TPTP Problem File: SLH0074^1.p
View Solutions
- Solve Problem
%------------------------------------------------------------------------------
% 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 : Universal_Hash_Families/0028_Field/prob_00206_007336__18327104_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1340 ( 641 unt; 69 typ; 0 def)
% Number of atoms : 3438 (1144 equ; 0 cnn)
% Maximal formula atoms : 12 ( 2 avg)
% Number of connectives : 9119 ( 287 ~; 74 |; 143 &;7299 @)
% ( 0 <=>;1316 =>; 0 <=; 0 <~>)
% Maximal formula depth : 18 ( 6 avg)
% Number of types : 11 ( 10 usr)
% Number of type conns : 284 ( 284 >; 0 *; 0 +; 0 <<)
% Number of symbols : 62 ( 59 usr; 9 con; 0-3 aty)
% Number of variables : 3093 ( 136 ^;2842 !; 115 ?;3093 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 14:42:17.011
%------------------------------------------------------------------------------
% Could-be-implicit typings (10)
thf(ty_n_t__Congruence__Opartial____object__Opartial____object____ext_It__Set__Oset_It__Int__Oint_J_Mt__Group__Omonoid__Omonoid____ext_It__Set__Oset_It__Int__Oint_J_Mt__Ring__Oring__Oring____ext_It__Set__Oset_It__Int__Oint_J_Mt__Product____Type__Ounit_J_J_J,type,
partia4934656038542163276t_unit: $tType ).
thf(ty_n_t__Congruence__Opartial____object__Opartial____object____ext_It__Nat__Onat_Mt__Group__Omonoid__Omonoid____ext_It__Nat__Onat_Mt__Ring__Oring__Oring____ext_It__Nat__Onat_Mt__Product____Type__Ounit_J_J_J,type,
partia4692342223508353374t_unit: $tType ).
thf(ty_n_t__Set__Oset_I_062_It__Set__Oset_It__Int__Oint_J_Mt__Nat__Onat_J_J,type,
set_set_int_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Int__Oint_J_J,type,
set_set_int: $tType ).
thf(ty_n_t__Set__Oset_It__Real__Oreal_J,type,
set_real: $tType ).
thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
set_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Int__Oint_J,type,
set_int: $tType ).
thf(ty_n_t__Real__Oreal,type,
real: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
thf(ty_n_t__Int__Oint,type,
int: $tType ).
% Explicit typings (59)
thf(sy_c_Archimedean__Field_Oceiling_001t__Real__Oreal,type,
archim7802044766580827645g_real: real > int ).
thf(sy_c_Archimedean__Field_Ofloor__ceiling__class_Ofloor_001t__Real__Oreal,type,
archim6058952711729229775r_real: real > int ).
thf(sy_c_Archimedean__Field_Ofrac_001t__Real__Oreal,type,
archim2898591450579166408c_real: real > real ).
thf(sy_c_Archimedean__Field_Oround_001t__Real__Oreal,type,
archim8280529875227126926d_real: real > int ).
thf(sy_c_Embedded__Algebras_Oembedded__algebra_001t__Nat__Onat_001t__Product____Type__Ounit,type,
embedd4772573186654392350t_unit: set_nat > partia4692342223508353374t_unit > $o ).
thf(sy_c_Embedded__Algebras_Oembedded__algebra_001t__Set__Oset_It__Int__Oint_J_001t__Product____Type__Ounit,type,
embedd5517612406894904972t_unit: set_set_int > partia4934656038542163276t_unit > $o ).
thf(sy_c_Field_Omod__ring,type,
mod_ring: nat > partia4692342223508353374t_unit ).
thf(sy_c_Field_Ozfact__iso__inv,type,
zfact_iso_inv: nat > set_int > 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_Oone__class_Oone_001t__Real__Oreal,type,
one_one_real: real ).
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_Oplus__class_Oplus_001t__Real__Oreal,type,
plus_plus_real: real > real > real ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Int__Oint,type,
times_times_int: int > int > int ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Nat__Onat,type,
times_times_nat: nat > nat > nat ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Real__Oreal,type,
times_times_real: real > real > real ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Int__Oint,type,
uminus_uminus_int: int > int ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Real__Oreal,type,
uminus_uminus_real: real > real ).
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_Groups_Ozero__class_Ozero_001t__Real__Oreal,type,
zero_zero_real: real ).
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_IntRing_OZFact,type,
zFact: int > partia4934656038542163276t_unit ).
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_Int_Oring__1__class_Oof__int_001t__Real__Oreal,type,
ring_1_of_int_real: int > real ).
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_Nat_Osemiring__1__class_Oof__nat_001t__Real__Oreal,type,
semiri5074537144036343181t_real: nat > real ).
thf(sy_c_Num_Oneg__numeral__class_Odbl__dec_001t__Int__Oint,type,
neg_nu3811975205180677377ec_int: int > int ).
thf(sy_c_Num_Oneg__numeral__class_Odbl__dec_001t__Real__Oreal,type,
neg_nu6075765906172075777c_real: real > real ).
thf(sy_c_Num_Oneg__numeral__class_Odbl__inc_001t__Int__Oint,type,
neg_nu5851722552734809277nc_int: int > int ).
thf(sy_c_Num_Oneg__numeral__class_Odbl__inc_001t__Real__Oreal,type,
neg_nu8295874005876285629c_real: real > real ).
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__Real__Oreal,type,
ord_less_real: real > real > $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__Real__Oreal,type,
ord_less_eq_real: real > real > $o ).
thf(sy_c_Power_Opower__class_Opower_001t__Int__Oint,type,
power_power_int: int > nat > int ).
thf(sy_c_Power_Opower__class_Opower_001t__Nat__Onat,type,
power_power_nat: nat > nat > nat ).
thf(sy_c_Power_Opower__class_Opower_001t__Real__Oreal,type,
power_power_real: real > nat > real ).
thf(sy_c_QuotRing_Oring__iso_001t__Set__Oset_It__Int__Oint_J_001t__Product____Type__Ounit_001t__Nat__Onat_001t__Product____Type__Ounit,type,
ring_i6162119212153773794t_unit: partia4934656038542163276t_unit > partia4692342223508353374t_unit > set_set_int_nat ).
thf(sy_c_Ring_Ocring_001t__Nat__Onat_001t__Product____Type__Ounit,type,
cring_4736503688146807908t_unit: partia4692342223508353374t_unit > $o ).
thf(sy_c_Ring_Ocring_001t__Set__Oset_It__Int__Oint_J_001t__Product____Type__Ounit,type,
cring_3079150759069666002t_unit: partia4934656038542163276t_unit > $o ).
thf(sy_c_Ring_Oring_001t__Nat__Onat_001t__Product____Type__Ounit,type,
ring_n9194430563101542159t_unit: partia4692342223508353374t_unit > $o ).
thf(sy_c_Ring_Oring_001t__Set__Oset_It__Int__Oint_J_001t__Product____Type__Ounit,type,
ring_s5316885176909347197t_unit: partia4934656038542163276t_unit > $o ).
thf(sy_c_Ring_Oring_Ozero_001t__Set__Oset_It__Int__Oint_J_001t__Product____Type__Ounit,type,
zero_s6269048424454532197t_unit: partia4934656038542163276t_unit > set_int ).
thf(sy_c_Ring__Characteristic_Ozfact__iso,type,
ring_zfact_iso: nat > nat > set_int ).
thf(sy_c_Rings_Odivide__class_Odivide_001t__Int__Oint,type,
divide_divide_int: int > int > int ).
thf(sy_c_Rings_Odivide__class_Odivide_001t__Nat__Onat,type,
divide_divide_nat: nat > nat > nat ).
thf(sy_c_Rings_Odivide__class_Odivide_001t__Real__Oreal,type,
divide_divide_real: real > real > real ).
thf(sy_c_Set_OCollect_001_062_It__Set__Oset_It__Int__Oint_J_Mt__Nat__Onat_J,type,
collect_set_int_nat: ( ( set_int > nat ) > $o ) > set_set_int_nat ).
thf(sy_c_Set_OCollect_001t__Real__Oreal,type,
collect_real: ( real > $o ) > set_real ).
thf(sy_c_member_001_062_It__Set__Oset_It__Int__Oint_J_Mt__Nat__Onat_J,type,
member_set_int_nat: ( set_int > nat ) > set_set_int_nat > $o ).
thf(sy_c_member_001t__Real__Oreal,type,
member_real: real > set_real > $o ).
thf(sy_v_n,type,
n: nat ).
% Relevant facts (1265)
thf(fact_0_cring_Oaxioms_I1_J,axiom,
! [R: partia4934656038542163276t_unit] :
( ( cring_3079150759069666002t_unit @ R )
=> ( ring_s5316885176909347197t_unit @ R ) ) ).
% cring.axioms(1)
thf(fact_1_cring_Oaxioms_I1_J,axiom,
! [R: partia4692342223508353374t_unit] :
( ( cring_4736503688146807908t_unit @ R )
=> ( ring_n9194430563101542159t_unit @ R ) ) ).
% cring.axioms(1)
thf(fact_2_ZFact__is__cring,axiom,
! [K: int] : ( cring_3079150759069666002t_unit @ ( zFact @ K ) ) ).
% ZFact_is_cring
thf(fact_3_of__nat__eq__iff,axiom,
! [M: nat,N: nat] :
( ( ( semiri1314217659103216013at_int @ M )
= ( semiri1314217659103216013at_int @ N ) )
= ( M = N ) ) ).
% of_nat_eq_iff
thf(fact_4_of__nat__eq__iff,axiom,
! [M: nat,N: nat] :
( ( ( semiri5074537144036343181t_real @ M )
= ( semiri5074537144036343181t_real @ N ) )
= ( M = N ) ) ).
% of_nat_eq_iff
thf(fact_5_n__ge__1,axiom,
ord_less_nat @ one_one_nat @ n ).
% n_ge_1
thf(fact_6_nat__int__comparison_I1_J,axiom,
( ( ^ [Y: nat,Z: nat] : ( Y = Z ) )
= ( ^ [A: nat,B: nat] :
( ( semiri1314217659103216013at_int @ A )
= ( semiri1314217659103216013at_int @ B ) ) ) ) ).
% nat_int_comparison(1)
thf(fact_7_int__if,axiom,
! [P: $o,A2: nat,B2: nat] :
( ( P
=> ( ( semiri1314217659103216013at_int @ ( if_nat @ P @ A2 @ B2 ) )
= ( semiri1314217659103216013at_int @ A2 ) ) )
& ( ~ P
=> ( ( semiri1314217659103216013at_int @ ( if_nat @ P @ A2 @ B2 ) )
= ( semiri1314217659103216013at_int @ B2 ) ) ) ) ).
% int_if
thf(fact_8_int__int__eq,axiom,
! [M: nat,N: nat] :
( ( ( semiri1314217659103216013at_int @ M )
= ( semiri1314217659103216013at_int @ N ) )
= ( M = N ) ) ).
% int_int_eq
thf(fact_9_round__of__nat,axiom,
! [N: nat] :
( ( archim8280529875227126926d_real @ ( semiri5074537144036343181t_real @ N ) )
= ( semiri1314217659103216013at_int @ N ) ) ).
% round_of_nat
thf(fact_10_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_11_of__int__of__nat__eq,axiom,
! [N: nat] :
( ( ring_1_of_int_real @ ( semiri1314217659103216013at_int @ N ) )
= ( semiri5074537144036343181t_real @ N ) ) ).
% of_int_of_nat_eq
thf(fact_12_ceiling__of__nat,axiom,
! [N: nat] :
( ( archim7802044766580827645g_real @ ( semiri5074537144036343181t_real @ N ) )
= ( semiri1314217659103216013at_int @ N ) ) ).
% ceiling_of_nat
thf(fact_13_floor__of__nat,axiom,
! [N: nat] :
( ( archim6058952711729229775r_real @ ( semiri5074537144036343181t_real @ N ) )
= ( semiri1314217659103216013at_int @ N ) ) ).
% floor_of_nat
thf(fact_14_of__int__eq__iff,axiom,
! [W: int,Z2: int] :
( ( ( ring_1_of_int_real @ W )
= ( ring_1_of_int_real @ Z2 ) )
= ( W = Z2 ) ) ).
% of_int_eq_iff
thf(fact_15_of__int__eq__iff,axiom,
! [W: int,Z2: int] :
( ( ( ring_1_of_int_int @ W )
= ( ring_1_of_int_int @ Z2 ) )
= ( W = Z2 ) ) ).
% of_int_eq_iff
thf(fact_16_of__int__less__iff,axiom,
! [W: int,Z2: int] :
( ( ord_less_int @ ( ring_1_of_int_int @ W ) @ ( ring_1_of_int_int @ Z2 ) )
= ( ord_less_int @ W @ Z2 ) ) ).
% of_int_less_iff
thf(fact_17_of__int__less__iff,axiom,
! [W: int,Z2: int] :
( ( ord_less_real @ ( ring_1_of_int_real @ W ) @ ( ring_1_of_int_real @ Z2 ) )
= ( ord_less_int @ W @ Z2 ) ) ).
% of_int_less_iff
thf(fact_18_of__int__1,axiom,
( ( ring_1_of_int_real @ one_one_int )
= one_one_real ) ).
% of_int_1
thf(fact_19_of__int__1,axiom,
( ( ring_1_of_int_int @ one_one_int )
= one_one_int ) ).
% of_int_1
thf(fact_20_of__int__eq__1__iff,axiom,
! [Z2: int] :
( ( ( ring_1_of_int_real @ Z2 )
= one_one_real )
= ( Z2 = one_one_int ) ) ).
% of_int_eq_1_iff
thf(fact_21_of__int__eq__1__iff,axiom,
! [Z2: int] :
( ( ( ring_1_of_int_int @ Z2 )
= one_one_int )
= ( Z2 = one_one_int ) ) ).
% of_int_eq_1_iff
thf(fact_22_floor__one,axiom,
( ( archim6058952711729229775r_real @ one_one_real )
= one_one_int ) ).
% floor_one
thf(fact_23_ceiling__one,axiom,
( ( archim7802044766580827645g_real @ one_one_real )
= one_one_int ) ).
% ceiling_one
thf(fact_24_floor__of__int,axiom,
! [Z2: int] :
( ( archim6058952711729229775r_real @ ( ring_1_of_int_real @ Z2 ) )
= Z2 ) ).
% floor_of_int
thf(fact_25_ceiling__of__int,axiom,
! [Z2: int] :
( ( archim7802044766580827645g_real @ ( ring_1_of_int_real @ Z2 ) )
= Z2 ) ).
% ceiling_of_int
thf(fact_26_round__1,axiom,
( ( archim8280529875227126926d_real @ one_one_real )
= one_one_int ) ).
% round_1
thf(fact_27_round__of__int,axiom,
! [N: int] :
( ( archim8280529875227126926d_real @ ( ring_1_of_int_real @ N ) )
= N ) ).
% round_of_int
thf(fact_28_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_29_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_30_of__nat__less__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% of_nat_less_iff
thf(fact_31_of__int__1__less__iff,axiom,
! [Z2: int] :
( ( ord_less_int @ one_one_int @ ( ring_1_of_int_int @ Z2 ) )
= ( ord_less_int @ one_one_int @ Z2 ) ) ).
% of_int_1_less_iff
thf(fact_32_of__int__1__less__iff,axiom,
! [Z2: int] :
( ( ord_less_real @ one_one_real @ ( ring_1_of_int_real @ Z2 ) )
= ( ord_less_int @ one_one_int @ Z2 ) ) ).
% of_int_1_less_iff
thf(fact_33_of__int__less__1__iff,axiom,
! [Z2: int] :
( ( ord_less_int @ ( ring_1_of_int_int @ Z2 ) @ one_one_int )
= ( ord_less_int @ Z2 @ one_one_int ) ) ).
% of_int_less_1_iff
thf(fact_34_of__int__less__1__iff,axiom,
! [Z2: int] :
( ( ord_less_real @ ( ring_1_of_int_real @ Z2 ) @ one_one_real )
= ( ord_less_int @ Z2 @ one_one_int ) ) ).
% of_int_less_1_iff
thf(fact_35_of__nat__1,axiom,
( ( semiri1316708129612266289at_nat @ one_one_nat )
= one_one_nat ) ).
% of_nat_1
thf(fact_36_of__nat__1,axiom,
( ( semiri1314217659103216013at_int @ one_one_nat )
= one_one_int ) ).
% of_nat_1
thf(fact_37_of__nat__1,axiom,
( ( semiri5074537144036343181t_real @ one_one_nat )
= one_one_real ) ).
% of_nat_1
thf(fact_38_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_39_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_40_of__nat__1__eq__iff,axiom,
! [N: nat] :
( ( one_one_real
= ( semiri5074537144036343181t_real @ N ) )
= ( N = one_one_nat ) ) ).
% of_nat_1_eq_iff
thf(fact_41_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_42_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_43_of__nat__eq__1__iff,axiom,
! [N: nat] :
( ( ( semiri5074537144036343181t_real @ N )
= one_one_real )
= ( N = one_one_nat ) ) ).
% of_nat_eq_1_iff
thf(fact_44_floor__less__one,axiom,
! [X: real] :
( ( ord_less_int @ ( archim6058952711729229775r_real @ X ) @ one_one_int )
= ( ord_less_real @ X @ one_one_real ) ) ).
% floor_less_one
thf(fact_45_one__less__ceiling,axiom,
! [X: real] :
( ( ord_less_int @ one_one_int @ ( archim7802044766580827645g_real @ X ) )
= ( ord_less_real @ one_one_real @ X ) ) ).
% one_less_ceiling
thf(fact_46_verit__comp__simplify1_I1_J,axiom,
! [A2: nat] :
~ ( ord_less_nat @ A2 @ A2 ) ).
% verit_comp_simplify1(1)
thf(fact_47_verit__comp__simplify1_I1_J,axiom,
! [A2: int] :
~ ( ord_less_int @ A2 @ A2 ) ).
% verit_comp_simplify1(1)
thf(fact_48_verit__comp__simplify1_I1_J,axiom,
! [A2: real] :
~ ( ord_less_real @ A2 @ A2 ) ).
% verit_comp_simplify1(1)
thf(fact_49_cring_Ois__cring,axiom,
! [R: partia4934656038542163276t_unit] :
( ( cring_3079150759069666002t_unit @ R )
=> ( cring_3079150759069666002t_unit @ R ) ) ).
% cring.is_cring
thf(fact_50_cring_Ois__cring,axiom,
! [R: partia4692342223508353374t_unit] :
( ( cring_4736503688146807908t_unit @ R )
=> ( cring_4736503688146807908t_unit @ R ) ) ).
% cring.is_cring
thf(fact_51_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_52_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_53_less__imp__of__nat__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) ) ) ).
% less_imp_of_nat_less
thf(fact_54_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_55_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_56_of__nat__less__imp__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) )
=> ( ord_less_nat @ M @ N ) ) ).
% of_nat_less_imp_less
thf(fact_57_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_58_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_not_refl
thf(fact_59_less__not__refl2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
=> ( M != N ) ) ).
% less_not_refl2
thf(fact_60_less__not__refl3,axiom,
! [S: nat,T: nat] :
( ( ord_less_nat @ S @ T )
=> ( S != T ) ) ).
% less_not_refl3
thf(fact_61_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_62_nat__less__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ! [M2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( P @ M2 ) )
=> ( P @ N2 ) )
=> ( P @ N ) ) ).
% nat_less_induct
thf(fact_63_infinite__descent,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ~ ( P @ N2 )
=> ? [M2: nat] :
( ( ord_less_nat @ M2 @ N2 )
& ~ ( P @ M2 ) ) )
=> ( P @ N ) ) ).
% infinite_descent
thf(fact_64_linorder__neqE__nat,axiom,
! [X: nat,Y2: nat] :
( ( X != Y2 )
=> ( ~ ( ord_less_nat @ X @ Y2 )
=> ( ord_less_nat @ Y2 @ X ) ) ) ).
% linorder_neqE_nat
thf(fact_65_mem__Collect__eq,axiom,
! [A2: set_int > nat,P: ( set_int > nat ) > $o] :
( ( member_set_int_nat @ A2 @ ( collect_set_int_nat @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_66_mem__Collect__eq,axiom,
! [A2: real,P: real > $o] :
( ( member_real @ A2 @ ( collect_real @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_67_Collect__mem__eq,axiom,
! [A3: set_set_int_nat] :
( ( collect_set_int_nat
@ ^ [X2: set_int > nat] : ( member_set_int_nat @ X2 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_68_Collect__mem__eq,axiom,
! [A3: set_real] :
( ( collect_real
@ ^ [X2: real] : ( member_real @ X2 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_69_ex__less__of__int,axiom,
! [X: real] :
? [Z3: int] : ( ord_less_real @ X @ ( ring_1_of_int_real @ Z3 ) ) ).
% ex_less_of_int
thf(fact_70_ex__of__int__less,axiom,
! [X: real] :
? [Z3: int] : ( ord_less_real @ ( ring_1_of_int_real @ Z3 ) @ X ) ).
% ex_of_int_less
thf(fact_71_floor__less__iff,axiom,
! [X: real,Z2: int] :
( ( ord_less_int @ ( archim6058952711729229775r_real @ X ) @ Z2 )
= ( ord_less_real @ X @ ( ring_1_of_int_real @ Z2 ) ) ) ).
% floor_less_iff
thf(fact_72_less__ceiling__iff,axiom,
! [Z2: int,X: real] :
( ( ord_less_int @ Z2 @ ( archim7802044766580827645g_real @ X ) )
= ( ord_less_real @ ( ring_1_of_int_real @ Z2 ) @ X ) ) ).
% less_ceiling_iff
thf(fact_73_floor__less__cancel,axiom,
! [X: real,Y2: real] :
( ( ord_less_int @ ( archim6058952711729229775r_real @ X ) @ ( archim6058952711729229775r_real @ Y2 ) )
=> ( ord_less_real @ X @ Y2 ) ) ).
% floor_less_cancel
thf(fact_74_ceiling__less__cancel,axiom,
! [X: real,Y2: real] :
( ( ord_less_int @ ( archim7802044766580827645g_real @ X ) @ ( archim7802044766580827645g_real @ Y2 ) )
=> ( ord_less_real @ X @ Y2 ) ) ).
% ceiling_less_cancel
thf(fact_75_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_76_of__nat__less__of__int__iff,axiom,
! [N: nat,X: int] :
( ( ord_less_real @ ( semiri5074537144036343181t_real @ N ) @ ( ring_1_of_int_real @ X ) )
= ( ord_less_int @ ( semiri1314217659103216013at_int @ N ) @ X ) ) ).
% of_nat_less_of_int_iff
thf(fact_77_reals__Archimedean2,axiom,
! [X: real] :
? [N2: nat] : ( ord_less_real @ X @ ( semiri5074537144036343181t_real @ N2 ) ) ).
% reals_Archimedean2
thf(fact_78_of__int__ceiling__cancel,axiom,
! [X: real] :
( ( ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ X ) )
= X )
= ( ? [N3: int] :
( X
= ( ring_1_of_int_real @ N3 ) ) ) ) ).
% of_int_ceiling_cancel
thf(fact_79_of__int__floor__cancel,axiom,
! [X: real] :
( ( ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ X ) )
= X )
= ( ? [N3: int] :
( X
= ( ring_1_of_int_real @ N3 ) ) ) ) ).
% of_int_floor_cancel
thf(fact_80_less__numeral__extra_I4_J,axiom,
~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).
% less_numeral_extra(4)
thf(fact_81_less__numeral__extra_I4_J,axiom,
~ ( ord_less_int @ one_one_int @ one_one_int ) ).
% less_numeral_extra(4)
thf(fact_82_less__numeral__extra_I4_J,axiom,
~ ( ord_less_real @ one_one_real @ one_one_real ) ).
% less_numeral_extra(4)
thf(fact_83_mod__ring__is__cring,axiom,
! [N: nat] :
( ( ord_less_nat @ one_one_nat @ N )
=> ( cring_4736503688146807908t_unit @ ( mod_ring @ N ) ) ) ).
% mod_ring_is_cring
thf(fact_84_one__natural_Orsp,axiom,
one_one_nat = one_one_nat ).
% one_natural.rsp
thf(fact_85_embedded__algebra_Oaxioms_I2_J,axiom,
! [K2: set_set_int,R: partia4934656038542163276t_unit] :
( ( embedd5517612406894904972t_unit @ K2 @ R )
=> ( ring_s5316885176909347197t_unit @ R ) ) ).
% embedded_algebra.axioms(2)
thf(fact_86_embedded__algebra_Oaxioms_I2_J,axiom,
! [K2: set_nat,R: partia4692342223508353374t_unit] :
( ( embedd4772573186654392350t_unit @ K2 @ R )
=> ( ring_n9194430563101542159t_unit @ R ) ) ).
% embedded_algebra.axioms(2)
thf(fact_87_one__reorient,axiom,
! [X: nat] :
( ( one_one_nat = X )
= ( X = one_one_nat ) ) ).
% one_reorient
thf(fact_88_one__reorient,axiom,
! [X: int] :
( ( one_one_int = X )
= ( X = one_one_int ) ) ).
% one_reorient
thf(fact_89_one__reorient,axiom,
! [X: real] :
( ( one_one_real = X )
= ( X = one_one_real ) ) ).
% one_reorient
thf(fact_90_minf_I7_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z3 )
=> ~ ( ord_less_nat @ T @ X3 ) ) ).
% minf(7)
thf(fact_91_minf_I7_J,axiom,
! [T: int] :
? [Z3: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z3 )
=> ~ ( ord_less_int @ T @ X3 ) ) ).
% minf(7)
thf(fact_92_minf_I7_J,axiom,
! [T: real] :
? [Z3: real] :
! [X3: real] :
( ( ord_less_real @ X3 @ Z3 )
=> ~ ( ord_less_real @ T @ X3 ) ) ).
% minf(7)
thf(fact_93_minf_I5_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z3 )
=> ( ord_less_nat @ X3 @ T ) ) ).
% minf(5)
thf(fact_94_minf_I5_J,axiom,
! [T: int] :
? [Z3: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z3 )
=> ( ord_less_int @ X3 @ T ) ) ).
% minf(5)
thf(fact_95_minf_I5_J,axiom,
! [T: real] :
? [Z3: real] :
! [X3: real] :
( ( ord_less_real @ X3 @ Z3 )
=> ( ord_less_real @ X3 @ T ) ) ).
% minf(5)
thf(fact_96_minf_I4_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z3 )
=> ( X3 != T ) ) ).
% minf(4)
thf(fact_97_minf_I4_J,axiom,
! [T: int] :
? [Z3: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z3 )
=> ( X3 != T ) ) ).
% minf(4)
thf(fact_98_minf_I4_J,axiom,
! [T: real] :
? [Z3: real] :
! [X3: real] :
( ( ord_less_real @ X3 @ Z3 )
=> ( X3 != T ) ) ).
% minf(4)
thf(fact_99_one__integer_Orsp,axiom,
one_one_int = one_one_int ).
% one_integer.rsp
thf(fact_100_nat__int__comparison_I2_J,axiom,
( ord_less_nat
= ( ^ [A: nat,B: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ) ).
% nat_int_comparison(2)
thf(fact_101_int__ops_I2_J,axiom,
( ( semiri1314217659103216013at_int @ one_one_nat )
= one_one_int ) ).
% int_ops(2)
thf(fact_102_pinf_I1_J,axiom,
! [P: nat > $o,P2: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z4: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z4 @ X4 )
=> ( ( P @ X4 )
= ( P2 @ X4 ) ) )
=> ( ? [Z4: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z4 @ X4 )
=> ( ( Q @ X4 )
= ( Q2 @ X4 ) ) )
=> ? [Z3: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z3 @ X3 )
=> ( ( ( P @ X3 )
& ( Q @ X3 ) )
= ( ( P2 @ X3 )
& ( Q2 @ X3 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_103_pinf_I1_J,axiom,
! [P: int > $o,P2: int > $o,Q: int > $o,Q2: int > $o] :
( ? [Z4: int] :
! [X4: int] :
( ( ord_less_int @ Z4 @ X4 )
=> ( ( P @ X4 )
= ( P2 @ X4 ) ) )
=> ( ? [Z4: int] :
! [X4: int] :
( ( ord_less_int @ Z4 @ X4 )
=> ( ( Q @ X4 )
= ( Q2 @ X4 ) ) )
=> ? [Z3: int] :
! [X3: int] :
( ( ord_less_int @ Z3 @ X3 )
=> ( ( ( P @ X3 )
& ( Q @ X3 ) )
= ( ( P2 @ X3 )
& ( Q2 @ X3 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_104_pinf_I1_J,axiom,
! [P: real > $o,P2: real > $o,Q: real > $o,Q2: real > $o] :
( ? [Z4: real] :
! [X4: real] :
( ( ord_less_real @ Z4 @ X4 )
=> ( ( P @ X4 )
= ( P2 @ X4 ) ) )
=> ( ? [Z4: real] :
! [X4: real] :
( ( ord_less_real @ Z4 @ X4 )
=> ( ( Q @ X4 )
= ( Q2 @ X4 ) ) )
=> ? [Z3: real] :
! [X3: real] :
( ( ord_less_real @ Z3 @ X3 )
=> ( ( ( P @ X3 )
& ( Q @ X3 ) )
= ( ( P2 @ X3 )
& ( Q2 @ X3 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_105_pinf_I2_J,axiom,
! [P: nat > $o,P2: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z4: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z4 @ X4 )
=> ( ( P @ X4 )
= ( P2 @ X4 ) ) )
=> ( ? [Z4: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z4 @ X4 )
=> ( ( Q @ X4 )
= ( Q2 @ X4 ) ) )
=> ? [Z3: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z3 @ X3 )
=> ( ( ( P @ X3 )
| ( Q @ X3 ) )
= ( ( P2 @ X3 )
| ( Q2 @ X3 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_106_pinf_I2_J,axiom,
! [P: int > $o,P2: int > $o,Q: int > $o,Q2: int > $o] :
( ? [Z4: int] :
! [X4: int] :
( ( ord_less_int @ Z4 @ X4 )
=> ( ( P @ X4 )
= ( P2 @ X4 ) ) )
=> ( ? [Z4: int] :
! [X4: int] :
( ( ord_less_int @ Z4 @ X4 )
=> ( ( Q @ X4 )
= ( Q2 @ X4 ) ) )
=> ? [Z3: int] :
! [X3: int] :
( ( ord_less_int @ Z3 @ X3 )
=> ( ( ( P @ X3 )
| ( Q @ X3 ) )
= ( ( P2 @ X3 )
| ( Q2 @ X3 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_107_pinf_I2_J,axiom,
! [P: real > $o,P2: real > $o,Q: real > $o,Q2: real > $o] :
( ? [Z4: real] :
! [X4: real] :
( ( ord_less_real @ Z4 @ X4 )
=> ( ( P @ X4 )
= ( P2 @ X4 ) ) )
=> ( ? [Z4: real] :
! [X4: real] :
( ( ord_less_real @ Z4 @ X4 )
=> ( ( Q @ X4 )
= ( Q2 @ X4 ) ) )
=> ? [Z3: real] :
! [X3: real] :
( ( ord_less_real @ Z3 @ X3 )
=> ( ( ( P @ X3 )
| ( Q @ X3 ) )
= ( ( P2 @ X3 )
| ( Q2 @ X3 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_108_pinf_I3_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z3 @ X3 )
=> ( X3 != T ) ) ).
% pinf(3)
thf(fact_109_pinf_I3_J,axiom,
! [T: int] :
? [Z3: int] :
! [X3: int] :
( ( ord_less_int @ Z3 @ X3 )
=> ( X3 != T ) ) ).
% pinf(3)
thf(fact_110_pinf_I3_J,axiom,
! [T: real] :
? [Z3: real] :
! [X3: real] :
( ( ord_less_real @ Z3 @ X3 )
=> ( X3 != T ) ) ).
% pinf(3)
thf(fact_111_pinf_I4_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z3 @ X3 )
=> ( X3 != T ) ) ).
% pinf(4)
thf(fact_112_pinf_I4_J,axiom,
! [T: int] :
? [Z3: int] :
! [X3: int] :
( ( ord_less_int @ Z3 @ X3 )
=> ( X3 != T ) ) ).
% pinf(4)
thf(fact_113_pinf_I4_J,axiom,
! [T: real] :
? [Z3: real] :
! [X3: real] :
( ( ord_less_real @ Z3 @ X3 )
=> ( X3 != T ) ) ).
% pinf(4)
thf(fact_114_pinf_I5_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z3 @ X3 )
=> ~ ( ord_less_nat @ X3 @ T ) ) ).
% pinf(5)
thf(fact_115_pinf_I5_J,axiom,
! [T: int] :
? [Z3: int] :
! [X3: int] :
( ( ord_less_int @ Z3 @ X3 )
=> ~ ( ord_less_int @ X3 @ T ) ) ).
% pinf(5)
thf(fact_116_pinf_I5_J,axiom,
! [T: real] :
? [Z3: real] :
! [X3: real] :
( ( ord_less_real @ Z3 @ X3 )
=> ~ ( ord_less_real @ X3 @ T ) ) ).
% pinf(5)
thf(fact_117_pinf_I7_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z3 @ X3 )
=> ( ord_less_nat @ T @ X3 ) ) ).
% pinf(7)
thf(fact_118_pinf_I7_J,axiom,
! [T: int] :
? [Z3: int] :
! [X3: int] :
( ( ord_less_int @ Z3 @ X3 )
=> ( ord_less_int @ T @ X3 ) ) ).
% pinf(7)
thf(fact_119_pinf_I7_J,axiom,
! [T: real] :
? [Z3: real] :
! [X3: real] :
( ( ord_less_real @ Z3 @ X3 )
=> ( ord_less_real @ T @ X3 ) ) ).
% pinf(7)
thf(fact_120_minf_I1_J,axiom,
! [P: nat > $o,P2: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z4: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z4 )
=> ( ( P @ X4 )
= ( P2 @ X4 ) ) )
=> ( ? [Z4: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z4 )
=> ( ( Q @ X4 )
= ( Q2 @ X4 ) ) )
=> ? [Z3: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z3 )
=> ( ( ( P @ X3 )
& ( Q @ X3 ) )
= ( ( P2 @ X3 )
& ( Q2 @ X3 ) ) ) ) ) ) ).
% minf(1)
thf(fact_121_minf_I1_J,axiom,
! [P: int > $o,P2: int > $o,Q: int > $o,Q2: int > $o] :
( ? [Z4: int] :
! [X4: int] :
( ( ord_less_int @ X4 @ Z4 )
=> ( ( P @ X4 )
= ( P2 @ X4 ) ) )
=> ( ? [Z4: int] :
! [X4: int] :
( ( ord_less_int @ X4 @ Z4 )
=> ( ( Q @ X4 )
= ( Q2 @ X4 ) ) )
=> ? [Z3: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z3 )
=> ( ( ( P @ X3 )
& ( Q @ X3 ) )
= ( ( P2 @ X3 )
& ( Q2 @ X3 ) ) ) ) ) ) ).
% minf(1)
thf(fact_122_minf_I1_J,axiom,
! [P: real > $o,P2: real > $o,Q: real > $o,Q2: real > $o] :
( ? [Z4: real] :
! [X4: real] :
( ( ord_less_real @ X4 @ Z4 )
=> ( ( P @ X4 )
= ( P2 @ X4 ) ) )
=> ( ? [Z4: real] :
! [X4: real] :
( ( ord_less_real @ X4 @ Z4 )
=> ( ( Q @ X4 )
= ( Q2 @ X4 ) ) )
=> ? [Z3: real] :
! [X3: real] :
( ( ord_less_real @ X3 @ Z3 )
=> ( ( ( P @ X3 )
& ( Q @ X3 ) )
= ( ( P2 @ X3 )
& ( Q2 @ X3 ) ) ) ) ) ) ).
% minf(1)
thf(fact_123_minf_I2_J,axiom,
! [P: nat > $o,P2: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z4: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z4 )
=> ( ( P @ X4 )
= ( P2 @ X4 ) ) )
=> ( ? [Z4: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z4 )
=> ( ( Q @ X4 )
= ( Q2 @ X4 ) ) )
=> ? [Z3: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z3 )
=> ( ( ( P @ X3 )
| ( Q @ X3 ) )
= ( ( P2 @ X3 )
| ( Q2 @ X3 ) ) ) ) ) ) ).
% minf(2)
thf(fact_124_minf_I2_J,axiom,
! [P: int > $o,P2: int > $o,Q: int > $o,Q2: int > $o] :
( ? [Z4: int] :
! [X4: int] :
( ( ord_less_int @ X4 @ Z4 )
=> ( ( P @ X4 )
= ( P2 @ X4 ) ) )
=> ( ? [Z4: int] :
! [X4: int] :
( ( ord_less_int @ X4 @ Z4 )
=> ( ( Q @ X4 )
= ( Q2 @ X4 ) ) )
=> ? [Z3: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z3 )
=> ( ( ( P @ X3 )
| ( Q @ X3 ) )
= ( ( P2 @ X3 )
| ( Q2 @ X3 ) ) ) ) ) ) ).
% minf(2)
thf(fact_125_minf_I2_J,axiom,
! [P: real > $o,P2: real > $o,Q: real > $o,Q2: real > $o] :
( ? [Z4: real] :
! [X4: real] :
( ( ord_less_real @ X4 @ Z4 )
=> ( ( P @ X4 )
= ( P2 @ X4 ) ) )
=> ( ? [Z4: real] :
! [X4: real] :
( ( ord_less_real @ X4 @ Z4 )
=> ( ( Q @ X4 )
= ( Q2 @ X4 ) ) )
=> ? [Z3: real] :
! [X3: real] :
( ( ord_less_real @ X3 @ Z3 )
=> ( ( ( P @ X3 )
| ( Q @ X3 ) )
= ( ( P2 @ X3 )
| ( Q2 @ X3 ) ) ) ) ) ) ).
