%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : ITP097^2 : 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 : n018.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 08:55:32 AM UTC 2026
% Result : Theorem 11.73s 3.15s
% Output : Refutation 11.73s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.11 % Problem : ITP097^2 : TPTP v9.2.1. Released v7.5.0.
% 0.12/0.12 % Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.15/0.32 % Computer : n018.cluster.edu
% 0.15/0.32 % Model : x86_64 x86_64
% 0.15/0.32 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.15/0.32 % Memory : 8042.1875MB
% 0.15/0.32 % OS : Linux 3.10.0-693.el7.x86_64
% 0.15/0.32 % CPULimit : 300
% 0.15/0.32 % WCLimit : 300
% 0.15/0.32 % DateTime : Tue Feb 24 08:26:16 EST 2026
% 0.15/0.33 % CPUTime :
% 0.15/0.33 Running /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.18/0.44 ---- Original DTF file ---
% 0.18/0.44 thf(spec,logic,$$dhol).
% 0.18/0.44 %------------------------------------------------------------------------------
% 0.18/0.44 % File : ITP097^2 : TPTP v9.2.1. Released v7.5.0.
% 0.18/0.44 % Domain : Interactive Theorem Proving
% 0.18/0.44 % Problem : Sledgehammer Liouville_Numbers problem prob_81__5865280_1
% 0.18/0.44 % Version : Especial.
% 0.18/0.44 % English :
% 0.18/0.44
% 0.18/0.44 % Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% 0.18/0.44 % : [Des21] Desharnais (2021), Email to Geoff Sutcliffe
% 0.18/0.44 % Source : [Des21]
% 0.18/0.44 % Names : Liouville_Numbers/prob_81__5865280_1 [Des21]
% 0.18/0.44
% 0.18/0.44 % Status : Theorem
% 0.18/0.44 % Rating : 0.00 v7.5.0
% 0.18/0.44 % Syntax : Number of formulae : 377 ( 152 unt; 51 typ; 0 def)
% 0.18/0.44 % Number of atoms : 775 ( 302 equ; 0 cnn)
% 0.18/0.44 % Maximal formula atoms : 13 ( 2 avg)
% 0.18/0.44 % Number of connectives : 2629 ( 44 ~; 9 |; 17 &;2278 @)
% 0.18/0.44 % ( 0 <=>; 281 =>; 0 <=; 0 <~>)
% 0.18/0.44 % Maximal formula depth : 17 ( 5 avg)
% 0.18/0.44 % Number of types : 5 ( 4 usr)
% 0.18/0.44 % Number of type conns : 69 ( 69 >; 0 *; 0 +; 0 <<)
% 0.18/0.44 % Number of symbols : 50 ( 47 usr; 5 con; 0-4 aty)
% 0.18/0.44 % Number of variables : 655 ( 22 ^; 581 !; 14 ?; 655 :)
% 0.18/0.44 % ( 38 !>; 0 ?*; 0 @-; 0 @+)
% 0.18/0.44 % SPC : TH1_THM_EQU_NAR
% 0.18/0.44
% 0.18/0.44 % Comments : This file was generated by Sledgehammer 2021-02-23 16:24:05.479
% 0.18/0.44 %------------------------------------------------------------------------------
% 0.18/0.44 % Could-be-implicit typings (5)
% 0.18/0.44 thf(ty_t_Real_Oreal,type,
% 0.18/0.44 real: $tType ).
% 0.18/0.44
% 0.18/0.44 thf(ty_t_Set_Oset,type,
% 0.18/0.44 set: $tType > $tType ).
% 0.18/0.44
% 0.18/0.44 thf(ty_t_Num_Onum,type,
% 0.18/0.44 num: $tType ).
% 0.18/0.44
% 0.18/0.44 thf(ty_t_Nat_Onat,type,
% 0.18/0.44 nat: $tType ).
% 0.18/0.44
% 0.18/0.44 thf(ty_t_Int_Oint,type,
% 0.18/0.44 int: $tType ).
% 0.18/0.44
% 0.18/0.44 % Explicit typings (46)
% 0.18/0.44 thf(sy_cl_HOL_Otype,type,
% 0.18/0.44 type:
% 0.18/0.44 !>[A: $tType] : $o ).
% 0.18/0.44
% 0.18/0.44 thf(sy_cl_Groups_Oone,type,
% 0.18/0.44 one:
% 0.18/0.44 !>[A: $tType] : $o ).
% 0.18/0.44
% 0.18/0.44 thf(sy_cl_Groups_Ozero,type,
% 0.18/0.44 zero:
% 0.18/0.44 !>[A: $tType] : $o ).
% 0.18/0.44
% 0.18/0.44 thf(sy_cl_Num_Onumeral,type,
% 0.18/0.44 numeral:
% 0.18/0.45 !>[A: $tType] : $o ).
% 0.18/0.45
% 0.18/0.45 thf(sy_cl_Power_Opower,type,
% 0.18/0.45 power:
% 0.18/0.45 !>[A: $tType] : $o ).
% 0.18/0.45
% 0.18/0.45 thf(sy_cl_Rings_Oring__1,type,
% 0.18/0.45 ring_1:
% 0.18/0.45 !>[A: $tType] : $o ).
% 0.18/0.45
% 0.18/0.45 thf(sy_cl_Orderings_Oord,type,
% 0.18/0.45 ord:
% 0.18/0.45 !>[A: $tType] : $o ).
% 0.18/0.45
% 0.18/0.45 thf(sy_cl_Nat_Oring__char__0,type,
% 0.18/0.45 ring_char_0:
% 0.18/0.45 !>[A: $tType] : $o ).
% 0.18/0.45
% 0.18/0.45 thf(sy_cl_Num_Oneg__numeral,type,
% 0.18/0.45 neg_numeral:
% 0.18/0.45 !>[A: $tType] : $o ).
% 0.18/0.45
% 0.18/0.45 thf(sy_cl_Orderings_Oorder,type,
% 0.18/0.45 order:
% 0.18/0.45 !>[A: $tType] : $o ).
% 0.18/0.45
% 0.18/0.45 thf(sy_cl_Rings_Osemiring__1,type,
% 0.18/0.45 semiring_1:
% 0.18/0.45 !>[A: $tType] : $o ).
% 0.18/0.45
% 0.18/0.45 thf(sy_cl_Transcendental_Oln,type,
% 0.18/0.45 ln:
% 0.18/0.45 !>[A: $tType] : $o ).
% 0.18/0.45
% 0.18/0.45 thf(sy_cl_Groups_Omonoid__mult,type,
% 0.18/0.45 monoid_mult:
% 0.18/0.45 !>[A: $tType] : $o ).
% 0.18/0.45
% 0.18/0.45 thf(sy_cl_Orderings_Olinorder,type,
% 0.18/0.45 linorder:
% 0.18/0.45 !>[A: $tType] : $o ).
% 0.18/0.45
% 0.18/0.45 thf(sy_cl_Orderings_Opreorder,type,
% 0.18/0.45 preorder:
% 0.18/0.45 !>[A: $tType] : $o ).
% 0.18/0.45
% 0.18/0.45 thf(sy_cl_Nat_Osemiring__char__0,type,
% 0.18/0.45 semiring_char_0:
% 0.18/0.45 !>[A: $tType] : $o ).
% 0.18/0.45
% 0.18/0.45 thf(sy_cl_Num_Osemiring__numeral,type,
% 0.18/0.45 semiring_numeral:
% 0.18/0.45 !>[A: $tType] : $o ).
% 0.18/0.45
% 0.18/0.45 thf(sy_cl_Rings_Olinordered__idom,type,
% 0.18/0.45 linordered_idom:
% 0.18/0.45 !>[A: $tType] : $o ).
% 0.18/0.45
% 0.18/0.45 thf(sy_cl_Rings_Olinordered__semidom,type,
% 0.18/0.45 linordered_semidom:
% 0.18/0.45 !>[A: $tType] : $o ).
% 0.18/0.45
% 0.18/0.45 thf(sy_cl_Archimedean__Field_Ofloor__ceiling,type,
% 0.18/0.45 archim1727834104eiling:
% 0.18/0.45 !>[A: $tType] : $o ).
% 0.18/0.45
% 0.18/0.45 thf(sy_cl_Rings_Olinordered__nonzero__semiring,type,
% 0.18/0.45 linord1659791738miring:
% 0.18/0.45 !>[A: $tType] : $o ).
% 0.18/0.45
% 0.18/0.45 thf(sy_cl_Rings_Osemiring__1__no__zero__divisors,type,
% 0.18/0.45 semiri134348788visors:
% 0.18/0.45 !>[A: $tType] : $o ).
% 0.18/0.45
% 0.18/0.45 thf(sy_cl_Archimedean__Field_Oarchimedean__field,type,
% 0.18/0.45 archim1804426504_field:
% 0.18/0.45 !>[A: $tType] : $o ).
% 0.18/0.45
% 0.18/0.45 thf(sy_cl_Groups_Ocanonically__ordered__monoid__add,type,
% 0.18/0.45 canoni770627133id_add:
% 0.18/0.45 !>[A: $tType] : $o ).
% 0.18/0.45
% 0.18/0.45 thf(sy_c_Archimedean__Field_Oceiling,type,
% 0.18/0.45 archimedean_ceiling:
% 0.18/0.45 !>[A: $tType] : ( A > int ) ).
% 0.18/0.45
% 0.18/0.45 thf(sy_c_Groups_Oone__class_Oone,type,
% 0.18/0.45 one_one:
% 0.18/0.45 !>[A: $tType] : A ).
% 0.18/0.45
% 0.18/0.45 thf(sy_c_Groups_Ozero__class_Ozero,type,
% 0.18/0.45 zero_zero:
% 0.18/0.45 !>[A: $tType] : A ).
% 0.18/0.45
% 0.18/0.45 thf(sy_c_If,type,
% 0.18/0.45 if:
% 0.18/0.45 !>[A: $tType] : ( $o > A > A > A ) ).
% 0.18/0.45
% 0.18/0.45 thf(sy_c_Int_Onat,type,
% 0.18/0.45 nat2: int > nat ).
% 0.18/0.45
% 0.18/0.45 thf(sy_c_Int_Oring__1__class_Oof__int,type,
% 0.18/0.45 ring_1_of_int:
% 0.18/0.45 !>[A: $tType] : ( int > A ) ).
% 0.18/0.45
% 0.18/0.45 thf(sy_c_Nat_OSuc,type,
% 0.18/0.45 suc: nat > nat ).
% 0.18/0.45
% 0.18/0.45 thf(sy_c_Nat_Osemiring__1__class_Oof__nat,type,
% 0.18/0.45 semiring_1_of_nat:
% 0.18/0.45 !>[A: $tType] : ( nat > A ) ).
% 0.18/0.45
% 0.18/0.45 thf(sy_c_Num_Oneg__numeral__class_Odbl,type,
% 0.18/0.45 neg_numeral_dbl:
% 0.18/0.45 !>[A: $tType] : ( A > A ) ).
% 0.18/0.45
% 0.18/0.45 thf(sy_c_Num_Onum_OBit0,type,
% 0.18/0.45 bit0: num > num ).
% 0.18/0.45
% 0.18/0.45 thf(sy_c_Num_Onum_OOne,type,
% 0.18/0.45 one2: num ).
% 0.18/0.45
% 0.18/0.45 thf(sy_c_Num_Onumeral__class_Onumeral,type,
% 0.18/0.45 numeral_numeral:
% 0.18/0.45 !>[A: $tType] : ( num > A ) ).
% 0.18/0.45
% 0.18/0.45 thf(sy_c_Num_Opow,type,
% 0.18/0.45 pow: num > num > num ).
% 0.18/0.45
% 0.18/0.45 thf(sy_c_Orderings_Oord__class_Oless,type,
% 0.18/0.45 ord_less:
% 0.18/0.45 !>[A: $tType] : ( A > A > $o ) ).
% 0.18/0.45
% 0.18/0.45 thf(sy_c_Orderings_Oord__class_Oless__eq,type,
% 0.18/0.45 ord_less_eq:
% 0.18/0.45 !>[A: $tType] : ( A > A > $o ) ).
% 0.18/0.45
% 0.18/0.45 thf(sy_c_Power_Opower__class_Opower,type,
% 0.18/0.45 power_power:
% 0.18/0.45 !>[A: $tType] : ( A > nat > A ) ).
% 0.18/0.45
% 0.18/0.45 thf(sy_c_Set_OCollect,type,
% 0.18/0.45 collect:
% 0.18/0.45 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
% 0.18/0.45
% 0.18/0.45 thf(sy_c_Transcendental_Olog,type,
% 0.18/0.45 log: real > real > real ).
% 0.18/0.45
% 0.18/0.45 thf(sy_c_Transcendental_Opowr,type,
% 0.18/0.45 powr:
% 0.18/0.45 !>[A: $tType] : ( A > A > A ) ).
% 0.18/0.45
% 0.18/0.45 thf(sy_c_member,type,
% 0.18/0.45 member:
% 0.18/0.45 !>[A: $tType] : ( A > ( set @ A ) > $o ) ).
% 0.18/0.45
% 0.18/0.45 thf(sy_v_d____,type,
% 0.18/0.45 d: int ).
% 0.18/0.45
% 0.18/0.45 thf(sy_v_n____,type,
% 0.18/0.45 n: nat ).
% 0.18/0.45
% 0.18/0.45 % Relevant facts (251)
% 0.18/0.45 thf(fact_0_numeral__power__le__of__int__cancel__iff,axiom,
% 0.18/0.45 ! [A: $tType] :
% 0.18/0.45 ( ( linordered_idom @ A )
% 0.18/0.45 => ! [X: num,N: nat,A2: int] :
% 0.18/0.45 ( ( ord_less_eq @ A @ ( power_power @ A @ ( numeral_numeral @ A @ X ) @ N ) @ ( ring_1_of_int @ A @ A2 ) )
% 0.18/0.45 = ( ord_less_eq @ int @ ( power_power @ int @ ( numeral_numeral @ int @ X ) @ N ) @ A2 ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % numeral_power_le_of_int_cancel_iff
% 0.18/0.45 thf(fact_1_of__int__le__numeral__power__cancel__iff,axiom,
% 0.18/0.45 ! [A: $tType] :
% 0.18/0.45 ( ( linordered_idom @ A )
% 0.18/0.45 => ! [A2: int,X: num,N: nat] :
% 0.18/0.45 ( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ A2 ) @ ( power_power @ A @ ( numeral_numeral @ A @ X ) @ N ) )
% 0.18/0.45 = ( ord_less_eq @ int @ A2 @ ( power_power @ int @ ( numeral_numeral @ int @ X ) @ N ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % of_int_le_numeral_power_cancel_iff
% 0.18/0.45 thf(fact_2_ceiling__numeral__power,axiom,
% 0.18/0.45 ! [A: $tType] :
% 0.18/0.45 ( ( archim1727834104eiling @ A )
% 0.18/0.45 => ! [X: num,N: nat] :
% 0.18/0.45 ( ( archimedean_ceiling @ A @ ( power_power @ A @ ( numeral_numeral @ A @ X ) @ N ) )
% 0.18/0.45 = ( power_power @ int @ ( numeral_numeral @ int @ X ) @ N ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % ceiling_numeral_power
% 0.18/0.45 thf(fact_3_numeral__power__eq__of__int__cancel__iff,axiom,
% 0.18/0.45 ! [A: $tType] :
% 0.18/0.45 ( ( ring_char_0 @ A )
% 0.18/0.45 => ! [X: num,N: nat,Y: int] :
% 0.18/0.45 ( ( ( power_power @ A @ ( numeral_numeral @ A @ X ) @ N )
% 0.18/0.45 = ( ring_1_of_int @ A @ Y ) )
% 0.18/0.45 = ( ( power_power @ int @ ( numeral_numeral @ int @ X ) @ N )
% 0.18/0.45 = Y ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % numeral_power_eq_of_int_cancel_iff
% 0.18/0.45 thf(fact_4_of__int__eq__numeral__power__cancel__iff,axiom,
% 0.18/0.45 ! [A: $tType] :
% 0.18/0.45 ( ( ring_char_0 @ A )
% 0.18/0.45 => ! [Y: int,X: num,N: nat] :
% 0.18/0.45 ( ( ( ring_1_of_int @ A @ Y )
% 0.18/0.45 = ( power_power @ A @ ( numeral_numeral @ A @ X ) @ N ) )
% 0.18/0.45 = ( Y
% 0.18/0.45 = ( power_power @ int @ ( numeral_numeral @ int @ X ) @ N ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % of_int_eq_numeral_power_cancel_iff
% 0.18/0.45 thf(fact_5_of__int__le__of__int__power__cancel__iff,axiom,
% 0.18/0.45 ! [A: $tType] :
% 0.18/0.45 ( ( linordered_idom @ A )
% 0.18/0.45 => ! [B: int,W: nat,X: int] :
% 0.18/0.45 ( ( ord_less_eq @ A @ ( power_power @ A @ ( ring_1_of_int @ A @ B ) @ W ) @ ( ring_1_of_int @ A @ X ) )
% 0.18/0.45 = ( ord_less_eq @ int @ ( power_power @ int @ B @ W ) @ X ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % of_int_le_of_int_power_cancel_iff
% 0.18/0.45 thf(fact_6_of__int__power__le__of__int__cancel__iff,axiom,
% 0.18/0.45 ! [A: $tType] :
% 0.18/0.45 ( ( linordered_idom @ A )
% 0.18/0.45 => ! [X: int,B: int,W: nat] :
% 0.18/0.45 ( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ X ) @ ( power_power @ A @ ( ring_1_of_int @ A @ B ) @ W ) )
% 0.18/0.45 = ( ord_less_eq @ int @ X @ ( power_power @ int @ B @ W ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % of_int_power_le_of_int_cancel_iff
% 0.18/0.45 thf(fact_7_ceiling__le__numeral,axiom,
% 0.18/0.45 ! [A: $tType] :
% 0.18/0.45 ( ( archim1727834104eiling @ A )
% 0.18/0.45 => ! [X: A,V: num] :
% 0.18/0.45 ( ( ord_less_eq @ int @ ( archimedean_ceiling @ A @ X ) @ ( numeral_numeral @ int @ V ) )
% 0.18/0.45 = ( ord_less_eq @ A @ X @ ( numeral_numeral @ A @ V ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % ceiling_le_numeral
% 0.18/0.45 thf(fact_8_of__int__le__numeral__iff,axiom,
% 0.18/0.45 ! [A: $tType] :
% 0.18/0.45 ( ( linordered_idom @ A )
% 0.18/0.45 => ! [Z: int,N: num] :
% 0.18/0.45 ( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ Z ) @ ( numeral_numeral @ A @ N ) )
% 0.18/0.45 = ( ord_less_eq @ int @ Z @ ( numeral_numeral @ int @ N ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % of_int_le_numeral_iff
% 0.18/0.45 thf(fact_9_of__int__numeral__le__iff,axiom,
% 0.18/0.45 ! [A: $tType] :
% 0.18/0.45 ( ( linordered_idom @ A )
% 0.18/0.45 => ! [N: num,Z: int] :
% 0.18/0.45 ( ( ord_less_eq @ A @ ( numeral_numeral @ A @ N ) @ ( ring_1_of_int @ A @ Z ) )
% 0.18/0.45 = ( ord_less_eq @ int @ ( numeral_numeral @ int @ N ) @ Z ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % of_int_numeral_le_iff
% 0.18/0.45 thf(fact_10_of__int__power,axiom,
% 0.18/0.45 ! [A: $tType] :
% 0.18/0.45 ( ( ring_1 @ A )
% 0.18/0.45 => ! [Z: int,N: nat] :
% 0.18/0.45 ( ( ring_1_of_int @ A @ ( power_power @ int @ Z @ N ) )
% 0.18/0.45 = ( power_power @ A @ ( ring_1_of_int @ A @ Z ) @ N ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % of_int_power
% 0.18/0.45 thf(fact_11_of__int__eq__of__int__power__cancel__iff,axiom,
% 0.18/0.45 ! [A: $tType] :
% 0.18/0.45 ( ( ring_char_0 @ A )
% 0.18/0.45 => ! [B: int,W: nat,X: int] :
% 0.18/0.45 ( ( ( power_power @ A @ ( ring_1_of_int @ A @ B ) @ W )
% 0.18/0.45 = ( ring_1_of_int @ A @ X ) )
% 0.18/0.45 = ( ( power_power @ int @ B @ W )
% 0.18/0.45 = X ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % of_int_eq_of_int_power_cancel_iff
% 0.18/0.45 thf(fact_12_of__int__eq__iff,axiom,
% 0.18/0.45 ! [A: $tType] :
% 0.18/0.45 ( ( ring_char_0 @ A )
% 0.18/0.45 => ! [W: int,Z: int] :
% 0.18/0.45 ( ( ( ring_1_of_int @ A @ W )
% 0.18/0.45 = ( ring_1_of_int @ A @ Z ) )
% 0.18/0.45 = ( W = Z ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % of_int_eq_iff
% 0.18/0.45 thf(fact_13_nat__numeral,axiom,
% 0.18/0.45 ! [K: num] :
% 0.18/0.45 ( ( nat2 @ ( numeral_numeral @ int @ K ) )
% 0.18/0.45 = ( numeral_numeral @ nat @ K ) ) ).
% 0.18/0.45
% 0.18/0.45 % nat_numeral
% 0.18/0.45 thf(fact_14_ceiling__of__int,axiom,
% 0.18/0.45 ! [A: $tType] :
% 0.18/0.45 ( ( archim1727834104eiling @ A )
% 0.18/0.45 => ! [Z: int] :
% 0.18/0.45 ( ( archimedean_ceiling @ A @ ( ring_1_of_int @ A @ Z ) )
% 0.18/0.45 = Z ) ) ).
% 0.18/0.45
% 0.18/0.45 % ceiling_of_int
% 0.18/0.45 thf(fact_15_of__int__ceiling__cancel,axiom,
% 0.18/0.45 ! [A: $tType] :
% 0.18/0.45 ( ( archim1727834104eiling @ A )
% 0.18/0.45 => ! [X: A] :
% 0.18/0.45 ( ( ( ring_1_of_int @ A @ ( archimedean_ceiling @ A @ X ) )
% 0.18/0.45 = X )
% 0.18/0.45 = ( ? [N2: int] :
% 0.18/0.45 ( X
% 0.18/0.45 = ( ring_1_of_int @ A @ N2 ) ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % of_int_ceiling_cancel
% 0.18/0.45 thf(fact_16_of__int__le__iff,axiom,
% 0.18/0.45 ! [A: $tType] :
% 0.18/0.45 ( ( linordered_idom @ A )
% 0.18/0.45 => ! [W: int,Z: int] :
% 0.18/0.45 ( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ W ) @ ( ring_1_of_int @ A @ Z ) )
% 0.18/0.45 = ( ord_less_eq @ int @ W @ Z ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % of_int_le_iff
% 0.18/0.45 thf(fact_17_of__int__numeral,axiom,
% 0.18/0.45 ! [A: $tType] :
% 0.18/0.45 ( ( ring_1 @ A )
% 0.18/0.45 => ! [K: num] :
% 0.18/0.45 ( ( ring_1_of_int @ A @ ( numeral_numeral @ int @ K ) )
% 0.18/0.45 = ( numeral_numeral @ A @ K ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % of_int_numeral
% 0.18/0.45 thf(fact_18_of__int__eq__numeral__iff,axiom,
% 0.18/0.45 ! [A: $tType] :
% 0.18/0.45 ( ( ring_char_0 @ A )
% 0.18/0.45 => ! [Z: int,N: num] :
% 0.18/0.45 ( ( ( ring_1_of_int @ A @ Z )
% 0.18/0.45 = ( numeral_numeral @ A @ N ) )
% 0.18/0.45 = ( Z
% 0.18/0.45 = ( numeral_numeral @ int @ N ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % of_int_eq_numeral_iff
% 0.18/0.45 thf(fact_19_ceiling__numeral,axiom,
% 0.18/0.45 ! [A: $tType] :
% 0.18/0.45 ( ( archim1727834104eiling @ A )
% 0.18/0.45 => ! [V: num] :
% 0.18/0.45 ( ( archimedean_ceiling @ A @ ( numeral_numeral @ A @ V ) )
% 0.18/0.45 = ( numeral_numeral @ int @ V ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % ceiling_numeral
% 0.18/0.45 thf(fact_20_of__int__power__eq__of__int__cancel__iff,axiom,
% 0.18/0.45 ! [A: $tType] :
% 0.18/0.45 ( ( ring_char_0 @ A )
% 0.18/0.45 => ! [X: int,B: int,W: nat] :
% 0.18/0.45 ( ( ( ring_1_of_int @ A @ X )
% 0.18/0.45 = ( power_power @ A @ ( ring_1_of_int @ A @ B ) @ W ) )
% 0.18/0.45 = ( X
% 0.18/0.45 = ( power_power @ int @ B @ W ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % of_int_power_eq_of_int_cancel_iff
% 0.18/0.45 thf(fact_21_numeral__power__eq__nat__cancel__iff,axiom,
% 0.18/0.45 ! [X: num,N: nat,Y: int] :
% 0.18/0.45 ( ( ( power_power @ nat @ ( numeral_numeral @ nat @ X ) @ N )
% 0.18/0.45 = ( nat2 @ Y ) )
% 0.18/0.45 = ( ( power_power @ int @ ( numeral_numeral @ int @ X ) @ N )
% 0.18/0.45 = Y ) ) ).
% 0.18/0.45
% 0.18/0.45 % numeral_power_eq_nat_cancel_iff
% 0.18/0.45 thf(fact_22_nat__eq__numeral__power__cancel__iff,axiom,
% 0.18/0.45 ! [Y: int,X: num,N: nat] :
% 0.18/0.45 ( ( ( nat2 @ Y )
% 0.18/0.45 = ( power_power @ nat @ ( numeral_numeral @ nat @ X ) @ N ) )
% 0.18/0.45 = ( Y
% 0.18/0.45 = ( power_power @ int @ ( numeral_numeral @ int @ X ) @ N ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % nat_eq_numeral_power_cancel_iff
% 0.18/0.45 thf(fact_23_numeral__power__le__nat__cancel__iff,axiom,
% 0.18/0.45 ! [X: num,N: nat,A2: int] :
% 0.18/0.45 ( ( ord_less_eq @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ X ) @ N ) @ ( nat2 @ A2 ) )
% 0.18/0.45 = ( ord_less_eq @ int @ ( power_power @ int @ ( numeral_numeral @ int @ X ) @ N ) @ A2 ) ) ).
% 0.18/0.45
% 0.18/0.45 % numeral_power_le_nat_cancel_iff
% 0.18/0.45 thf(fact_24_nat__le__numeral__power__cancel__iff,axiom,
% 0.18/0.45 ! [A2: int,X: num,N: nat] :
% 0.18/0.45 ( ( ord_less_eq @ nat @ ( nat2 @ A2 ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ X ) @ N ) )
% 0.18/0.45 = ( ord_less_eq @ int @ A2 @ ( power_power @ int @ ( numeral_numeral @ int @ X ) @ N ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % nat_le_numeral_power_cancel_iff
% 0.18/0.45 thf(fact_25_nat__mono,axiom,
% 0.18/0.45 ! [X: int,Y: int] :
% 0.18/0.45 ( ( ord_less_eq @ int @ X @ Y )
% 0.18/0.45 => ( ord_less_eq @ nat @ ( nat2 @ X ) @ ( nat2 @ Y ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % nat_mono
% 0.18/0.45 thf(fact_26_ex__le__of__int,axiom,
% 0.18/0.45 ! [A: $tType] :
% 0.18/0.45 ( ( archim1804426504_field @ A )
% 0.18/0.45 => ! [X: A] :
% 0.18/0.45 ? [Z2: int] : ( ord_less_eq @ A @ X @ ( ring_1_of_int @ A @ Z2 ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % ex_le_of_int
% 0.18/0.45 thf(fact_27_le__of__int__ceiling,axiom,
% 0.18/0.45 ! [A: $tType] :
% 0.18/0.45 ( ( archim1727834104eiling @ A )
% 0.18/0.45 => ! [X: A] : ( ord_less_eq @ A @ X @ ( ring_1_of_int @ A @ ( archimedean_ceiling @ A @ X ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % le_of_int_ceiling
% 0.18/0.45 thf(fact_28_ceiling__mono,axiom,
% 0.18/0.45 ! [A: $tType] :
% 0.18/0.45 ( ( archim1727834104eiling @ A )
% 0.18/0.45 => ! [Y: A,X: A] :
% 0.18/0.45 ( ( ord_less_eq @ A @ Y @ X )
% 0.18/0.45 => ( ord_less_eq @ int @ ( archimedean_ceiling @ A @ Y ) @ ( archimedean_ceiling @ A @ X ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % ceiling_mono
% 0.18/0.45 thf(fact_29_ceiling__le__iff,axiom,
% 0.18/0.45 ! [A: $tType] :
% 0.18/0.45 ( ( archim1727834104eiling @ A )
% 0.18/0.45 => ! [X: A,Z: int] :
% 0.18/0.45 ( ( ord_less_eq @ int @ ( archimedean_ceiling @ A @ X ) @ Z )
% 0.18/0.45 = ( ord_less_eq @ A @ X @ ( ring_1_of_int @ A @ Z ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % ceiling_le_iff
% 0.18/0.45 thf(fact_30_ceiling__le,axiom,
% 0.18/0.45 ! [A: $tType] :
% 0.18/0.45 ( ( archim1727834104eiling @ A )
% 0.18/0.45 => ! [X: A,A2: int] :
% 0.18/0.45 ( ( ord_less_eq @ A @ X @ ( ring_1_of_int @ A @ A2 ) )
% 0.18/0.45 => ( ord_less_eq @ int @ ( archimedean_ceiling @ A @ X ) @ A2 ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % ceiling_le
% 0.18/0.45 thf(fact_31__092_060open_062log_A2_A1_A_092_060le_062_Alog_A2_A_Ireal__of__int_Ad_J_092_060close_062,axiom,
% 0.18/0.45 ord_less_eq @ real @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( one_one @ real ) ) @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( ring_1_of_int @ real @ d ) ) ).
% 0.18/0.45
% 0.18/0.45 % \<open>log 2 1 \<le> log 2 (real_of_int d)\<close>
% 0.18/0.45 thf(fact_32_semiring__norm_I69_J,axiom,
% 0.18/0.45 ! [M: num] :
% 0.18/0.45 ~ ( ord_less_eq @ num @ ( bit0 @ M ) @ one2 ) ).
% 0.18/0.45
% 0.18/0.45 % semiring_norm(69)
% 0.18/0.45 thf(fact_33_semiring__norm_I83_J,axiom,
% 0.18/0.45 ! [N: num] :
% 0.18/0.45 ( one2
% 0.18/0.45 != ( bit0 @ N ) ) ).
% 0.18/0.45
% 0.18/0.45 % semiring_norm(83)
% 0.18/0.45 thf(fact_34_semiring__norm_I85_J,axiom,
% 0.18/0.45 ! [M: num] :
% 0.18/0.45 ( ( bit0 @ M )
% 0.18/0.45 != one2 ) ).
% 0.18/0.45
% 0.18/0.45 % semiring_norm(85)
% 0.18/0.45 thf(fact_35_numeral__le__iff,axiom,
% 0.18/0.45 ! [A: $tType] :
% 0.18/0.45 ( ( linord1659791738miring @ A )
% 0.18/0.45 => ! [M: num,N: num] :
% 0.18/0.45 ( ( ord_less_eq @ A @ ( numeral_numeral @ A @ M ) @ ( numeral_numeral @ A @ N ) )
% 0.18/0.45 = ( ord_less_eq @ num @ M @ N ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % numeral_le_iff
% 0.18/0.45 thf(fact_36__092_060open_062real__of__int_Ad_A_061_A2_Apowr_Alog_A2_A_Ireal__of__int_Ad_J_092_060close_062,axiom,
% 0.18/0.45 ( ( ring_1_of_int @ real @ d )
% 0.18/0.45 = ( powr @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( ring_1_of_int @ real @ d ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % \<open>real_of_int d = 2 powr log 2 (real_of_int d)\<close>
% 0.18/0.45 thf(fact_37_n__def,axiom,
% 0.18/0.45 ( n
% 0.18/0.45 = ( suc @ ( nat2 @ ( archimedean_ceiling @ real @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( ring_1_of_int @ real @ d ) ) ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % n_def
% 0.18/0.45 thf(fact_38_semiring__norm_I68_J,axiom,
% 0.18/0.45 ! [N: num] : ( ord_less_eq @ num @ one2 @ N ) ).
% 0.18/0.45
% 0.18/0.45 % semiring_norm(68)
% 0.18/0.45 thf(fact_39_semiring__norm_I71_J,axiom,
% 0.18/0.45 ! [M: num,N: num] :
% 0.18/0.45 ( ( ord_less_eq @ num @ ( bit0 @ M ) @ ( bit0 @ N ) )
% 0.18/0.45 = ( ord_less_eq @ num @ M @ N ) ) ).
% 0.18/0.45
% 0.18/0.45 % semiring_norm(71)
% 0.18/0.45 thf(fact_40_semiring__norm_I87_J,axiom,
% 0.18/0.45 ! [M: num,N: num] :
% 0.18/0.45 ( ( ( bit0 @ M )
% 0.18/0.45 = ( bit0 @ N ) )
% 0.18/0.45 = ( M = N ) ) ).
% 0.18/0.45
% 0.18/0.45 % semiring_norm(87)
% 0.18/0.45 thf(fact_41_verit__eq__simplify_I8_J,axiom,
% 0.18/0.45 ! [X2: num,Y2: num] :
% 0.18/0.45 ( ( ( bit0 @ X2 )
% 0.18/0.45 = ( bit0 @ Y2 ) )
% 0.18/0.45 = ( X2 = Y2 ) ) ).
% 0.18/0.45
% 0.18/0.45 % verit_eq_simplify(8)
% 0.18/0.45 thf(fact_42_numeral__eq__iff,axiom,
% 0.18/0.45 ! [A: $tType] :
% 0.18/0.45 ( ( semiring_char_0 @ A )
% 0.18/0.45 => ! [M: num,N: num] :
% 0.18/0.45 ( ( ( numeral_numeral @ A @ M )
% 0.18/0.45 = ( numeral_numeral @ A @ N ) )
% 0.18/0.45 = ( M = N ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % numeral_eq_iff
% 0.18/0.45 thf(fact_43_of__int__eq__1__iff,axiom,
% 0.18/0.45 ! [A: $tType] :
% 0.18/0.45 ( ( ring_char_0 @ A )
% 0.18/0.45 => ! [Z: int] :
% 0.18/0.45 ( ( ( ring_1_of_int @ A @ Z )
% 0.18/0.45 = ( one_one @ A ) )
% 0.18/0.45 = ( Z
% 0.18/0.45 = ( one_one @ int ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % of_int_eq_1_iff
% 0.18/0.45 thf(fact_44_mem__Collect__eq,axiom,
% 0.18/0.45 ! [A: $tType,A2: A,P: A > $o] :
% 0.18/0.45 ( ( member @ A @ A2 @ ( collect @ A @ P ) )
% 0.18/0.45 = ( P @ A2 ) ) ).
% 0.18/0.45
% 0.18/0.45 % mem_Collect_eq
% 0.18/0.45 thf(fact_45_Collect__mem__eq,axiom,
% 0.18/0.45 ! [A: $tType,A3: set @ A] :
% 0.18/0.45 ( ( collect @ A
% 0.18/0.45 @ ^ [X3: A] : ( member @ A @ X3 @ A3 ) )
% 0.18/0.45 = A3 ) ).
% 0.18/0.45
% 0.18/0.45 % Collect_mem_eq
% 0.18/0.45 thf(fact_46_Collect__cong,axiom,
% 0.18/0.45 ! [A: $tType,P: A > $o,Q: A > $o] :
% 0.18/0.45 ( ! [X4: A] :
% 0.18/0.45 ( ( P @ X4 )
% 0.18/0.45 = ( Q @ X4 ) )
% 0.18/0.45 => ( ( collect @ A @ P )
% 0.18/0.45 = ( collect @ A @ Q ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % Collect_cong
% 0.18/0.45 thf(fact_47_ext,axiom,
% 0.18/0.45 ! [B2: $tType,A: $tType,F: A > B2,G: A > B2] :
% 0.18/0.45 ( ! [X4: A] :
% 0.18/0.45 ( ( F @ X4 )
% 0.18/0.45 = ( G @ X4 ) )
% 0.18/0.45 => ( F = G ) ) ).
% 0.18/0.45
% 0.18/0.45 % ext
% 0.18/0.45 thf(fact_48_of__int__1,axiom,
% 0.18/0.45 ! [A: $tType] :
% 0.18/0.45 ( ( ring_1 @ A )
% 0.18/0.45 => ( ( ring_1_of_int @ A @ ( one_one @ int ) )
% 0.18/0.45 = ( one_one @ A ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % of_int_1
% 0.18/0.45 thf(fact_49_ceiling__one,axiom,
% 0.18/0.45 ! [A: $tType] :
% 0.18/0.45 ( ( archim1727834104eiling @ A )
% 0.18/0.45 => ( ( archimedean_ceiling @ A @ ( one_one @ A ) )
% 0.18/0.45 = ( one_one @ int ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % ceiling_one
% 0.18/0.45 thf(fact_50_numeral__eq__one__iff,axiom,
% 0.18/0.45 ! [A: $tType] :
% 0.18/0.45 ( ( semiring_char_0 @ A )
% 0.18/0.45 => ! [N: num] :
% 0.18/0.45 ( ( ( numeral_numeral @ A @ N )
% 0.18/0.45 = ( one_one @ A ) )
% 0.18/0.45 = ( N = one2 ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % numeral_eq_one_iff
% 0.18/0.45 thf(fact_51_one__eq__numeral__iff,axiom,
% 0.18/0.45 ! [A: $tType] :
% 0.18/0.45 ( ( semiring_char_0 @ A )
% 0.18/0.45 => ! [N: num] :
% 0.18/0.45 ( ( ( one_one @ A )
% 0.18/0.45 = ( numeral_numeral @ A @ N ) )
% 0.18/0.45 = ( one2 = N ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % one_eq_numeral_iff
% 0.18/0.45 thf(fact_52_of__int__1__le__iff,axiom,
% 0.18/0.45 ! [A: $tType] :
% 0.18/0.45 ( ( linordered_idom @ A )
% 0.18/0.45 => ! [Z: int] :
% 0.18/0.45 ( ( ord_less_eq @ A @ ( one_one @ A ) @ ( ring_1_of_int @ A @ Z ) )
% 0.18/0.45 = ( ord_less_eq @ int @ ( one_one @ int ) @ Z ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % of_int_1_le_iff
% 0.18/0.45 thf(fact_53_of__int__le__1__iff,axiom,
% 0.18/0.45 ! [A: $tType] :
% 0.18/0.45 ( ( linordered_idom @ A )
% 0.18/0.45 => ! [Z: int] :
% 0.18/0.45 ( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ Z ) @ ( one_one @ A ) )
% 0.18/0.45 = ( ord_less_eq @ int @ Z @ ( one_one @ int ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % of_int_le_1_iff
% 0.18/0.45 thf(fact_54_ceiling__le__one,axiom,
% 0.18/0.45 ! [A: $tType] :
% 0.18/0.45 ( ( archim1727834104eiling @ A )
% 0.18/0.45 => ! [X: A] :
% 0.18/0.45 ( ( ord_less_eq @ int @ ( archimedean_ceiling @ A @ X ) @ ( one_one @ int ) )
% 0.18/0.45 = ( ord_less_eq @ A @ X @ ( one_one @ A ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % ceiling_le_one
% 0.18/0.45 thf(fact_55_numeral__le__one__iff,axiom,
% 0.18/0.45 ! [A: $tType] :
% 0.18/0.45 ( ( linord1659791738miring @ A )
% 0.18/0.45 => ! [N: num] :
% 0.18/0.45 ( ( ord_less_eq @ A @ ( numeral_numeral @ A @ N ) @ ( one_one @ A ) )
% 0.18/0.45 = ( ord_less_eq @ num @ N @ one2 ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % numeral_le_one_iff
% 0.18/0.45 thf(fact_56_complete__real,axiom,
% 0.18/0.45 ! [S: set @ real] :
% 0.18/0.45 ( ? [X5: real] : ( member @ real @ X5 @ S )
% 0.18/0.45 => ( ? [Z3: real] :
% 0.18/0.45 ! [X4: real] :
% 0.18/0.45 ( ( member @ real @ X4 @ S )
% 0.18/0.45 => ( ord_less_eq @ real @ X4 @ Z3 ) )
% 0.18/0.45 => ? [Y3: real] :
% 0.18/0.45 ( ! [X5: real] :
% 0.18/0.45 ( ( member @ real @ X5 @ S )
% 0.18/0.45 => ( ord_less_eq @ real @ X5 @ Y3 ) )
% 0.18/0.45 & ! [Z3: real] :
% 0.18/0.45 ( ! [X4: real] :
% 0.18/0.45 ( ( member @ real @ X4 @ S )
% 0.18/0.45 => ( ord_less_eq @ real @ X4 @ Z3 ) )
% 0.18/0.45 => ( ord_less_eq @ real @ Y3 @ Z3 ) ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % complete_real
% 0.18/0.45 thf(fact_57_le__numeral__extra_I4_J,axiom,
% 0.18/0.45 ! [A: $tType] :
% 0.18/0.45 ( ( linord1659791738miring @ A )
% 0.18/0.45 => ( ord_less_eq @ A @ ( one_one @ A ) @ ( one_one @ A ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % le_numeral_extra(4)
% 0.18/0.45 thf(fact_58_one__le__numeral,axiom,
% 0.18/0.45 ! [A: $tType] :
% 0.18/0.45 ( ( linord1659791738miring @ A )
% 0.18/0.45 => ! [N: num] : ( ord_less_eq @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ N ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % one_le_numeral
% 0.18/0.45 thf(fact_59_numeral__One,axiom,
% 0.18/0.45 ! [A: $tType] :
% 0.18/0.45 ( ( numeral @ A )
% 0.18/0.45 => ( ( numeral_numeral @ A @ one2 )
% 0.18/0.45 = ( one_one @ A ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % numeral_One
% 0.18/0.45 thf(fact_60_verit__la__disequality,axiom,
% 0.18/0.45 ! [A: $tType] :
% 0.18/0.45 ( ( linorder @ A )
% 0.18/0.45 => ! [A2: A,B: A] :
% 0.18/0.45 ( ( A2 = B )
% 0.18/0.45 | ~ ( ord_less_eq @ A @ A2 @ B )
% 0.18/0.45 | ~ ( ord_less_eq @ A @ B @ A2 ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % verit_la_disequality
% 0.18/0.45 thf(fact_61_le__num__One__iff,axiom,
% 0.18/0.45 ! [X: num] :
% 0.18/0.45 ( ( ord_less_eq @ num @ X @ one2 )
% 0.18/0.45 = ( X = one2 ) ) ).
% 0.18/0.45
% 0.18/0.45 % le_num_One_iff
% 0.18/0.45 thf(fact_62_two__realpow__ge__one,axiom,
% 0.18/0.45 ! [N: nat] : ( ord_less_eq @ real @ ( one_one @ real ) @ ( power_power @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ N ) ) ).
% 0.18/0.45
% 0.18/0.45 % two_realpow_ge_one
% 0.18/0.45 thf(fact_63_verit__la__generic,axiom,
% 0.18/0.45 ! [A2: int,X: int] :
% 0.18/0.45 ( ( ord_less_eq @ int @ A2 @ X )
% 0.18/0.45 | ( A2 = X )
% 0.18/0.45 | ( ord_less_eq @ int @ X @ A2 ) ) ).
% 0.18/0.45
% 0.18/0.45 % verit_la_generic
% 0.18/0.45 thf(fact_64_verit__eq__simplify_I10_J,axiom,
% 0.18/0.45 ! [X2: num] :
% 0.18/0.45 ( one2
% 0.18/0.45 != ( bit0 @ X2 ) ) ).
% 0.18/0.45
% 0.18/0.45 % verit_eq_simplify(10)
% 0.18/0.45 thf(fact_65_nat__numeral__as__int,axiom,
% 0.18/0.45 ( ( numeral_numeral @ nat )
% 0.18/0.45 = ( ^ [I: num] : ( nat2 @ ( numeral_numeral @ int @ I ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % nat_numeral_as_int
% 0.18/0.45 thf(fact_66__092_060open_0622_Apowr_Areal_A_Inat_A_092_060lceil_062log_A2_A_Ireal__of__int_Ad_J_092_060rceil_062_J_A_061_Areal__of__int_A_I2_A_094_Anat_A_092_060lceil_062log_A2_A_Ireal__of__int_Ad_J_092_060rceil_062_J_092_060close_062,axiom,
% 0.18/0.45 ( ( powr @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ real @ ( nat2 @ ( archimedean_ceiling @ real @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( ring_1_of_int @ real @ d ) ) ) ) ) )
% 0.18/0.45 = ( ring_1_of_int @ real @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( nat2 @ ( archimedean_ceiling @ real @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( ring_1_of_int @ real @ d ) ) ) ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % \<open>2 powr real (nat \<lceil>log 2 (real_of_int d)\<rceil>) = real_of_int (2 ^ nat \<lceil>log 2 (real_of_int d)\<rceil>)\<close>
% 0.18/0.45 thf(fact_67__092_060open_0622_Apowr_Alog_A2_A_Ireal__of__int_Ad_J_A_092_060le_062_A2_Apowr_Areal_A_Inat_A_092_060lceil_062log_A2_A_Ireal__of__int_Ad_J_092_060rceil_062_J_092_060close_062,axiom,
% 0.18/0.45 ord_less_eq @ real @ ( powr @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( ring_1_of_int @ real @ d ) ) ) @ ( powr @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ real @ ( nat2 @ ( archimedean_ceiling @ real @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( ring_1_of_int @ real @ d ) ) ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % \<open>2 powr log 2 (real_of_int d) \<le> 2 powr real (nat \<lceil>log 2 (real_of_int d)\<rceil>)\<close>
% 0.18/0.45 thf(fact_68_numeral__powr__numeral__real,axiom,
% 0.18/0.45 ! [M: num,N: num] :
% 0.18/0.45 ( ( powr @ real @ ( numeral_numeral @ real @ M ) @ ( numeral_numeral @ real @ N ) )
% 0.18/0.45 = ( power_power @ real @ ( numeral_numeral @ real @ M ) @ ( numeral_numeral @ nat @ N ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % numeral_powr_numeral_real
% 0.18/0.45 thf(fact_69_powr__one__eq__one,axiom,
% 0.18/0.45 ! [A: $tType] :
% 0.18/0.45 ( ( ln @ A )
% 0.18/0.45 => ! [A2: A] :
% 0.18/0.45 ( ( powr @ A @ ( one_one @ A ) @ A2 )
% 0.18/0.45 = ( one_one @ A ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % powr_one_eq_one
% 0.18/0.45 thf(fact_70_self__le__ge2__pow,axiom,
% 0.18/0.45 ! [K: nat,M: nat] :
% 0.18/0.45 ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ K )
% 0.18/0.45 => ( ord_less_eq @ nat @ M @ ( power_power @ nat @ K @ M ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % self_le_ge2_pow
% 0.18/0.45 thf(fact_71_power2__nat__le__eq__le,axiom,
% 0.18/0.45 ! [M: nat,N: nat] :
% 0.18/0.45 ( ( ord_less_eq @ nat @ ( power_power @ nat @ M @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
% 0.18/0.45 = ( ord_less_eq @ nat @ M @ N ) ) ).
% 0.18/0.45
% 0.18/0.45 % power2_nat_le_eq_le
% 0.18/0.45 thf(fact_72_power2__nat__le__imp__le,axiom,
% 0.18/0.45 ! [M: nat,N: nat] :
% 0.18/0.45 ( ( ord_less_eq @ nat @ ( power_power @ nat @ M @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ N )
% 0.18/0.45 => ( ord_less_eq @ nat @ M @ N ) ) ).
% 0.18/0.45
% 0.18/0.45 % power2_nat_le_imp_le
% 0.18/0.45 thf(fact_73_Suc__le__mono,axiom,
% 0.18/0.45 ! [N: nat,M: nat] :
% 0.18/0.45 ( ( ord_less_eq @ nat @ ( suc @ N ) @ ( suc @ M ) )
% 0.18/0.45 = ( ord_less_eq @ nat @ N @ M ) ) ).
% 0.18/0.45
% 0.18/0.45 % Suc_le_mono
% 0.18/0.45 thf(fact_74_power__one__right,axiom,
% 0.18/0.45 ! [A: $tType] :
% 0.18/0.45 ( ( monoid_mult @ A )
% 0.18/0.45 => ! [A2: A] :
% 0.18/0.45 ( ( power_power @ A @ A2 @ ( one_one @ nat ) )
% 0.18/0.45 = A2 ) ) ).
% 0.18/0.45
% 0.18/0.45 % power_one_right
% 0.18/0.45 thf(fact_75_nat_Oinject,axiom,
% 0.18/0.45 ! [X2: nat,Y2: nat] :
% 0.18/0.45 ( ( ( suc @ X2 )
% 0.18/0.45 = ( suc @ Y2 ) )
% 0.18/0.45 = ( X2 = Y2 ) ) ).
% 0.18/0.45
% 0.18/0.45 % nat.inject
% 0.18/0.45 thf(fact_76_old_Onat_Oinject,axiom,
% 0.18/0.45 ! [Nat: nat,Nat2: nat] :
% 0.18/0.45 ( ( ( suc @ Nat )
% 0.18/0.45 = ( suc @ Nat2 ) )
% 0.18/0.45 = ( Nat = Nat2 ) ) ).
% 0.18/0.45
% 0.18/0.45 % old.nat.inject
% 0.18/0.45 thf(fact_77_of__nat__eq__iff,axiom,
% 0.18/0.45 ! [A: $tType] :
% 0.18/0.45 ( ( semiring_char_0 @ A )
% 0.18/0.45 => ! [M: nat,N: nat] :
% 0.18/0.45 ( ( ( semiring_1_of_nat @ A @ M )
% 0.18/0.45 = ( semiring_1_of_nat @ A @ N ) )
% 0.18/0.45 = ( M = N ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % of_nat_eq_iff
% 0.18/0.45 thf(fact_78_power__one,axiom,
% 0.18/0.45 ! [A: $tType] :
% 0.18/0.45 ( ( monoid_mult @ A )
% 0.18/0.45 => ! [N: nat] :
% 0.18/0.45 ( ( power_power @ A @ ( one_one @ A ) @ N )
% 0.18/0.45 = ( one_one @ A ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % power_one
% 0.18/0.45 thf(fact_79_of__nat__eq__1__iff,axiom,
% 0.18/0.45 ! [A: $tType] :
% 0.18/0.45 ( ( semiring_char_0 @ A )
% 0.18/0.45 => ! [N: nat] :
% 0.18/0.45 ( ( ( semiring_1_of_nat @ A @ N )
% 0.18/0.45 = ( one_one @ A ) )
% 0.18/0.45 = ( N
% 0.18/0.45 = ( one_one @ nat ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % of_nat_eq_1_iff
% 0.18/0.45 thf(fact_80_of__nat__1__eq__iff,axiom,
% 0.18/0.45 ! [A: $tType] :
% 0.18/0.45 ( ( semiring_char_0 @ A )
% 0.18/0.45 => ! [N: nat] :
% 0.18/0.45 ( ( ( one_one @ A )
% 0.18/0.45 = ( semiring_1_of_nat @ A @ N ) )
% 0.18/0.45 = ( N
% 0.18/0.45 = ( one_one @ nat ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % of_nat_1_eq_iff
% 0.18/0.45 thf(fact_81_of__nat__1,axiom,
% 0.18/0.45 ! [A: $tType] :
% 0.18/0.45 ( ( semiring_1 @ A )
% 0.18/0.45 => ( ( semiring_1_of_nat @ A @ ( one_one @ nat ) )
% 0.18/0.45 = ( one_one @ A ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % of_nat_1
% 0.18/0.45 thf(fact_82_of__int__of__nat__eq,axiom,
% 0.18/0.45 ! [A: $tType] :
% 0.18/0.45 ( ( ring_1 @ A )
% 0.18/0.45 => ! [N: nat] :
% 0.18/0.45 ( ( ring_1_of_int @ A @ ( semiring_1_of_nat @ int @ N ) )
% 0.18/0.45 = ( semiring_1_of_nat @ A @ N ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % of_int_of_nat_eq
% 0.18/0.45 thf(fact_83_ceiling__of__nat,axiom,
% 0.18/0.45 ! [A: $tType] :
% 0.18/0.45 ( ( archim1727834104eiling @ A )
% 0.18/0.45 => ! [N: nat] :
% 0.18/0.45 ( ( archimedean_ceiling @ A @ ( semiring_1_of_nat @ A @ N ) )
% 0.18/0.45 = ( semiring_1_of_nat @ int @ N ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % ceiling_of_nat
% 0.18/0.45 thf(fact_84_of__nat__le__iff,axiom,
% 0.18/0.45 ! [A: $tType] :
% 0.18/0.45 ( ( linord1659791738miring @ A )
% 0.18/0.45 => ! [M: nat,N: nat] :
% 0.18/0.45 ( ( ord_less_eq @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N ) )
% 0.18/0.45 = ( ord_less_eq @ nat @ M @ N ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % of_nat_le_iff
% 0.18/0.45 thf(fact_85_of__nat__numeral,axiom,
% 0.18/0.45 ! [A: $tType] :
% 0.18/0.45 ( ( semiring_1 @ A )
% 0.18/0.45 => ! [N: num] :
% 0.18/0.45 ( ( semiring_1_of_nat @ A @ ( numeral_numeral @ nat @ N ) )
% 0.18/0.45 = ( numeral_numeral @ A @ N ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % of_nat_numeral
% 0.18/0.45 thf(fact_86_of__nat__power__eq__of__nat__cancel__iff,axiom,
% 0.18/0.45 ! [A: $tType] :
% 0.18/0.45 ( ( semiring_char_0 @ A )
% 0.18/0.45 => ! [X: nat,B: nat,W: nat] :
% 0.18/0.45 ( ( ( semiring_1_of_nat @ A @ X )
% 0.18/0.45 = ( power_power @ A @ ( semiring_1_of_nat @ A @ B ) @ W ) )
% 0.18/0.45 = ( X
% 0.18/0.45 = ( power_power @ nat @ B @ W ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % of_nat_power_eq_of_nat_cancel_iff
% 0.18/0.45 thf(fact_87_of__nat__eq__of__nat__power__cancel__iff,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( semiring_char_0 @ A )
% 0.29/0.45 => ! [B: nat,W: nat,X: nat] :
% 0.29/0.45 ( ( ( power_power @ A @ ( semiring_1_of_nat @ A @ B ) @ W )
% 0.29/0.45 = ( semiring_1_of_nat @ A @ X ) )
% 0.29/0.45 = ( ( power_power @ nat @ B @ W )
% 0.29/0.45 = X ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % of_nat_eq_of_nat_power_cancel_iff
% 0.29/0.45 thf(fact_88_of__nat__power,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( semiring_1 @ A )
% 0.29/0.45 => ! [M: nat,N: nat] :
% 0.29/0.45 ( ( semiring_1_of_nat @ A @ ( power_power @ nat @ M @ N ) )
% 0.29/0.45 = ( power_power @ A @ ( semiring_1_of_nat @ A @ M ) @ N ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % of_nat_power
% 0.29/0.45 thf(fact_89_Suc__1,axiom,
% 0.29/0.45 ( ( suc @ ( one_one @ nat ) )
% 0.29/0.45 = ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % Suc_1
% 0.29/0.45 thf(fact_90_of__nat__power__le__of__nat__cancel__iff,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( linordered_semidom @ A )
% 0.29/0.45 => ! [X: nat,B: nat,W: nat] :
% 0.29/0.45 ( ( ord_less_eq @ A @ ( semiring_1_of_nat @ A @ X ) @ ( power_power @ A @ ( semiring_1_of_nat @ A @ B ) @ W ) )
% 0.29/0.45 = ( ord_less_eq @ nat @ X @ ( power_power @ nat @ B @ W ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % of_nat_power_le_of_nat_cancel_iff
% 0.29/0.45 thf(fact_91_of__nat__le__of__nat__power__cancel__iff,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( linordered_semidom @ A )
% 0.29/0.45 => ! [B: nat,W: nat,X: nat] :
% 0.29/0.45 ( ( ord_less_eq @ A @ ( power_power @ A @ ( semiring_1_of_nat @ A @ B ) @ W ) @ ( semiring_1_of_nat @ A @ X ) )
% 0.29/0.45 = ( ord_less_eq @ nat @ ( power_power @ nat @ B @ W ) @ X ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % of_nat_le_of_nat_power_cancel_iff
% 0.29/0.45 thf(fact_92_real__of__nat__eq__numeral__power__cancel__iff,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( semiring_char_0 @ A )
% 0.29/0.45 => ! [Y: nat,X: num,N: nat] :
% 0.29/0.45 ( ( ( semiring_1_of_nat @ A @ Y )
% 0.29/0.45 = ( power_power @ A @ ( numeral_numeral @ A @ X ) @ N ) )
% 0.29/0.45 = ( Y
% 0.29/0.45 = ( power_power @ nat @ ( numeral_numeral @ nat @ X ) @ N ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % real_of_nat_eq_numeral_power_cancel_iff
% 0.29/0.45 thf(fact_93_numeral__power__eq__of__nat__cancel__iff,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( semiring_char_0 @ A )
% 0.29/0.45 => ! [X: num,N: nat,Y: nat] :
% 0.29/0.45 ( ( ( power_power @ A @ ( numeral_numeral @ A @ X ) @ N )
% 0.29/0.45 = ( semiring_1_of_nat @ A @ Y ) )
% 0.29/0.45 = ( ( power_power @ nat @ ( numeral_numeral @ nat @ X ) @ N )
% 0.29/0.45 = Y ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % numeral_power_eq_of_nat_cancel_iff
% 0.29/0.45 thf(fact_94_numeral__le__real__of__nat__iff,axiom,
% 0.29/0.45 ! [N: num,M: nat] :
% 0.29/0.45 ( ( ord_less_eq @ real @ ( numeral_numeral @ real @ N ) @ ( semiring_1_of_nat @ real @ M ) )
% 0.29/0.45 = ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ N ) @ M ) ) ).
