%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : ITP130^1 : 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 : n009.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 08:55:47 AM UTC 2026
% Result : Theorem 2.50s 1.42s
% Output : Refutation 2.50s
% Verified :
% SZS Type : Refutation
% Derivation depth : 8
% Number of leaves : 5
% Syntax : Number of formulae : 25 ( 22 unt; 0 typ; 0 def)
% Number of atoms : 56 ( 28 equ; 0 cnn)
% Maximal formula atoms : 4 ( 2 avg)
% Number of connectives : 152 ( 10 ~; 3 |; 1 &; 137 @)
% ( 1 <=>; 0 =>; 0 <=; 0 <~>)
% Maximal formula depth : 7 ( 3 avg)
% Number of types : 4 ( 3 usr)
% Number of type conns : 0 ( 0 >; 0 *; 0 +; 0 <<)
% Number of symbols : 81 ( 78 usr; 12 con; 0-2 aty)
% Number of variables : 23 ( 0 ^; 23 !; 0 ?; 23 :)
% Comments :
%------------------------------------------------------------------------------
thf(type_def_5,type,
int: $tType ).
thf(type_def_6,type,
nat: $tType ).
thf(type_def_7,type,
real: $tType ).
thf(func_def_0,type,
real: $tType ).
thf(func_def_1,type,
nat: $tType ).
thf(func_def_2,type,
int: $tType ).
thf(func_def_3,type,
zero_zero_int: int ).
thf(func_def_4,type,
zero_zero_nat: nat ).
thf(func_def_5,type,
zero_zero_real: real ).
thf(func_def_6,type,
ring_1_of_int_int: int > int ).
thf(func_def_7,type,
ring_1_of_int_real: int > real ).
thf(func_def_8,type,
suc: nat > nat ).
thf(func_def_9,type,
nthRoo1352626670d_root: nat > nat > $o ).
thf(func_def_10,type,
ord_less_int: int > int > $o ).
thf(func_def_11,type,
ord_less_nat: nat > nat > $o ).
thf(func_def_12,type,
ord_less_real: real > real > $o ).
thf(func_def_13,type,
ord_less_eq_int: int > int > $o ).
thf(func_def_14,type,
ord_less_eq_nat: nat > nat > $o ).
thf(func_def_15,type,
ord_less_eq_real: real > real > $o ).
thf(func_def_16,type,
power_power_int: int > nat > int ).
thf(func_def_17,type,
power_power_nat: nat > nat > nat ).
thf(func_def_18,type,
power_power_real: real > nat > real ).
thf(func_def_19,type,
divide_divide_int: int > int > int ).
thf(func_def_20,type,
divide_divide_nat: nat > nat > nat ).
thf(func_def_21,type,
divide_divide_real: real > real > real ).
thf(func_def_22,type,
ny: real ).
thf(func_def_23,type,
na: int ).
thf(func_def_24,type,
p: nat ).
thf(func_def_25,type,
pm: nat ).
thf(func_def_26,type,
x: int ).
thf(func_def_27,type,
y: int ).
thf(func_def_28,type,
ya: int ).
thf(func_def_39,type,
sK0: ( nat > $o ) > nat ).
thf(func_def_40,type,
sK1: ( nat > real ) > nat ).
thf(func_def_41,type,
sK2: real > nat > real ).
thf(func_def_42,type,
sK3: ( nat > int ) > nat ).
thf(func_def_43,type,
sK4: nat > nat ).
thf(func_def_44,type,
sK5: ( nat > int ) > nat ).
thf(func_def_45,type,
sK6: real > real ).
thf(func_def_46,type,
sK7: real > int ).
thf(func_def_47,type,
sK8: real > int ).
thf(func_def_48,type,
sK9: ( nat > int ) > nat ).
thf(func_def_49,type,
sK10: ( nat > $o ) > nat ).
thf(func_def_50,type,
sK11: nat > nat ).
thf(func_def_51,type,
sK12: real > real ).
thf(func_def_52,type,
sK13: ( nat > int ) > nat ).
thf(func_def_53,type,
sK14: nat > real > real ).
thf(func_def_54,type,
sK15: nat > ( nat > $o ) > nat ).
thf(func_def_55,type,
sK16: nat > ( nat > $o ) > nat ).
thf(func_def_56,type,
sK17: ( nat > real ) > nat ).
thf(func_def_57,type,
sK18: nat > ( nat > $o ) > nat ).
thf(func_def_58,type,
sK19: nat > ( nat > $o ) > nat ).
thf(func_def_59,type,
sK20: nat > nat > nat ).
thf(func_def_60,type,
sK21: ( nat > $o ) > nat ).
thf(func_def_61,type,
sK22: ( nat > nat ) > nat ).
thf(func_def_62,type,
sK23: nat > ( nat > $o ) > nat ).
thf(func_def_63,type,
sK24: nat > ( nat > $o ) > nat ).
thf(func_def_64,type,
sK25: real > nat > real ).
thf(func_def_65,type,
sK26: nat > nat ).
thf(func_def_66,type,
sK27: nat > nat > nat ).
thf(func_def_67,type,
sK28: ( nat > $o ) > nat ).
thf(func_def_68,type,
sK29: nat > nat ).
thf(func_def_69,type,
sK30: nat > nat > nat ).
thf(func_def_70,type,
sK31: ( nat > $o ) > nat > nat ).
thf(func_def_71,type,
sK32: ( nat > $o ) > nat > nat ).
thf(func_def_72,type,
sK33: ( nat > $o ) > nat ).
thf(func_def_73,type,
sK34: nat > ( nat > $o ) > nat ).
thf(func_def_74,type,
sK35: nat > ( nat > $o ) > nat ).
thf(func_def_75,type,
sK36: ( nat > $o ) > nat ).
thf(func_def_76,type,
sK37: ( nat > nat > $o ) > nat ).
thf(func_def_77,type,
sK38: ( nat > nat > $o ) > nat ).
thf(func_def_78,type,
sK39: ( nat > nat > $o ) > nat ).
thf(func_def_79,type,
sK40: ( nat > nat > $o ) > nat ).
thf(func_def_80,type,
sK41: nat > nat > nat ).
thf(func_def_81,type,
sK42: ( nat > real ) > nat ).
thf(func_def_82,type,
sK43: ( nat > nat ) > nat ).
thf(func_def_83,type,
sK44: real > int ).
thf(func_def_84,type,
sK45: ( nat > real ) > nat ).
thf(func_def_85,type,
sK46: ( nat > $o ) > nat ).
thf(func_def_86,type,
sK47: ( nat > nat > $o ) > nat ).
thf(func_def_87,type,
sK48: ( nat > nat > $o ) > nat ).
thf(func_def_88,type,
sK49: ( nat > nat > $o ) > nat ).
thf(func_def_89,type,
sK50: ( nat > nat > $o ) > nat ).
thf(func_def_91,type,
ph52:
!>[X0: $tType] : X0 ).
thf(f1678,plain,
$false,
inference(subsumption_resolution,[status(thm)],[f1666,f1570]) ).
thf(f1570,plain,
( ( ord_less_int @ na @ ( power_power_int @ ya @ ( suc @ pm ) ) )
= $true ),
inference(definition_unfolding,[status(thm)],[f1358,f1496]) ).
thf(f1496,plain,
( p
= ( suc @ pm ) ),
inference(cnf_transformation,[status(thm)],[f87]) ).
thf(f87,axiom,
( p
= ( suc @ pm ) ),
file('/export/starexec/sandbox2/tmp/tmp.paLrYnjNlH/DTF2THF_32705.p',fact_86_p) ).
thf(f1358,plain,
( ( ord_less_int @ na @ ( power_power_int @ ya @ p ) )
= $true ),
inference(cnf_transformation,[status(thm)],[f627]) ).
thf(f627,plain,
( ( ord_less_int @ na @ ( power_power_int @ ya @ p ) )
= $true ),
inference(fool_elimination,[status(thm)],[f626]) ).
thf(f626,plain,
ord_less_int @ na @ ( power_power_int @ ya @ p ),
inference(rectify,[status(thm)],[f82]) ).
thf(f82,axiom,
ord_less_int @ na @ ( power_power_int @ ya @ p ),
file('/export/starexec/sandbox2/tmp/tmp.paLrYnjNlH/DTF2THF_32705.p',fact_81_yyn) ).
thf(f1666,plain,
( ( ord_less_int @ na @ ( power_power_int @ ya @ ( suc @ pm ) ) )
!= $true ),
inference(superposition,[status(thm)],[f1665,f1290]) ).
thf(f1290,plain,
! [X0: int,X1: int] :
( ( ord_less_real @ ( ring_1_of_int_real @ X1 ) @ ( ring_1_of_int_real @ X0 ) )
= ( ord_less_int @ X1 @ X0 ) ),
inference(cnf_transformation,[status(thm)],[f830]) ).
thf(f830,plain,
! [X0: int,X1: int] :
( ( ord_less_real @ ( ring_1_of_int_real @ X1 ) @ ( ring_1_of_int_real @ X0 ) )
= ( ord_less_int @ X1 @ X0 ) ),
inference(rectify,[status(thm)],[f12]) ).
thf(f12,axiom,
! [X3: int,X1: int] :
( ( ord_less_real @ ( ring_1_of_int_real @ X1 ) @ ( ring_1_of_int_real @ X3 ) )
= ( ord_less_int @ X1 @ X3 ) ),
file('/export/starexec/sandbox2/tmp/tmp.paLrYnjNlH/DTF2THF_32705.p',fact_11_of__int__less__iff) ).
thf(f1665,plain,
( $true
!= ( ord_less_real @ ( ring_1_of_int_real @ na ) @ ( ring_1_of_int_real @ ( power_power_int @ ya @ ( suc @ pm ) ) ) ) ),
inference(forward_demodulation,[status(thm)],[f1310,f1625]) ).
thf(f1625,plain,
! [X2: nat,X1: int] :
( ( ring_1_of_int_real @ ( power_power_int @ X1 @ X2 ) )
= ( power_power_real @ ( ring_1_of_int_real @ X1 ) @ X2 ) ),
inference(equality_resolution,[status(thm)],[f1238]) ).
thf(f1238,plain,
! [X2: nat,X0: int,X1: int] :
( ( ( ring_1_of_int_real @ X0 )
= ( power_power_real @ ( ring_1_of_int_real @ X1 ) @ X2 ) )
| ( ( power_power_int @ X1 @ X2 )
!= X0 ) ),
inference(cnf_transformation,[status(thm)],[f1060]) ).
thf(f1060,plain,
! [X0: int,X1: int,X2: nat] :
( ( ( ( power_power_int @ X1 @ X2 )
= X0 )
| ( ( ring_1_of_int_real @ X0 )
!= ( power_power_real @ ( ring_1_of_int_real @ X1 ) @ X2 ) ) )
& ( ( ( ring_1_of_int_real @ X0 )
= ( power_power_real @ ( ring_1_of_int_real @ X1 ) @ X2 ) )
| ( ( power_power_int @ X1 @ X2 )
!= X0 ) ) ),
inference(nnf_transformation,[status(thm)],[f727]) ).
thf(f727,plain,
! [X0: int,X1: int,X2: nat] :
( ( ( power_power_int @ X1 @ X2 )
= X0 )
<=> ( ( ring_1_of_int_real @ X0 )
= ( power_power_real @ ( ring_1_of_int_real @ X1 ) @ X2 ) ) ),
inference(fool_elimination,[status(thm)],[f726]) ).
thf(f726,plain,
! [X0: int,X1: int,X2: nat] :
( ( ( power_power_int @ X1 @ X2 )
= X0 )
= ( ( ring_1_of_int_real @ X0 )
= ( power_power_real @ ( ring_1_of_int_real @ X1 ) @ X2 ) ) ),
inference(rectify,[status(thm)],[f8]) ).
thf(f8,axiom,
! [X2: int,X0: int,X1: nat] :
( ( ( power_power_int @ X0 @ X1 )
= X2 )
= ( ( power_power_real @ ( ring_1_of_int_real @ X0 ) @ X1 )
= ( ring_1_of_int_real @ X2 ) ) ),
file('/export/starexec/sandbox2/tmp/tmp.paLrYnjNlH/DTF2THF_32705.p',fact_7_of__int__eq__of__int__power__cancel__iff) ).
thf(f1310,plain,
( ( ord_less_real @ ( ring_1_of_int_real @ na ) @ ( power_power_real @ ( ring_1_of_int_real @ ya ) @ ( suc @ pm ) ) )
!= $true ),
inference(cnf_transformation,[status(thm)],[f824]) ).
thf(f824,plain,
( ( ord_less_real @ ( ring_1_of_int_real @ na ) @ ( power_power_real @ ( ring_1_of_int_real @ ya ) @ ( suc @ pm ) ) )
!= $true ),
inference(flattening,[status(thm)],[f358]) ).
thf(f358,plain,
( ( ord_less_real @ ( ring_1_of_int_real @ na ) @ ( power_power_real @ ( ring_1_of_int_real @ ya ) @ ( suc @ pm ) ) )
!= $true ),
inference(fool_elimination,[status(thm)],[f357]) ).
thf(f357,plain,
~ ( ord_less_real @ ( ring_1_of_int_real @ na ) @ ( power_power_real @ ( ring_1_of_int_real @ ya ) @ ( suc @ pm ) ) ),
inference(rectify,[status(thm)],[f321]) ).
thf(f321,negated_conjecture,
~ ( ord_less_real @ ( ring_1_of_int_real @ na ) @ ( power_power_real @ ( ring_1_of_int_real @ ya ) @ ( suc @ pm ) ) ),
inference(negated_conjecture,[status(cth)],[f320]) ).
thf(f320,conjecture,
ord_less_real @ ( ring_1_of_int_real @ na ) @ ( power_power_real @ ( ring_1_of_int_real @ ya ) @ ( suc @ pm ) ),
file('/export/starexec/sandbox2/tmp/tmp.paLrYnjNlH/DTF2THF_32705.p',conj_0) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.11 % Problem : ITP130^1 : 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
% 0.12/0.30 % Computer : n009.cluster.edu
% 0.12/0.30 % Model : x86_64 x86_64
% 0.12/0.30 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.12/0.30 % Memory : 8042.1875MB
% 0.12/0.30 % OS : Linux 3.10.0-693.el7.x86_64
% 0.12/0.30 % CPULimit : 300
% 0.12/0.30 % WCLimit : 300
% 0.12/0.30 % DateTime : Tue Feb 24 08:27:10 EST 2026
% 0.12/0.30 % CPUTime :
% 0.12/0.30 Running /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.18/0.38 ---- Original DTF file ---
% 0.18/0.38 thf(spec,logic,$$dhol).
% 0.18/0.38 %------------------------------------------------------------------------------
% 0.18/0.38 % File : ITP130^1 : TPTP v9.2.1. Released v7.5.0.
% 0.18/0.38 % Domain : Interactive Theorem Proving
% 0.18/0.38 % Problem : Sledgehammer NthRoot_Impl problem prob_360__3277026_1
% 0.18/0.38 % Version : Especial.
% 0.18/0.38 % English :
% 0.18/0.38
% 0.18/0.38 % Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% 0.18/0.38 % : [Des21] Desharnais (2021), Email to Geoff Sutcliffe
% 0.18/0.38 % Source : [Des21]
% 0.18/0.38 % Names : NthRoot_Impl/prob_360__3277026_1 [Des21]
% 0.18/0.38
% 0.18/0.38 % Status : Theorem
% 0.18/0.38 % Rating : 0.11 v9.1.0, 0.12 v9.0.0, 0.20 v8.2.0, 0.15 v8.1.0, 0.18 v7.5.0
% 0.18/0.38 % Syntax : Number of formulae : 349 ( 158 unt; 29 typ; 0 def)
% 0.18/0.38 % Number of atoms : 838 ( 285 equ; 0 cnn)
% 0.18/0.38 % Maximal formula atoms : 7 ( 2 avg)
% 0.18/0.38 % Number of connectives : 2338 ( 77 ~; 22 |; 38 &;1860 @)
% 0.18/0.38 % ( 0 <=>; 341 =>; 0 <=; 0 <~>)
% 0.18/0.38 % Maximal formula depth : 16 ( 6 avg)
% 0.18/0.38 % Number of types : 4 ( 3 usr)
% 0.18/0.38 % Number of type conns : 67 ( 67 >; 0 *; 0 +; 0 <<)
% 0.18/0.38 % Number of symbols : 27 ( 26 usr; 10 con; 0-2 aty)
% 0.18/0.38 % Number of variables : 663 ( 7 ^; 631 !; 25 ?; 663 :)
% 0.18/0.38 % SPC : TH0_THM_EQU_NAR
% 0.18/0.38
% 0.18/0.38 % Comments : This file was generated by Sledgehammer 2021-02-23 15:41:39.250
% 0.18/0.38 %------------------------------------------------------------------------------
% 0.18/0.38 % Could-be-implicit typings (3)
% 0.18/0.38 thf(ty_n_t__Real__Oreal,type,
% 0.18/0.38 real: $tType ).
% 0.18/0.38
% 0.18/0.38 thf(ty_n_t__Nat__Onat,type,
% 0.18/0.38 nat: $tType ).
% 0.18/0.38
% 0.18/0.38 thf(ty_n_t__Int__Oint,type,
% 0.18/0.38 int: $tType ).
% 0.18/0.38
% 0.18/0.38 % Explicit typings (26)
% 0.18/0.38 thf(sy_c_Groups_Ozero__class_Ozero_001t__Int__Oint,type,
% 0.18/0.38 zero_zero_int: int ).
% 0.18/0.38
% 0.18/0.38 thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
% 0.18/0.38 zero_zero_nat: nat ).
% 0.18/0.38
% 0.18/0.38 thf(sy_c_Groups_Ozero__class_Ozero_001t__Real__Oreal,type,
% 0.18/0.38 zero_zero_real: real ).
% 0.18/0.38
% 0.18/0.38 thf(sy_c_Int_Oring__1__class_Oof__int_001t__Int__Oint,type,
% 0.18/0.38 ring_1_of_int_int: int > int ).
% 0.18/0.38
% 0.18/0.38 thf(sy_c_Int_Oring__1__class_Oof__int_001t__Real__Oreal,type,
% 0.18/0.38 ring_1_of_int_real: int > real ).
% 0.18/0.38
% 0.18/0.38 thf(sy_c_Nat_OSuc,type,
% 0.18/0.38 suc: nat > nat ).
% 0.18/0.38
% 0.18/0.38 thf(sy_c_NthRoot__Impl__Mirabelle__bcfbdygymo_Ofixed__root,type,
% 0.18/0.38 nthRoo1352626670d_root: nat > nat > $o ).
% 0.18/0.38
% 0.18/0.38 thf(sy_c_Orderings_Oord__class_Oless_001t__Int__Oint,type,
% 0.18/0.38 ord_less_int: int > int > $o ).
% 0.18/0.38
% 0.18/0.38 thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
% 0.18/0.38 ord_less_nat: nat > nat > $o ).
% 0.18/0.38
% 0.18/0.38 thf(sy_c_Orderings_Oord__class_Oless_001t__Real__Oreal,type,
% 0.18/0.38 ord_less_real: real > real > $o ).
% 0.18/0.38
% 0.18/0.38 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Int__Oint,type,
% 0.18/0.38 ord_less_eq_int: int > int > $o ).
% 0.18/0.38
% 0.18/0.38 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
% 0.18/0.38 ord_less_eq_nat: nat > nat > $o ).
% 0.18/0.38
% 0.18/0.38 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Real__Oreal,type,
% 0.18/0.38 ord_less_eq_real: real > real > $o ).
% 0.18/0.38
% 0.18/0.38 thf(sy_c_Power_Opower__class_Opower_001t__Int__Oint,type,
% 0.18/0.38 power_power_int: int > nat > int ).
% 0.18/0.38
% 0.18/0.38 thf(sy_c_Power_Opower__class_Opower_001t__Nat__Onat,type,
% 0.18/0.38 power_power_nat: nat > nat > nat ).
% 0.18/0.38
% 0.18/0.38 thf(sy_c_Power_Opower__class_Opower_001t__Real__Oreal,type,
% 0.18/0.38 power_power_real: real > nat > real ).
% 0.18/0.38
% 0.18/0.38 thf(sy_c_Rings_Odivide__class_Odivide_001t__Int__Oint,type,
% 0.18/0.38 divide_divide_int: int > int > int ).
% 0.18/0.38
% 0.18/0.38 thf(sy_c_Rings_Odivide__class_Odivide_001t__Nat__Onat,type,
% 0.18/0.38 divide_divide_nat: nat > nat > nat ).
% 0.18/0.38
% 0.18/0.38 thf(sy_c_Rings_Odivide__class_Odivide_001t__Real__Oreal,type,
% 0.18/0.38 divide_divide_real: real > real > real ).
% 0.18/0.38
% 0.18/0.38 thf(sy_v_NY____,type,
% 0.18/0.38 ny: real ).
% 0.18/0.38
% 0.18/0.38 thf(sy_v_na____,type,
% 0.18/0.38 na: int ).
% 0.18/0.38
% 0.18/0.38 thf(sy_v_p,type,
% 0.18/0.38 p: nat ).
% 0.18/0.38
% 0.18/0.38 thf(sy_v_pm,type,
% 0.18/0.38 pm: nat ).
% 0.18/0.38
% 0.18/0.38 thf(sy_v_x,type,
% 0.18/0.38 x: int ).
% 0.18/0.38
% 0.18/0.38 thf(sy_v_y_H____,type,
% 0.18/0.38 y: int ).
% 0.18/0.38
% 0.18/0.38 thf(sy_v_ya____,type,
% 0.18/0.38 ya: int ).
% 0.18/0.38
% 0.18/0.38 % Relevant facts (319)
% 0.18/0.38 thf(fact_0_n00,axiom,
% 0.18/0.38 na != zero_zero_int ).
% 0.18/0.38
% 0.18/0.38 % n00
% 0.18/0.38 thf(fact_1_of__int__less__of__int__power__cancel__iff,axiom,
% 0.18/0.38 ! [B: int,W: nat,X: int] :
% 0.18/0.38 ( ( ord_less_real @ ( power_power_real @ ( ring_1_of_int_real @ B ) @ W ) @ ( ring_1_of_int_real @ X ) )
% 0.18/0.38 = ( ord_less_int @ ( power_power_int @ B @ W ) @ X ) ) ).
% 0.18/0.38
% 0.18/0.38 % of_int_less_of_int_power_cancel_iff
% 0.18/0.38 thf(fact_2_of__int__less__of__int__power__cancel__iff,axiom,
% 0.18/0.38 ! [B: int,W: nat,X: int] :
% 0.18/0.38 ( ( ord_less_int @ ( power_power_int @ ( ring_1_of_int_int @ B ) @ W ) @ ( ring_1_of_int_int @ X ) )
% 0.18/0.38 = ( ord_less_int @ ( power_power_int @ B @ W ) @ X ) ) ).
% 0.18/0.38
% 0.18/0.38 % of_int_less_of_int_power_cancel_iff
% 0.18/0.38 thf(fact_3_of__int__power__less__of__int__cancel__iff,axiom,
% 0.18/0.38 ! [X: int,B: int,W: nat] :
% 0.18/0.38 ( ( ord_less_real @ ( ring_1_of_int_real @ X ) @ ( power_power_real @ ( ring_1_of_int_real @ B ) @ W ) )
% 0.18/0.38 = ( ord_less_int @ X @ ( power_power_int @ B @ W ) ) ) ).
% 0.18/0.38
% 0.18/0.38 % of_int_power_less_of_int_cancel_iff
% 0.18/0.38 thf(fact_4_of__int__power__less__of__int__cancel__iff,axiom,
% 0.18/0.38 ! [X: int,B: int,W: nat] :
% 0.18/0.38 ( ( ord_less_int @ ( ring_1_of_int_int @ X ) @ ( power_power_int @ ( ring_1_of_int_int @ B ) @ W ) )
% 0.18/0.38 = ( ord_less_int @ X @ ( power_power_int @ B @ W ) ) ) ).
% 0.18/0.38
% 0.18/0.38 % of_int_power_less_of_int_cancel_iff
% 0.18/0.38 thf(fact_5_of__int__power,axiom,
% 0.18/0.38 ! [Z: int,N: nat] :
% 0.18/0.38 ( ( ring_1_of_int_real @ ( power_power_int @ Z @ N ) )
% 0.18/0.38 = ( power_power_real @ ( ring_1_of_int_real @ Z ) @ N ) ) ).
% 0.18/0.38
% 0.18/0.38 % of_int_power
% 0.18/0.38 thf(fact_6_of__int__power,axiom,
% 0.18/0.38 ! [Z: int,N: nat] :
% 0.18/0.38 ( ( ring_1_of_int_int @ ( power_power_int @ Z @ N ) )
% 0.18/0.38 = ( power_power_int @ ( ring_1_of_int_int @ Z ) @ N ) ) ).
% 0.18/0.38
% 0.18/0.38 % of_int_power
% 0.18/0.38 thf(fact_7_of__int__eq__of__int__power__cancel__iff,axiom,
% 0.18/0.38 ! [B: int,W: nat,X: int] :
% 0.18/0.38 ( ( ( power_power_real @ ( ring_1_of_int_real @ B ) @ W )
% 0.18/0.38 = ( ring_1_of_int_real @ X ) )
% 0.18/0.38 = ( ( power_power_int @ B @ W )
% 0.18/0.38 = X ) ) ).
% 0.18/0.38
% 0.18/0.38 % of_int_eq_of_int_power_cancel_iff
% 0.18/0.38 thf(fact_8_of__int__eq__of__int__power__cancel__iff,axiom,
% 0.18/0.38 ! [B: int,W: nat,X: int] :
% 0.18/0.38 ( ( ( power_power_int @ ( ring_1_of_int_int @ B ) @ W )
% 0.18/0.38 = ( ring_1_of_int_int @ X ) )
% 0.18/0.38 = ( ( power_power_int @ B @ W )
% 0.18/0.38 = X ) ) ).
% 0.18/0.38
% 0.18/0.38 % of_int_eq_of_int_power_cancel_iff
% 0.18/0.38 thf(fact_9_of__int__power__eq__of__int__cancel__iff,axiom,
% 0.18/0.38 ! [X: int,B: int,W: nat] :
% 0.18/0.38 ( ( ( ring_1_of_int_real @ X )
% 0.18/0.38 = ( power_power_real @ ( ring_1_of_int_real @ B ) @ W ) )
% 0.18/0.38 = ( X
% 0.18/0.38 = ( power_power_int @ B @ W ) ) ) ).
% 0.18/0.38
% 0.18/0.38 % of_int_power_eq_of_int_cancel_iff
% 0.18/0.38 thf(fact_10_of__int__power__eq__of__int__cancel__iff,axiom,
% 0.18/0.38 ! [X: int,B: int,W: nat] :
% 0.18/0.38 ( ( ( ring_1_of_int_int @ X )
% 0.18/0.38 = ( power_power_int @ ( ring_1_of_int_int @ B ) @ W ) )
% 0.18/0.38 = ( X
% 0.18/0.38 = ( power_power_int @ B @ W ) ) ) ).
% 0.18/0.38
% 0.18/0.38 % of_int_power_eq_of_int_cancel_iff
% 0.18/0.38 thf(fact_11_of__int__less__iff,axiom,
% 0.18/0.38 ! [W: int,Z: int] :
% 0.18/0.38 ( ( ord_less_real @ ( ring_1_of_int_real @ W ) @ ( ring_1_of_int_real @ Z ) )
% 0.18/0.38 = ( ord_less_int @ W @ Z ) ) ).
% 0.18/0.38
% 0.18/0.38 % of_int_less_iff
% 0.18/0.38 thf(fact_12_of__int__less__iff,axiom,
% 0.18/0.38 ! [W: int,Z: int] :
% 0.18/0.38 ( ( ord_less_int @ ( ring_1_of_int_int @ W ) @ ( ring_1_of_int_int @ Z ) )
% 0.18/0.38 = ( ord_less_int @ W @ Z ) ) ).
% 0.18/0.38
% 0.18/0.38 % of_int_less_iff
% 0.18/0.38 thf(fact_13_n0,axiom,
% 0.18/0.38 ord_less_real @ zero_zero_real @ ( ring_1_of_int_real @ na ) ).
% 0.18/0.38
% 0.18/0.38 % n0
% 0.18/0.38 thf(fact_14_y0,axiom,
% 0.18/0.38 ord_less_real @ zero_zero_real @ ( ring_1_of_int_real @ ya ) ).
% 0.18/0.38
% 0.18/0.38 % y0
% 0.18/0.38 thf(fact_15_of__int__eq__iff,axiom,
% 0.18/0.38 ! [W: int,Z: int] :
% 0.18/0.38 ( ( ( ring_1_of_int_int @ W )
% 0.18/0.38 = ( ring_1_of_int_int @ Z ) )
% 0.18/0.38 = ( W = Z ) ) ).
% 0.18/0.38
% 0.18/0.38 % of_int_eq_iff
% 0.18/0.38 thf(fact_16_of__int__eq__iff,axiom,
% 0.18/0.38 ! [W: int,Z: int] :
% 0.18/0.38 ( ( ( ring_1_of_int_real @ W )
% 0.18/0.38 = ( ring_1_of_int_real @ Z ) )
% 0.18/0.38 = ( W = Z ) ) ).
% 0.18/0.38
% 0.18/0.38 % of_int_eq_iff
% 0.18/0.38 thf(fact_17_nat_Oinject,axiom,
% 0.18/0.38 ! [X2: nat,Y2: nat] :
% 0.18/0.38 ( ( ( suc @ X2 )
% 0.18/0.38 = ( suc @ Y2 ) )
% 0.18/0.38 = ( X2 = Y2 ) ) ).
% 0.18/0.38
% 0.18/0.38 % nat.inject
% 0.18/0.38 thf(fact_18_old_Onat_Oinject,axiom,
% 0.18/0.38 ! [Nat: nat,Nat2: nat] :
% 0.18/0.38 ( ( ( suc @ Nat )
% 0.18/0.38 = ( suc @ Nat2 ) )
% 0.18/0.38 = ( Nat = Nat2 ) ) ).
% 0.18/0.38
% 0.18/0.38 % old.nat.inject
% 0.18/0.38 thf(fact_19_y00,axiom,
% 0.18/0.38 ya != zero_zero_int ).
% 0.18/0.38
% 0.18/0.38 % y00
% 0.18/0.38 thf(fact_20__092_060open_0620_A_060_Ay_A_092_060or_062_Ay_A_061_A0_092_060close_062,axiom,
% 0.18/0.38 ( ( ord_less_int @ zero_zero_int @ ya )
% 0.18/0.38 | ( ya = zero_zero_int ) ) ).
% 0.18/0.38
% 0.18/0.38 % \<open>0 < y \<or> y = 0\<close>
% 0.18/0.38 thf(fact_21_Suc__less__eq,axiom,
% 0.18/0.38 ! [M: nat,N: nat] :
% 0.18/0.38 ( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
% 0.18/0.38 = ( ord_less_nat @ M @ N ) ) ).
% 0.18/0.38
% 0.18/0.38 % Suc_less_eq
% 0.18/0.38 thf(fact_22_Suc__mono,axiom,
% 0.18/0.38 ! [M: nat,N: nat] :
% 0.18/0.38 ( ( ord_less_nat @ M @ N )
% 0.18/0.38 => ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) ) ) ).
% 0.18/0.38
% 0.18/0.38 % Suc_mono
% 0.18/0.38 thf(fact_23_lessI,axiom,
% 0.18/0.38 ! [N: nat] : ( ord_less_nat @ N @ ( suc @ N ) ) ).
% 0.18/0.38
% 0.18/0.38 % lessI
% 0.18/0.38 thf(fact_24_not_I1_J,axiom,
% 0.18/0.38 ~ ( ( ord_less_int @ ya @ zero_zero_int )
% 0.18/0.38 | ( ord_less_int @ na @ zero_zero_int ) ) ).
% 0.18/0.38
% 0.18/0.38 % not(1)
% 0.18/0.38 thf(fact_25_not_I2_J,axiom,
% 0.18/0.38 ~ ( ( ord_less_int @ ya @ zero_zero_int )
% 0.18/0.38 | ( ord_less_int @ na @ zero_zero_int ) ) ).
% 0.18/0.38
% 0.18/0.38 % not(2)
% 0.18/0.38 thf(fact_26__C1_Oprems_C_I2_J,axiom,
% 0.18/0.38 ord_less_eq_int @ zero_zero_int @ ya ).
% 0.18/0.38
% 0.18/0.38 % "1.prems"(2)
% 0.18/0.38 thf(fact_27__C1_Oprems_C_I3_J,axiom,
% 0.18/0.38 ord_less_eq_int @ zero_zero_int @ na ).
% 0.18/0.38
% 0.18/0.38 % "1.prems"(3)
% 0.18/0.38 thf(fact_28_of__int__0,axiom,
% 0.18/0.38 ( ( ring_1_of_int_real @ zero_zero_int )
% 0.18/0.38 = zero_zero_real ) ).
% 0.18/0.38
% 0.18/0.38 % of_int_0
% 0.18/0.38 thf(fact_29_of__int__0,axiom,
% 0.18/0.38 ( ( ring_1_of_int_int @ zero_zero_int )
% 0.18/0.38 = zero_zero_int ) ).
% 0.18/0.38
% 0.18/0.38 % of_int_0
% 0.18/0.38 thf(fact_30_of__int__0__eq__iff,axiom,
% 0.18/0.38 ! [Z: int] :
% 0.18/0.38 ( ( zero_zero_real
% 0.18/0.38 = ( ring_1_of_int_real @ Z ) )
% 0.18/0.38 = ( Z = zero_zero_int ) ) ).
% 0.18/0.38
% 0.18/0.38 % of_int_0_eq_iff
% 0.18/0.38 thf(fact_31_of__int__0__eq__iff,axiom,
% 0.18/0.38 ! [Z: int] :
% 0.18/0.38 ( ( zero_zero_int
% 0.18/0.38 = ( ring_1_of_int_int @ Z ) )
% 0.18/0.38 = ( Z = zero_zero_int ) ) ).
% 0.18/0.38
% 0.18/0.38 % of_int_0_eq_iff
% 0.18/0.38 thf(fact_32_of__int__eq__0__iff,axiom,
% 0.18/0.38 ! [Z: int] :
% 0.18/0.38 ( ( ( ring_1_of_int_real @ Z )
% 0.18/0.38 = zero_zero_real )
% 0.18/0.38 = ( Z = zero_zero_int ) ) ).
% 0.18/0.38
% 0.18/0.38 % of_int_eq_0_iff
% 0.18/0.38 thf(fact_33_of__int__eq__0__iff,axiom,
% 0.18/0.38 ! [Z: int] :
% 0.18/0.38 ( ( ( ring_1_of_int_int @ Z )
% 0.18/0.38 = zero_zero_int )
% 0.18/0.38 = ( Z = zero_zero_int ) ) ).
% 0.18/0.38
% 0.18/0.38 % of_int_eq_0_iff
% 0.18/0.38 thf(fact_34_of__int__less__0__iff,axiom,
% 0.18/0.38 ! [Z: int] :
% 0.18/0.38 ( ( ord_less_real @ ( ring_1_of_int_real @ Z ) @ zero_zero_real )
% 0.18/0.38 = ( ord_less_int @ Z @ zero_zero_int ) ) ).
% 0.18/0.38
% 0.18/0.38 % of_int_less_0_iff
% 0.18/0.38 thf(fact_35_of__int__less__0__iff,axiom,
% 0.18/0.38 ! [Z: int] :
% 0.18/0.38 ( ( ord_less_int @ ( ring_1_of_int_int @ Z ) @ zero_zero_int )
% 0.18/0.38 = ( ord_less_int @ Z @ zero_zero_int ) ) ).
% 0.18/0.38
% 0.18/0.38 % of_int_less_0_iff
% 0.18/0.38 thf(fact_36_of__int__0__less__iff,axiom,
% 0.18/0.38 ! [Z: int] :
% 0.18/0.38 ( ( ord_less_real @ zero_zero_real @ ( ring_1_of_int_real @ Z ) )
% 0.18/0.38 = ( ord_less_int @ zero_zero_int @ Z ) ) ).
% 0.18/0.38
% 0.18/0.38 % of_int_0_less_iff
% 0.18/0.38 thf(fact_37_of__int__0__less__iff,axiom,
% 0.18/0.38 ! [Z: int] :
% 0.18/0.38 ( ( ord_less_int @ zero_zero_int @ ( ring_1_of_int_int @ Z ) )
% 0.18/0.38 = ( ord_less_int @ zero_zero_int @ Z ) ) ).
