%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : ITP094^2 : 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 : n005.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 08:55:30 AM UTC 2026
% Result : Theorem 2.68s 1.66s
% Output : Refutation 2.68s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : ITP094^2 : TPTP v9.2.1. Released v7.5.0.
% 0.12/0.13 % Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.17/0.34 % Computer : n005.cluster.edu
% 0.17/0.34 % Model : x86_64 x86_64
% 0.17/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.17/0.34 % Memory : 8042.1875MB
% 0.17/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.17/0.34 % CPULimit : 300
% 0.17/0.34 % WCLimit : 300
% 0.17/0.34 % DateTime : Tue Feb 24 08:26:14 EST 2026
% 0.17/0.34 % CPUTime :
% 0.17/0.34 Running /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.20/0.47 ---- Original DTF file ---
% 0.20/0.47 thf(spec,logic,$$dhol).
% 0.20/0.47 %------------------------------------------------------------------------------
% 0.20/0.47 % File : ITP094^2 : TPTP v9.2.1. Released v7.5.0.
% 0.20/0.47 % Domain : Interactive Theorem Proving
% 0.20/0.47 % Problem : Sledgehammer Liouville_Numbers problem prob_128__5866194_1
% 0.20/0.47 % Version : Especial.
% 0.20/0.47 % English :
% 0.20/0.47
% 0.20/0.47 % Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% 0.20/0.47 % : [Des21] Desharnais (2021), Email to Geoff Sutcliffe
% 0.20/0.47 % Source : [Des21]
% 0.20/0.47 % Names : Liouville_Numbers/prob_128__5866194_1 [Des21]
% 0.20/0.47
% 0.20/0.47 % Status : Theorem
% 0.20/0.47 % Rating : 0.00 v7.5.0
% 0.20/0.47 % Syntax : Number of formulae : 381 ( 99 unt; 64 typ; 0 def)
% 0.20/0.47 % Number of atoms : 856 ( 300 equ; 0 cnn)
% 0.20/0.47 % Maximal formula atoms : 7 ( 2 avg)
% 0.20/0.47 % Number of connectives : 2979 ( 132 ~; 11 |; 83 &;2362 @)
% 0.20/0.47 % ( 0 <=>; 391 =>; 0 <=; 0 <~>)
% 0.20/0.47 % Maximal formula depth : 19 ( 6 avg)
% 0.20/0.47 % Number of types : 3 ( 2 usr)
% 0.20/0.47 % Number of type conns : 108 ( 108 >; 0 *; 0 +; 0 <<)
% 0.20/0.47 % Number of symbols : 64 ( 62 usr; 4 con; 0-7 aty)
% 0.20/0.47 % Number of variables : 711 ( 41 ^; 579 !; 34 ?; 711 :)
% 0.20/0.47 % ( 57 !>; 0 ?*; 0 @-; 0 @+)
% 0.20/0.47 % SPC : TH1_THM_EQU_NAR
% 0.20/0.47
% 0.20/0.47 % Comments : This file was generated by Sledgehammer 2021-02-23 16:24:32.418
% 0.20/0.47 %------------------------------------------------------------------------------
% 0.20/0.47 % Could-be-implicit typings (4)
% 0.20/0.47 thf(ty_t_Polynomial_Opoly,type,
% 0.20/0.47 poly: $tType > $tType ).
% 0.20/0.47
% 0.20/0.47 thf(ty_t_Real_Oreal,type,
% 0.20/0.47 real: $tType ).
% 0.20/0.47
% 0.20/0.47 thf(ty_t_Set_Oset,type,
% 0.20/0.47 set: $tType > $tType ).
% 0.20/0.47
% 0.20/0.47 thf(ty_t_Nat_Onat,type,
% 0.20/0.47 nat: $tType ).
% 0.20/0.47
% 0.20/0.47 % Explicit typings (60)
% 0.20/0.47 thf(sy_cl_HOL_Otype,type,
% 0.20/0.47 type:
% 0.20/0.47 !>[A: $tType] : $o ).
% 0.20/0.47
% 0.20/0.47 thf(sy_cl_Groups_Oone,type,
% 0.20/0.47 one:
% 0.20/0.47 !>[A: $tType] : $o ).
% 0.20/0.47
% 0.20/0.47 thf(sy_cl_Rings_Oidom,type,
% 0.20/0.47 idom:
% 0.20/0.47 !>[A: $tType] : $o ).
% 0.20/0.47
% 0.20/0.47 thf(sy_cl_Groups_Ozero,type,
% 0.20/0.47 zero:
% 0.20/0.47 !>[A: $tType] : $o ).
% 0.20/0.47
% 0.20/0.47 thf(sy_cl_Fields_Ofield,type,
% 0.20/0.47 field:
% 0.20/0.47 !>[A: $tType] : $o ).
% 0.20/0.47
% 0.20/0.47 thf(sy_cl_Rings_Oring__1,type,
% 0.20/0.47 ring_1:
% 0.20/0.47 !>[A: $tType] : $o ).
% 0.20/0.47
% 0.20/0.47 thf(sy_cl_Nat_Oring__char__0,type,
% 0.20/0.47 ring_char_0:
% 0.20/0.47 !>[A: $tType] : $o ).
% 0.20/0.47
% 0.20/0.47 thf(sy_cl_Orderings_Oorder,type,
% 0.20/0.47 order:
% 0.20/0.47 !>[A: $tType] : $o ).
% 0.20/0.47
% 0.20/0.47 thf(sy_cl_Finite__Set_Ofinite,type,
% 0.20/0.47 finite_finite:
% 0.20/0.47 !>[A: $tType] : $o ).
% 0.20/0.47
% 0.20/0.47 thf(sy_cl_Rings_Ocomm__ring__1,type,
% 0.20/0.47 comm_ring_1:
% 0.20/0.47 !>[A: $tType] : $o ).
% 0.20/0.47
% 0.20/0.47 thf(sy_cl_Rings_Oidom__divide,type,
% 0.20/0.47 idom_divide:
% 0.20/0.47 !>[A: $tType] : $o ).
% 0.20/0.47
% 0.20/0.47 thf(sy_cl_Rings_Ozero__neq__one,type,
% 0.20/0.47 zero_neq_one:
% 0.20/0.47 !>[A: $tType] : $o ).
% 0.20/0.47
% 0.20/0.47 thf(sy_cl_Fields_Ofield__char__0,type,
% 0.20/0.47 field_char_0:
% 0.20/0.47 !>[A: $tType] : $o ).
% 0.20/0.47
% 0.20/0.47 thf(sy_cl_Nat_Osemiring__char__0,type,
% 0.20/0.47 semiring_char_0:
% 0.20/0.47 !>[A: $tType] : $o ).
% 0.20/0.47
% 0.20/0.47 thf(sy_cl_Orderings_Owellorder,type,
% 0.20/0.47 wellorder:
% 0.20/0.47 !>[A: $tType] : $o ).
% 0.20/0.47
% 0.20/0.47 thf(sy_cl_Rings_Ozero__less__one,type,
% 0.20/0.47 zero_less_one:
% 0.20/0.47 !>[A: $tType] : $o ).
% 0.20/0.47
% 0.20/0.47 thf(sy_cl_Fields_Odivision__ring,type,
% 0.20/0.47 division_ring:
% 0.20/0.47 !>[A: $tType] : $o ).
% 0.20/0.47
% 0.20/0.47 thf(sy_cl_Rings_Ocomm__semiring__0,type,
% 0.20/0.47 comm_semiring_0:
% 0.20/0.47 !>[A: $tType] : $o ).
% 0.20/0.47
% 0.20/0.47 thf(sy_cl_Rings_Ocomm__semiring__1,type,
% 0.20/0.47 comm_semiring_1:
% 0.20/0.47 !>[A: $tType] : $o ).
% 0.20/0.47
% 0.20/0.47 thf(sy_cl_Rings_Olinordered__idom,type,
% 0.20/0.47 linordered_idom:
% 0.20/0.47 !>[A: $tType] : $o ).
% 0.20/0.47
% 0.20/0.47 thf(sy_cl_Fields_Olinordered__field,type,
% 0.20/0.47 linordered_field:
% 0.20/0.47 !>[A: $tType] : $o ).
% 0.20/0.47
% 0.20/0.47 thf(sy_cl_Groups_Ocomm__monoid__mult,type,
% 0.20/0.47 comm_monoid_mult:
% 0.20/0.47 !>[A: $tType] : $o ).
% 0.20/0.47
% 0.20/0.47 thf(sy_cl_Rings_Oalgebraic__semidom,type,
% 0.20/0.47 algebraic_semidom:
% 0.20/0.47 !>[A: $tType] : $o ).
% 0.20/0.47
% 0.20/0.47 thf(sy_cl_Rings_Oring__no__zero__divisors,type,
% 0.20/0.47 ring_n68954251visors:
% 0.20/0.47 !>[A: $tType] : $o ).
% 0.20/0.47
% 0.20/0.47 thf(sy_cl_Rings_Osemiring__no__zero__divisors,type,
% 0.20/0.47 semiri1193490041visors:
% 0.20/0.47 !>[A: $tType] : $o ).
% 0.20/0.47
% 0.20/0.47 thf(sy_cl_Rings_Olinordered__nonzero__semiring,type,
% 0.20/0.47 linord1659791738miring:
% 0.20/0.47 !>[A: $tType] : $o ).
% 0.20/0.47
% 0.20/0.47 thf(sy_cl_Rings_Osemiring__1__no__zero__divisors,type,
% 0.20/0.47 semiri134348788visors:
% 0.20/0.47 !>[A: $tType] : $o ).
% 0.20/0.47
% 0.20/0.47 thf(sy_cl_Groups_Ocanonically__ordered__monoid__add,type,
% 0.20/0.47 canoni770627133id_add:
% 0.20/0.47 !>[A: $tType] : $o ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Fields_Oinverse__class_Oinverse,type,
% 0.20/0.47 inverse_inverse:
% 0.20/0.47 !>[A: $tType] : ( A > A ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Finite__Set_Ofinite,type,
% 0.20/0.47 finite_finite2:
% 0.20/0.47 !>[A: $tType] : ( ( set @ A ) > $o ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Groups_Oone__class_Oone,type,
% 0.20/0.47 one_one:
% 0.20/0.47 !>[A: $tType] : A ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Groups_Ozero__class_Ozero,type,
% 0.20/0.47 zero_zero:
% 0.20/0.47 !>[A: $tType] : A ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Int_Oring__1__class_OInts,type,
% 0.20/0.47 ring_1_Ints:
% 0.20/0.47 !>[A: $tType] : ( set @ A ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Nat_OSuc,type,
% 0.20/0.47 suc: nat > nat ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Orderings_Oord__class_Oless,type,
% 0.20/0.47 ord_less:
% 0.20/0.47 !>[A: $tType] : ( A > A > $o ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Orderings_Oord__class_Oless__eq,type,
% 0.20/0.47 ord_less_eq:
% 0.20/0.47 !>[A: $tType] : ( A > A > $o ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Polynomial_Oalgebraic,type,
% 0.20/0.47 algebraic:
% 0.20/0.47 !>[A: $tType] : ( A > $o ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Polynomial_Oalgebraic__int,type,
% 0.20/0.47 algebraic_int:
% 0.20/0.47 !>[A: $tType] : ( A > $o ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Polynomial_Ocr__poly,type,
% 0.20/0.47 cr_poly:
% 0.20/0.47 !>[A: $tType] : ( ( nat > A ) > ( poly @ A ) > $o ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Polynomial_Odegree,type,
% 0.20/0.47 degree:
% 0.20/0.47 !>[A: $tType] : ( ( poly @ A ) > nat ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Polynomial_Odivide__poly__main,type,
% 0.20/0.47 divide_poly_main:
% 0.20/0.47 !>[A: $tType] : ( A > ( poly @ A ) > ( poly @ A ) > ( poly @ A ) > nat > nat > ( poly @ A ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Polynomial_Ois__zero,type,
% 0.20/0.47 is_zero:
% 0.20/0.47 !>[A: $tType] : ( ( poly @ A ) > $o ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Polynomial_Oorder,type,
% 0.20/0.47 order2:
% 0.20/0.47 !>[A: $tType] : ( A > ( poly @ A ) > nat ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Polynomial_Opcr__poly,type,
% 0.20/0.47 pcr_poly:
% 0.20/0.47 !>[C: $tType,B: $tType] : ( ( C > B > $o ) > ( nat > C ) > ( poly @ B ) > $o ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Polynomial_Opderiv,type,
% 0.20/0.47 pderiv:
% 0.20/0.47 !>[A: $tType] : ( ( poly @ A ) > ( poly @ A ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Polynomial_Opoly,type,
% 0.20/0.47 poly2:
% 0.20/0.47 !>[A: $tType] : ( ( poly @ A ) > A > A ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Polynomial_Opoly_Ocoeff,type,
% 0.20/0.47 coeff:
% 0.20/0.47 !>[A: $tType] : ( ( poly @ A ) > nat > A ) ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_Polynomial_Opoly__cutoff,type,
% 0.32/0.47 poly_cutoff:
% 0.32/0.47 !>[A: $tType] : ( nat > ( poly @ A ) > ( poly @ A ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_Polynomial_Opoly__shift,type,
% 0.32/0.47 poly_shift:
% 0.32/0.47 !>[A: $tType] : ( nat > ( poly @ A ) > ( poly @ A ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_Polynomial_Opseudo__mod,type,
% 0.32/0.47 pseudo_mod:
% 0.32/0.47 !>[A: $tType] : ( ( poly @ A ) > ( poly @ A ) > ( poly @ A ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_Polynomial_Oreflect__poly,type,
% 0.32/0.47 reflect_poly:
% 0.32/0.47 !>[A: $tType] : ( ( poly @ A ) > ( poly @ A ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_Polynomial_Orsquarefree,type,
% 0.32/0.47 rsquarefree:
% 0.32/0.47 !>[A: $tType] : ( ( poly @ A ) > $o ) ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_Polynomial_Osynthetic__div,type,
% 0.32/0.47 synthetic_div:
% 0.32/0.47 !>[A: $tType] : ( ( poly @ A ) > A > ( poly @ A ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_Rat_Ofield__char__0__class_ORats,type,
% 0.32/0.47 field_char_0_Rats:
% 0.32/0.47 !>[A: $tType] : ( set @ A ) ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_Rings_Odvd__class_Odvd,type,
% 0.32/0.47 dvd_dvd:
% 0.32/0.47 !>[A: $tType] : ( A > A > $o ) ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_Set_OCollect,type,
% 0.32/0.47 collect:
% 0.32/0.47 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_member,type,
% 0.32/0.47 member:
% 0.32/0.47 !>[A: $tType] : ( A > ( set @ A ) > $o ) ).
% 0.32/0.47
% 0.32/0.47 thf(sy_v_n____,type,
% 0.32/0.47 n: nat ).
% 0.32/0.47
% 0.32/0.47 thf(sy_v_p____,type,
% 0.32/0.47 p: poly @ real ).
% 0.32/0.47
% 0.32/0.47 thf(sy_v_x,type,
% 0.32/0.47 x: real ).
% 0.32/0.47
% 0.32/0.47 % Relevant facts (252)
% 0.32/0.47 thf(fact_0_poly__roots__finite,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( ( comm_ring_1 @ A )
% 0.32/0.47 & ( ring_n68954251visors @ A ) )
% 0.32/0.47 => ! [P: poly @ A] :
% 0.32/0.47 ( ( P
% 0.32/0.47 != ( zero_zero @ ( poly @ A ) ) )
% 0.32/0.47 => ( finite_finite2 @ A
% 0.32/0.47 @ ( collect @ A
% 0.32/0.47 @ ^ [X: A] :
% 0.32/0.47 ( ( poly2 @ A @ P @ X )
% 0.32/0.47 = ( zero_zero @ A ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % poly_roots_finite
% 0.32/0.47 thf(fact_1_p_I2_J,axiom,
% 0.32/0.47 ( p
% 0.32/0.47 != ( zero_zero @ ( poly @ real ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % p(2)
% 0.32/0.47 thf(fact_2__092_060open_062pderiv_Ap_A_092_060noteq_062_A0_092_060close_062,axiom,
% 0.32/0.47 ( ( pderiv @ real @ p )
% 0.32/0.47 != ( zero_zero @ ( poly @ real ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % \<open>pderiv p \<noteq> 0\<close>
% 0.32/0.47 thf(fact_3_p_I3_J,axiom,
% 0.32/0.47 ( ( poly2 @ real @ p @ x )
% 0.32/0.47 = ( zero_zero @ real ) ) ).
% 0.32/0.47
% 0.32/0.47 % p(3)
% 0.32/0.47 thf(fact_4_poly__0,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( comm_semiring_0 @ A )
% 0.32/0.47 => ! [X2: A] :
% 0.32/0.47 ( ( poly2 @ A @ ( zero_zero @ ( poly @ A ) ) @ X2 )
% 0.32/0.47 = ( zero_zero @ A ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % poly_0
% 0.32/0.47 thf(fact_5_finite__Collect__conjI,axiom,
% 0.32/0.47 ! [A: $tType,P2: A > $o,Q: A > $o] :
% 0.32/0.47 ( ( ( finite_finite2 @ A @ ( collect @ A @ P2 ) )
% 0.32/0.47 | ( finite_finite2 @ A @ ( collect @ A @ Q ) ) )
% 0.32/0.47 => ( finite_finite2 @ A
% 0.32/0.47 @ ( collect @ A
% 0.32/0.47 @ ^ [X: A] :
% 0.32/0.47 ( ( P2 @ X )
% 0.32/0.47 & ( Q @ X ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % finite_Collect_conjI
% 0.32/0.47 thf(fact_6_finite__Collect__disjI,axiom,
% 0.32/0.47 ! [A: $tType,P2: A > $o,Q: A > $o] :
% 0.32/0.47 ( ( finite_finite2 @ A
% 0.32/0.47 @ ( collect @ A
% 0.32/0.47 @ ^ [X: A] :
% 0.32/0.47 ( ( P2 @ X )
% 0.32/0.47 | ( Q @ X ) ) ) )
% 0.32/0.47 = ( ( finite_finite2 @ A @ ( collect @ A @ P2 ) )
% 0.32/0.47 & ( finite_finite2 @ A @ ( collect @ A @ Q ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % finite_Collect_disjI
% 0.32/0.47 thf(fact_7_finite__code,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( finite_finite @ A )
% 0.32/0.47 => ( ( finite_finite2 @ A )
% 0.32/0.47 = ( ^ [A2: set @ A] : $true ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % finite_code
% 0.32/0.47 thf(fact_8_poly__all__0__iff__0,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( ( ring_char_0 @ A )
% 0.32/0.47 & ( comm_ring_1 @ A )
% 0.32/0.47 & ( ring_n68954251visors @ A ) )
% 0.32/0.47 => ! [P: poly @ A] :
% 0.32/0.47 ( ( ! [X: A] :
% 0.32/0.47 ( ( poly2 @ A @ P @ X )
% 0.32/0.47 = ( zero_zero @ A ) ) )
% 0.32/0.47 = ( P
% 0.32/0.47 = ( zero_zero @ ( poly @ A ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % poly_all_0_iff_0
% 0.32/0.47 thf(fact_9_rsquarefree__roots,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( field_char_0 @ A )
% 0.32/0.47 => ( ( rsquarefree @ A )
% 0.32/0.47 = ( ^ [P3: poly @ A] :
% 0.32/0.47 ! [A3: A] :
% 0.32/0.47 ~ ( ( ( poly2 @ A @ P3 @ A3 )
% 0.32/0.47 = ( zero_zero @ A ) )
% 0.32/0.47 & ( ( poly2 @ A @ ( pderiv @ A @ P3 ) @ A3 )
% 0.32/0.47 = ( zero_zero @ A ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % rsquarefree_roots
% 0.32/0.47 thf(fact_10_not__finite__existsD,axiom,
% 0.32/0.47 ! [A: $tType,P2: A > $o] :
% 0.32/0.47 ( ~ ( finite_finite2 @ A @ ( collect @ A @ P2 ) )
% 0.32/0.47 => ? [X_1: A] : ( P2 @ X_1 ) ) ).
% 0.32/0.47
% 0.32/0.47 % not_finite_existsD
% 0.32/0.47 thf(fact_11_pigeonhole__infinite__rel,axiom,
% 0.32/0.47 ! [B: $tType,A: $tType,A4: set @ A,B2: set @ B,R: A > B > $o] :
% 0.32/0.47 ( ~ ( finite_finite2 @ A @ A4 )
% 0.32/0.47 => ( ( finite_finite2 @ B @ B2 )
% 0.32/0.47 => ( ! [X3: A] :
% 0.32/0.47 ( ( member @ A @ X3 @ A4 )
% 0.32/0.47 => ? [Xa: B] :
% 0.32/0.47 ( ( member @ B @ Xa @ B2 )
% 0.32/0.47 & ( R @ X3 @ Xa ) ) )
% 0.32/0.47 => ? [X3: B] :
% 0.32/0.47 ( ( member @ B @ X3 @ B2 )
% 0.32/0.47 & ~ ( finite_finite2 @ A
% 0.32/0.47 @ ( collect @ A
% 0.32/0.47 @ ^ [A3: A] :
% 0.32/0.47 ( ( member @ A @ A3 @ A4 )
% 0.32/0.47 & ( R @ A3 @ X3 ) ) ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % pigeonhole_infinite_rel
% 0.32/0.47 thf(fact_12_fun__cong__unused__0,axiom,
% 0.32/0.47 ! [A: $tType,B: $tType,C: $tType] :
% 0.32/0.47 ( ( zero @ B )
% 0.32/0.47 => ! [F: ( A > B ) > C,G: C] :
% 0.32/0.47 ( ( F
% 0.32/0.47 = ( ^ [X: A > B] : G ) )
% 0.32/0.47 => ( ( F
% 0.32/0.47 @ ^ [X: A] : ( zero_zero @ B ) )
% 0.32/0.47 = G ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % fun_cong_unused_0
% 0.32/0.47 thf(fact_13_poly__eq__poly__eq__iff,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( ( ring_char_0 @ A )
% 0.32/0.47 & ( comm_ring_1 @ A )
% 0.32/0.47 & ( ring_n68954251visors @ A ) )
% 0.32/0.47 => ! [P: poly @ A,Q2: poly @ A] :
% 0.32/0.47 ( ( ( poly2 @ A @ P )
% 0.32/0.47 = ( poly2 @ A @ Q2 ) )
% 0.32/0.47 = ( P = Q2 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % poly_eq_poly_eq_iff
% 0.32/0.47 thf(fact_14_n__def,axiom,
% 0.32/0.47 ( n
% 0.32/0.47 = ( degree @ real @ p ) ) ).
% 0.32/0.47
% 0.32/0.47 % n_def
% 0.32/0.47 thf(fact_15_assms_I2_J,axiom,
% 0.32/0.47 algebraic @ real @ x ).
% 0.32/0.47
% 0.32/0.47 % assms(2)
% 0.32/0.47 thf(fact_16_irrationsl,axiom,
% 0.32/0.47 ~ ( member @ real @ x @ ( field_char_0_Rats @ real ) ) ).
% 0.32/0.47
% 0.32/0.47 % irrationsl
% 0.32/0.47 thf(fact_17_degree__0,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( zero @ A )
% 0.32/0.47 => ( ( degree @ A @ ( zero_zero @ ( poly @ A ) ) )
% 0.32/0.47 = ( zero_zero @ nat ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % degree_0
% 0.32/0.47 thf(fact_18_pderiv__0,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( ( comm_semiring_1 @ A )
% 0.32/0.47 & ( semiri1193490041visors @ A ) )
% 0.32/0.47 => ( ( pderiv @ A @ ( zero_zero @ ( poly @ A ) ) )
% 0.32/0.47 = ( zero_zero @ ( poly @ A ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % pderiv_0
% 0.32/0.47 thf(fact_19_pderiv__eq__0__iff,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( ( semiring_char_0 @ A )
% 0.32/0.47 & ( comm_semiring_1 @ A )
% 0.32/0.47 & ( semiri1193490041visors @ A ) )
% 0.32/0.47 => ! [P: poly @ A] :
% 0.32/0.47 ( ( ( pderiv @ A @ P )
% 0.32/0.47 = ( zero_zero @ ( poly @ A ) ) )
% 0.32/0.47 = ( ( degree @ A @ P )
% 0.32/0.47 = ( zero_zero @ nat ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % pderiv_eq_0_iff
% 0.32/0.47 thf(fact_20_zero__reorient,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( zero @ A )
% 0.32/0.47 => ! [X2: A] :
% 0.32/0.47 ( ( ( zero_zero @ A )
% 0.32/0.47 = X2 )
% 0.32/0.47 = ( X2
% 0.32/0.47 = ( zero_zero @ A ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % zero_reorient
% 0.32/0.47 thf(fact_21_finite,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( finite_finite @ A )
% 0.32/0.47 => ! [A4: set @ A] : ( finite_finite2 @ A @ A4 ) ) ).
% 0.32/0.47
% 0.32/0.47 % finite
% 0.32/0.47 thf(fact_22_finite__set__choice,axiom,
% 0.32/0.47 ! [B: $tType,A: $tType,A4: set @ A,P2: A > B > $o] :
% 0.32/0.47 ( ( finite_finite2 @ A @ A4 )
% 0.32/0.47 => ( ! [X3: A] :
% 0.32/0.47 ( ( member @ A @ X3 @ A4 )
% 0.32/0.47 => ? [X_12: B] : ( P2 @ X3 @ X_12 ) )
% 0.32/0.47 => ? [F2: A > B] :
% 0.32/0.47 ! [X4: A] :
% 0.32/0.47 ( ( member @ A @ X4 @ A4 )
% 0.32/0.47 => ( P2 @ X4 @ ( F2 @ X4 ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % finite_set_choice
% 0.32/0.47 thf(fact_23_is__zero__null,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( zero @ A )
% 0.32/0.47 => ( ( is_zero @ A )
% 0.32/0.47 = ( ^ [P3: poly @ A] :
% 0.32/0.47 ( P3
% 0.32/0.47 = ( zero_zero @ ( poly @ A ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % is_zero_null
% 0.32/0.47 thf(fact_24_poly__cutoff__0,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( zero @ A )
% 0.32/0.47 => ! [N: nat] :
% 0.32/0.47 ( ( poly_cutoff @ A @ N @ ( zero_zero @ ( poly @ A ) ) )
% 0.32/0.47 = ( zero_zero @ ( poly @ A ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % poly_cutoff_0
% 0.32/0.47 thf(fact_25_reflect__poly__at__0__eq__0__iff,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( comm_semiring_0 @ A )
% 0.32/0.47 => ! [P: poly @ A] :
% 0.32/0.47 ( ( ( poly2 @ A @ ( reflect_poly @ A @ P ) @ ( zero_zero @ A ) )
% 0.32/0.47 = ( zero_zero @ A ) )
% 0.32/0.47 = ( P
% 0.32/0.47 = ( zero_zero @ ( poly @ A ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % reflect_poly_at_0_eq_0_iff
% 0.32/0.47 thf(fact_26__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,
% 0.32/0.47 ~ ! [P4: poly @ real] :
% 0.32/0.47 ( ! [I: nat] : ( member @ real @ ( coeff @ real @ P4 @ I ) @ ( ring_1_Ints @ real ) )
% 0.32/0.47 => ( ( P4
% 0.32/0.47 != ( zero_zero @ ( poly @ real ) ) )
% 0.32/0.47 => ( ( poly2 @ real @ P4 @ x )
% 0.32/0.47 != ( zero_zero @ real ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % \<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>
% 0.32/0.47 thf(fact_27_synthetic__div__eq__0__iff,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( comm_semiring_0 @ A )
% 0.32/0.47 => ! [P: poly @ A,C2: A] :
% 0.32/0.47 ( ( ( synthetic_div @ A @ P @ C2 )
% 0.32/0.47 = ( zero_zero @ ( poly @ A ) ) )
% 0.32/0.47 = ( ( degree @ A @ P )
% 0.32/0.47 = ( zero_zero @ nat ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % synthetic_div_eq_0_iff
% 0.32/0.47 thf(fact_28_poly__shift__0,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( zero @ A )
% 0.32/0.47 => ! [N: nat] :
% 0.32/0.47 ( ( poly_shift @ A @ N @ ( zero_zero @ ( poly @ A ) ) )
% 0.32/0.47 = ( zero_zero @ ( poly @ A ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % poly_shift_0
% 0.32/0.47 thf(fact_29_order__root,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( idom @ A )
% 0.32/0.47 => ! [P: poly @ A,A5: A] :
% 0.32/0.47 ( ( ( poly2 @ A @ P @ A5 )
% 0.32/0.47 = ( zero_zero @ A ) )
% 0.32/0.47 = ( ( P
% 0.32/0.47 = ( zero_zero @ ( poly @ A ) ) )
% 0.32/0.47 | ( ( order2 @ A @ A5 @ P )
% 0.32/0.47 != ( zero_zero @ nat ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % order_root
% 0.32/0.47 thf(fact_30_leading__coeff__0__iff,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( zero @ A )
% 0.32/0.47 => ! [P: poly @ A] :
% 0.32/0.47 ( ( ( coeff @ A @ P @ ( degree @ A @ P ) )
% 0.32/0.47 = ( zero_zero @ A ) )
% 0.32/0.47 = ( P
% 0.32/0.47 = ( zero_zero @ ( poly @ A ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % leading_coeff_0_iff
% 0.32/0.47 thf(fact_31_divide__poly__main__0,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( idom_divide @ A )
% 0.32/0.47 => ! [R2: poly @ A,D: poly @ A,Dr: nat,N: nat] :
% 0.32/0.47 ( ( divide_poly_main @ A @ ( zero_zero @ A ) @ ( zero_zero @ ( poly @ A ) ) @ R2 @ D @ Dr @ N )
% 0.32/0.47 = ( zero_zero @ ( poly @ A ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % divide_poly_main_0
% 0.32/0.47 thf(fact_32_zero__poly_Otransfer,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( zero @ A )
% 0.32/0.47 => ( pcr_poly @ A @ A
% 0.32/0.47 @ ^ [Y: A,Z: A] : ( Y = Z )
% 0.32/0.47 @ ^ [Uu: nat] : ( zero_zero @ A )
% 0.32/0.47 @ ( zero_zero @ ( poly @ A ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % zero_poly.transfer
% 0.32/0.47 thf(fact_33_p_I1_J,axiom,
% 0.32/0.47 ! [I2: nat] : ( member @ real @ ( coeff @ real @ p @ I2 ) @ ( ring_1_Ints @ real ) ) ).
% 0.32/0.47
% 0.32/0.47 % p(1)
% 0.32/0.47 thf(fact_34_reflect__poly__0,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( zero @ A )
% 0.32/0.47 => ( ( reflect_poly @ A @ ( zero_zero @ ( poly @ A ) ) )
% 0.32/0.47 = ( zero_zero @ ( poly @ A ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % reflect_poly_0
% 0.32/0.47 thf(fact_35_synthetic__div__0,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( comm_semiring_0 @ A )
% 0.32/0.47 => ! [C2: A] :
% 0.32/0.47 ( ( synthetic_div @ A @ ( zero_zero @ ( poly @ A ) ) @ C2 )
% 0.32/0.47 = ( zero_zero @ ( poly @ A ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % synthetic_div_0
% 0.32/0.47 thf(fact_36_poly__shift__id,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( zero @ A )
% 0.32/0.47 => ( ( poly_shift @ A @ ( zero_zero @ nat ) )
% 0.32/0.47 = ( ^ [X: poly @ A] : X ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % poly_shift_id
% 0.32/0.47 thf(fact_37_coeff__0,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( zero @ A )
% 0.32/0.47 => ! [N: nat] :
% 0.32/0.47 ( ( coeff @ A @ ( zero_zero @ ( poly @ A ) ) @ N )
% 0.32/0.47 = ( zero_zero @ A ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % coeff_0
% 0.32/0.47 thf(fact_38_reflect__poly__reflect__poly,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( zero @ A )
% 0.32/0.47 => ! [P: poly @ A] :
% 0.32/0.47 ( ( ( coeff @ A @ P @ ( zero_zero @ nat ) )
% 0.32/0.47 != ( zero_zero @ A ) )
% 0.32/0.47 => ( ( reflect_poly @ A @ ( reflect_poly @ A @ P ) )
% 0.32/0.47 = P ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % reflect_poly_reflect_poly
% 0.32/0.47 thf(fact_39_coeff__0__reflect__poly,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( zero @ A )
% 0.32/0.47 => ! [P: poly @ A] :
% 0.32/0.47 ( ( coeff @ A @ ( reflect_poly @ A @ P ) @ ( zero_zero @ nat ) )
% 0.32/0.47 = ( coeff @ A @ P @ ( degree @ A @ P ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % coeff_0_reflect_poly
% 0.32/0.47 thf(fact_40_coeff__0__reflect__poly__0__iff,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( zero @ A )
% 0.32/0.47 => ! [P: poly @ A] :
% 0.32/0.47 ( ( ( coeff @ A @ ( reflect_poly @ A @ P ) @ ( zero_zero @ nat ) )
% 0.32/0.47 = ( zero_zero @ A ) )
% 0.32/0.47 = ( P
% 0.32/0.47 = ( zero_zero @ ( poly @ A ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % coeff_0_reflect_poly_0_iff
% 0.32/0.47 thf(fact_41_degree__reflect__poly__eq,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( zero @ A )
% 0.32/0.47 => ! [P: poly @ A] :
% 0.32/0.47 ( ( ( coeff @ A @ P @ ( zero_zero @ nat ) )
% 0.32/0.47 != ( zero_zero @ A ) )
% 0.32/0.47 => ( ( degree @ A @ ( reflect_poly @ A @ P ) )
% 0.32/0.47 = ( degree @ A @ P ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % degree_reflect_poly_eq
% 0.32/0.47 thf(fact_42_poly__reflect__poly__0,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( comm_semiring_0 @ A )
% 0.32/0.47 => ! [P: poly @ A] :
% 0.32/0.47 ( ( poly2 @ A @ ( reflect_poly @ A @ P ) @ ( zero_zero @ A ) )
% 0.32/0.47 = ( coeff @ A @ P @ ( degree @ A @ P ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % poly_reflect_poly_0
% 0.32/0.47 thf(fact_43_divide__poly__main_Osimps_I2_J,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( idom_divide @ A )
% 0.32/0.47 => ! [Lc: A,Q2: poly @ A,R2: poly @ A,D: poly @ A,Dr: nat] :
% 0.32/0.47 ( ( divide_poly_main @ A @ Lc @ Q2 @ R2 @ D @ Dr @ ( zero_zero @ nat ) )
% 0.32/0.47 = Q2 ) ) ).
% 0.32/0.47
% 0.32/0.47 % divide_poly_main.simps(2)
% 0.32/0.47 thf(fact_44_mem__Collect__eq,axiom,
% 0.32/0.47 ! [A: $tType,A5: A,P2: A > $o] :
% 0.32/0.47 ( ( member @ A @ A5 @ ( collect @ A @ P2 ) )
% 0.32/0.47 = ( P2 @ A5 ) ) ).
% 0.32/0.47
% 0.32/0.47 % mem_Collect_eq
% 0.32/0.47 thf(fact_45_Collect__mem__eq,axiom,
% 0.32/0.47 ! [A: $tType,A4: set @ A] :
% 0.32/0.47 ( ( collect @ A
% 0.32/0.47 @ ^ [X: A] : ( member @ A @ X @ A4 ) )
% 0.32/0.47 = A4 ) ).
% 0.32/0.47
% 0.32/0.47 % Collect_mem_eq
% 0.32/0.47 thf(fact_46_Collect__cong,axiom,
% 0.32/0.47 ! [A: $tType,P2: A > $o,Q: A > $o] :
% 0.32/0.47 ( ! [X3: A] :
% 0.32/0.47 ( ( P2 @ X3 )
% 0.32/0.47 = ( Q @ X3 ) )
% 0.32/0.47 => ( ( collect @ A @ P2 )
% 0.32/0.47 = ( collect @ A @ Q ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % Collect_cong
% 0.32/0.47 thf(fact_47_ext,axiom,
% 0.32/0.47 ! [B: $tType,A: $tType,F: A > B,G: A > B] :
% 0.32/0.47 ( ! [X3: A] :
% 0.32/0.47 ( ( F @ X3 )
% 0.32/0.47 = ( G @ X3 ) )
% 0.32/0.47 => ( F = G ) ) ).
% 0.32/0.47
% 0.32/0.47 % ext
% 0.32/0.47 thf(fact_48_poly__eqI,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( zero @ A )
% 0.32/0.47 => ! [P: poly @ A,Q2: poly @ A] :
% 0.32/0.47 ( ! [N2: nat] :
% 0.32/0.47 ( ( coeff @ A @ P @ N2 )
% 0.32/0.47 = ( coeff @ A @ Q2 @ N2 ) )
% 0.32/0.47 => ( P = Q2 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % poly_eqI
% 0.32/0.47 thf(fact_49_poly__eq__iff,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( zero @ A )
% 0.32/0.47 => ( ( ^ [Y: poly @ A,Z: poly @ A] : ( Y = Z ) )
% 0.32/0.47 = ( ^ [P3: poly @ A,Q3: poly @ A] :
% 0.32/0.47 ! [N3: nat] :
% 0.32/0.47 ( ( coeff @ A @ P3 @ N3 )
% 0.32/0.47 = ( coeff @ A @ Q3 @ N3 ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % poly_eq_iff
% 0.32/0.47 thf(fact_50_coeff__inject,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( zero @ A )
% 0.32/0.47 => ! [X2: poly @ A,Y2: poly @ A] :
% 0.32/0.47 ( ( ( coeff @ A @ X2 )
% 0.32/0.47 = ( coeff @ A @ Y2 ) )
% 0.32/0.47 = ( X2 = Y2 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % coeff_inject
% 0.32/0.47 thf(fact_51_algebraicE,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( field_char_0 @ A )
% 0.32/0.47 => ! [X2: A] :
% 0.32/0.47 ( ( algebraic @ A @ X2 )
% 0.32/0.47 => ~ ! [P4: poly @ A] :
% 0.32/0.47 ( ! [I: nat] : ( member @ A @ ( coeff @ A @ P4 @ I ) @ ( ring_1_Ints @ A ) )
% 0.32/0.47 => ( ( P4
% 0.32/0.47 != ( zero_zero @ ( poly @ A ) ) )
% 0.32/0.47 => ( ( poly2 @ A @ P4 @ X2 )
% 0.32/0.47 != ( zero_zero @ A ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % algebraicE
% 0.32/0.47 thf(fact_52_algebraicI,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( field_char_0 @ A )
% 0.32/0.47 => ! [P: poly @ A,X2: A] :
% 0.32/0.47 ( ! [I3: nat] : ( member @ A @ ( coeff @ A @ P @ I3 ) @ ( ring_1_Ints @ A ) )
% 0.32/0.47 => ( ( P
% 0.32/0.47 != ( zero_zero @ ( poly @ A ) ) )
% 0.32/0.47 => ( ( ( poly2 @ A @ P @ X2 )
% 0.32/0.47 = ( zero_zero @ A ) )
% 0.32/0.47 => ( algebraic @ A @ X2 ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % algebraicI
% 0.32/0.47 thf(fact_53_algebraic__def,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( field_char_0 @ A )
% 0.32/0.47 => ( ( algebraic @ A )
% 0.32/0.47 = ( ^ [X: A] :
% 0.32/0.47 ? [P3: poly @ A] :
% 0.32/0.47 ( ! [I4: nat] : ( member @ A @ ( coeff @ A @ P3 @ I4 ) @ ( ring_1_Ints @ A ) )
% 0.32/0.47 & ( P3
% 0.32/0.47 != ( zero_zero @ ( poly @ A ) ) )
% 0.32/0.47 & ( ( poly2 @ A @ P3 @ X )
% 0.32/0.47 = ( zero_zero @ A ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % algebraic_def
% 0.32/0.47 thf(fact_54_algebraic__altdef,axiom,
% 0.32/0.47 ! [B: $tType] :
% 0.32/0.47 ( ( field_char_0 @ B )
% 0.32/0.47 => ( ( algebraic @ B )
% 0.32/0.47 = ( ^ [X: B] :
% 0.32/0.47 ? [P3: poly @ B] :
% 0.32/0.47 ( ! [I4: nat] : ( member @ B @ ( coeff @ B @ P3 @ I4 ) @ ( field_char_0_Rats @ B ) )
% 0.32/0.47 & ( P3
% 0.32/0.47 != ( zero_zero @ ( poly @ B ) ) )
% 0.32/0.47 & ( ( poly2 @ B @ P3 @ X )
% 0.32/0.47 = ( zero_zero @ B ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % algebraic_altdef
% 0.32/0.47 thf(fact_55_zero__poly_Orep__eq,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( zero @ A )
% 0.32/0.47 => ( ( coeff @ A @ ( zero_zero @ ( poly @ A ) ) )
% 0.32/0.47 = ( ^ [Uu: nat] : ( zero_zero @ A ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % zero_poly.rep_eq
% 0.32/0.47 thf(fact_56_poly__0__coeff__0,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( comm_semiring_0 @ A )
% 0.32/0.47 => ! [P: poly @ A] :
% 0.32/0.47 ( ( poly2 @ A @ P @ ( zero_zero @ A ) )
% 0.32/0.47 = ( coeff @ A @ P @ ( zero_zero @ nat ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % poly_0_coeff_0
% 0.32/0.47 thf(fact_57_order__0I,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( idom @ A )
% 0.32/0.47 => ! [P: poly @ A,A5: A] :
% 0.32/0.47 ( ( ( poly2 @ A @ P @ A5 )
% 0.32/0.47 != ( zero_zero @ A ) )
% 0.32/0.47 => ( ( order2 @ A @ A5 @ P )
% 0.32/0.47 = ( zero_zero @ nat ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % order_0I
% 0.32/0.47 thf(fact_58_leading__coeff__neq__0,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( zero @ A )
% 0.32/0.47 => ! [P: poly @ A] :
% 0.32/0.47 ( ( P
% 0.32/0.47 != ( zero_zero @ ( poly @ A ) ) )
% 0.32/0.47 => ( ( coeff @ A @ P @ ( degree @ A @ P ) )
% 0.32/0.47 != ( zero_zero @ A ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % leading_coeff_neq_0
% 0.32/0.47 thf(fact_59_Rats__infinite,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( field_char_0 @ A )
% 0.32/0.47 => ~ ( finite_finite2 @ A @ ( field_char_0_Rats @ A ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % Rats_infinite
% 0.32/0.47 thf(fact_60_Rats__0,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( field_char_0 @ A )
% 0.32/0.47 => ( member @ A @ ( zero_zero @ A ) @ ( field_char_0_Rats @ A ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % Rats_0
% 0.32/0.47 thf(fact_61_Ints__0,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( ring_1 @ A )
% 0.32/0.47 => ( member @ A @ ( zero_zero @ A ) @ ( ring_1_Ints @ A ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % Ints_0
% 0.32/0.47 thf(fact_62_rsquarefree__def,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( idom @ A )
% 0.32/0.47 => ( ( rsquarefree @ A )
% 0.32/0.47 = ( ^ [P3: poly @ A] :
% 0.32/0.47 ( ( P3
% 0.32/0.47 != ( zero_zero @ ( poly @ A ) ) )
% 0.32/0.47 & ! [A3: A] :
% 0.32/0.47 ( ( ( order2 @ A @ A3 @ P3 )
% 0.32/0.47 = ( zero_zero @ nat ) )
% 0.32/0.47 | ( ( order2 @ A @ A3 @ P3 )
% 0.32/0.47 = ( one_one @ nat ) ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % rsquarefree_def
% 0.32/0.47 thf(fact_63_poly__cutoff__1,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( comm_semiring_1 @ A )
% 0.32/0.47 => ! [N: nat] :
% 0.32/0.47 ( ( ( N
% 0.32/0.47 = ( zero_zero @ nat ) )
% 0.32/0.47 => ( ( poly_cutoff @ A @ N @ ( one_one @ ( poly @ A ) ) )
% 0.32/0.47 = ( zero_zero @ ( poly @ A ) ) ) )
% 0.32/0.47 & ( ( N
% 0.32/0.47 != ( zero_zero @ nat ) )
% 0.32/0.47 => ( ( poly_cutoff @ A @ N @ ( one_one @ ( poly @ A ) ) )
% 0.32/0.47 = ( one_one @ ( poly @ A ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % poly_cutoff_1
% 0.32/0.47 thf(fact_64_order__pderiv2,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( field_char_0 @ A )
% 0.32/0.47 => ! [P: poly @ A,A5: A,N: nat] :
% 0.32/0.47 ( ( ( pderiv @ A @ P )
% 0.32/0.47 != ( zero_zero @ ( poly @ A ) ) )
% 0.32/0.47 => ( ( ( order2 @ A @ A5 @ P )
% 0.32/0.47 != ( zero_zero @ nat ) )
% 0.32/0.47 => ( ( ( order2 @ A @ A5 @ ( pderiv @ A @ P ) )
% 0.32/0.47 = N )
% 0.32/0.47 = ( ( order2 @ A @ A5 @ P )
% 0.32/0.47 = ( suc @ N ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % order_pderiv2
% 0.32/0.47 thf(fact_65_order__pderiv,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( field_char_0 @ A )
% 0.32/0.47 => ! [P: poly @ A,A5: A] :
% 0.32/0.47 ( ( ( pderiv @ A @ P )
% 0.32/0.47 != ( zero_zero @ ( poly @ A ) ) )
% 0.32/0.47 => ( ( ( order2 @ A @ A5 @ P )
% 0.32/0.47 != ( zero_zero @ nat ) )
% 0.32/0.47 => ( ( order2 @ A @ A5 @ P )
% 0.32/0.47 = ( suc @ ( order2 @ A @ A5 @ ( pderiv @ A @ P ) ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % order_pderiv
% 0.32/0.47 thf(fact_66_cr__poly__def,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( zero @ A )
% 0.32/0.47 => ( ( cr_poly @ A )
% 0.32/0.47 = ( ^ [X: nat > A,Y3: poly @ A] :
% 0.32/0.47 ( X
% 0.32/0.47 = ( coeff @ A @ Y3 ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % cr_poly_def
% 0.32/0.47 thf(fact_67_dvd__pderiv__iff,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( ( semiring_char_0 @ A )
% 0.32/0.47 & ( comm_semiring_1 @ A )
% 0.32/0.47 & ( semiri1193490041visors @ A ) )
% 0.32/0.47 => ! [P: poly @ A] :
% 0.32/0.47 ( ( dvd_dvd @ ( poly @ A ) @ P @ ( pderiv @ A @ P ) )
% 0.32/0.47 = ( ( degree @ A @ P )
% 0.32/0.47 = ( zero_zero @ nat ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % dvd_pderiv_iff
% 0.32/0.47 thf(fact_68_reflect__poly__1,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( comm_semiring_1 @ A )
% 0.32/0.47 => ( ( reflect_poly @ A @ ( one_one @ ( poly @ A ) ) )
% 0.32/0.47 = ( one_one @ ( poly @ A ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % reflect_poly_1
% 0.32/0.47 thf(fact_69_degree__1,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( comm_semiring_1 @ A )
% 0.32/0.47 => ( ( degree @ A @ ( one_one @ ( poly @ A ) ) )
% 0.32/0.47 = ( zero_zero @ nat ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % degree_1
% 0.32/0.47 thf(fact_70_poly__1,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( comm_semiring_1 @ A )
% 0.32/0.47 => ! [X2: A] :
% 0.32/0.47 ( ( poly2 @ A @ ( one_one @ ( poly @ A ) ) @ X2 )
% 0.32/0.47 = ( one_one @ A ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % poly_1
% 0.32/0.47 thf(fact_71_pderiv__1,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( ( comm_semiring_1 @ A )
% 0.32/0.47 & ( semiri1193490041visors @ A ) )
% 0.32/0.47 => ( ( pderiv @ A @ ( one_one @ ( poly @ A ) ) )
% 0.32/0.47 = ( zero_zero @ ( poly @ A ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % pderiv_1
% 0.32/0.47 thf(fact_72_order__1__eq__0,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( idom @ A )
% 0.32/0.47 => ! [X2: A] :
% 0.32/0.47 ( ( order2 @ A @ X2 @ ( one_one @ ( poly @ A ) ) )
% 0.32/0.47 = ( zero_zero @ nat ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % order_1_eq_0
% 0.32/0.47 thf(fact_73_lead__coeff__1,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( comm_semiring_1 @ A )
% 0.32/0.47 => ( ( coeff @ A @ ( one_one @ ( poly @ A ) ) @ ( degree @ A @ ( one_one @ ( poly @ A ) ) ) )
% 0.32/0.47 = ( one_one @ A ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % lead_coeff_1
% 0.32/0.47 thf(fact_74_one__reorient,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( one @ A )
% 0.32/0.47 => ! [X2: A] :
% 0.32/0.47 ( ( ( one_one @ A )
% 0.32/0.47 = X2 )
% 0.32/0.47 = ( X2
% 0.32/0.47 = ( one_one @ A ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % one_reorient
% 0.32/0.47 thf(fact_75_Ints__1,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( ring_1 @ A )
% 0.32/0.47 => ( member @ A @ ( one_one @ A ) @ ( ring_1_Ints @ A ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % Ints_1
% 0.32/0.47 thf(fact_76_Rats__1,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( field_char_0 @ A )
% 0.32/0.47 => ( member @ A @ ( one_one @ A ) @ ( field_char_0_Rats @ A ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % Rats_1
% 0.32/0.47 thf(fact_77_is__unit__iff__degree,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( field @ A )
% 0.32/0.47 => ! [P: poly @ A] :
% 0.32/0.47 ( ( P
% 0.32/0.47 != ( zero_zero @ ( poly @ A ) ) )
% 0.32/0.47 => ( ( dvd_dvd @ ( poly @ A ) @ P @ ( one_one @ ( poly @ A ) ) )
% 0.32/0.47 = ( ( degree @ A @ P )
% 0.32/0.47 = ( zero_zero @ nat ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % is_unit_iff_degree
% 0.32/0.47 thf(fact_78_not__dvd__pderiv,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( ( semiring_char_0 @ A )
% 0.32/0.47 & ( comm_semiring_1 @ A )
% 0.32/0.47 & ( semiri1193490041visors @ A ) )
% 0.32/0.47 => ! [P: poly @ A] :
% 0.32/0.47 ( ( ( degree @ A @ P )
% 0.32/0.47 != ( zero_zero @ nat ) )
% 0.32/0.47 => ~ ( dvd_dvd @ ( poly @ A ) @ P @ ( pderiv @ A @ P ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % not_dvd_pderiv
% 0.32/0.47 thf(fact_79_poly_Opcr__cr__eq,axiom,
% 0.32/0.47 ! [D2: $tType] :
% 0.32/0.47 ( ( zero @ D2 )
% 0.32/0.47 => ( ( pcr_poly @ D2 @ D2
% 0.32/0.47 @ ^ [Y: D2,Z: D2] : ( Y = Z ) )
% 0.32/0.47 = ( cr_poly @ D2 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % poly.pcr_cr_eq
% 0.32/0.47 thf(fact_80_poly__shift__1,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( comm_semiring_1 @ A )
% 0.32/0.47 => ! [N: nat] :
% 0.32/0.47 ( ( ( N
% 0.32/0.47 = ( zero_zero @ nat ) )
% 0.32/0.47 => ( ( poly_shift @ A @ N @ ( one_one @ ( poly @ A ) ) )
% 0.32/0.47 = ( one_one @ ( poly @ A ) ) ) )
% 0.32/0.47 & ( ( N
% 0.32/0.47 != ( zero_zero @ nat ) )
% 0.32/0.47 => ( ( poly_shift @ A @ N @ ( one_one @ ( poly @ A ) ) )
% 0.32/0.47 = ( zero_zero @ ( poly @ A ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % poly_shift_1
% 0.32/0.47 thf(fact_81_dvd__0__left__iff,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( comm_semiring_1 @ A )
% 0.32/0.47 => ! [A5: A] :
% 0.32/0.47 ( ( dvd_dvd @ A @ ( zero_zero @ A ) @ A5 )
% 0.32/0.47 = ( A5
% 0.32/0.47 = ( zero_zero @ A ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % dvd_0_left_iff
% 0.32/0.47 thf(fact_82_dvd__0__right,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( comm_semiring_1 @ A )
% 0.32/0.47 => ! [A5: A] : ( dvd_dvd @ A @ A5 @ ( zero_zero @ A ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % dvd_0_right
% 0.32/0.47 thf(fact_83_One__nat__def,axiom,
% 0.32/0.47 ( ( one_one @ nat )
% 0.32/0.47 = ( suc @ ( zero_zero @ nat ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % One_nat_def
% 0.32/0.47 thf(fact_84_not__is__unit__0,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( algebraic_semidom @ A )
% 0.32/0.47 => ~ ( dvd_dvd @ A @ ( zero_zero @ A ) @ ( one_one @ A ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % not_is_unit_0
% 0.32/0.47 thf(fact_85_nat_Oinject,axiom,
% 0.32/0.47 ! [X22: nat,Y22: nat] :
% 0.32/0.47 ( ( ( suc @ X22 )
% 0.32/0.47 = ( suc @ Y22 ) )
% 0.32/0.47 = ( X22 = Y22 ) ) ).
