%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : ITP063^1 : TPTP v9.2.1. Released v7.5.0.
% Transfm : none
% Format : tptp:raw
% Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% Computer : n026.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Wed Feb 25 08:55:15 AM UTC 2026
% Result : Theorem 2.53s 1.55s
% Output : Refutation 2.53s
% Verified :
% SZS Type : Refutation
% Derivation depth : 9
% Number of leaves : 4
% Syntax : Number of formulae : 24 ( 16 unt; 0 typ; 0 def)
% Number of atoms : 89 ( 25 equ; 0 cnn)
% Maximal formula atoms : 3 ( 3 avg)
% Number of connectives : 290 ( 17 ~; 9 |; 3 &; 258 @)
% ( 0 <=>; 3 =>; 0 <=; 0 <~>)
% Maximal formula depth : 12 ( 5 avg)
% Number of types : 3 ( 2 usr)
% Number of type conns : 0 ( 0 >; 0 *; 0 +; 0 <<)
% Number of symbols : 34 ( 31 usr; 15 con; 0-4 aty)
% Number of variables : 18 ( 0 ^; 18 !; 0 ?; 18 :)
% Comments :
%------------------------------------------------------------------------------
thf(type_def_5,type,
nat: $tType ).
thf(type_def_6,type,
real: $tType ).
thf(func_def_0,type,
real: $tType ).
thf(func_def_1,type,
nat: $tType ).
thf(func_def_2,type,
genClo1161277105lle_PC: nat > real > real ).
thf(func_def_3,type,
genClo1278781456e_beta: real ).
thf(func_def_4,type,
genClo872980712lle_mu: real ).
thf(func_def_5,type,
genClo1144207539le_rho: real ).
thf(func_def_6,type,
genClo1015804716orrect: nat > real > $o ).
thf(func_def_7,type,
genClo293725282kRead2: ( nat > real ) > ( nat > real ) > real > ( nat > $o ) > $o ).
thf(func_def_8,type,
genClo1650508560e_rmax: real ).
thf(func_def_9,type,
genClo1651033342e_rmin: real ).
thf(func_def_10,type,
genClo1163638703lle_te: nat > nat > real ).
thf(func_def_11,type,
abs_abs_real: real > real ).
thf(func_def_12,type,
minus_minus_nat: nat > nat > nat ).
thf(func_def_13,type,
minus_minus_real: real > real > real ).
thf(func_def_14,type,
one_one_nat: nat ).
thf(func_def_15,type,
one_one_real: real ).
thf(func_def_16,type,
plus_plus_nat: nat > nat > nat ).
thf(func_def_17,type,
plus_plus_real: real > real > real ).
thf(func_def_18,type,
times_times_nat: nat > nat > nat ).
thf(func_def_19,type,
times_times_real: real > real > real ).
thf(func_def_20,type,
zero_zero_nat: nat ).
thf(func_def_21,type,
zero_zero_real: real ).
thf(func_def_22,type,
ord_less_eq_nat: nat > nat > $o ).
thf(func_def_23,type,
ord_less_eq_real: real > real > $o ).
thf(func_def_24,type,
i: nat ).
thf(func_def_25,type,
l: nat ).
thf(func_def_26,type,
p: nat ).
thf(func_def_27,type,
q: nat ).
thf(func_def_41,type,
sK0: nat > nat > nat ).
thf(func_def_42,type,
sK1: nat > nat > nat ).
thf(func_def_43,type,
sK2: ( nat > $o ) > nat ).
thf(func_def_44,type,
sK3: ( nat > $o ) > nat > nat ).
thf(func_def_46,type,
ph5:
!>[X0: $tType] : X0 ).
thf(f1387,plain,
$false,
inference(subsumption_resolution,[status(thm)],[f1386,f1159]) ).
thf(f1159,plain,
( ( genClo1015804716orrect @ q @ ( genClo1163638703lle_te @ q @ ( plus_plus_nat @ i @ one_one_nat ) ) )
= $true ),
inference(cnf_transformation,[status(thm)],[f410]) ).
thf(f410,plain,
( ( genClo1015804716orrect @ q @ ( genClo1163638703lle_te @ q @ ( plus_plus_nat @ i @ one_one_nat ) ) )
= $true ),
inference(fool_elimination,[status(thm)],[f409]) ).
thf(f409,plain,
genClo1015804716orrect @ q @ ( genClo1163638703lle_te @ q @ ( plus_plus_nat @ i @ one_one_nat ) ),
inference(rectify,[status(thm)],[f3]) ).
thf(f3,axiom,
genClo1015804716orrect @ q @ ( genClo1163638703lle_te @ q @ ( plus_plus_nat @ i @ one_one_nat ) ),
file('/export/starexec/sandbox2/tmp/tmp.5TcjDfaA5h/DTF2THF_25639.p',fact_2_corr__q__tq) ).
thf(f1386,plain,
( ( genClo1015804716orrect @ q @ ( genClo1163638703lle_te @ q @ ( plus_plus_nat @ i @ one_one_nat ) ) )
!= $true ),
inference(subsumption_resolution,[status(thm)],[f1375,f1022]) ).
thf(f1022,plain,
( ( genClo1015804716orrect @ p @ ( genClo1163638703lle_te @ p @ ( plus_plus_nat @ i @ one_one_nat ) ) )
= $true ),
inference(cnf_transformation,[status(thm)],[f722]) ).
thf(f722,plain,
( ( genClo1015804716orrect @ p @ ( genClo1163638703lle_te @ p @ ( plus_plus_nat @ i @ one_one_nat ) ) )
= $true ),
inference(fool_elimination,[status(thm)],[f721]) ).
thf(f721,plain,
genClo1015804716orrect @ p @ ( genClo1163638703lle_te @ p @ ( plus_plus_nat @ i @ one_one_nat ) ),
inference(rectify,[status(thm)],[f2]) ).
thf(f2,axiom,
genClo1015804716orrect @ p @ ( genClo1163638703lle_te @ p @ ( plus_plus_nat @ i @ one_one_nat ) ),
file('/export/starexec/sandbox2/tmp/tmp.5TcjDfaA5h/DTF2THF_25639.p',fact_1_corr__p) ).
thf(f1375,plain,
( ( ( genClo1015804716orrect @ p @ ( genClo1163638703lle_te @ p @ ( plus_plus_nat @ i @ one_one_nat ) ) )
!= $true )
| ( ( genClo1015804716orrect @ q @ ( genClo1163638703lle_te @ q @ ( plus_plus_nat @ i @ one_one_nat ) ) )
!= $true ) ),
inference(trivial_inequality_removal,[status(thm)],[f1371]) ).
thf(f1371,plain,
( ( ( genClo1015804716orrect @ p @ ( genClo1163638703lle_te @ p @ ( plus_plus_nat @ i @ one_one_nat ) ) )
!= $true )
| ( ( genClo1015804716orrect @ q @ ( genClo1163638703lle_te @ q @ ( plus_plus_nat @ i @ one_one_nat ) ) )
!= $true )
| ( $true != $true ) ),
inference(superposition,[status(thm)],[f1125,f1170]) ).
thf(f1170,plain,
! [X2: nat,X0: nat,X1: nat] :
( ( ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( genClo1163638703lle_te @ X0 @ X2 ) @ ( genClo1163638703lle_te @ X1 @ X2 ) ) ) @ genClo1278781456e_beta )
= $true )
| ( ( genClo1015804716orrect @ X0 @ ( genClo1163638703lle_te @ X0 @ X2 ) )
!= $true )
| ( ( genClo1015804716orrect @ X1 @ ( genClo1163638703lle_te @ X1 @ X2 ) )
!= $true ) ),
inference(cnf_transformation,[status(thm)],[f909]) ).
thf(f909,plain,
! [X0: nat,X1: nat,X2: nat] :
( ( ( genClo1015804716orrect @ X1 @ ( genClo1163638703lle_te @ X1 @ X2 ) )
!= $true )
| ( ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( genClo1163638703lle_te @ X0 @ X2 ) @ ( genClo1163638703lle_te @ X1 @ X2 ) ) ) @ genClo1278781456e_beta )
= $true )
| ( ( genClo1015804716orrect @ X0 @ ( genClo1163638703lle_te @ X0 @ X2 ) )
!= $true ) ),
inference(flattening,[status(thm)],[f908]) ).
thf(f908,plain,
! [X0: nat,X1: nat,X2: nat] :
( ( ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( genClo1163638703lle_te @ X0 @ X2 ) @ ( genClo1163638703lle_te @ X1 @ X2 ) ) ) @ genClo1278781456e_beta )
= $true )
| ( ( genClo1015804716orrect @ X1 @ ( genClo1163638703lle_te @ X1 @ X2 ) )
!= $true )
| ( ( genClo1015804716orrect @ X0 @ ( genClo1163638703lle_te @ X0 @ X2 ) )
!= $true ) ),
inference(ennf_transformation,[status(thm)],[f620]) ).
thf(f620,plain,
! [X0: nat,X1: nat,X2: nat] :
( ( ( ( genClo1015804716orrect @ X1 @ ( genClo1163638703lle_te @ X1 @ X2 ) )
= $true )
& ( ( genClo1015804716orrect @ X0 @ ( genClo1163638703lle_te @ X0 @ X2 ) )
= $true ) )
=> ( ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( genClo1163638703lle_te @ X0 @ X2 ) @ ( genClo1163638703lle_te @ X1 @ X2 ) ) ) @ genClo1278781456e_beta )
= $true ) ),
inference(fool_elimination,[status(thm)],[f619]) ).
thf(f619,plain,
! [X0: nat,X1: nat,X2: nat] :
( ( ( genClo1015804716orrect @ X0 @ ( genClo1163638703lle_te @ X0 @ X2 ) )
& ( genClo1015804716orrect @ X1 @ ( genClo1163638703lle_te @ X1 @ X2 ) ) )
=> ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( genClo1163638703lle_te @ X0 @ X2 ) @ ( genClo1163638703lle_te @ X1 @ X2 ) ) ) @ genClo1278781456e_beta ) ),
inference(rectify,[status(thm)],[f1]) ).
thf(f1,axiom,
! [X0: nat,X1: nat,X2: nat] :
( ( ( genClo1015804716orrect @ X0 @ ( genClo1163638703lle_te @ X0 @ X2 ) )
& ( genClo1015804716orrect @ X1 @ ( genClo1163638703lle_te @ X1 @ X2 ) ) )
=> ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( genClo1163638703lle_te @ X0 @ X2 ) @ ( genClo1163638703lle_te @ X1 @ X2 ) ) ) @ genClo1278781456e_beta ) ),
file('/export/starexec/sandbox2/tmp/tmp.5TcjDfaA5h/DTF2THF_25639.p',fact_0_rts2b) ).
thf(f1125,plain,
( ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( genClo1163638703lle_te @ p @ ( plus_plus_nat @ i @ one_one_nat ) ) @ ( genClo1163638703lle_te @ q @ ( plus_plus_nat @ i @ one_one_nat ) ) ) ) @ genClo1278781456e_beta )
!= $true ),
inference(cnf_transformation,[status(thm)],[f729]) ).
thf(f729,plain,
( ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( genClo1163638703lle_te @ p @ ( plus_plus_nat @ i @ one_one_nat ) ) @ ( genClo1163638703lle_te @ q @ ( plus_plus_nat @ i @ one_one_nat ) ) ) ) @ genClo1278781456e_beta )
!= $true ),
inference(flattening,[status(thm)],[f672]) ).
thf(f672,plain,
( ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( genClo1163638703lle_te @ p @ ( plus_plus_nat @ i @ one_one_nat ) ) @ ( genClo1163638703lle_te @ q @ ( plus_plus_nat @ i @ one_one_nat ) ) ) ) @ genClo1278781456e_beta )
!= $true ),
inference(fool_elimination,[status(thm)],[f671]) ).
thf(f671,plain,
~ ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( genClo1163638703lle_te @ p @ ( plus_plus_nat @ i @ one_one_nat ) ) @ ( genClo1163638703lle_te @ q @ ( plus_plus_nat @ i @ one_one_nat ) ) ) ) @ genClo1278781456e_beta ),
inference(rectify,[status(thm)],[f339]) ).
thf(f339,negated_conjecture,
~ ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( genClo1163638703lle_te @ p @ ( plus_plus_nat @ i @ one_one_nat ) ) @ ( genClo1163638703lle_te @ q @ ( plus_plus_nat @ i @ one_one_nat ) ) ) ) @ genClo1278781456e_beta ),
inference(negated_conjecture,[status(cth)],[f338]) ).
thf(f338,conjecture,
ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( genClo1163638703lle_te @ p @ ( plus_plus_nat @ i @ one_one_nat ) ) @ ( genClo1163638703lle_te @ q @ ( plus_plus_nat @ i @ one_one_nat ) ) ) ) @ genClo1278781456e_beta,
file('/export/starexec/sandbox2/tmp/tmp.5TcjDfaA5h/DTF2THF_25639.p',conj_0) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.11 % Problem : ITP063^1 : TPTP v9.2.1. Released v7.5.0.
% 0.12/0.12 % Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.14/0.32 % Computer : n026.cluster.edu
% 0.14/0.32 % Model : x86_64 x86_64
% 0.14/0.32 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.14/0.32 % Memory : 8042.1875MB
% 0.14/0.32 % OS : Linux 3.10.0-693.el7.x86_64
% 0.14/0.32 % CPULimit : 300
% 0.14/0.32 % WCLimit : 300
% 0.14/0.32 % DateTime : Tue Feb 24 08:25:25 EST 2026
% 0.14/0.32 % CPUTime :
% 0.14/0.32 Running /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.17/0.44 ---- Original DTF file ---
% 0.17/0.45 thf(spec,logic,$$dhol).
% 0.17/0.45 %------------------------------------------------------------------------------
% 0.17/0.45 % File : ITP063^1 : TPTP v9.2.1. Released v7.5.0.
% 0.17/0.45 % Domain : Interactive Theorem Proving
% 0.17/0.45 % Problem : Sledgehammer GenClock problem prob_539__3247850_1
% 0.17/0.45 % Version : Especial.
% 0.17/0.45 % English :
% 0.17/0.45
% 0.17/0.45 % Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% 0.17/0.45 % : [Des21] Desharnais (2021), Email to Geoff Sutcliffe
% 0.17/0.45 % Source : [Des21]
% 0.17/0.45 % Names : GenClock/prob_539__3247850_1 [Des21]
% 0.17/0.45
% 0.17/0.45 % Status : Theorem
% 0.17/0.45 % Rating : 0.22 v9.1.0, 0.12 v9.0.0, 0.30 v8.2.0, 0.31 v8.1.0, 0.27 v7.5.0
% 0.17/0.45 % Syntax : Number of formulae : 366 ( 211 unt; 28 typ; 0 def)
% 0.17/0.45 % Number of atoms : 723 ( 408 equ; 0 cnn)
% 0.17/0.45 % Maximal formula atoms : 5 ( 2 avg)
% 0.17/0.45 % Number of connectives : 2297 ( 24 ~; 13 |; 27 &;2057 @)
% 0.17/0.45 % ( 0 <=>; 176 =>; 0 <=; 0 <~>)
% 0.17/0.45 % Maximal formula depth : 14 ( 5 avg)
% 0.17/0.45 % Number of types : 3 ( 2 usr)
% 0.17/0.45 % Number of type conns : 34 ( 34 >; 0 *; 0 +; 0 <<)
% 0.17/0.45 % Number of symbols : 27 ( 26 usr; 13 con; 0-4 aty)
% 0.17/0.45 % Number of variables : 760 ( 22 ^; 734 !; 4 ?; 760 :)
% 0.17/0.45 % SPC : TH0_THM_EQU_NAR
% 0.17/0.45
% 0.17/0.45 % Comments : This file was generated by Sledgehammer 2021-02-23 15:30:39.819
% 0.17/0.45 %------------------------------------------------------------------------------
% 0.17/0.45 % Could-be-implicit typings (2)
% 0.17/0.45 thf(ty_n_t__Real__Oreal,type,
% 0.17/0.45 real: $tType ).
% 0.17/0.45
% 0.17/0.45 thf(ty_n_t__Nat__Onat,type,
% 0.17/0.45 nat: $tType ).
% 0.17/0.45
% 0.17/0.45 % Explicit typings (26)
% 0.17/0.45 thf(sy_c_GenClock__Mirabelle__bsvkzpgbls_OPC,type,
% 0.17/0.45 genClo1161277105lle_PC: nat > real > real ).
% 0.17/0.45
% 0.17/0.45 thf(sy_c_GenClock__Mirabelle__bsvkzpgbls_O_092_060beta_062,type,
% 0.17/0.45 genClo1278781456e_beta: real ).
% 0.17/0.45
% 0.17/0.45 thf(sy_c_GenClock__Mirabelle__bsvkzpgbls_O_092_060mu_062,type,
% 0.17/0.45 genClo872980712lle_mu: real ).
% 0.17/0.45
% 0.17/0.45 thf(sy_c_GenClock__Mirabelle__bsvkzpgbls_O_092_060rho_062,type,
% 0.17/0.45 genClo1144207539le_rho: real ).
% 0.17/0.45
% 0.17/0.45 thf(sy_c_GenClock__Mirabelle__bsvkzpgbls_Ocorrect,type,
% 0.17/0.45 genClo1015804716orrect: nat > real > $o ).
% 0.17/0.45
% 0.17/0.45 thf(sy_c_GenClock__Mirabelle__bsvkzpgbls_OokRead2,type,
% 0.17/0.45 genClo293725282kRead2: ( nat > real ) > ( nat > real ) > real > ( nat > $o ) > $o ).
% 0.17/0.45
% 0.17/0.45 thf(sy_c_GenClock__Mirabelle__bsvkzpgbls_Ormax,type,
% 0.17/0.45 genClo1650508560e_rmax: real ).
% 0.17/0.45
% 0.17/0.45 thf(sy_c_GenClock__Mirabelle__bsvkzpgbls_Ormin,type,
% 0.17/0.45 genClo1651033342e_rmin: real ).
% 0.17/0.45
% 0.17/0.45 thf(sy_c_GenClock__Mirabelle__bsvkzpgbls_Ote,type,
% 0.17/0.45 genClo1163638703lle_te: nat > nat > real ).
% 0.17/0.45
% 0.17/0.45 thf(sy_c_Groups_Oabs__class_Oabs_001t__Real__Oreal,type,
% 0.17/0.45 abs_abs_real: real > real ).
% 0.17/0.45
% 0.17/0.45 thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat,type,
% 0.17/0.45 minus_minus_nat: nat > nat > nat ).
% 0.17/0.45
% 0.17/0.45 thf(sy_c_Groups_Ominus__class_Ominus_001t__Real__Oreal,type,
% 0.17/0.45 minus_minus_real: real > real > real ).
% 0.17/0.45
% 0.17/0.45 thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat,type,
% 0.17/0.45 one_one_nat: nat ).
% 0.17/0.45
% 0.17/0.45 thf(sy_c_Groups_Oone__class_Oone_001t__Real__Oreal,type,
% 0.17/0.45 one_one_real: real ).
% 0.17/0.45
% 0.17/0.45 thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat,type,
% 0.17/0.45 plus_plus_nat: nat > nat > nat ).
% 0.17/0.45
% 0.17/0.45 thf(sy_c_Groups_Oplus__class_Oplus_001t__Real__Oreal,type,
% 0.17/0.45 plus_plus_real: real > real > real ).
% 0.17/0.45
% 0.17/0.45 thf(sy_c_Groups_Otimes__class_Otimes_001t__Nat__Onat,type,
% 0.17/0.45 times_times_nat: nat > nat > nat ).
% 0.17/0.45
% 0.17/0.45 thf(sy_c_Groups_Otimes__class_Otimes_001t__Real__Oreal,type,
% 0.17/0.45 times_times_real: real > real > real ).
% 0.17/0.45
% 0.17/0.45 thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
% 0.17/0.45 zero_zero_nat: nat ).
% 0.17/0.45
% 0.17/0.45 thf(sy_c_Groups_Ozero__class_Ozero_001t__Real__Oreal,type,
% 0.17/0.45 zero_zero_real: real ).
% 0.17/0.45
% 0.17/0.45 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
% 0.17/0.45 ord_less_eq_nat: nat > nat > $o ).
% 0.17/0.45
% 0.17/0.45 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Real__Oreal,type,
% 0.17/0.45 ord_less_eq_real: real > real > $o ).
% 0.17/0.45
% 0.17/0.45 thf(sy_v_i,type,
% 0.17/0.45 i: nat ).
% 0.17/0.45
% 0.17/0.45 thf(sy_v_l,type,
% 0.17/0.45 l: nat ).
% 0.17/0.45
% 0.17/0.45 thf(sy_v_p,type,
% 0.17/0.45 p: nat ).
% 0.17/0.45
% 0.17/0.45 thf(sy_v_q,type,
% 0.17/0.45 q: nat ).
% 0.17/0.45
% 0.17/0.45 % Relevant facts (337)
% 0.17/0.45 thf(fact_0_rts2b,axiom,
% 0.17/0.45 ! [P: nat,Q: nat,I: nat] :
% 0.17/0.45 ( ( ( genClo1015804716orrect @ P @ ( genClo1163638703lle_te @ P @ I ) )
% 0.17/0.45 & ( genClo1015804716orrect @ Q @ ( genClo1163638703lle_te @ Q @ I ) ) )
% 0.17/0.45 => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( genClo1163638703lle_te @ P @ I ) @ ( genClo1163638703lle_te @ Q @ I ) ) ) @ genClo1278781456e_beta ) ) ).
% 0.17/0.45
% 0.17/0.45 % rts2b
% 0.17/0.45 thf(fact_1_corr__p,axiom,
% 0.17/0.45 genClo1015804716orrect @ p @ ( genClo1163638703lle_te @ p @ ( plus_plus_nat @ i @ one_one_nat ) ) ).
% 0.17/0.45
% 0.17/0.45 % corr_p
% 0.17/0.45 thf(fact_2_corr__q__tq,axiom,
% 0.17/0.45 genClo1015804716orrect @ q @ ( genClo1163638703lle_te @ q @ ( plus_plus_nat @ i @ one_one_nat ) ) ).
% 0.17/0.45
% 0.17/0.45 % corr_q_tq
% 0.17/0.45 thf(fact_3_ie,axiom,
% 0.17/0.45 ord_less_eq_real @ ( genClo1163638703lle_te @ q @ ( plus_plus_nat @ i @ one_one_nat ) ) @ ( genClo1163638703lle_te @ p @ ( plus_plus_nat @ i @ one_one_nat ) ) ).
% 0.17/0.45
% 0.17/0.45 % ie
% 0.17/0.45 thf(fact_4_corr__q,axiom,
% 0.17/0.45 genClo1015804716orrect @ q @ ( genClo1163638703lle_te @ p @ ( plus_plus_nat @ i @ one_one_nat ) ) ).
% 0.17/0.45
% 0.17/0.45 % corr_q
% 0.17/0.45 thf(fact_5_corr__l,axiom,
% 0.17/0.45 genClo1015804716orrect @ l @ ( genClo1163638703lle_te @ p @ ( plus_plus_nat @ i @ one_one_nat ) ) ).
% 0.17/0.45
% 0.17/0.45 % corr_l
% 0.17/0.45 thf(fact_6_posD,axiom,
% 0.17/0.45 ord_less_eq_real @ zero_zero_real @ ( minus_minus_real @ ( genClo1163638703lle_te @ p @ ( plus_plus_nat @ i @ one_one_nat ) ) @ ( genClo1163638703lle_te @ q @ ( plus_plus_nat @ i @ one_one_nat ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % posD
% 0.17/0.45 thf(fact_7_corr__l__tq,axiom,
% 0.17/0.45 genClo1015804716orrect @ l @ ( genClo1163638703lle_te @ q @ ( plus_plus_nat @ i @ one_one_nat ) ) ).
% 0.17/0.45
% 0.17/0.45 % corr_l_tq
% 0.17/0.45 thf(fact_8_le__add__diff__inverse,axiom,
% 0.17/0.45 ! [B: real,A: real] :
% 0.17/0.45 ( ( ord_less_eq_real @ B @ A )
% 0.17/0.45 => ( ( plus_plus_real @ B @ ( minus_minus_real @ A @ B ) )
% 0.17/0.45 = A ) ) ).
% 0.17/0.45
% 0.17/0.45 % le_add_diff_inverse
% 0.17/0.45 thf(fact_9_le__add__diff__inverse,axiom,
% 0.17/0.45 ! [B: nat,A: nat] :
% 0.17/0.45 ( ( ord_less_eq_nat @ B @ A )
% 0.17/0.45 => ( ( plus_plus_nat @ B @ ( minus_minus_nat @ A @ B ) )
% 0.17/0.45 = A ) ) ).
% 0.17/0.45
% 0.17/0.45 % le_add_diff_inverse
% 0.17/0.45 thf(fact_10_le__add__diff__inverse2,axiom,
% 0.17/0.45 ! [B: real,A: real] :
% 0.17/0.45 ( ( ord_less_eq_real @ B @ A )
% 0.17/0.45 => ( ( plus_plus_real @ ( minus_minus_real @ A @ B ) @ B )
% 0.17/0.45 = A ) ) ).
% 0.17/0.45
% 0.17/0.45 % le_add_diff_inverse2
% 0.17/0.45 thf(fact_11_le__add__diff__inverse2,axiom,
% 0.17/0.45 ! [B: nat,A: nat] :
% 0.17/0.45 ( ( ord_less_eq_nat @ B @ A )
% 0.17/0.45 => ( ( plus_plus_nat @ ( minus_minus_nat @ A @ B ) @ B )
% 0.17/0.45 = A ) ) ).
% 0.17/0.45
% 0.17/0.45 % le_add_diff_inverse2
% 0.17/0.45 thf(fact_12_rte,axiom,
% 0.17/0.45 ! [P2: nat,I2: nat] :
% 0.17/0.45 ( ( genClo1015804716orrect @ P2 @ ( genClo1163638703lle_te @ P2 @ ( plus_plus_nat @ I2 @ one_one_nat ) ) )
% 0.17/0.45 => ( ord_less_eq_real @ ( genClo1163638703lle_te @ P2 @ I2 ) @ ( genClo1163638703lle_te @ P2 @ ( plus_plus_nat @ I2 @ one_one_nat ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % rte
% 0.17/0.45 thf(fact_13_abs__1,axiom,
% 0.17/0.45 ( ( abs_abs_real @ one_one_real )
% 0.17/0.45 = one_one_real ) ).
% 0.17/0.45
% 0.17/0.45 % abs_1
% 0.17/0.45 thf(fact_14_abs__add__abs,axiom,
% 0.17/0.45 ! [A: real,B: real] :
% 0.17/0.45 ( ( abs_abs_real @ ( plus_plus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) )
% 0.17/0.45 = ( plus_plus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % abs_add_abs
% 0.17/0.45 thf(fact_15_add__diff__cancel,axiom,
% 0.17/0.45 ! [A: real,B: real] :
% 0.17/0.45 ( ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ B )
% 0.17/0.45 = A ) ).
% 0.17/0.45
% 0.17/0.45 % add_diff_cancel
% 0.17/0.45 thf(fact_16_diff__add__cancel,axiom,
% 0.17/0.45 ! [A: real,B: real] :
% 0.17/0.45 ( ( plus_plus_real @ ( minus_minus_real @ A @ B ) @ B )
% 0.17/0.45 = A ) ).
% 0.17/0.45
% 0.17/0.45 % diff_add_cancel
% 0.17/0.45 thf(fact_17_add__diff__cancel__left,axiom,
% 0.17/0.45 ! [C: real,A: real,B: real] :
% 0.17/0.45 ( ( minus_minus_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
% 0.17/0.45 = ( minus_minus_real @ A @ B ) ) ).
% 0.17/0.45
% 0.17/0.45 % add_diff_cancel_left
% 0.17/0.45 thf(fact_18_add__diff__cancel__left,axiom,
% 0.17/0.45 ! [C: nat,A: nat,B: nat] :
% 0.17/0.45 ( ( minus_minus_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
% 0.17/0.45 = ( minus_minus_nat @ A @ B ) ) ).
% 0.17/0.45
% 0.17/0.45 % add_diff_cancel_left
% 0.17/0.45 thf(fact_19_add__diff__cancel__left_H,axiom,
% 0.17/0.45 ! [A: real,B: real] :
% 0.17/0.45 ( ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ A )
% 0.17/0.45 = B ) ).
% 0.17/0.45
% 0.17/0.45 % add_diff_cancel_left'
% 0.17/0.45 thf(fact_20_add__diff__cancel__left_H,axiom,
% 0.17/0.45 ! [A: nat,B: nat] :
% 0.17/0.45 ( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ A )
% 0.17/0.45 = B ) ).
% 0.17/0.45
% 0.17/0.45 % add_diff_cancel_left'
% 0.17/0.45 thf(fact_21_add__diff__cancel__right,axiom,
% 0.17/0.45 ! [A: real,C: real,B: real] :
% 0.17/0.45 ( ( minus_minus_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
% 0.17/0.45 = ( minus_minus_real @ A @ B ) ) ).
% 0.17/0.45
% 0.17/0.45 % add_diff_cancel_right
% 0.17/0.45 thf(fact_22_add__diff__cancel__right,axiom,
% 0.17/0.45 ! [A: nat,C: nat,B: nat] :
% 0.17/0.45 ( ( minus_minus_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
% 0.17/0.45 = ( minus_minus_nat @ A @ B ) ) ).
% 0.17/0.45
% 0.17/0.45 % add_diff_cancel_right
% 0.17/0.45 thf(fact_23_add__right__cancel,axiom,
% 0.17/0.45 ! [B: nat,A: nat,C: nat] :
% 0.17/0.45 ( ( ( plus_plus_nat @ B @ A )
% 0.17/0.45 = ( plus_plus_nat @ C @ A ) )
% 0.17/0.45 = ( B = C ) ) ).
% 0.17/0.45
% 0.17/0.45 % add_right_cancel
% 0.17/0.45 thf(fact_24_add__right__cancel,axiom,
% 0.17/0.45 ! [B: real,A: real,C: real] :
% 0.17/0.45 ( ( ( plus_plus_real @ B @ A )
% 0.17/0.45 = ( plus_plus_real @ C @ A ) )
% 0.17/0.45 = ( B = C ) ) ).
% 0.17/0.45
% 0.17/0.45 % add_right_cancel
% 0.17/0.45 thf(fact_25_add__left__cancel,axiom,
% 0.17/0.45 ! [A: nat,B: nat,C: nat] :
% 0.17/0.45 ( ( ( plus_plus_nat @ A @ B )
% 0.17/0.45 = ( plus_plus_nat @ A @ C ) )
% 0.17/0.45 = ( B = C ) ) ).
% 0.17/0.45
% 0.17/0.45 % add_left_cancel
% 0.17/0.45 thf(fact_26_add__left__cancel,axiom,
% 0.17/0.45 ! [A: real,B: real,C: real] :
% 0.17/0.45 ( ( ( plus_plus_real @ A @ B )
% 0.17/0.45 = ( plus_plus_real @ A @ C ) )
% 0.17/0.45 = ( B = C ) ) ).
% 0.17/0.45
% 0.17/0.45 % add_left_cancel
% 0.17/0.45 thf(fact_27_abs__idempotent,axiom,
% 0.17/0.45 ! [A: real] :
% 0.17/0.45 ( ( abs_abs_real @ ( abs_abs_real @ A ) )
% 0.17/0.45 = ( abs_abs_real @ A ) ) ).
% 0.17/0.45
% 0.17/0.45 % abs_idempotent
% 0.17/0.45 thf(fact_28_abs__abs,axiom,
% 0.17/0.45 ! [A: real] :
% 0.17/0.45 ( ( abs_abs_real @ ( abs_abs_real @ A ) )
% 0.17/0.45 = ( abs_abs_real @ A ) ) ).
% 0.17/0.45
% 0.17/0.45 % abs_abs
% 0.17/0.45 thf(fact_29_le__zero__eq,axiom,
% 0.17/0.45 ! [N: nat] :
% 0.17/0.45 ( ( ord_less_eq_nat @ N @ zero_zero_nat )
% 0.17/0.45 = ( N = zero_zero_nat ) ) ).
% 0.17/0.45
% 0.17/0.45 % le_zero_eq
% 0.17/0.45 thf(fact_30_add__le__cancel__right,axiom,
% 0.17/0.45 ! [A: real,C: real,B: real] :
% 0.17/0.45 ( ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
% 0.17/0.45 = ( ord_less_eq_real @ A @ B ) ) ).
% 0.17/0.45
% 0.17/0.45 % add_le_cancel_right
% 0.17/0.45 thf(fact_31_add__le__cancel__right,axiom,
% 0.17/0.45 ! [A: nat,C: nat,B: nat] :
% 0.17/0.45 ( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
% 0.17/0.45 = ( ord_less_eq_nat @ A @ B ) ) ).
% 0.17/0.45
% 0.17/0.45 % add_le_cancel_right
% 0.17/0.45 thf(fact_32_add__le__cancel__left,axiom,
% 0.17/0.45 ! [C: real,A: real,B: real] :
% 0.17/0.45 ( ( ord_less_eq_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
% 0.17/0.45 = ( ord_less_eq_real @ A @ B ) ) ).
% 0.17/0.45
% 0.17/0.45 % add_le_cancel_left
% 0.17/0.45 thf(fact_33_add__le__cancel__left,axiom,
% 0.17/0.45 ! [C: nat,A: nat,B: nat] :
% 0.17/0.45 ( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
% 0.17/0.45 = ( ord_less_eq_nat @ A @ B ) ) ).
% 0.17/0.45
% 0.17/0.45 % add_le_cancel_left
% 0.17/0.45 thf(fact_34_zero__eq__add__iff__both__eq__0,axiom,
% 0.17/0.45 ! [X: nat,Y: nat] :
% 0.17/0.45 ( ( zero_zero_nat
% 0.17/0.45 = ( plus_plus_nat @ X @ Y ) )
% 0.17/0.45 = ( ( X = zero_zero_nat )
% 0.17/0.45 & ( Y = zero_zero_nat ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % zero_eq_add_iff_both_eq_0
% 0.17/0.45 thf(fact_35_add__eq__0__iff__both__eq__0,axiom,
% 0.17/0.45 ! [X: nat,Y: nat] :
% 0.17/0.45 ( ( ( plus_plus_nat @ X @ Y )
% 0.17/0.45 = zero_zero_nat )
% 0.17/0.45 = ( ( X = zero_zero_nat )
% 0.17/0.45 & ( Y = zero_zero_nat ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % add_eq_0_iff_both_eq_0
% 0.17/0.45 thf(fact_36_add__cancel__right__right,axiom,
% 0.17/0.45 ! [A: real,B: real] :
% 0.17/0.45 ( ( A
% 0.17/0.45 = ( plus_plus_real @ A @ B ) )
% 0.17/0.45 = ( B = zero_zero_real ) ) ).
% 0.17/0.45
% 0.17/0.45 % add_cancel_right_right
% 0.17/0.45 thf(fact_37_add__cancel__right__right,axiom,
% 0.17/0.45 ! [A: nat,B: nat] :
% 0.17/0.45 ( ( A
% 0.17/0.45 = ( plus_plus_nat @ A @ B ) )
% 0.17/0.45 = ( B = zero_zero_nat ) ) ).
% 0.17/0.45
% 0.17/0.45 % add_cancel_right_right
% 0.17/0.45 thf(fact_38_add__cancel__right__left,axiom,
% 0.17/0.45 ! [A: real,B: real] :
% 0.17/0.45 ( ( A
% 0.17/0.45 = ( plus_plus_real @ B @ A ) )
% 0.17/0.45 = ( B = zero_zero_real ) ) ).
% 0.17/0.45
% 0.17/0.45 % add_cancel_right_left
% 0.17/0.45 thf(fact_39_add__cancel__right__left,axiom,
% 0.17/0.45 ! [A: nat,B: nat] :
% 0.17/0.45 ( ( A
% 0.17/0.45 = ( plus_plus_nat @ B @ A ) )
% 0.17/0.45 = ( B = zero_zero_nat ) ) ).
% 0.17/0.45
% 0.17/0.45 % add_cancel_right_left
% 0.17/0.45 thf(fact_40_add__cancel__left__right,axiom,
% 0.17/0.45 ! [A: real,B: real] :
% 0.17/0.45 ( ( ( plus_plus_real @ A @ B )
% 0.17/0.45 = A )
% 0.17/0.45 = ( B = zero_zero_real ) ) ).
% 0.17/0.45
% 0.17/0.45 % add_cancel_left_right
% 0.17/0.45 thf(fact_41_add__cancel__left__right,axiom,
% 0.17/0.45 ! [A: nat,B: nat] :
% 0.17/0.45 ( ( ( plus_plus_nat @ A @ B )
% 0.17/0.45 = A )
% 0.17/0.45 = ( B = zero_zero_nat ) ) ).
% 0.17/0.45
% 0.17/0.45 % add_cancel_left_right
% 0.17/0.45 thf(fact_42_add__cancel__left__left,axiom,
% 0.17/0.45 ! [B: real,A: real] :
% 0.17/0.45 ( ( ( plus_plus_real @ B @ A )
% 0.17/0.45 = A )
% 0.17/0.45 = ( B = zero_zero_real ) ) ).
% 0.17/0.45
% 0.17/0.45 % add_cancel_left_left
% 0.17/0.45 thf(fact_43_add__cancel__left__left,axiom,
% 0.17/0.45 ! [B: nat,A: nat] :
% 0.17/0.45 ( ( ( plus_plus_nat @ B @ A )
% 0.17/0.45 = A )
% 0.17/0.45 = ( B = zero_zero_nat ) ) ).
