( sup_su6024340866399070445nt_int @ A @ B ) @ B )
= ( sup_su6024340866399070445nt_int @ A @ B ) ) ).
% sup.right_idem
thf(fact_73_sup__left__idem,axiom,
! [X: set_Pr958786334691620121nt_int,Y: set_Pr958786334691620121nt_int] :
( ( sup_su6024340866399070445nt_int @ X @ ( sup_su6024340866399070445nt_int @ X @ Y ) )
= ( sup_su6024340866399070445nt_int @ X @ Y ) ) ).
% sup_left_idem
thf(fact_74_sup_Oleft__idem,axiom,
! [A: set_Pr958786334691620121nt_int,B: set_Pr958786334691620121nt_int] :
( ( sup_su6024340866399070445nt_int @ A @ ( sup_su6024340866399070445nt_int @ A @ B ) )
= ( sup_su6024340866399070445nt_int @ A @ B ) ) ).
% sup.left_idem
thf(fact_75_sup__idem,axiom,
! [X: set_Pr958786334691620121nt_int] :
( ( sup_su6024340866399070445nt_int @ X @ X )
= X ) ).
% sup_idem
thf(fact_76_sup_Oidem,axiom,
! [A: set_Pr958786334691620121nt_int] :
( ( sup_su6024340866399070445nt_int @ A @ A )
= A ) ).
% sup.idem
thf(fact_77_old_Oprod_Oinject,axiom,
! [A: int,B: int,A4: int,B4: int] :
( ( ( product_Pair_int_int @ A @ B )
= ( product_Pair_int_int @ A4 @ B4 ) )
= ( ( A = A4 )
& ( B = B4 ) ) ) ).
% old.prod.inject
thf(fact_78_old_Oprod_Oinject,axiom,
! [A: product_prod_int_int,B: product_prod_int_int,A4: product_prod_int_int,B4: product_prod_int_int] :
( ( ( produc3646306378393792727nt_int @ A @ B )
= ( produc3646306378393792727nt_int @ A4 @ B4 ) )
= ( ( A = A4 )
& ( B = B4 ) ) ) ).
% old.prod.inject
thf(fact_79_old_Oprod_Oinject,axiom,
! [A: produc8551481072490612790e_term > option6357759511663192854e_term,B: product_prod_int_int,A4: produc8551481072490612790e_term > option6357759511663192854e_term,B4: product_prod_int_int] :
( ( ( produc5700946648718959541nt_int @ A @ B )
= ( produc5700946648718959541nt_int @ A4 @ B4 ) )
= ( ( A = A4 )
& ( B = B4 ) ) ) ).
% old.prod.inject
thf(fact_80_old_Oprod_Oinject,axiom,
! [A: int > option6357759511663192854e_term,B: product_prod_int_int,A4: int > option6357759511663192854e_term,B4: product_prod_int_int] :
( ( ( produc4305682042979456191nt_int @ A @ B )
= ( produc4305682042979456191nt_int @ A4 @ B4 ) )
= ( ( A = A4 )
& ( B = B4 ) ) ) ).
% old.prod.inject
thf(fact_81_old_Oprod_Oinject,axiom,
! [A: nat,B: set_int,A4: nat,B4: set_int] :
( ( ( produc29655638201817675et_int @ A @ B )
= ( produc29655638201817675et_int @ A4 @ B4 ) )
= ( ( A = A4 )
& ( B = B4 ) ) ) ).
% old.prod.inject
thf(fact_82_prod_Oinject,axiom,
! [X1: int,X22: int,Y1: int,Y2: int] :
( ( ( product_Pair_int_int @ X1 @ X22 )
= ( product_Pair_int_int @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X22 = Y2 ) ) ) ).
% prod.inject
thf(fact_83_prod_Oinject,axiom,
! [X1: product_prod_int_int,X22: product_prod_int_int,Y1: product_prod_int_int,Y2: product_prod_int_int] :
( ( ( produc3646306378393792727nt_int @ X1 @ X22 )
= ( produc3646306378393792727nt_int @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X22 = Y2 ) ) ) ).
% prod.inject
thf(fact_84_prod_Oinject,axiom,
! [X1: produc8551481072490612790e_term > option6357759511663192854e_term,X22: product_prod_int_int,Y1: produc8551481072490612790e_term > option6357759511663192854e_term,Y2: product_prod_int_int] :
( ( ( produc5700946648718959541nt_int @ X1 @ X22 )
= ( produc5700946648718959541nt_int @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X22 = Y2 ) ) ) ).
% prod.inject
thf(fact_85_prod_Oinject,axiom,
! [X1: int > option6357759511663192854e_term,X22: product_prod_int_int,Y1: int > option6357759511663192854e_term,Y2: product_prod_int_int] :
( ( ( produc4305682042979456191nt_int @ X1 @ X22 )
= ( produc4305682042979456191nt_int @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X22 = Y2 ) ) ) ).
% prod.inject
thf(fact_86_prod_Oinject,axiom,
! [X1: nat,X22: set_int,Y1: nat,Y2: set_int] :
( ( ( produc29655638201817675et_int @ X1 @ X22 )
= ( produc29655638201817675et_int @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X22 = Y2 ) ) ) ).
% prod.inject
thf(fact_87_step__checker_Ocases,axiom,
! [X: produc1219242969750017639nt_int] :
~ ! [I2: int,J2: int,I3: int,J3: int] :
( X
!= ( produc3646306378393792727nt_int @ ( product_Pair_int_int @ I2 @ J2 ) @ ( product_Pair_int_int @ I3 @ J3 ) ) ) ).
% step_checker.cases
thf(fact_88_old_Oprod_Oexhaust,axiom,
! [Y: product_prod_int_int] :
~ ! [A5: int,B5: int] :
( Y
!= ( product_Pair_int_int @ A5 @ B5 ) ) ).
% old.prod.exhaust
thf(fact_89_old_Oprod_Oexhaust,axiom,
! [Y: produc1219242969750017639nt_int] :
~ ! [A5: product_prod_int_int,B5: product_prod_int_int] :
( Y
!= ( produc3646306378393792727nt_int @ A5 @ B5 ) ) ).
% old.prod.exhaust
thf(fact_90_old_Oprod_Oexhaust,axiom,
! [Y: produc2285326912895808259nt_int] :
~ ! [A5: produc8551481072490612790e_term > option6357759511663192854e_term,B5: product_prod_int_int] :
( Y
!= ( produc5700946648718959541nt_int @ A5 @ B5 ) ) ).
% old.prod.exhaust
thf(fact_91_old_Oprod_Oexhaust,axiom,
! [Y: produc7773217078559923341nt_int] :
~ ! [A5: int > option6357759511663192854e_term,B5: product_prod_int_int] :
( Y
!= ( produc4305682042979456191nt_int @ A5 @ B5 ) ) ).
% old.prod.exhaust
thf(fact_92_old_Oprod_Oexhaust,axiom,
! [Y: produc9133624956312949779et_int] :
~ ! [A5: nat,B5: set_int] :
( Y
!= ( produc29655638201817675et_int @ A5 @ B5 ) ) ).
% old.prod.exhaust
thf(fact_93_surj__pair,axiom,
! [P2: product_prod_int_int] :
? [X3: int,Y3: int] :
( P2
= ( product_Pair_int_int @ X3 @ Y3 ) ) ).
% surj_pair
thf(fact_94_surj__pair,axiom,
! [P2: produc1219242969750017639nt_int] :
? [X3: product_prod_int_int,Y3: product_prod_int_int] :
( P2
= ( produc3646306378393792727nt_int @ X3 @ Y3 ) ) ).
% surj_pair
thf(fact_95_surj__pair,axiom,
! [P2: produc2285326912895808259nt_int] :
? [X3: produc8551481072490612790e_term > option6357759511663192854e_term,Y3: product_prod_int_int] :
( P2
= ( produc5700946648718959541nt_int @ X3 @ Y3 ) ) ).
% surj_pair
thf(fact_96_surj__pair,axiom,
! [P2: produc7773217078559923341nt_int] :
? [X3: int > option6357759511663192854e_term,Y3: product_prod_int_int] :
( P2
= ( produc4305682042979456191nt_int @ X3 @ Y3 ) ) ).
% surj_pair
thf(fact_97_surj__pair,axiom,
! [P2: produc9133624956312949779et_int] :
? [X3: nat,Y3: set_int] :
( P2
= ( produc29655638201817675et_int @ X3 @ Y3 ) ) ).
% surj_pair
thf(fact_98_prod__cases,axiom,
! [P: product_prod_int_int > $o,P2: product_prod_int_int] :
( ! [A5: int,B5: int] : ( P @ ( product_Pair_int_int @ A5 @ B5 ) )
=> ( P @ P2 ) ) ).
% prod_cases
thf(fact_99_prod__cases,axiom,
! [P: produc1219242969750017639nt_int > $o,P2: produc1219242969750017639nt_int] :
( ! [A5: product_prod_int_int,B5: product_prod_int_int] : ( P @ ( produc3646306378393792727nt_int @ A5 @ B5 ) )
=> ( P @ P2 ) ) ).
% prod_cases
thf(fact_100_prod__cases,axiom,
! [P: produc2285326912895808259nt_int > $o,P2: produc2285326912895808259nt_int] :
( ! [A5: produc8551481072490612790e_term > option6357759511663192854e_term,B5: product_prod_int_int] : ( P @ ( produc5700946648718959541nt_int @ A5 @ B5 ) )
=> ( P @ P2 ) ) ).
% prod_cases
thf(fact_101_prod__cases,axiom,
! [P: produc7773217078559923341nt_int > $o,P2: produc7773217078559923341nt_int] :
( ! [A5: int > option6357759511663192854e_term,B5: product_prod_int_int] : ( P @ ( produc4305682042979456191nt_int @ A5 @ B5 ) )
=> ( P @ P2 ) ) ).
% prod_cases
thf(fact_102_prod__cases,axiom,
! [P: produc9133624956312949779et_int > $o,P2: produc9133624956312949779et_int] :
( ! [A5: nat,B5: set_int] : ( P @ ( produc29655638201817675et_int @ A5 @ B5 ) )
=> ( P @ P2 ) ) ).
% prod_cases
thf(fact_103_Pair__inject,axiom,
! [A: int,B: int,A4: int,B4: int] :
( ( ( product_Pair_int_int @ A @ B )
= ( product_Pair_int_int @ A4 @ B4 ) )
=> ~ ( ( A = A4 )
=> ( B != B4 ) ) ) ).
% Pair_inject
thf(fact_104_Pair__inject,axiom,
! [A: product_prod_int_int,B: product_prod_int_int,A4: product_prod_int_int,B4: product_prod_int_int] :
( ( ( produc3646306378393792727nt_int @ A @ B )
= ( produc3646306378393792727nt_int @ A4 @ B4 ) )
=> ~ ( ( A = A4 )
=> ( B != B4 ) ) ) ).
% Pair_inject
thf(fact_105_Pair__inject,axiom,
! [A: produc8551481072490612790e_term > option6357759511663192854e_term,B: product_prod_int_int,A4: produc8551481072490612790e_term > option6357759511663192854e_term,B4: product_prod_int_int] :
( ( ( produc5700946648718959541nt_int @ A @ B )
= ( produc5700946648718959541nt_int @ A4 @ B4 ) )
=> ~ ( ( A = A4 )
=> ( B != B4 ) ) ) ).
% Pair_inject
thf(fact_106_Pair__inject,axiom,
! [A: int > option6357759511663192854e_term,B: product_prod_int_int,A4: int > option6357759511663192854e_term,B4: product_prod_int_int] :
( ( ( produc4305682042979456191nt_int @ A @ B )
= ( produc4305682042979456191nt_int @ A4 @ B4 ) )
=> ~ ( ( A = A4 )
=> ( B != B4 ) ) ) ).
% Pair_inject
thf(fact_107_Pair__inject,axiom,
! [A: nat,B: set_int,A4: nat,B4: set_int] :
( ( ( produc29655638201817675et_int @ A @ B )
= ( produc29655638201817675et_int @ A4 @ B4 ) )
=> ~ ( ( A = A4 )
=> ( B != B4 ) ) ) ).
% Pair_inject
thf(fact_108_prod__cases3,axiom,
! [Y: produc1219242969750017639nt_int] :
~ ! [A5: product_prod_int_int,B5: int,C2: int] :
( Y
!= ( produc3646306378393792727nt_int @ A5 @ ( product_Pair_int_int @ B5 @ C2 ) ) ) ).
% prod_cases3
thf(fact_109_prod__cases3,axiom,
! [Y: produc2285326912895808259nt_int] :
~ ! [A5: produc8551481072490612790e_term > option6357759511663192854e_term,B5: int,C2: int] :
( Y
!= ( produc5700946648718959541nt_int @ A5 @ ( product_Pair_int_int @ B5 @ C2 ) ) ) ).
% prod_cases3
thf(fact_110_prod__cases3,axiom,
! [Y: produc7773217078559923341nt_int] :
~ ! [A5: int > option6357759511663192854e_term,B5: int,C2: int] :
( Y
!= ( produc4305682042979456191nt_int @ A5 @ ( product_Pair_int_int @ B5 @ C2 ) ) ) ).
% prod_cases3
thf(fact_111_prod__induct3,axiom,
! [P: produc1219242969750017639nt_int > $o,X: produc1219242969750017639nt_int] :
( ! [A5: product_prod_int_int,B5: int,C2: int] : ( P @ ( produc3646306378393792727nt_int @ A5 @ ( product_Pair_int_int @ B5 @ C2 ) ) )
=> ( P @ X ) ) ).
% prod_induct3
thf(fact_112_prod__induct3,axiom,
! [P: produc2285326912895808259nt_int > $o,X: produc2285326912895808259nt_int] :
( ! [A5: produc8551481072490612790e_term > option6357759511663192854e_term,B5: int,C2: int] : ( P @ ( produc5700946648718959541nt_int @ A5 @ ( product_Pair_int_int @ B5 @ C2 ) ) )
=> ( P @ X ) ) ).
% prod_induct3
thf(fact_113_prod__induct3,axiom,
! [P: produc7773217078559923341nt_int > $o,X: produc7773217078559923341nt_int] :
( ! [A5: int > option6357759511663192854e_term,B5: int,C2: int] : ( P @ ( produc4305682042979456191nt_int @ A5 @ ( product_Pair_int_int @ B5 @ C2 ) ) )
=> ( P @ X ) ) ).
% prod_induct3
thf(fact_114_inf__sup__aci_I8_J,axiom,
! [X: set_Pr958786334691620121nt_int,Y: set_Pr958786334691620121nt_int] :
( ( sup_su6024340866399070445nt_int @ X @ ( sup_su6024340866399070445nt_int @ X @ Y ) )
= ( sup_su6024340866399070445nt_int @ X @ Y ) ) ).
% inf_sup_aci(8)
thf(fact_115_inf__sup__aci_I7_J,axiom,
! [X: set_Pr958786334691620121nt_int,Y: set_Pr958786334691620121nt_int,Z: set_Pr958786334691620121nt_int] :
( ( sup_su6024340866399070445nt_int @ X @ ( sup_su6024340866399070445nt_int @ Y @ Z ) )
= ( sup_su6024340866399070445nt_int @ Y @ ( sup_su6024340866399070445nt_int @ X @ Z ) ) ) ).
% inf_sup_aci(7)
thf(fact_116_inf__sup__aci_I6_J,axiom,
! [X: set_Pr958786334691620121nt_int,Y: set_Pr958786334691620121nt_int,Z: set_Pr958786334691620121nt_int] :
( ( sup_su6024340866399070445nt_int @ ( sup_su6024340866399070445nt_int @ X @ Y ) @ Z )
= ( sup_su6024340866399070445nt_int @ X @ ( sup_su6024340866399070445nt_int @ Y @ Z ) ) ) ).
% inf_sup_aci(6)
thf(fact_117_inf__sup__aci_I5_J,axiom,
( sup_su6024340866399070445nt_int
= ( ^ [X2: set_Pr958786334691620121nt_int,Y4: set_Pr958786334691620121nt_int] : ( sup_su6024340866399070445nt_int @ Y4 @ X2 ) ) ) ).
% inf_sup_aci(5)
thf(fact_118_sup_Oassoc,axiom,
! [A: set_Pr958786334691620121nt_int,B: set_Pr958786334691620121nt_int,C: set_Pr958786334691620121nt_int] :
( ( sup_su6024340866399070445nt_int @ ( sup_su6024340866399070445nt_int @ A @ B ) @ C )
= ( sup_su6024340866399070445nt_int @ A @ ( sup_su6024340866399070445nt_int @ B @ C ) ) ) ).
% sup.assoc
thf(fact_119_sup__assoc,axiom,
! [X: set_Pr958786334691620121nt_int,Y: set_Pr958786334691620121nt_int,Z: set_Pr958786334691620121nt_int] :
( ( sup_su6024340866399070445nt_int @ ( sup_su6024340866399070445nt_int @ X @ Y ) @ Z )
= ( sup_su6024340866399070445nt_int @ X @ ( sup_su6024340866399070445nt_int @ Y @ Z ) ) ) ).
% sup_assoc
thf(fact_120_sup_Ocommute,axiom,
( sup_su6024340866399070445nt_int
= ( ^ [A2: set_Pr958786334691620121nt_int,B2: set_Pr958786334691620121nt_int] : ( sup_su6024340866399070445nt_int @ B2 @ A2 ) ) ) ).
% sup.commute
thf(fact_121_sup__commute,axiom,
( sup_su6024340866399070445nt_int
= ( ^ [X2: set_Pr958786334691620121nt_int,Y4: set_Pr958786334691620121nt_int] : ( sup_su6024340866399070445nt_int @ Y4 @ X2 ) ) ) ).
% sup_commute
thf(fact_122_sup_Oleft__commute,axiom,
! [B: set_Pr958786334691620121nt_int,A: set_Pr958786334691620121nt_int,C: set_Pr958786334691620121nt_int] :
( ( sup_su6024340866399070445nt_int @ B @ ( sup_su6024340866399070445nt_int @ A @ C ) )
= ( sup_su6024340866399070445nt_int @ A @ ( sup_su6024340866399070445nt_int @ B @ C ) ) ) ).
% sup.left_commute
thf(fact_123_sup__left__commute,axiom,
! [X: set_Pr958786334691620121nt_int,Y: set_Pr958786334691620121nt_int,Z: set_Pr958786334691620121nt_int] :
( ( sup_su6024340866399070445nt_int @ X @ ( sup_su6024340866399070445nt_int @ Y @ Z ) )
= ( sup_su6024340866399070445nt_int @ Y @ ( sup_su6024340866399070445nt_int @ X @ Z ) ) ) ).
% sup_left_commute
thf(fact_124_plus__int__code_I2_J,axiom,
! [L: int] :
( ( plus_plus_int @ zero_zero_int @ L )
= L ) ).
% plus_int_code(2)
thf(fact_125_plus__int__code_I1_J,axiom,
! [K: int] :
( ( plus_plus_int @ K @ zero_zero_int )
= K ) ).
% plus_int_code(1)
thf(fact_126_UnE,axiom,
! [C: int,A3: set_int,B3: set_int] :
( ( member_int @ C @ ( sup_sup_set_int @ A3 @ B3 ) )
=> ( ~ ( member_int @ C @ A3 )
=> ( member_int @ C @ B3 ) ) ) ).
% UnE
thf(fact_127_UnE,axiom,
! [C: product_prod_int_int,A3: set_Pr958786334691620121nt_int,B3: set_Pr958786334691620121nt_int] :
( ( member5262025264175285858nt_int @ C @ ( sup_su6024340866399070445nt_int @ A3 @ B3 ) )
=> ( ~ ( member5262025264175285858nt_int @ C @ A3 )
=> ( member5262025264175285858nt_int @ C @ B3 ) ) ) ).
% UnE
thf(fact_128_UnI1,axiom,
! [C: int,A3: set_int,B3: set_int] :
( ( member_int @ C @ A3 )
=> ( member_int @ C @ ( sup_sup_set_int @ A3 @ B3 ) ) ) ).
% UnI1
thf(fact_129_UnI1,axiom,
! [C: product_prod_int_int,A3: set_Pr958786334691620121nt_int,B3: set_Pr958786334691620121nt_int] :
( ( member5262025264175285858nt_int @ C @ A3 )
=> ( member5262025264175285858nt_int @ C @ ( sup_su6024340866399070445nt_int @ A3 @ B3 ) ) ) ).
% UnI1
thf(fact_130_UnI2,axiom,
! [C: int,B3: set_int,A3: set_int] :
( ( member_int @ C @ B3 )
=> ( member_int @ C @ ( sup_sup_set_int @ A3 @ B3 ) ) ) ).
% UnI2
thf(fact_131_UnI2,axiom,
! [C: product_prod_int_int,B3: set_Pr958786334691620121nt_int,A3: set_Pr958786334691620121nt_int] :
( ( member5262025264175285858nt_int @ C @ B3 )
=> ( member5262025264175285858nt_int @ C @ ( sup_su6024340866399070445nt_int @ A3 @ B3 ) ) ) ).
% UnI2
thf(fact_132_bex__Un,axiom,
! [A3: set_Pr958786334691620121nt_int,B3: set_Pr958786334691620121nt_int,P: product_prod_int_int > $o] :
( ( ? [X2: product_prod_int_int] :
( ( member5262025264175285858nt_int @ X2 @ ( sup_su6024340866399070445nt_int @ A3 @ B3 ) )
& ( P @ X2 ) ) )
= ( ? [X2: product_prod_int_int] :
( ( member5262025264175285858nt_int @ X2 @ A3 )
& ( P @ X2 ) )
| ? [X2: product_prod_int_int] :
( ( member5262025264175285858nt_int @ X2 @ B3 )
& ( P @ X2 ) ) ) ) ).
% bex_Un
thf(fact_133_ball__Un,axiom,
! [A3: set_Pr958786334691620121nt_int,B3: set_Pr958786334691620121nt_int,P: product_prod_int_int > $o] :
( ( ! [X2: product_prod_int_int] :
( ( member5262025264175285858nt_int @ X2 @ ( sup_su6024340866399070445nt_int @ A3 @ B3 ) )
=> ( P @ X2 ) ) )
= ( ! [X2: product_prod_int_int] :
( ( member5262025264175285858nt_int @ X2 @ A3 )
=> ( P @ X2 ) )
& ! [X2: product_prod_int_int] :
( ( member5262025264175285858nt_int @ X2 @ B3 )
=> ( P @ X2 ) ) ) ) ).
% ball_Un
thf(fact_134_Un__assoc,axiom,
! [A3: set_Pr958786334691620121nt_int,B3: set_Pr958786334691620121nt_int,C3: set_Pr958786334691620121nt_int] :
( ( sup_su6024340866399070445nt_int @ ( sup_su6024340866399070445nt_int @ A3 @ B3 ) @ C3 )
= ( sup_su6024340866399070445nt_int @ A3 @ ( sup_su6024340866399070445nt_int @ B3 @ C3 ) ) ) ).
% Un_assoc
thf(fact_135_Un__absorb,axiom,
! [A3: set_Pr958786334691620121nt_int] :
( ( sup_su6024340866399070445nt_int @ A3 @ A3 )
= A3 ) ).
% Un_absorb
thf(fact_136_Un__commute,axiom,
( sup_su6024340866399070445nt_int
= ( ^ [A6: set_Pr958786334691620121nt_int,B6: set_Pr958786334691620121nt_int] : ( sup_su6024340866399070445nt_int @ B6 @ A6 ) ) ) ).
% Un_commute
thf(fact_137_Un__left__absorb,axiom,
! [A3: set_Pr958786334691620121nt_int,B3: set_Pr958786334691620121nt_int] :
( ( sup_su6024340866399070445nt_int @ A3 @ ( sup_su6024340866399070445nt_int @ A3 @ B3 ) )
= ( sup_su6024340866399070445nt_int @ A3 @ B3 ) ) ).
% Un_left_absorb
thf(fact_138_Un__left__commute,axiom,
! [A3: set_Pr958786334691620121nt_int,B3: set_Pr958786334691620121nt_int,C3: set_Pr958786334691620121nt_int] :
( ( sup_su6024340866399070445nt_int @ A3 @ ( sup_su6024340866399070445nt_int @ B3 @ C3 ) )
= ( sup_su6024340866399070445nt_int @ B3 @ ( sup_su6024340866399070445nt_int @ A3 @ C3 ) ) ) ).
% Un_left_commute
thf(fact_139_nat__int__comparison_I1_J,axiom,
( ( ^ [Y5: nat,Z2: nat] : ( Y5 = Z2 ) )
= ( ^ [A2: nat,B2: nat] :
( ( semiri1314217659103216013at_int @ A2 )
= ( semiri1314217659103216013at_int @ B2 ) ) ) ) ).
% nat_int_comparison(1)
thf(fact_140_int__if,axiom,
! [P: $o,A: nat,B: nat] :
( ( P
=> ( ( semiri1314217659103216013at_int @ ( if_nat @ P @ A @ B ) )
= ( semiri1314217659103216013at_int @ A ) ) )
& ( ~ P
=> ( ( semiri1314217659103216013at_int @ ( if_nat @ P @ A @ B ) )
= ( semiri1314217659103216013at_int @ B ) ) ) ) ).
% int_if
thf(fact_141_int__int__eq,axiom,
! [M: nat,N: nat] :
( ( ( semiri1314217659103216013at_int @ M )
= ( semiri1314217659103216013at_int @ N ) )
= ( M = N ) ) ).
% int_int_eq
thf(fact_142_verit__sum__simplify,axiom,
! [A: int] :
( ( plus_plus_int @ A @ zero_zero_int )
= A ) ).
% verit_sum_simplify
thf(fact_143_verit__sum__simplify,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% verit_sum_simplify
thf(fact_144_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_145_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_146_zadd__int__left,axiom,
! [M: nat,N: nat,Z: int] :
( ( plus_plus_int @ ( semiri1314217659103216013at_int @ M ) @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ Z ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ ( plus_plus_nat @ M @ N ) ) @ Z ) ) ).
% zadd_int_left
thf(fact_147_Euclid__induct,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A5: nat,B5: nat] :
( ( P @ A5 @ B5 )
= ( P @ B5 @ A5 ) )
=> ( ! [A5: nat] : ( P @ A5 @ zero_zero_nat )
=> ( ! [A5: nat,B5: nat] :
( ( P @ A5 @ B5 )
=> ( P @ A5 @ ( plus_plus_nat @ A5 @ B5 ) ) )
=> ( P @ A @ B ) ) ) ) ).
% Euclid_induct
thf(fact_148_add__0__iff,axiom,
! [B: int,A: int] :
( ( B
= ( plus_plus_int @ B @ A ) )
= ( A = zero_zero_int ) ) ).
% add_0_iff
thf(fact_149_add__0__iff,axiom,
! [B: nat,A: nat] :
( ( B
= ( plus_plus_nat @ B @ A ) )
= ( A = zero_zero_nat ) ) ).
% add_0_iff
thf(fact_150_transpose__trans__board,axiom,
! [K_1: int,K_2: int,B: set_Pr958786334691620121nt_int] :
( ( transpose_board @ ( trans_board @ ( product_Pair_int_int @ K_1 @ K_2 ) @ B ) )
= ( trans_board @ ( product_Pair_int_int @ K_2 @ K_1 ) @ ( transpose_board @ B ) ) ) ).
% transpose_trans_board
thf(fact_151_divides__aux__eq,axiom,
! [Q: int,R: int] :
( ( unique5329631941980267465ux_int @ ( product_Pair_int_int @ Q @ R ) )
= ( R = zero_zero_int ) ) ).
% divides_aux_eq
thf(fact_152_divides__aux__eq,axiom,
! [Q: nat,R: nat] :
( ( unique5332122412489317741ux_nat @ ( product_Pair_nat_nat @ Q @ R ) )
= ( R = zero_zero_nat ) ) ).
% divides_aux_eq
thf(fact_153_of__nat__le__0__iff,axiom,
! [M: nat] :
( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ M ) @ zero_zero_nat )
= ( M = zero_zero_nat ) ) ).
% of_nat_le_0_iff
thf(fact_154_of__nat__le__0__iff,axiom,
! [M: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ zero_zero_int )
= ( M = zero_zero_nat ) ) ).
% of_nat_le_0_iff
thf(fact_155_full__exhaustive__int_H_Ocases,axiom,
! [X: produc2285326912895808259nt_int] :
~ ! [F: produc8551481072490612790e_term > option6357759511663192854e_term,D: int,I2: int] :
( X
!= ( produc5700946648718959541nt_int @ F @ ( product_Pair_int_int @ D @ I2 ) ) ) ).
% full_exhaustive_int'.cases
thf(fact_156_exhaustive__int_H_Ocases,axiom,
! [X: produc7773217078559923341nt_int] :
~ ! [F: int > option6357759511663192854e_term,D: int,I2: int] :
( X
!= ( produc4305682042979456191nt_int @ F @ ( product_Pair_int_int @ D @ I2 ) ) ) ).
% exhaustive_int'.cases
thf(fact_157_small__lazy_H_Ocases,axiom,
! [X: product_prod_int_int] :
~ ! [D: int,I2: int] :
( X
!= ( product_Pair_int_int @ D @ I2 ) ) ).
% small_lazy'.cases
thf(fact_158_zero__integer_Orsp,axiom,
zero_zero_int = zero_zero_int ).
% zero_integer.rsp
thf(fact_159_bot__nat__0_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% bot_nat_0.extremum
thf(fact_160_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_161_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_162_Un__subset__iff,axiom,
! [A3: set_Pr958786334691620121nt_int,B3: set_Pr958786334691620121nt_int,C3: set_Pr958786334691620121nt_int] :
( ( ord_le2843351958646193337nt_int @ ( sup_su6024340866399070445nt_int @ A3 @ B3 ) @ C3 )
= ( ( ord_le2843351958646193337nt_int @ A3 @ C3 )
& ( ord_le2843351958646193337nt_int @ B3 @ C3 ) ) ) ).
% Un_subset_iff
thf(fact_163_le__zero__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_164_add__le__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_cancel_right
thf(fact_165_add__le__cancel__right,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
= ( ord_less_eq_int @ A @ B ) ) ).
% add_le_cancel_right
thf(fact_166_add__le__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
= ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_cancel_left
thf(fact_167_add__le__cancel__left,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
= ( ord_less_eq_int @ A @ B ) ) ).
% add_le_cancel_left
thf(fact_168_sup_Obounded__iff,axiom,
! [B: nat,C: nat,A: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ B @ C ) @ A )
= ( ( ord_less_eq_nat @ B @ A )
& ( ord_less_eq_nat @ C @ A ) ) ) ).
% sup.bounded_iff
thf(fact_169_sup_Obounded__iff,axiom,
! [B: int,C: int,A: int] :
( ( ord_less_eq_int @ ( sup_sup_int @ B @ C ) @ A )
= ( ( ord_less_eq_int @ B @ A )
& ( ord_less_eq_int @ C @ A ) ) ) ).
% sup.bounded_iff
thf(fact_170_sup_Obounded__iff,axiom,
! [B: set_Pr958786334691620121nt_int,C: set_Pr958786334691620121nt_int,A: set_Pr958786334691620121nt_int] :
( ( ord_le2843351958646193337nt_int @ ( sup_su6024340866399070445nt_int @ B @ C ) @ A )
= ( ( ord_le2843351958646193337nt_int @ B @ A )
& ( ord_le2843351958646193337nt_int @ C @ A ) ) ) ).
% sup.bounded_iff
thf(fact_171_le__sup__iff,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ X @ Y ) @ Z )
= ( ( ord_less_eq_nat @ X @ Z )
& ( ord_less_eq_nat @ Y @ Z ) ) ) ).
% le_sup_iff
thf(fact_172_le__sup__iff,axiom,
! [X: int,Y: int,Z: int] :
( ( ord_less_eq_int @ ( sup_sup_int @ X @ Y ) @ Z )
= ( ( ord_less_eq_int @ X @ Z )
& ( ord_less_eq_int @ Y @ Z ) ) ) ).
% le_sup_iff
thf(fact_173_le__sup__iff,axiom,
! [X: set_Pr958786334691620121nt_int,Y: set_Pr958786334691620121nt_int,Z: set_Pr958786334691620121nt_int] :
( ( ord_le2843351958646193337nt_int @ ( sup_su6024340866399070445nt_int @ X @ Y ) @ Z )
= ( ( ord_le2843351958646193337nt_int @ X @ Z )
& ( ord_le2843351958646193337nt_int @ Y @ Z ) ) ) ).
