%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : ITP151^1 : TPTP v9.2.1. Released v7.5.0.
% Transfm : none
% Format : tptp:raw
% Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% Computer : n008.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 08:55:57 AM UTC 2026
% Result : Theorem 2.69s 1.62s
% Output : Refutation 2.69s
% Verified :
% SZS Type : Refutation
% Derivation depth : 12
% Number of leaves : 6
% Syntax : Number of formulae : 31 ( 17 unt; 0 typ; 0 def)
% Number of atoms : 91 ( 46 equ; 0 cnn)
% Maximal formula atoms : 4 ( 2 avg)
% Number of connectives : 119 ( 15 ~; 12 |; 3 &; 84 @)
% ( 1 <=>; 4 =>; 0 <=; 0 <~>)
% Maximal formula depth : 6 ( 3 avg)
% Number of types : 4 ( 3 usr)
% Number of type conns : 0 ( 0 >; 0 *; 0 +; 0 <<)
% Number of symbols : 76 ( 73 usr; 6 con; 0-2 aty)
% Number of variables : 28 ( 0 ^; 22 !; 6 ?; 28 :)
% Comments :
%------------------------------------------------------------------------------
thf(type_def_5,type,
int: $tType ).
thf(type_def_7,type,
nat: $tType ).
thf(type_def_8,type,
polyno727731844poly_a: $tType ).
thf(func_def_0,type,
polyno727731844poly_a: $tType ).
thf(func_def_1,type,
nat: $tType ).
thf(func_def_2,type,
int: $tType ).
thf(func_def_3,type,
minus_minus_int: int > int > int ).
thf(func_def_4,type,
minus_minus_nat: nat > nat > nat ).
thf(func_def_5,type,
zero_zero_int: int ).
thf(func_def_6,type,
zero_zero_nat: nat ).
thf(func_def_7,type,
suc: nat > nat ).
thf(func_def_8,type,
semiri2019852685at_int: nat > int ).
thf(func_def_9,type,
semiri1382578993at_nat: nat > nat ).
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_eq_int: int > int > $o ).
thf(func_def_13,type,
ord_less_eq_nat: nat > nat > $o ).
thf(func_def_14,type,
polyno578545843gree_a: polyno727731844poly_a > nat ).
thf(func_def_15,type,
polyno1674775833reen_a: polyno727731844poly_a > nat > nat ).
thf(func_def_16,type,
polyno190918219poly_a: polyno727731844poly_a > $o ).
thf(func_def_17,type,
polyno1372495879olyh_a: polyno727731844poly_a > nat > $o ).
thf(func_def_18,type,
power_power_int: int > nat > int ).
thf(func_def_19,type,
power_power_nat: nat > nat > nat ).
thf(func_def_20,type,
n: nat ).
thf(func_def_21,type,
p: polyno727731844poly_a ).
thf(func_def_36,type,
sK0: ( nat > $o ) > nat ).
thf(func_def_37,type,
sK1: nat > nat ).
thf(func_def_38,type,
sK2: ( nat > nat ) > nat ).
thf(func_def_39,type,
sK3: ( nat > nat ) > nat ).
thf(func_def_40,type,
sK4: nat > nat ).
thf(func_def_41,type,
sK5: ( nat > $o ) > nat > nat ).
thf(func_def_42,type,
sK6: ( nat > $o ) > nat > nat ).
thf(func_def_43,type,
sK7: nat > nat ).
thf(func_def_44,type,
sK8: ( nat > nat ) > nat ).
thf(func_def_45,type,
sK9: ( nat > nat ) > nat ).
thf(func_def_46,type,
sK10: nat > nat > nat ).
thf(func_def_47,type,
sK11: ( nat > $o ) > nat ).
thf(func_def_48,type,
sK12: nat > nat > nat ).
thf(func_def_49,type,
sK13: ( nat > $o ) > nat ).
thf(func_def_50,type,
sK14: nat > nat > nat ).
thf(func_def_51,type,
sK15: ( nat > $o ) > nat ).
thf(func_def_52,type,
sK16: ( nat > $o ) > nat ).
thf(func_def_53,type,
sK17: ( nat > nat > $o ) > nat ).
thf(func_def_54,type,
sK18: ( nat > nat > $o ) > nat ).
thf(func_def_55,type,
sK19: ( nat > nat > $o ) > nat ).
thf(func_def_56,type,
sK20: ( nat > nat > $o ) > nat ).
thf(func_def_57,type,
sK21: ( nat > nat ) > nat ).
thf(func_def_58,type,
sK22: nat > ( nat > $o ) > nat ).
thf(func_def_59,type,
sK23: nat > ( nat > $o ) > nat ).
thf(func_def_60,type,
sK24: nat > nat > nat ).
thf(func_def_61,type,
sK25: ( nat > nat ) > nat ).
thf(func_def_62,type,
sK26: ( nat > nat ) > nat ).
thf(func_def_63,type,
sK27: ( nat > nat ) > nat ).
thf(func_def_64,type,
sK28: nat > nat ).
thf(func_def_65,type,
sK29: ( nat > nat ) > nat ).
thf(func_def_66,type,
sK30: ( nat > $o ) > nat > nat ).
thf(func_def_67,type,
sK31: ( nat > $o ) > nat > nat ).
thf(func_def_68,type,
sK32: ( nat > $o ) > nat ).
thf(func_def_69,type,
sK33: ( nat > nat > $o ) > nat ).
thf(func_def_70,type,
sK34: ( nat > nat > $o ) > nat ).
thf(func_def_71,type,
sK35: ( nat > nat > $o ) > nat ).
thf(func_def_72,type,
sK36: ( nat > nat > $o ) > nat ).
thf(func_def_73,type,
sK37: ( nat > nat > $o ) > nat ).
thf(func_def_74,type,
sK38: nat > nat ).
thf(func_def_75,type,
sK39: ( nat > nat > $o ) > nat ).
thf(func_def_76,type,
sK40: ( nat > nat > $o ) > nat ).
thf(func_def_77,type,
sK41: ( nat > nat > $o ) > nat ).
thf(func_def_78,type,
sK42: ( nat > nat > $o ) > nat ).
thf(func_def_79,type,
sK43: ( nat > nat ) > nat ).
thf(func_def_80,type,
sK44: ( nat > nat ) > nat ).
thf(func_def_81,type,
sK45: ( nat > $o ) > nat > nat ).
thf(func_def_82,type,
sK46: ( nat > $o ) > nat > nat ).
thf(func_def_83,type,
sK47: ( nat > $o ) > nat ).
thf(func_def_84,type,
sK48: nat > nat ).
thf(func_def_85,type,
sK49: ( nat > $o ) > nat ).
thf(func_def_86,type,
sK50: ( nat > $o ) > nat > nat ).
thf(func_def_87,type,
sK51: ( nat > $o ) > nat > nat ).
thf(func_def_88,type,
sK52: ( nat > nat ) > nat ).
thf(func_def_90,type,
ph54:
!>[X0: $tType] : X0 ).
thf(f1607,plain,
$false,
inference(subsumption_resolution,[status(thm)],[f1606,f1314]) ).
thf(f1314,plain,
( zero_zero_nat
!= ( polyno578545843gree_a @ p ) ),
inference(cnf_transformation,[status(thm)],[f926]) ).
thf(f926,plain,
( zero_zero_nat
!= ( polyno578545843gree_a @ p ) ),
inference(flattening,[status(thm)],[f352]) ).
thf(f352,negated_conjecture,
( zero_zero_nat
!= ( polyno578545843gree_a @ p ) ),
inference(negated_conjecture,[status(cth)],[f351]) ).
thf(f351,conjecture,
( zero_zero_nat
= ( polyno578545843gree_a @ p ) ),
file('/export/starexec/sandbox/tmp/tmp.4Dn3J3wAeS/DTF2THF_9327.p',conj_2) ).
thf(f1606,plain,
( zero_zero_nat
= ( polyno578545843gree_a @ p ) ),
inference(trivial_inequality_removal,[status(thm)],[f1605]) ).
thf(f1605,plain,
( ( $true != $true )
| ( zero_zero_nat
= ( polyno578545843gree_a @ p ) ) ),
inference(superposition,[status(thm)],[f1432,f1222]) ).
thf(f1222,plain,
( ( polyno1372495879olyh_a @ p @ n )
= $true ),
inference(cnf_transformation,[status(thm)],[f613]) ).
thf(f613,plain,
( ( polyno1372495879olyh_a @ p @ n )
= $true ),
inference(fool_elimination,[status(thm)],[f612]) ).
thf(f612,plain,
polyno1372495879olyh_a @ p @ n,
inference(rectify,[status(thm)],[f350]) ).
thf(f350,axiom,
polyno1372495879olyh_a @ p @ n,
file('/export/starexec/sandbox/tmp/tmp.4Dn3J3wAeS/DTF2THF_9327.p',conj_1) ).
thf(f1432,plain,
! [X0: polyno727731844poly_a] :
( ( $true
!= ( polyno1372495879olyh_a @ X0 @ n ) )
| ( zero_zero_nat
= ( polyno578545843gree_a @ X0 ) ) ),
inference(superposition,[status(thm)],[f1189,f1423]) ).
thf(f1423,plain,
( n
= ( suc @ ( sK7 @ n ) ) ),
inference(trivial_inequality_removal,[status(thm)],[f1412]) ).
thf(f1412,plain,
( ( $true != $true )
| ( n
= ( suc @ ( sK7 @ n ) ) ) ),
inference(superposition,[status(thm)],[f1216,f1256]) ).
thf(f1256,plain,
( ( ord_less_nat @ zero_zero_nat @ n )
= $true ),
inference(cnf_transformation,[status(thm)],[f475]) ).
thf(f475,plain,
( ( ord_less_nat @ zero_zero_nat @ n )
= $true ),
inference(fool_elimination,[status(thm)],[f474]) ).
thf(f474,plain,
ord_less_nat @ zero_zero_nat @ n,
inference(rectify,[status(thm)],[f349]) ).
thf(f349,axiom,
ord_less_nat @ zero_zero_nat @ n,
file('/export/starexec/sandbox/tmp/tmp.4Dn3J3wAeS/DTF2THF_9327.p',conj_0) ).
thf(f1216,plain,
! [X0: nat] :
( ( ( ord_less_nat @ zero_zero_nat @ X0 )
!= $true )
| ( ( suc @ ( sK7 @ X0 ) )
= X0 ) ),
inference(cnf_transformation,[status(thm)],[f1086]) ).
thf(f1086,plain,
! [X0: nat] :
( ( ( ( ord_less_nat @ zero_zero_nat @ X0 )
= $true )
| ! [X1: nat] :
( ( suc @ X1 )
!= X0 ) )
& ( ( ( suc @ ( sK7 @ X0 ) )
= X0 )
| ( ( ord_less_nat @ zero_zero_nat @ X0 )
!= $true ) ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK7])],[f1084,f1085]) ).
thf(f1085,plain,
! [X0: nat] :
( ? [X2: nat] :
( ( suc @ X2 )
= X0 )
=> ( ( suc @ ( sK7 @ X0 ) )
= X0 ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f1084,plain,
! [X0: nat] :
( ( ( ( ord_less_nat @ zero_zero_nat @ X0 )
= $true )
| ! [X1: nat] :
( ( suc @ X1 )
!= X0 ) )
& ( ? [X2: nat] :
( ( suc @ X2 )
= X0 )
| ( ( ord_less_nat @ zero_zero_nat @ X0 )
!= $true ) ) ),
inference(rectify,[status(thm)],[f1083]) ).
thf(f1083,plain,
! [X0: nat] :
( ( ( ( ord_less_nat @ zero_zero_nat @ X0 )
= $true )
| ! [X1: nat] :
( ( suc @ X1 )
!= X0 ) )
& ( ? [X1: nat] :
( ( suc @ X1 )
= X0 )
| ( ( ord_less_nat @ zero_zero_nat @ X0 )
!= $true ) ) ),
inference(nnf_transformation,[status(thm)],[f666]) ).
thf(f666,plain,
! [X0: nat] :
( ( ( ord_less_nat @ zero_zero_nat @ X0 )
= $true )
<=> ? [X1: nat] :
( ( suc @ X1 )
= X0 ) ),
inference(fool_elimination,[status(thm)],[f665]) ).
thf(f665,plain,
! [X0: nat] :
( ( ord_less_nat @ zero_zero_nat @ X0 )
= ( ? [X1: nat] :
( ( suc @ X1 )
= X0 ) ) ),
inference(rectify,[status(thm)],[f99]) ).
thf(f99,axiom,
! [X0: nat] :
( ( ord_less_nat @ zero_zero_nat @ X0 )
= ( ? [X32: nat] :
( ( suc @ X32 )
= X0 ) ) ),
file('/export/starexec/sandbox/tmp/tmp.4Dn3J3wAeS/DTF2THF_9327.p',fact_98_gr0__conv__Suc) ).
thf(f1189,plain,
! [X0: nat,X1: polyno727731844poly_a] :
( ( $true
!= ( polyno1372495879olyh_a @ X1 @ ( suc @ X0 ) ) )
| ( zero_zero_nat
= ( polyno578545843gree_a @ X1 ) ) ),
inference(cnf_transformation,[status(thm)],[f1056]) ).
thf(f1056,plain,
! [X0: nat,X1: polyno727731844poly_a] :
( ( $true
!= ( polyno1372495879olyh_a @ X1 @ ( suc @ X0 ) ) )
| ( zero_zero_nat
= ( polyno578545843gree_a @ X1 ) ) ),
inference(ennf_transformation,[status(thm)],[f650]) ).
thf(f650,plain,
! [X0: nat,X1: polyno727731844poly_a] :
( ( $true
= ( polyno1372495879olyh_a @ X1 @ ( suc @ X0 ) ) )
=> ( zero_zero_nat
= ( polyno578545843gree_a @ X1 ) ) ),
inference(fool_elimination,[status(thm)],[f649]) ).
thf(f649,plain,
! [X0: nat,X1: polyno727731844poly_a] :
( ( polyno1372495879olyh_a @ X1 @ ( suc @ X0 ) )
=> ( zero_zero_nat
= ( polyno578545843gree_a @ X1 ) ) ),
inference(rectify,[status(thm)],[f32]) ).
thf(f32,axiom,
! [X0: nat,X11: polyno727731844poly_a] :
( ( polyno1372495879olyh_a @ X11 @ ( suc @ X0 ) )
=> ( zero_zero_nat
= ( polyno578545843gree_a @ X11 ) ) ),
file('/export/starexec/sandbox/tmp/tmp.4Dn3J3wAeS/DTF2THF_9327.p',fact_31_degree__isnpolyh__Suc) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.11 % Problem : ITP151^1 : TPTP v9.2.1. Released v7.5.0.
% 0.00/0.13 % Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.15/0.33 % Computer : n008.cluster.edu
% 0.15/0.33 % Model : x86_64 x86_64
% 0.15/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.15/0.33 % Memory : 8042.1875MB
% 0.15/0.33 % OS : Linux 3.10.0-693.el7.x86_64
% 0.15/0.33 % CPULimit : 300
% 0.15/0.33 % WCLimit : 300
% 0.15/0.33 % DateTime : Tue Feb 24 08:28:09 EST 2026
% 0.15/0.34 % CPUTime :
% 0.15/0.34 Running /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.20/0.45 ---- Original DTF file ---
% 0.31/0.46 thf(spec,logic,$$dhol).
% 0.31/0.46 %------------------------------------------------------------------------------
% 0.31/0.46 % File : ITP151^1 : TPTP v9.2.1. Released v7.5.0.
% 0.31/0.46 % Domain : Interactive Theorem Proving
% 0.31/0.46 % Problem : Sledgehammer Polynomial_Expression problem prob_414__8350168_1
% 0.31/0.46 % Version : Especial.
% 0.31/0.46 % English :
% 0.31/0.46
% 0.31/0.46 % Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% 0.31/0.46 % : [Des21] Desharnais (2021), Email to Geoff Sutcliffe
% 0.31/0.46 % Source : [Des21]
% 0.31/0.46 % Names : Polynomial_Expression/prob_414__8350168_1 [Des21]
% 0.31/0.46
% 0.31/0.46 % Status : Theorem
% 0.31/0.46 % Rating : 0.33 v9.1.0, 0.38 v9.0.0, 0.40 v8.2.0, 0.23 v8.1.0, 0.18 v7.5.0
% 0.31/0.46 % Syntax : Number of formulae : 373 ( 151 unt; 22 typ; 0 def)
% 0.31/0.46 % Number of atoms : 966 ( 310 equ; 0 cnn)
% 0.31/0.46 % Maximal formula atoms : 12 ( 2 avg)
% 0.31/0.46 % Number of connectives : 2695 ( 134 ~; 20 |; 33 &;2048 @)
% 0.31/0.46 % ( 0 <=>; 460 =>; 0 <=; 0 <~>)
% 0.31/0.46 % Maximal formula depth : 18 ( 6 avg)
% 0.31/0.46 % Number of types : 4 ( 3 usr)
% 0.31/0.46 % Number of type conns : 94 ( 94 >; 0 *; 0 +; 0 <<)
% 0.31/0.46 % Number of symbols : 20 ( 19 usr; 4 con; 0-2 aty)
% 0.31/0.46 % Number of variables : 916 ( 39 ^; 855 !; 22 ?; 916 :)
% 0.31/0.46 % SPC : TH0_THM_EQU_NAR
% 0.31/0.46
% 0.31/0.46 % Comments : This file was generated by Sledgehammer 2021-02-23 15:31:04.009
% 0.31/0.46 %------------------------------------------------------------------------------
% 0.31/0.46 % Could-be-implicit typings (3)
% 0.31/0.46 thf(ty_n_t__Polynomial____Expression____Mirabelle____dwjuveeage__Opoly_Itf__a_J,type,
% 0.31/0.46 polyno727731844poly_a: $tType ).
% 0.31/0.46
% 0.31/0.46 thf(ty_n_t__Nat__Onat,type,
% 0.31/0.46 nat: $tType ).
% 0.31/0.46
% 0.31/0.46 thf(ty_n_t__Int__Oint,type,
% 0.31/0.46 int: $tType ).
% 0.31/0.46
% 0.31/0.46 % Explicit typings (19)
% 0.31/0.46 thf(sy_c_Groups_Ominus__class_Ominus_001t__Int__Oint,type,
% 0.31/0.46 minus_minus_int: int > int > int ).
% 0.31/0.46
% 0.31/0.46 thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat,type,
% 0.31/0.46 minus_minus_nat: nat > nat > nat ).
% 0.31/0.46
% 0.31/0.46 thf(sy_c_Groups_Ozero__class_Ozero_001t__Int__Oint,type,
% 0.31/0.46 zero_zero_int: int ).
% 0.31/0.46
% 0.31/0.46 thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
% 0.31/0.46 zero_zero_nat: nat ).
% 0.31/0.46
% 0.31/0.46 thf(sy_c_Nat_OSuc,type,
% 0.31/0.46 suc: nat > nat ).
% 0.31/0.46
% 0.31/0.46 thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Int__Oint,type,
% 0.31/0.46 semiri2019852685at_int: nat > int ).
% 0.31/0.46
% 0.31/0.46 thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Nat__Onat,type,
% 0.31/0.46 semiri1382578993at_nat: nat > nat ).
% 0.31/0.46
% 0.31/0.46 thf(sy_c_Orderings_Oord__class_Oless_001t__Int__Oint,type,
% 0.31/0.46 ord_less_int: int > int > $o ).
% 0.31/0.46
% 0.31/0.46 thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
% 0.31/0.46 ord_less_nat: nat > nat > $o ).
% 0.31/0.46
% 0.31/0.46 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Int__Oint,type,
% 0.31/0.46 ord_less_eq_int: int > int > $o ).
% 0.31/0.46
% 0.31/0.46 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
% 0.31/0.46 ord_less_eq_nat: nat > nat > $o ).
% 0.31/0.46
% 0.31/0.46 thf(sy_c_Polynomial__Expression__Mirabelle__dwjuveeage_Odegree_001tf__a,type,
% 0.31/0.46 polyno578545843gree_a: polyno727731844poly_a > nat ).
% 0.31/0.46
% 0.31/0.46 thf(sy_c_Polynomial__Expression__Mirabelle__dwjuveeage_Odegreen_001tf__a,type,
% 0.31/0.46 polyno1674775833reen_a: polyno727731844poly_a > nat > nat ).
% 0.31/0.46
% 0.31/0.46 thf(sy_c_Polynomial__Expression__Mirabelle__dwjuveeage_Oisnpoly_001tf__a,type,
% 0.31/0.46 polyno190918219poly_a: polyno727731844poly_a > $o ).
% 0.31/0.46
% 0.31/0.46 thf(sy_c_Polynomial__Expression__Mirabelle__dwjuveeage_Oisnpolyh_001tf__a,type,
% 0.31/0.46 polyno1372495879olyh_a: polyno727731844poly_a > nat > $o ).
% 0.31/0.46
% 0.31/0.46 thf(sy_c_Power_Opower__class_Opower_001t__Int__Oint,type,
% 0.31/0.46 power_power_int: int > nat > int ).
% 0.31/0.46
% 0.31/0.46 thf(sy_c_Power_Opower__class_Opower_001t__Nat__Onat,type,
% 0.31/0.46 power_power_nat: nat > nat > nat ).
% 0.31/0.46
% 0.31/0.46 thf(sy_v_n,type,
% 0.31/0.46 n: nat ).
% 0.31/0.46
% 0.31/0.46 thf(sy_v_p,type,
% 0.31/0.46 p: polyno727731844poly_a ).
% 0.31/0.46
% 0.31/0.46 % Relevant facts (348)
% 0.31/0.46 thf(fact_0_neq0__conv,axiom,
% 0.31/0.46 ! [N: nat] :
% 0.31/0.46 ( ( N != zero_zero_nat )
% 0.31/0.46 = ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 0.31/0.46
% 0.31/0.46 % neq0_conv
% 0.31/0.46 thf(fact_1_less__nat__zero__code,axiom,
% 0.31/0.46 ! [N: nat] :
% 0.31/0.46 ~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% 0.31/0.46
% 0.31/0.46 % less_nat_zero_code
% 0.31/0.46 thf(fact_2_bot__nat__0_Onot__eq__extremum,axiom,
% 0.31/0.46 ! [A: nat] :
% 0.31/0.46 ( ( A != zero_zero_nat )
% 0.31/0.46 = ( ord_less_nat @ zero_zero_nat @ A ) ) ).
% 0.31/0.46
% 0.31/0.46 % bot_nat_0.not_eq_extremum
% 0.31/0.46 thf(fact_3_not__gr__zero,axiom,
% 0.31/0.46 ! [N: nat] :
% 0.31/0.46 ( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
% 0.31/0.46 = ( N = zero_zero_nat ) ) ).
% 0.31/0.46
% 0.31/0.46 % not_gr_zero
% 0.31/0.46 thf(fact_4_gr0I,axiom,
% 0.31/0.46 ! [N: nat] :
% 0.31/0.46 ( ( N != zero_zero_nat )
% 0.31/0.46 => ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 0.31/0.46
% 0.31/0.46 % gr0I
% 0.31/0.46 thf(fact_5_not__gr0,axiom,
% 0.31/0.46 ! [N: nat] :
% 0.31/0.46 ( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
% 0.31/0.46 = ( N = zero_zero_nat ) ) ).
% 0.31/0.46
% 0.31/0.46 % not_gr0
% 0.31/0.46 thf(fact_6_not__less0,axiom,
% 0.31/0.46 ! [N: nat] :
% 0.31/0.46 ~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% 0.31/0.46
% 0.31/0.46 % not_less0
% 0.31/0.46 thf(fact_7_less__zeroE,axiom,
% 0.31/0.46 ! [N: nat] :
% 0.31/0.46 ~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% 0.31/0.46
% 0.31/0.46 % less_zeroE
% 0.31/0.46 thf(fact_8_gr__implies__not0,axiom,
% 0.31/0.46 ! [M: nat,N: nat] :
% 0.31/0.46 ( ( ord_less_nat @ M @ N )
% 0.31/0.46 => ( N != zero_zero_nat ) ) ).
% 0.31/0.46
% 0.31/0.46 % gr_implies_not0
% 0.31/0.46 thf(fact_9_infinite__descent0,axiom,
% 0.31/0.46 ! [P: nat > $o,N: nat] :
% 0.31/0.46 ( ( P @ zero_zero_nat )
% 0.31/0.46 => ( ! [N2: nat] :
% 0.31/0.46 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.31/0.46 => ( ~ ( P @ N2 )
% 0.31/0.46 => ? [M2: nat] :
% 0.31/0.46 ( ( ord_less_nat @ M2 @ N2 )
% 0.31/0.46 & ~ ( P @ M2 ) ) ) )
% 0.31/0.46 => ( P @ N ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % infinite_descent0
% 0.31/0.46 thf(fact_10_bot__nat__0_Oextremum__strict,axiom,
% 0.31/0.46 ! [A: nat] :
% 0.31/0.46 ~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% 0.31/0.46
% 0.31/0.46 % bot_nat_0.extremum_strict
% 0.31/0.46 thf(fact_11_zero__reorient,axiom,
% 0.31/0.46 ! [X: nat] :
% 0.31/0.46 ( ( zero_zero_nat = X )
% 0.31/0.46 = ( X = zero_zero_nat ) ) ).
% 0.31/0.46
% 0.31/0.46 % zero_reorient
% 0.31/0.46 thf(fact_12_zero__reorient,axiom,
% 0.31/0.46 ! [X: int] :
% 0.31/0.46 ( ( zero_zero_int = X )
% 0.31/0.46 = ( X = zero_zero_int ) ) ).
% 0.31/0.46
% 0.31/0.46 % zero_reorient
% 0.31/0.46 thf(fact_13_linorder__neqE__nat,axiom,
% 0.31/0.46 ! [X: nat,Y: nat] :
% 0.31/0.46 ( ( X != Y )
% 0.31/0.46 => ( ~ ( ord_less_nat @ X @ Y )
% 0.31/0.46 => ( ord_less_nat @ Y @ X ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % linorder_neqE_nat
% 0.31/0.46 thf(fact_14_infinite__descent,axiom,
% 0.31/0.46 ! [P: nat > $o,N: nat] :
% 0.31/0.46 ( ! [N2: nat] :
% 0.31/0.46 ( ~ ( P @ N2 )
% 0.31/0.46 => ? [M2: nat] :
% 0.31/0.46 ( ( ord_less_nat @ M2 @ N2 )
% 0.31/0.46 & ~ ( P @ M2 ) ) )
% 0.31/0.46 => ( P @ N ) ) ).
% 0.31/0.46
% 0.31/0.46 % infinite_descent
% 0.31/0.46 thf(fact_15_nat__less__induct,axiom,
% 0.31/0.46 ! [P: nat > $o,N: nat] :
% 0.31/0.46 ( ! [N2: nat] :
% 0.31/0.46 ( ! [M2: nat] :
% 0.31/0.46 ( ( ord_less_nat @ M2 @ N2 )
% 0.31/0.46 => ( P @ M2 ) )
% 0.31/0.46 => ( P @ N2 ) )
% 0.31/0.46 => ( P @ N ) ) ).
% 0.31/0.46
% 0.31/0.46 % nat_less_induct
% 0.31/0.46 thf(fact_16_less__irrefl__nat,axiom,
% 0.31/0.46 ! [N: nat] :
% 0.31/0.46 ~ ( ord_less_nat @ N @ N ) ).
% 0.31/0.46
% 0.31/0.46 % less_irrefl_nat
% 0.31/0.46 thf(fact_17_less__not__refl3,axiom,
% 0.31/0.46 ! [S: nat,T: nat] :
% 0.31/0.46 ( ( ord_less_nat @ S @ T )
% 0.31/0.46 => ( S != T ) ) ).
% 0.31/0.46
% 0.31/0.46 % less_not_refl3
% 0.31/0.46 thf(fact_18_less__not__refl2,axiom,
% 0.31/0.46 ! [N: nat,M: nat] :
% 0.31/0.46 ( ( ord_less_nat @ N @ M )
% 0.31/0.46 => ( M != N ) ) ).
% 0.31/0.46
% 0.31/0.46 % less_not_refl2
% 0.31/0.46 thf(fact_19_less__not__refl,axiom,
% 0.31/0.46 ! [N: nat] :
% 0.31/0.46 ~ ( ord_less_nat @ N @ N ) ).
% 0.31/0.46
% 0.31/0.46 % less_not_refl
% 0.31/0.46 thf(fact_20_nat__neq__iff,axiom,
% 0.31/0.46 ! [M: nat,N: nat] :
% 0.31/0.46 ( ( M != N )
% 0.31/0.46 = ( ( ord_less_nat @ M @ N )
% 0.31/0.46 | ( ord_less_nat @ N @ M ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % nat_neq_iff
% 0.31/0.46 thf(fact_21_zero__less__iff__neq__zero,axiom,
% 0.31/0.46 ! [N: nat] :
% 0.31/0.46 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.31/0.46 = ( N != zero_zero_nat ) ) ).
% 0.31/0.46
% 0.31/0.46 % zero_less_iff_neq_zero
% 0.31/0.46 thf(fact_22_gr__implies__not__zero,axiom,
% 0.31/0.46 ! [M: nat,N: nat] :
% 0.31/0.46 ( ( ord_less_nat @ M @ N )
% 0.31/0.46 => ( N != zero_zero_nat ) ) ).
% 0.31/0.46
% 0.31/0.46 % gr_implies_not_zero
% 0.31/0.46 thf(fact_23_not__less__zero,axiom,
% 0.31/0.46 ! [N: nat] :
% 0.31/0.46 ~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% 0.31/0.46
% 0.31/0.46 % not_less_zero
% 0.31/0.46 thf(fact_24_gr__zeroI,axiom,
% 0.31/0.46 ! [N: nat] :
% 0.31/0.46 ( ( N != zero_zero_nat )
% 0.31/0.46 => ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 0.31/0.46
% 0.31/0.46 % gr_zeroI
% 0.31/0.46 thf(fact_25_less__numeral__extra_I3_J,axiom,
% 0.31/0.46 ~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% 0.31/0.46
% 0.31/0.46 % less_numeral_extra(3)
% 0.31/0.46 thf(fact_26_less__numeral__extra_I3_J,axiom,
% 0.31/0.46 ~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% 0.31/0.46
% 0.31/0.46 % less_numeral_extra(3)
% 0.31/0.46 thf(fact_27_isnpoly__def,axiom,
% 0.31/0.46 ( polyno190918219poly_a
% 0.31/0.46 = ( ^ [P2: polyno727731844poly_a] : ( polyno1372495879olyh_a @ P2 @ zero_zero_nat ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % isnpoly_def
% 0.31/0.46 thf(fact_28_degreen__0,axiom,
% 0.31/0.46 ! [P3: polyno727731844poly_a,N: nat,M: nat] :
% 0.31/0.46 ( ( polyno1372495879olyh_a @ P3 @ N )
% 0.31/0.46 => ( ( ord_less_nat @ M @ N )
% 0.31/0.46 => ( ( polyno1674775833reen_a @ P3 @ M )
% 0.31/0.46 = zero_zero_nat ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % degreen_0
% 0.31/0.46 thf(fact_29_of__nat__0__less__iff,axiom,
% 0.31/0.46 ! [N: nat] :
% 0.31/0.46 ( ( ord_less_nat @ zero_zero_nat @ ( semiri1382578993at_nat @ N ) )
% 0.31/0.46 = ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 0.31/0.46
% 0.31/0.46 % of_nat_0_less_iff
% 0.31/0.46 thf(fact_30_of__nat__0__less__iff,axiom,
% 0.31/0.46 ! [N: nat] :
% 0.31/0.46 ( ( ord_less_int @ zero_zero_int @ ( semiri2019852685at_int @ N ) )
% 0.31/0.46 = ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 0.31/0.46
% 0.31/0.46 % of_nat_0_less_iff
% 0.31/0.46 thf(fact_31_degree__isnpolyh__Suc,axiom,
% 0.31/0.46 ! [P3: polyno727731844poly_a,N: nat] :
% 0.31/0.46 ( ( polyno1372495879olyh_a @ P3 @ ( suc @ N ) )
% 0.31/0.46 => ( ( polyno578545843gree_a @ P3 )
% 0.31/0.46 = zero_zero_nat ) ) ).
% 0.31/0.46
% 0.31/0.46 % degree_isnpolyh_Suc
% 0.31/0.46 thf(fact_32_zero__natural_Orsp,axiom,
% 0.31/0.46 zero_zero_nat = zero_zero_nat ).
% 0.31/0.46
% 0.31/0.46 % zero_natural.rsp
% 0.31/0.46 thf(fact_33_linorder__neqE__linordered__idom,axiom,
% 0.31/0.46 ! [X: int,Y: int] :
% 0.31/0.46 ( ( X != Y )
% 0.31/0.46 => ( ~ ( ord_less_int @ X @ Y )
% 0.31/0.46 => ( ord_less_int @ Y @ X ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % linorder_neqE_linordered_idom
% 0.31/0.46 thf(fact_34_dual__order_Ostrict__implies__not__eq,axiom,
% 0.31/0.46 ! [B: nat,A: nat] :
% 0.31/0.46 ( ( ord_less_nat @ B @ A )
% 0.31/0.46 => ( A != B ) ) ).
% 0.31/0.46
% 0.31/0.46 % dual_order.strict_implies_not_eq
% 0.31/0.46 thf(fact_35_dual__order_Ostrict__implies__not__eq,axiom,
% 0.31/0.46 ! [B: int,A: int] :
% 0.31/0.46 ( ( ord_less_int @ B @ A )
% 0.31/0.46 => ( A != B ) ) ).
% 0.31/0.46
% 0.31/0.46 % dual_order.strict_implies_not_eq
% 0.31/0.46 thf(fact_36_nat_Oinject,axiom,
% 0.31/0.46 ! [X2: nat,Y2: nat] :
% 0.31/0.46 ( ( ( suc @ X2 )
% 0.31/0.46 = ( suc @ Y2 ) )
% 0.31/0.46 = ( X2 = Y2 ) ) ).
% 0.31/0.46
% 0.31/0.46 % nat.inject
% 0.31/0.46 thf(fact_37_old_Onat_Oinject,axiom,
% 0.31/0.46 ! [Nat: nat,Nat2: nat] :
% 0.31/0.46 ( ( ( suc @ Nat )
% 0.31/0.46 = ( suc @ Nat2 ) )
% 0.31/0.46 = ( Nat = Nat2 ) ) ).
% 0.31/0.46
% 0.31/0.46 % old.nat.inject
% 0.31/0.46 thf(fact_38_of__nat__eq__iff,axiom,
% 0.31/0.46 ! [M: nat,N: nat] :
% 0.31/0.46 ( ( ( semiri2019852685at_int @ M )
% 0.31/0.46 = ( semiri2019852685at_int @ N ) )
% 0.31/0.46 = ( M = N ) ) ).
% 0.31/0.46
% 0.31/0.46 % of_nat_eq_iff
% 0.31/0.46 thf(fact_39_lessI,axiom,
% 0.31/0.46 ! [N: nat] : ( ord_less_nat @ N @ ( suc @ N ) ) ).
% 0.31/0.46
% 0.31/0.46 % lessI
% 0.31/0.46 thf(fact_40_Suc__mono,axiom,
% 0.31/0.46 ! [M: nat,N: nat] :
% 0.31/0.46 ( ( ord_less_nat @ M @ N )
% 0.31/0.46 => ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % Suc_mono
% 0.31/0.46 thf(fact_41_Suc__less__eq,axiom,
% 0.31/0.46 ! [M: nat,N: nat] :
% 0.31/0.46 ( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
% 0.31/0.46 = ( ord_less_nat @ M @ N ) ) ).
% 0.31/0.46
% 0.31/0.46 % Suc_less_eq
% 0.31/0.46 thf(fact_42_of__nat__0,axiom,
% 0.31/0.46 ( ( semiri1382578993at_nat @ zero_zero_nat )
% 0.31/0.46 = zero_zero_nat ) ).
% 0.31/0.46
% 0.31/0.46 % of_nat_0
% 0.31/0.46 thf(fact_43_of__nat__0,axiom,
% 0.31/0.46 ( ( semiri2019852685at_int @ zero_zero_nat )
% 0.31/0.46 = zero_zero_int ) ).
% 0.31/0.46
% 0.31/0.46 % of_nat_0
% 0.31/0.46 thf(fact_44_of__nat__0__eq__iff,axiom,
% 0.31/0.46 ! [N: nat] :
% 0.31/0.46 ( ( zero_zero_nat
% 0.31/0.46 = ( semiri1382578993at_nat @ N ) )
% 0.31/0.46 = ( zero_zero_nat = N ) ) ).
% 0.31/0.46
% 0.31/0.46 % of_nat_0_eq_iff
% 0.31/0.46 thf(fact_45_of__nat__0__eq__iff,axiom,
% 0.31/0.46 ! [N: nat] :
% 0.31/0.46 ( ( zero_zero_int
% 0.31/0.46 = ( semiri2019852685at_int @ N ) )
% 0.31/0.46 = ( zero_zero_nat = N ) ) ).
% 0.31/0.46
% 0.31/0.46 % of_nat_0_eq_iff
% 0.31/0.46 thf(fact_46_of__nat__eq__0__iff,axiom,
% 0.31/0.46 ! [M: nat] :
% 0.31/0.46 ( ( ( semiri1382578993at_nat @ M )
% 0.31/0.46 = zero_zero_nat )
% 0.31/0.46 = ( M = zero_zero_nat ) ) ).
% 0.31/0.46
% 0.31/0.46 % of_nat_eq_0_iff
% 0.31/0.46 thf(fact_47_of__nat__eq__0__iff,axiom,
% 0.31/0.46 ! [M: nat] :
% 0.31/0.46 ( ( ( semiri2019852685at_int @ M )
% 0.31/0.46 = zero_zero_int )
% 0.31/0.46 = ( M = zero_zero_nat ) ) ).
% 0.31/0.46
% 0.31/0.46 % of_nat_eq_0_iff
% 0.31/0.46 thf(fact_48_of__nat__less__iff,axiom,
% 0.31/0.46 ! [M: nat,N: nat] :
% 0.31/0.46 ( ( ord_less_nat @ ( semiri1382578993at_nat @ M ) @ ( semiri1382578993at_nat @ N ) )
% 0.31/0.46 = ( ord_less_nat @ M @ N ) ) ).
% 0.31/0.46
% 0.31/0.46 % of_nat_less_iff
% 0.31/0.46 thf(fact_49_of__nat__less__iff,axiom,
% 0.31/0.46 ! [M: nat,N: nat] :
% 0.31/0.46 ( ( ord_less_int @ ( semiri2019852685at_int @ M ) @ ( semiri2019852685at_int @ N ) )
% 0.31/0.46 = ( ord_less_nat @ M @ N ) ) ).
% 0.31/0.46
% 0.31/0.46 % of_nat_less_iff
% 0.31/0.46 thf(fact_50_less__Suc0,axiom,
% 0.31/0.46 ! [N: nat] :
% 0.31/0.46 ( ( ord_less_nat @ N @ ( suc @ zero_zero_nat ) )
% 0.31/0.46 = ( N = zero_zero_nat ) ) ).
% 0.31/0.46
% 0.31/0.46 % less_Suc0
% 0.31/0.46 thf(fact_51_zero__less__Suc,axiom,
% 0.31/0.46 ! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N ) ) ).
% 0.31/0.46
% 0.31/0.46 % zero_less_Suc
% 0.31/0.46 thf(fact_52_Suc__inject,axiom,
% 0.31/0.46 ! [X: nat,Y: nat] :
% 0.31/0.46 ( ( ( suc @ X )
% 0.31/0.46 = ( suc @ Y ) )
% 0.31/0.46 => ( X = Y ) ) ).
% 0.31/0.46
% 0.31/0.46 % Suc_inject
% 0.31/0.46 thf(fact_53_n__not__Suc__n,axiom,
% 0.31/0.46 ! [N: nat] :
% 0.31/0.46 ( N
% 0.31/0.46 != ( suc @ N ) ) ).
% 0.31/0.46
% 0.31/0.46 % n_not_Suc_n
% 0.31/0.46 thf(fact_54_of__nat__neq__0,axiom,
% 0.31/0.46 ! [N: nat] :
% 0.31/0.46 ( ( semiri1382578993at_nat @ ( suc @ N ) )
% 0.31/0.46 != zero_zero_nat ) ).
% 0.31/0.46
% 0.31/0.46 % of_nat_neq_0
% 0.31/0.46 thf(fact_55_of__nat__neq__0,axiom,
% 0.31/0.46 ! [N: nat] :
% 0.31/0.46 ( ( semiri2019852685at_int @ ( suc @ N ) )
% 0.31/0.46 != zero_zero_int ) ).
