%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : ITP195^2 : TPTP v9.2.1. Released v7.5.0.
% Transfm : none
% Format : tptp:raw
% Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% Computer : n021.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 08:56:19 AM UTC 2026
% Result : Theorem 3.82s 2.03s
% Output : Refutation 3.82s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.11 % Problem : ITP195^2 : TPTP v9.2.1. Released v7.5.0.
% 0.00/0.12 % Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.14/0.32 % Computer : n021.cluster.edu
% 0.14/0.32 % Model : x86_64 x86_64
% 0.14/0.32 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.14/0.32 % Memory : 8042.1875MB
% 0.14/0.32 % OS : Linux 3.10.0-693.el7.x86_64
% 0.14/0.32 % CPULimit : 300
% 0.14/0.32 % WCLimit : 300
% 0.14/0.32 % DateTime : Tue Feb 24 08:29:38 EST 2026
% 0.14/0.32 % CPUTime :
% 0.14/0.32 Running /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.18/0.43 ---- Original DTF file ---
% 0.18/0.44 thf(spec,logic,$$dhol).
% 0.18/0.44 %------------------------------------------------------------------------------
% 0.18/0.44 % File : ITP195^2 : TPTP v9.2.1. Released v7.5.0.
% 0.18/0.44 % Domain : Interactive Theorem Proving
% 0.18/0.44 % Problem : Sledgehammer Sturm_Tarski problem prob_450__5874512_1
% 0.18/0.44 % Version : Especial.
% 0.18/0.44 % English :
% 0.18/0.44
% 0.18/0.44 % Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% 0.18/0.44 % : [Des21] Desharnais (2021), Email to Geoff Sutcliffe
% 0.18/0.44 % Source : [Des21]
% 0.18/0.44 % Names : Sturm_Tarski/prob_450__5874512_1 [Des21]
% 0.18/0.44
% 0.18/0.44 % Status : Theorem
% 0.18/0.44 % Rating : 0.00 v7.5.0
% 0.18/0.44 % Syntax : Number of formulae : 394 ( 114 unt; 54 typ; 0 def)
% 0.18/0.44 % Number of atoms : 1007 ( 247 equ; 0 cnn)
% 0.18/0.44 % Maximal formula atoms : 13 ( 2 avg)
% 0.18/0.44 % Number of connectives : 3084 ( 121 ~; 28 |; 71 &;2354 @)
% 0.18/0.44 % ( 0 <=>; 510 =>; 0 <=; 0 <~>)
% 0.18/0.44 % Maximal formula depth : 21 ( 7 avg)
% 0.18/0.44 % Number of types : 4 ( 3 usr)
% 0.18/0.44 % Number of type conns : 112 ( 112 >; 0 *; 0 +; 0 <<)
% 0.18/0.44 % Number of symbols : 52 ( 51 usr; 2 con; 0-3 aty)
% 0.18/0.44 % Number of variables : 912 ( 46 ^; 778 !; 45 ?; 912 :)
% 0.18/0.44 % ( 43 !>; 0 ?*; 0 @-; 0 @+)
% 0.18/0.44 % SPC : TH1_THM_EQU_NAR
% 0.18/0.44
% 0.18/0.44 % Comments : This file was generated by Sledgehammer 2021-02-23 16:27:21.925
% 0.18/0.44 %------------------------------------------------------------------------------
% 0.18/0.44 % Could-be-implicit typings (5)
% 0.18/0.44 thf(ty_t_Polynomial_Opoly,type,
% 0.18/0.44 poly: $tType > $tType ).
% 0.18/0.44
% 0.18/0.44 thf(ty_t_Real_Oreal,type,
% 0.18/0.44 real: $tType ).
% 0.18/0.44
% 0.18/0.44 thf(ty_t_Set_Oset,type,
% 0.18/0.44 set: $tType > $tType ).
% 0.18/0.44
% 0.18/0.44 thf(ty_t_Nat_Onat,type,
% 0.18/0.44 nat: $tType ).
% 0.18/0.44
% 0.18/0.44 thf(ty_t_Int_Oint,type,
% 0.18/0.44 int: $tType ).
% 0.18/0.44
% 0.18/0.44 % Explicit typings (49)
% 0.18/0.44 thf(sy_cl_Rings_Olinordered__idom,type,
% 0.18/0.44 linordered_idom:
% 0.18/0.44 !>[A: $tType] : $o ).
% 0.18/0.44
% 0.18/0.44 thf(sy_cl_HOL_Otype,type,
% 0.18/0.44 type:
% 0.18/0.44 !>[A: $tType] : $o ).
% 0.18/0.44
% 0.18/0.44 thf(sy_cl_Groups_Oone,type,
% 0.18/0.44 one:
% 0.18/0.44 !>[A: $tType] : $o ).
% 0.18/0.44
% 0.18/0.44 thf(sy_cl_Rings_Oidom,type,
% 0.18/0.44 idom:
% 0.18/0.44 !>[A: $tType] : $o ).