% 0.29/0.45
% 0.29/0.45 % numeral_le_real_of_nat_iff
% 0.29/0.45 thf(fact_95_nat__ceiling__le__eq,axiom,
% 0.29/0.45 ! [X: real,A2: nat] :
% 0.29/0.45 ( ( ord_less_eq @ nat @ ( nat2 @ ( archimedean_ceiling @ real @ X ) ) @ A2 )
% 0.29/0.45 = ( ord_less_eq @ real @ X @ ( semiring_1_of_nat @ real @ A2 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % nat_ceiling_le_eq
% 0.29/0.45 thf(fact_96_of__nat__le__numeral__power__cancel__iff,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( linordered_semidom @ A )
% 0.29/0.45 => ! [X: nat,I2: num,N: nat] :
% 0.29/0.45 ( ( ord_less_eq @ A @ ( semiring_1_of_nat @ A @ X ) @ ( power_power @ A @ ( numeral_numeral @ A @ I2 ) @ N ) )
% 0.29/0.45 = ( ord_less_eq @ nat @ X @ ( power_power @ nat @ ( numeral_numeral @ nat @ I2 ) @ N ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % of_nat_le_numeral_power_cancel_iff
% 0.29/0.45 thf(fact_97_numeral__power__le__of__nat__cancel__iff,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( linordered_semidom @ A )
% 0.29/0.45 => ! [I2: num,N: nat,X: nat] :
% 0.29/0.45 ( ( ord_less_eq @ A @ ( power_power @ A @ ( numeral_numeral @ A @ I2 ) @ N ) @ ( semiring_1_of_nat @ A @ X ) )
% 0.29/0.45 = ( ord_less_eq @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ I2 ) @ N ) @ X ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % numeral_power_le_of_nat_cancel_iff
% 0.29/0.45 thf(fact_98_nat__one__as__int,axiom,
% 0.29/0.45 ( ( one_one @ nat )
% 0.29/0.45 = ( nat2 @ ( one_one @ int ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % nat_one_as_int
% 0.29/0.45 thf(fact_99_real__arch__simple,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( archim1804426504_field @ A )
% 0.29/0.45 => ! [X: A] :
% 0.29/0.45 ? [N3: nat] : ( ord_less_eq @ A @ X @ ( semiring_1_of_nat @ A @ N3 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % real_arch_simple
% 0.29/0.45 thf(fact_100_of__nat__mono,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( linord1659791738miring @ A )
% 0.29/0.45 => ! [I2: nat,J: nat] :
% 0.29/0.45 ( ( ord_less_eq @ nat @ I2 @ J )
% 0.29/0.45 => ( ord_less_eq @ A @ ( semiring_1_of_nat @ A @ I2 ) @ ( semiring_1_of_nat @ A @ J ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % of_nat_mono
% 0.29/0.45 thf(fact_101_numerals_I1_J,axiom,
% 0.29/0.45 ( ( numeral_numeral @ nat @ one2 )
% 0.29/0.45 = ( one_one @ nat ) ) ).
% 0.29/0.45
% 0.29/0.45 % numerals(1)
% 0.29/0.45 thf(fact_102_of__nat__ceiling,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( archim1727834104eiling @ A )
% 0.29/0.45 => ! [R: A] : ( ord_less_eq @ A @ R @ ( semiring_1_of_nat @ A @ ( nat2 @ ( archimedean_ceiling @ A @ R ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % of_nat_ceiling
% 0.29/0.45 thf(fact_103_Suc__inject,axiom,
% 0.29/0.45 ! [X: nat,Y: nat] :
% 0.29/0.45 ( ( ( suc @ X )
% 0.29/0.45 = ( suc @ Y ) )
% 0.29/0.45 => ( X = Y ) ) ).
% 0.29/0.45
% 0.29/0.45 % Suc_inject
% 0.29/0.45 thf(fact_104_n__not__Suc__n,axiom,
% 0.29/0.45 ! [N: nat] :
% 0.29/0.45 ( N
% 0.29/0.45 != ( suc @ N ) ) ).
% 0.29/0.45
% 0.29/0.45 % n_not_Suc_n
% 0.29/0.45 thf(fact_105_Nat_Oex__has__greatest__nat,axiom,
% 0.29/0.45 ! [P: nat > $o,K: nat,B: nat] :
% 0.29/0.45 ( ( P @ K )
% 0.29/0.45 => ( ! [Y3: nat] :
% 0.29/0.45 ( ( P @ Y3 )
% 0.29/0.45 => ( ord_less_eq @ nat @ Y3 @ B ) )
% 0.29/0.45 => ? [X4: nat] :
% 0.29/0.45 ( ( P @ X4 )
% 0.29/0.45 & ! [Y4: nat] :
% 0.29/0.45 ( ( P @ Y4 )
% 0.29/0.45 => ( ord_less_eq @ nat @ Y4 @ X4 ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % Nat.ex_has_greatest_nat
% 0.29/0.45 thf(fact_106_nat__le__linear,axiom,
% 0.29/0.45 ! [M: nat,N: nat] :
% 0.29/0.45 ( ( ord_less_eq @ nat @ M @ N )
% 0.29/0.45 | ( ord_less_eq @ nat @ N @ M ) ) ).
% 0.29/0.45
% 0.29/0.45 % nat_le_linear
% 0.29/0.45 thf(fact_107_le__antisym,axiom,
% 0.29/0.45 ! [M: nat,N: nat] :
% 0.29/0.45 ( ( ord_less_eq @ nat @ M @ N )
% 0.29/0.45 => ( ( ord_less_eq @ nat @ N @ M )
% 0.29/0.45 => ( M = N ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % le_antisym
% 0.29/0.45 thf(fact_108_eq__imp__le,axiom,
% 0.29/0.45 ! [M: nat,N: nat] :
% 0.29/0.45 ( ( M = N )
% 0.29/0.45 => ( ord_less_eq @ nat @ M @ N ) ) ).
% 0.29/0.45
% 0.29/0.45 % eq_imp_le
% 0.29/0.45 thf(fact_109_le__trans,axiom,
% 0.29/0.45 ! [I2: nat,J: nat,K: nat] :
% 0.29/0.45 ( ( ord_less_eq @ nat @ I2 @ J )
% 0.29/0.45 => ( ( ord_less_eq @ nat @ J @ K )
% 0.29/0.45 => ( ord_less_eq @ nat @ I2 @ K ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % le_trans
% 0.29/0.45 thf(fact_110_le__refl,axiom,
% 0.29/0.45 ! [N: nat] : ( ord_less_eq @ nat @ N @ N ) ).
% 0.29/0.45
% 0.29/0.45 % le_refl
% 0.29/0.45 thf(fact_111_real__nat__ceiling__ge,axiom,
% 0.29/0.45 ! [X: real] : ( ord_less_eq @ real @ X @ ( semiring_1_of_nat @ real @ ( nat2 @ ( archimedean_ceiling @ real @ X ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % real_nat_ceiling_ge
% 0.29/0.45 thf(fact_112_powr__powr__swap,axiom,
% 0.29/0.45 ! [X: real,A2: real,B: real] :
% 0.29/0.45 ( ( powr @ real @ ( powr @ real @ X @ A2 ) @ B )
% 0.29/0.45 = ( powr @ real @ ( powr @ real @ X @ B ) @ A2 ) ) ).
% 0.29/0.45
% 0.29/0.45 % powr_powr_swap
% 0.29/0.45 thf(fact_113_log2__of__power__eq,axiom,
% 0.29/0.45 ! [M: nat,N: nat] :
% 0.29/0.45 ( ( M
% 0.29/0.45 = ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) )
% 0.29/0.45 => ( ( semiring_1_of_nat @ real @ N )
% 0.29/0.45 = ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ real @ M ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % log2_of_power_eq
% 0.29/0.45 thf(fact_114_le__log2__of__power,axiom,
% 0.29/0.45 ! [N: nat,M: nat] :
% 0.29/0.45 ( ( ord_less_eq @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) @ M )
% 0.29/0.45 => ( ord_less_eq @ real @ ( semiring_1_of_nat @ real @ N ) @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ real @ M ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % le_log2_of_power
% 0.29/0.45 thf(fact_115_transitive__stepwise__le,axiom,
% 0.29/0.45 ! [M: nat,N: nat,R2: nat > nat > $o] :
% 0.29/0.45 ( ( ord_less_eq @ nat @ M @ N )
% 0.29/0.45 => ( ! [X4: nat] : ( R2 @ X4 @ X4 )
% 0.29/0.45 => ( ! [X4: nat,Y3: nat,Z2: nat] :
% 0.29/0.45 ( ( R2 @ X4 @ Y3 )
% 0.29/0.45 => ( ( R2 @ Y3 @ Z2 )
% 0.29/0.45 => ( R2 @ X4 @ Z2 ) ) )
% 0.29/0.45 => ( ! [N3: nat] : ( R2 @ N3 @ ( suc @ N3 ) )
% 0.29/0.45 => ( R2 @ M @ N ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % transitive_stepwise_le
% 0.29/0.45 thf(fact_116_nat__induct__at__least,axiom,
% 0.29/0.45 ! [M: nat,N: nat,P: nat > $o] :
% 0.29/0.45 ( ( ord_less_eq @ nat @ M @ N )
% 0.29/0.45 => ( ( P @ M )
% 0.29/0.45 => ( ! [N3: nat] :
% 0.29/0.45 ( ( ord_less_eq @ nat @ M @ N3 )
% 0.29/0.45 => ( ( P @ N3 )
% 0.29/0.45 => ( P @ ( suc @ N3 ) ) ) )
% 0.29/0.45 => ( P @ N ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % nat_induct_at_least
% 0.29/0.45 thf(fact_117_full__nat__induct,axiom,
% 0.29/0.45 ! [P: nat > $o,N: nat] :
% 0.29/0.45 ( ! [N3: nat] :
% 0.29/0.45 ( ! [M2: nat] :
% 0.29/0.45 ( ( ord_less_eq @ nat @ ( suc @ M2 ) @ N3 )
% 0.29/0.45 => ( P @ M2 ) )
% 0.29/0.45 => ( P @ N3 ) )
% 0.29/0.45 => ( P @ N ) ) ).
% 0.29/0.45
% 0.29/0.45 % full_nat_induct
% 0.29/0.45 thf(fact_118_not__less__eq__eq,axiom,
% 0.29/0.45 ! [M: nat,N: nat] :
% 0.29/0.45 ( ( ~ ( ord_less_eq @ nat @ M @ N ) )
% 0.29/0.45 = ( ord_less_eq @ nat @ ( suc @ N ) @ M ) ) ).
% 0.29/0.45
% 0.29/0.45 % not_less_eq_eq
% 0.29/0.45 thf(fact_119_Suc__n__not__le__n,axiom,
% 0.29/0.45 ! [N: nat] :
% 0.29/0.45 ~ ( ord_less_eq @ nat @ ( suc @ N ) @ N ) ).
% 0.29/0.45
% 0.29/0.45 % Suc_n_not_le_n
% 0.29/0.45 thf(fact_120_le__Suc__eq,axiom,
% 0.29/0.45 ! [M: nat,N: nat] :
% 0.29/0.45 ( ( ord_less_eq @ nat @ M @ ( suc @ N ) )
% 0.29/0.45 = ( ( ord_less_eq @ nat @ M @ N )
% 0.29/0.45 | ( M
% 0.29/0.45 = ( suc @ N ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % le_Suc_eq
% 0.29/0.45 thf(fact_121_Suc__le__D,axiom,
% 0.29/0.45 ! [N: nat,M3: nat] :
% 0.29/0.45 ( ( ord_less_eq @ nat @ ( suc @ N ) @ M3 )
% 0.29/0.45 => ? [M4: nat] :
% 0.29/0.45 ( M3
% 0.29/0.45 = ( suc @ M4 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % Suc_le_D
% 0.29/0.45 thf(fact_122_le__SucI,axiom,
% 0.29/0.45 ! [M: nat,N: nat] :
% 0.29/0.45 ( ( ord_less_eq @ nat @ M @ N )
% 0.29/0.45 => ( ord_less_eq @ nat @ M @ ( suc @ N ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % le_SucI
% 0.29/0.45 thf(fact_123_le__SucE,axiom,
% 0.29/0.45 ! [M: nat,N: nat] :
% 0.29/0.45 ( ( ord_less_eq @ nat @ M @ ( suc @ N ) )
% 0.29/0.45 => ( ~ ( ord_less_eq @ nat @ M @ N )
% 0.29/0.45 => ( M
% 0.29/0.45 = ( suc @ N ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % le_SucE
% 0.29/0.45 thf(fact_124_Suc__leD,axiom,
% 0.29/0.45 ! [M: nat,N: nat] :
% 0.29/0.45 ( ( ord_less_eq @ nat @ ( suc @ M ) @ N )
% 0.29/0.45 => ( ord_less_eq @ nat @ M @ N ) ) ).
% 0.29/0.45
% 0.29/0.45 % Suc_leD
% 0.29/0.45 thf(fact_125_one__le__power,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( linordered_semidom @ A )
% 0.29/0.45 => ! [A2: A,N: nat] :
% 0.29/0.45 ( ( ord_less_eq @ A @ ( one_one @ A ) @ A2 )
% 0.29/0.45 => ( ord_less_eq @ A @ ( one_one @ A ) @ ( power_power @ A @ A2 @ N ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % one_le_power
% 0.29/0.45 thf(fact_126_lift__Suc__antimono__le,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( order @ A )
% 0.29/0.45 => ! [F: nat > A,N: nat,N4: nat] :
% 0.29/0.45 ( ! [N3: nat] : ( ord_less_eq @ A @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
% 0.29/0.45 => ( ( ord_less_eq @ nat @ N @ N4 )
% 0.29/0.45 => ( ord_less_eq @ A @ ( F @ N4 ) @ ( F @ N ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % lift_Suc_antimono_le
% 0.29/0.45 thf(fact_127_lift__Suc__mono__le,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( order @ A )
% 0.29/0.45 => ! [F: nat > A,N: nat,N4: nat] :
% 0.29/0.45 ( ! [N3: nat] : ( ord_less_eq @ A @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
% 0.29/0.45 => ( ( ord_less_eq @ nat @ N @ N4 )
% 0.29/0.45 => ( ord_less_eq @ A @ ( F @ N ) @ ( F @ N4 ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % lift_Suc_mono_le
% 0.29/0.45 thf(fact_128_powr__mono,axiom,
% 0.29/0.45 ! [A2: real,B: real,X: real] :
% 0.29/0.45 ( ( ord_less_eq @ real @ A2 @ B )
% 0.29/0.45 => ( ( ord_less_eq @ real @ ( one_one @ real ) @ X )
% 0.29/0.45 => ( ord_less_eq @ real @ ( powr @ real @ X @ A2 ) @ ( powr @ real @ X @ B ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % powr_mono
% 0.29/0.45 thf(fact_129_power__increasing,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( linordered_semidom @ A )
% 0.29/0.45 => ! [N: nat,N5: nat,A2: A] :
% 0.29/0.45 ( ( ord_less_eq @ nat @ N @ N5 )
% 0.29/0.45 => ( ( ord_less_eq @ A @ ( one_one @ A ) @ A2 )
% 0.29/0.45 => ( ord_less_eq @ A @ ( power_power @ A @ A2 @ N ) @ ( power_power @ A @ A2 @ N5 ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % power_increasing
% 0.29/0.45 thf(fact_130_one__power2,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( semiring_1 @ A )
% 0.29/0.45 => ( ( power_power @ A @ ( one_one @ A ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
% 0.29/0.45 = ( one_one @ A ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % one_power2
% 0.29/0.45 thf(fact_131_dbl__simps_I3_J,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( neg_numeral @ A )
% 0.29/0.45 => ( ( neg_numeral_dbl @ A @ ( one_one @ A ) )
% 0.29/0.45 = ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % dbl_simps(3)
% 0.29/0.45 thf(fact_132_power__numeral,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( semiring_numeral @ A )
% 0.29/0.45 => ! [K: num,L: num] :
% 0.29/0.45 ( ( power_power @ A @ ( numeral_numeral @ A @ K ) @ ( numeral_numeral @ nat @ L ) )
% 0.29/0.45 = ( numeral_numeral @ A @ ( pow @ K @ L ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % power_numeral
% 0.29/0.45 thf(fact_133_order__refl,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( preorder @ A )
% 0.29/0.45 => ! [X: A] : ( ord_less_eq @ A @ X @ X ) ) ).
% 0.29/0.45
% 0.29/0.45 % order_refl
% 0.29/0.45 thf(fact_134_powr__numeral,axiom,
% 0.29/0.45 ! [X: real,N: num] :
% 0.29/0.45 ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
% 0.29/0.45 => ( ( powr @ real @ X @ ( numeral_numeral @ real @ N ) )
% 0.29/0.45 = ( power_power @ real @ X @ ( numeral_numeral @ nat @ N ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % powr_numeral
% 0.29/0.45 thf(fact_135_power2__eq__iff__nonneg,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( linordered_semidom @ A )
% 0.29/0.45 => ! [X: A,Y: A] :
% 0.29/0.45 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
% 0.29/0.45 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y )
% 0.29/0.45 => ( ( ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
% 0.29/0.45 = ( power_power @ A @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
% 0.29/0.45 = ( X = Y ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % power2_eq_iff_nonneg
% 0.29/0.45 thf(fact_136_le0,axiom,
% 0.29/0.45 ! [N: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ N ) ).
% 0.29/0.45
% 0.29/0.45 % le0
% 0.29/0.45 thf(fact_137_bot__nat__0_Oextremum,axiom,
% 0.29/0.45 ! [A2: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ A2 ) ).
% 0.29/0.45
% 0.29/0.45 % bot_nat_0.extremum
% 0.29/0.45 thf(fact_138_d_I2_J,axiom,
% 0.29/0.45 ord_less @ int @ ( zero_zero @ int ) @ d ).
% 0.29/0.45
% 0.29/0.45 % d(2)
% 0.29/0.45 thf(fact_139_le__zero__eq,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( canoni770627133id_add @ A )
% 0.29/0.45 => ! [N: A] :
% 0.29/0.45 ( ( ord_less_eq @ A @ N @ ( zero_zero @ A ) )
% 0.29/0.45 = ( N
% 0.29/0.45 = ( zero_zero @ A ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % le_zero_eq
% 0.29/0.45 thf(fact_140_of__nat__0,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( semiring_1 @ A )
% 0.29/0.45 => ( ( semiring_1_of_nat @ A @ ( zero_zero @ nat ) )
% 0.29/0.45 = ( zero_zero @ A ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % of_nat_0
% 0.29/0.45 thf(fact_141_of__nat__0__eq__iff,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( semiring_char_0 @ A )
% 0.29/0.45 => ! [N: nat] :
% 0.29/0.45 ( ( ( zero_zero @ A )
% 0.29/0.45 = ( semiring_1_of_nat @ A @ N ) )
% 0.29/0.45 = ( ( zero_zero @ nat )
% 0.29/0.45 = N ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % of_nat_0_eq_iff
% 0.29/0.45 thf(fact_142_of__nat__eq__0__iff,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( semiring_char_0 @ A )
% 0.29/0.45 => ! [M: nat] :
% 0.29/0.45 ( ( ( semiring_1_of_nat @ A @ M )
% 0.29/0.45 = ( zero_zero @ A ) )
% 0.29/0.45 = ( M
% 0.29/0.45 = ( zero_zero @ nat ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % of_nat_eq_0_iff
% 0.29/0.45 thf(fact_143_power__Suc0__right,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( monoid_mult @ A )
% 0.29/0.45 => ! [A2: A] :
% 0.29/0.45 ( ( power_power @ A @ A2 @ ( suc @ ( zero_zero @ nat ) ) )
% 0.29/0.45 = A2 ) ) ).
% 0.29/0.45
% 0.29/0.45 % power_Suc0_right
% 0.29/0.45 thf(fact_144_of__int__0,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( ring_1 @ A )
% 0.29/0.45 => ( ( ring_1_of_int @ A @ ( zero_zero @ int ) )
% 0.29/0.45 = ( zero_zero @ A ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % of_int_0
% 0.29/0.45 thf(fact_145_of__int__0__eq__iff,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( ring_char_0 @ A )
% 0.29/0.45 => ! [Z: int] :
% 0.29/0.45 ( ( ( zero_zero @ A )
% 0.29/0.45 = ( ring_1_of_int @ A @ Z ) )
% 0.29/0.45 = ( Z
% 0.29/0.45 = ( zero_zero @ int ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % of_int_0_eq_iff
% 0.29/0.45 thf(fact_146_of__int__eq__0__iff,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( ring_char_0 @ A )
% 0.29/0.45 => ! [Z: int] :
% 0.29/0.45 ( ( ( ring_1_of_int @ A @ Z )
% 0.29/0.45 = ( zero_zero @ A ) )
% 0.29/0.45 = ( Z
% 0.29/0.45 = ( zero_zero @ int ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % of_int_eq_0_iff
% 0.29/0.45 thf(fact_147_powr__eq__0__iff,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( ln @ A )
% 0.29/0.45 => ! [W: A,Z: A] :
% 0.29/0.45 ( ( ( powr @ A @ W @ Z )
% 0.29/0.45 = ( zero_zero @ A ) )
% 0.29/0.45 = ( W
% 0.29/0.45 = ( zero_zero @ A ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % powr_eq_0_iff
% 0.29/0.45 thf(fact_148_powr__0,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( ln @ A )
% 0.29/0.45 => ! [Z: A] :
% 0.29/0.45 ( ( powr @ A @ ( zero_zero @ A ) @ Z )
% 0.29/0.45 = ( zero_zero @ A ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % powr_0
% 0.29/0.45 thf(fact_149_ceiling__zero,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( archim1727834104eiling @ A )
% 0.29/0.45 => ( ( archimedean_ceiling @ A @ ( zero_zero @ A ) )
% 0.29/0.45 = ( zero_zero @ int ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % ceiling_zero
% 0.29/0.45 thf(fact_150_nat__le__0,axiom,
% 0.29/0.45 ! [Z: int] :
% 0.29/0.45 ( ( ord_less_eq @ int @ Z @ ( zero_zero @ int ) )
% 0.29/0.45 => ( ( nat2 @ Z )
% 0.29/0.45 = ( zero_zero @ nat ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % nat_le_0
% 0.29/0.45 thf(fact_151_nat__0__iff,axiom,
% 0.29/0.45 ! [I2: int] :
% 0.29/0.45 ( ( ( nat2 @ I2 )
% 0.29/0.45 = ( zero_zero @ nat ) )
% 0.29/0.45 = ( ord_less_eq @ int @ I2 @ ( zero_zero @ int ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % nat_0_iff
% 0.29/0.45 thf(fact_152_power__Suc__0,axiom,
% 0.29/0.45 ! [N: nat] :
% 0.29/0.45 ( ( power_power @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N )
% 0.29/0.45 = ( suc @ ( zero_zero @ nat ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % power_Suc_0
% 0.29/0.45 thf(fact_153_nat__power__eq__Suc__0__iff,axiom,
% 0.29/0.45 ! [X: nat,M: nat] :
% 0.29/0.45 ( ( ( power_power @ nat @ X @ M )
% 0.29/0.45 = ( suc @ ( zero_zero @ nat ) ) )
% 0.29/0.45 = ( ( M
% 0.29/0.45 = ( zero_zero @ nat ) )
% 0.29/0.45 | ( X
% 0.29/0.45 = ( suc @ ( zero_zero @ nat ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % nat_power_eq_Suc_0_iff
% 0.29/0.45 thf(fact_154_nat__int,axiom,
% 0.29/0.45 ! [N: nat] :
% 0.29/0.45 ( ( nat2 @ ( semiring_1_of_nat @ int @ N ) )
% 0.29/0.45 = N ) ).
% 0.29/0.45
% 0.29/0.45 % nat_int
% 0.29/0.45 thf(fact_155_dbl__simps_I2_J,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( neg_numeral @ A )
% 0.29/0.45 => ( ( neg_numeral_dbl @ A @ ( zero_zero @ A ) )
% 0.29/0.45 = ( zero_zero @ A ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % dbl_simps(2)
% 0.29/0.45 thf(fact_156_of__nat__le__0__iff,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( linord1659791738miring @ A )
% 0.29/0.45 => ! [M: nat] :
% 0.29/0.45 ( ( ord_less_eq @ A @ ( semiring_1_of_nat @ A @ M ) @ ( zero_zero @ A ) )
% 0.29/0.45 = ( M
% 0.29/0.45 = ( zero_zero @ nat ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % of_nat_le_0_iff
% 0.29/0.45 thf(fact_157_power__0__Suc,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( semiring_1 @ A )
% 0.29/0.45 => ! [N: nat] :
% 0.29/0.45 ( ( power_power @ A @ ( zero_zero @ A ) @ ( suc @ N ) )
% 0.29/0.45 = ( zero_zero @ A ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % power_0_Suc
% 0.29/0.45 thf(fact_158_power__zero__numeral,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( semiring_1 @ A )
% 0.29/0.45 => ! [K: num] :
% 0.29/0.45 ( ( power_power @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ nat @ K ) )
% 0.29/0.45 = ( zero_zero @ A ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % power_zero_numeral
% 0.29/0.45 thf(fact_159_powr__zero__eq__one,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( ln @ A )
% 0.29/0.45 => ! [X: A] :
% 0.29/0.45 ( ( ( X
% 0.29/0.45 = ( zero_zero @ A ) )
% 0.29/0.45 => ( ( powr @ A @ X @ ( zero_zero @ A ) )
% 0.29/0.45 = ( zero_zero @ A ) ) )
% 0.29/0.45 & ( ( X
% 0.29/0.45 != ( zero_zero @ A ) )
% 0.29/0.45 => ( ( powr @ A @ X @ ( zero_zero @ A ) )
% 0.29/0.45 = ( one_one @ A ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % powr_zero_eq_one
% 0.29/0.45 thf(fact_160_powr__nonneg__iff,axiom,
% 0.29/0.45 ! [A2: real,X: real] :
% 0.29/0.45 ( ( ord_less_eq @ real @ ( powr @ real @ A2 @ X ) @ ( zero_zero @ real ) )
% 0.29/0.45 = ( A2
% 0.29/0.45 = ( zero_zero @ real ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % powr_nonneg_iff
% 0.29/0.45 thf(fact_161_nat__1,axiom,
% 0.29/0.45 ( ( nat2 @ ( one_one @ int ) )
% 0.29/0.45 = ( suc @ ( zero_zero @ nat ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % nat_1
% 0.29/0.45 thf(fact_162_int__eq__iff__numeral,axiom,
% 0.29/0.45 ! [M: nat,V: num] :
% 0.29/0.45 ( ( ( semiring_1_of_nat @ int @ M )
% 0.29/0.45 = ( numeral_numeral @ int @ V ) )
% 0.29/0.45 = ( M
% 0.29/0.45 = ( numeral_numeral @ nat @ V ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % int_eq_iff_numeral
% 0.29/0.45 thf(fact_163_log__one,axiom,
% 0.29/0.45 ! [A2: real] :
% 0.29/0.45 ( ( log @ A2 @ ( one_one @ real ) )
% 0.29/0.45 = ( zero_zero @ real ) ) ).
% 0.29/0.45
% 0.29/0.45 % log_one
% 0.29/0.45 thf(fact_164_int__nat__eq,axiom,
% 0.29/0.45 ! [Z: int] :
% 0.29/0.45 ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z )
% 0.29/0.45 => ( ( semiring_1_of_nat @ int @ ( nat2 @ Z ) )
% 0.29/0.45 = Z ) )
% 0.29/0.45 & ( ~ ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z )
% 0.29/0.45 => ( ( semiring_1_of_nat @ int @ ( nat2 @ Z ) )
% 0.29/0.45 = ( zero_zero @ int ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % int_nat_eq
% 0.29/0.45 thf(fact_165_dbl__simps_I5_J,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( neg_numeral @ A )
% 0.29/0.45 => ! [K: num] :
% 0.29/0.45 ( ( neg_numeral_dbl @ A @ ( numeral_numeral @ A @ K ) )
% 0.29/0.45 = ( numeral_numeral @ A @ ( bit0 @ K ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % dbl_simps(5)
% 0.29/0.45 thf(fact_166_of__int__0__le__iff,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( linordered_idom @ A )
% 0.29/0.45 => ! [Z: int] :
% 0.29/0.45 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( ring_1_of_int @ A @ Z ) )
% 0.29/0.45 = ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % of_int_0_le_iff
% 0.29/0.45 thf(fact_167_of__int__le__0__iff,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( linordered_idom @ A )
% 0.29/0.45 => ! [Z: int] :
% 0.29/0.45 ( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ Z ) @ ( zero_zero @ A ) )
% 0.29/0.45 = ( ord_less_eq @ int @ Z @ ( zero_zero @ int ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % of_int_le_0_iff
% 0.29/0.45 thf(fact_168_ceiling__le__zero,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( archim1727834104eiling @ A )
% 0.29/0.45 => ! [X: A] :
% 0.29/0.45 ( ( ord_less_eq @ int @ ( archimedean_ceiling @ A @ X ) @ ( zero_zero @ int ) )
% 0.29/0.45 = ( ord_less_eq @ A @ X @ ( zero_zero @ A ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % ceiling_le_zero
% 0.29/0.45 thf(fact_169_powr__one,axiom,
% 0.29/0.45 ! [X: real] :
% 0.29/0.45 ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
% 0.29/0.45 => ( ( powr @ real @ X @ ( one_one @ real ) )
% 0.29/0.45 = X ) ) ).
% 0.29/0.45
% 0.29/0.45 % powr_one
% 0.29/0.45 thf(fact_170_powr__one__gt__zero__iff,axiom,
% 0.29/0.45 ! [X: real] :
% 0.29/0.45 ( ( ( powr @ real @ X @ ( one_one @ real ) )
% 0.29/0.45 = X )
% 0.29/0.45 = ( ord_less_eq @ real @ ( zero_zero @ real ) @ X ) ) ).
% 0.29/0.45
% 0.29/0.45 % powr_one_gt_zero_iff
% 0.29/0.45 thf(fact_171_of__nat__nat,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( ring_1 @ A )
% 0.29/0.45 => ! [Z: int] :
% 0.29/0.45 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z )
% 0.29/0.45 => ( ( semiring_1_of_nat @ A @ ( nat2 @ Z ) )
% 0.29/0.45 = ( ring_1_of_int @ A @ Z ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % of_nat_nat
% 0.29/0.45 thf(fact_172_zero__eq__power2,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( semiri134348788visors @ A )
% 0.29/0.45 => ! [A2: A] :
% 0.29/0.45 ( ( ( power_power @ A @ A2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
% 0.29/0.45 = ( zero_zero @ A ) )
% 0.29/0.45 = ( A2
% 0.29/0.45 = ( zero_zero @ A ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % zero_eq_power2
% 0.29/0.45 thf(fact_173_power2__less__eq__zero__iff,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( linordered_idom @ A )
% 0.29/0.45 => ! [A2: A] :
% 0.29/0.45 ( ( ord_less_eq @ A @ ( power_power @ A @ A2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( zero_zero @ A ) )
% 0.29/0.45 = ( A2
% 0.29/0.45 = ( zero_zero @ A ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % power2_less_eq_zero_iff
% 0.29/0.45 thf(fact_174_int__if,axiom,
% 0.29/0.45 ! [P: $o,A2: nat,B: nat] :
% 0.29/0.45 ( ( P
% 0.29/0.45 => ( ( semiring_1_of_nat @ int @ ( if @ nat @ P @ A2 @ B ) )
% 0.29/0.45 = ( semiring_1_of_nat @ int @ A2 ) ) )
% 0.29/0.45 & ( ~ P
% 0.29/0.45 => ( ( semiring_1_of_nat @ int @ ( if @ nat @ P @ A2 @ B ) )
% 0.29/0.45 = ( semiring_1_of_nat @ int @ B ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % int_if
% 0.29/0.45 thf(fact_175_nat__int__comparison_I1_J,axiom,
% 0.29/0.45 ( ( ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 ) )
% 0.29/0.45 = ( ^ [A4: nat,B3: nat] :
% 0.29/0.45 ( ( semiring_1_of_nat @ int @ A4 )
% 0.29/0.45 = ( semiring_1_of_nat @ int @ B3 ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % nat_int_comparison(1)
% 0.29/0.45 thf(fact_176_int__ops_I1_J,axiom,
% 0.29/0.45 ( ( semiring_1_of_nat @ int @ ( zero_zero @ nat ) )
% 0.29/0.45 = ( zero_zero @ int ) ) ).
% 0.29/0.45
% 0.29/0.45 % int_ops(1)
% 0.29/0.45 thf(fact_177_zero__le__imp__eq__int,axiom,
% 0.29/0.45 ! [K: int] :
% 0.29/0.45 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
% 0.29/0.45 => ? [N3: nat] :
% 0.29/0.45 ( K
% 0.29/0.45 = ( semiring_1_of_nat @ int @ N3 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % zero_le_imp_eq_int
% 0.29/0.45 thf(fact_178_nonneg__int__cases,axiom,
% 0.29/0.45 ! [K: int] :
% 0.29/0.45 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
% 0.29/0.45 => ~ ! [N3: nat] :
% 0.29/0.45 ( K
% 0.29/0.45 != ( semiring_1_of_nat @ int @ N3 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % nonneg_int_cases
% 0.29/0.45 thf(fact_179_int__int__eq,axiom,
% 0.29/0.45 ! [M: nat,N: nat] :
% 0.29/0.45 ( ( ( semiring_1_of_nat @ int @ M )
% 0.29/0.45 = ( semiring_1_of_nat @ int @ N ) )
% 0.29/0.45 = ( M = N ) ) ).
% 0.29/0.45
% 0.29/0.45 % int_int_eq
% 0.29/0.45 thf(fact_180_nat__eq__iff2,axiom,
% 0.29/0.45 ! [M: nat,W: int] :
% 0.29/0.45 ( ( M
% 0.29/0.45 = ( nat2 @ W ) )
% 0.29/0.45 = ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ W )
% 0.29/0.45 => ( W
% 0.29/0.45 = ( semiring_1_of_nat @ int @ M ) ) )
% 0.29/0.45 & ( ~ ( ord_less_eq @ int @ ( zero_zero @ int ) @ W )
% 0.29/0.45 => ( M
% 0.29/0.45 = ( zero_zero @ nat ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % nat_eq_iff2
% 0.29/0.45 thf(fact_181_nat__eq__iff,axiom,
% 0.29/0.45 ! [W: int,M: nat] :
% 0.29/0.45 ( ( ( nat2 @ W )
% 0.29/0.45 = M )
% 0.29/0.45 = ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ W )
% 0.29/0.45 => ( W
% 0.29/0.45 = ( semiring_1_of_nat @ int @ M ) ) )
% 0.29/0.45 & ( ~ ( ord_less_eq @ int @ ( zero_zero @ int ) @ W )
% 0.29/0.45 => ( M
% 0.29/0.45 = ( zero_zero @ nat ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % nat_eq_iff
% 0.29/0.45 thf(fact_182_zero__reorient,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( zero @ A )
% 0.29/0.45 => ! [X: A] :
% 0.29/0.45 ( ( ( zero_zero @ A )
% 0.29/0.45 = X )
% 0.29/0.45 = ( X
% 0.29/0.45 = ( zero_zero @ A ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % zero_reorient
% 0.29/0.45 thf(fact_183_zero__le,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( canoni770627133id_add @ A )
% 0.29/0.45 => ! [X: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ X ) ) ).
% 0.29/0.45
% 0.29/0.45 % zero_le
% 0.29/0.45 thf(fact_184_nat__zero__as__int,axiom,
% 0.29/0.45 ( ( zero_zero @ nat )
% 0.29/0.45 = ( nat2 @ ( zero_zero @ int ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % nat_zero_as_int
% 0.29/0.45 thf(fact_185_power__0__left,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( semiring_1 @ A )
% 0.29/0.45 => ! [N: nat] :
% 0.29/0.45 ( ( ( N
% 0.29/0.45 = ( zero_zero @ nat ) )
% 0.29/0.45 => ( ( power_power @ A @ ( zero_zero @ A ) @ N )
% 0.29/0.45 = ( one_one @ A ) ) )
% 0.29/0.45 & ( ( N
% 0.29/0.45 != ( zero_zero @ nat ) )
% 0.29/0.45 => ( ( power_power @ A @ ( zero_zero @ A ) @ N )
% 0.29/0.45 = ( zero_zero @ A ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % power_0_left
% 0.29/0.45 thf(fact_186_nat__0__le,axiom,
% 0.29/0.45 ! [Z: int] :
% 0.29/0.45 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z )
% 0.29/0.45 => ( ( semiring_1_of_nat @ int @ ( nat2 @ Z ) )
% 0.29/0.45 = Z ) ) ).
% 0.29/0.45
% 0.29/0.45 % nat_0_le
% 0.29/0.45 thf(fact_187_int__eq__iff,axiom,
% 0.29/0.45 ! [M: nat,Z: int] :
% 0.29/0.45 ( ( ( semiring_1_of_nat @ int @ M )
% 0.29/0.45 = Z )
% 0.29/0.45 = ( ( M
% 0.29/0.45 = ( nat2 @ Z ) )
% 0.29/0.45 & ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % int_eq_iff
% 0.29/0.45 thf(fact_188_int__ops_I2_J,axiom,
% 0.29/0.45 ( ( semiring_1_of_nat @ int @ ( one_one @ nat ) )
% 0.29/0.45 = ( one_one @ int ) ) ).
% 0.29/0.45
% 0.29/0.45 % int_ops(2)
% 0.29/0.45 thf(fact_189_le__numeral__extra_I3_J,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( linord1659791738miring @ A )
% 0.29/0.45 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( zero_zero @ A ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % le_numeral_extra(3)
% 0.29/0.45 thf(fact_190_zero__neq__numeral,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( semiring_char_0 @ A )
% 0.29/0.45 => ! [N: num] :
% 0.29/0.45 ( ( zero_zero @ A )
% 0.29/0.45 != ( numeral_numeral @ A @ N ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % zero_neq_numeral
% 0.29/0.45 thf(fact_191_power__not__zero,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( semiri134348788visors @ A )
% 0.29/0.45 => ! [A2: A,N: nat] :
% 0.29/0.45 ( ( A2
% 0.29/0.45 != ( zero_zero @ A ) )
% 0.29/0.45 => ( ( power_power @ A @ A2 @ N )
% 0.29/0.45 != ( zero_zero @ A ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % power_not_zero
% 0.29/0.45 thf(fact_192_exists__least__lemma,axiom,
% 0.29/0.45 ! [P: nat > $o] :
% 0.29/0.45 ( ~ ( P @ ( zero_zero @ nat ) )
% 0.29/0.45 => ( ? [X_1: nat] : ( P @ X_1 )
% 0.29/0.45 => ? [N3: nat] :
% 0.29/0.45 ( ~ ( P @ N3 )
% 0.29/0.45 & ( P @ ( suc @ N3 ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % exists_least_lemma
% 0.29/0.45 thf(fact_193_nat_Odistinct_I1_J,axiom,
% 0.29/0.45 ! [X2: nat] :
% 0.29/0.45 ( ( zero_zero @ nat )
% 0.29/0.45 != ( suc @ X2 ) ) ).
% 0.29/0.45
% 0.29/0.45 % nat.distinct(1)
% 0.29/0.45 thf(fact_194_old_Onat_Odistinct_I2_J,axiom,
% 0.29/0.45 ! [Nat3: nat] :
% 0.29/0.45 ( ( suc @ Nat3 )
% 0.29/0.45 != ( zero_zero @ nat ) ) ).
% 0.29/0.45
% 0.29/0.45 % old.nat.distinct(2)
% 0.29/0.45 thf(fact_195_old_Onat_Odistinct_I1_J,axiom,
% 0.29/0.45 ! [Nat2: nat] :
% 0.29/0.45 ( ( zero_zero @ nat )
% 0.29/0.45 != ( suc @ Nat2 ) ) ).
% 0.29/0.45
% 0.29/0.45 % old.nat.distinct(1)
% 0.29/0.45 thf(fact_196_nat_OdiscI,axiom,
% 0.29/0.45 ! [Nat: nat,X2: nat] :
% 0.29/0.45 ( ( Nat
% 0.29/0.45 = ( suc @ X2 ) )
% 0.29/0.45 => ( Nat
% 0.29/0.45 != ( zero_zero @ nat ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % nat.discI
% 0.29/0.45 thf(fact_197_nat__induct,axiom,
% 0.29/0.45 ! [P: nat > $o,N: nat] :
% 0.29/0.45 ( ( P @ ( zero_zero @ nat ) )
% 0.29/0.45 => ( ! [N3: nat] :
% 0.29/0.45 ( ( P @ N3 )
% 0.29/0.45 => ( P @ ( suc @ N3 ) ) )
% 0.29/0.45 => ( P @ N ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % nat_induct
% 0.29/0.45 thf(fact_198_One__nat__def,axiom,
% 0.29/0.45 ( ( one_one @ nat )
% 0.29/0.45 = ( suc @ ( zero_zero @ nat ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % One_nat_def
% 0.29/0.45 thf(fact_199_diff__induct,axiom,
% 0.29/0.45 ! [P: nat > nat > $o,M: nat,N: nat] :
% 0.29/0.45 ( ! [X4: nat] : ( P @ X4 @ ( zero_zero @ nat ) )
% 0.29/0.45 => ( ! [Y3: nat] : ( P @ ( zero_zero @ nat ) @ ( suc @ Y3 ) )
% 0.29/0.45 => ( ! [X4: nat,Y3: nat] :
% 0.29/0.45 ( ( P @ X4 @ Y3 )
% 0.29/0.45 => ( P @ ( suc @ X4 ) @ ( suc @ Y3 ) ) )
% 0.29/0.45 => ( P @ M @ N ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % diff_induct
% 0.29/0.45 thf(fact_200_zero__induct,axiom,
% 0.29/0.45 ! [P: nat > $o,K: nat] :
% 0.29/0.45 ( ( P @ K )
% 0.29/0.45 => ( ! [N3: nat] :
% 0.29/0.45 ( ( P @ ( suc @ N3 ) )
% 0.29/0.45 => ( P @ N3 ) )
% 0.29/0.45 => ( P @ ( zero_zero @ nat ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % zero_induct
% 0.29/0.45 thf(fact_201_Suc__neq__Zero,axiom,
% 0.29/0.45 ! [M: nat] :
% 0.29/0.45 ( ( suc @ M )
% 0.29/0.45 != ( zero_zero @ nat ) ) ).
% 0.29/0.45
% 0.29/0.45 % Suc_neq_Zero
% 0.29/0.45 thf(fact_202_Zero__neq__Suc,axiom,
% 0.29/0.45 ! [M: nat] :
% 0.29/0.45 ( ( zero_zero @ nat )
% 0.29/0.45 != ( suc @ M ) ) ).
% 0.29/0.45
% 0.29/0.45 % Zero_neq_Suc
% 0.29/0.45 thf(fact_203_Zero__not__Suc,axiom,
% 0.29/0.45 ! [M: nat] :
% 0.29/0.45 ( ( zero_zero @ nat )
% 0.29/0.45 != ( suc @ M ) ) ).
% 0.29/0.45
% 0.29/0.45 % Zero_not_Suc
% 0.29/0.45 thf(fact_204_old_Onat_Oexhaust,axiom,
% 0.29/0.45 ! [Y: nat] :
% 0.29/0.45 ( ( Y
% 0.29/0.45 != ( zero_zero @ nat ) )
% 0.29/0.45 => ~ ! [Nat4: nat] :
% 0.29/0.45 ( Y
% 0.29/0.45 != ( suc @ Nat4 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % old.nat.exhaust
% 0.29/0.45 thf(fact_205_old_Onat_Oinducts,axiom,
% 0.29/0.45 ! [P: nat > $o,Nat: nat] :
% 0.29/0.45 ( ( P @ ( zero_zero @ nat ) )
% 0.29/0.45 => ( ! [Nat4: nat] :
% 0.29/0.45 ( ( P @ Nat4 )
% 0.29/0.45 => ( P @ ( suc @ Nat4 ) ) )
% 0.29/0.45 => ( P @ Nat ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % old.nat.inducts
% 0.29/0.45 thf(fact_206_not0__implies__Suc,axiom,
% 0.29/0.45 ! [N: nat] :
% 0.29/0.45 ( ( N
% 0.29/0.45 != ( zero_zero @ nat ) )
% 0.29/0.45 => ? [M4: nat] :
% 0.29/0.45 ( N
% 0.29/0.45 = ( suc @ M4 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % not0_implies_Suc
% 0.29/0.45 thf(fact_207_bot__nat__0_Oextremum__uniqueI,axiom,
% 0.29/0.45 ! [A2: nat] :
% 0.29/0.45 ( ( ord_less_eq @ nat @ A2 @ ( zero_zero @ nat ) )
% 0.29/0.45 => ( A2
% 0.29/0.45 = ( zero_zero @ nat ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % bot_nat_0.extremum_uniqueI
% 0.29/0.45 thf(fact_208_bot__nat__0_Oextremum__unique,axiom,
% 0.29/0.45 ! [A2: nat] :
% 0.29/0.45 ( ( ord_less_eq @ nat @ A2 @ ( zero_zero @ nat ) )
% 0.29/0.45 = ( A2
% 0.29/0.45 = ( zero_zero @ nat ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % bot_nat_0.extremum_unique
% 0.29/0.45 thf(fact_209_le__0__eq,axiom,
% 0.29/0.45 ! [N: nat] :
% 0.29/0.45 ( ( ord_less_eq @ nat @ N @ ( zero_zero @ nat ) )
% 0.29/0.45 = ( N
% 0.29/0.45 = ( zero_zero @ nat ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % le_0_eq
% 0.29/0.45 thf(fact_210_less__eq__nat_Osimps_I1_J,axiom,
% 0.29/0.45 ! [N: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ N ) ).
% 0.29/0.45
% 0.29/0.45 % less_eq_nat.simps(1)
% 0.29/0.45 thf(fact_211_less__eq__int__code_I1_J,axiom,
% 0.29/0.45 ord_less_eq @ int @ ( zero_zero @ int ) @ ( zero_zero @ int ) ).
% 0.29/0.45
% 0.29/0.45 % less_eq_int_code(1)
% 0.29/0.45 thf(fact_212_le__nat__iff,axiom,
% 0.29/0.45 ! [K: int,N: nat] :
% 0.29/0.45 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
% 0.29/0.45 => ( ( ord_less_eq @ nat @ N @ ( nat2 @ K ) )
% 0.29/0.45 = ( ord_less_eq @ int @ ( semiring_1_of_nat @ int @ N ) @ K ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % le_nat_iff
% 0.29/0.45 thf(fact_213_of__int__nonneg,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( linordered_idom @ A )
% 0.29/0.45 => ! [Z: int] :
% 0.29/0.45 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z )
% 0.29/0.45 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( ring_1_of_int @ A @ Z ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % of_int_nonneg
% 0.29/0.45 thf(fact_214_nat__int__comparison_I3_J,axiom,
% 0.29/0.45 ( ( ord_less_eq @ nat )
% 0.29/0.45 = ( ^ [A4: nat,B3: nat] : ( ord_less_eq @ int @ ( semiring_1_of_nat @ int @ A4 ) @ ( semiring_1_of_nat @ int @ B3 ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % nat_int_comparison(3)
% 0.29/0.45 thf(fact_215_zle__int,axiom,
% 0.29/0.45 ! [M: nat,N: nat] :
% 0.29/0.45 ( ( ord_less_eq @ int @ ( semiring_1_of_nat @ int @ M ) @ ( semiring_1_of_nat @ int @ N ) )
% 0.29/0.45 = ( ord_less_eq @ nat @ M @ N ) ) ).
% 0.29/0.45
% 0.29/0.45 % zle_int
% 0.29/0.45 thf(fact_216_int__ops_I3_J,axiom,
% 0.29/0.45 ! [N: num] :
% 0.29/0.45 ( ( semiring_1_of_nat @ int @ ( numeral_numeral @ nat @ N ) )
% 0.29/0.45 = ( numeral_numeral @ int @ N ) ) ).
% 0.29/0.45
% 0.29/0.45 % int_ops(3)
% 0.29/0.45 thf(fact_217_zero__le__numeral,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( linord1659791738miring @ A )
% 0.29/0.45 => ! [N: num] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ N ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % zero_le_numeral
% 0.29/0.45 thf(fact_218_not__numeral__le__zero,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( linord1659791738miring @ A )
% 0.29/0.45 => ! [N: num] :
% 0.29/0.45 ~ ( ord_less_eq @ A @ ( numeral_numeral @ A @ N ) @ ( zero_zero @ A ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % not_numeral_le_zero
% 0.29/0.45 thf(fact_219_power__mono,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( linordered_semidom @ A )
% 0.29/0.45 => ! [A2: A,B: A,N: nat] :
% 0.29/0.45 ( ( ord_less_eq @ A @ A2 @ B )
% 0.29/0.45 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
% 0.29/0.45 => ( ord_less_eq @ A @ ( power_power @ A @ A2 @ N ) @ ( power_power @ A @ B @ N ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % power_mono
% 0.29/0.45 thf(fact_220_zero__le__power,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( linordered_semidom @ A )
% 0.29/0.45 => ! [A2: A,N: nat] :
% 0.29/0.45 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
% 0.29/0.45 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A2 @ N ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % zero_le_power
% 0.29/0.45 thf(fact_221_of__nat__0__le__iff,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( linord1659791738miring @ A )
% 0.29/0.45 => ! [N: nat] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( semiring_1_of_nat @ A @ N ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % of_nat_0_le_iff
% 0.29/0.45 thf(fact_222_power__0,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( power @ A )
% 0.29/0.45 => ! [A2: A] :
% 0.29/0.45 ( ( power_power @ A @ A2 @ ( zero_zero @ nat ) )
% 0.29/0.45 = ( one_one @ A ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % power_0
% 0.29/0.45 thf(fact_223_of__nat__neq__0,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( semiring_char_0 @ A )
% 0.29/0.45 => ! [N: nat] :
% 0.29/0.45 ( ( semiring_1_of_nat @ A @ ( suc @ N ) )
% 0.29/0.45 != ( zero_zero @ A ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % of_nat_neq_0
% 0.29/0.45 thf(fact_224_ex__nat,axiom,
% 0.29/0.45 ( ( ^ [P2: nat > $o] :
% 0.29/0.45 ? [X6: nat] : ( P2 @ X6 ) )
% 0.29/0.45 = ( ^ [P3: nat > $o] :
% 0.29/0.45 ? [X3: int] :
% 0.29/0.45 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X3 )
% 0.29/0.45 & ( P3 @ ( nat2 @ X3 ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % ex_nat
% 0.29/0.45 thf(fact_225_all__nat,axiom,
% 0.29/0.45 ( ( ^ [P2: nat > $o] :
% 0.29/0.45 ! [X6: nat] : ( P2 @ X6 ) )
% 0.29/0.45 = ( ^ [P3: nat > $o] :
% 0.29/0.45 ! [X3: int] :
% 0.29/0.45 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X3 )
% 0.29/0.45 => ( P3 @ ( nat2 @ X3 ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % all_nat
% 0.29/0.45 thf(fact_226_eq__nat__nat__iff,axiom,
% 0.29/0.45 ! [Z: int,Z5: int] :
% 0.29/0.45 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z )
% 0.29/0.45 => ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z5 )
% 0.29/0.45 => ( ( ( nat2 @ Z )
% 0.29/0.45 = ( nat2 @ Z5 ) )
% 0.29/0.45 = ( Z = Z5 ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % eq_nat_nat_iff
% 0.29/0.45 thf(fact_227_powr__mono2,axiom,
% 0.29/0.45 ! [A2: real,X: real,Y: real] :
% 0.29/0.45 ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ A2 )
% 0.29/0.45 => ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
% 0.29/0.45 => ( ( ord_less_eq @ real @ X @ Y )
% 0.29/0.45 => ( ord_less_eq @ real @ ( powr @ real @ X @ A2 ) @ ( powr @ real @ Y @ A2 ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % powr_mono2
% 0.29/0.45 thf(fact_228_powr__ge__pzero,axiom,
% 0.29/0.45 ! [X: real,Y: real] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( powr @ real @ X @ Y ) ) ).
% 0.29/0.45
% 0.29/0.45 % powr_ge_pzero
% 0.29/0.45 thf(fact_229_pow_Osimps_I1_J,axiom,
% 0.29/0.45 ! [X: num] :
% 0.29/0.45 ( ( pow @ X @ one2 )
% 0.29/0.45 = X ) ).