% 0.17/0.45
% 0.17/0.45 % add_cancel_left_left
% 0.17/0.45 thf(fact_44_double__zero__sym,axiom,
% 0.17/0.45 ! [A: real] :
% 0.17/0.45 ( ( zero_zero_real
% 0.17/0.45 = ( plus_plus_real @ A @ A ) )
% 0.17/0.45 = ( A = zero_zero_real ) ) ).
% 0.17/0.45
% 0.17/0.45 % double_zero_sym
% 0.17/0.45 thf(fact_45_double__zero,axiom,
% 0.17/0.45 ! [A: real] :
% 0.17/0.45 ( ( ( plus_plus_real @ A @ A )
% 0.17/0.45 = zero_zero_real )
% 0.17/0.45 = ( A = zero_zero_real ) ) ).
% 0.17/0.45
% 0.17/0.45 % double_zero
% 0.17/0.45 thf(fact_46_add_Oright__neutral,axiom,
% 0.17/0.45 ! [A: real] :
% 0.17/0.45 ( ( plus_plus_real @ A @ zero_zero_real )
% 0.17/0.45 = A ) ).
% 0.17/0.45
% 0.17/0.45 % add.right_neutral
% 0.17/0.45 thf(fact_47_add_Oright__neutral,axiom,
% 0.17/0.45 ! [A: nat] :
% 0.17/0.45 ( ( plus_plus_nat @ A @ zero_zero_nat )
% 0.17/0.45 = A ) ).
% 0.17/0.45
% 0.17/0.45 % add.right_neutral
% 0.17/0.45 thf(fact_48_add_Oleft__neutral,axiom,
% 0.17/0.45 ! [A: real] :
% 0.17/0.45 ( ( plus_plus_real @ zero_zero_real @ A )
% 0.17/0.45 = A ) ).
% 0.17/0.45
% 0.17/0.45 % add.left_neutral
% 0.17/0.45 thf(fact_49_add_Oleft__neutral,axiom,
% 0.17/0.45 ! [A: nat] :
% 0.17/0.45 ( ( plus_plus_nat @ zero_zero_nat @ A )
% 0.17/0.45 = A ) ).
% 0.17/0.45
% 0.17/0.45 % add.left_neutral
% 0.17/0.45 thf(fact_50_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
% 0.17/0.45 ! [A: real] :
% 0.17/0.45 ( ( minus_minus_real @ A @ A )
% 0.17/0.45 = zero_zero_real ) ).
% 0.17/0.45
% 0.17/0.45 % cancel_comm_monoid_add_class.diff_cancel
% 0.17/0.45 thf(fact_51_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
% 0.17/0.45 ! [A: nat] :
% 0.17/0.45 ( ( minus_minus_nat @ A @ A )
% 0.17/0.45 = zero_zero_nat ) ).
% 0.17/0.45
% 0.17/0.45 % cancel_comm_monoid_add_class.diff_cancel
% 0.17/0.45 thf(fact_52_diff__zero,axiom,
% 0.17/0.45 ! [A: real] :
% 0.17/0.45 ( ( minus_minus_real @ A @ zero_zero_real )
% 0.17/0.45 = A ) ).
% 0.17/0.45
% 0.17/0.45 % diff_zero
% 0.17/0.45 thf(fact_53_diff__zero,axiom,
% 0.17/0.45 ! [A: nat] :
% 0.17/0.45 ( ( minus_minus_nat @ A @ zero_zero_nat )
% 0.17/0.45 = A ) ).
% 0.17/0.45
% 0.17/0.45 % diff_zero
% 0.17/0.45 thf(fact_54_zero__diff,axiom,
% 0.17/0.45 ! [A: nat] :
% 0.17/0.45 ( ( minus_minus_nat @ zero_zero_nat @ A )
% 0.17/0.45 = zero_zero_nat ) ).
% 0.17/0.45
% 0.17/0.45 % zero_diff
% 0.17/0.45 thf(fact_55_diff__0__right,axiom,
% 0.17/0.45 ! [A: real] :
% 0.17/0.45 ( ( minus_minus_real @ A @ zero_zero_real )
% 0.17/0.45 = A ) ).
% 0.17/0.45
% 0.17/0.45 % diff_0_right
% 0.17/0.45 thf(fact_56_diff__self,axiom,
% 0.17/0.45 ! [A: real] :
% 0.17/0.45 ( ( minus_minus_real @ A @ A )
% 0.17/0.45 = zero_zero_real ) ).
% 0.17/0.45
% 0.17/0.45 % diff_self
% 0.17/0.45 thf(fact_57_add__diff__cancel__right_H,axiom,
% 0.17/0.45 ! [A: real,B: real] :
% 0.17/0.45 ( ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ B )
% 0.17/0.45 = A ) ).
% 0.17/0.45
% 0.17/0.45 % add_diff_cancel_right'
% 0.17/0.45 thf(fact_58_add__diff__cancel__right_H,axiom,
% 0.17/0.45 ! [A: nat,B: nat] :
% 0.17/0.45 ( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ B )
% 0.17/0.45 = A ) ).
% 0.17/0.45
% 0.17/0.45 % add_diff_cancel_right'
% 0.17/0.45 thf(fact_59_abs__zero,axiom,
% 0.17/0.45 ( ( abs_abs_real @ zero_zero_real )
% 0.17/0.45 = zero_zero_real ) ).
% 0.17/0.45
% 0.17/0.45 % abs_zero
% 0.17/0.45 thf(fact_60_abs__eq__0,axiom,
% 0.17/0.45 ! [A: real] :
% 0.17/0.45 ( ( ( abs_abs_real @ A )
% 0.17/0.45 = zero_zero_real )
% 0.17/0.45 = ( A = zero_zero_real ) ) ).
% 0.17/0.45
% 0.17/0.45 % abs_eq_0
% 0.17/0.45 thf(fact_61_abs__0__eq,axiom,
% 0.17/0.45 ! [A: real] :
% 0.17/0.45 ( ( zero_zero_real
% 0.17/0.45 = ( abs_abs_real @ A ) )
% 0.17/0.45 = ( A = zero_zero_real ) ) ).
% 0.17/0.45
% 0.17/0.45 % abs_0_eq
% 0.17/0.45 thf(fact_62_abs__0,axiom,
% 0.17/0.45 ( ( abs_abs_real @ zero_zero_real )
% 0.17/0.45 = zero_zero_real ) ).
% 0.17/0.45
% 0.17/0.45 % abs_0
% 0.17/0.45 thf(fact_63_zero__le__double__add__iff__zero__le__single__add,axiom,
% 0.17/0.45 ! [A: real] :
% 0.17/0.45 ( ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ A @ A ) )
% 0.17/0.45 = ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% 0.17/0.45
% 0.17/0.45 % zero_le_double_add_iff_zero_le_single_add
% 0.17/0.45 thf(fact_64_double__add__le__zero__iff__single__add__le__zero,axiom,
% 0.17/0.45 ! [A: real] :
% 0.17/0.45 ( ( ord_less_eq_real @ ( plus_plus_real @ A @ A ) @ zero_zero_real )
% 0.17/0.45 = ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% 0.17/0.45
% 0.17/0.45 % double_add_le_zero_iff_single_add_le_zero
% 0.17/0.45 thf(fact_65_le__add__same__cancel2,axiom,
% 0.17/0.45 ! [A: real,B: real] :
% 0.17/0.45 ( ( ord_less_eq_real @ A @ ( plus_plus_real @ B @ A ) )
% 0.17/0.45 = ( ord_less_eq_real @ zero_zero_real @ B ) ) ).
% 0.17/0.45
% 0.17/0.45 % le_add_same_cancel2
% 0.17/0.45 thf(fact_66_le__add__same__cancel2,axiom,
% 0.17/0.45 ! [A: nat,B: nat] :
% 0.17/0.45 ( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ B @ A ) )
% 0.17/0.45 = ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% 0.17/0.45
% 0.17/0.45 % le_add_same_cancel2
% 0.17/0.45 thf(fact_67_le__add__same__cancel1,axiom,
% 0.17/0.45 ! [A: real,B: real] :
% 0.17/0.45 ( ( ord_less_eq_real @ A @ ( plus_plus_real @ A @ B ) )
% 0.17/0.45 = ( ord_less_eq_real @ zero_zero_real @ B ) ) ).
% 0.17/0.45
% 0.17/0.45 % le_add_same_cancel1
% 0.17/0.45 thf(fact_68_le__add__same__cancel1,axiom,
% 0.17/0.45 ! [A: nat,B: nat] :
% 0.17/0.45 ( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ A @ B ) )
% 0.17/0.45 = ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% 0.17/0.45
% 0.17/0.45 % le_add_same_cancel1
% 0.17/0.45 thf(fact_69_add__le__same__cancel2,axiom,
% 0.17/0.45 ! [A: real,B: real] :
% 0.17/0.45 ( ( ord_less_eq_real @ ( plus_plus_real @ A @ B ) @ B )
% 0.17/0.45 = ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% 0.17/0.45
% 0.17/0.45 % add_le_same_cancel2
% 0.17/0.45 thf(fact_70_add__le__same__cancel2,axiom,
% 0.17/0.45 ! [A: nat,B: nat] :
% 0.17/0.45 ( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ B )
% 0.17/0.45 = ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).
% 0.17/0.45
% 0.17/0.45 % add_le_same_cancel2
% 0.17/0.45 thf(fact_71_add__le__same__cancel1,axiom,
% 0.17/0.45 ! [B: real,A: real] :
% 0.17/0.45 ( ( ord_less_eq_real @ ( plus_plus_real @ B @ A ) @ B )
% 0.17/0.45 = ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% 0.17/0.45
% 0.17/0.45 % add_le_same_cancel1
% 0.17/0.45 thf(fact_72_add__le__same__cancel1,axiom,
% 0.17/0.45 ! [B: nat,A: nat] :
% 0.17/0.45 ( ( ord_less_eq_nat @ ( plus_plus_nat @ B @ A ) @ B )
% 0.17/0.45 = ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).
% 0.17/0.45
% 0.17/0.45 % add_le_same_cancel1
% 0.17/0.45 thf(fact_73_diff__ge__0__iff__ge,axiom,
% 0.17/0.45 ! [A: real,B: real] :
% 0.17/0.45 ( ( ord_less_eq_real @ zero_zero_real @ ( minus_minus_real @ A @ B ) )
% 0.17/0.45 = ( ord_less_eq_real @ B @ A ) ) ).
% 0.17/0.45
% 0.17/0.45 % diff_ge_0_iff_ge
% 0.17/0.45 thf(fact_74_diff__add__zero,axiom,
% 0.17/0.45 ! [A: nat,B: nat] :
% 0.17/0.45 ( ( minus_minus_nat @ A @ ( plus_plus_nat @ A @ B ) )
% 0.17/0.45 = zero_zero_nat ) ).
% 0.17/0.45
% 0.17/0.45 % diff_add_zero
% 0.17/0.45 thf(fact_75_abs__le__zero__iff,axiom,
% 0.17/0.45 ! [A: real] :
% 0.17/0.45 ( ( ord_less_eq_real @ ( abs_abs_real @ A ) @ zero_zero_real )
% 0.17/0.45 = ( A = zero_zero_real ) ) ).
% 0.17/0.45
% 0.17/0.45 % abs_le_zero_iff
% 0.17/0.45 thf(fact_76_abs__le__self__iff,axiom,
% 0.17/0.45 ! [A: real] :
% 0.17/0.45 ( ( ord_less_eq_real @ ( abs_abs_real @ A ) @ A )
% 0.17/0.45 = ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% 0.17/0.45
% 0.17/0.45 % abs_le_self_iff
% 0.17/0.45 thf(fact_77_abs__of__nonneg,axiom,
% 0.17/0.45 ! [A: real] :
% 0.17/0.45 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.17/0.45 => ( ( abs_abs_real @ A )
% 0.17/0.45 = A ) ) ).
% 0.17/0.45
% 0.17/0.45 % abs_of_nonneg
% 0.17/0.45 thf(fact_78_zero__reorient,axiom,
% 0.17/0.45 ! [X: real] :
% 0.17/0.45 ( ( zero_zero_real = X )
% 0.17/0.45 = ( X = zero_zero_real ) ) ).
% 0.17/0.45
% 0.17/0.45 % zero_reorient
% 0.17/0.45 thf(fact_79_zero__reorient,axiom,
% 0.17/0.45 ! [X: nat] :
% 0.17/0.45 ( ( zero_zero_nat = X )
% 0.17/0.45 = ( X = zero_zero_nat ) ) ).
% 0.17/0.45
% 0.17/0.45 % zero_reorient
% 0.17/0.45 thf(fact_80_zero__le,axiom,
% 0.17/0.45 ! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).
% 0.17/0.45
% 0.17/0.45 % zero_le
% 0.17/0.45 thf(fact_81_add_Ogroup__left__neutral,axiom,
% 0.17/0.45 ! [A: real] :
% 0.17/0.45 ( ( plus_plus_real @ zero_zero_real @ A )
% 0.17/0.45 = A ) ).
% 0.17/0.45
% 0.17/0.45 % add.group_left_neutral
% 0.17/0.45 thf(fact_82_add_Ocomm__neutral,axiom,
% 0.17/0.45 ! [A: real] :
% 0.17/0.45 ( ( plus_plus_real @ A @ zero_zero_real )
% 0.17/0.45 = A ) ).
% 0.17/0.45
% 0.17/0.45 % add.comm_neutral
% 0.17/0.45 thf(fact_83_add_Ocomm__neutral,axiom,
% 0.17/0.45 ! [A: nat] :
% 0.17/0.45 ( ( plus_plus_nat @ A @ zero_zero_nat )
% 0.17/0.45 = A ) ).
% 0.17/0.45
% 0.17/0.45 % add.comm_neutral
% 0.17/0.45 thf(fact_84_comm__monoid__add__class_Oadd__0,axiom,
% 0.17/0.45 ! [A: real] :
% 0.17/0.45 ( ( plus_plus_real @ zero_zero_real @ A )
% 0.17/0.45 = A ) ).
% 0.17/0.45
% 0.17/0.45 % comm_monoid_add_class.add_0
% 0.17/0.45 thf(fact_85_comm__monoid__add__class_Oadd__0,axiom,
% 0.17/0.45 ! [A: nat] :
% 0.17/0.45 ( ( plus_plus_nat @ zero_zero_nat @ A )
% 0.17/0.45 = A ) ).
% 0.17/0.45
% 0.17/0.45 % comm_monoid_add_class.add_0
% 0.17/0.45 thf(fact_86_zero__neq__one,axiom,
% 0.17/0.45 zero_zero_real != one_one_real ).
% 0.17/0.45
% 0.17/0.45 % zero_neq_one
% 0.17/0.45 thf(fact_87_zero__neq__one,axiom,
% 0.17/0.45 zero_zero_nat != one_one_nat ).
% 0.17/0.45
% 0.17/0.45 % zero_neq_one
% 0.17/0.45 thf(fact_88_eq__iff__diff__eq__0,axiom,
% 0.17/0.45 ( ( ^ [Y2: real,Z: real] : ( Y2 = Z ) )
% 0.17/0.45 = ( ^ [A2: real,B2: real] :
% 0.17/0.45 ( ( minus_minus_real @ A2 @ B2 )
% 0.17/0.45 = zero_zero_real ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % eq_iff_diff_eq_0
% 0.17/0.45 thf(fact_89_abs__eq__0__iff,axiom,
% 0.17/0.45 ! [A: real] :
% 0.17/0.45 ( ( ( abs_abs_real @ A )
% 0.17/0.45 = zero_zero_real )
% 0.17/0.45 = ( A = zero_zero_real ) ) ).
% 0.17/0.45
% 0.17/0.45 % abs_eq_0_iff
% 0.17/0.45 thf(fact_90_add__nonpos__eq__0__iff,axiom,
% 0.17/0.45 ! [X: real,Y: real] :
% 0.17/0.45 ( ( ord_less_eq_real @ X @ zero_zero_real )
% 0.17/0.45 => ( ( ord_less_eq_real @ Y @ zero_zero_real )
% 0.17/0.45 => ( ( ( plus_plus_real @ X @ Y )
% 0.17/0.45 = zero_zero_real )
% 0.17/0.45 = ( ( X = zero_zero_real )
% 0.17/0.45 & ( Y = zero_zero_real ) ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % add_nonpos_eq_0_iff
% 0.17/0.45 thf(fact_91_add__nonpos__eq__0__iff,axiom,
% 0.17/0.45 ! [X: nat,Y: nat] :
% 0.17/0.45 ( ( ord_less_eq_nat @ X @ zero_zero_nat )
% 0.17/0.45 => ( ( ord_less_eq_nat @ Y @ zero_zero_nat )
% 0.17/0.45 => ( ( ( plus_plus_nat @ X @ Y )
% 0.17/0.45 = zero_zero_nat )
% 0.17/0.45 = ( ( X = zero_zero_nat )
% 0.17/0.45 & ( Y = zero_zero_nat ) ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % add_nonpos_eq_0_iff
% 0.17/0.45 thf(fact_92_add__nonneg__eq__0__iff,axiom,
% 0.17/0.45 ! [X: real,Y: real] :
% 0.17/0.45 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.17/0.45 => ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 0.17/0.45 => ( ( ( plus_plus_real @ X @ Y )
% 0.17/0.45 = zero_zero_real )
% 0.17/0.45 = ( ( X = zero_zero_real )
% 0.17/0.45 & ( Y = zero_zero_real ) ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % add_nonneg_eq_0_iff
% 0.17/0.45 thf(fact_93_add__nonneg__eq__0__iff,axiom,
% 0.17/0.45 ! [X: nat,Y: nat] :
% 0.17/0.45 ( ( ord_less_eq_nat @ zero_zero_nat @ X )
% 0.17/0.45 => ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
% 0.17/0.45 => ( ( ( plus_plus_nat @ X @ Y )
% 0.17/0.45 = zero_zero_nat )
% 0.17/0.45 = ( ( X = zero_zero_nat )
% 0.17/0.45 & ( Y = zero_zero_nat ) ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % add_nonneg_eq_0_iff
% 0.17/0.45 thf(fact_94_add__nonpos__nonpos,axiom,
% 0.17/0.45 ! [A: real,B: real] :
% 0.17/0.45 ( ( ord_less_eq_real @ A @ zero_zero_real )
% 0.17/0.45 => ( ( ord_less_eq_real @ B @ zero_zero_real )
% 0.17/0.45 => ( ord_less_eq_real @ ( plus_plus_real @ A @ B ) @ zero_zero_real ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % add_nonpos_nonpos
% 0.17/0.45 thf(fact_95_add__nonpos__nonpos,axiom,
% 0.17/0.45 ! [A: nat,B: nat] :
% 0.17/0.45 ( ( ord_less_eq_nat @ A @ zero_zero_nat )
% 0.17/0.45 => ( ( ord_less_eq_nat @ B @ zero_zero_nat )
% 0.17/0.45 => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % add_nonpos_nonpos
% 0.17/0.45 thf(fact_96_add__nonneg__nonneg,axiom,
% 0.17/0.45 ! [A: real,B: real] :
% 0.17/0.45 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.17/0.45 => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 0.17/0.45 => ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ A @ B ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % add_nonneg_nonneg
% 0.17/0.45 thf(fact_97_add__nonneg__nonneg,axiom,
% 0.17/0.45 ! [A: nat,B: nat] :
% 0.17/0.45 ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.17/0.45 => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 0.17/0.45 => ( ord_less_eq_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % add_nonneg_nonneg
% 0.17/0.45 thf(fact_98_add__increasing2,axiom,
% 0.17/0.45 ! [C: real,B: real,A: real] :
% 0.17/0.45 ( ( ord_less_eq_real @ zero_zero_real @ C )
% 0.17/0.45 => ( ( ord_less_eq_real @ B @ A )
% 0.17/0.45 => ( ord_less_eq_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % add_increasing2
% 0.17/0.45 thf(fact_99_add__increasing2,axiom,
% 0.17/0.45 ! [C: nat,B: nat,A: nat] :
% 0.17/0.45 ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 0.17/0.45 => ( ( ord_less_eq_nat @ B @ A )
% 0.17/0.45 => ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % add_increasing2
% 0.17/0.45 thf(fact_100_add__decreasing2,axiom,
% 0.17/0.45 ! [C: real,A: real,B: real] :
% 0.17/0.45 ( ( ord_less_eq_real @ C @ zero_zero_real )
% 0.17/0.45 => ( ( ord_less_eq_real @ A @ B )
% 0.17/0.45 => ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ B ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % add_decreasing2
% 0.17/0.45 thf(fact_101_add__decreasing2,axiom,
% 0.17/0.45 ! [C: nat,A: nat,B: nat] :
% 0.17/0.45 ( ( ord_less_eq_nat @ C @ zero_zero_nat )
% 0.17/0.45 => ( ( ord_less_eq_nat @ A @ B )
% 0.17/0.45 => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ B ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % add_decreasing2
% 0.17/0.45 thf(fact_102_add__increasing,axiom,
% 0.17/0.45 ! [A: real,B: real,C: real] :
% 0.17/0.45 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.17/0.45 => ( ( ord_less_eq_real @ B @ C )
% 0.17/0.45 => ( ord_less_eq_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % add_increasing
% 0.17/0.45 thf(fact_103_add__increasing,axiom,
% 0.17/0.45 ! [A: nat,B: nat,C: nat] :
% 0.17/0.45 ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.17/0.45 => ( ( ord_less_eq_nat @ B @ C )
% 0.17/0.45 => ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % add_increasing
% 0.17/0.45 thf(fact_104_add__decreasing,axiom,
% 0.17/0.45 ! [A: real,C: real,B: real] :
% 0.17/0.45 ( ( ord_less_eq_real @ A @ zero_zero_real )
% 0.17/0.45 => ( ( ord_less_eq_real @ C @ B )
% 0.17/0.45 => ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ B ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % add_decreasing
% 0.17/0.45 thf(fact_105_add__decreasing,axiom,
% 0.17/0.45 ! [A: nat,C: nat,B: nat] :
% 0.17/0.45 ( ( ord_less_eq_nat @ A @ zero_zero_nat )
% 0.17/0.45 => ( ( ord_less_eq_nat @ C @ B )
% 0.17/0.45 => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ B ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % add_decreasing
% 0.17/0.45 thf(fact_106_not__one__le__zero,axiom,
% 0.17/0.45 ~ ( ord_less_eq_real @ one_one_real @ zero_zero_real ) ).
% 0.17/0.45
% 0.17/0.45 % not_one_le_zero
% 0.17/0.45 thf(fact_107_not__one__le__zero,axiom,
% 0.17/0.45 ~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).
% 0.17/0.45
% 0.17/0.45 % not_one_le_zero
% 0.17/0.45 thf(fact_108_zero__le__one,axiom,
% 0.17/0.45 ord_less_eq_real @ zero_zero_real @ one_one_real ).
% 0.17/0.45
% 0.17/0.45 % zero_le_one
% 0.17/0.45 thf(fact_109_zero__le__one,axiom,
% 0.17/0.45 ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% 0.17/0.45
% 0.17/0.45 % zero_le_one
% 0.17/0.45 thf(fact_110_le__iff__diff__le__0,axiom,
% 0.17/0.45 ( ord_less_eq_real
% 0.17/0.45 = ( ^ [A2: real,B2: real] : ( ord_less_eq_real @ ( minus_minus_real @ A2 @ B2 ) @ zero_zero_real ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % le_iff_diff_le_0
% 0.17/0.45 thf(fact_111_abs__ge__zero,axiom,
% 0.17/0.45 ! [A: real] : ( ord_less_eq_real @ zero_zero_real @ ( abs_abs_real @ A ) ) ).
% 0.17/0.45
% 0.17/0.45 % abs_ge_zero
% 0.17/0.45 thf(fact_112_rts2a,axiom,
% 0.17/0.45 ! [P: nat,Q: nat,T: real,I: nat] :
% 0.17/0.45 ( ( ( genClo1015804716orrect @ P @ T )
% 0.17/0.45 & ( genClo1015804716orrect @ Q @ T )
% 0.17/0.45 & ( ord_less_eq_real @ ( plus_plus_real @ ( genClo1163638703lle_te @ Q @ I ) @ genClo1278781456e_beta ) @ T ) )
% 0.17/0.45 => ( ord_less_eq_real @ ( genClo1163638703lle_te @ P @ I ) @ T ) ) ).
% 0.17/0.45
% 0.17/0.45 % rts2a
% 0.17/0.45 thf(fact_113_add__right__imp__eq,axiom,
% 0.17/0.45 ! [B: nat,A: nat,C: nat] :
% 0.17/0.45 ( ( ( plus_plus_nat @ B @ A )
% 0.17/0.45 = ( plus_plus_nat @ C @ A ) )
% 0.17/0.45 => ( B = C ) ) ).
% 0.17/0.45
% 0.17/0.45 % add_right_imp_eq
% 0.17/0.45 thf(fact_114_add__right__imp__eq,axiom,
% 0.17/0.45 ! [B: real,A: real,C: real] :
% 0.17/0.45 ( ( ( plus_plus_real @ B @ A )
% 0.17/0.45 = ( plus_plus_real @ C @ A ) )
% 0.17/0.45 => ( B = C ) ) ).
% 0.17/0.45
% 0.17/0.45 % add_right_imp_eq
% 0.17/0.45 thf(fact_115_add__left__imp__eq,axiom,
% 0.17/0.45 ! [A: nat,B: nat,C: nat] :
% 0.17/0.45 ( ( ( plus_plus_nat @ A @ B )
% 0.17/0.45 = ( plus_plus_nat @ A @ C ) )
% 0.17/0.45 => ( B = C ) ) ).
% 0.17/0.45
% 0.17/0.45 % add_left_imp_eq
% 0.17/0.45 thf(fact_116_add__left__imp__eq,axiom,
% 0.17/0.45 ! [A: real,B: real,C: real] :
% 0.17/0.45 ( ( ( plus_plus_real @ A @ B )
% 0.17/0.45 = ( plus_plus_real @ A @ C ) )
% 0.17/0.45 => ( B = C ) ) ).
% 0.17/0.45
% 0.17/0.45 % add_left_imp_eq
% 0.17/0.45 thf(fact_117_add_Oleft__commute,axiom,
% 0.17/0.45 ! [B: nat,A: nat,C: nat] :
% 0.17/0.45 ( ( plus_plus_nat @ B @ ( plus_plus_nat @ A @ C ) )
% 0.17/0.45 = ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % add.left_commute
% 0.17/0.45 thf(fact_118_add_Oleft__commute,axiom,
% 0.17/0.45 ! [B: real,A: real,C: real] :
% 0.17/0.45 ( ( plus_plus_real @ B @ ( plus_plus_real @ A @ C ) )
% 0.17/0.45 = ( plus_plus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % add.left_commute
% 0.17/0.45 thf(fact_119_add_Ocommute,axiom,
% 0.17/0.45 ( plus_plus_nat
% 0.17/0.45 = ( ^ [A2: nat,B2: nat] : ( plus_plus_nat @ B2 @ A2 ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % add.commute
% 0.17/0.45 thf(fact_120_add_Ocommute,axiom,
% 0.17/0.45 ( plus_plus_real
% 0.17/0.45 = ( ^ [A2: real,B2: real] : ( plus_plus_real @ B2 @ A2 ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % add.commute
% 0.17/0.45 thf(fact_121_add_Oright__cancel,axiom,
% 0.17/0.45 ! [B: real,A: real,C: real] :
% 0.17/0.45 ( ( ( plus_plus_real @ B @ A )
% 0.17/0.45 = ( plus_plus_real @ C @ A ) )
% 0.17/0.45 = ( B = C ) ) ).
% 0.17/0.45
% 0.17/0.45 % add.right_cancel
% 0.17/0.45 thf(fact_122_add_Oleft__cancel,axiom,
% 0.17/0.45 ! [A: real,B: real,C: real] :
% 0.17/0.45 ( ( ( plus_plus_real @ A @ B )
% 0.17/0.45 = ( plus_plus_real @ A @ C ) )
% 0.17/0.45 = ( B = C ) ) ).
% 0.17/0.45
% 0.17/0.45 % add.left_cancel
% 0.17/0.45 thf(fact_123_add_Oassoc,axiom,
% 0.17/0.45 ! [A: nat,B: nat,C: nat] :
% 0.17/0.45 ( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
% 0.17/0.45 = ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % add.assoc
% 0.17/0.45 thf(fact_124_add_Oassoc,axiom,
% 0.17/0.45 ! [A: real,B: real,C: real] :
% 0.17/0.45 ( ( plus_plus_real @ ( plus_plus_real @ A @ B ) @ C )
% 0.17/0.45 = ( plus_plus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % add.assoc
% 0.17/0.45 thf(fact_125_group__cancel_Oadd2,axiom,
% 0.17/0.45 ! [B3: nat,K: nat,B: nat,A: nat] :
% 0.17/0.45 ( ( B3
% 0.17/0.45 = ( plus_plus_nat @ K @ B ) )
% 0.17/0.45 => ( ( plus_plus_nat @ A @ B3 )
% 0.17/0.45 = ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % group_cancel.add2
% 0.17/0.45 thf(fact_126_group__cancel_Oadd2,axiom,
% 0.17/0.45 ! [B3: real,K: real,B: real,A: real] :
% 0.17/0.45 ( ( B3
% 0.17/0.45 = ( plus_plus_real @ K @ B ) )
% 0.17/0.45 => ( ( plus_plus_real @ A @ B3 )
% 0.17/0.45 = ( plus_plus_real @ K @ ( plus_plus_real @ A @ B ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % group_cancel.add2
% 0.17/0.45 thf(fact_127_group__cancel_Oadd1,axiom,
% 0.17/0.45 ! [A3: nat,K: nat,A: nat,B: nat] :
% 0.17/0.45 ( ( A3
% 0.17/0.45 = ( plus_plus_nat @ K @ A ) )
% 0.17/0.45 => ( ( plus_plus_nat @ A3 @ B )
% 0.17/0.45 = ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % group_cancel.add1
% 0.17/0.45 thf(fact_128_group__cancel_Oadd1,axiom,
% 0.17/0.45 ! [A3: real,K: real,A: real,B: real] :
% 0.17/0.45 ( ( A3
% 0.17/0.45 = ( plus_plus_real @ K @ A ) )
% 0.17/0.45 => ( ( plus_plus_real @ A3 @ B )
% 0.17/0.45 = ( plus_plus_real @ K @ ( plus_plus_real @ A @ B ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % group_cancel.add1
% 0.17/0.45 thf(fact_129_add__mono__thms__linordered__semiring_I4_J,axiom,
% 0.17/0.45 ! [I2: nat,J: nat,K: nat,L: nat] :
% 0.17/0.45 ( ( ( I2 = J )
% 0.17/0.45 & ( K = L ) )
% 0.17/0.45 => ( ( plus_plus_nat @ I2 @ K )
% 0.17/0.45 = ( plus_plus_nat @ J @ L ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % add_mono_thms_linordered_semiring(4)
% 0.17/0.45 thf(fact_130_add__mono__thms__linordered__semiring_I4_J,axiom,
% 0.17/0.45 ! [I2: real,J: real,K: real,L: real] :
% 0.17/0.45 ( ( ( I2 = J )
% 0.17/0.45 & ( K = L ) )
% 0.17/0.45 => ( ( plus_plus_real @ I2 @ K )
% 0.17/0.45 = ( plus_plus_real @ J @ L ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % add_mono_thms_linordered_semiring(4)
% 0.17/0.45 thf(fact_131_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
% 0.17/0.45 ! [A: nat,B: nat,C: nat] :
% 0.17/0.45 ( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
% 0.17/0.45 = ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % ab_semigroup_add_class.add_ac(1)
% 0.17/0.45 thf(fact_132_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
% 0.17/0.45 ! [A: real,B: real,C: real] :
% 0.17/0.45 ( ( plus_plus_real @ ( plus_plus_real @ A @ B ) @ C )
% 0.17/0.45 = ( plus_plus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % ab_semigroup_add_class.add_ac(1)
% 0.17/0.45 thf(fact_133_one__reorient,axiom,
% 0.17/0.45 ! [X: nat] :
% 0.17/0.45 ( ( one_one_nat = X )
% 0.17/0.45 = ( X = one_one_nat ) ) ).
% 0.17/0.45
% 0.17/0.45 % one_reorient
% 0.17/0.45 thf(fact_134_one__reorient,axiom,
% 0.17/0.45 ! [X: real] :
% 0.17/0.45 ( ( one_one_real = X )
% 0.17/0.45 = ( X = one_one_real ) ) ).
% 0.17/0.45
% 0.17/0.45 % one_reorient
% 0.17/0.45 thf(fact_135_diff__right__commute,axiom,
% 0.17/0.45 ! [A: real,C: real,B: real] :
% 0.17/0.45 ( ( minus_minus_real @ ( minus_minus_real @ A @ C ) @ B )
% 0.17/0.45 = ( minus_minus_real @ ( minus_minus_real @ A @ B ) @ C ) ) ).
% 0.17/0.45
% 0.17/0.45 % diff_right_commute
% 0.17/0.45 thf(fact_136_diff__right__commute,axiom,
% 0.17/0.45 ! [A: nat,C: nat,B: nat] :
% 0.17/0.45 ( ( minus_minus_nat @ ( minus_minus_nat @ A @ C ) @ B )
% 0.17/0.45 = ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C ) ) ).
% 0.17/0.45
% 0.17/0.45 % diff_right_commute
% 0.17/0.45 thf(fact_137_diff__eq__diff__eq,axiom,
% 0.17/0.45 ! [A: real,B: real,C: real,D: real] :
% 0.17/0.45 ( ( ( minus_minus_real @ A @ B )
% 0.17/0.45 = ( minus_minus_real @ C @ D ) )
% 0.17/0.45 => ( ( A = B )
% 0.17/0.45 = ( C = D ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % diff_eq_diff_eq
% 0.17/0.45 thf(fact_138_beta__bound1,axiom,
% 0.17/0.45 ! [P2: nat,I2: nat,Q2: nat] :
% 0.17/0.45 ( ( genClo1015804716orrect @ P2 @ ( genClo1163638703lle_te @ P2 @ ( plus_plus_nat @ I2 @ one_one_nat ) ) )
% 0.17/0.45 => ( ( genClo1015804716orrect @ Q2 @ ( genClo1163638703lle_te @ P2 @ ( plus_plus_nat @ I2 @ one_one_nat ) ) )
% 0.17/0.45 => ( ord_less_eq_real @ zero_zero_real @ ( minus_minus_real @ ( genClo1163638703lle_te @ P2 @ ( plus_plus_nat @ I2 @ one_one_nat ) ) @ ( genClo1163638703lle_te @ Q2 @ I2 ) ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % beta_bound1
% 0.17/0.45 thf(fact_139_correct__closed,axiom,
% 0.17/0.45 ! [P: nat,S: real,T: real] :
% 0.17/0.45 ( ( ( ord_less_eq_real @ S @ T )
% 0.17/0.45 & ( genClo1015804716orrect @ P @ T ) )
% 0.17/0.45 => ( genClo1015804716orrect @ P @ S ) ) ).
% 0.17/0.45
% 0.17/0.45 % correct_closed
% 0.17/0.45 thf(fact_140_add__le__imp__le__right,axiom,
% 0.17/0.45 ! [A: real,C: real,B: real] :
% 0.17/0.45 ( ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
% 0.17/0.45 => ( ord_less_eq_real @ A @ B ) ) ).
% 0.17/0.45
% 0.17/0.45 % add_le_imp_le_right
% 0.17/0.45 thf(fact_141_add__le__imp__le__right,axiom,
% 0.17/0.45 ! [A: nat,C: nat,B: nat] :
% 0.17/0.45 ( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
% 0.17/0.45 => ( ord_less_eq_nat @ A @ B ) ) ).
% 0.17/0.45
% 0.17/0.45 % add_le_imp_le_right
% 0.17/0.45 thf(fact_142_add__le__imp__le__left,axiom,
% 0.17/0.45 ! [C: real,A: real,B: real] :
% 0.17/0.45 ( ( ord_less_eq_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
% 0.17/0.45 => ( ord_less_eq_real @ A @ B ) ) ).
% 0.17/0.45
% 0.17/0.45 % add_le_imp_le_left
% 0.17/0.45 thf(fact_143_add__le__imp__le__left,axiom,
% 0.17/0.45 ! [C: nat,A: nat,B: nat] :
% 0.17/0.45 ( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
% 0.17/0.45 => ( ord_less_eq_nat @ A @ B ) ) ).
