%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : ITP095^2 : 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 : n024.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 08:55:31 AM UTC 2026
% Result : Theorem 2.30s 1.20s
% Output : Refutation 2.30s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.06 % Problem : ITP095^2 : TPTP v9.2.1. Released v7.5.0.
% 0.00/0.07 % Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.07/0.25 % Computer : n024.cluster.edu
% 0.07/0.25 % Model : x86_64 x86_64
% 0.07/0.25 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.07/0.25 % Memory : 8042.1875MB
% 0.07/0.25 % OS : Linux 3.10.0-693.el7.x86_64
% 0.07/0.25 % CPULimit : 300
% 0.07/0.25 % WCLimit : 300
% 0.07/0.25 % DateTime : Tue Feb 24 03:26:10 EST 2026
% 0.07/0.25 % CPUTime :
% 0.07/0.25 Running /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.07/0.31 ---- Original DTF file ---
% 0.07/0.31 thf(spec,logic,$$dhol).
% 0.07/0.31 %------------------------------------------------------------------------------
% 0.07/0.31 % File : ITP095^2 : TPTP v9.2.1. Released v7.5.0.
% 0.07/0.31 % Domain : Interactive Theorem Proving
% 0.07/0.31 % Problem : Sledgehammer Liouville_Numbers problem prob_189__5867098_1
% 0.07/0.31 % Version : Especial.
% 0.07/0.31 % English :
% 0.07/0.31
% 0.07/0.31 % Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% 0.07/0.31 % : [Des21] Desharnais (2021), Email to Geoff Sutcliffe
% 0.07/0.31 % Source : [Des21]
% 0.07/0.31 % Names : Liouville_Numbers/prob_189__5867098_1 [Des21]
% 0.07/0.31
% 0.07/0.31 % Status : Theorem
% 0.07/0.31 % Rating : 0.00 v7.5.0
% 0.07/0.31 % Syntax : Number of formulae : 408 ( 95 unt; 62 typ; 0 def)
% 0.07/0.31 % Number of atoms : 1022 ( 287 equ; 0 cnn)
% 0.07/0.31 % Maximal formula atoms : 6 ( 2 avg)
% 0.07/0.31 % Number of connectives : 3839 ( 43 ~; 21 |; 43 &;3306 @)
% 0.07/0.31 % ( 0 <=>; 426 =>; 0 <=; 0 <~>)
% 0.07/0.31 % Maximal formula depth : 16 ( 6 avg)
% 0.07/0.31 % Number of types : 4 ( 3 usr)
% 0.07/0.31 % Number of type conns : 30 ( 30 >; 0 *; 0 +; 0 <<)
% 0.07/0.31 % Number of symbols : 60 ( 59 usr; 9 con; 0-3 aty)
% 0.07/0.31 % Number of variables : 761 ( 17 ^; 688 !; 8 ?; 761 :)
% 0.07/0.31 % ( 48 !>; 0 ?*; 0 @-; 0 @+)
% 0.07/0.31 % SPC : TH1_THM_EQU_NAR
% 0.07/0.31
% 0.07/0.31 % Comments : This file was generated by Sledgehammer 2021-02-23 16:25:38.390
% 0.07/0.31 %------------------------------------------------------------------------------
% 0.07/0.31 % Could-be-implicit typings (5)
% 0.07/0.31 thf(ty_t_Polynomial_Opoly,type,
% 0.07/0.31 poly: $tType > $tType ).
% 0.07/0.31
% 0.07/0.31 thf(ty_t_Real_Oreal,type,
% 0.07/0.31 real: $tType ).
% 0.07/0.31
% 0.07/0.31 thf(ty_t_Set_Oset,type,
% 0.07/0.31 set: $tType > $tType ).
% 0.07/0.31
% 0.07/0.31 thf(ty_t_Nat_Onat,type,
% 0.07/0.31 nat: $tType ).
% 0.07/0.31
% 0.07/0.31 thf(ty_t_Int_Oint,type,
% 0.07/0.31 int: $tType ).
% 0.07/0.31
% 0.07/0.31 % Explicit typings (57)
% 0.07/0.31 thf(sy_cl_Groups_Oab__group__add,type,
% 0.07/0.31 ab_group_add:
% 0.07/0.31 !>[A: $tType] : $o ).
% 0.07/0.31
% 0.07/0.31 thf(sy_cl_Rings_Oidom,type,
% 0.07/0.31 idom:
% 0.07/0.31 !>[A: $tType] : $o ).
% 0.07/0.31
% 0.07/0.31 thf(sy_cl_Groups_Oone,type,
% 0.07/0.31 one:
% 0.07/0.31 !>[A: $tType] : $o ).
% 0.07/0.31
% 0.07/0.31 thf(sy_cl_Groups_Ozero,type,
% 0.07/0.31 zero:
% 0.07/0.31 !>[A: $tType] : $o ).
% 0.07/0.31
% 0.07/0.31 thf(sy_cl_Power_Opower,type,
% 0.07/0.31 power:
% 0.07/0.31 !>[A: $tType] : $o ).
% 0.07/0.31
% 0.07/0.31 thf(sy_cl_Fields_Ofield,type,
% 0.07/0.31 field:
% 0.07/0.31 !>[A: $tType] : $o ).
% 0.07/0.31
% 0.07/0.31 thf(sy_cl_Rings_Oring__1,type,
% 0.07/0.31 ring_1:
% 0.07/0.31 !>[A: $tType] : $o ).
% 0.07/0.31
% 0.07/0.31 thf(sy_cl_Nat_Oring__char__0,type,
% 0.07/0.31 ring_char_0:
% 0.07/0.31 !>[A: $tType] : $o ).
% 0.07/0.31
% 0.07/0.31 thf(sy_cl_Rings_Ocomm__ring,type,
% 0.07/0.31 comm_ring:
% 0.07/0.31 !>[A: $tType] : $o ).
% 0.07/0.31
% 0.07/0.31 thf(sy_cl_Groups_Ogroup__add,type,
% 0.07/0.31 group_add:
% 0.07/0.31 !>[A: $tType] : $o ).
% 0.07/0.31
% 0.07/0.31 thf(sy_cl_Rings_Osemiring__1,type,
% 0.07/0.31 semiring_1:
% 0.07/0.31 !>[A: $tType] : $o ).
% 0.07/0.31
% 0.07/0.31 thf(sy_cl_Rings_Ocomm__ring__1,type,
% 0.07/0.31 comm_ring_1:
% 0.07/0.31 !>[A: $tType] : $o ).
% 0.07/0.31
% 0.07/0.31 thf(sy_cl_Rings_Oidom__divide,type,
% 0.07/0.31 idom_divide:
% 0.07/0.31 !>[A: $tType] : $o ).
% 0.07/0.31
% 0.07/0.31 thf(sy_cl_Groups_Omonoid__mult,type,
% 0.07/0.31 monoid_mult:
% 0.07/0.31 !>[A: $tType] : $o ).
% 0.07/0.31
% 0.07/0.31 thf(sy_cl_Fields_Ofield__char__0,type,
% 0.07/0.31 field_char_0:
% 0.07/0.31 !>[A: $tType] : $o ).
% 0.07/0.31
% 0.07/0.31 thf(sy_cl_Fields_Odivision__ring,type,
% 0.07/0.31 division_ring:
% 0.07/0.31 !>[A: $tType] : $o ).
% 0.07/0.31
% 0.07/0.31 thf(sy_cl_Fields_Ofield__abs__sgn,type,
% 0.07/0.31 field_abs_sgn:
% 0.07/0.31 !>[A: $tType] : $o ).
% 0.07/0.31
% 0.07/0.31 thf(sy_cl_Rings_Ocomm__semiring__0,type,
% 0.07/0.31 comm_semiring_0:
% 0.07/0.31 !>[A: $tType] : $o ).
% 0.07/0.31
% 0.07/0.31 thf(sy_cl_Rings_Ocomm__semiring__1,type,
% 0.07/0.31 comm_semiring_1:
% 0.07/0.31 !>[A: $tType] : $o ).
% 0.07/0.31
% 0.07/0.31 thf(sy_cl_Rings_Olinordered__idom,type,
% 0.07/0.31 linordered_idom:
% 0.07/0.31 !>[A: $tType] : $o ).
% 0.07/0.31
% 0.07/0.31 thf(sy_cl_Fields_Olinordered__field,type,
% 0.07/0.31 linordered_field:
% 0.07/0.31 !>[A: $tType] : $o ).
% 0.07/0.31
% 0.07/0.31 thf(sy_cl_Groups_Ocomm__monoid__diff,type,
% 0.07/0.31 comm_monoid_diff:
% 0.07/0.31 !>[A: $tType] : $o ).
% 0.07/0.31
% 0.07/0.31 thf(sy_cl_Rings_Olinordered__semidom,type,
% 0.07/0.31 linordered_semidom:
% 0.07/0.31 !>[A: $tType] : $o ).
% 0.07/0.31
% 0.07/0.31 thf(sy_cl_Groups_Oordered__ab__group__add,type,
% 0.07/0.31 ordered_ab_group_add:
% 0.07/0.31 !>[A: $tType] : $o ).
% 0.07/0.31
% 0.07/0.31 thf(sy_cl_Rings_Oring__no__zero__divisors,type,
% 0.07/0.31 ring_n68954251visors:
% 0.07/0.31 !>[A: $tType] : $o ).
% 0.07/0.31
% 0.07/0.31 thf(sy_cl_Groups_Ocancel__comm__monoid__add,type,
% 0.07/0.31 cancel1352612707id_add:
% 0.07/0.31 !>[A: $tType] : $o ).
% 0.07/0.31
% 0.07/0.31 thf(sy_cl_Groups_Ocancel__ab__semigroup__add,type,
% 0.07/0.31 cancel146912293up_add:
% 0.07/0.31 !>[A: $tType] : $o ).
% 0.07/0.31
% 0.07/0.31 thf(sy_cl_Groups_Oordered__ab__group__add__abs,type,
% 0.07/0.31 ordere142940540dd_abs:
% 0.07/0.31 !>[A: $tType] : $o ).
% 0.07/0.31
% 0.07/0.31 thf(sy_cl_Rings_Osemiring__no__zero__divisors,type,
% 0.07/0.31 semiri1193490041visors:
% 0.07/0.31 !>[A: $tType] : $o ).
% 0.07/0.31
% 0.07/0.31 thf(sy_cl_Rings_Osemiring__1__no__zero__divisors,type,
% 0.07/0.31 semiri134348788visors:
% 0.07/0.31 !>[A: $tType] : $o ).
% 0.07/0.31
% 0.07/0.31 thf(sy_cl_Groups_Ocanonically__ordered__monoid__add,type,
% 0.07/0.31 canoni770627133id_add:
% 0.07/0.31 !>[A: $tType] : $o ).
% 0.07/0.31
% 0.07/0.31 thf(sy_c_Groups_Oabs__class_Oabs,type,
% 0.07/0.31 abs_abs:
% 0.07/0.31 !>[A: $tType] : ( A > A ) ).
% 0.07/0.31
% 0.07/0.31 thf(sy_c_Groups_Ominus__class_Ominus,type,
% 0.07/0.31 minus_minus:
% 0.07/0.31 !>[A: $tType] : ( A > A > A ) ).
% 0.07/0.31
% 0.07/0.31 thf(sy_c_Groups_Oone__class_Oone,type,
% 0.07/0.31 one_one:
% 0.07/0.31 !>[A: $tType] : A ).
% 0.07/0.31
% 0.07/0.31 thf(sy_c_Groups_Ozero__class_Ozero,type,
% 0.07/0.31 zero_zero:
% 0.07/0.31 !>[A: $tType] : A ).
% 0.07/0.31
% 0.07/0.31 thf(sy_c_Int_Oring__1__class_OInts,type,
% 0.07/0.31 ring_1_Ints:
% 0.07/0.31 !>[A: $tType] : ( set @ A ) ).
% 0.07/0.31
% 0.07/0.31 thf(sy_c_Int_Oring__1__class_Oof__int,type,
% 0.07/0.31 ring_1_of_int:
% 0.07/0.31 !>[A: $tType] : ( int > A ) ).
% 0.07/0.31
% 0.07/0.31 thf(sy_c_Orderings_Oord__class_Oless,type,
% 0.07/0.31 ord_less:
% 0.07/0.31 !>[A: $tType] : ( A > A > $o ) ).
% 0.07/0.31
% 0.07/0.31 thf(sy_c_Orderings_Oord__class_Oless__eq,type,
% 0.07/0.31 ord_less_eq:
% 0.07/0.31 !>[A: $tType] : ( A > A > $o ) ).
% 0.07/0.31
% 0.07/0.31 thf(sy_c_Polynomial_Oalgebraic,type,
% 0.07/0.31 algebraic:
% 0.07/0.31 !>[A: $tType] : ( A > $o ) ).
% 0.07/0.31
% 0.07/0.31 thf(sy_c_Polynomial_Odegree,type,
% 0.07/0.31 degree:
% 0.07/0.31 !>[A: $tType] : ( ( poly @ A ) > nat ) ).
% 0.07/0.31
% 0.07/0.31 thf(sy_c_Polynomial_Opoly,type,
% 0.07/0.31 poly2:
% 0.07/0.31 !>[A: $tType] : ( ( poly @ A ) > A > A ) ).
% 0.07/0.31
% 0.07/0.31 thf(sy_c_Polynomial_Opoly_Ocoeff,type,
% 0.07/0.31 coeff:
% 0.07/0.31 !>[A: $tType] : ( ( poly @ A ) > nat > A ) ).
% 0.07/0.31
% 0.07/0.31 thf(sy_c_Power_Opower__class_Opower,type,
% 0.07/0.31 power_power:
% 0.07/0.31 !>[A: $tType] : ( A > nat > A ) ).
% 0.07/0.31
% 0.07/0.31 thf(sy_c_Rat_Ofield__char__0__class_ORats,type,
% 0.07/0.31 field_char_0_Rats:
% 0.07/0.31 !>[A: $tType] : ( set @ A ) ).
% 0.07/0.31
% 0.07/0.31 thf(sy_c_Rings_Odivide__class_Odivide,type,
% 0.07/0.31 divide_divide:
% 0.07/0.31 !>[A: $tType] : ( A > A > A ) ).
% 0.07/0.31
% 0.07/0.31 thf(sy_c_Set_OCollect,type,
% 0.07/0.31 collect:
% 0.07/0.31 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(sy_c_member,type,
% 0.07/0.31 member:
% 0.07/0.31 !>[A: $tType] : ( A > ( set @ A ) > $o ) ).
% 0.07/0.31
% 0.07/0.31 thf(sy_v_A____,type,
% 0.07/0.31 a: real ).
% 0.07/0.31
% 0.07/0.31 thf(sy_v_M____,type,
% 0.07/0.31 m: real ).
% 0.07/0.31
% 0.07/0.31 thf(sy_v_a____,type,
% 0.07/0.31 a2: int ).
% 0.07/0.31
% 0.07/0.31 thf(sy_v_b____,type,
% 0.07/0.31 b: int ).
% 0.07/0.31
% 0.07/0.31 thf(sy_v_n____,type,
% 0.07/0.31 n: nat ).
% 0.07/0.31
% 0.07/0.31 thf(sy_v_p____,type,
% 0.07/0.31 p: poly @ real ).
% 0.07/0.31
% 0.07/0.31 thf(sy_v_roots____,type,
% 0.07/0.31 roots: set @ real ).
% 0.07/0.31
% 0.07/0.31 thf(sy_v_thesis,type,
% 0.07/0.31 thesis: $o ).
% 0.07/0.31
% 0.07/0.31 thf(sy_v_x,type,
% 0.07/0.31 x: real ).
% 0.07/0.31
% 0.07/0.31 % Relevant facts (254)
% 0.07/0.31 thf(fact_0_assms_I2_J,axiom,
% 0.07/0.31 algebraic @ real @ x ).
% 0.07/0.31
% 0.07/0.31 % assms(2)
% 0.07/0.31 thf(fact_1_p_I2_J,axiom,
% 0.07/0.31 ( p
% 0.07/0.31 != ( zero_zero @ ( poly @ real ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % p(2)
% 0.07/0.31 thf(fact_2_p_I1_J,axiom,
% 0.07/0.31 ! [I: nat] : ( member @ real @ ( coeff @ real @ p @ I ) @ ( ring_1_Ints @ real ) ) ).
% 0.07/0.31
% 0.07/0.31 % p(1)
% 0.07/0.31 thf(fact_3_b,axiom,
% 0.07/0.31 ord_less @ int @ ( zero_zero @ int ) @ b ).
% 0.07/0.31
% 0.07/0.31 % b
% 0.07/0.31 thf(fact_4_p_I3_J,axiom,
% 0.07/0.31 ( ( poly2 @ real @ p @ x )
% 0.07/0.31 = ( zero_zero @ real ) ) ).
% 0.07/0.31
% 0.07/0.31 % p(3)
% 0.07/0.31 thf(fact_5_no__root,axiom,
% 0.07/0.31 ( ( poly2 @ real @ p @ ( divide_divide @ real @ ( ring_1_of_int @ real @ a2 ) @ ( ring_1_of_int @ real @ b ) ) )
% 0.07/0.31 != ( zero_zero @ real ) ) ).
% 0.07/0.31
% 0.07/0.31 % no_root
% 0.07/0.31 thf(fact_6_n__def,axiom,
% 0.07/0.31 ( n
% 0.07/0.31 = ( degree @ real @ p ) ) ).
% 0.07/0.31
% 0.07/0.31 % n_def
% 0.07/0.31 thf(fact_7_of__int__le__of__int__power__cancel__iff,axiom,
% 0.07/0.31 ! [A: $tType] :
% 0.07/0.31 ( ( linordered_idom @ A )
% 0.07/0.31 => ! [B: int,W: nat,X: int] :
% 0.07/0.31 ( ( ord_less_eq @ A @ ( power_power @ A @ ( ring_1_of_int @ A @ B ) @ W ) @ ( ring_1_of_int @ A @ X ) )
% 0.07/0.31 = ( ord_less_eq @ int @ ( power_power @ int @ B @ W ) @ X ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % of_int_le_of_int_power_cancel_iff
% 0.07/0.31 thf(fact_8_of__int__power__le__of__int__cancel__iff,axiom,
% 0.07/0.31 ! [A: $tType] :
% 0.07/0.31 ( ( linordered_idom @ A )
% 0.07/0.31 => ! [X: int,B: int,W: nat] :
% 0.07/0.31 ( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ X ) @ ( power_power @ A @ ( ring_1_of_int @ A @ B ) @ W ) )
% 0.07/0.31 = ( ord_less_eq @ int @ X @ ( power_power @ int @ B @ W ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % of_int_power_le_of_int_cancel_iff
% 0.07/0.31 thf(fact_9_of__int__1__le__iff,axiom,
% 0.07/0.31 ! [A: $tType] :
% 0.07/0.31 ( ( linordered_idom @ A )
% 0.07/0.31 => ! [Z: int] :
% 0.07/0.31 ( ( ord_less_eq @ A @ ( one_one @ A ) @ ( ring_1_of_int @ A @ Z ) )
% 0.07/0.31 = ( ord_less_eq @ int @ ( one_one @ int ) @ Z ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % of_int_1_le_iff
% 0.07/0.31 thf(fact_10_of__int__le__1__iff,axiom,
% 0.07/0.31 ! [A: $tType] :
% 0.07/0.31 ( ( linordered_idom @ A )
% 0.07/0.31 => ! [Z: int] :
% 0.07/0.31 ( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ Z ) @ ( one_one @ A ) )
% 0.07/0.31 = ( ord_less_eq @ int @ Z @ ( one_one @ int ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % of_int_le_1_iff
% 0.07/0.31 thf(fact_11_poly__power,axiom,
% 0.07/0.31 ! [A: $tType] :
% 0.07/0.31 ( ( comm_semiring_1 @ A )
% 0.07/0.31 => ! [P: poly @ A,N: nat,X: A] :
% 0.07/0.31 ( ( poly2 @ A @ ( power_power @ ( poly @ A ) @ P @ N ) @ X )
% 0.07/0.31 = ( power_power @ A @ ( poly2 @ A @ P @ X ) @ N ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % poly_power
% 0.07/0.31 thf(fact_12_poly__1,axiom,
% 0.07/0.31 ! [A: $tType] :
% 0.07/0.31 ( ( comm_semiring_1 @ A )
% 0.07/0.31 => ! [X: A] :
% 0.07/0.31 ( ( poly2 @ A @ ( one_one @ ( poly @ A ) ) @ X )
% 0.07/0.31 = ( one_one @ A ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % poly_1
% 0.07/0.31 thf(fact_13_of__int__abs,axiom,
% 0.07/0.31 ! [A: $tType] :
% 0.07/0.31 ( ( linordered_idom @ A )
% 0.07/0.31 => ! [X: int] :
% 0.07/0.31 ( ( ring_1_of_int @ A @ ( abs_abs @ int @ X ) )
% 0.07/0.31 = ( abs_abs @ A @ ( ring_1_of_int @ A @ X ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % of_int_abs
% 0.07/0.31 thf(fact_14_of__int__power,axiom,
% 0.07/0.31 ! [A: $tType] :
% 0.07/0.31 ( ( ring_1 @ A )
% 0.07/0.31 => ! [Z: int,N: nat] :
% 0.07/0.31 ( ( ring_1_of_int @ A @ ( power_power @ int @ Z @ N ) )
% 0.07/0.31 = ( power_power @ A @ ( ring_1_of_int @ A @ Z ) @ N ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % of_int_power
% 0.07/0.31 thf(fact_15_of__int__eq__of__int__power__cancel__iff,axiom,
% 0.07/0.31 ! [A: $tType] :
% 0.07/0.31 ( ( ring_char_0 @ A )
% 0.07/0.31 => ! [B: int,W: nat,X: int] :
% 0.07/0.31 ( ( ( power_power @ A @ ( ring_1_of_int @ A @ B ) @ W )
% 0.07/0.31 = ( ring_1_of_int @ A @ X ) )
% 0.07/0.31 = ( ( power_power @ int @ B @ W )
% 0.07/0.31 = X ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % of_int_eq_of_int_power_cancel_iff
% 0.07/0.31 thf(fact_16_of__int__power__eq__of__int__cancel__iff,axiom,
% 0.07/0.31 ! [A: $tType] :
% 0.07/0.31 ( ( ring_char_0 @ A )
% 0.07/0.31 => ! [X: int,B: int,W: nat] :
% 0.07/0.31 ( ( ( ring_1_of_int @ A @ X )
% 0.07/0.31 = ( power_power @ A @ ( ring_1_of_int @ A @ B ) @ W ) )
% 0.07/0.31 = ( X
% 0.07/0.31 = ( power_power @ int @ B @ W ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % of_int_power_eq_of_int_cancel_iff
% 0.07/0.31 thf(fact_17_of__int__1,axiom,
% 0.07/0.31 ! [A: $tType] :
% 0.07/0.31 ( ( ring_1 @ A )
% 0.07/0.31 => ( ( ring_1_of_int @ A @ ( one_one @ int ) )
% 0.07/0.31 = ( one_one @ A ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % of_int_1
% 0.07/0.31 thf(fact_18_of__int__eq__1__iff,axiom,
% 0.07/0.31 ! [A: $tType] :
% 0.07/0.31 ( ( ring_char_0 @ A )
% 0.07/0.31 => ! [Z: int] :
% 0.07/0.31 ( ( ( ring_1_of_int @ A @ Z )
% 0.07/0.31 = ( one_one @ A ) )
% 0.07/0.31 = ( Z
% 0.07/0.31 = ( one_one @ int ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % of_int_eq_1_iff
% 0.07/0.31 thf(fact_19_irrationsl,axiom,
% 0.07/0.31 ~ ( member @ real @ x @ ( field_char_0_Rats @ real ) ) ).
% 0.07/0.31
% 0.07/0.31 % irrationsl
% 0.07/0.31 thf(fact_20_of__int__eq__iff,axiom,
% 0.07/0.31 ! [A: $tType] :
% 0.07/0.31 ( ( ring_char_0 @ A )
% 0.07/0.31 => ! [W: int,Z: int] :
% 0.07/0.31 ( ( ( ring_1_of_int @ A @ W )
% 0.07/0.31 = ( ring_1_of_int @ A @ Z ) )
% 0.07/0.31 = ( W = Z ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % of_int_eq_iff
% 0.07/0.31 thf(fact_21__092_060open_062_092_060And_062thesisa_O_A_I_092_060And_062p_O_A_092_060lbrakk_062_092_060And_062i_O_Acoeff_Ap_Ai_A_092_060in_062_A_092_060int_062_059_Ap_A_092_060noteq_062_A0_059_Apoly_Ap_Ax_A_061_A0_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesisa_J_A_092_060Longrightarrow_062_Athesisa_092_060close_062,axiom,
% 0.07/0.31 ~ ! [P2: poly @ real] :
% 0.07/0.31 ( ! [I2: nat] : ( member @ real @ ( coeff @ real @ P2 @ I2 ) @ ( ring_1_Ints @ real ) )
% 0.07/0.31 => ( ( P2
% 0.07/0.31 != ( zero_zero @ ( poly @ real ) ) )
% 0.07/0.31 => ( ( poly2 @ real @ P2 @ x )
% 0.07/0.31 != ( zero_zero @ real ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % \<open>\<And>thesisa. (\<And>p. \<lbrakk>\<And>i. coeff p i \<in> \<int>; p \<noteq> 0; poly p x = 0\<rbrakk> \<Longrightarrow> thesisa) \<Longrightarrow> thesisa\<close>
% 0.07/0.31 thf(fact_22__092_060open_062x_A_092_060notin_062_Aroots_092_060close_062,axiom,
% 0.07/0.31 ~ ( member @ real @ x @ roots ) ).
% 0.07/0.31
% 0.07/0.31 % \<open>x \<notin> roots\<close>
% 0.07/0.31 thf(fact_23_of__int__eq__0__iff,axiom,
% 0.07/0.31 ! [A: $tType] :
% 0.07/0.31 ( ( ring_char_0 @ A )
% 0.07/0.31 => ! [Z: int] :
% 0.07/0.31 ( ( ( ring_1_of_int @ A @ Z )
% 0.07/0.31 = ( zero_zero @ A ) )
% 0.07/0.31 = ( Z
% 0.07/0.31 = ( zero_zero @ int ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % of_int_eq_0_iff
% 0.07/0.31 thf(fact_24_of__int__0__eq__iff,axiom,
% 0.07/0.31 ! [A: $tType] :
% 0.07/0.31 ( ( ring_char_0 @ A )
% 0.07/0.31 => ! [Z: int] :
% 0.07/0.31 ( ( ( zero_zero @ A )
% 0.07/0.31 = ( ring_1_of_int @ A @ Z ) )
% 0.07/0.31 = ( Z
% 0.07/0.31 = ( zero_zero @ int ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % of_int_0_eq_iff
% 0.07/0.31 thf(fact_25_of__int__0,axiom,
% 0.07/0.31 ! [A: $tType] :
% 0.07/0.31 ( ( ring_1 @ A )
% 0.07/0.31 => ( ( ring_1_of_int @ A @ ( zero_zero @ int ) )
% 0.07/0.31 = ( zero_zero @ A ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % of_int_0
% 0.07/0.31 thf(fact_26_of__int__le__iff,axiom,
% 0.07/0.31 ! [A: $tType] :
% 0.07/0.31 ( ( linordered_idom @ A )
% 0.07/0.31 => ! [W: int,Z: int] :
% 0.07/0.31 ( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ W ) @ ( ring_1_of_int @ A @ Z ) )
% 0.07/0.31 = ( ord_less_eq @ int @ W @ Z ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % of_int_le_iff
% 0.07/0.31 thf(fact_27_of__int__less__iff,axiom,
% 0.07/0.31 ! [A: $tType] :
% 0.07/0.31 ( ( linordered_idom @ A )
% 0.07/0.31 => ! [W: int,Z: int] :
% 0.07/0.31 ( ( ord_less @ A @ ( ring_1_of_int @ A @ W ) @ ( ring_1_of_int @ A @ Z ) )
% 0.07/0.31 = ( ord_less @ int @ W @ Z ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % of_int_less_iff
% 0.07/0.31 thf(fact_28_coeff__0,axiom,
% 0.07/0.31 ! [A: $tType] :
% 0.07/0.31 ( ( zero @ A )
% 0.07/0.31 => ! [N: nat] :
% 0.07/0.31 ( ( coeff @ A @ ( zero_zero @ ( poly @ A ) ) @ N )
% 0.07/0.31 = ( zero_zero @ A ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % coeff_0
% 0.07/0.31 thf(fact_29_poly__0,axiom,
% 0.07/0.31 ! [A: $tType] :
% 0.07/0.31 ( ( comm_semiring_0 @ A )
% 0.07/0.31 => ! [X: A] :
% 0.07/0.31 ( ( poly2 @ A @ ( zero_zero @ ( poly @ A ) ) @ X )
% 0.07/0.31 = ( zero_zero @ A ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % poly_0
% 0.07/0.31 thf(fact_30_lead__coeff__of__int,axiom,
% 0.07/0.31 ! [A: $tType] :
% 0.07/0.31 ( ( comm_ring_1 @ A )
% 0.07/0.31 => ! [K: int] :
% 0.07/0.31 ( ( coeff @ A @ ( ring_1_of_int @ ( poly @ A ) @ K ) @ ( degree @ A @ ( ring_1_of_int @ ( poly @ A ) @ K ) ) )
% 0.07/0.31 = ( ring_1_of_int @ A @ K ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % lead_coeff_of_int
% 0.07/0.31 thf(fact_31_leading__coeff__0__iff,axiom,
% 0.07/0.31 ! [A: $tType] :
% 0.07/0.31 ( ( zero @ A )
% 0.07/0.31 => ! [P: poly @ A] :
% 0.07/0.31 ( ( ( coeff @ A @ P @ ( degree @ A @ P ) )
% 0.07/0.31 = ( zero_zero @ A ) )
% 0.07/0.31 = ( P
% 0.07/0.31 = ( zero_zero @ ( poly @ A ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % leading_coeff_0_iff
% 0.07/0.31 thf(fact_32_zabs__less__one__iff,axiom,
% 0.07/0.31 ! [Z: int] :
% 0.07/0.31 ( ( ord_less @ int @ ( abs_abs @ int @ Z ) @ ( one_one @ int ) )
% 0.07/0.31 = ( Z
% 0.07/0.31 = ( zero_zero @ int ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % zabs_less_one_iff
% 0.07/0.31 thf(fact_33_lead__coeff__1,axiom,
% 0.07/0.31 ! [A: $tType] :
% 0.07/0.31 ( ( comm_semiring_1 @ A )
% 0.07/0.31 => ( ( coeff @ A @ ( one_one @ ( poly @ A ) ) @ ( degree @ A @ ( one_one @ ( poly @ A ) ) ) )
% 0.07/0.31 = ( one_one @ A ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % lead_coeff_1
% 0.07/0.31 thf(fact_34_of__int__le__0__iff,axiom,
% 0.07/0.31 ! [A: $tType] :
% 0.07/0.31 ( ( linordered_idom @ A )
% 0.07/0.31 => ! [Z: int] :
% 0.07/0.31 ( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ Z ) @ ( zero_zero @ A ) )
% 0.07/0.31 = ( ord_less_eq @ int @ Z @ ( zero_zero @ int ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % of_int_le_0_iff
% 0.07/0.31 thf(fact_35_of__int__0__le__iff,axiom,
% 0.07/0.31 ! [A: $tType] :
% 0.07/0.31 ( ( linordered_idom @ A )
% 0.07/0.31 => ! [Z: int] :
% 0.07/0.31 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( ring_1_of_int @ A @ Z ) )
% 0.07/0.31 = ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % of_int_0_le_iff
% 0.07/0.31 thf(fact_36_of__int__less__0__iff,axiom,
% 0.07/0.31 ! [A: $tType] :
% 0.07/0.31 ( ( linordered_idom @ A )
% 0.07/0.31 => ! [Z: int] :
% 0.07/0.31 ( ( ord_less @ A @ ( ring_1_of_int @ A @ Z ) @ ( zero_zero @ A ) )
% 0.07/0.31 = ( ord_less @ int @ Z @ ( zero_zero @ int ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % of_int_less_0_iff
% 0.07/0.31 thf(fact_37_of__int__0__less__iff,axiom,
% 0.07/0.31 ! [A: $tType] :
% 0.07/0.31 ( ( linordered_idom @ A )
% 0.07/0.31 => ! [Z: int] :
% 0.07/0.31 ( ( ord_less @ A @ ( zero_zero @ A ) @ ( ring_1_of_int @ A @ Z ) )
% 0.07/0.31 = ( ord_less @ int @ ( zero_zero @ int ) @ Z ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % of_int_0_less_iff
% 0.07/0.31 thf(fact_38_of__int__less__1__iff,axiom,
% 0.07/0.31 ! [A: $tType] :
% 0.07/0.31 ( ( linordered_idom @ A )
% 0.07/0.31 => ! [Z: int] :
% 0.07/0.31 ( ( ord_less @ A @ ( ring_1_of_int @ A @ Z ) @ ( one_one @ A ) )
% 0.07/0.31 = ( ord_less @ int @ Z @ ( one_one @ int ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % of_int_less_1_iff
% 0.07/0.31 thf(fact_39_of__int__1__less__iff,axiom,
% 0.07/0.31 ! [A: $tType] :
% 0.07/0.31 ( ( linordered_idom @ A )
% 0.07/0.31 => ! [Z: int] :
% 0.07/0.31 ( ( ord_less @ A @ ( one_one @ A ) @ ( ring_1_of_int @ A @ Z ) )
% 0.07/0.31 = ( ord_less @ int @ ( one_one @ int ) @ Z ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % of_int_1_less_iff
% 0.07/0.31 thf(fact_40_of__int__power__less__of__int__cancel__iff,axiom,
% 0.07/0.32 ! [A: $tType] :
% 0.07/0.32 ( ( linordered_idom @ A )
% 0.07/0.32 => ! [X: int,B: int,W: nat] :
% 0.07/0.32 ( ( ord_less @ A @ ( ring_1_of_int @ A @ X ) @ ( power_power @ A @ ( ring_1_of_int @ A @ B ) @ W ) )
% 0.07/0.32 = ( ord_less @ int @ X @ ( power_power @ int @ B @ W ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % of_int_power_less_of_int_cancel_iff
% 0.07/0.32 thf(fact_41_of__int__less__of__int__power__cancel__iff,axiom,
% 0.07/0.32 ! [A: $tType] :
% 0.07/0.32 ( ( linordered_idom @ A )
% 0.07/0.32 => ! [B: int,W: nat,X: int] :
% 0.07/0.32 ( ( ord_less @ A @ ( power_power @ A @ ( ring_1_of_int @ A @ B ) @ W ) @ ( ring_1_of_int @ A @ X ) )
% 0.07/0.32 = ( ord_less @ int @ ( power_power @ int @ B @ W ) @ X ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % of_int_less_of_int_power_cancel_iff
% 0.07/0.32 thf(fact_42_less__int__code_I1_J,axiom,
% 0.07/0.32 ~ ( ord_less @ int @ ( zero_zero @ int ) @ ( zero_zero @ int ) ) ).
% 0.07/0.32
% 0.07/0.32 % less_int_code(1)
% 0.07/0.32 thf(fact_43_less__eq__int__code_I1_J,axiom,
% 0.07/0.32 ord_less_eq @ int @ ( zero_zero @ int ) @ ( zero_zero @ int ) ).
% 0.07/0.32
% 0.07/0.32 % less_eq_int_code(1)
% 0.07/0.32 thf(fact_44_poly__eqI,axiom,
% 0.07/0.32 ! [A: $tType] :
% 0.07/0.32 ( ( zero @ A )
% 0.07/0.32 => ! [P: poly @ A,Q: poly @ A] :
% 0.07/0.32 ( ! [N2: nat] :
% 0.07/0.32 ( ( coeff @ A @ P @ N2 )
% 0.07/0.32 = ( coeff @ A @ Q @ N2 ) )
% 0.07/0.32 => ( P = Q ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % poly_eqI
% 0.07/0.32 thf(fact_45_mem__Collect__eq,axiom,
% 0.07/0.32 ! [A: $tType,A2: A,P3: A > $o] :
% 0.07/0.32 ( ( member @ A @ A2 @ ( collect @ A @ P3 ) )
% 0.07/0.32 = ( P3 @ A2 ) ) ).
% 0.07/0.32
% 0.07/0.32 % mem_Collect_eq
% 0.07/0.32 thf(fact_46_Collect__mem__eq,axiom,
% 0.07/0.32 ! [A: $tType,A3: set @ A] :
% 0.07/0.32 ( ( collect @ A
% 0.07/0.32 @ ^ [X2: A] : ( member @ A @ X2 @ A3 ) )
% 0.07/0.32 = A3 ) ).
% 0.07/0.32
% 0.07/0.32 % Collect_mem_eq
% 0.07/0.32 thf(fact_47_Collect__cong,axiom,
% 0.07/0.32 ! [A: $tType,P3: A > $o,Q2: A > $o] :
% 0.07/0.32 ( ! [X3: A] :
% 0.07/0.32 ( ( P3 @ X3 )
% 0.07/0.32 = ( Q2 @ X3 ) )
% 0.07/0.32 => ( ( collect @ A @ P3 )
% 0.07/0.32 = ( collect @ A @ Q2 ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % Collect_cong
% 0.07/0.32 thf(fact_48_ext,axiom,
% 0.07/0.32 ! [B2: $tType,A: $tType,F: A > B2,G: A > B2] :
% 0.07/0.32 ( ! [X3: A] :
% 0.07/0.32 ( ( F @ X3 )
% 0.07/0.32 = ( G @ X3 ) )
% 0.07/0.32 => ( F = G ) ) ).
% 0.07/0.32
% 0.07/0.32 % ext
% 0.07/0.32 thf(fact_49_poly__eq__iff,axiom,
% 0.07/0.32 ! [A: $tType] :
% 0.07/0.32 ( ( zero @ A )
% 0.07/0.32 => ( ( ^ [Y: poly @ A,Z2: poly @ A] : ( Y = Z2 ) )
% 0.07/0.32 = ( ^ [P4: poly @ A,Q3: poly @ A] :
% 0.07/0.32 ! [N3: nat] :
% 0.07/0.32 ( ( coeff @ A @ P4 @ N3 )
% 0.07/0.32 = ( coeff @ A @ Q3 @ N3 ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % poly_eq_iff
% 0.07/0.32 thf(fact_50_Ints__0,axiom,
% 0.07/0.32 ! [A: $tType] :
% 0.07/0.32 ( ( ring_1 @ A )
% 0.07/0.32 => ( member @ A @ ( zero_zero @ A ) @ ( ring_1_Ints @ A ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % Ints_0
% 0.07/0.32 thf(fact_51_zero__poly_Orep__eq,axiom,
% 0.07/0.32 ! [A: $tType] :
% 0.07/0.32 ( ( zero @ A )
% 0.07/0.32 => ( ( coeff @ A @ ( zero_zero @ ( poly @ A ) ) )
% 0.07/0.32 = ( ^ [Uu: nat] : ( zero_zero @ A ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % zero_poly.rep_eq
% 0.07/0.32 thf(fact_52_int__one__le__iff__zero__less,axiom,
% 0.07/0.32 ! [Z: int] :
% 0.07/0.32 ( ( ord_less_eq @ int @ ( one_one @ int ) @ Z )
% 0.07/0.32 = ( ord_less @ int @ ( zero_zero @ int ) @ Z ) ) ).
% 0.07/0.32
% 0.07/0.32 % int_one_le_iff_zero_less
% 0.07/0.32 thf(fact_53_coeff__inject,axiom,
% 0.07/0.32 ! [A: $tType] :
% 0.07/0.32 ( ( zero @ A )
% 0.07/0.32 => ! [X: poly @ A,Y2: poly @ A] :
% 0.07/0.32 ( ( ( coeff @ A @ X )
% 0.07/0.32 = ( coeff @ A @ Y2 ) )
% 0.07/0.32 = ( X = Y2 ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % coeff_inject
% 0.07/0.32 thf(fact_54_le__degree,axiom,
% 0.07/0.32 ! [A: $tType] :
% 0.07/0.32 ( ( zero @ A )
% 0.07/0.32 => ! [P: poly @ A,N: nat] :
% 0.07/0.32 ( ( ( coeff @ A @ P @ N )
% 0.07/0.32 != ( zero_zero @ A ) )
% 0.07/0.32 => ( ord_less_eq @ nat @ N @ ( degree @ A @ P ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % le_degree
% 0.07/0.32 thf(fact_55_poly__0__coeff__0,axiom,
% 0.07/0.32 ! [A: $tType] :
% 0.07/0.32 ( ( comm_semiring_0 @ A )
% 0.07/0.32 => ! [P: poly @ A] :
% 0.07/0.32 ( ( poly2 @ A @ P @ ( zero_zero @ A ) )
% 0.07/0.32 = ( coeff @ A @ P @ ( zero_zero @ nat ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % poly_0_coeff_0
% 0.07/0.32 thf(fact_56_algebraic__def,axiom,
% 0.07/0.32 ! [A: $tType] :
% 0.07/0.32 ( ( field_char_0 @ A )
% 0.07/0.32 => ( ( algebraic @ A )
% 0.07/0.32 = ( ^ [X2: A] :
% 0.07/0.32 ? [P4: poly @ A] :
% 0.07/0.32 ( ! [I3: nat] : ( member @ A @ ( coeff @ A @ P4 @ I3 ) @ ( ring_1_Ints @ A ) )
% 0.07/0.32 & ( P4
% 0.07/0.32 != ( zero_zero @ ( poly @ A ) ) )
% 0.07/0.32 & ( ( poly2 @ A @ P4 @ X2 )
% 0.07/0.32 = ( zero_zero @ A ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % algebraic_def
% 0.07/0.32 thf(fact_57_algebraicI,axiom,
% 0.07/0.32 ! [A: $tType] :
% 0.07/0.32 ( ( field_char_0 @ A )
% 0.07/0.32 => ! [P: poly @ A,X: A] :
% 0.07/0.32 ( ! [I4: nat] : ( member @ A @ ( coeff @ A @ P @ I4 ) @ ( ring_1_Ints @ A ) )
% 0.07/0.32 => ( ( P
% 0.07/0.32 != ( zero_zero @ ( poly @ A ) ) )
% 0.07/0.32 => ( ( ( poly2 @ A @ P @ X )
% 0.07/0.32 = ( zero_zero @ A ) )
% 0.07/0.32 => ( algebraic @ A @ X ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % algebraicI
% 0.07/0.32 thf(fact_58_algebraicE,axiom,
% 0.07/0.32 ! [A: $tType] :
% 0.07/0.32 ( ( field_char_0 @ A )
% 0.07/0.32 => ! [X: A] :
% 0.07/0.32 ( ( algebraic @ A @ X )
% 0.07/0.32 => ~ ! [P2: poly @ A] :
% 0.07/0.32 ( ! [I2: nat] : ( member @ A @ ( coeff @ A @ P2 @ I2 ) @ ( ring_1_Ints @ A ) )
% 0.07/0.32 => ( ( P2
% 0.07/0.32 != ( zero_zero @ ( poly @ A ) ) )
% 0.07/0.32 => ( ( poly2 @ A @ P2 @ X )
% 0.07/0.32 != ( zero_zero @ A ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % algebraicE
% 0.07/0.32 thf(fact_59_of__int__pos,axiom,
% 0.07/0.32 ! [A: $tType] :
% 0.07/0.32 ( ( linordered_idom @ A )
% 0.07/0.32 => ! [Z: int] :
% 0.07/0.32 ( ( ord_less @ int @ ( zero_zero @ int ) @ Z )
% 0.07/0.32 => ( ord_less @ A @ ( zero_zero @ A ) @ ( ring_1_of_int @ A @ Z ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % of_int_pos
% 0.07/0.32 thf(fact_60_leading__coeff__neq__0,axiom,
% 0.07/0.32 ! [A: $tType] :
% 0.07/0.32 ( ( zero @ A )
% 0.07/0.32 => ! [P: poly @ A] :
% 0.07/0.32 ( ( P
% 0.07/0.32 != ( zero_zero @ ( poly @ A ) ) )
% 0.07/0.32 => ( ( coeff @ A @ P @ ( degree @ A @ P ) )
% 0.07/0.32 != ( zero_zero @ A ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % leading_coeff_neq_0
% 0.07/0.32 thf(fact_61_poly__all__0__iff__0,axiom,
% 0.07/0.32 ! [A: $tType] :
% 0.07/0.32 ( ( ( ring_char_0 @ A )
% 0.07/0.32 & ( comm_ring_1 @ A )
% 0.07/0.32 & ( ring_n68954251visors @ A ) )
% 0.07/0.32 => ! [P: poly @ A] :
% 0.07/0.32 ( ( ! [X2: A] :
% 0.07/0.32 ( ( poly2 @ A @ P @ X2 )
% 0.07/0.32 = ( zero_zero @ A ) ) )
% 0.07/0.32 = ( P
% 0.07/0.32 = ( zero_zero @ ( poly @ A ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % poly_all_0_iff_0
% 0.07/0.32 thf(fact_62_Ints__1,axiom,
% 0.07/0.32 ! [A: $tType] :
% 0.07/0.32 ( ( ring_1 @ A )
% 0.07/0.32 => ( member @ A @ ( one_one @ A ) @ ( ring_1_Ints @ A ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % Ints_1
% 0.07/0.32 thf(fact_63_Ints__power,axiom,
% 0.07/0.32 ! [A: $tType] :
% 0.07/0.32 ( ( ring_1 @ A )
% 0.07/0.32 => ! [A2: A,N: nat] :
% 0.07/0.32 ( ( member @ A @ A2 @ ( ring_1_Ints @ A ) )
% 0.07/0.32 => ( member @ A @ ( power_power @ A @ A2 @ N ) @ ( ring_1_Ints @ A ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % Ints_power
% 0.07/0.32 thf(fact_64_Ints__abs,axiom,
% 0.07/0.32 ! [A: $tType] :
% 0.07/0.32 ( ( linordered_idom @ A )
% 0.07/0.32 => ! [A2: A] :
% 0.07/0.32 ( ( member @ A @ A2 @ ( ring_1_Ints @ A ) )
% 0.07/0.32 => ( member @ A @ ( abs_abs @ A @ A2 ) @ ( ring_1_Ints @ A ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % Ints_abs
% 0.07/0.32 thf(fact_65_Ints__of__int,axiom,
% 0.07/0.32 ! [A: $tType] :
% 0.07/0.32 ( ( ring_1 @ A )
% 0.07/0.32 => ! [Z: int] : ( member @ A @ ( ring_1_of_int @ A @ Z ) @ ( ring_1_Ints @ A ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % Ints_of_int
% 0.07/0.32 thf(fact_66_Ints__induct,axiom,
% 0.07/0.32 ! [A: $tType] :
% 0.07/0.32 ( ( ring_1 @ A )
% 0.07/0.32 => ! [Q: A,P3: A > $o] :
% 0.07/0.32 ( ( member @ A @ Q @ ( ring_1_Ints @ A ) )
% 0.07/0.32 => ( ! [Z3: int] : ( P3 @ ( ring_1_of_int @ A @ Z3 ) )
% 0.07/0.32 => ( P3 @ Q ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % Ints_induct
% 0.07/0.32 thf(fact_67_Ints__cases,axiom,
% 0.07/0.32 ! [A: $tType] :
% 0.07/0.32 ( ( ring_1 @ A )
% 0.07/0.32 => ! [Q: A] :
% 0.07/0.32 ( ( member @ A @ Q @ ( ring_1_Ints @ A ) )
% 0.07/0.32 => ~ ! [Z3: int] :
% 0.07/0.32 ( Q
% 0.07/0.32 != ( ring_1_of_int @ A @ Z3 ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % Ints_cases
% 0.07/0.32 thf(fact_68_Ints__nonzero__abs__less1,axiom,
% 0.07/0.32 ! [A: $tType] :
% 0.07/0.32 ( ( linordered_idom @ A )
% 0.07/0.32 => ! [X: A] :
% 0.07/0.32 ( ( member @ A @ X @ ( ring_1_Ints @ A ) )
% 0.07/0.32 => ( ( ord_less @ A @ ( abs_abs @ A @ X ) @ ( one_one @ A ) )
% 0.07/0.32 => ( X
% 0.07/0.32 = ( zero_zero @ A ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % Ints_nonzero_abs_less1
% 0.07/0.32 thf(fact_69_of__int__nonneg,axiom,
% 0.07/0.32 ! [A: $tType] :
% 0.07/0.32 ( ( linordered_idom @ A )
% 0.07/0.32 => ! [Z: int] :
% 0.07/0.32 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z )
% 0.07/0.32 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( ring_1_of_int @ A @ Z ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % of_int_nonneg
% 0.07/0.32 thf(fact_70_lead__coeff__power,axiom,
% 0.07/0.32 ! [A: $tType] :
% 0.07/0.32 ( ( ( comm_semiring_1 @ A )
% 0.07/0.32 & ( semiri1193490041visors @ A ) )
% 0.07/0.32 => ! [P: poly @ A,N: nat] :
% 0.07/0.32 ( ( coeff @ A @ ( power_power @ ( poly @ A ) @ P @ N ) @ ( degree @ A @ ( power_power @ ( poly @ A ) @ P @ N ) ) )
% 0.07/0.32 = ( power_power @ A @ ( coeff @ A @ P @ ( degree @ A @ P ) ) @ N ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % lead_coeff_power
% 0.07/0.32 thf(fact_71_Ints__nonzero__abs__ge1,axiom,
% 0.07/0.32 ! [A: $tType] :
% 0.07/0.32 ( ( linordered_idom @ A )
% 0.07/0.32 => ! [X: A] :
% 0.07/0.32 ( ( member @ A @ X @ ( ring_1_Ints @ A ) )
% 0.07/0.32 => ( ( X
% 0.07/0.32 != ( zero_zero @ A ) )
% 0.07/0.32 => ( ord_less_eq @ A @ ( one_one @ A ) @ ( abs_abs @ A @ X ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % Ints_nonzero_abs_ge1
% 0.07/0.32 thf(fact_72_poly__eq__poly__eq__iff,axiom,
% 0.07/0.32 ! [A: $tType] :
% 0.07/0.32 ( ( ( ring_char_0 @ A )
% 0.07/0.32 & ( comm_ring_1 @ A )
% 0.07/0.32 & ( ring_n68954251visors @ A ) )
% 0.07/0.32 => ! [P: poly @ A,Q: poly @ A] :
% 0.07/0.32 ( ( ( poly2 @ A @ P )
% 0.07/0.32 = ( poly2 @ A @ Q ) )
% 0.07/0.32 = ( P = Q ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % poly_eq_poly_eq_iff
% 0.07/0.32 thf(fact_73_of__int__lessD,axiom,
% 0.07/0.32 ! [A: $tType] :
% 0.07/0.32 ( ( linordered_idom @ A )
% 0.07/0.32 => ! [N: int,X: A] :
% 0.07/0.32 ( ( ord_less @ A @ ( abs_abs @ A @ ( ring_1_of_int @ A @ N ) ) @ X )
% 0.07/0.32 => ( ( N
% 0.07/0.32 = ( zero_zero @ int ) )
% 0.07/0.32 | ( ord_less @ A @ ( one_one @ A ) @ X ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % of_int_lessD
% 0.07/0.32 thf(fact_74_of__int__leD,axiom,
% 0.07/0.32 ! [A: $tType] :
% 0.07/0.32 ( ( linordered_idom @ A )
% 0.07/0.32 => ! [N: int,X: A] :
% 0.07/0.32 ( ( ord_less_eq @ A @ ( abs_abs @ A @ ( ring_1_of_int @ A @ N ) ) @ X )
% 0.07/0.32 => ( ( N
% 0.07/0.32 = ( zero_zero @ int ) )
% 0.07/0.32 | ( ord_less_eq @ A @ ( one_one @ A ) @ X ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % of_int_leD
% 0.07/0.32 thf(fact_75_int__poly__rat__no__root__ge,axiom,
% 0.07/0.32 ! [P: poly @ real,B: int,A2: int] :
% 0.07/0.32 ( ! [N2: nat] : ( member @ real @ ( coeff @ real @ P @ N2 ) @ ( ring_1_Ints @ real ) )
% 0.07/0.32 => ( ( ord_less @ int @ ( zero_zero @ int ) @ B )
% 0.07/0.32 => ( ( ( poly2 @ real @ P @ ( divide_divide @ real @ ( ring_1_of_int @ real @ A2 ) @ ( ring_1_of_int @ real @ B ) ) )
% 0.07/0.32 != ( zero_zero @ real ) )
% 0.07/0.32 => ( ord_less_eq @ real @ ( divide_divide @ real @ ( one_one @ real ) @ ( power_power @ real @ ( ring_1_of_int @ real @ B ) @ ( degree @ real @ P ) ) ) @ ( abs_abs @ real @ ( poly2 @ real @ P @ ( divide_divide @ real @ ( ring_1_of_int @ real @ A2 ) @ ( ring_1_of_int @ real @ B ) ) ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % int_poly_rat_no_root_ge
% 0.07/0.32 thf(fact_76_power__decreasing__iff,axiom,
% 0.07/0.32 ! [A: $tType] :
% 0.07/0.32 ( ( linordered_semidom @ A )
% 0.07/0.32 => ! [B: A,M: nat,N: nat] :
% 0.07/0.32 ( ( ord_less @ A @ ( zero_zero @ A ) @ B )
% 0.07/0.32 => ( ( ord_less @ A @ B @ ( one_one @ A ) )
% 0.07/0.32 => ( ( ord_less_eq @ A @ ( power_power @ A @ B @ M ) @ ( power_power @ A @ B @ N ) )
% 0.07/0.32 = ( ord_less_eq @ nat @ N @ M ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % power_decreasing_iff
% 0.07/0.32 thf(fact_77_le__divide__eq__1__pos,axiom,
% 0.07/0.32 ! [A: $tType] :
% 0.07/0.32 ( ( linordered_field @ A )
% 0.07/0.32 => ! [A2: A,B: A] :
% 0.07/0.32 ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
% 0.07/0.32 => ( ( ord_less_eq @ A @ ( one_one @ A ) @ ( divide_divide @ A @ B @ A2 ) )
% 0.07/0.32 = ( ord_less_eq @ A @ A2 @ B ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % le_divide_eq_1_pos
% 0.07/0.32 thf(fact_78_le__divide__eq__1__neg,axiom,
% 0.07/0.32 ! [A: $tType] :
% 0.07/0.32 ( ( linordered_field @ A )
% 0.07/0.32 => ! [A2: A,B: A] :
% 0.07/0.32 ( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
% 0.07/0.32 => ( ( ord_less_eq @ A @ ( one_one @ A ) @ ( divide_divide @ A @ B @ A2 ) )
% 0.07/0.32 = ( ord_less_eq @ A @ B @ A2 ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % le_divide_eq_1_neg
% 0.07/0.32 thf(fact_79_divide__le__eq__1__pos,axiom,
% 0.07/0.32 ! [A: $tType] :
% 0.07/0.32 ( ( linordered_field @ A )
% 0.07/0.32 => ! [A2: A,B: A] :
% 0.07/0.32 ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
% 0.07/0.32 => ( ( ord_less_eq @ A @ ( divide_divide @ A @ B @ A2 ) @ ( one_one @ A ) )
% 0.07/0.32 = ( ord_less_eq @ A @ B @ A2 ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % divide_le_eq_1_pos
% 0.07/0.32 thf(fact_80_divide__le__eq__1__neg,axiom,
% 0.07/0.32 ! [A: $tType] :
% 0.07/0.32 ( ( linordered_field @ A )
% 0.07/0.32 => ! [A2: A,B: A] :
% 0.07/0.32 ( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
% 0.07/0.32 => ( ( ord_less_eq @ A @ ( divide_divide @ A @ B @ A2 ) @ ( one_one @ A ) )
% 0.07/0.32 = ( ord_less_eq @ A @ A2 @ B ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % divide_le_eq_1_neg
% 0.07/0.32 thf(fact_81__092_060open_062A_A_P_Areal__of__int_Ab_A_094_An_A_092_060le_062_AA_A_P_A1_092_060close_062,axiom,
% 0.07/0.32 ord_less_eq @ real @ ( divide_divide @ real @ a @ ( power_power @ real @ ( ring_1_of_int @ real @ b ) @ n ) ) @ ( divide_divide @ real @ a @ ( one_one @ real ) ) ).
% 0.07/0.32
% 0.07/0.32 % \<open>A / real_of_int b ^ n \<le> A / 1\<close>
% 0.07/0.32 thf(fact_82_zero__less__power__abs__iff,axiom,
% 0.07/0.32 ! [A: $tType] :
% 0.07/0.32 ( ( linordered_idom @ A )
% 0.07/0.32 => ! [A2: A,N: nat] :
% 0.07/0.32 ( ( ord_less @ A @ ( zero_zero @ A ) @ ( power_power @ A @ ( abs_abs @ A @ A2 ) @ N ) )
% 0.07/0.32 = ( ( A2
% 0.07/0.32 != ( zero_zero @ A ) )
% 0.07/0.32 | ( N
% 0.07/0.32 = ( zero_zero @ nat ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % zero_less_power_abs_iff
% 0.07/0.32 thf(fact_83_zero__le__divide__abs__iff,axiom,
% 0.07/0.32 ! [A: $tType] :
% 0.07/0.32 ( ( linordered_field @ A )
% 0.07/0.32 => ! [A2: A,B: A] :
% 0.07/0.32 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ A2 @ ( abs_abs @ A @ B ) ) )
% 0.07/0.32 = ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
% 0.07/0.32 | ( B
% 0.07/0.32 = ( zero_zero @ A ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % zero_le_divide_abs_iff
% 0.07/0.32 thf(fact_84_divide__le__0__abs__iff,axiom,
% 0.07/0.32 ! [A: $tType] :
% 0.07/0.32 ( ( linordered_field @ A )
% 0.07/0.32 => ! [A2: A,B: A] :
% 0.07/0.32 ( ( ord_less_eq @ A @ ( divide_divide @ A @ A2 @ ( abs_abs @ A @ B ) ) @ ( zero_zero @ A ) )
% 0.07/0.32 = ( ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) )
% 0.07/0.32 | ( B
% 0.07/0.32 = ( zero_zero @ A ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % divide_le_0_abs_iff
% 0.07/0.32 thf(fact_85_A__pos,axiom,
% 0.07/0.32 ord_less @ real @ ( zero_zero @ real ) @ a ).