% 0.18/0.38
% 0.18/0.38 % of_int_0_less_iff
% 0.18/0.38 thf(fact_38_less__int__code_I1_J,axiom,
% 0.18/0.38 ~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% 0.18/0.38
% 0.18/0.38 % less_int_code(1)
% 0.18/0.38 thf(fact_39_not__less__less__Suc__eq,axiom,
% 0.18/0.38 ! [N: nat,M: nat] :
% 0.18/0.38 ( ~ ( ord_less_nat @ N @ M )
% 0.18/0.38 => ( ( ord_less_nat @ N @ ( suc @ M ) )
% 0.18/0.38 = ( N = M ) ) ) ).
% 0.18/0.38
% 0.18/0.38 % not_less_less_Suc_eq
% 0.18/0.38 thf(fact_40_strict__inc__induct,axiom,
% 0.18/0.38 ! [I: nat,J: nat,P: nat > $o] :
% 0.18/0.38 ( ( ord_less_nat @ I @ J )
% 0.18/0.38 => ( ! [I2: nat] :
% 0.18/0.38 ( ( J
% 0.18/0.38 = ( suc @ I2 ) )
% 0.18/0.38 => ( P @ I2 ) )
% 0.18/0.38 => ( ! [I2: nat] :
% 0.18/0.38 ( ( ord_less_nat @ I2 @ J )
% 0.18/0.38 => ( ( P @ ( suc @ I2 ) )
% 0.18/0.38 => ( P @ I2 ) ) )
% 0.18/0.38 => ( P @ I ) ) ) ) ).
% 0.18/0.38
% 0.18/0.38 % strict_inc_induct
% 0.18/0.38 thf(fact_41_less__Suc__induct,axiom,
% 0.18/0.38 ! [I: nat,J: nat,P: nat > nat > $o] :
% 0.18/0.38 ( ( ord_less_nat @ I @ J )
% 0.18/0.38 => ( ! [I2: nat] : ( P @ I2 @ ( suc @ I2 ) )
% 0.18/0.38 => ( ! [I2: nat,J2: nat,K: nat] :
% 0.18/0.38 ( ( ord_less_nat @ I2 @ J2 )
% 0.18/0.38 => ( ( ord_less_nat @ J2 @ K )
% 0.18/0.38 => ( ( P @ I2 @ J2 )
% 0.18/0.38 => ( ( P @ J2 @ K )
% 0.18/0.38 => ( P @ I2 @ K ) ) ) ) )
% 0.18/0.38 => ( P @ I @ J ) ) ) ) ).
% 0.18/0.38
% 0.18/0.38 % less_Suc_induct
% 0.18/0.38 thf(fact_42_less__trans__Suc,axiom,
% 0.18/0.38 ! [I: nat,J: nat,K2: nat] :
% 0.18/0.38 ( ( ord_less_nat @ I @ J )
% 0.18/0.38 => ( ( ord_less_nat @ J @ K2 )
% 0.18/0.38 => ( ord_less_nat @ ( suc @ I ) @ K2 ) ) ) ).
% 0.18/0.38
% 0.18/0.38 % less_trans_Suc
% 0.18/0.38 thf(fact_43_Suc__less__SucD,axiom,
% 0.18/0.38 ! [M: nat,N: nat] :
% 0.18/0.38 ( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
% 0.18/0.38 => ( ord_less_nat @ M @ N ) ) ).
% 0.18/0.38
% 0.18/0.38 % Suc_less_SucD
% 0.18/0.38 thf(fact_44_less__antisym,axiom,
% 0.18/0.38 ! [N: nat,M: nat] :
% 0.18/0.38 ( ~ ( ord_less_nat @ N @ M )
% 0.18/0.38 => ( ( ord_less_nat @ N @ ( suc @ M ) )
% 0.18/0.38 => ( M = N ) ) ) ).
% 0.18/0.38
% 0.18/0.38 % less_antisym
% 0.18/0.38 thf(fact_45_Suc__less__eq2,axiom,
% 0.18/0.38 ! [N: nat,M: nat] :
% 0.18/0.38 ( ( ord_less_nat @ ( suc @ N ) @ M )
% 0.18/0.38 = ( ? [M2: nat] :
% 0.18/0.38 ( ( M
% 0.18/0.38 = ( suc @ M2 ) )
% 0.18/0.38 & ( ord_less_nat @ N @ M2 ) ) ) ) ).
% 0.18/0.38
% 0.18/0.38 % Suc_less_eq2
% 0.18/0.38 thf(fact_46_All__less__Suc,axiom,
% 0.18/0.38 ! [N: nat,P: nat > $o] :
% 0.18/0.38 ( ( ! [I3: nat] :
% 0.18/0.38 ( ( ord_less_nat @ I3 @ ( suc @ N ) )
% 0.18/0.38 => ( P @ I3 ) ) )
% 0.18/0.38 = ( ( P @ N )
% 0.18/0.38 & ! [I3: nat] :
% 0.18/0.38 ( ( ord_less_nat @ I3 @ N )
% 0.18/0.38 => ( P @ I3 ) ) ) ) ).
% 0.18/0.38
% 0.18/0.38 % All_less_Suc
% 0.18/0.38 thf(fact_47_not__less__eq,axiom,
% 0.18/0.38 ! [M: nat,N: nat] :
% 0.18/0.38 ( ( ~ ( ord_less_nat @ M @ N ) )
% 0.18/0.38 = ( ord_less_nat @ N @ ( suc @ M ) ) ) ).
% 0.18/0.38
% 0.18/0.38 % not_less_eq
% 0.18/0.38 thf(fact_48_less__Suc__eq,axiom,
% 0.18/0.38 ! [M: nat,N: nat] :
% 0.18/0.38 ( ( ord_less_nat @ M @ ( suc @ N ) )
% 0.18/0.38 = ( ( ord_less_nat @ M @ N )
% 0.18/0.38 | ( M = N ) ) ) ).
% 0.18/0.38
% 0.18/0.38 % less_Suc_eq
% 0.18/0.38 thf(fact_49_Ex__less__Suc,axiom,
% 0.18/0.38 ! [N: nat,P: nat > $o] :
% 0.18/0.38 ( ( ? [I3: nat] :
% 0.18/0.38 ( ( ord_less_nat @ I3 @ ( suc @ N ) )
% 0.18/0.38 & ( P @ I3 ) ) )
% 0.18/0.38 = ( ( P @ N )
% 0.18/0.38 | ? [I3: nat] :
% 0.18/0.38 ( ( ord_less_nat @ I3 @ N )
% 0.18/0.38 & ( P @ I3 ) ) ) ) ).
% 0.18/0.38
% 0.18/0.38 % Ex_less_Suc
% 0.18/0.38 thf(fact_50_less__SucI,axiom,
% 0.18/0.38 ! [M: nat,N: nat] :
% 0.18/0.38 ( ( ord_less_nat @ M @ N )
% 0.18/0.38 => ( ord_less_nat @ M @ ( suc @ N ) ) ) ).
% 0.18/0.38
% 0.18/0.38 % less_SucI
% 0.18/0.38 thf(fact_51_less__SucE,axiom,
% 0.18/0.38 ! [M: nat,N: nat] :
% 0.18/0.38 ( ( ord_less_nat @ M @ ( suc @ N ) )
% 0.18/0.38 => ( ~ ( ord_less_nat @ M @ N )
% 0.18/0.38 => ( M = N ) ) ) ).
% 0.18/0.38
% 0.18/0.38 % less_SucE
% 0.18/0.38 thf(fact_52_Suc__lessI,axiom,
% 0.18/0.38 ! [M: nat,N: nat] :
% 0.18/0.38 ( ( ord_less_nat @ M @ N )
% 0.18/0.38 => ( ( ( suc @ M )
% 0.18/0.38 != N )
% 0.18/0.38 => ( ord_less_nat @ ( suc @ M ) @ N ) ) ) ).
% 0.18/0.38
% 0.18/0.38 % Suc_lessI
% 0.18/0.38 thf(fact_53_Suc__lessE,axiom,
% 0.18/0.38 ! [I: nat,K2: nat] :
% 0.18/0.38 ( ( ord_less_nat @ ( suc @ I ) @ K2 )
% 0.18/0.38 => ~ ! [J2: nat] :
% 0.18/0.38 ( ( ord_less_nat @ I @ J2 )
% 0.18/0.38 => ( K2
% 0.18/0.38 != ( suc @ J2 ) ) ) ) ).
% 0.18/0.38
% 0.18/0.38 % Suc_lessE
% 0.18/0.38 thf(fact_54_Suc__lessD,axiom,
% 0.18/0.38 ! [M: nat,N: nat] :
% 0.18/0.38 ( ( ord_less_nat @ ( suc @ M ) @ N )
% 0.18/0.38 => ( ord_less_nat @ M @ N ) ) ).
% 0.18/0.38
% 0.18/0.38 % Suc_lessD
% 0.18/0.38 thf(fact_55_Nat_OlessE,axiom,
% 0.18/0.38 ! [I: nat,K2: nat] :
% 0.18/0.38 ( ( ord_less_nat @ I @ K2 )
% 0.18/0.38 => ( ( K2
% 0.18/0.38 != ( suc @ I ) )
% 0.18/0.38 => ~ ! [J2: nat] :
% 0.18/0.38 ( ( ord_less_nat @ I @ J2 )
% 0.18/0.38 => ( K2
% 0.18/0.38 != ( suc @ J2 ) ) ) ) ) ).
% 0.18/0.38
% 0.18/0.38 % Nat.lessE
% 0.18/0.38 thf(fact_56_of__int__pos,axiom,
% 0.18/0.38 ! [Z: int] :
% 0.18/0.38 ( ( ord_less_int @ zero_zero_int @ Z )
% 0.18/0.38 => ( ord_less_real @ zero_zero_real @ ( ring_1_of_int_real @ Z ) ) ) ).
% 0.18/0.38
% 0.18/0.38 % of_int_pos
% 0.18/0.38 thf(fact_57_of__int__pos,axiom,
% 0.18/0.38 ! [Z: int] :
% 0.18/0.38 ( ( ord_less_int @ zero_zero_int @ Z )
% 0.18/0.38 => ( ord_less_int @ zero_zero_int @ ( ring_1_of_int_int @ Z ) ) ) ).
% 0.18/0.38
% 0.18/0.38 % of_int_pos
% 0.18/0.38 thf(fact_58_n__not__Suc__n,axiom,
% 0.18/0.38 ! [N: nat] :
% 0.18/0.38 ( N
% 0.18/0.38 != ( suc @ N ) ) ).
% 0.18/0.38
% 0.18/0.38 % n_not_Suc_n
% 0.18/0.38 thf(fact_59_Suc__inject,axiom,
% 0.18/0.38 ! [X: nat,Y: nat] :
% 0.18/0.38 ( ( ( suc @ X )
% 0.18/0.38 = ( suc @ Y ) )
% 0.18/0.38 => ( X = Y ) ) ).
% 0.18/0.38
% 0.18/0.38 % Suc_inject
% 0.18/0.38 thf(fact_60_lift__Suc__mono__less__iff,axiom,
% 0.18/0.38 ! [F: nat > real,N: nat,M: nat] :
% 0.18/0.38 ( ! [N2: nat] : ( ord_less_real @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
% 0.18/0.38 => ( ( ord_less_real @ ( F @ N ) @ ( F @ M ) )
% 0.18/0.38 = ( ord_less_nat @ N @ M ) ) ) ).
% 0.18/0.38
% 0.18/0.38 % lift_Suc_mono_less_iff
% 0.18/0.38 thf(fact_61_lift__Suc__mono__less__iff,axiom,
% 0.18/0.38 ! [F: nat > int,N: nat,M: nat] :
% 0.18/0.38 ( ! [N2: nat] : ( ord_less_int @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
% 0.18/0.38 => ( ( ord_less_int @ ( F @ N ) @ ( F @ M ) )
% 0.18/0.38 = ( ord_less_nat @ N @ M ) ) ) ).
% 0.18/0.38
% 0.18/0.38 % lift_Suc_mono_less_iff
% 0.18/0.38 thf(fact_62_lift__Suc__mono__less__iff,axiom,
% 0.18/0.38 ! [F: nat > nat,N: nat,M: nat] :
% 0.18/0.38 ( ! [N2: nat] : ( ord_less_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
% 0.18/0.38 => ( ( ord_less_nat @ ( F @ N ) @ ( F @ M ) )
% 0.18/0.38 = ( ord_less_nat @ N @ M ) ) ) ).
% 0.18/0.38
% 0.18/0.38 % lift_Suc_mono_less_iff
% 0.18/0.38 thf(fact_63_lift__Suc__mono__less,axiom,
% 0.18/0.38 ! [F: nat > real,N: nat,N3: nat] :
% 0.18/0.38 ( ! [N2: nat] : ( ord_less_real @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
% 0.18/0.38 => ( ( ord_less_nat @ N @ N3 )
% 0.18/0.38 => ( ord_less_real @ ( F @ N ) @ ( F @ N3 ) ) ) ) ).
% 0.18/0.38
% 0.18/0.38 % lift_Suc_mono_less
% 0.18/0.38 thf(fact_64_lift__Suc__mono__less,axiom,
% 0.18/0.38 ! [F: nat > int,N: nat,N3: nat] :
% 0.18/0.38 ( ! [N2: nat] : ( ord_less_int @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
% 0.18/0.38 => ( ( ord_less_nat @ N @ N3 )
% 0.18/0.38 => ( ord_less_int @ ( F @ N ) @ ( F @ N3 ) ) ) ) ).
% 0.18/0.38
% 0.18/0.38 % lift_Suc_mono_less
% 0.18/0.38 thf(fact_65_lift__Suc__mono__less,axiom,
% 0.18/0.38 ! [F: nat > nat,N: nat,N3: nat] :
% 0.18/0.38 ( ! [N2: nat] : ( ord_less_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
% 0.18/0.38 => ( ( ord_less_nat @ N @ N3 )
% 0.18/0.38 => ( ord_less_nat @ ( F @ N ) @ ( F @ N3 ) ) ) ) ).
% 0.18/0.38
% 0.18/0.38 % lift_Suc_mono_less
% 0.18/0.38 thf(fact_66_power__0__Suc,axiom,
% 0.18/0.38 ! [N: nat] :
% 0.18/0.38 ( ( power_power_real @ zero_zero_real @ ( suc @ N ) )
% 0.18/0.38 = zero_zero_real ) ).
% 0.18/0.38
% 0.18/0.38 % power_0_Suc
% 0.18/0.38 thf(fact_67_power__0__Suc,axiom,
% 0.18/0.38 ! [N: nat] :
% 0.18/0.38 ( ( power_power_int @ zero_zero_int @ ( suc @ N ) )
% 0.18/0.38 = zero_zero_int ) ).
% 0.18/0.38
% 0.18/0.38 % power_0_Suc
% 0.18/0.38 thf(fact_68_power__0__Suc,axiom,
% 0.18/0.38 ! [N: nat] :
% 0.18/0.38 ( ( power_power_nat @ zero_zero_nat @ ( suc @ N ) )
% 0.18/0.38 = zero_zero_nat ) ).
% 0.18/0.38
% 0.18/0.38 % power_0_Suc
% 0.18/0.38 thf(fact_69_NY0,axiom,
% 0.18/0.38 ord_less_real @ zero_zero_real @ ny ).
% 0.18/0.38
% 0.18/0.38 % NY0
% 0.18/0.38 thf(fact_70_power__eq__0__iff,axiom,
% 0.18/0.38 ! [A: real,N: nat] :
% 0.18/0.38 ( ( ( power_power_real @ A @ N )
% 0.18/0.38 = zero_zero_real )
% 0.18/0.38 = ( ( A = zero_zero_real )
% 0.18/0.38 & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 0.18/0.38
% 0.18/0.38 % power_eq_0_iff
% 0.18/0.38 thf(fact_71_power__eq__0__iff,axiom,
% 0.18/0.38 ! [A: int,N: nat] :
% 0.18/0.38 ( ( ( power_power_int @ A @ N )
% 0.18/0.38 = zero_zero_int )
% 0.18/0.38 = ( ( A = zero_zero_int )
% 0.18/0.38 & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 0.18/0.38
% 0.18/0.38 % power_eq_0_iff
% 0.18/0.38 thf(fact_72_power__eq__0__iff,axiom,
% 0.18/0.38 ! [A: nat,N: nat] :
% 0.18/0.38 ( ( ( power_power_nat @ A @ N )
% 0.18/0.38 = zero_zero_nat )
% 0.18/0.38 = ( ( A = zero_zero_nat )
% 0.18/0.38 & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 0.18/0.38
% 0.18/0.38 % power_eq_0_iff
% 0.18/0.38 thf(fact_73_not__gr__zero,axiom,
% 0.18/0.38 ! [N: nat] :
% 0.18/0.38 ( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
% 0.18/0.38 = ( N = zero_zero_nat ) ) ).
% 0.18/0.38
% 0.18/0.38 % not_gr_zero
% 0.18/0.38 thf(fact_74_realpow__pos__nth2,axiom,
% 0.18/0.38 ! [A: real,N: nat] :
% 0.18/0.38 ( ( ord_less_real @ zero_zero_real @ A )
% 0.18/0.38 => ? [R: real] :
% 0.18/0.38 ( ( ord_less_real @ zero_zero_real @ R )
% 0.18/0.38 & ( ( power_power_real @ R @ ( suc @ N ) )
% 0.18/0.38 = A ) ) ) ).
% 0.18/0.38
% 0.18/0.38 % realpow_pos_nth2
% 0.18/0.38 thf(fact_75_zero__less__power,axiom,
% 0.18/0.38 ! [A: real,N: nat] :
% 0.18/0.38 ( ( ord_less_real @ zero_zero_real @ A )
% 0.18/0.38 => ( ord_less_real @ zero_zero_real @ ( power_power_real @ A @ N ) ) ) ).
% 0.18/0.38
% 0.18/0.38 % zero_less_power
% 0.18/0.38 thf(fact_76_zero__less__power,axiom,
% 0.18/0.38 ! [A: int,N: nat] :
% 0.18/0.38 ( ( ord_less_int @ zero_zero_int @ A )
% 0.18/0.38 => ( ord_less_int @ zero_zero_int @ ( power_power_int @ A @ N ) ) ) ).
% 0.18/0.38
% 0.18/0.38 % zero_less_power
% 0.18/0.38 thf(fact_77_zero__less__power,axiom,
% 0.18/0.38 ! [A: nat,N: nat] :
% 0.18/0.38 ( ( ord_less_nat @ zero_zero_nat @ A )
% 0.18/0.38 => ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ A @ N ) ) ) ).
% 0.18/0.38
% 0.18/0.38 % zero_less_power
% 0.18/0.38 thf(fact_78_NY__def,axiom,
% 0.18/0.38 ( ny
% 0.18/0.38 = ( divide_divide_real @ ( ring_1_of_int_real @ na ) @ ( power_power_real @ ( ring_1_of_int_real @ ya ) @ pm ) ) ) ).
% 0.18/0.38
% 0.18/0.38 % NY_def
% 0.18/0.38 thf(fact_79_ex__of__int__less,axiom,
% 0.18/0.38 ! [X: real] :
% 0.18/0.38 ? [Z2: int] : ( ord_less_real @ ( ring_1_of_int_real @ Z2 ) @ X ) ).
% 0.18/0.38
% 0.18/0.38 % ex_of_int_less
% 0.18/0.38 thf(fact_80_ex__less__of__int,axiom,
% 0.18/0.38 ! [X: real] :
% 0.18/0.38 ? [Z2: int] : ( ord_less_real @ X @ ( ring_1_of_int_real @ Z2 ) ) ).
% 0.18/0.38
% 0.18/0.38 % ex_less_of_int
% 0.18/0.38 thf(fact_81_yyn,axiom,
% 0.18/0.38 ord_less_int @ na @ ( power_power_int @ ya @ p ) ).
% 0.18/0.38
% 0.18/0.38 % yyn
% 0.18/0.38 thf(fact_82__092_060open_062pm_A_061_A0_A_092_060Longrightarrow_062_AFalse_092_060close_062,axiom,
% 0.18/0.38 pm != zero_zero_nat ).
% 0.18/0.38
% 0.18/0.38 % \<open>pm = 0 \<Longrightarrow> False\<close>
% 0.18/0.38 thf(fact_83_p0,axiom,
% 0.18/0.38 p != zero_zero_nat ).
% 0.18/0.38
% 0.18/0.38 % p0
% 0.18/0.38 thf(fact_84_pm0,axiom,
% 0.18/0.38 ord_less_nat @ zero_zero_nat @ pm ).
% 0.18/0.38
% 0.18/0.38 % pm0
% 0.18/0.38 thf(fact_85__C1_Oprems_C_I1_J,axiom,
% 0.18/0.38 ord_less_eq_int @ na @ ( power_power_int @ ya @ p ) ).
% 0.18/0.38
% 0.18/0.38 % "1.prems"(1)
% 0.18/0.38 thf(fact_86_p,axiom,
% 0.18/0.38 ( p
% 0.18/0.38 = ( suc @ pm ) ) ).
% 0.18/0.38
% 0.18/0.38 % p
% 0.18/0.38 thf(fact_87_fixed__root__axioms,axiom,
% 0.18/0.38 nthRoo1352626670d_root @ p @ pm ).
% 0.18/0.38
% 0.18/0.38 % fixed_root_axioms
% 0.18/0.38 thf(fact_88_le__zero__eq,axiom,
% 0.18/0.38 ! [N: nat] :
% 0.18/0.38 ( ( ord_less_eq_nat @ N @ zero_zero_nat )
% 0.18/0.38 = ( N = zero_zero_nat ) ) ).
% 0.18/0.38
% 0.18/0.38 % le_zero_eq
% 0.18/0.38 thf(fact_89_nat__power__eq__Suc__0__iff,axiom,
% 0.18/0.38 ! [X: nat,M: nat] :
% 0.18/0.38 ( ( ( power_power_nat @ X @ M )
% 0.18/0.38 = ( suc @ zero_zero_nat ) )
% 0.18/0.38 = ( ( M = zero_zero_nat )
% 0.18/0.38 | ( X
% 0.18/0.38 = ( suc @ zero_zero_nat ) ) ) ) ).
% 0.18/0.38
% 0.18/0.38 % nat_power_eq_Suc_0_iff
% 0.18/0.38 thf(fact_90_power__Suc__0,axiom,
% 0.18/0.38 ! [N: nat] :
% 0.18/0.38 ( ( power_power_nat @ ( suc @ zero_zero_nat ) @ N )
% 0.18/0.38 = ( suc @ zero_zero_nat ) ) ).
% 0.18/0.38
% 0.18/0.38 % power_Suc_0
% 0.18/0.38 thf(fact_91_neq0__conv,axiom,
% 0.18/0.38 ! [N: nat] :
% 0.18/0.38 ( ( N != zero_zero_nat )
% 0.18/0.38 = ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 0.18/0.38
% 0.18/0.38 % neq0_conv
% 0.18/0.38 thf(fact_92_less__nat__zero__code,axiom,
% 0.18/0.38 ! [N: nat] :
% 0.18/0.38 ~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% 0.18/0.38
% 0.18/0.38 % less_nat_zero_code
% 0.18/0.38 thf(fact_93_bot__nat__0_Onot__eq__extremum,axiom,
% 0.18/0.38 ! [A: nat] :
% 0.18/0.38 ( ( A != zero_zero_nat )
% 0.18/0.38 = ( ord_less_nat @ zero_zero_nat @ A ) ) ).
% 0.18/0.38
% 0.18/0.38 % bot_nat_0.not_eq_extremum
% 0.18/0.38 thf(fact_94_nat__zero__less__power__iff,axiom,
% 0.18/0.38 ! [X: nat,N: nat] :
% 0.18/0.38 ( ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ X @ N ) )
% 0.18/0.38 = ( ( ord_less_nat @ zero_zero_nat @ X )
% 0.18/0.38 | ( N = zero_zero_nat ) ) ) ).
% 0.18/0.38
% 0.18/0.38 % nat_zero_less_power_iff
% 0.18/0.38 thf(fact_95_power__Suc0__right,axiom,
% 0.18/0.38 ! [A: real] :
% 0.18/0.38 ( ( power_power_real @ A @ ( suc @ zero_zero_nat ) )
% 0.18/0.38 = A ) ).
% 0.18/0.38
% 0.18/0.38 % power_Suc0_right
% 0.18/0.38 thf(fact_96_power__Suc0__right,axiom,
% 0.18/0.38 ! [A: int] :
% 0.18/0.38 ( ( power_power_int @ A @ ( suc @ zero_zero_nat ) )
% 0.18/0.38 = A ) ).
% 0.18/0.38
% 0.18/0.38 % power_Suc0_right
% 0.18/0.38 thf(fact_97_power__Suc0__right,axiom,
% 0.18/0.38 ! [A: nat] :
% 0.18/0.38 ( ( power_power_nat @ A @ ( suc @ zero_zero_nat ) )
% 0.18/0.38 = A ) ).
% 0.18/0.38
% 0.18/0.38 % power_Suc0_right
% 0.18/0.38 thf(fact_98_zero__less__Suc,axiom,
% 0.18/0.38 ! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N ) ) ).
% 0.18/0.38
% 0.18/0.38 % zero_less_Suc
% 0.18/0.38 thf(fact_99_less__Suc0,axiom,
% 0.18/0.38 ! [N: nat] :
% 0.18/0.38 ( ( ord_less_nat @ N @ ( suc @ zero_zero_nat ) )
% 0.18/0.38 = ( N = zero_zero_nat ) ) ).
% 0.18/0.38
% 0.18/0.38 % less_Suc0
% 0.18/0.38 thf(fact_100_of__int__le__iff,axiom,
% 0.18/0.38 ! [W: int,Z: int] :
% 0.18/0.38 ( ( ord_less_eq_real @ ( ring_1_of_int_real @ W ) @ ( ring_1_of_int_real @ Z ) )
% 0.18/0.38 = ( ord_less_eq_int @ W @ Z ) ) ).
% 0.18/0.38
% 0.18/0.38 % of_int_le_iff
% 0.18/0.38 thf(fact_101_of__int__le__iff,axiom,
% 0.18/0.38 ! [W: int,Z: int] :
% 0.18/0.38 ( ( ord_less_eq_int @ ( ring_1_of_int_int @ W ) @ ( ring_1_of_int_int @ Z ) )
% 0.18/0.38 = ( ord_less_eq_int @ W @ Z ) ) ).
% 0.18/0.38
% 0.18/0.38 % of_int_le_iff
% 0.18/0.38 thf(fact_102_power__mono__iff,axiom,
% 0.18/0.38 ! [A: real,B: real,N: nat] :
% 0.18/0.38 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.18/0.38 => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 0.18/0.38 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.18/0.38 => ( ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B @ N ) )
% 0.18/0.38 = ( ord_less_eq_real @ A @ B ) ) ) ) ) ).
% 0.18/0.38
% 0.18/0.38 % power_mono_iff
% 0.18/0.38 thf(fact_103_power__mono__iff,axiom,
% 0.18/0.38 ! [A: int,B: int,N: nat] :
% 0.18/0.38 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.18/0.38 => ( ( ord_less_eq_int @ zero_zero_int @ B )
% 0.18/0.38 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.18/0.38 => ( ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) )
% 0.18/0.38 = ( ord_less_eq_int @ A @ B ) ) ) ) ) ).
% 0.18/0.38
% 0.18/0.38 % power_mono_iff
% 0.18/0.38 thf(fact_104_power__mono__iff,axiom,
% 0.18/0.38 ! [A: nat,B: nat,N: nat] :
% 0.18/0.38 ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.18/0.38 => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 0.18/0.38 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.18/0.38 => ( ( ord_less_eq_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) )
% 0.18/0.38 = ( ord_less_eq_nat @ A @ B ) ) ) ) ) ).
% 0.18/0.38
% 0.18/0.38 % power_mono_iff
% 0.18/0.38 thf(fact_105_of__int__le__0__iff,axiom,
% 0.18/0.38 ! [Z: int] :
% 0.18/0.38 ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z ) @ zero_zero_real )
% 0.18/0.38 = ( ord_less_eq_int @ Z @ zero_zero_int ) ) ).
% 0.18/0.38
% 0.18/0.38 % of_int_le_0_iff
% 0.18/0.38 thf(fact_106_of__int__le__0__iff,axiom,
% 0.18/0.38 ! [Z: int] :
% 0.18/0.38 ( ( ord_less_eq_int @ ( ring_1_of_int_int @ Z ) @ zero_zero_int )
% 0.18/0.38 = ( ord_less_eq_int @ Z @ zero_zero_int ) ) ).
% 0.18/0.38
% 0.18/0.38 % of_int_le_0_iff
% 0.18/0.38 thf(fact_107_of__int__0__le__iff,axiom,
% 0.18/0.38 ! [Z: int] :
% 0.18/0.38 ( ( ord_less_eq_real @ zero_zero_real @ ( ring_1_of_int_real @ Z ) )
% 0.18/0.38 = ( ord_less_eq_int @ zero_zero_int @ Z ) ) ).
% 0.18/0.38
% 0.18/0.38 % of_int_0_le_iff
% 0.18/0.38 thf(fact_108_of__int__0__le__iff,axiom,
% 0.18/0.38 ! [Z: int] :
% 0.18/0.38 ( ( ord_less_eq_int @ zero_zero_int @ ( ring_1_of_int_int @ Z ) )
% 0.18/0.38 = ( ord_less_eq_int @ zero_zero_int @ Z ) ) ).
% 0.18/0.38
% 0.18/0.38 % of_int_0_le_iff
% 0.18/0.38 thf(fact_109_of__int__le__of__int__power__cancel__iff,axiom,
% 0.18/0.38 ! [B: int,W: nat,X: int] :
% 0.18/0.38 ( ( ord_less_eq_real @ ( power_power_real @ ( ring_1_of_int_real @ B ) @ W ) @ ( ring_1_of_int_real @ X ) )
% 0.18/0.38 = ( ord_less_eq_int @ ( power_power_int @ B @ W ) @ X ) ) ).
% 0.18/0.38
% 0.18/0.38 % of_int_le_of_int_power_cancel_iff
% 0.18/0.38 thf(fact_110_of__int__le__of__int__power__cancel__iff,axiom,
% 0.18/0.38 ! [B: int,W: nat,X: int] :
% 0.18/0.38 ( ( ord_less_eq_int @ ( power_power_int @ ( ring_1_of_int_int @ B ) @ W ) @ ( ring_1_of_int_int @ X ) )
% 0.18/0.38 = ( ord_less_eq_int @ ( power_power_int @ B @ W ) @ X ) ) ).
% 0.18/0.38
% 0.18/0.38 % of_int_le_of_int_power_cancel_iff
% 0.18/0.38 thf(fact_111_of__int__power__le__of__int__cancel__iff,axiom,
% 0.18/0.38 ! [X: int,B: int,W: nat] :
% 0.18/0.38 ( ( ord_less_eq_real @ ( ring_1_of_int_real @ X ) @ ( power_power_real @ ( ring_1_of_int_real @ B ) @ W ) )
% 0.18/0.38 = ( ord_less_eq_int @ X @ ( power_power_int @ B @ W ) ) ) ).
% 0.18/0.38
% 0.18/0.38 % of_int_power_le_of_int_cancel_iff
% 0.18/0.38 thf(fact_112_of__int__power__le__of__int__cancel__iff,axiom,
% 0.18/0.38 ! [X: int,B: int,W: nat] :
% 0.18/0.38 ( ( ord_less_eq_int @ ( ring_1_of_int_int @ X ) @ ( power_power_int @ ( ring_1_of_int_int @ B ) @ W ) )
% 0.18/0.38 = ( ord_less_eq_int @ X @ ( power_power_int @ B @ W ) ) ) ).
% 0.18/0.38
% 0.18/0.38 % of_int_power_le_of_int_cancel_iff
% 0.18/0.38 thf(fact_113_gr0I,axiom,
% 0.18/0.38 ! [N: nat] :
% 0.18/0.38 ( ( N != zero_zero_nat )
% 0.18/0.38 => ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 0.18/0.38
% 0.18/0.38 % gr0I
% 0.18/0.38 thf(fact_114_not__gr0,axiom,
% 0.18/0.38 ! [N: nat] :
% 0.18/0.38 ( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
% 0.18/0.38 = ( N = zero_zero_nat ) ) ).
% 0.18/0.38
% 0.18/0.38 % not_gr0
% 0.18/0.38 thf(fact_115_not__less0,axiom,
% 0.18/0.38 ! [N: nat] :
% 0.18/0.38 ~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% 0.18/0.38
% 0.18/0.38 % not_less0
% 0.18/0.38 thf(fact_116_less__zeroE,axiom,
% 0.18/0.38 ! [N: nat] :
% 0.18/0.38 ~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% 0.18/0.38
% 0.18/0.38 % less_zeroE
% 0.18/0.38 thf(fact_117_nat__neq__iff,axiom,
% 0.18/0.38 ! [M: nat,N: nat] :
% 0.18/0.38 ( ( M != N )
% 0.18/0.38 = ( ( ord_less_nat @ M @ N )
% 0.18/0.38 | ( ord_less_nat @ N @ M ) ) ) ).
% 0.18/0.38
% 0.18/0.38 % nat_neq_iff
% 0.18/0.38 thf(fact_118_less__not__refl,axiom,
% 0.18/0.38 ! [N: nat] :
% 0.18/0.38 ~ ( ord_less_nat @ N @ N ) ).
% 0.18/0.38
% 0.18/0.38 % less_not_refl
% 0.18/0.38 thf(fact_119_less__not__refl2,axiom,
% 0.18/0.38 ! [N: nat,M: nat] :
% 0.18/0.38 ( ( ord_less_nat @ N @ M )
% 0.18/0.38 => ( M != N ) ) ).
% 0.18/0.38
% 0.18/0.38 % less_not_refl2
% 0.18/0.38 thf(fact_120_less__not__refl3,axiom,
% 0.18/0.38 ! [S: nat,T: nat] :
% 0.18/0.38 ( ( ord_less_nat @ S @ T )
% 0.18/0.38 => ( S != T ) ) ).
% 0.18/0.38
% 0.18/0.38 % less_not_refl3
% 0.18/0.38 thf(fact_121_gr__implies__not0,axiom,
% 0.18/0.38 ! [M: nat,N: nat] :
% 0.18/0.38 ( ( ord_less_nat @ M @ N )
% 0.18/0.38 => ( N != zero_zero_nat ) ) ).
% 0.18/0.38
% 0.18/0.38 % gr_implies_not0
% 0.18/0.38 thf(fact_122_less__irrefl__nat,axiom,
% 0.18/0.38 ! [N: nat] :
% 0.18/0.38 ~ ( ord_less_nat @ N @ N ) ).
% 0.18/0.38
% 0.18/0.38 % less_irrefl_nat
% 0.18/0.38 thf(fact_123_nat__less__induct,axiom,
% 0.18/0.38 ! [P: nat > $o,N: nat] :
% 0.18/0.38 ( ! [N2: nat] :
% 0.18/0.38 ( ! [M3: nat] :
% 0.18/0.38 ( ( ord_less_nat @ M3 @ N2 )
% 0.18/0.38 => ( P @ M3 ) )
% 0.18/0.38 => ( P @ N2 ) )
% 0.18/0.38 => ( P @ N ) ) ).
% 0.18/0.38
% 0.18/0.38 % nat_less_induct
% 0.18/0.38 thf(fact_124_infinite__descent,axiom,
% 0.18/0.38 ! [P: nat > $o,N: nat] :
% 0.18/0.38 ( ! [N2: nat] :
% 0.18/0.38 ( ~ ( P @ N2 )
% 0.18/0.38 => ? [M3: nat] :
% 0.18/0.38 ( ( ord_less_nat @ M3 @ N2 )
% 0.18/0.38 & ~ ( P @ M3 ) ) )
% 0.18/0.38 => ( P @ N ) ) ).
% 0.18/0.38
% 0.18/0.38 % infinite_descent
% 0.18/0.38 thf(fact_125_infinite__descent0,axiom,
% 0.18/0.38 ! [P: nat > $o,N: nat] :
% 0.18/0.38 ( ( P @ zero_zero_nat )
% 0.18/0.38 => ( ! [N2: nat] :
% 0.18/0.38 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.18/0.38 => ( ~ ( P @ N2 )
% 0.18/0.38 => ? [M3: nat] :
% 0.18/0.38 ( ( ord_less_nat @ M3 @ N2 )
% 0.18/0.38 & ~ ( P @ M3 ) ) ) )
% 0.18/0.38 => ( P @ N ) ) ) ).
% 0.18/0.38
% 0.18/0.38 % infinite_descent0
% 0.18/0.38 thf(fact_126_linorder__neqE__nat,axiom,
% 0.18/0.38 ! [X: nat,Y: nat] :
% 0.18/0.38 ( ( X != Y )
% 0.18/0.38 => ( ~ ( ord_less_nat @ X @ Y )
% 0.18/0.38 => ( ord_less_nat @ Y @ X ) ) ) ).