% le_sup_iff
thf(fact_174_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_175_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_176_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_177_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_178_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_179_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_180_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_181_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_182_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_183_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_184_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_185_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_186_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_187_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_188_verit__comp__simplify1_I2_J,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_189_verit__comp__simplify1_I2_J,axiom,
! [A: int] : ( ord_less_eq_int @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_190_verit__comp__simplify1_I2_J,axiom,
! [A: set_Pr958786334691620121nt_int] : ( ord_le2843351958646193337nt_int @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_191_transpose__board2,axiom,
! [B: set_Pr958786334691620121nt_int] :
( ( transpose_board @ ( transpose_board @ B ) )
= B ) ).
% transpose_board2
thf(fact_192_of__nat__mono,axiom,
! [I: nat,J: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ I ) @ ( semiri1316708129612266289at_nat @ J ) ) ) ).
% of_nat_mono
thf(fact_193_of__nat__mono,axiom,
! [I: nat,J: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ I ) @ ( semiri1314217659103216013at_int @ J ) ) ) ).
% of_nat_mono
thf(fact_194_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_195_nat__int__comparison_I3_J,axiom,
( ord_less_eq_nat
= ( ^ [A2: nat,B2: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A2 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ) ).
% nat_int_comparison(3)
thf(fact_196_board__leq__subset,axiom,
! [N_1: nat,N_2: nat,M_1: nat,M_2: nat] :
( ( ( ord_less_eq_nat @ N_1 @ N_2 )
& ( ord_less_eq_nat @ M_1 @ M_2 ) )
=> ( ord_le2843351958646193337nt_int @ ( board @ N_1 @ M_1 ) @ ( board @ N_2 @ M_2 ) ) ) ).
% board_leq_subset
thf(fact_197_board__subset,axiom,
! [N_1: nat,N_2: nat,M_1: nat,M_2: nat] :
( ( ord_less_eq_nat @ N_1 @ N_2 )
=> ( ( ord_less_eq_nat @ M_1 @ M_2 )
=> ( ord_le2843351958646193337nt_int @ ( board @ N_1 @ M_1 ) @ ( board @ N_2 @ M_2 ) ) ) ) ).
% board_subset
thf(fact_198_zero__le,axiom,
! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).
% zero_le
thf(fact_199_add__le__imp__le__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_imp_le_right
thf(fact_200_add__le__imp__le__right,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
=> ( ord_less_eq_int @ A @ B ) ) ).
% add_le_imp_le_right
thf(fact_201_add__le__imp__le__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_imp_le_left
thf(fact_202_add__le__imp__le__left,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
=> ( ord_less_eq_int @ A @ B ) ) ).
% add_le_imp_le_left
thf(fact_203_le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [A2: nat,B2: nat] :
? [C4: nat] :
( B2
= ( plus_plus_nat @ A2 @ C4 ) ) ) ) ).
% le_iff_add
thf(fact_204_add__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).
% add_right_mono
thf(fact_205_add__right__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) ) ) ).
% add_right_mono
thf(fact_206_less__eqE,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ~ ! [C2: nat] :
( B
!= ( plus_plus_nat @ A @ C2 ) ) ) ).
% less_eqE
thf(fact_207_add__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) ) ) ).
% add_left_mono
thf(fact_208_add__left__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) ) ) ).
% add_left_mono
thf(fact_209_add__mono,axiom,
! [A: nat,B: nat,C: nat,D2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D2 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D2 ) ) ) ) ).
% add_mono
thf(fact_210_add__mono,axiom,
! [A: int,B: int,C: int,D2: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ C @ D2 )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D2 ) ) ) ) ).
% add_mono
thf(fact_211_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_212_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_eq_int @ I @ J )
& ( ord_less_eq_int @ K @ L ) )
=> ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_213_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( I = J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_214_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( I = J )
& ( ord_less_eq_int @ K @ L ) )
=> ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_215_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( K = L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_216_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_eq_int @ I @ J )
& ( K = L ) )
=> ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_217_sup_OcoboundedI2,axiom,
! [C: nat,B: nat,A: nat] :
( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_eq_nat @ C @ ( sup_sup_nat @ A @ B ) ) ) ).
% sup.coboundedI2
thf(fact_218_sup_OcoboundedI2,axiom,
! [C: int,B: int,A: int] :
( ( ord_less_eq_int @ C @ B )
=> ( ord_less_eq_int @ C @ ( sup_sup_int @ A @ B ) ) ) ).
% sup.coboundedI2
thf(fact_219_sup_OcoboundedI2,axiom,
! [C: set_Pr958786334691620121nt_int,B: set_Pr958786334691620121nt_int,A: set_Pr958786334691620121nt_int] :
( ( ord_le2843351958646193337nt_int @ C @ B )
=> ( ord_le2843351958646193337nt_int @ C @ ( sup_su6024340866399070445nt_int @ A @ B ) ) ) ).
% sup.coboundedI2
thf(fact_220_sup_OcoboundedI1,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ C @ A )
=> ( ord_less_eq_nat @ C @ ( sup_sup_nat @ A @ B ) ) ) ).
% sup.coboundedI1
thf(fact_221_sup_OcoboundedI1,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_eq_int @ C @ A )
=> ( ord_less_eq_int @ C @ ( sup_sup_int @ A @ B ) ) ) ).
% sup.coboundedI1
thf(fact_222_sup_OcoboundedI1,axiom,
! [C: set_Pr958786334691620121nt_int,A: set_Pr958786334691620121nt_int,B: set_Pr958786334691620121nt_int] :
( ( ord_le2843351958646193337nt_int @ C @ A )
=> ( ord_le2843351958646193337nt_int @ C @ ( sup_su6024340866399070445nt_int @ A @ B ) ) ) ).
% sup.coboundedI1
thf(fact_223_sup_Oabsorb__iff2,axiom,
( ord_less_eq_nat
= ( ^ [A2: nat,B2: nat] :
( ( sup_sup_nat @ A2 @ B2 )
= B2 ) ) ) ).
% sup.absorb_iff2
thf(fact_224_sup_Oabsorb__iff2,axiom,
( ord_less_eq_int
= ( ^ [A2: int,B2: int] :
( ( sup_sup_int @ A2 @ B2 )
= B2 ) ) ) ).
% sup.absorb_iff2
thf(fact_225_sup_Oabsorb__iff2,axiom,
( ord_le2843351958646193337nt_int
= ( ^ [A2: set_Pr958786334691620121nt_int,B2: set_Pr958786334691620121nt_int] :
( ( sup_su6024340866399070445nt_int @ A2 @ B2 )
= B2 ) ) ) ).
% sup.absorb_iff2
thf(fact_226_sup_Oabsorb__iff1,axiom,
( ord_less_eq_nat
= ( ^ [B2: nat,A2: nat] :
( ( sup_sup_nat @ A2 @ B2 )
= A2 ) ) ) ).
% sup.absorb_iff1
thf(fact_227_sup_Oabsorb__iff1,axiom,
( ord_less_eq_int
= ( ^ [B2: int,A2: int] :
( ( sup_sup_int @ A2 @ B2 )
= A2 ) ) ) ).
% sup.absorb_iff1
thf(fact_228_sup_Oabsorb__iff1,axiom,
( ord_le2843351958646193337nt_int
= ( ^ [B2: set_Pr958786334691620121nt_int,A2: set_Pr958786334691620121nt_int] :
( ( sup_su6024340866399070445nt_int @ A2 @ B2 )
= A2 ) ) ) ).
% sup.absorb_iff1
thf(fact_229_sup_Ocobounded2,axiom,
! [B: nat,A: nat] : ( ord_less_eq_nat @ B @ ( sup_sup_nat @ A @ B ) ) ).
% sup.cobounded2
thf(fact_230_sup_Ocobounded2,axiom,
! [B: int,A: int] : ( ord_less_eq_int @ B @ ( sup_sup_int @ A @ B ) ) ).
% sup.cobounded2
thf(fact_231_sup_Ocobounded2,axiom,
! [B: set_Pr958786334691620121nt_int,A: set_Pr958786334691620121nt_int] : ( ord_le2843351958646193337nt_int @ B @ ( sup_su6024340866399070445nt_int @ A @ B ) ) ).
% sup.cobounded2
thf(fact_232_sup_Ocobounded1,axiom,
! [A: nat,B: nat] : ( ord_less_eq_nat @ A @ ( sup_sup_nat @ A @ B ) ) ).
% sup.cobounded1
thf(fact_233_sup_Ocobounded1,axiom,
! [A: int,B: int] : ( ord_less_eq_int @ A @ ( sup_sup_int @ A @ B ) ) ).
% sup.cobounded1
thf(fact_234_sup_Ocobounded1,axiom,
! [A: set_Pr958786334691620121nt_int,B: set_Pr958786334691620121nt_int] : ( ord_le2843351958646193337nt_int @ A @ ( sup_su6024340866399070445nt_int @ A @ B ) ) ).
% sup.cobounded1
thf(fact_235_sup_Oorder__iff,axiom,
( ord_less_eq_nat
= ( ^ [B2: nat,A2: nat] :
( A2
= ( sup_sup_nat @ A2 @ B2 ) ) ) ) ).
% sup.order_iff
thf(fact_236_sup_Oorder__iff,axiom,
( ord_less_eq_int
= ( ^ [B2: int,A2: int] :
( A2
= ( sup_sup_int @ A2 @ B2 ) ) ) ) ).
% sup.order_iff
thf(fact_237_sup_Oorder__iff,axiom,
( ord_le2843351958646193337nt_int
= ( ^ [B2: set_Pr958786334691620121nt_int,A2: set_Pr958786334691620121nt_int] :
( A2
= ( sup_su6024340866399070445nt_int @ A2 @ B2 ) ) ) ) ).
% sup.order_iff
thf(fact_238_sup_OboundedI,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_eq_nat @ C @ A )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ B @ C ) @ A ) ) ) ).
% sup.boundedI
thf(fact_239_sup_OboundedI,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( ord_less_eq_int @ C @ A )
=> ( ord_less_eq_int @ ( sup_sup_int @ B @ C ) @ A ) ) ) ).
% sup.boundedI
thf(fact_240_sup_OboundedI,axiom,
! [B: set_Pr958786334691620121nt_int,A: set_Pr958786334691620121nt_int,C: set_Pr958786334691620121nt_int] :
( ( ord_le2843351958646193337nt_int @ B @ A )
=> ( ( ord_le2843351958646193337nt_int @ C @ A )
=> ( ord_le2843351958646193337nt_int @ ( sup_su6024340866399070445nt_int @ B @ C ) @ A ) ) ) ).
% sup.boundedI
thf(fact_241_sup_OboundedE,axiom,
! [B: nat,C: nat,A: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ B @ C ) @ A )
=> ~ ( ( ord_less_eq_nat @ B @ A )
=> ~ ( ord_less_eq_nat @ C @ A ) ) ) ).
% sup.boundedE
thf(fact_242_sup_OboundedE,axiom,
! [B: int,C: int,A: int] :
( ( ord_less_eq_int @ ( sup_sup_int @ B @ C ) @ A )
=> ~ ( ( ord_less_eq_int @ B @ A )
=> ~ ( ord_less_eq_int @ C @ A ) ) ) ).
% sup.boundedE
thf(fact_243_sup_OboundedE,axiom,
! [B: set_Pr958786334691620121nt_int,C: set_Pr958786334691620121nt_int,A: set_Pr958786334691620121nt_int] :
( ( ord_le2843351958646193337nt_int @ ( sup_su6024340866399070445nt_int @ B @ C ) @ A )
=> ~ ( ( ord_le2843351958646193337nt_int @ B @ A )
=> ~ ( ord_le2843351958646193337nt_int @ C @ A ) ) ) ).
% sup.boundedE
thf(fact_244_sup__absorb2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( sup_sup_nat @ X @ Y )
= Y ) ) ).
% sup_absorb2
thf(fact_245_sup__absorb2,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( sup_sup_int @ X @ Y )
= Y ) ) ).
% sup_absorb2
thf(fact_246_sup__absorb2,axiom,
! [X: set_Pr958786334691620121nt_int,Y: set_Pr958786334691620121nt_int] :
( ( ord_le2843351958646193337nt_int @ X @ Y )
=> ( ( sup_su6024340866399070445nt_int @ X @ Y )
= Y ) ) ).
% sup_absorb2
thf(fact_247_sup__absorb1,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ( ( sup_sup_nat @ X @ Y )
= X ) ) ).
% sup_absorb1
thf(fact_248_sup__absorb1,axiom,
! [Y: int,X: int] :
( ( ord_less_eq_int @ Y @ X )
=> ( ( sup_sup_int @ X @ Y )
= X ) ) ).
% sup_absorb1
thf(fact_249_sup__absorb1,axiom,
! [Y: set_Pr958786334691620121nt_int,X: set_Pr958786334691620121nt_int] :
( ( ord_le2843351958646193337nt_int @ Y @ X )
=> ( ( sup_su6024340866399070445nt_int @ X @ Y )
= X ) ) ).
% sup_absorb1
thf(fact_250_sup_Oabsorb2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( sup_sup_nat @ A @ B )
= B ) ) ).
% sup.absorb2
thf(fact_251_sup_Oabsorb2,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( sup_sup_int @ A @ B )
= B ) ) ).
% sup.absorb2
thf(fact_252_sup_Oabsorb2,axiom,
! [A: set_Pr958786334691620121nt_int,B: set_Pr958786334691620121nt_int] :
( ( ord_le2843351958646193337nt_int @ A @ B )
=> ( ( sup_su6024340866399070445nt_int @ A @ B )
= B ) ) ).
% sup.absorb2
thf(fact_253_sup_Oabsorb1,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( sup_sup_nat @ A @ B )
= A ) ) ).
% sup.absorb1
thf(fact_254_sup_Oabsorb1,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( sup_sup_int @ A @ B )
= A ) ) ).
% sup.absorb1
thf(fact_255_sup_Oabsorb1,axiom,
! [B: set_Pr958786334691620121nt_int,A: set_Pr958786334691620121nt_int] :
( ( ord_le2843351958646193337nt_int @ B @ A )
=> ( ( sup_su6024340866399070445nt_int @ A @ B )
= A ) ) ).
% sup.absorb1
thf(fact_256_sup__unique,axiom,
! [F2: nat > nat > nat,X: nat,Y: nat] :
( ! [X3: nat,Y3: nat] : ( ord_less_eq_nat @ X3 @ ( F2 @ X3 @ Y3 ) )
=> ( ! [X3: nat,Y3: nat] : ( ord_less_eq_nat @ Y3 @ ( F2 @ X3 @ Y3 ) )
=> ( ! [X3: nat,Y3: nat,Z3: nat] :
( ( ord_less_eq_nat @ Y3 @ X3 )
=> ( ( ord_less_eq_nat @ Z3 @ X3 )
=> ( ord_less_eq_nat @ ( F2 @ Y3 @ Z3 ) @ X3 ) ) )
=> ( ( sup_sup_nat @ X @ Y )
= ( F2 @ X @ Y ) ) ) ) ) ).
% sup_unique
thf(fact_257_sup__unique,axiom,
! [F2: int > int > int,X: int,Y: int] :
( ! [X3: int,Y3: int] : ( ord_less_eq_int @ X3 @ ( F2 @ X3 @ Y3 ) )
=> ( ! [X3: int,Y3: int] : ( ord_less_eq_int @ Y3 @ ( F2 @ X3 @ Y3 ) )
=> ( ! [X3: int,Y3: int,Z3: int] :
( ( ord_less_eq_int @ Y3 @ X3 )
=> ( ( ord_less_eq_int @ Z3 @ X3 )
=> ( ord_less_eq_int @ ( F2 @ Y3 @ Z3 ) @ X3 ) ) )
=> ( ( sup_sup_int @ X @ Y )
= ( F2 @ X @ Y ) ) ) ) ) ).
% sup_unique
thf(fact_258_sup__unique,axiom,
! [F2: set_Pr958786334691620121nt_int > set_Pr958786334691620121nt_int > set_Pr958786334691620121nt_int,X: set_Pr958786334691620121nt_int,Y: set_Pr958786334691620121nt_int] :
( ! [X3: set_Pr958786334691620121nt_int,Y3: set_Pr958786334691620121nt_int] : ( ord_le2843351958646193337nt_int @ X3 @ ( F2 @ X3 @ Y3 ) )
=> ( ! [X3: set_Pr958786334691620121nt_int,Y3: set_Pr958786334691620121nt_int] : ( ord_le2843351958646193337nt_int @ Y3 @ ( F2 @ X3 @ Y3 ) )
=> ( ! [X3: set_Pr958786334691620121nt_int,Y3: set_Pr958786334691620121nt_int,Z3: set_Pr958786334691620121nt_int] :
( ( ord_le2843351958646193337nt_int @ Y3 @ X3 )
=> ( ( ord_le2843351958646193337nt_int @ Z3 @ X3 )
=> ( ord_le2843351958646193337nt_int @ ( F2 @ Y3 @ Z3 ) @ X3 ) ) )
=> ( ( sup_su6024340866399070445nt_int @ X @ Y )
= ( F2 @ X @ Y ) ) ) ) ) ).
% sup_unique
thf(fact_259_sup_OorderI,axiom,
! [A: nat,B: nat] :
( ( A
= ( sup_sup_nat @ A @ B ) )
=> ( ord_less_eq_nat @ B @ A ) ) ).
% sup.orderI
thf(fact_260_sup_OorderI,axiom,
! [A: int,B: int] :
( ( A
= ( sup_sup_int @ A @ B ) )
=> ( ord_less_eq_int @ B @ A ) ) ).
% sup.orderI
thf(fact_261_sup_OorderI,axiom,
! [A: set_Pr958786334691620121nt_int,B: set_Pr958786334691620121nt_int] :
( ( A
= ( sup_su6024340866399070445nt_int @ A @ B ) )
=> ( ord_le2843351958646193337nt_int @ B @ A ) ) ).
% sup.orderI
thf(fact_262_sup_OorderE,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( A
= ( sup_sup_nat @ A @ B ) ) ) ).
% sup.orderE
thf(fact_263_sup_OorderE,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ B @ A )
=> ( A
= ( sup_sup_int @ A @ B ) ) ) ).
% sup.orderE
thf(fact_264_sup_OorderE,axiom,
! [B: set_Pr958786334691620121nt_int,A: set_Pr958786334691620121nt_int] :
( ( ord_le2843351958646193337nt_int @ B @ A )
=> ( A
= ( sup_su6024340866399070445nt_int @ A @ B ) ) ) ).
% sup.orderE
thf(fact_265_le__iff__sup,axiom,
( ord_less_eq_nat
= ( ^ [X2: nat,Y4: nat] :
( ( sup_sup_nat @ X2 @ Y4 )
= Y4 ) ) ) ).
% le_iff_sup
thf(fact_266_le__iff__sup,axiom,
( ord_less_eq_int
= ( ^ [X2: int,Y4: int] :
( ( sup_sup_int @ X2 @ Y4 )
= Y4 ) ) ) ).
% le_iff_sup
thf(fact_267_le__iff__sup,axiom,
( ord_le2843351958646193337nt_int
= ( ^ [X2: set_Pr958786334691620121nt_int,Y4: set_Pr958786334691620121nt_int] :
( ( sup_su6024340866399070445nt_int @ X2 @ Y4 )
= Y4 ) ) ) ).
% le_iff_sup
thf(fact_268_sup__least,axiom,
! [Y: nat,X: nat,Z: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ( ( ord_less_eq_nat @ Z @ X )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ Y @ Z ) @ X ) ) ) ).
% sup_least
thf(fact_269_sup__least,axiom,
! [Y: int,X: int,Z: int] :
( ( ord_less_eq_int @ Y @ X )
=> ( ( ord_less_eq_int @ Z @ X )
=> ( ord_less_eq_int @ ( sup_sup_int @ Y @ Z ) @ X ) ) ) ).
% sup_least
thf(fact_270_sup__least,axiom,
! [Y: set_Pr958786334691620121nt_int,X: set_Pr958786334691620121nt_int,Z: set_Pr958786334691620121nt_int] :
( ( ord_le2843351958646193337nt_int @ Y @ X )
=> ( ( ord_le2843351958646193337nt_int @ Z @ X )
=> ( ord_le2843351958646193337nt_int @ ( sup_su6024340866399070445nt_int @ Y @ Z ) @ X ) ) ) ).
% sup_least
thf(fact_271_sup__mono,axiom,
! [A: nat,C: nat,B: nat,D2: nat] :
( ( ord_less_eq_nat @ A @ C )
=> ( ( ord_less_eq_nat @ B @ D2 )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ A @ B ) @ ( sup_sup_nat @ C @ D2 ) ) ) ) ).
% sup_mono
thf(fact_272_sup__mono,axiom,
! [A: int,C: int,B: int,D2: int] :
( ( ord_less_eq_int @ A @ C )
=> ( ( ord_less_eq_int @ B @ D2 )
=> ( ord_less_eq_int @ ( sup_sup_int @ A @ B ) @ ( sup_sup_int @ C @ D2 ) ) ) ) ).
% sup_mono
thf(fact_273_sup__mono,axiom,
! [A: set_Pr958786334691620121nt_int,C: set_Pr958786334691620121nt_int,B: set_Pr958786334691620121nt_int,D2: set_Pr958786334691620121nt_int] :
( ( ord_le2843351958646193337nt_int @ A @ C )
=> ( ( ord_le2843351958646193337nt_int @ B @ D2 )
=> ( ord_le2843351958646193337nt_int @ ( sup_su6024340866399070445nt_int @ A @ B ) @ ( sup_su6024340866399070445nt_int @ C @ D2 ) ) ) ) ).
% sup_mono
thf(fact_274_sup_Omono,axiom,
! [C: nat,A: nat,D2: nat,B: nat] :
( ( ord_less_eq_nat @ C @ A )
=> ( ( ord_less_eq_nat @ D2 @ B )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ C @ D2 ) @ ( sup_sup_nat @ A @ B ) ) ) ) ).
% sup.mono
thf(fact_275_sup_Omono,axiom,
! [C: int,A: int,D2: int,B: int] :
( ( ord_less_eq_int @ C @ A )
=> ( ( ord_less_eq_int @ D2 @ B )
=> ( ord_less_eq_int @ ( sup_sup_int @ C @ D2 ) @ ( sup_sup_int @ A @ B ) ) ) ) ).
% sup.mono
thf(fact_276_sup_Omono,axiom,
! [C: set_Pr958786334691620121nt_int,A: set_Pr958786334691620121nt_int,D2: set_Pr958786334691620121nt_int,B: set_Pr958786334691620121nt_int] :
( ( ord_le2843351958646193337nt_int @ C @ A )
=> ( ( ord_le2843351958646193337nt_int @ D2 @ B )
=> ( ord_le2843351958646193337nt_int @ ( sup_su6024340866399070445nt_int @ C @ D2 ) @ ( sup_su6024340866399070445nt_int @ A @ B ) ) ) ) ).
% sup.mono
thf(fact_277_le__supI2,axiom,
! [X: nat,B: nat,A: nat] :
( ( ord_less_eq_nat @ X @ B )
=> ( ord_less_eq_nat @ X @ ( sup_sup_nat @ A @ B ) ) ) ).
% le_supI2
thf(fact_278_le__supI2,axiom,
! [X: int,B: int,A: int] :
( ( ord_less_eq_int @ X @ B )
=> ( ord_less_eq_int @ X @ ( sup_sup_int @ A @ B ) ) ) ).
% le_supI2
thf(fact_279_le__supI2,axiom,
! [X: set_Pr958786334691620121nt_int,B: set_Pr958786334691620121nt_int,A: set_Pr958786334691620121nt_int] :
( ( ord_le2843351958646193337nt_int @ X @ B )
=> ( ord_le2843351958646193337nt_int @ X @ ( sup_su6024340866399070445nt_int @ A @ B ) ) ) ).
% le_supI2
thf(fact_280_le__supI1,axiom,
! [X: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ X @ A )
=> ( ord_less_eq_nat @ X @ ( sup_sup_nat @ A @ B ) ) ) ).
% le_supI1
thf(fact_281_le__supI1,axiom,
! [X: int,A: int,B: int] :
( ( ord_less_eq_int @ X @ A )
=> ( ord_less_eq_int @ X @ ( sup_sup_int @ A @ B ) ) ) ).
% le_supI1
thf(fact_282_le__supI1,axiom,
! [X: set_Pr958786334691620121nt_int,A: set_Pr958786334691620121nt_int,B: set_Pr958786334691620121nt_int] :
( ( ord_le2843351958646193337nt_int @ X @ A )
=> ( ord_le2843351958646193337nt_int @ X @ ( sup_su6024340866399070445nt_int @ A @ B ) ) ) ).
% le_supI1
thf(fact_283_sup__ge2,axiom,
! [Y: nat,X: nat] : ( ord_less_eq_nat @ Y @ ( sup_sup_nat @ X @ Y ) ) ).
% sup_ge2
thf(fact_284_sup__ge2,axiom,
! [Y: int,X: int] : ( ord_less_eq_int @ Y @ ( sup_sup_int @ X @ Y ) ) ).
% sup_ge2
thf(fact_285_sup__ge2,axiom,
! [Y: set_Pr958786334691620121nt_int,X: set_Pr958786334691620121nt_int] : ( ord_le2843351958646193337nt_int @ Y @ ( sup_su6024340866399070445nt_int @ X @ Y ) ) ).
% sup_ge2
thf(fact_286_sup__ge1,axiom,
! [X: nat,Y: nat] : ( ord_less_eq_nat @ X @ ( sup_sup_nat @ X @ Y ) ) ).
% sup_ge1
thf(fact_287_sup__ge1,axiom,
! [X: int,Y: int] : ( ord_less_eq_int @ X @ ( sup_sup_int @ X @ Y ) ) ).
% sup_ge1
thf(fact_288_sup__ge1,axiom,
! [X: set_Pr958786334691620121nt_int,Y: set_Pr958786334691620121nt_int] : ( ord_le2843351958646193337nt_int @ X @ ( sup_su6024340866399070445nt_int @ X @ Y ) ) ).
% sup_ge1
thf(fact_289_le__supI,axiom,
! [A: nat,X: nat,B: nat] :
( ( ord_less_eq_nat @ A @ X )
=> ( ( ord_less_eq_nat @ B @ X )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ A @ B ) @ X ) ) ) ).
% le_supI
thf(fact_290_le__supI,axiom,
! [A: int,X: int,B: int] :
( ( ord_less_eq_int @ A @ X )
=> ( ( ord_less_eq_int @ B @ X )
=> ( ord_less_eq_int @ ( sup_sup_int @ A @ B ) @ X ) ) ) ).
% le_supI
thf(fact_291_le__supI,axiom,
! [A: set_Pr958786334691620121nt_int,X: set_Pr958786334691620121nt_int,B: set_Pr958786334691620121nt_int] :
( ( ord_le2843351958646193337nt_int @ A @ X )
=> ( ( ord_le2843351958646193337nt_int @ B @ X )
=> ( ord_le2843351958646193337nt_int @ ( sup_su6024340866399070445nt_int @ A @ B ) @ X ) ) ) ).
% le_supI
thf(fact_292_le__supE,axiom,
! [A: nat,B: nat,X: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ A @ B ) @ X )
=> ~ ( ( ord_less_eq_nat @ A @ X )
=> ~ ( ord_less_eq_nat @ B @ X ) ) ) ).
% le_supE
thf(fact_293_le__supE,axiom,
! [A: int,B: int,X: int] :
( ( ord_less_eq_int @ ( sup_sup_int @ A @ B ) @ X )
=> ~ ( ( ord_less_eq_int @ A @ X )
=> ~ ( ord_less_eq_int @ B @ X ) ) ) ).
% le_supE
thf(fact_294_le__supE,axiom,
! [A: set_Pr958786334691620121nt_int,B: set_Pr958786334691620121nt_int,X: set_Pr958786334691620121nt_int] :
( ( ord_le2843351958646193337nt_int @ ( sup_su6024340866399070445nt_int @ A @ B ) @ X )
=> ~ ( ( ord_le2843351958646193337nt_int @ A @ X )
=> ~ ( ord_le2843351958646193337nt_int @ B @ X ) ) ) ).
% le_supE
thf(fact_295_inf__sup__ord_I3_J,axiom,
! [X: nat,Y: nat] : ( ord_less_eq_nat @ X @ ( sup_sup_nat @ X @ Y ) ) ).
% inf_sup_ord(3)
thf(fact_296_inf__sup__ord_I3_J,axiom,
! [X: int,Y: int] : ( ord_less_eq_int @ X @ ( sup_sup_int @ X @ Y ) ) ).
% inf_sup_ord(3)
thf(fact_297_inf__sup__ord_I3_J,axiom,
! [X: set_Pr958786334691620121nt_int,Y: set_Pr958786334691620121nt_int] : ( ord_le2843351958646193337nt_int @ X @ ( sup_su6024340866399070445nt_int @ X @ Y ) ) ).
% inf_sup_ord(3)
thf(fact_298_inf__sup__ord_I4_J,axiom,
! [Y: nat,X: nat] : ( ord_less_eq_nat @ Y @ ( sup_sup_nat @ X @ Y ) ) ).
% inf_sup_ord(4)
thf(fact_299_inf__sup__ord_I4_J,axiom,
! [Y: int,X: int] : ( ord_less_eq_int @ Y @ ( sup_sup_int @ X @ Y ) ) ).
% inf_sup_ord(4)
thf(fact_300_inf__sup__ord_I4_J,axiom,
! [Y: set_Pr958786334691620121nt_int,X: set_Pr958786334691620121nt_int] : ( ord_le2843351958646193337nt_int @ Y @ ( sup_su6024340866399070445nt_int @ X @ Y ) ) ).
% inf_sup_ord(4)
thf(fact_301_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_302_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_303_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_304_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_305_less__eq__int__code_I1_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% less_eq_int_code(1)
thf(fact_306_nat__le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [M2: nat,N2: nat] :
? [K2: nat] :
( N2
= ( plus_plus_nat @ M2 @ K2 ) ) ) ) ).
% nat_le_iff_add
thf(fact_307_trans__le__add2,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ M @ J ) ) ) ).
% trans_le_add2
thf(fact_308_trans__le__add1,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ J @ M ) ) ) ).
% trans_le_add1
thf(fact_309_add__le__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% add_le_mono1
thf(fact_310_add__le__mono,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ K @ L )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_le_mono
thf(fact_311_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_312_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_313_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_314_le__add2,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M @ N ) ) ).
% le_add2
thf(fact_315_le__add1,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M ) ) ).
% le_add1
thf(fact_316_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_317_subset__Un__eq,axiom,
( ord_le2843351958646193337nt_int
= ( ^ [A6: set_Pr958786334691620121nt_int,B6: set_Pr958786334691620121nt_int] :
( ( sup_su6024340866399070445nt_int @ A6 @ B6 )
= B6 ) ) ) ).
% subset_Un_eq
thf(fact_318_subset__UnE,axiom,
! [C3: set_Pr958786334691620121nt_int,A3: set_Pr958786334691620121nt_int,B3: set_Pr958786334691620121nt_int] :
( ( ord_le2843351958646193337nt_int @ C3 @ ( sup_su6024340866399070445nt_int @ A3 @ B3 ) )
=> ~ ! [A7: set_Pr958786334691620121nt_int] :
( ( ord_le2843351958646193337nt_int @ A7 @ A3 )
=> ! [B7: set_Pr958786334691620121nt_int] :
( ( ord_le2843351958646193337nt_int @ B7 @ B3 )
=> ( C3
!= ( sup_su6024340866399070445nt_int @ A7 @ B7 ) ) ) ) ) ).
% subset_UnE
thf(fact_319_Un__absorb2,axiom,
! [B3: set_Pr958786334691620121nt_int,A3: set_Pr958786334691620121nt_int] :
( ( ord_le2843351958646193337nt_int @ B3 @ A3 )
=> ( ( sup_su6024340866399070445nt_int @ A3 @ B3 )
= A3 ) ) ).
% Un_absorb2
thf(fact_320_Un__absorb1,axiom,
! [A3: set_Pr958786334691620121nt_int,B3: set_Pr958786334691620121nt_int] :
( ( ord_le2843351958646193337nt_int @ A3 @ B3 )
=> ( ( sup_su6024340866399070445nt_int @ A3 @ B3 )
= B3 ) ) ).