% 0.17/0.45
% 0.17/0.45 % add_le_imp_le_left
% 0.17/0.45 thf(fact_144_le__iff__add,axiom,
% 0.17/0.45 ( ord_less_eq_nat
% 0.17/0.45 = ( ^ [A2: nat,B2: nat] :
% 0.17/0.45 ? [C2: nat] :
% 0.17/0.45 ( B2
% 0.17/0.45 = ( plus_plus_nat @ A2 @ C2 ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % le_iff_add
% 0.17/0.45 thf(fact_145_add__right__mono,axiom,
% 0.17/0.45 ! [A: real,B: real,C: real] :
% 0.17/0.45 ( ( ord_less_eq_real @ A @ B )
% 0.17/0.45 => ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % add_right_mono
% 0.17/0.45 thf(fact_146_add__right__mono,axiom,
% 0.17/0.45 ! [A: nat,B: nat,C: nat] :
% 0.17/0.45 ( ( ord_less_eq_nat @ A @ B )
% 0.17/0.45 => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % add_right_mono
% 0.17/0.45 thf(fact_147_less__eqE,axiom,
% 0.17/0.45 ! [A: nat,B: nat] :
% 0.17/0.45 ( ( ord_less_eq_nat @ A @ B )
% 0.17/0.45 => ~ ! [C3: nat] :
% 0.17/0.45 ( B
% 0.17/0.45 != ( plus_plus_nat @ A @ C3 ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % less_eqE
% 0.17/0.45 thf(fact_148_add__left__mono,axiom,
% 0.17/0.45 ! [A: real,B: real,C: real] :
% 0.17/0.45 ( ( ord_less_eq_real @ A @ B )
% 0.17/0.45 => ( ord_less_eq_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % add_left_mono
% 0.17/0.45 thf(fact_149_add__left__mono,axiom,
% 0.17/0.45 ! [A: nat,B: nat,C: nat] :
% 0.17/0.45 ( ( ord_less_eq_nat @ A @ B )
% 0.17/0.45 => ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % add_left_mono
% 0.17/0.45 thf(fact_150_add__mono,axiom,
% 0.17/0.45 ! [A: real,B: real,C: real,D: real] :
% 0.17/0.45 ( ( ord_less_eq_real @ A @ B )
% 0.17/0.45 => ( ( ord_less_eq_real @ C @ D )
% 0.17/0.45 => ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % add_mono
% 0.17/0.45 thf(fact_151_add__mono,axiom,
% 0.17/0.45 ! [A: nat,B: nat,C: nat,D: nat] :
% 0.17/0.45 ( ( ord_less_eq_nat @ A @ B )
% 0.17/0.45 => ( ( ord_less_eq_nat @ C @ D )
% 0.17/0.45 => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % add_mono
% 0.17/0.45 thf(fact_152_add__mono__thms__linordered__semiring_I1_J,axiom,
% 0.17/0.45 ! [I2: real,J: real,K: real,L: real] :
% 0.17/0.45 ( ( ( ord_less_eq_real @ I2 @ J )
% 0.17/0.45 & ( ord_less_eq_real @ K @ L ) )
% 0.17/0.45 => ( ord_less_eq_real @ ( plus_plus_real @ I2 @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % add_mono_thms_linordered_semiring(1)
% 0.17/0.45 thf(fact_153_add__mono__thms__linordered__semiring_I1_J,axiom,
% 0.17/0.45 ! [I2: nat,J: nat,K: nat,L: nat] :
% 0.17/0.45 ( ( ( ord_less_eq_nat @ I2 @ J )
% 0.17/0.45 & ( ord_less_eq_nat @ K @ L ) )
% 0.17/0.45 => ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % add_mono_thms_linordered_semiring(1)
% 0.17/0.45 thf(fact_154_add__mono__thms__linordered__semiring_I2_J,axiom,
% 0.17/0.45 ! [I2: real,J: real,K: real,L: real] :
% 0.17/0.45 ( ( ( I2 = J )
% 0.17/0.45 & ( ord_less_eq_real @ K @ L ) )
% 0.17/0.45 => ( ord_less_eq_real @ ( plus_plus_real @ I2 @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % add_mono_thms_linordered_semiring(2)
% 0.17/0.45 thf(fact_155_add__mono__thms__linordered__semiring_I2_J,axiom,
% 0.17/0.45 ! [I2: nat,J: nat,K: nat,L: nat] :
% 0.17/0.45 ( ( ( I2 = J )
% 0.17/0.45 & ( ord_less_eq_nat @ K @ L ) )
% 0.17/0.45 => ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % add_mono_thms_linordered_semiring(2)
% 0.17/0.45 thf(fact_156_add__mono__thms__linordered__semiring_I3_J,axiom,
% 0.17/0.45 ! [I2: real,J: real,K: real,L: real] :
% 0.17/0.45 ( ( ( ord_less_eq_real @ I2 @ J )
% 0.17/0.45 & ( K = L ) )
% 0.17/0.45 => ( ord_less_eq_real @ ( plus_plus_real @ I2 @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % add_mono_thms_linordered_semiring(3)
% 0.17/0.45 thf(fact_157_add__mono__thms__linordered__semiring_I3_J,axiom,
% 0.17/0.45 ! [I2: nat,J: nat,K: nat,L: nat] :
% 0.17/0.45 ( ( ( ord_less_eq_nat @ I2 @ J )
% 0.17/0.45 & ( K = L ) )
% 0.17/0.45 => ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % add_mono_thms_linordered_semiring(3)
% 0.17/0.45 thf(fact_158_diff__eq__diff__less__eq,axiom,
% 0.17/0.45 ! [A: real,B: real,C: real,D: real] :
% 0.17/0.45 ( ( ( minus_minus_real @ A @ B )
% 0.17/0.45 = ( minus_minus_real @ C @ D ) )
% 0.17/0.45 => ( ( ord_less_eq_real @ A @ B )
% 0.17/0.45 = ( ord_less_eq_real @ C @ D ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % diff_eq_diff_less_eq
% 0.17/0.45 thf(fact_159_diff__right__mono,axiom,
% 0.17/0.45 ! [A: real,B: real,C: real] :
% 0.17/0.45 ( ( ord_less_eq_real @ A @ B )
% 0.17/0.45 => ( ord_less_eq_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ C ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % diff_right_mono
% 0.17/0.45 thf(fact_160_diff__left__mono,axiom,
% 0.17/0.45 ! [B: real,A: real,C: real] :
% 0.17/0.45 ( ( ord_less_eq_real @ B @ A )
% 0.17/0.45 => ( ord_less_eq_real @ ( minus_minus_real @ C @ A ) @ ( minus_minus_real @ C @ B ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % diff_left_mono
% 0.17/0.45 thf(fact_161_diff__mono,axiom,
% 0.17/0.45 ! [A: real,B: real,D: real,C: real] :
% 0.17/0.45 ( ( ord_less_eq_real @ A @ B )
% 0.17/0.45 => ( ( ord_less_eq_real @ D @ C )
% 0.17/0.45 => ( ord_less_eq_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ D ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % diff_mono
% 0.17/0.45 thf(fact_162_add__implies__diff,axiom,
% 0.17/0.45 ! [C: real,B: real,A: real] :
% 0.17/0.45 ( ( ( plus_plus_real @ C @ B )
% 0.17/0.45 = A )
% 0.17/0.45 => ( C
% 0.17/0.45 = ( minus_minus_real @ A @ B ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % add_implies_diff
% 0.17/0.45 thf(fact_163_add__implies__diff,axiom,
% 0.17/0.45 ! [C: nat,B: nat,A: nat] :
% 0.17/0.45 ( ( ( plus_plus_nat @ C @ B )
% 0.17/0.45 = A )
% 0.17/0.45 => ( C
% 0.17/0.45 = ( minus_minus_nat @ A @ B ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % add_implies_diff
% 0.17/0.45 thf(fact_164_diff__diff__add,axiom,
% 0.17/0.45 ! [A: real,B: real,C: real] :
% 0.17/0.45 ( ( minus_minus_real @ ( minus_minus_real @ A @ B ) @ C )
% 0.17/0.45 = ( minus_minus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % diff_diff_add
% 0.17/0.45 thf(fact_165_diff__diff__add,axiom,
% 0.17/0.45 ! [A: nat,B: nat,C: nat] :
% 0.17/0.45 ( ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C )
% 0.17/0.45 = ( minus_minus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % diff_diff_add
% 0.17/0.45 thf(fact_166_diff__add__eq__diff__diff__swap,axiom,
% 0.17/0.45 ! [A: real,B: real,C: real] :
% 0.17/0.45 ( ( minus_minus_real @ A @ ( plus_plus_real @ B @ C ) )
% 0.17/0.45 = ( minus_minus_real @ ( minus_minus_real @ A @ C ) @ B ) ) ).
% 0.17/0.45
% 0.17/0.45 % diff_add_eq_diff_diff_swap
% 0.17/0.45 thf(fact_167_diff__add__eq,axiom,
% 0.17/0.45 ! [A: real,B: real,C: real] :
% 0.17/0.45 ( ( plus_plus_real @ ( minus_minus_real @ A @ B ) @ C )
% 0.17/0.45 = ( minus_minus_real @ ( plus_plus_real @ A @ C ) @ B ) ) ).
% 0.17/0.45
% 0.17/0.45 % diff_add_eq
% 0.17/0.45 thf(fact_168_diff__diff__eq2,axiom,
% 0.17/0.45 ! [A: real,B: real,C: real] :
% 0.17/0.45 ( ( minus_minus_real @ A @ ( minus_minus_real @ B @ C ) )
% 0.17/0.45 = ( minus_minus_real @ ( plus_plus_real @ A @ C ) @ B ) ) ).
% 0.17/0.45
% 0.17/0.45 % diff_diff_eq2
% 0.17/0.45 thf(fact_169_add__diff__eq,axiom,
% 0.17/0.45 ! [A: real,B: real,C: real] :
% 0.17/0.45 ( ( plus_plus_real @ A @ ( minus_minus_real @ B @ C ) )
% 0.17/0.45 = ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ C ) ) ).
% 0.17/0.45
% 0.17/0.45 % add_diff_eq
% 0.17/0.45 thf(fact_170_eq__diff__eq,axiom,
% 0.17/0.45 ! [A: real,C: real,B: real] :
% 0.17/0.45 ( ( A
% 0.17/0.45 = ( minus_minus_real @ C @ B ) )
% 0.17/0.45 = ( ( plus_plus_real @ A @ B )
% 0.17/0.45 = C ) ) ).
% 0.17/0.45
% 0.17/0.45 % eq_diff_eq
% 0.17/0.45 thf(fact_171_diff__eq__eq,axiom,
% 0.17/0.45 ! [A: real,B: real,C: real] :
% 0.17/0.45 ( ( ( minus_minus_real @ A @ B )
% 0.17/0.45 = C )
% 0.17/0.45 = ( A
% 0.17/0.45 = ( plus_plus_real @ C @ B ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % diff_eq_eq
% 0.17/0.45 thf(fact_172_group__cancel_Osub1,axiom,
% 0.17/0.45 ! [A3: real,K: real,A: real,B: real] :
% 0.17/0.45 ( ( A3
% 0.17/0.45 = ( plus_plus_real @ K @ A ) )
% 0.17/0.45 => ( ( minus_minus_real @ A3 @ B )
% 0.17/0.45 = ( plus_plus_real @ K @ ( minus_minus_real @ A @ B ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % group_cancel.sub1
% 0.17/0.45 thf(fact_173_abs__ge__self,axiom,
% 0.17/0.45 ! [A: real] : ( ord_less_eq_real @ A @ ( abs_abs_real @ A ) ) ).
% 0.17/0.45
% 0.17/0.45 % abs_ge_self
% 0.17/0.45 thf(fact_174_abs__le__D1,axiom,
% 0.17/0.45 ! [A: real,B: real] :
% 0.17/0.45 ( ( ord_less_eq_real @ ( abs_abs_real @ A ) @ B )
% 0.17/0.45 => ( ord_less_eq_real @ A @ B ) ) ).
% 0.17/0.45
% 0.17/0.45 % abs_le_D1
% 0.17/0.45 thf(fact_175_abs__one,axiom,
% 0.17/0.45 ( ( abs_abs_real @ one_one_real )
% 0.17/0.45 = one_one_real ) ).
% 0.17/0.45
% 0.17/0.45 % abs_one
% 0.17/0.45 thf(fact_176_abs__minus__commute,axiom,
% 0.17/0.45 ! [A: real,B: real] :
% 0.17/0.45 ( ( abs_abs_real @ ( minus_minus_real @ A @ B ) )
% 0.17/0.45 = ( abs_abs_real @ ( minus_minus_real @ B @ A ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % abs_minus_commute
% 0.17/0.45 thf(fact_177_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
% 0.17/0.45 ! [A: nat,B: nat,C: nat] :
% 0.17/0.45 ( ( ord_less_eq_nat @ A @ B )
% 0.17/0.45 => ( ( ord_less_eq_nat @ A @ B )
% 0.17/0.45 => ( ( ( minus_minus_nat @ B @ A )
% 0.17/0.45 = C )
% 0.17/0.45 = ( B
% 0.17/0.45 = ( plus_plus_nat @ C @ A ) ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add
% 0.17/0.45 thf(fact_178_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
% 0.17/0.45 ! [A: nat,B: nat] :
% 0.17/0.45 ( ( ord_less_eq_nat @ A @ B )
% 0.17/0.45 => ( ( plus_plus_nat @ A @ ( minus_minus_nat @ B @ A ) )
% 0.17/0.45 = B ) ) ).
% 0.17/0.45
% 0.17/0.45 % ordered_cancel_comm_monoid_diff_class.add_diff_inverse
% 0.17/0.45 thf(fact_179_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
% 0.17/0.45 ! [A: nat,B: nat,C: nat] :
% 0.17/0.45 ( ( ord_less_eq_nat @ A @ B )
% 0.17/0.45 => ( ( minus_minus_nat @ C @ ( minus_minus_nat @ B @ A ) )
% 0.17/0.45 = ( minus_minus_nat @ ( plus_plus_nat @ C @ A ) @ B ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % ordered_cancel_comm_monoid_diff_class.diff_diff_right
% 0.17/0.45 thf(fact_180_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
% 0.17/0.45 ! [A: nat,B: nat,C: nat] :
% 0.17/0.45 ( ( ord_less_eq_nat @ A @ B )
% 0.17/0.45 => ( ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A )
% 0.17/0.45 = ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ C ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
% 0.17/0.45 thf(fact_181_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
% 0.17/0.45 ! [A: nat,B: nat,C: nat] :
% 0.17/0.45 ( ( ord_less_eq_nat @ A @ B )
% 0.17/0.45 => ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ C )
% 0.17/0.45 = ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
% 0.17/0.45 thf(fact_182_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
% 0.17/0.45 ! [A: nat,B: nat,C: nat] :
% 0.17/0.45 ( ( ord_less_eq_nat @ A @ B )
% 0.17/0.45 => ( ( minus_minus_nat @ ( plus_plus_nat @ C @ B ) @ A )
% 0.17/0.45 = ( plus_plus_nat @ C @ ( minus_minus_nat @ B @ A ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % ordered_cancel_comm_monoid_diff_class.diff_add_assoc
% 0.17/0.45 thf(fact_183_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
% 0.17/0.45 ! [A: nat,B: nat,C: nat] :
% 0.17/0.45 ( ( ord_less_eq_nat @ A @ B )
% 0.17/0.45 => ( ( plus_plus_nat @ C @ ( minus_minus_nat @ B @ A ) )
% 0.17/0.45 = ( minus_minus_nat @ ( plus_plus_nat @ C @ B ) @ A ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % ordered_cancel_comm_monoid_diff_class.add_diff_assoc
% 0.17/0.45 thf(fact_184_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
% 0.17/0.45 ! [A: nat,B: nat,C: nat] :
% 0.17/0.45 ( ( ord_less_eq_nat @ A @ B )
% 0.17/0.45 => ( ( ord_less_eq_nat @ C @ ( minus_minus_nat @ B @ A ) )
% 0.17/0.45 = ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ B ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % ordered_cancel_comm_monoid_diff_class.le_diff_conv2
% 0.17/0.45 thf(fact_185_le__add__diff,axiom,
% 0.17/0.45 ! [A: nat,B: nat,C: nat] :
% 0.17/0.45 ( ( ord_less_eq_nat @ A @ B )
% 0.17/0.45 => ( ord_less_eq_nat @ C @ ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % le_add_diff
% 0.17/0.45 thf(fact_186_add__le__add__imp__diff__le,axiom,
% 0.17/0.45 ! [I2: real,K: real,N: real,J: real] :
% 0.17/0.45 ( ( ord_less_eq_real @ ( plus_plus_real @ I2 @ K ) @ N )
% 0.17/0.45 => ( ( ord_less_eq_real @ N @ ( plus_plus_real @ J @ K ) )
% 0.17/0.45 => ( ( ord_less_eq_real @ ( plus_plus_real @ I2 @ K ) @ N )
% 0.17/0.45 => ( ( ord_less_eq_real @ N @ ( plus_plus_real @ J @ K ) )
% 0.17/0.45 => ( ord_less_eq_real @ ( minus_minus_real @ N @ K ) @ J ) ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % add_le_add_imp_diff_le
% 0.17/0.45 thf(fact_187_add__le__add__imp__diff__le,axiom,
% 0.17/0.45 ! [I2: nat,K: nat,N: nat,J: nat] :
% 0.17/0.45 ( ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ N )
% 0.17/0.45 => ( ( ord_less_eq_nat @ N @ ( plus_plus_nat @ J @ K ) )
% 0.17/0.45 => ( ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ N )
% 0.17/0.45 => ( ( ord_less_eq_nat @ N @ ( plus_plus_nat @ J @ K ) )
% 0.17/0.45 => ( ord_less_eq_nat @ ( minus_minus_nat @ N @ K ) @ J ) ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % add_le_add_imp_diff_le
% 0.17/0.45 thf(fact_188_diff__add,axiom,
% 0.17/0.45 ! [A: nat,B: nat] :
% 0.17/0.45 ( ( ord_less_eq_nat @ A @ B )
% 0.17/0.45 => ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ A )
% 0.17/0.45 = B ) ) ).
% 0.17/0.45
% 0.17/0.45 % diff_add
% 0.17/0.45 thf(fact_189_add__le__imp__le__diff,axiom,
% 0.17/0.45 ! [I2: real,K: real,N: real] :
% 0.17/0.45 ( ( ord_less_eq_real @ ( plus_plus_real @ I2 @ K ) @ N )
% 0.17/0.45 => ( ord_less_eq_real @ I2 @ ( minus_minus_real @ N @ K ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % add_le_imp_le_diff
% 0.17/0.45 thf(fact_190_add__le__imp__le__diff,axiom,
% 0.17/0.45 ! [I2: nat,K: nat,N: nat] :
% 0.17/0.45 ( ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ N )
% 0.17/0.45 => ( ord_less_eq_nat @ I2 @ ( minus_minus_nat @ N @ K ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % add_le_imp_le_diff
% 0.17/0.45 thf(fact_191_le__diff__eq,axiom,
% 0.17/0.45 ! [A: real,C: real,B: real] :
% 0.17/0.45 ( ( ord_less_eq_real @ A @ ( minus_minus_real @ C @ B ) )
% 0.17/0.45 = ( ord_less_eq_real @ ( plus_plus_real @ A @ B ) @ C ) ) ).
% 0.17/0.45
% 0.17/0.45 % le_diff_eq
% 0.17/0.45 thf(fact_192_diff__le__eq,axiom,
% 0.17/0.45 ! [A: real,B: real,C: real] :
% 0.17/0.45 ( ( ord_less_eq_real @ ( minus_minus_real @ A @ B ) @ C )
% 0.17/0.45 = ( ord_less_eq_real @ A @ ( plus_plus_real @ C @ B ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % diff_le_eq
% 0.17/0.45 thf(fact_193_abs__triangle__ineq,axiom,
% 0.17/0.45 ! [A: real,B: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( plus_plus_real @ A @ B ) ) @ ( plus_plus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % abs_triangle_ineq
% 0.17/0.45 thf(fact_194_abs__triangle__ineq2__sym,axiom,
% 0.17/0.45 ! [A: real,B: real] : ( ord_less_eq_real @ ( minus_minus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) @ ( abs_abs_real @ ( minus_minus_real @ B @ A ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % abs_triangle_ineq2_sym
% 0.17/0.45 thf(fact_195_abs__triangle__ineq3,axiom,
% 0.17/0.45 ! [A: real,B: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) ) @ ( abs_abs_real @ ( minus_minus_real @ A @ B ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % abs_triangle_ineq3
% 0.17/0.45 thf(fact_196_abs__triangle__ineq2,axiom,
% 0.17/0.45 ! [A: real,B: real] : ( ord_less_eq_real @ ( minus_minus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) @ ( abs_abs_real @ ( minus_minus_real @ A @ B ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % abs_triangle_ineq2
% 0.17/0.45 thf(fact_197_abs__diff__triangle__ineq,axiom,
% 0.17/0.45 ! [A: real,B: real,C: real,D: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ ( plus_plus_real @ C @ D ) ) ) @ ( plus_plus_real @ ( abs_abs_real @ ( minus_minus_real @ A @ C ) ) @ ( abs_abs_real @ ( minus_minus_real @ B @ D ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % abs_diff_triangle_ineq
% 0.17/0.45 thf(fact_198_abs__triangle__ineq4,axiom,
% 0.17/0.45 ! [A: real,B: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ A @ B ) ) @ ( plus_plus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % abs_triangle_ineq4
% 0.17/0.45 thf(fact_199_abs__diff__le__iff,axiom,
% 0.17/0.45 ! [X: real,A: real,R: real] :
% 0.17/0.45 ( ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ X @ A ) ) @ R )
% 0.17/0.45 = ( ( ord_less_eq_real @ ( minus_minus_real @ A @ R ) @ X )
% 0.17/0.45 & ( ord_less_eq_real @ X @ ( plus_plus_real @ A @ R ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % abs_diff_le_iff
% 0.17/0.45 thf(fact_200_posX,axiom,
% 0.17/0.45 ord_less_eq_real @ zero_zero_real @ ( minus_minus_real @ ( genClo1161277105lle_PC @ l @ ( genClo1163638703lle_te @ p @ ( plus_plus_nat @ i @ one_one_nat ) ) ) @ ( genClo1161277105lle_PC @ l @ ( genClo1163638703lle_te @ q @ ( plus_plus_nat @ i @ one_one_nat ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % posX
% 0.17/0.45 thf(fact_201_diff__numeral__special_I9_J,axiom,
% 0.17/0.45 ( ( minus_minus_real @ one_one_real @ one_one_real )
% 0.17/0.45 = zero_zero_real ) ).
% 0.17/0.45
% 0.17/0.45 % diff_numeral_special(9)
% 0.17/0.45 thf(fact_202_beta__bound2,axiom,
% 0.17/0.45 ! [P2: nat,I2: nat,Q2: nat] :
% 0.17/0.45 ( ( genClo1015804716orrect @ P2 @ ( genClo1163638703lle_te @ P2 @ ( plus_plus_nat @ I2 @ one_one_nat ) ) )
% 0.17/0.45 => ( ( genClo1015804716orrect @ Q2 @ ( genClo1163638703lle_te @ Q2 @ I2 ) )
% 0.17/0.45 => ( ord_less_eq_real @ ( minus_minus_real @ ( genClo1163638703lle_te @ P2 @ ( plus_plus_nat @ I2 @ one_one_nat ) ) @ ( genClo1163638703lle_te @ Q2 @ I2 ) ) @ ( plus_plus_real @ genClo1650508560e_rmax @ genClo1278781456e_beta ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % beta_bound2
% 0.17/0.45 thf(fact_203_rts1d,axiom,
% 0.17/0.45 ! [P2: nat,I2: nat] :
% 0.17/0.45 ( ( genClo1015804716orrect @ P2 @ ( genClo1163638703lle_te @ P2 @ ( plus_plus_nat @ I2 @ one_one_nat ) ) )
% 0.17/0.45 => ( ord_less_eq_real @ genClo1651033342e_rmin @ ( minus_minus_real @ ( genClo1163638703lle_te @ P2 @ ( plus_plus_nat @ I2 @ one_one_nat ) ) @ ( genClo1163638703lle_te @ P2 @ I2 ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % rts1d
% 0.17/0.45 thf(fact_204_rts1,axiom,
% 0.17/0.45 ! [P: nat,T: real,I: nat] :
% 0.17/0.45 ( ( ( genClo1015804716orrect @ P @ T )
% 0.17/0.45 & ( ord_less_eq_real @ ( genClo1163638703lle_te @ P @ ( plus_plus_nat @ I @ one_one_nat ) ) @ T ) )
% 0.17/0.45 => ( ord_less_eq_real @ genClo1651033342e_rmin @ ( minus_minus_real @ T @ ( genClo1163638703lle_te @ P @ I ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % rts1
% 0.17/0.45 thf(fact_205_rts0d,axiom,
% 0.17/0.45 ! [P2: nat,I2: nat] :
% 0.17/0.45 ( ( genClo1015804716orrect @ P2 @ ( genClo1163638703lle_te @ P2 @ ( plus_plus_nat @ I2 @ one_one_nat ) ) )
% 0.17/0.45 => ( ord_less_eq_real @ ( minus_minus_real @ ( genClo1163638703lle_te @ P2 @ ( plus_plus_nat @ I2 @ one_one_nat ) ) @ ( genClo1163638703lle_te @ P2 @ I2 ) ) @ genClo1650508560e_rmax ) ) ).
% 0.17/0.45
% 0.17/0.45 % rts0d
% 0.17/0.45 thf(fact_206_rts0,axiom,
% 0.17/0.45 ! [P: nat,T: real,I: nat] :
% 0.17/0.45 ( ( ( genClo1015804716orrect @ P @ T )
% 0.17/0.45 & ( ord_less_eq_real @ T @ ( genClo1163638703lle_te @ P @ ( plus_plus_nat @ I @ one_one_nat ) ) ) )
% 0.17/0.45 => ( ord_less_eq_real @ ( minus_minus_real @ T @ ( genClo1163638703lle_te @ P @ I ) ) @ genClo1650508560e_rmax ) ) ).
% 0.17/0.45
% 0.17/0.45 % rts0
% 0.17/0.45 thf(fact_207_okRead2__def,axiom,
% 0.17/0.45 ( genClo293725282kRead2
% 0.17/0.45 = ( ^ [F: nat > real,G: nat > real,X2: real,Ppred: nat > $o] :
% 0.17/0.45 ! [P3: nat] :
% 0.17/0.45 ( ( Ppred @ P3 )
% 0.17/0.45 => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( F @ P3 ) @ ( G @ P3 ) ) ) @ X2 ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % okRead2_def
% 0.17/0.45 thf(fact_208_sin__bound__lemma,axiom,
% 0.17/0.45 ! [X: real,Y: real,U: real,V: real] :
% 0.17/0.45 ( ( X = Y )
% 0.17/0.45 => ( ( ord_less_eq_real @ ( abs_abs_real @ U ) @ V )
% 0.17/0.45 => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( plus_plus_real @ X @ U ) @ Y ) ) @ V ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % sin_bound_lemma
% 0.17/0.45 thf(fact_209_Nat_Oadd__diff__assoc,axiom,
% 0.17/0.45 ! [K: nat,J: nat,I2: nat] :
% 0.17/0.45 ( ( ord_less_eq_nat @ K @ J )
% 0.17/0.45 => ( ( plus_plus_nat @ I2 @ ( minus_minus_nat @ J @ K ) )
% 0.17/0.45 = ( minus_minus_nat @ ( plus_plus_nat @ I2 @ J ) @ K ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % Nat.add_diff_assoc
% 0.17/0.45 thf(fact_210_le0,axiom,
% 0.17/0.45 ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% 0.17/0.45
% 0.17/0.45 % le0
% 0.17/0.45 thf(fact_211_bot__nat__0_Oextremum,axiom,
% 0.17/0.45 ! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% 0.17/0.45
% 0.17/0.45 % bot_nat_0.extremum
% 0.17/0.45 thf(fact_212_add__is__0,axiom,
% 0.17/0.45 ! [M: nat,N: nat] :
% 0.17/0.45 ( ( ( plus_plus_nat @ M @ N )
% 0.17/0.45 = zero_zero_nat )
% 0.17/0.45 = ( ( M = zero_zero_nat )
% 0.17/0.45 & ( N = zero_zero_nat ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % add_is_0
% 0.17/0.45 thf(fact_213_Nat_Oadd__0__right,axiom,
% 0.17/0.45 ! [M: nat] :
% 0.17/0.45 ( ( plus_plus_nat @ M @ zero_zero_nat )
% 0.17/0.45 = M ) ).
% 0.17/0.45
% 0.17/0.45 % Nat.add_0_right
% 0.17/0.45 thf(fact_214_diff__0__eq__0,axiom,
% 0.17/0.45 ! [N: nat] :
% 0.17/0.45 ( ( minus_minus_nat @ zero_zero_nat @ N )
% 0.17/0.45 = zero_zero_nat ) ).
% 0.17/0.45
% 0.17/0.45 % diff_0_eq_0
% 0.17/0.45 thf(fact_215_diff__self__eq__0,axiom,
% 0.17/0.45 ! [M: nat] :
% 0.17/0.45 ( ( minus_minus_nat @ M @ M )
% 0.17/0.45 = zero_zero_nat ) ).
% 0.17/0.45
% 0.17/0.45 % diff_self_eq_0
% 0.17/0.45 thf(fact_216_nat__add__left__cancel__le,axiom,
% 0.17/0.45 ! [K: nat,M: nat,N: nat] :
% 0.17/0.45 ( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
% 0.17/0.45 = ( ord_less_eq_nat @ M @ N ) ) ).
% 0.17/0.45
% 0.17/0.45 % nat_add_left_cancel_le
% 0.17/0.45 thf(fact_217_diff__is__0__eq,axiom,
% 0.17/0.45 ! [M: nat,N: nat] :
% 0.17/0.45 ( ( ( minus_minus_nat @ M @ N )
% 0.17/0.45 = zero_zero_nat )
% 0.17/0.45 = ( ord_less_eq_nat @ M @ N ) ) ).
% 0.17/0.45
% 0.17/0.45 % diff_is_0_eq
% 0.17/0.45 thf(fact_218_diff__is__0__eq_H,axiom,
% 0.17/0.45 ! [M: nat,N: nat] :
% 0.17/0.45 ( ( ord_less_eq_nat @ M @ N )
% 0.17/0.45 => ( ( minus_minus_nat @ M @ N )
% 0.17/0.45 = zero_zero_nat ) ) ).
% 0.17/0.45
% 0.17/0.45 % diff_is_0_eq'
% 0.17/0.45 thf(fact_219_diff__diff__cancel,axiom,
% 0.17/0.45 ! [I2: nat,N: nat] :
% 0.17/0.45 ( ( ord_less_eq_nat @ I2 @ N )
% 0.17/0.45 => ( ( minus_minus_nat @ N @ ( minus_minus_nat @ N @ I2 ) )
% 0.17/0.45 = I2 ) ) ).
% 0.17/0.45
% 0.17/0.45 % diff_diff_cancel
% 0.17/0.45 thf(fact_220_diff__diff__left,axiom,
% 0.17/0.45 ! [I2: nat,J: nat,K: nat] :
% 0.17/0.45 ( ( minus_minus_nat @ ( minus_minus_nat @ I2 @ J ) @ K )
% 0.17/0.45 = ( minus_minus_nat @ I2 @ ( plus_plus_nat @ J @ K ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % diff_diff_left
% 0.17/0.45 thf(fact_221_Nat_Odiff__diff__right,axiom,
% 0.17/0.45 ! [K: nat,J: nat,I2: nat] :
% 0.17/0.45 ( ( ord_less_eq_nat @ K @ J )
% 0.17/0.45 => ( ( minus_minus_nat @ I2 @ ( minus_minus_nat @ J @ K ) )
% 0.17/0.45 = ( minus_minus_nat @ ( plus_plus_nat @ I2 @ K ) @ J ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % Nat.diff_diff_right
% 0.17/0.45 thf(fact_222_Nat_Oadd__diff__assoc2,axiom,
% 0.17/0.45 ! [K: nat,J: nat,I2: nat] :
% 0.17/0.45 ( ( ord_less_eq_nat @ K @ J )
% 0.17/0.45 => ( ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I2 )
% 0.17/0.45 = ( minus_minus_nat @ ( plus_plus_nat @ J @ I2 ) @ K ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % Nat.add_diff_assoc2
% 0.17/0.45 thf(fact_223_less__eq__nat_Osimps_I1_J,axiom,
% 0.17/0.45 ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% 0.17/0.45
% 0.17/0.45 % less_eq_nat.simps(1)
% 0.17/0.45 thf(fact_224_le__0__eq,axiom,
% 0.17/0.45 ! [N: nat] :
% 0.17/0.45 ( ( ord_less_eq_nat @ N @ zero_zero_nat )
% 0.17/0.45 = ( N = zero_zero_nat ) ) ).
% 0.17/0.45
% 0.17/0.45 % le_0_eq
% 0.17/0.45 thf(fact_225_le__refl,axiom,
% 0.17/0.45 ! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% 0.17/0.45
% 0.17/0.45 % le_refl
% 0.17/0.45 thf(fact_226_le__trans,axiom,
% 0.17/0.45 ! [I2: nat,J: nat,K: nat] :
% 0.17/0.45 ( ( ord_less_eq_nat @ I2 @ J )
% 0.17/0.45 => ( ( ord_less_eq_nat @ J @ K )
% 0.17/0.45 => ( ord_less_eq_nat @ I2 @ K ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % le_trans
% 0.17/0.45 thf(fact_227_eq__imp__le,axiom,
% 0.17/0.45 ! [M: nat,N: nat] :
% 0.17/0.45 ( ( M = N )
% 0.17/0.45 => ( ord_less_eq_nat @ M @ N ) ) ).
% 0.17/0.45
% 0.17/0.45 % eq_imp_le
% 0.17/0.45 thf(fact_228_le__antisym,axiom,
% 0.17/0.45 ! [M: nat,N: nat] :
% 0.17/0.45 ( ( ord_less_eq_nat @ M @ N )
% 0.17/0.45 => ( ( ord_less_eq_nat @ N @ M )
% 0.17/0.45 => ( M = N ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % le_antisym
% 0.17/0.45 thf(fact_229_eq__diff__iff,axiom,
% 0.17/0.45 ! [K: nat,M: nat,N: nat] :
% 0.17/0.45 ( ( ord_less_eq_nat @ K @ M )
% 0.17/0.45 => ( ( ord_less_eq_nat @ K @ N )
% 0.17/0.45 => ( ( ( minus_minus_nat @ M @ K )
% 0.17/0.45 = ( minus_minus_nat @ N @ K ) )
% 0.17/0.45 = ( M = N ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % eq_diff_iff
% 0.17/0.45 thf(fact_230_le__diff__iff,axiom,
% 0.17/0.45 ! [K: nat,M: nat,N: nat] :
% 0.17/0.45 ( ( ord_less_eq_nat @ K @ M )
% 0.17/0.45 => ( ( ord_less_eq_nat @ K @ N )
% 0.17/0.45 => ( ( ord_less_eq_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
% 0.17/0.45 = ( ord_less_eq_nat @ M @ N ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % le_diff_iff
% 0.17/0.45 thf(fact_231_diff__commute,axiom,
% 0.17/0.45 ! [I2: nat,J: nat,K: nat] :
% 0.17/0.45 ( ( minus_minus_nat @ ( minus_minus_nat @ I2 @ J ) @ K )
% 0.17/0.45 = ( minus_minus_nat @ ( minus_minus_nat @ I2 @ K ) @ J ) ) ).
% 0.17/0.45
% 0.17/0.45 % diff_commute
% 0.17/0.45 thf(fact_232_Nat_Odiff__diff__eq,axiom,
% 0.17/0.45 ! [K: nat,M: nat,N: nat] :
% 0.17/0.45 ( ( ord_less_eq_nat @ K @ M )
% 0.17/0.45 => ( ( ord_less_eq_nat @ K @ N )
% 0.17/0.45 => ( ( minus_minus_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
% 0.17/0.45 = ( minus_minus_nat @ M @ N ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % Nat.diff_diff_eq
% 0.17/0.45 thf(fact_233_diff__le__mono,axiom,
% 0.17/0.45 ! [M: nat,N: nat,L: nat] :
% 0.17/0.45 ( ( ord_less_eq_nat @ M @ N )
% 0.17/0.45 => ( ord_less_eq_nat @ ( minus_minus_nat @ M @ L ) @ ( minus_minus_nat @ N @ L ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % diff_le_mono
% 0.17/0.45 thf(fact_234_diff__le__self,axiom,
% 0.17/0.45 ! [M: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ M ) ).
% 0.17/0.45
% 0.17/0.45 % diff_le_self
% 0.17/0.45 thf(fact_235_le__diff__iff_H,axiom,
% 0.17/0.45 ! [A: nat,C: nat,B: nat] :
% 0.17/0.45 ( ( ord_less_eq_nat @ A @ C )
% 0.17/0.45 => ( ( ord_less_eq_nat @ B @ C )
% 0.17/0.45 => ( ( ord_less_eq_nat @ ( minus_minus_nat @ C @ A ) @ ( minus_minus_nat @ C @ B ) )
% 0.17/0.45 = ( ord_less_eq_nat @ B @ A ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % le_diff_iff'
% 0.17/0.45 thf(fact_236_diff__le__mono2,axiom,
% 0.17/0.45 ! [M: nat,N: nat,L: nat] :
% 0.17/0.45 ( ( ord_less_eq_nat @ M @ N )
% 0.17/0.45 => ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % diff_le_mono2
% 0.17/0.45 thf(fact_237_nat__le__linear,axiom,
% 0.17/0.45 ! [M: nat,N: nat] :
% 0.17/0.45 ( ( ord_less_eq_nat @ M @ N )
% 0.17/0.45 | ( ord_less_eq_nat @ N @ M ) ) ).
% 0.17/0.45
% 0.17/0.45 % nat_le_linear
% 0.17/0.45 thf(fact_238_diffs0__imp__equal,axiom,
% 0.17/0.45 ! [M: nat,N: nat] :
% 0.17/0.45 ( ( ( minus_minus_nat @ M @ N )
% 0.17/0.45 = zero_zero_nat )
% 0.17/0.45 => ( ( ( minus_minus_nat @ N @ M )
% 0.17/0.45 = zero_zero_nat )
% 0.17/0.45 => ( M = N ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % diffs0_imp_equal
% 0.17/0.45 thf(fact_239_minus__nat_Odiff__0,axiom,
% 0.17/0.45 ! [M: nat] :
% 0.17/0.45 ( ( minus_minus_nat @ M @ zero_zero_nat )
% 0.17/0.45 = M ) ).