% 0.18/0.38
% 0.18/0.38 % linorder_neqE_nat
% 0.18/0.38 thf(fact_127_bot__nat__0_Oextremum__strict,axiom,
% 0.18/0.38 ! [A: nat] :
% 0.18/0.38 ~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% 0.18/0.38
% 0.18/0.38 % bot_nat_0.extremum_strict
% 0.18/0.38 thf(fact_128_nat__power__less__imp__less,axiom,
% 0.18/0.38 ! [I: nat,M: nat,N: nat] :
% 0.18/0.38 ( ( ord_less_nat @ zero_zero_nat @ I )
% 0.18/0.38 => ( ( ord_less_nat @ ( power_power_nat @ I @ M ) @ ( power_power_nat @ I @ N ) )
% 0.18/0.38 => ( ord_less_nat @ M @ N ) ) ) ).
% 0.18/0.38
% 0.18/0.38 % nat_power_less_imp_less
% 0.18/0.38 thf(fact_129_lift__Suc__antimono__le,axiom,
% 0.18/0.38 ! [F: nat > real,N: nat,N3: nat] :
% 0.18/0.38 ( ! [N2: nat] : ( ord_less_eq_real @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
% 0.18/0.38 => ( ( ord_less_eq_nat @ N @ N3 )
% 0.18/0.38 => ( ord_less_eq_real @ ( F @ N3 ) @ ( F @ N ) ) ) ) ).
% 0.18/0.38
% 0.18/0.38 % lift_Suc_antimono_le
% 0.18/0.38 thf(fact_130_lift__Suc__antimono__le,axiom,
% 0.18/0.38 ! [F: nat > int,N: nat,N3: nat] :
% 0.18/0.38 ( ! [N2: nat] : ( ord_less_eq_int @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
% 0.18/0.38 => ( ( ord_less_eq_nat @ N @ N3 )
% 0.18/0.38 => ( ord_less_eq_int @ ( F @ N3 ) @ ( F @ N ) ) ) ) ).
% 0.18/0.38
% 0.18/0.38 % lift_Suc_antimono_le
% 0.18/0.38 thf(fact_131_lift__Suc__antimono__le,axiom,
% 0.18/0.38 ! [F: nat > nat,N: nat,N3: nat] :
% 0.18/0.38 ( ! [N2: nat] : ( ord_less_eq_nat @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
% 0.18/0.38 => ( ( ord_less_eq_nat @ N @ N3 )
% 0.18/0.38 => ( ord_less_eq_nat @ ( F @ N3 ) @ ( F @ N ) ) ) ) ).
% 0.18/0.38
% 0.18/0.38 % lift_Suc_antimono_le
% 0.18/0.38 thf(fact_132_lift__Suc__mono__le,axiom,
% 0.18/0.38 ! [F: nat > real,N: nat,N3: nat] :
% 0.18/0.38 ( ! [N2: nat] : ( ord_less_eq_real @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
% 0.18/0.38 => ( ( ord_less_eq_nat @ N @ N3 )
% 0.18/0.38 => ( ord_less_eq_real @ ( F @ N ) @ ( F @ N3 ) ) ) ) ).
% 0.18/0.38
% 0.18/0.38 % lift_Suc_mono_le
% 0.18/0.38 thf(fact_133_lift__Suc__mono__le,axiom,
% 0.18/0.38 ! [F: nat > int,N: nat,N3: nat] :
% 0.18/0.38 ( ! [N2: nat] : ( ord_less_eq_int @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
% 0.18/0.38 => ( ( ord_less_eq_nat @ N @ N3 )
% 0.18/0.38 => ( ord_less_eq_int @ ( F @ N ) @ ( F @ N3 ) ) ) ) ).
% 0.18/0.38
% 0.18/0.38 % lift_Suc_mono_le
% 0.18/0.38 thf(fact_134_lift__Suc__mono__le,axiom,
% 0.18/0.38 ! [F: nat > nat,N: nat,N3: nat] :
% 0.18/0.38 ( ! [N2: nat] : ( ord_less_eq_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
% 0.18/0.38 => ( ( ord_less_eq_nat @ N @ N3 )
% 0.18/0.38 => ( ord_less_eq_nat @ ( F @ N ) @ ( F @ N3 ) ) ) ) ).
% 0.18/0.38
% 0.18/0.38 % lift_Suc_mono_le
% 0.18/0.38 thf(fact_135_zero__le,axiom,
% 0.18/0.38 ! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).
% 0.18/0.38
% 0.18/0.38 % zero_le
% 0.18/0.38 thf(fact_136_power__divide,axiom,
% 0.18/0.38 ! [A: real,B: real,N: nat] :
% 0.18/0.38 ( ( power_power_real @ ( divide_divide_real @ A @ B ) @ N )
% 0.18/0.38 = ( divide_divide_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B @ N ) ) ) ).
% 0.18/0.38
% 0.18/0.38 % power_divide
% 0.18/0.38 thf(fact_137_ex__le__of__int,axiom,
% 0.18/0.38 ! [X: real] :
% 0.18/0.38 ? [Z2: int] : ( ord_less_eq_real @ X @ ( ring_1_of_int_real @ Z2 ) ) ).
% 0.18/0.38
% 0.18/0.38 % ex_le_of_int
% 0.18/0.38 thf(fact_138_exists__least__lemma,axiom,
% 0.18/0.38 ! [P: nat > $o] :
% 0.18/0.38 ( ~ ( P @ zero_zero_nat )
% 0.18/0.38 => ( ? [X_1: nat] : ( P @ X_1 )
% 0.18/0.38 => ? [N2: nat] :
% 0.18/0.38 ( ~ ( P @ N2 )
% 0.18/0.38 & ( P @ ( suc @ N2 ) ) ) ) ) ).
% 0.18/0.38
% 0.18/0.38 % exists_least_lemma
% 0.18/0.38 thf(fact_139_power__gt__expt,axiom,
% 0.18/0.38 ! [N: nat,K2: nat] :
% 0.18/0.38 ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N )
% 0.18/0.38 => ( ord_less_nat @ K2 @ ( power_power_nat @ N @ K2 ) ) ) ).
% 0.18/0.38
% 0.18/0.38 % power_gt_expt
% 0.18/0.38 thf(fact_140_less__eq__int__code_I1_J,axiom,
% 0.18/0.38 ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% 0.18/0.38
% 0.18/0.38 % less_eq_int_code(1)
% 0.18/0.38 thf(fact_141_nat_Odistinct_I1_J,axiom,
% 0.18/0.38 ! [X2: nat] :
% 0.18/0.38 ( zero_zero_nat
% 0.18/0.38 != ( suc @ X2 ) ) ).
% 0.18/0.38
% 0.18/0.38 % nat.distinct(1)
% 0.18/0.38 thf(fact_142_old_Onat_Odistinct_I2_J,axiom,
% 0.18/0.38 ! [Nat2: nat] :
% 0.18/0.38 ( ( suc @ Nat2 )
% 0.18/0.38 != zero_zero_nat ) ).
% 0.18/0.38
% 0.18/0.38 % old.nat.distinct(2)
% 0.18/0.38 thf(fact_143_old_Onat_Odistinct_I1_J,axiom,
% 0.18/0.38 ! [Nat2: nat] :
% 0.18/0.38 ( zero_zero_nat
% 0.18/0.38 != ( suc @ Nat2 ) ) ).
% 0.18/0.38
% 0.18/0.38 % old.nat.distinct(1)
% 0.18/0.38 thf(fact_144_nat_OdiscI,axiom,
% 0.18/0.38 ! [Nat: nat,X2: nat] :
% 0.18/0.38 ( ( Nat
% 0.18/0.38 = ( suc @ X2 ) )
% 0.18/0.38 => ( Nat != zero_zero_nat ) ) ).
% 0.18/0.38
% 0.18/0.38 % nat.discI
% 0.18/0.38 thf(fact_145_nat__induct,axiom,
% 0.18/0.38 ! [P: nat > $o,N: nat] :
% 0.18/0.38 ( ( P @ zero_zero_nat )
% 0.18/0.38 => ( ! [N2: nat] :
% 0.18/0.38 ( ( P @ N2 )
% 0.18/0.38 => ( P @ ( suc @ N2 ) ) )
% 0.18/0.38 => ( P @ N ) ) ) ).
% 0.18/0.38
% 0.18/0.38 % nat_induct
% 0.18/0.38 thf(fact_146_diff__induct,axiom,
% 0.18/0.38 ! [P: nat > nat > $o,M: nat,N: nat] :
% 0.18/0.38 ( ! [X3: nat] : ( P @ X3 @ zero_zero_nat )
% 0.18/0.38 => ( ! [Y3: nat] : ( P @ zero_zero_nat @ ( suc @ Y3 ) )
% 0.18/0.38 => ( ! [X3: nat,Y3: nat] :
% 0.18/0.38 ( ( P @ X3 @ Y3 )
% 0.18/0.38 => ( P @ ( suc @ X3 ) @ ( suc @ Y3 ) ) )
% 0.18/0.38 => ( P @ M @ N ) ) ) ) ).
% 0.18/0.38
% 0.18/0.38 % diff_induct
% 0.18/0.38 thf(fact_147_zero__induct,axiom,
% 0.18/0.38 ! [P: nat > $o,K2: nat] :
% 0.18/0.38 ( ( P @ K2 )
% 0.18/0.38 => ( ! [N2: nat] :
% 0.18/0.38 ( ( P @ ( suc @ N2 ) )
% 0.18/0.38 => ( P @ N2 ) )
% 0.18/0.38 => ( P @ zero_zero_nat ) ) ) ).
% 0.18/0.38
% 0.18/0.38 % zero_induct
% 0.18/0.38 thf(fact_148_Suc__neq__Zero,axiom,
% 0.18/0.38 ! [M: nat] :
% 0.18/0.38 ( ( suc @ M )
% 0.18/0.38 != zero_zero_nat ) ).
% 0.18/0.38
% 0.18/0.38 % Suc_neq_Zero
% 0.18/0.38 thf(fact_149_Zero__neq__Suc,axiom,
% 0.18/0.38 ! [M: nat] :
% 0.18/0.38 ( zero_zero_nat
% 0.18/0.38 != ( suc @ M ) ) ).
% 0.18/0.38
% 0.18/0.38 % Zero_neq_Suc
% 0.18/0.38 thf(fact_150_Zero__not__Suc,axiom,
% 0.18/0.38 ! [M: nat] :
% 0.18/0.38 ( zero_zero_nat
% 0.18/0.38 != ( suc @ M ) ) ).
% 0.18/0.38
% 0.18/0.38 % Zero_not_Suc
% 0.18/0.38 thf(fact_151_old_Onat_Oexhaust,axiom,
% 0.18/0.38 ! [Y: nat] :
% 0.18/0.38 ( ( Y != zero_zero_nat )
% 0.18/0.38 => ~ ! [Nat3: nat] :
% 0.18/0.38 ( Y
% 0.18/0.38 != ( suc @ Nat3 ) ) ) ).
% 0.18/0.38
% 0.18/0.38 % old.nat.exhaust
% 0.18/0.38 thf(fact_152_old_Onat_Oinducts,axiom,
% 0.18/0.38 ! [P: nat > $o,Nat: nat] :
% 0.18/0.38 ( ( P @ zero_zero_nat )
% 0.18/0.38 => ( ! [Nat3: nat] :
% 0.18/0.38 ( ( P @ Nat3 )
% 0.18/0.38 => ( P @ ( suc @ Nat3 ) ) )
% 0.18/0.38 => ( P @ Nat ) ) ) ).
% 0.18/0.38
% 0.18/0.38 % old.nat.inducts
% 0.18/0.38 thf(fact_153_not0__implies__Suc,axiom,
% 0.18/0.38 ! [N: nat] :
% 0.18/0.38 ( ( N != zero_zero_nat )
% 0.18/0.38 => ? [M4: nat] :
% 0.18/0.38 ( N
% 0.18/0.38 = ( suc @ M4 ) ) ) ).
% 0.18/0.38
% 0.18/0.38 % not0_implies_Suc
% 0.18/0.38 thf(fact_154_less__Suc__eq__0__disj,axiom,
% 0.18/0.38 ! [M: nat,N: nat] :
% 0.18/0.38 ( ( ord_less_nat @ M @ ( suc @ N ) )
% 0.18/0.38 = ( ( M = zero_zero_nat )
% 0.18/0.38 | ? [J3: nat] :
% 0.18/0.38 ( ( M
% 0.18/0.38 = ( suc @ J3 ) )
% 0.18/0.38 & ( ord_less_nat @ J3 @ N ) ) ) ) ).
% 0.18/0.38
% 0.18/0.38 % less_Suc_eq_0_disj
% 0.18/0.38 thf(fact_155_gr0__implies__Suc,axiom,
% 0.18/0.38 ! [N: nat] :
% 0.18/0.38 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.18/0.38 => ? [M4: nat] :
% 0.18/0.38 ( N
% 0.18/0.38 = ( suc @ M4 ) ) ) ).
% 0.18/0.38
% 0.18/0.38 % gr0_implies_Suc
% 0.18/0.38 thf(fact_156_All__less__Suc2,axiom,
% 0.18/0.38 ! [N: nat,P: nat > $o] :
% 0.18/0.38 ( ( ! [I3: nat] :
% 0.18/0.38 ( ( ord_less_nat @ I3 @ ( suc @ N ) )
% 0.18/0.38 => ( P @ I3 ) ) )
% 0.18/0.38 = ( ( P @ zero_zero_nat )
% 0.18/0.38 & ! [I3: nat] :
% 0.18/0.38 ( ( ord_less_nat @ I3 @ N )
% 0.18/0.38 => ( P @ ( suc @ I3 ) ) ) ) ) ).
% 0.18/0.38
% 0.18/0.38 % All_less_Suc2
% 0.18/0.38 thf(fact_157_gr0__conv__Suc,axiom,
% 0.18/0.38 ! [N: nat] :
% 0.18/0.38 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.18/0.38 = ( ? [M5: nat] :
% 0.18/0.38 ( N
% 0.18/0.38 = ( suc @ M5 ) ) ) ) ).
% 0.18/0.38
% 0.18/0.38 % gr0_conv_Suc
% 0.18/0.38 thf(fact_158_Ex__less__Suc2,axiom,
% 0.18/0.38 ! [N: nat,P: nat > $o] :
% 0.18/0.38 ( ( ? [I3: nat] :
% 0.18/0.38 ( ( ord_less_nat @ I3 @ ( suc @ N ) )
% 0.18/0.38 & ( P @ I3 ) ) )
% 0.18/0.38 = ( ( P @ zero_zero_nat )
% 0.18/0.38 | ? [I3: nat] :
% 0.18/0.38 ( ( ord_less_nat @ I3 @ N )
% 0.18/0.38 & ( P @ ( suc @ I3 ) ) ) ) ) ).
% 0.18/0.38
% 0.18/0.38 % Ex_less_Suc2
% 0.18/0.38 thf(fact_159_power__eq__imp__eq__base,axiom,
% 0.18/0.38 ! [A: real,N: nat,B: real] :
% 0.18/0.38 ( ( ( power_power_real @ A @ N )
% 0.18/0.38 = ( power_power_real @ B @ N ) )
% 0.18/0.38 => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.18/0.38 => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 0.18/0.38 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.18/0.38 => ( A = B ) ) ) ) ) ).
% 0.18/0.38
% 0.18/0.38 % power_eq_imp_eq_base
% 0.18/0.38 thf(fact_160_power__eq__imp__eq__base,axiom,
% 0.18/0.38 ! [A: int,N: nat,B: int] :
% 0.18/0.38 ( ( ( power_power_int @ A @ N )
% 0.18/0.38 = ( power_power_int @ B @ N ) )
% 0.18/0.38 => ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.18/0.38 => ( ( ord_less_eq_int @ zero_zero_int @ B )
% 0.18/0.38 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.18/0.38 => ( A = B ) ) ) ) ) ).
% 0.18/0.38
% 0.18/0.38 % power_eq_imp_eq_base
% 0.18/0.38 thf(fact_161_power__eq__imp__eq__base,axiom,
% 0.18/0.38 ! [A: nat,N: nat,B: nat] :
% 0.18/0.38 ( ( ( power_power_nat @ A @ N )
% 0.18/0.38 = ( power_power_nat @ B @ N ) )
% 0.18/0.38 => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.18/0.38 => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 0.18/0.38 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.18/0.38 => ( A = B ) ) ) ) ) ).
% 0.18/0.38
% 0.18/0.38 % power_eq_imp_eq_base
% 0.18/0.38 thf(fact_162_power__eq__iff__eq__base,axiom,
% 0.18/0.38 ! [N: nat,A: real,B: real] :
% 0.18/0.38 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.18/0.38 => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.18/0.38 => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 0.18/0.38 => ( ( ( power_power_real @ A @ N )
% 0.18/0.38 = ( power_power_real @ B @ N ) )
% 0.18/0.38 = ( A = B ) ) ) ) ) ).
% 0.18/0.38
% 0.18/0.38 % power_eq_iff_eq_base
% 0.18/0.38 thf(fact_163_power__eq__iff__eq__base,axiom,
% 0.18/0.38 ! [N: nat,A: int,B: int] :
% 0.18/0.38 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.18/0.38 => ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.18/0.38 => ( ( ord_less_eq_int @ zero_zero_int @ B )
% 0.18/0.38 => ( ( ( power_power_int @ A @ N )
% 0.18/0.38 = ( power_power_int @ B @ N ) )
% 0.18/0.38 = ( A = B ) ) ) ) ) ).
% 0.18/0.38
% 0.18/0.38 % power_eq_iff_eq_base
% 0.18/0.38 thf(fact_164_power__eq__iff__eq__base,axiom,
% 0.18/0.38 ! [N: nat,A: nat,B: nat] :
% 0.18/0.38 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.18/0.38 => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.18/0.38 => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 0.18/0.38 => ( ( ( power_power_nat @ A @ N )
% 0.18/0.38 = ( power_power_nat @ B @ N ) )
% 0.18/0.38 = ( A = B ) ) ) ) ) ).
% 0.18/0.38
% 0.18/0.38 % power_eq_iff_eq_base
% 0.18/0.38 thf(fact_165_zero__le__power,axiom,
% 0.18/0.38 ! [A: real,N: nat] :
% 0.18/0.38 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.18/0.38 => ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ N ) ) ) ).
% 0.18/0.38
% 0.18/0.38 % zero_le_power
% 0.18/0.38 thf(fact_166_zero__le__power,axiom,
% 0.18/0.38 ! [A: int,N: nat] :
% 0.18/0.38 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.18/0.38 => ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ N ) ) ) ).
% 0.18/0.38
% 0.18/0.38 % zero_le_power
% 0.18/0.38 thf(fact_167_zero__le__power,axiom,
% 0.18/0.38 ! [A: nat,N: nat] :
% 0.18/0.38 ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.18/0.38 => ( ord_less_eq_nat @ zero_zero_nat @ ( power_power_nat @ A @ N ) ) ) ).
% 0.18/0.38
% 0.18/0.38 % zero_le_power
% 0.18/0.38 thf(fact_168_power__mono,axiom,
% 0.18/0.38 ! [A: real,B: real,N: nat] :
% 0.18/0.38 ( ( ord_less_eq_real @ A @ B )
% 0.18/0.38 => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.18/0.38 => ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B @ N ) ) ) ) ).
% 0.18/0.38
% 0.18/0.38 % power_mono
% 0.18/0.38 thf(fact_169_power__mono,axiom,
% 0.18/0.38 ! [A: int,B: int,N: nat] :
% 0.18/0.38 ( ( ord_less_eq_int @ A @ B )
% 0.18/0.38 => ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.18/0.38 => ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) ) ) ) ).
% 0.18/0.38
% 0.18/0.38 % power_mono
% 0.18/0.38 thf(fact_170_power__mono,axiom,
% 0.18/0.38 ! [A: nat,B: nat,N: nat] :
% 0.18/0.38 ( ( ord_less_eq_nat @ A @ B )
% 0.18/0.38 => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.18/0.38 => ( ord_less_eq_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) ) ) ) ).
% 0.18/0.38
% 0.18/0.38 % power_mono
% 0.18/0.38 thf(fact_171_of__int__nonneg,axiom,
% 0.18/0.38 ! [Z: int] :
% 0.18/0.38 ( ( ord_less_eq_int @ zero_zero_int @ Z )
% 0.18/0.38 => ( ord_less_eq_real @ zero_zero_real @ ( ring_1_of_int_real @ Z ) ) ) ).
% 0.18/0.38
% 0.18/0.38 % of_int_nonneg
% 0.18/0.38 thf(fact_172_of__int__nonneg,axiom,
% 0.18/0.38 ! [Z: int] :
% 0.18/0.38 ( ( ord_less_eq_int @ zero_zero_int @ Z )
% 0.18/0.38 => ( ord_less_eq_int @ zero_zero_int @ ( ring_1_of_int_int @ Z ) ) ) ).
% 0.18/0.38
% 0.18/0.38 % of_int_nonneg
% 0.18/0.38 thf(fact_173_power__less__imp__less__base,axiom,
% 0.18/0.38 ! [A: real,N: nat,B: real] :
% 0.18/0.38 ( ( ord_less_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B @ N ) )
% 0.18/0.38 => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 0.18/0.38 => ( ord_less_real @ A @ B ) ) ) ).
% 0.18/0.38
% 0.18/0.38 % power_less_imp_less_base
% 0.18/0.38 thf(fact_174_power__less__imp__less__base,axiom,
% 0.18/0.38 ! [A: int,N: nat,B: int] :
% 0.18/0.38 ( ( ord_less_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) )
% 0.18/0.38 => ( ( ord_less_eq_int @ zero_zero_int @ B )
% 0.18/0.38 => ( ord_less_int @ A @ B ) ) ) ).
% 0.18/0.38
% 0.18/0.38 % power_less_imp_less_base
% 0.18/0.38 thf(fact_175_power__less__imp__less__base,axiom,
% 0.18/0.38 ! [A: nat,N: nat,B: nat] :
% 0.18/0.38 ( ( ord_less_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) )
% 0.18/0.38 => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 0.18/0.38 => ( ord_less_nat @ A @ B ) ) ) ).
% 0.18/0.38
% 0.18/0.38 % power_less_imp_less_base
% 0.18/0.38 thf(fact_176_power__strict__mono,axiom,
% 0.18/0.38 ! [A: real,B: real,N: nat] :
% 0.18/0.38 ( ( ord_less_real @ A @ B )
% 0.18/0.38 => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.18/0.38 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.18/0.38 => ( ord_less_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B @ N ) ) ) ) ) ).
% 0.18/0.38
% 0.18/0.38 % power_strict_mono
% 0.18/0.38 thf(fact_177_power__strict__mono,axiom,
% 0.18/0.38 ! [A: int,B: int,N: nat] :
% 0.18/0.38 ( ( ord_less_int @ A @ B )
% 0.18/0.38 => ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.18/0.38 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.18/0.38 => ( ord_less_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) ) ) ) ) ).
% 0.18/0.38
% 0.18/0.38 % power_strict_mono
% 0.18/0.38 thf(fact_178_power__strict__mono,axiom,
% 0.18/0.38 ! [A: nat,B: nat,N: nat] :
% 0.18/0.38 ( ( ord_less_nat @ A @ B )
% 0.18/0.38 => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.18/0.38 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.18/0.38 => ( ord_less_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) ) ) ) ) ).
% 0.18/0.38
% 0.18/0.38 % power_strict_mono
% 0.18/0.38 thf(fact_179_power__inject__base,axiom,
% 0.18/0.38 ! [A: real,N: nat,B: real] :
% 0.18/0.38 ( ( ( power_power_real @ A @ ( suc @ N ) )
% 0.18/0.38 = ( power_power_real @ B @ ( suc @ N ) ) )
% 0.18/0.38 => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.18/0.38 => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 0.18/0.38 => ( A = B ) ) ) ) ).
% 0.18/0.38
% 0.18/0.38 % power_inject_base
% 0.18/0.38 thf(fact_180_power__inject__base,axiom,
% 0.18/0.38 ! [A: int,N: nat,B: int] :
% 0.18/0.38 ( ( ( power_power_int @ A @ ( suc @ N ) )
% 0.18/0.38 = ( power_power_int @ B @ ( suc @ N ) ) )
% 0.18/0.38 => ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.18/0.38 => ( ( ord_less_eq_int @ zero_zero_int @ B )
% 0.18/0.38 => ( A = B ) ) ) ) ).
% 0.18/0.38
% 0.18/0.38 % power_inject_base
% 0.18/0.38 thf(fact_181_power__inject__base,axiom,
% 0.18/0.38 ! [A: nat,N: nat,B: nat] :
% 0.18/0.38 ( ( ( power_power_nat @ A @ ( suc @ N ) )
% 0.18/0.38 = ( power_power_nat @ B @ ( suc @ N ) ) )
% 0.18/0.38 => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.18/0.38 => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 0.18/0.38 => ( A = B ) ) ) ) ).
% 0.18/0.38
% 0.18/0.38 % power_inject_base
% 0.18/0.38 thf(fact_182_power__le__imp__le__base,axiom,
% 0.18/0.38 ! [A: real,N: nat,B: real] :
% 0.18/0.38 ( ( ord_less_eq_real @ ( power_power_real @ A @ ( suc @ N ) ) @ ( power_power_real @ B @ ( suc @ N ) ) )
% 0.18/0.38 => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 0.18/0.38 => ( ord_less_eq_real @ A @ B ) ) ) ).
% 0.18/0.38
% 0.18/0.38 % power_le_imp_le_base
% 0.18/0.38 thf(fact_183_power__le__imp__le__base,axiom,
% 0.18/0.38 ! [A: int,N: nat,B: int] :
% 0.18/0.38 ( ( ord_less_eq_int @ ( power_power_int @ A @ ( suc @ N ) ) @ ( power_power_int @ B @ ( suc @ N ) ) )
% 0.18/0.38 => ( ( ord_less_eq_int @ zero_zero_int @ B )
% 0.18/0.38 => ( ord_less_eq_int @ A @ B ) ) ) ).
% 0.18/0.38
% 0.18/0.38 % power_le_imp_le_base
% 0.18/0.38 thf(fact_184_power__le__imp__le__base,axiom,
% 0.18/0.38 ! [A: nat,N: nat,B: nat] :
% 0.18/0.38 ( ( ord_less_eq_nat @ ( power_power_nat @ A @ ( suc @ N ) ) @ ( power_power_nat @ B @ ( suc @ N ) ) )
% 0.18/0.38 => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 0.18/0.38 => ( ord_less_eq_nat @ A @ B ) ) ) ).
% 0.18/0.38
% 0.18/0.38 % power_le_imp_le_base
% 0.18/0.38 thf(fact_185_realpow__pos__nth,axiom,
% 0.18/0.38 ! [N: nat,A: real] :
% 0.18/0.38 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.18/0.38 => ( ( ord_less_real @ zero_zero_real @ A )
% 0.18/0.38 => ? [R: real] :
% 0.18/0.38 ( ( ord_less_real @ zero_zero_real @ R )
% 0.18/0.38 & ( ( power_power_real @ R @ N )
% 0.18/0.38 = A ) ) ) ) ).
% 0.18/0.38
% 0.18/0.38 % realpow_pos_nth
% 0.18/0.38 thf(fact_186_realpow__pos__nth__unique,axiom,
% 0.18/0.38 ! [N: nat,A: real] :
% 0.18/0.38 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.18/0.38 => ( ( ord_less_real @ zero_zero_real @ A )
% 0.18/0.38 => ? [X3: real] :
% 0.18/0.38 ( ( ord_less_real @ zero_zero_real @ X3 )
% 0.18/0.38 & ( ( power_power_real @ X3 @ N )
% 0.18/0.38 = A )
% 0.18/0.38 & ! [Y4: real] :
% 0.18/0.38 ( ( ( ord_less_real @ zero_zero_real @ Y4 )
% 0.18/0.38 & ( ( power_power_real @ Y4 @ N )
% 0.18/0.38 = A ) )
% 0.18/0.38 => ( Y4 = X3 ) ) ) ) ) ).
% 0.18/0.38
% 0.18/0.38 % realpow_pos_nth_unique
% 0.18/0.38 thf(fact_187_zero__reorient,axiom,
% 0.18/0.38 ! [X: int] :
% 0.18/0.38 ( ( zero_zero_int = X )
% 0.18/0.38 = ( X = zero_zero_int ) ) ).
% 0.18/0.38
% 0.18/0.38 % zero_reorient
% 0.18/0.38 thf(fact_188_zero__reorient,axiom,
% 0.18/0.38 ! [X: real] :
% 0.18/0.38 ( ( zero_zero_real = X )
% 0.18/0.38 = ( X = zero_zero_real ) ) ).
% 0.18/0.38
% 0.18/0.38 % zero_reorient
% 0.18/0.38 thf(fact_189_zero__reorient,axiom,
% 0.18/0.38 ! [X: nat] :
% 0.18/0.38 ( ( zero_zero_nat = X )
% 0.18/0.38 = ( X = zero_zero_nat ) ) ).
% 0.18/0.38
% 0.18/0.38 % zero_reorient
% 0.18/0.38 thf(fact_190_gr__zeroI,axiom,
% 0.18/0.38 ! [N: nat] :
% 0.18/0.38 ( ( N != zero_zero_nat )
% 0.18/0.38 => ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 0.18/0.38
% 0.18/0.38 % gr_zeroI
% 0.18/0.38 thf(fact_191_not__less__zero,axiom,
% 0.18/0.38 ! [N: nat] :
% 0.18/0.38 ~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% 0.18/0.38
% 0.18/0.38 % not_less_zero
% 0.18/0.38 thf(fact_192_gr__implies__not__zero,axiom,
% 0.18/0.38 ! [M: nat,N: nat] :
% 0.18/0.38 ( ( ord_less_nat @ M @ N )
% 0.18/0.38 => ( N != zero_zero_nat ) ) ).
% 0.18/0.38
% 0.18/0.38 % gr_implies_not_zero
% 0.18/0.38 thf(fact_193_zero__less__iff__neq__zero,axiom,
% 0.18/0.38 ! [N: nat] :
% 0.18/0.38 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.18/0.38 = ( N != zero_zero_nat ) ) ).
% 0.18/0.38
% 0.18/0.38 % zero_less_iff_neq_zero
% 0.18/0.38 thf(fact_194_power__not__zero,axiom,
% 0.18/0.38 ! [A: real,N: nat] :
% 0.18/0.38 ( ( A != zero_zero_real )
% 0.18/0.38 => ( ( power_power_real @ A @ N )
% 0.18/0.38 != zero_zero_real ) ) ).
% 0.18/0.38
% 0.18/0.38 % power_not_zero
% 0.18/0.38 thf(fact_195_power__not__zero,axiom,
% 0.18/0.38 ! [A: int,N: nat] :
% 0.18/0.38 ( ( A != zero_zero_int )
% 0.18/0.38 => ( ( power_power_int @ A @ N )
% 0.18/0.38 != zero_zero_int ) ) ).
% 0.18/0.38
% 0.18/0.38 % power_not_zero
% 0.18/0.38 thf(fact_196_power__not__zero,axiom,
% 0.18/0.38 ! [A: nat,N: nat] :
% 0.18/0.38 ( ( A != zero_zero_nat )
% 0.18/0.38 => ( ( power_power_nat @ A @ N )
% 0.18/0.38 != zero_zero_nat ) ) ).
% 0.18/0.38
% 0.18/0.38 % power_not_zero
% 0.18/0.38 thf(fact_197_zero__power,axiom,
% 0.18/0.38 ! [N: nat] :
% 0.18/0.38 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.18/0.38 => ( ( power_power_real @ zero_zero_real @ N )
% 0.18/0.38 = zero_zero_real ) ) ).
% 0.18/0.38
% 0.18/0.38 % zero_power
% 0.18/0.38 thf(fact_198_zero__power,axiom,
% 0.18/0.38 ! [N: nat] :
% 0.18/0.38 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.18/0.38 => ( ( power_power_int @ zero_zero_int @ N )
% 0.18/0.38 = zero_zero_int ) ) ).
% 0.18/0.38
% 0.18/0.38 % zero_power
% 0.18/0.38 thf(fact_199_zero__power,axiom,
% 0.18/0.38 ! [N: nat] :
% 0.18/0.38 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.18/0.38 => ( ( power_power_nat @ zero_zero_nat @ N )
% 0.18/0.38 = zero_zero_nat ) ) ).
% 0.18/0.38
% 0.18/0.38 % zero_power
% 0.18/0.38 thf(fact_200_yyn_H,axiom,
% 0.18/0.38 ord_less_int @ ( power_power_int @ y @ p ) @ na ).
% 0.18/0.38
% 0.18/0.38 % yyn'
% 0.18/0.38 thf(fact_201_y_H0,axiom,
% 0.18/0.38 ord_less_eq_int @ zero_zero_int @ y ).
% 0.18/0.38
% 0.18/0.38 % y'0
% 0.18/0.38 thf(fact_202_compare__pow__less__iff,axiom,
% 0.18/0.38 ! [P2: nat,X: real,Y: real] :
% 0.18/0.38 ( ( ord_less_nat @ zero_zero_nat @ P2 )
% 0.18/0.38 => ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.18/0.38 => ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 0.18/0.38 => ( ( ord_less_real @ ( power_power_real @ X @ P2 ) @ ( power_power_real @ Y @ P2 ) )
% 0.18/0.38 = ( ord_less_real @ X @ Y ) ) ) ) ) ).
% 0.18/0.38
% 0.18/0.38 % compare_pow_less_iff
% 0.18/0.38 thf(fact_203_compare__pow__less__iff,axiom,
% 0.18/0.38 ! [P2: nat,X: int,Y: int] :
% 0.18/0.38 ( ( ord_less_nat @ zero_zero_nat @ P2 )
% 0.18/0.38 => ( ( ord_less_eq_int @ zero_zero_int @ X )
% 0.18/0.38 => ( ( ord_less_eq_int @ zero_zero_int @ Y )
% 0.18/0.38 => ( ( ord_less_int @ ( power_power_int @ X @ P2 ) @ ( power_power_int @ Y @ P2 ) )
% 0.18/0.38 = ( ord_less_int @ X @ Y ) ) ) ) ) ).
% 0.18/0.38
% 0.18/0.38 % compare_pow_less_iff
% 0.18/0.38 thf(fact_204_compare__pow__le__iff,axiom,
% 0.18/0.38 ! [P2: nat,X: real,Y: real] :
% 0.18/0.38 ( ( ord_less_nat @ zero_zero_nat @ P2 )
% 0.18/0.38 => ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.18/0.38 => ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 0.18/0.38 => ( ( ord_less_eq_real @ ( power_power_real @ X @ P2 ) @ ( power_power_real @ Y @ P2 ) )
% 0.18/0.38 = ( ord_less_eq_real @ X @ Y ) ) ) ) ) ).
% 0.18/0.38
% 0.18/0.38 % compare_pow_le_iff
% 0.18/0.38 thf(fact_205_compare__pow__le__iff,axiom,
% 0.18/0.38 ! [P2: nat,X: int,Y: int] :
% 0.18/0.38 ( ( ord_less_nat @ zero_zero_nat @ P2 )
% 0.18/0.38 => ( ( ord_less_eq_int @ zero_zero_int @ X )
% 0.18/0.38 => ( ( ord_less_eq_int @ zero_zero_int @ Y )
% 0.18/0.38 => ( ( ord_less_eq_int @ ( power_power_int @ X @ P2 ) @ ( power_power_int @ Y @ P2 ) )
% 0.18/0.38 = ( ord_less_eq_int @ X @ Y ) ) ) ) ) ).
% 0.18/0.38
% 0.18/0.38 % compare_pow_le_iff
% 0.18/0.38 thf(fact_206_division__ring__divide__zero,axiom,
% 0.18/0.38 ! [A: real] :
% 0.18/0.38 ( ( divide_divide_real @ A @ zero_zero_real )
% 0.18/0.38 = zero_zero_real ) ).
% 0.18/0.38
% 0.18/0.38 % division_ring_divide_zero
% 0.18/0.38 thf(fact_207_bits__div__by__0,axiom,
% 0.18/0.38 ! [A: int] :
% 0.18/0.38 ( ( divide_divide_int @ A @ zero_zero_int )
% 0.18/0.38 = zero_zero_int ) ).
% 0.18/0.38
% 0.18/0.38 % bits_div_by_0
% 0.18/0.38 thf(fact_208_bits__div__by__0,axiom,
% 0.18/0.38 ! [A: nat] :
% 0.18/0.38 ( ( divide_divide_nat @ A @ zero_zero_nat )
% 0.18/0.38 = zero_zero_nat ) ).
% 0.18/0.38
% 0.18/0.38 % bits_div_by_0
% 0.18/0.38 thf(fact_209_divide__cancel__right,axiom,
% 0.18/0.38 ! [A: real,C: real,B: real] :
% 0.18/0.38 ( ( ( divide_divide_real @ A @ C )
% 0.18/0.38 = ( divide_divide_real @ B @ C ) )
% 0.18/0.38 = ( ( C = zero_zero_real )
% 0.18/0.38 | ( A = B ) ) ) ).