% minf(2)
thf(fact_126_minf_I3_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z3 )
=> ( X3 != T ) ) ).
% minf(3)
thf(fact_127_minf_I3_J,axiom,
! [T: int] :
? [Z3: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z3 )
=> ( X3 != T ) ) ).
% minf(3)
thf(fact_128_minf_I3_J,axiom,
! [T: real] :
? [Z3: real] :
! [X3: real] :
( ( ord_less_real @ X3 @ Z3 )
=> ( X3 != T ) ) ).
% minf(3)
thf(fact_129_zfact__iso__inv__is__ring__iso,axiom,
! [N: nat] :
( ( ord_less_nat @ one_one_nat @ N )
=> ( member_set_int_nat @ ( zfact_iso_inv @ N ) @ ( ring_i6162119212153773794t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ N ) ) @ ( mod_ring @ N ) ) ) ) ).
% zfact_iso_inv_is_ring_iso
thf(fact_130_ceiling__altdef,axiom,
( archim7802044766580827645g_real
= ( ^ [X2: real] :
( if_int
@ ( X2
= ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ X2 ) ) )
@ ( archim6058952711729229775r_real @ X2 )
@ ( plus_plus_int @ ( archim6058952711729229775r_real @ X2 ) @ one_one_int ) ) ) ) ).
% ceiling_altdef
thf(fact_131_dbl__dec__simps_I3_J,axiom,
( ( neg_nu3811975205180677377ec_int @ one_one_int )
= one_one_int ) ).
% dbl_dec_simps(3)
thf(fact_132_dbl__dec__simps_I3_J,axiom,
( ( neg_nu6075765906172075777c_real @ one_one_real )
= one_one_real ) ).
% dbl_dec_simps(3)
thf(fact_133_order__less__imp__not__less,axiom,
! [X: nat,Y2: nat] :
( ( ord_less_nat @ X @ Y2 )
=> ~ ( ord_less_nat @ Y2 @ X ) ) ).
% order_less_imp_not_less
thf(fact_134_order__less__imp__not__less,axiom,
! [X: int,Y2: int] :
( ( ord_less_int @ X @ Y2 )
=> ~ ( ord_less_int @ Y2 @ X ) ) ).
% order_less_imp_not_less
thf(fact_135_order__less__imp__not__less,axiom,
! [X: real,Y2: real] :
( ( ord_less_real @ X @ Y2 )
=> ~ ( ord_less_real @ Y2 @ X ) ) ).
% order_less_imp_not_less
thf(fact_136_order__less__imp__not__eq2,axiom,
! [X: nat,Y2: nat] :
( ( ord_less_nat @ X @ Y2 )
=> ( Y2 != X ) ) ).
% order_less_imp_not_eq2
thf(fact_137_order__less__imp__not__eq2,axiom,
! [X: int,Y2: int] :
( ( ord_less_int @ X @ Y2 )
=> ( Y2 != X ) ) ).
% order_less_imp_not_eq2
thf(fact_138_order__less__imp__not__eq2,axiom,
! [X: real,Y2: real] :
( ( ord_less_real @ X @ Y2 )
=> ( Y2 != X ) ) ).
% order_less_imp_not_eq2
thf(fact_139_order__less__imp__not__eq,axiom,
! [X: nat,Y2: nat] :
( ( ord_less_nat @ X @ Y2 )
=> ( X != Y2 ) ) ).
% order_less_imp_not_eq
thf(fact_140_order__less__imp__not__eq,axiom,
! [X: int,Y2: int] :
( ( ord_less_int @ X @ Y2 )
=> ( X != Y2 ) ) ).
% order_less_imp_not_eq
thf(fact_141_order__less__imp__not__eq,axiom,
! [X: real,Y2: real] :
( ( ord_less_real @ X @ Y2 )
=> ( X != Y2 ) ) ).
% order_less_imp_not_eq
thf(fact_142_linorder__less__linear,axiom,
! [X: nat,Y2: nat] :
( ( ord_less_nat @ X @ Y2 )
| ( X = Y2 )
| ( ord_less_nat @ Y2 @ X ) ) ).
% linorder_less_linear
thf(fact_143_linorder__less__linear,axiom,
! [X: int,Y2: int] :
( ( ord_less_int @ X @ Y2 )
| ( X = Y2 )
| ( ord_less_int @ Y2 @ X ) ) ).
% linorder_less_linear
thf(fact_144_linorder__less__linear,axiom,
! [X: real,Y2: real] :
( ( ord_less_real @ X @ Y2 )
| ( X = Y2 )
| ( ord_less_real @ Y2 @ X ) ) ).
% linorder_less_linear
thf(fact_145_order__less__imp__triv,axiom,
! [X: nat,Y2: nat,P: $o] :
( ( ord_less_nat @ X @ Y2 )
=> ( ( ord_less_nat @ Y2 @ X )
=> P ) ) ).
% order_less_imp_triv
thf(fact_146_order__less__imp__triv,axiom,
! [X: int,Y2: int,P: $o] :
( ( ord_less_int @ X @ Y2 )
=> ( ( ord_less_int @ Y2 @ X )
=> P ) ) ).
% order_less_imp_triv
thf(fact_147_order__less__imp__triv,axiom,
! [X: real,Y2: real,P: $o] :
( ( ord_less_real @ X @ Y2 )
=> ( ( ord_less_real @ Y2 @ X )
=> P ) ) ).
% order_less_imp_triv
thf(fact_148_order__less__not__sym,axiom,
! [X: nat,Y2: nat] :
( ( ord_less_nat @ X @ Y2 )
=> ~ ( ord_less_nat @ Y2 @ X ) ) ).
% order_less_not_sym
thf(fact_149_order__less__not__sym,axiom,
! [X: int,Y2: int] :
( ( ord_less_int @ X @ Y2 )
=> ~ ( ord_less_int @ Y2 @ X ) ) ).
% order_less_not_sym
thf(fact_150_order__less__not__sym,axiom,
! [X: real,Y2: real] :
( ( ord_less_real @ X @ Y2 )
=> ~ ( ord_less_real @ Y2 @ X ) ) ).
% order_less_not_sym
thf(fact_151_order__less__subst2,axiom,
! [A2: nat,B2: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ord_less_nat @ ( F @ B2 ) @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_nat @ X4 @ Y3 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_152_order__less__subst2,axiom,
! [A2: nat,B2: nat,F: nat > int,C: int] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ord_less_int @ ( F @ B2 ) @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_nat @ X4 @ Y3 )
=> ( ord_less_int @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_153_order__less__subst2,axiom,
! [A2: nat,B2: nat,F: nat > real,C: real] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ord_less_real @ ( F @ B2 ) @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_nat @ X4 @ Y3 )
=> ( ord_less_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_154_order__less__subst2,axiom,
! [A2: int,B2: int,F: int > nat,C: nat] :
( ( ord_less_int @ A2 @ B2 )
=> ( ( ord_less_nat @ ( F @ B2 ) @ C )
=> ( ! [X4: int,Y3: int] :
( ( ord_less_int @ X4 @ Y3 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_155_order__less__subst2,axiom,
! [A2: int,B2: int,F: int > int,C: int] :
( ( ord_less_int @ A2 @ B2 )
=> ( ( ord_less_int @ ( F @ B2 ) @ C )
=> ( ! [X4: int,Y3: int] :
( ( ord_less_int @ X4 @ Y3 )
=> ( ord_less_int @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_156_order__less__subst2,axiom,
! [A2: int,B2: int,F: int > real,C: real] :
( ( ord_less_int @ A2 @ B2 )
=> ( ( ord_less_real @ ( F @ B2 ) @ C )
=> ( ! [X4: int,Y3: int] :
( ( ord_less_int @ X4 @ Y3 )
=> ( ord_less_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_157_order__less__subst2,axiom,
! [A2: real,B2: real,F: real > nat,C: nat] :
( ( ord_less_real @ A2 @ B2 )
=> ( ( ord_less_nat @ ( F @ B2 ) @ C )
=> ( ! [X4: real,Y3: real] :
( ( ord_less_real @ X4 @ Y3 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_158_order__less__subst2,axiom,
! [A2: real,B2: real,F: real > int,C: int] :
( ( ord_less_real @ A2 @ B2 )
=> ( ( ord_less_int @ ( F @ B2 ) @ C )
=> ( ! [X4: real,Y3: real] :
( ( ord_less_real @ X4 @ Y3 )
=> ( ord_less_int @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_159_order__less__subst2,axiom,
! [A2: real,B2: real,F: real > real,C: real] :
( ( ord_less_real @ A2 @ B2 )
=> ( ( ord_less_real @ ( F @ B2 ) @ C )
=> ( ! [X4: real,Y3: real] :
( ( ord_less_real @ X4 @ Y3 )
=> ( ord_less_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_160_add__right__cancel,axiom,
! [B2: int,A2: int,C: int] :
( ( ( plus_plus_int @ B2 @ A2 )
= ( plus_plus_int @ C @ A2 ) )
= ( B2 = C ) ) ).
% add_right_cancel
thf(fact_161_add__right__cancel,axiom,
! [B2: nat,A2: nat,C: nat] :
( ( ( plus_plus_nat @ B2 @ A2 )
= ( plus_plus_nat @ C @ A2 ) )
= ( B2 = C ) ) ).
% add_right_cancel
thf(fact_162_add__right__cancel,axiom,
! [B2: real,A2: real,C: real] :
( ( ( plus_plus_real @ B2 @ A2 )
= ( plus_plus_real @ C @ A2 ) )
= ( B2 = C ) ) ).
% add_right_cancel
thf(fact_163_add__left__cancel,axiom,
! [A2: int,B2: int,C: int] :
( ( ( plus_plus_int @ A2 @ B2 )
= ( plus_plus_int @ A2 @ C ) )
= ( B2 = C ) ) ).
% add_left_cancel
thf(fact_164_add__left__cancel,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ( plus_plus_nat @ A2 @ B2 )
= ( plus_plus_nat @ A2 @ C ) )
= ( B2 = C ) ) ).
% add_left_cancel
thf(fact_165_add__left__cancel,axiom,
! [A2: real,B2: real,C: real] :
( ( ( plus_plus_real @ A2 @ B2 )
= ( plus_plus_real @ A2 @ C ) )
= ( B2 = C ) ) ).
% add_left_cancel
thf(fact_166_add__less__cancel__left,axiom,
! [C: nat,A2: nat,B2: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B2 ) )
= ( ord_less_nat @ A2 @ B2 ) ) ).
% add_less_cancel_left
thf(fact_167_add__less__cancel__left,axiom,
! [C: int,A2: int,B2: int] :
( ( ord_less_int @ ( plus_plus_int @ C @ A2 ) @ ( plus_plus_int @ C @ B2 ) )
= ( ord_less_int @ A2 @ B2 ) ) ).
% add_less_cancel_left
thf(fact_168_add__less__cancel__left,axiom,
! [C: real,A2: real,B2: real] :
( ( ord_less_real @ ( plus_plus_real @ C @ A2 ) @ ( plus_plus_real @ C @ B2 ) )
= ( ord_less_real @ A2 @ B2 ) ) ).
% add_less_cancel_left
thf(fact_169_add__less__cancel__right,axiom,
! [A2: nat,C: nat,B2: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B2 @ C ) )
= ( ord_less_nat @ A2 @ B2 ) ) ).
% add_less_cancel_right
thf(fact_170_add__less__cancel__right,axiom,
! [A2: int,C: int,B2: int] :
( ( ord_less_int @ ( plus_plus_int @ A2 @ C ) @ ( plus_plus_int @ B2 @ C ) )
= ( ord_less_int @ A2 @ B2 ) ) ).
% add_less_cancel_right
thf(fact_171_add__less__cancel__right,axiom,
! [A2: real,C: real,B2: real] :
( ( ord_less_real @ ( plus_plus_real @ A2 @ C ) @ ( plus_plus_real @ B2 @ C ) )
= ( ord_less_real @ A2 @ B2 ) ) ).
% add_less_cancel_right
thf(fact_172_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_173_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_174_of__nat__add,axiom,
! [M: nat,N: nat] :
( ( semiri5074537144036343181t_real @ ( plus_plus_nat @ M @ N ) )
= ( plus_plus_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) ) ) ).
% of_nat_add
thf(fact_175_of__int__add,axiom,
! [W: int,Z2: int] :
( ( ring_1_of_int_real @ ( plus_plus_int @ W @ Z2 ) )
= ( plus_plus_real @ ( ring_1_of_int_real @ W ) @ ( ring_1_of_int_real @ Z2 ) ) ) ).
% of_int_add
thf(fact_176_of__int__add,axiom,
! [W: int,Z2: int] :
( ( ring_1_of_int_int @ ( plus_plus_int @ W @ Z2 ) )
= ( plus_plus_int @ ( ring_1_of_int_int @ W ) @ ( ring_1_of_int_int @ Z2 ) ) ) ).
% of_int_add
thf(fact_177_ceiling__add__of__int,axiom,
! [X: real,Z2: int] :
( ( archim7802044766580827645g_real @ ( plus_plus_real @ X @ ( ring_1_of_int_real @ Z2 ) ) )
= ( plus_plus_int @ ( archim7802044766580827645g_real @ X ) @ Z2 ) ) ).
% ceiling_add_of_int
thf(fact_178_ceiling__add__one,axiom,
! [X: real] :
( ( archim7802044766580827645g_real @ ( plus_plus_real @ X @ one_one_real ) )
= ( plus_plus_int @ ( archim7802044766580827645g_real @ X ) @ one_one_int ) ) ).
% ceiling_add_one
thf(fact_179_add__right__imp__eq,axiom,
! [B2: int,A2: int,C: int] :
( ( ( plus_plus_int @ B2 @ A2 )
= ( plus_plus_int @ C @ A2 ) )
=> ( B2 = C ) ) ).
% add_right_imp_eq
thf(fact_180_add__right__imp__eq,axiom,
! [B2: nat,A2: nat,C: nat] :
( ( ( plus_plus_nat @ B2 @ A2 )
= ( plus_plus_nat @ C @ A2 ) )
=> ( B2 = C ) ) ).
% add_right_imp_eq
thf(fact_181_add__right__imp__eq,axiom,
! [B2: real,A2: real,C: real] :
( ( ( plus_plus_real @ B2 @ A2 )
= ( plus_plus_real @ C @ A2 ) )
=> ( B2 = C ) ) ).
% add_right_imp_eq
thf(fact_182_add__left__imp__eq,axiom,
! [A2: int,B2: int,C: int] :
( ( ( plus_plus_int @ A2 @ B2 )
= ( plus_plus_int @ A2 @ C ) )
=> ( B2 = C ) ) ).
% add_left_imp_eq
thf(fact_183_add__left__imp__eq,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ( plus_plus_nat @ A2 @ B2 )
= ( plus_plus_nat @ A2 @ C ) )
=> ( B2 = C ) ) ).
% add_left_imp_eq
thf(fact_184_add__left__imp__eq,axiom,
! [A2: real,B2: real,C: real] :
( ( ( plus_plus_real @ A2 @ B2 )
= ( plus_plus_real @ A2 @ C ) )
=> ( B2 = C ) ) ).
% add_left_imp_eq
thf(fact_185_add_Oleft__commute,axiom,
! [B2: int,A2: int,C: int] :
( ( plus_plus_int @ B2 @ ( plus_plus_int @ A2 @ C ) )
= ( plus_plus_int @ A2 @ ( plus_plus_int @ B2 @ C ) ) ) ).
% add.left_commute
thf(fact_186_add_Oleft__commute,axiom,
! [B2: nat,A2: nat,C: nat] :
( ( plus_plus_nat @ B2 @ ( plus_plus_nat @ A2 @ C ) )
= ( plus_plus_nat @ A2 @ ( plus_plus_nat @ B2 @ C ) ) ) ).
% add.left_commute
thf(fact_187_add_Oleft__commute,axiom,
! [B2: real,A2: real,C: real] :
( ( plus_plus_real @ B2 @ ( plus_plus_real @ A2 @ C ) )
= ( plus_plus_real @ A2 @ ( plus_plus_real @ B2 @ C ) ) ) ).
% add.left_commute
thf(fact_188_add_Ocommute,axiom,
( plus_plus_int
= ( ^ [A: int,B: int] : ( plus_plus_int @ B @ A ) ) ) ).
% add.commute
thf(fact_189_add_Ocommute,axiom,
( plus_plus_nat
= ( ^ [A: nat,B: nat] : ( plus_plus_nat @ B @ A ) ) ) ).
% add.commute
thf(fact_190_add_Ocommute,axiom,
( plus_plus_real
= ( ^ [A: real,B: real] : ( plus_plus_real @ B @ A ) ) ) ).
% add.commute
thf(fact_191_add_Oright__cancel,axiom,
! [B2: int,A2: int,C: int] :
( ( ( plus_plus_int @ B2 @ A2 )
= ( plus_plus_int @ C @ A2 ) )
= ( B2 = C ) ) ).
% add.right_cancel
thf(fact_192_add_Oright__cancel,axiom,
! [B2: real,A2: real,C: real] :
( ( ( plus_plus_real @ B2 @ A2 )
= ( plus_plus_real @ C @ A2 ) )
= ( B2 = C ) ) ).
% add.right_cancel
thf(fact_193_add_Oleft__cancel,axiom,
! [A2: int,B2: int,C: int] :
( ( ( plus_plus_int @ A2 @ B2 )
= ( plus_plus_int @ A2 @ C ) )
= ( B2 = C ) ) ).
% add.left_cancel
thf(fact_194_add_Oleft__cancel,axiom,
! [A2: real,B2: real,C: real] :
( ( ( plus_plus_real @ A2 @ B2 )
= ( plus_plus_real @ A2 @ C ) )
= ( B2 = C ) ) ).
% add.left_cancel
thf(fact_195_add_Oassoc,axiom,
! [A2: int,B2: int,C: int] :
( ( plus_plus_int @ ( plus_plus_int @ A2 @ B2 ) @ C )
= ( plus_plus_int @ A2 @ ( plus_plus_int @ B2 @ C ) ) ) ).
% add.assoc
thf(fact_196_add_Oassoc,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A2 @ B2 ) @ C )
= ( plus_plus_nat @ A2 @ ( plus_plus_nat @ B2 @ C ) ) ) ).
% add.assoc
thf(fact_197_add_Oassoc,axiom,
! [A2: real,B2: real,C: real] :
( ( plus_plus_real @ ( plus_plus_real @ A2 @ B2 ) @ C )
= ( plus_plus_real @ A2 @ ( plus_plus_real @ B2 @ C ) ) ) ).
% add.assoc
thf(fact_198_group__cancel_Oadd2,axiom,
! [B3: int,K: int,B2: int,A2: int] :
( ( B3
= ( plus_plus_int @ K @ B2 ) )
=> ( ( plus_plus_int @ A2 @ B3 )
= ( plus_plus_int @ K @ ( plus_plus_int @ A2 @ B2 ) ) ) ) ).
% group_cancel.add2
thf(fact_199_group__cancel_Oadd2,axiom,
! [B3: nat,K: nat,B2: nat,A2: nat] :
( ( B3
= ( plus_plus_nat @ K @ B2 ) )
=> ( ( plus_plus_nat @ A2 @ B3 )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A2 @ B2 ) ) ) ) ).
% group_cancel.add2
thf(fact_200_group__cancel_Oadd2,axiom,
! [B3: real,K: real,B2: real,A2: real] :
( ( B3
= ( plus_plus_real @ K @ B2 ) )
=> ( ( plus_plus_real @ A2 @ B3 )
= ( plus_plus_real @ K @ ( plus_plus_real @ A2 @ B2 ) ) ) ) ).
% group_cancel.add2
thf(fact_201_group__cancel_Oadd1,axiom,
! [A3: int,K: int,A2: int,B2: int] :
( ( A3
= ( plus_plus_int @ K @ A2 ) )
=> ( ( plus_plus_int @ A3 @ B2 )
= ( plus_plus_int @ K @ ( plus_plus_int @ A2 @ B2 ) ) ) ) ).
% group_cancel.add1
thf(fact_202_group__cancel_Oadd1,axiom,
! [A3: nat,K: nat,A2: nat,B2: nat] :
( ( A3
= ( plus_plus_nat @ K @ A2 ) )
=> ( ( plus_plus_nat @ A3 @ B2 )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A2 @ B2 ) ) ) ) ).
% group_cancel.add1
thf(fact_203_group__cancel_Oadd1,axiom,
! [A3: real,K: real,A2: real,B2: real] :
( ( A3
= ( plus_plus_real @ K @ A2 ) )
=> ( ( plus_plus_real @ A3 @ B2 )
= ( plus_plus_real @ K @ ( plus_plus_real @ A2 @ B2 ) ) ) ) ).
% group_cancel.add1
thf(fact_204_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_205_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_206_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I: real,J: real,K: real,L: real] :
( ( ( I = J )
& ( K = L ) )
=> ( ( plus_plus_real @ I @ K )
= ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_207_is__num__normalize_I1_J,axiom,
! [A2: int,B2: int,C: int] :
( ( plus_plus_int @ ( plus_plus_int @ A2 @ B2 ) @ C )
= ( plus_plus_int @ A2 @ ( plus_plus_int @ B2 @ C ) ) ) ).
% is_num_normalize(1)
thf(fact_208_is__num__normalize_I1_J,axiom,
! [A2: real,B2: real,C: real] :
( ( plus_plus_real @ ( plus_plus_real @ A2 @ B2 ) @ C )
= ( plus_plus_real @ A2 @ ( plus_plus_real @ B2 @ C ) ) ) ).
% is_num_normalize(1)
thf(fact_209_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A2: int,B2: int,C: int] :
( ( plus_plus_int @ ( plus_plus_int @ A2 @ B2 ) @ C )
= ( plus_plus_int @ A2 @ ( plus_plus_int @ B2 @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_210_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A2 @ B2 ) @ C )
= ( plus_plus_nat @ A2 @ ( plus_plus_nat @ B2 @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_211_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A2: real,B2: real,C: real] :
( ( plus_plus_real @ ( plus_plus_real @ A2 @ B2 ) @ C )
= ( plus_plus_real @ A2 @ ( plus_plus_real @ B2 @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_212_floor__add__int,axiom,
! [X: real,Z2: int] :
( ( plus_plus_int @ ( archim6058952711729229775r_real @ X ) @ Z2 )
= ( archim6058952711729229775r_real @ ( plus_plus_real @ X @ ( ring_1_of_int_real @ Z2 ) ) ) ) ).
% floor_add_int
thf(fact_213_int__add__floor,axiom,
! [Z2: int,X: real] :
( ( plus_plus_int @ Z2 @ ( archim6058952711729229775r_real @ X ) )
= ( archim6058952711729229775r_real @ ( plus_plus_real @ ( ring_1_of_int_real @ Z2 ) @ X ) ) ) ).
% int_add_floor
thf(fact_214_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_215_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_216_add__mono__thms__linordered__field_I5_J,axiom,
! [I: real,J: real,K: real,L: real] :
( ( ( ord_less_real @ I @ J )
& ( ord_less_real @ K @ L ) )
=> ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_217_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_218_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_219_add__mono__thms__linordered__field_I2_J,axiom,
! [I: real,J: real,K: real,L: real] :
( ( ( I = J )
& ( ord_less_real @ K @ L ) )
=> ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_220_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_221_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_222_add__mono__thms__linordered__field_I1_J,axiom,
! [I: real,J: real,K: real,L: real] :
( ( ( ord_less_real @ I @ J )
& ( K = L ) )
=> ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_223_add__strict__mono,axiom,
! [A2: nat,B2: nat,C: nat,D: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ord_less_nat @ C @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B2 @ D ) ) ) ) ).
% add_strict_mono
thf(fact_224_add__strict__mono,axiom,
! [A2: int,B2: int,C: int,D: int] :
( ( ord_less_int @ A2 @ B2 )
=> ( ( ord_less_int @ C @ D )
=> ( ord_less_int @ ( plus_plus_int @ A2 @ C ) @ ( plus_plus_int @ B2 @ D ) ) ) ) ).
% add_strict_mono
thf(fact_225_add__strict__mono,axiom,
! [A2: real,B2: real,C: real,D: real] :
( ( ord_less_real @ A2 @ B2 )
=> ( ( ord_less_real @ C @ D )
=> ( ord_less_real @ ( plus_plus_real @ A2 @ C ) @ ( plus_plus_real @ B2 @ D ) ) ) ) ).
% add_strict_mono
thf(fact_226_add__strict__left__mono,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ord_less_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B2 ) ) ) ).
% add_strict_left_mono
thf(fact_227_add__strict__left__mono,axiom,
! [A2: int,B2: int,C: int] :
( ( ord_less_int @ A2 @ B2 )
=> ( ord_less_int @ ( plus_plus_int @ C @ A2 ) @ ( plus_plus_int @ C @ B2 ) ) ) ).
% add_strict_left_mono
thf(fact_228_add__strict__left__mono,axiom,
! [A2: real,B2: real,C: real] :
( ( ord_less_real @ A2 @ B2 )
=> ( ord_less_real @ ( plus_plus_real @ C @ A2 ) @ ( plus_plus_real @ C @ B2 ) ) ) ).
% add_strict_left_mono
thf(fact_229_add__strict__right__mono,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ord_less_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B2 @ C ) ) ) ).
% add_strict_right_mono
thf(fact_230_add__strict__right__mono,axiom,
! [A2: int,B2: int,C: int] :
( ( ord_less_int @ A2 @ B2 )
=> ( ord_less_int @ ( plus_plus_int @ A2 @ C ) @ ( plus_plus_int @ B2 @ C ) ) ) ).
% add_strict_right_mono
thf(fact_231_add__strict__right__mono,axiom,
! [A2: real,B2: real,C: real] :
( ( ord_less_real @ A2 @ B2 )
=> ( ord_less_real @ ( plus_plus_real @ A2 @ C ) @ ( plus_plus_real @ B2 @ C ) ) ) ).
% add_strict_right_mono
thf(fact_232_add__less__imp__less__left,axiom,
! [C: nat,A2: nat,B2: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B2 ) )
=> ( ord_less_nat @ A2 @ B2 ) ) ).
% add_less_imp_less_left
thf(fact_233_add__less__imp__less__left,axiom,
! [C: int,A2: int,B2: int] :
( ( ord_less_int @ ( plus_plus_int @ C @ A2 ) @ ( plus_plus_int @ C @ B2 ) )
=> ( ord_less_int @ A2 @ B2 ) ) ).
% add_less_imp_less_left
thf(fact_234_add__less__imp__less__left,axiom,
! [C: real,A2: real,B2: real] :
( ( ord_less_real @ ( plus_plus_real @ C @ A2 ) @ ( plus_plus_real @ C @ B2 ) )
=> ( ord_less_real @ A2 @ B2 ) ) ).
% add_less_imp_less_left
thf(fact_235_add__less__imp__less__right,axiom,
! [A2: nat,C: nat,B2: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B2 @ C ) )
=> ( ord_less_nat @ A2 @ B2 ) ) ).
% add_less_imp_less_right
thf(fact_236_add__less__imp__less__right,axiom,
! [A2: int,C: int,B2: int] :
( ( ord_less_int @ ( plus_plus_int @ A2 @ C ) @ ( plus_plus_int @ B2 @ C ) )
=> ( ord_less_int @ A2 @ B2 ) ) ).
% add_less_imp_less_right
thf(fact_237_add__less__imp__less__right,axiom,
! [A2: real,C: real,B2: real] :
( ( ord_less_real @ ( plus_plus_real @ A2 @ C ) @ ( plus_plus_real @ B2 @ C ) )
=> ( ord_less_real @ A2 @ B2 ) ) ).
% add_less_imp_less_right
thf(fact_238_one__add__floor,axiom,
! [X: real] :
( ( plus_plus_int @ ( archim6058952711729229775r_real @ X ) @ one_one_int )
= ( archim6058952711729229775r_real @ ( plus_plus_real @ X @ one_one_real ) ) ) ).
% one_add_floor
thf(fact_239_zless__add1__eq,axiom,
! [W: int,Z2: int] :
( ( ord_less_int @ W @ ( plus_plus_int @ Z2 @ one_one_int ) )
= ( ( ord_less_int @ W @ Z2 )
| ( W = Z2 ) ) ) ).
% zless_add1_eq
thf(fact_240_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_241_lt__ex,axiom,
! [X: int] :
? [Y3: int] : ( ord_less_int @ Y3 @ X ) ).
% lt_ex
thf(fact_242_lt__ex,axiom,
! [X: real] :
? [Y3: real] : ( ord_less_real @ Y3 @ X ) ).
% lt_ex
thf(fact_243_gt__ex,axiom,
! [X: nat] :
? [X_1: nat] : ( ord_less_nat @ X @ X_1 ) ).
% gt_ex
thf(fact_244_gt__ex,axiom,
! [X: int] :
? [X_1: int] : ( ord_less_int @ X @ X_1 ) ).
% gt_ex
thf(fact_245_gt__ex,axiom,
! [X: real] :
? [X_1: real] : ( ord_less_real @ X @ X_1 ) ).
% gt_ex
thf(fact_246_dense,axiom,
! [X: real,Y2: real] :
( ( ord_less_real @ X @ Y2 )
=> ? [Z3: real] :
( ( ord_less_real @ X @ Z3 )
& ( ord_less_real @ Z3 @ Y2 ) ) ) ).
% dense
thf(fact_247_less__imp__neq,axiom,
! [X: nat,Y2: nat] :
( ( ord_less_nat @ X @ Y2 )
=> ( X != Y2 ) ) ).
% less_imp_neq
thf(fact_248_less__imp__neq,axiom,
! [X: int,Y2: int] :
( ( ord_less_int @ X @ Y2 )
=> ( X != Y2 ) ) ).
% less_imp_neq
thf(fact_249_less__imp__neq,axiom,
! [X: real,Y2: real] :
( ( ord_less_real @ X @ Y2 )
=> ( X != Y2 ) ) ).
% less_imp_neq
thf(fact_250_order_Oasym,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ~ ( ord_less_nat @ B2 @ A2 ) ) ).
% order.asym
thf(fact_251_order_Oasym,axiom,
! [A2: int,B2: int] :
( ( ord_less_int @ A2 @ B2 )
=> ~ ( ord_less_int @ B2 @ A2 ) ) ).
% order.asym
thf(fact_252_order_Oasym,axiom,
! [A2: real,B2: real] :
( ( ord_less_real @ A2 @ B2 )
=> ~ ( ord_less_real @ B2 @ A2 ) ) ).
% order.asym
thf(fact_253_ord__eq__less__trans,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( A2 = B2 )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ord_less_nat @ A2 @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_254_ord__eq__less__trans,axiom,
! [A2: int,B2: int,C: int] :
( ( A2 = B2 )
=> ( ( ord_less_int @ B2 @ C )
=> ( ord_less_int @ A2 @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_255_ord__eq__less__trans,axiom,
! [A2: real,B2: real,C: real] :
( ( A2 = B2 )
=> ( ( ord_less_real @ B2 @ C )
=> ( ord_less_real @ A2 @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_256_ord__less__eq__trans,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( B2 = C )
=> ( ord_less_nat @ A2 @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_257_ord__less__eq__trans,axiom,
! [A2: int,B2: int,C: int] :
( ( ord_less_int @ A2 @ B2 )
=> ( ( B2 = C )
=> ( ord_less_int @ A2 @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_258_ord__less__eq__trans,axiom,
! [A2: real,B2: real,C: real] :
( ( ord_less_real @ A2 @ B2 )
=> ( ( B2 = C )
=> ( ord_less_real @ A2 @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_259_less__induct,axiom,
! [P: nat > $o,A2: nat] :
( ! [X4: nat] :
( ! [Y4: nat] :
( ( ord_less_nat @ Y4 @ X4 )
=> ( P @ Y4 ) )
=> ( P @ X4 ) )
=> ( P @ A2 ) ) ).
% less_induct
thf(fact_260_antisym__conv3,axiom,
! [Y2: nat,X: nat] :
( ~ ( ord_less_nat @ Y2 @ X )
=> ( ( ~ ( ord_less_nat @ X @ Y2 ) )
= ( X = Y2 ) ) ) ).
% antisym_conv3
thf(fact_261_antisym__conv3,axiom,
! [Y2: int,X: int] :
( ~ ( ord_less_int @ Y2 @ X )
=> ( ( ~ ( ord_less_int @ X @ Y2 ) )
= ( X = Y2 ) ) ) ).
% antisym_conv3
thf(fact_262_antisym__conv3,axiom,
! [Y2: real,X: real] :
( ~ ( ord_less_real @ Y2 @ X )
=> ( ( ~ ( ord_less_real @ X @ Y2 ) )
= ( X = Y2 ) ) ) ).
% antisym_conv3
thf(fact_263_linorder__cases,axiom,
! [X: nat,Y2: nat] :
( ~ ( ord_less_nat @ X @ Y2 )
=> ( ( X != Y2 )
=> ( ord_less_nat @ Y2 @ X ) ) ) ).
% linorder_cases
thf(fact_264_linorder__cases,axiom,
! [X: int,Y2: int] :
( ~ ( ord_less_int @ X @ Y2 )
=> ( ( X != Y2 )
=> ( ord_less_int @ Y2 @ X ) ) ) ).
% linorder_cases
thf(fact_265_linorder__cases,axiom,
! [X: real,Y2: real] :
( ~ ( ord_less_real @ X @ Y2 )
=> ( ( X != Y2 )
=> ( ord_less_real @ Y2 @ X ) ) ) ).
% linorder_cases
thf(fact_266_dual__order_Oasym,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_nat @ B2 @ A2 )
=> ~ ( ord_less_nat @ A2 @ B2 ) ) ).
% dual_order.asym
thf(fact_267_dual__order_Oasym,axiom,
! [B2: int,A2: int] :
( ( ord_less_int @ B2 @ A2 )
=> ~ ( ord_less_int @ A2 @ B2 ) ) ).
% dual_order.asym
thf(fact_268_dual__order_Oasym,axiom,
! [B2: real,A2: real] :
( ( ord_less_real @ B2 @ A2 )
=> ~ ( ord_less_real @ A2 @ B2 ) ) ).
% dual_order.asym
thf(fact_269_dual__order_Oirrefl,axiom,
! [A2: nat] :
~ ( ord_less_nat @ A2 @ A2 ) ).
% dual_order.irrefl
thf(fact_270_dual__order_Oirrefl,axiom,
! [A2: int] :
~ ( ord_less_int @ A2 @ A2 ) ).
% dual_order.irrefl
thf(fact_271_dual__order_Oirrefl,axiom,
! [A2: real] :
~ ( ord_less_real @ A2 @ A2 ) ).
% dual_order.irrefl
thf(fact_272_exists__least__iff,axiom,
( ( ^ [P3: nat > $o] :
? [X5: nat] : ( P3 @ X5 ) )
= ( ^ [P4: nat > $o] :
? [N3: nat] :
( ( P4 @ N3 )
& ! [M3: nat] :
( ( ord_less_nat @ M3 @ N3 )
=> ~ ( P4 @ M3 ) ) ) ) ) ).
% exists_least_iff
thf(fact_273_linorder__less__wlog,axiom,
! [P: nat > nat > $o,A2: nat,B2: nat] :
( ! [A4: nat,B4: nat] :
( ( ord_less_nat @ A4 @ B4 )
=> ( P @ A4 @ B4 ) )
=> ( ! [A4: nat] : ( P @ A4 @ A4 )
=> ( ! [A4: nat,B4: nat] :
( ( P @ B4 @ A4 )
=> ( P @ A4 @ B4 ) )
=> ( P @ A2 @ B2 ) ) ) ) ).
% linorder_less_wlog
thf(fact_274_linorder__less__wlog,axiom,
! [P: int > int > $o,A2: int,B2: int] :
( ! [A4: int,B4: int] :
( ( ord_less_int @ A4 @ B4 )
=> ( P @ A4 @ B4 ) )
=> ( ! [A4: int] : ( P @ A4 @ A4 )
=> ( ! [A4: int,B4: int] :
( ( P @ B4 @ A4 )
=> ( P @ A4 @ B4 ) )
=> ( P @ A2 @ B2 ) ) ) ) ).
% linorder_less_wlog
thf(fact_275_linorder__less__wlog,axiom,
! [P: real > real > $o,A2: real,B2: real] :
( ! [A4: real,B4: real] :
( ( ord_less_real @ A4 @ B4 )
=> ( P @ A4 @ B4 ) )
=> ( ! [A4: real] : ( P @ A4 @ A4 )
=> ( ! [A4: real,B4: real] :
( ( P @ B4 @ A4 )
=> ( P @ A4 @ B4 ) )
=> ( P @ A2 @ B2 ) ) ) ) ).
% linorder_less_wlog
thf(fact_276_order_Ostrict__trans,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ord_less_nat @ A2 @ C ) ) ) ).
% order.strict_trans
thf(fact_277_order_Ostrict__trans,axiom,
! [A2: int,B2: int,C: int] :
( ( ord_less_int @ A2 @ B2 )
=> ( ( ord_less_int @ B2 @ C )
=> ( ord_less_int @ A2 @ C ) ) ) ).
% order.strict_trans
thf(fact_278_order_Ostrict__trans,axiom,
! [A2: real,B2: real,C: real] :
( ( ord_less_real @ A2 @ B2 )
=> ( ( ord_less_real @ B2 @ C )
=> ( ord_less_real @ A2 @ C ) ) ) ).