% 0.17/0.45
% 0.17/0.45 % minus_nat.diff_0
% 0.17/0.45 thf(fact_240_Nat_Oex__has__greatest__nat,axiom,
% 0.17/0.45 ! [P4: nat > $o,K: nat,B: nat] :
% 0.17/0.45 ( ( P4 @ K )
% 0.17/0.45 => ( ! [Y3: nat] :
% 0.17/0.45 ( ( P4 @ Y3 )
% 0.17/0.45 => ( ord_less_eq_nat @ Y3 @ B ) )
% 0.17/0.45 => ? [X3: nat] :
% 0.17/0.45 ( ( P4 @ X3 )
% 0.17/0.45 & ! [Y4: nat] :
% 0.17/0.45 ( ( P4 @ Y4 )
% 0.17/0.45 => ( ord_less_eq_nat @ Y4 @ X3 ) ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % Nat.ex_has_greatest_nat
% 0.17/0.45 thf(fact_241_bot__nat__0_Oextremum__unique,axiom,
% 0.17/0.45 ! [A: nat] :
% 0.17/0.45 ( ( ord_less_eq_nat @ A @ zero_zero_nat )
% 0.17/0.45 = ( A = zero_zero_nat ) ) ).
% 0.17/0.45
% 0.17/0.45 % bot_nat_0.extremum_unique
% 0.17/0.45 thf(fact_242_bot__nat__0_Oextremum__uniqueI,axiom,
% 0.17/0.45 ! [A: nat] :
% 0.17/0.45 ( ( ord_less_eq_nat @ A @ zero_zero_nat )
% 0.17/0.45 => ( A = zero_zero_nat ) ) ).
% 0.17/0.45
% 0.17/0.45 % bot_nat_0.extremum_uniqueI
% 0.17/0.45 thf(fact_243_plus__nat_Oadd__0,axiom,
% 0.17/0.45 ! [N: nat] :
% 0.17/0.45 ( ( plus_plus_nat @ zero_zero_nat @ N )
% 0.17/0.45 = N ) ).
% 0.17/0.45
% 0.17/0.45 % plus_nat.add_0
% 0.17/0.45 thf(fact_244_add__eq__self__zero,axiom,
% 0.17/0.45 ! [M: nat,N: nat] :
% 0.17/0.45 ( ( ( plus_plus_nat @ M @ N )
% 0.17/0.45 = M )
% 0.17/0.45 => ( N = zero_zero_nat ) ) ).
% 0.17/0.45
% 0.17/0.45 % add_eq_self_zero
% 0.17/0.45 thf(fact_245_diff__add__0,axiom,
% 0.17/0.45 ! [N: nat,M: nat] :
% 0.17/0.45 ( ( minus_minus_nat @ N @ ( plus_plus_nat @ N @ M ) )
% 0.17/0.45 = zero_zero_nat ) ).
% 0.17/0.45
% 0.17/0.45 % diff_add_0
% 0.17/0.45 thf(fact_246_synch0,axiom,
% 0.17/0.45 ! [P: nat] :
% 0.17/0.45 ( ( genClo1163638703lle_te @ P @ zero_zero_nat )
% 0.17/0.45 = zero_zero_real ) ).
% 0.17/0.45
% 0.17/0.45 % synch0
% 0.17/0.45 thf(fact_247_is__num__normalize_I1_J,axiom,
% 0.17/0.45 ! [A: real,B: real,C: real] :
% 0.17/0.45 ( ( plus_plus_real @ ( plus_plus_real @ A @ B ) @ C )
% 0.17/0.45 = ( plus_plus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % is_num_normalize(1)
% 0.17/0.45 thf(fact_248_add__leE,axiom,
% 0.17/0.45 ! [M: nat,K: nat,N: nat] :
% 0.17/0.45 ( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
% 0.17/0.45 => ~ ( ( ord_less_eq_nat @ M @ N )
% 0.17/0.45 => ~ ( ord_less_eq_nat @ K @ N ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % add_leE
% 0.17/0.45 thf(fact_249_le__add1,axiom,
% 0.17/0.45 ! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M ) ) ).
% 0.17/0.45
% 0.17/0.45 % le_add1
% 0.17/0.45 thf(fact_250_le__add2,axiom,
% 0.17/0.45 ! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M @ N ) ) ).
% 0.17/0.45
% 0.17/0.45 % le_add2
% 0.17/0.45 thf(fact_251_add__leD1,axiom,
% 0.17/0.45 ! [M: nat,K: nat,N: nat] :
% 0.17/0.45 ( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
% 0.17/0.45 => ( ord_less_eq_nat @ M @ N ) ) ).
% 0.17/0.45
% 0.17/0.45 % add_leD1
% 0.17/0.45 thf(fact_252_add__leD2,axiom,
% 0.17/0.45 ! [M: nat,K: nat,N: nat] :
% 0.17/0.45 ( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
% 0.17/0.45 => ( ord_less_eq_nat @ K @ N ) ) ).
% 0.17/0.45
% 0.17/0.45 % add_leD2
% 0.17/0.45 thf(fact_253_le__Suc__ex,axiom,
% 0.17/0.45 ! [K: nat,L: nat] :
% 0.17/0.45 ( ( ord_less_eq_nat @ K @ L )
% 0.17/0.45 => ? [N2: nat] :
% 0.17/0.45 ( L
% 0.17/0.45 = ( plus_plus_nat @ K @ N2 ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % le_Suc_ex
% 0.17/0.45 thf(fact_254_add__le__mono,axiom,
% 0.17/0.45 ! [I2: nat,J: nat,K: nat,L: nat] :
% 0.17/0.45 ( ( ord_less_eq_nat @ I2 @ J )
% 0.17/0.45 => ( ( ord_less_eq_nat @ K @ L )
% 0.17/0.45 => ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % add_le_mono
% 0.17/0.45 thf(fact_255_add__le__mono1,axiom,
% 0.17/0.45 ! [I2: nat,J: nat,K: nat] :
% 0.17/0.45 ( ( ord_less_eq_nat @ I2 @ J )
% 0.17/0.45 => ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % add_le_mono1
% 0.17/0.45 thf(fact_256_trans__le__add1,axiom,
% 0.17/0.45 ! [I2: nat,J: nat,M: nat] :
% 0.17/0.45 ( ( ord_less_eq_nat @ I2 @ J )
% 0.17/0.45 => ( ord_less_eq_nat @ I2 @ ( plus_plus_nat @ J @ M ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % trans_le_add1
% 0.17/0.45 thf(fact_257_trans__le__add2,axiom,
% 0.17/0.45 ! [I2: nat,J: nat,M: nat] :
% 0.17/0.45 ( ( ord_less_eq_nat @ I2 @ J )
% 0.17/0.45 => ( ord_less_eq_nat @ I2 @ ( plus_plus_nat @ M @ J ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % trans_le_add2
% 0.17/0.45 thf(fact_258_nat__le__iff__add,axiom,
% 0.17/0.45 ( ord_less_eq_nat
% 0.17/0.45 = ( ^ [M2: nat,N3: nat] :
% 0.17/0.45 ? [K2: nat] :
% 0.17/0.45 ( N3
% 0.17/0.45 = ( plus_plus_nat @ M2 @ K2 ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % nat_le_iff_add
% 0.17/0.45 thf(fact_259_Nat_Odiff__cancel,axiom,
% 0.17/0.45 ! [K: nat,M: nat,N: nat] :
% 0.17/0.45 ( ( minus_minus_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
% 0.17/0.45 = ( minus_minus_nat @ M @ N ) ) ).
% 0.17/0.45
% 0.17/0.45 % Nat.diff_cancel
% 0.17/0.45 thf(fact_260_diff__cancel2,axiom,
% 0.17/0.45 ! [M: nat,K: nat,N: nat] :
% 0.17/0.45 ( ( minus_minus_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N @ K ) )
% 0.17/0.45 = ( minus_minus_nat @ M @ N ) ) ).
% 0.17/0.45
% 0.17/0.45 % diff_cancel2
% 0.17/0.45 thf(fact_261_diff__add__inverse,axiom,
% 0.17/0.45 ! [N: nat,M: nat] :
% 0.17/0.45 ( ( minus_minus_nat @ ( plus_plus_nat @ N @ M ) @ N )
% 0.17/0.45 = M ) ).
% 0.17/0.45
% 0.17/0.45 % diff_add_inverse
% 0.17/0.45 thf(fact_262_diff__add__inverse2,axiom,
% 0.17/0.45 ! [M: nat,N: nat] :
% 0.17/0.45 ( ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ N )
% 0.17/0.45 = M ) ).
% 0.17/0.45
% 0.17/0.45 % diff_add_inverse2
% 0.17/0.45 thf(fact_263_Nat_Ole__imp__diff__is__add,axiom,
% 0.17/0.45 ! [I2: nat,J: nat,K: nat] :
% 0.17/0.45 ( ( ord_less_eq_nat @ I2 @ J )
% 0.17/0.45 => ( ( ( minus_minus_nat @ J @ I2 )
% 0.17/0.45 = K )
% 0.17/0.45 = ( J
% 0.17/0.45 = ( plus_plus_nat @ K @ I2 ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % Nat.le_imp_diff_is_add
% 0.17/0.45 thf(fact_264_Nat_Odiff__add__assoc2,axiom,
% 0.17/0.45 ! [K: nat,J: nat,I2: nat] :
% 0.17/0.45 ( ( ord_less_eq_nat @ K @ J )
% 0.17/0.45 => ( ( minus_minus_nat @ ( plus_plus_nat @ J @ I2 ) @ K )
% 0.17/0.45 = ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I2 ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % Nat.diff_add_assoc2
% 0.17/0.45 thf(fact_265_Nat_Odiff__add__assoc,axiom,
% 0.17/0.45 ! [K: nat,J: nat,I2: nat] :
% 0.17/0.45 ( ( ord_less_eq_nat @ K @ J )
% 0.17/0.45 => ( ( minus_minus_nat @ ( plus_plus_nat @ I2 @ J ) @ K )
% 0.17/0.45 = ( plus_plus_nat @ I2 @ ( minus_minus_nat @ J @ K ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % Nat.diff_add_assoc
% 0.17/0.45 thf(fact_266_Nat_Ole__diff__conv2,axiom,
% 0.17/0.45 ! [K: nat,J: nat,I2: nat] :
% 0.17/0.45 ( ( ord_less_eq_nat @ K @ J )
% 0.17/0.45 => ( ( ord_less_eq_nat @ I2 @ ( minus_minus_nat @ J @ K ) )
% 0.17/0.45 = ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ J ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % Nat.le_diff_conv2
% 0.17/0.45 thf(fact_267_le__diff__conv,axiom,
% 0.17/0.45 ! [J: nat,K: nat,I2: nat] :
% 0.17/0.45 ( ( ord_less_eq_nat @ ( minus_minus_nat @ J @ K ) @ I2 )
% 0.17/0.45 = ( ord_less_eq_nat @ J @ ( plus_plus_nat @ I2 @ K ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % le_diff_conv
% 0.17/0.45 thf(fact_268_PC__monotone,axiom,
% 0.17/0.45 ! [P: nat,S: real,T: real] :
% 0.17/0.45 ( ( ( genClo1015804716orrect @ P @ T )
% 0.17/0.45 & ( ord_less_eq_real @ S @ T ) )
% 0.17/0.45 => ( ord_less_eq_real @ ( genClo1161277105lle_PC @ P @ S ) @ ( genClo1161277105lle_PC @ P @ T ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % PC_monotone
% 0.17/0.45 thf(fact_269_eq__diff__eq_H,axiom,
% 0.17/0.45 ! [X: real,Y: real,Z2: real] :
% 0.17/0.45 ( ( X
% 0.17/0.45 = ( minus_minus_real @ Y @ Z2 ) )
% 0.17/0.45 = ( Y
% 0.17/0.45 = ( plus_plus_real @ X @ Z2 ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % eq_diff_eq'
% 0.17/0.45 thf(fact_270_le__numeral__extra_I3_J,axiom,
% 0.17/0.45 ord_less_eq_real @ zero_zero_real @ zero_zero_real ).
% 0.17/0.45
% 0.17/0.45 % le_numeral_extra(3)
% 0.17/0.45 thf(fact_271_le__numeral__extra_I3_J,axiom,
% 0.17/0.45 ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% 0.17/0.45
% 0.17/0.45 % le_numeral_extra(3)
% 0.17/0.45 thf(fact_272_nonoverlap,axiom,
% 0.17/0.45 ord_less_eq_real @ genClo1278781456e_beta @ genClo1651033342e_rmin ).
% 0.17/0.45
% 0.17/0.45 % nonoverlap
% 0.17/0.45 thf(fact_273_le__numeral__extra_I4_J,axiom,
% 0.17/0.45 ord_less_eq_real @ one_one_real @ one_one_real ).
% 0.17/0.45
% 0.17/0.45 % le_numeral_extra(4)
% 0.17/0.45 thf(fact_274_le__numeral__extra_I4_J,axiom,
% 0.17/0.45 ord_less_eq_nat @ one_one_nat @ one_one_nat ).
% 0.17/0.45
% 0.17/0.45 % le_numeral_extra(4)
% 0.17/0.45 thf(fact_275_init,axiom,
% 0.17/0.45 ! [P: nat] :
% 0.17/0.45 ( ( genClo1015804716orrect @ P @ zero_zero_real )
% 0.17/0.45 => ( ( ord_less_eq_real @ zero_zero_real @ ( genClo1161277105lle_PC @ P @ zero_zero_real ) )
% 0.17/0.45 & ( ord_less_eq_real @ ( genClo1161277105lle_PC @ P @ zero_zero_real ) @ genClo872980712lle_mu ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % init
% 0.17/0.45 thf(fact_276_bound1,axiom,
% 0.17/0.45 ord_less_eq_real @ ( minus_minus_real @ ( genClo1161277105lle_PC @ l @ ( genClo1163638703lle_te @ p @ ( plus_plus_nat @ i @ one_one_nat ) ) ) @ ( genClo1161277105lle_PC @ l @ ( genClo1163638703lle_te @ q @ ( plus_plus_nat @ i @ one_one_nat ) ) ) ) @ ( times_times_real @ ( minus_minus_real @ ( genClo1163638703lle_te @ p @ ( plus_plus_nat @ i @ one_one_nat ) ) @ ( genClo1163638703lle_te @ q @ ( plus_plus_nat @ i @ one_one_nat ) ) ) @ ( plus_plus_real @ one_one_real @ genClo1144207539le_rho ) ) ).
% 0.17/0.45
% 0.17/0.45 % bound1
% 0.17/0.45 thf(fact_277_mult__cancel__right,axiom,
% 0.17/0.45 ! [A: nat,C: nat,B: nat] :
% 0.17/0.45 ( ( ( times_times_nat @ A @ C )
% 0.17/0.45 = ( times_times_nat @ B @ C ) )
% 0.17/0.45 = ( ( C = zero_zero_nat )
% 0.17/0.45 | ( A = B ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % mult_cancel_right
% 0.17/0.45 thf(fact_278_mult__cancel__right,axiom,
% 0.17/0.45 ! [A: real,C: real,B: real] :
% 0.17/0.45 ( ( ( times_times_real @ A @ C )
% 0.17/0.45 = ( times_times_real @ B @ C ) )
% 0.17/0.45 = ( ( C = zero_zero_real )
% 0.17/0.45 | ( A = B ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % mult_cancel_right
% 0.17/0.45 thf(fact_279_mult__cancel__left,axiom,
% 0.17/0.45 ! [C: nat,A: nat,B: nat] :
% 0.17/0.45 ( ( ( times_times_nat @ C @ A )
% 0.17/0.45 = ( times_times_nat @ C @ B ) )
% 0.17/0.45 = ( ( C = zero_zero_nat )
% 0.17/0.45 | ( A = B ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % mult_cancel_left
% 0.17/0.45 thf(fact_280_mult__cancel__left,axiom,
% 0.17/0.45 ! [C: real,A: real,B: real] :
% 0.17/0.45 ( ( ( times_times_real @ C @ A )
% 0.17/0.45 = ( times_times_real @ C @ B ) )
% 0.17/0.45 = ( ( C = zero_zero_real )
% 0.17/0.45 | ( A = B ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % mult_cancel_left
% 0.17/0.45 thf(fact_281_mult__eq__0__iff,axiom,
% 0.17/0.45 ! [A: nat,B: nat] :
% 0.17/0.45 ( ( ( times_times_nat @ A @ B )
% 0.17/0.45 = zero_zero_nat )
% 0.17/0.45 = ( ( A = zero_zero_nat )
% 0.17/0.45 | ( B = zero_zero_nat ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % mult_eq_0_iff
% 0.17/0.45 thf(fact_282_mult__eq__0__iff,axiom,
% 0.17/0.45 ! [A: real,B: real] :
% 0.17/0.45 ( ( ( times_times_real @ A @ B )
% 0.17/0.45 = zero_zero_real )
% 0.17/0.45 = ( ( A = zero_zero_real )
% 0.17/0.45 | ( B = zero_zero_real ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % mult_eq_0_iff
% 0.17/0.45 thf(fact_283_mult__zero__right,axiom,
% 0.17/0.45 ! [A: nat] :
% 0.17/0.45 ( ( times_times_nat @ A @ zero_zero_nat )
% 0.17/0.45 = zero_zero_nat ) ).
% 0.17/0.45
% 0.17/0.45 % mult_zero_right
% 0.17/0.45 thf(fact_284_mult__zero__right,axiom,
% 0.17/0.45 ! [A: real] :
% 0.17/0.45 ( ( times_times_real @ A @ zero_zero_real )
% 0.17/0.45 = zero_zero_real ) ).
% 0.17/0.45
% 0.17/0.45 % mult_zero_right
% 0.17/0.45 thf(fact_285_mult__zero__left,axiom,
% 0.17/0.45 ! [A: nat] :
% 0.17/0.45 ( ( times_times_nat @ zero_zero_nat @ A )
% 0.17/0.45 = zero_zero_nat ) ).
% 0.17/0.45
% 0.17/0.45 % mult_zero_left
% 0.17/0.45 thf(fact_286_mult__zero__left,axiom,
% 0.17/0.45 ! [A: real] :
% 0.17/0.45 ( ( times_times_real @ zero_zero_real @ A )
% 0.17/0.45 = zero_zero_real ) ).
% 0.17/0.45
% 0.17/0.45 % mult_zero_left
% 0.17/0.45 thf(fact_287_mult_Oleft__neutral,axiom,
% 0.17/0.45 ! [A: nat] :
% 0.17/0.45 ( ( times_times_nat @ one_one_nat @ A )
% 0.17/0.45 = A ) ).
% 0.17/0.45
% 0.17/0.45 % mult.left_neutral
% 0.17/0.45 thf(fact_288_mult_Oleft__neutral,axiom,
% 0.17/0.45 ! [A: real] :
% 0.17/0.45 ( ( times_times_real @ one_one_real @ A )
% 0.17/0.45 = A ) ).
% 0.17/0.45
% 0.17/0.45 % mult.left_neutral
% 0.17/0.45 thf(fact_289_mult_Oright__neutral,axiom,
% 0.17/0.45 ! [A: nat] :
% 0.17/0.45 ( ( times_times_nat @ A @ one_one_nat )
% 0.17/0.45 = A ) ).
% 0.17/0.45
% 0.17/0.45 % mult.right_neutral
% 0.17/0.45 thf(fact_290_mult_Oright__neutral,axiom,
% 0.17/0.45 ! [A: real] :
% 0.17/0.45 ( ( times_times_real @ A @ one_one_real )
% 0.17/0.45 = A ) ).
% 0.17/0.45
% 0.17/0.45 % mult.right_neutral
% 0.17/0.45 thf(fact_291_abs__mult__self__eq,axiom,
% 0.17/0.45 ! [A: real] :
% 0.17/0.45 ( ( times_times_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ A ) )
% 0.17/0.45 = ( times_times_real @ A @ A ) ) ).
% 0.17/0.45
% 0.17/0.45 % abs_mult_self_eq
% 0.17/0.45 thf(fact_292_mult__cancel__left1,axiom,
% 0.17/0.45 ! [C: real,B: real] :
% 0.17/0.45 ( ( C
% 0.17/0.45 = ( times_times_real @ C @ B ) )
% 0.17/0.45 = ( ( C = zero_zero_real )
% 0.17/0.45 | ( B = one_one_real ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % mult_cancel_left1
% 0.17/0.45 thf(fact_293_mult__cancel__left2,axiom,
% 0.17/0.45 ! [C: real,A: real] :
% 0.17/0.45 ( ( ( times_times_real @ C @ A )
% 0.17/0.45 = C )
% 0.17/0.45 = ( ( C = zero_zero_real )
% 0.17/0.45 | ( A = one_one_real ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % mult_cancel_left2
% 0.17/0.45 thf(fact_294_mult__cancel__right1,axiom,
% 0.17/0.45 ! [C: real,B: real] :
% 0.17/0.45 ( ( C
% 0.17/0.45 = ( times_times_real @ B @ C ) )
% 0.17/0.45 = ( ( C = zero_zero_real )
% 0.17/0.45 | ( B = one_one_real ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % mult_cancel_right1
% 0.17/0.45 thf(fact_295_mult__cancel__right2,axiom,
% 0.17/0.45 ! [A: real,C: real] :
% 0.17/0.45 ( ( ( times_times_real @ A @ C )
% 0.17/0.45 = C )
% 0.17/0.45 = ( ( C = zero_zero_real )
% 0.17/0.45 | ( A = one_one_real ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % mult_cancel_right2
% 0.17/0.45 thf(fact_296_mult__right__cancel,axiom,
% 0.17/0.45 ! [C: nat,A: nat,B: nat] :
% 0.17/0.45 ( ( C != zero_zero_nat )
% 0.17/0.45 => ( ( ( times_times_nat @ A @ C )
% 0.17/0.45 = ( times_times_nat @ B @ C ) )
% 0.17/0.45 = ( A = B ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % mult_right_cancel
% 0.17/0.45 thf(fact_297_mult__right__cancel,axiom,
% 0.17/0.45 ! [C: real,A: real,B: real] :
% 0.17/0.45 ( ( C != zero_zero_real )
% 0.17/0.45 => ( ( ( times_times_real @ A @ C )
% 0.17/0.45 = ( times_times_real @ B @ C ) )
% 0.17/0.45 = ( A = B ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % mult_right_cancel
% 0.17/0.45 thf(fact_298_mult__left__cancel,axiom,
% 0.17/0.45 ! [C: nat,A: nat,B: nat] :
% 0.17/0.45 ( ( C != zero_zero_nat )
% 0.17/0.45 => ( ( ( times_times_nat @ C @ A )
% 0.17/0.45 = ( times_times_nat @ C @ B ) )
% 0.17/0.45 = ( A = B ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % mult_left_cancel
% 0.17/0.45 thf(fact_299_mult__left__cancel,axiom,
% 0.17/0.45 ! [C: real,A: real,B: real] :
% 0.17/0.45 ( ( C != zero_zero_real )
% 0.17/0.45 => ( ( ( times_times_real @ C @ A )
% 0.17/0.45 = ( times_times_real @ C @ B ) )
% 0.17/0.45 = ( A = B ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % mult_left_cancel
% 0.17/0.45 thf(fact_300_no__zero__divisors,axiom,
% 0.17/0.45 ! [A: nat,B: nat] :
% 0.17/0.45 ( ( A != zero_zero_nat )
% 0.17/0.45 => ( ( B != zero_zero_nat )
% 0.17/0.45 => ( ( times_times_nat @ A @ B )
% 0.17/0.45 != zero_zero_nat ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % no_zero_divisors
% 0.17/0.45 thf(fact_301_no__zero__divisors,axiom,
% 0.17/0.45 ! [A: real,B: real] :
% 0.17/0.45 ( ( A != zero_zero_real )
% 0.17/0.45 => ( ( B != zero_zero_real )
% 0.17/0.45 => ( ( times_times_real @ A @ B )
% 0.17/0.45 != zero_zero_real ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % no_zero_divisors
% 0.17/0.45 thf(fact_302_divisors__zero,axiom,
% 0.17/0.45 ! [A: nat,B: nat] :
% 0.17/0.45 ( ( ( times_times_nat @ A @ B )
% 0.17/0.45 = zero_zero_nat )
% 0.17/0.45 => ( ( A = zero_zero_nat )
% 0.17/0.45 | ( B = zero_zero_nat ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % divisors_zero
% 0.17/0.45 thf(fact_303_divisors__zero,axiom,
% 0.17/0.45 ! [A: real,B: real] :
% 0.17/0.45 ( ( ( times_times_real @ A @ B )
% 0.17/0.45 = zero_zero_real )
% 0.17/0.45 => ( ( A = zero_zero_real )
% 0.17/0.45 | ( B = zero_zero_real ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % divisors_zero
% 0.17/0.45 thf(fact_304_mult__not__zero,axiom,
% 0.17/0.45 ! [A: nat,B: nat] :
% 0.17/0.45 ( ( ( times_times_nat @ A @ B )
% 0.17/0.45 != zero_zero_nat )
% 0.17/0.45 => ( ( A != zero_zero_nat )
% 0.17/0.45 & ( B != zero_zero_nat ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % mult_not_zero
% 0.17/0.45 thf(fact_305_mult__not__zero,axiom,
% 0.17/0.45 ! [A: real,B: real] :
% 0.17/0.45 ( ( ( times_times_real @ A @ B )
% 0.17/0.45 != zero_zero_real )
% 0.17/0.45 => ( ( A != zero_zero_real )
% 0.17/0.45 & ( B != zero_zero_real ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % mult_not_zero
% 0.17/0.45 thf(fact_306_ring__class_Oring__distribs_I2_J,axiom,
% 0.17/0.45 ! [A: real,B: real,C: real] :
% 0.17/0.45 ( ( times_times_real @ ( plus_plus_real @ A @ B ) @ C )
% 0.17/0.45 = ( plus_plus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % ring_class.ring_distribs(2)
% 0.17/0.45 thf(fact_307_ring__class_Oring__distribs_I1_J,axiom,
% 0.17/0.45 ! [A: real,B: real,C: real] :
% 0.17/0.45 ( ( times_times_real @ A @ ( plus_plus_real @ B @ C ) )
% 0.17/0.45 = ( plus_plus_real @ ( times_times_real @ A @ B ) @ ( times_times_real @ A @ C ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % ring_class.ring_distribs(1)
% 0.17/0.45 thf(fact_308_comm__semiring__class_Odistrib,axiom,
% 0.17/0.45 ! [A: nat,B: nat,C: nat] :
% 0.17/0.45 ( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ C )
% 0.17/0.45 = ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % comm_semiring_class.distrib
% 0.17/0.45 thf(fact_309_comm__semiring__class_Odistrib,axiom,
% 0.17/0.45 ! [A: real,B: real,C: real] :
% 0.17/0.45 ( ( times_times_real @ ( plus_plus_real @ A @ B ) @ C )
% 0.17/0.45 = ( plus_plus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % comm_semiring_class.distrib
% 0.17/0.45 thf(fact_310_distrib__left,axiom,
% 0.17/0.45 ! [A: nat,B: nat,C: nat] :
% 0.17/0.45 ( ( times_times_nat @ A @ ( plus_plus_nat @ B @ C ) )
% 0.17/0.45 = ( plus_plus_nat @ ( times_times_nat @ A @ B ) @ ( times_times_nat @ A @ C ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % distrib_left
% 0.17/0.45 thf(fact_311_distrib__left,axiom,
% 0.17/0.45 ! [A: real,B: real,C: real] :
% 0.17/0.45 ( ( times_times_real @ A @ ( plus_plus_real @ B @ C ) )
% 0.17/0.45 = ( plus_plus_real @ ( times_times_real @ A @ B ) @ ( times_times_real @ A @ C ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % distrib_left
% 0.17/0.45 thf(fact_312_distrib__right,axiom,
% 0.17/0.45 ! [A: nat,B: nat,C: nat] :
% 0.17/0.45 ( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ C )
% 0.17/0.45 = ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % distrib_right
% 0.17/0.45 thf(fact_313_distrib__right,axiom,
% 0.17/0.45 ! [A: real,B: real,C: real] :
% 0.17/0.45 ( ( times_times_real @ ( plus_plus_real @ A @ B ) @ C )
% 0.17/0.45 = ( plus_plus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % distrib_right
% 0.17/0.45 thf(fact_314_combine__common__factor,axiom,
% 0.17/0.45 ! [A: nat,E: nat,B: nat,C: nat] :
% 0.17/0.45 ( ( plus_plus_nat @ ( times_times_nat @ A @ E ) @ ( plus_plus_nat @ ( times_times_nat @ B @ E ) @ C ) )
% 0.17/0.45 = ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ E ) @ C ) ) ).
% 0.17/0.45
% 0.17/0.45 % combine_common_factor
% 0.17/0.45 thf(fact_315_combine__common__factor,axiom,
% 0.17/0.45 ! [A: real,E: real,B: real,C: real] :
% 0.17/0.45 ( ( plus_plus_real @ ( times_times_real @ A @ E ) @ ( plus_plus_real @ ( times_times_real @ B @ E ) @ C ) )
% 0.17/0.45 = ( plus_plus_real @ ( times_times_real @ ( plus_plus_real @ A @ B ) @ E ) @ C ) ) ).
% 0.17/0.45
% 0.17/0.45 % combine_common_factor
% 0.17/0.45 thf(fact_316_comm__monoid__mult__class_Omult__1,axiom,
% 0.17/0.45 ! [A: nat] :
% 0.17/0.45 ( ( times_times_nat @ one_one_nat @ A )
% 0.17/0.45 = A ) ).
% 0.17/0.45
% 0.17/0.45 % comm_monoid_mult_class.mult_1
% 0.17/0.45 thf(fact_317_comm__monoid__mult__class_Omult__1,axiom,
% 0.17/0.45 ! [A: real] :
% 0.17/0.45 ( ( times_times_real @ one_one_real @ A )
% 0.17/0.45 = A ) ).
% 0.17/0.45
% 0.17/0.45 % comm_monoid_mult_class.mult_1
% 0.17/0.45 thf(fact_318_mult_Ocomm__neutral,axiom,
% 0.17/0.45 ! [A: nat] :
% 0.17/0.45 ( ( times_times_nat @ A @ one_one_nat )
% 0.17/0.45 = A ) ).
% 0.17/0.45
% 0.17/0.45 % mult.comm_neutral
% 0.17/0.45 thf(fact_319_mult_Ocomm__neutral,axiom,
% 0.17/0.45 ! [A: real] :
% 0.17/0.45 ( ( times_times_real @ A @ one_one_real )
% 0.17/0.45 = A ) ).
% 0.17/0.45
% 0.17/0.45 % mult.comm_neutral
% 0.17/0.45 thf(fact_320_left__diff__distrib,axiom,
% 0.17/0.45 ! [A: real,B: real,C: real] :
% 0.17/0.45 ( ( times_times_real @ ( minus_minus_real @ A @ B ) @ C )
% 0.17/0.45 = ( minus_minus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % left_diff_distrib
% 0.17/0.45 thf(fact_321_right__diff__distrib,axiom,
% 0.17/0.45 ! [A: real,B: real,C: real] :
% 0.17/0.45 ( ( times_times_real @ A @ ( minus_minus_real @ B @ C ) )
% 0.17/0.45 = ( minus_minus_real @ ( times_times_real @ A @ B ) @ ( times_times_real @ A @ C ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % right_diff_distrib
% 0.17/0.45 thf(fact_322_left__diff__distrib_H,axiom,
% 0.17/0.45 ! [B: nat,C: nat,A: nat] :
% 0.17/0.45 ( ( times_times_nat @ ( minus_minus_nat @ B @ C ) @ A )
% 0.17/0.45 = ( minus_minus_nat @ ( times_times_nat @ B @ A ) @ ( times_times_nat @ C @ A ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % left_diff_distrib'
% 0.17/0.45 thf(fact_323_left__diff__distrib_H,axiom,
% 0.17/0.45 ! [B: real,C: real,A: real] :
% 0.17/0.45 ( ( times_times_real @ ( minus_minus_real @ B @ C ) @ A )
% 0.17/0.45 = ( minus_minus_real @ ( times_times_real @ B @ A ) @ ( times_times_real @ C @ A ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % left_diff_distrib'
% 0.17/0.45 thf(fact_324_right__diff__distrib_H,axiom,
% 0.17/0.45 ! [A: nat,B: nat,C: nat] :
% 0.17/0.45 ( ( times_times_nat @ A @ ( minus_minus_nat @ B @ C ) )
% 0.17/0.45 = ( minus_minus_nat @ ( times_times_nat @ A @ B ) @ ( times_times_nat @ A @ C ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % right_diff_distrib'
% 0.17/0.45 thf(fact_325_right__diff__distrib_H,axiom,
% 0.17/0.45 ! [A: real,B: real,C: real] :
% 0.17/0.45 ( ( times_times_real @ A @ ( minus_minus_real @ B @ C ) )
% 0.17/0.45 = ( minus_minus_real @ ( times_times_real @ A @ B ) @ ( times_times_real @ A @ C ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % right_diff_distrib'
% 0.17/0.45 thf(fact_326_abs__mult,axiom,
% 0.17/0.45 ! [A: real,B: real] :
% 0.17/0.45 ( ( abs_abs_real @ ( times_times_real @ A @ B ) )
% 0.17/0.45 = ( times_times_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % abs_mult
% 0.17/0.45 thf(fact_327_mult_Oleft__commute,axiom,
% 0.17/0.45 ! [B: nat,A: nat,C: nat] :
% 0.17/0.45 ( ( times_times_nat @ B @ ( times_times_nat @ A @ C ) )
% 0.17/0.45 = ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % mult.left_commute
% 0.17/0.45 thf(fact_328_mult_Oleft__commute,axiom,
% 0.17/0.45 ! [B: real,A: real,C: real] :
% 0.17/0.45 ( ( times_times_real @ B @ ( times_times_real @ A @ C ) )
% 0.17/0.45 = ( times_times_real @ A @ ( times_times_real @ B @ C ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % mult.left_commute
% 0.17/0.45 thf(fact_329_mult_Ocommute,axiom,
% 0.17/0.45 ( times_times_nat
% 0.17/0.45 = ( ^ [A2: nat,B2: nat] : ( times_times_nat @ B2 @ A2 ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % mult.commute
% 0.17/0.45 thf(fact_330_mult_Ocommute,axiom,
% 0.17/0.45 ( times_times_real
% 0.17/0.45 = ( ^ [A2: real,B2: real] : ( times_times_real @ B2 @ A2 ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % mult.commute
% 0.17/0.45 thf(fact_331_mult_Oassoc,axiom,
% 0.17/0.45 ! [A: nat,B: nat,C: nat] :
% 0.17/0.45 ( ( times_times_nat @ ( times_times_nat @ A @ B ) @ C )
% 0.17/0.45 = ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % mult.assoc
% 0.17/0.45 thf(fact_332_mult_Oassoc,axiom,
% 0.17/0.45 ! [A: real,B: real,C: real] :
% 0.17/0.45 ( ( times_times_real @ ( times_times_real @ A @ B ) @ C )
% 0.17/0.45 = ( times_times_real @ A @ ( times_times_real @ B @ C ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % mult.assoc
% 0.17/0.45 thf(fact_333_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
% 0.17/0.45 ! [A: nat,B: nat,C: nat] :
% 0.17/0.45 ( ( times_times_nat @ ( times_times_nat @ A @ B ) @ C )
% 0.17/0.45 = ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % ab_semigroup_mult_class.mult_ac(1)
% 0.17/0.45 thf(fact_334_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
% 0.17/0.45 ! [A: real,B: real,C: real] :
% 0.17/0.45 ( ( times_times_real @ ( times_times_real @ A @ B ) @ C )
% 0.17/0.45 = ( times_times_real @ A @ ( times_times_real @ B @ C ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % ab_semigroup_mult_class.mult_ac(1)
% 0.17/0.45 thf(fact_335_mult__mono,axiom,
% 0.17/0.45 ! [A: real,B: real,C: real,D: real] :
% 0.17/0.45 ( ( ord_less_eq_real @ A @ B )
% 0.17/0.45 => ( ( ord_less_eq_real @ C @ D )
% 0.17/0.45 => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 0.17/0.45 => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 0.17/0.45 => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % mult_mono
% 0.17/0.45 thf(fact_336_mult__mono,axiom,
% 0.17/0.45 ! [A: nat,B: nat,C: nat,D: nat] :
% 0.17/0.45 ( ( ord_less_eq_nat @ A @ B )
% 0.17/0.45 => ( ( ord_less_eq_nat @ C @ D )
% 0.17/0.45 => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 0.17/0.45 => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 0.17/0.45 => ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % mult_mono
% 0.17/0.45
% 0.17/0.45 % Conjectures (1)
% 0.17/0.45 thf(conj_0,conjecture,
% 0.17/0.45 ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( genClo1163638703lle_te @ p @ ( plus_plus_nat @ i @ one_one_nat ) ) @ ( genClo1163638703lle_te @ q @ ( plus_plus_nat @ i @ one_one_nat ) ) ) ) @ genClo1278781456e_beta ).
% 0.17/0.45
% 0.17/0.45 %------------------------------------------------------------------------------
% 0.17/0.45 ------------------------
% 2.31/1.47 ---- Embedded in THF ---
% 2.31/1.47 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 2.31/1.47 %%% Generated on Tue Feb 24 08:25:26 EST 2026
% 2.31/1.47 %%% using '$$dhol' embedding, version 1.3.0.
% 2.31/1.47 %%% Logic specification used:
% 2.31/1.47 %%% thf(spec, logic, $$dhol).
% 2.31/1.47
% 2.31/1.47 % SZS output start ListOfTHF for /export/starexec/sandbox2/tmp/tmp.5TcjDfaA5h/DTF2DTF25639.p
% See solution above
% 2.31/1.47 ------------------------
% 2.31/1.49 ---- Cleaned THF ---
% 2.31/1.49 thf(ty_n_t__Real__Oreal,type,
% 2.31/1.49 real: $tType ).
% 2.31/1.49
% 2.31/1.49 thf(ty_n_t__Nat__Onat,type,
% 2.31/1.49 nat: $tType ).