% 0.18/0.38
% 0.18/0.38 % divide_cancel_right
% 0.18/0.38 thf(fact_210_divide__eq__0__iff,axiom,
% 0.18/0.38 ! [A: real,B: real] :
% 0.18/0.38 ( ( ( divide_divide_real @ A @ B )
% 0.18/0.38 = zero_zero_real )
% 0.18/0.38 = ( ( A = zero_zero_real )
% 0.18/0.38 | ( B = zero_zero_real ) ) ) ).
% 0.18/0.38
% 0.18/0.38 % divide_eq_0_iff
% 0.18/0.38 thf(fact_211_divide__cancel__left,axiom,
% 0.18/0.38 ! [C: real,A: real,B: real] :
% 0.18/0.38 ( ( ( divide_divide_real @ C @ A )
% 0.18/0.38 = ( divide_divide_real @ C @ B ) )
% 0.18/0.38 = ( ( C = zero_zero_real )
% 0.18/0.38 | ( A = B ) ) ) ).
% 0.18/0.38
% 0.18/0.38 % divide_cancel_left
% 0.18/0.38 thf(fact_212_bits__div__0,axiom,
% 0.18/0.38 ! [A: int] :
% 0.18/0.38 ( ( divide_divide_int @ zero_zero_int @ A )
% 0.18/0.38 = zero_zero_int ) ).
% 0.18/0.38
% 0.18/0.38 % bits_div_0
% 0.18/0.38 thf(fact_213_bits__div__0,axiom,
% 0.18/0.38 ! [A: nat] :
% 0.18/0.38 ( ( divide_divide_nat @ zero_zero_nat @ A )
% 0.18/0.38 = zero_zero_nat ) ).
% 0.18/0.38
% 0.18/0.38 % bits_div_0
% 0.18/0.38 thf(fact_214_bot__nat__0_Oextremum,axiom,
% 0.18/0.38 ! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% 0.18/0.38
% 0.18/0.38 % bot_nat_0.extremum
% 0.18/0.38 thf(fact_215_le0,axiom,
% 0.18/0.38 ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% 0.18/0.38
% 0.18/0.38 % le0
% 0.18/0.38 thf(fact_216_Suc__le__mono,axiom,
% 0.18/0.38 ! [N: nat,M: nat] :
% 0.18/0.38 ( ( ord_less_eq_nat @ ( suc @ N ) @ ( suc @ M ) )
% 0.18/0.38 = ( ord_less_eq_nat @ N @ M ) ) ).
% 0.18/0.38
% 0.18/0.38 % Suc_le_mono
% 0.18/0.38 thf(fact_217_le__refl,axiom,
% 0.18/0.38 ! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% 0.18/0.38
% 0.18/0.38 % le_refl
% 0.18/0.38 thf(fact_218_le__trans,axiom,
% 0.18/0.38 ! [I: nat,J: nat,K2: nat] :
% 0.18/0.38 ( ( ord_less_eq_nat @ I @ J )
% 0.18/0.38 => ( ( ord_less_eq_nat @ J @ K2 )
% 0.18/0.38 => ( ord_less_eq_nat @ I @ K2 ) ) ) ).
% 0.18/0.38
% 0.18/0.38 % le_trans
% 0.18/0.38 thf(fact_219_eq__imp__le,axiom,
% 0.18/0.38 ! [M: nat,N: nat] :
% 0.18/0.38 ( ( M = N )
% 0.18/0.38 => ( ord_less_eq_nat @ M @ N ) ) ).
% 0.18/0.38
% 0.18/0.38 % eq_imp_le
% 0.18/0.38 thf(fact_220_le__antisym,axiom,
% 0.18/0.38 ! [M: nat,N: nat] :
% 0.18/0.38 ( ( ord_less_eq_nat @ M @ N )
% 0.18/0.38 => ( ( ord_less_eq_nat @ N @ M )
% 0.18/0.38 => ( M = N ) ) ) ).
% 0.18/0.38
% 0.18/0.38 % le_antisym
% 0.18/0.38 thf(fact_221_nat__le__linear,axiom,
% 0.18/0.38 ! [M: nat,N: nat] :
% 0.18/0.38 ( ( ord_less_eq_nat @ M @ N )
% 0.18/0.38 | ( ord_less_eq_nat @ N @ M ) ) ).
% 0.18/0.38
% 0.18/0.38 % nat_le_linear
% 0.18/0.38 thf(fact_222_Nat_Oex__has__greatest__nat,axiom,
% 0.18/0.38 ! [P: nat > $o,K2: nat,B: nat] :
% 0.18/0.38 ( ( P @ K2 )
% 0.18/0.38 => ( ! [Y3: nat] :
% 0.18/0.38 ( ( P @ Y3 )
% 0.18/0.38 => ( ord_less_eq_nat @ Y3 @ B ) )
% 0.18/0.38 => ? [X3: nat] :
% 0.18/0.38 ( ( P @ X3 )
% 0.18/0.38 & ! [Y4: nat] :
% 0.18/0.38 ( ( P @ Y4 )
% 0.18/0.38 => ( ord_less_eq_nat @ Y4 @ X3 ) ) ) ) ) ).
% 0.18/0.38
% 0.18/0.38 % Nat.ex_has_greatest_nat
% 0.18/0.38 thf(fact_223_bot__nat__0_Oextremum__uniqueI,axiom,
% 0.18/0.38 ! [A: nat] :
% 0.18/0.38 ( ( ord_less_eq_nat @ A @ zero_zero_nat )
% 0.18/0.38 => ( A = zero_zero_nat ) ) ).
% 0.18/0.38
% 0.18/0.38 % bot_nat_0.extremum_uniqueI
% 0.18/0.38 thf(fact_224_bot__nat__0_Oextremum__unique,axiom,
% 0.18/0.38 ! [A: nat] :
% 0.18/0.38 ( ( ord_less_eq_nat @ A @ zero_zero_nat )
% 0.18/0.38 = ( A = zero_zero_nat ) ) ).
% 0.18/0.38
% 0.18/0.38 % bot_nat_0.extremum_unique
% 0.18/0.38 thf(fact_225_le__0__eq,axiom,
% 0.18/0.38 ! [N: nat] :
% 0.18/0.38 ( ( ord_less_eq_nat @ N @ zero_zero_nat )
% 0.18/0.38 = ( N = zero_zero_nat ) ) ).
% 0.18/0.38
% 0.18/0.38 % le_0_eq
% 0.18/0.38 thf(fact_226_less__eq__nat_Osimps_I1_J,axiom,
% 0.18/0.38 ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% 0.18/0.38
% 0.18/0.38 % less_eq_nat.simps(1)
% 0.18/0.38 thf(fact_227_transitive__stepwise__le,axiom,
% 0.18/0.38 ! [M: nat,N: nat,R2: nat > nat > $o] :
% 0.18/0.38 ( ( ord_less_eq_nat @ M @ N )
% 0.18/0.38 => ( ! [X3: nat] : ( R2 @ X3 @ X3 )
% 0.18/0.38 => ( ! [X3: nat,Y3: nat,Z2: nat] :
% 0.18/0.38 ( ( R2 @ X3 @ Y3 )
% 0.18/0.38 => ( ( R2 @ Y3 @ Z2 )
% 0.18/0.38 => ( R2 @ X3 @ Z2 ) ) )
% 0.18/0.38 => ( ! [N2: nat] : ( R2 @ N2 @ ( suc @ N2 ) )
% 0.18/0.38 => ( R2 @ M @ N ) ) ) ) ) ).
% 0.18/0.38
% 0.18/0.38 % transitive_stepwise_le
% 0.18/0.38 thf(fact_228_nat__induct__at__least,axiom,
% 0.18/0.38 ! [M: nat,N: nat,P: nat > $o] :
% 0.18/0.38 ( ( ord_less_eq_nat @ M @ N )
% 0.18/0.38 => ( ( P @ M )
% 0.18/0.38 => ( ! [N2: nat] :
% 0.18/0.38 ( ( ord_less_eq_nat @ M @ N2 )
% 0.18/0.38 => ( ( P @ N2 )
% 0.18/0.38 => ( P @ ( suc @ N2 ) ) ) )
% 0.18/0.38 => ( P @ N ) ) ) ) ).
% 0.18/0.38
% 0.18/0.38 % nat_induct_at_least
% 0.18/0.38 thf(fact_229_full__nat__induct,axiom,
% 0.18/0.38 ! [P: nat > $o,N: nat] :
% 0.18/0.38 ( ! [N2: nat] :
% 0.18/0.38 ( ! [M3: nat] :
% 0.18/0.38 ( ( ord_less_eq_nat @ ( suc @ M3 ) @ N2 )
% 0.18/0.38 => ( P @ M3 ) )
% 0.18/0.38 => ( P @ N2 ) )
% 0.18/0.38 => ( P @ N ) ) ).
% 0.18/0.38
% 0.18/0.38 % full_nat_induct
% 0.18/0.38 thf(fact_230_not__less__eq__eq,axiom,
% 0.18/0.38 ! [M: nat,N: nat] :
% 0.18/0.38 ( ( ~ ( ord_less_eq_nat @ M @ N ) )
% 0.18/0.38 = ( ord_less_eq_nat @ ( suc @ N ) @ M ) ) ).
% 0.18/0.38
% 0.18/0.38 % not_less_eq_eq
% 0.18/0.38 thf(fact_231_Suc__n__not__le__n,axiom,
% 0.18/0.38 ! [N: nat] :
% 0.18/0.38 ~ ( ord_less_eq_nat @ ( suc @ N ) @ N ) ).
% 0.18/0.38
% 0.18/0.38 % Suc_n_not_le_n
% 0.18/0.38 thf(fact_232_le__Suc__eq,axiom,
% 0.18/0.38 ! [M: nat,N: nat] :
% 0.18/0.38 ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
% 0.18/0.38 = ( ( ord_less_eq_nat @ M @ N )
% 0.18/0.38 | ( M
% 0.18/0.38 = ( suc @ N ) ) ) ) ).
% 0.18/0.38
% 0.18/0.38 % le_Suc_eq
% 0.18/0.38 thf(fact_233_Suc__le__D,axiom,
% 0.18/0.38 ! [N: nat,M6: nat] :
% 0.18/0.38 ( ( ord_less_eq_nat @ ( suc @ N ) @ M6 )
% 0.18/0.38 => ? [M4: nat] :
% 0.18/0.38 ( M6
% 0.18/0.38 = ( suc @ M4 ) ) ) ).
% 0.18/0.38
% 0.18/0.38 % Suc_le_D
% 0.18/0.38 thf(fact_234_le__SucI,axiom,
% 0.18/0.38 ! [M: nat,N: nat] :
% 0.18/0.38 ( ( ord_less_eq_nat @ M @ N )
% 0.18/0.38 => ( ord_less_eq_nat @ M @ ( suc @ N ) ) ) ).
% 0.18/0.38
% 0.18/0.38 % le_SucI
% 0.18/0.38 thf(fact_235_le__SucE,axiom,
% 0.18/0.38 ! [M: nat,N: nat] :
% 0.18/0.38 ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
% 0.18/0.38 => ( ~ ( ord_less_eq_nat @ M @ N )
% 0.18/0.38 => ( M
% 0.18/0.38 = ( suc @ N ) ) ) ) ).
% 0.18/0.38
% 0.18/0.38 % le_SucE
% 0.18/0.38 thf(fact_236_Suc__leD,axiom,
% 0.18/0.38 ! [M: nat,N: nat] :
% 0.18/0.38 ( ( ord_less_eq_nat @ ( suc @ M ) @ N )
% 0.18/0.38 => ( ord_less_eq_nat @ M @ N ) ) ).
% 0.18/0.38
% 0.18/0.38 % Suc_leD
% 0.18/0.38 thf(fact_237_less__mono__imp__le__mono,axiom,
% 0.18/0.38 ! [F: nat > nat,I: nat,J: nat] :
% 0.18/0.38 ( ! [I2: nat,J2: nat] :
% 0.18/0.38 ( ( ord_less_nat @ I2 @ J2 )
% 0.18/0.38 => ( ord_less_nat @ ( F @ I2 ) @ ( F @ J2 ) ) )
% 0.18/0.38 => ( ( ord_less_eq_nat @ I @ J )
% 0.18/0.38 => ( ord_less_eq_nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).
% 0.18/0.38
% 0.18/0.38 % less_mono_imp_le_mono
% 0.18/0.38 thf(fact_238_le__neq__implies__less,axiom,
% 0.18/0.38 ! [M: nat,N: nat] :
% 0.18/0.38 ( ( ord_less_eq_nat @ M @ N )
% 0.18/0.38 => ( ( M != N )
% 0.18/0.38 => ( ord_less_nat @ M @ N ) ) ) ).
% 0.18/0.38
% 0.18/0.38 % le_neq_implies_less
% 0.18/0.38 thf(fact_239_less__or__eq__imp__le,axiom,
% 0.18/0.38 ! [M: nat,N: nat] :
% 0.18/0.38 ( ( ( ord_less_nat @ M @ N )
% 0.18/0.38 | ( M = N ) )
% 0.18/0.38 => ( ord_less_eq_nat @ M @ N ) ) ).
% 0.18/0.38
% 0.18/0.38 % less_or_eq_imp_le
% 0.18/0.38 thf(fact_240_le__eq__less__or__eq,axiom,
% 0.18/0.38 ( ord_less_eq_nat
% 0.18/0.38 = ( ^ [M5: nat,N4: nat] :
% 0.18/0.38 ( ( ord_less_nat @ M5 @ N4 )
% 0.18/0.38 | ( M5 = N4 ) ) ) ) ).
% 0.18/0.38
% 0.18/0.38 % le_eq_less_or_eq
% 0.18/0.38 thf(fact_241_less__imp__le__nat,axiom,
% 0.18/0.38 ! [M: nat,N: nat] :
% 0.18/0.38 ( ( ord_less_nat @ M @ N )
% 0.18/0.38 => ( ord_less_eq_nat @ M @ N ) ) ).
% 0.18/0.38
% 0.18/0.38 % less_imp_le_nat
% 0.18/0.38 thf(fact_242_nat__less__le,axiom,
% 0.18/0.38 ( ord_less_nat
% 0.18/0.38 = ( ^ [M5: nat,N4: nat] :
% 0.18/0.38 ( ( ord_less_eq_nat @ M5 @ N4 )
% 0.18/0.38 & ( M5 != N4 ) ) ) ) ).
% 0.18/0.38
% 0.18/0.38 % nat_less_le
% 0.18/0.38 thf(fact_243_fixed__root_Op0,axiom,
% 0.18/0.38 ! [P2: nat,Pm: nat] :
% 0.18/0.38 ( ( nthRoo1352626670d_root @ P2 @ Pm )
% 0.18/0.38 => ( P2 != zero_zero_nat ) ) ).
% 0.18/0.38
% 0.18/0.38 % fixed_root.p0
% 0.18/0.38 thf(fact_244_fixed__root_Ointro,axiom,
% 0.18/0.38 ! [P2: nat,Pm: nat] :
% 0.18/0.38 ( ( P2
% 0.18/0.38 = ( suc @ Pm ) )
% 0.18/0.38 => ( nthRoo1352626670d_root @ P2 @ Pm ) ) ).
% 0.18/0.38
% 0.18/0.38 % fixed_root.intro
% 0.18/0.38 thf(fact_245_fixed__root__def,axiom,
% 0.18/0.38 ( nthRoo1352626670d_root
% 0.18/0.38 = ( ^ [P3: nat,Pm2: nat] :
% 0.18/0.38 ( P3
% 0.18/0.38 = ( suc @ Pm2 ) ) ) ) ).
% 0.18/0.38
% 0.18/0.38 % fixed_root_def
% 0.18/0.38 thf(fact_246_fixed__root_Op,axiom,
% 0.18/0.38 ! [P2: nat,Pm: nat] :
% 0.18/0.38 ( ( nthRoo1352626670d_root @ P2 @ Pm )
% 0.18/0.38 => ( P2
% 0.18/0.38 = ( suc @ Pm ) ) ) ).
% 0.18/0.38
% 0.18/0.38 % fixed_root.p
% 0.18/0.38 thf(fact_247_nat__one__le__power,axiom,
% 0.18/0.38 ! [I: nat,N: nat] :
% 0.18/0.38 ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ I )
% 0.18/0.38 => ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( power_power_nat @ I @ N ) ) ) ).
% 0.18/0.38
% 0.18/0.38 % nat_one_le_power
% 0.18/0.38 thf(fact_248_ex__least__nat__le,axiom,
% 0.18/0.38 ! [P: nat > $o,N: nat] :
% 0.18/0.38 ( ( P @ N )
% 0.18/0.38 => ( ~ ( P @ zero_zero_nat )
% 0.18/0.38 => ? [K: nat] :
% 0.18/0.38 ( ( ord_less_eq_nat @ K @ N )
% 0.18/0.38 & ! [I4: nat] :
% 0.18/0.38 ( ( ord_less_nat @ I4 @ K )
% 0.18/0.38 => ~ ( P @ I4 ) )
% 0.18/0.38 & ( P @ K ) ) ) ) ).
% 0.18/0.38
% 0.18/0.38 % ex_least_nat_le
% 0.18/0.38 thf(fact_249_Suc__leI,axiom,
% 0.18/0.38 ! [M: nat,N: nat] :
% 0.18/0.38 ( ( ord_less_nat @ M @ N )
% 0.18/0.38 => ( ord_less_eq_nat @ ( suc @ M ) @ N ) ) ).
% 0.18/0.38
% 0.18/0.38 % Suc_leI
% 0.18/0.38 thf(fact_250_Suc__le__eq,axiom,
% 0.18/0.38 ! [M: nat,N: nat] :
% 0.18/0.38 ( ( ord_less_eq_nat @ ( suc @ M ) @ N )
% 0.18/0.38 = ( ord_less_nat @ M @ N ) ) ).
% 0.18/0.38
% 0.18/0.38 % Suc_le_eq
% 0.18/0.38 thf(fact_251_dec__induct,axiom,
% 0.18/0.38 ! [I: nat,J: nat,P: nat > $o] :
% 0.18/0.38 ( ( ord_less_eq_nat @ I @ J )
% 0.18/0.38 => ( ( P @ I )
% 0.18/0.38 => ( ! [N2: nat] :
% 0.18/0.38 ( ( ord_less_eq_nat @ I @ N2 )
% 0.18/0.38 => ( ( ord_less_nat @ N2 @ J )
% 0.18/0.38 => ( ( P @ N2 )
% 0.18/0.38 => ( P @ ( suc @ N2 ) ) ) ) )
% 0.18/0.38 => ( P @ J ) ) ) ) ).
% 0.18/0.38
% 0.18/0.38 % dec_induct
% 0.18/0.38 thf(fact_252_inc__induct,axiom,
% 0.18/0.38 ! [I: nat,J: nat,P: nat > $o] :
% 0.18/0.38 ( ( ord_less_eq_nat @ I @ J )
% 0.18/0.38 => ( ( P @ J )
% 0.18/0.38 => ( ! [N2: nat] :
% 0.18/0.38 ( ( ord_less_eq_nat @ I @ N2 )
% 0.18/0.38 => ( ( ord_less_nat @ N2 @ J )
% 0.18/0.38 => ( ( P @ ( suc @ N2 ) )
% 0.18/0.38 => ( P @ N2 ) ) ) )
% 0.18/0.38 => ( P @ I ) ) ) ) ).
% 0.18/0.38
% 0.18/0.38 % inc_induct
% 0.18/0.38 thf(fact_253_Suc__le__lessD,axiom,
% 0.18/0.38 ! [M: nat,N: nat] :
% 0.18/0.38 ( ( ord_less_eq_nat @ ( suc @ M ) @ N )
% 0.18/0.38 => ( ord_less_nat @ M @ N ) ) ).
% 0.18/0.38
% 0.18/0.38 % Suc_le_lessD
% 0.18/0.38 thf(fact_254_le__less__Suc__eq,axiom,
% 0.18/0.38 ! [M: nat,N: nat] :
% 0.18/0.38 ( ( ord_less_eq_nat @ M @ N )
% 0.18/0.38 => ( ( ord_less_nat @ N @ ( suc @ M ) )
% 0.18/0.38 = ( N = M ) ) ) ).
% 0.18/0.38
% 0.18/0.38 % le_less_Suc_eq
% 0.18/0.38 thf(fact_255_less__Suc__eq__le,axiom,
% 0.18/0.38 ! [M: nat,N: nat] :
% 0.18/0.38 ( ( ord_less_nat @ M @ ( suc @ N ) )
% 0.18/0.38 = ( ord_less_eq_nat @ M @ N ) ) ).
% 0.18/0.38
% 0.18/0.38 % less_Suc_eq_le
% 0.18/0.38 thf(fact_256_less__eq__Suc__le,axiom,
% 0.18/0.38 ( ord_less_nat
% 0.18/0.38 = ( ^ [N4: nat] : ( ord_less_eq_nat @ ( suc @ N4 ) ) ) ) ).
% 0.18/0.38
% 0.18/0.38 % less_eq_Suc_le
% 0.18/0.38 thf(fact_257_le__imp__less__Suc,axiom,
% 0.18/0.38 ! [M: nat,N: nat] :
% 0.18/0.38 ( ( ord_less_eq_nat @ M @ N )
% 0.18/0.38 => ( ord_less_nat @ M @ ( suc @ N ) ) ) ).
% 0.18/0.38
% 0.18/0.38 % le_imp_less_Suc
% 0.18/0.38 thf(fact_258_linordered__field__no__ub,axiom,
% 0.18/0.38 ! [X4: real] :
% 0.18/0.38 ? [X_12: real] : ( ord_less_real @ X4 @ X_12 ) ).
% 0.18/0.38
% 0.18/0.38 % linordered_field_no_ub
% 0.18/0.38 thf(fact_259_linordered__field__no__lb,axiom,
% 0.18/0.38 ! [X4: real] :
% 0.18/0.38 ? [Y3: real] : ( ord_less_real @ Y3 @ X4 ) ).
% 0.18/0.38
% 0.18/0.38 % linordered_field_no_lb
% 0.18/0.38 thf(fact_260_ex__least__nat__less,axiom,
% 0.18/0.38 ! [P: nat > $o,N: nat] :
% 0.18/0.38 ( ( P @ N )
% 0.18/0.38 => ( ~ ( P @ zero_zero_nat )
% 0.18/0.38 => ? [K: nat] :
% 0.18/0.38 ( ( ord_less_nat @ K @ N )
% 0.18/0.38 & ! [I4: nat] :
% 0.18/0.38 ( ( ord_less_eq_nat @ I4 @ K )
% 0.18/0.38 => ~ ( P @ I4 ) )
% 0.18/0.38 & ( P @ ( suc @ K ) ) ) ) ) ).
% 0.18/0.38
% 0.18/0.38 % ex_least_nat_less
% 0.18/0.38 thf(fact_261_divide__right__mono__neg,axiom,
% 0.18/0.38 ! [A: real,B: real,C: real] :
% 0.18/0.38 ( ( ord_less_eq_real @ A @ B )
% 0.18/0.38 => ( ( ord_less_eq_real @ C @ zero_zero_real )
% 0.18/0.38 => ( ord_less_eq_real @ ( divide_divide_real @ B @ C ) @ ( divide_divide_real @ A @ C ) ) ) ) ).
% 0.18/0.38
% 0.18/0.38 % divide_right_mono_neg
% 0.18/0.38 thf(fact_262_divide__nonpos__nonpos,axiom,
% 0.18/0.38 ! [X: real,Y: real] :
% 0.18/0.38 ( ( ord_less_eq_real @ X @ zero_zero_real )
% 0.18/0.38 => ( ( ord_less_eq_real @ Y @ zero_zero_real )
% 0.18/0.38 => ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X @ Y ) ) ) ) ).
% 0.18/0.38
% 0.18/0.38 % divide_nonpos_nonpos
% 0.18/0.38 thf(fact_263_divide__nonpos__nonneg,axiom,
% 0.18/0.38 ! [X: real,Y: real] :
% 0.18/0.38 ( ( ord_less_eq_real @ X @ zero_zero_real )
% 0.18/0.38 => ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 0.18/0.38 => ( ord_less_eq_real @ ( divide_divide_real @ X @ Y ) @ zero_zero_real ) ) ) ).
% 0.18/0.38
% 0.18/0.38 % divide_nonpos_nonneg
% 0.18/0.38 thf(fact_264_divide__nonneg__nonpos,axiom,
% 0.18/0.38 ! [X: real,Y: real] :
% 0.18/0.38 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.18/0.38 => ( ( ord_less_eq_real @ Y @ zero_zero_real )
% 0.18/0.38 => ( ord_less_eq_real @ ( divide_divide_real @ X @ Y ) @ zero_zero_real ) ) ) ).
% 0.18/0.38
% 0.18/0.38 % divide_nonneg_nonpos
% 0.18/0.38 thf(fact_265_divide__nonneg__nonneg,axiom,
% 0.18/0.38 ! [X: real,Y: real] :
% 0.18/0.38 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.18/0.38 => ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 0.18/0.38 => ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X @ Y ) ) ) ) ).
% 0.18/0.38
% 0.18/0.38 % divide_nonneg_nonneg
% 0.18/0.38 thf(fact_266_zero__le__divide__iff,axiom,
% 0.18/0.38 ! [A: real,B: real] :
% 0.18/0.38 ( ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ A @ B ) )
% 0.18/0.38 = ( ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.18/0.38 & ( ord_less_eq_real @ zero_zero_real @ B ) )
% 0.18/0.38 | ( ( ord_less_eq_real @ A @ zero_zero_real )
% 0.18/0.38 & ( ord_less_eq_real @ B @ zero_zero_real ) ) ) ) ).
% 0.18/0.38
% 0.18/0.38 % zero_le_divide_iff
% 0.18/0.38 thf(fact_267_divide__right__mono,axiom,
% 0.18/0.38 ! [A: real,B: real,C: real] :
% 0.18/0.38 ( ( ord_less_eq_real @ A @ B )
% 0.18/0.38 => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 0.18/0.38 => ( ord_less_eq_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) ) ) ) ).
% 0.18/0.38
% 0.18/0.38 % divide_right_mono
% 0.18/0.38 thf(fact_268_divide__le__0__iff,axiom,
% 0.18/0.38 ! [A: real,B: real] :
% 0.18/0.38 ( ( ord_less_eq_real @ ( divide_divide_real @ A @ B ) @ zero_zero_real )
% 0.18/0.38 = ( ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.18/0.38 & ( ord_less_eq_real @ B @ zero_zero_real ) )
% 0.18/0.38 | ( ( ord_less_eq_real @ A @ zero_zero_real )
% 0.18/0.38 & ( ord_less_eq_real @ zero_zero_real @ B ) ) ) ) ).
% 0.18/0.38
% 0.18/0.38 % divide_le_0_iff
% 0.18/0.38 thf(fact_269_divide__neg__neg,axiom,
% 0.18/0.38 ! [X: real,Y: real] :
% 0.18/0.38 ( ( ord_less_real @ X @ zero_zero_real )
% 0.18/0.38 => ( ( ord_less_real @ Y @ zero_zero_real )
% 0.18/0.38 => ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ X @ Y ) ) ) ) ).
% 0.18/0.38
% 0.18/0.38 % divide_neg_neg
% 0.18/0.38 thf(fact_270_divide__neg__pos,axiom,
% 0.18/0.38 ! [X: real,Y: real] :
% 0.18/0.38 ( ( ord_less_real @ X @ zero_zero_real )
% 0.18/0.38 => ( ( ord_less_real @ zero_zero_real @ Y )
% 0.18/0.38 => ( ord_less_real @ ( divide_divide_real @ X @ Y ) @ zero_zero_real ) ) ) ).
% 0.18/0.38
% 0.18/0.38 % divide_neg_pos
% 0.18/0.38 thf(fact_271_divide__pos__neg,axiom,
% 0.18/0.38 ! [X: real,Y: real] :
% 0.18/0.38 ( ( ord_less_real @ zero_zero_real @ X )
% 0.18/0.38 => ( ( ord_less_real @ Y @ zero_zero_real )
% 0.18/0.38 => ( ord_less_real @ ( divide_divide_real @ X @ Y ) @ zero_zero_real ) ) ) ).
% 0.18/0.38
% 0.18/0.38 % divide_pos_neg
% 0.18/0.38 thf(fact_272_divide__pos__pos,axiom,
% 0.18/0.38 ! [X: real,Y: real] :
% 0.18/0.38 ( ( ord_less_real @ zero_zero_real @ X )
% 0.18/0.38 => ( ( ord_less_real @ zero_zero_real @ Y )
% 0.18/0.38 => ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ X @ Y ) ) ) ) ).
% 0.18/0.38
% 0.18/0.38 % divide_pos_pos
% 0.18/0.38 thf(fact_273_divide__less__0__iff,axiom,
% 0.18/0.38 ! [A: real,B: real] :
% 0.18/0.38 ( ( ord_less_real @ ( divide_divide_real @ A @ B ) @ zero_zero_real )
% 0.18/0.38 = ( ( ( ord_less_real @ zero_zero_real @ A )
% 0.18/0.38 & ( ord_less_real @ B @ zero_zero_real ) )
% 0.18/0.38 | ( ( ord_less_real @ A @ zero_zero_real )
% 0.18/0.38 & ( ord_less_real @ zero_zero_real @ B ) ) ) ) ).
% 0.18/0.38
% 0.18/0.38 % divide_less_0_iff
% 0.18/0.38 thf(fact_274_divide__less__cancel,axiom,
% 0.18/0.38 ! [A: real,C: real,B: real] :
% 0.18/0.38 ( ( ord_less_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) )
% 0.18/0.38 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.18/0.38 => ( ord_less_real @ A @ B ) )
% 0.18/0.38 & ( ( ord_less_real @ C @ zero_zero_real )
% 0.18/0.38 => ( ord_less_real @ B @ A ) )
% 0.18/0.38 & ( C != zero_zero_real ) ) ) ).
% 0.18/0.38
% 0.18/0.38 % divide_less_cancel
% 0.18/0.38 thf(fact_275_zero__less__divide__iff,axiom,
% 0.18/0.38 ! [A: real,B: real] :
% 0.18/0.38 ( ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ A @ B ) )
% 0.18/0.38 = ( ( ( ord_less_real @ zero_zero_real @ A )
% 0.18/0.38 & ( ord_less_real @ zero_zero_real @ B ) )
% 0.18/0.38 | ( ( ord_less_real @ A @ zero_zero_real )
% 0.18/0.38 & ( ord_less_real @ B @ zero_zero_real ) ) ) ) ).
% 0.18/0.38
% 0.18/0.38 % zero_less_divide_iff
% 0.18/0.38 thf(fact_276_divide__strict__right__mono,axiom,
% 0.18/0.38 ! [A: real,B: real,C: real] :
% 0.18/0.38 ( ( ord_less_real @ A @ B )
% 0.18/0.38 => ( ( ord_less_real @ zero_zero_real @ C )
% 0.18/0.38 => ( ord_less_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) ) ) ) ).
% 0.18/0.38
% 0.18/0.38 % divide_strict_right_mono
% 0.18/0.38 thf(fact_277_divide__strict__right__mono__neg,axiom,
% 0.18/0.38 ! [B: real,A: real,C: real] :
% 0.18/0.38 ( ( ord_less_real @ B @ A )
% 0.18/0.38 => ( ( ord_less_real @ C @ zero_zero_real )
% 0.18/0.38 => ( ord_less_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) ) ) ) ).
% 0.18/0.38
% 0.18/0.38 % divide_strict_right_mono_neg
% 0.18/0.38 thf(fact_278_frac__le,axiom,
% 0.18/0.38 ! [Y: real,X: real,W: real,Z: real] :
% 0.18/0.38 ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 0.18/0.38 => ( ( ord_less_eq_real @ X @ Y )
% 0.18/0.38 => ( ( ord_less_real @ zero_zero_real @ W )
% 0.18/0.38 => ( ( ord_less_eq_real @ W @ Z )
% 0.18/0.38 => ( ord_less_eq_real @ ( divide_divide_real @ X @ Z ) @ ( divide_divide_real @ Y @ W ) ) ) ) ) ) ).
% 0.18/0.38
% 0.18/0.38 % frac_le
% 0.18/0.38 thf(fact_279_frac__less,axiom,
% 0.18/0.38 ! [X: real,Y: real,W: real,Z: real] :
% 0.18/0.38 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.18/0.38 => ( ( ord_less_real @ X @ Y )
% 0.18/0.38 => ( ( ord_less_real @ zero_zero_real @ W )
% 0.18/0.38 => ( ( ord_less_eq_real @ W @ Z )
% 0.18/0.38 => ( ord_less_real @ ( divide_divide_real @ X @ Z ) @ ( divide_divide_real @ Y @ W ) ) ) ) ) ) ).
% 0.18/0.38
% 0.18/0.38 % frac_less
% 0.18/0.38 thf(fact_280_frac__less2,axiom,
% 0.18/0.38 ! [X: real,Y: real,W: real,Z: real] :
% 0.18/0.38 ( ( ord_less_real @ zero_zero_real @ X )
% 0.18/0.38 => ( ( ord_less_eq_real @ X @ Y )
% 0.18/0.38 => ( ( ord_less_real @ zero_zero_real @ W )
% 0.18/0.38 => ( ( ord_less_real @ W @ Z )
% 0.18/0.38 => ( ord_less_real @ ( divide_divide_real @ X @ Z ) @ ( divide_divide_real @ Y @ W ) ) ) ) ) ) ).
% 0.18/0.38
% 0.18/0.38 % frac_less2
% 0.18/0.38 thf(fact_281_divide__le__cancel,axiom,
% 0.18/0.38 ! [A: real,C: real,B: real] :
% 0.18/0.38 ( ( ord_less_eq_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) )
% 0.18/0.38 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.18/0.38 => ( ord_less_eq_real @ A @ B ) )
% 0.18/0.38 & ( ( ord_less_real @ C @ zero_zero_real )
% 0.18/0.38 => ( ord_less_eq_real @ B @ A ) ) ) ) ).
% 0.18/0.38
% 0.18/0.38 % divide_le_cancel
% 0.18/0.38 thf(fact_282_divide__nonneg__neg,axiom,
% 0.18/0.38 ! [X: real,Y: real] :
% 0.18/0.38 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.18/0.38 => ( ( ord_less_real @ Y @ zero_zero_real )
% 0.18/0.38 => ( ord_less_eq_real @ ( divide_divide_real @ X @ Y ) @ zero_zero_real ) ) ) ).
% 0.18/0.38
% 0.18/0.38 % divide_nonneg_neg
% 0.18/0.38 thf(fact_283_divide__nonneg__pos,axiom,
% 0.18/0.38 ! [X: real,Y: real] :
% 0.18/0.38 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.18/0.38 => ( ( ord_less_real @ zero_zero_real @ Y )
% 0.18/0.38 => ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X @ Y ) ) ) ) ).
% 0.18/0.38
% 0.18/0.38 % divide_nonneg_pos
% 0.18/0.38 thf(fact_284_divide__nonpos__neg,axiom,
% 0.18/0.38 ! [X: real,Y: real] :
% 0.18/0.38 ( ( ord_less_eq_real @ X @ zero_zero_real )
% 0.18/0.38 => ( ( ord_less_real @ Y @ zero_zero_real )
% 0.18/0.38 => ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X @ Y ) ) ) ) ).
% 0.18/0.38
% 0.18/0.38 % divide_nonpos_neg
% 0.18/0.38 thf(fact_285_divide__nonpos__pos,axiom,
% 0.18/0.38 ! [X: real,Y: real] :
% 0.18/0.38 ( ( ord_less_eq_real @ X @ zero_zero_real )
% 0.18/0.38 => ( ( ord_less_real @ zero_zero_real @ Y )
% 0.18/0.38 => ( ord_less_eq_real @ ( divide_divide_real @ X @ Y ) @ zero_zero_real ) ) ) ).
% 0.18/0.38
% 0.18/0.38 % divide_nonpos_pos
% 0.18/0.38 thf(fact_286_div__neg__neg__trivial,axiom,
% 0.18/0.38 ! [K2: int,L: int] :
% 0.18/0.38 ( ( ord_less_eq_int @ K2 @ zero_zero_int )
% 0.18/0.38 => ( ( ord_less_int @ L @ K2 )
% 0.18/0.38 => ( ( divide_divide_int @ K2 @ L )
% 0.18/0.38 = zero_zero_int ) ) ) ).