% 0.31/0.46
% 0.31/0.46 % of_nat_neq_0
% 0.31/0.46 thf(fact_56_polypow_Ocases,axiom,
% 0.31/0.46 ! [X: nat] :
% 0.31/0.46 ( ( X != zero_zero_nat )
% 0.31/0.46 => ~ ! [V: nat] :
% 0.31/0.46 ( X
% 0.31/0.46 != ( suc @ V ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % polypow.cases
% 0.31/0.46 thf(fact_57_nat_Odistinct_I1_J,axiom,
% 0.31/0.46 ! [X2: nat] :
% 0.31/0.46 ( zero_zero_nat
% 0.31/0.46 != ( suc @ X2 ) ) ).
% 0.31/0.46
% 0.31/0.46 % nat.distinct(1)
% 0.31/0.46 thf(fact_58_old_Onat_Odistinct_I2_J,axiom,
% 0.31/0.46 ! [Nat2: nat] :
% 0.31/0.46 ( ( suc @ Nat2 )
% 0.31/0.46 != zero_zero_nat ) ).
% 0.31/0.46
% 0.31/0.46 % old.nat.distinct(2)
% 0.31/0.46 thf(fact_59_old_Onat_Odistinct_I1_J,axiom,
% 0.31/0.46 ! [Nat2: nat] :
% 0.31/0.46 ( zero_zero_nat
% 0.31/0.46 != ( suc @ Nat2 ) ) ).
% 0.31/0.46
% 0.31/0.46 % old.nat.distinct(1)
% 0.31/0.46 thf(fact_60_nat_OdiscI,axiom,
% 0.31/0.46 ! [Nat: nat,X2: nat] :
% 0.31/0.46 ( ( Nat
% 0.31/0.46 = ( suc @ X2 ) )
% 0.31/0.46 => ( Nat != zero_zero_nat ) ) ).
% 0.31/0.46
% 0.31/0.46 % nat.discI
% 0.31/0.46 thf(fact_61_nat__induct,axiom,
% 0.31/0.46 ! [P: nat > $o,N: nat] :
% 0.31/0.46 ( ( P @ zero_zero_nat )
% 0.31/0.46 => ( ! [N2: nat] :
% 0.31/0.46 ( ( P @ N2 )
% 0.31/0.46 => ( P @ ( suc @ N2 ) ) )
% 0.31/0.46 => ( P @ N ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % nat_induct
% 0.31/0.46 thf(fact_62_diff__induct,axiom,
% 0.31/0.46 ! [P: nat > nat > $o,M: nat,N: nat] :
% 0.31/0.46 ( ! [X3: nat] : ( P @ X3 @ zero_zero_nat )
% 0.31/0.46 => ( ! [Y3: nat] : ( P @ zero_zero_nat @ ( suc @ Y3 ) )
% 0.31/0.46 => ( ! [X3: nat,Y3: nat] :
% 0.31/0.46 ( ( P @ X3 @ Y3 )
% 0.31/0.46 => ( P @ ( suc @ X3 ) @ ( suc @ Y3 ) ) )
% 0.31/0.46 => ( P @ M @ N ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % diff_induct
% 0.31/0.46 thf(fact_63_zero__induct,axiom,
% 0.31/0.46 ! [P: nat > $o,K: nat] :
% 0.31/0.46 ( ( P @ K )
% 0.31/0.46 => ( ! [N2: nat] :
% 0.31/0.46 ( ( P @ ( suc @ N2 ) )
% 0.31/0.46 => ( P @ N2 ) )
% 0.31/0.46 => ( P @ zero_zero_nat ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % zero_induct
% 0.31/0.46 thf(fact_64_Suc__neq__Zero,axiom,
% 0.31/0.46 ! [M: nat] :
% 0.31/0.46 ( ( suc @ M )
% 0.31/0.46 != zero_zero_nat ) ).
% 0.31/0.46
% 0.31/0.46 % Suc_neq_Zero
% 0.31/0.46 thf(fact_65_Zero__neq__Suc,axiom,
% 0.31/0.46 ! [M: nat] :
% 0.31/0.46 ( zero_zero_nat
% 0.31/0.46 != ( suc @ M ) ) ).
% 0.31/0.46
% 0.31/0.46 % Zero_neq_Suc
% 0.31/0.46 thf(fact_66_Zero__not__Suc,axiom,
% 0.31/0.46 ! [M: nat] :
% 0.31/0.46 ( zero_zero_nat
% 0.31/0.46 != ( suc @ M ) ) ).
% 0.31/0.46
% 0.31/0.46 % Zero_not_Suc
% 0.31/0.46 thf(fact_67_old_Onat_Oexhaust,axiom,
% 0.31/0.46 ! [Y: nat] :
% 0.31/0.46 ( ( Y != zero_zero_nat )
% 0.31/0.46 => ~ ! [Nat3: nat] :
% 0.31/0.46 ( Y
% 0.31/0.46 != ( suc @ Nat3 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % old.nat.exhaust
% 0.31/0.46 thf(fact_68_old_Onat_Oinducts,axiom,
% 0.31/0.46 ! [P: nat > $o,Nat: nat] :
% 0.31/0.46 ( ( P @ zero_zero_nat )
% 0.31/0.46 => ( ! [Nat3: nat] :
% 0.31/0.46 ( ( P @ Nat3 )
% 0.31/0.46 => ( P @ ( suc @ Nat3 ) ) )
% 0.31/0.46 => ( P @ Nat ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % old.nat.inducts
% 0.31/0.46 thf(fact_69_not0__implies__Suc,axiom,
% 0.31/0.46 ! [N: nat] :
% 0.31/0.46 ( ( N != zero_zero_nat )
% 0.31/0.46 => ? [M3: nat] :
% 0.31/0.46 ( N
% 0.31/0.46 = ( suc @ M3 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % not0_implies_Suc
% 0.31/0.46 thf(fact_70_Nat_OlessE,axiom,
% 0.31/0.46 ! [I: nat,K: nat] :
% 0.31/0.46 ( ( ord_less_nat @ I @ K )
% 0.31/0.46 => ( ( K
% 0.31/0.46 != ( suc @ I ) )
% 0.31/0.46 => ~ ! [J: nat] :
% 0.31/0.46 ( ( ord_less_nat @ I @ J )
% 0.31/0.46 => ( K
% 0.31/0.46 != ( suc @ J ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % Nat.lessE
% 0.31/0.46 thf(fact_71_Suc__lessD,axiom,
% 0.31/0.46 ! [M: nat,N: nat] :
% 0.31/0.46 ( ( ord_less_nat @ ( suc @ M ) @ N )
% 0.31/0.46 => ( ord_less_nat @ M @ N ) ) ).
% 0.31/0.46
% 0.31/0.46 % Suc_lessD
% 0.31/0.46 thf(fact_72_Suc__lessE,axiom,
% 0.31/0.46 ! [I: nat,K: nat] :
% 0.31/0.46 ( ( ord_less_nat @ ( suc @ I ) @ K )
% 0.31/0.46 => ~ ! [J: nat] :
% 0.31/0.46 ( ( ord_less_nat @ I @ J )
% 0.31/0.46 => ( K
% 0.31/0.46 != ( suc @ J ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % Suc_lessE
% 0.31/0.46 thf(fact_73_Suc__lessI,axiom,
% 0.31/0.46 ! [M: nat,N: nat] :
% 0.31/0.46 ( ( ord_less_nat @ M @ N )
% 0.31/0.46 => ( ( ( suc @ M )
% 0.31/0.46 != N )
% 0.31/0.46 => ( ord_less_nat @ ( suc @ M ) @ N ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % Suc_lessI
% 0.31/0.46 thf(fact_74_less__SucE,axiom,
% 0.31/0.46 ! [M: nat,N: nat] :
% 0.31/0.46 ( ( ord_less_nat @ M @ ( suc @ N ) )
% 0.31/0.46 => ( ~ ( ord_less_nat @ M @ N )
% 0.31/0.46 => ( M = N ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % less_SucE
% 0.31/0.46 thf(fact_75_less__SucI,axiom,
% 0.31/0.46 ! [M: nat,N: nat] :
% 0.31/0.46 ( ( ord_less_nat @ M @ N )
% 0.31/0.46 => ( ord_less_nat @ M @ ( suc @ N ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % less_SucI
% 0.31/0.46 thf(fact_76_Ex__less__Suc,axiom,
% 0.31/0.46 ! [N: nat,P: nat > $o] :
% 0.31/0.46 ( ( ? [I2: nat] :
% 0.31/0.46 ( ( ord_less_nat @ I2 @ ( suc @ N ) )
% 0.31/0.46 & ( P @ I2 ) ) )
% 0.31/0.46 = ( ( P @ N )
% 0.31/0.46 | ? [I2: nat] :
% 0.31/0.46 ( ( ord_less_nat @ I2 @ N )
% 0.31/0.46 & ( P @ I2 ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % Ex_less_Suc
% 0.31/0.46 thf(fact_77_less__Suc__eq,axiom,
% 0.31/0.46 ! [M: nat,N: nat] :
% 0.31/0.46 ( ( ord_less_nat @ M @ ( suc @ N ) )
% 0.31/0.46 = ( ( ord_less_nat @ M @ N )
% 0.31/0.46 | ( M = N ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % less_Suc_eq
% 0.31/0.46 thf(fact_78_not__less__eq,axiom,
% 0.31/0.46 ! [M: nat,N: nat] :
% 0.31/0.46 ( ( ~ ( ord_less_nat @ M @ N ) )
% 0.31/0.46 = ( ord_less_nat @ N @ ( suc @ M ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % not_less_eq
% 0.31/0.46 thf(fact_79_All__less__Suc,axiom,
% 0.31/0.46 ! [N: nat,P: nat > $o] :
% 0.31/0.46 ( ( ! [I2: nat] :
% 0.31/0.46 ( ( ord_less_nat @ I2 @ ( suc @ N ) )
% 0.31/0.46 => ( P @ I2 ) ) )
% 0.31/0.46 = ( ( P @ N )
% 0.31/0.46 & ! [I2: nat] :
% 0.31/0.46 ( ( ord_less_nat @ I2 @ N )
% 0.31/0.46 => ( P @ I2 ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % All_less_Suc
% 0.31/0.46 thf(fact_80_Suc__less__eq2,axiom,
% 0.31/0.46 ! [N: nat,M: nat] :
% 0.31/0.46 ( ( ord_less_nat @ ( suc @ N ) @ M )
% 0.31/0.46 = ( ? [M4: nat] :
% 0.31/0.46 ( ( M
% 0.31/0.46 = ( suc @ M4 ) )
% 0.31/0.46 & ( ord_less_nat @ N @ M4 ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % Suc_less_eq2
% 0.31/0.46 thf(fact_81_less__antisym,axiom,
% 0.31/0.46 ! [N: nat,M: nat] :
% 0.31/0.46 ( ~ ( ord_less_nat @ N @ M )
% 0.31/0.46 => ( ( ord_less_nat @ N @ ( suc @ M ) )
% 0.31/0.46 => ( M = N ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % less_antisym
% 0.31/0.46 thf(fact_82_Suc__less__SucD,axiom,
% 0.31/0.46 ! [M: nat,N: nat] :
% 0.31/0.46 ( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
% 0.31/0.46 => ( ord_less_nat @ M @ N ) ) ).
% 0.31/0.46
% 0.31/0.46 % Suc_less_SucD
% 0.31/0.46 thf(fact_83_less__trans__Suc,axiom,
% 0.31/0.46 ! [I: nat,J2: nat,K: nat] :
% 0.31/0.46 ( ( ord_less_nat @ I @ J2 )
% 0.31/0.46 => ( ( ord_less_nat @ J2 @ K )
% 0.31/0.46 => ( ord_less_nat @ ( suc @ I ) @ K ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % less_trans_Suc
% 0.31/0.46 thf(fact_84_less__Suc__induct,axiom,
% 0.31/0.46 ! [I: nat,J2: nat,P: nat > nat > $o] :
% 0.31/0.46 ( ( ord_less_nat @ I @ J2 )
% 0.31/0.46 => ( ! [I3: nat] : ( P @ I3 @ ( suc @ I3 ) )
% 0.31/0.46 => ( ! [I3: nat,J: nat,K2: nat] :
% 0.31/0.46 ( ( ord_less_nat @ I3 @ J )
% 0.31/0.46 => ( ( ord_less_nat @ J @ K2 )
% 0.31/0.46 => ( ( P @ I3 @ J )
% 0.31/0.46 => ( ( P @ J @ K2 )
% 0.31/0.46 => ( P @ I3 @ K2 ) ) ) ) )
% 0.31/0.46 => ( P @ I @ J2 ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % less_Suc_induct
% 0.31/0.46 thf(fact_85_strict__inc__induct,axiom,
% 0.31/0.46 ! [I: nat,J2: nat,P: nat > $o] :
% 0.31/0.46 ( ( ord_less_nat @ I @ J2 )
% 0.31/0.46 => ( ! [I3: nat] :
% 0.31/0.46 ( ( J2
% 0.31/0.46 = ( suc @ I3 ) )
% 0.31/0.46 => ( P @ I3 ) )
% 0.31/0.46 => ( ! [I3: nat] :
% 0.31/0.46 ( ( ord_less_nat @ I3 @ J2 )
% 0.31/0.46 => ( ( P @ ( suc @ I3 ) )
% 0.31/0.46 => ( P @ I3 ) ) )
% 0.31/0.46 => ( P @ I ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % strict_inc_induct
% 0.31/0.46 thf(fact_86_not__less__less__Suc__eq,axiom,
% 0.31/0.46 ! [N: nat,M: nat] :
% 0.31/0.46 ( ~ ( ord_less_nat @ N @ M )
% 0.31/0.46 => ( ( ord_less_nat @ N @ ( suc @ M ) )
% 0.31/0.46 = ( N = M ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % not_less_less_Suc_eq
% 0.31/0.46 thf(fact_87_of__nat__less__0__iff,axiom,
% 0.31/0.46 ! [M: nat] :
% 0.31/0.46 ~ ( ord_less_nat @ ( semiri1382578993at_nat @ M ) @ zero_zero_nat ) ).
% 0.31/0.46
% 0.31/0.46 % of_nat_less_0_iff
% 0.31/0.46 thf(fact_88_of__nat__less__0__iff,axiom,
% 0.31/0.46 ! [M: nat] :
% 0.31/0.46 ~ ( ord_less_int @ ( semiri2019852685at_int @ M ) @ zero_zero_int ) ).
% 0.31/0.46
% 0.31/0.46 % of_nat_less_0_iff
% 0.31/0.46 thf(fact_89_less__imp__of__nat__less,axiom,
% 0.31/0.46 ! [M: nat,N: nat] :
% 0.31/0.46 ( ( ord_less_nat @ M @ N )
% 0.31/0.46 => ( ord_less_nat @ ( semiri1382578993at_nat @ M ) @ ( semiri1382578993at_nat @ N ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % less_imp_of_nat_less
% 0.31/0.46 thf(fact_90_less__imp__of__nat__less,axiom,
% 0.31/0.46 ! [M: nat,N: nat] :
% 0.31/0.46 ( ( ord_less_nat @ M @ N )
% 0.31/0.46 => ( ord_less_int @ ( semiri2019852685at_int @ M ) @ ( semiri2019852685at_int @ N ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % less_imp_of_nat_less
% 0.31/0.46 thf(fact_91_of__nat__less__imp__less,axiom,
% 0.31/0.46 ! [M: nat,N: nat] :
% 0.31/0.46 ( ( ord_less_nat @ ( semiri1382578993at_nat @ M ) @ ( semiri1382578993at_nat @ N ) )
% 0.31/0.46 => ( ord_less_nat @ M @ N ) ) ).
% 0.31/0.46
% 0.31/0.46 % of_nat_less_imp_less
% 0.31/0.46 thf(fact_92_of__nat__less__imp__less,axiom,
% 0.31/0.46 ! [M: nat,N: nat] :
% 0.31/0.46 ( ( ord_less_int @ ( semiri2019852685at_int @ M ) @ ( semiri2019852685at_int @ N ) )
% 0.31/0.46 => ( ord_less_nat @ M @ N ) ) ).
% 0.31/0.46
% 0.31/0.46 % of_nat_less_imp_less
% 0.31/0.46 thf(fact_93_lift__Suc__mono__less,axiom,
% 0.31/0.46 ! [F: nat > nat,N: nat,N3: nat] :
% 0.31/0.46 ( ! [N2: nat] : ( ord_less_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
% 0.31/0.46 => ( ( ord_less_nat @ N @ N3 )
% 0.31/0.46 => ( ord_less_nat @ ( F @ N ) @ ( F @ N3 ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % lift_Suc_mono_less
% 0.31/0.46 thf(fact_94_lift__Suc__mono__less,axiom,
% 0.31/0.46 ! [F: nat > int,N: nat,N3: nat] :
% 0.31/0.46 ( ! [N2: nat] : ( ord_less_int @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
% 0.31/0.46 => ( ( ord_less_nat @ N @ N3 )
% 0.31/0.46 => ( ord_less_int @ ( F @ N ) @ ( F @ N3 ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % lift_Suc_mono_less
% 0.31/0.46 thf(fact_95_lift__Suc__mono__less__iff,axiom,
% 0.31/0.46 ! [F: nat > nat,N: nat,M: nat] :
% 0.31/0.46 ( ! [N2: nat] : ( ord_less_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
% 0.31/0.46 => ( ( ord_less_nat @ ( F @ N ) @ ( F @ M ) )
% 0.31/0.46 = ( ord_less_nat @ N @ M ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % lift_Suc_mono_less_iff
% 0.31/0.46 thf(fact_96_lift__Suc__mono__less__iff,axiom,
% 0.31/0.46 ! [F: nat > int,N: nat,M: nat] :
% 0.31/0.46 ( ! [N2: nat] : ( ord_less_int @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
% 0.31/0.46 => ( ( ord_less_int @ ( F @ N ) @ ( F @ M ) )
% 0.31/0.46 = ( ord_less_nat @ N @ M ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % lift_Suc_mono_less_iff
% 0.31/0.46 thf(fact_97_Ex__less__Suc2,axiom,
% 0.31/0.46 ! [N: nat,P: nat > $o] :
% 0.31/0.46 ( ( ? [I2: nat] :
% 0.31/0.46 ( ( ord_less_nat @ I2 @ ( suc @ N ) )
% 0.31/0.46 & ( P @ I2 ) ) )
% 0.31/0.46 = ( ( P @ zero_zero_nat )
% 0.31/0.46 | ? [I2: nat] :
% 0.31/0.46 ( ( ord_less_nat @ I2 @ N )
% 0.31/0.46 & ( P @ ( suc @ I2 ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % Ex_less_Suc2
% 0.31/0.46 thf(fact_98_gr0__conv__Suc,axiom,
% 0.31/0.46 ! [N: nat] :
% 0.31/0.46 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.31/0.46 = ( ? [M5: nat] :
% 0.31/0.46 ( N
% 0.31/0.46 = ( suc @ M5 ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % gr0_conv_Suc
% 0.31/0.46 thf(fact_99_All__less__Suc2,axiom,
% 0.31/0.46 ! [N: nat,P: nat > $o] :
% 0.31/0.46 ( ( ! [I2: nat] :
% 0.31/0.46 ( ( ord_less_nat @ I2 @ ( suc @ N ) )
% 0.31/0.46 => ( P @ I2 ) ) )
% 0.31/0.46 = ( ( P @ zero_zero_nat )
% 0.31/0.46 & ! [I2: nat] :
% 0.31/0.46 ( ( ord_less_nat @ I2 @ N )
% 0.31/0.46 => ( P @ ( suc @ I2 ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % All_less_Suc2
% 0.31/0.46 thf(fact_100_gr0__implies__Suc,axiom,
% 0.31/0.46 ! [N: nat] :
% 0.31/0.46 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.31/0.46 => ? [M3: nat] :
% 0.31/0.46 ( N
% 0.31/0.46 = ( suc @ M3 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % gr0_implies_Suc
% 0.31/0.46 thf(fact_101_less__Suc__eq__0__disj,axiom,
% 0.31/0.46 ! [M: nat,N: nat] :
% 0.31/0.46 ( ( ord_less_nat @ M @ ( suc @ N ) )
% 0.31/0.46 = ( ( M = zero_zero_nat )
% 0.31/0.46 | ? [J3: nat] :
% 0.31/0.46 ( ( M
% 0.31/0.46 = ( suc @ J3 ) )
% 0.31/0.46 & ( ord_less_nat @ J3 @ N ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % less_Suc_eq_0_disj
% 0.31/0.46 thf(fact_102_ord__eq__less__subst,axiom,
% 0.31/0.46 ! [A: nat,F: nat > nat,B: nat,C: nat] :
% 0.31/0.46 ( ( A
% 0.31/0.46 = ( F @ B ) )
% 0.31/0.46 => ( ( ord_less_nat @ B @ C )
% 0.31/0.46 => ( ! [X3: nat,Y3: nat] :
% 0.31/0.46 ( ( ord_less_nat @ X3 @ Y3 )
% 0.31/0.46 => ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
% 0.31/0.46 => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % ord_eq_less_subst
% 0.31/0.46 thf(fact_103_ord__eq__less__subst,axiom,
% 0.31/0.46 ! [A: int,F: nat > int,B: nat,C: nat] :
% 0.31/0.46 ( ( A
% 0.31/0.46 = ( F @ B ) )
% 0.31/0.46 => ( ( ord_less_nat @ B @ C )
% 0.31/0.46 => ( ! [X3: nat,Y3: nat] :
% 0.31/0.46 ( ( ord_less_nat @ X3 @ Y3 )
% 0.31/0.46 => ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
% 0.31/0.46 => ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % ord_eq_less_subst
% 0.31/0.46 thf(fact_104_ord__eq__less__subst,axiom,
% 0.31/0.46 ! [A: nat,F: int > nat,B: int,C: int] :
% 0.31/0.46 ( ( A
% 0.31/0.46 = ( F @ B ) )
% 0.31/0.46 => ( ( ord_less_int @ B @ C )
% 0.31/0.46 => ( ! [X3: int,Y3: int] :
% 0.31/0.46 ( ( ord_less_int @ X3 @ Y3 )
% 0.31/0.46 => ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
% 0.31/0.46 => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % ord_eq_less_subst
% 0.31/0.46 thf(fact_105_ord__eq__less__subst,axiom,
% 0.31/0.46 ! [A: int,F: int > int,B: int,C: int] :
% 0.31/0.46 ( ( A
% 0.31/0.46 = ( F @ B ) )
% 0.31/0.46 => ( ( ord_less_int @ B @ C )
% 0.31/0.46 => ( ! [X3: int,Y3: int] :
% 0.31/0.46 ( ( ord_less_int @ X3 @ Y3 )
% 0.31/0.46 => ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
% 0.31/0.46 => ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % ord_eq_less_subst
% 0.31/0.46 thf(fact_106_ord__less__eq__subst,axiom,
% 0.31/0.46 ! [A: nat,B: nat,F: nat > nat,C: nat] :
% 0.31/0.46 ( ( ord_less_nat @ A @ B )
% 0.31/0.46 => ( ( ( F @ B )
% 0.31/0.46 = C )
% 0.31/0.46 => ( ! [X3: nat,Y3: nat] :
% 0.31/0.46 ( ( ord_less_nat @ X3 @ Y3 )
% 0.31/0.46 => ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
% 0.31/0.46 => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % ord_less_eq_subst
% 0.31/0.46 thf(fact_107_ord__less__eq__subst,axiom,
% 0.31/0.46 ! [A: nat,B: nat,F: nat > int,C: int] :
% 0.31/0.46 ( ( ord_less_nat @ A @ B )
% 0.31/0.46 => ( ( ( F @ B )
% 0.31/0.46 = C )
% 0.31/0.46 => ( ! [X3: nat,Y3: nat] :
% 0.31/0.46 ( ( ord_less_nat @ X3 @ Y3 )
% 0.31/0.46 => ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
% 0.31/0.46 => ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % ord_less_eq_subst
% 0.31/0.46 thf(fact_108_ord__less__eq__subst,axiom,
% 0.31/0.46 ! [A: int,B: int,F: int > nat,C: nat] :
% 0.31/0.46 ( ( ord_less_int @ A @ B )
% 0.31/0.46 => ( ( ( F @ B )
% 0.31/0.46 = C )
% 0.31/0.46 => ( ! [X3: int,Y3: int] :
% 0.31/0.46 ( ( ord_less_int @ X3 @ Y3 )
% 0.31/0.46 => ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
% 0.31/0.46 => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % ord_less_eq_subst
% 0.31/0.46 thf(fact_109_ord__less__eq__subst,axiom,
% 0.31/0.46 ! [A: int,B: int,F: int > int,C: int] :
% 0.31/0.46 ( ( ord_less_int @ A @ B )
% 0.31/0.46 => ( ( ( F @ B )
% 0.31/0.46 = C )
% 0.31/0.46 => ( ! [X3: int,Y3: int] :
% 0.31/0.46 ( ( ord_less_int @ X3 @ Y3 )
% 0.31/0.46 => ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
% 0.31/0.46 => ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % ord_less_eq_subst
% 0.31/0.46 thf(fact_110_order__less__subst1,axiom,
% 0.31/0.46 ! [A: nat,F: nat > nat,B: nat,C: nat] :
% 0.31/0.46 ( ( ord_less_nat @ A @ ( F @ B ) )
% 0.31/0.46 => ( ( ord_less_nat @ B @ C )
% 0.31/0.46 => ( ! [X3: nat,Y3: nat] :
% 0.31/0.46 ( ( ord_less_nat @ X3 @ Y3 )
% 0.31/0.46 => ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
% 0.31/0.46 => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % order_less_subst1
% 0.31/0.46 thf(fact_111_order__less__subst1,axiom,
% 0.31/0.46 ! [A: nat,F: int > nat,B: int,C: int] :
% 0.31/0.46 ( ( ord_less_nat @ A @ ( F @ B ) )
% 0.31/0.46 => ( ( ord_less_int @ B @ C )
% 0.31/0.46 => ( ! [X3: int,Y3: int] :
% 0.31/0.46 ( ( ord_less_int @ X3 @ Y3 )
% 0.31/0.46 => ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
% 0.31/0.46 => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % order_less_subst1
% 0.31/0.46 thf(fact_112_order__less__subst1,axiom,
% 0.31/0.46 ! [A: int,F: nat > int,B: nat,C: nat] :
% 0.31/0.46 ( ( ord_less_int @ A @ ( F @ B ) )
% 0.31/0.46 => ( ( ord_less_nat @ B @ C )
% 0.31/0.46 => ( ! [X3: nat,Y3: nat] :
% 0.31/0.46 ( ( ord_less_nat @ X3 @ Y3 )
% 0.31/0.46 => ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
% 0.31/0.46 => ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % order_less_subst1
% 0.31/0.46 thf(fact_113_order__less__subst1,axiom,
% 0.31/0.46 ! [A: int,F: int > int,B: int,C: int] :
% 0.31/0.46 ( ( ord_less_int @ A @ ( F @ B ) )
% 0.31/0.46 => ( ( ord_less_int @ B @ C )
% 0.31/0.46 => ( ! [X3: int,Y3: int] :
% 0.31/0.46 ( ( ord_less_int @ X3 @ Y3 )
% 0.31/0.46 => ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
% 0.31/0.46 => ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % order_less_subst1
% 0.31/0.46 thf(fact_114_order__less__subst2,axiom,
% 0.31/0.46 ! [A: nat,B: nat,F: nat > nat,C: nat] :
% 0.31/0.46 ( ( ord_less_nat @ A @ B )
% 0.31/0.46 => ( ( ord_less_nat @ ( F @ B ) @ C )
% 0.31/0.46 => ( ! [X3: nat,Y3: nat] :
% 0.31/0.46 ( ( ord_less_nat @ X3 @ Y3 )
% 0.31/0.46 => ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
% 0.31/0.46 => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % order_less_subst2
% 0.31/0.46 thf(fact_115_order__less__subst2,axiom,
% 0.31/0.46 ! [A: nat,B: nat,F: nat > int,C: int] :
% 0.31/0.46 ( ( ord_less_nat @ A @ B )
% 0.31/0.46 => ( ( ord_less_int @ ( F @ B ) @ C )
% 0.31/0.46 => ( ! [X3: nat,Y3: nat] :
% 0.31/0.46 ( ( ord_less_nat @ X3 @ Y3 )
% 0.31/0.46 => ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
% 0.31/0.46 => ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % order_less_subst2
% 0.31/0.46 thf(fact_116_order__less__subst2,axiom,
% 0.31/0.46 ! [A: int,B: int,F: int > nat,C: nat] :
% 0.31/0.46 ( ( ord_less_int @ A @ B )
% 0.31/0.46 => ( ( ord_less_nat @ ( F @ B ) @ C )
% 0.31/0.46 => ( ! [X3: int,Y3: int] :
% 0.31/0.46 ( ( ord_less_int @ X3 @ Y3 )
% 0.31/0.46 => ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
% 0.31/0.46 => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % order_less_subst2
% 0.31/0.46 thf(fact_117_order__less__subst2,axiom,
% 0.31/0.46 ! [A: int,B: int,F: int > int,C: int] :
% 0.31/0.46 ( ( ord_less_int @ A @ B )
% 0.31/0.46 => ( ( ord_less_int @ ( F @ B ) @ C )
% 0.31/0.46 => ( ! [X3: int,Y3: int] :
% 0.31/0.46 ( ( ord_less_int @ X3 @ Y3 )
% 0.31/0.46 => ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
% 0.31/0.46 => ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % order_less_subst2
% 0.31/0.46 thf(fact_118_lt__ex,axiom,
% 0.31/0.46 ! [X: int] :
% 0.31/0.46 ? [Y3: int] : ( ord_less_int @ Y3 @ X ) ).
% 0.31/0.46
% 0.31/0.46 % lt_ex
% 0.31/0.46 thf(fact_119_gt__ex,axiom,
% 0.31/0.46 ! [X: nat] :
% 0.31/0.46 ? [X_1: nat] : ( ord_less_nat @ X @ X_1 ) ).
% 0.31/0.46
% 0.31/0.46 % gt_ex
% 0.31/0.46 thf(fact_120_gt__ex,axiom,
% 0.31/0.46 ! [X: int] :
% 0.31/0.46 ? [X_1: int] : ( ord_less_int @ X @ X_1 ) ).
% 0.31/0.46
% 0.31/0.46 % gt_ex
% 0.31/0.46 thf(fact_121_neqE,axiom,
% 0.31/0.46 ! [X: nat,Y: nat] :
% 0.31/0.46 ( ( X != Y )
% 0.31/0.46 => ( ~ ( ord_less_nat @ X @ Y )
% 0.31/0.46 => ( ord_less_nat @ Y @ X ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % neqE
% 0.31/0.46 thf(fact_122_neqE,axiom,
% 0.31/0.46 ! [X: int,Y: int] :
% 0.31/0.46 ( ( X != Y )
% 0.31/0.46 => ( ~ ( ord_less_int @ X @ Y )
% 0.31/0.46 => ( ord_less_int @ Y @ X ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % neqE
% 0.31/0.46 thf(fact_123_neq__iff,axiom,
% 0.31/0.46 ! [X: nat,Y: nat] :
% 0.31/0.46 ( ( X != Y )
% 0.31/0.46 = ( ( ord_less_nat @ X @ Y )
% 0.31/0.46 | ( ord_less_nat @ Y @ X ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % neq_iff
% 0.31/0.46 thf(fact_124_neq__iff,axiom,
% 0.31/0.46 ! [X: int,Y: int] :
% 0.31/0.46 ( ( X != Y )
% 0.31/0.46 = ( ( ord_less_int @ X @ Y )
% 0.31/0.46 | ( ord_less_int @ Y @ X ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % neq_iff
% 0.31/0.46 thf(fact_125_order_Oasym,axiom,
% 0.31/0.46 ! [A: nat,B: nat] :
% 0.31/0.46 ( ( ord_less_nat @ A @ B )
% 0.31/0.46 => ~ ( ord_less_nat @ B @ A ) ) ).
% 0.31/0.46
% 0.31/0.46 % order.asym
% 0.31/0.46 thf(fact_126_order_Oasym,axiom,
% 0.31/0.46 ! [A: int,B: int] :
% 0.31/0.46 ( ( ord_less_int @ A @ B )
% 0.31/0.46 => ~ ( ord_less_int @ B @ A ) ) ).
% 0.31/0.46
% 0.31/0.46 % order.asym
% 0.31/0.46 thf(fact_127_less__imp__neq,axiom,
% 0.31/0.46 ! [X: nat,Y: nat] :
% 0.31/0.46 ( ( ord_less_nat @ X @ Y )
% 0.31/0.46 => ( X != Y ) ) ).
% 0.31/0.46
% 0.31/0.46 % less_imp_neq
% 0.31/0.46 thf(fact_128_less__imp__neq,axiom,
% 0.31/0.46 ! [X: int,Y: int] :
% 0.31/0.46 ( ( ord_less_int @ X @ Y )
% 0.31/0.46 => ( X != Y ) ) ).
% 0.31/0.46
% 0.31/0.46 % less_imp_neq
% 0.31/0.46 thf(fact_129_less__asym,axiom,
% 0.31/0.46 ! [X: nat,Y: nat] :
% 0.31/0.46 ( ( ord_less_nat @ X @ Y )
% 0.31/0.46 => ~ ( ord_less_nat @ Y @ X ) ) ).
% 0.31/0.46
% 0.31/0.46 % less_asym
% 0.31/0.46 thf(fact_130_less__asym,axiom,
% 0.31/0.46 ! [X: int,Y: int] :
% 0.31/0.46 ( ( ord_less_int @ X @ Y )
% 0.31/0.46 => ~ ( ord_less_int @ Y @ X ) ) ).
% 0.31/0.46
% 0.31/0.46 % less_asym
% 0.31/0.46 thf(fact_131_less__asym_H,axiom,
% 0.31/0.46 ! [A: nat,B: nat] :
% 0.31/0.46 ( ( ord_less_nat @ A @ B )
% 0.31/0.46 => ~ ( ord_less_nat @ B @ A ) ) ).
% 0.31/0.46
% 0.31/0.46 % less_asym'
% 0.31/0.46 thf(fact_132_less__asym_H,axiom,
% 0.31/0.46 ! [A: int,B: int] :
% 0.31/0.46 ( ( ord_less_int @ A @ B )
% 0.31/0.46 => ~ ( ord_less_int @ B @ A ) ) ).
% 0.31/0.46
% 0.31/0.46 % less_asym'
% 0.31/0.46 thf(fact_133_less__trans,axiom,
% 0.31/0.46 ! [X: nat,Y: nat,Z: nat] :
% 0.31/0.46 ( ( ord_less_nat @ X @ Y )
% 0.31/0.46 => ( ( ord_less_nat @ Y @ Z )
% 0.31/0.46 => ( ord_less_nat @ X @ Z ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % less_trans
% 0.31/0.46 thf(fact_134_less__trans,axiom,
% 0.31/0.46 ! [X: int,Y: int,Z: int] :
% 0.31/0.46 ( ( ord_less_int @ X @ Y )
% 0.31/0.46 => ( ( ord_less_int @ Y @ Z )
% 0.31/0.46 => ( ord_less_int @ X @ Z ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % less_trans
% 0.31/0.46 thf(fact_135_less__linear,axiom,
% 0.31/0.46 ! [X: nat,Y: nat] :
% 0.31/0.46 ( ( ord_less_nat @ X @ Y )
% 0.31/0.46 | ( X = Y )
% 0.31/0.46 | ( ord_less_nat @ Y @ X ) ) ).
% 0.31/0.46
% 0.31/0.46 % less_linear
% 0.31/0.46 thf(fact_136_less__linear,axiom,
% 0.31/0.46 ! [X: int,Y: int] :
% 0.31/0.46 ( ( ord_less_int @ X @ Y )
% 0.31/0.46 | ( X = Y )
% 0.31/0.46 | ( ord_less_int @ Y @ X ) ) ).
% 0.31/0.46
% 0.31/0.46 % less_linear
% 0.31/0.46 thf(fact_137_less__irrefl,axiom,
% 0.31/0.46 ! [X: nat] :
% 0.31/0.46 ~ ( ord_less_nat @ X @ X ) ).
% 0.31/0.46
% 0.31/0.46 % less_irrefl
% 0.31/0.46 thf(fact_138_less__irrefl,axiom,
% 0.31/0.46 ! [X: int] :
% 0.31/0.46 ~ ( ord_less_int @ X @ X ) ).
% 0.31/0.46
% 0.31/0.46 % less_irrefl
% 0.31/0.46 thf(fact_139_ord__eq__less__trans,axiom,
% 0.31/0.46 ! [A: nat,B: nat,C: nat] :
% 0.31/0.46 ( ( A = B )
% 0.31/0.46 => ( ( ord_less_nat @ B @ C )
% 0.31/0.46 => ( ord_less_nat @ A @ C ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % ord_eq_less_trans
% 0.31/0.46 thf(fact_140_ord__eq__less__trans,axiom,
% 0.31/0.46 ! [A: int,B: int,C: int] :
% 0.31/0.46 ( ( A = B )
% 0.31/0.46 => ( ( ord_less_int @ B @ C )
% 0.31/0.46 => ( ord_less_int @ A @ C ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % ord_eq_less_trans
% 0.31/0.46 thf(fact_141_ord__less__eq__trans,axiom,
% 0.31/0.46 ! [A: nat,B: nat,C: nat] :
% 0.31/0.46 ( ( ord_less_nat @ A @ B )
% 0.31/0.46 => ( ( B = C )
% 0.31/0.46 => ( ord_less_nat @ A @ C ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % ord_less_eq_trans
% 0.31/0.46 thf(fact_142_ord__less__eq__trans,axiom,
% 0.31/0.46 ! [A: int,B: int,C: int] :
% 0.31/0.46 ( ( ord_less_int @ A @ B )
% 0.31/0.46 => ( ( B = C )
% 0.31/0.46 => ( ord_less_int @ A @ C ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % ord_less_eq_trans
% 0.31/0.46 thf(fact_143_dual__order_Oasym,axiom,
% 0.31/0.46 ! [B: nat,A: nat] :
% 0.31/0.46 ( ( ord_less_nat @ B @ A )
% 0.31/0.46 => ~ ( ord_less_nat @ A @ B ) ) ).
% 0.31/0.46
% 0.31/0.46 % dual_order.asym
% 0.31/0.46 thf(fact_144_dual__order_Oasym,axiom,
% 0.31/0.46 ! [B: int,A: int] :
% 0.31/0.46 ( ( ord_less_int @ B @ A )
% 0.31/0.46 => ~ ( ord_less_int @ A @ B ) ) ).
% 0.31/0.46
% 0.31/0.46 % dual_order.asym
% 0.31/0.46 thf(fact_145_less__imp__not__eq,axiom,
% 0.31/0.46 ! [X: nat,Y: nat] :
% 0.31/0.46 ( ( ord_less_nat @ X @ Y )
% 0.31/0.46 => ( X != Y ) ) ).
% 0.31/0.46
% 0.31/0.46 % less_imp_not_eq
% 0.31/0.46 thf(fact_146_less__imp__not__eq,axiom,
% 0.31/0.46 ! [X: int,Y: int] :
% 0.31/0.46 ( ( ord_less_int @ X @ Y )
% 0.31/0.46 => ( X != Y ) ) ).
% 0.31/0.46
% 0.31/0.46 % less_imp_not_eq
% 0.31/0.46 thf(fact_147_less__not__sym,axiom,
% 0.31/0.46 ! [X: nat,Y: nat] :
% 0.31/0.46 ( ( ord_less_nat @ X @ Y )
% 0.31/0.46 => ~ ( ord_less_nat @ Y @ X ) ) ).
% 0.31/0.46
% 0.31/0.46 % less_not_sym
% 0.31/0.46 thf(fact_148_less__not__sym,axiom,
% 0.31/0.46 ! [X: int,Y: int] :
% 0.31/0.46 ( ( ord_less_int @ X @ Y )
% 0.31/0.46 => ~ ( ord_less_int @ Y @ X ) ) ).
% 0.31/0.46
% 0.31/0.46 % less_not_sym
% 0.31/0.46 thf(fact_149_less__induct,axiom,
% 0.31/0.46 ! [P: nat > $o,A: nat] :
% 0.31/0.46 ( ! [X3: nat] :
% 0.31/0.46 ( ! [Y4: nat] :
% 0.31/0.46 ( ( ord_less_nat @ Y4 @ X3 )
% 0.31/0.46 => ( P @ Y4 ) )
% 0.31/0.46 => ( P @ X3 ) )
% 0.31/0.46 => ( P @ A ) ) ).