% 2.31/1.49
% 2.31/1.49 thf(sy_c_GenClock__Mirabelle__bsvkzpgbls_OPC,type,
% 2.31/1.49 genClo1161277105lle_PC: nat > real > real ).
% 2.31/1.49
% 2.31/1.49 thf(sy_c_GenClock__Mirabelle__bsvkzpgbls_O_092_060beta_062,type,
% 2.31/1.49 genClo1278781456e_beta: real ).
% 2.31/1.49
% 2.31/1.49 thf(sy_c_GenClock__Mirabelle__bsvkzpgbls_O_092_060mu_062,type,
% 2.31/1.49 genClo872980712lle_mu: real ).
% 2.31/1.49
% 2.31/1.49 thf(sy_c_GenClock__Mirabelle__bsvkzpgbls_O_092_060rho_062,type,
% 2.31/1.49 genClo1144207539le_rho: real ).
% 2.31/1.49
% 2.31/1.49 thf(sy_c_GenClock__Mirabelle__bsvkzpgbls_Ocorrect,type,
% 2.31/1.49 genClo1015804716orrect: nat > real > $o ).
% 2.31/1.49
% 2.31/1.49 thf(sy_c_GenClock__Mirabelle__bsvkzpgbls_OokRead2,type,
% 2.31/1.49 genClo293725282kRead2: ( nat > real ) > ( nat > real ) > real > ( nat > $o ) > $o ).
% 2.31/1.49
% 2.31/1.49 thf(sy_c_GenClock__Mirabelle__bsvkzpgbls_Ormax,type,
% 2.31/1.49 genClo1650508560e_rmax: real ).
% 2.31/1.49
% 2.31/1.49 thf(sy_c_GenClock__Mirabelle__bsvkzpgbls_Ormin,type,
% 2.31/1.49 genClo1651033342e_rmin: real ).
% 2.31/1.49
% 2.31/1.49 thf(sy_c_GenClock__Mirabelle__bsvkzpgbls_Ote,type,
% 2.31/1.49 genClo1163638703lle_te: nat > nat > real ).
% 2.31/1.49
% 2.31/1.49 thf(sy_c_Groups_Oabs__class_Oabs_001t__Real__Oreal,type,
% 2.31/1.49 abs_abs_real: real > real ).
% 2.31/1.49
% 2.31/1.49 thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat,type,
% 2.31/1.49 minus_minus_nat: nat > nat > nat ).
% 2.31/1.49
% 2.31/1.49 thf(sy_c_Groups_Ominus__class_Ominus_001t__Real__Oreal,type,
% 2.31/1.49 minus_minus_real: real > real > real ).
% 2.31/1.49
% 2.31/1.49 thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat,type,
% 2.31/1.49 one_one_nat: nat ).
% 2.31/1.49
% 2.31/1.49 thf(sy_c_Groups_Oone__class_Oone_001t__Real__Oreal,type,
% 2.31/1.49 one_one_real: real ).
% 2.31/1.49
% 2.31/1.49 thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat,type,
% 2.31/1.49 plus_plus_nat: nat > nat > nat ).
% 2.31/1.49
% 2.31/1.49 thf(sy_c_Groups_Oplus__class_Oplus_001t__Real__Oreal,type,
% 2.31/1.49 plus_plus_real: real > real > real ).
% 2.31/1.49
% 2.31/1.49 thf(sy_c_Groups_Otimes__class_Otimes_001t__Nat__Onat,type,
% 2.31/1.49 times_times_nat: nat > nat > nat ).
% 2.31/1.49
% 2.31/1.49 thf(sy_c_Groups_Otimes__class_Otimes_001t__Real__Oreal,type,
% 2.31/1.49 times_times_real: real > real > real ).
% 2.31/1.49
% 2.31/1.49 thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
% 2.31/1.49 zero_zero_nat: nat ).
% 2.31/1.49
% 2.31/1.49 thf(sy_c_Groups_Ozero__class_Ozero_001t__Real__Oreal,type,
% 2.31/1.49 zero_zero_real: real ).
% 2.31/1.49
% 2.31/1.49 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
% 2.31/1.49 ord_less_eq_nat: nat > nat > $o ).
% 2.31/1.49
% 2.31/1.49 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Real__Oreal,type,
% 2.31/1.49 ord_less_eq_real: real > real > $o ).
% 2.31/1.49
% 2.31/1.49 thf(sy_v_i,type,
% 2.31/1.49 i: nat ).
% 2.31/1.49
% 2.31/1.49 thf(sy_v_l,type,
% 2.31/1.49 l: nat ).
% 2.31/1.49
% 2.31/1.49 thf(sy_v_p,type,
% 2.31/1.49 p: nat ).
% 2.31/1.49
% 2.31/1.49 thf(sy_v_q,type,
% 2.31/1.49 q: nat ).
% 2.31/1.49
% 2.31/1.49 thf(fact_0_rts2b,axiom,
% 2.31/1.49 ! [P: nat,Q: nat,I: nat] :
% 2.31/1.49 ( ( ( genClo1015804716orrect @ P @ ( genClo1163638703lle_te @ P @ I ) )
% 2.31/1.49 & ( genClo1015804716orrect @ Q @ ( genClo1163638703lle_te @ Q @ I ) ) )
% 2.31/1.49 => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( genClo1163638703lle_te @ P @ I ) @ ( genClo1163638703lle_te @ Q @ I ) ) ) @ genClo1278781456e_beta ) ) ).
% 2.31/1.49
% 2.31/1.49 thf(fact_1_corr__p,axiom,
% 2.31/1.49 genClo1015804716orrect @ p @ ( genClo1163638703lle_te @ p @ ( plus_plus_nat @ i @ one_one_nat ) ) ).
% 2.31/1.49
% 2.31/1.49 thf(fact_2_corr__q__tq,axiom,
% 2.31/1.49 genClo1015804716orrect @ q @ ( genClo1163638703lle_te @ q @ ( plus_plus_nat @ i @ one_one_nat ) ) ).
% 2.31/1.49
% 2.31/1.49 thf(fact_3_ie,axiom,
% 2.31/1.49 ord_less_eq_real @ ( genClo1163638703lle_te @ q @ ( plus_plus_nat @ i @ one_one_nat ) ) @ ( genClo1163638703lle_te @ p @ ( plus_plus_nat @ i @ one_one_nat ) ) ).
% 2.31/1.49
% 2.31/1.49 thf(fact_4_corr__q,axiom,
% 2.31/1.49 genClo1015804716orrect @ q @ ( genClo1163638703lle_te @ p @ ( plus_plus_nat @ i @ one_one_nat ) ) ).
% 2.31/1.49
% 2.31/1.49 thf(fact_5_corr__l,axiom,
% 2.31/1.49 genClo1015804716orrect @ l @ ( genClo1163638703lle_te @ p @ ( plus_plus_nat @ i @ one_one_nat ) ) ).
% 2.31/1.49
% 2.31/1.49 thf(fact_6_posD,axiom,
% 2.31/1.49 ord_less_eq_real @ zero_zero_real @ ( minus_minus_real @ ( genClo1163638703lle_te @ p @ ( plus_plus_nat @ i @ one_one_nat ) ) @ ( genClo1163638703lle_te @ q @ ( plus_plus_nat @ i @ one_one_nat ) ) ) ).
% 2.31/1.49
% 2.31/1.49 thf(fact_7_corr__l__tq,axiom,
% 2.31/1.49 genClo1015804716orrect @ l @ ( genClo1163638703lle_te @ q @ ( plus_plus_nat @ i @ one_one_nat ) ) ).
% 2.31/1.49
% 2.31/1.49 thf(fact_8_le__add__diff__inverse,axiom,
% 2.31/1.49 ! [B: real,A: real] :
% 2.31/1.49 ( ( ord_less_eq_real @ B @ A )
% 2.31/1.49 => ( ( plus_plus_real @ B @ ( minus_minus_real @ A @ B ) )
% 2.31/1.49 = A ) ) ).
% 2.31/1.49
% 2.31/1.49 thf(fact_9_le__add__diff__inverse,axiom,
% 2.31/1.49 ! [B: nat,A: nat] :
% 2.31/1.49 ( ( ord_less_eq_nat @ B @ A )
% 2.31/1.49 => ( ( plus_plus_nat @ B @ ( minus_minus_nat @ A @ B ) )
% 2.31/1.49 = A ) ) ).
% 2.31/1.49
% 2.31/1.49 thf(fact_10_le__add__diff__inverse2,axiom,
% 2.31/1.49 ! [B: real,A: real] :
% 2.31/1.49 ( ( ord_less_eq_real @ B @ A )
% 2.31/1.49 => ( ( plus_plus_real @ ( minus_minus_real @ A @ B ) @ B )
% 2.31/1.49 = A ) ) ).
% 2.31/1.49
% 2.31/1.49 thf(fact_11_le__add__diff__inverse2,axiom,
% 2.31/1.49 ! [B: nat,A: nat] :
% 2.31/1.49 ( ( ord_less_eq_nat @ B @ A )
% 2.31/1.49 => ( ( plus_plus_nat @ ( minus_minus_nat @ A @ B ) @ B )
% 2.31/1.49 = A ) ) ).
% 2.31/1.49
% 2.31/1.49 thf(fact_12_rte,axiom,
% 2.31/1.49 ! [P2: nat,I2: nat] :
% 2.31/1.49 ( ( genClo1015804716orrect @ P2 @ ( genClo1163638703lle_te @ P2 @ ( plus_plus_nat @ I2 @ one_one_nat ) ) )
% 2.31/1.49 => ( ord_less_eq_real @ ( genClo1163638703lle_te @ P2 @ I2 ) @ ( genClo1163638703lle_te @ P2 @ ( plus_plus_nat @ I2 @ one_one_nat ) ) ) ) ).
% 2.31/1.49
% 2.31/1.49 thf(fact_13_abs__1,axiom,
% 2.31/1.49 ( ( abs_abs_real @ one_one_real )
% 2.31/1.49 = one_one_real ) ).
% 2.31/1.49
% 2.31/1.49 thf(fact_14_abs__add__abs,axiom,
% 2.31/1.49 ! [A: real,B: real] :
% 2.31/1.49 ( ( abs_abs_real @ ( plus_plus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) )
% 2.31/1.49 = ( plus_plus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) ) ).
% 2.31/1.49
% 2.31/1.49 thf(fact_15_add__diff__cancel,axiom,
% 2.31/1.49 ! [A: real,B: real] :
% 2.31/1.49 ( ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ B )
% 2.31/1.49 = A ) ).
% 2.31/1.49
% 2.31/1.49 thf(fact_16_diff__add__cancel,axiom,
% 2.31/1.49 ! [A: real,B: real] :
% 2.31/1.49 ( ( plus_plus_real @ ( minus_minus_real @ A @ B ) @ B )
% 2.31/1.49 = A ) ).
% 2.31/1.49
% 2.31/1.49 thf(fact_17_add__diff__cancel__left,axiom,
% 2.31/1.49 ! [C: real,A: real,B: real] :
% 2.31/1.49 ( ( minus_minus_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
% 2.31/1.49 = ( minus_minus_real @ A @ B ) ) ).
% 2.31/1.49
% 2.31/1.49 thf(fact_18_add__diff__cancel__left,axiom,
% 2.31/1.49 ! [C: nat,A: nat,B: nat] :
% 2.31/1.49 ( ( minus_minus_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
% 2.31/1.49 = ( minus_minus_nat @ A @ B ) ) ).
% 2.31/1.49
% 2.31/1.49 thf(fact_19_add__diff__cancel__left_H,axiom,
% 2.31/1.49 ! [A: real,B: real] :
% 2.31/1.49 ( ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ A )
% 2.31/1.49 = B ) ).
% 2.31/1.49
% 2.31/1.49 thf(fact_20_add__diff__cancel__left_H,axiom,
% 2.31/1.49 ! [A: nat,B: nat] :
% 2.31/1.50 ( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ A )
% 2.31/1.50 = B ) ).
% 2.31/1.50
% 2.31/1.50 thf(fact_21_add__diff__cancel__right,axiom,
% 2.31/1.50 ! [A: real,C: real,B: real] :
% 2.31/1.50 ( ( minus_minus_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
% 2.31/1.50 = ( minus_minus_real @ A @ B ) ) ).
% 2.31/1.50
% 2.31/1.50 thf(fact_22_add__diff__cancel__right,axiom,
% 2.31/1.50 ! [A: nat,C: nat,B: nat] :
% 2.31/1.50 ( ( minus_minus_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
% 2.31/1.50 = ( minus_minus_nat @ A @ B ) ) ).
% 2.31/1.50
% 2.31/1.50 thf(fact_23_add__right__cancel,axiom,
% 2.31/1.50 ! [B: nat,A: nat,C: nat] :
% 2.31/1.50 ( ( ( plus_plus_nat @ B @ A )
% 2.31/1.50 = ( plus_plus_nat @ C @ A ) )
% 2.31/1.50 = ( B = C ) ) ).
% 2.31/1.50
% 2.31/1.50 thf(fact_24_add__right__cancel,axiom,
% 2.31/1.50 ! [B: real,A: real,C: real] :
% 2.31/1.50 ( ( ( plus_plus_real @ B @ A )
% 2.31/1.50 = ( plus_plus_real @ C @ A ) )
% 2.31/1.50 = ( B = C ) ) ).
% 2.31/1.50
% 2.31/1.50 thf(fact_25_add__left__cancel,axiom,
% 2.31/1.50 ! [A: nat,B: nat,C: nat] :
% 2.31/1.50 ( ( ( plus_plus_nat @ A @ B )
% 2.31/1.50 = ( plus_plus_nat @ A @ C ) )
% 2.31/1.50 = ( B = C ) ) ).
% 2.31/1.50
% 2.31/1.50 thf(fact_26_add__left__cancel,axiom,
% 2.31/1.50 ! [A: real,B: real,C: real] :
% 2.31/1.50 ( ( ( plus_plus_real @ A @ B )
% 2.31/1.50 = ( plus_plus_real @ A @ C ) )
% 2.31/1.50 = ( B = C ) ) ).
% 2.31/1.50
% 2.31/1.50 thf(fact_27_abs__idempotent,axiom,
% 2.31/1.50 ! [A: real] :
% 2.31/1.50 ( ( abs_abs_real @ ( abs_abs_real @ A ) )
% 2.31/1.50 = ( abs_abs_real @ A ) ) ).
% 2.31/1.50
% 2.31/1.50 thf(fact_28_abs__abs,axiom,
% 2.31/1.50 ! [A: real] :
% 2.31/1.50 ( ( abs_abs_real @ ( abs_abs_real @ A ) )
% 2.31/1.50 = ( abs_abs_real @ A ) ) ).
% 2.31/1.50
% 2.31/1.50 thf(fact_29_le__zero__eq,axiom,
% 2.31/1.50 ! [N: nat] :
% 2.31/1.50 ( ( ord_less_eq_nat @ N @ zero_zero_nat )
% 2.31/1.50 = ( N = zero_zero_nat ) ) ).
% 2.31/1.50
% 2.31/1.50 thf(fact_30_add__le__cancel__right,axiom,
% 2.31/1.50 ! [A: real,C: real,B: real] :
% 2.31/1.50 ( ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
% 2.31/1.50 = ( ord_less_eq_real @ A @ B ) ) ).
% 2.31/1.50
% 2.31/1.50 thf(fact_31_add__le__cancel__right,axiom,
% 2.31/1.50 ! [A: nat,C: nat,B: nat] :
% 2.31/1.50 ( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
% 2.31/1.50 = ( ord_less_eq_nat @ A @ B ) ) ).
% 2.31/1.50
% 2.31/1.50 thf(fact_32_add__le__cancel__left,axiom,
% 2.31/1.50 ! [C: real,A: real,B: real] :
% 2.31/1.50 ( ( ord_less_eq_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
% 2.31/1.50 = ( ord_less_eq_real @ A @ B ) ) ).
% 2.31/1.50
% 2.31/1.50 thf(fact_33_add__le__cancel__left,axiom,
% 2.31/1.50 ! [C: nat,A: nat,B: nat] :
% 2.31/1.50 ( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
% 2.31/1.50 = ( ord_less_eq_nat @ A @ B ) ) ).
% 2.31/1.50
% 2.31/1.50 thf(fact_34_zero__eq__add__iff__both__eq__0,axiom,
% 2.31/1.50 ! [X: nat,Y: nat] :
% 2.31/1.50 ( ( zero_zero_nat
% 2.31/1.50 = ( plus_plus_nat @ X @ Y ) )
% 2.31/1.50 = ( ( X = zero_zero_nat )
% 2.31/1.50 & ( Y = zero_zero_nat ) ) ) ).
% 2.31/1.50
% 2.31/1.50 thf(fact_35_add__eq__0__iff__both__eq__0,axiom,
% 2.31/1.50 ! [X: nat,Y: nat] :
% 2.31/1.50 ( ( ( plus_plus_nat @ X @ Y )
% 2.31/1.50 = zero_zero_nat )
% 2.31/1.50 = ( ( X = zero_zero_nat )
% 2.31/1.50 & ( Y = zero_zero_nat ) ) ) ).
% 2.31/1.50
% 2.31/1.50 thf(fact_36_add__cancel__right__right,axiom,
% 2.31/1.50 ! [A: real,B: real] :
% 2.31/1.50 ( ( A
% 2.31/1.50 = ( plus_plus_real @ A @ B ) )
% 2.31/1.50 = ( B = zero_zero_real ) ) ).
% 2.31/1.50
% 2.31/1.50 thf(fact_37_add__cancel__right__right,axiom,
% 2.31/1.50 ! [A: nat,B: nat] :
% 2.31/1.50 ( ( A
% 2.31/1.50 = ( plus_plus_nat @ A @ B ) )
% 2.31/1.50 = ( B = zero_zero_nat ) ) ).
% 2.31/1.50
% 2.31/1.50 thf(fact_38_add__cancel__right__left,axiom,
% 2.31/1.50 ! [A: real,B: real] :
% 2.31/1.50 ( ( A
% 2.31/1.50 = ( plus_plus_real @ B @ A ) )
% 2.31/1.50 = ( B = zero_zero_real ) ) ).
% 2.31/1.50
% 2.31/1.50 thf(fact_39_add__cancel__right__left,axiom,
% 2.31/1.50 ! [A: nat,B: nat] :
% 2.31/1.50 ( ( A
% 2.31/1.50 = ( plus_plus_nat @ B @ A ) )
% 2.31/1.50 = ( B = zero_zero_nat ) ) ).
% 2.31/1.50
% 2.31/1.50 thf(fact_40_add__cancel__left__right,axiom,
% 2.31/1.50 ! [A: real,B: real] :
% 2.31/1.50 ( ( ( plus_plus_real @ A @ B )
% 2.31/1.50 = A )
% 2.31/1.50 = ( B = zero_zero_real ) ) ).
% 2.31/1.50
% 2.31/1.50 thf(fact_41_add__cancel__left__right,axiom,
% 2.31/1.50 ! [A: nat,B: nat] :
% 2.31/1.50 ( ( ( plus_plus_nat @ A @ B )
% 2.31/1.50 = A )
% 2.31/1.50 = ( B = zero_zero_nat ) ) ).
% 2.31/1.50
% 2.31/1.50 thf(fact_42_add__cancel__left__left,axiom,
% 2.31/1.50 ! [B: real,A: real] :
% 2.31/1.50 ( ( ( plus_plus_real @ B @ A )
% 2.31/1.50 = A )
% 2.31/1.50 = ( B = zero_zero_real ) ) ).
% 2.31/1.50
% 2.31/1.50 thf(fact_43_add__cancel__left__left,axiom,
% 2.31/1.50 ! [B: nat,A: nat] :
% 2.31/1.50 ( ( ( plus_plus_nat @ B @ A )
% 2.31/1.50 = A )
% 2.31/1.50 = ( B = zero_zero_nat ) ) ).
% 2.31/1.50
% 2.31/1.50 thf(fact_44_double__zero__sym,axiom,
% 2.31/1.50 ! [A: real] :
% 2.31/1.50 ( ( zero_zero_real
% 2.31/1.50 = ( plus_plus_real @ A @ A ) )
% 2.31/1.50 = ( A = zero_zero_real ) ) ).
% 2.31/1.50
% 2.31/1.50 thf(fact_45_double__zero,axiom,
% 2.31/1.50 ! [A: real] :
% 2.31/1.50 ( ( ( plus_plus_real @ A @ A )
% 2.31/1.50 = zero_zero_real )
% 2.31/1.50 = ( A = zero_zero_real ) ) ).
% 2.31/1.50
% 2.31/1.50 thf(fact_46_add_Oright__neutral,axiom,
% 2.31/1.50 ! [A: real] :
% 2.31/1.50 ( ( plus_plus_real @ A @ zero_zero_real )
% 2.31/1.50 = A ) ).
% 2.31/1.50
% 2.31/1.50 thf(fact_47_add_Oright__neutral,axiom,
% 2.31/1.50 ! [A: nat] :
% 2.31/1.50 ( ( plus_plus_nat @ A @ zero_zero_nat )
% 2.31/1.50 = A ) ).
% 2.31/1.50
% 2.31/1.50 thf(fact_48_add_Oleft__neutral,axiom,
% 2.31/1.50 ! [A: real] :
% 2.31/1.50 ( ( plus_plus_real @ zero_zero_real @ A )
% 2.31/1.50 = A ) ).
% 2.31/1.50
% 2.31/1.50 thf(fact_49_add_Oleft__neutral,axiom,
% 2.31/1.50 ! [A: nat] :
% 2.31/1.50 ( ( plus_plus_nat @ zero_zero_nat @ A )
% 2.31/1.50 = A ) ).
% 2.31/1.50
% 2.31/1.50 thf(fact_50_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
% 2.31/1.50 ! [A: real] :
% 2.31/1.50 ( ( minus_minus_real @ A @ A )
% 2.31/1.50 = zero_zero_real ) ).
% 2.31/1.50
% 2.31/1.50 thf(fact_51_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
% 2.31/1.50 ! [A: nat] :
% 2.31/1.50 ( ( minus_minus_nat @ A @ A )
% 2.31/1.50 = zero_zero_nat ) ).
% 2.31/1.50
% 2.31/1.50 thf(fact_52_diff__zero,axiom,
% 2.31/1.50 ! [A: real] :
% 2.31/1.50 ( ( minus_minus_real @ A @ zero_zero_real )
% 2.31/1.50 = A ) ).
% 2.31/1.50
% 2.31/1.50 thf(fact_53_diff__zero,axiom,
% 2.31/1.50 ! [A: nat] :
% 2.31/1.50 ( ( minus_minus_nat @ A @ zero_zero_nat )
% 2.31/1.50 = A ) ).
% 2.31/1.50
% 2.31/1.50 thf(fact_54_zero__diff,axiom,
% 2.31/1.50 ! [A: nat] :
% 2.31/1.50 ( ( minus_minus_nat @ zero_zero_nat @ A )
% 2.31/1.50 = zero_zero_nat ) ).
% 2.31/1.50
% 2.31/1.50 thf(fact_55_diff__0__right,axiom,
% 2.31/1.50 ! [A: real] :
% 2.31/1.50 ( ( minus_minus_real @ A @ zero_zero_real )
% 2.31/1.50 = A ) ).
% 2.31/1.50
% 2.31/1.50 thf(fact_56_diff__self,axiom,
% 2.31/1.50 ! [A: real] :
% 2.31/1.50 ( ( minus_minus_real @ A @ A )
% 2.31/1.50 = zero_zero_real ) ).
% 2.31/1.50
% 2.31/1.50 thf(fact_57_add__diff__cancel__right_H,axiom,
% 2.31/1.50 ! [A: real,B: real] :
% 2.31/1.50 ( ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ B )
% 2.31/1.50 = A ) ).
% 2.31/1.50
% 2.31/1.50 thf(fact_58_add__diff__cancel__right_H,axiom,
% 2.31/1.50 ! [A: nat,B: nat] :
% 2.31/1.50 ( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ B )
% 2.31/1.50 = A ) ).
% 2.31/1.50
% 2.31/1.50 thf(fact_59_abs__zero,axiom,
% 2.31/1.50 ( ( abs_abs_real @ zero_zero_real )
% 2.31/1.50 = zero_zero_real ) ).
% 2.31/1.50
% 2.31/1.50 thf(fact_60_abs__eq__0,axiom,
% 2.31/1.50 ! [A: real] :
% 2.31/1.50 ( ( ( abs_abs_real @ A )
% 2.31/1.50 = zero_zero_real )
% 2.31/1.50 = ( A = zero_zero_real ) ) ).
% 2.31/1.50
% 2.31/1.50 thf(fact_61_abs__0__eq,axiom,
% 2.31/1.50 ! [A: real] :
% 2.31/1.50 ( ( zero_zero_real
% 2.31/1.50 = ( abs_abs_real @ A ) )
% 2.31/1.50 = ( A = zero_zero_real ) ) ).
% 2.31/1.50
% 2.31/1.50 thf(fact_62_abs__0,axiom,
% 2.31/1.50 ( ( abs_abs_real @ zero_zero_real )
% 2.31/1.50 = zero_zero_real ) ).
% 2.31/1.50
% 2.31/1.50 thf(fact_63_zero__le__double__add__iff__zero__le__single__add,axiom,
% 2.31/1.50 ! [A: real] :
% 2.31/1.50 ( ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ A @ A ) )
% 2.31/1.50 = ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% 2.31/1.50
% 2.31/1.50 thf(fact_64_double__add__le__zero__iff__single__add__le__zero,axiom,
% 2.31/1.50 ! [A: real] :
% 2.31/1.50 ( ( ord_less_eq_real @ ( plus_plus_real @ A @ A ) @ zero_zero_real )
% 2.31/1.50 = ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% 2.31/1.50
% 2.31/1.50 thf(fact_65_le__add__same__cancel2,axiom,
% 2.31/1.50 ! [A: real,B: real] :
% 2.31/1.50 ( ( ord_less_eq_real @ A @ ( plus_plus_real @ B @ A ) )
% 2.31/1.50 = ( ord_less_eq_real @ zero_zero_real @ B ) ) ).
% 2.31/1.50
% 2.31/1.50 thf(fact_66_le__add__same__cancel2,axiom,
% 2.31/1.50 ! [A: nat,B: nat] :
% 2.31/1.50 ( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ B @ A ) )
% 2.31/1.50 = ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% 2.31/1.50
% 2.31/1.50 thf(fact_67_le__add__same__cancel1,axiom,
% 2.31/1.50 ! [A: real,B: real] :
% 2.31/1.50 ( ( ord_less_eq_real @ A @ ( plus_plus_real @ A @ B ) )
% 2.31/1.50 = ( ord_less_eq_real @ zero_zero_real @ B ) ) ).
% 2.31/1.50
% 2.31/1.50 thf(fact_68_le__add__same__cancel1,axiom,
% 2.31/1.50 ! [A: nat,B: nat] :
% 2.31/1.50 ( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ A @ B ) )
% 2.31/1.50 = ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% 2.31/1.50
% 2.31/1.50 thf(fact_69_add__le__same__cancel2,axiom,
% 2.31/1.50 ! [A: real,B: real] :
% 2.31/1.50 ( ( ord_less_eq_real @ ( plus_plus_real @ A @ B ) @ B )
% 2.31/1.50 = ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% 2.31/1.50
% 2.31/1.50 thf(fact_70_add__le__same__cancel2,axiom,
% 2.31/1.50 ! [A: nat,B: nat] :
% 2.31/1.50 ( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ B )
% 2.31/1.50 = ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).
% 2.31/1.50
% 2.31/1.50 thf(fact_71_add__le__same__cancel1,axiom,
% 2.31/1.50 ! [B: real,A: real] :
% 2.31/1.50 ( ( ord_less_eq_real @ ( plus_plus_real @ B @ A ) @ B )
% 2.31/1.50 = ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% 2.31/1.50
% 2.31/1.50 thf(fact_72_add__le__same__cancel1,axiom,
% 2.31/1.50 ! [B: nat,A: nat] :
% 2.31/1.50 ( ( ord_less_eq_nat @ ( plus_plus_nat @ B @ A ) @ B )
% 2.31/1.50 = ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).
% 2.31/1.50
% 2.31/1.50 thf(fact_73_diff__ge__0__iff__ge,axiom,
% 2.31/1.50 ! [A: real,B: real] :
% 2.31/1.50 ( ( ord_less_eq_real @ zero_zero_real @ ( minus_minus_real @ A @ B ) )
% 2.31/1.50 = ( ord_less_eq_real @ B @ A ) ) ).
% 2.31/1.50
% 2.31/1.50 thf(fact_74_diff__add__zero,axiom,
% 2.31/1.50 ! [A: nat,B: nat] :
% 2.31/1.50 ( ( minus_minus_nat @ A @ ( plus_plus_nat @ A @ B ) )
% 2.31/1.50 = zero_zero_nat ) ).
% 2.31/1.50
% 2.31/1.50 thf(fact_75_abs__le__zero__iff,axiom,
% 2.31/1.50 ! [A: real] :
% 2.31/1.50 ( ( ord_less_eq_real @ ( abs_abs_real @ A ) @ zero_zero_real )
% 2.31/1.50 = ( A = zero_zero_real ) ) ).
% 2.31/1.50
% 2.31/1.50 thf(fact_76_abs__le__self__iff,axiom,
% 2.31/1.50 ! [A: real] :
% 2.31/1.50 ( ( ord_less_eq_real @ ( abs_abs_real @ A ) @ A )
% 2.31/1.50 = ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% 2.31/1.50
% 2.31/1.50 thf(fact_77_abs__of__nonneg,axiom,
% 2.31/1.50 ! [A: real] :
% 2.31/1.50 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 2.31/1.50 => ( ( abs_abs_real @ A )
% 2.31/1.50 = A ) ) ).
% 2.31/1.50
% 2.31/1.50 thf(fact_78_zero__reorient,axiom,
% 2.31/1.50 ! [X: real] :
% 2.31/1.50 ( ( zero_zero_real = X )
% 2.31/1.50 = ( X = zero_zero_real ) ) ).
% 2.31/1.50
% 2.31/1.50 thf(fact_79_zero__reorient,axiom,
% 2.31/1.50 ! [X: nat] :
% 2.31/1.50 ( ( zero_zero_nat = X )
% 2.31/1.50 = ( X = zero_zero_nat ) ) ).
% 2.31/1.50
% 2.31/1.50 thf(fact_80_zero__le,axiom,
% 2.31/1.50 ! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).
% 2.31/1.50
% 2.31/1.50 thf(fact_81_add_Ogroup__left__neutral,axiom,
% 2.31/1.50 ! [A: real] :
% 2.31/1.50 ( ( plus_plus_real @ zero_zero_real @ A )
% 2.31/1.50 = A ) ).
% 2.31/1.50
% 2.31/1.50 thf(fact_82_add_Ocomm__neutral,axiom,
% 2.31/1.50 ! [A: real] :
% 2.31/1.50 ( ( plus_plus_real @ A @ zero_zero_real )
% 2.31/1.50 = A ) ).
% 2.31/1.50
% 2.31/1.50 thf(fact_83_add_Ocomm__neutral,axiom,
% 2.31/1.50 ! [A: nat] :
% 2.31/1.50 ( ( plus_plus_nat @ A @ zero_zero_nat )
% 2.31/1.50 = A ) ).
% 2.31/1.50
% 2.31/1.50 thf(fact_84_comm__monoid__add__class_Oadd__0,axiom,
% 2.31/1.50 ! [A: real] :
% 2.31/1.50 ( ( plus_plus_real @ zero_zero_real @ A )
% 2.31/1.50 = A ) ).
% 2.31/1.50
% 2.31/1.50 thf(fact_85_comm__monoid__add__class_Oadd__0,axiom,
% 2.31/1.50 ! [A: nat] :
% 2.31/1.50 ( ( plus_plus_nat @ zero_zero_nat @ A )
% 2.31/1.50 = A ) ).
% 2.31/1.50
% 2.31/1.50 thf(fact_86_zero__neq__one,axiom,
% 2.31/1.50 zero_zero_real != one_one_real ).
% 2.31/1.50
% 2.31/1.50 thf(fact_87_zero__neq__one,axiom,
% 2.31/1.50 zero_zero_nat != one_one_nat ).
% 2.31/1.50
% 2.31/1.50 thf(fact_88_eq__iff__diff__eq__0,axiom,
% 2.31/1.50 ( ( ^ [Y2: real,Z: real] : ( Y2 = Z ) )
% 2.31/1.50 = ( ^ [A2: real,B2: real] :
% 2.31/1.50 ( ( minus_minus_real @ A2 @ B2 )
% 2.31/1.50 = zero_zero_real ) ) ) ).
% 2.31/1.50
% 2.31/1.50 thf(fact_89_abs__eq__0__iff,axiom,
% 2.31/1.50 ! [A: real] :
% 2.31/1.50 ( ( ( abs_abs_real @ A )
% 2.31/1.50 = zero_zero_real )
% 2.31/1.50 = ( A = zero_zero_real ) ) ).
% 2.31/1.50
% 2.31/1.50 thf(fact_90_add__nonpos__eq__0__iff,axiom,
% 2.31/1.50 ! [X: real,Y: real] :
% 2.31/1.50 ( ( ord_less_eq_real @ X @ zero_zero_real )
% 2.31/1.50 => ( ( ord_less_eq_real @ Y @ zero_zero_real )
% 2.31/1.50 => ( ( ( plus_plus_real @ X @ Y )
% 2.31/1.50 = zero_zero_real )
% 2.31/1.50 = ( ( X = zero_zero_real )
% 2.31/1.50 & ( Y = zero_zero_real ) ) ) ) ) ).
% 2.31/1.50
% 2.31/1.50 thf(fact_91_add__nonpos__eq__0__iff,axiom,
% 2.31/1.50 ! [X: nat,Y: nat] :
% 2.31/1.50 ( ( ord_less_eq_nat @ X @ zero_zero_nat )
% 2.31/1.50 => ( ( ord_less_eq_nat @ Y @ zero_zero_nat )
% 2.31/1.50 => ( ( ( plus_plus_nat @ X @ Y )
% 2.31/1.50 = zero_zero_nat )
% 2.31/1.50 = ( ( X = zero_zero_nat )
% 2.31/1.50 & ( Y = zero_zero_nat ) ) ) ) ) ).
% 2.31/1.50
% 2.31/1.50 thf(fact_92_add__nonneg__eq__0__iff,axiom,
% 2.31/1.50 ! [X: real,Y: real] :
% 2.31/1.50 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 2.31/1.50 => ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 2.31/1.50 => ( ( ( plus_plus_real @ X @ Y )
% 2.31/1.50 = zero_zero_real )
% 2.31/1.50 = ( ( X = zero_zero_real )
% 2.31/1.50 & ( Y = zero_zero_real ) ) ) ) ) ).
% 2.31/1.50
% 2.31/1.50 thf(fact_93_add__nonneg__eq__0__iff,axiom,
% 2.31/1.50 ! [X: nat,Y: nat] :
% 2.31/1.50 ( ( ord_less_eq_nat @ zero_zero_nat @ X )
% 2.31/1.50 => ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
% 2.31/1.50 => ( ( ( plus_plus_nat @ X @ Y )
% 2.31/1.50 = zero_zero_nat )
% 2.31/1.50 = ( ( X = zero_zero_nat )
% 2.31/1.50 & ( Y = zero_zero_nat ) ) ) ) ) ).
% 2.31/1.50
% 2.31/1.50 thf(fact_94_add__nonpos__nonpos,axiom,
% 2.31/1.50 ! [A: real,B: real] :
% 2.31/1.50 ( ( ord_less_eq_real @ A @ zero_zero_real )
% 2.31/1.50 => ( ( ord_less_eq_real @ B @ zero_zero_real )
% 2.31/1.50 => ( ord_less_eq_real @ ( plus_plus_real @ A @ B ) @ zero_zero_real ) ) ) ).
% 2.31/1.50
% 2.31/1.50 thf(fact_95_add__nonpos__nonpos,axiom,
% 2.31/1.50 ! [A: nat,B: nat] :
% 2.31/1.50 ( ( ord_less_eq_nat @ A @ zero_zero_nat )
% 2.31/1.50 => ( ( ord_less_eq_nat @ B @ zero_zero_nat )
% 2.31/1.50 => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% 2.31/1.50
% 2.31/1.50 thf(fact_96_add__nonneg__nonneg,axiom,
% 2.31/1.50 ! [A: real,B: real] :
% 2.31/1.50 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 2.31/1.50 => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 2.31/1.50 => ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ A @ B ) ) ) ) ).
% 2.31/1.50
% 2.31/1.50 thf(fact_97_add__nonneg__nonneg,axiom,
% 2.31/1.50 ! [A: nat,B: nat] :
% 2.31/1.50 ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 2.31/1.50 => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 2.31/1.50 => ( ord_less_eq_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% 2.31/1.50
% 2.31/1.50 thf(fact_98_add__increasing2,axiom,
% 2.31/1.50 ! [C: real,B: real,A: real] :
% 2.31/1.50 ( ( ord_less_eq_real @ zero_zero_real @ C )
% 2.31/1.50 => ( ( ord_less_eq_real @ B @ A )
% 2.31/1.50 => ( ord_less_eq_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).
% 2.31/1.50
% 2.31/1.50 thf(fact_99_add__increasing2,axiom,
% 2.31/1.50 ! [C: nat,B: nat,A: nat] :
% 2.31/1.50 ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 2.31/1.50 => ( ( ord_less_eq_nat @ B @ A )
% 2.31/1.50 => ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% 2.53/1.50
% 2.53/1.50 thf(fact_100_add__decreasing2,axiom,
% 2.53/1.50 ! [C: real,A: real,B: real] :
% 2.53/1.50 ( ( ord_less_eq_real @ C @ zero_zero_real )
% 2.53/1.50 => ( ( ord_less_eq_real @ A @ B )
% 2.53/1.50 => ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ B ) ) ) ).