% 0.29/0.45
% 0.29/0.45 % pow.simps(1)
% 0.29/0.45 thf(fact_230_nat__le__iff,axiom,
% 0.29/0.45 ! [X: int,N: nat] :
% 0.29/0.45 ( ( ord_less_eq @ nat @ ( nat2 @ X ) @ N )
% 0.29/0.45 = ( ord_less_eq @ int @ X @ ( semiring_1_of_nat @ int @ N ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % nat_le_iff
% 0.29/0.45 thf(fact_231_le__funD,axiom,
% 0.29/0.45 ! [B2: $tType,A: $tType] :
% 0.29/0.45 ( ( ord @ B2 )
% 0.29/0.45 => ! [F: A > B2,G: A > B2,X: A] :
% 0.29/0.45 ( ( ord_less_eq @ ( A > B2 ) @ F @ G )
% 0.29/0.45 => ( ord_less_eq @ B2 @ ( F @ X ) @ ( G @ X ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % le_funD
% 0.29/0.45 thf(fact_232_le__funE,axiom,
% 0.29/0.45 ! [B2: $tType,A: $tType] :
% 0.29/0.45 ( ( ord @ B2 )
% 0.29/0.45 => ! [F: A > B2,G: A > B2,X: A] :
% 0.29/0.45 ( ( ord_less_eq @ ( A > B2 ) @ F @ G )
% 0.29/0.45 => ( ord_less_eq @ B2 @ ( F @ X ) @ ( G @ X ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % le_funE
% 0.29/0.45 thf(fact_233_le__funI,axiom,
% 0.29/0.45 ! [B2: $tType,A: $tType] :
% 0.29/0.45 ( ( ord @ B2 )
% 0.29/0.45 => ! [F: A > B2,G: A > B2] :
% 0.29/0.45 ( ! [X4: A] : ( ord_less_eq @ B2 @ ( F @ X4 ) @ ( G @ X4 ) )
% 0.29/0.45 => ( ord_less_eq @ ( A > B2 ) @ F @ G ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % le_funI
% 0.29/0.45 thf(fact_234_le__fun__def,axiom,
% 0.29/0.45 ! [B2: $tType,A: $tType] :
% 0.29/0.45 ( ( ord @ B2 )
% 0.29/0.45 => ( ( ord_less_eq @ ( A > B2 ) )
% 0.29/0.45 = ( ^ [F2: A > B2,G2: A > B2] :
% 0.29/0.45 ! [X3: A] : ( ord_less_eq @ B2 @ ( F2 @ X3 ) @ ( G2 @ X3 ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % le_fun_def
% 0.29/0.45 thf(fact_235_order__subst1,axiom,
% 0.29/0.45 ! [A: $tType,B2: $tType] :
% 0.29/0.45 ( ( ( order @ B2 )
% 0.29/0.45 & ( order @ A ) )
% 0.29/0.45 => ! [A2: A,F: B2 > A,B: B2,C: B2] :
% 0.29/0.45 ( ( ord_less_eq @ A @ A2 @ ( F @ B ) )
% 0.29/0.45 => ( ( ord_less_eq @ B2 @ B @ C )
% 0.29/0.45 => ( ! [X4: B2,Y3: B2] :
% 0.29/0.45 ( ( ord_less_eq @ B2 @ X4 @ Y3 )
% 0.29/0.45 => ( ord_less_eq @ A @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 0.29/0.45 => ( ord_less_eq @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % order_subst1
% 0.29/0.45 thf(fact_236_order__subst2,axiom,
% 0.29/0.45 ! [A: $tType,C2: $tType] :
% 0.29/0.45 ( ( ( order @ C2 )
% 0.29/0.45 & ( order @ A ) )
% 0.29/0.45 => ! [A2: A,B: A,F: A > C2,C: C2] :
% 0.29/0.45 ( ( ord_less_eq @ A @ A2 @ B )
% 0.29/0.45 => ( ( ord_less_eq @ C2 @ ( F @ B ) @ C )
% 0.29/0.45 => ( ! [X4: A,Y3: A] :
% 0.29/0.45 ( ( ord_less_eq @ A @ X4 @ Y3 )
% 0.29/0.45 => ( ord_less_eq @ C2 @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 0.29/0.45 => ( ord_less_eq @ C2 @ ( F @ A2 ) @ C ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % order_subst2
% 0.29/0.45 thf(fact_237_ord__eq__le__subst,axiom,
% 0.29/0.45 ! [A: $tType,B2: $tType] :
% 0.29/0.45 ( ( ( ord @ B2 )
% 0.29/0.45 & ( ord @ A ) )
% 0.29/0.45 => ! [A2: A,F: B2 > A,B: B2,C: B2] :
% 0.29/0.45 ( ( A2
% 0.29/0.45 = ( F @ B ) )
% 0.29/0.45 => ( ( ord_less_eq @ B2 @ B @ C )
% 0.29/0.45 => ( ! [X4: B2,Y3: B2] :
% 0.29/0.45 ( ( ord_less_eq @ B2 @ X4 @ Y3 )
% 0.29/0.45 => ( ord_less_eq @ A @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 0.29/0.45 => ( ord_less_eq @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % ord_eq_le_subst
% 0.29/0.45 thf(fact_238_ord__le__eq__subst,axiom,
% 0.29/0.45 ! [A: $tType,B2: $tType] :
% 0.29/0.45 ( ( ( ord @ B2 )
% 0.29/0.45 & ( ord @ A ) )
% 0.29/0.45 => ! [A2: A,B: A,F: A > B2,C: B2] :
% 0.29/0.45 ( ( ord_less_eq @ A @ A2 @ B )
% 0.29/0.45 => ( ( ( F @ B )
% 0.29/0.45 = C )
% 0.29/0.45 => ( ! [X4: A,Y3: A] :
% 0.29/0.45 ( ( ord_less_eq @ A @ X4 @ Y3 )
% 0.29/0.45 => ( ord_less_eq @ B2 @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 0.29/0.45 => ( ord_less_eq @ B2 @ ( F @ A2 ) @ C ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % ord_le_eq_subst
% 0.29/0.45 thf(fact_239_eq__iff,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( order @ A )
% 0.29/0.45 => ( ( ^ [Y5: A,Z4: A] : ( Y5 = Z4 ) )
% 0.29/0.45 = ( ^ [X3: A,Y6: A] :
% 0.29/0.45 ( ( ord_less_eq @ A @ X3 @ Y6 )
% 0.29/0.45 & ( ord_less_eq @ A @ Y6 @ X3 ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % eq_iff
% 0.29/0.45 thf(fact_240_antisym,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( order @ A )
% 0.29/0.45 => ! [X: A,Y: A] :
% 0.29/0.45 ( ( ord_less_eq @ A @ X @ Y )
% 0.29/0.45 => ( ( ord_less_eq @ A @ Y @ X )
% 0.29/0.45 => ( X = Y ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % antisym
% 0.29/0.45 thf(fact_241_linear,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( linorder @ A )
% 0.29/0.46 => ! [X: A,Y: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ X @ Y )
% 0.29/0.46 | ( ord_less_eq @ A @ Y @ X ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % linear
% 0.29/0.46 thf(fact_242_eq__refl,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( preorder @ A )
% 0.29/0.46 => ! [X: A,Y: A] :
% 0.29/0.46 ( ( X = Y )
% 0.29/0.46 => ( ord_less_eq @ A @ X @ Y ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % eq_refl
% 0.29/0.46 thf(fact_243_le__cases,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( linorder @ A )
% 0.29/0.46 => ! [X: A,Y: A] :
% 0.29/0.46 ( ~ ( ord_less_eq @ A @ X @ Y )
% 0.29/0.46 => ( ord_less_eq @ A @ Y @ X ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % le_cases
% 0.29/0.46 thf(fact_244_order_Otrans,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( order @ A )
% 0.29/0.46 => ! [A2: A,B: A,C: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ A2 @ B )
% 0.29/0.46 => ( ( ord_less_eq @ A @ B @ C )
% 0.29/0.46 => ( ord_less_eq @ A @ A2 @ C ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % order.trans
% 0.29/0.46 thf(fact_245_le__cases3,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( linorder @ A )
% 0.29/0.46 => ! [X: A,Y: A,Z: A] :
% 0.29/0.46 ( ( ( ord_less_eq @ A @ X @ Y )
% 0.29/0.46 => ~ ( ord_less_eq @ A @ Y @ Z ) )
% 0.29/0.46 => ( ( ( ord_less_eq @ A @ Y @ X )
% 0.29/0.46 => ~ ( ord_less_eq @ A @ X @ Z ) )
% 0.29/0.46 => ( ( ( ord_less_eq @ A @ X @ Z )
% 0.29/0.46 => ~ ( ord_less_eq @ A @ Z @ Y ) )
% 0.29/0.46 => ( ( ( ord_less_eq @ A @ Z @ Y )
% 0.29/0.46 => ~ ( ord_less_eq @ A @ Y @ X ) )
% 0.29/0.46 => ( ( ( ord_less_eq @ A @ Y @ Z )
% 0.29/0.46 => ~ ( ord_less_eq @ A @ Z @ X ) )
% 0.29/0.46 => ~ ( ( ord_less_eq @ A @ Z @ X )
% 0.29/0.46 => ~ ( ord_less_eq @ A @ X @ Y ) ) ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % le_cases3
% 0.29/0.46 thf(fact_246_antisym__conv,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( order @ A )
% 0.29/0.46 => ! [Y: A,X: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ Y @ X )
% 0.29/0.46 => ( ( ord_less_eq @ A @ X @ Y )
% 0.29/0.46 = ( X = Y ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % antisym_conv
% 0.29/0.46 thf(fact_247_order__class_Oorder_Oeq__iff,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( order @ A )
% 0.29/0.46 => ( ( ^ [Y5: A,Z4: A] : ( Y5 = Z4 ) )
% 0.29/0.46 = ( ^ [A4: A,B3: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ A4 @ B3 )
% 0.29/0.46 & ( ord_less_eq @ A @ B3 @ A4 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % order_class.order.eq_iff
% 0.29/0.46 thf(fact_248_ord__eq__le__trans,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( ord @ A )
% 0.29/0.46 => ! [A2: A,B: A,C: A] :
% 0.29/0.46 ( ( A2 = B )
% 0.29/0.46 => ( ( ord_less_eq @ A @ B @ C )
% 0.29/0.46 => ( ord_less_eq @ A @ A2 @ C ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % ord_eq_le_trans
% 0.29/0.46 thf(fact_249_ord__le__eq__trans,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( ord @ A )
% 0.29/0.46 => ! [A2: A,B: A,C: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ A2 @ B )
% 0.29/0.46 => ( ( B = C )
% 0.29/0.46 => ( ord_less_eq @ A @ A2 @ C ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % ord_le_eq_trans
% 0.29/0.46 thf(fact_250_order__class_Oorder_Oantisym,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( order @ A )
% 0.29/0.46 => ! [A2: A,B: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ A2 @ B )
% 0.29/0.46 => ( ( ord_less_eq @ A @ B @ A2 )
% 0.29/0.46 => ( A2 = B ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % order_class.order.antisym
% 0.29/0.46
% 0.29/0.46 % Type constructors (71)
% 0.29/0.46 thf(tcon_fun___Orderings_Opreorder,axiom,
% 0.29/0.46 ! [A5: $tType,A6: $tType] :
% 0.29/0.46 ( ( preorder @ A6 )
% 0.29/0.46 => ( preorder @ ( A5 > A6 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_fun___Orderings_Oorder,axiom,
% 0.29/0.46 ! [A5: $tType,A6: $tType] :
% 0.29/0.46 ( ( order @ A6 )
% 0.29/0.46 => ( order @ ( A5 > A6 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_fun___Orderings_Oord,axiom,
% 0.29/0.46 ! [A5: $tType,A6: $tType] :
% 0.29/0.46 ( ( ord @ A6 )
% 0.29/0.46 => ( ord @ ( A5 > A6 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Int_Oint___Rings_Osemiring__1__no__zero__divisors,axiom,
% 0.29/0.46 semiri134348788visors @ int ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Int_Oint___Rings_Olinordered__nonzero__semiring,axiom,
% 0.29/0.46 linord1659791738miring @ int ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Int_Oint___Rings_Olinordered__semidom,axiom,
% 0.29/0.46 linordered_semidom @ int ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Int_Oint___Rings_Olinordered__idom,axiom,
% 0.29/0.46 linordered_idom @ int ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Int_Oint___Num_Osemiring__numeral,axiom,
% 0.29/0.46 semiring_numeral @ int ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Int_Oint___Nat_Osemiring__char__0,axiom,
% 0.29/0.46 semiring_char_0 @ int ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Int_Oint___Orderings_Opreorder_1,axiom,
% 0.29/0.46 preorder @ int ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Int_Oint___Orderings_Olinorder,axiom,
% 0.29/0.46 linorder @ int ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Int_Oint___Groups_Omonoid__mult,axiom,
% 0.29/0.46 monoid_mult @ int ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Int_Oint___Rings_Osemiring__1,axiom,
% 0.29/0.46 semiring_1 @ int ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Int_Oint___Orderings_Oorder_2,axiom,
% 0.29/0.46 order @ int ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Int_Oint___Num_Oneg__numeral,axiom,
% 0.29/0.46 neg_numeral @ int ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Int_Oint___Nat_Oring__char__0,axiom,
% 0.29/0.46 ring_char_0 @ int ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Int_Oint___Orderings_Oord_3,axiom,
% 0.29/0.46 ord @ int ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Int_Oint___Rings_Oring__1,axiom,
% 0.29/0.46 ring_1 @ int ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Int_Oint___Power_Opower,axiom,
% 0.29/0.46 power @ int ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Int_Oint___Num_Onumeral,axiom,
% 0.29/0.46 numeral @ int ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Int_Oint___Groups_Ozero,axiom,
% 0.29/0.46 zero @ int ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Int_Oint___Groups_Oone,axiom,
% 0.29/0.46 one @ int ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Nat_Onat___Groups_Ocanonically__ordered__monoid__add,axiom,
% 0.29/0.46 canoni770627133id_add @ nat ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Nat_Onat___Rings_Osemiring__1__no__zero__divisors_4,axiom,
% 0.29/0.46 semiri134348788visors @ nat ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Nat_Onat___Rings_Olinordered__nonzero__semiring_5,axiom,
% 0.29/0.46 linord1659791738miring @ nat ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Nat_Onat___Rings_Olinordered__semidom_6,axiom,
% 0.29/0.46 linordered_semidom @ nat ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Nat_Onat___Num_Osemiring__numeral_7,axiom,
% 0.29/0.46 semiring_numeral @ nat ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Nat_Onat___Nat_Osemiring__char__0_8,axiom,
% 0.29/0.46 semiring_char_0 @ nat ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Nat_Onat___Orderings_Opreorder_9,axiom,
% 0.29/0.46 preorder @ nat ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Nat_Onat___Orderings_Olinorder_10,axiom,
% 0.29/0.46 linorder @ nat ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Nat_Onat___Groups_Omonoid__mult_11,axiom,
% 0.29/0.46 monoid_mult @ nat ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Nat_Onat___Rings_Osemiring__1_12,axiom,
% 0.29/0.46 semiring_1 @ nat ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Nat_Onat___Orderings_Oorder_13,axiom,
% 0.29/0.46 order @ nat ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Nat_Onat___Orderings_Oord_14,axiom,
% 0.29/0.46 ord @ nat ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Nat_Onat___Power_Opower_15,axiom,
% 0.29/0.46 power @ nat ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Nat_Onat___Num_Onumeral_16,axiom,
% 0.29/0.46 numeral @ nat ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Nat_Onat___Groups_Ozero_17,axiom,
% 0.29/0.46 zero @ nat ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Nat_Onat___Groups_Oone_18,axiom,
% 0.29/0.46 one @ nat ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Num_Onum___Orderings_Opreorder_19,axiom,
% 0.29/0.46 preorder @ num ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Num_Onum___Orderings_Olinorder_20,axiom,
% 0.29/0.46 linorder @ num ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Num_Onum___Orderings_Oorder_21,axiom,
% 0.29/0.46 order @ num ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Num_Onum___Orderings_Oord_22,axiom,
% 0.29/0.46 ord @ num ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Set_Oset___Orderings_Opreorder_23,axiom,
% 0.29/0.46 ! [A5: $tType] : ( preorder @ ( set @ A5 ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Set_Oset___Orderings_Oorder_24,axiom,
% 0.29/0.46 ! [A5: $tType] : ( order @ ( set @ A5 ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Set_Oset___Orderings_Oord_25,axiom,
% 0.29/0.46 ! [A5: $tType] : ( ord @ ( set @ A5 ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_HOL_Obool___Orderings_Opreorder_26,axiom,
% 0.29/0.46 preorder @ $o ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_HOL_Obool___Orderings_Olinorder_27,axiom,
% 0.29/0.46 linorder @ $o ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_HOL_Obool___Orderings_Oorder_28,axiom,
% 0.29/0.46 order @ $o ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_HOL_Obool___Orderings_Oord_29,axiom,
% 0.29/0.46 ord @ $o ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Real_Oreal___Archimedean__Field_Oarchimedean__field,axiom,
% 0.29/0.46 archim1804426504_field @ real ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Real_Oreal___Rings_Osemiring__1__no__zero__divisors_30,axiom,
% 0.29/0.46 semiri134348788visors @ real ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Real_Oreal___Rings_Olinordered__nonzero__semiring_31,axiom,
% 0.29/0.46 linord1659791738miring @ real ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Real_Oreal___Archimedean__Field_Ofloor__ceiling,axiom,
% 0.29/0.46 archim1727834104eiling @ real ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Real_Oreal___Rings_Olinordered__semidom_32,axiom,
% 0.29/0.46 linordered_semidom @ real ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Real_Oreal___Rings_Olinordered__idom_33,axiom,
% 0.29/0.46 linordered_idom @ real ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Real_Oreal___Num_Osemiring__numeral_34,axiom,
% 0.29/0.46 semiring_numeral @ real ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Real_Oreal___Nat_Osemiring__char__0_35,axiom,
% 0.29/0.46 semiring_char_0 @ real ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Real_Oreal___Orderings_Opreorder_36,axiom,
% 0.29/0.46 preorder @ real ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Real_Oreal___Orderings_Olinorder_37,axiom,
% 0.29/0.46 linorder @ real ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Real_Oreal___Groups_Omonoid__mult_38,axiom,
% 0.29/0.46 monoid_mult @ real ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Real_Oreal___Transcendental_Oln,axiom,
% 0.29/0.46 ln @ real ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Real_Oreal___Rings_Osemiring__1_39,axiom,
% 0.29/0.46 semiring_1 @ real ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Real_Oreal___Orderings_Oorder_40,axiom,
% 0.29/0.46 order @ real ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Real_Oreal___Num_Oneg__numeral_41,axiom,
% 0.29/0.46 neg_numeral @ real ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Real_Oreal___Nat_Oring__char__0_42,axiom,
% 0.29/0.46 ring_char_0 @ real ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Real_Oreal___Orderings_Oord_43,axiom,
% 0.29/0.46 ord @ real ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Real_Oreal___Rings_Oring__1_44,axiom,
% 0.29/0.46 ring_1 @ real ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Real_Oreal___Power_Opower_45,axiom,
% 0.29/0.46 power @ real ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Real_Oreal___Num_Onumeral_46,axiom,
% 0.29/0.46 numeral @ real ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Real_Oreal___Groups_Ozero_47,axiom,
% 0.29/0.46 zero @ real ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Real_Oreal___Groups_Oone_48,axiom,
% 0.29/0.46 one @ real ).
% 0.29/0.46
% 0.29/0.46 % Helper facts (3)
% 0.29/0.46 thf(help_If_3_1_T,axiom,
% 0.29/0.46 ! [P: $o] :
% 0.29/0.46 ( ( P = $true )
% 0.29/0.46 | ( P = $false ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(help_If_2_1_T,axiom,
% 0.29/0.46 ! [A: $tType,X: A,Y: A] :
% 0.29/0.46 ( ( if @ A @ $false @ X @ Y )
% 0.29/0.46 = Y ) ).
% 0.29/0.46
% 0.29/0.46 thf(help_If_1_1_T,axiom,
% 0.29/0.46 ! [A: $tType,X: A,Y: A] :
% 0.29/0.46 ( ( if @ A @ $true @ X @ Y )
% 0.29/0.46 = X ) ).
% 0.29/0.46
% 0.29/0.46 % Conjectures (1)
% 0.29/0.46 thf(conj_0,conjecture,
% 0.29/0.46 ord_less_eq @ int @ d @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( nat2 @ ( archimedean_ceiling @ real @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( ring_1_of_int @ real @ d ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 %------------------------------------------------------------------------------
% 0.29/0.46 ------------------------
% 2.64/1.76 ---- Embedded in THF ---
% 2.64/1.76 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 2.64/1.76 %%% Generated on Tue Feb 24 08:26:17 EST 2026
% 2.64/1.76 %%% using '$$dhol' embedding, version 1.3.0.
% 2.64/1.76 %%% Logic specification used:
% 2.64/1.76 %%% thf(spec, logic, $$dhol).
% 2.64/1.76
% 2.64/1.76 % SZS output start ListOfTHF for /export/starexec/sandbox/tmp/tmp.aWXyJVFJl3/DTF2DTF4485.p
% 2.64/1.76 thf(ty_t_Real_Oreal, type, real: $tType).
% 2.64/1.76 thf(ty_t_Set_Oset, type, set: ($tType > $tType)).
% 2.64/1.76 thf(ty_t_Num_Onum, type, num: $tType).
% 2.64/1.76 thf(ty_t_Nat_Onat, type, nat: $tType).
% 2.64/1.76 thf(ty_t_Int_Oint, type, int: $tType).
% 2.64/1.76 thf(sy_cl_HOL_Otype, type, type: (!> [A:$tType]: ($o))).
% 2.64/1.76 thf(sy_cl_Groups_Oone, type, one: (!> [A:$tType]: ($o))).
% 2.64/1.76 thf(sy_cl_Groups_Ozero, type, zero: (!> [A:$tType]: ($o))).
% 2.64/1.76 thf(sy_cl_Num_Onumeral, type, numeral: (!> [A:$tType]: ($o))).
% 2.64/1.76 thf(sy_cl_Power_Opower, type, power: (!> [A:$tType]: ($o))).
% 2.64/1.76 thf(sy_cl_Rings_Oring__1, type, ring_1: (!> [A:$tType]: ($o))).
% 2.64/1.76 thf(sy_cl_Orderings_Oord, type, ord: (!> [A:$tType]: ($o))).
% 2.64/1.76 thf(sy_cl_Nat_Oring__char__0, type, ring_char_0: (!> [A:$tType]: ($o))).
% 2.64/1.76 thf(sy_cl_Num_Oneg__numeral, type, neg_numeral: (!> [A:$tType]: ($o))).
% 2.64/1.76 thf(sy_cl_Orderings_Oorder, type, order: (!> [A:$tType]: ($o))).
% 2.64/1.76 thf(sy_cl_Rings_Osemiring__1, type, semiring_1: (!> [A:$tType]: ($o))).
% 2.64/1.76 thf(sy_cl_Transcendental_Oln, type, ln: (!> [A:$tType]: ($o))).
% 2.64/1.76 thf(sy_cl_Groups_Omonoid__mult, type, monoid_mult: (!> [A:$tType]: ($o))).
% 2.64/1.76 thf(sy_cl_Orderings_Olinorder, type, linorder: (!> [A:$tType]: ($o))).
% 2.64/1.76 thf(sy_cl_Orderings_Opreorder, type, preorder: (!> [A:$tType]: ($o))).
% 2.64/1.76 thf(sy_cl_Nat_Osemiring__char__0, type, semiring_char_0: (!> [A:$tType]: ($o))).
% 2.64/1.76 thf(sy_cl_Num_Osemiring__numeral, type, semiring_numeral: (!> [A:$tType]: ($o))).
% 2.64/1.76 thf(sy_cl_Rings_Olinordered__idom, type, linordered_idom: (!> [A:$tType]: ($o))).
% 2.64/1.76 thf(sy_cl_Rings_Olinordered__semidom, type, linordered_semidom: (!> [A:$tType]: ($o))).
% 2.64/1.76 thf(sy_cl_Archimedean__Field_Ofloor__ceiling, type, archim1727834104eiling: (!> [A:$tType]: ($o))).
% 2.64/1.76 thf(sy_cl_Rings_Olinordered__nonzero__semiring, type, linord1659791738miring: (!> [A:$tType]: ($o))).
% 2.64/1.76 thf(sy_cl_Rings_Osemiring__1__no__zero__divisors, type, semiri134348788visors: (!> [A:$tType]: ($o))).
% 2.64/1.76 thf(sy_cl_Archimedean__Field_Oarchimedean__field, type, archim1804426504_field: (!> [A:$tType]: ($o))).
% 2.64/1.76 thf(sy_cl_Groups_Ocanonically__ordered__monoid__add, type, canoni770627133id_add: (!> [A:$tType]: ($o))).
% 2.64/1.76 thf(sy_c_Archimedean__Field_Oceiling, type, archimedean_ceiling: (!> [A:$tType]: ((A > int)))).
% 2.64/1.76 thf(sy_c_Groups_Oone__class_Oone, type, one_one: (!> [A:$tType]: (A))).
% 2.64/1.76 thf(sy_c_Groups_Ozero__class_Ozero, type, zero_zero: (!> [A:$tType]: (A))).
% 2.64/1.76 thf(sy_c_If, type, if: (!> [A:$tType]: (($o > (A > (A > A)))))).
% 2.64/1.76 thf(sy_c_Int_Onat, type, nat2: (int > nat)).
% 2.64/1.76 thf(sy_c_Int_Oring__1__class_Oof__int, type, ring_1_of_int: (!> [A:$tType]: ((int > A)))).
% 2.64/1.76 thf(sy_c_Nat_OSuc, type, suc: (nat > nat)).
% 2.64/1.76 thf(sy_c_Nat_Osemiring__1__class_Oof__nat, type, semiring_1_of_nat: (!> [A:$tType]: ((nat > A)))).
% 2.64/1.76 thf(sy_c_Num_Oneg__numeral__class_Odbl, type, neg_numeral_dbl: (!> [A:$tType]: ((A > A)))).
% 2.64/1.76 thf(sy_c_Num_Onum_OBit0, type, bit0: (num > num)).
% 2.64/1.76 thf(sy_c_Num_Onum_OOne, type, one2: num).
% 2.64/1.76 thf(sy_c_Num_Onumeral__class_Onumeral, type, numeral_numeral: (!> [A:$tType]: ((num > A)))).
% 2.64/1.76 thf(sy_c_Num_Opow, type, pow: (num > (num > num))).
% 2.64/1.76 thf(sy_c_Orderings_Oord__class_Oless, type, ord_less: (!> [A:$tType]: ((A > (A > $o))))).
% 2.64/1.76 thf(sy_c_Orderings_Oord__class_Oless__eq, type, ord_less_eq: (!> [A:$tType]: ((A > (A > $o))))).
% 2.64/1.76 thf(sy_c_Power_Opower__class_Opower, type, power_power: (!> [A:$tType]: ((A > (nat > A))))).
% 2.64/1.76 thf(sy_c_Set_OCollect, type, collect: (!> [A:$tType]: (((A > $o) > (set @ A))))).
% 2.64/1.76 thf(sy_c_Transcendental_Olog, type, log: (real > (real > real))).
% 2.64/1.76 thf(sy_c_Transcendental_Opowr, type, powr: (!> [A:$tType]: ((A > (A > A))))).
% 2.64/1.76 thf(sy_c_member, type, member: (!> [A:$tType]: ((A > ((set @ A) > $o))))).
% 2.64/1.76 thf(sy_v_d____, type, d: int).
% 2.64/1.76 thf(sy_v_n____, type, n: nat).
% 2.64/1.76 thf(fact_0_numeral__power__le__of__int__cancel__iff, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [X:num,N:nat,A2:int]: (((((ord_less_eq @ A) @ (((power_power @ A) @ ((numeral_numeral @ A) @ X)) @ N)) @ ((ring_1_of_int @ A) @ A2)) = (((ord_less_eq @ int) @ (((power_power @ int) @ ((numeral_numeral @ int) @ X)) @ N)) @ A2)))))))).
% 2.64/1.76 thf(fact_1_of__int__le__numeral__power__cancel__iff, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [A2:int,X:num,N:nat]: (((((ord_less_eq @ A) @ ((ring_1_of_int @ A) @ A2)) @ (((power_power @ A) @ ((numeral_numeral @ A) @ X)) @ N)) = (((ord_less_eq @ int) @ A2) @ (((power_power @ int) @ ((numeral_numeral @ int) @ X)) @ N))))))))).
% 2.64/1.76 thf(fact_2_ceiling__numeral__power, axiom, (! [A:$tType]: (((archim1727834104eiling @ A) => (! [X:num,N:nat]: ((((archimedean_ceiling @ A) @ (((power_power @ A) @ ((numeral_numeral @ A) @ X)) @ N)) = (((power_power @ int) @ ((numeral_numeral @ int) @ X)) @ N)))))))).
% 2.64/1.76 thf(fact_3_numeral__power__eq__of__int__cancel__iff, axiom, (! [A:$tType]: (((ring_char_0 @ A) => (! [X:num,N:nat,Y:int]: ((((((power_power @ A) @ ((numeral_numeral @ A) @ X)) @ N) = ((ring_1_of_int @ A) @ Y)) = ((((power_power @ int) @ ((numeral_numeral @ int) @ X)) @ N) = Y)))))))).
% 2.64/1.76 thf(fact_4_of__int__eq__numeral__power__cancel__iff, axiom, (! [A:$tType]: (((ring_char_0 @ A) => (! [Y:int,X:num,N:nat]: (((((ring_1_of_int @ A) @ Y) = (((power_power @ A) @ ((numeral_numeral @ A) @ X)) @ N)) = (Y = (((power_power @ int) @ ((numeral_numeral @ int) @ X)) @ N))))))))).
% 2.64/1.76 thf(fact_5_of__int__le__of__int__power__cancel__iff, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [B:int,W:nat,X:int]: (((((ord_less_eq @ A) @ (((power_power @ A) @ ((ring_1_of_int @ A) @ B)) @ W)) @ ((ring_1_of_int @ A) @ X)) = (((ord_less_eq @ int) @ (((power_power @ int) @ B) @ W)) @ X)))))))).
% 2.64/1.76 thf(fact_6_of__int__power__le__of__int__cancel__iff, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [X:int,B:int,W:nat]: (((((ord_less_eq @ A) @ ((ring_1_of_int @ A) @ X)) @ (((power_power @ A) @ ((ring_1_of_int @ A) @ B)) @ W)) = (((ord_less_eq @ int) @ X) @ (((power_power @ int) @ B) @ W))))))))).
% 2.64/1.76 thf(fact_7_ceiling__le__numeral, axiom, (! [A:$tType]: (((archim1727834104eiling @ A) => (! [X:A,V:num]: (((((ord_less_eq @ int) @ ((archimedean_ceiling @ A) @ X)) @ ((numeral_numeral @ int) @ V)) = (((ord_less_eq @ A) @ X) @ ((numeral_numeral @ A) @ V))))))))).
% 2.64/1.76 thf(fact_8_of__int__le__numeral__iff, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [Z:int,N:num]: (((((ord_less_eq @ A) @ ((ring_1_of_int @ A) @ Z)) @ ((numeral_numeral @ A) @ N)) = (((ord_less_eq @ int) @ Z) @ ((numeral_numeral @ int) @ N))))))))).
% 2.64/1.76 thf(fact_9_of__int__numeral__le__iff, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [N:num,Z:int]: (((((ord_less_eq @ A) @ ((numeral_numeral @ A) @ N)) @ ((ring_1_of_int @ A) @ Z)) = (((ord_less_eq @ int) @ ((numeral_numeral @ int) @ N)) @ Z)))))))).
% 2.64/1.76 thf(fact_10_of__int__power, axiom, (! [A:$tType]: (((ring_1 @ A) => (! [Z:int,N:nat]: ((((ring_1_of_int @ A) @ (((power_power @ int) @ Z) @ N)) = (((power_power @ A) @ ((ring_1_of_int @ A) @ Z)) @ N)))))))).
% 2.64/1.76 thf(fact_11_of__int__eq__of__int__power__cancel__iff, axiom, (! [A:$tType]: (((ring_char_0 @ A) => (! [B:int,W:nat,X:int]: ((((((power_power @ A) @ ((ring_1_of_int @ A) @ B)) @ W) = ((ring_1_of_int @ A) @ X)) = ((((power_power @ int) @ B) @ W) = X)))))))).
% 2.64/1.76 thf(fact_12_of__int__eq__iff, axiom, (! [A:$tType]: (((ring_char_0 @ A) => (! [W:int,Z:int]: (((((ring_1_of_int @ A) @ W) = ((ring_1_of_int @ A) @ Z)) = (W = Z)))))))).
% 2.64/1.76 thf(fact_13_nat__numeral, axiom, (! [K:num]: (((nat2 @ ((numeral_numeral @ int) @ K)) = ((numeral_numeral @ nat) @ K))))).
% 2.64/1.76 thf(fact_14_ceiling__of__int, axiom, (! [A:$tType]: (((archim1727834104eiling @ A) => (! [Z:int]: ((((archimedean_ceiling @ A) @ ((ring_1_of_int @ A) @ Z)) = Z))))))).
% 2.64/1.76 thf(fact_15_of__int__ceiling__cancel, axiom, (! [A:$tType]: (((archim1727834104eiling @ A) => (! [X:A]: (((((ring_1_of_int @ A) @ ((archimedean_ceiling @ A) @ X)) = X) = (? [N2:int]: ((X = ((ring_1_of_int @ A) @ N2))))))))))).
% 2.64/1.76 thf(fact_16_of__int__le__iff, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [W:int,Z:int]: (((((ord_less_eq @ A) @ ((ring_1_of_int @ A) @ W)) @ ((ring_1_of_int @ A) @ Z)) = (((ord_less_eq @ int) @ W) @ Z)))))))).
% 2.64/1.76 thf(fact_17_of__int__numeral, axiom, (! [A:$tType]: (((ring_1 @ A) => (! [K:num]: ((((ring_1_of_int @ A) @ ((numeral_numeral @ int) @ K)) = ((numeral_numeral @ A) @ K)))))))).
% 2.64/1.76 thf(fact_18_of__int__eq__numeral__iff, axiom, (! [A:$tType]: (((ring_char_0 @ A) => (! [Z:int,N:num]: (((((ring_1_of_int @ A) @ Z) = ((numeral_numeral @ A) @ N)) = (Z = ((numeral_numeral @ int) @ N))))))))).
% 2.64/1.76 thf(fact_19_ceiling__numeral, axiom, (! [A:$tType]: (((archim1727834104eiling @ A) => (! [V:num]: ((((archimedean_ceiling @ A) @ ((numeral_numeral @ A) @ V)) = ((numeral_numeral @ int) @ V)))))))).
% 2.64/1.76 thf(fact_20_of__int__power__eq__of__int__cancel__iff, axiom, (! [A:$tType]: (((ring_char_0 @ A) => (! [X:int,B:int,W:nat]: (((((ring_1_of_int @ A) @ X) = (((power_power @ A) @ ((ring_1_of_int @ A) @ B)) @ W)) = (X = (((power_power @ int) @ B) @ W))))))))).
% 2.64/1.76 thf(fact_21_numeral__power__eq__nat__cancel__iff, axiom, (! [X:num,N:nat,Y:int]: ((((((power_power @ nat) @ ((numeral_numeral @ nat) @ X)) @ N) = (nat2 @ Y)) = ((((power_power @ int) @ ((numeral_numeral @ int) @ X)) @ N) = Y))))).
% 2.64/1.76 thf(fact_22_nat__eq__numeral__power__cancel__iff, axiom, (! [Y:int,X:num,N:nat]: ((((nat2 @ Y) = (((power_power @ nat) @ ((numeral_numeral @ nat) @ X)) @ N)) = (Y = (((power_power @ int) @ ((numeral_numeral @ int) @ X)) @ N)))))).
% 2.64/1.76 thf(fact_23_numeral__power__le__nat__cancel__iff, axiom, (! [X:num,N:nat,A2:int]: (((((ord_less_eq @ nat) @ (((power_power @ nat) @ ((numeral_numeral @ nat) @ X)) @ N)) @ (nat2 @ A2)) = (((ord_less_eq @ int) @ (((power_power @ int) @ ((numeral_numeral @ int) @ X)) @ N)) @ A2))))).
% 2.64/1.76 thf(fact_24_nat__le__numeral__power__cancel__iff, axiom, (! [A2:int,X:num,N:nat]: (((((ord_less_eq @ nat) @ (nat2 @ A2)) @ (((power_power @ nat) @ ((numeral_numeral @ nat) @ X)) @ N)) = (((ord_less_eq @ int) @ A2) @ (((power_power @ int) @ ((numeral_numeral @ int) @ X)) @ N)))))).
% 2.64/1.76 thf(fact_25_nat__mono, axiom, (! [X:int,Y:int]: (((((ord_less_eq @ int) @ X) @ Y) => (((ord_less_eq @ nat) @ (nat2 @ X)) @ (nat2 @ Y)))))).
% 2.64/1.76 thf(fact_26_ex__le__of__int, axiom, (! [A:$tType]: (((archim1804426504_field @ A) => (! [X:A]: ((? [Z2:int]: ((((ord_less_eq @ A) @ X) @ ((ring_1_of_int @ A) @ Z2)))))))))).
% 2.64/1.76 thf(fact_27_le__of__int__ceiling, axiom, (! [A:$tType]: (((archim1727834104eiling @ A) => (! [X:A]: ((((ord_less_eq @ A) @ X) @ ((ring_1_of_int @ A) @ ((archimedean_ceiling @ A) @ X))))))))).
% 2.64/1.76 thf(fact_28_ceiling__mono, axiom, (! [A:$tType]: (((archim1727834104eiling @ A) => (! [Y:A,X:A]: (((((ord_less_eq @ A) @ Y) @ X) => (((ord_less_eq @ int) @ ((archimedean_ceiling @ A) @ Y)) @ ((archimedean_ceiling @ A) @ X))))))))).
% 2.64/1.76 thf(fact_29_ceiling__le__iff, axiom, (! [A:$tType]: (((archim1727834104eiling @ A) => (! [X:A,Z:int]: (((((ord_less_eq @ int) @ ((archimedean_ceiling @ A) @ X)) @ Z) = (((ord_less_eq @ A) @ X) @ ((ring_1_of_int @ A) @ Z))))))))).
% 2.64/1.76 thf(fact_30_ceiling__le, axiom, (! [A:$tType]: (((archim1727834104eiling @ A) => (! [X:A,A2:int]: (((((ord_less_eq @ A) @ X) @ ((ring_1_of_int @ A) @ A2)) => (((ord_less_eq @ int) @ ((archimedean_ceiling @ A) @ X)) @ A2)))))))).
% 2.64/1.76 thf(fact_31__092_060open_062log_A2_A1_A_092_060le_062_Alog_A2_A_Ireal__of__int_Ad_J_092_060close_062, axiom, (((ord_less_eq @ real) @ ((log @ ((numeral_numeral @ real) @ (bit0 @ one2))) @ (one_one @ real))) @ ((log @ ((numeral_numeral @ real) @ (bit0 @ one2))) @ ((ring_1_of_int @ real) @ d)))).
% 2.64/1.76 thf(fact_32_semiring__norm_I69_J, axiom, (! [M:num]: ((~ ((((ord_less_eq @ num) @ (bit0 @ M)) @ one2)))))).
% 2.64/1.76 thf(fact_33_semiring__norm_I83_J, axiom, (! [N:num]: ((one2 != (bit0 @ N))))).
% 2.64/1.76 thf(fact_34_semiring__norm_I85_J, axiom, (! [M:num]: (((bit0 @ M) != one2)))).
% 2.64/1.76 thf(fact_35_numeral__le__iff, axiom, (! [A:$tType]: (((linord1659791738miring @ A) => (! [M:num,N:num]: (((((ord_less_eq @ A) @ ((numeral_numeral @ A) @ M)) @ ((numeral_numeral @ A) @ N)) = (((ord_less_eq @ num) @ M) @ N)))))))).
% 2.64/1.76 thf(fact_36__092_060open_062real__of__int_Ad_A_061_A2_Apowr_Alog_A2_A_Ireal__of__int_Ad_J_092_060close_062, axiom, (((ring_1_of_int @ real) @ d) = (((powr @ real) @ ((numeral_numeral @ real) @ (bit0 @ one2))) @ ((log @ ((numeral_numeral @ real) @ (bit0 @ one2))) @ ((ring_1_of_int @ real) @ d))))).
% 2.64/1.76 thf(fact_37_n__def, axiom, (n = (suc @ (nat2 @ ((archimedean_ceiling @ real) @ ((log @ ((numeral_numeral @ real) @ (bit0 @ one2))) @ ((ring_1_of_int @ real) @ d))))))).
% 2.64/1.76 thf(fact_38_semiring__norm_I68_J, axiom, (! [N:num]: ((((ord_less_eq @ num) @ one2) @ N)))).
% 2.64/1.76 thf(fact_39_semiring__norm_I71_J, axiom, (! [M:num,N:num]: (((((ord_less_eq @ num) @ (bit0 @ M)) @ (bit0 @ N)) = (((ord_less_eq @ num) @ M) @ N))))).
% 2.64/1.76 thf(fact_40_semiring__norm_I87_J, axiom, (! [M:num,N:num]: ((((bit0 @ M) = (bit0 @ N)) = (M = N))))).
% 2.64/1.76 thf(fact_41_verit__eq__simplify_I8_J, axiom, (! [X2:num,Y2:num]: ((((bit0 @ X2) = (bit0 @ Y2)) = (X2 = Y2))))).
% 2.64/1.76 thf(fact_42_numeral__eq__iff, axiom, (! [A:$tType]: (((semiring_char_0 @ A) => (! [M:num,N:num]: (((((numeral_numeral @ A) @ M) = ((numeral_numeral @ A) @ N)) = (M = N)))))))).
% 2.64/1.76 thf(fact_43_of__int__eq__1__iff, axiom, (! [A:$tType]: (((ring_char_0 @ A) => (! [Z:int]: (((((ring_1_of_int @ A) @ Z) = (one_one @ A)) = (Z = (one_one @ int))))))))).
% 2.64/1.76 thf(fact_44_mem__Collect__eq, axiom, (! [A:$tType,A2:A,P:(A > $o)]: (((((member @ A) @ A2) @ ((collect @ A) @ P)) = (P @ A2))))).
% 2.64/1.76 thf(fact_45_Collect__mem__eq, axiom, (! [A:$tType,A3:(set @ A)]: ((((collect @ A) @ (^ [X3:A]: ((((member @ A) @ X3) @ A3)))) = A3)))).
% 2.64/1.76 thf(fact_46_Collect__cong, axiom, (! [A:$tType,P:(A > $o),Q:(A > $o)]: (((! [X4:A]: (((P @ X4) = (Q @ X4)))) => (((collect @ A) @ P) = ((collect @ A) @ Q)))))).
% 2.64/1.76 thf(fact_47_ext, axiom, (! [B2:$tType,A:$tType,F:(A > B2),G:(A > B2)]: (((! [X4:A]: (((F @ X4) = (G @ X4)))) => (F = G))))).
% 2.64/1.76 thf(fact_48_of__int__1, axiom, (! [A:$tType]: (((ring_1 @ A) => (((ring_1_of_int @ A) @ (one_one @ int)) = (one_one @ A)))))).
% 2.64/1.76 thf(fact_49_ceiling__one, axiom, (! [A:$tType]: (((archim1727834104eiling @ A) => (((archimedean_ceiling @ A) @ (one_one @ A)) = (one_one @ int)))))).
% 2.64/1.76 thf(fact_50_numeral__eq__one__iff, axiom, (! [A:$tType]: (((semiring_char_0 @ A) => (! [N:num]: (((((numeral_numeral @ A) @ N) = (one_one @ A)) = (N = one2)))))))).
% 2.64/1.76 thf(fact_51_one__eq__numeral__iff, axiom, (! [A:$tType]: (((semiring_char_0 @ A) => (! [N:num]: ((((one_one @ A) = ((numeral_numeral @ A) @ N)) = (one2 = N)))))))).
% 2.64/1.76 thf(fact_52_of__int__1__le__iff, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [Z:int]: (((((ord_less_eq @ A) @ (one_one @ A)) @ ((ring_1_of_int @ A) @ Z)) = (((ord_less_eq @ int) @ (one_one @ int)) @ Z)))))))).
% 2.64/1.76 thf(fact_53_of__int__le__1__iff, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [Z:int]: (((((ord_less_eq @ A) @ ((ring_1_of_int @ A) @ Z)) @ (one_one @ A)) = (((ord_less_eq @ int) @ Z) @ (one_one @ int))))))))).
% 2.64/1.76 thf(fact_54_ceiling__le__one, axiom, (! [A:$tType]: (((archim1727834104eiling @ A) => (! [X:A]: (((((ord_less_eq @ int) @ ((archimedean_ceiling @ A) @ X)) @ (one_one @ int)) = (((ord_less_eq @ A) @ X) @ (one_one @ A))))))))).
% 2.64/1.76 thf(fact_55_numeral__le__one__iff, axiom, (! [A:$tType]: (((linord1659791738miring @ A) => (! [N:num]: (((((ord_less_eq @ A) @ ((numeral_numeral @ A) @ N)) @ (one_one @ A)) = (((ord_less_eq @ num) @ N) @ one2)))))))).
% 2.64/1.76 thf(fact_56_complete__real, axiom, (! [S:(set @ real)]: (((? [X5:real]: ((((member @ real) @ X5) @ S))) => ((? [Z3:real]: ((! [X4:real]: (((((member @ real) @ X4) @ S) => (((ord_less_eq @ real) @ X4) @ Z3)))))) => (? [Y3:real]: (((! [X5:real]: (((((member @ real) @ X5) @ S) => (((ord_less_eq @ real) @ X5) @ Y3)))) & (! [Z3:real]: (((! [X4:real]: (((((member @ real) @ X4) @ S) => (((ord_less_eq @ real) @ X4) @ Z3)))) => (((ord_less_eq @ real) @ Y3) @ Z3)))))))))))).
% 2.64/1.76 thf(fact_57_le__numeral__extra_I4_J, axiom, (! [A:$tType]: (((linord1659791738miring @ A) => (((ord_less_eq @ A) @ (one_one @ A)) @ (one_one @ A)))))).
% 2.64/1.76 thf(fact_58_one__le__numeral, axiom, (! [A:$tType]: (((linord1659791738miring @ A) => (! [N:num]: ((((ord_less_eq @ A) @ (one_one @ A)) @ ((numeral_numeral @ A) @ N)))))))).
% 2.64/1.76 thf(fact_59_numeral__One, axiom, (! [A:$tType]: (((numeral @ A) => (((numeral_numeral @ A) @ one2) = (one_one @ A)))))).
% 2.64/1.76 thf(fact_60_verit__la__disequality, axiom, (! [A:$tType]: (((linorder @ A) => (! [A2:A,B:A]: (((A2 = B) | ((~ ((((ord_less_eq @ A) @ A2) @ B))) | (~ ((((ord_less_eq @ A) @ B) @ A2))))))))))).
% 2.64/1.76 thf(fact_61_le__num__One__iff, axiom, (! [X:num]: (((((ord_less_eq @ num) @ X) @ one2) = (X = one2))))).
% 2.64/1.76 thf(fact_62_two__realpow__ge__one, axiom, (! [N:nat]: ((((ord_less_eq @ real) @ (one_one @ real)) @ (((power_power @ real) @ ((numeral_numeral @ real) @ (bit0 @ one2))) @ N))))).
% 2.64/1.76 thf(fact_63_verit__la__generic, axiom, (! [A2:int,X:int]: (((((ord_less_eq @ int) @ A2) @ X) | ((A2 = X) | (((ord_less_eq @ int) @ X) @ A2)))))).
% 2.64/1.76 thf(fact_64_verit__eq__simplify_I10_J, axiom, (! [X2:num]: ((one2 != (bit0 @ X2))))).
% 2.64/1.76 thf(fact_65_nat__numeral__as__int, axiom, ((numeral_numeral @ nat) = (^ [I:num]: ((nat2 @ ((numeral_numeral @ int) @ I)))))).
% 2.64/1.76 thf(fact_66__092_060open_0622_Apowr_Areal_A_Inat_A_092_060lceil_062log_A2_A_Ireal__of__int_Ad_J_092_060rceil_062_J_A_061_Areal__of__int_A_I2_A_094_Anat_A_092_060lceil_062log_A2_A_Ireal__of__int_Ad_J_092_060rceil_062_J_092_060close_062, axiom, ((((powr @ real) @ ((numeral_numeral @ real) @ (bit0 @ one2))) @ ((semiring_1_of_nat @ real) @ (nat2 @ ((archimedean_ceiling @ real) @ ((log @ ((numeral_numeral @ real) @ (bit0 @ one2))) @ ((ring_1_of_int @ real) @ d)))))) = ((ring_1_of_int @ real) @ (((power_power @ int) @ ((numeral_numeral @ int) @ (bit0 @ one2))) @ (nat2 @ ((archimedean_ceiling @ real) @ ((log @ ((numeral_numeral @ real) @ (bit0 @ one2))) @ ((ring_1_of_int @ real) @ d)))))))).
% 2.64/1.76 thf(fact_67__092_060open_0622_Apowr_Alog_A2_A_Ireal__of__int_Ad_J_A_092_060le_062_A2_Apowr_Areal_A_Inat_A_092_060lceil_062log_A2_A_Ireal__of__int_Ad_J_092_060rceil_062_J_092_060close_062, axiom, (((ord_less_eq @ real) @ (((powr @ real) @ ((numeral_numeral @ real) @ (bit0 @ one2))) @ ((log @ ((numeral_numeral @ real) @ (bit0 @ one2))) @ ((ring_1_of_int @ real) @ d)))) @ (((powr @ real) @ ((numeral_numeral @ real) @ (bit0 @ one2))) @ ((semiring_1_of_nat @ real) @ (nat2 @ ((archimedean_ceiling @ real) @ ((log @ ((numeral_numeral @ real) @ (bit0 @ one2))) @ ((ring_1_of_int @ real) @ d)))))))).
% 2.64/1.76 thf(fact_68_numeral__powr__numeral__real, axiom, (! [M:num,N:num]: (((((powr @ real) @ ((numeral_numeral @ real) @ M)) @ ((numeral_numeral @ real) @ N)) = (((power_power @ real) @ ((numeral_numeral @ real) @ M)) @ ((numeral_numeral @ nat) @ N)))))).
% 2.64/1.76 thf(fact_69_powr__one__eq__one, axiom, (! [A:$tType]: (((ln @ A) => (! [A2:A]: (((((powr @ A) @ (one_one @ A)) @ A2) = (one_one @ A)))))))).
% 2.64/1.76 thf(fact_70_self__le__ge2__pow, axiom, (! [K:nat,M:nat]: (((((ord_less_eq @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ K) => (((ord_less_eq @ nat) @ M) @ (((power_power @ nat) @ K) @ M)))))).
% 2.64/1.76 thf(fact_71_power2__nat__le__eq__le, axiom, (! [M:nat,N:nat]: (((((ord_less_eq @ nat) @ (((power_power @ nat) @ M) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))) @ (((power_power @ nat) @ N) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))) = (((ord_less_eq @ nat) @ M) @ N))))).
% 2.64/1.76 thf(fact_72_power2__nat__le__imp__le, axiom, (! [M:nat,N:nat]: (((((ord_less_eq @ nat) @ (((power_power @ nat) @ M) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))) @ N) => (((ord_less_eq @ nat) @ M) @ N))))).
% 2.64/1.76 thf(fact_73_Suc__le__mono, axiom, (! [N:nat,M:nat]: (((((ord_less_eq @ nat) @ (suc @ N)) @ (suc @ M)) = (((ord_less_eq @ nat) @ N) @ M))))).
% 2.64/1.76 thf(fact_74_power__one__right, axiom, (! [A:$tType]: (((monoid_mult @ A) => (! [A2:A]: (((((power_power @ A) @ A2) @ (one_one @ nat)) = A2))))))).
% 2.64/1.76 thf(fact_75_nat_Oinject, axiom, (! [X2:nat,Y2:nat]: ((((suc @ X2) = (suc @ Y2)) = (X2 = Y2))))).
% 2.64/1.76 thf(fact_76_old_Onat_Oinject, axiom, (! [Nat:nat,Nat2:nat]: ((((suc @ Nat) = (suc @ Nat2)) = (Nat = Nat2))))).
% 2.64/1.76 thf(fact_77_of__nat__eq__iff, axiom, (! [A:$tType]: (((semiring_char_0 @ A) => (! [M:nat,N:nat]: (((((semiring_1_of_nat @ A) @ M) = ((semiring_1_of_nat @ A) @ N)) = (M = N)))))))).
% 2.64/1.76 thf(fact_78_power__one, axiom, (! [A:$tType]: (((monoid_mult @ A) => (! [N:nat]: (((((power_power @ A) @ (one_one @ A)) @ N) = (one_one @ A)))))))).
% 2.64/1.76 thf(fact_79_of__nat__eq__1__iff, axiom, (! [A:$tType]: (((semiring_char_0 @ A) => (! [N:nat]: (((((semiring_1_of_nat @ A) @ N) = (one_one @ A)) = (N = (one_one @ nat))))))))).
% 2.64/1.76 thf(fact_80_of__nat__1__eq__iff, axiom, (! [A:$tType]: (((semiring_char_0 @ A) => (! [N:nat]: ((((one_one @ A) = ((semiring_1_of_nat @ A) @ N)) = (N = (one_one @ nat))))))))).
% 2.64/1.76 thf(fact_81_of__nat__1, axiom, (! [A:$tType]: (((semiring_1 @ A) => (((semiring_1_of_nat @ A) @ (one_one @ nat)) = (one_one @ A)))))).
% 2.64/1.76 thf(fact_82_of__int__of__nat__eq, axiom, (! [A:$tType]: (((ring_1 @ A) => (! [N:nat]: ((((ring_1_of_int @ A) @ ((semiring_1_of_nat @ int) @ N)) = ((semiring_1_of_nat @ A) @ N)))))))).
% 2.64/1.76 thf(fact_83_ceiling__of__nat, axiom, (! [A:$tType]: (((archim1727834104eiling @ A) => (! [N:nat]: ((((archimedean_ceiling @ A) @ ((semiring_1_of_nat @ A) @ N)) = ((semiring_1_of_nat @ int) @ N)))))))).
% 2.64/1.76 thf(fact_84_of__nat__le__iff, axiom, (! [A:$tType]: (((linord1659791738miring @ A) => (! [M:nat,N:nat]: (((((ord_less_eq @ A) @ ((semiring_1_of_nat @ A) @ M)) @ ((semiring_1_of_nat @ A) @ N)) = (((ord_less_eq @ nat) @ M) @ N)))))))).
% 2.64/1.76 thf(fact_85_of__nat__numeral, axiom, (! [A:$tType]: (((semiring_1 @ A) => (! [N:num]: ((((semiring_1_of_nat @ A) @ ((numeral_numeral @ nat) @ N)) = ((numeral_numeral @ A) @ N)))))))).
% 2.64/1.76 thf(fact_86_of__nat__power__eq__of__nat__cancel__iff, axiom, (! [A:$tType]: (((semiring_char_0 @ A) => (! [X:nat,B:nat,W:nat]: (((((semiring_1_of_nat @ A) @ X) = (((power_power @ A) @ ((semiring_1_of_nat @ A) @ B)) @ W)) = (X = (((power_power @ nat) @ B) @ W))))))))).
% 2.64/1.76 thf(fact_87_of__nat__eq__of__nat__power__cancel__iff, axiom, (! [A:$tType]: (((semiring_char_0 @ A) => (! [B:nat,W:nat,X:nat]: ((((((power_power @ A) @ ((semiring_1_of_nat @ A) @ B)) @ W) = ((semiring_1_of_nat @ A) @ X)) = ((((power_power @ nat) @ B) @ W) = X)))))))).
% 2.64/1.76 thf(fact_88_of__nat__power, axiom, (! [A:$tType]: (((semiring_1 @ A) => (! [M:nat,N:nat]: ((((semiring_1_of_nat @ A) @ (((power_power @ nat) @ M) @ N)) = (((power_power @ A) @ ((semiring_1_of_nat @ A) @ M)) @ N)))))))).
% 2.64/1.76 thf(fact_89_Suc__1, axiom, ((suc @ (one_one @ nat)) = ((numeral_numeral @ nat) @ (bit0 @ one2)))).
% 2.64/1.76 thf(fact_90_of__nat__power__le__of__nat__cancel__iff, axiom, (! [A:$tType]: (((linordered_semidom @ A) => (! [X:nat,B:nat,W:nat]: (((((ord_less_eq @ A) @ ((semiring_1_of_nat @ A) @ X)) @ (((power_power @ A) @ ((semiring_1_of_nat @ A) @ B)) @ W)) = (((ord_less_eq @ nat) @ X) @ (((power_power @ nat) @ B) @ W))))))))).
% 2.64/1.76 thf(fact_91_of__nat__le__of__nat__power__cancel__iff, axiom, (! [A:$tType]: (((linordered_semidom @ A) => (! [B:nat,W:nat,X:nat]: (((((ord_less_eq @ A) @ (((power_power @ A) @ ((semiring_1_of_nat @ A) @ B)) @ W)) @ ((semiring_1_of_nat @ A) @ X)) = (((ord_less_eq @ nat) @ (((power_power @ nat) @ B) @ W)) @ X)))))))).
% 2.64/1.76 thf(fact_92_real__of__nat__eq__numeral__power__cancel__iff, axiom, (! [A:$tType]: (((semiring_char_0 @ A) => (! [Y:nat,X:num,N:nat]: (((((semiring_1_of_nat @ A) @ Y) = (((power_power @ A) @ ((numeral_numeral @ A) @ X)) @ N)) = (Y = (((power_power @ nat) @ ((numeral_numeral @ nat) @ X)) @ N))))))))).
% 2.64/1.76 thf(fact_93_numeral__power__eq__of__nat__cancel__iff, axiom, (! [A:$tType]: (((semiring_char_0 @ A) => (! [X:num,N:nat,Y:nat]: ((((((power_power @ A) @ ((numeral_numeral @ A) @ X)) @ N) = ((semiring_1_of_nat @ A) @ Y)) = ((((power_power @ nat) @ ((numeral_numeral @ nat) @ X)) @ N) = Y)))))))).
% 2.64/1.76 thf(fact_94_numeral__le__real__of__nat__iff, axiom, (! [N:num,M:nat]: (((((ord_less_eq @ real) @ ((numeral_numeral @ real) @ N)) @ ((semiring_1_of_nat @ real) @ M)) = (((ord_less_eq @ nat) @ ((numeral_numeral @ nat) @ N)) @ M))))).
% 2.64/1.76 thf(fact_95_nat__ceiling__le__eq, axiom, (! [X:real,A2:nat]: (((((ord_less_eq @ nat) @ (nat2 @ ((archimedean_ceiling @ real) @ X))) @ A2) = (((ord_less_eq @ real) @ X) @ ((semiring_1_of_nat @ real) @ A2)))))).
% 2.64/1.76 thf(fact_96_of__nat__le__numeral__power__cancel__iff, axiom, (! [A:$tType]: (((linordered_semidom @ A) => (! [X:nat,I2:num,N:nat]: (((((ord_less_eq @ A) @ ((semiring_1_of_nat @ A) @ X)) @ (((power_power @ A) @ ((numeral_numeral @ A) @ I2)) @ N)) = (((ord_less_eq @ nat) @ X) @ (((power_power @ nat) @ ((numeral_numeral @ nat) @ I2)) @ N))))))))).
% 2.64/1.76 thf(fact_97_numeral__power__le__of__nat__cancel__iff, axiom, (! [A:$tType]: (((linordered_semidom @ A) => (! [I2:num,N:nat,X:nat]: (((((ord_less_eq @ A) @ (((power_power @ A) @ ((numeral_numeral @ A) @ I2)) @ N)) @ ((semiring_1_of_nat @ A) @ X)) = (((ord_less_eq @ nat) @ (((power_power @ nat) @ ((numeral_numeral @ nat) @ I2)) @ N)) @ X)))))))).
% 2.64/1.76 thf(fact_98_nat__one__as__int, axiom, ((one_one @ nat) = (nat2 @ (one_one @ int)))).
% 2.64/1.76 thf(fact_99_real__arch__simple, axiom, (! [A:$tType]: (((archim1804426504_field @ A) => (! [X:A]: ((? [N3:nat]: ((((ord_less_eq @ A) @ X) @ ((semiring_1_of_nat @ A) @ N3)))))))))).
% 2.64/1.76 thf(fact_100_of__nat__mono, axiom, (! [A:$tType]: (((linord1659791738miring @ A) => (! [I2:nat,J:nat]: (((((ord_less_eq @ nat) @ I2) @ J) => (((ord_less_eq @ A) @ ((semiring_1_of_nat @ A) @ I2)) @ ((semiring_1_of_nat @ A) @ J))))))))).