% Un_absorb1
thf(fact_321_Un__upper2,axiom,
! [B3: set_Pr958786334691620121nt_int,A3: set_Pr958786334691620121nt_int] : ( ord_le2843351958646193337nt_int @ B3 @ ( sup_su6024340866399070445nt_int @ A3 @ B3 ) ) ).
% Un_upper2
thf(fact_322_Un__upper1,axiom,
! [A3: set_Pr958786334691620121nt_int,B3: set_Pr958786334691620121nt_int] : ( ord_le2843351958646193337nt_int @ A3 @ ( sup_su6024340866399070445nt_int @ A3 @ B3 ) ) ).
% Un_upper1
thf(fact_323_Un__least,axiom,
! [A3: set_Pr958786334691620121nt_int,C3: set_Pr958786334691620121nt_int,B3: set_Pr958786334691620121nt_int] :
( ( ord_le2843351958646193337nt_int @ A3 @ C3 )
=> ( ( ord_le2843351958646193337nt_int @ B3 @ C3 )
=> ( ord_le2843351958646193337nt_int @ ( sup_su6024340866399070445nt_int @ A3 @ B3 ) @ C3 ) ) ) ).
% Un_least
thf(fact_324_Un__mono,axiom,
! [A3: set_Pr958786334691620121nt_int,C3: set_Pr958786334691620121nt_int,B3: set_Pr958786334691620121nt_int,D3: set_Pr958786334691620121nt_int] :
( ( ord_le2843351958646193337nt_int @ A3 @ C3 )
=> ( ( ord_le2843351958646193337nt_int @ B3 @ D3 )
=> ( ord_le2843351958646193337nt_int @ ( sup_su6024340866399070445nt_int @ A3 @ B3 ) @ ( sup_su6024340866399070445nt_int @ C3 @ D3 ) ) ) ) ).
% Un_mono
thf(fact_325_add__nonpos__eq__0__iff,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ Y @ zero_zero_nat )
=> ( ( ( plus_plus_nat @ X @ Y )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_326_add__nonpos__eq__0__iff,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ zero_zero_int )
=> ( ( ord_less_eq_int @ Y @ zero_zero_int )
=> ( ( ( plus_plus_int @ X @ Y )
= zero_zero_int )
= ( ( X = zero_zero_int )
& ( Y = zero_zero_int ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_327_add__nonneg__eq__0__iff,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ X )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
=> ( ( ( plus_plus_nat @ X @ Y )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_328_add__nonneg__eq__0__iff,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ( ( plus_plus_int @ X @ Y )
= zero_zero_int )
= ( ( X = zero_zero_int )
& ( Y = zero_zero_int ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_329_add__nonpos__nonpos,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_nonpos_nonpos
thf(fact_330_add__nonpos__nonpos,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( ord_less_eq_int @ B @ zero_zero_int )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).
% add_nonpos_nonpos
thf(fact_331_add__nonneg__nonneg,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_332_add__nonneg__nonneg,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_333_add__increasing2,axiom,
! [C: nat,B: nat,A: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ( ord_less_eq_nat @ B @ A )
=> ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_increasing2
thf(fact_334_add__increasing2,axiom,
! [C: int,B: int,A: int] :
( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ( ord_less_eq_int @ B @ A )
=> ( ord_less_eq_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).
% add_increasing2
thf(fact_335_add__decreasing2,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ C @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ B ) ) ) ).
% add_decreasing2
thf(fact_336_add__decreasing2,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ( ord_less_eq_int @ A @ B )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ B ) ) ) ).
% add_decreasing2
thf(fact_337_add__increasing,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_increasing
thf(fact_338_add__increasing,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ord_less_eq_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).
% add_increasing
thf(fact_339_add__decreasing,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ B ) ) ) ).
% add_decreasing
thf(fact_340_add__decreasing,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( ord_less_eq_int @ C @ B )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ B ) ) ) ).
% add_decreasing
thf(fact_341_of__nat__0__le__iff,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( semiri1316708129612266289at_nat @ N ) ) ).
% of_nat_0_le_iff
thf(fact_342_of__nat__0__le__iff,axiom,
! [N: nat] : ( ord_less_eq_int @ zero_zero_int @ ( semiri1314217659103216013at_int @ N ) ) ).
% of_nat_0_le_iff
thf(fact_343_transpose__board,axiom,
! [N: nat,M: nat] :
( ( transpose_board @ ( board @ N @ M ) )
= ( board @ M @ N ) ) ).
% transpose_board
thf(fact_344_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_345_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_346_zle__iff__zadd,axiom,
( ord_less_eq_int
= ( ^ [W: int,Z4: int] :
? [N2: nat] :
( Z4
= ( plus_plus_int @ W @ ( semiri1314217659103216013at_int @ N2 ) ) ) ) ) ).
% zle_iff_zadd
thf(fact_347_transpose__union,axiom,
! [B_1: set_Pr958786334691620121nt_int,B_2: set_Pr958786334691620121nt_int] :
( ( transpose_board @ ( sup_su6024340866399070445nt_int @ B_1 @ B_2 ) )
= ( sup_su6024340866399070445nt_int @ ( transpose_board @ B_1 ) @ ( transpose_board @ B_2 ) ) ) ).
% transpose_union
thf(fact_348_zero__natural_Orsp,axiom,
zero_zero_nat = zero_zero_nat ).
% zero_natural.rsp
thf(fact_349_order__refl,axiom,
! [X: nat] : ( ord_less_eq_nat @ X @ X ) ).
% order_refl
thf(fact_350_order__refl,axiom,
! [X: int] : ( ord_less_eq_int @ X @ X ) ).
% order_refl
thf(fact_351_order__refl,axiom,
! [X: set_Pr958786334691620121nt_int] : ( ord_le2843351958646193337nt_int @ X @ X ) ).
% order_refl
thf(fact_352_dual__order_Orefl,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% dual_order.refl
thf(fact_353_dual__order_Orefl,axiom,
! [A: int] : ( ord_less_eq_int @ A @ A ) ).
% dual_order.refl
thf(fact_354_dual__order_Orefl,axiom,
! [A: set_Pr958786334691620121nt_int] : ( ord_le2843351958646193337nt_int @ A @ A ) ).
% dual_order.refl
thf(fact_355_le__numeral__extra_I3_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% le_numeral_extra(3)
thf(fact_356_le__numeral__extra_I3_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% le_numeral_extra(3)
thf(fact_357_same__fstI,axiom,
! [P: nat > $o,X: nat,Y6: set_int,Y: set_int,R2: nat > set_Pr2522554150109002629et_int] :
( ( P @ X )
=> ( ( member2572552093476627150et_int @ ( produc6363374080413544029et_int @ Y6 @ Y ) @ ( R2 @ X ) )
=> ( member5126324565730479632et_int @ ( produc985091676681408599et_int @ ( produc29655638201817675et_int @ X @ Y6 ) @ ( produc29655638201817675et_int @ X @ Y ) ) @ ( same_fst_nat_set_int @ P @ R2 ) ) ) ) ).
% same_fstI
thf(fact_358_same__fstI,axiom,
! [P: int > $o,X: int,Y6: int,Y: int,R2: int > set_Pr958786334691620121nt_int] :
( ( P @ X )
=> ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ Y6 @ Y ) @ ( R2 @ X ) )
=> ( member8566619992076573584nt_int @ ( produc3646306378393792727nt_int @ ( product_Pair_int_int @ X @ Y6 ) @ ( product_Pair_int_int @ X @ Y ) ) @ ( same_fst_int_int @ P @ R2 ) ) ) ) ).
% same_fstI
thf(fact_359_same__fstI,axiom,
! [P: product_prod_int_int > $o,X: product_prod_int_int,Y6: product_prod_int_int,Y: product_prod_int_int,R2: product_prod_int_int > set_Pr2560585780119916871nt_int] :
( ( P @ X )
=> ( ( member8566619992076573584nt_int @ ( produc3646306378393792727nt_int @ Y6 @ Y ) @ ( R2 @ X ) )
=> ( member8053542592415931152nt_int @ ( produc1923601798490594135nt_int @ ( produc3646306378393792727nt_int @ X @ Y6 ) @ ( produc3646306378393792727nt_int @ X @ Y ) ) @ ( same_f2440470920016040620nt_int @ P @ R2 ) ) ) ) ).
% same_fstI
thf(fact_360_same__fstI,axiom,
! [P: ( produc8551481072490612790e_term > option6357759511663192854e_term ) > $o,X: produc8551481072490612790e_term > option6357759511663192854e_term,Y6: product_prod_int_int,Y: product_prod_int_int,R2: ( produc8551481072490612790e_term > option6357759511663192854e_term ) > set_Pr2560585780119916871nt_int] :
( ( P @ X )
=> ( ( member8566619992076573584nt_int @ ( produc3646306378393792727nt_int @ Y6 @ Y ) @ ( R2 @ X ) )
=> ( member6582457606847315088nt_int @ ( produc7601053194514725023nt_int @ ( produc5700946648718959541nt_int @ X @ Y6 ) @ ( produc5700946648718959541nt_int @ X @ Y ) ) @ ( same_f5472592420709775776nt_int @ P @ R2 ) ) ) ) ).
% same_fstI
thf(fact_361_same__fstI,axiom,
! [P: ( int > option6357759511663192854e_term ) > $o,X: int > option6357759511663192854e_term,Y6: product_prod_int_int,Y: product_prod_int_int,R2: ( int > option6357759511663192854e_term ) > set_Pr2560585780119916871nt_int] :
( ( P @ X )
=> ( ( member8566619992076573584nt_int @ ( produc3646306378393792727nt_int @ Y6 @ Y ) @ ( R2 @ X ) )
=> ( member4085533954029916580nt_int @ ( produc8406175334058502835nt_int @ ( produc4305682042979456191nt_int @ X @ Y6 ) @ ( produc4305682042979456191nt_int @ X @ Y ) ) @ ( same_f6662725367016992042nt_int @ P @ R2 ) ) ) ) ).
% same_fstI
thf(fact_362_relChain__def,axiom,
( bNF_Ca1968104039914474786nt_nat
= ( ^ [R3: set_Pr958786334691620121nt_int,As: int > nat] :
! [I4: int,J4: int] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ I4 @ J4 ) @ R3 )
=> ( ord_less_eq_nat @ ( As @ I4 ) @ ( As @ J4 ) ) ) ) ) ).
% relChain_def
thf(fact_363_relChain__def,axiom,
( bNF_Ca1643832818461744997nt_nat
= ( ^ [R3: set_Pr2560585780119916871nt_int,As: product_prod_int_int > nat] :
! [I4: product_prod_int_int,J4: product_prod_int_int] :
( ( member8566619992076573584nt_int @ ( produc3646306378393792727nt_int @ I4 @ J4 ) @ R3 )
=> ( ord_less_eq_nat @ ( As @ I4 ) @ ( As @ J4 ) ) ) ) ) ).
% relChain_def
thf(fact_364_relChain__def,axiom,
( bNF_Ca1965613569405424510nt_int
= ( ^ [R3: set_Pr958786334691620121nt_int,As: int > int] :
! [I4: int,J4: int] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ I4 @ J4 ) @ R3 )
=> ( ord_less_eq_int @ ( As @ I4 ) @ ( As @ J4 ) ) ) ) ) ).
% relChain_def
thf(fact_365_relChain__def,axiom,
( bNF_Ca1641342347952694721nt_int
= ( ^ [R3: set_Pr2560585780119916871nt_int,As: product_prod_int_int > int] :
! [I4: product_prod_int_int,J4: product_prod_int_int] :
( ( member8566619992076573584nt_int @ ( produc3646306378393792727nt_int @ I4 @ J4 ) @ R3 )
=> ( ord_less_eq_int @ ( As @ I4 ) @ ( As @ J4 ) ) ) ) ) ).
% relChain_def
thf(fact_366_relChain__def,axiom,
( bNF_Ca8719598144974034247nt_int
= ( ^ [R3: set_Pr958786334691620121nt_int,As: int > set_Pr958786334691620121nt_int] :
! [I4: int,J4: int] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ I4 @ J4 ) @ R3 )
=> ( ord_le2843351958646193337nt_int @ ( As @ I4 ) @ ( As @ J4 ) ) ) ) ) ).
% relChain_def
thf(fact_367_relChain__def,axiom,
( bNF_Ca5742924509254848324nt_int
= ( ^ [R3: set_Pr2560585780119916871nt_int,As: product_prod_int_int > set_Pr958786334691620121nt_int] :
! [I4: product_prod_int_int,J4: product_prod_int_int] :
( ( member8566619992076573584nt_int @ ( produc3646306378393792727nt_int @ I4 @ J4 ) @ R3 )
=> ( ord_le2843351958646193337nt_int @ ( As @ I4 ) @ ( As @ J4 ) ) ) ) ) ).
% relChain_def
thf(fact_368_conj__le__cong,axiom,
! [X: int,X4: int,P: $o,P3: $o] :
( ( X = X4 )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X4 )
=> ( P = P3 ) )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X )
& P )
= ( ( ord_less_eq_int @ zero_zero_int @ X4 )
& P3 ) ) ) ) ).
% conj_le_cong
thf(fact_369_imp__le__cong,axiom,
! [X: int,X4: int,P: $o,P3: $o] :
( ( X = X4 )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X4 )
=> ( P = P3 ) )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X )
=> P )
= ( ( ord_less_eq_int @ zero_zero_int @ X4 )
=> P3 ) ) ) ) ).
% imp_le_cong
thf(fact_370_boolean__algebra__cancel_Osup2,axiom,
! [B3: set_Pr958786334691620121nt_int,K: set_Pr958786334691620121nt_int,B: set_Pr958786334691620121nt_int,A: set_Pr958786334691620121nt_int] :
( ( B3
= ( sup_su6024340866399070445nt_int @ K @ B ) )
=> ( ( sup_su6024340866399070445nt_int @ A @ B3 )
= ( sup_su6024340866399070445nt_int @ K @ ( sup_su6024340866399070445nt_int @ A @ B ) ) ) ) ).
% boolean_algebra_cancel.sup2
thf(fact_371_boolean__algebra__cancel_Osup1,axiom,
! [A3: set_Pr958786334691620121nt_int,K: set_Pr958786334691620121nt_int,A: set_Pr958786334691620121nt_int,B: set_Pr958786334691620121nt_int] :
( ( A3
= ( sup_su6024340866399070445nt_int @ K @ A ) )
=> ( ( sup_su6024340866399070445nt_int @ A3 @ B )
= ( sup_su6024340866399070445nt_int @ K @ ( sup_su6024340866399070445nt_int @ A @ B ) ) ) ) ).
% boolean_algebra_cancel.sup1
thf(fact_372_subset__antisym,axiom,
! [A3: set_Pr958786334691620121nt_int,B3: set_Pr958786334691620121nt_int] :
( ( ord_le2843351958646193337nt_int @ A3 @ B3 )
=> ( ( ord_le2843351958646193337nt_int @ B3 @ A3 )
=> ( A3 = B3 ) ) ) ).
% subset_antisym
thf(fact_373_subsetI,axiom,
! [A3: set_int,B3: set_int] :
( ! [X3: int] :
( ( member_int @ X3 @ A3 )
=> ( member_int @ X3 @ B3 ) )
=> ( ord_less_eq_set_int @ A3 @ B3 ) ) ).
% subsetI
thf(fact_374_subsetI,axiom,
! [A3: set_Pr958786334691620121nt_int,B3: set_Pr958786334691620121nt_int] :
( ! [X3: product_prod_int_int] :
( ( member5262025264175285858nt_int @ X3 @ A3 )
=> ( member5262025264175285858nt_int @ X3 @ B3 ) )
=> ( ord_le2843351958646193337nt_int @ A3 @ B3 ) ) ).
% subsetI
thf(fact_375_Collect__mono__iff,axiom,
! [P: product_prod_int_int > $o,Q2: product_prod_int_int > $o] :
( ( ord_le2843351958646193337nt_int @ ( collec213857154873943460nt_int @ P ) @ ( collec213857154873943460nt_int @ Q2 ) )
= ( ! [X2: product_prod_int_int] :
( ( P @ X2 )
=> ( Q2 @ X2 ) ) ) ) ).
% Collect_mono_iff
thf(fact_376_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_377_set__eq__subset,axiom,
( ( ^ [Y5: set_Pr958786334691620121nt_int,Z2: set_Pr958786334691620121nt_int] : ( Y5 = Z2 ) )
= ( ^ [A6: set_Pr958786334691620121nt_int,B6: set_Pr958786334691620121nt_int] :
( ( ord_le2843351958646193337nt_int @ A6 @ B6 )
& ( ord_le2843351958646193337nt_int @ B6 @ A6 ) ) ) ) ).
% set_eq_subset
thf(fact_378_subset__trans,axiom,
! [A3: set_Pr958786334691620121nt_int,B3: set_Pr958786334691620121nt_int,C3: set_Pr958786334691620121nt_int] :
( ( ord_le2843351958646193337nt_int @ A3 @ B3 )
=> ( ( ord_le2843351958646193337nt_int @ B3 @ C3 )
=> ( ord_le2843351958646193337nt_int @ A3 @ C3 ) ) ) ).
% subset_trans
thf(fact_379_Collect__mono,axiom,
! [P: product_prod_int_int > $o,Q2: product_prod_int_int > $o] :
( ! [X3: product_prod_int_int] :
( ( P @ X3 )
=> ( Q2 @ X3 ) )
=> ( ord_le2843351958646193337nt_int @ ( collec213857154873943460nt_int @ P ) @ ( collec213857154873943460nt_int @ Q2 ) ) ) ).
% Collect_mono
thf(fact_380_subset__refl,axiom,
! [A3: set_Pr958786334691620121nt_int] : ( ord_le2843351958646193337nt_int @ A3 @ A3 ) ).
% subset_refl
thf(fact_381_subset__iff,axiom,
( ord_less_eq_set_int
= ( ^ [A6: set_int,B6: set_int] :
! [T: int] :
( ( member_int @ T @ A6 )
=> ( member_int @ T @ B6 ) ) ) ) ).
% subset_iff
thf(fact_382_subset__iff,axiom,
( ord_le2843351958646193337nt_int
= ( ^ [A6: set_Pr958786334691620121nt_int,B6: set_Pr958786334691620121nt_int] :
! [T: product_prod_int_int] :
( ( member5262025264175285858nt_int @ T @ A6 )
=> ( member5262025264175285858nt_int @ T @ B6 ) ) ) ) ).
% subset_iff
thf(fact_383_equalityD2,axiom,
! [A3: set_Pr958786334691620121nt_int,B3: set_Pr958786334691620121nt_int] :
( ( A3 = B3 )
=> ( ord_le2843351958646193337nt_int @ B3 @ A3 ) ) ).
% equalityD2
thf(fact_384_equalityD1,axiom,
! [A3: set_Pr958786334691620121nt_int,B3: set_Pr958786334691620121nt_int] :
( ( A3 = B3 )
=> ( ord_le2843351958646193337nt_int @ A3 @ B3 ) ) ).
% equalityD1
thf(fact_385_subset__eq,axiom,
( ord_less_eq_set_int
= ( ^ [A6: set_int,B6: set_int] :
! [X2: int] :
( ( member_int @ X2 @ A6 )
=> ( member_int @ X2 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_386_subset__eq,axiom,
( ord_le2843351958646193337nt_int
= ( ^ [A6: set_Pr958786334691620121nt_int,B6: set_Pr958786334691620121nt_int] :
! [X2: product_prod_int_int] :
( ( member5262025264175285858nt_int @ X2 @ A6 )
=> ( member5262025264175285858nt_int @ X2 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_387_equalityE,axiom,
! [A3: set_Pr958786334691620121nt_int,B3: set_Pr958786334691620121nt_int] :
( ( A3 = B3 )
=> ~ ( ( ord_le2843351958646193337nt_int @ A3 @ B3 )
=> ~ ( ord_le2843351958646193337nt_int @ B3 @ A3 ) ) ) ).
% equalityE
thf(fact_388_subsetD,axiom,
! [A3: set_int,B3: set_int,C: int] :
( ( ord_less_eq_set_int @ A3 @ B3 )
=> ( ( member_int @ C @ A3 )
=> ( member_int @ C @ B3 ) ) ) ).
% subsetD
thf(fact_389_subsetD,axiom,
! [A3: set_Pr958786334691620121nt_int,B3: set_Pr958786334691620121nt_int,C: product_prod_int_int] :
( ( ord_le2843351958646193337nt_int @ A3 @ B3 )
=> ( ( member5262025264175285858nt_int @ C @ A3 )
=> ( member5262025264175285858nt_int @ C @ B3 ) ) ) ).
% subsetD
thf(fact_390_in__mono,axiom,
! [A3: set_int,B3: set_int,X: int] :
( ( ord_less_eq_set_int @ A3 @ B3 )
=> ( ( member_int @ X @ A3 )
=> ( member_int @ X @ B3 ) ) ) ).
% in_mono
thf(fact_391_in__mono,axiom,
! [A3: set_Pr958786334691620121nt_int,B3: set_Pr958786334691620121nt_int,X: product_prod_int_int] :
( ( ord_le2843351958646193337nt_int @ A3 @ B3 )
=> ( ( member5262025264175285858nt_int @ X @ A3 )
=> ( member5262025264175285858nt_int @ X @ B3 ) ) ) ).
% in_mono
thf(fact_392_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K: nat,B: nat] :
( ( P @ K )
=> ( ! [Y3: nat] :
( ( P @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ B ) )
=> ? [X3: nat] :
( ( P @ X3 )
& ! [Y7: nat] :
( ( P @ Y7 )
=> ( ord_less_eq_nat @ Y7 @ X3 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_393_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_394_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_395_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% eq_imp_le
thf(fact_396_le__trans,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ J @ K )
=> ( ord_less_eq_nat @ I @ K ) ) ) ).
% le_trans
thf(fact_397_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_398_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_399_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_400_order__antisym__conv,axiom,
! [Y: set_Pr958786334691620121nt_int,X: set_Pr958786334691620121nt_int] :
( ( ord_le2843351958646193337nt_int @ Y @ X )
=> ( ( ord_le2843351958646193337nt_int @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_401_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_402_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_403_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F2: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F2 @ B )
= C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F2 @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_404_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F2: nat > int,C: int] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F2 @ B )
= C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_int @ ( F2 @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_405_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F2: nat > set_Pr958786334691620121nt_int,C: set_Pr958786334691620121nt_int] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F2 @ B )
= C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_le2843351958646193337nt_int @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_le2843351958646193337nt_int @ ( F2 @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_406_ord__le__eq__subst,axiom,
! [A: int,B: int,F2: int > nat,C: nat] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ( F2 @ B )
= C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F2 @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_407_ord__le__eq__subst,axiom,
! [A: int,B: int,F2: int > int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ( F2 @ B )
= C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_int @ ( F2 @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_408_ord__le__eq__subst,axiom,
! [A: int,B: int,F2: int > set_Pr958786334691620121nt_int,C: set_Pr958786334691620121nt_int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ( F2 @ B )
= C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_le2843351958646193337nt_int @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_le2843351958646193337nt_int @ ( F2 @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_409_ord__le__eq__subst,axiom,
! [A: set_Pr958786334691620121nt_int,B: set_Pr958786334691620121nt_int,F2: set_Pr958786334691620121nt_int > nat,C: nat] :
( ( ord_le2843351958646193337nt_int @ A @ B )
=> ( ( ( F2 @ B )
= C )
=> ( ! [X3: set_Pr958786334691620121nt_int,Y3: set_Pr958786334691620121nt_int] :
( ( ord_le2843351958646193337nt_int @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F2 @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_410_ord__le__eq__subst,axiom,
! [A: set_Pr958786334691620121nt_int,B: set_Pr958786334691620121nt_int,F2: set_Pr958786334691620121nt_int > int,C: int] :
( ( ord_le2843351958646193337nt_int @ A @ B )
=> ( ( ( F2 @ B )
= C )
=> ( ! [X3: set_Pr958786334691620121nt_int,Y3: set_Pr958786334691620121nt_int] :
( ( ord_le2843351958646193337nt_int @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_int @ ( F2 @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_411_ord__le__eq__subst,axiom,
! [A: set_Pr958786334691620121nt_int,B: set_Pr958786334691620121nt_int,F2: set_Pr958786334691620121nt_int > set_Pr958786334691620121nt_int,C: set_Pr958786334691620121nt_int] :
( ( ord_le2843351958646193337nt_int @ A @ B )
=> ( ( ( F2 @ B )
= C )
=> ( ! [X3: set_Pr958786334691620121nt_int,Y3: set_Pr958786334691620121nt_int] :
( ( ord_le2843351958646193337nt_int @ X3 @ Y3 )
=> ( ord_le2843351958646193337nt_int @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_le2843351958646193337nt_int @ ( F2 @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_412_ord__eq__le__subst,axiom,
! [A: nat,F2: nat > nat,B: nat,C: nat] :
( ( A
= ( F2 @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F2 @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_413_ord__eq__le__subst,axiom,
! [A: int,F2: nat > int,B: nat,C: nat] :
( ( A
= ( F2 @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_int @ A @ ( F2 @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_414_ord__eq__le__subst,axiom,
! [A: set_Pr958786334691620121nt_int,F2: nat > set_Pr958786334691620121nt_int,B: nat,C: nat] :
( ( A
= ( F2 @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_le2843351958646193337nt_int @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_le2843351958646193337nt_int @ A @ ( F2 @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_415_ord__eq__le__subst,axiom,
! [A: nat,F2: int > nat,B: int,C: int] :
( ( A
= ( F2 @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F2 @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_416_ord__eq__le__subst,axiom,
! [A: int,F2: int > int,B: int,C: int] :
( ( A
= ( F2 @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_int @ A @ ( F2 @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_417_ord__eq__le__subst,axiom,
! [A: set_Pr958786334691620121nt_int,F2: int > set_Pr958786334691620121nt_int,B: int,C: int] :
( ( A
= ( F2 @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_le2843351958646193337nt_int @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_le2843351958646193337nt_int @ A @ ( F2 @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_418_ord__eq__le__subst,axiom,
! [A: nat,F2: set_Pr958786334691620121nt_int > nat,B: set_Pr958786334691620121nt_int,C: set_Pr958786334691620121nt_int] :
( ( A
= ( F2 @ B ) )
=> ( ( ord_le2843351958646193337nt_int @ B @ C )
=> ( ! [X3: set_Pr958786334691620121nt_int,Y3: set_Pr958786334691620121nt_int] :
( ( ord_le2843351958646193337nt_int @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F2 @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_419_ord__eq__le__subst,axiom,
! [A: int,F2: set_Pr958786334691620121nt_int > int,B: set_Pr958786334691620121nt_int,C: set_Pr958786334691620121nt_int] :
( ( A
= ( F2 @ B ) )
=> ( ( ord_le2843351958646193337nt_int @ B @ C )
=> ( ! [X3: set_Pr958786334691620121nt_int,Y3: set_Pr958786334691620121nt_int] :
( ( ord_le2843351958646193337nt_int @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_int @ A @ ( F2 @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_420_ord__eq__le__subst,axiom,
! [A: set_Pr958786334691620121nt_int,F2: set_Pr958786334691620121nt_int > set_Pr958786334691620121nt_int,B: set_Pr958786334691620121nt_int,C: set_Pr958786334691620121nt_int] :
( ( A
= ( F2 @ B ) )
=> ( ( ord_le2843351958646193337nt_int @ B @ C )
=> ( ! [X3: set_Pr958786334691620121nt_int,Y3: set_Pr958786334691620121nt_int] :
( ( ord_le2843351958646193337nt_int @ X3 @ Y3 )
=> ( ord_le2843351958646193337nt_int @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_le2843351958646193337nt_int @ A @ ( F2 @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_421_linorder__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
| ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_linear
thf(fact_422_linorder__linear,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
| ( ord_less_eq_int @ Y @ X ) ) ).
% linorder_linear
thf(fact_423_order__eq__refl,axiom,
! [X: nat,Y: nat] :
( ( X = Y )
=> ( ord_less_eq_nat @ X @ Y ) ) ).
% order_eq_refl
thf(fact_424_order__eq__refl,axiom,
! [X: int,Y: int] :
( ( X = Y )
=> ( ord_less_eq_int @ X @ Y ) ) ).
% order_eq_refl
thf(fact_425_order__eq__refl,axiom,
! [X: set_Pr958786334691620121nt_int,Y: set_Pr958786334691620121nt_int] :
( ( X = Y )
=> ( ord_le2843351958646193337nt_int @ X @ Y ) ) ).
% order_eq_refl
thf(fact_426_order__subst2,axiom,
! [A: nat,B: nat,F2: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F2 @ B ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F2 @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_427_order__subst2,axiom,
! [A: nat,B: nat,F2: nat > int,C: int] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_int @ ( F2 @ B ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_int @ ( F2 @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_428_order__subst2,axiom,
! [A: nat,B: nat,F2: nat > set_Pr958786334691620121nt_int,C: set_Pr958786334691620121nt_int] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_le2843351958646193337nt_int @ ( F2 @ B ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_le2843351958646193337nt_int @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_le2843351958646193337nt_int @ ( F2 @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_429_order__subst2,axiom,
! [A: int,B: int,F2: int > nat,C: nat] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_nat @ ( F2 @ B ) @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F2 @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_430_order__subst2,axiom,
! [A: int,B: int,F2: int > int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ ( F2 @ B ) @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_int @ ( F2 @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_431_order__subst2,axiom,
! [A: int,B: int,F2: int > set_Pr958786334691620121nt_int,C: set_Pr958786334691620121nt_int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_le2843351958646193337nt_int @ ( F2 @ B ) @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_le2843351958646193337nt_int @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_le2843351958646193337nt_int @ ( F2 @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_432_order__subst2,axiom,
! [A: set_Pr958786334691620121nt_int,B: set_Pr958786334691620121nt_int,F2: set_Pr958786334691620121nt_int > nat,C: nat] :
( ( ord_le2843351958646193337nt_int @ A @ B )
=> ( ( ord_less_eq_nat @ ( F2 @ B ) @ C )
=> ( ! [X3: set_Pr958786334691620121nt_int,Y3: set_Pr958786334691620121nt_int] :
( ( ord_le2843351958646193337nt_int @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F2 @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_433_order__subst2,axiom,
! [A: set_Pr958786334691620121nt_int,B: set_Pr958786334691620121nt_int,F2: set_Pr958786334691620121nt_int > int,C: int] :
( ( ord_le2843351958646193337nt_int @ A @ B )
=> ( ( ord_less_eq_int @ ( F2 @ B ) @ C )
=> ( ! [X3: set_Pr958786334691620121nt_int,Y3: set_Pr958786334691620121nt_int] :
( ( ord_le2843351958646193337nt_int @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_int @ ( F2 @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_434_order__subst2,axiom,
! [A: set_Pr958786334691620121nt_int,B: set_Pr958786334691620121nt_int,F2: set_Pr958786334691620121nt_int > set_Pr958786334691620121nt_int,C: set_Pr958786334691620121nt_int] :
( ( ord_le2843351958646193337nt_int @ A @ B )
=> ( ( ord_le2843351958646193337nt_int @ ( F2 @ B ) @ C )
=> ( ! [X3: set_Pr958786334691620121nt_int,Y3: set_Pr958786334691620121nt_int] :
( ( ord_le2843351958646193337nt_int @ X3 @ Y3 )
=> ( ord_le2843351958646193337nt_int @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_le2843351958646193337nt_int @ ( F2 @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_435_order__subst1,axiom,
! [A: nat,F2: nat > nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( F2 @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F2 @ C ) ) ) ) ) ).
% order_subst1
thf(fact_436_order__subst1,axiom,
! [A: nat,F2: int > nat,B: int,C: int] :
( ( ord_less_eq_nat @ A @ ( F2 @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F2 @ C ) ) ) ) ) ).
% order_subst1
thf(fact_437_order__subst1,axiom,
! [A: nat,F2: set_Pr958786334691620121nt_int > nat,B: set_Pr958786334691620121nt_int,C: set_Pr958786334691620121nt_int] :
( ( ord_less_eq_nat @ A @ ( F2 @ B ) )
=> ( ( ord_le2843351958646193337nt_int @ B @ C )
=> ( ! [X3: set_Pr958786334691620121nt_int,Y3: set_Pr958786334691620121nt_int] :
( ( ord_le2843351958646193337nt_int @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F2 @ C ) ) ) ) ) ).
% order_subst1
thf(fact_438_order__subst1,axiom,
! [A: int,F2: nat > int,B: nat,C: nat] :
( ( ord_less_eq_int @ A @ ( F2 @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_int @ A @ ( F2 @ C ) ) ) ) ) ).