% order.strict_trans
thf(fact_279_not__less__iff__gr__or__eq,axiom,
! [X: nat,Y2: nat] :
( ( ~ ( ord_less_nat @ X @ Y2 ) )
= ( ( ord_less_nat @ Y2 @ X )
| ( X = Y2 ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_280_not__less__iff__gr__or__eq,axiom,
! [X: int,Y2: int] :
( ( ~ ( ord_less_int @ X @ Y2 ) )
= ( ( ord_less_int @ Y2 @ X )
| ( X = Y2 ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_281_not__less__iff__gr__or__eq,axiom,
! [X: real,Y2: real] :
( ( ~ ( ord_less_real @ X @ Y2 ) )
= ( ( ord_less_real @ Y2 @ X )
| ( X = Y2 ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_282_dual__order_Ostrict__trans,axiom,
! [B2: nat,A2: nat,C: nat] :
( ( ord_less_nat @ B2 @ A2 )
=> ( ( ord_less_nat @ C @ B2 )
=> ( ord_less_nat @ C @ A2 ) ) ) ).
% dual_order.strict_trans
thf(fact_283_dual__order_Ostrict__trans,axiom,
! [B2: int,A2: int,C: int] :
( ( ord_less_int @ B2 @ A2 )
=> ( ( ord_less_int @ C @ B2 )
=> ( ord_less_int @ C @ A2 ) ) ) ).
% dual_order.strict_trans
thf(fact_284_dual__order_Ostrict__trans,axiom,
! [B2: real,A2: real,C: real] :
( ( ord_less_real @ B2 @ A2 )
=> ( ( ord_less_real @ C @ B2 )
=> ( ord_less_real @ C @ A2 ) ) ) ).
% dual_order.strict_trans
thf(fact_285_order_Ostrict__implies__not__eq,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( A2 != B2 ) ) ).
% order.strict_implies_not_eq
thf(fact_286_order_Ostrict__implies__not__eq,axiom,
! [A2: int,B2: int] :
( ( ord_less_int @ A2 @ B2 )
=> ( A2 != B2 ) ) ).
% order.strict_implies_not_eq
thf(fact_287_order_Ostrict__implies__not__eq,axiom,
! [A2: real,B2: real] :
( ( ord_less_real @ A2 @ B2 )
=> ( A2 != B2 ) ) ).
% order.strict_implies_not_eq
thf(fact_288_dual__order_Ostrict__implies__not__eq,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_nat @ B2 @ A2 )
=> ( A2 != B2 ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_289_dual__order_Ostrict__implies__not__eq,axiom,
! [B2: int,A2: int] :
( ( ord_less_int @ B2 @ A2 )
=> ( A2 != B2 ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_290_dual__order_Ostrict__implies__not__eq,axiom,
! [B2: real,A2: real] :
( ( ord_less_real @ B2 @ A2 )
=> ( A2 != B2 ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_291_linorder__neqE,axiom,
! [X: nat,Y2: nat] :
( ( X != Y2 )
=> ( ~ ( ord_less_nat @ X @ Y2 )
=> ( ord_less_nat @ Y2 @ X ) ) ) ).
% linorder_neqE
thf(fact_292_linorder__neqE,axiom,
! [X: int,Y2: int] :
( ( X != Y2 )
=> ( ~ ( ord_less_int @ X @ Y2 )
=> ( ord_less_int @ Y2 @ X ) ) ) ).
% linorder_neqE
thf(fact_293_linorder__neqE,axiom,
! [X: real,Y2: real] :
( ( X != Y2 )
=> ( ~ ( ord_less_real @ X @ Y2 )
=> ( ord_less_real @ Y2 @ X ) ) ) ).
% linorder_neqE
thf(fact_294_order__less__asym,axiom,
! [X: nat,Y2: nat] :
( ( ord_less_nat @ X @ Y2 )
=> ~ ( ord_less_nat @ Y2 @ X ) ) ).
% order_less_asym
thf(fact_295_order__less__asym,axiom,
! [X: int,Y2: int] :
( ( ord_less_int @ X @ Y2 )
=> ~ ( ord_less_int @ Y2 @ X ) ) ).
% order_less_asym
thf(fact_296_order__less__asym,axiom,
! [X: real,Y2: real] :
( ( ord_less_real @ X @ Y2 )
=> ~ ( ord_less_real @ Y2 @ X ) ) ).
% order_less_asym
thf(fact_297_linorder__neq__iff,axiom,
! [X: nat,Y2: nat] :
( ( X != Y2 )
= ( ( ord_less_nat @ X @ Y2 )
| ( ord_less_nat @ Y2 @ X ) ) ) ).
% linorder_neq_iff
thf(fact_298_linorder__neq__iff,axiom,
! [X: int,Y2: int] :
( ( X != Y2 )
= ( ( ord_less_int @ X @ Y2 )
| ( ord_less_int @ Y2 @ X ) ) ) ).
% linorder_neq_iff
thf(fact_299_linorder__neq__iff,axiom,
! [X: real,Y2: real] :
( ( X != Y2 )
= ( ( ord_less_real @ X @ Y2 )
| ( ord_less_real @ Y2 @ X ) ) ) ).
% linorder_neq_iff
thf(fact_300_order__less__asym_H,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ~ ( ord_less_nat @ B2 @ A2 ) ) ).
% order_less_asym'
thf(fact_301_order__less__asym_H,axiom,
! [A2: int,B2: int] :
( ( ord_less_int @ A2 @ B2 )
=> ~ ( ord_less_int @ B2 @ A2 ) ) ).
% order_less_asym'
thf(fact_302_order__less__asym_H,axiom,
! [A2: real,B2: real] :
( ( ord_less_real @ A2 @ B2 )
=> ~ ( ord_less_real @ B2 @ A2 ) ) ).
% order_less_asym'
thf(fact_303_order__less__trans,axiom,
! [X: nat,Y2: nat,Z2: nat] :
( ( ord_less_nat @ X @ Y2 )
=> ( ( ord_less_nat @ Y2 @ Z2 )
=> ( ord_less_nat @ X @ Z2 ) ) ) ).
% order_less_trans
thf(fact_304_order__less__trans,axiom,
! [X: int,Y2: int,Z2: int] :
( ( ord_less_int @ X @ Y2 )
=> ( ( ord_less_int @ Y2 @ Z2 )
=> ( ord_less_int @ X @ Z2 ) ) ) ).
% order_less_trans
thf(fact_305_order__less__trans,axiom,
! [X: real,Y2: real,Z2: real] :
( ( ord_less_real @ X @ Y2 )
=> ( ( ord_less_real @ Y2 @ Z2 )
=> ( ord_less_real @ X @ Z2 ) ) ) ).
% order_less_trans
thf(fact_306_ord__eq__less__subst,axiom,
! [A2: nat,F: nat > nat,B2: nat,C: nat] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_nat @ X4 @ Y3 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_307_ord__eq__less__subst,axiom,
! [A2: int,F: nat > int,B2: nat,C: nat] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_nat @ X4 @ Y3 )
=> ( ord_less_int @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_308_ord__eq__less__subst,axiom,
! [A2: real,F: nat > real,B2: nat,C: nat] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_nat @ X4 @ Y3 )
=> ( ord_less_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_309_ord__eq__less__subst,axiom,
! [A2: nat,F: int > nat,B2: int,C: int] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_int @ B2 @ C )
=> ( ! [X4: int,Y3: int] :
( ( ord_less_int @ X4 @ Y3 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_310_ord__eq__less__subst,axiom,
! [A2: int,F: int > int,B2: int,C: int] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_int @ B2 @ C )
=> ( ! [X4: int,Y3: int] :
( ( ord_less_int @ X4 @ Y3 )
=> ( ord_less_int @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_311_ord__eq__less__subst,axiom,
! [A2: real,F: int > real,B2: int,C: int] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_int @ B2 @ C )
=> ( ! [X4: int,Y3: int] :
( ( ord_less_int @ X4 @ Y3 )
=> ( ord_less_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_312_ord__eq__less__subst,axiom,
! [A2: nat,F: real > nat,B2: real,C: real] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_real @ B2 @ C )
=> ( ! [X4: real,Y3: real] :
( ( ord_less_real @ X4 @ Y3 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_313_ord__eq__less__subst,axiom,
! [A2: int,F: real > int,B2: real,C: real] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_real @ B2 @ C )
=> ( ! [X4: real,Y3: real] :
( ( ord_less_real @ X4 @ Y3 )
=> ( ord_less_int @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_314_ord__eq__less__subst,axiom,
! [A2: real,F: real > real,B2: real,C: real] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_real @ B2 @ C )
=> ( ! [X4: real,Y3: real] :
( ( ord_less_real @ X4 @ Y3 )
=> ( ord_less_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_315_ord__less__eq__subst,axiom,
! [A2: nat,B2: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_nat @ X4 @ Y3 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_316_ord__less__eq__subst,axiom,
! [A2: nat,B2: nat,F: nat > int,C: int] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_nat @ X4 @ Y3 )
=> ( ord_less_int @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_317_ord__less__eq__subst,axiom,
! [A2: nat,B2: nat,F: nat > real,C: real] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_nat @ X4 @ Y3 )
=> ( ord_less_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_318_ord__less__eq__subst,axiom,
! [A2: int,B2: int,F: int > nat,C: nat] :
( ( ord_less_int @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X4: int,Y3: int] :
( ( ord_less_int @ X4 @ Y3 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_319_ord__less__eq__subst,axiom,
! [A2: int,B2: int,F: int > int,C: int] :
( ( ord_less_int @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X4: int,Y3: int] :
( ( ord_less_int @ X4 @ Y3 )
=> ( ord_less_int @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_320_ord__less__eq__subst,axiom,
! [A2: int,B2: int,F: int > real,C: real] :
( ( ord_less_int @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X4: int,Y3: int] :
( ( ord_less_int @ X4 @ Y3 )
=> ( ord_less_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_321_ord__less__eq__subst,axiom,
! [A2: real,B2: real,F: real > nat,C: nat] :
( ( ord_less_real @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X4: real,Y3: real] :
( ( ord_less_real @ X4 @ Y3 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_322_ord__less__eq__subst,axiom,
! [A2: real,B2: real,F: real > int,C: int] :
( ( ord_less_real @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X4: real,Y3: real] :
( ( ord_less_real @ X4 @ Y3 )
=> ( ord_less_int @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_323_ord__less__eq__subst,axiom,
! [A2: real,B2: real,F: real > real,C: real] :
( ( ord_less_real @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X4: real,Y3: real] :
( ( ord_less_real @ X4 @ Y3 )
=> ( ord_less_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_324_order__less__irrefl,axiom,
! [X: nat] :
~ ( ord_less_nat @ X @ X ) ).
% order_less_irrefl
thf(fact_325_order__less__irrefl,axiom,
! [X: int] :
~ ( ord_less_int @ X @ X ) ).
% order_less_irrefl
thf(fact_326_order__less__irrefl,axiom,
! [X: real] :
~ ( ord_less_real @ X @ X ) ).
% order_less_irrefl
thf(fact_327_order__less__subst1,axiom,
! [A2: nat,F: nat > nat,B2: nat,C: nat] :
( ( ord_less_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_nat @ X4 @ Y3 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_328_order__less__subst1,axiom,
! [A2: nat,F: int > nat,B2: int,C: int] :
( ( ord_less_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_int @ B2 @ C )
=> ( ! [X4: int,Y3: int] :
( ( ord_less_int @ X4 @ Y3 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_329_order__less__subst1,axiom,
! [A2: nat,F: real > nat,B2: real,C: real] :
( ( ord_less_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_real @ B2 @ C )
=> ( ! [X4: real,Y3: real] :
( ( ord_less_real @ X4 @ Y3 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_330_order__less__subst1,axiom,
! [A2: int,F: nat > int,B2: nat,C: nat] :
( ( ord_less_int @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_nat @ X4 @ Y3 )
=> ( ord_less_int @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_331_order__less__subst1,axiom,
! [A2: int,F: int > int,B2: int,C: int] :
( ( ord_less_int @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_int @ B2 @ C )
=> ( ! [X4: int,Y3: int] :
( ( ord_less_int @ X4 @ Y3 )
=> ( ord_less_int @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_332_order__less__subst1,axiom,
! [A2: int,F: real > int,B2: real,C: real] :
( ( ord_less_int @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_real @ B2 @ C )
=> ( ! [X4: real,Y3: real] :
( ( ord_less_real @ X4 @ Y3 )
=> ( ord_less_int @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_333_order__less__subst1,axiom,
! [A2: real,F: nat > real,B2: nat,C: nat] :
( ( ord_less_real @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_nat @ X4 @ Y3 )
=> ( ord_less_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_334_order__less__subst1,axiom,
! [A2: real,F: int > real,B2: int,C: int] :
( ( ord_less_real @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_int @ B2 @ C )
=> ( ! [X4: int,Y3: int] :
( ( ord_less_int @ X4 @ Y3 )
=> ( ord_less_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_335_order__less__subst1,axiom,
! [A2: real,F: real > real,B2: real,C: real] :
( ( ord_less_real @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_real @ B2 @ C )
=> ( ! [X4: real,Y3: real] :
( ( ord_less_real @ X4 @ Y3 )
=> ( ord_less_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_336_less__add__one,axiom,
! [A2: nat] : ( ord_less_nat @ A2 @ ( plus_plus_nat @ A2 @ one_one_nat ) ) ).
% less_add_one
thf(fact_337_less__add__one,axiom,
! [A2: int] : ( ord_less_int @ A2 @ ( plus_plus_int @ A2 @ one_one_int ) ) ).
% less_add_one
thf(fact_338_less__add__one,axiom,
! [A2: real] : ( ord_less_real @ A2 @ ( plus_plus_real @ A2 @ one_one_real ) ) ).
% less_add_one
thf(fact_339_add__mono1,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ord_less_nat @ ( plus_plus_nat @ A2 @ one_one_nat ) @ ( plus_plus_nat @ B2 @ one_one_nat ) ) ) ).
% add_mono1
thf(fact_340_add__mono1,axiom,
! [A2: int,B2: int] :
( ( ord_less_int @ A2 @ B2 )
=> ( ord_less_int @ ( plus_plus_int @ A2 @ one_one_int ) @ ( plus_plus_int @ B2 @ one_one_int ) ) ) ).
% add_mono1
thf(fact_341_add__mono1,axiom,
! [A2: real,B2: real] :
( ( ord_less_real @ A2 @ B2 )
=> ( ord_less_real @ ( plus_plus_real @ A2 @ one_one_real ) @ ( plus_plus_real @ B2 @ one_one_real ) ) ) ).
% add_mono1
thf(fact_342_floor__add,axiom,
! [X: real,Y2: real] :
( ( ( ord_less_real @ ( plus_plus_real @ ( archim2898591450579166408c_real @ X ) @ ( archim2898591450579166408c_real @ Y2 ) ) @ one_one_real )
=> ( ( archim6058952711729229775r_real @ ( plus_plus_real @ X @ Y2 ) )
= ( plus_plus_int @ ( archim6058952711729229775r_real @ X ) @ ( archim6058952711729229775r_real @ Y2 ) ) ) )
& ( ~ ( ord_less_real @ ( plus_plus_real @ ( archim2898591450579166408c_real @ X ) @ ( archim2898591450579166408c_real @ Y2 ) ) @ one_one_real )
=> ( ( archim6058952711729229775r_real @ ( plus_plus_real @ X @ Y2 ) )
= ( plus_plus_int @ ( plus_plus_int @ ( archim6058952711729229775r_real @ X ) @ ( archim6058952711729229775r_real @ Y2 ) ) @ one_one_int ) ) ) ) ).
% floor_add
thf(fact_343_ceiling__eq,axiom,
! [N: int,X: real] :
( ( ord_less_real @ ( ring_1_of_int_real @ N ) @ X )
=> ( ( ord_less_eq_real @ X @ ( plus_plus_real @ ( ring_1_of_int_real @ N ) @ one_one_real ) )
=> ( ( archim7802044766580827645g_real @ X )
= ( plus_plus_int @ N @ one_one_int ) ) ) ) ).
% ceiling_eq
thf(fact_344_less__floor__iff,axiom,
! [Z2: int,X: real] :
( ( ord_less_int @ Z2 @ ( archim6058952711729229775r_real @ X ) )
= ( ord_less_eq_real @ ( plus_plus_real @ ( ring_1_of_int_real @ Z2 ) @ one_one_real ) @ X ) ) ).
% less_floor_iff
thf(fact_345_floor__correct,axiom,
! [X: real] :
( ( ord_less_eq_real @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ X ) ) @ X )
& ( ord_less_real @ X @ ( ring_1_of_int_real @ ( plus_plus_int @ ( archim6058952711729229775r_real @ X ) @ one_one_int ) ) ) ) ).
% floor_correct
thf(fact_346_floor__le__iff,axiom,
! [X: real,Z2: int] :
( ( ord_less_eq_int @ ( archim6058952711729229775r_real @ X ) @ Z2 )
= ( ord_less_real @ X @ ( plus_plus_real @ ( ring_1_of_int_real @ Z2 ) @ one_one_real ) ) ) ).
% floor_le_iff
thf(fact_347_floor__split,axiom,
! [P: int > $o,T: real] :
( ( P @ ( archim6058952711729229775r_real @ T ) )
= ( ! [I3: int] :
( ( ( ord_less_eq_real @ ( ring_1_of_int_real @ I3 ) @ T )
& ( ord_less_real @ T @ ( plus_plus_real @ ( ring_1_of_int_real @ I3 ) @ one_one_real ) ) )
=> ( P @ I3 ) ) ) ) ).
% floor_split
thf(fact_348_order__refl,axiom,
! [X: nat] : ( ord_less_eq_nat @ X @ X ) ).
% order_refl
thf(fact_349_order__refl,axiom,
! [X: int] : ( ord_less_eq_int @ X @ X ) ).
% order_refl
thf(fact_350_order__refl,axiom,
! [X: real] : ( ord_less_eq_real @ X @ X ) ).
% order_refl
thf(fact_351_dual__order_Orefl,axiom,
! [A2: nat] : ( ord_less_eq_nat @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_352_dual__order_Orefl,axiom,
! [A2: int] : ( ord_less_eq_int @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_353_dual__order_Orefl,axiom,
! [A2: real] : ( ord_less_eq_real @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_354_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_355_add__le__cancel__left,axiom,
! [C: nat,A2: nat,B2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B2 ) )
= ( ord_less_eq_nat @ A2 @ B2 ) ) ).
% add_le_cancel_left
thf(fact_356_add__le__cancel__left,axiom,
! [C: int,A2: int,B2: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ C @ A2 ) @ ( plus_plus_int @ C @ B2 ) )
= ( ord_less_eq_int @ A2 @ B2 ) ) ).
% add_le_cancel_left
thf(fact_357_add__le__cancel__left,axiom,
! [C: real,A2: real,B2: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ C @ A2 ) @ ( plus_plus_real @ C @ B2 ) )
= ( ord_less_eq_real @ A2 @ B2 ) ) ).
% add_le_cancel_left
thf(fact_358_add__le__cancel__right,axiom,
! [A2: nat,C: nat,B2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B2 @ C ) )
= ( ord_less_eq_nat @ A2 @ B2 ) ) ).
% add_le_cancel_right
thf(fact_359_add__le__cancel__right,axiom,
! [A2: int,C: int,B2: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A2 @ C ) @ ( plus_plus_int @ B2 @ C ) )
= ( ord_less_eq_int @ A2 @ B2 ) ) ).
% add_le_cancel_right
thf(fact_360_add__le__cancel__right,axiom,
! [A2: real,C: real,B2: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ A2 @ C ) @ ( plus_plus_real @ B2 @ C ) )
= ( ord_less_eq_real @ A2 @ B2 ) ) ).
% add_le_cancel_right
thf(fact_361_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_362_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_363_of__nat__le__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% of_nat_le_iff
thf(fact_364_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_365_of__int__le__iff,axiom,
! [W: int,Z2: int] :
( ( ord_less_eq_int @ ( ring_1_of_int_int @ W ) @ ( ring_1_of_int_int @ Z2 ) )
= ( ord_less_eq_int @ W @ Z2 ) ) ).
% of_int_le_iff
thf(fact_366_of__int__le__iff,axiom,
! [W: int,Z2: int] :
( ( ord_less_eq_real @ ( ring_1_of_int_real @ W ) @ ( ring_1_of_int_real @ Z2 ) )
= ( ord_less_eq_int @ W @ Z2 ) ) ).
% of_int_le_iff
thf(fact_367_zle__add1__eq__le,axiom,
! [W: int,Z2: int] :
( ( ord_less_int @ W @ ( plus_plus_int @ Z2 @ one_one_int ) )
= ( ord_less_eq_int @ W @ Z2 ) ) ).
% zle_add1_eq_le
thf(fact_368_of__int__le__1__iff,axiom,
! [Z2: int] :
( ( ord_less_eq_int @ ( ring_1_of_int_int @ Z2 ) @ one_one_int )
= ( ord_less_eq_int @ Z2 @ one_one_int ) ) ).
% of_int_le_1_iff
thf(fact_369_of__int__le__1__iff,axiom,
! [Z2: int] :
( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z2 ) @ one_one_real )
= ( ord_less_eq_int @ Z2 @ one_one_int ) ) ).
% of_int_le_1_iff
thf(fact_370_of__int__1__le__iff,axiom,
! [Z2: int] :
( ( ord_less_eq_int @ one_one_int @ ( ring_1_of_int_int @ Z2 ) )
= ( ord_less_eq_int @ one_one_int @ Z2 ) ) ).
% of_int_1_le_iff
thf(fact_371_of__int__1__le__iff,axiom,
! [Z2: int] :
( ( ord_less_eq_real @ one_one_real @ ( ring_1_of_int_real @ Z2 ) )
= ( ord_less_eq_int @ one_one_int @ Z2 ) ) ).
% of_int_1_le_iff
thf(fact_372_one__le__floor,axiom,
! [X: real] :
( ( ord_less_eq_int @ one_one_int @ ( archim6058952711729229775r_real @ X ) )
= ( ord_less_eq_real @ one_one_real @ X ) ) ).
% one_le_floor
thf(fact_373_ceiling__le__one,axiom,
! [X: real] :
( ( ord_less_eq_int @ ( archim7802044766580827645g_real @ X ) @ one_one_int )
= ( ord_less_eq_real @ X @ one_one_real ) ) ).
% ceiling_le_one
thf(fact_374_int__le__real__less,axiom,
( ord_less_eq_int
= ( ^ [N3: int,M3: int] : ( ord_less_real @ ( ring_1_of_int_real @ N3 ) @ ( plus_plus_real @ ( ring_1_of_int_real @ M3 ) @ one_one_real ) ) ) ) ).
% int_le_real_less
thf(fact_375_nle__le,axiom,
! [A2: nat,B2: nat] :
( ( ~ ( ord_less_eq_nat @ A2 @ B2 ) )
= ( ( ord_less_eq_nat @ B2 @ A2 )
& ( B2 != A2 ) ) ) ).
% nle_le
thf(fact_376_nle__le,axiom,
! [A2: int,B2: int] :
( ( ~ ( ord_less_eq_int @ A2 @ B2 ) )
= ( ( ord_less_eq_int @ B2 @ A2 )
& ( B2 != A2 ) ) ) ).
% nle_le
thf(fact_377_nle__le,axiom,
! [A2: real,B2: real] :
( ( ~ ( ord_less_eq_real @ A2 @ B2 ) )
= ( ( ord_less_eq_real @ B2 @ A2 )
& ( B2 != A2 ) ) ) ).
% nle_le
thf(fact_378_le__cases3,axiom,
! [X: nat,Y2: nat,Z2: nat] :
( ( ( ord_less_eq_nat @ X @ Y2 )
=> ~ ( ord_less_eq_nat @ Y2 @ Z2 ) )
=> ( ( ( ord_less_eq_nat @ Y2 @ X )
=> ~ ( ord_less_eq_nat @ X @ Z2 ) )
=> ( ( ( ord_less_eq_nat @ X @ Z2 )
=> ~ ( ord_less_eq_nat @ Z2 @ Y2 ) )
=> ( ( ( ord_less_eq_nat @ Z2 @ Y2 )
=> ~ ( ord_less_eq_nat @ Y2 @ X ) )
=> ( ( ( ord_less_eq_nat @ Y2 @ Z2 )
=> ~ ( ord_less_eq_nat @ Z2 @ X ) )
=> ~ ( ( ord_less_eq_nat @ Z2 @ X )
=> ~ ( ord_less_eq_nat @ X @ Y2 ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_379_le__cases3,axiom,
! [X: int,Y2: int,Z2: int] :
( ( ( ord_less_eq_int @ X @ Y2 )
=> ~ ( ord_less_eq_int @ Y2 @ Z2 ) )
=> ( ( ( ord_less_eq_int @ Y2 @ X )
=> ~ ( ord_less_eq_int @ X @ Z2 ) )
=> ( ( ( ord_less_eq_int @ X @ Z2 )
=> ~ ( ord_less_eq_int @ Z2 @ Y2 ) )
=> ( ( ( ord_less_eq_int @ Z2 @ Y2 )
=> ~ ( ord_less_eq_int @ Y2 @ X ) )
=> ( ( ( ord_less_eq_int @ Y2 @ Z2 )
=> ~ ( ord_less_eq_int @ Z2 @ X ) )
=> ~ ( ( ord_less_eq_int @ Z2 @ X )
=> ~ ( ord_less_eq_int @ X @ Y2 ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_380_le__cases3,axiom,
! [X: real,Y2: real,Z2: real] :
( ( ( ord_less_eq_real @ X @ Y2 )
=> ~ ( ord_less_eq_real @ Y2 @ Z2 ) )
=> ( ( ( ord_less_eq_real @ Y2 @ X )
=> ~ ( ord_less_eq_real @ X @ Z2 ) )
=> ( ( ( ord_less_eq_real @ X @ Z2 )
=> ~ ( ord_less_eq_real @ Z2 @ Y2 ) )
=> ( ( ( ord_less_eq_real @ Z2 @ Y2 )
=> ~ ( ord_less_eq_real @ Y2 @ X ) )
=> ( ( ( ord_less_eq_real @ Y2 @ Z2 )
=> ~ ( ord_less_eq_real @ Z2 @ X ) )
=> ~ ( ( ord_less_eq_real @ Z2 @ X )
=> ~ ( ord_less_eq_real @ X @ Y2 ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_381_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y: nat,Z: nat] : ( Y = Z ) )
= ( ^ [X2: nat,Y5: nat] :
( ( ord_less_eq_nat @ X2 @ Y5 )
& ( ord_less_eq_nat @ Y5 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_382_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y: int,Z: int] : ( Y = Z ) )
= ( ^ [X2: int,Y5: int] :
( ( ord_less_eq_int @ X2 @ Y5 )
& ( ord_less_eq_int @ Y5 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_383_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y: real,Z: real] : ( Y = Z ) )
= ( ^ [X2: real,Y5: real] :
( ( ord_less_eq_real @ X2 @ Y5 )
& ( ord_less_eq_real @ Y5 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_384_ord__eq__le__trans,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( A2 = B2 )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_385_ord__eq__le__trans,axiom,
! [A2: int,B2: int,C: int] :
( ( A2 = B2 )
=> ( ( ord_less_eq_int @ B2 @ C )
=> ( ord_less_eq_int @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_386_ord__eq__le__trans,axiom,
! [A2: real,B2: real,C: real] :
( ( A2 = B2 )
=> ( ( ord_less_eq_real @ B2 @ C )
=> ( ord_less_eq_real @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_387_ord__le__eq__trans,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( B2 = C )
=> ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_388_ord__le__eq__trans,axiom,
! [A2: int,B2: int,C: int] :
( ( ord_less_eq_int @ A2 @ B2 )
=> ( ( B2 = C )
=> ( ord_less_eq_int @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_389_ord__le__eq__trans,axiom,
! [A2: real,B2: real,C: real] :
( ( ord_less_eq_real @ A2 @ B2 )
=> ( ( B2 = C )
=> ( ord_less_eq_real @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_390_order__antisym,axiom,
! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ( ord_less_eq_nat @ Y2 @ X )
=> ( X = Y2 ) ) ) ).
% order_antisym
thf(fact_391_order__antisym,axiom,
! [X: int,Y2: int] :
( ( ord_less_eq_int @ X @ Y2 )
=> ( ( ord_less_eq_int @ Y2 @ X )
=> ( X = Y2 ) ) ) ).
% order_antisym
thf(fact_392_order__antisym,axiom,
! [X: real,Y2: real] :
( ( ord_less_eq_real @ X @ Y2 )
=> ( ( ord_less_eq_real @ Y2 @ X )
=> ( X = Y2 ) ) ) ).
% order_antisym
thf(fact_393_order_Otrans,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% order.trans
thf(fact_394_order_Otrans,axiom,
! [A2: int,B2: int,C: int] :
( ( ord_less_eq_int @ A2 @ B2 )
=> ( ( ord_less_eq_int @ B2 @ C )
=> ( ord_less_eq_int @ A2 @ C ) ) ) ).
% order.trans
thf(fact_395_order_Otrans,axiom,
! [A2: real,B2: real,C: real] :
( ( ord_less_eq_real @ A2 @ B2 )
=> ( ( ord_less_eq_real @ B2 @ C )
=> ( ord_less_eq_real @ A2 @ C ) ) ) ).
% order.trans
thf(fact_396_order__trans,axiom,
! [X: nat,Y2: nat,Z2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ( ord_less_eq_nat @ Y2 @ Z2 )
=> ( ord_less_eq_nat @ X @ Z2 ) ) ) ).
% order_trans
thf(fact_397_order__trans,axiom,
! [X: int,Y2: int,Z2: int] :
( ( ord_less_eq_int @ X @ Y2 )
=> ( ( ord_less_eq_int @ Y2 @ Z2 )
=> ( ord_less_eq_int @ X @ Z2 ) ) ) ).
% order_trans
thf(fact_398_order__trans,axiom,
! [X: real,Y2: real,Z2: real] :
( ( ord_less_eq_real @ X @ Y2 )
=> ( ( ord_less_eq_real @ Y2 @ Z2 )
=> ( ord_less_eq_real @ X @ Z2 ) ) ) ).
% order_trans
thf(fact_399_linorder__wlog,axiom,
! [P: nat > nat > $o,A2: nat,B2: nat] :
( ! [A4: nat,B4: nat] :
( ( ord_less_eq_nat @ A4 @ B4 )
=> ( P @ A4 @ B4 ) )
=> ( ! [A4: nat,B4: nat] :
( ( P @ B4 @ A4 )
=> ( P @ A4 @ B4 ) )
=> ( P @ A2 @ B2 ) ) ) ).
% linorder_wlog
thf(fact_400_linorder__wlog,axiom,
! [P: int > int > $o,A2: int,B2: int] :
( ! [A4: int,B4: int] :
( ( ord_less_eq_int @ A4 @ B4 )
=> ( P @ A4 @ B4 ) )
=> ( ! [A4: int,B4: int] :
( ( P @ B4 @ A4 )
=> ( P @ A4 @ B4 ) )
=> ( P @ A2 @ B2 ) ) ) ).
% linorder_wlog
thf(fact_401_linorder__wlog,axiom,
! [P: real > real > $o,A2: real,B2: real] :
( ! [A4: real,B4: real] :
( ( ord_less_eq_real @ A4 @ B4 )
=> ( P @ A4 @ B4 ) )
=> ( ! [A4: real,B4: real] :
( ( P @ B4 @ A4 )
=> ( P @ A4 @ B4 ) )
=> ( P @ A2 @ B2 ) ) ) ).
% linorder_wlog
thf(fact_402_dual__order_Oeq__iff,axiom,
( ( ^ [Y: nat,Z: nat] : ( Y = Z ) )
= ( ^ [A: nat,B: nat] :
( ( ord_less_eq_nat @ B @ A )
& ( ord_less_eq_nat @ A @ B ) ) ) ) ).
% dual_order.eq_iff
thf(fact_403_dual__order_Oeq__iff,axiom,
( ( ^ [Y: int,Z: int] : ( Y = Z ) )
= ( ^ [A: int,B: int] :
( ( ord_less_eq_int @ B @ A )
& ( ord_less_eq_int @ A @ B ) ) ) ) ).
% dual_order.eq_iff
thf(fact_404_dual__order_Oeq__iff,axiom,
( ( ^ [Y: real,Z: real] : ( Y = Z ) )
= ( ^ [A: real,B: real] :
( ( ord_less_eq_real @ B @ A )
& ( ord_less_eq_real @ A @ B ) ) ) ) ).
% dual_order.eq_iff
thf(fact_405_dual__order_Oantisym,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
=> ( ( ord_less_eq_nat @ A2 @ B2 )
=> ( A2 = B2 ) ) ) ).
% dual_order.antisym
thf(fact_406_dual__order_Oantisym,axiom,
! [B2: int,A2: int] :
( ( ord_less_eq_int @ B2 @ A2 )
=> ( ( ord_less_eq_int @ A2 @ B2 )
=> ( A2 = B2 ) ) ) ).
% dual_order.antisym
thf(fact_407_dual__order_Oantisym,axiom,
! [B2: real,A2: real] :
( ( ord_less_eq_real @ B2 @ A2 )
=> ( ( ord_less_eq_real @ A2 @ B2 )
=> ( A2 = B2 ) ) ) ).
% dual_order.antisym
thf(fact_408_dual__order_Otrans,axiom,
! [B2: nat,A2: nat,C: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
=> ( ( ord_less_eq_nat @ C @ B2 )
=> ( ord_less_eq_nat @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_409_dual__order_Otrans,axiom,
! [B2: int,A2: int,C: int] :
( ( ord_less_eq_int @ B2 @ A2 )
=> ( ( ord_less_eq_int @ C @ B2 )
=> ( ord_less_eq_int @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_410_dual__order_Otrans,axiom,
! [B2: real,A2: real,C: real] :
( ( ord_less_eq_real @ B2 @ A2 )
=> ( ( ord_less_eq_real @ C @ B2 )
=> ( ord_less_eq_real @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_411_antisym,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% antisym
thf(fact_412_antisym,axiom,
! [A2: int,B2: int] :
( ( ord_less_eq_int @ A2 @ B2 )
=> ( ( ord_less_eq_int @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% antisym
thf(fact_413_antisym,axiom,
! [A2: real,B2: real] :
( ( ord_less_eq_real @ A2 @ B2 )
=> ( ( ord_less_eq_real @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% antisym
thf(fact_414_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y: nat,Z: nat] : ( Y = Z ) )
= ( ^ [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
& ( ord_less_eq_nat @ B @ A ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_415_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y: int,Z: int] : ( Y = Z ) )
= ( ^ [A: int,B: int] :
( ( ord_less_eq_int @ A @ B )
& ( ord_less_eq_int @ B @ A ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_416_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y: real,Z: real] : ( Y = Z ) )
= ( ^ [A: real,B: real] :
( ( ord_less_eq_real @ A @ B )
& ( ord_less_eq_real @ B @ A ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_417_order__subst1,axiom,
! [A2: nat,F: nat > nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_418_order__subst1,axiom,
! [A2: nat,F: int > nat,B2: int,C: int] :
( ( ord_less_eq_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_int @ B2 @ C )
=> ( ! [X4: int,Y3: int] :
( ( ord_less_eq_int @ X4 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_419_order__subst1,axiom,
! [A2: nat,F: real > nat,B2: real,C: real] :
( ( ord_less_eq_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_real @ B2 @ C )
=> ( ! [X4: real,Y3: real] :
( ( ord_less_eq_real @ X4 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_420_order__subst1,axiom,
! [A2: int,F: nat > int,B2: nat,C: nat] :
( ( ord_less_eq_int @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_421_order__subst1,axiom,
! [A2: int,F: int > int,B2: int,C: int] :
( ( ord_less_eq_int @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_int @ B2 @ C )
=> ( ! [X4: int,Y3: int] :
( ( ord_less_eq_int @ X4 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_422_order__subst1,axiom,
! [A2: int,F: real > int,B2: real,C: real] :
( ( ord_less_eq_int @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_real @ B2 @ C )
=> ( ! [X4: real,Y3: real] :
( ( ord_less_eq_real @ X4 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_423_order__subst1,axiom,
! [A2: real,F: nat > real,B2: nat,C: nat] :
( ( ord_less_eq_real @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_real @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_424_order__subst1,axiom,
! [A2: real,F: int > real,B2: int,C: int] :
( ( ord_less_eq_real @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_int @ B2 @ C )
=> ( ! [X4: int,Y3: int] :
( ( ord_less_eq_int @ X4 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_real @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_425_order__subst1,axiom,
! [A2: real,F: real > real,B2: real,C: real] :
( ( ord_less_eq_real @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_real @ B2 @ C )
=> ( ! [X4: real,Y3: real] :
( ( ord_less_eq_real @ X4 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_real @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_426_order__subst2,axiom,
! [A2: nat,B2: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ ( F @ B2 ) @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_427_order__subst2,axiom,
! [A2: nat,B2: nat,F: nat > int,C: int] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_int @ ( F @ B2 ) @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_428_order__subst2,axiom,
! [A2: nat,B2: nat,F: nat > real,C: real] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_real @ ( F @ B2 ) @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_real @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_429_order__subst2,axiom,
! [A2: int,B2: int,F: int > nat,C: nat] :
( ( ord_less_eq_int @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ ( F @ B2 ) @ C )
=> ( ! [X4: int,Y3: int] :
( ( ord_less_eq_int @ X4 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_430_order__subst2,axiom,
! [A2: int,B2: int,F: int > int,C: int] :
( ( ord_less_eq_int @ A2 @ B2 )
=> ( ( ord_less_eq_int @ ( F @ B2 ) @ C )
=> ( ! [X4: int,Y3: int] :
( ( ord_less_eq_int @ X4 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_431_order__subst2,axiom,
! [A2: int,B2: int,F: int > real,C: real] :
( ( ord_less_eq_int @ A2 @ B2 )
=> ( ( ord_less_eq_real @ ( F @ B2 ) @ C )
=> ( ! [X4: int,Y3: int] :
( ( ord_less_eq_int @ X4 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_real @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_432_order__subst2,axiom,
! [A2: real,B2: real,F: real > nat,C: nat] :
( ( ord_less_eq_real @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ ( F @ B2 ) @ C )
=> ( ! [X4: real,Y3: real] :
( ( ord_less_eq_real @ X4 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_433_order__subst2,axiom,
! [A2: real,B2: real,F: real > int,C: int] :
( ( ord_less_eq_real @ A2 @ B2 )
=> ( ( ord_less_eq_int @ ( F @ B2 ) @ C )
=> ( ! [X4: real,Y3: real] :
( ( ord_less_eq_real @ X4 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_434_order__subst2,axiom,
! [A2: real,B2: real,F: real > real,C: real] :
( ( ord_less_eq_real @ A2 @ B2 )
=> ( ( ord_less_eq_real @ ( F @ B2 ) @ C )
=> ( ! [X4: real,Y3: real] :
( ( ord_less_eq_real @ X4 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_real @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_435_order__eq__refl,axiom,
! [X: nat,Y2: nat] :
( ( X = Y2 )
=> ( ord_less_eq_nat @ X @ Y2 ) ) ).