% 2.64/1.76 thf(fact_101_numerals_I1_J, axiom, (((numeral_numeral @ nat) @ one2) = (one_one @ nat))).
% 2.64/1.76 thf(fact_102_of__nat__ceiling, axiom, (! [A:$tType]: (((archim1727834104eiling @ A) => (! [R:A]: ((((ord_less_eq @ A) @ R) @ ((semiring_1_of_nat @ A) @ (nat2 @ ((archimedean_ceiling @ A) @ R)))))))))).
% 2.64/1.76 thf(fact_103_Suc__inject, axiom, (! [X:nat,Y:nat]: ((((suc @ X) = (suc @ Y)) => (X = Y))))).
% 2.64/1.76 thf(fact_104_n__not__Suc__n, axiom, (! [N:nat]: ((N != (suc @ N))))).
% 2.64/1.76 thf(fact_105_Nat_Oex__has__greatest__nat, axiom, (! [P:(nat > $o),K:nat,B:nat]: (((P @ K) => ((! [Y3:nat]: (((P @ Y3) => (((ord_less_eq @ nat) @ Y3) @ B)))) => (? [X4:nat]: (((P @ X4) & (! [Y4:nat]: (((P @ Y4) => (((ord_less_eq @ nat) @ Y4) @ X4)))))))))))).
% 2.64/1.76 thf(fact_106_nat__le__linear, axiom, (! [M:nat,N:nat]: (((((ord_less_eq @ nat) @ M) @ N) | (((ord_less_eq @ nat) @ N) @ M))))).
% 2.64/1.76 thf(fact_107_le__antisym, axiom, (! [M:nat,N:nat]: (((((ord_less_eq @ nat) @ M) @ N) => ((((ord_less_eq @ nat) @ N) @ M) => (M = N)))))).
% 2.64/1.76 thf(fact_108_eq__imp__le, axiom, (! [M:nat,N:nat]: (((M = N) => (((ord_less_eq @ nat) @ M) @ N))))).
% 2.64/1.76 thf(fact_109_le__trans, axiom, (! [I2:nat,J:nat,K:nat]: (((((ord_less_eq @ nat) @ I2) @ J) => ((((ord_less_eq @ nat) @ J) @ K) => (((ord_less_eq @ nat) @ I2) @ K)))))).
% 2.64/1.76 thf(fact_110_le__refl, axiom, (! [N:nat]: ((((ord_less_eq @ nat) @ N) @ N)))).
% 2.64/1.76 thf(fact_111_real__nat__ceiling__ge, axiom, (! [X:real]: ((((ord_less_eq @ real) @ X) @ ((semiring_1_of_nat @ real) @ (nat2 @ ((archimedean_ceiling @ real) @ X))))))).
% 2.64/1.76 thf(fact_112_powr__powr__swap, axiom, (! [X:real,A2:real,B:real]: (((((powr @ real) @ (((powr @ real) @ X) @ A2)) @ B) = (((powr @ real) @ (((powr @ real) @ X) @ B)) @ A2))))).
% 2.64/1.76 thf(fact_113_log2__of__power__eq, axiom, (! [M:nat,N:nat]: (((M = (((power_power @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ N)) => (((semiring_1_of_nat @ real) @ N) = ((log @ ((numeral_numeral @ real) @ (bit0 @ one2))) @ ((semiring_1_of_nat @ real) @ M))))))).
% 2.64/1.76 thf(fact_114_le__log2__of__power, axiom, (! [N:nat,M:nat]: (((((ord_less_eq @ nat) @ (((power_power @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ N)) @ M) => (((ord_less_eq @ real) @ ((semiring_1_of_nat @ real) @ N)) @ ((log @ ((numeral_numeral @ real) @ (bit0 @ one2))) @ ((semiring_1_of_nat @ real) @ M))))))).
% 2.64/1.76 thf(fact_115_transitive__stepwise__le, axiom, (! [M:nat,N:nat,R2:(nat > (nat > $o))]: (((((ord_less_eq @ nat) @ M) @ N) => ((! [X4:nat]: (((R2 @ X4) @ X4))) => ((! [X4:nat,Y3:nat,Z2:nat]: ((((R2 @ X4) @ Y3) => (((R2 @ Y3) @ Z2) => ((R2 @ X4) @ Z2))))) => ((! [N3:nat]: (((R2 @ N3) @ (suc @ N3)))) => ((R2 @ M) @ N)))))))).
% 2.64/1.76 thf(fact_116_nat__induct__at__least, axiom, (! [M:nat,N:nat,P:(nat > $o)]: (((((ord_less_eq @ nat) @ M) @ N) => ((P @ M) => ((! [N3:nat]: (((((ord_less_eq @ nat) @ M) @ N3) => ((P @ N3) => (P @ (suc @ N3)))))) => (P @ N))))))).
% 2.64/1.76 thf(fact_117_full__nat__induct, axiom, (! [P:(nat > $o),N:nat]: (((! [N3:nat]: (((! [M2:nat]: (((((ord_less_eq @ nat) @ (suc @ M2)) @ N3) => (P @ M2)))) => (P @ N3)))) => (P @ N))))).
% 2.64/1.76 thf(fact_118_not__less__eq__eq, axiom, (! [M:nat,N:nat]: (((~ ((((ord_less_eq @ nat) @ M) @ N))) = (((ord_less_eq @ nat) @ (suc @ N)) @ M))))).
% 2.64/1.76 thf(fact_119_Suc__n__not__le__n, axiom, (! [N:nat]: ((~ ((((ord_less_eq @ nat) @ (suc @ N)) @ N)))))).
% 2.64/1.76 thf(fact_120_le__Suc__eq, axiom, (! [M:nat,N:nat]: (((((ord_less_eq @ nat) @ M) @ (suc @ N)) = ((((ord_less_eq @ nat) @ M) @ N) | (M = (suc @ N))))))).
% 2.64/1.76 thf(fact_121_Suc__le__D, axiom, (! [N:nat,M3:nat]: (((((ord_less_eq @ nat) @ (suc @ N)) @ M3) => (? [M4:nat]: ((M3 = (suc @ M4)))))))).
% 2.64/1.76 thf(fact_122_le__SucI, axiom, (! [M:nat,N:nat]: (((((ord_less_eq @ nat) @ M) @ N) => (((ord_less_eq @ nat) @ M) @ (suc @ N)))))).
% 2.64/1.76 thf(fact_123_le__SucE, axiom, (! [M:nat,N:nat]: (((((ord_less_eq @ nat) @ M) @ (suc @ N)) => ((~ ((((ord_less_eq @ nat) @ M) @ N))) => (M = (suc @ N))))))).
% 2.64/1.76 thf(fact_124_Suc__leD, axiom, (! [M:nat,N:nat]: (((((ord_less_eq @ nat) @ (suc @ M)) @ N) => (((ord_less_eq @ nat) @ M) @ N))))).
% 2.64/1.76 thf(fact_125_one__le__power, axiom, (! [A:$tType]: (((linordered_semidom @ A) => (! [A2:A,N:nat]: (((((ord_less_eq @ A) @ (one_one @ A)) @ A2) => (((ord_less_eq @ A) @ (one_one @ A)) @ (((power_power @ A) @ A2) @ N))))))))).
% 2.64/1.76 thf(fact_126_lift__Suc__antimono__le, axiom, (! [A:$tType]: (((order @ A) => (! [F:(nat > A),N:nat,N4:nat]: (((! [N3:nat]: ((((ord_less_eq @ A) @ (F @ (suc @ N3))) @ (F @ N3)))) => ((((ord_less_eq @ nat) @ N) @ N4) => (((ord_less_eq @ A) @ (F @ N4)) @ (F @ N)))))))))).
% 2.64/1.76 thf(fact_127_lift__Suc__mono__le, axiom, (! [A:$tType]: (((order @ A) => (! [F:(nat > A),N:nat,N4:nat]: (((! [N3:nat]: ((((ord_less_eq @ A) @ (F @ N3)) @ (F @ (suc @ N3))))) => ((((ord_less_eq @ nat) @ N) @ N4) => (((ord_less_eq @ A) @ (F @ N)) @ (F @ N4)))))))))).
% 2.64/1.76 thf(fact_128_powr__mono, axiom, (! [A2:real,B:real,X:real]: (((((ord_less_eq @ real) @ A2) @ B) => ((((ord_less_eq @ real) @ (one_one @ real)) @ X) => (((ord_less_eq @ real) @ (((powr @ real) @ X) @ A2)) @ (((powr @ real) @ X) @ B))))))).
% 2.64/1.76 thf(fact_129_power__increasing, axiom, (! [A:$tType]: (((linordered_semidom @ A) => (! [N:nat,N5:nat,A2:A]: (((((ord_less_eq @ nat) @ N) @ N5) => ((((ord_less_eq @ A) @ (one_one @ A)) @ A2) => (((ord_less_eq @ A) @ (((power_power @ A) @ A2) @ N)) @ (((power_power @ A) @ A2) @ N5)))))))))).
% 2.64/1.76 thf(fact_130_one__power2, axiom, (! [A:$tType]: (((semiring_1 @ A) => ((((power_power @ A) @ (one_one @ A)) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) = (one_one @ A)))))).
% 2.64/1.76 thf(fact_131_dbl__simps_I3_J, axiom, (! [A:$tType]: (((neg_numeral @ A) => (((neg_numeral_dbl @ A) @ (one_one @ A)) = ((numeral_numeral @ A) @ (bit0 @ one2))))))).
% 2.64/1.76 thf(fact_132_power__numeral, axiom, (! [A:$tType]: (((semiring_numeral @ A) => (! [K:num,L:num]: (((((power_power @ A) @ ((numeral_numeral @ A) @ K)) @ ((numeral_numeral @ nat) @ L)) = ((numeral_numeral @ A) @ ((pow @ K) @ L))))))))).
% 2.64/1.76 thf(fact_133_order__refl, axiom, (! [A:$tType]: (((preorder @ A) => (! [X:A]: ((((ord_less_eq @ A) @ X) @ X))))))).
% 2.64/1.76 thf(fact_134_powr__numeral, axiom, (! [X:real,N:num]: (((((ord_less_eq @ real) @ (zero_zero @ real)) @ X) => ((((powr @ real) @ X) @ ((numeral_numeral @ real) @ N)) = (((power_power @ real) @ X) @ ((numeral_numeral @ nat) @ N))))))).
% 2.64/1.76 thf(fact_135_power2__eq__iff__nonneg, axiom, (! [A:$tType]: (((linordered_semidom @ A) => (! [X:A,Y:A]: (((((ord_less_eq @ A) @ (zero_zero @ A)) @ X) => ((((ord_less_eq @ A) @ (zero_zero @ A)) @ Y) => (((((power_power @ A) @ X) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) = (((power_power @ A) @ Y) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))) = (X = Y)))))))))).
% 2.64/1.76 thf(fact_136_le0, axiom, (! [N:nat]: ((((ord_less_eq @ nat) @ (zero_zero @ nat)) @ N)))).
% 2.64/1.76 thf(fact_137_bot__nat__0_Oextremum, axiom, (! [A2:nat]: ((((ord_less_eq @ nat) @ (zero_zero @ nat)) @ A2)))).
% 2.64/1.76 thf(fact_138_d_I2_J, axiom, (((ord_less @ int) @ (zero_zero @ int)) @ d)).
% 2.64/1.76 thf(fact_139_le__zero__eq, axiom, (! [A:$tType]: (((canoni770627133id_add @ A) => (! [N:A]: (((((ord_less_eq @ A) @ N) @ (zero_zero @ A)) = (N = (zero_zero @ A))))))))).
% 2.64/1.76 thf(fact_140_of__nat__0, axiom, (! [A:$tType]: (((semiring_1 @ A) => (((semiring_1_of_nat @ A) @ (zero_zero @ nat)) = (zero_zero @ A)))))).
% 2.64/1.76 thf(fact_141_of__nat__0__eq__iff, axiom, (! [A:$tType]: (((semiring_char_0 @ A) => (! [N:nat]: ((((zero_zero @ A) = ((semiring_1_of_nat @ A) @ N)) = ((zero_zero @ nat) = N)))))))).
% 2.64/1.76 thf(fact_142_of__nat__eq__0__iff, axiom, (! [A:$tType]: (((semiring_char_0 @ A) => (! [M:nat]: (((((semiring_1_of_nat @ A) @ M) = (zero_zero @ A)) = (M = (zero_zero @ nat))))))))).
% 2.64/1.76 thf(fact_143_power__Suc0__right, axiom, (! [A:$tType]: (((monoid_mult @ A) => (! [A2:A]: (((((power_power @ A) @ A2) @ (suc @ (zero_zero @ nat))) = A2))))))).
% 2.64/1.76 thf(fact_144_of__int__0, axiom, (! [A:$tType]: (((ring_1 @ A) => (((ring_1_of_int @ A) @ (zero_zero @ int)) = (zero_zero @ A)))))).
% 2.64/1.76 thf(fact_145_of__int__0__eq__iff, axiom, (! [A:$tType]: (((ring_char_0 @ A) => (! [Z:int]: ((((zero_zero @ A) = ((ring_1_of_int @ A) @ Z)) = (Z = (zero_zero @ int))))))))).
% 2.64/1.76 thf(fact_146_of__int__eq__0__iff, axiom, (! [A:$tType]: (((ring_char_0 @ A) => (! [Z:int]: (((((ring_1_of_int @ A) @ Z) = (zero_zero @ A)) = (Z = (zero_zero @ int))))))))).
% 2.64/1.76 thf(fact_147_powr__eq__0__iff, axiom, (! [A:$tType]: (((ln @ A) => (! [W:A,Z:A]: ((((((powr @ A) @ W) @ Z) = (zero_zero @ A)) = (W = (zero_zero @ A))))))))).
% 2.64/1.76 thf(fact_148_powr__0, axiom, (! [A:$tType]: (((ln @ A) => (! [Z:A]: (((((powr @ A) @ (zero_zero @ A)) @ Z) = (zero_zero @ A)))))))).
% 2.64/1.76 thf(fact_149_ceiling__zero, axiom, (! [A:$tType]: (((archim1727834104eiling @ A) => (((archimedean_ceiling @ A) @ (zero_zero @ A)) = (zero_zero @ int)))))).
% 2.64/1.76 thf(fact_150_nat__le__0, axiom, (! [Z:int]: (((((ord_less_eq @ int) @ Z) @ (zero_zero @ int)) => ((nat2 @ Z) = (zero_zero @ nat)))))).
% 2.64/1.76 thf(fact_151_nat__0__iff, axiom, (! [I2:int]: ((((nat2 @ I2) = (zero_zero @ nat)) = (((ord_less_eq @ int) @ I2) @ (zero_zero @ int)))))).
% 2.64/1.76 thf(fact_152_power__Suc__0, axiom, (! [N:nat]: (((((power_power @ nat) @ (suc @ (zero_zero @ nat))) @ N) = (suc @ (zero_zero @ nat)))))).
% 2.64/1.76 thf(fact_153_nat__power__eq__Suc__0__iff, axiom, (! [X:nat,M:nat]: ((((((power_power @ nat) @ X) @ M) = (suc @ (zero_zero @ nat))) = ((M = (zero_zero @ nat)) | (X = (suc @ (zero_zero @ nat)))))))).
% 2.64/1.76 thf(fact_154_nat__int, axiom, (! [N:nat]: (((nat2 @ ((semiring_1_of_nat @ int) @ N)) = N)))).
% 2.64/1.76 thf(fact_155_dbl__simps_I2_J, axiom, (! [A:$tType]: (((neg_numeral @ A) => (((neg_numeral_dbl @ A) @ (zero_zero @ A)) = (zero_zero @ A)))))).
% 2.64/1.76 thf(fact_156_of__nat__le__0__iff, axiom, (! [A:$tType]: (((linord1659791738miring @ A) => (! [M:nat]: (((((ord_less_eq @ A) @ ((semiring_1_of_nat @ A) @ M)) @ (zero_zero @ A)) = (M = (zero_zero @ nat))))))))).
% 2.64/1.76 thf(fact_157_power__0__Suc, axiom, (! [A:$tType]: (((semiring_1 @ A) => (! [N:nat]: (((((power_power @ A) @ (zero_zero @ A)) @ (suc @ N)) = (zero_zero @ A)))))))).
% 2.64/1.76 thf(fact_158_power__zero__numeral, axiom, (! [A:$tType]: (((semiring_1 @ A) => (! [K:num]: (((((power_power @ A) @ (zero_zero @ A)) @ ((numeral_numeral @ nat) @ K)) = (zero_zero @ A)))))))).
% 2.64/1.76 thf(fact_159_powr__zero__eq__one, axiom, (! [A:$tType]: (((ln @ A) => (! [X:A]: ((((X = (zero_zero @ A)) => ((((powr @ A) @ X) @ (zero_zero @ A)) = (zero_zero @ A))) & ((X != (zero_zero @ A)) => ((((powr @ A) @ X) @ (zero_zero @ A)) = (one_one @ A)))))))))).
% 2.64/1.76 thf(fact_160_powr__nonneg__iff, axiom, (! [A2:real,X:real]: (((((ord_less_eq @ real) @ (((powr @ real) @ A2) @ X)) @ (zero_zero @ real)) = (A2 = (zero_zero @ real)))))).
% 2.64/1.76 thf(fact_161_nat__1, axiom, ((nat2 @ (one_one @ int)) = (suc @ (zero_zero @ nat)))).
% 2.64/1.76 thf(fact_162_int__eq__iff__numeral, axiom, (! [M:nat,V:num]: (((((semiring_1_of_nat @ int) @ M) = ((numeral_numeral @ int) @ V)) = (M = ((numeral_numeral @ nat) @ V)))))).
% 2.64/1.76 thf(fact_163_log__one, axiom, (! [A2:real]: ((((log @ A2) @ (one_one @ real)) = (zero_zero @ real))))).
% 2.64/1.76 thf(fact_164_int__nat__eq, axiom, (! [Z:int]: ((((((ord_less_eq @ int) @ (zero_zero @ int)) @ Z) => (((semiring_1_of_nat @ int) @ (nat2 @ Z)) = Z)) & ((~ ((((ord_less_eq @ int) @ (zero_zero @ int)) @ Z))) => (((semiring_1_of_nat @ int) @ (nat2 @ Z)) = (zero_zero @ int))))))).
% 2.64/1.76 thf(fact_165_dbl__simps_I5_J, axiom, (! [A:$tType]: (((neg_numeral @ A) => (! [K:num]: ((((neg_numeral_dbl @ A) @ ((numeral_numeral @ A) @ K)) = ((numeral_numeral @ A) @ (bit0 @ K))))))))).
% 2.64/1.76 thf(fact_166_of__int__0__le__iff, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [Z:int]: (((((ord_less_eq @ A) @ (zero_zero @ A)) @ ((ring_1_of_int @ A) @ Z)) = (((ord_less_eq @ int) @ (zero_zero @ int)) @ Z)))))))).
% 2.64/1.76 thf(fact_167_of__int__le__0__iff, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [Z:int]: (((((ord_less_eq @ A) @ ((ring_1_of_int @ A) @ Z)) @ (zero_zero @ A)) = (((ord_less_eq @ int) @ Z) @ (zero_zero @ int))))))))).
% 2.64/1.76 thf(fact_168_ceiling__le__zero, axiom, (! [A:$tType]: (((archim1727834104eiling @ A) => (! [X:A]: (((((ord_less_eq @ int) @ ((archimedean_ceiling @ A) @ X)) @ (zero_zero @ int)) = (((ord_less_eq @ A) @ X) @ (zero_zero @ A))))))))).
% 2.64/1.76 thf(fact_169_powr__one, axiom, (! [X:real]: (((((ord_less_eq @ real) @ (zero_zero @ real)) @ X) => ((((powr @ real) @ X) @ (one_one @ real)) = X))))).
% 2.64/1.76 thf(fact_170_powr__one__gt__zero__iff, axiom, (! [X:real]: ((((((powr @ real) @ X) @ (one_one @ real)) = X) = (((ord_less_eq @ real) @ (zero_zero @ real)) @ X))))).
% 2.64/1.76 thf(fact_171_of__nat__nat, axiom, (! [A:$tType]: (((ring_1 @ A) => (! [Z:int]: (((((ord_less_eq @ int) @ (zero_zero @ int)) @ Z) => (((semiring_1_of_nat @ A) @ (nat2 @ Z)) = ((ring_1_of_int @ A) @ Z))))))))).
% 2.64/1.76 thf(fact_172_zero__eq__power2, axiom, (! [A:$tType]: (((semiri134348788visors @ A) => (! [A2:A]: ((((((power_power @ A) @ A2) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) = (zero_zero @ A)) = (A2 = (zero_zero @ A))))))))).
% 2.64/1.76 thf(fact_173_power2__less__eq__zero__iff, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [A2:A]: (((((ord_less_eq @ A) @ (((power_power @ A) @ A2) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))) @ (zero_zero @ A)) = (A2 = (zero_zero @ A))))))))).
% 2.64/1.76 thf(fact_174_int__if, axiom, (! [P:$o,A2:nat,B:nat]: (((P => (((semiring_1_of_nat @ int) @ ((((if @ nat) @ P) @ A2) @ B)) = ((semiring_1_of_nat @ int) @ A2))) & ((~ (P)) => (((semiring_1_of_nat @ int) @ ((((if @ nat) @ P) @ A2) @ B)) = ((semiring_1_of_nat @ int) @ B))))))).
% 2.64/1.76 thf(fact_175_nat__int__comparison_I1_J, axiom, ((^ [Y5:nat,Z4:nat]: ((Y5 = Z4))) = (^ [A4:nat,B3:nat]: ((((semiring_1_of_nat @ int) @ A4) = ((semiring_1_of_nat @ int) @ B3)))))).
% 2.64/1.76 thf(fact_176_int__ops_I1_J, axiom, (((semiring_1_of_nat @ int) @ (zero_zero @ nat)) = (zero_zero @ int))).
% 2.64/1.76 thf(fact_177_zero__le__imp__eq__int, axiom, (! [K:int]: (((((ord_less_eq @ int) @ (zero_zero @ int)) @ K) => (? [N3:nat]: ((K = ((semiring_1_of_nat @ int) @ N3)))))))).
% 2.64/1.76 thf(fact_178_nonneg__int__cases, axiom, (! [K:int]: (((((ord_less_eq @ int) @ (zero_zero @ int)) @ K) => (~ ((! [N3:nat]: ((K != ((semiring_1_of_nat @ int) @ N3)))))))))).
% 2.64/1.76 thf(fact_179_int__int__eq, axiom, (! [M:nat,N:nat]: (((((semiring_1_of_nat @ int) @ M) = ((semiring_1_of_nat @ int) @ N)) = (M = N))))).
% 2.64/1.76 thf(fact_180_nat__eq__iff2, axiom, (! [M:nat,W:int]: (((M = (nat2 @ W)) = (((((ord_less_eq @ int) @ (zero_zero @ int)) @ W) => (W = ((semiring_1_of_nat @ int) @ M))) & ((~ ((((ord_less_eq @ int) @ (zero_zero @ int)) @ W))) => (M = (zero_zero @ nat)))))))).
% 2.64/1.76 thf(fact_181_nat__eq__iff, axiom, (! [W:int,M:nat]: ((((nat2 @ W) = M) = (((((ord_less_eq @ int) @ (zero_zero @ int)) @ W) => (W = ((semiring_1_of_nat @ int) @ M))) & ((~ ((((ord_less_eq @ int) @ (zero_zero @ int)) @ W))) => (M = (zero_zero @ nat)))))))).
% 2.64/1.76 thf(fact_182_zero__reorient, axiom, (! [A:$tType]: (((zero @ A) => (! [X:A]: ((((zero_zero @ A) = X) = (X = (zero_zero @ A))))))))).
% 2.64/1.76 thf(fact_183_zero__le, axiom, (! [A:$tType]: (((canoni770627133id_add @ A) => (! [X:A]: ((((ord_less_eq @ A) @ (zero_zero @ A)) @ X))))))).
% 2.64/1.76 thf(fact_184_nat__zero__as__int, axiom, ((zero_zero @ nat) = (nat2 @ (zero_zero @ int)))).
% 2.64/1.76 thf(fact_185_power__0__left, axiom, (! [A:$tType]: (((semiring_1 @ A) => (! [N:nat]: ((((N = (zero_zero @ nat)) => ((((power_power @ A) @ (zero_zero @ A)) @ N) = (one_one @ A))) & ((N != (zero_zero @ nat)) => ((((power_power @ A) @ (zero_zero @ A)) @ N) = (zero_zero @ A)))))))))).
% 2.64/1.76 thf(fact_186_nat__0__le, axiom, (! [Z:int]: (((((ord_less_eq @ int) @ (zero_zero @ int)) @ Z) => (((semiring_1_of_nat @ int) @ (nat2 @ Z)) = Z))))).
% 2.64/1.76 thf(fact_187_int__eq__iff, axiom, (! [M:nat,Z:int]: (((((semiring_1_of_nat @ int) @ M) = Z) = ((M = (nat2 @ Z)) & (((ord_less_eq @ int) @ (zero_zero @ int)) @ Z)))))).
% 2.64/1.76 thf(fact_188_int__ops_I2_J, axiom, (((semiring_1_of_nat @ int) @ (one_one @ nat)) = (one_one @ int))).
% 2.64/1.76 thf(fact_189_le__numeral__extra_I3_J, axiom, (! [A:$tType]: (((linord1659791738miring @ A) => (((ord_less_eq @ A) @ (zero_zero @ A)) @ (zero_zero @ A)))))).
% 2.64/1.76 thf(fact_190_zero__neq__numeral, axiom, (! [A:$tType]: (((semiring_char_0 @ A) => (! [N:num]: (((zero_zero @ A) != ((numeral_numeral @ A) @ N)))))))).
% 2.64/1.76 thf(fact_191_power__not__zero, axiom, (! [A:$tType]: (((semiri134348788visors @ A) => (! [A2:A,N:nat]: (((A2 != (zero_zero @ A)) => ((((power_power @ A) @ A2) @ N) != (zero_zero @ A))))))))).
% 2.64/1.76 thf(fact_192_exists__least__lemma, axiom, (! [P:(nat > $o)]: (((~ ((P @ (zero_zero @ nat)))) => ((? [X_1:nat]: ((P @ X_1))) => (? [N3:nat]: (((~ ((P @ N3))) & (P @ (suc @ N3)))))))))).
% 2.64/1.76 thf(fact_193_nat_Odistinct_I1_J, axiom, (! [X2:nat]: (((zero_zero @ nat) != (suc @ X2))))).
% 2.64/1.76 thf(fact_194_old_Onat_Odistinct_I2_J, axiom, (! [Nat3:nat]: (((suc @ Nat3) != (zero_zero @ nat))))).
% 2.64/1.76 thf(fact_195_old_Onat_Odistinct_I1_J, axiom, (! [Nat2:nat]: (((zero_zero @ nat) != (suc @ Nat2))))).
% 2.64/1.76 thf(fact_196_nat_OdiscI, axiom, (! [Nat:nat,X2:nat]: (((Nat = (suc @ X2)) => (Nat != (zero_zero @ nat)))))).
% 2.64/1.76 thf(fact_197_nat__induct, axiom, (! [P:(nat > $o),N:nat]: (((P @ (zero_zero @ nat)) => ((! [N3:nat]: (((P @ N3) => (P @ (suc @ N3))))) => (P @ N)))))).
% 2.64/1.76 thf(fact_198_One__nat__def, axiom, ((one_one @ nat) = (suc @ (zero_zero @ nat)))).
% 2.64/1.76 thf(fact_199_diff__induct, axiom, (! [P:(nat > (nat > $o)),M:nat,N:nat]: (((! [X4:nat]: (((P @ X4) @ (zero_zero @ nat)))) => ((! [Y3:nat]: (((P @ (zero_zero @ nat)) @ (suc @ Y3)))) => ((! [X4:nat,Y3:nat]: ((((P @ X4) @ Y3) => ((P @ (suc @ X4)) @ (suc @ Y3))))) => ((P @ M) @ N))))))).
% 2.64/1.76 thf(fact_200_zero__induct, axiom, (! [P:(nat > $o),K:nat]: (((P @ K) => ((! [N3:nat]: (((P @ (suc @ N3)) => (P @ N3)))) => (P @ (zero_zero @ nat))))))).
% 2.64/1.76 thf(fact_201_Suc__neq__Zero, axiom, (! [M:nat]: (((suc @ M) != (zero_zero @ nat))))).
% 2.64/1.76 thf(fact_202_Zero__neq__Suc, axiom, (! [M:nat]: (((zero_zero @ nat) != (suc @ M))))).
% 2.64/1.76 thf(fact_203_Zero__not__Suc, axiom, (! [M:nat]: (((zero_zero @ nat) != (suc @ M))))).
% 2.64/1.76 thf(fact_204_old_Onat_Oexhaust, axiom, (! [Y:nat]: (((Y != (zero_zero @ nat)) => (~ ((! [Nat4:nat]: ((Y != (suc @ Nat4)))))))))).
% 2.64/1.76 thf(fact_205_old_Onat_Oinducts, axiom, (! [P:(nat > $o),Nat:nat]: (((P @ (zero_zero @ nat)) => ((! [Nat4:nat]: (((P @ Nat4) => (P @ (suc @ Nat4))))) => (P @ Nat)))))).
% 2.64/1.76 thf(fact_206_not0__implies__Suc, axiom, (! [N:nat]: (((N != (zero_zero @ nat)) => (? [M4:nat]: ((N = (suc @ M4)))))))).
% 2.64/1.76 thf(fact_207_bot__nat__0_Oextremum__uniqueI, axiom, (! [A2:nat]: (((((ord_less_eq @ nat) @ A2) @ (zero_zero @ nat)) => (A2 = (zero_zero @ nat)))))).
% 2.64/1.76 thf(fact_208_bot__nat__0_Oextremum__unique, axiom, (! [A2:nat]: (((((ord_less_eq @ nat) @ A2) @ (zero_zero @ nat)) = (A2 = (zero_zero @ nat)))))).
% 2.64/1.76 thf(fact_209_le__0__eq, axiom, (! [N:nat]: (((((ord_less_eq @ nat) @ N) @ (zero_zero @ nat)) = (N = (zero_zero @ nat)))))).
% 2.64/1.76 thf(fact_210_less__eq__nat_Osimps_I1_J, axiom, (! [N:nat]: ((((ord_less_eq @ nat) @ (zero_zero @ nat)) @ N)))).
% 2.64/1.76 thf(fact_211_less__eq__int__code_I1_J, axiom, (((ord_less_eq @ int) @ (zero_zero @ int)) @ (zero_zero @ int))).
% 2.64/1.76 thf(fact_212_le__nat__iff, axiom, (! [K:int,N:nat]: (((((ord_less_eq @ int) @ (zero_zero @ int)) @ K) => ((((ord_less_eq @ nat) @ N) @ (nat2 @ K)) = (((ord_less_eq @ int) @ ((semiring_1_of_nat @ int) @ N)) @ K)))))).
% 2.64/1.76 thf(fact_213_of__int__nonneg, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [Z:int]: (((((ord_less_eq @ int) @ (zero_zero @ int)) @ Z) => (((ord_less_eq @ A) @ (zero_zero @ A)) @ ((ring_1_of_int @ A) @ Z))))))))).
% 2.64/1.76 thf(fact_214_nat__int__comparison_I3_J, axiom, ((ord_less_eq @ nat) = (^ [A4:nat,B3:nat]: ((((ord_less_eq @ int) @ ((semiring_1_of_nat @ int) @ A4)) @ ((semiring_1_of_nat @ int) @ B3)))))).
% 2.64/1.76 thf(fact_215_zle__int, axiom, (! [M:nat,N:nat]: (((((ord_less_eq @ int) @ ((semiring_1_of_nat @ int) @ M)) @ ((semiring_1_of_nat @ int) @ N)) = (((ord_less_eq @ nat) @ M) @ N))))).
% 2.64/1.76 thf(fact_216_int__ops_I3_J, axiom, (! [N:num]: ((((semiring_1_of_nat @ int) @ ((numeral_numeral @ nat) @ N)) = ((numeral_numeral @ int) @ N))))).
% 2.64/1.76 thf(fact_217_zero__le__numeral, axiom, (! [A:$tType]: (((linord1659791738miring @ A) => (! [N:num]: ((((ord_less_eq @ A) @ (zero_zero @ A)) @ ((numeral_numeral @ A) @ N)))))))).
% 2.64/1.76 thf(fact_218_not__numeral__le__zero, axiom, (! [A:$tType]: (((linord1659791738miring @ A) => (! [N:num]: ((~ ((((ord_less_eq @ A) @ ((numeral_numeral @ A) @ N)) @ (zero_zero @ A)))))))))).
% 2.64/1.76 thf(fact_219_power__mono, axiom, (! [A:$tType]: (((linordered_semidom @ A) => (! [A2:A,B:A,N:nat]: (((((ord_less_eq @ A) @ A2) @ B) => ((((ord_less_eq @ A) @ (zero_zero @ A)) @ A2) => (((ord_less_eq @ A) @ (((power_power @ A) @ A2) @ N)) @ (((power_power @ A) @ B) @ N)))))))))).
% 2.64/1.76 thf(fact_220_zero__le__power, axiom, (! [A:$tType]: (((linordered_semidom @ A) => (! [A2:A,N:nat]: (((((ord_less_eq @ A) @ (zero_zero @ A)) @ A2) => (((ord_less_eq @ A) @ (zero_zero @ A)) @ (((power_power @ A) @ A2) @ N))))))))).
% 2.64/1.76 thf(fact_221_of__nat__0__le__iff, axiom, (! [A:$tType]: (((linord1659791738miring @ A) => (! [N:nat]: ((((ord_less_eq @ A) @ (zero_zero @ A)) @ ((semiring_1_of_nat @ A) @ N)))))))).
% 2.64/1.76 thf(fact_222_power__0, axiom, (! [A:$tType]: (((power @ A) => (! [A2:A]: (((((power_power @ A) @ A2) @ (zero_zero @ nat)) = (one_one @ A)))))))).
% 2.64/1.76 thf(fact_223_of__nat__neq__0, axiom, (! [A:$tType]: (((semiring_char_0 @ A) => (! [N:nat]: ((((semiring_1_of_nat @ A) @ (suc @ N)) != (zero_zero @ A)))))))).
% 2.64/1.76 thf(fact_224_ex__nat, axiom, ((^ [P2:(nat > $o)]: ((? [X6:nat]: ((P2 @ X6))))) = (^ [P3:(nat > $o)]: ((? [X3:int]: (((((ord_less_eq @ int) @ (zero_zero @ int)) @ X3) & (P3 @ (nat2 @ X3))))))))).
% 2.64/1.76 thf(fact_225_all__nat, axiom, ((^ [P2:(nat > $o)]: ((! [X6:nat]: ((P2 @ X6))))) = (^ [P3:(nat > $o)]: ((! [X3:int]: (((((ord_less_eq @ int) @ (zero_zero @ int)) @ X3) => (P3 @ (nat2 @ X3))))))))).
% 2.64/1.76 thf(fact_226_eq__nat__nat__iff, axiom, (! [Z:int,Z5:int]: (((((ord_less_eq @ int) @ (zero_zero @ int)) @ Z) => ((((ord_less_eq @ int) @ (zero_zero @ int)) @ Z5) => (((nat2 @ Z) = (nat2 @ Z5)) = (Z = Z5))))))).
% 2.64/1.76 thf(fact_227_powr__mono2, axiom, (! [A2:real,X:real,Y:real]: (((((ord_less_eq @ real) @ (zero_zero @ real)) @ A2) => ((((ord_less_eq @ real) @ (zero_zero @ real)) @ X) => ((((ord_less_eq @ real) @ X) @ Y) => (((ord_less_eq @ real) @ (((powr @ real) @ X) @ A2)) @ (((powr @ real) @ Y) @ A2)))))))).
% 2.64/1.76 thf(fact_228_powr__ge__pzero, axiom, (! [X:real,Y:real]: ((((ord_less_eq @ real) @ (zero_zero @ real)) @ (((powr @ real) @ X) @ Y))))).
% 2.64/1.76 thf(fact_229_pow_Osimps_I1_J, axiom, (! [X:num]: ((((pow @ X) @ one2) = X)))).
% 2.64/1.76 thf(fact_230_nat__le__iff, axiom, (! [X:int,N:nat]: (((((ord_less_eq @ nat) @ (nat2 @ X)) @ N) = (((ord_less_eq @ int) @ X) @ ((semiring_1_of_nat @ int) @ N)))))).
% 2.64/1.76 thf(fact_231_le__funD, axiom, (! [B2:$tType,A:$tType]: (((ord @ B2) => (! [F:(A > B2),G:(A > B2),X:A]: (((((ord_less_eq @ (A > B2)) @ F) @ G) => (((ord_less_eq @ B2) @ (F @ X)) @ (G @ X))))))))).
% 2.64/1.76 thf(fact_232_le__funE, axiom, (! [B2:$tType,A:$tType]: (((ord @ B2) => (! [F:(A > B2),G:(A > B2),X:A]: (((((ord_less_eq @ (A > B2)) @ F) @ G) => (((ord_less_eq @ B2) @ (F @ X)) @ (G @ X))))))))).
% 2.64/1.76 thf(fact_233_le__funI, axiom, (! [B2:$tType,A:$tType]: (((ord @ B2) => (! [F:(A > B2),G:(A > B2)]: (((! [X4:A]: ((((ord_less_eq @ B2) @ (F @ X4)) @ (G @ X4)))) => (((ord_less_eq @ (A > B2)) @ F) @ G)))))))).
% 2.64/1.76 thf(fact_234_le__fun__def, axiom, (! [B2:$tType,A:$tType]: (((ord @ B2) => ((ord_less_eq @ (A > B2)) = (^ [F2:(A > B2),G2:(A > B2)]: ((! [X3:A]: ((((ord_less_eq @ B2) @ (F2 @ X3)) @ (G2 @ X3))))))))))).
% 2.64/1.76 thf(fact_235_order__subst1, axiom, (! [A:$tType,B2:$tType]: ((((order @ B2) & (order @ A)) => (! [A2:A,F:(B2 > A),B:B2,C:B2]: (((((ord_less_eq @ A) @ A2) @ (F @ B)) => ((((ord_less_eq @ B2) @ B) @ C) => ((! [X4:B2,Y3:B2]: (((((ord_less_eq @ B2) @ X4) @ Y3) => (((ord_less_eq @ A) @ (F @ X4)) @ (F @ Y3))))) => (((ord_less_eq @ A) @ A2) @ (F @ C))))))))))).
% 2.64/1.76 thf(fact_236_order__subst2, axiom, (! [A:$tType,C2:$tType]: ((((order @ C2) & (order @ A)) => (! [A2:A,B:A,F:(A > C2),C:C2]: (((((ord_less_eq @ A) @ A2) @ B) => ((((ord_less_eq @ C2) @ (F @ B)) @ C) => ((! [X4:A,Y3:A]: (((((ord_less_eq @ A) @ X4) @ Y3) => (((ord_less_eq @ C2) @ (F @ X4)) @ (F @ Y3))))) => (((ord_less_eq @ C2) @ (F @ A2)) @ C)))))))))).
% 2.64/1.76 thf(fact_237_ord__eq__le__subst, axiom, (! [A:$tType,B2:$tType]: ((((ord @ B2) & (ord @ A)) => (! [A2:A,F:(B2 > A),B:B2,C:B2]: (((A2 = (F @ B)) => ((((ord_less_eq @ B2) @ B) @ C) => ((! [X4:B2,Y3:B2]: (((((ord_less_eq @ B2) @ X4) @ Y3) => (((ord_less_eq @ A) @ (F @ X4)) @ (F @ Y3))))) => (((ord_less_eq @ A) @ A2) @ (F @ C))))))))))).
% 2.64/1.76 thf(fact_238_ord__le__eq__subst, axiom, (! [A:$tType,B2:$tType]: ((((ord @ B2) & (ord @ A)) => (! [A2:A,B:A,F:(A > B2),C:B2]: (((((ord_less_eq @ A) @ A2) @ B) => (((F @ B) = C) => ((! [X4:A,Y3:A]: (((((ord_less_eq @ A) @ X4) @ Y3) => (((ord_less_eq @ B2) @ (F @ X4)) @ (F @ Y3))))) => (((ord_less_eq @ B2) @ (F @ A2)) @ C)))))))))).
% 2.64/1.76 thf(fact_239_eq__iff, axiom, (! [A:$tType]: (((order @ A) => ((^ [Y5:A,Z4:A]: ((Y5 = Z4))) = (^ [X3:A,Y6:A]: (((((ord_less_eq @ A) @ X3) @ Y6) & (((ord_less_eq @ A) @ Y6) @ X3))))))))).
% 2.64/1.76 thf(fact_240_antisym, axiom, (! [A:$tType]: (((order @ A) => (! [X:A,Y:A]: (((((ord_less_eq @ A) @ X) @ Y) => ((((ord_less_eq @ A) @ Y) @ X) => (X = Y))))))))).
% 2.64/1.76 thf(fact_241_linear, axiom, (! [A:$tType]: (((linorder @ A) => (! [X:A,Y:A]: (((((ord_less_eq @ A) @ X) @ Y) | (((ord_less_eq @ A) @ Y) @ X)))))))).
% 2.64/1.76 thf(fact_242_eq__refl, axiom, (! [A:$tType]: (((preorder @ A) => (! [X:A,Y:A]: (((X = Y) => (((ord_less_eq @ A) @ X) @ Y)))))))).
% 2.64/1.76 thf(fact_243_le__cases, axiom, (! [A:$tType]: (((linorder @ A) => (! [X:A,Y:A]: (((~ ((((ord_less_eq @ A) @ X) @ Y))) => (((ord_less_eq @ A) @ Y) @ X)))))))).
% 2.64/1.76 thf(fact_244_order_Otrans, axiom, (! [A:$tType]: (((order @ A) => (! [A2:A,B:A,C:A]: (((((ord_less_eq @ A) @ A2) @ B) => ((((ord_less_eq @ A) @ B) @ C) => (((ord_less_eq @ A) @ A2) @ C))))))))).
% 2.64/1.76 thf(fact_245_le__cases3, axiom, (! [A:$tType]: (((linorder @ A) => (! [X:A,Y:A,Z:A]: ((((((ord_less_eq @ A) @ X) @ Y) => (~ ((((ord_less_eq @ A) @ Y) @ Z)))) => (((((ord_less_eq @ A) @ Y) @ X) => (~ ((((ord_less_eq @ A) @ X) @ Z)))) => (((((ord_less_eq @ A) @ X) @ Z) => (~ ((((ord_less_eq @ A) @ Z) @ Y)))) => (((((ord_less_eq @ A) @ Z) @ Y) => (~ ((((ord_less_eq @ A) @ Y) @ X)))) => (((((ord_less_eq @ A) @ Y) @ Z) => (~ ((((ord_less_eq @ A) @ Z) @ X)))) => (~ (((((ord_less_eq @ A) @ Z) @ X) => (~ ((((ord_less_eq @ A) @ X) @ Y))))))))))))))))).
% 2.64/1.76 thf(fact_246_antisym__conv, axiom, (! [A:$tType]: (((order @ A) => (! [Y:A,X:A]: (((((ord_less_eq @ A) @ Y) @ X) => ((((ord_less_eq @ A) @ X) @ Y) = (X = Y))))))))).
% 2.64/1.76 thf(fact_247_order__class_Oorder_Oeq__iff, axiom, (! [A:$tType]: (((order @ A) => ((^ [Y5:A,Z4:A]: ((Y5 = Z4))) = (^ [A4:A,B3:A]: (((((ord_less_eq @ A) @ A4) @ B3) & (((ord_less_eq @ A) @ B3) @ A4))))))))).
% 2.64/1.76 thf(fact_248_ord__eq__le__trans, axiom, (! [A:$tType]: (((ord @ A) => (! [A2:A,B:A,C:A]: (((A2 = B) => ((((ord_less_eq @ A) @ B) @ C) => (((ord_less_eq @ A) @ A2) @ C))))))))).
% 2.64/1.76 thf(fact_249_ord__le__eq__trans, axiom, (! [A:$tType]: (((ord @ A) => (! [A2:A,B:A,C:A]: (((((ord_less_eq @ A) @ A2) @ B) => ((B = C) => (((ord_less_eq @ A) @ A2) @ C))))))))).
% 2.64/1.76 thf(fact_250_order__class_Oorder_Oantisym, axiom, (! [A:$tType]: (((order @ A) => (! [A2:A,B:A]: (((((ord_less_eq @ A) @ A2) @ B) => ((((ord_less_eq @ A) @ B) @ A2) => (A2 = B))))))))).
% 2.64/1.76 thf(tcon_fun___Orderings_Opreorder, axiom, (! [A5:$tType,A6:$tType]: (((preorder @ A6) => (preorder @ (A5 > A6)))))).
% 2.64/1.76 thf(tcon_fun___Orderings_Oorder, axiom, (! [A5:$tType,A6:$tType]: (((order @ A6) => (order @ (A5 > A6)))))).
% 2.64/1.76 thf(tcon_fun___Orderings_Oord, axiom, (! [A5:$tType,A6:$tType]: (((ord @ A6) => (ord @ (A5 > A6)))))).
% 2.64/1.76 thf(tcon_Int_Oint___Rings_Osemiring__1__no__zero__divisors, axiom, (semiri134348788visors @ int)).
% 2.64/1.76 thf(tcon_Int_Oint___Rings_Olinordered__nonzero__semiring, axiom, (linord1659791738miring @ int)).
% 2.64/1.76 thf(tcon_Int_Oint___Rings_Olinordered__semidom, axiom, (linordered_semidom @ int)).
% 2.64/1.76 thf(tcon_Int_Oint___Rings_Olinordered__idom, axiom, (linordered_idom @ int)).
% 2.64/1.76 thf(tcon_Int_Oint___Num_Osemiring__numeral, axiom, (semiring_numeral @ int)).
% 2.64/1.76 thf(tcon_Int_Oint___Nat_Osemiring__char__0, axiom, (semiring_char_0 @ int)).
% 2.64/1.76 thf(tcon_Int_Oint___Orderings_Opreorder_1, axiom, (preorder @ int)).
% 2.64/1.76 thf(tcon_Int_Oint___Orderings_Olinorder, axiom, (linorder @ int)).
% 2.64/1.76 thf(tcon_Int_Oint___Groups_Omonoid__mult, axiom, (monoid_mult @ int)).
% 2.64/1.76 thf(tcon_Int_Oint___Rings_Osemiring__1, axiom, (semiring_1 @ int)).
% 2.64/1.76 thf(tcon_Int_Oint___Orderings_Oorder_2, axiom, (order @ int)).
% 2.64/1.76 thf(tcon_Int_Oint___Num_Oneg__numeral, axiom, (neg_numeral @ int)).
% 2.64/1.76 thf(tcon_Int_Oint___Nat_Oring__char__0, axiom, (ring_char_0 @ int)).
% 2.64/1.76 thf(tcon_Int_Oint___Orderings_Oord_3, axiom, (ord @ int)).
% 2.64/1.76 thf(tcon_Int_Oint___Rings_Oring__1, axiom, (ring_1 @ int)).
% 2.64/1.76 thf(tcon_Int_Oint___Power_Opower, axiom, (power @ int)).
% 2.64/1.76 thf(tcon_Int_Oint___Num_Onumeral, axiom, (numeral @ int)).
% 2.64/1.76 thf(tcon_Int_Oint___Groups_Ozero, axiom, (zero @ int)).
% 2.64/1.76 thf(tcon_Int_Oint___Groups_Oone, axiom, (one @ int)).
% 2.64/1.76 thf(tcon_Nat_Onat___Groups_Ocanonically__ordered__monoid__add, axiom, (canoni770627133id_add @ nat)).
% 2.64/1.76 thf(tcon_Nat_Onat___Rings_Osemiring__1__no__zero__divisors_4, axiom, (semiri134348788visors @ nat)).
% 2.64/1.76 thf(tcon_Nat_Onat___Rings_Olinordered__nonzero__semiring_5, axiom, (linord1659791738miring @ nat)).
% 2.64/1.76 thf(tcon_Nat_Onat___Rings_Olinordered__semidom_6, axiom, (linordered_semidom @ nat)).
% 2.64/1.76 thf(tcon_Nat_Onat___Num_Osemiring__numeral_7, axiom, (semiring_numeral @ nat)).
% 2.64/1.76 thf(tcon_Nat_Onat___Nat_Osemiring__char__0_8, axiom, (semiring_char_0 @ nat)).
% 2.64/1.76 thf(tcon_Nat_Onat___Orderings_Opreorder_9, axiom, (preorder @ nat)).
% 2.64/1.76 thf(tcon_Nat_Onat___Orderings_Olinorder_10, axiom, (linorder @ nat)).
% 2.64/1.76 thf(tcon_Nat_Onat___Groups_Omonoid__mult_11, axiom, (monoid_mult @ nat)).
% 2.64/1.76 thf(tcon_Nat_Onat___Rings_Osemiring__1_12, axiom, (semiring_1 @ nat)).
% 2.64/1.76 thf(tcon_Nat_Onat___Orderings_Oorder_13, axiom, (order @ nat)).
% 2.64/1.76 thf(tcon_Nat_Onat___Orderings_Oord_14, axiom, (ord @ nat)).
% 2.64/1.76 thf(tcon_Nat_Onat___Power_Opower_15, axiom, (power @ nat)).
% 2.64/1.76 thf(tcon_Nat_Onat___Num_Onumeral_16, axiom, (numeral @ nat)).
% 2.64/1.76 thf(tcon_Nat_Onat___Groups_Ozero_17, axiom, (zero @ nat)).
% 2.64/1.76 thf(tcon_Nat_Onat___Groups_Oone_18, axiom, (one @ nat)).
% 2.64/1.76 thf(tcon_Num_Onum___Orderings_Opreorder_19, axiom, (preorder @ num)).
% 2.64/1.76 thf(tcon_Num_Onum___Orderings_Olinorder_20, axiom, (linorder @ num)).
% 2.64/1.76 thf(tcon_Num_Onum___Orderings_Oorder_21, axiom, (order @ num)).
% 2.64/1.76 thf(tcon_Num_Onum___Orderings_Oord_22, axiom, (ord @ num)).
% 2.64/1.76 thf(tcon_Set_Oset___Orderings_Opreorder_23, axiom, (! [A5:$tType]: ((preorder @ (set @ A5))))).
% 2.64/1.76 thf(tcon_Set_Oset___Orderings_Oorder_24, axiom, (! [A5:$tType]: ((order @ (set @ A5))))).
% 2.64/1.76 thf(tcon_Set_Oset___Orderings_Oord_25, axiom, (! [A5:$tType]: ((ord @ (set @ A5))))).
% 2.64/1.76 thf(tcon_HOL_Obool___Orderings_Opreorder_26, axiom, (preorder @ $o)).
% 2.64/1.76 thf(tcon_HOL_Obool___Orderings_Olinorder_27, axiom, (linorder @ $o)).
% 2.64/1.76 thf(tcon_HOL_Obool___Orderings_Oorder_28, axiom, (order @ $o)).
% 2.64/1.76 thf(tcon_HOL_Obool___Orderings_Oord_29, axiom, (ord @ $o)).
% 2.64/1.76 thf(tcon_Real_Oreal___Archimedean__Field_Oarchimedean__field, axiom, (archim1804426504_field @ real)).
% 2.64/1.76 thf(tcon_Real_Oreal___Rings_Osemiring__1__no__zero__divisors_30, axiom, (semiri134348788visors @ real)).
% 2.64/1.76 thf(tcon_Real_Oreal___Rings_Olinordered__nonzero__semiring_31, axiom, (linord1659791738miring @ real)).
% 2.64/1.76 thf(tcon_Real_Oreal___Archimedean__Field_Ofloor__ceiling, axiom, (archim1727834104eiling @ real)).
% 2.64/1.76 thf(tcon_Real_Oreal___Rings_Olinordered__semidom_32, axiom, (linordered_semidom @ real)).
% 2.64/1.76 thf(tcon_Real_Oreal___Rings_Olinordered__idom_33, axiom, (linordered_idom @ real)).
% 2.64/1.76 thf(tcon_Real_Oreal___Num_Osemiring__numeral_34, axiom, (semiring_numeral @ real)).
% 2.64/1.76 thf(tcon_Real_Oreal___Nat_Osemiring__char__0_35, axiom, (semiring_char_0 @ real)).
% 2.64/1.76 thf(tcon_Real_Oreal___Orderings_Opreorder_36, axiom, (preorder @ real)).
% 2.64/1.76 thf(tcon_Real_Oreal___Orderings_Olinorder_37, axiom, (linorder @ real)).
% 2.64/1.76 thf(tcon_Real_Oreal___Groups_Omonoid__mult_38, axiom, (monoid_mult @ real)).
% 2.64/1.76 thf(tcon_Real_Oreal___Transcendental_Oln, axiom, (ln @ real)).
% 2.64/1.76 thf(tcon_Real_Oreal___Rings_Osemiring__1_39, axiom, (semiring_1 @ real)).
% 2.64/1.76 thf(tcon_Real_Oreal___Orderings_Oorder_40, axiom, (order @ real)).
% 2.64/1.76 thf(tcon_Real_Oreal___Num_Oneg__numeral_41, axiom, (neg_numeral @ real)).
% 2.64/1.76 thf(tcon_Real_Oreal___Nat_Oring__char__0_42, axiom, (ring_char_0 @ real)).
% 2.64/1.76 thf(tcon_Real_Oreal___Orderings_Oord_43, axiom, (ord @ real)).
% 2.64/1.76 thf(tcon_Real_Oreal___Rings_Oring__1_44, axiom, (ring_1 @ real)).
% 2.64/1.76 thf(tcon_Real_Oreal___Power_Opower_45, axiom, (power @ real)).
% 2.64/1.76 thf(tcon_Real_Oreal___Num_Onumeral_46, axiom, (numeral @ real)).
% 2.64/1.76 thf(tcon_Real_Oreal___Groups_Ozero_47, axiom, (zero @ real)).
% 2.64/1.76 thf(tcon_Real_Oreal___Groups_Oone_48, axiom, (one @ real)).
% 2.64/1.76 thf(help_If_3_1_T, axiom, (! [P:$o]: (((P = $true) | (P = $false))))).
% 2.64/1.76 thf(help_If_2_1_T, axiom, (! [A:$tType,X:A,Y:A]: ((((((if @ A) @ $false) @ X) @ Y) = Y)))).
% 2.64/1.76 thf(help_If_1_1_T, axiom, (! [A:$tType,X:A,Y:A]: ((((((if @ A) @ $true) @ X) @ Y) = X)))).
% 2.64/1.76 thf(conj_0, conjecture, (((ord_less_eq @ int) @ d) @ (((power_power @ int) @ ((numeral_numeral @ int) @ (bit0 @ one2))) @ (nat2 @ ((archimedean_ceiling @ real) @ ((log @ ((numeral_numeral @ real) @ (bit0 @ one2))) @ ((ring_1_of_int @ real) @ d))))))).
% 2.64/1.76 % SZS output end ListOfTHF for /export/starexec/sandbox/tmp/tmp.aWXyJVFJl3/DTF2DTF4485.p
% 2.64/1.76 ------------------------
% 2.64/1.79 ---- Cleaned THF ---
% 2.64/1.79 thf(ty_t_Real_Oreal,type,
% 2.64/1.79 real: $tType ).
% 2.64/1.79
% 2.64/1.79 thf(ty_t_Set_Oset,type,
% 2.64/1.79 set: $tType > $tType ).
% 2.64/1.79
% 2.64/1.79 thf(ty_t_Num_Onum,type,
% 2.64/1.79 num: $tType ).
% 2.64/1.79
% 2.64/1.79 thf(ty_t_Nat_Onat,type,
% 2.64/1.79 nat: $tType ).
% 2.64/1.79
% 2.64/1.79 thf(ty_t_Int_Oint,type,
% 2.64/1.79 int: $tType ).
% 2.64/1.79
% 2.64/1.79 thf(sy_cl_HOL_Otype,type,
% 2.64/1.79 type:
% 2.64/1.79 !>[A: $tType] : $o ).
% 2.64/1.79
% 2.64/1.79 thf(sy_cl_Groups_Oone,type,
% 2.64/1.79 one:
% 2.64/1.79 !>[A: $tType] : $o ).
% 2.64/1.79
% 2.64/1.79 thf(sy_cl_Groups_Ozero,type,
% 2.64/1.79 zero:
% 2.64/1.79 !>[A: $tType] : $o ).
% 2.64/1.79
% 2.64/1.79 thf(sy_cl_Num_Onumeral,type,
% 2.64/1.79 numeral:
% 2.64/1.79 !>[A: $tType] : $o ).
% 2.64/1.79
% 2.64/1.79 thf(sy_cl_Power_Opower,type,
% 2.64/1.79 power:
% 2.64/1.79 !>[A: $tType] : $o ).
% 2.64/1.79
% 2.64/1.79 thf(sy_cl_Rings_Oring__1,type,
% 2.64/1.79 ring_1:
% 2.64/1.79 !>[A: $tType] : $o ).