% 2.53/1.50
% 2.53/1.50 thf(fact_101_add__decreasing2,axiom,
% 2.53/1.50 ! [C: nat,A: nat,B: nat] :
% 2.53/1.50 ( ( ord_less_eq_nat @ C @ zero_zero_nat )
% 2.53/1.50 => ( ( ord_less_eq_nat @ A @ B )
% 2.53/1.50 => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ B ) ) ) ).
% 2.53/1.50
% 2.53/1.50 thf(fact_102_add__increasing,axiom,
% 2.53/1.50 ! [A: real,B: real,C: real] :
% 2.53/1.50 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 2.53/1.50 => ( ( ord_less_eq_real @ B @ C )
% 2.53/1.50 => ( ord_less_eq_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).
% 2.53/1.50
% 2.53/1.50 thf(fact_103_add__increasing,axiom,
% 2.53/1.50 ! [A: nat,B: nat,C: nat] :
% 2.53/1.50 ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 2.53/1.50 => ( ( ord_less_eq_nat @ B @ C )
% 2.53/1.50 => ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% 2.53/1.50
% 2.53/1.50 thf(fact_104_add__decreasing,axiom,
% 2.53/1.50 ! [A: real,C: real,B: real] :
% 2.53/1.50 ( ( ord_less_eq_real @ A @ zero_zero_real )
% 2.53/1.50 => ( ( ord_less_eq_real @ C @ B )
% 2.53/1.50 => ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ B ) ) ) ).
% 2.53/1.50
% 2.53/1.50 thf(fact_105_add__decreasing,axiom,
% 2.53/1.50 ! [A: nat,C: nat,B: nat] :
% 2.53/1.50 ( ( ord_less_eq_nat @ A @ zero_zero_nat )
% 2.53/1.50 => ( ( ord_less_eq_nat @ C @ B )
% 2.53/1.50 => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ B ) ) ) ).
% 2.53/1.50
% 2.53/1.50 thf(fact_106_not__one__le__zero,axiom,
% 2.53/1.50 ~ ( ord_less_eq_real @ one_one_real @ zero_zero_real ) ).
% 2.53/1.50
% 2.53/1.50 thf(fact_107_not__one__le__zero,axiom,
% 2.53/1.50 ~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).
% 2.53/1.50
% 2.53/1.50 thf(fact_108_zero__le__one,axiom,
% 2.53/1.50 ord_less_eq_real @ zero_zero_real @ one_one_real ).
% 2.53/1.50
% 2.53/1.50 thf(fact_109_zero__le__one,axiom,
% 2.53/1.50 ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% 2.53/1.50
% 2.53/1.50 thf(fact_110_le__iff__diff__le__0,axiom,
% 2.53/1.50 ( ord_less_eq_real
% 2.53/1.50 = ( ^ [A2: real,B2: real] : ( ord_less_eq_real @ ( minus_minus_real @ A2 @ B2 ) @ zero_zero_real ) ) ) ).
% 2.53/1.50
% 2.53/1.50 thf(fact_111_abs__ge__zero,axiom,
% 2.53/1.50 ! [A: real] : ( ord_less_eq_real @ zero_zero_real @ ( abs_abs_real @ A ) ) ).
% 2.53/1.50
% 2.53/1.50 thf(fact_112_rts2a,axiom,
% 2.53/1.50 ! [P: nat,Q: nat,T: real,I: nat] :
% 2.53/1.50 ( ( ( genClo1015804716orrect @ P @ T )
% 2.53/1.50 & ( genClo1015804716orrect @ Q @ T )
% 2.53/1.50 & ( ord_less_eq_real @ ( plus_plus_real @ ( genClo1163638703lle_te @ Q @ I ) @ genClo1278781456e_beta ) @ T ) )
% 2.53/1.50 => ( ord_less_eq_real @ ( genClo1163638703lle_te @ P @ I ) @ T ) ) ).
% 2.53/1.50
% 2.53/1.50 thf(fact_113_add__right__imp__eq,axiom,
% 2.53/1.50 ! [B: nat,A: nat,C: nat] :
% 2.53/1.50 ( ( ( plus_plus_nat @ B @ A )
% 2.53/1.50 = ( plus_plus_nat @ C @ A ) )
% 2.53/1.50 => ( B = C ) ) ).
% 2.53/1.50
% 2.53/1.50 thf(fact_114_add__right__imp__eq,axiom,
% 2.53/1.50 ! [B: real,A: real,C: real] :
% 2.53/1.50 ( ( ( plus_plus_real @ B @ A )
% 2.53/1.50 = ( plus_plus_real @ C @ A ) )
% 2.53/1.50 => ( B = C ) ) ).
% 2.53/1.50
% 2.53/1.50 thf(fact_115_add__left__imp__eq,axiom,
% 2.53/1.50 ! [A: nat,B: nat,C: nat] :
% 2.53/1.50 ( ( ( plus_plus_nat @ A @ B )
% 2.53/1.50 = ( plus_plus_nat @ A @ C ) )
% 2.53/1.50 => ( B = C ) ) ).
% 2.53/1.50
% 2.53/1.50 thf(fact_116_add__left__imp__eq,axiom,
% 2.53/1.50 ! [A: real,B: real,C: real] :
% 2.53/1.50 ( ( ( plus_plus_real @ A @ B )
% 2.53/1.50 = ( plus_plus_real @ A @ C ) )
% 2.53/1.50 => ( B = C ) ) ).
% 2.53/1.50
% 2.53/1.50 thf(fact_117_add_Oleft__commute,axiom,
% 2.53/1.50 ! [B: nat,A: nat,C: nat] :
% 2.53/1.50 ( ( plus_plus_nat @ B @ ( plus_plus_nat @ A @ C ) )
% 2.53/1.50 = ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% 2.53/1.50
% 2.53/1.50 thf(fact_118_add_Oleft__commute,axiom,
% 2.53/1.50 ! [B: real,A: real,C: real] :
% 2.53/1.50 ( ( plus_plus_real @ B @ ( plus_plus_real @ A @ C ) )
% 2.53/1.50 = ( plus_plus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).
% 2.53/1.50
% 2.53/1.50 thf(fact_119_add_Ocommute,axiom,
% 2.53/1.50 ( plus_plus_nat
% 2.53/1.50 = ( ^ [A2: nat,B2: nat] : ( plus_plus_nat @ B2 @ A2 ) ) ) ).
% 2.53/1.50
% 2.53/1.50 thf(fact_120_add_Ocommute,axiom,
% 2.53/1.50 ( plus_plus_real
% 2.53/1.50 = ( ^ [A2: real,B2: real] : ( plus_plus_real @ B2 @ A2 ) ) ) ).
% 2.53/1.50
% 2.53/1.50 thf(fact_121_add_Oright__cancel,axiom,
% 2.53/1.50 ! [B: real,A: real,C: real] :
% 2.53/1.50 ( ( ( plus_plus_real @ B @ A )
% 2.53/1.50 = ( plus_plus_real @ C @ A ) )
% 2.53/1.50 = ( B = C ) ) ).
% 2.53/1.50
% 2.53/1.50 thf(fact_122_add_Oleft__cancel,axiom,
% 2.53/1.50 ! [A: real,B: real,C: real] :
% 2.53/1.50 ( ( ( plus_plus_real @ A @ B )
% 2.53/1.50 = ( plus_plus_real @ A @ C ) )
% 2.53/1.50 = ( B = C ) ) ).
% 2.53/1.50
% 2.53/1.50 thf(fact_123_add_Oassoc,axiom,
% 2.53/1.50 ! [A: nat,B: nat,C: nat] :
% 2.53/1.50 ( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
% 2.53/1.50 = ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% 2.53/1.50
% 2.53/1.50 thf(fact_124_add_Oassoc,axiom,
% 2.53/1.50 ! [A: real,B: real,C: real] :
% 2.53/1.50 ( ( plus_plus_real @ ( plus_plus_real @ A @ B ) @ C )
% 2.53/1.50 = ( plus_plus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).
% 2.53/1.50
% 2.53/1.50 thf(fact_125_group__cancel_Oadd2,axiom,
% 2.53/1.50 ! [B3: nat,K: nat,B: nat,A: nat] :
% 2.53/1.50 ( ( B3
% 2.53/1.50 = ( plus_plus_nat @ K @ B ) )
% 2.53/1.50 => ( ( plus_plus_nat @ A @ B3 )
% 2.53/1.50 = ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% 2.53/1.50
% 2.53/1.50 thf(fact_126_group__cancel_Oadd2,axiom,
% 2.53/1.50 ! [B3: real,K: real,B: real,A: real] :
% 2.53/1.50 ( ( B3
% 2.53/1.50 = ( plus_plus_real @ K @ B ) )
% 2.53/1.50 => ( ( plus_plus_real @ A @ B3 )
% 2.53/1.50 = ( plus_plus_real @ K @ ( plus_plus_real @ A @ B ) ) ) ) ).
% 2.53/1.50
% 2.53/1.50 thf(fact_127_group__cancel_Oadd1,axiom,
% 2.53/1.50 ! [A3: nat,K: nat,A: nat,B: nat] :
% 2.53/1.50 ( ( A3
% 2.53/1.50 = ( plus_plus_nat @ K @ A ) )
% 2.53/1.50 => ( ( plus_plus_nat @ A3 @ B )
% 2.53/1.50 = ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% 2.53/1.50
% 2.53/1.50 thf(fact_128_group__cancel_Oadd1,axiom,
% 2.53/1.50 ! [A3: real,K: real,A: real,B: real] :
% 2.53/1.50 ( ( A3
% 2.53/1.50 = ( plus_plus_real @ K @ A ) )
% 2.53/1.50 => ( ( plus_plus_real @ A3 @ B )
% 2.53/1.50 = ( plus_plus_real @ K @ ( plus_plus_real @ A @ B ) ) ) ) ).
% 2.53/1.50
% 2.53/1.50 thf(fact_129_add__mono__thms__linordered__semiring_I4_J,axiom,
% 2.53/1.50 ! [I2: nat,J: nat,K: nat,L: nat] :
% 2.53/1.50 ( ( ( I2 = J )
% 2.53/1.50 & ( K = L ) )
% 2.53/1.50 => ( ( plus_plus_nat @ I2 @ K )
% 2.53/1.50 = ( plus_plus_nat @ J @ L ) ) ) ).
% 2.53/1.50
% 2.53/1.50 thf(fact_130_add__mono__thms__linordered__semiring_I4_J,axiom,
% 2.53/1.50 ! [I2: real,J: real,K: real,L: real] :
% 2.53/1.50 ( ( ( I2 = J )
% 2.53/1.50 & ( K = L ) )
% 2.53/1.50 => ( ( plus_plus_real @ I2 @ K )
% 2.53/1.50 = ( plus_plus_real @ J @ L ) ) ) ).
% 2.53/1.50
% 2.53/1.50 thf(fact_131_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
% 2.53/1.50 ! [A: nat,B: nat,C: nat] :
% 2.53/1.50 ( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
% 2.53/1.50 = ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% 2.53/1.50
% 2.53/1.50 thf(fact_132_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
% 2.53/1.50 ! [A: real,B: real,C: real] :
% 2.53/1.50 ( ( plus_plus_real @ ( plus_plus_real @ A @ B ) @ C )
% 2.53/1.50 = ( plus_plus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).
% 2.53/1.50
% 2.53/1.50 thf(fact_133_one__reorient,axiom,
% 2.53/1.50 ! [X: nat] :
% 2.53/1.50 ( ( one_one_nat = X )
% 2.53/1.50 = ( X = one_one_nat ) ) ).
% 2.53/1.50
% 2.53/1.50 thf(fact_134_one__reorient,axiom,
% 2.53/1.50 ! [X: real] :
% 2.53/1.50 ( ( one_one_real = X )
% 2.53/1.50 = ( X = one_one_real ) ) ).
% 2.53/1.50
% 2.53/1.50 thf(fact_135_diff__right__commute,axiom,
% 2.53/1.50 ! [A: real,C: real,B: real] :
% 2.53/1.50 ( ( minus_minus_real @ ( minus_minus_real @ A @ C ) @ B )
% 2.53/1.50 = ( minus_minus_real @ ( minus_minus_real @ A @ B ) @ C ) ) ).
% 2.53/1.50
% 2.53/1.50 thf(fact_136_diff__right__commute,axiom,
% 2.53/1.50 ! [A: nat,C: nat,B: nat] :
% 2.53/1.50 ( ( minus_minus_nat @ ( minus_minus_nat @ A @ C ) @ B )
% 2.53/1.50 = ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C ) ) ).
% 2.53/1.50
% 2.53/1.50 thf(fact_137_diff__eq__diff__eq,axiom,
% 2.53/1.50 ! [A: real,B: real,C: real,D: real] :
% 2.53/1.50 ( ( ( minus_minus_real @ A @ B )
% 2.53/1.50 = ( minus_minus_real @ C @ D ) )
% 2.53/1.50 => ( ( A = B )
% 2.53/1.50 = ( C = D ) ) ) ).
% 2.53/1.50
% 2.53/1.50 thf(fact_138_beta__bound1,axiom,
% 2.53/1.50 ! [P2: nat,I2: nat,Q2: nat] :
% 2.53/1.50 ( ( genClo1015804716orrect @ P2 @ ( genClo1163638703lle_te @ P2 @ ( plus_plus_nat @ I2 @ one_one_nat ) ) )
% 2.53/1.50 => ( ( genClo1015804716orrect @ Q2 @ ( genClo1163638703lle_te @ P2 @ ( plus_plus_nat @ I2 @ one_one_nat ) ) )
% 2.53/1.50 => ( ord_less_eq_real @ zero_zero_real @ ( minus_minus_real @ ( genClo1163638703lle_te @ P2 @ ( plus_plus_nat @ I2 @ one_one_nat ) ) @ ( genClo1163638703lle_te @ Q2 @ I2 ) ) ) ) ) ).
% 2.53/1.50
% 2.53/1.50 thf(fact_139_correct__closed,axiom,
% 2.53/1.50 ! [P: nat,S: real,T: real] :
% 2.53/1.50 ( ( ( ord_less_eq_real @ S @ T )
% 2.53/1.50 & ( genClo1015804716orrect @ P @ T ) )
% 2.53/1.50 => ( genClo1015804716orrect @ P @ S ) ) ).
% 2.53/1.50
% 2.53/1.50 thf(fact_140_add__le__imp__le__right,axiom,
% 2.53/1.50 ! [A: real,C: real,B: real] :
% 2.53/1.50 ( ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
% 2.53/1.50 => ( ord_less_eq_real @ A @ B ) ) ).
% 2.53/1.50
% 2.53/1.50 thf(fact_141_add__le__imp__le__right,axiom,
% 2.53/1.50 ! [A: nat,C: nat,B: nat] :
% 2.53/1.50 ( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
% 2.53/1.50 => ( ord_less_eq_nat @ A @ B ) ) ).
% 2.53/1.50
% 2.53/1.50 thf(fact_142_add__le__imp__le__left,axiom,
% 2.53/1.50 ! [C: real,A: real,B: real] :
% 2.53/1.50 ( ( ord_less_eq_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
% 2.53/1.50 => ( ord_less_eq_real @ A @ B ) ) ).
% 2.53/1.50
% 2.53/1.50 thf(fact_143_add__le__imp__le__left,axiom,
% 2.53/1.50 ! [C: nat,A: nat,B: nat] :
% 2.53/1.50 ( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
% 2.53/1.50 => ( ord_less_eq_nat @ A @ B ) ) ).
% 2.53/1.50
% 2.53/1.50 thf(fact_144_le__iff__add,axiom,
% 2.53/1.50 ( ord_less_eq_nat
% 2.53/1.50 = ( ^ [A2: nat,B2: nat] :
% 2.53/1.50 ? [C2: nat] :
% 2.53/1.50 ( B2
% 2.53/1.50 = ( plus_plus_nat @ A2 @ C2 ) ) ) ) ).
% 2.53/1.50
% 2.53/1.50 thf(fact_145_add__right__mono,axiom,
% 2.53/1.50 ! [A: real,B: real,C: real] :
% 2.53/1.50 ( ( ord_less_eq_real @ A @ B )
% 2.53/1.50 => ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) ) ) ).
% 2.53/1.50
% 2.53/1.50 thf(fact_146_add__right__mono,axiom,
% 2.53/1.50 ! [A: nat,B: nat,C: nat] :
% 2.53/1.50 ( ( ord_less_eq_nat @ A @ B )
% 2.53/1.50 => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).
% 2.53/1.50
% 2.53/1.50 thf(fact_147_less__eqE,axiom,
% 2.53/1.50 ! [A: nat,B: nat] :
% 2.53/1.50 ( ( ord_less_eq_nat @ A @ B )
% 2.53/1.50 => ~ ! [C3: nat] :
% 2.53/1.50 ( B
% 2.53/1.50 != ( plus_plus_nat @ A @ C3 ) ) ) ).
% 2.53/1.50
% 2.53/1.50 thf(fact_148_add__left__mono,axiom,
% 2.53/1.50 ! [A: real,B: real,C: real] :
% 2.53/1.50 ( ( ord_less_eq_real @ A @ B )
% 2.53/1.50 => ( ord_less_eq_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) ) ) ).
% 2.53/1.50
% 2.53/1.50 thf(fact_149_add__left__mono,axiom,
% 2.53/1.50 ! [A: nat,B: nat,C: nat] :
% 2.53/1.50 ( ( ord_less_eq_nat @ A @ B )
% 2.53/1.50 => ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) ) ) ).
% 2.53/1.50
% 2.53/1.50 thf(fact_150_add__mono,axiom,
% 2.53/1.50 ! [A: real,B: real,C: real,D: real] :
% 2.53/1.50 ( ( ord_less_eq_real @ A @ B )
% 2.53/1.50 => ( ( ord_less_eq_real @ C @ D )
% 2.53/1.50 => ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) ) ) ) ).
% 2.53/1.50
% 2.53/1.50 thf(fact_151_add__mono,axiom,
% 2.53/1.50 ! [A: nat,B: nat,C: nat,D: nat] :
% 2.53/1.50 ( ( ord_less_eq_nat @ A @ B )
% 2.53/1.50 => ( ( ord_less_eq_nat @ C @ D )
% 2.53/1.50 => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% 2.53/1.50
% 2.53/1.50 thf(fact_152_add__mono__thms__linordered__semiring_I1_J,axiom,
% 2.53/1.50 ! [I2: real,J: real,K: real,L: real] :
% 2.53/1.50 ( ( ( ord_less_eq_real @ I2 @ J )
% 2.53/1.50 & ( ord_less_eq_real @ K @ L ) )
% 2.53/1.50 => ( ord_less_eq_real @ ( plus_plus_real @ I2 @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% 2.53/1.50
% 2.53/1.50 thf(fact_153_add__mono__thms__linordered__semiring_I1_J,axiom,
% 2.53/1.50 ! [I2: nat,J: nat,K: nat,L: nat] :
% 2.53/1.50 ( ( ( ord_less_eq_nat @ I2 @ J )
% 2.53/1.50 & ( ord_less_eq_nat @ K @ L ) )
% 2.53/1.50 => ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% 2.53/1.50
% 2.53/1.50 thf(fact_154_add__mono__thms__linordered__semiring_I2_J,axiom,
% 2.53/1.50 ! [I2: real,J: real,K: real,L: real] :
% 2.53/1.50 ( ( ( I2 = J )
% 2.53/1.50 & ( ord_less_eq_real @ K @ L ) )
% 2.53/1.50 => ( ord_less_eq_real @ ( plus_plus_real @ I2 @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% 2.53/1.50
% 2.53/1.50 thf(fact_155_add__mono__thms__linordered__semiring_I2_J,axiom,
% 2.53/1.50 ! [I2: nat,J: nat,K: nat,L: nat] :
% 2.53/1.50 ( ( ( I2 = J )
% 2.53/1.50 & ( ord_less_eq_nat @ K @ L ) )
% 2.53/1.50 => ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% 2.53/1.50
% 2.53/1.50 thf(fact_156_add__mono__thms__linordered__semiring_I3_J,axiom,
% 2.53/1.50 ! [I2: real,J: real,K: real,L: real] :
% 2.53/1.50 ( ( ( ord_less_eq_real @ I2 @ J )
% 2.53/1.50 & ( K = L ) )
% 2.53/1.50 => ( ord_less_eq_real @ ( plus_plus_real @ I2 @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% 2.53/1.50
% 2.53/1.50 thf(fact_157_add__mono__thms__linordered__semiring_I3_J,axiom,
% 2.53/1.50 ! [I2: nat,J: nat,K: nat,L: nat] :
% 2.53/1.50 ( ( ( ord_less_eq_nat @ I2 @ J )
% 2.53/1.50 & ( K = L ) )
% 2.53/1.50 => ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% 2.53/1.50
% 2.53/1.50 thf(fact_158_diff__eq__diff__less__eq,axiom,
% 2.53/1.50 ! [A: real,B: real,C: real,D: real] :
% 2.53/1.50 ( ( ( minus_minus_real @ A @ B )
% 2.53/1.50 = ( minus_minus_real @ C @ D ) )
% 2.53/1.50 => ( ( ord_less_eq_real @ A @ B )
% 2.53/1.50 = ( ord_less_eq_real @ C @ D ) ) ) ).
% 2.53/1.50
% 2.53/1.50 thf(fact_159_diff__right__mono,axiom,
% 2.53/1.50 ! [A: real,B: real,C: real] :
% 2.53/1.50 ( ( ord_less_eq_real @ A @ B )
% 2.53/1.50 => ( ord_less_eq_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ C ) ) ) ).
% 2.53/1.50
% 2.53/1.50 thf(fact_160_diff__left__mono,axiom,
% 2.53/1.50 ! [B: real,A: real,C: real] :
% 2.53/1.50 ( ( ord_less_eq_real @ B @ A )
% 2.53/1.50 => ( ord_less_eq_real @ ( minus_minus_real @ C @ A ) @ ( minus_minus_real @ C @ B ) ) ) ).
% 2.53/1.50
% 2.53/1.50 thf(fact_161_diff__mono,axiom,
% 2.53/1.50 ! [A: real,B: real,D: real,C: real] :
% 2.53/1.50 ( ( ord_less_eq_real @ A @ B )
% 2.53/1.50 => ( ( ord_less_eq_real @ D @ C )
% 2.53/1.50 => ( ord_less_eq_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ D ) ) ) ) ).
% 2.53/1.50
% 2.53/1.50 thf(fact_162_add__implies__diff,axiom,
% 2.53/1.50 ! [C: real,B: real,A: real] :
% 2.53/1.50 ( ( ( plus_plus_real @ C @ B )
% 2.53/1.50 = A )
% 2.53/1.50 => ( C
% 2.53/1.50 = ( minus_minus_real @ A @ B ) ) ) ).
% 2.53/1.50
% 2.53/1.50 thf(fact_163_add__implies__diff,axiom,
% 2.53/1.50 ! [C: nat,B: nat,A: nat] :
% 2.53/1.50 ( ( ( plus_plus_nat @ C @ B )
% 2.53/1.50 = A )
% 2.53/1.50 => ( C
% 2.53/1.50 = ( minus_minus_nat @ A @ B ) ) ) ).
% 2.53/1.50
% 2.53/1.50 thf(fact_164_diff__diff__add,axiom,
% 2.53/1.50 ! [A: real,B: real,C: real] :
% 2.53/1.50 ( ( minus_minus_real @ ( minus_minus_real @ A @ B ) @ C )
% 2.53/1.50 = ( minus_minus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).
% 2.53/1.50
% 2.53/1.50 thf(fact_165_diff__diff__add,axiom,
% 2.53/1.50 ! [A: nat,B: nat,C: nat] :
% 2.53/1.50 ( ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C )
% 2.53/1.50 = ( minus_minus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% 2.53/1.50
% 2.53/1.50 thf(fact_166_diff__add__eq__diff__diff__swap,axiom,
% 2.53/1.50 ! [A: real,B: real,C: real] :
% 2.53/1.50 ( ( minus_minus_real @ A @ ( plus_plus_real @ B @ C ) )
% 2.53/1.50 = ( minus_minus_real @ ( minus_minus_real @ A @ C ) @ B ) ) ).
% 2.53/1.50
% 2.53/1.50 thf(fact_167_diff__add__eq,axiom,
% 2.53/1.50 ! [A: real,B: real,C: real] :
% 2.53/1.50 ( ( plus_plus_real @ ( minus_minus_real @ A @ B ) @ C )
% 2.53/1.50 = ( minus_minus_real @ ( plus_plus_real @ A @ C ) @ B ) ) ).
% 2.53/1.50
% 2.53/1.50 thf(fact_168_diff__diff__eq2,axiom,
% 2.53/1.50 ! [A: real,B: real,C: real] :
% 2.53/1.50 ( ( minus_minus_real @ A @ ( minus_minus_real @ B @ C ) )
% 2.53/1.50 = ( minus_minus_real @ ( plus_plus_real @ A @ C ) @ B ) ) ).
% 2.53/1.50
% 2.53/1.50 thf(fact_169_add__diff__eq,axiom,
% 2.53/1.50 ! [A: real,B: real,C: real] :
% 2.53/1.50 ( ( plus_plus_real @ A @ ( minus_minus_real @ B @ C ) )
% 2.53/1.50 = ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ C ) ) ).
% 2.53/1.50
% 2.53/1.50 thf(fact_170_eq__diff__eq,axiom,
% 2.53/1.50 ! [A: real,C: real,B: real] :
% 2.53/1.50 ( ( A
% 2.53/1.50 = ( minus_minus_real @ C @ B ) )
% 2.53/1.50 = ( ( plus_plus_real @ A @ B )
% 2.53/1.50 = C ) ) ).
% 2.53/1.50
% 2.53/1.50 thf(fact_171_diff__eq__eq,axiom,
% 2.53/1.50 ! [A: real,B: real,C: real] :
% 2.53/1.50 ( ( ( minus_minus_real @ A @ B )
% 2.53/1.50 = C )
% 2.53/1.50 = ( A
% 2.53/1.50 = ( plus_plus_real @ C @ B ) ) ) ).
% 2.53/1.50
% 2.53/1.50 thf(fact_172_group__cancel_Osub1,axiom,
% 2.53/1.50 ! [A3: real,K: real,A: real,B: real] :
% 2.53/1.50 ( ( A3
% 2.53/1.50 = ( plus_plus_real @ K @ A ) )
% 2.53/1.50 => ( ( minus_minus_real @ A3 @ B )
% 2.53/1.50 = ( plus_plus_real @ K @ ( minus_minus_real @ A @ B ) ) ) ) ).
% 2.53/1.50
% 2.53/1.50 thf(fact_173_abs__ge__self,axiom,
% 2.53/1.50 ! [A: real] : ( ord_less_eq_real @ A @ ( abs_abs_real @ A ) ) ).
% 2.53/1.50
% 2.53/1.50 thf(fact_174_abs__le__D1,axiom,
% 2.53/1.50 ! [A: real,B: real] :
% 2.53/1.50 ( ( ord_less_eq_real @ ( abs_abs_real @ A ) @ B )
% 2.53/1.50 => ( ord_less_eq_real @ A @ B ) ) ).
% 2.53/1.50
% 2.53/1.50 thf(fact_175_abs__one,axiom,
% 2.53/1.50 ( ( abs_abs_real @ one_one_real )
% 2.53/1.50 = one_one_real ) ).
% 2.53/1.50
% 2.53/1.50 thf(fact_176_abs__minus__commute,axiom,
% 2.53/1.50 ! [A: real,B: real] :
% 2.53/1.50 ( ( abs_abs_real @ ( minus_minus_real @ A @ B ) )
% 2.53/1.50 = ( abs_abs_real @ ( minus_minus_real @ B @ A ) ) ) ).
% 2.53/1.50
% 2.53/1.50 thf(fact_177_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
% 2.53/1.50 ! [A: nat,B: nat,C: nat] :
% 2.53/1.50 ( ( ord_less_eq_nat @ A @ B )
% 2.53/1.50 => ( ( ord_less_eq_nat @ A @ B )
% 2.53/1.50 => ( ( ( minus_minus_nat @ B @ A )
% 2.53/1.50 = C )
% 2.53/1.50 = ( B
% 2.53/1.50 = ( plus_plus_nat @ C @ A ) ) ) ) ) ).
% 2.53/1.50
% 2.53/1.50 thf(fact_178_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
% 2.53/1.50 ! [A: nat,B: nat] :
% 2.53/1.50 ( ( ord_less_eq_nat @ A @ B )
% 2.53/1.50 => ( ( plus_plus_nat @ A @ ( minus_minus_nat @ B @ A ) )
% 2.53/1.50 = B ) ) ).
% 2.53/1.50
% 2.53/1.50 thf(fact_179_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
% 2.53/1.50 ! [A: nat,B: nat,C: nat] :
% 2.53/1.50 ( ( ord_less_eq_nat @ A @ B )
% 2.53/1.50 => ( ( minus_minus_nat @ C @ ( minus_minus_nat @ B @ A ) )
% 2.53/1.50 = ( minus_minus_nat @ ( plus_plus_nat @ C @ A ) @ B ) ) ) ).
% 2.53/1.50
% 2.53/1.50 thf(fact_180_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
% 2.53/1.50 ! [A: nat,B: nat,C: nat] :
% 2.53/1.50 ( ( ord_less_eq_nat @ A @ B )
% 2.53/1.50 => ( ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A )
% 2.53/1.50 = ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ C ) ) ) ).
% 2.53/1.50
% 2.53/1.50 thf(fact_181_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
% 2.53/1.50 ! [A: nat,B: nat,C: nat] :
% 2.53/1.50 ( ( ord_less_eq_nat @ A @ B )
% 2.53/1.50 => ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ C )
% 2.53/1.50 = ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A ) ) ) ).
% 2.53/1.50
% 2.53/1.50 thf(fact_182_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
% 2.53/1.50 ! [A: nat,B: nat,C: nat] :
% 2.53/1.50 ( ( ord_less_eq_nat @ A @ B )
% 2.53/1.50 => ( ( minus_minus_nat @ ( plus_plus_nat @ C @ B ) @ A )
% 2.53/1.50 = ( plus_plus_nat @ C @ ( minus_minus_nat @ B @ A ) ) ) ) ).
% 2.53/1.50
% 2.53/1.50 thf(fact_183_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
% 2.53/1.50 ! [A: nat,B: nat,C: nat] :
% 2.53/1.50 ( ( ord_less_eq_nat @ A @ B )
% 2.53/1.50 => ( ( plus_plus_nat @ C @ ( minus_minus_nat @ B @ A ) )
% 2.53/1.50 = ( minus_minus_nat @ ( plus_plus_nat @ C @ B ) @ A ) ) ) ).
% 2.53/1.50
% 2.53/1.50 thf(fact_184_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
% 2.53/1.50 ! [A: nat,B: nat,C: nat] :
% 2.53/1.50 ( ( ord_less_eq_nat @ A @ B )
% 2.53/1.50 => ( ( ord_less_eq_nat @ C @ ( minus_minus_nat @ B @ A ) )
% 2.53/1.50 = ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ B ) ) ) ).
% 2.53/1.50
% 2.53/1.50 thf(fact_185_le__add__diff,axiom,
% 2.53/1.50 ! [A: nat,B: nat,C: nat] :
% 2.53/1.50 ( ( ord_less_eq_nat @ A @ B )
% 2.53/1.50 => ( ord_less_eq_nat @ C @ ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A ) ) ) ).
% 2.53/1.50
% 2.53/1.50 thf(fact_186_add__le__add__imp__diff__le,axiom,
% 2.53/1.50 ! [I2: real,K: real,N: real,J: real] :
% 2.53/1.50 ( ( ord_less_eq_real @ ( plus_plus_real @ I2 @ K ) @ N )
% 2.53/1.50 => ( ( ord_less_eq_real @ N @ ( plus_plus_real @ J @ K ) )
% 2.53/1.50 => ( ( ord_less_eq_real @ ( plus_plus_real @ I2 @ K ) @ N )
% 2.53/1.50 => ( ( ord_less_eq_real @ N @ ( plus_plus_real @ J @ K ) )
% 2.53/1.50 => ( ord_less_eq_real @ ( minus_minus_real @ N @ K ) @ J ) ) ) ) ) ).
% 2.53/1.50
% 2.53/1.50 thf(fact_187_add__le__add__imp__diff__le,axiom,
% 2.53/1.50 ! [I2: nat,K: nat,N: nat,J: nat] :
% 2.53/1.50 ( ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ N )
% 2.53/1.50 => ( ( ord_less_eq_nat @ N @ ( plus_plus_nat @ J @ K ) )
% 2.53/1.50 => ( ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ N )
% 2.53/1.50 => ( ( ord_less_eq_nat @ N @ ( plus_plus_nat @ J @ K ) )
% 2.53/1.50 => ( ord_less_eq_nat @ ( minus_minus_nat @ N @ K ) @ J ) ) ) ) ) ).
% 2.53/1.50
% 2.53/1.50 thf(fact_188_diff__add,axiom,
% 2.53/1.50 ! [A: nat,B: nat] :
% 2.53/1.50 ( ( ord_less_eq_nat @ A @ B )
% 2.53/1.50 => ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ A )
% 2.53/1.50 = B ) ) ).
% 2.53/1.50
% 2.53/1.50 thf(fact_189_add__le__imp__le__diff,axiom,
% 2.53/1.50 ! [I2: real,K: real,N: real] :
% 2.53/1.50 ( ( ord_less_eq_real @ ( plus_plus_real @ I2 @ K ) @ N )
% 2.53/1.50 => ( ord_less_eq_real @ I2 @ ( minus_minus_real @ N @ K ) ) ) ).
% 2.53/1.50
% 2.53/1.50 thf(fact_190_add__le__imp__le__diff,axiom,
% 2.53/1.50 ! [I2: nat,K: nat,N: nat] :
% 2.53/1.50 ( ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ N )
% 2.53/1.50 => ( ord_less_eq_nat @ I2 @ ( minus_minus_nat @ N @ K ) ) ) ).
% 2.53/1.50
% 2.53/1.50 thf(fact_191_le__diff__eq,axiom,
% 2.53/1.50 ! [A: real,C: real,B: real] :
% 2.53/1.50 ( ( ord_less_eq_real @ A @ ( minus_minus_real @ C @ B ) )
% 2.53/1.50 = ( ord_less_eq_real @ ( plus_plus_real @ A @ B ) @ C ) ) ).
% 2.53/1.50
% 2.53/1.50 thf(fact_192_diff__le__eq,axiom,
% 2.53/1.50 ! [A: real,B: real,C: real] :
% 2.53/1.50 ( ( ord_less_eq_real @ ( minus_minus_real @ A @ B ) @ C )
% 2.53/1.50 = ( ord_less_eq_real @ A @ ( plus_plus_real @ C @ B ) ) ) ).
% 2.53/1.50
% 2.53/1.50 thf(fact_193_abs__triangle__ineq,axiom,
% 2.53/1.50 ! [A: real,B: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( plus_plus_real @ A @ B ) ) @ ( plus_plus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) ) ).
% 2.53/1.50
% 2.53/1.50 thf(fact_194_abs__triangle__ineq2__sym,axiom,
% 2.53/1.50 ! [A: real,B: real] : ( ord_less_eq_real @ ( minus_minus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) @ ( abs_abs_real @ ( minus_minus_real @ B @ A ) ) ) ).
% 2.53/1.50
% 2.53/1.50 thf(fact_195_abs__triangle__ineq3,axiom,
% 2.53/1.50 ! [A: real,B: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) ) @ ( abs_abs_real @ ( minus_minus_real @ A @ B ) ) ) ).
% 2.53/1.50
% 2.53/1.50 thf(fact_196_abs__triangle__ineq2,axiom,
% 2.53/1.50 ! [A: real,B: real] : ( ord_less_eq_real @ ( minus_minus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) @ ( abs_abs_real @ ( minus_minus_real @ A @ B ) ) ) ).
% 2.53/1.50
% 2.53/1.50 thf(fact_197_abs__diff__triangle__ineq,axiom,
% 2.53/1.50 ! [A: real,B: real,C: real,D: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ ( plus_plus_real @ C @ D ) ) ) @ ( plus_plus_real @ ( abs_abs_real @ ( minus_minus_real @ A @ C ) ) @ ( abs_abs_real @ ( minus_minus_real @ B @ D ) ) ) ) ).
% 2.53/1.50
% 2.53/1.50 thf(fact_198_abs__triangle__ineq4,axiom,
% 2.53/1.50 ! [A: real,B: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ A @ B ) ) @ ( plus_plus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) ) ).
% 2.53/1.50
% 2.53/1.50 thf(fact_199_abs__diff__le__iff,axiom,
% 2.53/1.50 ! [X: real,A: real,R: real] :
% 2.53/1.50 ( ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ X @ A ) ) @ R )
% 2.53/1.50 = ( ( ord_less_eq_real @ ( minus_minus_real @ A @ R ) @ X )
% 2.53/1.50 & ( ord_less_eq_real @ X @ ( plus_plus_real @ A @ R ) ) ) ) ).
% 2.53/1.50
% 2.53/1.50 thf(fact_200_posX,axiom,
% 2.53/1.50 ord_less_eq_real @ zero_zero_real @ ( minus_minus_real @ ( genClo1161277105lle_PC @ l @ ( genClo1163638703lle_te @ p @ ( plus_plus_nat @ i @ one_one_nat ) ) ) @ ( genClo1161277105lle_PC @ l @ ( genClo1163638703lle_te @ q @ ( plus_plus_nat @ i @ one_one_nat ) ) ) ) ).
% 2.53/1.50
% 2.53/1.50 thf(fact_201_diff__numeral__special_I9_J,axiom,
% 2.53/1.50 ( ( minus_minus_real @ one_one_real @ one_one_real )
% 2.53/1.50 = zero_zero_real ) ).