% 0.18/0.38
% 0.18/0.38 % div_neg_neg_trivial
% 0.18/0.38 thf(fact_287_div__pos__pos__trivial,axiom,
% 0.18/0.38 ! [K2: int,L: int] :
% 0.18/0.38 ( ( ord_less_eq_int @ zero_zero_int @ K2 )
% 0.18/0.38 => ( ( ord_less_int @ K2 @ L )
% 0.18/0.38 => ( ( divide_divide_int @ K2 @ L )
% 0.18/0.38 = zero_zero_int ) ) ) ).
% 0.18/0.38
% 0.18/0.38 % div_pos_pos_trivial
% 0.18/0.38 thf(fact_288_x,axiom,
% 0.18/0.38 x = y ).
% 0.18/0.38
% 0.18/0.38 % x
% 0.18/0.38 thf(fact_289_div__by__Suc__0,axiom,
% 0.18/0.38 ! [M: nat] :
% 0.18/0.38 ( ( divide_divide_nat @ M @ ( suc @ zero_zero_nat ) )
% 0.18/0.38 = M ) ).
% 0.18/0.38
% 0.18/0.38 % div_by_Suc_0
% 0.18/0.38 thf(fact_290_div__less,axiom,
% 0.18/0.38 ! [M: nat,N: nat] :
% 0.18/0.38 ( ( ord_less_nat @ M @ N )
% 0.18/0.38 => ( ( divide_divide_nat @ M @ N )
% 0.18/0.38 = zero_zero_nat ) ) ).
% 0.18/0.38
% 0.18/0.38 % div_less
% 0.18/0.38 thf(fact_291_div__le__mono,axiom,
% 0.18/0.38 ! [M: nat,N: nat,K2: nat] :
% 0.18/0.38 ( ( ord_less_eq_nat @ M @ N )
% 0.18/0.38 => ( ord_less_eq_nat @ ( divide_divide_nat @ M @ K2 ) @ ( divide_divide_nat @ N @ K2 ) ) ) ).
% 0.18/0.38
% 0.18/0.38 % div_le_mono
% 0.18/0.38 thf(fact_292_div__le__dividend,axiom,
% 0.18/0.38 ! [M: nat,N: nat] : ( ord_less_eq_nat @ ( divide_divide_nat @ M @ N ) @ M ) ).
% 0.18/0.38
% 0.18/0.38 % div_le_dividend
% 0.18/0.38 thf(fact_293_Euclidean__Division_Odiv__eq__0__iff,axiom,
% 0.18/0.38 ! [M: nat,N: nat] :
% 0.18/0.38 ( ( ( divide_divide_nat @ M @ N )
% 0.18/0.38 = zero_zero_nat )
% 0.18/0.38 = ( ( ord_less_nat @ M @ N )
% 0.18/0.38 | ( N = zero_zero_nat ) ) ) ).
% 0.18/0.38
% 0.18/0.38 % Euclidean_Division.div_eq_0_iff
% 0.18/0.38 thf(fact_294_Suc__div__le__mono,axiom,
% 0.18/0.38 ! [M: nat,N: nat] : ( ord_less_eq_nat @ ( divide_divide_nat @ M @ N ) @ ( divide_divide_nat @ ( suc @ M ) @ N ) ) ).
% 0.18/0.38
% 0.18/0.38 % Suc_div_le_mono
% 0.18/0.38 thf(fact_295_div__greater__zero__iff,axiom,
% 0.18/0.38 ! [M: nat,N: nat] :
% 0.18/0.38 ( ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ M @ N ) )
% 0.18/0.38 = ( ( ord_less_eq_nat @ N @ M )
% 0.18/0.38 & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 0.18/0.38
% 0.18/0.38 % div_greater_zero_iff
% 0.18/0.38 thf(fact_296_div__le__mono2,axiom,
% 0.18/0.38 ! [M: nat,N: nat,K2: nat] :
% 0.18/0.38 ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.18/0.38 => ( ( ord_less_eq_nat @ M @ N )
% 0.18/0.38 => ( ord_less_eq_nat @ ( divide_divide_nat @ K2 @ N ) @ ( divide_divide_nat @ K2 @ M ) ) ) ) ).
% 0.18/0.38
% 0.18/0.38 % div_le_mono2
% 0.18/0.38 thf(fact_297_div__by__0,axiom,
% 0.18/0.38 ! [A: real] :
% 0.18/0.38 ( ( divide_divide_real @ A @ zero_zero_real )
% 0.18/0.38 = zero_zero_real ) ).
% 0.18/0.38
% 0.18/0.38 % div_by_0
% 0.18/0.38 thf(fact_298_div__by__0,axiom,
% 0.18/0.38 ! [A: int] :
% 0.18/0.38 ( ( divide_divide_int @ A @ zero_zero_int )
% 0.18/0.38 = zero_zero_int ) ).
% 0.18/0.38
% 0.18/0.38 % div_by_0
% 0.18/0.38 thf(fact_299_div__by__0,axiom,
% 0.18/0.38 ! [A: nat] :
% 0.18/0.38 ( ( divide_divide_nat @ A @ zero_zero_nat )
% 0.18/0.38 = zero_zero_nat ) ).
% 0.18/0.38
% 0.18/0.38 % div_by_0
% 0.18/0.38 thf(fact_300_div__0,axiom,
% 0.18/0.38 ! [A: real] :
% 0.18/0.38 ( ( divide_divide_real @ zero_zero_real @ A )
% 0.18/0.38 = zero_zero_real ) ).
% 0.18/0.38
% 0.18/0.38 % div_0
% 0.18/0.38 thf(fact_301_div__0,axiom,
% 0.18/0.38 ! [A: int] :
% 0.18/0.38 ( ( divide_divide_int @ zero_zero_int @ A )
% 0.18/0.38 = zero_zero_int ) ).
% 0.18/0.38
% 0.18/0.38 % div_0
% 0.18/0.38 thf(fact_302_div__0,axiom,
% 0.18/0.38 ! [A: nat] :
% 0.18/0.38 ( ( divide_divide_nat @ zero_zero_nat @ A )
% 0.18/0.38 = zero_zero_nat ) ).
% 0.18/0.38
% 0.18/0.38 % div_0
% 0.18/0.38 thf(fact_303_linorder__neqE__linordered__idom,axiom,
% 0.18/0.38 ! [X: real,Y: real] :
% 0.18/0.38 ( ( X != Y )
% 0.18/0.38 => ( ~ ( ord_less_real @ X @ Y )
% 0.18/0.38 => ( ord_less_real @ Y @ X ) ) ) ).
% 0.18/0.38
% 0.18/0.38 % linorder_neqE_linordered_idom
% 0.18/0.38 thf(fact_304_linorder__neqE__linordered__idom,axiom,
% 0.18/0.38 ! [X: int,Y: int] :
% 0.18/0.38 ( ( X != Y )
% 0.18/0.38 => ( ~ ( ord_less_int @ X @ Y )
% 0.18/0.38 => ( ord_less_int @ Y @ X ) ) ) ).
% 0.18/0.38
% 0.18/0.38 % linorder_neqE_linordered_idom
% 0.18/0.38 thf(fact_305_zdiv__mono1,axiom,
% 0.18/0.38 ! [A: int,A2: int,B: int] :
% 0.18/0.38 ( ( ord_less_eq_int @ A @ A2 )
% 0.18/0.38 => ( ( ord_less_int @ zero_zero_int @ B )
% 0.18/0.38 => ( ord_less_eq_int @ ( divide_divide_int @ A @ B ) @ ( divide_divide_int @ A2 @ B ) ) ) ) ).
% 0.18/0.38
% 0.18/0.38 % zdiv_mono1
% 0.18/0.38 thf(fact_306_zdiv__mono2,axiom,
% 0.18/0.38 ! [A: int,B2: int,B: int] :
% 0.18/0.38 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.18/0.38 => ( ( ord_less_int @ zero_zero_int @ B2 )
% 0.18/0.38 => ( ( ord_less_eq_int @ B2 @ B )
% 0.18/0.38 => ( ord_less_eq_int @ ( divide_divide_int @ A @ B ) @ ( divide_divide_int @ A @ B2 ) ) ) ) ) ).
% 0.18/0.38
% 0.18/0.38 % zdiv_mono2
% 0.18/0.38 thf(fact_307_pos__imp__zdiv__neg__iff,axiom,
% 0.18/0.38 ! [B: int,A: int] :
% 0.18/0.38 ( ( ord_less_int @ zero_zero_int @ B )
% 0.18/0.38 => ( ( ord_less_int @ ( divide_divide_int @ A @ B ) @ zero_zero_int )
% 0.18/0.38 = ( ord_less_int @ A @ zero_zero_int ) ) ) ).
% 0.18/0.38
% 0.18/0.38 % pos_imp_zdiv_neg_iff
% 0.18/0.38 thf(fact_308_neg__imp__zdiv__neg__iff,axiom,
% 0.18/0.38 ! [B: int,A: int] :
% 0.18/0.38 ( ( ord_less_int @ B @ zero_zero_int )
% 0.18/0.38 => ( ( ord_less_int @ ( divide_divide_int @ A @ B ) @ zero_zero_int )
% 0.18/0.38 = ( ord_less_int @ zero_zero_int @ A ) ) ) ).
% 0.18/0.38
% 0.18/0.38 % neg_imp_zdiv_neg_iff
% 0.18/0.38 thf(fact_309_div__neg__pos__less0,axiom,
% 0.18/0.38 ! [A: int,B: int] :
% 0.18/0.38 ( ( ord_less_int @ A @ zero_zero_int )
% 0.18/0.38 => ( ( ord_less_int @ zero_zero_int @ B )
% 0.18/0.38 => ( ord_less_int @ ( divide_divide_int @ A @ B ) @ zero_zero_int ) ) ) ).
% 0.18/0.38
% 0.18/0.38 % div_neg_pos_less0
% 0.18/0.38 thf(fact_310_div__positive,axiom,
% 0.18/0.38 ! [B: int,A: int] :
% 0.18/0.38 ( ( ord_less_int @ zero_zero_int @ B )
% 0.18/0.38 => ( ( ord_less_eq_int @ B @ A )
% 0.18/0.38 => ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ A @ B ) ) ) ) ).
% 0.18/0.38
% 0.18/0.38 % div_positive
% 0.18/0.38 thf(fact_311_div__positive,axiom,
% 0.18/0.38 ! [B: nat,A: nat] :
% 0.18/0.38 ( ( ord_less_nat @ zero_zero_nat @ B )
% 0.18/0.38 => ( ( ord_less_eq_nat @ B @ A )
% 0.18/0.38 => ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ A @ B ) ) ) ) ).
% 0.18/0.38
% 0.18/0.38 % div_positive
% 0.18/0.38 thf(fact_312_unique__euclidean__semiring__numeral__class_Odiv__less,axiom,
% 0.18/0.38 ! [A: int,B: int] :
% 0.18/0.38 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.18/0.38 => ( ( ord_less_int @ A @ B )
% 0.18/0.38 => ( ( divide_divide_int @ A @ B )
% 0.18/0.38 = zero_zero_int ) ) ) ).
% 0.18/0.38
% 0.18/0.38 % unique_euclidean_semiring_numeral_class.div_less
% 0.18/0.38 thf(fact_313_unique__euclidean__semiring__numeral__class_Odiv__less,axiom,
% 0.18/0.38 ! [A: nat,B: nat] :
% 0.18/0.38 ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.18/0.38 => ( ( ord_less_nat @ A @ B )
% 0.18/0.38 => ( ( divide_divide_nat @ A @ B )
% 0.18/0.38 = zero_zero_nat ) ) ) ).
% 0.18/0.38
% 0.18/0.38 % unique_euclidean_semiring_numeral_class.div_less
% 0.18/0.38 thf(fact_314_nonneg1__imp__zdiv__pos__iff,axiom,
% 0.18/0.38 ! [A: int,B: int] :
% 0.18/0.38 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.18/0.38 => ( ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ A @ B ) )
% 0.18/0.38 = ( ( ord_less_eq_int @ B @ A )
% 0.18/0.38 & ( ord_less_int @ zero_zero_int @ B ) ) ) ) ).
% 0.18/0.38
% 0.18/0.38 % nonneg1_imp_zdiv_pos_iff
% 0.18/0.38 thf(fact_315_pos__imp__zdiv__nonneg__iff,axiom,
% 0.18/0.38 ! [B: int,A: int] :
% 0.18/0.38 ( ( ord_less_int @ zero_zero_int @ B )
% 0.18/0.38 => ( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ A @ B ) )
% 0.18/0.38 = ( ord_less_eq_int @ zero_zero_int @ A ) ) ) ).
% 0.18/0.38
% 0.18/0.38 % pos_imp_zdiv_nonneg_iff
% 0.18/0.38 thf(fact_316_neg__imp__zdiv__nonneg__iff,axiom,
% 0.18/0.38 ! [B: int,A: int] :
% 0.18/0.38 ( ( ord_less_int @ B @ zero_zero_int )
% 0.18/0.38 => ( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ A @ B ) )
% 0.18/0.38 = ( ord_less_eq_int @ A @ zero_zero_int ) ) ) ).
% 0.18/0.38
% 0.18/0.38 % neg_imp_zdiv_nonneg_iff
% 0.18/0.38 thf(fact_317_pos__imp__zdiv__pos__iff,axiom,
% 0.18/0.38 ! [K2: int,I: int] :
% 0.18/0.38 ( ( ord_less_int @ zero_zero_int @ K2 )
% 0.18/0.38 => ( ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ I @ K2 ) )
% 0.18/0.38 = ( ord_less_eq_int @ K2 @ I ) ) ) ).
% 0.18/0.38
% 0.18/0.38 % pos_imp_zdiv_pos_iff
% 0.18/0.38 thf(fact_318_div__nonpos__pos__le0,axiom,
% 0.18/0.38 ! [A: int,B: int] :
% 0.18/0.38 ( ( ord_less_eq_int @ A @ zero_zero_int )
% 0.18/0.38 => ( ( ord_less_int @ zero_zero_int @ B )
% 0.18/0.38 => ( ord_less_eq_int @ ( divide_divide_int @ A @ B ) @ zero_zero_int ) ) ) ).
% 0.18/0.38
% 0.18/0.38 % div_nonpos_pos_le0
% 0.18/0.38
% 0.18/0.38 % Conjectures (1)
% 0.18/0.38 thf(conj_0,conjecture,
% 0.18/0.38 ord_less_real @ ( ring_1_of_int_real @ na ) @ ( power_power_real @ ( ring_1_of_int_real @ ya ) @ ( suc @ pm ) ) ).
% 0.18/0.38
% 0.18/0.38 %------------------------------------------------------------------------------
% 0.18/0.38 ------------------------
% 2.44/1.31 ---- Embedded in THF ---
% 2.44/1.32 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 2.44/1.32 %%% Generated on Tue Feb 24 08:27:11 EST 2026
% 2.44/1.32 %%% using '$$dhol' embedding, version 1.3.0.
% 2.44/1.32 %%% Logic specification used:
% 2.44/1.32 %%% thf(spec, logic, $$dhol).
% 2.44/1.32
% 2.44/1.32 % SZS output start ListOfTHF for /export/starexec/sandbox2/tmp/tmp.paLrYnjNlH/DTF2DTF32705.p
% See solution above
% 2.44/1.32 ------------------------
% 2.44/1.35 ---- Cleaned THF ---
% 2.44/1.35 thf(ty_n_t__Real__Oreal,type,
% 2.44/1.35 real: $tType ).
% 2.44/1.35
% 2.44/1.35 thf(ty_n_t__Nat__Onat,type,
% 2.44/1.35 nat: $tType ).
% 2.44/1.35
% 2.44/1.35 thf(ty_n_t__Int__Oint,type,
% 2.44/1.35 int: $tType ).
% 2.44/1.35
% 2.44/1.35 thf(sy_c_Groups_Ozero__class_Ozero_001t__Int__Oint,type,
% 2.44/1.35 zero_zero_int: int ).
% 2.44/1.35
% 2.44/1.35 thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
% 2.44/1.35 zero_zero_nat: nat ).
% 2.44/1.35
% 2.44/1.35 thf(sy_c_Groups_Ozero__class_Ozero_001t__Real__Oreal,type,
% 2.44/1.35 zero_zero_real: real ).
% 2.44/1.35
% 2.44/1.35 thf(sy_c_Int_Oring__1__class_Oof__int_001t__Int__Oint,type,
% 2.44/1.35 ring_1_of_int_int: int > int ).
% 2.44/1.35
% 2.44/1.35 thf(sy_c_Int_Oring__1__class_Oof__int_001t__Real__Oreal,type,
% 2.44/1.35 ring_1_of_int_real: int > real ).
% 2.44/1.35
% 2.44/1.35 thf(sy_c_Nat_OSuc,type,
% 2.44/1.35 suc: nat > nat ).
% 2.44/1.35
% 2.44/1.35 thf(sy_c_NthRoot__Impl__Mirabelle__bcfbdygymo_Ofixed__root,type,
% 2.44/1.35 nthRoo1352626670d_root: nat > nat > $o ).
% 2.44/1.35
% 2.44/1.35 thf(sy_c_Orderings_Oord__class_Oless_001t__Int__Oint,type,
% 2.44/1.35 ord_less_int: int > int > $o ).
% 2.44/1.35
% 2.44/1.35 thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
% 2.44/1.35 ord_less_nat: nat > nat > $o ).
% 2.44/1.35
% 2.44/1.35 thf(sy_c_Orderings_Oord__class_Oless_001t__Real__Oreal,type,
% 2.44/1.35 ord_less_real: real > real > $o ).
% 2.44/1.35
% 2.44/1.35 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Int__Oint,type,
% 2.44/1.35 ord_less_eq_int: int > int > $o ).
% 2.44/1.35
% 2.44/1.35 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
% 2.44/1.35 ord_less_eq_nat: nat > nat > $o ).
% 2.44/1.35
% 2.44/1.35 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Real__Oreal,type,
% 2.44/1.35 ord_less_eq_real: real > real > $o ).
% 2.44/1.35
% 2.44/1.35 thf(sy_c_Power_Opower__class_Opower_001t__Int__Oint,type,
% 2.44/1.35 power_power_int: int > nat > int ).
% 2.44/1.35
% 2.44/1.35 thf(sy_c_Power_Opower__class_Opower_001t__Nat__Onat,type,
% 2.44/1.35 power_power_nat: nat > nat > nat ).
% 2.44/1.35
% 2.44/1.35 thf(sy_c_Power_Opower__class_Opower_001t__Real__Oreal,type,
% 2.44/1.35 power_power_real: real > nat > real ).
% 2.44/1.35
% 2.44/1.35 thf(sy_c_Rings_Odivide__class_Odivide_001t__Int__Oint,type,
% 2.44/1.35 divide_divide_int: int > int > int ).
% 2.44/1.35
% 2.44/1.35 thf(sy_c_Rings_Odivide__class_Odivide_001t__Nat__Onat,type,
% 2.44/1.35 divide_divide_nat: nat > nat > nat ).
% 2.44/1.35
% 2.44/1.35 thf(sy_c_Rings_Odivide__class_Odivide_001t__Real__Oreal,type,
% 2.44/1.35 divide_divide_real: real > real > real ).
% 2.44/1.35
% 2.44/1.35 thf(sy_v_NY____,type,
% 2.44/1.35 ny: real ).
% 2.44/1.35
% 2.44/1.35 thf(sy_v_na____,type,
% 2.44/1.35 na: int ).
% 2.44/1.35
% 2.44/1.35 thf(sy_v_p,type,
% 2.44/1.35 p: nat ).
% 2.44/1.35
% 2.44/1.35 thf(sy_v_pm,type,
% 2.44/1.35 pm: nat ).
% 2.44/1.35
% 2.44/1.35 thf(sy_v_x,type,
% 2.44/1.35 x: int ).
% 2.44/1.35
% 2.44/1.35 thf(sy_v_y_H____,type,
% 2.44/1.35 y: int ).
% 2.44/1.35
% 2.44/1.35 thf(sy_v_ya____,type,
% 2.44/1.35 ya: int ).
% 2.44/1.35
% 2.44/1.35 thf(fact_0_n00,axiom,
% 2.44/1.35 na != zero_zero_int ).
% 2.44/1.35
% 2.44/1.35 thf(fact_1_of__int__less__of__int__power__cancel__iff,axiom,
% 2.44/1.35 ! [B: int,W: nat,X: int] :
% 2.44/1.35 ( ( ord_less_real @ ( power_power_real @ ( ring_1_of_int_real @ B ) @ W ) @ ( ring_1_of_int_real @ X ) )
% 2.44/1.35 = ( ord_less_int @ ( power_power_int @ B @ W ) @ X ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_2_of__int__less__of__int__power__cancel__iff,axiom,
% 2.44/1.35 ! [B: int,W: nat,X: int] :
% 2.44/1.35 ( ( ord_less_int @ ( power_power_int @ ( ring_1_of_int_int @ B ) @ W ) @ ( ring_1_of_int_int @ X ) )
% 2.44/1.35 = ( ord_less_int @ ( power_power_int @ B @ W ) @ X ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_3_of__int__power__less__of__int__cancel__iff,axiom,
% 2.44/1.35 ! [X: int,B: int,W: nat] :
% 2.44/1.35 ( ( ord_less_real @ ( ring_1_of_int_real @ X ) @ ( power_power_real @ ( ring_1_of_int_real @ B ) @ W ) )
% 2.44/1.35 = ( ord_less_int @ X @ ( power_power_int @ B @ W ) ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_4_of__int__power__less__of__int__cancel__iff,axiom,
% 2.44/1.35 ! [X: int,B: int,W: nat] :
% 2.44/1.35 ( ( ord_less_int @ ( ring_1_of_int_int @ X ) @ ( power_power_int @ ( ring_1_of_int_int @ B ) @ W ) )
% 2.44/1.35 = ( ord_less_int @ X @ ( power_power_int @ B @ W ) ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_5_of__int__power,axiom,
% 2.44/1.35 ! [Z: int,N: nat] :
% 2.44/1.35 ( ( ring_1_of_int_real @ ( power_power_int @ Z @ N ) )
% 2.44/1.35 = ( power_power_real @ ( ring_1_of_int_real @ Z ) @ N ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_6_of__int__power,axiom,
% 2.44/1.35 ! [Z: int,N: nat] :
% 2.44/1.35 ( ( ring_1_of_int_int @ ( power_power_int @ Z @ N ) )
% 2.44/1.35 = ( power_power_int @ ( ring_1_of_int_int @ Z ) @ N ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_7_of__int__eq__of__int__power__cancel__iff,axiom,
% 2.44/1.35 ! [B: int,W: nat,X: int] :
% 2.44/1.35 ( ( ( power_power_real @ ( ring_1_of_int_real @ B ) @ W )
% 2.44/1.35 = ( ring_1_of_int_real @ X ) )
% 2.44/1.35 = ( ( power_power_int @ B @ W )
% 2.44/1.35 = X ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_8_of__int__eq__of__int__power__cancel__iff,axiom,
% 2.44/1.35 ! [B: int,W: nat,X: int] :
% 2.44/1.35 ( ( ( power_power_int @ ( ring_1_of_int_int @ B ) @ W )
% 2.44/1.35 = ( ring_1_of_int_int @ X ) )
% 2.44/1.35 = ( ( power_power_int @ B @ W )
% 2.44/1.35 = X ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_9_of__int__power__eq__of__int__cancel__iff,axiom,
% 2.44/1.35 ! [X: int,B: int,W: nat] :
% 2.44/1.35 ( ( ( ring_1_of_int_real @ X )
% 2.44/1.35 = ( power_power_real @ ( ring_1_of_int_real @ B ) @ W ) )
% 2.44/1.35 = ( X
% 2.44/1.35 = ( power_power_int @ B @ W ) ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_10_of__int__power__eq__of__int__cancel__iff,axiom,
% 2.44/1.35 ! [X: int,B: int,W: nat] :
% 2.44/1.35 ( ( ( ring_1_of_int_int @ X )
% 2.44/1.35 = ( power_power_int @ ( ring_1_of_int_int @ B ) @ W ) )
% 2.44/1.35 = ( X
% 2.44/1.35 = ( power_power_int @ B @ W ) ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_11_of__int__less__iff,axiom,
% 2.44/1.35 ! [W: int,Z: int] :
% 2.44/1.35 ( ( ord_less_real @ ( ring_1_of_int_real @ W ) @ ( ring_1_of_int_real @ Z ) )
% 2.44/1.35 = ( ord_less_int @ W @ Z ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_12_of__int__less__iff,axiom,
% 2.44/1.35 ! [W: int,Z: int] :
% 2.44/1.35 ( ( ord_less_int @ ( ring_1_of_int_int @ W ) @ ( ring_1_of_int_int @ Z ) )
% 2.44/1.35 = ( ord_less_int @ W @ Z ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_13_n0,axiom,
% 2.44/1.35 ord_less_real @ zero_zero_real @ ( ring_1_of_int_real @ na ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_14_y0,axiom,
% 2.44/1.35 ord_less_real @ zero_zero_real @ ( ring_1_of_int_real @ ya ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_15_of__int__eq__iff,axiom,
% 2.44/1.35 ! [W: int,Z: int] :
% 2.44/1.35 ( ( ( ring_1_of_int_int @ W )
% 2.44/1.35 = ( ring_1_of_int_int @ Z ) )
% 2.44/1.35 = ( W = Z ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_16_of__int__eq__iff,axiom,
% 2.44/1.35 ! [W: int,Z: int] :
% 2.44/1.35 ( ( ( ring_1_of_int_real @ W )
% 2.44/1.35 = ( ring_1_of_int_real @ Z ) )
% 2.44/1.35 = ( W = Z ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_17_nat_Oinject,axiom,
% 2.44/1.35 ! [X2: nat,Y2: nat] :
% 2.44/1.35 ( ( ( suc @ X2 )
% 2.44/1.35 = ( suc @ Y2 ) )
% 2.44/1.35 = ( X2 = Y2 ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_18_old_Onat_Oinject,axiom,
% 2.44/1.35 ! [Nat: nat,Nat2: nat] :
% 2.44/1.35 ( ( ( suc @ Nat )
% 2.44/1.35 = ( suc @ Nat2 ) )
% 2.44/1.35 = ( Nat = Nat2 ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_19_y00,axiom,
% 2.44/1.35 ya != zero_zero_int ).
% 2.44/1.35
% 2.44/1.35 thf(fact_20__092_060open_0620_A_060_Ay_A_092_060or_062_Ay_A_061_A0_092_060close_062,axiom,
% 2.44/1.35 ( ( ord_less_int @ zero_zero_int @ ya )
% 2.44/1.35 | ( ya = zero_zero_int ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_21_Suc__less__eq,axiom,
% 2.44/1.35 ! [M: nat,N: nat] :
% 2.44/1.35 ( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
% 2.44/1.35 = ( ord_less_nat @ M @ N ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_22_Suc__mono,axiom,
% 2.44/1.35 ! [M: nat,N: nat] :
% 2.44/1.35 ( ( ord_less_nat @ M @ N )
% 2.44/1.35 => ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_23_lessI,axiom,
% 2.44/1.35 ! [N: nat] : ( ord_less_nat @ N @ ( suc @ N ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_24_not_I1_J,axiom,
% 2.44/1.35 ~ ( ( ord_less_int @ ya @ zero_zero_int )
% 2.44/1.35 | ( ord_less_int @ na @ zero_zero_int ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_25_not_I2_J,axiom,
% 2.44/1.35 ~ ( ( ord_less_int @ ya @ zero_zero_int )
% 2.44/1.35 | ( ord_less_int @ na @ zero_zero_int ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_26__C1_Oprems_C_I2_J,axiom,
% 2.44/1.35 ord_less_eq_int @ zero_zero_int @ ya ).
% 2.44/1.35
% 2.44/1.35 thf(fact_27__C1_Oprems_C_I3_J,axiom,
% 2.44/1.35 ord_less_eq_int @ zero_zero_int @ na ).
% 2.44/1.35
% 2.44/1.35 thf(fact_28_of__int__0,axiom,
% 2.44/1.35 ( ( ring_1_of_int_real @ zero_zero_int )
% 2.44/1.35 = zero_zero_real ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_29_of__int__0,axiom,
% 2.44/1.35 ( ( ring_1_of_int_int @ zero_zero_int )
% 2.44/1.35 = zero_zero_int ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_30_of__int__0__eq__iff,axiom,
% 2.44/1.35 ! [Z: int] :
% 2.44/1.35 ( ( zero_zero_real
% 2.44/1.35 = ( ring_1_of_int_real @ Z ) )
% 2.44/1.35 = ( Z = zero_zero_int ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_31_of__int__0__eq__iff,axiom,
% 2.44/1.35 ! [Z: int] :
% 2.44/1.35 ( ( zero_zero_int
% 2.44/1.35 = ( ring_1_of_int_int @ Z ) )
% 2.44/1.35 = ( Z = zero_zero_int ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_32_of__int__eq__0__iff,axiom,
% 2.44/1.35 ! [Z: int] :
% 2.44/1.35 ( ( ( ring_1_of_int_real @ Z )
% 2.44/1.35 = zero_zero_real )
% 2.44/1.35 = ( Z = zero_zero_int ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_33_of__int__eq__0__iff,axiom,
% 2.44/1.35 ! [Z: int] :
% 2.44/1.35 ( ( ( ring_1_of_int_int @ Z )
% 2.44/1.35 = zero_zero_int )
% 2.44/1.35 = ( Z = zero_zero_int ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_34_of__int__less__0__iff,axiom,
% 2.44/1.35 ! [Z: int] :
% 2.44/1.35 ( ( ord_less_real @ ( ring_1_of_int_real @ Z ) @ zero_zero_real )
% 2.44/1.35 = ( ord_less_int @ Z @ zero_zero_int ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_35_of__int__less__0__iff,axiom,
% 2.44/1.35 ! [Z: int] :
% 2.44/1.35 ( ( ord_less_int @ ( ring_1_of_int_int @ Z ) @ zero_zero_int )
% 2.44/1.35 = ( ord_less_int @ Z @ zero_zero_int ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_36_of__int__0__less__iff,axiom,
% 2.44/1.35 ! [Z: int] :
% 2.44/1.35 ( ( ord_less_real @ zero_zero_real @ ( ring_1_of_int_real @ Z ) )
% 2.44/1.35 = ( ord_less_int @ zero_zero_int @ Z ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_37_of__int__0__less__iff,axiom,
% 2.44/1.35 ! [Z: int] :
% 2.44/1.35 ( ( ord_less_int @ zero_zero_int @ ( ring_1_of_int_int @ Z ) )
% 2.44/1.35 = ( ord_less_int @ zero_zero_int @ Z ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_38_less__int__code_I1_J,axiom,
% 2.44/1.35 ~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_39_not__less__less__Suc__eq,axiom,
% 2.44/1.35 ! [N: nat,M: nat] :
% 2.44/1.35 ( ~ ( ord_less_nat @ N @ M )
% 2.44/1.35 => ( ( ord_less_nat @ N @ ( suc @ M ) )
% 2.44/1.35 = ( N = M ) ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_40_strict__inc__induct,axiom,
% 2.44/1.35 ! [I: nat,J: nat,P: nat > $o] :
% 2.44/1.35 ( ( ord_less_nat @ I @ J )
% 2.44/1.35 => ( ! [I2: nat] :
% 2.44/1.35 ( ( J
% 2.44/1.35 = ( suc @ I2 ) )
% 2.44/1.35 => ( P @ I2 ) )
% 2.44/1.35 => ( ! [I2: nat] :
% 2.44/1.35 ( ( ord_less_nat @ I2 @ J )
% 2.44/1.35 => ( ( P @ ( suc @ I2 ) )
% 2.44/1.35 => ( P @ I2 ) ) )
% 2.44/1.35 => ( P @ I ) ) ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_41_less__Suc__induct,axiom,
% 2.44/1.35 ! [I: nat,J: nat,P: nat > nat > $o] :
% 2.44/1.35 ( ( ord_less_nat @ I @ J )
% 2.44/1.35 => ( ! [I2: nat] : ( P @ I2 @ ( suc @ I2 ) )
% 2.44/1.35 => ( ! [I2: nat,J2: nat,K: nat] :
% 2.44/1.35 ( ( ord_less_nat @ I2 @ J2 )
% 2.44/1.35 => ( ( ord_less_nat @ J2 @ K )
% 2.44/1.35 => ( ( P @ I2 @ J2 )
% 2.44/1.35 => ( ( P @ J2 @ K )
% 2.44/1.35 => ( P @ I2 @ K ) ) ) ) )
% 2.44/1.35 => ( P @ I @ J ) ) ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_42_less__trans__Suc,axiom,
% 2.44/1.35 ! [I: nat,J: nat,K2: nat] :
% 2.44/1.35 ( ( ord_less_nat @ I @ J )
% 2.44/1.35 => ( ( ord_less_nat @ J @ K2 )
% 2.44/1.35 => ( ord_less_nat @ ( suc @ I ) @ K2 ) ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_43_Suc__less__SucD,axiom,
% 2.44/1.35 ! [M: nat,N: nat] :
% 2.44/1.35 ( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
% 2.44/1.35 => ( ord_less_nat @ M @ N ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_44_less__antisym,axiom,
% 2.44/1.35 ! [N: nat,M: nat] :
% 2.44/1.35 ( ~ ( ord_less_nat @ N @ M )
% 2.44/1.35 => ( ( ord_less_nat @ N @ ( suc @ M ) )
% 2.44/1.35 => ( M = N ) ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_45_Suc__less__eq2,axiom,
% 2.44/1.35 ! [N: nat,M: nat] :
% 2.44/1.35 ( ( ord_less_nat @ ( suc @ N ) @ M )
% 2.44/1.35 = ( ? [M2: nat] :
% 2.44/1.35 ( ( M
% 2.44/1.35 = ( suc @ M2 ) )
% 2.44/1.35 & ( ord_less_nat @ N @ M2 ) ) ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_46_All__less__Suc,axiom,
% 2.44/1.35 ! [N: nat,P: nat > $o] :
% 2.44/1.35 ( ( ! [I3: nat] :
% 2.44/1.35 ( ( ord_less_nat @ I3 @ ( suc @ N ) )
% 2.44/1.35 => ( P @ I3 ) ) )
% 2.44/1.35 = ( ( P @ N )
% 2.44/1.35 & ! [I3: nat] :
% 2.44/1.35 ( ( ord_less_nat @ I3 @ N )
% 2.44/1.35 => ( P @ I3 ) ) ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_47_not__less__eq,axiom,
% 2.44/1.35 ! [M: nat,N: nat] :
% 2.44/1.35 ( ( ~ ( ord_less_nat @ M @ N ) )
% 2.44/1.35 = ( ord_less_nat @ N @ ( suc @ M ) ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_48_less__Suc__eq,axiom,
% 2.44/1.35 ! [M: nat,N: nat] :
% 2.44/1.35 ( ( ord_less_nat @ M @ ( suc @ N ) )
% 2.44/1.35 = ( ( ord_less_nat @ M @ N )
% 2.44/1.35 | ( M = N ) ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_49_Ex__less__Suc,axiom,
% 2.44/1.35 ! [N: nat,P: nat > $o] :
% 2.44/1.35 ( ( ? [I3: nat] :
% 2.44/1.35 ( ( ord_less_nat @ I3 @ ( suc @ N ) )
% 2.44/1.35 & ( P @ I3 ) ) )
% 2.44/1.35 = ( ( P @ N )
% 2.44/1.35 | ? [I3: nat] :
% 2.44/1.35 ( ( ord_less_nat @ I3 @ N )
% 2.44/1.35 & ( P @ I3 ) ) ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_50_less__SucI,axiom,
% 2.44/1.35 ! [M: nat,N: nat] :
% 2.44/1.35 ( ( ord_less_nat @ M @ N )
% 2.44/1.35 => ( ord_less_nat @ M @ ( suc @ N ) ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_51_less__SucE,axiom,
% 2.44/1.35 ! [M: nat,N: nat] :
% 2.44/1.35 ( ( ord_less_nat @ M @ ( suc @ N ) )
% 2.44/1.35 => ( ~ ( ord_less_nat @ M @ N )
% 2.44/1.35 => ( M = N ) ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_52_Suc__lessI,axiom,
% 2.44/1.35 ! [M: nat,N: nat] :
% 2.44/1.35 ( ( ord_less_nat @ M @ N )
% 2.44/1.35 => ( ( ( suc @ M )
% 2.44/1.35 != N )
% 2.44/1.35 => ( ord_less_nat @ ( suc @ M ) @ N ) ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_53_Suc__lessE,axiom,
% 2.44/1.35 ! [I: nat,K2: nat] :
% 2.44/1.35 ( ( ord_less_nat @ ( suc @ I ) @ K2 )
% 2.44/1.35 => ~ ! [J2: nat] :
% 2.44/1.35 ( ( ord_less_nat @ I @ J2 )
% 2.44/1.35 => ( K2
% 2.44/1.35 != ( suc @ J2 ) ) ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_54_Suc__lessD,axiom,
% 2.44/1.35 ! [M: nat,N: nat] :
% 2.44/1.35 ( ( ord_less_nat @ ( suc @ M ) @ N )
% 2.44/1.35 => ( ord_less_nat @ M @ N ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_55_Nat_OlessE,axiom,
% 2.44/1.35 ! [I: nat,K2: nat] :
% 2.44/1.35 ( ( ord_less_nat @ I @ K2 )
% 2.44/1.35 => ( ( K2
% 2.44/1.35 != ( suc @ I ) )
% 2.44/1.35 => ~ ! [J2: nat] :
% 2.44/1.35 ( ( ord_less_nat @ I @ J2 )
% 2.44/1.35 => ( K2
% 2.44/1.35 != ( suc @ J2 ) ) ) ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_56_of__int__pos,axiom,
% 2.44/1.35 ! [Z: int] :
% 2.44/1.35 ( ( ord_less_int @ zero_zero_int @ Z )
% 2.44/1.35 => ( ord_less_real @ zero_zero_real @ ( ring_1_of_int_real @ Z ) ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_57_of__int__pos,axiom,
% 2.44/1.35 ! [Z: int] :
% 2.44/1.35 ( ( ord_less_int @ zero_zero_int @ Z )
% 2.44/1.35 => ( ord_less_int @ zero_zero_int @ ( ring_1_of_int_int @ Z ) ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_58_n__not__Suc__n,axiom,
% 2.44/1.35 ! [N: nat] :
% 2.44/1.35 ( N
% 2.44/1.35 != ( suc @ N ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_59_Suc__inject,axiom,
% 2.44/1.35 ! [X: nat,Y: nat] :
% 2.44/1.35 ( ( ( suc @ X )
% 2.44/1.35 = ( suc @ Y ) )
% 2.44/1.35 => ( X = Y ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_60_lift__Suc__mono__less__iff,axiom,
% 2.44/1.35 ! [F: nat > real,N: nat,M: nat] :
% 2.44/1.35 ( ! [N2: nat] : ( ord_less_real @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
% 2.44/1.35 => ( ( ord_less_real @ ( F @ N ) @ ( F @ M ) )
% 2.44/1.35 = ( ord_less_nat @ N @ M ) ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_61_lift__Suc__mono__less__iff,axiom,
% 2.44/1.35 ! [F: nat > int,N: nat,M: nat] :
% 2.44/1.35 ( ! [N2: nat] : ( ord_less_int @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
% 2.44/1.35 => ( ( ord_less_int @ ( F @ N ) @ ( F @ M ) )
% 2.44/1.35 = ( ord_less_nat @ N @ M ) ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_62_lift__Suc__mono__less__iff,axiom,
% 2.44/1.35 ! [F: nat > nat,N: nat,M: nat] :
% 2.44/1.35 ( ! [N2: nat] : ( ord_less_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
% 2.44/1.35 => ( ( ord_less_nat @ ( F @ N ) @ ( F @ M ) )
% 2.44/1.35 = ( ord_less_nat @ N @ M ) ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_63_lift__Suc__mono__less,axiom,
% 2.44/1.35 ! [F: nat > real,N: nat,N3: nat] :
% 2.44/1.35 ( ! [N2: nat] : ( ord_less_real @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
% 2.44/1.35 => ( ( ord_less_nat @ N @ N3 )
% 2.44/1.35 => ( ord_less_real @ ( F @ N ) @ ( F @ N3 ) ) ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_64_lift__Suc__mono__less,axiom,
% 2.44/1.35 ! [F: nat > int,N: nat,N3: nat] :
% 2.44/1.35 ( ! [N2: nat] : ( ord_less_int @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
% 2.44/1.35 => ( ( ord_less_nat @ N @ N3 )
% 2.44/1.35 => ( ord_less_int @ ( F @ N ) @ ( F @ N3 ) ) ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_65_lift__Suc__mono__less,axiom,
% 2.44/1.35 ! [F: nat > nat,N: nat,N3: nat] :
% 2.44/1.35 ( ! [N2: nat] : ( ord_less_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
% 2.44/1.35 => ( ( ord_less_nat @ N @ N3 )
% 2.44/1.35 => ( ord_less_nat @ ( F @ N ) @ ( F @ N3 ) ) ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_66_power__0__Suc,axiom,
% 2.44/1.35 ! [N: nat] :
% 2.44/1.35 ( ( power_power_real @ zero_zero_real @ ( suc @ N ) )
% 2.44/1.35 = zero_zero_real ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_67_power__0__Suc,axiom,
% 2.44/1.35 ! [N: nat] :
% 2.44/1.35 ( ( power_power_int @ zero_zero_int @ ( suc @ N ) )
% 2.44/1.35 = zero_zero_int ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_68_power__0__Suc,axiom,
% 2.44/1.35 ! [N: nat] :
% 2.44/1.35 ( ( power_power_nat @ zero_zero_nat @ ( suc @ N ) )
% 2.44/1.35 = zero_zero_nat ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_69_NY0,axiom,
% 2.44/1.35 ord_less_real @ zero_zero_real @ ny ).