% 0.18/0.44
% 0.18/0.44 thf(sy_cl_Groups_Ozero,type,
% 0.18/0.44 zero:
% 0.18/0.44 !>[A: $tType] : $o ).
% 0.18/0.44
% 0.18/0.44 thf(sy_cl_Orderings_Oord,type,
% 0.18/0.44 ord:
% 0.18/0.44 !>[A: $tType] : $o ).
% 0.18/0.44
% 0.18/0.44 thf(sy_cl_Nat_Oring__char__0,type,
% 0.18/0.44 ring_char_0:
% 0.18/0.44 !>[A: $tType] : $o ).
% 0.18/0.44
% 0.18/0.44 thf(sy_cl_Orderings_Oorder,type,
% 0.18/0.44 order:
% 0.18/0.44 !>[A: $tType] : $o ).
% 0.18/0.44
% 0.18/0.44 thf(sy_cl_Orderings_Ono__bot,type,
% 0.18/0.44 no_bot:
% 0.18/0.44 !>[A: $tType] : $o ).
% 0.18/0.44
% 0.18/0.44 thf(sy_cl_Orderings_Ono__top,type,
% 0.18/0.44 no_top:
% 0.18/0.44 !>[A: $tType] : $o ).
% 0.18/0.44
% 0.18/0.44 thf(sy_cl_Rings_Osemiring__1,type,
% 0.18/0.44 semiring_1:
% 0.18/0.44 !>[A: $tType] : $o ).
% 0.18/0.44
% 0.18/0.44 thf(sy_cl_Rings_Ocomm__ring__1,type,
% 0.18/0.44 comm_ring_1:
% 0.18/0.44 !>[A: $tType] : $o ).
% 0.18/0.44
% 0.18/0.44 thf(sy_cl_Orderings_Olinorder,type,
% 0.18/0.44 linorder:
% 0.18/0.44 !>[A: $tType] : $o ).
% 0.18/0.44
% 0.18/0.44 thf(sy_cl_Orderings_Opreorder,type,
% 0.18/0.44 preorder:
% 0.18/0.44 !>[A: $tType] : $o ).
% 0.18/0.44
% 0.18/0.44 thf(sy_cl_Fields_Ofield__char__0,type,
% 0.18/0.44 field_char_0:
% 0.18/0.44 !>[A: $tType] : $o ).
% 0.18/0.44
% 0.18/0.44 thf(sy_cl_Nat_Osemiring__char__0,type,
% 0.18/0.44 semiring_char_0:
% 0.18/0.44 !>[A: $tType] : $o ).
% 0.18/0.44
% 0.18/0.44 thf(sy_cl_Orderings_Owellorder,type,
% 0.18/0.44 wellorder:
% 0.18/0.44 !>[A: $tType] : $o ).
% 0.18/0.44
% 0.18/0.44 thf(sy_cl_Orderings_Odense__order,type,
% 0.18/0.44 dense_order:
% 0.18/0.44 !>[A: $tType] : $o ).
% 0.18/0.44
% 0.18/0.44 thf(sy_cl_Rings_Ocomm__semiring__0,type,
% 0.18/0.44 comm_semiring_0:
% 0.18/0.44 !>[A: $tType] : $o ).
% 0.18/0.44
% 0.18/0.44 thf(sy_cl_Rings_Ocomm__semiring__1,type,
% 0.18/0.44 comm_semiring_1:
% 0.18/0.44 !>[A: $tType] : $o ).
% 0.18/0.44
% 0.18/0.44 thf(sy_cl_Fields_Olinordered__field,type,
% 0.18/0.44 linordered_field:
% 0.18/0.44 !>[A: $tType] : $o ).
% 0.18/0.44
% 0.18/0.44 thf(sy_cl_Orderings_Odense__linorder,type,
% 0.18/0.44 dense_linorder:
% 0.18/0.44 !>[A: $tType] : $o ).
% 0.18/0.44
% 0.18/0.44 thf(sy_cl_Rings_Oring__no__zero__divisors,type,
% 0.18/0.44 ring_n68954251visors:
% 0.18/0.44 !>[A: $tType] : $o ).
% 0.18/0.44
% 0.18/0.44 thf(sy_cl_Rings_Osemiring__no__zero__divisors,type,
% 0.18/0.44 semiri1193490041visors:
% 0.18/0.44 !>[A: $tType] : $o ).
% 0.18/0.44
% 0.18/0.44 thf(sy_cl_Rings_Olinordered__nonzero__semiring,type,
% 0.18/0.44 linord1659791738miring:
% 0.18/0.44 !>[A: $tType] : $o ).
% 0.18/0.44
% 0.18/0.44 thf(sy_cl_Groups_Ocanonically__ordered__monoid__add,type,
% 0.18/0.44 canoni770627133id_add:
% 0.18/0.44 !>[A: $tType] : $o ).
