%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : ITP064^1 : 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 : n020.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 2.78s 1.61s
% Output : Refutation 2.78s
% Verified :
% SZS Type : Refutation
% Derivation depth : 12
% Number of leaves : 12
% Syntax : Number of formulae : 65 ( 21 unt; 0 typ; 4 def)
% Number of atoms : 246 ( 71 equ; 0 cnn)
% Maximal formula atoms : 4 ( 3 avg)
% Number of connectives : 444 ( 43 ~; 43 |; 3 &; 342 @)
% ( 4 <=>; 9 =>; 0 <=; 0 <~>)
% Maximal formula depth : 8 ( 4 avg)
% Number of types : 3 ( 2 usr)
% Number of type conns : 0 ( 0 >; 0 *; 0 +; 0 <<)
% Number of symbols : 72 ( 69 usr; 19 con; 0-4 aty)
% Number of variables : 50 ( 0 ^; 50 !; 0 ?; 50 :)
% 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,
genClo1160817912lle_IC: nat > nat > real > real ).
thf(func_def_3,type,
genClo721845095Lambda: real ).
thf(func_def_4,type,
genClo1278781456e_beta: real ).
thf(func_def_5,type,
genClo1400225944_theta: nat > nat > nat > real ).
thf(func_def_6,type,
genClo1015804716orrect: nat > real > $o ).
thf(func_def_7,type,
genClo293725281kRead1: ( nat > real ) > real > ( nat > $o ) > $o ).
thf(func_def_8,type,
genClo293725282kRead2: ( nat > real ) > ( nat > real ) > real > ( nat > $o ) > $o ).
thf(func_def_9,type,
genClo208577157axsync: nat > real > $o ).
thf(func_def_10,type,
genClo1650508560e_rmax: real ).
thf(func_def_11,type,
genClo1651033342e_rmin: real ).
thf(func_def_12,type,
genClo1163638703lle_te: nat > nat > real ).
thf(func_def_13,type,
abs_abs_real: real > real ).
thf(func_def_14,type,
minus_minus_nat: nat > nat > nat ).
thf(func_def_15,type,
minus_minus_real: real > real > real ).
thf(func_def_16,type,
one_one_nat: nat ).
thf(func_def_17,type,
one_one_real: real ).
thf(func_def_18,type,
plus_plus_nat: nat > nat > nat ).
thf(func_def_19,type,
plus_plus_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,
if_nat: $o > nat > nat > nat ).
thf(func_def_23,type,
if_real: $o > real > real > real ).
thf(func_def_24,type,
ord_less_eq_nat: nat > nat > $o ).
thf(func_def_25,type,
ord_less_eq_real: real > real > $o ).
thf(func_def_26,type,
ord_max_nat: nat > nat > nat ).
thf(func_def_27,type,
ord_max_real: real > real > real ).
thf(func_def_28,type,
delta_S: real ).
thf(func_def_29,type,
i: nat ).
thf(func_def_30,type,
l: nat ).
thf(func_def_31,type,
m: nat ).
thf(func_def_32,type,
p: nat ).
thf(func_def_47,type,
sP0: real > real > real > $o ).
thf(func_def_48,type,
sP1: real > real > real > $o ).
thf(func_def_49,type,
sP2: real > real > real > $o ).
thf(func_def_50,type,
sK3: ( real > real ) > real ).
thf(func_def_51,type,
sK4: ( real > real ) > real ).
thf(func_def_52,type,
sK5: ( real > nat ) > real ).
thf(func_def_53,type,
sK6: ( real > nat ) > real ).
thf(func_def_54,type,
sK7: ( real > nat ) > real ).
thf(func_def_55,type,
sK8: ( real > nat ) > real ).
thf(func_def_56,type,
sK9: ( nat > real ) > nat ).
thf(func_def_57,type,
sK10: ( nat > real ) > nat ).
thf(func_def_58,type,
sK11: ( real > real ) > real ).
thf(func_def_59,type,
sK12: ( real > real ) > real ).
thf(func_def_60,type,
sK13: ( real > real ) > real ).
thf(func_def_61,type,
sK14: ( real > real ) > real ).
thf(func_def_62,type,
sK15: ( nat > real ) > nat ).
thf(func_def_63,type,
sK16: ( nat > real ) > nat ).
thf(func_def_64,type,
sK17: nat > nat > nat ).
thf(func_def_65,type,
sK18: nat > nat > nat ).
thf(func_def_66,type,
sK19: ( real > nat ) > real ).
thf(func_def_67,type,
sK20: ( real > nat ) > real ).
thf(func_def_68,type,
sK21: ( nat > real ) > nat ).
thf(func_def_69,type,
sK22: ( nat > real ) > nat ).
thf(func_def_70,type,
sK23: ( real > real > $o ) > real ).
thf(func_def_71,type,
sK24: ( real > real > $o ) > real ).
thf(func_def_72,type,
sK25: ( real > real > $o ) > real ).
thf(func_def_73,type,
sK26: ( real > real > $o ) > real ).
thf(func_def_74,type,
sK27: ( real > real ) > real ).
thf(func_def_75,type,
sK28: ( real > real ) > real ).
thf(func_def_76,type,
sK29: ( real > nat ) > real ).
thf(func_def_77,type,
sK30: ( real > nat ) > real ).
thf(func_def_78,type,
sK31: ( nat > real ) > nat ).
thf(func_def_79,type,
sK32: ( nat > real ) > nat ).
thf(func_def_81,type,
ph34:
!>[X0: $tType] : X0 ).
thf(f1513,plain,
$false,
inference(avatar_sat_refutation,[status(thm)],[f1365,f1366,f1384,f1462,f1512]) ).
thf(f1512,plain,
spl33_3,
inference(avatar_contradiction_clause,[status(thm)],[f1511]) ).
thf(f1511,plain,
( $false
| spl33_3 ),
inference(subsumption_resolution,[status(thm)],[f1510,f1115]) ).
thf(f1115,plain,
( ( genClo1015804716orrect @ m @ ( genClo1163638703lle_te @ p @ ( plus_plus_nat @ i @ one_one_nat ) ) )
= $true ),
inference(cnf_transformation,[status(thm)],[f471]) ).
thf(f471,plain,
( ( genClo1015804716orrect @ m @ ( genClo1163638703lle_te @ p @ ( plus_plus_nat @ i @ one_one_nat ) ) )
= $true ),
inference(fool_elimination,[status(thm)],[f470]) ).
thf(f470,plain,
genClo1015804716orrect @ m @ ( genClo1163638703lle_te @ p @ ( plus_plus_nat @ i @ one_one_nat ) ),
inference(rectify,[status(thm)],[f2]) ).
thf(f2,axiom,
genClo1015804716orrect @ m @ ( genClo1163638703lle_te @ p @ ( plus_plus_nat @ i @ one_one_nat ) ),
file('/export/starexec/sandbox2/tmp/tmp.k9LPPfwlmN/DTF2THF_777.p',fact_1_corr__m) ).
thf(f1510,plain,
( ( ( genClo1015804716orrect @ m @ ( genClo1163638703lle_te @ p @ ( plus_plus_nat @ i @ one_one_nat ) ) )
!= $true )
| spl33_3 ),
inference(trivial_inequality_removal,[status(thm)],[f1472]) ).
thf(f1472,plain,
( ( $true != $true )
| ( ( genClo1015804716orrect @ m @ ( genClo1163638703lle_te @ p @ ( plus_plus_nat @ i @ one_one_nat ) ) )
!= $true )
| spl33_3 ),
inference(superposition,[status(thm)],[f1386,f1239]) ).
thf(f1239,plain,
( ( ord_less_eq_real @ ( genClo1163638703lle_te @ m @ i ) @ ( genClo1163638703lle_te @ p @ ( plus_plus_nat @ i @ one_one_nat ) ) )
= $true ),
inference(cnf_transformation,[status(thm)],[f571]) ).
thf(f571,plain,
( ( ord_less_eq_real @ ( genClo1163638703lle_te @ m @ i ) @ ( genClo1163638703lle_te @ p @ ( plus_plus_nat @ i @ one_one_nat ) ) )
= $true ),
inference(fool_elimination,[status(thm)],[f570]) ).
thf(f570,plain,
ord_less_eq_real @ ( genClo1163638703lle_te @ m @ i ) @ ( genClo1163638703lle_te @ p @ ( plus_plus_nat @ i @ one_one_nat ) ),
inference(rectify,[status(thm)],[f7]) ).
thf(f7,axiom,
ord_less_eq_real @ ( genClo1163638703lle_te @ m @ i ) @ ( genClo1163638703lle_te @ p @ ( plus_plus_nat @ i @ one_one_nat ) ),
file('/export/starexec/sandbox2/tmp/tmp.k9LPPfwlmN/DTF2THF_777.p',fact_6_tmi__le__tp) ).
thf(f1386,plain,
( ! [X0: real] :
( ( $true
!= ( ord_less_eq_real @ ( genClo1163638703lle_te @ m @ i ) @ X0 ) )
| ( ( genClo1015804716orrect @ m @ X0 )
!= $true ) )
| spl33_3 ),
inference(trivial_inequality_removal,[status(thm)],[f1385]) ).
thf(f1385,plain,
( ! [X0: real] :
( ( $true
!= ( ord_less_eq_real @ ( genClo1163638703lle_te @ m @ i ) @ X0 ) )
| ( $true != $true )
| ( ( genClo1015804716orrect @ m @ X0 )
!= $true ) )
| spl33_3 ),
inference(superposition,[status(thm)],[f1358,f1276]) ).
thf(f1276,plain,
! [X2: real,X0: nat,X1: real] :
( ( ( genClo1015804716orrect @ X0 @ X2 )
= $true )
| ( ( genClo1015804716orrect @ X0 @ X1 )
!= $true )
| ( ( ord_less_eq_real @ X2 @ X1 )
!= $true ) ),
inference(cnf_transformation,[status(thm)],[f933]) ).
thf(f933,plain,
! [X0: nat,X1: real,X2: real] :
( ( ( ord_less_eq_real @ X2 @ X1 )
!= $true )
| ( ( genClo1015804716orrect @ X0 @ X1 )
!= $true )
| ( ( genClo1015804716orrect @ X0 @ X2 )
= $true ) ),
inference(flattening,[status(thm)],[f932]) ).
thf(f932,plain,
! [X0: nat,X1: real,X2: real] :
( ( ( genClo1015804716orrect @ X0 @ X2 )
= $true )
| ( ( ord_less_eq_real @ X2 @ X1 )
!= $true )
| ( ( genClo1015804716orrect @ X0 @ X1 )
!= $true ) ),
inference(ennf_transformation,[status(thm)],[f709]) ).
thf(f709,plain,
! [X0: nat,X1: real,X2: real] :
( ( ( ( ord_less_eq_real @ X2 @ X1 )
= $true )
& ( ( genClo1015804716orrect @ X0 @ X1 )
= $true ) )
=> ( ( genClo1015804716orrect @ X0 @ X2 )
= $true ) ),
inference(fool_elimination,[status(thm)],[f708]) ).
thf(f708,plain,
! [X0: nat,X1: real,X2: real] :
( ( ( ord_less_eq_real @ X2 @ X1 )
& ( genClo1015804716orrect @ X0 @ X1 ) )
=> ( genClo1015804716orrect @ X0 @ X2 ) ),
inference(rectify,[status(thm)],[f1]) ).
thf(f1,axiom,
! [X0: nat,X2: real,X1: real] :
( ( ( ord_less_eq_real @ X1 @ X2 )
& ( genClo1015804716orrect @ X0 @ X2 ) )
=> ( genClo1015804716orrect @ X0 @ X1 ) ),
file('/export/starexec/sandbox2/tmp/tmp.k9LPPfwlmN/DTF2THF_777.p',fact_0_correct__closed) ).
thf(f1358,plain,
( ( ( genClo1015804716orrect @ m @ ( genClo1163638703lle_te @ m @ i ) )
!= $true )
| spl33_3 ),
inference(avatar_component_clause,[status(thm)],[f1356]) ).
thf(f1356,definition,
( spl33_3
<=> ( ( genClo1015804716orrect @ m @ ( genClo1163638703lle_te @ m @ i ) )
= $true ) ),
introduced(definition,[new_symbols(naming,[spl33_3])],[avatar_definition]) ).
thf(f1462,plain,
spl33_1,
inference(avatar_contradiction_clause,[status(thm)],[f1461]) ).
thf(f1461,plain,
( $false
| spl33_1 ),
inference(subsumption_resolution,[status(thm)],[f1454,f1115]) ).
thf(f1454,plain,
( ( ( genClo1015804716orrect @ m @ ( genClo1163638703lle_te @ p @ ( plus_plus_nat @ i @ one_one_nat ) ) )
!= $true )
| spl33_1 ),
inference(trivial_inequality_removal,[status(thm)],[f1421]) ).
thf(f1421,plain,
( ( $true != $true )
| ( ( genClo1015804716orrect @ m @ ( genClo1163638703lle_te @ p @ ( plus_plus_nat @ i @ one_one_nat ) ) )
!= $true )
| spl33_1 ),
inference(superposition,[status(thm)],[f1368,f1219]) ).
thf(f1219,plain,
( ( ord_less_eq_real @ ( genClo1163638703lle_te @ l @ i ) @ ( genClo1163638703lle_te @ p @ ( plus_plus_nat @ i @ one_one_nat ) ) )
= $true ),
inference(cnf_transformation,[status(thm)],[f823]) ).
thf(f823,plain,
( ( ord_less_eq_real @ ( genClo1163638703lle_te @ l @ i ) @ ( genClo1163638703lle_te @ p @ ( plus_plus_nat @ i @ one_one_nat ) ) )
= $true ),
inference(fool_elimination,[status(thm)],[f822]) ).
thf(f822,plain,
ord_less_eq_real @ ( genClo1163638703lle_te @ l @ i ) @ ( genClo1163638703lle_te @ p @ ( plus_plus_nat @ i @ one_one_nat ) ),
inference(rectify,[status(thm)],[f8]) ).
thf(f8,axiom,
ord_less_eq_real @ ( genClo1163638703lle_te @ l @ i ) @ ( genClo1163638703lle_te @ p @ ( plus_plus_nat @ i @ one_one_nat ) ),
file('/export/starexec/sandbox2/tmp/tmp.k9LPPfwlmN/DTF2THF_777.p',fact_7_tli__le__tp) ).
thf(f1368,plain,
( ! [X0: real] :
( ( ( ord_less_eq_real @ ( genClo1163638703lle_te @ l @ i ) @ X0 )
!= $true )
| ( ( genClo1015804716orrect @ m @ X0 )
!= $true ) )
| spl33_1 ),
inference(trivial_inequality_removal,[status(thm)],[f1367]) ).
thf(f1367,plain,
( ! [X0: real] :
( ( ( ord_less_eq_real @ ( genClo1163638703lle_te @ l @ i ) @ X0 )
!= $true )
| ( ( genClo1015804716orrect @ m @ X0 )
!= $true )
| ( $true != $true ) )
| spl33_1 ),
inference(superposition,[status(thm)],[f1349,f1276]) ).
thf(f1349,plain,
( ( ( genClo1015804716orrect @ m @ ( genClo1163638703lle_te @ l @ i ) )
!= $true )
| spl33_1 ),
inference(avatar_component_clause,[status(thm)],[f1347]) ).
thf(f1347,definition,
( spl33_1
<=> ( ( genClo1015804716orrect @ m @ ( genClo1163638703lle_te @ l @ i ) )
= $true ) ),
introduced(definition,[new_symbols(naming,[spl33_1])],[avatar_definition]) ).
thf(f1384,plain,
( spl33_2
| spl33_4 ),
inference(avatar_split_clause,[status(thm)],[f1379,f1360,f1351]) ).
thf(f1351,definition,
( spl33_2
<=> ( ( ord_less_eq_real @ ( genClo1163638703lle_te @ m @ i ) @ ( genClo1163638703lle_te @ l @ i ) )
= $true ) ),
introduced(definition,[new_symbols(naming,[spl33_2])],[avatar_definition]) ).
thf(f1360,definition,
( spl33_4
<=> ( ( ord_less_eq_real @ ( genClo1163638703lle_te @ l @ i ) @ ( genClo1163638703lle_te @ m @ i ) )
= $true ) ),
introduced(definition,[new_symbols(naming,[spl33_4])],[avatar_definition]) ).
thf(f1379,plain,
( ( ( ord_less_eq_real @ ( genClo1163638703lle_te @ m @ i ) @ ( genClo1163638703lle_te @ l @ i ) )
= $true )
| spl33_4 ),
inference(trivial_inequality_removal,[status(thm)],[f1372]) ).
thf(f1372,plain,
( ( ( ord_less_eq_real @ ( genClo1163638703lle_te @ m @ i ) @ ( genClo1163638703lle_te @ l @ i ) )
= $true )
| ( $true != $true )
| spl33_4 ),
inference(superposition,[status(thm)],[f1362,f1225]) ).
thf(f1225,plain,
! [X0: real,X1: real] :
( ( $true
= ( ord_less_eq_real @ X1 @ X0 ) )
| ( ( ord_less_eq_real @ X0 @ X1 )
= $true ) ),
inference(cnf_transformation,[status(thm)],[f861]) ).
thf(f861,plain,
! [X0: real,X1: real] :
( ( ( ord_less_eq_real @ X0 @ X1 )
= $true )
| ( $true
= ( ord_less_eq_real @ X1 @ X0 ) ) ),
inference(fool_elimination,[status(thm)],[f860]) ).
thf(f860,plain,
! [X0: real,X1: real] :
( ( ord_less_eq_real @ X0 @ X1 )
| ( ord_less_eq_real @ X1 @ X0 ) ),
inference(rectify,[status(thm)],[f49]) ).
thf(f49,axiom,
! [X10: real,X11: real] :
( ( ord_less_eq_real @ X10 @ X11 )
| ( ord_less_eq_real @ X11 @ X10 ) ),
file('/export/starexec/sandbox2/tmp/tmp.k9LPPfwlmN/DTF2THF_777.p',fact_48_linear) ).
thf(f1362,plain,
( ( ( ord_less_eq_real @ ( genClo1163638703lle_te @ l @ i ) @ ( genClo1163638703lle_te @ m @ i ) )
!= $true )
| spl33_4 ),
inference(avatar_component_clause,[status(thm)],[f1360]) ).
thf(f1366,plain,
( ~ spl33_3
| ~ spl33_4 ),
inference(avatar_split_clause,[status(thm)],[f1342,f1360,f1356]) ).
thf(f1342,plain,
( ( ( ord_less_eq_real @ ( genClo1163638703lle_te @ l @ i ) @ ( genClo1163638703lle_te @ m @ i ) )
!= $true )
| ( ( genClo1015804716orrect @ m @ ( genClo1163638703lle_te @ m @ i ) )
!= $true ) ),
inference(superposition,[status(thm)],[f1139,f1162]) ).
thf(f1162,plain,
! [X0: real,X1: real] :
( ( ( ord_max_real @ X1 @ X0 )
= X0 )
| ( $true
!= ( ord_less_eq_real @ X1 @ X0 ) ) ),
inference(cnf_transformation,[status(thm)],[f923]) ).
thf(f923,plain,
! [X0: real,X1: real] :
( ( ( ord_max_real @ X1 @ X0 )
= X0 )
| ( $true
!= ( ord_less_eq_real @ X1 @ X0 ) ) ),
inference(ennf_transformation,[status(thm)],[f833]) ).
thf(f833,plain,
! [X0: real,X1: real] :
( ( $true
= ( ord_less_eq_real @ X1 @ X0 ) )
=> ( ( ord_max_real @ X1 @ X0 )
= X0 ) ),
inference(fool_elimination,[status(thm)],[f832]) ).
thf(f832,plain,
! [X0: real,X1: real] :
( ( ord_less_eq_real @ X1 @ X0 )
=> ( ( ord_max_real @ X1 @ X0 )
= X0 ) ),
inference(rectify,[status(thm)],[f101]) ).
thf(f101,axiom,
! [X4: real,X3: real] :
( ( ord_less_eq_real @ X3 @ X4 )
=> ( ( ord_max_real @ X3 @ X4 )
= X4 ) ),
file('/export/starexec/sandbox2/tmp/tmp.k9LPPfwlmN/DTF2THF_777.p',fact_100_max_Oabsorb2) ).
thf(f1139,plain,
( ( genClo1015804716orrect @ m @ ( ord_max_real @ ( genClo1163638703lle_te @ l @ i ) @ ( genClo1163638703lle_te @ m @ i ) ) )
!= $true ),
inference(cnf_transformation,[status(thm)],[f862]) ).
thf(f862,plain,
( ( genClo1015804716orrect @ m @ ( ord_max_real @ ( genClo1163638703lle_te @ l @ i ) @ ( genClo1163638703lle_te @ m @ i ) ) )
!= $true ),
inference(flattening,[status(thm)],[f857]) ).
thf(f857,plain,
( ( genClo1015804716orrect @ m @ ( ord_max_real @ ( genClo1163638703lle_te @ l @ i ) @ ( genClo1163638703lle_te @ m @ i ) ) )
!= $true ),
inference(fool_elimination,[status(thm)],[f856]) ).
thf(f856,plain,
~ ( genClo1015804716orrect @ m @ ( ord_max_real @ ( genClo1163638703lle_te @ l @ i ) @ ( genClo1163638703lle_te @ m @ i ) ) ),
inference(rectify,[status(thm)],[f360]) ).
thf(f360,negated_conjecture,
~ ( genClo1015804716orrect @ m @ ( ord_max_real @ ( genClo1163638703lle_te @ l @ i ) @ ( genClo1163638703lle_te @ m @ i ) ) ),
inference(negated_conjecture,[status(cth)],[f359]) ).
thf(f359,conjecture,
genClo1015804716orrect @ m @ ( ord_max_real @ ( genClo1163638703lle_te @ l @ i ) @ ( genClo1163638703lle_te @ m @ i ) ),
file('/export/starexec/sandbox2/tmp/tmp.k9LPPfwlmN/DTF2THF_777.p',conj_0) ).
thf(f1365,plain,
( ~ spl33_2
| ~ spl33_1 ),
inference(avatar_split_clause,[status(thm)],[f1339,f1347,f1351]) ).
thf(f1339,plain,
( ( ( ord_less_eq_real @ ( genClo1163638703lle_te @ m @ i ) @ ( genClo1163638703lle_te @ l @ i ) )
!= $true )
| ( ( genClo1015804716orrect @ m @ ( genClo1163638703lle_te @ l @ i ) )
!= $true ) ),
inference(superposition,[status(thm)],[f1139,f1104]) ).
thf(f1104,plain,
! [X0: real,X1: real] :
( ( ( ord_max_real @ X0 @ X1 )
= X0 )
| ( $true
!= ( ord_less_eq_real @ X1 @ X0 ) ) ),
inference(cnf_transformation,[status(thm)],[f965]) ).
thf(f965,plain,
! [X0: real,X1: real] :
( ( ( ord_max_real @ X0 @ X1 )
= X0 )
| ( $true
!= ( ord_less_eq_real @ X1 @ X0 ) ) ),
inference(ennf_transformation,[status(thm)],[f603]) ).
thf(f603,plain,
! [X0: real,X1: real] :
( ( $true
= ( ord_less_eq_real @ X1 @ X0 ) )
=> ( ( ord_max_real @ X0 @ X1 )
= X0 ) ),
inference(fool_elimination,[status(thm)],[f602]) ).
thf(f602,plain,
! [X0: real,X1: real] :
( ( ord_less_eq_real @ X1 @ X0 )
=> ( ( ord_max_real @ X0 @ X1 )
= X0 ) ),
inference(rectify,[status(thm)],[f103]) ).
thf(f103,axiom,
! [X3: real,X4: real] :
( ( ord_less_eq_real @ X4 @ X3 )
=> ( ( ord_max_real @ X3 @ X4 )
= X3 ) ),
file('/export/starexec/sandbox2/tmp/tmp.k9LPPfwlmN/DTF2THF_777.p',fact_102_max_Oabsorb1) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.11 % Problem : ITP064^1 : TPTP v9.2.1. Released v7.5.0.
% 0.00/0.12 % Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.12/0.31 % Computer : n020.cluster.edu
% 0.12/0.31 % Model : x86_64 x86_64
% 0.12/0.31 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.12/0.31 % Memory : 8042.1875MB
% 0.12/0.31 % OS : Linux 3.10.0-693.el7.x86_64
% 0.12/0.31 % CPULimit : 300
% 0.12/0.31 % WCLimit : 300
% 0.12/0.31 % DateTime : Tue Feb 24 08:25:27 EST 2026
% 0.12/0.32 % CPUTime :
% 0.12/0.32 Running /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.16/0.40 ---- Original DTF file ---
% 0.16/0.40 thf(spec,logic,$$dhol).
% 0.16/0.40 %------------------------------------------------------------------------------
% 0.16/0.40 % File : ITP064^1 : TPTP v9.2.1. Released v7.5.0.
% 0.16/0.40 % Domain : Interactive Theorem Proving
% 0.16/0.40 % Problem : Sledgehammer GenClock problem prob_727__3251548_1
% 0.16/0.40 % Version : Especial.
% 0.16/0.40 % English :
% 0.16/0.40
% 0.16/0.40 % Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% 0.16/0.40 % : [Des21] Desharnais (2021), Email to Geoff Sutcliffe
% 0.16/0.40 % Source : [Des21]
% 0.16/0.40 % Names : GenClock/prob_727__3251548_1 [Des21]
% 0.16/0.40
% 0.16/0.40 % Status : Theorem
% 0.16/0.40 % Rating : 0.22 v9.1.0, 0.12 v9.0.0, 0.30 v8.2.0, 0.23 v8.1.0, 0.27 v7.5.0
% 0.16/0.40 % Syntax : Number of formulae : 392 ( 188 unt; 33 typ; 0 def)
% 0.16/0.40 % Number of atoms : 855 ( 341 equ; 0 cnn)
% 0.16/0.40 % Maximal formula atoms : 12 ( 2 avg)
% 0.16/0.40 % Number of connectives : 2769 ( 26 ~; 6 |; 29 &;2404 @)
% 0.16/0.40 % ( 0 <=>; 304 =>; 0 <=; 0 <~>)
% 0.16/0.40 % Maximal formula depth : 14 ( 6 avg)
% 0.16/0.40 % Number of types : 3 ( 2 usr)
% 0.16/0.40 % Number of type conns : 73 ( 73 >; 0 *; 0 +; 0 <<)
% 0.16/0.40 % Number of symbols : 34 ( 31 usr; 15 con; 0-4 aty)
% 0.16/0.40 % Number of variables : 937 ( 65 ^; 868 !; 4 ?; 937 :)
% 0.16/0.40 % SPC : TH0_THM_EQU_NAR
% 0.16/0.40
% 0.16/0.40 % Comments : This file was generated by Sledgehammer 2021-02-23 15:31:14.669
% 0.16/0.40 %------------------------------------------------------------------------------
% 0.16/0.40 % Could-be-implicit typings (2)
% 0.16/0.40 thf(ty_n_t__Real__Oreal,type,
% 0.16/0.40 real: $tType ).
% 0.16/0.40
% 0.16/0.40 thf(ty_n_t__Nat__Onat,type,
% 0.16/0.40 nat: $tType ).
% 0.16/0.40
% 0.16/0.40 % Explicit typings (31)
% 0.16/0.40 thf(sy_c_GenClock__Mirabelle__bsvkzpgbls_OIC,type,
% 0.16/0.40 genClo1160817912lle_IC: nat > nat > real > real ).
% 0.16/0.40
% 0.16/0.40 thf(sy_c_GenClock__Mirabelle__bsvkzpgbls_O_092_060Lambda_062,type,
% 0.16/0.40 genClo721845095Lambda: real ).
% 0.16/0.40
% 0.16/0.40 thf(sy_c_GenClock__Mirabelle__bsvkzpgbls_O_092_060beta_062,type,
% 0.16/0.40 genClo1278781456e_beta: real ).
% 0.16/0.40
% 0.16/0.40 thf(sy_c_GenClock__Mirabelle__bsvkzpgbls_O_092_060theta_062,type,
% 0.16/0.40 genClo1400225944_theta: nat > nat > nat > real ).
% 0.16/0.40
% 0.16/0.40 thf(sy_c_GenClock__Mirabelle__bsvkzpgbls_Ocorrect,type,
% 0.16/0.40 genClo1015804716orrect: nat > real > $o ).
% 0.16/0.40
% 0.16/0.40 thf(sy_c_GenClock__Mirabelle__bsvkzpgbls_OokRead1,type,
% 0.16/0.40 genClo293725281kRead1: ( nat > real ) > real > ( nat > $o ) > $o ).
% 0.16/0.40
% 0.16/0.40 thf(sy_c_GenClock__Mirabelle__bsvkzpgbls_OokRead2,type,
% 0.16/0.40 genClo293725282kRead2: ( nat > real ) > ( nat > real ) > real > ( nat > $o ) > $o ).
% 0.16/0.40
% 0.16/0.40 thf(sy_c_GenClock__Mirabelle__bsvkzpgbls_Ookmaxsync,type,
% 0.16/0.40 genClo208577157axsync: nat > real > $o ).
% 0.16/0.40
% 0.16/0.40 thf(sy_c_GenClock__Mirabelle__bsvkzpgbls_Ormax,type,
% 0.16/0.40 genClo1650508560e_rmax: real ).
% 0.16/0.40
% 0.16/0.40 thf(sy_c_GenClock__Mirabelle__bsvkzpgbls_Ormin,type,
% 0.16/0.40 genClo1651033342e_rmin: real ).
% 0.16/0.40
% 0.16/0.40 thf(sy_c_GenClock__Mirabelle__bsvkzpgbls_Ote,type,
% 0.16/0.40 genClo1163638703lle_te: nat > nat > real ).
% 0.16/0.40
% 0.16/0.40 thf(sy_c_Groups_Oabs__class_Oabs_001t__Real__Oreal,type,
% 0.16/0.40 abs_abs_real: real > real ).
% 0.16/0.40
% 0.16/0.40 thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat,type,
% 0.16/0.40 minus_minus_nat: nat > nat > nat ).
% 0.16/0.40
% 0.16/0.40 thf(sy_c_Groups_Ominus__class_Ominus_001t__Real__Oreal,type,
% 0.16/0.40 minus_minus_real: real > real > real ).
% 0.16/0.40
% 0.16/0.40 thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat,type,
% 0.16/0.40 one_one_nat: nat ).
% 0.16/0.40
% 0.16/0.40 thf(sy_c_Groups_Oone__class_Oone_001t__Real__Oreal,type,
% 0.16/0.40 one_one_real: real ).
% 0.16/0.40
% 0.16/0.40 thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat,type,
% 0.16/0.40 plus_plus_nat: nat > nat > nat ).
% 0.16/0.40
% 0.16/0.40 thf(sy_c_Groups_Oplus__class_Oplus_001t__Real__Oreal,type,
% 0.16/0.40 plus_plus_real: real > real > real ).
% 0.16/0.40
% 0.16/0.40 thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
% 0.16/0.40 zero_zero_nat: nat ).
% 0.16/0.40
% 0.16/0.40 thf(sy_c_Groups_Ozero__class_Ozero_001t__Real__Oreal,type,
% 0.16/0.40 zero_zero_real: real ).
% 0.16/0.40
% 0.16/0.40 thf(sy_c_If_001t__Nat__Onat,type,
% 0.16/0.40 if_nat: $o > nat > nat > nat ).
% 0.16/0.40
% 0.16/0.40 thf(sy_c_If_001t__Real__Oreal,type,
% 0.16/0.40 if_real: $o > real > real > real ).
% 0.16/0.40
% 0.16/0.40 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
% 0.16/0.40 ord_less_eq_nat: nat > nat > $o ).
% 0.16/0.40
% 0.16/0.40 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Real__Oreal,type,
% 0.16/0.40 ord_less_eq_real: real > real > $o ).
% 0.16/0.40
% 0.16/0.40 thf(sy_c_Orderings_Oord__class_Omax_001t__Nat__Onat,type,
% 0.16/0.40 ord_max_nat: nat > nat > nat ).
% 0.16/0.40
% 0.16/0.40 thf(sy_c_Orderings_Oord__class_Omax_001t__Real__Oreal,type,
% 0.16/0.40 ord_max_real: real > real > real ).
% 0.16/0.40
% 0.16/0.40 thf(sy_v__092_060delta_062S,type,
% 0.16/0.40 delta_S: real ).
% 0.16/0.40
% 0.16/0.40 thf(sy_v_i,type,
% 0.16/0.40 i: nat ).
% 0.16/0.40
% 0.16/0.40 thf(sy_v_l,type,
% 0.16/0.40 l: nat ).
% 0.16/0.40
% 0.16/0.40 thf(sy_v_m,type,
% 0.16/0.40 m: nat ).
% 0.16/0.40
% 0.16/0.40 thf(sy_v_p,type,
% 0.16/0.40 p: nat ).
% 0.16/0.40
% 0.16/0.40 % Relevant facts (353)
% 0.16/0.40 thf(fact_0_correct__closed,axiom,
% 0.16/0.40 ! [P: nat,S: real,T: real] :
% 0.16/0.40 ( ( ( ord_less_eq_real @ S @ T )
% 0.16/0.40 & ( genClo1015804716orrect @ P @ T ) )
% 0.16/0.40 => ( genClo1015804716orrect @ P @ S ) ) ).
% 0.16/0.40
% 0.16/0.40 % correct_closed
% 0.16/0.40 thf(fact_1_corr__m,axiom,
% 0.16/0.40 genClo1015804716orrect @ m @ ( genClo1163638703lle_te @ p @ ( plus_plus_nat @ i @ one_one_nat ) ) ).
% 0.16/0.40
% 0.16/0.40 % corr_m
% 0.16/0.40 thf(fact_2_corr__p,axiom,
% 0.16/0.40 genClo1015804716orrect @ p @ ( genClo1163638703lle_te @ p @ ( plus_plus_nat @ i @ one_one_nat ) ) ).
% 0.16/0.40
% 0.16/0.40 % corr_p
% 0.16/0.40 thf(fact_3_tml__le__tp,axiom,
% 0.16/0.40 ord_less_eq_real @ ( ord_max_real @ ( genClo1163638703lle_te @ l @ i ) @ ( genClo1163638703lle_te @ m @ i ) ) @ ( genClo1163638703lle_te @ p @ ( plus_plus_nat @ i @ one_one_nat ) ) ).
% 0.16/0.41
% 0.16/0.41 % tml_le_tp
% 0.16/0.41 thf(fact_4_corr__l,axiom,
% 0.16/0.41 genClo1015804716orrect @ l @ ( genClo1163638703lle_te @ p @ ( plus_plus_nat @ i @ one_one_nat ) ) ).
% 0.16/0.41
% 0.16/0.41 % corr_l
% 0.16/0.41 thf(fact_5_corr__l__tml,axiom,
% 0.16/0.41 genClo1015804716orrect @ l @ ( ord_max_real @ ( genClo1163638703lle_te @ l @ i ) @ ( genClo1163638703lle_te @ m @ i ) ) ).
% 0.16/0.41
% 0.16/0.41 % corr_l_tml
% 0.16/0.41 thf(fact_6_tmi__le__tp,axiom,
% 0.16/0.41 ord_less_eq_real @ ( genClo1163638703lle_te @ m @ i ) @ ( genClo1163638703lle_te @ p @ ( plus_plus_nat @ i @ one_one_nat ) ) ).
% 0.16/0.41
% 0.16/0.41 % tmi_le_tp
% 0.16/0.41 thf(fact_7_tli__le__tp,axiom,
% 0.16/0.41 ord_less_eq_real @ ( genClo1163638703lle_te @ l @ i ) @ ( genClo1163638703lle_te @ p @ ( plus_plus_nat @ i @ one_one_nat ) ) ).
% 0.16/0.41
% 0.16/0.41 % tli_le_tp
% 0.16/0.41 thf(fact_8_max_Oidem,axiom,
% 0.16/0.41 ! [A: real] :
% 0.16/0.41 ( ( ord_max_real @ A @ A )
% 0.16/0.41 = A ) ).
% 0.16/0.41
% 0.16/0.41 % max.idem
% 0.16/0.41 thf(fact_9_max_Oidem,axiom,
% 0.16/0.41 ! [A: nat] :
% 0.16/0.41 ( ( ord_max_nat @ A @ A )
% 0.16/0.41 = A ) ).
% 0.16/0.41
% 0.16/0.41 % max.idem
% 0.16/0.41 thf(fact_10_max_Oleft__idem,axiom,
% 0.16/0.41 ! [A: real,B: real] :
% 0.16/0.41 ( ( ord_max_real @ A @ ( ord_max_real @ A @ B ) )
% 0.16/0.41 = ( ord_max_real @ A @ B ) ) ).
% 0.16/0.41
% 0.16/0.41 % max.left_idem
% 0.16/0.41 thf(fact_11_max_Oleft__idem,axiom,
% 0.16/0.41 ! [A: nat,B: nat] :
% 0.16/0.41 ( ( ord_max_nat @ A @ ( ord_max_nat @ A @ B ) )
% 0.16/0.41 = ( ord_max_nat @ A @ B ) ) ).
% 0.16/0.41
% 0.16/0.41 % max.left_idem
% 0.16/0.41 thf(fact_12_max_Oright__idem,axiom,
% 0.16/0.41 ! [A: real,B: real] :
% 0.16/0.41 ( ( ord_max_real @ ( ord_max_real @ A @ B ) @ B )
% 0.16/0.41 = ( ord_max_real @ A @ B ) ) ).
% 0.16/0.41
% 0.16/0.41 % max.right_idem
% 0.16/0.41 thf(fact_13_max_Oright__idem,axiom,
% 0.16/0.41 ! [A: nat,B: nat] :
% 0.16/0.41 ( ( ord_max_nat @ ( ord_max_nat @ A @ B ) @ B )
% 0.16/0.41 = ( ord_max_nat @ A @ B ) ) ).
% 0.16/0.41
% 0.16/0.41 % max.right_idem
% 0.16/0.41 thf(fact_14_IC__bound,axiom,
% 0.16/0.41 ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( genClo1160817912lle_IC @ l @ i @ ( ord_max_real @ ( genClo1163638703lle_te @ l @ i ) @ ( genClo1163638703lle_te @ m @ i ) ) ) @ ( genClo1160817912lle_IC @ m @ i @ ( ord_max_real @ ( genClo1163638703lle_te @ l @ i ) @ ( genClo1163638703lle_te @ m @ i ) ) ) ) ) @ delta_S ).
% 0.16/0.41
% 0.16/0.41 % IC_bound
% 0.16/0.41 thf(fact_15_max_Oassoc,axiom,
% 0.16/0.41 ! [A: real,B: real,C: real] :
% 0.16/0.41 ( ( ord_max_real @ ( ord_max_real @ A @ B ) @ C )
% 0.16/0.41 = ( ord_max_real @ A @ ( ord_max_real @ B @ C ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % max.assoc
% 0.16/0.41 thf(fact_16_max_Oassoc,axiom,
% 0.16/0.41 ! [A: nat,B: nat,C: nat] :
% 0.16/0.41 ( ( ord_max_nat @ ( ord_max_nat @ A @ B ) @ C )
% 0.16/0.41 = ( ord_max_nat @ A @ ( ord_max_nat @ B @ C ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % max.assoc
% 0.16/0.41 thf(fact_17_max_Ocommute,axiom,
% 0.16/0.41 ( ord_max_real
% 0.16/0.41 = ( ^ [A2: real,B2: real] : ( ord_max_real @ B2 @ A2 ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % max.commute
% 0.16/0.41 thf(fact_18_max_Ocommute,axiom,
% 0.16/0.41 ( ord_max_nat
% 0.16/0.41 = ( ^ [A2: nat,B2: nat] : ( ord_max_nat @ B2 @ A2 ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % max.commute
% 0.16/0.41 thf(fact_19_max_Oleft__commute,axiom,
% 0.16/0.41 ! [B: real,A: real,C: real] :
% 0.16/0.41 ( ( ord_max_real @ B @ ( ord_max_real @ A @ C ) )
% 0.16/0.41 = ( ord_max_real @ A @ ( ord_max_real @ B @ C ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % max.left_commute
% 0.16/0.41 thf(fact_20_max_Oleft__commute,axiom,
% 0.16/0.41 ! [B: nat,A: nat,C: nat] :
% 0.16/0.41 ( ( ord_max_nat @ B @ ( ord_max_nat @ A @ C ) )
% 0.16/0.41 = ( ord_max_nat @ A @ ( ord_max_nat @ B @ C ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % max.left_commute
% 0.16/0.41 thf(fact_21_rte,axiom,
% 0.16/0.41 ! [P2: nat,I: nat] :
% 0.16/0.41 ( ( genClo1015804716orrect @ P2 @ ( genClo1163638703lle_te @ P2 @ ( plus_plus_nat @ I @ one_one_nat ) ) )
% 0.16/0.41 => ( ord_less_eq_real @ ( genClo1163638703lle_te @ P2 @ I ) @ ( genClo1163638703lle_te @ P2 @ ( plus_plus_nat @ I @ one_one_nat ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % rte
% 0.16/0.41 thf(fact_22_max_Obounded__iff,axiom,
% 0.16/0.41 ! [B: real,C: real,A: real] :
% 0.16/0.41 ( ( ord_less_eq_real @ ( ord_max_real @ B @ C ) @ A )
% 0.16/0.41 = ( ( ord_less_eq_real @ B @ A )
% 0.16/0.41 & ( ord_less_eq_real @ C @ A ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % max.bounded_iff
% 0.16/0.41 thf(fact_23_max_Obounded__iff,axiom,
% 0.16/0.41 ! [B: nat,C: nat,A: nat] :
% 0.16/0.41 ( ( ord_less_eq_nat @ ( ord_max_nat @ B @ C ) @ A )
% 0.16/0.41 = ( ( ord_less_eq_nat @ B @ A )
% 0.16/0.41 & ( ord_less_eq_nat @ C @ A ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % max.bounded_iff
% 0.16/0.41 thf(fact_24_max__absorb1,axiom,
% 0.16/0.41 ! [Y: real,X: real] :
% 0.16/0.41 ( ( ord_less_eq_real @ Y @ X )
% 0.16/0.41 => ( ( ord_max_real @ X @ Y )
% 0.16/0.41 = X ) ) ).