% order_subst1
thf(fact_439_order__subst1,axiom,
! [A: int,F2: int > int,B: int,C: int] :
( ( ord_less_eq_int @ A @ ( F2 @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_int @ A @ ( F2 @ C ) ) ) ) ) ).
% order_subst1
thf(fact_440_order__subst1,axiom,
! [A: int,F2: set_Pr958786334691620121nt_int > int,B: set_Pr958786334691620121nt_int,C: set_Pr958786334691620121nt_int] :
( ( ord_less_eq_int @ A @ ( F2 @ B ) )
=> ( ( ord_le2843351958646193337nt_int @ B @ C )
=> ( ! [X3: set_Pr958786334691620121nt_int,Y3: set_Pr958786334691620121nt_int] :
( ( ord_le2843351958646193337nt_int @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_int @ A @ ( F2 @ C ) ) ) ) ) ).
% order_subst1
thf(fact_441_order__subst1,axiom,
! [A: set_Pr958786334691620121nt_int,F2: nat > set_Pr958786334691620121nt_int,B: nat,C: nat] :
( ( ord_le2843351958646193337nt_int @ A @ ( F2 @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_le2843351958646193337nt_int @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_le2843351958646193337nt_int @ A @ ( F2 @ C ) ) ) ) ) ).
% order_subst1
thf(fact_442_order__subst1,axiom,
! [A: set_Pr958786334691620121nt_int,F2: int > set_Pr958786334691620121nt_int,B: int,C: int] :
( ( ord_le2843351958646193337nt_int @ A @ ( F2 @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_le2843351958646193337nt_int @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_le2843351958646193337nt_int @ A @ ( F2 @ C ) ) ) ) ) ).
% order_subst1
thf(fact_443_order__subst1,axiom,
! [A: set_Pr958786334691620121nt_int,F2: set_Pr958786334691620121nt_int > set_Pr958786334691620121nt_int,B: set_Pr958786334691620121nt_int,C: set_Pr958786334691620121nt_int] :
( ( ord_le2843351958646193337nt_int @ A @ ( F2 @ B ) )
=> ( ( ord_le2843351958646193337nt_int @ B @ C )
=> ( ! [X3: set_Pr958786334691620121nt_int,Y3: set_Pr958786334691620121nt_int] :
( ( ord_le2843351958646193337nt_int @ X3 @ Y3 )
=> ( ord_le2843351958646193337nt_int @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_le2843351958646193337nt_int @ A @ ( F2 @ C ) ) ) ) ) ).
% order_subst1
thf(fact_444_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: nat,Z2: nat] : ( Y5 = Z2 ) )
= ( ^ [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
& ( ord_less_eq_nat @ B2 @ A2 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_445_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: int,Z2: int] : ( Y5 = Z2 ) )
= ( ^ [A2: int,B2: int] :
( ( ord_less_eq_int @ A2 @ B2 )
& ( ord_less_eq_int @ B2 @ A2 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_446_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: set_Pr958786334691620121nt_int,Z2: set_Pr958786334691620121nt_int] : ( Y5 = Z2 ) )
= ( ^ [A2: set_Pr958786334691620121nt_int,B2: set_Pr958786334691620121nt_int] :
( ( ord_le2843351958646193337nt_int @ A2 @ B2 )
& ( ord_le2843351958646193337nt_int @ B2 @ A2 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_447_antisym,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_448_antisym,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_449_antisym,axiom,
! [A: set_Pr958786334691620121nt_int,B: set_Pr958786334691620121nt_int] :
( ( ord_le2843351958646193337nt_int @ A @ B )
=> ( ( ord_le2843351958646193337nt_int @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_450_dual__order_Otrans,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_eq_nat @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_451_dual__order_Otrans,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( ord_less_eq_int @ C @ B )
=> ( ord_less_eq_int @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_452_dual__order_Otrans,axiom,
! [B: set_Pr958786334691620121nt_int,A: set_Pr958786334691620121nt_int,C: set_Pr958786334691620121nt_int] :
( ( ord_le2843351958646193337nt_int @ B @ A )
=> ( ( ord_le2843351958646193337nt_int @ C @ B )
=> ( ord_le2843351958646193337nt_int @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_453_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_454_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_455_dual__order_Oantisym,axiom,
! [B: set_Pr958786334691620121nt_int,A: set_Pr958786334691620121nt_int] :
( ( ord_le2843351958646193337nt_int @ B @ A )
=> ( ( ord_le2843351958646193337nt_int @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_456_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: nat,Z2: nat] : ( Y5 = Z2 ) )
= ( ^ [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
& ( ord_less_eq_nat @ A2 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_457_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: int,Z2: int] : ( Y5 = Z2 ) )
= ( ^ [A2: int,B2: int] :
( ( ord_less_eq_int @ B2 @ A2 )
& ( ord_less_eq_int @ A2 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_458_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: set_Pr958786334691620121nt_int,Z2: set_Pr958786334691620121nt_int] : ( Y5 = Z2 ) )
= ( ^ [A2: set_Pr958786334691620121nt_int,B2: set_Pr958786334691620121nt_int] :
( ( ord_le2843351958646193337nt_int @ B2 @ A2 )
& ( ord_le2843351958646193337nt_int @ A2 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_459_linorder__wlog,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A5: nat,B5: nat] :
( ( ord_less_eq_nat @ A5 @ B5 )
=> ( P @ A5 @ B5 ) )
=> ( ! [A5: nat,B5: nat] :
( ( P @ B5 @ A5 )
=> ( P @ A5 @ B5 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_460_linorder__wlog,axiom,
! [P: int > int > $o,A: int,B: int] :
( ! [A5: int,B5: int] :
( ( ord_less_eq_int @ A5 @ B5 )
=> ( P @ A5 @ B5 ) )
=> ( ! [A5: int,B5: int] :
( ( P @ B5 @ A5 )
=> ( P @ A5 @ B5 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_461_order__trans,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z )
=> ( ord_less_eq_nat @ X @ Z ) ) ) ).
% order_trans
thf(fact_462_order__trans,axiom,
! [X: int,Y: int,Z: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ord_less_eq_int @ Y @ Z )
=> ( ord_less_eq_int @ X @ Z ) ) ) ).
% order_trans
thf(fact_463_order__trans,axiom,
! [X: set_Pr958786334691620121nt_int,Y: set_Pr958786334691620121nt_int,Z: set_Pr958786334691620121nt_int] :
( ( ord_le2843351958646193337nt_int @ X @ Y )
=> ( ( ord_le2843351958646193337nt_int @ Y @ Z )
=> ( ord_le2843351958646193337nt_int @ X @ Z ) ) ) ).
% order_trans
thf(fact_464_order_Otrans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% order.trans
thf(fact_465_order_Otrans,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ord_less_eq_int @ A @ C ) ) ) ).
% order.trans
thf(fact_466_order_Otrans,axiom,
! [A: set_Pr958786334691620121nt_int,B: set_Pr958786334691620121nt_int,C: set_Pr958786334691620121nt_int] :
( ( ord_le2843351958646193337nt_int @ A @ B )
=> ( ( ord_le2843351958646193337nt_int @ B @ C )
=> ( ord_le2843351958646193337nt_int @ A @ C ) ) ) ).
% order.trans
thf(fact_467_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_468_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_469_order__antisym,axiom,
! [X: set_Pr958786334691620121nt_int,Y: set_Pr958786334691620121nt_int] :
( ( ord_le2843351958646193337nt_int @ X @ Y )
=> ( ( ord_le2843351958646193337nt_int @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_470_ord__le__eq__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_471_ord__le__eq__trans,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_int @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_472_ord__le__eq__trans,axiom,
! [A: set_Pr958786334691620121nt_int,B: set_Pr958786334691620121nt_int,C: set_Pr958786334691620121nt_int] :
( ( ord_le2843351958646193337nt_int @ A @ B )
=> ( ( B = C )
=> ( ord_le2843351958646193337nt_int @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_473_ord__eq__le__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( A = B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_474_ord__eq__le__trans,axiom,
! [A: int,B: int,C: int] :
( ( A = B )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ord_less_eq_int @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_475_ord__eq__le__trans,axiom,
! [A: set_Pr958786334691620121nt_int,B: set_Pr958786334691620121nt_int,C: set_Pr958786334691620121nt_int] :
( ( A = B )
=> ( ( ord_le2843351958646193337nt_int @ B @ C )
=> ( ord_le2843351958646193337nt_int @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_476_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: nat,Z2: nat] : ( Y5 = Z2 ) )
= ( ^ [X2: nat,Y4: nat] :
( ( ord_less_eq_nat @ X2 @ Y4 )
& ( ord_less_eq_nat @ Y4 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_477_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: int,Z2: int] : ( Y5 = Z2 ) )
= ( ^ [X2: int,Y4: int] :
( ( ord_less_eq_int @ X2 @ Y4 )
& ( ord_less_eq_int @ Y4 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_478_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: set_Pr958786334691620121nt_int,Z2: set_Pr958786334691620121nt_int] : ( Y5 = Z2 ) )
= ( ^ [X2: set_Pr958786334691620121nt_int,Y4: set_Pr958786334691620121nt_int] :
( ( ord_le2843351958646193337nt_int @ X2 @ Y4 )
& ( ord_le2843351958646193337nt_int @ Y4 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_479_le__cases3,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ( ord_less_eq_nat @ X @ Y )
=> ~ ( ord_less_eq_nat @ Y @ Z ) )
=> ( ( ( ord_less_eq_nat @ Y @ X )
=> ~ ( ord_less_eq_nat @ X @ Z ) )
=> ( ( ( ord_less_eq_nat @ X @ Z )
=> ~ ( ord_less_eq_nat @ Z @ Y ) )
=> ( ( ( ord_less_eq_nat @ Z @ Y )
=> ~ ( ord_less_eq_nat @ Y @ X ) )
=> ( ( ( ord_less_eq_nat @ Y @ Z )
=> ~ ( ord_less_eq_nat @ Z @ X ) )
=> ~ ( ( ord_less_eq_nat @ Z @ X )
=> ~ ( ord_less_eq_nat @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_480_le__cases3,axiom,
! [X: int,Y: int,Z: int] :
( ( ( ord_less_eq_int @ X @ Y )
=> ~ ( ord_less_eq_int @ Y @ Z ) )
=> ( ( ( ord_less_eq_int @ Y @ X )
=> ~ ( ord_less_eq_int @ X @ Z ) )
=> ( ( ( ord_less_eq_int @ X @ Z )
=> ~ ( ord_less_eq_int @ Z @ Y ) )
=> ( ( ( ord_less_eq_int @ Z @ Y )
=> ~ ( ord_less_eq_int @ Y @ X ) )
=> ( ( ( ord_less_eq_int @ Y @ Z )
=> ~ ( ord_less_eq_int @ Z @ X ) )
=> ~ ( ( ord_less_eq_int @ Z @ X )
=> ~ ( ord_less_eq_int @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_481_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_482_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_483_is__num__normalize_I1_J,axiom,
! [A: int,B: int,C: int] :
( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C )
= ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% is_num_normalize(1)
thf(fact_484_subrelI,axiom,
! [R: set_Pr2560585780119916871nt_int,S: set_Pr2560585780119916871nt_int] :
( ! [X3: product_prod_int_int,Y3: product_prod_int_int] :
( ( member8566619992076573584nt_int @ ( produc3646306378393792727nt_int @ X3 @ Y3 ) @ R )
=> ( member8566619992076573584nt_int @ ( produc3646306378393792727nt_int @ X3 @ Y3 ) @ S ) )
=> ( ord_le6090609446090860775nt_int @ R @ S ) ) ).
% subrelI
thf(fact_485_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_486_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_487_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_488_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_489_of__int__0__le__iff,axiom,
! [Z: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( ring_1_of_int_int @ Z ) )
= ( ord_less_eq_int @ zero_zero_int @ Z ) ) ).
% of_int_0_le_iff
thf(fact_490_of__int__le__0__iff,axiom,
! [Z: int] :
( ( ord_less_eq_int @ ( ring_1_of_int_int @ Z ) @ zero_zero_int )
= ( ord_less_eq_int @ Z @ zero_zero_int ) ) ).
% of_int_le_0_iff
thf(fact_491_nat__add__distrib,axiom,
! [Z: int,Z5: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z )
=> ( ( ord_less_eq_int @ zero_zero_int @ Z5 )
=> ( ( nat2 @ ( plus_plus_int @ Z @ Z5 ) )
= ( plus_plus_nat @ ( nat2 @ Z ) @ ( nat2 @ Z5 ) ) ) ) ) ).
% nat_add_distrib
thf(fact_492_le__nat__iff,axiom,
! [K: int,N: nat] :
( ( ord_less_eq_int @ zero_zero_int @ K )
=> ( ( ord_less_eq_nat @ N @ ( nat2 @ K ) )
= ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ N ) @ K ) ) ) ).
% le_nat_iff
thf(fact_493_nat__eq__iff,axiom,
! [W2: int,M: nat] :
( ( ( nat2 @ W2 )
= M )
= ( ( ( ord_less_eq_int @ zero_zero_int @ W2 )
=> ( W2
= ( semiri1314217659103216013at_int @ M ) ) )
& ( ~ ( ord_less_eq_int @ zero_zero_int @ W2 )
=> ( M = zero_zero_nat ) ) ) ) ).
% nat_eq_iff
thf(fact_494_nat__eq__iff2,axiom,
! [M: nat,W2: int] :
( ( M
= ( nat2 @ W2 ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ W2 )
=> ( W2
= ( semiri1314217659103216013at_int @ M ) ) )
& ( ~ ( ord_less_eq_int @ zero_zero_int @ W2 )
=> ( M = zero_zero_nat ) ) ) ) ).
% nat_eq_iff2
thf(fact_495_nat__int,axiom,
! [N: nat] :
( ( nat2 @ ( semiri1314217659103216013at_int @ N ) )
= N ) ).
% nat_int
thf(fact_496_of__int__eq__0__iff,axiom,
! [Z: int] :
( ( ( ring_1_of_int_int @ Z )
= zero_zero_int )
= ( Z = zero_zero_int ) ) ).
% of_int_eq_0_iff
thf(fact_497_of__int__0__eq__iff,axiom,
! [Z: int] :
( ( zero_zero_int
= ( ring_1_of_int_int @ Z ) )
= ( Z = zero_zero_int ) ) ).
% of_int_0_eq_iff
thf(fact_498_of__int__0,axiom,
( ( ring_1_of_int_int @ zero_zero_int )
= zero_zero_int ) ).
% of_int_0
thf(fact_499_of__int__le__iff,axiom,
! [W2: int,Z: int] :
( ( ord_less_eq_int @ ( ring_1_of_int_int @ W2 ) @ ( ring_1_of_int_int @ Z ) )
= ( ord_less_eq_int @ W2 @ Z ) ) ).
% of_int_le_iff
thf(fact_500_of__int__add,axiom,
! [W2: int,Z: int] :
( ( ring_1_of_int_int @ ( plus_plus_int @ W2 @ Z ) )
= ( plus_plus_int @ ( ring_1_of_int_int @ W2 ) @ ( ring_1_of_int_int @ Z ) ) ) ).
% of_int_add
thf(fact_501_of__int__of__nat__eq,axiom,
! [N: nat] :
( ( ring_1_of_int_int @ ( semiri1314217659103216013at_int @ N ) )
= ( semiri1314217659103216013at_int @ N ) ) ).
% of_int_of_nat_eq
thf(fact_502_nat__le__0,axiom,
! [Z: int] :
( ( ord_less_eq_int @ Z @ zero_zero_int )
=> ( ( nat2 @ Z )
= zero_zero_nat ) ) ).
% nat_le_0
thf(fact_503_nat__0__iff,axiom,
! [I: int] :
( ( ( nat2 @ I )
= zero_zero_nat )
= ( ord_less_eq_int @ I @ zero_zero_int ) ) ).
% nat_0_iff
thf(fact_504_int__nat__eq,axiom,
! [Z: int] :
( ( ( ord_less_eq_int @ zero_zero_int @ Z )
=> ( ( semiri1314217659103216013at_int @ ( nat2 @ Z ) )
= Z ) )
& ( ~ ( ord_less_eq_int @ zero_zero_int @ Z )
=> ( ( semiri1314217659103216013at_int @ ( nat2 @ Z ) )
= zero_zero_int ) ) ) ).
% int_nat_eq
thf(fact_505_of__nat__nat,axiom,
! [Z: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z )
=> ( ( semiri1314217659103216013at_int @ ( nat2 @ Z ) )
= ( ring_1_of_int_int @ Z ) ) ) ).
% of_nat_nat
thf(fact_506_nat__zero__as__int,axiom,
( zero_zero_nat
= ( nat2 @ zero_zero_int ) ) ).
% nat_zero_as_int
thf(fact_507_nat__mono,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ord_less_eq_nat @ ( nat2 @ X ) @ ( nat2 @ Y ) ) ) ).
% nat_mono
thf(fact_508_eq__nat__nat__iff,axiom,
! [Z: int,Z5: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z )
=> ( ( ord_less_eq_int @ zero_zero_int @ Z5 )
=> ( ( ( nat2 @ Z )
= ( nat2 @ Z5 ) )
= ( Z = Z5 ) ) ) ) ).
% eq_nat_nat_iff
thf(fact_509_all__nat,axiom,
( ( ^ [P4: nat > $o] :
! [X5: nat] : ( P4 @ X5 ) )
= ( ^ [P5: nat > $o] :
! [X2: int] :
( ( ord_less_eq_int @ zero_zero_int @ X2 )
=> ( P5 @ ( nat2 @ X2 ) ) ) ) ) ).
% all_nat
thf(fact_510_ex__nat,axiom,
( ( ^ [P4: nat > $o] :
? [X5: nat] : ( P4 @ X5 ) )
= ( ^ [P5: nat > $o] :
? [X2: int] :
( ( ord_less_eq_int @ zero_zero_int @ X2 )
& ( P5 @ ( nat2 @ X2 ) ) ) ) ) ).
% ex_nat
thf(fact_511_nat__le__iff,axiom,
! [X: int,N: nat] :
( ( ord_less_eq_nat @ ( nat2 @ X ) @ N )
= ( ord_less_eq_int @ X @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% nat_le_iff
thf(fact_512_int__eq__iff,axiom,
! [M: nat,Z: int] :
( ( ( semiri1314217659103216013at_int @ M )
= Z )
= ( ( M
= ( nat2 @ Z ) )
& ( ord_less_eq_int @ zero_zero_int @ Z ) ) ) ).
% int_eq_iff
thf(fact_513_nat__0__le,axiom,
! [Z: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z )
=> ( ( semiri1314217659103216013at_int @ ( nat2 @ Z ) )
= Z ) ) ).
% nat_0_le
thf(fact_514_nat__int__add,axiom,
! [A: nat,B: nat] :
( ( nat2 @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) )
= ( plus_plus_nat @ A @ B ) ) ).
% nat_int_add
thf(fact_515_of__int__nonneg,axiom,
! [Z: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z )
=> ( ord_less_eq_int @ zero_zero_int @ ( ring_1_of_int_int @ Z ) ) ) ).
% of_int_nonneg
thf(fact_516_nat__le__eq__zle,axiom,
! [W2: int,Z: int] :
( ( ( ord_less_int @ zero_zero_int @ W2 )
| ( ord_less_eq_int @ zero_zero_int @ Z ) )
=> ( ( ord_less_eq_nat @ ( nat2 @ W2 ) @ ( nat2 @ Z ) )
= ( ord_less_eq_int @ W2 @ Z ) ) ) ).
% nat_le_eq_zle
thf(fact_517_split__nat,axiom,
! [P: nat > $o,I: int] :
( ( P @ ( nat2 @ I ) )
= ( ! [N2: nat] :
( ( I
= ( semiri1314217659103216013at_int @ N2 ) )
=> ( P @ N2 ) )
& ( ( ord_less_int @ I @ zero_zero_int )
=> ( P @ zero_zero_nat ) ) ) ) ).
% split_nat
thf(fact_518_nat__abs__triangle__ineq,axiom,
! [K: int,L: int] : ( ord_less_eq_nat @ ( nat2 @ ( abs_abs_int @ ( plus_plus_int @ K @ L ) ) ) @ ( plus_plus_nat @ ( nat2 @ ( abs_abs_int @ K ) ) @ ( nat2 @ ( abs_abs_int @ L ) ) ) ) ).
% nat_abs_triangle_ineq
thf(fact_519_nat__zminus__int,axiom,
! [N: nat] :
( ( nat2 @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) )
= zero_zero_nat ) ).
% nat_zminus_int
thf(fact_520_GreatestI2__order,axiom,
! [P: int > $o,X: int,Q2: int > $o] :
( ( P @ X )
=> ( ! [Y3: int] :
( ( P @ Y3 )
=> ( ord_less_eq_int @ Y3 @ X ) )
=> ( ! [X3: int] :
( ( P @ X3 )
=> ( ! [Y7: int] :
( ( P @ Y7 )
=> ( ord_less_eq_int @ Y7 @ X3 ) )
=> ( Q2 @ X3 ) ) )
=> ( Q2 @ ( order_Greatest_int @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_521_GreatestI2__order,axiom,
! [P: set_Pr958786334691620121nt_int > $o,X: set_Pr958786334691620121nt_int,Q2: set_Pr958786334691620121nt_int > $o] :
( ( P @ X )
=> ( ! [Y3: set_Pr958786334691620121nt_int] :
( ( P @ Y3 )
=> ( ord_le2843351958646193337nt_int @ Y3 @ X ) )
=> ( ! [X3: set_Pr958786334691620121nt_int] :
( ( P @ X3 )
=> ( ! [Y7: set_Pr958786334691620121nt_int] :
( ( P @ Y7 )
=> ( ord_le2843351958646193337nt_int @ Y7 @ X3 ) )
=> ( Q2 @ X3 ) ) )
=> ( Q2 @ ( order_3894005715824938610nt_int @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_522_GreatestI2__order,axiom,
! [P: nat > $o,X: nat,Q2: nat > $o] :
( ( P @ X )
=> ( ! [Y3: nat] :
( ( P @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ X ) )
=> ( ! [X3: nat] :
( ( P @ X3 )
=> ( ! [Y7: nat] :
( ( P @ Y7 )
=> ( ord_less_eq_nat @ Y7 @ X3 ) )
=> ( Q2 @ X3 ) ) )
=> ( Q2 @ ( order_Greatest_nat @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_523_Greatest__equality,axiom,
! [P: int > $o,X: int] :
( ( P @ X )
=> ( ! [Y3: int] :
( ( P @ Y3 )
=> ( ord_less_eq_int @ Y3 @ X ) )
=> ( ( order_Greatest_int @ P )
= X ) ) ) ).
% Greatest_equality
thf(fact_524_Greatest__equality,axiom,
! [P: set_Pr958786334691620121nt_int > $o,X: set_Pr958786334691620121nt_int] :
( ( P @ X )
=> ( ! [Y3: set_Pr958786334691620121nt_int] :
( ( P @ Y3 )
=> ( ord_le2843351958646193337nt_int @ Y3 @ X ) )
=> ( ( order_3894005715824938610nt_int @ P )
= X ) ) ) ).
% Greatest_equality
thf(fact_525_Greatest__equality,axiom,
! [P: nat > $o,X: nat] :
( ( P @ X )
=> ( ! [Y3: nat] :
( ( P @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ X ) )
=> ( ( order_Greatest_nat @ P )
= X ) ) ) ).
% Greatest_equality
thf(fact_526_add_Oinverse__inverse,axiom,
! [A: int] :
( ( uminus_uminus_int @ ( uminus_uminus_int @ A ) )
= A ) ).
% add.inverse_inverse
thf(fact_527_neg__equal__iff__equal,axiom,
! [A: int,B: int] :
( ( ( uminus_uminus_int @ A )
= ( uminus_uminus_int @ B ) )
= ( A = B ) ) ).
% neg_equal_iff_equal
thf(fact_528_verit__minus__simplify_I4_J,axiom,
! [B: int] :
( ( uminus_uminus_int @ ( uminus_uminus_int @ B ) )
= B ) ).
% verit_minus_simplify(4)
thf(fact_529_abs__idempotent,axiom,
! [A: int] :
( ( abs_abs_int @ ( abs_abs_int @ A ) )
= ( abs_abs_int @ A ) ) ).
% abs_idempotent
thf(fact_530_not__gr__zero,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_531_add__less__cancel__left,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
= ( ord_less_int @ A @ B ) ) ).
% add_less_cancel_left
thf(fact_532_add__less__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
= ( ord_less_nat @ A @ B ) ) ).
% add_less_cancel_left
thf(fact_533_add__less__cancel__right,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
= ( ord_less_int @ A @ B ) ) ).
% add_less_cancel_right
thf(fact_534_add__less__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( ord_less_nat @ A @ B ) ) ).
% add_less_cancel_right
thf(fact_535_compl__le__compl__iff,axiom,
! [X: set_Pr958786334691620121nt_int,Y: set_Pr958786334691620121nt_int] :
( ( ord_le2843351958646193337nt_int @ ( uminus6221592323253981072nt_int @ X ) @ ( uminus6221592323253981072nt_int @ Y ) )
= ( ord_le2843351958646193337nt_int @ Y @ X ) ) ).
% compl_le_compl_iff
thf(fact_536_neg__le__iff__le,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) )
= ( ord_less_eq_int @ A @ B ) ) ).
% neg_le_iff_le
thf(fact_537_add_Oinverse__neutral,axiom,
( ( uminus_uminus_int @ zero_zero_int )
= zero_zero_int ) ).
% add.inverse_neutral
thf(fact_538_neg__0__equal__iff__equal,axiom,
! [A: int] :
( ( zero_zero_int
= ( uminus_uminus_int @ A ) )
= ( zero_zero_int = A ) ) ).
% neg_0_equal_iff_equal
thf(fact_539_neg__equal__0__iff__equal,axiom,
! [A: int] :
( ( ( uminus_uminus_int @ A )
= zero_zero_int )
= ( A = zero_zero_int ) ) ).
% neg_equal_0_iff_equal
thf(fact_540_equal__neg__zero,axiom,
! [A: int] :
( ( A
= ( uminus_uminus_int @ A ) )
= ( A = zero_zero_int ) ) ).
% equal_neg_zero
thf(fact_541_neg__equal__zero,axiom,
! [A: int] :
( ( ( uminus_uminus_int @ A )
= A )
= ( A = zero_zero_int ) ) ).
% neg_equal_zero
thf(fact_542_neg__less__iff__less,axiom,
! [B: int,A: int] :
( ( ord_less_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) )
= ( ord_less_int @ A @ B ) ) ).
% neg_less_iff_less
thf(fact_543_add__minus__cancel,axiom,
! [A: int,B: int] :
( ( plus_plus_int @ A @ ( plus_plus_int @ ( uminus_uminus_int @ A ) @ B ) )
= B ) ).
% add_minus_cancel
thf(fact_544_minus__add__cancel,axiom,
! [A: int,B: int] :
( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ ( plus_plus_int @ A @ B ) )
= B ) ).
% minus_add_cancel
thf(fact_545_minus__add__distrib,axiom,
! [A: int,B: int] :
( ( uminus_uminus_int @ ( plus_plus_int @ A @ B ) )
= ( plus_plus_int @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B ) ) ) ).
% minus_add_distrib
thf(fact_546_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_547_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_548_abs__0__eq,axiom,
! [A: int] :
( ( zero_zero_int
= ( abs_abs_int @ A ) )
= ( A = zero_zero_int ) ) ).
% abs_0_eq
thf(fact_549_abs__eq__0,axiom,
! [A: int] :
( ( ( abs_abs_int @ A )
= zero_zero_int )
= ( A = zero_zero_int ) ) ).
% abs_eq_0
thf(fact_550_abs__zero,axiom,
( ( abs_abs_int @ zero_zero_int )
= zero_zero_int ) ).
% abs_zero
thf(fact_551_abs__add__abs,axiom,
! [A: int,B: int] :
( ( abs_abs_int @ ( plus_plus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) )
= ( plus_plus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) ) ).
% abs_add_abs
thf(fact_552_abs__minus__cancel,axiom,
! [A: int] :
( ( abs_abs_int @ ( uminus_uminus_int @ A ) )
= ( abs_abs_int @ A ) ) ).
% abs_minus_cancel
thf(fact_553_abs__of__nat,axiom,
! [N: nat] :
( ( abs_abs_int @ ( semiri1314217659103216013at_int @ N ) )
= ( semiri1314217659103216013at_int @ N ) ) ).
% abs_of_nat
thf(fact_554_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_555_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_556_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_557_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_558_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_559_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_560_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_561_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_562_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_563_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_564_neg__less__eq__nonneg,axiom,
! [A: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ A )
= ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% neg_less_eq_nonneg
thf(fact_565_less__eq__neg__nonpos,axiom,
! [A: int] :
( ( ord_less_eq_int @ A @ ( uminus_uminus_int @ A ) )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% less_eq_neg_nonpos
thf(fact_566_neg__le__0__iff__le,axiom,
! [A: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ zero_zero_int )
= ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% neg_le_0_iff_le
thf(fact_567_neg__0__le__iff__le,axiom,
! [A: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ A ) )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% neg_0_le_iff_le
thf(fact_568_less__neg__neg,axiom,
! [A: int] :
( ( ord_less_int @ A @ ( uminus_uminus_int @ A ) )
= ( ord_less_int @ A @ zero_zero_int ) ) ).
% less_neg_neg
thf(fact_569_neg__less__pos,axiom,
! [A: int] :
( ( ord_less_int @ ( uminus_uminus_int @ A ) @ A )
= ( ord_less_int @ zero_zero_int @ A ) ) ).
% neg_less_pos
thf(fact_570_neg__0__less__iff__less,axiom,
! [A: int] :
( ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ A ) )
= ( ord_less_int @ A @ zero_zero_int ) ) ).
% neg_0_less_iff_less
thf(fact_571_neg__less__0__iff__less,axiom,
! [A: int] :
( ( ord_less_int @ ( uminus_uminus_int @ A ) @ zero_zero_int )
= ( ord_less_int @ zero_zero_int @ A ) ) ).
% neg_less_0_iff_less
thf(fact_572_add_Oright__inverse,axiom,
! [A: int] :
( ( plus_plus_int @ A @ ( uminus_uminus_int @ A ) )
= zero_zero_int ) ).
% add.right_inverse
thf(fact_573_ab__left__minus,axiom,
! [A: int] :
( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ A )
= zero_zero_int ) ).
% ab_left_minus
thf(fact_574_abs__le__zero__iff,axiom,
! [A: int] :
( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ zero_zero_int )
= ( A = zero_zero_int ) ) ).
% abs_le_zero_iff
thf(fact_575_abs__le__self__iff,axiom,
! [A: int] :
( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ A )
= ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% abs_le_self_iff
thf(fact_576_abs__of__nonneg,axiom,
! [A: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( abs_abs_int @ A )
= A ) ) ).
% abs_of_nonneg
thf(fact_577_zero__less__abs__iff,axiom,
! [A: int] :
( ( ord_less_int @ zero_zero_int @ ( abs_abs_int @ A ) )
= ( A != zero_zero_int ) ) ).
% zero_less_abs_iff
thf(fact_578_negative__eq__positive,axiom,
! [N: nat,M: nat] :
( ( ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) )
= ( semiri1314217659103216013at_int @ M ) )
= ( ( N = zero_zero_nat )
& ( M = zero_zero_nat ) ) ) ).
% negative_eq_positive
thf(fact_579_of__int__less__iff,axiom,
! [W2: int,Z: int] :
( ( ord_less_int @ ( ring_1_of_int_int @ W2 ) @ ( ring_1_of_int_int @ Z ) )
= ( ord_less_int @ W2 @ Z ) ) ).
% of_int_less_iff
thf(fact_580_of__int__minus,axiom,
! [Z: int] :
( ( ring_1_of_int_int @ ( uminus_uminus_int @ Z ) )
= ( uminus_uminus_int @ ( ring_1_of_int_int @ Z ) ) ) ).
% of_int_minus
thf(fact_581_negative__zle,axiom,
! [N: nat,M: nat] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ ( semiri1314217659103216013at_int @ M ) ) ).
% negative_zle
thf(fact_582_of__int__abs,axiom,
! [X: int] :
( ( ring_1_of_int_int @ ( abs_abs_int @ X ) )
= ( abs_abs_int @ ( ring_1_of_int_int @ X ) ) ) ).