% 0.07/0.32
% 0.07/0.32 % A_pos
% 0.07/0.32 thf(fact_86_A__less_I1_J,axiom,
% 0.07/0.32 ord_less @ real @ a @ ( one_one @ real ) ).
% 0.07/0.32
% 0.07/0.32 % A_less(1)
% 0.07/0.32 thf(fact_87__092_060open_062A_A_092_060le_062_A1_092_060close_062,axiom,
% 0.07/0.32 ord_less_eq @ real @ a @ ( one_one @ real ) ).
% 0.07/0.32
% 0.07/0.32 % \<open>A \<le> 1\<close>
% 0.07/0.32 thf(fact_88_power__one__right,axiom,
% 0.07/0.32 ! [A: $tType] :
% 0.07/0.32 ( ( monoid_mult @ A )
% 0.07/0.32 => ! [A2: A] :
% 0.07/0.32 ( ( power_power @ A @ A2 @ ( one_one @ nat ) )
% 0.07/0.32 = A2 ) ) ).
% 0.07/0.32
% 0.07/0.32 % power_one_right
% 0.07/0.32 thf(fact_89_nat__zero__less__power__iff,axiom,
% 0.07/0.32 ! [X: nat,N: nat] :
% 0.07/0.32 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( power_power @ nat @ X @ N ) )
% 0.07/0.32 = ( ( ord_less @ nat @ ( zero_zero @ nat ) @ X )
% 0.07/0.32 | ( N
% 0.07/0.32 = ( zero_zero @ nat ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % nat_zero_less_power_iff
% 0.07/0.32 thf(fact_90_divide__eq__0__iff,axiom,
% 0.07/0.32 ! [A: $tType] :
% 0.07/0.32 ( ( field @ A )
% 0.07/0.32 => ! [A2: A,B: A] :
% 0.07/0.32 ( ( ( divide_divide @ A @ A2 @ B )
% 0.07/0.32 = ( zero_zero @ A ) )
% 0.07/0.32 = ( ( A2
% 0.07/0.32 = ( zero_zero @ A ) )
% 0.07/0.32 | ( B
% 0.07/0.32 = ( zero_zero @ A ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % divide_eq_0_iff
% 0.07/0.32 thf(fact_91_divide__cancel__left,axiom,
% 0.07/0.32 ! [A: $tType] :
% 0.07/0.32 ( ( field @ A )
% 0.07/0.32 => ! [C: A,A2: A,B: A] :
% 0.07/0.32 ( ( ( divide_divide @ A @ C @ A2 )
% 0.07/0.32 = ( divide_divide @ A @ C @ B ) )
% 0.07/0.32 = ( ( C
% 0.07/0.32 = ( zero_zero @ A ) )
% 0.07/0.32 | ( A2 = B ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % divide_cancel_left
% 0.07/0.32 thf(fact_92_divide__cancel__right,axiom,
% 0.07/0.32 ! [A: $tType] :
% 0.07/0.32 ( ( field @ A )
% 0.07/0.32 => ! [A2: A,C: A,B: A] :
% 0.07/0.32 ( ( ( divide_divide @ A @ A2 @ C )
% 0.07/0.32 = ( divide_divide @ A @ B @ C ) )
% 0.07/0.32 = ( ( C
% 0.07/0.32 = ( zero_zero @ A ) )
% 0.07/0.32 | ( A2 = B ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % divide_cancel_right
% 0.07/0.32 thf(fact_93_division__ring__divide__zero,axiom,
% 0.07/0.32 ! [A: $tType] :
% 0.07/0.32 ( ( division_ring @ A )
% 0.07/0.32 => ! [A2: A] :
% 0.07/0.32 ( ( divide_divide @ A @ A2 @ ( zero_zero @ A ) )
% 0.07/0.32 = ( zero_zero @ A ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % division_ring_divide_zero
% 0.07/0.32 thf(fact_94_power__one,axiom,
% 0.07/0.32 ! [A: $tType] :
% 0.07/0.32 ( ( monoid_mult @ A )
% 0.07/0.32 => ! [N: nat] :
% 0.07/0.32 ( ( power_power @ A @ ( one_one @ A ) @ N )
% 0.07/0.32 = ( one_one @ A ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % power_one
% 0.07/0.32 thf(fact_95_abs__divide,axiom,
% 0.07/0.32 ! [A: $tType] :
% 0.07/0.32 ( ( field_abs_sgn @ A )
% 0.07/0.32 => ! [A2: A,B: A] :
% 0.07/0.32 ( ( abs_abs @ A @ ( divide_divide @ A @ A2 @ B ) )
% 0.07/0.32 = ( divide_divide @ A @ ( abs_abs @ A @ A2 ) @ ( abs_abs @ A @ B ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % abs_divide
% 0.07/0.32 thf(fact_96_divide__eq__1__iff,axiom,
% 0.07/0.32 ! [A: $tType] :
% 0.07/0.32 ( ( field @ A )
% 0.07/0.32 => ! [A2: A,B: A] :
% 0.07/0.32 ( ( ( divide_divide @ A @ A2 @ B )
% 0.07/0.32 = ( one_one @ A ) )
% 0.07/0.32 = ( ( B
% 0.07/0.32 != ( zero_zero @ A ) )
% 0.07/0.32 & ( A2 = B ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % divide_eq_1_iff
% 0.07/0.32 thf(fact_97_one__eq__divide__iff,axiom,
% 0.07/0.32 ! [A: $tType] :
% 0.07/0.32 ( ( field @ A )
% 0.07/0.32 => ! [A2: A,B: A] :
% 0.07/0.32 ( ( ( one_one @ A )
% 0.07/0.32 = ( divide_divide @ A @ A2 @ B ) )
% 0.07/0.32 = ( ( B
% 0.07/0.32 != ( zero_zero @ A ) )
% 0.07/0.32 & ( A2 = B ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % one_eq_divide_iff
% 0.07/0.32 thf(fact_98_divide__self,axiom,
% 0.07/0.32 ! [A: $tType] :
% 0.07/0.32 ( ( division_ring @ A )
% 0.07/0.32 => ! [A2: A] :
% 0.07/0.32 ( ( A2
% 0.07/0.32 != ( zero_zero @ A ) )
% 0.07/0.32 => ( ( divide_divide @ A @ A2 @ A2 )
% 0.07/0.32 = ( one_one @ A ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % divide_self
% 0.07/0.32 thf(fact_99_divide__self__if,axiom,
% 0.07/0.32 ! [A: $tType] :
% 0.07/0.32 ( ( division_ring @ A )
% 0.07/0.32 => ! [A2: A] :
% 0.07/0.32 ( ( ( A2
% 0.07/0.32 = ( zero_zero @ A ) )
% 0.07/0.32 => ( ( divide_divide @ A @ A2 @ A2 )
% 0.07/0.32 = ( zero_zero @ A ) ) )
% 0.07/0.32 & ( ( A2
% 0.07/0.32 != ( zero_zero @ A ) )
% 0.07/0.32 => ( ( divide_divide @ A @ A2 @ A2 )
% 0.07/0.32 = ( one_one @ A ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % divide_self_if
% 0.07/0.32 thf(fact_100_divide__eq__eq__1,axiom,
% 0.07/0.32 ! [A: $tType] :
% 0.07/0.32 ( ( linordered_field @ A )
% 0.07/0.32 => ! [B: A,A2: A] :
% 0.07/0.32 ( ( ( divide_divide @ A @ B @ A2 )
% 0.07/0.32 = ( one_one @ A ) )
% 0.07/0.32 = ( ( A2
% 0.07/0.32 != ( zero_zero @ A ) )
% 0.07/0.32 & ( A2 = B ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % divide_eq_eq_1
% 0.07/0.32 thf(fact_101_eq__divide__eq__1,axiom,
% 0.07/0.32 ! [A: $tType] :
% 0.07/0.32 ( ( linordered_field @ A )
% 0.07/0.32 => ! [B: A,A2: A] :
% 0.07/0.32 ( ( ( one_one @ A )
% 0.07/0.32 = ( divide_divide @ A @ B @ A2 ) )
% 0.07/0.32 = ( ( A2
% 0.07/0.32 != ( zero_zero @ A ) )
% 0.07/0.32 & ( A2 = B ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % eq_divide_eq_1
% 0.07/0.32 thf(fact_102_one__divide__eq__0__iff,axiom,
% 0.07/0.32 ! [A: $tType] :
% 0.07/0.32 ( ( linordered_field @ A )
% 0.07/0.32 => ! [A2: A] :
% 0.07/0.32 ( ( ( divide_divide @ A @ ( one_one @ A ) @ A2 )
% 0.07/0.32 = ( zero_zero @ A ) )
% 0.07/0.32 = ( A2
% 0.07/0.32 = ( zero_zero @ A ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % one_divide_eq_0_iff
% 0.07/0.32 thf(fact_103_zero__eq__1__divide__iff,axiom,
% 0.07/0.32 ! [A: $tType] :
% 0.07/0.32 ( ( linordered_field @ A )
% 0.07/0.32 => ! [A2: A] :
% 0.07/0.32 ( ( ( zero_zero @ A )
% 0.07/0.32 = ( divide_divide @ A @ ( one_one @ A ) @ A2 ) )
% 0.07/0.32 = ( A2
% 0.07/0.32 = ( zero_zero @ A ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % zero_eq_1_divide_iff
% 0.07/0.32 thf(fact_104_power__inject__exp,axiom,
% 0.07/0.32 ! [A: $tType] :
% 0.07/0.32 ( ( linordered_semidom @ A )
% 0.07/0.32 => ! [A2: A,M: nat,N: nat] :
% 0.07/0.32 ( ( ord_less @ A @ ( one_one @ A ) @ A2 )
% 0.07/0.32 => ( ( ( power_power @ A @ A2 @ M )
% 0.07/0.32 = ( power_power @ A @ A2 @ N ) )
% 0.07/0.32 = ( M = N ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % power_inject_exp
% 0.07/0.32 thf(fact_105_power__strict__increasing__iff,axiom,
% 0.07/0.32 ! [A: $tType] :
% 0.07/0.32 ( ( linordered_semidom @ A )
% 0.07/0.32 => ! [B: A,X: nat,Y2: nat] :
% 0.07/0.32 ( ( ord_less @ A @ ( one_one @ A ) @ B )
% 0.07/0.32 => ( ( ord_less @ A @ ( power_power @ A @ B @ X ) @ ( power_power @ A @ B @ Y2 ) )
% 0.07/0.32 = ( ord_less @ nat @ X @ Y2 ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % power_strict_increasing_iff
% 0.07/0.32 thf(fact_106_power__eq__0__iff,axiom,
% 0.07/0.32 ! [A: $tType] :
% 0.07/0.32 ( ( semiri134348788visors @ A )
% 0.07/0.32 => ! [A2: A,N: nat] :
% 0.07/0.32 ( ( ( power_power @ A @ A2 @ N )
% 0.07/0.32 = ( zero_zero @ A ) )
% 0.07/0.32 = ( ( A2
% 0.07/0.32 = ( zero_zero @ A ) )
% 0.07/0.32 & ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % power_eq_0_iff
% 0.07/0.32 thf(fact_107_degree__0,axiom,
% 0.07/0.32 ! [A: $tType] :
% 0.07/0.32 ( ( zero @ A )
% 0.07/0.32 => ( ( degree @ A @ ( zero_zero @ ( poly @ A ) ) )
% 0.07/0.32 = ( zero_zero @ nat ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % degree_0
% 0.07/0.32 thf(fact_108_degree__1,axiom,
% 0.07/0.32 ! [A: $tType] :
% 0.07/0.32 ( ( comm_semiring_1 @ A )
% 0.07/0.32 => ( ( degree @ A @ ( one_one @ ( poly @ A ) ) )
% 0.07/0.32 = ( zero_zero @ nat ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % degree_1
% 0.07/0.32 thf(fact_109_degree__of__int,axiom,
% 0.07/0.32 ! [A: $tType] :
% 0.07/0.32 ( ( comm_ring_1 @ A )
% 0.07/0.32 => ! [K: int] :
% 0.07/0.32 ( ( degree @ A @ ( ring_1_of_int @ ( poly @ A ) @ K ) )
% 0.07/0.32 = ( zero_zero @ nat ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % degree_of_int
% 0.07/0.32 thf(fact_110_divide__le__0__1__iff,axiom,
% 0.07/0.32 ! [A: $tType] :
% 0.07/0.32 ( ( linordered_field @ A )
% 0.07/0.32 => ! [A2: A] :
% 0.07/0.32 ( ( ord_less_eq @ A @ ( divide_divide @ A @ ( one_one @ A ) @ A2 ) @ ( zero_zero @ A ) )
% 0.07/0.32 = ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % divide_le_0_1_iff
% 0.07/0.32 thf(fact_111_zero__le__divide__1__iff,axiom,
% 0.07/0.32 ! [A: $tType] :
% 0.07/0.32 ( ( linordered_field @ A )
% 0.07/0.32 => ! [A2: A] :
% 0.07/0.32 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ ( one_one @ A ) @ A2 ) )
% 0.07/0.32 = ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % zero_le_divide_1_iff
% 0.07/0.32 thf(fact_112_divide__less__0__1__iff,axiom,
% 0.07/0.32 ! [A: $tType] :
% 0.07/0.32 ( ( linordered_field @ A )
% 0.07/0.32 => ! [A2: A] :
% 0.07/0.32 ( ( ord_less @ A @ ( divide_divide @ A @ ( one_one @ A ) @ A2 ) @ ( zero_zero @ A ) )
% 0.07/0.32 = ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % divide_less_0_1_iff
% 0.07/0.32 thf(fact_113_divide__less__eq__1__neg,axiom,
% 0.07/0.32 ! [A: $tType] :
% 0.07/0.32 ( ( linordered_field @ A )
% 0.07/0.32 => ! [A2: A,B: A] :
% 0.07/0.32 ( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
% 0.07/0.32 => ( ( ord_less @ A @ ( divide_divide @ A @ B @ A2 ) @ ( one_one @ A ) )
% 0.07/0.32 = ( ord_less @ A @ A2 @ B ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % divide_less_eq_1_neg
% 0.07/0.32 thf(fact_114_divide__less__eq__1__pos,axiom,
% 0.07/0.32 ! [A: $tType] :
% 0.07/0.32 ( ( linordered_field @ A )
% 0.07/0.32 => ! [A2: A,B: A] :
% 0.07/0.32 ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
% 0.07/0.32 => ( ( ord_less @ A @ ( divide_divide @ A @ B @ A2 ) @ ( one_one @ A ) )
% 0.07/0.32 = ( ord_less @ A @ B @ A2 ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % divide_less_eq_1_pos
% 0.07/0.32 thf(fact_115_less__divide__eq__1__neg,axiom,
% 0.07/0.32 ! [A: $tType] :
% 0.07/0.32 ( ( linordered_field @ A )
% 0.07/0.32 => ! [A2: A,B: A] :
% 0.07/0.32 ( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
% 0.07/0.32 => ( ( ord_less @ A @ ( one_one @ A ) @ ( divide_divide @ A @ B @ A2 ) )
% 0.07/0.32 = ( ord_less @ A @ B @ A2 ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % less_divide_eq_1_neg
% 0.07/0.32 thf(fact_116_less__divide__eq__1__pos,axiom,
% 0.07/0.32 ! [A: $tType] :
% 0.07/0.32 ( ( linordered_field @ A )
% 0.07/0.32 => ! [A2: A,B: A] :
% 0.07/0.32 ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
% 0.07/0.32 => ( ( ord_less @ A @ ( one_one @ A ) @ ( divide_divide @ A @ B @ A2 ) )
% 0.07/0.32 = ( ord_less @ A @ A2 @ B ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % less_divide_eq_1_pos
% 0.07/0.32 thf(fact_117_zero__less__divide__1__iff,axiom,
% 0.07/0.32 ! [A: $tType] :
% 0.07/0.32 ( ( linordered_field @ A )
% 0.07/0.32 => ! [A2: A] :
% 0.07/0.32 ( ( ord_less @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ ( one_one @ A ) @ A2 ) )
% 0.07/0.32 = ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % zero_less_divide_1_iff
% 0.07/0.32 thf(fact_118_power__strict__decreasing__iff,axiom,
% 0.07/0.32 ! [A: $tType] :
% 0.07/0.32 ( ( linordered_semidom @ A )
% 0.07/0.32 => ! [B: A,M: nat,N: nat] :
% 0.07/0.32 ( ( ord_less @ A @ ( zero_zero @ A ) @ B )
% 0.07/0.32 => ( ( ord_less @ A @ B @ ( one_one @ A ) )
% 0.07/0.32 => ( ( ord_less @ A @ ( power_power @ A @ B @ M ) @ ( power_power @ A @ B @ N ) )
% 0.07/0.32 = ( ord_less @ nat @ N @ M ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % power_strict_decreasing_iff
% 0.07/0.32 thf(fact_119_power__mono__iff,axiom,
% 0.07/0.32 ! [A: $tType] :
% 0.07/0.32 ( ( linordered_semidom @ A )
% 0.07/0.32 => ! [A2: A,B: A,N: nat] :
% 0.07/0.32 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
% 0.07/0.32 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B )
% 0.07/0.32 => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 0.07/0.32 => ( ( ord_less_eq @ A @ ( power_power @ A @ A2 @ N ) @ ( power_power @ A @ B @ N ) )
% 0.07/0.32 = ( ord_less_eq @ A @ A2 @ B ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % power_mono_iff
% 0.07/0.32 thf(fact_120_power__increasing__iff,axiom,
% 0.07/0.32 ! [A: $tType] :
% 0.07/0.32 ( ( linordered_semidom @ A )
% 0.07/0.32 => ! [B: A,X: nat,Y2: nat] :
% 0.07/0.32 ( ( ord_less @ A @ ( one_one @ A ) @ B )
% 0.07/0.32 => ( ( ord_less_eq @ A @ ( power_power @ A @ B @ X ) @ ( power_power @ A @ B @ Y2 ) )
% 0.07/0.32 = ( ord_less_eq @ nat @ X @ Y2 ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % power_increasing_iff
% 0.07/0.32 thf(fact_121__092_060open_062_092_060not_062_AA_A_P_Areal__of__int_Ab_A_094_An_A_060_A_092_060bar_062x_A_N_Areal__of__int_Aa_A_P_Areal__of__int_Ab_092_060bar_062_092_060close_062,axiom,
% 0.07/0.32 ~ ( ord_less @ real @ ( divide_divide @ real @ a @ ( power_power @ real @ ( ring_1_of_int @ real @ b ) @ n ) ) @ ( abs_abs @ real @ ( minus_minus @ real @ x @ ( divide_divide @ real @ ( ring_1_of_int @ real @ a2 ) @ ( ring_1_of_int @ real @ b ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % \<open>\<not> A / real_of_int b ^ n < \<bar>x - real_of_int a / real_of_int b\<bar>\<close>
% 0.07/0.32 thf(fact_122_nat__power__less__imp__less,axiom,
% 0.07/0.32 ! [I: nat,M: nat,N: nat] :
% 0.07/0.32 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ I )
% 0.07/0.32 => ( ( ord_less @ nat @ ( power_power @ nat @ I @ M ) @ ( power_power @ nat @ I @ N ) )
% 0.07/0.32 => ( ord_less @ nat @ M @ N ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % nat_power_less_imp_less
% 0.07/0.32 thf(fact_123_div__poly__less,axiom,
% 0.07/0.32 ! [A: $tType] :
% 0.07/0.32 ( ( field @ A )
% 0.07/0.32 => ! [X: poly @ A,Y2: poly @ A] :
% 0.07/0.32 ( ( ord_less @ nat @ ( degree @ A @ X ) @ ( degree @ A @ Y2 ) )
% 0.07/0.32 => ( ( divide_divide @ ( poly @ A ) @ X @ Y2 )
% 0.07/0.32 = ( zero_zero @ ( poly @ A ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % div_poly_less
% 0.07/0.32 thf(fact_124_self__le__power,axiom,
% 0.07/0.32 ! [A: $tType] :
% 0.07/0.32 ( ( linordered_semidom @ A )
% 0.07/0.32 => ! [A2: A,N: nat] :
% 0.07/0.32 ( ( ord_less_eq @ A @ ( one_one @ A ) @ A2 )
% 0.07/0.32 => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 0.07/0.32 => ( ord_less_eq @ A @ A2 @ ( power_power @ A @ A2 @ N ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % self_le_power
% 0.07/0.32 thf(fact_125_divide__poly__0,axiom,
% 0.07/0.32 ! [A: $tType] :
% 0.07/0.32 ( ( idom_divide @ A )
% 0.07/0.32 => ! [F: poly @ A] :
% 0.07/0.32 ( ( divide_divide @ ( poly @ A ) @ F @ ( zero_zero @ ( poly @ A ) ) )
% 0.07/0.32 = ( zero_zero @ ( poly @ A ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % divide_poly_0
% 0.07/0.32 thf(fact_126_zero__power,axiom,
% 0.07/0.32 ! [A: $tType] :
% 0.07/0.32 ( ( semiring_1 @ A )
% 0.07/0.32 => ! [N: nat] :
% 0.07/0.32 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 0.07/0.32 => ( ( power_power @ A @ ( zero_zero @ A ) @ N )
% 0.07/0.32 = ( zero_zero @ A ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % zero_power
% 0.07/0.32 thf(fact_127_power__0,axiom,
% 0.07/0.32 ! [A: $tType] :
% 0.07/0.32 ( ( power @ A )
% 0.07/0.32 => ! [A2: A] :
% 0.07/0.32 ( ( power_power @ A @ A2 @ ( zero_zero @ nat ) )
% 0.07/0.32 = ( one_one @ A ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % power_0
% 0.07/0.32 thf(fact_128_power__eq__iff__eq__base,axiom,
% 0.07/0.32 ! [A: $tType] :
% 0.07/0.32 ( ( linordered_semidom @ A )
% 0.07/0.32 => ! [N: nat,A2: A,B: A] :
% 0.07/0.32 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 0.07/0.32 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
% 0.07/0.32 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B )
% 0.07/0.32 => ( ( ( power_power @ A @ A2 @ N )
% 0.07/0.32 = ( power_power @ A @ B @ N ) )
% 0.07/0.32 = ( A2 = B ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % power_eq_iff_eq_base
% 0.07/0.32 thf(fact_129_power__eq__imp__eq__base,axiom,
% 0.07/0.32 ! [A: $tType] :
% 0.07/0.32 ( ( linordered_semidom @ A )
% 0.07/0.32 => ! [A2: A,N: nat,B: A] :
% 0.07/0.32 ( ( ( power_power @ A @ A2 @ N )
% 0.07/0.32 = ( power_power @ A @ B @ N ) )
% 0.07/0.32 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
% 0.07/0.32 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B )
% 0.07/0.32 => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 0.07/0.32 => ( A2 = B ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % power_eq_imp_eq_base
% 0.07/0.32 thf(fact_130_one__less__power,axiom,
% 0.07/0.32 ! [A: $tType] :
% 0.07/0.32 ( ( linordered_semidom @ A )
% 0.07/0.32 => ! [A2: A,N: nat] :
% 0.07/0.32 ( ( ord_less @ A @ ( one_one @ A ) @ A2 )
% 0.07/0.32 => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 0.07/0.32 => ( ord_less @ A @ ( one_one @ A ) @ ( power_power @ A @ A2 @ N ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % one_less_power
% 0.07/0.32 thf(fact_131_poly__IVT__neg,axiom,
% 0.07/0.32 ! [A2: real,B: real,P: poly @ real] :
% 0.07/0.32 ( ( ord_less @ real @ A2 @ B )
% 0.07/0.32 => ( ( ord_less @ real @ ( zero_zero @ real ) @ ( poly2 @ real @ P @ A2 ) )
% 0.07/0.32 => ( ( ord_less @ real @ ( poly2 @ real @ P @ B ) @ ( zero_zero @ real ) )
% 0.07/0.32 => ? [X3: real] :
% 0.07/0.32 ( ( ord_less @ real @ A2 @ X3 )
% 0.07/0.32 & ( ord_less @ real @ X3 @ B )
% 0.07/0.32 & ( ( poly2 @ real @ P @ X3 )
% 0.07/0.32 = ( zero_zero @ real ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % poly_IVT_neg
% 0.07/0.32 thf(fact_132_poly__IVT__pos,axiom,
% 0.07/0.32 ! [A2: real,B: real,P: poly @ real] :
% 0.07/0.32 ( ( ord_less @ real @ A2 @ B )
% 0.07/0.32 => ( ( ord_less @ real @ ( poly2 @ real @ P @ A2 ) @ ( zero_zero @ real ) )
% 0.07/0.32 => ( ( ord_less @ real @ ( zero_zero @ real ) @ ( poly2 @ real @ P @ B ) )
% 0.07/0.32 => ? [X3: real] :
% 0.07/0.32 ( ( ord_less @ real @ A2 @ X3 )
% 0.07/0.32 & ( ord_less @ real @ X3 @ B )
% 0.07/0.32 & ( ( poly2 @ real @ P @ X3 )
% 0.07/0.32 = ( zero_zero @ real ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % poly_IVT_pos
% 0.07/0.32 thf(fact_133_linordered__field__no__ub,axiom,
% 0.07/0.32 ! [A: $tType] :
% 0.07/0.32 ( ( linordered_field @ A )
% 0.07/0.32 => ! [X4: A] :
% 0.07/0.32 ? [X_1: A] : ( ord_less @ A @ X4 @ X_1 ) ) ).
% 0.07/0.32
% 0.07/0.32 % linordered_field_no_ub
% 0.07/0.32 thf(fact_134_linordered__field__no__lb,axiom,
% 0.07/0.32 ! [A: $tType] :
% 0.07/0.32 ( ( linordered_field @ A )
% 0.07/0.32 => ! [X4: A] :
% 0.07/0.32 ? [Y3: A] : ( ord_less @ A @ Y3 @ X4 ) ) ).
% 0.07/0.32
% 0.07/0.32 % linordered_field_no_lb
% 0.07/0.32 thf(fact_135_coeff__eq__0,axiom,
% 0.07/0.32 ! [A: $tType] :
% 0.07/0.32 ( ( zero @ A )
% 0.07/0.32 => ! [P: poly @ A,N: nat] :
% 0.07/0.32 ( ( ord_less @ nat @ ( degree @ A @ P ) @ N )
% 0.07/0.32 => ( ( coeff @ A @ P @ N )
% 0.07/0.32 = ( zero_zero @ A ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % coeff_eq_0
% 0.07/0.32 thf(fact_136_less__degree__imp,axiom,
% 0.07/0.32 ! [A: $tType] :
% 0.07/0.32 ( ( zero @ A )
% 0.07/0.32 => ! [N: nat,P: poly @ A] :
% 0.07/0.32 ( ( ord_less @ nat @ N @ ( degree @ A @ P ) )
% 0.07/0.32 => ? [I4: nat] :
% 0.07/0.32 ( ( ord_less @ nat @ N @ I4 )
% 0.07/0.32 & ( ( coeff @ A @ P @ I4 )
% 0.07/0.32 != ( zero_zero @ A ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % less_degree_imp
% 0.07/0.32 thf(fact_137_degree__le,axiom,
% 0.07/0.32 ! [A: $tType] :
% 0.07/0.32 ( ( zero @ A )
% 0.07/0.32 => ! [N: nat,P: poly @ A] :
% 0.07/0.32 ( ! [I4: nat] :
% 0.07/0.32 ( ( ord_less @ nat @ N @ I4 )
% 0.07/0.32 => ( ( coeff @ A @ P @ I4 )
% 0.07/0.32 = ( zero_zero @ A ) ) )
% 0.07/0.32 => ( ord_less_eq @ nat @ ( degree @ A @ P ) @ N ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % degree_le
% 0.07/0.32 thf(fact_138_Liouville__Numbers__Misc_ORats__cases_H,axiom,
% 0.07/0.32 ! [X: real] :
% 0.07/0.32 ( ( member @ real @ X @ ( field_char_0_Rats @ real ) )
% 0.07/0.32 => ~ ! [P2: int,Q4: int] :
% 0.07/0.32 ( ( ord_less @ int @ ( zero_zero @ int ) @ Q4 )
% 0.07/0.32 => ( X
% 0.07/0.32 != ( divide_divide @ real @ ( ring_1_of_int @ real @ P2 ) @ ( ring_1_of_int @ real @ Q4 ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % Liouville_Numbers_Misc.Rats_cases'
% 0.07/0.32 thf(fact_139_coeff__0__power,axiom,
% 0.07/0.32 ! [A: $tType] :
% 0.07/0.32 ( ( comm_semiring_1 @ A )
% 0.07/0.32 => ! [P: poly @ A,N: nat] :
% 0.07/0.32 ( ( coeff @ A @ ( power_power @ ( poly @ A ) @ P @ N ) @ ( zero_zero @ nat ) )
% 0.07/0.32 = ( power_power @ A @ ( coeff @ A @ P @ ( zero_zero @ nat ) ) @ N ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % coeff_0_power
% 0.07/0.32 thf(fact_140_eq__zero__or__degree__less,axiom,
% 0.07/0.32 ! [A: $tType] :
% 0.07/0.32 ( ( zero @ A )
% 0.07/0.32 => ! [P: poly @ A,N: nat] :
% 0.07/0.32 ( ( ord_less_eq @ nat @ ( degree @ A @ P ) @ N )
% 0.07/0.32 => ( ( ( coeff @ A @ P @ N )
% 0.07/0.32 = ( zero_zero @ A ) )
% 0.07/0.32 => ( ( P
% 0.07/0.32 = ( zero_zero @ ( poly @ A ) ) )
% 0.07/0.32 | ( ord_less @ nat @ ( degree @ A @ P ) @ N ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % eq_zero_or_degree_less
% 0.07/0.32 thf(fact_141_power__not__zero,axiom,
% 0.07/0.32 ! [A: $tType] :
% 0.07/0.32 ( ( semiri134348788visors @ A )
% 0.07/0.32 => ! [A2: A,N: nat] :
% 0.07/0.32 ( ( A2
% 0.07/0.32 != ( zero_zero @ A ) )
% 0.07/0.32 => ( ( power_power @ A @ A2 @ N )
% 0.07/0.32 != ( zero_zero @ A ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % power_not_zero
% 0.07/0.32 thf(fact_142_power__divide,axiom,
% 0.07/0.32 ! [A: $tType] :
% 0.07/0.32 ( ( field @ A )
% 0.07/0.32 => ! [A2: A,B: A,N: nat] :
% 0.07/0.32 ( ( power_power @ A @ ( divide_divide @ A @ A2 @ B ) @ N )
% 0.07/0.32 = ( divide_divide @ A @ ( power_power @ A @ A2 @ N ) @ ( power_power @ A @ B @ N ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % power_divide
% 0.07/0.32 thf(fact_143_poly__pinfty__gt__lc,axiom,
% 0.07/0.32 ! [P: poly @ real] :
% 0.07/0.32 ( ( ord_less @ real @ ( zero_zero @ real ) @ ( coeff @ real @ P @ ( degree @ real @ P ) ) )
% 0.07/0.32 => ? [N2: real] :
% 0.07/0.32 ! [X4: real] :
% 0.07/0.32 ( ( ord_less_eq @ real @ N2 @ X4 )
% 0.07/0.32 => ( ord_less_eq @ real @ ( coeff @ real @ P @ ( degree @ real @ P ) ) @ ( poly2 @ real @ P @ X4 ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % poly_pinfty_gt_lc
% 0.07/0.32 thf(fact_144_power__abs,axiom,
% 0.07/0.32 ! [A: $tType] :
% 0.07/0.32 ( ( linordered_idom @ A )
% 0.07/0.32 => ! [A2: A,N: nat] :
% 0.07/0.32 ( ( abs_abs @ A @ ( power_power @ A @ A2 @ N ) )
% 0.07/0.32 = ( power_power @ A @ ( abs_abs @ A @ A2 ) @ N ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % power_abs
% 0.07/0.32 thf(fact_145_algebraic__altdef,axiom,
% 0.07/0.32 ! [B2: $tType] :
% 0.07/0.32 ( ( field_char_0 @ B2 )
% 0.07/0.32 => ( ( algebraic @ B2 )
% 0.07/0.32 = ( ^ [X2: B2] :
% 0.07/0.32 ? [P4: poly @ B2] :
% 0.07/0.32 ( ! [I3: nat] : ( member @ B2 @ ( coeff @ B2 @ P4 @ I3 ) @ ( field_char_0_Rats @ B2 ) )
% 0.07/0.32 & ( P4
% 0.07/0.32 != ( zero_zero @ ( poly @ B2 ) ) )
% 0.07/0.32 & ( ( poly2 @ B2 @ P4 @ X2 )
% 0.07/0.32 = ( zero_zero @ B2 ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % algebraic_altdef
% 0.07/0.32 thf(fact_146_divide__le__0__iff,axiom,
% 0.07/0.32 ! [A: $tType] :
% 0.07/0.32 ( ( linordered_field @ A )
% 0.07/0.32 => ! [A2: A,B: A] :
% 0.07/0.32 ( ( ord_less_eq @ A @ ( divide_divide @ A @ A2 @ B ) @ ( zero_zero @ A ) )
% 0.07/0.32 = ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
% 0.07/0.32 & ( ord_less_eq @ A @ B @ ( zero_zero @ A ) ) )
% 0.07/0.32 | ( ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) )
% 0.07/0.32 & ( ord_less_eq @ A @ ( zero_zero @ A ) @ B ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % divide_le_0_iff
% 0.07/0.32 thf(fact_147_divide__right__mono,axiom,
% 0.07/0.32 ! [A: $tType] :
% 0.07/0.32 ( ( linordered_field @ A )
% 0.07/0.32 => ! [A2: A,B: A,C: A] :
% 0.07/0.32 ( ( ord_less_eq @ A @ A2 @ B )
% 0.07/0.32 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C )
% 0.07/0.32 => ( ord_less_eq @ A @ ( divide_divide @ A @ A2 @ C ) @ ( divide_divide @ A @ B @ C ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % divide_right_mono
% 0.07/0.32 thf(fact_148_zero__le__divide__iff,axiom,
% 0.07/0.32 ! [A: $tType] :
% 0.07/0.32 ( ( linordered_field @ A )
% 0.07/0.32 => ! [A2: A,B: A] :
% 0.07/0.32 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ A2 @ B ) )
% 0.07/0.32 = ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
% 0.07/0.32 & ( ord_less_eq @ A @ ( zero_zero @ A ) @ B ) )
% 0.07/0.32 | ( ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) )
% 0.07/0.32 & ( ord_less_eq @ A @ B @ ( zero_zero @ A ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % zero_le_divide_iff
% 0.07/0.32 thf(fact_149_divide__nonneg__nonneg,axiom,
% 0.07/0.32 ! [A: $tType] :
% 0.07/0.32 ( ( linordered_field @ A )
% 0.07/0.32 => ! [X: A,Y2: A] :
% 0.07/0.32 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
% 0.07/0.32 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y2 )
% 0.07/0.32 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ X @ Y2 ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % divide_nonneg_nonneg
% 0.07/0.32 thf(fact_150_divide__nonneg__nonpos,axiom,
% 0.07/0.32 ! [A: $tType] :
% 0.07/0.32 ( ( linordered_field @ A )
% 0.07/0.32 => ! [X: A,Y2: A] :
% 0.07/0.32 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
% 0.07/0.32 => ( ( ord_less_eq @ A @ Y2 @ ( zero_zero @ A ) )
% 0.07/0.32 => ( ord_less_eq @ A @ ( divide_divide @ A @ X @ Y2 ) @ ( zero_zero @ A ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % divide_nonneg_nonpos
% 0.07/0.32 thf(fact_151_divide__nonpos__nonneg,axiom,
% 0.07/0.32 ! [A: $tType] :
% 0.07/0.32 ( ( linordered_field @ A )
% 0.07/0.32 => ! [X: A,Y2: A] :
% 0.07/0.32 ( ( ord_less_eq @ A @ X @ ( zero_zero @ A ) )
% 0.07/0.32 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y2 )
% 0.07/0.32 => ( ord_less_eq @ A @ ( divide_divide @ A @ X @ Y2 ) @ ( zero_zero @ A ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % divide_nonpos_nonneg
% 0.07/0.32 thf(fact_152_divide__nonpos__nonpos,axiom,
% 0.07/0.32 ! [A: $tType] :
% 0.07/0.32 ( ( linordered_field @ A )
% 0.07/0.32 => ! [X: A,Y2: A] :
% 0.07/0.32 ( ( ord_less_eq @ A @ X @ ( zero_zero @ A ) )
% 0.07/0.32 => ( ( ord_less_eq @ A @ Y2 @ ( zero_zero @ A ) )
% 0.07/0.32 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ X @ Y2 ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % divide_nonpos_nonpos
% 0.07/0.32 thf(fact_153_divide__right__mono__neg,axiom,
% 0.07/0.32 ! [A: $tType] :
% 0.07/0.32 ( ( linordered_field @ A )
% 0.07/0.32 => ! [A2: A,B: A,C: A] :
% 0.07/0.32 ( ( ord_less_eq @ A @ A2 @ B )
% 0.07/0.32 => ( ( ord_less_eq @ A @ C @ ( zero_zero @ A ) )
% 0.07/0.32 => ( ord_less_eq @ A @ ( divide_divide @ A @ B @ C ) @ ( divide_divide @ A @ A2 @ C ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % divide_right_mono_neg
% 0.07/0.32 thf(fact_154_power__mono,axiom,
% 0.07/0.32 ! [A: $tType] :
% 0.07/0.32 ( ( linordered_semidom @ A )
% 0.07/0.32 => ! [A2: A,B: A,N: nat] :
% 0.07/0.32 ( ( ord_less_eq @ A @ A2 @ B )
% 0.07/0.32 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
% 0.07/0.32 => ( ord_less_eq @ A @ ( power_power @ A @ A2 @ N ) @ ( power_power @ A @ B @ N ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % power_mono
% 0.07/0.32 thf(fact_155_zero__le__power,axiom,
% 0.07/0.32 ! [A: $tType] :
% 0.07/0.32 ( ( linordered_semidom @ A )
% 0.07/0.32 => ! [A2: A,N: nat] :
% 0.07/0.32 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
% 0.07/0.32 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A2 @ N ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % zero_le_power
% 0.07/0.32 thf(fact_156_divide__neg__neg,axiom,
% 0.07/0.32 ! [A: $tType] :
% 0.07/0.32 ( ( linordered_field @ A )
% 0.07/0.32 => ! [X: A,Y2: A] :
% 0.07/0.32 ( ( ord_less @ A @ X @ ( zero_zero @ A ) )
% 0.07/0.32 => ( ( ord_less @ A @ Y2 @ ( zero_zero @ A ) )
% 0.07/0.32 => ( ord_less @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ X @ Y2 ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % divide_neg_neg
% 0.07/0.32 thf(fact_157_divide__neg__pos,axiom,
% 0.07/0.32 ! [A: $tType] :
% 0.07/0.32 ( ( linordered_field @ A )
% 0.07/0.32 => ! [X: A,Y2: A] :
% 0.07/0.32 ( ( ord_less @ A @ X @ ( zero_zero @ A ) )
% 0.07/0.32 => ( ( ord_less @ A @ ( zero_zero @ A ) @ Y2 )
% 0.07/0.32 => ( ord_less @ A @ ( divide_divide @ A @ X @ Y2 ) @ ( zero_zero @ A ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % divide_neg_pos
% 0.07/0.32 thf(fact_158_divide__pos__neg,axiom,
% 0.07/0.32 ! [A: $tType] :
% 0.07/0.32 ( ( linordered_field @ A )
% 0.07/0.32 => ! [X: A,Y2: A] :
% 0.07/0.32 ( ( ord_less @ A @ ( zero_zero @ A ) @ X )
% 0.07/0.32 => ( ( ord_less @ A @ Y2 @ ( zero_zero @ A ) )
% 0.07/0.32 => ( ord_less @ A @ ( divide_divide @ A @ X @ Y2 ) @ ( zero_zero @ A ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % divide_pos_neg
% 0.07/0.32 thf(fact_159_divide__pos__pos,axiom,
% 0.07/0.32 ! [A: $tType] :
% 0.07/0.32 ( ( linordered_field @ A )
% 0.07/0.32 => ! [X: A,Y2: A] :
% 0.07/0.32 ( ( ord_less @ A @ ( zero_zero @ A ) @ X )
% 0.07/0.32 => ( ( ord_less @ A @ ( zero_zero @ A ) @ Y2 )
% 0.07/0.32 => ( ord_less @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ X @ Y2 ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % divide_pos_pos
% 0.07/0.32 thf(fact_160_divide__less__0__iff,axiom,
% 0.07/0.32 ! [A: $tType] :
% 0.07/0.32 ( ( linordered_field @ A )
% 0.07/0.32 => ! [A2: A,B: A] :
% 0.07/0.32 ( ( ord_less @ A @ ( divide_divide @ A @ A2 @ B ) @ ( zero_zero @ A ) )
% 0.07/0.32 = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
% 0.07/0.32 & ( ord_less @ A @ B @ ( zero_zero @ A ) ) )
% 0.07/0.32 | ( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
% 0.07/0.32 & ( ord_less @ A @ ( zero_zero @ A ) @ B ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % divide_less_0_iff
% 0.07/0.32 thf(fact_161_divide__less__cancel,axiom,
% 0.07/0.32 ! [A: $tType] :
% 0.07/0.32 ( ( linordered_field @ A )
% 0.07/0.32 => ! [A2: A,C: A,B: A] :
% 0.07/0.32 ( ( ord_less @ A @ ( divide_divide @ A @ A2 @ C ) @ ( divide_divide @ A @ B @ C ) )
% 0.07/0.32 = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C )
% 0.07/0.32 => ( ord_less @ A @ A2 @ B ) )
% 0.07/0.32 & ( ( ord_less @ A @ C @ ( zero_zero @ A ) )
% 0.07/0.32 => ( ord_less @ A @ B @ A2 ) )
% 0.07/0.32 & ( C
% 0.07/0.32 != ( zero_zero @ A ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % divide_less_cancel
% 0.07/0.32 thf(fact_162_zero__less__divide__iff,axiom,
% 0.07/0.32 ! [A: $tType] :
% 0.07/0.32 ( ( linordered_field @ A )
% 0.07/0.32 => ! [A2: A,B: A] :
% 0.07/0.32 ( ( ord_less @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ A2 @ B ) )
% 0.07/0.32 = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
% 0.07/0.32 & ( ord_less @ A @ ( zero_zero @ A ) @ B ) )
% 0.07/0.32 | ( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
% 0.07/0.32 & ( ord_less @ A @ B @ ( zero_zero @ A ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % zero_less_divide_iff
% 0.07/0.32 thf(fact_163_divide__strict__right__mono,axiom,
% 0.07/0.32 ! [A: $tType] :
% 0.07/0.32 ( ( linordered_field @ A )
% 0.07/0.32 => ! [A2: A,B: A,C: A] :
% 0.07/0.32 ( ( ord_less @ A @ A2 @ B )
% 0.07/0.32 => ( ( ord_less @ A @ ( zero_zero @ A ) @ C )
% 0.07/0.32 => ( ord_less @ A @ ( divide_divide @ A @ A2 @ C ) @ ( divide_divide @ A @ B @ C ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % divide_strict_right_mono
% 0.07/0.32 thf(fact_164_divide__strict__right__mono__neg,axiom,
% 0.07/0.32 ! [A: $tType] :
% 0.07/0.32 ( ( linordered_field @ A )
% 0.07/0.32 => ! [B: A,A2: A,C: A] :
% 0.07/0.32 ( ( ord_less @ A @ B @ A2 )
% 0.07/0.32 => ( ( ord_less @ A @ C @ ( zero_zero @ A ) )
% 0.07/0.32 => ( ord_less @ A @ ( divide_divide @ A @ A2 @ C ) @ ( divide_divide @ A @ B @ C ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % divide_strict_right_mono_neg
% 0.07/0.32 thf(fact_165_zero__less__power,axiom,
% 0.07/0.32 ! [A: $tType] :
% 0.07/0.32 ( ( linordered_semidom @ A )
% 0.07/0.32 => ! [A2: A,N: nat] :
% 0.07/0.32 ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
% 0.07/0.32 => ( ord_less @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A2 @ N ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % zero_less_power
% 0.07/0.32 thf(fact_166_right__inverse__eq,axiom,
% 0.07/0.32 ! [A: $tType] :
% 0.07/0.32 ( ( division_ring @ A )
% 0.07/0.32 => ! [B: A,A2: A] :
% 0.07/0.32 ( ( B
% 0.07/0.32 != ( zero_zero @ A ) )
% 0.07/0.32 => ( ( ( divide_divide @ A @ A2 @ B )
% 0.07/0.32 = ( one_one @ A ) )
% 0.07/0.32 = ( A2 = B ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % right_inverse_eq
% 0.07/0.32 thf(fact_167_power__0__left,axiom,
% 0.07/0.32 ! [A: $tType] :
% 0.07/0.32 ( ( semiring_1 @ A )
% 0.07/0.32 => ! [N: nat] :
% 0.07/0.32 ( ( ( N
% 0.07/0.32 = ( zero_zero @ nat ) )
% 0.07/0.32 => ( ( power_power @ A @ ( zero_zero @ A ) @ N )
% 0.07/0.32 = ( one_one @ A ) ) )
% 0.07/0.32 & ( ( N
% 0.07/0.32 != ( zero_zero @ nat ) )
% 0.07/0.32 => ( ( power_power @ A @ ( zero_zero @ A ) @ N )
% 0.07/0.32 = ( zero_zero @ A ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % power_0_left
% 0.07/0.32 thf(fact_168_one__le__power,axiom,
% 0.07/0.32 ! [A: $tType] :
% 0.07/0.32 ( ( linordered_semidom @ A )
% 0.07/0.32 => ! [A2: A,N: nat] :
% 0.07/0.32 ( ( ord_less_eq @ A @ ( one_one @ A ) @ A2 )
% 0.07/0.32 => ( ord_less_eq @ A @ ( one_one @ A ) @ ( power_power @ A @ A2 @ N ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % one_le_power
% 0.07/0.32 thf(fact_169_power__increasing,axiom,
% 0.07/0.32 ! [A: $tType] :
% 0.07/0.32 ( ( linordered_semidom @ A )
% 0.07/0.32 => ! [N: nat,N4: nat,A2: A] :
% 0.07/0.32 ( ( ord_less_eq @ nat @ N @ N4 )
% 0.07/0.32 => ( ( ord_less_eq @ A @ ( one_one @ A ) @ A2 )
% 0.07/0.32 => ( ord_less_eq @ A @ ( power_power @ A @ A2 @ N ) @ ( power_power @ A @ A2 @ N4 ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % power_increasing
% 0.07/0.32 thf(fact_170_power__less__imp__less__exp,axiom,
% 0.07/0.32 ! [A: $tType] :
% 0.07/0.32 ( ( linordered_semidom @ A )
% 0.07/0.32 => ! [A2: A,M: nat,N: nat] :
% 0.07/0.32 ( ( ord_less @ A @ ( one_one @ A ) @ A2 )
% 0.07/0.32 => ( ( ord_less @ A @ ( power_power @ A @ A2 @ M ) @ ( power_power @ A @ A2 @ N ) )
% 0.07/0.32 => ( ord_less @ nat @ M @ N ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % power_less_imp_less_exp
% 0.07/0.32 thf(fact_171_power__strict__increasing,axiom,
% 0.07/0.32 ! [A: $tType] :
% 0.07/0.32 ( ( linordered_semidom @ A )
% 0.07/0.32 => ! [N: nat,N4: nat,A2: A] :
% 0.07/0.32 ( ( ord_less @ nat @ N @ N4 )
% 0.07/0.32 => ( ( ord_less @ A @ ( one_one @ A ) @ A2 )
% 0.07/0.32 => ( ord_less @ A @ ( power_power @ A @ A2 @ N ) @ ( power_power @ A @ A2 @ N4 ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % power_strict_increasing
% 0.07/0.32 thf(fact_172_power__one__over,axiom,
% 0.07/0.32 ! [A: $tType] :
% 0.07/0.32 ( ( division_ring @ A )
% 0.07/0.32 => ! [A2: A,N: nat] :
% 0.07/0.32 ( ( power_power @ A @ ( divide_divide @ A @ ( one_one @ A ) @ A2 ) @ N )
% 0.07/0.32 = ( divide_divide @ A @ ( one_one @ A ) @ ( power_power @ A @ A2 @ N ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % power_one_over
% 0.07/0.32 thf(fact_173_nonzero__abs__divide,axiom,
% 0.07/0.32 ! [A: $tType] :
% 0.07/0.32 ( ( linordered_field @ A )
% 0.07/0.32 => ! [B: A,A2: A] :
% 0.07/0.32 ( ( B
% 0.07/0.32 != ( zero_zero @ A ) )
% 0.07/0.32 => ( ( abs_abs @ A @ ( divide_divide @ A @ A2 @ B ) )
% 0.07/0.32 = ( divide_divide @ A @ ( abs_abs @ A @ A2 ) @ ( abs_abs @ A @ B ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % nonzero_abs_divide
% 0.07/0.32 thf(fact_174_frac__le,axiom,
% 0.07/0.32 ! [A: $tType] :
% 0.07/0.32 ( ( linordered_field @ A )
% 0.07/0.32 => ! [Y2: A,X: A,W: A,Z: A] :
% 0.07/0.32 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y2 )
% 0.07/0.32 => ( ( ord_less_eq @ A @ X @ Y2 )
% 0.07/0.32 => ( ( ord_less @ A @ ( zero_zero @ A ) @ W )
% 0.07/0.32 => ( ( ord_less_eq @ A @ W @ Z )
% 0.07/0.32 => ( ord_less_eq @ A @ ( divide_divide @ A @ X @ Z ) @ ( divide_divide @ A @ Y2 @ W ) ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % frac_le
% 0.07/0.32 thf(fact_175_frac__less,axiom,
% 0.07/0.32 ! [A: $tType] :
% 0.07/0.32 ( ( linordered_field @ A )
% 0.07/0.32 => ! [X: A,Y2: A,W: A,Z: A] :
% 0.07/0.32 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
% 0.07/0.32 => ( ( ord_less @ A @ X @ Y2 )
% 0.07/0.32 => ( ( ord_less @ A @ ( zero_zero @ A ) @ W )
% 0.07/0.32 => ( ( ord_less_eq @ A @ W @ Z )
% 0.07/0.32 => ( ord_less @ A @ ( divide_divide @ A @ X @ Z ) @ ( divide_divide @ A @ Y2 @ W ) ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % frac_less
% 0.07/0.32 thf(fact_176_frac__less2,axiom,
% 0.07/0.32 ! [A: $tType] :
% 0.07/0.32 ( ( linordered_field @ A )
% 0.07/0.32 => ! [X: A,Y2: A,W: A,Z: A] :
% 0.07/0.32 ( ( ord_less @ A @ ( zero_zero @ A ) @ X )
% 0.07/0.32 => ( ( ord_less_eq @ A @ X @ Y2 )
% 0.07/0.32 => ( ( ord_less @ A @ ( zero_zero @ A ) @ W )
% 0.07/0.32 => ( ( ord_less @ A @ W @ Z )
% 0.07/0.32 => ( ord_less @ A @ ( divide_divide @ A @ X @ Z ) @ ( divide_divide @ A @ Y2 @ W ) ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % frac_less2
% 0.07/0.32 thf(fact_177_divide__le__cancel,axiom,
% 0.07/0.32 ! [A: $tType] :
% 0.07/0.32 ( ( linordered_field @ A )
% 0.07/0.32 => ! [A2: A,C: A,B: A] :
% 0.07/0.32 ( ( ord_less_eq @ A @ ( divide_divide @ A @ A2 @ C ) @ ( divide_divide @ A @ B @ C ) )
% 0.07/0.32 = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C )
% 0.07/0.32 => ( ord_less_eq @ A @ A2 @ B ) )
% 0.07/0.32 & ( ( ord_less @ A @ C @ ( zero_zero @ A ) )
% 0.07/0.32 => ( ord_less_eq @ A @ B @ A2 ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % divide_le_cancel
% 0.07/0.32 thf(fact_178_divide__nonneg__neg,axiom,
% 0.07/0.32 ! [A: $tType] :
% 0.07/0.32 ( ( linordered_field @ A )
% 0.07/0.32 => ! [X: A,Y2: A] :
% 0.07/0.32 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
% 0.07/0.32 => ( ( ord_less @ A @ Y2 @ ( zero_zero @ A ) )
% 0.07/0.32 => ( ord_less_eq @ A @ ( divide_divide @ A @ X @ Y2 ) @ ( zero_zero @ A ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % divide_nonneg_neg
% 0.07/0.32 thf(fact_179_divide__nonneg__pos,axiom,
% 0.07/0.32 ! [A: $tType] :
% 0.07/0.32 ( ( linordered_field @ A )
% 0.07/0.32 => ! [X: A,Y2: A] :
% 0.07/0.32 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
% 0.07/0.32 => ( ( ord_less @ A @ ( zero_zero @ A ) @ Y2 )
% 0.07/0.32 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ X @ Y2 ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % divide_nonneg_pos
% 0.07/0.32 thf(fact_180_divide__nonpos__neg,axiom,
% 0.07/0.32 ! [A: $tType] :
% 0.07/0.32 ( ( linordered_field @ A )
% 0.07/0.32 => ! [X: A,Y2: A] :
% 0.07/0.32 ( ( ord_less_eq @ A @ X @ ( zero_zero @ A ) )
% 0.07/0.32 => ( ( ord_less @ A @ Y2 @ ( zero_zero @ A ) )
% 0.07/0.32 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ X @ Y2 ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % divide_nonpos_neg
% 0.07/0.32 thf(fact_181_divide__nonpos__pos,axiom,
% 0.07/0.32 ! [A: $tType] :
% 0.07/0.32 ( ( linordered_field @ A )
% 0.07/0.32 => ! [X: A,Y2: A] :
% 0.07/0.32 ( ( ord_less_eq @ A @ X @ ( zero_zero @ A ) )
% 0.07/0.32 => ( ( ord_less @ A @ ( zero_zero @ A ) @ Y2 )
% 0.07/0.32 => ( ord_less_eq @ A @ ( divide_divide @ A @ X @ Y2 ) @ ( zero_zero @ A ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % divide_nonpos_pos
% 0.07/0.32 thf(fact_182_power__less__imp__less__base,axiom,
% 0.07/0.32 ! [A: $tType] :
% 0.07/0.32 ( ( linordered_semidom @ A )
% 0.07/0.32 => ! [A2: A,N: nat,B: A] :
% 0.07/0.32 ( ( ord_less @ A @ ( power_power @ A @ A2 @ N ) @ ( power_power @ A @ B @ N ) )
% 0.07/0.32 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B )
% 0.07/0.32 => ( ord_less @ A @ A2 @ B ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % power_less_imp_less_base
% 0.07/0.32 thf(fact_183_power__strict__mono,axiom,
% 0.07/0.32 ! [A: $tType] :
% 0.07/0.32 ( ( linordered_semidom @ A )
% 0.07/0.32 => ! [A2: A,B: A,N: nat] :
% 0.07/0.32 ( ( ord_less @ A @ A2 @ B )
% 0.07/0.32 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
% 0.07/0.32 => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 0.07/0.32 => ( ord_less @ A @ ( power_power @ A @ A2 @ N ) @ ( power_power @ A @ B @ N ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % power_strict_mono
% 0.07/0.32 thf(fact_184_power__le__one,axiom,
% 0.07/0.32 ! [A: $tType] :
% 0.07/0.32 ( ( linordered_semidom @ A )
% 0.07/0.32 => ! [A2: A,N: nat] :
% 0.07/0.32 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
% 0.07/0.32 => ( ( ord_less_eq @ A @ A2 @ ( one_one @ A ) )
% 0.07/0.32 => ( ord_less_eq @ A @ ( power_power @ A @ A2 @ N ) @ ( one_one @ A ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % power_le_one
% 0.07/0.32 thf(fact_185_power__decreasing,axiom,
% 0.07/0.32 ! [A: $tType] :
% 0.07/0.32 ( ( linordered_semidom @ A )
% 0.07/0.32 => ! [N: nat,N4: nat,A2: A] :
% 0.07/0.32 ( ( ord_less_eq @ nat @ N @ N4 )
% 0.07/0.32 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
% 0.07/0.32 => ( ( ord_less_eq @ A @ A2 @ ( one_one @ A ) )
% 0.07/0.32 => ( ord_less_eq @ A @ ( power_power @ A @ A2 @ N4 ) @ ( power_power @ A @ A2 @ N ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % power_decreasing
% 0.07/0.32 thf(fact_186_divide__less__eq__1,axiom,
% 0.07/0.32 ! [A: $tType] :
% 0.07/0.32 ( ( linordered_field @ A )
% 0.07/0.32 => ! [B: A,A2: A] :
% 0.07/0.32 ( ( ord_less @ A @ ( divide_divide @ A @ B @ A2 ) @ ( one_one @ A ) )
% 0.07/0.32 = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
% 0.07/0.32 & ( ord_less @ A @ B @ A2 ) )
% 0.07/0.32 | ( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
% 0.07/0.32 & ( ord_less @ A @ A2 @ B ) )
% 0.07/0.32 | ( A2
% 0.07/0.32 = ( zero_zero @ A ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % divide_less_eq_1
% 0.07/0.32 thf(fact_187_less__divide__eq__1,axiom,
% 0.07/0.32 ! [A: $tType] :
% 0.07/0.32 ( ( linordered_field @ A )
% 0.07/0.32 => ! [B: A,A2: A] :
% 0.07/0.32 ( ( ord_less @ A @ ( one_one @ A ) @ ( divide_divide @ A @ B @ A2 ) )
% 0.07/0.32 = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
% 0.07/0.32 & ( ord_less @ A @ A2 @ B ) )
% 0.07/0.32 | ( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
% 0.07/0.32 & ( ord_less @ A @ B @ A2 ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % less_divide_eq_1
% 0.07/0.32 thf(fact_188_power__strict__decreasing,axiom,
% 0.07/0.32 ! [A: $tType] :
% 0.07/0.32 ( ( linordered_semidom @ A )
% 0.07/0.32 => ! [N: nat,N4: nat,A2: A] :
% 0.07/0.32 ( ( ord_less @ nat @ N @ N4 )
% 0.07/0.32 => ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
% 0.07/0.32 => ( ( ord_less @ A @ A2 @ ( one_one @ A ) )
% 0.07/0.32 => ( ord_less @ A @ ( power_power @ A @ A2 @ N4 ) @ ( power_power @ A @ A2 @ N ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % power_strict_decreasing
% 0.07/0.32 thf(fact_189_power__le__imp__le__exp,axiom,
% 0.07/0.32 ! [A: $tType] :
% 0.07/0.32 ( ( linordered_semidom @ A )
% 0.07/0.32 => ! [A2: A,M: nat,N: nat] :
% 0.07/0.32 ( ( ord_less @ A @ ( one_one @ A ) @ A2 )
% 0.07/0.32 => ( ( ord_less_eq @ A @ ( power_power @ A @ A2 @ M ) @ ( power_power @ A @ A2 @ N ) )
% 0.07/0.32 => ( ord_less_eq @ nat @ M @ N ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % power_le_imp_le_exp
% 0.07/0.32 thf(fact_190_zero__le__power__abs,axiom,
% 0.07/0.32 ! [A: $tType] :
% 0.07/0.32 ( ( linordered_idom @ A )
% 0.07/0.32 => ! [A2: A,N: nat] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( power_power @ A @ ( abs_abs @ A @ A2 ) @ N ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % zero_le_power_abs
% 0.07/0.32 thf(fact_191_abs__div__pos,axiom,
% 0.07/0.32 ! [A: $tType] :
% 0.07/0.32 ( ( linordered_field @ A )
% 0.07/0.32 => ! [Y2: A,X: A] :
% 0.07/0.32 ( ( ord_less @ A @ ( zero_zero @ A ) @ Y2 )
% 0.07/0.32 => ( ( divide_divide @ A @ ( abs_abs @ A @ X ) @ Y2 )
% 0.07/0.32 = ( abs_abs @ A @ ( divide_divide @ A @ X @ Y2 ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % abs_div_pos
% 0.07/0.32 thf(fact_192_divide__le__eq__1,axiom,
% 0.07/0.32 ! [A: $tType] :
% 0.07/0.32 ( ( linordered_field @ A )
% 0.07/0.32 => ! [B: A,A2: A] :
% 0.07/0.32 ( ( ord_less_eq @ A @ ( divide_divide @ A @ B @ A2 ) @ ( one_one @ A ) )
% 0.07/0.32 = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
% 0.07/0.32 & ( ord_less_eq @ A @ B @ A2 ) )
% 0.07/0.32 | ( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
% 0.07/0.32 & ( ord_less_eq @ A @ A2 @ B ) )
% 0.07/0.32 | ( A2
% 0.07/0.32 = ( zero_zero @ A ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % divide_le_eq_1
% 0.07/0.32 thf(fact_193_le__divide__eq__1,axiom,
% 0.07/0.32 ! [A: $tType] :
% 0.07/0.32 ( ( linordered_field @ A )
% 0.07/0.32 => ! [B: A,A2: A] :
% 0.07/0.32 ( ( ord_less_eq @ A @ ( one_one @ A ) @ ( divide_divide @ A @ B @ A2 ) )
% 0.07/0.32 = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
% 0.07/0.32 & ( ord_less_eq @ A @ A2 @ B ) )
% 0.07/0.32 | ( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
% 0.07/0.32 & ( ord_less_eq @ A @ B @ A2 ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % le_divide_eq_1
% 0.07/0.32 thf(fact_194_A__less_I2_J,axiom,
% 0.07/0.32 ord_less @ real @ a @ ( divide_divide @ real @ ( one_one @ real ) @ m ) ).