% 0.18/0.44
% 0.18/0.44 thf(sy_cl_Conditionally__Complete__Lattices_Oconditionally__complete__linorder,type,
% 0.18/0.44 condit1037483654norder:
% 0.18/0.44 !>[A: $tType] : $o ).
% 0.18/0.44
% 0.18/0.44 thf(sy_c_Groups_Oone__class_Oone,type,
% 0.18/0.44 one_one:
% 0.18/0.44 !>[A: $tType] : A ).
% 0.18/0.44
% 0.18/0.44 thf(sy_c_Groups_Ozero__class_Ozero,type,
% 0.18/0.44 zero_zero:
% 0.18/0.44 !>[A: $tType] : A ).
% 0.18/0.44
% 0.18/0.44 thf(sy_c_Nat_OSuc,type,
% 0.18/0.44 suc: nat > nat ).
% 0.18/0.44
% 0.18/0.44 thf(sy_c_Nat_Osemiring__1__class_Oof__nat,type,
% 0.18/0.44 semiring_1_of_nat:
% 0.18/0.44 !>[A: $tType] : ( nat > A ) ).
% 0.18/0.44
% 0.18/0.44 thf(sy_c_Orderings_Oord__class_Oless,type,
% 0.18/0.44 ord_less:
% 0.18/0.44 !>[A: $tType] : ( A > A > $o ) ).
% 0.18/0.44
% 0.18/0.44 thf(sy_c_Orderings_Oord__class_Oless__eq,type,
% 0.18/0.44 ord_less_eq:
% 0.18/0.44 !>[A: $tType] : ( A > A > $o ) ).
% 0.18/0.44
% 0.18/0.44 thf(sy_c_Polynomial_Ois__zero,type,
% 0.18/0.44 is_zero:
% 0.18/0.44 !>[A: $tType] : ( ( poly @ A ) > $o ) ).
% 0.18/0.44
% 0.18/0.44 thf(sy_c_Polynomial_Oorder,type,
% 0.18/0.44 order2:
% 0.18/0.44 !>[A: $tType] : ( A > ( poly @ A ) > nat ) ).
% 0.18/0.44
% 0.18/0.44 thf(sy_c_Polynomial_Opderiv,type,
% 0.18/0.44 pderiv:
% 0.18/0.44 !>[A: $tType] : ( ( poly @ A ) > ( poly @ A ) ) ).
% 0.18/0.44
% 0.18/0.44 thf(sy_c_Polynomial_Opoly,type,
% 0.18/0.44 poly2:
% 0.18/0.44 !>[A: $tType] : ( ( poly @ A ) > A > A ) ).
% 0.18/0.44
% 0.18/0.44 thf(sy_c_Polynomial_Opoly_Ocoeff,type,
% 0.18/0.44 coeff:
% 0.18/0.44 !>[A: $tType] : ( ( poly @ A ) > nat > A ) ).
% 0.18/0.44
% 0.18/0.44 thf(sy_c_Polynomial_Opoly__cutoff,type,
% 0.18/0.44 poly_cutoff:
% 0.18/0.44 !>[A: $tType] : ( nat > ( poly @ A ) > ( poly @ A ) ) ).
% 0.18/0.44
% 0.18/0.44 thf(sy_c_Polynomial_Opoly__shift,type,
% 0.18/0.44 poly_shift:
% 0.18/0.44 !>[A: $tType] : ( nat > ( poly @ A ) > ( poly @ A ) ) ).
% 0.18/0.44
% 0.18/0.44 thf(sy_c_Polynomial_Oreflect__poly,type,
% 0.18/0.44 reflect_poly:
% 0.18/0.44 !>[A: $tType] : ( ( poly @ A ) > ( poly @ A ) ) ).
% 0.18/0.44
% 0.18/0.44 thf(sy_c_Polynomial_Orsquarefree,type,
% 0.18/0.44 rsquarefree:
% 0.18/0.44 !>[A: $tType] : ( ( poly @ A ) > $o ) ).
% 0.18/0.44
% 0.18/0.44 thf(sy_c_Set_OCollect,type,
% 0.18/0.44 collect:
% 0.18/0.44 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
% 0.18/0.44
% 0.18/0.44 thf(sy_c_Sturm__Tarski__Mirabelle__osyjafhhev_Ocross,type,
% 0.18/0.44 sturm_424270202_cross: ( poly @ real ) > real > real > int ).
% 0.18/0.44
% 0.18/0.44 thf(sy_c_Sturm__Tarski__Mirabelle__osyjafhhev_Osign__r__pos,type,
% 0.18/0.44 sturm_1700286437_r_pos: ( poly @ real ) > real > $o ).
% 0.18/0.44
% 0.18/0.44 thf(sy_c_Sturm__Tarski__Mirabelle__osyjafhhev_Ovariation,type,
% 0.18/0.44 sturm_1771227917iation: real > real > int ).