% 0.31/0.46
% 0.31/0.46 % less_induct
% 0.31/0.46 thf(fact_150_antisym__conv3,axiom,
% 0.31/0.46 ! [Y: nat,X: nat] :
% 0.31/0.46 ( ~ ( ord_less_nat @ Y @ X )
% 0.31/0.46 => ( ( ~ ( ord_less_nat @ X @ Y ) )
% 0.31/0.46 = ( X = Y ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % antisym_conv3
% 0.31/0.46 thf(fact_151_antisym__conv3,axiom,
% 0.31/0.46 ! [Y: int,X: int] :
% 0.31/0.46 ( ~ ( ord_less_int @ Y @ X )
% 0.31/0.46 => ( ( ~ ( ord_less_int @ X @ Y ) )
% 0.31/0.46 = ( X = Y ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % antisym_conv3
% 0.31/0.46 thf(fact_152_less__imp__not__eq2,axiom,
% 0.31/0.46 ! [X: nat,Y: nat] :
% 0.31/0.46 ( ( ord_less_nat @ X @ Y )
% 0.31/0.46 => ( Y != X ) ) ).
% 0.31/0.46
% 0.31/0.46 % less_imp_not_eq2
% 0.31/0.46 thf(fact_153_less__imp__not__eq2,axiom,
% 0.31/0.46 ! [X: int,Y: int] :
% 0.31/0.46 ( ( ord_less_int @ X @ Y )
% 0.31/0.46 => ( Y != X ) ) ).
% 0.31/0.46
% 0.31/0.46 % less_imp_not_eq2
% 0.31/0.46 thf(fact_154_less__imp__triv,axiom,
% 0.31/0.46 ! [X: nat,Y: nat,P: $o] :
% 0.31/0.46 ( ( ord_less_nat @ X @ Y )
% 0.31/0.46 => ( ( ord_less_nat @ Y @ X )
% 0.31/0.46 => P ) ) ).
% 0.31/0.46
% 0.31/0.46 % less_imp_triv
% 0.31/0.46 thf(fact_155_less__imp__triv,axiom,
% 0.31/0.46 ! [X: int,Y: int,P: $o] :
% 0.31/0.46 ( ( ord_less_int @ X @ Y )
% 0.31/0.46 => ( ( ord_less_int @ Y @ X )
% 0.31/0.46 => P ) ) ).
% 0.31/0.46
% 0.31/0.46 % less_imp_triv
% 0.31/0.46 thf(fact_156_linorder__cases,axiom,
% 0.31/0.46 ! [X: nat,Y: nat] :
% 0.31/0.46 ( ~ ( ord_less_nat @ X @ Y )
% 0.31/0.46 => ( ( X != Y )
% 0.31/0.46 => ( ord_less_nat @ Y @ X ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % linorder_cases
% 0.31/0.46 thf(fact_157_linorder__cases,axiom,
% 0.31/0.46 ! [X: int,Y: int] :
% 0.31/0.46 ( ~ ( ord_less_int @ X @ Y )
% 0.31/0.46 => ( ( X != Y )
% 0.31/0.46 => ( ord_less_int @ Y @ X ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % linorder_cases
% 0.31/0.46 thf(fact_158_dual__order_Oirrefl,axiom,
% 0.31/0.46 ! [A: nat] :
% 0.31/0.46 ~ ( ord_less_nat @ A @ A ) ).
% 0.31/0.46
% 0.31/0.46 % dual_order.irrefl
% 0.31/0.46 thf(fact_159_dual__order_Oirrefl,axiom,
% 0.31/0.46 ! [A: int] :
% 0.31/0.46 ~ ( ord_less_int @ A @ A ) ).
% 0.31/0.46
% 0.31/0.46 % dual_order.irrefl
% 0.31/0.46 thf(fact_160_order_Ostrict__trans,axiom,
% 0.31/0.46 ! [A: nat,B: nat,C: nat] :
% 0.31/0.46 ( ( ord_less_nat @ A @ B )
% 0.31/0.46 => ( ( ord_less_nat @ B @ C )
% 0.31/0.46 => ( ord_less_nat @ A @ C ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % order.strict_trans
% 0.31/0.46 thf(fact_161_order_Ostrict__trans,axiom,
% 0.31/0.46 ! [A: int,B: int,C: int] :
% 0.31/0.46 ( ( ord_less_int @ A @ B )
% 0.31/0.46 => ( ( ord_less_int @ B @ C )
% 0.31/0.46 => ( ord_less_int @ A @ C ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % order.strict_trans
% 0.31/0.46 thf(fact_162_less__imp__not__less,axiom,
% 0.31/0.46 ! [X: nat,Y: nat] :
% 0.31/0.46 ( ( ord_less_nat @ X @ Y )
% 0.31/0.46 => ~ ( ord_less_nat @ Y @ X ) ) ).
% 0.31/0.46
% 0.31/0.46 % less_imp_not_less
% 0.31/0.46 thf(fact_163_less__imp__not__less,axiom,
% 0.31/0.46 ! [X: int,Y: int] :
% 0.31/0.46 ( ( ord_less_int @ X @ Y )
% 0.31/0.46 => ~ ( ord_less_int @ Y @ X ) ) ).
% 0.31/0.46
% 0.31/0.46 % less_imp_not_less
% 0.31/0.46 thf(fact_164_exists__least__iff,axiom,
% 0.31/0.46 ( ( ^ [P4: nat > $o] :
% 0.31/0.46 ? [X4: nat] : ( P4 @ X4 ) )
% 0.31/0.46 = ( ^ [P5: nat > $o] :
% 0.31/0.46 ? [N4: nat] :
% 0.31/0.46 ( ( P5 @ N4 )
% 0.31/0.46 & ! [M5: nat] :
% 0.31/0.46 ( ( ord_less_nat @ M5 @ N4 )
% 0.31/0.46 => ~ ( P5 @ M5 ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % exists_least_iff
% 0.31/0.46 thf(fact_165_linorder__less__wlog,axiom,
% 0.31/0.46 ! [P: nat > nat > $o,A: nat,B: nat] :
% 0.31/0.46 ( ! [A2: nat,B2: nat] :
% 0.31/0.46 ( ( ord_less_nat @ A2 @ B2 )
% 0.31/0.46 => ( P @ A2 @ B2 ) )
% 0.31/0.46 => ( ! [A2: nat] : ( P @ A2 @ A2 )
% 0.31/0.46 => ( ! [A2: nat,B2: nat] :
% 0.31/0.46 ( ( P @ B2 @ A2 )
% 0.31/0.46 => ( P @ A2 @ B2 ) )
% 0.31/0.46 => ( P @ A @ B ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % linorder_less_wlog
% 0.31/0.46 thf(fact_166_linorder__less__wlog,axiom,
% 0.31/0.46 ! [P: int > int > $o,A: int,B: int] :
% 0.31/0.46 ( ! [A2: int,B2: int] :
% 0.31/0.46 ( ( ord_less_int @ A2 @ B2 )
% 0.31/0.46 => ( P @ A2 @ B2 ) )
% 0.31/0.46 => ( ! [A2: int] : ( P @ A2 @ A2 )
% 0.31/0.46 => ( ! [A2: int,B2: int] :
% 0.31/0.46 ( ( P @ B2 @ A2 )
% 0.31/0.46 => ( P @ A2 @ B2 ) )
% 0.31/0.46 => ( P @ A @ B ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % linorder_less_wlog
% 0.31/0.46 thf(fact_167_dual__order_Ostrict__trans,axiom,
% 0.31/0.46 ! [B: nat,A: nat,C: nat] :
% 0.31/0.46 ( ( ord_less_nat @ B @ A )
% 0.31/0.46 => ( ( ord_less_nat @ C @ B )
% 0.31/0.46 => ( ord_less_nat @ C @ A ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % dual_order.strict_trans
% 0.31/0.46 thf(fact_168_dual__order_Ostrict__trans,axiom,
% 0.31/0.46 ! [B: int,A: int,C: int] :
% 0.31/0.46 ( ( ord_less_int @ B @ A )
% 0.31/0.46 => ( ( ord_less_int @ C @ B )
% 0.31/0.46 => ( ord_less_int @ C @ A ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % dual_order.strict_trans
% 0.31/0.46 thf(fact_169_not__less__iff__gr__or__eq,axiom,
% 0.31/0.46 ! [X: nat,Y: nat] :
% 0.31/0.46 ( ( ~ ( ord_less_nat @ X @ Y ) )
% 0.31/0.46 = ( ( ord_less_nat @ Y @ X )
% 0.31/0.46 | ( X = Y ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % not_less_iff_gr_or_eq
% 0.31/0.46 thf(fact_170_not__less__iff__gr__or__eq,axiom,
% 0.31/0.46 ! [X: int,Y: int] :
% 0.31/0.46 ( ( ~ ( ord_less_int @ X @ Y ) )
% 0.31/0.46 = ( ( ord_less_int @ Y @ X )
% 0.31/0.46 | ( X = Y ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % not_less_iff_gr_or_eq
% 0.31/0.46 thf(fact_171_order_Ostrict__implies__not__eq,axiom,
% 0.31/0.46 ! [A: nat,B: nat] :
% 0.31/0.46 ( ( ord_less_nat @ A @ B )
% 0.31/0.46 => ( A != B ) ) ).
% 0.31/0.46
% 0.31/0.46 % order.strict_implies_not_eq
% 0.31/0.46 thf(fact_172_order_Ostrict__implies__not__eq,axiom,
% 0.31/0.46 ! [A: int,B: int] :
% 0.31/0.46 ( ( ord_less_int @ A @ B )
% 0.31/0.46 => ( A != B ) ) ).
% 0.31/0.46
% 0.31/0.46 % order.strict_implies_not_eq
% 0.31/0.46 thf(fact_173_pos__int__cases,axiom,
% 0.31/0.46 ! [K: int] :
% 0.31/0.46 ( ( ord_less_int @ zero_zero_int @ K )
% 0.31/0.46 => ~ ! [N2: nat] :
% 0.31/0.46 ( ( K
% 0.31/0.46 = ( semiri2019852685at_int @ N2 ) )
% 0.31/0.46 => ~ ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % pos_int_cases
% 0.31/0.46 thf(fact_174_zero__less__imp__eq__int,axiom,
% 0.31/0.46 ! [K: int] :
% 0.31/0.46 ( ( ord_less_int @ zero_zero_int @ K )
% 0.31/0.46 => ? [N2: nat] :
% 0.31/0.46 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.31/0.46 & ( K
% 0.31/0.46 = ( semiri2019852685at_int @ N2 ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % zero_less_imp_eq_int
% 0.31/0.46 thf(fact_175_exists__least__lemma,axiom,
% 0.31/0.46 ! [P: nat > $o] :
% 0.31/0.46 ( ~ ( P @ zero_zero_nat )
% 0.31/0.46 => ( ? [X_12: nat] : ( P @ X_12 )
% 0.31/0.46 => ? [N2: nat] :
% 0.31/0.46 ( ~ ( P @ N2 )
% 0.31/0.46 & ( P @ ( suc @ N2 ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % exists_least_lemma
% 0.31/0.46 thf(fact_176_of__nat__zero__less__power__iff,axiom,
% 0.31/0.46 ! [X: nat,N: nat] :
% 0.31/0.46 ( ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ ( semiri1382578993at_nat @ X ) @ N ) )
% 0.31/0.46 = ( ( ord_less_nat @ zero_zero_nat @ X )
% 0.31/0.46 | ( N = zero_zero_nat ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % of_nat_zero_less_power_iff
% 0.31/0.46 thf(fact_177_of__nat__zero__less__power__iff,axiom,
% 0.31/0.46 ! [X: nat,N: nat] :
% 0.31/0.46 ( ( ord_less_int @ zero_zero_int @ ( power_power_int @ ( semiri2019852685at_int @ X ) @ N ) )
% 0.31/0.46 = ( ( ord_less_nat @ zero_zero_nat @ X )
% 0.31/0.46 | ( N = zero_zero_nat ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % of_nat_zero_less_power_iff
% 0.31/0.46 thf(fact_178_of__nat__le__0__iff,axiom,
% 0.31/0.46 ! [M: nat] :
% 0.31/0.46 ( ( ord_less_eq_nat @ ( semiri1382578993at_nat @ M ) @ zero_zero_nat )
% 0.31/0.46 = ( M = zero_zero_nat ) ) ).
% 0.31/0.46
% 0.31/0.46 % of_nat_le_0_iff
% 0.31/0.46 thf(fact_179_of__nat__le__0__iff,axiom,
% 0.31/0.46 ! [M: nat] :
% 0.31/0.46 ( ( ord_less_eq_int @ ( semiri2019852685at_int @ M ) @ zero_zero_int )
% 0.31/0.46 = ( M = zero_zero_nat ) ) ).
% 0.31/0.46
% 0.31/0.46 % of_nat_le_0_iff
% 0.31/0.46 thf(fact_180_Suc__pred,axiom,
% 0.31/0.46 ! [N: nat] :
% 0.31/0.46 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.31/0.46 => ( ( suc @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) )
% 0.31/0.46 = N ) ) ).
% 0.31/0.46
% 0.31/0.46 % Suc_pred
% 0.31/0.46 thf(fact_181_order__refl,axiom,
% 0.31/0.46 ! [X: nat] : ( ord_less_eq_nat @ X @ X ) ).
% 0.31/0.46
% 0.31/0.46 % order_refl
% 0.31/0.46 thf(fact_182_order__refl,axiom,
% 0.31/0.46 ! [X: int] : ( ord_less_eq_int @ X @ X ) ).
% 0.31/0.46
% 0.31/0.46 % order_refl
% 0.31/0.46 thf(fact_183_bot__nat__0_Oextremum,axiom,
% 0.31/0.46 ! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% 0.31/0.46
% 0.31/0.46 % bot_nat_0.extremum
% 0.31/0.46 thf(fact_184_le0,axiom,
% 0.31/0.46 ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% 0.31/0.46
% 0.31/0.46 % le0
% 0.31/0.46 thf(fact_185_Suc__le__mono,axiom,
% 0.31/0.46 ! [N: nat,M: nat] :
% 0.31/0.46 ( ( ord_less_eq_nat @ ( suc @ N ) @ ( suc @ M ) )
% 0.31/0.46 = ( ord_less_eq_nat @ N @ M ) ) ).
% 0.31/0.46
% 0.31/0.46 % Suc_le_mono
% 0.31/0.46 thf(fact_186_diff__diff__cancel,axiom,
% 0.31/0.46 ! [I: nat,N: nat] :
% 0.31/0.46 ( ( ord_less_eq_nat @ I @ N )
% 0.31/0.46 => ( ( minus_minus_nat @ N @ ( minus_minus_nat @ N @ I ) )
% 0.31/0.46 = I ) ) ).
% 0.31/0.46
% 0.31/0.46 % diff_diff_cancel
% 0.31/0.46 thf(fact_187_le__zero__eq,axiom,
% 0.31/0.46 ! [N: nat] :
% 0.31/0.46 ( ( ord_less_eq_nat @ N @ zero_zero_nat )
% 0.31/0.46 = ( N = zero_zero_nat ) ) ).
% 0.31/0.46
% 0.31/0.46 % le_zero_eq
% 0.31/0.46 thf(fact_188_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
% 0.31/0.46 ! [A: int] :
% 0.31/0.46 ( ( minus_minus_int @ A @ A )
% 0.31/0.46 = zero_zero_int ) ).
% 0.31/0.46
% 0.31/0.46 % cancel_comm_monoid_add_class.diff_cancel
% 0.31/0.46 thf(fact_189_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
% 0.31/0.46 ! [A: nat] :
% 0.31/0.46 ( ( minus_minus_nat @ A @ A )
% 0.31/0.46 = zero_zero_nat ) ).
% 0.31/0.46
% 0.31/0.46 % cancel_comm_monoid_add_class.diff_cancel
% 0.31/0.46 thf(fact_190_diff__zero,axiom,
% 0.31/0.46 ! [A: int] :
% 0.31/0.46 ( ( minus_minus_int @ A @ zero_zero_int )
% 0.31/0.46 = A ) ).
% 0.31/0.46
% 0.31/0.46 % diff_zero
% 0.31/0.46 thf(fact_191_diff__zero,axiom,
% 0.31/0.46 ! [A: nat] :
% 0.31/0.46 ( ( minus_minus_nat @ A @ zero_zero_nat )
% 0.31/0.46 = A ) ).
% 0.31/0.46
% 0.31/0.46 % diff_zero
% 0.31/0.46 thf(fact_192_zero__diff,axiom,
% 0.31/0.46 ! [A: nat] :
% 0.31/0.46 ( ( minus_minus_nat @ zero_zero_nat @ A )
% 0.31/0.46 = zero_zero_nat ) ).
% 0.31/0.46
% 0.31/0.46 % zero_diff
% 0.31/0.46 thf(fact_193_diff__0__right,axiom,
% 0.31/0.46 ! [A: int] :
% 0.31/0.46 ( ( minus_minus_int @ A @ zero_zero_int )
% 0.31/0.46 = A ) ).
% 0.31/0.46
% 0.31/0.46 % diff_0_right
% 0.31/0.46 thf(fact_194_diff__self,axiom,
% 0.31/0.46 ! [A: int] :
% 0.31/0.46 ( ( minus_minus_int @ A @ A )
% 0.31/0.46 = zero_zero_int ) ).
% 0.31/0.46
% 0.31/0.46 % diff_self
% 0.31/0.46 thf(fact_195_of__nat__le__iff,axiom,
% 0.31/0.46 ! [M: nat,N: nat] :
% 0.31/0.46 ( ( ord_less_eq_nat @ ( semiri1382578993at_nat @ M ) @ ( semiri1382578993at_nat @ N ) )
% 0.31/0.46 = ( ord_less_eq_nat @ M @ N ) ) ).
% 0.31/0.46
% 0.31/0.46 % of_nat_le_iff
% 0.31/0.46 thf(fact_196_of__nat__le__iff,axiom,
% 0.31/0.46 ! [M: nat,N: nat] :
% 0.31/0.46 ( ( ord_less_eq_int @ ( semiri2019852685at_int @ M ) @ ( semiri2019852685at_int @ N ) )
% 0.31/0.46 = ( ord_less_eq_nat @ M @ N ) ) ).
% 0.31/0.46
% 0.31/0.46 % of_nat_le_iff
% 0.31/0.46 thf(fact_197_nat__power__eq__Suc__0__iff,axiom,
% 0.31/0.46 ! [X: nat,M: nat] :
% 0.31/0.46 ( ( ( power_power_nat @ X @ M )
% 0.31/0.46 = ( suc @ zero_zero_nat ) )
% 0.31/0.46 = ( ( M = zero_zero_nat )
% 0.31/0.46 | ( X
% 0.31/0.46 = ( suc @ zero_zero_nat ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % nat_power_eq_Suc_0_iff
% 0.31/0.46 thf(fact_198_power__Suc__0,axiom,
% 0.31/0.46 ! [N: nat] :
% 0.31/0.46 ( ( power_power_nat @ ( suc @ zero_zero_nat ) @ N )
% 0.31/0.46 = ( suc @ zero_zero_nat ) ) ).
% 0.31/0.46
% 0.31/0.46 % power_Suc_0
% 0.31/0.46 thf(fact_199_of__nat__power__eq__of__nat__cancel__iff,axiom,
% 0.31/0.46 ! [X: nat,B: nat,W: nat] :
% 0.31/0.46 ( ( ( semiri1382578993at_nat @ X )
% 0.31/0.46 = ( power_power_nat @ ( semiri1382578993at_nat @ B ) @ W ) )
% 0.31/0.46 = ( X
% 0.31/0.46 = ( power_power_nat @ B @ W ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % of_nat_power_eq_of_nat_cancel_iff
% 0.31/0.46 thf(fact_200_of__nat__power__eq__of__nat__cancel__iff,axiom,
% 0.31/0.46 ! [X: nat,B: nat,W: nat] :
% 0.31/0.46 ( ( ( semiri2019852685at_int @ X )
% 0.31/0.46 = ( power_power_int @ ( semiri2019852685at_int @ B ) @ W ) )
% 0.31/0.46 = ( X
% 0.31/0.46 = ( power_power_nat @ B @ W ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % of_nat_power_eq_of_nat_cancel_iff
% 0.31/0.46 thf(fact_201_of__nat__eq__of__nat__power__cancel__iff,axiom,
% 0.31/0.46 ! [B: nat,W: nat,X: nat] :
% 0.31/0.46 ( ( ( power_power_nat @ ( semiri1382578993at_nat @ B ) @ W )
% 0.31/0.46 = ( semiri1382578993at_nat @ X ) )
% 0.31/0.46 = ( ( power_power_nat @ B @ W )
% 0.31/0.46 = X ) ) ).
% 0.31/0.46
% 0.31/0.46 % of_nat_eq_of_nat_power_cancel_iff
% 0.31/0.46 thf(fact_202_of__nat__eq__of__nat__power__cancel__iff,axiom,
% 0.31/0.46 ! [B: nat,W: nat,X: nat] :
% 0.31/0.46 ( ( ( power_power_int @ ( semiri2019852685at_int @ B ) @ W )
% 0.31/0.46 = ( semiri2019852685at_int @ X ) )
% 0.31/0.46 = ( ( power_power_nat @ B @ W )
% 0.31/0.46 = X ) ) ).
% 0.31/0.46
% 0.31/0.46 % of_nat_eq_of_nat_power_cancel_iff
% 0.31/0.46 thf(fact_203_of__nat__power,axiom,
% 0.31/0.46 ! [M: nat,N: nat] :
% 0.31/0.46 ( ( semiri1382578993at_nat @ ( power_power_nat @ M @ N ) )
% 0.31/0.46 = ( power_power_nat @ ( semiri1382578993at_nat @ M ) @ N ) ) ).
% 0.31/0.46
% 0.31/0.46 % of_nat_power
% 0.31/0.46 thf(fact_204_of__nat__power,axiom,
% 0.31/0.46 ! [M: nat,N: nat] :
% 0.31/0.46 ( ( semiri2019852685at_int @ ( power_power_nat @ M @ N ) )
% 0.31/0.46 = ( power_power_int @ ( semiri2019852685at_int @ M ) @ N ) ) ).
% 0.31/0.46
% 0.31/0.46 % of_nat_power
% 0.31/0.46 thf(fact_205_nat__zero__less__power__iff,axiom,
% 0.31/0.46 ! [X: nat,N: nat] :
% 0.31/0.46 ( ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ X @ N ) )
% 0.31/0.46 = ( ( ord_less_nat @ zero_zero_nat @ X )
% 0.31/0.46 | ( N = zero_zero_nat ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % nat_zero_less_power_iff
% 0.31/0.46 thf(fact_206_diff__is__0__eq_H,axiom,
% 0.31/0.46 ! [M: nat,N: nat] :
% 0.31/0.46 ( ( ord_less_eq_nat @ M @ N )
% 0.31/0.46 => ( ( minus_minus_nat @ M @ N )
% 0.31/0.46 = zero_zero_nat ) ) ).
% 0.31/0.46
% 0.31/0.46 % diff_is_0_eq'
% 0.31/0.46 thf(fact_207_diff__is__0__eq,axiom,
% 0.31/0.46 ! [M: nat,N: nat] :
% 0.31/0.46 ( ( ( minus_minus_nat @ M @ N )
% 0.31/0.46 = zero_zero_nat )
% 0.31/0.46 = ( ord_less_eq_nat @ M @ N ) ) ).
% 0.31/0.46
% 0.31/0.46 % diff_is_0_eq
% 0.31/0.46 thf(fact_208_diff__self__eq__0,axiom,
% 0.31/0.46 ! [M: nat] :
% 0.31/0.46 ( ( minus_minus_nat @ M @ M )
% 0.31/0.46 = zero_zero_nat ) ).
% 0.31/0.46
% 0.31/0.46 % diff_self_eq_0
% 0.31/0.46 thf(fact_209_diff__0__eq__0,axiom,
% 0.31/0.46 ! [N: nat] :
% 0.31/0.46 ( ( minus_minus_nat @ zero_zero_nat @ N )
% 0.31/0.46 = zero_zero_nat ) ).
% 0.31/0.46
% 0.31/0.46 % diff_0_eq_0
% 0.31/0.46 thf(fact_210_Suc__diff__diff,axiom,
% 0.31/0.46 ! [M: nat,N: nat,K: nat] :
% 0.31/0.46 ( ( minus_minus_nat @ ( minus_minus_nat @ ( suc @ M ) @ N ) @ ( suc @ K ) )
% 0.31/0.46 = ( minus_minus_nat @ ( minus_minus_nat @ M @ N ) @ K ) ) ).
% 0.31/0.46
% 0.31/0.46 % Suc_diff_diff
% 0.31/0.46 thf(fact_211_diff__Suc__Suc,axiom,
% 0.31/0.46 ! [M: nat,N: nat] :
% 0.31/0.46 ( ( minus_minus_nat @ ( suc @ M ) @ ( suc @ N ) )
% 0.31/0.46 = ( minus_minus_nat @ M @ N ) ) ).
% 0.31/0.46
% 0.31/0.46 % diff_Suc_Suc
% 0.31/0.46 thf(fact_212_diff__ge__0__iff__ge,axiom,
% 0.31/0.46 ! [A: int,B: int] :
% 0.31/0.46 ( ( ord_less_eq_int @ zero_zero_int @ ( minus_minus_int @ A @ B ) )
% 0.31/0.46 = ( ord_less_eq_int @ B @ A ) ) ).
% 0.31/0.46
% 0.31/0.46 % diff_ge_0_iff_ge
% 0.31/0.46 thf(fact_213_diff__gt__0__iff__gt,axiom,
% 0.31/0.46 ! [A: int,B: int] :
% 0.31/0.46 ( ( ord_less_int @ zero_zero_int @ ( minus_minus_int @ A @ B ) )
% 0.31/0.46 = ( ord_less_int @ B @ A ) ) ).
% 0.31/0.46
% 0.31/0.46 % diff_gt_0_iff_gt
% 0.31/0.46 thf(fact_214_power__0__Suc,axiom,
% 0.31/0.46 ! [N: nat] :
% 0.31/0.46 ( ( power_power_int @ zero_zero_int @ ( suc @ N ) )
% 0.31/0.46 = zero_zero_int ) ).
% 0.31/0.46
% 0.31/0.46 % power_0_Suc
% 0.31/0.46 thf(fact_215_power__0__Suc,axiom,
% 0.31/0.46 ! [N: nat] :
% 0.31/0.46 ( ( power_power_nat @ zero_zero_nat @ ( suc @ N ) )
% 0.31/0.46 = zero_zero_nat ) ).
% 0.31/0.46
% 0.31/0.46 % power_0_Suc
% 0.31/0.46 thf(fact_216_of__nat__power__le__of__nat__cancel__iff,axiom,
% 0.31/0.46 ! [X: nat,B: nat,W: nat] :
% 0.31/0.46 ( ( ord_less_eq_nat @ ( semiri1382578993at_nat @ X ) @ ( power_power_nat @ ( semiri1382578993at_nat @ B ) @ W ) )
% 0.31/0.46 = ( ord_less_eq_nat @ X @ ( power_power_nat @ B @ W ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % of_nat_power_le_of_nat_cancel_iff
% 0.31/0.46 thf(fact_217_of__nat__power__le__of__nat__cancel__iff,axiom,
% 0.31/0.46 ! [X: nat,B: nat,W: nat] :
% 0.31/0.46 ( ( ord_less_eq_int @ ( semiri2019852685at_int @ X ) @ ( power_power_int @ ( semiri2019852685at_int @ B ) @ W ) )
% 0.31/0.46 = ( ord_less_eq_nat @ X @ ( power_power_nat @ B @ W ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % of_nat_power_le_of_nat_cancel_iff
% 0.31/0.46 thf(fact_218_of__nat__le__of__nat__power__cancel__iff,axiom,
% 0.31/0.46 ! [B: nat,W: nat,X: nat] :
% 0.31/0.46 ( ( ord_less_eq_nat @ ( power_power_nat @ ( semiri1382578993at_nat @ B ) @ W ) @ ( semiri1382578993at_nat @ X ) )
% 0.31/0.46 = ( ord_less_eq_nat @ ( power_power_nat @ B @ W ) @ X ) ) ).
% 0.31/0.46
% 0.31/0.46 % of_nat_le_of_nat_power_cancel_iff
% 0.31/0.46 thf(fact_219_of__nat__le__of__nat__power__cancel__iff,axiom,
% 0.31/0.46 ! [B: nat,W: nat,X: nat] :
% 0.31/0.46 ( ( ord_less_eq_int @ ( power_power_int @ ( semiri2019852685at_int @ B ) @ W ) @ ( semiri2019852685at_int @ X ) )
% 0.31/0.46 = ( ord_less_eq_nat @ ( power_power_nat @ B @ W ) @ X ) ) ).
% 0.31/0.46
% 0.31/0.46 % of_nat_le_of_nat_power_cancel_iff
% 0.31/0.46 thf(fact_220_power__Suc0__right,axiom,
% 0.31/0.46 ! [A: nat] :
% 0.31/0.46 ( ( power_power_nat @ A @ ( suc @ zero_zero_nat ) )
% 0.31/0.46 = A ) ).
% 0.31/0.46
% 0.31/0.46 % power_Suc0_right
% 0.31/0.46 thf(fact_221_zero__less__diff,axiom,
% 0.31/0.46 ! [N: nat,M: nat] :
% 0.31/0.46 ( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N @ M ) )
% 0.31/0.46 = ( ord_less_nat @ M @ N ) ) ).
% 0.31/0.46
% 0.31/0.46 % zero_less_diff
% 0.31/0.46 thf(fact_222_power__eq__0__iff,axiom,
% 0.31/0.46 ! [A: int,N: nat] :
% 0.31/0.46 ( ( ( power_power_int @ A @ N )
% 0.31/0.46 = zero_zero_int )
% 0.31/0.46 = ( ( A = zero_zero_int )
% 0.31/0.46 & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % power_eq_0_iff
% 0.31/0.46 thf(fact_223_power__eq__0__iff,axiom,
% 0.31/0.46 ! [A: nat,N: nat] :
% 0.31/0.46 ( ( ( power_power_nat @ A @ N )
% 0.31/0.46 = zero_zero_nat )
% 0.31/0.46 = ( ( A = zero_zero_nat )
% 0.31/0.46 & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % power_eq_0_iff
% 0.31/0.46 thf(fact_224_of__nat__less__of__nat__power__cancel__iff,axiom,
% 0.31/0.46 ! [B: nat,W: nat,X: nat] :
% 0.31/0.46 ( ( ord_less_nat @ ( power_power_nat @ ( semiri1382578993at_nat @ B ) @ W ) @ ( semiri1382578993at_nat @ X ) )
% 0.31/0.46 = ( ord_less_nat @ ( power_power_nat @ B @ W ) @ X ) ) ).
% 0.31/0.46
% 0.31/0.46 % of_nat_less_of_nat_power_cancel_iff
% 0.31/0.46 thf(fact_225_of__nat__less__of__nat__power__cancel__iff,axiom,
% 0.31/0.46 ! [B: nat,W: nat,X: nat] :
% 0.31/0.46 ( ( ord_less_int @ ( power_power_int @ ( semiri2019852685at_int @ B ) @ W ) @ ( semiri2019852685at_int @ X ) )
% 0.31/0.46 = ( ord_less_nat @ ( power_power_nat @ B @ W ) @ X ) ) ).
% 0.31/0.46
% 0.31/0.46 % of_nat_less_of_nat_power_cancel_iff
% 0.31/0.46 thf(fact_226_of__nat__power__less__of__nat__cancel__iff,axiom,
% 0.31/0.46 ! [X: nat,B: nat,W: nat] :
% 0.31/0.46 ( ( ord_less_nat @ ( semiri1382578993at_nat @ X ) @ ( power_power_nat @ ( semiri1382578993at_nat @ B ) @ W ) )
% 0.31/0.46 = ( ord_less_nat @ X @ ( power_power_nat @ B @ W ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % of_nat_power_less_of_nat_cancel_iff
% 0.31/0.46 thf(fact_227_of__nat__power__less__of__nat__cancel__iff,axiom,
% 0.31/0.46 ! [X: nat,B: nat,W: nat] :
% 0.31/0.46 ( ( ord_less_int @ ( semiri2019852685at_int @ X ) @ ( power_power_int @ ( semiri2019852685at_int @ B ) @ W ) )
% 0.31/0.46 = ( ord_less_nat @ X @ ( power_power_nat @ B @ W ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % of_nat_power_less_of_nat_cancel_iff
% 0.31/0.46 thf(fact_228_power__mono__iff,axiom,
% 0.31/0.46 ! [A: nat,B: nat,N: nat] :
% 0.31/0.46 ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.31/0.46 => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 0.31/0.46 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.31/0.46 => ( ( ord_less_eq_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) )
% 0.31/0.46 = ( ord_less_eq_nat @ A @ B ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % power_mono_iff
% 0.31/0.46 thf(fact_229_power__mono__iff,axiom,
% 0.31/0.46 ! [A: int,B: int,N: nat] :
% 0.31/0.46 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.31/0.46 => ( ( ord_less_eq_int @ zero_zero_int @ B )
% 0.31/0.46 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.31/0.46 => ( ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) )
% 0.31/0.46 = ( ord_less_eq_int @ A @ B ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % power_mono_iff
% 0.31/0.46 thf(fact_230_Suc__diff__le,axiom,
% 0.31/0.46 ! [N: nat,M: nat] :
% 0.31/0.46 ( ( ord_less_eq_nat @ N @ M )
% 0.31/0.46 => ( ( minus_minus_nat @ ( suc @ M ) @ N )
% 0.31/0.46 = ( suc @ ( minus_minus_nat @ M @ N ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % Suc_diff_le
% 0.31/0.46 thf(fact_231_nonneg__int__cases,axiom,
% 0.31/0.46 ! [K: int] :
% 0.31/0.46 ( ( ord_less_eq_int @ zero_zero_int @ K )
% 0.31/0.46 => ~ ! [N2: nat] :
% 0.31/0.46 ( K
% 0.31/0.46 != ( semiri2019852685at_int @ N2 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % nonneg_int_cases
% 0.31/0.46 thf(fact_232_zero__le__imp__eq__int,axiom,
% 0.31/0.46 ! [K: int] :
% 0.31/0.46 ( ( ord_less_eq_int @ zero_zero_int @ K )
% 0.31/0.46 => ? [N2: nat] :
% 0.31/0.46 ( K
% 0.31/0.46 = ( semiri2019852685at_int @ N2 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % zero_le_imp_eq_int
% 0.31/0.46 thf(fact_233_zero__integer_Orsp,axiom,
% 0.31/0.46 zero_zero_int = zero_zero_int ).
% 0.31/0.46
% 0.31/0.46 % zero_integer.rsp
% 0.31/0.46 thf(fact_234_lift__Suc__mono__le,axiom,
% 0.31/0.46 ! [F: nat > nat,N: nat,N3: nat] :
% 0.31/0.46 ( ! [N2: nat] : ( ord_less_eq_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
% 0.31/0.46 => ( ( ord_less_eq_nat @ N @ N3 )
% 0.31/0.46 => ( ord_less_eq_nat @ ( F @ N ) @ ( F @ N3 ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % lift_Suc_mono_le
% 0.31/0.46 thf(fact_235_lift__Suc__mono__le,axiom,
% 0.31/0.46 ! [F: nat > int,N: nat,N3: nat] :
% 0.31/0.46 ( ! [N2: nat] : ( ord_less_eq_int @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
% 0.31/0.46 => ( ( ord_less_eq_nat @ N @ N3 )
% 0.31/0.46 => ( ord_less_eq_int @ ( F @ N ) @ ( F @ N3 ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % lift_Suc_mono_le
% 0.31/0.46 thf(fact_236_lift__Suc__antimono__le,axiom,
% 0.31/0.46 ! [F: nat > nat,N: nat,N3: nat] :
% 0.31/0.46 ( ! [N2: nat] : ( ord_less_eq_nat @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
% 0.31/0.46 => ( ( ord_less_eq_nat @ N @ N3 )
% 0.31/0.46 => ( ord_less_eq_nat @ ( F @ N3 ) @ ( F @ N ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % lift_Suc_antimono_le
% 0.31/0.46 thf(fact_237_lift__Suc__antimono__le,axiom,
% 0.31/0.46 ! [F: nat > int,N: nat,N3: nat] :
% 0.31/0.46 ( ! [N2: nat] : ( ord_less_eq_int @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
% 0.31/0.46 => ( ( ord_less_eq_nat @ N @ N3 )
% 0.31/0.46 => ( ord_less_eq_int @ ( F @ N3 ) @ ( F @ N ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % lift_Suc_antimono_le
% 0.31/0.46 thf(fact_238_of__nat__diff,axiom,
% 0.31/0.46 ! [N: nat,M: nat] :
% 0.31/0.46 ( ( ord_less_eq_nat @ N @ M )
% 0.31/0.46 => ( ( semiri1382578993at_nat @ ( minus_minus_nat @ M @ N ) )
% 0.31/0.46 = ( minus_minus_nat @ ( semiri1382578993at_nat @ M ) @ ( semiri1382578993at_nat @ N ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % of_nat_diff
% 0.31/0.46 thf(fact_239_of__nat__diff,axiom,
% 0.31/0.46 ! [N: nat,M: nat] :
% 0.31/0.46 ( ( ord_less_eq_nat @ N @ M )
% 0.31/0.46 => ( ( semiri2019852685at_int @ ( minus_minus_nat @ M @ N ) )
% 0.31/0.46 = ( minus_minus_int @ ( semiri2019852685at_int @ M ) @ ( semiri2019852685at_int @ N ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % of_nat_diff
% 0.31/0.46 thf(fact_240_power__inject__base,axiom,
% 0.31/0.46 ! [A: nat,N: nat,B: nat] :
% 0.31/0.46 ( ( ( power_power_nat @ A @ ( suc @ N ) )
% 0.31/0.46 = ( power_power_nat @ B @ ( suc @ N ) ) )
% 0.31/0.46 => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.31/0.46 => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 0.31/0.46 => ( A = B ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % power_inject_base
% 0.31/0.46 thf(fact_241_power__inject__base,axiom,
% 0.31/0.46 ! [A: int,N: nat,B: int] :
% 0.31/0.46 ( ( ( power_power_int @ A @ ( suc @ N ) )
% 0.31/0.46 = ( power_power_int @ B @ ( suc @ N ) ) )
% 0.31/0.46 => ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.31/0.46 => ( ( ord_less_eq_int @ zero_zero_int @ B )
% 0.31/0.46 => ( A = B ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % power_inject_base
% 0.31/0.46 thf(fact_242_of__nat__mono,axiom,
% 0.31/0.46 ! [I: nat,J2: nat] :
% 0.31/0.46 ( ( ord_less_eq_nat @ I @ J2 )
% 0.31/0.46 => ( ord_less_eq_nat @ ( semiri1382578993at_nat @ I ) @ ( semiri1382578993at_nat @ J2 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % of_nat_mono
% 0.31/0.46 thf(fact_243_of__nat__mono,axiom,
% 0.31/0.46 ! [I: nat,J2: nat] :
% 0.31/0.46 ( ( ord_less_eq_nat @ I @ J2 )
% 0.31/0.46 => ( ord_less_eq_int @ ( semiri2019852685at_int @ I ) @ ( semiri2019852685at_int @ J2 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % of_nat_mono
% 0.31/0.46 thf(fact_244_power__le__imp__le__base,axiom,
% 0.31/0.46 ! [A: nat,N: nat,B: nat] :
% 0.31/0.46 ( ( ord_less_eq_nat @ ( power_power_nat @ A @ ( suc @ N ) ) @ ( power_power_nat @ B @ ( suc @ N ) ) )
% 0.31/0.46 => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 0.31/0.46 => ( ord_less_eq_nat @ A @ B ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % power_le_imp_le_base
% 0.31/0.46 thf(fact_245_power__le__imp__le__base,axiom,
% 0.31/0.46 ! [A: int,N: nat,B: int] :
% 0.31/0.46 ( ( ord_less_eq_int @ ( power_power_int @ A @ ( suc @ N ) ) @ ( power_power_int @ B @ ( suc @ N ) ) )
% 0.31/0.46 => ( ( ord_less_eq_int @ zero_zero_int @ B )
% 0.31/0.46 => ( ord_less_eq_int @ A @ B ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % power_le_imp_le_base
% 0.31/0.46 thf(fact_246_dual__order_Oantisym,axiom,
% 0.31/0.46 ! [B: nat,A: nat] :
% 0.31/0.46 ( ( ord_less_eq_nat @ B @ A )
% 0.31/0.46 => ( ( ord_less_eq_nat @ A @ B )
% 0.31/0.46 => ( A = B ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % dual_order.antisym
% 0.31/0.46 thf(fact_247_dual__order_Oantisym,axiom,
% 0.31/0.46 ! [B: int,A: int] :
% 0.31/0.46 ( ( ord_less_eq_int @ B @ A )
% 0.31/0.46 => ( ( ord_less_eq_int @ A @ B )
% 0.31/0.46 => ( A = B ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % dual_order.antisym
% 0.31/0.46 thf(fact_248_dual__order_Oeq__iff,axiom,
% 0.31/0.46 ( ( ^ [Y5: nat,Z2: nat] : ( Y5 = Z2 ) )
% 0.31/0.46 = ( ^ [A3: nat,B3: nat] :
% 0.31/0.46 ( ( ord_less_eq_nat @ B3 @ A3 )
% 0.31/0.46 & ( ord_less_eq_nat @ A3 @ B3 ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % dual_order.eq_iff
% 0.31/0.46 thf(fact_249_dual__order_Oeq__iff,axiom,
% 0.31/0.46 ( ( ^ [Y5: int,Z2: int] : ( Y5 = Z2 ) )
% 0.31/0.46 = ( ^ [A3: int,B3: int] :
% 0.31/0.46 ( ( ord_less_eq_int @ B3 @ A3 )
% 0.31/0.46 & ( ord_less_eq_int @ A3 @ B3 ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % dual_order.eq_iff
% 0.31/0.46 thf(fact_250_dual__order_Otrans,axiom,
% 0.31/0.46 ! [B: nat,A: nat,C: nat] :
% 0.31/0.46 ( ( ord_less_eq_nat @ B @ A )
% 0.31/0.46 => ( ( ord_less_eq_nat @ C @ B )
% 0.31/0.46 => ( ord_less_eq_nat @ C @ A ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % dual_order.trans
% 0.31/0.46 thf(fact_251_dual__order_Otrans,axiom,
% 0.31/0.46 ! [B: int,A: int,C: int] :
% 0.31/0.46 ( ( ord_less_eq_int @ B @ A )
% 0.31/0.46 => ( ( ord_less_eq_int @ C @ B )
% 0.31/0.46 => ( ord_less_eq_int @ C @ A ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % dual_order.trans
% 0.31/0.46 thf(fact_252_linorder__wlog,axiom,
% 0.31/0.46 ! [P: nat > nat > $o,A: nat,B: nat] :
% 0.31/0.46 ( ! [A2: nat,B2: nat] :
% 0.31/0.46 ( ( ord_less_eq_nat @ A2 @ B2 )
% 0.31/0.46 => ( P @ A2 @ B2 ) )
% 0.31/0.46 => ( ! [A2: nat,B2: nat] :
% 0.31/0.46 ( ( P @ B2 @ A2 )
% 0.31/0.46 => ( P @ A2 @ B2 ) )
% 0.31/0.46 => ( P @ A @ B ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % linorder_wlog
% 0.31/0.46 thf(fact_253_linorder__wlog,axiom,
% 0.31/0.46 ! [P: int > int > $o,A: int,B: int] :
% 0.31/0.46 ( ! [A2: int,B2: int] :
% 0.31/0.46 ( ( ord_less_eq_int @ A2 @ B2 )
% 0.31/0.46 => ( P @ A2 @ B2 ) )
% 0.31/0.46 => ( ! [A2: int,B2: int] :
% 0.31/0.46 ( ( P @ B2 @ A2 )
% 0.31/0.46 => ( P @ A2 @ B2 ) )
% 0.31/0.46 => ( P @ A @ B ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % linorder_wlog
% 0.31/0.46 thf(fact_254_dual__order_Orefl,axiom,
% 0.31/0.46 ! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% 0.31/0.46
% 0.31/0.46 % dual_order.refl
% 0.31/0.46 thf(fact_255_dual__order_Orefl,axiom,
% 0.31/0.46 ! [A: int] : ( ord_less_eq_int @ A @ A ) ).