% 0.07/0.32
% 0.07/0.32 % A_less(2)
% 0.07/0.32 thf(fact_195_ab,axiom,
% 0.07/0.32 ord_less_eq @ real @ ( abs_abs @ real @ ( minus_minus @ real @ x @ ( divide_divide @ real @ ( ring_1_of_int @ real @ a2 ) @ ( ring_1_of_int @ real @ b ) ) ) ) @ ( divide_divide @ real @ a @ ( power_power @ real @ ( ring_1_of_int @ real @ b ) @ n ) ) ).
% 0.07/0.32
% 0.07/0.32 % ab
% 0.07/0.32 thf(fact_196_ab_H,axiom,
% 0.07/0.32 ord_less_eq @ real @ ( abs_abs @ real @ ( minus_minus @ real @ x @ ( divide_divide @ real @ ( ring_1_of_int @ real @ a2 ) @ ( ring_1_of_int @ real @ b ) ) ) ) @ a ).
% 0.07/0.32
% 0.07/0.32 % ab'
% 0.07/0.32 thf(fact_197_div__pos__pos__trivial,axiom,
% 0.07/0.32 ! [K: int,L: int] :
% 0.07/0.32 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
% 0.07/0.32 => ( ( ord_less @ int @ K @ L )
% 0.07/0.32 => ( ( divide_divide @ int @ K @ L )
% 0.07/0.32 = ( zero_zero @ int ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % div_pos_pos_trivial
% 0.07/0.32 thf(fact_198_div__neg__neg__trivial,axiom,
% 0.07/0.32 ! [K: int,L: int] :
% 0.07/0.32 ( ( ord_less_eq @ int @ K @ ( zero_zero @ int ) )
% 0.07/0.32 => ( ( ord_less @ int @ L @ K )
% 0.07/0.32 => ( ( divide_divide @ int @ K @ L )
% 0.07/0.32 = ( zero_zero @ int ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % div_neg_neg_trivial
% 0.07/0.32 thf(fact_199_zero__less__abs__iff,axiom,
% 0.07/0.32 ! [A: $tType] :
% 0.07/0.32 ( ( ordere142940540dd_abs @ A )
% 0.07/0.32 => ! [A2: A] :
% 0.07/0.32 ( ( ord_less @ A @ ( zero_zero @ A ) @ ( abs_abs @ A @ A2 ) )
% 0.07/0.32 = ( A2
% 0.07/0.32 != ( zero_zero @ A ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % zero_less_abs_iff
% 0.07/0.32 thf(fact_200_M__pos,axiom,
% 0.07/0.32 ord_less @ real @ ( zero_zero @ real ) @ m ).
% 0.07/0.32
% 0.07/0.32 % M_pos
% 0.07/0.32 thf(fact_201_abs__idempotent,axiom,
% 0.07/0.32 ! [A: $tType] :
% 0.07/0.32 ( ( ordere142940540dd_abs @ A )
% 0.07/0.32 => ! [A2: A] :
% 0.07/0.32 ( ( abs_abs @ A @ ( abs_abs @ A @ A2 ) )
% 0.07/0.32 = ( abs_abs @ A @ A2 ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % abs_idempotent
% 0.07/0.32 thf(fact_202_le__zero__eq,axiom,
% 0.07/0.32 ! [A: $tType] :
% 0.07/0.32 ( ( canoni770627133id_add @ A )
% 0.07/0.32 => ! [N: A] :
% 0.07/0.32 ( ( ord_less_eq @ A @ N @ ( zero_zero @ A ) )
% 0.07/0.32 = ( N
% 0.07/0.32 = ( zero_zero @ A ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % le_zero_eq
% 0.07/0.32 thf(fact_203_not__gr__zero,axiom,
% 0.07/0.32 ! [A: $tType] :
% 0.07/0.32 ( ( canoni770627133id_add @ A )
% 0.07/0.32 => ! [N: A] :
% 0.07/0.32 ( ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ N ) )
% 0.07/0.32 = ( N
% 0.07/0.32 = ( zero_zero @ A ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % not_gr_zero
% 0.07/0.32 thf(fact_204_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
% 0.07/0.32 ! [A: $tType] :
% 0.07/0.32 ( ( cancel1352612707id_add @ A )
% 0.07/0.32 => ! [A2: A] :
% 0.07/0.32 ( ( minus_minus @ A @ A2 @ A2 )
% 0.07/0.32 = ( zero_zero @ A ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % cancel_comm_monoid_add_class.diff_cancel
% 0.07/0.32 thf(fact_205_diff__zero,axiom,
% 0.07/0.32 ! [A: $tType] :
% 0.07/0.32 ( ( cancel1352612707id_add @ A )
% 0.07/0.32 => ! [A2: A] :
% 0.07/0.32 ( ( minus_minus @ A @ A2 @ ( zero_zero @ A ) )
% 0.07/0.32 = A2 ) ) ).
% 0.07/0.32
% 0.07/0.32 % diff_zero
% 0.07/0.32 thf(fact_206_zero__diff,axiom,
% 0.07/0.32 ! [A: $tType] :
% 0.07/0.32 ( ( comm_monoid_diff @ A )
% 0.07/0.32 => ! [A2: A] :
% 0.07/0.32 ( ( minus_minus @ A @ ( zero_zero @ A ) @ A2 )
% 0.07/0.32 = ( zero_zero @ A ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % zero_diff
% 0.07/0.32 thf(fact_207_diff__0__right,axiom,
% 0.07/0.32 ! [A: $tType] :
% 0.07/0.32 ( ( group_add @ A )
% 0.07/0.32 => ! [A2: A] :
% 0.07/0.32 ( ( minus_minus @ A @ A2 @ ( zero_zero @ A ) )
% 0.07/0.32 = A2 ) ) ).
% 0.07/0.32
% 0.07/0.32 % diff_0_right
% 0.07/0.32 thf(fact_208_diff__self,axiom,
% 0.07/0.32 ! [A: $tType] :
% 0.07/0.32 ( ( group_add @ A )
% 0.07/0.32 => ! [A2: A] :
% 0.07/0.32 ( ( minus_minus @ A @ A2 @ A2 )
% 0.07/0.32 = ( zero_zero @ A ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % diff_self
% 0.07/0.32 thf(fact_209_abs__zero,axiom,
% 0.07/0.32 ! [A: $tType] :
% 0.07/0.32 ( ( ordere142940540dd_abs @ A )
% 0.07/0.32 => ( ( abs_abs @ A @ ( zero_zero @ A ) )
% 0.07/0.32 = ( zero_zero @ A ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % abs_zero
% 0.07/0.32 thf(fact_210_abs__eq__0,axiom,
% 0.07/0.32 ! [A: $tType] :
% 0.07/0.32 ( ( ordere142940540dd_abs @ A )
% 0.07/0.32 => ! [A2: A] :
% 0.07/0.32 ( ( ( abs_abs @ A @ A2 )
% 0.07/0.32 = ( zero_zero @ A ) )
% 0.07/0.32 = ( A2
% 0.07/0.32 = ( zero_zero @ A ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % abs_eq_0
% 0.07/0.32 thf(fact_211_abs__0__eq,axiom,
% 0.07/0.32 ! [A: $tType] :
% 0.07/0.32 ( ( ordere142940540dd_abs @ A )
% 0.07/0.32 => ! [A2: A] :
% 0.07/0.32 ( ( ( zero_zero @ A )
% 0.07/0.32 = ( abs_abs @ A @ A2 ) )
% 0.07/0.32 = ( A2
% 0.07/0.32 = ( zero_zero @ A ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % abs_0_eq
% 0.07/0.32 thf(fact_212_div__less,axiom,
% 0.07/0.32 ! [M: nat,N: nat] :
% 0.07/0.32 ( ( ord_less @ nat @ M @ N )
% 0.07/0.32 => ( ( divide_divide @ nat @ M @ N )
% 0.07/0.32 = ( zero_zero @ nat ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % div_less
% 0.07/0.32 thf(fact_213_of__int__diff,axiom,
% 0.07/0.32 ! [A: $tType] :
% 0.07/0.32 ( ( ring_1 @ A )
% 0.07/0.32 => ! [W: int,Z: int] :
% 0.07/0.32 ( ( ring_1_of_int @ A @ ( minus_minus @ int @ W @ Z ) )
% 0.07/0.32 = ( minus_minus @ A @ ( ring_1_of_int @ A @ W ) @ ( ring_1_of_int @ A @ Z ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % of_int_diff
% 0.07/0.32 thf(fact_214_coeff__diff,axiom,
% 0.07/0.32 ! [A: $tType] :
% 0.07/0.32 ( ( cancel1352612707id_add @ A )
% 0.07/0.32 => ! [P: poly @ A,Q: poly @ A,N: nat] :
% 0.07/0.32 ( ( coeff @ A @ ( minus_minus @ ( poly @ A ) @ P @ Q ) @ N )
% 0.07/0.32 = ( minus_minus @ A @ ( coeff @ A @ P @ N ) @ ( coeff @ A @ Q @ N ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % coeff_diff
% 0.07/0.32 thf(fact_215_poly__diff,axiom,
% 0.07/0.32 ! [A: $tType] :
% 0.07/0.32 ( ( comm_ring @ A )
% 0.07/0.32 => ! [P: poly @ A,Q: poly @ A,X: A] :
% 0.07/0.32 ( ( poly2 @ A @ ( minus_minus @ ( poly @ A ) @ P @ Q ) @ X )
% 0.07/0.32 = ( minus_minus @ A @ ( poly2 @ A @ P @ X ) @ ( poly2 @ A @ Q @ X ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % poly_diff
% 0.07/0.32 thf(fact_216_A__less_I3_J,axiom,
% 0.07/0.32 ! [X5: real] :
% 0.07/0.32 ( ( X5 != x )
% 0.07/0.32 => ( ( ( poly2 @ real @ p @ X5 )
% 0.07/0.32 = ( zero_zero @ real ) )
% 0.07/0.32 => ( ord_less @ real @ a @ ( abs_abs @ real @ ( minus_minus @ real @ X5 @ x ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % A_less(3)
% 0.07/0.32 thf(fact_217_diff__ge__0__iff__ge,axiom,
% 0.07/0.32 ! [A: $tType] :
% 0.07/0.32 ( ( ordered_ab_group_add @ A )
% 0.07/0.32 => ! [A2: A,B: A] :
% 0.07/0.32 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( minus_minus @ A @ A2 @ B ) )
% 0.07/0.32 = ( ord_less_eq @ A @ B @ A2 ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % diff_ge_0_iff_ge
% 0.07/0.32 thf(fact_218_diff__gt__0__iff__gt,axiom,
% 0.07/0.32 ! [A: $tType] :
% 0.07/0.32 ( ( ordered_ab_group_add @ A )
% 0.07/0.32 => ! [A2: A,B: A] :
% 0.07/0.32 ( ( ord_less @ A @ ( zero_zero @ A ) @ ( minus_minus @ A @ A2 @ B ) )
% 0.07/0.32 = ( ord_less @ A @ B @ A2 ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % diff_gt_0_iff_gt
% 0.07/0.32 thf(fact_219_abs__le__zero__iff,axiom,
% 0.07/0.32 ! [A: $tType] :
% 0.07/0.32 ( ( ordere142940540dd_abs @ A )
% 0.07/0.32 => ! [A2: A] :
% 0.07/0.32 ( ( ord_less_eq @ A @ ( abs_abs @ A @ A2 ) @ ( zero_zero @ A ) )
% 0.07/0.32 = ( A2
% 0.07/0.32 = ( zero_zero @ A ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % abs_le_zero_iff
% 0.07/0.32 thf(fact_220_abs__le__self__iff,axiom,
% 0.07/0.32 ! [A: $tType] :
% 0.07/0.32 ( ( ordere142940540dd_abs @ A )
% 0.07/0.32 => ! [A2: A] :
% 0.07/0.32 ( ( ord_less_eq @ A @ ( abs_abs @ A @ A2 ) @ A2 )
% 0.07/0.32 = ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % abs_le_self_iff
% 0.07/0.32 thf(fact_221_abs__of__nonneg,axiom,
% 0.07/0.32 ! [A: $tType] :
% 0.07/0.32 ( ( ordere142940540dd_abs @ A )
% 0.07/0.32 => ! [A2: A] :
% 0.07/0.32 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
% 0.07/0.32 => ( ( abs_abs @ A @ A2 )
% 0.07/0.32 = A2 ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % abs_of_nonneg
% 0.07/0.32 thf(fact_222_that,axiom,
% 0.07/0.32 ! [C: real,N: nat] :
% 0.07/0.32 ( ( ord_less @ real @ ( zero_zero @ real ) @ C )
% 0.07/0.32 => ( ! [P2: int,Q4: int] :
% 0.07/0.32 ( ( ord_less @ int @ ( zero_zero @ int ) @ Q4 )
% 0.07/0.32 => ( ord_less @ real @ ( divide_divide @ real @ C @ ( power_power @ real @ ( ring_1_of_int @ real @ Q4 ) @ N ) ) @ ( abs_abs @ real @ ( minus_minus @ real @ x @ ( divide_divide @ real @ ( ring_1_of_int @ real @ P2 ) @ ( ring_1_of_int @ real @ Q4 ) ) ) ) ) )
% 0.07/0.32 => thesis ) ) ).
% 0.07/0.32
% 0.07/0.32 % that
% 0.07/0.32 thf(fact_223_minus__poly_Orep__eq,axiom,
% 0.07/0.32 ! [A: $tType] :
% 0.07/0.32 ( ( cancel1352612707id_add @ A )
% 0.07/0.32 => ! [X: poly @ A,Xa: poly @ A] :
% 0.07/0.32 ( ( coeff @ A @ ( minus_minus @ ( poly @ A ) @ X @ Xa ) )
% 0.07/0.32 = ( ^ [N3: nat] : ( minus_minus @ A @ ( coeff @ A @ X @ N3 ) @ ( coeff @ A @ Xa @ N3 ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % minus_poly.rep_eq
% 0.07/0.32 thf(fact_224_eq__iff__diff__eq__0,axiom,
% 0.07/0.32 ! [A: $tType] :
% 0.07/0.32 ( ( group_add @ A )
% 0.07/0.32 => ( ( ^ [Y: A,Z2: A] : ( Y = Z2 ) )
% 0.07/0.32 = ( ^ [A4: A,B3: A] :
% 0.07/0.32 ( ( minus_minus @ A @ A4 @ B3 )
% 0.07/0.32 = ( zero_zero @ A ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % eq_iff_diff_eq_0
% 0.07/0.32 thf(fact_225_diff__mono,axiom,
% 0.07/0.32 ! [A: $tType] :
% 0.07/0.32 ( ( ordered_ab_group_add @ A )
% 0.07/0.32 => ! [A2: A,B: A,D: A,C: A] :
% 0.07/0.32 ( ( ord_less_eq @ A @ A2 @ B )
% 0.07/0.32 => ( ( ord_less_eq @ A @ D @ C )
% 0.07/0.32 => ( ord_less_eq @ A @ ( minus_minus @ A @ A2 @ C ) @ ( minus_minus @ A @ B @ D ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % diff_mono
% 0.07/0.32 thf(fact_226_diff__left__mono,axiom,
% 0.07/0.32 ! [A: $tType] :
% 0.07/0.32 ( ( ordered_ab_group_add @ A )
% 0.07/0.32 => ! [B: A,A2: A,C: A] :
% 0.07/0.32 ( ( ord_less_eq @ A @ B @ A2 )
% 0.07/0.32 => ( ord_less_eq @ A @ ( minus_minus @ A @ C @ A2 ) @ ( minus_minus @ A @ C @ B ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % diff_left_mono
% 0.07/0.32 thf(fact_227_diff__right__mono,axiom,
% 0.07/0.32 ! [A: $tType] :
% 0.07/0.32 ( ( ordered_ab_group_add @ A )
% 0.07/0.32 => ! [A2: A,B: A,C: A] :
% 0.07/0.32 ( ( ord_less_eq @ A @ A2 @ B )
% 0.07/0.32 => ( ord_less_eq @ A @ ( minus_minus @ A @ A2 @ C ) @ ( minus_minus @ A @ B @ C ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % diff_right_mono
% 0.07/0.32 thf(fact_228_diff__eq__diff__less__eq,axiom,
% 0.07/0.32 ! [A: $tType] :
% 0.07/0.32 ( ( ordered_ab_group_add @ A )
% 0.07/0.32 => ! [A2: A,B: A,C: A,D: A] :
% 0.07/0.32 ( ( ( minus_minus @ A @ A2 @ B )
% 0.07/0.32 = ( minus_minus @ A @ C @ D ) )
% 0.07/0.32 => ( ( ord_less_eq @ A @ A2 @ B )
% 0.07/0.32 = ( ord_less_eq @ A @ C @ D ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % diff_eq_diff_less_eq
% 0.07/0.32 thf(fact_229_diff__strict__right__mono,axiom,
% 0.07/0.32 ! [A: $tType] :
% 0.07/0.32 ( ( ordered_ab_group_add @ A )
% 0.07/0.32 => ! [A2: A,B: A,C: A] :
% 0.07/0.32 ( ( ord_less @ A @ A2 @ B )
% 0.07/0.32 => ( ord_less @ A @ ( minus_minus @ A @ A2 @ C ) @ ( minus_minus @ A @ B @ C ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % diff_strict_right_mono
% 0.07/0.32 thf(fact_230_diff__strict__left__mono,axiom,
% 0.07/0.32 ! [A: $tType] :
% 0.07/0.32 ( ( ordered_ab_group_add @ A )
% 0.07/0.32 => ! [B: A,A2: A,C: A] :
% 0.07/0.32 ( ( ord_less @ A @ B @ A2 )
% 0.07/0.32 => ( ord_less @ A @ ( minus_minus @ A @ C @ A2 ) @ ( minus_minus @ A @ C @ B ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % diff_strict_left_mono
% 0.07/0.32 thf(fact_231_diff__eq__diff__less,axiom,
% 0.07/0.32 ! [A: $tType] :
% 0.07/0.32 ( ( ordered_ab_group_add @ A )
% 0.07/0.32 => ! [A2: A,B: A,C: A,D: A] :
% 0.07/0.32 ( ( ( minus_minus @ A @ A2 @ B )
% 0.07/0.32 = ( minus_minus @ A @ C @ D ) )
% 0.07/0.32 => ( ( ord_less @ A @ A2 @ B )
% 0.07/0.32 = ( ord_less @ A @ C @ D ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % diff_eq_diff_less
% 0.07/0.32 thf(fact_232_diff__strict__mono,axiom,
% 0.07/0.32 ! [A: $tType] :
% 0.07/0.32 ( ( ordered_ab_group_add @ A )
% 0.07/0.32 => ! [A2: A,B: A,D: A,C: A] :
% 0.07/0.32 ( ( ord_less @ A @ A2 @ B )
% 0.07/0.32 => ( ( ord_less @ A @ D @ C )
% 0.07/0.32 => ( ord_less @ A @ ( minus_minus @ A @ A2 @ C ) @ ( minus_minus @ A @ B @ D ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % diff_strict_mono
% 0.07/0.32 thf(fact_233_abs__minus__commute,axiom,
% 0.07/0.32 ! [A: $tType] :
% 0.07/0.32 ( ( ordere142940540dd_abs @ A )
% 0.07/0.32 => ! [A2: A,B: A] :
% 0.07/0.32 ( ( abs_abs @ A @ ( minus_minus @ A @ A2 @ B ) )
% 0.07/0.32 = ( abs_abs @ A @ ( minus_minus @ A @ B @ A2 ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % abs_minus_commute
% 0.07/0.32 thf(fact_234_diff__eq__diff__eq,axiom,
% 0.07/0.32 ! [A: $tType] :
% 0.07/0.32 ( ( group_add @ A )
% 0.07/0.32 => ! [A2: A,B: A,C: A,D: A] :
% 0.07/0.32 ( ( ( minus_minus @ A @ A2 @ B )
% 0.07/0.32 = ( minus_minus @ A @ C @ D ) )
% 0.07/0.32 => ( ( A2 = B )
% 0.07/0.32 = ( C = D ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % diff_eq_diff_eq
% 0.07/0.32 thf(fact_235_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
% 0.07/0.32 ! [A: $tType] :
% 0.07/0.32 ( ( cancel146912293up_add @ A )
% 0.07/0.32 => ! [A2: A,C: A,B: A] :
% 0.07/0.32 ( ( minus_minus @ A @ ( minus_minus @ A @ A2 @ C ) @ B )
% 0.07/0.32 = ( minus_minus @ A @ ( minus_minus @ A @ A2 @ B ) @ C ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % cancel_ab_semigroup_add_class.diff_right_commute
% 0.07/0.32 thf(fact_236_diff__divide__distrib,axiom,
% 0.07/0.32 ! [A: $tType] :
% 0.07/0.32 ( ( division_ring @ A )
% 0.07/0.32 => ! [A2: A,B: A,C: A] :
% 0.07/0.32 ( ( divide_divide @ A @ ( minus_minus @ A @ A2 @ B ) @ C )
% 0.07/0.32 = ( minus_minus @ A @ ( divide_divide @ A @ A2 @ C ) @ ( divide_divide @ A @ B @ C ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % diff_divide_distrib
% 0.07/0.32 thf(fact_237_le__iff__diff__le__0,axiom,
% 0.07/0.32 ! [A: $tType] :
% 0.07/0.32 ( ( ordered_ab_group_add @ A )
% 0.07/0.32 => ( ( ord_less_eq @ A )
% 0.07/0.32 = ( ^ [A4: A,B3: A] : ( ord_less_eq @ A @ ( minus_minus @ A @ A4 @ B3 ) @ ( zero_zero @ A ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % le_iff_diff_le_0
% 0.07/0.32 thf(fact_238_less__iff__diff__less__0,axiom,
% 0.07/0.32 ! [A: $tType] :
% 0.07/0.32 ( ( ordered_ab_group_add @ A )
% 0.07/0.32 => ( ( ord_less @ A )
% 0.07/0.32 = ( ^ [A4: A,B3: A] : ( ord_less @ A @ ( minus_minus @ A @ A4 @ B3 ) @ ( zero_zero @ A ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % less_iff_diff_less_0
% 0.07/0.32 thf(fact_239_Ints__diff,axiom,
% 0.07/0.32 ! [A: $tType] :
% 0.07/0.32 ( ( ring_1 @ A )
% 0.07/0.32 => ! [A2: A,B: A] :
% 0.07/0.32 ( ( member @ A @ A2 @ ( ring_1_Ints @ A ) )
% 0.07/0.32 => ( ( member @ A @ B @ ( ring_1_Ints @ A ) )
% 0.07/0.32 => ( member @ A @ ( minus_minus @ A @ A2 @ B ) @ ( ring_1_Ints @ A ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % Ints_diff
% 0.07/0.32 thf(fact_240_abs__triangle__ineq2__sym,axiom,
% 0.07/0.32 ! [A: $tType] :
% 0.07/0.32 ( ( ordere142940540dd_abs @ A )
% 0.07/0.32 => ! [A2: A,B: A] : ( ord_less_eq @ A @ ( minus_minus @ A @ ( abs_abs @ A @ A2 ) @ ( abs_abs @ A @ B ) ) @ ( abs_abs @ A @ ( minus_minus @ A @ B @ A2 ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % abs_triangle_ineq2_sym
% 0.07/0.32 thf(fact_241_abs__triangle__ineq3,axiom,
% 0.07/0.32 ! [A: $tType] :
% 0.07/0.32 ( ( ordere142940540dd_abs @ A )
% 0.07/0.32 => ! [A2: A,B: A] : ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ ( abs_abs @ A @ A2 ) @ ( abs_abs @ A @ B ) ) ) @ ( abs_abs @ A @ ( minus_minus @ A @ A2 @ B ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % abs_triangle_ineq3
% 0.07/0.32 thf(fact_242_abs__triangle__ineq2,axiom,
% 0.07/0.32 ! [A: $tType] :
% 0.07/0.32 ( ( ordere142940540dd_abs @ A )
% 0.07/0.32 => ! [A2: A,B: A] : ( ord_less_eq @ A @ ( minus_minus @ A @ ( abs_abs @ A @ A2 ) @ ( abs_abs @ A @ B ) ) @ ( abs_abs @ A @ ( minus_minus @ A @ A2 @ B ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % abs_triangle_ineq2
% 0.07/0.32 thf(fact_243_zero__reorient,axiom,
% 0.07/0.32 ! [A: $tType] :
% 0.07/0.32 ( ( zero @ A )
% 0.07/0.32 => ! [X: A] :
% 0.07/0.32 ( ( ( zero_zero @ A )
% 0.07/0.32 = X )
% 0.07/0.32 = ( X
% 0.07/0.32 = ( zero_zero @ A ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % zero_reorient
% 0.07/0.32 thf(fact_244_one__reorient,axiom,
% 0.07/0.32 ! [A: $tType] :
% 0.07/0.32 ( ( one @ A )
% 0.07/0.32 => ! [X: A] :
% 0.07/0.32 ( ( ( one_one @ A )
% 0.07/0.32 = X )
% 0.07/0.32 = ( X
% 0.07/0.32 = ( one_one @ A ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % one_reorient
% 0.07/0.32 thf(fact_245_div__eq__dividend__iff,axiom,
% 0.07/0.32 ! [M: nat,N: nat] :
% 0.07/0.32 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
% 0.07/0.32 => ( ( ( divide_divide @ nat @ M @ N )
% 0.07/0.32 = M )
% 0.07/0.32 = ( N
% 0.07/0.32 = ( one_one @ nat ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % div_eq_dividend_iff
% 0.07/0.32 thf(fact_246_div__less__dividend,axiom,
% 0.07/0.32 ! [N: nat,M: nat] :
% 0.07/0.32 ( ( ord_less @ nat @ ( one_one @ nat ) @ N )
% 0.07/0.32 => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
% 0.07/0.32 => ( ord_less @ nat @ ( divide_divide @ nat @ M @ N ) @ M ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % div_less_dividend
% 0.07/0.32 thf(fact_247_Euclidean__Division_Odiv__eq__0__iff,axiom,
% 0.07/0.32 ! [M: nat,N: nat] :
% 0.07/0.32 ( ( ( divide_divide @ nat @ M @ N )
% 0.07/0.32 = ( zero_zero @ nat ) )
% 0.07/0.32 = ( ( ord_less @ nat @ M @ N )
% 0.07/0.32 | ( N
% 0.07/0.32 = ( zero_zero @ nat ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % Euclidean_Division.div_eq_0_iff
% 0.07/0.32 thf(fact_248_div__le__dividend,axiom,
% 0.07/0.32 ! [M: nat,N: nat] : ( ord_less_eq @ nat @ ( divide_divide @ nat @ M @ N ) @ M ) ).
% 0.07/0.32
% 0.07/0.32 % div_le_dividend
% 0.07/0.32 thf(fact_249_div__le__mono,axiom,
% 0.07/0.32 ! [M: nat,N: nat,K: nat] :
% 0.07/0.32 ( ( ord_less_eq @ nat @ M @ N )
% 0.07/0.32 => ( ord_less_eq @ nat @ ( divide_divide @ nat @ M @ K ) @ ( divide_divide @ nat @ N @ K ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % div_le_mono
% 0.07/0.32 thf(fact_250_Ints__eq__abs__less1,axiom,
% 0.07/0.32 ! [A: $tType] :
% 0.07/0.32 ( ( linordered_idom @ A )
% 0.07/0.32 => ! [X: A,Y2: A] :
% 0.07/0.32 ( ( member @ A @ X @ ( ring_1_Ints @ A ) )
% 0.07/0.32 => ( ( member @ A @ Y2 @ ( ring_1_Ints @ A ) )
% 0.07/0.32 => ( ( X = Y2 )
% 0.07/0.32 = ( ord_less @ A @ ( abs_abs @ A @ ( minus_minus @ A @ X @ Y2 ) ) @ ( one_one @ A ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % Ints_eq_abs_less1
% 0.07/0.32 thf(fact_251_zero__le,axiom,
% 0.07/0.32 ! [A: $tType] :
% 0.07/0.32 ( ( canoni770627133id_add @ A )
% 0.07/0.32 => ! [X: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ X ) ) ).
% 0.07/0.32
% 0.07/0.32 % zero_le
% 0.07/0.32 thf(fact_252_gr__zeroI,axiom,
% 0.07/0.32 ! [A: $tType] :
% 0.07/0.32 ( ( canoni770627133id_add @ A )
% 0.07/0.32 => ! [N: A] :
% 0.07/0.32 ( ( N
% 0.07/0.32 != ( zero_zero @ A ) )
% 0.07/0.32 => ( ord_less @ A @ ( zero_zero @ A ) @ N ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % gr_zeroI
% 0.07/0.32 thf(fact_253_not__less__zero,axiom,
% 0.07/0.32 ! [A: $tType] :
% 0.07/0.32 ( ( canoni770627133id_add @ A )
% 0.07/0.32 => ! [N: A] :
% 0.07/0.32 ~ ( ord_less @ A @ N @ ( zero_zero @ A ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % not_less_zero
% 0.07/0.32
% 0.07/0.32 % Type constructors (91)
% 0.07/0.32 thf(tcon_Polynomial_Opoly___Rings_Oidom,axiom,
% 0.07/0.32 ! [A5: $tType] :
% 0.07/0.32 ( ( idom @ A5 )
% 0.07/0.32 => ( idom @ ( poly @ A5 ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(tcon_Polynomial_Opoly___Groups_Oab__group__add,axiom,
% 0.07/0.32 ! [A5: $tType] :
% 0.07/0.32 ( ( ab_group_add @ A5 )
% 0.07/0.32 => ( ab_group_add @ ( poly @ A5 ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(tcon_Real_Oreal___Rings_Oidom_1,axiom,
% 0.07/0.32 idom @ real ).
% 0.07/0.32
% 0.07/0.32 thf(tcon_Real_Oreal___Groups_Oab__group__add_2,axiom,
% 0.07/0.32 ab_group_add @ real ).
% 0.07/0.32
% 0.07/0.32 thf(tcon_Int_Oint___Rings_Oidom_3,axiom,
% 0.07/0.32 idom @ int ).
% 0.07/0.32
% 0.07/0.32 thf(tcon_Int_Oint___Groups_Oab__group__add_4,axiom,
% 0.07/0.32 ab_group_add @ int ).
% 0.07/0.32
% 0.07/0.32 thf(tcon_Int_Oint___Rings_Osemiring__1__no__zero__divisors,axiom,
% 0.07/0.32 semiri134348788visors @ int ).
% 0.07/0.32
% 0.07/0.32 thf(tcon_Int_Oint___Rings_Osemiring__no__zero__divisors,axiom,
% 0.07/0.32 semiri1193490041visors @ int ).
% 0.07/0.32
% 0.07/0.32 thf(tcon_Int_Oint___Groups_Oordered__ab__group__add__abs,axiom,
% 0.07/0.32 ordere142940540dd_abs @ int ).
% 0.07/0.32
% 0.07/0.32 thf(tcon_Int_Oint___Groups_Ocancel__ab__semigroup__add,axiom,
% 0.07/0.32 cancel146912293up_add @ int ).
% 0.07/0.32
% 0.07/0.32 thf(tcon_Int_Oint___Groups_Ocancel__comm__monoid__add,axiom,
% 0.07/0.32 cancel1352612707id_add @ int ).
% 0.07/0.32
% 0.07/0.32 thf(tcon_Int_Oint___Rings_Oring__no__zero__divisors,axiom,
% 0.07/0.32 ring_n68954251visors @ int ).
% 0.07/0.32
% 0.07/0.32 thf(tcon_Int_Oint___Groups_Oordered__ab__group__add,axiom,
% 0.07/0.32 ordered_ab_group_add @ int ).
% 0.07/0.32
% 0.07/0.32 thf(tcon_Int_Oint___Rings_Olinordered__semidom,axiom,
% 0.07/0.32 linordered_semidom @ int ).
% 0.07/0.32
% 0.07/0.32 thf(tcon_Int_Oint___Rings_Olinordered__idom,axiom,
% 0.07/0.32 linordered_idom @ int ).
% 0.07/0.32
% 0.07/0.32 thf(tcon_Int_Oint___Rings_Ocomm__semiring__1,axiom,
% 0.07/0.32 comm_semiring_1 @ int ).
% 0.07/0.32
% 0.07/0.32 thf(tcon_Int_Oint___Rings_Ocomm__semiring__0,axiom,
% 0.07/0.32 comm_semiring_0 @ int ).
% 0.07/0.32
% 0.07/0.32 thf(tcon_Int_Oint___Groups_Omonoid__mult,axiom,
% 0.07/0.32 monoid_mult @ int ).
% 0.07/0.32
% 0.07/0.32 thf(tcon_Int_Oint___Rings_Oidom__divide,axiom,
% 0.07/0.32 idom_divide @ int ).
% 0.07/0.32
% 0.07/0.32 thf(tcon_Int_Oint___Rings_Ocomm__ring__1,axiom,
% 0.07/0.32 comm_ring_1 @ int ).
% 0.07/0.32
% 0.07/0.32 thf(tcon_Int_Oint___Rings_Osemiring__1,axiom,
% 0.07/0.32 semiring_1 @ int ).
% 0.07/0.32
% 0.07/0.32 thf(tcon_Int_Oint___Groups_Ogroup__add,axiom,
% 0.07/0.32 group_add @ int ).
% 0.07/0.32
% 0.07/0.32 thf(tcon_Int_Oint___Rings_Ocomm__ring,axiom,
% 0.07/0.32 comm_ring @ int ).
% 0.07/0.32
% 0.07/0.32 thf(tcon_Int_Oint___Nat_Oring__char__0,axiom,
% 0.07/0.32 ring_char_0 @ int ).
% 0.07/0.32
% 0.07/0.32 thf(tcon_Int_Oint___Rings_Oring__1,axiom,
% 0.07/0.32 ring_1 @ int ).
% 0.07/0.32
% 0.07/0.32 thf(tcon_Int_Oint___Power_Opower,axiom,
% 0.07/0.32 power @ int ).
% 0.07/0.32
% 0.07/0.32 thf(tcon_Int_Oint___Groups_Ozero,axiom,
% 0.07/0.32 zero @ int ).
% 0.07/0.32
% 0.07/0.32 thf(tcon_Int_Oint___Groups_Oone,axiom,
% 0.07/0.32 one @ int ).
% 0.07/0.32
% 0.07/0.32 thf(tcon_Nat_Onat___Groups_Ocanonically__ordered__monoid__add,axiom,
% 0.07/0.32 canoni770627133id_add @ nat ).
% 0.07/0.32
% 0.07/0.32 thf(tcon_Nat_Onat___Rings_Osemiring__1__no__zero__divisors_5,axiom,
% 0.07/0.32 semiri134348788visors @ nat ).
% 0.07/0.32
% 0.07/0.32 thf(tcon_Nat_Onat___Rings_Osemiring__no__zero__divisors_6,axiom,
% 0.07/0.32 semiri1193490041visors @ nat ).
% 0.07/0.32
% 0.07/0.32 thf(tcon_Nat_Onat___Groups_Ocancel__ab__semigroup__add_7,axiom,
% 0.07/0.32 cancel146912293up_add @ nat ).
% 0.07/0.32
% 0.07/0.32 thf(tcon_Nat_Onat___Groups_Ocancel__comm__monoid__add_8,axiom,
% 0.07/0.32 cancel1352612707id_add @ nat ).
% 0.07/0.32
% 0.07/0.32 thf(tcon_Nat_Onat___Rings_Olinordered__semidom_9,axiom,
% 0.07/0.32 linordered_semidom @ nat ).
% 0.07/0.32
% 0.07/0.32 thf(tcon_Nat_Onat___Groups_Ocomm__monoid__diff,axiom,
% 0.07/0.32 comm_monoid_diff @ nat ).
% 0.07/0.32
% 0.07/0.32 thf(tcon_Nat_Onat___Rings_Ocomm__semiring__1_10,axiom,
% 0.07/0.32 comm_semiring_1 @ nat ).
% 0.07/0.32
% 0.07/0.32 thf(tcon_Nat_Onat___Rings_Ocomm__semiring__0_11,axiom,
% 0.07/0.32 comm_semiring_0 @ nat ).
% 0.07/0.32
% 0.07/0.32 thf(tcon_Nat_Onat___Groups_Omonoid__mult_12,axiom,
% 0.07/0.32 monoid_mult @ nat ).
% 0.07/0.32
% 0.07/0.32 thf(tcon_Nat_Onat___Rings_Osemiring__1_13,axiom,
% 0.07/0.32 semiring_1 @ nat ).
% 0.07/0.32
% 0.07/0.32 thf(tcon_Nat_Onat___Power_Opower_14,axiom,
% 0.07/0.32 power @ nat ).
% 0.07/0.32
% 0.07/0.32 thf(tcon_Nat_Onat___Groups_Ozero_15,axiom,
% 0.07/0.32 zero @ nat ).
% 0.07/0.32
% 0.07/0.32 thf(tcon_Nat_Onat___Groups_Oone_16,axiom,
% 0.07/0.32 one @ nat ).
% 0.07/0.32
% 0.07/0.32 thf(tcon_Real_Oreal___Rings_Osemiring__1__no__zero__divisors_17,axiom,
% 0.07/0.32 semiri134348788visors @ real ).
% 0.07/0.32
% 0.07/0.32 thf(tcon_Real_Oreal___Rings_Osemiring__no__zero__divisors_18,axiom,
% 0.07/0.32 semiri1193490041visors @ real ).
% 0.07/0.32
% 0.07/0.32 thf(tcon_Real_Oreal___Groups_Oordered__ab__group__add__abs_19,axiom,
% 0.07/0.32 ordere142940540dd_abs @ real ).
% 0.07/0.32
% 0.07/0.32 thf(tcon_Real_Oreal___Groups_Ocancel__ab__semigroup__add_20,axiom,
% 0.07/0.32 cancel146912293up_add @ real ).
% 0.07/0.32
% 0.07/0.32 thf(tcon_Real_Oreal___Groups_Ocancel__comm__monoid__add_21,axiom,
% 0.07/0.32 cancel1352612707id_add @ real ).
% 0.07/0.32
% 0.07/0.32 thf(tcon_Real_Oreal___Rings_Oring__no__zero__divisors_22,axiom,
% 0.07/0.32 ring_n68954251visors @ real ).
% 0.07/0.32
% 0.07/0.32 thf(tcon_Real_Oreal___Groups_Oordered__ab__group__add_23,axiom,
% 0.07/0.32 ordered_ab_group_add @ real ).
% 0.07/0.32
% 0.07/0.32 thf(tcon_Real_Oreal___Rings_Olinordered__semidom_24,axiom,
% 0.07/0.32 linordered_semidom @ real ).
% 0.07/0.32
% 0.07/0.32 thf(tcon_Real_Oreal___Fields_Olinordered__field,axiom,
% 0.07/0.32 linordered_field @ real ).
% 0.07/0.32
% 0.07/0.32 thf(tcon_Real_Oreal___Rings_Olinordered__idom_25,axiom,
% 0.07/0.32 linordered_idom @ real ).
% 0.07/0.32
% 0.07/0.32 thf(tcon_Real_Oreal___Rings_Ocomm__semiring__1_26,axiom,
% 0.07/0.32 comm_semiring_1 @ real ).
% 0.07/0.32
% 0.07/0.32 thf(tcon_Real_Oreal___Rings_Ocomm__semiring__0_27,axiom,
% 0.07/0.32 comm_semiring_0 @ real ).
% 0.07/0.32
% 0.07/0.32 thf(tcon_Real_Oreal___Fields_Ofield__abs__sgn,axiom,
% 0.07/0.32 field_abs_sgn @ real ).
% 0.07/0.32
% 0.07/0.32 thf(tcon_Real_Oreal___Fields_Odivision__ring,axiom,
% 0.07/0.32 division_ring @ real ).
% 0.07/0.32
% 0.07/0.32 thf(tcon_Real_Oreal___Fields_Ofield__char__0,axiom,
% 0.07/0.32 field_char_0 @ real ).
% 0.07/0.32
% 0.07/0.32 thf(tcon_Real_Oreal___Groups_Omonoid__mult_28,axiom,
% 0.07/0.32 monoid_mult @ real ).
% 0.07/0.32
% 0.07/0.32 thf(tcon_Real_Oreal___Rings_Oidom__divide_29,axiom,
% 0.07/0.32 idom_divide @ real ).
% 0.07/0.32
% 0.07/0.32 thf(tcon_Real_Oreal___Rings_Ocomm__ring__1_30,axiom,
% 0.07/0.32 comm_ring_1 @ real ).
% 0.07/0.32
% 0.07/0.32 thf(tcon_Real_Oreal___Rings_Osemiring__1_31,axiom,
% 0.07/0.32 semiring_1 @ real ).
% 0.07/0.32
% 0.07/0.32 thf(tcon_Real_Oreal___Groups_Ogroup__add_32,axiom,
% 0.07/0.32 group_add @ real ).
% 0.07/0.32
% 0.07/0.32 thf(tcon_Real_Oreal___Rings_Ocomm__ring_33,axiom,
% 0.07/0.32 comm_ring @ real ).
% 0.07/0.32
% 0.07/0.32 thf(tcon_Real_Oreal___Nat_Oring__char__0_34,axiom,
% 0.07/0.32 ring_char_0 @ real ).
% 0.07/0.32
% 0.07/0.32 thf(tcon_Real_Oreal___Rings_Oring__1_35,axiom,
% 0.07/0.32 ring_1 @ real ).
% 0.07/0.32
% 0.07/0.32 thf(tcon_Real_Oreal___Fields_Ofield,axiom,
% 0.07/0.32 field @ real ).
% 0.07/0.32
% 0.07/0.32 thf(tcon_Real_Oreal___Power_Opower_36,axiom,
% 0.07/0.32 power @ real ).
% 0.07/0.32
% 0.07/0.32 thf(tcon_Real_Oreal___Groups_Ozero_37,axiom,
% 0.07/0.32 zero @ real ).
% 0.07/0.32
% 0.07/0.32 thf(tcon_Real_Oreal___Groups_Oone_38,axiom,
% 0.07/0.32 one @ real ).
% 0.07/0.32
% 0.07/0.32 thf(tcon_Polynomial_Opoly___Rings_Osemiring__1__no__zero__divisors_39,axiom,
% 0.07/0.32 ! [A5: $tType] :
% 0.07/0.32 ( ( ( comm_semiring_1 @ A5 )
% 0.07/0.32 & ( semiri134348788visors @ A5 ) )
% 0.07/0.32 => ( semiri134348788visors @ ( poly @ A5 ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(tcon_Polynomial_Opoly___Rings_Osemiring__no__zero__divisors_40,axiom,
% 0.07/0.32 ! [A5: $tType] :
% 0.07/0.32 ( ( ( comm_semiring_0 @ A5 )
% 0.07/0.32 & ( semiri1193490041visors @ A5 ) )
% 0.07/0.32 => ( semiri1193490041visors @ ( poly @ A5 ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(tcon_Polynomial_Opoly___Groups_Oordered__ab__group__add__abs_41,axiom,
% 0.07/0.32 ! [A5: $tType] :
% 0.07/0.32 ( ( linordered_idom @ A5 )
% 0.07/0.32 => ( ordere142940540dd_abs @ ( poly @ A5 ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(tcon_Polynomial_Opoly___Groups_Ocancel__ab__semigroup__add_42,axiom,
% 0.07/0.32 ! [A5: $tType] :
% 0.07/0.32 ( ( cancel1352612707id_add @ A5 )
% 0.07/0.32 => ( cancel146912293up_add @ ( poly @ A5 ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(tcon_Polynomial_Opoly___Groups_Ocancel__comm__monoid__add_43,axiom,
% 0.07/0.32 ! [A5: $tType] :
% 0.07/0.32 ( ( cancel1352612707id_add @ A5 )
% 0.07/0.32 => ( cancel1352612707id_add @ ( poly @ A5 ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(tcon_Polynomial_Opoly___Rings_Oring__no__zero__divisors_44,axiom,
% 0.07/0.32 ! [A5: $tType] :
% 0.07/0.32 ( ( idom @ A5 )
% 0.07/0.32 => ( ring_n68954251visors @ ( poly @ A5 ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(tcon_Polynomial_Opoly___Groups_Oordered__ab__group__add_45,axiom,
% 0.07/0.32 ! [A5: $tType] :
% 0.07/0.32 ( ( linordered_idom @ A5 )
% 0.07/0.32 => ( ordered_ab_group_add @ ( poly @ A5 ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(tcon_Polynomial_Opoly___Rings_Olinordered__semidom_46,axiom,
% 0.07/0.32 ! [A5: $tType] :
% 0.07/0.32 ( ( linordered_idom @ A5 )
% 0.07/0.32 => ( linordered_semidom @ ( poly @ A5 ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(tcon_Polynomial_Opoly___Rings_Olinordered__idom_47,axiom,
% 0.07/0.32 ! [A5: $tType] :
% 0.07/0.32 ( ( linordered_idom @ A5 )
% 0.07/0.32 => ( linordered_idom @ ( poly @ A5 ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(tcon_Polynomial_Opoly___Rings_Ocomm__semiring__1_48,axiom,
% 0.07/0.32 ! [A5: $tType] :
% 0.07/0.32 ( ( comm_semiring_1 @ A5 )
% 0.07/0.32 => ( comm_semiring_1 @ ( poly @ A5 ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(tcon_Polynomial_Opoly___Rings_Ocomm__semiring__0_49,axiom,
% 0.07/0.32 ! [A5: $tType] :
% 0.07/0.32 ( ( comm_semiring_0 @ A5 )
% 0.07/0.32 => ( comm_semiring_0 @ ( poly @ A5 ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(tcon_Polynomial_Opoly___Groups_Omonoid__mult_50,axiom,
% 0.07/0.32 ! [A5: $tType] :
% 0.07/0.32 ( ( comm_semiring_1 @ A5 )
% 0.07/0.32 => ( monoid_mult @ ( poly @ A5 ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(tcon_Polynomial_Opoly___Rings_Oidom__divide_51,axiom,
% 0.07/0.32 ! [A5: $tType] :
% 0.07/0.32 ( ( idom_divide @ A5 )
% 0.07/0.32 => ( idom_divide @ ( poly @ A5 ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(tcon_Polynomial_Opoly___Rings_Ocomm__ring__1_52,axiom,
% 0.07/0.32 ! [A5: $tType] :
% 0.07/0.32 ( ( comm_ring_1 @ A5 )
% 0.07/0.32 => ( comm_ring_1 @ ( poly @ A5 ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(tcon_Polynomial_Opoly___Rings_Osemiring__1_53,axiom,
% 0.07/0.32 ! [A5: $tType] :
% 0.07/0.32 ( ( comm_semiring_1 @ A5 )
% 0.07/0.32 => ( semiring_1 @ ( poly @ A5 ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(tcon_Polynomial_Opoly___Groups_Ogroup__add_54,axiom,
% 0.07/0.32 ! [A5: $tType] :
% 0.07/0.32 ( ( ab_group_add @ A5 )
% 0.07/0.32 => ( group_add @ ( poly @ A5 ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(tcon_Polynomial_Opoly___Rings_Ocomm__ring_55,axiom,
% 0.07/0.32 ! [A5: $tType] :
% 0.07/0.32 ( ( comm_ring @ A5 )
% 0.07/0.32 => ( comm_ring @ ( poly @ A5 ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(tcon_Polynomial_Opoly___Nat_Oring__char__0_56,axiom,
% 0.07/0.32 ! [A5: $tType] :
% 0.07/0.32 ( ( ( ring_char_0 @ A5 )
% 0.07/0.32 & ( comm_ring_1 @ A5 ) )
% 0.07/0.32 => ( ring_char_0 @ ( poly @ A5 ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(tcon_Polynomial_Opoly___Rings_Oring__1_57,axiom,
% 0.07/0.32 ! [A5: $tType] :
% 0.07/0.32 ( ( comm_ring_1 @ A5 )
% 0.07/0.32 => ( ring_1 @ ( poly @ A5 ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(tcon_Polynomial_Opoly___Power_Opower_58,axiom,
% 0.07/0.32 ! [A5: $tType] :
% 0.07/0.32 ( ( comm_semiring_1 @ A5 )
% 0.07/0.32 => ( power @ ( poly @ A5 ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(tcon_Polynomial_Opoly___Groups_Ozero_59,axiom,
% 0.07/0.32 ! [A5: $tType] :
% 0.07/0.32 ( ( zero @ A5 )
% 0.07/0.32 => ( zero @ ( poly @ A5 ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(tcon_Polynomial_Opoly___Groups_Oone_60,axiom,
% 0.07/0.32 ! [A5: $tType] :
% 0.07/0.32 ( ( comm_semiring_1 @ A5 )
% 0.07/0.32 => ( one @ ( poly @ A5 ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % Conjectures (1)
% 0.07/0.32 thf(conj_0,conjecture,
% 0.07/0.32 ord_less_eq @ real @ ( divide_divide @ real @ ( one_one @ real ) @ ( power_power @ real @ ( ring_1_of_int @ real @ b ) @ ( degree @ real @ p ) ) ) @ ( abs_abs @ real @ ( poly2 @ real @ p @ ( divide_divide @ real @ ( ring_1_of_int @ real @ a2 ) @ ( ring_1_of_int @ real @ b ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 %------------------------------------------------------------------------------
% 0.07/0.32 ------------------------
% 2.25/1.13 ---- Embedded in THF ---
% 2.25/1.13 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 2.25/1.13 %%% Generated on Tue Feb 24 03:26:11 EST 2026
% 2.25/1.13 %%% using '$$dhol' embedding, version 1.3.0.
% 2.25/1.13 %%% Logic specification used:
% 2.25/1.13 %%% thf(spec, logic, $$dhol).
% 2.25/1.13
% 2.25/1.13 % SZS output start ListOfTHF for /export/starexec/sandbox/tmp/tmp.qtQCAE7Wjp/DTF2DTF32390.p
% 2.25/1.13 thf(ty_t_Polynomial_Opoly, type, poly: ($tType > $tType)).
% 2.25/1.13 thf(ty_t_Real_Oreal, type, real: $tType).
% 2.25/1.13 thf(ty_t_Set_Oset, type, set: ($tType > $tType)).
% 2.25/1.13 thf(ty_t_Nat_Onat, type, nat: $tType).
% 2.25/1.13 thf(ty_t_Int_Oint, type, int: $tType).
% 2.25/1.13 thf(sy_cl_Groups_Oab__group__add, type, ab_group_add: (!> [A:$tType]: ($o))).
% 2.25/1.13 thf(sy_cl_Rings_Oidom, type, idom: (!> [A:$tType]: ($o))).
% 2.25/1.13 thf(sy_cl_Groups_Oone, type, one: (!> [A:$tType]: ($o))).
% 2.25/1.13 thf(sy_cl_Groups_Ozero, type, zero: (!> [A:$tType]: ($o))).
% 2.25/1.13 thf(sy_cl_Power_Opower, type, power: (!> [A:$tType]: ($o))).
% 2.25/1.13 thf(sy_cl_Fields_Ofield, type, field: (!> [A:$tType]: ($o))).
% 2.25/1.13 thf(sy_cl_Rings_Oring__1, type, ring_1: (!> [A:$tType]: ($o))).
% 2.25/1.13 thf(sy_cl_Nat_Oring__char__0, type, ring_char_0: (!> [A:$tType]: ($o))).
% 2.25/1.13 thf(sy_cl_Rings_Ocomm__ring, type, comm_ring: (!> [A:$tType]: ($o))).
% 2.25/1.13 thf(sy_cl_Groups_Ogroup__add, type, group_add: (!> [A:$tType]: ($o))).
% 2.25/1.13 thf(sy_cl_Rings_Osemiring__1, type, semiring_1: (!> [A:$tType]: ($o))).
% 2.25/1.13 thf(sy_cl_Rings_Ocomm__ring__1, type, comm_ring_1: (!> [A:$tType]: ($o))).
% 2.25/1.13 thf(sy_cl_Rings_Oidom__divide, type, idom_divide: (!> [A:$tType]: ($o))).
% 2.25/1.13 thf(sy_cl_Groups_Omonoid__mult, type, monoid_mult: (!> [A:$tType]: ($o))).
% 2.25/1.13 thf(sy_cl_Fields_Ofield__char__0, type, field_char_0: (!> [A:$tType]: ($o))).
% 2.25/1.13 thf(sy_cl_Fields_Odivision__ring, type, division_ring: (!> [A:$tType]: ($o))).
% 2.25/1.13 thf(sy_cl_Fields_Ofield__abs__sgn, type, field_abs_sgn: (!> [A:$tType]: ($o))).
% 2.25/1.13 thf(sy_cl_Rings_Ocomm__semiring__0, type, comm_semiring_0: (!> [A:$tType]: ($o))).
% 2.25/1.13 thf(sy_cl_Rings_Ocomm__semiring__1, type, comm_semiring_1: (!> [A:$tType]: ($o))).
% 2.25/1.13 thf(sy_cl_Rings_Olinordered__idom, type, linordered_idom: (!> [A:$tType]: ($o))).