% order_eq_refl
thf(fact_436_order__eq__refl,axiom,
! [X: int,Y2: int] :
( ( X = Y2 )
=> ( ord_less_eq_int @ X @ Y2 ) ) ).
% order_eq_refl
thf(fact_437_order__eq__refl,axiom,
! [X: real,Y2: real] :
( ( X = Y2 )
=> ( ord_less_eq_real @ X @ Y2 ) ) ).
% order_eq_refl
thf(fact_438_linorder__linear,axiom,
! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
| ( ord_less_eq_nat @ Y2 @ X ) ) ).
% linorder_linear
thf(fact_439_linorder__linear,axiom,
! [X: int,Y2: int] :
( ( ord_less_eq_int @ X @ Y2 )
| ( ord_less_eq_int @ Y2 @ X ) ) ).
% linorder_linear
thf(fact_440_linorder__linear,axiom,
! [X: real,Y2: real] :
( ( ord_less_eq_real @ X @ Y2 )
| ( ord_less_eq_real @ Y2 @ X ) ) ).
% linorder_linear
thf(fact_441_ord__eq__le__subst,axiom,
! [A2: nat,F: nat > nat,B2: nat,C: nat] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_442_ord__eq__le__subst,axiom,
! [A2: int,F: nat > int,B2: nat,C: nat] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_443_ord__eq__le__subst,axiom,
! [A2: real,F: nat > real,B2: nat,C: nat] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_real @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_444_ord__eq__le__subst,axiom,
! [A2: nat,F: int > nat,B2: int,C: int] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_int @ B2 @ C )
=> ( ! [X4: int,Y3: int] :
( ( ord_less_eq_int @ X4 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_445_ord__eq__le__subst,axiom,
! [A2: int,F: int > int,B2: int,C: int] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_int @ B2 @ C )
=> ( ! [X4: int,Y3: int] :
( ( ord_less_eq_int @ X4 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_446_ord__eq__le__subst,axiom,
! [A2: real,F: int > real,B2: int,C: int] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_int @ B2 @ C )
=> ( ! [X4: int,Y3: int] :
( ( ord_less_eq_int @ X4 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_real @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_447_ord__eq__le__subst,axiom,
! [A2: nat,F: real > nat,B2: real,C: real] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_real @ B2 @ C )
=> ( ! [X4: real,Y3: real] :
( ( ord_less_eq_real @ X4 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_448_ord__eq__le__subst,axiom,
! [A2: int,F: real > int,B2: real,C: real] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_real @ B2 @ C )
=> ( ! [X4: real,Y3: real] :
( ( ord_less_eq_real @ X4 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_449_ord__eq__le__subst,axiom,
! [A2: real,F: real > real,B2: real,C: real] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_real @ B2 @ C )
=> ( ! [X4: real,Y3: real] :
( ( ord_less_eq_real @ X4 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_real @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_450_ord__le__eq__subst,axiom,
! [A2: nat,B2: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_451_ord__le__eq__subst,axiom,
! [A2: nat,B2: nat,F: nat > int,C: int] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_452_ord__le__eq__subst,axiom,
! [A2: nat,B2: nat,F: nat > real,C: real] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_real @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_453_ord__le__eq__subst,axiom,
! [A2: int,B2: int,F: int > nat,C: nat] :
( ( ord_less_eq_int @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X4: int,Y3: int] :
( ( ord_less_eq_int @ X4 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_454_ord__le__eq__subst,axiom,
! [A2: int,B2: int,F: int > int,C: int] :
( ( ord_less_eq_int @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X4: int,Y3: int] :
( ( ord_less_eq_int @ X4 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_455_ord__le__eq__subst,axiom,
! [A2: int,B2: int,F: int > real,C: real] :
( ( ord_less_eq_int @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X4: int,Y3: int] :
( ( ord_less_eq_int @ X4 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_real @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_456_ord__le__eq__subst,axiom,
! [A2: real,B2: real,F: real > nat,C: nat] :
( ( ord_less_eq_real @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X4: real,Y3: real] :
( ( ord_less_eq_real @ X4 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_457_ord__le__eq__subst,axiom,
! [A2: real,B2: real,F: real > int,C: int] :
( ( ord_less_eq_real @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X4: real,Y3: real] :
( ( ord_less_eq_real @ X4 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_458_ord__le__eq__subst,axiom,
! [A2: real,B2: real,F: real > real,C: real] :
( ( ord_less_eq_real @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X4: real,Y3: real] :
( ( ord_less_eq_real @ X4 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_real @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_459_linorder__le__cases,axiom,
! [X: nat,Y2: nat] :
( ~ ( ord_less_eq_nat @ X @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ X ) ) ).
% linorder_le_cases
thf(fact_460_linorder__le__cases,axiom,
! [X: int,Y2: int] :
( ~ ( ord_less_eq_int @ X @ Y2 )
=> ( ord_less_eq_int @ Y2 @ X ) ) ).
% linorder_le_cases
thf(fact_461_linorder__le__cases,axiom,
! [X: real,Y2: real] :
( ~ ( ord_less_eq_real @ X @ Y2 )
=> ( ord_less_eq_real @ Y2 @ X ) ) ).
% linorder_le_cases
thf(fact_462_order__antisym__conv,axiom,
! [Y2: nat,X: nat] :
( ( ord_less_eq_nat @ Y2 @ X )
=> ( ( ord_less_eq_nat @ X @ Y2 )
= ( X = Y2 ) ) ) ).
% order_antisym_conv
thf(fact_463_order__antisym__conv,axiom,
! [Y2: int,X: int] :
( ( ord_less_eq_int @ Y2 @ X )
=> ( ( ord_less_eq_int @ X @ Y2 )
= ( X = Y2 ) ) ) ).
% order_antisym_conv
thf(fact_464_order__antisym__conv,axiom,
! [Y2: real,X: real] :
( ( ord_less_eq_real @ Y2 @ X )
=> ( ( ord_less_eq_real @ X @ Y2 )
= ( X = Y2 ) ) ) ).
% order_antisym_conv
thf(fact_465_nat__le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [M3: nat,N3: nat] :
? [K3: nat] :
( N3
= ( plus_plus_nat @ M3 @ K3 ) ) ) ) ).
% nat_le_iff_add
thf(fact_466_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_467_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_468_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_469_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_470_le__Suc__ex,axiom,
! [K: nat,L: nat] :
( ( ord_less_eq_nat @ K @ L )
=> ? [N2: nat] :
( L
= ( plus_plus_nat @ K @ N2 ) ) ) ).
% le_Suc_ex
thf(fact_471_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_472_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_473_le__add2,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M @ N ) ) ).
% le_add2
thf(fact_474_le__add1,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M ) ) ).
% le_add1
thf(fact_475_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_476_nat__less__real__le,axiom,
( ord_less_nat
= ( ^ [N3: nat,M3: nat] : ( ord_less_eq_real @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ N3 ) @ one_one_real ) @ ( semiri5074537144036343181t_real @ M3 ) ) ) ) ).
% nat_less_real_le
thf(fact_477_mono__nat__linear__lb,axiom,
! [F: nat > nat,M: nat,K: nat] :
( ! [M4: nat,N2: nat] :
( ( ord_less_nat @ M4 @ N2 )
=> ( ord_less_nat @ ( F @ M4 ) @ ( F @ N2 ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M ) @ K ) @ ( F @ ( plus_plus_nat @ M @ K ) ) ) ) ).
% mono_nat_linear_lb
thf(fact_478_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_479_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_480_of__nat__mono,axiom,
! [I: nat,J: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ I ) @ ( semiri5074537144036343181t_real @ J ) ) ) ).
% of_nat_mono
thf(fact_481_verit__comp__simplify1_I2_J,axiom,
! [A2: nat] : ( ord_less_eq_nat @ A2 @ A2 ) ).
% verit_comp_simplify1(2)
thf(fact_482_verit__comp__simplify1_I2_J,axiom,
! [A2: int] : ( ord_less_eq_int @ A2 @ A2 ) ).
% verit_comp_simplify1(2)
thf(fact_483_verit__comp__simplify1_I2_J,axiom,
! [A2: real] : ( ord_less_eq_real @ A2 @ A2 ) ).
% verit_comp_simplify1(2)
thf(fact_484_verit__la__generic,axiom,
! [A2: int,X: int] :
( ( ord_less_eq_int @ A2 @ X )
| ( A2 = X )
| ( ord_less_eq_int @ X @ A2 ) ) ).
% verit_la_generic
thf(fact_485_verit__la__disequality,axiom,
! [A2: nat,B2: nat] :
( ( A2 = B2 )
| ~ ( ord_less_eq_nat @ A2 @ B2 )
| ~ ( ord_less_eq_nat @ B2 @ A2 ) ) ).
% verit_la_disequality
thf(fact_486_verit__la__disequality,axiom,
! [A2: int,B2: int] :
( ( A2 = B2 )
| ~ ( ord_less_eq_int @ A2 @ B2 )
| ~ ( ord_less_eq_int @ B2 @ A2 ) ) ).
% verit_la_disequality
thf(fact_487_verit__la__disequality,axiom,
! [A2: real,B2: real] :
( ( A2 = B2 )
| ~ ( ord_less_eq_real @ A2 @ B2 )
| ~ ( ord_less_eq_real @ B2 @ A2 ) ) ).
% verit_la_disequality
thf(fact_488_int__less__real__le,axiom,
( ord_less_int
= ( ^ [N3: int,M3: int] : ( ord_less_eq_real @ ( plus_plus_real @ ( ring_1_of_int_real @ N3 ) @ one_one_real ) @ ( ring_1_of_int_real @ M3 ) ) ) ) ).
% int_less_real_le
thf(fact_489_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_490_nat__int__comparison_I3_J,axiom,
( ord_less_eq_nat
= ( ^ [A: nat,B: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ) ).
% nat_int_comparison(3)
thf(fact_491_floor__mono,axiom,
! [X: real,Y2: real] :
( ( ord_less_eq_real @ X @ Y2 )
=> ( ord_less_eq_int @ ( archim6058952711729229775r_real @ X ) @ ( archim6058952711729229775r_real @ Y2 ) ) ) ).
% floor_mono
thf(fact_492_ceiling__mono,axiom,
! [Y2: real,X: real] :
( ( ord_less_eq_real @ Y2 @ X )
=> ( ord_less_eq_int @ ( archim7802044766580827645g_real @ Y2 ) @ ( archim7802044766580827645g_real @ X ) ) ) ).
% ceiling_mono
thf(fact_493_round__mono,axiom,
! [X: real,Y2: real] :
( ( ord_less_eq_real @ X @ Y2 )
=> ( ord_less_eq_int @ ( archim8280529875227126926d_real @ X ) @ ( archim8280529875227126926d_real @ Y2 ) ) ) ).
% round_mono
thf(fact_494_le__floor__iff,axiom,
! [Z2: int,X: real] :
( ( ord_less_eq_int @ Z2 @ ( archim6058952711729229775r_real @ X ) )
= ( ord_less_eq_real @ ( ring_1_of_int_real @ Z2 ) @ X ) ) ).
% le_floor_iff
thf(fact_495_ceiling__le,axiom,
! [X: real,A2: int] :
( ( ord_less_eq_real @ X @ ( ring_1_of_int_real @ A2 ) )
=> ( ord_less_eq_int @ ( archim7802044766580827645g_real @ X ) @ A2 ) ) ).
% ceiling_le
thf(fact_496_ceiling__le__iff,axiom,
! [X: real,Z2: int] :
( ( ord_less_eq_int @ ( archim7802044766580827645g_real @ X ) @ Z2 )
= ( ord_less_eq_real @ X @ ( ring_1_of_int_real @ Z2 ) ) ) ).
% ceiling_le_iff
thf(fact_497_pinf_I6_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z3 @ X3 )
=> ~ ( ord_less_eq_nat @ X3 @ T ) ) ).
% pinf(6)
thf(fact_498_pinf_I6_J,axiom,
! [T: int] :
? [Z3: int] :
! [X3: int] :
( ( ord_less_int @ Z3 @ X3 )
=> ~ ( ord_less_eq_int @ X3 @ T ) ) ).
% pinf(6)
thf(fact_499_pinf_I6_J,axiom,
! [T: real] :
? [Z3: real] :
! [X3: real] :
( ( ord_less_real @ Z3 @ X3 )
=> ~ ( ord_less_eq_real @ X3 @ T ) ) ).
% pinf(6)
thf(fact_500_pinf_I8_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z3 @ X3 )
=> ( ord_less_eq_nat @ T @ X3 ) ) ).
% pinf(8)
thf(fact_501_pinf_I8_J,axiom,
! [T: int] :
? [Z3: int] :
! [X3: int] :
( ( ord_less_int @ Z3 @ X3 )
=> ( ord_less_eq_int @ T @ X3 ) ) ).
% pinf(8)
thf(fact_502_pinf_I8_J,axiom,
! [T: real] :
? [Z3: real] :
! [X3: real] :
( ( ord_less_real @ Z3 @ X3 )
=> ( ord_less_eq_real @ T @ X3 ) ) ).
% pinf(8)
thf(fact_503_minf_I6_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z3 )
=> ( ord_less_eq_nat @ X3 @ T ) ) ).
% minf(6)
thf(fact_504_minf_I6_J,axiom,
! [T: int] :
? [Z3: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z3 )
=> ( ord_less_eq_int @ X3 @ T ) ) ).
% minf(6)
thf(fact_505_minf_I6_J,axiom,
! [T: real] :
? [Z3: real] :
! [X3: real] :
( ( ord_less_real @ X3 @ Z3 )
=> ( ord_less_eq_real @ X3 @ T ) ) ).
% minf(6)
thf(fact_506_minf_I8_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z3 )
=> ~ ( ord_less_eq_nat @ T @ X3 ) ) ).
% minf(8)
thf(fact_507_minf_I8_J,axiom,
! [T: int] :
? [Z3: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z3 )
=> ~ ( ord_less_eq_int @ T @ X3 ) ) ).
% minf(8)
thf(fact_508_minf_I8_J,axiom,
! [T: real] :
? [Z3: real] :
! [X3: real] :
( ( ord_less_real @ X3 @ Z3 )
=> ~ ( ord_less_eq_real @ T @ X3 ) ) ).
% minf(8)
thf(fact_509_verit__comp__simplify1_I3_J,axiom,
! [B5: nat,A5: nat] :
( ( ~ ( ord_less_eq_nat @ B5 @ A5 ) )
= ( ord_less_nat @ A5 @ B5 ) ) ).
% verit_comp_simplify1(3)
thf(fact_510_verit__comp__simplify1_I3_J,axiom,
! [B5: int,A5: int] :
( ( ~ ( ord_less_eq_int @ B5 @ A5 ) )
= ( ord_less_int @ A5 @ B5 ) ) ).
% verit_comp_simplify1(3)
thf(fact_511_verit__comp__simplify1_I3_J,axiom,
! [B5: real,A5: real] :
( ( ~ ( ord_less_eq_real @ B5 @ A5 ) )
= ( ord_less_real @ A5 @ B5 ) ) ).
% verit_comp_simplify1(3)
thf(fact_512_order__le__imp__less__or__eq,axiom,
! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ( ord_less_nat @ X @ Y2 )
| ( X = Y2 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_513_order__le__imp__less__or__eq,axiom,
! [X: int,Y2: int] :
( ( ord_less_eq_int @ X @ Y2 )
=> ( ( ord_less_int @ X @ Y2 )
| ( X = Y2 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_514_order__le__imp__less__or__eq,axiom,
! [X: real,Y2: real] :
( ( ord_less_eq_real @ X @ Y2 )
=> ( ( ord_less_real @ X @ Y2 )
| ( X = Y2 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_515_linorder__le__less__linear,axiom,
! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
| ( ord_less_nat @ Y2 @ X ) ) ).
% linorder_le_less_linear
thf(fact_516_linorder__le__less__linear,axiom,
! [X: int,Y2: int] :
( ( ord_less_eq_int @ X @ Y2 )
| ( ord_less_int @ Y2 @ X ) ) ).
% linorder_le_less_linear
thf(fact_517_linorder__le__less__linear,axiom,
! [X: real,Y2: real] :
( ( ord_less_eq_real @ X @ Y2 )
| ( ord_less_real @ Y2 @ X ) ) ).
% linorder_le_less_linear
thf(fact_518_order__less__le__subst2,axiom,
! [A2: nat,B2: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ ( F @ B2 ) @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_nat @ X4 @ Y3 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_519_order__less__le__subst2,axiom,
! [A2: int,B2: int,F: int > nat,C: nat] :
( ( ord_less_int @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ ( F @ B2 ) @ C )
=> ( ! [X4: int,Y3: int] :
( ( ord_less_int @ X4 @ Y3 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_520_order__less__le__subst2,axiom,
! [A2: real,B2: real,F: real > nat,C: nat] :
( ( ord_less_real @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ ( F @ B2 ) @ C )
=> ( ! [X4: real,Y3: real] :
( ( ord_less_real @ X4 @ Y3 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_521_order__less__le__subst2,axiom,
! [A2: nat,B2: nat,F: nat > int,C: int] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ord_less_eq_int @ ( F @ B2 ) @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_nat @ X4 @ Y3 )
=> ( ord_less_int @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_522_order__less__le__subst2,axiom,
! [A2: int,B2: int,F: int > int,C: int] :
( ( ord_less_int @ A2 @ B2 )
=> ( ( ord_less_eq_int @ ( F @ B2 ) @ C )
=> ( ! [X4: int,Y3: int] :
( ( ord_less_int @ X4 @ Y3 )
=> ( ord_less_int @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_523_order__less__le__subst2,axiom,
! [A2: real,B2: real,F: real > int,C: int] :
( ( ord_less_real @ A2 @ B2 )
=> ( ( ord_less_eq_int @ ( F @ B2 ) @ C )
=> ( ! [X4: real,Y3: real] :
( ( ord_less_real @ X4 @ Y3 )
=> ( ord_less_int @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_524_order__less__le__subst2,axiom,
! [A2: nat,B2: nat,F: nat > real,C: real] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ord_less_eq_real @ ( F @ B2 ) @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_nat @ X4 @ Y3 )
=> ( ord_less_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_525_order__less__le__subst2,axiom,
! [A2: int,B2: int,F: int > real,C: real] :
( ( ord_less_int @ A2 @ B2 )
=> ( ( ord_less_eq_real @ ( F @ B2 ) @ C )
=> ( ! [X4: int,Y3: int] :
( ( ord_less_int @ X4 @ Y3 )
=> ( ord_less_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_526_order__less__le__subst2,axiom,
! [A2: real,B2: real,F: real > real,C: real] :
( ( ord_less_real @ A2 @ B2 )
=> ( ( ord_less_eq_real @ ( F @ B2 ) @ C )
=> ( ! [X4: real,Y3: real] :
( ( ord_less_real @ X4 @ Y3 )
=> ( ord_less_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_527_order__less__le__subst1,axiom,
! [A2: nat,F: nat > nat,B2: nat,C: nat] :
( ( ord_less_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_528_order__less__le__subst1,axiom,
! [A2: int,F: nat > int,B2: nat,C: nat] :
( ( ord_less_int @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_529_order__less__le__subst1,axiom,
! [A2: real,F: nat > real,B2: nat,C: nat] :
( ( ord_less_real @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_530_order__less__le__subst1,axiom,
! [A2: nat,F: int > nat,B2: int,C: int] :
( ( ord_less_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_int @ B2 @ C )
=> ( ! [X4: int,Y3: int] :
( ( ord_less_eq_int @ X4 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_531_order__less__le__subst1,axiom,
! [A2: int,F: int > int,B2: int,C: int] :
( ( ord_less_int @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_int @ B2 @ C )
=> ( ! [X4: int,Y3: int] :
( ( ord_less_eq_int @ X4 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_532_order__less__le__subst1,axiom,
! [A2: real,F: int > real,B2: int,C: int] :
( ( ord_less_real @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_int @ B2 @ C )
=> ( ! [X4: int,Y3: int] :
( ( ord_less_eq_int @ X4 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_533_order__less__le__subst1,axiom,
! [A2: nat,F: real > nat,B2: real,C: real] :
( ( ord_less_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_real @ B2 @ C )
=> ( ! [X4: real,Y3: real] :
( ( ord_less_eq_real @ X4 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_534_order__less__le__subst1,axiom,
! [A2: int,F: real > int,B2: real,C: real] :
( ( ord_less_int @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_real @ B2 @ C )
=> ( ! [X4: real,Y3: real] :
( ( ord_less_eq_real @ X4 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_535_order__less__le__subst1,axiom,
! [A2: real,F: real > real,B2: real,C: real] :
( ( ord_less_real @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_real @ B2 @ C )
=> ( ! [X4: real,Y3: real] :
( ( ord_less_eq_real @ X4 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_536_order__le__less__subst2,axiom,
! [A2: nat,B2: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_nat @ ( F @ B2 ) @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_537_order__le__less__subst2,axiom,
! [A2: nat,B2: nat,F: nat > int,C: int] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_int @ ( F @ B2 ) @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_538_order__le__less__subst2,axiom,
! [A2: nat,B2: nat,F: nat > real,C: real] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_real @ ( F @ B2 ) @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_539_order__le__less__subst2,axiom,
! [A2: int,B2: int,F: int > nat,C: nat] :
( ( ord_less_eq_int @ A2 @ B2 )
=> ( ( ord_less_nat @ ( F @ B2 ) @ C )
=> ( ! [X4: int,Y3: int] :
( ( ord_less_eq_int @ X4 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_540_order__le__less__subst2,axiom,
! [A2: int,B2: int,F: int > int,C: int] :
( ( ord_less_eq_int @ A2 @ B2 )
=> ( ( ord_less_int @ ( F @ B2 ) @ C )
=> ( ! [X4: int,Y3: int] :
( ( ord_less_eq_int @ X4 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_541_order__le__less__subst2,axiom,
! [A2: int,B2: int,F: int > real,C: real] :
( ( ord_less_eq_int @ A2 @ B2 )
=> ( ( ord_less_real @ ( F @ B2 ) @ C )
=> ( ! [X4: int,Y3: int] :
( ( ord_less_eq_int @ X4 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_542_order__le__less__subst2,axiom,
! [A2: real,B2: real,F: real > nat,C: nat] :
( ( ord_less_eq_real @ A2 @ B2 )
=> ( ( ord_less_nat @ ( F @ B2 ) @ C )
=> ( ! [X4: real,Y3: real] :
( ( ord_less_eq_real @ X4 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_543_order__le__less__subst2,axiom,
! [A2: real,B2: real,F: real > int,C: int] :
( ( ord_less_eq_real @ A2 @ B2 )
=> ( ( ord_less_int @ ( F @ B2 ) @ C )
=> ( ! [X4: real,Y3: real] :
( ( ord_less_eq_real @ X4 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_544_order__le__less__subst2,axiom,
! [A2: real,B2: real,F: real > real,C: real] :
( ( ord_less_eq_real @ A2 @ B2 )
=> ( ( ord_less_real @ ( F @ B2 ) @ C )
=> ( ! [X4: real,Y3: real] :
( ( ord_less_eq_real @ X4 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_545_order__le__less__subst1,axiom,
! [A2: nat,F: nat > nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_nat @ X4 @ Y3 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_546_order__le__less__subst1,axiom,
! [A2: nat,F: int > nat,B2: int,C: int] :
( ( ord_less_eq_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_int @ B2 @ C )
=> ( ! [X4: int,Y3: int] :
( ( ord_less_int @ X4 @ Y3 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_547_order__le__less__subst1,axiom,
! [A2: nat,F: real > nat,B2: real,C: real] :
( ( ord_less_eq_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_real @ B2 @ C )
=> ( ! [X4: real,Y3: real] :
( ( ord_less_real @ X4 @ Y3 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_548_order__le__less__subst1,axiom,
! [A2: int,F: nat > int,B2: nat,C: nat] :
( ( ord_less_eq_int @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_nat @ X4 @ Y3 )
=> ( ord_less_int @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_549_order__le__less__subst1,axiom,
! [A2: int,F: int > int,B2: int,C: int] :
( ( ord_less_eq_int @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_int @ B2 @ C )
=> ( ! [X4: int,Y3: int] :
( ( ord_less_int @ X4 @ Y3 )
=> ( ord_less_int @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_550_order__le__less__subst1,axiom,
! [A2: int,F: real > int,B2: real,C: real] :
( ( ord_less_eq_int @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_real @ B2 @ C )
=> ( ! [X4: real,Y3: real] :
( ( ord_less_real @ X4 @ Y3 )
=> ( ord_less_int @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_551_order__le__less__subst1,axiom,
! [A2: real,F: nat > real,B2: nat,C: nat] :
( ( ord_less_eq_real @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_nat @ X4 @ Y3 )
=> ( ord_less_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_552_order__le__less__subst1,axiom,
! [A2: real,F: int > real,B2: int,C: int] :
( ( ord_less_eq_real @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_int @ B2 @ C )
=> ( ! [X4: int,Y3: int] :
( ( ord_less_int @ X4 @ Y3 )
=> ( ord_less_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_553_order__le__less__subst1,axiom,
! [A2: real,F: real > real,B2: real,C: real] :
( ( ord_less_eq_real @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_real @ B2 @ C )
=> ( ! [X4: real,Y3: real] :
( ( ord_less_real @ X4 @ Y3 )
=> ( ord_less_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_554_order__less__le__trans,axiom,
! [X: nat,Y2: nat,Z2: nat] :
( ( ord_less_nat @ X @ Y2 )
=> ( ( ord_less_eq_nat @ Y2 @ Z2 )
=> ( ord_less_nat @ X @ Z2 ) ) ) ).
% order_less_le_trans
thf(fact_555_order__less__le__trans,axiom,
! [X: int,Y2: int,Z2: int] :
( ( ord_less_int @ X @ Y2 )
=> ( ( ord_less_eq_int @ Y2 @ Z2 )
=> ( ord_less_int @ X @ Z2 ) ) ) ).
% order_less_le_trans
thf(fact_556_order__less__le__trans,axiom,
! [X: real,Y2: real,Z2: real] :
( ( ord_less_real @ X @ Y2 )
=> ( ( ord_less_eq_real @ Y2 @ Z2 )
=> ( ord_less_real @ X @ Z2 ) ) ) ).
% order_less_le_trans
thf(fact_557_order__le__less__trans,axiom,
! [X: nat,Y2: nat,Z2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ( ord_less_nat @ Y2 @ Z2 )
=> ( ord_less_nat @ X @ Z2 ) ) ) ).
% order_le_less_trans
thf(fact_558_order__le__less__trans,axiom,
! [X: int,Y2: int,Z2: int] :
( ( ord_less_eq_int @ X @ Y2 )
=> ( ( ord_less_int @ Y2 @ Z2 )
=> ( ord_less_int @ X @ Z2 ) ) ) ).
% order_le_less_trans
thf(fact_559_order__le__less__trans,axiom,
! [X: real,Y2: real,Z2: real] :
( ( ord_less_eq_real @ X @ Y2 )
=> ( ( ord_less_real @ Y2 @ Z2 )
=> ( ord_less_real @ X @ Z2 ) ) ) ).
% order_le_less_trans
thf(fact_560_order__neq__le__trans,axiom,
! [A2: nat,B2: nat] :
( ( A2 != B2 )
=> ( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ord_less_nat @ A2 @ B2 ) ) ) ).
% order_neq_le_trans
thf(fact_561_order__neq__le__trans,axiom,
! [A2: int,B2: int] :
( ( A2 != B2 )
=> ( ( ord_less_eq_int @ A2 @ B2 )
=> ( ord_less_int @ A2 @ B2 ) ) ) ).
% order_neq_le_trans
thf(fact_562_order__neq__le__trans,axiom,
! [A2: real,B2: real] :
( ( A2 != B2 )
=> ( ( ord_less_eq_real @ A2 @ B2 )
=> ( ord_less_real @ A2 @ B2 ) ) ) ).
% order_neq_le_trans
thf(fact_563_order__le__neq__trans,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( A2 != B2 )
=> ( ord_less_nat @ A2 @ B2 ) ) ) ).
% order_le_neq_trans
thf(fact_564_order__le__neq__trans,axiom,
! [A2: int,B2: int] :
( ( ord_less_eq_int @ A2 @ B2 )
=> ( ( A2 != B2 )
=> ( ord_less_int @ A2 @ B2 ) ) ) ).
% order_le_neq_trans
thf(fact_565_order__le__neq__trans,axiom,
! [A2: real,B2: real] :
( ( ord_less_eq_real @ A2 @ B2 )
=> ( ( A2 != B2 )
=> ( ord_less_real @ A2 @ B2 ) ) ) ).
% order_le_neq_trans
thf(fact_566_order__less__imp__le,axiom,
! [X: nat,Y2: nat] :
( ( ord_less_nat @ X @ Y2 )
=> ( ord_less_eq_nat @ X @ Y2 ) ) ).
% order_less_imp_le
thf(fact_567_order__less__imp__le,axiom,
! [X: int,Y2: int] :
( ( ord_less_int @ X @ Y2 )
=> ( ord_less_eq_int @ X @ Y2 ) ) ).
% order_less_imp_le
thf(fact_568_order__less__imp__le,axiom,
! [X: real,Y2: real] :
( ( ord_less_real @ X @ Y2 )
=> ( ord_less_eq_real @ X @ Y2 ) ) ).
% order_less_imp_le
thf(fact_569_linorder__not__less,axiom,
! [X: nat,Y2: nat] :
( ( ~ ( ord_less_nat @ X @ Y2 ) )
= ( ord_less_eq_nat @ Y2 @ X ) ) ).
% linorder_not_less
thf(fact_570_linorder__not__less,axiom,
! [X: int,Y2: int] :
( ( ~ ( ord_less_int @ X @ Y2 ) )
= ( ord_less_eq_int @ Y2 @ X ) ) ).
% linorder_not_less
thf(fact_571_linorder__not__less,axiom,
! [X: real,Y2: real] :
( ( ~ ( ord_less_real @ X @ Y2 ) )
= ( ord_less_eq_real @ Y2 @ X ) ) ).
% linorder_not_less
thf(fact_572_linorder__not__le,axiom,
! [X: nat,Y2: nat] :
( ( ~ ( ord_less_eq_nat @ X @ Y2 ) )
= ( ord_less_nat @ Y2 @ X ) ) ).
% linorder_not_le
thf(fact_573_linorder__not__le,axiom,
! [X: int,Y2: int] :
( ( ~ ( ord_less_eq_int @ X @ Y2 ) )
= ( ord_less_int @ Y2 @ X ) ) ).
% linorder_not_le
thf(fact_574_linorder__not__le,axiom,
! [X: real,Y2: real] :
( ( ~ ( ord_less_eq_real @ X @ Y2 ) )
= ( ord_less_real @ Y2 @ X ) ) ).
% linorder_not_le
thf(fact_575_order__less__le,axiom,
( ord_less_nat
= ( ^ [X2: nat,Y5: nat] :
( ( ord_less_eq_nat @ X2 @ Y5 )
& ( X2 != Y5 ) ) ) ) ).
% order_less_le
thf(fact_576_order__less__le,axiom,
( ord_less_int
= ( ^ [X2: int,Y5: int] :
( ( ord_less_eq_int @ X2 @ Y5 )
& ( X2 != Y5 ) ) ) ) ).
% order_less_le
thf(fact_577_order__less__le,axiom,
( ord_less_real
= ( ^ [X2: real,Y5: real] :
( ( ord_less_eq_real @ X2 @ Y5 )
& ( X2 != Y5 ) ) ) ) ).
% order_less_le
thf(fact_578_order__le__less,axiom,
( ord_less_eq_nat
= ( ^ [X2: nat,Y5: nat] :
( ( ord_less_nat @ X2 @ Y5 )
| ( X2 = Y5 ) ) ) ) ).
% order_le_less
thf(fact_579_order__le__less,axiom,
( ord_less_eq_int
= ( ^ [X2: int,Y5: int] :
( ( ord_less_int @ X2 @ Y5 )
| ( X2 = Y5 ) ) ) ) ).
% order_le_less
thf(fact_580_order__le__less,axiom,
( ord_less_eq_real
= ( ^ [X2: real,Y5: real] :
( ( ord_less_real @ X2 @ Y5 )
| ( X2 = Y5 ) ) ) ) ).
% order_le_less
thf(fact_581_dual__order_Ostrict__implies__order,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_nat @ B2 @ A2 )
=> ( ord_less_eq_nat @ B2 @ A2 ) ) ).
% dual_order.strict_implies_order
thf(fact_582_dual__order_Ostrict__implies__order,axiom,
! [B2: int,A2: int] :
( ( ord_less_int @ B2 @ A2 )
=> ( ord_less_eq_int @ B2 @ A2 ) ) ).
% dual_order.strict_implies_order
thf(fact_583_dual__order_Ostrict__implies__order,axiom,
! [B2: real,A2: real] :
( ( ord_less_real @ B2 @ A2 )
=> ( ord_less_eq_real @ B2 @ A2 ) ) ).
% dual_order.strict_implies_order
thf(fact_584_order_Ostrict__implies__order,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ord_less_eq_nat @ A2 @ B2 ) ) ).
% order.strict_implies_order
thf(fact_585_order_Ostrict__implies__order,axiom,
! [A2: int,B2: int] :
( ( ord_less_int @ A2 @ B2 )
=> ( ord_less_eq_int @ A2 @ B2 ) ) ).
% order.strict_implies_order
thf(fact_586_order_Ostrict__implies__order,axiom,
! [A2: real,B2: real] :
( ( ord_less_real @ A2 @ B2 )
=> ( ord_less_eq_real @ A2 @ B2 ) ) ).
% order.strict_implies_order
thf(fact_587_dual__order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
& ~ ( ord_less_eq_nat @ A @ B ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_588_dual__order_Ostrict__iff__not,axiom,
( ord_less_int
= ( ^ [B: int,A: int] :
( ( ord_less_eq_int @ B @ A )
& ~ ( ord_less_eq_int @ A @ B ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_589_dual__order_Ostrict__iff__not,axiom,
( ord_less_real
= ( ^ [B: real,A: real] :
( ( ord_less_eq_real @ B @ A )
& ~ ( ord_less_eq_real @ A @ B ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_590_dual__order_Ostrict__trans2,axiom,
! [B2: nat,A2: nat,C: nat] :
( ( ord_less_nat @ B2 @ A2 )
=> ( ( ord_less_eq_nat @ C @ B2 )
=> ( ord_less_nat @ C @ A2 ) ) ) ).
% dual_order.strict_trans2
thf(fact_591_dual__order_Ostrict__trans2,axiom,
! [B2: int,A2: int,C: int] :
( ( ord_less_int @ B2 @ A2 )
=> ( ( ord_less_eq_int @ C @ B2 )
=> ( ord_less_int @ C @ A2 ) ) ) ).
% dual_order.strict_trans2
thf(fact_592_dual__order_Ostrict__trans2,axiom,
! [B2: real,A2: real,C: real] :
( ( ord_less_real @ B2 @ A2 )
=> ( ( ord_less_eq_real @ C @ B2 )
=> ( ord_less_real @ C @ A2 ) ) ) ).
% dual_order.strict_trans2
thf(fact_593_dual__order_Ostrict__trans1,axiom,
! [B2: nat,A2: nat,C: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
=> ( ( ord_less_nat @ C @ B2 )
=> ( ord_less_nat @ C @ A2 ) ) ) ).
% dual_order.strict_trans1
thf(fact_594_dual__order_Ostrict__trans1,axiom,
! [B2: int,A2: int,C: int] :
( ( ord_less_eq_int @ B2 @ A2 )
=> ( ( ord_less_int @ C @ B2 )
=> ( ord_less_int @ C @ A2 ) ) ) ).
% dual_order.strict_trans1
thf(fact_595_dual__order_Ostrict__trans1,axiom,
! [B2: real,A2: real,C: real] :
( ( ord_less_eq_real @ B2 @ A2 )
=> ( ( ord_less_real @ C @ B2 )
=> ( ord_less_real @ C @ A2 ) ) ) ).
% dual_order.strict_trans1
thf(fact_596_dual__order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
& ( A != B ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_597_dual__order_Ostrict__iff__order,axiom,
( ord_less_int
= ( ^ [B: int,A: int] :
( ( ord_less_eq_int @ B @ A )
& ( A != B ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_598_dual__order_Ostrict__iff__order,axiom,
( ord_less_real
= ( ^ [B: real,A: real] :
( ( ord_less_eq_real @ B @ A )
& ( A != B ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_599_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
| ( A = B ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_600_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_int
= ( ^ [B: int,A: int] :
( ( ord_less_int @ B @ A )
| ( A = B ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_601_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_real
= ( ^ [B: real,A: real] :
( ( ord_less_real @ B @ A )
| ( A = B ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_602_dense__le__bounded,axiom,
! [X: real,Y2: real,Z2: real] :
( ( ord_less_real @ X @ Y2 )
=> ( ! [W2: real] :
( ( ord_less_real @ X @ W2 )
=> ( ( ord_less_real @ W2 @ Y2 )
=> ( ord_less_eq_real @ W2 @ Z2 ) ) )
=> ( ord_less_eq_real @ Y2 @ Z2 ) ) ) ).