% 0.18/0.44
% 0.18/0.44 thf(sy_c_member,type,
% 0.18/0.44 member:
% 0.18/0.44 !>[A: $tType] : ( A > ( set @ A ) > $o ) ).
% 0.18/0.44
% 0.18/0.44 thf(sy_v_p,type,
% 0.18/0.44 p: poly @ real ).
% 0.18/0.44
% 0.18/0.44 thf(sy_v_x,type,
% 0.18/0.44 x: real ).
% 0.18/0.44
% 0.18/0.44 % Relevant facts (252)
% 0.18/0.44 thf(fact_0_False,axiom,
% 0.18/0.44 ( ( poly2 @ real @ p @ x )
% 0.18/0.44 != ( zero_zero @ real ) ) ).
% 0.18/0.44
% 0.18/0.44 % False
% 0.18/0.44 thf(fact_1__092_060open_062sign__r__pos_Ap_Ax_A_092_060Longrightarrow_062_A0_A_060_Apoly_Ap_Ax_092_060close_062,axiom,
% 0.18/0.44 ( ( sturm_1700286437_r_pos @ p @ x )
% 0.18/0.44 => ( ord_less @ real @ ( zero_zero @ real ) @ ( poly2 @ real @ p @ x ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % \<open>sign_r_pos p x \<Longrightarrow> 0 < poly p x\<close>
% 0.18/0.44 thf(fact_2__092_060open_0620_A_060_Apoly_Ap_Ax_A_092_060Longrightarrow_062_Asign__r__pos_Ap_Ax_092_060close_062,axiom,
% 0.18/0.44 ( ( ord_less @ real @ ( zero_zero @ real ) @ ( poly2 @ real @ p @ x ) )
% 0.18/0.44 => ( sturm_1700286437_r_pos @ p @ x ) ) ).
% 0.18/0.44
% 0.18/0.44 % \<open>0 < poly p x \<Longrightarrow> sign_r_pos p x\<close>
% 0.18/0.44 thf(fact_3_assms,axiom,
% 0.18/0.44 ( p
% 0.18/0.44 != ( zero_zero @ ( poly @ real ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % assms
% 0.18/0.44 thf(fact_4_poly__0,axiom,
% 0.18/0.44 ! [A: $tType] :
% 0.18/0.44 ( ( comm_semiring_0 @ A )
% 0.18/0.44 => ! [X: A] :
% 0.18/0.44 ( ( poly2 @ A @ ( zero_zero @ ( poly @ A ) ) @ X )
% 0.18/0.44 = ( zero_zero @ A ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % poly_0
% 0.18/0.44 thf(fact_5_not__gr__zero,axiom,
% 0.18/0.44 ! [A: $tType] :
% 0.18/0.44 ( ( canoni770627133id_add @ A )
% 0.18/0.44 => ! [N: A] :
% 0.18/0.44 ( ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ N ) )
% 0.18/0.44 = ( N
% 0.18/0.44 = ( zero_zero @ A ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % not_gr_zero
% 0.18/0.44 thf(fact_6_poly__IVT__neg,axiom,
% 0.18/0.44 ! [A2: real,B: real,P: poly @ real] :
% 0.18/0.44 ( ( ord_less @ real @ A2 @ B )
% 0.18/0.44 => ( ( ord_less @ real @ ( zero_zero @ real ) @ ( poly2 @ real @ P @ A2 ) )
% 0.18/0.44 => ( ( ord_less @ real @ ( poly2 @ real @ P @ B ) @ ( zero_zero @ real ) )
% 0.18/0.44 => ? [X2: real] :
% 0.18/0.44 ( ( ord_less @ real @ A2 @ X2 )
% 0.18/0.44 & ( ord_less @ real @ X2 @ B )
% 0.18/0.44 & ( ( poly2 @ real @ P @ X2 )
% 0.18/0.44 = ( zero_zero @ real ) ) ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % poly_IVT_neg
% 0.18/0.44 thf(fact_7_poly__IVT__pos,axiom,
% 0.18/0.44 ! [A2: real,B: real,P: poly @ real] :
% 0.18/0.44 ( ( ord_less @ real @ A2 @ B )
% 0.18/0.44 => ( ( ord_less @ real @ ( poly2 @ real @ P @ A2 ) @ ( zero_zero @ real ) )
% 0.18/0.44 => ( ( ord_less @ real @ ( zero_zero @ real ) @ ( poly2 @ real @ P @ B ) )
% 0.18/0.44 => ? [X2: real] :
% 0.18/0.44 ( ( ord_less @ real @ A2 @ X2 )
% 0.18/0.44 & ( ord_less @ real @ X2 @ B )
% 0.18/0.44 & ( ( poly2 @ real @ P @ X2 )
% 0.18/0.44 = ( zero_zero @ real ) ) ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % poly_IVT_pos
% 0.18/0.44 thf(fact_8_poly__all__0__iff__0,axiom,
% 0.18/0.44 ! [A: $tType] :
% 0.18/0.44 ( ( ( ring_char_0 @ A )
% 0.18/0.44 & ( comm_ring_1 @ A )
% 0.18/0.44 & ( ring_n68954251visors @ A ) )
% 0.18/0.44 => ! [P: poly @ A] :
% 0.18/0.44 ( ( ! [X3: A] :
% 0.18/0.44 ( ( poly2 @ A @ P @ X3 )