% 0.32/0.47
% 0.32/0.47 % nat.inject
% 0.32/0.47 thf(fact_86_old_Onat_Oinject,axiom,
% 0.32/0.47 ! [Nat: nat,Nat2: nat] :
% 0.32/0.47 ( ( ( suc @ Nat )
% 0.32/0.47 = ( suc @ Nat2 ) )
% 0.32/0.47 = ( Nat = Nat2 ) ) ).
% 0.32/0.47
% 0.32/0.47 % old.nat.inject
% 0.32/0.47 thf(fact_87_nat__dvd__1__iff__1,axiom,
% 0.32/0.47 ! [M: nat] :
% 0.32/0.47 ( ( dvd_dvd @ nat @ M @ ( one_one @ nat ) )
% 0.32/0.47 = ( M
% 0.32/0.47 = ( one_one @ nat ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % nat_dvd_1_iff_1
% 0.32/0.47 thf(fact_88_dvd__1__iff__1,axiom,
% 0.32/0.47 ! [M: nat] :
% 0.32/0.47 ( ( dvd_dvd @ nat @ M @ ( suc @ ( zero_zero @ nat ) ) )
% 0.32/0.47 = ( M
% 0.32/0.47 = ( suc @ ( zero_zero @ nat ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % dvd_1_iff_1
% 0.32/0.47 thf(fact_89_dvd__1__left,axiom,
% 0.32/0.47 ! [K: nat] : ( dvd_dvd @ nat @ ( suc @ ( zero_zero @ nat ) ) @ K ) ).
% 0.32/0.47
% 0.32/0.47 % dvd_1_left
% 0.32/0.47 thf(fact_90_dvd__trans,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( comm_monoid_mult @ A )
% 0.32/0.47 => ! [A5: A,B3: A,C2: A] :
% 0.32/0.47 ( ( dvd_dvd @ A @ A5 @ B3 )
% 0.32/0.47 => ( ( dvd_dvd @ A @ B3 @ C2 )
% 0.32/0.47 => ( dvd_dvd @ A @ A5 @ C2 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % dvd_trans
% 0.32/0.47 thf(fact_91_dvd__refl,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( comm_monoid_mult @ A )
% 0.32/0.47 => ! [A5: A] : ( dvd_dvd @ A @ A5 @ A5 ) ) ).
% 0.32/0.47
% 0.32/0.47 % dvd_refl
% 0.32/0.47 thf(fact_92_n__not__Suc__n,axiom,
% 0.32/0.47 ! [N: nat] :
% 0.32/0.47 ( N
% 0.32/0.47 != ( suc @ N ) ) ).
% 0.32/0.47
% 0.32/0.47 % n_not_Suc_n
% 0.32/0.47 thf(fact_93_Suc__inject,axiom,
% 0.32/0.47 ! [X2: nat,Y2: nat] :
% 0.32/0.47 ( ( ( suc @ X2 )
% 0.32/0.47 = ( suc @ Y2 ) )
% 0.32/0.47 => ( X2 = Y2 ) ) ).
% 0.32/0.47
% 0.32/0.47 % Suc_inject
% 0.32/0.47 thf(fact_94_zero__neq__one,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( zero_neq_one @ A )
% 0.32/0.47 => ( ( zero_zero @ A )
% 0.32/0.47 != ( one_one @ A ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % zero_neq_one
% 0.32/0.47 thf(fact_95_dvd__0__left,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( comm_semiring_1 @ A )
% 0.32/0.47 => ! [A5: A] :
% 0.32/0.47 ( ( dvd_dvd @ A @ ( zero_zero @ A ) @ A5 )
% 0.32/0.47 => ( A5
% 0.32/0.47 = ( zero_zero @ A ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % dvd_0_left
% 0.32/0.47 thf(fact_96_dvd__unit__imp__unit,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( algebraic_semidom @ A )
% 0.32/0.47 => ! [A5: A,B3: A] :
% 0.32/0.47 ( ( dvd_dvd @ A @ A5 @ B3 )
% 0.32/0.47 => ( ( dvd_dvd @ A @ B3 @ ( one_one @ A ) )
% 0.32/0.47 => ( dvd_dvd @ A @ A5 @ ( one_one @ A ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % dvd_unit_imp_unit
% 0.32/0.47 thf(fact_97_unit__imp__dvd,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( algebraic_semidom @ A )
% 0.32/0.47 => ! [B3: A,A5: A] :
% 0.32/0.47 ( ( dvd_dvd @ A @ B3 @ ( one_one @ A ) )
% 0.32/0.47 => ( dvd_dvd @ A @ B3 @ A5 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % unit_imp_dvd
% 0.32/0.47 thf(fact_98_one__dvd,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( comm_monoid_mult @ A )
% 0.32/0.47 => ! [A5: A] : ( dvd_dvd @ A @ ( one_one @ A ) @ A5 ) ) ).
% 0.32/0.47
% 0.32/0.47 % one_dvd
% 0.32/0.47 thf(fact_99_not0__implies__Suc,axiom,
% 0.32/0.47 ! [N: nat] :
% 0.32/0.47 ( ( N
% 0.32/0.47 != ( zero_zero @ nat ) )
% 0.32/0.47 => ? [M2: nat] :
% 0.32/0.47 ( N
% 0.32/0.47 = ( suc @ M2 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % not0_implies_Suc
% 0.32/0.47 thf(fact_100_old_Onat_Oinducts,axiom,
% 0.32/0.47 ! [P2: nat > $o,Nat: nat] :
% 0.32/0.47 ( ( P2 @ ( zero_zero @ nat ) )
% 0.32/0.47 => ( ! [Nat3: nat] :
% 0.32/0.47 ( ( P2 @ Nat3 )
% 0.32/0.47 => ( P2 @ ( suc @ Nat3 ) ) )
% 0.32/0.47 => ( P2 @ Nat ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % old.nat.inducts
% 0.32/0.47 thf(fact_101_old_Onat_Oexhaust,axiom,
% 0.32/0.47 ! [Y2: nat] :
% 0.32/0.47 ( ( Y2
% 0.32/0.47 != ( zero_zero @ nat ) )
% 0.32/0.47 => ~ ! [Nat3: nat] :
% 0.32/0.47 ( Y2
% 0.32/0.47 != ( suc @ Nat3 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % old.nat.exhaust
% 0.32/0.47 thf(fact_102_Zero__not__Suc,axiom,
% 0.32/0.47 ! [M: nat] :
% 0.32/0.47 ( ( zero_zero @ nat )
% 0.32/0.47 != ( suc @ M ) ) ).
% 0.32/0.47
% 0.32/0.47 % Zero_not_Suc
% 0.32/0.47 thf(fact_103_Zero__neq__Suc,axiom,
% 0.32/0.47 ! [M: nat] :
% 0.32/0.47 ( ( zero_zero @ nat )
% 0.32/0.47 != ( suc @ M ) ) ).
% 0.32/0.47
% 0.32/0.47 % Zero_neq_Suc
% 0.32/0.47 thf(fact_104_Suc__neq__Zero,axiom,
% 0.32/0.47 ! [M: nat] :
% 0.32/0.47 ( ( suc @ M )
% 0.32/0.47 != ( zero_zero @ nat ) ) ).
% 0.32/0.47
% 0.32/0.47 % Suc_neq_Zero
% 0.32/0.47 thf(fact_105_zero__induct,axiom,
% 0.32/0.47 ! [P2: nat > $o,K: nat] :
% 0.32/0.47 ( ( P2 @ K )
% 0.32/0.47 => ( ! [N2: nat] :
% 0.32/0.47 ( ( P2 @ ( suc @ N2 ) )
% 0.32/0.47 => ( P2 @ N2 ) )
% 0.32/0.47 => ( P2 @ ( zero_zero @ nat ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % zero_induct
% 0.32/0.47 thf(fact_106_diff__induct,axiom,
% 0.32/0.47 ! [P2: nat > nat > $o,M: nat,N: nat] :
% 0.32/0.47 ( ! [X3: nat] : ( P2 @ X3 @ ( zero_zero @ nat ) )
% 0.32/0.47 => ( ! [Y4: nat] : ( P2 @ ( zero_zero @ nat ) @ ( suc @ Y4 ) )
% 0.32/0.47 => ( ! [X3: nat,Y4: nat] :
% 0.32/0.47 ( ( P2 @ X3 @ Y4 )
% 0.32/0.47 => ( P2 @ ( suc @ X3 ) @ ( suc @ Y4 ) ) )
% 0.32/0.47 => ( P2 @ M @ N ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % diff_induct
% 0.32/0.47 thf(fact_107_nat__induct,axiom,
% 0.32/0.47 ! [P2: nat > $o,N: nat] :
% 0.32/0.47 ( ( P2 @ ( zero_zero @ nat ) )
% 0.32/0.47 => ( ! [N2: nat] :
% 0.32/0.47 ( ( P2 @ N2 )
% 0.32/0.47 => ( P2 @ ( suc @ N2 ) ) )
% 0.32/0.47 => ( P2 @ N ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % nat_induct
% 0.32/0.47 thf(fact_108_nat_OdiscI,axiom,
% 0.32/0.47 ! [Nat: nat,X22: nat] :
% 0.32/0.47 ( ( Nat
% 0.32/0.47 = ( suc @ X22 ) )
% 0.32/0.47 => ( Nat
% 0.32/0.47 != ( zero_zero @ nat ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % nat.discI
% 0.32/0.47 thf(fact_109_old_Onat_Odistinct_I1_J,axiom,
% 0.32/0.47 ! [Nat2: nat] :
% 0.32/0.47 ( ( zero_zero @ nat )
% 0.32/0.47 != ( suc @ Nat2 ) ) ).
% 0.32/0.47
% 0.32/0.47 % old.nat.distinct(1)
% 0.32/0.47 thf(fact_110_old_Onat_Odistinct_I2_J,axiom,
% 0.32/0.47 ! [Nat4: nat] :
% 0.32/0.47 ( ( suc @ Nat4 )
% 0.32/0.47 != ( zero_zero @ nat ) ) ).
% 0.32/0.47
% 0.32/0.47 % old.nat.distinct(2)
% 0.32/0.47 thf(fact_111_nat_Odistinct_I1_J,axiom,
% 0.32/0.47 ! [X22: nat] :
% 0.32/0.47 ( ( zero_zero @ nat )
% 0.32/0.47 != ( suc @ X22 ) ) ).
% 0.32/0.47
% 0.32/0.47 % nat.distinct(1)
% 0.32/0.47 thf(fact_112_algebraic__int_Oinducts,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( field @ A )
% 0.32/0.47 => ! [X2: A,P2: A > $o] :
% 0.32/0.47 ( ( algebraic_int @ A @ X2 )
% 0.32/0.47 => ( ! [P4: poly @ A,X3: A] :
% 0.32/0.47 ( ( ( coeff @ A @ P4 @ ( degree @ A @ P4 ) )
% 0.32/0.47 = ( one_one @ A ) )
% 0.32/0.47 => ( ! [I: nat] : ( member @ A @ ( coeff @ A @ P4 @ I ) @ ( ring_1_Ints @ A ) )
% 0.32/0.47 => ( ( ( poly2 @ A @ P4 @ X3 )
% 0.32/0.47 = ( zero_zero @ A ) )
% 0.32/0.47 => ( P2 @ X3 ) ) ) )
% 0.32/0.47 => ( P2 @ X2 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % algebraic_int.inducts
% 0.32/0.47 thf(fact_113_algebraic__int_Ointros,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( field @ A )
% 0.32/0.47 => ! [P: poly @ A,X2: A] :
% 0.32/0.47 ( ( ( coeff @ A @ P @ ( degree @ A @ P ) )
% 0.32/0.47 = ( one_one @ A ) )
% 0.32/0.47 => ( ! [I3: nat] : ( member @ A @ ( coeff @ A @ P @ I3 ) @ ( ring_1_Ints @ A ) )
% 0.32/0.47 => ( ( ( poly2 @ A @ P @ X2 )
% 0.32/0.47 = ( zero_zero @ A ) )
% 0.32/0.47 => ( algebraic_int @ A @ X2 ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % algebraic_int.intros
% 0.32/0.47 thf(fact_114_algebraic__int_Osimps,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( field @ A )
% 0.32/0.47 => ( ( algebraic_int @ A )
% 0.32/0.47 = ( ^ [A3: A] :
% 0.32/0.47 ? [P3: poly @ A,X: A] :
% 0.32/0.47 ( ( A3 = X )
% 0.32/0.47 & ( ( coeff @ A @ P3 @ ( degree @ A @ P3 ) )
% 0.32/0.47 = ( one_one @ A ) )
% 0.32/0.47 & ! [I4: nat] : ( member @ A @ ( coeff @ A @ P3 @ I4 ) @ ( ring_1_Ints @ A ) )
% 0.32/0.47 & ( ( poly2 @ A @ P3 @ X )
% 0.32/0.47 = ( zero_zero @ A ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % algebraic_int.simps
% 0.32/0.47 thf(fact_115_algebraic__int_Ocases,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( field @ A )
% 0.32/0.47 => ! [A5: A] :
% 0.32/0.47 ( ( algebraic_int @ A @ A5 )
% 0.32/0.47 => ~ ! [P4: poly @ A] :
% 0.32/0.47 ( ( ( coeff @ A @ P4 @ ( degree @ A @ P4 ) )
% 0.32/0.47 = ( one_one @ A ) )
% 0.32/0.47 => ( ! [I: nat] : ( member @ A @ ( coeff @ A @ P4 @ I ) @ ( ring_1_Ints @ A ) )
% 0.32/0.47 => ( ( poly2 @ A @ P4 @ A5 )
% 0.32/0.47 != ( zero_zero @ A ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % algebraic_int.cases
% 0.32/0.47 thf(fact_116_exists__least__lemma,axiom,
% 0.32/0.47 ! [P2: nat > $o] :
% 0.32/0.47 ( ~ ( P2 @ ( zero_zero @ nat ) )
% 0.32/0.47 => ( ? [X_12: nat] : ( P2 @ X_12 )
% 0.32/0.47 => ? [N2: nat] :
% 0.32/0.47 ( ~ ( P2 @ N2 )
% 0.32/0.47 & ( P2 @ ( suc @ N2 ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % exists_least_lemma
% 0.32/0.47 thf(fact_117_dependent__nat__choice,axiom,
% 0.32/0.47 ! [A: $tType,P2: nat > A > $o,Q: nat > A > A > $o] :
% 0.32/0.47 ( ? [X_12: A] : ( P2 @ ( zero_zero @ nat ) @ X_12 )
% 0.32/0.47 => ( ! [X3: A,N2: nat] :
% 0.32/0.47 ( ( P2 @ N2 @ X3 )
% 0.32/0.47 => ? [Y5: A] :
% 0.32/0.47 ( ( P2 @ ( suc @ N2 ) @ Y5 )
% 0.32/0.47 & ( Q @ N2 @ X3 @ Y5 ) ) )
% 0.32/0.47 => ? [F2: nat > A] :
% 0.32/0.47 ! [N4: nat] :
% 0.32/0.47 ( ( P2 @ N4 @ ( F2 @ N4 ) )
% 0.32/0.47 & ( Q @ N4 @ ( F2 @ N4 ) @ ( F2 @ ( suc @ N4 ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % dependent_nat_choice
% 0.32/0.47 thf(fact_118_algebraic__int__0,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( field @ A )
% 0.32/0.47 => ( algebraic_int @ A @ ( zero_zero @ A ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % algebraic_int_0
% 0.32/0.47 thf(fact_119_algebraic__int__1,axiom,
% 0.32/0.47 ! [B: $tType] :
% 0.32/0.47 ( ( field @ B )
% 0.32/0.47 => ( algebraic_int @ B @ ( one_one @ B ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % algebraic_int_1
% 0.32/0.47 thf(fact_120_dvd__antisym,axiom,
% 0.32/0.47 ! [M: nat,N: nat] :
% 0.32/0.47 ( ( dvd_dvd @ nat @ M @ N )
% 0.32/0.47 => ( ( dvd_dvd @ nat @ N @ M )
% 0.32/0.47 => ( M = N ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % dvd_antisym
% 0.32/0.47 thf(fact_121_int__imp__algebraic__int,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( field @ A )
% 0.32/0.47 => ! [X2: A] :
% 0.32/0.47 ( ( member @ A @ X2 @ ( ring_1_Ints @ A ) )
% 0.32/0.47 => ( algebraic_int @ A @ X2 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % int_imp_algebraic_int
% 0.32/0.47 thf(fact_122_algebraic__int__imp__algebraic,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( field_char_0 @ A )
% 0.32/0.47 => ! [X2: A] :
% 0.32/0.47 ( ( algebraic_int @ A @ X2 )
% 0.32/0.47 => ( algebraic @ A @ X2 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % algebraic_int_imp_algebraic
% 0.32/0.47 thf(fact_123_rational__algebraic__int__is__int,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( field_char_0 @ A )
% 0.32/0.47 => ! [X2: A] :
% 0.32/0.47 ( ( algebraic_int @ A @ X2 )
% 0.32/0.47 => ( ( member @ A @ X2 @ ( field_char_0_Rats @ A ) )
% 0.32/0.47 => ( member @ A @ X2 @ ( ring_1_Ints @ A ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % rational_algebraic_int_is_int
% 0.32/0.47 thf(fact_124_algebraic__int__inverse,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( field @ A )
% 0.32/0.47 => ! [P: poly @ A,X2: A] :
% 0.32/0.47 ( ( ( poly2 @ A @ P @ X2 )
% 0.32/0.47 = ( zero_zero @ A ) )
% 0.32/0.47 => ( ! [I3: nat] : ( member @ A @ ( coeff @ A @ P @ I3 ) @ ( ring_1_Ints @ A ) )
% 0.32/0.47 => ( ( ( coeff @ A @ P @ ( zero_zero @ nat ) )
% 0.32/0.47 = ( one_one @ A ) )
% 0.32/0.47 => ( algebraic_int @ A @ ( inverse_inverse @ A @ X2 ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % algebraic_int_inverse
% 0.32/0.47 thf(fact_125_algebraic__int__root,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( field @ A )
% 0.32/0.47 => ! [Y2: A,P: poly @ A,X2: A] :
% 0.32/0.47 ( ( algebraic_int @ A @ Y2 )
% 0.32/0.47 => ( ( ( poly2 @ A @ P @ X2 )
% 0.32/0.47 = Y2 )
% 0.32/0.47 => ( ! [I3: nat] : ( member @ A @ ( coeff @ A @ P @ I3 ) @ ( ring_1_Ints @ A ) )
% 0.32/0.47 => ( ( ( coeff @ A @ P @ ( degree @ A @ P ) )
% 0.32/0.47 = ( one_one @ A ) )
% 0.32/0.47 => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( degree @ A @ P ) )
% 0.32/0.47 => ( algebraic_int @ A @ X2 ) ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % algebraic_int_root
% 0.32/0.47 thf(fact_126_gcd__nat_Oextremum__uniqueI,axiom,
% 0.32/0.47 ! [A5: nat] :
% 0.32/0.47 ( ( dvd_dvd @ nat @ ( zero_zero @ nat ) @ A5 )
% 0.32/0.47 => ( A5
% 0.32/0.47 = ( zero_zero @ nat ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % gcd_nat.extremum_uniqueI
% 0.32/0.47 thf(fact_127_gcd__nat_Onot__eq__extremum,axiom,
% 0.32/0.47 ! [A5: nat] :
% 0.32/0.47 ( ( A5
% 0.32/0.47 != ( zero_zero @ nat ) )
% 0.32/0.47 = ( ( dvd_dvd @ nat @ A5 @ ( zero_zero @ nat ) )
% 0.32/0.47 & ( A5
% 0.32/0.47 != ( zero_zero @ nat ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % gcd_nat.not_eq_extremum
% 0.32/0.47 thf(fact_128_finite__Collect__less__nat,axiom,
% 0.32/0.47 ! [K: nat] :
% 0.32/0.47 ( finite_finite2 @ nat
% 0.32/0.47 @ ( collect @ nat
% 0.32/0.47 @ ^ [N3: nat] : ( ord_less @ nat @ N3 @ K ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % finite_Collect_less_nat
% 0.32/0.47 thf(fact_129_not__gr__zero,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( canoni770627133id_add @ A )
% 0.32/0.47 => ! [N: A] :
% 0.32/0.47 ( ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ N ) )
% 0.32/0.47 = ( N
% 0.32/0.47 = ( zero_zero @ A ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % not_gr_zero
% 0.32/0.47 thf(fact_130_bot__nat__0_Onot__eq__extremum,axiom,
% 0.32/0.47 ! [A5: nat] :
% 0.32/0.47 ( ( A5
% 0.32/0.47 != ( zero_zero @ nat ) )
% 0.32/0.47 = ( ord_less @ nat @ ( zero_zero @ nat ) @ A5 ) ) ).
% 0.32/0.47
% 0.32/0.47 % bot_nat_0.not_eq_extremum
% 0.32/0.47 thf(fact_131_less__nat__zero__code,axiom,
% 0.32/0.47 ! [N: nat] :
% 0.32/0.47 ~ ( ord_less @ nat @ N @ ( zero_zero @ nat ) ) ).
% 0.32/0.47
% 0.32/0.47 % less_nat_zero_code
% 0.32/0.47 thf(fact_132_neq0__conv,axiom,
% 0.32/0.47 ! [N: nat] :
% 0.32/0.47 ( ( N
% 0.32/0.47 != ( zero_zero @ nat ) )
% 0.32/0.47 = ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ).
% 0.32/0.47
% 0.32/0.47 % neq0_conv
% 0.32/0.47 thf(fact_133_lessI,axiom,
% 0.32/0.47 ! [N: nat] : ( ord_less @ nat @ N @ ( suc @ N ) ) ).
% 0.32/0.47
% 0.32/0.47 % lessI
% 0.32/0.47 thf(fact_134_Suc__mono,axiom,
% 0.32/0.47 ! [M: nat,N: nat] :
% 0.32/0.47 ( ( ord_less @ nat @ M @ N )
% 0.32/0.47 => ( ord_less @ nat @ ( suc @ M ) @ ( suc @ N ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % Suc_mono
% 0.32/0.47 thf(fact_135_Suc__less__eq,axiom,
% 0.32/0.47 ! [M: nat,N: nat] :
% 0.32/0.47 ( ( ord_less @ nat @ ( suc @ M ) @ ( suc @ N ) )
% 0.32/0.47 = ( ord_less @ nat @ M @ N ) ) ).
% 0.32/0.47
% 0.32/0.47 % Suc_less_eq
% 0.32/0.47 thf(fact_136_zero__less__Suc,axiom,
% 0.32/0.47 ! [N: nat] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( suc @ N ) ) ).
% 0.32/0.47
% 0.32/0.47 % zero_less_Suc
% 0.32/0.47 thf(fact_137_less__Suc0,axiom,
% 0.32/0.47 ! [N: nat] :
% 0.32/0.47 ( ( ord_less @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) )
% 0.32/0.47 = ( N
% 0.32/0.47 = ( zero_zero @ nat ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % less_Suc0
% 0.32/0.47 thf(fact_138_less__one,axiom,
% 0.32/0.47 ! [N: nat] :
% 0.32/0.47 ( ( ord_less @ nat @ N @ ( one_one @ nat ) )
% 0.32/0.47 = ( N
% 0.32/0.47 = ( zero_zero @ nat ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % less_one
% 0.32/0.47 thf(fact_139_zero__less__iff__neq__zero,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( canoni770627133id_add @ A )
% 0.32/0.47 => ! [N: A] :
% 0.32/0.47 ( ( ord_less @ A @ ( zero_zero @ A ) @ N )
% 0.32/0.47 = ( N
% 0.32/0.47 != ( zero_zero @ A ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % zero_less_iff_neq_zero
% 0.32/0.47 thf(fact_140_gr__implies__not__zero,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( canoni770627133id_add @ A )
% 0.32/0.47 => ! [M: A,N: A] :
% 0.32/0.47 ( ( ord_less @ A @ M @ N )
% 0.32/0.47 => ( N
% 0.32/0.47 != ( zero_zero @ A ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % gr_implies_not_zero
% 0.32/0.47 thf(fact_141_not__less__zero,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( canoni770627133id_add @ A )
% 0.32/0.47 => ! [N: A] :
% 0.32/0.47 ~ ( ord_less @ A @ N @ ( zero_zero @ A ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % not_less_zero
% 0.32/0.47 thf(fact_142_gr__zeroI,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( canoni770627133id_add @ A )
% 0.32/0.47 => ! [N: A] :
% 0.32/0.47 ( ( N
% 0.32/0.47 != ( zero_zero @ A ) )
% 0.32/0.47 => ( ord_less @ A @ ( zero_zero @ A ) @ N ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % gr_zeroI
% 0.32/0.47 thf(fact_143_gr0I,axiom,
% 0.32/0.47 ! [N: nat] :
% 0.32/0.47 ( ( N
% 0.32/0.47 != ( zero_zero @ nat ) )
% 0.32/0.47 => ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ).
% 0.32/0.47
% 0.32/0.47 % gr0I
% 0.32/0.47 thf(fact_144_not__gr0,axiom,
% 0.32/0.47 ! [N: nat] :
% 0.32/0.47 ( ( ~ ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) )
% 0.32/0.47 = ( N
% 0.32/0.47 = ( zero_zero @ nat ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % not_gr0
% 0.32/0.47 thf(fact_145_not__less0,axiom,
% 0.32/0.47 ! [N: nat] :
% 0.32/0.47 ~ ( ord_less @ nat @ N @ ( zero_zero @ nat ) ) ).
% 0.32/0.47
% 0.32/0.47 % not_less0
% 0.32/0.47 thf(fact_146_less__zeroE,axiom,
% 0.32/0.47 ! [N: nat] :
% 0.32/0.47 ~ ( ord_less @ nat @ N @ ( zero_zero @ nat ) ) ).
% 0.32/0.47
% 0.32/0.47 % less_zeroE
% 0.32/0.47 thf(fact_147_gr__implies__not0,axiom,
% 0.32/0.47 ! [M: nat,N: nat] :
% 0.32/0.47 ( ( ord_less @ nat @ M @ N )
% 0.32/0.47 => ( N
% 0.32/0.47 != ( zero_zero @ nat ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % gr_implies_not0
% 0.32/0.47 thf(fact_148_infinite__descent0,axiom,
% 0.32/0.47 ! [P2: nat > $o,N: nat] :
% 0.32/0.47 ( ( P2 @ ( zero_zero @ nat ) )
% 0.32/0.47 => ( ! [N2: nat] :
% 0.32/0.47 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
% 0.32/0.47 => ( ~ ( P2 @ N2 )
% 0.32/0.47 => ? [M3: nat] :
% 0.32/0.47 ( ( ord_less @ nat @ M3 @ N2 )
% 0.32/0.47 & ~ ( P2 @ M3 ) ) ) )
% 0.32/0.47 => ( P2 @ N ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % infinite_descent0
% 0.32/0.47 thf(fact_149_bot__nat__0_Oextremum__strict,axiom,
% 0.32/0.47 ! [A5: nat] :
% 0.32/0.47 ~ ( ord_less @ nat @ A5 @ ( zero_zero @ nat ) ) ).
% 0.32/0.47
% 0.32/0.47 % bot_nat_0.extremum_strict
% 0.32/0.47 thf(fact_150_infinite__descent0__measure,axiom,
% 0.32/0.47 ! [A: $tType,V: A > nat,P2: A > $o,X2: A] :
% 0.32/0.47 ( ! [X3: A] :
% 0.32/0.47 ( ( ( V @ X3 )
% 0.32/0.47 = ( zero_zero @ nat ) )
% 0.32/0.47 => ( P2 @ X3 ) )
% 0.32/0.47 => ( ! [X3: A] :
% 0.32/0.47 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( V @ X3 ) )
% 0.32/0.47 => ( ~ ( P2 @ X3 )
% 0.32/0.47 => ? [Y5: A] :
% 0.32/0.47 ( ( ord_less @ nat @ ( V @ Y5 ) @ ( V @ X3 ) )
% 0.32/0.47 & ~ ( P2 @ Y5 ) ) ) )
% 0.32/0.47 => ( P2 @ X2 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % infinite_descent0_measure
% 0.32/0.47 thf(fact_151_not__less__less__Suc__eq,axiom,
% 0.32/0.47 ! [N: nat,M: nat] :
% 0.32/0.47 ( ~ ( ord_less @ nat @ N @ M )
% 0.32/0.47 => ( ( ord_less @ nat @ N @ ( suc @ M ) )
% 0.32/0.47 = ( N = M ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % not_less_less_Suc_eq
% 0.32/0.47 thf(fact_152_strict__inc__induct,axiom,
% 0.32/0.47 ! [I2: nat,J: nat,P2: nat > $o] :
% 0.32/0.47 ( ( ord_less @ nat @ I2 @ J )
% 0.32/0.47 => ( ! [I3: nat] :
% 0.32/0.47 ( ( J
% 0.32/0.47 = ( suc @ I3 ) )
% 0.32/0.47 => ( P2 @ I3 ) )
% 0.32/0.47 => ( ! [I3: nat] :
% 0.32/0.47 ( ( ord_less @ nat @ I3 @ J )
% 0.32/0.47 => ( ( P2 @ ( suc @ I3 ) )
% 0.32/0.47 => ( P2 @ I3 ) ) )
% 0.32/0.47 => ( P2 @ I2 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % strict_inc_induct
% 0.32/0.47 thf(fact_153_less__Suc__induct,axiom,
% 0.32/0.47 ! [I2: nat,J: nat,P2: nat > nat > $o] :
% 0.32/0.47 ( ( ord_less @ nat @ I2 @ J )
% 0.32/0.47 => ( ! [I3: nat] : ( P2 @ I3 @ ( suc @ I3 ) )
% 0.32/0.47 => ( ! [I3: nat,J2: nat,K2: nat] :
% 0.32/0.47 ( ( ord_less @ nat @ I3 @ J2 )
% 0.32/0.47 => ( ( ord_less @ nat @ J2 @ K2 )
% 0.32/0.47 => ( ( P2 @ I3 @ J2 )
% 0.32/0.47 => ( ( P2 @ J2 @ K2 )
% 0.32/0.47 => ( P2 @ I3 @ K2 ) ) ) ) )
% 0.32/0.47 => ( P2 @ I2 @ J ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % less_Suc_induct
% 0.32/0.47 thf(fact_154_less__trans__Suc,axiom,
% 0.32/0.47 ! [I2: nat,J: nat,K: nat] :
% 0.32/0.47 ( ( ord_less @ nat @ I2 @ J )
% 0.32/0.47 => ( ( ord_less @ nat @ J @ K )
% 0.32/0.47 => ( ord_less @ nat @ ( suc @ I2 ) @ K ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % less_trans_Suc
% 0.32/0.47 thf(fact_155_Suc__less__SucD,axiom,
% 0.32/0.47 ! [M: nat,N: nat] :
% 0.32/0.47 ( ( ord_less @ nat @ ( suc @ M ) @ ( suc @ N ) )
% 0.32/0.47 => ( ord_less @ nat @ M @ N ) ) ).
% 0.32/0.47
% 0.32/0.47 % Suc_less_SucD
% 0.32/0.47 thf(fact_156_less__antisym,axiom,
% 0.32/0.47 ! [N: nat,M: nat] :
% 0.32/0.47 ( ~ ( ord_less @ nat @ N @ M )
% 0.32/0.47 => ( ( ord_less @ nat @ N @ ( suc @ M ) )
% 0.32/0.47 => ( M = N ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % less_antisym
% 0.32/0.47 thf(fact_157_Suc__less__eq2,axiom,
% 0.32/0.47 ! [N: nat,M: nat] :
% 0.32/0.47 ( ( ord_less @ nat @ ( suc @ N ) @ M )
% 0.32/0.47 = ( ? [M4: nat] :
% 0.32/0.47 ( ( M
% 0.32/0.47 = ( suc @ M4 ) )
% 0.32/0.47 & ( ord_less @ nat @ N @ M4 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % Suc_less_eq2
% 0.32/0.47 thf(fact_158_All__less__Suc,axiom,
% 0.32/0.47 ! [N: nat,P2: nat > $o] :
% 0.32/0.47 ( ( ! [I4: nat] :
% 0.32/0.47 ( ( ord_less @ nat @ I4 @ ( suc @ N ) )
% 0.32/0.47 => ( P2 @ I4 ) ) )
% 0.32/0.47 = ( ( P2 @ N )
% 0.32/0.47 & ! [I4: nat] :
% 0.32/0.47 ( ( ord_less @ nat @ I4 @ N )
% 0.32/0.47 => ( P2 @ I4 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % All_less_Suc
% 0.32/0.47 thf(fact_159_not__less__eq,axiom,
% 0.32/0.47 ! [M: nat,N: nat] :
% 0.32/0.47 ( ( ~ ( ord_less @ nat @ M @ N ) )
% 0.32/0.47 = ( ord_less @ nat @ N @ ( suc @ M ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % not_less_eq
% 0.32/0.47 thf(fact_160_less__Suc__eq,axiom,
% 0.32/0.47 ! [M: nat,N: nat] :
% 0.32/0.47 ( ( ord_less @ nat @ M @ ( suc @ N ) )
% 0.32/0.47 = ( ( ord_less @ nat @ M @ N )
% 0.32/0.47 | ( M = N ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % less_Suc_eq
% 0.32/0.47 thf(fact_161_Ex__less__Suc,axiom,
% 0.32/0.47 ! [N: nat,P2: nat > $o] :
% 0.32/0.47 ( ( ? [I4: nat] :
% 0.32/0.47 ( ( ord_less @ nat @ I4 @ ( suc @ N ) )
% 0.32/0.47 & ( P2 @ I4 ) ) )
% 0.32/0.47 = ( ( P2 @ N )
% 0.32/0.47 | ? [I4: nat] :
% 0.32/0.47 ( ( ord_less @ nat @ I4 @ N )
% 0.32/0.47 & ( P2 @ I4 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % Ex_less_Suc
% 0.32/0.47 thf(fact_162_less__SucI,axiom,
% 0.32/0.47 ! [M: nat,N: nat] :
% 0.32/0.47 ( ( ord_less @ nat @ M @ N )
% 0.32/0.47 => ( ord_less @ nat @ M @ ( suc @ N ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % less_SucI
% 0.32/0.47 thf(fact_163_less__SucE,axiom,
% 0.32/0.47 ! [M: nat,N: nat] :
% 0.32/0.47 ( ( ord_less @ nat @ M @ ( suc @ N ) )
% 0.32/0.47 => ( ~ ( ord_less @ nat @ M @ N )
% 0.32/0.47 => ( M = N ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % less_SucE
% 0.32/0.47 thf(fact_164_Suc__lessI,axiom,
% 0.32/0.47 ! [M: nat,N: nat] :
% 0.32/0.47 ( ( ord_less @ nat @ M @ N )
% 0.32/0.47 => ( ( ( suc @ M )
% 0.32/0.47 != N )
% 0.32/0.47 => ( ord_less @ nat @ ( suc @ M ) @ N ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % Suc_lessI
% 0.32/0.47 thf(fact_165_Suc__lessE,axiom,
% 0.32/0.47 ! [I2: nat,K: nat] :
% 0.32/0.47 ( ( ord_less @ nat @ ( suc @ I2 ) @ K )
% 0.32/0.47 => ~ ! [J2: nat] :
% 0.32/0.47 ( ( ord_less @ nat @ I2 @ J2 )
% 0.32/0.47 => ( K
% 0.32/0.47 != ( suc @ J2 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % Suc_lessE
% 0.32/0.47 thf(fact_166_Suc__lessD,axiom,
% 0.32/0.47 ! [M: nat,N: nat] :
% 0.32/0.47 ( ( ord_less @ nat @ ( suc @ M ) @ N )
% 0.32/0.47 => ( ord_less @ nat @ M @ N ) ) ).
% 0.32/0.47
% 0.32/0.47 % Suc_lessD
% 0.32/0.47 thf(fact_167_Nat_OlessE,axiom,
% 0.32/0.47 ! [I2: nat,K: nat] :
% 0.32/0.47 ( ( ord_less @ nat @ I2 @ K )
% 0.32/0.47 => ( ( K
% 0.32/0.47 != ( suc @ I2 ) )
% 0.32/0.47 => ~ ! [J2: nat] :
% 0.32/0.47 ( ( ord_less @ nat @ I2 @ J2 )
% 0.32/0.47 => ( K
% 0.32/0.47 != ( suc @ J2 ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % Nat.lessE
% 0.32/0.47 thf(fact_168_Rats__inverse,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( field_char_0 @ A )
% 0.32/0.47 => ! [A5: A] :
% 0.32/0.47 ( ( member @ A @ A5 @ ( field_char_0_Rats @ A ) )
% 0.32/0.47 => ( member @ A @ ( inverse_inverse @ A @ A5 ) @ ( field_char_0_Rats @ A ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % Rats_inverse
% 0.32/0.47 thf(fact_169_nat__neq__iff,axiom,
% 0.32/0.47 ! [M: nat,N: nat] :
% 0.32/0.47 ( ( M != N )
% 0.32/0.47 = ( ( ord_less @ nat @ M @ N )
% 0.32/0.47 | ( ord_less @ nat @ N @ M ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % nat_neq_iff
% 0.32/0.47 thf(fact_170_less__not__refl,axiom,
% 0.32/0.47 ! [N: nat] :
% 0.32/0.47 ~ ( ord_less @ nat @ N @ N ) ).
% 0.32/0.47
% 0.32/0.47 % less_not_refl
% 0.32/0.47 thf(fact_171_less__not__refl2,axiom,
% 0.32/0.47 ! [N: nat,M: nat] :
% 0.32/0.47 ( ( ord_less @ nat @ N @ M )
% 0.32/0.47 => ( M != N ) ) ).
% 0.32/0.47
% 0.32/0.47 % less_not_refl2
% 0.32/0.47 thf(fact_172_less__not__refl3,axiom,
% 0.32/0.47 ! [S: nat,T: nat] :
% 0.32/0.47 ( ( ord_less @ nat @ S @ T )
% 0.32/0.47 => ( S != T ) ) ).
% 0.32/0.47
% 0.32/0.47 % less_not_refl3
% 0.32/0.47 thf(fact_173_measure__induct,axiom,
% 0.32/0.47 ! [B: $tType,A: $tType] :
% 0.32/0.47 ( ( wellorder @ B )
% 0.32/0.47 => ! [F: A > B,P2: A > $o,A5: A] :
% 0.32/0.47 ( ! [X3: A] :
% 0.32/0.47 ( ! [Y5: A] :
% 0.32/0.47 ( ( ord_less @ B @ ( F @ Y5 ) @ ( F @ X3 ) )
% 0.32/0.47 => ( P2 @ Y5 ) )
% 0.32/0.47 => ( P2 @ X3 ) )
% 0.32/0.47 => ( P2 @ A5 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % measure_induct
% 0.32/0.47 thf(fact_174_less__irrefl__nat,axiom,
% 0.32/0.47 ! [N: nat] :
% 0.32/0.47 ~ ( ord_less @ nat @ N @ N ) ).
% 0.32/0.47
% 0.32/0.47 % less_irrefl_nat
% 0.32/0.47 thf(fact_175_nat__less__induct,axiom,
% 0.32/0.47 ! [P2: nat > $o,N: nat] :
% 0.32/0.47 ( ! [N2: nat] :
% 0.32/0.47 ( ! [M3: nat] :
% 0.32/0.47 ( ( ord_less @ nat @ M3 @ N2 )
% 0.32/0.47 => ( P2 @ M3 ) )
% 0.32/0.47 => ( P2 @ N2 ) )
% 0.32/0.47 => ( P2 @ N ) ) ).
% 0.32/0.47
% 0.32/0.47 % nat_less_induct
% 0.32/0.47 thf(fact_176_infinite__descent,axiom,
% 0.32/0.47 ! [P2: nat > $o,N: nat] :
% 0.32/0.47 ( ! [N2: nat] :
% 0.32/0.47 ( ~ ( P2 @ N2 )
% 0.32/0.47 => ? [M3: nat] :
% 0.32/0.47 ( ( ord_less @ nat @ M3 @ N2 )
% 0.32/0.47 & ~ ( P2 @ M3 ) ) )
% 0.32/0.47 => ( P2 @ N ) ) ).
% 0.32/0.47
% 0.32/0.47 % infinite_descent
% 0.32/0.47 thf(fact_177_linorder__neqE__nat,axiom,
% 0.32/0.47 ! [X2: nat,Y2: nat] :
% 0.32/0.47 ( ( X2 != Y2 )
% 0.32/0.47 => ( ~ ( ord_less @ nat @ X2 @ Y2 )
% 0.32/0.47 => ( ord_less @ nat @ Y2 @ X2 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % linorder_neqE_nat
% 0.32/0.47 thf(fact_178_measure__induct__rule,axiom,
% 0.32/0.47 ! [B: $tType,A: $tType] :
% 0.32/0.47 ( ( wellorder @ B )
% 0.32/0.47 => ! [F: A > B,P2: A > $o,A5: A] :
% 0.32/0.47 ( ! [X3: A] :
% 0.32/0.47 ( ! [Y5: A] :
% 0.32/0.47 ( ( ord_less @ B @ ( F @ Y5 ) @ ( F @ X3 ) )
% 0.32/0.47 => ( P2 @ Y5 ) )
% 0.32/0.47 => ( P2 @ X3 ) )
% 0.32/0.47 => ( P2 @ A5 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % measure_induct_rule
% 0.32/0.47 thf(fact_179_infinite__descent__measure,axiom,
% 0.32/0.47 ! [A: $tType,P2: A > $o,V: A > nat,X2: A] :
% 0.32/0.47 ( ! [X3: A] :
% 0.32/0.47 ( ~ ( P2 @ X3 )
% 0.32/0.47 => ? [Y5: A] :
% 0.32/0.47 ( ( ord_less @ nat @ ( V @ Y5 ) @ ( V @ X3 ) )
% 0.32/0.47 & ~ ( P2 @ Y5 ) ) )
% 0.32/0.47 => ( P2 @ X2 ) ) ).
% 0.32/0.47
% 0.32/0.47 % infinite_descent_measure
% 0.32/0.47 thf(fact_180_linorder__neqE__linordered__idom,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( linordered_idom @ A )
% 0.32/0.47 => ! [X2: A,Y2: A] :
% 0.32/0.47 ( ( X2 != Y2 )
% 0.32/0.47 => ( ~ ( ord_less @ A @ X2 @ Y2 )
% 0.32/0.47 => ( ord_less @ A @ Y2 @ X2 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % linorder_neqE_linordered_idom
% 0.32/0.47 thf(fact_181_lift__Suc__mono__less__iff,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( order @ A )
% 0.32/0.47 => ! [F: nat > A,N: nat,M: nat] :
% 0.32/0.47 ( ! [N2: nat] : ( ord_less @ A @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
% 0.32/0.47 => ( ( ord_less @ A @ ( F @ N ) @ ( F @ M ) )
% 0.32/0.47 = ( ord_less @ nat @ N @ M ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % lift_Suc_mono_less_iff
% 0.32/0.47 thf(fact_182_lift__Suc__mono__less,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( order @ A )
% 0.32/0.47 => ! [F: nat > A,N: nat,N5: nat] :
% 0.32/0.47 ( ! [N2: nat] : ( ord_less @ A @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
% 0.32/0.47 => ( ( ord_less @ nat @ N @ N5 )
% 0.32/0.47 => ( ord_less @ A @ ( F @ N ) @ ( F @ N5 ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % lift_Suc_mono_less
% 0.32/0.47 thf(fact_183_dvd__pos__nat,axiom,
% 0.32/0.47 ! [N: nat,M: nat] :
% 0.32/0.47 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 0.32/0.47 => ( ( dvd_dvd @ nat @ M @ N )
% 0.32/0.47 => ( ord_less @ nat @ ( zero_zero @ nat ) @ M ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % dvd_pos_nat
% 0.32/0.47 thf(fact_184_finite__divisors__nat,axiom,
% 0.32/0.47 ! [M: nat] :
% 0.32/0.47 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
% 0.32/0.47 => ( finite_finite2 @ nat
% 0.32/0.47 @ ( collect @ nat
% 0.32/0.47 @ ^ [D3: nat] : ( dvd_dvd @ nat @ D3 @ M ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % finite_divisors_nat
% 0.32/0.47 thf(fact_185_not__one__less__zero,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( linord1659791738miring @ A )
% 0.32/0.47 => ~ ( ord_less @ A @ ( one_one @ A ) @ ( zero_zero @ A ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % not_one_less_zero
% 0.32/0.47 thf(fact_186_zero__less__one,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( zero_less_one @ A )
% 0.32/0.47 => ( ord_less @ A @ ( zero_zero @ A ) @ ( one_one @ A ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % zero_less_one
% 0.32/0.47 thf(fact_187_less__Suc__eq__0__disj,axiom,
% 0.32/0.47 ! [M: nat,N: nat] :
% 0.32/0.47 ( ( ord_less @ nat @ M @ ( suc @ N ) )
% 0.32/0.47 = ( ( M
% 0.32/0.47 = ( zero_zero @ nat ) )
% 0.32/0.47 | ? [J3: nat] :
% 0.32/0.47 ( ( M
% 0.32/0.47 = ( suc @ J3 ) )
% 0.32/0.47 & ( ord_less @ nat @ J3 @ N ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % less_Suc_eq_0_disj
% 0.32/0.47 thf(fact_188_gr0__implies__Suc,axiom,
% 0.32/0.47 ! [N: nat] :
% 0.32/0.47 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 0.32/0.47 => ? [M2: nat] :
% 0.32/0.47 ( N
% 0.32/0.47 = ( suc @ M2 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % gr0_implies_Suc
% 0.32/0.47 thf(fact_189_All__less__Suc2,axiom,
% 0.32/0.47 ! [N: nat,P2: nat > $o] :
% 0.32/0.47 ( ( ! [I4: nat] :
% 0.32/0.47 ( ( ord_less @ nat @ I4 @ ( suc @ N ) )
% 0.32/0.47 => ( P2 @ I4 ) ) )
% 0.32/0.47 = ( ( P2 @ ( zero_zero @ nat ) )
% 0.32/0.47 & ! [I4: nat] :
% 0.32/0.47 ( ( ord_less @ nat @ I4 @ N )
% 0.32/0.47 => ( P2 @ ( suc @ I4 ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % All_less_Suc2
% 0.32/0.47 thf(fact_190_gr0__conv__Suc,axiom,
% 0.32/0.47 ! [N: nat] :
% 0.32/0.47 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 0.32/0.47 = ( ? [M5: nat] :
% 0.32/0.47 ( N
% 0.32/0.47 = ( suc @ M5 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % gr0_conv_Suc
% 0.32/0.47 thf(fact_191_Ex__less__Suc2,axiom,
% 0.32/0.47 ! [N: nat,P2: nat > $o] :
% 0.32/0.47 ( ( ? [I4: nat] :
% 0.32/0.47 ( ( ord_less @ nat @ I4 @ ( suc @ N ) )
% 0.32/0.47 & ( P2 @ I4 ) ) )
% 0.32/0.47 = ( ( P2 @ ( zero_zero @ nat ) )
% 0.32/0.47 | ? [I4: nat] :
% 0.32/0.47 ( ( ord_less @ nat @ I4 @ N )
% 0.32/0.47 & ( P2 @ ( suc @ I4 ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % Ex_less_Suc2
% 0.32/0.47 thf(fact_192_nat__dvd__not__less,axiom,
% 0.32/0.47 ! [M: nat,N: nat] :
% 0.32/0.47 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
% 0.32/0.47 => ( ( ord_less @ nat @ M @ N )
% 0.32/0.47 => ~ ( dvd_dvd @ nat @ N @ M ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % nat_dvd_not_less
% 0.32/0.47 thf(fact_193_nat__induct__non__zero,axiom,
% 0.32/0.47 ! [N: nat,P2: nat > $o] :
% 0.32/0.47 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 0.32/0.47 => ( ( P2 @ ( one_one @ nat ) )
% 0.32/0.47 => ( ! [N2: nat] :
% 0.32/0.47 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
% 0.32/0.47 => ( ( P2 @ N2 )
% 0.32/0.47 => ( P2 @ ( suc @ N2 ) ) ) )
% 0.32/0.47 => ( P2 @ N ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % nat_induct_non_zero
% 0.32/0.47 thf(fact_194_less__degree__imp,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( zero @ A )
% 0.32/0.47 => ! [N: nat,P: poly @ A] :
% 0.32/0.47 ( ( ord_less @ nat @ N @ ( degree @ A @ P ) )
% 0.32/0.47 => ? [I3: nat] :
% 0.32/0.47 ( ( ord_less @ nat @ N @ I3 )
% 0.32/0.47 & ( ( coeff @ A @ P @ I3 )
% 0.32/0.47 != ( zero_zero @ A ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % less_degree_imp
% 0.32/0.47 thf(fact_195_coeff__eq__0,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( zero @ A )
% 0.32/0.47 => ! [P: poly @ A,N: nat] :
% 0.32/0.47 ( ( ord_less @ nat @ ( degree @ A @ P ) @ N )
% 0.32/0.47 => ( ( coeff @ A @ P @ N )
% 0.32/0.47 = ( zero_zero @ A ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % coeff_eq_0
% 0.32/0.47 thf(fact_196_gcd__nat_Onot__eq__order__implies__strict,axiom,
% 0.32/0.47 ! [A5: nat,B3: nat] :
% 0.32/0.47 ( ( A5 != B3 )
% 0.32/0.47 => ( ( dvd_dvd @ nat @ A5 @ B3 )
% 0.32/0.47 => ( ( dvd_dvd @ nat @ A5 @ B3 )
% 0.32/0.47 & ( A5 != B3 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % gcd_nat.not_eq_order_implies_strict
% 0.32/0.47 thf(fact_197_gcd__nat_Ostrict__implies__not__eq,axiom,
% 0.32/0.47 ! [A5: nat,B3: nat] :
% 0.32/0.47 ( ( ( dvd_dvd @ nat @ A5 @ B3 )
% 0.32/0.47 & ( A5 != B3 ) )
% 0.32/0.47 => ( A5 != B3 ) ) ).