% 2.53/1.50
% 2.53/1.50 thf(fact_202_beta__bound2,axiom,
% 2.53/1.50 ! [P2: nat,I2: nat,Q2: nat] :
% 2.53/1.50 ( ( genClo1015804716orrect @ P2 @ ( genClo1163638703lle_te @ P2 @ ( plus_plus_nat @ I2 @ one_one_nat ) ) )
% 2.53/1.50 => ( ( genClo1015804716orrect @ Q2 @ ( genClo1163638703lle_te @ Q2 @ I2 ) )
% 2.53/1.50 => ( ord_less_eq_real @ ( minus_minus_real @ ( genClo1163638703lle_te @ P2 @ ( plus_plus_nat @ I2 @ one_one_nat ) ) @ ( genClo1163638703lle_te @ Q2 @ I2 ) ) @ ( plus_plus_real @ genClo1650508560e_rmax @ genClo1278781456e_beta ) ) ) ) ).
% 2.53/1.50
% 2.53/1.50 thf(fact_203_rts1d,axiom,
% 2.53/1.50 ! [P2: nat,I2: nat] :
% 2.53/1.50 ( ( genClo1015804716orrect @ P2 @ ( genClo1163638703lle_te @ P2 @ ( plus_plus_nat @ I2 @ one_one_nat ) ) )
% 2.53/1.50 => ( ord_less_eq_real @ genClo1651033342e_rmin @ ( minus_minus_real @ ( genClo1163638703lle_te @ P2 @ ( plus_plus_nat @ I2 @ one_one_nat ) ) @ ( genClo1163638703lle_te @ P2 @ I2 ) ) ) ) ).
% 2.53/1.50
% 2.53/1.50 thf(fact_204_rts1,axiom,
% 2.53/1.50 ! [P: nat,T: real,I: nat] :
% 2.53/1.50 ( ( ( genClo1015804716orrect @ P @ T )
% 2.53/1.50 & ( ord_less_eq_real @ ( genClo1163638703lle_te @ P @ ( plus_plus_nat @ I @ one_one_nat ) ) @ T ) )
% 2.53/1.50 => ( ord_less_eq_real @ genClo1651033342e_rmin @ ( minus_minus_real @ T @ ( genClo1163638703lle_te @ P @ I ) ) ) ) ).
% 2.53/1.50
% 2.53/1.50 thf(fact_205_rts0d,axiom,
% 2.53/1.50 ! [P2: nat,I2: nat] :
% 2.53/1.50 ( ( genClo1015804716orrect @ P2 @ ( genClo1163638703lle_te @ P2 @ ( plus_plus_nat @ I2 @ one_one_nat ) ) )
% 2.53/1.50 => ( ord_less_eq_real @ ( minus_minus_real @ ( genClo1163638703lle_te @ P2 @ ( plus_plus_nat @ I2 @ one_one_nat ) ) @ ( genClo1163638703lle_te @ P2 @ I2 ) ) @ genClo1650508560e_rmax ) ) ).
% 2.53/1.50
% 2.53/1.50 thf(fact_206_rts0,axiom,
% 2.53/1.50 ! [P: nat,T: real,I: nat] :
% 2.53/1.50 ( ( ( genClo1015804716orrect @ P @ T )
% 2.53/1.50 & ( ord_less_eq_real @ T @ ( genClo1163638703lle_te @ P @ ( plus_plus_nat @ I @ one_one_nat ) ) ) )
% 2.53/1.50 => ( ord_less_eq_real @ ( minus_minus_real @ T @ ( genClo1163638703lle_te @ P @ I ) ) @ genClo1650508560e_rmax ) ) ).
% 2.53/1.50
% 2.53/1.50 thf(fact_207_okRead2__def,axiom,
% 2.53/1.50 ( genClo293725282kRead2
% 2.53/1.50 = ( ^ [F: nat > real,G: nat > real,X2: real,Ppred: nat > $o] :
% 2.53/1.50 ! [P3: nat] :
% 2.53/1.50 ( ( Ppred @ P3 )
% 2.53/1.50 => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( F @ P3 ) @ ( G @ P3 ) ) ) @ X2 ) ) ) ) ).
% 2.53/1.50
% 2.53/1.50 thf(fact_208_sin__bound__lemma,axiom,
% 2.53/1.50 ! [X: real,Y: real,U: real,V: real] :
% 2.53/1.50 ( ( X = Y )
% 2.53/1.50 => ( ( ord_less_eq_real @ ( abs_abs_real @ U ) @ V )
% 2.53/1.50 => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( plus_plus_real @ X @ U ) @ Y ) ) @ V ) ) ) ).
% 2.53/1.50
% 2.53/1.50 thf(fact_209_Nat_Oadd__diff__assoc,axiom,
% 2.53/1.50 ! [K: nat,J: nat,I2: nat] :
% 2.53/1.50 ( ( ord_less_eq_nat @ K @ J )
% 2.53/1.50 => ( ( plus_plus_nat @ I2 @ ( minus_minus_nat @ J @ K ) )
% 2.53/1.50 = ( minus_minus_nat @ ( plus_plus_nat @ I2 @ J ) @ K ) ) ) ).
% 2.53/1.50
% 2.53/1.50 thf(fact_210_le0,axiom,
% 2.53/1.50 ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% 2.53/1.50
% 2.53/1.50 thf(fact_211_bot__nat__0_Oextremum,axiom,
% 2.53/1.50 ! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% 2.53/1.50
% 2.53/1.50 thf(fact_212_add__is__0,axiom,
% 2.53/1.50 ! [M: nat,N: nat] :
% 2.53/1.50 ( ( ( plus_plus_nat @ M @ N )
% 2.53/1.50 = zero_zero_nat )
% 2.53/1.50 = ( ( M = zero_zero_nat )
% 2.53/1.50 & ( N = zero_zero_nat ) ) ) ).
% 2.53/1.50
% 2.53/1.50 thf(fact_213_Nat_Oadd__0__right,axiom,
% 2.53/1.50 ! [M: nat] :
% 2.53/1.50 ( ( plus_plus_nat @ M @ zero_zero_nat )
% 2.53/1.50 = M ) ).
% 2.53/1.50
% 2.53/1.50 thf(fact_214_diff__0__eq__0,axiom,
% 2.53/1.50 ! [N: nat] :
% 2.53/1.50 ( ( minus_minus_nat @ zero_zero_nat @ N )
% 2.53/1.50 = zero_zero_nat ) ).
% 2.53/1.50
% 2.53/1.50 thf(fact_215_diff__self__eq__0,axiom,
% 2.53/1.50 ! [M: nat] :
% 2.53/1.50 ( ( minus_minus_nat @ M @ M )
% 2.53/1.50 = zero_zero_nat ) ).
% 2.53/1.50
% 2.53/1.50 thf(fact_216_nat__add__left__cancel__le,axiom,
% 2.53/1.50 ! [K: nat,M: nat,N: nat] :
% 2.53/1.50 ( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
% 2.53/1.50 = ( ord_less_eq_nat @ M @ N ) ) ).
% 2.53/1.50
% 2.53/1.50 thf(fact_217_diff__is__0__eq,axiom,
% 2.53/1.50 ! [M: nat,N: nat] :
% 2.53/1.50 ( ( ( minus_minus_nat @ M @ N )
% 2.53/1.50 = zero_zero_nat )
% 2.53/1.50 = ( ord_less_eq_nat @ M @ N ) ) ).
% 2.53/1.50
% 2.53/1.50 thf(fact_218_diff__is__0__eq_H,axiom,
% 2.53/1.50 ! [M: nat,N: nat] :
% 2.53/1.50 ( ( ord_less_eq_nat @ M @ N )
% 2.53/1.50 => ( ( minus_minus_nat @ M @ N )
% 2.53/1.50 = zero_zero_nat ) ) ).
% 2.53/1.50
% 2.53/1.50 thf(fact_219_diff__diff__cancel,axiom,
% 2.53/1.50 ! [I2: nat,N: nat] :
% 2.53/1.50 ( ( ord_less_eq_nat @ I2 @ N )
% 2.53/1.50 => ( ( minus_minus_nat @ N @ ( minus_minus_nat @ N @ I2 ) )
% 2.53/1.50 = I2 ) ) ).
% 2.53/1.50
% 2.53/1.50 thf(fact_220_diff__diff__left,axiom,
% 2.53/1.50 ! [I2: nat,J: nat,K: nat] :
% 2.53/1.50 ( ( minus_minus_nat @ ( minus_minus_nat @ I2 @ J ) @ K )
% 2.53/1.50 = ( minus_minus_nat @ I2 @ ( plus_plus_nat @ J @ K ) ) ) ).
% 2.53/1.50
% 2.53/1.50 thf(fact_221_Nat_Odiff__diff__right,axiom,
% 2.53/1.50 ! [K: nat,J: nat,I2: nat] :
% 2.53/1.50 ( ( ord_less_eq_nat @ K @ J )
% 2.53/1.50 => ( ( minus_minus_nat @ I2 @ ( minus_minus_nat @ J @ K ) )
% 2.53/1.50 = ( minus_minus_nat @ ( plus_plus_nat @ I2 @ K ) @ J ) ) ) ).
% 2.53/1.50
% 2.53/1.50 thf(fact_222_Nat_Oadd__diff__assoc2,axiom,
% 2.53/1.50 ! [K: nat,J: nat,I2: nat] :
% 2.53/1.50 ( ( ord_less_eq_nat @ K @ J )
% 2.53/1.50 => ( ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I2 )
% 2.53/1.50 = ( minus_minus_nat @ ( plus_plus_nat @ J @ I2 ) @ K ) ) ) ).
% 2.53/1.50
% 2.53/1.50 thf(fact_223_less__eq__nat_Osimps_I1_J,axiom,
% 2.53/1.50 ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% 2.53/1.50
% 2.53/1.50 thf(fact_224_le__0__eq,axiom,
% 2.53/1.50 ! [N: nat] :
% 2.53/1.50 ( ( ord_less_eq_nat @ N @ zero_zero_nat )
% 2.53/1.50 = ( N = zero_zero_nat ) ) ).
% 2.53/1.50
% 2.53/1.50 thf(fact_225_le__refl,axiom,
% 2.53/1.50 ! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% 2.53/1.50
% 2.53/1.50 thf(fact_226_le__trans,axiom,
% 2.53/1.50 ! [I2: nat,J: nat,K: nat] :
% 2.53/1.50 ( ( ord_less_eq_nat @ I2 @ J )
% 2.53/1.50 => ( ( ord_less_eq_nat @ J @ K )
% 2.53/1.50 => ( ord_less_eq_nat @ I2 @ K ) ) ) ).
% 2.53/1.50
% 2.53/1.50 thf(fact_227_eq__imp__le,axiom,
% 2.53/1.50 ! [M: nat,N: nat] :
% 2.53/1.50 ( ( M = N )
% 2.53/1.50 => ( ord_less_eq_nat @ M @ N ) ) ).
% 2.53/1.50
% 2.53/1.50 thf(fact_228_le__antisym,axiom,
% 2.53/1.50 ! [M: nat,N: nat] :
% 2.53/1.50 ( ( ord_less_eq_nat @ M @ N )
% 2.53/1.50 => ( ( ord_less_eq_nat @ N @ M )
% 2.53/1.50 => ( M = N ) ) ) ).
% 2.53/1.50
% 2.53/1.50 thf(fact_229_eq__diff__iff,axiom,
% 2.53/1.50 ! [K: nat,M: nat,N: nat] :
% 2.53/1.50 ( ( ord_less_eq_nat @ K @ M )
% 2.53/1.50 => ( ( ord_less_eq_nat @ K @ N )
% 2.53/1.50 => ( ( ( minus_minus_nat @ M @ K )
% 2.53/1.50 = ( minus_minus_nat @ N @ K ) )
% 2.53/1.50 = ( M = N ) ) ) ) ).
% 2.53/1.50
% 2.53/1.50 thf(fact_230_le__diff__iff,axiom,
% 2.53/1.50 ! [K: nat,M: nat,N: nat] :
% 2.53/1.50 ( ( ord_less_eq_nat @ K @ M )
% 2.53/1.50 => ( ( ord_less_eq_nat @ K @ N )
% 2.53/1.50 => ( ( ord_less_eq_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
% 2.53/1.50 = ( ord_less_eq_nat @ M @ N ) ) ) ) ).
% 2.53/1.50
% 2.53/1.50 thf(fact_231_diff__commute,axiom,
% 2.53/1.50 ! [I2: nat,J: nat,K: nat] :
% 2.53/1.50 ( ( minus_minus_nat @ ( minus_minus_nat @ I2 @ J ) @ K )
% 2.53/1.50 = ( minus_minus_nat @ ( minus_minus_nat @ I2 @ K ) @ J ) ) ).
% 2.53/1.50
% 2.53/1.50 thf(fact_232_Nat_Odiff__diff__eq,axiom,
% 2.53/1.50 ! [K: nat,M: nat,N: nat] :
% 2.53/1.50 ( ( ord_less_eq_nat @ K @ M )
% 2.53/1.50 => ( ( ord_less_eq_nat @ K @ N )
% 2.53/1.50 => ( ( minus_minus_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
% 2.53/1.50 = ( minus_minus_nat @ M @ N ) ) ) ) ).
% 2.53/1.50
% 2.53/1.50 thf(fact_233_diff__le__mono,axiom,
% 2.53/1.50 ! [M: nat,N: nat,L: nat] :
% 2.53/1.50 ( ( ord_less_eq_nat @ M @ N )
% 2.53/1.50 => ( ord_less_eq_nat @ ( minus_minus_nat @ M @ L ) @ ( minus_minus_nat @ N @ L ) ) ) ).
% 2.53/1.50
% 2.53/1.50 thf(fact_234_diff__le__self,axiom,
% 2.53/1.50 ! [M: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ M ) ).
% 2.53/1.50
% 2.53/1.50 thf(fact_235_le__diff__iff_H,axiom,
% 2.53/1.50 ! [A: nat,C: nat,B: nat] :
% 2.53/1.50 ( ( ord_less_eq_nat @ A @ C )
% 2.53/1.50 => ( ( ord_less_eq_nat @ B @ C )
% 2.53/1.50 => ( ( ord_less_eq_nat @ ( minus_minus_nat @ C @ A ) @ ( minus_minus_nat @ C @ B ) )
% 2.53/1.50 = ( ord_less_eq_nat @ B @ A ) ) ) ) ).
% 2.53/1.50
% 2.53/1.50 thf(fact_236_diff__le__mono2,axiom,
% 2.53/1.50 ! [M: nat,N: nat,L: nat] :
% 2.53/1.50 ( ( ord_less_eq_nat @ M @ N )
% 2.53/1.50 => ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ).
% 2.53/1.50
% 2.53/1.50 thf(fact_237_nat__le__linear,axiom,
% 2.53/1.50 ! [M: nat,N: nat] :
% 2.53/1.50 ( ( ord_less_eq_nat @ M @ N )
% 2.53/1.50 | ( ord_less_eq_nat @ N @ M ) ) ).
% 2.53/1.50
% 2.53/1.50 thf(fact_238_diffs0__imp__equal,axiom,
% 2.53/1.50 ! [M: nat,N: nat] :
% 2.53/1.50 ( ( ( minus_minus_nat @ M @ N )
% 2.53/1.50 = zero_zero_nat )
% 2.53/1.50 => ( ( ( minus_minus_nat @ N @ M )
% 2.53/1.50 = zero_zero_nat )
% 2.53/1.50 => ( M = N ) ) ) ).
% 2.53/1.50
% 2.53/1.50 thf(fact_239_minus__nat_Odiff__0,axiom,
% 2.53/1.50 ! [M: nat] :
% 2.53/1.50 ( ( minus_minus_nat @ M @ zero_zero_nat )
% 2.53/1.50 = M ) ).
% 2.53/1.50
% 2.53/1.50 thf(fact_240_Nat_Oex__has__greatest__nat,axiom,
% 2.53/1.50 ! [P4: nat > $o,K: nat,B: nat] :
% 2.53/1.50 ( ( P4 @ K )
% 2.53/1.50 => ( ! [Y3: nat] :
% 2.53/1.50 ( ( P4 @ Y3 )
% 2.53/1.50 => ( ord_less_eq_nat @ Y3 @ B ) )
% 2.53/1.50 => ? [X3: nat] :
% 2.53/1.50 ( ( P4 @ X3 )
% 2.53/1.50 & ! [Y4: nat] :
% 2.53/1.50 ( ( P4 @ Y4 )
% 2.53/1.50 => ( ord_less_eq_nat @ Y4 @ X3 ) ) ) ) ) ).
% 2.53/1.50
% 2.53/1.50 thf(fact_241_bot__nat__0_Oextremum__unique,axiom,
% 2.53/1.50 ! [A: nat] :
% 2.53/1.50 ( ( ord_less_eq_nat @ A @ zero_zero_nat )
% 2.53/1.50 = ( A = zero_zero_nat ) ) ).
% 2.53/1.50
% 2.53/1.50 thf(fact_242_bot__nat__0_Oextremum__uniqueI,axiom,
% 2.53/1.50 ! [A: nat] :
% 2.53/1.50 ( ( ord_less_eq_nat @ A @ zero_zero_nat )
% 2.53/1.50 => ( A = zero_zero_nat ) ) ).
% 2.53/1.50
% 2.53/1.50 thf(fact_243_plus__nat_Oadd__0,axiom,
% 2.53/1.50 ! [N: nat] :
% 2.53/1.50 ( ( plus_plus_nat @ zero_zero_nat @ N )
% 2.53/1.50 = N ) ).
% 2.53/1.50
% 2.53/1.50 thf(fact_244_add__eq__self__zero,axiom,
% 2.53/1.50 ! [M: nat,N: nat] :
% 2.53/1.50 ( ( ( plus_plus_nat @ M @ N )
% 2.53/1.50 = M )
% 2.53/1.50 => ( N = zero_zero_nat ) ) ).
% 2.53/1.50
% 2.53/1.50 thf(fact_245_diff__add__0,axiom,
% 2.53/1.50 ! [N: nat,M: nat] :
% 2.53/1.50 ( ( minus_minus_nat @ N @ ( plus_plus_nat @ N @ M ) )
% 2.53/1.50 = zero_zero_nat ) ).
% 2.53/1.50
% 2.53/1.50 thf(fact_246_synch0,axiom,
% 2.53/1.50 ! [P: nat] :
% 2.53/1.50 ( ( genClo1163638703lle_te @ P @ zero_zero_nat )
% 2.53/1.50 = zero_zero_real ) ).
% 2.53/1.50
% 2.53/1.50 thf(fact_247_is__num__normalize_I1_J,axiom,
% 2.53/1.50 ! [A: real,B: real,C: real] :
% 2.53/1.50 ( ( plus_plus_real @ ( plus_plus_real @ A @ B ) @ C )
% 2.53/1.50 = ( plus_plus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).
% 2.53/1.50
% 2.53/1.50 thf(fact_248_add__leE,axiom,
% 2.53/1.50 ! [M: nat,K: nat,N: nat] :
% 2.53/1.50 ( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
% 2.53/1.50 => ~ ( ( ord_less_eq_nat @ M @ N )
% 2.53/1.50 => ~ ( ord_less_eq_nat @ K @ N ) ) ) ).
% 2.53/1.50
% 2.53/1.50 thf(fact_249_le__add1,axiom,
% 2.53/1.50 ! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M ) ) ).
% 2.53/1.50
% 2.53/1.50 thf(fact_250_le__add2,axiom,
% 2.53/1.50 ! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M @ N ) ) ).
% 2.53/1.50
% 2.53/1.50 thf(fact_251_add__leD1,axiom,
% 2.53/1.50 ! [M: nat,K: nat,N: nat] :
% 2.53/1.50 ( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
% 2.53/1.50 => ( ord_less_eq_nat @ M @ N ) ) ).
% 2.53/1.50
% 2.53/1.50 thf(fact_252_add__leD2,axiom,
% 2.53/1.50 ! [M: nat,K: nat,N: nat] :
% 2.53/1.50 ( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
% 2.53/1.50 => ( ord_less_eq_nat @ K @ N ) ) ).
% 2.53/1.50
% 2.53/1.50 thf(fact_253_le__Suc__ex,axiom,
% 2.53/1.50 ! [K: nat,L: nat] :
% 2.53/1.50 ( ( ord_less_eq_nat @ K @ L )
% 2.53/1.50 => ? [N2: nat] :
% 2.53/1.50 ( L
% 2.53/1.50 = ( plus_plus_nat @ K @ N2 ) ) ) ).
% 2.53/1.50
% 2.53/1.50 thf(fact_254_add__le__mono,axiom,
% 2.53/1.50 ! [I2: nat,J: nat,K: nat,L: nat] :
% 2.53/1.50 ( ( ord_less_eq_nat @ I2 @ J )
% 2.53/1.50 => ( ( ord_less_eq_nat @ K @ L )
% 2.53/1.50 => ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% 2.53/1.50
% 2.53/1.50 thf(fact_255_add__le__mono1,axiom,
% 2.53/1.50 ! [I2: nat,J: nat,K: nat] :
% 2.53/1.50 ( ( ord_less_eq_nat @ I2 @ J )
% 2.53/1.50 => ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% 2.53/1.50
% 2.53/1.50 thf(fact_256_trans__le__add1,axiom,
% 2.53/1.50 ! [I2: nat,J: nat,M: nat] :
% 2.53/1.50 ( ( ord_less_eq_nat @ I2 @ J )
% 2.53/1.50 => ( ord_less_eq_nat @ I2 @ ( plus_plus_nat @ J @ M ) ) ) ).
% 2.53/1.50
% 2.53/1.50 thf(fact_257_trans__le__add2,axiom,
% 2.53/1.50 ! [I2: nat,J: nat,M: nat] :
% 2.53/1.50 ( ( ord_less_eq_nat @ I2 @ J )
% 2.53/1.50 => ( ord_less_eq_nat @ I2 @ ( plus_plus_nat @ M @ J ) ) ) ).
% 2.53/1.50
% 2.53/1.50 thf(fact_258_nat__le__iff__add,axiom,
% 2.53/1.50 ( ord_less_eq_nat
% 2.53/1.50 = ( ^ [M2: nat,N3: nat] :
% 2.53/1.50 ? [K2: nat] :
% 2.53/1.50 ( N3
% 2.53/1.50 = ( plus_plus_nat @ M2 @ K2 ) ) ) ) ).
% 2.53/1.50
% 2.53/1.50 thf(fact_259_Nat_Odiff__cancel,axiom,
% 2.53/1.50 ! [K: nat,M: nat,N: nat] :
% 2.53/1.50 ( ( minus_minus_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
% 2.53/1.50 = ( minus_minus_nat @ M @ N ) ) ).
% 2.53/1.50
% 2.53/1.50 thf(fact_260_diff__cancel2,axiom,
% 2.53/1.50 ! [M: nat,K: nat,N: nat] :
% 2.53/1.50 ( ( minus_minus_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N @ K ) )
% 2.53/1.50 = ( minus_minus_nat @ M @ N ) ) ).
% 2.53/1.50
% 2.53/1.50 thf(fact_261_diff__add__inverse,axiom,
% 2.53/1.50 ! [N: nat,M: nat] :
% 2.53/1.50 ( ( minus_minus_nat @ ( plus_plus_nat @ N @ M ) @ N )
% 2.53/1.50 = M ) ).
% 2.53/1.50
% 2.53/1.50 thf(fact_262_diff__add__inverse2,axiom,
% 2.53/1.50 ! [M: nat,N: nat] :
% 2.53/1.50 ( ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ N )
% 2.53/1.50 = M ) ).
% 2.53/1.50
% 2.53/1.50 thf(fact_263_Nat_Ole__imp__diff__is__add,axiom,
% 2.53/1.50 ! [I2: nat,J: nat,K: nat] :
% 2.53/1.50 ( ( ord_less_eq_nat @ I2 @ J )
% 2.53/1.50 => ( ( ( minus_minus_nat @ J @ I2 )
% 2.53/1.50 = K )
% 2.53/1.50 = ( J
% 2.53/1.50 = ( plus_plus_nat @ K @ I2 ) ) ) ) ).
% 2.53/1.50
% 2.53/1.50 thf(fact_264_Nat_Odiff__add__assoc2,axiom,
% 2.53/1.50 ! [K: nat,J: nat,I2: nat] :
% 2.53/1.50 ( ( ord_less_eq_nat @ K @ J )
% 2.53/1.50 => ( ( minus_minus_nat @ ( plus_plus_nat @ J @ I2 ) @ K )
% 2.53/1.50 = ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I2 ) ) ) ).
% 2.53/1.50
% 2.53/1.50 thf(fact_265_Nat_Odiff__add__assoc,axiom,
% 2.53/1.50 ! [K: nat,J: nat,I2: nat] :
% 2.53/1.50 ( ( ord_less_eq_nat @ K @ J )
% 2.53/1.50 => ( ( minus_minus_nat @ ( plus_plus_nat @ I2 @ J ) @ K )
% 2.53/1.50 = ( plus_plus_nat @ I2 @ ( minus_minus_nat @ J @ K ) ) ) ) ).
% 2.53/1.50
% 2.53/1.50 thf(fact_266_Nat_Ole__diff__conv2,axiom,
% 2.53/1.50 ! [K: nat,J: nat,I2: nat] :
% 2.53/1.50 ( ( ord_less_eq_nat @ K @ J )
% 2.53/1.50 => ( ( ord_less_eq_nat @ I2 @ ( minus_minus_nat @ J @ K ) )
% 2.53/1.50 = ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ J ) ) ) ).
% 2.53/1.50
% 2.53/1.50 thf(fact_267_le__diff__conv,axiom,
% 2.53/1.50 ! [J: nat,K: nat,I2: nat] :
% 2.53/1.50 ( ( ord_less_eq_nat @ ( minus_minus_nat @ J @ K ) @ I2 )
% 2.53/1.50 = ( ord_less_eq_nat @ J @ ( plus_plus_nat @ I2 @ K ) ) ) ).
% 2.53/1.50
% 2.53/1.50 thf(fact_268_PC__monotone,axiom,
% 2.53/1.50 ! [P: nat,S: real,T: real] :
% 2.53/1.50 ( ( ( genClo1015804716orrect @ P @ T )
% 2.53/1.50 & ( ord_less_eq_real @ S @ T ) )
% 2.53/1.50 => ( ord_less_eq_real @ ( genClo1161277105lle_PC @ P @ S ) @ ( genClo1161277105lle_PC @ P @ T ) ) ) ).
% 2.53/1.50
% 2.53/1.50 thf(fact_269_eq__diff__eq_H,axiom,
% 2.53/1.50 ! [X: real,Y: real,Z2: real] :
% 2.53/1.50 ( ( X
% 2.53/1.50 = ( minus_minus_real @ Y @ Z2 ) )
% 2.53/1.50 = ( Y
% 2.53/1.50 = ( plus_plus_real @ X @ Z2 ) ) ) ).
% 2.53/1.50
% 2.53/1.50 thf(fact_270_le__numeral__extra_I3_J,axiom,
% 2.53/1.50 ord_less_eq_real @ zero_zero_real @ zero_zero_real ).
% 2.53/1.50
% 2.53/1.50 thf(fact_271_le__numeral__extra_I3_J,axiom,
% 2.53/1.50 ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% 2.53/1.50
% 2.53/1.50 thf(fact_272_nonoverlap,axiom,
% 2.53/1.50 ord_less_eq_real @ genClo1278781456e_beta @ genClo1651033342e_rmin ).
% 2.53/1.50
% 2.53/1.50 thf(fact_273_le__numeral__extra_I4_J,axiom,
% 2.53/1.50 ord_less_eq_real @ one_one_real @ one_one_real ).
% 2.53/1.50
% 2.53/1.50 thf(fact_274_le__numeral__extra_I4_J,axiom,
% 2.53/1.50 ord_less_eq_nat @ one_one_nat @ one_one_nat ).
% 2.53/1.50
% 2.53/1.50 thf(fact_275_init,axiom,
% 2.53/1.50 ! [P: nat] :
% 2.53/1.50 ( ( genClo1015804716orrect @ P @ zero_zero_real )
% 2.53/1.50 => ( ( ord_less_eq_real @ zero_zero_real @ ( genClo1161277105lle_PC @ P @ zero_zero_real ) )
% 2.53/1.50 & ( ord_less_eq_real @ ( genClo1161277105lle_PC @ P @ zero_zero_real ) @ genClo872980712lle_mu ) ) ) ).
% 2.53/1.50
% 2.53/1.50 thf(fact_276_bound1,axiom,
% 2.53/1.50 ord_less_eq_real @ ( minus_minus_real @ ( genClo1161277105lle_PC @ l @ ( genClo1163638703lle_te @ p @ ( plus_plus_nat @ i @ one_one_nat ) ) ) @ ( genClo1161277105lle_PC @ l @ ( genClo1163638703lle_te @ q @ ( plus_plus_nat @ i @ one_one_nat ) ) ) ) @ ( times_times_real @ ( minus_minus_real @ ( genClo1163638703lle_te @ p @ ( plus_plus_nat @ i @ one_one_nat ) ) @ ( genClo1163638703lle_te @ q @ ( plus_plus_nat @ i @ one_one_nat ) ) ) @ ( plus_plus_real @ one_one_real @ genClo1144207539le_rho ) ) ).
% 2.53/1.50
% 2.53/1.50 thf(fact_277_mult__cancel__right,axiom,
% 2.53/1.50 ! [A: nat,C: nat,B: nat] :
% 2.53/1.50 ( ( ( times_times_nat @ A @ C )
% 2.53/1.50 = ( times_times_nat @ B @ C ) )
% 2.53/1.50 = ( ( C = zero_zero_nat )
% 2.53/1.50 | ( A = B ) ) ) ).
% 2.53/1.50
% 2.53/1.50 thf(fact_278_mult__cancel__right,axiom,
% 2.53/1.50 ! [A: real,C: real,B: real] :
% 2.53/1.50 ( ( ( times_times_real @ A @ C )
% 2.53/1.50 = ( times_times_real @ B @ C ) )
% 2.53/1.50 = ( ( C = zero_zero_real )
% 2.53/1.50 | ( A = B ) ) ) ).
% 2.53/1.50
% 2.53/1.50 thf(fact_279_mult__cancel__left,axiom,
% 2.53/1.50 ! [C: nat,A: nat,B: nat] :
% 2.53/1.50 ( ( ( times_times_nat @ C @ A )
% 2.53/1.50 = ( times_times_nat @ C @ B ) )
% 2.53/1.50 = ( ( C = zero_zero_nat )
% 2.53/1.50 | ( A = B ) ) ) ).
% 2.53/1.50
% 2.53/1.50 thf(fact_280_mult__cancel__left,axiom,
% 2.53/1.50 ! [C: real,A: real,B: real] :
% 2.53/1.50 ( ( ( times_times_real @ C @ A )
% 2.53/1.50 = ( times_times_real @ C @ B ) )
% 2.53/1.50 = ( ( C = zero_zero_real )
% 2.53/1.50 | ( A = B ) ) ) ).
% 2.53/1.50
% 2.53/1.50 thf(fact_281_mult__eq__0__iff,axiom,
% 2.53/1.50 ! [A: nat,B: nat] :
% 2.53/1.50 ( ( ( times_times_nat @ A @ B )
% 2.53/1.50 = zero_zero_nat )
% 2.53/1.50 = ( ( A = zero_zero_nat )
% 2.53/1.50 | ( B = zero_zero_nat ) ) ) ).
% 2.53/1.50
% 2.53/1.50 thf(fact_282_mult__eq__0__iff,axiom,
% 2.53/1.50 ! [A: real,B: real] :
% 2.53/1.50 ( ( ( times_times_real @ A @ B )
% 2.53/1.50 = zero_zero_real )
% 2.53/1.50 = ( ( A = zero_zero_real )
% 2.53/1.50 | ( B = zero_zero_real ) ) ) ).
% 2.53/1.50
% 2.53/1.50 thf(fact_283_mult__zero__right,axiom,
% 2.53/1.50 ! [A: nat] :
% 2.53/1.50 ( ( times_times_nat @ A @ zero_zero_nat )
% 2.53/1.50 = zero_zero_nat ) ).
% 2.53/1.50
% 2.53/1.50 thf(fact_284_mult__zero__right,axiom,
% 2.53/1.50 ! [A: real] :
% 2.53/1.50 ( ( times_times_real @ A @ zero_zero_real )
% 2.53/1.50 = zero_zero_real ) ).
% 2.53/1.50
% 2.53/1.50 thf(fact_285_mult__zero__left,axiom,
% 2.53/1.50 ! [A: nat] :
% 2.53/1.50 ( ( times_times_nat @ zero_zero_nat @ A )
% 2.53/1.50 = zero_zero_nat ) ).
% 2.53/1.50
% 2.53/1.50 thf(fact_286_mult__zero__left,axiom,
% 2.53/1.50 ! [A: real] :
% 2.53/1.50 ( ( times_times_real @ zero_zero_real @ A )
% 2.53/1.50 = zero_zero_real ) ).
% 2.53/1.50
% 2.53/1.50 thf(fact_287_mult_Oleft__neutral,axiom,
% 2.53/1.50 ! [A: nat] :
% 2.53/1.50 ( ( times_times_nat @ one_one_nat @ A )
% 2.53/1.50 = A ) ).
% 2.53/1.50
% 2.53/1.50 thf(fact_288_mult_Oleft__neutral,axiom,
% 2.53/1.50 ! [A: real] :
% 2.53/1.50 ( ( times_times_real @ one_one_real @ A )
% 2.53/1.50 = A ) ).
% 2.53/1.50
% 2.53/1.50 thf(fact_289_mult_Oright__neutral,axiom,
% 2.53/1.50 ! [A: nat] :
% 2.53/1.50 ( ( times_times_nat @ A @ one_one_nat )
% 2.53/1.50 = A ) ).
% 2.53/1.50
% 2.53/1.50 thf(fact_290_mult_Oright__neutral,axiom,
% 2.53/1.50 ! [A: real] :
% 2.53/1.50 ( ( times_times_real @ A @ one_one_real )
% 2.53/1.50 = A ) ).
% 2.53/1.50
% 2.53/1.50 thf(fact_291_abs__mult__self__eq,axiom,
% 2.53/1.50 ! [A: real] :
% 2.53/1.50 ( ( times_times_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ A ) )
% 2.53/1.50 = ( times_times_real @ A @ A ) ) ).
% 2.53/1.50
% 2.53/1.50 thf(fact_292_mult__cancel__left1,axiom,
% 2.53/1.50 ! [C: real,B: real] :
% 2.53/1.50 ( ( C
% 2.53/1.50 = ( times_times_real @ C @ B ) )
% 2.53/1.50 = ( ( C = zero_zero_real )
% 2.53/1.50 | ( B = one_one_real ) ) ) ).
% 2.53/1.50
% 2.53/1.50 thf(fact_293_mult__cancel__left2,axiom,
% 2.53/1.50 ! [C: real,A: real] :
% 2.53/1.50 ( ( ( times_times_real @ C @ A )
% 2.53/1.50 = C )
% 2.53/1.50 = ( ( C = zero_zero_real )
% 2.53/1.50 | ( A = one_one_real ) ) ) ).
% 2.53/1.50
% 2.53/1.50 thf(fact_294_mult__cancel__right1,axiom,
% 2.53/1.50 ! [C: real,B: real] :
% 2.53/1.50 ( ( C
% 2.53/1.50 = ( times_times_real @ B @ C ) )
% 2.53/1.50 = ( ( C = zero_zero_real )
% 2.53/1.50 | ( B = one_one_real ) ) ) ).
% 2.53/1.50
% 2.53/1.50 thf(fact_295_mult__cancel__right2,axiom,
% 2.53/1.50 ! [A: real,C: real] :
% 2.53/1.50 ( ( ( times_times_real @ A @ C )
% 2.53/1.50 = C )
% 2.53/1.50 = ( ( C = zero_zero_real )
% 2.53/1.50 | ( A = one_one_real ) ) ) ).
% 2.53/1.50
% 2.53/1.50 thf(fact_296_mult__right__cancel,axiom,
% 2.53/1.50 ! [C: nat,A: nat,B: nat] :
% 2.53/1.50 ( ( C != zero_zero_nat )
% 2.53/1.50 => ( ( ( times_times_nat @ A @ C )
% 2.53/1.50 = ( times_times_nat @ B @ C ) )
% 2.53/1.50 = ( A = B ) ) ) ).
% 2.53/1.50
% 2.53/1.50 thf(fact_297_mult__right__cancel,axiom,
% 2.53/1.50 ! [C: real,A: real,B: real] :
% 2.53/1.50 ( ( C != zero_zero_real )
% 2.53/1.50 => ( ( ( times_times_real @ A @ C )
% 2.53/1.50 = ( times_times_real @ B @ C ) )
% 2.53/1.50 = ( A = B ) ) ) ).
% 2.53/1.50
% 2.53/1.50 thf(fact_298_mult__left__cancel,axiom,
% 2.53/1.50 ! [C: nat,A: nat,B: nat] :
% 2.53/1.50 ( ( C != zero_zero_nat )
% 2.53/1.50 => ( ( ( times_times_nat @ C @ A )
% 2.53/1.50 = ( times_times_nat @ C @ B ) )
% 2.53/1.50 = ( A = B ) ) ) ).
% 2.53/1.50
% 2.53/1.50 thf(fact_299_mult__left__cancel,axiom,
% 2.53/1.50 ! [C: real,A: real,B: real] :
% 2.53/1.50 ( ( C != zero_zero_real )
% 2.53/1.50 => ( ( ( times_times_real @ C @ A )
% 2.53/1.50 = ( times_times_real @ C @ B ) )
% 2.53/1.50 = ( A = B ) ) ) ).