% 0.31/0.46
% 0.31/0.46 % dual_order.refl
% 0.31/0.46 thf(fact_256_order__trans,axiom,
% 0.31/0.46 ! [X: nat,Y: nat,Z: nat] :
% 0.31/0.46 ( ( ord_less_eq_nat @ X @ Y )
% 0.31/0.46 => ( ( ord_less_eq_nat @ Y @ Z )
% 0.31/0.46 => ( ord_less_eq_nat @ X @ Z ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % order_trans
% 0.31/0.46 thf(fact_257_order__trans,axiom,
% 0.31/0.46 ! [X: int,Y: int,Z: int] :
% 0.31/0.46 ( ( ord_less_eq_int @ X @ Y )
% 0.31/0.46 => ( ( ord_less_eq_int @ Y @ Z )
% 0.31/0.46 => ( ord_less_eq_int @ X @ Z ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % order_trans
% 0.31/0.46 thf(fact_258_order__class_Oorder_Oantisym,axiom,
% 0.31/0.46 ! [A: nat,B: nat] :
% 0.31/0.46 ( ( ord_less_eq_nat @ A @ B )
% 0.31/0.46 => ( ( ord_less_eq_nat @ B @ A )
% 0.31/0.46 => ( A = B ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % order_class.order.antisym
% 0.31/0.46 thf(fact_259_order__class_Oorder_Oantisym,axiom,
% 0.31/0.46 ! [A: int,B: int] :
% 0.31/0.46 ( ( ord_less_eq_int @ A @ B )
% 0.31/0.46 => ( ( ord_less_eq_int @ B @ A )
% 0.31/0.46 => ( A = B ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % order_class.order.antisym
% 0.31/0.46 thf(fact_260_ord__le__eq__trans,axiom,
% 0.31/0.46 ! [A: nat,B: nat,C: nat] :
% 0.31/0.46 ( ( ord_less_eq_nat @ A @ B )
% 0.31/0.46 => ( ( B = C )
% 0.31/0.46 => ( ord_less_eq_nat @ A @ C ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % ord_le_eq_trans
% 0.31/0.46 thf(fact_261_ord__le__eq__trans,axiom,
% 0.31/0.46 ! [A: int,B: int,C: int] :
% 0.31/0.46 ( ( ord_less_eq_int @ A @ B )
% 0.31/0.46 => ( ( B = C )
% 0.31/0.46 => ( ord_less_eq_int @ A @ C ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % ord_le_eq_trans
% 0.31/0.46 thf(fact_262_ord__eq__le__trans,axiom,
% 0.31/0.46 ! [A: nat,B: nat,C: nat] :
% 0.31/0.46 ( ( A = B )
% 0.31/0.46 => ( ( ord_less_eq_nat @ B @ C )
% 0.31/0.46 => ( ord_less_eq_nat @ A @ C ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % ord_eq_le_trans
% 0.31/0.46 thf(fact_263_ord__eq__le__trans,axiom,
% 0.31/0.46 ! [A: int,B: int,C: int] :
% 0.31/0.46 ( ( A = B )
% 0.31/0.46 => ( ( ord_less_eq_int @ B @ C )
% 0.31/0.46 => ( ord_less_eq_int @ A @ C ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % ord_eq_le_trans
% 0.31/0.46 thf(fact_264_order__class_Oorder_Oeq__iff,axiom,
% 0.31/0.46 ( ( ^ [Y5: nat,Z2: nat] : ( Y5 = Z2 ) )
% 0.31/0.46 = ( ^ [A3: nat,B3: nat] :
% 0.31/0.46 ( ( ord_less_eq_nat @ A3 @ B3 )
% 0.31/0.46 & ( ord_less_eq_nat @ B3 @ A3 ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % order_class.order.eq_iff
% 0.31/0.46 thf(fact_265_order__class_Oorder_Oeq__iff,axiom,
% 0.31/0.46 ( ( ^ [Y5: int,Z2: int] : ( Y5 = Z2 ) )
% 0.31/0.46 = ( ^ [A3: int,B3: int] :
% 0.31/0.46 ( ( ord_less_eq_int @ A3 @ B3 )
% 0.31/0.46 & ( ord_less_eq_int @ B3 @ A3 ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % order_class.order.eq_iff
% 0.31/0.46 thf(fact_266_antisym__conv,axiom,
% 0.31/0.46 ! [Y: nat,X: nat] :
% 0.31/0.46 ( ( ord_less_eq_nat @ Y @ X )
% 0.31/0.46 => ( ( ord_less_eq_nat @ X @ Y )
% 0.31/0.46 = ( X = Y ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % antisym_conv
% 0.31/0.46 thf(fact_267_antisym__conv,axiom,
% 0.31/0.46 ! [Y: int,X: int] :
% 0.31/0.46 ( ( ord_less_eq_int @ Y @ X )
% 0.31/0.46 => ( ( ord_less_eq_int @ X @ Y )
% 0.31/0.46 = ( X = Y ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % antisym_conv
% 0.31/0.46 thf(fact_268_le__cases3,axiom,
% 0.31/0.46 ! [X: nat,Y: nat,Z: nat] :
% 0.31/0.46 ( ( ( ord_less_eq_nat @ X @ Y )
% 0.31/0.46 => ~ ( ord_less_eq_nat @ Y @ Z ) )
% 0.31/0.46 => ( ( ( ord_less_eq_nat @ Y @ X )
% 0.31/0.46 => ~ ( ord_less_eq_nat @ X @ Z ) )
% 0.31/0.46 => ( ( ( ord_less_eq_nat @ X @ Z )
% 0.31/0.46 => ~ ( ord_less_eq_nat @ Z @ Y ) )
% 0.31/0.46 => ( ( ( ord_less_eq_nat @ Z @ Y )
% 0.31/0.46 => ~ ( ord_less_eq_nat @ Y @ X ) )
% 0.31/0.46 => ( ( ( ord_less_eq_nat @ Y @ Z )
% 0.31/0.46 => ~ ( ord_less_eq_nat @ Z @ X ) )
% 0.31/0.46 => ~ ( ( ord_less_eq_nat @ Z @ X )
% 0.31/0.46 => ~ ( ord_less_eq_nat @ X @ Y ) ) ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % le_cases3
% 0.31/0.46 thf(fact_269_le__cases3,axiom,
% 0.31/0.46 ! [X: int,Y: int,Z: int] :
% 0.31/0.46 ( ( ( ord_less_eq_int @ X @ Y )
% 0.31/0.46 => ~ ( ord_less_eq_int @ Y @ Z ) )
% 0.31/0.46 => ( ( ( ord_less_eq_int @ Y @ X )
% 0.31/0.46 => ~ ( ord_less_eq_int @ X @ Z ) )
% 0.31/0.46 => ( ( ( ord_less_eq_int @ X @ Z )
% 0.31/0.46 => ~ ( ord_less_eq_int @ Z @ Y ) )
% 0.31/0.46 => ( ( ( ord_less_eq_int @ Z @ Y )
% 0.31/0.46 => ~ ( ord_less_eq_int @ Y @ X ) )
% 0.31/0.46 => ( ( ( ord_less_eq_int @ Y @ Z )
% 0.31/0.46 => ~ ( ord_less_eq_int @ Z @ X ) )
% 0.31/0.46 => ~ ( ( ord_less_eq_int @ Z @ X )
% 0.31/0.46 => ~ ( ord_less_eq_int @ X @ Y ) ) ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % le_cases3
% 0.31/0.46 thf(fact_270_order_Otrans,axiom,
% 0.31/0.46 ! [A: nat,B: nat,C: nat] :
% 0.31/0.46 ( ( ord_less_eq_nat @ A @ B )
% 0.31/0.46 => ( ( ord_less_eq_nat @ B @ C )
% 0.31/0.46 => ( ord_less_eq_nat @ A @ C ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % order.trans
% 0.31/0.46 thf(fact_271_order_Otrans,axiom,
% 0.31/0.46 ! [A: int,B: int,C: int] :
% 0.31/0.46 ( ( ord_less_eq_int @ A @ B )
% 0.31/0.46 => ( ( ord_less_eq_int @ B @ C )
% 0.31/0.46 => ( ord_less_eq_int @ A @ C ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % order.trans
% 0.31/0.46 thf(fact_272_le__cases,axiom,
% 0.31/0.46 ! [X: nat,Y: nat] :
% 0.31/0.46 ( ~ ( ord_less_eq_nat @ X @ Y )
% 0.31/0.46 => ( ord_less_eq_nat @ Y @ X ) ) ).
% 0.31/0.46
% 0.31/0.46 % le_cases
% 0.31/0.46 thf(fact_273_le__cases,axiom,
% 0.31/0.46 ! [X: int,Y: int] :
% 0.31/0.46 ( ~ ( ord_less_eq_int @ X @ Y )
% 0.31/0.46 => ( ord_less_eq_int @ Y @ X ) ) ).
% 0.31/0.46
% 0.31/0.46 % le_cases
% 0.31/0.46 thf(fact_274_eq__refl,axiom,
% 0.31/0.46 ! [X: nat,Y: nat] :
% 0.31/0.46 ( ( X = Y )
% 0.31/0.46 => ( ord_less_eq_nat @ X @ Y ) ) ).
% 0.31/0.46
% 0.31/0.46 % eq_refl
% 0.31/0.46 thf(fact_275_eq__refl,axiom,
% 0.31/0.46 ! [X: int,Y: int] :
% 0.31/0.46 ( ( X = Y )
% 0.31/0.46 => ( ord_less_eq_int @ X @ Y ) ) ).
% 0.31/0.46
% 0.31/0.46 % eq_refl
% 0.31/0.46 thf(fact_276_linear,axiom,
% 0.31/0.46 ! [X: nat,Y: nat] :
% 0.31/0.46 ( ( ord_less_eq_nat @ X @ Y )
% 0.31/0.46 | ( ord_less_eq_nat @ Y @ X ) ) ).
% 0.31/0.46
% 0.31/0.46 % linear
% 0.31/0.46 thf(fact_277_linear,axiom,
% 0.31/0.46 ! [X: int,Y: int] :
% 0.31/0.46 ( ( ord_less_eq_int @ X @ Y )
% 0.31/0.46 | ( ord_less_eq_int @ Y @ X ) ) ).
% 0.31/0.46
% 0.31/0.46 % linear
% 0.31/0.46 thf(fact_278_antisym,axiom,
% 0.31/0.46 ! [X: nat,Y: nat] :
% 0.31/0.46 ( ( ord_less_eq_nat @ X @ Y )
% 0.31/0.46 => ( ( ord_less_eq_nat @ Y @ X )
% 0.31/0.46 => ( X = Y ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % antisym
% 0.31/0.46 thf(fact_279_antisym,axiom,
% 0.31/0.46 ! [X: int,Y: int] :
% 0.31/0.46 ( ( ord_less_eq_int @ X @ Y )
% 0.31/0.46 => ( ( ord_less_eq_int @ Y @ X )
% 0.31/0.46 => ( X = Y ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % antisym
% 0.31/0.46 thf(fact_280_eq__iff,axiom,
% 0.31/0.46 ( ( ^ [Y5: nat,Z2: nat] : ( Y5 = Z2 ) )
% 0.31/0.46 = ( ^ [X5: nat,Y6: nat] :
% 0.31/0.46 ( ( ord_less_eq_nat @ X5 @ Y6 )
% 0.31/0.46 & ( ord_less_eq_nat @ Y6 @ X5 ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % eq_iff
% 0.31/0.46 thf(fact_281_eq__iff,axiom,
% 0.31/0.46 ( ( ^ [Y5: int,Z2: int] : ( Y5 = Z2 ) )
% 0.31/0.46 = ( ^ [X5: int,Y6: int] :
% 0.31/0.46 ( ( ord_less_eq_int @ X5 @ Y6 )
% 0.31/0.46 & ( ord_less_eq_int @ Y6 @ X5 ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % eq_iff
% 0.31/0.46 thf(fact_282_ord__le__eq__subst,axiom,
% 0.31/0.46 ! [A: nat,B: nat,F: nat > nat,C: nat] :
% 0.31/0.46 ( ( ord_less_eq_nat @ A @ B )
% 0.31/0.46 => ( ( ( F @ B )
% 0.31/0.46 = C )
% 0.31/0.46 => ( ! [X3: nat,Y3: nat] :
% 0.31/0.46 ( ( ord_less_eq_nat @ X3 @ Y3 )
% 0.31/0.46 => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
% 0.31/0.46 => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % ord_le_eq_subst
% 0.31/0.46 thf(fact_283_ord__le__eq__subst,axiom,
% 0.31/0.46 ! [A: nat,B: nat,F: nat > int,C: int] :
% 0.31/0.46 ( ( ord_less_eq_nat @ A @ B )
% 0.31/0.46 => ( ( ( F @ B )
% 0.31/0.46 = C )
% 0.31/0.46 => ( ! [X3: nat,Y3: nat] :
% 0.31/0.46 ( ( ord_less_eq_nat @ X3 @ Y3 )
% 0.31/0.46 => ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
% 0.31/0.46 => ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % ord_le_eq_subst
% 0.31/0.46 thf(fact_284_ord__le__eq__subst,axiom,
% 0.31/0.46 ! [A: int,B: int,F: int > nat,C: nat] :
% 0.31/0.46 ( ( ord_less_eq_int @ A @ B )
% 0.31/0.46 => ( ( ( F @ B )
% 0.31/0.46 = C )
% 0.31/0.46 => ( ! [X3: int,Y3: int] :
% 0.31/0.46 ( ( ord_less_eq_int @ X3 @ Y3 )
% 0.31/0.46 => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
% 0.31/0.46 => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % ord_le_eq_subst
% 0.31/0.46 thf(fact_285_ord__le__eq__subst,axiom,
% 0.31/0.46 ! [A: int,B: int,F: int > int,C: int] :
% 0.31/0.46 ( ( ord_less_eq_int @ A @ B )
% 0.31/0.46 => ( ( ( F @ B )
% 0.31/0.46 = C )
% 0.31/0.46 => ( ! [X3: int,Y3: int] :
% 0.31/0.46 ( ( ord_less_eq_int @ X3 @ Y3 )
% 0.31/0.46 => ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
% 0.31/0.46 => ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % ord_le_eq_subst
% 0.31/0.46 thf(fact_286_ord__eq__le__subst,axiom,
% 0.31/0.46 ! [A: nat,F: nat > nat,B: nat,C: nat] :
% 0.31/0.46 ( ( A
% 0.31/0.46 = ( F @ B ) )
% 0.31/0.46 => ( ( ord_less_eq_nat @ B @ C )
% 0.31/0.46 => ( ! [X3: nat,Y3: nat] :
% 0.31/0.46 ( ( ord_less_eq_nat @ X3 @ Y3 )
% 0.31/0.46 => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
% 0.31/0.46 => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % ord_eq_le_subst
% 0.31/0.46 thf(fact_287_ord__eq__le__subst,axiom,
% 0.31/0.46 ! [A: int,F: nat > int,B: nat,C: nat] :
% 0.31/0.46 ( ( A
% 0.31/0.46 = ( F @ B ) )
% 0.31/0.46 => ( ( ord_less_eq_nat @ B @ C )
% 0.31/0.46 => ( ! [X3: nat,Y3: nat] :
% 0.31/0.46 ( ( ord_less_eq_nat @ X3 @ Y3 )
% 0.31/0.46 => ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
% 0.31/0.46 => ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % ord_eq_le_subst
% 0.31/0.46 thf(fact_288_ord__eq__le__subst,axiom,
% 0.31/0.46 ! [A: nat,F: int > nat,B: int,C: int] :
% 0.31/0.46 ( ( A
% 0.31/0.46 = ( F @ B ) )
% 0.31/0.46 => ( ( ord_less_eq_int @ B @ C )
% 0.31/0.46 => ( ! [X3: int,Y3: int] :
% 0.31/0.46 ( ( ord_less_eq_int @ X3 @ Y3 )
% 0.31/0.46 => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
% 0.31/0.46 => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % ord_eq_le_subst
% 0.31/0.46 thf(fact_289_ord__eq__le__subst,axiom,
% 0.31/0.46 ! [A: int,F: int > int,B: int,C: int] :
% 0.31/0.46 ( ( A
% 0.31/0.46 = ( F @ B ) )
% 0.31/0.46 => ( ( ord_less_eq_int @ B @ C )
% 0.31/0.46 => ( ! [X3: int,Y3: int] :
% 0.31/0.46 ( ( ord_less_eq_int @ X3 @ Y3 )
% 0.31/0.46 => ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
% 0.31/0.46 => ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % ord_eq_le_subst
% 0.31/0.46 thf(fact_290_order__subst2,axiom,
% 0.31/0.46 ! [A: nat,B: nat,F: nat > nat,C: nat] :
% 0.31/0.46 ( ( ord_less_eq_nat @ A @ B )
% 0.31/0.46 => ( ( ord_less_eq_nat @ ( F @ B ) @ C )
% 0.31/0.46 => ( ! [X3: nat,Y3: nat] :
% 0.31/0.46 ( ( ord_less_eq_nat @ X3 @ Y3 )
% 0.31/0.46 => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
% 0.31/0.46 => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % order_subst2
% 0.31/0.46 thf(fact_291_order__subst2,axiom,
% 0.31/0.46 ! [A: nat,B: nat,F: nat > int,C: int] :
% 0.31/0.46 ( ( ord_less_eq_nat @ A @ B )
% 0.31/0.46 => ( ( ord_less_eq_int @ ( F @ B ) @ C )
% 0.31/0.46 => ( ! [X3: nat,Y3: nat] :
% 0.31/0.46 ( ( ord_less_eq_nat @ X3 @ Y3 )
% 0.31/0.46 => ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
% 0.31/0.46 => ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % order_subst2
% 0.31/0.46 thf(fact_292_order__subst2,axiom,
% 0.31/0.46 ! [A: int,B: int,F: int > nat,C: nat] :
% 0.31/0.46 ( ( ord_less_eq_int @ A @ B )
% 0.31/0.46 => ( ( ord_less_eq_nat @ ( F @ B ) @ C )
% 0.31/0.46 => ( ! [X3: int,Y3: int] :
% 0.31/0.46 ( ( ord_less_eq_int @ X3 @ Y3 )
% 0.31/0.46 => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
% 0.31/0.46 => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % order_subst2
% 0.31/0.46 thf(fact_293_order__subst2,axiom,
% 0.31/0.46 ! [A: int,B: int,F: int > int,C: int] :
% 0.31/0.46 ( ( ord_less_eq_int @ A @ B )
% 0.31/0.46 => ( ( ord_less_eq_int @ ( F @ B ) @ C )
% 0.31/0.46 => ( ! [X3: int,Y3: int] :
% 0.31/0.46 ( ( ord_less_eq_int @ X3 @ Y3 )
% 0.31/0.46 => ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
% 0.31/0.46 => ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % order_subst2
% 0.31/0.46 thf(fact_294_order__subst1,axiom,
% 0.31/0.46 ! [A: nat,F: nat > nat,B: nat,C: nat] :
% 0.31/0.46 ( ( ord_less_eq_nat @ A @ ( F @ B ) )
% 0.31/0.46 => ( ( ord_less_eq_nat @ B @ C )
% 0.31/0.46 => ( ! [X3: nat,Y3: nat] :
% 0.31/0.46 ( ( ord_less_eq_nat @ X3 @ Y3 )
% 0.31/0.46 => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
% 0.31/0.46 => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % order_subst1
% 0.31/0.46 thf(fact_295_order__subst1,axiom,
% 0.31/0.46 ! [A: nat,F: int > nat,B: int,C: int] :
% 0.31/0.46 ( ( ord_less_eq_nat @ A @ ( F @ B ) )
% 0.31/0.46 => ( ( ord_less_eq_int @ B @ C )
% 0.31/0.46 => ( ! [X3: int,Y3: int] :
% 0.31/0.46 ( ( ord_less_eq_int @ X3 @ Y3 )
% 0.31/0.46 => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
% 0.31/0.46 => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % order_subst1
% 0.31/0.46 thf(fact_296_order__subst1,axiom,
% 0.31/0.46 ! [A: int,F: nat > int,B: nat,C: nat] :
% 0.31/0.46 ( ( ord_less_eq_int @ A @ ( F @ B ) )
% 0.31/0.46 => ( ( ord_less_eq_nat @ B @ C )
% 0.31/0.46 => ( ! [X3: nat,Y3: nat] :
% 0.31/0.46 ( ( ord_less_eq_nat @ X3 @ Y3 )
% 0.31/0.46 => ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
% 0.31/0.46 => ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % order_subst1
% 0.31/0.46 thf(fact_297_order__subst1,axiom,
% 0.31/0.46 ! [A: int,F: int > int,B: int,C: int] :
% 0.31/0.46 ( ( ord_less_eq_int @ A @ ( F @ B ) )
% 0.31/0.46 => ( ( ord_less_eq_int @ B @ C )
% 0.31/0.46 => ( ! [X3: int,Y3: int] :
% 0.31/0.46 ( ( ord_less_eq_int @ X3 @ Y3 )
% 0.31/0.46 => ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
% 0.31/0.46 => ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % order_subst1
% 0.31/0.46 thf(fact_298_int__int__eq,axiom,
% 0.31/0.46 ! [M: nat,N: nat] :
% 0.31/0.46 ( ( ( semiri2019852685at_int @ M )
% 0.31/0.46 = ( semiri2019852685at_int @ N ) )
% 0.31/0.46 = ( M = N ) ) ).
% 0.31/0.46
% 0.31/0.46 % int_int_eq
% 0.31/0.46 thf(fact_299_zle__int,axiom,
% 0.31/0.46 ! [M: nat,N: nat] :
% 0.31/0.46 ( ( ord_less_eq_int @ ( semiri2019852685at_int @ M ) @ ( semiri2019852685at_int @ N ) )
% 0.31/0.46 = ( ord_less_eq_nat @ M @ N ) ) ).
% 0.31/0.46
% 0.31/0.46 % zle_int
% 0.31/0.46 thf(fact_300_le__iff__diff__le__0,axiom,
% 0.31/0.46 ( ord_less_eq_int
% 0.31/0.46 = ( ^ [A3: int,B3: int] : ( ord_less_eq_int @ ( minus_minus_int @ A3 @ B3 ) @ zero_zero_int ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % le_iff_diff_le_0
% 0.31/0.46 thf(fact_301_diff__less__mono,axiom,
% 0.31/0.46 ! [A: nat,B: nat,C: nat] :
% 0.31/0.46 ( ( ord_less_nat @ A @ B )
% 0.31/0.46 => ( ( ord_less_eq_nat @ C @ A )
% 0.31/0.46 => ( ord_less_nat @ ( minus_minus_nat @ A @ C ) @ ( minus_minus_nat @ B @ C ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % diff_less_mono
% 0.31/0.46 thf(fact_302_less__diff__iff,axiom,
% 0.31/0.46 ! [K: nat,M: nat,N: nat] :
% 0.31/0.46 ( ( ord_less_eq_nat @ K @ M )
% 0.31/0.46 => ( ( ord_less_eq_nat @ K @ N )
% 0.31/0.46 => ( ( ord_less_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
% 0.31/0.46 = ( ord_less_nat @ M @ N ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % less_diff_iff
% 0.31/0.46 thf(fact_303_power__not__zero,axiom,
% 0.31/0.46 ! [A: int,N: nat] :
% 0.31/0.46 ( ( A != zero_zero_int )
% 0.31/0.46 => ( ( power_power_int @ A @ N )
% 0.31/0.46 != zero_zero_int ) ) ).
% 0.31/0.46
% 0.31/0.46 % power_not_zero
% 0.31/0.46 thf(fact_304_power__not__zero,axiom,
% 0.31/0.46 ! [A: nat,N: nat] :
% 0.31/0.46 ( ( A != zero_zero_nat )
% 0.31/0.46 => ( ( power_power_nat @ A @ N )
% 0.31/0.46 != zero_zero_nat ) ) ).
% 0.31/0.46
% 0.31/0.46 % power_not_zero
% 0.31/0.46 thf(fact_305_power__less__imp__less__base,axiom,
% 0.31/0.46 ! [A: nat,N: nat,B: nat] :
% 0.31/0.46 ( ( ord_less_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) )
% 0.31/0.46 => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 0.31/0.46 => ( ord_less_nat @ A @ B ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % power_less_imp_less_base
% 0.31/0.46 thf(fact_306_power__less__imp__less__base,axiom,
% 0.31/0.46 ! [A: int,N: nat,B: int] :
% 0.31/0.46 ( ( ord_less_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) )
% 0.31/0.46 => ( ( ord_less_eq_int @ zero_zero_int @ B )
% 0.31/0.46 => ( ord_less_int @ A @ B ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % power_less_imp_less_base
% 0.31/0.46 thf(fact_307_diff__right__commute,axiom,
% 0.31/0.46 ! [A: nat,C: nat,B: nat] :
% 0.31/0.46 ( ( minus_minus_nat @ ( minus_minus_nat @ A @ C ) @ B )
% 0.31/0.46 = ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C ) ) ).
% 0.31/0.46
% 0.31/0.46 % diff_right_commute
% 0.31/0.46 thf(fact_308_diff__eq__diff__less__eq,axiom,
% 0.31/0.46 ! [A: int,B: int,C: int,D: int] :
% 0.31/0.46 ( ( ( minus_minus_int @ A @ B )
% 0.31/0.46 = ( minus_minus_int @ C @ D ) )
% 0.31/0.46 => ( ( ord_less_eq_int @ A @ B )
% 0.31/0.46 = ( ord_less_eq_int @ C @ D ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % diff_eq_diff_less_eq
% 0.31/0.46 thf(fact_309_diff__right__mono,axiom,
% 0.31/0.46 ! [A: int,B: int,C: int] :
% 0.31/0.46 ( ( ord_less_eq_int @ A @ B )
% 0.31/0.46 => ( ord_less_eq_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ C ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % diff_right_mono
% 0.31/0.46 thf(fact_310_diff__left__mono,axiom,
% 0.31/0.46 ! [B: int,A: int,C: int] :
% 0.31/0.46 ( ( ord_less_eq_int @ B @ A )
% 0.31/0.46 => ( ord_less_eq_int @ ( minus_minus_int @ C @ A ) @ ( minus_minus_int @ C @ B ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % diff_left_mono
% 0.31/0.46 thf(fact_311_zero__le__power,axiom,
% 0.31/0.46 ! [A: nat,N: nat] :
% 0.31/0.46 ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.31/0.46 => ( ord_less_eq_nat @ zero_zero_nat @ ( power_power_nat @ A @ N ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % zero_le_power
% 0.31/0.46 thf(fact_312_zero__le__power,axiom,
% 0.31/0.46 ! [A: int,N: nat] :
% 0.31/0.46 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.31/0.46 => ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ N ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % zero_le_power
% 0.31/0.46 thf(fact_313_diff__mono,axiom,
% 0.31/0.46 ! [A: int,B: int,D: int,C: int] :
% 0.31/0.46 ( ( ord_less_eq_int @ A @ B )
% 0.31/0.46 => ( ( ord_less_eq_int @ D @ C )
% 0.31/0.46 => ( ord_less_eq_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ D ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % diff_mono
% 0.31/0.46 thf(fact_314_power__mono,axiom,
% 0.31/0.46 ! [A: nat,B: nat,N: nat] :
% 0.31/0.46 ( ( ord_less_eq_nat @ A @ B )
% 0.31/0.46 => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.31/0.46 => ( ord_less_eq_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % power_mono
% 0.31/0.46 thf(fact_315_power__mono,axiom,
% 0.31/0.46 ! [A: int,B: int,N: nat] :
% 0.31/0.46 ( ( ord_less_eq_int @ A @ B )
% 0.31/0.46 => ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.31/0.46 => ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % power_mono
% 0.31/0.46 thf(fact_316_diff__le__mono2,axiom,
% 0.31/0.46 ! [M: nat,N: nat,L: nat] :
% 0.31/0.46 ( ( ord_less_eq_nat @ M @ N )
% 0.31/0.46 => ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % diff_le_mono2
% 0.31/0.46 thf(fact_317_le__diff__iff_H,axiom,
% 0.31/0.46 ! [A: nat,C: nat,B: nat] :
% 0.31/0.46 ( ( ord_less_eq_nat @ A @ C )
% 0.31/0.46 => ( ( ord_less_eq_nat @ B @ C )
% 0.31/0.46 => ( ( ord_less_eq_nat @ ( minus_minus_nat @ C @ A ) @ ( minus_minus_nat @ C @ B ) )
% 0.31/0.46 = ( ord_less_eq_nat @ B @ A ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % le_diff_iff'
% 0.31/0.46 thf(fact_318_diff__le__self,axiom,
% 0.31/0.46 ! [M: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ M ) ).
% 0.31/0.46
% 0.31/0.46 % diff_le_self
% 0.31/0.46 thf(fact_319_diff__le__mono,axiom,
% 0.31/0.46 ! [M: nat,N: nat,L: nat] :
% 0.31/0.46 ( ( ord_less_eq_nat @ M @ N )
% 0.31/0.46 => ( ord_less_eq_nat @ ( minus_minus_nat @ M @ L ) @ ( minus_minus_nat @ N @ L ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % diff_le_mono
% 0.31/0.46 thf(fact_320_Nat_Odiff__diff__eq,axiom,
% 0.31/0.46 ! [K: nat,M: nat,N: nat] :
% 0.31/0.46 ( ( ord_less_eq_nat @ K @ M )
% 0.31/0.46 => ( ( ord_less_eq_nat @ K @ N )
% 0.31/0.46 => ( ( minus_minus_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
% 0.31/0.46 = ( minus_minus_nat @ M @ N ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % Nat.diff_diff_eq
% 0.31/0.46 thf(fact_321_diff__commute,axiom,
% 0.31/0.46 ! [I: nat,J2: nat,K: nat] :
% 0.31/0.46 ( ( minus_minus_nat @ ( minus_minus_nat @ I @ J2 ) @ K )
% 0.31/0.46 = ( minus_minus_nat @ ( minus_minus_nat @ I @ K ) @ J2 ) ) ).
% 0.31/0.46
% 0.31/0.46 % diff_commute
% 0.31/0.46 thf(fact_322_le__diff__iff,axiom,
% 0.31/0.46 ! [K: nat,M: nat,N: nat] :
% 0.31/0.46 ( ( ord_less_eq_nat @ K @ M )
% 0.31/0.46 => ( ( ord_less_eq_nat @ K @ N )
% 0.31/0.46 => ( ( ord_less_eq_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
% 0.31/0.46 = ( ord_less_eq_nat @ M @ N ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % le_diff_iff
% 0.31/0.46 thf(fact_323_eq__diff__iff,axiom,
% 0.31/0.46 ! [K: nat,M: nat,N: nat] :
% 0.31/0.46 ( ( ord_less_eq_nat @ K @ M )
% 0.31/0.46 => ( ( ord_less_eq_nat @ K @ N )
% 0.31/0.46 => ( ( ( minus_minus_nat @ M @ K )
% 0.31/0.46 = ( minus_minus_nat @ N @ K ) )
% 0.31/0.46 = ( M = N ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % eq_diff_iff
% 0.31/0.46 thf(fact_324_less__eq__int__code_I1_J,axiom,
% 0.31/0.46 ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% 0.31/0.46
% 0.31/0.46 % less_eq_int_code(1)
% 0.31/0.46 thf(fact_325_less__int__code_I1_J,axiom,
% 0.31/0.46 ~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% 0.31/0.46
% 0.31/0.46 % less_int_code(1)
% 0.31/0.46 thf(fact_326_nat__one__le__power,axiom,
% 0.31/0.46 ! [I: nat,N: nat] :
% 0.31/0.46 ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ I )
% 0.31/0.46 => ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( power_power_nat @ I @ N ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % nat_one_le_power
% 0.31/0.46 thf(fact_327_power__eq__imp__eq__base,axiom,
% 0.31/0.46 ! [A: nat,N: nat,B: nat] :
% 0.31/0.46 ( ( ( power_power_nat @ A @ N )
% 0.31/0.46 = ( power_power_nat @ B @ N ) )
% 0.31/0.46 => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.31/0.46 => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 0.31/0.46 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.31/0.46 => ( A = B ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % power_eq_imp_eq_base
% 0.31/0.46 thf(fact_328_power__eq__imp__eq__base,axiom,
% 0.31/0.46 ! [A: int,N: nat,B: int] :
% 0.31/0.46 ( ( ( power_power_int @ A @ N )
% 0.31/0.46 = ( power_power_int @ B @ N ) )
% 0.31/0.46 => ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.31/0.46 => ( ( ord_less_eq_int @ zero_zero_int @ B )
% 0.31/0.46 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.31/0.46 => ( A = B ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % power_eq_imp_eq_base
% 0.31/0.46 thf(fact_329_power__eq__iff__eq__base,axiom,
% 0.31/0.46 ! [N: nat,A: nat,B: nat] :
% 0.31/0.46 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.31/0.46 => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.31/0.46 => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 0.31/0.46 => ( ( ( power_power_nat @ A @ N )
% 0.31/0.46 = ( power_power_nat @ B @ N ) )
% 0.31/0.46 = ( A = B ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % power_eq_iff_eq_base
% 0.31/0.46 thf(fact_330_power__eq__iff__eq__base,axiom,
% 0.31/0.46 ! [N: nat,A: int,B: int] :
% 0.31/0.46 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.31/0.46 => ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.31/0.46 => ( ( ord_less_eq_int @ zero_zero_int @ B )
% 0.31/0.46 => ( ( ( power_power_int @ A @ N )
% 0.31/0.46 = ( power_power_int @ B @ N ) )
% 0.31/0.46 = ( A = B ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % power_eq_iff_eq_base
% 0.31/0.46 thf(fact_331_eq__iff__diff__eq__0,axiom,
% 0.31/0.46 ( ( ^ [Y5: int,Z2: int] : ( Y5 = Z2 ) )
% 0.31/0.46 = ( ^ [A3: int,B3: int] :
% 0.31/0.46 ( ( minus_minus_int @ A3 @ B3 )
% 0.31/0.46 = zero_zero_int ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % eq_iff_diff_eq_0
% 0.31/0.46 thf(fact_332_diff__strict__right__mono,axiom,
% 0.31/0.46 ! [A: int,B: int,C: int] :
% 0.31/0.46 ( ( ord_less_int @ A @ B )
% 0.31/0.46 => ( ord_less_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ C ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % diff_strict_right_mono
% 0.31/0.46 thf(fact_333_diff__strict__left__mono,axiom,
% 0.31/0.46 ! [B: int,A: int,C: int] :
% 0.31/0.46 ( ( ord_less_int @ B @ A )
% 0.31/0.46 => ( ord_less_int @ ( minus_minus_int @ C @ A ) @ ( minus_minus_int @ C @ B ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % diff_strict_left_mono
% 0.31/0.46 thf(fact_334_diff__eq__diff__less,axiom,
% 0.31/0.46 ! [A: int,B: int,C: int,D: int] :
% 0.31/0.46 ( ( ( minus_minus_int @ A @ B )
% 0.31/0.46 = ( minus_minus_int @ C @ D ) )
% 0.31/0.46 => ( ( ord_less_int @ A @ B )
% 0.31/0.46 = ( ord_less_int @ C @ D ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % diff_eq_diff_less
% 0.31/0.46 thf(fact_335_diff__strict__mono,axiom,
% 0.31/0.46 ! [A: int,B: int,D: int,C: int] :
% 0.31/0.46 ( ( ord_less_int @ A @ B )
% 0.31/0.46 => ( ( ord_less_int @ D @ C )
% 0.31/0.46 => ( ord_less_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ D ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % diff_strict_mono
% 0.31/0.46 thf(fact_336_zero__le,axiom,
% 0.31/0.46 ! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).