% 2.25/1.13 thf(sy_cl_Fields_Olinordered__field, type, linordered_field: (!> [A:$tType]: ($o))).
% 2.25/1.13 thf(sy_cl_Groups_Ocomm__monoid__diff, type, comm_monoid_diff: (!> [A:$tType]: ($o))).
% 2.25/1.13 thf(sy_cl_Rings_Olinordered__semidom, type, linordered_semidom: (!> [A:$tType]: ($o))).
% 2.25/1.13 thf(sy_cl_Groups_Oordered__ab__group__add, type, ordered_ab_group_add: (!> [A:$tType]: ($o))).
% 2.25/1.13 thf(sy_cl_Rings_Oring__no__zero__divisors, type, ring_n68954251visors: (!> [A:$tType]: ($o))).
% 2.25/1.13 thf(sy_cl_Groups_Ocancel__comm__monoid__add, type, cancel1352612707id_add: (!> [A:$tType]: ($o))).
% 2.25/1.13 thf(sy_cl_Groups_Ocancel__ab__semigroup__add, type, cancel146912293up_add: (!> [A:$tType]: ($o))).
% 2.25/1.13 thf(sy_cl_Groups_Oordered__ab__group__add__abs, type, ordere142940540dd_abs: (!> [A:$tType]: ($o))).
% 2.25/1.13 thf(sy_cl_Rings_Osemiring__no__zero__divisors, type, semiri1193490041visors: (!> [A:$tType]: ($o))).
% 2.25/1.13 thf(sy_cl_Rings_Osemiring__1__no__zero__divisors, type, semiri134348788visors: (!> [A:$tType]: ($o))).
% 2.25/1.13 thf(sy_cl_Groups_Ocanonically__ordered__monoid__add, type, canoni770627133id_add: (!> [A:$tType]: ($o))).
% 2.25/1.13 thf(sy_c_Groups_Oabs__class_Oabs, type, abs_abs: (!> [A:$tType]: ((A > A)))).
% 2.25/1.13 thf(sy_c_Groups_Ominus__class_Ominus, type, minus_minus: (!> [A:$tType]: ((A > (A > A))))).
% 2.25/1.13 thf(sy_c_Groups_Oone__class_Oone, type, one_one: (!> [A:$tType]: (A))).
% 2.25/1.13 thf(sy_c_Groups_Ozero__class_Ozero, type, zero_zero: (!> [A:$tType]: (A))).
% 2.25/1.13 thf(sy_c_Int_Oring__1__class_OInts, type, ring_1_Ints: (!> [A:$tType]: ((set @ A)))).
% 2.25/1.13 thf(sy_c_Int_Oring__1__class_Oof__int, type, ring_1_of_int: (!> [A:$tType]: ((int > A)))).
% 2.25/1.13 thf(sy_c_Orderings_Oord__class_Oless, type, ord_less: (!> [A:$tType]: ((A > (A > $o))))).
% 2.25/1.13 thf(sy_c_Orderings_Oord__class_Oless__eq, type, ord_less_eq: (!> [A:$tType]: ((A > (A > $o))))).
% 2.25/1.13 thf(sy_c_Polynomial_Oalgebraic, type, algebraic: (!> [A:$tType]: ((A > $o)))).
% 2.25/1.13 thf(sy_c_Polynomial_Odegree, type, degree: (!> [A:$tType]: (((poly @ A) > nat)))).
% 2.25/1.13 thf(sy_c_Polynomial_Opoly, type, poly2: (!> [A:$tType]: (((poly @ A) > (A > A))))).
% 2.25/1.13 thf(sy_c_Polynomial_Opoly_Ocoeff, type, coeff: (!> [A:$tType]: (((poly @ A) > (nat > A))))).
% 2.25/1.13 thf(sy_c_Power_Opower__class_Opower, type, power_power: (!> [A:$tType]: ((A > (nat > A))))).
% 2.25/1.13 thf(sy_c_Rat_Ofield__char__0__class_ORats, type, field_char_0_Rats: (!> [A:$tType]: ((set @ A)))).
% 2.25/1.13 thf(sy_c_Rings_Odivide__class_Odivide, type, divide_divide: (!> [A:$tType]: ((A > (A > A))))).
% 2.25/1.13 thf(sy_c_Set_OCollect, type, collect: (!> [A:$tType]: (((A > $o) > (set @ A))))).
% 2.25/1.13 thf(sy_c_member, type, member: (!> [A:$tType]: ((A > ((set @ A) > $o))))).
% 2.25/1.13 thf(sy_v_A____, type, a: real).
% 2.25/1.13 thf(sy_v_M____, type, m: real).
% 2.25/1.13 thf(sy_v_a____, type, a2: int).
% 2.25/1.13 thf(sy_v_b____, type, b: int).
% 2.25/1.13 thf(sy_v_n____, type, n: nat).
% 2.25/1.13 thf(sy_v_p____, type, p: (poly @ real)).
% 2.25/1.13 thf(sy_v_roots____, type, roots: (set @ real)).
% 2.25/1.13 thf(sy_v_thesis, type, thesis: $o).
% 2.25/1.13 thf(sy_v_x, type, x: real).
% 2.25/1.13 thf(fact_0_assms_I2_J, axiom, ((algebraic @ real) @ x)).
% 2.25/1.13 thf(fact_1_p_I2_J, axiom, (p != (zero_zero @ (poly @ real)))).
% 2.25/1.13 thf(fact_2_p_I1_J, axiom, (! [I:nat]: ((((member @ real) @ (((coeff @ real) @ p) @ I)) @ (ring_1_Ints @ real))))).
% 2.25/1.13 thf(fact_3_b, axiom, (((ord_less @ int) @ (zero_zero @ int)) @ b)).
% 2.25/1.13 thf(fact_4_p_I3_J, axiom, ((((poly2 @ real) @ p) @ x) = (zero_zero @ real))).
% 2.25/1.13 thf(fact_5_no__root, axiom, ((((poly2 @ real) @ p) @ (((divide_divide @ real) @ ((ring_1_of_int @ real) @ a2)) @ ((ring_1_of_int @ real) @ b))) != (zero_zero @ real))).
% 2.25/1.13 thf(fact_6_n__def, axiom, (n = ((degree @ real) @ p))).
% 2.25/1.13 thf(fact_7_of__int__le__of__int__power__cancel__iff, axiom, (! [A:$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)))))))).
% 2.25/1.13 thf(fact_8_of__int__power__le__of__int__cancel__iff, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [X:int,B:int,W:nat]: (((((ord_less_eq @ A) @ ((ring_1_of_int @ A) @ X)) @ (((power_power @ A) @ ((ring_1_of_int @ A) @ B)) @ W)) = (((ord_less_eq @ int) @ X) @ (((power_power @ int) @ B) @ W))))))))).
% 2.25/1.13 thf(fact_9_of__int__1__le__iff, axiom, (! [A:$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)))))))).
% 2.25/1.13 thf(fact_10_of__int__le__1__iff, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [Z:int]: (((((ord_less_eq @ A) @ ((ring_1_of_int @ A) @ Z)) @ (one_one @ A)) = (((ord_less_eq @ int) @ Z) @ (one_one @ int))))))))).
% 2.25/1.13 thf(fact_11_poly__power, axiom, (! [A:$tType]: (((comm_semiring_1 @ A) => (! [P:(poly @ A),N:nat,X:A]: (((((poly2 @ A) @ (((power_power @ (poly @ A)) @ P) @ N)) @ X) = (((power_power @ A) @ (((poly2 @ A) @ P) @ X)) @ N)))))))).
% 2.25/1.13 thf(fact_12_poly__1, axiom, (! [A:$tType]: (((comm_semiring_1 @ A) => (! [X:A]: (((((poly2 @ A) @ (one_one @ (poly @ A))) @ X) = (one_one @ A)))))))).
% 2.25/1.13 thf(fact_13_of__int__abs, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [X:int]: ((((ring_1_of_int @ A) @ ((abs_abs @ int) @ X)) = ((abs_abs @ A) @ ((ring_1_of_int @ A) @ X))))))))).
% 2.25/1.13 thf(fact_14_of__int__power, axiom, (! [A:$tType]: (((ring_1 @ A) => (! [Z:int,N:nat]: ((((ring_1_of_int @ A) @ (((power_power @ int) @ Z) @ N)) = (((power_power @ A) @ ((ring_1_of_int @ A) @ Z)) @ N)))))))).
% 2.25/1.13 thf(fact_15_of__int__eq__of__int__power__cancel__iff, axiom, (! [A:$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)))))))).
% 2.25/1.13 thf(fact_16_of__int__power__eq__of__int__cancel__iff, axiom, (! [A:$tType]: (((ring_char_0 @ A) => (! [X:int,B:int,W:nat]: (((((ring_1_of_int @ A) @ X) = (((power_power @ A) @ ((ring_1_of_int @ A) @ B)) @ W)) = (X = (((power_power @ int) @ B) @ W))))))))).
% 2.25/1.13 thf(fact_17_of__int__1, axiom, (! [A:$tType]: (((ring_1 @ A) => (((ring_1_of_int @ A) @ (one_one @ int)) = (one_one @ A)))))).
% 2.25/1.13 thf(fact_18_of__int__eq__1__iff, axiom, (! [A:$tType]: (((ring_char_0 @ A) => (! [Z:int]: (((((ring_1_of_int @ A) @ Z) = (one_one @ A)) = (Z = (one_one @ int))))))))).
% 2.25/1.13 thf(fact_19_irrationsl, axiom, (~ ((((member @ real) @ x) @ (field_char_0_Rats @ real))))).
% 2.25/1.13 thf(fact_20_of__int__eq__iff, axiom, (! [A:$tType]: (((ring_char_0 @ A) => (! [W:int,Z:int]: (((((ring_1_of_int @ A) @ W) = ((ring_1_of_int @ A) @ Z)) = (W = Z)))))))).
% 2.25/1.13 thf(fact_21__092_060open_062_092_060And_062thesisa_O_A_I_092_060And_062p_O_A_092_060lbrakk_062_092_060And_062i_O_Acoeff_Ap_Ai_A_092_060in_062_A_092_060int_062_059_Ap_A_092_060noteq_062_A0_059_Apoly_Ap_Ax_A_061_A0_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesisa_J_A_092_060Longrightarrow_062_Athesisa_092_060close_062, axiom, (~ ((! [P2:(poly @ real)]: (((! [I2:nat]: ((((member @ real) @ (((coeff @ real) @ P2) @ I2)) @ (ring_1_Ints @ real)))) => ((P2 != (zero_zero @ (poly @ real))) => ((((poly2 @ real) @ P2) @ x) != (zero_zero @ real))))))))).
% 2.25/1.13 thf(fact_22__092_060open_062x_A_092_060notin_062_Aroots_092_060close_062, axiom, (~ ((((member @ real) @ x) @ roots)))).
% 2.25/1.13 thf(fact_23_of__int__eq__0__iff, axiom, (! [A:$tType]: (((ring_char_0 @ A) => (! [Z:int]: (((((ring_1_of_int @ A) @ Z) = (zero_zero @ A)) = (Z = (zero_zero @ int))))))))).
% 2.25/1.13 thf(fact_24_of__int__0__eq__iff, axiom, (! [A:$tType]: (((ring_char_0 @ A) => (! [Z:int]: ((((zero_zero @ A) = ((ring_1_of_int @ A) @ Z)) = (Z = (zero_zero @ int))))))))).
% 2.25/1.13 thf(fact_25_of__int__0, axiom, (! [A:$tType]: (((ring_1 @ A) => (((ring_1_of_int @ A) @ (zero_zero @ int)) = (zero_zero @ A)))))).
% 2.25/1.13 thf(fact_26_of__int__le__iff, axiom, (! [A:$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)))))))).
% 2.25/1.13 thf(fact_27_of__int__less__iff, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [W:int,Z:int]: (((((ord_less @ A) @ ((ring_1_of_int @ A) @ W)) @ ((ring_1_of_int @ A) @ Z)) = (((ord_less @ int) @ W) @ Z)))))))).
% 2.25/1.13 thf(fact_28_coeff__0, axiom, (! [A:$tType]: (((zero @ A) => (! [N:nat]: (((((coeff @ A) @ (zero_zero @ (poly @ A))) @ N) = (zero_zero @ A)))))))).
% 2.25/1.13 thf(fact_29_poly__0, axiom, (! [A:$tType]: (((comm_semiring_0 @ A) => (! [X:A]: (((((poly2 @ A) @ (zero_zero @ (poly @ A))) @ X) = (zero_zero @ A)))))))).
% 2.25/1.13 thf(fact_30_lead__coeff__of__int, axiom, (! [A:$tType]: (((comm_ring_1 @ A) => (! [K:int]: (((((coeff @ A) @ ((ring_1_of_int @ (poly @ A)) @ K)) @ ((degree @ A) @ ((ring_1_of_int @ (poly @ A)) @ K))) = ((ring_1_of_int @ A) @ K)))))))).
% 2.25/1.13 thf(fact_31_leading__coeff__0__iff, axiom, (! [A:$tType]: (((zero @ A) => (! [P:(poly @ A)]: ((((((coeff @ A) @ P) @ ((degree @ A) @ P)) = (zero_zero @ A)) = (P = (zero_zero @ (poly @ A)))))))))).
% 2.25/1.13 thf(fact_32_zabs__less__one__iff, axiom, (! [Z:int]: (((((ord_less @ int) @ ((abs_abs @ int) @ Z)) @ (one_one @ int)) = (Z = (zero_zero @ int)))))).
% 2.25/1.13 thf(fact_33_lead__coeff__1, axiom, (! [A:$tType]: (((comm_semiring_1 @ A) => ((((coeff @ A) @ (one_one @ (poly @ A))) @ ((degree @ A) @ (one_one @ (poly @ A)))) = (one_one @ A)))))).
% 2.25/1.13 thf(fact_34_of__int__le__0__iff, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [Z:int]: (((((ord_less_eq @ A) @ ((ring_1_of_int @ A) @ Z)) @ (zero_zero @ A)) = (((ord_less_eq @ int) @ Z) @ (zero_zero @ int))))))))).
% 2.25/1.13 thf(fact_35_of__int__0__le__iff, axiom, (! [A:$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)))))))).
% 2.25/1.13 thf(fact_36_of__int__less__0__iff, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [Z:int]: (((((ord_less @ A) @ ((ring_1_of_int @ A) @ Z)) @ (zero_zero @ A)) = (((ord_less @ int) @ Z) @ (zero_zero @ int))))))))).
% 2.25/1.13 thf(fact_37_of__int__0__less__iff, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [Z:int]: (((((ord_less @ A) @ (zero_zero @ A)) @ ((ring_1_of_int @ A) @ Z)) = (((ord_less @ int) @ (zero_zero @ int)) @ Z)))))))).
% 2.25/1.13 thf(fact_38_of__int__less__1__iff, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [Z:int]: (((((ord_less @ A) @ ((ring_1_of_int @ A) @ Z)) @ (one_one @ A)) = (((ord_less @ int) @ Z) @ (one_one @ int))))))))).
% 2.25/1.13 thf(fact_39_of__int__1__less__iff, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [Z:int]: (((((ord_less @ A) @ (one_one @ A)) @ ((ring_1_of_int @ A) @ Z)) = (((ord_less @ int) @ (one_one @ int)) @ Z)))))))).
% 2.25/1.13 thf(fact_40_of__int__power__less__of__int__cancel__iff, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [X:int,B:int,W:nat]: (((((ord_less @ A) @ ((ring_1_of_int @ A) @ X)) @ (((power_power @ A) @ ((ring_1_of_int @ A) @ B)) @ W)) = (((ord_less @ int) @ X) @ (((power_power @ int) @ B) @ W))))))))).
% 2.25/1.13 thf(fact_41_of__int__less__of__int__power__cancel__iff, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [B:int,W:nat,X:int]: (((((ord_less @ A) @ (((power_power @ A) @ ((ring_1_of_int @ A) @ B)) @ W)) @ ((ring_1_of_int @ A) @ X)) = (((ord_less @ int) @ (((power_power @ int) @ B) @ W)) @ X)))))))).
% 2.25/1.13 thf(fact_42_less__int__code_I1_J, axiom, (~ ((((ord_less @ int) @ (zero_zero @ int)) @ (zero_zero @ int))))).
% 2.25/1.13 thf(fact_43_less__eq__int__code_I1_J, axiom, (((ord_less_eq @ int) @ (zero_zero @ int)) @ (zero_zero @ int))).
% 2.25/1.13 thf(fact_44_poly__eqI, axiom, (! [A:$tType]: (((zero @ A) => (! [P:(poly @ A),Q:(poly @ A)]: (((! [N2:nat]: (((((coeff @ A) @ P) @ N2) = (((coeff @ A) @ Q) @ N2)))) => (P = Q)))))))).
% 2.25/1.13 thf(fact_45_mem__Collect__eq, axiom, (! [A:$tType,A2:A,P3:(A > $o)]: (((((member @ A) @ A2) @ ((collect @ A) @ P3)) = (P3 @ A2))))).
% 2.25/1.13 thf(fact_46_Collect__mem__eq, axiom, (! [A:$tType,A3:(set @ A)]: ((((collect @ A) @ (^ [X2:A]: ((((member @ A) @ X2) @ A3)))) = A3)))).
% 2.25/1.13 thf(fact_47_Collect__cong, axiom, (! [A:$tType,P3:(A > $o),Q2:(A > $o)]: (((! [X3:A]: (((P3 @ X3) = (Q2 @ X3)))) => (((collect @ A) @ P3) = ((collect @ A) @ Q2)))))).
% 2.25/1.13 thf(fact_48_ext, axiom, (! [B2:$tType,A:$tType,F:(A > B2),G:(A > B2)]: (((! [X3:A]: (((F @ X3) = (G @ X3)))) => (F = G))))).
% 2.25/1.13 thf(fact_49_poly__eq__iff, axiom, (! [A:$tType]: (((zero @ A) => ((^ [Y:(poly @ A),Z2:(poly @ A)]: ((Y = Z2))) = (^ [P4:(poly @ A),Q3:(poly @ A)]: ((! [N3:nat]: (((((coeff @ A) @ P4) @ N3) = (((coeff @ A) @ Q3) @ N3))))))))))).
% 2.25/1.13 thf(fact_50_Ints__0, axiom, (! [A:$tType]: (((ring_1 @ A) => (((member @ A) @ (zero_zero @ A)) @ (ring_1_Ints @ A)))))).
% 2.25/1.13 thf(fact_51_zero__poly_Orep__eq, axiom, (! [A:$tType]: (((zero @ A) => (((coeff @ A) @ (zero_zero @ (poly @ A))) = (^ [Uu:nat]: ((zero_zero @ A)))))))).
% 2.25/1.13 thf(fact_52_int__one__le__iff__zero__less, axiom, (! [Z:int]: (((((ord_less_eq @ int) @ (one_one @ int)) @ Z) = (((ord_less @ int) @ (zero_zero @ int)) @ Z))))).
% 2.25/1.13 thf(fact_53_coeff__inject, axiom, (! [A:$tType]: (((zero @ A) => (! [X:(poly @ A),Y2:(poly @ A)]: (((((coeff @ A) @ X) = ((coeff @ A) @ Y2)) = (X = Y2)))))))).
% 2.25/1.13 thf(fact_54_le__degree, axiom, (! [A:$tType]: (((zero @ A) => (! [P:(poly @ A),N:nat]: ((((((coeff @ A) @ P) @ N) != (zero_zero @ A)) => (((ord_less_eq @ nat) @ N) @ ((degree @ A) @ P))))))))).
% 2.25/1.13 thf(fact_55_poly__0__coeff__0, axiom, (! [A:$tType]: (((comm_semiring_0 @ A) => (! [P:(poly @ A)]: (((((poly2 @ A) @ P) @ (zero_zero @ A)) = (((coeff @ A) @ P) @ (zero_zero @ nat))))))))).
% 2.25/1.13 thf(fact_56_algebraic__def, axiom, (! [A:$tType]: (((field_char_0 @ A) => ((algebraic @ A) = (^ [X2:A]: ((? [P4:(poly @ A)]: (((! [I3:nat]: ((((member @ A) @ (((coeff @ A) @ P4) @ I3)) @ (ring_1_Ints @ A)))) & ((P4 != (zero_zero @ (poly @ A))) & ((((poly2 @ A) @ P4) @ X2) = (zero_zero @ A))))))))))))).
% 2.25/1.13 thf(fact_57_algebraicI, axiom, (! [A:$tType]: (((field_char_0 @ A) => (! [P:(poly @ A),X:A]: (((! [I4:nat]: ((((member @ A) @ (((coeff @ A) @ P) @ I4)) @ (ring_1_Ints @ A)))) => ((P != (zero_zero @ (poly @ A))) => (((((poly2 @ A) @ P) @ X) = (zero_zero @ A)) => ((algebraic @ A) @ X)))))))))).
% 2.25/1.13 thf(fact_58_algebraicE, axiom, (! [A:$tType]: (((field_char_0 @ A) => (! [X:A]: ((((algebraic @ A) @ X) => (~ ((! [P2:(poly @ A)]: (((! [I2:nat]: ((((member @ A) @ (((coeff @ A) @ P2) @ I2)) @ (ring_1_Ints @ A)))) => ((P2 != (zero_zero @ (poly @ A))) => ((((poly2 @ A) @ P2) @ X) != (zero_zero @ A))))))))))))))).
% 2.25/1.13 thf(fact_59_of__int__pos, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [Z:int]: (((((ord_less @ int) @ (zero_zero @ int)) @ Z) => (((ord_less @ A) @ (zero_zero @ A)) @ ((ring_1_of_int @ A) @ Z))))))))).
% 2.25/1.13 thf(fact_60_leading__coeff__neq__0, axiom, (! [A:$tType]: (((zero @ A) => (! [P:(poly @ A)]: (((P != (zero_zero @ (poly @ A))) => ((((coeff @ A) @ P) @ ((degree @ A) @ P)) != (zero_zero @ A))))))))).
% 2.25/1.13 thf(fact_61_poly__all__0__iff__0, axiom, (! [A:$tType]: ((((ring_char_0 @ A) & ((comm_ring_1 @ A) & (ring_n68954251visors @ A))) => (! [P:(poly @ A)]: (((! [X2:A]: (((((poly2 @ A) @ P) @ X2) = (zero_zero @ A)))) = (P = (zero_zero @ (poly @ A)))))))))).
% 2.25/1.13 thf(fact_62_Ints__1, axiom, (! [A:$tType]: (((ring_1 @ A) => (((member @ A) @ (one_one @ A)) @ (ring_1_Ints @ A)))))).
% 2.25/1.13 thf(fact_63_Ints__power, axiom, (! [A:$tType]: (((ring_1 @ A) => (! [A2:A,N:nat]: (((((member @ A) @ A2) @ (ring_1_Ints @ A)) => (((member @ A) @ (((power_power @ A) @ A2) @ N)) @ (ring_1_Ints @ A))))))))).
% 2.25/1.13 thf(fact_64_Ints__abs, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [A2:A]: (((((member @ A) @ A2) @ (ring_1_Ints @ A)) => (((member @ A) @ ((abs_abs @ A) @ A2)) @ (ring_1_Ints @ A))))))))).
% 2.25/1.13 thf(fact_65_Ints__of__int, axiom, (! [A:$tType]: (((ring_1 @ A) => (! [Z:int]: ((((member @ A) @ ((ring_1_of_int @ A) @ Z)) @ (ring_1_Ints @ A)))))))).
% 2.25/1.13 thf(fact_66_Ints__induct, axiom, (! [A:$tType]: (((ring_1 @ A) => (! [Q:A,P3:(A > $o)]: (((((member @ A) @ Q) @ (ring_1_Ints @ A)) => ((! [Z3:int]: ((P3 @ ((ring_1_of_int @ A) @ Z3)))) => (P3 @ Q))))))))).
% 2.25/1.13 thf(fact_67_Ints__cases, axiom, (! [A:$tType]: (((ring_1 @ A) => (! [Q:A]: (((((member @ A) @ Q) @ (ring_1_Ints @ A)) => (~ ((! [Z3:int]: ((Q != ((ring_1_of_int @ A) @ Z3))))))))))))).
% 2.25/1.13 thf(fact_68_Ints__nonzero__abs__less1, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [X:A]: (((((member @ A) @ X) @ (ring_1_Ints @ A)) => ((((ord_less @ A) @ ((abs_abs @ A) @ X)) @ (one_one @ A)) => (X = (zero_zero @ A)))))))))).
% 2.25/1.13 thf(fact_69_of__int__nonneg, axiom, (! [A:$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))))))))).
% 2.25/1.13 thf(fact_70_lead__coeff__power, axiom, (! [A:$tType]: ((((comm_semiring_1 @ A) & (semiri1193490041visors @ A)) => (! [P:(poly @ A),N:nat]: (((((coeff @ A) @ (((power_power @ (poly @ A)) @ P) @ N)) @ ((degree @ A) @ (((power_power @ (poly @ A)) @ P) @ N))) = (((power_power @ A) @ (((coeff @ A) @ P) @ ((degree @ A) @ P))) @ N)))))))).
% 2.25/1.13 thf(fact_71_Ints__nonzero__abs__ge1, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [X:A]: (((((member @ A) @ X) @ (ring_1_Ints @ A)) => ((X != (zero_zero @ A)) => (((ord_less_eq @ A) @ (one_one @ A)) @ ((abs_abs @ A) @ X)))))))))).
% 2.25/1.13 thf(fact_72_poly__eq__poly__eq__iff, axiom, (! [A:$tType]: ((((ring_char_0 @ A) & ((comm_ring_1 @ A) & (ring_n68954251visors @ A))) => (! [P:(poly @ A),Q:(poly @ A)]: (((((poly2 @ A) @ P) = ((poly2 @ A) @ Q)) = (P = Q)))))))).
% 2.25/1.13 thf(fact_73_of__int__lessD, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [N:int,X:A]: (((((ord_less @ A) @ ((abs_abs @ A) @ ((ring_1_of_int @ A) @ N))) @ X) => ((N = (zero_zero @ int)) | (((ord_less @ A) @ (one_one @ A)) @ X))))))))).
% 2.25/1.13 thf(fact_74_of__int__leD, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [N:int,X:A]: (((((ord_less_eq @ A) @ ((abs_abs @ A) @ ((ring_1_of_int @ A) @ N))) @ X) => ((N = (zero_zero @ int)) | (((ord_less_eq @ A) @ (one_one @ A)) @ X))))))))).
% 2.25/1.13 thf(fact_75_int__poly__rat__no__root__ge, axiom, (! [P:(poly @ real),B:int,A2:int]: (((! [N2:nat]: ((((member @ real) @ (((coeff @ real) @ P) @ N2)) @ (ring_1_Ints @ real)))) => ((((ord_less @ int) @ (zero_zero @ int)) @ B) => (((((poly2 @ real) @ P) @ (((divide_divide @ real) @ ((ring_1_of_int @ real) @ A2)) @ ((ring_1_of_int @ real) @ B))) != (zero_zero @ real)) => (((ord_less_eq @ real) @ (((divide_divide @ real) @ (one_one @ real)) @ (((power_power @ real) @ ((ring_1_of_int @ real) @ B)) @ ((degree @ real) @ P)))) @ ((abs_abs @ real) @ (((poly2 @ real) @ P) @ (((divide_divide @ real) @ ((ring_1_of_int @ real) @ A2)) @ ((ring_1_of_int @ real) @ B))))))))))).
% 2.25/1.13 thf(fact_76_power__decreasing__iff, axiom, (! [A:$tType]: (((linordered_semidom @ A) => (! [B:A,M:nat,N:nat]: (((((ord_less @ A) @ (zero_zero @ A)) @ B) => ((((ord_less @ A) @ B) @ (one_one @ A)) => ((((ord_less_eq @ A) @ (((power_power @ A) @ B) @ M)) @ (((power_power @ A) @ B) @ N)) = (((ord_less_eq @ nat) @ N) @ M)))))))))).
% 2.25/1.13 thf(fact_77_le__divide__eq__1__pos, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A2:A,B:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ A2) => ((((ord_less_eq @ A) @ (one_one @ A)) @ (((divide_divide @ A) @ B) @ A2)) = (((ord_less_eq @ A) @ A2) @ B))))))))).
% 2.25/1.13 thf(fact_78_le__divide__eq__1__neg, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A2:A,B:A]: (((((ord_less @ A) @ A2) @ (zero_zero @ A)) => ((((ord_less_eq @ A) @ (one_one @ A)) @ (((divide_divide @ A) @ B) @ A2)) = (((ord_less_eq @ A) @ B) @ A2))))))))).
% 2.25/1.13 thf(fact_79_divide__le__eq__1__pos, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A2:A,B:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ A2) => ((((ord_less_eq @ A) @ (((divide_divide @ A) @ B) @ A2)) @ (one_one @ A)) = (((ord_less_eq @ A) @ B) @ A2))))))))).
% 2.25/1.13 thf(fact_80_divide__le__eq__1__neg, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A2:A,B:A]: (((((ord_less @ A) @ A2) @ (zero_zero @ A)) => ((((ord_less_eq @ A) @ (((divide_divide @ A) @ B) @ A2)) @ (one_one @ A)) = (((ord_less_eq @ A) @ A2) @ B))))))))).
% 2.25/1.13 thf(fact_81__092_060open_062A_A_P_Areal__of__int_Ab_A_094_An_A_092_060le_062_AA_A_P_A1_092_060close_062, axiom, (((ord_less_eq @ real) @ (((divide_divide @ real) @ a) @ (((power_power @ real) @ ((ring_1_of_int @ real) @ b)) @ n))) @ (((divide_divide @ real) @ a) @ (one_one @ real)))).
% 2.25/1.13 thf(fact_82_zero__less__power__abs__iff, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [A2:A,N:nat]: (((((ord_less @ A) @ (zero_zero @ A)) @ (((power_power @ A) @ ((abs_abs @ A) @ A2)) @ N)) = ((A2 != (zero_zero @ A)) | (N = (zero_zero @ nat)))))))))).
% 2.25/1.13 thf(fact_83_zero__le__divide__abs__iff, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A2:A,B:A]: (((((ord_less_eq @ A) @ (zero_zero @ A)) @ (((divide_divide @ A) @ A2) @ ((abs_abs @ A) @ B))) = ((((ord_less_eq @ A) @ (zero_zero @ A)) @ A2) | (B = (zero_zero @ A)))))))))).
% 2.25/1.13 thf(fact_84_divide__le__0__abs__iff, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A2:A,B:A]: (((((ord_less_eq @ A) @ (((divide_divide @ A) @ A2) @ ((abs_abs @ A) @ B))) @ (zero_zero @ A)) = ((((ord_less_eq @ A) @ A2) @ (zero_zero @ A)) | (B = (zero_zero @ A)))))))))).
% 2.25/1.13 thf(fact_85_A__pos, axiom, (((ord_less @ real) @ (zero_zero @ real)) @ a)).
% 2.25/1.13 thf(fact_86_A__less_I1_J, axiom, (((ord_less @ real) @ a) @ (one_one @ real))).
% 2.25/1.13 thf(fact_87__092_060open_062A_A_092_060le_062_A1_092_060close_062, axiom, (((ord_less_eq @ real) @ a) @ (one_one @ real))).
% 2.25/1.13 thf(fact_88_power__one__right, axiom, (! [A:$tType]: (((monoid_mult @ A) => (! [A2:A]: (((((power_power @ A) @ A2) @ (one_one @ nat)) = A2))))))).
% 2.25/1.13 thf(fact_89_nat__zero__less__power__iff, axiom, (! [X:nat,N:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ (((power_power @ nat) @ X) @ N)) = ((((ord_less @ nat) @ (zero_zero @ nat)) @ X) | (N = (zero_zero @ nat))))))).
% 2.25/1.13 thf(fact_90_divide__eq__0__iff, axiom, (! [A:$tType]: (((field @ A) => (! [A2:A,B:A]: ((((((divide_divide @ A) @ A2) @ B) = (zero_zero @ A)) = ((A2 = (zero_zero @ A)) | (B = (zero_zero @ A)))))))))).
% 2.25/1.13 thf(fact_91_divide__cancel__left, axiom, (! [A:$tType]: (((field @ A) => (! [C:A,A2:A,B:A]: ((((((divide_divide @ A) @ C) @ A2) = (((divide_divide @ A) @ C) @ B)) = ((C = (zero_zero @ A)) | (A2 = B))))))))).
% 2.25/1.13 thf(fact_92_divide__cancel__right, axiom, (! [A:$tType]: (((field @ A) => (! [A2:A,C:A,B:A]: ((((((divide_divide @ A) @ A2) @ C) = (((divide_divide @ A) @ B) @ C)) = ((C = (zero_zero @ A)) | (A2 = B))))))))).
% 2.25/1.13 thf(fact_93_division__ring__divide__zero, axiom, (! [A:$tType]: (((division_ring @ A) => (! [A2:A]: (((((divide_divide @ A) @ A2) @ (zero_zero @ A)) = (zero_zero @ A)))))))).
% 2.25/1.13 thf(fact_94_power__one, axiom, (! [A:$tType]: (((monoid_mult @ A) => (! [N:nat]: (((((power_power @ A) @ (one_one @ A)) @ N) = (one_one @ A)))))))).
% 2.25/1.13 thf(fact_95_abs__divide, axiom, (! [A:$tType]: (((field_abs_sgn @ A) => (! [A2:A,B:A]: ((((abs_abs @ A) @ (((divide_divide @ A) @ A2) @ B)) = (((divide_divide @ A) @ ((abs_abs @ A) @ A2)) @ ((abs_abs @ A) @ B))))))))).
% 2.25/1.13 thf(fact_96_divide__eq__1__iff, axiom, (! [A:$tType]: (((field @ A) => (! [A2:A,B:A]: ((((((divide_divide @ A) @ A2) @ B) = (one_one @ A)) = ((B != (zero_zero @ A)) & (A2 = B))))))))).
% 2.25/1.13 thf(fact_97_one__eq__divide__iff, axiom, (! [A:$tType]: (((field @ A) => (! [A2:A,B:A]: ((((one_one @ A) = (((divide_divide @ A) @ A2) @ B)) = ((B != (zero_zero @ A)) & (A2 = B))))))))).
% 2.25/1.13 thf(fact_98_divide__self, axiom, (! [A:$tType]: (((division_ring @ A) => (! [A2:A]: (((A2 != (zero_zero @ A)) => ((((divide_divide @ A) @ A2) @ A2) = (one_one @ A))))))))).
% 2.25/1.13 thf(fact_99_divide__self__if, axiom, (! [A:$tType]: (((division_ring @ A) => (! [A2:A]: ((((A2 = (zero_zero @ A)) => ((((divide_divide @ A) @ A2) @ A2) = (zero_zero @ A))) & ((A2 != (zero_zero @ A)) => ((((divide_divide @ A) @ A2) @ A2) = (one_one @ A)))))))))).
% 2.25/1.13 thf(fact_100_divide__eq__eq__1, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [B:A,A2:A]: ((((((divide_divide @ A) @ B) @ A2) = (one_one @ A)) = ((A2 != (zero_zero @ A)) & (A2 = B))))))))).
% 2.25/1.13 thf(fact_101_eq__divide__eq__1, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [B:A,A2:A]: ((((one_one @ A) = (((divide_divide @ A) @ B) @ A2)) = ((A2 != (zero_zero @ A)) & (A2 = B))))))))).
% 2.25/1.13 thf(fact_102_one__divide__eq__0__iff, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A2:A]: ((((((divide_divide @ A) @ (one_one @ A)) @ A2) = (zero_zero @ A)) = (A2 = (zero_zero @ A))))))))).
% 2.25/1.13 thf(fact_103_zero__eq__1__divide__iff, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A2:A]: ((((zero_zero @ A) = (((divide_divide @ A) @ (one_one @ A)) @ A2)) = (A2 = (zero_zero @ A))))))))).
% 2.25/1.13 thf(fact_104_power__inject__exp, axiom, (! [A:$tType]: (((linordered_semidom @ A) => (! [A2:A,M:nat,N:nat]: (((((ord_less @ A) @ (one_one @ A)) @ A2) => (((((power_power @ A) @ A2) @ M) = (((power_power @ A) @ A2) @ N)) = (M = N))))))))).
% 2.25/1.13 thf(fact_105_power__strict__increasing__iff, axiom, (! [A:$tType]: (((linordered_semidom @ A) => (! [B:A,X:nat,Y2:nat]: (((((ord_less @ A) @ (one_one @ A)) @ B) => ((((ord_less @ A) @ (((power_power @ A) @ B) @ X)) @ (((power_power @ A) @ B) @ Y2)) = (((ord_less @ nat) @ X) @ Y2))))))))).
% 2.25/1.13 thf(fact_106_power__eq__0__iff, axiom, (! [A:$tType]: (((semiri134348788visors @ A) => (! [A2:A,N:nat]: ((((((power_power @ A) @ A2) @ N) = (zero_zero @ A)) = ((A2 = (zero_zero @ A)) & (((ord_less @ nat) @ (zero_zero @ nat)) @ N))))))))).
% 2.25/1.13 thf(fact_107_degree__0, axiom, (! [A:$tType]: (((zero @ A) => (((degree @ A) @ (zero_zero @ (poly @ A))) = (zero_zero @ nat)))))).
% 2.25/1.13 thf(fact_108_degree__1, axiom, (! [A:$tType]: (((comm_semiring_1 @ A) => (((degree @ A) @ (one_one @ (poly @ A))) = (zero_zero @ nat)))))).
% 2.25/1.13 thf(fact_109_degree__of__int, axiom, (! [A:$tType]: (((comm_ring_1 @ A) => (! [K:int]: ((((degree @ A) @ ((ring_1_of_int @ (poly @ A)) @ K)) = (zero_zero @ nat)))))))).
% 2.25/1.13 thf(fact_110_divide__le__0__1__iff, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A2:A]: (((((ord_less_eq @ A) @ (((divide_divide @ A) @ (one_one @ A)) @ A2)) @ (zero_zero @ A)) = (((ord_less_eq @ A) @ A2) @ (zero_zero @ A))))))))).
% 2.25/1.13 thf(fact_111_zero__le__divide__1__iff, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A2:A]: (((((ord_less_eq @ A) @ (zero_zero @ A)) @ (((divide_divide @ A) @ (one_one @ A)) @ A2)) = (((ord_less_eq @ A) @ (zero_zero @ A)) @ A2)))))))).
% 2.25/1.13 thf(fact_112_divide__less__0__1__iff, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A2:A]: (((((ord_less @ A) @ (((divide_divide @ A) @ (one_one @ A)) @ A2)) @ (zero_zero @ A)) = (((ord_less @ A) @ A2) @ (zero_zero @ A))))))))).
% 2.25/1.13 thf(fact_113_divide__less__eq__1__neg, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A2:A,B:A]: (((((ord_less @ A) @ A2) @ (zero_zero @ A)) => ((((ord_less @ A) @ (((divide_divide @ A) @ B) @ A2)) @ (one_one @ A)) = (((ord_less @ A) @ A2) @ B))))))))).
% 2.25/1.13 thf(fact_114_divide__less__eq__1__pos, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A2:A,B:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ A2) => ((((ord_less @ A) @ (((divide_divide @ A) @ B) @ A2)) @ (one_one @ A)) = (((ord_less @ A) @ B) @ A2))))))))).
% 2.25/1.13 thf(fact_115_less__divide__eq__1__neg, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A2:A,B:A]: (((((ord_less @ A) @ A2) @ (zero_zero @ A)) => ((((ord_less @ A) @ (one_one @ A)) @ (((divide_divide @ A) @ B) @ A2)) = (((ord_less @ A) @ B) @ A2))))))))).
% 2.25/1.13 thf(fact_116_less__divide__eq__1__pos, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A2:A,B:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ A2) => ((((ord_less @ A) @ (one_one @ A)) @ (((divide_divide @ A) @ B) @ A2)) = (((ord_less @ A) @ A2) @ B))))))))).
% 2.25/1.13 thf(fact_117_zero__less__divide__1__iff, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A2:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ (((divide_divide @ A) @ (one_one @ A)) @ A2)) = (((ord_less @ A) @ (zero_zero @ A)) @ A2)))))))).
% 2.25/1.13 thf(fact_118_power__strict__decreasing__iff, axiom, (! [A:$tType]: (((linordered_semidom @ A) => (! [B:A,M:nat,N:nat]: (((((ord_less @ A) @ (zero_zero @ A)) @ B) => ((((ord_less @ A) @ B) @ (one_one @ A)) => ((((ord_less @ A) @ (((power_power @ A) @ B) @ M)) @ (((power_power @ A) @ B) @ N)) = (((ord_less @ nat) @ N) @ M)))))))))).
% 2.25/1.13 thf(fact_119_power__mono__iff, axiom, (! [A:$tType]: (((linordered_semidom @ A) => (! [A2:A,B:A,N:nat]: (((((ord_less_eq @ A) @ (zero_zero @ A)) @ A2) => ((((ord_less_eq @ A) @ (zero_zero @ A)) @ B) => ((((ord_less @ nat) @ (zero_zero @ nat)) @ N) => ((((ord_less_eq @ A) @ (((power_power @ A) @ A2) @ N)) @ (((power_power @ A) @ B) @ N)) = (((ord_less_eq @ A) @ A2) @ B))))))))))).
% 2.25/1.13 thf(fact_120_power__increasing__iff, axiom, (! [A:$tType]: (((linordered_semidom @ A) => (! [B:A,X:nat,Y2:nat]: (((((ord_less @ A) @ (one_one @ A)) @ B) => ((((ord_less_eq @ A) @ (((power_power @ A) @ B) @ X)) @ (((power_power @ A) @ B) @ Y2)) = (((ord_less_eq @ nat) @ X) @ Y2))))))))).
% 2.25/1.13 thf(fact_121__092_060open_062_092_060not_062_AA_A_P_Areal__of__int_Ab_A_094_An_A_060_A_092_060bar_062x_A_N_Areal__of__int_Aa_A_P_Areal__of__int_Ab_092_060bar_062_092_060close_062, axiom, (~ ((((ord_less @ real) @ (((divide_divide @ real) @ a) @ (((power_power @ real) @ ((ring_1_of_int @ real) @ b)) @ n))) @ ((abs_abs @ real) @ (((minus_minus @ real) @ x) @ (((divide_divide @ real) @ ((ring_1_of_int @ real) @ a2)) @ ((ring_1_of_int @ real) @ b)))))))).
% 2.25/1.13 thf(fact_122_nat__power__less__imp__less, axiom, (! [I:nat,M:nat,N:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ I) => ((((ord_less @ nat) @ (((power_power @ nat) @ I) @ M)) @ (((power_power @ nat) @ I) @ N)) => (((ord_less @ nat) @ M) @ N)))))).
% 2.25/1.13 thf(fact_123_div__poly__less, axiom, (! [A:$tType]: (((field @ A) => (! [X:(poly @ A),Y2:(poly @ A)]: (((((ord_less @ nat) @ ((degree @ A) @ X)) @ ((degree @ A) @ Y2)) => ((((divide_divide @ (poly @ A)) @ X) @ Y2) = (zero_zero @ (poly @ A)))))))))).
% 2.25/1.13 thf(fact_124_self__le__power, axiom, (! [A:$tType]: (((linordered_semidom @ A) => (! [A2:A,N:nat]: (((((ord_less_eq @ A) @ (one_one @ A)) @ A2) => ((((ord_less @ nat) @ (zero_zero @ nat)) @ N) => (((ord_less_eq @ A) @ A2) @ (((power_power @ A) @ A2) @ N)))))))))).
% 2.25/1.13 thf(fact_125_divide__poly__0, axiom, (! [A:$tType]: (((idom_divide @ A) => (! [F:(poly @ A)]: (((((divide_divide @ (poly @ A)) @ F) @ (zero_zero @ (poly @ A))) = (zero_zero @ (poly @ A))))))))).
% 2.25/1.13 thf(fact_126_zero__power, axiom, (! [A:$tType]: (((semiring_1 @ A) => (! [N:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ N) => ((((power_power @ A) @ (zero_zero @ A)) @ N) = (zero_zero @ A))))))))).
% 2.25/1.13 thf(fact_127_power__0, axiom, (! [A:$tType]: (((power @ A) => (! [A2:A]: (((((power_power @ A) @ A2) @ (zero_zero @ nat)) = (one_one @ A)))))))).
% 2.25/1.13 thf(fact_128_power__eq__iff__eq__base, axiom, (! [A:$tType]: (((linordered_semidom @ A) => (! [N:nat,A2:A,B:A]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ N) => ((((ord_less_eq @ A) @ (zero_zero @ A)) @ A2) => ((((ord_less_eq @ A) @ (zero_zero @ A)) @ B) => (((((power_power @ A) @ A2) @ N) = (((power_power @ A) @ B) @ N)) = (A2 = B))))))))))).
% 2.25/1.13 thf(fact_129_power__eq__imp__eq__base, axiom, (! [A:$tType]: (((linordered_semidom @ A) => (! [A2:A,N:nat,B:A]: ((((((power_power @ A) @ A2) @ N) = (((power_power @ A) @ B) @ N)) => ((((ord_less_eq @ A) @ (zero_zero @ A)) @ A2) => ((((ord_less_eq @ A) @ (zero_zero @ A)) @ B) => ((((ord_less @ nat) @ (zero_zero @ nat)) @ N) => (A2 = B))))))))))).
% 2.25/1.13 thf(fact_130_one__less__power, axiom, (! [A:$tType]: (((linordered_semidom @ A) => (! [A2:A,N:nat]: (((((ord_less @ A) @ (one_one @ A)) @ A2) => ((((ord_less @ nat) @ (zero_zero @ nat)) @ N) => (((ord_less @ A) @ (one_one @ A)) @ (((power_power @ A) @ A2) @ N)))))))))).
% 2.25/1.13 thf(fact_131_poly__IVT__neg, axiom, (! [A2:real,B:real,P:(poly @ real)]: (((((ord_less @ real) @ A2) @ B) => ((((ord_less @ real) @ (zero_zero @ real)) @ (((poly2 @ real) @ P) @ A2)) => ((((ord_less @ real) @ (((poly2 @ real) @ P) @ B)) @ (zero_zero @ real)) => (? [X3:real]: (((((ord_less @ real) @ A2) @ X3) & ((((ord_less @ real) @ X3) @ B) & ((((poly2 @ real) @ P) @ X3) = (zero_zero @ real)))))))))))).
% 2.25/1.13 thf(fact_132_poly__IVT__pos, axiom, (! [A2:real,B:real,P:(poly @ real)]: (((((ord_less @ real) @ A2) @ B) => ((((ord_less @ real) @ (((poly2 @ real) @ P) @ A2)) @ (zero_zero @ real)) => ((((ord_less @ real) @ (zero_zero @ real)) @ (((poly2 @ real) @ P) @ B)) => (? [X3:real]: (((((ord_less @ real) @ A2) @ X3) & ((((ord_less @ real) @ X3) @ B) & ((((poly2 @ real) @ P) @ X3) = (zero_zero @ real)))))))))))).
% 2.25/1.13 thf(fact_133_linordered__field__no__ub, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [X4:A]: ((? [X_1:A]: ((((ord_less @ A) @ X4) @ X_1))))))))).
% 2.25/1.13 thf(fact_134_linordered__field__no__lb, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [X4:A]: ((? [Y3:A]: ((((ord_less @ A) @ Y3) @ X4))))))))).
% 2.25/1.13 thf(fact_135_coeff__eq__0, axiom, (! [A:$tType]: (((zero @ A) => (! [P:(poly @ A),N:nat]: (((((ord_less @ nat) @ ((degree @ A) @ P)) @ N) => ((((coeff @ A) @ P) @ N) = (zero_zero @ A))))))))).
% 2.25/1.13 thf(fact_136_less__degree__imp, axiom, (! [A:$tType]: (((zero @ A) => (! [N:nat,P:(poly @ A)]: (((((ord_less @ nat) @ N) @ ((degree @ A) @ P)) => (? [I4:nat]: (((((ord_less @ nat) @ N) @ I4) & ((((coeff @ A) @ P) @ I4) != (zero_zero @ A)))))))))))).
% 2.25/1.13 thf(fact_137_degree__le, axiom, (! [A:$tType]: (((zero @ A) => (! [N:nat,P:(poly @ A)]: (((! [I4:nat]: (((((ord_less @ nat) @ N) @ I4) => ((((coeff @ A) @ P) @ I4) = (zero_zero @ A))))) => (((ord_less_eq @ nat) @ ((degree @ A) @ P)) @ N)))))))).
% 2.25/1.13 thf(fact_138_Liouville__Numbers__Misc_ORats__cases_H, axiom, (! [X:real]: (((((member @ real) @ X) @ (field_char_0_Rats @ real)) => (~ ((! [P2:int,Q4:int]: (((((ord_less @ int) @ (zero_zero @ int)) @ Q4) => (X != (((divide_divide @ real) @ ((ring_1_of_int @ real) @ P2)) @ ((ring_1_of_int @ real) @ Q4)))))))))))).
% 2.25/1.13 thf(fact_139_coeff__0__power, axiom, (! [A:$tType]: (((comm_semiring_1 @ A) => (! [P:(poly @ A),N:nat]: (((((coeff @ A) @ (((power_power @ (poly @ A)) @ P) @ N)) @ (zero_zero @ nat)) = (((power_power @ A) @ (((coeff @ A) @ P) @ (zero_zero @ nat))) @ N)))))))).
% 2.25/1.13 thf(fact_140_eq__zero__or__degree__less, axiom, (! [A:$tType]: (((zero @ A) => (! [P:(poly @ A),N:nat]: (((((ord_less_eq @ nat) @ ((degree @ A) @ P)) @ N) => (((((coeff @ A) @ P) @ N) = (zero_zero @ A)) => ((P = (zero_zero @ (poly @ A))) | (((ord_less @ nat) @ ((degree @ A) @ P)) @ N)))))))))).
% 2.25/1.13 thf(fact_141_power__not__zero, axiom, (! [A:$tType]: (((semiri134348788visors @ A) => (! [A2:A,N:nat]: (((A2 != (zero_zero @ A)) => ((((power_power @ A) @ A2) @ N) != (zero_zero @ A))))))))).
% 2.25/1.13 thf(fact_142_power__divide, axiom, (! [A:$tType]: (((field @ A) => (! [A2:A,B:A,N:nat]: (((((power_power @ A) @ (((divide_divide @ A) @ A2) @ B)) @ N) = (((divide_divide @ A) @ (((power_power @ A) @ A2) @ N)) @ (((power_power @ A) @ B) @ N))))))))).
% 2.25/1.13 thf(fact_143_poly__pinfty__gt__lc, axiom, (! [P:(poly @ real)]: (((((ord_less @ real) @ (zero_zero @ real)) @ (((coeff @ real) @ P) @ ((degree @ real) @ P))) => (? [N2:real]: ((! [X4:real]: (((((ord_less_eq @ real) @ N2) @ X4) => (((ord_less_eq @ real) @ (((coeff @ real) @ P) @ ((degree @ real) @ P))) @ (((poly2 @ real) @ P) @ X4))))))))))).
% 2.25/1.13 thf(fact_144_power__abs, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [A2:A,N:nat]: ((((abs_abs @ A) @ (((power_power @ A) @ A2) @ N)) = (((power_power @ A) @ ((abs_abs @ A) @ A2)) @ N)))))))).
% 2.25/1.13 thf(fact_145_algebraic__altdef, axiom, (! [B2:$tType]: (((field_char_0 @ B2) => ((algebraic @ B2) = (^ [X2:B2]: ((? [P4:(poly @ B2)]: (((! [I3:nat]: ((((member @ B2) @ (((coeff @ B2) @ P4) @ I3)) @ (field_char_0_Rats @ B2)))) & ((P4 != (zero_zero @ (poly @ B2))) & ((((poly2 @ B2) @ P4) @ X2) = (zero_zero @ B2))))))))))))).
% 2.25/1.13 thf(fact_146_divide__le__0__iff, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A2:A,B:A]: (((((ord_less_eq @ A) @ (((divide_divide @ A) @ A2) @ B)) @ (zero_zero @ A)) = (((((ord_less_eq @ A) @ (zero_zero @ A)) @ A2) & (((ord_less_eq @ A) @ B) @ (zero_zero @ A))) | ((((ord_less_eq @ A) @ A2) @ (zero_zero @ A)) & (((ord_less_eq @ A) @ (zero_zero @ A)) @ B)))))))))).
% 2.25/1.13 thf(fact_147_divide__right__mono, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A2:A,B:A,C:A]: (((((ord_less_eq @ A) @ A2) @ B) => ((((ord_less_eq @ A) @ (zero_zero @ A)) @ C) => (((ord_less_eq @ A) @ (((divide_divide @ A) @ A2) @ C)) @ (((divide_divide @ A) @ B) @ C)))))))))).
% 2.25/1.13 thf(fact_148_zero__le__divide__iff, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A2:A,B:A]: (((((ord_less_eq @ A) @ (zero_zero @ A)) @ (((divide_divide @ A) @ A2) @ B)) = (((((ord_less_eq @ A) @ (zero_zero @ A)) @ A2) & (((ord_less_eq @ A) @ (zero_zero @ A)) @ B)) | ((((ord_less_eq @ A) @ A2) @ (zero_zero @ A)) & (((ord_less_eq @ A) @ B) @ (zero_zero @ A))))))))))).
% 2.25/1.13 thf(fact_149_divide__nonneg__nonneg, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [X:A,Y2:A]: (((((ord_less_eq @ A) @ (zero_zero @ A)) @ X) => ((((ord_less_eq @ A) @ (zero_zero @ A)) @ Y2) => (((ord_less_eq @ A) @ (zero_zero @ A)) @ (((divide_divide @ A) @ X) @ Y2)))))))))).
% 2.25/1.13 thf(fact_150_divide__nonneg__nonpos, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [X:A,Y2:A]: (((((ord_less_eq @ A) @ (zero_zero @ A)) @ X) => ((((ord_less_eq @ A) @ Y2) @ (zero_zero @ A)) => (((ord_less_eq @ A) @ (((divide_divide @ A) @ X) @ Y2)) @ (zero_zero @ A)))))))))).
% 2.25/1.13 thf(fact_151_divide__nonpos__nonneg, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [X:A,Y2:A]: (((((ord_less_eq @ A) @ X) @ (zero_zero @ A)) => ((((ord_less_eq @ A) @ (zero_zero @ A)) @ Y2) => (((ord_less_eq @ A) @ (((divide_divide @ A) @ X) @ Y2)) @ (zero_zero @ A)))))))))).
% 2.25/1.13 thf(fact_152_divide__nonpos__nonpos, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [X:A,Y2:A]: (((((ord_less_eq @ A) @ X) @ (zero_zero @ A)) => ((((ord_less_eq @ A) @ Y2) @ (zero_zero @ A)) => (((ord_less_eq @ A) @ (zero_zero @ A)) @ (((divide_divide @ A) @ X) @ Y2)))))))))).
% 2.25/1.13 thf(fact_153_divide__right__mono__neg, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A2:A,B:A,C:A]: (((((ord_less_eq @ A) @ A2) @ B) => ((((ord_less_eq @ A) @ C) @ (zero_zero @ A)) => (((ord_less_eq @ A) @ (((divide_divide @ A) @ B) @ C)) @ (((divide_divide @ A) @ A2) @ C)))))))))).
% 2.25/1.13 thf(fact_154_power__mono, axiom, (! [A:$tType]: (((linordered_semidom @ A) => (! [A2:A,B:A,N:nat]: (((((ord_less_eq @ A) @ A2) @ B) => ((((ord_less_eq @ A) @ (zero_zero @ A)) @ A2) => (((ord_less_eq @ A) @ (((power_power @ A) @ A2) @ N)) @ (((power_power @ A) @ B) @ N)))))))))).
% 2.25/1.13 thf(fact_155_zero__le__power, axiom, (! [A:$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))))))))).
% 2.25/1.13 thf(fact_156_divide__neg__neg, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [X:A,Y2:A]: (((((ord_less @ A) @ X) @ (zero_zero @ A)) => ((((ord_less @ A) @ Y2) @ (zero_zero @ A)) => (((ord_less @ A) @ (zero_zero @ A)) @ (((divide_divide @ A) @ X) @ Y2)))))))))).
% 2.25/1.13 thf(fact_157_divide__neg__pos, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [X:A,Y2:A]: (((((ord_less @ A) @ X) @ (zero_zero @ A)) => ((((ord_less @ A) @ (zero_zero @ A)) @ Y2) => (((ord_less @ A) @ (((divide_divide @ A) @ X) @ Y2)) @ (zero_zero @ A)))))))))).
% 2.25/1.13 thf(fact_158_divide__pos__neg, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [X:A,Y2:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ X) => ((((ord_less @ A) @ Y2) @ (zero_zero @ A)) => (((ord_less @ A) @ (((divide_divide @ A) @ X) @ Y2)) @ (zero_zero @ A)))))))))).
% 2.25/1.13 thf(fact_159_divide__pos__pos, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [X:A,Y2:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ X) => ((((ord_less @ A) @ (zero_zero @ A)) @ Y2) => (((ord_less @ A) @ (zero_zero @ A)) @ (((divide_divide @ A) @ X) @ Y2)))))))))).
% 2.25/1.13 thf(fact_160_divide__less__0__iff, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A2:A,B:A]: (((((ord_less @ A) @ (((divide_divide @ A) @ A2) @ B)) @ (zero_zero @ A)) = (((((ord_less @ A) @ (zero_zero @ A)) @ A2) & (((ord_less @ A) @ B) @ (zero_zero @ A))) | ((((ord_less @ A) @ A2) @ (zero_zero @ A)) & (((ord_less @ A) @ (zero_zero @ A)) @ B)))))))))).