% 2.53/1.50
% 2.53/1.50 thf(fact_300_no__zero__divisors,axiom,
% 2.53/1.50 ! [A: nat,B: nat] :
% 2.53/1.50 ( ( A != zero_zero_nat )
% 2.53/1.50 => ( ( B != zero_zero_nat )
% 2.53/1.50 => ( ( times_times_nat @ A @ B )
% 2.53/1.50 != zero_zero_nat ) ) ) ).
% 2.53/1.50
% 2.53/1.50 thf(fact_301_no__zero__divisors,axiom,
% 2.53/1.50 ! [A: real,B: real] :
% 2.53/1.50 ( ( A != zero_zero_real )
% 2.53/1.50 => ( ( B != zero_zero_real )
% 2.53/1.50 => ( ( times_times_real @ A @ B )
% 2.53/1.50 != zero_zero_real ) ) ) ).
% 2.53/1.50
% 2.53/1.50 thf(fact_302_divisors__zero,axiom,
% 2.53/1.50 ! [A: nat,B: nat] :
% 2.53/1.50 ( ( ( times_times_nat @ A @ B )
% 2.53/1.50 = zero_zero_nat )
% 2.53/1.50 => ( ( A = zero_zero_nat )
% 2.53/1.50 | ( B = zero_zero_nat ) ) ) ).
% 2.53/1.50
% 2.53/1.50 thf(fact_303_divisors__zero,axiom,
% 2.53/1.50 ! [A: real,B: real] :
% 2.53/1.50 ( ( ( times_times_real @ A @ B )
% 2.53/1.50 = zero_zero_real )
% 2.53/1.50 => ( ( A = zero_zero_real )
% 2.53/1.50 | ( B = zero_zero_real ) ) ) ).
% 2.53/1.50
% 2.53/1.50 thf(fact_304_mult__not__zero,axiom,
% 2.53/1.50 ! [A: nat,B: nat] :
% 2.53/1.50 ( ( ( times_times_nat @ A @ B )
% 2.53/1.50 != zero_zero_nat )
% 2.53/1.50 => ( ( A != zero_zero_nat )
% 2.53/1.50 & ( B != zero_zero_nat ) ) ) ).
% 2.53/1.50
% 2.53/1.50 thf(fact_305_mult__not__zero,axiom,
% 2.53/1.50 ! [A: real,B: real] :
% 2.53/1.50 ( ( ( times_times_real @ A @ B )
% 2.53/1.50 != zero_zero_real )
% 2.53/1.50 => ( ( A != zero_zero_real )
% 2.53/1.50 & ( B != zero_zero_real ) ) ) ).
% 2.53/1.50
% 2.53/1.50 thf(fact_306_ring__class_Oring__distribs_I2_J,axiom,
% 2.53/1.50 ! [A: real,B: real,C: real] :
% 2.53/1.50 ( ( times_times_real @ ( plus_plus_real @ A @ B ) @ C )
% 2.53/1.50 = ( plus_plus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ).
% 2.53/1.50
% 2.53/1.50 thf(fact_307_ring__class_Oring__distribs_I1_J,axiom,
% 2.53/1.50 ! [A: real,B: real,C: real] :
% 2.53/1.50 ( ( times_times_real @ A @ ( plus_plus_real @ B @ C ) )
% 2.53/1.50 = ( plus_plus_real @ ( times_times_real @ A @ B ) @ ( times_times_real @ A @ C ) ) ) ).
% 2.53/1.50
% 2.53/1.50 thf(fact_308_comm__semiring__class_Odistrib,axiom,
% 2.53/1.50 ! [A: nat,B: nat,C: nat] :
% 2.53/1.50 ( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ C )
% 2.53/1.50 = ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ).
% 2.53/1.50
% 2.53/1.50 thf(fact_309_comm__semiring__class_Odistrib,axiom,
% 2.53/1.50 ! [A: real,B: real,C: real] :
% 2.53/1.50 ( ( times_times_real @ ( plus_plus_real @ A @ B ) @ C )
% 2.53/1.50 = ( plus_plus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ).
% 2.53/1.50
% 2.53/1.50 thf(fact_310_distrib__left,axiom,
% 2.53/1.50 ! [A: nat,B: nat,C: nat] :
% 2.53/1.50 ( ( times_times_nat @ A @ ( plus_plus_nat @ B @ C ) )
% 2.53/1.50 = ( plus_plus_nat @ ( times_times_nat @ A @ B ) @ ( times_times_nat @ A @ C ) ) ) ).
% 2.53/1.50
% 2.53/1.50 thf(fact_311_distrib__left,axiom,
% 2.53/1.50 ! [A: real,B: real,C: real] :
% 2.53/1.50 ( ( times_times_real @ A @ ( plus_plus_real @ B @ C ) )
% 2.53/1.50 = ( plus_plus_real @ ( times_times_real @ A @ B ) @ ( times_times_real @ A @ C ) ) ) ).
% 2.53/1.50
% 2.53/1.50 thf(fact_312_distrib__right,axiom,
% 2.53/1.50 ! [A: nat,B: nat,C: nat] :
% 2.53/1.50 ( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ C )
% 2.53/1.50 = ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ).
% 2.53/1.50
% 2.53/1.50 thf(fact_313_distrib__right,axiom,
% 2.53/1.50 ! [A: real,B: real,C: real] :
% 2.53/1.50 ( ( times_times_real @ ( plus_plus_real @ A @ B ) @ C )
% 2.53/1.50 = ( plus_plus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ).
% 2.53/1.50
% 2.53/1.50 thf(fact_314_combine__common__factor,axiom,
% 2.53/1.50 ! [A: nat,E: nat,B: nat,C: nat] :
% 2.53/1.50 ( ( plus_plus_nat @ ( times_times_nat @ A @ E ) @ ( plus_plus_nat @ ( times_times_nat @ B @ E ) @ C ) )
% 2.53/1.50 = ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ E ) @ C ) ) ).
% 2.53/1.50
% 2.53/1.50 thf(fact_315_combine__common__factor,axiom,
% 2.53/1.50 ! [A: real,E: real,B: real,C: real] :
% 2.53/1.50 ( ( plus_plus_real @ ( times_times_real @ A @ E ) @ ( plus_plus_real @ ( times_times_real @ B @ E ) @ C ) )
% 2.53/1.50 = ( plus_plus_real @ ( times_times_real @ ( plus_plus_real @ A @ B ) @ E ) @ C ) ) ).
% 2.53/1.50
% 2.53/1.50 thf(fact_316_comm__monoid__mult__class_Omult__1,axiom,
% 2.53/1.50 ! [A: nat] :
% 2.53/1.50 ( ( times_times_nat @ one_one_nat @ A )
% 2.53/1.50 = A ) ).
% 2.53/1.50
% 2.53/1.50 thf(fact_317_comm__monoid__mult__class_Omult__1,axiom,
% 2.53/1.50 ! [A: real] :
% 2.53/1.50 ( ( times_times_real @ one_one_real @ A )
% 2.53/1.50 = A ) ).
% 2.53/1.50
% 2.53/1.50 thf(fact_318_mult_Ocomm__neutral,axiom,
% 2.53/1.50 ! [A: nat] :
% 2.53/1.50 ( ( times_times_nat @ A @ one_one_nat )
% 2.53/1.50 = A ) ).
% 2.53/1.50
% 2.53/1.50 thf(fact_319_mult_Ocomm__neutral,axiom,
% 2.53/1.50 ! [A: real] :
% 2.53/1.50 ( ( times_times_real @ A @ one_one_real )
% 2.53/1.50 = A ) ).
% 2.53/1.50
% 2.53/1.50 thf(fact_320_left__diff__distrib,axiom,
% 2.53/1.50 ! [A: real,B: real,C: real] :
% 2.53/1.50 ( ( times_times_real @ ( minus_minus_real @ A @ B ) @ C )
% 2.53/1.50 = ( minus_minus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ).
% 2.53/1.50
% 2.53/1.50 thf(fact_321_right__diff__distrib,axiom,
% 2.53/1.50 ! [A: real,B: real,C: real] :
% 2.53/1.50 ( ( times_times_real @ A @ ( minus_minus_real @ B @ C ) )
% 2.53/1.50 = ( minus_minus_real @ ( times_times_real @ A @ B ) @ ( times_times_real @ A @ C ) ) ) ).
% 2.53/1.50
% 2.53/1.50 thf(fact_322_left__diff__distrib_H,axiom,
% 2.53/1.50 ! [B: nat,C: nat,A: nat] :
% 2.53/1.50 ( ( times_times_nat @ ( minus_minus_nat @ B @ C ) @ A )
% 2.53/1.50 = ( minus_minus_nat @ ( times_times_nat @ B @ A ) @ ( times_times_nat @ C @ A ) ) ) ).
% 2.53/1.50
% 2.53/1.50 thf(fact_323_left__diff__distrib_H,axiom,
% 2.53/1.50 ! [B: real,C: real,A: real] :
% 2.53/1.50 ( ( times_times_real @ ( minus_minus_real @ B @ C ) @ A )
% 2.53/1.50 = ( minus_minus_real @ ( times_times_real @ B @ A ) @ ( times_times_real @ C @ A ) ) ) ).
% 2.53/1.50
% 2.53/1.50 thf(fact_324_right__diff__distrib_H,axiom,
% 2.53/1.50 ! [A: nat,B: nat,C: nat] :
% 2.53/1.50 ( ( times_times_nat @ A @ ( minus_minus_nat @ B @ C ) )
% 2.53/1.50 = ( minus_minus_nat @ ( times_times_nat @ A @ B ) @ ( times_times_nat @ A @ C ) ) ) ).
% 2.53/1.50
% 2.53/1.50 thf(fact_325_right__diff__distrib_H,axiom,
% 2.53/1.50 ! [A: real,B: real,C: real] :
% 2.53/1.50 ( ( times_times_real @ A @ ( minus_minus_real @ B @ C ) )
% 2.53/1.50 = ( minus_minus_real @ ( times_times_real @ A @ B ) @ ( times_times_real @ A @ C ) ) ) ).
% 2.53/1.50
% 2.53/1.50 thf(fact_326_abs__mult,axiom,
% 2.53/1.50 ! [A: real,B: real] :
% 2.53/1.50 ( ( abs_abs_real @ ( times_times_real @ A @ B ) )
% 2.53/1.50 = ( times_times_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) ) ).
% 2.53/1.50
% 2.53/1.50 thf(fact_327_mult_Oleft__commute,axiom,
% 2.53/1.50 ! [B: nat,A: nat,C: nat] :
% 2.53/1.50 ( ( times_times_nat @ B @ ( times_times_nat @ A @ C ) )
% 2.53/1.50 = ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% 2.53/1.50
% 2.53/1.50 thf(fact_328_mult_Oleft__commute,axiom,
% 2.53/1.50 ! [B: real,A: real,C: real] :
% 2.53/1.50 ( ( times_times_real @ B @ ( times_times_real @ A @ C ) )
% 2.53/1.50 = ( times_times_real @ A @ ( times_times_real @ B @ C ) ) ) ).
% 2.53/1.50
% 2.53/1.50 thf(fact_329_mult_Ocommute,axiom,
% 2.53/1.50 ( times_times_nat
% 2.53/1.50 = ( ^ [A2: nat,B2: nat] : ( times_times_nat @ B2 @ A2 ) ) ) ).
% 2.53/1.50
% 2.53/1.50 thf(fact_330_mult_Ocommute,axiom,
% 2.53/1.50 ( times_times_real
% 2.53/1.50 = ( ^ [A2: real,B2: real] : ( times_times_real @ B2 @ A2 ) ) ) ).
% 2.53/1.50
% 2.53/1.50 thf(fact_331_mult_Oassoc,axiom,
% 2.53/1.50 ! [A: nat,B: nat,C: nat] :
% 2.53/1.50 ( ( times_times_nat @ ( times_times_nat @ A @ B ) @ C )
% 2.53/1.50 = ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% 2.53/1.50
% 2.53/1.50 thf(fact_332_mult_Oassoc,axiom,
% 2.53/1.50 ! [A: real,B: real,C: real] :
% 2.53/1.50 ( ( times_times_real @ ( times_times_real @ A @ B ) @ C )
% 2.53/1.50 = ( times_times_real @ A @ ( times_times_real @ B @ C ) ) ) ).
% 2.53/1.50
% 2.53/1.50 thf(fact_333_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
% 2.53/1.50 ! [A: nat,B: nat,C: nat] :
% 2.53/1.50 ( ( times_times_nat @ ( times_times_nat @ A @ B ) @ C )
% 2.53/1.50 = ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% 2.53/1.50
% 2.53/1.50 thf(fact_334_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
% 2.53/1.50 ! [A: real,B: real,C: real] :
% 2.53/1.50 ( ( times_times_real @ ( times_times_real @ A @ B ) @ C )
% 2.53/1.50 = ( times_times_real @ A @ ( times_times_real @ B @ C ) ) ) ).
% 2.53/1.50
% 2.53/1.50 thf(fact_335_mult__mono,axiom,
% 2.53/1.50 ! [A: real,B: real,C: real,D: real] :
% 2.53/1.50 ( ( ord_less_eq_real @ A @ B )
% 2.53/1.50 => ( ( ord_less_eq_real @ C @ D )
% 2.53/1.50 => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 2.53/1.50 => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 2.53/1.50 => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).
% 2.53/1.50
% 2.53/1.50 thf(fact_336_mult__mono,axiom,
% 2.53/1.50 ! [A: nat,B: nat,C: nat,D: nat] :
% 2.53/1.50 ( ( ord_less_eq_nat @ A @ B )
% 2.53/1.50 => ( ( ord_less_eq_nat @ C @ D )
% 2.53/1.50 => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 2.53/1.50 => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 2.53/1.50 => ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% 2.53/1.50
% 2.53/1.50 thf(conj_0,conjecture,
% 2.53/1.50 ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( genClo1163638703lle_te @ p @ ( plus_plus_nat @ i @ one_one_nat ) ) @ ( genClo1163638703lle_te @ q @ ( plus_plus_nat @ i @ one_one_nat ) ) ) ) @ genClo1278781456e_beta ).
% 2.53/1.50 ------------------------
% 2.53/1.52 % (25821)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_25639 for (2999ds/2Mi)
% 2.53/1.52 % (25820)dis+1010_1:1_au=on:cbe=off:chr=on:fsr=off:hfsq=on:nm=64:sos=theory:sp=weighted_frequency:i=27:si=on:rtra=on_0 on DTF2THF_25639 for (2999ds/27Mi)
% 2.53/1.52 % (25824)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_25639 for (2999ds/18Mi)
% 2.53/1.52 % (25818)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_25639 for (2999ds/183Mi)
% 2.53/1.52 % (25819)lrs+10_1:1_c=on:cnfonf=conj_eager:fd=off:fe=off:kws=frequency:spb=intro:i=4:si=on:rtra=on_0 on DTF2THF_25639 for (2999ds/4Mi)
% 2.53/1.52 % (25822)lrs+1002_1:128_aac=none:au=on:cnfonf=lazy_not_gen_be_off:sos=all:i=2:si=on:rtra=on_0 on DTF2THF_25639 for (2999ds/2Mi)
% 2.53/1.52 % (25821)Instruction limit reached!
% 2.53/1.52 % (25821)------------------------------
% 2.53/1.52 % (25821)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.53/1.52 % (25821)Termination reason: Unknown
% 2.53/1.52 % (25823)lrs+1002_1:1_au=on:bd=off:e2e=on:sd=2:sos=on:ss=axioms:i=275:si=on:rtra=on_0 on DTF2THF_25639 for (2999ds/275Mi)
% 2.53/1.52 % (25821)Termination phase: shuffling
% 2.53/1.52
% 2.53/1.52 % (25821)Memory used [KB]: 1279
% 2.53/1.52 % (25821)Time elapsed: 0.002 s
% 2.53/1.52 % (25821)Instructions burned: 2 (million)
% 2.53/1.52 % (25821)------------------------------
% 2.53/1.52 % (25821)------------------------------
% 2.53/1.52 % (25822)Instruction limit reached!
% 2.53/1.52 % (25822)------------------------------
% 2.53/1.52 % (25822)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.53/1.52 % (25822)Termination reason: Unknown
% 2.53/1.52 % (25822)Termination phase: shuffling
% 2.53/1.52
% 2.53/1.52 % (25822)Memory used [KB]: 1279
% 2.53/1.52 % (25822)Time elapsed: 0.002 s
% 2.53/1.52 % (25822)Instructions burned: 3 (million)
% 2.53/1.52 % (25822)------------------------------
% 2.53/1.52 % (25822)------------------------------
% 2.53/1.52 % (25819)Instruction limit reached!
% 2.53/1.52 % (25819)------------------------------
% 2.53/1.52 % (25819)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.53/1.52 % (25819)Termination reason: Unknown
% 2.53/1.52 % (25819)Termination phase: shuffling
% 2.53/1.52
% 2.53/1.52 % (25819)Memory used [KB]: 1407
% 2.53/1.52 % (25819)Time elapsed: 0.003 s
% 2.53/1.52 % (25819)Instructions burned: 7 (million)
% 2.53/1.52 % (25819)------------------------------
% 2.53/1.52 % (25819)------------------------------
% 2.53/1.52 % (25824)Instruction limit reached!
% 2.53/1.52 % (25824)------------------------------
% 2.53/1.52 % (25824)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.53/1.52 % (25824)Termination reason: Unknown
% 2.53/1.52 % (25824)Termination phase: Property scanning
% 2.53/1.52
% 2.53/1.52 % (25824)Memory used [KB]: 1535
% 2.53/1.52 % (25824)Time elapsed: 0.007 s
% 2.53/1.52 % (25824)Instructions burned: 19 (million)
% 2.53/1.52 % (25824)------------------------------
% 2.53/1.52 % (25824)------------------------------
% 2.53/1.53 % (25820)Instruction limit reached!
% 2.53/1.53 % (25820)------------------------------
% 2.53/1.53 % (25820)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.53/1.53 % (25820)Termination reason: Unknown
% 2.53/1.53 % (25820)Termination phase: Property scanning
% 2.53/1.53
% 2.53/1.53 % (25820)Memory used [KB]: 1535
% 2.53/1.53 % (25820)Time elapsed: 0.010 s
% 2.53/1.53 % (25820)Instructions burned: 30 (million)
% 2.53/1.53 % (25820)------------------------------
% 2.53/1.53 % (25820)------------------------------
% 2.53/1.53 % (25825)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_25639 for (2999ds/3Mi)
% 2.53/1.53 % (25825)Instruction limit reached!
% 2.53/1.53 % (25825)------------------------------
% 2.53/1.53 % (25825)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.53/1.53 % (25825)Termination reason: Unknown
% 2.53/1.53 % (25825)Termination phase: shuffling
% 2.53/1.53
% 2.53/1.53 % (25825)Memory used [KB]: 1279
% 2.53/1.53 % (25825)Time elapsed: 0.002 s
% 2.53/1.53 % (25826)lrs+1002_1:1_cnfonf=lazy_not_be_gen:hud=14:prag=on:sp=weighted_frequency:tnu=1:i=37:si=on:rtra=on_0 on DTF2THF_25639 for (2999ds/37Mi)
% 2.53/1.53 % (25825)Instructions burned: 3 (million)
% 2.53/1.53 % (25825)------------------------------
% 2.53/1.53 % (25825)------------------------------
% 2.53/1.53 % (25827)lrs+2_16:1_acc=model:au=on:bd=off:c=on:e2e=on:nm=2:sos=all:i=15:si=on:rtra=on_0 on DTF2THF_25639 for (2999ds/15Mi)
% 2.53/1.53 % (25828)dis+21_1:1_cbe=off:cnfonf=off:fs=off:fsr=off:hud=1:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_25639 for (2999ds/3Mi)
% 2.53/1.53 % (25828)Instruction limit reached!
% 2.53/1.53 % (25828)------------------------------
% 2.53/1.53 % (25828)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.53/1.53 % (25828)Termination reason: Unknown
% 2.53/1.53 % (25828)Termination phase: shuffling
% 2.53/1.53
% 2.53/1.53 % (25828)Memory used [KB]: 1279
% 2.53/1.53 % (25828)Time elapsed: 0.002 s
% 2.53/1.53 % (25828)Instructions burned: 3 (million)
% 2.53/1.53 % (25828)------------------------------
% 2.53/1.53 % (25828)------------------------------
% 2.53/1.53 % (25827)Instruction limit reached!
% 2.53/1.53 % (25827)------------------------------
% 2.53/1.53 % (25827)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.53/1.53 % (25827)Termination reason: Unknown
% 2.53/1.53 % (25827)Termination phase: shuffling
% 2.53/1.53
% 2.53/1.53 % (25827)Memory used [KB]: 1535
% 2.53/1.53 % (25827)Time elapsed: 0.007 s
% 2.53/1.53 % (25827)Instructions burned: 15 (million)
% 2.53/1.53 % (25827)------------------------------
% 2.53/1.53 % (25827)------------------------------
% 2.53/1.53 % (25829)lrs+1002_1:1_aac=none:au=on:cnfonf=lazy_gen:plsq=on:plsqc=1:plsqr=4203469,65536:i=1041:si=on:rtra=on_0 on DTF2THF_25639 for (2999ds/1041Mi)
% 2.53/1.54 % (25830)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_25639 for (2999ds/7Mi)
% 2.53/1.54 % (25826)Instruction limit reached!
% 2.53/1.54 % (25826)------------------------------
% 2.53/1.54 % (25826)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.53/1.54 % (25826)Termination reason: Unknown
% 2.53/1.54 % (25826)Termination phase: shuffling
% 2.53/1.54
% 2.53/1.54 % (25826)Memory used [KB]: 2046
% 2.53/1.54 % (25826)Time elapsed: 0.011 s
% 2.53/1.54 % (25826)Instructions burned: 38 (million)
% 2.53/1.54 % (25826)------------------------------
% 2.53/1.54 % (25826)------------------------------
% 2.53/1.54 % (25830)Instruction limit reached!
% 2.53/1.54 % (25830)------------------------------
% 2.53/1.54 % (25830)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.53/1.54 % (25830)Termination reason: Unknown
% 2.53/1.54 % (25830)Termination phase: shuffling
% 2.53/1.54
% 2.53/1.54 % (25830)Memory used [KB]: 1407
% 2.53/1.54 % (25830)Time elapsed: 0.004 s
% 2.53/1.54 % (25830)Instructions burned: 9 (million)
% 2.53/1.54 % (25830)------------------------------
% 2.53/1.54 % (25830)------------------------------
% 2.53/1.54 % (25831)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_25639 for (2999ds/16Mi)
% 2.53/1.54 % (25823)First to succeed.
% 2.53/1.54 % (25832)lrs+21_1:1_au=on:cnfonf=off:fd=preordered:fe=off:fsr=off:hud=11:inj=on:kws=precedence:s2pl=no:sp=weighted_frequency:tgt=full:i=3:si=on:rtra=on_0 on DTF2THF_25639 for (2999ds/3Mi)
% 2.53/1.55 % (25823)Refutation found. Thanks to Tanya!
% 2.53/1.55 % SZS status Theorem for DTF2THF_25639
% 2.53/1.55 % SZS output start Proof for DTF2THF_25639
% 2.53/1.55 thf(type_def_5, type, nat: $tType).
% 2.53/1.55 thf(type_def_6, type, real: $tType).
% 2.53/1.55 thf(func_def_0, type, real: $tType).
% 2.53/1.55 thf(func_def_1, type, nat: $tType).
% 2.53/1.55 thf(func_def_2, type, genClo1161277105lle_PC: nat > real > real).
% 2.53/1.55 thf(func_def_3, type, genClo1278781456e_beta: real).
% 2.53/1.55 thf(func_def_4, type, genClo872980712lle_mu: real).
% 2.53/1.55 thf(func_def_5, type, genClo1144207539le_rho: real).
% 2.53/1.55 thf(func_def_6, type, genClo1015804716orrect: nat > real > $o).
% 2.53/1.55 thf(func_def_7, type, genClo293725282kRead2: (nat > real) > (nat > real) > real > (nat > $o) > $o).
% 2.53/1.55 thf(func_def_8, type, genClo1650508560e_rmax: real).
% 2.53/1.55 thf(func_def_9, type, genClo1651033342e_rmin: real).
% 2.53/1.55 thf(func_def_10, type, genClo1163638703lle_te: nat > nat > real).
% 2.53/1.55 thf(func_def_11, type, abs_abs_real: real > real).
% 2.53/1.55 thf(func_def_12, type, minus_minus_nat: nat > nat > nat).
% 2.53/1.55 thf(func_def_13, type, minus_minus_real: real > real > real).
% 2.53/1.55 thf(func_def_14, type, one_one_nat: nat).
% 2.53/1.55 thf(func_def_15, type, one_one_real: real).
% 2.53/1.55 thf(func_def_16, type, plus_plus_nat: nat > nat > nat).
% 2.53/1.55 thf(func_def_17, type, plus_plus_real: real > real > real).
% 2.53/1.55 thf(func_def_18, type, times_times_nat: nat > nat > nat).
% 2.53/1.55 thf(func_def_19, type, times_times_real: real > real > real).
% 2.53/1.55 thf(func_def_20, type, zero_zero_nat: nat).
% 2.53/1.55 thf(func_def_21, type, zero_zero_real: real).
% 2.53/1.55 thf(func_def_22, type, ord_less_eq_nat: nat > nat > $o).
% 2.53/1.55 thf(func_def_23, type, ord_less_eq_real: real > real > $o).
% 2.53/1.55 thf(func_def_24, type, i: nat).
% 2.53/1.55 thf(func_def_25, type, l: nat).
% 2.53/1.55 thf(func_def_26, type, p: nat).
% 2.53/1.55 thf(func_def_27, type, q: nat).
% 2.53/1.55 thf(func_def_41, type, sK0: nat > nat > nat).
% 2.53/1.55 thf(func_def_42, type, sK1: nat > nat > nat).
% 2.53/1.55 thf(func_def_43, type, sK2: (nat > $o) > nat).
% 2.53/1.55 thf(func_def_44, type, sK3: (nat > $o) > nat > nat).
% 2.53/1.55 thf(func_def_46, type, ph5: !>[X0: $tType]:(X0)).
% 2.53/1.55 thf(f1387,plain,(
% 2.53/1.55 $false),
% 2.53/1.55 inference(subsumption_resolution,[status(thm)],[f1386,f1159])).
% 2.53/1.55 thf(f1159,plain,(
% 2.53/1.55 ((genClo1015804716orrect @ q @ (genClo1163638703lle_te @ q @ (plus_plus_nat @ i @ one_one_nat))) = $true)),
% 2.53/1.55 inference(cnf_transformation,[status(thm)],[f410])).
% 2.53/1.55 thf(f410,plain,(
% 2.53/1.55 ((genClo1015804716orrect @ q @ (genClo1163638703lle_te @ q @ (plus_plus_nat @ i @ one_one_nat))) = $true)),
% 2.53/1.55 inference(fool_elimination,[status(thm)],[f409])).
% 2.53/1.55 thf(f409,plain,(
% 2.53/1.55 (genClo1015804716orrect @ q @ (genClo1163638703lle_te @ q @ (plus_plus_nat @ i @ one_one_nat)))),
% 2.53/1.55 inference(rectify,[status(thm)],[f3])).
% 2.53/1.55 thf(f3,axiom,(
% 2.53/1.55 (genClo1015804716orrect @ q @ (genClo1163638703lle_te @ q @ (plus_plus_nat @ i @ one_one_nat)))),
% 2.53/1.55 file('/export/starexec/sandbox2/tmp/tmp.5TcjDfaA5h/DTF2THF_25639.p',fact_2_corr__q__tq)).
% 2.53/1.55 thf(f1386,plain,(
% 2.53/1.55 ((genClo1015804716orrect @ q @ (genClo1163638703lle_te @ q @ (plus_plus_nat @ i @ one_one_nat))) != $true)),
% 2.53/1.55 inference(subsumption_resolution,[status(thm)],[f1375,f1022])).
% 2.53/1.55 thf(f1022,plain,(
% 2.53/1.55 ((genClo1015804716orrect @ p @ (genClo1163638703lle_te @ p @ (plus_plus_nat @ i @ one_one_nat))) = $true)),
% 2.53/1.55 inference(cnf_transformation,[status(thm)],[f722])).
% 2.53/1.55 thf(f722,plain,(
% 2.53/1.55 ((genClo1015804716orrect @ p @ (genClo1163638703lle_te @ p @ (plus_plus_nat @ i @ one_one_nat))) = $true)),
% 2.53/1.55 inference(fool_elimination,[status(thm)],[f721])).
% 2.53/1.55 thf(f721,plain,(
% 2.53/1.55 (genClo1015804716orrect @ p @ (genClo1163638703lle_te @ p @ (plus_plus_nat @ i @ one_one_nat)))),
% 2.53/1.55 inference(rectify,[status(thm)],[f2])).
% 2.53/1.55 thf(f2,axiom,(
% 2.53/1.55 (genClo1015804716orrect @ p @ (genClo1163638703lle_te @ p @ (plus_plus_nat @ i @ one_one_nat)))),
% 2.53/1.55 file('/export/starexec/sandbox2/tmp/tmp.5TcjDfaA5h/DTF2THF_25639.p',fact_1_corr__p)).
% 2.53/1.55 thf(f1375,plain,(
% 2.53/1.55 ((genClo1015804716orrect @ p @ (genClo1163638703lle_te @ p @ (plus_plus_nat @ i @ one_one_nat))) != $true) | ((genClo1015804716orrect @ q @ (genClo1163638703lle_te @ q @ (plus_plus_nat @ i @ one_one_nat))) != $true)),
% 2.53/1.55 inference(trivial_inequality_removal,[status(thm)],[f1371])).
% 2.53/1.55 thf(f1371,plain,(
% 2.53/1.55 ((genClo1015804716orrect @ p @ (genClo1163638703lle_te @ p @ (plus_plus_nat @ i @ one_one_nat))) != $true) | ((genClo1015804716orrect @ q @ (genClo1163638703lle_te @ q @ (plus_plus_nat @ i @ one_one_nat))) != $true) | ($true != $true)),
% 2.53/1.55 inference(superposition,[status(thm)],[f1125,f1170])).
% 2.53/1.55 thf(f1170,plain,(
% 2.53/1.55 ( ! [X2 : nat,X0 : nat,X1 : nat] : (((ord_less_eq_real @ (abs_abs_real @ (minus_minus_real @ (genClo1163638703lle_te @ X0 @ X2) @ (genClo1163638703lle_te @ X1 @ X2))) @ genClo1278781456e_beta) = $true) | ((genClo1015804716orrect @ X0 @ (genClo1163638703lle_te @ X0 @ X2)) != $true) | ((genClo1015804716orrect @ X1 @ (genClo1163638703lle_te @ X1 @ X2)) != $true)) )),
% 2.53/1.55 inference(cnf_transformation,[status(thm)],[f909])).
% 2.53/1.55 thf(f909,plain,(
% 2.53/1.55 ! [X0 : nat,X1 : nat,X2 : nat] : (((genClo1015804716orrect @ X1 @ (genClo1163638703lle_te @ X1 @ X2)) != $true) | ((ord_less_eq_real @ (abs_abs_real @ (minus_minus_real @ (genClo1163638703lle_te @ X0 @ X2) @ (genClo1163638703lle_te @ X1 @ X2))) @ genClo1278781456e_beta) = $true) | ((genClo1015804716orrect @ X0 @ (genClo1163638703lle_te @ X0 @ X2)) != $true))),
% 2.53/1.55 inference(flattening,[status(thm)],[f908])).
% 2.53/1.55 thf(f908,plain,(
% 2.53/1.55 ! [X0 : nat,X1 : nat,X2 : nat] : (((ord_less_eq_real @ (abs_abs_real @ (minus_minus_real @ (genClo1163638703lle_te @ X0 @ X2) @ (genClo1163638703lle_te @ X1 @ X2))) @ genClo1278781456e_beta) = $true) | (((genClo1015804716orrect @ X1 @ (genClo1163638703lle_te @ X1 @ X2)) != $true) | ((genClo1015804716orrect @ X0 @ (genClo1163638703lle_te @ X0 @ X2)) != $true)))),
% 2.53/1.55 inference(ennf_transformation,[status(thm)],[f620])).
% 2.53/1.55 thf(f620,plain,(
% 2.53/1.55 ! [X0 : nat,X1 : nat,X2 : nat] : ((((genClo1015804716orrect @ X1 @ (genClo1163638703lle_te @ X1 @ X2)) = $true) & ((genClo1015804716orrect @ X0 @ (genClo1163638703lle_te @ X0 @ X2)) = $true)) => ((ord_less_eq_real @ (abs_abs_real @ (minus_minus_real @ (genClo1163638703lle_te @ X0 @ X2) @ (genClo1163638703lle_te @ X1 @ X2))) @ genClo1278781456e_beta) = $true))),
% 2.53/1.55 inference(fool_elimination,[status(thm)],[f619])).
% 2.53/1.55 thf(f619,plain,(
% 2.53/1.55 ! [X0 : nat,X1 : nat,X2 : nat] : (((genClo1015804716orrect @ X0 @ (genClo1163638703lle_te @ X0 @ X2)) & (genClo1015804716orrect @ X1 @ (genClo1163638703lle_te @ X1 @ X2))) => (ord_less_eq_real @ (abs_abs_real @ (minus_minus_real @ (genClo1163638703lle_te @ X0 @ X2) @ (genClo1163638703lle_te @ X1 @ X2))) @ genClo1278781456e_beta))),
% 2.53/1.55 inference(rectify,[status(thm)],[f1])).
% 2.53/1.55 thf(f1,axiom,(
% 2.53/1.55 ! [X0 : nat,X1 : nat,X2 : nat] : (((genClo1015804716orrect @ X0 @ (genClo1163638703lle_te @ X0 @ X2)) & (genClo1015804716orrect @ X1 @ (genClo1163638703lle_te @ X1 @ X2))) => (ord_less_eq_real @ (abs_abs_real @ (minus_minus_real @ (genClo1163638703lle_te @ X0 @ X2) @ (genClo1163638703lle_te @ X1 @ X2))) @ genClo1278781456e_beta))),
% 2.53/1.55 file('/export/starexec/sandbox2/tmp/tmp.5TcjDfaA5h/DTF2THF_25639.p',fact_0_rts2b)).
% 2.53/1.55 thf(f1125,plain,(
% 2.53/1.55 ((ord_less_eq_real @ (abs_abs_real @ (minus_minus_real @ (genClo1163638703lle_te @ p @ (plus_plus_nat @ i @ one_one_nat)) @ (genClo1163638703lle_te @ q @ (plus_plus_nat @ i @ one_one_nat)))) @ genClo1278781456e_beta) != $true)),
% 2.53/1.55 inference(cnf_transformation,[status(thm)],[f729])).
% 2.53/1.55 thf(f729,plain,(
% 2.53/1.55 ((ord_less_eq_real @ (abs_abs_real @ (minus_minus_real @ (genClo1163638703lle_te @ p @ (plus_plus_nat @ i @ one_one_nat)) @ (genClo1163638703lle_te @ q @ (plus_plus_nat @ i @ one_one_nat)))) @ genClo1278781456e_beta) != $true)),
% 2.53/1.55 inference(flattening,[status(thm)],[f672])).
% 2.53/1.55 thf(f672,plain,(
% 2.53/1.55 ~((ord_less_eq_real @ (abs_abs_real @ (minus_minus_real @ (genClo1163638703lle_te @ p @ (plus_plus_nat @ i @ one_one_nat)) @ (genClo1163638703lle_te @ q @ (plus_plus_nat @ i @ one_one_nat)))) @ genClo1278781456e_beta) = $true)),
% 2.53/1.55 inference(fool_elimination,[status(thm)],[f671])).
% 2.53/1.55 thf(f671,plain,(
% 2.53/1.55 ~(ord_less_eq_real @ (abs_abs_real @ (minus_minus_real @ (genClo1163638703lle_te @ p @ (plus_plus_nat @ i @ one_one_nat)) @ (genClo1163638703lle_te @ q @ (plus_plus_nat @ i @ one_one_nat)))) @ genClo1278781456e_beta)),
% 2.53/1.55 inference(rectify,[status(thm)],[f339])).
% 2.53/1.55 thf(f339,negated_conjecture,(
% 2.53/1.55 ~(ord_less_eq_real @ (abs_abs_real @ (minus_minus_real @ (genClo1163638703lle_te @ p @ (plus_plus_nat @ i @ one_one_nat)) @ (genClo1163638703lle_te @ q @ (plus_plus_nat @ i @ one_one_nat)))) @ genClo1278781456e_beta)),
% 2.53/1.55 inference(negated_conjecture,[status(cth)],[f338])).
% 2.53/1.55 thf(f338,conjecture,(
% 2.53/1.55 (ord_less_eq_real @ (abs_abs_real @ (minus_minus_real @ (genClo1163638703lle_te @ p @ (plus_plus_nat @ i @ one_one_nat)) @ (genClo1163638703lle_te @ q @ (plus_plus_nat @ i @ one_one_nat)))) @ genClo1278781456e_beta)),
% 2.53/1.55 file('/export/starexec/sandbox2/tmp/tmp.5TcjDfaA5h/DTF2THF_25639.p',conj_0)).
% 2.53/1.55 % SZS output end Proof for DTF2THF_25639
% 2.53/1.55 % (25823)------------------------------
% 2.53/1.55 % (25823)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.53/1.55 % (25823)Termination reason: Refutation
% 2.53/1.55
% 2.53/1.55 % (25823)Memory used [KB]: 6524
% 2.53/1.55 % (25823)Time elapsed: 0.028 s
% 2.53/1.55 % (25823)Instructions burned: 77 (million)
% 2.53/1.55 % (25823)------------------------------
% 2.53/1.55 % (25823)------------------------------
% 2.53/1.55 % (25815)Success in time 0.042 s
%------------------------------------------------------------------------------