% 0.31/0.46
% 0.31/0.46 % zero_le
% 0.31/0.46 thf(fact_337_le__numeral__extra_I3_J,axiom,
% 0.31/0.46 ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% 0.31/0.46
% 0.31/0.46 % le_numeral_extra(3)
% 0.31/0.46 thf(fact_338_le__numeral__extra_I3_J,axiom,
% 0.31/0.46 ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% 0.31/0.46
% 0.31/0.46 % le_numeral_extra(3)
% 0.31/0.46 thf(fact_339_complete__interval,axiom,
% 0.31/0.46 ! [A: nat,B: nat,P: nat > $o] :
% 0.31/0.46 ( ( ord_less_nat @ A @ B )
% 0.31/0.46 => ( ( P @ A )
% 0.31/0.46 => ( ~ ( P @ B )
% 0.31/0.46 => ? [C2: nat] :
% 0.31/0.46 ( ( ord_less_eq_nat @ A @ C2 )
% 0.31/0.46 & ( ord_less_eq_nat @ C2 @ B )
% 0.31/0.46 & ! [X6: nat] :
% 0.31/0.46 ( ( ( ord_less_eq_nat @ A @ X6 )
% 0.31/0.46 & ( ord_less_nat @ X6 @ C2 ) )
% 0.31/0.46 => ( P @ X6 ) )
% 0.31/0.46 & ! [D2: nat] :
% 0.31/0.46 ( ! [X3: nat] :
% 0.31/0.46 ( ( ( ord_less_eq_nat @ A @ X3 )
% 0.31/0.46 & ( ord_less_nat @ X3 @ D2 ) )
% 0.31/0.46 => ( P @ X3 ) )
% 0.31/0.46 => ( ord_less_eq_nat @ D2 @ C2 ) ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % complete_interval
% 0.31/0.46 thf(fact_340_complete__interval,axiom,
% 0.31/0.46 ! [A: int,B: int,P: int > $o] :
% 0.31/0.46 ( ( ord_less_int @ A @ B )
% 0.31/0.46 => ( ( P @ A )
% 0.31/0.46 => ( ~ ( P @ B )
% 0.31/0.46 => ? [C2: int] :
% 0.31/0.46 ( ( ord_less_eq_int @ A @ C2 )
% 0.31/0.46 & ( ord_less_eq_int @ C2 @ B )
% 0.31/0.46 & ! [X6: int] :
% 0.31/0.46 ( ( ( ord_less_eq_int @ A @ X6 )
% 0.31/0.46 & ( ord_less_int @ X6 @ C2 ) )
% 0.31/0.46 => ( P @ X6 ) )
% 0.31/0.46 & ! [D2: int] :
% 0.31/0.46 ( ! [X3: int] :
% 0.31/0.46 ( ( ( ord_less_eq_int @ A @ X3 )
% 0.31/0.46 & ( ord_less_int @ X3 @ D2 ) )
% 0.31/0.46 => ( P @ X3 ) )
% 0.31/0.46 => ( ord_less_eq_int @ D2 @ C2 ) ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % complete_interval
% 0.31/0.46 thf(fact_341_order_Onot__eq__order__implies__strict,axiom,
% 0.31/0.46 ! [A: nat,B: nat] :
% 0.31/0.46 ( ( A != B )
% 0.31/0.46 => ( ( ord_less_eq_nat @ A @ B )
% 0.31/0.46 => ( ord_less_nat @ A @ B ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % order.not_eq_order_implies_strict
% 0.31/0.46 thf(fact_342_order_Onot__eq__order__implies__strict,axiom,
% 0.31/0.46 ! [A: int,B: int] :
% 0.31/0.46 ( ( A != B )
% 0.31/0.46 => ( ( ord_less_eq_int @ A @ B )
% 0.31/0.46 => ( ord_less_int @ A @ B ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % order.not_eq_order_implies_strict
% 0.31/0.46 thf(fact_343_dual__order_Ostrict__implies__order,axiom,
% 0.31/0.46 ! [B: nat,A: nat] :
% 0.31/0.46 ( ( ord_less_nat @ B @ A )
% 0.31/0.46 => ( ord_less_eq_nat @ B @ A ) ) ).
% 0.31/0.46
% 0.31/0.46 % dual_order.strict_implies_order
% 0.31/0.46 thf(fact_344_dual__order_Ostrict__implies__order,axiom,
% 0.31/0.46 ! [B: int,A: int] :
% 0.31/0.46 ( ( ord_less_int @ B @ A )
% 0.31/0.46 => ( ord_less_eq_int @ B @ A ) ) ).
% 0.31/0.46
% 0.31/0.46 % dual_order.strict_implies_order
% 0.31/0.46 thf(fact_345_dual__order_Ostrict__iff__order,axiom,
% 0.31/0.46 ( ord_less_nat
% 0.31/0.46 = ( ^ [B3: nat,A3: nat] :
% 0.31/0.46 ( ( ord_less_eq_nat @ B3 @ A3 )
% 0.31/0.46 & ( A3 != B3 ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % dual_order.strict_iff_order
% 0.31/0.46 thf(fact_346_dual__order_Ostrict__iff__order,axiom,
% 0.31/0.46 ( ord_less_int
% 0.31/0.46 = ( ^ [B3: int,A3: int] :
% 0.31/0.46 ( ( ord_less_eq_int @ B3 @ A3 )
% 0.31/0.46 & ( A3 != B3 ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % dual_order.strict_iff_order
% 0.31/0.46 thf(fact_347_dual__order_Oorder__iff__strict,axiom,
% 0.31/0.46 ( ord_less_eq_int
% 0.31/0.46 = ( ^ [B3: int,A3: int] :
% 0.31/0.46 ( ( ord_less_int @ B3 @ A3 )
% 0.31/0.46 | ( A3 = B3 ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % dual_order.order_iff_strict
% 0.31/0.46
% 0.31/0.46 % Conjectures (3)
% 0.31/0.46 thf(conj_0,hypothesis,
% 0.31/0.46 ord_less_nat @ zero_zero_nat @ n ).
% 0.31/0.46
% 0.31/0.46 thf(conj_1,hypothesis,
% 0.31/0.46 polyno1372495879olyh_a @ p @ n ).
% 0.31/0.46
% 0.31/0.46 thf(conj_2,conjecture,
% 0.31/0.46 ( ( polyno578545843gree_a @ p )
% 0.31/0.46 = zero_zero_nat ) ).
% 0.31/0.46
% 0.31/0.46 %------------------------------------------------------------------------------
% 0.31/0.46 ------------------------
% 2.48/1.48 ---- Embedded in THF ---
% 2.48/1.48 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 2.48/1.48 %%% Generated on Tue Feb 24 08:28:10 EST 2026
% 2.48/1.48 %%% using '$$dhol' embedding, version 1.3.0.
% 2.48/1.48 %%% Logic specification used:
% 2.48/1.48 %%% thf(spec, logic, $$dhol).
% 2.48/1.48
% 2.48/1.48 % SZS output start ListOfTHF for /export/starexec/sandbox/tmp/tmp.4Dn3J3wAeS/DTF2DTF9327.p
% See solution above
% 2.48/1.48 ------------------------
% 2.66/1.50 ---- Cleaned THF ---
% 2.66/1.50 thf(ty_n_t__Polynomial____Expression____Mirabelle____dwjuveeage__Opoly_Itf__a_J,type,
% 2.66/1.50 polyno727731844poly_a: $tType ).
% 2.66/1.50
% 2.66/1.50 thf(ty_n_t__Nat__Onat,type,
% 2.66/1.50 nat: $tType ).
% 2.66/1.50
% 2.66/1.50 thf(ty_n_t__Int__Oint,type,
% 2.66/1.50 int: $tType ).
% 2.66/1.50
% 2.66/1.50 thf(sy_c_Groups_Ominus__class_Ominus_001t__Int__Oint,type,
% 2.66/1.50 minus_minus_int: int > int > int ).
% 2.66/1.50
% 2.66/1.50 thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat,type,
% 2.66/1.50 minus_minus_nat: nat > nat > nat ).
% 2.66/1.50
% 2.66/1.50 thf(sy_c_Groups_Ozero__class_Ozero_001t__Int__Oint,type,
% 2.66/1.50 zero_zero_int: int ).
% 2.66/1.50
% 2.66/1.50 thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
% 2.66/1.50 zero_zero_nat: nat ).
% 2.66/1.50
% 2.66/1.50 thf(sy_c_Nat_OSuc,type,
% 2.66/1.50 suc: nat > nat ).
% 2.66/1.50
% 2.66/1.50 thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Int__Oint,type,
% 2.66/1.50 semiri2019852685at_int: nat > int ).
% 2.66/1.50
% 2.66/1.50 thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Nat__Onat,type,
% 2.66/1.50 semiri1382578993at_nat: nat > nat ).
% 2.66/1.50
% 2.66/1.50 thf(sy_c_Orderings_Oord__class_Oless_001t__Int__Oint,type,
% 2.66/1.50 ord_less_int: int > int > $o ).
% 2.66/1.50
% 2.66/1.50 thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
% 2.66/1.50 ord_less_nat: nat > nat > $o ).
% 2.66/1.50
% 2.66/1.50 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Int__Oint,type,
% 2.66/1.50 ord_less_eq_int: int > int > $o ).
% 2.66/1.50
% 2.66/1.50 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
% 2.66/1.50 ord_less_eq_nat: nat > nat > $o ).
% 2.66/1.50
% 2.66/1.50 thf(sy_c_Polynomial__Expression__Mirabelle__dwjuveeage_Odegree_001tf__a,type,
% 2.66/1.50 polyno578545843gree_a: polyno727731844poly_a > nat ).
% 2.66/1.50
% 2.66/1.50 thf(sy_c_Polynomial__Expression__Mirabelle__dwjuveeage_Odegreen_001tf__a,type,
% 2.66/1.50 polyno1674775833reen_a: polyno727731844poly_a > nat > nat ).
% 2.66/1.50
% 2.66/1.50 thf(sy_c_Polynomial__Expression__Mirabelle__dwjuveeage_Oisnpoly_001tf__a,type,
% 2.66/1.50 polyno190918219poly_a: polyno727731844poly_a > $o ).
% 2.66/1.50
% 2.66/1.50 thf(sy_c_Polynomial__Expression__Mirabelle__dwjuveeage_Oisnpolyh_001tf__a,type,
% 2.66/1.50 polyno1372495879olyh_a: polyno727731844poly_a > nat > $o ).
% 2.66/1.50
% 2.66/1.50 thf(sy_c_Power_Opower__class_Opower_001t__Int__Oint,type,
% 2.66/1.50 power_power_int: int > nat > int ).
% 2.66/1.50
% 2.66/1.50 thf(sy_c_Power_Opower__class_Opower_001t__Nat__Onat,type,
% 2.66/1.50 power_power_nat: nat > nat > nat ).
% 2.66/1.50
% 2.66/1.50 thf(sy_v_n,type,
% 2.66/1.50 n: nat ).
% 2.66/1.50
% 2.66/1.50 thf(sy_v_p,type,
% 2.66/1.50 p: polyno727731844poly_a ).
% 2.66/1.50
% 2.66/1.50 thf(fact_0_neq0__conv,axiom,
% 2.66/1.50 ! [N: nat] :
% 2.66/1.50 ( ( N != zero_zero_nat )
% 2.66/1.50 = ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 2.66/1.50
% 2.66/1.50 thf(fact_1_less__nat__zero__code,axiom,
% 2.66/1.50 ! [N: nat] :
% 2.66/1.50 ~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% 2.66/1.50
% 2.66/1.50 thf(fact_2_bot__nat__0_Onot__eq__extremum,axiom,
% 2.66/1.50 ! [A: nat] :
% 2.66/1.50 ( ( A != zero_zero_nat )
% 2.66/1.50 = ( ord_less_nat @ zero_zero_nat @ A ) ) ).
% 2.66/1.50
% 2.66/1.50 thf(fact_3_not__gr__zero,axiom,
% 2.66/1.50 ! [N: nat] :
% 2.66/1.50 ( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
% 2.66/1.50 = ( N = zero_zero_nat ) ) ).
% 2.66/1.50
% 2.66/1.50 thf(fact_4_gr0I,axiom,
% 2.66/1.50 ! [N: nat] :
% 2.66/1.50 ( ( N != zero_zero_nat )
% 2.66/1.50 => ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 2.66/1.50
% 2.66/1.50 thf(fact_5_not__gr0,axiom,
% 2.66/1.50 ! [N: nat] :
% 2.66/1.50 ( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
% 2.66/1.50 = ( N = zero_zero_nat ) ) ).
% 2.66/1.50
% 2.66/1.50 thf(fact_6_not__less0,axiom,
% 2.66/1.50 ! [N: nat] :
% 2.66/1.50 ~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% 2.66/1.50
% 2.66/1.50 thf(fact_7_less__zeroE,axiom,
% 2.66/1.50 ! [N: nat] :
% 2.66/1.50 ~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% 2.66/1.50
% 2.66/1.50 thf(fact_8_gr__implies__not0,axiom,
% 2.66/1.50 ! [M: nat,N: nat] :
% 2.66/1.50 ( ( ord_less_nat @ M @ N )
% 2.66/1.50 => ( N != zero_zero_nat ) ) ).
% 2.66/1.50
% 2.66/1.50 thf(fact_9_infinite__descent0,axiom,
% 2.66/1.50 ! [P: nat > $o,N: nat] :
% 2.66/1.50 ( ( P @ zero_zero_nat )
% 2.66/1.50 => ( ! [N2: nat] :
% 2.66/1.50 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 2.66/1.50 => ( ~ ( P @ N2 )
% 2.66/1.50 => ? [M2: nat] :
% 2.66/1.50 ( ( ord_less_nat @ M2 @ N2 )
% 2.66/1.50 & ~ ( P @ M2 ) ) ) )
% 2.66/1.50 => ( P @ N ) ) ) ).
% 2.66/1.50
% 2.66/1.50 thf(fact_10_bot__nat__0_Oextremum__strict,axiom,
% 2.66/1.50 ! [A: nat] :
% 2.66/1.50 ~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% 2.66/1.50
% 2.66/1.50 thf(fact_11_zero__reorient,axiom,
% 2.66/1.50 ! [X: nat] :
% 2.66/1.50 ( ( zero_zero_nat = X )
% 2.66/1.50 = ( X = zero_zero_nat ) ) ).
% 2.66/1.50
% 2.66/1.50 thf(fact_12_zero__reorient,axiom,
% 2.66/1.50 ! [X: int] :
% 2.66/1.50 ( ( zero_zero_int = X )
% 2.66/1.50 = ( X = zero_zero_int ) ) ).
% 2.66/1.50
% 2.66/1.50 thf(fact_13_linorder__neqE__nat,axiom,
% 2.66/1.50 ! [X: nat,Y: nat] :
% 2.66/1.50 ( ( X != Y )
% 2.66/1.50 => ( ~ ( ord_less_nat @ X @ Y )
% 2.66/1.50 => ( ord_less_nat @ Y @ X ) ) ) ).
% 2.66/1.50
% 2.66/1.50 thf(fact_14_infinite__descent,axiom,
% 2.66/1.50 ! [P: nat > $o,N: nat] :
% 2.66/1.50 ( ! [N2: nat] :
% 2.66/1.50 ( ~ ( P @ N2 )
% 2.66/1.50 => ? [M2: nat] :
% 2.66/1.50 ( ( ord_less_nat @ M2 @ N2 )
% 2.66/1.50 & ~ ( P @ M2 ) ) )
% 2.66/1.50 => ( P @ N ) ) ).
% 2.66/1.50
% 2.66/1.50 thf(fact_15_nat__less__induct,axiom,
% 2.66/1.50 ! [P: nat > $o,N: nat] :
% 2.66/1.50 ( ! [N2: nat] :
% 2.66/1.50 ( ! [M2: nat] :
% 2.66/1.50 ( ( ord_less_nat @ M2 @ N2 )
% 2.66/1.50 => ( P @ M2 ) )
% 2.66/1.50 => ( P @ N2 ) )
% 2.66/1.50 => ( P @ N ) ) ).
% 2.66/1.50
% 2.66/1.50 thf(fact_16_less__irrefl__nat,axiom,
% 2.66/1.50 ! [N: nat] :
% 2.66/1.50 ~ ( ord_less_nat @ N @ N ) ).
% 2.66/1.50
% 2.66/1.50 thf(fact_17_less__not__refl3,axiom,
% 2.66/1.50 ! [S: nat,T: nat] :
% 2.66/1.50 ( ( ord_less_nat @ S @ T )
% 2.66/1.50 => ( S != T ) ) ).
% 2.66/1.50
% 2.66/1.50 thf(fact_18_less__not__refl2,axiom,
% 2.66/1.50 ! [N: nat,M: nat] :
% 2.66/1.50 ( ( ord_less_nat @ N @ M )
% 2.66/1.50 => ( M != N ) ) ).
% 2.66/1.50
% 2.66/1.50 thf(fact_19_less__not__refl,axiom,
% 2.66/1.50 ! [N: nat] :
% 2.66/1.50 ~ ( ord_less_nat @ N @ N ) ).
% 2.66/1.50
% 2.66/1.50 thf(fact_20_nat__neq__iff,axiom,
% 2.66/1.50 ! [M: nat,N: nat] :
% 2.66/1.50 ( ( M != N )
% 2.66/1.50 = ( ( ord_less_nat @ M @ N )
% 2.66/1.50 | ( ord_less_nat @ N @ M ) ) ) ).
% 2.66/1.50
% 2.66/1.50 thf(fact_21_zero__less__iff__neq__zero,axiom,
% 2.66/1.51 ! [N: nat] :
% 2.66/1.51 ( ( ord_less_nat @ zero_zero_nat @ N )
% 2.66/1.51 = ( N != zero_zero_nat ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_22_gr__implies__not__zero,axiom,
% 2.66/1.51 ! [M: nat,N: nat] :
% 2.66/1.51 ( ( ord_less_nat @ M @ N )
% 2.66/1.51 => ( N != zero_zero_nat ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_23_not__less__zero,axiom,
% 2.66/1.51 ! [N: nat] :
% 2.66/1.51 ~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_24_gr__zeroI,axiom,
% 2.66/1.51 ! [N: nat] :
% 2.66/1.51 ( ( N != zero_zero_nat )
% 2.66/1.51 => ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_25_less__numeral__extra_I3_J,axiom,
% 2.66/1.51 ~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_26_less__numeral__extra_I3_J,axiom,
% 2.66/1.51 ~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_27_isnpoly__def,axiom,
% 2.66/1.51 ( polyno190918219poly_a
% 2.66/1.51 = ( ^ [P2: polyno727731844poly_a] : ( polyno1372495879olyh_a @ P2 @ zero_zero_nat ) ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_28_degreen__0,axiom,
% 2.66/1.51 ! [P3: polyno727731844poly_a,N: nat,M: nat] :
% 2.66/1.51 ( ( polyno1372495879olyh_a @ P3 @ N )
% 2.66/1.51 => ( ( ord_less_nat @ M @ N )
% 2.66/1.51 => ( ( polyno1674775833reen_a @ P3 @ M )
% 2.66/1.51 = zero_zero_nat ) ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_29_of__nat__0__less__iff,axiom,
% 2.66/1.51 ! [N: nat] :
% 2.66/1.51 ( ( ord_less_nat @ zero_zero_nat @ ( semiri1382578993at_nat @ N ) )
% 2.66/1.51 = ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_30_of__nat__0__less__iff,axiom,
% 2.66/1.51 ! [N: nat] :
% 2.66/1.51 ( ( ord_less_int @ zero_zero_int @ ( semiri2019852685at_int @ N ) )
% 2.66/1.51 = ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_31_degree__isnpolyh__Suc,axiom,
% 2.66/1.51 ! [P3: polyno727731844poly_a,N: nat] :
% 2.66/1.51 ( ( polyno1372495879olyh_a @ P3 @ ( suc @ N ) )
% 2.66/1.51 => ( ( polyno578545843gree_a @ P3 )
% 2.66/1.51 = zero_zero_nat ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_32_zero__natural_Orsp,axiom,
% 2.66/1.51 zero_zero_nat = zero_zero_nat ).
% 2.66/1.51
% 2.66/1.51 thf(fact_33_linorder__neqE__linordered__idom,axiom,
% 2.66/1.51 ! [X: int,Y: int] :
% 2.66/1.51 ( ( X != Y )
% 2.66/1.51 => ( ~ ( ord_less_int @ X @ Y )
% 2.66/1.51 => ( ord_less_int @ Y @ X ) ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_34_dual__order_Ostrict__implies__not__eq,axiom,
% 2.66/1.51 ! [B: nat,A: nat] :
% 2.66/1.51 ( ( ord_less_nat @ B @ A )
% 2.66/1.51 => ( A != B ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_35_dual__order_Ostrict__implies__not__eq,axiom,
% 2.66/1.51 ! [B: int,A: int] :
% 2.66/1.51 ( ( ord_less_int @ B @ A )
% 2.66/1.51 => ( A != B ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_36_nat_Oinject,axiom,
% 2.66/1.51 ! [X2: nat,Y2: nat] :
% 2.66/1.51 ( ( ( suc @ X2 )
% 2.66/1.51 = ( suc @ Y2 ) )
% 2.66/1.51 = ( X2 = Y2 ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_37_old_Onat_Oinject,axiom,
% 2.66/1.51 ! [Nat: nat,Nat2: nat] :
% 2.66/1.51 ( ( ( suc @ Nat )
% 2.66/1.51 = ( suc @ Nat2 ) )
% 2.66/1.51 = ( Nat = Nat2 ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_38_of__nat__eq__iff,axiom,
% 2.66/1.51 ! [M: nat,N: nat] :
% 2.66/1.51 ( ( ( semiri2019852685at_int @ M )
% 2.66/1.51 = ( semiri2019852685at_int @ N ) )
% 2.66/1.51 = ( M = N ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_39_lessI,axiom,
% 2.66/1.51 ! [N: nat] : ( ord_less_nat @ N @ ( suc @ N ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_40_Suc__mono,axiom,
% 2.66/1.51 ! [M: nat,N: nat] :
% 2.66/1.51 ( ( ord_less_nat @ M @ N )
% 2.66/1.51 => ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_41_Suc__less__eq,axiom,
% 2.66/1.51 ! [M: nat,N: nat] :
% 2.66/1.51 ( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
% 2.66/1.51 = ( ord_less_nat @ M @ N ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_42_of__nat__0,axiom,
% 2.66/1.51 ( ( semiri1382578993at_nat @ zero_zero_nat )
% 2.66/1.51 = zero_zero_nat ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_43_of__nat__0,axiom,
% 2.66/1.51 ( ( semiri2019852685at_int @ zero_zero_nat )
% 2.66/1.51 = zero_zero_int ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_44_of__nat__0__eq__iff,axiom,
% 2.66/1.51 ! [N: nat] :
% 2.66/1.51 ( ( zero_zero_nat
% 2.66/1.51 = ( semiri1382578993at_nat @ N ) )
% 2.66/1.51 = ( zero_zero_nat = N ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_45_of__nat__0__eq__iff,axiom,
% 2.66/1.51 ! [N: nat] :
% 2.66/1.51 ( ( zero_zero_int
% 2.66/1.51 = ( semiri2019852685at_int @ N ) )
% 2.66/1.51 = ( zero_zero_nat = N ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_46_of__nat__eq__0__iff,axiom,
% 2.66/1.51 ! [M: nat] :
% 2.66/1.51 ( ( ( semiri1382578993at_nat @ M )
% 2.66/1.51 = zero_zero_nat )
% 2.66/1.51 = ( M = zero_zero_nat ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_47_of__nat__eq__0__iff,axiom,
% 2.66/1.51 ! [M: nat] :
% 2.66/1.51 ( ( ( semiri2019852685at_int @ M )
% 2.66/1.51 = zero_zero_int )
% 2.66/1.51 = ( M = zero_zero_nat ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_48_of__nat__less__iff,axiom,
% 2.66/1.51 ! [M: nat,N: nat] :
% 2.66/1.51 ( ( ord_less_nat @ ( semiri1382578993at_nat @ M ) @ ( semiri1382578993at_nat @ N ) )
% 2.66/1.51 = ( ord_less_nat @ M @ N ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_49_of__nat__less__iff,axiom,
% 2.66/1.51 ! [M: nat,N: nat] :
% 2.66/1.51 ( ( ord_less_int @ ( semiri2019852685at_int @ M ) @ ( semiri2019852685at_int @ N ) )
% 2.66/1.51 = ( ord_less_nat @ M @ N ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_50_less__Suc0,axiom,
% 2.66/1.51 ! [N: nat] :
% 2.66/1.51 ( ( ord_less_nat @ N @ ( suc @ zero_zero_nat ) )
% 2.66/1.51 = ( N = zero_zero_nat ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_51_zero__less__Suc,axiom,
% 2.66/1.51 ! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_52_Suc__inject,axiom,
% 2.66/1.51 ! [X: nat,Y: nat] :
% 2.66/1.51 ( ( ( suc @ X )
% 2.66/1.51 = ( suc @ Y ) )
% 2.66/1.51 => ( X = Y ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_53_n__not__Suc__n,axiom,
% 2.66/1.51 ! [N: nat] :
% 2.66/1.51 ( N
% 2.66/1.51 != ( suc @ N ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_54_of__nat__neq__0,axiom,
% 2.66/1.51 ! [N: nat] :
% 2.66/1.51 ( ( semiri1382578993at_nat @ ( suc @ N ) )
% 2.66/1.51 != zero_zero_nat ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_55_of__nat__neq__0,axiom,
% 2.66/1.51 ! [N: nat] :
% 2.66/1.51 ( ( semiri2019852685at_int @ ( suc @ N ) )
% 2.66/1.51 != zero_zero_int ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_56_polypow_Ocases,axiom,
% 2.66/1.51 ! [X: nat] :
% 2.66/1.51 ( ( X != zero_zero_nat )
% 2.66/1.51 => ~ ! [V: nat] :
% 2.66/1.51 ( X
% 2.66/1.51 != ( suc @ V ) ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_57_nat_Odistinct_I1_J,axiom,
% 2.66/1.51 ! [X2: nat] :
% 2.66/1.51 ( zero_zero_nat
% 2.66/1.51 != ( suc @ X2 ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_58_old_Onat_Odistinct_I2_J,axiom,
% 2.66/1.51 ! [Nat2: nat] :
% 2.66/1.51 ( ( suc @ Nat2 )
% 2.66/1.51 != zero_zero_nat ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_59_old_Onat_Odistinct_I1_J,axiom,
% 2.66/1.51 ! [Nat2: nat] :
% 2.66/1.51 ( zero_zero_nat
% 2.66/1.51 != ( suc @ Nat2 ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_60_nat_OdiscI,axiom,
% 2.66/1.51 ! [Nat: nat,X2: nat] :
% 2.66/1.51 ( ( Nat
% 2.66/1.51 = ( suc @ X2 ) )
% 2.66/1.51 => ( Nat != zero_zero_nat ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_61_nat__induct,axiom,
% 2.66/1.51 ! [P: nat > $o,N: nat] :
% 2.66/1.51 ( ( P @ zero_zero_nat )
% 2.66/1.51 => ( ! [N2: nat] :
% 2.66/1.51 ( ( P @ N2 )
% 2.66/1.51 => ( P @ ( suc @ N2 ) ) )
% 2.66/1.51 => ( P @ N ) ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_62_diff__induct,axiom,
% 2.66/1.51 ! [P: nat > nat > $o,M: nat,N: nat] :
% 2.66/1.51 ( ! [X3: nat] : ( P @ X3 @ zero_zero_nat )
% 2.66/1.51 => ( ! [Y3: nat] : ( P @ zero_zero_nat @ ( suc @ Y3 ) )
% 2.66/1.51 => ( ! [X3: nat,Y3: nat] :
% 2.66/1.51 ( ( P @ X3 @ Y3 )
% 2.66/1.51 => ( P @ ( suc @ X3 ) @ ( suc @ Y3 ) ) )
% 2.66/1.51 => ( P @ M @ N ) ) ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_63_zero__induct,axiom,
% 2.66/1.51 ! [P: nat > $o,K: nat] :
% 2.66/1.51 ( ( P @ K )
% 2.66/1.51 => ( ! [N2: nat] :
% 2.66/1.51 ( ( P @ ( suc @ N2 ) )
% 2.66/1.51 => ( P @ N2 ) )
% 2.66/1.51 => ( P @ zero_zero_nat ) ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_64_Suc__neq__Zero,axiom,
% 2.66/1.51 ! [M: nat] :
% 2.66/1.51 ( ( suc @ M )
% 2.66/1.51 != zero_zero_nat ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_65_Zero__neq__Suc,axiom,
% 2.66/1.51 ! [M: nat] :
% 2.66/1.51 ( zero_zero_nat
% 2.66/1.51 != ( suc @ M ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_66_Zero__not__Suc,axiom,
% 2.66/1.51 ! [M: nat] :
% 2.66/1.51 ( zero_zero_nat
% 2.66/1.51 != ( suc @ M ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_67_old_Onat_Oexhaust,axiom,
% 2.66/1.51 ! [Y: nat] :
% 2.66/1.51 ( ( Y != zero_zero_nat )
% 2.66/1.51 => ~ ! [Nat3: nat] :
% 2.66/1.51 ( Y
% 2.66/1.51 != ( suc @ Nat3 ) ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_68_old_Onat_Oinducts,axiom,
% 2.66/1.51 ! [P: nat > $o,Nat: nat] :
% 2.66/1.51 ( ( P @ zero_zero_nat )
% 2.66/1.51 => ( ! [Nat3: nat] :
% 2.66/1.51 ( ( P @ Nat3 )
% 2.66/1.51 => ( P @ ( suc @ Nat3 ) ) )
% 2.66/1.51 => ( P @ Nat ) ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_69_not0__implies__Suc,axiom,
% 2.66/1.51 ! [N: nat] :
% 2.66/1.51 ( ( N != zero_zero_nat )
% 2.66/1.51 => ? [M3: nat] :
% 2.66/1.51 ( N
% 2.66/1.51 = ( suc @ M3 ) ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_70_Nat_OlessE,axiom,
% 2.66/1.51 ! [I: nat,K: nat] :
% 2.66/1.51 ( ( ord_less_nat @ I @ K )
% 2.66/1.51 => ( ( K
% 2.66/1.51 != ( suc @ I ) )
% 2.66/1.51 => ~ ! [J: nat] :
% 2.66/1.51 ( ( ord_less_nat @ I @ J )
% 2.66/1.51 => ( K
% 2.66/1.51 != ( suc @ J ) ) ) ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_71_Suc__lessD,axiom,
% 2.66/1.51 ! [M: nat,N: nat] :
% 2.66/1.51 ( ( ord_less_nat @ ( suc @ M ) @ N )
% 2.66/1.51 => ( ord_less_nat @ M @ N ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_72_Suc__lessE,axiom,
% 2.66/1.51 ! [I: nat,K: nat] :
% 2.66/1.51 ( ( ord_less_nat @ ( suc @ I ) @ K )
% 2.66/1.51 => ~ ! [J: nat] :
% 2.66/1.51 ( ( ord_less_nat @ I @ J )
% 2.66/1.51 => ( K
% 2.66/1.51 != ( suc @ J ) ) ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_73_Suc__lessI,axiom,
% 2.66/1.51 ! [M: nat,N: nat] :
% 2.66/1.51 ( ( ord_less_nat @ M @ N )
% 2.66/1.51 => ( ( ( suc @ M )
% 2.66/1.51 != N )
% 2.66/1.51 => ( ord_less_nat @ ( suc @ M ) @ N ) ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_74_less__SucE,axiom,
% 2.66/1.51 ! [M: nat,N: nat] :
% 2.66/1.51 ( ( ord_less_nat @ M @ ( suc @ N ) )
% 2.66/1.51 => ( ~ ( ord_less_nat @ M @ N )
% 2.66/1.51 => ( M = N ) ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_75_less__SucI,axiom,
% 2.66/1.51 ! [M: nat,N: nat] :
% 2.66/1.51 ( ( ord_less_nat @ M @ N )
% 2.66/1.51 => ( ord_less_nat @ M @ ( suc @ N ) ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_76_Ex__less__Suc,axiom,
% 2.66/1.51 ! [N: nat,P: nat > $o] :
% 2.66/1.51 ( ( ? [I2: nat] :
% 2.66/1.51 ( ( ord_less_nat @ I2 @ ( suc @ N ) )
% 2.66/1.51 & ( P @ I2 ) ) )
% 2.66/1.51 = ( ( P @ N )
% 2.66/1.51 | ? [I2: nat] :
% 2.66/1.51 ( ( ord_less_nat @ I2 @ N )
% 2.66/1.51 & ( P @ I2 ) ) ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_77_less__Suc__eq,axiom,
% 2.66/1.51 ! [M: nat,N: nat] :
% 2.66/1.51 ( ( ord_less_nat @ M @ ( suc @ N ) )
% 2.66/1.51 = ( ( ord_less_nat @ M @ N )
% 2.66/1.51 | ( M = N ) ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_78_not__less__eq,axiom,
% 2.66/1.51 ! [M: nat,N: nat] :
% 2.66/1.51 ( ( ~ ( ord_less_nat @ M @ N ) )
% 2.66/1.51 = ( ord_less_nat @ N @ ( suc @ M ) ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_79_All__less__Suc,axiom,
% 2.66/1.51 ! [N: nat,P: nat > $o] :
% 2.66/1.51 ( ( ! [I2: nat] :
% 2.66/1.51 ( ( ord_less_nat @ I2 @ ( suc @ N ) )
% 2.66/1.51 => ( P @ I2 ) ) )
% 2.66/1.51 = ( ( P @ N )
% 2.66/1.51 & ! [I2: nat] :
% 2.66/1.51 ( ( ord_less_nat @ I2 @ N )
% 2.66/1.51 => ( P @ I2 ) ) ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_80_Suc__less__eq2,axiom,
% 2.66/1.51 ! [N: nat,M: nat] :
% 2.66/1.51 ( ( ord_less_nat @ ( suc @ N ) @ M )
% 2.66/1.51 = ( ? [M4: nat] :
% 2.66/1.51 ( ( M
% 2.66/1.51 = ( suc @ M4 ) )
% 2.66/1.51 & ( ord_less_nat @ N @ M4 ) ) ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_81_less__antisym,axiom,
% 2.66/1.51 ! [N: nat,M: nat] :
% 2.66/1.51 ( ~ ( ord_less_nat @ N @ M )
% 2.66/1.51 => ( ( ord_less_nat @ N @ ( suc @ M ) )
% 2.66/1.51 => ( M = N ) ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_82_Suc__less__SucD,axiom,
% 2.66/1.51 ! [M: nat,N: nat] :
% 2.66/1.51 ( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
% 2.66/1.51 => ( ord_less_nat @ M @ N ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_83_less__trans__Suc,axiom,
% 2.66/1.51 ! [I: nat,J2: nat,K: nat] :
% 2.66/1.51 ( ( ord_less_nat @ I @ J2 )
% 2.66/1.51 => ( ( ord_less_nat @ J2 @ K )
% 2.66/1.51 => ( ord_less_nat @ ( suc @ I ) @ K ) ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_84_less__Suc__induct,axiom,
% 2.66/1.51 ! [I: nat,J2: nat,P: nat > nat > $o] :
% 2.66/1.51 ( ( ord_less_nat @ I @ J2 )
% 2.66/1.51 => ( ! [I3: nat] : ( P @ I3 @ ( suc @ I3 ) )
% 2.66/1.51 => ( ! [I3: nat,J: nat,K2: nat] :
% 2.66/1.51 ( ( ord_less_nat @ I3 @ J )
% 2.66/1.51 => ( ( ord_less_nat @ J @ K2 )
% 2.66/1.51 => ( ( P @ I3 @ J )
% 2.66/1.51 => ( ( P @ J @ K2 )
% 2.66/1.51 => ( P @ I3 @ K2 ) ) ) ) )
% 2.66/1.51 => ( P @ I @ J2 ) ) ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_85_strict__inc__induct,axiom,
% 2.66/1.51 ! [I: nat,J2: nat,P: nat > $o] :
% 2.66/1.51 ( ( ord_less_nat @ I @ J2 )
% 2.66/1.51 => ( ! [I3: nat] :
% 2.66/1.51 ( ( J2
% 2.66/1.51 = ( suc @ I3 ) )
% 2.66/1.51 => ( P @ I3 ) )
% 2.66/1.51 => ( ! [I3: nat] :
% 2.66/1.51 ( ( ord_less_nat @ I3 @ J2 )
% 2.66/1.51 => ( ( P @ ( suc @ I3 ) )
% 2.66/1.51 => ( P @ I3 ) ) )
% 2.66/1.51 => ( P @ I ) ) ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_86_not__less__less__Suc__eq,axiom,
% 2.66/1.51 ! [N: nat,M: nat] :
% 2.66/1.51 ( ~ ( ord_less_nat @ N @ M )
% 2.66/1.51 => ( ( ord_less_nat @ N @ ( suc @ M ) )
% 2.66/1.51 = ( N = M ) ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_87_of__nat__less__0__iff,axiom,
% 2.66/1.51 ! [M: nat] :
% 2.66/1.51 ~ ( ord_less_nat @ ( semiri1382578993at_nat @ M ) @ zero_zero_nat ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_88_of__nat__less__0__iff,axiom,
% 2.66/1.51 ! [M: nat] :
% 2.66/1.51 ~ ( ord_less_int @ ( semiri2019852685at_int @ M ) @ zero_zero_int ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_89_less__imp__of__nat__less,axiom,
% 2.66/1.51 ! [M: nat,N: nat] :
% 2.66/1.51 ( ( ord_less_nat @ M @ N )
% 2.66/1.51 => ( ord_less_nat @ ( semiri1382578993at_nat @ M ) @ ( semiri1382578993at_nat @ N ) ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_90_less__imp__of__nat__less,axiom,
% 2.66/1.51 ! [M: nat,N: nat] :
% 2.66/1.51 ( ( ord_less_nat @ M @ N )
% 2.66/1.51 => ( ord_less_int @ ( semiri2019852685at_int @ M ) @ ( semiri2019852685at_int @ N ) ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_91_of__nat__less__imp__less,axiom,
% 2.66/1.51 ! [M: nat,N: nat] :
% 2.66/1.51 ( ( ord_less_nat @ ( semiri1382578993at_nat @ M ) @ ( semiri1382578993at_nat @ N ) )
% 2.66/1.51 => ( ord_less_nat @ M @ N ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_92_of__nat__less__imp__less,axiom,
% 2.66/1.51 ! [M: nat,N: nat] :
% 2.66/1.51 ( ( ord_less_int @ ( semiri2019852685at_int @ M ) @ ( semiri2019852685at_int @ N ) )
% 2.66/1.51 => ( ord_less_nat @ M @ N ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_93_lift__Suc__mono__less,axiom,
% 2.66/1.51 ! [F: nat > nat,N: nat,N3: nat] :
% 2.66/1.51 ( ! [N2: nat] : ( ord_less_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
% 2.66/1.51 => ( ( ord_less_nat @ N @ N3 )
% 2.66/1.51 => ( ord_less_nat @ ( F @ N ) @ ( F @ N3 ) ) ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_94_lift__Suc__mono__less,axiom,
% 2.66/1.51 ! [F: nat > int,N: nat,N3: nat] :
% 2.66/1.51 ( ! [N2: nat] : ( ord_less_int @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
% 2.66/1.51 => ( ( ord_less_nat @ N @ N3 )
% 2.66/1.51 => ( ord_less_int @ ( F @ N ) @ ( F @ N3 ) ) ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_95_lift__Suc__mono__less__iff,axiom,
% 2.66/1.51 ! [F: nat > nat,N: nat,M: nat] :
% 2.66/1.51 ( ! [N2: nat] : ( ord_less_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
% 2.66/1.51 => ( ( ord_less_nat @ ( F @ N ) @ ( F @ M ) )
% 2.66/1.51 = ( ord_less_nat @ N @ M ) ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_96_lift__Suc__mono__less__iff,axiom,
% 2.66/1.51 ! [F: nat > int,N: nat,M: nat] :
% 2.66/1.51 ( ! [N2: nat] : ( ord_less_int @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
% 2.66/1.51 => ( ( ord_less_int @ ( F @ N ) @ ( F @ M ) )
% 2.66/1.51 = ( ord_less_nat @ N @ M ) ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_97_Ex__less__Suc2,axiom,
% 2.66/1.51 ! [N: nat,P: nat > $o] :
% 2.66/1.51 ( ( ? [I2: nat] :
% 2.66/1.51 ( ( ord_less_nat @ I2 @ ( suc @ N ) )
% 2.66/1.51 & ( P @ I2 ) ) )
% 2.66/1.51 = ( ( P @ zero_zero_nat )
% 2.66/1.51 | ? [I2: nat] :
% 2.66/1.51 ( ( ord_less_nat @ I2 @ N )
% 2.66/1.51 & ( P @ ( suc @ I2 ) ) ) ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_98_gr0__conv__Suc,axiom,
% 2.66/1.51 ! [N: nat] :
% 2.66/1.51 ( ( ord_less_nat @ zero_zero_nat @ N )
% 2.66/1.51 = ( ? [M5: nat] :
% 2.66/1.51 ( N
% 2.66/1.51 = ( suc @ M5 ) ) ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_99_All__less__Suc2,axiom,
% 2.66/1.51 ! [N: nat,P: nat > $o] :
% 2.66/1.51 ( ( ! [I2: nat] :
% 2.66/1.51 ( ( ord_less_nat @ I2 @ ( suc @ N ) )
% 2.66/1.51 => ( P @ I2 ) ) )
% 2.66/1.51 = ( ( P @ zero_zero_nat )
% 2.66/1.51 & ! [I2: nat] :
% 2.66/1.51 ( ( ord_less_nat @ I2 @ N )
% 2.66/1.51 => ( P @ ( suc @ I2 ) ) ) ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_100_gr0__implies__Suc,axiom,
% 2.66/1.51 ! [N: nat] :
% 2.66/1.51 ( ( ord_less_nat @ zero_zero_nat @ N )
% 2.66/1.51 => ? [M3: nat] :
% 2.66/1.51 ( N
% 2.66/1.51 = ( suc @ M3 ) ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_101_less__Suc__eq__0__disj,axiom,
% 2.66/1.51 ! [M: nat,N: nat] :
% 2.66/1.51 ( ( ord_less_nat @ M @ ( suc @ N ) )
% 2.66/1.51 = ( ( M = zero_zero_nat )
% 2.66/1.51 | ? [J3: nat] :
% 2.66/1.51 ( ( M
% 2.66/1.51 = ( suc @ J3 ) )
% 2.66/1.51 & ( ord_less_nat @ J3 @ N ) ) ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_102_ord__eq__less__subst,axiom,
% 2.66/1.51 ! [A: nat,F: nat > nat,B: nat,C: nat] :
% 2.66/1.51 ( ( A
% 2.66/1.51 = ( F @ B ) )
% 2.66/1.51 => ( ( ord_less_nat @ B @ C )
% 2.66/1.51 => ( ! [X3: nat,Y3: nat] :
% 2.66/1.51 ( ( ord_less_nat @ X3 @ Y3 )
% 2.66/1.51 => ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
% 2.66/1.51 => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_103_ord__eq__less__subst,axiom,
% 2.66/1.51 ! [A: int,F: nat > int,B: nat,C: nat] :
% 2.66/1.51 ( ( A
% 2.66/1.51 = ( F @ B ) )
% 2.66/1.51 => ( ( ord_less_nat @ B @ C )
% 2.66/1.51 => ( ! [X3: nat,Y3: nat] :
% 2.66/1.51 ( ( ord_less_nat @ X3 @ Y3 )
% 2.66/1.51 => ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
% 2.66/1.51 => ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_104_ord__eq__less__subst,axiom,
% 2.66/1.51 ! [A: nat,F: int > nat,B: int,C: int] :
% 2.66/1.51 ( ( A
% 2.66/1.51 = ( F @ B ) )
% 2.66/1.51 => ( ( ord_less_int @ B @ C )
% 2.66/1.51 => ( ! [X3: int,Y3: int] :
% 2.66/1.51 ( ( ord_less_int @ X3 @ Y3 )
% 2.66/1.51 => ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
% 2.66/1.51 => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_105_ord__eq__less__subst,axiom,
% 2.66/1.51 ! [A: int,F: int > int,B: int,C: int] :
% 2.66/1.51 ( ( A
% 2.66/1.51 = ( F @ B ) )
% 2.66/1.51 => ( ( ord_less_int @ B @ C )
% 2.66/1.51 => ( ! [X3: int,Y3: int] :
% 2.66/1.51 ( ( ord_less_int @ X3 @ Y3 )
% 2.66/1.51 => ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