% 2.44/1.35
% 2.44/1.35 thf(fact_70_power__eq__0__iff,axiom,
% 2.44/1.35 ! [A: real,N: nat] :
% 2.44/1.35 ( ( ( power_power_real @ A @ N )
% 2.44/1.35 = zero_zero_real )
% 2.44/1.35 = ( ( A = zero_zero_real )
% 2.44/1.35 & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_71_power__eq__0__iff,axiom,
% 2.44/1.35 ! [A: int,N: nat] :
% 2.44/1.35 ( ( ( power_power_int @ A @ N )
% 2.44/1.35 = zero_zero_int )
% 2.44/1.35 = ( ( A = zero_zero_int )
% 2.44/1.35 & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_72_power__eq__0__iff,axiom,
% 2.44/1.35 ! [A: nat,N: nat] :
% 2.44/1.35 ( ( ( power_power_nat @ A @ N )
% 2.44/1.35 = zero_zero_nat )
% 2.44/1.35 = ( ( A = zero_zero_nat )
% 2.44/1.35 & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_73_not__gr__zero,axiom,
% 2.44/1.35 ! [N: nat] :
% 2.44/1.35 ( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
% 2.44/1.35 = ( N = zero_zero_nat ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_74_realpow__pos__nth2,axiom,
% 2.44/1.35 ! [A: real,N: nat] :
% 2.44/1.35 ( ( ord_less_real @ zero_zero_real @ A )
% 2.44/1.35 => ? [R: real] :
% 2.44/1.35 ( ( ord_less_real @ zero_zero_real @ R )
% 2.44/1.35 & ( ( power_power_real @ R @ ( suc @ N ) )
% 2.44/1.35 = A ) ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_75_zero__less__power,axiom,
% 2.44/1.35 ! [A: real,N: nat] :
% 2.44/1.35 ( ( ord_less_real @ zero_zero_real @ A )
% 2.44/1.35 => ( ord_less_real @ zero_zero_real @ ( power_power_real @ A @ N ) ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_76_zero__less__power,axiom,
% 2.44/1.35 ! [A: int,N: nat] :
% 2.44/1.35 ( ( ord_less_int @ zero_zero_int @ A )
% 2.44/1.35 => ( ord_less_int @ zero_zero_int @ ( power_power_int @ A @ N ) ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_77_zero__less__power,axiom,
% 2.44/1.35 ! [A: nat,N: nat] :
% 2.44/1.35 ( ( ord_less_nat @ zero_zero_nat @ A )
% 2.44/1.35 => ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ A @ N ) ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_78_NY__def,axiom,
% 2.44/1.35 ( ny
% 2.44/1.35 = ( divide_divide_real @ ( ring_1_of_int_real @ na ) @ ( power_power_real @ ( ring_1_of_int_real @ ya ) @ pm ) ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_79_ex__of__int__less,axiom,
% 2.44/1.35 ! [X: real] :
% 2.44/1.35 ? [Z2: int] : ( ord_less_real @ ( ring_1_of_int_real @ Z2 ) @ X ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_80_ex__less__of__int,axiom,
% 2.44/1.35 ! [X: real] :
% 2.44/1.35 ? [Z2: int] : ( ord_less_real @ X @ ( ring_1_of_int_real @ Z2 ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_81_yyn,axiom,
% 2.44/1.35 ord_less_int @ na @ ( power_power_int @ ya @ p ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_82__092_060open_062pm_A_061_A0_A_092_060Longrightarrow_062_AFalse_092_060close_062,axiom,
% 2.44/1.35 pm != zero_zero_nat ).
% 2.44/1.35
% 2.44/1.35 thf(fact_83_p0,axiom,
% 2.44/1.35 p != zero_zero_nat ).
% 2.44/1.35
% 2.44/1.35 thf(fact_84_pm0,axiom,
% 2.44/1.35 ord_less_nat @ zero_zero_nat @ pm ).
% 2.44/1.35
% 2.44/1.35 thf(fact_85__C1_Oprems_C_I1_J,axiom,
% 2.44/1.35 ord_less_eq_int @ na @ ( power_power_int @ ya @ p ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_86_p,axiom,
% 2.44/1.35 ( p
% 2.44/1.35 = ( suc @ pm ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_87_fixed__root__axioms,axiom,
% 2.44/1.35 nthRoo1352626670d_root @ p @ pm ).
% 2.44/1.35
% 2.44/1.35 thf(fact_88_le__zero__eq,axiom,
% 2.44/1.35 ! [N: nat] :
% 2.44/1.35 ( ( ord_less_eq_nat @ N @ zero_zero_nat )
% 2.44/1.35 = ( N = zero_zero_nat ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_89_nat__power__eq__Suc__0__iff,axiom,
% 2.44/1.35 ! [X: nat,M: nat] :
% 2.44/1.35 ( ( ( power_power_nat @ X @ M )
% 2.44/1.35 = ( suc @ zero_zero_nat ) )
% 2.44/1.35 = ( ( M = zero_zero_nat )
% 2.44/1.35 | ( X
% 2.44/1.35 = ( suc @ zero_zero_nat ) ) ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_90_power__Suc__0,axiom,
% 2.44/1.35 ! [N: nat] :
% 2.44/1.35 ( ( power_power_nat @ ( suc @ zero_zero_nat ) @ N )
% 2.44/1.35 = ( suc @ zero_zero_nat ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_91_neq0__conv,axiom,
% 2.44/1.35 ! [N: nat] :
% 2.44/1.35 ( ( N != zero_zero_nat )
% 2.44/1.35 = ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_92_less__nat__zero__code,axiom,
% 2.44/1.35 ! [N: nat] :
% 2.44/1.35 ~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_93_bot__nat__0_Onot__eq__extremum,axiom,
% 2.44/1.35 ! [A: nat] :
% 2.44/1.35 ( ( A != zero_zero_nat )
% 2.44/1.35 = ( ord_less_nat @ zero_zero_nat @ A ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_94_nat__zero__less__power__iff,axiom,
% 2.44/1.35 ! [X: nat,N: nat] :
% 2.44/1.35 ( ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ X @ N ) )
% 2.44/1.35 = ( ( ord_less_nat @ zero_zero_nat @ X )
% 2.44/1.35 | ( N = zero_zero_nat ) ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_95_power__Suc0__right,axiom,
% 2.44/1.35 ! [A: real] :
% 2.44/1.35 ( ( power_power_real @ A @ ( suc @ zero_zero_nat ) )
% 2.44/1.35 = A ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_96_power__Suc0__right,axiom,
% 2.44/1.35 ! [A: int] :
% 2.44/1.35 ( ( power_power_int @ A @ ( suc @ zero_zero_nat ) )
% 2.44/1.35 = A ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_97_power__Suc0__right,axiom,
% 2.44/1.35 ! [A: nat] :
% 2.44/1.35 ( ( power_power_nat @ A @ ( suc @ zero_zero_nat ) )
% 2.44/1.35 = A ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_98_zero__less__Suc,axiom,
% 2.44/1.35 ! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_99_less__Suc0,axiom,
% 2.44/1.35 ! [N: nat] :
% 2.44/1.35 ( ( ord_less_nat @ N @ ( suc @ zero_zero_nat ) )
% 2.44/1.35 = ( N = zero_zero_nat ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_100_of__int__le__iff,axiom,
% 2.44/1.35 ! [W: int,Z: int] :
% 2.44/1.35 ( ( ord_less_eq_real @ ( ring_1_of_int_real @ W ) @ ( ring_1_of_int_real @ Z ) )
% 2.44/1.35 = ( ord_less_eq_int @ W @ Z ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_101_of__int__le__iff,axiom,
% 2.44/1.35 ! [W: int,Z: int] :
% 2.44/1.35 ( ( ord_less_eq_int @ ( ring_1_of_int_int @ W ) @ ( ring_1_of_int_int @ Z ) )
% 2.44/1.35 = ( ord_less_eq_int @ W @ Z ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_102_power__mono__iff,axiom,
% 2.44/1.35 ! [A: real,B: real,N: nat] :
% 2.44/1.35 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 2.44/1.35 => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 2.44/1.35 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 2.44/1.35 => ( ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B @ N ) )
% 2.44/1.35 = ( ord_less_eq_real @ A @ B ) ) ) ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_103_power__mono__iff,axiom,
% 2.44/1.35 ! [A: int,B: int,N: nat] :
% 2.44/1.35 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 2.44/1.35 => ( ( ord_less_eq_int @ zero_zero_int @ B )
% 2.44/1.35 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 2.44/1.35 => ( ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) )
% 2.44/1.35 = ( ord_less_eq_int @ A @ B ) ) ) ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_104_power__mono__iff,axiom,
% 2.44/1.35 ! [A: nat,B: nat,N: nat] :
% 2.44/1.35 ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 2.44/1.35 => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 2.44/1.35 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 2.44/1.35 => ( ( ord_less_eq_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) )
% 2.44/1.35 = ( ord_less_eq_nat @ A @ B ) ) ) ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_105_of__int__le__0__iff,axiom,
% 2.44/1.35 ! [Z: int] :
% 2.44/1.35 ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z ) @ zero_zero_real )
% 2.44/1.35 = ( ord_less_eq_int @ Z @ zero_zero_int ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_106_of__int__le__0__iff,axiom,
% 2.44/1.35 ! [Z: int] :
% 2.44/1.35 ( ( ord_less_eq_int @ ( ring_1_of_int_int @ Z ) @ zero_zero_int )
% 2.44/1.35 = ( ord_less_eq_int @ Z @ zero_zero_int ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_107_of__int__0__le__iff,axiom,
% 2.44/1.35 ! [Z: int] :
% 2.44/1.35 ( ( ord_less_eq_real @ zero_zero_real @ ( ring_1_of_int_real @ Z ) )
% 2.44/1.35 = ( ord_less_eq_int @ zero_zero_int @ Z ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_108_of__int__0__le__iff,axiom,
% 2.44/1.35 ! [Z: int] :
% 2.44/1.35 ( ( ord_less_eq_int @ zero_zero_int @ ( ring_1_of_int_int @ Z ) )
% 2.44/1.35 = ( ord_less_eq_int @ zero_zero_int @ Z ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_109_of__int__le__of__int__power__cancel__iff,axiom,
% 2.44/1.35 ! [B: int,W: nat,X: int] :
% 2.44/1.35 ( ( ord_less_eq_real @ ( power_power_real @ ( ring_1_of_int_real @ B ) @ W ) @ ( ring_1_of_int_real @ X ) )
% 2.44/1.35 = ( ord_less_eq_int @ ( power_power_int @ B @ W ) @ X ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_110_of__int__le__of__int__power__cancel__iff,axiom,
% 2.44/1.35 ! [B: int,W: nat,X: int] :
% 2.44/1.35 ( ( ord_less_eq_int @ ( power_power_int @ ( ring_1_of_int_int @ B ) @ W ) @ ( ring_1_of_int_int @ X ) )
% 2.44/1.35 = ( ord_less_eq_int @ ( power_power_int @ B @ W ) @ X ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_111_of__int__power__le__of__int__cancel__iff,axiom,
% 2.44/1.35 ! [X: int,B: int,W: nat] :
% 2.44/1.35 ( ( ord_less_eq_real @ ( ring_1_of_int_real @ X ) @ ( power_power_real @ ( ring_1_of_int_real @ B ) @ W ) )
% 2.44/1.35 = ( ord_less_eq_int @ X @ ( power_power_int @ B @ W ) ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_112_of__int__power__le__of__int__cancel__iff,axiom,
% 2.44/1.35 ! [X: int,B: int,W: nat] :
% 2.44/1.35 ( ( ord_less_eq_int @ ( ring_1_of_int_int @ X ) @ ( power_power_int @ ( ring_1_of_int_int @ B ) @ W ) )
% 2.44/1.35 = ( ord_less_eq_int @ X @ ( power_power_int @ B @ W ) ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_113_gr0I,axiom,
% 2.44/1.35 ! [N: nat] :
% 2.44/1.35 ( ( N != zero_zero_nat )
% 2.44/1.35 => ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_114_not__gr0,axiom,
% 2.44/1.35 ! [N: nat] :
% 2.44/1.35 ( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
% 2.44/1.35 = ( N = zero_zero_nat ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_115_not__less0,axiom,
% 2.44/1.35 ! [N: nat] :
% 2.44/1.35 ~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_116_less__zeroE,axiom,
% 2.44/1.35 ! [N: nat] :
% 2.44/1.35 ~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_117_nat__neq__iff,axiom,
% 2.44/1.35 ! [M: nat,N: nat] :
% 2.44/1.35 ( ( M != N )
% 2.44/1.35 = ( ( ord_less_nat @ M @ N )
% 2.44/1.35 | ( ord_less_nat @ N @ M ) ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_118_less__not__refl,axiom,
% 2.44/1.35 ! [N: nat] :
% 2.44/1.35 ~ ( ord_less_nat @ N @ N ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_119_less__not__refl2,axiom,
% 2.44/1.35 ! [N: nat,M: nat] :
% 2.44/1.35 ( ( ord_less_nat @ N @ M )
% 2.44/1.35 => ( M != N ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_120_less__not__refl3,axiom,
% 2.44/1.35 ! [S: nat,T: nat] :
% 2.44/1.35 ( ( ord_less_nat @ S @ T )
% 2.44/1.35 => ( S != T ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_121_gr__implies__not0,axiom,
% 2.44/1.35 ! [M: nat,N: nat] :
% 2.44/1.35 ( ( ord_less_nat @ M @ N )
% 2.44/1.35 => ( N != zero_zero_nat ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_122_less__irrefl__nat,axiom,
% 2.44/1.35 ! [N: nat] :
% 2.44/1.35 ~ ( ord_less_nat @ N @ N ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_123_nat__less__induct,axiom,
% 2.44/1.35 ! [P: nat > $o,N: nat] :
% 2.44/1.35 ( ! [N2: nat] :
% 2.44/1.35 ( ! [M3: nat] :
% 2.44/1.35 ( ( ord_less_nat @ M3 @ N2 )
% 2.44/1.35 => ( P @ M3 ) )
% 2.44/1.35 => ( P @ N2 ) )
% 2.44/1.35 => ( P @ N ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_124_infinite__descent,axiom,
% 2.44/1.35 ! [P: nat > $o,N: nat] :
% 2.44/1.35 ( ! [N2: nat] :
% 2.44/1.35 ( ~ ( P @ N2 )
% 2.44/1.35 => ? [M3: nat] :
% 2.44/1.35 ( ( ord_less_nat @ M3 @ N2 )
% 2.44/1.35 & ~ ( P @ M3 ) ) )
% 2.44/1.35 => ( P @ N ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_125_infinite__descent0,axiom,
% 2.44/1.35 ! [P: nat > $o,N: nat] :
% 2.44/1.35 ( ( P @ zero_zero_nat )
% 2.44/1.35 => ( ! [N2: nat] :
% 2.44/1.35 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 2.44/1.35 => ( ~ ( P @ N2 )
% 2.44/1.35 => ? [M3: nat] :
% 2.44/1.35 ( ( ord_less_nat @ M3 @ N2 )
% 2.44/1.35 & ~ ( P @ M3 ) ) ) )
% 2.44/1.35 => ( P @ N ) ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_126_linorder__neqE__nat,axiom,
% 2.44/1.35 ! [X: nat,Y: nat] :
% 2.44/1.35 ( ( X != Y )
% 2.44/1.35 => ( ~ ( ord_less_nat @ X @ Y )
% 2.44/1.35 => ( ord_less_nat @ Y @ X ) ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_127_bot__nat__0_Oextremum__strict,axiom,
% 2.44/1.35 ! [A: nat] :
% 2.44/1.35 ~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_128_nat__power__less__imp__less,axiom,
% 2.44/1.35 ! [I: nat,M: nat,N: nat] :
% 2.44/1.35 ( ( ord_less_nat @ zero_zero_nat @ I )
% 2.44/1.35 => ( ( ord_less_nat @ ( power_power_nat @ I @ M ) @ ( power_power_nat @ I @ N ) )
% 2.44/1.35 => ( ord_less_nat @ M @ N ) ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_129_lift__Suc__antimono__le,axiom,
% 2.44/1.35 ! [F: nat > real,N: nat,N3: nat] :
% 2.44/1.35 ( ! [N2: nat] : ( ord_less_eq_real @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
% 2.44/1.35 => ( ( ord_less_eq_nat @ N @ N3 )
% 2.44/1.35 => ( ord_less_eq_real @ ( F @ N3 ) @ ( F @ N ) ) ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_130_lift__Suc__antimono__le,axiom,
% 2.44/1.35 ! [F: nat > int,N: nat,N3: nat] :
% 2.44/1.35 ( ! [N2: nat] : ( ord_less_eq_int @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
% 2.44/1.35 => ( ( ord_less_eq_nat @ N @ N3 )
% 2.44/1.35 => ( ord_less_eq_int @ ( F @ N3 ) @ ( F @ N ) ) ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_131_lift__Suc__antimono__le,axiom,
% 2.44/1.35 ! [F: nat > nat,N: nat,N3: nat] :
% 2.44/1.35 ( ! [N2: nat] : ( ord_less_eq_nat @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
% 2.44/1.35 => ( ( ord_less_eq_nat @ N @ N3 )
% 2.44/1.35 => ( ord_less_eq_nat @ ( F @ N3 ) @ ( F @ N ) ) ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_132_lift__Suc__mono__le,axiom,
% 2.44/1.35 ! [F: nat > real,N: nat,N3: nat] :
% 2.44/1.35 ( ! [N2: nat] : ( ord_less_eq_real @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
% 2.44/1.35 => ( ( ord_less_eq_nat @ N @ N3 )
% 2.44/1.35 => ( ord_less_eq_real @ ( F @ N ) @ ( F @ N3 ) ) ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_133_lift__Suc__mono__le,axiom,
% 2.44/1.35 ! [F: nat > int,N: nat,N3: nat] :
% 2.44/1.35 ( ! [N2: nat] : ( ord_less_eq_int @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
% 2.44/1.35 => ( ( ord_less_eq_nat @ N @ N3 )
% 2.44/1.35 => ( ord_less_eq_int @ ( F @ N ) @ ( F @ N3 ) ) ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_134_lift__Suc__mono__le,axiom,
% 2.44/1.35 ! [F: nat > nat,N: nat,N3: nat] :
% 2.44/1.35 ( ! [N2: nat] : ( ord_less_eq_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
% 2.44/1.35 => ( ( ord_less_eq_nat @ N @ N3 )
% 2.44/1.35 => ( ord_less_eq_nat @ ( F @ N ) @ ( F @ N3 ) ) ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_135_zero__le,axiom,
% 2.44/1.35 ! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_136_power__divide,axiom,
% 2.44/1.35 ! [A: real,B: real,N: nat] :
% 2.44/1.35 ( ( power_power_real @ ( divide_divide_real @ A @ B ) @ N )
% 2.44/1.35 = ( divide_divide_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B @ N ) ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_137_ex__le__of__int,axiom,
% 2.44/1.35 ! [X: real] :
% 2.44/1.35 ? [Z2: int] : ( ord_less_eq_real @ X @ ( ring_1_of_int_real @ Z2 ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_138_exists__least__lemma,axiom,
% 2.44/1.35 ! [P: nat > $o] :
% 2.44/1.35 ( ~ ( P @ zero_zero_nat )
% 2.44/1.35 => ( ? [X_1: nat] : ( P @ X_1 )
% 2.44/1.35 => ? [N2: nat] :
% 2.44/1.35 ( ~ ( P @ N2 )
% 2.44/1.35 & ( P @ ( suc @ N2 ) ) ) ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_139_power__gt__expt,axiom,
% 2.44/1.35 ! [N: nat,K2: nat] :
% 2.44/1.35 ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N )
% 2.44/1.35 => ( ord_less_nat @ K2 @ ( power_power_nat @ N @ K2 ) ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_140_less__eq__int__code_I1_J,axiom,
% 2.44/1.35 ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% 2.44/1.35
% 2.44/1.35 thf(fact_141_nat_Odistinct_I1_J,axiom,
% 2.44/1.35 ! [X2: nat] :
% 2.44/1.35 ( zero_zero_nat
% 2.44/1.35 != ( suc @ X2 ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_142_old_Onat_Odistinct_I2_J,axiom,
% 2.44/1.35 ! [Nat2: nat] :
% 2.44/1.35 ( ( suc @ Nat2 )
% 2.44/1.35 != zero_zero_nat ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_143_old_Onat_Odistinct_I1_J,axiom,
% 2.44/1.35 ! [Nat2: nat] :
% 2.44/1.35 ( zero_zero_nat
% 2.44/1.35 != ( suc @ Nat2 ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_144_nat_OdiscI,axiom,
% 2.44/1.35 ! [Nat: nat,X2: nat] :
% 2.44/1.35 ( ( Nat
% 2.44/1.35 = ( suc @ X2 ) )
% 2.44/1.35 => ( Nat != zero_zero_nat ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_145_nat__induct,axiom,
% 2.44/1.35 ! [P: nat > $o,N: nat] :
% 2.44/1.35 ( ( P @ zero_zero_nat )
% 2.44/1.35 => ( ! [N2: nat] :
% 2.44/1.35 ( ( P @ N2 )
% 2.44/1.35 => ( P @ ( suc @ N2 ) ) )
% 2.44/1.35 => ( P @ N ) ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_146_diff__induct,axiom,
% 2.44/1.35 ! [P: nat > nat > $o,M: nat,N: nat] :
% 2.44/1.35 ( ! [X3: nat] : ( P @ X3 @ zero_zero_nat )
% 2.44/1.35 => ( ! [Y3: nat] : ( P @ zero_zero_nat @ ( suc @ Y3 ) )
% 2.44/1.35 => ( ! [X3: nat,Y3: nat] :
% 2.44/1.35 ( ( P @ X3 @ Y3 )
% 2.44/1.35 => ( P @ ( suc @ X3 ) @ ( suc @ Y3 ) ) )
% 2.44/1.35 => ( P @ M @ N ) ) ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_147_zero__induct,axiom,
% 2.44/1.35 ! [P: nat > $o,K2: nat] :
% 2.44/1.35 ( ( P @ K2 )
% 2.44/1.35 => ( ! [N2: nat] :
% 2.44/1.35 ( ( P @ ( suc @ N2 ) )
% 2.44/1.35 => ( P @ N2 ) )
% 2.44/1.35 => ( P @ zero_zero_nat ) ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_148_Suc__neq__Zero,axiom,
% 2.44/1.35 ! [M: nat] :
% 2.44/1.35 ( ( suc @ M )
% 2.44/1.35 != zero_zero_nat ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_149_Zero__neq__Suc,axiom,
% 2.44/1.35 ! [M: nat] :
% 2.44/1.35 ( zero_zero_nat
% 2.44/1.35 != ( suc @ M ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_150_Zero__not__Suc,axiom,
% 2.44/1.35 ! [M: nat] :
% 2.44/1.35 ( zero_zero_nat
% 2.44/1.35 != ( suc @ M ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_151_old_Onat_Oexhaust,axiom,
% 2.44/1.35 ! [Y: nat] :
% 2.44/1.35 ( ( Y != zero_zero_nat )
% 2.44/1.35 => ~ ! [Nat3: nat] :
% 2.44/1.35 ( Y
% 2.44/1.35 != ( suc @ Nat3 ) ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_152_old_Onat_Oinducts,axiom,
% 2.44/1.35 ! [P: nat > $o,Nat: nat] :
% 2.44/1.35 ( ( P @ zero_zero_nat )
% 2.44/1.35 => ( ! [Nat3: nat] :
% 2.44/1.35 ( ( P @ Nat3 )
% 2.44/1.35 => ( P @ ( suc @ Nat3 ) ) )
% 2.44/1.35 => ( P @ Nat ) ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_153_not0__implies__Suc,axiom,
% 2.44/1.35 ! [N: nat] :
% 2.44/1.35 ( ( N != zero_zero_nat )
% 2.44/1.35 => ? [M4: nat] :
% 2.44/1.35 ( N
% 2.44/1.35 = ( suc @ M4 ) ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_154_less__Suc__eq__0__disj,axiom,
% 2.44/1.35 ! [M: nat,N: nat] :
% 2.44/1.35 ( ( ord_less_nat @ M @ ( suc @ N ) )
% 2.44/1.35 = ( ( M = zero_zero_nat )
% 2.44/1.35 | ? [J3: nat] :
% 2.44/1.35 ( ( M
% 2.44/1.35 = ( suc @ J3 ) )
% 2.44/1.35 & ( ord_less_nat @ J3 @ N ) ) ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_155_gr0__implies__Suc,axiom,
% 2.44/1.35 ! [N: nat] :
% 2.44/1.35 ( ( ord_less_nat @ zero_zero_nat @ N )
% 2.44/1.35 => ? [M4: nat] :
% 2.44/1.35 ( N
% 2.44/1.35 = ( suc @ M4 ) ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_156_All__less__Suc2,axiom,
% 2.44/1.35 ! [N: nat,P: nat > $o] :
% 2.44/1.35 ( ( ! [I3: nat] :
% 2.44/1.35 ( ( ord_less_nat @ I3 @ ( suc @ N ) )
% 2.44/1.35 => ( P @ I3 ) ) )
% 2.44/1.35 = ( ( P @ zero_zero_nat )
% 2.44/1.35 & ! [I3: nat] :
% 2.44/1.35 ( ( ord_less_nat @ I3 @ N )
% 2.44/1.35 => ( P @ ( suc @ I3 ) ) ) ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_157_gr0__conv__Suc,axiom,
% 2.44/1.35 ! [N: nat] :
% 2.44/1.35 ( ( ord_less_nat @ zero_zero_nat @ N )
% 2.44/1.35 = ( ? [M5: nat] :
% 2.44/1.35 ( N
% 2.44/1.35 = ( suc @ M5 ) ) ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_158_Ex__less__Suc2,axiom,
% 2.44/1.35 ! [N: nat,P: nat > $o] :
% 2.44/1.35 ( ( ? [I3: nat] :
% 2.44/1.35 ( ( ord_less_nat @ I3 @ ( suc @ N ) )
% 2.44/1.35 & ( P @ I3 ) ) )
% 2.44/1.35 = ( ( P @ zero_zero_nat )
% 2.44/1.35 | ? [I3: nat] :
% 2.44/1.35 ( ( ord_less_nat @ I3 @ N )
% 2.44/1.35 & ( P @ ( suc @ I3 ) ) ) ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_159_power__eq__imp__eq__base,axiom,
% 2.44/1.35 ! [A: real,N: nat,B: real] :
% 2.44/1.35 ( ( ( power_power_real @ A @ N )
% 2.44/1.35 = ( power_power_real @ B @ N ) )
% 2.44/1.35 => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 2.44/1.35 => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 2.44/1.35 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 2.44/1.35 => ( A = B ) ) ) ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_160_power__eq__imp__eq__base,axiom,
% 2.44/1.35 ! [A: int,N: nat,B: int] :
% 2.44/1.35 ( ( ( power_power_int @ A @ N )
% 2.44/1.35 = ( power_power_int @ B @ N ) )
% 2.44/1.35 => ( ( ord_less_eq_int @ zero_zero_int @ A )
% 2.44/1.35 => ( ( ord_less_eq_int @ zero_zero_int @ B )
% 2.44/1.35 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 2.44/1.35 => ( A = B ) ) ) ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_161_power__eq__imp__eq__base,axiom,
% 2.44/1.35 ! [A: nat,N: nat,B: nat] :
% 2.44/1.35 ( ( ( power_power_nat @ A @ N )
% 2.44/1.35 = ( power_power_nat @ B @ N ) )
% 2.44/1.35 => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 2.44/1.35 => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 2.44/1.35 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 2.44/1.35 => ( A = B ) ) ) ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_162_power__eq__iff__eq__base,axiom,
% 2.44/1.35 ! [N: nat,A: real,B: real] :
% 2.44/1.35 ( ( ord_less_nat @ zero_zero_nat @ N )
% 2.44/1.35 => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 2.44/1.35 => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 2.44/1.35 => ( ( ( power_power_real @ A @ N )
% 2.44/1.35 = ( power_power_real @ B @ N ) )
% 2.44/1.35 = ( A = B ) ) ) ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_163_power__eq__iff__eq__base,axiom,
% 2.44/1.35 ! [N: nat,A: int,B: int] :
% 2.44/1.35 ( ( ord_less_nat @ zero_zero_nat @ N )
% 2.44/1.35 => ( ( ord_less_eq_int @ zero_zero_int @ A )
% 2.44/1.35 => ( ( ord_less_eq_int @ zero_zero_int @ B )
% 2.44/1.35 => ( ( ( power_power_int @ A @ N )
% 2.44/1.35 = ( power_power_int @ B @ N ) )
% 2.44/1.35 = ( A = B ) ) ) ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_164_power__eq__iff__eq__base,axiom,
% 2.44/1.35 ! [N: nat,A: nat,B: nat] :
% 2.44/1.35 ( ( ord_less_nat @ zero_zero_nat @ N )
% 2.44/1.35 => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 2.44/1.35 => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 2.44/1.35 => ( ( ( power_power_nat @ A @ N )
% 2.44/1.35 = ( power_power_nat @ B @ N ) )
% 2.44/1.35 = ( A = B ) ) ) ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_165_zero__le__power,axiom,
% 2.44/1.35 ! [A: real,N: nat] :
% 2.44/1.35 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 2.44/1.35 => ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ N ) ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_166_zero__le__power,axiom,
% 2.44/1.35 ! [A: int,N: nat] :
% 2.44/1.35 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 2.44/1.35 => ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ N ) ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_167_zero__le__power,axiom,
% 2.44/1.35 ! [A: nat,N: nat] :
% 2.44/1.35 ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 2.44/1.35 => ( ord_less_eq_nat @ zero_zero_nat @ ( power_power_nat @ A @ N ) ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_168_power__mono,axiom,
% 2.44/1.35 ! [A: real,B: real,N: nat] :
% 2.44/1.35 ( ( ord_less_eq_real @ A @ B )
% 2.44/1.35 => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 2.44/1.35 => ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B @ N ) ) ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_169_power__mono,axiom,
% 2.44/1.35 ! [A: int,B: int,N: nat] :
% 2.44/1.35 ( ( ord_less_eq_int @ A @ B )
% 2.44/1.35 => ( ( ord_less_eq_int @ zero_zero_int @ A )
% 2.44/1.35 => ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) ) ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_170_power__mono,axiom,
% 2.44/1.35 ! [A: nat,B: nat,N: nat] :
% 2.44/1.35 ( ( ord_less_eq_nat @ A @ B )
% 2.44/1.35 => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 2.44/1.35 => ( ord_less_eq_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) ) ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_171_of__int__nonneg,axiom,
% 2.44/1.35 ! [Z: int] :
% 2.44/1.35 ( ( ord_less_eq_int @ zero_zero_int @ Z )
% 2.44/1.35 => ( ord_less_eq_real @ zero_zero_real @ ( ring_1_of_int_real @ Z ) ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_172_of__int__nonneg,axiom,
% 2.44/1.35 ! [Z: int] :
% 2.44/1.35 ( ( ord_less_eq_int @ zero_zero_int @ Z )
% 2.44/1.35 => ( ord_less_eq_int @ zero_zero_int @ ( ring_1_of_int_int @ Z ) ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_173_power__less__imp__less__base,axiom,
% 2.44/1.35 ! [A: real,N: nat,B: real] :
% 2.44/1.35 ( ( ord_less_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B @ N ) )
% 2.44/1.35 => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 2.44/1.35 => ( ord_less_real @ A @ B ) ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_174_power__less__imp__less__base,axiom,
% 2.44/1.35 ! [A: int,N: nat,B: int] :
% 2.44/1.35 ( ( ord_less_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) )
% 2.44/1.35 => ( ( ord_less_eq_int @ zero_zero_int @ B )
% 2.44/1.35 => ( ord_less_int @ A @ B ) ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_175_power__less__imp__less__base,axiom,
% 2.44/1.35 ! [A: nat,N: nat,B: nat] :
% 2.44/1.35 ( ( ord_less_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) )
% 2.44/1.35 => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 2.44/1.35 => ( ord_less_nat @ A @ B ) ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_176_power__strict__mono,axiom,
% 2.44/1.35 ! [A: real,B: real,N: nat] :
% 2.44/1.35 ( ( ord_less_real @ A @ B )
% 2.44/1.35 => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 2.44/1.35 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 2.44/1.35 => ( ord_less_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B @ N ) ) ) ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_177_power__strict__mono,axiom,
% 2.44/1.35 ! [A: int,B: int,N: nat] :
% 2.44/1.35 ( ( ord_less_int @ A @ B )
% 2.44/1.35 => ( ( ord_less_eq_int @ zero_zero_int @ A )
% 2.44/1.35 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 2.44/1.35 => ( ord_less_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) ) ) ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_178_power__strict__mono,axiom,
% 2.44/1.35 ! [A: nat,B: nat,N: nat] :
% 2.44/1.35 ( ( ord_less_nat @ A @ B )
% 2.44/1.35 => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 2.44/1.35 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 2.44/1.35 => ( ord_less_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) ) ) ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_179_power__inject__base,axiom,
% 2.44/1.35 ! [A: real,N: nat,B: real] :
% 2.44/1.35 ( ( ( power_power_real @ A @ ( suc @ N ) )
% 2.44/1.35 = ( power_power_real @ B @ ( suc @ N ) ) )
% 2.44/1.35 => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 2.44/1.35 => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 2.44/1.35 => ( A = B ) ) ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_180_power__inject__base,axiom,
% 2.44/1.35 ! [A: int,N: nat,B: int] :
% 2.44/1.35 ( ( ( power_power_int @ A @ ( suc @ N ) )
% 2.44/1.35 = ( power_power_int @ B @ ( suc @ N ) ) )
% 2.44/1.35 => ( ( ord_less_eq_int @ zero_zero_int @ A )
% 2.44/1.35 => ( ( ord_less_eq_int @ zero_zero_int @ B )
% 2.44/1.35 => ( A = B ) ) ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_181_power__inject__base,axiom,
% 2.44/1.35 ! [A: nat,N: nat,B: nat] :
% 2.44/1.35 ( ( ( power_power_nat @ A @ ( suc @ N ) )
% 2.44/1.35 = ( power_power_nat @ B @ ( suc @ N ) ) )
% 2.44/1.35 => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 2.44/1.35 => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 2.44/1.35 => ( A = B ) ) ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_182_power__le__imp__le__base,axiom,
% 2.44/1.35 ! [A: real,N: nat,B: real] :
% 2.44/1.35 ( ( ord_less_eq_real @ ( power_power_real @ A @ ( suc @ N ) ) @ ( power_power_real @ B @ ( suc @ N ) ) )
% 2.44/1.35 => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 2.44/1.35 => ( ord_less_eq_real @ A @ B ) ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_183_power__le__imp__le__base,axiom,
% 2.44/1.35 ! [A: int,N: nat,B: int] :
% 2.44/1.35 ( ( ord_less_eq_int @ ( power_power_int @ A @ ( suc @ N ) ) @ ( power_power_int @ B @ ( suc @ N ) ) )
% 2.44/1.35 => ( ( ord_less_eq_int @ zero_zero_int @ B )
% 2.44/1.35 => ( ord_less_eq_int @ A @ B ) ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_184_power__le__imp__le__base,axiom,
% 2.44/1.35 ! [A: nat,N: nat,B: nat] :
% 2.44/1.35 ( ( ord_less_eq_nat @ ( power_power_nat @ A @ ( suc @ N ) ) @ ( power_power_nat @ B @ ( suc @ N ) ) )
% 2.44/1.35 => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 2.44/1.35 => ( ord_less_eq_nat @ A @ B ) ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_185_realpow__pos__nth,axiom,
% 2.44/1.35 ! [N: nat,A: real] :
% 2.44/1.35 ( ( ord_less_nat @ zero_zero_nat @ N )
% 2.44/1.35 => ( ( ord_less_real @ zero_zero_real @ A )
% 2.44/1.35 => ? [R: real] :
% 2.44/1.35 ( ( ord_less_real @ zero_zero_real @ R )
% 2.44/1.35 & ( ( power_power_real @ R @ N )
% 2.44/1.35 = A ) ) ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_186_realpow__pos__nth__unique,axiom,
% 2.44/1.35 ! [N: nat,A: real] :
% 2.44/1.35 ( ( ord_less_nat @ zero_zero_nat @ N )
% 2.44/1.35 => ( ( ord_less_real @ zero_zero_real @ A )
% 2.44/1.35 => ? [X3: real] :
% 2.44/1.35 ( ( ord_less_real @ zero_zero_real @ X3 )
% 2.44/1.35 & ( ( power_power_real @ X3 @ N )
% 2.44/1.35 = A )
% 2.44/1.35 & ! [Y4: real] :
% 2.44/1.35 ( ( ( ord_less_real @ zero_zero_real @ Y4 )
% 2.44/1.35 & ( ( power_power_real @ Y4 @ N )
% 2.44/1.35 = A ) )
% 2.44/1.35 => ( Y4 = X3 ) ) ) ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_187_zero__reorient,axiom,
% 2.44/1.35 ! [X: int] :
% 2.44/1.35 ( ( zero_zero_int = X )
% 2.44/1.35 = ( X = zero_zero_int ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_188_zero__reorient,axiom,
% 2.44/1.35 ! [X: real] :
% 2.44/1.35 ( ( zero_zero_real = X )
% 2.44/1.35 = ( X = zero_zero_real ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_189_zero__reorient,axiom,
% 2.44/1.35 ! [X: nat] :
% 2.44/1.35 ( ( zero_zero_nat = X )
% 2.44/1.35 = ( X = zero_zero_nat ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_190_gr__zeroI,axiom,
% 2.44/1.35 ! [N: nat] :
% 2.44/1.35 ( ( N != zero_zero_nat )
% 2.44/1.35 => ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_191_not__less__zero,axiom,
% 2.44/1.35 ! [N: nat] :
% 2.44/1.35 ~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_192_gr__implies__not__zero,axiom,
% 2.44/1.35 ! [M: nat,N: nat] :
% 2.44/1.35 ( ( ord_less_nat @ M @ N )
% 2.44/1.35 => ( N != zero_zero_nat ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_193_zero__less__iff__neq__zero,axiom,
% 2.44/1.35 ! [N: nat] :
% 2.44/1.35 ( ( ord_less_nat @ zero_zero_nat @ N )
% 2.44/1.35 = ( N != zero_zero_nat ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_194_power__not__zero,axiom,
% 2.44/1.35 ! [A: real,N: nat] :
% 2.44/1.35 ( ( A != zero_zero_real )
% 2.44/1.35 => ( ( power_power_real @ A @ N )
% 2.44/1.35 != zero_zero_real ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_195_power__not__zero,axiom,
% 2.44/1.35 ! [A: int,N: nat] :
% 2.44/1.35 ( ( A != zero_zero_int )
% 2.44/1.35 => ( ( power_power_int @ A @ N )
% 2.44/1.35 != zero_zero_int ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_196_power__not__zero,axiom,
% 2.44/1.35 ! [A: nat,N: nat] :
% 2.44/1.35 ( ( A != zero_zero_nat )
% 2.44/1.35 => ( ( power_power_nat @ A @ N )
% 2.44/1.35 != zero_zero_nat ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_197_zero__power,axiom,
% 2.44/1.35 ! [N: nat] :
% 2.44/1.35 ( ( ord_less_nat @ zero_zero_nat @ N )
% 2.44/1.35 => ( ( power_power_real @ zero_zero_real @ N )
% 2.44/1.35 = zero_zero_real ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_198_zero__power,axiom,
% 2.44/1.35 ! [N: nat] :
% 2.44/1.35 ( ( ord_less_nat @ zero_zero_nat @ N )
% 2.44/1.35 => ( ( power_power_int @ zero_zero_int @ N )
% 2.44/1.35 = zero_zero_int ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_199_zero__power,axiom,
% 2.44/1.35 ! [N: nat] :
% 2.44/1.35 ( ( ord_less_nat @ zero_zero_nat @ N )
% 2.44/1.35 => ( ( power_power_nat @ zero_zero_nat @ N )
% 2.44/1.35 = zero_zero_nat ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_200_yyn_H,axiom,
% 2.44/1.35 ord_less_int @ ( power_power_int @ y @ p ) @ na ).