% dense_le_bounded
thf(fact_603_dense__ge__bounded,axiom,
! [Z2: real,X: real,Y2: real] :
( ( ord_less_real @ Z2 @ X )
=> ( ! [W2: real] :
( ( ord_less_real @ Z2 @ W2 )
=> ( ( ord_less_real @ W2 @ X )
=> ( ord_less_eq_real @ Y2 @ W2 ) ) )
=> ( ord_less_eq_real @ Y2 @ Z2 ) ) ) ).
% dense_ge_bounded
thf(fact_604_order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
& ~ ( ord_less_eq_nat @ B @ A ) ) ) ) ).
% order.strict_iff_not
thf(fact_605_order_Ostrict__iff__not,axiom,
( ord_less_int
= ( ^ [A: int,B: int] :
( ( ord_less_eq_int @ A @ B )
& ~ ( ord_less_eq_int @ B @ A ) ) ) ) ).
% order.strict_iff_not
thf(fact_606_order_Ostrict__iff__not,axiom,
( ord_less_real
= ( ^ [A: real,B: real] :
( ( ord_less_eq_real @ A @ B )
& ~ ( ord_less_eq_real @ B @ A ) ) ) ) ).
% order.strict_iff_not
thf(fact_607_order_Ostrict__trans2,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ord_less_nat @ A2 @ C ) ) ) ).
% order.strict_trans2
thf(fact_608_order_Ostrict__trans2,axiom,
! [A2: int,B2: int,C: int] :
( ( ord_less_int @ A2 @ B2 )
=> ( ( ord_less_eq_int @ B2 @ C )
=> ( ord_less_int @ A2 @ C ) ) ) ).
% order.strict_trans2
thf(fact_609_order_Ostrict__trans2,axiom,
! [A2: real,B2: real,C: real] :
( ( ord_less_real @ A2 @ B2 )
=> ( ( ord_less_eq_real @ B2 @ C )
=> ( ord_less_real @ A2 @ C ) ) ) ).
% order.strict_trans2
thf(fact_610_order_Ostrict__trans1,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ord_less_nat @ A2 @ C ) ) ) ).
% order.strict_trans1
thf(fact_611_order_Ostrict__trans1,axiom,
! [A2: int,B2: int,C: int] :
( ( ord_less_eq_int @ A2 @ B2 )
=> ( ( ord_less_int @ B2 @ C )
=> ( ord_less_int @ A2 @ C ) ) ) ).
% order.strict_trans1
thf(fact_612_order_Ostrict__trans1,axiom,
! [A2: real,B2: real,C: real] :
( ( ord_less_eq_real @ A2 @ B2 )
=> ( ( ord_less_real @ B2 @ C )
=> ( ord_less_real @ A2 @ C ) ) ) ).
% order.strict_trans1
thf(fact_613_order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
& ( A != B ) ) ) ) ).
% order.strict_iff_order
thf(fact_614_order_Ostrict__iff__order,axiom,
( ord_less_int
= ( ^ [A: int,B: int] :
( ( ord_less_eq_int @ A @ B )
& ( A != B ) ) ) ) ).
% order.strict_iff_order
thf(fact_615_order_Ostrict__iff__order,axiom,
( ord_less_real
= ( ^ [A: real,B: real] :
( ( ord_less_eq_real @ A @ B )
& ( A != B ) ) ) ) ).
% order.strict_iff_order
thf(fact_616_order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
| ( A = B ) ) ) ) ).
% order.order_iff_strict
thf(fact_617_order_Oorder__iff__strict,axiom,
( ord_less_eq_int
= ( ^ [A: int,B: int] :
( ( ord_less_int @ A @ B )
| ( A = B ) ) ) ) ).
% order.order_iff_strict
thf(fact_618_order_Oorder__iff__strict,axiom,
( ord_less_eq_real
= ( ^ [A: real,B: real] :
( ( ord_less_real @ A @ B )
| ( A = B ) ) ) ) ).
% order.order_iff_strict
thf(fact_619_not__le__imp__less,axiom,
! [Y2: nat,X: nat] :
( ~ ( ord_less_eq_nat @ Y2 @ X )
=> ( ord_less_nat @ X @ Y2 ) ) ).
% not_le_imp_less
thf(fact_620_not__le__imp__less,axiom,
! [Y2: int,X: int] :
( ~ ( ord_less_eq_int @ Y2 @ X )
=> ( ord_less_int @ X @ Y2 ) ) ).
% not_le_imp_less
thf(fact_621_not__le__imp__less,axiom,
! [Y2: real,X: real] :
( ~ ( ord_less_eq_real @ Y2 @ X )
=> ( ord_less_real @ X @ Y2 ) ) ).
% not_le_imp_less
thf(fact_622_less__le__not__le,axiom,
( ord_less_nat
= ( ^ [X2: nat,Y5: nat] :
( ( ord_less_eq_nat @ X2 @ Y5 )
& ~ ( ord_less_eq_nat @ Y5 @ X2 ) ) ) ) ).
% less_le_not_le
thf(fact_623_less__le__not__le,axiom,
( ord_less_int
= ( ^ [X2: int,Y5: int] :
( ( ord_less_eq_int @ X2 @ Y5 )
& ~ ( ord_less_eq_int @ Y5 @ X2 ) ) ) ) ).
% less_le_not_le
thf(fact_624_less__le__not__le,axiom,
( ord_less_real
= ( ^ [X2: real,Y5: real] :
( ( ord_less_eq_real @ X2 @ Y5 )
& ~ ( ord_less_eq_real @ Y5 @ X2 ) ) ) ) ).
% less_le_not_le
thf(fact_625_dense__le,axiom,
! [Y2: real,Z2: real] :
( ! [X4: real] :
( ( ord_less_real @ X4 @ Y2 )
=> ( ord_less_eq_real @ X4 @ Z2 ) )
=> ( ord_less_eq_real @ Y2 @ Z2 ) ) ).
% dense_le
thf(fact_626_dense__ge,axiom,
! [Z2: real,Y2: real] :
( ! [X4: real] :
( ( ord_less_real @ Z2 @ X4 )
=> ( ord_less_eq_real @ Y2 @ X4 ) )
=> ( ord_less_eq_real @ Y2 @ Z2 ) ) ).
% dense_ge
thf(fact_627_antisym__conv2,axiom,
! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ( ~ ( ord_less_nat @ X @ Y2 ) )
= ( X = Y2 ) ) ) ).
% antisym_conv2
thf(fact_628_antisym__conv2,axiom,
! [X: int,Y2: int] :
( ( ord_less_eq_int @ X @ Y2 )
=> ( ( ~ ( ord_less_int @ X @ Y2 ) )
= ( X = Y2 ) ) ) ).
% antisym_conv2
thf(fact_629_antisym__conv2,axiom,
! [X: real,Y2: real] :
( ( ord_less_eq_real @ X @ Y2 )
=> ( ( ~ ( ord_less_real @ X @ Y2 ) )
= ( X = Y2 ) ) ) ).
% antisym_conv2
thf(fact_630_antisym__conv1,axiom,
! [X: nat,Y2: nat] :
( ~ ( ord_less_nat @ X @ Y2 )
=> ( ( ord_less_eq_nat @ X @ Y2 )
= ( X = Y2 ) ) ) ).
% antisym_conv1
thf(fact_631_antisym__conv1,axiom,
! [X: int,Y2: int] :
( ~ ( ord_less_int @ X @ Y2 )
=> ( ( ord_less_eq_int @ X @ Y2 )
= ( X = Y2 ) ) ) ).
% antisym_conv1
thf(fact_632_antisym__conv1,axiom,
! [X: real,Y2: real] :
( ~ ( ord_less_real @ X @ Y2 )
=> ( ( ord_less_eq_real @ X @ Y2 )
= ( X = Y2 ) ) ) ).
% antisym_conv1
thf(fact_633_nless__le,axiom,
! [A2: nat,B2: nat] :
( ( ~ ( ord_less_nat @ A2 @ B2 ) )
= ( ~ ( ord_less_eq_nat @ A2 @ B2 )
| ( A2 = B2 ) ) ) ).
% nless_le
thf(fact_634_nless__le,axiom,
! [A2: int,B2: int] :
( ( ~ ( ord_less_int @ A2 @ B2 ) )
= ( ~ ( ord_less_eq_int @ A2 @ B2 )
| ( A2 = B2 ) ) ) ).
% nless_le
thf(fact_635_nless__le,axiom,
! [A2: real,B2: real] :
( ( ~ ( ord_less_real @ A2 @ B2 ) )
= ( ~ ( ord_less_eq_real @ A2 @ B2 )
| ( A2 = B2 ) ) ) ).
% nless_le
thf(fact_636_leI,axiom,
! [X: nat,Y2: nat] :
( ~ ( ord_less_nat @ X @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ X ) ) ).
% leI
thf(fact_637_leI,axiom,
! [X: int,Y2: int] :
( ~ ( ord_less_int @ X @ Y2 )
=> ( ord_less_eq_int @ Y2 @ X ) ) ).
% leI
thf(fact_638_leI,axiom,
! [X: real,Y2: real] :
( ~ ( ord_less_real @ X @ Y2 )
=> ( ord_less_eq_real @ Y2 @ X ) ) ).
% leI
thf(fact_639_leD,axiom,
! [Y2: nat,X: nat] :
( ( ord_less_eq_nat @ Y2 @ X )
=> ~ ( ord_less_nat @ X @ Y2 ) ) ).
% leD
thf(fact_640_leD,axiom,
! [Y2: int,X: int] :
( ( ord_less_eq_int @ Y2 @ X )
=> ~ ( ord_less_int @ X @ Y2 ) ) ).
% leD
thf(fact_641_leD,axiom,
! [Y2: real,X: real] :
( ( ord_less_eq_real @ Y2 @ X )
=> ~ ( ord_less_real @ X @ Y2 ) ) ).
% leD
thf(fact_642_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_643_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_644_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I: real,J: real,K: real,L: real] :
( ( ( ord_less_eq_real @ I @ J )
& ( K = L ) )
=> ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_645_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_646_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_647_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I: real,J: real,K: real,L: real] :
( ( ( I = J )
& ( ord_less_eq_real @ K @ L ) )
=> ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_648_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_649_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_650_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I: real,J: real,K: real,L: real] :
( ( ( ord_less_eq_real @ I @ J )
& ( ord_less_eq_real @ K @ L ) )
=> ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_651_add__mono,axiom,
! [A2: nat,B2: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B2 @ D ) ) ) ) ).
% add_mono
thf(fact_652_add__mono,axiom,
! [A2: int,B2: int,C: int,D: int] :
( ( ord_less_eq_int @ A2 @ B2 )
=> ( ( ord_less_eq_int @ C @ D )
=> ( ord_less_eq_int @ ( plus_plus_int @ A2 @ C ) @ ( plus_plus_int @ B2 @ D ) ) ) ) ).
% add_mono
thf(fact_653_add__mono,axiom,
! [A2: real,B2: real,C: real,D: real] :
( ( ord_less_eq_real @ A2 @ B2 )
=> ( ( ord_less_eq_real @ C @ D )
=> ( ord_less_eq_real @ ( plus_plus_real @ A2 @ C ) @ ( plus_plus_real @ B2 @ D ) ) ) ) ).
% add_mono
thf(fact_654_add__left__mono,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B2 ) ) ) ).
% add_left_mono
thf(fact_655_add__left__mono,axiom,
! [A2: int,B2: int,C: int] :
( ( ord_less_eq_int @ A2 @ B2 )
=> ( ord_less_eq_int @ ( plus_plus_int @ C @ A2 ) @ ( plus_plus_int @ C @ B2 ) ) ) ).
% add_left_mono
thf(fact_656_add__left__mono,axiom,
! [A2: real,B2: real,C: real] :
( ( ord_less_eq_real @ A2 @ B2 )
=> ( ord_less_eq_real @ ( plus_plus_real @ C @ A2 ) @ ( plus_plus_real @ C @ B2 ) ) ) ).
% add_left_mono
thf(fact_657_less__eqE,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ~ ! [C2: nat] :
( B2
!= ( plus_plus_nat @ A2 @ C2 ) ) ) ).
% less_eqE
thf(fact_658_add__right__mono,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B2 @ C ) ) ) ).
% add_right_mono
thf(fact_659_add__right__mono,axiom,
! [A2: int,B2: int,C: int] :
( ( ord_less_eq_int @ A2 @ B2 )
=> ( ord_less_eq_int @ ( plus_plus_int @ A2 @ C ) @ ( plus_plus_int @ B2 @ C ) ) ) ).
% add_right_mono
thf(fact_660_add__right__mono,axiom,
! [A2: real,B2: real,C: real] :
( ( ord_less_eq_real @ A2 @ B2 )
=> ( ord_less_eq_real @ ( plus_plus_real @ A2 @ C ) @ ( plus_plus_real @ B2 @ C ) ) ) ).
% add_right_mono
thf(fact_661_le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [A: nat,B: nat] :
? [C3: nat] :
( B
= ( plus_plus_nat @ A @ C3 ) ) ) ) ).
% le_iff_add
thf(fact_662_add__le__imp__le__left,axiom,
! [C: nat,A2: nat,B2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B2 ) )
=> ( ord_less_eq_nat @ A2 @ B2 ) ) ).
% add_le_imp_le_left
thf(fact_663_add__le__imp__le__left,axiom,
! [C: int,A2: int,B2: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ C @ A2 ) @ ( plus_plus_int @ C @ B2 ) )
=> ( ord_less_eq_int @ A2 @ B2 ) ) ).
% add_le_imp_le_left
thf(fact_664_add__le__imp__le__left,axiom,
! [C: real,A2: real,B2: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ C @ A2 ) @ ( plus_plus_real @ C @ B2 ) )
=> ( ord_less_eq_real @ A2 @ B2 ) ) ).
% add_le_imp_le_left
thf(fact_665_add__le__imp__le__right,axiom,
! [A2: nat,C: nat,B2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B2 @ C ) )
=> ( ord_less_eq_nat @ A2 @ B2 ) ) ).
% add_le_imp_le_right
thf(fact_666_add__le__imp__le__right,axiom,
! [A2: int,C: int,B2: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A2 @ C ) @ ( plus_plus_int @ B2 @ C ) )
=> ( ord_less_eq_int @ A2 @ B2 ) ) ).
% add_le_imp_le_right
thf(fact_667_add__le__imp__le__right,axiom,
! [A2: real,C: real,B2: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ A2 @ C ) @ ( plus_plus_real @ B2 @ C ) )
=> ( ord_less_eq_real @ A2 @ B2 ) ) ).
% add_le_imp_le_right
thf(fact_668_le__numeral__extra_I4_J,axiom,
ord_less_eq_nat @ one_one_nat @ one_one_nat ).
% le_numeral_extra(4)
thf(fact_669_le__numeral__extra_I4_J,axiom,
ord_less_eq_int @ one_one_int @ one_one_int ).
% le_numeral_extra(4)
thf(fact_670_le__numeral__extra_I4_J,axiom,
ord_less_eq_real @ one_one_real @ one_one_real ).
% le_numeral_extra(4)
thf(fact_671_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_672_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_673_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_674_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_675_not__add__less2,axiom,
! [J: nat,I: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ J @ I ) @ I ) ).
% not_add_less2
thf(fact_676_not__add__less1,axiom,
! [I: nat,J: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ I ) ).
% not_add_less1
thf(fact_677_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_678_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_679_real__arch__simple,axiom,
! [X: real] :
? [N2: nat] : ( ord_less_eq_real @ X @ ( semiri5074537144036343181t_real @ N2 ) ) ).
% real_arch_simple
thf(fact_680_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I: nat,J: nat] :
( ! [I2: nat,J2: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ord_less_nat @ ( F @ I2 ) @ ( F @ J2 ) ) )
=> ( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_681_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_682_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_683_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M3: nat,N3: nat] :
( ( ord_less_nat @ M3 @ N3 )
| ( M3 = N3 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_684_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_685_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M3: nat,N3: nat] :
( ( ord_less_eq_nat @ M3 @ N3 )
& ( M3 != N3 ) ) ) ) ).
% nat_less_le
thf(fact_686_ex__le__of__int,axiom,
! [X: real] :
? [Z3: int] : ( ord_less_eq_real @ X @ ( ring_1_of_int_real @ Z3 ) ) ).
% ex_le_of_int
thf(fact_687_int_Olless__eq,axiom,
( ord_less_int
= ( ^ [X2: int,Y5: int] :
( ( ord_less_eq_int @ X2 @ Y5 )
& ( X2 != Y5 ) ) ) ) ).
% int.lless_eq
thf(fact_688_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_689_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_690_add__mono__thms__linordered__field_I4_J,axiom,
! [I: real,J: real,K: real,L: real] :
( ( ( ord_less_eq_real @ I @ J )
& ( ord_less_real @ K @ L ) )
=> ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_691_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_692_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_693_add__mono__thms__linordered__field_I3_J,axiom,
! [I: real,J: real,K: real,L: real] :
( ( ( ord_less_real @ I @ J )
& ( ord_less_eq_real @ K @ L ) )
=> ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_694_add__le__less__mono,axiom,
! [A2: nat,B2: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_nat @ C @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B2 @ D ) ) ) ) ).
% add_le_less_mono
thf(fact_695_add__le__less__mono,axiom,
! [A2: int,B2: int,C: int,D: int] :
( ( ord_less_eq_int @ A2 @ B2 )
=> ( ( ord_less_int @ C @ D )
=> ( ord_less_int @ ( plus_plus_int @ A2 @ C ) @ ( plus_plus_int @ B2 @ D ) ) ) ) ).
% add_le_less_mono
thf(fact_696_add__le__less__mono,axiom,
! [A2: real,B2: real,C: real,D: real] :
( ( ord_less_eq_real @ A2 @ B2 )
=> ( ( ord_less_real @ C @ D )
=> ( ord_less_real @ ( plus_plus_real @ A2 @ C ) @ ( plus_plus_real @ B2 @ D ) ) ) ) ).
% add_le_less_mono
thf(fact_697_add__less__le__mono,axiom,
! [A2: nat,B2: nat,C: nat,D: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B2 @ D ) ) ) ) ).
% add_less_le_mono
thf(fact_698_add__less__le__mono,axiom,
! [A2: int,B2: int,C: int,D: int] :
( ( ord_less_int @ A2 @ B2 )
=> ( ( ord_less_eq_int @ C @ D )
=> ( ord_less_int @ ( plus_plus_int @ A2 @ C ) @ ( plus_plus_int @ B2 @ D ) ) ) ) ).
% add_less_le_mono
thf(fact_699_add__less__le__mono,axiom,
! [A2: real,B2: real,C: real,D: real] :
( ( ord_less_real @ A2 @ B2 )
=> ( ( ord_less_eq_real @ C @ D )
=> ( ord_less_real @ ( plus_plus_real @ A2 @ C ) @ ( plus_plus_real @ B2 @ D ) ) ) ) ).
% add_less_le_mono
thf(fact_700_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_701_zadd__int__left,axiom,
! [M: nat,N: nat,Z2: int] :
( ( plus_plus_int @ ( semiri1314217659103216013at_int @ M ) @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ Z2 ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ ( plus_plus_nat @ M @ N ) ) @ Z2 ) ) ).
% zadd_int_left
thf(fact_702_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_703_int__ops_I5_J,axiom,
! [A2: nat,B2: nat] :
( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ A2 @ B2 ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ A2 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ).
% int_ops(5)
thf(fact_704_zle__iff__zadd,axiom,
( ord_less_eq_int
= ( ^ [W3: int,Z5: int] :
? [N3: nat] :
( Z5
= ( plus_plus_int @ W3 @ ( semiri1314217659103216013at_int @ N3 ) ) ) ) ) ).
% zle_iff_zadd
thf(fact_705_of__int__floor__le,axiom,
! [X: real] : ( ord_less_eq_real @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ X ) ) @ X ) ).
% of_int_floor_le
thf(fact_706_le__of__int__ceiling,axiom,
! [X: real] : ( ord_less_eq_real @ X @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ X ) ) ) ).
% le_of_int_ceiling
thf(fact_707_floor__le__ceiling,axiom,
! [X: real] : ( ord_less_eq_int @ ( archim6058952711729229775r_real @ X ) @ ( archim7802044766580827645g_real @ X ) ) ).
% floor_le_ceiling
thf(fact_708_frac__lt__1,axiom,
! [X: real] : ( ord_less_real @ ( archim2898591450579166408c_real @ X ) @ one_one_real ) ).
% frac_lt_1
thf(fact_709_frac__1__eq,axiom,
! [X: real] :
( ( archim2898591450579166408c_real @ ( plus_plus_real @ X @ one_one_real ) )
= ( archim2898591450579166408c_real @ X ) ) ).
% frac_1_eq
thf(fact_710_floor__le__round,axiom,
! [X: real] : ( ord_less_eq_int @ ( archim6058952711729229775r_real @ X ) @ ( archim8280529875227126926d_real @ X ) ) ).
% floor_le_round
thf(fact_711_ceiling__ge__round,axiom,
! [X: real] : ( ord_less_eq_int @ ( archim8280529875227126926d_real @ X ) @ ( archim7802044766580827645g_real @ X ) ) ).
% ceiling_ge_round
thf(fact_712_linorder__neqE__linordered__idom,axiom,
! [X: int,Y2: int] :
( ( X != Y2 )
=> ( ~ ( ord_less_int @ X @ Y2 )
=> ( ord_less_int @ Y2 @ X ) ) ) ).
% linorder_neqE_linordered_idom
thf(fact_713_linorder__neqE__linordered__idom,axiom,
! [X: real,Y2: real] :
( ( X != Y2 )
=> ( ~ ( ord_less_real @ X @ Y2 )
=> ( ord_less_real @ Y2 @ X ) ) ) ).
% linorder_neqE_linordered_idom
thf(fact_714_add1__zle__eq,axiom,
! [W: int,Z2: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ W @ one_one_int ) @ Z2 )
= ( ord_less_int @ W @ Z2 ) ) ).
% add1_zle_eq
thf(fact_715_zless__imp__add1__zle,axiom,
! [W: int,Z2: int] :
( ( ord_less_int @ W @ Z2 )
=> ( ord_less_eq_int @ ( plus_plus_int @ W @ one_one_int ) @ Z2 ) ) ).
% zless_imp_add1_zle
thf(fact_716_le__floor__add,axiom,
! [X: real,Y2: real] : ( ord_less_eq_int @ ( plus_plus_int @ ( archim6058952711729229775r_real @ X ) @ ( archim6058952711729229775r_real @ Y2 ) ) @ ( archim6058952711729229775r_real @ ( plus_plus_real @ X @ Y2 ) ) ) ).
% le_floor_add
thf(fact_717_ceiling__add__le,axiom,
! [X: real,Y2: real] : ( ord_less_eq_int @ ( archim7802044766580827645g_real @ ( plus_plus_real @ X @ Y2 ) ) @ ( plus_plus_int @ ( archim7802044766580827645g_real @ X ) @ ( archim7802044766580827645g_real @ Y2 ) ) ) ).
% ceiling_add_le
thf(fact_718_floor__exists1,axiom,
! [X: real] :
? [X4: int] :
( ( ord_less_eq_real @ ( ring_1_of_int_real @ X4 ) @ X )
& ( ord_less_real @ X @ ( ring_1_of_int_real @ ( plus_plus_int @ X4 @ one_one_int ) ) )
& ! [Y4: int] :
( ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Y4 ) @ X )
& ( ord_less_real @ X @ ( ring_1_of_int_real @ ( plus_plus_int @ Y4 @ one_one_int ) ) ) )
=> ( Y4 = X4 ) ) ) ).
% floor_exists1
thf(fact_719_floor__exists,axiom,
! [X: real] :
? [Z3: int] :
( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z3 ) @ X )
& ( ord_less_real @ X @ ( ring_1_of_int_real @ ( plus_plus_int @ Z3 @ one_one_int ) ) ) ) ).
% floor_exists
thf(fact_720_of__int__ceiling__le__add__one,axiom,
! [R2: real] : ( ord_less_eq_real @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ R2 ) ) @ ( plus_plus_real @ R2 @ one_one_real ) ) ).
% of_int_ceiling_le_add_one
thf(fact_721_floor__unique,axiom,
! [Z2: int,X: real] :
( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z2 ) @ X )
=> ( ( ord_less_real @ X @ ( plus_plus_real @ ( ring_1_of_int_real @ Z2 ) @ one_one_real ) )
=> ( ( archim6058952711729229775r_real @ X )
= Z2 ) ) ) ).
% floor_unique
thf(fact_722_floor__eq__iff,axiom,
! [X: real,A2: int] :
( ( ( archim6058952711729229775r_real @ X )
= A2 )
= ( ( ord_less_eq_real @ ( ring_1_of_int_real @ A2 ) @ X )
& ( ord_less_real @ X @ ( plus_plus_real @ ( ring_1_of_int_real @ A2 ) @ one_one_real ) ) ) ) ).
% floor_eq_iff
thf(fact_723_complete__interval,axiom,
! [A2: nat,B2: nat,P: nat > $o] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( P @ A2 )
=> ( ~ ( P @ B2 )
=> ? [C2: nat] :
( ( ord_less_eq_nat @ A2 @ C2 )
& ( ord_less_eq_nat @ C2 @ B2 )
& ! [X3: nat] :
( ( ( ord_less_eq_nat @ A2 @ X3 )
& ( ord_less_nat @ X3 @ C2 ) )
=> ( P @ X3 ) )
& ! [D2: nat] :
( ! [X4: nat] :
( ( ( ord_less_eq_nat @ A2 @ X4 )
& ( ord_less_nat @ X4 @ D2 ) )
=> ( P @ X4 ) )
=> ( ord_less_eq_nat @ D2 @ C2 ) ) ) ) ) ) ).
% complete_interval
thf(fact_724_complete__interval,axiom,
! [A2: int,B2: int,P: int > $o] :
( ( ord_less_int @ A2 @ B2 )
=> ( ( P @ A2 )
=> ( ~ ( P @ B2 )
=> ? [C2: int] :
( ( ord_less_eq_int @ A2 @ C2 )
& ( ord_less_eq_int @ C2 @ B2 )
& ! [X3: int] :
( ( ( ord_less_eq_int @ A2 @ X3 )
& ( ord_less_int @ X3 @ C2 ) )
=> ( P @ X3 ) )
& ! [D2: int] :
( ! [X4: int] :
( ( ( ord_less_eq_int @ A2 @ X4 )
& ( ord_less_int @ X4 @ D2 ) )
=> ( P @ X4 ) )
=> ( ord_less_eq_int @ D2 @ C2 ) ) ) ) ) ) ).
% complete_interval
thf(fact_725_complete__interval,axiom,
! [A2: real,B2: real,P: real > $o] :
( ( ord_less_real @ A2 @ B2 )
=> ( ( P @ A2 )
=> ( ~ ( P @ B2 )
=> ? [C2: real] :
( ( ord_less_eq_real @ A2 @ C2 )
& ( ord_less_eq_real @ C2 @ B2 )
& ! [X3: real] :
( ( ( ord_less_eq_real @ A2 @ X3 )
& ( ord_less_real @ X3 @ C2 ) )
=> ( P @ X3 ) )
& ! [D2: real] :
( ! [X4: real] :
( ( ( ord_less_eq_real @ A2 @ X4 )
& ( ord_less_real @ X4 @ D2 ) )
=> ( P @ X4 ) )
=> ( ord_less_eq_real @ D2 @ C2 ) ) ) ) ) ) ).
% complete_interval
thf(fact_726_zero__less__floor,axiom,
! [X: real] :
( ( ord_less_int @ zero_zero_int @ ( archim6058952711729229775r_real @ X ) )
= ( ord_less_eq_real @ one_one_real @ X ) ) ).
% zero_less_floor
thf(fact_727_ceiling__less__one,axiom,
! [X: real] :
( ( ord_less_int @ ( archim7802044766580827645g_real @ X ) @ one_one_int )
= ( ord_less_eq_real @ X @ zero_zero_real ) ) ).
% ceiling_less_one
thf(fact_728_floor__le__zero,axiom,
! [X: real] :
( ( ord_less_eq_int @ ( archim6058952711729229775r_real @ X ) @ zero_zero_int )
= ( ord_less_real @ X @ one_one_real ) ) ).
% floor_le_zero
thf(fact_729_one__le__ceiling,axiom,
! [X: real] :
( ( ord_less_eq_int @ one_one_int @ ( archim7802044766580827645g_real @ X ) )
= ( ord_less_real @ zero_zero_real @ X ) ) ).
% one_le_ceiling
thf(fact_730_bot__nat__0_Onot__eq__extremum,axiom,
! [A2: nat] :
( ( A2 != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ A2 ) ) ).
% bot_nat_0.not_eq_extremum
thf(fact_731_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% neq0_conv
thf(fact_732_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_733_bot__nat__0_Oextremum,axiom,
! [A2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A2 ) ).
% bot_nat_0.extremum
thf(fact_734_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_735_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_736_Nat_Oadd__0__right,axiom,
! [M: nat] :
( ( plus_plus_nat @ M @ zero_zero_nat )
= M ) ).
% Nat.add_0_right
thf(fact_737_le__zero__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_738_not__gr__zero,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_739_add__0,axiom,
! [A2: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A2 )
= A2 ) ).
% add_0
thf(fact_740_add__0,axiom,
! [A2: int] :
( ( plus_plus_int @ zero_zero_int @ A2 )
= A2 ) ).
% add_0
thf(fact_741_add__0,axiom,
! [A2: real] :
( ( plus_plus_real @ zero_zero_real @ A2 )
= A2 ) ).
% add_0
thf(fact_742_zero__eq__add__iff__both__eq__0,axiom,
! [X: nat,Y2: nat] :
( ( zero_zero_nat
= ( plus_plus_nat @ X @ Y2 ) )
= ( ( X = zero_zero_nat )
& ( Y2 = zero_zero_nat ) ) ) ).
% zero_eq_add_iff_both_eq_0
thf(fact_743_add__eq__0__iff__both__eq__0,axiom,
! [X: nat,Y2: nat] :
( ( ( plus_plus_nat @ X @ Y2 )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y2 = zero_zero_nat ) ) ) ).
% add_eq_0_iff_both_eq_0
thf(fact_744_add__cancel__right__right,axiom,
! [A2: nat,B2: nat] :
( ( A2
= ( plus_plus_nat @ A2 @ B2 ) )
= ( B2 = zero_zero_nat ) ) ).
% add_cancel_right_right
thf(fact_745_add__cancel__right__right,axiom,
! [A2: int,B2: int] :
( ( A2
= ( plus_plus_int @ A2 @ B2 ) )
= ( B2 = zero_zero_int ) ) ).
% add_cancel_right_right
thf(fact_746_add__cancel__right__right,axiom,
! [A2: real,B2: real] :
( ( A2
= ( plus_plus_real @ A2 @ B2 ) )
= ( B2 = zero_zero_real ) ) ).
% add_cancel_right_right
thf(fact_747_add__cancel__right__left,axiom,
! [A2: nat,B2: nat] :
( ( A2
= ( plus_plus_nat @ B2 @ A2 ) )
= ( B2 = zero_zero_nat ) ) ).
% add_cancel_right_left
thf(fact_748_add__cancel__right__left,axiom,
! [A2: int,B2: int] :
( ( A2
= ( plus_plus_int @ B2 @ A2 ) )
= ( B2 = zero_zero_int ) ) ).
% add_cancel_right_left
thf(fact_749_add__cancel__right__left,axiom,
! [A2: real,B2: real] :
( ( A2
= ( plus_plus_real @ B2 @ A2 ) )
= ( B2 = zero_zero_real ) ) ).
% add_cancel_right_left
thf(fact_750_add__cancel__left__right,axiom,
! [A2: nat,B2: nat] :
( ( ( plus_plus_nat @ A2 @ B2 )
= A2 )
= ( B2 = zero_zero_nat ) ) ).
% add_cancel_left_right
thf(fact_751_add__cancel__left__right,axiom,
! [A2: int,B2: int] :
( ( ( plus_plus_int @ A2 @ B2 )
= A2 )
= ( B2 = zero_zero_int ) ) ).
% add_cancel_left_right
thf(fact_752_add__cancel__left__right,axiom,
! [A2: real,B2: real] :
( ( ( plus_plus_real @ A2 @ B2 )
= A2 )
= ( B2 = zero_zero_real ) ) ).
% add_cancel_left_right
thf(fact_753_add__cancel__left__left,axiom,
! [B2: nat,A2: nat] :
( ( ( plus_plus_nat @ B2 @ A2 )
= A2 )
= ( B2 = zero_zero_nat ) ) ).
% add_cancel_left_left
thf(fact_754_add__cancel__left__left,axiom,
! [B2: int,A2: int] :
( ( ( plus_plus_int @ B2 @ A2 )
= A2 )
= ( B2 = zero_zero_int ) ) ).
% add_cancel_left_left
thf(fact_755_add__cancel__left__left,axiom,
! [B2: real,A2: real] :
( ( ( plus_plus_real @ B2 @ A2 )
= A2 )
= ( B2 = zero_zero_real ) ) ).
% add_cancel_left_left
thf(fact_756_double__zero__sym,axiom,
! [A2: int] :
( ( zero_zero_int
= ( plus_plus_int @ A2 @ A2 ) )
= ( A2 = zero_zero_int ) ) ).
% double_zero_sym
thf(fact_757_double__zero__sym,axiom,
! [A2: real] :
( ( zero_zero_real
= ( plus_plus_real @ A2 @ A2 ) )
= ( A2 = zero_zero_real ) ) ).
% double_zero_sym
thf(fact_758_add_Oright__neutral,axiom,
! [A2: nat] :
( ( plus_plus_nat @ A2 @ zero_zero_nat )
= A2 ) ).
% add.right_neutral
thf(fact_759_add_Oright__neutral,axiom,
! [A2: int] :
( ( plus_plus_int @ A2 @ zero_zero_int )
= A2 ) ).
% add.right_neutral
thf(fact_760_add_Oright__neutral,axiom,
! [A2: real] :
( ( plus_plus_real @ A2 @ zero_zero_real )
= A2 ) ).
% add.right_neutral
thf(fact_761_double__eq__0__iff,axiom,
! [A2: int] :
( ( ( plus_plus_int @ A2 @ A2 )
= zero_zero_int )
= ( A2 = zero_zero_int ) ) ).
% double_eq_0_iff
thf(fact_762_double__eq__0__iff,axiom,
! [A2: real] :
( ( ( plus_plus_real @ A2 @ A2 )
= zero_zero_real )
= ( A2 = zero_zero_real ) ) ).
% double_eq_0_iff
thf(fact_763_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_764_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_765_of__nat__eq__0__iff,axiom,
! [M: nat] :
( ( ( semiri5074537144036343181t_real @ M )
= zero_zero_real )
= ( M = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
thf(fact_766_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_767_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_768_of__nat__0__eq__iff,axiom,
! [N: nat] :
( ( zero_zero_real
= ( semiri5074537144036343181t_real @ N ) )
= ( zero_zero_nat = N ) ) ).
% of_nat_0_eq_iff
thf(fact_769_of__nat__0,axiom,
( ( semiri1316708129612266289at_nat @ zero_zero_nat )
= zero_zero_nat ) ).
% of_nat_0
thf(fact_770_of__nat__0,axiom,
( ( semiri1314217659103216013at_int @ zero_zero_nat )
= zero_zero_int ) ).
% of_nat_0
thf(fact_771_of__nat__0,axiom,
( ( semiri5074537144036343181t_real @ zero_zero_nat )
= zero_zero_real ) ).
% of_nat_0
thf(fact_772_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_773_less__one,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ one_one_nat )
= ( N = zero_zero_nat ) ) ).
% less_one
thf(fact_774_zero__le__double__add__iff__zero__le__single__add,axiom,
! [A2: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A2 @ A2 ) )
= ( ord_less_eq_int @ zero_zero_int @ A2 ) ) ).
% zero_le_double_add_iff_zero_le_single_add
thf(fact_775_zero__le__double__add__iff__zero__le__single__add,axiom,
! [A2: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ A2 @ A2 ) )
= ( ord_less_eq_real @ zero_zero_real @ A2 ) ) ).
% zero_le_double_add_iff_zero_le_single_add
thf(fact_776_double__add__le__zero__iff__single__add__le__zero,axiom,
! [A2: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A2 @ A2 ) @ zero_zero_int )
= ( ord_less_eq_int @ A2 @ zero_zero_int ) ) ).
% double_add_le_zero_iff_single_add_le_zero
thf(fact_777_double__add__le__zero__iff__single__add__le__zero,axiom,
! [A2: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ A2 @ A2 ) @ zero_zero_real )
= ( ord_less_eq_real @ A2 @ zero_zero_real ) ) ).
% double_add_le_zero_iff_single_add_le_zero
thf(fact_778_le__add__same__cancel2,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ ( plus_plus_nat @ B2 @ A2 ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B2 ) ) ).
% le_add_same_cancel2
thf(fact_779_le__add__same__cancel2,axiom,
! [A2: int,B2: int] :
( ( ord_less_eq_int @ A2 @ ( plus_plus_int @ B2 @ A2 ) )
= ( ord_less_eq_int @ zero_zero_int @ B2 ) ) ).