% 2.66/1.51 => ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_106_ord__less__eq__subst,axiom,
% 2.66/1.51 ! [A: nat,B: nat,F: nat > nat,C: nat] :
% 2.66/1.51 ( ( ord_less_nat @ A @ B )
% 2.66/1.51 => ( ( ( F @ B )
% 2.66/1.51 = C )
% 2.66/1.51 => ( ! [X3: nat,Y3: nat] :
% 2.66/1.51 ( ( ord_less_nat @ X3 @ Y3 )
% 2.66/1.51 => ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
% 2.66/1.51 => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_107_ord__less__eq__subst,axiom,
% 2.66/1.51 ! [A: nat,B: nat,F: nat > int,C: int] :
% 2.66/1.51 ( ( ord_less_nat @ A @ B )
% 2.66/1.51 => ( ( ( F @ B )
% 2.66/1.51 = C )
% 2.66/1.51 => ( ! [X3: nat,Y3: nat] :
% 2.66/1.51 ( ( ord_less_nat @ X3 @ Y3 )
% 2.66/1.51 => ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
% 2.66/1.51 => ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_108_ord__less__eq__subst,axiom,
% 2.66/1.51 ! [A: int,B: int,F: int > nat,C: nat] :
% 2.66/1.51 ( ( ord_less_int @ A @ B )
% 2.66/1.51 => ( ( ( F @ B )
% 2.66/1.51 = C )
% 2.66/1.51 => ( ! [X3: int,Y3: int] :
% 2.66/1.51 ( ( ord_less_int @ X3 @ Y3 )
% 2.66/1.51 => ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
% 2.66/1.51 => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_109_ord__less__eq__subst,axiom,
% 2.66/1.51 ! [A: int,B: int,F: int > int,C: int] :
% 2.66/1.51 ( ( ord_less_int @ A @ B )
% 2.66/1.51 => ( ( ( F @ B )
% 2.66/1.51 = C )
% 2.66/1.51 => ( ! [X3: int,Y3: int] :
% 2.66/1.51 ( ( ord_less_int @ X3 @ Y3 )
% 2.66/1.51 => ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
% 2.66/1.51 => ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_110_order__less__subst1,axiom,
% 2.66/1.51 ! [A: nat,F: nat > nat,B: nat,C: nat] :
% 2.66/1.51 ( ( ord_less_nat @ A @ ( F @ B ) )
% 2.66/1.51 => ( ( ord_less_nat @ B @ C )
% 2.66/1.51 => ( ! [X3: nat,Y3: nat] :
% 2.66/1.51 ( ( ord_less_nat @ X3 @ Y3 )
% 2.66/1.51 => ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
% 2.66/1.51 => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_111_order__less__subst1,axiom,
% 2.66/1.51 ! [A: nat,F: int > nat,B: int,C: int] :
% 2.66/1.51 ( ( ord_less_nat @ A @ ( F @ B ) )
% 2.66/1.51 => ( ( ord_less_int @ B @ C )
% 2.66/1.51 => ( ! [X3: int,Y3: int] :
% 2.66/1.51 ( ( ord_less_int @ X3 @ Y3 )
% 2.66/1.51 => ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
% 2.66/1.51 => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_112_order__less__subst1,axiom,
% 2.66/1.51 ! [A: int,F: nat > int,B: nat,C: nat] :
% 2.66/1.51 ( ( ord_less_int @ A @ ( F @ B ) )
% 2.66/1.51 => ( ( ord_less_nat @ B @ C )
% 2.66/1.51 => ( ! [X3: nat,Y3: nat] :
% 2.66/1.51 ( ( ord_less_nat @ X3 @ Y3 )
% 2.66/1.51 => ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
% 2.66/1.51 => ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_113_order__less__subst1,axiom,
% 2.66/1.51 ! [A: int,F: int > int,B: int,C: int] :
% 2.66/1.51 ( ( ord_less_int @ A @ ( F @ B ) )
% 2.66/1.51 => ( ( ord_less_int @ B @ C )
% 2.66/1.51 => ( ! [X3: int,Y3: int] :
% 2.66/1.51 ( ( ord_less_int @ X3 @ Y3 )
% 2.66/1.51 => ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
% 2.66/1.51 => ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_114_order__less__subst2,axiom,
% 2.66/1.51 ! [A: nat,B: nat,F: nat > nat,C: nat] :
% 2.66/1.51 ( ( ord_less_nat @ A @ B )
% 2.66/1.51 => ( ( ord_less_nat @ ( F @ B ) @ C )
% 2.66/1.51 => ( ! [X3: nat,Y3: nat] :
% 2.66/1.51 ( ( ord_less_nat @ X3 @ Y3 )
% 2.66/1.51 => ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
% 2.66/1.51 => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_115_order__less__subst2,axiom,
% 2.66/1.51 ! [A: nat,B: nat,F: nat > int,C: int] :
% 2.66/1.51 ( ( ord_less_nat @ A @ B )
% 2.66/1.51 => ( ( ord_less_int @ ( F @ B ) @ C )
% 2.66/1.51 => ( ! [X3: nat,Y3: nat] :
% 2.66/1.51 ( ( ord_less_nat @ X3 @ Y3 )
% 2.66/1.51 => ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
% 2.66/1.51 => ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_116_order__less__subst2,axiom,
% 2.66/1.51 ! [A: int,B: int,F: int > nat,C: nat] :
% 2.66/1.51 ( ( ord_less_int @ A @ B )
% 2.66/1.51 => ( ( ord_less_nat @ ( F @ B ) @ C )
% 2.66/1.51 => ( ! [X3: int,Y3: int] :
% 2.66/1.51 ( ( ord_less_int @ X3 @ Y3 )
% 2.66/1.51 => ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
% 2.66/1.51 => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_117_order__less__subst2,axiom,
% 2.66/1.51 ! [A: int,B: int,F: int > int,C: int] :
% 2.66/1.51 ( ( ord_less_int @ A @ B )
% 2.66/1.51 => ( ( ord_less_int @ ( F @ B ) @ C )
% 2.66/1.51 => ( ! [X3: int,Y3: int] :
% 2.66/1.51 ( ( ord_less_int @ X3 @ Y3 )
% 2.66/1.51 => ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
% 2.66/1.51 => ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_118_lt__ex,axiom,
% 2.66/1.51 ! [X: int] :
% 2.66/1.51 ? [Y3: int] : ( ord_less_int @ Y3 @ X ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_119_gt__ex,axiom,
% 2.66/1.51 ! [X: nat] :
% 2.66/1.51 ? [X_1: nat] : ( ord_less_nat @ X @ X_1 ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_120_gt__ex,axiom,
% 2.66/1.51 ! [X: int] :
% 2.66/1.51 ? [X_1: int] : ( ord_less_int @ X @ X_1 ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_121_neqE,axiom,
% 2.66/1.51 ! [X: nat,Y: nat] :
% 2.66/1.51 ( ( X != Y )
% 2.66/1.51 => ( ~ ( ord_less_nat @ X @ Y )
% 2.66/1.51 => ( ord_less_nat @ Y @ X ) ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_122_neqE,axiom,
% 2.66/1.51 ! [X: int,Y: int] :
% 2.66/1.51 ( ( X != Y )
% 2.66/1.51 => ( ~ ( ord_less_int @ X @ Y )
% 2.66/1.51 => ( ord_less_int @ Y @ X ) ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_123_neq__iff,axiom,
% 2.66/1.51 ! [X: nat,Y: nat] :
% 2.66/1.51 ( ( X != Y )
% 2.66/1.51 = ( ( ord_less_nat @ X @ Y )
% 2.66/1.51 | ( ord_less_nat @ Y @ X ) ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_124_neq__iff,axiom,
% 2.66/1.51 ! [X: int,Y: int] :
% 2.66/1.51 ( ( X != Y )
% 2.66/1.51 = ( ( ord_less_int @ X @ Y )
% 2.66/1.51 | ( ord_less_int @ Y @ X ) ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_125_order_Oasym,axiom,
% 2.66/1.51 ! [A: nat,B: nat] :
% 2.66/1.51 ( ( ord_less_nat @ A @ B )
% 2.66/1.51 => ~ ( ord_less_nat @ B @ A ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_126_order_Oasym,axiom,
% 2.66/1.51 ! [A: int,B: int] :
% 2.66/1.51 ( ( ord_less_int @ A @ B )
% 2.66/1.51 => ~ ( ord_less_int @ B @ A ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_127_less__imp__neq,axiom,
% 2.66/1.51 ! [X: nat,Y: nat] :
% 2.66/1.51 ( ( ord_less_nat @ X @ Y )
% 2.66/1.51 => ( X != Y ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_128_less__imp__neq,axiom,
% 2.66/1.51 ! [X: int,Y: int] :
% 2.66/1.51 ( ( ord_less_int @ X @ Y )
% 2.66/1.51 => ( X != Y ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_129_less__asym,axiom,
% 2.66/1.51 ! [X: nat,Y: nat] :
% 2.66/1.51 ( ( ord_less_nat @ X @ Y )
% 2.66/1.51 => ~ ( ord_less_nat @ Y @ X ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_130_less__asym,axiom,
% 2.66/1.51 ! [X: int,Y: int] :
% 2.66/1.51 ( ( ord_less_int @ X @ Y )
% 2.66/1.51 => ~ ( ord_less_int @ Y @ X ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_131_less__asym_H,axiom,
% 2.66/1.51 ! [A: nat,B: nat] :
% 2.66/1.51 ( ( ord_less_nat @ A @ B )
% 2.66/1.51 => ~ ( ord_less_nat @ B @ A ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_132_less__asym_H,axiom,
% 2.66/1.51 ! [A: int,B: int] :
% 2.66/1.51 ( ( ord_less_int @ A @ B )
% 2.66/1.51 => ~ ( ord_less_int @ B @ A ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_133_less__trans,axiom,
% 2.66/1.51 ! [X: nat,Y: nat,Z: nat] :
% 2.66/1.51 ( ( ord_less_nat @ X @ Y )
% 2.66/1.51 => ( ( ord_less_nat @ Y @ Z )
% 2.66/1.51 => ( ord_less_nat @ X @ Z ) ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_134_less__trans,axiom,
% 2.66/1.51 ! [X: int,Y: int,Z: int] :
% 2.66/1.51 ( ( ord_less_int @ X @ Y )
% 2.66/1.51 => ( ( ord_less_int @ Y @ Z )
% 2.66/1.51 => ( ord_less_int @ X @ Z ) ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_135_less__linear,axiom,
% 2.66/1.51 ! [X: nat,Y: nat] :
% 2.66/1.51 ( ( ord_less_nat @ X @ Y )
% 2.66/1.51 | ( X = Y )
% 2.66/1.51 | ( ord_less_nat @ Y @ X ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_136_less__linear,axiom,
% 2.66/1.51 ! [X: int,Y: int] :
% 2.66/1.51 ( ( ord_less_int @ X @ Y )
% 2.66/1.51 | ( X = Y )
% 2.66/1.51 | ( ord_less_int @ Y @ X ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_137_less__irrefl,axiom,
% 2.66/1.51 ! [X: nat] :
% 2.66/1.51 ~ ( ord_less_nat @ X @ X ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_138_less__irrefl,axiom,
% 2.66/1.51 ! [X: int] :
% 2.66/1.51 ~ ( ord_less_int @ X @ X ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_139_ord__eq__less__trans,axiom,
% 2.66/1.51 ! [A: nat,B: nat,C: nat] :
% 2.66/1.51 ( ( A = B )
% 2.66/1.51 => ( ( ord_less_nat @ B @ C )
% 2.66/1.51 => ( ord_less_nat @ A @ C ) ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_140_ord__eq__less__trans,axiom,
% 2.66/1.51 ! [A: int,B: int,C: int] :
% 2.66/1.51 ( ( A = B )
% 2.66/1.51 => ( ( ord_less_int @ B @ C )
% 2.66/1.51 => ( ord_less_int @ A @ C ) ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_141_ord__less__eq__trans,axiom,
% 2.66/1.51 ! [A: nat,B: nat,C: nat] :
% 2.66/1.51 ( ( ord_less_nat @ A @ B )
% 2.66/1.51 => ( ( B = C )
% 2.66/1.51 => ( ord_less_nat @ A @ C ) ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_142_ord__less__eq__trans,axiom,
% 2.66/1.51 ! [A: int,B: int,C: int] :
% 2.66/1.51 ( ( ord_less_int @ A @ B )
% 2.66/1.51 => ( ( B = C )
% 2.66/1.51 => ( ord_less_int @ A @ C ) ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_143_dual__order_Oasym,axiom,
% 2.66/1.51 ! [B: nat,A: nat] :
% 2.66/1.51 ( ( ord_less_nat @ B @ A )
% 2.66/1.51 => ~ ( ord_less_nat @ A @ B ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_144_dual__order_Oasym,axiom,
% 2.66/1.51 ! [B: int,A: int] :
% 2.66/1.51 ( ( ord_less_int @ B @ A )
% 2.66/1.51 => ~ ( ord_less_int @ A @ B ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_145_less__imp__not__eq,axiom,
% 2.66/1.51 ! [X: nat,Y: nat] :
% 2.66/1.51 ( ( ord_less_nat @ X @ Y )
% 2.66/1.51 => ( X != Y ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_146_less__imp__not__eq,axiom,
% 2.66/1.51 ! [X: int,Y: int] :
% 2.66/1.51 ( ( ord_less_int @ X @ Y )
% 2.66/1.51 => ( X != Y ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_147_less__not__sym,axiom,
% 2.66/1.51 ! [X: nat,Y: nat] :
% 2.66/1.51 ( ( ord_less_nat @ X @ Y )
% 2.66/1.51 => ~ ( ord_less_nat @ Y @ X ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_148_less__not__sym,axiom,
% 2.66/1.51 ! [X: int,Y: int] :
% 2.66/1.51 ( ( ord_less_int @ X @ Y )
% 2.66/1.51 => ~ ( ord_less_int @ Y @ X ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_149_less__induct,axiom,
% 2.66/1.51 ! [P: nat > $o,A: nat] :
% 2.66/1.51 ( ! [X3: nat] :
% 2.66/1.51 ( ! [Y4: nat] :
% 2.66/1.51 ( ( ord_less_nat @ Y4 @ X3 )
% 2.66/1.51 => ( P @ Y4 ) )
% 2.66/1.51 => ( P @ X3 ) )
% 2.66/1.51 => ( P @ A ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_150_antisym__conv3,axiom,
% 2.66/1.51 ! [Y: nat,X: nat] :
% 2.66/1.51 ( ~ ( ord_less_nat @ Y @ X )
% 2.66/1.51 => ( ( ~ ( ord_less_nat @ X @ Y ) )
% 2.66/1.51 = ( X = Y ) ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_151_antisym__conv3,axiom,
% 2.66/1.51 ! [Y: int,X: int] :
% 2.66/1.51 ( ~ ( ord_less_int @ Y @ X )
% 2.66/1.51 => ( ( ~ ( ord_less_int @ X @ Y ) )
% 2.66/1.51 = ( X = Y ) ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_152_less__imp__not__eq2,axiom,
% 2.66/1.51 ! [X: nat,Y: nat] :
% 2.66/1.51 ( ( ord_less_nat @ X @ Y )
% 2.66/1.51 => ( Y != X ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_153_less__imp__not__eq2,axiom,
% 2.66/1.51 ! [X: int,Y: int] :
% 2.66/1.51 ( ( ord_less_int @ X @ Y )
% 2.66/1.51 => ( Y != X ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_154_less__imp__triv,axiom,
% 2.66/1.51 ! [X: nat,Y: nat,P: $o] :
% 2.66/1.51 ( ( ord_less_nat @ X @ Y )
% 2.66/1.51 => ( ( ord_less_nat @ Y @ X )
% 2.66/1.51 => P ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_155_less__imp__triv,axiom,
% 2.66/1.51 ! [X: int,Y: int,P: $o] :
% 2.66/1.51 ( ( ord_less_int @ X @ Y )
% 2.66/1.51 => ( ( ord_less_int @ Y @ X )
% 2.66/1.51 => P ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_156_linorder__cases,axiom,
% 2.66/1.51 ! [X: nat,Y: nat] :
% 2.66/1.51 ( ~ ( ord_less_nat @ X @ Y )
% 2.66/1.51 => ( ( X != Y )
% 2.66/1.51 => ( ord_less_nat @ Y @ X ) ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_157_linorder__cases,axiom,
% 2.66/1.51 ! [X: int,Y: int] :
% 2.66/1.51 ( ~ ( ord_less_int @ X @ Y )
% 2.66/1.51 => ( ( X != Y )
% 2.66/1.51 => ( ord_less_int @ Y @ X ) ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_158_dual__order_Oirrefl,axiom,
% 2.66/1.51 ! [A: nat] :
% 2.66/1.51 ~ ( ord_less_nat @ A @ A ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_159_dual__order_Oirrefl,axiom,
% 2.66/1.51 ! [A: int] :
% 2.66/1.51 ~ ( ord_less_int @ A @ A ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_160_order_Ostrict__trans,axiom,
% 2.66/1.51 ! [A: nat,B: nat,C: nat] :
% 2.66/1.51 ( ( ord_less_nat @ A @ B )
% 2.66/1.51 => ( ( ord_less_nat @ B @ C )
% 2.66/1.51 => ( ord_less_nat @ A @ C ) ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_161_order_Ostrict__trans,axiom,
% 2.66/1.51 ! [A: int,B: int,C: int] :
% 2.66/1.51 ( ( ord_less_int @ A @ B )
% 2.66/1.51 => ( ( ord_less_int @ B @ C )
% 2.66/1.51 => ( ord_less_int @ A @ C ) ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_162_less__imp__not__less,axiom,
% 2.66/1.51 ! [X: nat,Y: nat] :
% 2.66/1.51 ( ( ord_less_nat @ X @ Y )
% 2.66/1.51 => ~ ( ord_less_nat @ Y @ X ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_163_less__imp__not__less,axiom,
% 2.66/1.51 ! [X: int,Y: int] :
% 2.66/1.51 ( ( ord_less_int @ X @ Y )
% 2.66/1.51 => ~ ( ord_less_int @ Y @ X ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_164_exists__least__iff,axiom,
% 2.66/1.51 ( ( ^ [P4: nat > $o] :
% 2.66/1.51 ? [X4: nat] : ( P4 @ X4 ) )
% 2.66/1.51 = ( ^ [P5: nat > $o] :
% 2.66/1.51 ? [N4: nat] :
% 2.66/1.51 ( ( P5 @ N4 )
% 2.66/1.51 & ! [M5: nat] :
% 2.66/1.51 ( ( ord_less_nat @ M5 @ N4 )
% 2.66/1.51 => ~ ( P5 @ M5 ) ) ) ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_165_linorder__less__wlog,axiom,
% 2.66/1.51 ! [P: nat > nat > $o,A: nat,B: nat] :
% 2.66/1.51 ( ! [A2: nat,B2: nat] :
% 2.66/1.51 ( ( ord_less_nat @ A2 @ B2 )
% 2.66/1.51 => ( P @ A2 @ B2 ) )
% 2.66/1.51 => ( ! [A2: nat] : ( P @ A2 @ A2 )
% 2.66/1.51 => ( ! [A2: nat,B2: nat] :
% 2.66/1.51 ( ( P @ B2 @ A2 )
% 2.66/1.51 => ( P @ A2 @ B2 ) )
% 2.66/1.51 => ( P @ A @ B ) ) ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_166_linorder__less__wlog,axiom,
% 2.66/1.51 ! [P: int > int > $o,A: int,B: int] :
% 2.66/1.51 ( ! [A2: int,B2: int] :
% 2.66/1.51 ( ( ord_less_int @ A2 @ B2 )
% 2.66/1.51 => ( P @ A2 @ B2 ) )
% 2.66/1.51 => ( ! [A2: int] : ( P @ A2 @ A2 )
% 2.66/1.51 => ( ! [A2: int,B2: int] :
% 2.66/1.51 ( ( P @ B2 @ A2 )
% 2.66/1.51 => ( P @ A2 @ B2 ) )
% 2.66/1.51 => ( P @ A @ B ) ) ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_167_dual__order_Ostrict__trans,axiom,
% 2.66/1.51 ! [B: nat,A: nat,C: nat] :
% 2.66/1.51 ( ( ord_less_nat @ B @ A )
% 2.66/1.51 => ( ( ord_less_nat @ C @ B )
% 2.66/1.51 => ( ord_less_nat @ C @ A ) ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_168_dual__order_Ostrict__trans,axiom,
% 2.66/1.51 ! [B: int,A: int,C: int] :
% 2.66/1.51 ( ( ord_less_int @ B @ A )
% 2.66/1.51 => ( ( ord_less_int @ C @ B )
% 2.66/1.51 => ( ord_less_int @ C @ A ) ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_169_not__less__iff__gr__or__eq,axiom,
% 2.66/1.51 ! [X: nat,Y: nat] :
% 2.66/1.51 ( ( ~ ( ord_less_nat @ X @ Y ) )
% 2.66/1.51 = ( ( ord_less_nat @ Y @ X )
% 2.66/1.51 | ( X = Y ) ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_170_not__less__iff__gr__or__eq,axiom,
% 2.66/1.51 ! [X: int,Y: int] :
% 2.66/1.51 ( ( ~ ( ord_less_int @ X @ Y ) )
% 2.66/1.51 = ( ( ord_less_int @ Y @ X )
% 2.66/1.51 | ( X = Y ) ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_171_order_Ostrict__implies__not__eq,axiom,
% 2.66/1.51 ! [A: nat,B: nat] :
% 2.66/1.51 ( ( ord_less_nat @ A @ B )
% 2.66/1.51 => ( A != B ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_172_order_Ostrict__implies__not__eq,axiom,
% 2.66/1.51 ! [A: int,B: int] :
% 2.66/1.51 ( ( ord_less_int @ A @ B )
% 2.66/1.51 => ( A != B ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_173_pos__int__cases,axiom,
% 2.66/1.51 ! [K: int] :
% 2.66/1.51 ( ( ord_less_int @ zero_zero_int @ K )
% 2.66/1.51 => ~ ! [N2: nat] :
% 2.66/1.51 ( ( K
% 2.66/1.51 = ( semiri2019852685at_int @ N2 ) )
% 2.66/1.51 => ~ ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_174_zero__less__imp__eq__int,axiom,
% 2.66/1.51 ! [K: int] :
% 2.66/1.51 ( ( ord_less_int @ zero_zero_int @ K )
% 2.66/1.51 => ? [N2: nat] :
% 2.66/1.51 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 2.66/1.51 & ( K
% 2.66/1.51 = ( semiri2019852685at_int @ N2 ) ) ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_175_exists__least__lemma,axiom,
% 2.66/1.51 ! [P: nat > $o] :
% 2.66/1.51 ( ~ ( P @ zero_zero_nat )
% 2.66/1.51 => ( ? [X_12: nat] : ( P @ X_12 )
% 2.66/1.51 => ? [N2: nat] :
% 2.66/1.51 ( ~ ( P @ N2 )
% 2.66/1.51 & ( P @ ( suc @ N2 ) ) ) ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_176_of__nat__zero__less__power__iff,axiom,
% 2.66/1.51 ! [X: nat,N: nat] :
% 2.66/1.51 ( ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ ( semiri1382578993at_nat @ X ) @ N ) )
% 2.66/1.51 = ( ( ord_less_nat @ zero_zero_nat @ X )
% 2.66/1.51 | ( N = zero_zero_nat ) ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_177_of__nat__zero__less__power__iff,axiom,
% 2.66/1.51 ! [X: nat,N: nat] :
% 2.66/1.51 ( ( ord_less_int @ zero_zero_int @ ( power_power_int @ ( semiri2019852685at_int @ X ) @ N ) )
% 2.66/1.51 = ( ( ord_less_nat @ zero_zero_nat @ X )
% 2.66/1.51 | ( N = zero_zero_nat ) ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_178_of__nat__le__0__iff,axiom,
% 2.66/1.51 ! [M: nat] :
% 2.66/1.51 ( ( ord_less_eq_nat @ ( semiri1382578993at_nat @ M ) @ zero_zero_nat )
% 2.66/1.51 = ( M = zero_zero_nat ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_179_of__nat__le__0__iff,axiom,
% 2.66/1.51 ! [M: nat] :
% 2.66/1.51 ( ( ord_less_eq_int @ ( semiri2019852685at_int @ M ) @ zero_zero_int )
% 2.66/1.51 = ( M = zero_zero_nat ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_180_Suc__pred,axiom,
% 2.66/1.51 ! [N: nat] :
% 2.66/1.51 ( ( ord_less_nat @ zero_zero_nat @ N )
% 2.66/1.51 => ( ( suc @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) )
% 2.66/1.51 = N ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_181_order__refl,axiom,
% 2.66/1.51 ! [X: nat] : ( ord_less_eq_nat @ X @ X ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_182_order__refl,axiom,
% 2.66/1.51 ! [X: int] : ( ord_less_eq_int @ X @ X ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_183_bot__nat__0_Oextremum,axiom,
% 2.66/1.51 ! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_184_le0,axiom,
% 2.66/1.51 ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_185_Suc__le__mono,axiom,
% 2.66/1.51 ! [N: nat,M: nat] :
% 2.66/1.51 ( ( ord_less_eq_nat @ ( suc @ N ) @ ( suc @ M ) )
% 2.66/1.51 = ( ord_less_eq_nat @ N @ M ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_186_diff__diff__cancel,axiom,
% 2.66/1.51 ! [I: nat,N: nat] :
% 2.66/1.51 ( ( ord_less_eq_nat @ I @ N )
% 2.66/1.51 => ( ( minus_minus_nat @ N @ ( minus_minus_nat @ N @ I ) )
% 2.66/1.51 = I ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_187_le__zero__eq,axiom,
% 2.66/1.51 ! [N: nat] :
% 2.66/1.51 ( ( ord_less_eq_nat @ N @ zero_zero_nat )
% 2.66/1.51 = ( N = zero_zero_nat ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_188_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
% 2.66/1.51 ! [A: int] :
% 2.66/1.51 ( ( minus_minus_int @ A @ A )
% 2.66/1.51 = zero_zero_int ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_189_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
% 2.66/1.51 ! [A: nat] :
% 2.66/1.51 ( ( minus_minus_nat @ A @ A )
% 2.66/1.51 = zero_zero_nat ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_190_diff__zero,axiom,
% 2.66/1.51 ! [A: int] :
% 2.66/1.51 ( ( minus_minus_int @ A @ zero_zero_int )
% 2.66/1.51 = A ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_191_diff__zero,axiom,
% 2.66/1.51 ! [A: nat] :
% 2.66/1.51 ( ( minus_minus_nat @ A @ zero_zero_nat )
% 2.66/1.51 = A ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_192_zero__diff,axiom,
% 2.66/1.51 ! [A: nat] :
% 2.66/1.51 ( ( minus_minus_nat @ zero_zero_nat @ A )
% 2.66/1.51 = zero_zero_nat ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_193_diff__0__right,axiom,
% 2.66/1.51 ! [A: int] :
% 2.66/1.51 ( ( minus_minus_int @ A @ zero_zero_int )
% 2.66/1.51 = A ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_194_diff__self,axiom,
% 2.66/1.51 ! [A: int] :
% 2.66/1.51 ( ( minus_minus_int @ A @ A )
% 2.66/1.51 = zero_zero_int ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_195_of__nat__le__iff,axiom,
% 2.66/1.51 ! [M: nat,N: nat] :
% 2.66/1.51 ( ( ord_less_eq_nat @ ( semiri1382578993at_nat @ M ) @ ( semiri1382578993at_nat @ N ) )
% 2.66/1.51 = ( ord_less_eq_nat @ M @ N ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_196_of__nat__le__iff,axiom,
% 2.66/1.51 ! [M: nat,N: nat] :
% 2.66/1.51 ( ( ord_less_eq_int @ ( semiri2019852685at_int @ M ) @ ( semiri2019852685at_int @ N ) )
% 2.66/1.51 = ( ord_less_eq_nat @ M @ N ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_197_nat__power__eq__Suc__0__iff,axiom,
% 2.66/1.51 ! [X: nat,M: nat] :
% 2.66/1.51 ( ( ( power_power_nat @ X @ M )
% 2.66/1.51 = ( suc @ zero_zero_nat ) )
% 2.66/1.51 = ( ( M = zero_zero_nat )
% 2.66/1.51 | ( X
% 2.66/1.51 = ( suc @ zero_zero_nat ) ) ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_198_power__Suc__0,axiom,
% 2.66/1.51 ! [N: nat] :
% 2.66/1.51 ( ( power_power_nat @ ( suc @ zero_zero_nat ) @ N )
% 2.66/1.51 = ( suc @ zero_zero_nat ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_199_of__nat__power__eq__of__nat__cancel__iff,axiom,
% 2.66/1.51 ! [X: nat,B: nat,W: nat] :
% 2.66/1.51 ( ( ( semiri1382578993at_nat @ X )
% 2.66/1.51 = ( power_power_nat @ ( semiri1382578993at_nat @ B ) @ W ) )
% 2.66/1.51 = ( X
% 2.66/1.51 = ( power_power_nat @ B @ W ) ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_200_of__nat__power__eq__of__nat__cancel__iff,axiom,
% 2.66/1.51 ! [X: nat,B: nat,W: nat] :
% 2.66/1.51 ( ( ( semiri2019852685at_int @ X )
% 2.66/1.51 = ( power_power_int @ ( semiri2019852685at_int @ B ) @ W ) )
% 2.66/1.51 = ( X
% 2.66/1.51 = ( power_power_nat @ B @ W ) ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_201_of__nat__eq__of__nat__power__cancel__iff,axiom,
% 2.66/1.51 ! [B: nat,W: nat,X: nat] :
% 2.66/1.51 ( ( ( power_power_nat @ ( semiri1382578993at_nat @ B ) @ W )
% 2.66/1.51 = ( semiri1382578993at_nat @ X ) )
% 2.66/1.51 = ( ( power_power_nat @ B @ W )
% 2.66/1.51 = X ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_202_of__nat__eq__of__nat__power__cancel__iff,axiom,
% 2.66/1.51 ! [B: nat,W: nat,X: nat] :
% 2.66/1.51 ( ( ( power_power_int @ ( semiri2019852685at_int @ B ) @ W )
% 2.66/1.51 = ( semiri2019852685at_int @ X ) )
% 2.66/1.51 = ( ( power_power_nat @ B @ W )
% 2.66/1.51 = X ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_203_of__nat__power,axiom,
% 2.66/1.51 ! [M: nat,N: nat] :
% 2.66/1.51 ( ( semiri1382578993at_nat @ ( power_power_nat @ M @ N ) )
% 2.66/1.51 = ( power_power_nat @ ( semiri1382578993at_nat @ M ) @ N ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_204_of__nat__power,axiom,
% 2.66/1.51 ! [M: nat,N: nat] :
% 2.66/1.51 ( ( semiri2019852685at_int @ ( power_power_nat @ M @ N ) )
% 2.66/1.51 = ( power_power_int @ ( semiri2019852685at_int @ M ) @ N ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_205_nat__zero__less__power__iff,axiom,
% 2.66/1.51 ! [X: nat,N: nat] :
% 2.66/1.51 ( ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ X @ N ) )
% 2.66/1.51 = ( ( ord_less_nat @ zero_zero_nat @ X )
% 2.66/1.51 | ( N = zero_zero_nat ) ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_206_diff__is__0__eq_H,axiom,
% 2.66/1.51 ! [M: nat,N: nat] :
% 2.66/1.51 ( ( ord_less_eq_nat @ M @ N )
% 2.66/1.51 => ( ( minus_minus_nat @ M @ N )
% 2.66/1.51 = zero_zero_nat ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_207_diff__is__0__eq,axiom,
% 2.66/1.51 ! [M: nat,N: nat] :
% 2.66/1.51 ( ( ( minus_minus_nat @ M @ N )
% 2.66/1.51 = zero_zero_nat )
% 2.66/1.51 = ( ord_less_eq_nat @ M @ N ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_208_diff__self__eq__0,axiom,
% 2.66/1.51 ! [M: nat] :
% 2.66/1.51 ( ( minus_minus_nat @ M @ M )
% 2.66/1.51 = zero_zero_nat ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_209_diff__0__eq__0,axiom,
% 2.66/1.51 ! [N: nat] :
% 2.66/1.51 ( ( minus_minus_nat @ zero_zero_nat @ N )
% 2.66/1.51 = zero_zero_nat ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_210_Suc__diff__diff,axiom,
% 2.66/1.51 ! [M: nat,N: nat,K: nat] :
% 2.66/1.51 ( ( minus_minus_nat @ ( minus_minus_nat @ ( suc @ M ) @ N ) @ ( suc @ K ) )
% 2.66/1.51 = ( minus_minus_nat @ ( minus_minus_nat @ M @ N ) @ K ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_211_diff__Suc__Suc,axiom,
% 2.66/1.51 ! [M: nat,N: nat] :
% 2.66/1.51 ( ( minus_minus_nat @ ( suc @ M ) @ ( suc @ N ) )
% 2.66/1.51 = ( minus_minus_nat @ M @ N ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_212_diff__ge__0__iff__ge,axiom,
% 2.66/1.51 ! [A: int,B: int] :
% 2.66/1.51 ( ( ord_less_eq_int @ zero_zero_int @ ( minus_minus_int @ A @ B ) )
% 2.66/1.51 = ( ord_less_eq_int @ B @ A ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_213_diff__gt__0__iff__gt,axiom,
% 2.66/1.51 ! [A: int,B: int] :
% 2.66/1.51 ( ( ord_less_int @ zero_zero_int @ ( minus_minus_int @ A @ B ) )
% 2.66/1.51 = ( ord_less_int @ B @ A ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_214_power__0__Suc,axiom,
% 2.66/1.51 ! [N: nat] :
% 2.66/1.51 ( ( power_power_int @ zero_zero_int @ ( suc @ N ) )
% 2.66/1.51 = zero_zero_int ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_215_power__0__Suc,axiom,
% 2.66/1.51 ! [N: nat] :
% 2.66/1.51 ( ( power_power_nat @ zero_zero_nat @ ( suc @ N ) )
% 2.66/1.51 = zero_zero_nat ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_216_of__nat__power__le__of__nat__cancel__iff,axiom,
% 2.66/1.51 ! [X: nat,B: nat,W: nat] :
% 2.66/1.51 ( ( ord_less_eq_nat @ ( semiri1382578993at_nat @ X ) @ ( power_power_nat @ ( semiri1382578993at_nat @ B ) @ W ) )
% 2.66/1.51 = ( ord_less_eq_nat @ X @ ( power_power_nat @ B @ W ) ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_217_of__nat__power__le__of__nat__cancel__iff,axiom,
% 2.66/1.51 ! [X: nat,B: nat,W: nat] :
% 2.66/1.51 ( ( ord_less_eq_int @ ( semiri2019852685at_int @ X ) @ ( power_power_int @ ( semiri2019852685at_int @ B ) @ W ) )
% 2.66/1.51 = ( ord_less_eq_nat @ X @ ( power_power_nat @ B @ W ) ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_218_of__nat__le__of__nat__power__cancel__iff,axiom,
% 2.66/1.51 ! [B: nat,W: nat,X: nat] :
% 2.66/1.51 ( ( ord_less_eq_nat @ ( power_power_nat @ ( semiri1382578993at_nat @ B ) @ W ) @ ( semiri1382578993at_nat @ X ) )
% 2.66/1.51 = ( ord_less_eq_nat @ ( power_power_nat @ B @ W ) @ X ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_219_of__nat__le__of__nat__power__cancel__iff,axiom,
% 2.66/1.51 ! [B: nat,W: nat,X: nat] :
% 2.66/1.51 ( ( ord_less_eq_int @ ( power_power_int @ ( semiri2019852685at_int @ B ) @ W ) @ ( semiri2019852685at_int @ X ) )
% 2.66/1.51 = ( ord_less_eq_nat @ ( power_power_nat @ B @ W ) @ X ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_220_power__Suc0__right,axiom,
% 2.66/1.51 ! [A: nat] :
% 2.66/1.51 ( ( power_power_nat @ A @ ( suc @ zero_zero_nat ) )
% 2.66/1.51 = A ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_221_zero__less__diff,axiom,
% 2.66/1.51 ! [N: nat,M: nat] :
% 2.66/1.51 ( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N @ M ) )
% 2.66/1.51 = ( ord_less_nat @ M @ N ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_222_power__eq__0__iff,axiom,
% 2.66/1.51 ! [A: int,N: nat] :
% 2.66/1.51 ( ( ( power_power_int @ A @ N )
% 2.66/1.51 = zero_zero_int )
% 2.66/1.51 = ( ( A = zero_zero_int )
% 2.66/1.51 & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_223_power__eq__0__iff,axiom,
% 2.66/1.51 ! [A: nat,N: nat] :
% 2.66/1.51 ( ( ( power_power_nat @ A @ N )
% 2.66/1.51 = zero_zero_nat )
% 2.66/1.51 = ( ( A = zero_zero_nat )
% 2.66/1.51 & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_224_of__nat__less__of__nat__power__cancel__iff,axiom,
% 2.66/1.51 ! [B: nat,W: nat,X: nat] :
% 2.66/1.51 ( ( ord_less_nat @ ( power_power_nat @ ( semiri1382578993at_nat @ B ) @ W ) @ ( semiri1382578993at_nat @ X ) )
% 2.66/1.51 = ( ord_less_nat @ ( power_power_nat @ B @ W ) @ X ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_225_of__nat__less__of__nat__power__cancel__iff,axiom,
% 2.66/1.51 ! [B: nat,W: nat,X: nat] :
% 2.66/1.51 ( ( ord_less_int @ ( power_power_int @ ( semiri2019852685at_int @ B ) @ W ) @ ( semiri2019852685at_int @ X ) )
% 2.66/1.51 = ( ord_less_nat @ ( power_power_nat @ B @ W ) @ X ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_226_of__nat__power__less__of__nat__cancel__iff,axiom,
% 2.66/1.51 ! [X: nat,B: nat,W: nat] :
% 2.66/1.51 ( ( ord_less_nat @ ( semiri1382578993at_nat @ X ) @ ( power_power_nat @ ( semiri1382578993at_nat @ B ) @ W ) )
% 2.66/1.51 = ( ord_less_nat @ X @ ( power_power_nat @ B @ W ) ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_227_of__nat__power__less__of__nat__cancel__iff,axiom,
% 2.66/1.51 ! [X: nat,B: nat,W: nat] :
% 2.66/1.51 ( ( ord_less_int @ ( semiri2019852685at_int @ X ) @ ( power_power_int @ ( semiri2019852685at_int @ B ) @ W ) )
% 2.66/1.51 = ( ord_less_nat @ X @ ( power_power_nat @ B @ W ) ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_228_power__mono__iff,axiom,
% 2.66/1.51 ! [A: nat,B: nat,N: nat] :
% 2.66/1.51 ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 2.66/1.51 => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 2.66/1.51 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 2.66/1.51 => ( ( ord_less_eq_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) )
% 2.66/1.51 = ( ord_less_eq_nat @ A @ B ) ) ) ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_229_power__mono__iff,axiom,
% 2.66/1.51 ! [A: int,B: int,N: nat] :
% 2.66/1.51 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 2.66/1.51 => ( ( ord_less_eq_int @ zero_zero_int @ B )
% 2.66/1.51 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 2.66/1.51 => ( ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) )
% 2.66/1.51 = ( ord_less_eq_int @ A @ B ) ) ) ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_230_Suc__diff__le,axiom,
% 2.66/1.51 ! [N: nat,M: nat] :
% 2.66/1.51 ( ( ord_less_eq_nat @ N @ M )
% 2.66/1.51 => ( ( minus_minus_nat @ ( suc @ M ) @ N )
% 2.66/1.51 = ( suc @ ( minus_minus_nat @ M @ N ) ) ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_231_nonneg__int__cases,axiom,
% 2.66/1.51 ! [K: int] :
% 2.66/1.51 ( ( ord_less_eq_int @ zero_zero_int @ K )
% 2.66/1.51 => ~ ! [N2: nat] :
% 2.66/1.51 ( K
% 2.66/1.51 != ( semiri2019852685at_int @ N2 ) ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_232_zero__le__imp__eq__int,axiom,
% 2.66/1.51 ! [K: int] :
% 2.66/1.51 ( ( ord_less_eq_int @ zero_zero_int @ K )
% 2.66/1.51 => ? [N2: nat] :
% 2.66/1.51 ( K
% 2.66/1.51 = ( semiri2019852685at_int @ N2 ) ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_233_zero__integer_Orsp,axiom,
% 2.66/1.51 zero_zero_int = zero_zero_int ).