% 0.32/0.47
% 0.32/0.47 % gcd_nat.strict_implies_not_eq
% 0.32/0.47 thf(fact_198_gcd__nat_Ostrict__implies__order,axiom,
% 0.32/0.47 ! [A5: nat,B3: nat] :
% 0.32/0.47 ( ( ( dvd_dvd @ nat @ A5 @ B3 )
% 0.32/0.47 & ( A5 != B3 ) )
% 0.32/0.47 => ( dvd_dvd @ nat @ A5 @ B3 ) ) ).
% 0.32/0.47
% 0.32/0.47 % gcd_nat.strict_implies_order
% 0.32/0.47 thf(fact_199_gcd__nat_Ostrict__iff__order,axiom,
% 0.32/0.47 ! [A5: nat,B3: nat] :
% 0.32/0.47 ( ( ( dvd_dvd @ nat @ A5 @ B3 )
% 0.32/0.47 & ( A5 != B3 ) )
% 0.32/0.47 = ( ( dvd_dvd @ nat @ A5 @ B3 )
% 0.32/0.47 & ( A5 != B3 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % gcd_nat.strict_iff_order
% 0.32/0.47 thf(fact_200_gcd__nat_Oorder__iff__strict,axiom,
% 0.32/0.47 ( ( dvd_dvd @ nat )
% 0.32/0.47 = ( ^ [A3: nat,B4: nat] :
% 0.32/0.47 ( ( ( dvd_dvd @ nat @ A3 @ B4 )
% 0.32/0.47 & ( A3 != B4 ) )
% 0.32/0.47 | ( A3 = B4 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % gcd_nat.order_iff_strict
% 0.32/0.47 thf(fact_201_gcd__nat_Ostrict__trans2,axiom,
% 0.32/0.47 ! [A5: nat,B3: nat,C2: nat] :
% 0.32/0.47 ( ( ( dvd_dvd @ nat @ A5 @ B3 )
% 0.32/0.47 & ( A5 != B3 ) )
% 0.32/0.47 => ( ( dvd_dvd @ nat @ B3 @ C2 )
% 0.32/0.47 => ( ( dvd_dvd @ nat @ A5 @ C2 )
% 0.32/0.47 & ( A5 != C2 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % gcd_nat.strict_trans2
% 0.32/0.47 thf(fact_202_gcd__nat_Ostrict__trans1,axiom,
% 0.32/0.47 ! [A5: nat,B3: nat,C2: nat] :
% 0.32/0.47 ( ( dvd_dvd @ nat @ A5 @ B3 )
% 0.32/0.47 => ( ( ( dvd_dvd @ nat @ B3 @ C2 )
% 0.32/0.47 & ( B3 != C2 ) )
% 0.32/0.47 => ( ( dvd_dvd @ nat @ A5 @ C2 )
% 0.32/0.47 & ( A5 != C2 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % gcd_nat.strict_trans1
% 0.32/0.47 thf(fact_203_gcd__nat_Ostrict__trans,axiom,
% 0.32/0.47 ! [A5: nat,B3: nat,C2: nat] :
% 0.32/0.47 ( ( ( dvd_dvd @ nat @ A5 @ B3 )
% 0.32/0.47 & ( A5 != B3 ) )
% 0.32/0.47 => ( ( ( dvd_dvd @ nat @ B3 @ C2 )
% 0.32/0.47 & ( B3 != C2 ) )
% 0.32/0.47 => ( ( dvd_dvd @ nat @ A5 @ C2 )
% 0.32/0.47 & ( A5 != C2 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % gcd_nat.strict_trans
% 0.32/0.47 thf(fact_204_gcd__nat_Oantisym,axiom,
% 0.32/0.47 ! [A5: nat,B3: nat] :
% 0.32/0.47 ( ( dvd_dvd @ nat @ A5 @ B3 )
% 0.32/0.47 => ( ( dvd_dvd @ nat @ B3 @ A5 )
% 0.32/0.47 => ( A5 = B3 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % gcd_nat.antisym
% 0.32/0.47 thf(fact_205_gcd__nat_Oirrefl,axiom,
% 0.32/0.47 ! [A5: nat] :
% 0.32/0.47 ~ ( ( dvd_dvd @ nat @ A5 @ A5 )
% 0.32/0.47 & ( A5 != A5 ) ) ).
% 0.32/0.47
% 0.32/0.47 % gcd_nat.irrefl
% 0.32/0.47 thf(fact_206_gcd__nat_Oeq__iff,axiom,
% 0.32/0.47 ( ( ^ [Y: nat,Z: nat] : ( Y = Z ) )
% 0.32/0.47 = ( ^ [A3: nat,B4: nat] :
% 0.32/0.47 ( ( dvd_dvd @ nat @ A3 @ B4 )
% 0.32/0.47 & ( dvd_dvd @ nat @ B4 @ A3 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % gcd_nat.eq_iff
% 0.32/0.47 thf(fact_207_gcd__nat_Otrans,axiom,
% 0.32/0.47 ! [A5: nat,B3: nat,C2: nat] :
% 0.32/0.47 ( ( dvd_dvd @ nat @ A5 @ B3 )
% 0.32/0.47 => ( ( dvd_dvd @ nat @ B3 @ C2 )
% 0.32/0.47 => ( dvd_dvd @ nat @ A5 @ C2 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % gcd_nat.trans
% 0.32/0.47 thf(fact_208_gcd__nat_Orefl,axiom,
% 0.32/0.47 ! [A5: nat] : ( dvd_dvd @ nat @ A5 @ A5 ) ).
% 0.32/0.47
% 0.32/0.47 % gcd_nat.refl
% 0.32/0.47 thf(fact_209_gcd__nat_Oasym,axiom,
% 0.32/0.47 ! [A5: nat,B3: nat] :
% 0.32/0.47 ( ( ( dvd_dvd @ nat @ A5 @ B3 )
% 0.32/0.47 & ( A5 != B3 ) )
% 0.32/0.47 => ~ ( ( dvd_dvd @ nat @ B3 @ A5 )
% 0.32/0.47 & ( B3 != A5 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % gcd_nat.asym
% 0.32/0.47 thf(fact_210_coeff__poly__cutoff,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( zero @ A )
% 0.32/0.47 => ! [K: nat,N: nat,P: poly @ A] :
% 0.32/0.47 ( ( ( ord_less @ nat @ K @ N )
% 0.32/0.47 => ( ( coeff @ A @ ( poly_cutoff @ A @ N @ P ) @ K )
% 0.32/0.47 = ( coeff @ A @ P @ K ) ) )
% 0.32/0.47 & ( ~ ( ord_less @ nat @ K @ N )
% 0.32/0.47 => ( ( coeff @ A @ ( poly_cutoff @ A @ N @ P ) @ K )
% 0.32/0.47 = ( zero_zero @ A ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % coeff_poly_cutoff
% 0.32/0.47 thf(fact_211_gcd__nat_Oextremum,axiom,
% 0.32/0.47 ! [A5: nat] : ( dvd_dvd @ nat @ A5 @ ( zero_zero @ nat ) ) ).
% 0.32/0.47
% 0.32/0.47 % gcd_nat.extremum
% 0.32/0.47 thf(fact_212_gcd__nat_Oextremum__strict,axiom,
% 0.32/0.47 ! [A5: nat] :
% 0.32/0.47 ~ ( ( dvd_dvd @ nat @ ( zero_zero @ nat ) @ A5 )
% 0.32/0.47 & ( ( zero_zero @ nat )
% 0.32/0.47 != A5 ) ) ).
% 0.32/0.47
% 0.32/0.47 % gcd_nat.extremum_strict
% 0.32/0.47 thf(fact_213_gcd__nat_Oextremum__unique,axiom,
% 0.32/0.47 ! [A5: nat] :
% 0.32/0.47 ( ( dvd_dvd @ nat @ ( zero_zero @ nat ) @ A5 )
% 0.32/0.47 = ( A5
% 0.32/0.47 = ( zero_zero @ nat ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % gcd_nat.extremum_unique
% 0.32/0.47 thf(fact_214_inverse__positive__iff__positive,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( linordered_field @ A )
% 0.32/0.47 => ! [A5: A] :
% 0.32/0.47 ( ( ord_less @ A @ ( zero_zero @ A ) @ ( inverse_inverse @ A @ A5 ) )
% 0.32/0.47 = ( ord_less @ A @ ( zero_zero @ A ) @ A5 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % inverse_positive_iff_positive
% 0.32/0.47 thf(fact_215_inverse__negative__iff__negative,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( linordered_field @ A )
% 0.32/0.47 => ! [A5: A] :
% 0.32/0.47 ( ( ord_less @ A @ ( inverse_inverse @ A @ A5 ) @ ( zero_zero @ A ) )
% 0.32/0.47 = ( ord_less @ A @ A5 @ ( zero_zero @ A ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % inverse_negative_iff_negative
% 0.32/0.47 thf(fact_216_inverse__less__iff__less__neg,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( linordered_field @ A )
% 0.32/0.47 => ! [A5: A,B3: A] :
% 0.32/0.47 ( ( ord_less @ A @ A5 @ ( zero_zero @ A ) )
% 0.32/0.47 => ( ( ord_less @ A @ B3 @ ( zero_zero @ A ) )
% 0.32/0.47 => ( ( ord_less @ A @ ( inverse_inverse @ A @ A5 ) @ ( inverse_inverse @ A @ B3 ) )
% 0.32/0.47 = ( ord_less @ A @ B3 @ A5 ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % inverse_less_iff_less_neg
% 0.32/0.47 thf(fact_217_inverse__less__iff__less,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( linordered_field @ A )
% 0.32/0.47 => ! [A5: A,B3: A] :
% 0.32/0.47 ( ( ord_less @ A @ ( zero_zero @ A ) @ A5 )
% 0.32/0.47 => ( ( ord_less @ A @ ( zero_zero @ A ) @ B3 )
% 0.32/0.47 => ( ( ord_less @ A @ ( inverse_inverse @ A @ A5 ) @ ( inverse_inverse @ A @ B3 ) )
% 0.32/0.47 = ( ord_less @ A @ B3 @ A5 ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % inverse_less_iff_less
% 0.32/0.47 thf(fact_218_inverse__zero,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( division_ring @ A )
% 0.32/0.47 => ( ( inverse_inverse @ A @ ( zero_zero @ A ) )
% 0.32/0.47 = ( zero_zero @ A ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % inverse_zero
% 0.32/0.47 thf(fact_219_inverse__nonzero__iff__nonzero,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( division_ring @ A )
% 0.32/0.47 => ! [A5: A] :
% 0.32/0.47 ( ( ( inverse_inverse @ A @ A5 )
% 0.32/0.47 = ( zero_zero @ A ) )
% 0.32/0.47 = ( A5
% 0.32/0.47 = ( zero_zero @ A ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % inverse_nonzero_iff_nonzero
% 0.32/0.47 thf(fact_220_inverse__eq__1__iff,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( field @ A )
% 0.32/0.47 => ! [X2: A] :
% 0.32/0.47 ( ( ( inverse_inverse @ A @ X2 )
% 0.32/0.47 = ( one_one @ A ) )
% 0.32/0.47 = ( X2
% 0.32/0.47 = ( one_one @ A ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % inverse_eq_1_iff
% 0.32/0.47 thf(fact_221_inverse__1,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( division_ring @ A )
% 0.32/0.47 => ( ( inverse_inverse @ A @ ( one_one @ A ) )
% 0.32/0.47 = ( one_one @ A ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % inverse_1
% 0.32/0.47 thf(fact_222_finite__psubset__induct,axiom,
% 0.32/0.47 ! [A: $tType,A4: set @ A,P2: ( set @ A ) > $o] :
% 0.32/0.47 ( ( finite_finite2 @ A @ A4 )
% 0.32/0.47 => ( ! [A6: set @ A] :
% 0.32/0.47 ( ( finite_finite2 @ A @ A6 )
% 0.32/0.47 => ( ! [B5: set @ A] :
% 0.32/0.47 ( ( ord_less @ ( set @ A ) @ B5 @ A6 )
% 0.32/0.47 => ( P2 @ B5 ) )
% 0.32/0.47 => ( P2 @ A6 ) ) )
% 0.32/0.47 => ( P2 @ A4 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % finite_psubset_induct
% 0.32/0.47 thf(fact_223_strict__subset__divisors__dvd,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( comm_monoid_mult @ A )
% 0.32/0.47 => ! [A5: A,B3: A] :
% 0.32/0.47 ( ( ord_less @ ( set @ A )
% 0.32/0.47 @ ( collect @ A
% 0.32/0.47 @ ^ [C3: A] : ( dvd_dvd @ A @ C3 @ A5 ) )
% 0.32/0.47 @ ( collect @ A
% 0.32/0.47 @ ^ [C3: A] : ( dvd_dvd @ A @ C3 @ B3 ) ) )
% 0.32/0.47 = ( ( dvd_dvd @ A @ A5 @ B3 )
% 0.32/0.47 & ~ ( dvd_dvd @ A @ B3 @ A5 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % strict_subset_divisors_dvd
% 0.32/0.47 thf(fact_224_poly__IVT__pos,axiom,
% 0.32/0.47 ! [A5: real,B3: real,P: poly @ real] :
% 0.32/0.47 ( ( ord_less @ real @ A5 @ B3 )
% 0.32/0.47 => ( ( ord_less @ real @ ( poly2 @ real @ P @ A5 ) @ ( zero_zero @ real ) )
% 0.32/0.47 => ( ( ord_less @ real @ ( zero_zero @ real ) @ ( poly2 @ real @ P @ B3 ) )
% 0.32/0.47 => ? [X3: real] :
% 0.32/0.47 ( ( ord_less @ real @ A5 @ X3 )
% 0.32/0.47 & ( ord_less @ real @ X3 @ B3 )
% 0.32/0.47 & ( ( poly2 @ real @ P @ X3 )
% 0.32/0.47 = ( zero_zero @ real ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % poly_IVT_pos
% 0.32/0.47 thf(fact_225_poly__IVT__neg,axiom,
% 0.32/0.47 ! [A5: real,B3: real,P: poly @ real] :
% 0.32/0.47 ( ( ord_less @ real @ A5 @ B3 )
% 0.32/0.47 => ( ( ord_less @ real @ ( zero_zero @ real ) @ ( poly2 @ real @ P @ A5 ) )
% 0.32/0.47 => ( ( ord_less @ real @ ( poly2 @ real @ P @ B3 ) @ ( zero_zero @ real ) )
% 0.32/0.47 => ? [X3: real] :
% 0.32/0.47 ( ( ord_less @ real @ A5 @ X3 )
% 0.32/0.47 & ( ord_less @ real @ X3 @ B3 )
% 0.32/0.47 & ( ( poly2 @ real @ P @ X3 )
% 0.32/0.47 = ( zero_zero @ real ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % poly_IVT_neg
% 0.32/0.47 thf(fact_226_dvd__field__iff,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( field @ A )
% 0.32/0.47 => ( ( dvd_dvd @ A )
% 0.32/0.47 = ( ^ [A3: A,B4: A] :
% 0.32/0.47 ( ( A3
% 0.32/0.47 = ( zero_zero @ A ) )
% 0.32/0.47 => ( B4
% 0.32/0.47 = ( zero_zero @ A ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % dvd_field_iff
% 0.32/0.47 thf(fact_227_field__class_Ofield__inverse__zero,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( field @ A )
% 0.32/0.47 => ( ( inverse_inverse @ A @ ( zero_zero @ A ) )
% 0.32/0.47 = ( zero_zero @ A ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % field_class.field_inverse_zero
% 0.32/0.47 thf(fact_228_inverse__zero__imp__zero,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( division_ring @ A )
% 0.32/0.47 => ! [A5: A] :
% 0.32/0.47 ( ( ( inverse_inverse @ A @ A5 )
% 0.32/0.47 = ( zero_zero @ A ) )
% 0.32/0.47 => ( A5
% 0.32/0.47 = ( zero_zero @ A ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % inverse_zero_imp_zero
% 0.32/0.47 thf(fact_229_nonzero__inverse__eq__imp__eq,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( division_ring @ A )
% 0.32/0.47 => ! [A5: A,B3: A] :
% 0.32/0.47 ( ( ( inverse_inverse @ A @ A5 )
% 0.32/0.47 = ( inverse_inverse @ A @ B3 ) )
% 0.32/0.47 => ( ( A5
% 0.32/0.47 != ( zero_zero @ A ) )
% 0.32/0.47 => ( ( B3
% 0.32/0.47 != ( zero_zero @ A ) )
% 0.32/0.47 => ( A5 = B3 ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % nonzero_inverse_eq_imp_eq
% 0.32/0.47 thf(fact_230_nonzero__inverse__inverse__eq,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( division_ring @ A )
% 0.32/0.47 => ! [A5: A] :
% 0.32/0.47 ( ( A5
% 0.32/0.47 != ( zero_zero @ A ) )
% 0.32/0.47 => ( ( inverse_inverse @ A @ ( inverse_inverse @ A @ A5 ) )
% 0.32/0.47 = A5 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % nonzero_inverse_inverse_eq
% 0.32/0.47 thf(fact_231_nonzero__imp__inverse__nonzero,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( division_ring @ A )
% 0.32/0.47 => ! [A5: A] :
% 0.32/0.47 ( ( A5
% 0.32/0.47 != ( zero_zero @ A ) )
% 0.32/0.47 => ( ( inverse_inverse @ A @ A5 )
% 0.32/0.47 != ( zero_zero @ A ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % nonzero_imp_inverse_nonzero
% 0.32/0.47 thf(fact_232_inverse__less__imp__less,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( linordered_field @ A )
% 0.32/0.47 => ! [A5: A,B3: A] :
% 0.32/0.47 ( ( ord_less @ A @ ( inverse_inverse @ A @ A5 ) @ ( inverse_inverse @ A @ B3 ) )
% 0.32/0.47 => ( ( ord_less @ A @ ( zero_zero @ A ) @ A5 )
% 0.32/0.47 => ( ord_less @ A @ B3 @ A5 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % inverse_less_imp_less
% 0.32/0.47 thf(fact_233_less__imp__inverse__less,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( linordered_field @ A )
% 0.32/0.47 => ! [A5: A,B3: A] :
% 0.32/0.47 ( ( ord_less @ A @ A5 @ B3 )
% 0.32/0.47 => ( ( ord_less @ A @ ( zero_zero @ A ) @ A5 )
% 0.32/0.47 => ( ord_less @ A @ ( inverse_inverse @ A @ B3 ) @ ( inverse_inverse @ A @ A5 ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % less_imp_inverse_less
% 0.32/0.47 thf(fact_234_inverse__less__imp__less__neg,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( linordered_field @ A )
% 0.32/0.47 => ! [A5: A,B3: A] :
% 0.32/0.47 ( ( ord_less @ A @ ( inverse_inverse @ A @ A5 ) @ ( inverse_inverse @ A @ B3 ) )
% 0.32/0.47 => ( ( ord_less @ A @ B3 @ ( zero_zero @ A ) )
% 0.32/0.47 => ( ord_less @ A @ B3 @ A5 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % inverse_less_imp_less_neg
% 0.32/0.47 thf(fact_235_less__imp__inverse__less__neg,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( linordered_field @ A )
% 0.32/0.47 => ! [A5: A,B3: A] :
% 0.32/0.47 ( ( ord_less @ A @ A5 @ B3 )
% 0.32/0.47 => ( ( ord_less @ A @ B3 @ ( zero_zero @ A ) )
% 0.32/0.47 => ( ord_less @ A @ ( inverse_inverse @ A @ B3 ) @ ( inverse_inverse @ A @ A5 ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % less_imp_inverse_less_neg
% 0.32/0.47 thf(fact_236_inverse__negative__imp__negative,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( linordered_field @ A )
% 0.32/0.47 => ! [A5: A] :
% 0.32/0.47 ( ( ord_less @ A @ ( inverse_inverse @ A @ A5 ) @ ( zero_zero @ A ) )
% 0.32/0.47 => ( ( A5
% 0.32/0.47 != ( zero_zero @ A ) )
% 0.32/0.47 => ( ord_less @ A @ A5 @ ( zero_zero @ A ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % inverse_negative_imp_negative
% 0.32/0.47 thf(fact_237_inverse__positive__imp__positive,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( linordered_field @ A )
% 0.32/0.47 => ! [A5: A] :
% 0.32/0.47 ( ( ord_less @ A @ ( zero_zero @ A ) @ ( inverse_inverse @ A @ A5 ) )
% 0.32/0.47 => ( ( A5
% 0.32/0.47 != ( zero_zero @ A ) )
% 0.32/0.47 => ( ord_less @ A @ ( zero_zero @ A ) @ A5 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % inverse_positive_imp_positive
% 0.32/0.47 thf(fact_238_negative__imp__inverse__negative,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( linordered_field @ A )
% 0.32/0.47 => ! [A5: A] :
% 0.32/0.47 ( ( ord_less @ A @ A5 @ ( zero_zero @ A ) )
% 0.32/0.47 => ( ord_less @ A @ ( inverse_inverse @ A @ A5 ) @ ( zero_zero @ A ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % negative_imp_inverse_negative
% 0.32/0.47 thf(fact_239_positive__imp__inverse__positive,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( linordered_field @ A )
% 0.32/0.47 => ! [A5: A] :
% 0.32/0.47 ( ( ord_less @ A @ ( zero_zero @ A ) @ A5 )
% 0.32/0.47 => ( ord_less @ A @ ( zero_zero @ A ) @ ( inverse_inverse @ A @ A5 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % positive_imp_inverse_positive
% 0.32/0.47 thf(fact_240_one__less__inverse,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( linordered_field @ A )
% 0.32/0.47 => ! [A5: A] :
% 0.32/0.47 ( ( ord_less @ A @ ( zero_zero @ A ) @ A5 )
% 0.32/0.47 => ( ( ord_less @ A @ A5 @ ( one_one @ A ) )
% 0.32/0.47 => ( ord_less @ A @ ( one_one @ A ) @ ( inverse_inverse @ A @ A5 ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % one_less_inverse
% 0.32/0.47 thf(fact_241_one__less__inverse__iff,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( linordered_field @ A )
% 0.32/0.47 => ! [X2: A] :
% 0.32/0.47 ( ( ord_less @ A @ ( one_one @ A ) @ ( inverse_inverse @ A @ X2 ) )
% 0.32/0.47 = ( ( ord_less @ A @ ( zero_zero @ A ) @ X2 )
% 0.32/0.47 & ( ord_less @ A @ X2 @ ( one_one @ A ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % one_less_inverse_iff
% 0.32/0.47 thf(fact_242_less__numeral__extra_I1_J,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( linord1659791738miring @ A )
% 0.32/0.47 => ( ord_less @ A @ ( zero_zero @ A ) @ ( one_one @ A ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % less_numeral_extra(1)
% 0.32/0.47 thf(fact_243_finite__M__bounded__by__nat,axiom,
% 0.32/0.47 ! [P2: nat > $o,I2: nat] :
% 0.32/0.47 ( finite_finite2 @ nat
% 0.32/0.47 @ ( collect @ nat
% 0.32/0.47 @ ^ [K3: nat] :
% 0.32/0.47 ( ( P2 @ K3 )
% 0.32/0.47 & ( ord_less @ nat @ K3 @ I2 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % finite_M_bounded_by_nat
% 0.32/0.47 thf(fact_244_less__numeral__extra_I3_J,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( linord1659791738miring @ A )
% 0.32/0.47 => ~ ( ord_less @ A @ ( zero_zero @ A ) @ ( zero_zero @ A ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % less_numeral_extra(3)
% 0.32/0.47 thf(fact_245_less__numeral__extra_I4_J,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( linord1659791738miring @ A )
% 0.32/0.47 => ~ ( ord_less @ A @ ( one_one @ A ) @ ( one_one @ A ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % less_numeral_extra(4)
% 0.32/0.47 thf(fact_246_Rats__dense__in__real,axiom,
% 0.32/0.47 ! [X2: real,Y2: real] :
% 0.32/0.47 ( ( ord_less @ real @ X2 @ Y2 )
% 0.32/0.47 => ? [X3: real] :
% 0.32/0.47 ( ( member @ real @ X3 @ ( field_char_0_Rats @ real ) )
% 0.32/0.47 & ( ord_less @ real @ X2 @ X3 )
% 0.32/0.47 & ( ord_less @ real @ X3 @ Y2 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % Rats_dense_in_real
% 0.32/0.47 thf(fact_247_Rats__no__bot__less,axiom,
% 0.32/0.47 ! [X2: real] :
% 0.32/0.47 ? [X3: real] :
% 0.32/0.47 ( ( member @ real @ X3 @ ( field_char_0_Rats @ real ) )
% 0.32/0.47 & ( ord_less @ real @ X3 @ X2 ) ) ).
% 0.32/0.47
% 0.32/0.47 % Rats_no_bot_less
% 0.32/0.47 thf(fact_248_field__lbound__gt__zero,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( linordered_field @ A )
% 0.32/0.47 => ! [D1: A,D22: A] :
% 0.32/0.47 ( ( ord_less @ A @ ( zero_zero @ A ) @ D1 )
% 0.32/0.47 => ( ( ord_less @ A @ ( zero_zero @ A ) @ D22 )
% 0.32/0.47 => ? [E: A] :
% 0.32/0.47 ( ( ord_less @ A @ ( zero_zero @ A ) @ E )
% 0.32/0.47 & ( ord_less @ A @ E @ D1 )
% 0.32/0.47 & ( ord_less @ A @ E @ D22 ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % field_lbound_gt_zero
% 0.32/0.47 thf(fact_249_pseudo__mod_I2_J,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( ( comm_ring_1 @ A )
% 0.32/0.47 & ( semiri134348788visors @ A ) )
% 0.32/0.47 => ! [G: poly @ A,F: poly @ A] :
% 0.32/0.47 ( ( G
% 0.32/0.47 != ( zero_zero @ ( poly @ A ) ) )
% 0.32/0.47 => ( ( ( pseudo_mod @ A @ F @ G )
% 0.32/0.47 = ( zero_zero @ ( poly @ A ) ) )
% 0.32/0.47 | ( ord_less @ nat @ ( degree @ A @ ( pseudo_mod @ A @ F @ G ) ) @ ( degree @ A @ G ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % pseudo_mod(2)
% 0.32/0.47 thf(fact_250_poly__pinfty__gt__lc,axiom,
% 0.32/0.47 ! [P: poly @ real] :
% 0.32/0.47 ( ( ord_less @ real @ ( zero_zero @ real ) @ ( coeff @ real @ P @ ( degree @ real @ P ) ) )
% 0.32/0.47 => ? [N2: real] :
% 0.32/0.47 ! [X4: real] :
% 0.32/0.47 ( ( ord_less_eq @ real @ N2 @ X4 )
% 0.32/0.47 => ( ord_less_eq @ real @ ( coeff @ real @ P @ ( degree @ real @ P ) ) @ ( poly2 @ real @ P @ X4 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % poly_pinfty_gt_lc
% 0.32/0.47 thf(fact_251_le__zero__eq,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( canoni770627133id_add @ A )
% 0.32/0.47 => ! [N: A] :
% 0.32/0.47 ( ( ord_less_eq @ A @ N @ ( zero_zero @ A ) )
% 0.32/0.47 = ( N
% 0.32/0.47 = ( zero_zero @ A ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % le_zero_eq
% 0.32/0.47
% 0.32/0.47 % Type constructors (64)
% 0.32/0.47 thf(tcon_fun___Finite__Set_Ofinite,axiom,
% 0.32/0.47 ! [A7: $tType,A8: $tType] :
% 0.32/0.47 ( ( ( finite_finite @ A7 )
% 0.32/0.47 & ( finite_finite @ A8 ) )
% 0.32/0.47 => ( finite_finite @ ( A7 > A8 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_fun___Orderings_Oorder,axiom,
% 0.32/0.47 ! [A7: $tType,A8: $tType] :
% 0.32/0.47 ( ( order @ A8 )
% 0.32/0.47 => ( order @ ( A7 > A8 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_Nat_Onat___Groups_Ocanonically__ordered__monoid__add,axiom,
% 0.32/0.47 canoni770627133id_add @ nat ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_Nat_Onat___Rings_Osemiring__1__no__zero__divisors,axiom,
% 0.32/0.47 semiri134348788visors @ nat ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_Nat_Onat___Rings_Olinordered__nonzero__semiring,axiom,
% 0.32/0.47 linord1659791738miring @ nat ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_Nat_Onat___Rings_Osemiring__no__zero__divisors,axiom,
% 0.32/0.47 semiri1193490041visors @ nat ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_Nat_Onat___Rings_Oalgebraic__semidom,axiom,
% 0.32/0.47 algebraic_semidom @ nat ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_Nat_Onat___Groups_Ocomm__monoid__mult,axiom,
% 0.32/0.47 comm_monoid_mult @ nat ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_Nat_Onat___Rings_Ocomm__semiring__1,axiom,
% 0.32/0.47 comm_semiring_1 @ nat ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_Nat_Onat___Rings_Ocomm__semiring__0,axiom,
% 0.32/0.47 comm_semiring_0 @ nat ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_Nat_Onat___Rings_Ozero__less__one,axiom,
% 0.32/0.47 zero_less_one @ nat ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_Nat_Onat___Orderings_Owellorder,axiom,
% 0.32/0.47 wellorder @ nat ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_Nat_Onat___Nat_Osemiring__char__0,axiom,
% 0.32/0.47 semiring_char_0 @ nat ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_Nat_Onat___Rings_Ozero__neq__one,axiom,
% 0.32/0.47 zero_neq_one @ nat ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_Nat_Onat___Orderings_Oorder_1,axiom,
% 0.32/0.47 order @ nat ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_Nat_Onat___Groups_Ozero,axiom,
% 0.32/0.47 zero @ nat ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_Nat_Onat___Groups_Oone,axiom,
% 0.32/0.47 one @ nat ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_Set_Oset___Finite__Set_Ofinite_2,axiom,
% 0.32/0.47 ! [A7: $tType] :
% 0.32/0.47 ( ( finite_finite @ A7 )
% 0.32/0.47 => ( finite_finite @ ( set @ A7 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_Set_Oset___Orderings_Oorder_3,axiom,
% 0.32/0.47 ! [A7: $tType] : ( order @ ( set @ A7 ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_HOL_Obool___Finite__Set_Ofinite_4,axiom,
% 0.32/0.47 finite_finite @ $o ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_HOL_Obool___Orderings_Oorder_5,axiom,
% 0.32/0.47 order @ $o ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_Real_Oreal___Rings_Osemiring__1__no__zero__divisors_6,axiom,
% 0.32/0.47 semiri134348788visors @ real ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_Real_Oreal___Rings_Olinordered__nonzero__semiring_7,axiom,
% 0.32/0.47 linord1659791738miring @ real ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_Real_Oreal___Rings_Osemiring__no__zero__divisors_8,axiom,
% 0.32/0.47 semiri1193490041visors @ real ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_Real_Oreal___Rings_Oring__no__zero__divisors,axiom,
% 0.32/0.47 ring_n68954251visors @ real ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_Real_Oreal___Groups_Ocomm__monoid__mult_9,axiom,
% 0.32/0.47 comm_monoid_mult @ real ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_Real_Oreal___Fields_Olinordered__field,axiom,
% 0.32/0.47 linordered_field @ real ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_Real_Oreal___Rings_Olinordered__idom,axiom,
% 0.32/0.47 linordered_idom @ real ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_Real_Oreal___Rings_Ocomm__semiring__1_10,axiom,
% 0.32/0.47 comm_semiring_1 @ real ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_Real_Oreal___Rings_Ocomm__semiring__0_11,axiom,
% 0.32/0.47 comm_semiring_0 @ real ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_Real_Oreal___Fields_Odivision__ring,axiom,
% 0.32/0.47 division_ring @ real ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_Real_Oreal___Rings_Ozero__less__one_12,axiom,
% 0.32/0.47 zero_less_one @ real ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_Real_Oreal___Nat_Osemiring__char__0_13,axiom,
% 0.32/0.47 semiring_char_0 @ real ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_Real_Oreal___Fields_Ofield__char__0,axiom,
% 0.32/0.47 field_char_0 @ real ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_Real_Oreal___Rings_Ozero__neq__one_14,axiom,
% 0.32/0.47 zero_neq_one @ real ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_Real_Oreal___Rings_Oidom__divide,axiom,
% 0.32/0.47 idom_divide @ real ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_Real_Oreal___Rings_Ocomm__ring__1,axiom,
% 0.32/0.47 comm_ring_1 @ real ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_Real_Oreal___Orderings_Oorder_15,axiom,
% 0.32/0.47 order @ real ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_Real_Oreal___Nat_Oring__char__0,axiom,
% 0.32/0.47 ring_char_0 @ real ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_Real_Oreal___Rings_Oring__1,axiom,
% 0.32/0.47 ring_1 @ real ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_Real_Oreal___Fields_Ofield,axiom,
% 0.32/0.47 field @ real ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_Real_Oreal___Groups_Ozero_16,axiom,
% 0.32/0.47 zero @ real ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_Real_Oreal___Rings_Oidom,axiom,
% 0.32/0.47 idom @ real ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_Real_Oreal___Groups_Oone_17,axiom,
% 0.32/0.47 one @ real ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_Polynomial_Opoly___Rings_Osemiring__1__no__zero__divisors_18,axiom,
% 0.32/0.47 ! [A7: $tType] :
% 0.32/0.47 ( ( ( comm_semiring_1 @ A7 )
% 0.32/0.47 & ( semiri134348788visors @ A7 ) )
% 0.32/0.47 => ( semiri134348788visors @ ( poly @ A7 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_Polynomial_Opoly___Rings_Olinordered__nonzero__semiring_19,axiom,
% 0.32/0.47 ! [A7: $tType] :
% 0.32/0.47 ( ( linordered_idom @ A7 )
% 0.32/0.47 => ( linord1659791738miring @ ( poly @ A7 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_Polynomial_Opoly___Rings_Osemiring__no__zero__divisors_20,axiom,
% 0.32/0.47 ! [A7: $tType] :
% 0.32/0.47 ( ( ( comm_semiring_0 @ A7 )
% 0.32/0.47 & ( semiri1193490041visors @ A7 ) )
% 0.32/0.47 => ( semiri1193490041visors @ ( poly @ A7 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_Polynomial_Opoly___Rings_Oring__no__zero__divisors_21,axiom,
% 0.32/0.47 ! [A7: $tType] :
% 0.32/0.47 ( ( idom @ A7 )
% 0.32/0.47 => ( ring_n68954251visors @ ( poly @ A7 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_Polynomial_Opoly___Rings_Oalgebraic__semidom_22,axiom,
% 0.32/0.47 ! [A7: $tType] :
% 0.32/0.47 ( ( idom_divide @ A7 )
% 0.32/0.47 => ( algebraic_semidom @ ( poly @ A7 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_Polynomial_Opoly___Groups_Ocomm__monoid__mult_23,axiom,
% 0.32/0.47 ! [A7: $tType] :
% 0.32/0.47 ( ( comm_semiring_1 @ A7 )
% 0.32/0.47 => ( comm_monoid_mult @ ( poly @ A7 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_Polynomial_Opoly___Rings_Olinordered__idom_24,axiom,
% 0.32/0.47 ! [A7: $tType] :
% 0.32/0.47 ( ( linordered_idom @ A7 )
% 0.32/0.47 => ( linordered_idom @ ( poly @ A7 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_Polynomial_Opoly___Rings_Ocomm__semiring__1_25,axiom,
% 0.32/0.47 ! [A7: $tType] :
% 0.32/0.47 ( ( comm_semiring_1 @ A7 )
% 0.32/0.47 => ( comm_semiring_1 @ ( poly @ A7 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_Polynomial_Opoly___Rings_Ocomm__semiring__0_26,axiom,
% 0.32/0.47 ! [A7: $tType] :
% 0.32/0.47 ( ( comm_semiring_0 @ A7 )
% 0.32/0.47 => ( comm_semiring_0 @ ( poly @ A7 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_Polynomial_Opoly___Rings_Ozero__less__one_27,axiom,
% 0.32/0.47 ! [A7: $tType] :
% 0.32/0.47 ( ( linordered_idom @ A7 )
% 0.32/0.47 => ( zero_less_one @ ( poly @ A7 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_Polynomial_Opoly___Nat_Osemiring__char__0_28,axiom,
% 0.32/0.47 ! [A7: $tType] :
% 0.32/0.47 ( ( ( ring_char_0 @ A7 )
% 0.32/0.47 & ( comm_ring_1 @ A7 ) )
% 0.32/0.47 => ( semiring_char_0 @ ( poly @ A7 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_Polynomial_Opoly___Rings_Ozero__neq__one_29,axiom,
% 0.32/0.47 ! [A7: $tType] :
% 0.32/0.47 ( ( comm_semiring_1 @ A7 )
% 0.32/0.47 => ( zero_neq_one @ ( poly @ A7 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_Polynomial_Opoly___Rings_Oidom__divide_30,axiom,
% 0.32/0.47 ! [A7: $tType] :
% 0.32/0.47 ( ( idom_divide @ A7 )
% 0.32/0.47 => ( idom_divide @ ( poly @ A7 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_Polynomial_Opoly___Rings_Ocomm__ring__1_31,axiom,
% 0.32/0.47 ! [A7: $tType] :
% 0.32/0.47 ( ( comm_ring_1 @ A7 )
% 0.32/0.47 => ( comm_ring_1 @ ( poly @ A7 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_Polynomial_Opoly___Orderings_Oorder_32,axiom,
% 0.32/0.47 ! [A7: $tType] :
% 0.32/0.47 ( ( linordered_idom @ A7 )
% 0.32/0.47 => ( order @ ( poly @ A7 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_Polynomial_Opoly___Nat_Oring__char__0_33,axiom,
% 0.32/0.47 ! [A7: $tType] :
% 0.32/0.47 ( ( ( ring_char_0 @ A7 )
% 0.32/0.47 & ( comm_ring_1 @ A7 ) )
% 0.32/0.47 => ( ring_char_0 @ ( poly @ A7 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_Polynomial_Opoly___Rings_Oring__1_34,axiom,
% 0.32/0.47 ! [A7: $tType] :
% 0.32/0.47 ( ( comm_ring_1 @ A7 )
% 0.32/0.47 => ( ring_1 @ ( poly @ A7 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_Polynomial_Opoly___Groups_Ozero_35,axiom,
% 0.32/0.47 ! [A7: $tType] :
% 0.32/0.47 ( ( zero @ A7 )
% 0.32/0.47 => ( zero @ ( poly @ A7 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_Polynomial_Opoly___Rings_Oidom_36,axiom,
% 0.32/0.47 ! [A7: $tType] :
% 0.32/0.47 ( ( idom @ A7 )
% 0.32/0.47 => ( idom @ ( poly @ A7 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_Polynomial_Opoly___Groups_Oone_37,axiom,
% 0.32/0.47 ! [A7: $tType] :
% 0.32/0.47 ( ( comm_semiring_1 @ A7 )
% 0.32/0.47 => ( one @ ( poly @ A7 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % Conjectures (1)
% 0.32/0.47 thf(conj_0,conjecture,
% 0.32/0.47 ( finite_finite2 @ real
% 0.32/0.47 @ ( collect @ real
% 0.32/0.47 @ ^ [X: real] :
% 0.32/0.47 ( ( poly2 @ real @ ( pderiv @ real @ p ) @ X )
% 0.32/0.47 = ( zero_zero @ real ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 %------------------------------------------------------------------------------
% 0.32/0.47 ------------------------
% 0.32/1.50 ---- Embedded in THF ---
% 0.32/1.50 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 0.32/1.50 %%% Generated on Tue Feb 24 08:26:15 EST 2026
% 0.32/1.50 %%% using '$$dhol' embedding, version 1.3.0.
% 0.32/1.50 %%% Logic specification used:
% 0.32/1.50 %%% thf(spec, logic, $$dhol).
% 0.32/1.50
% 0.32/1.50 % SZS output start ListOfTHF for /export/starexec/sandbox/tmp/tmp.AWHgCUKr9y/DTF2DTF8563.p
% 0.32/1.50 thf(ty_t_Polynomial_Opoly, type, poly: ($tType > $tType)).
% 0.32/1.50 thf(ty_t_Real_Oreal, type, real: $tType).
% 0.32/1.50 thf(ty_t_Set_Oset, type, set: ($tType > $tType)).
% 0.32/1.50 thf(ty_t_Nat_Onat, type, nat: $tType).
% 0.32/1.50 thf(sy_cl_HOL_Otype, type, type: (!> [A:$tType]: ($o))).
% 0.32/1.50 thf(sy_cl_Groups_Oone, type, one: (!> [A:$tType]: ($o))).
% 0.32/1.50 thf(sy_cl_Rings_Oidom, type, idom: (!> [A:$tType]: ($o))).
% 0.32/1.50 thf(sy_cl_Groups_Ozero, type, zero: (!> [A:$tType]: ($o))).
% 0.32/1.50 thf(sy_cl_Fields_Ofield, type, field: (!> [A:$tType]: ($o))).
% 0.32/1.50 thf(sy_cl_Rings_Oring__1, type, ring_1: (!> [A:$tType]: ($o))).
% 0.32/1.50 thf(sy_cl_Nat_Oring__char__0, type, ring_char_0: (!> [A:$tType]: ($o))).
% 0.32/1.50 thf(sy_cl_Orderings_Oorder, type, order: (!> [A:$tType]: ($o))).
% 0.32/1.50 thf(sy_cl_Finite__Set_Ofinite, type, finite_finite: (!> [A:$tType]: ($o))).
% 0.32/1.50 thf(sy_cl_Rings_Ocomm__ring__1, type, comm_ring_1: (!> [A:$tType]: ($o))).
% 0.32/1.50 thf(sy_cl_Rings_Oidom__divide, type, idom_divide: (!> [A:$tType]: ($o))).
% 0.32/1.50 thf(sy_cl_Rings_Ozero__neq__one, type, zero_neq_one: (!> [A:$tType]: ($o))).
% 0.32/1.50 thf(sy_cl_Fields_Ofield__char__0, type, field_char_0: (!> [A:$tType]: ($o))).
% 0.32/1.50 thf(sy_cl_Nat_Osemiring__char__0, type, semiring_char_0: (!> [A:$tType]: ($o))).
% 0.32/1.50 thf(sy_cl_Orderings_Owellorder, type, wellorder: (!> [A:$tType]: ($o))).
% 0.32/1.50 thf(sy_cl_Rings_Ozero__less__one, type, zero_less_one: (!> [A:$tType]: ($o))).
% 0.32/1.50 thf(sy_cl_Fields_Odivision__ring, type, division_ring: (!> [A:$tType]: ($o))).
% 0.32/1.50 thf(sy_cl_Rings_Ocomm__semiring__0, type, comm_semiring_0: (!> [A:$tType]: ($o))).
% 0.32/1.50 thf(sy_cl_Rings_Ocomm__semiring__1, type, comm_semiring_1: (!> [A:$tType]: ($o))).
% 0.32/1.50 thf(sy_cl_Rings_Olinordered__idom, type, linordered_idom: (!> [A:$tType]: ($o))).
% 0.32/1.50 thf(sy_cl_Fields_Olinordered__field, type, linordered_field: (!> [A:$tType]: ($o))).
% 0.32/1.50 thf(sy_cl_Groups_Ocomm__monoid__mult, type, comm_monoid_mult: (!> [A:$tType]: ($o))).
% 0.32/1.50 thf(sy_cl_Rings_Oalgebraic__semidom, type, algebraic_semidom: (!> [A:$tType]: ($o))).
% 0.32/1.50 thf(sy_cl_Rings_Oring__no__zero__divisors, type, ring_n68954251visors: (!> [A:$tType]: ($o))).
% 0.32/1.50 thf(sy_cl_Rings_Osemiring__no__zero__divisors, type, semiri1193490041visors: (!> [A:$tType]: ($o))).
% 0.32/1.50 thf(sy_cl_Rings_Olinordered__nonzero__semiring, type, linord1659791738miring: (!> [A:$tType]: ($o))).
% 0.32/1.50 thf(sy_cl_Rings_Osemiring__1__no__zero__divisors, type, semiri134348788visors: (!> [A:$tType]: ($o))).
% 0.32/1.50 thf(sy_cl_Groups_Ocanonically__ordered__monoid__add, type, canoni770627133id_add: (!> [A:$tType]: ($o))).
% 0.32/1.50 thf(sy_c_Fields_Oinverse__class_Oinverse, type, inverse_inverse: (!> [A:$tType]: ((A > A)))).
% 0.32/1.50 thf(sy_c_Finite__Set_Ofinite, type, finite_finite2: (!> [A:$tType]: (((set @ A) > $o)))).
% 0.32/1.50 thf(sy_c_Groups_Oone__class_Oone, type, one_one: (!> [A:$tType]: (A))).
% 0.32/1.50 thf(sy_c_Groups_Ozero__class_Ozero, type, zero_zero: (!> [A:$tType]: (A))).
% 0.32/1.50 thf(sy_c_Int_Oring__1__class_OInts, type, ring_1_Ints: (!> [A:$tType]: ((set @ A)))).
% 0.32/1.50 thf(sy_c_Nat_OSuc, type, suc: (nat > nat)).
% 0.32/1.50 thf(sy_c_Orderings_Oord__class_Oless, type, ord_less: (!> [A:$tType]: ((A > (A > $o))))).
% 0.32/1.50 thf(sy_c_Orderings_Oord__class_Oless__eq, type, ord_less_eq: (!> [A:$tType]: ((A > (A > $o))))).
% 0.32/1.50 thf(sy_c_Polynomial_Oalgebraic, type, algebraic: (!> [A:$tType]: ((A > $o)))).
% 0.32/1.50 thf(sy_c_Polynomial_Oalgebraic__int, type, algebraic_int: (!> [A:$tType]: ((A > $o)))).
% 0.32/1.50 thf(sy_c_Polynomial_Ocr__poly, type, cr_poly: (!> [A:$tType]: (((nat > A) > ((poly @ A) > $o))))).
% 0.32/1.50 thf(sy_c_Polynomial_Odegree, type, degree: (!> [A:$tType]: (((poly @ A) > nat)))).
% 0.32/1.50 thf(sy_c_Polynomial_Odivide__poly__main, type, divide_poly_main: (!> [A:$tType]: ((A > ((poly @ A) > ((poly @ A) > ((poly @ A) > (nat > (nat > (poly @ A)))))))))).
% 0.32/1.50 thf(sy_c_Polynomial_Ois__zero, type, is_zero: (!> [A:$tType]: (((poly @ A) > $o)))).
% 0.32/1.50 thf(sy_c_Polynomial_Oorder, type, order2: (!> [A:$tType]: ((A > ((poly @ A) > nat))))).
% 0.32/1.50 thf(sy_c_Polynomial_Opcr__poly, type, pcr_poly: (!> [C:$tType,B:$tType]: (((C > (B > $o)) > ((nat > C) > ((poly @ B) > $o)))))).
% 0.32/1.50 thf(sy_c_Polynomial_Opderiv, type, pderiv: (!> [A:$tType]: (((poly @ A) > (poly @ A))))).
% 0.32/1.50 thf(sy_c_Polynomial_Opoly, type, poly2: (!> [A:$tType]: (((poly @ A) > (A > A))))).
% 0.32/1.50 thf(sy_c_Polynomial_Opoly_Ocoeff, type, coeff: (!> [A:$tType]: (((poly @ A) > (nat > A))))).
% 0.32/1.50 thf(sy_c_Polynomial_Opoly__cutoff, type, poly_cutoff: (!> [A:$tType]: ((nat > ((poly @ A) > (poly @ A)))))).
% 0.32/1.50 thf(sy_c_Polynomial_Opoly__shift, type, poly_shift: (!> [A:$tType]: ((nat > ((poly @ A) > (poly @ A)))))).
% 0.32/1.50 thf(sy_c_Polynomial_Opseudo__mod, type, pseudo_mod: (!> [A:$tType]: (((poly @ A) > ((poly @ A) > (poly @ A)))))).
% 0.32/1.50 thf(sy_c_Polynomial_Oreflect__poly, type, reflect_poly: (!> [A:$tType]: (((poly @ A) > (poly @ A))))).
% 0.32/1.50 thf(sy_c_Polynomial_Orsquarefree, type, rsquarefree: (!> [A:$tType]: (((poly @ A) > $o)))).
% 0.32/1.50 thf(sy_c_Polynomial_Osynthetic__div, type, synthetic_div: (!> [A:$tType]: (((poly @ A) > (A > (poly @ A)))))).
% 0.32/1.50 thf(sy_c_Rat_Ofield__char__0__class_ORats, type, field_char_0_Rats: (!> [A:$tType]: ((set @ A)))).
% 0.32/1.50 thf(sy_c_Rings_Odvd__class_Odvd, type, dvd_dvd: (!> [A:$tType]: ((A > (A > $o))))).
% 0.32/1.50 thf(sy_c_Set_OCollect, type, collect: (!> [A:$tType]: (((A > $o) > (set @ A))))).
% 0.32/1.50 thf(sy_c_member, type, member: (!> [A:$tType]: ((A > ((set @ A) > $o))))).
% 0.32/1.50 thf(sy_v_n____, type, n: nat).
% 0.32/1.50 thf(sy_v_p____, type, p: (poly @ real)).
% 0.32/1.50 thf(sy_v_x, type, x: real).
% 0.32/1.50 thf(fact_0_poly__roots__finite, axiom, (! [A:$tType]: ((((comm_ring_1 @ A) & (ring_n68954251visors @ A)) => (! [P:(poly @ A)]: (((P != (zero_zero @ (poly @ A))) => ((finite_finite2 @ A) @ ((collect @ A) @ (^ [X:A]: (((((poly2 @ A) @ P) @ X) = (zero_zero @ A))))))))))))).
% 0.32/1.50 thf(fact_1_p_I2_J, axiom, (p != (zero_zero @ (poly @ real)))).
% 0.32/1.50 thf(fact_2__092_060open_062pderiv_Ap_A_092_060noteq_062_A0_092_060close_062, axiom, (((pderiv @ real) @ p) != (zero_zero @ (poly @ real)))).
% 0.32/1.50 thf(fact_3_p_I3_J, axiom, ((((poly2 @ real) @ p) @ x) = (zero_zero @ real))).
% 0.32/1.50 thf(fact_4_poly__0, axiom, (! [A:$tType]: (((comm_semiring_0 @ A) => (! [X2:A]: (((((poly2 @ A) @ (zero_zero @ (poly @ A))) @ X2) = (zero_zero @ A)))))))).
% 0.32/1.50 thf(fact_5_finite__Collect__conjI, axiom, (! [A:$tType,P2:(A > $o),Q:(A > $o)]: (((((finite_finite2 @ A) @ ((collect @ A) @ P2)) | ((finite_finite2 @ A) @ ((collect @ A) @ Q))) => ((finite_finite2 @ A) @ ((collect @ A) @ (^ [X:A]: (((P2 @ X) & (Q @ X)))))))))).