% 0.16/0.41
% 0.16/0.41 % max_absorb1
% 0.16/0.41 thf(fact_25_max__absorb1,axiom,
% 0.16/0.41 ! [Y: nat,X: nat] :
% 0.16/0.41 ( ( ord_less_eq_nat @ Y @ X )
% 0.16/0.41 => ( ( ord_max_nat @ X @ Y )
% 0.16/0.41 = X ) ) ).
% 0.16/0.41
% 0.16/0.41 % max_absorb1
% 0.16/0.41 thf(fact_26_order__refl,axiom,
% 0.16/0.41 ! [X: real] : ( ord_less_eq_real @ X @ X ) ).
% 0.16/0.41
% 0.16/0.41 % order_refl
% 0.16/0.41 thf(fact_27_order__refl,axiom,
% 0.16/0.41 ! [X: nat] : ( ord_less_eq_nat @ X @ X ) ).
% 0.16/0.41
% 0.16/0.41 % order_refl
% 0.16/0.41 thf(fact_28_order__subst1,axiom,
% 0.16/0.41 ! [A: real,F: real > real,B: real,C: real] :
% 0.16/0.41 ( ( ord_less_eq_real @ A @ ( F @ B ) )
% 0.16/0.41 => ( ( ord_less_eq_real @ B @ C )
% 0.16/0.41 => ( ! [X2: real,Y2: real] :
% 0.16/0.41 ( ( ord_less_eq_real @ X2 @ Y2 )
% 0.16/0.41 => ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 0.16/0.41 => ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % order_subst1
% 0.16/0.41 thf(fact_29_order__subst1,axiom,
% 0.16/0.41 ! [A: real,F: nat > real,B: nat,C: nat] :
% 0.16/0.41 ( ( ord_less_eq_real @ A @ ( F @ B ) )
% 0.16/0.41 => ( ( ord_less_eq_nat @ B @ C )
% 0.16/0.41 => ( ! [X2: nat,Y2: nat] :
% 0.16/0.41 ( ( ord_less_eq_nat @ X2 @ Y2 )
% 0.16/0.41 => ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 0.16/0.41 => ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % order_subst1
% 0.16/0.41 thf(fact_30_order__subst1,axiom,
% 0.16/0.41 ! [A: nat,F: real > nat,B: real,C: real] :
% 0.16/0.41 ( ( ord_less_eq_nat @ A @ ( F @ B ) )
% 0.16/0.41 => ( ( ord_less_eq_real @ B @ C )
% 0.16/0.41 => ( ! [X2: real,Y2: real] :
% 0.16/0.41 ( ( ord_less_eq_real @ X2 @ Y2 )
% 0.16/0.41 => ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 0.16/0.41 => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % order_subst1
% 0.16/0.41 thf(fact_31_order__subst1,axiom,
% 0.16/0.41 ! [A: nat,F: nat > nat,B: nat,C: nat] :
% 0.16/0.41 ( ( ord_less_eq_nat @ A @ ( F @ B ) )
% 0.16/0.41 => ( ( ord_less_eq_nat @ B @ C )
% 0.16/0.41 => ( ! [X2: nat,Y2: nat] :
% 0.16/0.41 ( ( ord_less_eq_nat @ X2 @ Y2 )
% 0.16/0.41 => ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 0.16/0.41 => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % order_subst1
% 0.16/0.41 thf(fact_32_order__subst2,axiom,
% 0.16/0.41 ! [A: real,B: real,F: real > real,C: real] :
% 0.16/0.41 ( ( ord_less_eq_real @ A @ B )
% 0.16/0.41 => ( ( ord_less_eq_real @ ( F @ B ) @ C )
% 0.16/0.41 => ( ! [X2: real,Y2: real] :
% 0.16/0.41 ( ( ord_less_eq_real @ X2 @ Y2 )
% 0.16/0.41 => ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 0.16/0.41 => ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % order_subst2
% 0.16/0.41 thf(fact_33_order__subst2,axiom,
% 0.16/0.41 ! [A: real,B: real,F: real > nat,C: nat] :
% 0.16/0.41 ( ( ord_less_eq_real @ A @ B )
% 0.16/0.41 => ( ( ord_less_eq_nat @ ( F @ B ) @ C )
% 0.16/0.41 => ( ! [X2: real,Y2: real] :
% 0.16/0.41 ( ( ord_less_eq_real @ X2 @ Y2 )
% 0.16/0.41 => ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 0.16/0.41 => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % order_subst2
% 0.16/0.41 thf(fact_34_order__subst2,axiom,
% 0.16/0.41 ! [A: nat,B: nat,F: nat > real,C: real] :
% 0.16/0.41 ( ( ord_less_eq_nat @ A @ B )
% 0.16/0.41 => ( ( ord_less_eq_real @ ( F @ B ) @ C )
% 0.16/0.41 => ( ! [X2: nat,Y2: nat] :
% 0.16/0.41 ( ( ord_less_eq_nat @ X2 @ Y2 )
% 0.16/0.41 => ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 0.16/0.41 => ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % order_subst2
% 0.16/0.41 thf(fact_35_order__subst2,axiom,
% 0.16/0.41 ! [A: nat,B: nat,F: nat > nat,C: nat] :
% 0.16/0.41 ( ( ord_less_eq_nat @ A @ B )
% 0.16/0.41 => ( ( ord_less_eq_nat @ ( F @ B ) @ C )
% 0.16/0.41 => ( ! [X2: nat,Y2: nat] :
% 0.16/0.41 ( ( ord_less_eq_nat @ X2 @ Y2 )
% 0.16/0.41 => ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 0.16/0.41 => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % order_subst2
% 0.16/0.41 thf(fact_36_ord__eq__le__subst,axiom,
% 0.16/0.41 ! [A: real,F: real > real,B: real,C: real] :
% 0.16/0.41 ( ( A
% 0.16/0.41 = ( F @ B ) )
% 0.16/0.41 => ( ( ord_less_eq_real @ B @ C )
% 0.16/0.41 => ( ! [X2: real,Y2: real] :
% 0.16/0.41 ( ( ord_less_eq_real @ X2 @ Y2 )
% 0.16/0.41 => ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 0.16/0.41 => ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % ord_eq_le_subst
% 0.16/0.41 thf(fact_37_ord__eq__le__subst,axiom,
% 0.16/0.41 ! [A: nat,F: real > nat,B: real,C: real] :
% 0.16/0.41 ( ( A
% 0.16/0.41 = ( F @ B ) )
% 0.16/0.41 => ( ( ord_less_eq_real @ B @ C )
% 0.16/0.41 => ( ! [X2: real,Y2: real] :
% 0.16/0.41 ( ( ord_less_eq_real @ X2 @ Y2 )
% 0.16/0.41 => ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 0.16/0.41 => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % ord_eq_le_subst
% 0.16/0.41 thf(fact_38_ord__eq__le__subst,axiom,
% 0.16/0.41 ! [A: real,F: nat > real,B: nat,C: nat] :
% 0.16/0.41 ( ( A
% 0.16/0.41 = ( F @ B ) )
% 0.16/0.41 => ( ( ord_less_eq_nat @ B @ C )
% 0.16/0.41 => ( ! [X2: nat,Y2: nat] :
% 0.16/0.41 ( ( ord_less_eq_nat @ X2 @ Y2 )
% 0.16/0.41 => ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 0.16/0.41 => ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % ord_eq_le_subst
% 0.16/0.41 thf(fact_39_ord__eq__le__subst,axiom,
% 0.16/0.41 ! [A: nat,F: nat > nat,B: nat,C: nat] :
% 0.16/0.41 ( ( A
% 0.16/0.41 = ( F @ B ) )
% 0.16/0.41 => ( ( ord_less_eq_nat @ B @ C )
% 0.16/0.41 => ( ! [X2: nat,Y2: nat] :
% 0.16/0.41 ( ( ord_less_eq_nat @ X2 @ Y2 )
% 0.16/0.41 => ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 0.16/0.41 => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % ord_eq_le_subst
% 0.16/0.41 thf(fact_40_ord__le__eq__subst,axiom,
% 0.16/0.41 ! [A: real,B: real,F: real > real,C: real] :
% 0.16/0.41 ( ( ord_less_eq_real @ A @ B )
% 0.16/0.41 => ( ( ( F @ B )
% 0.16/0.41 = C )
% 0.16/0.41 => ( ! [X2: real,Y2: real] :
% 0.16/0.41 ( ( ord_less_eq_real @ X2 @ Y2 )
% 0.16/0.41 => ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 0.16/0.41 => ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % ord_le_eq_subst
% 0.16/0.41 thf(fact_41_ord__le__eq__subst,axiom,
% 0.16/0.41 ! [A: real,B: real,F: real > nat,C: nat] :
% 0.16/0.41 ( ( ord_less_eq_real @ A @ B )
% 0.16/0.41 => ( ( ( F @ B )
% 0.16/0.41 = C )
% 0.16/0.41 => ( ! [X2: real,Y2: real] :
% 0.16/0.41 ( ( ord_less_eq_real @ X2 @ Y2 )
% 0.16/0.41 => ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 0.16/0.41 => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % ord_le_eq_subst
% 0.16/0.41 thf(fact_42_ord__le__eq__subst,axiom,
% 0.16/0.41 ! [A: nat,B: nat,F: nat > real,C: real] :
% 0.16/0.41 ( ( ord_less_eq_nat @ A @ B )
% 0.16/0.41 => ( ( ( F @ B )
% 0.16/0.41 = C )
% 0.16/0.41 => ( ! [X2: nat,Y2: nat] :
% 0.16/0.41 ( ( ord_less_eq_nat @ X2 @ Y2 )
% 0.16/0.41 => ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 0.16/0.41 => ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % ord_le_eq_subst
% 0.16/0.41 thf(fact_43_ord__le__eq__subst,axiom,
% 0.16/0.41 ! [A: nat,B: nat,F: nat > nat,C: nat] :
% 0.16/0.41 ( ( ord_less_eq_nat @ A @ B )
% 0.16/0.41 => ( ( ( F @ B )
% 0.16/0.41 = C )
% 0.16/0.41 => ( ! [X2: nat,Y2: nat] :
% 0.16/0.41 ( ( ord_less_eq_nat @ X2 @ Y2 )
% 0.16/0.41 => ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 0.16/0.41 => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % ord_le_eq_subst
% 0.16/0.41 thf(fact_44_eq__iff,axiom,
% 0.16/0.41 ( ( ^ [Y3: real,Z: real] : ( Y3 = Z ) )
% 0.16/0.41 = ( ^ [X3: real,Y4: real] :
% 0.16/0.41 ( ( ord_less_eq_real @ X3 @ Y4 )
% 0.16/0.41 & ( ord_less_eq_real @ Y4 @ X3 ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % eq_iff
% 0.16/0.41 thf(fact_45_eq__iff,axiom,
% 0.16/0.41 ( ( ^ [Y3: nat,Z: nat] : ( Y3 = Z ) )
% 0.16/0.41 = ( ^ [X3: nat,Y4: nat] :
% 0.16/0.41 ( ( ord_less_eq_nat @ X3 @ Y4 )
% 0.16/0.41 & ( ord_less_eq_nat @ Y4 @ X3 ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % eq_iff
% 0.16/0.41 thf(fact_46_antisym,axiom,
% 0.16/0.41 ! [X: real,Y: real] :
% 0.16/0.41 ( ( ord_less_eq_real @ X @ Y )
% 0.16/0.41 => ( ( ord_less_eq_real @ Y @ X )
% 0.16/0.41 => ( X = Y ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % antisym
% 0.16/0.41 thf(fact_47_antisym,axiom,
% 0.16/0.41 ! [X: nat,Y: nat] :
% 0.16/0.41 ( ( ord_less_eq_nat @ X @ Y )
% 0.16/0.41 => ( ( ord_less_eq_nat @ Y @ X )
% 0.16/0.41 => ( X = Y ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % antisym
% 0.16/0.41 thf(fact_48_linear,axiom,
% 0.16/0.41 ! [X: real,Y: real] :
% 0.16/0.41 ( ( ord_less_eq_real @ X @ Y )
% 0.16/0.41 | ( ord_less_eq_real @ Y @ X ) ) ).
% 0.16/0.41
% 0.16/0.41 % linear
% 0.16/0.41 thf(fact_49_linear,axiom,
% 0.16/0.41 ! [X: nat,Y: nat] :
% 0.16/0.41 ( ( ord_less_eq_nat @ X @ Y )
% 0.16/0.41 | ( ord_less_eq_nat @ Y @ X ) ) ).
% 0.16/0.41
% 0.16/0.41 % linear
% 0.16/0.41 thf(fact_50_eq__refl,axiom,
% 0.16/0.41 ! [X: real,Y: real] :
% 0.16/0.41 ( ( X = Y )
% 0.16/0.41 => ( ord_less_eq_real @ X @ Y ) ) ).
% 0.16/0.41
% 0.16/0.41 % eq_refl
% 0.16/0.41 thf(fact_51_eq__refl,axiom,
% 0.16/0.41 ! [X: nat,Y: nat] :
% 0.16/0.41 ( ( X = Y )
% 0.16/0.41 => ( ord_less_eq_nat @ X @ Y ) ) ).
% 0.16/0.41
% 0.16/0.41 % eq_refl
% 0.16/0.41 thf(fact_52_le__cases,axiom,
% 0.16/0.41 ! [X: real,Y: real] :
% 0.16/0.41 ( ~ ( ord_less_eq_real @ X @ Y )
% 0.16/0.41 => ( ord_less_eq_real @ Y @ X ) ) ).
% 0.16/0.41
% 0.16/0.41 % le_cases
% 0.16/0.41 thf(fact_53_le__cases,axiom,
% 0.16/0.41 ! [X: nat,Y: nat] :
% 0.16/0.41 ( ~ ( ord_less_eq_nat @ X @ Y )
% 0.16/0.41 => ( ord_less_eq_nat @ Y @ X ) ) ).
% 0.16/0.41
% 0.16/0.41 % le_cases
% 0.16/0.41 thf(fact_54_order_Otrans,axiom,
% 0.16/0.41 ! [A: real,B: real,C: real] :
% 0.16/0.41 ( ( ord_less_eq_real @ A @ B )
% 0.16/0.41 => ( ( ord_less_eq_real @ B @ C )
% 0.16/0.41 => ( ord_less_eq_real @ A @ C ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % order.trans
% 0.16/0.41 thf(fact_55_order_Otrans,axiom,
% 0.16/0.41 ! [A: nat,B: nat,C: nat] :
% 0.16/0.41 ( ( ord_less_eq_nat @ A @ B )
% 0.16/0.41 => ( ( ord_less_eq_nat @ B @ C )
% 0.16/0.41 => ( ord_less_eq_nat @ A @ C ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % order.trans
% 0.16/0.41 thf(fact_56_le__cases3,axiom,
% 0.16/0.41 ! [X: real,Y: real,Z2: real] :
% 0.16/0.41 ( ( ( ord_less_eq_real @ X @ Y )
% 0.16/0.41 => ~ ( ord_less_eq_real @ Y @ Z2 ) )
% 0.16/0.41 => ( ( ( ord_less_eq_real @ Y @ X )
% 0.16/0.41 => ~ ( ord_less_eq_real @ X @ Z2 ) )
% 0.16/0.41 => ( ( ( ord_less_eq_real @ X @ Z2 )
% 0.16/0.41 => ~ ( ord_less_eq_real @ Z2 @ Y ) )
% 0.16/0.41 => ( ( ( ord_less_eq_real @ Z2 @ Y )
% 0.16/0.41 => ~ ( ord_less_eq_real @ Y @ X ) )
% 0.16/0.41 => ( ( ( ord_less_eq_real @ Y @ Z2 )
% 0.16/0.41 => ~ ( ord_less_eq_real @ Z2 @ X ) )
% 0.16/0.41 => ~ ( ( ord_less_eq_real @ Z2 @ X )
% 0.16/0.41 => ~ ( ord_less_eq_real @ X @ Y ) ) ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % le_cases3
% 0.16/0.41 thf(fact_57_le__cases3,axiom,
% 0.16/0.41 ! [X: nat,Y: nat,Z2: nat] :
% 0.16/0.41 ( ( ( ord_less_eq_nat @ X @ Y )
% 0.16/0.41 => ~ ( ord_less_eq_nat @ Y @ Z2 ) )
% 0.16/0.41 => ( ( ( ord_less_eq_nat @ Y @ X )
% 0.16/0.41 => ~ ( ord_less_eq_nat @ X @ Z2 ) )
% 0.16/0.41 => ( ( ( ord_less_eq_nat @ X @ Z2 )
% 0.16/0.41 => ~ ( ord_less_eq_nat @ Z2 @ Y ) )
% 0.16/0.41 => ( ( ( ord_less_eq_nat @ Z2 @ Y )
% 0.16/0.41 => ~ ( ord_less_eq_nat @ Y @ X ) )
% 0.16/0.41 => ( ( ( ord_less_eq_nat @ Y @ Z2 )
% 0.16/0.41 => ~ ( ord_less_eq_nat @ Z2 @ X ) )
% 0.16/0.41 => ~ ( ( ord_less_eq_nat @ Z2 @ X )
% 0.16/0.41 => ~ ( ord_less_eq_nat @ X @ Y ) ) ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % le_cases3
% 0.16/0.41 thf(fact_58_antisym__conv,axiom,
% 0.16/0.41 ! [Y: real,X: real] :
% 0.16/0.41 ( ( ord_less_eq_real @ Y @ X )
% 0.16/0.41 => ( ( ord_less_eq_real @ X @ Y )
% 0.16/0.41 = ( X = Y ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % antisym_conv
% 0.16/0.41 thf(fact_59_antisym__conv,axiom,
% 0.16/0.41 ! [Y: nat,X: nat] :
% 0.16/0.41 ( ( ord_less_eq_nat @ Y @ X )
% 0.16/0.41 => ( ( ord_less_eq_nat @ X @ Y )
% 0.16/0.41 = ( X = Y ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % antisym_conv
% 0.16/0.41 thf(fact_60_order__class_Oorder_Oeq__iff,axiom,
% 0.16/0.41 ( ( ^ [Y3: real,Z: real] : ( Y3 = Z ) )
% 0.16/0.41 = ( ^ [A2: real,B2: real] :
% 0.16/0.41 ( ( ord_less_eq_real @ A2 @ B2 )
% 0.16/0.41 & ( ord_less_eq_real @ B2 @ A2 ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % order_class.order.eq_iff
% 0.16/0.41 thf(fact_61_order__class_Oorder_Oeq__iff,axiom,
% 0.16/0.41 ( ( ^ [Y3: nat,Z: nat] : ( Y3 = Z ) )
% 0.16/0.41 = ( ^ [A2: nat,B2: nat] :
% 0.16/0.41 ( ( ord_less_eq_nat @ A2 @ B2 )
% 0.16/0.41 & ( ord_less_eq_nat @ B2 @ A2 ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % order_class.order.eq_iff
% 0.16/0.41 thf(fact_62_ord__eq__le__trans,axiom,
% 0.16/0.41 ! [A: real,B: real,C: real] :
% 0.16/0.41 ( ( A = B )
% 0.16/0.41 => ( ( ord_less_eq_real @ B @ C )
% 0.16/0.41 => ( ord_less_eq_real @ A @ C ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % ord_eq_le_trans
% 0.16/0.41 thf(fact_63_ord__eq__le__trans,axiom,
% 0.16/0.41 ! [A: nat,B: nat,C: nat] :
% 0.16/0.41 ( ( A = B )
% 0.16/0.41 => ( ( ord_less_eq_nat @ B @ C )
% 0.16/0.41 => ( ord_less_eq_nat @ A @ C ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % ord_eq_le_trans
% 0.16/0.41 thf(fact_64_ord__le__eq__trans,axiom,
% 0.16/0.41 ! [A: real,B: real,C: real] :
% 0.16/0.41 ( ( ord_less_eq_real @ A @ B )
% 0.16/0.41 => ( ( B = C )
% 0.16/0.41 => ( ord_less_eq_real @ A @ C ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % ord_le_eq_trans
% 0.16/0.41 thf(fact_65_ord__le__eq__trans,axiom,
% 0.16/0.41 ! [A: nat,B: nat,C: nat] :
% 0.16/0.41 ( ( ord_less_eq_nat @ A @ B )
% 0.16/0.41 => ( ( B = C )
% 0.16/0.41 => ( ord_less_eq_nat @ A @ C ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % ord_le_eq_trans
% 0.16/0.41 thf(fact_66_order__class_Oorder_Oantisym,axiom,
% 0.16/0.41 ! [A: real,B: real] :
% 0.16/0.41 ( ( ord_less_eq_real @ A @ B )
% 0.16/0.41 => ( ( ord_less_eq_real @ B @ A )
% 0.16/0.41 => ( A = B ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % order_class.order.antisym
% 0.16/0.41 thf(fact_67_order__class_Oorder_Oantisym,axiom,
% 0.16/0.41 ! [A: nat,B: nat] :
% 0.16/0.41 ( ( ord_less_eq_nat @ A @ B )
% 0.16/0.41 => ( ( ord_less_eq_nat @ B @ A )
% 0.16/0.41 => ( A = B ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % order_class.order.antisym
% 0.16/0.41 thf(fact_68_order__trans,axiom,
% 0.16/0.41 ! [X: real,Y: real,Z2: real] :
% 0.16/0.41 ( ( ord_less_eq_real @ X @ Y )
% 0.16/0.41 => ( ( ord_less_eq_real @ Y @ Z2 )
% 0.16/0.41 => ( ord_less_eq_real @ X @ Z2 ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % order_trans
% 0.16/0.41 thf(fact_69_order__trans,axiom,
% 0.16/0.41 ! [X: nat,Y: nat,Z2: nat] :
% 0.16/0.41 ( ( ord_less_eq_nat @ X @ Y )
% 0.16/0.41 => ( ( ord_less_eq_nat @ Y @ Z2 )
% 0.16/0.41 => ( ord_less_eq_nat @ X @ Z2 ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % order_trans
% 0.16/0.41 thf(fact_70_dual__order_Orefl,axiom,
% 0.16/0.41 ! [A: real] : ( ord_less_eq_real @ A @ A ) ).
% 0.16/0.41
% 0.16/0.41 % dual_order.refl
% 0.16/0.41 thf(fact_71_dual__order_Orefl,axiom,
% 0.16/0.41 ! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% 0.16/0.41
% 0.16/0.41 % dual_order.refl
% 0.16/0.41 thf(fact_72_linorder__wlog,axiom,
% 0.16/0.41 ! [P3: real > real > $o,A: real,B: real] :
% 0.16/0.41 ( ! [A3: real,B3: real] :
% 0.16/0.41 ( ( ord_less_eq_real @ A3 @ B3 )
% 0.16/0.41 => ( P3 @ A3 @ B3 ) )
% 0.16/0.41 => ( ! [A3: real,B3: real] :
% 0.16/0.41 ( ( P3 @ B3 @ A3 )
% 0.16/0.41 => ( P3 @ A3 @ B3 ) )
% 0.16/0.41 => ( P3 @ A @ B ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % linorder_wlog
% 0.16/0.41 thf(fact_73_linorder__wlog,axiom,
% 0.16/0.41 ! [P3: nat > nat > $o,A: nat,B: nat] :
% 0.16/0.41 ( ! [A3: nat,B3: nat] :
% 0.16/0.41 ( ( ord_less_eq_nat @ A3 @ B3 )
% 0.16/0.41 => ( P3 @ A3 @ B3 ) )
% 0.16/0.41 => ( ! [A3: nat,B3: nat] :
% 0.16/0.41 ( ( P3 @ B3 @ A3 )
% 0.16/0.41 => ( P3 @ A3 @ B3 ) )
% 0.16/0.41 => ( P3 @ A @ B ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % linorder_wlog
% 0.16/0.41 thf(fact_74_dual__order_Otrans,axiom,
% 0.16/0.41 ! [B: real,A: real,C: real] :
% 0.16/0.41 ( ( ord_less_eq_real @ B @ A )
% 0.16/0.41 => ( ( ord_less_eq_real @ C @ B )
% 0.16/0.41 => ( ord_less_eq_real @ C @ A ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % dual_order.trans
% 0.16/0.41 thf(fact_75_dual__order_Otrans,axiom,
% 0.16/0.41 ! [B: nat,A: nat,C: nat] :
% 0.16/0.41 ( ( ord_less_eq_nat @ B @ A )
% 0.16/0.41 => ( ( ord_less_eq_nat @ C @ B )
% 0.16/0.41 => ( ord_less_eq_nat @ C @ A ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % dual_order.trans
% 0.16/0.41 thf(fact_76_dual__order_Oeq__iff,axiom,
% 0.16/0.41 ( ( ^ [Y3: real,Z: real] : ( Y3 = Z ) )
% 0.16/0.41 = ( ^ [A2: real,B2: real] :
% 0.16/0.41 ( ( ord_less_eq_real @ B2 @ A2 )
% 0.16/0.41 & ( ord_less_eq_real @ A2 @ B2 ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % dual_order.eq_iff
% 0.16/0.41 thf(fact_77_dual__order_Oeq__iff,axiom,
% 0.16/0.41 ( ( ^ [Y3: nat,Z: nat] : ( Y3 = Z ) )
% 0.16/0.41 = ( ^ [A2: nat,B2: nat] :
% 0.16/0.41 ( ( ord_less_eq_nat @ B2 @ A2 )
% 0.16/0.41 & ( ord_less_eq_nat @ A2 @ B2 ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % dual_order.eq_iff
% 0.16/0.41 thf(fact_78_dual__order_Oantisym,axiom,
% 0.16/0.41 ! [B: real,A: real] :
% 0.16/0.41 ( ( ord_less_eq_real @ B @ A )
% 0.16/0.41 => ( ( ord_less_eq_real @ A @ B )
% 0.16/0.41 => ( A = B ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % dual_order.antisym
% 0.16/0.41 thf(fact_79_dual__order_Oantisym,axiom,
% 0.16/0.41 ! [B: nat,A: nat] :
% 0.16/0.41 ( ( ord_less_eq_nat @ B @ A )
% 0.16/0.41 => ( ( ord_less_eq_nat @ A @ B )
% 0.16/0.41 => ( A = B ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % dual_order.antisym
% 0.16/0.41 thf(fact_80_max_OcoboundedI2,axiom,
% 0.16/0.41 ! [C: real,B: real,A: real] :
% 0.16/0.41 ( ( ord_less_eq_real @ C @ B )
% 0.16/0.41 => ( ord_less_eq_real @ C @ ( ord_max_real @ A @ B ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % max.coboundedI2
% 0.16/0.41 thf(fact_81_max_OcoboundedI2,axiom,
% 0.16/0.41 ! [C: nat,B: nat,A: nat] :
% 0.16/0.41 ( ( ord_less_eq_nat @ C @ B )
% 0.16/0.41 => ( ord_less_eq_nat @ C @ ( ord_max_nat @ A @ B ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % max.coboundedI2
% 0.16/0.41 thf(fact_82_max_OcoboundedI1,axiom,
% 0.16/0.41 ! [C: real,A: real,B: real] :
% 0.16/0.41 ( ( ord_less_eq_real @ C @ A )
% 0.16/0.41 => ( ord_less_eq_real @ C @ ( ord_max_real @ A @ B ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % max.coboundedI1
% 0.16/0.41 thf(fact_83_max_OcoboundedI1,axiom,
% 0.16/0.41 ! [C: nat,A: nat,B: nat] :
% 0.16/0.41 ( ( ord_less_eq_nat @ C @ A )
% 0.16/0.41 => ( ord_less_eq_nat @ C @ ( ord_max_nat @ A @ B ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % max.coboundedI1
% 0.16/0.41 thf(fact_84_max_Oabsorb__iff2,axiom,
% 0.16/0.41 ( ord_less_eq_real
% 0.16/0.41 = ( ^ [A2: real,B2: real] :
% 0.16/0.41 ( ( ord_max_real @ A2 @ B2 )
% 0.16/0.41 = B2 ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % max.absorb_iff2
% 0.16/0.41 thf(fact_85_max_Oabsorb__iff2,axiom,
% 0.16/0.41 ( ord_less_eq_nat
% 0.16/0.41 = ( ^ [A2: nat,B2: nat] :
% 0.16/0.41 ( ( ord_max_nat @ A2 @ B2 )
% 0.16/0.41 = B2 ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % max.absorb_iff2
% 0.16/0.41 thf(fact_86_max_Oabsorb__iff1,axiom,
% 0.16/0.41 ( ord_less_eq_real
% 0.16/0.41 = ( ^ [B2: real,A2: real] :
% 0.16/0.41 ( ( ord_max_real @ A2 @ B2 )
% 0.16/0.41 = A2 ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % max.absorb_iff1
% 0.16/0.41 thf(fact_87_max_Oabsorb__iff1,axiom,
% 0.16/0.41 ( ord_less_eq_nat
% 0.16/0.41 = ( ^ [B2: nat,A2: nat] :
% 0.16/0.41 ( ( ord_max_nat @ A2 @ B2 )
% 0.16/0.41 = A2 ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % max.absorb_iff1
% 0.16/0.41 thf(fact_88_le__max__iff__disj,axiom,
% 0.16/0.41 ! [Z2: real,X: real,Y: real] :
% 0.16/0.41 ( ( ord_less_eq_real @ Z2 @ ( ord_max_real @ X @ Y ) )
% 0.16/0.41 = ( ( ord_less_eq_real @ Z2 @ X )
% 0.16/0.41 | ( ord_less_eq_real @ Z2 @ Y ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % le_max_iff_disj
% 0.16/0.41 thf(fact_89_le__max__iff__disj,axiom,
% 0.16/0.41 ! [Z2: nat,X: nat,Y: nat] :
% 0.16/0.41 ( ( ord_less_eq_nat @ Z2 @ ( ord_max_nat @ X @ Y ) )
% 0.16/0.41 = ( ( ord_less_eq_nat @ Z2 @ X )
% 0.16/0.41 | ( ord_less_eq_nat @ Z2 @ Y ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % le_max_iff_disj
% 0.16/0.41 thf(fact_90_max_Ocobounded2,axiom,
% 0.16/0.41 ! [B: real,A: real] : ( ord_less_eq_real @ B @ ( ord_max_real @ A @ B ) ) ).
% 0.16/0.41
% 0.16/0.41 % max.cobounded2
% 0.16/0.41 thf(fact_91_max_Ocobounded2,axiom,
% 0.16/0.41 ! [B: nat,A: nat] : ( ord_less_eq_nat @ B @ ( ord_max_nat @ A @ B ) ) ).
% 0.16/0.41
% 0.16/0.41 % max.cobounded2
% 0.16/0.41 thf(fact_92_max_Ocobounded1,axiom,
% 0.16/0.41 ! [A: real,B: real] : ( ord_less_eq_real @ A @ ( ord_max_real @ A @ B ) ) ).
% 0.16/0.41
% 0.16/0.41 % max.cobounded1
% 0.16/0.41 thf(fact_93_max_Ocobounded1,axiom,
% 0.16/0.41 ! [A: nat,B: nat] : ( ord_less_eq_nat @ A @ ( ord_max_nat @ A @ B ) ) ).
% 0.16/0.41
% 0.16/0.41 % max.cobounded1
% 0.16/0.41 thf(fact_94_max_Oorder__iff,axiom,
% 0.16/0.41 ( ord_less_eq_real
% 0.16/0.41 = ( ^ [B2: real,A2: real] :
% 0.16/0.41 ( A2
% 0.16/0.41 = ( ord_max_real @ A2 @ B2 ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % max.order_iff
% 0.16/0.41 thf(fact_95_max_Oorder__iff,axiom,
% 0.16/0.41 ( ord_less_eq_nat
% 0.16/0.41 = ( ^ [B2: nat,A2: nat] :
% 0.16/0.41 ( A2
% 0.16/0.41 = ( ord_max_nat @ A2 @ B2 ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % max.order_iff
% 0.16/0.41 thf(fact_96_max_OboundedI,axiom,
% 0.16/0.41 ! [B: real,A: real,C: real] :
% 0.16/0.41 ( ( ord_less_eq_real @ B @ A )
% 0.16/0.41 => ( ( ord_less_eq_real @ C @ A )
% 0.16/0.41 => ( ord_less_eq_real @ ( ord_max_real @ B @ C ) @ A ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % max.boundedI
% 0.16/0.41 thf(fact_97_max_OboundedI,axiom,
% 0.16/0.41 ! [B: nat,A: nat,C: nat] :
% 0.16/0.41 ( ( ord_less_eq_nat @ B @ A )
% 0.16/0.41 => ( ( ord_less_eq_nat @ C @ A )
% 0.16/0.41 => ( ord_less_eq_nat @ ( ord_max_nat @ B @ C ) @ A ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % max.boundedI
% 0.16/0.41 thf(fact_98_max_OboundedE,axiom,
% 0.16/0.41 ! [B: real,C: real,A: real] :
% 0.16/0.41 ( ( ord_less_eq_real @ ( ord_max_real @ B @ C ) @ A )
% 0.16/0.41 => ~ ( ( ord_less_eq_real @ B @ A )
% 0.16/0.41 => ~ ( ord_less_eq_real @ C @ A ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % max.boundedE
% 0.16/0.41 thf(fact_99_max_OboundedE,axiom,
% 0.16/0.41 ! [B: nat,C: nat,A: nat] :
% 0.16/0.41 ( ( ord_less_eq_nat @ ( ord_max_nat @ B @ C ) @ A )
% 0.16/0.41 => ~ ( ( ord_less_eq_nat @ B @ A )
% 0.16/0.41 => ~ ( ord_less_eq_nat @ C @ A ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % max.boundedE
% 0.16/0.41 thf(fact_100_max_Oabsorb2,axiom,
% 0.16/0.41 ! [A: real,B: real] :
% 0.16/0.41 ( ( ord_less_eq_real @ A @ B )
% 0.16/0.41 => ( ( ord_max_real @ A @ B )
% 0.16/0.41 = B ) ) ).
% 0.16/0.41
% 0.16/0.41 % max.absorb2
% 0.16/0.41 thf(fact_101_max_Oabsorb2,axiom,
% 0.16/0.41 ! [A: nat,B: nat] :
% 0.16/0.41 ( ( ord_less_eq_nat @ A @ B )
% 0.16/0.41 => ( ( ord_max_nat @ A @ B )
% 0.16/0.41 = B ) ) ).
% 0.16/0.41
% 0.16/0.41 % max.absorb2
% 0.16/0.41 thf(fact_102_max_Oabsorb1,axiom,
% 0.16/0.41 ! [B: real,A: real] :
% 0.16/0.41 ( ( ord_less_eq_real @ B @ A )
% 0.16/0.41 => ( ( ord_max_real @ A @ B )
% 0.16/0.41 = A ) ) ).
% 0.16/0.41
% 0.16/0.41 % max.absorb1
% 0.16/0.41 thf(fact_103_max_Oabsorb1,axiom,
% 0.16/0.41 ! [B: nat,A: nat] :
% 0.16/0.41 ( ( ord_less_eq_nat @ B @ A )
% 0.16/0.41 => ( ( ord_max_nat @ A @ B )
% 0.16/0.41 = A ) ) ).
% 0.16/0.41
% 0.16/0.41 % max.absorb1
% 0.16/0.41 thf(fact_104_max_OorderI,axiom,
% 0.16/0.41 ! [A: real,B: real] :
% 0.16/0.41 ( ( A
% 0.16/0.41 = ( ord_max_real @ A @ B ) )
% 0.16/0.41 => ( ord_less_eq_real @ B @ A ) ) ).
% 0.16/0.41
% 0.16/0.41 % max.orderI
% 0.16/0.41 thf(fact_105_max_OorderI,axiom,
% 0.16/0.41 ! [A: nat,B: nat] :
% 0.16/0.41 ( ( A
% 0.16/0.41 = ( ord_max_nat @ A @ B ) )
% 0.16/0.41 => ( ord_less_eq_nat @ B @ A ) ) ).
% 0.16/0.41
% 0.16/0.41 % max.orderI
% 0.16/0.41 thf(fact_106_max_OorderE,axiom,
% 0.16/0.41 ! [B: real,A: real] :
% 0.16/0.41 ( ( ord_less_eq_real @ B @ A )
% 0.16/0.41 => ( A
% 0.16/0.41 = ( ord_max_real @ A @ B ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % max.orderE
% 0.16/0.41 thf(fact_107_max_OorderE,axiom,
% 0.16/0.41 ! [B: nat,A: nat] :
% 0.16/0.41 ( ( ord_less_eq_nat @ B @ A )
% 0.16/0.41 => ( A
% 0.16/0.41 = ( ord_max_nat @ A @ B ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % max.orderE
% 0.16/0.41 thf(fact_108_max_Omono,axiom,
% 0.16/0.41 ! [C: real,A: real,D: real,B: real] :
% 0.16/0.41 ( ( ord_less_eq_real @ C @ A )
% 0.16/0.41 => ( ( ord_less_eq_real @ D @ B )
% 0.16/0.41 => ( ord_less_eq_real @ ( ord_max_real @ C @ D ) @ ( ord_max_real @ A @ B ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % max.mono
% 0.16/0.41 thf(fact_109_max_Omono,axiom,
% 0.16/0.41 ! [C: nat,A: nat,D: nat,B: nat] :
% 0.16/0.41 ( ( ord_less_eq_nat @ C @ A )
% 0.16/0.41 => ( ( ord_less_eq_nat @ D @ B )
% 0.16/0.41 => ( ord_less_eq_nat @ ( ord_max_nat @ C @ D ) @ ( ord_max_nat @ A @ B ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % max.mono
% 0.16/0.41 thf(fact_110_max__def,axiom,
% 0.16/0.41 ( ord_max_real
% 0.16/0.41 = ( ^ [A2: real,B2: real] : ( if_real @ ( ord_less_eq_real @ A2 @ B2 ) @ B2 @ A2 ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % max_def
% 0.16/0.41 thf(fact_111_max__def,axiom,
% 0.16/0.41 ( ord_max_nat
% 0.16/0.41 = ( ^ [A2: nat,B2: nat] : ( if_nat @ ( ord_less_eq_nat @ A2 @ B2 ) @ B2 @ A2 ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % max_def
% 0.16/0.41 thf(fact_112_max__absorb2,axiom,
% 0.16/0.41 ! [X: real,Y: real] :
% 0.16/0.41 ( ( ord_less_eq_real @ X @ Y )
% 0.16/0.41 => ( ( ord_max_real @ X @ Y )
% 0.16/0.41 = Y ) ) ).
% 0.16/0.41
% 0.16/0.41 % max_absorb2
% 0.16/0.41 thf(fact_113_max__absorb2,axiom,
% 0.16/0.41 ! [X: nat,Y: nat] :
% 0.16/0.41 ( ( ord_less_eq_nat @ X @ Y )
% 0.16/0.41 => ( ( ord_max_nat @ X @ Y )
% 0.16/0.41 = Y ) ) ).
% 0.16/0.41
% 0.16/0.41 % max_absorb2
% 0.16/0.41 thf(fact_114_okmaxsync__def,axiom,
% 0.16/0.41 ( genClo208577157axsync
% 0.16/0.41 = ( ^ [I2: nat,X3: real] :
% 0.16/0.41 ! [P4: nat,Q: nat] :
% 0.16/0.41 ( ( ( genClo1015804716orrect @ P4 @ ( ord_max_real @ ( genClo1163638703lle_te @ P4 @ I2 ) @ ( genClo1163638703lle_te @ Q @ I2 ) ) )
% 0.16/0.41 & ( genClo1015804716orrect @ Q @ ( ord_max_real @ ( genClo1163638703lle_te @ P4 @ I2 ) @ ( genClo1163638703lle_te @ Q @ I2 ) ) ) )
% 0.16/0.41 => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( genClo1160817912lle_IC @ P4 @ I2 @ ( ord_max_real @ ( genClo1163638703lle_te @ P4 @ I2 ) @ ( genClo1163638703lle_te @ Q @ I2 ) ) ) @ ( genClo1160817912lle_IC @ Q @ I2 @ ( ord_max_real @ ( genClo1163638703lle_te @ P4 @ I2 ) @ ( genClo1163638703lle_te @ Q @ I2 ) ) ) ) ) @ X3 ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % okmaxsync_def
% 0.16/0.41 thf(fact_115_le__add__diff__inverse,axiom,
% 0.16/0.41 ! [B: real,A: real] :
% 0.16/0.41 ( ( ord_less_eq_real @ B @ A )
% 0.16/0.41 => ( ( plus_plus_real @ B @ ( minus_minus_real @ A @ B ) )
% 0.16/0.41 = A ) ) ).
% 0.16/0.41
% 0.16/0.41 % le_add_diff_inverse
% 0.16/0.41 thf(fact_116_le__add__diff__inverse,axiom,
% 0.16/0.41 ! [B: nat,A: nat] :
% 0.16/0.41 ( ( ord_less_eq_nat @ B @ A )
% 0.16/0.41 => ( ( plus_plus_nat @ B @ ( minus_minus_nat @ A @ B ) )
% 0.16/0.41 = A ) ) ).