% of_int_abs
thf(fact_583_of__nat__0__less__iff,axiom,
! [N: nat] :
( ( ord_less_int @ zero_zero_int @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% of_nat_0_less_iff
thf(fact_584_of__nat__0__less__iff,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( semiri1316708129612266289at_nat @ N ) )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% of_nat_0_less_iff
thf(fact_585_abs__of__nonpos,axiom,
! [A: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( abs_abs_int @ A )
= ( uminus_uminus_int @ A ) ) ) ).
% abs_of_nonpos
thf(fact_586_of__int__less__0__iff,axiom,
! [Z: int] :
( ( ord_less_int @ ( ring_1_of_int_int @ Z ) @ zero_zero_int )
= ( ord_less_int @ Z @ zero_zero_int ) ) ).
% of_int_less_0_iff
thf(fact_587_of__int__0__less__iff,axiom,
! [Z: int] :
( ( ord_less_int @ zero_zero_int @ ( ring_1_of_int_int @ Z ) )
= ( ord_less_int @ zero_zero_int @ Z ) ) ).
% of_int_0_less_iff
thf(fact_588_abs__ge__minus__self,axiom,
! [A: int] : ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ ( abs_abs_int @ A ) ) ).
% abs_ge_minus_self
thf(fact_589_abs__le__iff,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ B )
= ( ( ord_less_eq_int @ A @ B )
& ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ B ) ) ) ).
% abs_le_iff
thf(fact_590_abs__le__D2,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ B )
=> ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ B ) ) ).
% abs_le_D2
thf(fact_591_abs__leI,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ B )
=> ( ord_less_eq_int @ ( abs_abs_int @ A ) @ B ) ) ) ).
% abs_leI
thf(fact_592_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_593_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_594_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_595_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_596_not__int__zless__negative,axiom,
! [N: nat,M: nat] :
~ ( ord_less_int @ ( semiri1314217659103216013at_int @ N ) @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ M ) ) ) ).
% not_int_zless_negative
thf(fact_597_equation__minus__iff,axiom,
! [A: int,B: int] :
( ( A
= ( uminus_uminus_int @ B ) )
= ( B
= ( uminus_uminus_int @ A ) ) ) ).
% equation_minus_iff
thf(fact_598_minus__equation__iff,axiom,
! [A: int,B: int] :
( ( ( uminus_uminus_int @ A )
= B )
= ( ( uminus_uminus_int @ B )
= A ) ) ).
% minus_equation_iff
thf(fact_599_abs__of__neg,axiom,
! [A: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( abs_abs_int @ A )
= ( uminus_uminus_int @ A ) ) ) ).
% abs_of_neg
thf(fact_600_less__minus__iff,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ ( uminus_uminus_int @ B ) )
= ( ord_less_int @ B @ ( uminus_uminus_int @ A ) ) ) ).
% less_minus_iff
thf(fact_601_minus__less__iff,axiom,
! [A: int,B: int] :
( ( ord_less_int @ ( uminus_uminus_int @ A ) @ B )
= ( ord_less_int @ ( uminus_uminus_int @ B ) @ A ) ) ).
% minus_less_iff
thf(fact_602_abs__of__pos,axiom,
! [A: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( abs_abs_int @ A )
= A ) ) ).
% abs_of_pos
thf(fact_603_abs__not__less__zero,axiom,
! [A: int] :
~ ( ord_less_int @ ( abs_abs_int @ A ) @ zero_zero_int ) ).
% abs_not_less_zero
thf(fact_604_verit__comp__simplify1_I1_J,axiom,
! [A: int] :
~ ( ord_less_int @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_605_verit__comp__simplify1_I1_J,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_606_verit__negate__coefficient_I3_J,axiom,
! [A: int,B: int] :
( ( A = B )
=> ( ( uminus_uminus_int @ A )
= ( uminus_uminus_int @ B ) ) ) ).
% verit_negate_coefficient(3)
thf(fact_607_verit__negate__coefficient_I2_J,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) ) ) ).
% verit_negate_coefficient(2)
thf(fact_608_abs__if__raw,axiom,
( abs_abs_int
= ( ^ [A2: int] : ( if_int @ ( ord_less_int @ A2 @ zero_zero_int ) @ ( uminus_uminus_int @ A2 ) @ A2 ) ) ) ).
% abs_if_raw
thf(fact_609_zabs__def,axiom,
( abs_abs_int
= ( ^ [I4: int] : ( if_int @ ( ord_less_int @ I4 @ zero_zero_int ) @ ( uminus_uminus_int @ I4 ) @ I4 ) ) ) ).
% zabs_def
thf(fact_610_minf_I7_J,axiom,
! [T2: int] :
? [Z3: int] :
! [X6: int] :
( ( ord_less_int @ X6 @ Z3 )
=> ~ ( ord_less_int @ T2 @ X6 ) ) ).
% minf(7)
thf(fact_611_minf_I7_J,axiom,
! [T2: nat] :
? [Z3: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z3 )
=> ~ ( ord_less_nat @ T2 @ X6 ) ) ).
% minf(7)
thf(fact_612_minf_I5_J,axiom,
! [T2: int] :
? [Z3: int] :
! [X6: int] :
( ( ord_less_int @ X6 @ Z3 )
=> ( ord_less_int @ X6 @ T2 ) ) ).
% minf(5)
thf(fact_613_minf_I5_J,axiom,
! [T2: nat] :
? [Z3: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z3 )
=> ( ord_less_nat @ X6 @ T2 ) ) ).
% minf(5)
thf(fact_614_minf_I4_J,axiom,
! [T2: int] :
? [Z3: int] :
! [X6: int] :
( ( ord_less_int @ X6 @ Z3 )
=> ( X6 != T2 ) ) ).
% minf(4)
thf(fact_615_minf_I4_J,axiom,
! [T2: nat] :
? [Z3: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z3 )
=> ( X6 != T2 ) ) ).
% minf(4)
thf(fact_616_minf_I3_J,axiom,
! [T2: int] :
? [Z3: int] :
! [X6: int] :
( ( ord_less_int @ X6 @ Z3 )
=> ( X6 != T2 ) ) ).
% minf(3)
thf(fact_617_minf_I3_J,axiom,
! [T2: nat] :
? [Z3: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z3 )
=> ( X6 != T2 ) ) ).
% minf(3)
thf(fact_618_minf_I2_J,axiom,
! [P: int > $o,P3: int > $o,Q2: int > $o,Q3: int > $o] :
( ? [Z6: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z6 )
=> ( ( P @ X3 )
= ( P3 @ X3 ) ) )
=> ( ? [Z6: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z6 )
=> ( ( Q2 @ X3 )
= ( Q3 @ X3 ) ) )
=> ? [Z3: int] :
! [X6: int] :
( ( ord_less_int @ X6 @ Z3 )
=> ( ( ( P @ X6 )
| ( Q2 @ X6 ) )
= ( ( P3 @ X6 )
| ( Q3 @ X6 ) ) ) ) ) ) ).
% minf(2)
thf(fact_619_minf_I2_J,axiom,
! [P: nat > $o,P3: nat > $o,Q2: nat > $o,Q3: nat > $o] :
( ? [Z6: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z6 )
=> ( ( P @ X3 )
= ( P3 @ X3 ) ) )
=> ( ? [Z6: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z6 )
=> ( ( Q2 @ X3 )
= ( Q3 @ X3 ) ) )
=> ? [Z3: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z3 )
=> ( ( ( P @ X6 )
| ( Q2 @ X6 ) )
= ( ( P3 @ X6 )
| ( Q3 @ X6 ) ) ) ) ) ) ).
% minf(2)
thf(fact_620_minf_I1_J,axiom,
! [P: int > $o,P3: int > $o,Q2: int > $o,Q3: int > $o] :
( ? [Z6: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z6 )
=> ( ( P @ X3 )
= ( P3 @ X3 ) ) )
=> ( ? [Z6: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z6 )
=> ( ( Q2 @ X3 )
= ( Q3 @ X3 ) ) )
=> ? [Z3: int] :
! [X6: int] :
( ( ord_less_int @ X6 @ Z3 )
=> ( ( ( P @ X6 )
& ( Q2 @ X6 ) )
= ( ( P3 @ X6 )
& ( Q3 @ X6 ) ) ) ) ) ) ).
% minf(1)
thf(fact_621_minf_I1_J,axiom,
! [P: nat > $o,P3: nat > $o,Q2: nat > $o,Q3: nat > $o] :
( ? [Z6: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z6 )
=> ( ( P @ X3 )
= ( P3 @ X3 ) ) )
=> ( ? [Z6: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z6 )
=> ( ( Q2 @ X3 )
= ( Q3 @ X3 ) ) )
=> ? [Z3: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z3 )
=> ( ( ( P @ X6 )
& ( Q2 @ X6 ) )
= ( ( P3 @ X6 )
& ( Q3 @ X6 ) ) ) ) ) ) ).
% minf(1)
thf(fact_622_pinf_I7_J,axiom,
! [T2: int] :
? [Z3: int] :
! [X6: int] :
( ( ord_less_int @ Z3 @ X6 )
=> ( ord_less_int @ T2 @ X6 ) ) ).
% pinf(7)
thf(fact_623_pinf_I7_J,axiom,
! [T2: nat] :
? [Z3: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z3 @ X6 )
=> ( ord_less_nat @ T2 @ X6 ) ) ).
% pinf(7)
thf(fact_624_pinf_I5_J,axiom,
! [T2: int] :
? [Z3: int] :
! [X6: int] :
( ( ord_less_int @ Z3 @ X6 )
=> ~ ( ord_less_int @ X6 @ T2 ) ) ).
% pinf(5)
thf(fact_625_pinf_I5_J,axiom,
! [T2: nat] :
? [Z3: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z3 @ X6 )
=> ~ ( ord_less_nat @ X6 @ T2 ) ) ).
% pinf(5)
thf(fact_626_pinf_I4_J,axiom,
! [T2: int] :
? [Z3: int] :
! [X6: int] :
( ( ord_less_int @ Z3 @ X6 )
=> ( X6 != T2 ) ) ).
% pinf(4)
thf(fact_627_pinf_I4_J,axiom,
! [T2: nat] :
? [Z3: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z3 @ X6 )
=> ( X6 != T2 ) ) ).
% pinf(4)
thf(fact_628_pinf_I3_J,axiom,
! [T2: int] :
? [Z3: int] :
! [X6: int] :
( ( ord_less_int @ Z3 @ X6 )
=> ( X6 != T2 ) ) ).
% pinf(3)
thf(fact_629_pinf_I3_J,axiom,
! [T2: nat] :
? [Z3: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z3 @ X6 )
=> ( X6 != T2 ) ) ).
% pinf(3)
thf(fact_630_pinf_I2_J,axiom,
! [P: int > $o,P3: int > $o,Q2: int > $o,Q3: int > $o] :
( ? [Z6: int] :
! [X3: int] :
( ( ord_less_int @ Z6 @ X3 )
=> ( ( P @ X3 )
= ( P3 @ X3 ) ) )
=> ( ? [Z6: int] :
! [X3: int] :
( ( ord_less_int @ Z6 @ X3 )
=> ( ( Q2 @ X3 )
= ( Q3 @ X3 ) ) )
=> ? [Z3: int] :
! [X6: int] :
( ( ord_less_int @ Z3 @ X6 )
=> ( ( ( P @ X6 )
| ( Q2 @ X6 ) )
= ( ( P3 @ X6 )
| ( Q3 @ X6 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_631_pinf_I2_J,axiom,
! [P: nat > $o,P3: nat > $o,Q2: nat > $o,Q3: nat > $o] :
( ? [Z6: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z6 @ X3 )
=> ( ( P @ X3 )
= ( P3 @ X3 ) ) )
=> ( ? [Z6: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z6 @ X3 )
=> ( ( Q2 @ X3 )
= ( Q3 @ X3 ) ) )
=> ? [Z3: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z3 @ X6 )
=> ( ( ( P @ X6 )
| ( Q2 @ X6 ) )
= ( ( P3 @ X6 )
| ( Q3 @ X6 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_632_pinf_I1_J,axiom,
! [P: int > $o,P3: int > $o,Q2: int > $o,Q3: int > $o] :
( ? [Z6: int] :
! [X3: int] :
( ( ord_less_int @ Z6 @ X3 )
=> ( ( P @ X3 )
= ( P3 @ X3 ) ) )
=> ( ? [Z6: int] :
! [X3: int] :
( ( ord_less_int @ Z6 @ X3 )
=> ( ( Q2 @ X3 )
= ( Q3 @ X3 ) ) )
=> ? [Z3: int] :
! [X6: int] :
( ( ord_less_int @ Z3 @ X6 )
=> ( ( ( P @ X6 )
& ( Q2 @ X6 ) )
= ( ( P3 @ X6 )
& ( Q3 @ X6 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_633_pinf_I1_J,axiom,
! [P: nat > $o,P3: nat > $o,Q2: nat > $o,Q3: nat > $o] :
( ? [Z6: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z6 @ X3 )
=> ( ( P @ X3 )
= ( P3 @ X3 ) ) )
=> ( ? [Z6: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z6 @ X3 )
=> ( ( Q2 @ X3 )
= ( Q3 @ X3 ) ) )
=> ? [Z3: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z3 @ X6 )
=> ( ( ( P @ X6 )
& ( Q2 @ X6 ) )
= ( ( P3 @ X6 )
& ( Q3 @ X6 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_634_lt__ex,axiom,
! [X: int] :
? [Y3: int] : ( ord_less_int @ Y3 @ X ) ).
% lt_ex
thf(fact_635_gt__ex,axiom,
! [X: int] :
? [X_1: int] : ( ord_less_int @ X @ X_1 ) ).
% gt_ex
thf(fact_636_gt__ex,axiom,
! [X: nat] :
? [X_1: nat] : ( ord_less_nat @ X @ X_1 ) ).
% gt_ex
thf(fact_637_less__imp__neq,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_638_less__imp__neq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_639_order_Oasym,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ~ ( ord_less_int @ B @ A ) ) ).
% order.asym
thf(fact_640_order_Oasym,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% order.asym
thf(fact_641_ord__eq__less__trans,axiom,
! [A: int,B: int,C: int] :
( ( A = B )
=> ( ( ord_less_int @ B @ C )
=> ( ord_less_int @ A @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_642_ord__eq__less__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( A = B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_643_ord__less__eq__trans,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( B = C )
=> ( ord_less_int @ A @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_644_ord__less__eq__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( B = C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_645_less__induct,axiom,
! [P: nat > $o,A: nat] :
( ! [X3: nat] :
( ! [Y7: nat] :
( ( ord_less_nat @ Y7 @ X3 )
=> ( P @ Y7 ) )
=> ( P @ X3 ) )
=> ( P @ A ) ) ).
% less_induct
thf(fact_646_antisym__conv3,axiom,
! [Y: int,X: int] :
( ~ ( ord_less_int @ Y @ X )
=> ( ( ~ ( ord_less_int @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_647_antisym__conv3,axiom,
! [Y: nat,X: nat] :
( ~ ( ord_less_nat @ Y @ X )
=> ( ( ~ ( ord_less_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_648_linorder__cases,axiom,
! [X: int,Y: int] :
( ~ ( ord_less_int @ X @ Y )
=> ( ( X != Y )
=> ( ord_less_int @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_649_linorder__cases,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ( X != Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_650_dual__order_Oasym,axiom,
! [B: int,A: int] :
( ( ord_less_int @ B @ A )
=> ~ ( ord_less_int @ A @ B ) ) ).
% dual_order.asym
thf(fact_651_dual__order_Oasym,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ~ ( ord_less_nat @ A @ B ) ) ).
% dual_order.asym
thf(fact_652_dual__order_Oirrefl,axiom,
! [A: int] :
~ ( ord_less_int @ A @ A ) ).
% dual_order.irrefl
thf(fact_653_dual__order_Oirrefl,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% dual_order.irrefl
thf(fact_654_exists__least__iff,axiom,
( ( ^ [P4: nat > $o] :
? [X5: nat] : ( P4 @ X5 ) )
= ( ^ [P5: nat > $o] :
? [N2: nat] :
( ( P5 @ N2 )
& ! [M2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ~ ( P5 @ M2 ) ) ) ) ) ).
% exists_least_iff
thf(fact_655_linorder__less__wlog,axiom,
! [P: int > int > $o,A: int,B: int] :
( ! [A5: int,B5: int] :
( ( ord_less_int @ A5 @ B5 )
=> ( P @ A5 @ B5 ) )
=> ( ! [A5: int] : ( P @ A5 @ A5 )
=> ( ! [A5: int,B5: int] :
( ( P @ B5 @ A5 )
=> ( P @ A5 @ B5 ) )
=> ( P @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_656_linorder__less__wlog,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A5: nat,B5: nat] :
( ( ord_less_nat @ A5 @ B5 )
=> ( P @ A5 @ B5 ) )
=> ( ! [A5: nat] : ( P @ A5 @ A5 )
=> ( ! [A5: nat,B5: nat] :
( ( P @ B5 @ A5 )
=> ( P @ A5 @ B5 ) )
=> ( P @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_657_order_Ostrict__trans,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ B @ C )
=> ( ord_less_int @ A @ C ) ) ) ).
% order.strict_trans
thf(fact_658_order_Ostrict__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans
thf(fact_659_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_660_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_661_dual__order_Ostrict__trans,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_int @ B @ A )
=> ( ( ord_less_int @ C @ B )
=> ( ord_less_int @ C @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_662_dual__order_Ostrict__trans,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( ord_less_nat @ C @ B )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_663_order_Ostrict__implies__not__eq,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_664_order_Ostrict__implies__not__eq,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_665_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_666_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_667_linorder__neqE,axiom,
! [X: int,Y: int] :
( ( X != Y )
=> ( ~ ( ord_less_int @ X @ Y )
=> ( ord_less_int @ Y @ X ) ) ) ).
% linorder_neqE
thf(fact_668_linorder__neqE,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neqE
thf(fact_669_order__less__asym,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ~ ( ord_less_int @ Y @ X ) ) ).
% order_less_asym
thf(fact_670_order__less__asym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_asym
thf(fact_671_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_672_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_673_order__less__asym_H,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ~ ( ord_less_int @ B @ A ) ) ).
% order_less_asym'
thf(fact_674_order__less__asym_H,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% order_less_asym'
thf(fact_675_order__less__trans,axiom,
! [X: int,Y: int,Z: int] :
( ( ord_less_int @ X @ Y )
=> ( ( ord_less_int @ Y @ Z )
=> ( ord_less_int @ X @ Z ) ) ) ).
% order_less_trans
thf(fact_676_order__less__trans,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ Z )
=> ( ord_less_nat @ X @ Z ) ) ) ).
% order_less_trans
thf(fact_677_ord__eq__less__subst,axiom,
! [A: int,F2: int > int,B: int,C: int] :
( ( A
= ( F2 @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_int @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_int @ A @ ( F2 @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_678_ord__eq__less__subst,axiom,
! [A: nat,F2: int > nat,B: int,C: int] :
( ( A
= ( F2 @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F2 @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_679_ord__eq__less__subst,axiom,
! [A: int,F2: nat > int,B: nat,C: nat] :
( ( A
= ( F2 @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_int @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_int @ A @ ( F2 @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_680_ord__eq__less__subst,axiom,
! [A: nat,F2: nat > nat,B: nat,C: nat] :
( ( A
= ( F2 @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F2 @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_681_ord__less__eq__subst,axiom,
! [A: int,B: int,F2: int > int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ( F2 @ B )
= C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_int @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_int @ ( F2 @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_682_ord__less__eq__subst,axiom,
! [A: int,B: int,F2: int > nat,C: nat] :
( ( ord_less_int @ A @ B )
=> ( ( ( F2 @ B )
= C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_nat @ ( F2 @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_683_ord__less__eq__subst,axiom,
! [A: nat,B: nat,F2: nat > int,C: int] :
( ( ord_less_nat @ A @ B )
=> ( ( ( F2 @ B )
= C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_int @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_int @ ( F2 @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_684_ord__less__eq__subst,axiom,
! [A: nat,B: nat,F2: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ( F2 @ B )
= C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_nat @ ( F2 @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_685_order__less__irrefl,axiom,
! [X: int] :
~ ( ord_less_int @ X @ X ) ).
% order_less_irrefl
thf(fact_686_order__less__irrefl,axiom,
! [X: nat] :
~ ( ord_less_nat @ X @ X ) ).
% order_less_irrefl
thf(fact_687_order__less__subst1,axiom,
! [A: int,F2: int > int,B: int,C: int] :
( ( ord_less_int @ A @ ( F2 @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_int @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_int @ A @ ( F2 @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_688_order__less__subst1,axiom,
! [A: int,F2: nat > int,B: nat,C: nat] :
( ( ord_less_int @ A @ ( F2 @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_int @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_int @ A @ ( F2 @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_689_order__less__subst1,axiom,
! [A: nat,F2: int > nat,B: int,C: int] :
( ( ord_less_nat @ A @ ( F2 @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F2 @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_690_order__less__subst1,axiom,
! [A: nat,F2: nat > nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ ( F2 @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F2 @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_691_order__less__subst2,axiom,
! [A: int,B: int,F2: int > int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ ( F2 @ B ) @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_int @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_int @ ( F2 @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_692_order__less__subst2,axiom,
! [A: int,B: int,F2: int > nat,C: nat] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_nat @ ( F2 @ B ) @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_nat @ ( F2 @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_693_order__less__subst2,axiom,
! [A: nat,B: nat,F2: nat > int,C: int] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_int @ ( F2 @ B ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_int @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_int @ ( F2 @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_694_order__less__subst2,axiom,
! [A: nat,B: nat,F2: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ ( F2 @ B ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_nat @ ( F2 @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_695_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_696_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_697_order__less__imp__triv,axiom,
! [X: int,Y: int,P: $o] :
( ( ord_less_int @ X @ Y )
=> ( ( ord_less_int @ Y @ X )
=> P ) ) ).
% order_less_imp_triv
thf(fact_698_order__less__imp__triv,axiom,
! [X: nat,Y: nat,P: $o] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ X )
=> P ) ) ).
% order_less_imp_triv
thf(fact_699_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_700_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_701_order__less__imp__not__eq,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_702_order__less__imp__not__eq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_703_order__less__imp__not__eq2,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_704_order__less__imp__not__eq2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_705_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_706_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_707_abs__minus__le__zero,axiom,
! [A: int] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( abs_abs_int @ A ) ) @ zero_zero_int ) ).
% abs_minus_le_zero
thf(fact_708_abs__le__D1,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ B )
=> ( ord_less_eq_int @ A @ B ) ) ).
% abs_le_D1
thf(fact_709_abs__ge__self,axiom,
! [A: int] : ( ord_less_eq_int @ A @ ( abs_abs_int @ A ) ) ).
% abs_ge_self
thf(fact_710_compl__mono,axiom,
! [X: set_Pr958786334691620121nt_int,Y: set_Pr958786334691620121nt_int] :
( ( ord_le2843351958646193337nt_int @ X @ Y )
=> ( ord_le2843351958646193337nt_int @ ( uminus6221592323253981072nt_int @ Y ) @ ( uminus6221592323253981072nt_int @ X ) ) ) ).
% compl_mono
thf(fact_711_compl__le__swap1,axiom,
! [Y: set_Pr958786334691620121nt_int,X: set_Pr958786334691620121nt_int] :
( ( ord_le2843351958646193337nt_int @ Y @ ( uminus6221592323253981072nt_int @ X ) )
=> ( ord_le2843351958646193337nt_int @ X @ ( uminus6221592323253981072nt_int @ Y ) ) ) ).
% compl_le_swap1
thf(fact_712_compl__le__swap2,axiom,
! [Y: set_Pr958786334691620121nt_int,X: set_Pr958786334691620121nt_int] :
( ( ord_le2843351958646193337nt_int @ ( uminus6221592323253981072nt_int @ Y ) @ X )
=> ( ord_le2843351958646193337nt_int @ ( uminus6221592323253981072nt_int @ X ) @ Y ) ) ).
% compl_le_swap2
thf(fact_713_le__minus__iff,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ ( uminus_uminus_int @ B ) )
= ( ord_less_eq_int @ B @ ( uminus_uminus_int @ A ) ) ) ).
% le_minus_iff
thf(fact_714_minus__le__iff,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ B )
= ( ord_less_eq_int @ ( uminus_uminus_int @ B ) @ A ) ) ).
% minus_le_iff
thf(fact_715_le__imp__neg__le,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ord_less_eq_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) ) ) ).
% le_imp_neg_le
thf(fact_716_is__num__normalize_I8_J,axiom,
! [A: int,B: int] :
( ( uminus_uminus_int @ ( plus_plus_int @ A @ B ) )
= ( plus_plus_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) ) ) ).
% is_num_normalize(8)
thf(fact_717_group__cancel_Oneg1,axiom,
! [A3: int,K: int,A: int] :
( ( A3
= ( plus_plus_int @ K @ A ) )
=> ( ( uminus_uminus_int @ A3 )
= ( plus_plus_int @ ( uminus_uminus_int @ K ) @ ( uminus_uminus_int @ A ) ) ) ) ).
% group_cancel.neg1
thf(fact_718_add_Oinverse__distrib__swap,axiom,
! [A: int,B: int] :
( ( uminus_uminus_int @ ( plus_plus_int @ A @ B ) )
= ( plus_plus_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) ) ) ).
% add.inverse_distrib_swap
thf(fact_719_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_720_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_721_order__le__imp__less__or__eq,axiom,
! [X: set_Pr958786334691620121nt_int,Y: set_Pr958786334691620121nt_int] :
( ( ord_le2843351958646193337nt_int @ X @ Y )
=> ( ( ord_le7563427860532173253nt_int @ X @ Y )
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_722_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_723_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_724_order__less__le__subst2,axiom,
! [A: int,B: int,F2: int > nat,C: nat] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_nat @ ( F2 @ B ) @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_nat @ ( F2 @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_725_order__less__le__subst2,axiom,
! [A: nat,B: nat,F2: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F2 @ B ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_nat @ ( F2 @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_726_order__less__le__subst2,axiom,
! [A: int,B: int,F2: int > int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_int @ ( F2 @ B ) @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_int @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_int @ ( F2 @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_727_order__less__le__subst2,axiom,
! [A: nat,B: nat,F2: nat > int,C: int] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_int @ ( F2 @ B ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_int @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_int @ ( F2 @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_728_order__less__le__subst2,axiom,
! [A: int,B: int,F2: int > set_Pr958786334691620121nt_int,C: set_Pr958786334691620121nt_int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_le2843351958646193337nt_int @ ( F2 @ B ) @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_le7563427860532173253nt_int @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_le7563427860532173253nt_int @ ( F2 @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_729_order__less__le__subst2,axiom,
! [A: nat,B: nat,F2: nat > set_Pr958786334691620121nt_int,C: set_Pr958786334691620121nt_int] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_le2843351958646193337nt_int @ ( F2 @ B ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_le7563427860532173253nt_int @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_le7563427860532173253nt_int @ ( F2 @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_730_order__less__le__subst1,axiom,
! [A: nat,F2: nat > nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ ( F2 @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F2 @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_731_order__less__le__subst1,axiom,
! [A: int,F2: nat > int,B: nat,C: nat] :
( ( ord_less_int @ A @ ( F2 @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_int @ A @ ( F2 @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_732_order__less__le__subst1,axiom,
! [A: set_Pr958786334691620121nt_int,F2: nat > set_Pr958786334691620121nt_int,B: nat,C: nat] :
( ( ord_le7563427860532173253nt_int @ A @ ( F2 @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_le2843351958646193337nt_int @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_le7563427860532173253nt_int @ A @ ( F2 @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_733_order__less__le__subst1,axiom,
! [A: nat,F2: int > nat,B: int,C: int] :
( ( ord_less_nat @ A @ ( F2 @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F2 @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_734_order__less__le__subst1,axiom,
! [A: int,F2: int > int,B: int,C: int] :
( ( ord_less_int @ A @ ( F2 @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_int @ A @ ( F2 @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_735_order__less__le__subst1,axiom,
! [A: set_Pr958786334691620121nt_int,F2: int > set_Pr958786334691620121nt_int,B: int,C: int] :
( ( ord_le7563427860532173253nt_int @ A @ ( F2 @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_le2843351958646193337nt_int @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_le7563427860532173253nt_int @ A @ ( F2 @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_736_order__less__le__subst1,axiom,
! [A: nat,F2: set_Pr958786334691620121nt_int > nat,B: set_Pr958786334691620121nt_int,C: set_Pr958786334691620121nt_int] :
( ( ord_less_nat @ A @ ( F2 @ B ) )
=> ( ( ord_le2843351958646193337nt_int @ B @ C )
=> ( ! [X3: set_Pr958786334691620121nt_int,Y3: set_Pr958786334691620121nt_int] :
( ( ord_le2843351958646193337nt_int @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F2 @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_737_order__less__le__subst1,axiom,
! [A: int,F2: set_Pr958786334691620121nt_int > int,B: set_Pr958786334691620121nt_int,C: set_Pr958786334691620121nt_int] :
( ( ord_less_int @ A @ ( F2 @ B ) )
=> ( ( ord_le2843351958646193337nt_int @ B @ C )
=> ( ! [X3: set_Pr958786334691620121nt_int,Y3: set_Pr958786334691620121nt_int] :
( ( ord_le2843351958646193337nt_int @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_int @ A @ ( F2 @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_738_order__less__le__subst1,axiom,
! [A: set_Pr958786334691620121nt_int,F2: set_Pr958786334691620121nt_int > set_Pr958786334691620121nt_int,B: set_Pr958786334691620121nt_int,C: set_Pr958786334691620121nt_int] :
( ( ord_le7563427860532173253nt_int @ A @ ( F2 @ B ) )
=> ( ( ord_le2843351958646193337nt_int @ B @ C )
=> ( ! [X3: set_Pr958786334691620121nt_int,Y3: set_Pr958786334691620121nt_int] :
( ( ord_le2843351958646193337nt_int @ X3 @ Y3 )
=> ( ord_le2843351958646193337nt_int @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_le7563427860532173253nt_int @ A @ ( F2 @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_739_order__le__less__subst2,axiom,
! [A: nat,B: nat,F2: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ ( F2 @ B ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_nat @ ( F2 @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_740_order__le__less__subst2,axiom,
! [A: nat,B: nat,F2: nat > int,C: int] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_int @ ( F2 @ B ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_int @ ( F2 @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_741_order__le__less__subst2,axiom,
! [A: nat,B: nat,F2: nat > set_Pr958786334691620121nt_int,C: set_Pr958786334691620121nt_int] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_le7563427860532173253nt_int @ ( F2 @ B ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_le2843351958646193337nt_int @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_le7563427860532173253nt_int @ ( F2 @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_742_order__le__less__subst2,axiom,
! [A: int,B: int,F2: int > nat,C: nat] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_nat @ ( F2 @ B ) @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_nat @ ( F2 @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_743_order__le__less__subst2,axiom,
! [A: int,B: int,F2: int > int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_int @ ( F2 @ B ) @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_int @ ( F2 @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_744_order__le__less__subst2,axiom,
! [A: int,B: int,F2: int > set_Pr958786334691620121nt_int,C: set_Pr958786334691620121nt_int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_le7563427860532173253nt_int @ ( F2 @ B ) @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_le2843351958646193337nt_int @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_le7563427860532173253nt_int @ ( F2 @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_745_order__le__less__subst2,axiom,
! [A: set_Pr958786334691620121nt_int,B: set_Pr958786334691620121nt_int,F2: set_Pr958786334691620121nt_int > nat,C: nat] :
( ( ord_le2843351958646193337nt_int @ A @ B )
=> ( ( ord_less_nat @ ( F2 @ B ) @ C )
=> ( ! [X3: set_Pr958786334691620121nt_int,Y3: set_Pr958786334691620121nt_int] :
( ( ord_le2843351958646193337nt_int @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_nat @ ( F2 @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_746_order__le__less__subst2,axiom,
! [A: set_Pr958786334691620121nt_int,B: set_Pr958786334691620121nt_int,F2: set_Pr958786334691620121nt_int > int,C: int] :
( ( ord_le2843351958646193337nt_int @ A @ B )
=> ( ( ord_less_int @ ( F2 @ B ) @ C )
=> ( ! [X3: set_Pr958786334691620121nt_int,Y3: set_Pr958786334691620121nt_int] :
( ( ord_le2843351958646193337nt_int @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_int @ ( F2 @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_747_order__le__less__subst2,axiom,
! [A: set_Pr958786334691620121nt_int,B: set_Pr958786334691620121nt_int,F2: set_Pr958786334691620121nt_int > set_Pr958786334691620121nt_int,C: set_Pr958786334691620121nt_int] :
( ( ord_le2843351958646193337nt_int @ A @ B )
=> ( ( ord_le7563427860532173253nt_int @ ( F2 @ B ) @ C )
=> ( ! [X3: set_Pr958786334691620121nt_int,Y3: set_Pr958786334691620121nt_int] :
( ( ord_le2843351958646193337nt_int @ X3 @ Y3 )
=> ( ord_le2843351958646193337nt_int @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_le7563427860532173253nt_int @ ( F2 @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_748_order__le__less__subst1,axiom,
! [A: nat,F2: int > nat,B: int,C: int] :
( ( ord_less_eq_nat @ A @ ( F2 @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F2 @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_749_order__le__less__subst1,axiom,
! [A: nat,F2: nat > nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( F2 @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F2 @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_750_order__le__less__subst1,axiom,
! [A: int,F2: int > int,B: int,C: int] :
( ( ord_less_eq_int @ A @ ( F2 @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_int @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_int @ A @ ( F2 @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_751_order__le__less__subst1,axiom,
! [A: int,F2: nat > int,B: nat,C: nat] :
( ( ord_less_eq_int @ A @ ( F2 @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_int @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_int @ A @ ( F2 @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_752_order__le__less__subst1,axiom,
! [A: set_Pr958786334691620121nt_int,F2: int > set_Pr958786334691620121nt_int,B: int,C: int] :
( ( ord_le2843351958646193337nt_int @ A @ ( F2 @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_le7563427860532173253nt_int @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_le7563427860532173253nt_int @ A @ ( F2 @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_753_order__le__less__subst1,axiom,
! [A: set_Pr958786334691620121nt_int,F2: nat > set_Pr958786334691620121nt_int,B: nat,C: nat] :
( ( ord_le2843351958646193337nt_int @ A @ ( F2 @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_le7563427860532173253nt_int @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_le7563427860532173253nt_int @ A @ ( F2 @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_754_order__less__le__trans,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z )
=> ( ord_less_nat @ X @ Z ) ) ) ).