% 2.25/1.13 thf(fact_161_divide__less__cancel, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A2:A,C:A,B:A]: (((((ord_less @ A) @ (((divide_divide @ A) @ A2) @ C)) @ (((divide_divide @ A) @ B) @ C)) = (((((ord_less @ A) @ (zero_zero @ A)) @ C) => (((ord_less @ A) @ A2) @ B)) & (((((ord_less @ A) @ C) @ (zero_zero @ A)) => (((ord_less @ A) @ B) @ A2)) & (C != (zero_zero @ A))))))))))).
% 2.25/1.13 thf(fact_162_zero__less__divide__iff, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A2:A,B:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ (((divide_divide @ A) @ A2) @ B)) = (((((ord_less @ A) @ (zero_zero @ A)) @ A2) & (((ord_less @ A) @ (zero_zero @ A)) @ B)) | ((((ord_less @ A) @ A2) @ (zero_zero @ A)) & (((ord_less @ A) @ B) @ (zero_zero @ A))))))))))).
% 2.25/1.13 thf(fact_163_divide__strict__right__mono, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A2:A,B:A,C:A]: (((((ord_less @ A) @ A2) @ B) => ((((ord_less @ A) @ (zero_zero @ A)) @ C) => (((ord_less @ A) @ (((divide_divide @ A) @ A2) @ C)) @ (((divide_divide @ A) @ B) @ C)))))))))).
% 2.25/1.13 thf(fact_164_divide__strict__right__mono__neg, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [B:A,A2:A,C:A]: (((((ord_less @ A) @ B) @ A2) => ((((ord_less @ A) @ C) @ (zero_zero @ A)) => (((ord_less @ A) @ (((divide_divide @ A) @ A2) @ C)) @ (((divide_divide @ A) @ B) @ C)))))))))).
% 2.25/1.13 thf(fact_165_zero__less__power, axiom, (! [A:$tType]: (((linordered_semidom @ A) => (! [A2:A,N:nat]: (((((ord_less @ A) @ (zero_zero @ A)) @ A2) => (((ord_less @ A) @ (zero_zero @ A)) @ (((power_power @ A) @ A2) @ N))))))))).
% 2.25/1.13 thf(fact_166_right__inverse__eq, axiom, (! [A:$tType]: (((division_ring @ A) => (! [B:A,A2:A]: (((B != (zero_zero @ A)) => (((((divide_divide @ A) @ A2) @ B) = (one_one @ A)) = (A2 = B))))))))).
% 2.25/1.13 thf(fact_167_power__0__left, axiom, (! [A:$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)))))))))).
% 2.25/1.13 thf(fact_168_one__le__power, axiom, (! [A:$tType]: (((linordered_semidom @ A) => (! [A2:A,N:nat]: (((((ord_less_eq @ A) @ (one_one @ A)) @ A2) => (((ord_less_eq @ A) @ (one_one @ A)) @ (((power_power @ A) @ A2) @ N))))))))).
% 2.25/1.13 thf(fact_169_power__increasing, axiom, (! [A:$tType]: (((linordered_semidom @ A) => (! [N:nat,N4:nat,A2:A]: (((((ord_less_eq @ nat) @ N) @ N4) => ((((ord_less_eq @ A) @ (one_one @ A)) @ A2) => (((ord_less_eq @ A) @ (((power_power @ A) @ A2) @ N)) @ (((power_power @ A) @ A2) @ N4)))))))))).
% 2.25/1.13 thf(fact_170_power__less__imp__less__exp, axiom, (! [A:$tType]: (((linordered_semidom @ A) => (! [A2:A,M:nat,N:nat]: (((((ord_less @ A) @ (one_one @ A)) @ A2) => ((((ord_less @ A) @ (((power_power @ A) @ A2) @ M)) @ (((power_power @ A) @ A2) @ N)) => (((ord_less @ nat) @ M) @ N))))))))).
% 2.25/1.13 thf(fact_171_power__strict__increasing, axiom, (! [A:$tType]: (((linordered_semidom @ A) => (! [N:nat,N4:nat,A2:A]: (((((ord_less @ nat) @ N) @ N4) => ((((ord_less @ A) @ (one_one @ A)) @ A2) => (((ord_less @ A) @ (((power_power @ A) @ A2) @ N)) @ (((power_power @ A) @ A2) @ N4)))))))))).
% 2.25/1.13 thf(fact_172_power__one__over, axiom, (! [A:$tType]: (((division_ring @ A) => (! [A2:A,N:nat]: (((((power_power @ A) @ (((divide_divide @ A) @ (one_one @ A)) @ A2)) @ N) = (((divide_divide @ A) @ (one_one @ A)) @ (((power_power @ A) @ A2) @ N))))))))).
% 2.25/1.13 thf(fact_173_nonzero__abs__divide, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [B:A,A2:A]: (((B != (zero_zero @ A)) => (((abs_abs @ A) @ (((divide_divide @ A) @ A2) @ B)) = (((divide_divide @ A) @ ((abs_abs @ A) @ A2)) @ ((abs_abs @ A) @ B)))))))))).
% 2.25/1.13 thf(fact_174_frac__le, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [Y2:A,X:A,W:A,Z:A]: (((((ord_less_eq @ A) @ (zero_zero @ A)) @ Y2) => ((((ord_less_eq @ A) @ X) @ Y2) => ((((ord_less @ A) @ (zero_zero @ A)) @ W) => ((((ord_less_eq @ A) @ W) @ Z) => (((ord_less_eq @ A) @ (((divide_divide @ A) @ X) @ Z)) @ (((divide_divide @ A) @ Y2) @ W)))))))))))).
% 2.25/1.13 thf(fact_175_frac__less, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [X:A,Y2:A,W:A,Z:A]: (((((ord_less_eq @ A) @ (zero_zero @ A)) @ X) => ((((ord_less @ A) @ X) @ Y2) => ((((ord_less @ A) @ (zero_zero @ A)) @ W) => ((((ord_less_eq @ A) @ W) @ Z) => (((ord_less @ A) @ (((divide_divide @ A) @ X) @ Z)) @ (((divide_divide @ A) @ Y2) @ W)))))))))))).
% 2.25/1.13 thf(fact_176_frac__less2, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [X:A,Y2:A,W:A,Z:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ X) => ((((ord_less_eq @ A) @ X) @ Y2) => ((((ord_less @ A) @ (zero_zero @ A)) @ W) => ((((ord_less @ A) @ W) @ Z) => (((ord_less @ A) @ (((divide_divide @ A) @ X) @ Z)) @ (((divide_divide @ A) @ Y2) @ W)))))))))))).
% 2.25/1.13 thf(fact_177_divide__le__cancel, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A2:A,C:A,B:A]: (((((ord_less_eq @ A) @ (((divide_divide @ A) @ A2) @ C)) @ (((divide_divide @ A) @ B) @ C)) = (((((ord_less @ A) @ (zero_zero @ A)) @ C) => (((ord_less_eq @ A) @ A2) @ B)) & ((((ord_less @ A) @ C) @ (zero_zero @ A)) => (((ord_less_eq @ A) @ B) @ A2)))))))))).
% 2.25/1.13 thf(fact_178_divide__nonneg__neg, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [X:A,Y2:A]: (((((ord_less_eq @ A) @ (zero_zero @ A)) @ X) => ((((ord_less @ A) @ Y2) @ (zero_zero @ A)) => (((ord_less_eq @ A) @ (((divide_divide @ A) @ X) @ Y2)) @ (zero_zero @ A)))))))))).
% 2.25/1.13 thf(fact_179_divide__nonneg__pos, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [X:A,Y2:A]: (((((ord_less_eq @ A) @ (zero_zero @ A)) @ X) => ((((ord_less @ A) @ (zero_zero @ A)) @ Y2) => (((ord_less_eq @ A) @ (zero_zero @ A)) @ (((divide_divide @ A) @ X) @ Y2)))))))))).
% 2.25/1.13 thf(fact_180_divide__nonpos__neg, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [X:A,Y2:A]: (((((ord_less_eq @ A) @ X) @ (zero_zero @ A)) => ((((ord_less @ A) @ Y2) @ (zero_zero @ A)) => (((ord_less_eq @ A) @ (zero_zero @ A)) @ (((divide_divide @ A) @ X) @ Y2)))))))))).
% 2.25/1.13 thf(fact_181_divide__nonpos__pos, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [X:A,Y2:A]: (((((ord_less_eq @ A) @ X) @ (zero_zero @ A)) => ((((ord_less @ A) @ (zero_zero @ A)) @ Y2) => (((ord_less_eq @ A) @ (((divide_divide @ A) @ X) @ Y2)) @ (zero_zero @ A)))))))))).
% 2.25/1.13 thf(fact_182_power__less__imp__less__base, axiom, (! [A:$tType]: (((linordered_semidom @ A) => (! [A2:A,N:nat,B:A]: (((((ord_less @ A) @ (((power_power @ A) @ A2) @ N)) @ (((power_power @ A) @ B) @ N)) => ((((ord_less_eq @ A) @ (zero_zero @ A)) @ B) => (((ord_less @ A) @ A2) @ B))))))))).
% 2.25/1.13 thf(fact_183_power__strict__mono, axiom, (! [A:$tType]: (((linordered_semidom @ A) => (! [A2:A,B:A,N:nat]: (((((ord_less @ A) @ A2) @ B) => ((((ord_less_eq @ A) @ (zero_zero @ A)) @ A2) => ((((ord_less @ nat) @ (zero_zero @ nat)) @ N) => (((ord_less @ A) @ (((power_power @ A) @ A2) @ N)) @ (((power_power @ A) @ B) @ N))))))))))).
% 2.25/1.13 thf(fact_184_power__le__one, axiom, (! [A:$tType]: (((linordered_semidom @ A) => (! [A2:A,N:nat]: (((((ord_less_eq @ A) @ (zero_zero @ A)) @ A2) => ((((ord_less_eq @ A) @ A2) @ (one_one @ A)) => (((ord_less_eq @ A) @ (((power_power @ A) @ A2) @ N)) @ (one_one @ A)))))))))).
% 2.25/1.13 thf(fact_185_power__decreasing, axiom, (! [A:$tType]: (((linordered_semidom @ A) => (! [N:nat,N4:nat,A2:A]: (((((ord_less_eq @ nat) @ N) @ N4) => ((((ord_less_eq @ A) @ (zero_zero @ A)) @ A2) => ((((ord_less_eq @ A) @ A2) @ (one_one @ A)) => (((ord_less_eq @ A) @ (((power_power @ A) @ A2) @ N4)) @ (((power_power @ A) @ A2) @ N))))))))))).
% 2.25/1.13 thf(fact_186_divide__less__eq__1, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [B:A,A2:A]: (((((ord_less @ A) @ (((divide_divide @ A) @ B) @ A2)) @ (one_one @ A)) = (((((ord_less @ A) @ (zero_zero @ A)) @ A2) & (((ord_less @ A) @ B) @ A2)) | (((((ord_less @ A) @ A2) @ (zero_zero @ A)) & (((ord_less @ A) @ A2) @ B)) | (A2 = (zero_zero @ A))))))))))).
% 2.25/1.13 thf(fact_187_less__divide__eq__1, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [B:A,A2:A]: (((((ord_less @ A) @ (one_one @ A)) @ (((divide_divide @ A) @ B) @ A2)) = (((((ord_less @ A) @ (zero_zero @ A)) @ A2) & (((ord_less @ A) @ A2) @ B)) | ((((ord_less @ A) @ A2) @ (zero_zero @ A)) & (((ord_less @ A) @ B) @ A2)))))))))).
% 2.25/1.13 thf(fact_188_power__strict__decreasing, axiom, (! [A:$tType]: (((linordered_semidom @ A) => (! [N:nat,N4:nat,A2:A]: (((((ord_less @ nat) @ N) @ N4) => ((((ord_less @ A) @ (zero_zero @ A)) @ A2) => ((((ord_less @ A) @ A2) @ (one_one @ A)) => (((ord_less @ A) @ (((power_power @ A) @ A2) @ N4)) @ (((power_power @ A) @ A2) @ N))))))))))).
% 2.25/1.13 thf(fact_189_power__le__imp__le__exp, axiom, (! [A:$tType]: (((linordered_semidom @ A) => (! [A2:A,M:nat,N:nat]: (((((ord_less @ A) @ (one_one @ A)) @ A2) => ((((ord_less_eq @ A) @ (((power_power @ A) @ A2) @ M)) @ (((power_power @ A) @ A2) @ N)) => (((ord_less_eq @ nat) @ M) @ N))))))))).
% 2.25/1.13 thf(fact_190_zero__le__power__abs, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [A2:A,N:nat]: ((((ord_less_eq @ A) @ (zero_zero @ A)) @ (((power_power @ A) @ ((abs_abs @ A) @ A2)) @ N)))))))).
% 2.25/1.13 thf(fact_191_abs__div__pos, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [Y2:A,X:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ Y2) => ((((divide_divide @ A) @ ((abs_abs @ A) @ X)) @ Y2) = ((abs_abs @ A) @ (((divide_divide @ A) @ X) @ Y2)))))))))).
% 2.25/1.13 thf(fact_192_divide__le__eq__1, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [B:A,A2:A]: (((((ord_less_eq @ A) @ (((divide_divide @ A) @ B) @ A2)) @ (one_one @ A)) = (((((ord_less @ A) @ (zero_zero @ A)) @ A2) & (((ord_less_eq @ A) @ B) @ A2)) | (((((ord_less @ A) @ A2) @ (zero_zero @ A)) & (((ord_less_eq @ A) @ A2) @ B)) | (A2 = (zero_zero @ A))))))))))).
% 2.25/1.13 thf(fact_193_le__divide__eq__1, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [B:A,A2:A]: (((((ord_less_eq @ A) @ (one_one @ A)) @ (((divide_divide @ A) @ B) @ A2)) = (((((ord_less @ A) @ (zero_zero @ A)) @ A2) & (((ord_less_eq @ A) @ A2) @ B)) | ((((ord_less @ A) @ A2) @ (zero_zero @ A)) & (((ord_less_eq @ A) @ B) @ A2)))))))))).
% 2.25/1.13 thf(fact_194_A__less_I2_J, axiom, (((ord_less @ real) @ a) @ (((divide_divide @ real) @ (one_one @ real)) @ m))).
% 2.25/1.13 thf(fact_195_ab, axiom, (((ord_less_eq @ real) @ ((abs_abs @ real) @ (((minus_minus @ real) @ x) @ (((divide_divide @ real) @ ((ring_1_of_int @ real) @ a2)) @ ((ring_1_of_int @ real) @ b))))) @ (((divide_divide @ real) @ a) @ (((power_power @ real) @ ((ring_1_of_int @ real) @ b)) @ n)))).
% 2.25/1.13 thf(fact_196_ab_H, axiom, (((ord_less_eq @ real) @ ((abs_abs @ real) @ (((minus_minus @ real) @ x) @ (((divide_divide @ real) @ ((ring_1_of_int @ real) @ a2)) @ ((ring_1_of_int @ real) @ b))))) @ a)).
% 2.25/1.13 thf(fact_197_div__pos__pos__trivial, axiom, (! [K:int,L:int]: (((((ord_less_eq @ int) @ (zero_zero @ int)) @ K) => ((((ord_less @ int) @ K) @ L) => ((((divide_divide @ int) @ K) @ L) = (zero_zero @ int))))))).
% 2.25/1.13 thf(fact_198_div__neg__neg__trivial, axiom, (! [K:int,L:int]: (((((ord_less_eq @ int) @ K) @ (zero_zero @ int)) => ((((ord_less @ int) @ L) @ K) => ((((divide_divide @ int) @ K) @ L) = (zero_zero @ int))))))).
% 2.25/1.13 thf(fact_199_zero__less__abs__iff, axiom, (! [A:$tType]: (((ordere142940540dd_abs @ A) => (! [A2:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ ((abs_abs @ A) @ A2)) = (A2 != (zero_zero @ A))))))))).
% 2.25/1.13 thf(fact_200_M__pos, axiom, (((ord_less @ real) @ (zero_zero @ real)) @ m)).
% 2.25/1.13 thf(fact_201_abs__idempotent, axiom, (! [A:$tType]: (((ordere142940540dd_abs @ A) => (! [A2:A]: ((((abs_abs @ A) @ ((abs_abs @ A) @ A2)) = ((abs_abs @ A) @ A2)))))))).
% 2.25/1.13 thf(fact_202_le__zero__eq, axiom, (! [A:$tType]: (((canoni770627133id_add @ A) => (! [N:A]: (((((ord_less_eq @ A) @ N) @ (zero_zero @ A)) = (N = (zero_zero @ A))))))))).
% 2.25/1.13 thf(fact_203_not__gr__zero, axiom, (! [A:$tType]: (((canoni770627133id_add @ A) => (! [N:A]: (((~ ((((ord_less @ A) @ (zero_zero @ A)) @ N))) = (N = (zero_zero @ A))))))))).
% 2.25/1.13 thf(fact_204_cancel__comm__monoid__add__class_Odiff__cancel, axiom, (! [A:$tType]: (((cancel1352612707id_add @ A) => (! [A2:A]: (((((minus_minus @ A) @ A2) @ A2) = (zero_zero @ A)))))))).
% 2.25/1.13 thf(fact_205_diff__zero, axiom, (! [A:$tType]: (((cancel1352612707id_add @ A) => (! [A2:A]: (((((minus_minus @ A) @ A2) @ (zero_zero @ A)) = A2))))))).
% 2.25/1.13 thf(fact_206_zero__diff, axiom, (! [A:$tType]: (((comm_monoid_diff @ A) => (! [A2:A]: (((((minus_minus @ A) @ (zero_zero @ A)) @ A2) = (zero_zero @ A)))))))).
% 2.25/1.13 thf(fact_207_diff__0__right, axiom, (! [A:$tType]: (((group_add @ A) => (! [A2:A]: (((((minus_minus @ A) @ A2) @ (zero_zero @ A)) = A2))))))).
% 2.25/1.13 thf(fact_208_diff__self, axiom, (! [A:$tType]: (((group_add @ A) => (! [A2:A]: (((((minus_minus @ A) @ A2) @ A2) = (zero_zero @ A)))))))).
% 2.25/1.13 thf(fact_209_abs__zero, axiom, (! [A:$tType]: (((ordere142940540dd_abs @ A) => (((abs_abs @ A) @ (zero_zero @ A)) = (zero_zero @ A)))))).
% 2.25/1.13 thf(fact_210_abs__eq__0, axiom, (! [A:$tType]: (((ordere142940540dd_abs @ A) => (! [A2:A]: (((((abs_abs @ A) @ A2) = (zero_zero @ A)) = (A2 = (zero_zero @ A))))))))).
% 2.25/1.13 thf(fact_211_abs__0__eq, axiom, (! [A:$tType]: (((ordere142940540dd_abs @ A) => (! [A2:A]: ((((zero_zero @ A) = ((abs_abs @ A) @ A2)) = (A2 = (zero_zero @ A))))))))).
% 2.25/1.13 thf(fact_212_div__less, axiom, (! [M:nat,N:nat]: (((((ord_less @ nat) @ M) @ N) => ((((divide_divide @ nat) @ M) @ N) = (zero_zero @ nat)))))).
% 2.25/1.13 thf(fact_213_of__int__diff, axiom, (! [A:$tType]: (((ring_1 @ A) => (! [W:int,Z:int]: ((((ring_1_of_int @ A) @ (((minus_minus @ int) @ W) @ Z)) = (((minus_minus @ A) @ ((ring_1_of_int @ A) @ W)) @ ((ring_1_of_int @ A) @ Z))))))))).
% 2.25/1.13 thf(fact_214_coeff__diff, axiom, (! [A:$tType]: (((cancel1352612707id_add @ A) => (! [P:(poly @ A),Q:(poly @ A),N:nat]: (((((coeff @ A) @ (((minus_minus @ (poly @ A)) @ P) @ Q)) @ N) = (((minus_minus @ A) @ (((coeff @ A) @ P) @ N)) @ (((coeff @ A) @ Q) @ N))))))))).
% 2.25/1.13 thf(fact_215_poly__diff, axiom, (! [A:$tType]: (((comm_ring @ A) => (! [P:(poly @ A),Q:(poly @ A),X:A]: (((((poly2 @ A) @ (((minus_minus @ (poly @ A)) @ P) @ Q)) @ X) = (((minus_minus @ A) @ (((poly2 @ A) @ P) @ X)) @ (((poly2 @ A) @ Q) @ X))))))))).
% 2.25/1.13 thf(fact_216_A__less_I3_J, axiom, (! [X5:real]: (((X5 != x) => (((((poly2 @ real) @ p) @ X5) = (zero_zero @ real)) => (((ord_less @ real) @ a) @ ((abs_abs @ real) @ (((minus_minus @ real) @ X5) @ x)))))))).
% 2.25/1.13 thf(fact_217_diff__ge__0__iff__ge, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [A2:A,B:A]: (((((ord_less_eq @ A) @ (zero_zero @ A)) @ (((minus_minus @ A) @ A2) @ B)) = (((ord_less_eq @ A) @ B) @ A2)))))))).
% 2.25/1.13 thf(fact_218_diff__gt__0__iff__gt, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [A2:A,B:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ (((minus_minus @ A) @ A2) @ B)) = (((ord_less @ A) @ B) @ A2)))))))).
% 2.25/1.13 thf(fact_219_abs__le__zero__iff, axiom, (! [A:$tType]: (((ordere142940540dd_abs @ A) => (! [A2:A]: (((((ord_less_eq @ A) @ ((abs_abs @ A) @ A2)) @ (zero_zero @ A)) = (A2 = (zero_zero @ A))))))))).
% 2.25/1.13 thf(fact_220_abs__le__self__iff, axiom, (! [A:$tType]: (((ordere142940540dd_abs @ A) => (! [A2:A]: (((((ord_less_eq @ A) @ ((abs_abs @ A) @ A2)) @ A2) = (((ord_less_eq @ A) @ (zero_zero @ A)) @ A2)))))))).
% 2.25/1.13 thf(fact_221_abs__of__nonneg, axiom, (! [A:$tType]: (((ordere142940540dd_abs @ A) => (! [A2:A]: (((((ord_less_eq @ A) @ (zero_zero @ A)) @ A2) => (((abs_abs @ A) @ A2) = A2)))))))).
% 2.25/1.13 thf(fact_222_that, axiom, (! [C:real,N:nat]: (((((ord_less @ real) @ (zero_zero @ real)) @ C) => ((! [P2:int,Q4:int]: (((((ord_less @ int) @ (zero_zero @ int)) @ Q4) => (((ord_less @ real) @ (((divide_divide @ real) @ C) @ (((power_power @ real) @ ((ring_1_of_int @ real) @ Q4)) @ N))) @ ((abs_abs @ real) @ (((minus_minus @ real) @ x) @ (((divide_divide @ real) @ ((ring_1_of_int @ real) @ P2)) @ ((ring_1_of_int @ real) @ Q4)))))))) => thesis))))).
% 2.25/1.13 thf(fact_223_minus__poly_Orep__eq, axiom, (! [A:$tType]: (((cancel1352612707id_add @ A) => (! [X:(poly @ A),Xa:(poly @ A)]: ((((coeff @ A) @ (((minus_minus @ (poly @ A)) @ X) @ Xa)) = (^ [N3:nat]: ((((minus_minus @ A) @ (((coeff @ A) @ X) @ N3)) @ (((coeff @ A) @ Xa) @ N3))))))))))).
% 2.25/1.13 thf(fact_224_eq__iff__diff__eq__0, axiom, (! [A:$tType]: (((group_add @ A) => ((^ [Y:A,Z2:A]: ((Y = Z2))) = (^ [A4:A,B3:A]: (((((minus_minus @ A) @ A4) @ B3) = (zero_zero @ A))))))))).
% 2.25/1.13 thf(fact_225_diff__mono, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [A2:A,B:A,D:A,C:A]: (((((ord_less_eq @ A) @ A2) @ B) => ((((ord_less_eq @ A) @ D) @ C) => (((ord_less_eq @ A) @ (((minus_minus @ A) @ A2) @ C)) @ (((minus_minus @ A) @ B) @ D)))))))))).
% 2.25/1.13 thf(fact_226_diff__left__mono, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [B:A,A2:A,C:A]: (((((ord_less_eq @ A) @ B) @ A2) => (((ord_less_eq @ A) @ (((minus_minus @ A) @ C) @ A2)) @ (((minus_minus @ A) @ C) @ B))))))))).
% 2.25/1.13 thf(fact_227_diff__right__mono, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [A2:A,B:A,C:A]: (((((ord_less_eq @ A) @ A2) @ B) => (((ord_less_eq @ A) @ (((minus_minus @ A) @ A2) @ C)) @ (((minus_minus @ A) @ B) @ C))))))))).
% 2.25/1.13 thf(fact_228_diff__eq__diff__less__eq, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [A2:A,B:A,C:A,D:A]: ((((((minus_minus @ A) @ A2) @ B) = (((minus_minus @ A) @ C) @ D)) => ((((ord_less_eq @ A) @ A2) @ B) = (((ord_less_eq @ A) @ C) @ D))))))))).
% 2.25/1.13 thf(fact_229_diff__strict__right__mono, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [A2:A,B:A,C:A]: (((((ord_less @ A) @ A2) @ B) => (((ord_less @ A) @ (((minus_minus @ A) @ A2) @ C)) @ (((minus_minus @ A) @ B) @ C))))))))).
% 2.25/1.13 thf(fact_230_diff__strict__left__mono, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [B:A,A2:A,C:A]: (((((ord_less @ A) @ B) @ A2) => (((ord_less @ A) @ (((minus_minus @ A) @ C) @ A2)) @ (((minus_minus @ A) @ C) @ B))))))))).
% 2.25/1.13 thf(fact_231_diff__eq__diff__less, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [A2:A,B:A,C:A,D:A]: ((((((minus_minus @ A) @ A2) @ B) = (((minus_minus @ A) @ C) @ D)) => ((((ord_less @ A) @ A2) @ B) = (((ord_less @ A) @ C) @ D))))))))).
% 2.25/1.13 thf(fact_232_diff__strict__mono, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [A2:A,B:A,D:A,C:A]: (((((ord_less @ A) @ A2) @ B) => ((((ord_less @ A) @ D) @ C) => (((ord_less @ A) @ (((minus_minus @ A) @ A2) @ C)) @ (((minus_minus @ A) @ B) @ D)))))))))).
% 2.25/1.13 thf(fact_233_abs__minus__commute, axiom, (! [A:$tType]: (((ordere142940540dd_abs @ A) => (! [A2:A,B:A]: ((((abs_abs @ A) @ (((minus_minus @ A) @ A2) @ B)) = ((abs_abs @ A) @ (((minus_minus @ A) @ B) @ A2))))))))).
% 2.25/1.13 thf(fact_234_diff__eq__diff__eq, axiom, (! [A:$tType]: (((group_add @ A) => (! [A2:A,B:A,C:A,D:A]: ((((((minus_minus @ A) @ A2) @ B) = (((minus_minus @ A) @ C) @ D)) => ((A2 = B) = (C = D))))))))).
% 2.25/1.13 thf(fact_235_cancel__ab__semigroup__add__class_Odiff__right__commute, axiom, (! [A:$tType]: (((cancel146912293up_add @ A) => (! [A2:A,C:A,B:A]: (((((minus_minus @ A) @ (((minus_minus @ A) @ A2) @ C)) @ B) = (((minus_minus @ A) @ (((minus_minus @ A) @ A2) @ B)) @ C)))))))).
% 2.25/1.13 thf(fact_236_diff__divide__distrib, axiom, (! [A:$tType]: (((division_ring @ A) => (! [A2:A,B:A,C:A]: (((((divide_divide @ A) @ (((minus_minus @ A) @ A2) @ B)) @ C) = (((minus_minus @ A) @ (((divide_divide @ A) @ A2) @ C)) @ (((divide_divide @ A) @ B) @ C))))))))).
% 2.25/1.13 thf(fact_237_le__iff__diff__le__0, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => ((ord_less_eq @ A) = (^ [A4:A,B3:A]: ((((ord_less_eq @ A) @ (((minus_minus @ A) @ A4) @ B3)) @ (zero_zero @ A))))))))).
% 2.25/1.13 thf(fact_238_less__iff__diff__less__0, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => ((ord_less @ A) = (^ [A4:A,B3:A]: ((((ord_less @ A) @ (((minus_minus @ A) @ A4) @ B3)) @ (zero_zero @ A))))))))).
% 2.25/1.13 thf(fact_239_Ints__diff, axiom, (! [A:$tType]: (((ring_1 @ A) => (! [A2:A,B:A]: (((((member @ A) @ A2) @ (ring_1_Ints @ A)) => ((((member @ A) @ B) @ (ring_1_Ints @ A)) => (((member @ A) @ (((minus_minus @ A) @ A2) @ B)) @ (ring_1_Ints @ A)))))))))).
% 2.25/1.13 thf(fact_240_abs__triangle__ineq2__sym, axiom, (! [A:$tType]: (((ordere142940540dd_abs @ A) => (! [A2:A,B:A]: ((((ord_less_eq @ A) @ (((minus_minus @ A) @ ((abs_abs @ A) @ A2)) @ ((abs_abs @ A) @ B))) @ ((abs_abs @ A) @ (((minus_minus @ A) @ B) @ A2))))))))).
% 2.25/1.13 thf(fact_241_abs__triangle__ineq3, axiom, (! [A:$tType]: (((ordere142940540dd_abs @ A) => (! [A2:A,B:A]: ((((ord_less_eq @ A) @ ((abs_abs @ A) @ (((minus_minus @ A) @ ((abs_abs @ A) @ A2)) @ ((abs_abs @ A) @ B)))) @ ((abs_abs @ A) @ (((minus_minus @ A) @ A2) @ B))))))))).
% 2.25/1.13 thf(fact_242_abs__triangle__ineq2, axiom, (! [A:$tType]: (((ordere142940540dd_abs @ A) => (! [A2:A,B:A]: ((((ord_less_eq @ A) @ (((minus_minus @ A) @ ((abs_abs @ A) @ A2)) @ ((abs_abs @ A) @ B))) @ ((abs_abs @ A) @ (((minus_minus @ A) @ A2) @ B))))))))).
% 2.25/1.13 thf(fact_243_zero__reorient, axiom, (! [A:$tType]: (((zero @ A) => (! [X:A]: ((((zero_zero @ A) = X) = (X = (zero_zero @ A))))))))).
% 2.25/1.13 thf(fact_244_one__reorient, axiom, (! [A:$tType]: (((one @ A) => (! [X:A]: ((((one_one @ A) = X) = (X = (one_one @ A))))))))).
% 2.25/1.13 thf(fact_245_div__eq__dividend__iff, axiom, (! [M:nat,N:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ M) => (((((divide_divide @ nat) @ M) @ N) = M) = (N = (one_one @ nat))))))).
% 2.25/1.13 thf(fact_246_div__less__dividend, axiom, (! [N:nat,M:nat]: (((((ord_less @ nat) @ (one_one @ nat)) @ N) => ((((ord_less @ nat) @ (zero_zero @ nat)) @ M) => (((ord_less @ nat) @ (((divide_divide @ nat) @ M) @ N)) @ M)))))).
% 2.25/1.13 thf(fact_247_Euclidean__Division_Odiv__eq__0__iff, axiom, (! [M:nat,N:nat]: ((((((divide_divide @ nat) @ M) @ N) = (zero_zero @ nat)) = ((((ord_less @ nat) @ M) @ N) | (N = (zero_zero @ nat))))))).
% 2.25/1.13 thf(fact_248_div__le__dividend, axiom, (! [M:nat,N:nat]: ((((ord_less_eq @ nat) @ (((divide_divide @ nat) @ M) @ N)) @ M)))).
% 2.25/1.13 thf(fact_249_div__le__mono, axiom, (! [M:nat,N:nat,K:nat]: (((((ord_less_eq @ nat) @ M) @ N) => (((ord_less_eq @ nat) @ (((divide_divide @ nat) @ M) @ K)) @ (((divide_divide @ nat) @ N) @ K)))))).
% 2.25/1.13 thf(fact_250_Ints__eq__abs__less1, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [X:A,Y2:A]: (((((member @ A) @ X) @ (ring_1_Ints @ A)) => ((((member @ A) @ Y2) @ (ring_1_Ints @ A)) => ((X = Y2) = (((ord_less @ A) @ ((abs_abs @ A) @ (((minus_minus @ A) @ X) @ Y2))) @ (one_one @ A))))))))))).
% 2.25/1.13 thf(fact_251_zero__le, axiom, (! [A:$tType]: (((canoni770627133id_add @ A) => (! [X:A]: ((((ord_less_eq @ A) @ (zero_zero @ A)) @ X))))))).
% 2.25/1.13 thf(fact_252_gr__zeroI, axiom, (! [A:$tType]: (((canoni770627133id_add @ A) => (! [N:A]: (((N != (zero_zero @ A)) => (((ord_less @ A) @ (zero_zero @ A)) @ N)))))))).
% 2.25/1.13 thf(fact_253_not__less__zero, axiom, (! [A:$tType]: (((canoni770627133id_add @ A) => (! [N:A]: ((~ ((((ord_less @ A) @ N) @ (zero_zero @ A)))))))))).
% 2.25/1.13 thf(tcon_Polynomial_Opoly___Rings_Oidom, axiom, (! [A5:$tType]: (((idom @ A5) => (idom @ (poly @ A5)))))).
% 2.25/1.13 thf(tcon_Polynomial_Opoly___Groups_Oab__group__add, axiom, (! [A5:$tType]: (((ab_group_add @ A5) => (ab_group_add @ (poly @ A5)))))).
% 2.25/1.13 thf(tcon_Real_Oreal___Rings_Oidom_1, axiom, (idom @ real)).
% 2.25/1.13 thf(tcon_Real_Oreal___Groups_Oab__group__add_2, axiom, (ab_group_add @ real)).
% 2.25/1.13 thf(tcon_Int_Oint___Rings_Oidom_3, axiom, (idom @ int)).
% 2.25/1.13 thf(tcon_Int_Oint___Groups_Oab__group__add_4, axiom, (ab_group_add @ int)).
% 2.25/1.13 thf(tcon_Int_Oint___Rings_Osemiring__1__no__zero__divisors, axiom, (semiri134348788visors @ int)).
% 2.25/1.13 thf(tcon_Int_Oint___Rings_Osemiring__no__zero__divisors, axiom, (semiri1193490041visors @ int)).
% 2.25/1.13 thf(tcon_Int_Oint___Groups_Oordered__ab__group__add__abs, axiom, (ordere142940540dd_abs @ int)).
% 2.25/1.13 thf(tcon_Int_Oint___Groups_Ocancel__ab__semigroup__add, axiom, (cancel146912293up_add @ int)).
% 2.25/1.13 thf(tcon_Int_Oint___Groups_Ocancel__comm__monoid__add, axiom, (cancel1352612707id_add @ int)).
% 2.25/1.13 thf(tcon_Int_Oint___Rings_Oring__no__zero__divisors, axiom, (ring_n68954251visors @ int)).
% 2.25/1.13 thf(tcon_Int_Oint___Groups_Oordered__ab__group__add, axiom, (ordered_ab_group_add @ int)).
% 2.25/1.13 thf(tcon_Int_Oint___Rings_Olinordered__semidom, axiom, (linordered_semidom @ int)).
% 2.25/1.13 thf(tcon_Int_Oint___Rings_Olinordered__idom, axiom, (linordered_idom @ int)).
% 2.25/1.13 thf(tcon_Int_Oint___Rings_Ocomm__semiring__1, axiom, (comm_semiring_1 @ int)).
% 2.25/1.13 thf(tcon_Int_Oint___Rings_Ocomm__semiring__0, axiom, (comm_semiring_0 @ int)).
% 2.25/1.13 thf(tcon_Int_Oint___Groups_Omonoid__mult, axiom, (monoid_mult @ int)).
% 2.25/1.13 thf(tcon_Int_Oint___Rings_Oidom__divide, axiom, (idom_divide @ int)).
% 2.25/1.13 thf(tcon_Int_Oint___Rings_Ocomm__ring__1, axiom, (comm_ring_1 @ int)).
% 2.25/1.13 thf(tcon_Int_Oint___Rings_Osemiring__1, axiom, (semiring_1 @ int)).
% 2.25/1.13 thf(tcon_Int_Oint___Groups_Ogroup__add, axiom, (group_add @ int)).
% 2.25/1.13 thf(tcon_Int_Oint___Rings_Ocomm__ring, axiom, (comm_ring @ int)).
% 2.25/1.13 thf(tcon_Int_Oint___Nat_Oring__char__0, axiom, (ring_char_0 @ int)).
% 2.25/1.13 thf(tcon_Int_Oint___Rings_Oring__1, axiom, (ring_1 @ int)).
% 2.25/1.13 thf(tcon_Int_Oint___Power_Opower, axiom, (power @ int)).
% 2.25/1.13 thf(tcon_Int_Oint___Groups_Ozero, axiom, (zero @ int)).
% 2.25/1.13 thf(tcon_Int_Oint___Groups_Oone, axiom, (one @ int)).
% 2.25/1.13 thf(tcon_Nat_Onat___Groups_Ocanonically__ordered__monoid__add, axiom, (canoni770627133id_add @ nat)).
% 2.25/1.13 thf(tcon_Nat_Onat___Rings_Osemiring__1__no__zero__divisors_5, axiom, (semiri134348788visors @ nat)).
% 2.25/1.13 thf(tcon_Nat_Onat___Rings_Osemiring__no__zero__divisors_6, axiom, (semiri1193490041visors @ nat)).
% 2.25/1.13 thf(tcon_Nat_Onat___Groups_Ocancel__ab__semigroup__add_7, axiom, (cancel146912293up_add @ nat)).
% 2.25/1.13 thf(tcon_Nat_Onat___Groups_Ocancel__comm__monoid__add_8, axiom, (cancel1352612707id_add @ nat)).
% 2.25/1.13 thf(tcon_Nat_Onat___Rings_Olinordered__semidom_9, axiom, (linordered_semidom @ nat)).
% 2.25/1.13 thf(tcon_Nat_Onat___Groups_Ocomm__monoid__diff, axiom, (comm_monoid_diff @ nat)).
% 2.25/1.13 thf(tcon_Nat_Onat___Rings_Ocomm__semiring__1_10, axiom, (comm_semiring_1 @ nat)).
% 2.25/1.13 thf(tcon_Nat_Onat___Rings_Ocomm__semiring__0_11, axiom, (comm_semiring_0 @ nat)).
% 2.25/1.13 thf(tcon_Nat_Onat___Groups_Omonoid__mult_12, axiom, (monoid_mult @ nat)).
% 2.25/1.13 thf(tcon_Nat_Onat___Rings_Osemiring__1_13, axiom, (semiring_1 @ nat)).
% 2.25/1.13 thf(tcon_Nat_Onat___Power_Opower_14, axiom, (power @ nat)).
% 2.25/1.13 thf(tcon_Nat_Onat___Groups_Ozero_15, axiom, (zero @ nat)).
% 2.25/1.13 thf(tcon_Nat_Onat___Groups_Oone_16, axiom, (one @ nat)).
% 2.25/1.13 thf(tcon_Real_Oreal___Rings_Osemiring__1__no__zero__divisors_17, axiom, (semiri134348788visors @ real)).
% 2.25/1.13 thf(tcon_Real_Oreal___Rings_Osemiring__no__zero__divisors_18, axiom, (semiri1193490041visors @ real)).
% 2.25/1.13 thf(tcon_Real_Oreal___Groups_Oordered__ab__group__add__abs_19, axiom, (ordere142940540dd_abs @ real)).
% 2.25/1.13 thf(tcon_Real_Oreal___Groups_Ocancel__ab__semigroup__add_20, axiom, (cancel146912293up_add @ real)).
% 2.25/1.13 thf(tcon_Real_Oreal___Groups_Ocancel__comm__monoid__add_21, axiom, (cancel1352612707id_add @ real)).
% 2.25/1.13 thf(tcon_Real_Oreal___Rings_Oring__no__zero__divisors_22, axiom, (ring_n68954251visors @ real)).
% 2.25/1.13 thf(tcon_Real_Oreal___Groups_Oordered__ab__group__add_23, axiom, (ordered_ab_group_add @ real)).
% 2.25/1.13 thf(tcon_Real_Oreal___Rings_Olinordered__semidom_24, axiom, (linordered_semidom @ real)).
% 2.25/1.13 thf(tcon_Real_Oreal___Fields_Olinordered__field, axiom, (linordered_field @ real)).
% 2.25/1.13 thf(tcon_Real_Oreal___Rings_Olinordered__idom_25, axiom, (linordered_idom @ real)).
% 2.25/1.13 thf(tcon_Real_Oreal___Rings_Ocomm__semiring__1_26, axiom, (comm_semiring_1 @ real)).
% 2.25/1.13 thf(tcon_Real_Oreal___Rings_Ocomm__semiring__0_27, axiom, (comm_semiring_0 @ real)).
% 2.25/1.13 thf(tcon_Real_Oreal___Fields_Ofield__abs__sgn, axiom, (field_abs_sgn @ real)).
% 2.25/1.13 thf(tcon_Real_Oreal___Fields_Odivision__ring, axiom, (division_ring @ real)).
% 2.25/1.13 thf(tcon_Real_Oreal___Fields_Ofield__char__0, axiom, (field_char_0 @ real)).
% 2.25/1.13 thf(tcon_Real_Oreal___Groups_Omonoid__mult_28, axiom, (monoid_mult @ real)).
% 2.25/1.13 thf(tcon_Real_Oreal___Rings_Oidom__divide_29, axiom, (idom_divide @ real)).
% 2.25/1.13 thf(tcon_Real_Oreal___Rings_Ocomm__ring__1_30, axiom, (comm_ring_1 @ real)).
% 2.25/1.13 thf(tcon_Real_Oreal___Rings_Osemiring__1_31, axiom, (semiring_1 @ real)).
% 2.25/1.13 thf(tcon_Real_Oreal___Groups_Ogroup__add_32, axiom, (group_add @ real)).
% 2.25/1.13 thf(tcon_Real_Oreal___Rings_Ocomm__ring_33, axiom, (comm_ring @ real)).
% 2.25/1.13 thf(tcon_Real_Oreal___Nat_Oring__char__0_34, axiom, (ring_char_0 @ real)).
% 2.25/1.13 thf(tcon_Real_Oreal___Rings_Oring__1_35, axiom, (ring_1 @ real)).
% 2.25/1.13 thf(tcon_Real_Oreal___Fields_Ofield, axiom, (field @ real)).
% 2.25/1.13 thf(tcon_Real_Oreal___Power_Opower_36, axiom, (power @ real)).
% 2.25/1.13 thf(tcon_Real_Oreal___Groups_Ozero_37, axiom, (zero @ real)).
% 2.25/1.13 thf(tcon_Real_Oreal___Groups_Oone_38, axiom, (one @ real)).
% 2.25/1.13 thf(tcon_Polynomial_Opoly___Rings_Osemiring__1__no__zero__divisors_39, axiom, (! [A5:$tType]: ((((comm_semiring_1 @ A5) & (semiri134348788visors @ A5)) => (semiri134348788visors @ (poly @ A5)))))).
% 2.25/1.13 thf(tcon_Polynomial_Opoly___Rings_Osemiring__no__zero__divisors_40, axiom, (! [A5:$tType]: ((((comm_semiring_0 @ A5) & (semiri1193490041visors @ A5)) => (semiri1193490041visors @ (poly @ A5)))))).
% 2.25/1.13 thf(tcon_Polynomial_Opoly___Groups_Oordered__ab__group__add__abs_41, axiom, (! [A5:$tType]: (((linordered_idom @ A5) => (ordere142940540dd_abs @ (poly @ A5)))))).
% 2.25/1.13 thf(tcon_Polynomial_Opoly___Groups_Ocancel__ab__semigroup__add_42, axiom, (! [A5:$tType]: (((cancel1352612707id_add @ A5) => (cancel146912293up_add @ (poly @ A5)))))).
% 2.25/1.13 thf(tcon_Polynomial_Opoly___Groups_Ocancel__comm__monoid__add_43, axiom, (! [A5:$tType]: (((cancel1352612707id_add @ A5) => (cancel1352612707id_add @ (poly @ A5)))))).
% 2.25/1.13 thf(tcon_Polynomial_Opoly___Rings_Oring__no__zero__divisors_44, axiom, (! [A5:$tType]: (((idom @ A5) => (ring_n68954251visors @ (poly @ A5)))))).
% 2.25/1.13 thf(tcon_Polynomial_Opoly___Groups_Oordered__ab__group__add_45, axiom, (! [A5:$tType]: (((linordered_idom @ A5) => (ordered_ab_group_add @ (poly @ A5)))))).
% 2.25/1.13 thf(tcon_Polynomial_Opoly___Rings_Olinordered__semidom_46, axiom, (! [A5:$tType]: (((linordered_idom @ A5) => (linordered_semidom @ (poly @ A5)))))).
% 2.25/1.13 thf(tcon_Polynomial_Opoly___Rings_Olinordered__idom_47, axiom, (! [A5:$tType]: (((linordered_idom @ A5) => (linordered_idom @ (poly @ A5)))))).
% 2.25/1.13 thf(tcon_Polynomial_Opoly___Rings_Ocomm__semiring__1_48, axiom, (! [A5:$tType]: (((comm_semiring_1 @ A5) => (comm_semiring_1 @ (poly @ A5)))))).
% 2.25/1.13 thf(tcon_Polynomial_Opoly___Rings_Ocomm__semiring__0_49, axiom, (! [A5:$tType]: (((comm_semiring_0 @ A5) => (comm_semiring_0 @ (poly @ A5)))))).
% 2.25/1.13 thf(tcon_Polynomial_Opoly___Groups_Omonoid__mult_50, axiom, (! [A5:$tType]: (((comm_semiring_1 @ A5) => (monoid_mult @ (poly @ A5)))))).
% 2.25/1.13 thf(tcon_Polynomial_Opoly___Rings_Oidom__divide_51, axiom, (! [A5:$tType]: (((idom_divide @ A5) => (idom_divide @ (poly @ A5)))))).
% 2.25/1.13 thf(tcon_Polynomial_Opoly___Rings_Ocomm__ring__1_52, axiom, (! [A5:$tType]: (((comm_ring_1 @ A5) => (comm_ring_1 @ (poly @ A5)))))).
% 2.25/1.13 thf(tcon_Polynomial_Opoly___Rings_Osemiring__1_53, axiom, (! [A5:$tType]: (((comm_semiring_1 @ A5) => (semiring_1 @ (poly @ A5)))))).
% 2.25/1.13 thf(tcon_Polynomial_Opoly___Groups_Ogroup__add_54, axiom, (! [A5:$tType]: (((ab_group_add @ A5) => (group_add @ (poly @ A5)))))).
% 2.25/1.13 thf(tcon_Polynomial_Opoly___Rings_Ocomm__ring_55, axiom, (! [A5:$tType]: (((comm_ring @ A5) => (comm_ring @ (poly @ A5)))))).
% 2.25/1.13 thf(tcon_Polynomial_Opoly___Nat_Oring__char__0_56, axiom, (! [A5:$tType]: ((((ring_char_0 @ A5) & (comm_ring_1 @ A5)) => (ring_char_0 @ (poly @ A5)))))).
% 2.25/1.13 thf(tcon_Polynomial_Opoly___Rings_Oring__1_57, axiom, (! [A5:$tType]: (((comm_ring_1 @ A5) => (ring_1 @ (poly @ A5)))))).
% 2.25/1.13 thf(tcon_Polynomial_Opoly___Power_Opower_58, axiom, (! [A5:$tType]: (((comm_semiring_1 @ A5) => (power @ (poly @ A5)))))).
% 2.25/1.13 thf(tcon_Polynomial_Opoly___Groups_Ozero_59, axiom, (! [A5:$tType]: (((zero @ A5) => (zero @ (poly @ A5)))))).
% 2.25/1.13 thf(tcon_Polynomial_Opoly___Groups_Oone_60, axiom, (! [A5:$tType]: (((comm_semiring_1 @ A5) => (one @ (poly @ A5)))))).
% 2.25/1.13 thf(conj_0, conjecture, (((ord_less_eq @ real) @ (((divide_divide @ real) @ (one_one @ real)) @ (((power_power @ real) @ ((ring_1_of_int @ real) @ b)) @ ((degree @ real) @ p)))) @ ((abs_abs @ real) @ (((poly2 @ real) @ p) @ (((divide_divide @ real) @ ((ring_1_of_int @ real) @ a2)) @ ((ring_1_of_int @ real) @ b)))))).
% 2.25/1.13 % SZS output end ListOfTHF for /export/starexec/sandbox/tmp/tmp.qtQCAE7Wjp/DTF2DTF32390.p
% 2.25/1.13 ------------------------
% 2.25/1.15 ---- Cleaned THF ---
% 2.25/1.15 thf(ty_t_Polynomial_Opoly,type,
% 2.25/1.15 poly: $tType > $tType ).
% 2.25/1.15
% 2.25/1.15 thf(ty_t_Real_Oreal,type,
% 2.25/1.15 real: $tType ).
% 2.25/1.15
% 2.25/1.15 thf(ty_t_Set_Oset,type,
% 2.25/1.15 set: $tType > $tType ).
% 2.25/1.15
% 2.25/1.15 thf(ty_t_Nat_Onat,type,
% 2.25/1.15 nat: $tType ).
% 2.25/1.15
% 2.25/1.15 thf(ty_t_Int_Oint,type,
% 2.25/1.15 int: $tType ).
% 2.25/1.15
% 2.25/1.15 thf(sy_cl_Groups_Oab__group__add,type,
% 2.25/1.15 ab_group_add:
% 2.25/1.15 !>[A: $tType] : $o ).
% 2.25/1.15
% 2.25/1.15 thf(sy_cl_Rings_Oidom,type,
% 2.25/1.15 idom:
% 2.25/1.15 !>[A: $tType] : $o ).
% 2.25/1.15
% 2.25/1.15 thf(sy_cl_Groups_Oone,type,
% 2.25/1.15 one:
% 2.25/1.15 !>[A: $tType] : $o ).
% 2.25/1.15
% 2.25/1.15 thf(sy_cl_Groups_Ozero,type,
% 2.25/1.15 zero:
% 2.25/1.15 !>[A: $tType] : $o ).
% 2.25/1.15
% 2.25/1.15 thf(sy_cl_Power_Opower,type,
% 2.25/1.15 power:
% 2.25/1.15 !>[A: $tType] : $o ).
% 2.25/1.15
% 2.25/1.15 thf(sy_cl_Fields_Ofield,type,
% 2.25/1.15 field:
% 2.25/1.15 !>[A: $tType] : $o ).
% 2.25/1.15
% 2.25/1.15 thf(sy_cl_Rings_Oring__1,type,
% 2.25/1.15 ring_1:
% 2.25/1.15 !>[A: $tType] : $o ).
% 2.25/1.15
% 2.25/1.15 thf(sy_cl_Nat_Oring__char__0,type,
% 2.25/1.15 ring_char_0:
% 2.25/1.15 !>[A: $tType] : $o ).
% 2.25/1.15
% 2.25/1.15 thf(sy_cl_Rings_Ocomm__ring,type,
% 2.25/1.15 comm_ring:
% 2.25/1.15 !>[A: $tType] : $o ).
% 2.25/1.15
% 2.25/1.15 thf(sy_cl_Groups_Ogroup__add,type,
% 2.25/1.15 group_add:
% 2.25/1.15 !>[A: $tType] : $o ).
% 2.25/1.15
% 2.25/1.15 thf(sy_cl_Rings_Osemiring__1,type,
% 2.25/1.15 semiring_1:
% 2.25/1.15 !>[A: $tType] : $o ).
% 2.25/1.15
% 2.25/1.15 thf(sy_cl_Rings_Ocomm__ring__1,type,
% 2.25/1.15 comm_ring_1:
% 2.25/1.15 !>[A: $tType] : $o ).
% 2.25/1.15
% 2.25/1.15 thf(sy_cl_Rings_Oidom__divide,type,
% 2.25/1.15 idom_divide:
% 2.25/1.15 !>[A: $tType] : $o ).
% 2.25/1.15
% 2.25/1.15 thf(sy_cl_Groups_Omonoid__mult,type,
% 2.25/1.15 monoid_mult:
% 2.25/1.15 !>[A: $tType] : $o ).
% 2.25/1.15
% 2.25/1.15 thf(sy_cl_Fields_Ofield__char__0,type,
% 2.25/1.15 field_char_0:
% 2.25/1.15 !>[A: $tType] : $o ).
% 2.25/1.15
% 2.25/1.15 thf(sy_cl_Fields_Odivision__ring,type,
% 2.25/1.15 division_ring:
% 2.25/1.15 !>[A: $tType] : $o ).
% 2.25/1.15
% 2.25/1.15 thf(sy_cl_Fields_Ofield__abs__sgn,type,
% 2.25/1.15 field_abs_sgn:
% 2.25/1.15 !>[A: $tType] : $o ).
% 2.25/1.15
% 2.25/1.15 thf(sy_cl_Rings_Ocomm__semiring__0,type,
% 2.25/1.15 comm_semiring_0:
% 2.25/1.15 !>[A: $tType] : $o ).
% 2.25/1.15
% 2.25/1.15 thf(sy_cl_Rings_Ocomm__semiring__1,type,
% 2.25/1.15 comm_semiring_1:
% 2.25/1.15 !>[A: $tType] : $o ).
% 2.25/1.15
% 2.25/1.15 thf(sy_cl_Rings_Olinordered__idom,type,
% 2.25/1.15 linordered_idom:
% 2.25/1.15 !>[A: $tType] : $o ).
% 2.25/1.15
% 2.25/1.15 thf(sy_cl_Fields_Olinordered__field,type,
% 2.25/1.15 linordered_field:
% 2.25/1.15 !>[A: $tType] : $o ).
% 2.25/1.15
% 2.25/1.15 thf(sy_cl_Groups_Ocomm__monoid__diff,type,
% 2.25/1.15 comm_monoid_diff:
% 2.25/1.15 !>[A: $tType] : $o ).
% 2.25/1.15
% 2.25/1.15 thf(sy_cl_Rings_Olinordered__semidom,type,
% 2.25/1.15 linordered_semidom:
% 2.25/1.15 !>[A: $tType] : $o ).
% 2.25/1.15
% 2.25/1.15 thf(sy_cl_Groups_Oordered__ab__group__add,type,
% 2.25/1.15 ordered_ab_group_add:
% 2.25/1.15 !>[A: $tType] : $o ).
% 2.25/1.15
% 2.25/1.15 thf(sy_cl_Rings_Oring__no__zero__divisors,type,
% 2.25/1.15 ring_n68954251visors:
% 2.25/1.15 !>[A: $tType] : $o ).
% 2.25/1.15
% 2.25/1.15 thf(sy_cl_Groups_Ocancel__comm__monoid__add,type,
% 2.25/1.15 cancel1352612707id_add:
% 2.25/1.15 !>[A: $tType] : $o ).
% 2.25/1.15
% 2.25/1.15 thf(sy_cl_Groups_Ocancel__ab__semigroup__add,type,
% 2.25/1.15 cancel146912293up_add:
% 2.25/1.15 !>[A: $tType] : $o ).
% 2.25/1.15
% 2.25/1.15 thf(sy_cl_Groups_Oordered__ab__group__add__abs,type,
% 2.25/1.15 ordere142940540dd_abs:
% 2.25/1.15 !>[A: $tType] : $o ).
% 2.25/1.15
% 2.25/1.15 thf(sy_cl_Rings_Osemiring__no__zero__divisors,type,
% 2.25/1.15 semiri1193490041visors:
% 2.25/1.15 !>[A: $tType] : $o ).
% 2.25/1.15
% 2.25/1.15 thf(sy_cl_Rings_Osemiring__1__no__zero__divisors,type,
% 2.25/1.15 semiri134348788visors:
% 2.25/1.15 !>[A: $tType] : $o ).
% 2.25/1.15
% 2.25/1.15 thf(sy_cl_Groups_Ocanonically__ordered__monoid__add,type,
% 2.25/1.15 canoni770627133id_add:
% 2.25/1.15 !>[A: $tType] : $o ).
% 2.25/1.15
% 2.25/1.15 thf(sy_c_Groups_Oabs__class_Oabs,type,
% 2.25/1.15 abs_abs:
% 2.25/1.15 !>[A: $tType] : ( A > A ) ).
% 2.25/1.15
% 2.25/1.15 thf(sy_c_Groups_Ominus__class_Ominus,type,
% 2.25/1.15 minus_minus:
% 2.25/1.15 !>[A: $tType] : ( A > A > A ) ).
% 2.25/1.15
% 2.25/1.15 thf(sy_c_Groups_Oone__class_Oone,type,
% 2.25/1.15 one_one:
% 2.25/1.15 !>[A: $tType] : A ).
% 2.25/1.15
% 2.25/1.15 thf(sy_c_Groups_Ozero__class_Ozero,type,
% 2.25/1.15 zero_zero:
% 2.25/1.15 !>[A: $tType] : A ).
% 2.25/1.15
% 2.25/1.15 thf(sy_c_Int_Oring__1__class_OInts,type,
% 2.25/1.15 ring_1_Ints:
% 2.25/1.15 !>[A: $tType] : ( set @ A ) ).
% 2.25/1.15
% 2.25/1.15 thf(sy_c_Int_Oring__1__class_Oof__int,type,
% 2.25/1.15 ring_1_of_int:
% 2.25/1.15 !>[A: $tType] : ( int > A ) ).