% 0.16/0.41
% 0.16/0.41 % le_add_diff_inverse
% 0.16/0.41 thf(fact_117_le__add__diff__inverse2,axiom,
% 0.16/0.41 ! [B: real,A: real] :
% 0.16/0.41 ( ( ord_less_eq_real @ B @ A )
% 0.16/0.41 => ( ( plus_plus_real @ ( minus_minus_real @ A @ B ) @ B )
% 0.16/0.41 = A ) ) ).
% 0.16/0.41
% 0.16/0.41 % le_add_diff_inverse2
% 0.16/0.41 thf(fact_118_le__add__diff__inverse2,axiom,
% 0.16/0.41 ! [B: nat,A: nat] :
% 0.16/0.41 ( ( ord_less_eq_nat @ B @ A )
% 0.16/0.41 => ( ( plus_plus_nat @ ( minus_minus_nat @ A @ B ) @ B )
% 0.16/0.41 = A ) ) ).
% 0.16/0.41
% 0.16/0.41 % le_add_diff_inverse2
% 0.16/0.41 thf(fact_119_abs__1,axiom,
% 0.16/0.41 ( ( abs_abs_real @ one_one_real )
% 0.16/0.41 = one_one_real ) ).
% 0.16/0.41
% 0.16/0.41 % abs_1
% 0.16/0.41 thf(fact_120_abs__add__abs,axiom,
% 0.16/0.41 ! [A: real,B: real] :
% 0.16/0.41 ( ( abs_abs_real @ ( plus_plus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) )
% 0.16/0.41 = ( plus_plus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % abs_add_abs
% 0.16/0.41 thf(fact_121_add__diff__cancel,axiom,
% 0.16/0.41 ! [A: real,B: real] :
% 0.16/0.41 ( ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ B )
% 0.16/0.41 = A ) ).
% 0.16/0.41
% 0.16/0.41 % add_diff_cancel
% 0.16/0.41 thf(fact_122_diff__add__cancel,axiom,
% 0.16/0.41 ! [A: real,B: real] :
% 0.16/0.41 ( ( plus_plus_real @ ( minus_minus_real @ A @ B ) @ B )
% 0.16/0.41 = A ) ).
% 0.16/0.41
% 0.16/0.41 % diff_add_cancel
% 0.16/0.41 thf(fact_123_add__diff__cancel__left,axiom,
% 0.16/0.41 ! [C: real,A: real,B: real] :
% 0.16/0.41 ( ( minus_minus_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
% 0.16/0.41 = ( minus_minus_real @ A @ B ) ) ).
% 0.16/0.41
% 0.16/0.41 % add_diff_cancel_left
% 0.16/0.41 thf(fact_124_add__diff__cancel__left,axiom,
% 0.16/0.41 ! [C: nat,A: nat,B: nat] :
% 0.16/0.41 ( ( minus_minus_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
% 0.16/0.41 = ( minus_minus_nat @ A @ B ) ) ).
% 0.16/0.41
% 0.16/0.41 % add_diff_cancel_left
% 0.16/0.41 thf(fact_125_add__diff__cancel__left_H,axiom,
% 0.16/0.41 ! [A: real,B: real] :
% 0.16/0.41 ( ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ A )
% 0.16/0.41 = B ) ).
% 0.16/0.41
% 0.16/0.41 % add_diff_cancel_left'
% 0.16/0.41 thf(fact_126_add__diff__cancel__left_H,axiom,
% 0.16/0.41 ! [A: nat,B: nat] :
% 0.16/0.41 ( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ A )
% 0.16/0.41 = B ) ).
% 0.16/0.41
% 0.16/0.41 % add_diff_cancel_left'
% 0.16/0.41 thf(fact_127_add__diff__cancel__right,axiom,
% 0.16/0.41 ! [A: real,C: real,B: real] :
% 0.16/0.41 ( ( minus_minus_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
% 0.16/0.41 = ( minus_minus_real @ A @ B ) ) ).
% 0.16/0.41
% 0.16/0.41 % add_diff_cancel_right
% 0.16/0.41 thf(fact_128_add__diff__cancel__right,axiom,
% 0.16/0.41 ! [A: nat,C: nat,B: nat] :
% 0.16/0.41 ( ( minus_minus_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
% 0.16/0.41 = ( minus_minus_nat @ A @ B ) ) ).
% 0.16/0.41
% 0.16/0.41 % add_diff_cancel_right
% 0.16/0.41 thf(fact_129_add__right__cancel,axiom,
% 0.16/0.41 ! [B: nat,A: nat,C: nat] :
% 0.16/0.41 ( ( ( plus_plus_nat @ B @ A )
% 0.16/0.41 = ( plus_plus_nat @ C @ A ) )
% 0.16/0.41 = ( B = C ) ) ).
% 0.16/0.41
% 0.16/0.41 % add_right_cancel
% 0.16/0.41 thf(fact_130_add__right__cancel,axiom,
% 0.16/0.41 ! [B: real,A: real,C: real] :
% 0.16/0.41 ( ( ( plus_plus_real @ B @ A )
% 0.16/0.41 = ( plus_plus_real @ C @ A ) )
% 0.16/0.41 = ( B = C ) ) ).
% 0.16/0.41
% 0.16/0.41 % add_right_cancel
% 0.16/0.41 thf(fact_131_add__left__cancel,axiom,
% 0.16/0.41 ! [A: nat,B: nat,C: nat] :
% 0.16/0.41 ( ( ( plus_plus_nat @ A @ B )
% 0.16/0.41 = ( plus_plus_nat @ A @ C ) )
% 0.16/0.41 = ( B = C ) ) ).
% 0.16/0.41
% 0.16/0.41 % add_left_cancel
% 0.16/0.41 thf(fact_132_add__left__cancel,axiom,
% 0.16/0.41 ! [A: real,B: real,C: real] :
% 0.16/0.41 ( ( ( plus_plus_real @ A @ B )
% 0.16/0.41 = ( plus_plus_real @ A @ C ) )
% 0.16/0.41 = ( B = C ) ) ).
% 0.16/0.41
% 0.16/0.41 % add_left_cancel
% 0.16/0.41 thf(fact_133_abs__idempotent,axiom,
% 0.16/0.41 ! [A: real] :
% 0.16/0.41 ( ( abs_abs_real @ ( abs_abs_real @ A ) )
% 0.16/0.41 = ( abs_abs_real @ A ) ) ).
% 0.16/0.41
% 0.16/0.41 % abs_idempotent
% 0.16/0.41 thf(fact_134_abs__abs,axiom,
% 0.16/0.41 ! [A: real] :
% 0.16/0.41 ( ( abs_abs_real @ ( abs_abs_real @ A ) )
% 0.16/0.41 = ( abs_abs_real @ A ) ) ).
% 0.16/0.41
% 0.16/0.41 % abs_abs
% 0.16/0.41 thf(fact_135_add__le__cancel__right,axiom,
% 0.16/0.41 ! [A: real,C: real,B: real] :
% 0.16/0.41 ( ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
% 0.16/0.41 = ( ord_less_eq_real @ A @ B ) ) ).
% 0.16/0.41
% 0.16/0.41 % add_le_cancel_right
% 0.16/0.41 thf(fact_136_add__le__cancel__right,axiom,
% 0.16/0.41 ! [A: nat,C: nat,B: nat] :
% 0.16/0.41 ( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
% 0.16/0.41 = ( ord_less_eq_nat @ A @ B ) ) ).
% 0.16/0.41
% 0.16/0.41 % add_le_cancel_right
% 0.16/0.41 thf(fact_137_add__le__cancel__left,axiom,
% 0.16/0.41 ! [C: real,A: real,B: real] :
% 0.16/0.41 ( ( ord_less_eq_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
% 0.16/0.41 = ( ord_less_eq_real @ A @ B ) ) ).
% 0.16/0.41
% 0.16/0.41 % add_le_cancel_left
% 0.16/0.41 thf(fact_138_add__le__cancel__left,axiom,
% 0.16/0.41 ! [C: nat,A: nat,B: nat] :
% 0.16/0.41 ( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
% 0.16/0.41 = ( ord_less_eq_nat @ A @ B ) ) ).
% 0.16/0.41
% 0.16/0.41 % add_le_cancel_left
% 0.16/0.41 thf(fact_139_add__diff__cancel__right_H,axiom,
% 0.16/0.41 ! [A: real,B: real] :
% 0.16/0.41 ( ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ B )
% 0.16/0.41 = A ) ).
% 0.16/0.41
% 0.16/0.41 % add_diff_cancel_right'
% 0.16/0.41 thf(fact_140_add__diff__cancel__right_H,axiom,
% 0.16/0.41 ! [A: nat,B: nat] :
% 0.16/0.41 ( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ B )
% 0.16/0.41 = A ) ).
% 0.16/0.41
% 0.16/0.41 % add_diff_cancel_right'
% 0.16/0.41 thf(fact_141_add__right__imp__eq,axiom,
% 0.16/0.41 ! [B: nat,A: nat,C: nat] :
% 0.16/0.41 ( ( ( plus_plus_nat @ B @ A )
% 0.16/0.41 = ( plus_plus_nat @ C @ A ) )
% 0.16/0.41 => ( B = C ) ) ).
% 0.16/0.41
% 0.16/0.41 % add_right_imp_eq
% 0.16/0.41 thf(fact_142_add__right__imp__eq,axiom,
% 0.16/0.41 ! [B: real,A: real,C: real] :
% 0.16/0.41 ( ( ( plus_plus_real @ B @ A )
% 0.16/0.41 = ( plus_plus_real @ C @ A ) )
% 0.16/0.41 => ( B = C ) ) ).
% 0.16/0.41
% 0.16/0.41 % add_right_imp_eq
% 0.16/0.41 thf(fact_143_add__left__imp__eq,axiom,
% 0.16/0.41 ! [A: nat,B: nat,C: nat] :
% 0.16/0.41 ( ( ( plus_plus_nat @ A @ B )
% 0.16/0.41 = ( plus_plus_nat @ A @ C ) )
% 0.16/0.41 => ( B = C ) ) ).
% 0.16/0.41
% 0.16/0.41 % add_left_imp_eq
% 0.16/0.41 thf(fact_144_add__left__imp__eq,axiom,
% 0.16/0.41 ! [A: real,B: real,C: real] :
% 0.16/0.41 ( ( ( plus_plus_real @ A @ B )
% 0.16/0.41 = ( plus_plus_real @ A @ C ) )
% 0.16/0.41 => ( B = C ) ) ).
% 0.16/0.41
% 0.16/0.41 % add_left_imp_eq
% 0.16/0.41 thf(fact_145_add_Oleft__commute,axiom,
% 0.16/0.41 ! [B: nat,A: nat,C: nat] :
% 0.16/0.41 ( ( plus_plus_nat @ B @ ( plus_plus_nat @ A @ C ) )
% 0.16/0.41 = ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % add.left_commute
% 0.16/0.41 thf(fact_146_add_Oleft__commute,axiom,
% 0.16/0.41 ! [B: real,A: real,C: real] :
% 0.16/0.41 ( ( plus_plus_real @ B @ ( plus_plus_real @ A @ C ) )
% 0.16/0.41 = ( plus_plus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % add.left_commute
% 0.16/0.41 thf(fact_147_add_Ocommute,axiom,
% 0.16/0.41 ( plus_plus_nat
% 0.16/0.41 = ( ^ [A2: nat,B2: nat] : ( plus_plus_nat @ B2 @ A2 ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % add.commute
% 0.16/0.41 thf(fact_148_add_Ocommute,axiom,
% 0.16/0.41 ( plus_plus_real
% 0.16/0.41 = ( ^ [A2: real,B2: real] : ( plus_plus_real @ B2 @ A2 ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % add.commute
% 0.16/0.41 thf(fact_149_add_Oright__cancel,axiom,
% 0.16/0.41 ! [B: real,A: real,C: real] :
% 0.16/0.41 ( ( ( plus_plus_real @ B @ A )
% 0.16/0.41 = ( plus_plus_real @ C @ A ) )
% 0.16/0.41 = ( B = C ) ) ).
% 0.16/0.41
% 0.16/0.41 % add.right_cancel
% 0.16/0.41 thf(fact_150_add_Oleft__cancel,axiom,
% 0.16/0.41 ! [A: real,B: real,C: real] :
% 0.16/0.41 ( ( ( plus_plus_real @ A @ B )
% 0.16/0.41 = ( plus_plus_real @ A @ C ) )
% 0.16/0.41 = ( B = C ) ) ).
% 0.16/0.41
% 0.16/0.41 % add.left_cancel
% 0.16/0.41 thf(fact_151_add_Oassoc,axiom,
% 0.16/0.41 ! [A: nat,B: nat,C: nat] :
% 0.16/0.41 ( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
% 0.16/0.41 = ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % add.assoc
% 0.16/0.41 thf(fact_152_add_Oassoc,axiom,
% 0.16/0.41 ! [A: real,B: real,C: real] :
% 0.16/0.41 ( ( plus_plus_real @ ( plus_plus_real @ A @ B ) @ C )
% 0.16/0.41 = ( plus_plus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % add.assoc
% 0.16/0.41 thf(fact_153_group__cancel_Oadd2,axiom,
% 0.16/0.41 ! [B4: nat,K: nat,B: nat,A: nat] :
% 0.16/0.41 ( ( B4
% 0.16/0.41 = ( plus_plus_nat @ K @ B ) )
% 0.16/0.41 => ( ( plus_plus_nat @ A @ B4 )
% 0.16/0.41 = ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % group_cancel.add2
% 0.16/0.41 thf(fact_154_group__cancel_Oadd2,axiom,
% 0.16/0.41 ! [B4: real,K: real,B: real,A: real] :
% 0.16/0.41 ( ( B4
% 0.16/0.41 = ( plus_plus_real @ K @ B ) )
% 0.16/0.41 => ( ( plus_plus_real @ A @ B4 )
% 0.16/0.41 = ( plus_plus_real @ K @ ( plus_plus_real @ A @ B ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % group_cancel.add2
% 0.16/0.41 thf(fact_155_group__cancel_Oadd1,axiom,
% 0.16/0.41 ! [A4: nat,K: nat,A: nat,B: nat] :
% 0.16/0.41 ( ( A4
% 0.16/0.41 = ( plus_plus_nat @ K @ A ) )
% 0.16/0.41 => ( ( plus_plus_nat @ A4 @ B )
% 0.16/0.41 = ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % group_cancel.add1
% 0.16/0.41 thf(fact_156_group__cancel_Oadd1,axiom,
% 0.16/0.41 ! [A4: real,K: real,A: real,B: real] :
% 0.16/0.41 ( ( A4
% 0.16/0.41 = ( plus_plus_real @ K @ A ) )
% 0.16/0.41 => ( ( plus_plus_real @ A4 @ B )
% 0.16/0.41 = ( plus_plus_real @ K @ ( plus_plus_real @ A @ B ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % group_cancel.add1
% 0.16/0.41 thf(fact_157_add__mono__thms__linordered__semiring_I4_J,axiom,
% 0.16/0.41 ! [I: nat,J: nat,K: nat,L: nat] :
% 0.16/0.41 ( ( ( I = J )
% 0.16/0.41 & ( K = L ) )
% 0.16/0.41 => ( ( plus_plus_nat @ I @ K )
% 0.16/0.41 = ( plus_plus_nat @ J @ L ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % add_mono_thms_linordered_semiring(4)
% 0.16/0.41 thf(fact_158_add__mono__thms__linordered__semiring_I4_J,axiom,
% 0.16/0.41 ! [I: real,J: real,K: real,L: real] :
% 0.16/0.41 ( ( ( I = J )
% 0.16/0.41 & ( K = L ) )
% 0.16/0.41 => ( ( plus_plus_real @ I @ K )
% 0.16/0.41 = ( plus_plus_real @ J @ L ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % add_mono_thms_linordered_semiring(4)
% 0.16/0.41 thf(fact_159_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
% 0.16/0.41 ! [A: nat,B: nat,C: nat] :
% 0.16/0.41 ( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
% 0.16/0.41 = ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % ab_semigroup_add_class.add_ac(1)
% 0.16/0.41 thf(fact_160_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
% 0.16/0.41 ! [A: real,B: real,C: real] :
% 0.16/0.41 ( ( plus_plus_real @ ( plus_plus_real @ A @ B ) @ C )
% 0.16/0.41 = ( plus_plus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % ab_semigroup_add_class.add_ac(1)
% 0.16/0.41 thf(fact_161_one__reorient,axiom,
% 0.16/0.41 ! [X: nat] :
% 0.16/0.41 ( ( one_one_nat = X )
% 0.16/0.41 = ( X = one_one_nat ) ) ).
% 0.16/0.41
% 0.16/0.41 % one_reorient
% 0.16/0.41 thf(fact_162_diff__right__commute,axiom,
% 0.16/0.41 ! [A: real,C: real,B: real] :
% 0.16/0.41 ( ( minus_minus_real @ ( minus_minus_real @ A @ C ) @ B )
% 0.16/0.41 = ( minus_minus_real @ ( minus_minus_real @ A @ B ) @ C ) ) ).
% 0.16/0.41
% 0.16/0.41 % diff_right_commute
% 0.16/0.41 thf(fact_163_diff__right__commute,axiom,
% 0.16/0.41 ! [A: nat,C: nat,B: nat] :
% 0.16/0.41 ( ( minus_minus_nat @ ( minus_minus_nat @ A @ C ) @ B )
% 0.16/0.41 = ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C ) ) ).
% 0.16/0.41
% 0.16/0.41 % diff_right_commute
% 0.16/0.41 thf(fact_164_diff__eq__diff__eq,axiom,
% 0.16/0.41 ! [A: real,B: real,C: real,D: real] :
% 0.16/0.41 ( ( ( minus_minus_real @ A @ B )
% 0.16/0.41 = ( minus_minus_real @ C @ D ) )
% 0.16/0.41 => ( ( A = B )
% 0.16/0.41 = ( C = D ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % diff_eq_diff_eq
% 0.16/0.41 thf(fact_165_add__le__imp__le__right,axiom,
% 0.16/0.41 ! [A: real,C: real,B: real] :
% 0.16/0.41 ( ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
% 0.16/0.41 => ( ord_less_eq_real @ A @ B ) ) ).
% 0.16/0.41
% 0.16/0.41 % add_le_imp_le_right
% 0.16/0.41 thf(fact_166_add__le__imp__le__right,axiom,
% 0.16/0.41 ! [A: nat,C: nat,B: nat] :
% 0.16/0.41 ( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
% 0.16/0.41 => ( ord_less_eq_nat @ A @ B ) ) ).
% 0.16/0.41
% 0.16/0.41 % add_le_imp_le_right
% 0.16/0.41 thf(fact_167_add__le__imp__le__left,axiom,
% 0.16/0.41 ! [C: real,A: real,B: real] :
% 0.16/0.41 ( ( ord_less_eq_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
% 0.16/0.41 => ( ord_less_eq_real @ A @ B ) ) ).
% 0.16/0.41
% 0.16/0.41 % add_le_imp_le_left
% 0.16/0.41 thf(fact_168_add__le__imp__le__left,axiom,
% 0.16/0.41 ! [C: nat,A: nat,B: nat] :
% 0.16/0.41 ( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
% 0.16/0.41 => ( ord_less_eq_nat @ A @ B ) ) ).
% 0.16/0.41
% 0.16/0.41 % add_le_imp_le_left
% 0.16/0.41 thf(fact_169_le__iff__add,axiom,
% 0.16/0.41 ( ord_less_eq_nat
% 0.16/0.41 = ( ^ [A2: nat,B2: nat] :
% 0.16/0.41 ? [C2: nat] :
% 0.16/0.41 ( B2
% 0.16/0.41 = ( plus_plus_nat @ A2 @ C2 ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % le_iff_add
% 0.16/0.41 thf(fact_170_add__right__mono,axiom,
% 0.16/0.41 ! [A: real,B: real,C: real] :
% 0.16/0.41 ( ( ord_less_eq_real @ A @ B )
% 0.16/0.41 => ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % add_right_mono
% 0.16/0.41 thf(fact_171_add__right__mono,axiom,
% 0.16/0.41 ! [A: nat,B: nat,C: nat] :
% 0.16/0.41 ( ( ord_less_eq_nat @ A @ B )
% 0.16/0.41 => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % add_right_mono
% 0.16/0.41 thf(fact_172_less__eqE,axiom,
% 0.16/0.41 ! [A: nat,B: nat] :
% 0.16/0.41 ( ( ord_less_eq_nat @ A @ B )
% 0.16/0.41 => ~ ! [C3: nat] :
% 0.16/0.41 ( B
% 0.16/0.41 != ( plus_plus_nat @ A @ C3 ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % less_eqE
% 0.16/0.41 thf(fact_173_add__left__mono,axiom,
% 0.16/0.41 ! [A: real,B: real,C: real] :
% 0.16/0.41 ( ( ord_less_eq_real @ A @ B )
% 0.16/0.41 => ( ord_less_eq_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % add_left_mono
% 0.16/0.41 thf(fact_174_add__left__mono,axiom,
% 0.16/0.41 ! [A: nat,B: nat,C: nat] :
% 0.16/0.41 ( ( ord_less_eq_nat @ A @ B )
% 0.16/0.41 => ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % add_left_mono
% 0.16/0.41 thf(fact_175_add__mono,axiom,
% 0.16/0.41 ! [A: real,B: real,C: real,D: real] :
% 0.16/0.41 ( ( ord_less_eq_real @ A @ B )
% 0.16/0.41 => ( ( ord_less_eq_real @ C @ D )
% 0.16/0.41 => ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % add_mono
% 0.16/0.41 thf(fact_176_add__mono,axiom,
% 0.16/0.41 ! [A: nat,B: nat,C: nat,D: nat] :
% 0.16/0.41 ( ( ord_less_eq_nat @ A @ B )
% 0.16/0.41 => ( ( ord_less_eq_nat @ C @ D )
% 0.16/0.41 => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % add_mono
% 0.16/0.41 thf(fact_177_add__mono__thms__linordered__semiring_I1_J,axiom,
% 0.16/0.41 ! [I: real,J: real,K: real,L: real] :
% 0.16/0.41 ( ( ( ord_less_eq_real @ I @ J )
% 0.16/0.41 & ( ord_less_eq_real @ K @ L ) )
% 0.16/0.41 => ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % add_mono_thms_linordered_semiring(1)
% 0.16/0.41 thf(fact_178_add__mono__thms__linordered__semiring_I1_J,axiom,
% 0.16/0.41 ! [I: nat,J: nat,K: nat,L: nat] :
% 0.16/0.41 ( ( ( ord_less_eq_nat @ I @ J )
% 0.16/0.41 & ( ord_less_eq_nat @ K @ L ) )
% 0.16/0.41 => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % add_mono_thms_linordered_semiring(1)
% 0.16/0.41 thf(fact_179_add__mono__thms__linordered__semiring_I2_J,axiom,
% 0.16/0.41 ! [I: real,J: real,K: real,L: real] :
% 0.16/0.41 ( ( ( I = J )
% 0.16/0.41 & ( ord_less_eq_real @ K @ L ) )
% 0.16/0.41 => ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % add_mono_thms_linordered_semiring(2)
% 0.16/0.41 thf(fact_180_add__mono__thms__linordered__semiring_I2_J,axiom,
% 0.16/0.41 ! [I: nat,J: nat,K: nat,L: nat] :
% 0.16/0.41 ( ( ( I = J )
% 0.16/0.41 & ( ord_less_eq_nat @ K @ L ) )
% 0.16/0.41 => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % add_mono_thms_linordered_semiring(2)
% 0.16/0.41 thf(fact_181_add__mono__thms__linordered__semiring_I3_J,axiom,
% 0.16/0.41 ! [I: real,J: real,K: real,L: real] :
% 0.16/0.41 ( ( ( ord_less_eq_real @ I @ J )
% 0.16/0.41 & ( K = L ) )
% 0.16/0.41 => ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % add_mono_thms_linordered_semiring(3)
% 0.16/0.41 thf(fact_182_add__mono__thms__linordered__semiring_I3_J,axiom,
% 0.16/0.41 ! [I: nat,J: nat,K: nat,L: nat] :
% 0.16/0.41 ( ( ( ord_less_eq_nat @ I @ J )
% 0.16/0.41 & ( K = L ) )
% 0.16/0.41 => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % add_mono_thms_linordered_semiring(3)
% 0.16/0.41 thf(fact_183_diff__eq__diff__less__eq,axiom,
% 0.16/0.41 ! [A: real,B: real,C: real,D: real] :
% 0.16/0.41 ( ( ( minus_minus_real @ A @ B )
% 0.16/0.41 = ( minus_minus_real @ C @ D ) )
% 0.16/0.41 => ( ( ord_less_eq_real @ A @ B )
% 0.16/0.41 = ( ord_less_eq_real @ C @ D ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % diff_eq_diff_less_eq
% 0.16/0.41 thf(fact_184_diff__right__mono,axiom,
% 0.16/0.41 ! [A: real,B: real,C: real] :
% 0.16/0.41 ( ( ord_less_eq_real @ A @ B )
% 0.16/0.41 => ( ord_less_eq_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ C ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % diff_right_mono
% 0.16/0.41 thf(fact_185_diff__left__mono,axiom,
% 0.16/0.41 ! [B: real,A: real,C: real] :
% 0.16/0.41 ( ( ord_less_eq_real @ B @ A )
% 0.16/0.41 => ( ord_less_eq_real @ ( minus_minus_real @ C @ A ) @ ( minus_minus_real @ C @ B ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % diff_left_mono
% 0.16/0.41 thf(fact_186_diff__mono,axiom,
% 0.16/0.41 ! [A: real,B: real,D: real,C: real] :
% 0.16/0.41 ( ( ord_less_eq_real @ A @ B )
% 0.16/0.41 => ( ( ord_less_eq_real @ D @ C )
% 0.16/0.41 => ( ord_less_eq_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ D ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % diff_mono
% 0.16/0.41 thf(fact_187_add__implies__diff,axiom,
% 0.16/0.41 ! [C: real,B: real,A: real] :
% 0.16/0.41 ( ( ( plus_plus_real @ C @ B )
% 0.16/0.41 = A )
% 0.16/0.41 => ( C
% 0.16/0.41 = ( minus_minus_real @ A @ B ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % add_implies_diff
% 0.16/0.41 thf(fact_188_add__implies__diff,axiom,
% 0.16/0.41 ! [C: nat,B: nat,A: nat] :
% 0.16/0.41 ( ( ( plus_plus_nat @ C @ B )
% 0.16/0.41 = A )
% 0.16/0.41 => ( C
% 0.16/0.41 = ( minus_minus_nat @ A @ B ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % add_implies_diff
% 0.16/0.41 thf(fact_189_diff__diff__add,axiom,
% 0.16/0.41 ! [A: real,B: real,C: real] :
% 0.16/0.41 ( ( minus_minus_real @ ( minus_minus_real @ A @ B ) @ C )
% 0.16/0.41 = ( minus_minus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % diff_diff_add
% 0.16/0.41 thf(fact_190_diff__diff__add,axiom,
% 0.16/0.41 ! [A: nat,B: nat,C: nat] :
% 0.16/0.41 ( ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C )
% 0.16/0.41 = ( minus_minus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % diff_diff_add
% 0.16/0.41 thf(fact_191_diff__add__eq__diff__diff__swap,axiom,
% 0.16/0.41 ! [A: real,B: real,C: real] :
% 0.16/0.41 ( ( minus_minus_real @ A @ ( plus_plus_real @ B @ C ) )
% 0.16/0.41 = ( minus_minus_real @ ( minus_minus_real @ A @ C ) @ B ) ) ).
% 0.16/0.41
% 0.16/0.41 % diff_add_eq_diff_diff_swap
% 0.16/0.41 thf(fact_192_diff__add__eq,axiom,
% 0.16/0.41 ! [A: real,B: real,C: real] :
% 0.16/0.41 ( ( plus_plus_real @ ( minus_minus_real @ A @ B ) @ C )
% 0.16/0.41 = ( minus_minus_real @ ( plus_plus_real @ A @ C ) @ B ) ) ).
% 0.16/0.41
% 0.16/0.41 % diff_add_eq
% 0.16/0.41 thf(fact_193_diff__diff__eq2,axiom,
% 0.16/0.41 ! [A: real,B: real,C: real] :
% 0.16/0.41 ( ( minus_minus_real @ A @ ( minus_minus_real @ B @ C ) )
% 0.16/0.41 = ( minus_minus_real @ ( plus_plus_real @ A @ C ) @ B ) ) ).
% 0.16/0.41
% 0.16/0.41 % diff_diff_eq2
% 0.16/0.41 thf(fact_194_add__diff__eq,axiom,
% 0.16/0.41 ! [A: real,B: real,C: real] :
% 0.16/0.41 ( ( plus_plus_real @ A @ ( minus_minus_real @ B @ C ) )
% 0.16/0.41 = ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ C ) ) ).
% 0.16/0.41
% 0.16/0.41 % add_diff_eq
% 0.16/0.41 thf(fact_195_eq__diff__eq,axiom,
% 0.16/0.41 ! [A: real,C: real,B: real] :
% 0.16/0.41 ( ( A
% 0.16/0.41 = ( minus_minus_real @ C @ B ) )
% 0.16/0.41 = ( ( plus_plus_real @ A @ B )
% 0.16/0.41 = C ) ) ).
% 0.16/0.41
% 0.16/0.41 % eq_diff_eq
% 0.16/0.41 thf(fact_196_diff__eq__eq,axiom,
% 0.16/0.41 ! [A: real,B: real,C: real] :
% 0.16/0.41 ( ( ( minus_minus_real @ A @ B )
% 0.16/0.41 = C )
% 0.16/0.41 = ( A
% 0.16/0.41 = ( plus_plus_real @ C @ B ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % diff_eq_eq
% 0.16/0.41 thf(fact_197_group__cancel_Osub1,axiom,
% 0.16/0.41 ! [A4: real,K: real,A: real,B: real] :
% 0.16/0.41 ( ( A4
% 0.16/0.41 = ( plus_plus_real @ K @ A ) )
% 0.16/0.41 => ( ( minus_minus_real @ A4 @ B )
% 0.16/0.41 = ( plus_plus_real @ K @ ( minus_minus_real @ A @ B ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % group_cancel.sub1
% 0.16/0.41 thf(fact_198_abs__ge__self,axiom,
% 0.16/0.41 ! [A: real] : ( ord_less_eq_real @ A @ ( abs_abs_real @ A ) ) ).
% 0.16/0.41
% 0.16/0.41 % abs_ge_self
% 0.16/0.41 thf(fact_199_abs__le__D1,axiom,
% 0.16/0.41 ! [A: real,B: real] :
% 0.16/0.41 ( ( ord_less_eq_real @ ( abs_abs_real @ A ) @ B )
% 0.16/0.41 => ( ord_less_eq_real @ A @ B ) ) ).
% 0.16/0.41
% 0.16/0.41 % abs_le_D1
% 0.16/0.41 thf(fact_200_abs__one,axiom,
% 0.16/0.41 ( ( abs_abs_real @ one_one_real )
% 0.16/0.41 = one_one_real ) ).
% 0.16/0.41
% 0.16/0.41 % abs_one
% 0.16/0.41 thf(fact_201_abs__minus__commute,axiom,
% 0.16/0.41 ! [A: real,B: real] :
% 0.16/0.41 ( ( abs_abs_real @ ( minus_minus_real @ A @ B ) )
% 0.16/0.41 = ( abs_abs_real @ ( minus_minus_real @ B @ A ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % abs_minus_commute
% 0.16/0.41 thf(fact_202_max__add__distrib__left,axiom,
% 0.16/0.41 ! [X: real,Y: real,Z2: real] :
% 0.16/0.41 ( ( plus_plus_real @ ( ord_max_real @ X @ Y ) @ Z2 )
% 0.16/0.41 = ( ord_max_real @ ( plus_plus_real @ X @ Z2 ) @ ( plus_plus_real @ Y @ Z2 ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % max_add_distrib_left
% 0.16/0.41 thf(fact_203_max__add__distrib__left,axiom,
% 0.16/0.41 ! [X: nat,Y: nat,Z2: nat] :
% 0.16/0.41 ( ( plus_plus_nat @ ( ord_max_nat @ X @ Y ) @ Z2 )
% 0.16/0.41 = ( ord_max_nat @ ( plus_plus_nat @ X @ Z2 ) @ ( plus_plus_nat @ Y @ Z2 ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % max_add_distrib_left
% 0.16/0.41 thf(fact_204_max__add__distrib__right,axiom,
% 0.16/0.41 ! [X: real,Y: real,Z2: real] :
% 0.16/0.41 ( ( plus_plus_real @ X @ ( ord_max_real @ Y @ Z2 ) )
% 0.16/0.41 = ( ord_max_real @ ( plus_plus_real @ X @ Y ) @ ( plus_plus_real @ X @ Z2 ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % max_add_distrib_right
% 0.16/0.41 thf(fact_205_max__add__distrib__right,axiom,
% 0.16/0.41 ! [X: nat,Y: nat,Z2: nat] :
% 0.16/0.41 ( ( plus_plus_nat @ X @ ( ord_max_nat @ Y @ Z2 ) )
% 0.16/0.41 = ( ord_max_nat @ ( plus_plus_nat @ X @ Y ) @ ( plus_plus_nat @ X @ Z2 ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % max_add_distrib_right
% 0.16/0.41 thf(fact_206_max__diff__distrib__left,axiom,
% 0.16/0.41 ! [X: real,Y: real,Z2: real] :
% 0.16/0.41 ( ( minus_minus_real @ ( ord_max_real @ X @ Y ) @ Z2 )
% 0.16/0.41 = ( ord_max_real @ ( minus_minus_real @ X @ Z2 ) @ ( minus_minus_real @ Y @ Z2 ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % max_diff_distrib_left
% 0.16/0.41 thf(fact_207_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
% 0.16/0.41 ! [A: nat,B: nat,C: nat] :
% 0.16/0.41 ( ( ord_less_eq_nat @ A @ B )
% 0.16/0.41 => ( ( ord_less_eq_nat @ A @ B )
% 0.16/0.41 => ( ( ( minus_minus_nat @ B @ A )
% 0.16/0.41 = C )
% 0.16/0.41 = ( B
% 0.16/0.41 = ( plus_plus_nat @ C @ A ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add
% 0.16/0.41 thf(fact_208_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
% 0.16/0.41 ! [A: nat,B: nat] :
% 0.16/0.41 ( ( ord_less_eq_nat @ A @ B )
% 0.16/0.41 => ( ( plus_plus_nat @ A @ ( minus_minus_nat @ B @ A ) )
% 0.16/0.41 = B ) ) ).
% 0.16/0.41
% 0.16/0.41 % ordered_cancel_comm_monoid_diff_class.add_diff_inverse
% 0.16/0.41 thf(fact_209_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
% 0.16/0.41 ! [A: nat,B: nat,C: nat] :
% 0.16/0.41 ( ( ord_less_eq_nat @ A @ B )
% 0.16/0.41 => ( ( minus_minus_nat @ C @ ( minus_minus_nat @ B @ A ) )
% 0.16/0.41 = ( minus_minus_nat @ ( plus_plus_nat @ C @ A ) @ B ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % ordered_cancel_comm_monoid_diff_class.diff_diff_right
% 0.16/0.41 thf(fact_210_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
% 0.16/0.41 ! [A: nat,B: nat,C: nat] :
% 0.16/0.41 ( ( ord_less_eq_nat @ A @ B )
% 0.16/0.41 => ( ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A )
% 0.16/0.41 = ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ C ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
% 0.16/0.41 thf(fact_211_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
% 0.16/0.41 ! [A: nat,B: nat,C: nat] :
% 0.16/0.41 ( ( ord_less_eq_nat @ A @ B )
% 0.16/0.41 => ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ C )
% 0.16/0.41 = ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
% 0.16/0.41 thf(fact_212_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
% 0.16/0.41 ! [A: nat,B: nat,C: nat] :
% 0.16/0.41 ( ( ord_less_eq_nat @ A @ B )
% 0.16/0.41 => ( ( minus_minus_nat @ ( plus_plus_nat @ C @ B ) @ A )
% 0.16/0.41 = ( plus_plus_nat @ C @ ( minus_minus_nat @ B @ A ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % ordered_cancel_comm_monoid_diff_class.diff_add_assoc
% 0.16/0.41 thf(fact_213_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
% 0.16/0.41 ! [A: nat,B: nat,C: nat] :
% 0.16/0.41 ( ( ord_less_eq_nat @ A @ B )
% 0.16/0.41 => ( ( plus_plus_nat @ C @ ( minus_minus_nat @ B @ A ) )
% 0.16/0.41 = ( minus_minus_nat @ ( plus_plus_nat @ C @ B ) @ A ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % ordered_cancel_comm_monoid_diff_class.add_diff_assoc
% 0.16/0.41 thf(fact_214_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
% 0.16/0.41 ! [A: nat,B: nat,C: nat] :
% 0.16/0.41 ( ( ord_less_eq_nat @ A @ B )
% 0.16/0.41 => ( ( ord_less_eq_nat @ C @ ( minus_minus_nat @ B @ A ) )
% 0.16/0.41 = ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ B ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % ordered_cancel_comm_monoid_diff_class.le_diff_conv2
% 0.16/0.41 thf(fact_215_le__add__diff,axiom,
% 0.16/0.41 ! [A: nat,B: nat,C: nat] :
% 0.16/0.41 ( ( ord_less_eq_nat @ A @ B )
% 0.16/0.41 => ( ord_less_eq_nat @ C @ ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % le_add_diff
% 0.16/0.41 thf(fact_216_add__le__add__imp__diff__le,axiom,
% 0.16/0.41 ! [I: real,K: real,N: real,J: real] :
% 0.16/0.41 ( ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ N )
% 0.16/0.41 => ( ( ord_less_eq_real @ N @ ( plus_plus_real @ J @ K ) )
% 0.16/0.41 => ( ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ N )
% 0.16/0.41 => ( ( ord_less_eq_real @ N @ ( plus_plus_real @ J @ K ) )
% 0.16/0.41 => ( ord_less_eq_real @ ( minus_minus_real @ N @ K ) @ J ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % add_le_add_imp_diff_le
% 0.16/0.41 thf(fact_217_add__le__add__imp__diff__le,axiom,
% 0.16/0.41 ! [I: nat,K: nat,N: nat,J: nat] :
% 0.16/0.41 ( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N )
% 0.16/0.41 => ( ( ord_less_eq_nat @ N @ ( plus_plus_nat @ J @ K ) )
% 0.16/0.41 => ( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N )
% 0.16/0.41 => ( ( ord_less_eq_nat @ N @ ( plus_plus_nat @ J @ K ) )
% 0.16/0.41 => ( ord_less_eq_nat @ ( minus_minus_nat @ N @ K ) @ J ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % add_le_add_imp_diff_le
% 0.16/0.41 thf(fact_218_diff__add,axiom,
% 0.16/0.41 ! [A: nat,B: nat] :
% 0.16/0.41 ( ( ord_less_eq_nat @ A @ B )
% 0.16/0.41 => ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ A )
% 0.16/0.41 = B ) ) ).
% 0.16/0.41
% 0.16/0.41 % diff_add
% 0.16/0.41 thf(fact_219_add__le__imp__le__diff,axiom,
% 0.16/0.41 ! [I: real,K: real,N: real] :
% 0.16/0.41 ( ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ N )
% 0.16/0.41 => ( ord_less_eq_real @ I @ ( minus_minus_real @ N @ K ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % add_le_imp_le_diff
% 0.16/0.41 thf(fact_220_add__le__imp__le__diff,axiom,
% 0.16/0.41 ! [I: nat,K: nat,N: nat] :
% 0.16/0.41 ( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N )
% 0.16/0.41 => ( ord_less_eq_nat @ I @ ( minus_minus_nat @ N @ K ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % add_le_imp_le_diff
% 0.16/0.41 thf(fact_221_le__diff__eq,axiom,
% 0.16/0.41 ! [A: real,C: real,B: real] :
% 0.16/0.41 ( ( ord_less_eq_real @ A @ ( minus_minus_real @ C @ B ) )
% 0.16/0.41 = ( ord_less_eq_real @ ( plus_plus_real @ A @ B ) @ C ) ) ).