% order_less_le_trans
thf(fact_755_order__less__le__trans,axiom,
! [X: int,Y: int,Z: int] :
( ( ord_less_int @ X @ Y )
=> ( ( ord_less_eq_int @ Y @ Z )
=> ( ord_less_int @ X @ Z ) ) ) ).
% order_less_le_trans
thf(fact_756_order__less__le__trans,axiom,
! [X: set_Pr958786334691620121nt_int,Y: set_Pr958786334691620121nt_int,Z: set_Pr958786334691620121nt_int] :
( ( ord_le7563427860532173253nt_int @ X @ Y )
=> ( ( ord_le2843351958646193337nt_int @ Y @ Z )
=> ( ord_le7563427860532173253nt_int @ X @ Z ) ) ) ).
% order_less_le_trans
thf(fact_757_order__le__less__trans,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ Z )
=> ( ord_less_nat @ X @ Z ) ) ) ).
% order_le_less_trans
thf(fact_758_order__le__less__trans,axiom,
! [X: int,Y: int,Z: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ord_less_int @ Y @ Z )
=> ( ord_less_int @ X @ Z ) ) ) ).
% order_le_less_trans
thf(fact_759_order__le__less__trans,axiom,
! [X: set_Pr958786334691620121nt_int,Y: set_Pr958786334691620121nt_int,Z: set_Pr958786334691620121nt_int] :
( ( ord_le2843351958646193337nt_int @ X @ Y )
=> ( ( ord_le7563427860532173253nt_int @ Y @ Z )
=> ( ord_le7563427860532173253nt_int @ X @ Z ) ) ) ).
% order_le_less_trans
thf(fact_760_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_761_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_762_order__neq__le__trans,axiom,
! [A: set_Pr958786334691620121nt_int,B: set_Pr958786334691620121nt_int] :
( ( A != B )
=> ( ( ord_le2843351958646193337nt_int @ A @ B )
=> ( ord_le7563427860532173253nt_int @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_763_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_764_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_765_order__le__neq__trans,axiom,
! [A: set_Pr958786334691620121nt_int,B: set_Pr958786334691620121nt_int] :
( ( ord_le2843351958646193337nt_int @ A @ B )
=> ( ( A != B )
=> ( ord_le7563427860532173253nt_int @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_766_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_767_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_768_order__less__imp__le,axiom,
! [X: set_Pr958786334691620121nt_int,Y: set_Pr958786334691620121nt_int] :
( ( ord_le7563427860532173253nt_int @ X @ Y )
=> ( ord_le2843351958646193337nt_int @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_769_linorder__not__less,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X @ Y ) )
= ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_not_less
thf(fact_770_linorder__not__less,axiom,
! [X: int,Y: int] :
( ( ~ ( ord_less_int @ X @ Y ) )
= ( ord_less_eq_int @ Y @ X ) ) ).
% linorder_not_less
thf(fact_771_linorder__not__le,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_eq_nat @ X @ Y ) )
= ( ord_less_nat @ Y @ X ) ) ).
% linorder_not_le
thf(fact_772_linorder__not__le,axiom,
! [X: int,Y: int] :
( ( ~ ( ord_less_eq_int @ X @ Y ) )
= ( ord_less_int @ Y @ X ) ) ).
% linorder_not_le
thf(fact_773_order__less__le,axiom,
( ord_less_nat
= ( ^ [X2: nat,Y4: nat] :
( ( ord_less_eq_nat @ X2 @ Y4 )
& ( X2 != Y4 ) ) ) ) ).
% order_less_le
thf(fact_774_order__less__le,axiom,
( ord_less_int
= ( ^ [X2: int,Y4: int] :
( ( ord_less_eq_int @ X2 @ Y4 )
& ( X2 != Y4 ) ) ) ) ).
% order_less_le
thf(fact_775_order__less__le,axiom,
( ord_le7563427860532173253nt_int
= ( ^ [X2: set_Pr958786334691620121nt_int,Y4: set_Pr958786334691620121nt_int] :
( ( ord_le2843351958646193337nt_int @ X2 @ Y4 )
& ( X2 != Y4 ) ) ) ) ).
% order_less_le
thf(fact_776_order__le__less,axiom,
( ord_less_eq_nat
= ( ^ [X2: nat,Y4: nat] :
( ( ord_less_nat @ X2 @ Y4 )
| ( X2 = Y4 ) ) ) ) ).
% order_le_less
thf(fact_777_order__le__less,axiom,
( ord_less_eq_int
= ( ^ [X2: int,Y4: int] :
( ( ord_less_int @ X2 @ Y4 )
| ( X2 = Y4 ) ) ) ) ).
% order_le_less
thf(fact_778_order__le__less,axiom,
( ord_le2843351958646193337nt_int
= ( ^ [X2: set_Pr958786334691620121nt_int,Y4: set_Pr958786334691620121nt_int] :
( ( ord_le7563427860532173253nt_int @ X2 @ Y4 )
| ( X2 = Y4 ) ) ) ) ).
% order_le_less
thf(fact_779_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_780_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_781_dual__order_Ostrict__implies__order,axiom,
! [B: set_Pr958786334691620121nt_int,A: set_Pr958786334691620121nt_int] :
( ( ord_le7563427860532173253nt_int @ B @ A )
=> ( ord_le2843351958646193337nt_int @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_782_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_783_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_784_order_Ostrict__implies__order,axiom,
! [A: set_Pr958786334691620121nt_int,B: set_Pr958786334691620121nt_int] :
( ( ord_le7563427860532173253nt_int @ A @ B )
=> ( ord_le2843351958646193337nt_int @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_785_dual__order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [B2: nat,A2: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
& ~ ( ord_less_eq_nat @ A2 @ B2 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_786_dual__order_Ostrict__iff__not,axiom,
( ord_less_int
= ( ^ [B2: int,A2: int] :
( ( ord_less_eq_int @ B2 @ A2 )
& ~ ( ord_less_eq_int @ A2 @ B2 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_787_dual__order_Ostrict__iff__not,axiom,
( ord_le7563427860532173253nt_int
= ( ^ [B2: set_Pr958786334691620121nt_int,A2: set_Pr958786334691620121nt_int] :
( ( ord_le2843351958646193337nt_int @ B2 @ A2 )
& ~ ( ord_le2843351958646193337nt_int @ A2 @ B2 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_788_dual__order_Ostrict__trans2,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_789_dual__order_Ostrict__trans2,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_int @ B @ A )
=> ( ( ord_less_eq_int @ C @ B )
=> ( ord_less_int @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_790_dual__order_Ostrict__trans2,axiom,
! [B: set_Pr958786334691620121nt_int,A: set_Pr958786334691620121nt_int,C: set_Pr958786334691620121nt_int] :
( ( ord_le7563427860532173253nt_int @ B @ A )
=> ( ( ord_le2843351958646193337nt_int @ C @ B )
=> ( ord_le7563427860532173253nt_int @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_791_dual__order_Ostrict__trans1,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_nat @ C @ B )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_792_dual__order_Ostrict__trans1,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( ord_less_int @ C @ B )
=> ( ord_less_int @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_793_dual__order_Ostrict__trans1,axiom,
! [B: set_Pr958786334691620121nt_int,A: set_Pr958786334691620121nt_int,C: set_Pr958786334691620121nt_int] :
( ( ord_le2843351958646193337nt_int @ B @ A )
=> ( ( ord_le7563427860532173253nt_int @ C @ B )
=> ( ord_le7563427860532173253nt_int @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_794_dual__order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [B2: nat,A2: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
& ( A2 != B2 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_795_dual__order_Ostrict__iff__order,axiom,
( ord_less_int
= ( ^ [B2: int,A2: int] :
( ( ord_less_eq_int @ B2 @ A2 )
& ( A2 != B2 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_796_dual__order_Ostrict__iff__order,axiom,
( ord_le7563427860532173253nt_int
= ( ^ [B2: set_Pr958786334691620121nt_int,A2: set_Pr958786334691620121nt_int] :
( ( ord_le2843351958646193337nt_int @ B2 @ A2 )
& ( A2 != B2 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_797_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [B2: nat,A2: nat] :
( ( ord_less_nat @ B2 @ A2 )
| ( A2 = B2 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_798_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_int
= ( ^ [B2: int,A2: int] :
( ( ord_less_int @ B2 @ A2 )
| ( A2 = B2 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_799_dual__order_Oorder__iff__strict,axiom,
( ord_le2843351958646193337nt_int
= ( ^ [B2: set_Pr958786334691620121nt_int,A2: set_Pr958786334691620121nt_int] :
( ( ord_le7563427860532173253nt_int @ B2 @ A2 )
| ( A2 = B2 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_800_order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
& ~ ( ord_less_eq_nat @ B2 @ A2 ) ) ) ) ).
% order.strict_iff_not
thf(fact_801_order_Ostrict__iff__not,axiom,
( ord_less_int
= ( ^ [A2: int,B2: int] :
( ( ord_less_eq_int @ A2 @ B2 )
& ~ ( ord_less_eq_int @ B2 @ A2 ) ) ) ) ).
% order.strict_iff_not
thf(fact_802_order_Ostrict__iff__not,axiom,
( ord_le7563427860532173253nt_int
= ( ^ [A2: set_Pr958786334691620121nt_int,B2: set_Pr958786334691620121nt_int] :
( ( ord_le2843351958646193337nt_int @ A2 @ B2 )
& ~ ( ord_le2843351958646193337nt_int @ B2 @ A2 ) ) ) ) ).
% order.strict_iff_not
thf(fact_803_order_Ostrict__trans2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_804_order_Ostrict__trans2,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ord_less_int @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_805_order_Ostrict__trans2,axiom,
! [A: set_Pr958786334691620121nt_int,B: set_Pr958786334691620121nt_int,C: set_Pr958786334691620121nt_int] :
( ( ord_le7563427860532173253nt_int @ A @ B )
=> ( ( ord_le2843351958646193337nt_int @ B @ C )
=> ( ord_le7563427860532173253nt_int @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_806_order_Ostrict__trans1,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_807_order_Ostrict__trans1,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_int @ B @ C )
=> ( ord_less_int @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_808_order_Ostrict__trans1,axiom,
! [A: set_Pr958786334691620121nt_int,B: set_Pr958786334691620121nt_int,C: set_Pr958786334691620121nt_int] :
( ( ord_le2843351958646193337nt_int @ A @ B )
=> ( ( ord_le7563427860532173253nt_int @ B @ C )
=> ( ord_le7563427860532173253nt_int @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_809_order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
& ( A2 != B2 ) ) ) ) ).
% order.strict_iff_order
thf(fact_810_order_Ostrict__iff__order,axiom,
( ord_less_int
= ( ^ [A2: int,B2: int] :
( ( ord_less_eq_int @ A2 @ B2 )
& ( A2 != B2 ) ) ) ) ).
% order.strict_iff_order
thf(fact_811_order_Ostrict__iff__order,axiom,
( ord_le7563427860532173253nt_int
= ( ^ [A2: set_Pr958786334691620121nt_int,B2: set_Pr958786334691620121nt_int] :
( ( ord_le2843351958646193337nt_int @ A2 @ B2 )
& ( A2 != B2 ) ) ) ) ).
% order.strict_iff_order
thf(fact_812_order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [A2: nat,B2: nat] :
( ( ord_less_nat @ A2 @ B2 )
| ( A2 = B2 ) ) ) ) ).
% order.order_iff_strict
thf(fact_813_order_Oorder__iff__strict,axiom,
( ord_less_eq_int
= ( ^ [A2: int,B2: int] :
( ( ord_less_int @ A2 @ B2 )
| ( A2 = B2 ) ) ) ) ).
% order.order_iff_strict
thf(fact_814_order_Oorder__iff__strict,axiom,
( ord_le2843351958646193337nt_int
= ( ^ [A2: set_Pr958786334691620121nt_int,B2: set_Pr958786334691620121nt_int] :
( ( ord_le7563427860532173253nt_int @ A2 @ B2 )
| ( A2 = B2 ) ) ) ) ).
% order.order_iff_strict
thf(fact_815_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_816_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_817_less__le__not__le,axiom,
( ord_less_nat
= ( ^ [X2: nat,Y4: nat] :
( ( ord_less_eq_nat @ X2 @ Y4 )
& ~ ( ord_less_eq_nat @ Y4 @ X2 ) ) ) ) ).
% less_le_not_le
thf(fact_818_less__le__not__le,axiom,
( ord_less_int
= ( ^ [X2: int,Y4: int] :
( ( ord_less_eq_int @ X2 @ Y4 )
& ~ ( ord_less_eq_int @ Y4 @ X2 ) ) ) ) ).
% less_le_not_le
thf(fact_819_less__le__not__le,axiom,
( ord_le7563427860532173253nt_int
= ( ^ [X2: set_Pr958786334691620121nt_int,Y4: set_Pr958786334691620121nt_int] :
( ( ord_le2843351958646193337nt_int @ X2 @ Y4 )
& ~ ( ord_le2843351958646193337nt_int @ Y4 @ X2 ) ) ) ) ).
% less_le_not_le
thf(fact_820_antisym__conv2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ~ ( ord_less_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_821_antisym__conv2,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ~ ( ord_less_int @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_822_antisym__conv2,axiom,
! [X: set_Pr958786334691620121nt_int,Y: set_Pr958786334691620121nt_int] :
( ( ord_le2843351958646193337nt_int @ X @ Y )
=> ( ( ~ ( ord_le7563427860532173253nt_int @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_823_antisym__conv1,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_824_antisym__conv1,axiom,
! [X: int,Y: int] :
( ~ ( ord_less_int @ X @ Y )
=> ( ( ord_less_eq_int @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_825_antisym__conv1,axiom,
! [X: set_Pr958786334691620121nt_int,Y: set_Pr958786334691620121nt_int] :
( ~ ( ord_le7563427860532173253nt_int @ X @ Y )
=> ( ( ord_le2843351958646193337nt_int @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_826_nless__le,axiom,
! [A: nat,B: nat] :
( ( ~ ( ord_less_nat @ A @ B ) )
= ( ~ ( ord_less_eq_nat @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_827_nless__le,axiom,
! [A: int,B: int] :
( ( ~ ( ord_less_int @ A @ B ) )
= ( ~ ( ord_less_eq_int @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_828_nless__le,axiom,
! [A: set_Pr958786334691620121nt_int,B: set_Pr958786334691620121nt_int] :
( ( ~ ( ord_le7563427860532173253nt_int @ A @ B ) )
= ( ~ ( ord_le2843351958646193337nt_int @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_829_leI,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_eq_nat @ Y @ X ) ) ).
% leI
thf(fact_830_leI,axiom,
! [X: int,Y: int] :
( ~ ( ord_less_int @ X @ Y )
=> ( ord_less_eq_int @ Y @ X ) ) ).
% leI
thf(fact_831_leD,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ~ ( ord_less_nat @ X @ Y ) ) ).
% leD
thf(fact_832_leD,axiom,
! [Y: int,X: int] :
( ( ord_less_eq_int @ Y @ X )
=> ~ ( ord_less_int @ X @ Y ) ) ).
% leD
thf(fact_833_leD,axiom,
! [Y: set_Pr958786334691620121nt_int,X: set_Pr958786334691620121nt_int] :
( ( ord_le2843351958646193337nt_int @ Y @ X )
=> ~ ( ord_le7563427860532173253nt_int @ X @ Y ) ) ).
% leD
thf(fact_834_pinf_I6_J,axiom,
! [T2: nat] :
? [Z3: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z3 @ X6 )
=> ~ ( ord_less_eq_nat @ X6 @ T2 ) ) ).
% pinf(6)
thf(fact_835_pinf_I6_J,axiom,
! [T2: int] :
? [Z3: int] :
! [X6: int] :
( ( ord_less_int @ Z3 @ X6 )
=> ~ ( ord_less_eq_int @ X6 @ T2 ) ) ).
% pinf(6)
thf(fact_836_pinf_I8_J,axiom,
! [T2: nat] :
? [Z3: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z3 @ X6 )
=> ( ord_less_eq_nat @ T2 @ X6 ) ) ).
% pinf(8)
thf(fact_837_pinf_I8_J,axiom,
! [T2: int] :
? [Z3: int] :
! [X6: int] :
( ( ord_less_int @ Z3 @ X6 )
=> ( ord_less_eq_int @ T2 @ X6 ) ) ).
% pinf(8)
thf(fact_838_minf_I6_J,axiom,
! [T2: nat] :
? [Z3: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z3 )
=> ( ord_less_eq_nat @ X6 @ T2 ) ) ).
% minf(6)
thf(fact_839_minf_I6_J,axiom,
! [T2: int] :
? [Z3: int] :
! [X6: int] :
( ( ord_less_int @ X6 @ Z3 )
=> ( ord_less_eq_int @ X6 @ T2 ) ) ).
% minf(6)
thf(fact_840_minf_I8_J,axiom,
! [T2: nat] :
? [Z3: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z3 )
=> ~ ( ord_less_eq_nat @ T2 @ X6 ) ) ).
% minf(8)
thf(fact_841_minf_I8_J,axiom,
! [T2: int] :
? [Z3: int] :
! [X6: int] :
( ( ord_less_int @ X6 @ Z3 )
=> ~ ( ord_less_eq_int @ T2 @ X6 ) ) ).
% minf(8)
thf(fact_842_verit__comp__simplify1_I3_J,axiom,
! [B4: nat,A4: nat] :
( ( ~ ( ord_less_eq_nat @ B4 @ A4 ) )
= ( ord_less_nat @ A4 @ B4 ) ) ).
% verit_comp_simplify1(3)
thf(fact_843_verit__comp__simplify1_I3_J,axiom,
! [B4: int,A4: int] :
( ( ~ ( ord_less_eq_int @ B4 @ A4 ) )
= ( ord_less_int @ A4 @ B4 ) ) ).
% verit_comp_simplify1(3)
thf(fact_844_less__numeral__extra_I3_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_numeral_extra(3)
thf(fact_845_less__numeral__extra_I3_J,axiom,
~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% less_numeral_extra(3)
thf(fact_846_gr__zeroI,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr_zeroI
thf(fact_847_not__less__zero,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less_zero
thf(fact_848_gr__implies__not__zero,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_849_zero__less__iff__neq__zero,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
= ( N != zero_zero_nat ) ) ).
% zero_less_iff_neq_zero
thf(fact_850_add__mono__thms__linordered__field_I5_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_int @ I @ J )
& ( ord_less_int @ K @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_851_add__mono__thms__linordered__field_I5_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I @ J )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_852_add__mono__thms__linordered__field_I2_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( I = J )
& ( ord_less_int @ K @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_853_add__mono__thms__linordered__field_I2_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( I = J )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_854_add__mono__thms__linordered__field_I1_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_int @ I @ J )
& ( K = L ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_855_add__mono__thms__linordered__field_I1_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I @ J )
& ( K = L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_856_add__strict__mono,axiom,
! [A: int,B: int,C: int,D2: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ C @ D2 )
=> ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D2 ) ) ) ) ).
% add_strict_mono
thf(fact_857_add__strict__mono,axiom,
! [A: nat,B: nat,C: nat,D2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D2 )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D2 ) ) ) ) ).
% add_strict_mono
thf(fact_858_add__strict__left__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) ) ) ).
% add_strict_left_mono
thf(fact_859_add__strict__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) ) ) ).
% add_strict_left_mono
thf(fact_860_add__strict__right__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) ) ) ).
% add_strict_right_mono
thf(fact_861_add__strict__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).
% add_strict_right_mono
thf(fact_862_add__less__imp__less__left,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
=> ( ord_less_int @ A @ B ) ) ).
% add_less_imp_less_left
thf(fact_863_add__less__imp__less__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
=> ( ord_less_nat @ A @ B ) ) ).
% add_less_imp_less_left
thf(fact_864_add__less__imp__less__right,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
=> ( ord_less_int @ A @ B ) ) ).
% add_less_imp_less_right
thf(fact_865_add__less__imp__less__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
=> ( ord_less_nat @ A @ B ) ) ).
% add_less_imp_less_right
thf(fact_866_uminus__int__code_I1_J,axiom,
( ( uminus_uminus_int @ zero_zero_int )
= zero_zero_int ) ).
% uminus_int_code(1)
thf(fact_867_less__supI1,axiom,
! [X: set_Pr958786334691620121nt_int,A: set_Pr958786334691620121nt_int,B: set_Pr958786334691620121nt_int] :
( ( ord_le7563427860532173253nt_int @ X @ A )
=> ( ord_le7563427860532173253nt_int @ X @ ( sup_su6024340866399070445nt_int @ A @ B ) ) ) ).
% less_supI1
thf(fact_868_less__supI1,axiom,
! [X: int,A: int,B: int] :
( ( ord_less_int @ X @ A )
=> ( ord_less_int @ X @ ( sup_sup_int @ A @ B ) ) ) ).
% less_supI1
thf(fact_869_less__supI1,axiom,
! [X: nat,A: nat,B: nat] :
( ( ord_less_nat @ X @ A )
=> ( ord_less_nat @ X @ ( sup_sup_nat @ A @ B ) ) ) ).
% less_supI1
thf(fact_870_less__supI2,axiom,
! [X: set_Pr958786334691620121nt_int,B: set_Pr958786334691620121nt_int,A: set_Pr958786334691620121nt_int] :
( ( ord_le7563427860532173253nt_int @ X @ B )
=> ( ord_le7563427860532173253nt_int @ X @ ( sup_su6024340866399070445nt_int @ A @ B ) ) ) ).
% less_supI2
thf(fact_871_less__supI2,axiom,
! [X: int,B: int,A: int] :
( ( ord_less_int @ X @ B )
=> ( ord_less_int @ X @ ( sup_sup_int @ A @ B ) ) ) ).
% less_supI2
thf(fact_872_less__supI2,axiom,
! [X: nat,B: nat,A: nat] :
( ( ord_less_nat @ X @ B )
=> ( ord_less_nat @ X @ ( sup_sup_nat @ A @ B ) ) ) ).
% less_supI2
thf(fact_873_sup_Oabsorb3,axiom,
! [B: set_Pr958786334691620121nt_int,A: set_Pr958786334691620121nt_int] :
( ( ord_le7563427860532173253nt_int @ B @ A )
=> ( ( sup_su6024340866399070445nt_int @ A @ B )
= A ) ) ).
% sup.absorb3
thf(fact_874_sup_Oabsorb3,axiom,
! [B: int,A: int] :
( ( ord_less_int @ B @ A )
=> ( ( sup_sup_int @ A @ B )
= A ) ) ).
% sup.absorb3
thf(fact_875_sup_Oabsorb3,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( sup_sup_nat @ A @ B )
= A ) ) ).
% sup.absorb3
thf(fact_876_sup_Oabsorb4,axiom,
! [A: set_Pr958786334691620121nt_int,B: set_Pr958786334691620121nt_int] :
( ( ord_le7563427860532173253nt_int @ A @ B )
=> ( ( sup_su6024340866399070445nt_int @ A @ B )
= B ) ) ).
% sup.absorb4
thf(fact_877_sup_Oabsorb4,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ( ( sup_sup_int @ A @ B )
= B ) ) ).
% sup.absorb4
thf(fact_878_sup_Oabsorb4,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( sup_sup_nat @ A @ B )
= B ) ) ).
% sup.absorb4
thf(fact_879_sup_Ostrict__boundedE,axiom,
! [B: set_Pr958786334691620121nt_int,C: set_Pr958786334691620121nt_int,A: set_Pr958786334691620121nt_int] :
( ( ord_le7563427860532173253nt_int @ ( sup_su6024340866399070445nt_int @ B @ C ) @ A )
=> ~ ( ( ord_le7563427860532173253nt_int @ B @ A )
=> ~ ( ord_le7563427860532173253nt_int @ C @ A ) ) ) ).
% sup.strict_boundedE
thf(fact_880_sup_Ostrict__boundedE,axiom,
! [B: int,C: int,A: int] :
( ( ord_less_int @ ( sup_sup_int @ B @ C ) @ A )
=> ~ ( ( ord_less_int @ B @ A )
=> ~ ( ord_less_int @ C @ A ) ) ) ).
% sup.strict_boundedE
thf(fact_881_sup_Ostrict__boundedE,axiom,
! [B: nat,C: nat,A: nat] :
( ( ord_less_nat @ ( sup_sup_nat @ B @ C ) @ A )
=> ~ ( ( ord_less_nat @ B @ A )
=> ~ ( ord_less_nat @ C @ A ) ) ) ).
% sup.strict_boundedE
thf(fact_882_sup_Ostrict__order__iff,axiom,
( ord_le7563427860532173253nt_int
= ( ^ [B2: set_Pr958786334691620121nt_int,A2: set_Pr958786334691620121nt_int] :
( ( A2
= ( sup_su6024340866399070445nt_int @ A2 @ B2 ) )
& ( A2 != B2 ) ) ) ) ).
% sup.strict_order_iff
thf(fact_883_sup_Ostrict__order__iff,axiom,
( ord_less_int
= ( ^ [B2: int,A2: int] :
( ( A2
= ( sup_sup_int @ A2 @ B2 ) )
& ( A2 != B2 ) ) ) ) ).
% sup.strict_order_iff
thf(fact_884_sup_Ostrict__order__iff,axiom,
( ord_less_nat
= ( ^ [B2: nat,A2: nat] :
( ( A2
= ( sup_sup_nat @ A2 @ B2 ) )
& ( A2 != B2 ) ) ) ) ).
% sup.strict_order_iff
thf(fact_885_sup_Ostrict__coboundedI1,axiom,
! [C: set_Pr958786334691620121nt_int,A: set_Pr958786334691620121nt_int,B: set_Pr958786334691620121nt_int] :
( ( ord_le7563427860532173253nt_int @ C @ A )
=> ( ord_le7563427860532173253nt_int @ C @ ( sup_su6024340866399070445nt_int @ A @ B ) ) ) ).
% sup.strict_coboundedI1
thf(fact_886_sup_Ostrict__coboundedI1,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ C @ A )
=> ( ord_less_int @ C @ ( sup_sup_int @ A @ B ) ) ) ).
% sup.strict_coboundedI1
thf(fact_887_sup_Ostrict__coboundedI1,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_nat @ C @ A )
=> ( ord_less_nat @ C @ ( sup_sup_nat @ A @ B ) ) ) ).
% sup.strict_coboundedI1
thf(fact_888_sup_Ostrict__coboundedI2,axiom,
! [C: set_Pr958786334691620121nt_int,B: set_Pr958786334691620121nt_int,A: set_Pr958786334691620121nt_int] :
( ( ord_le7563427860532173253nt_int @ C @ B )
=> ( ord_le7563427860532173253nt_int @ C @ ( sup_su6024340866399070445nt_int @ A @ B ) ) ) ).
% sup.strict_coboundedI2
thf(fact_889_sup_Ostrict__coboundedI2,axiom,
! [C: int,B: int,A: int] :
( ( ord_less_int @ C @ B )
=> ( ord_less_int @ C @ ( sup_sup_int @ A @ B ) ) ) ).
% sup.strict_coboundedI2
thf(fact_890_sup_Ostrict__coboundedI2,axiom,
! [C: nat,B: nat,A: nat] :
( ( ord_less_nat @ C @ B )
=> ( ord_less_nat @ C @ ( sup_sup_nat @ A @ B ) ) ) ).
% sup.strict_coboundedI2
thf(fact_891_less__int__code_I1_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_int_code(1)
thf(fact_892_int__cases2,axiom,
! [Z: int] :
( ! [N3: nat] :
( Z
!= ( semiri1314217659103216013at_int @ N3 ) )
=> ~ ! [N3: nat] :
( Z
!= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N3 ) ) ) ) ).
% int_cases2
thf(fact_893_of__int__of__nat,axiom,
( ring_1_of_int_int
= ( ^ [K2: int] : ( if_int @ ( ord_less_int @ K2 @ zero_zero_int ) @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( nat2 @ ( uminus_uminus_int @ K2 ) ) ) ) @ ( semiri1314217659103216013at_int @ ( nat2 @ K2 ) ) ) ) ) ).
% of_int_of_nat
thf(fact_894_of__int__pos,axiom,
! [Z: int] :
( ( ord_less_int @ zero_zero_int @ Z )
=> ( ord_less_int @ zero_zero_int @ ( ring_1_of_int_int @ Z ) ) ) ).
% of_int_pos
thf(fact_895_of__nat__less__of__int__iff,axiom,
! [N: nat,X: int] :
( ( ord_less_int @ ( semiri1314217659103216013at_int @ N ) @ ( ring_1_of_int_int @ X ) )
= ( ord_less_int @ ( semiri1314217659103216013at_int @ N ) @ X ) ) ).
% of_nat_less_of_int_iff
thf(fact_896_abs__ge__zero,axiom,
! [A: int] : ( ord_less_eq_int @ zero_zero_int @ ( abs_abs_int @ A ) ) ).
% abs_ge_zero
thf(fact_897_abs__triangle__ineq,axiom,
! [A: int,B: int] : ( ord_less_eq_int @ ( abs_abs_int @ ( plus_plus_int @ A @ B ) ) @ ( plus_plus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) ) ).
% abs_triangle_ineq
thf(fact_898_add__eq__0__iff,axiom,
! [A: int,B: int] :
( ( ( plus_plus_int @ A @ B )
= zero_zero_int )
= ( B
= ( uminus_uminus_int @ A ) ) ) ).
% add_eq_0_iff
thf(fact_899_ab__group__add__class_Oab__left__minus,axiom,
! [A: int] :
( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ A )
= zero_zero_int ) ).
% ab_group_add_class.ab_left_minus
thf(fact_900_add_Oinverse__unique,axiom,
! [A: int,B: int] :
( ( ( plus_plus_int @ A @ B )
= zero_zero_int )
=> ( ( uminus_uminus_int @ A )
= B ) ) ).
% add.inverse_unique
thf(fact_901_eq__neg__iff__add__eq__0,axiom,
! [A: int,B: int] :
( ( A
= ( uminus_uminus_int @ B ) )
= ( ( plus_plus_int @ A @ B )
= zero_zero_int ) ) ).
% eq_neg_iff_add_eq_0
thf(fact_902_neg__eq__iff__add__eq__0,axiom,
! [A: int,B: int] :
( ( ( uminus_uminus_int @ A )
= B )
= ( ( plus_plus_int @ A @ B )
= zero_zero_int ) ) ).
% neg_eq_iff_add_eq_0
thf(fact_903_add__less__le__mono,axiom,
! [A: nat,B: nat,C: nat,D2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D2 )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D2 ) ) ) ) ).