% 0.18/0.44 = ( zero_zero @ A ) ) )
% 0.18/0.44 = ( P
% 0.18/0.44 = ( zero_zero @ ( poly @ A ) ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % poly_all_0_iff_0
% 0.18/0.44 thf(fact_9_less__numeral__extra_I3_J,axiom,
% 0.18/0.44 ! [A: $tType] :
% 0.18/0.44 ( ( linord1659791738miring @ A )
% 0.18/0.44 => ~ ( ord_less @ A @ ( zero_zero @ A ) @ ( zero_zero @ A ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % less_numeral_extra(3)
% 0.18/0.44 thf(fact_10_field__lbound__gt__zero,axiom,
% 0.18/0.44 ! [A: $tType] :
% 0.18/0.44 ( ( linordered_field @ A )
% 0.18/0.44 => ! [D1: A,D2: A] :
% 0.18/0.44 ( ( ord_less @ A @ ( zero_zero @ A ) @ D1 )
% 0.18/0.44 => ( ( ord_less @ A @ ( zero_zero @ A ) @ D2 )
% 0.18/0.44 => ? [E: A] :
% 0.18/0.44 ( ( ord_less @ A @ ( zero_zero @ A ) @ E )
% 0.18/0.44 & ( ord_less @ A @ E @ D1 )
% 0.18/0.44 & ( ord_less @ A @ E @ D2 ) ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % field_lbound_gt_zero
% 0.18/0.44 thf(fact_11_gr__zeroI,axiom,
% 0.18/0.44 ! [A: $tType] :
% 0.18/0.44 ( ( canoni770627133id_add @ A )
% 0.18/0.44 => ! [N: A] :
% 0.18/0.44 ( ( N
% 0.18/0.44 != ( zero_zero @ A ) )
% 0.18/0.44 => ( ord_less @ A @ ( zero_zero @ A ) @ N ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % gr_zeroI
% 0.18/0.44 thf(fact_12_not__less__zero,axiom,
% 0.18/0.44 ! [A: $tType] :
% 0.18/0.44 ( ( canoni770627133id_add @ A )
% 0.18/0.44 => ! [N: A] :
% 0.18/0.44 ~ ( ord_less @ A @ N @ ( zero_zero @ A ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % not_less_zero
% 0.18/0.44 thf(fact_13_gr__implies__not__zero,axiom,
% 0.18/0.44 ! [A: $tType] :
% 0.18/0.44 ( ( canoni770627133id_add @ A )
% 0.18/0.44 => ! [M: A,N: A] :
% 0.18/0.44 ( ( ord_less @ A @ M @ N )
% 0.18/0.44 => ( N
% 0.18/0.44 != ( zero_zero @ A ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % gr_implies_not_zero
% 0.18/0.44 thf(fact_14_zero__less__iff__neq__zero,axiom,
% 0.18/0.44 ! [A: $tType] :
% 0.18/0.44 ( ( canoni770627133id_add @ A )
% 0.18/0.44 => ! [N: A] :
% 0.18/0.44 ( ( ord_less @ A @ ( zero_zero @ A ) @ N )
% 0.18/0.44 = ( N
% 0.18/0.44 != ( zero_zero @ A ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % zero_less_iff_neq_zero
% 0.18/0.44 thf(fact_15_pderiv__0,axiom,
% 0.18/0.44 ! [A: $tType] :
% 0.18/0.44 ( ( ( comm_semiring_1 @ A )
% 0.18/0.44 & ( semiri1193490041visors @ A ) )
% 0.18/0.44 => ( ( pderiv @ A @ ( zero_zero @ ( poly @ A ) ) )
% 0.18/0.44 = ( zero_zero @ ( poly @ A ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % pderiv_0
% 0.18/0.44 thf(fact_16_zero__reorient,axiom,
% 0.18/0.44 ! [A: $tType] :
% 0.18/0.44 ( ( zero @ A )
% 0.18/0.44 => ! [X: A] :
% 0.18/0.44 ( ( ( zero_zero @ A )
% 0.18/0.44 = X )
% 0.18/0.44 = ( X
% 0.18/0.44 = ( zero_zero @ A ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % zero_reorient
% 0.18/0.44 thf(fact_17_poly__eq__poly__eq__iff,axiom,
% 0.18/0.44 ! [A: $tType] :
% 0.18/0.44 ( ( ( ring_char_0 @ A )
% 0.18/0.44 & ( comm_ring_1 @ A )
% 0.18/0.44 & ( ring_n68954251visors @ A ) )
% 0.18/0.44 => ! [P: poly @ A,Q: poly @ A] :
% 0.18/0.44 ( ( ( poly2 @ A @ P )
% 0.18/0.44 = ( poly2 @ A @ Q ) )
% 0.18/0.44 = ( P = Q ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % poly_eq_poly_eq_iff
% 0.18/0.44 thf(fact_18_cross__0,axiom,
% 0.18/0.44 ! [A2: real,B: real] :
% 0.18/0.44 ( ( sturm_424270202_cross @ ( zero_zero @ ( poly @ real ) ) @ A2 @ B )
% 0.18/0.44 = ( zero_zero @ int ) ) ).