% 0.32/1.50 thf(fact_6_finite__Collect__disjI, axiom, (! [A:$tType,P2:(A > $o),Q:(A > $o)]: ((((finite_finite2 @ A) @ ((collect @ A) @ (^ [X:A]: (((P2 @ X) | (Q @ X)))))) = (((finite_finite2 @ A) @ ((collect @ A) @ P2)) & ((finite_finite2 @ A) @ ((collect @ A) @ Q))))))).
% 0.32/1.50 thf(fact_7_finite__code, axiom, (! [A:$tType]: (((finite_finite @ A) => ((finite_finite2 @ A) = (^ [A2:(set @ A)]: ($true))))))).
% 0.32/1.50 thf(fact_8_poly__all__0__iff__0, axiom, (! [A:$tType]: ((((ring_char_0 @ A) & ((comm_ring_1 @ A) & (ring_n68954251visors @ A))) => (! [P:(poly @ A)]: (((! [X:A]: (((((poly2 @ A) @ P) @ X) = (zero_zero @ A)))) = (P = (zero_zero @ (poly @ A)))))))))).
% 0.32/1.50 thf(fact_9_rsquarefree__roots, axiom, (! [A:$tType]: (((field_char_0 @ A) => ((rsquarefree @ A) = (^ [P3:(poly @ A)]: ((! [A3:A]: ((~ ((((((poly2 @ A) @ P3) @ A3) = (zero_zero @ A)) & ((((poly2 @ A) @ ((pderiv @ A) @ P3)) @ A3) = (zero_zero @ A)))))))))))))).
% 0.32/1.50 thf(fact_10_not__finite__existsD, axiom, (! [A:$tType,P2:(A > $o)]: (((~ (((finite_finite2 @ A) @ ((collect @ A) @ P2)))) => (? [X_1:A]: ((P2 @ X_1))))))).
% 0.32/1.50 thf(fact_11_pigeonhole__infinite__rel, axiom, (! [B:$tType,A:$tType,A4:(set @ A),B2:(set @ B),R:(A > (B > $o))]: (((~ (((finite_finite2 @ A) @ A4))) => (((finite_finite2 @ B) @ B2) => ((! [X3:A]: (((((member @ A) @ X3) @ A4) => (? [Xa:B]: (((((member @ B) @ Xa) @ B2) & ((R @ X3) @ Xa))))))) => (? [X3:B]: (((((member @ B) @ X3) @ B2) & (~ (((finite_finite2 @ A) @ ((collect @ A) @ (^ [A3:A]: (((((member @ A) @ A3) @ A4) & ((R @ A3) @ X3))))))))))))))))).
% 0.32/1.50 thf(fact_12_fun__cong__unused__0, axiom, (! [A:$tType,B:$tType,C:$tType]: (((zero @ B) => (! [F:((A > B) > C),G:C]: (((F = (^ [X:(A > B)]: (G))) => ((F @ (^ [X:A]: ((zero_zero @ B)))) = G)))))))).
% 0.32/1.50 thf(fact_13_poly__eq__poly__eq__iff, axiom, (! [A:$tType]: ((((ring_char_0 @ A) & ((comm_ring_1 @ A) & (ring_n68954251visors @ A))) => (! [P:(poly @ A),Q2:(poly @ A)]: (((((poly2 @ A) @ P) = ((poly2 @ A) @ Q2)) = (P = Q2)))))))).
% 0.32/1.50 thf(fact_14_n__def, axiom, (n = ((degree @ real) @ p))).
% 0.32/1.50 thf(fact_15_assms_I2_J, axiom, ((algebraic @ real) @ x)).
% 0.32/1.50 thf(fact_16_irrationsl, axiom, (~ ((((member @ real) @ x) @ (field_char_0_Rats @ real))))).
% 0.32/1.50 thf(fact_17_degree__0, axiom, (! [A:$tType]: (((zero @ A) => (((degree @ A) @ (zero_zero @ (poly @ A))) = (zero_zero @ nat)))))).
% 0.32/1.50 thf(fact_18_pderiv__0, axiom, (! [A:$tType]: ((((comm_semiring_1 @ A) & (semiri1193490041visors @ A)) => (((pderiv @ A) @ (zero_zero @ (poly @ A))) = (zero_zero @ (poly @ A))))))).
% 0.32/1.50 thf(fact_19_pderiv__eq__0__iff, axiom, (! [A:$tType]: ((((semiring_char_0 @ A) & ((comm_semiring_1 @ A) & (semiri1193490041visors @ A))) => (! [P:(poly @ A)]: (((((pderiv @ A) @ P) = (zero_zero @ (poly @ A))) = (((degree @ A) @ P) = (zero_zero @ nat))))))))).
% 0.32/1.50 thf(fact_20_zero__reorient, axiom, (! [A:$tType]: (((zero @ A) => (! [X2:A]: ((((zero_zero @ A) = X2) = (X2 = (zero_zero @ A))))))))).
% 0.32/1.50 thf(fact_21_finite, axiom, (! [A:$tType]: (((finite_finite @ A) => (! [A4:(set @ A)]: (((finite_finite2 @ A) @ A4))))))).
% 0.32/1.50 thf(fact_22_finite__set__choice, axiom, (! [B:$tType,A:$tType,A4:(set @ A),P2:(A > (B > $o))]: ((((finite_finite2 @ A) @ A4) => ((! [X3:A]: (((((member @ A) @ X3) @ A4) => (? [X_12:B]: (((P2 @ X3) @ X_12)))))) => (? [F2:(A > B)]: ((! [X4:A]: (((((member @ A) @ X4) @ A4) => ((P2 @ X4) @ (F2 @ X4)))))))))))).
% 0.32/1.50 thf(fact_23_is__zero__null, axiom, (! [A:$tType]: (((zero @ A) => ((is_zero @ A) = (^ [P3:(poly @ A)]: ((P3 = (zero_zero @ (poly @ A)))))))))).
% 0.32/1.50 thf(fact_24_poly__cutoff__0, axiom, (! [A:$tType]: (((zero @ A) => (! [N:nat]: (((((poly_cutoff @ A) @ N) @ (zero_zero @ (poly @ A))) = (zero_zero @ (poly @ A))))))))).
% 0.32/1.50 thf(fact_25_reflect__poly__at__0__eq__0__iff, axiom, (! [A:$tType]: (((comm_semiring_0 @ A) => (! [P:(poly @ A)]: ((((((poly2 @ A) @ ((reflect_poly @ A) @ P)) @ (zero_zero @ A)) = (zero_zero @ A)) = (P = (zero_zero @ (poly @ A)))))))))).
% 0.32/1.50 thf(fact_26__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, (~ ((! [P4:(poly @ real)]: (((! [I:nat]: ((((member @ real) @ (((coeff @ real) @ P4) @ I)) @ (ring_1_Ints @ real)))) => ((P4 != (zero_zero @ (poly @ real))) => ((((poly2 @ real) @ P4) @ x) != (zero_zero @ real))))))))).
% 0.32/1.50 thf(fact_27_synthetic__div__eq__0__iff, axiom, (! [A:$tType]: (((comm_semiring_0 @ A) => (! [P:(poly @ A),C2:A]: ((((((synthetic_div @ A) @ P) @ C2) = (zero_zero @ (poly @ A))) = (((degree @ A) @ P) = (zero_zero @ nat))))))))).
% 0.32/1.50 thf(fact_28_poly__shift__0, axiom, (! [A:$tType]: (((zero @ A) => (! [N:nat]: (((((poly_shift @ A) @ N) @ (zero_zero @ (poly @ A))) = (zero_zero @ (poly @ A))))))))).
% 0.32/1.50 thf(fact_29_order__root, axiom, (! [A:$tType]: (((idom @ A) => (! [P:(poly @ A),A5:A]: ((((((poly2 @ A) @ P) @ A5) = (zero_zero @ A)) = ((P = (zero_zero @ (poly @ A))) | ((((order2 @ A) @ A5) @ P) != (zero_zero @ nat)))))))))).
% 0.32/1.50 thf(fact_30_leading__coeff__0__iff, axiom, (! [A:$tType]: (((zero @ A) => (! [P:(poly @ A)]: ((((((coeff @ A) @ P) @ ((degree @ A) @ P)) = (zero_zero @ A)) = (P = (zero_zero @ (poly @ A)))))))))).
% 0.32/1.50 thf(fact_31_divide__poly__main__0, axiom, (! [A:$tType]: (((idom_divide @ A) => (! [R2:(poly @ A),D:(poly @ A),Dr:nat,N:nat]: (((((((((divide_poly_main @ A) @ (zero_zero @ A)) @ (zero_zero @ (poly @ A))) @ R2) @ D) @ Dr) @ N) = (zero_zero @ (poly @ A))))))))).
% 0.32/1.50 thf(fact_32_zero__poly_Otransfer, axiom, (! [A:$tType]: (((zero @ A) => (((((pcr_poly @ A) @ A) @ (^ [Y:A,Z:A]: ((Y = Z)))) @ (^ [Uu:nat]: ((zero_zero @ A)))) @ (zero_zero @ (poly @ A))))))).
% 0.32/1.50 thf(fact_33_p_I1_J, axiom, (! [I2:nat]: ((((member @ real) @ (((coeff @ real) @ p) @ I2)) @ (ring_1_Ints @ real))))).
% 0.32/1.50 thf(fact_34_reflect__poly__0, axiom, (! [A:$tType]: (((zero @ A) => (((reflect_poly @ A) @ (zero_zero @ (poly @ A))) = (zero_zero @ (poly @ A))))))).
% 0.32/1.50 thf(fact_35_synthetic__div__0, axiom, (! [A:$tType]: (((comm_semiring_0 @ A) => (! [C2:A]: (((((synthetic_div @ A) @ (zero_zero @ (poly @ A))) @ C2) = (zero_zero @ (poly @ A))))))))).
% 0.32/1.50 thf(fact_36_poly__shift__id, axiom, (! [A:$tType]: (((zero @ A) => (((poly_shift @ A) @ (zero_zero @ nat)) = (^ [X:(poly @ A)]: (X))))))).
% 0.32/1.50 thf(fact_37_coeff__0, axiom, (! [A:$tType]: (((zero @ A) => (! [N:nat]: (((((coeff @ A) @ (zero_zero @ (poly @ A))) @ N) = (zero_zero @ A)))))))).
% 0.32/1.50 thf(fact_38_reflect__poly__reflect__poly, axiom, (! [A:$tType]: (((zero @ A) => (! [P:(poly @ A)]: ((((((coeff @ A) @ P) @ (zero_zero @ nat)) != (zero_zero @ A)) => (((reflect_poly @ A) @ ((reflect_poly @ A) @ P)) = P)))))))).
% 0.32/1.50 thf(fact_39_coeff__0__reflect__poly, axiom, (! [A:$tType]: (((zero @ A) => (! [P:(poly @ A)]: (((((coeff @ A) @ ((reflect_poly @ A) @ P)) @ (zero_zero @ nat)) = (((coeff @ A) @ P) @ ((degree @ A) @ P))))))))).
% 0.32/1.50 thf(fact_40_coeff__0__reflect__poly__0__iff, axiom, (! [A:$tType]: (((zero @ A) => (! [P:(poly @ A)]: ((((((coeff @ A) @ ((reflect_poly @ A) @ P)) @ (zero_zero @ nat)) = (zero_zero @ A)) = (P = (zero_zero @ (poly @ A)))))))))).
% 0.32/1.50 thf(fact_41_degree__reflect__poly__eq, axiom, (! [A:$tType]: (((zero @ A) => (! [P:(poly @ A)]: ((((((coeff @ A) @ P) @ (zero_zero @ nat)) != (zero_zero @ A)) => (((degree @ A) @ ((reflect_poly @ A) @ P)) = ((degree @ A) @ P))))))))).
% 0.32/1.50 thf(fact_42_poly__reflect__poly__0, axiom, (! [A:$tType]: (((comm_semiring_0 @ A) => (! [P:(poly @ A)]: (((((poly2 @ A) @ ((reflect_poly @ A) @ P)) @ (zero_zero @ A)) = (((coeff @ A) @ P) @ ((degree @ A) @ P))))))))).
% 0.32/1.50 thf(fact_43_divide__poly__main_Osimps_I2_J, axiom, (! [A:$tType]: (((idom_divide @ A) => (! [Lc:A,Q2:(poly @ A),R2:(poly @ A),D:(poly @ A),Dr:nat]: (((((((((divide_poly_main @ A) @ Lc) @ Q2) @ R2) @ D) @ Dr) @ (zero_zero @ nat)) = Q2))))))).
% 0.32/1.50 thf(fact_44_mem__Collect__eq, axiom, (! [A:$tType,A5:A,P2:(A > $o)]: (((((member @ A) @ A5) @ ((collect @ A) @ P2)) = (P2 @ A5))))).
% 0.32/1.50 thf(fact_45_Collect__mem__eq, axiom, (! [A:$tType,A4:(set @ A)]: ((((collect @ A) @ (^ [X:A]: ((((member @ A) @ X) @ A4)))) = A4)))).
% 0.32/1.50 thf(fact_46_Collect__cong, axiom, (! [A:$tType,P2:(A > $o),Q:(A > $o)]: (((! [X3:A]: (((P2 @ X3) = (Q @ X3)))) => (((collect @ A) @ P2) = ((collect @ A) @ Q)))))).
% 0.32/1.50 thf(fact_47_ext, axiom, (! [B:$tType,A:$tType,F:(A > B),G:(A > B)]: (((! [X3:A]: (((F @ X3) = (G @ X3)))) => (F = G))))).
% 0.32/1.50 thf(fact_48_poly__eqI, axiom, (! [A:$tType]: (((zero @ A) => (! [P:(poly @ A),Q2:(poly @ A)]: (((! [N2:nat]: (((((coeff @ A) @ P) @ N2) = (((coeff @ A) @ Q2) @ N2)))) => (P = Q2)))))))).
% 0.32/1.50 thf(fact_49_poly__eq__iff, axiom, (! [A:$tType]: (((zero @ A) => ((^ [Y:(poly @ A),Z:(poly @ A)]: ((Y = Z))) = (^ [P3:(poly @ A),Q3:(poly @ A)]: ((! [N3:nat]: (((((coeff @ A) @ P3) @ N3) = (((coeff @ A) @ Q3) @ N3))))))))))).
% 0.32/1.50 thf(fact_50_coeff__inject, axiom, (! [A:$tType]: (((zero @ A) => (! [X2:(poly @ A),Y2:(poly @ A)]: (((((coeff @ A) @ X2) = ((coeff @ A) @ Y2)) = (X2 = Y2)))))))).
% 0.32/1.50 thf(fact_51_algebraicE, axiom, (! [A:$tType]: (((field_char_0 @ A) => (! [X2:A]: ((((algebraic @ A) @ X2) => (~ ((! [P4:(poly @ A)]: (((! [I:nat]: ((((member @ A) @ (((coeff @ A) @ P4) @ I)) @ (ring_1_Ints @ A)))) => ((P4 != (zero_zero @ (poly @ A))) => ((((poly2 @ A) @ P4) @ X2) != (zero_zero @ A))))))))))))))).
% 0.32/1.50 thf(fact_52_algebraicI, axiom, (! [A:$tType]: (((field_char_0 @ A) => (! [P:(poly @ A),X2:A]: (((! [I3:nat]: ((((member @ A) @ (((coeff @ A) @ P) @ I3)) @ (ring_1_Ints @ A)))) => ((P != (zero_zero @ (poly @ A))) => (((((poly2 @ A) @ P) @ X2) = (zero_zero @ A)) => ((algebraic @ A) @ X2)))))))))).
% 0.32/1.50 thf(fact_53_algebraic__def, axiom, (! [A:$tType]: (((field_char_0 @ A) => ((algebraic @ A) = (^ [X:A]: ((? [P3:(poly @ A)]: (((! [I4:nat]: ((((member @ A) @ (((coeff @ A) @ P3) @ I4)) @ (ring_1_Ints @ A)))) & ((P3 != (zero_zero @ (poly @ A))) & ((((poly2 @ A) @ P3) @ X) = (zero_zero @ A))))))))))))).
% 0.32/1.50 thf(fact_54_algebraic__altdef, axiom, (! [B:$tType]: (((field_char_0 @ B) => ((algebraic @ B) = (^ [X:B]: ((? [P3:(poly @ B)]: (((! [I4:nat]: ((((member @ B) @ (((coeff @ B) @ P3) @ I4)) @ (field_char_0_Rats @ B)))) & ((P3 != (zero_zero @ (poly @ B))) & ((((poly2 @ B) @ P3) @ X) = (zero_zero @ B))))))))))))).
% 0.32/1.50 thf(fact_55_zero__poly_Orep__eq, axiom, (! [A:$tType]: (((zero @ A) => (((coeff @ A) @ (zero_zero @ (poly @ A))) = (^ [Uu:nat]: ((zero_zero @ A)))))))).
% 0.32/1.50 thf(fact_56_poly__0__coeff__0, axiom, (! [A:$tType]: (((comm_semiring_0 @ A) => (! [P:(poly @ A)]: (((((poly2 @ A) @ P) @ (zero_zero @ A)) = (((coeff @ A) @ P) @ (zero_zero @ nat))))))))).
% 0.32/1.50 thf(fact_57_order__0I, axiom, (! [A:$tType]: (((idom @ A) => (! [P:(poly @ A),A5:A]: ((((((poly2 @ A) @ P) @ A5) != (zero_zero @ A)) => ((((order2 @ A) @ A5) @ P) = (zero_zero @ nat))))))))).
% 0.32/1.50 thf(fact_58_leading__coeff__neq__0, axiom, (! [A:$tType]: (((zero @ A) => (! [P:(poly @ A)]: (((P != (zero_zero @ (poly @ A))) => ((((coeff @ A) @ P) @ ((degree @ A) @ P)) != (zero_zero @ A))))))))).
% 0.32/1.50 thf(fact_59_Rats__infinite, axiom, (! [A:$tType]: (((field_char_0 @ A) => (~ (((finite_finite2 @ A) @ (field_char_0_Rats @ A)))))))).
% 0.32/1.50 thf(fact_60_Rats__0, axiom, (! [A:$tType]: (((field_char_0 @ A) => (((member @ A) @ (zero_zero @ A)) @ (field_char_0_Rats @ A)))))).
% 0.32/1.50 thf(fact_61_Ints__0, axiom, (! [A:$tType]: (((ring_1 @ A) => (((member @ A) @ (zero_zero @ A)) @ (ring_1_Ints @ A)))))).
% 0.32/1.50 thf(fact_62_rsquarefree__def, axiom, (! [A:$tType]: (((idom @ A) => ((rsquarefree @ A) = (^ [P3:(poly @ A)]: (((P3 != (zero_zero @ (poly @ A))) & (! [A3:A]: ((((((order2 @ A) @ A3) @ P3) = (zero_zero @ nat)) | ((((order2 @ A) @ A3) @ P3) = (one_one @ nat))))))))))))).
% 0.32/1.50 thf(fact_63_poly__cutoff__1, axiom, (! [A:$tType]: (((comm_semiring_1 @ A) => (! [N:nat]: ((((N = (zero_zero @ nat)) => ((((poly_cutoff @ A) @ N) @ (one_one @ (poly @ A))) = (zero_zero @ (poly @ A)))) & ((N != (zero_zero @ nat)) => ((((poly_cutoff @ A) @ N) @ (one_one @ (poly @ A))) = (one_one @ (poly @ A))))))))))).
% 0.32/1.50 thf(fact_64_order__pderiv2, axiom, (! [A:$tType]: (((field_char_0 @ A) => (! [P:(poly @ A),A5:A,N:nat]: (((((pderiv @ A) @ P) != (zero_zero @ (poly @ A))) => (((((order2 @ A) @ A5) @ P) != (zero_zero @ nat)) => (((((order2 @ A) @ A5) @ ((pderiv @ A) @ P)) = N) = ((((order2 @ A) @ A5) @ P) = (suc @ N))))))))))).
% 0.32/1.50 thf(fact_65_order__pderiv, axiom, (! [A:$tType]: (((field_char_0 @ A) => (! [P:(poly @ A),A5:A]: (((((pderiv @ A) @ P) != (zero_zero @ (poly @ A))) => (((((order2 @ A) @ A5) @ P) != (zero_zero @ nat)) => ((((order2 @ A) @ A5) @ P) = (suc @ (((order2 @ A) @ A5) @ ((pderiv @ A) @ P)))))))))))).
% 0.32/1.50 thf(fact_66_cr__poly__def, axiom, (! [A:$tType]: (((zero @ A) => ((cr_poly @ A) = (^ [X:(nat > A),Y3:(poly @ A)]: ((X = ((coeff @ A) @ Y3))))))))).
% 0.32/1.50 thf(fact_67_dvd__pderiv__iff, axiom, (! [A:$tType]: ((((semiring_char_0 @ A) & ((comm_semiring_1 @ A) & (semiri1193490041visors @ A))) => (! [P:(poly @ A)]: (((((dvd_dvd @ (poly @ A)) @ P) @ ((pderiv @ A) @ P)) = (((degree @ A) @ P) = (zero_zero @ nat))))))))).
% 0.32/1.50 thf(fact_68_reflect__poly__1, axiom, (! [A:$tType]: (((comm_semiring_1 @ A) => (((reflect_poly @ A) @ (one_one @ (poly @ A))) = (one_one @ (poly @ A))))))).
% 0.32/1.50 thf(fact_69_degree__1, axiom, (! [A:$tType]: (((comm_semiring_1 @ A) => (((degree @ A) @ (one_one @ (poly @ A))) = (zero_zero @ nat)))))).
% 0.32/1.50 thf(fact_70_poly__1, axiom, (! [A:$tType]: (((comm_semiring_1 @ A) => (! [X2:A]: (((((poly2 @ A) @ (one_one @ (poly @ A))) @ X2) = (one_one @ A)))))))).
% 0.32/1.50 thf(fact_71_pderiv__1, axiom, (! [A:$tType]: ((((comm_semiring_1 @ A) & (semiri1193490041visors @ A)) => (((pderiv @ A) @ (one_one @ (poly @ A))) = (zero_zero @ (poly @ A))))))).
% 0.32/1.50 thf(fact_72_order__1__eq__0, axiom, (! [A:$tType]: (((idom @ A) => (! [X2:A]: (((((order2 @ A) @ X2) @ (one_one @ (poly @ A))) = (zero_zero @ nat)))))))).
% 0.32/1.50 thf(fact_73_lead__coeff__1, axiom, (! [A:$tType]: (((comm_semiring_1 @ A) => ((((coeff @ A) @ (one_one @ (poly @ A))) @ ((degree @ A) @ (one_one @ (poly @ A)))) = (one_one @ A)))))).
% 0.32/1.50 thf(fact_74_one__reorient, axiom, (! [A:$tType]: (((one @ A) => (! [X2:A]: ((((one_one @ A) = X2) = (X2 = (one_one @ A))))))))).
% 0.32/1.50 thf(fact_75_Ints__1, axiom, (! [A:$tType]: (((ring_1 @ A) => (((member @ A) @ (one_one @ A)) @ (ring_1_Ints @ A)))))).
% 0.32/1.50 thf(fact_76_Rats__1, axiom, (! [A:$tType]: (((field_char_0 @ A) => (((member @ A) @ (one_one @ A)) @ (field_char_0_Rats @ A)))))).
% 0.32/1.50 thf(fact_77_is__unit__iff__degree, axiom, (! [A:$tType]: (((field @ A) => (! [P:(poly @ A)]: (((P != (zero_zero @ (poly @ A))) => ((((dvd_dvd @ (poly @ A)) @ P) @ (one_one @ (poly @ A))) = (((degree @ A) @ P) = (zero_zero @ nat)))))))))).
% 0.32/1.50 thf(fact_78_not__dvd__pderiv, axiom, (! [A:$tType]: ((((semiring_char_0 @ A) & ((comm_semiring_1 @ A) & (semiri1193490041visors @ A))) => (! [P:(poly @ A)]: (((((degree @ A) @ P) != (zero_zero @ nat)) => (~ ((((dvd_dvd @ (poly @ A)) @ P) @ ((pderiv @ A) @ P))))))))))).
% 0.32/1.50 thf(fact_79_poly_Opcr__cr__eq, axiom, (! [D2:$tType]: (((zero @ D2) => ((((pcr_poly @ D2) @ D2) @ (^ [Y:D2,Z:D2]: ((Y = Z)))) = (cr_poly @ D2)))))).
% 0.32/1.50 thf(fact_80_poly__shift__1, axiom, (! [A:$tType]: (((comm_semiring_1 @ A) => (! [N:nat]: ((((N = (zero_zero @ nat)) => ((((poly_shift @ A) @ N) @ (one_one @ (poly @ A))) = (one_one @ (poly @ A)))) & ((N != (zero_zero @ nat)) => ((((poly_shift @ A) @ N) @ (one_one @ (poly @ A))) = (zero_zero @ (poly @ A))))))))))).
% 0.32/1.50 thf(fact_81_dvd__0__left__iff, axiom, (! [A:$tType]: (((comm_semiring_1 @ A) => (! [A5:A]: (((((dvd_dvd @ A) @ (zero_zero @ A)) @ A5) = (A5 = (zero_zero @ A))))))))).
% 0.32/1.50 thf(fact_82_dvd__0__right, axiom, (! [A:$tType]: (((comm_semiring_1 @ A) => (! [A5:A]: ((((dvd_dvd @ A) @ A5) @ (zero_zero @ A)))))))).
% 0.32/1.50 thf(fact_83_One__nat__def, axiom, ((one_one @ nat) = (suc @ (zero_zero @ nat)))).
% 0.32/1.50 thf(fact_84_not__is__unit__0, axiom, (! [A:$tType]: (((algebraic_semidom @ A) => (~ ((((dvd_dvd @ A) @ (zero_zero @ A)) @ (one_one @ A)))))))).
% 0.32/1.50 thf(fact_85_nat_Oinject, axiom, (! [X22:nat,Y22:nat]: ((((suc @ X22) = (suc @ Y22)) = (X22 = Y22))))).
% 0.32/1.50 thf(fact_86_old_Onat_Oinject, axiom, (! [Nat:nat,Nat2:nat]: ((((suc @ Nat) = (suc @ Nat2)) = (Nat = Nat2))))).
% 0.32/1.50 thf(fact_87_nat__dvd__1__iff__1, axiom, (! [M:nat]: (((((dvd_dvd @ nat) @ M) @ (one_one @ nat)) = (M = (one_one @ nat)))))).
% 0.32/1.50 thf(fact_88_dvd__1__iff__1, axiom, (! [M:nat]: (((((dvd_dvd @ nat) @ M) @ (suc @ (zero_zero @ nat))) = (M = (suc @ (zero_zero @ nat))))))).
% 0.32/1.50 thf(fact_89_dvd__1__left, axiom, (! [K:nat]: ((((dvd_dvd @ nat) @ (suc @ (zero_zero @ nat))) @ K)))).
% 0.32/1.50 thf(fact_90_dvd__trans, axiom, (! [A:$tType]: (((comm_monoid_mult @ A) => (! [A5:A,B3:A,C2:A]: (((((dvd_dvd @ A) @ A5) @ B3) => ((((dvd_dvd @ A) @ B3) @ C2) => (((dvd_dvd @ A) @ A5) @ C2))))))))).
% 0.32/1.50 thf(fact_91_dvd__refl, axiom, (! [A:$tType]: (((comm_monoid_mult @ A) => (! [A5:A]: ((((dvd_dvd @ A) @ A5) @ A5))))))).
% 0.32/1.50 thf(fact_92_n__not__Suc__n, axiom, (! [N:nat]: ((N != (suc @ N))))).
% 0.32/1.50 thf(fact_93_Suc__inject, axiom, (! [X2:nat,Y2:nat]: ((((suc @ X2) = (suc @ Y2)) => (X2 = Y2))))).
% 0.32/1.50 thf(fact_94_zero__neq__one, axiom, (! [A:$tType]: (((zero_neq_one @ A) => ((zero_zero @ A) != (one_one @ A)))))).
% 0.32/1.50 thf(fact_95_dvd__0__left, axiom, (! [A:$tType]: (((comm_semiring_1 @ A) => (! [A5:A]: (((((dvd_dvd @ A) @ (zero_zero @ A)) @ A5) => (A5 = (zero_zero @ A))))))))).
% 0.32/1.50 thf(fact_96_dvd__unit__imp__unit, axiom, (! [A:$tType]: (((algebraic_semidom @ A) => (! [A5:A,B3:A]: (((((dvd_dvd @ A) @ A5) @ B3) => ((((dvd_dvd @ A) @ B3) @ (one_one @ A)) => (((dvd_dvd @ A) @ A5) @ (one_one @ A)))))))))).
% 0.32/1.50 thf(fact_97_unit__imp__dvd, axiom, (! [A:$tType]: (((algebraic_semidom @ A) => (! [B3:A,A5:A]: (((((dvd_dvd @ A) @ B3) @ (one_one @ A)) => (((dvd_dvd @ A) @ B3) @ A5)))))))).
% 0.32/1.50 thf(fact_98_one__dvd, axiom, (! [A:$tType]: (((comm_monoid_mult @ A) => (! [A5:A]: ((((dvd_dvd @ A) @ (one_one @ A)) @ A5))))))).
% 0.32/1.50 thf(fact_99_not0__implies__Suc, axiom, (! [N:nat]: (((N != (zero_zero @ nat)) => (? [M2:nat]: ((N = (suc @ M2)))))))).
% 0.32/1.50 thf(fact_100_old_Onat_Oinducts, axiom, (! [P2:(nat > $o),Nat:nat]: (((P2 @ (zero_zero @ nat)) => ((! [Nat3:nat]: (((P2 @ Nat3) => (P2 @ (suc @ Nat3))))) => (P2 @ Nat)))))).
% 0.32/1.50 thf(fact_101_old_Onat_Oexhaust, axiom, (! [Y2:nat]: (((Y2 != (zero_zero @ nat)) => (~ ((! [Nat3:nat]: ((Y2 != (suc @ Nat3)))))))))).
% 0.32/1.50 thf(fact_102_Zero__not__Suc, axiom, (! [M:nat]: (((zero_zero @ nat) != (suc @ M))))).
% 0.32/1.50 thf(fact_103_Zero__neq__Suc, axiom, (! [M:nat]: (((zero_zero @ nat) != (suc @ M))))).
% 0.32/1.50 thf(fact_104_Suc__neq__Zero, axiom, (! [M:nat]: (((suc @ M) != (zero_zero @ nat))))).
% 0.32/1.50 thf(fact_105_zero__induct, axiom, (! [P2:(nat > $o),K:nat]: (((P2 @ K) => ((! [N2:nat]: (((P2 @ (suc @ N2)) => (P2 @ N2)))) => (P2 @ (zero_zero @ nat))))))).
% 0.32/1.50 thf(fact_106_diff__induct, axiom, (! [P2:(nat > (nat > $o)),M:nat,N:nat]: (((! [X3:nat]: (((P2 @ X3) @ (zero_zero @ nat)))) => ((! [Y4:nat]: (((P2 @ (zero_zero @ nat)) @ (suc @ Y4)))) => ((! [X3:nat,Y4:nat]: ((((P2 @ X3) @ Y4) => ((P2 @ (suc @ X3)) @ (suc @ Y4))))) => ((P2 @ M) @ N))))))).
% 0.32/1.50 thf(fact_107_nat__induct, axiom, (! [P2:(nat > $o),N:nat]: (((P2 @ (zero_zero @ nat)) => ((! [N2:nat]: (((P2 @ N2) => (P2 @ (suc @ N2))))) => (P2 @ N)))))).
% 0.32/1.50 thf(fact_108_nat_OdiscI, axiom, (! [Nat:nat,X22:nat]: (((Nat = (suc @ X22)) => (Nat != (zero_zero @ nat)))))).
% 0.32/1.50 thf(fact_109_old_Onat_Odistinct_I1_J, axiom, (! [Nat2:nat]: (((zero_zero @ nat) != (suc @ Nat2))))).
% 0.32/1.50 thf(fact_110_old_Onat_Odistinct_I2_J, axiom, (! [Nat4:nat]: (((suc @ Nat4) != (zero_zero @ nat))))).
% 0.32/1.50 thf(fact_111_nat_Odistinct_I1_J, axiom, (! [X22:nat]: (((zero_zero @ nat) != (suc @ X22))))).
% 0.32/1.50 thf(fact_112_algebraic__int_Oinducts, axiom, (! [A:$tType]: (((field @ A) => (! [X2:A,P2:(A > $o)]: ((((algebraic_int @ A) @ X2) => ((! [P4:(poly @ A),X3:A]: ((((((coeff @ A) @ P4) @ ((degree @ A) @ P4)) = (one_one @ A)) => ((! [I:nat]: ((((member @ A) @ (((coeff @ A) @ P4) @ I)) @ (ring_1_Ints @ A)))) => (((((poly2 @ A) @ P4) @ X3) = (zero_zero @ A)) => (P2 @ X3)))))) => (P2 @ X2))))))))).
% 0.32/1.50 thf(fact_113_algebraic__int_Ointros, axiom, (! [A:$tType]: (((field @ A) => (! [P:(poly @ A),X2:A]: ((((((coeff @ A) @ P) @ ((degree @ A) @ P)) = (one_one @ A)) => ((! [I3:nat]: ((((member @ A) @ (((coeff @ A) @ P) @ I3)) @ (ring_1_Ints @ A)))) => (((((poly2 @ A) @ P) @ X2) = (zero_zero @ A)) => ((algebraic_int @ A) @ X2)))))))))).
% 0.32/1.50 thf(fact_114_algebraic__int_Osimps, axiom, (! [A:$tType]: (((field @ A) => ((algebraic_int @ A) = (^ [A3:A]: ((? [P3:(poly @ A),X:A]: (((A3 = X) & (((((coeff @ A) @ P3) @ ((degree @ A) @ P3)) = (one_one @ A)) & ((! [I4:nat]: ((((member @ A) @ (((coeff @ A) @ P3) @ I4)) @ (ring_1_Ints @ A)))) & ((((poly2 @ A) @ P3) @ X) = (zero_zero @ A)))))))))))))).
% 0.32/1.50 thf(fact_115_algebraic__int_Ocases, axiom, (! [A:$tType]: (((field @ A) => (! [A5:A]: ((((algebraic_int @ A) @ A5) => (~ ((! [P4:(poly @ A)]: ((((((coeff @ A) @ P4) @ ((degree @ A) @ P4)) = (one_one @ A)) => ((! [I:nat]: ((((member @ A) @ (((coeff @ A) @ P4) @ I)) @ (ring_1_Ints @ A)))) => ((((poly2 @ A) @ P4) @ A5) != (zero_zero @ A))))))))))))))).
% 0.32/1.50 thf(fact_116_exists__least__lemma, axiom, (! [P2:(nat > $o)]: (((~ ((P2 @ (zero_zero @ nat)))) => ((? [X_12:nat]: ((P2 @ X_12))) => (? [N2:nat]: (((~ ((P2 @ N2))) & (P2 @ (suc @ N2)))))))))).
% 0.32/1.50 thf(fact_117_dependent__nat__choice, axiom, (! [A:$tType,P2:(nat > (A > $o)),Q:(nat > (A > (A > $o)))]: (((? [X_12:A]: (((P2 @ (zero_zero @ nat)) @ X_12))) => ((! [X3:A,N2:nat]: ((((P2 @ N2) @ X3) => (? [Y5:A]: ((((P2 @ (suc @ N2)) @ Y5) & (((Q @ N2) @ X3) @ Y5))))))) => (? [F2:(nat > A)]: ((! [N4:nat]: ((((P2 @ N4) @ (F2 @ N4)) & (((Q @ N4) @ (F2 @ N4)) @ (F2 @ (suc @ N4))))))))))))).
% 0.32/1.50 thf(fact_118_algebraic__int__0, axiom, (! [A:$tType]: (((field @ A) => ((algebraic_int @ A) @ (zero_zero @ A)))))).
% 0.32/1.50 thf(fact_119_algebraic__int__1, axiom, (! [B:$tType]: (((field @ B) => ((algebraic_int @ B) @ (one_one @ B)))))).
% 0.32/1.50 thf(fact_120_dvd__antisym, axiom, (! [M:nat,N:nat]: (((((dvd_dvd @ nat) @ M) @ N) => ((((dvd_dvd @ nat) @ N) @ M) => (M = N)))))).
% 0.32/1.50 thf(fact_121_int__imp__algebraic__int, axiom, (! [A:$tType]: (((field @ A) => (! [X2:A]: (((((member @ A) @ X2) @ (ring_1_Ints @ A)) => ((algebraic_int @ A) @ X2)))))))).
% 0.32/1.50 thf(fact_122_algebraic__int__imp__algebraic, axiom, (! [A:$tType]: (((field_char_0 @ A) => (! [X2:A]: ((((algebraic_int @ A) @ X2) => ((algebraic @ A) @ X2)))))))).
% 0.32/1.50 thf(fact_123_rational__algebraic__int__is__int, axiom, (! [A:$tType]: (((field_char_0 @ A) => (! [X2:A]: ((((algebraic_int @ A) @ X2) => ((((member @ A) @ X2) @ (field_char_0_Rats @ A)) => (((member @ A) @ X2) @ (ring_1_Ints @ A)))))))))).
% 0.32/1.50 thf(fact_124_algebraic__int__inverse, axiom, (! [A:$tType]: (((field @ A) => (! [P:(poly @ A),X2:A]: ((((((poly2 @ A) @ P) @ X2) = (zero_zero @ A)) => ((! [I3:nat]: ((((member @ A) @ (((coeff @ A) @ P) @ I3)) @ (ring_1_Ints @ A)))) => (((((coeff @ A) @ P) @ (zero_zero @ nat)) = (one_one @ A)) => ((algebraic_int @ A) @ ((inverse_inverse @ A) @ X2))))))))))).
% 0.32/1.50 thf(fact_125_algebraic__int__root, axiom, (! [A:$tType]: (((field @ A) => (! [Y2:A,P:(poly @ A),X2:A]: ((((algebraic_int @ A) @ Y2) => (((((poly2 @ A) @ P) @ X2) = Y2) => ((! [I3:nat]: ((((member @ A) @ (((coeff @ A) @ P) @ I3)) @ (ring_1_Ints @ A)))) => (((((coeff @ A) @ P) @ ((degree @ A) @ P)) = (one_one @ A)) => ((((ord_less @ nat) @ (zero_zero @ nat)) @ ((degree @ A) @ P)) => ((algebraic_int @ A) @ X2)))))))))))).
% 0.32/1.50 thf(fact_126_gcd__nat_Oextremum__uniqueI, axiom, (! [A5:nat]: (((((dvd_dvd @ nat) @ (zero_zero @ nat)) @ A5) => (A5 = (zero_zero @ nat)))))).
% 0.32/1.50 thf(fact_127_gcd__nat_Onot__eq__extremum, axiom, (! [A5:nat]: (((A5 != (zero_zero @ nat)) = ((((dvd_dvd @ nat) @ A5) @ (zero_zero @ nat)) & (A5 != (zero_zero @ nat))))))).
% 0.32/1.50 thf(fact_128_finite__Collect__less__nat, axiom, (! [K:nat]: (((finite_finite2 @ nat) @ ((collect @ nat) @ (^ [N3:nat]: ((((ord_less @ nat) @ N3) @ K)))))))).
% 0.32/1.50 thf(fact_129_not__gr__zero, axiom, (! [A:$tType]: (((canoni770627133id_add @ A) => (! [N:A]: (((~ ((((ord_less @ A) @ (zero_zero @ A)) @ N))) = (N = (zero_zero @ A))))))))).
% 0.32/1.50 thf(fact_130_bot__nat__0_Onot__eq__extremum, axiom, (! [A5:nat]: (((A5 != (zero_zero @ nat)) = (((ord_less @ nat) @ (zero_zero @ nat)) @ A5))))).
% 0.32/1.50 thf(fact_131_less__nat__zero__code, axiom, (! [N:nat]: ((~ ((((ord_less @ nat) @ N) @ (zero_zero @ nat))))))).
% 0.32/1.50 thf(fact_132_neq0__conv, axiom, (! [N:nat]: (((N != (zero_zero @ nat)) = (((ord_less @ nat) @ (zero_zero @ nat)) @ N))))).
% 0.32/1.50 thf(fact_133_lessI, axiom, (! [N:nat]: ((((ord_less @ nat) @ N) @ (suc @ N))))).
% 0.32/1.50 thf(fact_134_Suc__mono, axiom, (! [M:nat,N:nat]: (((((ord_less @ nat) @ M) @ N) => (((ord_less @ nat) @ (suc @ M)) @ (suc @ N)))))).
% 0.32/1.50 thf(fact_135_Suc__less__eq, axiom, (! [M:nat,N:nat]: (((((ord_less @ nat) @ (suc @ M)) @ (suc @ N)) = (((ord_less @ nat) @ M) @ N))))).
% 0.32/1.50 thf(fact_136_zero__less__Suc, axiom, (! [N:nat]: ((((ord_less @ nat) @ (zero_zero @ nat)) @ (suc @ N))))).
% 0.32/1.50 thf(fact_137_less__Suc0, axiom, (! [N:nat]: (((((ord_less @ nat) @ N) @ (suc @ (zero_zero @ nat))) = (N = (zero_zero @ nat)))))).
% 0.32/1.50 thf(fact_138_less__one, axiom, (! [N:nat]: (((((ord_less @ nat) @ N) @ (one_one @ nat)) = (N = (zero_zero @ nat)))))).
% 0.32/1.50 thf(fact_139_zero__less__iff__neq__zero, axiom, (! [A:$tType]: (((canoni770627133id_add @ A) => (! [N:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ N) = (N != (zero_zero @ A))))))))).
% 0.32/1.50 thf(fact_140_gr__implies__not__zero, axiom, (! [A:$tType]: (((canoni770627133id_add @ A) => (! [M:A,N:A]: (((((ord_less @ A) @ M) @ N) => (N != (zero_zero @ A))))))))).
% 0.32/1.50 thf(fact_141_not__less__zero, axiom, (! [A:$tType]: (((canoni770627133id_add @ A) => (! [N:A]: ((~ ((((ord_less @ A) @ N) @ (zero_zero @ A)))))))))).
% 0.32/1.50 thf(fact_142_gr__zeroI, axiom, (! [A:$tType]: (((canoni770627133id_add @ A) => (! [N:A]: (((N != (zero_zero @ A)) => (((ord_less @ A) @ (zero_zero @ A)) @ N)))))))).
% 0.32/1.50 thf(fact_143_gr0I, axiom, (! [N:nat]: (((N != (zero_zero @ nat)) => (((ord_less @ nat) @ (zero_zero @ nat)) @ N))))).
% 0.32/1.50 thf(fact_144_not__gr0, axiom, (! [N:nat]: (((~ ((((ord_less @ nat) @ (zero_zero @ nat)) @ N))) = (N = (zero_zero @ nat)))))).
% 0.32/1.50 thf(fact_145_not__less0, axiom, (! [N:nat]: ((~ ((((ord_less @ nat) @ N) @ (zero_zero @ nat))))))).
% 0.32/1.50 thf(fact_146_less__zeroE, axiom, (! [N:nat]: ((~ ((((ord_less @ nat) @ N) @ (zero_zero @ nat))))))).
% 0.32/1.50 thf(fact_147_gr__implies__not0, axiom, (! [M:nat,N:nat]: (((((ord_less @ nat) @ M) @ N) => (N != (zero_zero @ nat)))))).
% 0.32/1.50 thf(fact_148_infinite__descent0, axiom, (! [P2:(nat > $o),N:nat]: (((P2 @ (zero_zero @ nat)) => ((! [N2:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ N2) => ((~ ((P2 @ N2))) => (? [M3:nat]: (((((ord_less @ nat) @ M3) @ N2) & (~ ((P2 @ M3)))))))))) => (P2 @ N)))))).
% 0.32/1.50 thf(fact_149_bot__nat__0_Oextremum__strict, axiom, (! [A5:nat]: ((~ ((((ord_less @ nat) @ A5) @ (zero_zero @ nat))))))).
% 0.32/1.50 thf(fact_150_infinite__descent0__measure, axiom, (! [A:$tType,V:(A > nat),P2:(A > $o),X2:A]: (((! [X3:A]: ((((V @ X3) = (zero_zero @ nat)) => (P2 @ X3)))) => ((! [X3:A]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ (V @ X3)) => ((~ ((P2 @ X3))) => (? [Y5:A]: (((((ord_less @ nat) @ (V @ Y5)) @ (V @ X3)) & (~ ((P2 @ Y5)))))))))) => (P2 @ X2)))))).
% 0.32/1.50 thf(fact_151_not__less__less__Suc__eq, axiom, (! [N:nat,M:nat]: (((~ ((((ord_less @ nat) @ N) @ M))) => ((((ord_less @ nat) @ N) @ (suc @ M)) = (N = M)))))).
% 0.32/1.50 thf(fact_152_strict__inc__induct, axiom, (! [I2:nat,J:nat,P2:(nat > $o)]: (((((ord_less @ nat) @ I2) @ J) => ((! [I3:nat]: (((J = (suc @ I3)) => (P2 @ I3)))) => ((! [I3:nat]: (((((ord_less @ nat) @ I3) @ J) => ((P2 @ (suc @ I3)) => (P2 @ I3))))) => (P2 @ I2))))))).
% 0.32/1.50 thf(fact_153_less__Suc__induct, axiom, (! [I2:nat,J:nat,P2:(nat > (nat > $o))]: (((((ord_less @ nat) @ I2) @ J) => ((! [I3:nat]: (((P2 @ I3) @ (suc @ I3)))) => ((! [I3:nat,J2:nat,K2:nat]: (((((ord_less @ nat) @ I3) @ J2) => ((((ord_less @ nat) @ J2) @ K2) => (((P2 @ I3) @ J2) => (((P2 @ J2) @ K2) => ((P2 @ I3) @ K2))))))) => ((P2 @ I2) @ J))))))).
% 0.32/1.50 thf(fact_154_less__trans__Suc, axiom, (! [I2:nat,J:nat,K:nat]: (((((ord_less @ nat) @ I2) @ J) => ((((ord_less @ nat) @ J) @ K) => (((ord_less @ nat) @ (suc @ I2)) @ K)))))).
% 0.32/1.50 thf(fact_155_Suc__less__SucD, axiom, (! [M:nat,N:nat]: (((((ord_less @ nat) @ (suc @ M)) @ (suc @ N)) => (((ord_less @ nat) @ M) @ N))))).
% 0.32/1.50 thf(fact_156_less__antisym, axiom, (! [N:nat,M:nat]: (((~ ((((ord_less @ nat) @ N) @ M))) => ((((ord_less @ nat) @ N) @ (suc @ M)) => (M = N)))))).
% 0.32/1.50 thf(fact_157_Suc__less__eq2, axiom, (! [N:nat,M:nat]: (((((ord_less @ nat) @ (suc @ N)) @ M) = (? [M4:nat]: (((M = (suc @ M4)) & (((ord_less @ nat) @ N) @ M4)))))))).
% 0.32/1.50 thf(fact_158_All__less__Suc, axiom, (! [N:nat,P2:(nat > $o)]: (((! [I4:nat]: (((((ord_less @ nat) @ I4) @ (suc @ N)) => (P2 @ I4)))) = ((P2 @ N) & (! [I4:nat]: (((((ord_less @ nat) @ I4) @ N) => (P2 @ I4))))))))).
% 0.32/1.50 thf(fact_159_not__less__eq, axiom, (! [M:nat,N:nat]: (((~ ((((ord_less @ nat) @ M) @ N))) = (((ord_less @ nat) @ N) @ (suc @ M)))))).
% 0.32/1.50 thf(fact_160_less__Suc__eq, axiom, (! [M:nat,N:nat]: (((((ord_less @ nat) @ M) @ (suc @ N)) = ((((ord_less @ nat) @ M) @ N) | (M = N)))))).
% 0.32/1.50 thf(fact_161_Ex__less__Suc, axiom, (! [N:nat,P2:(nat > $o)]: (((? [I4:nat]: (((((ord_less @ nat) @ I4) @ (suc @ N)) & (P2 @ I4)))) = ((P2 @ N) | (? [I4:nat]: (((((ord_less @ nat) @ I4) @ N) & (P2 @ I4))))))))).
% 0.32/1.50 thf(fact_162_less__SucI, axiom, (! [M:nat,N:nat]: (((((ord_less @ nat) @ M) @ N) => (((ord_less @ nat) @ M) @ (suc @ N)))))).
% 0.32/1.50 thf(fact_163_less__SucE, axiom, (! [M:nat,N:nat]: (((((ord_less @ nat) @ M) @ (suc @ N)) => ((~ ((((ord_less @ nat) @ M) @ N))) => (M = N)))))).
% 0.32/1.50 thf(fact_164_Suc__lessI, axiom, (! [M:nat,N:nat]: (((((ord_less @ nat) @ M) @ N) => (((suc @ M) != N) => (((ord_less @ nat) @ (suc @ M)) @ N)))))).
% 0.32/1.50 thf(fact_165_Suc__lessE, axiom, (! [I2:nat,K:nat]: (((((ord_less @ nat) @ (suc @ I2)) @ K) => (~ ((! [J2:nat]: (((((ord_less @ nat) @ I2) @ J2) => (K != (suc @ J2))))))))))).
% 0.32/1.50 thf(fact_166_Suc__lessD, axiom, (! [M:nat,N:nat]: (((((ord_less @ nat) @ (suc @ M)) @ N) => (((ord_less @ nat) @ M) @ N))))).
% 0.32/1.50 thf(fact_167_Nat_OlessE, axiom, (! [I2:nat,K:nat]: (((((ord_less @ nat) @ I2) @ K) => ((K != (suc @ I2)) => (~ ((! [J2:nat]: (((((ord_less @ nat) @ I2) @ J2) => (K != (suc @ J2)))))))))))).
% 0.32/1.50 thf(fact_168_Rats__inverse, axiom, (! [A:$tType]: (((field_char_0 @ A) => (! [A5:A]: (((((member @ A) @ A5) @ (field_char_0_Rats @ A)) => (((member @ A) @ ((inverse_inverse @ A) @ A5)) @ (field_char_0_Rats @ A))))))))).
% 0.32/1.50 thf(fact_169_nat__neq__iff, axiom, (! [M:nat,N:nat]: (((M != N) = ((((ord_less @ nat) @ M) @ N) | (((ord_less @ nat) @ N) @ M)))))).
% 0.32/1.50 thf(fact_170_less__not__refl, axiom, (! [N:nat]: ((~ ((((ord_less @ nat) @ N) @ N)))))).
% 0.32/1.50 thf(fact_171_less__not__refl2, axiom, (! [N:nat,M:nat]: (((((ord_less @ nat) @ N) @ M) => (M != N))))).
% 0.32/1.50 thf(fact_172_less__not__refl3, axiom, (! [S:nat,T:nat]: (((((ord_less @ nat) @ S) @ T) => (S != T))))).
% 0.32/1.50 thf(fact_173_measure__induct, axiom, (! [B:$tType,A:$tType]: (((wellorder @ B) => (! [F:(A > B),P2:(A > $o),A5:A]: (((! [X3:A]: (((! [Y5:A]: (((((ord_less @ B) @ (F @ Y5)) @ (F @ X3)) => (P2 @ Y5)))) => (P2 @ X3)))) => (P2 @ A5)))))))).