% 0.16/0.41
% 0.16/0.41 % le_diff_eq
% 0.16/0.41 thf(fact_222_diff__le__eq,axiom,
% 0.16/0.41 ! [A: real,B: real,C: real] :
% 0.16/0.41 ( ( ord_less_eq_real @ ( minus_minus_real @ A @ B ) @ C )
% 0.16/0.41 = ( ord_less_eq_real @ A @ ( plus_plus_real @ C @ B ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % diff_le_eq
% 0.16/0.41 thf(fact_223_abs__triangle__ineq,axiom,
% 0.16/0.41 ! [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 ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % abs_triangle_ineq
% 0.16/0.41 thf(fact_224_abs__triangle__ineq2__sym,axiom,
% 0.16/0.41 ! [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 ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % abs_triangle_ineq2_sym
% 0.16/0.41 thf(fact_225_abs__triangle__ineq3,axiom,
% 0.16/0.41 ! [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 ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % abs_triangle_ineq3
% 0.16/0.41 thf(fact_226_abs__triangle__ineq2,axiom,
% 0.16/0.41 ! [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 ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % abs_triangle_ineq2
% 0.16/0.41 thf(fact_227_abs__diff__triangle__ineq,axiom,
% 0.16/0.41 ! [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 ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % abs_diff_triangle_ineq
% 0.16/0.41 thf(fact_228_abs__triangle__ineq4,axiom,
% 0.16/0.41 ! [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 ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % abs_triangle_ineq4
% 0.16/0.41 thf(fact_229_abs__diff__le__iff,axiom,
% 0.16/0.41 ! [X: real,A: real,R: real] :
% 0.16/0.41 ( ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ X @ A ) ) @ R )
% 0.16/0.41 = ( ( ord_less_eq_real @ ( minus_minus_real @ A @ R ) @ X )
% 0.16/0.41 & ( ord_less_eq_real @ X @ ( plus_plus_real @ A @ R ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % abs_diff_le_iff
% 0.16/0.41 thf(fact_230_rts1,axiom,
% 0.16/0.41 ! [P: nat,T: real,I3: nat] :
% 0.16/0.41 ( ( ( genClo1015804716orrect @ P @ T )
% 0.16/0.41 & ( ord_less_eq_real @ ( genClo1163638703lle_te @ P @ ( plus_plus_nat @ I3 @ one_one_nat ) ) @ T ) )
% 0.16/0.41 => ( ord_less_eq_real @ genClo1651033342e_rmin @ ( minus_minus_real @ T @ ( genClo1163638703lle_te @ P @ I3 ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % rts1
% 0.16/0.41 thf(fact_231_rts1d,axiom,
% 0.16/0.41 ! [P2: nat,I: nat] :
% 0.16/0.41 ( ( genClo1015804716orrect @ P2 @ ( genClo1163638703lle_te @ P2 @ ( plus_plus_nat @ I @ one_one_nat ) ) )
% 0.16/0.41 => ( ord_less_eq_real @ genClo1651033342e_rmin @ ( minus_minus_real @ ( genClo1163638703lle_te @ P2 @ ( plus_plus_nat @ I @ one_one_nat ) ) @ ( genClo1163638703lle_te @ P2 @ I ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % rts1d
% 0.16/0.41 thf(fact_232_rts0,axiom,
% 0.16/0.41 ! [P: nat,T: real,I3: nat] :
% 0.16/0.41 ( ( ( genClo1015804716orrect @ P @ T )
% 0.16/0.41 & ( ord_less_eq_real @ T @ ( genClo1163638703lle_te @ P @ ( plus_plus_nat @ I3 @ one_one_nat ) ) ) )
% 0.16/0.41 => ( ord_less_eq_real @ ( minus_minus_real @ T @ ( genClo1163638703lle_te @ P @ I3 ) ) @ genClo1650508560e_rmax ) ) ).
% 0.16/0.41
% 0.16/0.41 % rts0
% 0.16/0.41 thf(fact_233_rts0d,axiom,
% 0.16/0.41 ! [P2: nat,I: nat] :
% 0.16/0.41 ( ( genClo1015804716orrect @ P2 @ ( genClo1163638703lle_te @ P2 @ ( plus_plus_nat @ I @ one_one_nat ) ) )
% 0.16/0.41 => ( ord_less_eq_real @ ( minus_minus_real @ ( genClo1163638703lle_te @ P2 @ ( plus_plus_nat @ I @ one_one_nat ) ) @ ( genClo1163638703lle_te @ P2 @ I ) ) @ genClo1650508560e_rmax ) ) ).
% 0.16/0.41
% 0.16/0.41 % rts0d
% 0.16/0.41 thf(fact_234_okRead2__def,axiom,
% 0.16/0.41 ( genClo293725282kRead2
% 0.16/0.41 = ( ^ [F2: nat > real,G: nat > real,X3: real,Ppred: nat > $o] :
% 0.16/0.41 ! [P4: nat] :
% 0.16/0.41 ( ( Ppred @ P4 )
% 0.16/0.41 => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( F2 @ P4 ) @ ( G @ P4 ) ) ) @ X3 ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % okRead2_def
% 0.16/0.41 thf(fact_235_sin__bound__lemma,axiom,
% 0.16/0.41 ! [X: real,Y: real,U: real,V: real] :
% 0.16/0.41 ( ( X = Y )
% 0.16/0.41 => ( ( ord_less_eq_real @ ( abs_abs_real @ U ) @ V )
% 0.16/0.41 => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( plus_plus_real @ X @ U ) @ Y ) ) @ V ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % sin_bound_lemma
% 0.16/0.41 thf(fact_236_eq__diff__eq_H,axiom,
% 0.16/0.41 ! [X: real,Y: real,Z2: real] :
% 0.16/0.41 ( ( X
% 0.16/0.41 = ( minus_minus_real @ Y @ Z2 ) )
% 0.16/0.41 = ( Y
% 0.16/0.41 = ( plus_plus_real @ X @ Z2 ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % eq_diff_eq'
% 0.16/0.41 thf(fact_237_beta__bound2,axiom,
% 0.16/0.41 ! [P2: nat,I: nat,Q2: nat] :
% 0.16/0.41 ( ( genClo1015804716orrect @ P2 @ ( genClo1163638703lle_te @ P2 @ ( plus_plus_nat @ I @ one_one_nat ) ) )
% 0.16/0.41 => ( ( genClo1015804716orrect @ Q2 @ ( genClo1163638703lle_te @ Q2 @ I ) )
% 0.16/0.41 => ( ord_less_eq_real @ ( minus_minus_real @ ( genClo1163638703lle_te @ P2 @ ( plus_plus_nat @ I @ one_one_nat ) ) @ ( genClo1163638703lle_te @ Q2 @ I ) ) @ ( plus_plus_real @ genClo1650508560e_rmax @ genClo1278781456e_beta ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % beta_bound2
% 0.16/0.41 thf(fact_238_rts2b,axiom,
% 0.16/0.41 ! [P: nat,Q3: nat,I3: nat] :
% 0.16/0.41 ( ( ( genClo1015804716orrect @ P @ ( genClo1163638703lle_te @ P @ I3 ) )
% 0.16/0.41 & ( genClo1015804716orrect @ Q3 @ ( genClo1163638703lle_te @ Q3 @ I3 ) ) )
% 0.16/0.41 => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( genClo1163638703lle_te @ P @ I3 ) @ ( genClo1163638703lle_te @ Q3 @ I3 ) ) ) @ genClo1278781456e_beta ) ) ).
% 0.16/0.41
% 0.16/0.41 % rts2b
% 0.16/0.41 thf(fact_239_nat__add__left__cancel__le,axiom,
% 0.16/0.41 ! [K: nat,M: nat,N: nat] :
% 0.16/0.41 ( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
% 0.16/0.41 = ( ord_less_eq_nat @ M @ N ) ) ).
% 0.16/0.41
% 0.16/0.41 % nat_add_left_cancel_le
% 0.16/0.41 thf(fact_240_diff__diff__cancel,axiom,
% 0.16/0.41 ! [I: nat,N: nat] :
% 0.16/0.41 ( ( ord_less_eq_nat @ I @ N )
% 0.16/0.41 => ( ( minus_minus_nat @ N @ ( minus_minus_nat @ N @ I ) )
% 0.16/0.41 = I ) ) ).
% 0.16/0.41
% 0.16/0.41 % diff_diff_cancel
% 0.16/0.41 thf(fact_241_diff__diff__left,axiom,
% 0.16/0.41 ! [I: nat,J: nat,K: nat] :
% 0.16/0.41 ( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
% 0.16/0.41 = ( minus_minus_nat @ I @ ( plus_plus_nat @ J @ K ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % diff_diff_left
% 0.16/0.41 thf(fact_242_Nat_Odiff__diff__right,axiom,
% 0.16/0.41 ! [K: nat,J: nat,I: nat] :
% 0.16/0.41 ( ( ord_less_eq_nat @ K @ J )
% 0.16/0.41 => ( ( minus_minus_nat @ I @ ( minus_minus_nat @ J @ K ) )
% 0.16/0.41 = ( minus_minus_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % Nat.diff_diff_right
% 0.16/0.41 thf(fact_243_Nat_Oadd__diff__assoc2,axiom,
% 0.16/0.41 ! [K: nat,J: nat,I: nat] :
% 0.16/0.41 ( ( ord_less_eq_nat @ K @ J )
% 0.16/0.41 => ( ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I )
% 0.16/0.41 = ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % Nat.add_diff_assoc2
% 0.16/0.41 thf(fact_244_Nat_Oadd__diff__assoc,axiom,
% 0.16/0.41 ! [K: nat,J: nat,I: nat] :
% 0.16/0.41 ( ( ord_less_eq_nat @ K @ J )
% 0.16/0.41 => ( ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K ) )
% 0.16/0.41 = ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % Nat.add_diff_assoc
% 0.16/0.41 thf(fact_245_Nat_Oex__has__greatest__nat,axiom,
% 0.16/0.41 ! [P3: nat > $o,K: nat,B: nat] :
% 0.16/0.41 ( ( P3 @ K )
% 0.16/0.41 => ( ! [Y2: nat] :
% 0.16/0.41 ( ( P3 @ Y2 )
% 0.16/0.41 => ( ord_less_eq_nat @ Y2 @ B ) )
% 0.16/0.41 => ? [X2: nat] :
% 0.16/0.41 ( ( P3 @ X2 )
% 0.16/0.41 & ! [Y5: nat] :
% 0.16/0.41 ( ( P3 @ Y5 )
% 0.16/0.41 => ( ord_less_eq_nat @ Y5 @ X2 ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % Nat.ex_has_greatest_nat
% 0.16/0.41 thf(fact_246_nat__le__linear,axiom,
% 0.16/0.41 ! [M: nat,N: nat] :
% 0.16/0.41 ( ( ord_less_eq_nat @ M @ N )
% 0.16/0.41 | ( ord_less_eq_nat @ N @ M ) ) ).
% 0.16/0.41
% 0.16/0.41 % nat_le_linear
% 0.16/0.41 thf(fact_247_diff__le__mono2,axiom,
% 0.16/0.41 ! [M: nat,N: nat,L: nat] :
% 0.16/0.41 ( ( ord_less_eq_nat @ M @ N )
% 0.16/0.41 => ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % diff_le_mono2
% 0.16/0.41 thf(fact_248_le__diff__iff_H,axiom,
% 0.16/0.41 ! [A: nat,C: nat,B: nat] :
% 0.16/0.41 ( ( ord_less_eq_nat @ A @ C )
% 0.16/0.41 => ( ( ord_less_eq_nat @ B @ C )
% 0.16/0.41 => ( ( ord_less_eq_nat @ ( minus_minus_nat @ C @ A ) @ ( minus_minus_nat @ C @ B ) )
% 0.16/0.41 = ( ord_less_eq_nat @ B @ A ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % le_diff_iff'
% 0.16/0.41 thf(fact_249_diff__le__self,axiom,
% 0.16/0.41 ! [M: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ M ) ).
% 0.16/0.41
% 0.16/0.41 % diff_le_self
% 0.16/0.41 thf(fact_250_diff__le__mono,axiom,
% 0.16/0.41 ! [M: nat,N: nat,L: nat] :
% 0.16/0.41 ( ( ord_less_eq_nat @ M @ N )
% 0.16/0.41 => ( ord_less_eq_nat @ ( minus_minus_nat @ M @ L ) @ ( minus_minus_nat @ N @ L ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % diff_le_mono
% 0.16/0.41 thf(fact_251_Nat_Odiff__diff__eq,axiom,
% 0.16/0.41 ! [K: nat,M: nat,N: nat] :
% 0.16/0.41 ( ( ord_less_eq_nat @ K @ M )
% 0.16/0.41 => ( ( ord_less_eq_nat @ K @ N )
% 0.16/0.41 => ( ( minus_minus_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
% 0.16/0.41 = ( minus_minus_nat @ M @ N ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % Nat.diff_diff_eq
% 0.16/0.41 thf(fact_252_le__diff__iff,axiom,
% 0.16/0.41 ! [K: nat,M: nat,N: nat] :
% 0.16/0.41 ( ( ord_less_eq_nat @ K @ M )
% 0.16/0.41 => ( ( ord_less_eq_nat @ K @ N )
% 0.16/0.41 => ( ( ord_less_eq_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
% 0.16/0.41 = ( ord_less_eq_nat @ M @ N ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % le_diff_iff
% 0.16/0.41 thf(fact_253_eq__diff__iff,axiom,
% 0.16/0.41 ! [K: nat,M: nat,N: nat] :
% 0.16/0.41 ( ( ord_less_eq_nat @ K @ M )
% 0.16/0.41 => ( ( ord_less_eq_nat @ K @ N )
% 0.16/0.41 => ( ( ( minus_minus_nat @ M @ K )
% 0.16/0.41 = ( minus_minus_nat @ N @ K ) )
% 0.16/0.41 = ( M = N ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % eq_diff_iff
% 0.16/0.41 thf(fact_254_le__antisym,axiom,
% 0.16/0.41 ! [M: nat,N: nat] :
% 0.16/0.41 ( ( ord_less_eq_nat @ M @ N )
% 0.16/0.41 => ( ( ord_less_eq_nat @ N @ M )
% 0.16/0.41 => ( M = N ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % le_antisym
% 0.16/0.41 thf(fact_255_eq__imp__le,axiom,
% 0.16/0.41 ! [M: nat,N: nat] :
% 0.16/0.41 ( ( M = N )
% 0.16/0.41 => ( ord_less_eq_nat @ M @ N ) ) ).
% 0.16/0.41
% 0.16/0.41 % eq_imp_le
% 0.16/0.41 thf(fact_256_le__trans,axiom,
% 0.16/0.41 ! [I: nat,J: nat,K: nat] :
% 0.16/0.41 ( ( ord_less_eq_nat @ I @ J )
% 0.16/0.41 => ( ( ord_less_eq_nat @ J @ K )
% 0.16/0.41 => ( ord_less_eq_nat @ I @ K ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % le_trans
% 0.16/0.41 thf(fact_257_le__refl,axiom,
% 0.16/0.41 ! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% 0.16/0.41
% 0.16/0.41 % le_refl
% 0.16/0.41 thf(fact_258_Nat_Ole__imp__diff__is__add,axiom,
% 0.16/0.41 ! [I: nat,J: nat,K: nat] :
% 0.16/0.41 ( ( ord_less_eq_nat @ I @ J )
% 0.16/0.41 => ( ( ( minus_minus_nat @ J @ I )
% 0.16/0.41 = K )
% 0.16/0.41 = ( J
% 0.16/0.41 = ( plus_plus_nat @ K @ I ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % Nat.le_imp_diff_is_add
% 0.16/0.41 thf(fact_259_nat__minus__add__max,axiom,
% 0.16/0.41 ! [N: nat,M: nat] :
% 0.16/0.41 ( ( plus_plus_nat @ ( minus_minus_nat @ N @ M ) @ M )
% 0.16/0.41 = ( ord_max_nat @ N @ M ) ) ).
% 0.16/0.41
% 0.16/0.41 % nat_minus_add_max
% 0.16/0.41 thf(fact_260_diff__add__inverse2,axiom,
% 0.16/0.41 ! [M: nat,N: nat] :
% 0.16/0.41 ( ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ N )
% 0.16/0.41 = M ) ).
% 0.16/0.41
% 0.16/0.41 % diff_add_inverse2
% 0.16/0.41 thf(fact_261_diff__add__inverse,axiom,
% 0.16/0.41 ! [N: nat,M: nat] :
% 0.16/0.41 ( ( minus_minus_nat @ ( plus_plus_nat @ N @ M ) @ N )
% 0.16/0.41 = M ) ).
% 0.16/0.41
% 0.16/0.41 % diff_add_inverse
% 0.16/0.41 thf(fact_262_Nat_Odiff__add__assoc2,axiom,
% 0.16/0.41 ! [K: nat,J: nat,I: nat] :
% 0.16/0.41 ( ( ord_less_eq_nat @ K @ J )
% 0.16/0.41 => ( ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K )
% 0.16/0.41 = ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % Nat.diff_add_assoc2
% 0.16/0.41 thf(fact_263_Nat_Odiff__add__assoc,axiom,
% 0.16/0.41 ! [K: nat,J: nat,I: nat] :
% 0.16/0.41 ( ( ord_less_eq_nat @ K @ J )
% 0.16/0.41 => ( ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K )
% 0.16/0.41 = ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % Nat.diff_add_assoc
% 0.16/0.41 thf(fact_264_Nat_Ole__diff__conv2,axiom,
% 0.16/0.41 ! [K: nat,J: nat,I: nat] :
% 0.16/0.41 ( ( ord_less_eq_nat @ K @ J )
% 0.16/0.41 => ( ( ord_less_eq_nat @ I @ ( minus_minus_nat @ J @ K ) )
% 0.16/0.41 = ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % Nat.le_diff_conv2
% 0.16/0.41 thf(fact_265_le__diff__conv,axiom,
% 0.16/0.41 ! [J: nat,K: nat,I: nat] :
% 0.16/0.41 ( ( ord_less_eq_nat @ ( minus_minus_nat @ J @ K ) @ I )
% 0.16/0.41 = ( ord_less_eq_nat @ J @ ( plus_plus_nat @ I @ K ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % le_diff_conv
% 0.16/0.41 thf(fact_266_diff__cancel2,axiom,
% 0.16/0.41 ! [M: nat,K: nat,N: nat] :
% 0.16/0.41 ( ( minus_minus_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N @ K ) )
% 0.16/0.41 = ( minus_minus_nat @ M @ N ) ) ).
% 0.16/0.41
% 0.16/0.41 % diff_cancel2
% 0.16/0.41 thf(fact_267_Nat_Odiff__cancel,axiom,
% 0.16/0.41 ! [K: nat,M: nat,N: nat] :
% 0.16/0.41 ( ( minus_minus_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
% 0.16/0.41 = ( minus_minus_nat @ M @ N ) ) ).
% 0.16/0.41
% 0.16/0.41 % Nat.diff_cancel
% 0.16/0.41 thf(fact_268_nat__add__max__right,axiom,
% 0.16/0.41 ! [M: nat,N: nat,Q2: nat] :
% 0.16/0.41 ( ( plus_plus_nat @ M @ ( ord_max_nat @ N @ Q2 ) )
% 0.16/0.41 = ( ord_max_nat @ ( plus_plus_nat @ M @ N ) @ ( plus_plus_nat @ M @ Q2 ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % nat_add_max_right
% 0.16/0.41 thf(fact_269_nat__add__max__left,axiom,
% 0.16/0.41 ! [M: nat,N: nat,Q2: nat] :
% 0.16/0.41 ( ( plus_plus_nat @ ( ord_max_nat @ M @ N ) @ Q2 )
% 0.16/0.41 = ( ord_max_nat @ ( plus_plus_nat @ M @ Q2 ) @ ( plus_plus_nat @ N @ Q2 ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % nat_add_max_left
% 0.16/0.41 thf(fact_270_nat__le__iff__add,axiom,
% 0.16/0.41 ( ord_less_eq_nat
% 0.16/0.41 = ( ^ [M2: nat,N2: nat] :
% 0.16/0.41 ? [K2: nat] :
% 0.16/0.41 ( N2
% 0.16/0.41 = ( plus_plus_nat @ M2 @ K2 ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % nat_le_iff_add
% 0.16/0.41 thf(fact_271_trans__le__add2,axiom,
% 0.16/0.41 ! [I: nat,J: nat,M: nat] :
% 0.16/0.41 ( ( ord_less_eq_nat @ I @ J )
% 0.16/0.41 => ( ord_less_eq_nat @ I @ ( plus_plus_nat @ M @ J ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % trans_le_add2
% 0.16/0.41 thf(fact_272_trans__le__add1,axiom,
% 0.16/0.41 ! [I: nat,J: nat,M: nat] :
% 0.16/0.41 ( ( ord_less_eq_nat @ I @ J )
% 0.16/0.41 => ( ord_less_eq_nat @ I @ ( plus_plus_nat @ J @ M ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % trans_le_add1
% 0.16/0.41 thf(fact_273_add__le__mono1,axiom,
% 0.16/0.41 ! [I: nat,J: nat,K: nat] :
% 0.16/0.41 ( ( ord_less_eq_nat @ I @ J )
% 0.16/0.41 => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % add_le_mono1
% 0.16/0.41 thf(fact_274_add__le__mono,axiom,
% 0.16/0.41 ! [I: nat,J: nat,K: nat,L: nat] :
% 0.16/0.41 ( ( ord_less_eq_nat @ I @ J )
% 0.16/0.41 => ( ( ord_less_eq_nat @ K @ L )
% 0.16/0.41 => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % add_le_mono
% 0.16/0.41 thf(fact_275_le__Suc__ex,axiom,
% 0.16/0.41 ! [K: nat,L: nat] :
% 0.16/0.41 ( ( ord_less_eq_nat @ K @ L )
% 0.16/0.41 => ? [N3: nat] :
% 0.16/0.41 ( L
% 0.16/0.41 = ( plus_plus_nat @ K @ N3 ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % le_Suc_ex
% 0.16/0.41 thf(fact_276_add__leD2,axiom,
% 0.16/0.41 ! [M: nat,K: nat,N: nat] :
% 0.16/0.41 ( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
% 0.16/0.41 => ( ord_less_eq_nat @ K @ N ) ) ).
% 0.16/0.41
% 0.16/0.41 % add_leD2
% 0.16/0.41 thf(fact_277_add__leD1,axiom,
% 0.16/0.41 ! [M: nat,K: nat,N: nat] :
% 0.16/0.41 ( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
% 0.16/0.41 => ( ord_less_eq_nat @ M @ N ) ) ).
% 0.16/0.41
% 0.16/0.41 % add_leD1
% 0.16/0.41 thf(fact_278_le__add2,axiom,
% 0.16/0.41 ! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M @ N ) ) ).
% 0.16/0.41
% 0.16/0.41 % le_add2
% 0.16/0.41 thf(fact_279_le__add1,axiom,
% 0.16/0.41 ! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M ) ) ).
% 0.16/0.41
% 0.16/0.41 % le_add1
% 0.16/0.41 thf(fact_280_add__leE,axiom,
% 0.16/0.41 ! [M: nat,K: nat,N: nat] :
% 0.16/0.41 ( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
% 0.16/0.41 => ~ ( ( ord_less_eq_nat @ M @ N )
% 0.16/0.41 => ~ ( ord_less_eq_nat @ K @ N ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % add_leE
% 0.16/0.41 thf(fact_281_nonoverlap,axiom,
% 0.16/0.41 ord_less_eq_real @ genClo1278781456e_beta @ genClo1651033342e_rmin ).
% 0.16/0.41
% 0.16/0.41 % nonoverlap
% 0.16/0.41 thf(fact_282_rts2a,axiom,
% 0.16/0.41 ! [P: nat,Q3: nat,T: real,I3: nat] :
% 0.16/0.41 ( ( ( genClo1015804716orrect @ P @ T )
% 0.16/0.41 & ( genClo1015804716orrect @ Q3 @ T )
% 0.16/0.41 & ( ord_less_eq_real @ ( plus_plus_real @ ( genClo1163638703lle_te @ Q3 @ I3 ) @ genClo1278781456e_beta ) @ T ) )
% 0.16/0.41 => ( ord_less_eq_real @ ( genClo1163638703lle_te @ P @ I3 ) @ T ) ) ).
% 0.16/0.41
% 0.16/0.41 % rts2a
% 0.16/0.41 thf(fact_283_readerror,axiom,
% 0.16/0.41 ! [P: nat,Q3: nat,I3: nat] :
% 0.16/0.41 ( ( ( genClo1015804716orrect @ P @ ( genClo1163638703lle_te @ P @ ( plus_plus_nat @ I3 @ one_one_nat ) ) )
% 0.16/0.41 & ( genClo1015804716orrect @ Q3 @ ( genClo1163638703lle_te @ P @ ( plus_plus_nat @ I3 @ one_one_nat ) ) ) )
% 0.16/0.41 => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( genClo1400225944_theta @ P @ ( plus_plus_nat @ I3 @ one_one_nat ) @ Q3 ) @ ( genClo1160817912lle_IC @ Q3 @ I3 @ ( genClo1163638703lle_te @ P @ ( plus_plus_nat @ I3 @ one_one_nat ) ) ) ) ) @ genClo721845095Lambda ) ) ).
% 0.16/0.41
% 0.16/0.41 % readerror
% 0.16/0.41 thf(fact_284_okRead1__def,axiom,
% 0.16/0.41 ( genClo293725281kRead1
% 0.16/0.41 = ( ^ [F2: nat > real,X3: real,Ppred: nat > $o] :
% 0.16/0.41 ! [L2: nat,M2: nat] :
% 0.16/0.41 ( ( ( Ppred @ L2 )
% 0.16/0.41 & ( Ppred @ M2 ) )
% 0.16/0.41 => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( F2 @ L2 ) @ ( F2 @ M2 ) ) ) @ X3 ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % okRead1_def
% 0.16/0.41 thf(fact_285_beta__bound1,axiom,
% 0.16/0.41 ! [P2: nat,I: nat,Q2: nat] :
% 0.16/0.41 ( ( genClo1015804716orrect @ P2 @ ( genClo1163638703lle_te @ P2 @ ( plus_plus_nat @ I @ one_one_nat ) ) )
% 0.16/0.41 => ( ( genClo1015804716orrect @ Q2 @ ( genClo1163638703lle_te @ P2 @ ( plus_plus_nat @ I @ one_one_nat ) ) )
% 0.16/0.41 => ( ord_less_eq_real @ zero_zero_real @ ( minus_minus_real @ ( genClo1163638703lle_te @ P2 @ ( plus_plus_nat @ I @ one_one_nat ) ) @ ( genClo1163638703lle_te @ Q2 @ I ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % beta_bound1
% 0.16/0.41 thf(fact_286_le__zero__eq,axiom,
% 0.16/0.41 ! [N: nat] :
% 0.16/0.41 ( ( ord_less_eq_nat @ N @ zero_zero_nat )
% 0.16/0.41 = ( N = zero_zero_nat ) ) ).
% 0.16/0.41
% 0.16/0.41 % le_zero_eq
% 0.16/0.41 thf(fact_287_zero__eq__add__iff__both__eq__0,axiom,
% 0.16/0.41 ! [X: nat,Y: nat] :
% 0.16/0.41 ( ( zero_zero_nat
% 0.16/0.41 = ( plus_plus_nat @ X @ Y ) )
% 0.16/0.41 = ( ( X = zero_zero_nat )
% 0.16/0.41 & ( Y = zero_zero_nat ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % zero_eq_add_iff_both_eq_0
% 0.16/0.41 thf(fact_288_add__eq__0__iff__both__eq__0,axiom,
% 0.16/0.41 ! [X: nat,Y: nat] :
% 0.16/0.41 ( ( ( plus_plus_nat @ X @ Y )
% 0.16/0.41 = zero_zero_nat )
% 0.16/0.41 = ( ( X = zero_zero_nat )
% 0.16/0.41 & ( Y = zero_zero_nat ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % add_eq_0_iff_both_eq_0
% 0.16/0.41 thf(fact_289_add__cancel__right__right,axiom,
% 0.16/0.41 ! [A: nat,B: nat] :
% 0.16/0.41 ( ( A
% 0.16/0.41 = ( plus_plus_nat @ A @ B ) )
% 0.16/0.41 = ( B = zero_zero_nat ) ) ).
% 0.16/0.41
% 0.16/0.41 % add_cancel_right_right
% 0.16/0.41 thf(fact_290_add__cancel__right__right,axiom,
% 0.16/0.41 ! [A: real,B: real] :
% 0.16/0.41 ( ( A
% 0.16/0.41 = ( plus_plus_real @ A @ B ) )
% 0.16/0.41 = ( B = zero_zero_real ) ) ).
% 0.16/0.41
% 0.16/0.41 % add_cancel_right_right
% 0.16/0.41 thf(fact_291_add__cancel__right__left,axiom,
% 0.16/0.41 ! [A: nat,B: nat] :
% 0.16/0.41 ( ( A
% 0.16/0.41 = ( plus_plus_nat @ B @ A ) )
% 0.16/0.41 = ( B = zero_zero_nat ) ) ).
% 0.16/0.41
% 0.16/0.41 % add_cancel_right_left
% 0.16/0.41 thf(fact_292_add__cancel__right__left,axiom,
% 0.16/0.41 ! [A: real,B: real] :
% 0.16/0.41 ( ( A
% 0.16/0.41 = ( plus_plus_real @ B @ A ) )
% 0.16/0.41 = ( B = zero_zero_real ) ) ).
% 0.16/0.41
% 0.16/0.41 % add_cancel_right_left
% 0.16/0.41 thf(fact_293_add__cancel__left__right,axiom,
% 0.16/0.41 ! [A: nat,B: nat] :
% 0.16/0.41 ( ( ( plus_plus_nat @ A @ B )
% 0.16/0.41 = A )
% 0.16/0.41 = ( B = zero_zero_nat ) ) ).
% 0.16/0.41
% 0.16/0.41 % add_cancel_left_right
% 0.16/0.41 thf(fact_294_add__cancel__left__right,axiom,
% 0.16/0.41 ! [A: real,B: real] :
% 0.16/0.41 ( ( ( plus_plus_real @ A @ B )
% 0.16/0.41 = A )
% 0.16/0.41 = ( B = zero_zero_real ) ) ).
% 0.16/0.41
% 0.16/0.41 % add_cancel_left_right
% 0.16/0.41 thf(fact_295_add__cancel__left__left,axiom,
% 0.16/0.41 ! [B: nat,A: nat] :
% 0.16/0.41 ( ( ( plus_plus_nat @ B @ A )
% 0.16/0.41 = A )
% 0.16/0.41 = ( B = zero_zero_nat ) ) ).
% 0.16/0.41
% 0.16/0.41 % add_cancel_left_left
% 0.16/0.41 thf(fact_296_add__cancel__left__left,axiom,
% 0.16/0.41 ! [B: real,A: real] :
% 0.16/0.41 ( ( ( plus_plus_real @ B @ A )
% 0.16/0.41 = A )
% 0.16/0.41 = ( B = zero_zero_real ) ) ).
% 0.16/0.41
% 0.16/0.41 % add_cancel_left_left
% 0.16/0.41 thf(fact_297_double__zero__sym,axiom,
% 0.16/0.41 ! [A: real] :
% 0.16/0.41 ( ( zero_zero_real
% 0.16/0.41 = ( plus_plus_real @ A @ A ) )
% 0.16/0.41 = ( A = zero_zero_real ) ) ).
% 0.16/0.41
% 0.16/0.41 % double_zero_sym
% 0.16/0.41 thf(fact_298_double__zero,axiom,
% 0.16/0.41 ! [A: real] :
% 0.16/0.41 ( ( ( plus_plus_real @ A @ A )
% 0.16/0.41 = zero_zero_real )
% 0.16/0.41 = ( A = zero_zero_real ) ) ).
% 0.16/0.41
% 0.16/0.41 % double_zero
% 0.16/0.41 thf(fact_299_add_Oright__neutral,axiom,
% 0.16/0.41 ! [A: nat] :
% 0.16/0.41 ( ( plus_plus_nat @ A @ zero_zero_nat )
% 0.16/0.41 = A ) ).
% 0.16/0.41
% 0.16/0.41 % add.right_neutral
% 0.16/0.41 thf(fact_300_add_Oright__neutral,axiom,
% 0.16/0.41 ! [A: real] :
% 0.16/0.41 ( ( plus_plus_real @ A @ zero_zero_real )
% 0.16/0.41 = A ) ).
% 0.16/0.41
% 0.16/0.41 % add.right_neutral
% 0.16/0.41 thf(fact_301_add_Oleft__neutral,axiom,
% 0.16/0.41 ! [A: nat] :
% 0.16/0.41 ( ( plus_plus_nat @ zero_zero_nat @ A )
% 0.16/0.41 = A ) ).
% 0.16/0.41
% 0.16/0.41 % add.left_neutral
% 0.16/0.41 thf(fact_302_add_Oleft__neutral,axiom,
% 0.16/0.41 ! [A: real] :
% 0.16/0.41 ( ( plus_plus_real @ zero_zero_real @ A )
% 0.16/0.41 = A ) ).
% 0.16/0.41
% 0.16/0.41 % add.left_neutral
% 0.16/0.41 thf(fact_303_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
% 0.16/0.41 ! [A: real] :
% 0.16/0.41 ( ( minus_minus_real @ A @ A )
% 0.16/0.41 = zero_zero_real ) ).
% 0.16/0.41
% 0.16/0.41 % cancel_comm_monoid_add_class.diff_cancel
% 0.16/0.41 thf(fact_304_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
% 0.16/0.41 ! [A: nat] :
% 0.16/0.41 ( ( minus_minus_nat @ A @ A )
% 0.16/0.41 = zero_zero_nat ) ).
% 0.16/0.41
% 0.16/0.41 % cancel_comm_monoid_add_class.diff_cancel
% 0.16/0.41 thf(fact_305_diff__zero,axiom,
% 0.16/0.41 ! [A: real] :
% 0.16/0.41 ( ( minus_minus_real @ A @ zero_zero_real )
% 0.16/0.41 = A ) ).
% 0.16/0.41
% 0.16/0.41 % diff_zero
% 0.16/0.41 thf(fact_306_diff__zero,axiom,
% 0.16/0.41 ! [A: nat] :
% 0.16/0.41 ( ( minus_minus_nat @ A @ zero_zero_nat )
% 0.16/0.41 = A ) ).
% 0.16/0.41
% 0.16/0.41 % diff_zero
% 0.16/0.41 thf(fact_307_zero__diff,axiom,
% 0.16/0.41 ! [A: nat] :
% 0.16/0.41 ( ( minus_minus_nat @ zero_zero_nat @ A )
% 0.16/0.41 = zero_zero_nat ) ).
% 0.16/0.41
% 0.16/0.41 % zero_diff
% 0.16/0.41 thf(fact_308_diff__0__right,axiom,
% 0.16/0.41 ! [A: real] :
% 0.16/0.41 ( ( minus_minus_real @ A @ zero_zero_real )
% 0.16/0.41 = A ) ).
% 0.16/0.41
% 0.16/0.41 % diff_0_right
% 0.16/0.41 thf(fact_309_diff__self,axiom,
% 0.16/0.41 ! [A: real] :
% 0.16/0.41 ( ( minus_minus_real @ A @ A )
% 0.16/0.41 = zero_zero_real ) ).
% 0.16/0.41
% 0.16/0.41 % diff_self
% 0.16/0.41 thf(fact_310_abs__zero,axiom,
% 0.16/0.41 ( ( abs_abs_real @ zero_zero_real )
% 0.16/0.41 = zero_zero_real ) ).
% 0.16/0.41
% 0.16/0.41 % abs_zero
% 0.16/0.41 thf(fact_311_abs__eq__0,axiom,
% 0.16/0.41 ! [A: real] :
% 0.16/0.41 ( ( ( abs_abs_real @ A )
% 0.16/0.41 = zero_zero_real )
% 0.16/0.41 = ( A = zero_zero_real ) ) ).
% 0.16/0.41
% 0.16/0.41 % abs_eq_0
% 0.16/0.41 thf(fact_312_abs__0__eq,axiom,
% 0.16/0.41 ! [A: real] :
% 0.16/0.41 ( ( zero_zero_real
% 0.16/0.41 = ( abs_abs_real @ A ) )
% 0.16/0.41 = ( A = zero_zero_real ) ) ).
% 0.16/0.41
% 0.16/0.41 % abs_0_eq
% 0.16/0.41 thf(fact_313_abs__0,axiom,
% 0.16/0.41 ( ( abs_abs_real @ zero_zero_real )
% 0.16/0.41 = zero_zero_real ) ).
% 0.16/0.41
% 0.16/0.41 % abs_0
% 0.16/0.41 thf(fact_314_add__le__same__cancel1,axiom,
% 0.16/0.41 ! [B: real,A: real] :
% 0.16/0.41 ( ( ord_less_eq_real @ ( plus_plus_real @ B @ A ) @ B )
% 0.16/0.41 = ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% 0.16/0.41
% 0.16/0.41 % add_le_same_cancel1
% 0.16/0.41 thf(fact_315_add__le__same__cancel1,axiom,
% 0.16/0.41 ! [B: nat,A: nat] :
% 0.16/0.41 ( ( ord_less_eq_nat @ ( plus_plus_nat @ B @ A ) @ B )
% 0.16/0.41 = ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).
% 0.16/0.41
% 0.16/0.41 % add_le_same_cancel1
% 0.16/0.41 thf(fact_316_add__le__same__cancel2,axiom,
% 0.16/0.41 ! [A: real,B: real] :
% 0.16/0.41 ( ( ord_less_eq_real @ ( plus_plus_real @ A @ B ) @ B )
% 0.16/0.41 = ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% 0.16/0.41
% 0.16/0.41 % add_le_same_cancel2
% 0.16/0.41 thf(fact_317_add__le__same__cancel2,axiom,
% 0.16/0.41 ! [A: nat,B: nat] :
% 0.16/0.41 ( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ B )
% 0.16/0.41 = ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).
% 0.16/0.41
% 0.16/0.41 % add_le_same_cancel2
% 0.16/0.41 thf(fact_318_le__add__same__cancel1,axiom,
% 0.16/0.41 ! [A: real,B: real] :
% 0.16/0.41 ( ( ord_less_eq_real @ A @ ( plus_plus_real @ A @ B ) )
% 0.16/0.41 = ( ord_less_eq_real @ zero_zero_real @ B ) ) ).
% 0.16/0.41
% 0.16/0.41 % le_add_same_cancel1
% 0.16/0.41 thf(fact_319_le__add__same__cancel1,axiom,
% 0.16/0.41 ! [A: nat,B: nat] :
% 0.16/0.41 ( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ A @ B ) )
% 0.16/0.41 = ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% 0.16/0.41
% 0.16/0.41 % le_add_same_cancel1
% 0.16/0.41 thf(fact_320_le__add__same__cancel2,axiom,
% 0.16/0.41 ! [A: real,B: real] :
% 0.16/0.41 ( ( ord_less_eq_real @ A @ ( plus_plus_real @ B @ A ) )
% 0.16/0.41 = ( ord_less_eq_real @ zero_zero_real @ B ) ) ).
% 0.16/0.41
% 0.16/0.41 % le_add_same_cancel2
% 0.16/0.41 thf(fact_321_le__add__same__cancel2,axiom,
% 0.16/0.41 ! [A: nat,B: nat] :
% 0.16/0.41 ( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ B @ A ) )
% 0.16/0.41 = ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% 0.16/0.41
% 0.16/0.41 % le_add_same_cancel2
% 0.16/0.41 thf(fact_322_double__add__le__zero__iff__single__add__le__zero,axiom,
% 0.16/0.41 ! [A: real] :
% 0.16/0.41 ( ( ord_less_eq_real @ ( plus_plus_real @ A @ A ) @ zero_zero_real )
% 0.16/0.41 = ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% 0.16/0.41
% 0.16/0.41 % double_add_le_zero_iff_single_add_le_zero
% 0.16/0.41 thf(fact_323_zero__le__double__add__iff__zero__le__single__add,axiom,
% 0.16/0.41 ! [A: real] :
% 0.16/0.41 ( ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ A @ A ) )
% 0.16/0.41 = ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% 0.16/0.41
% 0.16/0.41 % zero_le_double_add_iff_zero_le_single_add
% 0.16/0.41 thf(fact_324_diff__ge__0__iff__ge,axiom,
% 0.16/0.41 ! [A: real,B: real] :
% 0.16/0.41 ( ( ord_less_eq_real @ zero_zero_real @ ( minus_minus_real @ A @ B ) )
% 0.16/0.41 = ( ord_less_eq_real @ B @ A ) ) ).
% 0.16/0.41
% 0.16/0.41 % diff_ge_0_iff_ge
% 0.16/0.41 thf(fact_325_diff__add__zero,axiom,
% 0.16/0.41 ! [A: nat,B: nat] :
% 0.16/0.41 ( ( minus_minus_nat @ A @ ( plus_plus_nat @ A @ B ) )
% 0.16/0.41 = zero_zero_nat ) ).
% 0.16/0.41
% 0.16/0.41 % diff_add_zero
% 0.16/0.41 thf(fact_326_abs__of__nonneg,axiom,
% 0.16/0.41 ! [A: real] :
% 0.16/0.41 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.16/0.41 => ( ( abs_abs_real @ A )
% 0.16/0.41 = A ) ) ).
% 0.16/0.41
% 0.16/0.41 % abs_of_nonneg
% 0.16/0.41 thf(fact_327_abs__le__self__iff,axiom,
% 0.16/0.41 ! [A: real] :
% 0.16/0.41 ( ( ord_less_eq_real @ ( abs_abs_real @ A ) @ A )
% 0.16/0.41 = ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% 0.16/0.41
% 0.16/0.41 % abs_le_self_iff
% 0.16/0.41 thf(fact_328_abs__le__zero__iff,axiom,
% 0.16/0.41 ! [A: real] :
% 0.16/0.41 ( ( ord_less_eq_real @ ( abs_abs_real @ A ) @ zero_zero_real )
% 0.16/0.41 = ( A = zero_zero_real ) ) ).
% 0.16/0.41
% 0.16/0.41 % abs_le_zero_iff
% 0.16/0.41 thf(fact_329_zero__le,axiom,
% 0.16/0.41 ! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).
% 0.16/0.41
% 0.16/0.41 % zero_le
% 0.16/0.41 thf(fact_330_diff__commute,axiom,
% 0.16/0.41 ! [I: nat,J: nat,K: nat] :
% 0.16/0.41 ( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
% 0.16/0.41 = ( minus_minus_nat @ ( minus_minus_nat @ I @ K ) @ J ) ) ).
% 0.16/0.41
% 0.16/0.41 % diff_commute
% 0.16/0.41 thf(fact_331_zero__reorient,axiom,
% 0.16/0.41 ! [X: real] :
% 0.16/0.41 ( ( zero_zero_real = X )
% 0.16/0.41 = ( X = zero_zero_real ) ) ).