% 0.18/0.44
% 0.18/0.44 % cross_0
% 0.18/0.44 thf(fact_19_rsquarefree__roots,axiom,
% 0.18/0.44 ! [A: $tType] :
% 0.18/0.44 ( ( field_char_0 @ A )
% 0.18/0.44 => ( ( rsquarefree @ A )
% 0.18/0.44 = ( ^ [P2: poly @ A] :
% 0.18/0.44 ! [A3: A] :
% 0.18/0.44 ~ ( ( ( poly2 @ A @ P2 @ A3 )
% 0.18/0.44 = ( zero_zero @ A ) )
% 0.18/0.44 & ( ( poly2 @ A @ ( pderiv @ A @ P2 ) @ A3 )
% 0.18/0.44 = ( zero_zero @ A ) ) ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % rsquarefree_roots
% 0.18/0.44 thf(fact_20_cross__no__root,axiom,
% 0.18/0.44 ! [A2: real,B: real,P: poly @ real] :
% 0.18/0.44 ( ( ord_less @ real @ A2 @ B )
% 0.18/0.44 => ( ! [X2: real] :
% 0.18/0.44 ( ( ( ord_less @ real @ A2 @ X2 )
% 0.18/0.44 & ( ord_less @ real @ X2 @ B ) )
% 0.18/0.44 => ( ( poly2 @ real @ P @ X2 )
% 0.18/0.44 != ( zero_zero @ real ) ) )
% 0.18/0.44 => ( ( sturm_424270202_cross @ P @ A2 @ B )
% 0.18/0.44 = ( zero_zero @ int ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % cross_no_root
% 0.18/0.44 thf(fact_21_is__zero__null,axiom,
% 0.18/0.44 ! [A: $tType] :
% 0.18/0.44 ( ( zero @ A )
% 0.18/0.44 => ( ( is_zero @ A )
% 0.18/0.44 = ( ^ [P2: poly @ A] :
% 0.18/0.44 ( P2
% 0.18/0.44 = ( zero_zero @ ( poly @ A ) ) ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % is_zero_null
% 0.18/0.44 thf(fact_22_poly__cutoff__0,axiom,
% 0.18/0.44 ! [A: $tType] :
% 0.18/0.44 ( ( zero @ A )
% 0.18/0.44 => ! [N: nat] :
% 0.18/0.44 ( ( poly_cutoff @ A @ N @ ( zero_zero @ ( poly @ A ) ) )
% 0.18/0.44 = ( zero_zero @ ( poly @ A ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % poly_cutoff_0
% 0.18/0.44 thf(fact_23_reflect__poly__at__0__eq__0__iff,axiom,
% 0.18/0.44 ! [A: $tType] :
% 0.18/0.44 ( ( comm_semiring_0 @ A )
% 0.18/0.44 => ! [P: poly @ A] :
% 0.18/0.44 ( ( ( poly2 @ A @ ( reflect_poly @ A @ P ) @ ( zero_zero @ A ) )
% 0.18/0.44 = ( zero_zero @ A ) )
% 0.18/0.44 = ( P
% 0.18/0.44 = ( zero_zero @ ( poly @ A ) ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % reflect_poly_at_0_eq_0_iff
% 0.18/0.44 thf(fact_24_next__non__root__interval,axiom,
% 0.18/0.44 ! [P: poly @ real,Lb: real] :
% 0.18/0.44 ( ( P
% 0.18/0.44 != ( zero_zero @ ( poly @ real ) ) )
% 0.18/0.44 => ~ ! [Ub: real] :
% 0.18/0.44 ( ( ord_less @ real @ Lb @ Ub )
% 0.18/0.44 => ~ ! [Z: real] :
% 0.18/0.44 ( ( ( ord_less @ real @ Lb @ Z )
% 0.18/0.44 & ( ord_less_eq @ real @ Z @ Ub ) )
% 0.18/0.44 => ( ( poly2 @ real @ P @ Z )
% 0.18/0.44 != ( zero_zero @ real ) ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % next_non_root_interval
% 0.18/0.44 thf(fact_25_last__non__root__interval,axiom,
% 0.18/0.44 ! [P: poly @ real,Ub2: real] :
% 0.18/0.44 ( ( P
% 0.18/0.44 != ( zero_zero @ ( poly @ real ) ) )
% 0.18/0.44 => ~ ! [Lb2: real] :
% 0.18/0.44 ( ( ord_less @ real @ Lb2 @ Ub2 )
% 0.18/0.44 => ~ ! [Z: real] :
% 0.18/0.44 ( ( ( ord_less_eq @ real @ Lb2 @ Z )
% 0.18/0.44 & ( ord_less @ real @ Z @ Ub2 ) )