% 0.32/1.50 thf(fact_174_less__irrefl__nat, axiom, (! [N:nat]: ((~ ((((ord_less @ nat) @ N) @ N)))))).
% 0.32/1.50 thf(fact_175_nat__less__induct, axiom, (! [P2:(nat > $o),N:nat]: (((! [N2:nat]: (((! [M3:nat]: (((((ord_less @ nat) @ M3) @ N2) => (P2 @ M3)))) => (P2 @ N2)))) => (P2 @ N))))).
% 0.32/1.50 thf(fact_176_infinite__descent, axiom, (! [P2:(nat > $o),N:nat]: (((! [N2:nat]: (((~ ((P2 @ N2))) => (? [M3:nat]: (((((ord_less @ nat) @ M3) @ N2) & (~ ((P2 @ M3))))))))) => (P2 @ N))))).
% 0.32/1.50 thf(fact_177_linorder__neqE__nat, axiom, (! [X2:nat,Y2:nat]: (((X2 != Y2) => ((~ ((((ord_less @ nat) @ X2) @ Y2))) => (((ord_less @ nat) @ Y2) @ X2)))))).
% 0.32/1.50 thf(fact_178_measure__induct__rule, axiom, (! [B:$tType,A:$tType]: (((wellorder @ B) => (! [F:(A > B),P2:(A > $o),A5:A]: (((! [X3:A]: (((! [Y5:A]: (((((ord_less @ B) @ (F @ Y5)) @ (F @ X3)) => (P2 @ Y5)))) => (P2 @ X3)))) => (P2 @ A5)))))))).
% 0.32/1.50 thf(fact_179_infinite__descent__measure, axiom, (! [A:$tType,P2:(A > $o),V:(A > nat),X2:A]: (((! [X3:A]: (((~ ((P2 @ X3))) => (? [Y5:A]: (((((ord_less @ nat) @ (V @ Y5)) @ (V @ X3)) & (~ ((P2 @ Y5))))))))) => (P2 @ X2))))).
% 0.32/1.50 thf(fact_180_linorder__neqE__linordered__idom, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [X2:A,Y2:A]: (((X2 != Y2) => ((~ ((((ord_less @ A) @ X2) @ Y2))) => (((ord_less @ A) @ Y2) @ X2))))))))).
% 0.32/1.50 thf(fact_181_lift__Suc__mono__less__iff, axiom, (! [A:$tType]: (((order @ A) => (! [F:(nat > A),N:nat,M:nat]: (((! [N2:nat]: ((((ord_less @ A) @ (F @ N2)) @ (F @ (suc @ N2))))) => ((((ord_less @ A) @ (F @ N)) @ (F @ M)) = (((ord_less @ nat) @ N) @ M))))))))).
% 0.32/1.50 thf(fact_182_lift__Suc__mono__less, axiom, (! [A:$tType]: (((order @ A) => (! [F:(nat > A),N:nat,N5:nat]: (((! [N2:nat]: ((((ord_less @ A) @ (F @ N2)) @ (F @ (suc @ N2))))) => ((((ord_less @ nat) @ N) @ N5) => (((ord_less @ A) @ (F @ N)) @ (F @ N5)))))))))).
% 0.32/1.50 thf(fact_183_dvd__pos__nat, axiom, (! [N:nat,M:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ N) => ((((dvd_dvd @ nat) @ M) @ N) => (((ord_less @ nat) @ (zero_zero @ nat)) @ M)))))).
% 0.32/1.50 thf(fact_184_finite__divisors__nat, axiom, (! [M:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ M) => ((finite_finite2 @ nat) @ ((collect @ nat) @ (^ [D3:nat]: ((((dvd_dvd @ nat) @ D3) @ M))))))))).
% 0.32/1.50 thf(fact_185_not__one__less__zero, axiom, (! [A:$tType]: (((linord1659791738miring @ A) => (~ ((((ord_less @ A) @ (one_one @ A)) @ (zero_zero @ A)))))))).
% 0.32/1.50 thf(fact_186_zero__less__one, axiom, (! [A:$tType]: (((zero_less_one @ A) => (((ord_less @ A) @ (zero_zero @ A)) @ (one_one @ A)))))).
% 0.32/1.50 thf(fact_187_less__Suc__eq__0__disj, axiom, (! [M:nat,N:nat]: (((((ord_less @ nat) @ M) @ (suc @ N)) = ((M = (zero_zero @ nat)) | (? [J3:nat]: (((M = (suc @ J3)) & (((ord_less @ nat) @ J3) @ N))))))))).
% 0.32/1.50 thf(fact_188_gr0__implies__Suc, axiom, (! [N:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ N) => (? [M2:nat]: ((N = (suc @ M2)))))))).
% 0.32/1.50 thf(fact_189_All__less__Suc2, axiom, (! [N:nat,P2:(nat > $o)]: (((! [I4:nat]: (((((ord_less @ nat) @ I4) @ (suc @ N)) => (P2 @ I4)))) = ((P2 @ (zero_zero @ nat)) & (! [I4:nat]: (((((ord_less @ nat) @ I4) @ N) => (P2 @ (suc @ I4)))))))))).
% 0.32/1.50 thf(fact_190_gr0__conv__Suc, axiom, (! [N:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ N) = (? [M5:nat]: ((N = (suc @ M5)))))))).
% 0.32/1.50 thf(fact_191_Ex__less__Suc2, axiom, (! [N:nat,P2:(nat > $o)]: (((? [I4:nat]: (((((ord_less @ nat) @ I4) @ (suc @ N)) & (P2 @ I4)))) = ((P2 @ (zero_zero @ nat)) | (? [I4:nat]: (((((ord_less @ nat) @ I4) @ N) & (P2 @ (suc @ I4)))))))))).
% 0.32/1.50 thf(fact_192_nat__dvd__not__less, axiom, (! [M:nat,N:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ M) => ((((ord_less @ nat) @ M) @ N) => (~ ((((dvd_dvd @ nat) @ N) @ M)))))))).
% 0.32/1.50 thf(fact_193_nat__induct__non__zero, axiom, (! [N:nat,P2:(nat > $o)]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ N) => ((P2 @ (one_one @ nat)) => ((! [N2:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ N2) => ((P2 @ N2) => (P2 @ (suc @ N2)))))) => (P2 @ N))))))).
% 0.32/1.50 thf(fact_194_less__degree__imp, axiom, (! [A:$tType]: (((zero @ A) => (! [N:nat,P:(poly @ A)]: (((((ord_less @ nat) @ N) @ ((degree @ A) @ P)) => (? [I3:nat]: (((((ord_less @ nat) @ N) @ I3) & ((((coeff @ A) @ P) @ I3) != (zero_zero @ A)))))))))))).
% 0.32/1.50 thf(fact_195_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))))))))).
% 0.32/1.50 thf(fact_196_gcd__nat_Onot__eq__order__implies__strict, axiom, (! [A5:nat,B3:nat]: (((A5 != B3) => ((((dvd_dvd @ nat) @ A5) @ B3) => ((((dvd_dvd @ nat) @ A5) @ B3) & (A5 != B3))))))).
% 0.32/1.50 thf(fact_197_gcd__nat_Ostrict__implies__not__eq, axiom, (! [A5:nat,B3:nat]: ((((((dvd_dvd @ nat) @ A5) @ B3) & (A5 != B3)) => (A5 != B3))))).
% 0.32/1.50 thf(fact_198_gcd__nat_Ostrict__implies__order, axiom, (! [A5:nat,B3:nat]: ((((((dvd_dvd @ nat) @ A5) @ B3) & (A5 != B3)) => (((dvd_dvd @ nat) @ A5) @ B3))))).
% 0.32/1.50 thf(fact_199_gcd__nat_Ostrict__iff__order, axiom, (! [A5:nat,B3:nat]: ((((((dvd_dvd @ nat) @ A5) @ B3) & (A5 != B3)) = ((((dvd_dvd @ nat) @ A5) @ B3) & (A5 != B3)))))).
% 0.32/1.50 thf(fact_200_gcd__nat_Oorder__iff__strict, axiom, ((dvd_dvd @ nat) = (^ [A3:nat,B4:nat]: ((((((dvd_dvd @ nat) @ A3) @ B4) & (A3 != B4)) | (A3 = B4)))))).
% 0.32/1.50 thf(fact_201_gcd__nat_Ostrict__trans2, axiom, (! [A5:nat,B3:nat,C2:nat]: ((((((dvd_dvd @ nat) @ A5) @ B3) & (A5 != B3)) => ((((dvd_dvd @ nat) @ B3) @ C2) => ((((dvd_dvd @ nat) @ A5) @ C2) & (A5 != C2))))))).
% 0.32/1.50 thf(fact_202_gcd__nat_Ostrict__trans1, axiom, (! [A5:nat,B3:nat,C2:nat]: (((((dvd_dvd @ nat) @ A5) @ B3) => (((((dvd_dvd @ nat) @ B3) @ C2) & (B3 != C2)) => ((((dvd_dvd @ nat) @ A5) @ C2) & (A5 != C2))))))).
% 0.32/1.50 thf(fact_203_gcd__nat_Ostrict__trans, axiom, (! [A5:nat,B3:nat,C2:nat]: ((((((dvd_dvd @ nat) @ A5) @ B3) & (A5 != B3)) => (((((dvd_dvd @ nat) @ B3) @ C2) & (B3 != C2)) => ((((dvd_dvd @ nat) @ A5) @ C2) & (A5 != C2))))))).
% 0.32/1.50 thf(fact_204_gcd__nat_Oantisym, axiom, (! [A5:nat,B3:nat]: (((((dvd_dvd @ nat) @ A5) @ B3) => ((((dvd_dvd @ nat) @ B3) @ A5) => (A5 = B3)))))).
% 0.32/1.50 thf(fact_205_gcd__nat_Oirrefl, axiom, (! [A5:nat]: ((~ (((((dvd_dvd @ nat) @ A5) @ A5) & (A5 != A5))))))).
% 0.32/1.50 thf(fact_206_gcd__nat_Oeq__iff, axiom, ((^ [Y:nat,Z:nat]: ((Y = Z))) = (^ [A3:nat,B4:nat]: (((((dvd_dvd @ nat) @ A3) @ B4) & (((dvd_dvd @ nat) @ B4) @ A3)))))).
% 0.32/1.50 thf(fact_207_gcd__nat_Otrans, axiom, (! [A5:nat,B3:nat,C2:nat]: (((((dvd_dvd @ nat) @ A5) @ B3) => ((((dvd_dvd @ nat) @ B3) @ C2) => (((dvd_dvd @ nat) @ A5) @ C2)))))).
% 0.32/1.50 thf(fact_208_gcd__nat_Orefl, axiom, (! [A5:nat]: ((((dvd_dvd @ nat) @ A5) @ A5)))).
% 0.32/1.50 thf(fact_209_gcd__nat_Oasym, axiom, (! [A5:nat,B3:nat]: ((((((dvd_dvd @ nat) @ A5) @ B3) & (A5 != B3)) => (~ (((((dvd_dvd @ nat) @ B3) @ A5) & (B3 != A5)))))))).
% 0.32/1.50 thf(fact_210_coeff__poly__cutoff, axiom, (! [A:$tType]: (((zero @ A) => (! [K:nat,N:nat,P:(poly @ A)]: ((((((ord_less @ nat) @ K) @ N) => ((((coeff @ A) @ (((poly_cutoff @ A) @ N) @ P)) @ K) = (((coeff @ A) @ P) @ K))) & ((~ ((((ord_less @ nat) @ K) @ N))) => ((((coeff @ A) @ (((poly_cutoff @ A) @ N) @ P)) @ K) = (zero_zero @ A)))))))))).
% 0.32/1.50 thf(fact_211_gcd__nat_Oextremum, axiom, (! [A5:nat]: ((((dvd_dvd @ nat) @ A5) @ (zero_zero @ nat))))).
% 0.32/1.50 thf(fact_212_gcd__nat_Oextremum__strict, axiom, (! [A5:nat]: ((~ (((((dvd_dvd @ nat) @ (zero_zero @ nat)) @ A5) & ((zero_zero @ nat) != A5))))))).
% 0.32/1.50 thf(fact_213_gcd__nat_Oextremum__unique, axiom, (! [A5:nat]: (((((dvd_dvd @ nat) @ (zero_zero @ nat)) @ A5) = (A5 = (zero_zero @ nat)))))).
% 0.32/1.50 thf(fact_214_inverse__positive__iff__positive, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A5:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ ((inverse_inverse @ A) @ A5)) = (((ord_less @ A) @ (zero_zero @ A)) @ A5)))))))).
% 0.32/1.50 thf(fact_215_inverse__negative__iff__negative, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A5:A]: (((((ord_less @ A) @ ((inverse_inverse @ A) @ A5)) @ (zero_zero @ A)) = (((ord_less @ A) @ A5) @ (zero_zero @ A))))))))).
% 0.32/1.50 thf(fact_216_inverse__less__iff__less__neg, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A5:A,B3:A]: (((((ord_less @ A) @ A5) @ (zero_zero @ A)) => ((((ord_less @ A) @ B3) @ (zero_zero @ A)) => ((((ord_less @ A) @ ((inverse_inverse @ A) @ A5)) @ ((inverse_inverse @ A) @ B3)) = (((ord_less @ A) @ B3) @ A5)))))))))).
% 0.32/1.50 thf(fact_217_inverse__less__iff__less, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A5:A,B3:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ A5) => ((((ord_less @ A) @ (zero_zero @ A)) @ B3) => ((((ord_less @ A) @ ((inverse_inverse @ A) @ A5)) @ ((inverse_inverse @ A) @ B3)) = (((ord_less @ A) @ B3) @ A5)))))))))).
% 0.32/1.50 thf(fact_218_inverse__zero, axiom, (! [A:$tType]: (((division_ring @ A) => (((inverse_inverse @ A) @ (zero_zero @ A)) = (zero_zero @ A)))))).
% 0.32/1.50 thf(fact_219_inverse__nonzero__iff__nonzero, axiom, (! [A:$tType]: (((division_ring @ A) => (! [A5:A]: (((((inverse_inverse @ A) @ A5) = (zero_zero @ A)) = (A5 = (zero_zero @ A))))))))).
% 0.32/1.50 thf(fact_220_inverse__eq__1__iff, axiom, (! [A:$tType]: (((field @ A) => (! [X2:A]: (((((inverse_inverse @ A) @ X2) = (one_one @ A)) = (X2 = (one_one @ A))))))))).
% 0.32/1.50 thf(fact_221_inverse__1, axiom, (! [A:$tType]: (((division_ring @ A) => (((inverse_inverse @ A) @ (one_one @ A)) = (one_one @ A)))))).
% 0.32/1.50 thf(fact_222_finite__psubset__induct, axiom, (! [A:$tType,A4:(set @ A),P2:((set @ A) > $o)]: ((((finite_finite2 @ A) @ A4) => ((! [A6:(set @ A)]: ((((finite_finite2 @ A) @ A6) => ((! [B5:(set @ A)]: (((((ord_less @ (set @ A)) @ B5) @ A6) => (P2 @ B5)))) => (P2 @ A6))))) => (P2 @ A4)))))).
% 0.32/1.50 thf(fact_223_strict__subset__divisors__dvd, axiom, (! [A:$tType]: (((comm_monoid_mult @ A) => (! [A5:A,B3:A]: (((((ord_less @ (set @ A)) @ ((collect @ A) @ (^ [C3:A]: ((((dvd_dvd @ A) @ C3) @ A5))))) @ ((collect @ A) @ (^ [C3:A]: ((((dvd_dvd @ A) @ C3) @ B3))))) = ((((dvd_dvd @ A) @ A5) @ B3) & (~ ((((dvd_dvd @ A) @ B3) @ A5))))))))))).
% 0.32/1.50 thf(fact_224_poly__IVT__pos, axiom, (! [A5:real,B3:real,P:(poly @ real)]: (((((ord_less @ real) @ A5) @ B3) => ((((ord_less @ real) @ (((poly2 @ real) @ P) @ A5)) @ (zero_zero @ real)) => ((((ord_less @ real) @ (zero_zero @ real)) @ (((poly2 @ real) @ P) @ B3)) => (? [X3:real]: (((((ord_less @ real) @ A5) @ X3) & ((((ord_less @ real) @ X3) @ B3) & ((((poly2 @ real) @ P) @ X3) = (zero_zero @ real)))))))))))).
% 0.32/1.50 thf(fact_225_poly__IVT__neg, axiom, (! [A5:real,B3:real,P:(poly @ real)]: (((((ord_less @ real) @ A5) @ B3) => ((((ord_less @ real) @ (zero_zero @ real)) @ (((poly2 @ real) @ P) @ A5)) => ((((ord_less @ real) @ (((poly2 @ real) @ P) @ B3)) @ (zero_zero @ real)) => (? [X3:real]: (((((ord_less @ real) @ A5) @ X3) & ((((ord_less @ real) @ X3) @ B3) & ((((poly2 @ real) @ P) @ X3) = (zero_zero @ real)))))))))))).
% 0.32/1.50 thf(fact_226_dvd__field__iff, axiom, (! [A:$tType]: (((field @ A) => ((dvd_dvd @ A) = (^ [A3:A,B4:A]: (((A3 = (zero_zero @ A)) => (B4 = (zero_zero @ A)))))))))).
% 0.32/1.50 thf(fact_227_field__class_Ofield__inverse__zero, axiom, (! [A:$tType]: (((field @ A) => (((inverse_inverse @ A) @ (zero_zero @ A)) = (zero_zero @ A)))))).
% 0.32/1.50 thf(fact_228_inverse__zero__imp__zero, axiom, (! [A:$tType]: (((division_ring @ A) => (! [A5:A]: (((((inverse_inverse @ A) @ A5) = (zero_zero @ A)) => (A5 = (zero_zero @ A))))))))).
% 0.32/1.50 thf(fact_229_nonzero__inverse__eq__imp__eq, axiom, (! [A:$tType]: (((division_ring @ A) => (! [A5:A,B3:A]: (((((inverse_inverse @ A) @ A5) = ((inverse_inverse @ A) @ B3)) => ((A5 != (zero_zero @ A)) => ((B3 != (zero_zero @ A)) => (A5 = B3)))))))))).
% 0.32/1.50 thf(fact_230_nonzero__inverse__inverse__eq, axiom, (! [A:$tType]: (((division_ring @ A) => (! [A5:A]: (((A5 != (zero_zero @ A)) => (((inverse_inverse @ A) @ ((inverse_inverse @ A) @ A5)) = A5)))))))).
% 0.32/1.50 thf(fact_231_nonzero__imp__inverse__nonzero, axiom, (! [A:$tType]: (((division_ring @ A) => (! [A5:A]: (((A5 != (zero_zero @ A)) => (((inverse_inverse @ A) @ A5) != (zero_zero @ A))))))))).
% 0.32/1.50 thf(fact_232_inverse__less__imp__less, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A5:A,B3:A]: (((((ord_less @ A) @ ((inverse_inverse @ A) @ A5)) @ ((inverse_inverse @ A) @ B3)) => ((((ord_less @ A) @ (zero_zero @ A)) @ A5) => (((ord_less @ A) @ B3) @ A5))))))))).
% 0.32/1.50 thf(fact_233_less__imp__inverse__less, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A5:A,B3:A]: (((((ord_less @ A) @ A5) @ B3) => ((((ord_less @ A) @ (zero_zero @ A)) @ A5) => (((ord_less @ A) @ ((inverse_inverse @ A) @ B3)) @ ((inverse_inverse @ A) @ A5)))))))))).
% 0.32/1.50 thf(fact_234_inverse__less__imp__less__neg, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A5:A,B3:A]: (((((ord_less @ A) @ ((inverse_inverse @ A) @ A5)) @ ((inverse_inverse @ A) @ B3)) => ((((ord_less @ A) @ B3) @ (zero_zero @ A)) => (((ord_less @ A) @ B3) @ A5))))))))).
% 0.32/1.50 thf(fact_235_less__imp__inverse__less__neg, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A5:A,B3:A]: (((((ord_less @ A) @ A5) @ B3) => ((((ord_less @ A) @ B3) @ (zero_zero @ A)) => (((ord_less @ A) @ ((inverse_inverse @ A) @ B3)) @ ((inverse_inverse @ A) @ A5)))))))))).
% 0.32/1.50 thf(fact_236_inverse__negative__imp__negative, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A5:A]: (((((ord_less @ A) @ ((inverse_inverse @ A) @ A5)) @ (zero_zero @ A)) => ((A5 != (zero_zero @ A)) => (((ord_less @ A) @ A5) @ (zero_zero @ A)))))))))).
% 0.32/1.50 thf(fact_237_inverse__positive__imp__positive, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A5:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ ((inverse_inverse @ A) @ A5)) => ((A5 != (zero_zero @ A)) => (((ord_less @ A) @ (zero_zero @ A)) @ A5))))))))).
% 0.32/1.50 thf(fact_238_negative__imp__inverse__negative, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A5:A]: (((((ord_less @ A) @ A5) @ (zero_zero @ A)) => (((ord_less @ A) @ ((inverse_inverse @ A) @ A5)) @ (zero_zero @ A))))))))).
% 0.32/1.50 thf(fact_239_positive__imp__inverse__positive, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A5:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ A5) => (((ord_less @ A) @ (zero_zero @ A)) @ ((inverse_inverse @ A) @ A5))))))))).
% 0.32/1.50 thf(fact_240_one__less__inverse, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A5:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ A5) => ((((ord_less @ A) @ A5) @ (one_one @ A)) => (((ord_less @ A) @ (one_one @ A)) @ ((inverse_inverse @ A) @ A5)))))))))).
% 0.32/1.50 thf(fact_241_one__less__inverse__iff, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [X2:A]: (((((ord_less @ A) @ (one_one @ A)) @ ((inverse_inverse @ A) @ X2)) = ((((ord_less @ A) @ (zero_zero @ A)) @ X2) & (((ord_less @ A) @ X2) @ (one_one @ A)))))))))).
% 0.32/1.50 thf(fact_242_less__numeral__extra_I1_J, axiom, (! [A:$tType]: (((linord1659791738miring @ A) => (((ord_less @ A) @ (zero_zero @ A)) @ (one_one @ A)))))).
% 0.32/1.50 thf(fact_243_finite__M__bounded__by__nat, axiom, (! [P2:(nat > $o),I2:nat]: (((finite_finite2 @ nat) @ ((collect @ nat) @ (^ [K3:nat]: (((P2 @ K3) & (((ord_less @ nat) @ K3) @ I2))))))))).
% 0.32/1.50 thf(fact_244_less__numeral__extra_I3_J, axiom, (! [A:$tType]: (((linord1659791738miring @ A) => (~ ((((ord_less @ A) @ (zero_zero @ A)) @ (zero_zero @ A)))))))).
% 0.32/1.50 thf(fact_245_less__numeral__extra_I4_J, axiom, (! [A:$tType]: (((linord1659791738miring @ A) => (~ ((((ord_less @ A) @ (one_one @ A)) @ (one_one @ A)))))))).
% 0.32/1.50 thf(fact_246_Rats__dense__in__real, axiom, (! [X2:real,Y2:real]: (((((ord_less @ real) @ X2) @ Y2) => (? [X3:real]: (((((member @ real) @ X3) @ (field_char_0_Rats @ real)) & ((((ord_less @ real) @ X2) @ X3) & (((ord_less @ real) @ X3) @ Y2))))))))).
% 0.32/1.50 thf(fact_247_Rats__no__bot__less, axiom, (! [X2:real]: ((? [X3:real]: (((((member @ real) @ X3) @ (field_char_0_Rats @ real)) & (((ord_less @ real) @ X3) @ X2))))))).
% 0.32/1.50 thf(fact_248_field__lbound__gt__zero, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [D1:A,D22:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ D1) => ((((ord_less @ A) @ (zero_zero @ A)) @ D22) => (? [E:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ E) & ((((ord_less @ A) @ E) @ D1) & (((ord_less @ A) @ E) @ D22))))))))))))).
% 0.32/1.50 thf(fact_249_pseudo__mod_I2_J, axiom, (! [A:$tType]: ((((comm_ring_1 @ A) & (semiri134348788visors @ A)) => (! [G:(poly @ A),F:(poly @ A)]: (((G != (zero_zero @ (poly @ A))) => (((((pseudo_mod @ A) @ F) @ G) = (zero_zero @ (poly @ A))) | (((ord_less @ nat) @ ((degree @ A) @ (((pseudo_mod @ A) @ F) @ G))) @ ((degree @ A) @ G)))))))))).
% 0.32/1.50 thf(fact_250_poly__pinfty__gt__lc, axiom, (! [P:(poly @ real)]: (((((ord_less @ real) @ (zero_zero @ real)) @ (((coeff @ real) @ P) @ ((degree @ real) @ P))) => (? [N2:real]: ((! [X4:real]: (((((ord_less_eq @ real) @ N2) @ X4) => (((ord_less_eq @ real) @ (((coeff @ real) @ P) @ ((degree @ real) @ P))) @ (((poly2 @ real) @ P) @ X4))))))))))).
% 0.32/1.50 thf(fact_251_le__zero__eq, axiom, (! [A:$tType]: (((canoni770627133id_add @ A) => (! [N:A]: (((((ord_less_eq @ A) @ N) @ (zero_zero @ A)) = (N = (zero_zero @ A))))))))).
% 0.32/1.50 thf(tcon_fun___Finite__Set_Ofinite, axiom, (! [A7:$tType,A8:$tType]: ((((finite_finite @ A7) & (finite_finite @ A8)) => (finite_finite @ (A7 > A8)))))).
% 0.32/1.50 thf(tcon_fun___Orderings_Oorder, axiom, (! [A7:$tType,A8:$tType]: (((order @ A8) => (order @ (A7 > A8)))))).
% 0.32/1.50 thf(tcon_Nat_Onat___Groups_Ocanonically__ordered__monoid__add, axiom, (canoni770627133id_add @ nat)).
% 0.32/1.50 thf(tcon_Nat_Onat___Rings_Osemiring__1__no__zero__divisors, axiom, (semiri134348788visors @ nat)).
% 0.32/1.50 thf(tcon_Nat_Onat___Rings_Olinordered__nonzero__semiring, axiom, (linord1659791738miring @ nat)).
% 0.32/1.50 thf(tcon_Nat_Onat___Rings_Osemiring__no__zero__divisors, axiom, (semiri1193490041visors @ nat)).
% 0.32/1.50 thf(tcon_Nat_Onat___Rings_Oalgebraic__semidom, axiom, (algebraic_semidom @ nat)).
% 0.32/1.50 thf(tcon_Nat_Onat___Groups_Ocomm__monoid__mult, axiom, (comm_monoid_mult @ nat)).
% 0.32/1.50 thf(tcon_Nat_Onat___Rings_Ocomm__semiring__1, axiom, (comm_semiring_1 @ nat)).
% 0.32/1.50 thf(tcon_Nat_Onat___Rings_Ocomm__semiring__0, axiom, (comm_semiring_0 @ nat)).
% 0.32/1.50 thf(tcon_Nat_Onat___Rings_Ozero__less__one, axiom, (zero_less_one @ nat)).
% 0.32/1.50 thf(tcon_Nat_Onat___Orderings_Owellorder, axiom, (wellorder @ nat)).
% 0.32/1.50 thf(tcon_Nat_Onat___Nat_Osemiring__char__0, axiom, (semiring_char_0 @ nat)).
% 0.32/1.50 thf(tcon_Nat_Onat___Rings_Ozero__neq__one, axiom, (zero_neq_one @ nat)).
% 0.32/1.50 thf(tcon_Nat_Onat___Orderings_Oorder_1, axiom, (order @ nat)).
% 0.32/1.50 thf(tcon_Nat_Onat___Groups_Ozero, axiom, (zero @ nat)).
% 0.32/1.50 thf(tcon_Nat_Onat___Groups_Oone, axiom, (one @ nat)).
% 0.32/1.50 thf(tcon_Set_Oset___Finite__Set_Ofinite_2, axiom, (! [A7:$tType]: (((finite_finite @ A7) => (finite_finite @ (set @ A7)))))).
% 0.32/1.50 thf(tcon_Set_Oset___Orderings_Oorder_3, axiom, (! [A7:$tType]: ((order @ (set @ A7))))).
% 0.32/1.50 thf(tcon_HOL_Obool___Finite__Set_Ofinite_4, axiom, (finite_finite @ $o)).
% 0.32/1.50 thf(tcon_HOL_Obool___Orderings_Oorder_5, axiom, (order @ $o)).
% 0.32/1.50 thf(tcon_Real_Oreal___Rings_Osemiring__1__no__zero__divisors_6, axiom, (semiri134348788visors @ real)).
% 0.32/1.50 thf(tcon_Real_Oreal___Rings_Olinordered__nonzero__semiring_7, axiom, (linord1659791738miring @ real)).
% 0.32/1.50 thf(tcon_Real_Oreal___Rings_Osemiring__no__zero__divisors_8, axiom, (semiri1193490041visors @ real)).
% 0.32/1.50 thf(tcon_Real_Oreal___Rings_Oring__no__zero__divisors, axiom, (ring_n68954251visors @ real)).
% 0.32/1.50 thf(tcon_Real_Oreal___Groups_Ocomm__monoid__mult_9, axiom, (comm_monoid_mult @ real)).
% 0.32/1.50 thf(tcon_Real_Oreal___Fields_Olinordered__field, axiom, (linordered_field @ real)).
% 0.32/1.50 thf(tcon_Real_Oreal___Rings_Olinordered__idom, axiom, (linordered_idom @ real)).
% 0.32/1.50 thf(tcon_Real_Oreal___Rings_Ocomm__semiring__1_10, axiom, (comm_semiring_1 @ real)).
% 0.32/1.50 thf(tcon_Real_Oreal___Rings_Ocomm__semiring__0_11, axiom, (comm_semiring_0 @ real)).
% 0.32/1.50 thf(tcon_Real_Oreal___Fields_Odivision__ring, axiom, (division_ring @ real)).
% 0.32/1.50 thf(tcon_Real_Oreal___Rings_Ozero__less__one_12, axiom, (zero_less_one @ real)).
% 0.32/1.50 thf(tcon_Real_Oreal___Nat_Osemiring__char__0_13, axiom, (semiring_char_0 @ real)).
% 0.32/1.50 thf(tcon_Real_Oreal___Fields_Ofield__char__0, axiom, (field_char_0 @ real)).
% 0.32/1.50 thf(tcon_Real_Oreal___Rings_Ozero__neq__one_14, axiom, (zero_neq_one @ real)).
% 0.32/1.50 thf(tcon_Real_Oreal___Rings_Oidom__divide, axiom, (idom_divide @ real)).
% 0.32/1.50 thf(tcon_Real_Oreal___Rings_Ocomm__ring__1, axiom, (comm_ring_1 @ real)).
% 0.32/1.50 thf(tcon_Real_Oreal___Orderings_Oorder_15, axiom, (order @ real)).
% 0.32/1.50 thf(tcon_Real_Oreal___Nat_Oring__char__0, axiom, (ring_char_0 @ real)).
% 0.32/1.50 thf(tcon_Real_Oreal___Rings_Oring__1, axiom, (ring_1 @ real)).
% 0.32/1.50 thf(tcon_Real_Oreal___Fields_Ofield, axiom, (field @ real)).
% 0.32/1.50 thf(tcon_Real_Oreal___Groups_Ozero_16, axiom, (zero @ real)).
% 0.32/1.50 thf(tcon_Real_Oreal___Rings_Oidom, axiom, (idom @ real)).
% 0.32/1.50 thf(tcon_Real_Oreal___Groups_Oone_17, axiom, (one @ real)).
% 0.32/1.50 thf(tcon_Polynomial_Opoly___Rings_Osemiring__1__no__zero__divisors_18, axiom, (! [A7:$tType]: ((((comm_semiring_1 @ A7) & (semiri134348788visors @ A7)) => (semiri134348788visors @ (poly @ A7)))))).
% 0.32/1.50 thf(tcon_Polynomial_Opoly___Rings_Olinordered__nonzero__semiring_19, axiom, (! [A7:$tType]: (((linordered_idom @ A7) => (linord1659791738miring @ (poly @ A7)))))).
% 0.32/1.50 thf(tcon_Polynomial_Opoly___Rings_Osemiring__no__zero__divisors_20, axiom, (! [A7:$tType]: ((((comm_semiring_0 @ A7) & (semiri1193490041visors @ A7)) => (semiri1193490041visors @ (poly @ A7)))))).
% 0.32/1.50 thf(tcon_Polynomial_Opoly___Rings_Oring__no__zero__divisors_21, axiom, (! [A7:$tType]: (((idom @ A7) => (ring_n68954251visors @ (poly @ A7)))))).
% 0.32/1.50 thf(tcon_Polynomial_Opoly___Rings_Oalgebraic__semidom_22, axiom, (! [A7:$tType]: (((idom_divide @ A7) => (algebraic_semidom @ (poly @ A7)))))).
% 0.32/1.50 thf(tcon_Polynomial_Opoly___Groups_Ocomm__monoid__mult_23, axiom, (! [A7:$tType]: (((comm_semiring_1 @ A7) => (comm_monoid_mult @ (poly @ A7)))))).
% 0.32/1.50 thf(tcon_Polynomial_Opoly___Rings_Olinordered__idom_24, axiom, (! [A7:$tType]: (((linordered_idom @ A7) => (linordered_idom @ (poly @ A7)))))).
% 0.32/1.50 thf(tcon_Polynomial_Opoly___Rings_Ocomm__semiring__1_25, axiom, (! [A7:$tType]: (((comm_semiring_1 @ A7) => (comm_semiring_1 @ (poly @ A7)))))).
% 0.32/1.50 thf(tcon_Polynomial_Opoly___Rings_Ocomm__semiring__0_26, axiom, (! [A7:$tType]: (((comm_semiring_0 @ A7) => (comm_semiring_0 @ (poly @ A7)))))).
% 0.32/1.50 thf(tcon_Polynomial_Opoly___Rings_Ozero__less__one_27, axiom, (! [A7:$tType]: (((linordered_idom @ A7) => (zero_less_one @ (poly @ A7)))))).
% 0.32/1.50 thf(tcon_Polynomial_Opoly___Nat_Osemiring__char__0_28, axiom, (! [A7:$tType]: ((((ring_char_0 @ A7) & (comm_ring_1 @ A7)) => (semiring_char_0 @ (poly @ A7)))))).
% 0.32/1.50 thf(tcon_Polynomial_Opoly___Rings_Ozero__neq__one_29, axiom, (! [A7:$tType]: (((comm_semiring_1 @ A7) => (zero_neq_one @ (poly @ A7)))))).
% 0.32/1.50 thf(tcon_Polynomial_Opoly___Rings_Oidom__divide_30, axiom, (! [A7:$tType]: (((idom_divide @ A7) => (idom_divide @ (poly @ A7)))))).
% 0.32/1.50 thf(tcon_Polynomial_Opoly___Rings_Ocomm__ring__1_31, axiom, (! [A7:$tType]: (((comm_ring_1 @ A7) => (comm_ring_1 @ (poly @ A7)))))).
% 0.32/1.50 thf(tcon_Polynomial_Opoly___Orderings_Oorder_32, axiom, (! [A7:$tType]: (((linordered_idom @ A7) => (order @ (poly @ A7)))))).
% 0.32/1.50 thf(tcon_Polynomial_Opoly___Nat_Oring__char__0_33, axiom, (! [A7:$tType]: ((((ring_char_0 @ A7) & (comm_ring_1 @ A7)) => (ring_char_0 @ (poly @ A7)))))).
% 0.32/1.50 thf(tcon_Polynomial_Opoly___Rings_Oring__1_34, axiom, (! [A7:$tType]: (((comm_ring_1 @ A7) => (ring_1 @ (poly @ A7)))))).
% 0.32/1.50 thf(tcon_Polynomial_Opoly___Groups_Ozero_35, axiom, (! [A7:$tType]: (((zero @ A7) => (zero @ (poly @ A7)))))).
% 0.32/1.50 thf(tcon_Polynomial_Opoly___Rings_Oidom_36, axiom, (! [A7:$tType]: (((idom @ A7) => (idom @ (poly @ A7)))))).
% 0.32/1.50 thf(tcon_Polynomial_Opoly___Groups_Oone_37, axiom, (! [A7:$tType]: (((comm_semiring_1 @ A7) => (one @ (poly @ A7)))))).
% 0.32/1.50 thf(conj_0, conjecture, ((finite_finite2 @ real) @ ((collect @ real) @ (^ [X:real]: (((((poly2 @ real) @ ((pderiv @ real) @ p)) @ X) = (zero_zero @ real))))))).
% 0.32/1.50 % SZS output end ListOfTHF for /export/starexec/sandbox/tmp/tmp.AWHgCUKr9y/DTF2DTF8563.p
% 0.32/1.50 ------------------------
% 0.32/1.53 ---- Cleaned THF ---
% 0.32/1.53 thf(ty_t_Polynomial_Opoly,type,
% 0.32/1.53 poly: $tType > $tType ).
% 0.32/1.53
% 0.32/1.53 thf(ty_t_Real_Oreal,type,
% 0.32/1.53 real: $tType ).
% 0.32/1.53
% 0.32/1.53 thf(ty_t_Set_Oset,type,
% 0.32/1.53 set: $tType > $tType ).
% 0.32/1.53
% 0.32/1.53 thf(ty_t_Nat_Onat,type,
% 0.32/1.53 nat: $tType ).
% 0.32/1.53
% 0.32/1.53 thf(sy_cl_HOL_Otype,type,
% 0.32/1.53 type:
% 0.32/1.53 !>[A: $tType] : $o ).
% 0.32/1.53
% 0.32/1.53 thf(sy_cl_Groups_Oone,type,
% 0.32/1.53 one:
% 0.32/1.53 !>[A: $tType] : $o ).
% 0.32/1.53
% 0.32/1.53 thf(sy_cl_Rings_Oidom,type,
% 0.32/1.53 idom:
% 0.32/1.53 !>[A: $tType] : $o ).
% 0.32/1.53
% 0.32/1.53 thf(sy_cl_Groups_Ozero,type,
% 0.32/1.53 zero:
% 0.32/1.53 !>[A: $tType] : $o ).
% 0.32/1.53
% 0.32/1.53 thf(sy_cl_Fields_Ofield,type,
% 0.32/1.53 field:
% 0.32/1.53 !>[A: $tType] : $o ).
% 0.32/1.53
% 0.32/1.53 thf(sy_cl_Rings_Oring__1,type,
% 0.32/1.53 ring_1:
% 0.32/1.53 !>[A: $tType] : $o ).
% 0.32/1.53
% 0.32/1.53 thf(sy_cl_Nat_Oring__char__0,type,
% 0.32/1.53 ring_char_0:
% 0.32/1.53 !>[A: $tType] : $o ).
% 0.32/1.53
% 0.32/1.53 thf(sy_cl_Orderings_Oorder,type,
% 0.32/1.53 order:
% 0.32/1.53 !>[A: $tType] : $o ).
% 0.32/1.53
% 0.32/1.53 thf(sy_cl_Finite__Set_Ofinite,type,
% 0.32/1.53 finite_finite:
% 0.32/1.53 !>[A: $tType] : $o ).
% 0.32/1.53
% 0.32/1.53 thf(sy_cl_Rings_Ocomm__ring__1,type,
% 0.32/1.53 comm_ring_1:
% 0.32/1.53 !>[A: $tType] : $o ).
% 0.32/1.53
% 0.32/1.53 thf(sy_cl_Rings_Oidom__divide,type,
% 0.32/1.53 idom_divide:
% 0.32/1.53 !>[A: $tType] : $o ).
% 0.32/1.53
% 0.32/1.53 thf(sy_cl_Rings_Ozero__neq__one,type,
% 0.32/1.53 zero_neq_one:
% 0.32/1.53 !>[A: $tType] : $o ).
% 0.32/1.53
% 0.32/1.53 thf(sy_cl_Fields_Ofield__char__0,type,
% 0.32/1.53 field_char_0:
% 0.32/1.53 !>[A: $tType] : $o ).
% 0.32/1.53
% 0.32/1.53 thf(sy_cl_Nat_Osemiring__char__0,type,
% 0.32/1.53 semiring_char_0:
% 0.32/1.53 !>[A: $tType] : $o ).
% 0.32/1.53
% 0.32/1.53 thf(sy_cl_Orderings_Owellorder,type,
% 0.32/1.53 wellorder:
% 0.32/1.53 !>[A: $tType] : $o ).
% 0.32/1.53
% 0.32/1.53 thf(sy_cl_Rings_Ozero__less__one,type,
% 0.32/1.53 zero_less_one:
% 0.32/1.53 !>[A: $tType] : $o ).
% 0.32/1.53
% 0.32/1.53 thf(sy_cl_Fields_Odivision__ring,type,
% 0.32/1.53 division_ring:
% 0.32/1.53 !>[A: $tType] : $o ).
% 0.32/1.53
% 0.32/1.53 thf(sy_cl_Rings_Ocomm__semiring__0,type,
% 0.32/1.53 comm_semiring_0:
% 0.32/1.53 !>[A: $tType] : $o ).
% 0.32/1.53
% 0.32/1.53 thf(sy_cl_Rings_Ocomm__semiring__1,type,
% 0.32/1.53 comm_semiring_1:
% 0.32/1.53 !>[A: $tType] : $o ).
% 0.32/1.53
% 0.32/1.53 thf(sy_cl_Rings_Olinordered__idom,type,
% 0.32/1.53 linordered_idom:
% 0.32/1.53 !>[A: $tType] : $o ).
% 0.32/1.53
% 0.32/1.53 thf(sy_cl_Fields_Olinordered__field,type,
% 0.32/1.53 linordered_field:
% 0.32/1.53 !>[A: $tType] : $o ).
% 0.32/1.53
% 0.32/1.53 thf(sy_cl_Groups_Ocomm__monoid__mult,type,
% 0.32/1.53 comm_monoid_mult:
% 0.32/1.53 !>[A: $tType] : $o ).
% 0.32/1.53
% 0.32/1.53 thf(sy_cl_Rings_Oalgebraic__semidom,type,
% 0.32/1.53 algebraic_semidom:
% 0.32/1.53 !>[A: $tType] : $o ).
% 0.32/1.53
% 0.32/1.53 thf(sy_cl_Rings_Oring__no__zero__divisors,type,
% 0.32/1.53 ring_n68954251visors:
% 0.32/1.53 !>[A: $tType] : $o ).
% 0.32/1.53
% 0.32/1.53 thf(sy_cl_Rings_Osemiring__no__zero__divisors,type,
% 0.32/1.53 semiri1193490041visors:
% 0.32/1.53 !>[A: $tType] : $o ).
% 0.32/1.53
% 0.32/1.53 thf(sy_cl_Rings_Olinordered__nonzero__semiring,type,
% 0.32/1.53 linord1659791738miring:
% 0.32/1.53 !>[A: $tType] : $o ).
% 0.32/1.53
% 0.32/1.53 thf(sy_cl_Rings_Osemiring__1__no__zero__divisors,type,
% 0.32/1.53 semiri134348788visors:
% 0.32/1.53 !>[A: $tType] : $o ).
% 0.32/1.53
% 0.32/1.53 thf(sy_cl_Groups_Ocanonically__ordered__monoid__add,type,
% 0.32/1.53 canoni770627133id_add:
% 0.32/1.53 !>[A: $tType] : $o ).
% 0.32/1.53
% 0.32/1.53 thf(sy_c_Fields_Oinverse__class_Oinverse,type,
% 0.32/1.53 inverse_inverse:
% 0.32/1.53 !>[A: $tType] : ( A > A ) ).
% 0.32/1.53
% 0.32/1.53 thf(sy_c_Finite__Set_Ofinite,type,
% 0.32/1.53 finite_finite2:
% 0.32/1.53 !>[A: $tType] : ( ( set @ A ) > $o ) ).
% 0.32/1.53
% 0.32/1.53 thf(sy_c_Groups_Oone__class_Oone,type,
% 0.32/1.53 one_one:
% 0.32/1.53 !>[A: $tType] : A ).
% 0.32/1.53
% 0.32/1.53 thf(sy_c_Groups_Ozero__class_Ozero,type,
% 0.32/1.53 zero_zero:
% 0.32/1.53 !>[A: $tType] : A ).
% 0.32/1.53
% 0.32/1.53 thf(sy_c_Int_Oring__1__class_OInts,type,
% 0.32/1.53 ring_1_Ints:
% 0.32/1.53 !>[A: $tType] : ( set @ A ) ).
% 0.32/1.53
% 0.32/1.53 thf(sy_c_Nat_OSuc,type,
% 0.32/1.53 suc: nat > nat ).
% 0.32/1.53
% 0.32/1.53 thf(sy_c_Orderings_Oord__class_Oless,type,
% 0.32/1.53 ord_less:
% 0.32/1.53 !>[A: $tType] : ( A > A > $o ) ).
% 0.32/1.53
% 0.32/1.53 thf(sy_c_Orderings_Oord__class_Oless__eq,type,
% 0.32/1.53 ord_less_eq:
% 0.32/1.53 !>[A: $tType] : ( A > A > $o ) ).
% 0.32/1.53
% 0.32/1.53 thf(sy_c_Polynomial_Oalgebraic,type,
% 0.32/1.53 algebraic:
% 0.32/1.53 !>[A: $tType] : ( A > $o ) ).
% 0.32/1.53
% 0.32/1.53 thf(sy_c_Polynomial_Oalgebraic__int,type,
% 0.32/1.53 algebraic_int:
% 0.32/1.53 !>[A: $tType] : ( A > $o ) ).
% 0.32/1.53
% 0.32/1.53 thf(sy_c_Polynomial_Ocr__poly,type,
% 0.32/1.53 cr_poly:
% 0.32/1.53 !>[A: $tType] : ( ( nat > A ) > ( poly @ A ) > $o ) ).
% 0.32/1.53
% 0.32/1.53 thf(sy_c_Polynomial_Odegree,type,
% 0.32/1.53 degree:
% 0.32/1.53 !>[A: $tType] : ( ( poly @ A ) > nat ) ).
% 0.32/1.53
% 0.32/1.53 thf(sy_c_Polynomial_Odivide__poly__main,type,
% 0.32/1.53 divide_poly_main:
% 0.32/1.53 !>[A: $tType] : ( A > ( poly @ A ) > ( poly @ A ) > ( poly @ A ) > nat > nat > ( poly @ A ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(sy_c_Polynomial_Ois__zero,type,
% 0.32/1.53 is_zero:
% 0.32/1.53 !>[A: $tType] : ( ( poly @ A ) > $o ) ).
% 0.32/1.53
% 0.32/1.53 thf(sy_c_Polynomial_Oorder,type,
% 0.32/1.53 order2:
% 0.32/1.53 !>[A: $tType] : ( A > ( poly @ A ) > nat ) ).
% 0.32/1.53
% 0.32/1.53 thf(sy_c_Polynomial_Opcr__poly,type,
% 0.32/1.53 pcr_poly:
% 0.32/1.53 !>[C: $tType,B: $tType] : ( ( C > B > $o ) > ( nat > C ) > ( poly @ B ) > $o ) ).
% 0.32/1.53
% 0.32/1.53 thf(sy_c_Polynomial_Opderiv,type,
% 0.32/1.53 pderiv:
% 0.32/1.53 !>[A: $tType] : ( ( poly @ A ) > ( poly @ A ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(sy_c_Polynomial_Opoly,type,
% 0.32/1.53 poly2:
% 0.32/1.53 !>[A: $tType] : ( ( poly @ A ) > A > A ) ).
% 0.32/1.53
% 0.32/1.53 thf(sy_c_Polynomial_Opoly_Ocoeff,type,
% 0.32/1.53 coeff:
% 0.32/1.53 !>[A: $tType] : ( ( poly @ A ) > nat > A ) ).
% 0.32/1.53
% 0.32/1.53 thf(sy_c_Polynomial_Opoly__cutoff,type,
% 0.32/1.53 poly_cutoff:
% 0.32/1.53 !>[A: $tType] : ( nat > ( poly @ A ) > ( poly @ A ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(sy_c_Polynomial_Opoly__shift,type,
% 0.32/1.53 poly_shift:
% 0.32/1.53 !>[A: $tType] : ( nat > ( poly @ A ) > ( poly @ A ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(sy_c_Polynomial_Opseudo__mod,type,
% 0.32/1.53 pseudo_mod:
% 0.32/1.53 !>[A: $tType] : ( ( poly @ A ) > ( poly @ A ) > ( poly @ A ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(sy_c_Polynomial_Oreflect__poly,type,
% 0.32/1.53 reflect_poly:
% 0.32/1.53 !>[A: $tType] : ( ( poly @ A ) > ( poly @ A ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(sy_c_Polynomial_Orsquarefree,type,
% 0.32/1.53 rsquarefree:
% 0.32/1.53 !>[A: $tType] : ( ( poly @ A ) > $o ) ).
% 0.32/1.53
% 0.32/1.53 thf(sy_c_Polynomial_Osynthetic__div,type,
% 0.32/1.53 synthetic_div:
% 0.32/1.53 !>[A: $tType] : ( ( poly @ A ) > A > ( poly @ A ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(sy_c_Rat_Ofield__char__0__class_ORats,type,
% 0.32/1.53 field_char_0_Rats:
% 0.32/1.53 !>[A: $tType] : ( set @ A ) ).
% 0.32/1.53
% 0.32/1.53 thf(sy_c_Rings_Odvd__class_Odvd,type,
% 0.32/1.53 dvd_dvd:
% 0.32/1.53 !>[A: $tType] : ( A > A > $o ) ).
% 0.32/1.53
% 0.32/1.53 thf(sy_c_Set_OCollect,type,
% 0.32/1.53 collect:
% 0.32/1.53 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(sy_c_member,type,
% 0.32/1.53 member:
% 0.32/1.53 !>[A: $tType] : ( A > ( set @ A ) > $o ) ).
% 0.32/1.53
% 0.32/1.53 thf(sy_v_n____,type,
% 0.32/1.53 n: nat ).
% 0.32/1.53
% 0.32/1.53 thf(sy_v_p____,type,
% 0.32/1.53 p: poly @ real ).
% 0.32/1.53
% 0.32/1.53 thf(sy_v_x,type,
% 0.32/1.53 x: real ).