% 0.16/0.41
% 0.16/0.41 % zero_reorient
% 0.16/0.41 thf(fact_332_abs__eq__0__iff,axiom,
% 0.16/0.41 ! [A: real] :
% 0.16/0.41 ( ( ( abs_abs_real @ A )
% 0.16/0.41 = zero_zero_real )
% 0.16/0.41 = ( A = zero_zero_real ) ) ).
% 0.16/0.41
% 0.16/0.41 % abs_eq_0_iff
% 0.16/0.41 thf(fact_333_eq__iff__diff__eq__0,axiom,
% 0.16/0.41 ( ( ^ [Y3: real,Z: real] : ( Y3 = Z ) )
% 0.16/0.41 = ( ^ [A2: real,B2: real] :
% 0.16/0.41 ( ( minus_minus_real @ A2 @ B2 )
% 0.16/0.41 = zero_zero_real ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % eq_iff_diff_eq_0
% 0.16/0.41 thf(fact_334_zero__neq__one,axiom,
% 0.16/0.41 zero_zero_nat != one_one_nat ).
% 0.16/0.41
% 0.16/0.41 % zero_neq_one
% 0.16/0.41 thf(fact_335_zero__neq__one,axiom,
% 0.16/0.41 zero_zero_real != one_one_real ).
% 0.16/0.41
% 0.16/0.41 % zero_neq_one
% 0.16/0.41 thf(fact_336_add_Ogroup__left__neutral,axiom,
% 0.16/0.41 ! [A: real] :
% 0.16/0.41 ( ( plus_plus_real @ zero_zero_real @ A )
% 0.16/0.41 = A ) ).
% 0.16/0.41
% 0.16/0.41 % add.group_left_neutral
% 0.16/0.41 thf(fact_337_add_Ocomm__neutral,axiom,
% 0.16/0.41 ! [A: nat] :
% 0.16/0.41 ( ( plus_plus_nat @ A @ zero_zero_nat )
% 0.16/0.41 = A ) ).
% 0.16/0.41
% 0.16/0.41 % add.comm_neutral
% 0.16/0.41 thf(fact_338_add_Ocomm__neutral,axiom,
% 0.16/0.41 ! [A: real] :
% 0.16/0.41 ( ( plus_plus_real @ A @ zero_zero_real )
% 0.16/0.41 = A ) ).
% 0.16/0.41
% 0.16/0.41 % add.comm_neutral
% 0.16/0.41 thf(fact_339_comm__monoid__add__class_Oadd__0,axiom,
% 0.16/0.41 ! [A: nat] :
% 0.16/0.41 ( ( plus_plus_nat @ zero_zero_nat @ A )
% 0.16/0.41 = A ) ).
% 0.16/0.41
% 0.16/0.41 % comm_monoid_add_class.add_0
% 0.16/0.41 thf(fact_340_comm__monoid__add__class_Oadd__0,axiom,
% 0.16/0.41 ! [A: real] :
% 0.16/0.41 ( ( plus_plus_real @ zero_zero_real @ A )
% 0.16/0.41 = A ) ).
% 0.16/0.41
% 0.16/0.41 % comm_monoid_add_class.add_0
% 0.16/0.41 thf(fact_341_add__decreasing,axiom,
% 0.16/0.41 ! [A: real,C: real,B: real] :
% 0.16/0.41 ( ( ord_less_eq_real @ A @ zero_zero_real )
% 0.16/0.41 => ( ( ord_less_eq_real @ C @ B )
% 0.16/0.41 => ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ B ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % add_decreasing
% 0.16/0.41 thf(fact_342_add__decreasing,axiom,
% 0.16/0.41 ! [A: nat,C: nat,B: nat] :
% 0.16/0.41 ( ( ord_less_eq_nat @ A @ zero_zero_nat )
% 0.16/0.41 => ( ( ord_less_eq_nat @ C @ B )
% 0.16/0.41 => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ B ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % add_decreasing
% 0.16/0.41 thf(fact_343_add__increasing,axiom,
% 0.16/0.41 ! [A: real,B: real,C: real] :
% 0.16/0.41 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.16/0.41 => ( ( ord_less_eq_real @ B @ C )
% 0.16/0.41 => ( ord_less_eq_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % add_increasing
% 0.16/0.41 thf(fact_344_add__increasing,axiom,
% 0.16/0.41 ! [A: nat,B: nat,C: nat] :
% 0.16/0.41 ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.16/0.41 => ( ( ord_less_eq_nat @ B @ C )
% 0.16/0.41 => ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % add_increasing
% 0.16/0.41 thf(fact_345_add__decreasing2,axiom,
% 0.16/0.41 ! [C: real,A: real,B: real] :
% 0.16/0.41 ( ( ord_less_eq_real @ C @ zero_zero_real )
% 0.16/0.41 => ( ( ord_less_eq_real @ A @ B )
% 0.16/0.41 => ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ B ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % add_decreasing2
% 0.16/0.41 thf(fact_346_add__decreasing2,axiom,
% 0.16/0.41 ! [C: nat,A: nat,B: nat] :
% 0.16/0.41 ( ( ord_less_eq_nat @ C @ zero_zero_nat )
% 0.16/0.41 => ( ( ord_less_eq_nat @ A @ B )
% 0.16/0.41 => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ B ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % add_decreasing2
% 0.16/0.41 thf(fact_347_add__increasing2,axiom,
% 0.16/0.41 ! [C: real,B: real,A: real] :
% 0.16/0.41 ( ( ord_less_eq_real @ zero_zero_real @ C )
% 0.16/0.41 => ( ( ord_less_eq_real @ B @ A )
% 0.16/0.41 => ( ord_less_eq_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % add_increasing2
% 0.16/0.41 thf(fact_348_add__increasing2,axiom,
% 0.16/0.41 ! [C: nat,B: nat,A: nat] :
% 0.16/0.41 ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 0.16/0.41 => ( ( ord_less_eq_nat @ B @ A )
% 0.16/0.41 => ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % add_increasing2
% 0.16/0.41 thf(fact_349_add__nonneg__nonneg,axiom,
% 0.16/0.41 ! [A: real,B: real] :
% 0.16/0.41 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.16/0.41 => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 0.16/0.41 => ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ A @ B ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % add_nonneg_nonneg
% 0.16/0.41 thf(fact_350_add__nonneg__nonneg,axiom,
% 0.16/0.41 ! [A: nat,B: nat] :
% 0.16/0.41 ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.16/0.41 => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 0.16/0.41 => ( ord_less_eq_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % add_nonneg_nonneg
% 0.16/0.41 thf(fact_351_add__nonpos__nonpos,axiom,
% 0.16/0.41 ! [A: real,B: real] :
% 0.16/0.41 ( ( ord_less_eq_real @ A @ zero_zero_real )
% 0.16/0.41 => ( ( ord_less_eq_real @ B @ zero_zero_real )
% 0.16/0.41 => ( ord_less_eq_real @ ( plus_plus_real @ A @ B ) @ zero_zero_real ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % add_nonpos_nonpos
% 0.16/0.41 thf(fact_352_add__nonpos__nonpos,axiom,
% 0.16/0.41 ! [A: nat,B: nat] :
% 0.16/0.41 ( ( ord_less_eq_nat @ A @ zero_zero_nat )
% 0.16/0.41 => ( ( ord_less_eq_nat @ B @ zero_zero_nat )
% 0.16/0.41 => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % add_nonpos_nonpos
% 0.16/0.41
% 0.16/0.41 % Helper facts (5)
% 0.16/0.41 thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
% 0.16/0.41 ! [X: nat,Y: nat] :
% 0.16/0.41 ( ( if_nat @ $false @ X @ Y )
% 0.16/0.41 = Y ) ).
% 0.16/0.41
% 0.16/0.41 thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
% 0.16/0.41 ! [X: nat,Y: nat] :
% 0.16/0.41 ( ( if_nat @ $true @ X @ Y )
% 0.16/0.41 = X ) ).
% 0.16/0.41
% 0.16/0.41 thf(help_If_3_1_If_001t__Real__Oreal_T,axiom,
% 0.16/0.41 ! [P3: $o] :
% 0.16/0.41 ( ( P3 = $true )
% 0.16/0.41 | ( P3 = $false ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(help_If_2_1_If_001t__Real__Oreal_T,axiom,
% 0.16/0.41 ! [X: real,Y: real] :
% 0.16/0.41 ( ( if_real @ $false @ X @ Y )
% 0.16/0.41 = Y ) ).
% 0.16/0.41
% 0.16/0.41 thf(help_If_1_1_If_001t__Real__Oreal_T,axiom,
% 0.16/0.41 ! [X: real,Y: real] :
% 0.16/0.41 ( ( if_real @ $true @ X @ Y )
% 0.16/0.41 = X ) ).
% 0.16/0.41
% 0.16/0.41 % Conjectures (1)
% 0.16/0.41 thf(conj_0,conjecture,
% 0.16/0.41 genClo1015804716orrect @ m @ ( ord_max_real @ ( genClo1163638703lle_te @ l @ i ) @ ( genClo1163638703lle_te @ m @ i ) ) ).
% 0.16/0.41
% 0.16/0.41 %------------------------------------------------------------------------------
% 0.16/0.41 ------------------------
% 2.51/1.47 ---- Embedded in THF ---
% 2.51/1.48 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 2.51/1.48 %%% Generated on Tue Feb 24 08:25:28 EST 2026
% 2.51/1.48 %%% using '$$dhol' embedding, version 1.3.0.
% 2.51/1.48 %%% Logic specification used:
% 2.51/1.48 %%% thf(spec, logic, $$dhol).
% 2.51/1.48
% 2.51/1.48 % SZS output start ListOfTHF for /export/starexec/sandbox2/tmp/tmp.k9LPPfwlmN/DTF2DTF777.p
% See solution above
% 2.51/1.48 ------------------------
% 2.74/1.52 ---- Cleaned THF ---
% 2.74/1.52 thf(ty_n_t__Real__Oreal,type,
% 2.74/1.52 real: $tType ).
% 2.74/1.52
% 2.74/1.52 thf(ty_n_t__Nat__Onat,type,
% 2.74/1.52 nat: $tType ).
% 2.74/1.52
% 2.74/1.52 thf(sy_c_GenClock__Mirabelle__bsvkzpgbls_OIC,type,
% 2.74/1.52 genClo1160817912lle_IC: nat > nat > real > real ).
% 2.74/1.52
% 2.74/1.52 thf(sy_c_GenClock__Mirabelle__bsvkzpgbls_O_092_060Lambda_062,type,
% 2.74/1.52 genClo721845095Lambda: real ).
% 2.74/1.52
% 2.74/1.52 thf(sy_c_GenClock__Mirabelle__bsvkzpgbls_O_092_060beta_062,type,
% 2.74/1.52 genClo1278781456e_beta: real ).
% 2.74/1.52
% 2.74/1.52 thf(sy_c_GenClock__Mirabelle__bsvkzpgbls_O_092_060theta_062,type,
% 2.74/1.52 genClo1400225944_theta: nat > nat > nat > real ).
% 2.74/1.52
% 2.74/1.52 thf(sy_c_GenClock__Mirabelle__bsvkzpgbls_Ocorrect,type,
% 2.74/1.52 genClo1015804716orrect: nat > real > $o ).
% 2.74/1.52
% 2.74/1.52 thf(sy_c_GenClock__Mirabelle__bsvkzpgbls_OokRead1,type,
% 2.74/1.52 genClo293725281kRead1: ( nat > real ) > real > ( nat > $o ) > $o ).
% 2.74/1.52
% 2.74/1.52 thf(sy_c_GenClock__Mirabelle__bsvkzpgbls_OokRead2,type,
% 2.74/1.52 genClo293725282kRead2: ( nat > real ) > ( nat > real ) > real > ( nat > $o ) > $o ).
% 2.74/1.52
% 2.74/1.52 thf(sy_c_GenClock__Mirabelle__bsvkzpgbls_Ookmaxsync,type,
% 2.74/1.52 genClo208577157axsync: nat > real > $o ).
% 2.74/1.52
% 2.74/1.52 thf(sy_c_GenClock__Mirabelle__bsvkzpgbls_Ormax,type,
% 2.74/1.52 genClo1650508560e_rmax: real ).
% 2.74/1.52
% 2.74/1.52 thf(sy_c_GenClock__Mirabelle__bsvkzpgbls_Ormin,type,
% 2.74/1.52 genClo1651033342e_rmin: real ).
% 2.74/1.52
% 2.74/1.52 thf(sy_c_GenClock__Mirabelle__bsvkzpgbls_Ote,type,
% 2.74/1.52 genClo1163638703lle_te: nat > nat > real ).
% 2.74/1.52
% 2.74/1.52 thf(sy_c_Groups_Oabs__class_Oabs_001t__Real__Oreal,type,
% 2.74/1.52 abs_abs_real: real > real ).
% 2.74/1.52
% 2.74/1.52 thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat,type,
% 2.74/1.52 minus_minus_nat: nat > nat > nat ).
% 2.74/1.52
% 2.74/1.52 thf(sy_c_Groups_Ominus__class_Ominus_001t__Real__Oreal,type,
% 2.74/1.52 minus_minus_real: real > real > real ).
% 2.74/1.52
% 2.74/1.52 thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat,type,
% 2.74/1.52 one_one_nat: nat ).
% 2.74/1.52
% 2.74/1.52 thf(sy_c_Groups_Oone__class_Oone_001t__Real__Oreal,type,
% 2.74/1.52 one_one_real: real ).
% 2.74/1.52
% 2.74/1.52 thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat,type,
% 2.74/1.52 plus_plus_nat: nat > nat > nat ).
% 2.74/1.52
% 2.74/1.52 thf(sy_c_Groups_Oplus__class_Oplus_001t__Real__Oreal,type,
% 2.74/1.52 plus_plus_real: real > real > real ).
% 2.74/1.52
% 2.74/1.52 thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
% 2.74/1.52 zero_zero_nat: nat ).
% 2.74/1.52
% 2.74/1.52 thf(sy_c_Groups_Ozero__class_Ozero_001t__Real__Oreal,type,
% 2.74/1.52 zero_zero_real: real ).
% 2.74/1.52
% 2.74/1.52 thf(sy_c_If_001t__Nat__Onat,type,
% 2.74/1.52 if_nat: $o > nat > nat > nat ).
% 2.74/1.52
% 2.74/1.52 thf(sy_c_If_001t__Real__Oreal,type,
% 2.74/1.52 if_real: $o > real > real > real ).
% 2.74/1.52
% 2.74/1.52 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
% 2.74/1.52 ord_less_eq_nat: nat > nat > $o ).
% 2.74/1.52
% 2.74/1.52 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Real__Oreal,type,
% 2.74/1.52 ord_less_eq_real: real > real > $o ).
% 2.74/1.52
% 2.74/1.52 thf(sy_c_Orderings_Oord__class_Omax_001t__Nat__Onat,type,
% 2.74/1.52 ord_max_nat: nat > nat > nat ).
% 2.74/1.52
% 2.74/1.52 thf(sy_c_Orderings_Oord__class_Omax_001t__Real__Oreal,type,
% 2.74/1.52 ord_max_real: real > real > real ).
% 2.74/1.52
% 2.74/1.52 thf(sy_v__092_060delta_062S,type,
% 2.74/1.52 delta_S: real ).
% 2.74/1.52
% 2.74/1.52 thf(sy_v_i,type,
% 2.74/1.52 i: nat ).
% 2.74/1.52
% 2.74/1.52 thf(sy_v_l,type,
% 2.74/1.52 l: nat ).
% 2.74/1.52
% 2.74/1.52 thf(sy_v_m,type,
% 2.74/1.52 m: nat ).
% 2.74/1.52
% 2.74/1.52 thf(sy_v_p,type,
% 2.74/1.52 p: nat ).
% 2.74/1.52
% 2.74/1.52 thf(fact_0_correct__closed,axiom,
% 2.74/1.52 ! [P: nat,S: real,T: real] :
% 2.74/1.52 ( ( ( ord_less_eq_real @ S @ T )
% 2.74/1.52 & ( genClo1015804716orrect @ P @ T ) )
% 2.74/1.52 => ( genClo1015804716orrect @ P @ S ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_1_corr__m,axiom,
% 2.74/1.52 genClo1015804716orrect @ m @ ( genClo1163638703lle_te @ p @ ( plus_plus_nat @ i @ one_one_nat ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_2_corr__p,axiom,
% 2.74/1.52 genClo1015804716orrect @ p @ ( genClo1163638703lle_te @ p @ ( plus_plus_nat @ i @ one_one_nat ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_3_tml__le__tp,axiom,
% 2.74/1.52 ord_less_eq_real @ ( ord_max_real @ ( genClo1163638703lle_te @ l @ i ) @ ( genClo1163638703lle_te @ m @ i ) ) @ ( genClo1163638703lle_te @ p @ ( plus_plus_nat @ i @ one_one_nat ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_4_corr__l,axiom,
% 2.74/1.52 genClo1015804716orrect @ l @ ( genClo1163638703lle_te @ p @ ( plus_plus_nat @ i @ one_one_nat ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_5_corr__l__tml,axiom,
% 2.74/1.52 genClo1015804716orrect @ l @ ( ord_max_real @ ( genClo1163638703lle_te @ l @ i ) @ ( genClo1163638703lle_te @ m @ i ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_6_tmi__le__tp,axiom,
% 2.74/1.52 ord_less_eq_real @ ( genClo1163638703lle_te @ m @ i ) @ ( genClo1163638703lle_te @ p @ ( plus_plus_nat @ i @ one_one_nat ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_7_tli__le__tp,axiom,
% 2.74/1.52 ord_less_eq_real @ ( genClo1163638703lle_te @ l @ i ) @ ( genClo1163638703lle_te @ p @ ( plus_plus_nat @ i @ one_one_nat ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_8_max_Oidem,axiom,
% 2.74/1.52 ! [A: real] :
% 2.74/1.52 ( ( ord_max_real @ A @ A )
% 2.74/1.52 = A ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_9_max_Oidem,axiom,
% 2.74/1.52 ! [A: nat] :
% 2.74/1.52 ( ( ord_max_nat @ A @ A )
% 2.74/1.52 = A ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_10_max_Oleft__idem,axiom,
% 2.74/1.52 ! [A: real,B: real] :
% 2.74/1.52 ( ( ord_max_real @ A @ ( ord_max_real @ A @ B ) )
% 2.74/1.52 = ( ord_max_real @ A @ B ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_11_max_Oleft__idem,axiom,
% 2.74/1.52 ! [A: nat,B: nat] :
% 2.74/1.52 ( ( ord_max_nat @ A @ ( ord_max_nat @ A @ B ) )
% 2.74/1.52 = ( ord_max_nat @ A @ B ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_12_max_Oright__idem,axiom,
% 2.74/1.52 ! [A: real,B: real] :
% 2.74/1.52 ( ( ord_max_real @ ( ord_max_real @ A @ B ) @ B )
% 2.74/1.52 = ( ord_max_real @ A @ B ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_13_max_Oright__idem,axiom,
% 2.74/1.52 ! [A: nat,B: nat] :
% 2.74/1.52 ( ( ord_max_nat @ ( ord_max_nat @ A @ B ) @ B )
% 2.74/1.52 = ( ord_max_nat @ A @ B ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_14_IC__bound,axiom,
% 2.74/1.52 ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( genClo1160817912lle_IC @ l @ i @ ( ord_max_real @ ( genClo1163638703lle_te @ l @ i ) @ ( genClo1163638703lle_te @ m @ i ) ) ) @ ( genClo1160817912lle_IC @ m @ i @ ( ord_max_real @ ( genClo1163638703lle_te @ l @ i ) @ ( genClo1163638703lle_te @ m @ i ) ) ) ) ) @ delta_S ).
% 2.74/1.52
% 2.74/1.52 thf(fact_15_max_Oassoc,axiom,
% 2.74/1.52 ! [A: real,B: real,C: real] :
% 2.74/1.52 ( ( ord_max_real @ ( ord_max_real @ A @ B ) @ C )
% 2.74/1.52 = ( ord_max_real @ A @ ( ord_max_real @ B @ C ) ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_16_max_Oassoc,axiom,
% 2.74/1.52 ! [A: nat,B: nat,C: nat] :
% 2.74/1.52 ( ( ord_max_nat @ ( ord_max_nat @ A @ B ) @ C )
% 2.74/1.52 = ( ord_max_nat @ A @ ( ord_max_nat @ B @ C ) ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_17_max_Ocommute,axiom,
% 2.74/1.52 ( ord_max_real
% 2.74/1.52 = ( ^ [A2: real,B2: real] : ( ord_max_real @ B2 @ A2 ) ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_18_max_Ocommute,axiom,
% 2.74/1.52 ( ord_max_nat
% 2.74/1.52 = ( ^ [A2: nat,B2: nat] : ( ord_max_nat @ B2 @ A2 ) ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_19_max_Oleft__commute,axiom,
% 2.74/1.52 ! [B: real,A: real,C: real] :
% 2.74/1.52 ( ( ord_max_real @ B @ ( ord_max_real @ A @ C ) )
% 2.74/1.52 = ( ord_max_real @ A @ ( ord_max_real @ B @ C ) ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_20_max_Oleft__commute,axiom,
% 2.74/1.52 ! [B: nat,A: nat,C: nat] :
% 2.74/1.52 ( ( ord_max_nat @ B @ ( ord_max_nat @ A @ C ) )
% 2.74/1.52 = ( ord_max_nat @ A @ ( ord_max_nat @ B @ C ) ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_21_rte,axiom,
% 2.74/1.52 ! [P2: nat,I: nat] :
% 2.74/1.52 ( ( genClo1015804716orrect @ P2 @ ( genClo1163638703lle_te @ P2 @ ( plus_plus_nat @ I @ one_one_nat ) ) )
% 2.74/1.52 => ( ord_less_eq_real @ ( genClo1163638703lle_te @ P2 @ I ) @ ( genClo1163638703lle_te @ P2 @ ( plus_plus_nat @ I @ one_one_nat ) ) ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_22_max_Obounded__iff,axiom,
% 2.74/1.52 ! [B: real,C: real,A: real] :
% 2.74/1.52 ( ( ord_less_eq_real @ ( ord_max_real @ B @ C ) @ A )
% 2.74/1.52 = ( ( ord_less_eq_real @ B @ A )
% 2.74/1.52 & ( ord_less_eq_real @ C @ A ) ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_23_max_Obounded__iff,axiom,
% 2.74/1.52 ! [B: nat,C: nat,A: nat] :
% 2.74/1.52 ( ( ord_less_eq_nat @ ( ord_max_nat @ B @ C ) @ A )
% 2.74/1.52 = ( ( ord_less_eq_nat @ B @ A )
% 2.74/1.52 & ( ord_less_eq_nat @ C @ A ) ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_24_max__absorb1,axiom,
% 2.74/1.52 ! [Y: real,X: real] :
% 2.74/1.52 ( ( ord_less_eq_real @ Y @ X )
% 2.74/1.52 => ( ( ord_max_real @ X @ Y )
% 2.74/1.52 = X ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_25_max__absorb1,axiom,
% 2.74/1.52 ! [Y: nat,X: nat] :
% 2.74/1.52 ( ( ord_less_eq_nat @ Y @ X )
% 2.74/1.52 => ( ( ord_max_nat @ X @ Y )
% 2.74/1.52 = X ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_26_order__refl,axiom,
% 2.74/1.52 ! [X: real] : ( ord_less_eq_real @ X @ X ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_27_order__refl,axiom,
% 2.74/1.52 ! [X: nat] : ( ord_less_eq_nat @ X @ X ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_28_order__subst1,axiom,
% 2.74/1.52 ! [A: real,F: real > real,B: real,C: real] :
% 2.74/1.52 ( ( ord_less_eq_real @ A @ ( F @ B ) )
% 2.74/1.52 => ( ( ord_less_eq_real @ B @ C )
% 2.74/1.52 => ( ! [X2: real,Y2: real] :
% 2.74/1.52 ( ( ord_less_eq_real @ X2 @ Y2 )
% 2.74/1.52 => ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 2.74/1.52 => ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_29_order__subst1,axiom,
% 2.74/1.52 ! [A: real,F: nat > real,B: nat,C: nat] :
% 2.74/1.52 ( ( ord_less_eq_real @ A @ ( F @ B ) )
% 2.74/1.52 => ( ( ord_less_eq_nat @ B @ C )
% 2.74/1.52 => ( ! [X2: nat,Y2: nat] :
% 2.74/1.52 ( ( ord_less_eq_nat @ X2 @ Y2 )
% 2.74/1.52 => ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 2.74/1.52 => ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_30_order__subst1,axiom,
% 2.74/1.52 ! [A: nat,F: real > nat,B: real,C: real] :
% 2.74/1.52 ( ( ord_less_eq_nat @ A @ ( F @ B ) )
% 2.74/1.52 => ( ( ord_less_eq_real @ B @ C )
% 2.74/1.52 => ( ! [X2: real,Y2: real] :
% 2.74/1.52 ( ( ord_less_eq_real @ X2 @ Y2 )
% 2.74/1.52 => ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 2.74/1.52 => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_31_order__subst1,axiom,
% 2.74/1.52 ! [A: nat,F: nat > nat,B: nat,C: nat] :
% 2.74/1.52 ( ( ord_less_eq_nat @ A @ ( F @ B ) )
% 2.74/1.52 => ( ( ord_less_eq_nat @ B @ C )
% 2.74/1.52 => ( ! [X2: nat,Y2: nat] :
% 2.74/1.52 ( ( ord_less_eq_nat @ X2 @ Y2 )
% 2.74/1.52 => ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 2.74/1.52 => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_32_order__subst2,axiom,
% 2.74/1.52 ! [A: real,B: real,F: real > real,C: real] :
% 2.74/1.52 ( ( ord_less_eq_real @ A @ B )
% 2.74/1.52 => ( ( ord_less_eq_real @ ( F @ B ) @ C )
% 2.74/1.52 => ( ! [X2: real,Y2: real] :
% 2.74/1.52 ( ( ord_less_eq_real @ X2 @ Y2 )
% 2.74/1.52 => ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 2.74/1.52 => ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_33_order__subst2,axiom,
% 2.74/1.52 ! [A: real,B: real,F: real > nat,C: nat] :
% 2.74/1.52 ( ( ord_less_eq_real @ A @ B )
% 2.74/1.52 => ( ( ord_less_eq_nat @ ( F @ B ) @ C )
% 2.74/1.52 => ( ! [X2: real,Y2: real] :
% 2.74/1.52 ( ( ord_less_eq_real @ X2 @ Y2 )
% 2.74/1.52 => ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 2.74/1.52 => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_34_order__subst2,axiom,
% 2.74/1.52 ! [A: nat,B: nat,F: nat > real,C: real] :
% 2.74/1.52 ( ( ord_less_eq_nat @ A @ B )
% 2.74/1.52 => ( ( ord_less_eq_real @ ( F @ B ) @ C )
% 2.74/1.52 => ( ! [X2: nat,Y2: nat] :
% 2.74/1.52 ( ( ord_less_eq_nat @ X2 @ Y2 )
% 2.74/1.52 => ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 2.74/1.52 => ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_35_order__subst2,axiom,
% 2.74/1.52 ! [A: nat,B: nat,F: nat > nat,C: nat] :
% 2.74/1.52 ( ( ord_less_eq_nat @ A @ B )
% 2.74/1.52 => ( ( ord_less_eq_nat @ ( F @ B ) @ C )
% 2.74/1.52 => ( ! [X2: nat,Y2: nat] :
% 2.74/1.52 ( ( ord_less_eq_nat @ X2 @ Y2 )
% 2.74/1.52 => ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 2.74/1.52 => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_36_ord__eq__le__subst,axiom,
% 2.74/1.52 ! [A: real,F: real > real,B: real,C: real] :
% 2.74/1.52 ( ( A
% 2.74/1.52 = ( F @ B ) )
% 2.74/1.52 => ( ( ord_less_eq_real @ B @ C )
% 2.74/1.52 => ( ! [X2: real,Y2: real] :
% 2.74/1.52 ( ( ord_less_eq_real @ X2 @ Y2 )
% 2.74/1.52 => ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 2.74/1.52 => ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_37_ord__eq__le__subst,axiom,
% 2.74/1.52 ! [A: nat,F: real > nat,B: real,C: real] :
% 2.74/1.52 ( ( A
% 2.74/1.52 = ( F @ B ) )
% 2.74/1.52 => ( ( ord_less_eq_real @ B @ C )
% 2.74/1.52 => ( ! [X2: real,Y2: real] :
% 2.74/1.52 ( ( ord_less_eq_real @ X2 @ Y2 )
% 2.74/1.52 => ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 2.74/1.52 => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_38_ord__eq__le__subst,axiom,
% 2.74/1.52 ! [A: real,F: nat > real,B: nat,C: nat] :
% 2.74/1.52 ( ( A
% 2.74/1.52 = ( F @ B ) )
% 2.74/1.52 => ( ( ord_less_eq_nat @ B @ C )
% 2.74/1.52 => ( ! [X2: nat,Y2: nat] :
% 2.74/1.52 ( ( ord_less_eq_nat @ X2 @ Y2 )
% 2.74/1.52 => ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 2.74/1.52 => ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_39_ord__eq__le__subst,axiom,
% 2.74/1.52 ! [A: nat,F: nat > nat,B: nat,C: nat] :
% 2.74/1.52 ( ( A
% 2.74/1.52 = ( F @ B ) )
% 2.74/1.52 => ( ( ord_less_eq_nat @ B @ C )
% 2.74/1.52 => ( ! [X2: nat,Y2: nat] :
% 2.74/1.52 ( ( ord_less_eq_nat @ X2 @ Y2 )
% 2.74/1.52 => ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 2.74/1.52 => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_40_ord__le__eq__subst,axiom,
% 2.74/1.52 ! [A: real,B: real,F: real > real,C: real] :
% 2.74/1.52 ( ( ord_less_eq_real @ A @ B )
% 2.74/1.52 => ( ( ( F @ B )
% 2.74/1.52 = C )
% 2.74/1.52 => ( ! [X2: real,Y2: real] :
% 2.74/1.52 ( ( ord_less_eq_real @ X2 @ Y2 )
% 2.74/1.52 => ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 2.74/1.52 => ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_41_ord__le__eq__subst,axiom,
% 2.74/1.52 ! [A: real,B: real,F: real > nat,C: nat] :
% 2.74/1.52 ( ( ord_less_eq_real @ A @ B )
% 2.74/1.52 => ( ( ( F @ B )
% 2.74/1.52 = C )
% 2.74/1.52 => ( ! [X2: real,Y2: real] :
% 2.74/1.52 ( ( ord_less_eq_real @ X2 @ Y2 )
% 2.74/1.52 => ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 2.74/1.52 => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_42_ord__le__eq__subst,axiom,
% 2.74/1.52 ! [A: nat,B: nat,F: nat > real,C: real] :
% 2.74/1.52 ( ( ord_less_eq_nat @ A @ B )
% 2.74/1.52 => ( ( ( F @ B )
% 2.74/1.52 = C )
% 2.74/1.52 => ( ! [X2: nat,Y2: nat] :
% 2.74/1.52 ( ( ord_less_eq_nat @ X2 @ Y2 )
% 2.74/1.52 => ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 2.74/1.52 => ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_43_ord__le__eq__subst,axiom,
% 2.74/1.52 ! [A: nat,B: nat,F: nat > nat,C: nat] :
% 2.74/1.52 ( ( ord_less_eq_nat @ A @ B )
% 2.74/1.52 => ( ( ( F @ B )
% 2.74/1.52 = C )
% 2.74/1.52 => ( ! [X2: nat,Y2: nat] :
% 2.74/1.52 ( ( ord_less_eq_nat @ X2 @ Y2 )
% 2.74/1.52 => ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 2.74/1.52 => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_44_eq__iff,axiom,
% 2.74/1.52 ( ( ^ [Y3: real,Z: real] : ( Y3 = Z ) )
% 2.74/1.52 = ( ^ [X3: real,Y4: real] :
% 2.74/1.52 ( ( ord_less_eq_real @ X3 @ Y4 )
% 2.74/1.52 & ( ord_less_eq_real @ Y4 @ X3 ) ) ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_45_eq__iff,axiom,
% 2.74/1.52 ( ( ^ [Y3: nat,Z: nat] : ( Y3 = Z ) )
% 2.74/1.52 = ( ^ [X3: nat,Y4: nat] :
% 2.74/1.52 ( ( ord_less_eq_nat @ X3 @ Y4 )
% 2.74/1.52 & ( ord_less_eq_nat @ Y4 @ X3 ) ) ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_46_antisym,axiom,
% 2.74/1.52 ! [X: real,Y: real] :
% 2.74/1.52 ( ( ord_less_eq_real @ X @ Y )
% 2.74/1.52 => ( ( ord_less_eq_real @ Y @ X )
% 2.74/1.52 => ( X = Y ) ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_47_antisym,axiom,
% 2.74/1.52 ! [X: nat,Y: nat] :
% 2.74/1.52 ( ( ord_less_eq_nat @ X @ Y )
% 2.74/1.52 => ( ( ord_less_eq_nat @ Y @ X )
% 2.74/1.52 => ( X = Y ) ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_48_linear,axiom,
% 2.74/1.52 ! [X: real,Y: real] :
% 2.74/1.52 ( ( ord_less_eq_real @ X @ Y )
% 2.74/1.52 | ( ord_less_eq_real @ Y @ X ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_49_linear,axiom,
% 2.74/1.52 ! [X: nat,Y: nat] :
% 2.74/1.52 ( ( ord_less_eq_nat @ X @ Y )
% 2.74/1.52 | ( ord_less_eq_nat @ Y @ X ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_50_eq__refl,axiom,
% 2.74/1.52 ! [X: real,Y: real] :
% 2.74/1.52 ( ( X = Y )
% 2.74/1.52 => ( ord_less_eq_real @ X @ Y ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_51_eq__refl,axiom,
% 2.74/1.52 ! [X: nat,Y: nat] :
% 2.74/1.52 ( ( X = Y )
% 2.74/1.52 => ( ord_less_eq_nat @ X @ Y ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_52_le__cases,axiom,
% 2.74/1.52 ! [X: real,Y: real] :
% 2.74/1.52 ( ~ ( ord_less_eq_real @ X @ Y )
% 2.74/1.52 => ( ord_less_eq_real @ Y @ X ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_53_le__cases,axiom,
% 2.74/1.52 ! [X: nat,Y: nat] :
% 2.74/1.52 ( ~ ( ord_less_eq_nat @ X @ Y )
% 2.74/1.52 => ( ord_less_eq_nat @ Y @ X ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_54_order_Otrans,axiom,
% 2.74/1.52 ! [A: real,B: real,C: real] :
% 2.74/1.52 ( ( ord_less_eq_real @ A @ B )
% 2.74/1.52 => ( ( ord_less_eq_real @ B @ C )
% 2.74/1.52 => ( ord_less_eq_real @ A @ C ) ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_55_order_Otrans,axiom,
% 2.74/1.52 ! [A: nat,B: nat,C: nat] :
% 2.74/1.52 ( ( ord_less_eq_nat @ A @ B )
% 2.74/1.52 => ( ( ord_less_eq_nat @ B @ C )
% 2.74/1.52 => ( ord_less_eq_nat @ A @ C ) ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_56_le__cases3,axiom,
% 2.74/1.52 ! [X: real,Y: real,Z2: real] :
% 2.74/1.52 ( ( ( ord_less_eq_real @ X @ Y )
% 2.74/1.52 => ~ ( ord_less_eq_real @ Y @ Z2 ) )
% 2.74/1.52 => ( ( ( ord_less_eq_real @ Y @ X )
% 2.74/1.52 => ~ ( ord_less_eq_real @ X @ Z2 ) )
% 2.74/1.52 => ( ( ( ord_less_eq_real @ X @ Z2 )
% 2.74/1.52 => ~ ( ord_less_eq_real @ Z2 @ Y ) )
% 2.74/1.52 => ( ( ( ord_less_eq_real @ Z2 @ Y )
% 2.74/1.52 => ~ ( ord_less_eq_real @ Y @ X ) )
% 2.74/1.52 => ( ( ( ord_less_eq_real @ Y @ Z2 )
% 2.74/1.52 => ~ ( ord_less_eq_real @ Z2 @ X ) )
% 2.74/1.52 => ~ ( ( ord_less_eq_real @ Z2 @ X )
% 2.74/1.52 => ~ ( ord_less_eq_real @ X @ Y ) ) ) ) ) ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_57_le__cases3,axiom,
% 2.74/1.52 ! [X: nat,Y: nat,Z2: nat] :
% 2.74/1.52 ( ( ( ord_less_eq_nat @ X @ Y )
% 2.74/1.52 => ~ ( ord_less_eq_nat @ Y @ Z2 ) )
% 2.74/1.52 => ( ( ( ord_less_eq_nat @ Y @ X )
% 2.74/1.52 => ~ ( ord_less_eq_nat @ X @ Z2 ) )
% 2.74/1.52 => ( ( ( ord_less_eq_nat @ X @ Z2 )
% 2.74/1.52 => ~ ( ord_less_eq_nat @ Z2 @ Y ) )
% 2.74/1.52 => ( ( ( ord_less_eq_nat @ Z2 @ Y )
% 2.74/1.52 => ~ ( ord_less_eq_nat @ Y @ X ) )
% 2.74/1.52 => ( ( ( ord_less_eq_nat @ Y @ Z2 )
% 2.74/1.52 => ~ ( ord_less_eq_nat @ Z2 @ X ) )
% 2.74/1.52 => ~ ( ( ord_less_eq_nat @ Z2 @ X )
% 2.74/1.52 => ~ ( ord_less_eq_nat @ X @ Y ) ) ) ) ) ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_58_antisym__conv,axiom,
% 2.74/1.52 ! [Y: real,X: real] :
% 2.74/1.52 ( ( ord_less_eq_real @ Y @ X )
% 2.74/1.52 => ( ( ord_less_eq_real @ X @ Y )
% 2.74/1.52 = ( X = Y ) ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_59_antisym__conv,axiom,
% 2.74/1.52 ! [Y: nat,X: nat] :
% 2.74/1.52 ( ( ord_less_eq_nat @ Y @ X )
% 2.74/1.52 => ( ( ord_less_eq_nat @ X @ Y )
% 2.74/1.52 = ( X = Y ) ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_60_order__class_Oorder_Oeq__iff,axiom,
% 2.74/1.52 ( ( ^ [Y3: real,Z: real] : ( Y3 = Z ) )
% 2.74/1.52 = ( ^ [A2: real,B2: real] :
% 2.74/1.52 ( ( ord_less_eq_real @ A2 @ B2 )
% 2.74/1.52 & ( ord_less_eq_real @ B2 @ A2 ) ) ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_61_order__class_Oorder_Oeq__iff,axiom,
% 2.74/1.52 ( ( ^ [Y3: nat,Z: nat] : ( Y3 = Z ) )
% 2.74/1.52 = ( ^ [A2: nat,B2: nat] :
% 2.74/1.52 ( ( ord_less_eq_nat @ A2 @ B2 )
% 2.74/1.52 & ( ord_less_eq_nat @ B2 @ A2 ) ) ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_62_ord__eq__le__trans,axiom,
% 2.74/1.52 ! [A: real,B: real,C: real] :
% 2.74/1.52 ( ( A = B )
% 2.74/1.52 => ( ( ord_less_eq_real @ B @ C )
% 2.74/1.52 => ( ord_less_eq_real @ A @ C ) ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_63_ord__eq__le__trans,axiom,
% 2.74/1.52 ! [A: nat,B: nat,C: nat] :
% 2.74/1.52 ( ( A = B )
% 2.74/1.52 => ( ( ord_less_eq_nat @ B @ C )
% 2.74/1.52 => ( ord_less_eq_nat @ A @ C ) ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_64_ord__le__eq__trans,axiom,
% 2.74/1.52 ! [A: real,B: real,C: real] :
% 2.74/1.52 ( ( ord_less_eq_real @ A @ B )
% 2.74/1.52 => ( ( B = C )
% 2.74/1.52 => ( ord_less_eq_real @ A @ C ) ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_65_ord__le__eq__trans,axiom,
% 2.74/1.52 ! [A: nat,B: nat,C: nat] :
% 2.74/1.52 ( ( ord_less_eq_nat @ A @ B )
% 2.74/1.52 => ( ( B = C )
% 2.74/1.52 => ( ord_less_eq_nat @ A @ C ) ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_66_order__class_Oorder_Oantisym,axiom,
% 2.74/1.52 ! [A: real,B: real] :
% 2.74/1.52 ( ( ord_less_eq_real @ A @ B )
% 2.74/1.52 => ( ( ord_less_eq_real @ B @ A )
% 2.74/1.52 => ( A = B ) ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_67_order__class_Oorder_Oantisym,axiom,
% 2.74/1.52 ! [A: nat,B: nat] :
% 2.74/1.52 ( ( ord_less_eq_nat @ A @ B )
% 2.74/1.52 => ( ( ord_less_eq_nat @ B @ A )
% 2.74/1.52 => ( A = B ) ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_68_order__trans,axiom,
% 2.74/1.52 ! [X: real,Y: real,Z2: real] :
% 2.74/1.52 ( ( ord_less_eq_real @ X @ Y )
% 2.74/1.52 => ( ( ord_less_eq_real @ Y @ Z2 )
% 2.74/1.52 => ( ord_less_eq_real @ X @ Z2 ) ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_69_order__trans,axiom,
% 2.74/1.52 ! [X: nat,Y: nat,Z2: nat] :
% 2.74/1.52 ( ( ord_less_eq_nat @ X @ Y )
% 2.74/1.52 => ( ( ord_less_eq_nat @ Y @ Z2 )
% 2.74/1.52 => ( ord_less_eq_nat @ X @ Z2 ) ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_70_dual__order_Orefl,axiom,
% 2.74/1.52 ! [A: real] : ( ord_less_eq_real @ A @ A ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_71_dual__order_Orefl,axiom,
% 2.74/1.52 ! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_72_linorder__wlog,axiom,
% 2.74/1.52 ! [P3: real > real > $o,A: real,B: real] :
% 2.74/1.52 ( ! [A3: real,B3: real] :
% 2.74/1.52 ( ( ord_less_eq_real @ A3 @ B3 )
% 2.74/1.52 => ( P3 @ A3 @ B3 ) )
% 2.74/1.52 => ( ! [A3: real,B3: real] :
% 2.74/1.52 ( ( P3 @ B3 @ A3 )
% 2.74/1.52 => ( P3 @ A3 @ B3 ) )
% 2.74/1.52 => ( P3 @ A @ B ) ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_73_linorder__wlog,axiom,
% 2.74/1.52 ! [P3: nat > nat > $o,A: nat,B: nat] :
% 2.74/1.52 ( ! [A3: nat,B3: nat] :
% 2.74/1.52 ( ( ord_less_eq_nat @ A3 @ B3 )
% 2.74/1.52 => ( P3 @ A3 @ B3 ) )
% 2.74/1.52 => ( ! [A3: nat,B3: nat] :
% 2.74/1.52 ( ( P3 @ B3 @ A3 )
% 2.74/1.52 => ( P3 @ A3 @ B3 ) )
% 2.74/1.52 => ( P3 @ A @ B ) ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_74_dual__order_Otrans,axiom,
% 2.74/1.52 ! [B: real,A: real,C: real] :
% 2.74/1.52 ( ( ord_less_eq_real @ B @ A )
% 2.74/1.52 => ( ( ord_less_eq_real @ C @ B )
% 2.74/1.52 => ( ord_less_eq_real @ C @ A ) ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_75_dual__order_Otrans,axiom,
% 2.74/1.52 ! [B: nat,A: nat,C: nat] :
% 2.74/1.52 ( ( ord_less_eq_nat @ B @ A )
% 2.74/1.52 => ( ( ord_less_eq_nat @ C @ B )
% 2.74/1.52 => ( ord_less_eq_nat @ C @ A ) ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_76_dual__order_Oeq__iff,axiom,
% 2.74/1.52 ( ( ^ [Y3: real,Z: real] : ( Y3 = Z ) )
% 2.74/1.52 = ( ^ [A2: real,B2: real] :
% 2.74/1.52 ( ( ord_less_eq_real @ B2 @ A2 )
% 2.74/1.52 & ( ord_less_eq_real @ A2 @ B2 ) ) ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_77_dual__order_Oeq__iff,axiom,
% 2.74/1.52 ( ( ^ [Y3: nat,Z: nat] : ( Y3 = Z ) )
% 2.74/1.52 = ( ^ [A2: nat,B2: nat] :
% 2.74/1.52 ( ( ord_less_eq_nat @ B2 @ A2 )
% 2.74/1.52 & ( ord_less_eq_nat @ A2 @ B2 ) ) ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_78_dual__order_Oantisym,axiom,
% 2.74/1.52 ! [B: real,A: real] :
% 2.74/1.52 ( ( ord_less_eq_real @ B @ A )
% 2.74/1.52 => ( ( ord_less_eq_real @ A @ B )
% 2.74/1.52 => ( A = B ) ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_79_dual__order_Oantisym,axiom,
% 2.74/1.52 ! [B: nat,A: nat] :
% 2.74/1.52 ( ( ord_less_eq_nat @ B @ A )
% 2.74/1.52 => ( ( ord_less_eq_nat @ A @ B )
% 2.74/1.52 => ( A = B ) ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_80_max_OcoboundedI2,axiom,
% 2.74/1.52 ! [C: real,B: real,A: real] :
% 2.74/1.52 ( ( ord_less_eq_real @ C @ B )
% 2.74/1.52 => ( ord_less_eq_real @ C @ ( ord_max_real @ A @ B ) ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_81_max_OcoboundedI2,axiom,
% 2.74/1.52 ! [C: nat,B: nat,A: nat] :
% 2.74/1.52 ( ( ord_less_eq_nat @ C @ B )