% 0.18/0.44 => ( ( poly2 @ real @ P @ Z )
% 0.18/0.44 != ( zero_zero @ real ) ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % last_non_root_interval
% 0.18/0.44 thf(fact_26_poly__shift__0,axiom,
% 0.18/0.44 ! [A: $tType] :
% 0.18/0.44 ( ( zero @ A )
% 0.18/0.44 => ! [N: nat] :
% 0.18/0.44 ( ( poly_shift @ A @ N @ ( zero_zero @ ( poly @ A ) ) )
% 0.18/0.44 = ( zero_zero @ ( poly @ A ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % poly_shift_0
% 0.18/0.44 thf(fact_27_order__root,axiom,
% 0.18/0.44 ! [A: $tType] :
% 0.18/0.44 ( ( idom @ A )
% 0.18/0.44 => ! [P: poly @ A,A2: A] :
% 0.18/0.44 ( ( ( poly2 @ A @ P @ A2 )
% 0.18/0.44 = ( zero_zero @ A ) )
% 0.18/0.44 = ( ( P
% 0.18/0.44 = ( zero_zero @ ( poly @ A ) ) )
% 0.18/0.44 | ( ( order2 @ A @ A2 @ P )
% 0.18/0.44 != ( zero_zero @ nat ) ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % order_root
% 0.18/0.44 thf(fact_28_le__zero__eq,axiom,
% 0.18/0.44 ! [A: $tType] :
% 0.18/0.44 ( ( canoni770627133id_add @ A )
% 0.18/0.44 => ! [N: A] :
% 0.18/0.44 ( ( ord_less_eq @ A @ N @ ( zero_zero @ A ) )
% 0.18/0.44 = ( N
% 0.18/0.44 = ( zero_zero @ A ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % le_zero_eq
% 0.18/0.44 thf(fact_29_reflect__poly__0,axiom,
% 0.18/0.44 ! [A: $tType] :
% 0.18/0.44 ( ( zero @ A )
% 0.18/0.44 => ( ( reflect_poly @ A @ ( zero_zero @ ( poly @ A ) ) )
% 0.18/0.44 = ( zero_zero @ ( poly @ A ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % reflect_poly_0
% 0.18/0.44 thf(fact_30_poly__shift__id,axiom,
% 0.18/0.44 ! [A: $tType] :
% 0.18/0.44 ( ( zero @ A )
% 0.18/0.44 => ( ( poly_shift @ A @ ( zero_zero @ nat ) )
% 0.18/0.44 = ( ^ [X3: poly @ A] : X3 ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % poly_shift_id
% 0.18/0.44 thf(fact_31_complete__real,axiom,
% 0.18/0.44 ! [S: set @ real] :
% 0.18/0.44 ( ? [X4: real] : ( member @ real @ X4 @ S )
% 0.18/0.44 => ( ? [Z: real] :
% 0.18/0.44 ! [X2: real] :
% 0.18/0.44 ( ( member @ real @ X2 @ S )
% 0.18/0.44 => ( ord_less_eq @ real @ X2 @ Z ) )
% 0.18/0.44 => ? [Y: real] :
% 0.18/0.44 ( ! [X4: real] :
% 0.18/0.44 ( ( member @ real @ X4 @ S )
% 0.18/0.44 => ( ord_less_eq @ real @ X4 @ Y ) )
% 0.18/0.44 & ! [Z: real] :
% 0.18/0.44 ( ! [X2: real] :
% 0.18/0.44 ( ( member @ real @ X2 @ S )
% 0.18/0.44 => ( ord_less_eq @ real @ X2 @ Z ) )
% 0.18/0.44 => ( ord_less_eq @ real @ Y @ Z ) ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % complete_real
% 0.18/0.44 thf(fact_32_le__numeral__extra_I3_J,axiom,
% 0.18/0.44 ! [A: $tType] :
% 0.18/0.44 ( ( linord1659791738miring @ A )
% 0.18/0.44 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( zero_zero @ A ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % le_numeral_extra(3)
% 0.18/0.44 thf(fact_33_zero__le,axiom,
% 0.18/0.44 ! [A: $tType] :
% 0.18/0.44 ( ( canoni770627133id_add @ A )
% 0.18/0.44 => ! [X: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ X ) ) ).