% 2.25/1.15
% 2.25/1.15 thf(sy_c_Orderings_Oord__class_Oless,type,
% 2.25/1.15 ord_less:
% 2.25/1.15 !>[A: $tType] : ( A > A > $o ) ).
% 2.25/1.15
% 2.25/1.15 thf(sy_c_Orderings_Oord__class_Oless__eq,type,
% 2.25/1.15 ord_less_eq:
% 2.25/1.15 !>[A: $tType] : ( A > A > $o ) ).
% 2.25/1.15
% 2.25/1.15 thf(sy_c_Polynomial_Oalgebraic,type,
% 2.25/1.15 algebraic:
% 2.25/1.15 !>[A: $tType] : ( A > $o ) ).
% 2.25/1.15
% 2.25/1.15 thf(sy_c_Polynomial_Odegree,type,
% 2.25/1.15 degree:
% 2.25/1.15 !>[A: $tType] : ( ( poly @ A ) > nat ) ).
% 2.25/1.15
% 2.25/1.15 thf(sy_c_Polynomial_Opoly,type,
% 2.25/1.15 poly2:
% 2.25/1.15 !>[A: $tType] : ( ( poly @ A ) > A > A ) ).
% 2.25/1.15
% 2.25/1.15 thf(sy_c_Polynomial_Opoly_Ocoeff,type,
% 2.25/1.15 coeff:
% 2.25/1.15 !>[A: $tType] : ( ( poly @ A ) > nat > A ) ).
% 2.25/1.15
% 2.25/1.15 thf(sy_c_Power_Opower__class_Opower,type,
% 2.25/1.15 power_power:
% 2.25/1.15 !>[A: $tType] : ( A > nat > A ) ).
% 2.25/1.15
% 2.25/1.15 thf(sy_c_Rat_Ofield__char__0__class_ORats,type,
% 2.25/1.15 field_char_0_Rats:
% 2.25/1.15 !>[A: $tType] : ( set @ A ) ).
% 2.25/1.15
% 2.25/1.15 thf(sy_c_Rings_Odivide__class_Odivide,type,
% 2.25/1.15 divide_divide:
% 2.25/1.15 !>[A: $tType] : ( A > A > A ) ).
% 2.25/1.15
% 2.25/1.15 thf(sy_c_Set_OCollect,type,
% 2.25/1.15 collect:
% 2.25/1.15 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
% 2.25/1.15
% 2.25/1.15 thf(sy_c_member,type,
% 2.25/1.15 member:
% 2.25/1.15 !>[A: $tType] : ( A > ( set @ A ) > $o ) ).
% 2.25/1.15
% 2.25/1.15 thf(sy_v_A____,type,
% 2.25/1.15 a: real ).
% 2.25/1.15
% 2.25/1.15 thf(sy_v_M____,type,
% 2.25/1.15 m: real ).
% 2.25/1.15
% 2.25/1.15 thf(sy_v_a____,type,
% 2.25/1.15 a2: int ).
% 2.25/1.15
% 2.25/1.15 thf(sy_v_b____,type,
% 2.25/1.15 b: int ).
% 2.25/1.15
% 2.25/1.15 thf(sy_v_n____,type,
% 2.25/1.15 n: nat ).
% 2.25/1.15
% 2.25/1.15 thf(sy_v_p____,type,
% 2.25/1.15 p: poly @ real ).
% 2.25/1.15
% 2.25/1.15 thf(sy_v_roots____,type,
% 2.25/1.15 roots: set @ real ).
% 2.25/1.15
% 2.25/1.15 thf(sy_v_thesis,type,
% 2.25/1.15 thesis: $o ).
% 2.25/1.15
% 2.25/1.15 thf(sy_v_x,type,
% 2.25/1.15 x: real ).
% 2.25/1.15
% 2.25/1.15 thf(fact_0_assms_I2_J,axiom,
% 2.25/1.15 algebraic @ real @ x ).
% 2.25/1.15
% 2.25/1.15 thf(fact_1_p_I2_J,axiom,
% 2.25/1.15 ( p
% 2.25/1.15 != ( zero_zero @ ( poly @ real ) ) ) ).
% 2.25/1.15
% 2.25/1.15 thf(fact_2_p_I1_J,axiom,
% 2.25/1.15 ! [I: nat] : ( member @ real @ ( coeff @ real @ p @ I ) @ ( ring_1_Ints @ real ) ) ).
% 2.25/1.15
% 2.25/1.15 thf(fact_3_b,axiom,
% 2.25/1.15 ord_less @ int @ ( zero_zero @ int ) @ b ).
% 2.25/1.15
% 2.25/1.15 thf(fact_4_p_I3_J,axiom,
% 2.25/1.15 ( ( poly2 @ real @ p @ x )
% 2.25/1.15 = ( zero_zero @ real ) ) ).
% 2.25/1.15
% 2.25/1.15 thf(fact_5_no__root,axiom,
% 2.25/1.15 ( ( poly2 @ real @ p @ ( divide_divide @ real @ ( ring_1_of_int @ real @ a2 ) @ ( ring_1_of_int @ real @ b ) ) )
% 2.25/1.15 != ( zero_zero @ real ) ) ).
% 2.25/1.15
% 2.25/1.15 thf(fact_6_n__def,axiom,
% 2.25/1.15 ( n
% 2.25/1.15 = ( degree @ real @ p ) ) ).
% 2.25/1.15
% 2.25/1.15 thf(fact_7_of__int__le__of__int__power__cancel__iff,axiom,
% 2.25/1.15 ! [A: $tType] :
% 2.25/1.15 ( ( linordered_idom @ A )
% 2.25/1.15 => ! [B: int,W: nat,X: int] :
% 2.25/1.15 ( ( ord_less_eq @ A @ ( power_power @ A @ ( ring_1_of_int @ A @ B ) @ W ) @ ( ring_1_of_int @ A @ X ) )
% 2.25/1.15 = ( ord_less_eq @ int @ ( power_power @ int @ B @ W ) @ X ) ) ) ).
% 2.25/1.15
% 2.25/1.15 thf(fact_8_of__int__power__le__of__int__cancel__iff,axiom,
% 2.25/1.15 ! [A: $tType] :
% 2.25/1.15 ( ( linordered_idom @ A )
% 2.25/1.15 => ! [X: int,B: int,W: nat] :
% 2.25/1.16 ( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ X ) @ ( power_power @ A @ ( ring_1_of_int @ A @ B ) @ W ) )
% 2.25/1.16 = ( ord_less_eq @ int @ X @ ( power_power @ int @ B @ W ) ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_9_of__int__1__le__iff,axiom,
% 2.25/1.16 ! [A: $tType] :
% 2.25/1.16 ( ( linordered_idom @ A )
% 2.25/1.16 => ! [Z: int] :
% 2.25/1.16 ( ( ord_less_eq @ A @ ( one_one @ A ) @ ( ring_1_of_int @ A @ Z ) )
% 2.25/1.16 = ( ord_less_eq @ int @ ( one_one @ int ) @ Z ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_10_of__int__le__1__iff,axiom,
% 2.25/1.16 ! [A: $tType] :
% 2.25/1.16 ( ( linordered_idom @ A )
% 2.25/1.16 => ! [Z: int] :
% 2.25/1.16 ( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ Z ) @ ( one_one @ A ) )
% 2.25/1.16 = ( ord_less_eq @ int @ Z @ ( one_one @ int ) ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_11_poly__power,axiom,
% 2.25/1.16 ! [A: $tType] :
% 2.25/1.16 ( ( comm_semiring_1 @ A )
% 2.25/1.16 => ! [P: poly @ A,N: nat,X: A] :
% 2.25/1.16 ( ( poly2 @ A @ ( power_power @ ( poly @ A ) @ P @ N ) @ X )
% 2.25/1.16 = ( power_power @ A @ ( poly2 @ A @ P @ X ) @ N ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_12_poly__1,axiom,
% 2.25/1.16 ! [A: $tType] :
% 2.25/1.16 ( ( comm_semiring_1 @ A )
% 2.25/1.16 => ! [X: A] :
% 2.25/1.16 ( ( poly2 @ A @ ( one_one @ ( poly @ A ) ) @ X )
% 2.25/1.16 = ( one_one @ A ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_13_of__int__abs,axiom,
% 2.25/1.16 ! [A: $tType] :
% 2.25/1.16 ( ( linordered_idom @ A )
% 2.25/1.16 => ! [X: int] :
% 2.25/1.16 ( ( ring_1_of_int @ A @ ( abs_abs @ int @ X ) )
% 2.25/1.16 = ( abs_abs @ A @ ( ring_1_of_int @ A @ X ) ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_14_of__int__power,axiom,
% 2.25/1.16 ! [A: $tType] :
% 2.25/1.16 ( ( ring_1 @ A )
% 2.25/1.16 => ! [Z: int,N: nat] :
% 2.25/1.16 ( ( ring_1_of_int @ A @ ( power_power @ int @ Z @ N ) )
% 2.25/1.16 = ( power_power @ A @ ( ring_1_of_int @ A @ Z ) @ N ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_15_of__int__eq__of__int__power__cancel__iff,axiom,
% 2.25/1.16 ! [A: $tType] :
% 2.25/1.16 ( ( ring_char_0 @ A )
% 2.25/1.16 => ! [B: int,W: nat,X: int] :
% 2.25/1.16 ( ( ( power_power @ A @ ( ring_1_of_int @ A @ B ) @ W )
% 2.25/1.16 = ( ring_1_of_int @ A @ X ) )
% 2.25/1.16 = ( ( power_power @ int @ B @ W )
% 2.25/1.16 = X ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_16_of__int__power__eq__of__int__cancel__iff,axiom,
% 2.25/1.16 ! [A: $tType] :
% 2.25/1.16 ( ( ring_char_0 @ A )
% 2.25/1.16 => ! [X: int,B: int,W: nat] :
% 2.25/1.16 ( ( ( ring_1_of_int @ A @ X )
% 2.25/1.16 = ( power_power @ A @ ( ring_1_of_int @ A @ B ) @ W ) )
% 2.25/1.16 = ( X
% 2.25/1.16 = ( power_power @ int @ B @ W ) ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_17_of__int__1,axiom,
% 2.25/1.16 ! [A: $tType] :
% 2.25/1.16 ( ( ring_1 @ A )
% 2.25/1.16 => ( ( ring_1_of_int @ A @ ( one_one @ int ) )
% 2.25/1.16 = ( one_one @ A ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_18_of__int__eq__1__iff,axiom,
% 2.25/1.16 ! [A: $tType] :
% 2.25/1.16 ( ( ring_char_0 @ A )
% 2.25/1.16 => ! [Z: int] :
% 2.25/1.16 ( ( ( ring_1_of_int @ A @ Z )
% 2.25/1.16 = ( one_one @ A ) )
% 2.25/1.16 = ( Z
% 2.25/1.16 = ( one_one @ int ) ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_19_irrationsl,axiom,
% 2.25/1.16 ~ ( member @ real @ x @ ( field_char_0_Rats @ real ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_20_of__int__eq__iff,axiom,
% 2.25/1.16 ! [A: $tType] :
% 2.25/1.16 ( ( ring_char_0 @ A )
% 2.25/1.16 => ! [W: int,Z: int] :
% 2.25/1.16 ( ( ( ring_1_of_int @ A @ W )
% 2.25/1.16 = ( ring_1_of_int @ A @ Z ) )
% 2.25/1.16 = ( W = Z ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_21__092_060open_062_092_060And_062thesisa_O_A_I_092_060And_062p_O_A_092_060lbrakk_062_092_060And_062i_O_Acoeff_Ap_Ai_A_092_060in_062_A_092_060int_062_059_Ap_A_092_060noteq_062_A0_059_Apoly_Ap_Ax_A_061_A0_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesisa_J_A_092_060Longrightarrow_062_Athesisa_092_060close_062,axiom,
% 2.25/1.16 ~ ! [P2: poly @ real] :
% 2.25/1.16 ( ! [I2: nat] : ( member @ real @ ( coeff @ real @ P2 @ I2 ) @ ( ring_1_Ints @ real ) )
% 2.25/1.16 => ( ( P2
% 2.25/1.16 != ( zero_zero @ ( poly @ real ) ) )
% 2.25/1.16 => ( ( poly2 @ real @ P2 @ x )
% 2.25/1.16 != ( zero_zero @ real ) ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_22__092_060open_062x_A_092_060notin_062_Aroots_092_060close_062,axiom,
% 2.25/1.16 ~ ( member @ real @ x @ roots ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_23_of__int__eq__0__iff,axiom,
% 2.25/1.16 ! [A: $tType] :
% 2.25/1.16 ( ( ring_char_0 @ A )
% 2.25/1.16 => ! [Z: int] :
% 2.25/1.16 ( ( ( ring_1_of_int @ A @ Z )
% 2.25/1.16 = ( zero_zero @ A ) )
% 2.25/1.16 = ( Z
% 2.25/1.16 = ( zero_zero @ int ) ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_24_of__int__0__eq__iff,axiom,
% 2.25/1.16 ! [A: $tType] :
% 2.25/1.16 ( ( ring_char_0 @ A )
% 2.25/1.16 => ! [Z: int] :
% 2.25/1.16 ( ( ( zero_zero @ A )
% 2.25/1.16 = ( ring_1_of_int @ A @ Z ) )
% 2.25/1.16 = ( Z
% 2.25/1.16 = ( zero_zero @ int ) ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_25_of__int__0,axiom,
% 2.25/1.16 ! [A: $tType] :
% 2.25/1.16 ( ( ring_1 @ A )
% 2.25/1.16 => ( ( ring_1_of_int @ A @ ( zero_zero @ int ) )
% 2.25/1.16 = ( zero_zero @ A ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_26_of__int__le__iff,axiom,
% 2.25/1.16 ! [A: $tType] :
% 2.25/1.16 ( ( linordered_idom @ A )
% 2.25/1.16 => ! [W: int,Z: int] :
% 2.25/1.16 ( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ W ) @ ( ring_1_of_int @ A @ Z ) )
% 2.25/1.16 = ( ord_less_eq @ int @ W @ Z ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_27_of__int__less__iff,axiom,
% 2.25/1.16 ! [A: $tType] :
% 2.25/1.16 ( ( linordered_idom @ A )
% 2.25/1.16 => ! [W: int,Z: int] :
% 2.25/1.16 ( ( ord_less @ A @ ( ring_1_of_int @ A @ W ) @ ( ring_1_of_int @ A @ Z ) )
% 2.25/1.16 = ( ord_less @ int @ W @ Z ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_28_coeff__0,axiom,
% 2.25/1.16 ! [A: $tType] :
% 2.25/1.16 ( ( zero @ A )
% 2.25/1.16 => ! [N: nat] :
% 2.25/1.16 ( ( coeff @ A @ ( zero_zero @ ( poly @ A ) ) @ N )
% 2.25/1.16 = ( zero_zero @ A ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_29_poly__0,axiom,
% 2.25/1.16 ! [A: $tType] :
% 2.25/1.16 ( ( comm_semiring_0 @ A )
% 2.25/1.16 => ! [X: A] :
% 2.25/1.16 ( ( poly2 @ A @ ( zero_zero @ ( poly @ A ) ) @ X )
% 2.25/1.16 = ( zero_zero @ A ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_30_lead__coeff__of__int,axiom,
% 2.25/1.16 ! [A: $tType] :
% 2.25/1.16 ( ( comm_ring_1 @ A )
% 2.25/1.16 => ! [K: int] :
% 2.25/1.16 ( ( coeff @ A @ ( ring_1_of_int @ ( poly @ A ) @ K ) @ ( degree @ A @ ( ring_1_of_int @ ( poly @ A ) @ K ) ) )
% 2.25/1.16 = ( ring_1_of_int @ A @ K ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_31_leading__coeff__0__iff,axiom,
% 2.25/1.16 ! [A: $tType] :
% 2.25/1.16 ( ( zero @ A )
% 2.25/1.16 => ! [P: poly @ A] :
% 2.25/1.16 ( ( ( coeff @ A @ P @ ( degree @ A @ P ) )
% 2.25/1.16 = ( zero_zero @ A ) )
% 2.25/1.16 = ( P
% 2.25/1.16 = ( zero_zero @ ( poly @ A ) ) ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_32_zabs__less__one__iff,axiom,
% 2.25/1.16 ! [Z: int] :
% 2.25/1.16 ( ( ord_less @ int @ ( abs_abs @ int @ Z ) @ ( one_one @ int ) )
% 2.25/1.16 = ( Z
% 2.25/1.16 = ( zero_zero @ int ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_33_lead__coeff__1,axiom,
% 2.25/1.16 ! [A: $tType] :
% 2.25/1.16 ( ( comm_semiring_1 @ A )
% 2.25/1.16 => ( ( coeff @ A @ ( one_one @ ( poly @ A ) ) @ ( degree @ A @ ( one_one @ ( poly @ A ) ) ) )
% 2.25/1.16 = ( one_one @ A ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_34_of__int__le__0__iff,axiom,
% 2.25/1.16 ! [A: $tType] :
% 2.25/1.16 ( ( linordered_idom @ A )
% 2.25/1.16 => ! [Z: int] :
% 2.25/1.16 ( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ Z ) @ ( zero_zero @ A ) )
% 2.25/1.16 = ( ord_less_eq @ int @ Z @ ( zero_zero @ int ) ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_35_of__int__0__le__iff,axiom,
% 2.25/1.16 ! [A: $tType] :
% 2.25/1.16 ( ( linordered_idom @ A )
% 2.25/1.16 => ! [Z: int] :
% 2.25/1.16 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( ring_1_of_int @ A @ Z ) )
% 2.25/1.16 = ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_36_of__int__less__0__iff,axiom,
% 2.25/1.16 ! [A: $tType] :
% 2.25/1.16 ( ( linordered_idom @ A )
% 2.25/1.16 => ! [Z: int] :
% 2.25/1.16 ( ( ord_less @ A @ ( ring_1_of_int @ A @ Z ) @ ( zero_zero @ A ) )
% 2.25/1.16 = ( ord_less @ int @ Z @ ( zero_zero @ int ) ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_37_of__int__0__less__iff,axiom,
% 2.25/1.16 ! [A: $tType] :
% 2.25/1.16 ( ( linordered_idom @ A )
% 2.25/1.16 => ! [Z: int] :
% 2.25/1.16 ( ( ord_less @ A @ ( zero_zero @ A ) @ ( ring_1_of_int @ A @ Z ) )
% 2.25/1.16 = ( ord_less @ int @ ( zero_zero @ int ) @ Z ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_38_of__int__less__1__iff,axiom,
% 2.25/1.16 ! [A: $tType] :
% 2.25/1.16 ( ( linordered_idom @ A )
% 2.25/1.16 => ! [Z: int] :
% 2.25/1.16 ( ( ord_less @ A @ ( ring_1_of_int @ A @ Z ) @ ( one_one @ A ) )
% 2.25/1.16 = ( ord_less @ int @ Z @ ( one_one @ int ) ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_39_of__int__1__less__iff,axiom,
% 2.25/1.16 ! [A: $tType] :
% 2.25/1.16 ( ( linordered_idom @ A )
% 2.25/1.16 => ! [Z: int] :
% 2.25/1.16 ( ( ord_less @ A @ ( one_one @ A ) @ ( ring_1_of_int @ A @ Z ) )
% 2.25/1.16 = ( ord_less @ int @ ( one_one @ int ) @ Z ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_40_of__int__power__less__of__int__cancel__iff,axiom,
% 2.25/1.16 ! [A: $tType] :
% 2.25/1.16 ( ( linordered_idom @ A )
% 2.25/1.16 => ! [X: int,B: int,W: nat] :
% 2.25/1.16 ( ( ord_less @ A @ ( ring_1_of_int @ A @ X ) @ ( power_power @ A @ ( ring_1_of_int @ A @ B ) @ W ) )
% 2.25/1.16 = ( ord_less @ int @ X @ ( power_power @ int @ B @ W ) ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_41_of__int__less__of__int__power__cancel__iff,axiom,
% 2.25/1.16 ! [A: $tType] :
% 2.25/1.16 ( ( linordered_idom @ A )
% 2.25/1.16 => ! [B: int,W: nat,X: int] :
% 2.25/1.16 ( ( ord_less @ A @ ( power_power @ A @ ( ring_1_of_int @ A @ B ) @ W ) @ ( ring_1_of_int @ A @ X ) )
% 2.25/1.16 = ( ord_less @ int @ ( power_power @ int @ B @ W ) @ X ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_42_less__int__code_I1_J,axiom,
% 2.25/1.16 ~ ( ord_less @ int @ ( zero_zero @ int ) @ ( zero_zero @ int ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_43_less__eq__int__code_I1_J,axiom,
% 2.25/1.16 ord_less_eq @ int @ ( zero_zero @ int ) @ ( zero_zero @ int ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_44_poly__eqI,axiom,
% 2.25/1.16 ! [A: $tType] :
% 2.25/1.16 ( ( zero @ A )
% 2.25/1.16 => ! [P: poly @ A,Q: poly @ A] :
% 2.25/1.16 ( ! [N2: nat] :
% 2.25/1.16 ( ( coeff @ A @ P @ N2 )
% 2.25/1.16 = ( coeff @ A @ Q @ N2 ) )
% 2.25/1.16 => ( P = Q ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_45_mem__Collect__eq,axiom,
% 2.25/1.16 ! [A: $tType,A2: A,P3: A > $o] :
% 2.25/1.16 ( ( member @ A @ A2 @ ( collect @ A @ P3 ) )
% 2.25/1.16 = ( P3 @ A2 ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_46_Collect__mem__eq,axiom,
% 2.25/1.16 ! [A: $tType,A3: set @ A] :
% 2.25/1.16 ( ( collect @ A
% 2.25/1.16 @ ^ [X2: A] : ( member @ A @ X2 @ A3 ) )
% 2.25/1.16 = A3 ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_47_Collect__cong,axiom,
% 2.25/1.16 ! [A: $tType,P3: A > $o,Q2: A > $o] :
% 2.25/1.16 ( ! [X3: A] :
% 2.25/1.16 ( ( P3 @ X3 )
% 2.25/1.16 = ( Q2 @ X3 ) )
% 2.25/1.16 => ( ( collect @ A @ P3 )
% 2.25/1.16 = ( collect @ A @ Q2 ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_48_ext,axiom,
% 2.25/1.16 ! [B2: $tType,A: $tType,F: A > B2,G: A > B2] :
% 2.25/1.16 ( ! [X3: A] :
% 2.25/1.16 ( ( F @ X3 )
% 2.25/1.16 = ( G @ X3 ) )
% 2.25/1.16 => ( F = G ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_49_poly__eq__iff,axiom,
% 2.25/1.16 ! [A: $tType] :
% 2.25/1.16 ( ( zero @ A )
% 2.25/1.16 => ( ( ^ [Y: poly @ A,Z2: poly @ A] : ( Y = Z2 ) )
% 2.25/1.16 = ( ^ [P4: poly @ A,Q3: poly @ A] :
% 2.25/1.16 ! [N3: nat] :
% 2.25/1.16 ( ( coeff @ A @ P4 @ N3 )
% 2.25/1.16 = ( coeff @ A @ Q3 @ N3 ) ) ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_50_Ints__0,axiom,
% 2.25/1.16 ! [A: $tType] :
% 2.25/1.16 ( ( ring_1 @ A )
% 2.25/1.16 => ( member @ A @ ( zero_zero @ A ) @ ( ring_1_Ints @ A ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_51_zero__poly_Orep__eq,axiom,
% 2.25/1.16 ! [A: $tType] :
% 2.25/1.16 ( ( zero @ A )
% 2.25/1.16 => ( ( coeff @ A @ ( zero_zero @ ( poly @ A ) ) )
% 2.25/1.16 = ( ^ [Uu: nat] : ( zero_zero @ A ) ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_52_int__one__le__iff__zero__less,axiom,
% 2.25/1.16 ! [Z: int] :
% 2.25/1.16 ( ( ord_less_eq @ int @ ( one_one @ int ) @ Z )
% 2.25/1.16 = ( ord_less @ int @ ( zero_zero @ int ) @ Z ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_53_coeff__inject,axiom,
% 2.25/1.16 ! [A: $tType] :
% 2.25/1.16 ( ( zero @ A )
% 2.25/1.16 => ! [X: poly @ A,Y2: poly @ A] :
% 2.25/1.16 ( ( ( coeff @ A @ X )
% 2.25/1.16 = ( coeff @ A @ Y2 ) )
% 2.25/1.16 = ( X = Y2 ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_54_le__degree,axiom,
% 2.25/1.16 ! [A: $tType] :
% 2.25/1.16 ( ( zero @ A )
% 2.25/1.16 => ! [P: poly @ A,N: nat] :
% 2.25/1.16 ( ( ( coeff @ A @ P @ N )
% 2.25/1.16 != ( zero_zero @ A ) )
% 2.25/1.16 => ( ord_less_eq @ nat @ N @ ( degree @ A @ P ) ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_55_poly__0__coeff__0,axiom,
% 2.25/1.16 ! [A: $tType] :
% 2.25/1.16 ( ( comm_semiring_0 @ A )
% 2.25/1.16 => ! [P: poly @ A] :
% 2.25/1.16 ( ( poly2 @ A @ P @ ( zero_zero @ A ) )
% 2.25/1.16 = ( coeff @ A @ P @ ( zero_zero @ nat ) ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_56_algebraic__def,axiom,
% 2.25/1.16 ! [A: $tType] :
% 2.25/1.16 ( ( field_char_0 @ A )
% 2.25/1.16 => ( ( algebraic @ A )
% 2.25/1.16 = ( ^ [X2: A] :
% 2.25/1.16 ? [P4: poly @ A] :
% 2.25/1.16 ( ! [I3: nat] : ( member @ A @ ( coeff @ A @ P4 @ I3 ) @ ( ring_1_Ints @ A ) )
% 2.25/1.16 & ( P4
% 2.25/1.16 != ( zero_zero @ ( poly @ A ) ) )
% 2.25/1.16 & ( ( poly2 @ A @ P4 @ X2 )
% 2.25/1.16 = ( zero_zero @ A ) ) ) ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_57_algebraicI,axiom,
% 2.25/1.16 ! [A: $tType] :
% 2.25/1.16 ( ( field_char_0 @ A )
% 2.25/1.16 => ! [P: poly @ A,X: A] :
% 2.25/1.16 ( ! [I4: nat] : ( member @ A @ ( coeff @ A @ P @ I4 ) @ ( ring_1_Ints @ A ) )
% 2.25/1.16 => ( ( P
% 2.25/1.16 != ( zero_zero @ ( poly @ A ) ) )
% 2.25/1.16 => ( ( ( poly2 @ A @ P @ X )
% 2.25/1.16 = ( zero_zero @ A ) )
% 2.25/1.16 => ( algebraic @ A @ X ) ) ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_58_algebraicE,axiom,
% 2.25/1.16 ! [A: $tType] :
% 2.25/1.16 ( ( field_char_0 @ A )
% 2.25/1.16 => ! [X: A] :
% 2.25/1.16 ( ( algebraic @ A @ X )
% 2.25/1.16 => ~ ! [P2: poly @ A] :
% 2.25/1.16 ( ! [I2: nat] : ( member @ A @ ( coeff @ A @ P2 @ I2 ) @ ( ring_1_Ints @ A ) )
% 2.25/1.16 => ( ( P2
% 2.25/1.16 != ( zero_zero @ ( poly @ A ) ) )
% 2.25/1.16 => ( ( poly2 @ A @ P2 @ X )
% 2.25/1.16 != ( zero_zero @ A ) ) ) ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_59_of__int__pos,axiom,
% 2.25/1.16 ! [A: $tType] :
% 2.25/1.16 ( ( linordered_idom @ A )
% 2.25/1.16 => ! [Z: int] :
% 2.25/1.16 ( ( ord_less @ int @ ( zero_zero @ int ) @ Z )
% 2.25/1.16 => ( ord_less @ A @ ( zero_zero @ A ) @ ( ring_1_of_int @ A @ Z ) ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_60_leading__coeff__neq__0,axiom,
% 2.25/1.16 ! [A: $tType] :
% 2.25/1.16 ( ( zero @ A )
% 2.25/1.16 => ! [P: poly @ A] :
% 2.25/1.16 ( ( P
% 2.25/1.16 != ( zero_zero @ ( poly @ A ) ) )
% 2.25/1.16 => ( ( coeff @ A @ P @ ( degree @ A @ P ) )
% 2.25/1.16 != ( zero_zero @ A ) ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_61_poly__all__0__iff__0,axiom,
% 2.25/1.16 ! [A: $tType] :
% 2.25/1.16 ( ( ( ring_char_0 @ A )
% 2.25/1.16 & ( comm_ring_1 @ A )
% 2.25/1.16 & ( ring_n68954251visors @ A ) )
% 2.25/1.16 => ! [P: poly @ A] :
% 2.25/1.16 ( ( ! [X2: A] :
% 2.25/1.16 ( ( poly2 @ A @ P @ X2 )
% 2.25/1.16 = ( zero_zero @ A ) ) )
% 2.25/1.16 = ( P
% 2.25/1.16 = ( zero_zero @ ( poly @ A ) ) ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_62_Ints__1,axiom,
% 2.25/1.16 ! [A: $tType] :
% 2.25/1.16 ( ( ring_1 @ A )
% 2.25/1.16 => ( member @ A @ ( one_one @ A ) @ ( ring_1_Ints @ A ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_63_Ints__power,axiom,
% 2.25/1.16 ! [A: $tType] :
% 2.25/1.16 ( ( ring_1 @ A )
% 2.25/1.16 => ! [A2: A,N: nat] :
% 2.25/1.16 ( ( member @ A @ A2 @ ( ring_1_Ints @ A ) )
% 2.25/1.16 => ( member @ A @ ( power_power @ A @ A2 @ N ) @ ( ring_1_Ints @ A ) ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_64_Ints__abs,axiom,
% 2.25/1.16 ! [A: $tType] :
% 2.25/1.16 ( ( linordered_idom @ A )
% 2.25/1.16 => ! [A2: A] :
% 2.25/1.16 ( ( member @ A @ A2 @ ( ring_1_Ints @ A ) )
% 2.25/1.16 => ( member @ A @ ( abs_abs @ A @ A2 ) @ ( ring_1_Ints @ A ) ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_65_Ints__of__int,axiom,
% 2.25/1.16 ! [A: $tType] :
% 2.25/1.16 ( ( ring_1 @ A )
% 2.25/1.16 => ! [Z: int] : ( member @ A @ ( ring_1_of_int @ A @ Z ) @ ( ring_1_Ints @ A ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_66_Ints__induct,axiom,
% 2.25/1.16 ! [A: $tType] :
% 2.25/1.16 ( ( ring_1 @ A )
% 2.25/1.16 => ! [Q: A,P3: A > $o] :
% 2.25/1.16 ( ( member @ A @ Q @ ( ring_1_Ints @ A ) )
% 2.25/1.16 => ( ! [Z3: int] : ( P3 @ ( ring_1_of_int @ A @ Z3 ) )
% 2.25/1.16 => ( P3 @ Q ) ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_67_Ints__cases,axiom,
% 2.25/1.16 ! [A: $tType] :
% 2.25/1.16 ( ( ring_1 @ A )
% 2.25/1.16 => ! [Q: A] :
% 2.25/1.16 ( ( member @ A @ Q @ ( ring_1_Ints @ A ) )
% 2.25/1.16 => ~ ! [Z3: int] :
% 2.25/1.16 ( Q
% 2.25/1.16 != ( ring_1_of_int @ A @ Z3 ) ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_68_Ints__nonzero__abs__less1,axiom,
% 2.25/1.16 ! [A: $tType] :
% 2.25/1.16 ( ( linordered_idom @ A )
% 2.25/1.16 => ! [X: A] :
% 2.25/1.16 ( ( member @ A @ X @ ( ring_1_Ints @ A ) )
% 2.25/1.16 => ( ( ord_less @ A @ ( abs_abs @ A @ X ) @ ( one_one @ A ) )
% 2.25/1.16 => ( X
% 2.25/1.16 = ( zero_zero @ A ) ) ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_69_of__int__nonneg,axiom,
% 2.25/1.16 ! [A: $tType] :
% 2.25/1.16 ( ( linordered_idom @ A )
% 2.25/1.16 => ! [Z: int] :
% 2.25/1.16 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z )
% 2.25/1.16 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( ring_1_of_int @ A @ Z ) ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_70_lead__coeff__power,axiom,
% 2.25/1.16 ! [A: $tType] :
% 2.25/1.16 ( ( ( comm_semiring_1 @ A )
% 2.25/1.16 & ( semiri1193490041visors @ A ) )
% 2.25/1.16 => ! [P: poly @ A,N: nat] :
% 2.25/1.16 ( ( coeff @ A @ ( power_power @ ( poly @ A ) @ P @ N ) @ ( degree @ A @ ( power_power @ ( poly @ A ) @ P @ N ) ) )
% 2.25/1.16 = ( power_power @ A @ ( coeff @ A @ P @ ( degree @ A @ P ) ) @ N ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_71_Ints__nonzero__abs__ge1,axiom,
% 2.25/1.16 ! [A: $tType] :
% 2.25/1.16 ( ( linordered_idom @ A )
% 2.25/1.16 => ! [X: A] :
% 2.25/1.16 ( ( member @ A @ X @ ( ring_1_Ints @ A ) )
% 2.25/1.16 => ( ( X
% 2.25/1.16 != ( zero_zero @ A ) )
% 2.25/1.16 => ( ord_less_eq @ A @ ( one_one @ A ) @ ( abs_abs @ A @ X ) ) ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_72_poly__eq__poly__eq__iff,axiom,
% 2.25/1.16 ! [A: $tType] :
% 2.25/1.16 ( ( ( ring_char_0 @ A )
% 2.25/1.16 & ( comm_ring_1 @ A )
% 2.25/1.16 & ( ring_n68954251visors @ A ) )
% 2.25/1.16 => ! [P: poly @ A,Q: poly @ A] :
% 2.25/1.16 ( ( ( poly2 @ A @ P )
% 2.25/1.16 = ( poly2 @ A @ Q ) )
% 2.25/1.16 = ( P = Q ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_73_of__int__lessD,axiom,
% 2.25/1.16 ! [A: $tType] :
% 2.25/1.16 ( ( linordered_idom @ A )
% 2.25/1.16 => ! [N: int,X: A] :
% 2.25/1.16 ( ( ord_less @ A @ ( abs_abs @ A @ ( ring_1_of_int @ A @ N ) ) @ X )
% 2.25/1.16 => ( ( N
% 2.25/1.16 = ( zero_zero @ int ) )
% 2.25/1.16 | ( ord_less @ A @ ( one_one @ A ) @ X ) ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_74_of__int__leD,axiom,
% 2.25/1.16 ! [A: $tType] :
% 2.25/1.16 ( ( linordered_idom @ A )
% 2.25/1.16 => ! [N: int,X: A] :
% 2.25/1.16 ( ( ord_less_eq @ A @ ( abs_abs @ A @ ( ring_1_of_int @ A @ N ) ) @ X )
% 2.25/1.16 => ( ( N
% 2.25/1.16 = ( zero_zero @ int ) )
% 2.25/1.16 | ( ord_less_eq @ A @ ( one_one @ A ) @ X ) ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_75_int__poly__rat__no__root__ge,axiom,
% 2.25/1.16 ! [P: poly @ real,B: int,A2: int] :
% 2.25/1.16 ( ! [N2: nat] : ( member @ real @ ( coeff @ real @ P @ N2 ) @ ( ring_1_Ints @ real ) )
% 2.25/1.16 => ( ( ord_less @ int @ ( zero_zero @ int ) @ B )
% 2.25/1.16 => ( ( ( poly2 @ real @ P @ ( divide_divide @ real @ ( ring_1_of_int @ real @ A2 ) @ ( ring_1_of_int @ real @ B ) ) )
% 2.25/1.16 != ( zero_zero @ real ) )
% 2.25/1.16 => ( ord_less_eq @ real @ ( divide_divide @ real @ ( one_one @ real ) @ ( power_power @ real @ ( ring_1_of_int @ real @ B ) @ ( degree @ real @ P ) ) ) @ ( abs_abs @ real @ ( poly2 @ real @ P @ ( divide_divide @ real @ ( ring_1_of_int @ real @ A2 ) @ ( ring_1_of_int @ real @ B ) ) ) ) ) ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_76_power__decreasing__iff,axiom,
% 2.25/1.16 ! [A: $tType] :
% 2.25/1.16 ( ( linordered_semidom @ A )
% 2.25/1.16 => ! [B: A,M: nat,N: nat] :
% 2.25/1.16 ( ( ord_less @ A @ ( zero_zero @ A ) @ B )
% 2.25/1.16 => ( ( ord_less @ A @ B @ ( one_one @ A ) )
% 2.25/1.16 => ( ( ord_less_eq @ A @ ( power_power @ A @ B @ M ) @ ( power_power @ A @ B @ N ) )
% 2.25/1.16 = ( ord_less_eq @ nat @ N @ M ) ) ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_77_le__divide__eq__1__pos,axiom,
% 2.25/1.16 ! [A: $tType] :
% 2.25/1.16 ( ( linordered_field @ A )
% 2.25/1.16 => ! [A2: A,B: A] :
% 2.25/1.16 ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
% 2.25/1.16 => ( ( ord_less_eq @ A @ ( one_one @ A ) @ ( divide_divide @ A @ B @ A2 ) )
% 2.25/1.16 = ( ord_less_eq @ A @ A2 @ B ) ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_78_le__divide__eq__1__neg,axiom,
% 2.25/1.16 ! [A: $tType] :
% 2.25/1.16 ( ( linordered_field @ A )
% 2.25/1.16 => ! [A2: A,B: A] :
% 2.25/1.16 ( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
% 2.25/1.16 => ( ( ord_less_eq @ A @ ( one_one @ A ) @ ( divide_divide @ A @ B @ A2 ) )
% 2.25/1.16 = ( ord_less_eq @ A @ B @ A2 ) ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_79_divide__le__eq__1__pos,axiom,
% 2.25/1.16 ! [A: $tType] :
% 2.25/1.16 ( ( linordered_field @ A )
% 2.25/1.16 => ! [A2: A,B: A] :
% 2.25/1.16 ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
% 2.25/1.16 => ( ( ord_less_eq @ A @ ( divide_divide @ A @ B @ A2 ) @ ( one_one @ A ) )
% 2.25/1.16 = ( ord_less_eq @ A @ B @ A2 ) ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_80_divide__le__eq__1__neg,axiom,
% 2.25/1.16 ! [A: $tType] :
% 2.25/1.16 ( ( linordered_field @ A )
% 2.25/1.16 => ! [A2: A,B: A] :
% 2.25/1.16 ( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
% 2.25/1.16 => ( ( ord_less_eq @ A @ ( divide_divide @ A @ B @ A2 ) @ ( one_one @ A ) )
% 2.25/1.16 = ( ord_less_eq @ A @ A2 @ B ) ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_81__092_060open_062A_A_P_Areal__of__int_Ab_A_094_An_A_092_060le_062_AA_A_P_A1_092_060close_062,axiom,
% 2.25/1.16 ord_less_eq @ real @ ( divide_divide @ real @ a @ ( power_power @ real @ ( ring_1_of_int @ real @ b ) @ n ) ) @ ( divide_divide @ real @ a @ ( one_one @ real ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_82_zero__less__power__abs__iff,axiom,
% 2.25/1.16 ! [A: $tType] :
% 2.25/1.16 ( ( linordered_idom @ A )
% 2.25/1.16 => ! [A2: A,N: nat] :
% 2.25/1.16 ( ( ord_less @ A @ ( zero_zero @ A ) @ ( power_power @ A @ ( abs_abs @ A @ A2 ) @ N ) )
% 2.25/1.16 = ( ( A2
% 2.25/1.16 != ( zero_zero @ A ) )
% 2.25/1.16 | ( N
% 2.25/1.16 = ( zero_zero @ nat ) ) ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_83_zero__le__divide__abs__iff,axiom,
% 2.25/1.16 ! [A: $tType] :
% 2.25/1.16 ( ( linordered_field @ A )
% 2.25/1.16 => ! [A2: A,B: A] :
% 2.25/1.16 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ A2 @ ( abs_abs @ A @ B ) ) )
% 2.25/1.16 = ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
% 2.25/1.16 | ( B
% 2.25/1.16 = ( zero_zero @ A ) ) ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_84_divide__le__0__abs__iff,axiom,
% 2.25/1.16 ! [A: $tType] :
% 2.25/1.16 ( ( linordered_field @ A )
% 2.25/1.16 => ! [A2: A,B: A] :
% 2.25/1.16 ( ( ord_less_eq @ A @ ( divide_divide @ A @ A2 @ ( abs_abs @ A @ B ) ) @ ( zero_zero @ A ) )
% 2.25/1.16 = ( ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) )
% 2.25/1.16 | ( B
% 2.25/1.16 = ( zero_zero @ A ) ) ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_85_A__pos,axiom,
% 2.25/1.16 ord_less @ real @ ( zero_zero @ real ) @ a ).