% 2.64/1.79
% 2.64/1.79 thf(sy_cl_Orderings_Oord,type,
% 2.64/1.79 ord:
% 2.64/1.79 !>[A: $tType] : $o ).
% 2.64/1.79
% 2.64/1.79 thf(sy_cl_Nat_Oring__char__0,type,
% 2.64/1.79 ring_char_0:
% 2.64/1.79 !>[A: $tType] : $o ).
% 2.64/1.79
% 2.64/1.79 thf(sy_cl_Num_Oneg__numeral,type,
% 2.64/1.79 neg_numeral:
% 2.64/1.79 !>[A: $tType] : $o ).
% 2.64/1.79
% 2.64/1.79 thf(sy_cl_Orderings_Oorder,type,
% 2.64/1.79 order:
% 2.64/1.79 !>[A: $tType] : $o ).
% 2.64/1.79
% 2.64/1.79 thf(sy_cl_Rings_Osemiring__1,type,
% 2.64/1.79 semiring_1:
% 2.64/1.79 !>[A: $tType] : $o ).
% 2.64/1.79
% 2.64/1.79 thf(sy_cl_Transcendental_Oln,type,
% 2.64/1.79 ln:
% 2.64/1.79 !>[A: $tType] : $o ).
% 2.64/1.79
% 2.64/1.79 thf(sy_cl_Groups_Omonoid__mult,type,
% 2.64/1.79 monoid_mult:
% 2.64/1.79 !>[A: $tType] : $o ).
% 2.64/1.79
% 2.64/1.79 thf(sy_cl_Orderings_Olinorder,type,
% 2.64/1.79 linorder:
% 2.64/1.79 !>[A: $tType] : $o ).
% 2.64/1.79
% 2.64/1.79 thf(sy_cl_Orderings_Opreorder,type,
% 2.64/1.79 preorder:
% 2.64/1.79 !>[A: $tType] : $o ).
% 2.64/1.79
% 2.64/1.79 thf(sy_cl_Nat_Osemiring__char__0,type,
% 2.64/1.79 semiring_char_0:
% 2.64/1.79 !>[A: $tType] : $o ).
% 2.64/1.79
% 2.64/1.79 thf(sy_cl_Num_Osemiring__numeral,type,
% 2.64/1.79 semiring_numeral:
% 2.64/1.79 !>[A: $tType] : $o ).
% 2.64/1.79
% 2.64/1.79 thf(sy_cl_Rings_Olinordered__idom,type,
% 2.64/1.79 linordered_idom:
% 2.64/1.79 !>[A: $tType] : $o ).
% 2.64/1.79
% 2.64/1.79 thf(sy_cl_Rings_Olinordered__semidom,type,
% 2.64/1.79 linordered_semidom:
% 2.64/1.79 !>[A: $tType] : $o ).
% 2.64/1.79
% 2.64/1.79 thf(sy_cl_Archimedean__Field_Ofloor__ceiling,type,
% 2.64/1.79 archim1727834104eiling:
% 2.64/1.79 !>[A: $tType] : $o ).
% 2.64/1.79
% 2.64/1.79 thf(sy_cl_Rings_Olinordered__nonzero__semiring,type,
% 2.64/1.79 linord1659791738miring:
% 2.64/1.79 !>[A: $tType] : $o ).
% 2.64/1.79
% 2.64/1.79 thf(sy_cl_Rings_Osemiring__1__no__zero__divisors,type,
% 2.64/1.79 semiri134348788visors:
% 2.64/1.79 !>[A: $tType] : $o ).
% 2.64/1.79
% 2.64/1.79 thf(sy_cl_Archimedean__Field_Oarchimedean__field,type,
% 2.64/1.79 archim1804426504_field:
% 2.64/1.79 !>[A: $tType] : $o ).
% 2.64/1.79
% 2.64/1.79 thf(sy_cl_Groups_Ocanonically__ordered__monoid__add,type,
% 2.64/1.79 canoni770627133id_add:
% 2.64/1.79 !>[A: $tType] : $o ).
% 2.64/1.79
% 2.64/1.79 thf(sy_c_Archimedean__Field_Oceiling,type,
% 2.64/1.79 archimedean_ceiling:
% 2.64/1.79 !>[A: $tType] : ( A > int ) ).
% 2.64/1.79
% 2.64/1.79 thf(sy_c_Groups_Oone__class_Oone,type,
% 2.64/1.79 one_one:
% 2.64/1.79 !>[A: $tType] : A ).
% 2.64/1.79
% 2.64/1.79 thf(sy_c_Groups_Ozero__class_Ozero,type,
% 2.64/1.79 zero_zero:
% 2.64/1.79 !>[A: $tType] : A ).
% 2.64/1.79
% 2.64/1.79 thf(sy_c_If,type,
% 2.64/1.79 if:
% 2.64/1.79 !>[A: $tType] : ( $o > A > A > A ) ).
% 2.64/1.79
% 2.64/1.79 thf(sy_c_Int_Onat,type,
% 2.64/1.79 nat2: int > nat ).
% 2.64/1.79
% 2.64/1.79 thf(sy_c_Int_Oring__1__class_Oof__int,type,
% 2.64/1.79 ring_1_of_int:
% 2.64/1.79 !>[A: $tType] : ( int > A ) ).
% 2.64/1.79
% 2.64/1.79 thf(sy_c_Nat_OSuc,type,
% 2.64/1.79 suc: nat > nat ).
% 2.64/1.79
% 2.64/1.79 thf(sy_c_Nat_Osemiring__1__class_Oof__nat,type,
% 2.64/1.79 semiring_1_of_nat:
% 2.64/1.79 !>[A: $tType] : ( nat > A ) ).
% 2.64/1.79
% 2.64/1.79 thf(sy_c_Num_Oneg__numeral__class_Odbl,type,
% 2.64/1.79 neg_numeral_dbl:
% 2.64/1.79 !>[A: $tType] : ( A > A ) ).
% 2.64/1.79
% 2.64/1.79 thf(sy_c_Num_Onum_OBit0,type,
% 2.64/1.79 bit0: num > num ).
% 2.64/1.79
% 2.64/1.79 thf(sy_c_Num_Onum_OOne,type,
% 2.64/1.79 one2: num ).
% 2.64/1.79
% 2.64/1.79 thf(sy_c_Num_Onumeral__class_Onumeral,type,
% 2.64/1.79 numeral_numeral:
% 2.64/1.79 !>[A: $tType] : ( num > A ) ).
% 2.64/1.79
% 2.64/1.79 thf(sy_c_Num_Opow,type,
% 2.64/1.79 pow: num > num > num ).
% 2.64/1.79
% 2.64/1.79 thf(sy_c_Orderings_Oord__class_Oless,type,
% 2.64/1.79 ord_less:
% 2.64/1.79 !>[A: $tType] : ( A > A > $o ) ).
% 2.64/1.79
% 2.64/1.79 thf(sy_c_Orderings_Oord__class_Oless__eq,type,
% 2.64/1.79 ord_less_eq:
% 2.64/1.79 !>[A: $tType] : ( A > A > $o ) ).
% 2.64/1.79
% 2.64/1.79 thf(sy_c_Power_Opower__class_Opower,type,
% 2.64/1.79 power_power:
% 2.64/1.79 !>[A: $tType] : ( A > nat > A ) ).
% 2.64/1.79
% 2.64/1.79 thf(sy_c_Set_OCollect,type,
% 2.64/1.79 collect:
% 2.64/1.79 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(sy_c_Transcendental_Olog,type,
% 2.64/1.79 log: real > real > real ).
% 2.64/1.79
% 2.64/1.79 thf(sy_c_Transcendental_Opowr,type,
% 2.64/1.79 powr:
% 2.64/1.79 !>[A: $tType] : ( A > A > A ) ).
% 2.64/1.79
% 2.64/1.79 thf(sy_c_member,type,
% 2.64/1.79 member:
% 2.64/1.79 !>[A: $tType] : ( A > ( set @ A ) > $o ) ).
% 2.64/1.79
% 2.64/1.79 thf(sy_v_d____,type,
% 2.64/1.79 d: int ).
% 2.64/1.79
% 2.64/1.79 thf(sy_v_n____,type,
% 2.64/1.79 n: nat ).
% 2.64/1.79
% 2.64/1.79 thf(fact_0_numeral__power__le__of__int__cancel__iff,axiom,
% 2.64/1.79 ! [A: $tType] :
% 2.64/1.79 ( ( linordered_idom @ A )
% 2.64/1.79 => ! [X: num,N: nat,A2: int] :
% 2.64/1.79 ( ( ord_less_eq @ A @ ( power_power @ A @ ( numeral_numeral @ A @ X ) @ N ) @ ( ring_1_of_int @ A @ A2 ) )
% 2.64/1.79 = ( ord_less_eq @ int @ ( power_power @ int @ ( numeral_numeral @ int @ X ) @ N ) @ A2 ) ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_1_of__int__le__numeral__power__cancel__iff,axiom,
% 2.64/1.79 ! [A: $tType] :
% 2.64/1.79 ( ( linordered_idom @ A )
% 2.64/1.79 => ! [A2: int,X: num,N: nat] :
% 2.64/1.79 ( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ A2 ) @ ( power_power @ A @ ( numeral_numeral @ A @ X ) @ N ) )
% 2.64/1.79 = ( ord_less_eq @ int @ A2 @ ( power_power @ int @ ( numeral_numeral @ int @ X ) @ N ) ) ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_2_ceiling__numeral__power,axiom,
% 2.64/1.79 ! [A: $tType] :
% 2.64/1.79 ( ( archim1727834104eiling @ A )
% 2.64/1.79 => ! [X: num,N: nat] :
% 2.64/1.79 ( ( archimedean_ceiling @ A @ ( power_power @ A @ ( numeral_numeral @ A @ X ) @ N ) )
% 2.64/1.79 = ( power_power @ int @ ( numeral_numeral @ int @ X ) @ N ) ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_3_numeral__power__eq__of__int__cancel__iff,axiom,
% 2.64/1.79 ! [A: $tType] :
% 2.64/1.79 ( ( ring_char_0 @ A )
% 2.64/1.79 => ! [X: num,N: nat,Y: int] :
% 2.64/1.79 ( ( ( power_power @ A @ ( numeral_numeral @ A @ X ) @ N )
% 2.64/1.79 = ( ring_1_of_int @ A @ Y ) )
% 2.64/1.79 = ( ( power_power @ int @ ( numeral_numeral @ int @ X ) @ N )
% 2.64/1.79 = Y ) ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_4_of__int__eq__numeral__power__cancel__iff,axiom,
% 2.64/1.79 ! [A: $tType] :
% 2.64/1.79 ( ( ring_char_0 @ A )
% 2.64/1.79 => ! [Y: int,X: num,N: nat] :
% 2.64/1.79 ( ( ( ring_1_of_int @ A @ Y )
% 2.64/1.79 = ( power_power @ A @ ( numeral_numeral @ A @ X ) @ N ) )
% 2.64/1.79 = ( Y
% 2.64/1.79 = ( power_power @ int @ ( numeral_numeral @ int @ X ) @ N ) ) ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_5_of__int__le__of__int__power__cancel__iff,axiom,
% 2.64/1.79 ! [A: $tType] :
% 2.64/1.79 ( ( linordered_idom @ A )
% 2.64/1.79 => ! [B: int,W: nat,X: int] :
% 2.64/1.79 ( ( ord_less_eq @ A @ ( power_power @ A @ ( ring_1_of_int @ A @ B ) @ W ) @ ( ring_1_of_int @ A @ X ) )
% 2.64/1.79 = ( ord_less_eq @ int @ ( power_power @ int @ B @ W ) @ X ) ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_6_of__int__power__le__of__int__cancel__iff,axiom,
% 2.64/1.79 ! [A: $tType] :
% 2.64/1.79 ( ( linordered_idom @ A )
% 2.64/1.79 => ! [X: int,B: int,W: nat] :
% 2.64/1.79 ( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ X ) @ ( power_power @ A @ ( ring_1_of_int @ A @ B ) @ W ) )
% 2.64/1.79 = ( ord_less_eq @ int @ X @ ( power_power @ int @ B @ W ) ) ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_7_ceiling__le__numeral,axiom,
% 2.64/1.79 ! [A: $tType] :
% 2.64/1.79 ( ( archim1727834104eiling @ A )
% 2.64/1.79 => ! [X: A,V: num] :
% 2.64/1.79 ( ( ord_less_eq @ int @ ( archimedean_ceiling @ A @ X ) @ ( numeral_numeral @ int @ V ) )
% 2.64/1.79 = ( ord_less_eq @ A @ X @ ( numeral_numeral @ A @ V ) ) ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_8_of__int__le__numeral__iff,axiom,
% 2.64/1.79 ! [A: $tType] :
% 2.64/1.79 ( ( linordered_idom @ A )
% 2.64/1.79 => ! [Z: int,N: num] :
% 2.64/1.79 ( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ Z ) @ ( numeral_numeral @ A @ N ) )
% 2.64/1.79 = ( ord_less_eq @ int @ Z @ ( numeral_numeral @ int @ N ) ) ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_9_of__int__numeral__le__iff,axiom,
% 2.64/1.79 ! [A: $tType] :
% 2.64/1.79 ( ( linordered_idom @ A )
% 2.64/1.79 => ! [N: num,Z: int] :
% 2.64/1.79 ( ( ord_less_eq @ A @ ( numeral_numeral @ A @ N ) @ ( ring_1_of_int @ A @ Z ) )
% 2.64/1.79 = ( ord_less_eq @ int @ ( numeral_numeral @ int @ N ) @ Z ) ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_10_of__int__power,axiom,
% 2.64/1.79 ! [A: $tType] :
% 2.64/1.79 ( ( ring_1 @ A )
% 2.64/1.79 => ! [Z: int,N: nat] :
% 2.64/1.79 ( ( ring_1_of_int @ A @ ( power_power @ int @ Z @ N ) )
% 2.64/1.79 = ( power_power @ A @ ( ring_1_of_int @ A @ Z ) @ N ) ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_11_of__int__eq__of__int__power__cancel__iff,axiom,
% 2.64/1.79 ! [A: $tType] :
% 2.64/1.79 ( ( ring_char_0 @ A )
% 2.64/1.79 => ! [B: int,W: nat,X: int] :
% 2.64/1.79 ( ( ( power_power @ A @ ( ring_1_of_int @ A @ B ) @ W )
% 2.64/1.79 = ( ring_1_of_int @ A @ X ) )
% 2.64/1.79 = ( ( power_power @ int @ B @ W )
% 2.64/1.79 = X ) ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_12_of__int__eq__iff,axiom,
% 2.64/1.79 ! [A: $tType] :
% 2.64/1.79 ( ( ring_char_0 @ A )
% 2.64/1.79 => ! [W: int,Z: int] :
% 2.64/1.79 ( ( ( ring_1_of_int @ A @ W )
% 2.64/1.79 = ( ring_1_of_int @ A @ Z ) )
% 2.64/1.79 = ( W = Z ) ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_13_nat__numeral,axiom,
% 2.64/1.79 ! [K: num] :
% 2.64/1.79 ( ( nat2 @ ( numeral_numeral @ int @ K ) )
% 2.64/1.79 = ( numeral_numeral @ nat @ K ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_14_ceiling__of__int,axiom,
% 2.64/1.79 ! [A: $tType] :
% 2.64/1.79 ( ( archim1727834104eiling @ A )
% 2.64/1.79 => ! [Z: int] :
% 2.64/1.79 ( ( archimedean_ceiling @ A @ ( ring_1_of_int @ A @ Z ) )
% 2.64/1.79 = Z ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_15_of__int__ceiling__cancel,axiom,
% 2.64/1.79 ! [A: $tType] :
% 2.64/1.79 ( ( archim1727834104eiling @ A )
% 2.64/1.79 => ! [X: A] :
% 2.64/1.79 ( ( ( ring_1_of_int @ A @ ( archimedean_ceiling @ A @ X ) )
% 2.64/1.79 = X )
% 2.64/1.79 = ( ? [N2: int] :
% 2.64/1.79 ( X
% 2.64/1.79 = ( ring_1_of_int @ A @ N2 ) ) ) ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_16_of__int__le__iff,axiom,
% 2.64/1.79 ! [A: $tType] :
% 2.64/1.79 ( ( linordered_idom @ A )
% 2.64/1.79 => ! [W: int,Z: int] :
% 2.64/1.79 ( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ W ) @ ( ring_1_of_int @ A @ Z ) )
% 2.64/1.79 = ( ord_less_eq @ int @ W @ Z ) ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_17_of__int__numeral,axiom,
% 2.64/1.79 ! [A: $tType] :
% 2.64/1.79 ( ( ring_1 @ A )
% 2.64/1.79 => ! [K: num] :
% 2.64/1.79 ( ( ring_1_of_int @ A @ ( numeral_numeral @ int @ K ) )
% 2.64/1.79 = ( numeral_numeral @ A @ K ) ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_18_of__int__eq__numeral__iff,axiom,
% 2.64/1.79 ! [A: $tType] :
% 2.64/1.79 ( ( ring_char_0 @ A )
% 2.64/1.79 => ! [Z: int,N: num] :
% 2.64/1.79 ( ( ( ring_1_of_int @ A @ Z )
% 2.64/1.79 = ( numeral_numeral @ A @ N ) )
% 2.64/1.79 = ( Z
% 2.64/1.79 = ( numeral_numeral @ int @ N ) ) ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_19_ceiling__numeral,axiom,
% 2.64/1.79 ! [A: $tType] :
% 2.64/1.79 ( ( archim1727834104eiling @ A )
% 2.64/1.79 => ! [V: num] :
% 2.64/1.79 ( ( archimedean_ceiling @ A @ ( numeral_numeral @ A @ V ) )
% 2.64/1.79 = ( numeral_numeral @ int @ V ) ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_20_of__int__power__eq__of__int__cancel__iff,axiom,
% 2.64/1.79 ! [A: $tType] :
% 2.64/1.79 ( ( ring_char_0 @ A )
% 2.64/1.79 => ! [X: int,B: int,W: nat] :
% 2.64/1.79 ( ( ( ring_1_of_int @ A @ X )
% 2.64/1.79 = ( power_power @ A @ ( ring_1_of_int @ A @ B ) @ W ) )
% 2.64/1.79 = ( X
% 2.64/1.79 = ( power_power @ int @ B @ W ) ) ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_21_numeral__power__eq__nat__cancel__iff,axiom,
% 2.64/1.79 ! [X: num,N: nat,Y: int] :
% 2.64/1.79 ( ( ( power_power @ nat @ ( numeral_numeral @ nat @ X ) @ N )
% 2.64/1.79 = ( nat2 @ Y ) )
% 2.64/1.79 = ( ( power_power @ int @ ( numeral_numeral @ int @ X ) @ N )
% 2.64/1.79 = Y ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_22_nat__eq__numeral__power__cancel__iff,axiom,
% 2.64/1.79 ! [Y: int,X: num,N: nat] :
% 2.64/1.79 ( ( ( nat2 @ Y )
% 2.64/1.79 = ( power_power @ nat @ ( numeral_numeral @ nat @ X ) @ N ) )
% 2.64/1.79 = ( Y
% 2.64/1.79 = ( power_power @ int @ ( numeral_numeral @ int @ X ) @ N ) ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_23_numeral__power__le__nat__cancel__iff,axiom,
% 2.64/1.79 ! [X: num,N: nat,A2: int] :
% 2.64/1.79 ( ( ord_less_eq @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ X ) @ N ) @ ( nat2 @ A2 ) )
% 2.64/1.79 = ( ord_less_eq @ int @ ( power_power @ int @ ( numeral_numeral @ int @ X ) @ N ) @ A2 ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_24_nat__le__numeral__power__cancel__iff,axiom,
% 2.64/1.79 ! [A2: int,X: num,N: nat] :
% 2.64/1.79 ( ( ord_less_eq @ nat @ ( nat2 @ A2 ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ X ) @ N ) )
% 2.64/1.79 = ( ord_less_eq @ int @ A2 @ ( power_power @ int @ ( numeral_numeral @ int @ X ) @ N ) ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_25_nat__mono,axiom,
% 2.64/1.79 ! [X: int,Y: int] :
% 2.64/1.79 ( ( ord_less_eq @ int @ X @ Y )
% 2.64/1.79 => ( ord_less_eq @ nat @ ( nat2 @ X ) @ ( nat2 @ Y ) ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_26_ex__le__of__int,axiom,
% 2.64/1.79 ! [A: $tType] :
% 2.64/1.79 ( ( archim1804426504_field @ A )
% 2.64/1.79 => ! [X: A] :
% 2.64/1.79 ? [Z2: int] : ( ord_less_eq @ A @ X @ ( ring_1_of_int @ A @ Z2 ) ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_27_le__of__int__ceiling,axiom,
% 2.64/1.79 ! [A: $tType] :
% 2.64/1.79 ( ( archim1727834104eiling @ A )
% 2.64/1.79 => ! [X: A] : ( ord_less_eq @ A @ X @ ( ring_1_of_int @ A @ ( archimedean_ceiling @ A @ X ) ) ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_28_ceiling__mono,axiom,
% 2.64/1.79 ! [A: $tType] :
% 2.64/1.79 ( ( archim1727834104eiling @ A )
% 2.64/1.79 => ! [Y: A,X: A] :
% 2.64/1.79 ( ( ord_less_eq @ A @ Y @ X )
% 2.64/1.79 => ( ord_less_eq @ int @ ( archimedean_ceiling @ A @ Y ) @ ( archimedean_ceiling @ A @ X ) ) ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_29_ceiling__le__iff,axiom,
% 2.64/1.79 ! [A: $tType] :
% 2.64/1.79 ( ( archim1727834104eiling @ A )
% 2.64/1.79 => ! [X: A,Z: int] :
% 2.64/1.79 ( ( ord_less_eq @ int @ ( archimedean_ceiling @ A @ X ) @ Z )
% 2.64/1.79 = ( ord_less_eq @ A @ X @ ( ring_1_of_int @ A @ Z ) ) ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_30_ceiling__le,axiom,
% 2.64/1.79 ! [A: $tType] :
% 2.64/1.79 ( ( archim1727834104eiling @ A )
% 2.64/1.79 => ! [X: A,A2: int] :
% 2.64/1.79 ( ( ord_less_eq @ A @ X @ ( ring_1_of_int @ A @ A2 ) )
% 2.64/1.79 => ( ord_less_eq @ int @ ( archimedean_ceiling @ A @ X ) @ A2 ) ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_31__092_060open_062log_A2_A1_A_092_060le_062_Alog_A2_A_Ireal__of__int_Ad_J_092_060close_062,axiom,
% 2.64/1.79 ord_less_eq @ real @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( one_one @ real ) ) @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( ring_1_of_int @ real @ d ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_32_semiring__norm_I69_J,axiom,
% 2.64/1.79 ! [M: num] :
% 2.64/1.79 ~ ( ord_less_eq @ num @ ( bit0 @ M ) @ one2 ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_33_semiring__norm_I83_J,axiom,
% 2.64/1.79 ! [N: num] :
% 2.64/1.79 ( one2
% 2.64/1.79 != ( bit0 @ N ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_34_semiring__norm_I85_J,axiom,
% 2.64/1.79 ! [M: num] :
% 2.64/1.79 ( ( bit0 @ M )
% 2.64/1.79 != one2 ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_35_numeral__le__iff,axiom,
% 2.64/1.79 ! [A: $tType] :
% 2.64/1.79 ( ( linord1659791738miring @ A )
% 2.64/1.79 => ! [M: num,N: num] :
% 2.64/1.79 ( ( ord_less_eq @ A @ ( numeral_numeral @ A @ M ) @ ( numeral_numeral @ A @ N ) )
% 2.64/1.79 = ( ord_less_eq @ num @ M @ N ) ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_36__092_060open_062real__of__int_Ad_A_061_A2_Apowr_Alog_A2_A_Ireal__of__int_Ad_J_092_060close_062,axiom,
% 2.64/1.79 ( ( ring_1_of_int @ real @ d )
% 2.64/1.79 = ( powr @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( ring_1_of_int @ real @ d ) ) ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_37_n__def,axiom,
% 2.64/1.79 ( n
% 2.64/1.79 = ( suc @ ( nat2 @ ( archimedean_ceiling @ real @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( ring_1_of_int @ real @ d ) ) ) ) ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_38_semiring__norm_I68_J,axiom,
% 2.64/1.79 ! [N: num] : ( ord_less_eq @ num @ one2 @ N ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_39_semiring__norm_I71_J,axiom,
% 2.64/1.79 ! [M: num,N: num] :
% 2.64/1.79 ( ( ord_less_eq @ num @ ( bit0 @ M ) @ ( bit0 @ N ) )
% 2.64/1.79 = ( ord_less_eq @ num @ M @ N ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_40_semiring__norm_I87_J,axiom,
% 2.64/1.79 ! [M: num,N: num] :
% 2.64/1.79 ( ( ( bit0 @ M )
% 2.64/1.79 = ( bit0 @ N ) )
% 2.64/1.79 = ( M = N ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_41_verit__eq__simplify_I8_J,axiom,
% 2.64/1.79 ! [X2: num,Y2: num] :
% 2.64/1.79 ( ( ( bit0 @ X2 )
% 2.64/1.79 = ( bit0 @ Y2 ) )
% 2.64/1.79 = ( X2 = Y2 ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_42_numeral__eq__iff,axiom,
% 2.64/1.79 ! [A: $tType] :
% 2.64/1.79 ( ( semiring_char_0 @ A )
% 2.64/1.79 => ! [M: num,N: num] :
% 2.64/1.79 ( ( ( numeral_numeral @ A @ M )
% 2.64/1.79 = ( numeral_numeral @ A @ N ) )
% 2.64/1.79 = ( M = N ) ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_43_of__int__eq__1__iff,axiom,
% 2.64/1.79 ! [A: $tType] :
% 2.64/1.79 ( ( ring_char_0 @ A )
% 2.64/1.79 => ! [Z: int] :
% 2.64/1.79 ( ( ( ring_1_of_int @ A @ Z )
% 2.64/1.79 = ( one_one @ A ) )
% 2.64/1.79 = ( Z
% 2.64/1.79 = ( one_one @ int ) ) ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_44_mem__Collect__eq,axiom,
% 2.64/1.79 ! [A: $tType,A2: A,P: A > $o] :
% 2.64/1.79 ( ( member @ A @ A2 @ ( collect @ A @ P ) )
% 2.64/1.79 = ( P @ A2 ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_45_Collect__mem__eq,axiom,
% 2.64/1.79 ! [A: $tType,A3: set @ A] :
% 2.64/1.79 ( ( collect @ A
% 2.64/1.79 @ ^ [X3: A] : ( member @ A @ X3 @ A3 ) )
% 2.64/1.79 = A3 ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_46_Collect__cong,axiom,
% 2.64/1.79 ! [A: $tType,P: A > $o,Q: A > $o] :
% 2.64/1.79 ( ! [X4: A] :
% 2.64/1.79 ( ( P @ X4 )
% 2.64/1.79 = ( Q @ X4 ) )
% 2.64/1.79 => ( ( collect @ A @ P )
% 2.64/1.79 = ( collect @ A @ Q ) ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_47_ext,axiom,
% 2.64/1.79 ! [B2: $tType,A: $tType,F: A > B2,G: A > B2] :
% 2.64/1.79 ( ! [X4: A] :
% 2.64/1.79 ( ( F @ X4 )
% 2.64/1.79 = ( G @ X4 ) )
% 2.64/1.79 => ( F = G ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_48_of__int__1,axiom,
% 2.64/1.79 ! [A: $tType] :
% 2.64/1.79 ( ( ring_1 @ A )
% 2.64/1.79 => ( ( ring_1_of_int @ A @ ( one_one @ int ) )
% 2.64/1.79 = ( one_one @ A ) ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_49_ceiling__one,axiom,
% 2.64/1.79 ! [A: $tType] :
% 2.64/1.79 ( ( archim1727834104eiling @ A )
% 2.64/1.79 => ( ( archimedean_ceiling @ A @ ( one_one @ A ) )
% 2.64/1.79 = ( one_one @ int ) ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_50_numeral__eq__one__iff,axiom,
% 2.64/1.79 ! [A: $tType] :
% 2.64/1.79 ( ( semiring_char_0 @ A )
% 2.64/1.79 => ! [N: num] :
% 2.64/1.79 ( ( ( numeral_numeral @ A @ N )
% 2.64/1.79 = ( one_one @ A ) )
% 2.64/1.79 = ( N = one2 ) ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_51_one__eq__numeral__iff,axiom,
% 2.64/1.79 ! [A: $tType] :
% 2.64/1.79 ( ( semiring_char_0 @ A )
% 2.64/1.79 => ! [N: num] :
% 2.64/1.79 ( ( ( one_one @ A )
% 2.64/1.79 = ( numeral_numeral @ A @ N ) )
% 2.64/1.79 = ( one2 = N ) ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_52_of__int__1__le__iff,axiom,
% 2.64/1.79 ! [A: $tType] :
% 2.64/1.79 ( ( linordered_idom @ A )
% 2.64/1.79 => ! [Z: int] :
% 2.64/1.79 ( ( ord_less_eq @ A @ ( one_one @ A ) @ ( ring_1_of_int @ A @ Z ) )
% 2.64/1.79 = ( ord_less_eq @ int @ ( one_one @ int ) @ Z ) ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_53_of__int__le__1__iff,axiom,
% 2.64/1.79 ! [A: $tType] :
% 2.64/1.79 ( ( linordered_idom @ A )
% 2.64/1.79 => ! [Z: int] :
% 2.64/1.79 ( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ Z ) @ ( one_one @ A ) )
% 2.64/1.79 = ( ord_less_eq @ int @ Z @ ( one_one @ int ) ) ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_54_ceiling__le__one,axiom,
% 2.64/1.79 ! [A: $tType] :
% 2.64/1.79 ( ( archim1727834104eiling @ A )
% 2.64/1.79 => ! [X: A] :
% 2.64/1.79 ( ( ord_less_eq @ int @ ( archimedean_ceiling @ A @ X ) @ ( one_one @ int ) )
% 2.64/1.79 = ( ord_less_eq @ A @ X @ ( one_one @ A ) ) ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_55_numeral__le__one__iff,axiom,
% 2.64/1.79 ! [A: $tType] :
% 2.64/1.79 ( ( linord1659791738miring @ A )
% 2.64/1.79 => ! [N: num] :
% 2.64/1.79 ( ( ord_less_eq @ A @ ( numeral_numeral @ A @ N ) @ ( one_one @ A ) )
% 2.64/1.79 = ( ord_less_eq @ num @ N @ one2 ) ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_56_complete__real,axiom,
% 2.64/1.79 ! [S: set @ real] :
% 2.64/1.79 ( ? [X5: real] : ( member @ real @ X5 @ S )
% 2.64/1.79 => ( ? [Z3: real] :
% 2.64/1.79 ! [X4: real] :
% 2.64/1.79 ( ( member @ real @ X4 @ S )
% 2.64/1.79 => ( ord_less_eq @ real @ X4 @ Z3 ) )
% 2.64/1.79 => ? [Y3: real] :
% 2.64/1.79 ( ! [X5: real] :
% 2.64/1.79 ( ( member @ real @ X5 @ S )
% 2.64/1.79 => ( ord_less_eq @ real @ X5 @ Y3 ) )
% 2.64/1.79 & ! [Z3: real] :
% 2.64/1.79 ( ! [X4: real] :
% 2.64/1.79 ( ( member @ real @ X4 @ S )
% 2.64/1.79 => ( ord_less_eq @ real @ X4 @ Z3 ) )
% 2.64/1.79 => ( ord_less_eq @ real @ Y3 @ Z3 ) ) ) ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_57_le__numeral__extra_I4_J,axiom,
% 2.64/1.79 ! [A: $tType] :
% 2.64/1.79 ( ( linord1659791738miring @ A )
% 2.64/1.79 => ( ord_less_eq @ A @ ( one_one @ A ) @ ( one_one @ A ) ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_58_one__le__numeral,axiom,
% 2.64/1.79 ! [A: $tType] :
% 2.64/1.79 ( ( linord1659791738miring @ A )
% 2.64/1.79 => ! [N: num] : ( ord_less_eq @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ N ) ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_59_numeral__One,axiom,
% 2.64/1.79 ! [A: $tType] :
% 2.64/1.79 ( ( numeral @ A )
% 2.64/1.79 => ( ( numeral_numeral @ A @ one2 )
% 2.64/1.79 = ( one_one @ A ) ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_60_verit__la__disequality,axiom,
% 2.64/1.79 ! [A: $tType] :
% 2.64/1.79 ( ( linorder @ A )
% 2.64/1.79 => ! [A2: A,B: A] :
% 2.64/1.79 ( ( A2 = B )
% 2.64/1.79 | ~ ( ord_less_eq @ A @ A2 @ B )
% 2.64/1.79 | ~ ( ord_less_eq @ A @ B @ A2 ) ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_61_le__num__One__iff,axiom,
% 2.64/1.79 ! [X: num] :
% 2.64/1.79 ( ( ord_less_eq @ num @ X @ one2 )
% 2.64/1.79 = ( X = one2 ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_62_two__realpow__ge__one,axiom,
% 2.64/1.79 ! [N: nat] : ( ord_less_eq @ real @ ( one_one @ real ) @ ( power_power @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ N ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_63_verit__la__generic,axiom,
% 2.64/1.79 ! [A2: int,X: int] :
% 2.64/1.79 ( ( ord_less_eq @ int @ A2 @ X )
% 2.64/1.79 | ( A2 = X )
% 2.64/1.79 | ( ord_less_eq @ int @ X @ A2 ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_64_verit__eq__simplify_I10_J,axiom,
% 2.64/1.79 ! [X2: num] :
% 2.64/1.79 ( one2
% 2.64/1.79 != ( bit0 @ X2 ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_65_nat__numeral__as__int,axiom,
% 2.64/1.79 ( ( numeral_numeral @ nat )
% 2.64/1.79 = ( ^ [I: num] : ( nat2 @ ( numeral_numeral @ int @ I ) ) ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_66__092_060open_0622_Apowr_Areal_A_Inat_A_092_060lceil_062log_A2_A_Ireal__of__int_Ad_J_092_060rceil_062_J_A_061_Areal__of__int_A_I2_A_094_Anat_A_092_060lceil_062log_A2_A_Ireal__of__int_Ad_J_092_060rceil_062_J_092_060close_062,axiom,
% 2.64/1.79 ( ( powr @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ real @ ( nat2 @ ( archimedean_ceiling @ real @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( ring_1_of_int @ real @ d ) ) ) ) ) )
% 2.64/1.79 = ( ring_1_of_int @ real @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( nat2 @ ( archimedean_ceiling @ real @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( ring_1_of_int @ real @ d ) ) ) ) ) ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_67__092_060open_0622_Apowr_Alog_A2_A_Ireal__of__int_Ad_J_A_092_060le_062_A2_Apowr_Areal_A_Inat_A_092_060lceil_062log_A2_A_Ireal__of__int_Ad_J_092_060rceil_062_J_092_060close_062,axiom,
% 2.64/1.79 ord_less_eq @ real @ ( powr @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( ring_1_of_int @ real @ d ) ) ) @ ( powr @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ real @ ( nat2 @ ( archimedean_ceiling @ real @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( ring_1_of_int @ real @ d ) ) ) ) ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_68_numeral__powr__numeral__real,axiom,
% 2.64/1.79 ! [M: num,N: num] :
% 2.64/1.79 ( ( powr @ real @ ( numeral_numeral @ real @ M ) @ ( numeral_numeral @ real @ N ) )
% 2.64/1.79 = ( power_power @ real @ ( numeral_numeral @ real @ M ) @ ( numeral_numeral @ nat @ N ) ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_69_powr__one__eq__one,axiom,
% 2.64/1.79 ! [A: $tType] :
% 2.64/1.79 ( ( ln @ A )
% 2.64/1.79 => ! [A2: A] :
% 2.64/1.79 ( ( powr @ A @ ( one_one @ A ) @ A2 )
% 2.64/1.79 = ( one_one @ A ) ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_70_self__le__ge2__pow,axiom,
% 2.64/1.79 ! [K: nat,M: nat] :
% 2.64/1.79 ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ K )
% 2.64/1.79 => ( ord_less_eq @ nat @ M @ ( power_power @ nat @ K @ M ) ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_71_power2__nat__le__eq__le,axiom,
% 2.64/1.79 ! [M: nat,N: nat] :
% 2.64/1.79 ( ( ord_less_eq @ nat @ ( power_power @ nat @ M @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
% 2.64/1.79 = ( ord_less_eq @ nat @ M @ N ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_72_power2__nat__le__imp__le,axiom,
% 2.64/1.79 ! [M: nat,N: nat] :
% 2.64/1.79 ( ( ord_less_eq @ nat @ ( power_power @ nat @ M @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ N )
% 2.64/1.79 => ( ord_less_eq @ nat @ M @ N ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_73_Suc__le__mono,axiom,
% 2.64/1.79 ! [N: nat,M: nat] :
% 2.64/1.79 ( ( ord_less_eq @ nat @ ( suc @ N ) @ ( suc @ M ) )
% 2.64/1.79 = ( ord_less_eq @ nat @ N @ M ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_74_power__one__right,axiom,
% 2.64/1.79 ! [A: $tType] :
% 2.64/1.79 ( ( monoid_mult @ A )
% 2.64/1.79 => ! [A2: A] :
% 2.64/1.79 ( ( power_power @ A @ A2 @ ( one_one @ nat ) )
% 2.64/1.79 = A2 ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_75_nat_Oinject,axiom,
% 2.64/1.79 ! [X2: nat,Y2: nat] :
% 2.64/1.79 ( ( ( suc @ X2 )
% 2.64/1.79 = ( suc @ Y2 ) )
% 2.64/1.79 = ( X2 = Y2 ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_76_old_Onat_Oinject,axiom,
% 2.64/1.79 ! [Nat: nat,Nat2: nat] :
% 2.64/1.79 ( ( ( suc @ Nat )
% 2.64/1.79 = ( suc @ Nat2 ) )
% 2.64/1.79 = ( Nat = Nat2 ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_77_of__nat__eq__iff,axiom,
% 2.64/1.79 ! [A: $tType] :
% 2.64/1.79 ( ( semiring_char_0 @ A )
% 2.64/1.79 => ! [M: nat,N: nat] :
% 2.64/1.79 ( ( ( semiring_1_of_nat @ A @ M )
% 2.64/1.79 = ( semiring_1_of_nat @ A @ N ) )
% 2.64/1.79 = ( M = N ) ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_78_power__one,axiom,
% 2.64/1.79 ! [A: $tType] :
% 2.64/1.79 ( ( monoid_mult @ A )
% 2.64/1.79 => ! [N: nat] :
% 2.64/1.79 ( ( power_power @ A @ ( one_one @ A ) @ N )
% 2.64/1.79 = ( one_one @ A ) ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_79_of__nat__eq__1__iff,axiom,
% 2.64/1.79 ! [A: $tType] :
% 2.64/1.79 ( ( semiring_char_0 @ A )
% 2.64/1.79 => ! [N: nat] :
% 2.64/1.79 ( ( ( semiring_1_of_nat @ A @ N )
% 2.64/1.79 = ( one_one @ A ) )
% 2.64/1.79 = ( N
% 2.64/1.79 = ( one_one @ nat ) ) ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_80_of__nat__1__eq__iff,axiom,
% 2.64/1.79 ! [A: $tType] :
% 2.64/1.79 ( ( semiring_char_0 @ A )
% 2.64/1.79 => ! [N: nat] :
% 2.64/1.79 ( ( ( one_one @ A )
% 2.64/1.79 = ( semiring_1_of_nat @ A @ N ) )
% 2.64/1.79 = ( N
% 2.64/1.79 = ( one_one @ nat ) ) ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_81_of__nat__1,axiom,
% 2.64/1.79 ! [A: $tType] :
% 2.64/1.79 ( ( semiring_1 @ A )
% 2.64/1.79 => ( ( semiring_1_of_nat @ A @ ( one_one @ nat ) )
% 2.64/1.79 = ( one_one @ A ) ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_82_of__int__of__nat__eq,axiom,
% 2.64/1.79 ! [A: $tType] :
% 2.64/1.79 ( ( ring_1 @ A )
% 2.64/1.79 => ! [N: nat] :
% 2.64/1.79 ( ( ring_1_of_int @ A @ ( semiring_1_of_nat @ int @ N ) )
% 2.64/1.79 = ( semiring_1_of_nat @ A @ N ) ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_83_ceiling__of__nat,axiom,
% 2.64/1.79 ! [A: $tType] :
% 2.64/1.79 ( ( archim1727834104eiling @ A )
% 2.64/1.79 => ! [N: nat] :
% 2.64/1.79 ( ( archimedean_ceiling @ A @ ( semiring_1_of_nat @ A @ N ) )
% 2.64/1.79 = ( semiring_1_of_nat @ int @ N ) ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_84_of__nat__le__iff,axiom,
% 2.64/1.79 ! [A: $tType] :
% 2.64/1.79 ( ( linord1659791738miring @ A )
% 2.64/1.79 => ! [M: nat,N: nat] :
% 2.64/1.79 ( ( ord_less_eq @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N ) )
% 2.64/1.79 = ( ord_less_eq @ nat @ M @ N ) ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_85_of__nat__numeral,axiom,
% 2.64/1.79 ! [A: $tType] :
% 2.64/1.79 ( ( semiring_1 @ A )
% 2.64/1.79 => ! [N: num] :
% 2.64/1.79 ( ( semiring_1_of_nat @ A @ ( numeral_numeral @ nat @ N ) )
% 2.64/1.79 = ( numeral_numeral @ A @ N ) ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_86_of__nat__power__eq__of__nat__cancel__iff,axiom,
% 2.64/1.79 ! [A: $tType] :
% 2.64/1.79 ( ( semiring_char_0 @ A )
% 2.64/1.79 => ! [X: nat,B: nat,W: nat] :
% 2.64/1.79 ( ( ( semiring_1_of_nat @ A @ X )
% 2.64/1.79 = ( power_power @ A @ ( semiring_1_of_nat @ A @ B ) @ W ) )
% 2.64/1.79 = ( X
% 2.64/1.79 = ( power_power @ nat @ B @ W ) ) ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_87_of__nat__eq__of__nat__power__cancel__iff,axiom,
% 2.64/1.79 ! [A: $tType] :
% 2.64/1.79 ( ( semiring_char_0 @ A )
% 2.64/1.79 => ! [B: nat,W: nat,X: nat] :
% 2.64/1.79 ( ( ( power_power @ A @ ( semiring_1_of_nat @ A @ B ) @ W )
% 2.64/1.79 = ( semiring_1_of_nat @ A @ X ) )
% 2.64/1.79 = ( ( power_power @ nat @ B @ W )
% 2.64/1.79 = X ) ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_88_of__nat__power,axiom,
% 2.64/1.79 ! [A: $tType] :
% 2.64/1.79 ( ( semiring_1 @ A )
% 2.64/1.79 => ! [M: nat,N: nat] :
% 2.64/1.79 ( ( semiring_1_of_nat @ A @ ( power_power @ nat @ M @ N ) )
% 2.64/1.79 = ( power_power @ A @ ( semiring_1_of_nat @ A @ M ) @ N ) ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_89_Suc__1,axiom,
% 2.64/1.79 ( ( suc @ ( one_one @ nat ) )
% 2.64/1.79 = ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_90_of__nat__power__le__of__nat__cancel__iff,axiom,
% 2.64/1.79 ! [A: $tType] :
% 2.64/1.79 ( ( linordered_semidom @ A )
% 2.64/1.79 => ! [X: nat,B: nat,W: nat] :
% 2.64/1.79 ( ( ord_less_eq @ A @ ( semiring_1_of_nat @ A @ X ) @ ( power_power @ A @ ( semiring_1_of_nat @ A @ B ) @ W ) )
% 2.64/1.79 = ( ord_less_eq @ nat @ X @ ( power_power @ nat @ B @ W ) ) ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_91_of__nat__le__of__nat__power__cancel__iff,axiom,
% 2.64/1.79 ! [A: $tType] :
% 2.64/1.79 ( ( linordered_semidom @ A )
% 2.64/1.79 => ! [B: nat,W: nat,X: nat] :
% 2.64/1.79 ( ( ord_less_eq @ A @ ( power_power @ A @ ( semiring_1_of_nat @ A @ B ) @ W ) @ ( semiring_1_of_nat @ A @ X ) )
% 2.64/1.79 = ( ord_less_eq @ nat @ ( power_power @ nat @ B @ W ) @ X ) ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_92_real__of__nat__eq__numeral__power__cancel__iff,axiom,
% 2.64/1.79 ! [A: $tType] :
% 2.64/1.79 ( ( semiring_char_0 @ A )
% 2.64/1.79 => ! [Y: nat,X: num,N: nat] :
% 2.64/1.79 ( ( ( semiring_1_of_nat @ A @ Y )
% 2.64/1.79 = ( power_power @ A @ ( numeral_numeral @ A @ X ) @ N ) )
% 2.64/1.79 = ( Y
% 2.64/1.79 = ( power_power @ nat @ ( numeral_numeral @ nat @ X ) @ N ) ) ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_93_numeral__power__eq__of__nat__cancel__iff,axiom,
% 2.64/1.79 ! [A: $tType] :
% 2.64/1.79 ( ( semiring_char_0 @ A )
% 2.64/1.79 => ! [X: num,N: nat,Y: nat] :
% 2.64/1.79 ( ( ( power_power @ A @ ( numeral_numeral @ A @ X ) @ N )
% 2.64/1.79 = ( semiring_1_of_nat @ A @ Y ) )
% 2.64/1.79 = ( ( power_power @ nat @ ( numeral_numeral @ nat @ X ) @ N )
% 2.64/1.79 = Y ) ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_94_numeral__le__real__of__nat__iff,axiom,
% 2.64/1.79 ! [N: num,M: nat] :
% 2.64/1.79 ( ( ord_less_eq @ real @ ( numeral_numeral @ real @ N ) @ ( semiring_1_of_nat @ real @ M ) )
% 2.64/1.79 = ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ N ) @ M ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_95_nat__ceiling__le__eq,axiom,
% 2.64/1.79 ! [X: real,A2: nat] :
% 2.64/1.79 ( ( ord_less_eq @ nat @ ( nat2 @ ( archimedean_ceiling @ real @ X ) ) @ A2 )
% 2.64/1.79 = ( ord_less_eq @ real @ X @ ( semiring_1_of_nat @ real @ A2 ) ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_96_of__nat__le__numeral__power__cancel__iff,axiom,
% 2.64/1.79 ! [A: $tType] :
% 2.64/1.79 ( ( linordered_semidom @ A )
% 2.64/1.79 => ! [X: nat,I2: num,N: nat] :
% 2.64/1.79 ( ( ord_less_eq @ A @ ( semiring_1_of_nat @ A @ X ) @ ( power_power @ A @ ( numeral_numeral @ A @ I2 ) @ N ) )
% 2.64/1.79 = ( ord_less_eq @ nat @ X @ ( power_power @ nat @ ( numeral_numeral @ nat @ I2 ) @ N ) ) ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_97_numeral__power__le__of__nat__cancel__iff,axiom,
% 2.64/1.79 ! [A: $tType] :
% 2.64/1.79 ( ( linordered_semidom @ A )
% 2.64/1.79 => ! [I2: num,N: nat,X: nat] :
% 2.64/1.79 ( ( ord_less_eq @ A @ ( power_power @ A @ ( numeral_numeral @ A @ I2 ) @ N ) @ ( semiring_1_of_nat @ A @ X ) )
% 2.64/1.79 = ( ord_less_eq @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ I2 ) @ N ) @ X ) ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_98_nat__one__as__int,axiom,
% 2.64/1.79 ( ( one_one @ nat )
% 2.64/1.79 = ( nat2 @ ( one_one @ int ) ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_99_real__arch__simple,axiom,
% 2.64/1.79 ! [A: $tType] :
% 2.64/1.79 ( ( archim1804426504_field @ A )
% 2.64/1.79 => ! [X: A] :
% 2.64/1.79 ? [N3: nat] : ( ord_less_eq @ A @ X @ ( semiring_1_of_nat @ A @ N3 ) ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_100_of__nat__mono,axiom,
% 2.64/1.79 ! [A: $tType] :
% 2.64/1.79 ( ( linord1659791738miring @ A )
% 2.64/1.79 => ! [I2: nat,J: nat] :
% 2.64/1.79 ( ( ord_less_eq @ nat @ I2 @ J )
% 2.64/1.79 => ( ord_less_eq @ A @ ( semiring_1_of_nat @ A @ I2 ) @ ( semiring_1_of_nat @ A @ J ) ) ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_101_numerals_I1_J,axiom,
% 2.64/1.79 ( ( numeral_numeral @ nat @ one2 )
% 2.64/1.79 = ( one_one @ nat ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_102_of__nat__ceiling,axiom,
% 2.64/1.79 ! [A: $tType] :
% 2.64/1.79 ( ( archim1727834104eiling @ A )
% 2.64/1.79 => ! [R: A] : ( ord_less_eq @ A @ R @ ( semiring_1_of_nat @ A @ ( nat2 @ ( archimedean_ceiling @ A @ R ) ) ) ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_103_Suc__inject,axiom,
% 2.64/1.79 ! [X: nat,Y: nat] :
% 2.64/1.79 ( ( ( suc @ X )
% 2.64/1.79 = ( suc @ Y ) )
% 2.64/1.79 => ( X = Y ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_104_n__not__Suc__n,axiom,
% 2.64/1.79 ! [N: nat] :
% 2.64/1.79 ( N
% 2.64/1.79 != ( suc @ N ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_105_Nat_Oex__has__greatest__nat,axiom,
% 2.64/1.79 ! [P: nat > $o,K: nat,B: nat] :
% 2.64/1.79 ( ( P @ K )
% 2.64/1.79 => ( ! [Y3: nat] :
% 2.64/1.79 ( ( P @ Y3 )
% 2.64/1.79 => ( ord_less_eq @ nat @ Y3 @ B ) )
% 2.64/1.79 => ? [X4: nat] :
% 2.64/1.79 ( ( P @ X4 )
% 2.64/1.79 & ! [Y4: nat] :
% 2.64/1.79 ( ( P @ Y4 )
% 2.64/1.79 => ( ord_less_eq @ nat @ Y4 @ X4 ) ) ) ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_106_nat__le__linear,axiom,
% 2.64/1.79 ! [M: nat,N: nat] :
% 2.64/1.79 ( ( ord_less_eq @ nat @ M @ N )
% 2.64/1.79 | ( ord_less_eq @ nat @ N @ M ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_107_le__antisym,axiom,
% 2.64/1.79 ! [M: nat,N: nat] :
% 2.64/1.79 ( ( ord_less_eq @ nat @ M @ N )
% 2.64/1.79 => ( ( ord_less_eq @ nat @ N @ M )
% 2.64/1.79 => ( M = N ) ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_108_eq__imp__le,axiom,
% 2.64/1.79 ! [M: nat,N: nat] :
% 2.64/1.79 ( ( M = N )
% 2.64/1.79 => ( ord_less_eq @ nat @ M @ N ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_109_le__trans,axiom,
% 2.64/1.79 ! [I2: nat,J: nat,K: nat] :
% 2.64/1.79 ( ( ord_less_eq @ nat @ I2 @ J )
% 2.64/1.79 => ( ( ord_less_eq @ nat @ J @ K )
% 2.64/1.79 => ( ord_less_eq @ nat @ I2 @ K ) ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_110_le__refl,axiom,
% 2.64/1.79 ! [N: nat] : ( ord_less_eq @ nat @ N @ N ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_111_real__nat__ceiling__ge,axiom,
% 2.64/1.79 ! [X: real] : ( ord_less_eq @ real @ X @ ( semiring_1_of_nat @ real @ ( nat2 @ ( archimedean_ceiling @ real @ X ) ) ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_112_powr__powr__swap,axiom,
% 2.64/1.79 ! [X: real,A2: real,B: real] :
% 2.64/1.79 ( ( powr @ real @ ( powr @ real @ X @ A2 ) @ B )
% 2.64/1.79 = ( powr @ real @ ( powr @ real @ X @ B ) @ A2 ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_113_log2__of__power__eq,axiom,
% 2.64/1.79 ! [M: nat,N: nat] :
% 2.64/1.79 ( ( M
% 2.64/1.79 = ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) )
% 2.64/1.79 => ( ( semiring_1_of_nat @ real @ N )
% 2.64/1.79 = ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ real @ M ) ) ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_114_le__log2__of__power,axiom,
% 2.64/1.79 ! [N: nat,M: nat] :
% 2.64/1.79 ( ( ord_less_eq @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) @ M )
% 2.64/1.79 => ( ord_less_eq @ real @ ( semiring_1_of_nat @ real @ N ) @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ real @ M ) ) ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_115_transitive__stepwise__le,axiom,
% 2.64/1.79 ! [M: nat,N: nat,R2: nat > nat > $o] :
% 2.64/1.79 ( ( ord_less_eq @ nat @ M @ N )
% 2.64/1.79 => ( ! [X4: nat] : ( R2 @ X4 @ X4 )
% 2.64/1.79 => ( ! [X4: nat,Y3: nat,Z2: nat] :
% 2.64/1.79 ( ( R2 @ X4 @ Y3 )
% 2.64/1.79 => ( ( R2 @ Y3 @ Z2 )
% 2.64/1.79 => ( R2 @ X4 @ Z2 ) ) )
% 2.64/1.79 => ( ! [N3: nat] : ( R2 @ N3 @ ( suc @ N3 ) )
% 2.64/1.79 => ( R2 @ M @ N ) ) ) ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_116_nat__induct__at__least,axiom,
% 2.64/1.79 ! [M: nat,N: nat,P: nat > $o] :
% 2.64/1.79 ( ( ord_less_eq @ nat @ M @ N )
% 2.64/1.79 => ( ( P @ M )
% 2.64/1.79 => ( ! [N3: nat] :
% 2.64/1.79 ( ( ord_less_eq @ nat @ M @ N3 )
% 2.64/1.79 => ( ( P @ N3 )
% 2.64/1.79 => ( P @ ( suc @ N3 ) ) ) )
% 2.64/1.79 => ( P @ N ) ) ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_117_full__nat__induct,axiom,
% 2.64/1.79 ! [P: nat > $o,N: nat] :
% 2.64/1.79 ( ! [N3: nat] :
% 2.64/1.79 ( ! [M2: nat] :
% 2.64/1.79 ( ( ord_less_eq @ nat @ ( suc @ M2 ) @ N3 )
% 2.64/1.79 => ( P @ M2 ) )
% 2.64/1.79 => ( P @ N3 ) )
% 2.64/1.79 => ( P @ N ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_118_not__less__eq__eq,axiom,
% 2.64/1.79 ! [M: nat,N: nat] :
% 2.64/1.79 ( ( ~ ( ord_less_eq @ nat @ M @ N ) )
% 2.64/1.79 = ( ord_less_eq @ nat @ ( suc @ N ) @ M ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_119_Suc__n__not__le__n,axiom,
% 2.64/1.79 ! [N: nat] :
% 2.64/1.79 ~ ( ord_less_eq @ nat @ ( suc @ N ) @ N ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_120_le__Suc__eq,axiom,
% 2.64/1.79 ! [M: nat,N: nat] :
% 2.64/1.79 ( ( ord_less_eq @ nat @ M @ ( suc @ N ) )
% 2.64/1.79 = ( ( ord_less_eq @ nat @ M @ N )
% 2.64/1.79 | ( M
% 2.64/1.79 = ( suc @ N ) ) ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_121_Suc__le__D,axiom,
% 2.64/1.79 ! [N: nat,M3: nat] :
% 2.64/1.79 ( ( ord_less_eq @ nat @ ( suc @ N ) @ M3 )
% 2.64/1.79 => ? [M4: nat] :
% 2.64/1.79 ( M3
% 2.64/1.79 = ( suc @ M4 ) ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_122_le__SucI,axiom,
% 2.64/1.79 ! [M: nat,N: nat] :
% 2.64/1.79 ( ( ord_less_eq @ nat @ M @ N )
% 2.64/1.79 => ( ord_less_eq @ nat @ M @ ( suc @ N ) ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_123_le__SucE,axiom,
% 2.64/1.79 ! [M: nat,N: nat] :
% 2.64/1.79 ( ( ord_less_eq @ nat @ M @ ( suc @ N ) )
% 2.64/1.79 => ( ~ ( ord_less_eq @ nat @ M @ N )
% 2.64/1.79 => ( M
% 2.64/1.79 = ( suc @ N ) ) ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_124_Suc__leD,axiom,
% 2.64/1.79 ! [M: nat,N: nat] :
% 2.64/1.79 ( ( ord_less_eq @ nat @ ( suc @ M ) @ N )
% 2.64/1.79 => ( ord_less_eq @ nat @ M @ N ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_125_one__le__power,axiom,
% 2.64/1.79 ! [A: $tType] :
% 2.64/1.79 ( ( linordered_semidom @ A )
% 2.64/1.79 => ! [A2: A,N: nat] :
% 2.64/1.79 ( ( ord_less_eq @ A @ ( one_one @ A ) @ A2 )
% 2.64/1.79 => ( ord_less_eq @ A @ ( one_one @ A ) @ ( power_power @ A @ A2 @ N ) ) ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_126_lift__Suc__antimono__le,axiom,
% 2.64/1.79 ! [A: $tType] :
% 2.64/1.79 ( ( order @ A )
% 2.64/1.79 => ! [F: nat > A,N: nat,N4: nat] :
% 2.64/1.79 ( ! [N3: nat] : ( ord_less_eq @ A @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
% 2.64/1.79 => ( ( ord_less_eq @ nat @ N @ N4 )
% 2.64/1.79 => ( ord_less_eq @ A @ ( F @ N4 ) @ ( F @ N ) ) ) ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_127_lift__Suc__mono__le,axiom,
% 2.64/1.79 ! [A: $tType] :
% 2.64/1.79 ( ( order @ A )
% 2.64/1.79 => ! [F: nat > A,N: nat,N4: nat] :
% 2.64/1.79 ( ! [N3: nat] : ( ord_less_eq @ A @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
% 2.64/1.79 => ( ( ord_less_eq @ nat @ N @ N4 )
% 2.64/1.79 => ( ord_less_eq @ A @ ( F @ N ) @ ( F @ N4 ) ) ) ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_128_powr__mono,axiom,
% 2.64/1.79 ! [A2: real,B: real,X: real] :
% 2.64/1.79 ( ( ord_less_eq @ real @ A2 @ B )
% 2.64/1.79 => ( ( ord_less_eq @ real @ ( one_one @ real ) @ X )
% 2.64/1.79 => ( ord_less_eq @ real @ ( powr @ real @ X @ A2 ) @ ( powr @ real @ X @ B ) ) ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_129_power__increasing,axiom,
% 2.64/1.79 ! [A: $tType] :
% 2.64/1.79 ( ( linordered_semidom @ A )
% 2.64/1.79 => ! [N: nat,N5: nat,A2: A] :
% 2.64/1.79 ( ( ord_less_eq @ nat @ N @ N5 )
% 2.64/1.79 => ( ( ord_less_eq @ A @ ( one_one @ A ) @ A2 )
% 2.64/1.79 => ( ord_less_eq @ A @ ( power_power @ A @ A2 @ N ) @ ( power_power @ A @ A2 @ N5 ) ) ) ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_130_one__power2,axiom,
% 2.64/1.79 ! [A: $tType] :
% 2.64/1.79 ( ( semiring_1 @ A )
% 2.64/1.79 => ( ( power_power @ A @ ( one_one @ A ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
% 2.64/1.79 = ( one_one @ A ) ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_131_dbl__simps_I3_J,axiom,
% 2.64/1.79 ! [A: $tType] :
% 2.64/1.79 ( ( neg_numeral @ A )
% 2.64/1.79 => ( ( neg_numeral_dbl @ A @ ( one_one @ A ) )
% 2.64/1.79 = ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_132_power__numeral,axiom,
% 2.64/1.79 ! [A: $tType] :
% 2.64/1.79 ( ( semiring_numeral @ A )
% 2.64/1.79 => ! [K: num,L: num] :
% 2.64/1.79 ( ( power_power @ A @ ( numeral_numeral @ A @ K ) @ ( numeral_numeral @ nat @ L ) )
% 2.64/1.79 = ( numeral_numeral @ A @ ( pow @ K @ L ) ) ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_133_order__refl,axiom,
% 2.64/1.79 ! [A: $tType] :
% 2.64/1.79 ( ( preorder @ A )
% 2.64/1.79 => ! [X: A] : ( ord_less_eq @ A @ X @ X ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_134_powr__numeral,axiom,
% 2.64/1.79 ! [X: real,N: num] :
% 2.64/1.79 ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
% 2.64/1.79 => ( ( powr @ real @ X @ ( numeral_numeral @ real @ N ) )
% 2.64/1.79 = ( power_power @ real @ X @ ( numeral_numeral @ nat @ N ) ) ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_135_power2__eq__iff__nonneg,axiom,
% 2.64/1.79 ! [A: $tType] :
% 2.64/1.79 ( ( linordered_semidom @ A )
% 2.64/1.79 => ! [X: A,Y: A] :
% 2.64/1.79 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
% 2.64/1.79 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y )
% 2.64/1.79 => ( ( ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
% 2.64/1.79 = ( power_power @ A @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
% 2.64/1.79 = ( X = Y ) ) ) ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_136_le0,axiom,
% 2.64/1.79 ! [N: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ N ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_137_bot__nat__0_Oextremum,axiom,
% 2.64/1.79 ! [A2: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ A2 ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_138_d_I2_J,axiom,
% 2.64/1.79 ord_less @ int @ ( zero_zero @ int ) @ d ).