% add_less_le_mono
thf(fact_904_add__less__le__mono,axiom,
! [A: int,B: int,C: int,D2: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_int @ C @ D2 )
=> ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D2 ) ) ) ) ).
% add_less_le_mono
thf(fact_905_add__le__less__mono,axiom,
! [A: nat,B: nat,C: nat,D2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D2 )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D2 ) ) ) ) ).
% add_le_less_mono
thf(fact_906_add__le__less__mono,axiom,
! [A: int,B: int,C: int,D2: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_int @ C @ D2 )
=> ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D2 ) ) ) ) ).
% add_le_less_mono
thf(fact_907_add__mono__thms__linordered__field_I3_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I @ J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_908_add__mono__thms__linordered__field_I3_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_int @ I @ J )
& ( ord_less_eq_int @ K @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_909_add__mono__thms__linordered__field_I4_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_910_add__mono__thms__linordered__field_I4_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_eq_int @ I @ J )
& ( ord_less_int @ K @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_911_add__neg__neg,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).
% add_neg_neg
thf(fact_912_add__neg__neg,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ zero_zero_nat )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_neg_neg
thf(fact_913_add__pos__pos,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).
% add_pos_pos
thf(fact_914_add__pos__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_pos_pos
thf(fact_915_canonically__ordered__monoid__add__class_OlessE,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ! [C2: nat] :
( ( B
= ( plus_plus_nat @ A @ C2 ) )
=> ( C2 = zero_zero_nat ) ) ) ).
% canonically_ordered_monoid_add_class.lessE
thf(fact_916_pos__add__strict,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ B @ C )
=> ( ord_less_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).
% pos_add_strict
thf(fact_917_pos__add__strict,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% pos_add_strict
thf(fact_918_of__nat__less__0__iff,axiom,
! [M: nat] :
~ ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ zero_zero_int ) ).
% of_nat_less_0_iff
thf(fact_919_of__nat__less__0__iff,axiom,
! [M: nat] :
~ ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ zero_zero_nat ) ).
% of_nat_less_0_iff
thf(fact_920_GreatestI__nat,axiom,
! [P: nat > $o,K: nat,B: nat] :
( ( P @ K )
=> ( ! [Y3: nat] :
( ( P @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ B ) )
=> ( P @ ( order_Greatest_nat @ P ) ) ) ) ).
% GreatestI_nat
thf(fact_921_Greatest__le__nat,axiom,
! [P: nat > $o,K: nat,B: nat] :
( ( P @ K )
=> ( ! [Y3: nat] :
( ( P @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ B ) )
=> ( ord_less_eq_nat @ K @ ( order_Greatest_nat @ P ) ) ) ) ).
% Greatest_le_nat
thf(fact_922_GreatestI__ex__nat,axiom,
! [P: nat > $o,B: nat] :
( ? [X_12: nat] : ( P @ X_12 )
=> ( ! [Y3: nat] :
( ( P @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ B ) )
=> ( P @ ( order_Greatest_nat @ P ) ) ) ) ).
% GreatestI_ex_nat
thf(fact_923_add__neg__nonpos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_neg_nonpos
thf(fact_924_add__neg__nonpos,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_eq_int @ B @ zero_zero_int )
=> ( ord_less_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).
% add_neg_nonpos
thf(fact_925_add__nonneg__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_nonneg_pos
thf(fact_926_add__nonneg__pos,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).
% add_nonneg_pos
thf(fact_927_add__nonpos__neg,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_nonpos_neg
thf(fact_928_add__nonpos__neg,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).
% add_nonpos_neg
thf(fact_929_add__pos__nonneg,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_pos_nonneg
thf(fact_930_add__pos__nonneg,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).
% add_pos_nonneg
thf(fact_931_add__strict__increasing,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_strict_increasing
thf(fact_932_add__strict__increasing,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ord_less_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).
% add_strict_increasing
thf(fact_933_add__strict__increasing2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_strict_increasing2
thf(fact_934_add__strict__increasing2,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ B @ C )
=> ( ord_less_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).
% add_strict_increasing2
thf(fact_935_int__zle__neg,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ N ) @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ M ) ) )
= ( ( N = zero_zero_nat )
& ( M = zero_zero_nat ) ) ) ).
% int_zle_neg
thf(fact_936_negative__zle__0,axiom,
! [N: nat] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ zero_zero_int ) ).
% negative_zle_0
thf(fact_937_nonpos__int__cases,axiom,
! [K: int] :
( ( ord_less_eq_int @ K @ zero_zero_int )
=> ~ ! [N3: nat] :
( K
!= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N3 ) ) ) ) ).
% nonpos_int_cases
thf(fact_938_abs__0,axiom,
( ( abs_abs_int @ zero_zero_int )
= zero_zero_int ) ).
% abs_0
thf(fact_939_abs__if,axiom,
( abs_abs_int
= ( ^ [A2: int] : ( if_int @ ( ord_less_int @ A2 @ zero_zero_int ) @ ( uminus_uminus_int @ A2 ) @ A2 ) ) ) ).
% abs_if
thf(fact_940_abs__eq__iff_H,axiom,
! [A: int,B: int] :
( ( ( abs_abs_int @ A )
= B )
= ( ( ord_less_eq_int @ zero_zero_int @ B )
& ( ( A = B )
| ( A
= ( uminus_uminus_int @ B ) ) ) ) ) ).
% abs_eq_iff'
thf(fact_941_eq__abs__iff_H,axiom,
! [A: int,B: int] :
( ( A
= ( abs_abs_int @ B ) )
= ( ( ord_less_eq_int @ zero_zero_int @ A )
& ( ( B = A )
| ( B
= ( uminus_uminus_int @ A ) ) ) ) ) ).
% eq_abs_iff'
thf(fact_942_Compl__anti__mono,axiom,
! [A3: set_Pr958786334691620121nt_int,B3: set_Pr958786334691620121nt_int] :
( ( ord_le2843351958646193337nt_int @ A3 @ B3 )
=> ( ord_le2843351958646193337nt_int @ ( uminus6221592323253981072nt_int @ B3 ) @ ( uminus6221592323253981072nt_int @ A3 ) ) ) ).
% Compl_anti_mono
thf(fact_943_Compl__subset__Compl__iff,axiom,
! [A3: set_Pr958786334691620121nt_int,B3: set_Pr958786334691620121nt_int] :
( ( ord_le2843351958646193337nt_int @ ( uminus6221592323253981072nt_int @ A3 ) @ ( uminus6221592323253981072nt_int @ B3 ) )
= ( ord_le2843351958646193337nt_int @ B3 @ A3 ) ) ).
% Compl_subset_Compl_iff
thf(fact_944_psubsetI,axiom,
! [A3: set_Pr958786334691620121nt_int,B3: set_Pr958786334691620121nt_int] :
( ( ord_le2843351958646193337nt_int @ A3 @ B3 )
=> ( ( A3 != B3 )
=> ( ord_le7563427860532173253nt_int @ A3 @ B3 ) ) ) ).
% psubsetI
thf(fact_945_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_946_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% neq0_conv
thf(fact_947_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_948_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_949_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_950_zless__nat__conj,axiom,
! [W2: int,Z: int] :
( ( ord_less_nat @ ( nat2 @ W2 ) @ ( nat2 @ Z ) )
= ( ( ord_less_int @ zero_zero_int @ Z )
& ( ord_less_int @ W2 @ Z ) ) ) ).
% zless_nat_conj
thf(fact_951_zero__less__nat__eq,axiom,
! [Z: int] :
( ( ord_less_nat @ zero_zero_nat @ ( nat2 @ Z ) )
= ( ord_less_int @ zero_zero_int @ Z ) ) ).
% zero_less_nat_eq
thf(fact_952_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_953_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_not_refl
thf(fact_954_less__not__refl2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
=> ( M != N ) ) ).
% less_not_refl2
thf(fact_955_less__not__refl3,axiom,
! [S: nat,T2: nat] :
( ( ord_less_nat @ S @ T2 )
=> ( S != T2 ) ) ).
% less_not_refl3
thf(fact_956_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_957_nat__less__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N3: nat] :
( ! [M3: nat] :
( ( ord_less_nat @ M3 @ N3 )
=> ( P @ M3 ) )
=> ( P @ N3 ) )
=> ( P @ N ) ) ).
% nat_less_induct
thf(fact_958_infinite__descent,axiom,
! [P: nat > $o,N: nat] :
( ! [N3: nat] :
( ~ ( P @ N3 )
=> ? [M3: nat] :
( ( ord_less_nat @ M3 @ N3 )
& ~ ( P @ M3 ) ) )
=> ( P @ N ) ) ).
% infinite_descent
thf(fact_959_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_960_psubsetE,axiom,
! [A3: set_Pr958786334691620121nt_int,B3: set_Pr958786334691620121nt_int] :
( ( ord_le7563427860532173253nt_int @ A3 @ B3 )
=> ~ ( ( ord_le2843351958646193337nt_int @ A3 @ B3 )
=> ( ord_le2843351958646193337nt_int @ B3 @ A3 ) ) ) ).
% psubsetE
thf(fact_961_psubset__eq,axiom,
( ord_le7563427860532173253nt_int
= ( ^ [A6: set_Pr958786334691620121nt_int,B6: set_Pr958786334691620121nt_int] :
( ( ord_le2843351958646193337nt_int @ A6 @ B6 )
& ( A6 != B6 ) ) ) ) ).
% psubset_eq
thf(fact_962_psubset__imp__subset,axiom,
! [A3: set_Pr958786334691620121nt_int,B3: set_Pr958786334691620121nt_int] :
( ( ord_le7563427860532173253nt_int @ A3 @ B3 )
=> ( ord_le2843351958646193337nt_int @ A3 @ B3 ) ) ).
% psubset_imp_subset
thf(fact_963_psubset__subset__trans,axiom,
! [A3: set_Pr958786334691620121nt_int,B3: set_Pr958786334691620121nt_int,C3: set_Pr958786334691620121nt_int] :
( ( ord_le7563427860532173253nt_int @ A3 @ B3 )
=> ( ( ord_le2843351958646193337nt_int @ B3 @ C3 )
=> ( ord_le7563427860532173253nt_int @ A3 @ C3 ) ) ) ).
% psubset_subset_trans
thf(fact_964_subset__not__subset__eq,axiom,
( ord_le7563427860532173253nt_int
= ( ^ [A6: set_Pr958786334691620121nt_int,B6: set_Pr958786334691620121nt_int] :
( ( ord_le2843351958646193337nt_int @ A6 @ B6 )
& ~ ( ord_le2843351958646193337nt_int @ B6 @ A6 ) ) ) ) ).
% subset_not_subset_eq
thf(fact_965_subset__psubset__trans,axiom,
! [A3: set_Pr958786334691620121nt_int,B3: set_Pr958786334691620121nt_int,C3: set_Pr958786334691620121nt_int] :
( ( ord_le2843351958646193337nt_int @ A3 @ B3 )
=> ( ( ord_le7563427860532173253nt_int @ B3 @ C3 )
=> ( ord_le7563427860532173253nt_int @ A3 @ C3 ) ) ) ).
% subset_psubset_trans
thf(fact_966_subset__iff__psubset__eq,axiom,
( ord_le2843351958646193337nt_int
= ( ^ [A6: set_Pr958786334691620121nt_int,B6: set_Pr958786334691620121nt_int] :
( ( ord_le7563427860532173253nt_int @ A6 @ B6 )
| ( A6 = B6 ) ) ) ) ).
% subset_iff_psubset_eq
thf(fact_967_bot__nat__0_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_968_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr0I
thf(fact_969_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr0
thf(fact_970_not__less0,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less0
thf(fact_971_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_zeroE
thf(fact_972_gr__implies__not0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_973_infinite__descent0,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ~ ( P @ N3 )
=> ? [M3: nat] :
( ( ord_less_nat @ M3 @ N3 )
& ~ ( P @ M3 ) ) ) )
=> ( P @ N ) ) ) ).
% infinite_descent0
thf(fact_974_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
& ( M2 != N2 ) ) ) ) ).
% nat_less_le
thf(fact_975_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_976_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_977_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_978_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_979_less__mono__imp__le__mono,axiom,
! [F2: nat > nat,I: nat,J: nat] :
( ! [I2: nat,J2: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ord_less_nat @ ( F2 @ I2 ) @ ( F2 @ J2 ) ) )
=> ( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( F2 @ I ) @ ( F2 @ J ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_980_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_981_trans__less__add2,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ M @ J ) ) ) ).
% trans_less_add2
thf(fact_982_trans__less__add1,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ J @ M ) ) ) ).
% trans_less_add1
thf(fact_983_add__less__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% add_less_mono1
thf(fact_984_not__add__less2,axiom,
! [J: nat,I: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ J @ I ) @ I ) ).
% not_add_less2
thf(fact_985_not__add__less1,axiom,
! [I: nat,J: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ I ) ).
% not_add_less1
thf(fact_986_add__less__mono,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ K @ L )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_less_mono
thf(fact_987_add__lessD1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ K )
=> ( ord_less_nat @ I @ K ) ) ).
% add_lessD1
thf(fact_988_ex__least__nat__le,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K3: nat] :
( ( ord_less_eq_nat @ K3 @ N )
& ! [I5: nat] :
( ( ord_less_nat @ I5 @ K3 )
=> ~ ( P @ I5 ) )
& ( P @ K3 ) ) ) ) ).
% ex_least_nat_le
thf(fact_989_less__imp__add__positive,axiom,
! [I: nat,J: nat] :
( ( ord_less_nat @ I @ J )
=> ? [K3: nat] :
( ( ord_less_nat @ zero_zero_nat @ K3 )
& ( ( plus_plus_nat @ I @ K3 )
= J ) ) ) ).
% less_imp_add_positive
thf(fact_990_mono__nat__linear__lb,axiom,
! [F2: nat > nat,M: nat,K: nat] :
( ! [M4: nat,N3: nat] :
( ( ord_less_nat @ M4 @ N3 )
=> ( ord_less_nat @ ( F2 @ M4 ) @ ( F2 @ N3 ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( F2 @ M ) @ K ) @ ( F2 @ ( plus_plus_nat @ M @ K ) ) ) ) ).
% mono_nat_linear_lb
thf(fact_991_nat__int__comparison_I2_J,axiom,
( ord_less_nat
= ( ^ [A2: nat,B2: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A2 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ) ).
% nat_int_comparison(2)
thf(fact_992_int__cases4,axiom,
! [M: int] :
( ! [N3: nat] :
( M
!= ( semiri1314217659103216013at_int @ N3 ) )
=> ~ ! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( M
!= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N3 ) ) ) ) ) ).
% int_cases4
thf(fact_993_nat__mono__iff,axiom,
! [Z: int,W2: int] :
( ( ord_less_int @ zero_zero_int @ Z )
=> ( ( ord_less_nat @ ( nat2 @ W2 ) @ ( nat2 @ Z ) )
= ( ord_less_int @ W2 @ Z ) ) ) ).
% nat_mono_iff
thf(fact_994_zless__nat__eq__int__zless,axiom,
! [M: nat,Z: int] :
( ( ord_less_nat @ M @ ( nat2 @ Z ) )
= ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ Z ) ) ).
% zless_nat_eq_int_zless
thf(fact_995_zero__less__imp__eq__int,axiom,
! [K: int] :
( ( ord_less_int @ zero_zero_int @ K )
=> ? [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
& ( K
= ( semiri1314217659103216013at_int @ N3 ) ) ) ) ).
% zero_less_imp_eq_int
thf(fact_996_pos__int__cases,axiom,
! [K: int] :
( ( ord_less_int @ zero_zero_int @ K )
=> ~ ! [N3: nat] :
( ( K
= ( semiri1314217659103216013at_int @ N3 ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N3 ) ) ) ).
% pos_int_cases
thf(fact_997_int__cases3,axiom,
! [K: int] :
( ( K != zero_zero_int )
=> ( ! [N3: nat] :
( ( K
= ( semiri1314217659103216013at_int @ N3 ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N3 ) )
=> ~ ! [N3: nat] :
( ( K
= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N3 ) ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N3 ) ) ) ) ).
% int_cases3
thf(fact_998_nat__less__eq__zless,axiom,
! [W2: int,Z: int] :
( ( ord_less_eq_int @ zero_zero_int @ W2 )
=> ( ( ord_less_nat @ ( nat2 @ W2 ) @ ( nat2 @ Z ) )
= ( ord_less_int @ W2 @ Z ) ) ) ).
% nat_less_eq_zless
thf(fact_999_neg__int__cases,axiom,
! [K: int] :
( ( ord_less_int @ K @ zero_zero_int )
=> ~ ! [N3: nat] :
( ( K
= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N3 ) ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N3 ) ) ) ).
% neg_int_cases
thf(fact_1000_nat__less__iff,axiom,
! [W2: int,M: nat] :
( ( ord_less_eq_int @ zero_zero_int @ W2 )
=> ( ( ord_less_nat @ ( nat2 @ W2 ) @ M )
= ( ord_less_int @ W2 @ ( semiri1314217659103216013at_int @ M ) ) ) ) ).
% nat_less_iff
thf(fact_1001_abs__eq__0__iff,axiom,
! [A: int] :
( ( ( abs_abs_int @ A )
= zero_zero_int )
= ( A = zero_zero_int ) ) ).
% abs_eq_0_iff
thf(fact_1002_add__less__zeroD,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ ( plus_plus_int @ X @ Y ) @ zero_zero_int )
=> ( ( ord_less_int @ X @ zero_zero_int )
| ( ord_less_int @ Y @ zero_zero_int ) ) ) ).
% add_less_zeroD
thf(fact_1003_nat__descend__induct,axiom,
! [N: nat,P: nat > $o,M: nat] :
( ! [K3: nat] :
( ( ord_less_nat @ N @ K3 )
=> ( P @ K3 ) )
=> ( ! [K3: nat] :
( ( ord_less_eq_nat @ K3 @ N )
=> ( ! [I5: nat] :
( ( ord_less_nat @ K3 @ I5 )
=> ( P @ I5 ) )
=> ( P @ K3 ) ) )
=> ( P @ M ) ) ) ).
% nat_descend_induct
thf(fact_1004_complete__interval,axiom,
! [A: nat,B: nat,P: nat > $o] :
( ( ord_less_nat @ A @ B )
=> ( ( P @ A )
=> ( ~ ( P @ B )
=> ? [C2: nat] :
( ( ord_less_eq_nat @ A @ C2 )
& ( ord_less_eq_nat @ C2 @ B )
& ! [X6: nat] :
( ( ( ord_less_eq_nat @ A @ X6 )
& ( ord_less_nat @ X6 @ C2 ) )
=> ( P @ X6 ) )
& ! [D4: nat] :
( ! [X3: nat] :
( ( ( ord_less_eq_nat @ A @ X3 )
& ( ord_less_nat @ X3 @ D4 ) )
=> ( P @ X3 ) )
=> ( ord_less_eq_nat @ D4 @ C2 ) ) ) ) ) ) ).
% complete_interval
thf(fact_1005_complete__interval,axiom,
! [A: int,B: int,P: int > $o] :
( ( ord_less_int @ A @ B )
=> ( ( P @ A )
=> ( ~ ( P @ B )
=> ? [C2: int] :
( ( ord_less_eq_int @ A @ C2 )
& ( ord_less_eq_int @ C2 @ B )
& ! [X6: int] :
( ( ( ord_less_eq_int @ A @ X6 )
& ( ord_less_int @ X6 @ C2 ) )
=> ( P @ X6 ) )
& ! [D4: int] :
( ! [X3: int] :
( ( ( ord_less_eq_int @ A @ X3 )
& ( ord_less_int @ X3 @ D4 ) )
=> ( P @ X3 ) )
=> ( ord_less_eq_int @ D4 @ C2 ) ) ) ) ) ) ).
% complete_interval
thf(fact_1006_ComplI,axiom,
! [C: product_prod_int_int,A3: set_Pr958786334691620121nt_int] :
( ~ ( member5262025264175285858nt_int @ C @ A3 )
=> ( member5262025264175285858nt_int @ C @ ( uminus6221592323253981072nt_int @ A3 ) ) ) ).
% ComplI
thf(fact_1007_ComplI,axiom,
! [C: int,A3: set_int] :
( ~ ( member_int @ C @ A3 )
=> ( member_int @ C @ ( uminus1532241313380277803et_int @ A3 ) ) ) ).
% ComplI
thf(fact_1008_Compl__iff,axiom,
! [C: product_prod_int_int,A3: set_Pr958786334691620121nt_int] :
( ( member5262025264175285858nt_int @ C @ ( uminus6221592323253981072nt_int @ A3 ) )
= ( ~ ( member5262025264175285858nt_int @ C @ A3 ) ) ) ).
% Compl_iff
thf(fact_1009_Compl__iff,axiom,
! [C: int,A3: set_int] :
( ( member_int @ C @ ( uminus1532241313380277803et_int @ A3 ) )
= ( ~ ( member_int @ C @ A3 ) ) ) ).
% Compl_iff
thf(fact_1010_ComplD,axiom,
! [C: product_prod_int_int,A3: set_Pr958786334691620121nt_int] :
( ( member5262025264175285858nt_int @ C @ ( uminus6221592323253981072nt_int @ A3 ) )
=> ~ ( member5262025264175285858nt_int @ C @ A3 ) ) ).
% ComplD
thf(fact_1011_ComplD,axiom,
! [C: int,A3: set_int] :
( ( member_int @ C @ ( uminus1532241313380277803et_int @ A3 ) )
=> ~ ( member_int @ C @ A3 ) ) ).
% ComplD
thf(fact_1012_psubsetD,axiom,
! [A3: set_Pr958786334691620121nt_int,B3: set_Pr958786334691620121nt_int,C: product_prod_int_int] :
( ( ord_le7563427860532173253nt_int @ A3 @ B3 )
=> ( ( member5262025264175285858nt_int @ C @ A3 )
=> ( member5262025264175285858nt_int @ C @ B3 ) ) ) ).
% psubsetD
thf(fact_1013_psubsetD,axiom,
! [A3: set_int,B3: set_int,C: int] :
( ( ord_less_set_int @ A3 @ B3 )
=> ( ( member_int @ C @ A3 )
=> ( member_int @ C @ B3 ) ) ) ).
% psubsetD
thf(fact_1014_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_1015_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_1016_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_1017_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_1018_of__nat__1,axiom,
( ( semiri1314217659103216013at_int @ one_one_nat )
= one_one_int ) ).
% of_nat_1
thf(fact_1019_of__nat__1,axiom,
( ( semiri1316708129612266289at_nat @ one_one_nat )
= one_one_nat ) ).
% of_nat_1
thf(fact_1020_less__one,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ one_one_nat )
= ( N = zero_zero_nat ) ) ).
% less_one
thf(fact_1021_of__int__1,axiom,
( ( ring_1_of_int_int @ one_one_int )
= one_one_int ) ).
% of_int_1
thf(fact_1022_of__int__eq__1__iff,axiom,
! [Z: int] :
( ( ( ring_1_of_int_int @ Z )
= one_one_int )
= ( Z = one_one_int ) ) ).
% of_int_eq_1_iff
thf(fact_1023_abs__neg__one,axiom,
( ( abs_abs_int @ ( uminus_uminus_int @ one_one_int ) )
= one_one_int ) ).
% abs_neg_one
thf(fact_1024_zle__add1__eq__le,axiom,
! [W2: int,Z: int] :
( ( ord_less_int @ W2 @ ( plus_plus_int @ Z @ one_one_int ) )
= ( ord_less_eq_int @ W2 @ Z ) ) ).
% zle_add1_eq_le
thf(fact_1025_zabs__less__one__iff,axiom,
! [Z: int] :
( ( ord_less_int @ ( abs_abs_int @ Z ) @ one_one_int )
= ( Z = zero_zero_int ) ) ).
% zabs_less_one_iff
thf(fact_1026_add__neg__numeral__special_I7_J,axiom,
( ( plus_plus_int @ one_one_int @ ( uminus_uminus_int @ one_one_int ) )
= zero_zero_int ) ).
% add_neg_numeral_special(7)
thf(fact_1027_add__neg__numeral__special_I8_J,axiom,
( ( plus_plus_int @ ( uminus_uminus_int @ one_one_int ) @ one_one_int )
= zero_zero_int ) ).
% add_neg_numeral_special(8)
thf(fact_1028_of__int__1__le__iff,axiom,
! [Z: int] :
( ( ord_less_eq_int @ one_one_int @ ( ring_1_of_int_int @ Z ) )
= ( ord_less_eq_int @ one_one_int @ Z ) ) ).
% of_int_1_le_iff
thf(fact_1029_of__int__le__1__iff,axiom,
! [Z: int] :
( ( ord_less_eq_int @ ( ring_1_of_int_int @ Z ) @ one_one_int )
= ( ord_less_eq_int @ Z @ one_one_int ) ) ).
% of_int_le_1_iff
thf(fact_1030_of__int__less__1__iff,axiom,
! [Z: int] :
( ( ord_less_int @ ( ring_1_of_int_int @ Z ) @ one_one_int )
= ( ord_less_int @ Z @ one_one_int ) ) ).
% of_int_less_1_iff
thf(fact_1031_of__int__1__less__iff,axiom,
! [Z: int] :
( ( ord_less_int @ one_one_int @ ( ring_1_of_int_int @ Z ) )
= ( ord_less_int @ one_one_int @ Z ) ) ).
% of_int_1_less_iff
thf(fact_1032_less__numeral__extra_I4_J,axiom,
~ ( ord_less_int @ one_one_int @ one_one_int ) ).
% less_numeral_extra(4)
thf(fact_1033_less__numeral__extra_I4_J,axiom,
~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).
% less_numeral_extra(4)
thf(fact_1034_one__neq__neg__one,axiom,
( one_one_int
!= ( uminus_uminus_int @ one_one_int ) ) ).
% one_neq_neg_one
thf(fact_1035_int__ops_I2_J,axiom,
( ( semiri1314217659103216013at_int @ one_one_nat )
= one_one_int ) ).
% int_ops(2)
thf(fact_1036_one__reorient,axiom,
! [X: nat] :
( ( one_one_nat = X )
= ( X = one_one_nat ) ) ).
% one_reorient
thf(fact_1037_one__reorient,axiom,
! [X: int] :
( ( one_one_int = X )
= ( X = one_one_int ) ) ).
% one_reorient
thf(fact_1038_nat__one__as__int,axiom,
( one_one_nat
= ( nat2 @ one_one_int ) ) ).
% nat_one_as_int
thf(fact_1039_le__numeral__extra_I4_J,axiom,
ord_less_eq_nat @ one_one_nat @ one_one_nat ).
% le_numeral_extra(4)
thf(fact_1040_le__numeral__extra_I4_J,axiom,
ord_less_eq_int @ one_one_int @ one_one_int ).
% le_numeral_extra(4)
thf(fact_1041_zero__neq__one,axiom,
zero_zero_int != one_one_int ).
% zero_neq_one
thf(fact_1042_zero__neq__one,axiom,
zero_zero_nat != one_one_nat ).
% zero_neq_one
thf(fact_1043_not__one__le__zero,axiom,
~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_le_zero
thf(fact_1044_not__one__le__zero,axiom,
~ ( ord_less_eq_int @ one_one_int @ zero_zero_int ) ).
% not_one_le_zero
thf(fact_1045_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% linordered_nonzero_semiring_class.zero_le_one
thf(fact_1046_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
ord_less_eq_int @ zero_zero_int @ one_one_int ).
% linordered_nonzero_semiring_class.zero_le_one
thf(fact_1047_zero__less__one__class_Ozero__le__one,axiom,
ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one_class.zero_le_one
thf(fact_1048_zero__less__one__class_Ozero__le__one,axiom,
ord_less_eq_int @ zero_zero_int @ one_one_int ).
% zero_less_one_class.zero_le_one
thf(fact_1049_not__one__less__zero,axiom,
~ ( ord_less_int @ one_one_int @ zero_zero_int ) ).
% not_one_less_zero
thf(fact_1050_not__one__less__zero,axiom,
~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_less_zero
thf(fact_1051_zero__less__one,axiom,
ord_less_int @ zero_zero_int @ one_one_int ).
% zero_less_one
thf(fact_1052_zero__less__one,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one
thf(fact_1053_less__numeral__extra_I1_J,axiom,
ord_less_int @ zero_zero_int @ one_one_int ).
% less_numeral_extra(1)
thf(fact_1054_less__numeral__extra_I1_J,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% less_numeral_extra(1)
thf(fact_1055_less__add__one,axiom,
! [A: int] : ( ord_less_int @ A @ ( plus_plus_int @ A @ one_one_int ) ) ).
% less_add_one
thf(fact_1056_less__add__one,axiom,
! [A: nat] : ( ord_less_nat @ A @ ( plus_plus_nat @ A @ one_one_nat ) ) ).
% less_add_one
thf(fact_1057_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_1058_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_1059_le__minus__one__simps_I2_J,axiom,
ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ one_one_int ).
% le_minus_one_simps(2)
thf(fact_1060_le__minus__one__simps_I4_J,axiom,
~ ( ord_less_eq_int @ one_one_int @ ( uminus_uminus_int @ one_one_int ) ) ).
% le_minus_one_simps(4)
thf(fact_1061_zero__neq__neg__one,axiom,
( zero_zero_int
!= ( uminus_uminus_int @ one_one_int ) ) ).
% zero_neq_neg_one
thf(fact_1062_less__minus__one__simps_I2_J,axiom,
ord_less_int @ ( uminus_uminus_int @ one_one_int ) @ one_one_int ).
% less_minus_one_simps(2)
thf(fact_1063_less__minus__one__simps_I4_J,axiom,
~ ( ord_less_int @ one_one_int @ ( uminus_uminus_int @ one_one_int ) ) ).
% less_minus_one_simps(4)
thf(fact_1064_odd__nonzero,axiom,
! [Z: int] :
( ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z ) @ Z )
!= zero_zero_int ) ).
% odd_nonzero
thf(fact_1065_int__ge__induct,axiom,
! [K: int,I: int,P: int > $o] :
( ( ord_less_eq_int @ K @ I )
=> ( ( P @ K )
=> ( ! [I2: int] :
( ( ord_less_eq_int @ K @ I2 )
=> ( ( P @ I2 )
=> ( P @ ( plus_plus_int @ I2 @ one_one_int ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_ge_induct
thf(fact_1066_int__gr__induct,axiom,
! [K: int,I: int,P: int > $o] :
( ( ord_less_int @ K @ I )
=> ( ( P @ ( plus_plus_int @ K @ one_one_int ) )
=> ( ! [I2: int] :
( ( ord_less_int @ K @ I2 )
=> ( ( P @ I2 )
=> ( P @ ( plus_plus_int @ I2 @ one_one_int ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_gr_induct
thf(fact_1067_zless__add1__eq,axiom,
! [W2: int,Z: int] :
( ( ord_less_int @ W2 @ ( plus_plus_int @ Z @ one_one_int ) )
= ( ( ord_less_int @ W2 @ Z )
| ( W2 = Z ) ) ) ).
% zless_add1_eq
thf(fact_1068_zero__less__two,axiom,
ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ one_one_int ) ).
% zero_less_two
thf(fact_1069_zero__less__two,axiom,
ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ).
% zero_less_two
thf(fact_1070_le__minus__one__simps_I1_J,axiom,
ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ zero_zero_int ).
% le_minus_one_simps(1)
thf(fact_1071_le__minus__one__simps_I3_J,axiom,
~ ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ one_one_int ) ) ).
% le_minus_one_simps(3)
thf(fact_1072_less__minus__one__simps_I3_J,axiom,
~ ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ one_one_int ) ) ).
% less_minus_one_simps(3)
thf(fact_1073_less__minus__one__simps_I1_J,axiom,
ord_less_int @ ( uminus_uminus_int @ one_one_int ) @ zero_zero_int ).
% less_minus_one_simps(1)
thf(fact_1074_int__one__le__iff__zero__less,axiom,
! [Z: int] :
( ( ord_less_eq_int @ one_one_int @ Z )
= ( ord_less_int @ zero_zero_int @ Z ) ) ).
% int_one_le_iff_zero_less
thf(fact_1075_odd__less__0__iff,axiom,
! [Z: int] :
( ( ord_less_int @ ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z ) @ Z ) @ zero_zero_int )
= ( ord_less_int @ Z @ zero_zero_int ) ) ).