% 2.25/1.16
% 2.25/1.16 thf(fact_86_A__less_I1_J,axiom,
% 2.25/1.16 ord_less @ real @ a @ ( one_one @ real ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_87__092_060open_062A_A_092_060le_062_A1_092_060close_062,axiom,
% 2.25/1.16 ord_less_eq @ real @ a @ ( one_one @ real ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_88_power__one__right,axiom,
% 2.25/1.16 ! [A: $tType] :
% 2.25/1.16 ( ( monoid_mult @ A )
% 2.25/1.16 => ! [A2: A] :
% 2.25/1.16 ( ( power_power @ A @ A2 @ ( one_one @ nat ) )
% 2.25/1.16 = A2 ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_89_nat__zero__less__power__iff,axiom,
% 2.25/1.16 ! [X: nat,N: nat] :
% 2.25/1.16 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( power_power @ nat @ X @ N ) )
% 2.25/1.16 = ( ( ord_less @ nat @ ( zero_zero @ nat ) @ X )
% 2.25/1.16 | ( N
% 2.25/1.16 = ( zero_zero @ nat ) ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_90_divide__eq__0__iff,axiom,
% 2.25/1.16 ! [A: $tType] :
% 2.25/1.16 ( ( field @ A )
% 2.25/1.16 => ! [A2: A,B: A] :
% 2.25/1.16 ( ( ( divide_divide @ A @ A2 @ B )
% 2.25/1.16 = ( zero_zero @ A ) )
% 2.25/1.16 = ( ( A2
% 2.25/1.16 = ( zero_zero @ A ) )
% 2.25/1.16 | ( B
% 2.25/1.16 = ( zero_zero @ A ) ) ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_91_divide__cancel__left,axiom,
% 2.25/1.16 ! [A: $tType] :
% 2.25/1.16 ( ( field @ A )
% 2.25/1.16 => ! [C: A,A2: A,B: A] :
% 2.25/1.16 ( ( ( divide_divide @ A @ C @ A2 )
% 2.25/1.16 = ( divide_divide @ A @ C @ B ) )
% 2.25/1.16 = ( ( C
% 2.25/1.16 = ( zero_zero @ A ) )
% 2.25/1.16 | ( A2 = B ) ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_92_divide__cancel__right,axiom,
% 2.25/1.16 ! [A: $tType] :
% 2.25/1.16 ( ( field @ A )
% 2.25/1.16 => ! [A2: A,C: A,B: A] :
% 2.25/1.16 ( ( ( divide_divide @ A @ A2 @ C )
% 2.25/1.16 = ( divide_divide @ A @ B @ C ) )
% 2.25/1.16 = ( ( C
% 2.25/1.16 = ( zero_zero @ A ) )
% 2.25/1.16 | ( A2 = B ) ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_93_division__ring__divide__zero,axiom,
% 2.25/1.16 ! [A: $tType] :
% 2.25/1.16 ( ( division_ring @ A )
% 2.25/1.16 => ! [A2: A] :
% 2.25/1.16 ( ( divide_divide @ A @ A2 @ ( zero_zero @ A ) )
% 2.25/1.16 = ( zero_zero @ A ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_94_power__one,axiom,
% 2.25/1.16 ! [A: $tType] :
% 2.25/1.16 ( ( monoid_mult @ A )
% 2.25/1.16 => ! [N: nat] :
% 2.25/1.16 ( ( power_power @ A @ ( one_one @ A ) @ N )
% 2.25/1.16 = ( one_one @ A ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_95_abs__divide,axiom,
% 2.25/1.16 ! [A: $tType] :
% 2.25/1.16 ( ( field_abs_sgn @ A )
% 2.25/1.16 => ! [A2: A,B: A] :
% 2.25/1.16 ( ( abs_abs @ A @ ( divide_divide @ A @ A2 @ B ) )
% 2.25/1.16 = ( divide_divide @ A @ ( abs_abs @ A @ A2 ) @ ( abs_abs @ A @ B ) ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_96_divide__eq__1__iff,axiom,
% 2.25/1.16 ! [A: $tType] :
% 2.25/1.16 ( ( field @ A )
% 2.25/1.16 => ! [A2: A,B: A] :
% 2.25/1.16 ( ( ( divide_divide @ A @ A2 @ B )
% 2.25/1.16 = ( one_one @ A ) )
% 2.25/1.16 = ( ( B
% 2.25/1.16 != ( zero_zero @ A ) )
% 2.25/1.16 & ( A2 = B ) ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_97_one__eq__divide__iff,axiom,
% 2.25/1.16 ! [A: $tType] :
% 2.25/1.16 ( ( field @ A )
% 2.25/1.16 => ! [A2: A,B: A] :
% 2.25/1.16 ( ( ( one_one @ A )
% 2.25/1.16 = ( divide_divide @ A @ A2 @ B ) )
% 2.25/1.16 = ( ( B
% 2.25/1.16 != ( zero_zero @ A ) )
% 2.25/1.16 & ( A2 = B ) ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_98_divide__self,axiom,
% 2.25/1.16 ! [A: $tType] :
% 2.25/1.16 ( ( division_ring @ A )
% 2.25/1.16 => ! [A2: A] :
% 2.25/1.16 ( ( A2
% 2.25/1.16 != ( zero_zero @ A ) )
% 2.25/1.16 => ( ( divide_divide @ A @ A2 @ A2 )
% 2.25/1.16 = ( one_one @ A ) ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_99_divide__self__if,axiom,
% 2.25/1.16 ! [A: $tType] :
% 2.25/1.16 ( ( division_ring @ A )
% 2.25/1.16 => ! [A2: A] :
% 2.25/1.16 ( ( ( A2
% 2.25/1.16 = ( zero_zero @ A ) )
% 2.25/1.16 => ( ( divide_divide @ A @ A2 @ A2 )
% 2.25/1.16 = ( zero_zero @ A ) ) )
% 2.25/1.16 & ( ( A2
% 2.25/1.16 != ( zero_zero @ A ) )
% 2.25/1.16 => ( ( divide_divide @ A @ A2 @ A2 )
% 2.25/1.16 = ( one_one @ A ) ) ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_100_divide__eq__eq__1,axiom,
% 2.25/1.16 ! [A: $tType] :
% 2.25/1.16 ( ( linordered_field @ A )
% 2.25/1.16 => ! [B: A,A2: A] :
% 2.25/1.16 ( ( ( divide_divide @ A @ B @ A2 )
% 2.25/1.16 = ( one_one @ A ) )
% 2.25/1.16 = ( ( A2
% 2.25/1.16 != ( zero_zero @ A ) )
% 2.25/1.16 & ( A2 = B ) ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_101_eq__divide__eq__1,axiom,
% 2.25/1.16 ! [A: $tType] :
% 2.25/1.16 ( ( linordered_field @ A )
% 2.25/1.16 => ! [B: A,A2: A] :
% 2.25/1.16 ( ( ( one_one @ A )
% 2.25/1.16 = ( divide_divide @ A @ B @ A2 ) )
% 2.25/1.16 = ( ( A2
% 2.25/1.16 != ( zero_zero @ A ) )
% 2.25/1.16 & ( A2 = B ) ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_102_one__divide__eq__0__iff,axiom,
% 2.25/1.16 ! [A: $tType] :
% 2.25/1.16 ( ( linordered_field @ A )
% 2.25/1.16 => ! [A2: A] :
% 2.25/1.16 ( ( ( divide_divide @ A @ ( one_one @ A ) @ A2 )
% 2.25/1.16 = ( zero_zero @ A ) )
% 2.25/1.16 = ( A2
% 2.25/1.16 = ( zero_zero @ A ) ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_103_zero__eq__1__divide__iff,axiom,
% 2.25/1.16 ! [A: $tType] :
% 2.25/1.16 ( ( linordered_field @ A )
% 2.25/1.16 => ! [A2: A] :
% 2.25/1.16 ( ( ( zero_zero @ A )
% 2.25/1.16 = ( divide_divide @ A @ ( one_one @ A ) @ A2 ) )
% 2.25/1.16 = ( A2
% 2.25/1.16 = ( zero_zero @ A ) ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_104_power__inject__exp,axiom,
% 2.25/1.16 ! [A: $tType] :
% 2.25/1.16 ( ( linordered_semidom @ A )
% 2.25/1.16 => ! [A2: A,M: nat,N: nat] :
% 2.25/1.16 ( ( ord_less @ A @ ( one_one @ A ) @ A2 )
% 2.25/1.16 => ( ( ( power_power @ A @ A2 @ M )
% 2.25/1.16 = ( power_power @ A @ A2 @ N ) )
% 2.25/1.16 = ( M = N ) ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_105_power__strict__increasing__iff,axiom,
% 2.25/1.16 ! [A: $tType] :
% 2.25/1.16 ( ( linordered_semidom @ A )
% 2.25/1.16 => ! [B: A,X: nat,Y2: nat] :
% 2.25/1.16 ( ( ord_less @ A @ ( one_one @ A ) @ B )
% 2.25/1.16 => ( ( ord_less @ A @ ( power_power @ A @ B @ X ) @ ( power_power @ A @ B @ Y2 ) )
% 2.25/1.16 = ( ord_less @ nat @ X @ Y2 ) ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_106_power__eq__0__iff,axiom,
% 2.25/1.16 ! [A: $tType] :
% 2.25/1.16 ( ( semiri134348788visors @ A )
% 2.25/1.16 => ! [A2: A,N: nat] :
% 2.25/1.16 ( ( ( power_power @ A @ A2 @ N )
% 2.25/1.16 = ( zero_zero @ A ) )
% 2.25/1.16 = ( ( A2
% 2.25/1.16 = ( zero_zero @ A ) )
% 2.25/1.16 & ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_107_degree__0,axiom,
% 2.25/1.16 ! [A: $tType] :
% 2.25/1.16 ( ( zero @ A )
% 2.25/1.16 => ( ( degree @ A @ ( zero_zero @ ( poly @ A ) ) )
% 2.25/1.16 = ( zero_zero @ nat ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_108_degree__1,axiom,
% 2.25/1.16 ! [A: $tType] :
% 2.25/1.16 ( ( comm_semiring_1 @ A )
% 2.25/1.16 => ( ( degree @ A @ ( one_one @ ( poly @ A ) ) )
% 2.25/1.16 = ( zero_zero @ nat ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_109_degree__of__int,axiom,
% 2.25/1.16 ! [A: $tType] :
% 2.25/1.16 ( ( comm_ring_1 @ A )
% 2.25/1.16 => ! [K: int] :
% 2.25/1.16 ( ( degree @ A @ ( ring_1_of_int @ ( poly @ A ) @ K ) )
% 2.25/1.16 = ( zero_zero @ nat ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_110_divide__le__0__1__iff,axiom,
% 2.25/1.16 ! [A: $tType] :
% 2.25/1.16 ( ( linordered_field @ A )
% 2.25/1.16 => ! [A2: A] :
% 2.25/1.16 ( ( ord_less_eq @ A @ ( divide_divide @ A @ ( one_one @ A ) @ A2 ) @ ( zero_zero @ A ) )
% 2.25/1.16 = ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_111_zero__le__divide__1__iff,axiom,
% 2.25/1.16 ! [A: $tType] :
% 2.25/1.16 ( ( linordered_field @ A )
% 2.25/1.16 => ! [A2: A] :
% 2.25/1.16 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ ( one_one @ A ) @ A2 ) )
% 2.25/1.16 = ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_112_divide__less__0__1__iff,axiom,
% 2.25/1.16 ! [A: $tType] :
% 2.25/1.16 ( ( linordered_field @ A )
% 2.25/1.16 => ! [A2: A] :
% 2.25/1.16 ( ( ord_less @ A @ ( divide_divide @ A @ ( one_one @ A ) @ A2 ) @ ( zero_zero @ A ) )
% 2.25/1.16 = ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_113_divide__less__eq__1__neg,axiom,
% 2.25/1.16 ! [A: $tType] :
% 2.25/1.16 ( ( linordered_field @ A )
% 2.25/1.16 => ! [A2: A,B: A] :
% 2.25/1.16 ( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
% 2.25/1.16 => ( ( ord_less @ A @ ( divide_divide @ A @ B @ A2 ) @ ( one_one @ A ) )
% 2.25/1.16 = ( ord_less @ A @ A2 @ B ) ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_114_divide__less__eq__1__pos,axiom,
% 2.25/1.16 ! [A: $tType] :
% 2.25/1.16 ( ( linordered_field @ A )
% 2.25/1.16 => ! [A2: A,B: A] :
% 2.25/1.16 ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
% 2.25/1.16 => ( ( ord_less @ A @ ( divide_divide @ A @ B @ A2 ) @ ( one_one @ A ) )
% 2.25/1.16 = ( ord_less @ A @ B @ A2 ) ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_115_less__divide__eq__1__neg,axiom,
% 2.25/1.16 ! [A: $tType] :
% 2.25/1.16 ( ( linordered_field @ A )
% 2.25/1.16 => ! [A2: A,B: A] :
% 2.25/1.16 ( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
% 2.25/1.16 => ( ( ord_less @ A @ ( one_one @ A ) @ ( divide_divide @ A @ B @ A2 ) )
% 2.25/1.16 = ( ord_less @ A @ B @ A2 ) ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_116_less__divide__eq__1__pos,axiom,
% 2.25/1.16 ! [A: $tType] :
% 2.25/1.16 ( ( linordered_field @ A )
% 2.25/1.16 => ! [A2: A,B: A] :
% 2.25/1.16 ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
% 2.25/1.16 => ( ( ord_less @ A @ ( one_one @ A ) @ ( divide_divide @ A @ B @ A2 ) )
% 2.25/1.16 = ( ord_less @ A @ A2 @ B ) ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_117_zero__less__divide__1__iff,axiom,
% 2.25/1.16 ! [A: $tType] :
% 2.25/1.16 ( ( linordered_field @ A )
% 2.25/1.16 => ! [A2: A] :
% 2.25/1.16 ( ( ord_less @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ ( one_one @ A ) @ A2 ) )
% 2.25/1.16 = ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_118_power__strict__decreasing__iff,axiom,
% 2.25/1.16 ! [A: $tType] :
% 2.25/1.16 ( ( linordered_semidom @ A )
% 2.25/1.16 => ! [B: A,M: nat,N: nat] :
% 2.25/1.16 ( ( ord_less @ A @ ( zero_zero @ A ) @ B )
% 2.25/1.16 => ( ( ord_less @ A @ B @ ( one_one @ A ) )
% 2.25/1.16 => ( ( ord_less @ A @ ( power_power @ A @ B @ M ) @ ( power_power @ A @ B @ N ) )
% 2.25/1.16 = ( ord_less @ nat @ N @ M ) ) ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_119_power__mono__iff,axiom,
% 2.25/1.16 ! [A: $tType] :
% 2.25/1.16 ( ( linordered_semidom @ A )
% 2.25/1.16 => ! [A2: A,B: A,N: nat] :
% 2.25/1.16 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
% 2.25/1.16 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B )
% 2.25/1.16 => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 2.25/1.16 => ( ( ord_less_eq @ A @ ( power_power @ A @ A2 @ N ) @ ( power_power @ A @ B @ N ) )
% 2.25/1.16 = ( ord_less_eq @ A @ A2 @ B ) ) ) ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_120_power__increasing__iff,axiom,
% 2.25/1.16 ! [A: $tType] :
% 2.25/1.16 ( ( linordered_semidom @ A )
% 2.25/1.16 => ! [B: A,X: nat,Y2: nat] :
% 2.25/1.16 ( ( ord_less @ A @ ( one_one @ A ) @ B )
% 2.25/1.16 => ( ( ord_less_eq @ A @ ( power_power @ A @ B @ X ) @ ( power_power @ A @ B @ Y2 ) )
% 2.25/1.16 = ( ord_less_eq @ nat @ X @ Y2 ) ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_121__092_060open_062_092_060not_062_AA_A_P_Areal__of__int_Ab_A_094_An_A_060_A_092_060bar_062x_A_N_Areal__of__int_Aa_A_P_Areal__of__int_Ab_092_060bar_062_092_060close_062,axiom,
% 2.25/1.16 ~ ( ord_less @ real @ ( divide_divide @ real @ a @ ( power_power @ real @ ( ring_1_of_int @ real @ b ) @ n ) ) @ ( abs_abs @ real @ ( minus_minus @ real @ x @ ( divide_divide @ real @ ( ring_1_of_int @ real @ a2 ) @ ( ring_1_of_int @ real @ b ) ) ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_122_nat__power__less__imp__less,axiom,
% 2.25/1.16 ! [I: nat,M: nat,N: nat] :
% 2.25/1.16 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ I )
% 2.25/1.16 => ( ( ord_less @ nat @ ( power_power @ nat @ I @ M ) @ ( power_power @ nat @ I @ N ) )
% 2.25/1.16 => ( ord_less @ nat @ M @ N ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_123_div__poly__less,axiom,
% 2.25/1.16 ! [A: $tType] :
% 2.25/1.16 ( ( field @ A )
% 2.25/1.16 => ! [X: poly @ A,Y2: poly @ A] :
% 2.25/1.16 ( ( ord_less @ nat @ ( degree @ A @ X ) @ ( degree @ A @ Y2 ) )
% 2.25/1.16 => ( ( divide_divide @ ( poly @ A ) @ X @ Y2 )
% 2.25/1.16 = ( zero_zero @ ( poly @ A ) ) ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_124_self__le__power,axiom,
% 2.25/1.16 ! [A: $tType] :
% 2.25/1.16 ( ( linordered_semidom @ A )
% 2.25/1.16 => ! [A2: A,N: nat] :
% 2.25/1.16 ( ( ord_less_eq @ A @ ( one_one @ A ) @ A2 )
% 2.25/1.16 => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 2.25/1.16 => ( ord_less_eq @ A @ A2 @ ( power_power @ A @ A2 @ N ) ) ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_125_divide__poly__0,axiom,
% 2.25/1.16 ! [A: $tType] :
% 2.25/1.16 ( ( idom_divide @ A )
% 2.25/1.16 => ! [F: poly @ A] :
% 2.25/1.16 ( ( divide_divide @ ( poly @ A ) @ F @ ( zero_zero @ ( poly @ A ) ) )
% 2.25/1.16 = ( zero_zero @ ( poly @ A ) ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_126_zero__power,axiom,
% 2.25/1.16 ! [A: $tType] :
% 2.25/1.16 ( ( semiring_1 @ A )
% 2.25/1.16 => ! [N: nat] :
% 2.25/1.16 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 2.25/1.16 => ( ( power_power @ A @ ( zero_zero @ A ) @ N )
% 2.25/1.16 = ( zero_zero @ A ) ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_127_power__0,axiom,
% 2.25/1.16 ! [A: $tType] :
% 2.25/1.16 ( ( power @ A )
% 2.25/1.16 => ! [A2: A] :
% 2.25/1.16 ( ( power_power @ A @ A2 @ ( zero_zero @ nat ) )
% 2.25/1.16 = ( one_one @ A ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_128_power__eq__iff__eq__base,axiom,
% 2.25/1.16 ! [A: $tType] :
% 2.25/1.16 ( ( linordered_semidom @ A )
% 2.25/1.16 => ! [N: nat,A2: A,B: A] :
% 2.25/1.16 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 2.25/1.16 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
% 2.25/1.16 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B )
% 2.25/1.16 => ( ( ( power_power @ A @ A2 @ N )
% 2.25/1.16 = ( power_power @ A @ B @ N ) )
% 2.25/1.16 = ( A2 = B ) ) ) ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_129_power__eq__imp__eq__base,axiom,
% 2.25/1.16 ! [A: $tType] :
% 2.25/1.16 ( ( linordered_semidom @ A )
% 2.25/1.16 => ! [A2: A,N: nat,B: A] :
% 2.25/1.16 ( ( ( power_power @ A @ A2 @ N )
% 2.25/1.16 = ( power_power @ A @ B @ N ) )
% 2.25/1.16 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
% 2.25/1.16 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B )
% 2.25/1.16 => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 2.25/1.16 => ( A2 = B ) ) ) ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_130_one__less__power,axiom,
% 2.25/1.16 ! [A: $tType] :
% 2.25/1.16 ( ( linordered_semidom @ A )
% 2.25/1.16 => ! [A2: A,N: nat] :
% 2.25/1.16 ( ( ord_less @ A @ ( one_one @ A ) @ A2 )
% 2.25/1.16 => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 2.25/1.16 => ( ord_less @ A @ ( one_one @ A ) @ ( power_power @ A @ A2 @ N ) ) ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_131_poly__IVT__neg,axiom,
% 2.25/1.16 ! [A2: real,B: real,P: poly @ real] :
% 2.25/1.16 ( ( ord_less @ real @ A2 @ B )
% 2.25/1.16 => ( ( ord_less @ real @ ( zero_zero @ real ) @ ( poly2 @ real @ P @ A2 ) )
% 2.25/1.16 => ( ( ord_less @ real @ ( poly2 @ real @ P @ B ) @ ( zero_zero @ real ) )
% 2.25/1.16 => ? [X3: real] :
% 2.25/1.16 ( ( ord_less @ real @ A2 @ X3 )
% 2.25/1.16 & ( ord_less @ real @ X3 @ B )
% 2.25/1.16 & ( ( poly2 @ real @ P @ X3 )
% 2.25/1.16 = ( zero_zero @ real ) ) ) ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_132_poly__IVT__pos,axiom,
% 2.25/1.16 ! [A2: real,B: real,P: poly @ real] :
% 2.25/1.16 ( ( ord_less @ real @ A2 @ B )
% 2.25/1.16 => ( ( ord_less @ real @ ( poly2 @ real @ P @ A2 ) @ ( zero_zero @ real ) )
% 2.25/1.16 => ( ( ord_less @ real @ ( zero_zero @ real ) @ ( poly2 @ real @ P @ B ) )
% 2.25/1.16 => ? [X3: real] :
% 2.25/1.16 ( ( ord_less @ real @ A2 @ X3 )
% 2.25/1.16 & ( ord_less @ real @ X3 @ B )
% 2.25/1.16 & ( ( poly2 @ real @ P @ X3 )
% 2.25/1.16 = ( zero_zero @ real ) ) ) ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_133_linordered__field__no__ub,axiom,
% 2.25/1.16 ! [A: $tType] :
% 2.25/1.16 ( ( linordered_field @ A )
% 2.25/1.16 => ! [X4: A] :
% 2.25/1.16 ? [X_1: A] : ( ord_less @ A @ X4 @ X_1 ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_134_linordered__field__no__lb,axiom,
% 2.25/1.16 ! [A: $tType] :
% 2.25/1.16 ( ( linordered_field @ A )
% 2.25/1.16 => ! [X4: A] :
% 2.25/1.16 ? [Y3: A] : ( ord_less @ A @ Y3 @ X4 ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_135_coeff__eq__0,axiom,
% 2.25/1.16 ! [A: $tType] :
% 2.25/1.16 ( ( zero @ A )
% 2.25/1.16 => ! [P: poly @ A,N: nat] :
% 2.25/1.16 ( ( ord_less @ nat @ ( degree @ A @ P ) @ N )
% 2.25/1.16 => ( ( coeff @ A @ P @ N )
% 2.25/1.16 = ( zero_zero @ A ) ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_136_less__degree__imp,axiom,
% 2.25/1.16 ! [A: $tType] :
% 2.25/1.16 ( ( zero @ A )
% 2.25/1.16 => ! [N: nat,P: poly @ A] :
% 2.25/1.16 ( ( ord_less @ nat @ N @ ( degree @ A @ P ) )
% 2.25/1.16 => ? [I4: nat] :
% 2.25/1.16 ( ( ord_less @ nat @ N @ I4 )
% 2.25/1.16 & ( ( coeff @ A @ P @ I4 )
% 2.25/1.16 != ( zero_zero @ A ) ) ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_137_degree__le,axiom,
% 2.25/1.16 ! [A: $tType] :
% 2.25/1.16 ( ( zero @ A )
% 2.25/1.16 => ! [N: nat,P: poly @ A] :
% 2.25/1.16 ( ! [I4: nat] :
% 2.25/1.16 ( ( ord_less @ nat @ N @ I4 )
% 2.25/1.16 => ( ( coeff @ A @ P @ I4 )
% 2.25/1.16 = ( zero_zero @ A ) ) )
% 2.25/1.16 => ( ord_less_eq @ nat @ ( degree @ A @ P ) @ N ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_138_Liouville__Numbers__Misc_ORats__cases_H,axiom,
% 2.25/1.16 ! [X: real] :
% 2.25/1.16 ( ( member @ real @ X @ ( field_char_0_Rats @ real ) )
% 2.25/1.16 => ~ ! [P2: int,Q4: int] :
% 2.25/1.16 ( ( ord_less @ int @ ( zero_zero @ int ) @ Q4 )
% 2.25/1.16 => ( X
% 2.25/1.16 != ( divide_divide @ real @ ( ring_1_of_int @ real @ P2 ) @ ( ring_1_of_int @ real @ Q4 ) ) ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_139_coeff__0__power,axiom,
% 2.25/1.16 ! [A: $tType] :
% 2.25/1.16 ( ( comm_semiring_1 @ A )
% 2.25/1.16 => ! [P: poly @ A,N: nat] :
% 2.25/1.16 ( ( coeff @ A @ ( power_power @ ( poly @ A ) @ P @ N ) @ ( zero_zero @ nat ) )
% 2.25/1.16 = ( power_power @ A @ ( coeff @ A @ P @ ( zero_zero @ nat ) ) @ N ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_140_eq__zero__or__degree__less,axiom,
% 2.25/1.16 ! [A: $tType] :
% 2.25/1.16 ( ( zero @ A )
% 2.25/1.16 => ! [P: poly @ A,N: nat] :
% 2.25/1.16 ( ( ord_less_eq @ nat @ ( degree @ A @ P ) @ N )
% 2.25/1.16 => ( ( ( coeff @ A @ P @ N )
% 2.25/1.16 = ( zero_zero @ A ) )
% 2.25/1.16 => ( ( P
% 2.25/1.16 = ( zero_zero @ ( poly @ A ) ) )
% 2.25/1.16 | ( ord_less @ nat @ ( degree @ A @ P ) @ N ) ) ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_141_power__not__zero,axiom,
% 2.25/1.16 ! [A: $tType] :
% 2.25/1.16 ( ( semiri134348788visors @ A )
% 2.25/1.16 => ! [A2: A,N: nat] :
% 2.25/1.16 ( ( A2
% 2.25/1.16 != ( zero_zero @ A ) )
% 2.25/1.16 => ( ( power_power @ A @ A2 @ N )
% 2.25/1.16 != ( zero_zero @ A ) ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_142_power__divide,axiom,
% 2.25/1.16 ! [A: $tType] :
% 2.25/1.16 ( ( field @ A )
% 2.25/1.16 => ! [A2: A,B: A,N: nat] :
% 2.25/1.16 ( ( power_power @ A @ ( divide_divide @ A @ A2 @ B ) @ N )
% 2.25/1.16 = ( divide_divide @ A @ ( power_power @ A @ A2 @ N ) @ ( power_power @ A @ B @ N ) ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_143_poly__pinfty__gt__lc,axiom,
% 2.25/1.16 ! [P: poly @ real] :
% 2.25/1.16 ( ( ord_less @ real @ ( zero_zero @ real ) @ ( coeff @ real @ P @ ( degree @ real @ P ) ) )
% 2.25/1.16 => ? [N2: real] :
% 2.25/1.16 ! [X4: real] :
% 2.25/1.16 ( ( ord_less_eq @ real @ N2 @ X4 )
% 2.25/1.16 => ( ord_less_eq @ real @ ( coeff @ real @ P @ ( degree @ real @ P ) ) @ ( poly2 @ real @ P @ X4 ) ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_144_power__abs,axiom,
% 2.25/1.16 ! [A: $tType] :
% 2.25/1.16 ( ( linordered_idom @ A )
% 2.25/1.16 => ! [A2: A,N: nat] :
% 2.25/1.16 ( ( abs_abs @ A @ ( power_power @ A @ A2 @ N ) )
% 2.25/1.16 = ( power_power @ A @ ( abs_abs @ A @ A2 ) @ N ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_145_algebraic__altdef,axiom,
% 2.25/1.16 ! [B2: $tType] :
% 2.25/1.16 ( ( field_char_0 @ B2 )
% 2.25/1.16 => ( ( algebraic @ B2 )
% 2.25/1.16 = ( ^ [X2: B2] :
% 2.25/1.16 ? [P4: poly @ B2] :
% 2.25/1.16 ( ! [I3: nat] : ( member @ B2 @ ( coeff @ B2 @ P4 @ I3 ) @ ( field_char_0_Rats @ B2 ) )
% 2.25/1.16 & ( P4
% 2.25/1.16 != ( zero_zero @ ( poly @ B2 ) ) )
% 2.25/1.16 & ( ( poly2 @ B2 @ P4 @ X2 )
% 2.25/1.16 = ( zero_zero @ B2 ) ) ) ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_146_divide__le__0__iff,axiom,
% 2.25/1.16 ! [A: $tType] :
% 2.25/1.16 ( ( linordered_field @ A )
% 2.25/1.16 => ! [A2: A,B: A] :
% 2.25/1.16 ( ( ord_less_eq @ A @ ( divide_divide @ A @ A2 @ B ) @ ( zero_zero @ A ) )
% 2.25/1.16 = ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
% 2.25/1.16 & ( ord_less_eq @ A @ B @ ( zero_zero @ A ) ) )
% 2.25/1.16 | ( ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) )
% 2.25/1.16 & ( ord_less_eq @ A @ ( zero_zero @ A ) @ B ) ) ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_147_divide__right__mono,axiom,
% 2.25/1.16 ! [A: $tType] :
% 2.25/1.16 ( ( linordered_field @ A )
% 2.25/1.16 => ! [A2: A,B: A,C: A] :
% 2.25/1.16 ( ( ord_less_eq @ A @ A2 @ B )
% 2.25/1.16 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C )
% 2.25/1.16 => ( ord_less_eq @ A @ ( divide_divide @ A @ A2 @ C ) @ ( divide_divide @ A @ B @ C ) ) ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_148_zero__le__divide__iff,axiom,
% 2.25/1.16 ! [A: $tType] :
% 2.25/1.16 ( ( linordered_field @ A )
% 2.25/1.16 => ! [A2: A,B: A] :
% 2.25/1.16 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ A2 @ B ) )
% 2.25/1.16 = ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
% 2.25/1.16 & ( ord_less_eq @ A @ ( zero_zero @ A ) @ B ) )
% 2.25/1.16 | ( ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) )
% 2.25/1.16 & ( ord_less_eq @ A @ B @ ( zero_zero @ A ) ) ) ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_149_divide__nonneg__nonneg,axiom,
% 2.25/1.16 ! [A: $tType] :
% 2.25/1.16 ( ( linordered_field @ A )
% 2.25/1.16 => ! [X: A,Y2: A] :
% 2.25/1.16 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
% 2.25/1.16 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y2 )
% 2.25/1.16 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ X @ Y2 ) ) ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_150_divide__nonneg__nonpos,axiom,
% 2.25/1.16 ! [A: $tType] :
% 2.25/1.16 ( ( linordered_field @ A )
% 2.25/1.16 => ! [X: A,Y2: A] :
% 2.25/1.16 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
% 2.25/1.16 => ( ( ord_less_eq @ A @ Y2 @ ( zero_zero @ A ) )
% 2.25/1.16 => ( ord_less_eq @ A @ ( divide_divide @ A @ X @ Y2 ) @ ( zero_zero @ A ) ) ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_151_divide__nonpos__nonneg,axiom,
% 2.25/1.16 ! [A: $tType] :
% 2.25/1.16 ( ( linordered_field @ A )
% 2.25/1.16 => ! [X: A,Y2: A] :
% 2.25/1.16 ( ( ord_less_eq @ A @ X @ ( zero_zero @ A ) )
% 2.25/1.16 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y2 )
% 2.25/1.16 => ( ord_less_eq @ A @ ( divide_divide @ A @ X @ Y2 ) @ ( zero_zero @ A ) ) ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_152_divide__nonpos__nonpos,axiom,
% 2.25/1.16 ! [A: $tType] :
% 2.25/1.16 ( ( linordered_field @ A )
% 2.25/1.16 => ! [X: A,Y2: A] :
% 2.25/1.16 ( ( ord_less_eq @ A @ X @ ( zero_zero @ A ) )
% 2.25/1.16 => ( ( ord_less_eq @ A @ Y2 @ ( zero_zero @ A ) )
% 2.25/1.16 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ X @ Y2 ) ) ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_153_divide__right__mono__neg,axiom,
% 2.25/1.16 ! [A: $tType] :
% 2.25/1.16 ( ( linordered_field @ A )
% 2.25/1.16 => ! [A2: A,B: A,C: A] :
% 2.25/1.16 ( ( ord_less_eq @ A @ A2 @ B )
% 2.25/1.16 => ( ( ord_less_eq @ A @ C @ ( zero_zero @ A ) )
% 2.25/1.16 => ( ord_less_eq @ A @ ( divide_divide @ A @ B @ C ) @ ( divide_divide @ A @ A2 @ C ) ) ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_154_power__mono,axiom,
% 2.25/1.16 ! [A: $tType] :
% 2.25/1.16 ( ( linordered_semidom @ A )
% 2.25/1.16 => ! [A2: A,B: A,N: nat] :
% 2.25/1.16 ( ( ord_less_eq @ A @ A2 @ B )
% 2.25/1.16 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
% 2.25/1.16 => ( ord_less_eq @ A @ ( power_power @ A @ A2 @ N ) @ ( power_power @ A @ B @ N ) ) ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_155_zero__le__power,axiom,
% 2.25/1.16 ! [A: $tType] :
% 2.25/1.16 ( ( linordered_semidom @ A )
% 2.25/1.16 => ! [A2: A,N: nat] :
% 2.25/1.16 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
% 2.25/1.16 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A2 @ N ) ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_156_divide__neg__neg,axiom,
% 2.25/1.16 ! [A: $tType] :
% 2.25/1.16 ( ( linordered_field @ A )
% 2.25/1.16 => ! [X: A,Y2: A] :
% 2.25/1.16 ( ( ord_less @ A @ X @ ( zero_zero @ A ) )
% 2.25/1.16 => ( ( ord_less @ A @ Y2 @ ( zero_zero @ A ) )
% 2.25/1.16 => ( ord_less @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ X @ Y2 ) ) ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_157_divide__neg__pos,axiom,
% 2.25/1.16 ! [A: $tType] :
% 2.25/1.16 ( ( linordered_field @ A )
% 2.25/1.16 => ! [X: A,Y2: A] :
% 2.25/1.16 ( ( ord_less @ A @ X @ ( zero_zero @ A ) )
% 2.25/1.16 => ( ( ord_less @ A @ ( zero_zero @ A ) @ Y2 )
% 2.25/1.16 => ( ord_less @ A @ ( divide_divide @ A @ X @ Y2 ) @ ( zero_zero @ A ) ) ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_158_divide__pos__neg,axiom,
% 2.25/1.16 ! [A: $tType] :
% 2.25/1.16 ( ( linordered_field @ A )
% 2.25/1.16 => ! [X: A,Y2: A] :
% 2.25/1.16 ( ( ord_less @ A @ ( zero_zero @ A ) @ X )
% 2.25/1.16 => ( ( ord_less @ A @ Y2 @ ( zero_zero @ A ) )
% 2.25/1.16 => ( ord_less @ A @ ( divide_divide @ A @ X @ Y2 ) @ ( zero_zero @ A ) ) ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_159_divide__pos__pos,axiom,
% 2.25/1.16 ! [A: $tType] :
% 2.25/1.16 ( ( linordered_field @ A )
% 2.25/1.16 => ! [X: A,Y2: A] :
% 2.25/1.16 ( ( ord_less @ A @ ( zero_zero @ A ) @ X )
% 2.25/1.16 => ( ( ord_less @ A @ ( zero_zero @ A ) @ Y2 )
% 2.25/1.16 => ( ord_less @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ X @ Y2 ) ) ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_160_divide__less__0__iff,axiom,
% 2.25/1.16 ! [A: $tType] :
% 2.25/1.16 ( ( linordered_field @ A )
% 2.25/1.16 => ! [A2: A,B: A] :
% 2.25/1.16 ( ( ord_less @ A @ ( divide_divide @ A @ A2 @ B ) @ ( zero_zero @ A ) )
% 2.25/1.16 = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
% 2.25/1.16 & ( ord_less @ A @ B @ ( zero_zero @ A ) ) )
% 2.25/1.16 | ( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
% 2.25/1.16 & ( ord_less @ A @ ( zero_zero @ A ) @ B ) ) ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_161_divide__less__cancel,axiom,
% 2.25/1.16 ! [A: $tType] :
% 2.25/1.16 ( ( linordered_field @ A )
% 2.25/1.16 => ! [A2: A,C: A,B: A] :
% 2.25/1.16 ( ( ord_less @ A @ ( divide_divide @ A @ A2 @ C ) @ ( divide_divide @ A @ B @ C ) )
% 2.25/1.16 = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C )
% 2.25/1.16 => ( ord_less @ A @ A2 @ B ) )
% 2.25/1.16 & ( ( ord_less @ A @ C @ ( zero_zero @ A ) )
% 2.25/1.16 => ( ord_less @ A @ B @ A2 ) )
% 2.25/1.16 & ( C
% 2.25/1.16 != ( zero_zero @ A ) ) ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_162_zero__less__divide__iff,axiom,
% 2.25/1.16 ! [A: $tType] :
% 2.25/1.16 ( ( linordered_field @ A )
% 2.25/1.16 => ! [A2: A,B: A] :
% 2.25/1.16 ( ( ord_less @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ A2 @ B ) )
% 2.25/1.16 = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
% 2.25/1.16 & ( ord_less @ A @ ( zero_zero @ A ) @ B ) )
% 2.25/1.16 | ( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
% 2.25/1.16 & ( ord_less @ A @ B @ ( zero_zero @ A ) ) ) ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_163_divide__strict__right__mono,axiom,
% 2.25/1.16 ! [A: $tType] :
% 2.25/1.16 ( ( linordered_field @ A )
% 2.25/1.16 => ! [A2: A,B: A,C: A] :
% 2.25/1.16 ( ( ord_less @ A @ A2 @ B )
% 2.25/1.16 => ( ( ord_less @ A @ ( zero_zero @ A ) @ C )
% 2.25/1.16 => ( ord_less @ A @ ( divide_divide @ A @ A2 @ C ) @ ( divide_divide @ A @ B @ C ) ) ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_164_divide__strict__right__mono__neg,axiom,
% 2.25/1.16 ! [A: $tType] :
% 2.25/1.16 ( ( linordered_field @ A )
% 2.25/1.16 => ! [B: A,A2: A,C: A] :
% 2.25/1.16 ( ( ord_less @ A @ B @ A2 )
% 2.25/1.16 => ( ( ord_less @ A @ C @ ( zero_zero @ A ) )
% 2.25/1.16 => ( ord_less @ A @ ( divide_divide @ A @ A2 @ C ) @ ( divide_divide @ A @ B @ C ) ) ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_165_zero__less__power,axiom,
% 2.25/1.16 ! [A: $tType] :
% 2.25/1.16 ( ( linordered_semidom @ A )
% 2.25/1.16 => ! [A2: A,N: nat] :
% 2.25/1.16 ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
% 2.25/1.16 => ( ord_less @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A2 @ N ) ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_166_right__inverse__eq,axiom,
% 2.25/1.16 ! [A: $tType] :
% 2.25/1.16 ( ( division_ring @ A )
% 2.25/1.16 => ! [B: A,A2: A] :
% 2.25/1.16 ( ( B
% 2.25/1.16 != ( zero_zero @ A ) )
% 2.25/1.16 => ( ( ( divide_divide @ A @ A2 @ B )
% 2.25/1.16 = ( one_one @ A ) )
% 2.25/1.16 = ( A2 = B ) ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_167_power__0__left,axiom,
% 2.25/1.16 ! [A: $tType] :
% 2.25/1.16 ( ( semiring_1 @ A )
% 2.25/1.16 => ! [N: nat] :
% 2.25/1.16 ( ( ( N
% 2.25/1.16 = ( zero_zero @ nat ) )
% 2.25/1.16 => ( ( power_power @ A @ ( zero_zero @ A ) @ N )
% 2.25/1.16 = ( one_one @ A ) ) )
% 2.25/1.16 & ( ( N
% 2.25/1.16 != ( zero_zero @ nat ) )
% 2.25/1.16 => ( ( power_power @ A @ ( zero_zero @ A ) @ N )
% 2.25/1.16 = ( zero_zero @ A ) ) ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_168_one__le__power,axiom,
% 2.25/1.16 ! [A: $tType] :
% 2.25/1.16 ( ( linordered_semidom @ A )
% 2.25/1.16 => ! [A2: A,N: nat] :
% 2.25/1.16 ( ( ord_less_eq @ A @ ( one_one @ A ) @ A2 )
% 2.25/1.16 => ( ord_less_eq @ A @ ( one_one @ A ) @ ( power_power @ A @ A2 @ N ) ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_169_power__increasing,axiom,
% 2.25/1.16 ! [A: $tType] :
% 2.25/1.16 ( ( linordered_semidom @ A )
% 2.25/1.16 => ! [N: nat,N4: nat,A2: A] :
% 2.25/1.16 ( ( ord_less_eq @ nat @ N @ N4 )
% 2.25/1.16 => ( ( ord_less_eq @ A @ ( one_one @ A ) @ A2 )
% 2.25/1.16 => ( ord_less_eq @ A @ ( power_power @ A @ A2 @ N ) @ ( power_power @ A @ A2 @ N4 ) ) ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_170_power__less__imp__less__exp,axiom,
% 2.25/1.16 ! [A: $tType] :
% 2.25/1.16 ( ( linordered_semidom @ A )
% 2.25/1.16 => ! [A2: A,M: nat,N: nat] :
% 2.25/1.16 ( ( ord_less @ A @ ( one_one @ A ) @ A2 )
% 2.25/1.16 => ( ( ord_less @ A @ ( power_power @ A @ A2 @ M ) @ ( power_power @ A @ A2 @ N ) )
% 2.25/1.16 => ( ord_less @ nat @ M @ N ) ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_171_power__strict__increasing,axiom,
% 2.25/1.16 ! [A: $tType] :
% 2.25/1.16 ( ( linordered_semidom @ A )
% 2.25/1.16 => ! [N: nat,N4: nat,A2: A] :
% 2.25/1.16 ( ( ord_less @ nat @ N @ N4 )
% 2.25/1.16 => ( ( ord_less @ A @ ( one_one @ A ) @ A2 )
% 2.25/1.16 => ( ord_less @ A @ ( power_power @ A @ A2 @ N ) @ ( power_power @ A @ A2 @ N4 ) ) ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_172_power__one__over,axiom,
% 2.25/1.16 ! [A: $tType] :
% 2.25/1.16 ( ( division_ring @ A )
% 2.25/1.16 => ! [A2: A,N: nat] :
% 2.25/1.16 ( ( power_power @ A @ ( divide_divide @ A @ ( one_one @ A ) @ A2 ) @ N )
% 2.25/1.16 = ( divide_divide @ A @ ( one_one @ A ) @ ( power_power @ A @ A2 @ N ) ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_173_nonzero__abs__divide,axiom,
% 2.25/1.16 ! [A: $tType] :
% 2.25/1.16 ( ( linordered_field @ A )
% 2.25/1.16 => ! [B: A,A2: A] :
% 2.25/1.16 ( ( B
% 2.25/1.16 != ( zero_zero @ A ) )
% 2.25/1.16 => ( ( abs_abs @ A @ ( divide_divide @ A @ A2 @ B ) )
% 2.25/1.16 = ( divide_divide @ A @ ( abs_abs @ A @ A2 ) @ ( abs_abs @ A @ B ) ) ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_174_frac__le,axiom,
% 2.25/1.16 ! [A: $tType] :
% 2.25/1.16 ( ( linordered_field @ A )
% 2.25/1.16 => ! [Y2: A,X: A,W: A,Z: A] :
% 2.25/1.16 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y2 )
% 2.25/1.16 => ( ( ord_less_eq @ A @ X @ Y2 )
% 2.25/1.16 => ( ( ord_less @ A @ ( zero_zero @ A ) @ W )
% 2.25/1.16 => ( ( ord_less_eq @ A @ W @ Z )
% 2.25/1.16 => ( ord_less_eq @ A @ ( divide_divide @ A @ X @ Z ) @ ( divide_divide @ A @ Y2 @ W ) ) ) ) ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_175_frac__less,axiom,
% 2.25/1.16 ! [A: $tType] :
% 2.25/1.16 ( ( linordered_field @ A )
% 2.25/1.16 => ! [X: A,Y2: A,W: A,Z: A] :
% 2.25/1.16 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
% 2.25/1.16 => ( ( ord_less @ A @ X @ Y2 )
% 2.25/1.16 => ( ( ord_less @ A @ ( zero_zero @ A ) @ W )
% 2.25/1.16 => ( ( ord_less_eq @ A @ W @ Z )
% 2.25/1.16 => ( ord_less @ A @ ( divide_divide @ A @ X @ Z ) @ ( divide_divide @ A @ Y2 @ W ) ) ) ) ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_176_frac__less2,axiom,
% 2.25/1.16 ! [A: $tType] :
% 2.25/1.16 ( ( linordered_field @ A )
% 2.25/1.16 => ! [X: A,Y2: A,W: A,Z: A] :
% 2.25/1.16 ( ( ord_less @ A @ ( zero_zero @ A ) @ X )
% 2.25/1.16 => ( ( ord_less_eq @ A @ X @ Y2 )
% 2.25/1.16 => ( ( ord_less @ A @ ( zero_zero @ A ) @ W )
% 2.25/1.16 => ( ( ord_less @ A @ W @ Z )
% 2.25/1.16 => ( ord_less @ A @ ( divide_divide @ A @ X @ Z ) @ ( divide_divide @ A @ Y2 @ W ) ) ) ) ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_177_divide__le__cancel,axiom,
% 2.25/1.16 ! [A: $tType] :
% 2.25/1.16 ( ( linordered_field @ A )
% 2.25/1.16 => ! [A2: A,C: A,B: A] :
% 2.25/1.16 ( ( ord_less_eq @ A @ ( divide_divide @ A @ A2 @ C ) @ ( divide_divide @ A @ B @ C ) )
% 2.25/1.16 = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C )
% 2.25/1.16 => ( ord_less_eq @ A @ A2 @ B ) )
% 2.25/1.16 & ( ( ord_less @ A @ C @ ( zero_zero @ A ) )
% 2.25/1.16 => ( ord_less_eq @ A @ B @ A2 ) ) ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_178_divide__nonneg__neg,axiom,
% 2.25/1.16 ! [A: $tType] :
% 2.25/1.16 ( ( linordered_field @ A )
% 2.25/1.16 => ! [X: A,Y2: A] :
% 2.25/1.16 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
% 2.25/1.16 => ( ( ord_less @ A @ Y2 @ ( zero_zero @ A ) )
% 2.25/1.16 => ( ord_less_eq @ A @ ( divide_divide @ A @ X @ Y2 ) @ ( zero_zero @ A ) ) ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_179_divide__nonneg__pos,axiom,
% 2.25/1.16 ! [A: $tType] :
% 2.25/1.16 ( ( linordered_field @ A )
% 2.25/1.16 => ! [X: A,Y2: A] :
% 2.25/1.16 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
% 2.25/1.16 => ( ( ord_less @ A @ ( zero_zero @ A ) @ Y2 )
% 2.25/1.16 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ X @ Y2 ) ) ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_180_divide__nonpos__neg,axiom,
% 2.25/1.16 ! [A: $tType] :
% 2.25/1.16 ( ( linordered_field @ A )
% 2.25/1.16 => ! [X: A,Y2: A] :
% 2.25/1.16 ( ( ord_less_eq @ A @ X @ ( zero_zero @ A ) )
% 2.25/1.16 => ( ( ord_less @ A @ Y2 @ ( zero_zero @ A ) )
% 2.25/1.16 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ X @ Y2 ) ) ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_181_divide__nonpos__pos,axiom,
% 2.25/1.16 ! [A: $tType] :
% 2.25/1.16 ( ( linordered_field @ A )
% 2.25/1.16 => ! [X: A,Y2: A] :
% 2.25/1.16 ( ( ord_less_eq @ A @ X @ ( zero_zero @ A ) )
% 2.25/1.16 => ( ( ord_less @ A @ ( zero_zero @ A ) @ Y2 )
% 2.25/1.16 => ( ord_less_eq @ A @ ( divide_divide @ A @ X @ Y2 ) @ ( zero_zero @ A ) ) ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_182_power__less__imp__less__base,axiom,
% 2.25/1.16 ! [A: $tType] :
% 2.25/1.16 ( ( linordered_semidom @ A )
% 2.25/1.16 => ! [A2: A,N: nat,B: A] :
% 2.25/1.16 ( ( ord_less @ A @ ( power_power @ A @ A2 @ N ) @ ( power_power @ A @ B @ N ) )
% 2.25/1.16 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B )
% 2.25/1.16 => ( ord_less @ A @ A2 @ B ) ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_183_power__strict__mono,axiom,
% 2.25/1.16 ! [A: $tType] :
% 2.25/1.16 ( ( linordered_semidom @ A )
% 2.25/1.16 => ! [A2: A,B: A,N: nat] :
% 2.25/1.16 ( ( ord_less @ A @ A2 @ B )
% 2.25/1.16 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
% 2.25/1.16 => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 2.25/1.16 => ( ord_less @ A @ ( power_power @ A @ A2 @ N ) @ ( power_power @ A @ B @ N ) ) ) ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_184_power__le__one,axiom,
% 2.25/1.16 ! [A: $tType] :
% 2.25/1.16 ( ( linordered_semidom @ A )
% 2.25/1.16 => ! [A2: A,N: nat] :
% 2.25/1.16 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
% 2.25/1.16 => ( ( ord_less_eq @ A @ A2 @ ( one_one @ A ) )
% 2.25/1.16 => ( ord_less_eq @ A @ ( power_power @ A @ A2 @ N ) @ ( one_one @ A ) ) ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_185_power__decreasing,axiom,
% 2.25/1.16 ! [A: $tType] :
% 2.25/1.16 ( ( linordered_semidom @ A )
% 2.25/1.16 => ! [N: nat,N4: nat,A2: A] :
% 2.25/1.16 ( ( ord_less_eq @ nat @ N @ N4 )
% 2.25/1.16 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
% 2.25/1.16 => ( ( ord_less_eq @ A @ A2 @ ( one_one @ A ) )
% 2.25/1.16 => ( ord_less_eq @ A @ ( power_power @ A @ A2 @ N4 ) @ ( power_power @ A @ A2 @ N ) ) ) ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_186_divide__less__eq__1,axiom,
% 2.25/1.16 ! [A: $tType] :
% 2.25/1.16 ( ( linordered_field @ A )
% 2.25/1.16 => ! [B: A,A2: A] :
% 2.25/1.16 ( ( ord_less @ A @ ( divide_divide @ A @ B @ A2 ) @ ( one_one @ A ) )
% 2.25/1.16 = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
% 2.25/1.16 & ( ord_less @ A @ B @ A2 ) )
% 2.25/1.16 | ( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
% 2.25/1.16 & ( ord_less @ A @ A2 @ B ) )
% 2.25/1.16 | ( A2
% 2.25/1.16 = ( zero_zero @ A ) ) ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_187_less__divide__eq__1,axiom,
% 2.25/1.16 ! [A: $tType] :
% 2.25/1.16 ( ( linordered_field @ A )
% 2.25/1.16 => ! [B: A,A2: A] :
% 2.25/1.16 ( ( ord_less @ A @ ( one_one @ A ) @ ( divide_divide @ A @ B @ A2 ) )
% 2.25/1.16 = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
% 2.25/1.16 & ( ord_less @ A @ A2 @ B ) )
% 2.25/1.16 | ( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
% 2.25/1.16 & ( ord_less @ A @ B @ A2 ) ) ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_188_power__strict__decreasing,axiom,
% 2.25/1.16 ! [A: $tType] :
% 2.25/1.16 ( ( linordered_semidom @ A )
% 2.25/1.16 => ! [N: nat,N4: nat,A2: A] :
% 2.25/1.16 ( ( ord_less @ nat @ N @ N4 )
% 2.25/1.16 => ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
% 2.25/1.16 => ( ( ord_less @ A @ A2 @ ( one_one @ A ) )
% 2.25/1.16 => ( ord_less @ A @ ( power_power @ A @ A2 @ N4 ) @ ( power_power @ A @ A2 @ N ) ) ) ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_189_power__le__imp__le__exp,axiom,
% 2.25/1.16 ! [A: $tType] :
% 2.25/1.16 ( ( linordered_semidom @ A )
% 2.25/1.16 => ! [A2: A,M: nat,N: nat] :
% 2.25/1.16 ( ( ord_less @ A @ ( one_one @ A ) @ A2 )
% 2.25/1.16 => ( ( ord_less_eq @ A @ ( power_power @ A @ A2 @ M ) @ ( power_power @ A @ A2 @ N ) )
% 2.25/1.16 => ( ord_less_eq @ nat @ M @ N ) ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_190_zero__le__power__abs,axiom,
% 2.25/1.16 ! [A: $tType] :
% 2.25/1.16 ( ( linordered_idom @ A )
% 2.25/1.16 => ! [A2: A,N: nat] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( power_power @ A @ ( abs_abs @ A @ A2 ) @ N ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_191_abs__div__pos,axiom,
% 2.25/1.16 ! [A: $tType] :
% 2.25/1.16 ( ( linordered_field @ A )
% 2.25/1.16 => ! [Y2: A,X: A] :
% 2.25/1.16 ( ( ord_less @ A @ ( zero_zero @ A ) @ Y2 )
% 2.25/1.16 => ( ( divide_divide @ A @ ( abs_abs @ A @ X ) @ Y2 )
% 2.25/1.16 = ( abs_abs @ A @ ( divide_divide @ A @ X @ Y2 ) ) ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_192_divide__le__eq__1,axiom,
% 2.25/1.16 ! [A: $tType] :
% 2.25/1.16 ( ( linordered_field @ A )
% 2.25/1.16 => ! [B: A,A2: A] :
% 2.25/1.16 ( ( ord_less_eq @ A @ ( divide_divide @ A @ B @ A2 ) @ ( one_one @ A ) )
% 2.25/1.16 = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
% 2.25/1.16 & ( ord_less_eq @ A @ B @ A2 ) )
% 2.25/1.16 | ( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
% 2.25/1.16 & ( ord_less_eq @ A @ A2 @ B ) )
% 2.25/1.16 | ( A2
% 2.25/1.16 = ( zero_zero @ A ) ) ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_193_le__divide__eq__1,axiom,
% 2.25/1.16 ! [A: $tType] :
% 2.25/1.16 ( ( linordered_field @ A )
% 2.25/1.16 => ! [B: A,A2: A] :
% 2.25/1.16 ( ( ord_less_eq @ A @ ( one_one @ A ) @ ( divide_divide @ A @ B @ A2 ) )
% 2.25/1.16 = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
% 2.25/1.16 & ( ord_less_eq @ A @ A2 @ B ) )
% 2.25/1.16 | ( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
% 2.25/1.16 & ( ord_less_eq @ A @ B @ A2 ) ) ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_194_A__less_I2_J,axiom,
% 2.25/1.16 ord_less @ real @ a @ ( divide_divide @ real @ ( one_one @ real ) @ m ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_195_ab,axiom,
% 2.25/1.16 ord_less_eq @ real @ ( abs_abs @ real @ ( minus_minus @ real @ x @ ( divide_divide @ real @ ( ring_1_of_int @ real @ a2 ) @ ( ring_1_of_int @ real @ b ) ) ) ) @ ( divide_divide @ real @ a @ ( power_power @ real @ ( ring_1_of_int @ real @ b ) @ n ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_196_ab_H,axiom,
% 2.25/1.16 ord_less_eq @ real @ ( abs_abs @ real @ ( minus_minus @ real @ x @ ( divide_divide @ real @ ( ring_1_of_int @ real @ a2 ) @ ( ring_1_of_int @ real @ b ) ) ) ) @ a ).
% 2.25/1.16
% 2.25/1.16 thf(fact_197_div__pos__pos__trivial,axiom,
% 2.25/1.16 ! [K: int,L: int] :
% 2.25/1.16 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
% 2.25/1.16 => ( ( ord_less @ int @ K @ L )
% 2.25/1.16 => ( ( divide_divide @ int @ K @ L )
% 2.25/1.16 = ( zero_zero @ int ) ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_198_div__neg__neg__trivial,axiom,
% 2.25/1.16 ! [K: int,L: int] :
% 2.25/1.16 ( ( ord_less_eq @ int @ K @ ( zero_zero @ int ) )
% 2.25/1.16 => ( ( ord_less @ int @ L @ K )
% 2.25/1.16 => ( ( divide_divide @ int @ K @ L )
% 2.25/1.16 = ( zero_zero @ int ) ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_199_zero__less__abs__iff,axiom,
% 2.25/1.16 ! [A: $tType] :
% 2.25/1.16 ( ( ordere142940540dd_abs @ A )
% 2.25/1.16 => ! [A2: A] :
% 2.25/1.16 ( ( ord_less @ A @ ( zero_zero @ A ) @ ( abs_abs @ A @ A2 ) )
% 2.25/1.16 = ( A2
% 2.25/1.16 != ( zero_zero @ A ) ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_200_M__pos,axiom,
% 2.25/1.16 ord_less @ real @ ( zero_zero @ real ) @ m ).
% 2.25/1.16
% 2.25/1.16 thf(fact_201_abs__idempotent,axiom,
% 2.25/1.16 ! [A: $tType] :
% 2.25/1.16 ( ( ordere142940540dd_abs @ A )
% 2.25/1.16 => ! [A2: A] :
% 2.25/1.16 ( ( abs_abs @ A @ ( abs_abs @ A @ A2 ) )
% 2.25/1.16 = ( abs_abs @ A @ A2 ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_202_le__zero__eq,axiom,
% 2.25/1.16 ! [A: $tType] :
% 2.25/1.16 ( ( canoni770627133id_add @ A )
% 2.25/1.16 => ! [N: A] :
% 2.25/1.16 ( ( ord_less_eq @ A @ N @ ( zero_zero @ A ) )
% 2.25/1.16 = ( N
% 2.25/1.16 = ( zero_zero @ A ) ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_203_not__gr__zero,axiom,
% 2.25/1.16 ! [A: $tType] :
% 2.25/1.16 ( ( canoni770627133id_add @ A )
% 2.25/1.16 => ! [N: A] :
% 2.25/1.16 ( ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ N ) )
% 2.25/1.16 = ( N
% 2.25/1.16 = ( zero_zero @ A ) ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_204_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
% 2.25/1.16 ! [A: $tType] :
% 2.25/1.16 ( ( cancel1352612707id_add @ A )
% 2.25/1.16 => ! [A2: A] :
% 2.25/1.16 ( ( minus_minus @ A @ A2 @ A2 )
% 2.25/1.16 = ( zero_zero @ A ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_205_diff__zero,axiom,
% 2.25/1.16 ! [A: $tType] :
% 2.25/1.16 ( ( cancel1352612707id_add @ A )
% 2.25/1.16 => ! [A2: A] :
% 2.25/1.16 ( ( minus_minus @ A @ A2 @ ( zero_zero @ A ) )
% 2.25/1.16 = A2 ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_206_zero__diff,axiom,
% 2.25/1.16 ! [A: $tType] :
% 2.25/1.16 ( ( comm_monoid_diff @ A )
% 2.25/1.16 => ! [A2: A] :
% 2.25/1.16 ( ( minus_minus @ A @ ( zero_zero @ A ) @ A2 )
% 2.25/1.16 = ( zero_zero @ A ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_207_diff__0__right,axiom,
% 2.25/1.16 ! [A: $tType] :
% 2.25/1.16 ( ( group_add @ A )
% 2.25/1.16 => ! [A2: A] :
% 2.25/1.16 ( ( minus_minus @ A @ A2 @ ( zero_zero @ A ) )
% 2.25/1.16 = A2 ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_208_diff__self,axiom,
% 2.25/1.16 ! [A: $tType] :
% 2.25/1.16 ( ( group_add @ A )
% 2.25/1.16 => ! [A2: A] :
% 2.25/1.16 ( ( minus_minus @ A @ A2 @ A2 )
% 2.25/1.16 = ( zero_zero @ A ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_209_abs__zero,axiom,
% 2.25/1.16 ! [A: $tType] :
% 2.25/1.16 ( ( ordere142940540dd_abs @ A )
% 2.25/1.16 => ( ( abs_abs @ A @ ( zero_zero @ A ) )
% 2.25/1.16 = ( zero_zero @ A ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_210_abs__eq__0,axiom,
% 2.25/1.16 ! [A: $tType] :
% 2.25/1.16 ( ( ordere142940540dd_abs @ A )
% 2.25/1.16 => ! [A2: A] :
% 2.25/1.16 ( ( ( abs_abs @ A @ A2 )
% 2.25/1.16 = ( zero_zero @ A ) )
% 2.25/1.16 = ( A2
% 2.25/1.16 = ( zero_zero @ A ) ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_211_abs__0__eq,axiom,
% 2.25/1.16 ! [A: $tType] :
% 2.25/1.16 ( ( ordere142940540dd_abs @ A )
% 2.25/1.16 => ! [A2: A] :
% 2.25/1.16 ( ( ( zero_zero @ A )
% 2.25/1.16 = ( abs_abs @ A @ A2 ) )
% 2.25/1.16 = ( A2
% 2.25/1.16 = ( zero_zero @ A ) ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_212_div__less,axiom,
% 2.25/1.16 ! [M: nat,N: nat] :
% 2.25/1.16 ( ( ord_less @ nat @ M @ N )
% 2.25/1.16 => ( ( divide_divide @ nat @ M @ N )
% 2.25/1.16 = ( zero_zero @ nat ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_213_of__int__diff,axiom,
% 2.25/1.16 ! [A: $tType] :
% 2.25/1.16 ( ( ring_1 @ A )
% 2.25/1.16 => ! [W: int,Z: int] :
% 2.25/1.16 ( ( ring_1_of_int @ A @ ( minus_minus @ int @ W @ Z ) )
% 2.25/1.16 = ( minus_minus @ A @ ( ring_1_of_int @ A @ W ) @ ( ring_1_of_int @ A @ Z ) ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_214_coeff__diff,axiom,
% 2.25/1.16 ! [A: $tType] :
% 2.25/1.16 ( ( cancel1352612707id_add @ A )
% 2.25/1.16 => ! [P: poly @ A,Q: poly @ A,N: nat] :
% 2.25/1.16 ( ( coeff @ A @ ( minus_minus @ ( poly @ A ) @ P @ Q ) @ N )
% 2.25/1.16 = ( minus_minus @ A @ ( coeff @ A @ P @ N ) @ ( coeff @ A @ Q @ N ) ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_215_poly__diff,axiom,
% 2.25/1.16 ! [A: $tType] :
% 2.25/1.16 ( ( comm_ring @ A )
% 2.25/1.16 => ! [P: poly @ A,Q: poly @ A,X: A] :
% 2.25/1.16 ( ( poly2 @ A @ ( minus_minus @ ( poly @ A ) @ P @ Q ) @ X )
% 2.25/1.16 = ( minus_minus @ A @ ( poly2 @ A @ P @ X ) @ ( poly2 @ A @ Q @ X ) ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_216_A__less_I3_J,axiom,
% 2.25/1.16 ! [X5: real] :
% 2.25/1.16 ( ( X5 != x )
% 2.25/1.16 => ( ( ( poly2 @ real @ p @ X5 )
% 2.25/1.16 = ( zero_zero @ real ) )
% 2.25/1.16 => ( ord_less @ real @ a @ ( abs_abs @ real @ ( minus_minus @ real @ X5 @ x ) ) ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_217_diff__ge__0__iff__ge,axiom,
% 2.25/1.16 ! [A: $tType] :
% 2.25/1.16 ( ( ordered_ab_group_add @ A )
% 2.25/1.16 => ! [A2: A,B: A] :
% 2.25/1.16 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( minus_minus @ A @ A2 @ B ) )
% 2.25/1.16 = ( ord_less_eq @ A @ B @ A2 ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_218_diff__gt__0__iff__gt,axiom,
% 2.25/1.16 ! [A: $tType] :
% 2.25/1.16 ( ( ordered_ab_group_add @ A )
% 2.25/1.16 => ! [A2: A,B: A] :
% 2.25/1.16 ( ( ord_less @ A @ ( zero_zero @ A ) @ ( minus_minus @ A @ A2 @ B ) )
% 2.25/1.16 = ( ord_less @ A @ B @ A2 ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_219_abs__le__zero__iff,axiom,
% 2.25/1.16 ! [A: $tType] :
% 2.25/1.16 ( ( ordere142940540dd_abs @ A )
% 2.25/1.16 => ! [A2: A] :
% 2.25/1.16 ( ( ord_less_eq @ A @ ( abs_abs @ A @ A2 ) @ ( zero_zero @ A ) )
% 2.25/1.16 = ( A2
% 2.25/1.16 = ( zero_zero @ A ) ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_220_abs__le__self__iff,axiom,
% 2.25/1.16 ! [A: $tType] :
% 2.25/1.16 ( ( ordere142940540dd_abs @ A )
% 2.25/1.16 => ! [A2: A] :
% 2.25/1.16 ( ( ord_less_eq @ A @ ( abs_abs @ A @ A2 ) @ A2 )
% 2.25/1.16 = ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_221_abs__of__nonneg,axiom,
% 2.25/1.16 ! [A: $tType] :
% 2.25/1.16 ( ( ordere142940540dd_abs @ A )
% 2.25/1.16 => ! [A2: A] :
% 2.25/1.16 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
% 2.25/1.16 => ( ( abs_abs @ A @ A2 )
% 2.25/1.16 = A2 ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_222_that,axiom,
% 2.25/1.16 ! [C: real,N: nat] :
% 2.25/1.16 ( ( ord_less @ real @ ( zero_zero @ real ) @ C )
% 2.25/1.16 => ( ! [P2: int,Q4: int] :
% 2.25/1.16 ( ( ord_less @ int @ ( zero_zero @ int ) @ Q4 )
% 2.25/1.16 => ( ord_less @ real @ ( divide_divide @ real @ C @ ( power_power @ real @ ( ring_1_of_int @ real @ Q4 ) @ N ) ) @ ( abs_abs @ real @ ( minus_minus @ real @ x @ ( divide_divide @ real @ ( ring_1_of_int @ real @ P2 ) @ ( ring_1_of_int @ real @ Q4 ) ) ) ) ) )
% 2.25/1.16 => thesis ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_223_minus__poly_Orep__eq,axiom,
% 2.25/1.16 ! [A: $tType] :
% 2.25/1.16 ( ( cancel1352612707id_add @ A )
% 2.25/1.16 => ! [X: poly @ A,Xa: poly @ A] :
% 2.25/1.16 ( ( coeff @ A @ ( minus_minus @ ( poly @ A ) @ X @ Xa ) )
% 2.25/1.16 = ( ^ [N3: nat] : ( minus_minus @ A @ ( coeff @ A @ X @ N3 ) @ ( coeff @ A @ Xa @ N3 ) ) ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_224_eq__iff__diff__eq__0,axiom,
% 2.25/1.16 ! [A: $tType] :
% 2.25/1.16 ( ( group_add @ A )
% 2.25/1.16 => ( ( ^ [Y: A,Z2: A] : ( Y = Z2 ) )
% 2.25/1.16 = ( ^ [A4: A,B3: A] :
% 2.25/1.16 ( ( minus_minus @ A @ A4 @ B3 )
% 2.25/1.16 = ( zero_zero @ A ) ) ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_225_diff__mono,axiom,
% 2.25/1.16 ! [A: $tType] :
% 2.25/1.16 ( ( ordered_ab_group_add @ A )
% 2.25/1.16 => ! [A2: A,B: A,D: A,C: A] :
% 2.25/1.16 ( ( ord_less_eq @ A @ A2 @ B )
% 2.25/1.16 => ( ( ord_less_eq @ A @ D @ C )
% 2.25/1.16 => ( ord_less_eq @ A @ ( minus_minus @ A @ A2 @ C ) @ ( minus_minus @ A @ B @ D ) ) ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_226_diff__left__mono,axiom,
% 2.25/1.16 ! [A: $tType] :
% 2.25/1.16 ( ( ordered_ab_group_add @ A )
% 2.25/1.16 => ! [B: A,A2: A,C: A] :
% 2.25/1.16 ( ( ord_less_eq @ A @ B @ A2 )
% 2.25/1.16 => ( ord_less_eq @ A @ ( minus_minus @ A @ C @ A2 ) @ ( minus_minus @ A @ C @ B ) ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_227_diff__right__mono,axiom,
% 2.25/1.16 ! [A: $tType] :
% 2.25/1.16 ( ( ordered_ab_group_add @ A )
% 2.25/1.16 => ! [A2: A,B: A,C: A] :
% 2.25/1.16 ( ( ord_less_eq @ A @ A2 @ B )
% 2.25/1.16 => ( ord_less_eq @ A @ ( minus_minus @ A @ A2 @ C ) @ ( minus_minus @ A @ B @ C ) ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_228_diff__eq__diff__less__eq,axiom,
% 2.25/1.16 ! [A: $tType] :
% 2.25/1.16 ( ( ordered_ab_group_add @ A )
% 2.25/1.16 => ! [A2: A,B: A,C: A,D: A] :
% 2.25/1.16 ( ( ( minus_minus @ A @ A2 @ B )
% 2.25/1.16 = ( minus_minus @ A @ C @ D ) )
% 2.25/1.16 => ( ( ord_less_eq @ A @ A2 @ B )
% 2.25/1.16 = ( ord_less_eq @ A @ C @ D ) ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_229_diff__strict__right__mono,axiom,
% 2.25/1.16 ! [A: $tType] :
% 2.25/1.16 ( ( ordered_ab_group_add @ A )
% 2.25/1.16 => ! [A2: A,B: A,C: A] :
% 2.25/1.16 ( ( ord_less @ A @ A2 @ B )
% 2.25/1.16 => ( ord_less @ A @ ( minus_minus @ A @ A2 @ C ) @ ( minus_minus @ A @ B @ C ) ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_230_diff__strict__left__mono,axiom,
% 2.25/1.16 ! [A: $tType] :
% 2.25/1.16 ( ( ordered_ab_group_add @ A )
% 2.25/1.16 => ! [B: A,A2: A,C: A] :
% 2.25/1.16 ( ( ord_less @ A @ B @ A2 )
% 2.25/1.16 => ( ord_less @ A @ ( minus_minus @ A @ C @ A2 ) @ ( minus_minus @ A @ C @ B ) ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_231_diff__eq__diff__less,axiom,
% 2.25/1.16 ! [A: $tType] :
% 2.25/1.16 ( ( ordered_ab_group_add @ A )
% 2.25/1.16 => ! [A2: A,B: A,C: A,D: A] :
% 2.25/1.16 ( ( ( minus_minus @ A @ A2 @ B )
% 2.25/1.16 = ( minus_minus @ A @ C @ D ) )
% 2.25/1.16 => ( ( ord_less @ A @ A2 @ B )
% 2.25/1.16 = ( ord_less @ A @ C @ D ) ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_232_diff__strict__mono,axiom,
% 2.25/1.16 ! [A: $tType] :
% 2.25/1.16 ( ( ordered_ab_group_add @ A )
% 2.25/1.16 => ! [A2: A,B: A,D: A,C: A] :
% 2.25/1.16 ( ( ord_less @ A @ A2 @ B )
% 2.25/1.16 => ( ( ord_less @ A @ D @ C )
% 2.25/1.16 => ( ord_less @ A @ ( minus_minus @ A @ A2 @ C ) @ ( minus_minus @ A @ B @ D ) ) ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_233_abs__minus__commute,axiom,
% 2.25/1.16 ! [A: $tType] :
% 2.25/1.16 ( ( ordere142940540dd_abs @ A )
% 2.25/1.16 => ! [A2: A,B: A] :
% 2.25/1.16 ( ( abs_abs @ A @ ( minus_minus @ A @ A2 @ B ) )
% 2.25/1.16 = ( abs_abs @ A @ ( minus_minus @ A @ B @ A2 ) ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_234_diff__eq__diff__eq,axiom,
% 2.25/1.16 ! [A: $tType] :
% 2.25/1.16 ( ( group_add @ A )
% 2.25/1.16 => ! [A2: A,B: A,C: A,D: A] :
% 2.25/1.16 ( ( ( minus_minus @ A @ A2 @ B )
% 2.25/1.16 = ( minus_minus @ A @ C @ D ) )
% 2.25/1.16 => ( ( A2 = B )
% 2.25/1.16 = ( C = D ) ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_235_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
% 2.25/1.16 ! [A: $tType] :
% 2.25/1.16 ( ( cancel146912293up_add @ A )
% 2.25/1.16 => ! [A2: A,C: A,B: A] :
% 2.25/1.16 ( ( minus_minus @ A @ ( minus_minus @ A @ A2 @ C ) @ B )
% 2.25/1.16 = ( minus_minus @ A @ ( minus_minus @ A @ A2 @ B ) @ C ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_236_diff__divide__distrib,axiom,
% 2.25/1.16 ! [A: $tType] :
% 2.25/1.16 ( ( division_ring @ A )
% 2.25/1.16 => ! [A2: A,B: A,C: A] :
% 2.25/1.16 ( ( divide_divide @ A @ ( minus_minus @ A @ A2 @ B ) @ C )
% 2.25/1.16 = ( minus_minus @ A @ ( divide_divide @ A @ A2 @ C ) @ ( divide_divide @ A @ B @ C ) ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_237_le__iff__diff__le__0,axiom,
% 2.25/1.16 ! [A: $tType] :
% 2.25/1.16 ( ( ordered_ab_group_add @ A )
% 2.25/1.16 => ( ( ord_less_eq @ A )
% 2.25/1.16 = ( ^ [A4: A,B3: A] : ( ord_less_eq @ A @ ( minus_minus @ A @ A4 @ B3 ) @ ( zero_zero @ A ) ) ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_238_less__iff__diff__less__0,axiom,
% 2.25/1.16 ! [A: $tType] :
% 2.25/1.16 ( ( ordered_ab_group_add @ A )
% 2.25/1.16 => ( ( ord_less @ A )
% 2.25/1.16 = ( ^ [A4: A,B3: A] : ( ord_less @ A @ ( minus_minus @ A @ A4 @ B3 ) @ ( zero_zero @ A ) ) ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_239_Ints__diff,axiom,
% 2.25/1.16 ! [A: $tType] :
% 2.25/1.16 ( ( ring_1 @ A )
% 2.25/1.16 => ! [A2: A,B: A] :
% 2.25/1.16 ( ( member @ A @ A2 @ ( ring_1_Ints @ A ) )
% 2.25/1.16 => ( ( member @ A @ B @ ( ring_1_Ints @ A ) )
% 2.25/1.16 => ( member @ A @ ( minus_minus @ A @ A2 @ B ) @ ( ring_1_Ints @ A ) ) ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_240_abs__triangle__ineq2__sym,axiom,
% 2.25/1.16 ! [A: $tType] :
% 2.25/1.16 ( ( ordere142940540dd_abs @ A )
% 2.25/1.16 => ! [A2: A,B: A] : ( ord_less_eq @ A @ ( minus_minus @ A @ ( abs_abs @ A @ A2 ) @ ( abs_abs @ A @ B ) ) @ ( abs_abs @ A @ ( minus_minus @ A @ B @ A2 ) ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_241_abs__triangle__ineq3,axiom,
% 2.25/1.16 ! [A: $tType] :
% 2.25/1.16 ( ( ordere142940540dd_abs @ A )
% 2.25/1.16 => ! [A2: A,B: A] : ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ ( abs_abs @ A @ A2 ) @ ( abs_abs @ A @ B ) ) ) @ ( abs_abs @ A @ ( minus_minus @ A @ A2 @ B ) ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_242_abs__triangle__ineq2,axiom,
% 2.25/1.16 ! [A: $tType] :
% 2.25/1.16 ( ( ordere142940540dd_abs @ A )
% 2.25/1.16 => ! [A2: A,B: A] : ( ord_less_eq @ A @ ( minus_minus @ A @ ( abs_abs @ A @ A2 ) @ ( abs_abs @ A @ B ) ) @ ( abs_abs @ A @ ( minus_minus @ A @ A2 @ B ) ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_243_zero__reorient,axiom,
% 2.25/1.16 ! [A: $tType] :
% 2.25/1.16 ( ( zero @ A )
% 2.25/1.16 => ! [X: A] :
% 2.25/1.16 ( ( ( zero_zero @ A )
% 2.25/1.16 = X )
% 2.25/1.16 = ( X
% 2.25/1.16 = ( zero_zero @ A ) ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_244_one__reorient,axiom,
% 2.25/1.16 ! [A: $tType] :
% 2.25/1.16 ( ( one @ A )
% 2.25/1.16 => ! [X: A] :
% 2.25/1.16 ( ( ( one_one @ A )
% 2.25/1.16 = X )
% 2.25/1.16 = ( X
% 2.25/1.16 = ( one_one @ A ) ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_245_div__eq__dividend__iff,axiom,
% 2.25/1.16 ! [M: nat,N: nat] :
% 2.25/1.16 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
% 2.25/1.16 => ( ( ( divide_divide @ nat @ M @ N )
% 2.25/1.16 = M )
% 2.25/1.16 = ( N
% 2.25/1.16 = ( one_one @ nat ) ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_246_div__less__dividend,axiom,
% 2.25/1.16 ! [N: nat,M: nat] :
% 2.25/1.16 ( ( ord_less @ nat @ ( one_one @ nat ) @ N )
% 2.25/1.16 => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
% 2.25/1.16 => ( ord_less @ nat @ ( divide_divide @ nat @ M @ N ) @ M ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_247_Euclidean__Division_Odiv__eq__0__iff,axiom,
% 2.25/1.16 ! [M: nat,N: nat] :
% 2.25/1.16 ( ( ( divide_divide @ nat @ M @ N )
% 2.25/1.16 = ( zero_zero @ nat ) )
% 2.25/1.16 = ( ( ord_less @ nat @ M @ N )
% 2.25/1.16 | ( N
% 2.25/1.16 = ( zero_zero @ nat ) ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_248_div__le__dividend,axiom,
% 2.25/1.16 ! [M: nat,N: nat] : ( ord_less_eq @ nat @ ( divide_divide @ nat @ M @ N ) @ M ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_249_div__le__mono,axiom,
% 2.25/1.16 ! [M: nat,N: nat,K: nat] :
% 2.25/1.16 ( ( ord_less_eq @ nat @ M @ N )
% 2.25/1.16 => ( ord_less_eq @ nat @ ( divide_divide @ nat @ M @ K ) @ ( divide_divide @ nat @ N @ K ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_250_Ints__eq__abs__less1,axiom,
% 2.25/1.16 ! [A: $tType] :
% 2.25/1.16 ( ( linordered_idom @ A )
% 2.25/1.16 => ! [X: A,Y2: A] :
% 2.25/1.16 ( ( member @ A @ X @ ( ring_1_Ints @ A ) )
% 2.25/1.16 => ( ( member @ A @ Y2 @ ( ring_1_Ints @ A ) )
% 2.25/1.16 => ( ( X = Y2 )
% 2.25/1.16 = ( ord_less @ A @ ( abs_abs @ A @ ( minus_minus @ A @ X @ Y2 ) ) @ ( one_one @ A ) ) ) ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_251_zero__le,axiom,
% 2.25/1.16 ! [A: $tType] :
% 2.25/1.16 ( ( canoni770627133id_add @ A )
% 2.25/1.16 => ! [X: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ X ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_252_gr__zeroI,axiom,
% 2.25/1.16 ! [A: $tType] :
% 2.25/1.16 ( ( canoni770627133id_add @ A )
% 2.25/1.16 => ! [N: A] :
% 2.25/1.16 ( ( N
% 2.25/1.16 != ( zero_zero @ A ) )
% 2.25/1.16 => ( ord_less @ A @ ( zero_zero @ A ) @ N ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(fact_253_not__less__zero,axiom,
% 2.25/1.16 ! [A: $tType] :
% 2.25/1.16 ( ( canoni770627133id_add @ A )
% 2.25/1.16 => ! [N: A] :
% 2.25/1.16 ~ ( ord_less @ A @ N @ ( zero_zero @ A ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(tcon_Polynomial_Opoly___Rings_Oidom,axiom,
% 2.25/1.16 ! [A5: $tType] :
% 2.25/1.16 ( ( idom @ A5 )
% 2.25/1.16 => ( idom @ ( poly @ A5 ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(tcon_Polynomial_Opoly___Groups_Oab__group__add,axiom,
% 2.25/1.16 ! [A5: $tType] :
% 2.25/1.16 ( ( ab_group_add @ A5 )
% 2.25/1.16 => ( ab_group_add @ ( poly @ A5 ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(tcon_Real_Oreal___Rings_Oidom_1,axiom,
% 2.25/1.16 idom @ real ).