% 2.66/1.51
% 2.66/1.51 thf(fact_234_lift__Suc__mono__le,axiom,
% 2.66/1.51 ! [F: nat > nat,N: nat,N3: nat] :
% 2.66/1.51 ( ! [N2: nat] : ( ord_less_eq_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
% 2.66/1.51 => ( ( ord_less_eq_nat @ N @ N3 )
% 2.66/1.51 => ( ord_less_eq_nat @ ( F @ N ) @ ( F @ N3 ) ) ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_235_lift__Suc__mono__le,axiom,
% 2.66/1.51 ! [F: nat > int,N: nat,N3: nat] :
% 2.66/1.51 ( ! [N2: nat] : ( ord_less_eq_int @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
% 2.66/1.51 => ( ( ord_less_eq_nat @ N @ N3 )
% 2.66/1.51 => ( ord_less_eq_int @ ( F @ N ) @ ( F @ N3 ) ) ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_236_lift__Suc__antimono__le,axiom,
% 2.66/1.51 ! [F: nat > nat,N: nat,N3: nat] :
% 2.66/1.51 ( ! [N2: nat] : ( ord_less_eq_nat @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
% 2.66/1.51 => ( ( ord_less_eq_nat @ N @ N3 )
% 2.66/1.51 => ( ord_less_eq_nat @ ( F @ N3 ) @ ( F @ N ) ) ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_237_lift__Suc__antimono__le,axiom,
% 2.66/1.51 ! [F: nat > int,N: nat,N3: nat] :
% 2.66/1.51 ( ! [N2: nat] : ( ord_less_eq_int @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
% 2.66/1.51 => ( ( ord_less_eq_nat @ N @ N3 )
% 2.66/1.51 => ( ord_less_eq_int @ ( F @ N3 ) @ ( F @ N ) ) ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_238_of__nat__diff,axiom,
% 2.66/1.51 ! [N: nat,M: nat] :
% 2.66/1.51 ( ( ord_less_eq_nat @ N @ M )
% 2.66/1.51 => ( ( semiri1382578993at_nat @ ( minus_minus_nat @ M @ N ) )
% 2.66/1.51 = ( minus_minus_nat @ ( semiri1382578993at_nat @ M ) @ ( semiri1382578993at_nat @ N ) ) ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_239_of__nat__diff,axiom,
% 2.66/1.51 ! [N: nat,M: nat] :
% 2.66/1.51 ( ( ord_less_eq_nat @ N @ M )
% 2.66/1.51 => ( ( semiri2019852685at_int @ ( minus_minus_nat @ M @ N ) )
% 2.66/1.51 = ( minus_minus_int @ ( semiri2019852685at_int @ M ) @ ( semiri2019852685at_int @ N ) ) ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_240_power__inject__base,axiom,
% 2.66/1.51 ! [A: nat,N: nat,B: nat] :
% 2.66/1.51 ( ( ( power_power_nat @ A @ ( suc @ N ) )
% 2.66/1.51 = ( power_power_nat @ B @ ( suc @ N ) ) )
% 2.66/1.51 => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 2.66/1.51 => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 2.66/1.51 => ( A = B ) ) ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_241_power__inject__base,axiom,
% 2.66/1.51 ! [A: int,N: nat,B: int] :
% 2.66/1.51 ( ( ( power_power_int @ A @ ( suc @ N ) )
% 2.66/1.51 = ( power_power_int @ B @ ( suc @ N ) ) )
% 2.66/1.51 => ( ( ord_less_eq_int @ zero_zero_int @ A )
% 2.66/1.51 => ( ( ord_less_eq_int @ zero_zero_int @ B )
% 2.66/1.51 => ( A = B ) ) ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_242_of__nat__mono,axiom,
% 2.66/1.51 ! [I: nat,J2: nat] :
% 2.66/1.51 ( ( ord_less_eq_nat @ I @ J2 )
% 2.66/1.51 => ( ord_less_eq_nat @ ( semiri1382578993at_nat @ I ) @ ( semiri1382578993at_nat @ J2 ) ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_243_of__nat__mono,axiom,
% 2.66/1.51 ! [I: nat,J2: nat] :
% 2.66/1.51 ( ( ord_less_eq_nat @ I @ J2 )
% 2.66/1.51 => ( ord_less_eq_int @ ( semiri2019852685at_int @ I ) @ ( semiri2019852685at_int @ J2 ) ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_244_power__le__imp__le__base,axiom,
% 2.66/1.51 ! [A: nat,N: nat,B: nat] :
% 2.66/1.51 ( ( ord_less_eq_nat @ ( power_power_nat @ A @ ( suc @ N ) ) @ ( power_power_nat @ B @ ( suc @ N ) ) )
% 2.66/1.51 => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 2.66/1.51 => ( ord_less_eq_nat @ A @ B ) ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_245_power__le__imp__le__base,axiom,
% 2.66/1.51 ! [A: int,N: nat,B: int] :
% 2.66/1.51 ( ( ord_less_eq_int @ ( power_power_int @ A @ ( suc @ N ) ) @ ( power_power_int @ B @ ( suc @ N ) ) )
% 2.66/1.51 => ( ( ord_less_eq_int @ zero_zero_int @ B )
% 2.66/1.51 => ( ord_less_eq_int @ A @ B ) ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_246_dual__order_Oantisym,axiom,
% 2.66/1.51 ! [B: nat,A: nat] :
% 2.66/1.51 ( ( ord_less_eq_nat @ B @ A )
% 2.66/1.51 => ( ( ord_less_eq_nat @ A @ B )
% 2.66/1.51 => ( A = B ) ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_247_dual__order_Oantisym,axiom,
% 2.66/1.51 ! [B: int,A: int] :
% 2.66/1.51 ( ( ord_less_eq_int @ B @ A )
% 2.66/1.51 => ( ( ord_less_eq_int @ A @ B )
% 2.66/1.51 => ( A = B ) ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_248_dual__order_Oeq__iff,axiom,
% 2.66/1.51 ( ( ^ [Y5: nat,Z2: nat] : ( Y5 = Z2 ) )
% 2.66/1.51 = ( ^ [A3: nat,B3: nat] :
% 2.66/1.51 ( ( ord_less_eq_nat @ B3 @ A3 )
% 2.66/1.51 & ( ord_less_eq_nat @ A3 @ B3 ) ) ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_249_dual__order_Oeq__iff,axiom,
% 2.66/1.51 ( ( ^ [Y5: int,Z2: int] : ( Y5 = Z2 ) )
% 2.66/1.51 = ( ^ [A3: int,B3: int] :
% 2.66/1.51 ( ( ord_less_eq_int @ B3 @ A3 )
% 2.66/1.51 & ( ord_less_eq_int @ A3 @ B3 ) ) ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_250_dual__order_Otrans,axiom,
% 2.66/1.51 ! [B: nat,A: nat,C: nat] :
% 2.66/1.51 ( ( ord_less_eq_nat @ B @ A )
% 2.66/1.51 => ( ( ord_less_eq_nat @ C @ B )
% 2.66/1.51 => ( ord_less_eq_nat @ C @ A ) ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_251_dual__order_Otrans,axiom,
% 2.66/1.51 ! [B: int,A: int,C: int] :
% 2.66/1.51 ( ( ord_less_eq_int @ B @ A )
% 2.66/1.51 => ( ( ord_less_eq_int @ C @ B )
% 2.66/1.51 => ( ord_less_eq_int @ C @ A ) ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_252_linorder__wlog,axiom,
% 2.66/1.51 ! [P: nat > nat > $o,A: nat,B: nat] :
% 2.66/1.51 ( ! [A2: nat,B2: nat] :
% 2.66/1.51 ( ( ord_less_eq_nat @ A2 @ B2 )
% 2.66/1.51 => ( P @ A2 @ B2 ) )
% 2.66/1.51 => ( ! [A2: nat,B2: nat] :
% 2.66/1.51 ( ( P @ B2 @ A2 )
% 2.66/1.51 => ( P @ A2 @ B2 ) )
% 2.66/1.51 => ( P @ A @ B ) ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_253_linorder__wlog,axiom,
% 2.66/1.51 ! [P: int > int > $o,A: int,B: int] :
% 2.66/1.51 ( ! [A2: int,B2: int] :
% 2.66/1.51 ( ( ord_less_eq_int @ A2 @ B2 )
% 2.66/1.51 => ( P @ A2 @ B2 ) )
% 2.66/1.51 => ( ! [A2: int,B2: int] :
% 2.66/1.51 ( ( P @ B2 @ A2 )
% 2.66/1.51 => ( P @ A2 @ B2 ) )
% 2.66/1.51 => ( P @ A @ B ) ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_254_dual__order_Orefl,axiom,
% 2.66/1.51 ! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_255_dual__order_Orefl,axiom,
% 2.66/1.51 ! [A: int] : ( ord_less_eq_int @ A @ A ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_256_order__trans,axiom,
% 2.66/1.51 ! [X: nat,Y: nat,Z: nat] :
% 2.66/1.51 ( ( ord_less_eq_nat @ X @ Y )
% 2.66/1.51 => ( ( ord_less_eq_nat @ Y @ Z )
% 2.66/1.51 => ( ord_less_eq_nat @ X @ Z ) ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_257_order__trans,axiom,
% 2.66/1.51 ! [X: int,Y: int,Z: int] :
% 2.66/1.51 ( ( ord_less_eq_int @ X @ Y )
% 2.66/1.51 => ( ( ord_less_eq_int @ Y @ Z )
% 2.66/1.51 => ( ord_less_eq_int @ X @ Z ) ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_258_order__class_Oorder_Oantisym,axiom,
% 2.66/1.51 ! [A: nat,B: nat] :
% 2.66/1.51 ( ( ord_less_eq_nat @ A @ B )
% 2.66/1.51 => ( ( ord_less_eq_nat @ B @ A )
% 2.66/1.51 => ( A = B ) ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_259_order__class_Oorder_Oantisym,axiom,
% 2.66/1.51 ! [A: int,B: int] :
% 2.66/1.51 ( ( ord_less_eq_int @ A @ B )
% 2.66/1.51 => ( ( ord_less_eq_int @ B @ A )
% 2.66/1.51 => ( A = B ) ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_260_ord__le__eq__trans,axiom,
% 2.66/1.51 ! [A: nat,B: nat,C: nat] :
% 2.66/1.51 ( ( ord_less_eq_nat @ A @ B )
% 2.66/1.51 => ( ( B = C )
% 2.66/1.51 => ( ord_less_eq_nat @ A @ C ) ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_261_ord__le__eq__trans,axiom,
% 2.66/1.51 ! [A: int,B: int,C: int] :
% 2.66/1.51 ( ( ord_less_eq_int @ A @ B )
% 2.66/1.51 => ( ( B = C )
% 2.66/1.51 => ( ord_less_eq_int @ A @ C ) ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_262_ord__eq__le__trans,axiom,
% 2.66/1.51 ! [A: nat,B: nat,C: nat] :
% 2.66/1.51 ( ( A = B )
% 2.66/1.51 => ( ( ord_less_eq_nat @ B @ C )
% 2.66/1.51 => ( ord_less_eq_nat @ A @ C ) ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_263_ord__eq__le__trans,axiom,
% 2.66/1.51 ! [A: int,B: int,C: int] :
% 2.66/1.51 ( ( A = B )
% 2.66/1.51 => ( ( ord_less_eq_int @ B @ C )
% 2.66/1.51 => ( ord_less_eq_int @ A @ C ) ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_264_order__class_Oorder_Oeq__iff,axiom,
% 2.66/1.51 ( ( ^ [Y5: nat,Z2: nat] : ( Y5 = Z2 ) )
% 2.66/1.51 = ( ^ [A3: nat,B3: nat] :
% 2.66/1.51 ( ( ord_less_eq_nat @ A3 @ B3 )
% 2.66/1.51 & ( ord_less_eq_nat @ B3 @ A3 ) ) ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_265_order__class_Oorder_Oeq__iff,axiom,
% 2.66/1.51 ( ( ^ [Y5: int,Z2: int] : ( Y5 = Z2 ) )
% 2.66/1.51 = ( ^ [A3: int,B3: int] :
% 2.66/1.51 ( ( ord_less_eq_int @ A3 @ B3 )
% 2.66/1.51 & ( ord_less_eq_int @ B3 @ A3 ) ) ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_266_antisym__conv,axiom,
% 2.66/1.51 ! [Y: nat,X: nat] :
% 2.66/1.51 ( ( ord_less_eq_nat @ Y @ X )
% 2.66/1.51 => ( ( ord_less_eq_nat @ X @ Y )
% 2.66/1.51 = ( X = Y ) ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_267_antisym__conv,axiom,
% 2.66/1.51 ! [Y: int,X: int] :
% 2.66/1.51 ( ( ord_less_eq_int @ Y @ X )
% 2.66/1.51 => ( ( ord_less_eq_int @ X @ Y )
% 2.66/1.51 = ( X = Y ) ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_268_le__cases3,axiom,
% 2.66/1.51 ! [X: nat,Y: nat,Z: nat] :
% 2.66/1.51 ( ( ( ord_less_eq_nat @ X @ Y )
% 2.66/1.51 => ~ ( ord_less_eq_nat @ Y @ Z ) )
% 2.66/1.51 => ( ( ( ord_less_eq_nat @ Y @ X )
% 2.66/1.51 => ~ ( ord_less_eq_nat @ X @ Z ) )
% 2.66/1.51 => ( ( ( ord_less_eq_nat @ X @ Z )
% 2.66/1.51 => ~ ( ord_less_eq_nat @ Z @ Y ) )
% 2.66/1.51 => ( ( ( ord_less_eq_nat @ Z @ Y )
% 2.66/1.51 => ~ ( ord_less_eq_nat @ Y @ X ) )
% 2.66/1.51 => ( ( ( ord_less_eq_nat @ Y @ Z )
% 2.66/1.51 => ~ ( ord_less_eq_nat @ Z @ X ) )
% 2.66/1.51 => ~ ( ( ord_less_eq_nat @ Z @ X )
% 2.66/1.51 => ~ ( ord_less_eq_nat @ X @ Y ) ) ) ) ) ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_269_le__cases3,axiom,
% 2.66/1.51 ! [X: int,Y: int,Z: int] :
% 2.66/1.51 ( ( ( ord_less_eq_int @ X @ Y )
% 2.66/1.51 => ~ ( ord_less_eq_int @ Y @ Z ) )
% 2.66/1.51 => ( ( ( ord_less_eq_int @ Y @ X )
% 2.66/1.51 => ~ ( ord_less_eq_int @ X @ Z ) )
% 2.66/1.51 => ( ( ( ord_less_eq_int @ X @ Z )
% 2.66/1.51 => ~ ( ord_less_eq_int @ Z @ Y ) )
% 2.66/1.51 => ( ( ( ord_less_eq_int @ Z @ Y )
% 2.66/1.51 => ~ ( ord_less_eq_int @ Y @ X ) )
% 2.66/1.51 => ( ( ( ord_less_eq_int @ Y @ Z )
% 2.66/1.51 => ~ ( ord_less_eq_int @ Z @ X ) )
% 2.66/1.51 => ~ ( ( ord_less_eq_int @ Z @ X )
% 2.66/1.51 => ~ ( ord_less_eq_int @ X @ Y ) ) ) ) ) ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_270_order_Otrans,axiom,
% 2.66/1.51 ! [A: nat,B: nat,C: nat] :
% 2.66/1.51 ( ( ord_less_eq_nat @ A @ B )
% 2.66/1.51 => ( ( ord_less_eq_nat @ B @ C )
% 2.66/1.51 => ( ord_less_eq_nat @ A @ C ) ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_271_order_Otrans,axiom,
% 2.66/1.51 ! [A: int,B: int,C: int] :
% 2.66/1.51 ( ( ord_less_eq_int @ A @ B )
% 2.66/1.51 => ( ( ord_less_eq_int @ B @ C )
% 2.66/1.51 => ( ord_less_eq_int @ A @ C ) ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_272_le__cases,axiom,
% 2.66/1.51 ! [X: nat,Y: nat] :
% 2.66/1.51 ( ~ ( ord_less_eq_nat @ X @ Y )
% 2.66/1.51 => ( ord_less_eq_nat @ Y @ X ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_273_le__cases,axiom,
% 2.66/1.51 ! [X: int,Y: int] :
% 2.66/1.51 ( ~ ( ord_less_eq_int @ X @ Y )
% 2.66/1.51 => ( ord_less_eq_int @ Y @ X ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_274_eq__refl,axiom,
% 2.66/1.51 ! [X: nat,Y: nat] :
% 2.66/1.51 ( ( X = Y )
% 2.66/1.51 => ( ord_less_eq_nat @ X @ Y ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_275_eq__refl,axiom,
% 2.66/1.51 ! [X: int,Y: int] :
% 2.66/1.51 ( ( X = Y )
% 2.66/1.51 => ( ord_less_eq_int @ X @ Y ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_276_linear,axiom,
% 2.66/1.51 ! [X: nat,Y: nat] :
% 2.66/1.51 ( ( ord_less_eq_nat @ X @ Y )
% 2.66/1.51 | ( ord_less_eq_nat @ Y @ X ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_277_linear,axiom,
% 2.66/1.51 ! [X: int,Y: int] :
% 2.66/1.51 ( ( ord_less_eq_int @ X @ Y )
% 2.66/1.51 | ( ord_less_eq_int @ Y @ X ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_278_antisym,axiom,
% 2.66/1.51 ! [X: nat,Y: nat] :
% 2.66/1.51 ( ( ord_less_eq_nat @ X @ Y )
% 2.66/1.51 => ( ( ord_less_eq_nat @ Y @ X )
% 2.66/1.51 => ( X = Y ) ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_279_antisym,axiom,
% 2.66/1.51 ! [X: int,Y: int] :
% 2.66/1.51 ( ( ord_less_eq_int @ X @ Y )
% 2.66/1.51 => ( ( ord_less_eq_int @ Y @ X )
% 2.66/1.51 => ( X = Y ) ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_280_eq__iff,axiom,
% 2.66/1.51 ( ( ^ [Y5: nat,Z2: nat] : ( Y5 = Z2 ) )
% 2.66/1.51 = ( ^ [X5: nat,Y6: nat] :
% 2.66/1.51 ( ( ord_less_eq_nat @ X5 @ Y6 )
% 2.66/1.51 & ( ord_less_eq_nat @ Y6 @ X5 ) ) ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_281_eq__iff,axiom,
% 2.66/1.51 ( ( ^ [Y5: int,Z2: int] : ( Y5 = Z2 ) )
% 2.66/1.51 = ( ^ [X5: int,Y6: int] :
% 2.66/1.51 ( ( ord_less_eq_int @ X5 @ Y6 )
% 2.66/1.51 & ( ord_less_eq_int @ Y6 @ X5 ) ) ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_282_ord__le__eq__subst,axiom,
% 2.66/1.51 ! [A: nat,B: nat,F: nat > nat,C: nat] :
% 2.66/1.51 ( ( ord_less_eq_nat @ A @ B )
% 2.66/1.51 => ( ( ( F @ B )
% 2.66/1.51 = C )
% 2.66/1.51 => ( ! [X3: nat,Y3: nat] :
% 2.66/1.51 ( ( ord_less_eq_nat @ X3 @ Y3 )
% 2.66/1.51 => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
% 2.66/1.51 => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_283_ord__le__eq__subst,axiom,
% 2.66/1.51 ! [A: nat,B: nat,F: nat > int,C: int] :
% 2.66/1.51 ( ( ord_less_eq_nat @ A @ B )
% 2.66/1.51 => ( ( ( F @ B )
% 2.66/1.51 = C )
% 2.66/1.51 => ( ! [X3: nat,Y3: nat] :
% 2.66/1.51 ( ( ord_less_eq_nat @ X3 @ Y3 )
% 2.66/1.51 => ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
% 2.66/1.51 => ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_284_ord__le__eq__subst,axiom,
% 2.66/1.51 ! [A: int,B: int,F: int > nat,C: nat] :
% 2.66/1.51 ( ( ord_less_eq_int @ A @ B )
% 2.66/1.51 => ( ( ( F @ B )
% 2.66/1.51 = C )
% 2.66/1.51 => ( ! [X3: int,Y3: int] :
% 2.66/1.51 ( ( ord_less_eq_int @ X3 @ Y3 )
% 2.66/1.51 => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
% 2.66/1.51 => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_285_ord__le__eq__subst,axiom,
% 2.66/1.51 ! [A: int,B: int,F: int > int,C: int] :
% 2.66/1.51 ( ( ord_less_eq_int @ A @ B )
% 2.66/1.51 => ( ( ( F @ B )
% 2.66/1.51 = C )
% 2.66/1.51 => ( ! [X3: int,Y3: int] :
% 2.66/1.51 ( ( ord_less_eq_int @ X3 @ Y3 )
% 2.66/1.51 => ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
% 2.66/1.51 => ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_286_ord__eq__le__subst,axiom,
% 2.66/1.51 ! [A: nat,F: nat > nat,B: nat,C: nat] :
% 2.66/1.51 ( ( A
% 2.66/1.51 = ( F @ B ) )
% 2.66/1.51 => ( ( ord_less_eq_nat @ B @ C )
% 2.66/1.51 => ( ! [X3: nat,Y3: nat] :
% 2.66/1.51 ( ( ord_less_eq_nat @ X3 @ Y3 )
% 2.66/1.51 => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
% 2.66/1.51 => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_287_ord__eq__le__subst,axiom,
% 2.66/1.51 ! [A: int,F: nat > int,B: nat,C: nat] :
% 2.66/1.51 ( ( A
% 2.66/1.51 = ( F @ B ) )
% 2.66/1.51 => ( ( ord_less_eq_nat @ B @ C )
% 2.66/1.51 => ( ! [X3: nat,Y3: nat] :
% 2.66/1.51 ( ( ord_less_eq_nat @ X3 @ Y3 )
% 2.66/1.51 => ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
% 2.66/1.51 => ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_288_ord__eq__le__subst,axiom,
% 2.66/1.51 ! [A: nat,F: int > nat,B: int,C: int] :
% 2.66/1.51 ( ( A
% 2.66/1.51 = ( F @ B ) )
% 2.66/1.51 => ( ( ord_less_eq_int @ B @ C )
% 2.66/1.51 => ( ! [X3: int,Y3: int] :
% 2.66/1.51 ( ( ord_less_eq_int @ X3 @ Y3 )
% 2.66/1.51 => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
% 2.66/1.51 => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_289_ord__eq__le__subst,axiom,
% 2.66/1.51 ! [A: int,F: int > int,B: int,C: int] :
% 2.66/1.51 ( ( A
% 2.66/1.51 = ( F @ B ) )
% 2.66/1.51 => ( ( ord_less_eq_int @ B @ C )
% 2.66/1.51 => ( ! [X3: int,Y3: int] :
% 2.66/1.51 ( ( ord_less_eq_int @ X3 @ Y3 )
% 2.66/1.51 => ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
% 2.66/1.51 => ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_290_order__subst2,axiom,
% 2.66/1.51 ! [A: nat,B: nat,F: nat > nat,C: nat] :
% 2.66/1.51 ( ( ord_less_eq_nat @ A @ B )
% 2.66/1.51 => ( ( ord_less_eq_nat @ ( F @ B ) @ C )
% 2.66/1.51 => ( ! [X3: nat,Y3: nat] :
% 2.66/1.51 ( ( ord_less_eq_nat @ X3 @ Y3 )
% 2.66/1.51 => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
% 2.66/1.51 => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_291_order__subst2,axiom,
% 2.66/1.51 ! [A: nat,B: nat,F: nat > int,C: int] :
% 2.66/1.51 ( ( ord_less_eq_nat @ A @ B )
% 2.66/1.51 => ( ( ord_less_eq_int @ ( F @ B ) @ C )
% 2.66/1.51 => ( ! [X3: nat,Y3: nat] :
% 2.66/1.51 ( ( ord_less_eq_nat @ X3 @ Y3 )
% 2.66/1.51 => ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
% 2.66/1.51 => ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_292_order__subst2,axiom,
% 2.66/1.51 ! [A: int,B: int,F: int > nat,C: nat] :
% 2.66/1.51 ( ( ord_less_eq_int @ A @ B )
% 2.66/1.51 => ( ( ord_less_eq_nat @ ( F @ B ) @ C )
% 2.66/1.51 => ( ! [X3: int,Y3: int] :
% 2.66/1.51 ( ( ord_less_eq_int @ X3 @ Y3 )
% 2.66/1.51 => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
% 2.66/1.51 => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_293_order__subst2,axiom,
% 2.66/1.51 ! [A: int,B: int,F: int > int,C: int] :
% 2.66/1.51 ( ( ord_less_eq_int @ A @ B )
% 2.66/1.51 => ( ( ord_less_eq_int @ ( F @ B ) @ C )
% 2.66/1.51 => ( ! [X3: int,Y3: int] :
% 2.66/1.51 ( ( ord_less_eq_int @ X3 @ Y3 )
% 2.66/1.51 => ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
% 2.66/1.51 => ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_294_order__subst1,axiom,
% 2.66/1.51 ! [A: nat,F: nat > nat,B: nat,C: nat] :
% 2.66/1.51 ( ( ord_less_eq_nat @ A @ ( F @ B ) )
% 2.66/1.51 => ( ( ord_less_eq_nat @ B @ C )
% 2.66/1.51 => ( ! [X3: nat,Y3: nat] :
% 2.66/1.51 ( ( ord_less_eq_nat @ X3 @ Y3 )
% 2.66/1.51 => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
% 2.66/1.51 => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_295_order__subst1,axiom,
% 2.66/1.51 ! [A: nat,F: int > nat,B: int,C: int] :
% 2.66/1.51 ( ( ord_less_eq_nat @ A @ ( F @ B ) )
% 2.66/1.51 => ( ( ord_less_eq_int @ B @ C )
% 2.66/1.51 => ( ! [X3: int,Y3: int] :
% 2.66/1.51 ( ( ord_less_eq_int @ X3 @ Y3 )
% 2.66/1.51 => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
% 2.66/1.51 => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_296_order__subst1,axiom,
% 2.66/1.51 ! [A: int,F: nat > int,B: nat,C: nat] :
% 2.66/1.51 ( ( ord_less_eq_int @ A @ ( F @ B ) )
% 2.66/1.51 => ( ( ord_less_eq_nat @ B @ C )
% 2.66/1.51 => ( ! [X3: nat,Y3: nat] :
% 2.66/1.51 ( ( ord_less_eq_nat @ X3 @ Y3 )
% 2.66/1.51 => ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
% 2.66/1.51 => ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_297_order__subst1,axiom,
% 2.66/1.51 ! [A: int,F: int > int,B: int,C: int] :
% 2.66/1.51 ( ( ord_less_eq_int @ A @ ( F @ B ) )
% 2.66/1.51 => ( ( ord_less_eq_int @ B @ C )
% 2.66/1.51 => ( ! [X3: int,Y3: int] :
% 2.66/1.51 ( ( ord_less_eq_int @ X3 @ Y3 )
% 2.66/1.51 => ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
% 2.66/1.51 => ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_298_int__int__eq,axiom,
% 2.66/1.51 ! [M: nat,N: nat] :
% 2.66/1.51 ( ( ( semiri2019852685at_int @ M )
% 2.66/1.51 = ( semiri2019852685at_int @ N ) )
% 2.66/1.51 = ( M = N ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_299_zle__int,axiom,
% 2.66/1.51 ! [M: nat,N: nat] :
% 2.66/1.51 ( ( ord_less_eq_int @ ( semiri2019852685at_int @ M ) @ ( semiri2019852685at_int @ N ) )
% 2.66/1.51 = ( ord_less_eq_nat @ M @ N ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_300_le__iff__diff__le__0,axiom,
% 2.66/1.51 ( ord_less_eq_int
% 2.66/1.51 = ( ^ [A3: int,B3: int] : ( ord_less_eq_int @ ( minus_minus_int @ A3 @ B3 ) @ zero_zero_int ) ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_301_diff__less__mono,axiom,
% 2.66/1.51 ! [A: nat,B: nat,C: nat] :
% 2.66/1.51 ( ( ord_less_nat @ A @ B )
% 2.66/1.51 => ( ( ord_less_eq_nat @ C @ A )
% 2.66/1.51 => ( ord_less_nat @ ( minus_minus_nat @ A @ C ) @ ( minus_minus_nat @ B @ C ) ) ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_302_less__diff__iff,axiom,
% 2.66/1.51 ! [K: nat,M: nat,N: nat] :
% 2.66/1.51 ( ( ord_less_eq_nat @ K @ M )
% 2.66/1.51 => ( ( ord_less_eq_nat @ K @ N )
% 2.66/1.51 => ( ( ord_less_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
% 2.66/1.51 = ( ord_less_nat @ M @ N ) ) ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_303_power__not__zero,axiom,
% 2.66/1.51 ! [A: int,N: nat] :
% 2.66/1.51 ( ( A != zero_zero_int )
% 2.66/1.51 => ( ( power_power_int @ A @ N )
% 2.66/1.51 != zero_zero_int ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_304_power__not__zero,axiom,
% 2.66/1.51 ! [A: nat,N: nat] :
% 2.66/1.51 ( ( A != zero_zero_nat )
% 2.66/1.51 => ( ( power_power_nat @ A @ N )
% 2.66/1.51 != zero_zero_nat ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_305_power__less__imp__less__base,axiom,
% 2.66/1.51 ! [A: nat,N: nat,B: nat] :
% 2.66/1.51 ( ( ord_less_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) )
% 2.66/1.51 => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 2.66/1.51 => ( ord_less_nat @ A @ B ) ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_306_power__less__imp__less__base,axiom,
% 2.66/1.51 ! [A: int,N: nat,B: int] :
% 2.66/1.51 ( ( ord_less_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) )
% 2.66/1.51 => ( ( ord_less_eq_int @ zero_zero_int @ B )
% 2.66/1.51 => ( ord_less_int @ A @ B ) ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_307_diff__right__commute,axiom,
% 2.66/1.51 ! [A: nat,C: nat,B: nat] :
% 2.66/1.51 ( ( minus_minus_nat @ ( minus_minus_nat @ A @ C ) @ B )
% 2.66/1.51 = ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_308_diff__eq__diff__less__eq,axiom,
% 2.66/1.51 ! [A: int,B: int,C: int,D: int] :
% 2.66/1.51 ( ( ( minus_minus_int @ A @ B )
% 2.66/1.51 = ( minus_minus_int @ C @ D ) )
% 2.66/1.51 => ( ( ord_less_eq_int @ A @ B )
% 2.66/1.51 = ( ord_less_eq_int @ C @ D ) ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_309_diff__right__mono,axiom,
% 2.66/1.51 ! [A: int,B: int,C: int] :
% 2.66/1.51 ( ( ord_less_eq_int @ A @ B )
% 2.66/1.51 => ( ord_less_eq_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ C ) ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_310_diff__left__mono,axiom,
% 2.66/1.51 ! [B: int,A: int,C: int] :
% 2.66/1.51 ( ( ord_less_eq_int @ B @ A )
% 2.66/1.51 => ( ord_less_eq_int @ ( minus_minus_int @ C @ A ) @ ( minus_minus_int @ C @ B ) ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_311_zero__le__power,axiom,
% 2.66/1.51 ! [A: nat,N: nat] :
% 2.66/1.51 ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 2.66/1.51 => ( ord_less_eq_nat @ zero_zero_nat @ ( power_power_nat @ A @ N ) ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_312_zero__le__power,axiom,
% 2.66/1.51 ! [A: int,N: nat] :
% 2.66/1.51 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 2.66/1.51 => ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ N ) ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_313_diff__mono,axiom,
% 2.66/1.51 ! [A: int,B: int,D: int,C: int] :
% 2.66/1.51 ( ( ord_less_eq_int @ A @ B )
% 2.66/1.51 => ( ( ord_less_eq_int @ D @ C )
% 2.66/1.51 => ( ord_less_eq_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ D ) ) ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_314_power__mono,axiom,
% 2.66/1.51 ! [A: nat,B: nat,N: nat] :
% 2.66/1.51 ( ( ord_less_eq_nat @ A @ B )
% 2.66/1.51 => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 2.66/1.51 => ( ord_less_eq_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) ) ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_315_power__mono,axiom,
% 2.66/1.51 ! [A: int,B: int,N: nat] :
% 2.66/1.51 ( ( ord_less_eq_int @ A @ B )
% 2.66/1.51 => ( ( ord_less_eq_int @ zero_zero_int @ A )
% 2.66/1.51 => ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) ) ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_316_diff__le__mono2,axiom,
% 2.66/1.51 ! [M: nat,N: nat,L: nat] :
% 2.66/1.51 ( ( ord_less_eq_nat @ M @ N )
% 2.66/1.51 => ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_317_le__diff__iff_H,axiom,
% 2.66/1.51 ! [A: nat,C: nat,B: nat] :
% 2.66/1.51 ( ( ord_less_eq_nat @ A @ C )
% 2.66/1.51 => ( ( ord_less_eq_nat @ B @ C )
% 2.66/1.51 => ( ( ord_less_eq_nat @ ( minus_minus_nat @ C @ A ) @ ( minus_minus_nat @ C @ B ) )
% 2.66/1.51 = ( ord_less_eq_nat @ B @ A ) ) ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_318_diff__le__self,axiom,
% 2.66/1.51 ! [M: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ M ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_319_diff__le__mono,axiom,
% 2.66/1.51 ! [M: nat,N: nat,L: nat] :
% 2.66/1.51 ( ( ord_less_eq_nat @ M @ N )
% 2.66/1.51 => ( ord_less_eq_nat @ ( minus_minus_nat @ M @ L ) @ ( minus_minus_nat @ N @ L ) ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_320_Nat_Odiff__diff__eq,axiom,
% 2.66/1.51 ! [K: nat,M: nat,N: nat] :
% 2.66/1.51 ( ( ord_less_eq_nat @ K @ M )
% 2.66/1.51 => ( ( ord_less_eq_nat @ K @ N )
% 2.66/1.51 => ( ( minus_minus_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
% 2.66/1.51 = ( minus_minus_nat @ M @ N ) ) ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_321_diff__commute,axiom,
% 2.66/1.51 ! [I: nat,J2: nat,K: nat] :
% 2.66/1.51 ( ( minus_minus_nat @ ( minus_minus_nat @ I @ J2 ) @ K )
% 2.66/1.51 = ( minus_minus_nat @ ( minus_minus_nat @ I @ K ) @ J2 ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_322_le__diff__iff,axiom,
% 2.66/1.51 ! [K: nat,M: nat,N: nat] :
% 2.66/1.51 ( ( ord_less_eq_nat @ K @ M )
% 2.66/1.51 => ( ( ord_less_eq_nat @ K @ N )
% 2.66/1.51 => ( ( ord_less_eq_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
% 2.66/1.51 = ( ord_less_eq_nat @ M @ N ) ) ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_323_eq__diff__iff,axiom,
% 2.66/1.51 ! [K: nat,M: nat,N: nat] :
% 2.66/1.51 ( ( ord_less_eq_nat @ K @ M )
% 2.66/1.51 => ( ( ord_less_eq_nat @ K @ N )
% 2.66/1.51 => ( ( ( minus_minus_nat @ M @ K )
% 2.66/1.51 = ( minus_minus_nat @ N @ K ) )
% 2.66/1.51 = ( M = N ) ) ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_324_less__eq__int__code_I1_J,axiom,
% 2.66/1.51 ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% 2.66/1.51
% 2.66/1.51 thf(fact_325_less__int__code_I1_J,axiom,
% 2.66/1.51 ~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_326_nat__one__le__power,axiom,
% 2.66/1.51 ! [I: nat,N: nat] :
% 2.66/1.51 ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ I )
% 2.66/1.51 => ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( power_power_nat @ I @ N ) ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_327_power__eq__imp__eq__base,axiom,
% 2.66/1.51 ! [A: nat,N: nat,B: nat] :
% 2.66/1.51 ( ( ( power_power_nat @ A @ N )
% 2.66/1.51 = ( power_power_nat @ B @ N ) )
% 2.66/1.51 => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 2.66/1.51 => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 2.66/1.51 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 2.66/1.51 => ( A = B ) ) ) ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_328_power__eq__imp__eq__base,axiom,
% 2.66/1.51 ! [A: int,N: nat,B: int] :
% 2.66/1.51 ( ( ( power_power_int @ A @ N )
% 2.66/1.51 = ( power_power_int @ B @ N ) )
% 2.66/1.51 => ( ( ord_less_eq_int @ zero_zero_int @ A )
% 2.66/1.51 => ( ( ord_less_eq_int @ zero_zero_int @ B )
% 2.66/1.51 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 2.66/1.51 => ( A = B ) ) ) ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_329_power__eq__iff__eq__base,axiom,
% 2.66/1.51 ! [N: nat,A: nat,B: nat] :
% 2.66/1.51 ( ( ord_less_nat @ zero_zero_nat @ N )
% 2.66/1.51 => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 2.66/1.51 => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 2.66/1.51 => ( ( ( power_power_nat @ A @ N )
% 2.66/1.51 = ( power_power_nat @ B @ N ) )
% 2.66/1.51 = ( A = B ) ) ) ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_330_power__eq__iff__eq__base,axiom,
% 2.66/1.51 ! [N: nat,A: int,B: int] :
% 2.66/1.51 ( ( ord_less_nat @ zero_zero_nat @ N )
% 2.66/1.51 => ( ( ord_less_eq_int @ zero_zero_int @ A )
% 2.66/1.51 => ( ( ord_less_eq_int @ zero_zero_int @ B )
% 2.66/1.51 => ( ( ( power_power_int @ A @ N )
% 2.66/1.51 = ( power_power_int @ B @ N ) )
% 2.66/1.51 = ( A = B ) ) ) ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_331_eq__iff__diff__eq__0,axiom,
% 2.66/1.51 ( ( ^ [Y5: int,Z2: int] : ( Y5 = Z2 ) )
% 2.66/1.51 = ( ^ [A3: int,B3: int] :
% 2.66/1.51 ( ( minus_minus_int @ A3 @ B3 )
% 2.66/1.51 = zero_zero_int ) ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_332_diff__strict__right__mono,axiom,
% 2.66/1.51 ! [A: int,B: int,C: int] :
% 2.66/1.51 ( ( ord_less_int @ A @ B )
% 2.66/1.51 => ( ord_less_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ C ) ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_333_diff__strict__left__mono,axiom,
% 2.66/1.51 ! [B: int,A: int,C: int] :
% 2.66/1.51 ( ( ord_less_int @ B @ A )
% 2.66/1.51 => ( ord_less_int @ ( minus_minus_int @ C @ A ) @ ( minus_minus_int @ C @ B ) ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_334_diff__eq__diff__less,axiom,
% 2.66/1.51 ! [A: int,B: int,C: int,D: int] :
% 2.66/1.51 ( ( ( minus_minus_int @ A @ B )
% 2.66/1.51 = ( minus_minus_int @ C @ D ) )
% 2.66/1.51 => ( ( ord_less_int @ A @ B )
% 2.66/1.51 = ( ord_less_int @ C @ D ) ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_335_diff__strict__mono,axiom,
% 2.66/1.51 ! [A: int,B: int,D: int,C: int] :
% 2.66/1.51 ( ( ord_less_int @ A @ B )
% 2.66/1.51 => ( ( ord_less_int @ D @ C )
% 2.66/1.51 => ( ord_less_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ D ) ) ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_336_zero__le,axiom,
% 2.66/1.51 ! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_337_le__numeral__extra_I3_J,axiom,
% 2.66/1.51 ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% 2.66/1.51
% 2.66/1.51 thf(fact_338_le__numeral__extra_I3_J,axiom,
% 2.66/1.51 ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% 2.66/1.51
% 2.66/1.51 thf(fact_339_complete__interval,axiom,
% 2.66/1.51 ! [A: nat,B: nat,P: nat > $o] :
% 2.66/1.51 ( ( ord_less_nat @ A @ B )
% 2.66/1.51 => ( ( P @ A )
% 2.66/1.51 => ( ~ ( P @ B )
% 2.66/1.51 => ? [C2: nat] :
% 2.66/1.51 ( ( ord_less_eq_nat @ A @ C2 )
% 2.66/1.51 & ( ord_less_eq_nat @ C2 @ B )
% 2.66/1.51 & ! [X6: nat] :
% 2.66/1.51 ( ( ( ord_less_eq_nat @ A @ X6 )
% 2.66/1.51 & ( ord_less_nat @ X6 @ C2 ) )
% 2.66/1.51 => ( P @ X6 ) )
% 2.66/1.51 & ! [D2: nat] :
% 2.66/1.51 ( ! [X3: nat] :
% 2.66/1.51 ( ( ( ord_less_eq_nat @ A @ X3 )
% 2.66/1.51 & ( ord_less_nat @ X3 @ D2 ) )
% 2.66/1.51 => ( P @ X3 ) )
% 2.66/1.51 => ( ord_less_eq_nat @ D2 @ C2 ) ) ) ) ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_340_complete__interval,axiom,
% 2.66/1.51 ! [A: int,B: int,P: int > $o] :
% 2.66/1.51 ( ( ord_less_int @ A @ B )
% 2.66/1.51 => ( ( P @ A )
% 2.66/1.51 => ( ~ ( P @ B )
% 2.66/1.51 => ? [C2: int] :
% 2.66/1.51 ( ( ord_less_eq_int @ A @ C2 )
% 2.66/1.51 & ( ord_less_eq_int @ C2 @ B )
% 2.66/1.51 & ! [X6: int] :
% 2.66/1.51 ( ( ( ord_less_eq_int @ A @ X6 )
% 2.66/1.51 & ( ord_less_int @ X6 @ C2 ) )
% 2.66/1.51 => ( P @ X6 ) )
% 2.66/1.51 & ! [D2: int] :
% 2.66/1.51 ( ! [X3: int] :
% 2.66/1.51 ( ( ( ord_less_eq_int @ A @ X3 )
% 2.66/1.51 & ( ord_less_int @ X3 @ D2 ) )
% 2.66/1.51 => ( P @ X3 ) )
% 2.66/1.51 => ( ord_less_eq_int @ D2 @ C2 ) ) ) ) ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_341_order_Onot__eq__order__implies__strict,axiom,
% 2.66/1.51 ! [A: nat,B: nat] :
% 2.66/1.51 ( ( A != B )
% 2.66/1.51 => ( ( ord_less_eq_nat @ A @ B )
% 2.66/1.51 => ( ord_less_nat @ A @ B ) ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_342_order_Onot__eq__order__implies__strict,axiom,
% 2.66/1.51 ! [A: int,B: int] :
% 2.66/1.51 ( ( A != B )
% 2.66/1.51 => ( ( ord_less_eq_int @ A @ B )
% 2.66/1.51 => ( ord_less_int @ A @ B ) ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_343_dual__order_Ostrict__implies__order,axiom,
% 2.66/1.51 ! [B: nat,A: nat] :
% 2.66/1.51 ( ( ord_less_nat @ B @ A )
% 2.66/1.51 => ( ord_less_eq_nat @ B @ A ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_344_dual__order_Ostrict__implies__order,axiom,
% 2.66/1.51 ! [B: int,A: int] :
% 2.66/1.51 ( ( ord_less_int @ B @ A )
% 2.66/1.51 => ( ord_less_eq_int @ B @ A ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_345_dual__order_Ostrict__iff__order,axiom,
% 2.66/1.51 ( ord_less_nat
% 2.66/1.51 = ( ^ [B3: nat,A3: nat] :
% 2.66/1.51 ( ( ord_less_eq_nat @ B3 @ A3 )
% 2.66/1.51 & ( A3 != B3 ) ) ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_346_dual__order_Ostrict__iff__order,axiom,
% 2.66/1.51 ( ord_less_int
% 2.66/1.51 = ( ^ [B3: int,A3: int] :
% 2.66/1.51 ( ( ord_less_eq_int @ B3 @ A3 )
% 2.66/1.51 & ( A3 != B3 ) ) ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(fact_347_dual__order_Oorder__iff__strict,axiom,
% 2.66/1.51 ( ord_less_eq_int
% 2.66/1.51 = ( ^ [B3: int,A3: int] :
% 2.66/1.51 ( ( ord_less_int @ B3 @ A3 )
% 2.66/1.51 | ( A3 = B3 ) ) ) ) ).