% 2.74/1.52 => ( ord_less_eq_nat @ C @ ( ord_max_nat @ A @ B ) ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_82_max_OcoboundedI1,axiom,
% 2.74/1.52 ! [C: real,A: real,B: real] :
% 2.74/1.52 ( ( ord_less_eq_real @ C @ A )
% 2.74/1.52 => ( ord_less_eq_real @ C @ ( ord_max_real @ A @ B ) ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_83_max_OcoboundedI1,axiom,
% 2.74/1.52 ! [C: nat,A: nat,B: nat] :
% 2.74/1.52 ( ( ord_less_eq_nat @ C @ A )
% 2.74/1.52 => ( ord_less_eq_nat @ C @ ( ord_max_nat @ A @ B ) ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_84_max_Oabsorb__iff2,axiom,
% 2.74/1.52 ( ord_less_eq_real
% 2.74/1.52 = ( ^ [A2: real,B2: real] :
% 2.74/1.52 ( ( ord_max_real @ A2 @ B2 )
% 2.74/1.52 = B2 ) ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_85_max_Oabsorb__iff2,axiom,
% 2.74/1.52 ( ord_less_eq_nat
% 2.74/1.52 = ( ^ [A2: nat,B2: nat] :
% 2.74/1.52 ( ( ord_max_nat @ A2 @ B2 )
% 2.74/1.52 = B2 ) ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_86_max_Oabsorb__iff1,axiom,
% 2.74/1.52 ( ord_less_eq_real
% 2.74/1.52 = ( ^ [B2: real,A2: real] :
% 2.74/1.52 ( ( ord_max_real @ A2 @ B2 )
% 2.74/1.52 = A2 ) ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_87_max_Oabsorb__iff1,axiom,
% 2.74/1.52 ( ord_less_eq_nat
% 2.74/1.52 = ( ^ [B2: nat,A2: nat] :
% 2.74/1.52 ( ( ord_max_nat @ A2 @ B2 )
% 2.74/1.52 = A2 ) ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_88_le__max__iff__disj,axiom,
% 2.74/1.52 ! [Z2: real,X: real,Y: real] :
% 2.74/1.52 ( ( ord_less_eq_real @ Z2 @ ( ord_max_real @ X @ Y ) )
% 2.74/1.52 = ( ( ord_less_eq_real @ Z2 @ X )
% 2.74/1.52 | ( ord_less_eq_real @ Z2 @ Y ) ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_89_le__max__iff__disj,axiom,
% 2.74/1.52 ! [Z2: nat,X: nat,Y: nat] :
% 2.74/1.52 ( ( ord_less_eq_nat @ Z2 @ ( ord_max_nat @ X @ Y ) )
% 2.74/1.52 = ( ( ord_less_eq_nat @ Z2 @ X )
% 2.74/1.52 | ( ord_less_eq_nat @ Z2 @ Y ) ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_90_max_Ocobounded2,axiom,
% 2.74/1.52 ! [B: real,A: real] : ( ord_less_eq_real @ B @ ( ord_max_real @ A @ B ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_91_max_Ocobounded2,axiom,
% 2.74/1.52 ! [B: nat,A: nat] : ( ord_less_eq_nat @ B @ ( ord_max_nat @ A @ B ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_92_max_Ocobounded1,axiom,
% 2.74/1.52 ! [A: real,B: real] : ( ord_less_eq_real @ A @ ( ord_max_real @ A @ B ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_93_max_Ocobounded1,axiom,
% 2.74/1.52 ! [A: nat,B: nat] : ( ord_less_eq_nat @ A @ ( ord_max_nat @ A @ B ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_94_max_Oorder__iff,axiom,
% 2.74/1.52 ( ord_less_eq_real
% 2.74/1.52 = ( ^ [B2: real,A2: real] :
% 2.74/1.52 ( A2
% 2.74/1.52 = ( ord_max_real @ A2 @ B2 ) ) ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_95_max_Oorder__iff,axiom,
% 2.74/1.52 ( ord_less_eq_nat
% 2.74/1.52 = ( ^ [B2: nat,A2: nat] :
% 2.74/1.52 ( A2
% 2.74/1.52 = ( ord_max_nat @ A2 @ B2 ) ) ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_96_max_OboundedI,axiom,
% 2.74/1.52 ! [B: real,A: real,C: real] :
% 2.74/1.52 ( ( ord_less_eq_real @ B @ A )
% 2.74/1.52 => ( ( ord_less_eq_real @ C @ A )
% 2.74/1.52 => ( ord_less_eq_real @ ( ord_max_real @ B @ C ) @ A ) ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_97_max_OboundedI,axiom,
% 2.74/1.52 ! [B: nat,A: nat,C: nat] :
% 2.74/1.52 ( ( ord_less_eq_nat @ B @ A )
% 2.74/1.52 => ( ( ord_less_eq_nat @ C @ A )
% 2.74/1.52 => ( ord_less_eq_nat @ ( ord_max_nat @ B @ C ) @ A ) ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_98_max_OboundedE,axiom,
% 2.74/1.52 ! [B: real,C: real,A: real] :
% 2.74/1.52 ( ( ord_less_eq_real @ ( ord_max_real @ B @ C ) @ A )
% 2.74/1.52 => ~ ( ( ord_less_eq_real @ B @ A )
% 2.74/1.52 => ~ ( ord_less_eq_real @ C @ A ) ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_99_max_OboundedE,axiom,
% 2.74/1.52 ! [B: nat,C: nat,A: nat] :
% 2.74/1.52 ( ( ord_less_eq_nat @ ( ord_max_nat @ B @ C ) @ A )
% 2.74/1.52 => ~ ( ( ord_less_eq_nat @ B @ A )
% 2.74/1.52 => ~ ( ord_less_eq_nat @ C @ A ) ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_100_max_Oabsorb2,axiom,
% 2.74/1.52 ! [A: real,B: real] :
% 2.74/1.52 ( ( ord_less_eq_real @ A @ B )
% 2.74/1.52 => ( ( ord_max_real @ A @ B )
% 2.74/1.52 = B ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_101_max_Oabsorb2,axiom,
% 2.74/1.52 ! [A: nat,B: nat] :
% 2.74/1.52 ( ( ord_less_eq_nat @ A @ B )
% 2.74/1.52 => ( ( ord_max_nat @ A @ B )
% 2.74/1.52 = B ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_102_max_Oabsorb1,axiom,
% 2.74/1.52 ! [B: real,A: real] :
% 2.74/1.52 ( ( ord_less_eq_real @ B @ A )
% 2.74/1.52 => ( ( ord_max_real @ A @ B )
% 2.74/1.52 = A ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_103_max_Oabsorb1,axiom,
% 2.74/1.52 ! [B: nat,A: nat] :
% 2.74/1.52 ( ( ord_less_eq_nat @ B @ A )
% 2.74/1.52 => ( ( ord_max_nat @ A @ B )
% 2.74/1.52 = A ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_104_max_OorderI,axiom,
% 2.74/1.52 ! [A: real,B: real] :
% 2.74/1.52 ( ( A
% 2.74/1.52 = ( ord_max_real @ A @ B ) )
% 2.74/1.52 => ( ord_less_eq_real @ B @ A ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_105_max_OorderI,axiom,
% 2.74/1.52 ! [A: nat,B: nat] :
% 2.74/1.52 ( ( A
% 2.74/1.52 = ( ord_max_nat @ A @ B ) )
% 2.74/1.52 => ( ord_less_eq_nat @ B @ A ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_106_max_OorderE,axiom,
% 2.74/1.52 ! [B: real,A: real] :
% 2.74/1.52 ( ( ord_less_eq_real @ B @ A )
% 2.74/1.52 => ( A
% 2.74/1.52 = ( ord_max_real @ A @ B ) ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_107_max_OorderE,axiom,
% 2.74/1.52 ! [B: nat,A: nat] :
% 2.74/1.52 ( ( ord_less_eq_nat @ B @ A )
% 2.74/1.52 => ( A
% 2.74/1.52 = ( ord_max_nat @ A @ B ) ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_108_max_Omono,axiom,
% 2.74/1.52 ! [C: real,A: real,D: real,B: real] :
% 2.74/1.52 ( ( ord_less_eq_real @ C @ A )
% 2.74/1.52 => ( ( ord_less_eq_real @ D @ B )
% 2.74/1.52 => ( ord_less_eq_real @ ( ord_max_real @ C @ D ) @ ( ord_max_real @ A @ B ) ) ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_109_max_Omono,axiom,
% 2.74/1.52 ! [C: nat,A: nat,D: nat,B: nat] :
% 2.74/1.52 ( ( ord_less_eq_nat @ C @ A )
% 2.74/1.52 => ( ( ord_less_eq_nat @ D @ B )
% 2.74/1.52 => ( ord_less_eq_nat @ ( ord_max_nat @ C @ D ) @ ( ord_max_nat @ A @ B ) ) ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_110_max__def,axiom,
% 2.74/1.52 ( ord_max_real
% 2.74/1.52 = ( ^ [A2: real,B2: real] : ( if_real @ ( ord_less_eq_real @ A2 @ B2 ) @ B2 @ A2 ) ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_111_max__def,axiom,
% 2.74/1.52 ( ord_max_nat
% 2.74/1.52 = ( ^ [A2: nat,B2: nat] : ( if_nat @ ( ord_less_eq_nat @ A2 @ B2 ) @ B2 @ A2 ) ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_112_max__absorb2,axiom,
% 2.74/1.52 ! [X: real,Y: real] :
% 2.74/1.52 ( ( ord_less_eq_real @ X @ Y )
% 2.74/1.52 => ( ( ord_max_real @ X @ Y )
% 2.74/1.52 = Y ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_113_max__absorb2,axiom,
% 2.74/1.52 ! [X: nat,Y: nat] :
% 2.74/1.52 ( ( ord_less_eq_nat @ X @ Y )
% 2.74/1.52 => ( ( ord_max_nat @ X @ Y )
% 2.74/1.52 = Y ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_114_okmaxsync__def,axiom,
% 2.74/1.52 ( genClo208577157axsync
% 2.74/1.52 = ( ^ [I2: nat,X3: real] :
% 2.74/1.52 ! [P4: nat,Q: nat] :
% 2.74/1.52 ( ( ( genClo1015804716orrect @ P4 @ ( ord_max_real @ ( genClo1163638703lle_te @ P4 @ I2 ) @ ( genClo1163638703lle_te @ Q @ I2 ) ) )
% 2.74/1.52 & ( genClo1015804716orrect @ Q @ ( ord_max_real @ ( genClo1163638703lle_te @ P4 @ I2 ) @ ( genClo1163638703lle_te @ Q @ I2 ) ) ) )
% 2.74/1.52 => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( genClo1160817912lle_IC @ P4 @ I2 @ ( ord_max_real @ ( genClo1163638703lle_te @ P4 @ I2 ) @ ( genClo1163638703lle_te @ Q @ I2 ) ) ) @ ( genClo1160817912lle_IC @ Q @ I2 @ ( ord_max_real @ ( genClo1163638703lle_te @ P4 @ I2 ) @ ( genClo1163638703lle_te @ Q @ I2 ) ) ) ) ) @ X3 ) ) ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_115_le__add__diff__inverse,axiom,
% 2.74/1.52 ! [B: real,A: real] :
% 2.74/1.52 ( ( ord_less_eq_real @ B @ A )
% 2.74/1.52 => ( ( plus_plus_real @ B @ ( minus_minus_real @ A @ B ) )
% 2.74/1.52 = A ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_116_le__add__diff__inverse,axiom,
% 2.74/1.52 ! [B: nat,A: nat] :
% 2.74/1.52 ( ( ord_less_eq_nat @ B @ A )
% 2.74/1.52 => ( ( plus_plus_nat @ B @ ( minus_minus_nat @ A @ B ) )
% 2.74/1.52 = A ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_117_le__add__diff__inverse2,axiom,
% 2.74/1.52 ! [B: real,A: real] :
% 2.74/1.52 ( ( ord_less_eq_real @ B @ A )
% 2.74/1.52 => ( ( plus_plus_real @ ( minus_minus_real @ A @ B ) @ B )
% 2.74/1.52 = A ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_118_le__add__diff__inverse2,axiom,
% 2.74/1.52 ! [B: nat,A: nat] :
% 2.74/1.52 ( ( ord_less_eq_nat @ B @ A )
% 2.74/1.52 => ( ( plus_plus_nat @ ( minus_minus_nat @ A @ B ) @ B )
% 2.74/1.52 = A ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_119_abs__1,axiom,
% 2.74/1.52 ( ( abs_abs_real @ one_one_real )
% 2.74/1.52 = one_one_real ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_120_abs__add__abs,axiom,
% 2.74/1.52 ! [A: real,B: real] :
% 2.74/1.52 ( ( abs_abs_real @ ( plus_plus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) )
% 2.74/1.52 = ( plus_plus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_121_add__diff__cancel,axiom,
% 2.74/1.52 ! [A: real,B: real] :
% 2.74/1.52 ( ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ B )
% 2.74/1.52 = A ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_122_diff__add__cancel,axiom,
% 2.74/1.52 ! [A: real,B: real] :
% 2.74/1.52 ( ( plus_plus_real @ ( minus_minus_real @ A @ B ) @ B )
% 2.74/1.52 = A ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_123_add__diff__cancel__left,axiom,
% 2.74/1.52 ! [C: real,A: real,B: real] :
% 2.74/1.52 ( ( minus_minus_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
% 2.74/1.52 = ( minus_minus_real @ A @ B ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_124_add__diff__cancel__left,axiom,
% 2.74/1.52 ! [C: nat,A: nat,B: nat] :
% 2.74/1.52 ( ( minus_minus_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
% 2.74/1.52 = ( minus_minus_nat @ A @ B ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_125_add__diff__cancel__left_H,axiom,
% 2.74/1.52 ! [A: real,B: real] :
% 2.74/1.52 ( ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ A )
% 2.74/1.52 = B ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_126_add__diff__cancel__left_H,axiom,
% 2.74/1.52 ! [A: nat,B: nat] :
% 2.74/1.52 ( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ A )
% 2.74/1.52 = B ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_127_add__diff__cancel__right,axiom,
% 2.74/1.52 ! [A: real,C: real,B: real] :
% 2.74/1.52 ( ( minus_minus_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
% 2.74/1.52 = ( minus_minus_real @ A @ B ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_128_add__diff__cancel__right,axiom,
% 2.74/1.52 ! [A: nat,C: nat,B: nat] :
% 2.74/1.52 ( ( minus_minus_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
% 2.74/1.52 = ( minus_minus_nat @ A @ B ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_129_add__right__cancel,axiom,
% 2.74/1.52 ! [B: nat,A: nat,C: nat] :
% 2.74/1.52 ( ( ( plus_plus_nat @ B @ A )
% 2.74/1.52 = ( plus_plus_nat @ C @ A ) )
% 2.74/1.52 = ( B = C ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_130_add__right__cancel,axiom,
% 2.74/1.52 ! [B: real,A: real,C: real] :
% 2.74/1.52 ( ( ( plus_plus_real @ B @ A )
% 2.74/1.52 = ( plus_plus_real @ C @ A ) )
% 2.74/1.52 = ( B = C ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_131_add__left__cancel,axiom,
% 2.74/1.52 ! [A: nat,B: nat,C: nat] :
% 2.74/1.52 ( ( ( plus_plus_nat @ A @ B )
% 2.74/1.52 = ( plus_plus_nat @ A @ C ) )
% 2.74/1.52 = ( B = C ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_132_add__left__cancel,axiom,
% 2.74/1.52 ! [A: real,B: real,C: real] :
% 2.74/1.52 ( ( ( plus_plus_real @ A @ B )
% 2.74/1.52 = ( plus_plus_real @ A @ C ) )
% 2.74/1.52 = ( B = C ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_133_abs__idempotent,axiom,
% 2.74/1.52 ! [A: real] :
% 2.74/1.52 ( ( abs_abs_real @ ( abs_abs_real @ A ) )
% 2.74/1.52 = ( abs_abs_real @ A ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_134_abs__abs,axiom,
% 2.74/1.52 ! [A: real] :
% 2.74/1.52 ( ( abs_abs_real @ ( abs_abs_real @ A ) )
% 2.74/1.52 = ( abs_abs_real @ A ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_135_add__le__cancel__right,axiom,
% 2.74/1.52 ! [A: real,C: real,B: real] :
% 2.74/1.52 ( ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
% 2.74/1.52 = ( ord_less_eq_real @ A @ B ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_136_add__le__cancel__right,axiom,
% 2.74/1.52 ! [A: nat,C: nat,B: nat] :
% 2.74/1.52 ( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
% 2.74/1.52 = ( ord_less_eq_nat @ A @ B ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_137_add__le__cancel__left,axiom,
% 2.74/1.52 ! [C: real,A: real,B: real] :
% 2.74/1.52 ( ( ord_less_eq_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
% 2.74/1.52 = ( ord_less_eq_real @ A @ B ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_138_add__le__cancel__left,axiom,
% 2.74/1.52 ! [C: nat,A: nat,B: nat] :
% 2.74/1.52 ( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
% 2.74/1.52 = ( ord_less_eq_nat @ A @ B ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_139_add__diff__cancel__right_H,axiom,
% 2.74/1.52 ! [A: real,B: real] :
% 2.74/1.52 ( ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ B )
% 2.74/1.52 = A ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_140_add__diff__cancel__right_H,axiom,
% 2.74/1.52 ! [A: nat,B: nat] :
% 2.74/1.52 ( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ B )
% 2.74/1.52 = A ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_141_add__right__imp__eq,axiom,
% 2.74/1.52 ! [B: nat,A: nat,C: nat] :
% 2.74/1.52 ( ( ( plus_plus_nat @ B @ A )
% 2.74/1.52 = ( plus_plus_nat @ C @ A ) )
% 2.74/1.52 => ( B = C ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_142_add__right__imp__eq,axiom,
% 2.74/1.52 ! [B: real,A: real,C: real] :
% 2.74/1.52 ( ( ( plus_plus_real @ B @ A )
% 2.74/1.52 = ( plus_plus_real @ C @ A ) )
% 2.74/1.52 => ( B = C ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_143_add__left__imp__eq,axiom,
% 2.74/1.52 ! [A: nat,B: nat,C: nat] :
% 2.74/1.52 ( ( ( plus_plus_nat @ A @ B )
% 2.74/1.52 = ( plus_plus_nat @ A @ C ) )
% 2.74/1.52 => ( B = C ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_144_add__left__imp__eq,axiom,
% 2.74/1.52 ! [A: real,B: real,C: real] :
% 2.74/1.52 ( ( ( plus_plus_real @ A @ B )
% 2.74/1.52 = ( plus_plus_real @ A @ C ) )
% 2.74/1.52 => ( B = C ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_145_add_Oleft__commute,axiom,
% 2.74/1.52 ! [B: nat,A: nat,C: nat] :
% 2.74/1.52 ( ( plus_plus_nat @ B @ ( plus_plus_nat @ A @ C ) )
% 2.74/1.52 = ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_146_add_Oleft__commute,axiom,
% 2.74/1.52 ! [B: real,A: real,C: real] :
% 2.74/1.52 ( ( plus_plus_real @ B @ ( plus_plus_real @ A @ C ) )
% 2.74/1.52 = ( plus_plus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_147_add_Ocommute,axiom,
% 2.74/1.52 ( plus_plus_nat
% 2.74/1.52 = ( ^ [A2: nat,B2: nat] : ( plus_plus_nat @ B2 @ A2 ) ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_148_add_Ocommute,axiom,
% 2.74/1.52 ( plus_plus_real
% 2.74/1.52 = ( ^ [A2: real,B2: real] : ( plus_plus_real @ B2 @ A2 ) ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_149_add_Oright__cancel,axiom,
% 2.74/1.52 ! [B: real,A: real,C: real] :
% 2.74/1.52 ( ( ( plus_plus_real @ B @ A )
% 2.74/1.52 = ( plus_plus_real @ C @ A ) )
% 2.74/1.52 = ( B = C ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_150_add_Oleft__cancel,axiom,
% 2.74/1.52 ! [A: real,B: real,C: real] :
% 2.74/1.52 ( ( ( plus_plus_real @ A @ B )
% 2.74/1.52 = ( plus_plus_real @ A @ C ) )
% 2.74/1.52 = ( B = C ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_151_add_Oassoc,axiom,
% 2.74/1.52 ! [A: nat,B: nat,C: nat] :
% 2.74/1.52 ( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
% 2.74/1.52 = ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_152_add_Oassoc,axiom,
% 2.74/1.52 ! [A: real,B: real,C: real] :
% 2.74/1.52 ( ( plus_plus_real @ ( plus_plus_real @ A @ B ) @ C )
% 2.74/1.52 = ( plus_plus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_153_group__cancel_Oadd2,axiom,
% 2.74/1.52 ! [B4: nat,K: nat,B: nat,A: nat] :
% 2.74/1.52 ( ( B4
% 2.74/1.52 = ( plus_plus_nat @ K @ B ) )
% 2.74/1.52 => ( ( plus_plus_nat @ A @ B4 )
% 2.74/1.52 = ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_154_group__cancel_Oadd2,axiom,
% 2.74/1.52 ! [B4: real,K: real,B: real,A: real] :
% 2.74/1.52 ( ( B4
% 2.74/1.52 = ( plus_plus_real @ K @ B ) )
% 2.74/1.52 => ( ( plus_plus_real @ A @ B4 )
% 2.74/1.52 = ( plus_plus_real @ K @ ( plus_plus_real @ A @ B ) ) ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_155_group__cancel_Oadd1,axiom,
% 2.74/1.52 ! [A4: nat,K: nat,A: nat,B: nat] :
% 2.74/1.52 ( ( A4
% 2.74/1.52 = ( plus_plus_nat @ K @ A ) )
% 2.74/1.52 => ( ( plus_plus_nat @ A4 @ B )
% 2.74/1.52 = ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_156_group__cancel_Oadd1,axiom,
% 2.74/1.52 ! [A4: real,K: real,A: real,B: real] :
% 2.74/1.52 ( ( A4
% 2.74/1.52 = ( plus_plus_real @ K @ A ) )
% 2.74/1.52 => ( ( plus_plus_real @ A4 @ B )
% 2.74/1.52 = ( plus_plus_real @ K @ ( plus_plus_real @ A @ B ) ) ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_157_add__mono__thms__linordered__semiring_I4_J,axiom,
% 2.74/1.52 ! [I: nat,J: nat,K: nat,L: nat] :
% 2.74/1.52 ( ( ( I = J )
% 2.74/1.52 & ( K = L ) )
% 2.74/1.52 => ( ( plus_plus_nat @ I @ K )
% 2.74/1.52 = ( plus_plus_nat @ J @ L ) ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_158_add__mono__thms__linordered__semiring_I4_J,axiom,
% 2.74/1.52 ! [I: real,J: real,K: real,L: real] :
% 2.74/1.52 ( ( ( I = J )
% 2.74/1.52 & ( K = L ) )
% 2.74/1.52 => ( ( plus_plus_real @ I @ K )
% 2.74/1.52 = ( plus_plus_real @ J @ L ) ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_159_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
% 2.74/1.52 ! [A: nat,B: nat,C: nat] :
% 2.74/1.52 ( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
% 2.74/1.52 = ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_160_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
% 2.74/1.52 ! [A: real,B: real,C: real] :
% 2.74/1.52 ( ( plus_plus_real @ ( plus_plus_real @ A @ B ) @ C )
% 2.74/1.52 = ( plus_plus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_161_one__reorient,axiom,
% 2.74/1.52 ! [X: nat] :
% 2.74/1.52 ( ( one_one_nat = X )
% 2.74/1.52 = ( X = one_one_nat ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_162_diff__right__commute,axiom,
% 2.74/1.52 ! [A: real,C: real,B: real] :
% 2.74/1.52 ( ( minus_minus_real @ ( minus_minus_real @ A @ C ) @ B )
% 2.74/1.52 = ( minus_minus_real @ ( minus_minus_real @ A @ B ) @ C ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_163_diff__right__commute,axiom,
% 2.74/1.52 ! [A: nat,C: nat,B: nat] :
% 2.74/1.52 ( ( minus_minus_nat @ ( minus_minus_nat @ A @ C ) @ B )
% 2.74/1.52 = ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_164_diff__eq__diff__eq,axiom,
% 2.74/1.52 ! [A: real,B: real,C: real,D: real] :
% 2.74/1.52 ( ( ( minus_minus_real @ A @ B )
% 2.74/1.52 = ( minus_minus_real @ C @ D ) )
% 2.74/1.52 => ( ( A = B )
% 2.74/1.52 = ( C = D ) ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_165_add__le__imp__le__right,axiom,
% 2.74/1.52 ! [A: real,C: real,B: real] :
% 2.74/1.52 ( ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
% 2.74/1.52 => ( ord_less_eq_real @ A @ B ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_166_add__le__imp__le__right,axiom,
% 2.74/1.52 ! [A: nat,C: nat,B: nat] :
% 2.74/1.52 ( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
% 2.74/1.52 => ( ord_less_eq_nat @ A @ B ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_167_add__le__imp__le__left,axiom,
% 2.74/1.52 ! [C: real,A: real,B: real] :
% 2.74/1.52 ( ( ord_less_eq_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
% 2.74/1.52 => ( ord_less_eq_real @ A @ B ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_168_add__le__imp__le__left,axiom,
% 2.74/1.52 ! [C: nat,A: nat,B: nat] :
% 2.74/1.52 ( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
% 2.74/1.52 => ( ord_less_eq_nat @ A @ B ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_169_le__iff__add,axiom,
% 2.74/1.52 ( ord_less_eq_nat
% 2.74/1.52 = ( ^ [A2: nat,B2: nat] :
% 2.74/1.52 ? [C2: nat] :
% 2.74/1.52 ( B2
% 2.74/1.52 = ( plus_plus_nat @ A2 @ C2 ) ) ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_170_add__right__mono,axiom,
% 2.74/1.52 ! [A: real,B: real,C: real] :
% 2.74/1.52 ( ( ord_less_eq_real @ A @ B )
% 2.74/1.52 => ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_171_add__right__mono,axiom,
% 2.74/1.52 ! [A: nat,B: nat,C: nat] :
% 2.74/1.52 ( ( ord_less_eq_nat @ A @ B )
% 2.74/1.52 => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_172_less__eqE,axiom,
% 2.74/1.52 ! [A: nat,B: nat] :
% 2.74/1.52 ( ( ord_less_eq_nat @ A @ B )
% 2.74/1.52 => ~ ! [C3: nat] :
% 2.74/1.52 ( B
% 2.74/1.52 != ( plus_plus_nat @ A @ C3 ) ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_173_add__left__mono,axiom,
% 2.74/1.52 ! [A: real,B: real,C: real] :
% 2.74/1.52 ( ( ord_less_eq_real @ A @ B )
% 2.74/1.52 => ( ord_less_eq_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_174_add__left__mono,axiom,
% 2.74/1.52 ! [A: nat,B: nat,C: nat] :
% 2.74/1.52 ( ( ord_less_eq_nat @ A @ B )
% 2.74/1.52 => ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_175_add__mono,axiom,
% 2.74/1.52 ! [A: real,B: real,C: real,D: real] :
% 2.74/1.52 ( ( ord_less_eq_real @ A @ B )
% 2.74/1.52 => ( ( ord_less_eq_real @ C @ D )
% 2.74/1.52 => ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) ) ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_176_add__mono,axiom,
% 2.74/1.52 ! [A: nat,B: nat,C: nat,D: nat] :
% 2.74/1.52 ( ( ord_less_eq_nat @ A @ B )
% 2.74/1.52 => ( ( ord_less_eq_nat @ C @ D )
% 2.74/1.52 => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_177_add__mono__thms__linordered__semiring_I1_J,axiom,
% 2.74/1.52 ! [I: real,J: real,K: real,L: real] :
% 2.74/1.52 ( ( ( ord_less_eq_real @ I @ J )
% 2.74/1.52 & ( ord_less_eq_real @ K @ L ) )
% 2.74/1.52 => ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_178_add__mono__thms__linordered__semiring_I1_J,axiom,
% 2.74/1.52 ! [I: nat,J: nat,K: nat,L: nat] :
% 2.74/1.52 ( ( ( ord_less_eq_nat @ I @ J )
% 2.74/1.52 & ( ord_less_eq_nat @ K @ L ) )
% 2.74/1.52 => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_179_add__mono__thms__linordered__semiring_I2_J,axiom,
% 2.74/1.52 ! [I: real,J: real,K: real,L: real] :
% 2.74/1.52 ( ( ( I = J )
% 2.74/1.52 & ( ord_less_eq_real @ K @ L ) )
% 2.74/1.52 => ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_180_add__mono__thms__linordered__semiring_I2_J,axiom,
% 2.74/1.52 ! [I: nat,J: nat,K: nat,L: nat] :
% 2.74/1.52 ( ( ( I = J )
% 2.74/1.52 & ( ord_less_eq_nat @ K @ L ) )
% 2.74/1.52 => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_181_add__mono__thms__linordered__semiring_I3_J,axiom,
% 2.74/1.52 ! [I: real,J: real,K: real,L: real] :
% 2.74/1.52 ( ( ( ord_less_eq_real @ I @ J )
% 2.74/1.52 & ( K = L ) )
% 2.74/1.52 => ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_182_add__mono__thms__linordered__semiring_I3_J,axiom,
% 2.74/1.52 ! [I: nat,J: nat,K: nat,L: nat] :
% 2.74/1.52 ( ( ( ord_less_eq_nat @ I @ J )
% 2.74/1.52 & ( K = L ) )
% 2.74/1.52 => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_183_diff__eq__diff__less__eq,axiom,
% 2.74/1.52 ! [A: real,B: real,C: real,D: real] :
% 2.74/1.52 ( ( ( minus_minus_real @ A @ B )
% 2.74/1.52 = ( minus_minus_real @ C @ D ) )
% 2.74/1.52 => ( ( ord_less_eq_real @ A @ B )
% 2.74/1.52 = ( ord_less_eq_real @ C @ D ) ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_184_diff__right__mono,axiom,
% 2.74/1.52 ! [A: real,B: real,C: real] :
% 2.74/1.52 ( ( ord_less_eq_real @ A @ B )
% 2.74/1.52 => ( ord_less_eq_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ C ) ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_185_diff__left__mono,axiom,
% 2.74/1.52 ! [B: real,A: real,C: real] :
% 2.74/1.52 ( ( ord_less_eq_real @ B @ A )
% 2.74/1.52 => ( ord_less_eq_real @ ( minus_minus_real @ C @ A ) @ ( minus_minus_real @ C @ B ) ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_186_diff__mono,axiom,
% 2.74/1.52 ! [A: real,B: real,D: real,C: real] :
% 2.74/1.52 ( ( ord_less_eq_real @ A @ B )
% 2.74/1.52 => ( ( ord_less_eq_real @ D @ C )
% 2.74/1.52 => ( ord_less_eq_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ D ) ) ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_187_add__implies__diff,axiom,
% 2.74/1.52 ! [C: real,B: real,A: real] :
% 2.74/1.52 ( ( ( plus_plus_real @ C @ B )
% 2.74/1.52 = A )
% 2.74/1.52 => ( C
% 2.74/1.52 = ( minus_minus_real @ A @ B ) ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_188_add__implies__diff,axiom,
% 2.74/1.52 ! [C: nat,B: nat,A: nat] :
% 2.74/1.52 ( ( ( plus_plus_nat @ C @ B )
% 2.74/1.52 = A )
% 2.74/1.52 => ( C
% 2.74/1.52 = ( minus_minus_nat @ A @ B ) ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_189_diff__diff__add,axiom,
% 2.74/1.52 ! [A: real,B: real,C: real] :
% 2.74/1.52 ( ( minus_minus_real @ ( minus_minus_real @ A @ B ) @ C )
% 2.74/1.52 = ( minus_minus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_190_diff__diff__add,axiom,
% 2.74/1.52 ! [A: nat,B: nat,C: nat] :
% 2.74/1.52 ( ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C )
% 2.74/1.52 = ( minus_minus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_191_diff__add__eq__diff__diff__swap,axiom,
% 2.74/1.52 ! [A: real,B: real,C: real] :
% 2.74/1.52 ( ( minus_minus_real @ A @ ( plus_plus_real @ B @ C ) )
% 2.74/1.52 = ( minus_minus_real @ ( minus_minus_real @ A @ C ) @ B ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_192_diff__add__eq,axiom,
% 2.74/1.52 ! [A: real,B: real,C: real] :
% 2.74/1.52 ( ( plus_plus_real @ ( minus_minus_real @ A @ B ) @ C )
% 2.74/1.52 = ( minus_minus_real @ ( plus_plus_real @ A @ C ) @ B ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_193_diff__diff__eq2,axiom,
% 2.74/1.52 ! [A: real,B: real,C: real] :
% 2.74/1.52 ( ( minus_minus_real @ A @ ( minus_minus_real @ B @ C ) )
% 2.74/1.52 = ( minus_minus_real @ ( plus_plus_real @ A @ C ) @ B ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_194_add__diff__eq,axiom,
% 2.74/1.52 ! [A: real,B: real,C: real] :
% 2.74/1.52 ( ( plus_plus_real @ A @ ( minus_minus_real @ B @ C ) )
% 2.74/1.52 = ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ C ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_195_eq__diff__eq,axiom,
% 2.74/1.52 ! [A: real,C: real,B: real] :
% 2.74/1.52 ( ( A
% 2.74/1.52 = ( minus_minus_real @ C @ B ) )
% 2.74/1.52 = ( ( plus_plus_real @ A @ B )
% 2.74/1.52 = C ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_196_diff__eq__eq,axiom,
% 2.74/1.52 ! [A: real,B: real,C: real] :
% 2.74/1.52 ( ( ( minus_minus_real @ A @ B )
% 2.74/1.52 = C )
% 2.74/1.52 = ( A
% 2.74/1.52 = ( plus_plus_real @ C @ B ) ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_197_group__cancel_Osub1,axiom,
% 2.74/1.52 ! [A4: real,K: real,A: real,B: real] :
% 2.74/1.52 ( ( A4
% 2.74/1.52 = ( plus_plus_real @ K @ A ) )
% 2.74/1.52 => ( ( minus_minus_real @ A4 @ B )
% 2.74/1.52 = ( plus_plus_real @ K @ ( minus_minus_real @ A @ B ) ) ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_198_abs__ge__self,axiom,
% 2.74/1.52 ! [A: real] : ( ord_less_eq_real @ A @ ( abs_abs_real @ A ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_199_abs__le__D1,axiom,
% 2.74/1.52 ! [A: real,B: real] :
% 2.74/1.52 ( ( ord_less_eq_real @ ( abs_abs_real @ A ) @ B )
% 2.74/1.52 => ( ord_less_eq_real @ A @ B ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_200_abs__one,axiom,
% 2.74/1.52 ( ( abs_abs_real @ one_one_real )
% 2.74/1.52 = one_one_real ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_201_abs__minus__commute,axiom,
% 2.74/1.52 ! [A: real,B: real] :
% 2.74/1.52 ( ( abs_abs_real @ ( minus_minus_real @ A @ B ) )
% 2.74/1.52 = ( abs_abs_real @ ( minus_minus_real @ B @ A ) ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_202_max__add__distrib__left,axiom,
% 2.74/1.52 ! [X: real,Y: real,Z2: real] :
% 2.74/1.52 ( ( plus_plus_real @ ( ord_max_real @ X @ Y ) @ Z2 )
% 2.74/1.52 = ( ord_max_real @ ( plus_plus_real @ X @ Z2 ) @ ( plus_plus_real @ Y @ Z2 ) ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_203_max__add__distrib__left,axiom,
% 2.74/1.52 ! [X: nat,Y: nat,Z2: nat] :
% 2.74/1.52 ( ( plus_plus_nat @ ( ord_max_nat @ X @ Y ) @ Z2 )
% 2.74/1.52 = ( ord_max_nat @ ( plus_plus_nat @ X @ Z2 ) @ ( plus_plus_nat @ Y @ Z2 ) ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_204_max__add__distrib__right,axiom,
% 2.74/1.52 ! [X: real,Y: real,Z2: real] :
% 2.74/1.52 ( ( plus_plus_real @ X @ ( ord_max_real @ Y @ Z2 ) )
% 2.74/1.52 = ( ord_max_real @ ( plus_plus_real @ X @ Y ) @ ( plus_plus_real @ X @ Z2 ) ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_205_max__add__distrib__right,axiom,
% 2.74/1.52 ! [X: nat,Y: nat,Z2: nat] :
% 2.74/1.52 ( ( plus_plus_nat @ X @ ( ord_max_nat @ Y @ Z2 ) )
% 2.74/1.52 = ( ord_max_nat @ ( plus_plus_nat @ X @ Y ) @ ( plus_plus_nat @ X @ Z2 ) ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_206_max__diff__distrib__left,axiom,
% 2.74/1.52 ! [X: real,Y: real,Z2: real] :
% 2.74/1.52 ( ( minus_minus_real @ ( ord_max_real @ X @ Y ) @ Z2 )
% 2.74/1.52 = ( ord_max_real @ ( minus_minus_real @ X @ Z2 ) @ ( minus_minus_real @ Y @ Z2 ) ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_207_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
% 2.74/1.52 ! [A: nat,B: nat,C: nat] :
% 2.74/1.52 ( ( ord_less_eq_nat @ A @ B )
% 2.74/1.52 => ( ( ord_less_eq_nat @ A @ B )
% 2.74/1.52 => ( ( ( minus_minus_nat @ B @ A )
% 2.74/1.52 = C )
% 2.74/1.52 = ( B
% 2.74/1.52 = ( plus_plus_nat @ C @ A ) ) ) ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_208_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
% 2.74/1.52 ! [A: nat,B: nat] :
% 2.74/1.52 ( ( ord_less_eq_nat @ A @ B )
% 2.74/1.52 => ( ( plus_plus_nat @ A @ ( minus_minus_nat @ B @ A ) )
% 2.74/1.52 = B ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_209_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
% 2.74/1.52 ! [A: nat,B: nat,C: nat] :
% 2.74/1.52 ( ( ord_less_eq_nat @ A @ B )
% 2.74/1.52 => ( ( minus_minus_nat @ C @ ( minus_minus_nat @ B @ A ) )
% 2.74/1.52 = ( minus_minus_nat @ ( plus_plus_nat @ C @ A ) @ B ) ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_210_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
% 2.74/1.52 ! [A: nat,B: nat,C: nat] :
% 2.74/1.52 ( ( ord_less_eq_nat @ A @ B )
% 2.74/1.52 => ( ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A )
% 2.74/1.52 = ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ C ) ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_211_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
% 2.74/1.52 ! [A: nat,B: nat,C: nat] :
% 2.74/1.52 ( ( ord_less_eq_nat @ A @ B )
% 2.74/1.52 => ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ C )
% 2.74/1.52 = ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A ) ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_212_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
% 2.74/1.52 ! [A: nat,B: nat,C: nat] :
% 2.74/1.52 ( ( ord_less_eq_nat @ A @ B )
% 2.74/1.52 => ( ( minus_minus_nat @ ( plus_plus_nat @ C @ B ) @ A )
% 2.74/1.52 = ( plus_plus_nat @ C @ ( minus_minus_nat @ B @ A ) ) ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_213_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
% 2.74/1.52 ! [A: nat,B: nat,C: nat] :
% 2.74/1.52 ( ( ord_less_eq_nat @ A @ B )
% 2.74/1.52 => ( ( plus_plus_nat @ C @ ( minus_minus_nat @ B @ A ) )
% 2.74/1.52 = ( minus_minus_nat @ ( plus_plus_nat @ C @ B ) @ A ) ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_214_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
% 2.74/1.52 ! [A: nat,B: nat,C: nat] :
% 2.74/1.52 ( ( ord_less_eq_nat @ A @ B )
% 2.74/1.52 => ( ( ord_less_eq_nat @ C @ ( minus_minus_nat @ B @ A ) )
% 2.74/1.52 = ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ B ) ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_215_le__add__diff,axiom,
% 2.74/1.52 ! [A: nat,B: nat,C: nat] :
% 2.74/1.52 ( ( ord_less_eq_nat @ A @ B )
% 2.74/1.52 => ( ord_less_eq_nat @ C @ ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A ) ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_216_add__le__add__imp__diff__le,axiom,
% 2.74/1.52 ! [I: real,K: real,N: real,J: real] :
% 2.74/1.52 ( ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ N )
% 2.74/1.52 => ( ( ord_less_eq_real @ N @ ( plus_plus_real @ J @ K ) )
% 2.74/1.52 => ( ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ N )
% 2.74/1.52 => ( ( ord_less_eq_real @ N @ ( plus_plus_real @ J @ K ) )
% 2.74/1.52 => ( ord_less_eq_real @ ( minus_minus_real @ N @ K ) @ J ) ) ) ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_217_add__le__add__imp__diff__le,axiom,
% 2.74/1.52 ! [I: nat,K: nat,N: nat,J: nat] :
% 2.74/1.52 ( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N )