% 2.25/1.16
% 2.25/1.16 thf(tcon_Real_Oreal___Groups_Oab__group__add_2,axiom,
% 2.25/1.16 ab_group_add @ real ).
% 2.25/1.16
% 2.25/1.16 thf(tcon_Int_Oint___Rings_Oidom_3,axiom,
% 2.25/1.16 idom @ int ).
% 2.25/1.16
% 2.25/1.16 thf(tcon_Int_Oint___Groups_Oab__group__add_4,axiom,
% 2.25/1.16 ab_group_add @ int ).
% 2.25/1.16
% 2.25/1.16 thf(tcon_Int_Oint___Rings_Osemiring__1__no__zero__divisors,axiom,
% 2.25/1.16 semiri134348788visors @ int ).
% 2.25/1.16
% 2.25/1.16 thf(tcon_Int_Oint___Rings_Osemiring__no__zero__divisors,axiom,
% 2.25/1.16 semiri1193490041visors @ int ).
% 2.25/1.16
% 2.25/1.16 thf(tcon_Int_Oint___Groups_Oordered__ab__group__add__abs,axiom,
% 2.25/1.16 ordere142940540dd_abs @ int ).
% 2.25/1.16
% 2.25/1.16 thf(tcon_Int_Oint___Groups_Ocancel__ab__semigroup__add,axiom,
% 2.25/1.16 cancel146912293up_add @ int ).
% 2.25/1.16
% 2.25/1.16 thf(tcon_Int_Oint___Groups_Ocancel__comm__monoid__add,axiom,
% 2.25/1.16 cancel1352612707id_add @ int ).
% 2.25/1.16
% 2.25/1.16 thf(tcon_Int_Oint___Rings_Oring__no__zero__divisors,axiom,
% 2.25/1.16 ring_n68954251visors @ int ).
% 2.25/1.16
% 2.25/1.16 thf(tcon_Int_Oint___Groups_Oordered__ab__group__add,axiom,
% 2.25/1.16 ordered_ab_group_add @ int ).
% 2.25/1.16
% 2.25/1.16 thf(tcon_Int_Oint___Rings_Olinordered__semidom,axiom,
% 2.25/1.16 linordered_semidom @ int ).
% 2.25/1.16
% 2.25/1.16 thf(tcon_Int_Oint___Rings_Olinordered__idom,axiom,
% 2.25/1.16 linordered_idom @ int ).
% 2.25/1.16
% 2.25/1.16 thf(tcon_Int_Oint___Rings_Ocomm__semiring__1,axiom,
% 2.25/1.16 comm_semiring_1 @ int ).
% 2.25/1.16
% 2.25/1.16 thf(tcon_Int_Oint___Rings_Ocomm__semiring__0,axiom,
% 2.25/1.16 comm_semiring_0 @ int ).
% 2.25/1.16
% 2.25/1.16 thf(tcon_Int_Oint___Groups_Omonoid__mult,axiom,
% 2.25/1.16 monoid_mult @ int ).
% 2.25/1.16
% 2.25/1.16 thf(tcon_Int_Oint___Rings_Oidom__divide,axiom,
% 2.25/1.16 idom_divide @ int ).
% 2.25/1.16
% 2.25/1.16 thf(tcon_Int_Oint___Rings_Ocomm__ring__1,axiom,
% 2.25/1.16 comm_ring_1 @ int ).
% 2.25/1.16
% 2.25/1.16 thf(tcon_Int_Oint___Rings_Osemiring__1,axiom,
% 2.25/1.16 semiring_1 @ int ).
% 2.25/1.16
% 2.25/1.16 thf(tcon_Int_Oint___Groups_Ogroup__add,axiom,
% 2.25/1.16 group_add @ int ).
% 2.25/1.16
% 2.25/1.16 thf(tcon_Int_Oint___Rings_Ocomm__ring,axiom,
% 2.25/1.16 comm_ring @ int ).
% 2.25/1.16
% 2.25/1.16 thf(tcon_Int_Oint___Nat_Oring__char__0,axiom,
% 2.25/1.16 ring_char_0 @ int ).
% 2.25/1.16
% 2.25/1.16 thf(tcon_Int_Oint___Rings_Oring__1,axiom,
% 2.25/1.16 ring_1 @ int ).
% 2.25/1.16
% 2.25/1.16 thf(tcon_Int_Oint___Power_Opower,axiom,
% 2.25/1.16 power @ int ).
% 2.25/1.16
% 2.25/1.16 thf(tcon_Int_Oint___Groups_Ozero,axiom,
% 2.25/1.16 zero @ int ).
% 2.25/1.16
% 2.25/1.16 thf(tcon_Int_Oint___Groups_Oone,axiom,
% 2.25/1.16 one @ int ).
% 2.25/1.16
% 2.25/1.16 thf(tcon_Nat_Onat___Groups_Ocanonically__ordered__monoid__add,axiom,
% 2.25/1.16 canoni770627133id_add @ nat ).
% 2.25/1.16
% 2.25/1.16 thf(tcon_Nat_Onat___Rings_Osemiring__1__no__zero__divisors_5,axiom,
% 2.25/1.16 semiri134348788visors @ nat ).
% 2.25/1.16
% 2.25/1.16 thf(tcon_Nat_Onat___Rings_Osemiring__no__zero__divisors_6,axiom,
% 2.25/1.16 semiri1193490041visors @ nat ).
% 2.25/1.16
% 2.25/1.16 thf(tcon_Nat_Onat___Groups_Ocancel__ab__semigroup__add_7,axiom,
% 2.25/1.16 cancel146912293up_add @ nat ).
% 2.25/1.16
% 2.25/1.16 thf(tcon_Nat_Onat___Groups_Ocancel__comm__monoid__add_8,axiom,
% 2.25/1.16 cancel1352612707id_add @ nat ).
% 2.25/1.16
% 2.25/1.16 thf(tcon_Nat_Onat___Rings_Olinordered__semidom_9,axiom,
% 2.25/1.16 linordered_semidom @ nat ).
% 2.25/1.16
% 2.25/1.16 thf(tcon_Nat_Onat___Groups_Ocomm__monoid__diff,axiom,
% 2.25/1.16 comm_monoid_diff @ nat ).
% 2.25/1.16
% 2.25/1.16 thf(tcon_Nat_Onat___Rings_Ocomm__semiring__1_10,axiom,
% 2.25/1.16 comm_semiring_1 @ nat ).
% 2.25/1.16
% 2.25/1.16 thf(tcon_Nat_Onat___Rings_Ocomm__semiring__0_11,axiom,
% 2.25/1.16 comm_semiring_0 @ nat ).
% 2.25/1.16
% 2.25/1.16 thf(tcon_Nat_Onat___Groups_Omonoid__mult_12,axiom,
% 2.25/1.16 monoid_mult @ nat ).
% 2.25/1.16
% 2.25/1.16 thf(tcon_Nat_Onat___Rings_Osemiring__1_13,axiom,
% 2.25/1.16 semiring_1 @ nat ).
% 2.25/1.16
% 2.25/1.16 thf(tcon_Nat_Onat___Power_Opower_14,axiom,
% 2.25/1.16 power @ nat ).
% 2.25/1.16
% 2.25/1.16 thf(tcon_Nat_Onat___Groups_Ozero_15,axiom,
% 2.25/1.16 zero @ nat ).
% 2.25/1.16
% 2.25/1.16 thf(tcon_Nat_Onat___Groups_Oone_16,axiom,
% 2.25/1.16 one @ nat ).
% 2.25/1.16
% 2.25/1.16 thf(tcon_Real_Oreal___Rings_Osemiring__1__no__zero__divisors_17,axiom,
% 2.25/1.16 semiri134348788visors @ real ).
% 2.25/1.16
% 2.25/1.16 thf(tcon_Real_Oreal___Rings_Osemiring__no__zero__divisors_18,axiom,
% 2.25/1.16 semiri1193490041visors @ real ).
% 2.25/1.16
% 2.25/1.16 thf(tcon_Real_Oreal___Groups_Oordered__ab__group__add__abs_19,axiom,
% 2.25/1.16 ordere142940540dd_abs @ real ).
% 2.25/1.16
% 2.25/1.16 thf(tcon_Real_Oreal___Groups_Ocancel__ab__semigroup__add_20,axiom,
% 2.25/1.16 cancel146912293up_add @ real ).
% 2.25/1.16
% 2.25/1.16 thf(tcon_Real_Oreal___Groups_Ocancel__comm__monoid__add_21,axiom,
% 2.25/1.16 cancel1352612707id_add @ real ).
% 2.25/1.16
% 2.25/1.16 thf(tcon_Real_Oreal___Rings_Oring__no__zero__divisors_22,axiom,
% 2.25/1.16 ring_n68954251visors @ real ).
% 2.25/1.16
% 2.25/1.16 thf(tcon_Real_Oreal___Groups_Oordered__ab__group__add_23,axiom,
% 2.25/1.16 ordered_ab_group_add @ real ).
% 2.25/1.16
% 2.25/1.16 thf(tcon_Real_Oreal___Rings_Olinordered__semidom_24,axiom,
% 2.25/1.16 linordered_semidom @ real ).
% 2.25/1.16
% 2.25/1.16 thf(tcon_Real_Oreal___Fields_Olinordered__field,axiom,
% 2.25/1.16 linordered_field @ real ).
% 2.25/1.16
% 2.25/1.16 thf(tcon_Real_Oreal___Rings_Olinordered__idom_25,axiom,
% 2.25/1.16 linordered_idom @ real ).
% 2.25/1.16
% 2.25/1.16 thf(tcon_Real_Oreal___Rings_Ocomm__semiring__1_26,axiom,
% 2.25/1.16 comm_semiring_1 @ real ).
% 2.25/1.16
% 2.25/1.16 thf(tcon_Real_Oreal___Rings_Ocomm__semiring__0_27,axiom,
% 2.25/1.16 comm_semiring_0 @ real ).
% 2.25/1.16
% 2.25/1.16 thf(tcon_Real_Oreal___Fields_Ofield__abs__sgn,axiom,
% 2.25/1.16 field_abs_sgn @ real ).
% 2.25/1.16
% 2.25/1.16 thf(tcon_Real_Oreal___Fields_Odivision__ring,axiom,
% 2.25/1.16 division_ring @ real ).
% 2.25/1.16
% 2.25/1.16 thf(tcon_Real_Oreal___Fields_Ofield__char__0,axiom,
% 2.25/1.16 field_char_0 @ real ).
% 2.25/1.16
% 2.25/1.16 thf(tcon_Real_Oreal___Groups_Omonoid__mult_28,axiom,
% 2.25/1.16 monoid_mult @ real ).
% 2.25/1.16
% 2.25/1.16 thf(tcon_Real_Oreal___Rings_Oidom__divide_29,axiom,
% 2.25/1.16 idom_divide @ real ).
% 2.25/1.16
% 2.25/1.16 thf(tcon_Real_Oreal___Rings_Ocomm__ring__1_30,axiom,
% 2.25/1.16 comm_ring_1 @ real ).
% 2.25/1.16
% 2.25/1.16 thf(tcon_Real_Oreal___Rings_Osemiring__1_31,axiom,
% 2.25/1.16 semiring_1 @ real ).
% 2.25/1.16
% 2.25/1.16 thf(tcon_Real_Oreal___Groups_Ogroup__add_32,axiom,
% 2.25/1.16 group_add @ real ).
% 2.25/1.16
% 2.25/1.16 thf(tcon_Real_Oreal___Rings_Ocomm__ring_33,axiom,
% 2.25/1.16 comm_ring @ real ).
% 2.25/1.16
% 2.25/1.16 thf(tcon_Real_Oreal___Nat_Oring__char__0_34,axiom,
% 2.25/1.16 ring_char_0 @ real ).
% 2.25/1.16
% 2.25/1.16 thf(tcon_Real_Oreal___Rings_Oring__1_35,axiom,
% 2.25/1.16 ring_1 @ real ).
% 2.25/1.16
% 2.25/1.16 thf(tcon_Real_Oreal___Fields_Ofield,axiom,
% 2.25/1.16 field @ real ).
% 2.25/1.16
% 2.25/1.16 thf(tcon_Real_Oreal___Power_Opower_36,axiom,
% 2.25/1.16 power @ real ).
% 2.25/1.16
% 2.25/1.16 thf(tcon_Real_Oreal___Groups_Ozero_37,axiom,
% 2.25/1.16 zero @ real ).
% 2.25/1.16
% 2.25/1.16 thf(tcon_Real_Oreal___Groups_Oone_38,axiom,
% 2.25/1.16 one @ real ).
% 2.25/1.16
% 2.25/1.16 thf(tcon_Polynomial_Opoly___Rings_Osemiring__1__no__zero__divisors_39,axiom,
% 2.25/1.16 ! [A5: $tType] :
% 2.25/1.16 ( ( ( comm_semiring_1 @ A5 )
% 2.25/1.16 & ( semiri134348788visors @ A5 ) )
% 2.25/1.16 => ( semiri134348788visors @ ( poly @ A5 ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(tcon_Polynomial_Opoly___Rings_Osemiring__no__zero__divisors_40,axiom,
% 2.25/1.16 ! [A5: $tType] :
% 2.25/1.16 ( ( ( comm_semiring_0 @ A5 )
% 2.25/1.16 & ( semiri1193490041visors @ A5 ) )
% 2.25/1.16 => ( semiri1193490041visors @ ( poly @ A5 ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(tcon_Polynomial_Opoly___Groups_Oordered__ab__group__add__abs_41,axiom,
% 2.25/1.16 ! [A5: $tType] :
% 2.25/1.16 ( ( linordered_idom @ A5 )
% 2.25/1.16 => ( ordere142940540dd_abs @ ( poly @ A5 ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(tcon_Polynomial_Opoly___Groups_Ocancel__ab__semigroup__add_42,axiom,
% 2.25/1.16 ! [A5: $tType] :
% 2.25/1.16 ( ( cancel1352612707id_add @ A5 )
% 2.25/1.16 => ( cancel146912293up_add @ ( poly @ A5 ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(tcon_Polynomial_Opoly___Groups_Ocancel__comm__monoid__add_43,axiom,
% 2.25/1.16 ! [A5: $tType] :
% 2.25/1.16 ( ( cancel1352612707id_add @ A5 )
% 2.25/1.16 => ( cancel1352612707id_add @ ( poly @ A5 ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(tcon_Polynomial_Opoly___Rings_Oring__no__zero__divisors_44,axiom,
% 2.25/1.16 ! [A5: $tType] :
% 2.25/1.16 ( ( idom @ A5 )
% 2.25/1.16 => ( ring_n68954251visors @ ( poly @ A5 ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(tcon_Polynomial_Opoly___Groups_Oordered__ab__group__add_45,axiom,
% 2.25/1.16 ! [A5: $tType] :
% 2.25/1.16 ( ( linordered_idom @ A5 )
% 2.25/1.16 => ( ordered_ab_group_add @ ( poly @ A5 ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(tcon_Polynomial_Opoly___Rings_Olinordered__semidom_46,axiom,
% 2.25/1.16 ! [A5: $tType] :
% 2.25/1.16 ( ( linordered_idom @ A5 )
% 2.25/1.16 => ( linordered_semidom @ ( poly @ A5 ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(tcon_Polynomial_Opoly___Rings_Olinordered__idom_47,axiom,
% 2.25/1.16 ! [A5: $tType] :
% 2.25/1.16 ( ( linordered_idom @ A5 )
% 2.25/1.16 => ( linordered_idom @ ( poly @ A5 ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(tcon_Polynomial_Opoly___Rings_Ocomm__semiring__1_48,axiom,
% 2.25/1.16 ! [A5: $tType] :
% 2.25/1.16 ( ( comm_semiring_1 @ A5 )
% 2.25/1.16 => ( comm_semiring_1 @ ( poly @ A5 ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(tcon_Polynomial_Opoly___Rings_Ocomm__semiring__0_49,axiom,
% 2.25/1.16 ! [A5: $tType] :
% 2.25/1.16 ( ( comm_semiring_0 @ A5 )
% 2.25/1.16 => ( comm_semiring_0 @ ( poly @ A5 ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(tcon_Polynomial_Opoly___Groups_Omonoid__mult_50,axiom,
% 2.25/1.16 ! [A5: $tType] :
% 2.25/1.16 ( ( comm_semiring_1 @ A5 )
% 2.25/1.16 => ( monoid_mult @ ( poly @ A5 ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(tcon_Polynomial_Opoly___Rings_Oidom__divide_51,axiom,
% 2.25/1.16 ! [A5: $tType] :
% 2.25/1.16 ( ( idom_divide @ A5 )
% 2.25/1.16 => ( idom_divide @ ( poly @ A5 ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(tcon_Polynomial_Opoly___Rings_Ocomm__ring__1_52,axiom,
% 2.25/1.16 ! [A5: $tType] :
% 2.25/1.16 ( ( comm_ring_1 @ A5 )
% 2.25/1.16 => ( comm_ring_1 @ ( poly @ A5 ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(tcon_Polynomial_Opoly___Rings_Osemiring__1_53,axiom,
% 2.25/1.16 ! [A5: $tType] :
% 2.25/1.16 ( ( comm_semiring_1 @ A5 )
% 2.25/1.16 => ( semiring_1 @ ( poly @ A5 ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(tcon_Polynomial_Opoly___Groups_Ogroup__add_54,axiom,
% 2.25/1.16 ! [A5: $tType] :
% 2.25/1.16 ( ( ab_group_add @ A5 )
% 2.25/1.16 => ( group_add @ ( poly @ A5 ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(tcon_Polynomial_Opoly___Rings_Ocomm__ring_55,axiom,
% 2.25/1.16 ! [A5: $tType] :
% 2.25/1.16 ( ( comm_ring @ A5 )
% 2.25/1.16 => ( comm_ring @ ( poly @ A5 ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(tcon_Polynomial_Opoly___Nat_Oring__char__0_56,axiom,
% 2.25/1.16 ! [A5: $tType] :
% 2.25/1.16 ( ( ( ring_char_0 @ A5 )
% 2.25/1.16 & ( comm_ring_1 @ A5 ) )
% 2.25/1.16 => ( ring_char_0 @ ( poly @ A5 ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(tcon_Polynomial_Opoly___Rings_Oring__1_57,axiom,
% 2.25/1.16 ! [A5: $tType] :
% 2.25/1.16 ( ( comm_ring_1 @ A5 )
% 2.25/1.16 => ( ring_1 @ ( poly @ A5 ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(tcon_Polynomial_Opoly___Power_Opower_58,axiom,
% 2.25/1.16 ! [A5: $tType] :
% 2.25/1.16 ( ( comm_semiring_1 @ A5 )
% 2.25/1.16 => ( power @ ( poly @ A5 ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(tcon_Polynomial_Opoly___Groups_Ozero_59,axiom,
% 2.25/1.16 ! [A5: $tType] :
% 2.25/1.16 ( ( zero @ A5 )
% 2.25/1.16 => ( zero @ ( poly @ A5 ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(tcon_Polynomial_Opoly___Groups_Oone_60,axiom,
% 2.25/1.16 ! [A5: $tType] :
% 2.25/1.16 ( ( comm_semiring_1 @ A5 )
% 2.25/1.16 => ( one @ ( poly @ A5 ) ) ) ).
% 2.25/1.16
% 2.25/1.16 thf(conj_0,conjecture,
% 2.25/1.16 ord_less_eq @ real @ ( divide_divide @ real @ ( one_one @ real ) @ ( power_power @ real @ ( ring_1_of_int @ real @ b ) @ ( degree @ real @ p ) ) ) @ ( abs_abs @ real @ ( poly2 @ real @ p @ ( divide_divide @ real @ ( ring_1_of_int @ real @ a2 ) @ ( ring_1_of_int @ real @ b ) ) ) ) ).
% 2.25/1.16 ------------------------
% 2.25/1.18 % (32588)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_32390 for (2999ds/275Mi)
% 2.25/1.18 % (32584)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_32390 for (2999ds/4Mi)
% 2.25/1.18 % (32585)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_32390 for (2999ds/27Mi)
% 2.25/1.18 % (32587)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_32390 for (2999ds/2Mi)
% 2.25/1.18 % (32586)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_32390 for (2999ds/2Mi)
% 2.25/1.18 % (32587)Instruction limit reached!
% 2.25/1.18 % (32587)------------------------------
% 2.25/1.18 % (32587)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.25/1.18 % (32587)Termination reason: Unknown
% 2.25/1.18 % (32587)Termination phase: shuffling
% 2.25/1.18
% 2.25/1.18 % (32587)Memory used [KB]: 1279
% 2.25/1.18 % (32587)Time elapsed: 0.002 s
% 2.25/1.18 % (32587)Instructions burned: 2 (million)
% 2.25/1.18 % (32587)------------------------------
% 2.25/1.18 % (32587)------------------------------
% 2.25/1.18 % (32589)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_32390 for (2999ds/18Mi)
% 2.25/1.18 % (32584)Instruction limit reached!
% 2.25/1.18 % (32584)------------------------------
% 2.25/1.18 % (32584)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.25/1.18 % (32584)Termination reason: Unknown
% 2.25/1.18 % (32584)Termination phase: shuffling
% 2.25/1.18
% 2.25/1.18 % (32584)Memory used [KB]: 1407
% 2.25/1.18 % (32584)Time elapsed: 0.003 s
% 2.25/1.18 % (32584)Instructions burned: 5 (million)
% 2.25/1.18 % (32584)------------------------------
% 2.25/1.18 % (32584)------------------------------
% 2.25/1.18 % (32586)Instruction limit reached!
% 2.25/1.18 % (32586)------------------------------
% 2.25/1.18 % (32586)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.25/1.18 % (32586)Termination reason: Unknown
% 2.25/1.18 % (32586)Termination phase: shuffling
% 2.25/1.18
% 2.25/1.18 % (32586)Memory used [KB]: 1407
% 2.25/1.18 % (32586)Time elapsed: 0.003 s
% 2.25/1.18 % (32586)Instructions burned: 3 (million)
% 2.25/1.18 % (32586)------------------------------
% 2.25/1.18 % (32586)------------------------------
% 2.25/1.18 % (32583)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_32390 for (2999ds/183Mi)
% 2.25/1.19 % (32589)Instruction limit reached!
% 2.25/1.19 % (32589)------------------------------
% 2.25/1.19 % (32589)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.25/1.19 % (32589)Termination reason: Unknown
% 2.25/1.19 % (32589)Termination phase: shuffling
% 2.25/1.19
% 2.25/1.19 % (32589)Memory used [KB]: 1663
% 2.25/1.19 % (32589)Time elapsed: 0.008 s
% 2.25/1.19 % (32589)Instructions burned: 19 (million)
% 2.25/1.19 % (32589)------------------------------
% 2.25/1.19 % (32589)------------------------------
% 2.25/1.19 % (32585)Instruction limit reached!
% 2.25/1.19 % (32585)------------------------------
% 2.25/1.19 % (32585)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.25/1.19 % (32585)Termination reason: Unknown
% 2.25/1.19 % (32585)Termination phase: Property scanning
% 2.25/1.19
% 2.25/1.19 % (32585)Memory used [KB]: 1791
% 2.25/1.19 % (32585)Time elapsed: 0.010 s
% 2.25/1.19 % (32585)Instructions burned: 27 (million)
% 2.25/1.19 % (32585)------------------------------
% 2.25/1.19 % (32585)------------------------------
% 2.25/1.19 % (32590)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_32390 for (2999ds/3Mi)
% 2.25/1.19 % (32592)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_32390 for (2999ds/15Mi)
% 2.25/1.19 % (32590)Instruction limit reached!
% 2.25/1.19 % (32590)------------------------------
% 2.25/1.19 % (32590)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.25/1.19 % (32590)Termination reason: Unknown
% 2.25/1.19 % (32590)Termination phase: shuffling
% 2.25/1.19
% 2.25/1.19 % (32590)Memory used [KB]: 1407
% 2.25/1.19 % (32590)Time elapsed: 0.002 s
% 2.25/1.19 % (32590)Instructions burned: 3 (million)
% 2.25/1.19 % (32590)------------------------------
% 2.25/1.19 % (32590)------------------------------
% 2.25/1.19 % (32591)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_32390 for (2999ds/37Mi)
% 2.25/1.19 % (32592)Instruction limit reached!
% 2.25/1.19 % (32592)------------------------------
% 2.25/1.19 % (32592)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.25/1.19 % (32592)Termination reason: Unknown
% 2.25/1.19 % (32592)Termination phase: shuffling
% 2.25/1.19
% 2.25/1.19 % (32592)Memory used [KB]: 1663
% 2.25/1.19 % (32592)Time elapsed: 0.006 s
% 2.25/1.19 % (32592)Instructions burned: 15 (million)
% 2.25/1.19 % (32592)------------------------------
% 2.25/1.19 % (32592)------------------------------
% 2.25/1.20 % (32593)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_32390 for (2999ds/3Mi)
% 2.30/1.20 % (32594)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_32390 for (2999ds/1041Mi)
% 2.30/1.20 % (32593)Instruction limit reached!
% 2.30/1.20 % (32593)------------------------------
% 2.30/1.20 % (32593)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.30/1.20 % (32593)Termination reason: Unknown
% 2.30/1.20 % (32593)Termination phase: shuffling
% 2.30/1.20
% 2.30/1.20 % (32593)Memory used [KB]: 1407
% 2.30/1.20 % (32593)Time elapsed: 0.002 s
% 2.30/1.20 % (32593)Instructions burned: 3 (million)
% 2.30/1.20 % (32593)------------------------------
% 2.30/1.20 % (32593)------------------------------
% 2.30/1.20 % (32595)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_32390 for (2999ds/7Mi)
% 2.30/1.20 % (32591)Instruction limit reached!
% 2.30/1.20 % (32591)------------------------------
% 2.30/1.20 % (32591)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.30/1.20 % (32591)Termination reason: Unknown
% 2.30/1.20 % (32591)Termination phase: Property scanning
% 2.30/1.20
% 2.30/1.20 % (32591)Memory used [KB]: 2046
% 2.30/1.20 % (32591)Time elapsed: 0.012 s
% 2.30/1.20 % (32591)Instructions burned: 37 (million)
% 2.30/1.20 % (32591)------------------------------
% 2.30/1.20 % (32591)------------------------------
% 2.30/1.20 % (32595)Instruction limit reached!
% 2.30/1.20 % (32595)------------------------------
% 2.30/1.20 % (32595)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.30/1.20 % (32595)Termination reason: Unknown
% 2.30/1.20 % (32595)Termination phase: shuffling
% 2.30/1.20
% 2.30/1.20 % (32595)Memory used [KB]: 1535
% 2.30/1.20 % (32595)Time elapsed: 0.004 s
% 2.30/1.20 % (32595)Instructions burned: 8 (million)
% 2.30/1.20 % (32595)------------------------------
% 2.30/1.20 % (32595)------------------------------
% 2.30/1.20 % (32588)First to succeed.
% 2.30/1.20 % (32596)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_32390 for (2999ds/16Mi)
% 2.30/1.20 % (32588)Refutation found. Thanks to Tanya!
% 2.30/1.20 % SZS status Theorem for DTF2THF_32390
% 2.30/1.20 % SZS output start Proof for DTF2THF_32390
% 2.30/1.20 thf(type_def_6, type, set: ($tType) > $tType).
% 2.30/1.20 thf(type_def_7, type, int: $tType).
% 2.30/1.20 thf(type_def_8, type, poly: ($tType) > $tType).
% 2.30/1.20 thf(type_def_9, type, nat: $tType).
% 2.30/1.20 thf(type_def_10, type, real: $tType).
% 2.30/1.20 thf(func_def_0, type, poly: ($tType) > $tType).
% 2.30/1.20 thf(func_def_1, type, real: $tType).
% 2.30/1.20 thf(func_def_2, type, set: ($tType) > $tType).
% 2.30/1.20 thf(func_def_3, type, nat: $tType).
% 2.30/1.20 thf(func_def_4, type, int: $tType).
% 2.30/1.20 thf(func_def_5, type, ab_group_add: !>[X0: $tType]:($o)).
% 2.30/1.20 thf(func_def_6, type, idom: !>[X0: $tType]:($o)).
% 2.30/1.20 thf(func_def_7, type, one: !>[X0: $tType]:($o)).
% 2.30/1.20 thf(func_def_8, type, zero: !>[X0: $tType]:($o)).
% 2.30/1.20 thf(func_def_9, type, power: !>[X0: $tType]:($o)).
% 2.30/1.20 thf(func_def_10, type, field: !>[X0: $tType]:($o)).
% 2.30/1.20 thf(func_def_11, type, ring_1: !>[X0: $tType]:($o)).
% 2.30/1.20 thf(func_def_12, type, ring_char_0: !>[X0: $tType]:($o)).
% 2.30/1.20 thf(func_def_13, type, comm_ring: !>[X0: $tType]:($o)).
% 2.30/1.20 thf(func_def_14, type, group_add: !>[X0: $tType]:($o)).
% 2.30/1.20 thf(func_def_15, type, semiring_1: !>[X0: $tType]:($o)).
% 2.30/1.20 thf(func_def_16, type, comm_ring_1: !>[X0: $tType]:($o)).
% 2.30/1.20 thf(func_def_17, type, idom_divide: !>[X0: $tType]:($o)).
% 2.30/1.20 thf(func_def_18, type, monoid_mult: !>[X0: $tType]:($o)).
% 2.30/1.20 thf(func_def_19, type, field_char_0: !>[X0: $tType]:($o)).
% 2.30/1.20 thf(func_def_20, type, division_ring: !>[X0: $tType]:($o)).
% 2.30/1.20 thf(func_def_21, type, field_abs_sgn: !>[X0: $tType]:($o)).
% 2.30/1.20 thf(func_def_22, type, comm_semiring_0: !>[X0: $tType]:($o)).
% 2.30/1.20 thf(func_def_23, type, comm_semiring_1: !>[X0: $tType]:($o)).
% 2.30/1.20 thf(func_def_24, type, linordered_idom: !>[X0: $tType]:($o)).
% 2.30/1.20 thf(func_def_25, type, linordered_field: !>[X0: $tType]:($o)).
% 2.30/1.20 thf(func_def_26, type, comm_monoid_diff: !>[X0: $tType]:($o)).
% 2.30/1.20 thf(func_def_27, type, linordered_semidom: !>[X0: $tType]:($o)).
% 2.30/1.20 thf(func_def_28, type, ordered_ab_group_add: !>[X0: $tType]:($o)).
% 2.30/1.20 thf(func_def_29, type, ring_n68954251visors: !>[X0: $tType]:($o)).
% 2.30/1.20 thf(func_def_30, type, cancel1352612707id_add: !>[X0: $tType]:($o)).
% 2.30/1.20 thf(func_def_31, type, cancel146912293up_add: !>[X0: $tType]:($o)).
% 2.30/1.20 thf(func_def_32, type, ordere142940540dd_abs: !>[X0: $tType]:($o)).
% 2.30/1.20 thf(func_def_33, type, semiri1193490041visors: !>[X0: $tType]:($o)).
% 2.30/1.20 thf(func_def_34, type, semiri134348788visors: !>[X0: $tType]:($o)).
% 2.30/1.20 thf(func_def_35, type, canoni770627133id_add: !>[X0: $tType]:($o)).
% 2.30/1.20 thf(func_def_36, type, abs_abs: !>[X0: $tType]:(X0 > X0)).
% 2.30/1.20 thf(func_def_37, type, minus_minus: !>[X0: $tType]:(X0 > X0 > X0)).
% 2.30/1.20 thf(func_def_38, type, one_one: !>[X0: $tType]:(X0)).
% 2.30/1.20 thf(func_def_39, type, zero_zero: !>[X0: $tType]:(X0)).
% 2.30/1.20 thf(func_def_40, type, ring_1_Ints: !>[X0: $tType]:(set @ X0)).
% 2.30/1.20 thf(func_def_41, type, ring_1_of_int: !>[X0: $tType]:(int > X0)).
% 2.30/1.20 thf(func_def_42, type, ord_less: !>[X0: $tType]:(X0 > X0 > $o)).
% 2.30/1.20 thf(func_def_43, type, ord_less_eq: !>[X0: $tType]:(X0 > X0 > $o)).
% 2.30/1.20 thf(func_def_44, type, algebraic: !>[X0: $tType]:(X0 > $o)).
% 2.30/1.20 thf(func_def_45, type, degree: !>[X0: $tType]:(poly @ X0 > nat)).
% 2.30/1.20 thf(func_def_46, type, poly2: !>[X0: $tType]:(poly @ X0 > X0 > X0)).
% 2.30/1.20 thf(func_def_47, type, coeff: !>[X0: $tType]:(poly @ X0 > nat > X0)).
% 2.30/1.20 thf(func_def_48, type, power_power: !>[X0: $tType]:(X0 > nat > X0)).
% 2.30/1.20 thf(func_def_49, type, field_char_0_Rats: !>[X0: $tType]:(set @ X0)).
% 2.30/1.20 thf(func_def_50, type, divide_divide: !>[X0: $tType]:(X0 > X0 > X0)).
% 2.30/1.20 thf(func_def_51, type, collect: !>[X0: $tType]:((X0 > $o) > set @ X0)).
% 2.30/1.20 thf(func_def_52, type, member: !>[X0: $tType]:(X0 > set @ X0 > $o)).
% 2.30/1.20 thf(func_def_53, type, a: real).
% 2.30/1.20 thf(func_def_54, type, m: real).
% 2.30/1.20 thf(func_def_55, type, a2: int).
% 2.30/1.20 thf(func_def_56, type, b: int).
% 2.30/1.20 thf(func_def_57, type, n: nat).
% 2.30/1.20 thf(func_def_58, type, p: poly @ real).
% 2.30/1.20 thf(func_def_59, type, roots: set @ real).
% 2.30/1.20 thf(func_def_61, type, x: real).
% 2.30/1.20 thf(func_def_74, type, sK0: poly @ real > real > real > real).
% 2.30/1.20 thf(func_def_75, type, sK1: !>[X0: $tType]:(X0 > X0)).
% 2.30/1.20 thf(func_def_76, type, sK2: !>[X0: $tType]:(poly @ X0 > nat > nat)).
% 2.30/1.20 thf(func_def_77, type, sK3: poly @ real).
% 2.30/1.20 thf(func_def_78, type, sK4: !>[X0: $tType]:(poly @ X0 > poly @ X0 > nat)).
% 2.30/1.20 thf(func_def_79, type, sK5: !>[X0: $tType]:(X0 > X0)).
% 2.30/1.20 thf(func_def_80, type, sK6: !>[X0: $tType]:(poly @ X0 > nat > nat)).
% 2.30/1.20 thf(func_def_81, type, sK7: !>[X0: $tType, X1: $tType]:((X1 > X0) > (X1 > X0) > X1)).
% 2.30/1.20 thf(func_def_82, type, sK8: poly @ real > real > real > real).
% 2.30/1.20 thf(func_def_83, type, sK9: poly @ real > real).
% 2.30/1.20 thf(func_def_84, type, sK10: poly @ real > nat).
% 2.30/1.20 thf(func_def_86, type, ph12: !>[X0: $tType]:(X0)).
% 2.30/1.20 thf(f1414,plain,(
% 2.30/1.20 $false),
% 2.30/1.20 inference(subsumption_resolution,[status(thm)],[f1413,f1209])).
% 2.30/1.20 thf(f1209,plain,(
% 2.30/1.20 ( ! [X0 : nat] : (((member @ real @ (coeff @ real @ p @ X0) @ ring_1_Ints @ real) = $true)) )),
% 2.30/1.20 inference(cnf_transformation,[status(thm)],[f1004])).
% 2.30/1.20 thf(f1004,plain,(
% 2.30/1.20 ! [X0 : nat] : ((member @ real @ (coeff @ real @ p @ X0) @ ring_1_Ints @ real) = $true)),
% 2.30/1.20 inference(fool_elimination,[status(thm)],[f1003])).
% 2.30/1.20 thf(f1003,plain,(
% 2.30/1.20 ! [X0 : nat] : (member @ real @ (coeff @ real @ p @ X0) @ ring_1_Ints @ real)),
% 2.30/1.20 inference(rectify,[status(thm)],[f3])).
% 2.30/1.20 thf(f3,axiom,(
% 2.30/1.20 ! [X1 : nat] : (member @ real @ (coeff @ real @ p @ X1) @ ring_1_Ints @ real)),
% 2.30/1.20 file('/export/starexec/sandbox/tmp/tmp.qtQCAE7Wjp/DTF2THF_32390.p',fact_2_p_I1_J)).
% 2.30/1.20 thf(f1413,plain,(
% 2.30/1.20 ((member @ real @ (coeff @ real @ p @ (sK10 @ p)) @ ring_1_Ints @ real) != $true)),
% 2.30/1.20 inference(subsumption_resolution,[status(thm)],[f1412,f1317])).
% 2.30/1.20 thf(f1317,plain,(
% 2.30/1.20 (zero_zero @ real != (poly2 @ real @ p @ (divide_divide @ real @ (ring_1_of_int @ real @ a2) @ (ring_1_of_int @ real @ b))))),
% 2.30/1.20 inference(cnf_transformation,[status(thm)],[f6])).
% 2.30/1.20 thf(f6,axiom,(
% 2.30/1.20 (zero_zero @ real != (poly2 @ real @ p @ (divide_divide @ real @ (ring_1_of_int @ real @ a2) @ (ring_1_of_int @ real @ b))))),
% 2.30/1.20 file('/export/starexec/sandbox/tmp/tmp.qtQCAE7Wjp/DTF2THF_32390.p',fact_5_no__root)).
% 2.30/1.20 thf(f1412,plain,(
% 2.30/1.20 (zero_zero @ real = (poly2 @ real @ p @ (divide_divide @ real @ (ring_1_of_int @ real @ a2) @ (ring_1_of_int @ real @ b)))) | ((member @ real @ (coeff @ real @ p @ (sK10 @ p)) @ ring_1_Ints @ real) != $true)),
% 2.30/1.20 inference(subsumption_resolution,[status(thm)],[f1411,f1318])).
% 2.30/1.20 thf(f1318,plain,(
% 2.30/1.20 ((ord_less @ int @ zero_zero @ int @ b) = $true)),
% 2.30/1.20 inference(cnf_transformation,[status(thm)],[f668])).
% 2.30/1.20 thf(f668,plain,(
% 2.30/1.20 ((ord_less @ int @ zero_zero @ int @ b) = $true)),
% 2.30/1.20 inference(fool_elimination,[status(thm)],[f667])).
% 2.30/1.20 thf(f667,plain,(
% 2.30/1.20 (ord_less @ int @ zero_zero @ int @ b)),
% 2.30/1.20 inference(rectify,[status(thm)],[f4])).
% 2.30/1.20 thf(f4,axiom,(
% 2.30/1.20 (ord_less @ int @ zero_zero @ int @ b)),
% 2.30/1.20 file('/export/starexec/sandbox/tmp/tmp.qtQCAE7Wjp/DTF2THF_32390.p',fact_3_b)).
% 2.30/1.20 thf(f1411,plain,(
% 2.30/1.20 ((ord_less @ int @ zero_zero @ int @ b) != $true) | ((member @ real @ (coeff @ real @ p @ (sK10 @ p)) @ ring_1_Ints @ real) != $true) | (zero_zero @ real = (poly2 @ real @ p @ (divide_divide @ real @ (ring_1_of_int @ real @ a2) @ (ring_1_of_int @ real @ b))))),
% 2.30/1.20 inference(trivial_inequality_removal,[status(thm)],[f1410])).
% 2.30/1.20 thf(f1410,plain,(
% 2.30/1.20 ((ord_less @ int @ zero_zero @ int @ b) != $true) | ((member @ real @ (coeff @ real @ p @ (sK10 @ p)) @ ring_1_Ints @ real) != $true) | ($true != $true) | (zero_zero @ real = (poly2 @ real @ p @ (divide_divide @ real @ (ring_1_of_int @ real @ a2) @ (ring_1_of_int @ real @ b))))),
% 2.30/1.20 inference(superposition,[status(thm)],[f1271,f1344])).
% 2.30/1.20 thf(f1344,plain,(
% 2.30/1.20 ( ! [X2 : int,X0 : poly @ real,X1 : int] : (((ord_less_eq @ real @ (divide_divide @ real @ one_one @ real @ (power_power @ real @ (ring_1_of_int @ real @ X1) @ (degree @ real @ X0))) @ (abs_abs @ real @ (poly2 @ real @ X0 @ (divide_divide @ real @ (ring_1_of_int @ real @ X2) @ (ring_1_of_int @ real @ X1))))) = $true) | ($true != (member @ real @ (coeff @ real @ X0 @ (sK10 @ X0)) @ ring_1_Ints @ real)) | ($true != (ord_less @ int @ zero_zero @ int @ X1)) | (zero_zero @ real = (poly2 @ real @ X0 @ (divide_divide @ real @ (ring_1_of_int @ real @ X2) @ (ring_1_of_int @ real @ X1))))) )),
% 2.30/1.20 inference(cnf_transformation,[status(thm)],[f1195])).
% 2.30/1.20 thf(f1195,plain,(
% 2.30/1.20 ! [X0 : poly @ real,X1 : int,X2 : int] : ((zero_zero @ real = (poly2 @ real @ X0 @ (divide_divide @ real @ (ring_1_of_int @ real @ X2) @ (ring_1_of_int @ real @ X1)))) | ($true != (member @ real @ (coeff @ real @ X0 @ (sK10 @ X0)) @ ring_1_Ints @ real)) | ((ord_less_eq @ real @ (divide_divide @ real @ one_one @ real @ (power_power @ real @ (ring_1_of_int @ real @ X1) @ (degree @ real @ X0))) @ (abs_abs @ real @ (poly2 @ real @ X0 @ (divide_divide @ real @ (ring_1_of_int @ real @ X2) @ (ring_1_of_int @ real @ X1))))) = $true) | ($true != (ord_less @ int @ zero_zero @ int @ X1)))),
% 2.30/1.20 inference(skolemisation,[status(esa),new_symbols(skolem,[sK10])],[f1086,f1194])).
% 2.30/1.20 thf(f1194,plain,(
% 2.30/1.20 ! [X0 : poly @ real] : (? [X3 : nat] : ((member @ real @ (coeff @ real @ X0 @ X3) @ ring_1_Ints @ real) != $true) => ($true != (member @ real @ (coeff @ real @ X0 @ (sK10 @ X0)) @ ring_1_Ints @ real)))),
% 2.30/1.20 introduced(definition,[],[choice_axiom])).
% 2.30/1.20 thf(f1086,plain,(
% 2.30/1.20 ! [X0 : poly @ real,X1 : int,X2 : int] : ((zero_zero @ real = (poly2 @ real @ X0 @ (divide_divide @ real @ (ring_1_of_int @ real @ X2) @ (ring_1_of_int @ real @ X1)))) | ? [X3 : nat] : ((member @ real @ (coeff @ real @ X0 @ X3) @ ring_1_Ints @ real) != $true) | ((ord_less_eq @ real @ (divide_divide @ real @ one_one @ real @ (power_power @ real @ (ring_1_of_int @ real @ X1) @ (degree @ real @ X0))) @ (abs_abs @ real @ (poly2 @ real @ X0 @ (divide_divide @ real @ (ring_1_of_int @ real @ X2) @ (ring_1_of_int @ real @ X1))))) = $true) | ($true != (ord_less @ int @ zero_zero @ int @ X1)))),
% 2.30/1.20 inference(flattening,[status(thm)],[f1085])).
% 2.30/1.20 thf(f1085,plain,(
% 2.30/1.20 ! [X0 : poly @ real,X1 : int,X2 : int] : (((((ord_less_eq @ real @ (divide_divide @ real @ one_one @ real @ (power_power @ real @ (ring_1_of_int @ real @ X1) @ (degree @ real @ X0))) @ (abs_abs @ real @ (poly2 @ real @ X0 @ (divide_divide @ real @ (ring_1_of_int @ real @ X2) @ (ring_1_of_int @ real @ X1))))) = $true) | (zero_zero @ real = (poly2 @ real @ X0 @ (divide_divide @ real @ (ring_1_of_int @ real @ X2) @ (ring_1_of_int @ real @ X1))))) | ($true != (ord_less @ int @ zero_zero @ int @ X1))) | ? [X3 : nat] : ((member @ real @ (coeff @ real @ X0 @ X3) @ ring_1_Ints @ real) != $true))),
% 2.30/1.20 inference(ennf_transformation,[status(thm)],[f962])).
% 2.30/1.20 thf(f962,plain,(
% 2.30/1.20 ! [X0 : poly @ real,X1 : int,X2 : int] : (! [X3 : nat] : ((member @ real @ (coeff @ real @ X0 @ X3) @ ring_1_Ints @ real) = $true) => (($true = (ord_less @ int @ zero_zero @ int @ X1)) => ((zero_zero @ real != (poly2 @ real @ X0 @ (divide_divide @ real @ (ring_1_of_int @ real @ X2) @ (ring_1_of_int @ real @ X1)))) => ((ord_less_eq @ real @ (divide_divide @ real @ one_one @ real @ (power_power @ real @ (ring_1_of_int @ real @ X1) @ (degree @ real @ X0))) @ (abs_abs @ real @ (poly2 @ real @ X0 @ (divide_divide @ real @ (ring_1_of_int @ real @ X2) @ (ring_1_of_int @ real @ X1))))) = $true))))),
% 2.30/1.20 inference(fool_elimination,[status(thm)],[f961])).
% 2.30/1.20 thf(f961,plain,(
% 2.30/1.20 ! [X0 : poly @ real,X1 : int,X2 : int] : (! [X3 : nat] : (member @ real @ (coeff @ real @ X0 @ X3) @ ring_1_Ints @ real) => ((ord_less @ int @ zero_zero @ int @ X1) => ((zero_zero @ real != (poly2 @ real @ X0 @ (divide_divide @ real @ (ring_1_of_int @ real @ X2) @ (ring_1_of_int @ real @ X1)))) => (ord_less_eq @ real @ (divide_divide @ real @ one_one @ real @ (power_power @ real @ (ring_1_of_int @ real @ X1) @ (degree @ real @ X0))) @ (abs_abs @ real @ (poly2 @ real @ X0 @ (divide_divide @ real @ (ring_1_of_int @ real @ X2) @ (ring_1_of_int @ real @ X1))))))))),
% 2.30/1.20 inference(rectify,[status(thm)],[f76])).
% 2.30/1.20 thf(f76,axiom,(
% 2.30/1.20 ! [X6 : poly @ real,X2 : int,X13 : int] : (! [X12 : nat] : (member @ real @ (coeff @ real @ X6 @ X12) @ ring_1_Ints @ real) => ((ord_less @ int @ zero_zero @ int @ X2) => ((zero_zero @ real != (poly2 @ real @ X6 @ (divide_divide @ real @ (ring_1_of_int @ real @ X13) @ (ring_1_of_int @ real @ X2)))) => (ord_less_eq @ real @ (divide_divide @ real @ one_one @ real @ (power_power @ real @ (ring_1_of_int @ real @ X2) @ (degree @ real @ X6))) @ (abs_abs @ real @ (poly2 @ real @ X6 @ (divide_divide @ real @ (ring_1_of_int @ real @ X13) @ (ring_1_of_int @ real @ X2))))))))),
% 2.30/1.20 file('/export/starexec/sandbox/tmp/tmp.qtQCAE7Wjp/DTF2THF_32390.p',fact_75_int__poly__rat__no__root__ge)).
% 2.30/1.20 thf(f1271,plain,(
% 2.30/1.20 ((ord_less_eq @ real @ (divide_divide @ real @ one_one @ real @ (power_power @ real @ (ring_1_of_int @ real @ b) @ (degree @ real @ p))) @ (abs_abs @ real @ (poly2 @ real @ p @ (divide_divide @ real @ (ring_1_of_int @ real @ a2) @ (ring_1_of_int @ real @ b))))) != $true)),
% 2.30/1.20 inference(cnf_transformation,[status(thm)],[f1025])).
% 2.30/1.20 thf(f1025,plain,(
% 2.30/1.20 ((ord_less_eq @ real @ (divide_divide @ real @ one_one @ real @ (power_power @ real @ (ring_1_of_int @ real @ b) @ (degree @ real @ p))) @ (abs_abs @ real @ (poly2 @ real @ p @ (divide_divide @ real @ (ring_1_of_int @ real @ a2) @ (ring_1_of_int @ real @ b))))) != $true)),
% 2.30/1.20 inference(flattening,[status(thm)],[f672])).
% 2.30/1.20 thf(f672,plain,(
% 2.30/1.20 ~((ord_less_eq @ real @ (divide_divide @ real @ one_one @ real @ (power_power @ real @ (ring_1_of_int @ real @ b) @ (degree @ real @ p))) @ (abs_abs @ real @ (poly2 @ real @ p @ (divide_divide @ real @ (ring_1_of_int @ real @ a2) @ (ring_1_of_int @ real @ b))))) = $true)),
% 2.30/1.20 inference(fool_elimination,[status(thm)],[f671])).
% 2.30/1.20 thf(f671,plain,(
% 2.30/1.20 ~(ord_less_eq @ real @ (divide_divide @ real @ one_one @ real @ (power_power @ real @ (ring_1_of_int @ real @ b) @ (degree @ real @ p))) @ (abs_abs @ real @ (poly2 @ real @ p @ (divide_divide @ real @ (ring_1_of_int @ real @ a2) @ (ring_1_of_int @ real @ b)))))),
% 2.30/1.20 inference(rectify,[status(thm)],[f347])).
% 2.30/1.20 thf(f347,negated_conjecture,(
% 2.30/1.20 ~(ord_less_eq @ real @ (divide_divide @ real @ one_one @ real @ (power_power @ real @ (ring_1_of_int @ real @ b) @ (degree @ real @ p))) @ (abs_abs @ real @ (poly2 @ real @ p @ (divide_divide @ real @ (ring_1_of_int @ real @ a2) @ (ring_1_of_int @ real @ b)))))),
% 2.30/1.20 inference(negated_conjecture,[status(cth)],[f346])).
% 2.30/1.20 thf(f346,conjecture,(
% 2.30/1.20 (ord_less_eq @ real @ (divide_divide @ real @ one_one @ real @ (power_power @ real @ (ring_1_of_int @ real @ b) @ (degree @ real @ p))) @ (abs_abs @ real @ (poly2 @ real @ p @ (divide_divide @ real @ (ring_1_of_int @ real @ a2) @ (ring_1_of_int @ real @ b)))))),
% 2.30/1.20 file('/export/starexec/sandbox/tmp/tmp.qtQCAE7Wjp/DTF2THF_32390.p',conj_0)).
% 2.30/1.20 % SZS output end Proof for DTF2THF_32390
% 2.30/1.20 % (32588)------------------------------
% 2.30/1.20 % (32588)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.30/1.20 % (32588)Termination reason: Refutation
% 2.30/1.20
% 2.30/1.20 % (32588)Memory used [KB]: 6652
% 2.30/1.20 % (32588)Time elapsed: 0.026 s
% 2.30/1.20 % (32588)Instructions burned: 72 (million)
% 2.30/1.20 % (32588)------------------------------
% 2.30/1.20 % (32588)------------------------------
% 2.30/1.20 % (32582)Success in time 0.053 s
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