% 0.32/1.53
% 0.32/1.53 thf(fact_0_poly__roots__finite,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( ( comm_ring_1 @ A )
% 0.32/1.53 & ( ring_n68954251visors @ A ) )
% 0.32/1.53 => ! [P: poly @ A] :
% 0.32/1.53 ( ( P
% 0.32/1.53 != ( zero_zero @ ( poly @ A ) ) )
% 0.32/1.53 => ( finite_finite2 @ A
% 0.32/1.53 @ ( collect @ A
% 0.32/1.53 @ ^ [X: A] :
% 0.32/1.53 ( ( poly2 @ A @ P @ X )
% 0.32/1.53 = ( zero_zero @ A ) ) ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_1_p_I2_J,axiom,
% 0.32/1.53 ( p
% 0.32/1.53 != ( zero_zero @ ( poly @ real ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_2__092_060open_062pderiv_Ap_A_092_060noteq_062_A0_092_060close_062,axiom,
% 0.32/1.53 ( ( pderiv @ real @ p )
% 0.32/1.53 != ( zero_zero @ ( poly @ real ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_3_p_I3_J,axiom,
% 0.32/1.53 ( ( poly2 @ real @ p @ x )
% 0.32/1.53 = ( zero_zero @ real ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_4_poly__0,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( comm_semiring_0 @ A )
% 0.32/1.53 => ! [X2: A] :
% 0.32/1.53 ( ( poly2 @ A @ ( zero_zero @ ( poly @ A ) ) @ X2 )
% 0.32/1.53 = ( zero_zero @ A ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_5_finite__Collect__conjI,axiom,
% 0.32/1.53 ! [A: $tType,P2: A > $o,Q: A > $o] :
% 0.32/1.53 ( ( ( finite_finite2 @ A @ ( collect @ A @ P2 ) )
% 0.32/1.53 | ( finite_finite2 @ A @ ( collect @ A @ Q ) ) )
% 0.32/1.53 => ( finite_finite2 @ A
% 0.32/1.53 @ ( collect @ A
% 0.32/1.53 @ ^ [X: A] :
% 0.32/1.53 ( ( P2 @ X )
% 0.32/1.53 & ( Q @ X ) ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_6_finite__Collect__disjI,axiom,
% 0.32/1.53 ! [A: $tType,P2: A > $o,Q: A > $o] :
% 0.32/1.53 ( ( finite_finite2 @ A
% 0.32/1.53 @ ( collect @ A
% 0.32/1.53 @ ^ [X: A] :
% 0.32/1.53 ( ( P2 @ X )
% 0.32/1.53 | ( Q @ X ) ) ) )
% 0.32/1.53 = ( ( finite_finite2 @ A @ ( collect @ A @ P2 ) )
% 0.32/1.53 & ( finite_finite2 @ A @ ( collect @ A @ Q ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_7_finite__code,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( finite_finite @ A )
% 0.32/1.53 => ( ( finite_finite2 @ A )
% 0.32/1.53 = ( ^ [A2: set @ A] : $true ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_8_poly__all__0__iff__0,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( ( ring_char_0 @ A )
% 0.32/1.53 & ( comm_ring_1 @ A )
% 0.32/1.53 & ( ring_n68954251visors @ A ) )
% 0.32/1.53 => ! [P: poly @ A] :
% 0.32/1.53 ( ( ! [X: A] :
% 0.32/1.53 ( ( poly2 @ A @ P @ X )
% 0.32/1.53 = ( zero_zero @ A ) ) )
% 0.32/1.53 = ( P
% 0.32/1.53 = ( zero_zero @ ( poly @ A ) ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_9_rsquarefree__roots,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( field_char_0 @ A )
% 0.32/1.53 => ( ( rsquarefree @ A )
% 0.32/1.53 = ( ^ [P3: poly @ A] :
% 0.32/1.53 ! [A3: A] :
% 0.32/1.53 ~ ( ( ( poly2 @ A @ P3 @ A3 )
% 0.32/1.53 = ( zero_zero @ A ) )
% 0.32/1.53 & ( ( poly2 @ A @ ( pderiv @ A @ P3 ) @ A3 )
% 0.32/1.53 = ( zero_zero @ A ) ) ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_10_not__finite__existsD,axiom,
% 0.32/1.53 ! [A: $tType,P2: A > $o] :
% 0.32/1.53 ( ~ ( finite_finite2 @ A @ ( collect @ A @ P2 ) )
% 0.32/1.53 => ? [X_1: A] : ( P2 @ X_1 ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_11_pigeonhole__infinite__rel,axiom,
% 0.32/1.53 ! [B: $tType,A: $tType,A4: set @ A,B2: set @ B,R: A > B > $o] :
% 0.32/1.53 ( ~ ( finite_finite2 @ A @ A4 )
% 0.32/1.53 => ( ( finite_finite2 @ B @ B2 )
% 0.32/1.53 => ( ! [X3: A] :
% 0.32/1.53 ( ( member @ A @ X3 @ A4 )
% 0.32/1.53 => ? [Xa: B] :
% 0.32/1.53 ( ( member @ B @ Xa @ B2 )
% 0.32/1.53 & ( R @ X3 @ Xa ) ) )
% 0.32/1.53 => ? [X3: B] :
% 0.32/1.53 ( ( member @ B @ X3 @ B2 )
% 0.32/1.53 & ~ ( finite_finite2 @ A
% 0.32/1.53 @ ( collect @ A
% 0.32/1.53 @ ^ [A3: A] :
% 0.32/1.53 ( ( member @ A @ A3 @ A4 )
% 0.32/1.53 & ( R @ A3 @ X3 ) ) ) ) ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_12_fun__cong__unused__0,axiom,
% 0.32/1.53 ! [A: $tType,B: $tType,C: $tType] :
% 0.32/1.53 ( ( zero @ B )
% 0.32/1.53 => ! [F: ( A > B ) > C,G: C] :
% 0.32/1.53 ( ( F
% 0.32/1.53 = ( ^ [X: A > B] : G ) )
% 0.32/1.53 => ( ( F
% 0.32/1.53 @ ^ [X: A] : ( zero_zero @ B ) )
% 0.32/1.53 = G ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_13_poly__eq__poly__eq__iff,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( ( ring_char_0 @ A )
% 0.32/1.53 & ( comm_ring_1 @ A )
% 0.32/1.53 & ( ring_n68954251visors @ A ) )
% 0.32/1.53 => ! [P: poly @ A,Q2: poly @ A] :
% 0.32/1.53 ( ( ( poly2 @ A @ P )
% 0.32/1.53 = ( poly2 @ A @ Q2 ) )
% 0.32/1.53 = ( P = Q2 ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_14_n__def,axiom,
% 0.32/1.53 ( n
% 0.32/1.53 = ( degree @ real @ p ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_15_assms_I2_J,axiom,
% 0.32/1.53 algebraic @ real @ x ).
% 0.32/1.53
% 0.32/1.53 thf(fact_16_irrationsl,axiom,
% 0.32/1.53 ~ ( member @ real @ x @ ( field_char_0_Rats @ real ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_17_degree__0,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( zero @ A )
% 0.32/1.53 => ( ( degree @ A @ ( zero_zero @ ( poly @ A ) ) )
% 0.32/1.53 = ( zero_zero @ nat ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_18_pderiv__0,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( ( comm_semiring_1 @ A )
% 0.32/1.53 & ( semiri1193490041visors @ A ) )
% 0.32/1.53 => ( ( pderiv @ A @ ( zero_zero @ ( poly @ A ) ) )
% 0.32/1.53 = ( zero_zero @ ( poly @ A ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_19_pderiv__eq__0__iff,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( ( semiring_char_0 @ A )
% 0.32/1.53 & ( comm_semiring_1 @ A )
% 0.32/1.53 & ( semiri1193490041visors @ A ) )
% 0.32/1.53 => ! [P: poly @ A] :
% 0.32/1.53 ( ( ( pderiv @ A @ P )
% 0.32/1.53 = ( zero_zero @ ( poly @ A ) ) )
% 0.32/1.53 = ( ( degree @ A @ P )
% 0.32/1.53 = ( zero_zero @ nat ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_20_zero__reorient,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( zero @ A )
% 0.32/1.53 => ! [X2: A] :
% 0.32/1.53 ( ( ( zero_zero @ A )
% 0.32/1.53 = X2 )
% 0.32/1.53 = ( X2
% 0.32/1.53 = ( zero_zero @ A ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_21_finite,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( finite_finite @ A )
% 0.32/1.53 => ! [A4: set @ A] : ( finite_finite2 @ A @ A4 ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_22_finite__set__choice,axiom,
% 0.32/1.53 ! [B: $tType,A: $tType,A4: set @ A,P2: A > B > $o] :
% 0.32/1.53 ( ( finite_finite2 @ A @ A4 )
% 0.32/1.53 => ( ! [X3: A] :
% 0.32/1.53 ( ( member @ A @ X3 @ A4 )
% 0.32/1.53 => ? [X_12: B] : ( P2 @ X3 @ X_12 ) )
% 0.32/1.53 => ? [F2: A > B] :
% 0.32/1.53 ! [X4: A] :
% 0.32/1.53 ( ( member @ A @ X4 @ A4 )
% 0.32/1.53 => ( P2 @ X4 @ ( F2 @ X4 ) ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_23_is__zero__null,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( zero @ A )
% 0.32/1.53 => ( ( is_zero @ A )
% 0.32/1.53 = ( ^ [P3: poly @ A] :
% 0.32/1.53 ( P3
% 0.32/1.53 = ( zero_zero @ ( poly @ A ) ) ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_24_poly__cutoff__0,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( zero @ A )
% 0.32/1.53 => ! [N: nat] :
% 0.32/1.53 ( ( poly_cutoff @ A @ N @ ( zero_zero @ ( poly @ A ) ) )
% 0.32/1.53 = ( zero_zero @ ( poly @ A ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_25_reflect__poly__at__0__eq__0__iff,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( comm_semiring_0 @ A )
% 0.32/1.53 => ! [P: poly @ A] :
% 0.32/1.53 ( ( ( poly2 @ A @ ( reflect_poly @ A @ P ) @ ( zero_zero @ A ) )
% 0.32/1.53 = ( zero_zero @ A ) )
% 0.32/1.53 = ( P
% 0.32/1.53 = ( zero_zero @ ( poly @ A ) ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_26__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,
% 0.32/1.53 ~ ! [P4: poly @ real] :
% 0.32/1.53 ( ! [I: nat] : ( member @ real @ ( coeff @ real @ P4 @ I ) @ ( ring_1_Ints @ real ) )
% 0.32/1.53 => ( ( P4
% 0.32/1.53 != ( zero_zero @ ( poly @ real ) ) )
% 0.32/1.53 => ( ( poly2 @ real @ P4 @ x )
% 0.32/1.53 != ( zero_zero @ real ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_27_synthetic__div__eq__0__iff,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( comm_semiring_0 @ A )
% 0.32/1.53 => ! [P: poly @ A,C2: A] :
% 0.32/1.53 ( ( ( synthetic_div @ A @ P @ C2 )
% 0.32/1.53 = ( zero_zero @ ( poly @ A ) ) )
% 0.32/1.53 = ( ( degree @ A @ P )
% 0.32/1.53 = ( zero_zero @ nat ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_28_poly__shift__0,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( zero @ A )
% 0.32/1.53 => ! [N: nat] :
% 0.32/1.53 ( ( poly_shift @ A @ N @ ( zero_zero @ ( poly @ A ) ) )
% 0.32/1.53 = ( zero_zero @ ( poly @ A ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_29_order__root,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( idom @ A )
% 0.32/1.53 => ! [P: poly @ A,A5: A] :
% 0.32/1.53 ( ( ( poly2 @ A @ P @ A5 )
% 0.32/1.53 = ( zero_zero @ A ) )
% 0.32/1.53 = ( ( P
% 0.32/1.53 = ( zero_zero @ ( poly @ A ) ) )
% 0.32/1.53 | ( ( order2 @ A @ A5 @ P )
% 0.32/1.53 != ( zero_zero @ nat ) ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_30_leading__coeff__0__iff,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( zero @ A )
% 0.32/1.53 => ! [P: poly @ A] :
% 0.32/1.53 ( ( ( coeff @ A @ P @ ( degree @ A @ P ) )
% 0.32/1.53 = ( zero_zero @ A ) )
% 0.32/1.53 = ( P
% 0.32/1.53 = ( zero_zero @ ( poly @ A ) ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_31_divide__poly__main__0,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( idom_divide @ A )
% 0.32/1.53 => ! [R2: poly @ A,D: poly @ A,Dr: nat,N: nat] :
% 0.32/1.53 ( ( divide_poly_main @ A @ ( zero_zero @ A ) @ ( zero_zero @ ( poly @ A ) ) @ R2 @ D @ Dr @ N )
% 0.32/1.53 = ( zero_zero @ ( poly @ A ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_32_zero__poly_Otransfer,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( zero @ A )
% 0.32/1.53 => ( pcr_poly @ A @ A
% 0.32/1.53 @ ^ [Y: A,Z: A] : ( Y = Z )
% 0.32/1.53 @ ^ [Uu: nat] : ( zero_zero @ A )
% 0.32/1.53 @ ( zero_zero @ ( poly @ A ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_33_p_I1_J,axiom,
% 0.32/1.53 ! [I2: nat] : ( member @ real @ ( coeff @ real @ p @ I2 ) @ ( ring_1_Ints @ real ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_34_reflect__poly__0,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( zero @ A )
% 0.32/1.53 => ( ( reflect_poly @ A @ ( zero_zero @ ( poly @ A ) ) )
% 0.32/1.53 = ( zero_zero @ ( poly @ A ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_35_synthetic__div__0,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( comm_semiring_0 @ A )
% 0.32/1.53 => ! [C2: A] :
% 0.32/1.53 ( ( synthetic_div @ A @ ( zero_zero @ ( poly @ A ) ) @ C2 )
% 0.32/1.53 = ( zero_zero @ ( poly @ A ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_36_poly__shift__id,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( zero @ A )
% 0.32/1.53 => ( ( poly_shift @ A @ ( zero_zero @ nat ) )
% 0.32/1.53 = ( ^ [X: poly @ A] : X ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_37_coeff__0,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( zero @ A )
% 0.32/1.53 => ! [N: nat] :
% 0.32/1.53 ( ( coeff @ A @ ( zero_zero @ ( poly @ A ) ) @ N )
% 0.32/1.53 = ( zero_zero @ A ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_38_reflect__poly__reflect__poly,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( zero @ A )
% 0.32/1.53 => ! [P: poly @ A] :
% 0.32/1.53 ( ( ( coeff @ A @ P @ ( zero_zero @ nat ) )
% 0.32/1.53 != ( zero_zero @ A ) )
% 0.32/1.53 => ( ( reflect_poly @ A @ ( reflect_poly @ A @ P ) )
% 0.32/1.53 = P ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_39_coeff__0__reflect__poly,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( zero @ A )
% 0.32/1.53 => ! [P: poly @ A] :
% 0.32/1.53 ( ( coeff @ A @ ( reflect_poly @ A @ P ) @ ( zero_zero @ nat ) )
% 0.32/1.53 = ( coeff @ A @ P @ ( degree @ A @ P ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_40_coeff__0__reflect__poly__0__iff,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( zero @ A )
% 0.32/1.53 => ! [P: poly @ A] :
% 0.32/1.53 ( ( ( coeff @ A @ ( reflect_poly @ A @ P ) @ ( zero_zero @ nat ) )
% 0.32/1.53 = ( zero_zero @ A ) )
% 0.32/1.53 = ( P
% 0.32/1.53 = ( zero_zero @ ( poly @ A ) ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_41_degree__reflect__poly__eq,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( zero @ A )
% 0.32/1.53 => ! [P: poly @ A] :
% 0.32/1.53 ( ( ( coeff @ A @ P @ ( zero_zero @ nat ) )
% 0.32/1.53 != ( zero_zero @ A ) )
% 0.32/1.53 => ( ( degree @ A @ ( reflect_poly @ A @ P ) )
% 0.32/1.53 = ( degree @ A @ P ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_42_poly__reflect__poly__0,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( comm_semiring_0 @ A )
% 0.32/1.53 => ! [P: poly @ A] :
% 0.32/1.53 ( ( poly2 @ A @ ( reflect_poly @ A @ P ) @ ( zero_zero @ A ) )
% 0.32/1.53 = ( coeff @ A @ P @ ( degree @ A @ P ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_43_divide__poly__main_Osimps_I2_J,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( idom_divide @ A )
% 0.32/1.53 => ! [Lc: A,Q2: poly @ A,R2: poly @ A,D: poly @ A,Dr: nat] :
% 0.32/1.53 ( ( divide_poly_main @ A @ Lc @ Q2 @ R2 @ D @ Dr @ ( zero_zero @ nat ) )
% 0.32/1.53 = Q2 ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_44_mem__Collect__eq,axiom,
% 0.32/1.53 ! [A: $tType,A5: A,P2: A > $o] :
% 0.32/1.53 ( ( member @ A @ A5 @ ( collect @ A @ P2 ) )
% 0.32/1.53 = ( P2 @ A5 ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_45_Collect__mem__eq,axiom,
% 0.32/1.53 ! [A: $tType,A4: set @ A] :
% 0.32/1.53 ( ( collect @ A
% 0.32/1.53 @ ^ [X: A] : ( member @ A @ X @ A4 ) )
% 0.32/1.53 = A4 ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_46_Collect__cong,axiom,
% 0.32/1.53 ! [A: $tType,P2: A > $o,Q: A > $o] :
% 0.32/1.53 ( ! [X3: A] :
% 0.32/1.53 ( ( P2 @ X3 )
% 0.32/1.53 = ( Q @ X3 ) )
% 0.32/1.53 => ( ( collect @ A @ P2 )
% 0.32/1.53 = ( collect @ A @ Q ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_47_ext,axiom,
% 0.32/1.53 ! [B: $tType,A: $tType,F: A > B,G: A > B] :
% 0.32/1.53 ( ! [X3: A] :
% 0.32/1.53 ( ( F @ X3 )
% 0.32/1.53 = ( G @ X3 ) )
% 0.32/1.53 => ( F = G ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_48_poly__eqI,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( zero @ A )
% 0.32/1.53 => ! [P: poly @ A,Q2: poly @ A] :
% 0.32/1.53 ( ! [N2: nat] :
% 0.32/1.53 ( ( coeff @ A @ P @ N2 )
% 0.32/1.53 = ( coeff @ A @ Q2 @ N2 ) )
% 0.32/1.53 => ( P = Q2 ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_49_poly__eq__iff,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( zero @ A )
% 0.32/1.53 => ( ( ^ [Y: poly @ A,Z: poly @ A] : ( Y = Z ) )
% 0.32/1.53 = ( ^ [P3: poly @ A,Q3: poly @ A] :
% 0.32/1.53 ! [N3: nat] :
% 0.32/1.53 ( ( coeff @ A @ P3 @ N3 )
% 0.32/1.53 = ( coeff @ A @ Q3 @ N3 ) ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_50_coeff__inject,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( zero @ A )
% 0.32/1.53 => ! [X2: poly @ A,Y2: poly @ A] :
% 0.32/1.53 ( ( ( coeff @ A @ X2 )
% 0.32/1.53 = ( coeff @ A @ Y2 ) )
% 0.32/1.53 = ( X2 = Y2 ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_51_algebraicE,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( field_char_0 @ A )
% 0.32/1.53 => ! [X2: A] :
% 0.32/1.53 ( ( algebraic @ A @ X2 )
% 0.32/1.53 => ~ ! [P4: poly @ A] :
% 0.32/1.53 ( ! [I: nat] : ( member @ A @ ( coeff @ A @ P4 @ I ) @ ( ring_1_Ints @ A ) )
% 0.32/1.53 => ( ( P4
% 0.32/1.53 != ( zero_zero @ ( poly @ A ) ) )
% 0.32/1.53 => ( ( poly2 @ A @ P4 @ X2 )
% 0.32/1.53 != ( zero_zero @ A ) ) ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_52_algebraicI,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( field_char_0 @ A )
% 0.32/1.53 => ! [P: poly @ A,X2: A] :
% 0.32/1.53 ( ! [I3: nat] : ( member @ A @ ( coeff @ A @ P @ I3 ) @ ( ring_1_Ints @ A ) )
% 0.32/1.53 => ( ( P
% 0.32/1.53 != ( zero_zero @ ( poly @ A ) ) )
% 0.32/1.53 => ( ( ( poly2 @ A @ P @ X2 )
% 0.32/1.53 = ( zero_zero @ A ) )
% 0.32/1.53 => ( algebraic @ A @ X2 ) ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_53_algebraic__def,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( field_char_0 @ A )
% 0.32/1.53 => ( ( algebraic @ A )
% 0.32/1.53 = ( ^ [X: A] :
% 0.32/1.53 ? [P3: poly @ A] :
% 0.32/1.53 ( ! [I4: nat] : ( member @ A @ ( coeff @ A @ P3 @ I4 ) @ ( ring_1_Ints @ A ) )
% 0.32/1.53 & ( P3
% 0.32/1.53 != ( zero_zero @ ( poly @ A ) ) )
% 0.32/1.53 & ( ( poly2 @ A @ P3 @ X )
% 0.32/1.53 = ( zero_zero @ A ) ) ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_54_algebraic__altdef,axiom,
% 0.32/1.53 ! [B: $tType] :
% 0.32/1.53 ( ( field_char_0 @ B )
% 0.32/1.53 => ( ( algebraic @ B )
% 0.32/1.53 = ( ^ [X: B] :
% 0.32/1.53 ? [P3: poly @ B] :
% 0.32/1.53 ( ! [I4: nat] : ( member @ B @ ( coeff @ B @ P3 @ I4 ) @ ( field_char_0_Rats @ B ) )
% 0.32/1.53 & ( P3
% 0.32/1.53 != ( zero_zero @ ( poly @ B ) ) )
% 0.32/1.53 & ( ( poly2 @ B @ P3 @ X )
% 0.32/1.53 = ( zero_zero @ B ) ) ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_55_zero__poly_Orep__eq,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( zero @ A )
% 0.32/1.53 => ( ( coeff @ A @ ( zero_zero @ ( poly @ A ) ) )
% 0.32/1.53 = ( ^ [Uu: nat] : ( zero_zero @ A ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_56_poly__0__coeff__0,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( comm_semiring_0 @ A )
% 0.32/1.53 => ! [P: poly @ A] :
% 0.32/1.53 ( ( poly2 @ A @ P @ ( zero_zero @ A ) )
% 0.32/1.53 = ( coeff @ A @ P @ ( zero_zero @ nat ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_57_order__0I,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( idom @ A )
% 0.32/1.53 => ! [P: poly @ A,A5: A] :
% 0.32/1.53 ( ( ( poly2 @ A @ P @ A5 )
% 0.32/1.53 != ( zero_zero @ A ) )
% 0.32/1.53 => ( ( order2 @ A @ A5 @ P )
% 0.32/1.53 = ( zero_zero @ nat ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_58_leading__coeff__neq__0,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( zero @ A )
% 0.32/1.53 => ! [P: poly @ A] :
% 0.32/1.53 ( ( P
% 0.32/1.53 != ( zero_zero @ ( poly @ A ) ) )
% 0.32/1.53 => ( ( coeff @ A @ P @ ( degree @ A @ P ) )
% 0.32/1.53 != ( zero_zero @ A ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_59_Rats__infinite,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( field_char_0 @ A )
% 0.32/1.53 => ~ ( finite_finite2 @ A @ ( field_char_0_Rats @ A ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_60_Rats__0,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( field_char_0 @ A )
% 0.32/1.53 => ( member @ A @ ( zero_zero @ A ) @ ( field_char_0_Rats @ A ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_61_Ints__0,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( ring_1 @ A )
% 0.32/1.53 => ( member @ A @ ( zero_zero @ A ) @ ( ring_1_Ints @ A ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_62_rsquarefree__def,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( idom @ A )
% 0.32/1.53 => ( ( rsquarefree @ A )
% 0.32/1.53 = ( ^ [P3: poly @ A] :
% 0.32/1.53 ( ( P3
% 0.32/1.53 != ( zero_zero @ ( poly @ A ) ) )
% 0.32/1.53 & ! [A3: A] :
% 0.32/1.53 ( ( ( order2 @ A @ A3 @ P3 )
% 0.32/1.53 = ( zero_zero @ nat ) )
% 0.32/1.53 | ( ( order2 @ A @ A3 @ P3 )
% 0.32/1.53 = ( one_one @ nat ) ) ) ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_63_poly__cutoff__1,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( comm_semiring_1 @ A )
% 0.32/1.53 => ! [N: nat] :
% 0.32/1.53 ( ( ( N
% 0.32/1.53 = ( zero_zero @ nat ) )
% 0.32/1.53 => ( ( poly_cutoff @ A @ N @ ( one_one @ ( poly @ A ) ) )
% 0.32/1.53 = ( zero_zero @ ( poly @ A ) ) ) )
% 0.32/1.53 & ( ( N
% 0.32/1.53 != ( zero_zero @ nat ) )
% 0.32/1.53 => ( ( poly_cutoff @ A @ N @ ( one_one @ ( poly @ A ) ) )
% 0.32/1.53 = ( one_one @ ( poly @ A ) ) ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_64_order__pderiv2,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( field_char_0 @ A )
% 0.32/1.53 => ! [P: poly @ A,A5: A,N: nat] :
% 0.32/1.53 ( ( ( pderiv @ A @ P )
% 0.32/1.53 != ( zero_zero @ ( poly @ A ) ) )
% 0.32/1.53 => ( ( ( order2 @ A @ A5 @ P )
% 0.32/1.53 != ( zero_zero @ nat ) )
% 0.32/1.53 => ( ( ( order2 @ A @ A5 @ ( pderiv @ A @ P ) )
% 0.32/1.53 = N )
% 0.32/1.53 = ( ( order2 @ A @ A5 @ P )
% 0.32/1.53 = ( suc @ N ) ) ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_65_order__pderiv,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( field_char_0 @ A )
% 0.32/1.53 => ! [P: poly @ A,A5: A] :
% 0.32/1.53 ( ( ( pderiv @ A @ P )
% 0.32/1.53 != ( zero_zero @ ( poly @ A ) ) )
% 0.32/1.53 => ( ( ( order2 @ A @ A5 @ P )
% 0.32/1.53 != ( zero_zero @ nat ) )
% 0.32/1.53 => ( ( order2 @ A @ A5 @ P )
% 0.32/1.53 = ( suc @ ( order2 @ A @ A5 @ ( pderiv @ A @ P ) ) ) ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_66_cr__poly__def,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( zero @ A )
% 0.32/1.53 => ( ( cr_poly @ A )
% 0.32/1.53 = ( ^ [X: nat > A,Y3: poly @ A] :
% 0.32/1.53 ( X
% 0.32/1.53 = ( coeff @ A @ Y3 ) ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_67_dvd__pderiv__iff,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( ( semiring_char_0 @ A )
% 0.32/1.53 & ( comm_semiring_1 @ A )
% 0.32/1.53 & ( semiri1193490041visors @ A ) )
% 0.32/1.53 => ! [P: poly @ A] :
% 0.32/1.53 ( ( dvd_dvd @ ( poly @ A ) @ P @ ( pderiv @ A @ P ) )
% 0.32/1.53 = ( ( degree @ A @ P )
% 0.32/1.53 = ( zero_zero @ nat ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_68_reflect__poly__1,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( comm_semiring_1 @ A )
% 0.32/1.53 => ( ( reflect_poly @ A @ ( one_one @ ( poly @ A ) ) )
% 0.32/1.53 = ( one_one @ ( poly @ A ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_69_degree__1,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( comm_semiring_1 @ A )
% 0.32/1.53 => ( ( degree @ A @ ( one_one @ ( poly @ A ) ) )
% 0.32/1.53 = ( zero_zero @ nat ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_70_poly__1,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( comm_semiring_1 @ A )
% 0.32/1.53 => ! [X2: A] :
% 0.32/1.53 ( ( poly2 @ A @ ( one_one @ ( poly @ A ) ) @ X2 )
% 0.32/1.53 = ( one_one @ A ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_71_pderiv__1,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( ( comm_semiring_1 @ A )
% 0.32/1.53 & ( semiri1193490041visors @ A ) )
% 0.32/1.53 => ( ( pderiv @ A @ ( one_one @ ( poly @ A ) ) )
% 0.32/1.53 = ( zero_zero @ ( poly @ A ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_72_order__1__eq__0,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( idom @ A )
% 0.32/1.53 => ! [X2: A] :
% 0.32/1.53 ( ( order2 @ A @ X2 @ ( one_one @ ( poly @ A ) ) )
% 0.32/1.53 = ( zero_zero @ nat ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_73_lead__coeff__1,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( comm_semiring_1 @ A )
% 0.32/1.53 => ( ( coeff @ A @ ( one_one @ ( poly @ A ) ) @ ( degree @ A @ ( one_one @ ( poly @ A ) ) ) )
% 0.32/1.53 = ( one_one @ A ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_74_one__reorient,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( one @ A )
% 0.32/1.53 => ! [X2: A] :
% 0.32/1.53 ( ( ( one_one @ A )
% 0.32/1.53 = X2 )
% 0.32/1.53 = ( X2
% 0.32/1.53 = ( one_one @ A ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_75_Ints__1,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( ring_1 @ A )
% 0.32/1.53 => ( member @ A @ ( one_one @ A ) @ ( ring_1_Ints @ A ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_76_Rats__1,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( field_char_0 @ A )
% 0.32/1.53 => ( member @ A @ ( one_one @ A ) @ ( field_char_0_Rats @ A ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_77_is__unit__iff__degree,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( field @ A )
% 0.32/1.53 => ! [P: poly @ A] :
% 0.32/1.53 ( ( P
% 0.32/1.53 != ( zero_zero @ ( poly @ A ) ) )
% 0.32/1.53 => ( ( dvd_dvd @ ( poly @ A ) @ P @ ( one_one @ ( poly @ A ) ) )
% 0.32/1.53 = ( ( degree @ A @ P )
% 0.32/1.53 = ( zero_zero @ nat ) ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_78_not__dvd__pderiv,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( ( semiring_char_0 @ A )
% 0.32/1.53 & ( comm_semiring_1 @ A )
% 0.32/1.53 & ( semiri1193490041visors @ A ) )
% 0.32/1.53 => ! [P: poly @ A] :
% 0.32/1.53 ( ( ( degree @ A @ P )
% 0.32/1.53 != ( zero_zero @ nat ) )
% 0.32/1.53 => ~ ( dvd_dvd @ ( poly @ A ) @ P @ ( pderiv @ A @ P ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_79_poly_Opcr__cr__eq,axiom,
% 0.32/1.53 ! [D2: $tType] :
% 0.32/1.53 ( ( zero @ D2 )
% 0.32/1.53 => ( ( pcr_poly @ D2 @ D2
% 0.32/1.53 @ ^ [Y: D2,Z: D2] : ( Y = Z ) )
% 0.32/1.53 = ( cr_poly @ D2 ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_80_poly__shift__1,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( comm_semiring_1 @ A )
% 0.32/1.53 => ! [N: nat] :
% 0.32/1.53 ( ( ( N
% 0.32/1.53 = ( zero_zero @ nat ) )
% 0.32/1.53 => ( ( poly_shift @ A @ N @ ( one_one @ ( poly @ A ) ) )
% 0.32/1.53 = ( one_one @ ( poly @ A ) ) ) )
% 0.32/1.53 & ( ( N
% 0.32/1.53 != ( zero_zero @ nat ) )
% 0.32/1.53 => ( ( poly_shift @ A @ N @ ( one_one @ ( poly @ A ) ) )
% 0.32/1.53 = ( zero_zero @ ( poly @ A ) ) ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_81_dvd__0__left__iff,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( comm_semiring_1 @ A )
% 0.32/1.53 => ! [A5: A] :
% 0.32/1.53 ( ( dvd_dvd @ A @ ( zero_zero @ A ) @ A5 )
% 0.32/1.53 = ( A5
% 0.32/1.53 = ( zero_zero @ A ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_82_dvd__0__right,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( comm_semiring_1 @ A )
% 0.32/1.53 => ! [A5: A] : ( dvd_dvd @ A @ A5 @ ( zero_zero @ A ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_83_One__nat__def,axiom,
% 0.32/1.53 ( ( one_one @ nat )
% 0.32/1.53 = ( suc @ ( zero_zero @ nat ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_84_not__is__unit__0,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( algebraic_semidom @ A )
% 0.32/1.53 => ~ ( dvd_dvd @ A @ ( zero_zero @ A ) @ ( one_one @ A ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_85_nat_Oinject,axiom,
% 0.32/1.53 ! [X22: nat,Y22: nat] :
% 0.32/1.53 ( ( ( suc @ X22 )
% 0.32/1.53 = ( suc @ Y22 ) )
% 0.32/1.53 = ( X22 = Y22 ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_86_old_Onat_Oinject,axiom,
% 0.32/1.53 ! [Nat: nat,Nat2: nat] :
% 0.32/1.53 ( ( ( suc @ Nat )
% 0.32/1.53 = ( suc @ Nat2 ) )
% 0.32/1.53 = ( Nat = Nat2 ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_87_nat__dvd__1__iff__1,axiom,
% 0.32/1.53 ! [M: nat] :
% 0.32/1.53 ( ( dvd_dvd @ nat @ M @ ( one_one @ nat ) )
% 0.32/1.53 = ( M
% 0.32/1.53 = ( one_one @ nat ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_88_dvd__1__iff__1,axiom,
% 0.32/1.53 ! [M: nat] :
% 0.32/1.53 ( ( dvd_dvd @ nat @ M @ ( suc @ ( zero_zero @ nat ) ) )
% 0.32/1.53 = ( M
% 0.32/1.53 = ( suc @ ( zero_zero @ nat ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_89_dvd__1__left,axiom,
% 0.32/1.53 ! [K: nat] : ( dvd_dvd @ nat @ ( suc @ ( zero_zero @ nat ) ) @ K ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_90_dvd__trans,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( comm_monoid_mult @ A )
% 0.32/1.53 => ! [A5: A,B3: A,C2: A] :
% 0.32/1.53 ( ( dvd_dvd @ A @ A5 @ B3 )
% 0.32/1.53 => ( ( dvd_dvd @ A @ B3 @ C2 )
% 0.32/1.53 => ( dvd_dvd @ A @ A5 @ C2 ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_91_dvd__refl,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( comm_monoid_mult @ A )
% 0.32/1.53 => ! [A5: A] : ( dvd_dvd @ A @ A5 @ A5 ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_92_n__not__Suc__n,axiom,
% 0.32/1.53 ! [N: nat] :
% 0.32/1.53 ( N
% 0.32/1.53 != ( suc @ N ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_93_Suc__inject,axiom,
% 0.32/1.53 ! [X2: nat,Y2: nat] :
% 0.32/1.53 ( ( ( suc @ X2 )
% 0.32/1.53 = ( suc @ Y2 ) )
% 0.32/1.53 => ( X2 = Y2 ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_94_zero__neq__one,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( zero_neq_one @ A )
% 0.32/1.53 => ( ( zero_zero @ A )
% 0.32/1.53 != ( one_one @ A ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_95_dvd__0__left,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( comm_semiring_1 @ A )
% 0.32/1.53 => ! [A5: A] :
% 0.32/1.53 ( ( dvd_dvd @ A @ ( zero_zero @ A ) @ A5 )
% 0.32/1.53 => ( A5
% 0.32/1.53 = ( zero_zero @ A ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_96_dvd__unit__imp__unit,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( algebraic_semidom @ A )
% 0.32/1.53 => ! [A5: A,B3: A] :
% 0.32/1.53 ( ( dvd_dvd @ A @ A5 @ B3 )
% 0.32/1.53 => ( ( dvd_dvd @ A @ B3 @ ( one_one @ A ) )
% 0.32/1.53 => ( dvd_dvd @ A @ A5 @ ( one_one @ A ) ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_97_unit__imp__dvd,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( algebraic_semidom @ A )
% 0.32/1.53 => ! [B3: A,A5: A] :
% 0.32/1.53 ( ( dvd_dvd @ A @ B3 @ ( one_one @ A ) )
% 0.32/1.53 => ( dvd_dvd @ A @ B3 @ A5 ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_98_one__dvd,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( comm_monoid_mult @ A )
% 0.32/1.53 => ! [A5: A] : ( dvd_dvd @ A @ ( one_one @ A ) @ A5 ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_99_not0__implies__Suc,axiom,
% 0.32/1.53 ! [N: nat] :
% 0.32/1.53 ( ( N
% 0.32/1.53 != ( zero_zero @ nat ) )
% 0.32/1.53 => ? [M2: nat] :
% 0.32/1.53 ( N
% 0.32/1.53 = ( suc @ M2 ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_100_old_Onat_Oinducts,axiom,
% 0.32/1.53 ! [P2: nat > $o,Nat: nat] :
% 0.32/1.53 ( ( P2 @ ( zero_zero @ nat ) )
% 0.32/1.53 => ( ! [Nat3: nat] :
% 0.32/1.53 ( ( P2 @ Nat3 )
% 0.32/1.53 => ( P2 @ ( suc @ Nat3 ) ) )
% 0.32/1.53 => ( P2 @ Nat ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_101_old_Onat_Oexhaust,axiom,
% 0.32/1.53 ! [Y2: nat] :
% 0.32/1.53 ( ( Y2
% 0.32/1.53 != ( zero_zero @ nat ) )
% 0.32/1.53 => ~ ! [Nat3: nat] :
% 0.32/1.53 ( Y2
% 0.32/1.53 != ( suc @ Nat3 ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_102_Zero__not__Suc,axiom,
% 0.32/1.53 ! [M: nat] :
% 0.32/1.53 ( ( zero_zero @ nat )
% 0.32/1.53 != ( suc @ M ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_103_Zero__neq__Suc,axiom,
% 0.32/1.53 ! [M: nat] :
% 0.32/1.53 ( ( zero_zero @ nat )
% 0.32/1.53 != ( suc @ M ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_104_Suc__neq__Zero,axiom,
% 0.32/1.53 ! [M: nat] :
% 0.32/1.53 ( ( suc @ M )
% 0.32/1.53 != ( zero_zero @ nat ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_105_zero__induct,axiom,
% 0.32/1.53 ! [P2: nat > $o,K: nat] :
% 0.32/1.53 ( ( P2 @ K )
% 0.32/1.53 => ( ! [N2: nat] :
% 0.32/1.53 ( ( P2 @ ( suc @ N2 ) )
% 0.32/1.53 => ( P2 @ N2 ) )
% 0.32/1.53 => ( P2 @ ( zero_zero @ nat ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_106_diff__induct,axiom,
% 0.32/1.53 ! [P2: nat > nat > $o,M: nat,N: nat] :
% 0.32/1.53 ( ! [X3: nat] : ( P2 @ X3 @ ( zero_zero @ nat ) )
% 0.32/1.53 => ( ! [Y4: nat] : ( P2 @ ( zero_zero @ nat ) @ ( suc @ Y4 ) )
% 0.32/1.53 => ( ! [X3: nat,Y4: nat] :
% 0.32/1.53 ( ( P2 @ X3 @ Y4 )
% 0.32/1.53 => ( P2 @ ( suc @ X3 ) @ ( suc @ Y4 ) ) )
% 0.32/1.53 => ( P2 @ M @ N ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_107_nat__induct,axiom,
% 0.32/1.53 ! [P2: nat > $o,N: nat] :
% 0.32/1.53 ( ( P2 @ ( zero_zero @ nat ) )
% 0.32/1.53 => ( ! [N2: nat] :
% 0.32/1.53 ( ( P2 @ N2 )
% 0.32/1.53 => ( P2 @ ( suc @ N2 ) ) )
% 0.32/1.53 => ( P2 @ N ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_108_nat_OdiscI,axiom,
% 0.32/1.53 ! [Nat: nat,X22: nat] :
% 0.32/1.53 ( ( Nat
% 0.32/1.53 = ( suc @ X22 ) )
% 0.32/1.53 => ( Nat
% 0.32/1.53 != ( zero_zero @ nat ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_109_old_Onat_Odistinct_I1_J,axiom,
% 0.32/1.53 ! [Nat2: nat] :
% 0.32/1.53 ( ( zero_zero @ nat )
% 0.32/1.53 != ( suc @ Nat2 ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_110_old_Onat_Odistinct_I2_J,axiom,
% 0.32/1.53 ! [Nat4: nat] :
% 0.32/1.53 ( ( suc @ Nat4 )
% 0.32/1.53 != ( zero_zero @ nat ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_111_nat_Odistinct_I1_J,axiom,
% 0.32/1.53 ! [X22: nat] :
% 0.32/1.53 ( ( zero_zero @ nat )
% 0.32/1.53 != ( suc @ X22 ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_112_algebraic__int_Oinducts,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( field @ A )
% 0.32/1.53 => ! [X2: A,P2: A > $o] :
% 0.32/1.53 ( ( algebraic_int @ A @ X2 )
% 0.32/1.53 => ( ! [P4: poly @ A,X3: A] :
% 0.32/1.53 ( ( ( coeff @ A @ P4 @ ( degree @ A @ P4 ) )
% 0.32/1.53 = ( one_one @ A ) )
% 0.32/1.53 => ( ! [I: nat] : ( member @ A @ ( coeff @ A @ P4 @ I ) @ ( ring_1_Ints @ A ) )
% 0.32/1.53 => ( ( ( poly2 @ A @ P4 @ X3 )
% 0.32/1.53 = ( zero_zero @ A ) )
% 0.32/1.53 => ( P2 @ X3 ) ) ) )
% 0.32/1.53 => ( P2 @ X2 ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_113_algebraic__int_Ointros,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( field @ A )
% 0.32/1.53 => ! [P: poly @ A,X2: A] :
% 0.32/1.53 ( ( ( coeff @ A @ P @ ( degree @ A @ P ) )
% 0.32/1.53 = ( one_one @ A ) )
% 0.32/1.53 => ( ! [I3: nat] : ( member @ A @ ( coeff @ A @ P @ I3 ) @ ( ring_1_Ints @ A ) )
% 0.32/1.53 => ( ( ( poly2 @ A @ P @ X2 )
% 0.32/1.53 = ( zero_zero @ A ) )
% 0.32/1.53 => ( algebraic_int @ A @ X2 ) ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_114_algebraic__int_Osimps,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( field @ A )
% 0.32/1.53 => ( ( algebraic_int @ A )
% 0.32/1.53 = ( ^ [A3: A] :
% 0.32/1.53 ? [P3: poly @ A,X: A] :
% 0.32/1.53 ( ( A3 = X )
% 0.32/1.53 & ( ( coeff @ A @ P3 @ ( degree @ A @ P3 ) )
% 0.32/1.53 = ( one_one @ A ) )
% 0.32/1.53 & ! [I4: nat] : ( member @ A @ ( coeff @ A @ P3 @ I4 ) @ ( ring_1_Ints @ A ) )
% 0.32/1.53 & ( ( poly2 @ A @ P3 @ X )
% 0.32/1.53 = ( zero_zero @ A ) ) ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_115_algebraic__int_Ocases,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( field @ A )
% 0.32/1.53 => ! [A5: A] :
% 0.32/1.53 ( ( algebraic_int @ A @ A5 )
% 0.32/1.53 => ~ ! [P4: poly @ A] :
% 0.32/1.53 ( ( ( coeff @ A @ P4 @ ( degree @ A @ P4 ) )
% 0.32/1.53 = ( one_one @ A ) )
% 0.32/1.53 => ( ! [I: nat] : ( member @ A @ ( coeff @ A @ P4 @ I ) @ ( ring_1_Ints @ A ) )
% 0.32/1.53 => ( ( poly2 @ A @ P4 @ A5 )
% 0.32/1.53 != ( zero_zero @ A ) ) ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_116_exists__least__lemma,axiom,
% 0.32/1.53 ! [P2: nat > $o] :
% 0.32/1.53 ( ~ ( P2 @ ( zero_zero @ nat ) )
% 0.32/1.53 => ( ? [X_12: nat] : ( P2 @ X_12 )
% 0.32/1.53 => ? [N2: nat] :
% 0.32/1.53 ( ~ ( P2 @ N2 )
% 0.32/1.53 & ( P2 @ ( suc @ N2 ) ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_117_dependent__nat__choice,axiom,
% 0.32/1.53 ! [A: $tType,P2: nat > A > $o,Q: nat > A > A > $o] :
% 0.32/1.53 ( ? [X_12: A] : ( P2 @ ( zero_zero @ nat ) @ X_12 )
% 0.32/1.53 => ( ! [X3: A,N2: nat] :
% 0.32/1.53 ( ( P2 @ N2 @ X3 )
% 0.32/1.53 => ? [Y5: A] :
% 0.32/1.53 ( ( P2 @ ( suc @ N2 ) @ Y5 )
% 0.32/1.53 & ( Q @ N2 @ X3 @ Y5 ) ) )
% 0.32/1.53 => ? [F2: nat > A] :
% 0.32/1.53 ! [N4: nat] :
% 0.32/1.53 ( ( P2 @ N4 @ ( F2 @ N4 ) )
% 0.32/1.53 & ( Q @ N4 @ ( F2 @ N4 ) @ ( F2 @ ( suc @ N4 ) ) ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_118_algebraic__int__0,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( field @ A )
% 0.32/1.53 => ( algebraic_int @ A @ ( zero_zero @ A ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_119_algebraic__int__1,axiom,
% 0.32/1.53 ! [B: $tType] :
% 0.32/1.53 ( ( field @ B )
% 0.32/1.53 => ( algebraic_int @ B @ ( one_one @ B ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_120_dvd__antisym,axiom,
% 0.32/1.53 ! [M: nat,N: nat] :
% 0.32/1.53 ( ( dvd_dvd @ nat @ M @ N )
% 0.32/1.53 => ( ( dvd_dvd @ nat @ N @ M )
% 0.32/1.53 => ( M = N ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_121_int__imp__algebraic__int,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( field @ A )
% 0.32/1.53 => ! [X2: A] :
% 0.32/1.53 ( ( member @ A @ X2 @ ( ring_1_Ints @ A ) )
% 0.32/1.53 => ( algebraic_int @ A @ X2 ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_122_algebraic__int__imp__algebraic,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( field_char_0 @ A )
% 0.32/1.53 => ! [X2: A] :
% 0.32/1.53 ( ( algebraic_int @ A @ X2 )
% 0.32/1.53 => ( algebraic @ A @ X2 ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_123_rational__algebraic__int__is__int,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( field_char_0 @ A )
% 0.32/1.53 => ! [X2: A] :
% 0.32/1.53 ( ( algebraic_int @ A @ X2 )
% 0.32/1.53 => ( ( member @ A @ X2 @ ( field_char_0_Rats @ A ) )
% 0.32/1.53 => ( member @ A @ X2 @ ( ring_1_Ints @ A ) ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_124_algebraic__int__inverse,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( field @ A )
% 0.32/1.53 => ! [P: poly @ A,X2: A] :
% 0.32/1.53 ( ( ( poly2 @ A @ P @ X2 )
% 0.32/1.53 = ( zero_zero @ A ) )
% 0.32/1.53 => ( ! [I3: nat] : ( member @ A @ ( coeff @ A @ P @ I3 ) @ ( ring_1_Ints @ A ) )
% 0.32/1.53 => ( ( ( coeff @ A @ P @ ( zero_zero @ nat ) )
% 0.32/1.53 = ( one_one @ A ) )
% 0.32/1.53 => ( algebraic_int @ A @ ( inverse_inverse @ A @ X2 ) ) ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_125_algebraic__int__root,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( field @ A )
% 0.32/1.53 => ! [Y2: A,P: poly @ A,X2: A] :
% 0.32/1.53 ( ( algebraic_int @ A @ Y2 )
% 0.32/1.53 => ( ( ( poly2 @ A @ P @ X2 )
% 0.32/1.53 = Y2 )
% 0.32/1.53 => ( ! [I3: nat] : ( member @ A @ ( coeff @ A @ P @ I3 ) @ ( ring_1_Ints @ A ) )
% 0.32/1.53 => ( ( ( coeff @ A @ P @ ( degree @ A @ P ) )
% 0.32/1.53 = ( one_one @ A ) )
% 0.32/1.53 => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( degree @ A @ P ) )
% 0.32/1.53 => ( algebraic_int @ A @ X2 ) ) ) ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_126_gcd__nat_Oextremum__uniqueI,axiom,
% 0.32/1.53 ! [A5: nat] :
% 0.32/1.53 ( ( dvd_dvd @ nat @ ( zero_zero @ nat ) @ A5 )
% 0.32/1.53 => ( A5
% 0.32/1.53 = ( zero_zero @ nat ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_127_gcd__nat_Onot__eq__extremum,axiom,
% 0.32/1.53 ! [A5: nat] :
% 0.32/1.53 ( ( A5
% 0.32/1.53 != ( zero_zero @ nat ) )
% 0.32/1.53 = ( ( dvd_dvd @ nat @ A5 @ ( zero_zero @ nat ) )
% 0.32/1.53 & ( A5
% 0.32/1.53 != ( zero_zero @ nat ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_128_finite__Collect__less__nat,axiom,
% 0.32/1.53 ! [K: nat] :
% 0.32/1.53 ( finite_finite2 @ nat
% 0.32/1.53 @ ( collect @ nat
% 0.32/1.53 @ ^ [N3: nat] : ( ord_less @ nat @ N3 @ K ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_129_not__gr__zero,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( canoni770627133id_add @ A )
% 0.32/1.53 => ! [N: A] :
% 0.32/1.53 ( ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ N ) )
% 0.32/1.53 = ( N
% 0.32/1.53 = ( zero_zero @ A ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_130_bot__nat__0_Onot__eq__extremum,axiom,
% 0.32/1.53 ! [A5: nat] :
% 0.32/1.53 ( ( A5
% 0.32/1.53 != ( zero_zero @ nat ) )
% 0.32/1.53 = ( ord_less @ nat @ ( zero_zero @ nat ) @ A5 ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_131_less__nat__zero__code,axiom,
% 0.32/1.53 ! [N: nat] :
% 0.32/1.53 ~ ( ord_less @ nat @ N @ ( zero_zero @ nat ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_132_neq0__conv,axiom,
% 0.32/1.53 ! [N: nat] :
% 0.32/1.53 ( ( N
% 0.32/1.53 != ( zero_zero @ nat ) )
% 0.32/1.53 = ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_133_lessI,axiom,
% 0.32/1.53 ! [N: nat] : ( ord_less @ nat @ N @ ( suc @ N ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_134_Suc__mono,axiom,
% 0.32/1.53 ! [M: nat,N: nat] :
% 0.32/1.53 ( ( ord_less @ nat @ M @ N )