% 2.64/1.79
% 2.64/1.79 thf(fact_139_le__zero__eq,axiom,
% 2.64/1.79 ! [A: $tType] :
% 2.64/1.79 ( ( canoni770627133id_add @ A )
% 2.64/1.79 => ! [N: A] :
% 2.64/1.79 ( ( ord_less_eq @ A @ N @ ( zero_zero @ A ) )
% 2.64/1.79 = ( N
% 2.64/1.79 = ( zero_zero @ A ) ) ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_140_of__nat__0,axiom,
% 2.64/1.79 ! [A: $tType] :
% 2.64/1.79 ( ( semiring_1 @ A )
% 2.64/1.79 => ( ( semiring_1_of_nat @ A @ ( zero_zero @ nat ) )
% 2.64/1.79 = ( zero_zero @ A ) ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_141_of__nat__0__eq__iff,axiom,
% 2.64/1.79 ! [A: $tType] :
% 2.64/1.79 ( ( semiring_char_0 @ A )
% 2.64/1.79 => ! [N: nat] :
% 2.64/1.79 ( ( ( zero_zero @ A )
% 2.64/1.79 = ( semiring_1_of_nat @ A @ N ) )
% 2.64/1.79 = ( ( zero_zero @ nat )
% 2.64/1.79 = N ) ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_142_of__nat__eq__0__iff,axiom,
% 2.64/1.79 ! [A: $tType] :
% 2.64/1.79 ( ( semiring_char_0 @ A )
% 2.64/1.79 => ! [M: nat] :
% 2.64/1.79 ( ( ( semiring_1_of_nat @ A @ M )
% 2.64/1.79 = ( zero_zero @ A ) )
% 2.64/1.79 = ( M
% 2.64/1.79 = ( zero_zero @ nat ) ) ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_143_power__Suc0__right,axiom,
% 2.64/1.79 ! [A: $tType] :
% 2.64/1.79 ( ( monoid_mult @ A )
% 2.64/1.79 => ! [A2: A] :
% 2.64/1.79 ( ( power_power @ A @ A2 @ ( suc @ ( zero_zero @ nat ) ) )
% 2.64/1.79 = A2 ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_144_of__int__0,axiom,
% 2.64/1.79 ! [A: $tType] :
% 2.64/1.79 ( ( ring_1 @ A )
% 2.64/1.79 => ( ( ring_1_of_int @ A @ ( zero_zero @ int ) )
% 2.64/1.79 = ( zero_zero @ A ) ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_145_of__int__0__eq__iff,axiom,
% 2.64/1.79 ! [A: $tType] :
% 2.64/1.79 ( ( ring_char_0 @ A )
% 2.64/1.79 => ! [Z: int] :
% 2.64/1.79 ( ( ( zero_zero @ A )
% 2.64/1.79 = ( ring_1_of_int @ A @ Z ) )
% 2.64/1.79 = ( Z
% 2.64/1.79 = ( zero_zero @ int ) ) ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_146_of__int__eq__0__iff,axiom,
% 2.64/1.79 ! [A: $tType] :
% 2.64/1.79 ( ( ring_char_0 @ A )
% 2.64/1.79 => ! [Z: int] :
% 2.64/1.79 ( ( ( ring_1_of_int @ A @ Z )
% 2.64/1.79 = ( zero_zero @ A ) )
% 2.64/1.79 = ( Z
% 2.64/1.79 = ( zero_zero @ int ) ) ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_147_powr__eq__0__iff,axiom,
% 2.64/1.79 ! [A: $tType] :
% 2.64/1.79 ( ( ln @ A )
% 2.64/1.79 => ! [W: A,Z: A] :
% 2.64/1.79 ( ( ( powr @ A @ W @ Z )
% 2.64/1.79 = ( zero_zero @ A ) )
% 2.64/1.79 = ( W
% 2.64/1.79 = ( zero_zero @ A ) ) ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_148_powr__0,axiom,
% 2.64/1.79 ! [A: $tType] :
% 2.64/1.79 ( ( ln @ A )
% 2.64/1.79 => ! [Z: A] :
% 2.64/1.79 ( ( powr @ A @ ( zero_zero @ A ) @ Z )
% 2.64/1.79 = ( zero_zero @ A ) ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_149_ceiling__zero,axiom,
% 2.64/1.79 ! [A: $tType] :
% 2.64/1.79 ( ( archim1727834104eiling @ A )
% 2.64/1.79 => ( ( archimedean_ceiling @ A @ ( zero_zero @ A ) )
% 2.64/1.79 = ( zero_zero @ int ) ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_150_nat__le__0,axiom,
% 2.64/1.79 ! [Z: int] :
% 2.64/1.79 ( ( ord_less_eq @ int @ Z @ ( zero_zero @ int ) )
% 2.64/1.79 => ( ( nat2 @ Z )
% 2.64/1.79 = ( zero_zero @ nat ) ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_151_nat__0__iff,axiom,
% 2.64/1.79 ! [I2: int] :
% 2.64/1.79 ( ( ( nat2 @ I2 )
% 2.64/1.79 = ( zero_zero @ nat ) )
% 2.64/1.79 = ( ord_less_eq @ int @ I2 @ ( zero_zero @ int ) ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_152_power__Suc__0,axiom,
% 2.64/1.79 ! [N: nat] :
% 2.64/1.79 ( ( power_power @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N )
% 2.64/1.79 = ( suc @ ( zero_zero @ nat ) ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_153_nat__power__eq__Suc__0__iff,axiom,
% 2.64/1.79 ! [X: nat,M: nat] :
% 2.64/1.79 ( ( ( power_power @ nat @ X @ M )
% 2.64/1.79 = ( suc @ ( zero_zero @ nat ) ) )
% 2.64/1.79 = ( ( M
% 2.64/1.79 = ( zero_zero @ nat ) )
% 2.64/1.79 | ( X
% 2.64/1.79 = ( suc @ ( zero_zero @ nat ) ) ) ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_154_nat__int,axiom,
% 2.64/1.79 ! [N: nat] :
% 2.64/1.79 ( ( nat2 @ ( semiring_1_of_nat @ int @ N ) )
% 2.64/1.79 = N ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_155_dbl__simps_I2_J,axiom,
% 2.64/1.79 ! [A: $tType] :
% 2.64/1.79 ( ( neg_numeral @ A )
% 2.64/1.79 => ( ( neg_numeral_dbl @ A @ ( zero_zero @ A ) )
% 2.64/1.79 = ( zero_zero @ A ) ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_156_of__nat__le__0__iff,axiom,
% 2.64/1.79 ! [A: $tType] :
% 2.64/1.79 ( ( linord1659791738miring @ A )
% 2.64/1.79 => ! [M: nat] :
% 2.64/1.79 ( ( ord_less_eq @ A @ ( semiring_1_of_nat @ A @ M ) @ ( zero_zero @ A ) )
% 2.64/1.79 = ( M
% 2.64/1.79 = ( zero_zero @ nat ) ) ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_157_power__0__Suc,axiom,
% 2.64/1.79 ! [A: $tType] :
% 2.64/1.79 ( ( semiring_1 @ A )
% 2.64/1.79 => ! [N: nat] :
% 2.64/1.79 ( ( power_power @ A @ ( zero_zero @ A ) @ ( suc @ N ) )
% 2.64/1.79 = ( zero_zero @ A ) ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_158_power__zero__numeral,axiom,
% 2.64/1.79 ! [A: $tType] :
% 2.64/1.79 ( ( semiring_1 @ A )
% 2.64/1.79 => ! [K: num] :
% 2.64/1.79 ( ( power_power @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ nat @ K ) )
% 2.64/1.79 = ( zero_zero @ A ) ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_159_powr__zero__eq__one,axiom,
% 2.64/1.79 ! [A: $tType] :
% 2.64/1.79 ( ( ln @ A )
% 2.64/1.79 => ! [X: A] :
% 2.64/1.79 ( ( ( X
% 2.64/1.79 = ( zero_zero @ A ) )
% 2.64/1.79 => ( ( powr @ A @ X @ ( zero_zero @ A ) )
% 2.64/1.79 = ( zero_zero @ A ) ) )
% 2.64/1.79 & ( ( X
% 2.64/1.79 != ( zero_zero @ A ) )
% 2.64/1.79 => ( ( powr @ A @ X @ ( zero_zero @ A ) )
% 2.64/1.79 = ( one_one @ A ) ) ) ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_160_powr__nonneg__iff,axiom,
% 2.64/1.79 ! [A2: real,X: real] :
% 2.64/1.79 ( ( ord_less_eq @ real @ ( powr @ real @ A2 @ X ) @ ( zero_zero @ real ) )
% 2.64/1.79 = ( A2
% 2.64/1.79 = ( zero_zero @ real ) ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_161_nat__1,axiom,
% 2.64/1.79 ( ( nat2 @ ( one_one @ int ) )
% 2.64/1.79 = ( suc @ ( zero_zero @ nat ) ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_162_int__eq__iff__numeral,axiom,
% 2.64/1.79 ! [M: nat,V: num] :
% 2.64/1.79 ( ( ( semiring_1_of_nat @ int @ M )
% 2.64/1.79 = ( numeral_numeral @ int @ V ) )
% 2.64/1.79 = ( M
% 2.64/1.79 = ( numeral_numeral @ nat @ V ) ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_163_log__one,axiom,
% 2.64/1.79 ! [A2: real] :
% 2.64/1.79 ( ( log @ A2 @ ( one_one @ real ) )
% 2.64/1.79 = ( zero_zero @ real ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_164_int__nat__eq,axiom,
% 2.64/1.79 ! [Z: int] :
% 2.64/1.79 ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z )
% 2.64/1.79 => ( ( semiring_1_of_nat @ int @ ( nat2 @ Z ) )
% 2.64/1.79 = Z ) )
% 2.64/1.79 & ( ~ ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z )
% 2.64/1.79 => ( ( semiring_1_of_nat @ int @ ( nat2 @ Z ) )
% 2.64/1.79 = ( zero_zero @ int ) ) ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_165_dbl__simps_I5_J,axiom,
% 2.64/1.79 ! [A: $tType] :
% 2.64/1.79 ( ( neg_numeral @ A )
% 2.64/1.79 => ! [K: num] :
% 2.64/1.79 ( ( neg_numeral_dbl @ A @ ( numeral_numeral @ A @ K ) )
% 2.64/1.79 = ( numeral_numeral @ A @ ( bit0 @ K ) ) ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_166_of__int__0__le__iff,axiom,
% 2.64/1.79 ! [A: $tType] :
% 2.64/1.79 ( ( linordered_idom @ A )
% 2.64/1.79 => ! [Z: int] :
% 2.64/1.79 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( ring_1_of_int @ A @ Z ) )
% 2.64/1.79 = ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z ) ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_167_of__int__le__0__iff,axiom,
% 2.64/1.79 ! [A: $tType] :
% 2.64/1.79 ( ( linordered_idom @ A )
% 2.64/1.79 => ! [Z: int] :
% 2.64/1.79 ( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ Z ) @ ( zero_zero @ A ) )
% 2.64/1.79 = ( ord_less_eq @ int @ Z @ ( zero_zero @ int ) ) ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_168_ceiling__le__zero,axiom,
% 2.64/1.79 ! [A: $tType] :
% 2.64/1.79 ( ( archim1727834104eiling @ A )
% 2.64/1.79 => ! [X: A] :
% 2.64/1.79 ( ( ord_less_eq @ int @ ( archimedean_ceiling @ A @ X ) @ ( zero_zero @ int ) )
% 2.64/1.79 = ( ord_less_eq @ A @ X @ ( zero_zero @ A ) ) ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_169_powr__one,axiom,
% 2.64/1.79 ! [X: real] :
% 2.64/1.79 ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
% 2.64/1.79 => ( ( powr @ real @ X @ ( one_one @ real ) )
% 2.64/1.79 = X ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_170_powr__one__gt__zero__iff,axiom,
% 2.64/1.79 ! [X: real] :
% 2.64/1.79 ( ( ( powr @ real @ X @ ( one_one @ real ) )
% 2.64/1.79 = X )
% 2.64/1.79 = ( ord_less_eq @ real @ ( zero_zero @ real ) @ X ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_171_of__nat__nat,axiom,
% 2.64/1.79 ! [A: $tType] :
% 2.64/1.79 ( ( ring_1 @ A )
% 2.64/1.79 => ! [Z: int] :
% 2.64/1.79 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z )
% 2.64/1.79 => ( ( semiring_1_of_nat @ A @ ( nat2 @ Z ) )
% 2.64/1.79 = ( ring_1_of_int @ A @ Z ) ) ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_172_zero__eq__power2,axiom,
% 2.64/1.79 ! [A: $tType] :
% 2.64/1.79 ( ( semiri134348788visors @ A )
% 2.64/1.79 => ! [A2: A] :
% 2.64/1.79 ( ( ( power_power @ A @ A2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
% 2.64/1.79 = ( zero_zero @ A ) )
% 2.64/1.79 = ( A2
% 2.64/1.79 = ( zero_zero @ A ) ) ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_173_power2__less__eq__zero__iff,axiom,
% 2.64/1.79 ! [A: $tType] :
% 2.64/1.79 ( ( linordered_idom @ A )
% 2.64/1.79 => ! [A2: A] :
% 2.64/1.79 ( ( ord_less_eq @ A @ ( power_power @ A @ A2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( zero_zero @ A ) )
% 2.64/1.79 = ( A2
% 2.64/1.79 = ( zero_zero @ A ) ) ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_174_int__if,axiom,
% 2.64/1.79 ! [P: $o,A2: nat,B: nat] :
% 2.64/1.79 ( ( P
% 2.64/1.79 => ( ( semiring_1_of_nat @ int @ ( if @ nat @ P @ A2 @ B ) )
% 2.64/1.79 = ( semiring_1_of_nat @ int @ A2 ) ) )
% 2.64/1.79 & ( ~ P
% 2.64/1.79 => ( ( semiring_1_of_nat @ int @ ( if @ nat @ P @ A2 @ B ) )
% 2.64/1.79 = ( semiring_1_of_nat @ int @ B ) ) ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_175_nat__int__comparison_I1_J,axiom,
% 2.64/1.79 ( ( ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 ) )
% 2.64/1.79 = ( ^ [A4: nat,B3: nat] :
% 2.64/1.79 ( ( semiring_1_of_nat @ int @ A4 )
% 2.64/1.79 = ( semiring_1_of_nat @ int @ B3 ) ) ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_176_int__ops_I1_J,axiom,
% 2.64/1.79 ( ( semiring_1_of_nat @ int @ ( zero_zero @ nat ) )
% 2.64/1.79 = ( zero_zero @ int ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_177_zero__le__imp__eq__int,axiom,
% 2.64/1.79 ! [K: int] :
% 2.64/1.79 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
% 2.64/1.79 => ? [N3: nat] :
% 2.64/1.79 ( K
% 2.64/1.79 = ( semiring_1_of_nat @ int @ N3 ) ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_178_nonneg__int__cases,axiom,
% 2.64/1.79 ! [K: int] :
% 2.64/1.79 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
% 2.64/1.79 => ~ ! [N3: nat] :
% 2.64/1.79 ( K
% 2.64/1.79 != ( semiring_1_of_nat @ int @ N3 ) ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_179_int__int__eq,axiom,
% 2.64/1.79 ! [M: nat,N: nat] :
% 2.64/1.79 ( ( ( semiring_1_of_nat @ int @ M )
% 2.64/1.79 = ( semiring_1_of_nat @ int @ N ) )
% 2.64/1.79 = ( M = N ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_180_nat__eq__iff2,axiom,
% 2.64/1.79 ! [M: nat,W: int] :
% 2.64/1.79 ( ( M
% 2.64/1.79 = ( nat2 @ W ) )
% 2.64/1.79 = ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ W )
% 2.64/1.79 => ( W
% 2.64/1.79 = ( semiring_1_of_nat @ int @ M ) ) )
% 2.64/1.79 & ( ~ ( ord_less_eq @ int @ ( zero_zero @ int ) @ W )
% 2.64/1.79 => ( M
% 2.64/1.79 = ( zero_zero @ nat ) ) ) ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_181_nat__eq__iff,axiom,
% 2.64/1.79 ! [W: int,M: nat] :
% 2.64/1.79 ( ( ( nat2 @ W )
% 2.64/1.79 = M )
% 2.64/1.79 = ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ W )
% 2.64/1.79 => ( W
% 2.64/1.79 = ( semiring_1_of_nat @ int @ M ) ) )
% 2.64/1.79 & ( ~ ( ord_less_eq @ int @ ( zero_zero @ int ) @ W )
% 2.64/1.79 => ( M
% 2.64/1.79 = ( zero_zero @ nat ) ) ) ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_182_zero__reorient,axiom,
% 2.64/1.79 ! [A: $tType] :
% 2.64/1.79 ( ( zero @ A )
% 2.64/1.79 => ! [X: A] :
% 2.64/1.79 ( ( ( zero_zero @ A )
% 2.64/1.79 = X )
% 2.64/1.79 = ( X
% 2.64/1.79 = ( zero_zero @ A ) ) ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_183_zero__le,axiom,
% 2.64/1.79 ! [A: $tType] :
% 2.64/1.79 ( ( canoni770627133id_add @ A )
% 2.64/1.79 => ! [X: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ X ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_184_nat__zero__as__int,axiom,
% 2.64/1.79 ( ( zero_zero @ nat )
% 2.64/1.79 = ( nat2 @ ( zero_zero @ int ) ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_185_power__0__left,axiom,
% 2.64/1.79 ! [A: $tType] :
% 2.64/1.79 ( ( semiring_1 @ A )
% 2.64/1.79 => ! [N: nat] :
% 2.64/1.79 ( ( ( N
% 2.64/1.79 = ( zero_zero @ nat ) )
% 2.64/1.79 => ( ( power_power @ A @ ( zero_zero @ A ) @ N )
% 2.64/1.79 = ( one_one @ A ) ) )
% 2.64/1.79 & ( ( N
% 2.64/1.79 != ( zero_zero @ nat ) )
% 2.64/1.79 => ( ( power_power @ A @ ( zero_zero @ A ) @ N )
% 2.64/1.79 = ( zero_zero @ A ) ) ) ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_186_nat__0__le,axiom,
% 2.64/1.79 ! [Z: int] :
% 2.64/1.79 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z )
% 2.64/1.79 => ( ( semiring_1_of_nat @ int @ ( nat2 @ Z ) )
% 2.64/1.79 = Z ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_187_int__eq__iff,axiom,
% 2.64/1.79 ! [M: nat,Z: int] :
% 2.64/1.79 ( ( ( semiring_1_of_nat @ int @ M )
% 2.64/1.79 = Z )
% 2.64/1.79 = ( ( M
% 2.64/1.79 = ( nat2 @ Z ) )
% 2.64/1.79 & ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z ) ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_188_int__ops_I2_J,axiom,
% 2.64/1.79 ( ( semiring_1_of_nat @ int @ ( one_one @ nat ) )
% 2.64/1.79 = ( one_one @ int ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_189_le__numeral__extra_I3_J,axiom,
% 2.64/1.79 ! [A: $tType] :
% 2.64/1.79 ( ( linord1659791738miring @ A )
% 2.64/1.79 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( zero_zero @ A ) ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_190_zero__neq__numeral,axiom,
% 2.64/1.79 ! [A: $tType] :
% 2.64/1.79 ( ( semiring_char_0 @ A )
% 2.64/1.79 => ! [N: num] :
% 2.64/1.79 ( ( zero_zero @ A )
% 2.64/1.79 != ( numeral_numeral @ A @ N ) ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_191_power__not__zero,axiom,
% 2.64/1.79 ! [A: $tType] :
% 2.64/1.79 ( ( semiri134348788visors @ A )
% 2.64/1.79 => ! [A2: A,N: nat] :
% 2.64/1.79 ( ( A2
% 2.64/1.79 != ( zero_zero @ A ) )
% 2.64/1.79 => ( ( power_power @ A @ A2 @ N )
% 2.64/1.79 != ( zero_zero @ A ) ) ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_192_exists__least__lemma,axiom,
% 2.64/1.79 ! [P: nat > $o] :
% 2.64/1.79 ( ~ ( P @ ( zero_zero @ nat ) )
% 2.64/1.79 => ( ? [X_1: nat] : ( P @ X_1 )
% 2.64/1.79 => ? [N3: nat] :
% 2.64/1.79 ( ~ ( P @ N3 )
% 2.64/1.79 & ( P @ ( suc @ N3 ) ) ) ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_193_nat_Odistinct_I1_J,axiom,
% 2.64/1.79 ! [X2: nat] :
% 2.64/1.79 ( ( zero_zero @ nat )
% 2.64/1.79 != ( suc @ X2 ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_194_old_Onat_Odistinct_I2_J,axiom,
% 2.64/1.79 ! [Nat3: nat] :
% 2.64/1.79 ( ( suc @ Nat3 )
% 2.64/1.79 != ( zero_zero @ nat ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_195_old_Onat_Odistinct_I1_J,axiom,
% 2.64/1.79 ! [Nat2: nat] :
% 2.64/1.79 ( ( zero_zero @ nat )
% 2.64/1.79 != ( suc @ Nat2 ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_196_nat_OdiscI,axiom,
% 2.64/1.79 ! [Nat: nat,X2: nat] :
% 2.64/1.79 ( ( Nat
% 2.64/1.79 = ( suc @ X2 ) )
% 2.64/1.79 => ( Nat
% 2.64/1.79 != ( zero_zero @ nat ) ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_197_nat__induct,axiom,
% 2.64/1.79 ! [P: nat > $o,N: nat] :
% 2.64/1.79 ( ( P @ ( zero_zero @ nat ) )
% 2.64/1.79 => ( ! [N3: nat] :
% 2.64/1.79 ( ( P @ N3 )
% 2.64/1.79 => ( P @ ( suc @ N3 ) ) )
% 2.64/1.79 => ( P @ N ) ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_198_One__nat__def,axiom,
% 2.64/1.79 ( ( one_one @ nat )
% 2.64/1.79 = ( suc @ ( zero_zero @ nat ) ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_199_diff__induct,axiom,
% 2.64/1.79 ! [P: nat > nat > $o,M: nat,N: nat] :
% 2.64/1.79 ( ! [X4: nat] : ( P @ X4 @ ( zero_zero @ nat ) )
% 2.64/1.79 => ( ! [Y3: nat] : ( P @ ( zero_zero @ nat ) @ ( suc @ Y3 ) )
% 2.64/1.79 => ( ! [X4: nat,Y3: nat] :
% 2.64/1.79 ( ( P @ X4 @ Y3 )
% 2.64/1.79 => ( P @ ( suc @ X4 ) @ ( suc @ Y3 ) ) )
% 2.64/1.79 => ( P @ M @ N ) ) ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_200_zero__induct,axiom,
% 2.64/1.79 ! [P: nat > $o,K: nat] :
% 2.64/1.79 ( ( P @ K )
% 2.64/1.79 => ( ! [N3: nat] :
% 2.64/1.79 ( ( P @ ( suc @ N3 ) )
% 2.64/1.79 => ( P @ N3 ) )
% 2.64/1.79 => ( P @ ( zero_zero @ nat ) ) ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_201_Suc__neq__Zero,axiom,
% 2.64/1.79 ! [M: nat] :
% 2.64/1.79 ( ( suc @ M )
% 2.64/1.79 != ( zero_zero @ nat ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_202_Zero__neq__Suc,axiom,
% 2.64/1.79 ! [M: nat] :
% 2.64/1.79 ( ( zero_zero @ nat )
% 2.64/1.79 != ( suc @ M ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_203_Zero__not__Suc,axiom,
% 2.64/1.79 ! [M: nat] :
% 2.64/1.79 ( ( zero_zero @ nat )
% 2.64/1.79 != ( suc @ M ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_204_old_Onat_Oexhaust,axiom,
% 2.64/1.79 ! [Y: nat] :
% 2.64/1.79 ( ( Y
% 2.64/1.79 != ( zero_zero @ nat ) )
% 2.64/1.79 => ~ ! [Nat4: nat] :
% 2.64/1.79 ( Y
% 2.64/1.79 != ( suc @ Nat4 ) ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_205_old_Onat_Oinducts,axiom,
% 2.64/1.79 ! [P: nat > $o,Nat: nat] :
% 2.64/1.79 ( ( P @ ( zero_zero @ nat ) )
% 2.64/1.79 => ( ! [Nat4: nat] :
% 2.64/1.79 ( ( P @ Nat4 )
% 2.64/1.79 => ( P @ ( suc @ Nat4 ) ) )
% 2.64/1.79 => ( P @ Nat ) ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_206_not0__implies__Suc,axiom,
% 2.64/1.79 ! [N: nat] :
% 2.64/1.79 ( ( N
% 2.64/1.79 != ( zero_zero @ nat ) )
% 2.64/1.79 => ? [M4: nat] :
% 2.64/1.79 ( N
% 2.64/1.79 = ( suc @ M4 ) ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_207_bot__nat__0_Oextremum__uniqueI,axiom,
% 2.64/1.79 ! [A2: nat] :
% 2.64/1.79 ( ( ord_less_eq @ nat @ A2 @ ( zero_zero @ nat ) )
% 2.64/1.79 => ( A2
% 2.64/1.79 = ( zero_zero @ nat ) ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_208_bot__nat__0_Oextremum__unique,axiom,
% 2.64/1.79 ! [A2: nat] :
% 2.64/1.79 ( ( ord_less_eq @ nat @ A2 @ ( zero_zero @ nat ) )
% 2.64/1.79 = ( A2
% 2.64/1.79 = ( zero_zero @ nat ) ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_209_le__0__eq,axiom,
% 2.64/1.79 ! [N: nat] :
% 2.64/1.79 ( ( ord_less_eq @ nat @ N @ ( zero_zero @ nat ) )
% 2.64/1.79 = ( N
% 2.64/1.79 = ( zero_zero @ nat ) ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_210_less__eq__nat_Osimps_I1_J,axiom,
% 2.64/1.79 ! [N: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ N ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_211_less__eq__int__code_I1_J,axiom,
% 2.64/1.79 ord_less_eq @ int @ ( zero_zero @ int ) @ ( zero_zero @ int ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_212_le__nat__iff,axiom,
% 2.64/1.79 ! [K: int,N: nat] :
% 2.64/1.79 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
% 2.64/1.79 => ( ( ord_less_eq @ nat @ N @ ( nat2 @ K ) )
% 2.64/1.79 = ( ord_less_eq @ int @ ( semiring_1_of_nat @ int @ N ) @ K ) ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_213_of__int__nonneg,axiom,
% 2.64/1.79 ! [A: $tType] :
% 2.64/1.79 ( ( linordered_idom @ A )
% 2.64/1.79 => ! [Z: int] :
% 2.64/1.79 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z )
% 2.64/1.79 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( ring_1_of_int @ A @ Z ) ) ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_214_nat__int__comparison_I3_J,axiom,
% 2.64/1.79 ( ( ord_less_eq @ nat )
% 2.64/1.79 = ( ^ [A4: nat,B3: nat] : ( ord_less_eq @ int @ ( semiring_1_of_nat @ int @ A4 ) @ ( semiring_1_of_nat @ int @ B3 ) ) ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_215_zle__int,axiom,
% 2.64/1.79 ! [M: nat,N: nat] :
% 2.64/1.79 ( ( ord_less_eq @ int @ ( semiring_1_of_nat @ int @ M ) @ ( semiring_1_of_nat @ int @ N ) )
% 2.64/1.79 = ( ord_less_eq @ nat @ M @ N ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_216_int__ops_I3_J,axiom,
% 2.64/1.79 ! [N: num] :
% 2.64/1.79 ( ( semiring_1_of_nat @ int @ ( numeral_numeral @ nat @ N ) )
% 2.64/1.79 = ( numeral_numeral @ int @ N ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_217_zero__le__numeral,axiom,
% 2.64/1.79 ! [A: $tType] :
% 2.64/1.79 ( ( linord1659791738miring @ A )
% 2.64/1.79 => ! [N: num] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ N ) ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_218_not__numeral__le__zero,axiom,
% 2.64/1.79 ! [A: $tType] :
% 2.64/1.79 ( ( linord1659791738miring @ A )
% 2.64/1.79 => ! [N: num] :
% 2.64/1.79 ~ ( ord_less_eq @ A @ ( numeral_numeral @ A @ N ) @ ( zero_zero @ A ) ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_219_power__mono,axiom,
% 2.64/1.79 ! [A: $tType] :
% 2.64/1.79 ( ( linordered_semidom @ A )
% 2.64/1.79 => ! [A2: A,B: A,N: nat] :
% 2.64/1.79 ( ( ord_less_eq @ A @ A2 @ B )
% 2.64/1.79 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
% 2.64/1.79 => ( ord_less_eq @ A @ ( power_power @ A @ A2 @ N ) @ ( power_power @ A @ B @ N ) ) ) ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_220_zero__le__power,axiom,
% 2.64/1.79 ! [A: $tType] :
% 2.64/1.79 ( ( linordered_semidom @ A )
% 2.64/1.79 => ! [A2: A,N: nat] :
% 2.64/1.79 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
% 2.64/1.79 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A2 @ N ) ) ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_221_of__nat__0__le__iff,axiom,
% 2.64/1.79 ! [A: $tType] :
% 2.64/1.79 ( ( linord1659791738miring @ A )
% 2.64/1.79 => ! [N: nat] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( semiring_1_of_nat @ A @ N ) ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_222_power__0,axiom,
% 2.64/1.79 ! [A: $tType] :
% 2.64/1.79 ( ( power @ A )
% 2.64/1.79 => ! [A2: A] :
% 2.64/1.79 ( ( power_power @ A @ A2 @ ( zero_zero @ nat ) )
% 2.64/1.79 = ( one_one @ A ) ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_223_of__nat__neq__0,axiom,
% 2.64/1.79 ! [A: $tType] :
% 2.64/1.79 ( ( semiring_char_0 @ A )
% 2.64/1.79 => ! [N: nat] :
% 2.64/1.79 ( ( semiring_1_of_nat @ A @ ( suc @ N ) )
% 2.64/1.79 != ( zero_zero @ A ) ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_224_ex__nat,axiom,
% 2.64/1.79 ( ( ^ [P2: nat > $o] :
% 2.64/1.79 ? [X6: nat] : ( P2 @ X6 ) )
% 2.64/1.79 = ( ^ [P3: nat > $o] :
% 2.64/1.79 ? [X3: int] :
% 2.64/1.79 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X3 )
% 2.64/1.79 & ( P3 @ ( nat2 @ X3 ) ) ) ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_225_all__nat,axiom,
% 2.64/1.79 ( ( ^ [P2: nat > $o] :
% 2.64/1.79 ! [X6: nat] : ( P2 @ X6 ) )
% 2.64/1.79 = ( ^ [P3: nat > $o] :
% 2.64/1.79 ! [X3: int] :
% 2.64/1.79 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X3 )
% 2.64/1.79 => ( P3 @ ( nat2 @ X3 ) ) ) ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_226_eq__nat__nat__iff,axiom,
% 2.64/1.79 ! [Z: int,Z5: int] :
% 2.64/1.79 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z )
% 2.64/1.79 => ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z5 )
% 2.64/1.79 => ( ( ( nat2 @ Z )
% 2.64/1.79 = ( nat2 @ Z5 ) )
% 2.64/1.79 = ( Z = Z5 ) ) ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_227_powr__mono2,axiom,
% 2.64/1.79 ! [A2: real,X: real,Y: real] :
% 2.64/1.79 ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ A2 )
% 2.64/1.79 => ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
% 2.64/1.79 => ( ( ord_less_eq @ real @ X @ Y )
% 2.64/1.79 => ( ord_less_eq @ real @ ( powr @ real @ X @ A2 ) @ ( powr @ real @ Y @ A2 ) ) ) ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_228_powr__ge__pzero,axiom,
% 2.64/1.79 ! [X: real,Y: real] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( powr @ real @ X @ Y ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_229_pow_Osimps_I1_J,axiom,
% 2.64/1.79 ! [X: num] :
% 2.64/1.79 ( ( pow @ X @ one2 )
% 2.64/1.79 = X ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_230_nat__le__iff,axiom,
% 2.64/1.79 ! [X: int,N: nat] :
% 2.64/1.79 ( ( ord_less_eq @ nat @ ( nat2 @ X ) @ N )
% 2.64/1.79 = ( ord_less_eq @ int @ X @ ( semiring_1_of_nat @ int @ N ) ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_231_le__funD,axiom,
% 2.64/1.79 ! [B2: $tType,A: $tType] :
% 2.64/1.79 ( ( ord @ B2 )
% 2.64/1.79 => ! [F: A > B2,G: A > B2,X: A] :
% 2.64/1.79 ( ( ord_less_eq @ ( A > B2 ) @ F @ G )
% 2.64/1.79 => ( ord_less_eq @ B2 @ ( F @ X ) @ ( G @ X ) ) ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_232_le__funE,axiom,
% 2.64/1.79 ! [B2: $tType,A: $tType] :
% 2.64/1.79 ( ( ord @ B2 )
% 2.64/1.79 => ! [F: A > B2,G: A > B2,X: A] :
% 2.64/1.79 ( ( ord_less_eq @ ( A > B2 ) @ F @ G )
% 2.64/1.79 => ( ord_less_eq @ B2 @ ( F @ X ) @ ( G @ X ) ) ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_233_le__funI,axiom,
% 2.64/1.79 ! [B2: $tType,A: $tType] :
% 2.64/1.79 ( ( ord @ B2 )
% 2.64/1.79 => ! [F: A > B2,G: A > B2] :
% 2.64/1.79 ( ! [X4: A] : ( ord_less_eq @ B2 @ ( F @ X4 ) @ ( G @ X4 ) )
% 2.64/1.79 => ( ord_less_eq @ ( A > B2 ) @ F @ G ) ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_234_le__fun__def,axiom,
% 2.64/1.79 ! [B2: $tType,A: $tType] :
% 2.64/1.79 ( ( ord @ B2 )
% 2.64/1.79 => ( ( ord_less_eq @ ( A > B2 ) )
% 2.64/1.79 = ( ^ [F2: A > B2,G2: A > B2] :
% 2.64/1.79 ! [X3: A] : ( ord_less_eq @ B2 @ ( F2 @ X3 ) @ ( G2 @ X3 ) ) ) ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_235_order__subst1,axiom,
% 2.64/1.79 ! [A: $tType,B2: $tType] :
% 2.64/1.79 ( ( ( order @ B2 )
% 2.64/1.79 & ( order @ A ) )
% 2.64/1.79 => ! [A2: A,F: B2 > A,B: B2,C: B2] :
% 2.64/1.79 ( ( ord_less_eq @ A @ A2 @ ( F @ B ) )
% 2.64/1.79 => ( ( ord_less_eq @ B2 @ B @ C )
% 2.64/1.79 => ( ! [X4: B2,Y3: B2] :
% 2.64/1.79 ( ( ord_less_eq @ B2 @ X4 @ Y3 )
% 2.64/1.79 => ( ord_less_eq @ A @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 2.64/1.79 => ( ord_less_eq @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_236_order__subst2,axiom,
% 2.64/1.79 ! [A: $tType,C2: $tType] :
% 2.64/1.79 ( ( ( order @ C2 )
% 2.64/1.79 & ( order @ A ) )
% 2.64/1.79 => ! [A2: A,B: A,F: A > C2,C: C2] :
% 2.64/1.79 ( ( ord_less_eq @ A @ A2 @ B )
% 2.64/1.79 => ( ( ord_less_eq @ C2 @ ( F @ B ) @ C )
% 2.64/1.79 => ( ! [X4: A,Y3: A] :
% 2.64/1.79 ( ( ord_less_eq @ A @ X4 @ Y3 )
% 2.64/1.79 => ( ord_less_eq @ C2 @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 2.64/1.79 => ( ord_less_eq @ C2 @ ( F @ A2 ) @ C ) ) ) ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_237_ord__eq__le__subst,axiom,
% 2.64/1.79 ! [A: $tType,B2: $tType] :
% 2.64/1.79 ( ( ( ord @ B2 )
% 2.64/1.79 & ( ord @ A ) )
% 2.64/1.79 => ! [A2: A,F: B2 > A,B: B2,C: B2] :
% 2.64/1.79 ( ( A2
% 2.64/1.79 = ( F @ B ) )
% 2.64/1.79 => ( ( ord_less_eq @ B2 @ B @ C )
% 2.64/1.79 => ( ! [X4: B2,Y3: B2] :
% 2.64/1.79 ( ( ord_less_eq @ B2 @ X4 @ Y3 )
% 2.64/1.79 => ( ord_less_eq @ A @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 2.64/1.79 => ( ord_less_eq @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_238_ord__le__eq__subst,axiom,
% 2.64/1.79 ! [A: $tType,B2: $tType] :
% 2.64/1.79 ( ( ( ord @ B2 )
% 2.64/1.79 & ( ord @ A ) )
% 2.64/1.79 => ! [A2: A,B: A,F: A > B2,C: B2] :
% 2.64/1.79 ( ( ord_less_eq @ A @ A2 @ B )
% 2.64/1.79 => ( ( ( F @ B )
% 2.64/1.79 = C )
% 2.64/1.79 => ( ! [X4: A,Y3: A] :
% 2.64/1.79 ( ( ord_less_eq @ A @ X4 @ Y3 )
% 2.64/1.79 => ( ord_less_eq @ B2 @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 2.64/1.79 => ( ord_less_eq @ B2 @ ( F @ A2 ) @ C ) ) ) ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_239_eq__iff,axiom,
% 2.64/1.79 ! [A: $tType] :
% 2.64/1.79 ( ( order @ A )
% 2.64/1.79 => ( ( ^ [Y5: A,Z4: A] : ( Y5 = Z4 ) )
% 2.64/1.79 = ( ^ [X3: A,Y6: A] :
% 2.64/1.79 ( ( ord_less_eq @ A @ X3 @ Y6 )
% 2.64/1.79 & ( ord_less_eq @ A @ Y6 @ X3 ) ) ) ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_240_antisym,axiom,
% 2.64/1.79 ! [A: $tType] :
% 2.64/1.79 ( ( order @ A )
% 2.64/1.79 => ! [X: A,Y: A] :
% 2.64/1.79 ( ( ord_less_eq @ A @ X @ Y )
% 2.64/1.79 => ( ( ord_less_eq @ A @ Y @ X )
% 2.64/1.79 => ( X = Y ) ) ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_241_linear,axiom,
% 2.64/1.79 ! [A: $tType] :
% 2.64/1.79 ( ( linorder @ A )
% 2.64/1.79 => ! [X: A,Y: A] :
% 2.64/1.79 ( ( ord_less_eq @ A @ X @ Y )
% 2.64/1.79 | ( ord_less_eq @ A @ Y @ X ) ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_242_eq__refl,axiom,
% 2.64/1.79 ! [A: $tType] :
% 2.64/1.79 ( ( preorder @ A )
% 2.64/1.79 => ! [X: A,Y: A] :
% 2.64/1.79 ( ( X = Y )
% 2.64/1.79 => ( ord_less_eq @ A @ X @ Y ) ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_243_le__cases,axiom,
% 2.64/1.79 ! [A: $tType] :
% 2.64/1.79 ( ( linorder @ A )
% 2.64/1.79 => ! [X: A,Y: A] :
% 2.64/1.79 ( ~ ( ord_less_eq @ A @ X @ Y )
% 2.64/1.79 => ( ord_less_eq @ A @ Y @ X ) ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_244_order_Otrans,axiom,
% 2.64/1.79 ! [A: $tType] :
% 2.64/1.79 ( ( order @ A )
% 2.64/1.79 => ! [A2: A,B: A,C: A] :
% 2.64/1.79 ( ( ord_less_eq @ A @ A2 @ B )
% 2.64/1.79 => ( ( ord_less_eq @ A @ B @ C )
% 2.64/1.79 => ( ord_less_eq @ A @ A2 @ C ) ) ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_245_le__cases3,axiom,
% 2.64/1.79 ! [A: $tType] :
% 2.64/1.79 ( ( linorder @ A )
% 2.64/1.79 => ! [X: A,Y: A,Z: A] :
% 2.64/1.79 ( ( ( ord_less_eq @ A @ X @ Y )
% 2.64/1.79 => ~ ( ord_less_eq @ A @ Y @ Z ) )
% 2.64/1.79 => ( ( ( ord_less_eq @ A @ Y @ X )
% 2.64/1.79 => ~ ( ord_less_eq @ A @ X @ Z ) )
% 2.64/1.79 => ( ( ( ord_less_eq @ A @ X @ Z )
% 2.64/1.79 => ~ ( ord_less_eq @ A @ Z @ Y ) )
% 2.64/1.79 => ( ( ( ord_less_eq @ A @ Z @ Y )
% 2.64/1.79 => ~ ( ord_less_eq @ A @ Y @ X ) )
% 2.64/1.79 => ( ( ( ord_less_eq @ A @ Y @ Z )
% 2.64/1.79 => ~ ( ord_less_eq @ A @ Z @ X ) )
% 2.64/1.79 => ~ ( ( ord_less_eq @ A @ Z @ X )
% 2.64/1.79 => ~ ( ord_less_eq @ A @ X @ Y ) ) ) ) ) ) ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_246_antisym__conv,axiom,
% 2.64/1.79 ! [A: $tType] :
% 2.64/1.79 ( ( order @ A )
% 2.64/1.79 => ! [Y: A,X: A] :
% 2.64/1.79 ( ( ord_less_eq @ A @ Y @ X )
% 2.64/1.79 => ( ( ord_less_eq @ A @ X @ Y )
% 2.64/1.79 = ( X = Y ) ) ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_247_order__class_Oorder_Oeq__iff,axiom,
% 2.64/1.79 ! [A: $tType] :
% 2.64/1.79 ( ( order @ A )
% 2.64/1.79 => ( ( ^ [Y5: A,Z4: A] : ( Y5 = Z4 ) )
% 2.64/1.79 = ( ^ [A4: A,B3: A] :
% 2.64/1.79 ( ( ord_less_eq @ A @ A4 @ B3 )
% 2.64/1.79 & ( ord_less_eq @ A @ B3 @ A4 ) ) ) ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_248_ord__eq__le__trans,axiom,
% 2.64/1.79 ! [A: $tType] :
% 2.64/1.79 ( ( ord @ A )
% 2.64/1.79 => ! [A2: A,B: A,C: A] :
% 2.64/1.79 ( ( A2 = B )
% 2.64/1.79 => ( ( ord_less_eq @ A @ B @ C )
% 2.64/1.79 => ( ord_less_eq @ A @ A2 @ C ) ) ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_249_ord__le__eq__trans,axiom,
% 2.64/1.79 ! [A: $tType] :
% 2.64/1.79 ( ( ord @ A )
% 2.64/1.79 => ! [A2: A,B: A,C: A] :
% 2.64/1.79 ( ( ord_less_eq @ A @ A2 @ B )
% 2.64/1.79 => ( ( B = C )
% 2.64/1.79 => ( ord_less_eq @ A @ A2 @ C ) ) ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(fact_250_order__class_Oorder_Oantisym,axiom,
% 2.64/1.79 ! [A: $tType] :
% 2.64/1.79 ( ( order @ A )
% 2.64/1.79 => ! [A2: A,B: A] :
% 2.64/1.79 ( ( ord_less_eq @ A @ A2 @ B )
% 2.64/1.79 => ( ( ord_less_eq @ A @ B @ A2 )
% 2.64/1.79 => ( A2 = B ) ) ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(tcon_fun___Orderings_Opreorder,axiom,
% 2.64/1.79 ! [A5: $tType,A6: $tType] :
% 2.64/1.79 ( ( preorder @ A6 )
% 2.64/1.79 => ( preorder @ ( A5 > A6 ) ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(tcon_fun___Orderings_Oorder,axiom,
% 2.64/1.79 ! [A5: $tType,A6: $tType] :
% 2.64/1.79 ( ( order @ A6 )
% 2.64/1.79 => ( order @ ( A5 > A6 ) ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(tcon_fun___Orderings_Oord,axiom,
% 2.64/1.79 ! [A5: $tType,A6: $tType] :
% 2.64/1.79 ( ( ord @ A6 )
% 2.64/1.79 => ( ord @ ( A5 > A6 ) ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(tcon_Int_Oint___Rings_Osemiring__1__no__zero__divisors,axiom,
% 2.64/1.79 semiri134348788visors @ int ).
% 2.64/1.79
% 2.64/1.79 thf(tcon_Int_Oint___Rings_Olinordered__nonzero__semiring,axiom,
% 2.64/1.79 linord1659791738miring @ int ).
% 2.64/1.79
% 2.64/1.79 thf(tcon_Int_Oint___Rings_Olinordered__semidom,axiom,
% 2.64/1.79 linordered_semidom @ int ).
% 2.64/1.79
% 2.64/1.79 thf(tcon_Int_Oint___Rings_Olinordered__idom,axiom,
% 2.64/1.79 linordered_idom @ int ).
% 2.64/1.79
% 2.64/1.79 thf(tcon_Int_Oint___Num_Osemiring__numeral,axiom,
% 2.64/1.79 semiring_numeral @ int ).
% 2.64/1.79
% 2.64/1.79 thf(tcon_Int_Oint___Nat_Osemiring__char__0,axiom,
% 2.64/1.79 semiring_char_0 @ int ).
% 2.64/1.79
% 2.64/1.79 thf(tcon_Int_Oint___Orderings_Opreorder_1,axiom,
% 2.64/1.79 preorder @ int ).
% 2.64/1.79
% 2.64/1.79 thf(tcon_Int_Oint___Orderings_Olinorder,axiom,
% 2.64/1.79 linorder @ int ).
% 2.64/1.79
% 2.64/1.79 thf(tcon_Int_Oint___Groups_Omonoid__mult,axiom,
% 2.64/1.79 monoid_mult @ int ).
% 2.64/1.79
% 2.64/1.79 thf(tcon_Int_Oint___Rings_Osemiring__1,axiom,
% 2.64/1.79 semiring_1 @ int ).
% 2.64/1.79
% 2.64/1.79 thf(tcon_Int_Oint___Orderings_Oorder_2,axiom,
% 2.64/1.79 order @ int ).
% 2.64/1.79
% 2.64/1.79 thf(tcon_Int_Oint___Num_Oneg__numeral,axiom,
% 2.64/1.79 neg_numeral @ int ).
% 2.64/1.79
% 2.64/1.79 thf(tcon_Int_Oint___Nat_Oring__char__0,axiom,
% 2.64/1.79 ring_char_0 @ int ).
% 2.64/1.79
% 2.64/1.79 thf(tcon_Int_Oint___Orderings_Oord_3,axiom,
% 2.64/1.79 ord @ int ).
% 2.64/1.79
% 2.64/1.79 thf(tcon_Int_Oint___Rings_Oring__1,axiom,
% 2.64/1.79 ring_1 @ int ).
% 2.64/1.79
% 2.64/1.79 thf(tcon_Int_Oint___Power_Opower,axiom,
% 2.64/1.79 power @ int ).
% 2.64/1.79
% 2.64/1.79 thf(tcon_Int_Oint___Num_Onumeral,axiom,
% 2.64/1.79 numeral @ int ).
% 2.64/1.79
% 2.64/1.79 thf(tcon_Int_Oint___Groups_Ozero,axiom,
% 2.64/1.79 zero @ int ).
% 2.64/1.79
% 2.64/1.79 thf(tcon_Int_Oint___Groups_Oone,axiom,
% 2.64/1.79 one @ int ).
% 2.64/1.79
% 2.64/1.79 thf(tcon_Nat_Onat___Groups_Ocanonically__ordered__monoid__add,axiom,
% 2.64/1.79 canoni770627133id_add @ nat ).
% 2.64/1.79
% 2.64/1.79 thf(tcon_Nat_Onat___Rings_Osemiring__1__no__zero__divisors_4,axiom,
% 2.64/1.79 semiri134348788visors @ nat ).
% 2.64/1.79
% 2.64/1.79 thf(tcon_Nat_Onat___Rings_Olinordered__nonzero__semiring_5,axiom,
% 2.64/1.79 linord1659791738miring @ nat ).
% 2.64/1.79
% 2.64/1.79 thf(tcon_Nat_Onat___Rings_Olinordered__semidom_6,axiom,
% 2.64/1.79 linordered_semidom @ nat ).
% 2.64/1.79
% 2.64/1.79 thf(tcon_Nat_Onat___Num_Osemiring__numeral_7,axiom,
% 2.64/1.79 semiring_numeral @ nat ).
% 2.64/1.79
% 2.64/1.79 thf(tcon_Nat_Onat___Nat_Osemiring__char__0_8,axiom,
% 2.64/1.79 semiring_char_0 @ nat ).
% 2.64/1.79
% 2.64/1.79 thf(tcon_Nat_Onat___Orderings_Opreorder_9,axiom,
% 2.64/1.79 preorder @ nat ).
% 2.64/1.79
% 2.64/1.79 thf(tcon_Nat_Onat___Orderings_Olinorder_10,axiom,
% 2.64/1.79 linorder @ nat ).
% 2.64/1.79
% 2.64/1.79 thf(tcon_Nat_Onat___Groups_Omonoid__mult_11,axiom,
% 2.64/1.79 monoid_mult @ nat ).
% 2.64/1.79
% 2.64/1.79 thf(tcon_Nat_Onat___Rings_Osemiring__1_12,axiom,
% 2.64/1.79 semiring_1 @ nat ).
% 2.64/1.79
% 2.64/1.79 thf(tcon_Nat_Onat___Orderings_Oorder_13,axiom,
% 2.64/1.79 order @ nat ).
% 2.64/1.79
% 2.64/1.79 thf(tcon_Nat_Onat___Orderings_Oord_14,axiom,
% 2.64/1.79 ord @ nat ).
% 2.64/1.79
% 2.64/1.79 thf(tcon_Nat_Onat___Power_Opower_15,axiom,
% 2.64/1.79 power @ nat ).
% 2.64/1.79
% 2.64/1.79 thf(tcon_Nat_Onat___Num_Onumeral_16,axiom,
% 2.64/1.79 numeral @ nat ).
% 2.64/1.79
% 2.64/1.79 thf(tcon_Nat_Onat___Groups_Ozero_17,axiom,
% 2.64/1.79 zero @ nat ).
% 2.64/1.79
% 2.64/1.79 thf(tcon_Nat_Onat___Groups_Oone_18,axiom,
% 2.64/1.79 one @ nat ).
% 2.64/1.79
% 2.64/1.79 thf(tcon_Num_Onum___Orderings_Opreorder_19,axiom,
% 2.64/1.79 preorder @ num ).
% 2.64/1.79
% 2.64/1.79 thf(tcon_Num_Onum___Orderings_Olinorder_20,axiom,
% 2.64/1.79 linorder @ num ).
% 2.64/1.79
% 2.64/1.79 thf(tcon_Num_Onum___Orderings_Oorder_21,axiom,
% 2.64/1.79 order @ num ).
% 2.64/1.79
% 2.64/1.79 thf(tcon_Num_Onum___Orderings_Oord_22,axiom,
% 2.64/1.79 ord @ num ).
% 2.64/1.79
% 2.64/1.79 thf(tcon_Set_Oset___Orderings_Opreorder_23,axiom,
% 2.64/1.79 ! [A5: $tType] : ( preorder @ ( set @ A5 ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(tcon_Set_Oset___Orderings_Oorder_24,axiom,
% 2.64/1.79 ! [A5: $tType] : ( order @ ( set @ A5 ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(tcon_Set_Oset___Orderings_Oord_25,axiom,
% 2.64/1.79 ! [A5: $tType] : ( ord @ ( set @ A5 ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(tcon_HOL_Obool___Orderings_Opreorder_26,axiom,
% 2.64/1.79 preorder @ $o ).
% 2.64/1.79
% 2.64/1.79 thf(tcon_HOL_Obool___Orderings_Olinorder_27,axiom,
% 2.64/1.79 linorder @ $o ).
% 2.64/1.79
% 2.64/1.79 thf(tcon_HOL_Obool___Orderings_Oorder_28,axiom,
% 2.64/1.79 order @ $o ).
% 2.64/1.79
% 2.64/1.79 thf(tcon_HOL_Obool___Orderings_Oord_29,axiom,
% 2.64/1.79 ord @ $o ).
% 2.64/1.79
% 2.64/1.79 thf(tcon_Real_Oreal___Archimedean__Field_Oarchimedean__field,axiom,
% 2.64/1.79 archim1804426504_field @ real ).
% 2.64/1.79
% 2.64/1.79 thf(tcon_Real_Oreal___Rings_Osemiring__1__no__zero__divisors_30,axiom,
% 2.64/1.79 semiri134348788visors @ real ).
% 2.64/1.79
% 2.64/1.79 thf(tcon_Real_Oreal___Rings_Olinordered__nonzero__semiring_31,axiom,
% 2.64/1.79 linord1659791738miring @ real ).