% 2.74/1.52 => ( ( ord_less_eq_nat @ N @ ( plus_plus_nat @ J @ K ) )
% 2.74/1.52 => ( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N )
% 2.74/1.52 => ( ( ord_less_eq_nat @ N @ ( plus_plus_nat @ J @ K ) )
% 2.74/1.52 => ( ord_less_eq_nat @ ( minus_minus_nat @ N @ K ) @ J ) ) ) ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_218_diff__add,axiom,
% 2.74/1.52 ! [A: nat,B: nat] :
% 2.74/1.52 ( ( ord_less_eq_nat @ A @ B )
% 2.74/1.52 => ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ A )
% 2.74/1.52 = B ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_219_add__le__imp__le__diff,axiom,
% 2.74/1.52 ! [I: real,K: real,N: real] :
% 2.74/1.52 ( ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ N )
% 2.74/1.52 => ( ord_less_eq_real @ I @ ( minus_minus_real @ N @ K ) ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_220_add__le__imp__le__diff,axiom,
% 2.74/1.52 ! [I: nat,K: nat,N: nat] :
% 2.74/1.52 ( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N )
% 2.74/1.52 => ( ord_less_eq_nat @ I @ ( minus_minus_nat @ N @ K ) ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_221_le__diff__eq,axiom,
% 2.74/1.52 ! [A: real,C: real,B: real] :
% 2.74/1.52 ( ( ord_less_eq_real @ A @ ( minus_minus_real @ C @ B ) )
% 2.74/1.52 = ( ord_less_eq_real @ ( plus_plus_real @ A @ B ) @ C ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_222_diff__le__eq,axiom,
% 2.74/1.52 ! [A: real,B: real,C: real] :
% 2.74/1.52 ( ( ord_less_eq_real @ ( minus_minus_real @ A @ B ) @ C )
% 2.74/1.52 = ( ord_less_eq_real @ A @ ( plus_plus_real @ C @ B ) ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_223_abs__triangle__ineq,axiom,
% 2.74/1.52 ! [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 ) ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_224_abs__triangle__ineq2__sym,axiom,
% 2.74/1.52 ! [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 ) ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_225_abs__triangle__ineq3,axiom,
% 2.74/1.52 ! [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 ) ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_226_abs__triangle__ineq2,axiom,
% 2.74/1.52 ! [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 ) ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_227_abs__diff__triangle__ineq,axiom,
% 2.74/1.52 ! [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 ) ) ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_228_abs__triangle__ineq4,axiom,
% 2.74/1.52 ! [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 ) ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_229_abs__diff__le__iff,axiom,
% 2.74/1.52 ! [X: real,A: real,R: real] :
% 2.74/1.52 ( ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ X @ A ) ) @ R )
% 2.74/1.52 = ( ( ord_less_eq_real @ ( minus_minus_real @ A @ R ) @ X )
% 2.74/1.52 & ( ord_less_eq_real @ X @ ( plus_plus_real @ A @ R ) ) ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_230_rts1,axiom,
% 2.74/1.52 ! [P: nat,T: real,I3: nat] :
% 2.74/1.52 ( ( ( genClo1015804716orrect @ P @ T )
% 2.74/1.52 & ( ord_less_eq_real @ ( genClo1163638703lle_te @ P @ ( plus_plus_nat @ I3 @ one_one_nat ) ) @ T ) )
% 2.74/1.52 => ( ord_less_eq_real @ genClo1651033342e_rmin @ ( minus_minus_real @ T @ ( genClo1163638703lle_te @ P @ I3 ) ) ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_231_rts1d,axiom,
% 2.74/1.52 ! [P2: nat,I: nat] :
% 2.74/1.52 ( ( genClo1015804716orrect @ P2 @ ( genClo1163638703lle_te @ P2 @ ( plus_plus_nat @ I @ one_one_nat ) ) )
% 2.74/1.52 => ( ord_less_eq_real @ genClo1651033342e_rmin @ ( minus_minus_real @ ( genClo1163638703lle_te @ P2 @ ( plus_plus_nat @ I @ one_one_nat ) ) @ ( genClo1163638703lle_te @ P2 @ I ) ) ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_232_rts0,axiom,
% 2.74/1.52 ! [P: nat,T: real,I3: nat] :
% 2.74/1.52 ( ( ( genClo1015804716orrect @ P @ T )
% 2.74/1.52 & ( ord_less_eq_real @ T @ ( genClo1163638703lle_te @ P @ ( plus_plus_nat @ I3 @ one_one_nat ) ) ) )
% 2.74/1.52 => ( ord_less_eq_real @ ( minus_minus_real @ T @ ( genClo1163638703lle_te @ P @ I3 ) ) @ genClo1650508560e_rmax ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_233_rts0d,axiom,
% 2.74/1.52 ! [P2: nat,I: nat] :
% 2.74/1.52 ( ( genClo1015804716orrect @ P2 @ ( genClo1163638703lle_te @ P2 @ ( plus_plus_nat @ I @ one_one_nat ) ) )
% 2.74/1.52 => ( ord_less_eq_real @ ( minus_minus_real @ ( genClo1163638703lle_te @ P2 @ ( plus_plus_nat @ I @ one_one_nat ) ) @ ( genClo1163638703lle_te @ P2 @ I ) ) @ genClo1650508560e_rmax ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_234_okRead2__def,axiom,
% 2.74/1.52 ( genClo293725282kRead2
% 2.74/1.52 = ( ^ [F2: nat > real,G: nat > real,X3: real,Ppred: nat > $o] :
% 2.74/1.52 ! [P4: nat] :
% 2.74/1.52 ( ( Ppred @ P4 )
% 2.74/1.52 => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( F2 @ P4 ) @ ( G @ P4 ) ) ) @ X3 ) ) ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_235_sin__bound__lemma,axiom,
% 2.74/1.52 ! [X: real,Y: real,U: real,V: real] :
% 2.74/1.52 ( ( X = Y )
% 2.74/1.52 => ( ( ord_less_eq_real @ ( abs_abs_real @ U ) @ V )
% 2.74/1.52 => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( plus_plus_real @ X @ U ) @ Y ) ) @ V ) ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_236_eq__diff__eq_H,axiom,
% 2.74/1.52 ! [X: real,Y: real,Z2: real] :
% 2.74/1.52 ( ( X
% 2.74/1.52 = ( minus_minus_real @ Y @ Z2 ) )
% 2.74/1.52 = ( Y
% 2.74/1.52 = ( plus_plus_real @ X @ Z2 ) ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_237_beta__bound2,axiom,
% 2.74/1.52 ! [P2: nat,I: nat,Q2: nat] :
% 2.74/1.52 ( ( genClo1015804716orrect @ P2 @ ( genClo1163638703lle_te @ P2 @ ( plus_plus_nat @ I @ one_one_nat ) ) )
% 2.74/1.52 => ( ( genClo1015804716orrect @ Q2 @ ( genClo1163638703lle_te @ Q2 @ I ) )
% 2.74/1.52 => ( ord_less_eq_real @ ( minus_minus_real @ ( genClo1163638703lle_te @ P2 @ ( plus_plus_nat @ I @ one_one_nat ) ) @ ( genClo1163638703lle_te @ Q2 @ I ) ) @ ( plus_plus_real @ genClo1650508560e_rmax @ genClo1278781456e_beta ) ) ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_238_rts2b,axiom,
% 2.74/1.52 ! [P: nat,Q3: nat,I3: nat] :
% 2.74/1.52 ( ( ( genClo1015804716orrect @ P @ ( genClo1163638703lle_te @ P @ I3 ) )
% 2.74/1.52 & ( genClo1015804716orrect @ Q3 @ ( genClo1163638703lle_te @ Q3 @ I3 ) ) )
% 2.74/1.52 => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( genClo1163638703lle_te @ P @ I3 ) @ ( genClo1163638703lle_te @ Q3 @ I3 ) ) ) @ genClo1278781456e_beta ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_239_nat__add__left__cancel__le,axiom,
% 2.74/1.52 ! [K: nat,M: nat,N: nat] :
% 2.74/1.52 ( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
% 2.74/1.52 = ( ord_less_eq_nat @ M @ N ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_240_diff__diff__cancel,axiom,
% 2.74/1.52 ! [I: nat,N: nat] :
% 2.74/1.52 ( ( ord_less_eq_nat @ I @ N )
% 2.74/1.52 => ( ( minus_minus_nat @ N @ ( minus_minus_nat @ N @ I ) )
% 2.74/1.52 = I ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_241_diff__diff__left,axiom,
% 2.74/1.52 ! [I: nat,J: nat,K: nat] :
% 2.74/1.52 ( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
% 2.74/1.52 = ( minus_minus_nat @ I @ ( plus_plus_nat @ J @ K ) ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_242_Nat_Odiff__diff__right,axiom,
% 2.74/1.52 ! [K: nat,J: nat,I: nat] :
% 2.74/1.52 ( ( ord_less_eq_nat @ K @ J )
% 2.74/1.52 => ( ( minus_minus_nat @ I @ ( minus_minus_nat @ J @ K ) )
% 2.74/1.52 = ( minus_minus_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_243_Nat_Oadd__diff__assoc2,axiom,
% 2.74/1.52 ! [K: nat,J: nat,I: nat] :
% 2.74/1.52 ( ( ord_less_eq_nat @ K @ J )
% 2.74/1.52 => ( ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I )
% 2.74/1.52 = ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K ) ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_244_Nat_Oadd__diff__assoc,axiom,
% 2.74/1.52 ! [K: nat,J: nat,I: nat] :
% 2.74/1.52 ( ( ord_less_eq_nat @ K @ J )
% 2.74/1.52 => ( ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K ) )
% 2.74/1.52 = ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K ) ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_245_Nat_Oex__has__greatest__nat,axiom,
% 2.74/1.52 ! [P3: nat > $o,K: nat,B: nat] :
% 2.74/1.52 ( ( P3 @ K )
% 2.74/1.52 => ( ! [Y2: nat] :
% 2.74/1.52 ( ( P3 @ Y2 )
% 2.74/1.52 => ( ord_less_eq_nat @ Y2 @ B ) )
% 2.74/1.52 => ? [X2: nat] :
% 2.74/1.52 ( ( P3 @ X2 )
% 2.74/1.52 & ! [Y5: nat] :
% 2.74/1.52 ( ( P3 @ Y5 )
% 2.74/1.52 => ( ord_less_eq_nat @ Y5 @ X2 ) ) ) ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_246_nat__le__linear,axiom,
% 2.74/1.52 ! [M: nat,N: nat] :
% 2.74/1.52 ( ( ord_less_eq_nat @ M @ N )
% 2.74/1.52 | ( ord_less_eq_nat @ N @ M ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_247_diff__le__mono2,axiom,
% 2.74/1.52 ! [M: nat,N: nat,L: nat] :
% 2.74/1.52 ( ( ord_less_eq_nat @ M @ N )
% 2.74/1.52 => ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_248_le__diff__iff_H,axiom,
% 2.74/1.52 ! [A: nat,C: nat,B: nat] :
% 2.74/1.52 ( ( ord_less_eq_nat @ A @ C )
% 2.74/1.52 => ( ( ord_less_eq_nat @ B @ C )
% 2.74/1.52 => ( ( ord_less_eq_nat @ ( minus_minus_nat @ C @ A ) @ ( minus_minus_nat @ C @ B ) )
% 2.74/1.52 = ( ord_less_eq_nat @ B @ A ) ) ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_249_diff__le__self,axiom,
% 2.74/1.52 ! [M: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ M ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_250_diff__le__mono,axiom,
% 2.74/1.52 ! [M: nat,N: nat,L: nat] :
% 2.74/1.52 ( ( ord_less_eq_nat @ M @ N )
% 2.74/1.52 => ( ord_less_eq_nat @ ( minus_minus_nat @ M @ L ) @ ( minus_minus_nat @ N @ L ) ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_251_Nat_Odiff__diff__eq,axiom,
% 2.74/1.52 ! [K: nat,M: nat,N: nat] :
% 2.74/1.52 ( ( ord_less_eq_nat @ K @ M )
% 2.74/1.52 => ( ( ord_less_eq_nat @ K @ N )
% 2.74/1.52 => ( ( minus_minus_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
% 2.74/1.52 = ( minus_minus_nat @ M @ N ) ) ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_252_le__diff__iff,axiom,
% 2.74/1.52 ! [K: nat,M: nat,N: nat] :
% 2.74/1.52 ( ( ord_less_eq_nat @ K @ M )
% 2.74/1.52 => ( ( ord_less_eq_nat @ K @ N )
% 2.74/1.52 => ( ( ord_less_eq_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
% 2.74/1.52 = ( ord_less_eq_nat @ M @ N ) ) ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_253_eq__diff__iff,axiom,
% 2.74/1.52 ! [K: nat,M: nat,N: nat] :
% 2.74/1.52 ( ( ord_less_eq_nat @ K @ M )
% 2.74/1.52 => ( ( ord_less_eq_nat @ K @ N )
% 2.74/1.52 => ( ( ( minus_minus_nat @ M @ K )
% 2.74/1.52 = ( minus_minus_nat @ N @ K ) )
% 2.74/1.52 = ( M = N ) ) ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_254_le__antisym,axiom,
% 2.74/1.52 ! [M: nat,N: nat] :
% 2.74/1.52 ( ( ord_less_eq_nat @ M @ N )
% 2.74/1.52 => ( ( ord_less_eq_nat @ N @ M )
% 2.74/1.52 => ( M = N ) ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_255_eq__imp__le,axiom,
% 2.74/1.52 ! [M: nat,N: nat] :
% 2.74/1.52 ( ( M = N )
% 2.74/1.52 => ( ord_less_eq_nat @ M @ N ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_256_le__trans,axiom,
% 2.74/1.52 ! [I: nat,J: nat,K: nat] :
% 2.74/1.52 ( ( ord_less_eq_nat @ I @ J )
% 2.74/1.52 => ( ( ord_less_eq_nat @ J @ K )
% 2.74/1.52 => ( ord_less_eq_nat @ I @ K ) ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_257_le__refl,axiom,
% 2.74/1.52 ! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_258_Nat_Ole__imp__diff__is__add,axiom,
% 2.74/1.52 ! [I: nat,J: nat,K: nat] :
% 2.74/1.52 ( ( ord_less_eq_nat @ I @ J )
% 2.74/1.52 => ( ( ( minus_minus_nat @ J @ I )
% 2.74/1.52 = K )
% 2.74/1.52 = ( J
% 2.74/1.52 = ( plus_plus_nat @ K @ I ) ) ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_259_nat__minus__add__max,axiom,
% 2.74/1.52 ! [N: nat,M: nat] :
% 2.74/1.52 ( ( plus_plus_nat @ ( minus_minus_nat @ N @ M ) @ M )
% 2.74/1.52 = ( ord_max_nat @ N @ M ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_260_diff__add__inverse2,axiom,
% 2.74/1.52 ! [M: nat,N: nat] :
% 2.74/1.52 ( ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ N )
% 2.74/1.52 = M ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_261_diff__add__inverse,axiom,
% 2.74/1.52 ! [N: nat,M: nat] :
% 2.74/1.52 ( ( minus_minus_nat @ ( plus_plus_nat @ N @ M ) @ N )
% 2.74/1.52 = M ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_262_Nat_Odiff__add__assoc2,axiom,
% 2.74/1.52 ! [K: nat,J: nat,I: nat] :
% 2.74/1.52 ( ( ord_less_eq_nat @ K @ J )
% 2.74/1.52 => ( ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K )
% 2.74/1.52 = ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I ) ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_263_Nat_Odiff__add__assoc,axiom,
% 2.74/1.52 ! [K: nat,J: nat,I: nat] :
% 2.74/1.52 ( ( ord_less_eq_nat @ K @ J )
% 2.74/1.52 => ( ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K )
% 2.74/1.52 = ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K ) ) ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_264_Nat_Ole__diff__conv2,axiom,
% 2.74/1.52 ! [K: nat,J: nat,I: nat] :
% 2.74/1.52 ( ( ord_less_eq_nat @ K @ J )
% 2.74/1.52 => ( ( ord_less_eq_nat @ I @ ( minus_minus_nat @ J @ K ) )
% 2.74/1.52 = ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_265_le__diff__conv,axiom,
% 2.74/1.52 ! [J: nat,K: nat,I: nat] :
% 2.74/1.52 ( ( ord_less_eq_nat @ ( minus_minus_nat @ J @ K ) @ I )
% 2.74/1.52 = ( ord_less_eq_nat @ J @ ( plus_plus_nat @ I @ K ) ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_266_diff__cancel2,axiom,
% 2.74/1.52 ! [M: nat,K: nat,N: nat] :
% 2.74/1.52 ( ( minus_minus_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N @ K ) )
% 2.74/1.52 = ( minus_minus_nat @ M @ N ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_267_Nat_Odiff__cancel,axiom,
% 2.74/1.52 ! [K: nat,M: nat,N: nat] :
% 2.74/1.52 ( ( minus_minus_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
% 2.74/1.52 = ( minus_minus_nat @ M @ N ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_268_nat__add__max__right,axiom,
% 2.74/1.52 ! [M: nat,N: nat,Q2: nat] :
% 2.74/1.52 ( ( plus_plus_nat @ M @ ( ord_max_nat @ N @ Q2 ) )
% 2.74/1.52 = ( ord_max_nat @ ( plus_plus_nat @ M @ N ) @ ( plus_plus_nat @ M @ Q2 ) ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_269_nat__add__max__left,axiom,
% 2.74/1.52 ! [M: nat,N: nat,Q2: nat] :
% 2.74/1.52 ( ( plus_plus_nat @ ( ord_max_nat @ M @ N ) @ Q2 )
% 2.74/1.52 = ( ord_max_nat @ ( plus_plus_nat @ M @ Q2 ) @ ( plus_plus_nat @ N @ Q2 ) ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_270_nat__le__iff__add,axiom,
% 2.74/1.52 ( ord_less_eq_nat
% 2.74/1.52 = ( ^ [M2: nat,N2: nat] :
% 2.74/1.52 ? [K2: nat] :
% 2.74/1.52 ( N2
% 2.74/1.52 = ( plus_plus_nat @ M2 @ K2 ) ) ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_271_trans__le__add2,axiom,
% 2.74/1.52 ! [I: nat,J: nat,M: nat] :
% 2.74/1.52 ( ( ord_less_eq_nat @ I @ J )
% 2.74/1.52 => ( ord_less_eq_nat @ I @ ( plus_plus_nat @ M @ J ) ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_272_trans__le__add1,axiom,
% 2.74/1.52 ! [I: nat,J: nat,M: nat] :
% 2.74/1.52 ( ( ord_less_eq_nat @ I @ J )
% 2.74/1.52 => ( ord_less_eq_nat @ I @ ( plus_plus_nat @ J @ M ) ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_273_add__le__mono1,axiom,
% 2.74/1.52 ! [I: nat,J: nat,K: nat] :
% 2.74/1.52 ( ( ord_less_eq_nat @ I @ J )
% 2.74/1.52 => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_274_add__le__mono,axiom,
% 2.74/1.52 ! [I: nat,J: nat,K: nat,L: nat] :
% 2.74/1.52 ( ( ord_less_eq_nat @ I @ J )
% 2.74/1.52 => ( ( ord_less_eq_nat @ K @ L )
% 2.74/1.52 => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_275_le__Suc__ex,axiom,
% 2.74/1.52 ! [K: nat,L: nat] :
% 2.74/1.52 ( ( ord_less_eq_nat @ K @ L )
% 2.74/1.52 => ? [N3: nat] :
% 2.74/1.52 ( L
% 2.74/1.52 = ( plus_plus_nat @ K @ N3 ) ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_276_add__leD2,axiom,
% 2.74/1.52 ! [M: nat,K: nat,N: nat] :
% 2.74/1.52 ( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
% 2.74/1.52 => ( ord_less_eq_nat @ K @ N ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_277_add__leD1,axiom,
% 2.74/1.52 ! [M: nat,K: nat,N: nat] :
% 2.74/1.52 ( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
% 2.74/1.52 => ( ord_less_eq_nat @ M @ N ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_278_le__add2,axiom,
% 2.74/1.52 ! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M @ N ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_279_le__add1,axiom,
% 2.74/1.52 ! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_280_add__leE,axiom,
% 2.74/1.52 ! [M: nat,K: nat,N: nat] :
% 2.74/1.52 ( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
% 2.74/1.52 => ~ ( ( ord_less_eq_nat @ M @ N )
% 2.74/1.52 => ~ ( ord_less_eq_nat @ K @ N ) ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_281_nonoverlap,axiom,
% 2.74/1.52 ord_less_eq_real @ genClo1278781456e_beta @ genClo1651033342e_rmin ).
% 2.74/1.52
% 2.74/1.52 thf(fact_282_rts2a,axiom,
% 2.74/1.52 ! [P: nat,Q3: nat,T: real,I3: nat] :
% 2.74/1.52 ( ( ( genClo1015804716orrect @ P @ T )
% 2.74/1.52 & ( genClo1015804716orrect @ Q3 @ T )
% 2.74/1.52 & ( ord_less_eq_real @ ( plus_plus_real @ ( genClo1163638703lle_te @ Q3 @ I3 ) @ genClo1278781456e_beta ) @ T ) )
% 2.74/1.52 => ( ord_less_eq_real @ ( genClo1163638703lle_te @ P @ I3 ) @ T ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_283_readerror,axiom,
% 2.74/1.52 ! [P: nat,Q3: nat,I3: nat] :
% 2.74/1.52 ( ( ( genClo1015804716orrect @ P @ ( genClo1163638703lle_te @ P @ ( plus_plus_nat @ I3 @ one_one_nat ) ) )
% 2.74/1.52 & ( genClo1015804716orrect @ Q3 @ ( genClo1163638703lle_te @ P @ ( plus_plus_nat @ I3 @ one_one_nat ) ) ) )
% 2.74/1.52 => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( genClo1400225944_theta @ P @ ( plus_plus_nat @ I3 @ one_one_nat ) @ Q3 ) @ ( genClo1160817912lle_IC @ Q3 @ I3 @ ( genClo1163638703lle_te @ P @ ( plus_plus_nat @ I3 @ one_one_nat ) ) ) ) ) @ genClo721845095Lambda ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_284_okRead1__def,axiom,
% 2.74/1.52 ( genClo293725281kRead1
% 2.74/1.52 = ( ^ [F2: nat > real,X3: real,Ppred: nat > $o] :
% 2.74/1.52 ! [L2: nat,M2: nat] :
% 2.74/1.52 ( ( ( Ppred @ L2 )
% 2.74/1.52 & ( Ppred @ M2 ) )
% 2.74/1.52 => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( F2 @ L2 ) @ ( F2 @ M2 ) ) ) @ X3 ) ) ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_285_beta__bound1,axiom,
% 2.74/1.52 ! [P2: nat,I: nat,Q2: nat] :
% 2.74/1.52 ( ( genClo1015804716orrect @ P2 @ ( genClo1163638703lle_te @ P2 @ ( plus_plus_nat @ I @ one_one_nat ) ) )
% 2.74/1.52 => ( ( genClo1015804716orrect @ Q2 @ ( genClo1163638703lle_te @ P2 @ ( plus_plus_nat @ I @ one_one_nat ) ) )
% 2.74/1.52 => ( ord_less_eq_real @ zero_zero_real @ ( minus_minus_real @ ( genClo1163638703lle_te @ P2 @ ( plus_plus_nat @ I @ one_one_nat ) ) @ ( genClo1163638703lle_te @ Q2 @ I ) ) ) ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_286_le__zero__eq,axiom,
% 2.74/1.52 ! [N: nat] :
% 2.74/1.52 ( ( ord_less_eq_nat @ N @ zero_zero_nat )
% 2.74/1.52 = ( N = zero_zero_nat ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_287_zero__eq__add__iff__both__eq__0,axiom,
% 2.74/1.52 ! [X: nat,Y: nat] :
% 2.74/1.52 ( ( zero_zero_nat
% 2.74/1.52 = ( plus_plus_nat @ X @ Y ) )
% 2.74/1.52 = ( ( X = zero_zero_nat )
% 2.74/1.52 & ( Y = zero_zero_nat ) ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_288_add__eq__0__iff__both__eq__0,axiom,
% 2.74/1.52 ! [X: nat,Y: nat] :
% 2.74/1.52 ( ( ( plus_plus_nat @ X @ Y )
% 2.74/1.52 = zero_zero_nat )
% 2.74/1.52 = ( ( X = zero_zero_nat )
% 2.74/1.52 & ( Y = zero_zero_nat ) ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_289_add__cancel__right__right,axiom,
% 2.74/1.52 ! [A: nat,B: nat] :
% 2.74/1.52 ( ( A
% 2.74/1.52 = ( plus_plus_nat @ A @ B ) )
% 2.74/1.52 = ( B = zero_zero_nat ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_290_add__cancel__right__right,axiom,
% 2.74/1.52 ! [A: real,B: real] :
% 2.74/1.52 ( ( A
% 2.74/1.52 = ( plus_plus_real @ A @ B ) )
% 2.74/1.52 = ( B = zero_zero_real ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_291_add__cancel__right__left,axiom,
% 2.74/1.52 ! [A: nat,B: nat] :
% 2.74/1.52 ( ( A
% 2.74/1.52 = ( plus_plus_nat @ B @ A ) )
% 2.74/1.52 = ( B = zero_zero_nat ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_292_add__cancel__right__left,axiom,
% 2.74/1.52 ! [A: real,B: real] :
% 2.74/1.52 ( ( A
% 2.74/1.52 = ( plus_plus_real @ B @ A ) )
% 2.74/1.52 = ( B = zero_zero_real ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_293_add__cancel__left__right,axiom,
% 2.74/1.52 ! [A: nat,B: nat] :
% 2.74/1.52 ( ( ( plus_plus_nat @ A @ B )
% 2.74/1.52 = A )
% 2.74/1.52 = ( B = zero_zero_nat ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_294_add__cancel__left__right,axiom,
% 2.74/1.52 ! [A: real,B: real] :
% 2.74/1.52 ( ( ( plus_plus_real @ A @ B )
% 2.74/1.52 = A )
% 2.74/1.52 = ( B = zero_zero_real ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_295_add__cancel__left__left,axiom,
% 2.74/1.52 ! [B: nat,A: nat] :
% 2.74/1.52 ( ( ( plus_plus_nat @ B @ A )
% 2.74/1.52 = A )
% 2.74/1.52 = ( B = zero_zero_nat ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_296_add__cancel__left__left,axiom,
% 2.74/1.52 ! [B: real,A: real] :
% 2.74/1.52 ( ( ( plus_plus_real @ B @ A )
% 2.74/1.52 = A )
% 2.74/1.52 = ( B = zero_zero_real ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_297_double__zero__sym,axiom,
% 2.74/1.52 ! [A: real] :
% 2.74/1.52 ( ( zero_zero_real
% 2.74/1.52 = ( plus_plus_real @ A @ A ) )
% 2.74/1.52 = ( A = zero_zero_real ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_298_double__zero,axiom,
% 2.74/1.52 ! [A: real] :
% 2.74/1.52 ( ( ( plus_plus_real @ A @ A )
% 2.74/1.52 = zero_zero_real )
% 2.74/1.52 = ( A = zero_zero_real ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_299_add_Oright__neutral,axiom,
% 2.74/1.52 ! [A: nat] :
% 2.74/1.52 ( ( plus_plus_nat @ A @ zero_zero_nat )
% 2.74/1.52 = A ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_300_add_Oright__neutral,axiom,
% 2.74/1.52 ! [A: real] :
% 2.74/1.52 ( ( plus_plus_real @ A @ zero_zero_real )
% 2.74/1.52 = A ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_301_add_Oleft__neutral,axiom,
% 2.74/1.52 ! [A: nat] :
% 2.74/1.52 ( ( plus_plus_nat @ zero_zero_nat @ A )
% 2.74/1.52 = A ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_302_add_Oleft__neutral,axiom,
% 2.74/1.52 ! [A: real] :
% 2.74/1.52 ( ( plus_plus_real @ zero_zero_real @ A )
% 2.74/1.52 = A ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_303_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
% 2.74/1.52 ! [A: real] :
% 2.74/1.52 ( ( minus_minus_real @ A @ A )
% 2.74/1.52 = zero_zero_real ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_304_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
% 2.74/1.52 ! [A: nat] :
% 2.74/1.52 ( ( minus_minus_nat @ A @ A )
% 2.74/1.52 = zero_zero_nat ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_305_diff__zero,axiom,
% 2.74/1.52 ! [A: real] :
% 2.74/1.52 ( ( minus_minus_real @ A @ zero_zero_real )
% 2.74/1.52 = A ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_306_diff__zero,axiom,
% 2.74/1.52 ! [A: nat] :
% 2.74/1.52 ( ( minus_minus_nat @ A @ zero_zero_nat )
% 2.74/1.52 = A ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_307_zero__diff,axiom,
% 2.74/1.52 ! [A: nat] :
% 2.74/1.52 ( ( minus_minus_nat @ zero_zero_nat @ A )
% 2.74/1.52 = zero_zero_nat ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_308_diff__0__right,axiom,
% 2.74/1.52 ! [A: real] :
% 2.74/1.52 ( ( minus_minus_real @ A @ zero_zero_real )
% 2.74/1.52 = A ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_309_diff__self,axiom,
% 2.74/1.52 ! [A: real] :
% 2.74/1.52 ( ( minus_minus_real @ A @ A )
% 2.74/1.52 = zero_zero_real ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_310_abs__zero,axiom,
% 2.74/1.52 ( ( abs_abs_real @ zero_zero_real )
% 2.74/1.52 = zero_zero_real ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_311_abs__eq__0,axiom,
% 2.74/1.52 ! [A: real] :
% 2.74/1.52 ( ( ( abs_abs_real @ A )
% 2.74/1.52 = zero_zero_real )
% 2.74/1.52 = ( A = zero_zero_real ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_312_abs__0__eq,axiom,
% 2.74/1.52 ! [A: real] :
% 2.74/1.52 ( ( zero_zero_real
% 2.74/1.52 = ( abs_abs_real @ A ) )
% 2.74/1.52 = ( A = zero_zero_real ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_313_abs__0,axiom,
% 2.74/1.52 ( ( abs_abs_real @ zero_zero_real )
% 2.74/1.52 = zero_zero_real ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_314_add__le__same__cancel1,axiom,
% 2.74/1.52 ! [B: real,A: real] :
% 2.74/1.52 ( ( ord_less_eq_real @ ( plus_plus_real @ B @ A ) @ B )
% 2.74/1.52 = ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_315_add__le__same__cancel1,axiom,
% 2.74/1.52 ! [B: nat,A: nat] :
% 2.74/1.52 ( ( ord_less_eq_nat @ ( plus_plus_nat @ B @ A ) @ B )
% 2.74/1.52 = ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_316_add__le__same__cancel2,axiom,
% 2.74/1.52 ! [A: real,B: real] :
% 2.74/1.52 ( ( ord_less_eq_real @ ( plus_plus_real @ A @ B ) @ B )
% 2.74/1.52 = ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_317_add__le__same__cancel2,axiom,
% 2.74/1.52 ! [A: nat,B: nat] :
% 2.74/1.52 ( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ B )
% 2.74/1.52 = ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_318_le__add__same__cancel1,axiom,
% 2.74/1.52 ! [A: real,B: real] :
% 2.74/1.52 ( ( ord_less_eq_real @ A @ ( plus_plus_real @ A @ B ) )
% 2.74/1.52 = ( ord_less_eq_real @ zero_zero_real @ B ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_319_le__add__same__cancel1,axiom,
% 2.74/1.52 ! [A: nat,B: nat] :
% 2.74/1.52 ( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ A @ B ) )
% 2.74/1.52 = ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_320_le__add__same__cancel2,axiom,
% 2.74/1.52 ! [A: real,B: real] :
% 2.74/1.52 ( ( ord_less_eq_real @ A @ ( plus_plus_real @ B @ A ) )
% 2.74/1.52 = ( ord_less_eq_real @ zero_zero_real @ B ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_321_le__add__same__cancel2,axiom,
% 2.74/1.52 ! [A: nat,B: nat] :
% 2.74/1.52 ( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ B @ A ) )
% 2.74/1.52 = ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_322_double__add__le__zero__iff__single__add__le__zero,axiom,
% 2.74/1.52 ! [A: real] :
% 2.74/1.52 ( ( ord_less_eq_real @ ( plus_plus_real @ A @ A ) @ zero_zero_real )
% 2.74/1.52 = ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_323_zero__le__double__add__iff__zero__le__single__add,axiom,
% 2.74/1.52 ! [A: real] :
% 2.74/1.52 ( ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ A @ A ) )
% 2.74/1.52 = ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_324_diff__ge__0__iff__ge,axiom,
% 2.74/1.52 ! [A: real,B: real] :
% 2.74/1.52 ( ( ord_less_eq_real @ zero_zero_real @ ( minus_minus_real @ A @ B ) )
% 2.74/1.52 = ( ord_less_eq_real @ B @ A ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_325_diff__add__zero,axiom,
% 2.74/1.52 ! [A: nat,B: nat] :
% 2.74/1.52 ( ( minus_minus_nat @ A @ ( plus_plus_nat @ A @ B ) )
% 2.74/1.52 = zero_zero_nat ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_326_abs__of__nonneg,axiom,
% 2.74/1.52 ! [A: real] :
% 2.74/1.52 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 2.74/1.52 => ( ( abs_abs_real @ A )
% 2.74/1.52 = A ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_327_abs__le__self__iff,axiom,
% 2.74/1.52 ! [A: real] :
% 2.74/1.52 ( ( ord_less_eq_real @ ( abs_abs_real @ A ) @ A )
% 2.74/1.52 = ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_328_abs__le__zero__iff,axiom,
% 2.74/1.52 ! [A: real] :
% 2.74/1.52 ( ( ord_less_eq_real @ ( abs_abs_real @ A ) @ zero_zero_real )
% 2.74/1.52 = ( A = zero_zero_real ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_329_zero__le,axiom,
% 2.74/1.52 ! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_330_diff__commute,axiom,
% 2.74/1.52 ! [I: nat,J: nat,K: nat] :
% 2.74/1.52 ( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
% 2.74/1.52 = ( minus_minus_nat @ ( minus_minus_nat @ I @ K ) @ J ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_331_zero__reorient,axiom,
% 2.74/1.52 ! [X: real] :
% 2.74/1.52 ( ( zero_zero_real = X )
% 2.74/1.52 = ( X = zero_zero_real ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_332_abs__eq__0__iff,axiom,
% 2.74/1.52 ! [A: real] :
% 2.74/1.52 ( ( ( abs_abs_real @ A )
% 2.74/1.52 = zero_zero_real )
% 2.74/1.52 = ( A = zero_zero_real ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_333_eq__iff__diff__eq__0,axiom,
% 2.74/1.52 ( ( ^ [Y3: real,Z: real] : ( Y3 = Z ) )
% 2.74/1.52 = ( ^ [A2: real,B2: real] :
% 2.74/1.52 ( ( minus_minus_real @ A2 @ B2 )
% 2.74/1.52 = zero_zero_real ) ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_334_zero__neq__one,axiom,
% 2.74/1.52 zero_zero_nat != one_one_nat ).
% 2.74/1.52
% 2.74/1.52 thf(fact_335_zero__neq__one,axiom,
% 2.74/1.52 zero_zero_real != one_one_real ).
% 2.74/1.52
% 2.74/1.52 thf(fact_336_add_Ogroup__left__neutral,axiom,
% 2.74/1.52 ! [A: real] :
% 2.74/1.52 ( ( plus_plus_real @ zero_zero_real @ A )
% 2.74/1.52 = A ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_337_add_Ocomm__neutral,axiom,
% 2.74/1.52 ! [A: nat] :
% 2.74/1.52 ( ( plus_plus_nat @ A @ zero_zero_nat )
% 2.74/1.52 = A ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_338_add_Ocomm__neutral,axiom,
% 2.74/1.52 ! [A: real] :
% 2.74/1.52 ( ( plus_plus_real @ A @ zero_zero_real )
% 2.74/1.52 = A ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_339_comm__monoid__add__class_Oadd__0,axiom,
% 2.74/1.52 ! [A: nat] :
% 2.74/1.52 ( ( plus_plus_nat @ zero_zero_nat @ A )
% 2.74/1.52 = A ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_340_comm__monoid__add__class_Oadd__0,axiom,
% 2.74/1.52 ! [A: real] :
% 2.74/1.52 ( ( plus_plus_real @ zero_zero_real @ A )
% 2.74/1.52 = A ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_341_add__decreasing,axiom,
% 2.74/1.52 ! [A: real,C: real,B: real] :
% 2.74/1.52 ( ( ord_less_eq_real @ A @ zero_zero_real )
% 2.74/1.52 => ( ( ord_less_eq_real @ C @ B )
% 2.74/1.52 => ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ B ) ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_342_add__decreasing,axiom,
% 2.74/1.52 ! [A: nat,C: nat,B: nat] :
% 2.74/1.52 ( ( ord_less_eq_nat @ A @ zero_zero_nat )
% 2.74/1.52 => ( ( ord_less_eq_nat @ C @ B )
% 2.74/1.52 => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ B ) ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_343_add__increasing,axiom,
% 2.74/1.52 ! [A: real,B: real,C: real] :
% 2.74/1.52 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 2.74/1.52 => ( ( ord_less_eq_real @ B @ C )
% 2.74/1.52 => ( ord_less_eq_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_344_add__increasing,axiom,
% 2.74/1.52 ! [A: nat,B: nat,C: nat] :
% 2.74/1.52 ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 2.74/1.52 => ( ( ord_less_eq_nat @ B @ C )
% 2.74/1.52 => ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_345_add__decreasing2,axiom,
% 2.74/1.52 ! [C: real,A: real,B: real] :
% 2.74/1.52 ( ( ord_less_eq_real @ C @ zero_zero_real )
% 2.74/1.52 => ( ( ord_less_eq_real @ A @ B )
% 2.74/1.52 => ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ B ) ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_346_add__decreasing2,axiom,
% 2.74/1.52 ! [C: nat,A: nat,B: nat] :
% 2.74/1.52 ( ( ord_less_eq_nat @ C @ zero_zero_nat )
% 2.74/1.52 => ( ( ord_less_eq_nat @ A @ B )
% 2.74/1.52 => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ B ) ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_347_add__increasing2,axiom,
% 2.74/1.52 ! [C: real,B: real,A: real] :
% 2.74/1.52 ( ( ord_less_eq_real @ zero_zero_real @ C )
% 2.74/1.52 => ( ( ord_less_eq_real @ B @ A )
% 2.74/1.52 => ( ord_less_eq_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_348_add__increasing2,axiom,
% 2.74/1.52 ! [C: nat,B: nat,A: nat] :
% 2.74/1.52 ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 2.74/1.52 => ( ( ord_less_eq_nat @ B @ A )
% 2.74/1.52 => ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_349_add__nonneg__nonneg,axiom,
% 2.74/1.52 ! [A: real,B: real] :
% 2.74/1.52 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 2.74/1.52 => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 2.74/1.52 => ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ A @ B ) ) ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_350_add__nonneg__nonneg,axiom,
% 2.74/1.52 ! [A: nat,B: nat] :
% 2.74/1.52 ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 2.74/1.52 => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 2.74/1.52 => ( ord_less_eq_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_351_add__nonpos__nonpos,axiom,
% 2.74/1.52 ! [A: real,B: real] :
% 2.74/1.52 ( ( ord_less_eq_real @ A @ zero_zero_real )
% 2.74/1.52 => ( ( ord_less_eq_real @ B @ zero_zero_real )
% 2.74/1.52 => ( ord_less_eq_real @ ( plus_plus_real @ A @ B ) @ zero_zero_real ) ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(fact_352_add__nonpos__nonpos,axiom,
% 2.74/1.52 ! [A: nat,B: nat] :
% 2.74/1.52 ( ( ord_less_eq_nat @ A @ zero_zero_nat )
% 2.74/1.52 => ( ( ord_less_eq_nat @ B @ zero_zero_nat )
% 2.74/1.52 => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
% 2.74/1.52 ! [X: nat,Y: nat] :
% 2.74/1.52 ( ( if_nat @ $false @ X @ Y )
% 2.74/1.52 = Y ) ).