% 0.32/1.53 => ( ord_less @ nat @ ( suc @ M ) @ ( suc @ N ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_135_Suc__less__eq,axiom,
% 0.32/1.53 ! [M: nat,N: nat] :
% 0.32/1.53 ( ( ord_less @ nat @ ( suc @ M ) @ ( suc @ N ) )
% 0.32/1.53 = ( ord_less @ nat @ M @ N ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_136_zero__less__Suc,axiom,
% 0.32/1.53 ! [N: nat] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( suc @ N ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_137_less__Suc0,axiom,
% 0.32/1.53 ! [N: nat] :
% 0.32/1.53 ( ( ord_less @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) )
% 0.32/1.53 = ( N
% 0.32/1.53 = ( zero_zero @ nat ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_138_less__one,axiom,
% 0.32/1.53 ! [N: nat] :
% 0.32/1.53 ( ( ord_less @ nat @ N @ ( one_one @ nat ) )
% 0.32/1.53 = ( N
% 0.32/1.53 = ( zero_zero @ nat ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_139_zero__less__iff__neq__zero,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( canoni770627133id_add @ A )
% 0.32/1.53 => ! [N: A] :
% 0.32/1.53 ( ( ord_less @ A @ ( zero_zero @ A ) @ N )
% 0.32/1.53 = ( N
% 0.32/1.53 != ( zero_zero @ A ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_140_gr__implies__not__zero,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( canoni770627133id_add @ A )
% 0.32/1.53 => ! [M: A,N: A] :
% 0.32/1.53 ( ( ord_less @ A @ M @ N )
% 0.32/1.53 => ( N
% 0.32/1.53 != ( zero_zero @ A ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_141_not__less__zero,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( canoni770627133id_add @ A )
% 0.32/1.53 => ! [N: A] :
% 0.32/1.53 ~ ( ord_less @ A @ N @ ( zero_zero @ A ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_142_gr__zeroI,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( canoni770627133id_add @ A )
% 0.32/1.53 => ! [N: A] :
% 0.32/1.53 ( ( N
% 0.32/1.53 != ( zero_zero @ A ) )
% 0.32/1.53 => ( ord_less @ A @ ( zero_zero @ A ) @ N ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_143_gr0I,axiom,
% 0.32/1.53 ! [N: nat] :
% 0.32/1.53 ( ( N
% 0.32/1.53 != ( zero_zero @ nat ) )
% 0.32/1.53 => ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_144_not__gr0,axiom,
% 0.32/1.53 ! [N: nat] :
% 0.32/1.53 ( ( ~ ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) )
% 0.32/1.53 = ( N
% 0.32/1.53 = ( zero_zero @ nat ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_145_not__less0,axiom,
% 0.32/1.53 ! [N: nat] :
% 0.32/1.53 ~ ( ord_less @ nat @ N @ ( zero_zero @ nat ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_146_less__zeroE,axiom,
% 0.32/1.53 ! [N: nat] :
% 0.32/1.53 ~ ( ord_less @ nat @ N @ ( zero_zero @ nat ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_147_gr__implies__not0,axiom,
% 0.32/1.53 ! [M: nat,N: nat] :
% 0.32/1.53 ( ( ord_less @ nat @ M @ N )
% 0.32/1.53 => ( N
% 0.32/1.53 != ( zero_zero @ nat ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_148_infinite__descent0,axiom,
% 0.32/1.53 ! [P2: nat > $o,N: nat] :
% 0.32/1.53 ( ( P2 @ ( zero_zero @ nat ) )
% 0.32/1.53 => ( ! [N2: nat] :
% 0.32/1.53 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
% 0.32/1.53 => ( ~ ( P2 @ N2 )
% 0.32/1.53 => ? [M3: nat] :
% 0.32/1.53 ( ( ord_less @ nat @ M3 @ N2 )
% 0.32/1.53 & ~ ( P2 @ M3 ) ) ) )
% 0.32/1.53 => ( P2 @ N ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_149_bot__nat__0_Oextremum__strict,axiom,
% 0.32/1.53 ! [A5: nat] :
% 0.32/1.53 ~ ( ord_less @ nat @ A5 @ ( zero_zero @ nat ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_150_infinite__descent0__measure,axiom,
% 0.32/1.53 ! [A: $tType,V: A > nat,P2: A > $o,X2: A] :
% 0.32/1.53 ( ! [X3: A] :
% 0.32/1.53 ( ( ( V @ X3 )
% 0.32/1.53 = ( zero_zero @ nat ) )
% 0.32/1.53 => ( P2 @ X3 ) )
% 0.32/1.53 => ( ! [X3: A] :
% 0.32/1.53 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( V @ X3 ) )
% 0.32/1.53 => ( ~ ( P2 @ X3 )
% 0.32/1.53 => ? [Y5: A] :
% 0.32/1.53 ( ( ord_less @ nat @ ( V @ Y5 ) @ ( V @ X3 ) )
% 0.32/1.53 & ~ ( P2 @ Y5 ) ) ) )
% 0.32/1.53 => ( P2 @ X2 ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_151_not__less__less__Suc__eq,axiom,
% 0.32/1.53 ! [N: nat,M: nat] :
% 0.32/1.53 ( ~ ( ord_less @ nat @ N @ M )
% 0.32/1.53 => ( ( ord_less @ nat @ N @ ( suc @ M ) )
% 0.32/1.53 = ( N = M ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_152_strict__inc__induct,axiom,
% 0.32/1.53 ! [I2: nat,J: nat,P2: nat > $o] :
% 0.32/1.53 ( ( ord_less @ nat @ I2 @ J )
% 0.32/1.53 => ( ! [I3: nat] :
% 0.32/1.53 ( ( J
% 0.32/1.53 = ( suc @ I3 ) )
% 0.32/1.53 => ( P2 @ I3 ) )
% 0.32/1.53 => ( ! [I3: nat] :
% 0.32/1.53 ( ( ord_less @ nat @ I3 @ J )
% 0.32/1.53 => ( ( P2 @ ( suc @ I3 ) )
% 0.32/1.53 => ( P2 @ I3 ) ) )
% 0.32/1.53 => ( P2 @ I2 ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_153_less__Suc__induct,axiom,
% 0.32/1.53 ! [I2: nat,J: nat,P2: nat > nat > $o] :
% 0.32/1.53 ( ( ord_less @ nat @ I2 @ J )
% 0.32/1.53 => ( ! [I3: nat] : ( P2 @ I3 @ ( suc @ I3 ) )
% 0.32/1.53 => ( ! [I3: nat,J2: nat,K2: nat] :
% 0.32/1.53 ( ( ord_less @ nat @ I3 @ J2 )
% 0.32/1.53 => ( ( ord_less @ nat @ J2 @ K2 )
% 0.32/1.53 => ( ( P2 @ I3 @ J2 )
% 0.32/1.53 => ( ( P2 @ J2 @ K2 )
% 0.32/1.53 => ( P2 @ I3 @ K2 ) ) ) ) )
% 0.32/1.53 => ( P2 @ I2 @ J ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_154_less__trans__Suc,axiom,
% 0.32/1.53 ! [I2: nat,J: nat,K: nat] :
% 0.32/1.53 ( ( ord_less @ nat @ I2 @ J )
% 0.32/1.53 => ( ( ord_less @ nat @ J @ K )
% 0.32/1.53 => ( ord_less @ nat @ ( suc @ I2 ) @ K ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_155_Suc__less__SucD,axiom,
% 0.32/1.53 ! [M: nat,N: nat] :
% 0.32/1.53 ( ( ord_less @ nat @ ( suc @ M ) @ ( suc @ N ) )
% 0.32/1.53 => ( ord_less @ nat @ M @ N ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_156_less__antisym,axiom,
% 0.32/1.53 ! [N: nat,M: nat] :
% 0.32/1.53 ( ~ ( ord_less @ nat @ N @ M )
% 0.32/1.53 => ( ( ord_less @ nat @ N @ ( suc @ M ) )
% 0.32/1.53 => ( M = N ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_157_Suc__less__eq2,axiom,
% 0.32/1.53 ! [N: nat,M: nat] :
% 0.32/1.53 ( ( ord_less @ nat @ ( suc @ N ) @ M )
% 0.32/1.53 = ( ? [M4: nat] :
% 0.32/1.53 ( ( M
% 0.32/1.53 = ( suc @ M4 ) )
% 0.32/1.53 & ( ord_less @ nat @ N @ M4 ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_158_All__less__Suc,axiom,
% 0.32/1.53 ! [N: nat,P2: nat > $o] :
% 0.32/1.53 ( ( ! [I4: nat] :
% 0.32/1.53 ( ( ord_less @ nat @ I4 @ ( suc @ N ) )
% 0.32/1.53 => ( P2 @ I4 ) ) )
% 0.32/1.53 = ( ( P2 @ N )
% 0.32/1.53 & ! [I4: nat] :
% 0.32/1.53 ( ( ord_less @ nat @ I4 @ N )
% 0.32/1.53 => ( P2 @ I4 ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_159_not__less__eq,axiom,
% 0.32/1.53 ! [M: nat,N: nat] :
% 0.32/1.53 ( ( ~ ( ord_less @ nat @ M @ N ) )
% 0.32/1.53 = ( ord_less @ nat @ N @ ( suc @ M ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_160_less__Suc__eq,axiom,
% 0.32/1.53 ! [M: nat,N: nat] :
% 0.32/1.53 ( ( ord_less @ nat @ M @ ( suc @ N ) )
% 0.32/1.53 = ( ( ord_less @ nat @ M @ N )
% 0.32/1.53 | ( M = N ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_161_Ex__less__Suc,axiom,
% 0.32/1.53 ! [N: nat,P2: nat > $o] :
% 0.32/1.53 ( ( ? [I4: nat] :
% 0.32/1.53 ( ( ord_less @ nat @ I4 @ ( suc @ N ) )
% 0.32/1.53 & ( P2 @ I4 ) ) )
% 0.32/1.53 = ( ( P2 @ N )
% 0.32/1.53 | ? [I4: nat] :
% 0.32/1.53 ( ( ord_less @ nat @ I4 @ N )
% 0.32/1.53 & ( P2 @ I4 ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_162_less__SucI,axiom,
% 0.32/1.53 ! [M: nat,N: nat] :
% 0.32/1.53 ( ( ord_less @ nat @ M @ N )
% 0.32/1.53 => ( ord_less @ nat @ M @ ( suc @ N ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_163_less__SucE,axiom,
% 0.32/1.53 ! [M: nat,N: nat] :
% 0.32/1.53 ( ( ord_less @ nat @ M @ ( suc @ N ) )
% 0.32/1.53 => ( ~ ( ord_less @ nat @ M @ N )
% 0.32/1.53 => ( M = N ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_164_Suc__lessI,axiom,
% 0.32/1.53 ! [M: nat,N: nat] :
% 0.32/1.53 ( ( ord_less @ nat @ M @ N )
% 0.32/1.53 => ( ( ( suc @ M )
% 0.32/1.53 != N )
% 0.32/1.53 => ( ord_less @ nat @ ( suc @ M ) @ N ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_165_Suc__lessE,axiom,
% 0.32/1.53 ! [I2: nat,K: nat] :
% 0.32/1.53 ( ( ord_less @ nat @ ( suc @ I2 ) @ K )
% 0.32/1.53 => ~ ! [J2: nat] :
% 0.32/1.53 ( ( ord_less @ nat @ I2 @ J2 )
% 0.32/1.53 => ( K
% 0.32/1.53 != ( suc @ J2 ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_166_Suc__lessD,axiom,
% 0.32/1.53 ! [M: nat,N: nat] :
% 0.32/1.53 ( ( ord_less @ nat @ ( suc @ M ) @ N )
% 0.32/1.53 => ( ord_less @ nat @ M @ N ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_167_Nat_OlessE,axiom,
% 0.32/1.53 ! [I2: nat,K: nat] :
% 0.32/1.53 ( ( ord_less @ nat @ I2 @ K )
% 0.32/1.53 => ( ( K
% 0.32/1.53 != ( suc @ I2 ) )
% 0.32/1.53 => ~ ! [J2: nat] :
% 0.32/1.53 ( ( ord_less @ nat @ I2 @ J2 )
% 0.32/1.53 => ( K
% 0.32/1.53 != ( suc @ J2 ) ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_168_Rats__inverse,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( field_char_0 @ A )
% 0.32/1.53 => ! [A5: A] :
% 0.32/1.53 ( ( member @ A @ A5 @ ( field_char_0_Rats @ A ) )
% 0.32/1.53 => ( member @ A @ ( inverse_inverse @ A @ A5 ) @ ( field_char_0_Rats @ A ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_169_nat__neq__iff,axiom,
% 0.32/1.53 ! [M: nat,N: nat] :
% 0.32/1.53 ( ( M != N )
% 0.32/1.53 = ( ( ord_less @ nat @ M @ N )
% 0.32/1.53 | ( ord_less @ nat @ N @ M ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_170_less__not__refl,axiom,
% 0.32/1.53 ! [N: nat] :
% 0.32/1.53 ~ ( ord_less @ nat @ N @ N ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_171_less__not__refl2,axiom,
% 0.32/1.53 ! [N: nat,M: nat] :
% 0.32/1.53 ( ( ord_less @ nat @ N @ M )
% 0.32/1.53 => ( M != N ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_172_less__not__refl3,axiom,
% 0.32/1.53 ! [S: nat,T: nat] :
% 0.32/1.53 ( ( ord_less @ nat @ S @ T )
% 0.32/1.53 => ( S != T ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_173_measure__induct,axiom,
% 0.32/1.53 ! [B: $tType,A: $tType] :
% 0.32/1.53 ( ( wellorder @ B )
% 0.32/1.53 => ! [F: A > B,P2: A > $o,A5: A] :
% 0.32/1.53 ( ! [X3: A] :
% 0.32/1.53 ( ! [Y5: A] :
% 0.32/1.53 ( ( ord_less @ B @ ( F @ Y5 ) @ ( F @ X3 ) )
% 0.32/1.53 => ( P2 @ Y5 ) )
% 0.32/1.53 => ( P2 @ X3 ) )
% 0.32/1.53 => ( P2 @ A5 ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_174_less__irrefl__nat,axiom,
% 0.32/1.53 ! [N: nat] :
% 0.32/1.53 ~ ( ord_less @ nat @ N @ N ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_175_nat__less__induct,axiom,
% 0.32/1.53 ! [P2: nat > $o,N: nat] :
% 0.32/1.53 ( ! [N2: nat] :
% 0.32/1.53 ( ! [M3: nat] :
% 0.32/1.53 ( ( ord_less @ nat @ M3 @ N2 )
% 0.32/1.53 => ( P2 @ M3 ) )
% 0.32/1.53 => ( P2 @ N2 ) )
% 0.32/1.53 => ( P2 @ N ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_176_infinite__descent,axiom,
% 0.32/1.53 ! [P2: nat > $o,N: nat] :
% 0.32/1.53 ( ! [N2: nat] :
% 0.32/1.53 ( ~ ( P2 @ N2 )
% 0.32/1.53 => ? [M3: nat] :
% 0.32/1.53 ( ( ord_less @ nat @ M3 @ N2 )
% 0.32/1.53 & ~ ( P2 @ M3 ) ) )
% 0.32/1.53 => ( P2 @ N ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_177_linorder__neqE__nat,axiom,
% 0.32/1.53 ! [X2: nat,Y2: nat] :
% 0.32/1.53 ( ( X2 != Y2 )
% 0.32/1.53 => ( ~ ( ord_less @ nat @ X2 @ Y2 )
% 0.32/1.53 => ( ord_less @ nat @ Y2 @ X2 ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_178_measure__induct__rule,axiom,
% 0.32/1.53 ! [B: $tType,A: $tType] :
% 0.32/1.53 ( ( wellorder @ B )
% 0.32/1.53 => ! [F: A > B,P2: A > $o,A5: A] :
% 0.32/1.53 ( ! [X3: A] :
% 0.32/1.53 ( ! [Y5: A] :
% 0.32/1.53 ( ( ord_less @ B @ ( F @ Y5 ) @ ( F @ X3 ) )
% 0.32/1.53 => ( P2 @ Y5 ) )
% 0.32/1.53 => ( P2 @ X3 ) )
% 0.32/1.53 => ( P2 @ A5 ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_179_infinite__descent__measure,axiom,
% 0.32/1.53 ! [A: $tType,P2: A > $o,V: A > nat,X2: A] :
% 0.32/1.53 ( ! [X3: A] :
% 0.32/1.53 ( ~ ( P2 @ X3 )
% 0.32/1.53 => ? [Y5: A] :
% 0.32/1.53 ( ( ord_less @ nat @ ( V @ Y5 ) @ ( V @ X3 ) )
% 0.32/1.53 & ~ ( P2 @ Y5 ) ) )
% 0.32/1.53 => ( P2 @ X2 ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_180_linorder__neqE__linordered__idom,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( linordered_idom @ A )
% 0.32/1.53 => ! [X2: A,Y2: A] :
% 0.32/1.53 ( ( X2 != Y2 )
% 0.32/1.53 => ( ~ ( ord_less @ A @ X2 @ Y2 )
% 0.32/1.53 => ( ord_less @ A @ Y2 @ X2 ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_181_lift__Suc__mono__less__iff,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( order @ A )
% 0.32/1.53 => ! [F: nat > A,N: nat,M: nat] :
% 0.32/1.53 ( ! [N2: nat] : ( ord_less @ A @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
% 0.32/1.53 => ( ( ord_less @ A @ ( F @ N ) @ ( F @ M ) )
% 0.32/1.53 = ( ord_less @ nat @ N @ M ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_182_lift__Suc__mono__less,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( order @ A )
% 0.32/1.53 => ! [F: nat > A,N: nat,N5: nat] :
% 0.32/1.53 ( ! [N2: nat] : ( ord_less @ A @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
% 0.32/1.53 => ( ( ord_less @ nat @ N @ N5 )
% 0.32/1.53 => ( ord_less @ A @ ( F @ N ) @ ( F @ N5 ) ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_183_dvd__pos__nat,axiom,
% 0.32/1.53 ! [N: nat,M: nat] :
% 0.32/1.53 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 0.32/1.53 => ( ( dvd_dvd @ nat @ M @ N )
% 0.32/1.53 => ( ord_less @ nat @ ( zero_zero @ nat ) @ M ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_184_finite__divisors__nat,axiom,
% 0.32/1.53 ! [M: nat] :
% 0.32/1.53 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
% 0.32/1.53 => ( finite_finite2 @ nat
% 0.32/1.53 @ ( collect @ nat
% 0.32/1.53 @ ^ [D3: nat] : ( dvd_dvd @ nat @ D3 @ M ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_185_not__one__less__zero,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( linord1659791738miring @ A )
% 0.32/1.53 => ~ ( ord_less @ A @ ( one_one @ A ) @ ( zero_zero @ A ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_186_zero__less__one,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( zero_less_one @ A )
% 0.32/1.53 => ( ord_less @ A @ ( zero_zero @ A ) @ ( one_one @ A ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_187_less__Suc__eq__0__disj,axiom,
% 0.32/1.53 ! [M: nat,N: nat] :
% 0.32/1.53 ( ( ord_less @ nat @ M @ ( suc @ N ) )
% 0.32/1.53 = ( ( M
% 0.32/1.53 = ( zero_zero @ nat ) )
% 0.32/1.53 | ? [J3: nat] :
% 0.32/1.53 ( ( M
% 0.32/1.53 = ( suc @ J3 ) )
% 0.32/1.53 & ( ord_less @ nat @ J3 @ N ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_188_gr0__implies__Suc,axiom,
% 0.32/1.53 ! [N: nat] :
% 0.32/1.53 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 0.32/1.53 => ? [M2: nat] :
% 0.32/1.53 ( N
% 0.32/1.53 = ( suc @ M2 ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_189_All__less__Suc2,axiom,
% 0.32/1.53 ! [N: nat,P2: nat > $o] :
% 0.32/1.53 ( ( ! [I4: nat] :
% 0.32/1.53 ( ( ord_less @ nat @ I4 @ ( suc @ N ) )
% 0.32/1.53 => ( P2 @ I4 ) ) )
% 0.32/1.53 = ( ( P2 @ ( zero_zero @ nat ) )
% 0.32/1.53 & ! [I4: nat] :
% 0.32/1.53 ( ( ord_less @ nat @ I4 @ N )
% 0.32/1.53 => ( P2 @ ( suc @ I4 ) ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_190_gr0__conv__Suc,axiom,
% 0.32/1.53 ! [N: nat] :
% 0.32/1.53 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 0.32/1.53 = ( ? [M5: nat] :
% 0.32/1.53 ( N
% 0.32/1.53 = ( suc @ M5 ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_191_Ex__less__Suc2,axiom,
% 0.32/1.53 ! [N: nat,P2: nat > $o] :
% 0.32/1.53 ( ( ? [I4: nat] :
% 0.32/1.53 ( ( ord_less @ nat @ I4 @ ( suc @ N ) )
% 0.32/1.53 & ( P2 @ I4 ) ) )
% 0.32/1.53 = ( ( P2 @ ( zero_zero @ nat ) )
% 0.32/1.53 | ? [I4: nat] :
% 0.32/1.53 ( ( ord_less @ nat @ I4 @ N )
% 0.32/1.53 & ( P2 @ ( suc @ I4 ) ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_192_nat__dvd__not__less,axiom,
% 0.32/1.53 ! [M: nat,N: nat] :
% 0.32/1.53 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
% 0.32/1.53 => ( ( ord_less @ nat @ M @ N )
% 0.32/1.53 => ~ ( dvd_dvd @ nat @ N @ M ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_193_nat__induct__non__zero,axiom,
% 0.32/1.53 ! [N: nat,P2: nat > $o] :
% 0.32/1.53 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 0.32/1.53 => ( ( P2 @ ( one_one @ nat ) )
% 0.32/1.53 => ( ! [N2: nat] :
% 0.32/1.53 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
% 0.32/1.53 => ( ( P2 @ N2 )
% 0.32/1.53 => ( P2 @ ( suc @ N2 ) ) ) )
% 0.32/1.53 => ( P2 @ N ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_194_less__degree__imp,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( zero @ A )
% 0.32/1.53 => ! [N: nat,P: poly @ A] :
% 0.32/1.53 ( ( ord_less @ nat @ N @ ( degree @ A @ P ) )
% 0.32/1.53 => ? [I3: nat] :
% 0.32/1.53 ( ( ord_less @ nat @ N @ I3 )
% 0.32/1.53 & ( ( coeff @ A @ P @ I3 )
% 0.32/1.53 != ( zero_zero @ A ) ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_195_coeff__eq__0,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( zero @ A )
% 0.32/1.53 => ! [P: poly @ A,N: nat] :
% 0.32/1.53 ( ( ord_less @ nat @ ( degree @ A @ P ) @ N )
% 0.32/1.53 => ( ( coeff @ A @ P @ N )
% 0.32/1.53 = ( zero_zero @ A ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_196_gcd__nat_Onot__eq__order__implies__strict,axiom,
% 0.32/1.53 ! [A5: nat,B3: nat] :
% 0.32/1.53 ( ( A5 != B3 )
% 0.32/1.53 => ( ( dvd_dvd @ nat @ A5 @ B3 )
% 0.32/1.53 => ( ( dvd_dvd @ nat @ A5 @ B3 )
% 0.32/1.53 & ( A5 != B3 ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_197_gcd__nat_Ostrict__implies__not__eq,axiom,
% 0.32/1.53 ! [A5: nat,B3: nat] :
% 0.32/1.53 ( ( ( dvd_dvd @ nat @ A5 @ B3 )
% 0.32/1.53 & ( A5 != B3 ) )
% 0.32/1.53 => ( A5 != B3 ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_198_gcd__nat_Ostrict__implies__order,axiom,
% 0.32/1.53 ! [A5: nat,B3: nat] :
% 0.32/1.53 ( ( ( dvd_dvd @ nat @ A5 @ B3 )
% 0.32/1.53 & ( A5 != B3 ) )
% 0.32/1.53 => ( dvd_dvd @ nat @ A5 @ B3 ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_199_gcd__nat_Ostrict__iff__order,axiom,
% 0.32/1.53 ! [A5: nat,B3: nat] :
% 0.32/1.53 ( ( ( dvd_dvd @ nat @ A5 @ B3 )
% 0.32/1.53 & ( A5 != B3 ) )
% 0.32/1.53 = ( ( dvd_dvd @ nat @ A5 @ B3 )
% 0.32/1.53 & ( A5 != B3 ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_200_gcd__nat_Oorder__iff__strict,axiom,
% 0.32/1.53 ( ( dvd_dvd @ nat )
% 0.32/1.53 = ( ^ [A3: nat,B4: nat] :
% 0.32/1.53 ( ( ( dvd_dvd @ nat @ A3 @ B4 )
% 0.32/1.53 & ( A3 != B4 ) )
% 0.32/1.53 | ( A3 = B4 ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_201_gcd__nat_Ostrict__trans2,axiom,
% 0.32/1.53 ! [A5: nat,B3: nat,C2: nat] :
% 0.32/1.53 ( ( ( dvd_dvd @ nat @ A5 @ B3 )
% 0.32/1.53 & ( A5 != B3 ) )
% 0.32/1.53 => ( ( dvd_dvd @ nat @ B3 @ C2 )
% 0.32/1.53 => ( ( dvd_dvd @ nat @ A5 @ C2 )
% 0.32/1.53 & ( A5 != C2 ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_202_gcd__nat_Ostrict__trans1,axiom,
% 0.32/1.53 ! [A5: nat,B3: nat,C2: nat] :
% 0.32/1.53 ( ( dvd_dvd @ nat @ A5 @ B3 )
% 0.32/1.53 => ( ( ( dvd_dvd @ nat @ B3 @ C2 )
% 0.32/1.53 & ( B3 != C2 ) )
% 0.32/1.53 => ( ( dvd_dvd @ nat @ A5 @ C2 )
% 0.32/1.53 & ( A5 != C2 ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_203_gcd__nat_Ostrict__trans,axiom,
% 0.32/1.53 ! [A5: nat,B3: nat,C2: nat] :
% 0.32/1.53 ( ( ( dvd_dvd @ nat @ A5 @ B3 )
% 0.32/1.53 & ( A5 != B3 ) )
% 0.32/1.53 => ( ( ( dvd_dvd @ nat @ B3 @ C2 )
% 0.32/1.53 & ( B3 != C2 ) )
% 0.32/1.53 => ( ( dvd_dvd @ nat @ A5 @ C2 )
% 0.32/1.53 & ( A5 != C2 ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_204_gcd__nat_Oantisym,axiom,
% 0.32/1.53 ! [A5: nat,B3: nat] :
% 0.32/1.53 ( ( dvd_dvd @ nat @ A5 @ B3 )
% 0.32/1.53 => ( ( dvd_dvd @ nat @ B3 @ A5 )
% 0.32/1.53 => ( A5 = B3 ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_205_gcd__nat_Oirrefl,axiom,
% 0.32/1.53 ! [A5: nat] :
% 0.32/1.53 ~ ( ( dvd_dvd @ nat @ A5 @ A5 )
% 0.32/1.53 & ( A5 != A5 ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_206_gcd__nat_Oeq__iff,axiom,
% 0.32/1.53 ( ( ^ [Y: nat,Z: nat] : ( Y = Z ) )
% 0.32/1.53 = ( ^ [A3: nat,B4: nat] :
% 0.32/1.53 ( ( dvd_dvd @ nat @ A3 @ B4 )
% 0.32/1.53 & ( dvd_dvd @ nat @ B4 @ A3 ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_207_gcd__nat_Otrans,axiom,
% 0.32/1.53 ! [A5: nat,B3: nat,C2: nat] :
% 0.32/1.53 ( ( dvd_dvd @ nat @ A5 @ B3 )
% 0.32/1.53 => ( ( dvd_dvd @ nat @ B3 @ C2 )
% 0.32/1.53 => ( dvd_dvd @ nat @ A5 @ C2 ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_208_gcd__nat_Orefl,axiom,
% 0.32/1.53 ! [A5: nat] : ( dvd_dvd @ nat @ A5 @ A5 ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_209_gcd__nat_Oasym,axiom,
% 0.32/1.53 ! [A5: nat,B3: nat] :
% 0.32/1.53 ( ( ( dvd_dvd @ nat @ A5 @ B3 )
% 0.32/1.53 & ( A5 != B3 ) )
% 0.32/1.53 => ~ ( ( dvd_dvd @ nat @ B3 @ A5 )
% 0.32/1.53 & ( B3 != A5 ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_210_coeff__poly__cutoff,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( zero @ A )
% 0.32/1.53 => ! [K: nat,N: nat,P: poly @ A] :
% 0.32/1.53 ( ( ( ord_less @ nat @ K @ N )
% 0.32/1.53 => ( ( coeff @ A @ ( poly_cutoff @ A @ N @ P ) @ K )
% 0.32/1.53 = ( coeff @ A @ P @ K ) ) )
% 0.32/1.53 & ( ~ ( ord_less @ nat @ K @ N )
% 0.32/1.53 => ( ( coeff @ A @ ( poly_cutoff @ A @ N @ P ) @ K )
% 0.32/1.53 = ( zero_zero @ A ) ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_211_gcd__nat_Oextremum,axiom,
% 0.32/1.53 ! [A5: nat] : ( dvd_dvd @ nat @ A5 @ ( zero_zero @ nat ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_212_gcd__nat_Oextremum__strict,axiom,
% 0.32/1.53 ! [A5: nat] :
% 0.32/1.53 ~ ( ( dvd_dvd @ nat @ ( zero_zero @ nat ) @ A5 )
% 0.32/1.53 & ( ( zero_zero @ nat )
% 0.32/1.53 != A5 ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_213_gcd__nat_Oextremum__unique,axiom,
% 0.32/1.53 ! [A5: nat] :
% 0.32/1.53 ( ( dvd_dvd @ nat @ ( zero_zero @ nat ) @ A5 )
% 0.32/1.53 = ( A5
% 0.32/1.53 = ( zero_zero @ nat ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_214_inverse__positive__iff__positive,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( linordered_field @ A )
% 0.32/1.53 => ! [A5: A] :
% 0.32/1.53 ( ( ord_less @ A @ ( zero_zero @ A ) @ ( inverse_inverse @ A @ A5 ) )
% 0.32/1.53 = ( ord_less @ A @ ( zero_zero @ A ) @ A5 ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_215_inverse__negative__iff__negative,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( linordered_field @ A )
% 0.32/1.53 => ! [A5: A] :
% 0.32/1.53 ( ( ord_less @ A @ ( inverse_inverse @ A @ A5 ) @ ( zero_zero @ A ) )
% 0.32/1.53 = ( ord_less @ A @ A5 @ ( zero_zero @ A ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_216_inverse__less__iff__less__neg,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( linordered_field @ A )
% 0.32/1.53 => ! [A5: A,B3: A] :
% 0.32/1.53 ( ( ord_less @ A @ A5 @ ( zero_zero @ A ) )
% 0.32/1.53 => ( ( ord_less @ A @ B3 @ ( zero_zero @ A ) )
% 0.32/1.53 => ( ( ord_less @ A @ ( inverse_inverse @ A @ A5 ) @ ( inverse_inverse @ A @ B3 ) )
% 0.32/1.53 = ( ord_less @ A @ B3 @ A5 ) ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_217_inverse__less__iff__less,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( linordered_field @ A )
% 0.32/1.53 => ! [A5: A,B3: A] :
% 0.32/1.53 ( ( ord_less @ A @ ( zero_zero @ A ) @ A5 )
% 0.32/1.53 => ( ( ord_less @ A @ ( zero_zero @ A ) @ B3 )
% 0.32/1.53 => ( ( ord_less @ A @ ( inverse_inverse @ A @ A5 ) @ ( inverse_inverse @ A @ B3 ) )
% 0.32/1.53 = ( ord_less @ A @ B3 @ A5 ) ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_218_inverse__zero,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( division_ring @ A )
% 0.32/1.53 => ( ( inverse_inverse @ A @ ( zero_zero @ A ) )
% 0.32/1.53 = ( zero_zero @ A ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_219_inverse__nonzero__iff__nonzero,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( division_ring @ A )
% 0.32/1.53 => ! [A5: A] :
% 0.32/1.53 ( ( ( inverse_inverse @ A @ A5 )
% 0.32/1.53 = ( zero_zero @ A ) )
% 0.32/1.53 = ( A5
% 0.32/1.53 = ( zero_zero @ A ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_220_inverse__eq__1__iff,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( field @ A )
% 0.32/1.53 => ! [X2: A] :
% 0.32/1.53 ( ( ( inverse_inverse @ A @ X2 )
% 0.32/1.53 = ( one_one @ A ) )
% 0.32/1.53 = ( X2
% 0.32/1.53 = ( one_one @ A ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_221_inverse__1,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( division_ring @ A )
% 0.32/1.53 => ( ( inverse_inverse @ A @ ( one_one @ A ) )
% 0.32/1.53 = ( one_one @ A ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_222_finite__psubset__induct,axiom,
% 0.32/1.53 ! [A: $tType,A4: set @ A,P2: ( set @ A ) > $o] :
% 0.32/1.53 ( ( finite_finite2 @ A @ A4 )
% 0.32/1.53 => ( ! [A6: set @ A] :
% 0.32/1.53 ( ( finite_finite2 @ A @ A6 )
% 0.32/1.53 => ( ! [B5: set @ A] :
% 0.32/1.53 ( ( ord_less @ ( set @ A ) @ B5 @ A6 )
% 0.32/1.53 => ( P2 @ B5 ) )
% 0.32/1.53 => ( P2 @ A6 ) ) )
% 0.32/1.53 => ( P2 @ A4 ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_223_strict__subset__divisors__dvd,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( comm_monoid_mult @ A )
% 0.32/1.53 => ! [A5: A,B3: A] :
% 0.32/1.53 ( ( ord_less @ ( set @ A )
% 0.32/1.53 @ ( collect @ A
% 0.32/1.53 @ ^ [C3: A] : ( dvd_dvd @ A @ C3 @ A5 ) )
% 0.32/1.53 @ ( collect @ A
% 0.32/1.53 @ ^ [C3: A] : ( dvd_dvd @ A @ C3 @ B3 ) ) )
% 0.32/1.53 = ( ( dvd_dvd @ A @ A5 @ B3 )
% 0.32/1.53 & ~ ( dvd_dvd @ A @ B3 @ A5 ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_224_poly__IVT__pos,axiom,
% 0.32/1.53 ! [A5: real,B3: real,P: poly @ real] :
% 0.32/1.53 ( ( ord_less @ real @ A5 @ B3 )
% 0.32/1.53 => ( ( ord_less @ real @ ( poly2 @ real @ P @ A5 ) @ ( zero_zero @ real ) )
% 0.32/1.53 => ( ( ord_less @ real @ ( zero_zero @ real ) @ ( poly2 @ real @ P @ B3 ) )
% 0.32/1.53 => ? [X3: real] :
% 0.32/1.53 ( ( ord_less @ real @ A5 @ X3 )
% 0.32/1.53 & ( ord_less @ real @ X3 @ B3 )
% 0.32/1.53 & ( ( poly2 @ real @ P @ X3 )
% 0.32/1.53 = ( zero_zero @ real ) ) ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_225_poly__IVT__neg,axiom,
% 0.32/1.53 ! [A5: real,B3: real,P: poly @ real] :
% 0.32/1.53 ( ( ord_less @ real @ A5 @ B3 )
% 0.32/1.53 => ( ( ord_less @ real @ ( zero_zero @ real ) @ ( poly2 @ real @ P @ A5 ) )
% 0.32/1.53 => ( ( ord_less @ real @ ( poly2 @ real @ P @ B3 ) @ ( zero_zero @ real ) )
% 0.32/1.53 => ? [X3: real] :
% 0.32/1.53 ( ( ord_less @ real @ A5 @ X3 )
% 0.32/1.53 & ( ord_less @ real @ X3 @ B3 )
% 0.32/1.53 & ( ( poly2 @ real @ P @ X3 )
% 0.32/1.53 = ( zero_zero @ real ) ) ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_226_dvd__field__iff,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( field @ A )
% 0.32/1.53 => ( ( dvd_dvd @ A )
% 0.32/1.53 = ( ^ [A3: A,B4: A] :
% 0.32/1.53 ( ( A3
% 0.32/1.53 = ( zero_zero @ A ) )
% 0.32/1.53 => ( B4
% 0.32/1.53 = ( zero_zero @ A ) ) ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_227_field__class_Ofield__inverse__zero,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( field @ A )
% 0.32/1.53 => ( ( inverse_inverse @ A @ ( zero_zero @ A ) )
% 0.32/1.53 = ( zero_zero @ A ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_228_inverse__zero__imp__zero,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( division_ring @ A )
% 0.32/1.53 => ! [A5: A] :
% 0.32/1.53 ( ( ( inverse_inverse @ A @ A5 )
% 0.32/1.53 = ( zero_zero @ A ) )
% 0.32/1.53 => ( A5
% 0.32/1.53 = ( zero_zero @ A ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_229_nonzero__inverse__eq__imp__eq,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( division_ring @ A )
% 0.32/1.53 => ! [A5: A,B3: A] :
% 0.32/1.53 ( ( ( inverse_inverse @ A @ A5 )
% 0.32/1.53 = ( inverse_inverse @ A @ B3 ) )
% 0.32/1.53 => ( ( A5
% 0.32/1.53 != ( zero_zero @ A ) )
% 0.32/1.53 => ( ( B3
% 0.32/1.53 != ( zero_zero @ A ) )
% 0.32/1.53 => ( A5 = B3 ) ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_230_nonzero__inverse__inverse__eq,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( division_ring @ A )
% 0.32/1.53 => ! [A5: A] :
% 0.32/1.53 ( ( A5
% 0.32/1.53 != ( zero_zero @ A ) )
% 0.32/1.53 => ( ( inverse_inverse @ A @ ( inverse_inverse @ A @ A5 ) )
% 0.32/1.53 = A5 ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_231_nonzero__imp__inverse__nonzero,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( division_ring @ A )
% 0.32/1.53 => ! [A5: A] :
% 0.32/1.53 ( ( A5
% 0.32/1.53 != ( zero_zero @ A ) )
% 0.32/1.53 => ( ( inverse_inverse @ A @ A5 )
% 0.32/1.53 != ( zero_zero @ A ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_232_inverse__less__imp__less,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( linordered_field @ A )
% 0.32/1.53 => ! [A5: A,B3: A] :
% 0.32/1.53 ( ( ord_less @ A @ ( inverse_inverse @ A @ A5 ) @ ( inverse_inverse @ A @ B3 ) )
% 0.32/1.53 => ( ( ord_less @ A @ ( zero_zero @ A ) @ A5 )
% 0.32/1.53 => ( ord_less @ A @ B3 @ A5 ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_233_less__imp__inverse__less,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( linordered_field @ A )
% 0.32/1.53 => ! [A5: A,B3: A] :
% 0.32/1.53 ( ( ord_less @ A @ A5 @ B3 )
% 0.32/1.53 => ( ( ord_less @ A @ ( zero_zero @ A ) @ A5 )
% 0.32/1.53 => ( ord_less @ A @ ( inverse_inverse @ A @ B3 ) @ ( inverse_inverse @ A @ A5 ) ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_234_inverse__less__imp__less__neg,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( linordered_field @ A )
% 0.32/1.53 => ! [A5: A,B3: A] :
% 0.32/1.53 ( ( ord_less @ A @ ( inverse_inverse @ A @ A5 ) @ ( inverse_inverse @ A @ B3 ) )
% 0.32/1.53 => ( ( ord_less @ A @ B3 @ ( zero_zero @ A ) )
% 0.32/1.53 => ( ord_less @ A @ B3 @ A5 ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_235_less__imp__inverse__less__neg,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( linordered_field @ A )
% 0.32/1.53 => ! [A5: A,B3: A] :
% 0.32/1.53 ( ( ord_less @ A @ A5 @ B3 )
% 0.32/1.53 => ( ( ord_less @ A @ B3 @ ( zero_zero @ A ) )
% 0.32/1.53 => ( ord_less @ A @ ( inverse_inverse @ A @ B3 ) @ ( inverse_inverse @ A @ A5 ) ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_236_inverse__negative__imp__negative,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( linordered_field @ A )
% 0.32/1.53 => ! [A5: A] :
% 0.32/1.53 ( ( ord_less @ A @ ( inverse_inverse @ A @ A5 ) @ ( zero_zero @ A ) )
% 0.32/1.53 => ( ( A5
% 0.32/1.53 != ( zero_zero @ A ) )
% 0.32/1.53 => ( ord_less @ A @ A5 @ ( zero_zero @ A ) ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_237_inverse__positive__imp__positive,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( linordered_field @ A )
% 0.32/1.53 => ! [A5: A] :
% 0.32/1.53 ( ( ord_less @ A @ ( zero_zero @ A ) @ ( inverse_inverse @ A @ A5 ) )
% 0.32/1.53 => ( ( A5
% 0.32/1.53 != ( zero_zero @ A ) )
% 0.32/1.53 => ( ord_less @ A @ ( zero_zero @ A ) @ A5 ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_238_negative__imp__inverse__negative,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( linordered_field @ A )
% 0.32/1.53 => ! [A5: A] :
% 0.32/1.53 ( ( ord_less @ A @ A5 @ ( zero_zero @ A ) )
% 0.32/1.53 => ( ord_less @ A @ ( inverse_inverse @ A @ A5 ) @ ( zero_zero @ A ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_239_positive__imp__inverse__positive,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( linordered_field @ A )
% 0.32/1.53 => ! [A5: A] :
% 0.32/1.53 ( ( ord_less @ A @ ( zero_zero @ A ) @ A5 )
% 0.32/1.53 => ( ord_less @ A @ ( zero_zero @ A ) @ ( inverse_inverse @ A @ A5 ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_240_one__less__inverse,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( linordered_field @ A )
% 0.32/1.53 => ! [A5: A] :
% 0.32/1.53 ( ( ord_less @ A @ ( zero_zero @ A ) @ A5 )
% 0.32/1.53 => ( ( ord_less @ A @ A5 @ ( one_one @ A ) )
% 0.32/1.53 => ( ord_less @ A @ ( one_one @ A ) @ ( inverse_inverse @ A @ A5 ) ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_241_one__less__inverse__iff,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( linordered_field @ A )
% 0.32/1.53 => ! [X2: A] :
% 0.32/1.53 ( ( ord_less @ A @ ( one_one @ A ) @ ( inverse_inverse @ A @ X2 ) )
% 0.32/1.53 = ( ( ord_less @ A @ ( zero_zero @ A ) @ X2 )
% 0.32/1.53 & ( ord_less @ A @ X2 @ ( one_one @ A ) ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_242_less__numeral__extra_I1_J,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( linord1659791738miring @ A )
% 0.32/1.53 => ( ord_less @ A @ ( zero_zero @ A ) @ ( one_one @ A ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_243_finite__M__bounded__by__nat,axiom,
% 0.32/1.53 ! [P2: nat > $o,I2: nat] :
% 0.32/1.53 ( finite_finite2 @ nat
% 0.32/1.53 @ ( collect @ nat
% 0.32/1.53 @ ^ [K3: nat] :
% 0.32/1.53 ( ( P2 @ K3 )
% 0.32/1.53 & ( ord_less @ nat @ K3 @ I2 ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_244_less__numeral__extra_I3_J,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( linord1659791738miring @ A )
% 0.32/1.53 => ~ ( ord_less @ A @ ( zero_zero @ A ) @ ( zero_zero @ A ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_245_less__numeral__extra_I4_J,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( linord1659791738miring @ A )
% 0.32/1.53 => ~ ( ord_less @ A @ ( one_one @ A ) @ ( one_one @ A ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_246_Rats__dense__in__real,axiom,
% 0.32/1.53 ! [X2: real,Y2: real] :
% 0.32/1.53 ( ( ord_less @ real @ X2 @ Y2 )
% 0.32/1.53 => ? [X3: real] :
% 0.32/1.53 ( ( member @ real @ X3 @ ( field_char_0_Rats @ real ) )
% 0.32/1.53 & ( ord_less @ real @ X2 @ X3 )
% 0.32/1.53 & ( ord_less @ real @ X3 @ Y2 ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_247_Rats__no__bot__less,axiom,
% 0.32/1.53 ! [X2: real] :
% 0.32/1.53 ? [X3: real] :
% 0.32/1.53 ( ( member @ real @ X3 @ ( field_char_0_Rats @ real ) )
% 0.32/1.53 & ( ord_less @ real @ X3 @ X2 ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_248_field__lbound__gt__zero,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( linordered_field @ A )
% 0.32/1.53 => ! [D1: A,D22: A] :
% 0.32/1.53 ( ( ord_less @ A @ ( zero_zero @ A ) @ D1 )
% 0.32/1.53 => ( ( ord_less @ A @ ( zero_zero @ A ) @ D22 )
% 0.32/1.53 => ? [E: A] :
% 0.32/1.53 ( ( ord_less @ A @ ( zero_zero @ A ) @ E )
% 0.32/1.53 & ( ord_less @ A @ E @ D1 )
% 0.32/1.53 & ( ord_less @ A @ E @ D22 ) ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_249_pseudo__mod_I2_J,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( ( comm_ring_1 @ A )
% 0.32/1.53 & ( semiri134348788visors @ A ) )
% 0.32/1.53 => ! [G: poly @ A,F: poly @ A] :
% 0.32/1.53 ( ( G
% 0.32/1.53 != ( zero_zero @ ( poly @ A ) ) )
% 0.32/1.53 => ( ( ( pseudo_mod @ A @ F @ G )
% 0.32/1.53 = ( zero_zero @ ( poly @ A ) ) )
% 0.32/1.53 | ( ord_less @ nat @ ( degree @ A @ ( pseudo_mod @ A @ F @ G ) ) @ ( degree @ A @ G ) ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_250_poly__pinfty__gt__lc,axiom,
% 0.32/1.53 ! [P: poly @ real] :
% 0.32/1.53 ( ( ord_less @ real @ ( zero_zero @ real ) @ ( coeff @ real @ P @ ( degree @ real @ P ) ) )
% 0.32/1.53 => ? [N2: real] :
% 0.32/1.53 ! [X4: real] :
% 0.32/1.53 ( ( ord_less_eq @ real @ N2 @ X4 )
% 0.32/1.53 => ( ord_less_eq @ real @ ( coeff @ real @ P @ ( degree @ real @ P ) ) @ ( poly2 @ real @ P @ X4 ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_251_le__zero__eq,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( canoni770627133id_add @ A )
% 0.32/1.53 => ! [N: A] :
% 0.32/1.53 ( ( ord_less_eq @ A @ N @ ( zero_zero @ A ) )
% 0.32/1.53 = ( N
% 0.32/1.53 = ( zero_zero @ A ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(tcon_fun___Finite__Set_Ofinite,axiom,
% 0.32/1.53 ! [A7: $tType,A8: $tType] :
% 0.32/1.53 ( ( ( finite_finite @ A7 )
% 0.32/1.53 & ( finite_finite @ A8 ) )
% 0.32/1.53 => ( finite_finite @ ( A7 > A8 ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(tcon_fun___Orderings_Oorder,axiom,
% 0.32/1.53 ! [A7: $tType,A8: $tType] :
% 0.32/1.53 ( ( order @ A8 )
% 0.32/1.53 => ( order @ ( A7 > A8 ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(tcon_Nat_Onat___Groups_Ocanonically__ordered__monoid__add,axiom,
% 0.32/1.53 canoni770627133id_add @ nat ).
% 0.32/1.53
% 0.32/1.53 thf(tcon_Nat_Onat___Rings_Osemiring__1__no__zero__divisors,axiom,
% 0.32/1.53 semiri134348788visors @ nat ).
% 0.32/1.53
% 0.32/1.53 thf(tcon_Nat_Onat___Rings_Olinordered__nonzero__semiring,axiom,
% 0.32/1.53 linord1659791738miring @ nat ).
% 0.32/1.53
% 0.32/1.53 thf(tcon_Nat_Onat___Rings_Osemiring__no__zero__divisors,axiom,
% 0.32/1.53 semiri1193490041visors @ nat ).
% 0.32/1.53
% 0.32/1.53 thf(tcon_Nat_Onat___Rings_Oalgebraic__semidom,axiom,
% 0.32/1.53 algebraic_semidom @ nat ).
% 0.32/1.53
% 0.32/1.53 thf(tcon_Nat_Onat___Groups_Ocomm__monoid__mult,axiom,
% 0.32/1.53 comm_monoid_mult @ nat ).
% 0.32/1.53
% 0.32/1.53 thf(tcon_Nat_Onat___Rings_Ocomm__semiring__1,axiom,
% 0.32/1.53 comm_semiring_1 @ nat ).
% 0.32/1.53
% 0.32/1.53 thf(tcon_Nat_Onat___Rings_Ocomm__semiring__0,axiom,
% 0.32/1.53 comm_semiring_0 @ nat ).
% 0.32/1.53
% 0.32/1.53 thf(tcon_Nat_Onat___Rings_Ozero__less__one,axiom,
% 0.32/1.53 zero_less_one @ nat ).
% 0.32/1.53
% 0.32/1.53 thf(tcon_Nat_Onat___Orderings_Owellorder,axiom,
% 0.32/1.53 wellorder @ nat ).
% 0.32/1.53
% 0.32/1.53 thf(tcon_Nat_Onat___Nat_Osemiring__char__0,axiom,
% 0.32/1.53 semiring_char_0 @ nat ).
% 0.32/1.53
% 0.32/1.53 thf(tcon_Nat_Onat___Rings_Ozero__neq__one,axiom,
% 0.32/1.53 zero_neq_one @ nat ).
% 0.32/1.53
% 0.32/1.53 thf(tcon_Nat_Onat___Orderings_Oorder_1,axiom,
% 0.32/1.53 order @ nat ).
% 0.32/1.53
% 0.32/1.53 thf(tcon_Nat_Onat___Groups_Ozero,axiom,
% 0.32/1.53 zero @ nat ).
% 0.32/1.53
% 0.32/1.53 thf(tcon_Nat_Onat___Groups_Oone,axiom,
% 0.32/1.53 one @ nat ).
% 0.32/1.53
% 0.32/1.53 thf(tcon_Set_Oset___Finite__Set_Ofinite_2,axiom,
% 0.32/1.53 ! [A7: $tType] :
% 0.32/1.53 ( ( finite_finite @ A7 )
% 0.32/1.53 => ( finite_finite @ ( set @ A7 ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(tcon_Set_Oset___Orderings_Oorder_3,axiom,
% 0.32/1.53 ! [A7: $tType] : ( order @ ( set @ A7 ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(tcon_HOL_Obool___Finite__Set_Ofinite_4,axiom,
% 0.32/1.53 finite_finite @ $o ).
% 0.32/1.53
% 0.32/1.53 thf(tcon_HOL_Obool___Orderings_Oorder_5,axiom,
% 0.32/1.53 order @ $o ).
% 0.32/1.53
% 0.32/1.53 thf(tcon_Real_Oreal___Rings_Osemiring__1__no__zero__divisors_6,axiom,
% 0.32/1.53 semiri134348788visors @ real ).
% 0.32/1.53
% 0.32/1.53 thf(tcon_Real_Oreal___Rings_Olinordered__nonzero__semiring_7,axiom,
% 0.32/1.53 linord1659791738miring @ real ).
% 0.32/1.53
% 0.32/1.53 thf(tcon_Real_Oreal___Rings_Osemiring__no__zero__divisors_8,axiom,
% 0.32/1.53 semiri1193490041visors @ real ).
% 0.32/1.53
% 0.32/1.53 thf(tcon_Real_Oreal___Rings_Oring__no__zero__divisors,axiom,
% 0.32/1.53 ring_n68954251visors @ real ).
% 0.32/1.53
% 0.32/1.53 thf(tcon_Real_Oreal___Groups_Ocomm__monoid__mult_9,axiom,
% 0.32/1.53 comm_monoid_mult @ real ).
% 0.32/1.53
% 0.32/1.53 thf(tcon_Real_Oreal___Fields_Olinordered__field,axiom,
% 0.32/1.53 linordered_field @ real ).
% 0.32/1.53
% 0.32/1.53 thf(tcon_Real_Oreal___Rings_Olinordered__idom,axiom,
% 0.32/1.53 linordered_idom @ real ).
% 0.32/1.53
% 0.32/1.53 thf(tcon_Real_Oreal___Rings_Ocomm__semiring__1_10,axiom,
% 0.32/1.53 comm_semiring_1 @ real ).
% 0.32/1.53
% 0.32/1.53 thf(tcon_Real_Oreal___Rings_Ocomm__semiring__0_11,axiom,
% 0.32/1.53 comm_semiring_0 @ real ).
% 0.32/1.53
% 0.32/1.53 thf(tcon_Real_Oreal___Fields_Odivision__ring,axiom,
% 0.32/1.53 division_ring @ real ).
% 0.32/1.53
% 0.32/1.53 thf(tcon_Real_Oreal___Rings_Ozero__less__one_12,axiom,
% 0.32/1.53 zero_less_one @ real ).
% 0.32/1.53
% 0.32/1.53 thf(tcon_Real_Oreal___Nat_Osemiring__char__0_13,axiom,
% 0.32/1.53 semiring_char_0 @ real ).
% 0.32/1.53
% 0.32/1.53 thf(tcon_Real_Oreal___Fields_Ofield__char__0,axiom,
% 0.32/1.53 field_char_0 @ real ).
% 0.32/1.53
% 0.32/1.53 thf(tcon_Real_Oreal___Rings_Ozero__neq__one_14,axiom,
% 0.32/1.53 zero_neq_one @ real ).