% 2.64/1.79
% 2.64/1.79 thf(tcon_Real_Oreal___Archimedean__Field_Ofloor__ceiling,axiom,
% 2.64/1.79 archim1727834104eiling @ real ).
% 2.64/1.79
% 2.64/1.79 thf(tcon_Real_Oreal___Rings_Olinordered__semidom_32,axiom,
% 2.64/1.79 linordered_semidom @ real ).
% 2.64/1.79
% 2.64/1.79 thf(tcon_Real_Oreal___Rings_Olinordered__idom_33,axiom,
% 2.64/1.79 linordered_idom @ real ).
% 2.64/1.79
% 2.64/1.79 thf(tcon_Real_Oreal___Num_Osemiring__numeral_34,axiom,
% 2.64/1.79 semiring_numeral @ real ).
% 2.64/1.79
% 2.64/1.79 thf(tcon_Real_Oreal___Nat_Osemiring__char__0_35,axiom,
% 2.64/1.79 semiring_char_0 @ real ).
% 2.64/1.79
% 2.64/1.79 thf(tcon_Real_Oreal___Orderings_Opreorder_36,axiom,
% 2.64/1.79 preorder @ real ).
% 2.64/1.79
% 2.64/1.79 thf(tcon_Real_Oreal___Orderings_Olinorder_37,axiom,
% 2.64/1.79 linorder @ real ).
% 2.64/1.79
% 2.64/1.79 thf(tcon_Real_Oreal___Groups_Omonoid__mult_38,axiom,
% 2.64/1.79 monoid_mult @ real ).
% 2.64/1.79
% 2.64/1.79 thf(tcon_Real_Oreal___Transcendental_Oln,axiom,
% 2.64/1.79 ln @ real ).
% 2.64/1.79
% 2.64/1.79 thf(tcon_Real_Oreal___Rings_Osemiring__1_39,axiom,
% 2.64/1.79 semiring_1 @ real ).
% 2.64/1.79
% 2.64/1.79 thf(tcon_Real_Oreal___Orderings_Oorder_40,axiom,
% 2.64/1.79 order @ real ).
% 2.64/1.79
% 2.64/1.79 thf(tcon_Real_Oreal___Num_Oneg__numeral_41,axiom,
% 2.64/1.79 neg_numeral @ real ).
% 2.64/1.79
% 2.64/1.79 thf(tcon_Real_Oreal___Nat_Oring__char__0_42,axiom,
% 2.64/1.79 ring_char_0 @ real ).
% 2.64/1.79
% 2.64/1.79 thf(tcon_Real_Oreal___Orderings_Oord_43,axiom,
% 2.64/1.79 ord @ real ).
% 2.64/1.79
% 2.64/1.79 thf(tcon_Real_Oreal___Rings_Oring__1_44,axiom,
% 2.64/1.79 ring_1 @ real ).
% 2.64/1.79
% 2.64/1.79 thf(tcon_Real_Oreal___Power_Opower_45,axiom,
% 2.64/1.79 power @ real ).
% 2.64/1.79
% 2.64/1.79 thf(tcon_Real_Oreal___Num_Onumeral_46,axiom,
% 2.64/1.79 numeral @ real ).
% 2.64/1.79
% 2.64/1.79 thf(tcon_Real_Oreal___Groups_Ozero_47,axiom,
% 2.64/1.79 zero @ real ).
% 2.64/1.79
% 2.64/1.79 thf(tcon_Real_Oreal___Groups_Oone_48,axiom,
% 2.64/1.79 one @ real ).
% 2.64/1.79
% 2.64/1.79 thf(help_If_3_1_T,axiom,
% 2.64/1.79 ! [P: $o] :
% 2.64/1.79 ( ( P = $true )
% 2.64/1.79 | ( P = $false ) ) ).
% 2.64/1.79
% 2.64/1.79 thf(help_If_2_1_T,axiom,
% 2.64/1.79 ! [A: $tType,X: A,Y: A] :
% 2.64/1.79 ( ( if @ A @ $false @ X @ Y )
% 2.64/1.79 = Y ) ).
% 2.64/1.79
% 2.64/1.79 thf(help_If_1_1_T,axiom,
% 2.64/1.79 ! [A: $tType,X: A,Y: A] :
% 2.64/1.79 ( ( if @ A @ $true @ X @ Y )
% 2.64/1.79 = X ) ).
% 2.64/1.79
% 2.64/1.79 thf(conj_0,conjecture,
% 2.64/1.79 ord_less_eq @ int @ d @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( nat2 @ ( archimedean_ceiling @ real @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( ring_1_of_int @ real @ d ) ) ) ) ) ).
% 2.64/1.79 ------------------------
% 2.82/1.81 % (4592)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_4485 for (2999ds/183Mi)
% 2.82/1.81 % (4594)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_4485 for (2999ds/27Mi)
% 2.82/1.81 % (4595)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_4485 for (2999ds/2Mi)
% 2.82/1.82 % (4597)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_4485 for (2999ds/275Mi)
% 2.82/1.82 % (4596)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_4485 for (2999ds/2Mi)
% 2.82/1.82 % (4594)Instruction limit reached!
% 2.82/1.82 % (4594)------------------------------
% 2.82/1.82 % (4594)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.82/1.82 % (4594)Termination reason: Unknown
% 2.82/1.82 % (4594)Termination phase: Preprocessing 3
% 2.82/1.82
% 2.82/1.82 % (4594)Memory used [KB]: 1791
% 2.82/1.82 % (4594)Time elapsed: 0.010 s
% 2.82/1.82 % (4594)Instructions burned: 28 (million)
% 2.82/1.82 % (4594)------------------------------
% 2.82/1.82 % (4594)------------------------------
% 2.82/1.82 % (4598)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_4485 for (2999ds/18Mi)
% 2.82/1.82 % (4595)Instruction limit reached!
% 2.82/1.82 % (4595)------------------------------
% 2.82/1.82 % (4595)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.82/1.82 % (4595)Termination reason: Unknown
% 2.82/1.82 % (4596)Instruction limit reached!
% 2.82/1.82 % (4596)------------------------------
% 2.82/1.82 % (4596)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.82/1.82 % (4596)Termination reason: Unknown
% 2.82/1.82 % (4596)Termination phase: shuffling
% 2.82/1.82
% 2.82/1.82 % (4596)Memory used [KB]: 1279
% 2.82/1.82 % (4596)Time elapsed: 0.003 s
% 2.82/1.82 % (4596)Instructions burned: 2 (million)
% 2.82/1.82 % (4596)------------------------------
% 2.82/1.82 % (4596)------------------------------
% 2.82/1.82 % (4595)Termination phase: shuffling
% 2.82/1.82
% 2.82/1.82 % (4595)Memory used [KB]: 1279
% 2.82/1.82 % (4595)Time elapsed: 0.003 s
% 2.82/1.82 % (4595)Instructions burned: 3 (million)
% 2.82/1.82 % (4595)------------------------------
% 2.82/1.82 % (4595)------------------------------
% 2.82/1.82 % (4593)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_4485 for (2999ds/4Mi)
% 2.82/1.82 % (4593)Instruction limit reached!
% 2.82/1.82 % (4593)------------------------------
% 2.82/1.82 % (4593)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.82/1.82 % (4593)Termination reason: Unknown
% 2.82/1.82 % (4593)Termination phase: shuffling
% 2.82/1.82
% 2.82/1.82 % (4593)Memory used [KB]: 1407
% 2.82/1.82 % (4593)Time elapsed: 0.004 s
% 2.82/1.82 % (4593)Instructions burned: 5 (million)
% 2.82/1.82 % (4593)------------------------------
% 2.82/1.82 % (4593)------------------------------
% 2.82/1.82 % (4599)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_4485 for (2999ds/3Mi)
% 2.82/1.83 % (4598)Instruction limit reached!
% 2.82/1.83 % (4598)------------------------------
% 2.82/1.83 % (4598)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.82/1.83 % (4598)Termination reason: Unknown
% 2.82/1.83 % (4598)Termination phase: Property scanning
% 2.82/1.83
% 2.82/1.83 % (4598)Memory used [KB]: 1663
% 2.82/1.83 % (4598)Time elapsed: 0.012 s
% 2.82/1.83 % (4598)Instructions burned: 18 (million)
% 2.82/1.83 % (4598)------------------------------
% 2.82/1.83 % (4598)------------------------------
% 2.82/1.83 % (4599)Instruction limit reached!
% 2.82/1.83 % (4599)------------------------------
% 2.82/1.83 % (4599)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.82/1.83 % (4599)Termination reason: Unknown
% 2.82/1.83 % (4599)Termination phase: shuffling
% 2.82/1.83
% 2.82/1.83 % (4599)Memory used [KB]: 1407
% 2.82/1.83 % (4599)Time elapsed: 0.002 s
% 2.82/1.83 % (4599)Instructions burned: 4 (million)
% 2.82/1.83 % (4599)------------------------------
% 2.82/1.83 % (4599)------------------------------
% 2.82/1.83 % (4600)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_4485 for (2999ds/37Mi)
% 2.82/1.84 % (4604)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_4485 for (2999ds/7Mi)
% 2.82/1.84 % (4602)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_4485 for (2999ds/3Mi)
% 2.82/1.84 % (4604)Instruction limit reached!
% 2.82/1.84 % (4604)------------------------------
% 2.82/1.84 % (4604)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.82/1.84 % (4604)Termination reason: Unknown
% 2.82/1.84 % (4604)Termination phase: shuffling
% 2.82/1.84
% 2.82/1.84 % (4604)Memory used [KB]: 1407
% 2.82/1.84 % (4604)Time elapsed: 0.003 s
% 2.82/1.84 % (4604)Instructions burned: 7 (million)
% 2.82/1.84 % (4604)------------------------------
% 2.82/1.84 % (4604)------------------------------
% 2.82/1.84 % (4602)Instruction limit reached!
% 2.82/1.84 % (4602)------------------------------
% 2.82/1.84 % (4602)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.82/1.84 % (4602)Termination reason: Unknown
% 2.82/1.84 % (4602)Termination phase: shuffling
% 2.82/1.84
% 2.82/1.84 % (4602)Memory used [KB]: 1407
% 2.82/1.84 % (4602)Time elapsed: 0.003 s
% 2.82/1.84 % (4602)Instructions burned: 3 (million)
% 2.82/1.84 % (4602)------------------------------
% 2.82/1.84 % (4602)------------------------------
% 2.82/1.84 % (4601)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_4485 for (2999ds/15Mi)
% 2.82/1.84 % (4597)Refutation not found, incomplete strategy
% 2.82/1.84 % (4597)------------------------------
% 2.82/1.84 % (4597)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.82/1.84 % (4597)Termination reason: Refutation not found, incomplete strategy
% 2.82/1.84
% 2.82/1.84
% 2.82/1.84 % (4597)Memory used [KB]: 6396
% 2.82/1.84 % (4597)Time elapsed: 0.031 s
% 2.82/1.84 % (4597)Instructions burned: 52 (million)
% 2.82/1.84 % (4597)------------------------------
% 2.82/1.84 % (4597)------------------------------
% 2.82/1.85 % (4605)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_4485 for (2999ds/16Mi)
% 2.82/1.85 % (4603)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_4485 for (2999ds/1041Mi)
% 2.82/1.85 % (4601)Instruction limit reached!
% 2.82/1.85 % (4601)------------------------------
% 2.82/1.85 % (4601)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.82/1.85 % (4601)Termination reason: Unknown
% 2.82/1.85 % (4601)Termination phase: shuffling
% 2.82/1.85
% 2.82/1.85 % (4601)Memory used [KB]: 1663
% 2.82/1.85 % (4601)Time elapsed: 0.010 s
% 2.82/1.85 % (4601)Instructions burned: 15 (million)
% 2.82/1.85 % (4601)------------------------------
% 2.82/1.85 % (4601)------------------------------
% 2.82/1.85 % (4605)Instruction limit reached!
% 2.82/1.85 % (4605)------------------------------
% 2.82/1.85 % (4605)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.82/1.85 % (4605)Termination reason: Unknown
% 2.82/1.85 % (4605)Termination phase: Property scanning
% 2.82/1.85
% 2.82/1.85 % (4605)Memory used [KB]: 1663
% 2.82/1.85 % (4605)Time elapsed: 0.006 s
% 2.82/1.85 % (4605)Instructions burned: 16 (million)
% 2.82/1.85 % (4605)------------------------------
% 2.82/1.85 % (4605)------------------------------
% 2.82/1.85 % (4600)Instruction limit reached!
% 2.82/1.85 % (4600)------------------------------
% 2.82/1.85 % (4600)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.82/1.85 % (4600)Termination reason: Unknown
% 2.82/1.85 % (4600)Termination phase: Property scanning
% 2.82/1.85
% 2.82/1.85 % (4600)Memory used [KB]: 2046
% 2.82/1.85 % (4600)Time elapsed: 0.020 s
% 2.82/1.85 % (4600)Instructions burned: 38 (million)
% 2.82/1.85 % (4600)------------------------------
% 2.82/1.85 % (4600)------------------------------
% 2.82/1.86 % (4606)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_4485 for (2999ds/3Mi)
% 2.82/1.86 % (4606)Instruction limit reached!
% 2.82/1.86 % (4606)------------------------------
% 2.82/1.86 % (4606)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.82/1.86 % (4606)Termination reason: Unknown
% 2.82/1.86 % (4606)Termination phase: shuffling
% 2.82/1.86
% 2.82/1.86 % (4606)Memory used [KB]: 1407
% 2.82/1.86 % (4606)Time elapsed: 0.003 s
% 2.82/1.86 % (4606)Instructions burned: 3 (million)
% 2.82/1.86 % (4606)------------------------------
% 2.82/1.86 % (4606)------------------------------
% 2.82/1.86 % (4609)dis+1002_1:1_add=large:cnfonf=lazy_pi_sigma_gen:fe=off:prag=on:i=3:si=on:rtra=on_0 on DTF2THF_4485 for (2999ds/3Mi)
% 2.82/1.86 % (4609)Instruction limit reached!
% 2.82/1.86 % (4609)------------------------------
% 2.82/1.86 % (4609)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.82/1.86 % (4609)Termination reason: Unknown
% 2.82/1.86 % (4609)Termination phase: shuffling
% 2.82/1.86
% 2.82/1.86 % (4609)Memory used [KB]: 1407
% 2.82/1.86 % (4609)Time elapsed: 0.002 s
% 2.82/1.86 % (4609)Instructions burned: 4 (million)
% 2.82/1.86 % (4609)------------------------------
% 2.82/1.86 % (4609)------------------------------
% 2.82/1.86 % (4592)Instruction limit reached!
% 2.82/1.86 % (4592)------------------------------
% 2.82/1.86 % (4592)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.82/1.86 % (4592)Termination reason: Unknown
% 2.82/1.86 % (4592)Termination phase: Saturation
% 2.82/1.86
% 2.82/1.86 % (4592)Memory used [KB]: 6908
% 2.82/1.86 % (4592)Time elapsed: 0.058 s
% 2.82/1.86 % (4592)Instructions burned: 185 (million)
% 2.82/1.86 % (4592)------------------------------
% 2.82/1.86 % (4592)------------------------------
% 2.82/1.86 % (4607)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_4485 for (2999ds/3Mi)
% 2.82/1.87 % (4607)Instruction limit reached!
% 2.82/1.87 % (4607)------------------------------
% 2.82/1.87 % (4607)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.82/1.87 % (4607)Termination reason: Unknown
% 2.82/1.87 % (4607)Termination phase: shuffling
% 2.82/1.87
% 2.82/1.87 % (4607)Memory used [KB]: 1279
% 2.82/1.87 % (4607)Time elapsed: 0.003 s
% 2.82/1.87 % (4607)Instructions burned: 3 (million)
% 2.82/1.87 % (4607)------------------------------
% 2.82/1.87 % (4607)------------------------------
% 2.82/1.87 % (4608)lrs+10_1:1_cnfonf=off:cs=on:hud=3:prag=on:sup=off:i=7:si=on:rtra=on_0 on DTF2THF_4485 for (2999ds/7Mi)
% 2.82/1.87 % (4612)lrs+10_1:1_e2e=on:sd=1:sgt=8:ss=axioms:i=710:si=on:rtra=on_0 on DTF2THF_4485 for (2999ds/710Mi)
% 2.82/1.87 % (4613)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_4485 for (2999ds/6Mi)
% 2.82/1.87 % (4608)Instruction limit reached!
% 2.82/1.87 % (4608)------------------------------
% 2.82/1.87 % (4608)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.82/1.87 % (4608)Termination reason: Unknown
% 2.82/1.87 % (4608)Termination phase: shuffling
% 2.82/1.87
% 2.82/1.87 % (4608)Memory used [KB]: 1407
% 2.82/1.87 % (4608)Time elapsed: 0.006 s
% 2.82/1.87 % (4608)Instructions burned: 7 (million)
% 2.82/1.87 % (4608)------------------------------
% 2.82/1.87 % (4608)------------------------------
% 2.82/1.87 % (4613)Instruction limit reached!
% 2.82/1.87 % (4613)------------------------------
% 2.82/1.87 % (4613)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.82/1.87 % (4613)Termination reason: Unknown
% 2.82/1.87 % (4613)Termination phase: shuffling
% 2.82/1.87
% 2.82/1.87 % (4613)Memory used [KB]: 1407
% 2.82/1.87 % (4613)Time elapsed: 0.003 s
% 2.82/1.87 % (4613)Instructions burned: 6 (million)
% 2.82/1.87 % (4613)------------------------------
% 2.82/1.87 % (4613)------------------------------
% 2.82/1.88 % (4610)dis+1004_1:1_cha=on:cs=on:fe=off:hud=1:i=4:si=on:rtra=on_0 on DTF2THF_4485 for (2999ds/4Mi)
% 2.82/1.88 % (4611)lrs+1002_1:1_anc=all_dependent:au=on:cbe=off:fde=unused:ntd=on:i=18:si=on:rtra=on_0 on DTF2THF_4485 for (2999ds/18Mi)
% 2.82/1.88 % (4610)Instruction limit reached!
% 2.82/1.88 % (4610)------------------------------
% 2.82/1.88 % (4610)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.82/1.88 % (4610)Termination reason: Unknown
% 2.82/1.88 % (4610)Termination phase: shuffling
% 2.82/1.88
% 2.82/1.88 % (4610)Memory used [KB]: 1407
% 2.82/1.88 % (4610)Time elapsed: 0.004 s
% 2.82/1.88 % (4610)Instructions burned: 5 (million)
% 2.82/1.88 % (4610)------------------------------
% 2.82/1.88 % (4610)------------------------------
% 2.82/1.88 % (4614)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_4485 for (2999ds/902Mi)
% 2.82/1.88 % (4616)dis+10_1:1_cnfonf=lazy_gen:fe=off:i=5:si=on:rtra=on_0 on DTF2THF_4485 for (2999ds/5Mi)
% 2.82/1.89 % (4616)Instruction limit reached!
% 2.82/1.89 % (4616)------------------------------
% 2.82/1.89 % (4616)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.82/1.89 % (4616)Termination reason: Unknown
% 2.82/1.89 % (4616)Termination phase: shuffling
% 2.82/1.89
% 2.82/1.89 % (4616)Memory used [KB]: 1407
% 2.82/1.89 % (4616)Time elapsed: 0.003 s
% 2.82/1.89 % (4616)Instructions burned: 8 (million)
% 2.82/1.89 % (4616)------------------------------
% 2.82/1.89 % (4616)------------------------------
% 2.82/1.89 % (4611)Instruction limit reached!
% 2.82/1.89 % (4611)------------------------------
% 2.82/1.89 % (4611)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.82/1.89 % (4611)Termination reason: Unknown
% 2.82/1.89 % (4611)Termination phase: Property scanning
% 2.82/1.89
% 2.82/1.89 % (4611)Memory used [KB]: 1663
% 2.82/1.89 % (4611)Time elapsed: 0.011 s
% 2.82/1.89 % (4611)Instructions burned: 18 (million)
% 2.82/1.89 % (4611)------------------------------
% 2.82/1.89 % (4611)------------------------------
% 2.82/1.89 % (4615)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_4485 for (2999ds/21Mi)
% 2.82/1.90 % (4618)lrs+1002_1:128_au=on:c=on:fsr=off:piset=equals:i=377:si=on:rtra=on_0 on DTF2THF_4485 for (2999ds/377Mi)
% 2.82/1.90 % (4617)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_4485 for (2999ds/6Mi)
% 2.82/1.90 % (4617)Instruction limit reached!
% 2.82/1.90 % (4617)------------------------------
% 2.82/1.90 % (4617)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.82/1.90 % (4617)Termination reason: Unknown
% 2.82/1.90 % (4617)Termination phase: shuffling
% 2.82/1.90
% 2.82/1.90 % (4617)Memory used [KB]: 1407
% 2.82/1.90 % (4617)Time elapsed: 0.005 s
% 2.82/1.90 % (4617)Instructions burned: 6 (million)
% 2.82/1.90 % (4617)------------------------------
% 2.82/1.90 % (4617)------------------------------
% 2.82/1.90 % (4615)Instruction limit reached!
% 2.82/1.90 % (4615)------------------------------
% 2.82/1.90 % (4615)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.82/1.90 % (4615)Termination reason: Unknown
% 2.82/1.90 % (4615)Termination phase: Property scanning
% 2.82/1.90
% 2.82/1.90 % (4615)Memory used [KB]: 1663
% 2.82/1.90 % (4615)Time elapsed: 0.014 s
% 2.82/1.90 % (4615)Instructions burned: 21 (million)
% 2.82/1.90 % (4615)------------------------------
% 2.82/1.90 % (4615)------------------------------
% 2.82/1.90 % (4619)dis+1010_1:4_atotf=0.2:c=on:cbe=off:cnfonf=lazy_simp:fe=off:ins=2:ntd=on:s2a=on:s2at=5.0:sgt=5:ss=axioms:st=1.5:i=779:si=on:rtra=on_0 on DTF2THF_4485 for (2999ds/779Mi)
% 2.82/1.92 % (4620)lrs+10_1:1_cnfonf=lazy_not_be_gen:ntd=on:sp=const_min:ss=axioms:sup=off:i=19:si=on:rtra=on_0 on DTF2THF_4485 for (2999ds/19Mi)
% 2.82/1.92 % (4621)lrs+1010_1:1_au=on:s2a=on:sd=1:sgt=50:ss=axioms:i=879:si=on:rtra=on_0 on DTF2THF_4485 for (2998ds/879Mi)
% 2.82/1.93 % (4620)Instruction limit reached!
% 2.82/1.93 % (4620)------------------------------
% 2.82/1.93 % (4620)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.82/1.93 % (4620)Termination reason: Unknown
% 2.82/1.93 % (4620)Termination phase: shuffling
% 2.82/1.93
% 2.82/1.93 % (4620)Memory used [KB]: 1663
% 2.82/1.93 % (4620)Time elapsed: 0.012 s
% 2.82/1.93 % (4620)Instructions burned: 19 (million)
% 2.82/1.93 % (4620)------------------------------
% 2.82/1.93 % (4620)------------------------------
% 2.82/1.95 % (4622)dis+1002_1:128_acc=on:er=filter:i=17:si=on:rtra=on_0 on DTF2THF_4485 for (2998ds/17Mi)
% 2.82/1.96 % (4622)Instruction limit reached!
% 2.82/1.96 % (4622)------------------------------
% 2.82/1.96 % (4622)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.82/1.96 % (4622)Termination reason: Unknown
% 2.82/1.96 % (4622)Termination phase: Property scanning
% 2.82/1.96
% 2.82/1.96 % (4622)Memory used [KB]: 1663
% 2.82/1.96 % (4622)Time elapsed: 0.011 s
% 2.82/1.96 % (4622)Instructions burned: 17 (million)
% 2.82/1.96 % (4622)------------------------------
% 2.82/1.96 % (4622)------------------------------
% 2.82/1.97 % (4623)ott+21_1:1_apa=on:au=on:cnfonf=off:sos=on:i=3:si=on:rtra=on_0 on DTF2THF_4485 for (2998ds/3Mi)
% 2.82/1.97 % (4623)Instruction limit reached!
% 2.82/1.97 % (4623)------------------------------
% 2.82/1.97 % (4623)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.82/1.97 % (4623)Termination reason: Unknown
% 2.82/1.97 % (4623)Termination phase: shuffling
% 2.82/1.97
% 2.82/1.97 % (4623)Memory used [KB]: 1407
% 2.82/1.97 % (4623)Time elapsed: 0.003 s
% 2.82/1.97 % (4623)Instructions burned: 5 (million)
% 2.82/1.97 % (4623)------------------------------
% 2.82/1.97 % (4623)------------------------------
% 2.82/1.98 % (4624)lrs+1010_1:8_cnfonf=off:hud=1:inj=on:tnu=5:i=30:si=on:rtra=on_0 on DTF2THF_4485 for (2998ds/30Mi)
% 2.82/1.99 % (4624)Instruction limit reached!
% 2.82/1.99 % (4624)------------------------------
% 2.82/1.99 % (4624)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.82/1.99 % (4624)Termination reason: Unknown
% 2.82/1.99 % (4624)Termination phase: Preprocessing 3
% 2.82/1.99
% 2.82/1.99 % (4624)Memory used [KB]: 1791
% 2.82/1.99 % (4624)Time elapsed: 0.011 s
% 2.82/1.99 % (4624)Instructions burned: 32 (million)
% 2.82/1.99 % (4624)------------------------------
% 2.82/1.99 % (4624)------------------------------
% 2.82/2.00 % (4625)dis+10_1:1_ixr=off:plsq=on:plsqc=1:plsqr=32,1:s2a=on:i=127:si=on:rtra=on_0 on DTF2THF_4485 for (2998ds/127Mi)
% 4.21/2.04 % (4625)Instruction limit reached!
% 4.21/2.04 % (4625)------------------------------
% 4.21/2.04 % (4625)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 4.21/2.04 % (4625)Termination reason: Unknown
% 4.21/2.04 % (4625)Termination phase: Saturation
% 4.21/2.04
% 4.21/2.04 % (4625)Memory used [KB]: 6780
% 4.21/2.04 % (4625)Time elapsed: 0.043 s
% 4.21/2.04 % (4625)Instructions burned: 128 (million)
% 4.21/2.04 % (4625)------------------------------
% 4.21/2.04 % (4625)------------------------------
% 4.21/2.05 % (4618)Instruction limit reached!
% 4.21/2.05 % (4618)------------------------------
% 4.21/2.05 % (4618)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 4.21/2.05 % (4618)Termination reason: Unknown
% 4.21/2.05 % (4618)Termination phase: Saturation
% 4.21/2.05
% 4.21/2.05 % (4618)Memory used [KB]: 7419
% 4.21/2.05 % (4618)Time elapsed: 0.168 s
% 4.21/2.05 % (4618)Instructions burned: 378 (million)
% 4.21/2.05 % (4618)------------------------------
% 4.21/2.05 % (4618)------------------------------
% 4.21/2.05 % (4626)lrs+1002_1:1_au=on:cbe=off:cnfonf=conj_eager:cond=on:hi=on:i=100:si=on:rtra=on_0 on DTF2THF_4485 for (2997ds/100Mi)
% 4.21/2.06 % (4627)dis+10_1:1_anc=none:cnfonf=lazy_gen:fd=preordered:fe=off:hud=10:ins=3:ixr=off:nwc=5.0:plsq=on:plsqc=1:plsqr=32,1:sp=const_frequency:uhcvi=on:i=3:si=on:rtra=on_0 on DTF2THF_4485 for (2997ds/3Mi)
% 4.21/2.06 % (4627)Instruction limit reached!
% 4.21/2.06 % (4627)------------------------------
% 4.21/2.06 % (4627)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 4.21/2.06 % (4627)Termination reason: Unknown
% 4.21/2.06 % (4627)Termination phase: shuffling
% 4.21/2.06
% 4.21/2.06 % (4627)Memory used [KB]: 1407
% 4.21/2.06 % (4627)Time elapsed: 0.003 s
% 4.21/2.06 % (4627)Instructions burned: 5 (million)
% 4.21/2.06 % (4627)------------------------------
% 4.21/2.06 % (4627)------------------------------
% 4.21/2.07 % (4628)lrs+10_8:1_au=on:avsq=on:e2e=on:ins=3:s2a=on:s2at=3.0:ss=axioms:i=20:si=on:rtra=on_0 on DTF2THF_4485 for (2997ds/20Mi)
% 4.21/2.08 % (4626)Instruction limit reached!
% 4.21/2.08 % (4626)------------------------------
% 4.21/2.08 % (4626)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 4.21/2.08 % (4626)Termination reason: Unknown
% 4.21/2.08 % (4626)Termination phase: Saturation
% 4.21/2.08
% 4.21/2.08 % (4626)Memory used [KB]: 6524
% 4.21/2.08 % (4626)Time elapsed: 0.028 s
% 4.21/2.08 % (4626)Instructions burned: 102 (million)
% 4.21/2.08 % (4626)------------------------------
% 4.21/2.08 % (4626)------------------------------
% 4.21/2.08 % (4628)Instruction limit reached!
% 4.21/2.08 % (4628)------------------------------
% 4.21/2.08 % (4628)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 4.21/2.08 % (4628)Termination reason: Unknown
% 4.21/2.08 % (4628)Termination phase: Property scanning
% 4.21/2.08
% 4.21/2.08 % (4628)Memory used [KB]: 1663
% 4.21/2.08 % (4628)Time elapsed: 0.008 s
% 4.21/2.08 % (4628)Instructions burned: 22 (million)
% 4.21/2.08 % (4628)------------------------------
% 4.21/2.08 % (4628)------------------------------
% 4.21/2.09 % (4629)dis+1002_1:1_cbe=off:hud=5:nm=4:plsq=on:plsqr=7,1:prag=on:sp=const_max:tnu=1:i=86:si=on:rtra=on_0 on DTF2THF_4485 for (2997ds/86Mi)
% 4.21/2.09 % (4630)lrs+1010_1:1_au=on:cbe=off:nm=2:ntd=on:sd=2:ss=axioms:st=5.0:i=107:si=on:rtra=on_0 on DTF2THF_4485 for (2997ds/107Mi)
% 4.21/2.10 % (4612)Instruction limit reached!
% 4.21/2.10 % (4612)------------------------------
% 4.21/2.10 % (4612)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 4.21/2.10 % (4612)Termination reason: Unknown
% 4.21/2.10 % (4612)Termination phase: Saturation
% 4.21/2.10
% 4.21/2.10 % (4612)Memory used [KB]: 7803
% 4.21/2.10 % (4612)Time elapsed: 0.226 s
% 4.21/2.10 % (4612)Instructions burned: 713 (million)
% 4.21/2.10 % (4612)------------------------------
% 4.21/2.10 % (4612)------------------------------
% 4.21/2.11 % (4631)lrs+2_1:1024_cnfonf=lazy_gen:fe=off:hud=15:plsq=on:plsqc=1:plsqr=32,1:i=39:si=on:rtra=on_0 on DTF2THF_4485 for (2997ds/39Mi)
% 4.88/2.11 % (4629)Instruction limit reached!
% 4.88/2.11 % (4629)------------------------------
% 4.88/2.11 % (4629)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 4.88/2.11 % (4629)Termination reason: Unknown
% 4.88/2.11 % (4629)Termination phase: Saturation
% 4.88/2.11
% 4.88/2.11 % (4629)Memory used [KB]: 6652
% 4.88/2.11 % (4629)Time elapsed: 0.027 s
% 4.88/2.11 % (4629)Instructions burned: 86 (million)
% 4.88/2.11 % (4629)------------------------------
% 4.88/2.11 % (4629)------------------------------
% 4.88/2.12 % (4631)Instruction limit reached!
% 4.88/2.12 % (4631)------------------------------
% 4.88/2.12 % (4631)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 4.88/2.12 % (4631)Termination reason: Unknown
% 4.88/2.12 % (4631)Termination phase: Property scanning
% 4.88/2.12
% 4.88/2.12 % (4631)Memory used [KB]: 2046
% 4.88/2.12 % (4631)Time elapsed: 0.013 s
% 4.88/2.12 % (4631)Instructions burned: 42 (million)
% 4.88/2.12 % (4631)------------------------------
% 4.88/2.12 % (4631)------------------------------
% 4.88/2.12 % (4632)dis+10_1:1_cnfonf=lazy_not_gen:fsr=off:kws=precedence:nwc=5.0:s2a=on:ss=axioms:st=1.5:i=448:si=on:rtra=on_0 on DTF2THF_4485 for (2996ds/448Mi)
% 4.88/2.13 % (4633)lrs+10_1:512_au=on:fde=unused:lma=on:nm=32:plsq=on:plsqc=1:plsqr=16121663,131072:sfv=off:sp=const_max:ss=axioms:st=3.0:tgt=full:i=46:si=on:rtra=on_0 on DTF2THF_4485 for (2996ds/46Mi)
% 4.88/2.14 % (4630)Instruction limit reached!
% 4.88/2.14 % (4630)------------------------------
% 4.88/2.14 % (4630)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 4.88/2.14 % (4630)Termination reason: Unknown
% 4.88/2.14 % (4630)Termination phase: Saturation
% 4.88/2.14
% 4.88/2.14 % (4630)Memory used [KB]: 6780
% 4.88/2.14 % (4630)Time elapsed: 0.053 s
% 4.88/2.14 % (4630)Instructions burned: 108 (million)
% 4.88/2.14 % (4630)------------------------------
% 4.88/2.14 % (4630)------------------------------
% 4.88/2.14 % (4633)Instruction limit reached!
% 4.88/2.14 % (4633)------------------------------
% 4.88/2.14 % (4633)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 4.88/2.14 % (4633)Termination reason: Unknown
% 4.88/2.14 % (4633)Termination phase: Property scanning
% 4.88/2.14
% 4.88/2.14 % (4633)Memory used [KB]: 1791
% 4.88/2.14 % (4633)Time elapsed: 0.016 s
% 4.88/2.14 % (4633)Instructions burned: 47 (million)
% 4.88/2.14 % (4633)------------------------------
% 4.88/2.14 % (4633)------------------------------
% 4.88/2.15 % (4634)lrs+10_1:10_au=on:av=off:cbe=off:cnfonf=lazy_pi_sigma_gen:ntd=on:plsq=on:plsqc=1:plsqr=32,1:i=98:si=on:rtra=on_0 on DTF2THF_4485 for (2996ds/98Mi)
% 4.88/2.15 % (4635)ott+1002_1:1_apa=on:au=on:bd=off:cnfonf=off:fd=off:sos=on:sp=weighted_frequency:i=507:si=on:rtra=on_0 on DTF2THF_4485 for (2996ds/507Mi)
% 4.88/2.19 % (4634)Instruction limit reached!
% 4.88/2.19 % (4634)------------------------------
% 4.88/2.19 % (4634)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 4.88/2.19 % (4634)Termination reason: Unknown
% 4.88/2.19 % (4634)Termination phase: Saturation
% 4.88/2.19
% 4.88/2.19 % (4634)Memory used [KB]: 2046
% 4.88/2.19 % (4634)Time elapsed: 0.045 s
% 4.88/2.19 % (4634)Instructions burned: 99 (million)
% 4.88/2.19 % (4634)------------------------------
% 4.88/2.19 % (4634)------------------------------
% 5.39/2.21 % (4636)dis+1010_2:3_amm=off:fd=preordered:ixr=off:nm=0:pe=on:piset=equals:prag=on:sac=on:tgt=ground:i=149:si=on:rtra=on_0 on DTF2THF_4485 for (2996ds/149Mi)
% 5.39/2.24 % (4619)Instruction limit reached!
% 5.39/2.24 % (4619)------------------------------
% 5.39/2.24 % (4619)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.39/2.24 % (4619)Termination reason: Unknown
% 5.39/2.24 % (4619)Termination phase: Saturation
% 5.39/2.24
% 5.39/2.24 % (4619)Memory used [KB]: 7419
% 5.39/2.24 % (4619)Time elapsed: 0.339 s
% 5.39/2.24 % (4619)Instructions burned: 780 (million)
% 5.39/2.24 % (4619)------------------------------
% 5.39/2.24 % (4619)------------------------------
% 5.39/2.25 % (4637)lrs+21_1:16_au=on:bd=off:piset=and:i=39:si=on:rtra=on_0 on DTF2THF_4485 for (2995ds/39Mi)
% 5.39/2.26 % (4637)Instruction limit reached!
% 5.39/2.26 % (4637)------------------------------
% 5.39/2.26 % (4637)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.39/2.26 % (4637)Termination reason: Unknown
% 5.39/2.26 % (4637)Termination phase: Property scanning
% 5.39/2.26
% 5.39/2.26 % (4637)Memory used [KB]: 1791
% 5.39/2.26 % (4637)Time elapsed: 0.013 s
% 5.39/2.26 % (4637)Instructions burned: 40 (million)
% 5.39/2.26 % (4637)------------------------------
% 5.39/2.26 % (4637)------------------------------
% 5.39/2.27 % (4632)Instruction limit reached!
% 5.39/2.27 % (4632)------------------------------
% 5.39/2.27 % (4632)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.39/2.27 % (4632)Termination reason: Unknown
% 5.39/2.27 % (4632)Termination phase: Saturation
% 5.39/2.27
% 5.39/2.27 % (4632)Memory used [KB]: 7419
% 5.39/2.27 % (4632)Time elapsed: 0.166 s
% 5.39/2.27 % (4632)Instructions burned: 450 (million)
% 5.39/2.27 % (4632)------------------------------
% 5.39/2.27 % (4632)------------------------------
% 5.39/2.27 % (4638)lrs+1004_1:1_cnfonf=lazy_pi_sigma_gen:fe=off:gs=on:hud=10:prag=on:tnu=1:i=6:si=on:rtra=on_0 on DTF2THF_4485 for (2995ds/6Mi)
% 5.39/2.27 % (4639)dis+1002_1:1_aac=none:au=on:e2e=on:nm=2:nwc=10.0:sd=1:ss=axioms:st=3.0:i=1087:si=on:rtra=on_0 on DTF2THF_4485 for (2995ds/1087Mi)
% 5.39/2.28 % (4638)Instruction limit reached!
% 5.39/2.28 % (4638)------------------------------
% 5.39/2.28 % (4638)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.39/2.28 % (4638)Termination reason: Unknown
% 5.39/2.28 % (4638)Termination phase: shuffling
% 5.39/2.28
% 5.39/2.28 % (4638)Memory used [KB]: 1407
% 5.39/2.28 % (4638)Time elapsed: 0.003 s
% 5.39/2.28 % (4638)Instructions burned: 7 (million)
% 5.39/2.28 % (4638)------------------------------
% 5.39/2.28 % (4638)------------------------------
% 5.39/2.28 % (4640)dis+10_1:1_aac=none:cs=on:i=47:si=on:rtra=on_0 on DTF2THF_4485 for (2995ds/47Mi)
% 5.94/2.30 % (4636)Instruction limit reached!
% 5.94/2.30 % (4636)------------------------------
% 5.94/2.30 % (4636)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.94/2.30 % (4636)Termination reason: Unknown
% 5.94/2.30 % (4636)Termination phase: Saturation
% 5.94/2.30
% 5.94/2.30 % (4636)Memory used [KB]: 7036
% 5.94/2.30 % (4636)Time elapsed: 0.110 s
% 5.94/2.30 % (4636)Instructions burned: 149 (million)
% 5.94/2.30 % (4636)------------------------------
% 5.94/2.30 % (4636)------------------------------
% 5.94/2.30 % (4640)Instruction limit reached!
% 5.94/2.30 % (4640)------------------------------
% 5.94/2.30 % (4640)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.94/2.30 % (4640)Termination reason: Unknown
% 5.94/2.30 % (4640)Termination phase: Property scanning
% 5.94/2.30
% 5.94/2.30 % (4640)Memory used [KB]: 1791
% 5.94/2.30 % (4640)Time elapsed: 0.015 s
% 5.94/2.30 % (4640)Instructions burned: 47 (million)
% 5.94/2.30 % (4640)------------------------------
% 5.94/2.30 % (4640)------------------------------
% 5.94/2.31 % (4642)dis+10_1:1_av=off:bd=off:cnfonf=lazy_gen:fe=off:hud=10:plsq=on:plsqc=1:plsqr=32,1:prag=on:tnu=1:i=432:si=on:rtra=on_0 on DTF2THF_4485 for (2994ds/432Mi)
% 5.94/2.31 % (4635)Instruction limit reached!
% 5.94/2.31 % (4635)------------------------------
% 5.94/2.31 % (4635)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.94/2.31 % (4635)Termination reason: Unknown
% 5.94/2.31 % (4635)Termination phase: Saturation
% 5.94/2.31
% 5.94/2.31 % (4635)Memory used [KB]: 7547
% 5.94/2.31 % (4635)Time elapsed: 0.178 s
% 5.94/2.31 % (4635)Instructions burned: 508 (million)
% 5.94/2.31 % (4635)------------------------------
% 5.94/2.31 % (4635)------------------------------
% 5.94/2.32 % (4641)lrs+1010_1:128_av=off:cbe=off:cnfonf=lazy_simp:fsr=off:ntd=on:ss=axioms:st=1.5:i=44:si=on:rtra=on_0 on DTF2THF_4485 for (2994ds/44Mi)
% 5.94/2.32 % (4643)lrs+10_7:1_cnfonf=lazy_gen:fe=off:nwc=5.0:plsq=on:i=62:si=on:rtra=on_0 on DTF2THF_4485 for (2994ds/62Mi)
% 5.94/2.33 % (4621)Instruction limit reached!
% 5.94/2.33 % (4621)------------------------------
% 5.94/2.33 % (4621)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.94/2.33 % (4621)Termination reason: Unknown
% 5.94/2.33 % (4621)Termination phase: Saturation
% 5.94/2.33
% 5.94/2.33 % (4621)Memory used [KB]: 7931
% 5.94/2.33 % (4621)Time elapsed: 0.407 s
% 5.94/2.33 % (4621)Instructions burned: 882 (million)
% 5.94/2.33 % (4621)------------------------------
% 5.94/2.33 % (4621)------------------------------
% 5.94/2.33 % (4641)Instruction limit reached!
% 5.94/2.33 % (4641)------------------------------
% 5.94/2.33 % (4641)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.94/2.33 % (4641)Termination reason: Unknown
% 5.94/2.33 % (4641)Termination phase: Property scanning
% 5.94/2.33
% 5.94/2.33 % (4641)Memory used [KB]: 2046
% 5.94/2.33 % (4641)Time elapsed: 0.019 s
% 5.94/2.33 % (4641)Instructions burned: 47 (million)
% 5.94/2.33 % (4641)------------------------------
% 5.94/2.33 % (4641)------------------------------
% 5.94/2.33 % (4603)Instruction limit reached!
% 5.94/2.33 % (4603)------------------------------
% 5.94/2.33 % (4603)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.94/2.33 % (4603)Termination reason: Unknown
% 5.94/2.33 % (4603)Termination phase: Saturation
% 5.94/2.33
% 5.94/2.33 % (4603)Memory used [KB]: 9722
% 5.94/2.33 % (4603)Time elapsed: 0.490 s
% 5.94/2.33 % (4603)Instructions burned: 1041 (million)
% 5.94/2.33 % (4603)------------------------------
% 5.94/2.33 % (4603)------------------------------
% 5.94/2.34 % (4643)Instruction limit reached!
% 5.94/2.34 % (4643)------------------------------
% 5.94/2.34 % (4643)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.94/2.34 % (4643)Termination reason: Unknown
% 5.94/2.34 % (4643)Termination phase: Property scanning
% 5.94/2.34
% 5.94/2.34 % (4643)Memory used [KB]: 2046
% 5.94/2.34 % (4643)Time elapsed: 0.018 s
% 5.94/2.34 % (4643)Instructions burned: 64 (million)
% 5.94/2.34 % (4643)------------------------------
% 5.94/2.34 % (4643)------------------------------
% 5.94/2.34 % (4644)lrs+1010_1:16_add=large:amm=off:anc=all_dependent:chr=on:fd=off:ins=2:sp=const_max:tnu=5:uhcvi=on:i=31:si=on:rtra=on_0 on DTF2THF_4485 for (2994ds/31Mi)
% 5.94/2.34 % (4645)dis+10_1:128_hud=1:i=96:si=on:rtra=on_0 on DTF2THF_4485 for (2994ds/96Mi)
% 5.94/2.35 % (4644)Instruction limit reached!
% 5.94/2.35 % (4644)------------------------------
% 5.94/2.35 % (4644)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.94/2.35 % (4644)Termination reason: Unknown
% 5.94/2.35 % (4644)Termination phase: Preprocessing 3
% 5.94/2.35
% 5.94/2.35 % (4644)Memory used [KB]: 1791
% 5.94/2.35 % (4644)Time elapsed: 0.011 s
% 5.94/2.35 % (4644)Instructions burned: 32 (million)
% 5.94/2.35 % (4644)------------------------------
% 5.94/2.35 % (4644)------------------------------
% 5.94/2.35 % (4646)lrs+1002_1:4_au=on:cbe=off:cnfonf=lazy_pi_sigma_gen:hud=15:inj=on:nwc=4.0:s2a=on:sac=on:sp=reverse_arity:i=453:si=on:rtra=on_0 on DTF2THF_4485 for (2994ds/453Mi)
% 5.94/2.36 % (4648)dis+1002_1:1_bd=off:fsr=off:ntd=on:s2a=on:ss=axioms:i=83:si=on:rtra=on_0 on DTF2THF_4485 for (2994ds/83Mi)
% 5.94/2.36 % (4647)lrs-1004_1:1_au=on:bsd=on:cbe=off:cnfonf=off:fe=axiom:fsd=on:i=71:si=on:rtra=on_0 on DTF2THF_4485 for (2994ds/71Mi)
% 5.94/2.37 % (4645)Instruction limit reached!
% 5.94/2.37 % (4645)------------------------------
% 5.94/2.37 % (4645)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.94/2.37 % (4645)Termination reason: Unknown
% 5.94/2.37 % (4645)Termination phase: Saturation
% 5.94/2.37
% 5.94/2.37 % (4645)Memory used [KB]: 6652
% 5.94/2.37 % (4645)Time elapsed: 0.031 s
% 5.94/2.37 % (4645)Instructions burned: 96 (million)
% 5.94/2.37 % (4645)------------------------------
% 5.94/2.37 % (4645)------------------------------
% 5.94/2.38 % (4614)Instruction limit reached!
% 5.94/2.38 % (4614)------------------------------
% 5.94/2.38 % (4614)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.94/2.38 % (4614)Termination reason: Unknown
% 5.94/2.38 % (4614)Termination phase: Saturation
% 5.94/2.38
% 5.94/2.38 % (4614)Memory used [KB]: 8059
% 5.94/2.38 % (4614)Time elapsed: 0.494 s
% 5.94/2.38 % (4614)Instructions burned: 903 (million)
% 5.94/2.38 % (4614)------------------------------
% 5.94/2.38 % (4614)------------------------------
% 5.94/2.38 % (4649)lrs+10_1:128_cnfonf=lazy_not_gen_be_off:hud=1:ins=1:i=11:si=on:rtra=on_0 on DTF2THF_4485 for (2994ds/11Mi)
% 5.94/2.38 % (4648)Instruction limit reached!
% 5.94/2.38 % (4648)------------------------------
% 5.94/2.38 % (4648)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.94/2.38 % (4648)Termination reason: Unknown
% 5.94/2.38 % (4648)Termination phase: Saturation
% 5.94/2.38
% 5.94/2.38 % (4648)Memory used [KB]: 6524
% 5.94/2.38 % (4648)Time elapsed: 0.028 s
% 5.94/2.38 % (4648)Instructions burned: 84 (million)
% 5.94/2.38 % (4648)------------------------------
% 5.94/2.38 % (4648)------------------------------
% 5.94/2.39 % (4649)Instruction limit reached!
% 5.94/2.39 % (4649)------------------------------
% 5.94/2.39 % (4649)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.94/2.39 % (4649)Termination reason: Unknown
% 5.94/2.39 % (4649)Termination phase: shuffling
% 5.94/2.39
% 5.94/2.39 % (4649)Memory used [KB]: 1535
% 5.94/2.39 % (4649)Time elapsed: 0.004 s
% 5.94/2.39 % (4649)Instructions burned: 11 (million)
% 5.94/2.39 % (4649)------------------------------
% 5.94/2.39 % (4649)------------------------------
% 5.94/2.39 % (4651)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_4485 for (2994ds/6Mi)
% 5.94/2.40 % (4652)dis+10_1:1_avsq=on:cnfonf=off:e2e=on:nm=32:i=150:si=on:rtra=on_0 on DTF2THF_4485 for (2994ds/150Mi)
% 5.94/2.40 % (4650)lrs+1010_1:1_au=on:cbe=off:cnfonf=lazy_not_gen_be_off:fsr=off:spb=non_intro:i=41:si=on:rtra=on_0 on DTF2THF_4485 for (2994ds/41Mi)
% 5.94/2.40 % (4651)Instruction limit reached!
% 5.94/2.40 % (4651)------------------------------
% 5.94/2.40 % (4651)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.58/2.40 % (4651)Termination reason: Unknown
% 6.58/2.40 % (4651)Termination phase: shuffling
% 6.58/2.40
% 6.58/2.40 % (4651)Memory used [KB]: 1407
% 6.58/2.40 % (4651)Time elapsed: 0.003 s
% 6.58/2.40 % (4651)Instructions burned: 6 (million)
% 6.58/2.40 % (4651)------------------------------
% 6.58/2.40 % (4651)------------------------------
% 6.58/2.40 % (4647)Instruction limit reached!
% 6.58/2.40 % (4647)------------------------------
% 6.58/2.40 % (4647)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.58/2.40 % (4647)Termination reason: Unknown
% 6.58/2.40 % (4647)Termination phase: Saturation
% 6.58/2.40
% 6.58/2.40 % (4647)Memory used [KB]: 6524
% 6.58/2.40 % (4647)Time elapsed: 0.040 s
% 6.58/2.40 % (4647)Instructions burned: 71 (million)
% 6.58/2.40 % (4647)------------------------------
% 6.58/2.40 % (4647)------------------------------
% 6.58/2.41 % (4653)ott+1010_1:1_au=on:s2a=on:sd=7:ss=axioms:st=1.5:i=343:si=on:rtra=on_0 on DTF2THF_4485 for (2993ds/343Mi)
% 6.58/2.42 % (4650)Instruction limit reached!
% 6.58/2.42 % (4650)------------------------------
% 6.58/2.42 % (4650)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.58/2.42 % (4650)Termination reason: Unknown
% 6.58/2.42 % (4650)Termination phase: Property scanning
% 6.58/2.42
% 6.58/2.42 % (4650)Memory used [KB]: 2046
% 6.58/2.42 % (4650)Time elapsed: 0.022 s
% 6.58/2.42 % (4650)Instructions burned: 41 (million)
% 6.58/2.42 % (4650)------------------------------
% 6.58/2.42 % (4650)------------------------------
% 6.58/2.42 % (4654)ott+1010_1:4_cbe=off:chr=on:hfsq=on:hfsqr=1,16:hud=3:sac=on:sp=unary_frequency:ss=axioms:st=5.0:tnu=2:i=109:si=on:rtra=on_0 on DTF2THF_4485 for (2993ds/109Mi)
% 6.58/2.44 % (4655)dis+10_1:1_au=on:c=on:e2e=on:fsr=off:sos=all:sp=const_frequency:ss=axioms:i=253:si=on:rtra=on_0 on DTF2THF_4485 for (2993ds/253Mi)
% 6.58/2.44 % (4652)Instruction limit reached!
% 6.58/2.44 % (4652)------------------------------
% 6.58/2.44 % (4652)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.58/2.44 % (4652)Termination reason: Unknown
% 6.58/2.44 % (4652)Termination phase: Saturation
% 6.58/2.44
% 6.58/2.44 % (4652)Memory used [KB]: 6908
% 6.58/2.44 % (4652)Time elapsed: 0.050 s
% 6.58/2.44 % (4652)Instructions burned: 151 (million)
% 6.58/2.44 % (4652)------------------------------
% 6.58/2.44 % (4652)------------------------------
% 6.58/2.44 % (4642)Instruction limit reached!
% 6.58/2.44 % (4642)------------------------------
% 6.58/2.44 % (4642)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.58/2.44 % (4642)Termination reason: Unknown
% 6.58/2.44 % (4642)Termination phase: Saturation
% 6.58/2.44
% 6.58/2.44 % (4642)Memory used [KB]: 3837
% 6.58/2.44 % (4642)Time elapsed: 0.138 s
% 6.58/2.45 % (4642)Instructions burned: 432 (million)
% 6.58/2.45 % (4642)------------------------------
% 6.58/2.45 % (4642)------------------------------
% 6.58/2.45 % (4656)lrs+10_1:1_au=on:cnfonf=off:sd=1:sgt=16:ss=axioms:i=140:si=on:rtra=on_0 on DTF2THF_4485 for (2993ds/140Mi)
% 6.58/2.45 % (4657)dis+1010_16:1_acc=model:au=on:avsq=on:cond=fast:plsq=on:plsqc=2:plsqr=9267,262144:i=614:si=on:rtra=on_0 on DTF2THF_4485 for (2993ds/614Mi)
% 6.58/2.48 % (4654)Instruction limit reached!
% 6.58/2.48 % (4654)------------------------------
% 6.58/2.48 % (4654)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.58/2.48 % (4654)Termination reason: Unknown
% 6.58/2.48 % (4654)Termination phase: Saturation
% 6.58/2.48
% 6.58/2.48 % (4654)Memory used [KB]: 6780
% 6.58/2.48 % (4654)Time elapsed: 0.061 s
% 6.58/2.48 % (4654)Instructions burned: 109 (million)
% 6.58/2.48 % (4654)------------------------------
% 6.58/2.48 % (4654)------------------------------
% 6.96/2.50 % (4646)Instruction limit reached!
% 6.96/2.50 % (4646)------------------------------
% 6.96/2.50 % (4646)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.96/2.50 % (4646)Termination reason: Unknown
% 6.96/2.50 % (4646)Termination phase: Saturation
% 6.96/2.50
% 6.96/2.50 % (4646)Memory used [KB]: 7547
% 6.96/2.50 % (4646)Time elapsed: 0.144 s
% 6.96/2.50 % (4646)Instructions burned: 454 (million)
% 6.96/2.50 % (4646)------------------------------
% 6.96/2.50 % (4646)------------------------------
% 6.96/2.50 % (4658)lrs+1002_1:1_e2e=on:fde=unused:fe=axiom:prag=on:sd=1:sos=on:ss=axioms:st=2.0:i=202:si=on:rtra=on_0 on DTF2THF_4485 for (2993ds/202Mi)
% 6.96/2.50 % (4656)Instruction limit reached!
% 6.96/2.50 % (4656)------------------------------
% 6.96/2.50 % (4656)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.96/2.50 % (4656)Termination reason: Unknown
% 6.96/2.50 % (4656)Termination phase: Saturation
% 6.96/2.50
% 6.96/2.50 % (4656)Memory used [KB]: 6652
% 6.96/2.50 % (4656)Time elapsed: 0.046 s
% 6.96/2.50 % (4656)Instructions burned: 140 (million)
% 6.96/2.50 % (4656)------------------------------
% 6.96/2.50 % (4656)------------------------------
% 6.96/2.51 % (4659)dis+1002_1:1_au=on:cs=on:ins=3:kws=inv_frequency:nwc=10.0:ss=axioms:st=3.0:i=339:si=on:rtra=on_0 on DTF2THF_4485 for (2993ds/339Mi)
% 6.96/2.51 % (4660)dis+10_1:1_anc=none:cnfonf=lazy_gen:fd=preordered:fe=off:hud=10:ins=3:ixr=off:nwc=5.0:plsq=on:plsqc=1:plsqr=32,1:sp=const_frequency:uhcvi=on:i=283:si=on:rtra=on_0 on DTF2THF_4485 for (2992ds/283Mi)
% 6.96/2.52 % (4653)Instruction limit reached!
% 6.96/2.52 % (4653)------------------------------
% 6.96/2.52 % (4653)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.96/2.52 % (4653)Termination reason: Unknown
% 6.96/2.52 % (4653)Termination phase: Saturation
% 6.96/2.52
% 6.96/2.52 % (4653)Memory used [KB]: 7036
% 6.96/2.52 % (4653)Time elapsed: 0.112 s
% 6.96/2.52 % (4653)Instructions burned: 344 (million)
% 6.96/2.52 % (4653)------------------------------
% 6.96/2.52 % (4653)------------------------------
% 6.96/2.53 % (4661)dis+2_1:1_cha=on:hud=7:nwc=3.0:prag=on:tnu=2:i=134:si=on:rtra=on_0 on DTF2THF_4485 for (2992ds/134Mi)
% 6.96/2.53 % (4658)Refutation not found, incomplete strategy
% 6.96/2.53 % (4658)------------------------------
% 6.96/2.53 % (4658)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.96/2.53 % (4658)Termination reason: Refutation not found, incomplete strategy
% 6.96/2.53
% 6.96/2.53
% 6.96/2.53 % (4658)Memory used [KB]: 6396
% 6.96/2.53 % (4658)Time elapsed: 0.031 s
% 6.96/2.53 % (4658)Instructions burned: 50 (million)
% 6.96/2.53 % (4658)------------------------------
% 6.96/2.53 % (4658)------------------------------
% 6.96/2.55 % (4662)dis+1010_1:64_cnfonf=lazy_simp:fd=off:i=298:si=on:rtra=on_0 on DTF2THF_4485 for (2992ds/298Mi)
% 6.96/2.57 % (4661)Instruction limit reached!