% 2.44/1.35
% 2.44/1.35 thf(fact_201_y_H0,axiom,
% 2.44/1.35 ord_less_eq_int @ zero_zero_int @ y ).
% 2.44/1.35
% 2.44/1.35 thf(fact_202_compare__pow__less__iff,axiom,
% 2.44/1.35 ! [P2: nat,X: real,Y: real] :
% 2.44/1.35 ( ( ord_less_nat @ zero_zero_nat @ P2 )
% 2.44/1.35 => ( ( ord_less_eq_real @ zero_zero_real @ X )
% 2.44/1.35 => ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 2.44/1.35 => ( ( ord_less_real @ ( power_power_real @ X @ P2 ) @ ( power_power_real @ Y @ P2 ) )
% 2.44/1.35 = ( ord_less_real @ X @ Y ) ) ) ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_203_compare__pow__less__iff,axiom,
% 2.44/1.35 ! [P2: nat,X: int,Y: int] :
% 2.44/1.35 ( ( ord_less_nat @ zero_zero_nat @ P2 )
% 2.44/1.35 => ( ( ord_less_eq_int @ zero_zero_int @ X )
% 2.44/1.35 => ( ( ord_less_eq_int @ zero_zero_int @ Y )
% 2.44/1.35 => ( ( ord_less_int @ ( power_power_int @ X @ P2 ) @ ( power_power_int @ Y @ P2 ) )
% 2.44/1.35 = ( ord_less_int @ X @ Y ) ) ) ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_204_compare__pow__le__iff,axiom,
% 2.44/1.35 ! [P2: nat,X: real,Y: real] :
% 2.44/1.35 ( ( ord_less_nat @ zero_zero_nat @ P2 )
% 2.44/1.35 => ( ( ord_less_eq_real @ zero_zero_real @ X )
% 2.44/1.35 => ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 2.44/1.35 => ( ( ord_less_eq_real @ ( power_power_real @ X @ P2 ) @ ( power_power_real @ Y @ P2 ) )
% 2.44/1.35 = ( ord_less_eq_real @ X @ Y ) ) ) ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_205_compare__pow__le__iff,axiom,
% 2.44/1.35 ! [P2: nat,X: int,Y: int] :
% 2.44/1.35 ( ( ord_less_nat @ zero_zero_nat @ P2 )
% 2.44/1.35 => ( ( ord_less_eq_int @ zero_zero_int @ X )
% 2.44/1.35 => ( ( ord_less_eq_int @ zero_zero_int @ Y )
% 2.44/1.35 => ( ( ord_less_eq_int @ ( power_power_int @ X @ P2 ) @ ( power_power_int @ Y @ P2 ) )
% 2.44/1.35 = ( ord_less_eq_int @ X @ Y ) ) ) ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_206_division__ring__divide__zero,axiom,
% 2.44/1.35 ! [A: real] :
% 2.44/1.35 ( ( divide_divide_real @ A @ zero_zero_real )
% 2.44/1.35 = zero_zero_real ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_207_bits__div__by__0,axiom,
% 2.44/1.35 ! [A: int] :
% 2.44/1.35 ( ( divide_divide_int @ A @ zero_zero_int )
% 2.44/1.35 = zero_zero_int ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_208_bits__div__by__0,axiom,
% 2.44/1.35 ! [A: nat] :
% 2.44/1.35 ( ( divide_divide_nat @ A @ zero_zero_nat )
% 2.44/1.35 = zero_zero_nat ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_209_divide__cancel__right,axiom,
% 2.44/1.35 ! [A: real,C: real,B: real] :
% 2.44/1.35 ( ( ( divide_divide_real @ A @ C )
% 2.44/1.35 = ( divide_divide_real @ B @ C ) )
% 2.44/1.35 = ( ( C = zero_zero_real )
% 2.44/1.35 | ( A = B ) ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_210_divide__eq__0__iff,axiom,
% 2.44/1.35 ! [A: real,B: real] :
% 2.44/1.35 ( ( ( divide_divide_real @ A @ B )
% 2.44/1.35 = zero_zero_real )
% 2.44/1.35 = ( ( A = zero_zero_real )
% 2.44/1.35 | ( B = zero_zero_real ) ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_211_divide__cancel__left,axiom,
% 2.44/1.35 ! [C: real,A: real,B: real] :
% 2.44/1.35 ( ( ( divide_divide_real @ C @ A )
% 2.44/1.35 = ( divide_divide_real @ C @ B ) )
% 2.44/1.35 = ( ( C = zero_zero_real )
% 2.44/1.35 | ( A = B ) ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_212_bits__div__0,axiom,
% 2.44/1.35 ! [A: int] :
% 2.44/1.35 ( ( divide_divide_int @ zero_zero_int @ A )
% 2.44/1.35 = zero_zero_int ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_213_bits__div__0,axiom,
% 2.44/1.35 ! [A: nat] :
% 2.44/1.35 ( ( divide_divide_nat @ zero_zero_nat @ A )
% 2.44/1.35 = zero_zero_nat ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_214_bot__nat__0_Oextremum,axiom,
% 2.44/1.35 ! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_215_le0,axiom,
% 2.44/1.35 ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_216_Suc__le__mono,axiom,
% 2.44/1.35 ! [N: nat,M: nat] :
% 2.44/1.35 ( ( ord_less_eq_nat @ ( suc @ N ) @ ( suc @ M ) )
% 2.44/1.35 = ( ord_less_eq_nat @ N @ M ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_217_le__refl,axiom,
% 2.44/1.35 ! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_218_le__trans,axiom,
% 2.44/1.35 ! [I: nat,J: nat,K2: nat] :
% 2.44/1.35 ( ( ord_less_eq_nat @ I @ J )
% 2.44/1.35 => ( ( ord_less_eq_nat @ J @ K2 )
% 2.44/1.35 => ( ord_less_eq_nat @ I @ K2 ) ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_219_eq__imp__le,axiom,
% 2.44/1.35 ! [M: nat,N: nat] :
% 2.44/1.35 ( ( M = N )
% 2.44/1.35 => ( ord_less_eq_nat @ M @ N ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_220_le__antisym,axiom,
% 2.44/1.35 ! [M: nat,N: nat] :
% 2.44/1.35 ( ( ord_less_eq_nat @ M @ N )
% 2.44/1.35 => ( ( ord_less_eq_nat @ N @ M )
% 2.44/1.35 => ( M = N ) ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_221_nat__le__linear,axiom,
% 2.44/1.35 ! [M: nat,N: nat] :
% 2.44/1.35 ( ( ord_less_eq_nat @ M @ N )
% 2.44/1.35 | ( ord_less_eq_nat @ N @ M ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_222_Nat_Oex__has__greatest__nat,axiom,
% 2.44/1.35 ! [P: nat > $o,K2: nat,B: nat] :
% 2.44/1.35 ( ( P @ K2 )
% 2.44/1.35 => ( ! [Y3: nat] :
% 2.44/1.35 ( ( P @ Y3 )
% 2.44/1.35 => ( ord_less_eq_nat @ Y3 @ B ) )
% 2.44/1.35 => ? [X3: nat] :
% 2.44/1.35 ( ( P @ X3 )
% 2.44/1.35 & ! [Y4: nat] :
% 2.44/1.35 ( ( P @ Y4 )
% 2.44/1.35 => ( ord_less_eq_nat @ Y4 @ X3 ) ) ) ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_223_bot__nat__0_Oextremum__uniqueI,axiom,
% 2.44/1.35 ! [A: nat] :
% 2.44/1.35 ( ( ord_less_eq_nat @ A @ zero_zero_nat )
% 2.44/1.35 => ( A = zero_zero_nat ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_224_bot__nat__0_Oextremum__unique,axiom,
% 2.44/1.35 ! [A: nat] :
% 2.44/1.35 ( ( ord_less_eq_nat @ A @ zero_zero_nat )
% 2.44/1.35 = ( A = zero_zero_nat ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_225_le__0__eq,axiom,
% 2.44/1.35 ! [N: nat] :
% 2.44/1.35 ( ( ord_less_eq_nat @ N @ zero_zero_nat )
% 2.44/1.35 = ( N = zero_zero_nat ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_226_less__eq__nat_Osimps_I1_J,axiom,
% 2.44/1.35 ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_227_transitive__stepwise__le,axiom,
% 2.44/1.35 ! [M: nat,N: nat,R2: nat > nat > $o] :
% 2.44/1.35 ( ( ord_less_eq_nat @ M @ N )
% 2.44/1.35 => ( ! [X3: nat] : ( R2 @ X3 @ X3 )
% 2.44/1.35 => ( ! [X3: nat,Y3: nat,Z2: nat] :
% 2.44/1.35 ( ( R2 @ X3 @ Y3 )
% 2.44/1.35 => ( ( R2 @ Y3 @ Z2 )
% 2.44/1.35 => ( R2 @ X3 @ Z2 ) ) )
% 2.44/1.35 => ( ! [N2: nat] : ( R2 @ N2 @ ( suc @ N2 ) )
% 2.44/1.35 => ( R2 @ M @ N ) ) ) ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_228_nat__induct__at__least,axiom,
% 2.44/1.35 ! [M: nat,N: nat,P: nat > $o] :
% 2.44/1.35 ( ( ord_less_eq_nat @ M @ N )
% 2.44/1.35 => ( ( P @ M )
% 2.44/1.35 => ( ! [N2: nat] :
% 2.44/1.35 ( ( ord_less_eq_nat @ M @ N2 )
% 2.44/1.35 => ( ( P @ N2 )
% 2.44/1.35 => ( P @ ( suc @ N2 ) ) ) )
% 2.44/1.35 => ( P @ N ) ) ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_229_full__nat__induct,axiom,
% 2.44/1.35 ! [P: nat > $o,N: nat] :
% 2.44/1.35 ( ! [N2: nat] :
% 2.44/1.35 ( ! [M3: nat] :
% 2.44/1.35 ( ( ord_less_eq_nat @ ( suc @ M3 ) @ N2 )
% 2.44/1.35 => ( P @ M3 ) )
% 2.44/1.35 => ( P @ N2 ) )
% 2.44/1.35 => ( P @ N ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_230_not__less__eq__eq,axiom,
% 2.44/1.35 ! [M: nat,N: nat] :
% 2.44/1.35 ( ( ~ ( ord_less_eq_nat @ M @ N ) )
% 2.44/1.35 = ( ord_less_eq_nat @ ( suc @ N ) @ M ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_231_Suc__n__not__le__n,axiom,
% 2.44/1.35 ! [N: nat] :
% 2.44/1.35 ~ ( ord_less_eq_nat @ ( suc @ N ) @ N ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_232_le__Suc__eq,axiom,
% 2.44/1.35 ! [M: nat,N: nat] :
% 2.44/1.35 ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
% 2.44/1.35 = ( ( ord_less_eq_nat @ M @ N )
% 2.44/1.35 | ( M
% 2.44/1.35 = ( suc @ N ) ) ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_233_Suc__le__D,axiom,
% 2.44/1.35 ! [N: nat,M6: nat] :
% 2.44/1.35 ( ( ord_less_eq_nat @ ( suc @ N ) @ M6 )
% 2.44/1.35 => ? [M4: nat] :
% 2.44/1.35 ( M6
% 2.44/1.35 = ( suc @ M4 ) ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_234_le__SucI,axiom,
% 2.44/1.35 ! [M: nat,N: nat] :
% 2.44/1.35 ( ( ord_less_eq_nat @ M @ N )
% 2.44/1.35 => ( ord_less_eq_nat @ M @ ( suc @ N ) ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_235_le__SucE,axiom,
% 2.44/1.35 ! [M: nat,N: nat] :
% 2.44/1.35 ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
% 2.44/1.35 => ( ~ ( ord_less_eq_nat @ M @ N )
% 2.44/1.35 => ( M
% 2.44/1.35 = ( suc @ N ) ) ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_236_Suc__leD,axiom,
% 2.44/1.35 ! [M: nat,N: nat] :
% 2.44/1.35 ( ( ord_less_eq_nat @ ( suc @ M ) @ N )
% 2.44/1.35 => ( ord_less_eq_nat @ M @ N ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_237_less__mono__imp__le__mono,axiom,
% 2.44/1.35 ! [F: nat > nat,I: nat,J: nat] :
% 2.44/1.35 ( ! [I2: nat,J2: nat] :
% 2.44/1.35 ( ( ord_less_nat @ I2 @ J2 )
% 2.44/1.35 => ( ord_less_nat @ ( F @ I2 ) @ ( F @ J2 ) ) )
% 2.44/1.35 => ( ( ord_less_eq_nat @ I @ J )
% 2.44/1.35 => ( ord_less_eq_nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_238_le__neq__implies__less,axiom,
% 2.44/1.35 ! [M: nat,N: nat] :
% 2.44/1.35 ( ( ord_less_eq_nat @ M @ N )
% 2.44/1.35 => ( ( M != N )
% 2.44/1.35 => ( ord_less_nat @ M @ N ) ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_239_less__or__eq__imp__le,axiom,
% 2.44/1.35 ! [M: nat,N: nat] :
% 2.44/1.35 ( ( ( ord_less_nat @ M @ N )
% 2.44/1.35 | ( M = N ) )
% 2.44/1.35 => ( ord_less_eq_nat @ M @ N ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_240_le__eq__less__or__eq,axiom,
% 2.44/1.35 ( ord_less_eq_nat
% 2.44/1.35 = ( ^ [M5: nat,N4: nat] :
% 2.44/1.35 ( ( ord_less_nat @ M5 @ N4 )
% 2.44/1.35 | ( M5 = N4 ) ) ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_241_less__imp__le__nat,axiom,
% 2.44/1.35 ! [M: nat,N: nat] :
% 2.44/1.35 ( ( ord_less_nat @ M @ N )
% 2.44/1.35 => ( ord_less_eq_nat @ M @ N ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_242_nat__less__le,axiom,
% 2.44/1.35 ( ord_less_nat
% 2.44/1.35 = ( ^ [M5: nat,N4: nat] :
% 2.44/1.35 ( ( ord_less_eq_nat @ M5 @ N4 )
% 2.44/1.35 & ( M5 != N4 ) ) ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_243_fixed__root_Op0,axiom,
% 2.44/1.35 ! [P2: nat,Pm: nat] :
% 2.44/1.35 ( ( nthRoo1352626670d_root @ P2 @ Pm )
% 2.44/1.35 => ( P2 != zero_zero_nat ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_244_fixed__root_Ointro,axiom,
% 2.44/1.35 ! [P2: nat,Pm: nat] :
% 2.44/1.35 ( ( P2
% 2.44/1.35 = ( suc @ Pm ) )
% 2.44/1.35 => ( nthRoo1352626670d_root @ P2 @ Pm ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_245_fixed__root__def,axiom,
% 2.44/1.35 ( nthRoo1352626670d_root
% 2.44/1.35 = ( ^ [P3: nat,Pm2: nat] :
% 2.44/1.35 ( P3
% 2.44/1.35 = ( suc @ Pm2 ) ) ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_246_fixed__root_Op,axiom,
% 2.44/1.35 ! [P2: nat,Pm: nat] :
% 2.44/1.35 ( ( nthRoo1352626670d_root @ P2 @ Pm )
% 2.44/1.35 => ( P2
% 2.44/1.35 = ( suc @ Pm ) ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_247_nat__one__le__power,axiom,
% 2.44/1.35 ! [I: nat,N: nat] :
% 2.44/1.35 ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ I )
% 2.44/1.35 => ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( power_power_nat @ I @ N ) ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_248_ex__least__nat__le,axiom,
% 2.44/1.35 ! [P: nat > $o,N: nat] :
% 2.44/1.35 ( ( P @ N )
% 2.44/1.35 => ( ~ ( P @ zero_zero_nat )
% 2.44/1.35 => ? [K: nat] :
% 2.44/1.35 ( ( ord_less_eq_nat @ K @ N )
% 2.44/1.35 & ! [I4: nat] :
% 2.44/1.35 ( ( ord_less_nat @ I4 @ K )
% 2.44/1.35 => ~ ( P @ I4 ) )
% 2.44/1.35 & ( P @ K ) ) ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_249_Suc__leI,axiom,
% 2.44/1.35 ! [M: nat,N: nat] :
% 2.44/1.35 ( ( ord_less_nat @ M @ N )
% 2.44/1.35 => ( ord_less_eq_nat @ ( suc @ M ) @ N ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_250_Suc__le__eq,axiom,
% 2.44/1.35 ! [M: nat,N: nat] :
% 2.44/1.35 ( ( ord_less_eq_nat @ ( suc @ M ) @ N )
% 2.44/1.35 = ( ord_less_nat @ M @ N ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_251_dec__induct,axiom,
% 2.44/1.35 ! [I: nat,J: nat,P: nat > $o] :
% 2.44/1.35 ( ( ord_less_eq_nat @ I @ J )
% 2.44/1.35 => ( ( P @ I )
% 2.44/1.35 => ( ! [N2: nat] :
% 2.44/1.35 ( ( ord_less_eq_nat @ I @ N2 )
% 2.44/1.35 => ( ( ord_less_nat @ N2 @ J )
% 2.44/1.35 => ( ( P @ N2 )
% 2.44/1.35 => ( P @ ( suc @ N2 ) ) ) ) )
% 2.44/1.35 => ( P @ J ) ) ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_252_inc__induct,axiom,
% 2.44/1.35 ! [I: nat,J: nat,P: nat > $o] :
% 2.44/1.35 ( ( ord_less_eq_nat @ I @ J )
% 2.44/1.35 => ( ( P @ J )
% 2.44/1.35 => ( ! [N2: nat] :
% 2.44/1.35 ( ( ord_less_eq_nat @ I @ N2 )
% 2.44/1.35 => ( ( ord_less_nat @ N2 @ J )
% 2.44/1.35 => ( ( P @ ( suc @ N2 ) )
% 2.44/1.35 => ( P @ N2 ) ) ) )
% 2.44/1.35 => ( P @ I ) ) ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_253_Suc__le__lessD,axiom,
% 2.44/1.35 ! [M: nat,N: nat] :
% 2.44/1.35 ( ( ord_less_eq_nat @ ( suc @ M ) @ N )
% 2.44/1.35 => ( ord_less_nat @ M @ N ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_254_le__less__Suc__eq,axiom,
% 2.44/1.35 ! [M: nat,N: nat] :
% 2.44/1.35 ( ( ord_less_eq_nat @ M @ N )
% 2.44/1.35 => ( ( ord_less_nat @ N @ ( suc @ M ) )
% 2.44/1.35 = ( N = M ) ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_255_less__Suc__eq__le,axiom,
% 2.44/1.35 ! [M: nat,N: nat] :
% 2.44/1.35 ( ( ord_less_nat @ M @ ( suc @ N ) )
% 2.44/1.35 = ( ord_less_eq_nat @ M @ N ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_256_less__eq__Suc__le,axiom,
% 2.44/1.35 ( ord_less_nat
% 2.44/1.35 = ( ^ [N4: nat] : ( ord_less_eq_nat @ ( suc @ N4 ) ) ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_257_le__imp__less__Suc,axiom,
% 2.44/1.35 ! [M: nat,N: nat] :
% 2.44/1.35 ( ( ord_less_eq_nat @ M @ N )
% 2.44/1.35 => ( ord_less_nat @ M @ ( suc @ N ) ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_258_linordered__field__no__ub,axiom,
% 2.44/1.35 ! [X4: real] :
% 2.44/1.35 ? [X_12: real] : ( ord_less_real @ X4 @ X_12 ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_259_linordered__field__no__lb,axiom,
% 2.44/1.35 ! [X4: real] :
% 2.44/1.35 ? [Y3: real] : ( ord_less_real @ Y3 @ X4 ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_260_ex__least__nat__less,axiom,
% 2.44/1.35 ! [P: nat > $o,N: nat] :
% 2.44/1.35 ( ( P @ N )
% 2.44/1.35 => ( ~ ( P @ zero_zero_nat )
% 2.44/1.35 => ? [K: nat] :
% 2.44/1.35 ( ( ord_less_nat @ K @ N )
% 2.44/1.35 & ! [I4: nat] :
% 2.44/1.35 ( ( ord_less_eq_nat @ I4 @ K )
% 2.44/1.35 => ~ ( P @ I4 ) )
% 2.44/1.35 & ( P @ ( suc @ K ) ) ) ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_261_divide__right__mono__neg,axiom,
% 2.44/1.35 ! [A: real,B: real,C: real] :
% 2.44/1.35 ( ( ord_less_eq_real @ A @ B )
% 2.44/1.35 => ( ( ord_less_eq_real @ C @ zero_zero_real )
% 2.44/1.35 => ( ord_less_eq_real @ ( divide_divide_real @ B @ C ) @ ( divide_divide_real @ A @ C ) ) ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_262_divide__nonpos__nonpos,axiom,
% 2.44/1.35 ! [X: real,Y: real] :
% 2.44/1.35 ( ( ord_less_eq_real @ X @ zero_zero_real )
% 2.44/1.35 => ( ( ord_less_eq_real @ Y @ zero_zero_real )
% 2.44/1.35 => ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X @ Y ) ) ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_263_divide__nonpos__nonneg,axiom,
% 2.44/1.35 ! [X: real,Y: real] :
% 2.44/1.35 ( ( ord_less_eq_real @ X @ zero_zero_real )
% 2.44/1.35 => ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 2.44/1.35 => ( ord_less_eq_real @ ( divide_divide_real @ X @ Y ) @ zero_zero_real ) ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_264_divide__nonneg__nonpos,axiom,
% 2.44/1.35 ! [X: real,Y: real] :
% 2.44/1.35 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 2.44/1.35 => ( ( ord_less_eq_real @ Y @ zero_zero_real )
% 2.44/1.35 => ( ord_less_eq_real @ ( divide_divide_real @ X @ Y ) @ zero_zero_real ) ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_265_divide__nonneg__nonneg,axiom,
% 2.44/1.35 ! [X: real,Y: real] :
% 2.44/1.35 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 2.44/1.35 => ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 2.44/1.35 => ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X @ Y ) ) ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_266_zero__le__divide__iff,axiom,
% 2.44/1.35 ! [A: real,B: real] :
% 2.44/1.35 ( ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ A @ B ) )
% 2.44/1.35 = ( ( ( ord_less_eq_real @ zero_zero_real @ A )
% 2.44/1.35 & ( ord_less_eq_real @ zero_zero_real @ B ) )
% 2.44/1.35 | ( ( ord_less_eq_real @ A @ zero_zero_real )
% 2.44/1.35 & ( ord_less_eq_real @ B @ zero_zero_real ) ) ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_267_divide__right__mono,axiom,
% 2.44/1.35 ! [A: real,B: real,C: real] :
% 2.44/1.35 ( ( ord_less_eq_real @ A @ B )
% 2.44/1.35 => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 2.44/1.35 => ( ord_less_eq_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) ) ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_268_divide__le__0__iff,axiom,
% 2.44/1.35 ! [A: real,B: real] :
% 2.44/1.35 ( ( ord_less_eq_real @ ( divide_divide_real @ A @ B ) @ zero_zero_real )
% 2.44/1.35 = ( ( ( ord_less_eq_real @ zero_zero_real @ A )
% 2.44/1.35 & ( ord_less_eq_real @ B @ zero_zero_real ) )
% 2.44/1.35 | ( ( ord_less_eq_real @ A @ zero_zero_real )
% 2.44/1.35 & ( ord_less_eq_real @ zero_zero_real @ B ) ) ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_269_divide__neg__neg,axiom,
% 2.44/1.35 ! [X: real,Y: real] :
% 2.44/1.35 ( ( ord_less_real @ X @ zero_zero_real )
% 2.44/1.35 => ( ( ord_less_real @ Y @ zero_zero_real )
% 2.44/1.35 => ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ X @ Y ) ) ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_270_divide__neg__pos,axiom,
% 2.44/1.35 ! [X: real,Y: real] :
% 2.44/1.35 ( ( ord_less_real @ X @ zero_zero_real )
% 2.44/1.35 => ( ( ord_less_real @ zero_zero_real @ Y )
% 2.44/1.35 => ( ord_less_real @ ( divide_divide_real @ X @ Y ) @ zero_zero_real ) ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_271_divide__pos__neg,axiom,
% 2.44/1.35 ! [X: real,Y: real] :
% 2.44/1.35 ( ( ord_less_real @ zero_zero_real @ X )
% 2.44/1.35 => ( ( ord_less_real @ Y @ zero_zero_real )
% 2.44/1.35 => ( ord_less_real @ ( divide_divide_real @ X @ Y ) @ zero_zero_real ) ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_272_divide__pos__pos,axiom,
% 2.44/1.35 ! [X: real,Y: real] :
% 2.44/1.35 ( ( ord_less_real @ zero_zero_real @ X )
% 2.44/1.35 => ( ( ord_less_real @ zero_zero_real @ Y )
% 2.44/1.35 => ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ X @ Y ) ) ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_273_divide__less__0__iff,axiom,
% 2.44/1.35 ! [A: real,B: real] :
% 2.44/1.35 ( ( ord_less_real @ ( divide_divide_real @ A @ B ) @ zero_zero_real )
% 2.44/1.35 = ( ( ( ord_less_real @ zero_zero_real @ A )
% 2.44/1.35 & ( ord_less_real @ B @ zero_zero_real ) )
% 2.44/1.35 | ( ( ord_less_real @ A @ zero_zero_real )
% 2.44/1.35 & ( ord_less_real @ zero_zero_real @ B ) ) ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_274_divide__less__cancel,axiom,
% 2.44/1.35 ! [A: real,C: real,B: real] :
% 2.44/1.35 ( ( ord_less_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) )
% 2.44/1.35 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 2.44/1.35 => ( ord_less_real @ A @ B ) )
% 2.44/1.35 & ( ( ord_less_real @ C @ zero_zero_real )
% 2.44/1.35 => ( ord_less_real @ B @ A ) )
% 2.44/1.35 & ( C != zero_zero_real ) ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_275_zero__less__divide__iff,axiom,
% 2.44/1.35 ! [A: real,B: real] :
% 2.44/1.35 ( ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ A @ B ) )
% 2.44/1.35 = ( ( ( ord_less_real @ zero_zero_real @ A )
% 2.44/1.35 & ( ord_less_real @ zero_zero_real @ B ) )
% 2.44/1.35 | ( ( ord_less_real @ A @ zero_zero_real )
% 2.44/1.35 & ( ord_less_real @ B @ zero_zero_real ) ) ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_276_divide__strict__right__mono,axiom,
% 2.44/1.35 ! [A: real,B: real,C: real] :
% 2.44/1.35 ( ( ord_less_real @ A @ B )
% 2.44/1.35 => ( ( ord_less_real @ zero_zero_real @ C )
% 2.44/1.35 => ( ord_less_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) ) ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_277_divide__strict__right__mono__neg,axiom,
% 2.44/1.35 ! [B: real,A: real,C: real] :
% 2.44/1.35 ( ( ord_less_real @ B @ A )
% 2.44/1.35 => ( ( ord_less_real @ C @ zero_zero_real )
% 2.44/1.35 => ( ord_less_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) ) ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_278_frac__le,axiom,
% 2.44/1.35 ! [Y: real,X: real,W: real,Z: real] :
% 2.44/1.35 ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 2.44/1.35 => ( ( ord_less_eq_real @ X @ Y )
% 2.44/1.35 => ( ( ord_less_real @ zero_zero_real @ W )
% 2.44/1.35 => ( ( ord_less_eq_real @ W @ Z )
% 2.44/1.35 => ( ord_less_eq_real @ ( divide_divide_real @ X @ Z ) @ ( divide_divide_real @ Y @ W ) ) ) ) ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_279_frac__less,axiom,
% 2.44/1.35 ! [X: real,Y: real,W: real,Z: real] :
% 2.44/1.35 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 2.44/1.35 => ( ( ord_less_real @ X @ Y )
% 2.44/1.35 => ( ( ord_less_real @ zero_zero_real @ W )
% 2.44/1.35 => ( ( ord_less_eq_real @ W @ Z )
% 2.44/1.35 => ( ord_less_real @ ( divide_divide_real @ X @ Z ) @ ( divide_divide_real @ Y @ W ) ) ) ) ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_280_frac__less2,axiom,
% 2.44/1.35 ! [X: real,Y: real,W: real,Z: real] :
% 2.44/1.35 ( ( ord_less_real @ zero_zero_real @ X )
% 2.44/1.35 => ( ( ord_less_eq_real @ X @ Y )
% 2.44/1.35 => ( ( ord_less_real @ zero_zero_real @ W )
% 2.44/1.35 => ( ( ord_less_real @ W @ Z )
% 2.44/1.35 => ( ord_less_real @ ( divide_divide_real @ X @ Z ) @ ( divide_divide_real @ Y @ W ) ) ) ) ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_281_divide__le__cancel,axiom,
% 2.44/1.35 ! [A: real,C: real,B: real] :
% 2.44/1.35 ( ( ord_less_eq_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) )
% 2.44/1.35 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 2.44/1.35 => ( ord_less_eq_real @ A @ B ) )
% 2.44/1.35 & ( ( ord_less_real @ C @ zero_zero_real )
% 2.44/1.35 => ( ord_less_eq_real @ B @ A ) ) ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_282_divide__nonneg__neg,axiom,
% 2.44/1.35 ! [X: real,Y: real] :
% 2.44/1.35 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 2.44/1.35 => ( ( ord_less_real @ Y @ zero_zero_real )
% 2.44/1.35 => ( ord_less_eq_real @ ( divide_divide_real @ X @ Y ) @ zero_zero_real ) ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_283_divide__nonneg__pos,axiom,
% 2.44/1.35 ! [X: real,Y: real] :
% 2.44/1.35 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 2.44/1.35 => ( ( ord_less_real @ zero_zero_real @ Y )
% 2.44/1.35 => ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X @ Y ) ) ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_284_divide__nonpos__neg,axiom,
% 2.44/1.35 ! [X: real,Y: real] :
% 2.44/1.35 ( ( ord_less_eq_real @ X @ zero_zero_real )
% 2.44/1.35 => ( ( ord_less_real @ Y @ zero_zero_real )
% 2.44/1.35 => ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X @ Y ) ) ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_285_divide__nonpos__pos,axiom,
% 2.44/1.35 ! [X: real,Y: real] :
% 2.44/1.35 ( ( ord_less_eq_real @ X @ zero_zero_real )
% 2.44/1.35 => ( ( ord_less_real @ zero_zero_real @ Y )
% 2.44/1.35 => ( ord_less_eq_real @ ( divide_divide_real @ X @ Y ) @ zero_zero_real ) ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_286_div__neg__neg__trivial,axiom,
% 2.44/1.35 ! [K2: int,L: int] :
% 2.44/1.35 ( ( ord_less_eq_int @ K2 @ zero_zero_int )
% 2.44/1.35 => ( ( ord_less_int @ L @ K2 )
% 2.44/1.35 => ( ( divide_divide_int @ K2 @ L )
% 2.44/1.35 = zero_zero_int ) ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_287_div__pos__pos__trivial,axiom,
% 2.44/1.35 ! [K2: int,L: int] :
% 2.44/1.35 ( ( ord_less_eq_int @ zero_zero_int @ K2 )
% 2.44/1.35 => ( ( ord_less_int @ K2 @ L )
% 2.44/1.35 => ( ( divide_divide_int @ K2 @ L )
% 2.44/1.35 = zero_zero_int ) ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_288_x,axiom,
% 2.44/1.35 x = y ).