% le_add_same_cancel2
thf(fact_780_le__add__same__cancel2,axiom,
! [A2: real,B2: real] :
( ( ord_less_eq_real @ A2 @ ( plus_plus_real @ B2 @ A2 ) )
= ( ord_less_eq_real @ zero_zero_real @ B2 ) ) ).
% le_add_same_cancel2
thf(fact_781_le__add__same__cancel1,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ ( plus_plus_nat @ A2 @ B2 ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B2 ) ) ).
% le_add_same_cancel1
thf(fact_782_le__add__same__cancel1,axiom,
! [A2: int,B2: int] :
( ( ord_less_eq_int @ A2 @ ( plus_plus_int @ A2 @ B2 ) )
= ( ord_less_eq_int @ zero_zero_int @ B2 ) ) ).
% le_add_same_cancel1
thf(fact_783_le__add__same__cancel1,axiom,
! [A2: real,B2: real] :
( ( ord_less_eq_real @ A2 @ ( plus_plus_real @ A2 @ B2 ) )
= ( ord_less_eq_real @ zero_zero_real @ B2 ) ) ).
% le_add_same_cancel1
thf(fact_784_add__le__same__cancel2,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ B2 ) @ B2 )
= ( ord_less_eq_nat @ A2 @ zero_zero_nat ) ) ).
% add_le_same_cancel2
thf(fact_785_add__le__same__cancel2,axiom,
! [A2: int,B2: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A2 @ B2 ) @ B2 )
= ( ord_less_eq_int @ A2 @ zero_zero_int ) ) ).
% add_le_same_cancel2
thf(fact_786_add__le__same__cancel2,axiom,
! [A2: real,B2: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ A2 @ B2 ) @ B2 )
= ( ord_less_eq_real @ A2 @ zero_zero_real ) ) ).
% add_le_same_cancel2
thf(fact_787_add__le__same__cancel1,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ B2 @ A2 ) @ B2 )
= ( ord_less_eq_nat @ A2 @ zero_zero_nat ) ) ).
% add_le_same_cancel1
thf(fact_788_add__le__same__cancel1,axiom,
! [B2: int,A2: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ B2 @ A2 ) @ B2 )
= ( ord_less_eq_int @ A2 @ zero_zero_int ) ) ).
% add_le_same_cancel1
thf(fact_789_add__le__same__cancel1,axiom,
! [B2: real,A2: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ B2 @ A2 ) @ B2 )
= ( ord_less_eq_real @ A2 @ zero_zero_real ) ) ).
% add_le_same_cancel1
thf(fact_790_zero__less__double__add__iff__zero__less__single__add,axiom,
! [A2: int] :
( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A2 @ A2 ) )
= ( ord_less_int @ zero_zero_int @ A2 ) ) ).
% zero_less_double_add_iff_zero_less_single_add
thf(fact_791_zero__less__double__add__iff__zero__less__single__add,axiom,
! [A2: real] :
( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A2 @ A2 ) )
= ( ord_less_real @ zero_zero_real @ A2 ) ) ).
% zero_less_double_add_iff_zero_less_single_add
thf(fact_792_double__add__less__zero__iff__single__add__less__zero,axiom,
! [A2: int] :
( ( ord_less_int @ ( plus_plus_int @ A2 @ A2 ) @ zero_zero_int )
= ( ord_less_int @ A2 @ zero_zero_int ) ) ).
% double_add_less_zero_iff_single_add_less_zero
thf(fact_793_double__add__less__zero__iff__single__add__less__zero,axiom,
! [A2: real] :
( ( ord_less_real @ ( plus_plus_real @ A2 @ A2 ) @ zero_zero_real )
= ( ord_less_real @ A2 @ zero_zero_real ) ) ).
% double_add_less_zero_iff_single_add_less_zero
thf(fact_794_less__add__same__cancel2,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_nat @ A2 @ ( plus_plus_nat @ B2 @ A2 ) )
= ( ord_less_nat @ zero_zero_nat @ B2 ) ) ).
% less_add_same_cancel2
thf(fact_795_less__add__same__cancel2,axiom,
! [A2: int,B2: int] :
( ( ord_less_int @ A2 @ ( plus_plus_int @ B2 @ A2 ) )
= ( ord_less_int @ zero_zero_int @ B2 ) ) ).
% less_add_same_cancel2
thf(fact_796_less__add__same__cancel2,axiom,
! [A2: real,B2: real] :
( ( ord_less_real @ A2 @ ( plus_plus_real @ B2 @ A2 ) )
= ( ord_less_real @ zero_zero_real @ B2 ) ) ).
% less_add_same_cancel2
thf(fact_797_less__add__same__cancel1,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_nat @ A2 @ ( plus_plus_nat @ A2 @ B2 ) )
= ( ord_less_nat @ zero_zero_nat @ B2 ) ) ).
% less_add_same_cancel1
thf(fact_798_less__add__same__cancel1,axiom,
! [A2: int,B2: int] :
( ( ord_less_int @ A2 @ ( plus_plus_int @ A2 @ B2 ) )
= ( ord_less_int @ zero_zero_int @ B2 ) ) ).
% less_add_same_cancel1
thf(fact_799_less__add__same__cancel1,axiom,
! [A2: real,B2: real] :
( ( ord_less_real @ A2 @ ( plus_plus_real @ A2 @ B2 ) )
= ( ord_less_real @ zero_zero_real @ B2 ) ) ).
% less_add_same_cancel1
thf(fact_800_add__less__same__cancel2,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A2 @ B2 ) @ B2 )
= ( ord_less_nat @ A2 @ zero_zero_nat ) ) ).
% add_less_same_cancel2
thf(fact_801_add__less__same__cancel2,axiom,
! [A2: int,B2: int] :
( ( ord_less_int @ ( plus_plus_int @ A2 @ B2 ) @ B2 )
= ( ord_less_int @ A2 @ zero_zero_int ) ) ).
% add_less_same_cancel2
thf(fact_802_add__less__same__cancel2,axiom,
! [A2: real,B2: real] :
( ( ord_less_real @ ( plus_plus_real @ A2 @ B2 ) @ B2 )
= ( ord_less_real @ A2 @ zero_zero_real ) ) ).
% add_less_same_cancel2
thf(fact_803_add__less__same__cancel1,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ B2 @ A2 ) @ B2 )
= ( ord_less_nat @ A2 @ zero_zero_nat ) ) ).
% add_less_same_cancel1
thf(fact_804_add__less__same__cancel1,axiom,
! [B2: int,A2: int] :
( ( ord_less_int @ ( plus_plus_int @ B2 @ A2 ) @ B2 )
= ( ord_less_int @ A2 @ zero_zero_int ) ) ).
% add_less_same_cancel1
thf(fact_805_add__less__same__cancel1,axiom,
! [B2: real,A2: real] :
( ( ord_less_real @ ( plus_plus_real @ B2 @ A2 ) @ B2 )
= ( ord_less_real @ A2 @ zero_zero_real ) ) ).
% add_less_same_cancel1
thf(fact_806_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_807_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_808_of__nat__le__0__iff,axiom,
! [M: nat] :
( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ M ) @ zero_zero_real )
= ( M = zero_zero_nat ) ) ).
% of_nat_le_0_iff
thf(fact_809_of__int__eq__0__iff,axiom,
! [Z2: int] :
( ( ( ring_1_of_int_real @ Z2 )
= zero_zero_real )
= ( Z2 = zero_zero_int ) ) ).
% of_int_eq_0_iff
thf(fact_810_of__int__eq__0__iff,axiom,
! [Z2: int] :
( ( ( ring_1_of_int_int @ Z2 )
= zero_zero_int )
= ( Z2 = zero_zero_int ) ) ).
% of_int_eq_0_iff
thf(fact_811_of__int__0__eq__iff,axiom,
! [Z2: int] :
( ( zero_zero_real
= ( ring_1_of_int_real @ Z2 ) )
= ( Z2 = zero_zero_int ) ) ).
% of_int_0_eq_iff
thf(fact_812_of__int__0__eq__iff,axiom,
! [Z2: int] :
( ( zero_zero_int
= ( ring_1_of_int_int @ Z2 ) )
= ( Z2 = zero_zero_int ) ) ).
% of_int_0_eq_iff
thf(fact_813_of__int__0,axiom,
( ( ring_1_of_int_real @ zero_zero_int )
= zero_zero_real ) ).
% of_int_0
thf(fact_814_of__int__0,axiom,
( ( ring_1_of_int_int @ zero_zero_int )
= zero_zero_int ) ).
% of_int_0
thf(fact_815_floor__zero,axiom,
( ( archim6058952711729229775r_real @ zero_zero_real )
= zero_zero_int ) ).
% floor_zero
thf(fact_816_ceiling__zero,axiom,
( ( archim7802044766580827645g_real @ zero_zero_real )
= zero_zero_int ) ).
% ceiling_zero
thf(fact_817_frac__1,axiom,
( ( archim2898591450579166408c_real @ one_one_real )
= zero_zero_real ) ).
% frac_1
thf(fact_818_frac__of__int,axiom,
! [Z2: int] :
( ( archim2898591450579166408c_real @ ( ring_1_of_int_real @ Z2 ) )
= zero_zero_real ) ).
% frac_of_int
thf(fact_819_round__0,axiom,
( ( archim8280529875227126926d_real @ zero_zero_real )
= zero_zero_int ) ).
% round_0
thf(fact_820_floor__frac,axiom,
! [R2: real] :
( ( archim6058952711729229775r_real @ ( archim2898591450579166408c_real @ R2 ) )
= zero_zero_int ) ).
% floor_frac
thf(fact_821_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_822_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_823_of__nat__0__less__iff,axiom,
! [N: nat] :
( ( ord_less_real @ zero_zero_real @ ( semiri5074537144036343181t_real @ N ) )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% of_nat_0_less_iff
thf(fact_824_of__int__le__0__iff,axiom,
! [Z2: int] :
( ( ord_less_eq_int @ ( ring_1_of_int_int @ Z2 ) @ zero_zero_int )
= ( ord_less_eq_int @ Z2 @ zero_zero_int ) ) ).
% of_int_le_0_iff
thf(fact_825_of__int__le__0__iff,axiom,
! [Z2: int] :
( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z2 ) @ zero_zero_real )
= ( ord_less_eq_int @ Z2 @ zero_zero_int ) ) ).
% of_int_le_0_iff
thf(fact_826_of__int__0__le__iff,axiom,
! [Z2: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( ring_1_of_int_int @ Z2 ) )
= ( ord_less_eq_int @ zero_zero_int @ Z2 ) ) ).
% of_int_0_le_iff
thf(fact_827_of__int__0__le__iff,axiom,
! [Z2: int] :
( ( ord_less_eq_real @ zero_zero_real @ ( ring_1_of_int_real @ Z2 ) )
= ( ord_less_eq_int @ zero_zero_int @ Z2 ) ) ).
% of_int_0_le_iff
thf(fact_828_of__int__0__less__iff,axiom,
! [Z2: int] :
( ( ord_less_int @ zero_zero_int @ ( ring_1_of_int_int @ Z2 ) )
= ( ord_less_int @ zero_zero_int @ Z2 ) ) ).
% of_int_0_less_iff
thf(fact_829_of__int__0__less__iff,axiom,
! [Z2: int] :
( ( ord_less_real @ zero_zero_real @ ( ring_1_of_int_real @ Z2 ) )
= ( ord_less_int @ zero_zero_int @ Z2 ) ) ).
% of_int_0_less_iff
thf(fact_830_of__int__less__0__iff,axiom,
! [Z2: int] :
( ( ord_less_int @ ( ring_1_of_int_int @ Z2 ) @ zero_zero_int )
= ( ord_less_int @ Z2 @ zero_zero_int ) ) ).
% of_int_less_0_iff
thf(fact_831_of__int__less__0__iff,axiom,
! [Z2: int] :
( ( ord_less_real @ ( ring_1_of_int_real @ Z2 ) @ zero_zero_real )
= ( ord_less_int @ Z2 @ zero_zero_int ) ) ).
% of_int_less_0_iff
thf(fact_832_zero__le__floor,axiom,
! [X: real] :
( ( ord_less_eq_int @ zero_zero_int @ ( archim6058952711729229775r_real @ X ) )
= ( ord_less_eq_real @ zero_zero_real @ X ) ) ).
% zero_le_floor
thf(fact_833_ceiling__le__zero,axiom,
! [X: real] :
( ( ord_less_eq_int @ ( archim7802044766580827645g_real @ X ) @ zero_zero_int )
= ( ord_less_eq_real @ X @ zero_zero_real ) ) ).
% ceiling_le_zero
thf(fact_834_floor__less__zero,axiom,
! [X: real] :
( ( ord_less_int @ ( archim6058952711729229775r_real @ X ) @ zero_zero_int )
= ( ord_less_real @ X @ zero_zero_real ) ) ).
% floor_less_zero
thf(fact_835_zero__less__ceiling,axiom,
! [X: real] :
( ( ord_less_int @ zero_zero_int @ ( archim7802044766580827645g_real @ X ) )
= ( ord_less_real @ zero_zero_real @ X ) ) ).
% zero_less_ceiling
thf(fact_836_floor__eq,axiom,
! [N: int,X: real] :
( ( ord_less_real @ ( ring_1_of_int_real @ N ) @ X )
=> ( ( ord_less_real @ X @ ( plus_plus_real @ ( ring_1_of_int_real @ N ) @ one_one_real ) )
=> ( ( archim6058952711729229775r_real @ X )
= N ) ) ) ).
% floor_eq
thf(fact_837_floor__eq2,axiom,
! [N: int,X: real] :
( ( ord_less_eq_real @ ( ring_1_of_int_real @ N ) @ X )
=> ( ( ord_less_real @ X @ ( plus_plus_real @ ( ring_1_of_int_real @ N ) @ one_one_real ) )
=> ( ( archim6058952711729229775r_real @ X )
= N ) ) ) ).
% floor_eq2
thf(fact_838_nat__le__real__less,axiom,
( ord_less_eq_nat
= ( ^ [N3: nat,M3: nat] : ( ord_less_real @ ( semiri5074537144036343181t_real @ N3 ) @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ M3 ) @ one_one_real ) ) ) ) ).
% nat_le_real_less
thf(fact_839_real__of__int__floor__add__one__ge,axiom,
! [R2: real] : ( ord_less_eq_real @ R2 @ ( plus_plus_real @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ R2 ) ) @ one_one_real ) ) ).
% real_of_int_floor_add_one_ge
thf(fact_840_real__of__int__floor__add__one__gt,axiom,
! [R2: real] : ( ord_less_real @ R2 @ ( plus_plus_real @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ R2 ) ) @ one_one_real ) ) ).
% real_of_int_floor_add_one_gt
thf(fact_841_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_842_bot__nat__0_Oextremum__unique,axiom,
! [A2: nat] :
( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
= ( A2 = zero_zero_nat ) ) ).
% bot_nat_0.extremum_unique
thf(fact_843_bot__nat__0_Oextremum__uniqueI,axiom,
! [A2: nat] :
( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
=> ( A2 = zero_zero_nat ) ) ).
% bot_nat_0.extremum_uniqueI
thf(fact_844_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_845_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_846_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_847_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% eq_imp_le
thf(fact_848_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_849_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_850_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K: nat,B2: nat] :
( ( P @ K )
=> ( ! [Y3: nat] :
( ( P @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ B2 ) )
=> ? [X4: nat] :
( ( P @ X4 )
& ! [Y4: nat] :
( ( P @ Y4 )
=> ( ord_less_eq_nat @ Y4 @ X4 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_851_bounded__Max__nat,axiom,
! [P: nat > $o,X: nat,M5: nat] :
( ( P @ X )
=> ( ! [X4: nat] :
( ( P @ X4 )
=> ( ord_less_eq_nat @ X4 @ M5 ) )
=> ~ ! [M4: nat] :
( ( P @ M4 )
=> ~ ! [X3: nat] :
( ( P @ X3 )
=> ( ord_less_eq_nat @ X3 @ M4 ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_852_complete__real,axiom,
! [S2: set_real] :
( ? [X3: real] : ( member_real @ X3 @ S2 )
=> ( ? [Z4: real] :
! [X4: real] :
( ( member_real @ X4 @ S2 )
=> ( ord_less_eq_real @ X4 @ Z4 ) )
=> ? [Y3: real] :
( ! [X3: real] :
( ( member_real @ X3 @ S2 )
=> ( ord_less_eq_real @ X3 @ Y3 ) )
& ! [Z4: real] :
( ! [X4: real] :
( ( member_real @ X4 @ S2 )
=> ( ord_less_eq_real @ X4 @ Z4 ) )
=> ( ord_less_eq_real @ Y3 @ Z4 ) ) ) ) ) ).
% complete_real
thf(fact_853_less__eq__real__def,axiom,
( ord_less_eq_real
= ( ^ [X2: real,Y5: real] :
( ( ord_less_real @ X2 @ Y5 )
| ( X2 = Y5 ) ) ) ) ).
% less_eq_real_def
thf(fact_854_int__ops_I1_J,axiom,
( ( semiri1314217659103216013at_int @ zero_zero_nat )
= zero_zero_int ) ).
% int_ops(1)
thf(fact_855_zero__reorient,axiom,
! [X: nat] :
( ( zero_zero_nat = X )
= ( X = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_856_zero__reorient,axiom,
! [X: int] :
( ( zero_zero_int = X )
= ( X = zero_zero_int ) ) ).
% zero_reorient
thf(fact_857_zero__reorient,axiom,
! [X: real] :
( ( zero_zero_real = X )
= ( X = zero_zero_real ) ) ).
% zero_reorient
thf(fact_858_zero__integer_Orsp,axiom,
zero_zero_int = zero_zero_int ).
% zero_integer.rsp
thf(fact_859_zero__le,axiom,
! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).
% zero_le
thf(fact_860_le__numeral__extra_I3_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% le_numeral_extra(3)
thf(fact_861_le__numeral__extra_I3_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% le_numeral_extra(3)
thf(fact_862_le__numeral__extra_I3_J,axiom,
ord_less_eq_real @ zero_zero_real @ zero_zero_real ).
% le_numeral_extra(3)
thf(fact_863_less__numeral__extra_I3_J,axiom,
~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% less_numeral_extra(3)
thf(fact_864_less__numeral__extra_I3_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_numeral_extra(3)
thf(fact_865_less__numeral__extra_I3_J,axiom,
~ ( ord_less_real @ zero_zero_real @ zero_zero_real ) ).
% less_numeral_extra(3)
thf(fact_866_gr__zeroI,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr_zeroI
thf(fact_867_not__less__zero,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less_zero
thf(fact_868_gr__implies__not__zero,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_869_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_870_field__lbound__gt__zero,axiom,
! [D1: real,D22: real] :
( ( ord_less_real @ zero_zero_real @ D1 )
=> ( ( ord_less_real @ zero_zero_real @ D22 )
=> ? [E: real] :
( ( ord_less_real @ zero_zero_real @ E )
& ( ord_less_real @ E @ D1 )
& ( ord_less_real @ E @ D22 ) ) ) ) ).
% field_lbound_gt_zero
thf(fact_871_add_Ogroup__left__neutral,axiom,
! [A2: int] :
( ( plus_plus_int @ zero_zero_int @ A2 )
= A2 ) ).
% add.group_left_neutral
thf(fact_872_add_Ogroup__left__neutral,axiom,
! [A2: real] :
( ( plus_plus_real @ zero_zero_real @ A2 )
= A2 ) ).
% add.group_left_neutral
thf(fact_873_add_Ocomm__neutral,axiom,
! [A2: nat] :
( ( plus_plus_nat @ A2 @ zero_zero_nat )
= A2 ) ).
% add.comm_neutral
thf(fact_874_add_Ocomm__neutral,axiom,
! [A2: int] :
( ( plus_plus_int @ A2 @ zero_zero_int )
= A2 ) ).
% add.comm_neutral
thf(fact_875_add_Ocomm__neutral,axiom,
! [A2: real] :
( ( plus_plus_real @ A2 @ zero_zero_real )
= A2 ) ).
% add.comm_neutral
thf(fact_876_comm__monoid__add__class_Oadd__0,axiom,
! [A2: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A2 )
= A2 ) ).
% comm_monoid_add_class.add_0
thf(fact_877_comm__monoid__add__class_Oadd__0,axiom,
! [A2: int] :
( ( plus_plus_int @ zero_zero_int @ A2 )
= A2 ) ).
% comm_monoid_add_class.add_0
thf(fact_878_comm__monoid__add__class_Oadd__0,axiom,
! [A2: real] :
( ( plus_plus_real @ zero_zero_real @ A2 )
= A2 ) ).
% comm_monoid_add_class.add_0
thf(fact_879_verit__sum__simplify,axiom,
! [A2: nat] :
( ( plus_plus_nat @ A2 @ zero_zero_nat )
= A2 ) ).
% verit_sum_simplify
thf(fact_880_verit__sum__simplify,axiom,
! [A2: int] :
( ( plus_plus_int @ A2 @ zero_zero_int )
= A2 ) ).
% verit_sum_simplify
thf(fact_881_verit__sum__simplify,axiom,
! [A2: real] :
( ( plus_plus_real @ A2 @ zero_zero_real )
= A2 ) ).
% verit_sum_simplify
thf(fact_882_zero__neq__one,axiom,
zero_zero_nat != one_one_nat ).
% zero_neq_one
thf(fact_883_zero__neq__one,axiom,
zero_zero_int != one_one_int ).
% zero_neq_one
thf(fact_884_zero__neq__one,axiom,
zero_zero_real != one_one_real ).
% zero_neq_one
thf(fact_885_bot__nat__0_Oextremum__strict,axiom,
! [A2: nat] :
~ ( ord_less_nat @ A2 @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_886_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr0I
thf(fact_887_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr0
thf(fact_888_not__less0,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less0
thf(fact_889_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_zeroE
thf(fact_890_gr__implies__not0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_891_infinite__descent0,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ~ ( P @ N2 )
=> ? [M2: nat] :
( ( ord_less_nat @ M2 @ N2 )
& ~ ( P @ M2 ) ) ) )
=> ( P @ N ) ) ) ).
% infinite_descent0
thf(fact_892_ex__least__nat__le,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K4: nat] :
( ( ord_less_eq_nat @ K4 @ N )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ K4 )
=> ~ ( P @ I4 ) )
& ( P @ K4 ) ) ) ) ).
% ex_least_nat_le
thf(fact_893_plus__int__code_I1_J,axiom,
! [K: int] :
( ( plus_plus_int @ K @ zero_zero_int )
= K ) ).
% plus_int_code(1)
thf(fact_894_plus__int__code_I2_J,axiom,
! [L: int] :
( ( plus_plus_int @ zero_zero_int @ L )
= L ) ).
% plus_int_code(2)
thf(fact_895_int_Ozero__not__one,axiom,
zero_zero_int != one_one_int ).
% int.zero_not_one
thf(fact_896_conj__le__cong,axiom,
! [X: int,X6: int,P: $o,P2: $o] :
( ( X = X6 )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X6 )
=> ( P = P2 ) )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X )
& P )
= ( ( ord_less_eq_int @ zero_zero_int @ X6 )
& P2 ) ) ) ) ).
% conj_le_cong
thf(fact_897_imp__le__cong,axiom,
! [X: int,X6: int,P: $o,P2: $o] :
( ( X = X6 )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X6 )
=> ( P = P2 ) )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X )
=> P )
= ( ( ord_less_eq_int @ zero_zero_int @ X6 )
=> P2 ) ) ) ) ).
% imp_le_cong
thf(fact_898_less__eq__int__code_I1_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% less_eq_int_code(1)
thf(fact_899_plus__nat_Oadd__0,axiom,
! [N: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N )
= N ) ).
% plus_nat.add_0
thf(fact_900_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_901_less__int__code_I1_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_int_code(1)
thf(fact_902_of__int__nonneg,axiom,
! [Z2: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z2 )
=> ( ord_less_eq_int @ zero_zero_int @ ( ring_1_of_int_int @ Z2 ) ) ) ).
% of_int_nonneg
thf(fact_903_of__int__nonneg,axiom,
! [Z2: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z2 )
=> ( ord_less_eq_real @ zero_zero_real @ ( ring_1_of_int_real @ Z2 ) ) ) ).
% of_int_nonneg
thf(fact_904_of__int__pos,axiom,
! [Z2: int] :
( ( ord_less_int @ zero_zero_int @ Z2 )
=> ( ord_less_int @ zero_zero_int @ ( ring_1_of_int_int @ Z2 ) ) ) ).
% of_int_pos
thf(fact_905_of__int__pos,axiom,
! [Z2: int] :
( ( ord_less_int @ zero_zero_int @ Z2 )
=> ( ord_less_real @ zero_zero_real @ ( ring_1_of_int_real @ Z2 ) ) ) ).
% of_int_pos
thf(fact_906_pos__int__cases,axiom,
! [K: int] :
( ( ord_less_int @ zero_zero_int @ K )
=> ~ ! [N2: nat] :
( ( K
= ( semiri1314217659103216013at_int @ N2 ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% pos_int_cases
thf(fact_907_zero__less__imp__eq__int,axiom,
! [K: int] :
( ( ord_less_int @ zero_zero_int @ K )
=> ? [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
& ( K
= ( semiri1314217659103216013at_int @ N2 ) ) ) ) ).
% zero_less_imp_eq_int
thf(fact_908_add__decreasing,axiom,
! [A2: nat,C: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ C @ B2 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C ) @ B2 ) ) ) ).
% add_decreasing
thf(fact_909_add__decreasing,axiom,
! [A2: int,C: int,B2: int] :
( ( ord_less_eq_int @ A2 @ zero_zero_int )
=> ( ( ord_less_eq_int @ C @ B2 )
=> ( ord_less_eq_int @ ( plus_plus_int @ A2 @ C ) @ B2 ) ) ) ).
% add_decreasing
thf(fact_910_add__decreasing,axiom,
! [A2: real,C: real,B2: real] :
( ( ord_less_eq_real @ A2 @ zero_zero_real )
=> ( ( ord_less_eq_real @ C @ B2 )
=> ( ord_less_eq_real @ ( plus_plus_real @ A2 @ C ) @ B2 ) ) ) ).
% add_decreasing
thf(fact_911_add__increasing,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ord_less_eq_nat @ B2 @ ( plus_plus_nat @ A2 @ C ) ) ) ) ).
% add_increasing
thf(fact_912_add__increasing,axiom,
! [A2: int,B2: int,C: int] :
( ( ord_less_eq_int @ zero_zero_int @ A2 )
=> ( ( ord_less_eq_int @ B2 @ C )
=> ( ord_less_eq_int @ B2 @ ( plus_plus_int @ A2 @ C ) ) ) ) ).
% add_increasing
thf(fact_913_add__increasing,axiom,
! [A2: real,B2: real,C: real] :
( ( ord_less_eq_real @ zero_zero_real @ A2 )
=> ( ( ord_less_eq_real @ B2 @ C )
=> ( ord_less_eq_real @ B2 @ ( plus_plus_real @ A2 @ C ) ) ) ) ).
% add_increasing
thf(fact_914_add__decreasing2,axiom,
! [C: nat,A2: nat,B2: nat] :
( ( ord_less_eq_nat @ C @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C ) @ B2 ) ) ) ).
% add_decreasing2
thf(fact_915_add__decreasing2,axiom,
! [C: int,A2: int,B2: int] :
( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ( ord_less_eq_int @ A2 @ B2 )
=> ( ord_less_eq_int @ ( plus_plus_int @ A2 @ C ) @ B2 ) ) ) ).
% add_decreasing2
thf(fact_916_add__decreasing2,axiom,
! [C: real,A2: real,B2: real] :
( ( ord_less_eq_real @ C @ zero_zero_real )
=> ( ( ord_less_eq_real @ A2 @ B2 )
=> ( ord_less_eq_real @ ( plus_plus_real @ A2 @ C ) @ B2 ) ) ) ).
% add_decreasing2
thf(fact_917_add__increasing2,axiom,
! [C: nat,B2: nat,A2: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ( ord_less_eq_nat @ B2 @ A2 )
=> ( ord_less_eq_nat @ B2 @ ( plus_plus_nat @ A2 @ C ) ) ) ) ).
% add_increasing2
thf(fact_918_add__increasing2,axiom,
! [C: int,B2: int,A2: int] :
( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ( ord_less_eq_int @ B2 @ A2 )
=> ( ord_less_eq_int @ B2 @ ( plus_plus_int @ A2 @ C ) ) ) ) ).
% add_increasing2
thf(fact_919_add__increasing2,axiom,
! [C: real,B2: real,A2: real] :
( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ( ord_less_eq_real @ B2 @ A2 )
=> ( ord_less_eq_real @ B2 @ ( plus_plus_real @ A2 @ C ) ) ) ) ).
% add_increasing2
thf(fact_920_add__nonneg__nonneg,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( plus_plus_nat @ A2 @ B2 ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_921_add__nonneg__nonneg,axiom,
! [A2: int,B2: int] :
( ( ord_less_eq_int @ zero_zero_int @ A2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ B2 )
=> ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A2 @ B2 ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_922_add__nonneg__nonneg,axiom,
! [A2: real,B2: real] :
( ( ord_less_eq_real @ zero_zero_real @ A2 )
=> ( ( ord_less_eq_real @ zero_zero_real @ B2 )
=> ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ A2 @ B2 ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_923_add__nonpos__nonpos,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ B2 @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ B2 ) @ zero_zero_nat ) ) ) ).
% add_nonpos_nonpos
thf(fact_924_add__nonpos__nonpos,axiom,
! [A2: int,B2: int] :
( ( ord_less_eq_int @ A2 @ zero_zero_int )
=> ( ( ord_less_eq_int @ B2 @ zero_zero_int )
=> ( ord_less_eq_int @ ( plus_plus_int @ A2 @ B2 ) @ zero_zero_int ) ) ) ).
% add_nonpos_nonpos
thf(fact_925_add__nonpos__nonpos,axiom,
! [A2: real,B2: real] :
( ( ord_less_eq_real @ A2 @ zero_zero_real )
=> ( ( ord_less_eq_real @ B2 @ zero_zero_real )
=> ( ord_less_eq_real @ ( plus_plus_real @ A2 @ B2 ) @ zero_zero_real ) ) ) ).
% add_nonpos_nonpos
thf(fact_926_add__nonneg__eq__0__iff,axiom,
! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ X )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ Y2 )
=> ( ( ( plus_plus_nat @ X @ Y2 )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y2 = zero_zero_nat ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_927_add__nonneg__eq__0__iff,axiom,
! [X: int,Y2: int] :
( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y2 )
=> ( ( ( plus_plus_int @ X @ Y2 )
= zero_zero_int )
= ( ( X = zero_zero_int )
& ( Y2 = zero_zero_int ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_928_add__nonneg__eq__0__iff,axiom,
! [X: real,Y2: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y2 )
=> ( ( ( plus_plus_real @ X @ Y2 )
= zero_zero_real )
= ( ( X = zero_zero_real )
& ( Y2 = zero_zero_real ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_929_add__nonpos__eq__0__iff,axiom,
! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ Y2 @ zero_zero_nat )
=> ( ( ( plus_plus_nat @ X @ Y2 )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y2 = zero_zero_nat ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_930_add__nonpos__eq__0__iff,axiom,
! [X: int,Y2: int] :
( ( ord_less_eq_int @ X @ zero_zero_int )
=> ( ( ord_less_eq_int @ Y2 @ zero_zero_int )
=> ( ( ( plus_plus_int @ X @ Y2 )
= zero_zero_int )
= ( ( X = zero_zero_int )
& ( Y2 = zero_zero_int ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_931_add__nonpos__eq__0__iff,axiom,
! [X: real,Y2: real] :
( ( ord_less_eq_real @ X @ zero_zero_real )
=> ( ( ord_less_eq_real @ Y2 @ zero_zero_real )
=> ( ( ( plus_plus_real @ X @ Y2 )
= zero_zero_real )
= ( ( X = zero_zero_real )
& ( Y2 = zero_zero_real ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_932_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_933_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_934_zero__less__one__class_Ozero__le__one,axiom,
ord_less_eq_real @ zero_zero_real @ one_one_real ).
% zero_less_one_class.zero_le_one
thf(fact_935_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_936_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_937_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
ord_less_eq_real @ zero_zero_real @ one_one_real ).
% linordered_nonzero_semiring_class.zero_le_one
thf(fact_938_not__one__le__zero,axiom,
~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_le_zero
thf(fact_939_not__one__le__zero,axiom,
~ ( ord_less_eq_int @ one_one_int @ zero_zero_int ) ).
% not_one_le_zero
thf(fact_940_not__one__le__zero,axiom,
~ ( ord_less_eq_real @ one_one_real @ zero_zero_real ) ).
% not_one_le_zero
thf(fact_941_pos__add__strict,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ord_less_nat @ B2 @ ( plus_plus_nat @ A2 @ C ) ) ) ) ).
% pos_add_strict
thf(fact_942_pos__add__strict,axiom,
! [A2: int,B2: int,C: int] :
( ( ord_less_int @ zero_zero_int @ A2 )
=> ( ( ord_less_int @ B2 @ C )
=> ( ord_less_int @ B2 @ ( plus_plus_int @ A2 @ C ) ) ) ) ).
% pos_add_strict
thf(fact_943_pos__add__strict,axiom,
! [A2: real,B2: real,C: real] :
( ( ord_less_real @ zero_zero_real @ A2 )
=> ( ( ord_less_real @ B2 @ C )
=> ( ord_less_real @ B2 @ ( plus_plus_real @ A2 @ C ) ) ) ) ).
% pos_add_strict
thf(fact_944_canonically__ordered__monoid__add__class_OlessE,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ~ ! [C2: nat] :
( ( B2
= ( plus_plus_nat @ A2 @ C2 ) )
=> ( C2 = zero_zero_nat ) ) ) ).
% canonically_ordered_monoid_add_class.lessE
thf(fact_945_add__pos__pos,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_nat @ zero_zero_nat @ B2 )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A2 @ B2 ) ) ) ) ).
% add_pos_pos
thf(fact_946_add__pos__pos,axiom,
! [A2: int,B2: int] :
( ( ord_less_int @ zero_zero_int @ A2 )
=> ( ( ord_less_int @ zero_zero_int @ B2 )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A2 @ B2 ) ) ) ) ).
% add_pos_pos
thf(fact_947_add__pos__pos,axiom,
! [A2: real,B2: real] :
( ( ord_less_real @ zero_zero_real @ A2 )
=> ( ( ord_less_real @ zero_zero_real @ B2 )
=> ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A2 @ B2 ) ) ) ) ).
% add_pos_pos
thf(fact_948_add__neg__neg,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_nat @ A2 @ zero_zero_nat )
=> ( ( ord_less_nat @ B2 @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A2 @ B2 ) @ zero_zero_nat ) ) ) ).
% add_neg_neg
thf(fact_949_add__neg__neg,axiom,
! [A2: int,B2: int] :
( ( ord_less_int @ A2 @ zero_zero_int )
=> ( ( ord_less_int @ B2 @ zero_zero_int )
=> ( ord_less_int @ ( plus_plus_int @ A2 @ B2 ) @ zero_zero_int ) ) ) ).
% add_neg_neg
thf(fact_950_add__neg__neg,axiom,
! [A2: real,B2: real] :
( ( ord_less_real @ A2 @ zero_zero_real )
=> ( ( ord_less_real @ B2 @ zero_zero_real )
=> ( ord_less_real @ ( plus_plus_real @ A2 @ B2 ) @ zero_zero_real ) ) ) ).
% add_neg_neg
thf(fact_951_add__less__zeroD,axiom,
! [X: int,Y2: int] :
( ( ord_less_int @ ( plus_plus_int @ X @ Y2 ) @ zero_zero_int )
=> ( ( ord_less_int @ X @ zero_zero_int )
| ( ord_less_int @ Y2 @ zero_zero_int ) ) ) ).
% add_less_zeroD
thf(fact_952_add__less__zeroD,axiom,
! [X: real,Y2: real] :
( ( ord_less_real @ ( plus_plus_real @ X @ Y2 ) @ zero_zero_real )
=> ( ( ord_less_real @ X @ zero_zero_real )
| ( ord_less_real @ Y2 @ zero_zero_real ) ) ) ).
% add_less_zeroD
thf(fact_953_less__numeral__extra_I1_J,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% less_numeral_extra(1)
thf(fact_954_less__numeral__extra_I1_J,axiom,
ord_less_int @ zero_zero_int @ one_one_int ).
% less_numeral_extra(1)
thf(fact_955_less__numeral__extra_I1_J,axiom,
ord_less_real @ zero_zero_real @ one_one_real ).
% less_numeral_extra(1)
thf(fact_956_zero__less__one,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one
thf(fact_957_zero__less__one,axiom,
ord_less_int @ zero_zero_int @ one_one_int ).
% zero_less_one
thf(fact_958_zero__less__one,axiom,
ord_less_real @ zero_zero_real @ one_one_real ).
% zero_less_one
thf(fact_959_not__one__less__zero,axiom,
~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_less_zero
thf(fact_960_not__one__less__zero,axiom,
~ ( ord_less_int @ one_one_int @ zero_zero_int ) ).
% not_one_less_zero
thf(fact_961_not__one__less__zero,axiom,
~ ( ord_less_real @ one_one_real @ zero_zero_real ) ).
% not_one_less_zero
thf(fact_962_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_963_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_964_of__nat__0__le__iff,axiom,
! [N: nat] : ( ord_less_eq_real @ zero_zero_real @ ( semiri5074537144036343181t_real @ N ) ) ).
% of_nat_0_le_iff
thf(fact_965_of__nat__less__0__iff,axiom,
! [M: nat] :
~ ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ zero_zero_nat ) ).