% 2.74/1.52
% 2.74/1.52 thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
% 2.74/1.52 ! [X: nat,Y: nat] :
% 2.74/1.52 ( ( if_nat @ $true @ X @ Y )
% 2.74/1.52 = X ) ).
% 2.74/1.52
% 2.74/1.52 thf(help_If_3_1_If_001t__Real__Oreal_T,axiom,
% 2.74/1.52 ! [P3: $o] :
% 2.74/1.52 ( ( P3 = $true )
% 2.74/1.52 | ( P3 = $false ) ) ).
% 2.74/1.52
% 2.74/1.52 thf(help_If_2_1_If_001t__Real__Oreal_T,axiom,
% 2.74/1.52 ! [X: real,Y: real] :
% 2.74/1.52 ( ( if_real @ $false @ X @ Y )
% 2.74/1.52 = Y ) ).
% 2.74/1.52
% 2.74/1.52 thf(help_If_1_1_If_001t__Real__Oreal_T,axiom,
% 2.74/1.52 ! [X: real,Y: real] :
% 2.74/1.52 ( ( if_real @ $true @ X @ Y )
% 2.74/1.52 = X ) ).
% 2.74/1.52
% 2.74/1.52 thf(conj_0,conjecture,
% 2.74/1.52 genClo1015804716orrect @ m @ ( ord_max_real @ ( genClo1163638703lle_te @ l @ i ) @ ( genClo1163638703lle_te @ m @ i ) ) ).
% 2.74/1.52 ------------------------
% 2.74/1.55 % (881)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_777 for (2999ds/183Mi)
% 2.74/1.56 % (883)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_777 for (2999ds/27Mi)
% 2.74/1.56 % (885)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_777 for (2999ds/2Mi)
% 2.74/1.56 % (884)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_777 for (2999ds/2Mi)
% 2.74/1.56 % (886)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_777 for (2999ds/275Mi)
% 2.74/1.56 % (882)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_777 for (2999ds/4Mi)
% 2.74/1.56 % (887)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_777 for (2999ds/18Mi)
% 2.74/1.56 % (884)Instruction limit reached!
% 2.74/1.56 % (884)------------------------------
% 2.74/1.56 % (884)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.74/1.56 % (884)Termination reason: Unknown
% 2.74/1.56 % (884)Termination phase: shuffling
% 2.74/1.56 % (885)Instruction limit reached!
% 2.74/1.56 % (885)------------------------------
% 2.74/1.56 % (885)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.74/1.56 % (885)Termination reason: Unknown
% 2.74/1.56 % (885)Termination phase: shuffling
% 2.74/1.56
% 2.74/1.56 % (885)Memory used [KB]: 1407
% 2.74/1.56 % (885)Time elapsed: 0.003 s
% 2.74/1.56 % (885)Instructions burned: 4 (million)
% 2.74/1.56 % (885)------------------------------
% 2.74/1.56 % (885)------------------------------
% 2.74/1.56
% 2.74/1.56 % (884)Memory used [KB]: 1407
% 2.74/1.56 % (884)Time elapsed: 0.003 s
% 2.74/1.56 % (884)Instructions burned: 3 (million)
% 2.74/1.56 % (884)------------------------------
% 2.74/1.56 % (884)------------------------------
% 2.74/1.56 % (882)Instruction limit reached!
% 2.74/1.56 % (882)------------------------------
% 2.74/1.56 % (882)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.74/1.56 % (882)Termination reason: Unknown
% 2.74/1.56 % (882)Termination phase: shuffling
% 2.74/1.56
% 2.74/1.56 % (882)Memory used [KB]: 1407
% 2.74/1.56 % (882)Time elapsed: 0.004 s
% 2.74/1.56 % (882)Instructions burned: 5 (million)
% 2.74/1.56 % (882)------------------------------
% 2.74/1.56 % (882)------------------------------
% 2.74/1.57 % (887)Instruction limit reached!
% 2.74/1.57 % (887)------------------------------
% 2.74/1.57 % (887)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.74/1.57 % (887)Termination reason: Unknown
% 2.74/1.57 % (887)Termination phase: shuffling
% 2.74/1.57
% 2.74/1.57 % (887)Memory used [KB]: 1663
% 2.74/1.57 % (887)Time elapsed: 0.011 s
% 2.74/1.57 % (887)Instructions burned: 19 (million)
% 2.74/1.57 % (887)------------------------------
% 2.74/1.57 % (887)------------------------------
% 2.74/1.57 % (883)Instruction limit reached!
% 2.74/1.57 % (883)------------------------------
% 2.74/1.57 % (883)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.74/1.57 % (883)Termination reason: Unknown
% 2.74/1.57 % (883)Termination phase: Property scanning
% 2.74/1.57
% 2.74/1.57 % (883)Memory used [KB]: 1663
% 2.74/1.57 % (883)Time elapsed: 0.015 s
% 2.74/1.57 % (883)Instructions burned: 28 (million)
% 2.74/1.57 % (883)------------------------------
% 2.74/1.57 % (883)------------------------------
% 2.74/1.57 % (889)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_777 for (2999ds/37Mi)
% 2.74/1.57 % (888)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_777 for (2999ds/3Mi)
% 2.74/1.58 % (888)Instruction limit reached!
% 2.74/1.58 % (888)------------------------------
% 2.74/1.58 % (888)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.74/1.58 % (888)Termination reason: Unknown
% 2.74/1.58 % (888)Termination phase: shuffling
% 2.74/1.58
% 2.74/1.58 % (888)Memory used [KB]: 1407
% 2.74/1.58 % (888)Time elapsed: 0.003 s
% 2.74/1.58 % (888)Instructions burned: 3 (million)
% 2.74/1.58 % (888)------------------------------
% 2.74/1.58 % (888)------------------------------
% 2.74/1.58 % (890)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_777 for (2999ds/15Mi)
% 2.74/1.58 % (891)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_777 for (2999ds/3Mi)
% 2.74/1.58 % (891)Instruction limit reached!
% 2.74/1.58 % (891)------------------------------
% 2.74/1.58 % (891)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.74/1.58 % (891)Termination reason: Unknown
% 2.74/1.58 % (891)Termination phase: shuffling
% 2.74/1.58
% 2.74/1.58 % (891)Memory used [KB]: 1407
% 2.74/1.58 % (891)Time elapsed: 0.003 s
% 2.74/1.58 % (891)Instructions burned: 3 (million)
% 2.74/1.58 % (890)Instruction limit reached!
% 2.74/1.58 % (890)------------------------------
% 2.74/1.58 % (890)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.74/1.58 % (890)Termination reason: Unknown
% 2.74/1.58 % (890)Termination phase: shuffling
% 2.74/1.58
% 2.74/1.58 % (890)Memory used [KB]: 1663
% 2.74/1.58 % (890)Time elapsed: 0.009 s
% 2.74/1.58 % (890)Instructions burned: 15 (million)
% 2.74/1.58 % (890)------------------------------
% 2.74/1.58 % (890)------------------------------
% 2.74/1.58 % (891)------------------------------
% 2.74/1.58 % (891)------------------------------
% 2.74/1.59 % (892)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_777 for (2999ds/1041Mi)
% 2.74/1.59 % (889)Instruction limit reached!
% 2.74/1.59 % (889)------------------------------
% 2.74/1.59 % (889)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.74/1.59 % (889)Termination reason: Unknown
% 2.74/1.59 % (889)Termination phase: shuffling
% 2.74/1.59
% 2.74/1.59 % (889)Memory used [KB]: 2174
% 2.74/1.59 % (889)Time elapsed: 0.019 s
% 2.74/1.59 % (889)Instructions burned: 38 (million)
% 2.74/1.59 % (889)------------------------------
% 2.74/1.59 % (889)------------------------------
% 2.74/1.59 % (893)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_777 for (2999ds/7Mi)
% 2.74/1.60 % (893)Instruction limit reached!
% 2.74/1.60 % (893)------------------------------
% 2.74/1.60 % (893)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.74/1.60 % (893)Termination reason: Unknown
% 2.74/1.60 % (893)Termination phase: shuffling
% 2.74/1.60
% 2.74/1.60 % (893)Memory used [KB]: 1535
% 2.74/1.60 % (893)Time elapsed: 0.005 s
% 2.74/1.60 % (893)Instructions burned: 7 (million)
% 2.74/1.60 % (893)------------------------------
% 2.74/1.60 % (893)------------------------------
% 2.78/1.60 % (894)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_777 for (2999ds/16Mi)
% 2.78/1.60 % (895)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_777 for (2999ds/3Mi)
% 2.78/1.60 % (895)Instruction limit reached!
% 2.78/1.60 % (895)------------------------------
% 2.78/1.60 % (895)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.78/1.60 % (895)Termination reason: Unknown
% 2.78/1.60 % (895)Termination phase: shuffling
% 2.78/1.60
% 2.78/1.60 % (895)Memory used [KB]: 1407
% 2.78/1.60 % (895)Time elapsed: 0.003 s
% 2.78/1.60 % (895)Instructions burned: 3 (million)
% 2.78/1.60 % (895)------------------------------
% 2.78/1.60 % (895)------------------------------
% 2.78/1.61 % (881)Instruction limit reached!
% 2.78/1.61 % (881)------------------------------
% 2.78/1.61 % (881)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.78/1.61 % (881)Termination reason: Unknown
% 2.78/1.61 % (881)Termination phase: Saturation
% 2.78/1.61
% 2.78/1.61 % (881)Memory used [KB]: 7036
% 2.78/1.61 % (881)Time elapsed: 0.057 s
% 2.78/1.61 % (881)Instructions burned: 186 (million)
% 2.78/1.61 % (881)------------------------------
% 2.78/1.61 % (881)------------------------------
% 2.78/1.61 % (896)lrs+2_1:1_apa=on:au=on:bd=preordered:cnfonf=off:cs=on:ixr=off:sos=on:i=3:si=on:rtra=on_0 on DTF2THF_777 for (2999ds/3Mi)
% 2.78/1.61 % (894)Instruction limit reached!
% 2.78/1.61 % (894)------------------------------
% 2.78/1.61 % (894)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.78/1.61 % (894)Termination reason: Unknown
% 2.78/1.61 % (894)Termination phase: shuffling
% 2.78/1.61
% 2.78/1.61 % (894)Memory used [KB]: 1663
% 2.78/1.61 % (894)Time elapsed: 0.010 s
% 2.78/1.61 % (894)Instructions burned: 16 (million)
% 2.78/1.61 % (894)------------------------------
% 2.78/1.61 % (894)------------------------------
% 2.78/1.61 % (896)Instruction limit reached!
% 2.78/1.61 % (896)------------------------------
% 2.78/1.61 % (896)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.78/1.61 % (896)Termination reason: Unknown
% 2.78/1.61 % (896)Termination phase: shuffling
% 2.78/1.61
% 2.78/1.61 % (896)Memory used [KB]: 1407
% 2.78/1.61 % (896)Time elapsed: 0.003 s
% 2.78/1.61 % (896)Instructions burned: 3 (million)
% 2.78/1.61 % (896)------------------------------
% 2.78/1.61 % (896)------------------------------
% 2.78/1.61 % (897)lrs+10_1:1_cnfonf=off:cs=on:hud=3:prag=on:sup=off:i=7:si=on:rtra=on_0 on DTF2THF_777 for (2999ds/7Mi)
% 2.78/1.61 % (886)First to succeed.
% 2.78/1.61 % (886)Refutation found. Thanks to Tanya!
% 2.78/1.61 % SZS status Theorem for DTF2THF_777
% 2.78/1.61 % SZS output start Proof for DTF2THF_777
% 2.78/1.61 thf(type_def_5, type, nat: $tType).
% 2.78/1.61 thf(type_def_6, type, real: $tType).
% 2.78/1.61 thf(func_def_0, type, real: $tType).
% 2.78/1.61 thf(func_def_1, type, nat: $tType).
% 2.78/1.61 thf(func_def_2, type, genClo1160817912lle_IC: nat > nat > real > real).
% 2.78/1.61 thf(func_def_3, type, genClo721845095Lambda: real).
% 2.78/1.61 thf(func_def_4, type, genClo1278781456e_beta: real).
% 2.78/1.61 thf(func_def_5, type, genClo1400225944_theta: nat > nat > nat > real).
% 2.78/1.61 thf(func_def_6, type, genClo1015804716orrect: nat > real > $o).
% 2.78/1.61 thf(func_def_7, type, genClo293725281kRead1: (nat > real) > real > (nat > $o) > $o).
% 2.78/1.61 thf(func_def_8, type, genClo293725282kRead2: (nat > real) > (nat > real) > real > (nat > $o) > $o).
% 2.78/1.61 thf(func_def_9, type, genClo208577157axsync: nat > real > $o).
% 2.78/1.61 thf(func_def_10, type, genClo1650508560e_rmax: real).
% 2.78/1.61 thf(func_def_11, type, genClo1651033342e_rmin: real).
% 2.78/1.61 thf(func_def_12, type, genClo1163638703lle_te: nat > nat > real).
% 2.78/1.61 thf(func_def_13, type, abs_abs_real: real > real).
% 2.78/1.61 thf(func_def_14, type, minus_minus_nat: nat > nat > nat).
% 2.78/1.61 thf(func_def_15, type, minus_minus_real: real > real > real).
% 2.78/1.61 thf(func_def_16, type, one_one_nat: nat).
% 2.78/1.61 thf(func_def_17, type, one_one_real: real).
% 2.78/1.61 thf(func_def_18, type, plus_plus_nat: nat > nat > nat).
% 2.78/1.61 thf(func_def_19, type, plus_plus_real: real > real > real).
% 2.78/1.61 thf(func_def_20, type, zero_zero_nat: nat).
% 2.78/1.61 thf(func_def_21, type, zero_zero_real: real).
% 2.78/1.61 thf(func_def_22, type, if_nat: $o > nat > nat > nat).
% 2.78/1.61 thf(func_def_23, type, if_real: $o > real > real > real).
% 2.78/1.61 thf(func_def_24, type, ord_less_eq_nat: nat > nat > $o).
% 2.78/1.61 thf(func_def_25, type, ord_less_eq_real: real > real > $o).
% 2.78/1.61 thf(func_def_26, type, ord_max_nat: nat > nat > nat).
% 2.78/1.61 thf(func_def_27, type, ord_max_real: real > real > real).
% 2.78/1.61 thf(func_def_28, type, delta_S: real).
% 2.78/1.61 thf(func_def_29, type, i: nat).
% 2.78/1.61 thf(func_def_30, type, l: nat).
% 2.78/1.61 thf(func_def_31, type, m: nat).
% 2.78/1.61 thf(func_def_32, type, p: nat).
% 2.78/1.61 thf(func_def_47, type, sP0: real > real > real > $o).
% 2.78/1.61 thf(func_def_48, type, sP1: real > real > real > $o).
% 2.78/1.61 thf(func_def_49, type, sP2: real > real > real > $o).
% 2.78/1.61 thf(func_def_50, type, sK3: (real > real) > real).
% 2.78/1.61 thf(func_def_51, type, sK4: (real > real) > real).
% 2.78/1.61 thf(func_def_52, type, sK5: (real > nat) > real).
% 2.78/1.61 thf(func_def_53, type, sK6: (real > nat) > real).
% 2.78/1.61 thf(func_def_54, type, sK7: (real > nat) > real).
% 2.78/1.61 thf(func_def_55, type, sK8: (real > nat) > real).
% 2.78/1.61 thf(func_def_56, type, sK9: (nat > real) > nat).
% 2.78/1.61 thf(func_def_57, type, sK10: (nat > real) > nat).
% 2.78/1.61 thf(func_def_58, type, sK11: (real > real) > real).
% 2.78/1.61 thf(func_def_59, type, sK12: (real > real) > real).
% 2.78/1.61 thf(func_def_60, type, sK13: (real > real) > real).
% 2.78/1.61 thf(func_def_61, type, sK14: (real > real) > real).
% 2.78/1.61 thf(func_def_62, type, sK15: (nat > real) > nat).
% 2.78/1.61 thf(func_def_63, type, sK16: (nat > real) > nat).
% 2.78/1.61 thf(func_def_64, type, sK17: nat > nat > nat).
% 2.78/1.61 thf(func_def_65, type, sK18: nat > nat > nat).
% 2.78/1.61 thf(func_def_66, type, sK19: (real > nat) > real).
% 2.78/1.61 thf(func_def_67, type, sK20: (real > nat) > real).
% 2.78/1.61 thf(func_def_68, type, sK21: (nat > real) > nat).
% 2.78/1.61 thf(func_def_69, type, sK22: (nat > real) > nat).
% 2.78/1.61 thf(func_def_70, type, sK23: (real > real > $o) > real).
% 2.78/1.61 thf(func_def_71, type, sK24: (real > real > $o) > real).
% 2.78/1.61 thf(func_def_72, type, sK25: (real > real > $o) > real).
% 2.78/1.61 thf(func_def_73, type, sK26: (real > real > $o) > real).
% 2.78/1.61 thf(func_def_74, type, sK27: (real > real) > real).
% 2.78/1.61 thf(func_def_75, type, sK28: (real > real) > real).
% 2.78/1.61 thf(func_def_76, type, sK29: (real > nat) > real).
% 2.78/1.61 thf(func_def_77, type, sK30: (real > nat) > real).
% 2.78/1.61 thf(func_def_78, type, sK31: (nat > real) > nat).
% 2.78/1.61 thf(func_def_79, type, sK32: (nat > real) > nat).
% 2.78/1.61 thf(func_def_81, type, ph34: !>[X0: $tType]:(X0)).
% 2.78/1.61 thf(f1513,plain,(
% 2.78/1.61 $false),
% 2.78/1.61 inference(avatar_sat_refutation,[status(thm)],[f1365,f1366,f1384,f1462,f1512])).
% 2.78/1.61 thf(f1512,plain,(
% 2.78/1.61 spl33_3),
% 2.78/1.61 inference(avatar_contradiction_clause,[status(thm)],[f1511])).
% 2.78/1.61 thf(f1511,plain,(
% 2.78/1.61 $false | spl33_3),
% 2.78/1.61 inference(subsumption_resolution,[status(thm)],[f1510,f1115])).
% 2.78/1.61 thf(f1115,plain,(
% 2.78/1.61 ((genClo1015804716orrect @ m @ (genClo1163638703lle_te @ p @ (plus_plus_nat @ i @ one_one_nat))) = $true)),
% 2.78/1.61 inference(cnf_transformation,[status(thm)],[f471])).
% 2.78/1.61 thf(f471,plain,(
% 2.78/1.61 ((genClo1015804716orrect @ m @ (genClo1163638703lle_te @ p @ (plus_plus_nat @ i @ one_one_nat))) = $true)),
% 2.78/1.61 inference(fool_elimination,[status(thm)],[f470])).
% 2.78/1.61 thf(f470,plain,(
% 2.78/1.61 (genClo1015804716orrect @ m @ (genClo1163638703lle_te @ p @ (plus_plus_nat @ i @ one_one_nat)))),
% 2.78/1.61 inference(rectify,[status(thm)],[f2])).
% 2.78/1.61 thf(f2,axiom,(
% 2.78/1.61 (genClo1015804716orrect @ m @ (genClo1163638703lle_te @ p @ (plus_plus_nat @ i @ one_one_nat)))),
% 2.78/1.61 file('/export/starexec/sandbox2/tmp/tmp.k9LPPfwlmN/DTF2THF_777.p',fact_1_corr__m)).
% 2.78/1.61 thf(f1510,plain,(
% 2.78/1.61 ((genClo1015804716orrect @ m @ (genClo1163638703lle_te @ p @ (plus_plus_nat @ i @ one_one_nat))) != $true) | spl33_3),
% 2.78/1.61 inference(trivial_inequality_removal,[status(thm)],[f1472])).
% 2.78/1.61 thf(f1472,plain,(
% 2.78/1.61 ($true != $true) | ((genClo1015804716orrect @ m @ (genClo1163638703lle_te @ p @ (plus_plus_nat @ i @ one_one_nat))) != $true) | spl33_3),
% 2.78/1.61 inference(superposition,[status(thm)],[f1386,f1239])).
% 2.78/1.61 thf(f1239,plain,(
% 2.78/1.61 ((ord_less_eq_real @ (genClo1163638703lle_te @ m @ i) @ (genClo1163638703lle_te @ p @ (plus_plus_nat @ i @ one_one_nat))) = $true)),
% 2.78/1.61 inference(cnf_transformation,[status(thm)],[f571])).
% 2.78/1.61 thf(f571,plain,(
% 2.78/1.61 ((ord_less_eq_real @ (genClo1163638703lle_te @ m @ i) @ (genClo1163638703lle_te @ p @ (plus_plus_nat @ i @ one_one_nat))) = $true)),
% 2.78/1.61 inference(fool_elimination,[status(thm)],[f570])).
% 2.78/1.61 thf(f570,plain,(
% 2.78/1.61 (ord_less_eq_real @ (genClo1163638703lle_te @ m @ i) @ (genClo1163638703lle_te @ p @ (plus_plus_nat @ i @ one_one_nat)))),
% 2.78/1.61 inference(rectify,[status(thm)],[f7])).
% 2.78/1.61 thf(f7,axiom,(
% 2.78/1.61 (ord_less_eq_real @ (genClo1163638703lle_te @ m @ i) @ (genClo1163638703lle_te @ p @ (plus_plus_nat @ i @ one_one_nat)))),
% 2.78/1.61 file('/export/starexec/sandbox2/tmp/tmp.k9LPPfwlmN/DTF2THF_777.p',fact_6_tmi__le__tp)).
% 2.78/1.61 thf(f1386,plain,(
% 2.78/1.61 ( ! [X0 : real] : (($true != (ord_less_eq_real @ (genClo1163638703lle_te @ m @ i) @ X0)) | ((genClo1015804716orrect @ m @ X0) != $true)) ) | spl33_3),
% 2.78/1.61 inference(trivial_inequality_removal,[status(thm)],[f1385])).
% 2.78/1.61 thf(f1385,plain,(
% 2.78/1.61 ( ! [X0 : real] : (($true != (ord_less_eq_real @ (genClo1163638703lle_te @ m @ i) @ X0)) | ($true != $true) | ((genClo1015804716orrect @ m @ X0) != $true)) ) | spl33_3),
% 2.78/1.61 inference(superposition,[status(thm)],[f1358,f1276])).
% 2.78/1.61 thf(f1276,plain,(
% 2.78/1.61 ( ! [X2 : real,X0 : nat,X1 : real] : (((genClo1015804716orrect @ X0 @ X2) = $true) | ((genClo1015804716orrect @ X0 @ X1) != $true) | ((ord_less_eq_real @ X2 @ X1) != $true)) )),
% 2.78/1.61 inference(cnf_transformation,[status(thm)],[f933])).
% 2.78/1.61 thf(f933,plain,(
% 2.78/1.61 ! [X0 : nat,X1 : real,X2 : real] : (((ord_less_eq_real @ X2 @ X1) != $true) | ((genClo1015804716orrect @ X0 @ X1) != $true) | ((genClo1015804716orrect @ X0 @ X2) = $true))),
% 2.78/1.61 inference(flattening,[status(thm)],[f932])).
% 2.78/1.61 thf(f932,plain,(
% 2.78/1.61 ! [X0 : nat,X1 : real,X2 : real] : (((genClo1015804716orrect @ X0 @ X2) = $true) | (((ord_less_eq_real @ X2 @ X1) != $true) | ((genClo1015804716orrect @ X0 @ X1) != $true)))),
% 2.78/1.62 inference(ennf_transformation,[status(thm)],[f709])).
% 2.78/1.62 thf(f709,plain,(
% 2.78/1.62 ! [X0 : nat,X1 : real,X2 : real] : ((((ord_less_eq_real @ X2 @ X1) = $true) & ((genClo1015804716orrect @ X0 @ X1) = $true)) => ((genClo1015804716orrect @ X0 @ X2) = $true))),
% 2.78/1.62 inference(fool_elimination,[status(thm)],[f708])).
% 2.78/1.62 thf(f708,plain,(
% 2.78/1.62 ! [X0 : nat,X1 : real,X2 : real] : (((ord_less_eq_real @ X2 @ X1) & (genClo1015804716orrect @ X0 @ X1)) => (genClo1015804716orrect @ X0 @ X2))),
% 2.78/1.62 inference(rectify,[status(thm)],[f1])).
% 2.78/1.62 thf(f1,axiom,(
% 2.78/1.62 ! [X0 : nat,X2 : real,X1 : real] : (((ord_less_eq_real @ X1 @ X2) & (genClo1015804716orrect @ X0 @ X2)) => (genClo1015804716orrect @ X0 @ X1))),
% 2.78/1.62 file('/export/starexec/sandbox2/tmp/tmp.k9LPPfwlmN/DTF2THF_777.p',fact_0_correct__closed)).
% 2.78/1.62 thf(f1358,plain,(
% 2.78/1.62 ((genClo1015804716orrect @ m @ (genClo1163638703lle_te @ m @ i)) != $true) | spl33_3),
% 2.78/1.62 inference(avatar_component_clause,[status(thm)],[f1356])).
% 2.78/1.62 thf(f1356,definition,(
% 2.78/1.62 spl33_3 <=> ((genClo1015804716orrect @ m @ (genClo1163638703lle_te @ m @ i)) = $true)),
% 2.78/1.62 introduced(definition,[new_symbols(naming,[spl33_3])],[avatar_definition])).
% 2.78/1.62 thf(f1462,plain,(
% 2.78/1.62 spl33_1),
% 2.78/1.62 inference(avatar_contradiction_clause,[status(thm)],[f1461])).
% 2.78/1.62 thf(f1461,plain,(
% 2.78/1.62 $false | spl33_1),
% 2.78/1.62 inference(subsumption_resolution,[status(thm)],[f1454,f1115])).
% 2.78/1.62 thf(f1454,plain,(
% 2.78/1.62 ((genClo1015804716orrect @ m @ (genClo1163638703lle_te @ p @ (plus_plus_nat @ i @ one_one_nat))) != $true) | spl33_1),
% 2.78/1.62 inference(trivial_inequality_removal,[status(thm)],[f1421])).
% 2.78/1.62 thf(f1421,plain,(
% 2.78/1.62 ($true != $true) | ((genClo1015804716orrect @ m @ (genClo1163638703lle_te @ p @ (plus_plus_nat @ i @ one_one_nat))) != $true) | spl33_1),
% 2.78/1.62 inference(superposition,[status(thm)],[f1368,f1219])).
% 2.78/1.62 thf(f1219,plain,(
% 2.78/1.62 ((ord_less_eq_real @ (genClo1163638703lle_te @ l @ i) @ (genClo1163638703lle_te @ p @ (plus_plus_nat @ i @ one_one_nat))) = $true)),
% 2.78/1.62 inference(cnf_transformation,[status(thm)],[f823])).
% 2.78/1.62 thf(f823,plain,(
% 2.78/1.62 ((ord_less_eq_real @ (genClo1163638703lle_te @ l @ i) @ (genClo1163638703lle_te @ p @ (plus_plus_nat @ i @ one_one_nat))) = $true)),
% 2.78/1.62 inference(fool_elimination,[status(thm)],[f822])).
% 2.78/1.62 thf(f822,plain,(
% 2.78/1.62 (ord_less_eq_real @ (genClo1163638703lle_te @ l @ i) @ (genClo1163638703lle_te @ p @ (plus_plus_nat @ i @ one_one_nat)))),
% 2.78/1.62 inference(rectify,[status(thm)],[f8])).
% 2.78/1.62 thf(f8,axiom,(
% 2.78/1.62 (ord_less_eq_real @ (genClo1163638703lle_te @ l @ i) @ (genClo1163638703lle_te @ p @ (plus_plus_nat @ i @ one_one_nat)))),
% 2.78/1.62 file('/export/starexec/sandbox2/tmp/tmp.k9LPPfwlmN/DTF2THF_777.p',fact_7_tli__le__tp)).
% 2.78/1.62 thf(f1368,plain,(
% 2.78/1.62 ( ! [X0 : real] : (((ord_less_eq_real @ (genClo1163638703lle_te @ l @ i) @ X0) != $true) | ((genClo1015804716orrect @ m @ X0) != $true)) ) | spl33_1),
% 2.78/1.62 inference(trivial_inequality_removal,[status(thm)],[f1367])).
% 2.78/1.62 thf(f1367,plain,(
% 2.78/1.62 ( ! [X0 : real] : (((ord_less_eq_real @ (genClo1163638703lle_te @ l @ i) @ X0) != $true) | ((genClo1015804716orrect @ m @ X0) != $true) | ($true != $true)) ) | spl33_1),
% 2.78/1.62 inference(superposition,[status(thm)],[f1349,f1276])).
% 2.78/1.62 thf(f1349,plain,(
% 2.78/1.62 ((genClo1015804716orrect @ m @ (genClo1163638703lle_te @ l @ i)) != $true) | spl33_1),
% 2.78/1.62 inference(avatar_component_clause,[status(thm)],[f1347])).
% 2.78/1.62 thf(f1347,definition,(
% 2.78/1.62 spl33_1 <=> ((genClo1015804716orrect @ m @ (genClo1163638703lle_te @ l @ i)) = $true)),
% 2.78/1.62 introduced(definition,[new_symbols(naming,[spl33_1])],[avatar_definition])).
% 2.78/1.62 thf(f1384,plain,(
% 2.78/1.62 spl33_2 | spl33_4),
% 2.78/1.62 inference(avatar_split_clause,[status(thm)],[f1379,f1360,f1351])).
% 2.78/1.62 thf(f1351,definition,(
% 2.78/1.62 spl33_2 <=> ((ord_less_eq_real @ (genClo1163638703lle_te @ m @ i) @ (genClo1163638703lle_te @ l @ i)) = $true)),
% 2.78/1.62 introduced(definition,[new_symbols(naming,[spl33_2])],[avatar_definition])).
% 2.78/1.62 thf(f1360,definition,(
% 2.78/1.62 spl33_4 <=> ((ord_less_eq_real @ (genClo1163638703lle_te @ l @ i) @ (genClo1163638703lle_te @ m @ i)) = $true)),
% 2.78/1.62 introduced(definition,[new_symbols(naming,[spl33_4])],[avatar_definition])).
% 2.78/1.62 thf(f1379,plain,(
% 2.78/1.62 ((ord_less_eq_real @ (genClo1163638703lle_te @ m @ i) @ (genClo1163638703lle_te @ l @ i)) = $true) | spl33_4),
% 2.78/1.62 inference(trivial_inequality_removal,[status(thm)],[f1372])).
% 2.78/1.62 thf(f1372,plain,(
% 2.78/1.62 ((ord_less_eq_real @ (genClo1163638703lle_te @ m @ i) @ (genClo1163638703lle_te @ l @ i)) = $true) | ($true != $true) | spl33_4),
% 2.78/1.62 inference(superposition,[status(thm)],[f1362,f1225])).
% 2.78/1.62 thf(f1225,plain,(
% 2.78/1.62 ( ! [X0 : real,X1 : real] : (($true = (ord_less_eq_real @ X1 @ X0)) | ((ord_less_eq_real @ X0 @ X1) = $true)) )),
% 2.78/1.62 inference(cnf_transformation,[status(thm)],[f861])).
% 2.78/1.62 thf(f861,plain,(
% 2.78/1.62 ! [X0 : real,X1 : real] : (((ord_less_eq_real @ X0 @ X1) = $true) | ($true = (ord_less_eq_real @ X1 @ X0)))),
% 2.78/1.62 inference(fool_elimination,[status(thm)],[f860])).
% 2.78/1.62 thf(f860,plain,(
% 2.78/1.62 ! [X0 : real,X1 : real] : ((ord_less_eq_real @ X0 @ X1) | (ord_less_eq_real @ X1 @ X0))),
% 2.78/1.62 inference(rectify,[status(thm)],[f49])).
% 2.78/1.62 thf(f49,axiom,(
% 2.78/1.62 ! [X10 : real,X11 : real] : ((ord_less_eq_real @ X10 @ X11) | (ord_less_eq_real @ X11 @ X10))),
% 2.78/1.62 file('/export/starexec/sandbox2/tmp/tmp.k9LPPfwlmN/DTF2THF_777.p',fact_48_linear)).
% 2.78/1.62 thf(f1362,plain,(
% 2.78/1.62 ((ord_less_eq_real @ (genClo1163638703lle_te @ l @ i) @ (genClo1163638703lle_te @ m @ i)) != $true) | spl33_4),
% 2.78/1.62 inference(avatar_component_clause,[status(thm)],[f1360])).
% 2.78/1.62 thf(f1366,plain,(
% 2.78/1.62 ~spl33_3 | ~spl33_4),
% 2.78/1.62 inference(avatar_split_clause,[status(thm)],[f1342,f1360,f1356])).
% 2.78/1.62 thf(f1342,plain,(
% 2.78/1.62 ((ord_less_eq_real @ (genClo1163638703lle_te @ l @ i) @ (genClo1163638703lle_te @ m @ i)) != $true) | ((genClo1015804716orrect @ m @ (genClo1163638703lle_te @ m @ i)) != $true)),
% 2.78/1.62 inference(superposition,[status(thm)],[f1139,f1162])).
% 2.78/1.62 thf(f1162,plain,(
% 2.78/1.62 ( ! [X0 : real,X1 : real] : (((ord_max_real @ X1 @ X0) = X0) | ($true != (ord_less_eq_real @ X1 @ X0))) )),
% 2.78/1.62 inference(cnf_transformation,[status(thm)],[f923])).
% 2.78/1.62 thf(f923,plain,(
% 2.78/1.62 ! [X0 : real,X1 : real] : (((ord_max_real @ X1 @ X0) = X0) | ($true != (ord_less_eq_real @ X1 @ X0)))),
% 2.78/1.62 inference(ennf_transformation,[status(thm)],[f833])).
% 2.78/1.62 thf(f833,plain,(
% 2.78/1.62 ! [X0 : real,X1 : real] : (($true = (ord_less_eq_real @ X1 @ X0)) => ((ord_max_real @ X1 @ X0) = X0))),
% 2.78/1.62 inference(fool_elimination,[status(thm)],[f832])).
% 2.78/1.62 thf(f832,plain,(
% 2.78/1.62 ! [X0 : real,X1 : real] : ((ord_less_eq_real @ X1 @ X0) => ((ord_max_real @ X1 @ X0) = X0))),
% 2.78/1.62 inference(rectify,[status(thm)],[f101])).
% 2.78/1.62 thf(f101,axiom,(
% 2.78/1.62 ! [X4 : real,X3 : real] : ((ord_less_eq_real @ X3 @ X4) => ((ord_max_real @ X3 @ X4) = X4))),
% 2.78/1.62 file('/export/starexec/sandbox2/tmp/tmp.k9LPPfwlmN/DTF2THF_777.p',fact_100_max_Oabsorb2)).
% 2.78/1.62 thf(f1139,plain,(
% 2.78/1.62 ((genClo1015804716orrect @ m @ (ord_max_real @ (genClo1163638703lle_te @ l @ i) @ (genClo1163638703lle_te @ m @ i))) != $true)),
% 2.78/1.62 inference(cnf_transformation,[status(thm)],[f862])).
% 2.78/1.62 thf(f862,plain,(
% 2.78/1.62 ((genClo1015804716orrect @ m @ (ord_max_real @ (genClo1163638703lle_te @ l @ i) @ (genClo1163638703lle_te @ m @ i))) != $true)),
% 2.78/1.62 inference(flattening,[status(thm)],[f857])).
% 2.78/1.62 thf(f857,plain,(
% 2.78/1.62 ~((genClo1015804716orrect @ m @ (ord_max_real @ (genClo1163638703lle_te @ l @ i) @ (genClo1163638703lle_te @ m @ i))) = $true)),
% 2.78/1.62 inference(fool_elimination,[status(thm)],[f856])).
% 2.78/1.62 thf(f856,plain,(
% 2.78/1.62 ~(genClo1015804716orrect @ m @ (ord_max_real @ (genClo1163638703lle_te @ l @ i) @ (genClo1163638703lle_te @ m @ i)))),
% 2.78/1.62 inference(rectify,[status(thm)],[f360])).
% 2.78/1.62 thf(f360,negated_conjecture,(
% 2.78/1.62 ~(genClo1015804716orrect @ m @ (ord_max_real @ (genClo1163638703lle_te @ l @ i) @ (genClo1163638703lle_te @ m @ i)))),
% 2.78/1.62 inference(negated_conjecture,[status(cth)],[f359])).
% 2.78/1.62 thf(f359,conjecture,(
% 2.78/1.62 (genClo1015804716orrect @ m @ (ord_max_real @ (genClo1163638703lle_te @ l @ i) @ (genClo1163638703lle_te @ m @ i)))),
% 2.78/1.62 file('/export/starexec/sandbox2/tmp/tmp.k9LPPfwlmN/DTF2THF_777.p',conj_0)).
% 2.78/1.62 thf(f1365,plain,(
% 2.78/1.62 ~spl33_2 | ~spl33_1),
% 2.78/1.62 inference(avatar_split_clause,[status(thm)],[f1339,f1347,f1351])).
% 2.78/1.62 thf(f1339,plain,(
% 2.78/1.62 ((ord_less_eq_real @ (genClo1163638703lle_te @ m @ i) @ (genClo1163638703lle_te @ l @ i)) != $true) | ((genClo1015804716orrect @ m @ (genClo1163638703lle_te @ l @ i)) != $true)),
% 2.78/1.62 inference(superposition,[status(thm)],[f1139,f1104])).
% 2.78/1.62 thf(f1104,plain,(
% 2.78/1.62 ( ! [X0 : real,X1 : real] : (((ord_max_real @ X0 @ X1) = X0) | ($true != (ord_less_eq_real @ X1 @ X0))) )),
% 2.78/1.62 inference(cnf_transformation,[status(thm)],[f965])).
% 2.78/1.62 thf(f965,plain,(
% 2.78/1.62 ! [X0 : real,X1 : real] : (((ord_max_real @ X0 @ X1) = X0) | ($true != (ord_less_eq_real @ X1 @ X0)))),
% 2.78/1.62 inference(ennf_transformation,[status(thm)],[f603])).
% 2.78/1.62 thf(f603,plain,(
% 2.78/1.62 ! [X0 : real,X1 : real] : (($true = (ord_less_eq_real @ X1 @ X0)) => ((ord_max_real @ X0 @ X1) = X0))),
% 2.78/1.62 inference(fool_elimination,[status(thm)],[f602])).
% 2.78/1.62 thf(f602,plain,(
% 2.78/1.62 ! [X0 : real,X1 : real] : ((ord_less_eq_real @ X1 @ X0) => ((ord_max_real @ X0 @ X1) = X0))),
% 2.78/1.62 inference(rectify,[status(thm)],[f103])).
% 2.78/1.62 thf(f103,axiom,(
% 2.78/1.62 ! [X3 : real,X4 : real] : ((ord_less_eq_real @ X4 @ X3) => ((ord_max_real @ X3 @ X4) = X3))),
% 2.78/1.62 file('/export/starexec/sandbox2/tmp/tmp.k9LPPfwlmN/DTF2THF_777.p',fact_102_max_Oabsorb1)).
% 2.78/1.62 % SZS output end Proof for DTF2THF_777
% 2.78/1.62 % (886)------------------------------
% 2.78/1.62 % (886)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.78/1.62 % (886)Termination reason: Refutation
% 2.78/1.62
% 2.78/1.62 % (886)Memory used [KB]: 6780
% 2.78/1.62 % (886)Time elapsed: 0.058 s
% 2.78/1.62 % (886)Instructions burned: 112 (million)
% 2.78/1.62 % (886)------------------------------
% 2.78/1.62 % (886)------------------------------
% 2.78/1.62 % (880)Success in time 0.083 s
%------------------------------------------------------------------------------