% 2.66/1.51
% 2.66/1.51 thf(conj_0,hypothesis,
% 2.66/1.51 ord_less_nat @ zero_zero_nat @ n ).
% 2.66/1.51
% 2.66/1.51 thf(conj_1,hypothesis,
% 2.66/1.51 polyno1372495879olyh_a @ p @ n ).
% 2.66/1.51
% 2.66/1.51 thf(conj_2,conjecture,
% 2.66/1.51 ( ( polyno578545843gree_a @ p )
% 2.66/1.51 = zero_zero_nat ) ).
% 2.66/1.51 ------------------------
% 2.66/1.54 % (9429)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_9327 for (2999ds/2Mi)
% 2.66/1.54 % (9428)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_9327 for (2999ds/27Mi)
% 2.66/1.54 % (9427)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_9327 for (2999ds/4Mi)
% 2.66/1.54 % (9429)Instruction limit reached!
% 2.66/1.54 % (9429)------------------------------
% 2.66/1.54 % (9429)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.66/1.54 % (9429)Termination reason: Unknown
% 2.66/1.54 % (9429)Termination phase: shuffling
% 2.66/1.54
% 2.66/1.54 % (9429)Memory used [KB]: 1279
% 2.66/1.54 % (9429)Time elapsed: 0.002 s
% 2.66/1.54 % (9429)Instructions burned: 2 (million)
% 2.66/1.54 % (9429)------------------------------
% 2.66/1.54 % (9429)------------------------------
% 2.66/1.54 % (9427)Instruction limit reached!
% 2.66/1.54 % (9427)------------------------------
% 2.66/1.54 % (9427)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.66/1.54 % (9427)Termination reason: Unknown
% 2.66/1.54 % (9427)Termination phase: shuffling
% 2.66/1.54
% 2.66/1.54 % (9427)Memory used [KB]: 1407
% 2.66/1.54 % (9427)Time elapsed: 0.003 s
% 2.66/1.54 % (9427)Instructions burned: 6 (million)
% 2.66/1.54 % (9427)------------------------------
% 2.66/1.54 % (9427)------------------------------
% 2.66/1.54 % (9426)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_9327 for (2999ds/183Mi)
% 2.66/1.54 % (9430)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_9327 for (2999ds/2Mi)
% 2.66/1.54 % (9431)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_9327 for (2999ds/275Mi)
% 2.66/1.54 % (9432)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_9327 for (2999ds/18Mi)
% 2.66/1.55 % (9428)Instruction limit reached!
% 2.66/1.55 % (9428)------------------------------
% 2.66/1.55 % (9428)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.66/1.55 % (9428)Termination reason: Unknown
% 2.66/1.55 % (9428)Termination phase: Property scanning
% 2.66/1.55
% 2.66/1.55 % (9428)Memory used [KB]: 1663
% 2.66/1.55 % (9428)Time elapsed: 0.010 s
% 2.66/1.55 % (9428)Instructions burned: 28 (million)
% 2.66/1.55 % (9428)------------------------------
% 2.66/1.55 % (9428)------------------------------
% 2.66/1.55 % (9430)Instruction limit reached!
% 2.66/1.55 % (9430)------------------------------
% 2.66/1.55 % (9430)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.66/1.55 % (9430)Termination reason: Unknown
% 2.66/1.55 % (9430)Termination phase: shuffling
% 2.66/1.55
% 2.66/1.55 % (9430)Memory used [KB]: 1407
% 2.66/1.55 % (9430)Time elapsed: 0.004 s
% 2.66/1.55 % (9430)Instructions burned: 3 (million)
% 2.66/1.55 % (9430)------------------------------
% 2.66/1.55 % (9430)------------------------------
% 2.66/1.55 % (9433)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_9327 for (2999ds/3Mi)
% 2.66/1.55 % (9433)Instruction limit reached!
% 2.66/1.55 % (9433)------------------------------
% 2.66/1.55 % (9433)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.66/1.55 % (9433)Termination reason: Unknown
% 2.66/1.55 % (9433)Termination phase: shuffling
% 2.66/1.55
% 2.66/1.55 % (9433)Memory used [KB]: 1279
% 2.66/1.55 % (9433)Time elapsed: 0.002 s
% 2.66/1.55 % (9433)Instructions burned: 3 (million)
% 2.66/1.55 % (9433)------------------------------
% 2.66/1.55 % (9433)------------------------------
% 2.66/1.55 % (9434)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_9327 for (2999ds/37Mi)
% 2.66/1.55 % (9432)Instruction limit reached!
% 2.66/1.55 % (9432)------------------------------
% 2.66/1.55 % (9432)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.66/1.55 % (9432)Termination reason: Unknown
% 2.66/1.55 % (9432)Termination phase: shuffling
% 2.66/1.55
% 2.66/1.55 % (9432)Memory used [KB]: 1663
% 2.66/1.55 % (9432)Time elapsed: 0.011 s
% 2.66/1.55 % (9432)Instructions burned: 18 (million)
% 2.66/1.55 % (9432)------------------------------
% 2.66/1.55 % (9432)------------------------------
% 2.66/1.55 % (9435)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_9327 for (2999ds/15Mi)
% 2.66/1.56 % (9437)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_9327 for (2999ds/1041Mi)
% 2.66/1.56 % (9435)Instruction limit reached!
% 2.66/1.56 % (9435)------------------------------
% 2.66/1.56 % (9435)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.66/1.56 % (9435)Termination reason: Unknown
% 2.66/1.56 % (9435)Termination phase: shuffling
% 2.66/1.56
% 2.66/1.56 % (9435)Memory used [KB]: 1663
% 2.66/1.56 % (9435)Time elapsed: 0.006 s
% 2.66/1.56 % (9435)Instructions burned: 16 (million)
% 2.66/1.56 % (9435)------------------------------
% 2.66/1.56 % (9435)------------------------------
% 2.66/1.56 % (9434)Instruction limit reached!
% 2.66/1.56 % (9434)------------------------------
% 2.66/1.56 % (9434)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.66/1.56 % (9434)Termination reason: Unknown
% 2.66/1.56 % (9434)Termination phase: shuffling
% 2.66/1.56
% 2.66/1.56 % (9434)Memory used [KB]: 2046
% 2.66/1.56 % (9434)Time elapsed: 0.013 s
% 2.66/1.56 % (9434)Instructions burned: 39 (million)
% 2.66/1.56 % (9434)------------------------------
% 2.66/1.56 % (9434)------------------------------
% 2.66/1.56 % (9438)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_9327 for (2999ds/7Mi)
% 2.66/1.56 % (9436)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_9327 for (2999ds/3Mi)
% 2.66/1.57 % (9436)Instruction limit reached!
% 2.66/1.57 % (9436)------------------------------
% 2.66/1.57 % (9436)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.66/1.57 % (9436)Termination reason: Unknown
% 2.66/1.57 % (9436)Termination phase: shuffling
% 2.66/1.57
% 2.66/1.57 % (9436)Memory used [KB]: 1279
% 2.66/1.57 % (9438)Instruction limit reached!
% 2.66/1.57 % (9438)------------------------------
% 2.66/1.57 % (9438)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.66/1.57 % (9436)Time elapsed: 0.004 s
% 2.66/1.57 % (9436)Instructions burned: 3 (million)
% 2.66/1.57 % (9436)------------------------------
% 2.66/1.57 % (9436)------------------------------
% 2.66/1.57 % (9438)Termination reason: Unknown
% 2.66/1.57 % (9438)Termination phase: shuffling
% 2.66/1.57
% 2.66/1.57 % (9438)Memory used [KB]: 1535
% 2.66/1.57 % (9438)Time elapsed: 0.004 s
% 2.66/1.57 % (9438)Instructions burned: 9 (million)
% 2.66/1.57 % (9438)------------------------------
% 2.66/1.57 % (9438)------------------------------
% 2.66/1.57 % (9439)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_9327 for (2999ds/16Mi)
% 2.66/1.57 % (9440)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_9327 for (2999ds/3Mi)
% 2.66/1.57 % (9440)Instruction limit reached!
% 2.66/1.57 % (9440)------------------------------
% 2.66/1.57 % (9440)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.66/1.57 % (9440)Termination reason: Unknown
% 2.66/1.57 % (9440)Termination phase: shuffling
% 2.66/1.57
% 2.66/1.57 % (9440)Memory used [KB]: 1279
% 2.66/1.57 % (9440)Time elapsed: 0.003 s
% 2.66/1.57 % (9440)Instructions burned: 3 (million)
% 2.66/1.57 % (9440)------------------------------
% 2.66/1.57 % (9440)------------------------------
% 2.66/1.57 % (9439)Instruction limit reached!
% 2.66/1.57 % (9439)------------------------------
% 2.66/1.57 % (9439)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.66/1.57 % (9439)Termination reason: Unknown
% 2.66/1.57 % (9439)Termination phase: shuffling
% 2.66/1.57
% 2.66/1.57 % (9439)Memory used [KB]: 1663
% 2.66/1.57 % (9439)Time elapsed: 0.006 s
% 2.66/1.57 % (9439)Instructions burned: 16 (million)
% 2.66/1.57 % (9439)------------------------------
% 2.66/1.57 % (9439)------------------------------
% 2.66/1.58 % (9441)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_9327 for (2999ds/3Mi)
% 2.66/1.58 % (9441)Instruction limit reached!
% 2.66/1.58 % (9441)------------------------------
% 2.66/1.58 % (9441)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.66/1.58 % (9441)Termination reason: Unknown
% 2.66/1.58 % (9441)Termination phase: shuffling
% 2.66/1.58
% 2.66/1.58 % (9441)Memory used [KB]: 1279
% 2.66/1.58 % (9441)Time elapsed: 0.002 s
% 2.66/1.58 % (9441)Instructions burned: 4 (million)
% 2.66/1.58 % (9441)------------------------------
% 2.66/1.58 % (9441)------------------------------
% 2.66/1.58 % (9443)dis+1002_1:1_add=large:cnfonf=lazy_pi_sigma_gen:fe=off:prag=on:i=3:si=on:rtra=on_0 on DTF2THF_9327 for (2999ds/3Mi)
% 2.66/1.58 % (9444)dis+1004_1:1_cha=on:cs=on:fe=off:hud=1:i=4:si=on:rtra=on_0 on DTF2THF_9327 for (2999ds/4Mi)
% 2.66/1.58 % (9443)Instruction limit reached!
% 2.66/1.58 % (9443)------------------------------
% 2.66/1.58 % (9443)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.66/1.58 % (9443)Termination reason: Unknown
% 2.66/1.58 % (9443)Termination phase: shuffling
% 2.66/1.58
% 2.66/1.58 % (9443)Memory used [KB]: 1407
% 2.66/1.58 % (9443)Time elapsed: 0.003 s
% 2.66/1.58 % (9443)Instructions burned: 4 (million)
% 2.66/1.58 % (9443)------------------------------
% 2.66/1.58 % (9443)------------------------------
% 2.66/1.58 % (9442)lrs+10_1:1_cnfonf=off:cs=on:hud=3:prag=on:sup=off:i=7:si=on:rtra=on_0 on DTF2THF_9327 for (2999ds/7Mi)
% 2.66/1.59 % (9444)Instruction limit reached!
% 2.66/1.59 % (9444)------------------------------
% 2.66/1.59 % (9444)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.66/1.59 % (9444)Termination reason: Unknown
% 2.66/1.59 % (9444)Termination phase: shuffling
% 2.66/1.59
% 2.66/1.59 % (9444)Memory used [KB]: 1407
% 2.66/1.59 % (9444)Time elapsed: 0.003 s
% 2.66/1.59 % (9444)Instructions burned: 6 (million)
% 2.66/1.59 % (9444)------------------------------
% 2.66/1.59 % (9444)------------------------------
% 2.66/1.59 % (9445)lrs+1002_1:1_anc=all_dependent:au=on:cbe=off:fde=unused:ntd=on:i=18:si=on:rtra=on_0 on DTF2THF_9327 for (2999ds/18Mi)
% 2.66/1.59 % (9442)Instruction limit reached!
% 2.66/1.59 % (9442)------------------------------
% 2.66/1.59 % (9442)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.66/1.59 % (9442)Termination reason: Unknown
% 2.66/1.59 % (9442)Termination phase: shuffling
% 2.66/1.59
% 2.66/1.59 % (9442)Memory used [KB]: 1407
% 2.66/1.59 % (9442)Time elapsed: 0.006 s
% 2.66/1.59 % (9442)Instructions burned: 7 (million)
% 2.66/1.59 % (9442)------------------------------
% 2.66/1.59 % (9442)------------------------------
% 2.66/1.59 % (9445)Instruction limit reached!
% 2.66/1.59 % (9445)------------------------------
% 2.66/1.59 % (9445)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.66/1.59 % (9445)Termination reason: Unknown
% 2.66/1.59 % (9445)Termination phase: shuffling
% 2.66/1.59
% 2.66/1.59 % (9445)Memory used [KB]: 1663
% 2.66/1.59 % (9445)Time elapsed: 0.007 s
% 2.66/1.59 % (9445)Instructions burned: 19 (million)
% 2.66/1.59 % (9445)------------------------------
% 2.66/1.59 % (9445)------------------------------
% 2.66/1.59 % (9446)lrs+10_1:1_e2e=on:sd=1:sgt=8:ss=axioms:i=710:si=on:rtra=on_0 on DTF2THF_9327 for (2999ds/710Mi)
% 2.66/1.59 % (9447)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_9327 for (2999ds/6Mi)
% 2.69/1.60 % (9447)Instruction limit reached!
% 2.69/1.60 % (9447)------------------------------
% 2.69/1.60 % (9447)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.69/1.60 % (9447)Termination reason: Unknown
% 2.69/1.60 % (9447)Termination phase: shuffling
% 2.69/1.60
% 2.69/1.60 % (9447)Memory used [KB]: 1407
% 2.69/1.60 % (9447)Time elapsed: 0.003 s
% 2.69/1.60 % (9447)Instructions burned: 7 (million)
% 2.69/1.60 % (9447)------------------------------
% 2.69/1.60 % (9447)------------------------------
% 2.69/1.60 % (9449)dis+21_1:8_apa=on:cnfonf=off:fd=off:fsr=off:hud=0:ins=1:kws=inv_frequency:nwc=10.0:ss=axioms:st=5.0:i=21:si=on:rtra=on_0 on DTF2THF_9327 for (2999ds/21Mi)
% 2.69/1.60 % (9448)dis+1002_5:1_au=on:bd=off:e2e=on:fde=none:fs=off:fsr=off:sos=on:i=902:si=on:rtra=on_0 on DTF2THF_9327 for (2999ds/902Mi)
% 2.69/1.61 % (9450)dis+10_1:1_cnfonf=lazy_gen:fe=off:i=5:si=on:rtra=on_0 on DTF2THF_9327 for (2999ds/5Mi)
% 2.69/1.61 % (9450)Instruction limit reached!
% 2.69/1.61 % (9450)------------------------------
% 2.69/1.61 % (9450)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.69/1.61 % (9450)Termination reason: Unknown
% 2.69/1.61 % (9450)Termination phase: shuffling
% 2.69/1.61
% 2.69/1.61 % (9450)Memory used [KB]: 1407
% 2.69/1.61 % (9450)Time elapsed: 0.003 s
% 2.69/1.61 % (9450)Instructions burned: 7 (million)
% 2.69/1.61 % (9450)------------------------------
% 2.69/1.61 % (9450)------------------------------
% 2.69/1.61 % (9449)Instruction limit reached!
% 2.69/1.61 % (9449)------------------------------
% 2.69/1.61 % (9449)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.69/1.61 % (9449)Termination reason: Unknown
% 2.69/1.61 % (9449)Termination phase: shuffling
% 2.69/1.61
% 2.69/1.61 % (9449)Memory used [KB]: 1663
% 2.69/1.61 % (9449)Time elapsed: 0.008 s
% 2.69/1.61 % (9449)Instructions burned: 22 (million)
% 2.69/1.61 % (9449)------------------------------
% 2.69/1.61 % (9449)------------------------------
% 2.69/1.62 % (9451)lrs+2_1:1_cnfonf=lazy_not_gen_be_off:cs=on:fe=off:hud=10:inj=on:ins=3:plsq=on:plsqc=1:sd=10:ss=axioms:tnu=1:i=6:si=on:rtra=on_0 on DTF2THF_9327 for (2999ds/6Mi)
% 2.69/1.62 % (9452)lrs+1002_1:128_au=on:c=on:fsr=off:piset=equals:i=377:si=on:rtra=on_0 on DTF2THF_9327 for (2999ds/377Mi)
% 2.69/1.62 % (9431)First to succeed.
% 2.69/1.62 % (9451)Instruction limit reached!
% 2.69/1.62 % (9451)------------------------------
% 2.69/1.62 % (9451)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.69/1.62 % (9451)Termination reason: Unknown
% 2.69/1.62 % (9451)Termination phase: shuffling
% 2.69/1.62
% 2.69/1.62 % (9451)Memory used [KB]: 1407
% 2.69/1.62 % (9451)Time elapsed: 0.003 s
% 2.69/1.62 % (9451)Instructions burned: 7 (million)
% 2.69/1.62 % (9451)------------------------------
% 2.69/1.62 % (9451)------------------------------
% 2.69/1.62 % (9431)Refutation found. Thanks to Tanya!
% 2.69/1.62 % SZS status Theorem for DTF2THF_9327
% 2.69/1.62 % SZS output start Proof for DTF2THF_9327
% 2.69/1.62 thf(type_def_5, type, int: $tType).
% 2.69/1.62 thf(type_def_7, type, nat: $tType).
% 2.69/1.62 thf(type_def_8, type, polyno727731844poly_a: $tType).
% 2.69/1.62 thf(func_def_0, type, polyno727731844poly_a: $tType).
% 2.69/1.62 thf(func_def_1, type, nat: $tType).
% 2.69/1.62 thf(func_def_2, type, int: $tType).
% 2.69/1.62 thf(func_def_3, type, minus_minus_int: int > int > int).
% 2.69/1.62 thf(func_def_4, type, minus_minus_nat: nat > nat > nat).
% 2.69/1.62 thf(func_def_5, type, zero_zero_int: int).
% 2.69/1.62 thf(func_def_6, type, zero_zero_nat: nat).
% 2.69/1.62 thf(func_def_7, type, suc: nat > nat).
% 2.69/1.62 thf(func_def_8, type, semiri2019852685at_int: nat > int).
% 2.69/1.62 thf(func_def_9, type, semiri1382578993at_nat: nat > nat).
% 2.69/1.62 thf(func_def_10, type, ord_less_int: int > int > $o).
% 2.69/1.62 thf(func_def_11, type, ord_less_nat: nat > nat > $o).
% 2.69/1.62 thf(func_def_12, type, ord_less_eq_int: int > int > $o).
% 2.69/1.62 thf(func_def_13, type, ord_less_eq_nat: nat > nat > $o).
% 2.69/1.62 thf(func_def_14, type, polyno578545843gree_a: polyno727731844poly_a > nat).
% 2.69/1.62 thf(func_def_15, type, polyno1674775833reen_a: polyno727731844poly_a > nat > nat).
% 2.69/1.62 thf(func_def_16, type, polyno190918219poly_a: polyno727731844poly_a > $o).
% 2.69/1.62 thf(func_def_17, type, polyno1372495879olyh_a: polyno727731844poly_a > nat > $o).
% 2.69/1.62 thf(func_def_18, type, power_power_int: int > nat > int).
% 2.69/1.62 thf(func_def_19, type, power_power_nat: nat > nat > nat).
% 2.69/1.62 thf(func_def_20, type, n: nat).
% 2.69/1.62 thf(func_def_21, type, p: polyno727731844poly_a).
% 2.69/1.62 thf(func_def_36, type, sK0: (nat > $o) > nat).
% 2.69/1.62 thf(func_def_37, type, sK1: nat > nat).
% 2.69/1.62 thf(func_def_38, type, sK2: (nat > nat) > nat).
% 2.69/1.62 thf(func_def_39, type, sK3: (nat > nat) > nat).
% 2.69/1.62 thf(func_def_40, type, sK4: nat > nat).
% 2.69/1.62 thf(func_def_41, type, sK5: (nat > $o) > nat > nat).
% 2.69/1.62 thf(func_def_42, type, sK6: (nat > $o) > nat > nat).
% 2.69/1.62 thf(func_def_43, type, sK7: nat > nat).
% 2.69/1.62 thf(func_def_44, type, sK8: (nat > nat) > nat).
% 2.69/1.62 thf(func_def_45, type, sK9: (nat > nat) > nat).
% 2.69/1.62 thf(func_def_46, type, sK10: nat > nat > nat).
% 2.69/1.62 thf(func_def_47, type, sK11: (nat > $o) > nat).
% 2.69/1.62 thf(func_def_48, type, sK12: nat > nat > nat).
% 2.69/1.62 thf(func_def_49, type, sK13: (nat > $o) > nat).
% 2.69/1.62 thf(func_def_50, type, sK14: nat > nat > nat).
% 2.69/1.62 thf(func_def_51, type, sK15: (nat > $o) > nat).
% 2.69/1.62 thf(func_def_52, type, sK16: (nat > $o) > nat).
% 2.69/1.62 thf(func_def_53, type, sK17: (nat > nat > $o) > nat).
% 2.69/1.62 thf(func_def_54, type, sK18: (nat > nat > $o) > nat).
% 2.69/1.62 thf(func_def_55, type, sK19: (nat > nat > $o) > nat).
% 2.69/1.62 thf(func_def_56, type, sK20: (nat > nat > $o) > nat).
% 2.69/1.62 thf(func_def_57, type, sK21: (nat > nat) > nat).
% 2.69/1.62 thf(func_def_58, type, sK22: nat > (nat > $o) > nat).
% 2.69/1.62 thf(func_def_59, type, sK23: nat > (nat > $o) > nat).
% 2.69/1.62 thf(func_def_60, type, sK24: nat > nat > nat).
% 2.69/1.62 thf(func_def_61, type, sK25: (nat > nat) > nat).
% 2.69/1.62 thf(func_def_62, type, sK26: (nat > nat) > nat).
% 2.69/1.62 thf(func_def_63, type, sK27: (nat > nat) > nat).
% 2.69/1.62 thf(func_def_64, type, sK28: nat > nat).
% 2.69/1.62 thf(func_def_65, type, sK29: (nat > nat) > nat).
% 2.69/1.62 thf(func_def_66, type, sK30: (nat > $o) > nat > nat).
% 2.69/1.62 thf(func_def_67, type, sK31: (nat > $o) > nat > nat).
% 2.69/1.62 thf(func_def_68, type, sK32: (nat > $o) > nat).
% 2.69/1.62 thf(func_def_69, type, sK33: (nat > nat > $o) > nat).
% 2.69/1.62 thf(func_def_70, type, sK34: (nat > nat > $o) > nat).
% 2.69/1.62 thf(func_def_71, type, sK35: (nat > nat > $o) > nat).
% 2.69/1.62 thf(func_def_72, type, sK36: (nat > nat > $o) > nat).
% 2.69/1.62 thf(func_def_73, type, sK37: (nat > nat > $o) > nat).
% 2.69/1.62 thf(func_def_74, type, sK38: nat > nat).
% 2.69/1.62 thf(func_def_75, type, sK39: (nat > nat > $o) > nat).
% 2.69/1.62 thf(func_def_76, type, sK40: (nat > nat > $o) > nat).
% 2.69/1.62 thf(func_def_77, type, sK41: (nat > nat > $o) > nat).
% 2.69/1.62 thf(func_def_78, type, sK42: (nat > nat > $o) > nat).
% 2.69/1.62 thf(func_def_79, type, sK43: (nat > nat) > nat).
% 2.69/1.62 thf(func_def_80, type, sK44: (nat > nat) > nat).
% 2.69/1.62 thf(func_def_81, type, sK45: (nat > $o) > nat > nat).
% 2.69/1.62 thf(func_def_82, type, sK46: (nat > $o) > nat > nat).
% 2.69/1.62 thf(func_def_83, type, sK47: (nat > $o) > nat).
% 2.69/1.62 thf(func_def_84, type, sK48: nat > nat).
% 2.69/1.62 thf(func_def_85, type, sK49: (nat > $o) > nat).
% 2.69/1.62 thf(func_def_86, type, sK50: (nat > $o) > nat > nat).
% 2.69/1.62 thf(func_def_87, type, sK51: (nat > $o) > nat > nat).
% 2.69/1.62 thf(func_def_88, type, sK52: (nat > nat) > nat).
% 2.69/1.62 thf(func_def_90, type, ph54: !>[X0: $tType]:(X0)).
% 2.69/1.62 thf(f1607,plain,(
% 2.69/1.62 $false),
% 2.69/1.62 inference(subsumption_resolution,[status(thm)],[f1606,f1314])).
% 2.69/1.62 thf(f1314,plain,(
% 2.69/1.62 (zero_zero_nat != (polyno578545843gree_a @ p))),
% 2.69/1.62 inference(cnf_transformation,[status(thm)],[f926])).
% 2.69/1.62 thf(f926,plain,(
% 2.69/1.62 (zero_zero_nat != (polyno578545843gree_a @ p))),
% 2.69/1.62 inference(flattening,[status(thm)],[f352])).
% 2.69/1.62 thf(f352,negated_conjecture,(
% 2.69/1.62 ~(zero_zero_nat = (polyno578545843gree_a @ p))),
% 2.69/1.62 inference(negated_conjecture,[status(cth)],[f351])).
% 2.69/1.62 thf(f351,conjecture,(
% 2.69/1.62 (zero_zero_nat = (polyno578545843gree_a @ p))),
% 2.69/1.62 file('/export/starexec/sandbox/tmp/tmp.4Dn3J3wAeS/DTF2THF_9327.p',conj_2)).
% 2.69/1.62 thf(f1606,plain,(
% 2.69/1.62 (zero_zero_nat = (polyno578545843gree_a @ p))),
% 2.69/1.62 inference(trivial_inequality_removal,[status(thm)],[f1605])).
% 2.69/1.62 thf(f1605,plain,(
% 2.69/1.62 ($true != $true) | (zero_zero_nat = (polyno578545843gree_a @ p))),
% 2.69/1.62 inference(superposition,[status(thm)],[f1432,f1222])).
% 2.69/1.62 thf(f1222,plain,(
% 2.69/1.62 ((polyno1372495879olyh_a @ p @ n) = $true)),
% 2.69/1.62 inference(cnf_transformation,[status(thm)],[f613])).
% 2.69/1.62 thf(f613,plain,(
% 2.69/1.62 ((polyno1372495879olyh_a @ p @ n) = $true)),
% 2.69/1.62 inference(fool_elimination,[status(thm)],[f612])).
% 2.69/1.62 thf(f612,plain,(
% 2.69/1.62 (polyno1372495879olyh_a @ p @ n)),
% 2.69/1.62 inference(rectify,[status(thm)],[f350])).
% 2.69/1.62 thf(f350,axiom,(
% 2.69/1.62 (polyno1372495879olyh_a @ p @ n)),
% 2.69/1.62 file('/export/starexec/sandbox/tmp/tmp.4Dn3J3wAeS/DTF2THF_9327.p',conj_1)).
% 2.69/1.62 thf(f1432,plain,(
% 2.69/1.62 ( ! [X0 : polyno727731844poly_a] : (($true != (polyno1372495879olyh_a @ X0 @ n)) | (zero_zero_nat = (polyno578545843gree_a @ X0))) )),
% 2.69/1.62 inference(superposition,[status(thm)],[f1189,f1423])).
% 2.69/1.62 thf(f1423,plain,(
% 2.69/1.62 (n = (suc @ (sK7 @ n)))),
% 2.69/1.62 inference(trivial_inequality_removal,[status(thm)],[f1412])).
% 2.69/1.62 thf(f1412,plain,(
% 2.69/1.62 ($true != $true) | (n = (suc @ (sK7 @ n)))),
% 2.69/1.62 inference(superposition,[status(thm)],[f1216,f1256])).
% 2.69/1.62 thf(f1256,plain,(
% 2.69/1.62 ((ord_less_nat @ zero_zero_nat @ n) = $true)),
% 2.69/1.62 inference(cnf_transformation,[status(thm)],[f475])).
% 2.69/1.62 thf(f475,plain,(
% 2.69/1.62 ((ord_less_nat @ zero_zero_nat @ n) = $true)),
% 2.69/1.62 inference(fool_elimination,[status(thm)],[f474])).
% 2.69/1.62 thf(f474,plain,(
% 2.69/1.62 (ord_less_nat @ zero_zero_nat @ n)),
% 2.69/1.62 inference(rectify,[status(thm)],[f349])).
% 2.69/1.62 thf(f349,axiom,(
% 2.69/1.62 (ord_less_nat @ zero_zero_nat @ n)),
% 2.69/1.62 file('/export/starexec/sandbox/tmp/tmp.4Dn3J3wAeS/DTF2THF_9327.p',conj_0)).
% 2.69/1.62 thf(f1216,plain,(
% 2.69/1.62 ( ! [X0 : nat] : (((ord_less_nat @ zero_zero_nat @ X0) != $true) | ((suc @ (sK7 @ X0)) = X0)) )),
% 2.69/1.62 inference(cnf_transformation,[status(thm)],[f1086])).
% 2.69/1.62 thf(f1086,plain,(
% 2.69/1.62 ! [X0 : nat] : ((((ord_less_nat @ zero_zero_nat @ X0) = $true) | ! [X1 : nat] : ((suc @ X1) != X0)) & (((suc @ (sK7 @ X0)) = X0) | ((ord_less_nat @ zero_zero_nat @ X0) != $true)))),
% 2.69/1.62 inference(skolemisation,[status(esa),new_symbols(skolem,[sK7])],[f1084,f1085])).
% 2.69/1.62 thf(f1085,plain,(
% 2.69/1.62 ! [X0 : nat] : (? [X2 : nat] : ((suc @ X2) = X0) => ((suc @ (sK7 @ X0)) = X0))),
% 2.69/1.62 introduced(definition,[],[choice_axiom])).
% 2.69/1.62 thf(f1084,plain,(
% 2.69/1.62 ! [X0 : nat] : ((((ord_less_nat @ zero_zero_nat @ X0) = $true) | ! [X1 : nat] : ((suc @ X1) != X0)) & (? [X2 : nat] : ((suc @ X2) = X0) | ((ord_less_nat @ zero_zero_nat @ X0) != $true)))),
% 2.69/1.62 inference(rectify,[status(thm)],[f1083])).
% 2.69/1.62 thf(f1083,plain,(
% 2.69/1.62 ! [X0 : nat] : ((((ord_less_nat @ zero_zero_nat @ X0) = $true) | ! [X1 : nat] : ((suc @ X1) != X0)) & (? [X1 : nat] : ((suc @ X1) = X0) | ((ord_less_nat @ zero_zero_nat @ X0) != $true)))),
% 2.69/1.62 inference(nnf_transformation,[status(thm)],[f666])).
% 2.69/1.62 thf(f666,plain,(
% 2.69/1.62 ! [X0 : nat] : (((ord_less_nat @ zero_zero_nat @ X0) = $true) <=> ? [X1 : nat] : ((suc @ X1) = X0))),
% 2.69/1.62 inference(fool_elimination,[status(thm)],[f665])).
% 2.69/1.62 thf(f665,plain,(
% 2.69/1.62 ! [X0 : nat] : ((ord_less_nat @ zero_zero_nat @ X0) = ? [X1 : nat] : ((suc @ X1) = X0))),
% 2.69/1.62 inference(rectify,[status(thm)],[f99])).
% 2.69/1.62 thf(f99,axiom,(
% 2.69/1.62 ! [X0 : nat] : ((ord_less_nat @ zero_zero_nat @ X0) = ? [X32 : nat] : ((suc @ X32) = X0))),
% 2.69/1.62 file('/export/starexec/sandbox/tmp/tmp.4Dn3J3wAeS/DTF2THF_9327.p',fact_98_gr0__conv__Suc)).
% 2.69/1.62 thf(f1189,plain,(
% 2.69/1.62 ( ! [X0 : nat,X1 : polyno727731844poly_a] : (($true != (polyno1372495879olyh_a @ X1 @ (suc @ X0))) | (zero_zero_nat = (polyno578545843gree_a @ X1))) )),
% 2.69/1.62 inference(cnf_transformation,[status(thm)],[f1056])).
% 2.69/1.62 thf(f1056,plain,(
% 2.69/1.62 ! [X0 : nat,X1 : polyno727731844poly_a] : (($true != (polyno1372495879olyh_a @ X1 @ (suc @ X0))) | (zero_zero_nat = (polyno578545843gree_a @ X1)))),
% 2.69/1.62 inference(ennf_transformation,[status(thm)],[f650])).
% 2.69/1.62 thf(f650,plain,(
% 2.69/1.62 ! [X0 : nat,X1 : polyno727731844poly_a] : (($true = (polyno1372495879olyh_a @ X1 @ (suc @ X0))) => (zero_zero_nat = (polyno578545843gree_a @ X1)))),
% 2.69/1.62 inference(fool_elimination,[status(thm)],[f649])).
% 2.69/1.62 thf(f649,plain,(
% 2.69/1.62 ! [X0 : nat,X1 : polyno727731844poly_a] : ((polyno1372495879olyh_a @ X1 @ (suc @ X0)) => (zero_zero_nat = (polyno578545843gree_a @ X1)))),
% 2.69/1.62 inference(rectify,[status(thm)],[f32])).
% 2.69/1.62 thf(f32,axiom,(
% 2.69/1.62 ! [X0 : nat,X11 : polyno727731844poly_a] : ((polyno1372495879olyh_a @ X11 @ (suc @ X0)) => (zero_zero_nat = (polyno578545843gree_a @ X11)))),
% 2.69/1.62 file('/export/starexec/sandbox/tmp/tmp.4Dn3J3wAeS/DTF2THF_9327.p',fact_31_degree__isnpolyh__Suc)).
% 2.69/1.62 % SZS output end Proof for DTF2THF_9327
% 2.69/1.62 % (9431)------------------------------
% 2.69/1.62 % (9431)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.69/1.62 % (9431)Termination reason: Refutation
% 2.69/1.62
% 2.69/1.62 % (9431)Memory used [KB]: 6780
% 2.69/1.62 % (9431)Time elapsed: 0.076 s
% 2.69/1.62 % (9431)Instructions burned: 128 (million)
% 2.69/1.62 % (9431)------------------------------
% 2.69/1.62 % (9431)------------------------------
% 2.69/1.62 % (9425)Success in time 0.092 s
%------------------------------------------------------------------------------