% odd_less_0_iff
thf(fact_1076_zless__imp__add1__zle,axiom,
! [W2: int,Z: int] :
( ( ord_less_int @ W2 @ Z )
=> ( ord_less_eq_int @ ( plus_plus_int @ W2 @ one_one_int ) @ Z ) ) ).
% zless_imp_add1_zle
thf(fact_1077_add1__zle__eq,axiom,
! [W2: int,Z: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ W2 @ one_one_int ) @ Z )
= ( ord_less_int @ W2 @ Z ) ) ).
% add1_zle_eq
thf(fact_1078_abs__add__one__gt__zero,axiom,
! [X: int] : ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ ( abs_abs_int @ X ) ) ) ).
% abs_add_one_gt_zero
thf(fact_1079_of__int__leD,axiom,
! [N: int,X: int] :
( ( ord_less_eq_int @ ( abs_abs_int @ ( ring_1_of_int_int @ N ) ) @ X )
=> ( ( N = zero_zero_int )
| ( ord_less_eq_int @ one_one_int @ X ) ) ) ).
% of_int_leD
thf(fact_1080_of__int__lessD,axiom,
! [N: int,X: int] :
( ( ord_less_int @ ( abs_abs_int @ ( ring_1_of_int_int @ N ) ) @ X )
=> ( ( N = zero_zero_int )
| ( ord_less_int @ one_one_int @ X ) ) ) ).
% of_int_lessD
thf(fact_1081_le__imp__0__less,axiom,
! [Z: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ Z ) ) ) ).
% le_imp_0_less
thf(fact_1082_dbl__dec__simps_I2_J,axiom,
( ( neg_nu3811975205180677377ec_int @ zero_zero_int )
= ( uminus_uminus_int @ one_one_int ) ) ).
% dbl_dec_simps(2)
thf(fact_1083_mirror2__board__id,axiom,
! [M: nat,N: nat] :
( ( mirror2_board @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ M ) @ one_one_int ) @ ( board @ N @ M ) )
= ( board @ N @ M ) ) ).
% mirror2_board_id
thf(fact_1084_mirror1__board__id,axiom,
! [N: nat,M: nat] :
( ( mirror1_board @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) @ ( board @ N @ M ) )
= ( board @ N @ M ) ) ).
% mirror1_board_id
thf(fact_1085_dbl__dec__simps_I3_J,axiom,
( ( neg_nu3811975205180677377ec_int @ one_one_int )
= one_one_int ) ).
% dbl_dec_simps(3)
thf(fact_1086_one__natural_Orsp,axiom,
one_one_nat = one_one_nat ).
% one_natural.rsp
thf(fact_1087_one__integer_Orsp,axiom,
one_one_int = one_one_int ).
% one_integer.rsp
thf(fact_1088_board__exec__leq,axiom,
! [I: int,J: int,N: nat,M: nat] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ I @ J ) @ ( board_exec @ N @ M ) )
= ( ( ord_less_eq_int @ one_one_int @ I )
& ( ord_less_eq_int @ I @ ( semiri1314217659103216013at_int @ N ) )
& ( ord_less_eq_int @ one_one_int @ J )
& ( ord_less_eq_int @ J @ ( semiri1314217659103216013at_int @ M ) ) ) ) ).
% board_exec_leq
thf(fact_1089_board__exec__correct,axiom,
board = board_exec ).
% board_exec_correct
thf(fact_1090_bounded__Max__nat,axiom,
! [P: nat > $o,X: nat,M5: nat] :
( ( P @ X )
=> ( ! [X3: nat] :
( ( P @ X3 )
=> ( ord_less_eq_nat @ X3 @ M5 ) )
=> ~ ! [M4: nat] :
( ( P @ M4 )
=> ~ ! [X6: nat] :
( ( P @ X6 )
=> ( ord_less_eq_nat @ X6 @ M4 ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_1091_nat0__intermed__int__val,axiom,
! [N: nat,F2: nat > int,K: int] :
( ! [I2: nat] :
( ( ord_less_nat @ I2 @ N )
=> ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F2 @ ( plus_plus_nat @ I2 @ one_one_nat ) ) @ ( F2 @ I2 ) ) ) @ one_one_int ) )
=> ( ( ord_less_eq_int @ ( F2 @ zero_zero_nat ) @ K )
=> ( ( ord_less_eq_int @ K @ ( F2 @ N ) )
=> ? [I2: nat] :
( ( ord_less_eq_nat @ I2 @ N )
& ( ( F2 @ I2 )
= K ) ) ) ) ) ).
% nat0_intermed_int_val
thf(fact_1092_bezw__0,axiom,
! [X: nat] :
( ( bezw @ X @ zero_zero_nat )
= ( product_Pair_int_int @ one_one_int @ zero_zero_int ) ) ).
% bezw_0
thf(fact_1093_board__exec__aux__leq__mem,axiom,
! [I: int,J: int,K: nat,M5: set_int] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ I @ J ) @ ( board_exec_aux @ K @ M5 ) )
= ( ( ord_less_eq_int @ one_one_int @ I )
& ( ord_less_eq_int @ I @ ( semiri1314217659103216013at_int @ K ) )
& ( member_int @ J @ M5 ) ) ) ).
% board_exec_aux_leq_mem
thf(fact_1094_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_1095_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_1096_diff__zero,axiom,
! [A: int] :
( ( minus_minus_int @ A @ zero_zero_int )
= A ) ).
% diff_zero
thf(fact_1097_diff__zero,axiom,
! [A: nat] :
( ( minus_minus_nat @ A @ zero_zero_nat )
= A ) ).
% diff_zero
thf(fact_1098_zero__diff,axiom,
! [A: nat] :
( ( minus_minus_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% zero_diff
thf(fact_1099_diff__0__right,axiom,
! [A: int] :
( ( minus_minus_int @ A @ zero_zero_int )
= A ) ).
% diff_0_right
thf(fact_1100_diff__self,axiom,
! [A: int] :
( ( minus_minus_int @ A @ A )
= zero_zero_int ) ).
% diff_self
thf(fact_1101_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_1102_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_1103_add__diff__cancel__right,axiom,
! [A: int,C: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
= ( minus_minus_int @ A @ B ) ) ).
% add_diff_cancel_right
thf(fact_1104_add__diff__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( minus_minus_nat @ A @ B ) ) ).
% add_diff_cancel_right
thf(fact_1105_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_1106_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_1107_add__diff__cancel__left,axiom,
! [C: int,A: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
= ( minus_minus_int @ A @ B ) ) ).
% add_diff_cancel_left
thf(fact_1108_add__diff__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
= ( minus_minus_nat @ A @ B ) ) ).
% add_diff_cancel_left
thf(fact_1109_diff__add__cancel,axiom,
! [A: int,B: int] :
( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ B )
= A ) ).
% diff_add_cancel
thf(fact_1110_add__diff__cancel,axiom,
! [A: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ B )
= A ) ).
% add_diff_cancel
thf(fact_1111_minus__diff__eq,axiom,
! [A: int,B: int] :
( ( uminus_uminus_int @ ( minus_minus_int @ A @ B ) )
= ( minus_minus_int @ B @ A ) ) ).
% minus_diff_eq
thf(fact_1112_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_1113_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_1114_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_1115_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_1116_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_1117_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_1118_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_1119_diff__numeral__special_I9_J,axiom,
( ( minus_minus_int @ one_one_int @ one_one_int )
= zero_zero_int ) ).
% diff_numeral_special(9)
thf(fact_1120_diff__0,axiom,
! [A: int] :
( ( minus_minus_int @ zero_zero_int @ A )
= ( uminus_uminus_int @ A ) ) ).
% diff_0
thf(fact_1121_verit__minus__simplify_I3_J,axiom,
! [B: int] :
( ( minus_minus_int @ zero_zero_int @ B )
= ( uminus_uminus_int @ B ) ) ).
% verit_minus_simplify(3)
thf(fact_1122_diff__minus__eq__add,axiom,
! [A: int,B: int] :
( ( minus_minus_int @ A @ ( uminus_uminus_int @ B ) )
= ( plus_plus_int @ A @ B ) ) ).
% diff_minus_eq_add
thf(fact_1123_uminus__add__conv__diff,axiom,
! [A: int,B: int] :
( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ B )
= ( minus_minus_int @ B @ A ) ) ).
% uminus_add_conv_diff
thf(fact_1124_of__int__diff,axiom,
! [W2: int,Z: int] :
( ( ring_1_of_int_int @ ( minus_minus_int @ W2 @ Z ) )
= ( minus_minus_int @ ( ring_1_of_int_int @ W2 ) @ ( ring_1_of_int_int @ Z ) ) ) ).
% of_int_diff
thf(fact_1125_diff__numeral__special_I12_J,axiom,
( ( minus_minus_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ one_one_int ) )
= zero_zero_int ) ).
% diff_numeral_special(12)
thf(fact_1126_zle__diff1__eq,axiom,
! [W2: int,Z: int] :
( ( ord_less_eq_int @ W2 @ ( minus_minus_int @ Z @ one_one_int ) )
= ( ord_less_int @ W2 @ Z ) ) ).
% zle_diff1_eq
thf(fact_1127_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_1128_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_1129_diff__eq__diff__less__eq,axiom,
! [A: int,B: int,C: int,D2: int] :
( ( ( minus_minus_int @ A @ B )
= ( minus_minus_int @ C @ D2 ) )
=> ( ( ord_less_eq_int @ A @ B )
= ( ord_less_eq_int @ C @ D2 ) ) ) ).
% diff_eq_diff_less_eq
thf(fact_1130_diff__right__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ord_less_eq_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ C ) ) ) ).
% diff_right_mono
thf(fact_1131_diff__left__mono,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ord_less_eq_int @ ( minus_minus_int @ C @ A ) @ ( minus_minus_int @ C @ B ) ) ) ).
% diff_left_mono
thf(fact_1132_diff__mono,axiom,
! [A: int,B: int,D2: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ D2 @ C )
=> ( ord_less_eq_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ D2 ) ) ) ) ).
% diff_mono
thf(fact_1133_eq__iff__diff__eq__0,axiom,
( ( ^ [Y5: int,Z2: int] : ( Y5 = Z2 ) )
= ( ^ [A2: int,B2: int] :
( ( minus_minus_int @ A2 @ B2 )
= zero_zero_int ) ) ) ).
% eq_iff_diff_eq_0
thf(fact_1134_minus__int__code_I1_J,axiom,
! [K: int] :
( ( minus_minus_int @ K @ zero_zero_int )
= K ) ).
% minus_int_code(1)
thf(fact_1135_int__diff__cases,axiom,
! [Z: int] :
~ ! [M4: nat,N3: nat] :
( Z
!= ( minus_minus_int @ ( semiri1314217659103216013at_int @ M4 ) @ ( semiri1314217659103216013at_int @ N3 ) ) ) ).
% int_diff_cases
thf(fact_1136_minus__int__code_I2_J,axiom,
! [L: int] :
( ( minus_minus_int @ zero_zero_int @ L )
= ( uminus_uminus_int @ L ) ) ).
% minus_int_code(2)
thf(fact_1137_int__le__induct,axiom,
! [I: int,K: int,P: int > $o] :
( ( ord_less_eq_int @ I @ K )
=> ( ( P @ K )
=> ( ! [I2: int] :
( ( ord_less_eq_int @ I2 @ K )
=> ( ( P @ I2 )
=> ( P @ ( minus_minus_int @ I2 @ one_one_int ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_le_induct
thf(fact_1138_int__less__induct,axiom,
! [I: int,K: int,P: int > $o] :
( ( ord_less_int @ I @ K )
=> ( ( P @ ( minus_minus_int @ K @ one_one_int ) )
=> ( ! [I2: int] :
( ( ord_less_int @ I2 @ K )
=> ( ( P @ I2 )
=> ( P @ ( minus_minus_int @ I2 @ one_one_int ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_less_induct
thf(fact_1139_int__induct,axiom,
! [P: int > $o,K: int,I: int] :
( ( P @ K )
=> ( ! [I2: int] :
( ( ord_less_eq_int @ K @ I2 )
=> ( ( P @ I2 )
=> ( P @ ( plus_plus_int @ I2 @ one_one_int ) ) ) )
=> ( ! [I2: int] :
( ( ord_less_eq_int @ I2 @ K )
=> ( ( P @ I2 )
=> ( P @ ( minus_minus_int @ I2 @ one_one_int ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_induct
thf(fact_1140_nat__ivt__aux,axiom,
! [N: nat,F2: nat > int,K: int] :
( ! [I2: nat] :
( ( ord_less_nat @ I2 @ N )
=> ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F2 @ ( suc @ I2 ) ) @ ( F2 @ I2 ) ) ) @ one_one_int ) )
=> ( ( ord_less_eq_int @ ( F2 @ zero_zero_nat ) @ K )
=> ( ( ord_less_eq_int @ K @ ( F2 @ N ) )
=> ? [I2: nat] :
( ( ord_less_eq_nat @ I2 @ N )
& ( ( F2 @ I2 )
= K ) ) ) ) ) ).
% nat_ivt_aux
thf(fact_1141_decr__lemma,axiom,
! [D2: int,X: int,Z: int] :
( ( ord_less_int @ zero_zero_int @ D2 )
=> ( ord_less_int @ ( minus_minus_int @ X @ ( times_times_int @ ( plus_plus_int @ ( abs_abs_int @ ( minus_minus_int @ X @ Z ) ) @ one_one_int ) @ D2 ) ) @ Z ) ) ).
% decr_lemma
thf(fact_1142_old_Onat_Oinject,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( suc @ Nat )
= ( suc @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% old.nat.inject
thf(fact_1143_nat_Oinject,axiom,
! [X22: nat,Y2: nat] :
( ( ( suc @ X22 )
= ( suc @ Y2 ) )
= ( X22 = Y2 ) ) ).
% nat.inject
thf(fact_1144_lessI,axiom,
! [N: nat] : ( ord_less_nat @ N @ ( suc @ N ) ) ).
% lessI
thf(fact_1145_Suc__mono,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) ) ) ).
% Suc_mono
thf(fact_1146_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_1147_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_1148_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_1149_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_1150_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_1151_diff__0__eq__0,axiom,
! [N: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_1152_diff__self__eq__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ M )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_1153_diff__diff__cancel,axiom,
! [I: nat,N: nat] :
( ( ord_less_eq_nat @ I @ N )
=> ( ( minus_minus_nat @ N @ ( minus_minus_nat @ N @ I ) )
= I ) ) ).
% diff_diff_cancel
thf(fact_1154_diff__diff__left,axiom,
! [I: nat,J: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
= ( minus_minus_nat @ I @ ( plus_plus_nat @ J @ K ) ) ) ).
% diff_diff_left
thf(fact_1155_less__Suc0,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ ( suc @ zero_zero_nat ) )
= ( N = zero_zero_nat ) ) ).
% less_Suc0
thf(fact_1156_zero__less__Suc,axiom,
! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N ) ) ).
% zero_less_Suc
thf(fact_1157_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_1158_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_1159_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_1160_Nat_Oadd__diff__assoc,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K ) ) ) ).
% Nat.add_diff_assoc
thf(fact_1161_Nat_Oadd__diff__assoc2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I )
= ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K ) ) ) ).
% Nat.add_diff_assoc2
thf(fact_1162_Nat_Odiff__diff__right,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ) ).
% Nat.diff_diff_right
thf(fact_1163_diff__Suc__1,axiom,
! [N: nat] :
( ( minus_minus_nat @ ( suc @ N ) @ one_one_nat )
= N ) ).
% diff_Suc_1
thf(fact_1164_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_1165_diff__Suc__diff__eq1,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ I @ ( suc @ ( minus_minus_nat @ J @ K ) ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ K ) @ ( suc @ J ) ) ) ) ).
% diff_Suc_diff_eq1
thf(fact_1166_diff__Suc__diff__eq2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( suc @ ( minus_minus_nat @ J @ K ) ) @ I )
= ( minus_minus_nat @ ( suc @ J ) @ ( plus_plus_nat @ K @ I ) ) ) ) ).
% diff_Suc_diff_eq2
thf(fact_1167_nat__1,axiom,
( ( nat2 @ one_one_int )
= ( suc @ zero_zero_nat ) ) ).
% nat_1
thf(fact_1168_negative__zless,axiom,
! [N: nat,M: nat] : ( ord_less_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) ) @ ( semiri1314217659103216013at_int @ M ) ) ).
% negative_zless
thf(fact_1169_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_1170_one__less__nat__eq,axiom,
! [Z: int] :
( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ ( nat2 @ Z ) )
= ( ord_less_int @ one_one_int @ Z ) ) ).
% one_less_nat_eq
thf(fact_1171_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_1172_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_1173_minus__nat_Odiff__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ zero_zero_nat )
= M ) ).
% minus_nat.diff_0
thf(fact_1174_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_1175_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_1176_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_1177_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_1178_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_1179_diff__le__self,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ M ) ).
% diff_le_self
thf(fact_1180_le__diff__iff_H,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ A @ C )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ C @ A ) @ ( minus_minus_nat @ C @ B ) )
= ( ord_less_eq_nat @ B @ A ) ) ) ) ).
% le_diff_iff'
thf(fact_1181_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_1182_diff__add__inverse2,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ N )
= M ) ).
% diff_add_inverse2
thf(fact_1183_diff__add__inverse,axiom,
! [N: nat,M: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ N @ M ) @ N )
= M ) ).
% diff_add_inverse
thf(fact_1184_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_1185_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_1186_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_1187_diff__commute,axiom,
! [I: nat,J: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
= ( minus_minus_nat @ ( minus_minus_nat @ I @ K ) @ J ) ) ).
% diff_commute
thf(fact_1188_zero__induct__lemma,axiom,
! [P: nat > $o,K: nat,I: nat] :
( ( P @ K )
=> ( ! [N3: nat] :
( ( P @ ( suc @ N3 ) )
=> ( P @ N3 ) )
=> ( P @ ( minus_minus_nat @ K @ I ) ) ) ) ).
% zero_induct_lemma
thf(fact_1189_int__distrib_I4_J,axiom,
! [W2: int,Z1: int,Z22: int] :
( ( times_times_int @ W2 @ ( minus_minus_int @ Z1 @ Z22 ) )
= ( minus_minus_int @ ( times_times_int @ W2 @ Z1 ) @ ( times_times_int @ W2 @ Z22 ) ) ) ).
% int_distrib(4)
thf(fact_1190_int__distrib_I3_J,axiom,
! [Z1: int,Z22: int,W2: int] :
( ( times_times_int @ ( minus_minus_int @ Z1 @ Z22 ) @ W2 )
= ( minus_minus_int @ ( times_times_int @ Z1 @ W2 ) @ ( times_times_int @ Z22 @ W2 ) ) ) ).
% int_distrib(3)
thf(fact_1191_less__imp__diff__less,axiom,
! [J: nat,K: nat,N: nat] :
( ( ord_less_nat @ J @ K )
=> ( ord_less_nat @ ( minus_minus_nat @ J @ N ) @ K ) ) ).
% less_imp_diff_less
thf(fact_1192_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_1193_diff__less__Suc,axiom,
! [M: nat,N: nat] : ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ ( suc @ M ) ) ).
% diff_less_Suc
thf(fact_1194_diff__Suc__less,axiom,
! [N: nat,I: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_nat @ ( minus_minus_nat @ N @ ( suc @ I ) ) @ N ) ) ).
% diff_Suc_less
thf(fact_1195_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_1196_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_1197_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_1198_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_1199_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_1200_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_1201_diff__less__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ A )
=> ( ord_less_nat @ ( minus_minus_nat @ A @ C ) @ ( minus_minus_nat @ B @ C ) ) ) ) ).
% diff_less_mono
thf(fact_1202_less__Suc__eq__0__disj,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
= ( ( M = zero_zero_nat )
| ? [J4: nat] :
( ( M
= ( suc @ J4 ) )
& ( ord_less_nat @ J4 @ N ) ) ) ) ).
% less_Suc_eq_0_disj
thf(fact_1203_gr0__implies__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ? [M4: nat] :
( N
= ( suc @ M4 ) ) ) ).
% gr0_implies_Suc
thf(fact_1204_All__less__Suc2,axiom,
! [N: nat,P: nat > $o] :
( ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( suc @ N ) )
=> ( P @ I4 ) ) )
= ( ( P @ zero_zero_nat )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ N )
=> ( P @ ( suc @ I4 ) ) ) ) ) ).
% All_less_Suc2
thf(fact_1205_gr0__conv__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
= ( ? [M2: nat] :
( N
= ( suc @ M2 ) ) ) ) ).
% gr0_conv_Suc
thf(fact_1206_Ex__less__Suc2,axiom,
! [N: nat,P: nat > $o] :
( ( ? [I4: nat] :
( ( ord_less_nat @ I4 @ ( suc @ N ) )
& ( P @ I4 ) ) )
= ( ( P @ zero_zero_nat )
| ? [I4: nat] :
( ( ord_less_nat @ I4 @ N )
& ( P @ ( suc @ I4 ) ) ) ) ) ).
% Ex_less_Suc2
thf(fact_1207_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_1208_less__diff__conv,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ).
% less_diff_conv
thf(fact_1209_Nat_Ole__imp__diff__is__add,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ( minus_minus_nat @ J @ I )
= K )
= ( J
= ( plus_plus_nat @ K @ I ) ) ) ) ).
% Nat.le_imp_diff_is_add
thf(fact_1210_Nat_Odiff__add__assoc2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K )
= ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I ) ) ) ).
% Nat.diff_add_assoc2
thf(fact_1211_Nat_Odiff__add__assoc,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K )
= ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K ) ) ) ) ).
% Nat.diff_add_assoc
thf(fact_1212_Nat_Ole__diff__conv2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( ord_less_eq_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ) ).
% Nat.le_diff_conv2
thf(fact_1213_le__diff__conv,axiom,
! [J: nat,K: nat,I: nat] :
( ( ord_less_eq_nat @ ( minus_minus_nat @ J @ K ) @ I )
= ( ord_less_eq_nat @ J @ ( plus_plus_nat @ I @ K ) ) ) ).
% le_diff_conv
thf(fact_1214_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_1215_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_1216_Nat_OlessE,axiom,
! [I: nat,K: nat] :
( ( ord_less_nat @ I @ K )
=> ( ( K
!= ( suc @ I ) )
=> ~ ! [J2: nat] :
( ( ord_less_nat @ I @ J2 )
=> ( K
!= ( suc @ J2 ) ) ) ) ) ).
% Nat.lessE
thf(fact_1217_Suc__lessD,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ N )
=> ( ord_less_nat @ M @ N ) ) ).
% Suc_lessD
thf(fact_1218_Suc__lessE,axiom,
! [I: nat,K: nat] :
( ( ord_less_nat @ ( suc @ I ) @ K )
=> ~ ! [J2: nat] :
( ( ord_less_nat @ I @ J2 )
=> ( K
!= ( suc @ J2 ) ) ) ) ).
% Suc_lessE
thf(fact_1219_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_1220_less__SucE,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
=> ( ~ ( ord_less_nat @ M @ N )
=> ( M = N ) ) ) ).
% less_SucE
thf(fact_1221_less__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ M @ ( suc @ N ) ) ) ).
% less_SucI
thf(fact_1222_Ex__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ? [I4: nat] :
( ( ord_less_nat @ I4 @ ( suc @ N ) )
& ( P @ I4 ) ) )
= ( ( P @ N )
| ? [I4: nat] :
( ( ord_less_nat @ I4 @ N )
& ( P @ I4 ) ) ) ) ).
% Ex_less_Suc
thf(fact_1223_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_1224_not__less__eq,axiom,
! [M: nat,N: nat] :
( ( ~ ( ord_less_nat @ M @ N ) )
= ( ord_less_nat @ N @ ( suc @ M ) ) ) ).
% not_less_eq
thf(fact_1225_All__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( suc @ N ) )
=> ( P @ I4 ) ) )
= ( ( P @ N )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ N )
=> ( P @ I4 ) ) ) ) ).
% All_less_Suc
thf(fact_1226_Suc__less__eq2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ ( suc @ N ) @ M )
= ( ? [M6: nat] :
( ( M
= ( suc @ M6 ) )
& ( ord_less_nat @ N @ M6 ) ) ) ) ).
% Suc_less_eq2
thf(fact_1227_less__antisym,axiom,
! [N: nat,M: nat] :
( ~ ( ord_less_nat @ N @ M )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
=> ( M = N ) ) ) ).
% less_antisym
thf(fact_1228_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_1229_less__trans__Suc,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ J @ K )
=> ( ord_less_nat @ ( suc @ I ) @ K ) ) ) ).
% less_trans_Suc
thf(fact_1230_less__Suc__induct,axiom,
! [I: nat,J: nat,P: nat > nat > $o] :
( ( ord_less_nat @ I @ J )
=> ( ! [I2: nat] : ( P @ I2 @ ( suc @ I2 ) )
=> ( ! [I2: nat,J2: nat,K3: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ( ord_less_nat @ J2 @ K3 )
=> ( ( P @ I2 @ J2 )
=> ( ( P @ J2 @ K3 )
=> ( P @ I2 @ K3 ) ) ) ) )
=> ( P @ I @ J ) ) ) ) ).
% less_Suc_induct
thf(fact_1231_strict__inc__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_nat @ I @ J )
=> ( ! [I2: nat] :
( ( J
= ( suc @ I2 ) )
=> ( P @ I2 ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ( P @ ( suc @ I2 ) )
=> ( P @ I2 ) ) )
=> ( P @ I ) ) ) ) ).
% strict_inc_induct
thf(fact_1232_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_1233_n__not__Suc__n,axiom,
! [N: nat] :
( N
!= ( suc @ N ) ) ).
% n_not_Suc_n
thf(fact_1234_Suc__inject,axiom,
! [X: nat,Y: nat] :
( ( ( suc @ X )
= ( suc @ Y ) )
=> ( X = Y ) ) ).
% Suc_inject
thf(fact_1235_int__distrib_I2_J,axiom,
! [W2: int,Z1: int,Z22: int] :
( ( times_times_int @ W2 @ ( plus_plus_int @ Z1 @ Z22 ) )
= ( plus_plus_int @ ( times_times_int @ W2 @ Z1 ) @ ( times_times_int @ W2 @ Z22 ) ) ) ).
% int_distrib(2)
thf(fact_1236_int__distrib_I1_J,axiom,
! [Z1: int,Z22: int,W2: int] :
( ( times_times_int @ ( plus_plus_int @ Z1 @ Z22 ) @ W2 )
= ( plus_plus_int @ ( times_times_int @ Z1 @ W2 ) @ ( times_times_int @ Z22 @ W2 ) ) ) ).
% int_distrib(1)
thf(fact_1237_times__int__code_I2_J,axiom,
! [L: int] :
( ( times_times_int @ zero_zero_int @ L )
= zero_zero_int ) ).
% times_int_code(2)
thf(fact_1238_times__int__code_I1_J,axiom,
! [K: int] :
( ( times_times_int @ K @ zero_zero_int )
= zero_zero_int ) ).
% times_int_code(1)
thf(fact_1239_nat__arith_Osuc1,axiom,
! [A3: nat,K: nat,A: nat] :
( ( A3
= ( plus_plus_nat @ K @ A ) )
=> ( ( suc @ A3 )
= ( plus_plus_nat @ K @ ( suc @ A ) ) ) ) ).
% nat_arith.suc1
thf(fact_1240_add__Suc,axiom,
! [M: nat,N: nat] :
( ( plus_plus_nat @ ( suc @ M ) @ N )
= ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).
% add_Suc
thf(fact_1241_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_1242_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,Z3: nat] :
( ( R2 @ X3 @ Y3 )
=> ( ( R2 @ Y3 @ Z3 )
=> ( R2 @ X3 @ Z3 ) ) )
=> ( ! [N3: nat] : ( R2 @ N3 @ ( suc @ N3 ) )
=> ( R2 @ M @ N ) ) ) ) ) ).
% transitive_stepwise_le
thf(fact_1243_nat__induct__at__least,axiom,
! [M: nat,N: nat,P: nat > $o] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( P @ M )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ M @ N3 )
=> ( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct_at_least
thf(fact_1244_full__nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N3: nat] :
( ! [M3: nat] :
( ( ord_less_eq_nat @ ( suc @ M3 ) @ N3 )
=> ( P @ M3 ) )
=> ( P @ N3 ) )
=> ( P @ N ) ) ).
% full_nat_induct
thf(fact_1245_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_1246_Suc__n__not__le__n,axiom,
! [N: nat] :
~ ( ord_less_eq_nat @ ( suc @ N ) @ N ) ).
% Suc_n_not_le_n
thf(fact_1247_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_1248_Suc__le__D,axiom,
! [N: nat,M7: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ M7 )
=> ? [M4: nat] :
( M7
= ( suc @ M4 ) ) ) ).
% Suc_le_D
thf(fact_1249_le__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ M @ ( suc @ N ) ) ) ).
% le_SucI
thf(fact_1250_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_1251_Suc__leD,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% Suc_leD
thf(fact_1252_board__exec__aux_Ocases,axiom,
! [X: produc9133624956312949779et_int] :
( ! [M8: set_int] :
( X
!= ( produc29655638201817675et_int @ zero_zero_nat @ M8 ) )
=> ~ ! [V: nat,M8: set_int] :
( X
!= ( produc29655638201817675et_int @ ( suc @ V ) @ M8 ) ) ) ).
% board_exec_aux.cases
thf(fact_1253_not0__implies__Suc,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ? [M4: nat] :
( N
= ( suc @ M4 ) ) ) ).
% not0_implies_Suc
thf(fact_1254_Zero__not__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
% Zero_not_Suc
thf(fact_1255_Zero__neq__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
% Zero_neq_Suc
thf(fact_1256_Suc__neq__Zero,axiom,
! [M: nat] :
( ( suc @ M )
!= zero_zero_nat ) ).
% Suc_neq_Zero
thf(fact_1257_zero__induct,axiom,
! [P: nat > $o,K: nat] :
( ( P @ K )
=> ( ! [N3: nat] :
( ( P @ ( suc @ N3 ) )
=> ( P @ N3 ) )
=> ( P @ zero_zero_nat ) ) ) ).
% zero_induct
thf(fact_1258_diff__induct,axiom,
! [P: nat > nat > $o,M: nat,N: nat] :
( ! [X3: nat] : ( P @ X3 @ zero_zero_nat )
=> ( ! [Y3: nat] : ( P @ zero_zero_nat @ ( suc @ Y3 ) )
=> ( ! [X3: nat,Y3: nat] :
( ( P @ X3 @ Y3 )
=> ( P @ ( suc @ X3 ) @ ( suc @ Y3 ) ) )
=> ( P @ M @ N ) ) ) ) ).
% diff_induct
thf(fact_1259_nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N3: nat] :
( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) )
=> ( P @ N ) ) ) ).
% nat_induct
thf(fact_1260_row__exec_Ocases,axiom,
! [X: nat] :
( ( X != zero_zero_nat )
=> ~ ! [V: nat] :
( X
!= ( suc @ V ) ) ) ).
% row_exec.cases
thf(fact_1261_old_Onat_Oexhaust,axiom,
! [Y: nat] :
( ( Y != zero_zero_nat )
=> ~ ! [Nat3: nat] :
( Y
!= ( suc @ Nat3 ) ) ) ).
% old.nat.exhaust
thf(fact_1262_nat_OdiscI,axiom,
! [Nat: nat,X22: nat] :
( ( Nat
= ( suc @ X22 ) )
=> ( Nat != zero_zero_nat ) ) ).
% nat.discI
thf(fact_1263_old_Onat_Odistinct_I1_J,axiom,
! [Nat2: nat] :
( zero_zero_nat
!= ( suc @ Nat2 ) ) ).
% old.nat.distinct(1)
thf(fact_1264_old_Onat_Odistinct_I2_J,axiom,
! [Nat2: nat] :
( ( suc @ Nat2 )
!= zero_zero_nat ) ).
% old.nat.distinct(2)
thf(fact_1265_nat_Odistinct_I1_J,axiom,
! [X22: nat] :
( zero_zero_nat
!= ( suc @ X22 ) ) ).
% nat.distinct(1)
% 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,
! [P: $o] :
( ( P = $true )
| ( P = $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 (1)
thf(conj_0,conjecture,
( ( sup_su6024340866399070445nt_int @ ( board @ n_1 @ m ) @ ( trans_board @ ( product_Pair_int_int @ ( semiri1314217659103216013at_int @ n_1 ) @ zero_zero_int ) @ ( board @ n_2 @ m ) ) )
= ( board @ ( plus_plus_nat @ n_1 @ n_2 ) @ m ) ) ).
%------------------------------------------------------------------------------