% 0.32/1.53
% 0.32/1.53 thf(tcon_Real_Oreal___Rings_Oidom__divide,axiom,
% 0.32/1.53 idom_divide @ real ).
% 0.32/1.53
% 0.32/1.53 thf(tcon_Real_Oreal___Rings_Ocomm__ring__1,axiom,
% 0.32/1.53 comm_ring_1 @ real ).
% 0.32/1.53
% 0.32/1.53 thf(tcon_Real_Oreal___Orderings_Oorder_15,axiom,
% 0.32/1.53 order @ real ).
% 0.32/1.53
% 0.32/1.53 thf(tcon_Real_Oreal___Nat_Oring__char__0,axiom,
% 0.32/1.53 ring_char_0 @ real ).
% 0.32/1.53
% 0.32/1.53 thf(tcon_Real_Oreal___Rings_Oring__1,axiom,
% 0.32/1.53 ring_1 @ real ).
% 0.32/1.53
% 0.32/1.53 thf(tcon_Real_Oreal___Fields_Ofield,axiom,
% 0.32/1.53 field @ real ).
% 0.32/1.53
% 0.32/1.53 thf(tcon_Real_Oreal___Groups_Ozero_16,axiom,
% 0.32/1.53 zero @ real ).
% 0.32/1.53
% 0.32/1.53 thf(tcon_Real_Oreal___Rings_Oidom,axiom,
% 0.32/1.53 idom @ real ).
% 0.32/1.53
% 0.32/1.53 thf(tcon_Real_Oreal___Groups_Oone_17,axiom,
% 0.32/1.53 one @ real ).
% 0.32/1.53
% 0.32/1.53 thf(tcon_Polynomial_Opoly___Rings_Osemiring__1__no__zero__divisors_18,axiom,
% 0.32/1.53 ! [A7: $tType] :
% 0.32/1.53 ( ( ( comm_semiring_1 @ A7 )
% 0.32/1.53 & ( semiri134348788visors @ A7 ) )
% 0.32/1.53 => ( semiri134348788visors @ ( poly @ A7 ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(tcon_Polynomial_Opoly___Rings_Olinordered__nonzero__semiring_19,axiom,
% 0.32/1.53 ! [A7: $tType] :
% 0.32/1.53 ( ( linordered_idom @ A7 )
% 0.32/1.53 => ( linord1659791738miring @ ( poly @ A7 ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(tcon_Polynomial_Opoly___Rings_Osemiring__no__zero__divisors_20,axiom,
% 0.32/1.53 ! [A7: $tType] :
% 0.32/1.53 ( ( ( comm_semiring_0 @ A7 )
% 0.32/1.53 & ( semiri1193490041visors @ A7 ) )
% 0.32/1.53 => ( semiri1193490041visors @ ( poly @ A7 ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(tcon_Polynomial_Opoly___Rings_Oring__no__zero__divisors_21,axiom,
% 0.32/1.53 ! [A7: $tType] :
% 0.32/1.53 ( ( idom @ A7 )
% 0.32/1.53 => ( ring_n68954251visors @ ( poly @ A7 ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(tcon_Polynomial_Opoly___Rings_Oalgebraic__semidom_22,axiom,
% 0.32/1.53 ! [A7: $tType] :
% 0.32/1.53 ( ( idom_divide @ A7 )
% 0.32/1.53 => ( algebraic_semidom @ ( poly @ A7 ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(tcon_Polynomial_Opoly___Groups_Ocomm__monoid__mult_23,axiom,
% 0.32/1.53 ! [A7: $tType] :
% 0.32/1.53 ( ( comm_semiring_1 @ A7 )
% 0.32/1.53 => ( comm_monoid_mult @ ( poly @ A7 ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(tcon_Polynomial_Opoly___Rings_Olinordered__idom_24,axiom,
% 0.32/1.53 ! [A7: $tType] :
% 0.32/1.53 ( ( linordered_idom @ A7 )
% 0.32/1.53 => ( linordered_idom @ ( poly @ A7 ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(tcon_Polynomial_Opoly___Rings_Ocomm__semiring__1_25,axiom,
% 0.32/1.53 ! [A7: $tType] :
% 0.32/1.53 ( ( comm_semiring_1 @ A7 )
% 0.32/1.53 => ( comm_semiring_1 @ ( poly @ A7 ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(tcon_Polynomial_Opoly___Rings_Ocomm__semiring__0_26,axiom,
% 0.32/1.53 ! [A7: $tType] :
% 0.32/1.53 ( ( comm_semiring_0 @ A7 )
% 0.32/1.53 => ( comm_semiring_0 @ ( poly @ A7 ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(tcon_Polynomial_Opoly___Rings_Ozero__less__one_27,axiom,
% 0.32/1.53 ! [A7: $tType] :
% 0.32/1.53 ( ( linordered_idom @ A7 )
% 0.32/1.53 => ( zero_less_one @ ( poly @ A7 ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(tcon_Polynomial_Opoly___Nat_Osemiring__char__0_28,axiom,
% 0.32/1.53 ! [A7: $tType] :
% 0.32/1.53 ( ( ( ring_char_0 @ A7 )
% 0.32/1.53 & ( comm_ring_1 @ A7 ) )
% 0.32/1.53 => ( semiring_char_0 @ ( poly @ A7 ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(tcon_Polynomial_Opoly___Rings_Ozero__neq__one_29,axiom,
% 0.32/1.53 ! [A7: $tType] :
% 0.32/1.53 ( ( comm_semiring_1 @ A7 )
% 0.32/1.53 => ( zero_neq_one @ ( poly @ A7 ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(tcon_Polynomial_Opoly___Rings_Oidom__divide_30,axiom,
% 0.32/1.53 ! [A7: $tType] :
% 0.32/1.53 ( ( idom_divide @ A7 )
% 0.32/1.53 => ( idom_divide @ ( poly @ A7 ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(tcon_Polynomial_Opoly___Rings_Ocomm__ring__1_31,axiom,
% 0.32/1.53 ! [A7: $tType] :
% 0.32/1.53 ( ( comm_ring_1 @ A7 )
% 0.32/1.53 => ( comm_ring_1 @ ( poly @ A7 ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(tcon_Polynomial_Opoly___Orderings_Oorder_32,axiom,
% 0.32/1.53 ! [A7: $tType] :
% 0.32/1.53 ( ( linordered_idom @ A7 )
% 0.32/1.53 => ( order @ ( poly @ A7 ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(tcon_Polynomial_Opoly___Nat_Oring__char__0_33,axiom,
% 0.32/1.53 ! [A7: $tType] :
% 0.32/1.53 ( ( ( ring_char_0 @ A7 )
% 0.32/1.53 & ( comm_ring_1 @ A7 ) )
% 0.32/1.53 => ( ring_char_0 @ ( poly @ A7 ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(tcon_Polynomial_Opoly___Rings_Oring__1_34,axiom,
% 0.32/1.53 ! [A7: $tType] :
% 0.32/1.53 ( ( comm_ring_1 @ A7 )
% 0.32/1.53 => ( ring_1 @ ( poly @ A7 ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(tcon_Polynomial_Opoly___Groups_Ozero_35,axiom,
% 0.32/1.53 ! [A7: $tType] :
% 0.32/1.53 ( ( zero @ A7 )
% 0.32/1.53 => ( zero @ ( poly @ A7 ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(tcon_Polynomial_Opoly___Rings_Oidom_36,axiom,
% 0.32/1.53 ! [A7: $tType] :
% 0.32/1.53 ( ( idom @ A7 )
% 0.32/1.53 => ( idom @ ( poly @ A7 ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(tcon_Polynomial_Opoly___Groups_Oone_37,axiom,
% 0.32/1.53 ! [A7: $tType] :
% 0.32/1.53 ( ( comm_semiring_1 @ A7 )
% 0.32/1.53 => ( one @ ( poly @ A7 ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(conj_0,conjecture,
% 0.32/1.53 ( finite_finite2 @ real
% 0.32/1.53 @ ( collect @ real
% 0.32/1.53 @ ^ [X: real] :
% 0.32/1.53 ( ( poly2 @ real @ ( pderiv @ real @ p ) @ X )
% 0.32/1.53 = ( zero_zero @ real ) ) ) ) ).
% 0.32/1.53 ------------------------
% 0.32/1.55 % (8897)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_8563 for (2999ds/27Mi)
% 0.32/1.55 % (8896)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_8563 for (2999ds/4Mi)
% 0.32/1.55 % (8896)Instruction limit reached!
% 0.32/1.55 % (8896)------------------------------
% 0.32/1.55 % (8896)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 0.32/1.55 % (8896)Termination reason: Unknown
% 0.32/1.55 % (8896)Termination phase: shuffling
% 0.32/1.55
% 0.32/1.55 % (8896)Memory used [KB]: 1407
% 0.32/1.55 % (8896)Time elapsed: 0.003 s
% 0.32/1.55 % (8896)Instructions burned: 6 (million)
% 0.32/1.55 % (8896)------------------------------
% 0.32/1.55 % (8896)------------------------------
% 0.32/1.56 % (8899)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_8563 for (2999ds/2Mi)
% 0.32/1.56 % (8895)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_8563 for (2999ds/183Mi)
% 0.32/1.56 % (8898)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_8563 for (2999ds/2Mi)
% 0.32/1.56 % (8900)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_8563 for (2999ds/275Mi)
% 0.32/1.56 % (8901)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_8563 for (2999ds/18Mi)
% 0.32/1.56 % (8897)Instruction limit reached!
% 0.32/1.56 % (8897)------------------------------
% 0.32/1.56 % (8897)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 0.32/1.56 % (8897)Termination reason: Unknown
% 0.32/1.56 % (8897)Termination phase: Property scanning
% 0.32/1.56
% 0.32/1.56 % (8897)Memory used [KB]: 1663
% 0.32/1.56 % (8897)Time elapsed: 0.010 s
% 0.32/1.56 % (8897)Instructions burned: 27 (million)
% 0.32/1.56 % (8897)------------------------------
% 0.32/1.56 % (8897)------------------------------
% 0.32/1.56 % (8899)Instruction limit reached!
% 0.32/1.56 % (8899)------------------------------
% 0.32/1.56 % (8899)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 0.32/1.56 % (8899)Termination reason: Unknown
% 0.32/1.56 % (8899)Termination phase: shuffling
% 0.32/1.56
% 0.32/1.56 % (8899)Memory used [KB]: 1279
% 0.32/1.56 % (8899)Time elapsed: 0.003 s
% 0.32/1.56 % (8899)Instructions burned: 2 (million)
% 0.32/1.56 % (8899)------------------------------
% 0.32/1.56 % (8899)------------------------------
% 0.32/1.56 % (8898)Instruction limit reached!
% 0.32/1.56 % (8898)------------------------------
% 0.32/1.56 % (8898)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 0.32/1.56 % (8898)Termination reason: Unknown
% 0.32/1.56 % (8898)Termination phase: shuffling
% 0.32/1.56
% 0.32/1.56 % (8898)Memory used [KB]: 1279
% 0.32/1.56 % (8898)Time elapsed: 0.004 s
% 0.32/1.56 % (8898)Instructions burned: 3 (million)
% 0.32/1.56 % (8898)------------------------------
% 0.32/1.56 % (8898)------------------------------
% 0.32/1.56 % (8902)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_8563 for (2999ds/3Mi)
% 0.32/1.57 % (8902)Instruction limit reached!
% 0.32/1.57 % (8902)------------------------------
% 0.32/1.57 % (8902)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 0.32/1.57 % (8902)Termination reason: Unknown
% 0.32/1.57 % (8902)Termination phase: shuffling
% 0.32/1.57
% 0.32/1.57 % (8902)Memory used [KB]: 1407
% 0.32/1.57 % (8902)Time elapsed: 0.003 s
% 0.32/1.57 % (8902)Instructions burned: 3 (million)
% 0.32/1.57 % (8902)------------------------------
% 0.32/1.57 % (8902)------------------------------
% 0.32/1.57 % (8903)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_8563 for (2999ds/37Mi)
% 0.32/1.57 % (8901)Instruction limit reached!
% 0.32/1.57 % (8901)------------------------------
% 0.32/1.57 % (8901)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 0.32/1.57 % (8901)Termination reason: Unknown
% 0.32/1.57 % (8901)Termination phase: Property scanning
% 0.32/1.57
% 0.32/1.57 % (8901)Memory used [KB]: 1663
% 0.32/1.57 % (8901)Time elapsed: 0.014 s
% 0.32/1.57 % (8901)Instructions burned: 18 (million)
% 0.32/1.57 % (8901)------------------------------
% 0.32/1.57 % (8901)------------------------------
% 0.32/1.58 % (8906)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_8563 for (2999ds/1041Mi)
% 0.32/1.58 % (8904)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_8563 for (2999ds/15Mi)
% 0.32/1.58 % (8905)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_8563 for (2999ds/3Mi)
% 0.32/1.58 % (8903)Instruction limit reached!
% 0.32/1.58 % (8903)------------------------------
% 0.32/1.58 % (8903)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 0.32/1.58 % (8903)Termination reason: Unknown
% 0.32/1.58 % (8903)Termination phase: Property scanning
% 0.32/1.58
% 0.32/1.58 % (8903)Memory used [KB]: 2046
% 0.32/1.58 % (8903)Time elapsed: 0.012 s
% 0.32/1.58 % (8903)Instructions burned: 39 (million)
% 0.32/1.58 % (8903)------------------------------
% 0.32/1.58 % (8903)------------------------------
% 0.32/1.58 % (8905)Instruction limit reached!
% 0.32/1.58 % (8905)------------------------------
% 0.32/1.58 % (8905)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 0.32/1.58 % (8905)Termination reason: Unknown
% 0.32/1.58 % (8905)Termination phase: shuffling
% 0.32/1.58
% 0.32/1.58 % (8905)Memory used [KB]: 1407
% 0.32/1.58 % (8905)Time elapsed: 0.004 s
% 0.32/1.58 % (8905)Instructions burned: 3 (million)
% 0.32/1.58 % (8905)------------------------------
% 0.32/1.58 % (8905)------------------------------
% 0.32/1.58 % (8904)Instruction limit reached!
% 0.32/1.58 % (8904)------------------------------
% 0.32/1.58 % (8904)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 0.32/1.58 % (8904)Termination reason: Unknown
% 0.32/1.58 % (8904)Termination phase: shuffling
% 0.32/1.58
% 0.32/1.58 % (8904)Memory used [KB]: 1663
% 0.32/1.58 % (8904)Time elapsed: 0.008 s
% 0.32/1.58 % (8904)Instructions burned: 15 (million)
% 0.32/1.58 % (8904)------------------------------
% 0.32/1.58 % (8904)------------------------------
% 0.32/1.59 % (8907)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_8563 for (2999ds/7Mi)
% 0.32/1.59 % (8911)lrs+2_1:1_apa=on:au=on:bd=preordered:cnfonf=off:cs=on:ixr=off:sos=on:i=3:si=on:rtra=on_0 on DTF2THF_8563 for (2999ds/3Mi)
% 0.32/1.59 % (8911)Instruction limit reached!
% 0.32/1.59 % (8911)------------------------------
% 0.32/1.59 % (8911)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 0.32/1.59 % (8911)Termination reason: Unknown
% 0.32/1.59 % (8911)Termination phase: shuffling
% 0.32/1.59
% 0.32/1.59 % (8911)Memory used [KB]: 1279
% 0.32/1.59 % (8911)Time elapsed: 0.002 s
% 0.32/1.59 % (8911)Instructions burned: 3 (million)
% 0.32/1.59 % (8911)------------------------------
% 0.32/1.59 % (8911)------------------------------
% 0.32/1.59 % (8907)Instruction limit reached!
% 0.32/1.59 % (8907)------------------------------
% 0.32/1.59 % (8907)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 0.32/1.59 % (8907)Termination reason: Unknown
% 0.32/1.59 % (8907)Termination phase: shuffling
% 0.32/1.59
% 0.32/1.59 % (8907)Memory used [KB]: 1535
% 0.32/1.59 % (8907)Time elapsed: 0.007 s
% 0.32/1.59 % (8907)Instructions burned: 8 (million)
% 0.32/1.59 % (8907)------------------------------
% 0.32/1.59 % (8907)------------------------------
% 2.68/1.60 % (8909)lrs+21_1:1_au=on:cnfonf=off:fd=preordered:fe=off:fsr=off:hud=11:inj=on:kws=precedence:s2pl=no:sp=weighted_frequency:tgt=full:i=3:si=on:rtra=on_0 on DTF2THF_8563 for (2999ds/3Mi)
% 2.68/1.60 % (8908)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_8563 for (2999ds/16Mi)
% 2.68/1.60 % (8909)Instruction limit reached!
% 2.68/1.60 % (8909)------------------------------
% 2.68/1.60 % (8909)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.68/1.60 % (8909)Termination reason: Unknown
% 2.68/1.60 % (8909)Termination phase: shuffling
% 2.68/1.60
% 2.68/1.60 % (8909)Memory used [KB]: 1407
% 2.68/1.60 % (8909)Time elapsed: 0.004 s
% 2.68/1.60 % (8909)Instructions burned: 4 (million)
% 2.68/1.60 % (8909)------------------------------
% 2.68/1.60 % (8909)------------------------------
% 2.68/1.60 % (8912)lrs+10_1:1_cnfonf=off:cs=on:hud=3:prag=on:sup=off:i=7:si=on:rtra=on_0 on DTF2THF_8563 for (2999ds/7Mi)
% 2.68/1.61 % (8912)Instruction limit reached!
% 2.68/1.61 % (8912)------------------------------
% 2.68/1.61 % (8912)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.68/1.61 % (8912)Termination reason: Unknown
% 2.68/1.61 % (8912)Termination phase: shuffling
% 2.68/1.61
% 2.68/1.61 % (8912)Memory used [KB]: 1535
% 2.68/1.61 % (8912)Time elapsed: 0.004 s
% 2.68/1.61 % (8912)Instructions burned: 8 (million)
% 2.68/1.61 % (8912)------------------------------
% 2.68/1.61 % (8912)------------------------------
% 2.68/1.61 % (8908)Instruction limit reached!
% 2.68/1.61 % (8908)------------------------------
% 2.68/1.61 % (8908)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.68/1.61 % (8908)Termination reason: Unknown
% 2.68/1.61 % (8908)Termination phase: shuffling
% 2.68/1.61
% 2.68/1.61 % (8908)Memory used [KB]: 1663
% 2.68/1.61 % (8908)Time elapsed: 0.012 s
% 2.68/1.61 % (8908)Instructions burned: 17 (million)
% 2.68/1.61 % (8908)------------------------------
% 2.68/1.61 % (8908)------------------------------
% 2.68/1.61 % (8913)dis+1002_1:1_add=large:cnfonf=lazy_pi_sigma_gen:fe=off:prag=on:i=3:si=on:rtra=on_0 on DTF2THF_8563 for (2999ds/3Mi)
% 2.68/1.61 % (8913)Instruction limit reached!
% 2.68/1.61 % (8913)------------------------------
% 2.68/1.61 % (8913)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.68/1.61 % (8913)Termination reason: Unknown
% 2.68/1.61 % (8913)Termination phase: shuffling
% 2.68/1.61
% 2.68/1.61 % (8913)Memory used [KB]: 1407
% 2.68/1.61 % (8913)Time elapsed: 0.004 s
% 2.68/1.61 % (8913)Instructions burned: 4 (million)
% 2.68/1.61 % (8913)------------------------------
% 2.68/1.61 % (8913)------------------------------
% 2.68/1.62 % (8915)lrs+1002_1:1_anc=all_dependent:au=on:cbe=off:fde=unused:ntd=on:i=18:si=on:rtra=on_0 on DTF2THF_8563 for (2999ds/18Mi)
% 2.68/1.62 % (8914)dis+1004_1:1_cha=on:cs=on:fe=off:hud=1:i=4:si=on:rtra=on_0 on DTF2THF_8563 for (2999ds/4Mi)
% 2.68/1.62 % (8914)Instruction limit reached!
% 2.68/1.62 % (8914)------------------------------
% 2.68/1.62 % (8914)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.68/1.62 % (8914)Termination reason: Unknown
% 2.68/1.62 % (8914)Termination phase: shuffling
% 2.68/1.62
% 2.68/1.62 % (8914)Memory used [KB]: 1407
% 2.68/1.62 % (8914)Time elapsed: 0.003 s
% 2.68/1.62 % (8914)Instructions burned: 6 (million)
% 2.68/1.62 % (8914)------------------------------
% 2.68/1.62 % (8914)------------------------------
% 2.68/1.62 % (8915)Instruction limit reached!
% 2.68/1.62 % (8915)------------------------------
% 2.68/1.62 % (8915)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.68/1.62 % (8915)Termination reason: Unknown
% 2.68/1.62 % (8915)Termination phase: shuffling
% 2.68/1.62
% 2.68/1.62 % (8915)Memory used [KB]: 1663
% 2.68/1.62 % (8915)Time elapsed: 0.007 s
% 2.68/1.62 % (8915)Instructions burned: 18 (million)
% 2.68/1.62 % (8915)------------------------------
% 2.68/1.62 % (8915)------------------------------
% 2.68/1.62 % (8916)lrs+10_1:1_e2e=on:sd=1:sgt=8:ss=axioms:i=710:si=on:rtra=on_0 on DTF2THF_8563 for (2999ds/710Mi)
% 2.68/1.63 % (8918)dis+1002_5:1_au=on:bd=off:e2e=on:fde=none:fs=off:fsr=off:sos=on:i=902:si=on:rtra=on_0 on DTF2THF_8563 for (2999ds/902Mi)
% 2.68/1.63 % (8919)dis+21_1:8_apa=on:cnfonf=off:fd=off:fsr=off:hud=0:ins=1:kws=inv_frequency:nwc=10.0:ss=axioms:st=5.0:i=21:si=on:rtra=on_0 on DTF2THF_8563 for (2999ds/21Mi)
% 2.68/1.63 % (8917)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_8563 for (2999ds/6Mi)
% 2.68/1.64 % (8917)Instruction limit reached!
% 2.68/1.64 % (8917)------------------------------
% 2.68/1.64 % (8917)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.68/1.64 % (8917)Termination reason: Unknown
% 2.68/1.64 % (8917)Termination phase: shuffling
% 2.68/1.64
% 2.68/1.64 % (8917)Memory used [KB]: 1407
% 2.68/1.64 % (8917)Time elapsed: 0.006 s
% 2.68/1.64 % (8917)Instructions burned: 7 (million)
% 2.68/1.64 % (8917)------------------------------
% 2.68/1.64 % (8917)------------------------------
% 2.68/1.64 % (8919)Instruction limit reached!
% 2.68/1.64 % (8919)------------------------------
% 2.68/1.64 % (8919)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.68/1.64 % (8919)Termination reason: Unknown
% 2.68/1.64 % (8919)Termination phase: Property scanning
% 2.68/1.64
% 2.68/1.64 % (8919)Memory used [KB]: 1791
% 2.68/1.64 % (8919)Time elapsed: 0.008 s
% 2.68/1.64 % (8919)Instructions burned: 22 (million)
% 2.68/1.64 % (8919)------------------------------
% 2.68/1.64 % (8919)------------------------------
% 2.68/1.65 % (8921)lrs+2_1:1_cnfonf=lazy_not_gen_be_off:cs=on:fe=off:hud=10:inj=on:ins=3:plsq=on:plsqc=1:sd=10:ss=axioms:tnu=1:i=6:si=on:rtra=on_0 on DTF2THF_8563 for (2999ds/6Mi)
% 2.68/1.65 % (8921)Instruction limit reached!
% 2.68/1.65 % (8921)------------------------------
% 2.68/1.65 % (8921)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.68/1.65 % (8921)Termination reason: Unknown
% 2.68/1.65 % (8921)Termination phase: shuffling
% 2.68/1.65
% 2.68/1.65 % (8921)Memory used [KB]: 1407
% 2.68/1.65 % (8921)Time elapsed: 0.003 s
% 2.68/1.65 % (8921)Instructions burned: 7 (million)
% 2.68/1.65 % (8921)------------------------------
% 2.68/1.65 % (8921)------------------------------
% 2.68/1.65 % (8920)dis+10_1:1_cnfonf=lazy_gen:fe=off:i=5:si=on:rtra=on_0 on DTF2THF_8563 for (2999ds/5Mi)
% 2.68/1.66 % (8920)Instruction limit reached!
% 2.68/1.66 % (8920)------------------------------
% 2.68/1.66 % (8920)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.68/1.66 % (8920)Termination reason: Unknown
% 2.68/1.66 % (8920)Termination phase: shuffling
% 2.68/1.66
% 2.68/1.66 % (8920)Memory used [KB]: 1407
% 2.68/1.66 % (8920)Time elapsed: 0.005 s
% 2.68/1.66 % (8920)Instructions burned: 6 (million)
% 2.68/1.66 % (8920)------------------------------
% 2.68/1.66 % (8920)------------------------------
% 2.68/1.66 % (8922)lrs+1002_1:128_au=on:c=on:fsr=off:piset=equals:i=377:si=on:rtra=on_0 on DTF2THF_8563 for (2998ds/377Mi)
% 2.68/1.66 % (8918)First to succeed.
% 2.68/1.66 % (8918)Refutation found. Thanks to Tanya!
% 2.68/1.66 % SZS status Theorem for DTF2THF_8563
% 2.68/1.66 % SZS output start Proof for DTF2THF_8563
% 2.68/1.66 thf(type_def_6, type, set: ($tType) > $tType).
% 2.68/1.66 thf(type_def_7, type, nat: $tType).
% 2.68/1.66 thf(type_def_8, type, poly: ($tType) > $tType).
% 2.68/1.66 thf(type_def_9, type, real: $tType).
% 2.68/1.66 thf(func_def_0, type, poly: ($tType) > $tType).
% 2.68/1.66 thf(func_def_1, type, real: $tType).
% 2.68/1.66 thf(func_def_2, type, set: ($tType) > $tType).
% 2.68/1.66 thf(func_def_3, type, nat: $tType).
% 2.68/1.66 thf(func_def_4, type, type: !>[X0: $tType]:($o)).
% 2.68/1.66 thf(func_def_5, type, one: !>[X0: $tType]:($o)).
% 2.68/1.66 thf(func_def_6, type, idom: !>[X0: $tType]:($o)).
% 2.68/1.66 thf(func_def_7, type, zero: !>[X0: $tType]:($o)).
% 2.68/1.66 thf(func_def_8, type, field: !>[X0: $tType]:($o)).
% 2.68/1.66 thf(func_def_9, type, ring_1: !>[X0: $tType]:($o)).
% 2.68/1.66 thf(func_def_10, type, ring_char_0: !>[X0: $tType]:($o)).
% 2.68/1.66 thf(func_def_11, type, order: !>[X0: $tType]:($o)).
% 2.68/1.66 thf(func_def_12, type, finite_finite: !>[X0: $tType]:($o)).
% 2.68/1.66 thf(func_def_13, type, comm_ring_1: !>[X0: $tType]:($o)).
% 2.68/1.66 thf(func_def_14, type, idom_divide: !>[X0: $tType]:($o)).
% 2.68/1.66 thf(func_def_15, type, zero_neq_one: !>[X0: $tType]:($o)).
% 2.68/1.66 thf(func_def_16, type, field_char_0: !>[X0: $tType]:($o)).
% 2.68/1.66 thf(func_def_17, type, semiring_char_0: !>[X0: $tType]:($o)).
% 2.68/1.66 thf(func_def_18, type, wellorder: !>[X0: $tType]:($o)).
% 2.68/1.66 thf(func_def_19, type, zero_less_one: !>[X0: $tType]:($o)).
% 2.68/1.66 thf(func_def_20, type, division_ring: !>[X0: $tType]:($o)).
% 2.68/1.66 thf(func_def_21, type, comm_semiring_0: !>[X0: $tType]:($o)).
% 2.68/1.66 thf(func_def_22, type, comm_semiring_1: !>[X0: $tType]:($o)).
% 2.68/1.66 thf(func_def_23, type, linordered_idom: !>[X0: $tType]:($o)).
% 2.68/1.66 thf(func_def_24, type, linordered_field: !>[X0: $tType]:($o)).
% 2.68/1.66 thf(func_def_25, type, comm_monoid_mult: !>[X0: $tType]:($o)).
% 2.68/1.66 thf(func_def_26, type, algebraic_semidom: !>[X0: $tType]:($o)).
% 2.68/1.66 thf(func_def_27, type, ring_n68954251visors: !>[X0: $tType]:($o)).
% 2.68/1.66 thf(func_def_28, type, semiri1193490041visors: !>[X0: $tType]:($o)).
% 2.68/1.66 thf(func_def_29, type, linord1659791738miring: !>[X0: $tType]:($o)).
% 2.68/1.66 thf(func_def_30, type, semiri134348788visors: !>[X0: $tType]:($o)).
% 2.68/1.66 thf(func_def_31, type, canoni770627133id_add: !>[X0: $tType]:($o)).
% 2.68/1.66 thf(func_def_32, type, inverse_inverse: !>[X0: $tType]:(X0 > X0)).
% 2.68/1.66 thf(func_def_33, type, finite_finite2: !>[X0: $tType]:(set @ X0 > $o)).
% 2.68/1.66 thf(func_def_34, type, one_one: !>[X0: $tType]:(X0)).
% 2.68/1.66 thf(func_def_35, type, zero_zero: !>[X0: $tType]:(X0)).
% 2.68/1.66 thf(func_def_36, type, ring_1_Ints: !>[X0: $tType]:(set @ X0)).
% 2.68/1.66 thf(func_def_37, type, suc: nat > nat).
% 2.68/1.66 thf(func_def_38, type, ord_less: !>[X0: $tType]:(X0 > X0 > $o)).
% 2.68/1.66 thf(func_def_39, type, ord_less_eq: !>[X0: $tType]:(X0 > X0 > $o)).
% 2.68/1.66 thf(func_def_40, type, algebraic: !>[X0: $tType]:(X0 > $o)).
% 2.68/1.66 thf(func_def_41, type, algebraic_int: !>[X0: $tType]:(X0 > $o)).
% 2.68/1.66 thf(func_def_42, type, cr_poly: !>[X0: $tType]:((nat > X0) > poly @ X0 > $o)).
% 2.68/1.66 thf(func_def_43, type, degree: !>[X0: $tType]:(poly @ X0 > nat)).
% 2.68/1.66 thf(func_def_44, type, divide_poly_main: !>[X0: $tType]:(X0 > poly @ X0 > poly @ X0 > poly @ X0 > nat > nat > poly @ X0)).
% 2.68/1.66 thf(func_def_45, type, is_zero: !>[X0: $tType]:(poly @ X0 > $o)).
% 2.68/1.66 thf(func_def_46, type, order2: !>[X0: $tType]:(X0 > poly @ X0 > nat)).
% 2.68/1.66 thf(func_def_47, type, pcr_poly: !>[X0: $tType, X1: $tType]:((X0 > X1 > $o) > (nat > X0) > poly @ X1 > $o)).
% 2.68/1.66 thf(func_def_48, type, pderiv: !>[X0: $tType]:(poly @ X0 > poly @ X0)).
% 2.68/1.66 thf(func_def_49, type, poly2: !>[X0: $tType]:(poly @ X0 > X0 > X0)).
% 2.68/1.66 thf(func_def_50, type, coeff: !>[X0: $tType]:(poly @ X0 > nat > X0)).
% 2.68/1.66 thf(func_def_51, type, poly_cutoff: !>[X0: $tType]:(nat > poly @ X0 > poly @ X0)).
% 2.68/1.66 thf(func_def_52, type, poly_shift: !>[X0: $tType]:(nat > poly @ X0 > poly @ X0)).
% 2.68/1.66 thf(func_def_53, type, pseudo_mod: !>[X0: $tType]:(poly @ X0 > poly @ X0 > poly @ X0)).
% 2.68/1.66 thf(func_def_54, type, reflect_poly: !>[X0: $tType]:(poly @ X0 > poly @ X0)).
% 2.68/1.66 thf(func_def_55, type, rsquarefree: !>[X0: $tType]:(poly @ X0 > $o)).
% 2.68/1.66 thf(func_def_56, type, synthetic_div: !>[X0: $tType]:(poly @ X0 > X0 > poly @ X0)).
% 2.68/1.66 thf(func_def_57, type, field_char_0_Rats: !>[X0: $tType]:(set @ X0)).
% 2.68/1.66 thf(func_def_58, type, dvd_dvd: !>[X0: $tType]:(X0 > X0 > $o)).
% 2.68/1.66 thf(func_def_59, type, collect: !>[X0: $tType]:((X0 > $o) > set @ X0)).
% 2.68/1.66 thf(func_def_60, type, member: !>[X0: $tType]:(X0 > set @ X0 > $o)).
% 2.68/1.66 thf(func_def_61, type, n: nat).
% 2.68/1.66 thf(func_def_62, type, p: poly @ real).
% 2.68/1.66 thf(func_def_63, type, x: real).
% 2.68/1.66 thf(func_def_78, type, sK0: poly @ real > real > real > real).
% 2.68/1.66 thf(func_def_79, type, sK1: !>[X0: $tType, X1: $tType]:((X1 > $o) > (X1 > X0) > X1)).
% 2.68/1.66 thf(func_def_80, type, sK2: real > real).
% 2.68/1.66 thf(func_def_81, type, sK3: (nat > nat > $o) > nat).
% 2.68/1.66 thf(func_def_82, type, sK4: (nat > nat > $o) > nat).
% 2.68/1.66 thf(func_def_83, type, sK5: (nat > nat > $o) > nat).
% 2.68/1.66 thf(func_def_84, type, sK6: (nat > nat > $o) > nat).
% 2.68/1.66 thf(func_def_85, type, sK7: (nat > $o) > nat).
% 2.68/1.66 thf(func_def_86, type, sK8: (nat > $o) > nat).
% 2.68/1.66 thf(func_def_87, type, sK9: !>[X0: $tType]:((X0 > $o) > (X0 > $o) > X0)).
% 2.68/1.66 thf(func_def_88, type, sK10: (nat > $o) > nat > nat).
% 2.68/1.66 thf(func_def_89, type, sK11: (nat > $o) > nat > nat).
% 2.68/1.66 thf(func_def_90, type, sK12: !>[X0: $tType]:(poly @ X0 > nat > nat)).
% 2.68/1.66 thf(func_def_91, type, sK13: real > real > real).
% 2.68/1.66 thf(func_def_92, type, sK14: !>[X0: $tType]:(poly @ X0 > nat)).
% 2.68/1.66 thf(func_def_93, type, sK15: (nat > $o) > nat).
% 2.68/1.66 thf(func_def_94, type, sK16: nat > nat > nat).
% 2.68/1.66 thf(func_def_95, type, sK17: (nat > $o) > nat).
% 2.68/1.66 thf(func_def_96, type, sK18: !>[X0: $tType, X1: $tType]:((X1 > X0 > $o) > set @ X1 > X1)).
% 2.68/1.66 thf(func_def_97, type, sK19: !>[X0: $tType, X1: $tType]:((X1 > X0 > $o) > set @ X1 > X1 > X0)).
% 2.68/1.66 thf(func_def_98, type, sK20: !>[X0: $tType]:((nat > X0 > X0 > $o) > (nat > X0 > $o) > nat > X0)).
% 2.68/1.66 thf(func_def_99, type, sK21: !>[X0: $tType]:((nat > X0 > X0 > $o) > (nat > X0 > $o) > X0)).
% 2.68/1.66 thf(func_def_100, type, sK22: !>[X0: $tType]:((nat > X0 > X0 > $o) > (nat > X0 > $o) > nat)).
% 2.68/1.66 thf(func_def_101, type, sK23: (nat > $o) > nat > nat).
% 2.68/1.66 thf(func_def_102, type, sK24: (nat > $o) > nat > nat).
% 2.68/1.66 thf(func_def_103, type, sK25: (nat > nat > $o) > nat).
% 2.68/1.66 thf(func_def_104, type, sK26: (nat > nat > $o) > nat).
% 2.68/1.66 thf(func_def_105, type, sK27: (nat > nat > $o) > nat).
% 2.68/1.66 thf(func_def_106, type, sK28: (nat > nat > $o) > nat).
% 2.68/1.66 thf(func_def_107, type, sK29: (nat > $o) > nat).
% 2.68/1.66 thf(func_def_108, type, sK30: !>[X0: $tType]:((nat > X0) > nat)).
% 2.68/1.66 thf(func_def_109, type, sK31: !>[X0: $tType]:((X0 > $o) > poly @ X0)).
% 2.68/1.66 thf(func_def_110, type, sK32: !>[X0: $tType]:((X0 > $o) > X0)).
% 2.68/1.66 thf(func_def_111, type, sK33: (nat > $o) > nat > nat).
% 2.68/1.66 thf(func_def_112, type, sK34: (nat > $o) > nat > nat).
% 2.68/1.66 thf(func_def_113, type, sK35: !>[X0: $tType]:((X0 > nat) > (X0 > $o) > X0)).
% 2.68/1.66 thf(func_def_114, type, sK36: (nat > $o) > nat).
% 2.68/1.66 thf(func_def_115, type, sK37: nat > nat).
% 2.68/1.66 thf(func_def_116, type, sK38: !>[X0: $tType]:((set @ X0 > $o) > set @ X0)).
% 2.68/1.66 thf(func_def_117, type, sK39: poly @ real).
% 2.68/1.66 thf(func_def_118, type, sK40: !>[X0: $tType]:((X0 > $o) > (X0 > nat) > X0)).
% 2.68/1.66 thf(func_def_119, type, sK41: !>[X0: $tType]:((X0 > $o) > (X0 > nat) > X0)).
% 2.68/1.66 thf(func_def_120, type, sK42: !>[X0: $tType]:((X0 > $o) > X0)).
% 2.68/1.66 thf(func_def_121, type, sK43: !>[X0: $tType, X1: $tType]:((X1 > X0) > (X1 > X0) > X1)).
% 2.68/1.66 thf(func_def_122, type, sK44: nat > nat).
% 2.68/1.66 thf(func_def_123, type, sK45: !>[X0: $tType]:(X0 > poly @ X0)).
% 2.68/1.66 thf(func_def_124, type, sK46: !>[X0: $tType]:(X0 > poly @ X0)).
% 2.68/1.66 thf(func_def_125, type, sK47: poly @ real > real > real > real).
% 2.68/1.66 thf(func_def_126, type, sK48: nat > nat).
% 2.68/1.66 thf(func_def_127, type, sK49: !>[X0: $tType]:(poly @ X0 > poly @ X0 > nat)).
% 2.68/1.66 thf(func_def_128, type, sK50: !>[X0: $tType]:((nat > X0) > nat)).
% 2.68/1.66 thf(func_def_129, type, sK51: !>[X0: $tType]:(poly @ X0 > X0)).
% 2.68/1.66 thf(func_def_130, type, sK52: nat > nat > nat).
% 2.68/1.66 thf(func_def_131, type, sK53: poly @ real > real).
% 2.68/1.66 thf(func_def_132, type, sK54: !>[X0: $tType]:(X0 > X0 > X0)).
% 2.68/1.66 thf(func_def_133, type, sK55: nat > nat > nat).
% 2.68/1.66 thf(func_def_134, type, sK56: !>[X0: $tType]:(poly @ X0 > nat)).
% 2.68/1.66 thf(func_def_135, type, sK57: nat > nat > nat).
% 2.68/1.66 thf(func_def_136, type, sK58: (nat > $o) > nat).
% 2.68/1.66 thf(func_def_137, type, sK59: !>[X0: $tType]:(poly @ X0 > nat)).
% 2.68/1.66 thf(func_def_138, type, sK60: nat > nat).
% 2.68/1.66 thf(func_def_139, type, sK61: (nat > $o) > nat).
% 2.68/1.66 thf(func_def_140, type, sK62: !>[X0: $tType, X1: $tType]:((X1 > X0 > $o) > set @ X0 > set @ X1 > X1)).
% 2.68/1.66 thf(func_def_141, type, sK63: !>[X0: $tType, X1: $tType]:((X1 > X0 > $o) > set @ X0 > set @ X1 > X0)).
% 2.68/1.66 thf(func_def_142, type, sK64: !>[X0: $tType, X1: $tType]:((X1 > $o) > (X1 > X0) > X1)).
% 2.68/1.66 thf(func_def_143, type, sK65: (nat > $o) > nat > nat).
% 2.68/1.66 thf(func_def_144, type, sK66: (nat > $o) > nat > nat).
% 2.68/1.66 thf(func_def_145, type, sK67: (nat > $o) > nat > nat).
% 2.68/1.66 thf(func_def_146, type, sK68: (nat > $o) > nat > nat).
% 2.68/1.66 thf(func_def_147, type, sK69: !>[X0: $tType]:(poly @ X0 > nat)).
% 2.68/1.66 thf(func_def_149, type, ph71: !>[X0: $tType]:(X0)).
% 2.68/1.66 thf(f1999,plain,(
% 2.68/1.66 $false),
% 2.68/1.66 inference(trivial_inequality_removal,[status(thm)],[f1989])).
% 2.68/1.66 thf(f1989,plain,(
% 2.68/1.66 (zero_zero @ poly @ real != zero_zero @ poly @ real)),
% 2.68/1.66 inference(superposition,[status(thm)],[f1867,f1988])).
% 2.68/1.66 thf(f1988,plain,(
% 2.68/1.66 (zero_zero @ poly @ real = (pderiv @ real @ p))),
% 2.68/1.66 inference(trivial_inequality_removal,[status(thm)],[f1987])).
% 2.68/1.66 thf(f1987,plain,(
% 2.68/1.66 (zero_zero @ poly @ real = (pderiv @ real @ p)) | ($true != $true)),
% 2.68/1.66 inference(forward_demodulation,[status(thm)],[f1986,f1452])).
% 2.68/1.66 thf(f1452,plain,(
% 2.68/1.66 (ring_n68954251visors @ real = $true)),
% 2.68/1.66 inference(cnf_transformation,[status(thm)],[f377])).
% 2.68/1.66 thf(f377,plain,(
% 2.68/1.66 (ring_n68954251visors @ real = $true)),
% 2.68/1.66 inference(fool_elimination,[status(thm)],[f376])).
% 2.68/1.66 thf(f376,plain,(
% 2.68/1.66 ring_n68954251visors @ real),
% 2.68/1.66 inference(rectify,[status(thm)],[f277])).
% 2.68/1.66 thf(f277,axiom,(
% 2.68/1.66 ring_n68954251visors @ real),
% 2.68/1.66 file('/export/starexec/sandbox/tmp/tmp.AWHgCUKr9y/DTF2THF_8563.p',tcon_Real_Oreal___Rings_Oring__no__zero__divisors)).
% 2.68/1.66 thf(f1986,plain,(
% 2.68/1.66 (ring_n68954251visors @ real != $true) | (zero_zero @ poly @ real = (pderiv @ real @ p))),
% 2.68/1.66 inference(trivial_inequality_removal,[status(thm)],[f1985])).
% 2.68/1.66 thf(f1985,plain,(
% 2.68/1.66 (zero_zero @ poly @ real = (pderiv @ real @ p)) | ($true != $true) | (ring_n68954251visors @ real != $true)),
% 2.68/1.66 inference(forward_demodulation,[status(thm)],[f1981,f1772])).
% 2.68/1.66 thf(f1772,plain,(
% 2.68/1.66 (comm_ring_1 @ real = $true)),
% 2.68/1.66 inference(cnf_transformation,[status(thm)],[f883])).
% 2.68/1.66 thf(f883,plain,(
% 2.68/1.66 (comm_ring_1 @ real = $true)),
% 2.68/1.66 inference(fool_elimination,[status(thm)],[f882])).
% 2.68/1.66 thf(f882,plain,(
% 2.68/1.66 comm_ring_1 @ real),
% 2.68/1.66 inference(rectify,[status(thm)],[f289])).
% 2.68/1.66 thf(f289,axiom,(
% 2.68/1.66 comm_ring_1 @ real),
% 2.68/1.66 file('/export/starexec/sandbox/tmp/tmp.AWHgCUKr9y/DTF2THF_8563.p',tcon_Real_Oreal___Rings_Ocomm__ring__1)).
% 2.68/1.66 thf(f1981,plain,(
% 2.68/1.66 (comm_ring_1 @ real != $true) | (ring_n68954251visors @ real != $true) | (zero_zero @ poly @ real = (pderiv @ real @ p))),
% 2.68/1.66 inference(trivial_inequality_removal,[status(thm)],[f1967])).
% 2.68/1.66 thf(f1967,plain,(
% 2.68/1.66 (comm_ring_1 @ real != $true) | (ring_n68954251visors @ real != $true) | (zero_zero @ poly @ real = (pderiv @ real @ p)) | ($true != $true)),
% 2.68/1.66 inference(superposition,[status(thm)],[f1518,f1756])).
% 2.68/1.66 thf(f1756,plain,(
% 2.68/1.66 ( ! [X0 : $tType,X1 : poly @ X0] : (((finite_finite2 @ X0 @ (collect @ X0 @ (^[Y0 : X0]: (zero_zero @ X0 = (poly2 @ X0 @ X1 @ Y0))))) = $true) | (comm_ring_1 @ X0 != $true) | (zero_zero @ poly @ X0 = X1) | (ring_n68954251visors @ X0 != $true)) )),
% 2.68/1.66 inference(cnf_transformation,[status(thm)],[f1115])).
% 2.68/1.66 thf(f1115,plain,(
% 2.68/1.66 ! [X0 : $tType] : ((comm_ring_1 @ X0 != $true) | ! [X1 : poly @ X0] : ((zero_zero @ poly @ X0 = X1) | ((finite_finite2 @ X0 @ (collect @ X0 @ (^[Y0 : X0]: (zero_zero @ X0 = (poly2 @ X0 @ X1 @ Y0))))) = $true)) | (ring_n68954251visors @ X0 != $true))),
% 2.68/1.66 inference(flattening,[status(thm)],[f1114])).
% 2.68/1.66 thf(f1114,plain,(
% 2.68/1.66 ! [X0 : $tType] : (! [X1 : poly @ X0] : ((zero_zero @ poly @ X0 = X1) | ((finite_finite2 @ X0 @ (collect @ X0 @ (^[Y0 : X0]: (zero_zero @ X0 = (poly2 @ X0 @ X1 @ Y0))))) = $true)) | ((ring_n68954251visors @ X0 != $true) | (comm_ring_1 @ X0 != $true)))),
% 2.68/1.66 inference(ennf_transformation,[status(thm)],[f719])).
% 2.68/1.66 thf(f719,plain,(
% 2.68/1.66 ! [X0 : $tType] : (((ring_n68954251visors @ X0 = $true) & (comm_ring_1 @ X0 = $true)) => ! [X1 : poly @ X0] : ((zero_zero @ poly @ X0 != X1) => ((finite_finite2 @ X0 @ (collect @ X0 @ (^[Y0 : X0]: (zero_zero @ X0 = (poly2 @ X0 @ X1 @ Y0))))) = $true)))),
% 2.68/1.66 inference(fool_elimination,[status(thm)],[f718])).
% 2.68/1.66 thf(f718,plain,(
% 2.68/1.66 ! [X0 : $tType] : ((ring_n68954251visors @ X0 & comm_ring_1 @ X0) => ! [X1 : poly @ X0] : ((zero_zero @ poly @ X0 != X1) => (finite_finite2 @ X0 @ (collect @ X0 @ (^[X2 : X0] : ((zero_zero @ X0 = (poly2 @ X0 @ X1 @ X2))))))))),
% 2.68/1.66 inference(rectify,[status(thm)],[f1])).
% 2.68/1.66 thf(f1,axiom,(
% 2.68/1.66 ! [X0 : $tType] : ((ring_n68954251visors @ X0 & comm_ring_1 @ X0) => ! [X3 : poly @ X0] : ((zero_zero @ poly @ X0 != X3) => (finite_finite2 @ X0 @ (collect @ X0 @ (^[X4 : X0] : (((poly2 @ X0 @ X3 @ X4) = zero_zero @ X0)))))))),
% 2.68/1.66 file('/export/starexec/sandbox/tmp/tmp.AWHgCUKr9y/DTF2THF_8563.p',fact_0_poly__roots__finite)).
% 2.68/1.66 thf(f1518,plain,(
% 2.68/1.66 ((finite_finite2 @ real @ (collect @ real @ (^[Y0 : real]: (zero_zero @ real = (poly2 @ real @ (pderiv @ real @ p) @ Y0))))) != $true)),
% 2.68/1.66 inference(cnf_transformation,[status(thm)],[f947])).
% 2.68/1.66 thf(f947,plain,(
% 2.68/1.66 ((finite_finite2 @ real @ (collect @ real @ (^[Y0 : real]: (zero_zero @ real = (poly2 @ real @ (pderiv @ real @ p) @ Y0))))) != $true)),
% 2.68/1.66 inference(flattening,[status(thm)],[f755])).
% 2.68/1.66 thf(f755,plain,(
% 2.68/1.66 ~((finite_finite2 @ real @ (collect @ real @ (^[Y0 : real]: (zero_zero @ real = (poly2 @ real @ (pderiv @ real @ p) @ Y0))))) = $true)),
% 2.68/1.66 inference(fool_elimination,[status(thm)],[f754])).
% 2.68/1.66 thf(f754,plain,(
% 2.68/1.66 ~(finite_finite2 @ real @ (collect @ real @ (^[X0 : real] : ((zero_zero @ real = (poly2 @ real @ (pderiv @ real @ p) @ X0))))))),
% 2.68/1.66 inference(rectify,[status(thm)],[f318])).
% 2.68/1.66 thf(f318,negated_conjecture,(
% 2.68/1.66 ~(finite_finite2 @ real @ (collect @ real @ (^[X4 : real] : ((zero_zero @ real = (poly2 @ real @ (pderiv @ real @ p) @ X4))))))),
% 2.68/1.66 inference(negated_conjecture,[status(cth)],[f317])).
% 2.68/1.66 thf(f317,conjecture,(
% 2.68/1.66 (finite_finite2 @ real @ (collect @ real @ (^[X4 : real] : ((zero_zero @ real = (poly2 @ real @ (pderiv @ real @ p) @ X4))))))),
% 2.68/1.66 file('/export/starexec/sandbox/tmp/tmp.AWHgCUKr9y/DTF2THF_8563.p',conj_0)).
% 2.68/1.66 thf(f1867,plain,(
% 2.68/1.66 (zero_zero @ poly @ real != (pderiv @ real @ p))),
% 2.68/1.66 inference(cnf_transformation,[status(thm)],[f3])).
% 2.68/1.66 thf(f3,axiom,(
% 2.68/1.66 (zero_zero @ poly @ real != (pderiv @ real @ p))),
% 2.68/1.66 file('/export/starexec/sandbox/tmp/tmp.AWHgCUKr9y/DTF2THF_8563.p',fact_2__092_060open_062pderiv_Ap_A_092_060noteq_062_A0_092_060close_062)).
% 2.68/1.66 % SZS output end Proof for DTF2THF_8563
% 2.68/1.66 % (8918)------------------------------
% 2.68/1.66 % (8918)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.68/1.66 % (8918)Termination reason: Refutation
% 2.68/1.66
% 2.68/1.66 % (8918)Memory used [KB]: 7036
% 2.68/1.66 % (8918)Time elapsed: 0.034 s
% 2.68/1.66 % (8918)Instructions burned: 95 (million)
% 2.68/1.66 % (8918)------------------------------
% 2.68/1.66 % (8918)------------------------------
% 2.68/1.66 % (8894)Success in time 0.119 s
%------------------------------------------------------------------------------