% of_nat_less_0_iff
thf(fact_966_of__nat__less__0__iff,axiom,
! [M: nat] :
~ ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ zero_zero_int ) ).
% of_nat_less_0_iff
thf(fact_967_of__nat__less__0__iff,axiom,
! [M: nat] :
~ ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ zero_zero_real ) ).
% of_nat_less_0_iff
thf(fact_968_odd__nonzero,axiom,
! [Z2: int] :
( ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z2 ) @ Z2 )
!= zero_zero_int ) ).
% odd_nonzero
thf(fact_969_less__imp__add__positive,axiom,
! [I: nat,J: nat] :
( ( ord_less_nat @ I @ J )
=> ? [K4: nat] :
( ( ord_less_nat @ zero_zero_nat @ K4 )
& ( ( plus_plus_nat @ I @ K4 )
= J ) ) ) ).
% less_imp_add_positive
thf(fact_970_nonneg__int__cases,axiom,
! [K: int] :
( ( ord_less_eq_int @ zero_zero_int @ K )
=> ~ ! [N2: nat] :
( K
!= ( semiri1314217659103216013at_int @ N2 ) ) ) ).
% nonneg_int_cases
thf(fact_971_zero__le__imp__eq__int,axiom,
! [K: int] :
( ( ord_less_eq_int @ zero_zero_int @ K )
=> ? [N2: nat] :
( K
= ( semiri1314217659103216013at_int @ N2 ) ) ) ).
% zero_le_imp_eq_int
thf(fact_972_frac__ge__0,axiom,
! [X: real] : ( ord_less_eq_real @ zero_zero_real @ ( archim2898591450579166408c_real @ X ) ) ).
% frac_ge_0
thf(fact_973_ex__gt__or__lt,axiom,
! [A2: real] :
? [B4: real] :
( ( ord_less_real @ A2 @ B4 )
| ( ord_less_real @ B4 @ A2 ) ) ).
% ex_gt_or_lt
thf(fact_974_add__neg__nonpos,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_nat @ A2 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ B2 @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A2 @ B2 ) @ zero_zero_nat ) ) ) ).
% add_neg_nonpos
thf(fact_975_add__neg__nonpos,axiom,
! [A2: int,B2: int] :
( ( ord_less_int @ A2 @ zero_zero_int )
=> ( ( ord_less_eq_int @ B2 @ zero_zero_int )
=> ( ord_less_int @ ( plus_plus_int @ A2 @ B2 ) @ zero_zero_int ) ) ) ).
% add_neg_nonpos
thf(fact_976_add__neg__nonpos,axiom,
! [A2: real,B2: real] :
( ( ord_less_real @ A2 @ zero_zero_real )
=> ( ( ord_less_eq_real @ B2 @ zero_zero_real )
=> ( ord_less_real @ ( plus_plus_real @ A2 @ B2 ) @ zero_zero_real ) ) ) ).
% add_neg_nonpos
thf(fact_977_add__nonneg__pos,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_nat @ zero_zero_nat @ B2 )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A2 @ B2 ) ) ) ) ).
% add_nonneg_pos
thf(fact_978_add__nonneg__pos,axiom,
! [A2: int,B2: int] :
( ( ord_less_eq_int @ zero_zero_int @ A2 )
=> ( ( ord_less_int @ zero_zero_int @ B2 )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A2 @ B2 ) ) ) ) ).
% add_nonneg_pos
thf(fact_979_add__nonneg__pos,axiom,
! [A2: real,B2: real] :
( ( ord_less_eq_real @ zero_zero_real @ A2 )
=> ( ( ord_less_real @ zero_zero_real @ B2 )
=> ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A2 @ B2 ) ) ) ) ).
% add_nonneg_pos
thf(fact_980_add__nonpos__neg,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
=> ( ( ord_less_nat @ B2 @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A2 @ B2 ) @ zero_zero_nat ) ) ) ).
% add_nonpos_neg
thf(fact_981_add__nonpos__neg,axiom,
! [A2: int,B2: int] :
( ( ord_less_eq_int @ A2 @ zero_zero_int )
=> ( ( ord_less_int @ B2 @ zero_zero_int )
=> ( ord_less_int @ ( plus_plus_int @ A2 @ B2 ) @ zero_zero_int ) ) ) ).
% add_nonpos_neg
thf(fact_982_add__nonpos__neg,axiom,
! [A2: real,B2: real] :
( ( ord_less_eq_real @ A2 @ zero_zero_real )
=> ( ( ord_less_real @ B2 @ zero_zero_real )
=> ( ord_less_real @ ( plus_plus_real @ A2 @ B2 ) @ zero_zero_real ) ) ) ).
% add_nonpos_neg
thf(fact_983_add__pos__nonneg,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A2 @ B2 ) ) ) ) ).
% add_pos_nonneg
thf(fact_984_add__pos__nonneg,axiom,
! [A2: int,B2: int] :
( ( ord_less_int @ zero_zero_int @ A2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ B2 )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A2 @ B2 ) ) ) ) ).
% add_pos_nonneg
thf(fact_985_add__pos__nonneg,axiom,
! [A2: real,B2: real] :
( ( ord_less_real @ zero_zero_real @ A2 )
=> ( ( ord_less_eq_real @ zero_zero_real @ B2 )
=> ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A2 @ B2 ) ) ) ) ).
% add_pos_nonneg
thf(fact_986_add__strict__increasing,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ord_less_nat @ B2 @ ( plus_plus_nat @ A2 @ C ) ) ) ) ).
% add_strict_increasing
thf(fact_987_add__strict__increasing,axiom,
! [A2: int,B2: int,C: int] :
( ( ord_less_int @ zero_zero_int @ A2 )
=> ( ( ord_less_eq_int @ B2 @ C )
=> ( ord_less_int @ B2 @ ( plus_plus_int @ A2 @ C ) ) ) ) ).
% add_strict_increasing
thf(fact_988_add__strict__increasing,axiom,
! [A2: real,B2: real,C: real] :
( ( ord_less_real @ zero_zero_real @ A2 )
=> ( ( ord_less_eq_real @ B2 @ C )
=> ( ord_less_real @ B2 @ ( plus_plus_real @ A2 @ C ) ) ) ) ).
% add_strict_increasing
thf(fact_989_add__strict__increasing2,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ord_less_nat @ B2 @ ( plus_plus_nat @ A2 @ C ) ) ) ) ).
% add_strict_increasing2
thf(fact_990_add__strict__increasing2,axiom,
! [A2: int,B2: int,C: int] :
( ( ord_less_eq_int @ zero_zero_int @ A2 )
=> ( ( ord_less_int @ B2 @ C )
=> ( ord_less_int @ B2 @ ( plus_plus_int @ A2 @ C ) ) ) ) ).
% add_strict_increasing2
thf(fact_991_add__strict__increasing2,axiom,
! [A2: real,B2: real,C: real] :
( ( ord_less_eq_real @ zero_zero_real @ A2 )
=> ( ( ord_less_real @ B2 @ C )
=> ( ord_less_real @ B2 @ ( plus_plus_real @ A2 @ C ) ) ) ) ).
% add_strict_increasing2
thf(fact_992_zero__less__two,axiom,
ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ).
% zero_less_two
thf(fact_993_zero__less__two,axiom,
ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ one_one_int ) ).
% zero_less_two
thf(fact_994_zero__less__two,axiom,
ord_less_real @ zero_zero_real @ ( plus_plus_real @ one_one_real @ one_one_real ) ).
% zero_less_two
thf(fact_995_odd__less__0__iff,axiom,
! [Z2: int] :
( ( ord_less_int @ ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z2 ) @ Z2 ) @ zero_zero_int )
= ( ord_less_int @ Z2 @ zero_zero_int ) ) ).
% odd_less_0_iff
thf(fact_996_int__one__le__iff__zero__less,axiom,
! [Z2: int] :
( ( ord_less_eq_int @ one_one_int @ Z2 )
= ( ord_less_int @ zero_zero_int @ Z2 ) ) ).
% int_one_le_iff_zero_less
thf(fact_997_frac__eq,axiom,
! [X: real] :
( ( ( archim2898591450579166408c_real @ X )
= X )
= ( ( ord_less_eq_real @ zero_zero_real @ X )
& ( ord_less_real @ X @ one_one_real ) ) ) ).
% frac_eq
thf(fact_998_le__imp__0__less,axiom,
! [Z2: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z2 )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ Z2 ) ) ) ).
% le_imp_0_less
thf(fact_999_field__le__epsilon,axiom,
! [X: real,Y2: real] :
( ! [E: real] :
( ( ord_less_real @ zero_zero_real @ E )
=> ( ord_less_eq_real @ X @ ( plus_plus_real @ Y2 @ E ) ) )
=> ( ord_less_eq_real @ X @ Y2 ) ) ).
% field_le_epsilon
thf(fact_1000_nat__descend__induct,axiom,
! [N: nat,P: nat > $o,M: nat] :
( ! [K4: nat] :
( ( ord_less_nat @ N @ K4 )
=> ( P @ K4 ) )
=> ( ! [K4: nat] :
( ( ord_less_eq_nat @ K4 @ N )
=> ( ! [I4: nat] :
( ( ord_less_nat @ K4 @ I4 )
=> ( P @ I4 ) )
=> ( P @ K4 ) ) )
=> ( P @ M ) ) ) ).
% nat_descend_induct
thf(fact_1001_add__0__iff,axiom,
! [B2: nat,A2: nat] :
( ( B2
= ( plus_plus_nat @ B2 @ A2 ) )
= ( A2 = zero_zero_nat ) ) ).
% add_0_iff
thf(fact_1002_add__0__iff,axiom,
! [B2: int,A2: int] :
( ( B2
= ( plus_plus_int @ B2 @ A2 ) )
= ( A2 = zero_zero_int ) ) ).
% add_0_iff
thf(fact_1003_add__0__iff,axiom,
! [B2: real,A2: real] :
( ( B2
= ( plus_plus_real @ B2 @ A2 ) )
= ( A2 = zero_zero_real ) ) ).
% add_0_iff
thf(fact_1004_ceiling__less__zero,axiom,
! [X: real] :
( ( ord_less_int @ ( archim7802044766580827645g_real @ X ) @ zero_zero_int )
= ( ord_less_eq_real @ X @ ( uminus_uminus_real @ one_one_real ) ) ) ).
% ceiling_less_zero
thf(fact_1005_zero__le__ceiling,axiom,
! [X: real] :
( ( ord_less_eq_int @ zero_zero_int @ ( archim7802044766580827645g_real @ X ) )
= ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X ) ) ).
% zero_le_ceiling
thf(fact_1006_dbl__inc__simps_I2_J,axiom,
( ( neg_nu5851722552734809277nc_int @ zero_zero_int )
= one_one_int ) ).
% dbl_inc_simps(2)
thf(fact_1007_dbl__inc__simps_I2_J,axiom,
( ( neg_nu8295874005876285629c_real @ zero_zero_real )
= one_one_real ) ).
% dbl_inc_simps(2)
thf(fact_1008_verit__minus__simplify_I4_J,axiom,
! [B2: int] :
( ( uminus_uminus_int @ ( uminus_uminus_int @ B2 ) )
= B2 ) ).
% verit_minus_simplify(4)
thf(fact_1009_verit__minus__simplify_I4_J,axiom,
! [B2: real] :
( ( uminus_uminus_real @ ( uminus_uminus_real @ B2 ) )
= B2 ) ).
% verit_minus_simplify(4)
thf(fact_1010_add_Oinverse__inverse,axiom,
! [A2: int] :
( ( uminus_uminus_int @ ( uminus_uminus_int @ A2 ) )
= A2 ) ).
% add.inverse_inverse
thf(fact_1011_add_Oinverse__inverse,axiom,
! [A2: real] :
( ( uminus_uminus_real @ ( uminus_uminus_real @ A2 ) )
= A2 ) ).
% add.inverse_inverse
thf(fact_1012_neg__equal__iff__equal,axiom,
! [A2: int,B2: int] :
( ( ( uminus_uminus_int @ A2 )
= ( uminus_uminus_int @ B2 ) )
= ( A2 = B2 ) ) ).
% neg_equal_iff_equal
thf(fact_1013_neg__equal__iff__equal,axiom,
! [A2: real,B2: real] :
( ( ( uminus_uminus_real @ A2 )
= ( uminus_uminus_real @ B2 ) )
= ( A2 = B2 ) ) ).
% neg_equal_iff_equal
thf(fact_1014_int_Ominus__zero,axiom,
( ( uminus_uminus_int @ zero_zero_int )
= zero_zero_int ) ).
% int.minus_zero
thf(fact_1015_real__add__minus__iff,axiom,
! [X: real,A2: real] :
( ( ( plus_plus_real @ X @ ( uminus_uminus_real @ A2 ) )
= zero_zero_real )
= ( X = A2 ) ) ).
% real_add_minus_iff
thf(fact_1016_neg__le__iff__le,axiom,
! [B2: int,A2: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ B2 ) @ ( uminus_uminus_int @ A2 ) )
= ( ord_less_eq_int @ A2 @ B2 ) ) ).
% neg_le_iff_le
thf(fact_1017_neg__le__iff__le,axiom,
! [B2: real,A2: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ B2 ) @ ( uminus_uminus_real @ A2 ) )
= ( ord_less_eq_real @ A2 @ B2 ) ) ).
% neg_le_iff_le
thf(fact_1018_add_Oinverse__neutral,axiom,
( ( uminus_uminus_int @ zero_zero_int )
= zero_zero_int ) ).
% add.inverse_neutral
thf(fact_1019_add_Oinverse__neutral,axiom,
( ( uminus_uminus_real @ zero_zero_real )
= zero_zero_real ) ).
% add.inverse_neutral
thf(fact_1020_neg__0__equal__iff__equal,axiom,
! [A2: int] :
( ( zero_zero_int
= ( uminus_uminus_int @ A2 ) )
= ( zero_zero_int = A2 ) ) ).
% neg_0_equal_iff_equal
thf(fact_1021_neg__0__equal__iff__equal,axiom,
! [A2: real] :
( ( zero_zero_real
= ( uminus_uminus_real @ A2 ) )
= ( zero_zero_real = A2 ) ) ).
% neg_0_equal_iff_equal
thf(fact_1022_neg__equal__0__iff__equal,axiom,
! [A2: int] :
( ( ( uminus_uminus_int @ A2 )
= zero_zero_int )
= ( A2 = zero_zero_int ) ) ).
% neg_equal_0_iff_equal
thf(fact_1023_neg__equal__0__iff__equal,axiom,
! [A2: real] :
( ( ( uminus_uminus_real @ A2 )
= zero_zero_real )
= ( A2 = zero_zero_real ) ) ).
% neg_equal_0_iff_equal
thf(fact_1024_equal__neg__zero,axiom,
! [A2: int] :
( ( A2
= ( uminus_uminus_int @ A2 ) )
= ( A2 = zero_zero_int ) ) ).
% equal_neg_zero
thf(fact_1025_equal__neg__zero,axiom,
! [A2: real] :
( ( A2
= ( uminus_uminus_real @ A2 ) )
= ( A2 = zero_zero_real ) ) ).
% equal_neg_zero
thf(fact_1026_neg__equal__zero,axiom,
! [A2: int] :
( ( ( uminus_uminus_int @ A2 )
= A2 )
= ( A2 = zero_zero_int ) ) ).
% neg_equal_zero
thf(fact_1027_neg__equal__zero,axiom,
! [A2: real] :
( ( ( uminus_uminus_real @ A2 )
= A2 )
= ( A2 = zero_zero_real ) ) ).
% neg_equal_zero
thf(fact_1028_neg__less__iff__less,axiom,
! [B2: int,A2: int] :
( ( ord_less_int @ ( uminus_uminus_int @ B2 ) @ ( uminus_uminus_int @ A2 ) )
= ( ord_less_int @ A2 @ B2 ) ) ).
% neg_less_iff_less
thf(fact_1029_neg__less__iff__less,axiom,
! [B2: real,A2: real] :
( ( ord_less_real @ ( uminus_uminus_real @ B2 ) @ ( uminus_uminus_real @ A2 ) )
= ( ord_less_real @ A2 @ B2 ) ) ).
% neg_less_iff_less
thf(fact_1030_add__minus__cancel,axiom,
! [A2: int,B2: int] :
( ( plus_plus_int @ A2 @ ( plus_plus_int @ ( uminus_uminus_int @ A2 ) @ B2 ) )
= B2 ) ).
% add_minus_cancel
thf(fact_1031_add__minus__cancel,axiom,
! [A2: real,B2: real] :
( ( plus_plus_real @ A2 @ ( plus_plus_real @ ( uminus_uminus_real @ A2 ) @ B2 ) )
= B2 ) ).
% add_minus_cancel
thf(fact_1032_minus__add__cancel,axiom,
! [A2: int,B2: int] :
( ( plus_plus_int @ ( uminus_uminus_int @ A2 ) @ ( plus_plus_int @ A2 @ B2 ) )
= B2 ) ).
% minus_add_cancel
thf(fact_1033_minus__add__cancel,axiom,
! [A2: real,B2: real] :
( ( plus_plus_real @ ( uminus_uminus_real @ A2 ) @ ( plus_plus_real @ A2 @ B2 ) )
= B2 ) ).
% minus_add_cancel
thf(fact_1034_minus__add__distrib,axiom,
! [A2: int,B2: int] :
( ( uminus_uminus_int @ ( plus_plus_int @ A2 @ B2 ) )
= ( plus_plus_int @ ( uminus_uminus_int @ A2 ) @ ( uminus_uminus_int @ B2 ) ) ) ).
% minus_add_distrib
thf(fact_1035_minus__add__distrib,axiom,
! [A2: real,B2: real] :
( ( uminus_uminus_real @ ( plus_plus_real @ A2 @ B2 ) )
= ( plus_plus_real @ ( uminus_uminus_real @ A2 ) @ ( uminus_uminus_real @ B2 ) ) ) ).
% minus_add_distrib
thf(fact_1036_of__int__minus,axiom,
! [Z2: int] :
( ( ring_1_of_int_int @ ( uminus_uminus_int @ Z2 ) )
= ( uminus_uminus_int @ ( ring_1_of_int_int @ Z2 ) ) ) ).
% of_int_minus
thf(fact_1037_of__int__minus,axiom,
! [Z2: int] :
( ( ring_1_of_int_real @ ( uminus_uminus_int @ Z2 ) )
= ( uminus_uminus_real @ ( ring_1_of_int_real @ Z2 ) ) ) ).
% of_int_minus
thf(fact_1038_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_1039_negative__zle,axiom,
! [N: nat,M: nat] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ ( semiri1314217659103216013at_int @ M ) ) ).
% negative_zle
thf(fact_1040_neg__less__eq__nonneg,axiom,
! [A2: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ A2 ) @ A2 )
= ( ord_less_eq_int @ zero_zero_int @ A2 ) ) ).
% neg_less_eq_nonneg
thf(fact_1041_neg__less__eq__nonneg,axiom,
! [A2: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ A2 ) @ A2 )
= ( ord_less_eq_real @ zero_zero_real @ A2 ) ) ).
% neg_less_eq_nonneg
thf(fact_1042_less__eq__neg__nonpos,axiom,
! [A2: int] :
( ( ord_less_eq_int @ A2 @ ( uminus_uminus_int @ A2 ) )
= ( ord_less_eq_int @ A2 @ zero_zero_int ) ) ).
% less_eq_neg_nonpos
thf(fact_1043_less__eq__neg__nonpos,axiom,
! [A2: real] :
( ( ord_less_eq_real @ A2 @ ( uminus_uminus_real @ A2 ) )
= ( ord_less_eq_real @ A2 @ zero_zero_real ) ) ).
% less_eq_neg_nonpos
thf(fact_1044_neg__le__0__iff__le,axiom,
! [A2: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ A2 ) @ zero_zero_int )
= ( ord_less_eq_int @ zero_zero_int @ A2 ) ) ).
% neg_le_0_iff_le
thf(fact_1045_neg__le__0__iff__le,axiom,
! [A2: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ A2 ) @ zero_zero_real )
= ( ord_less_eq_real @ zero_zero_real @ A2 ) ) ).
% neg_le_0_iff_le
thf(fact_1046_neg__0__le__iff__le,axiom,
! [A2: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ A2 ) )
= ( ord_less_eq_int @ A2 @ zero_zero_int ) ) ).
% neg_0_le_iff_le
thf(fact_1047_neg__0__le__iff__le,axiom,
! [A2: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( uminus_uminus_real @ A2 ) )
= ( ord_less_eq_real @ A2 @ zero_zero_real ) ) ).
% neg_0_le_iff_le
thf(fact_1048_neg__less__0__iff__less,axiom,
! [A2: int] :
( ( ord_less_int @ ( uminus_uminus_int @ A2 ) @ zero_zero_int )
= ( ord_less_int @ zero_zero_int @ A2 ) ) ).
% neg_less_0_iff_less
thf(fact_1049_neg__less__0__iff__less,axiom,
! [A2: real] :
( ( ord_less_real @ ( uminus_uminus_real @ A2 ) @ zero_zero_real )
= ( ord_less_real @ zero_zero_real @ A2 ) ) ).
% neg_less_0_iff_less
thf(fact_1050_neg__0__less__iff__less,axiom,
! [A2: int] :
( ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ A2 ) )
= ( ord_less_int @ A2 @ zero_zero_int ) ) ).
% neg_0_less_iff_less
thf(fact_1051_neg__0__less__iff__less,axiom,
! [A2: real] :
( ( ord_less_real @ zero_zero_real @ ( uminus_uminus_real @ A2 ) )
= ( ord_less_real @ A2 @ zero_zero_real ) ) ).
% neg_0_less_iff_less
thf(fact_1052_neg__less__pos,axiom,
! [A2: int] :
( ( ord_less_int @ ( uminus_uminus_int @ A2 ) @ A2 )
= ( ord_less_int @ zero_zero_int @ A2 ) ) ).
% neg_less_pos
thf(fact_1053_neg__less__pos,axiom,
! [A2: real] :
( ( ord_less_real @ ( uminus_uminus_real @ A2 ) @ A2 )
= ( ord_less_real @ zero_zero_real @ A2 ) ) ).
% neg_less_pos
thf(fact_1054_less__neg__neg,axiom,
! [A2: int] :
( ( ord_less_int @ A2 @ ( uminus_uminus_int @ A2 ) )
= ( ord_less_int @ A2 @ zero_zero_int ) ) ).
% less_neg_neg
thf(fact_1055_less__neg__neg,axiom,
! [A2: real] :
( ( ord_less_real @ A2 @ ( uminus_uminus_real @ A2 ) )
= ( ord_less_real @ A2 @ zero_zero_real ) ) ).
% less_neg_neg
thf(fact_1056_ab__left__minus,axiom,
! [A2: int] :
( ( plus_plus_int @ ( uminus_uminus_int @ A2 ) @ A2 )
= zero_zero_int ) ).
% ab_left_minus
thf(fact_1057_ab__left__minus,axiom,
! [A2: real] :
( ( plus_plus_real @ ( uminus_uminus_real @ A2 ) @ A2 )
= zero_zero_real ) ).
% ab_left_minus
thf(fact_1058_add_Oright__inverse,axiom,
! [A2: int] :
( ( plus_plus_int @ A2 @ ( uminus_uminus_int @ A2 ) )
= zero_zero_int ) ).
% add.right_inverse
thf(fact_1059_add_Oright__inverse,axiom,
! [A2: real] :
( ( plus_plus_real @ A2 @ ( uminus_uminus_real @ A2 ) )
= zero_zero_real ) ).
% add.right_inverse
thf(fact_1060_floor__uminus__of__int,axiom,
! [Z2: int] :
( ( archim6058952711729229775r_real @ ( uminus_uminus_real @ ( ring_1_of_int_real @ Z2 ) ) )
= ( uminus_uminus_int @ Z2 ) ) ).
% floor_uminus_of_int
thf(fact_1061_dbl__inc__simps_I4_J,axiom,
( ( neg_nu5851722552734809277nc_int @ ( uminus_uminus_int @ one_one_int ) )
= ( uminus_uminus_int @ one_one_int ) ) ).
% dbl_inc_simps(4)
thf(fact_1062_dbl__inc__simps_I4_J,axiom,
( ( neg_nu8295874005876285629c_real @ ( uminus_uminus_real @ one_one_real ) )
= ( uminus_uminus_real @ one_one_real ) ) ).
% dbl_inc_simps(4)
thf(fact_1063_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_1064_add__neg__numeral__special_I7_J,axiom,
( ( plus_plus_real @ one_one_real @ ( uminus_uminus_real @ one_one_real ) )
= zero_zero_real ) ).
% add_neg_numeral_special(7)
thf(fact_1065_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_1066_add__neg__numeral__special_I8_J,axiom,
( ( plus_plus_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real )
= zero_zero_real ) ).
% add_neg_numeral_special(8)
thf(fact_1067_dbl__dec__simps_I2_J,axiom,
( ( neg_nu3811975205180677377ec_int @ zero_zero_int )
= ( uminus_uminus_int @ one_one_int ) ) ).
% dbl_dec_simps(2)
thf(fact_1068_dbl__dec__simps_I2_J,axiom,
( ( neg_nu6075765906172075777c_real @ zero_zero_real )
= ( uminus_uminus_real @ one_one_real ) ) ).
% dbl_dec_simps(2)
thf(fact_1069_zero__natural_Orsp,axiom,
zero_zero_nat = zero_zero_nat ).
% zero_natural.rsp
thf(fact_1070_verit__negate__coefficient_I3_J,axiom,
! [A2: int,B2: int] :
( ( A2 = B2 )
=> ( ( uminus_uminus_int @ A2 )
= ( uminus_uminus_int @ B2 ) ) ) ).
% verit_negate_coefficient(3)
thf(fact_1071_verit__negate__coefficient_I3_J,axiom,
! [A2: real,B2: real] :
( ( A2 = B2 )
=> ( ( uminus_uminus_real @ A2 )
= ( uminus_uminus_real @ B2 ) ) ) ).
% verit_negate_coefficient(3)
thf(fact_1072_equation__minus__iff,axiom,
! [A2: int,B2: int] :
( ( A2
= ( uminus_uminus_int @ B2 ) )
= ( B2
= ( uminus_uminus_int @ A2 ) ) ) ).
% equation_minus_iff
thf(fact_1073_equation__minus__iff,axiom,
! [A2: real,B2: real] :
( ( A2
= ( uminus_uminus_real @ B2 ) )
= ( B2
= ( uminus_uminus_real @ A2 ) ) ) ).
% equation_minus_iff
thf(fact_1074_minus__equation__iff,axiom,
! [A2: int,B2: int] :
( ( ( uminus_uminus_int @ A2 )
= B2 )
= ( ( uminus_uminus_int @ B2 )
= A2 ) ) ).
% minus_equation_iff
thf(fact_1075_minus__equation__iff,axiom,
! [A2: real,B2: real] :
( ( ( uminus_uminus_real @ A2 )
= B2 )
= ( ( uminus_uminus_real @ B2 )
= A2 ) ) ).
% minus_equation_iff
thf(fact_1076_ceiling__minus,axiom,
! [X: real] :
( ( archim7802044766580827645g_real @ ( uminus_uminus_real @ X ) )
= ( uminus_uminus_int @ ( archim6058952711729229775r_real @ X ) ) ) ).
% ceiling_minus
thf(fact_1077_floor__minus,axiom,
! [X: real] :
( ( archim6058952711729229775r_real @ ( uminus_uminus_real @ X ) )
= ( uminus_uminus_int @ ( archim7802044766580827645g_real @ X ) ) ) ).
% floor_minus
thf(fact_1078_ceiling__def,axiom,
( archim7802044766580827645g_real
= ( ^ [X2: real] : ( uminus_uminus_int @ ( archim6058952711729229775r_real @ ( uminus_uminus_real @ X2 ) ) ) ) ) ).
% ceiling_def
thf(fact_1079_le__minus__iff,axiom,
! [A2: int,B2: int] :
( ( ord_less_eq_int @ A2 @ ( uminus_uminus_int @ B2 ) )
= ( ord_less_eq_int @ B2 @ ( uminus_uminus_int @ A2 ) ) ) ).
% le_minus_iff
thf(fact_1080_le__minus__iff,axiom,
! [A2: real,B2: real] :
( ( ord_less_eq_real @ A2 @ ( uminus_uminus_real @ B2 ) )
= ( ord_less_eq_real @ B2 @ ( uminus_uminus_real @ A2 ) ) ) ).
% le_minus_iff
thf(fact_1081_minus__le__iff,axiom,
! [A2: int,B2: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ A2 ) @ B2 )
= ( ord_less_eq_int @ ( uminus_uminus_int @ B2 ) @ A2 ) ) ).
% minus_le_iff
thf(fact_1082_minus__le__iff,axiom,
! [A2: real,B2: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ A2 ) @ B2 )
= ( ord_less_eq_real @ ( uminus_uminus_real @ B2 ) @ A2 ) ) ).
% minus_le_iff
thf(fact_1083_le__imp__neg__le,axiom,
! [A2: int,B2: int] :
( ( ord_less_eq_int @ A2 @ B2 )
=> ( ord_less_eq_int @ ( uminus_uminus_int @ B2 ) @ ( uminus_uminus_int @ A2 ) ) ) ).
% le_imp_neg_le
thf(fact_1084_le__imp__neg__le,axiom,
! [A2: real,B2: real] :
( ( ord_less_eq_real @ A2 @ B2 )
=> ( ord_less_eq_real @ ( uminus_uminus_real @ B2 ) @ ( uminus_uminus_real @ A2 ) ) ) ).
% le_imp_neg_le
thf(fact_1085_minus__less__iff,axiom,
! [A2: int,B2: int] :
( ( ord_less_int @ ( uminus_uminus_int @ A2 ) @ B2 )
= ( ord_less_int @ ( uminus_uminus_int @ B2 ) @ A2 ) ) ).
% minus_less_iff
thf(fact_1086_minus__less__iff,axiom,
! [A2: real,B2: real] :
( ( ord_less_real @ ( uminus_uminus_real @ A2 ) @ B2 )
= ( ord_less_real @ ( uminus_uminus_real @ B2 ) @ A2 ) ) ).
% minus_less_iff
thf(fact_1087_less__minus__iff,axiom,
! [A2: int,B2: int] :
( ( ord_less_int @ A2 @ ( uminus_uminus_int @ B2 ) )
= ( ord_less_int @ B2 @ ( uminus_uminus_int @ A2 ) ) ) ).
% less_minus_iff
thf(fact_1088_less__minus__iff,axiom,
! [A2: real,B2: real] :
( ( ord_less_real @ A2 @ ( uminus_uminus_real @ B2 ) )
= ( ord_less_real @ B2 @ ( uminus_uminus_real @ A2 ) ) ) ).
% less_minus_iff
thf(fact_1089_verit__negate__coefficient_I2_J,axiom,
! [A2: int,B2: int] :
( ( ord_less_int @ A2 @ B2 )
=> ( ord_less_int @ ( uminus_uminus_int @ B2 ) @ ( uminus_uminus_int @ A2 ) ) ) ).
% verit_negate_coefficient(2)
thf(fact_1090_verit__negate__coefficient_I2_J,axiom,
! [A2: real,B2: real] :
( ( ord_less_real @ A2 @ B2 )
=> ( ord_less_real @ ( uminus_uminus_real @ B2 ) @ ( uminus_uminus_real @ A2 ) ) ) ).
% verit_negate_coefficient(2)
thf(fact_1091_is__num__normalize_I8_J,axiom,
! [A2: int,B2: int] :
( ( uminus_uminus_int @ ( plus_plus_int @ A2 @ B2 ) )
= ( plus_plus_int @ ( uminus_uminus_int @ B2 ) @ ( uminus_uminus_int @ A2 ) ) ) ).
% is_num_normalize(8)
thf(fact_1092_is__num__normalize_I8_J,axiom,
! [A2: real,B2: real] :
( ( uminus_uminus_real @ ( plus_plus_real @ A2 @ B2 ) )
= ( plus_plus_real @ ( uminus_uminus_real @ B2 ) @ ( uminus_uminus_real @ A2 ) ) ) ).
% is_num_normalize(8)
thf(fact_1093_group__cancel_Oneg1,axiom,
! [A3: int,K: int,A2: int] :
( ( A3
= ( plus_plus_int @ K @ A2 ) )
=> ( ( uminus_uminus_int @ A3 )
= ( plus_plus_int @ ( uminus_uminus_int @ K ) @ ( uminus_uminus_int @ A2 ) ) ) ) ).
% group_cancel.neg1
thf(fact_1094_group__cancel_Oneg1,axiom,
! [A3: real,K: real,A2: real] :
( ( A3
= ( plus_plus_real @ K @ A2 ) )
=> ( ( uminus_uminus_real @ A3 )
= ( plus_plus_real @ ( uminus_uminus_real @ K ) @ ( uminus_uminus_real @ A2 ) ) ) ) ).
% group_cancel.neg1
thf(fact_1095_add_Oinverse__distrib__swap,axiom,
! [A2: int,B2: int] :
( ( uminus_uminus_int @ ( plus_plus_int @ A2 @ B2 ) )
= ( plus_plus_int @ ( uminus_uminus_int @ B2 ) @ ( uminus_uminus_int @ A2 ) ) ) ).
% add.inverse_distrib_swap
thf(fact_1096_add_Oinverse__distrib__swap,axiom,
! [A2: real,B2: real] :
( ( uminus_uminus_real @ ( plus_plus_real @ A2 @ B2 ) )
= ( plus_plus_real @ ( uminus_uminus_real @ B2 ) @ ( uminus_uminus_real @ A2 ) ) ) ).
% add.inverse_distrib_swap
thf(fact_1097_one__neq__neg__one,axiom,
( one_one_int
!= ( uminus_uminus_int @ one_one_int ) ) ).
% one_neq_neg_one
thf(fact_1098_one__neq__neg__one,axiom,
( one_one_real
!= ( uminus_uminus_real @ one_one_real ) ) ).
% one_neq_neg_one
thf(fact_1099_int__cases2,axiom,
! [Z2: int] :
( ! [N2: nat] :
( Z2
!= ( semiri1314217659103216013at_int @ N2 ) )
=> ~ ! [N2: nat] :
( Z2
!= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) ) ) ).
% int_cases2
thf(fact_1100_add__eq__0__iff,axiom,
! [A2: int,B2: int] :
( ( ( plus_plus_int @ A2 @ B2 )
= zero_zero_int )
= ( B2
= ( uminus_uminus_int @ A2 ) ) ) ).
% add_eq_0_iff
thf(fact_1101_add__eq__0__iff,axiom,
! [A2: real,B2: real] :
( ( ( plus_plus_real @ A2 @ B2 )
= zero_zero_real )
= ( B2
= ( uminus_uminus_real @ A2 ) ) ) ).
% add_eq_0_iff
thf(fact_1102_ab__group__add__class_Oab__left__minus,axiom,
! [A2: int] :
( ( plus_plus_int @ ( uminus_uminus_int @ A2 ) @ A2 )
= zero_zero_int ) ).
% ab_group_add_class.ab_left_minus
thf(fact_1103_ab__group__add__class_Oab__left__minus,axiom,
! [A2: real] :
( ( plus_plus_real @ ( uminus_uminus_real @ A2 ) @ A2 )
= zero_zero_real ) ).
% ab_group_add_class.ab_left_minus
thf(fact_1104_add_Oinverse__unique,axiom,
! [A2: int,B2: int] :
( ( ( plus_plus_int @ A2 @ B2 )
= zero_zero_int )
=> ( ( uminus_uminus_int @ A2 )
= B2 ) ) ).
% add.inverse_unique
thf(fact_1105_add_Oinverse__unique,axiom,
! [A2: real,B2: real] :
( ( ( plus_plus_real @ A2 @ B2 )
= zero_zero_real )
=> ( ( uminus_uminus_real @ A2 )
= B2 ) ) ).
% add.inverse_unique
thf(fact_1106_eq__neg__iff__add__eq__0,axiom,
! [A2: int,B2: int] :
( ( A2
= ( uminus_uminus_int @ B2 ) )
= ( ( plus_plus_int @ A2 @ B2 )
= zero_zero_int ) ) ).
% eq_neg_iff_add_eq_0
thf(fact_1107_eq__neg__iff__add__eq__0,axiom,
! [A2: real,B2: real] :
( ( A2
= ( uminus_uminus_real @ B2 ) )
= ( ( plus_plus_real @ A2 @ B2 )
= zero_zero_real ) ) ).
% eq_neg_iff_add_eq_0
thf(fact_1108_neg__eq__iff__add__eq__0,axiom,
! [A2: int,B2: int] :
( ( ( uminus_uminus_int @ A2 )
= B2 )
= ( ( plus_plus_int @ A2 @ B2 )
= zero_zero_int ) ) ).
% neg_eq_iff_add_eq_0
thf(fact_1109_neg__eq__iff__add__eq__0,axiom,
! [A2: real,B2: real] :
( ( ( uminus_uminus_real @ A2 )
= B2 )
= ( ( plus_plus_real @ A2 @ B2 )
= zero_zero_real ) ) ).
% neg_eq_iff_add_eq_0
thf(fact_1110_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_1111_le__minus__one__simps_I2_J,axiom,
ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real ).
% le_minus_one_simps(2)
thf(fact_1112_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_1113_le__minus__one__simps_I4_J,axiom,
~ ( ord_less_eq_real @ one_one_real @ ( uminus_uminus_real @ one_one_real ) ) ).
% le_minus_one_simps(4)
thf(fact_1114_zero__neq__neg__one,axiom,
( zero_zero_int
!= ( uminus_uminus_int @ one_one_int ) ) ).
% zero_neq_neg_one
thf(fact_1115_zero__neq__neg__one,axiom,
( zero_zero_real
!= ( uminus_uminus_real @ one_one_real ) ) ).