% 2.44/1.35
% 2.44/1.35 thf(fact_289_div__by__Suc__0,axiom,
% 2.44/1.35 ! [M: nat] :
% 2.44/1.35 ( ( divide_divide_nat @ M @ ( suc @ zero_zero_nat ) )
% 2.44/1.35 = M ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_290_div__less,axiom,
% 2.44/1.35 ! [M: nat,N: nat] :
% 2.44/1.35 ( ( ord_less_nat @ M @ N )
% 2.44/1.35 => ( ( divide_divide_nat @ M @ N )
% 2.44/1.35 = zero_zero_nat ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_291_div__le__mono,axiom,
% 2.44/1.35 ! [M: nat,N: nat,K2: nat] :
% 2.44/1.35 ( ( ord_less_eq_nat @ M @ N )
% 2.44/1.35 => ( ord_less_eq_nat @ ( divide_divide_nat @ M @ K2 ) @ ( divide_divide_nat @ N @ K2 ) ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_292_div__le__dividend,axiom,
% 2.44/1.35 ! [M: nat,N: nat] : ( ord_less_eq_nat @ ( divide_divide_nat @ M @ N ) @ M ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_293_Euclidean__Division_Odiv__eq__0__iff,axiom,
% 2.44/1.35 ! [M: nat,N: nat] :
% 2.44/1.35 ( ( ( divide_divide_nat @ M @ N )
% 2.44/1.35 = zero_zero_nat )
% 2.44/1.35 = ( ( ord_less_nat @ M @ N )
% 2.44/1.35 | ( N = zero_zero_nat ) ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_294_Suc__div__le__mono,axiom,
% 2.44/1.35 ! [M: nat,N: nat] : ( ord_less_eq_nat @ ( divide_divide_nat @ M @ N ) @ ( divide_divide_nat @ ( suc @ M ) @ N ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_295_div__greater__zero__iff,axiom,
% 2.44/1.35 ! [M: nat,N: nat] :
% 2.44/1.35 ( ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ M @ N ) )
% 2.44/1.35 = ( ( ord_less_eq_nat @ N @ M )
% 2.44/1.35 & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_296_div__le__mono2,axiom,
% 2.44/1.35 ! [M: nat,N: nat,K2: nat] :
% 2.44/1.35 ( ( ord_less_nat @ zero_zero_nat @ M )
% 2.44/1.35 => ( ( ord_less_eq_nat @ M @ N )
% 2.44/1.35 => ( ord_less_eq_nat @ ( divide_divide_nat @ K2 @ N ) @ ( divide_divide_nat @ K2 @ M ) ) ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_297_div__by__0,axiom,
% 2.44/1.35 ! [A: real] :
% 2.44/1.35 ( ( divide_divide_real @ A @ zero_zero_real )
% 2.44/1.35 = zero_zero_real ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_298_div__by__0,axiom,
% 2.44/1.35 ! [A: int] :
% 2.44/1.35 ( ( divide_divide_int @ A @ zero_zero_int )
% 2.44/1.35 = zero_zero_int ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_299_div__by__0,axiom,
% 2.44/1.35 ! [A: nat] :
% 2.44/1.35 ( ( divide_divide_nat @ A @ zero_zero_nat )
% 2.44/1.35 = zero_zero_nat ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_300_div__0,axiom,
% 2.44/1.35 ! [A: real] :
% 2.44/1.35 ( ( divide_divide_real @ zero_zero_real @ A )
% 2.44/1.35 = zero_zero_real ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_301_div__0,axiom,
% 2.44/1.35 ! [A: int] :
% 2.44/1.35 ( ( divide_divide_int @ zero_zero_int @ A )
% 2.44/1.35 = zero_zero_int ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_302_div__0,axiom,
% 2.44/1.35 ! [A: nat] :
% 2.44/1.35 ( ( divide_divide_nat @ zero_zero_nat @ A )
% 2.44/1.35 = zero_zero_nat ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_303_linorder__neqE__linordered__idom,axiom,
% 2.44/1.35 ! [X: real,Y: real] :
% 2.44/1.35 ( ( X != Y )
% 2.44/1.35 => ( ~ ( ord_less_real @ X @ Y )
% 2.44/1.35 => ( ord_less_real @ Y @ X ) ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_304_linorder__neqE__linordered__idom,axiom,
% 2.44/1.35 ! [X: int,Y: int] :
% 2.44/1.35 ( ( X != Y )
% 2.44/1.35 => ( ~ ( ord_less_int @ X @ Y )
% 2.44/1.35 => ( ord_less_int @ Y @ X ) ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_305_zdiv__mono1,axiom,
% 2.44/1.35 ! [A: int,A2: int,B: int] :
% 2.44/1.35 ( ( ord_less_eq_int @ A @ A2 )
% 2.44/1.35 => ( ( ord_less_int @ zero_zero_int @ B )
% 2.44/1.35 => ( ord_less_eq_int @ ( divide_divide_int @ A @ B ) @ ( divide_divide_int @ A2 @ B ) ) ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_306_zdiv__mono2,axiom,
% 2.44/1.35 ! [A: int,B2: int,B: int] :
% 2.44/1.35 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 2.44/1.35 => ( ( ord_less_int @ zero_zero_int @ B2 )
% 2.44/1.35 => ( ( ord_less_eq_int @ B2 @ B )
% 2.44/1.35 => ( ord_less_eq_int @ ( divide_divide_int @ A @ B ) @ ( divide_divide_int @ A @ B2 ) ) ) ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_307_pos__imp__zdiv__neg__iff,axiom,
% 2.44/1.35 ! [B: int,A: int] :
% 2.44/1.35 ( ( ord_less_int @ zero_zero_int @ B )
% 2.44/1.35 => ( ( ord_less_int @ ( divide_divide_int @ A @ B ) @ zero_zero_int )
% 2.44/1.35 = ( ord_less_int @ A @ zero_zero_int ) ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_308_neg__imp__zdiv__neg__iff,axiom,
% 2.44/1.35 ! [B: int,A: int] :
% 2.44/1.35 ( ( ord_less_int @ B @ zero_zero_int )
% 2.44/1.35 => ( ( ord_less_int @ ( divide_divide_int @ A @ B ) @ zero_zero_int )
% 2.44/1.35 = ( ord_less_int @ zero_zero_int @ A ) ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_309_div__neg__pos__less0,axiom,
% 2.44/1.35 ! [A: int,B: int] :
% 2.44/1.35 ( ( ord_less_int @ A @ zero_zero_int )
% 2.44/1.35 => ( ( ord_less_int @ zero_zero_int @ B )
% 2.44/1.35 => ( ord_less_int @ ( divide_divide_int @ A @ B ) @ zero_zero_int ) ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_310_div__positive,axiom,
% 2.44/1.35 ! [B: int,A: int] :
% 2.44/1.35 ( ( ord_less_int @ zero_zero_int @ B )
% 2.44/1.35 => ( ( ord_less_eq_int @ B @ A )
% 2.44/1.35 => ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ A @ B ) ) ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_311_div__positive,axiom,
% 2.44/1.35 ! [B: nat,A: nat] :
% 2.44/1.35 ( ( ord_less_nat @ zero_zero_nat @ B )
% 2.44/1.35 => ( ( ord_less_eq_nat @ B @ A )
% 2.44/1.35 => ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ A @ B ) ) ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_312_unique__euclidean__semiring__numeral__class_Odiv__less,axiom,
% 2.44/1.35 ! [A: int,B: int] :
% 2.44/1.35 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 2.44/1.35 => ( ( ord_less_int @ A @ B )
% 2.44/1.35 => ( ( divide_divide_int @ A @ B )
% 2.44/1.35 = zero_zero_int ) ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_313_unique__euclidean__semiring__numeral__class_Odiv__less,axiom,
% 2.44/1.35 ! [A: nat,B: nat] :
% 2.44/1.35 ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 2.44/1.35 => ( ( ord_less_nat @ A @ B )
% 2.44/1.35 => ( ( divide_divide_nat @ A @ B )
% 2.44/1.35 = zero_zero_nat ) ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_314_nonneg1__imp__zdiv__pos__iff,axiom,
% 2.44/1.35 ! [A: int,B: int] :
% 2.44/1.35 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 2.44/1.35 => ( ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ A @ B ) )
% 2.44/1.35 = ( ( ord_less_eq_int @ B @ A )
% 2.44/1.35 & ( ord_less_int @ zero_zero_int @ B ) ) ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_315_pos__imp__zdiv__nonneg__iff,axiom,
% 2.44/1.35 ! [B: int,A: int] :
% 2.44/1.35 ( ( ord_less_int @ zero_zero_int @ B )
% 2.44/1.35 => ( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ A @ B ) )
% 2.44/1.35 = ( ord_less_eq_int @ zero_zero_int @ A ) ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_316_neg__imp__zdiv__nonneg__iff,axiom,
% 2.44/1.35 ! [B: int,A: int] :
% 2.44/1.35 ( ( ord_less_int @ B @ zero_zero_int )
% 2.44/1.35 => ( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ A @ B ) )
% 2.44/1.35 = ( ord_less_eq_int @ A @ zero_zero_int ) ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_317_pos__imp__zdiv__pos__iff,axiom,
% 2.44/1.35 ! [K2: int,I: int] :
% 2.44/1.35 ( ( ord_less_int @ zero_zero_int @ K2 )
% 2.44/1.35 => ( ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ I @ K2 ) )
% 2.44/1.35 = ( ord_less_eq_int @ K2 @ I ) ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(fact_318_div__nonpos__pos__le0,axiom,
% 2.44/1.35 ! [A: int,B: int] :
% 2.44/1.35 ( ( ord_less_eq_int @ A @ zero_zero_int )
% 2.44/1.35 => ( ( ord_less_int @ zero_zero_int @ B )
% 2.44/1.35 => ( ord_less_eq_int @ ( divide_divide_int @ A @ B ) @ zero_zero_int ) ) ) ).
% 2.44/1.35
% 2.44/1.35 thf(conj_0,conjecture,
% 2.44/1.35 ord_less_real @ ( ring_1_of_int_real @ na ) @ ( power_power_real @ ( ring_1_of_int_real @ ya ) @ ( suc @ pm ) ) ).
% 2.44/1.35 ------------------------
% 2.44/1.37 % (342)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_32705 for (2999ds/183Mi)
% 2.44/1.38 % (345)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_32705 for (2999ds/2Mi)
% 2.44/1.38 % (343)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_32705 for (2999ds/4Mi)
% 2.44/1.38 % (346)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_32705 for (2999ds/2Mi)
% 2.44/1.38 % (347)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_32705 for (2999ds/275Mi)
% 2.44/1.38 % (348)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_32705 for (2999ds/18Mi)
% 2.44/1.38 % (344)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_32705 for (2999ds/27Mi)
% 2.44/1.38 % (345)Instruction limit reached!
% 2.44/1.38 % (345)------------------------------
% 2.44/1.38 % (345)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.44/1.38 % (346)Instruction limit reached!
% 2.44/1.38 % (346)------------------------------
% 2.44/1.38 % (346)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.44/1.38 % (346)Termination reason: Unknown
% 2.44/1.38 % (346)Termination phase: shuffling
% 2.44/1.38
% 2.44/1.38 % (346)Memory used [KB]: 1279
% 2.44/1.38 % (346)Time elapsed: 0.003 s
% 2.44/1.38 % (346)Instructions burned: 3 (million)
% 2.44/1.38 % (346)------------------------------
% 2.44/1.38 % (346)------------------------------
% 2.44/1.38 % (345)Termination reason: Unknown
% 2.44/1.38 % (345)Termination phase: shuffling
% 2.44/1.38
% 2.44/1.38 % (345)Memory used [KB]: 1279
% 2.44/1.38 % (345)Time elapsed: 0.003 s
% 2.44/1.38 % (345)Instructions burned: 3 (million)
% 2.44/1.38 % (345)------------------------------
% 2.44/1.38 % (345)------------------------------
% 2.44/1.38 % (343)Instruction limit reached!
% 2.44/1.38 % (343)------------------------------
% 2.44/1.38 % (343)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.44/1.38 % (343)Termination reason: Unknown
% 2.44/1.38 % (343)Termination phase: shuffling
% 2.44/1.38
% 2.44/1.38 % (343)Memory used [KB]: 1407
% 2.44/1.38 % (343)Time elapsed: 0.004 s
% 2.44/1.38 % (343)Instructions burned: 5 (million)
% 2.44/1.38 % (343)------------------------------
% 2.44/1.38 % (343)------------------------------
% 2.44/1.39 % (348)Instruction limit reached!
% 2.44/1.39 % (348)------------------------------
% 2.44/1.39 % (348)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.44/1.39 % (348)Termination reason: Unknown
% 2.44/1.39 % (348)Termination phase: Property scanning
% 2.44/1.39
% 2.44/1.39 % (348)Memory used [KB]: 1663
% 2.44/1.39 % (348)Time elapsed: 0.011 s
% 2.44/1.39 % (348)Instructions burned: 19 (million)
% 2.44/1.39 % (348)------------------------------
% 2.44/1.39 % (348)------------------------------
% 2.44/1.39 % (349)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_32705 for (2999ds/3Mi)
% 2.44/1.39 % (344)Instruction limit reached!
% 2.44/1.39 % (344)------------------------------
% 2.44/1.39 % (344)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.44/1.39 % (344)Termination reason: Unknown
% 2.44/1.39 % (344)Termination phase: Property scanning
% 2.44/1.39
% 2.44/1.39 % (344)Memory used [KB]: 1535
% 2.44/1.39 % (344)Time elapsed: 0.015 s
% 2.44/1.39 % (344)Instructions burned: 27 (million)
% 2.44/1.39 % (344)------------------------------
% 2.44/1.39 % (344)------------------------------
% 2.44/1.39 % (349)Instruction limit reached!
% 2.44/1.39 % (349)------------------------------
% 2.44/1.39 % (349)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.44/1.39 % (349)Termination reason: Unknown
% 2.44/1.39 % (349)Termination phase: shuffling
% 2.44/1.39
% 2.44/1.39 % (349)Memory used [KB]: 1407
% 2.44/1.39 % (349)Time elapsed: 0.003 s
% 2.44/1.39 % (349)Instructions burned: 6 (million)
% 2.44/1.39 % (349)------------------------------
% 2.44/1.39 % (349)------------------------------
% 2.44/1.39 % (350)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_32705 for (2999ds/37Mi)
% 2.50/1.40 % (351)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_32705 for (2999ds/15Mi)
% 2.50/1.40 % (354)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_32705 for (2999ds/7Mi)
% 2.50/1.40 % (351)Instruction limit reached!
% 2.50/1.40 % (351)------------------------------
% 2.50/1.40 % (351)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.50/1.40 % (351)Termination reason: Unknown
% 2.50/1.40 % (351)Termination phase: shuffling
% 2.50/1.40
% 2.50/1.40 % (351)Memory used [KB]: 1535
% 2.50/1.40 % (351)Time elapsed: 0.010 s
% 2.50/1.40 % (351)Instructions burned: 16 (million)
% 2.50/1.40 % (351)------------------------------
% 2.50/1.40 % (351)------------------------------
% 2.50/1.40 % (352)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_32705 for (2999ds/3Mi)
% 2.50/1.41 % (354)Instruction limit reached!
% 2.50/1.41 % (354)------------------------------
% 2.50/1.41 % (354)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.50/1.41 % (354)Termination reason: Unknown
% 2.50/1.41 % (354)Termination phase: shuffling
% 2.50/1.41
% 2.50/1.41 % (354)Memory used [KB]: 1407
% 2.50/1.41 % (354)Time elapsed: 0.006 s
% 2.50/1.41 % (354)Instructions burned: 8 (million)
% 2.50/1.41 % (354)------------------------------
% 2.50/1.41 % (354)------------------------------
% 2.50/1.41 % (352)Instruction limit reached!
% 2.50/1.41 % (352)------------------------------
% 2.50/1.41 % (352)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.50/1.41 % (352)Termination reason: Unknown
% 2.50/1.41 % (352)Termination phase: shuffling
% 2.50/1.41
% 2.50/1.41 % (352)Memory used [KB]: 1279
% 2.50/1.41 % (352)Time elapsed: 0.003 s
% 2.50/1.41 % (352)Instructions burned: 3 (million)
% 2.50/1.41 % (352)------------------------------
% 2.50/1.41 % (352)------------------------------
% 2.50/1.41 % (353)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_32705 for (2999ds/1041Mi)
% 2.50/1.41 % (350)Instruction limit reached!
% 2.50/1.41 % (350)------------------------------
% 2.50/1.41 % (350)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.50/1.41 % (350)Termination reason: Unknown
% 2.50/1.41 % (350)Termination phase: shuffling
% 2.50/1.41
% 2.50/1.41 % (350)Memory used [KB]: 2046
% 2.50/1.41 % (350)Time elapsed: 0.018 s
% 2.50/1.41 % (350)Instructions burned: 37 (million)
% 2.50/1.41 % (350)------------------------------
% 2.50/1.41 % (350)------------------------------
% 2.50/1.42 % (355)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_32705 for (2999ds/16Mi)
% 2.50/1.42 % (347)First to succeed.
% 2.50/1.42 % (356)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_32705 for (2999ds/3Mi)
% 2.50/1.42 % (357)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_32705 for (2999ds/3Mi)
% 2.50/1.42 % (356)Instruction limit reached!
% 2.50/1.42 % (356)------------------------------
% 2.50/1.42 % (356)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.50/1.42 % (356)Termination reason: Unknown
% 2.50/1.42 % (356)Termination phase: shuffling
% 2.50/1.42
% 2.50/1.42 % (356)Memory used [KB]: 1279
% 2.50/1.42 % (356)Time elapsed: 0.003 s
% 2.50/1.42 % (356)Instructions burned: 3 (million)
% 2.50/1.42 % (356)------------------------------
% 2.50/1.42 % (356)------------------------------
% 2.50/1.42 % (347)Refutation found. Thanks to Tanya!
% 2.50/1.42 % SZS status Theorem for DTF2THF_32705
% 2.50/1.42 % SZS output start Proof for DTF2THF_32705
% 2.50/1.42 thf(type_def_5, type, int: $tType).
% 2.50/1.42 thf(type_def_6, type, nat: $tType).
% 2.50/1.42 thf(type_def_7, type, real: $tType).
% 2.50/1.42 thf(func_def_0, type, real: $tType).
% 2.50/1.42 thf(func_def_1, type, nat: $tType).
% 2.50/1.42 thf(func_def_2, type, int: $tType).
% 2.50/1.42 thf(func_def_3, type, zero_zero_int: int).
% 2.50/1.42 thf(func_def_4, type, zero_zero_nat: nat).
% 2.50/1.42 thf(func_def_5, type, zero_zero_real: real).
% 2.50/1.42 thf(func_def_6, type, ring_1_of_int_int: int > int).
% 2.50/1.42 thf(func_def_7, type, ring_1_of_int_real: int > real).
% 2.50/1.42 thf(func_def_8, type, suc: nat > nat).
% 2.50/1.42 thf(func_def_9, type, nthRoo1352626670d_root: nat > nat > $o).
% 2.50/1.42 thf(func_def_10, type, ord_less_int: int > int > $o).
% 2.50/1.42 thf(func_def_11, type, ord_less_nat: nat > nat > $o).
% 2.50/1.42 thf(func_def_12, type, ord_less_real: real > real > $o).
% 2.50/1.42 thf(func_def_13, type, ord_less_eq_int: int > int > $o).
% 2.50/1.42 thf(func_def_14, type, ord_less_eq_nat: nat > nat > $o).
% 2.50/1.42 thf(func_def_15, type, ord_less_eq_real: real > real > $o).
% 2.50/1.42 thf(func_def_16, type, power_power_int: int > nat > int).
% 2.50/1.42 thf(func_def_17, type, power_power_nat: nat > nat > nat).
% 2.50/1.42 thf(func_def_18, type, power_power_real: real > nat > real).
% 2.50/1.42 thf(func_def_19, type, divide_divide_int: int > int > int).
% 2.50/1.42 thf(func_def_20, type, divide_divide_nat: nat > nat > nat).
% 2.50/1.42 thf(func_def_21, type, divide_divide_real: real > real > real).
% 2.50/1.42 thf(func_def_22, type, ny: real).
% 2.50/1.42 thf(func_def_23, type, na: int).
% 2.50/1.42 thf(func_def_24, type, p: nat).
% 2.50/1.42 thf(func_def_25, type, pm: nat).
% 2.50/1.42 thf(func_def_26, type, x: int).
% 2.50/1.42 thf(func_def_27, type, y: int).
% 2.50/1.42 thf(func_def_28, type, ya: int).
% 2.50/1.42 thf(func_def_39, type, sK0: (nat > $o) > nat).
% 2.50/1.42 thf(func_def_40, type, sK1: (nat > real) > nat).
% 2.50/1.42 thf(func_def_41, type, sK2: real > nat > real).
% 2.50/1.42 thf(func_def_42, type, sK3: (nat > int) > nat).
% 2.50/1.42 thf(func_def_43, type, sK4: nat > nat).
% 2.50/1.42 thf(func_def_44, type, sK5: (nat > int) > nat).
% 2.50/1.42 thf(func_def_45, type, sK6: real > real).
% 2.50/1.42 thf(func_def_46, type, sK7: real > int).
% 2.50/1.42 thf(func_def_47, type, sK8: real > int).
% 2.50/1.42 thf(func_def_48, type, sK9: (nat > int) > nat).
% 2.50/1.42 thf(func_def_49, type, sK10: (nat > $o) > nat).
% 2.50/1.42 thf(func_def_50, type, sK11: nat > nat).
% 2.50/1.42 thf(func_def_51, type, sK12: real > real).
% 2.50/1.42 thf(func_def_52, type, sK13: (nat > int) > nat).
% 2.50/1.42 thf(func_def_53, type, sK14: nat > real > real).
% 2.50/1.42 thf(func_def_54, type, sK15: nat > (nat > $o) > nat).
% 2.50/1.42 thf(func_def_55, type, sK16: nat > (nat > $o) > nat).
% 2.50/1.42 thf(func_def_56, type, sK17: (nat > real) > nat).
% 2.50/1.42 thf(func_def_57, type, sK18: nat > (nat > $o) > nat).
% 2.50/1.42 thf(func_def_58, type, sK19: nat > (nat > $o) > nat).
% 2.50/1.42 thf(func_def_59, type, sK20: nat > nat > nat).
% 2.50/1.42 thf(func_def_60, type, sK21: (nat > $o) > nat).
% 2.50/1.42 thf(func_def_61, type, sK22: (nat > nat) > nat).
% 2.50/1.42 thf(func_def_62, type, sK23: nat > (nat > $o) > nat).
% 2.50/1.42 thf(func_def_63, type, sK24: nat > (nat > $o) > nat).
% 2.50/1.42 thf(func_def_64, type, sK25: real > nat > real).
% 2.50/1.42 thf(func_def_65, type, sK26: nat > nat).
% 2.50/1.42 thf(func_def_66, type, sK27: nat > nat > nat).
% 2.50/1.42 thf(func_def_67, type, sK28: (nat > $o) > nat).
% 2.50/1.42 thf(func_def_68, type, sK29: nat > nat).
% 2.50/1.42 thf(func_def_69, type, sK30: nat > nat > nat).
% 2.50/1.42 thf(func_def_70, type, sK31: (nat > $o) > nat > nat).
% 2.50/1.42 thf(func_def_71, type, sK32: (nat > $o) > nat > nat).
% 2.50/1.42 thf(func_def_72, type, sK33: (nat > $o) > nat).
% 2.50/1.42 thf(func_def_73, type, sK34: nat > (nat > $o) > nat).
% 2.50/1.42 thf(func_def_74, type, sK35: nat > (nat > $o) > nat).
% 2.50/1.42 thf(func_def_75, type, sK36: (nat > $o) > nat).
% 2.50/1.42 thf(func_def_76, type, sK37: (nat > nat > $o) > nat).
% 2.50/1.42 thf(func_def_77, type, sK38: (nat > nat > $o) > nat).
% 2.50/1.42 thf(func_def_78, type, sK39: (nat > nat > $o) > nat).
% 2.50/1.42 thf(func_def_79, type, sK40: (nat > nat > $o) > nat).
% 2.50/1.42 thf(func_def_80, type, sK41: nat > nat > nat).
% 2.50/1.42 thf(func_def_81, type, sK42: (nat > real) > nat).
% 2.50/1.42 thf(func_def_82, type, sK43: (nat > nat) > nat).
% 2.50/1.42 thf(func_def_83, type, sK44: real > int).
% 2.50/1.42 thf(func_def_84, type, sK45: (nat > real) > nat).
% 2.50/1.42 thf(func_def_85, type, sK46: (nat > $o) > nat).
% 2.50/1.42 thf(func_def_86, type, sK47: (nat > nat > $o) > nat).
% 2.50/1.42 thf(func_def_87, type, sK48: (nat > nat > $o) > nat).
% 2.50/1.42 thf(func_def_88, type, sK49: (nat > nat > $o) > nat).
% 2.50/1.42 thf(func_def_89, type, sK50: (nat > nat > $o) > nat).
% 2.50/1.42 thf(func_def_91, type, ph52: !>[X0: $tType]:(X0)).
% 2.50/1.42 thf(f1678,plain,(
% 2.50/1.42 $false),
% 2.50/1.42 inference(subsumption_resolution,[status(thm)],[f1666,f1570])).
% 2.50/1.42 thf(f1570,plain,(
% 2.50/1.42 ((ord_less_int @ na @ (power_power_int @ ya @ (suc @ pm))) = $true)),
% 2.50/1.42 inference(definition_unfolding,[status(thm)],[f1358,f1496])).
% 2.50/1.42 thf(f1496,plain,(
% 2.50/1.42 (p = (suc @ pm))),
% 2.50/1.42 inference(cnf_transformation,[status(thm)],[f87])).
% 2.50/1.42 thf(f87,axiom,(
% 2.50/1.42 (p = (suc @ pm))),
% 2.50/1.42 file('/export/starexec/sandbox2/tmp/tmp.paLrYnjNlH/DTF2THF_32705.p',fact_86_p)).
% 2.50/1.42 thf(f1358,plain,(
% 2.50/1.42 ((ord_less_int @ na @ (power_power_int @ ya @ p)) = $true)),
% 2.50/1.42 inference(cnf_transformation,[status(thm)],[f627])).
% 2.50/1.42 thf(f627,plain,(
% 2.50/1.42 ((ord_less_int @ na @ (power_power_int @ ya @ p)) = $true)),
% 2.50/1.42 inference(fool_elimination,[status(thm)],[f626])).
% 2.50/1.42 thf(f626,plain,(
% 2.50/1.42 (ord_less_int @ na @ (power_power_int @ ya @ p))),
% 2.50/1.42 inference(rectify,[status(thm)],[f82])).
% 2.50/1.42 thf(f82,axiom,(
% 2.50/1.42 (ord_less_int @ na @ (power_power_int @ ya @ p))),
% 2.50/1.42 file('/export/starexec/sandbox2/tmp/tmp.paLrYnjNlH/DTF2THF_32705.p',fact_81_yyn)).
% 2.50/1.42 thf(f1666,plain,(
% 2.50/1.42 ((ord_less_int @ na @ (power_power_int @ ya @ (suc @ pm))) != $true)),
% 2.50/1.42 inference(superposition,[status(thm)],[f1665,f1290])).
% 2.50/1.42 thf(f1290,plain,(
% 2.50/1.42 ( ! [X0 : int,X1 : int] : (((ord_less_real @ (ring_1_of_int_real @ X1) @ (ring_1_of_int_real @ X0)) = (ord_less_int @ X1 @ X0))) )),
% 2.50/1.42 inference(cnf_transformation,[status(thm)],[f830])).
% 2.50/1.42 thf(f830,plain,(
% 2.50/1.42 ! [X0 : int,X1 : int] : ((ord_less_real @ (ring_1_of_int_real @ X1) @ (ring_1_of_int_real @ X0)) = (ord_less_int @ X1 @ X0))),
% 2.50/1.42 inference(rectify,[status(thm)],[f12])).
% 2.50/1.42 thf(f12,axiom,(
% 2.50/1.42 ! [X3 : int,X1 : int] : ((ord_less_real @ (ring_1_of_int_real @ X1) @ (ring_1_of_int_real @ X3)) = (ord_less_int @ X1 @ X3))),
% 2.50/1.42 file('/export/starexec/sandbox2/tmp/tmp.paLrYnjNlH/DTF2THF_32705.p',fact_11_of__int__less__iff)).
% 2.50/1.42 thf(f1665,plain,(
% 2.50/1.42 ($true != (ord_less_real @ (ring_1_of_int_real @ na) @ (ring_1_of_int_real @ (power_power_int @ ya @ (suc @ pm)))))),
% 2.50/1.42 inference(forward_demodulation,[status(thm)],[f1310,f1625])).
% 2.50/1.42 thf(f1625,plain,(
% 2.50/1.42 ( ! [X2 : nat,X1 : int] : (((ring_1_of_int_real @ (power_power_int @ X1 @ X2)) = (power_power_real @ (ring_1_of_int_real @ X1) @ X2))) )),
% 2.50/1.42 inference(equality_resolution,[status(thm)],[f1238])).
% 2.50/1.42 thf(f1238,plain,(
% 2.50/1.42 ( ! [X2 : nat,X0 : int,X1 : int] : (((ring_1_of_int_real @ X0) = (power_power_real @ (ring_1_of_int_real @ X1) @ X2)) | ((power_power_int @ X1 @ X2) != X0)) )),
% 2.50/1.42 inference(cnf_transformation,[status(thm)],[f1060])).
% 2.50/1.42 thf(f1060,plain,(
% 2.50/1.42 ! [X0 : int,X1 : int,X2 : nat] : ((((power_power_int @ X1 @ X2) = X0) | ((ring_1_of_int_real @ X0) != (power_power_real @ (ring_1_of_int_real @ X1) @ X2))) & (((ring_1_of_int_real @ X0) = (power_power_real @ (ring_1_of_int_real @ X1) @ X2)) | ((power_power_int @ X1 @ X2) != X0)))),
% 2.50/1.42 inference(nnf_transformation,[status(thm)],[f727])).
% 2.50/1.42 thf(f727,plain,(
% 2.50/1.42 ! [X0 : int,X1 : int,X2 : nat] : (((power_power_int @ X1 @ X2) = X0) <=> ((ring_1_of_int_real @ X0) = (power_power_real @ (ring_1_of_int_real @ X1) @ X2)))),
% 2.50/1.42 inference(fool_elimination,[status(thm)],[f726])).
% 2.50/1.42 thf(f726,plain,(
% 2.50/1.42 ! [X0 : int,X1 : int,X2 : nat] : (((power_power_int @ X1 @ X2) = X0) = ((ring_1_of_int_real @ X0) = (power_power_real @ (ring_1_of_int_real @ X1) @ X2)))),
% 2.50/1.42 inference(rectify,[status(thm)],[f8])).
% 2.50/1.42 thf(f8,axiom,(
% 2.50/1.42 ! [X2 : int,X0 : int,X1 : nat] : (((power_power_int @ X0 @ X1) = X2) = ((power_power_real @ (ring_1_of_int_real @ X0) @ X1) = (ring_1_of_int_real @ X2)))),
% 2.50/1.42 file('/export/starexec/sandbox2/tmp/tmp.paLrYnjNlH/DTF2THF_32705.p',fact_7_of__int__eq__of__int__power__cancel__iff)).
% 2.50/1.42 thf(f1310,plain,(
% 2.50/1.42 ((ord_less_real @ (ring_1_of_int_real @ na) @ (power_power_real @ (ring_1_of_int_real @ ya) @ (suc @ pm))) != $true)),
% 2.50/1.42 inference(cnf_transformation,[status(thm)],[f824])).
% 2.50/1.42 thf(f824,plain,(
% 2.50/1.42 ((ord_less_real @ (ring_1_of_int_real @ na) @ (power_power_real @ (ring_1_of_int_real @ ya) @ (suc @ pm))) != $true)),
% 2.50/1.42 inference(flattening,[status(thm)],[f358])).
% 2.50/1.42 thf(f358,plain,(
% 2.50/1.42 ~((ord_less_real @ (ring_1_of_int_real @ na) @ (power_power_real @ (ring_1_of_int_real @ ya) @ (suc @ pm))) = $true)),
% 2.50/1.42 inference(fool_elimination,[status(thm)],[f357])).
% 2.50/1.42 thf(f357,plain,(
% 2.50/1.42 ~(ord_less_real @ (ring_1_of_int_real @ na) @ (power_power_real @ (ring_1_of_int_real @ ya) @ (suc @ pm)))),
% 2.50/1.42 inference(rectify,[status(thm)],[f321])).
% 2.50/1.42 thf(f321,negated_conjecture,(
% 2.50/1.42 ~(ord_less_real @ (ring_1_of_int_real @ na) @ (power_power_real @ (ring_1_of_int_real @ ya) @ (suc @ pm)))),
% 2.50/1.42 inference(negated_conjecture,[status(cth)],[f320])).
% 2.50/1.42 thf(f320,conjecture,(
% 2.50/1.42 (ord_less_real @ (ring_1_of_int_real @ na) @ (power_power_real @ (ring_1_of_int_real @ ya) @ (suc @ pm)))),
% 2.50/1.42 file('/export/starexec/sandbox2/tmp/tmp.paLrYnjNlH/DTF2THF_32705.p',conj_0)).
% 2.50/1.42 % SZS output end Proof for DTF2THF_32705
% 2.50/1.42 % (347)------------------------------
% 2.50/1.42 % (347)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.50/1.42 % (347)Termination reason: Refutation
% 2.50/1.42
% 2.50/1.42 % (347)Memory used [KB]: 6780
% 2.50/1.42 % (347)Time elapsed: 0.046 s
% 2.50/1.42 % (347)Instructions burned: 90 (million)
% 2.50/1.42 % (347)------------------------------
% 2.50/1.42 % (347)------------------------------
% 2.50/1.42 % (341)Success in time 0.066 s
%------------------------------------------------------------------------------