% 6.96/2.57 % (4661)------------------------------
% 6.96/2.57 % (4661)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.96/2.57 % (4661)Termination reason: Unknown
% 6.96/2.57 % (4661)Termination phase: Saturation
% 6.96/2.57
% 6.96/2.57 % (4661)Memory used [KB]: 6780
% 6.96/2.57 % (4661)Time elapsed: 0.043 s
% 6.96/2.57 % (4661)Instructions burned: 134 (million)
% 6.96/2.57 % (4661)------------------------------
% 6.96/2.57 % (4661)------------------------------
% 6.96/2.57 % (4655)Instruction limit reached!
% 6.96/2.57 % (4655)------------------------------
% 6.96/2.57 % (4655)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.96/2.57 % (4655)Termination reason: Unknown
% 6.96/2.57 % (4655)Termination phase: Saturation
% 6.96/2.57
% 6.96/2.57 % (4655)Memory used [KB]: 7164
% 6.96/2.57 % (4655)Time elapsed: 0.140 s
% 6.96/2.57 % (4655)Instructions burned: 255 (million)
% 6.96/2.57 % (4655)------------------------------
% 6.96/2.57 % (4655)------------------------------
% 6.96/2.58 % (4663)lrs+2_1:8_erd=off:plsq=on:plsqr=32,1:sos=on:i=192:si=on:rtra=on_0 on DTF2THF_4485 for (2992ds/192Mi)
% 6.96/2.58 % (4664)lrs+10_1:1_bd=off:hud=1:nm=0:prag=on:sd=2:sgt=32:sos=all:ss=axioms:i=258:si=on:rtra=on_0 on DTF2THF_4485 for (2992ds/258Mi)
% 7.56/2.60 % (4660)Instruction limit reached!
% 7.56/2.60 % (4660)------------------------------
% 7.56/2.60 % (4660)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.56/2.60 % (4660)Termination reason: Unknown
% 7.56/2.60 % (4660)Termination phase: Saturation
% 7.56/2.60
% 7.56/2.60 % (4660)Memory used [KB]: 7547
% 7.56/2.60 % (4660)Time elapsed: 0.091 s
% 7.56/2.60 % (4660)Instructions burned: 285 (million)
% 7.56/2.60 % (4660)------------------------------
% 7.56/2.60 % (4660)------------------------------
% 7.56/2.61 % (4665)lrs+1002_1:1024_au=on:cond=fast:ntd=on:piset=and:plsq=on:plsqc=1:plsqr=183491,1048576:sac=on:tgt=ground:i=90:si=on:rtra=on_0 on DTF2THF_4485 for (2992ds/90Mi)
% 7.56/2.63 % (4659)Instruction limit reached!
% 7.56/2.63 % (4659)------------------------------
% 7.56/2.63 % (4659)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.56/2.63 % (4659)Termination reason: Unknown
% 7.56/2.63 % (4659)Termination phase: Saturation
% 7.56/2.63
% 7.56/2.63 % (4659)Memory used [KB]: 7675
% 7.56/2.63 % (4659)Time elapsed: 0.123 s
% 7.56/2.63 % (4659)Instructions burned: 340 (million)
% 7.56/2.63 % (4659)------------------------------
% 7.56/2.63 % (4659)------------------------------
% 7.56/2.63 % (4639)Instruction limit reached!
% 7.56/2.63 % (4639)------------------------------
% 7.56/2.63 % (4639)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.56/2.63 % (4639)Termination reason: Unknown
% 7.56/2.63 % (4639)Termination phase: Saturation
% 7.56/2.63
% 7.56/2.63 % (4639)Memory used [KB]: 8315
% 7.56/2.63 % (4639)Time elapsed: 0.376 s
% 7.56/2.63 % (4639)Instructions burned: 1088 (million)
% 7.56/2.63 % (4639)------------------------------
% 7.56/2.63 % (4639)------------------------------
% 7.56/2.64 % (4663)Instruction limit reached!
% 7.56/2.64 % (4663)------------------------------
% 7.56/2.64 % (4663)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.56/2.64 % (4663)Termination reason: Unknown
% 7.56/2.64 % (4663)Termination phase: Saturation
% 7.56/2.64
% 7.56/2.64 % (4663)Memory used [KB]: 7547
% 7.56/2.64 % (4663)Time elapsed: 0.060 s
% 7.56/2.64 % (4663)Instructions burned: 193 (million)
% 7.56/2.64 % (4663)------------------------------
% 7.56/2.64 % (4663)------------------------------
% 7.56/2.64 % (4666)dis+1004_16:1_avsq=on:avsqr=1,16:cnfonf=lazy_gen:cs=on:fe=off:tgt=ground:tnu=1:i=231:si=on:rtra=on_0 on DTF2THF_4485 for (2991ds/231Mi)
% 7.56/2.64 % (4665)Instruction limit reached!
% 7.56/2.64 % (4665)------------------------------
% 7.56/2.64 % (4665)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.56/2.64 % (4665)Termination reason: Unknown
% 7.56/2.64 % (4665)Termination phase: Saturation
% 7.56/2.64
% 7.56/2.64 % (4665)Memory used [KB]: 6652
% 7.56/2.64 % (4665)Time elapsed: 0.029 s
% 7.56/2.64 % (4665)Instructions burned: 93 (million)
% 7.56/2.64 % (4665)------------------------------
% 7.56/2.64 % (4665)------------------------------
% 7.56/2.64 % (4657)Instruction limit reached!
% 7.56/2.64 % (4657)------------------------------
% 7.56/2.64 % (4657)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.56/2.64 % (4657)Termination reason: Unknown
% 7.56/2.64 % (4657)Termination phase: Saturation
% 7.56/2.64
% 7.56/2.64 % (4657)Memory used [KB]: 8187
% 7.56/2.64 % (4657)Time elapsed: 0.186 s
% 7.56/2.64 % (4657)Instructions burned: 614 (million)
% 7.56/2.64 % (4657)------------------------------
% 7.56/2.64 % (4657)------------------------------
% 7.56/2.65 % (4668)lrs+1010_1:16_add=large:amm=off:anc=all_dependent:chr=on:fd=off:ins=2:sp=const_max:tnu=5:uhcvi=on:i=31:si=on:rtra=on_0 on DTF2THF_4485 for (2991ds/31Mi)
% 7.56/2.65 % (4667)lrs+1002_1:1_au=on:s2a=on:sgt=32:slsq=on:slsqc=1:slsql=off:ss=axioms:i=458:si=on:rtra=on_0 on DTF2THF_4485 for (2991ds/458Mi)
% 7.56/2.65 % (4669)lrs+10_1:1_cnfonf=lazy_gen:kws=inv_precedence:nwc=10.0:s2a=on:sp=weighted_frequency:ss=axioms:i=297:si=on:rtra=on_0 on DTF2THF_4485 for (2991ds/297Mi)
% 7.56/2.65 % (4670)dis+2_1:4_amm=off:au=on:cbe=off:cnfonf=lazy_gen:hfaw=0:hfsq=on:hfsql=off:plsq=on:plsqc=1:plsqr=35,4:sac=on:i=987:si=on:rtra=on_0 on DTF2THF_4485 for (2991ds/987Mi)
% 7.56/2.66 % (4668)Instruction limit reached!
% 7.56/2.66 % (4668)------------------------------
% 7.56/2.66 % (4668)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.56/2.66 % (4668)Termination reason: Unknown
% 7.56/2.66 % (4668)Termination phase: Preprocessing 3
% 7.56/2.66
% 7.56/2.66 % (4668)Memory used [KB]: 1791
% 7.56/2.66 % (4668)Time elapsed: 0.011 s
% 7.56/2.66 % (4668)Instructions burned: 33 (million)
% 7.56/2.66 % (4668)------------------------------
% 7.56/2.66 % (4668)------------------------------
% 7.56/2.66 % (4664)Instruction limit reached!
% 7.56/2.66 % (4664)------------------------------
% 7.56/2.66 % (4664)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.56/2.66 % (4664)Termination reason: Unknown
% 7.56/2.66 % (4664)Termination phase: Saturation
% 7.56/2.66
% 7.56/2.66 % (4664)Memory used [KB]: 7291
% 7.56/2.66 % (4664)Time elapsed: 0.081 s
% 7.56/2.66 % (4664)Instructions burned: 258 (million)
% 7.56/2.66 % (4664)------------------------------
% 7.56/2.66 % (4664)------------------------------
% 7.56/2.66 % (4662)Instruction limit reached!
% 7.56/2.66 % (4662)------------------------------
% 7.56/2.66 % (4662)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.56/2.66 % (4662)Termination reason: Unknown
% 7.56/2.66 % (4662)Termination phase: Saturation
% 7.56/2.66
% 7.56/2.66 % (4662)Memory used [KB]: 7675
% 7.56/2.66 % (4662)Time elapsed: 0.119 s
% 7.56/2.66 % (4662)Instructions burned: 300 (million)
% 7.56/2.66 % (4662)------------------------------
% 7.56/2.66 % (4662)------------------------------
% 7.56/2.67 % (4671)lrs+10_1:1024_avsq=on:avsqc=4:avsqr=584411,1048576:bd=off:bet=on:br=off:chr=on:fsd=on:fsdmm=1:hud=10:nm=0:ntd=on:nwc=9.0:piset=or:plsq=on:plsqc=2:plsql=on:plsqr=852219,1048576:rawr=on:sp=const_frequency:spb=units:i=182:si=on:rtra=on_0 on DTF2THF_4485 for (2991ds/182Mi)
% 7.56/2.67 % (4672)dis+10_1:1_au=on:e2e=on:fde=none:nwc=10.0:s2a=on:s2at=2.0:i=182:si=on:rtra=on_0 on DTF2THF_4485 for (2991ds/182Mi)
% 7.56/2.67 % (4673)dis+10_1:8_cha=on:cnfonf=lazy_pi_sigma_gen:fe=axiom:hud=5:plsq=on:plsqr=32,1:prag=on:sos=all:i=154:si=on:rtra=on_0 on DTF2THF_4485 for (2991ds/154Mi)
% 8.71/2.70 % (4666)Instruction limit reached!
% 8.71/2.70 % (4666)------------------------------
% 8.71/2.70 % (4666)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 8.71/2.70 % (4666)Termination reason: Unknown
% 8.71/2.70 % (4666)Termination phase: Saturation
% 8.71/2.70
% 8.71/2.70 % (4666)Memory used [KB]: 7164
% 8.71/2.70 % (4666)Time elapsed: 0.066 s
% 8.71/2.70 % (4666)Instructions burned: 232 (million)
% 8.71/2.70 % (4666)------------------------------
% 8.71/2.70 % (4666)------------------------------
% 8.71/2.71 % (4674)dis+1003_1:64_atotf=0.4:au=on:fsr=off:hflw=5:hfsq=on:hfsqc=20:hfsql=off:plsq=on:plsqc=1:plsqr=32,1:i=79:si=on:rtra=on_0 on DTF2THF_4485 for (2991ds/79Mi)
% 8.71/2.72 % (4673)Instruction limit reached!
% 8.71/2.72 % (4673)------------------------------
% 8.71/2.72 % (4673)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 8.71/2.72 % (4673)Termination reason: Unknown
% 8.71/2.72 % (4673)Termination phase: Saturation
% 8.71/2.72
% 8.71/2.72 % (4673)Memory used [KB]: 7291
% 8.71/2.72 % (4673)Time elapsed: 0.044 s
% 8.71/2.72 % (4673)Instructions burned: 154 (million)
% 8.71/2.72 % (4673)------------------------------
% 8.71/2.72 % (4673)------------------------------
% 8.71/2.72 % (4671)Instruction limit reached!
% 8.71/2.72 % (4671)------------------------------
% 8.71/2.72 % (4671)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 8.71/2.72 % (4671)Termination reason: Unknown
% 8.71/2.72 % (4671)Termination phase: Saturation
% 8.71/2.72
% 8.71/2.72 % (4671)Memory used [KB]: 6908
% 8.71/2.72 % (4671)Time elapsed: 0.057 s
% 8.71/2.72 % (4671)Instructions burned: 183 (million)
% 8.71/2.72 % (4671)------------------------------
% 8.71/2.72 % (4671)------------------------------
% 8.71/2.73 % (4675)dis+1010_5:2_anc=all_dependent:awrs=converge:bd=off:bet=on:c=on:cha=on:cnfonf=lazy_not_gen_be_off:fe=axiom:hud=5:kws=precedence:piset=and:prag=on:s2a=on:sac=on:slsq=on:slsqc=1:sp=reverse_arity:ss=axioms:st=1.5:i=216:si=on:rtra=on_0 on DTF2THF_4485 for (2990ds/216Mi)
% 8.71/2.73 % (4672)Instruction limit reached!
% 8.71/2.73 % (4672)------------------------------
% 8.71/2.73 % (4672)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 8.71/2.73 % (4672)Termination reason: Unknown
% 8.71/2.73 % (4672)Termination phase: Saturation
% 8.71/2.73
% 8.71/2.73 % (4672)Memory used [KB]: 7036
% 8.71/2.73 % (4672)Time elapsed: 0.059 s
% 8.71/2.73 % (4672)Instructions burned: 183 (million)
% 8.71/2.73 % (4672)------------------------------
% 8.71/2.73 % (4672)------------------------------
% 8.71/2.73 % (4676)lrs+1002_1:1_fe=off:hud=10:ins=2:sac=on:sp=const_frequency:i=114:si=on:rtra=on_0 on DTF2THF_4485 for (2990ds/114Mi)
% 8.71/2.74 % (4674)Instruction limit reached!
% 8.71/2.74 % (4674)------------------------------
% 8.71/2.74 % (4674)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 8.71/2.74 % (4674)Termination reason: Unknown
% 8.71/2.74 % (4674)Termination phase: Saturation
% 8.71/2.74
% 8.71/2.74 % (4674)Memory used [KB]: 6652
% 8.71/2.74 % (4674)Time elapsed: 0.025 s
% 8.71/2.74 % (4674)Instructions burned: 80 (million)
% 8.71/2.74 % (4674)------------------------------
% 8.71/2.74 % (4674)------------------------------
% 8.71/2.74 % (4669)Instruction limit reached!
% 8.71/2.74 % (4669)------------------------------
% 8.71/2.74 % (4669)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 8.71/2.74 % (4669)Termination reason: Unknown
% 8.71/2.74 % (4669)Termination phase: Saturation
% 8.71/2.74
% 8.71/2.74 % (4669)Memory used [KB]: 7164
% 8.71/2.74 % (4669)Time elapsed: 0.093 s
% 8.71/2.74 % (4669)Instructions burned: 297 (million)
% 8.71/2.74 % (4669)------------------------------
% 8.71/2.74 % (4669)------------------------------
% 8.71/2.74 % (4677)dis+1002_1:1_acc=on:au=on:bd=off:e2e=on:sgt=8:ss=axioms:i=922:si=on:rtra=on_0 on DTF2THF_4485 for (2990ds/922Mi)
% 8.71/2.75 % (4678)lrs+10_1:128_au=on:bd=off:cnfonf=lazy_not_gen_be_off:nwc=10.0:piset=and:i=182:si=on:rtra=on_0 on DTF2THF_4485 for (2990ds/182Mi)
% 8.71/2.75 % (4679)ott+1010_1:1024_au=on:av=off:c=on:cnfonf=conj_eager:piset=equals:s2a=on:sp=occurrence:spb=units:ss=axioms:i=164:si=on:rtra=on_0 on DTF2THF_4485 for (2990ds/164Mi)
% 8.71/2.77 % (4676)Instruction limit reached!
% 8.71/2.77 % (4676)------------------------------
% 8.71/2.77 % (4676)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 8.71/2.77 % (4676)Termination reason: Unknown
% 8.71/2.77 % (4676)Termination phase: Saturation
% 8.71/2.77
% 8.71/2.77 % (4676)Memory used [KB]: 6780
% 8.71/2.77 % (4676)Time elapsed: 0.035 s
% 8.71/2.77 % (4676)Instructions burned: 114 (million)
% 8.71/2.77 % (4676)------------------------------
% 8.71/2.77 % (4676)------------------------------
% 8.71/2.78 % (4680)lrs+1004_1:1_cnfonf=lazy_pi_sigma_gen:fe=off:gs=on:hud=10:prag=on:tnu=1:i=50:si=on:rtra=on_0 on DTF2THF_4485 for (2990ds/50Mi)
% 8.71/2.79 % (4680)Instruction limit reached!
% 8.71/2.79 % (4680)------------------------------
% 8.71/2.79 % (4680)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 8.71/2.79 % (4680)Termination reason: Unknown
% 8.71/2.79 % (4680)Termination phase: Preprocessing 3
% 8.71/2.79
% 8.71/2.79 % (4680)Memory used [KB]: 2046
% 8.71/2.79 % (4680)Time elapsed: 0.015 s
% 8.71/2.79 % (4680)Instructions burned: 51 (million)
% 8.71/2.79 % (4680)------------------------------
% 8.71/2.79 % (4680)------------------------------
% 8.71/2.79 % (4675)Instruction limit reached!
% 8.71/2.79 % (4675)------------------------------
% 8.71/2.79 % (4675)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 8.71/2.79 % (4675)Termination reason: Unknown
% 8.71/2.79 % (4675)Termination phase: Saturation
% 8.71/2.79
% 8.71/2.79 % (4675)Memory used [KB]: 7036
% 8.71/2.79 % (4675)Time elapsed: 0.067 s
% 8.71/2.79 % (4675)Instructions burned: 218 (million)
% 8.71/2.79 % (4675)------------------------------
% 8.71/2.79 % (4675)------------------------------
% 9.24/2.80 % (4678)Instruction limit reached!
% 9.24/2.80 % (4678)------------------------------
% 9.24/2.80 % (4678)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.24/2.80 % (4678)Termination reason: Unknown
% 9.24/2.80 % (4678)Termination phase: Saturation
% 9.24/2.80
% 9.24/2.80 % (4678)Memory used [KB]: 6908
% 9.24/2.80 % (4678)Time elapsed: 0.053 s
% 9.24/2.80 % (4678)Instructions burned: 184 (million)
% 9.24/2.80 % (4678)------------------------------
% 9.24/2.80 % (4678)------------------------------
% 9.24/2.80 % (4681)dis-30_1:1_acc=on:bd=off:fsr=off:plsq=on:plsqr=32,1:i=368:si=on:rtra=on_0 on DTF2THF_4485 for (2990ds/368Mi)
% 9.24/2.80 % (4679)Instruction limit reached!
% 9.24/2.80 % (4679)------------------------------
% 9.24/2.80 % (4679)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.24/2.80 % (4679)Termination reason: Unknown
% 9.24/2.80 % (4679)Termination phase: Saturation
% 9.24/2.80
% 9.24/2.80 % (4679)Memory used [KB]: 2430
% 9.24/2.80 % (4679)Time elapsed: 0.050 s
% 9.24/2.80 % (4679)Instructions burned: 166 (million)
% 9.24/2.80 % (4679)------------------------------
% 9.24/2.80 % (4679)------------------------------
% 9.24/2.80 % (4682)dis+2_1:1_bd=off:e2e=on:hud=1:nm=3:piset=or:prag=on:s2a=on:sp=const_max:i=1021:si=on:rtra=on_0 on DTF2THF_4485 for (2990ds/1021Mi)
% 9.24/2.81 % (4683)dis+10_1:1_cnfonf=lazy_gen:fe=off:plsq=on:plsqc=1:plsqr=32,1:prag=on:i=267:si=on:rtra=on_0 on DTF2THF_4485 for (2989ds/267Mi)
% 9.24/2.81 % (4684)dis+21_1:5_au=on:ntd=on:nwc=5.0:s2a=on:ss=axioms:st=5.0:i=204:si=on:rtra=on_0 on DTF2THF_4485 for (2989ds/204Mi)
% 9.24/2.88 % (4684)Instruction limit reached!
% 9.24/2.88 % (4684)------------------------------
% 9.24/2.88 % (4684)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.24/2.88 % (4684)Termination reason: Unknown
% 9.24/2.88 % (4684)Termination phase: Saturation
% 9.24/2.88
% 9.24/2.88 % (4684)Memory used [KB]: 6908
% 9.24/2.88 % (4684)Time elapsed: 0.068 s
% 9.24/2.88 % (4684)Instructions burned: 205 (million)
% 9.24/2.88 % (4684)------------------------------
% 9.24/2.88 % (4684)------------------------------
% 9.24/2.89 % (4685)dis+2_1:1024_atotf=0.0802881:au=on:bs=unit_only:cnfonf=lazy_not_gen_be_off:fde=none:nwc=10.0:prag=on:i=26:si=on:rtra=on_0 on DTF2THF_4485 for (2989ds/26Mi)
% 9.24/2.89 % (4667)Instruction limit reached!
% 9.24/2.89 % (4667)------------------------------
% 9.24/2.89 % (4667)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.24/2.89 % (4667)Termination reason: Unknown
% 9.24/2.89 % (4667)Termination phase: Saturation
% 9.24/2.89
% 9.24/2.89 % (4667)Memory used [KB]: 7931
% 9.24/2.89 % (4667)Time elapsed: 0.248 s
% 9.24/2.89 % (4667)Instructions burned: 460 (million)
% 9.24/2.89 % (4667)------------------------------
% 9.24/2.89 % (4667)------------------------------
% 9.24/2.89 % (4685)Instruction limit reached!
% 9.24/2.89 % (4685)------------------------------
% 9.24/2.89 % (4685)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.24/2.89 % (4685)Termination reason: Unknown
% 9.24/2.89 % (4685)Termination phase: shuffling
% 9.24/2.89
% 9.24/2.89 % (4685)Memory used [KB]: 1791
% 9.24/2.89 % (4685)Time elapsed: 0.009 s
% 9.24/2.89 % (4685)Instructions burned: 27 (million)
% 9.24/2.89 % (4685)------------------------------
% 9.24/2.89 % (4685)------------------------------
% 9.24/2.89 % (4683)Instruction limit reached!
% 9.24/2.89 % (4683)------------------------------
% 9.24/2.89 % (4683)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.24/2.89 % (4683)Termination reason: Unknown
% 9.24/2.89 % (4683)Termination phase: Saturation
% 9.24/2.89
% 9.24/2.89 % (4683)Memory used [KB]: 7675
% 9.24/2.89 % (4683)Time elapsed: 0.088 s
% 9.24/2.89 % (4683)Instructions burned: 268 (million)
% 9.24/2.89 % (4683)------------------------------
% 9.24/2.89 % (4683)------------------------------
% 9.82/2.90 % (4686)dis+1002_1:1_bd=preordered:c=on:cbe=off:cnfonf=lazy_not_gen_be_off:hud=10:ixr=off:prag=on:i=53:si=on:rtra=on_0 on DTF2THF_4485 for (2988ds/53Mi)
% 9.82/2.90 % (4687)dis+1002_1:1_aac=none:au=on:e2e=on:nm=2:nwc=10.0:sd=1:ss=axioms:st=3.0:i=4436:si=on:rtra=on_0 on DTF2THF_4485 for (2988ds/4436Mi)
% 9.82/2.90 % (4688)dis+1010_1:1_acc=model:avsq=on:ixr=off:prag=on:s2a=on:sac=on:i=1539:si=on:rtra=on_0 on DTF2THF_4485 for (2988ds/1539Mi)
% 9.82/2.92 % (4686)Instruction limit reached!
% 9.82/2.92 % (4686)------------------------------
% 9.82/2.92 % (4686)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.82/2.92 % (4686)Termination reason: Unknown
% 9.82/2.92 % (4686)Termination phase: Property scanning
% 9.82/2.92
% 9.82/2.92 % (4686)Memory used [KB]: 2046
% 9.82/2.92 % (4686)Time elapsed: 0.016 s
% 9.82/2.92 % (4686)Instructions burned: 55 (million)
% 9.82/2.92 % (4686)------------------------------
% 9.82/2.92 % (4686)------------------------------
% 9.82/2.92 % (4681)Instruction limit reached!
% 9.82/2.92 % (4681)------------------------------
% 9.82/2.92 % (4681)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.82/2.92 % (4681)Termination reason: Unknown
% 9.82/2.92 % (4681)Termination phase: Saturation
% 9.82/2.92
% 9.82/2.92 % (4681)Memory used [KB]: 8443
% 9.82/2.92 % (4681)Time elapsed: 0.148 s
% 9.82/2.92 % (4681)Instructions burned: 370 (million)
% 9.82/2.92 % (4681)------------------------------
% 9.82/2.92 % (4681)------------------------------
% 9.82/2.93 % (4689)lrs+2_1:1_bd=off:e2e=on:fde=none:kws=inv_precedence:plsq=on:plsqr=32,1:sos=on:i=408:si=on:rtra=on_0 on DTF2THF_4485 for (2988ds/408Mi)
% 9.82/2.93 % (4690)lrs+1002_1:1_aac=none:au=on:cnfonf=lazy_gen:plsq=on:plsqc=1:plsqr=4203469,65536:i=5205:si=on:rtra=on_0 on DTF2THF_4485 for (2988ds/5205Mi)
% 9.82/3.01 % (4670)Instruction limit reached!
% 9.82/3.01 % (4670)------------------------------
% 9.82/3.01 % (4670)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.82/3.01 % (4670)Termination reason: Unknown
% 9.82/3.01 % (4670)Termination phase: Saturation
% 9.82/3.01
% 9.82/3.01 % (4670)Memory used [KB]: 8827
% 9.82/3.01 % (4670)Time elapsed: 0.361 s
% 9.82/3.01 % (4670)Instructions burned: 989 (million)
% 9.82/3.01 % (4670)------------------------------
% 9.82/3.01 % (4670)------------------------------
% 11.20/3.02 % (4691)dis+1010_1:4_au=on:av=off:cbe=off:nwc=10.0:s2a=on:s2at=3.0:sd=2:ss=axioms:st=5.0:i=1913:si=on:rtra=on_0 on DTF2THF_4485 for (2987ds/1913Mi)
% 11.20/3.04 % (4677)Instruction limit reached!
% 11.20/3.04 % (4677)------------------------------
% 11.20/3.04 % (4677)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 11.20/3.04 % (4677)Termination reason: Unknown
% 11.20/3.04 % (4677)Termination phase: Saturation
% 11.20/3.04
% 11.20/3.04 % (4677)Memory used [KB]: 7931
% 11.20/3.04 % (4677)Time elapsed: 0.300 s
% 11.20/3.04 % (4677)Instructions burned: 922 (million)
% 11.20/3.04 % (4677)------------------------------
% 11.20/3.04 % (4677)------------------------------
% 11.20/3.05 % (4692)lrs+2_1:1_bd=off:cbe=off:piset=equals:sd=1:sgt=20:sos=all:ss=axioms:i=4350:si=on:rtra=on_0 on DTF2THF_4485 for (2987ds/4350Mi)
% 11.20/3.05 % (4689)Instruction limit reached!
% 11.20/3.05 % (4689)------------------------------
% 11.20/3.05 % (4689)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 11.20/3.05 % (4689)Termination reason: Unknown
% 11.20/3.05 % (4689)Termination phase: Saturation
% 11.20/3.05
% 11.20/3.05 % (4689)Memory used [KB]: 7419
% 11.20/3.05 % (4689)Time elapsed: 0.127 s
% 11.20/3.05 % (4689)Instructions burned: 409 (million)
% 11.20/3.05 % (4689)------------------------------
% 11.20/3.05 % (4689)------------------------------
% 11.20/3.06 % (4693)ott+1010_1:4_cbe=off:chr=on:hfsq=on:hfsqr=1,16:hud=3:sac=on:sp=unary_frequency:ss=axioms:st=5.0:tnu=2:i=68:si=on:rtra=on_0 on DTF2THF_4485 for (2987ds/68Mi)
% 11.20/3.08 % (4693)Instruction limit reached!
% 11.20/3.08 % (4693)------------------------------
% 11.20/3.08 % (4693)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 11.20/3.08 % (4693)Termination reason: Unknown
% 11.20/3.08 % (4693)Termination phase: Saturation
% 11.20/3.08
% 11.20/3.08 % (4693)Memory used [KB]: 6396
% 11.20/3.08 % (4693)Time elapsed: 0.022 s
% 11.20/3.08 % (4693)Instructions burned: 71 (million)
% 11.20/3.08 % (4693)------------------------------
% 11.20/3.08 % (4693)------------------------------
% 11.20/3.09 % (4695)dis+10_1:1_au=on:nwc=5.0:s2a=on:s2at=2.0:sac=on:sd=7:ss=axioms:st=2.0:i=358:si=on:rtra=on_0 on DTF2THF_4485 for (2987ds/358Mi)
% 11.73/3.13 % (4682)Instruction limit reached!
% 11.73/3.13 % (4682)------------------------------
% 11.73/3.13 % (4682)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 11.73/3.13 % (4682)Termination reason: Unknown
% 11.73/3.13 % (4682)Termination phase: Saturation
% 11.73/3.13
% 11.73/3.13 % (4682)Memory used [KB]: 8443
% 11.73/3.13 % (4682)Time elapsed: 0.352 s
% 11.73/3.13 % (4682)Instructions burned: 1023 (million)
% 11.73/3.13 % (4682)------------------------------
% 11.73/3.13 % (4682)------------------------------
% 11.73/3.14 % (4696)lrs+1010_1:16_add=large:amm=off:anc=all_dependent:chr=on:fd=off:ins=2:sp=const_max:tnu=5:uhcvi=on:i=31:si=on:rtra=on_0 on DTF2THF_4485 for (2986ds/31Mi)
% 11.73/3.15 % (4696)Instruction limit reached!
% 11.73/3.15 % (4696)------------------------------
% 11.73/3.15 % (4696)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 11.73/3.15 % (4696)Termination reason: Unknown
% 11.73/3.15 % (4696)Termination phase: Preprocessing 3
% 11.73/3.15
% 11.73/3.15 % (4696)Memory used [KB]: 1791
% 11.73/3.15 % (4696)Time elapsed: 0.011 s
% 11.73/3.15 % (4696)Instructions burned: 31 (million)
% 11.73/3.15 % (4696)------------------------------
% 11.73/3.15 % (4696)------------------------------
% 11.73/3.15 % (4695)First to succeed.
% 11.73/3.15 % (4695)Refutation found. Thanks to Tanya!
% 11.73/3.15 % SZS status Theorem for DTF2THF_4485
% 11.73/3.15 % SZS output start Proof for DTF2THF_4485
% 11.73/3.15 thf(type_def_5, type, int: $tType).
% 11.73/3.15 thf(type_def_7, type, nat: $tType).
% 11.73/3.15 thf(type_def_8, type, num: $tType).
% 11.73/3.15 thf(type_def_9, type, set: ($tType) > $tType).
% 11.73/3.15 thf(type_def_10, type, real: $tType).
% 11.73/3.15 thf(func_def_0, type, real: $tType).
% 11.73/3.15 thf(func_def_1, type, set: ($tType) > $tType).
% 11.73/3.15 thf(func_def_2, type, num: $tType).
% 11.73/3.15 thf(func_def_3, type, nat: $tType).
% 11.73/3.15 thf(func_def_4, type, int: $tType).
% 11.73/3.15 thf(func_def_5, type, type: !>[X0: $tType]:($o)).
% 11.73/3.15 thf(func_def_6, type, one: !>[X0: $tType]:($o)).
% 11.73/3.15 thf(func_def_7, type, zero: !>[X0: $tType]:($o)).
% 11.73/3.15 thf(func_def_8, type, numeral: !>[X0: $tType]:($o)).
% 11.73/3.15 thf(func_def_9, type, power: !>[X0: $tType]:($o)).
% 11.73/3.15 thf(func_def_10, type, ring_1: !>[X0: $tType]:($o)).
% 11.73/3.15 thf(func_def_11, type, ord: !>[X0: $tType]:($o)).
% 11.73/3.15 thf(func_def_12, type, ring_char_0: !>[X0: $tType]:($o)).
% 11.73/3.15 thf(func_def_13, type, neg_numeral: !>[X0: $tType]:($o)).
% 11.73/3.15 thf(func_def_14, type, order: !>[X0: $tType]:($o)).
% 11.73/3.15 thf(func_def_15, type, semiring_1: !>[X0: $tType]:($o)).
% 11.73/3.15 thf(func_def_16, type, ln: !>[X0: $tType]:($o)).
% 11.73/3.15 thf(func_def_17, type, monoid_mult: !>[X0: $tType]:($o)).
% 11.73/3.15 thf(func_def_18, type, linorder: !>[X0: $tType]:($o)).
% 11.73/3.15 thf(func_def_19, type, preorder: !>[X0: $tType]:($o)).
% 11.73/3.15 thf(func_def_20, type, semiring_char_0: !>[X0: $tType]:($o)).
% 11.73/3.15 thf(func_def_21, type, semiring_numeral: !>[X0: $tType]:($o)).
% 11.73/3.15 thf(func_def_22, type, linordered_idom: !>[X0: $tType]:($o)).
% 11.73/3.15 thf(func_def_23, type, linordered_semidom: !>[X0: $tType]:($o)).
% 11.73/3.15 thf(func_def_24, type, archim1727834104eiling: !>[X0: $tType]:($o)).
% 11.73/3.15 thf(func_def_25, type, linord1659791738miring: !>[X0: $tType]:($o)).
% 11.73/3.15 thf(func_def_26, type, semiri134348788visors: !>[X0: $tType]:($o)).
% 11.73/3.15 thf(func_def_27, type, archim1804426504_field: !>[X0: $tType]:($o)).
% 11.73/3.15 thf(func_def_28, type, canoni770627133id_add: !>[X0: $tType]:($o)).
% 11.73/3.15 thf(func_def_29, type, archimedean_ceiling: !>[X0: $tType]:(X0 > int)).
% 11.73/3.15 thf(func_def_30, type, one_one: !>[X0: $tType]:(X0)).
% 11.73/3.15 thf(func_def_31, type, zero_zero: !>[X0: $tType]:(X0)).
% 11.73/3.15 thf(func_def_32, type, if: !>[X0: $tType]:($o > X0 > X0 > X0)).
% 11.73/3.15 thf(func_def_33, type, nat2: int > nat).
% 11.73/3.15 thf(func_def_34, type, ring_1_of_int: !>[X0: $tType]:(int > X0)).
% 11.73/3.15 thf(func_def_35, type, suc: nat > nat).
% 11.73/3.15 thf(func_def_36, type, semiring_1_of_nat: !>[X0: $tType]:(nat > X0)).
% 11.73/3.15 thf(func_def_37, type, neg_numeral_dbl: !>[X0: $tType]:(X0 > X0)).
% 11.73/3.15 thf(func_def_38, type, bit0: num > num).
% 11.73/3.15 thf(func_def_39, type, one2: num).
% 11.73/3.15 thf(func_def_40, type, numeral_numeral: !>[X0: $tType]:(num > X0)).
% 11.73/3.15 thf(func_def_41, type, pow: num > num > num).
% 11.73/3.15 thf(func_def_42, type, ord_less: !>[X0: $tType]:(X0 > X0 > $o)).
% 11.73/3.15 thf(func_def_43, type, ord_less_eq: !>[X0: $tType]:(X0 > X0 > $o)).
% 11.73/3.15 thf(func_def_44, type, power_power: !>[X0: $tType]:(X0 > nat > X0)).
% 11.73/3.15 thf(func_def_45, type, collect: !>[X0: $tType]:((X0 > $o) > set @ X0)).
% 11.73/3.15 thf(func_def_46, type, log: real > real > real).
% 11.73/3.15 thf(func_def_47, type, powr: !>[X0: $tType]:(X0 > X0 > X0)).
% 11.73/3.15 thf(func_def_48, type, member: !>[X0: $tType]:(X0 > set @ X0 > $o)).
% 11.73/3.15 thf(func_def_49, type, d: int).
% 11.73/3.15 thf(func_def_50, type, n: nat).
% 11.73/3.15 thf(func_def_65, type, sK0: (nat > $o) > nat > nat).
% 11.73/3.15 thf(func_def_66, type, sK1: nat > nat).
% 11.73/3.15 thf(func_def_67, type, sK2: nat > nat).
% 11.73/3.15 thf(func_def_68, type, sK3: (nat > $o) > nat).
% 11.73/3.15 thf(func_def_69, type, sK4: (nat > nat > $o) > nat).
% 11.73/3.15 thf(func_def_70, type, sK5: (nat > nat > $o) > nat).
% 11.73/3.15 thf(func_def_71, type, sK6: (nat > nat > $o) > nat).
% 11.73/3.15 thf(func_def_72, type, sK7: (nat > nat > $o) > nat).
% 11.73/3.15 thf(func_def_73, type, sK8: (nat > nat > $o) > nat).
% 11.73/3.15 thf(func_def_74, type, sK9: int > nat).
% 11.73/3.15 thf(func_def_75, type, sK10: (nat > $o) > nat).
% 11.73/3.15 thf(func_def_76, type, sK11: nat > (nat > $o) > nat).
% 11.73/3.15 thf(func_def_77, type, sK12: (nat > nat > $o) > nat).
% 11.73/3.15 thf(func_def_78, type, sK13: (nat > nat > $o) > nat).
% 11.73/3.15 thf(func_def_79, type, sK14: (nat > nat > $o) > nat).
% 11.73/3.15 thf(func_def_80, type, sK15: (nat > nat > $o) > nat).
% 11.73/3.15 thf(func_def_81, type, sK16: !>[X0: $tType, X1: $tType]:((X1 > X0) > (X1 > X0) > X1)).
% 11.73/3.15 thf(func_def_82, type, sK17: (nat > $o) > nat).
% 11.73/3.15 thf(func_def_83, type, sK18: nat > nat).
% 11.73/3.15 thf(func_def_84, type, sK19: (nat > $o) > nat).
% 11.73/3.15 thf(func_def_85, type, sK20: (nat > $o) > nat).
% 11.73/3.15 thf(func_def_86, type, sK21: int > nat).
% 11.73/3.15 thf(func_def_87, type, sK22: (nat > $o) > nat).
% 11.73/3.15 thf(func_def_89, type, ph24: !>[X0: $tType]:(X0)).
% 11.73/3.15 thf(f1791,plain,(
% 11.73/3.15 $false),
% 11.73/3.15 inference(subsumption_resolution,[status(thm)],[f1790,f1077])).
% 11.73/3.15 thf(f1077,plain,(
% 11.73/3.15 ((ord_less_eq @ int @ d @ (power_power @ int @ (numeral_numeral @ int @ (bit0 @ one2)) @ (nat2 @ (archimedean_ceiling @ real @ (log @ (numeral_numeral @ real @ (bit0 @ one2)) @ (ring_1_of_int @ real @ d)))))) != $true)),
% 11.73/3.15 inference(cnf_transformation,[status(thm)],[f889])).
% 11.73/3.15 thf(f889,plain,(
% 11.73/3.15 ((ord_less_eq @ int @ d @ (power_power @ int @ (numeral_numeral @ int @ (bit0 @ one2)) @ (nat2 @ (archimedean_ceiling @ real @ (log @ (numeral_numeral @ real @ (bit0 @ one2)) @ (ring_1_of_int @ real @ d)))))) != $true)),
% 11.73/3.15 inference(flattening,[status(thm)],[f592])).
% 11.73/3.15 thf(f592,plain,(
% 11.73/3.15 ~((ord_less_eq @ int @ d @ (power_power @ int @ (numeral_numeral @ int @ (bit0 @ one2)) @ (nat2 @ (archimedean_ceiling @ real @ (log @ (numeral_numeral @ real @ (bit0 @ one2)) @ (ring_1_of_int @ real @ d)))))) = $true)),
% 11.73/3.15 inference(fool_elimination,[status(thm)],[f591])).
% 11.73/3.15 thf(f591,plain,(
% 11.73/3.15 ~(ord_less_eq @ int @ d @ (power_power @ int @ (numeral_numeral @ int @ (bit0 @ one2)) @ (nat2 @ (archimedean_ceiling @ real @ (log @ (numeral_numeral @ real @ (bit0 @ one2)) @ (ring_1_of_int @ real @ d))))))),
% 11.73/3.15 inference(rectify,[status(thm)],[f327])).
% 11.73/3.15 thf(f327,negated_conjecture,(
% 11.73/3.15 ~(ord_less_eq @ int @ d @ (power_power @ int @ (numeral_numeral @ int @ (bit0 @ one2)) @ (nat2 @ (archimedean_ceiling @ real @ (log @ (numeral_numeral @ real @ (bit0 @ one2)) @ (ring_1_of_int @ real @ d))))))),
% 11.73/3.15 inference(negated_conjecture,[status(cth)],[f326])).
% 11.73/3.15 thf(f326,conjecture,(
% 11.73/3.15 (ord_less_eq @ int @ d @ (power_power @ int @ (numeral_numeral @ int @ (bit0 @ one2)) @ (nat2 @ (archimedean_ceiling @ real @ (log @ (numeral_numeral @ real @ (bit0 @ one2)) @ (ring_1_of_int @ real @ d))))))),
% 11.73/3.15 file('/export/starexec/sandbox/tmp/tmp.aWXyJVFJl3/DTF2THF_4485.p',conj_0)).
% 11.73/3.15 thf(f1790,plain,(
% 11.73/3.15 ((ord_less_eq @ int @ d @ (power_power @ int @ (numeral_numeral @ int @ (bit0 @ one2)) @ (nat2 @ (archimedean_ceiling @ real @ (log @ (numeral_numeral @ real @ (bit0 @ one2)) @ (ring_1_of_int @ real @ d)))))) = $true)),
% 11.73/3.15 inference(forward_demodulation,[status(thm)],[f1789,f1442])).
% 11.73/3.15 thf(f1442,plain,(
% 11.73/3.15 ( ! [X0 : int] : (((archimedean_ceiling @ real @ (ring_1_of_int @ real @ X0)) = X0)) )),
% 11.73/3.15 inference(trivial_inequality_removal,[status(thm)],[f1441])).
% 11.73/3.15 thf(f1441,plain,(
% 11.73/3.15 ( ! [X0 : int] : (((archimedean_ceiling @ real @ (ring_1_of_int @ real @ X0)) = X0) | ($true != $true)) )),
% 11.73/3.15 inference(superposition,[status(thm)],[f1121,f1124])).
% 11.73/3.15 thf(f1124,plain,(
% 11.73/3.15 (archim1727834104eiling @ real = $true)),
% 11.73/3.15 inference(cnf_transformation,[status(thm)],[f554])).
% 11.73/3.15 thf(f554,plain,(
% 11.73/3.15 (archim1727834104eiling @ real = $true)),
% 11.73/3.15 inference(fool_elimination,[status(thm)],[f553])).
% 11.73/3.15 thf(f553,plain,(
% 11.73/3.15 archim1727834104eiling @ real),
% 11.73/3.15 inference(rectify,[status(thm)],[f304])).
% 11.73/3.15 thf(f304,axiom,(
% 11.73/3.15 archim1727834104eiling @ real),
% 11.73/3.15 file('/export/starexec/sandbox/tmp/tmp.aWXyJVFJl3/DTF2THF_4485.p',tcon_Real_Oreal___Archimedean__Field_Ofloor__ceiling)).
% 11.73/3.15 thf(f1121,plain,(
% 11.73/3.15 ( ! [X0 : $tType,X1 : int] : ((archim1727834104eiling @ X0 != $true) | ((archimedean_ceiling @ X0 @ (ring_1_of_int @ X0 @ X1)) = X1)) )),
% 11.73/3.15 inference(cnf_transformation,[status(thm)],[f978])).
% 11.73/3.15 thf(f978,plain,(
% 11.73/3.15 ! [X0 : $tType] : (! [X1 : int] : ((archimedean_ceiling @ X0 @ (ring_1_of_int @ X0 @ X1)) = X1) | (archim1727834104eiling @ X0 != $true))),
% 11.73/3.15 inference(ennf_transformation,[status(thm)],[f670])).
% 11.73/3.15 thf(f670,plain,(
% 11.73/3.15 ! [X0 : $tType] : ((archim1727834104eiling @ X0 = $true) => ! [X1 : int] : ((archimedean_ceiling @ X0 @ (ring_1_of_int @ X0 @ X1)) = X1))),
% 11.73/3.15 inference(fool_elimination,[status(thm)],[f669])).
% 11.73/3.15 thf(f669,plain,(
% 11.73/3.15 ! [X0 : $tType] : (archim1727834104eiling @ X0 => ! [X1 : int] : ((archimedean_ceiling @ X0 @ (ring_1_of_int @ X0 @ X1)) = X1))),
% 11.73/3.15 inference(rectify,[status(thm)],[f15])).
% 11.73/3.15 thf(f15,axiom,(
% 11.73/3.15 ! [X0 : $tType] : (archim1727834104eiling @ X0 => ! [X8 : int] : ((archimedean_ceiling @ X0 @ (ring_1_of_int @ X0 @ X8)) = X8))),
% 11.73/3.15 file('/export/starexec/sandbox/tmp/tmp.aWXyJVFJl3/DTF2THF_4485.p',fact_14_ceiling__of__int)).
% 11.73/3.15 thf(f1789,plain,(
% 11.73/3.15 ((ord_less_eq @ int @ (archimedean_ceiling @ real @ (ring_1_of_int @ real @ d)) @ (power_power @ int @ (numeral_numeral @ int @ (bit0 @ one2)) @ (nat2 @ (archimedean_ceiling @ real @ (log @ (numeral_numeral @ real @ (bit0 @ one2)) @ (ring_1_of_int @ real @ d)))))) = $true)),
% 11.73/3.15 inference(subsumption_resolution,[status(thm)],[f1788,f1124])).
% 11.73/3.15 thf(f1788,plain,(
% 11.73/3.15 (archim1727834104eiling @ real != $true) | ((ord_less_eq @ int @ (archimedean_ceiling @ real @ (ring_1_of_int @ real @ d)) @ (power_power @ int @ (numeral_numeral @ int @ (bit0 @ one2)) @ (nat2 @ (archimedean_ceiling @ real @ (log @ (numeral_numeral @ real @ (bit0 @ one2)) @ (ring_1_of_int @ real @ d)))))) = $true)),
% 11.73/3.15 inference(trivial_inequality_removal,[status(thm)],[f1786])).
% 11.73/3.15 thf(f1786,plain,(
% 11.73/3.15 ((ord_less_eq @ int @ (archimedean_ceiling @ real @ (ring_1_of_int @ real @ d)) @ (power_power @ int @ (numeral_numeral @ int @ (bit0 @ one2)) @ (nat2 @ (archimedean_ceiling @ real @ (log @ (numeral_numeral @ real @ (bit0 @ one2)) @ (ring_1_of_int @ real @ d)))))) = $true) | ($true != $true) | (archim1727834104eiling @ real != $true)),
% 11.73/3.15 inference(superposition,[status(thm)],[f1117,f1780])).
% 11.73/3.15 thf(f1780,plain,(
% 11.73/3.15 ($true = (ord_less_eq @ real @ (ring_1_of_int @ real @ d) @ (ring_1_of_int @ real @ (power_power @ int @ (numeral_numeral @ int @ (bit0 @ one2)) @ (nat2 @ (archimedean_ceiling @ real @ (log @ (numeral_numeral @ real @ (bit0 @ one2)) @ (ring_1_of_int @ real @ d))))))))),
% 11.73/3.15 inference(backward_demodulation,[status(thm)],[f1765,f1081])).
% 11.73/3.15 thf(f1081,plain,(
% 11.73/3.15 ((powr @ real @ (numeral_numeral @ real @ (bit0 @ one2)) @ (semiring_1_of_nat @ real @ (nat2 @ (archimedean_ceiling @ real @ (log @ (numeral_numeral @ real @ (bit0 @ one2)) @ (ring_1_of_int @ real @ d)))))) = (ring_1_of_int @ real @ (power_power @ int @ (numeral_numeral @ int @ (bit0 @ one2)) @ (nat2 @ (archimedean_ceiling @ real @ (log @ (numeral_numeral @ real @ (bit0 @ one2)) @ (ring_1_of_int @ real @ d)))))))),
% 11.73/3.15 inference(cnf_transformation,[status(thm)],[f67])).
% 11.73/3.15 thf(f67,axiom,(
% 11.73/3.15 ((powr @ real @ (numeral_numeral @ real @ (bit0 @ one2)) @ (semiring_1_of_nat @ real @ (nat2 @ (archimedean_ceiling @ real @ (log @ (numeral_numeral @ real @ (bit0 @ one2)) @ (ring_1_of_int @ real @ d)))))) = (ring_1_of_int @ real @ (power_power @ int @ (numeral_numeral @ int @ (bit0 @ one2)) @ (nat2 @ (archimedean_ceiling @ real @ (log @ (numeral_numeral @ real @ (bit0 @ one2)) @ (ring_1_of_int @ real @ d)))))))),
% 11.73/3.15 file('/export/starexec/sandbox/tmp/tmp.aWXyJVFJl3/DTF2THF_4485.p',fact_66__092_060open_0622_Apowr_Areal_A_Inat_A_092_060lceil_062log_A2_A_Ireal__of__int_Ad_J_092_060rceil_062_J_A_061_Areal__of__int_A_I2_A_094_Anat_A_092_060lceil_062log_A2_A_Ireal__of__int_Ad_J_092_060rceil_062_J_092_060close_062)).
% 11.73/3.15 thf(f1765,plain,(
% 11.73/3.15 ($true = (ord_less_eq @ real @ (ring_1_of_int @ real @ d) @ (powr @ real @ (numeral_numeral @ real @ (bit0 @ one2)) @ (semiring_1_of_nat @ real @ (nat2 @ (archimedean_ceiling @ real @ (log @ (numeral_numeral @ real @ (bit0 @ one2)) @ (ring_1_of_int @ real @ d))))))))),
% 11.73/3.15 inference(forward_demodulation,[status(thm)],[f1220,f1080])).
% 11.73/3.15 thf(f1080,plain,(
% 11.73/3.15 ((ring_1_of_int @ real @ d) = (powr @ real @ (numeral_numeral @ real @ (bit0 @ one2)) @ (log @ (numeral_numeral @ real @ (bit0 @ one2)) @ (ring_1_of_int @ real @ d))))),
% 11.73/3.15 inference(cnf_transformation,[status(thm)],[f37])).
% 11.73/3.15 thf(f37,axiom,(
% 11.73/3.15 ((ring_1_of_int @ real @ d) = (powr @ real @ (numeral_numeral @ real @ (bit0 @ one2)) @ (log @ (numeral_numeral @ real @ (bit0 @ one2)) @ (ring_1_of_int @ real @ d))))),
% 11.73/3.15 file('/export/starexec/sandbox/tmp/tmp.aWXyJVFJl3/DTF2THF_4485.p',fact_36__092_060open_062real__of__int_Ad_A_061_A2_Apowr_Alog_A2_A_Ireal__of__int_Ad_J_092_060close_062)).
% 11.73/3.15 thf(f1220,plain,(
% 11.73/3.15 ((ord_less_eq @ real @ (powr @ real @ (numeral_numeral @ real @ (bit0 @ one2)) @ (log @ (numeral_numeral @ real @ (bit0 @ one2)) @ (ring_1_of_int @ real @ d))) @ (powr @ real @ (numeral_numeral @ real @ (bit0 @ one2)) @ (semiring_1_of_nat @ real @ (nat2 @ (archimedean_ceiling @ real @ (log @ (numeral_numeral @ real @ (bit0 @ one2)) @ (ring_1_of_int @ real @ d))))))) = $true)),
% 11.73/3.15 inference(cnf_transformation,[status(thm)],[f546])).
% 11.73/3.15 thf(f546,plain,(
% 11.73/3.15 ((ord_less_eq @ real @ (powr @ real @ (numeral_numeral @ real @ (bit0 @ one2)) @ (log @ (numeral_numeral @ real @ (bit0 @ one2)) @ (ring_1_of_int @ real @ d))) @ (powr @ real @ (numeral_numeral @ real @ (bit0 @ one2)) @ (semiring_1_of_nat @ real @ (nat2 @ (archimedean_ceiling @ real @ (log @ (numeral_numeral @ real @ (bit0 @ one2)) @ (ring_1_of_int @ real @ d))))))) = $true)),
% 11.73/3.15 inference(fool_elimination,[status(thm)],[f545])).
% 11.73/3.15 thf(f545,plain,(
% 11.73/3.15 (ord_less_eq @ real @ (powr @ real @ (numeral_numeral @ real @ (bit0 @ one2)) @ (log @ (numeral_numeral @ real @ (bit0 @ one2)) @ (ring_1_of_int @ real @ d))) @ (powr @ real @ (numeral_numeral @ real @ (bit0 @ one2)) @ (semiring_1_of_nat @ real @ (nat2 @ (archimedean_ceiling @ real @ (log @ (numeral_numeral @ real @ (bit0 @ one2)) @ (ring_1_of_int @ real @ d)))))))),
% 11.73/3.15 inference(rectify,[status(thm)],[f68])).
% 11.73/3.15 thf(f68,axiom,(
% 11.73/3.15 (ord_less_eq @ real @ (powr @ real @ (numeral_numeral @ real @ (bit0 @ one2)) @ (log @ (numeral_numeral @ real @ (bit0 @ one2)) @ (ring_1_of_int @ real @ d))) @ (powr @ real @ (numeral_numeral @ real @ (bit0 @ one2)) @ (semiring_1_of_nat @ real @ (nat2 @ (archimedean_ceiling @ real @ (log @ (numeral_numeral @ real @ (bit0 @ one2)) @ (ring_1_of_int @ real @ d)))))))),
% 11.73/3.15 file('/export/starexec/sandbox/tmp/tmp.aWXyJVFJl3/DTF2THF_4485.p',fact_67__092_060open_0622_Apowr_Alog_A2_A_Ireal__of__int_Ad_J_A_092_060le_062_A2_Apowr_Areal_A_Inat_A_092_060lceil_062log_A2_A_Ireal__of__int_Ad_J_092_060rceil_062_J_092_060close_062)).
% 11.73/3.15 thf(f1117,plain,(
% 11.73/3.15 ( ! [X0 : $tType,X2 : int,X1 : X0] : (($true != (ord_less_eq @ X0 @ X1 @ (ring_1_of_int @ X0 @ X2))) | ((ord_less_eq @ int @ (archimedean_ceiling @ X0 @ X1) @ X2) = $true) | (archim1727834104eiling @ X0 != $true)) )),
% 11.73/3.15 inference(cnf_transformation,[status(thm)],[f976])).
% 11.73/3.15 thf(f976,plain,(
% 11.73/3.15 ! [X0 : $tType] : ((archim1727834104eiling @ X0 != $true) | ! [X1 : X0,X2 : int] : (((ord_less_eq @ int @ (archimedean_ceiling @ X0 @ X1) @ X2) = $true) | ($true != (ord_less_eq @ X0 @ X1 @ (ring_1_of_int @ X0 @ X2)))))),
% 11.73/3.15 inference(ennf_transformation,[status(thm)],[f376])).
% 11.73/3.15 thf(f376,plain,(
% 11.73/3.15 ! [X0 : $tType] : ((archim1727834104eiling @ X0 = $true) => ! [X1 : X0,X2 : int] : (($true = (ord_less_eq @ X0 @ X1 @ (ring_1_of_int @ X0 @ X2))) => ((ord_less_eq @ int @ (archimedean_ceiling @ X0 @ X1) @ X2) = $true)))),
% 11.73/3.15 inference(fool_elimination,[status(thm)],[f375])).
% 11.73/3.15 thf(f375,plain,(
% 11.73/3.15 ! [X0 : $tType] : (archim1727834104eiling @ X0 => ! [X1 : X0,X2 : int] : ((ord_less_eq @ X0 @ X1 @ (ring_1_of_int @ X0 @ X2)) => (ord_less_eq @ int @ (archimedean_ceiling @ X0 @ X1) @ X2)))),
% 11.73/3.15 inference(rectify,[status(thm)],[f31])).
% 11.73/3.15 thf(f31,axiom,(
% 11.73/3.15 ! [X0 : $tType] : (archim1727834104eiling @ X0 => ! [X1 : X0,X3 : int] : ((ord_less_eq @ X0 @ X1 @ (ring_1_of_int @ X0 @ X3)) => (ord_less_eq @ int @ (archimedean_ceiling @ X0 @ X1) @ X3)))),
% 11.73/3.15 file('/export/starexec/sandbox/tmp/tmp.aWXyJVFJl3/DTF2THF_4485.p',fact_30_ceiling__le)).
% 11.73/3.15 % SZS output end Proof for DTF2THF_4485
% 11.73/3.15 % (4695)------------------------------
% 11.73/3.15 % (4695)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 11.73/3.15 % (4695)Termination reason: Refutation
% 11.73/3.15
% 11.73/3.15 % (4695)Memory used [KB]: 6780
% 11.73/3.15 % (4695)Time elapsed: 0.059 s
% 11.73/3.15 % (4695)Instructions burned: 173 (million)
% 11.73/3.15 % (4695)------------------------------
% 11.73/3.15 % (4695)------------------------------
% 11.73/3.15 % (4591)Success in time 1.357 s
%------------------------------------------------------------------------------