% zero_neq_neg_one
thf(fact_1116_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_1117_less__minus__one__simps_I2_J,axiom,
ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real ).
% less_minus_one_simps(2)
thf(fact_1118_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_1119_less__minus__one__simps_I4_J,axiom,
~ ( ord_less_real @ one_one_real @ ( uminus_uminus_real @ one_one_real ) ) ).
% less_minus_one_simps(4)
thf(fact_1120_int__cases4,axiom,
! [M: int] :
( ! [N2: nat] :
( M
!= ( semiri1314217659103216013at_int @ N2 ) )
=> ~ ! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( M
!= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) ) ) ) ).
% int_cases4
thf(fact_1121_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_1122_real__add__less__0__iff,axiom,
! [X: real,Y2: real] :
( ( ord_less_real @ ( plus_plus_real @ X @ Y2 ) @ zero_zero_real )
= ( ord_less_real @ Y2 @ ( uminus_uminus_real @ X ) ) ) ).
% real_add_less_0_iff
thf(fact_1123_real__0__less__add__iff,axiom,
! [X: real,Y2: real] :
( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ X @ Y2 ) )
= ( ord_less_real @ ( uminus_uminus_real @ X ) @ Y2 ) ) ).
% real_0_less_add_iff
thf(fact_1124_real__0__le__add__iff,axiom,
! [X: real,Y2: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ X @ Y2 ) )
= ( ord_less_eq_real @ ( uminus_uminus_real @ X ) @ Y2 ) ) ).
% real_0_le_add_iff
thf(fact_1125_real__add__le__0__iff,axiom,
! [X: real,Y2: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ X @ Y2 ) @ zero_zero_real )
= ( ord_less_eq_real @ Y2 @ ( uminus_uminus_real @ X ) ) ) ).
% real_add_le_0_iff
thf(fact_1126_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_1127_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_1128_le__minus__one__simps_I1_J,axiom,
ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ zero_zero_real ).
% le_minus_one_simps(1)
thf(fact_1129_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_1130_le__minus__one__simps_I3_J,axiom,
~ ( ord_less_eq_real @ zero_zero_real @ ( uminus_uminus_real @ one_one_real ) ) ).
% le_minus_one_simps(3)
thf(fact_1131_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_1132_less__minus__one__simps_I1_J,axiom,
ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ zero_zero_real ).
% less_minus_one_simps(1)
thf(fact_1133_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_1134_less__minus__one__simps_I3_J,axiom,
~ ( ord_less_real @ zero_zero_real @ ( uminus_uminus_real @ one_one_real ) ) ).
% less_minus_one_simps(3)
thf(fact_1135_int__cases3,axiom,
! [K: int] :
( ( K != zero_zero_int )
=> ( ! [N2: nat] :
( ( K
= ( semiri1314217659103216013at_int @ N2 ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N2 ) )
=> ~ ! [N2: nat] :
( ( K
= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ) ).
% int_cases3
thf(fact_1136_negative__zle__0,axiom,
! [N: nat] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ zero_zero_int ) ).
% negative_zle_0
thf(fact_1137_nonpos__int__cases,axiom,
! [K: int] :
( ( ord_less_eq_int @ K @ zero_zero_int )
=> ~ ! [N2: nat] :
( K
!= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) ) ) ).
% nonpos_int_cases
thf(fact_1138_neg__int__cases,axiom,
! [K: int] :
( ( ord_less_int @ K @ zero_zero_int )
=> ~ ! [N2: nat] :
( ( K
= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% neg_int_cases
thf(fact_1139_Euclid__induct,axiom,
! [P: nat > nat > $o,A2: nat,B2: nat] :
( ! [A4: nat,B4: nat] :
( ( P @ A4 @ B4 )
= ( P @ B4 @ A4 ) )
=> ( ! [A4: nat] : ( P @ A4 @ zero_zero_nat )
=> ( ! [A4: nat,B4: nat] :
( ( P @ A4 @ B4 )
=> ( P @ A4 @ ( plus_plus_nat @ A4 @ B4 ) ) )
=> ( P @ A2 @ B2 ) ) ) ) ).
% Euclid_induct
thf(fact_1140_zfact__iso__inv__0,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( zfact_iso_inv @ N @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ N ) ) ) )
= zero_zero_nat ) ) ).
% zfact_iso_inv_0
thf(fact_1141_real__archimedian__rdiv__eq__0,axiom,
! [X: real,C: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ! [M4: nat] :
( ( ord_less_nat @ zero_zero_nat @ M4 )
=> ( ord_less_eq_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ M4 ) @ X ) @ C ) )
=> ( X = zero_zero_real ) ) ) ) ).
% real_archimedian_rdiv_eq_0
thf(fact_1142_nat__int,axiom,
! [N: nat] :
( ( nat2 @ ( semiri1314217659103216013at_int @ N ) )
= N ) ).
% nat_int
thf(fact_1143_nat__0__iff,axiom,
! [I: int] :
( ( ( nat2 @ I )
= zero_zero_nat )
= ( ord_less_eq_int @ I @ zero_zero_int ) ) ).
% nat_0_iff
thf(fact_1144_nat__le__0,axiom,
! [Z2: int] :
( ( ord_less_eq_int @ Z2 @ zero_zero_int )
=> ( ( nat2 @ Z2 )
= zero_zero_nat ) ) ).
% nat_le_0
thf(fact_1145_zless__nat__conj,axiom,
! [W: int,Z2: int] :
( ( ord_less_nat @ ( nat2 @ W ) @ ( nat2 @ Z2 ) )
= ( ( ord_less_int @ zero_zero_int @ Z2 )
& ( ord_less_int @ W @ Z2 ) ) ) ).
% zless_nat_conj
thf(fact_1146_int__nat__eq,axiom,
! [Z2: int] :
( ( ( ord_less_eq_int @ zero_zero_int @ Z2 )
=> ( ( semiri1314217659103216013at_int @ ( nat2 @ Z2 ) )
= Z2 ) )
& ( ~ ( ord_less_eq_int @ zero_zero_int @ Z2 )
=> ( ( semiri1314217659103216013at_int @ ( nat2 @ Z2 ) )
= zero_zero_int ) ) ) ).
% int_nat_eq
thf(fact_1147_nat__zminus__int,axiom,
! [N: nat] :
( ( nat2 @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) )
= zero_zero_nat ) ).
% nat_zminus_int
thf(fact_1148_nat__ceiling__le__eq,axiom,
! [X: real,A2: nat] :
( ( ord_less_eq_nat @ ( nat2 @ ( archim7802044766580827645g_real @ X ) ) @ A2 )
= ( ord_less_eq_real @ X @ ( semiri5074537144036343181t_real @ A2 ) ) ) ).
% nat_ceiling_le_eq
thf(fact_1149_zero__less__nat__eq,axiom,
! [Z2: int] :
( ( ord_less_nat @ zero_zero_nat @ ( nat2 @ Z2 ) )
= ( ord_less_int @ zero_zero_int @ Z2 ) ) ).
% zero_less_nat_eq
thf(fact_1150_nat__zero__as__int,axiom,
( zero_zero_nat
= ( nat2 @ zero_zero_int ) ) ).
% nat_zero_as_int
thf(fact_1151_eq__nat__nat__iff,axiom,
! [Z2: int,Z6: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ Z6 )
=> ( ( ( nat2 @ Z2 )
= ( nat2 @ Z6 ) )
= ( Z2 = Z6 ) ) ) ) ).
% eq_nat_nat_iff
thf(fact_1152_all__nat,axiom,
( ( ^ [P3: nat > $o] :
! [X5: nat] : ( P3 @ X5 ) )
= ( ^ [P4: nat > $o] :
! [X2: int] :
( ( ord_less_eq_int @ zero_zero_int @ X2 )
=> ( P4 @ ( nat2 @ X2 ) ) ) ) ) ).
% all_nat
thf(fact_1153_ex__nat,axiom,
( ( ^ [P3: nat > $o] :
? [X5: nat] : ( P3 @ X5 ) )
= ( ^ [P4: nat > $o] :
? [X2: int] :
( ( ord_less_eq_int @ zero_zero_int @ X2 )
& ( P4 @ ( nat2 @ X2 ) ) ) ) ) ).
% ex_nat
thf(fact_1154_nat__mono,axiom,
! [X: int,Y2: int] :
( ( ord_less_eq_int @ X @ Y2 )
=> ( ord_less_eq_nat @ ( nat2 @ X ) @ ( nat2 @ Y2 ) ) ) ).
% nat_mono
thf(fact_1155_nat__one__as__int,axiom,
( one_one_nat
= ( nat2 @ one_one_int ) ) ).
% nat_one_as_int
thf(fact_1156_real__nat__ceiling__ge,axiom,
! [X: real] : ( ord_less_eq_real @ X @ ( semiri5074537144036343181t_real @ ( nat2 @ ( archim7802044766580827645g_real @ X ) ) ) ) ).
% real_nat_ceiling_ge
thf(fact_1157_real__minus__mult__self__le,axiom,
! [U: real,X: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( times_times_real @ U @ U ) ) @ ( times_times_real @ X @ X ) ) ).
% real_minus_mult_self_le
thf(fact_1158_nat__mono__iff,axiom,
! [Z2: int,W: int] :
( ( ord_less_int @ zero_zero_int @ Z2 )
=> ( ( ord_less_nat @ ( nat2 @ W ) @ ( nat2 @ Z2 ) )
= ( ord_less_int @ W @ Z2 ) ) ) ).
% nat_mono_iff
thf(fact_1159_nat__0__le,axiom,
! [Z2: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z2 )
=> ( ( semiri1314217659103216013at_int @ ( nat2 @ Z2 ) )
= Z2 ) ) ).
% nat_0_le
thf(fact_1160_int__eq__iff,axiom,
! [M: nat,Z2: int] :
( ( ( semiri1314217659103216013at_int @ M )
= Z2 )
= ( ( M
= ( nat2 @ Z2 ) )
& ( ord_less_eq_int @ zero_zero_int @ Z2 ) ) ) ).
% int_eq_iff
thf(fact_1161_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_1162_nat__int__add,axiom,
! [A2: nat,B2: nat] :
( ( nat2 @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ A2 ) @ ( semiri1314217659103216013at_int @ B2 ) ) )
= ( plus_plus_nat @ A2 @ B2 ) ) ).
% nat_int_add
thf(fact_1163_zless__nat__eq__int__zless,axiom,
! [M: nat,Z2: int] :
( ( ord_less_nat @ M @ ( nat2 @ Z2 ) )
= ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ Z2 ) ) ).
% zless_nat_eq_int_zless
thf(fact_1164_reals__Archimedean3,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ! [Y4: real] :
? [N2: nat] : ( ord_less_real @ Y4 @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ X ) ) ) ).
% reals_Archimedean3
thf(fact_1165_nat__add__distrib,axiom,
! [Z2: int,Z6: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ Z6 )
=> ( ( nat2 @ ( plus_plus_int @ Z2 @ Z6 ) )
= ( plus_plus_nat @ ( nat2 @ Z2 ) @ ( nat2 @ Z6 ) ) ) ) ) ).
% nat_add_distrib
thf(fact_1166_nat__eq__iff2,axiom,
! [M: nat,W: int] :
( ( M
= ( nat2 @ W ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ W )
=> ( W
= ( semiri1314217659103216013at_int @ M ) ) )
& ( ~ ( ord_less_eq_int @ zero_zero_int @ W )
=> ( M = zero_zero_nat ) ) ) ) ).
% nat_eq_iff2
thf(fact_1167_nat__eq__iff,axiom,
! [W: int,M: nat] :
( ( ( nat2 @ W )
= M )
= ( ( ( ord_less_eq_int @ zero_zero_int @ W )
=> ( W
= ( semiri1314217659103216013at_int @ M ) ) )
& ( ~ ( ord_less_eq_int @ zero_zero_int @ W )
=> ( M = zero_zero_nat ) ) ) ) ).
% nat_eq_iff
thf(fact_1168_nat__less__eq__zless,axiom,
! [W: int,Z2: int] :
( ( ord_less_eq_int @ zero_zero_int @ W )
=> ( ( ord_less_nat @ ( nat2 @ W ) @ ( nat2 @ Z2 ) )
= ( ord_less_int @ W @ Z2 ) ) ) ).
% nat_less_eq_zless
thf(fact_1169_nat__le__eq__zle,axiom,
! [W: int,Z2: int] :
( ( ( ord_less_int @ zero_zero_int @ W )
| ( ord_less_eq_int @ zero_zero_int @ Z2 ) )
=> ( ( ord_less_eq_nat @ ( nat2 @ W ) @ ( nat2 @ Z2 ) )
= ( ord_less_eq_int @ W @ Z2 ) ) ) ).
% nat_le_eq_zle
thf(fact_1170_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_1171_split__nat,axiom,
! [P: nat > $o,I: int] :
( ( P @ ( nat2 @ I ) )
= ( ! [N3: nat] :
( ( I
= ( semiri1314217659103216013at_int @ N3 ) )
=> ( P @ N3 ) )
& ( ( ord_less_int @ I @ zero_zero_int )
=> ( P @ zero_zero_nat ) ) ) ) ).
% split_nat
thf(fact_1172_nat__floor__neg,axiom,
! [X: real] :
( ( ord_less_eq_real @ X @ zero_zero_real )
=> ( ( nat2 @ ( archim6058952711729229775r_real @ X ) )
= zero_zero_nat ) ) ).
% nat_floor_neg
thf(fact_1173_le__nat__floor,axiom,
! [X: nat,A2: real] :
( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ X ) @ A2 )
=> ( ord_less_eq_nat @ X @ ( nat2 @ ( archim6058952711729229775r_real @ A2 ) ) ) ) ).
% le_nat_floor
thf(fact_1174_nat__less__iff,axiom,
! [W: int,M: nat] :
( ( ord_less_eq_int @ zero_zero_int @ W )
=> ( ( ord_less_nat @ ( nat2 @ W ) @ M )
= ( ord_less_int @ W @ ( semiri1314217659103216013at_int @ M ) ) ) ) ).
% nat_less_iff
thf(fact_1175_not__real__square__gt__zero,axiom,
! [X: real] :
( ( ~ ( ord_less_real @ zero_zero_real @ ( times_times_real @ X @ X ) ) )
= ( X = zero_zero_real ) ) ).
% not_real_square_gt_zero
thf(fact_1176_zfact__iso__0,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ring_zfact_iso @ N @ zero_zero_nat )
= ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ N ) ) ) ) ) ).
% zfact_iso_0
thf(fact_1177_mult__is__0,axiom,
! [M: nat,N: nat] :
( ( ( times_times_nat @ M @ N )
= zero_zero_nat )
= ( ( M = zero_zero_nat )
| ( N = zero_zero_nat ) ) ) ).
% mult_is_0
thf(fact_1178_mult__0__right,axiom,
! [M: nat] :
( ( times_times_nat @ M @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_0_right
thf(fact_1179_mult__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( times_times_nat @ K @ M )
= ( times_times_nat @ K @ N ) )
= ( ( M = N )
| ( K = zero_zero_nat ) ) ) ).
% mult_cancel1
thf(fact_1180_mult__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ( times_times_nat @ M @ K )
= ( times_times_nat @ N @ K ) )
= ( ( M = N )
| ( K = zero_zero_nat ) ) ) ).
% mult_cancel2
thf(fact_1181_nat__1__eq__mult__iff,axiom,
! [M: nat,N: nat] :
( ( one_one_nat
= ( times_times_nat @ M @ N ) )
= ( ( M = one_one_nat )
& ( N = one_one_nat ) ) ) ).
% nat_1_eq_mult_iff
thf(fact_1182_nat__mult__eq__1__iff,axiom,
! [M: nat,N: nat] :
( ( ( times_times_nat @ M @ N )
= one_one_nat )
= ( ( M = one_one_nat )
& ( N = one_one_nat ) ) ) ).
% nat_mult_eq_1_iff
thf(fact_1183_nat__0__less__mult__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ M @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ M )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% nat_0_less_mult_iff
thf(fact_1184_mult__less__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
& ( ord_less_nat @ M @ N ) ) ) ).
% mult_less_cancel2
thf(fact_1185_mult__le__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_eq_nat @ M @ N ) ) ) ).
% mult_le_cancel2
thf(fact_1186_nat__mult__distrib,axiom,
! [Z2: int,Z6: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z2 )
=> ( ( nat2 @ ( times_times_int @ Z2 @ Z6 ) )
= ( times_times_nat @ ( nat2 @ Z2 ) @ ( nat2 @ Z6 ) ) ) ) ).
% nat_mult_distrib
thf(fact_1187_mult__0,axiom,
! [N: nat] :
( ( times_times_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% mult_0
thf(fact_1188_times__int__code_I2_J,axiom,
! [L: int] :
( ( times_times_int @ zero_zero_int @ L )
= zero_zero_int ) ).
% times_int_code(2)
thf(fact_1189_times__int__code_I1_J,axiom,
! [K: int] :
( ( times_times_int @ K @ zero_zero_int )
= zero_zero_int ) ).
% times_int_code(1)
thf(fact_1190_int__ops_I7_J,axiom,
! [A2: nat,B2: nat] :
( ( semiri1314217659103216013at_int @ ( times_times_nat @ A2 @ B2 ) )
= ( times_times_int @ ( semiri1314217659103216013at_int @ A2 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ).
% int_ops(7)
thf(fact_1191_nat__mult__1,axiom,
! [N: nat] :
( ( times_times_nat @ one_one_nat @ N )
= N ) ).
% nat_mult_1
thf(fact_1192_nat__mult__1__right,axiom,
! [N: nat] :
( ( times_times_nat @ N @ one_one_nat )
= N ) ).
% nat_mult_1_right
thf(fact_1193_add__mult__distrib,axiom,
! [M: nat,N: nat,K: nat] :
( ( times_times_nat @ ( plus_plus_nat @ M @ N ) @ K )
= ( plus_plus_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) ) ) ).
% add_mult_distrib
thf(fact_1194_add__mult__distrib2,axiom,
! [K: nat,M: nat,N: nat] :
( ( times_times_nat @ K @ ( plus_plus_nat @ M @ N ) )
= ( plus_plus_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) ) ) ).
% add_mult_distrib2
thf(fact_1195_le__cube,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ ( times_times_nat @ M @ M ) ) ) ).
% le_cube
thf(fact_1196_le__square,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ M ) ) ).
% le_square
thf(fact_1197_mult__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 @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ L ) ) ) ) ).
% mult_le_mono
thf(fact_1198_mult__le__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ K ) ) ) ).
% mult_le_mono1
thf(fact_1199_mult__le__mono2,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( times_times_nat @ K @ I ) @ ( times_times_nat @ K @ J ) ) ) ).
% mult_le_mono2
thf(fact_1200_int__distrib_I1_J,axiom,
! [Z1: int,Z22: int,W: int] :
( ( times_times_int @ ( plus_plus_int @ Z1 @ Z22 ) @ W )
= ( plus_plus_int @ ( times_times_int @ Z1 @ W ) @ ( times_times_int @ Z22 @ W ) ) ) ).
% int_distrib(1)
thf(fact_1201_int__distrib_I2_J,axiom,
! [W: int,Z1: int,Z22: int] :
( ( times_times_int @ W @ ( plus_plus_int @ Z1 @ Z22 ) )
= ( plus_plus_int @ ( times_times_int @ W @ Z1 ) @ ( times_times_int @ W @ Z22 ) ) ) ).
% int_distrib(2)
thf(fact_1202_nat__mult__distrib__neg,axiom,
! [Z2: int,Z6: int] :
( ( ord_less_eq_int @ Z2 @ zero_zero_int )
=> ( ( nat2 @ ( times_times_int @ Z2 @ Z6 ) )
= ( times_times_nat @ ( nat2 @ ( uminus_uminus_int @ Z2 ) ) @ ( nat2 @ ( uminus_uminus_int @ Z6 ) ) ) ) ) ).
% nat_mult_distrib_neg
thf(fact_1203_mult__less__mono2,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_nat @ ( times_times_nat @ K @ I ) @ ( times_times_nat @ K @ J ) ) ) ) ).
% mult_less_mono2
thf(fact_1204_mult__less__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ K ) ) ) ) ).
% mult_less_mono1
thf(fact_1205_mult__eq__self__implies__10,axiom,
! [M: nat,N: nat] :
( ( M
= ( times_times_nat @ M @ N ) )
=> ( ( N = one_one_nat )
| ( M = zero_zero_nat ) ) ) ).
% mult_eq_self_implies_10
thf(fact_1206_zmult__zless__mono2,axiom,
! [I: int,J: int,K: int] :
( ( ord_less_int @ I @ J )
=> ( ( ord_less_int @ zero_zero_int @ K )
=> ( ord_less_int @ ( times_times_int @ K @ I ) @ ( times_times_int @ K @ J ) ) ) ) ).
% zmult_zless_mono2
thf(fact_1207_zmult__eq__1__iff,axiom,
! [M: int,N: int] :
( ( ( times_times_int @ M @ N )
= one_one_int )
= ( ( ( M = one_one_int )
& ( N = one_one_int ) )
| ( ( M
= ( uminus_uminus_int @ one_one_int ) )
& ( N
= ( uminus_uminus_int @ one_one_int ) ) ) ) ) ).
% zmult_eq_1_iff
thf(fact_1208_pos__zmult__eq__1__iff__lemma,axiom,
! [M: int,N: int] :
( ( ( times_times_int @ M @ N )
= one_one_int )
=> ( ( M = one_one_int )
| ( M
= ( uminus_uminus_int @ one_one_int ) ) ) ) ).
% pos_zmult_eq_1_iff_lemma
thf(fact_1209_pos__zmult__eq__1__iff,axiom,
! [M: int,N: int] :
( ( ord_less_int @ zero_zero_int @ M )
=> ( ( ( times_times_int @ M @ N )
= one_one_int )
= ( ( M = one_one_int )
& ( N = one_one_int ) ) ) ) ).
% pos_zmult_eq_1_iff
thf(fact_1210_incr__mult__lemma,axiom,
! [D: int,P: int > $o,K: int] :
( ( ord_less_int @ zero_zero_int @ D )
=> ( ! [X4: int] :
( ( P @ X4 )
=> ( P @ ( plus_plus_int @ X4 @ D ) ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ K )
=> ! [X3: int] :
( ( P @ X3 )
=> ( P @ ( plus_plus_int @ X3 @ ( times_times_int @ K @ D ) ) ) ) ) ) ) ).
% incr_mult_lemma
thf(fact_1211_zmult__zless__mono2__lemma,axiom,
! [I: int,J: int,K: nat] :
( ( ord_less_int @ I @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ K ) @ I ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ K ) @ J ) ) ) ) ).
% zmult_zless_mono2_lemma
thf(fact_1212_nat__mult__le__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_eq_nat @ M @ N ) ) ) ).
% nat_mult_le_cancel_disj
thf(fact_1213_nat__mult__less__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
& ( ord_less_nat @ M @ N ) ) ) ).
% nat_mult_less_cancel_disj
thf(fact_1214_nat__mult__le__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ord_less_eq_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ) ).
% nat_mult_le_cancel1
thf(fact_1215_nat__mult__eq__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( times_times_nat @ K @ M )
= ( times_times_nat @ K @ N ) )
= ( ( K = zero_zero_nat )
| ( M = N ) ) ) ).
% nat_mult_eq_cancel_disj
thf(fact_1216_left__add__mult__distrib,axiom,
! [I: nat,U: nat,J: nat,K: nat] :
( ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ K ) )
= ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ I @ J ) @ U ) @ K ) ) ).
% left_add_mult_distrib
thf(fact_1217_nat__mult__less__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ord_less_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ord_less_nat @ M @ N ) ) ) ).
% nat_mult_less_cancel1
thf(fact_1218_nat__mult__eq__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ( times_times_nat @ K @ M )
= ( times_times_nat @ K @ N ) )
= ( M = N ) ) ) ).
% nat_mult_eq_cancel1
thf(fact_1219_Bernoulli__inequality,axiom,
! [X: real,N: nat] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
=> ( ord_less_eq_real @ ( plus_plus_real @ one_one_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ X ) ) @ ( power_power_real @ ( plus_plus_real @ one_one_real @ X ) @ N ) ) ) ).
% Bernoulli_inequality
thf(fact_1220_floor__divide__real__eq__div,axiom,
! [B2: int,A2: real] :
( ( ord_less_eq_int @ zero_zero_int @ B2 )
=> ( ( archim6058952711729229775r_real @ ( divide_divide_real @ A2 @ ( ring_1_of_int_real @ B2 ) ) )
= ( divide_divide_int @ ( archim6058952711729229775r_real @ A2 ) @ B2 ) ) ) ).
% floor_divide_real_eq_div
thf(fact_1221_real__arch__pow,axiom,
! [X: real,Y2: real] :
( ( ord_less_real @ one_one_real @ X )
=> ? [N2: nat] : ( ord_less_real @ Y2 @ ( power_power_real @ X @ N2 ) ) ) ).
% real_arch_pow
thf(fact_1222_int__ops_I8_J,axiom,
! [A2: nat,B2: nat] :
( ( semiri1314217659103216013at_int @ ( divide_divide_nat @ A2 @ B2 ) )
= ( divide_divide_int @ ( semiri1314217659103216013at_int @ A2 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ).
% int_ops(8)
thf(fact_1223_real__of__int__div4,axiom,
! [N: int,X: int] : ( ord_less_eq_real @ ( ring_1_of_int_real @ ( divide_divide_int @ N @ X ) ) @ ( divide_divide_real @ ( ring_1_of_int_real @ N ) @ ( ring_1_of_int_real @ X ) ) ) ).
% real_of_int_div4
thf(fact_1224_real__of__nat__div4,axiom,
! [N: nat,X: nat] : ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ ( divide_divide_nat @ N @ X ) ) @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri5074537144036343181t_real @ X ) ) ) ).
% real_of_nat_div4
thf(fact_1225_nat__mult__div__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( K = zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= zero_zero_nat ) )
& ( ( K != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( divide_divide_nat @ M @ N ) ) ) ) ).
% nat_mult_div_cancel_disj
thf(fact_1226_nat__mult__div__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( divide_divide_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( divide_divide_nat @ M @ N ) ) ) ).
% nat_mult_div_cancel1
thf(fact_1227_real__arch__pow__inv,axiom,
! [Y2: real,X: real] :
( ( ord_less_real @ zero_zero_real @ Y2 )
=> ( ( ord_less_real @ X @ one_one_real )
=> ? [N2: nat] : ( ord_less_real @ ( power_power_real @ X @ N2 ) @ Y2 ) ) ) ).
% real_arch_pow_inv
thf(fact_1228_realpow__pos__nth__unique,axiom,
! [N: nat,A2: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_real @ zero_zero_real @ A2 )
=> ? [X4: real] :
( ( ord_less_real @ zero_zero_real @ X4 )
& ( ( power_power_real @ X4 @ N )
= A2 )
& ! [Y4: real] :
( ( ( ord_less_real @ zero_zero_real @ Y4 )
& ( ( power_power_real @ Y4 @ N )
= A2 ) )
=> ( Y4 = X4 ) ) ) ) ) ).
% realpow_pos_nth_unique
thf(fact_1229_realpow__pos__nth,axiom,
! [N: nat,A2: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_real @ zero_zero_real @ A2 )
=> ? [R3: real] :
( ( ord_less_real @ zero_zero_real @ R3 )
& ( ( power_power_real @ R3 @ N )
= A2 ) ) ) ) ).
% realpow_pos_nth
thf(fact_1230_verit__less__mono__div__int2,axiom,
! [A3: int,B3: int,N: int] :
( ( ord_less_eq_int @ A3 @ B3 )
=> ( ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ N ) )
=> ( ord_less_eq_int @ ( divide_divide_int @ B3 @ N ) @ ( divide_divide_int @ A3 @ N ) ) ) ) ).
% verit_less_mono_div_int2
thf(fact_1231_int_Onat__pow__one,axiom,
! [N: nat] :
( ( power_power_int @ one_one_int @ N )
= one_one_int ) ).
% int.nat_pow_one
thf(fact_1232_div__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ( divide_divide_nat @ M @ N )
= zero_zero_nat ) ) ).
% div_less
thf(fact_1233_nat__zero__less__power__iff,axiom,
! [X: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ X @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ X )
| ( N = zero_zero_nat ) ) ) ).
% nat_zero_less_power_iff
thf(fact_1234_int_Onat__pow__0,axiom,
! [X: int] :
( ( power_power_int @ X @ zero_zero_nat )
= one_one_int ) ).
% int.nat_pow_0
thf(fact_1235_div__mult__self__is__m,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( divide_divide_nat @ ( times_times_nat @ M @ N ) @ N )
= M ) ) ).
% div_mult_self_is_m
thf(fact_1236_div__mult__self1__is__m,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( divide_divide_nat @ ( times_times_nat @ N @ M ) @ N )
= M ) ) ).
% div_mult_self1_is_m
thf(fact_1237_div__neg__neg__trivial,axiom,
! [K: int,L: int] :
( ( ord_less_eq_int @ K @ zero_zero_int )
=> ( ( ord_less_int @ L @ K )
=> ( ( divide_divide_int @ K @ L )
= zero_zero_int ) ) ) ).
% div_neg_neg_trivial
thf(fact_1238_div__pos__pos__trivial,axiom,
! [K: int,L: int] :
( ( ord_less_eq_int @ zero_zero_int @ K )
=> ( ( ord_less_int @ K @ L )
=> ( ( divide_divide_int @ K @ L )
= zero_zero_int ) ) ) ).
% div_pos_pos_trivial
thf(fact_1239_nat__power__less__imp__less,axiom,
! [I: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ I )
=> ( ( ord_less_nat @ ( power_power_nat @ I @ M ) @ ( power_power_nat @ I @ N ) )
=> ( ord_less_nat @ M @ N ) ) ) ).
% nat_power_less_imp_less
thf(fact_1240_int_Onat__pow__zero,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ( power_power_int @ zero_zero_int @ N )
= zero_zero_int ) ) ).
% int.nat_pow_zero
thf(fact_1241_nat__power__eq,axiom,
! [Z2: int,N: nat] :
( ( ord_less_eq_int @ zero_zero_int @ Z2 )
=> ( ( nat2 @ ( power_power_int @ Z2 @ N ) )
= ( power_power_nat @ ( nat2 @ Z2 ) @ N ) ) ) ).
% nat_power_eq
thf(fact_1242_div__le__mono,axiom,
! [M: nat,N: nat,K: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( divide_divide_nat @ M @ K ) @ ( divide_divide_nat @ N @ K ) ) ) ).
% div_le_mono
thf(fact_1243_div__le__dividend,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( divide_divide_nat @ M @ N ) @ M ) ).
% div_le_dividend
thf(fact_1244_zdiv__int,axiom,
! [M: nat,N: nat] :
( ( semiri1314217659103216013at_int @ ( divide_divide_nat @ M @ N ) )
= ( divide_divide_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% zdiv_int
thf(fact_1245_Euclidean__Division_Odiv__eq__0__iff,axiom,
! [M: nat,N: nat] :
( ( ( divide_divide_nat @ M @ N )
= zero_zero_nat )
= ( ( ord_less_nat @ M @ N )
| ( N = zero_zero_nat ) ) ) ).
% Euclidean_Division.div_eq_0_iff
thf(fact_1246_less__mult__imp__div__less,axiom,
! [M: nat,I: nat,N: nat] :
( ( ord_less_nat @ M @ ( times_times_nat @ I @ N ) )
=> ( ord_less_nat @ ( divide_divide_nat @ M @ N ) @ I ) ) ).
% less_mult_imp_div_less
thf(fact_1247_div__neg__pos__less0,axiom,
! [A2: int,B2: int] :
( ( ord_less_int @ A2 @ zero_zero_int )
=> ( ( ord_less_int @ zero_zero_int @ B2 )
=> ( ord_less_int @ ( divide_divide_int @ A2 @ B2 ) @ zero_zero_int ) ) ) ).
% div_neg_pos_less0
thf(fact_1248_neg__imp__zdiv__neg__iff,axiom,
! [B2: int,A2: int] :
( ( ord_less_int @ B2 @ zero_zero_int )
=> ( ( ord_less_int @ ( divide_divide_int @ A2 @ B2 ) @ zero_zero_int )
= ( ord_less_int @ zero_zero_int @ A2 ) ) ) ).
% neg_imp_zdiv_neg_iff
thf(fact_1249_pos__imp__zdiv__neg__iff,axiom,
! [B2: int,A2: int] :
( ( ord_less_int @ zero_zero_int @ B2 )
=> ( ( ord_less_int @ ( divide_divide_int @ A2 @ B2 ) @ zero_zero_int )
= ( ord_less_int @ A2 @ zero_zero_int ) ) ) ).
% pos_imp_zdiv_neg_iff
thf(fact_1250_div__times__less__eq__dividend,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( times_times_nat @ ( divide_divide_nat @ M @ N ) @ N ) @ M ) ).
% div_times_less_eq_dividend
thf(fact_1251_times__div__less__eq__dividend,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ ( times_times_nat @ N @ ( divide_divide_nat @ M @ N ) ) @ M ) ).
% times_div_less_eq_dividend
thf(fact_1252_div__le__mono2,axiom,
! [M: nat,N: nat,K: nat] :
( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( divide_divide_nat @ K @ N ) @ ( divide_divide_nat @ K @ M ) ) ) ) ).
% div_le_mono2
thf(fact_1253_div__greater__zero__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ M @ N ) )
= ( ( ord_less_eq_nat @ N @ M )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% div_greater_zero_iff
thf(fact_1254_div__eq__dividend__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ( ( divide_divide_nat @ M @ N )
= M )
= ( N = one_one_nat ) ) ) ).
% div_eq_dividend_iff
thf(fact_1255_div__less__dividend,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ one_one_nat @ N )
=> ( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ord_less_nat @ ( divide_divide_nat @ M @ N ) @ M ) ) ) ).
% div_less_dividend
thf(fact_1256_zdiv__zmult2__eq,axiom,
! [C: int,A2: int,B2: int] :
( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ( divide_divide_int @ A2 @ ( times_times_int @ B2 @ C ) )
= ( divide_divide_int @ ( divide_divide_int @ A2 @ B2 ) @ C ) ) ) ).
% zdiv_zmult2_eq
thf(fact_1257_div__less__iff__less__mult,axiom,
! [Q3: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ Q3 )
=> ( ( ord_less_nat @ ( divide_divide_nat @ M @ Q3 ) @ N )
= ( ord_less_nat @ M @ ( times_times_nat @ N @ Q3 ) ) ) ) ).
% div_less_iff_less_mult
thf(fact_1258_int__div__less__self,axiom,
! [X: int,K: int] :
( ( ord_less_int @ zero_zero_int @ X )
=> ( ( ord_less_int @ one_one_int @ K )
=> ( ord_less_int @ ( divide_divide_int @ X @ K ) @ X ) ) ) ).
% int_div_less_self
thf(fact_1259_nonneg1__imp__zdiv__pos__iff,axiom,
! [A2: int,B2: int] :
( ( ord_less_eq_int @ zero_zero_int @ A2 )
=> ( ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ A2 @ B2 ) )
= ( ( ord_less_eq_int @ B2 @ A2 )
& ( ord_less_int @ zero_zero_int @ B2 ) ) ) ) ).
% nonneg1_imp_zdiv_pos_iff
thf(fact_1260_pos__imp__zdiv__nonneg__iff,axiom,
! [B2: int,A2: int] :
( ( ord_less_int @ zero_zero_int @ B2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ A2 @ B2 ) )
= ( ord_less_eq_int @ zero_zero_int @ A2 ) ) ) ).
% pos_imp_zdiv_nonneg_iff
thf(fact_1261_neg__imp__zdiv__nonneg__iff,axiom,
! [B2: int,A2: int] :
( ( ord_less_int @ B2 @ zero_zero_int )
=> ( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ A2 @ B2 ) )
= ( ord_less_eq_int @ A2 @ zero_zero_int ) ) ) ).
% neg_imp_zdiv_nonneg_iff
thf(fact_1262_pos__imp__zdiv__pos__iff,axiom,
! [K: int,I: int] :
( ( ord_less_int @ zero_zero_int @ K )
=> ( ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ I @ K ) )
= ( ord_less_eq_int @ K @ I ) ) ) ).
% pos_imp_zdiv_pos_iff
thf(fact_1263_div__nonpos__pos__le0,axiom,
! [A2: int,B2: int] :
( ( ord_less_eq_int @ A2 @ zero_zero_int )
=> ( ( ord_less_int @ zero_zero_int @ B2 )
=> ( ord_less_eq_int @ ( divide_divide_int @ A2 @ B2 ) @ zero_zero_int ) ) ) ).
% div_nonpos_pos_le0
thf(fact_1264_div__nonneg__neg__le0,axiom,
! [A2: int,B2: int] :
( ( ord_less_eq_int @ zero_zero_int @ A2 )
=> ( ( ord_less_int @ B2 @ zero_zero_int )
=> ( ord_less_eq_int @ ( divide_divide_int @ A2 @ B2 ) @ zero_zero_int ) ) ) ).
% div_nonneg_neg_le0
% Helper facts (5)
thf(help_If_2_1_If_001t__Int__Oint_T,axiom,
! [X: int,Y2: int] :
( ( if_int @ $false @ X @ Y2 )
= Y2 ) ).
thf(help_If_1_1_If_001t__Int__Oint_T,axiom,
! [X: int,Y2: int] :
( ( if_int @ $true @ X @ Y2 )
= 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,Y2: nat] :
( ( if_nat @ $false @ X @ Y2 )
= Y2 ) ).
thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y2: nat] :
( ( if_nat @ $true @ X @ Y2 )
= X ) ).
% Conjectures (1)
thf(conj_0,conjecture,
ring_s5316885176909347197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ).
%------------------------------------------------------------------------------