% 0.18/0.44
% 0.18/0.44 % zero_le
% 0.18/0.44 thf(fact_34_less__eq__real__def,axiom,
% 0.18/0.44 ( ( ord_less_eq @ real )
% 0.18/0.44 = ( ^ [X3: real,Y2: real] :
% 0.18/0.44 ( ( ord_less @ real @ X3 @ Y2 )
% 0.18/0.44 | ( X3 = Y2 ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % less_eq_real_def
% 0.18/0.44 thf(fact_35_order__0I,axiom,
% 0.18/0.44 ! [A: $tType] :
% 0.18/0.44 ( ( idom @ A )
% 0.18/0.44 => ! [P: poly @ A,A2: A] :
% 0.18/0.44 ( ( ( poly2 @ A @ P @ A2 )
% 0.18/0.44 != ( zero_zero @ A ) )
% 0.18/0.44 => ( ( order2 @ A @ A2 @ P )
% 0.18/0.44 = ( zero_zero @ nat ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % order_0I
% 0.18/0.44 thf(fact_36_not__eq__pos__or__neg__iff__1,axiom,
% 0.18/0.44 ! [Lb: real,Ub2: real,P: poly @ real] :
% 0.18/0.44 ( ( ! [Z2: real] :
% 0.18/0.44 ( ( ( ord_less @ real @ Lb @ Z2 )
% 0.18/0.44 & ( ord_less_eq @ real @ Z2 @ Ub2 ) )
% 0.18/0.44 => ( ( poly2 @ real @ P @ Z2 )
% 0.18/0.44 != ( zero_zero @ real ) ) ) )
% 0.18/0.44 = ( ! [Z2: real] :
% 0.18/0.44 ( ( ( ord_less @ real @ Lb @ Z2 )
% 0.18/0.44 & ( ord_less_eq @ real @ Z2 @ Ub2 ) )
% 0.18/0.44 => ( ord_less @ real @ ( zero_zero @ real ) @ ( poly2 @ real @ P @ Z2 ) ) )
% 0.18/0.44 | ! [Z2: real] :
% 0.18/0.44 ( ( ( ord_less @ real @ Lb @ Z2 )
% 0.18/0.44 & ( ord_less_eq @ real @ Z2 @ Ub2 ) )
% 0.18/0.44 => ( ord_less @ real @ ( poly2 @ real @ P @ Z2 ) @ ( zero_zero @ real ) ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % not_eq_pos_or_neg_iff_1
% 0.18/0.44 thf(fact_37_not__eq__pos__or__neg__iff__2,axiom,
% 0.18/0.44 ! [Lb: real,Ub2: real,P: poly @ real] :
% 0.18/0.44 ( ( ! [Z2: real] :
% 0.18/0.44 ( ( ( ord_less_eq @ real @ Lb @ Z2 )
% 0.18/0.44 & ( ord_less @ real @ Z2 @ Ub2 ) )
% 0.18/0.44 => ( ( poly2 @ real @ P @ Z2 )
% 0.18/0.44 != ( zero_zero @ real ) ) ) )
% 0.18/0.44 = ( ! [Z2: real] :
% 0.18/0.44 ( ( ( ord_less_eq @ real @ Lb @ Z2 )
% 0.18/0.44 & ( ord_less @ real @ Z2 @ Ub2 ) )
% 0.18/0.44 => ( ord_less @ real @ ( zero_zero @ real ) @ ( poly2 @ real @ P @ Z2 ) ) )
% 0.18/0.44 | ! [Z2: real] :
% 0.18/0.44 ( ( ( ord_less_eq @ real @ Lb @ Z2 )
% 0.18/0.44 & ( ord_less @ real @ Z2 @ Ub2 ) )
% 0.18/0.44 => ( ord_less @ real @ ( poly2 @ real @ P @ Z2 ) @ ( zero_zero @ real ) ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % not_eq_pos_or_neg_iff_2
% 0.18/0.44 thf(fact_38_neq0__conv,axiom,
% 0.18/0.44 ! [N: nat] :
% 0.18/0.44 ( ( N
% 0.18/0.44 != ( zero_zero @ nat ) )
% 0.18/0.44 = ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ).
% 0.18/0.44
% 0.18/0.44 % neq0_conv
% 0.18/0.44 thf(fact_39_less__nat__zero__code,axiom,
% 0.18/0.44 ! [N: nat] :
% 0.18/0.44 ~ ( ord_less @ nat @ N @ ( zero_zero @ nat ) ) ).
% 0.18/0.44
% 0.18/0.44 % less_nat_zero_code
% 0.18/0.44 thf(fact_40_bot__nat__0_Onot__eq__extremum,axiom,
% 0.18/0.44 ! [A2: nat] :
% 0.18/0.44 ( ( A2
% 0.18/0.44 != ( zero_zero @ nat ) )
% 0.18/0.44 = ( ord_less @ nat @ ( zero_zero @ nat ) @ A2 ) ) ).
% 0.18/0.44
% 0.18/0.44 % bot_nat_0.not_eq_extremum
% 0.18/0.44 thf(fact_41_order__refl,axiom,
% 0.18/0.44 ! [A: $tType] :
% 0.18/0.44 ( ( preorder @ A )
% 0.18/0.44 => ! [X: A] : ( ord_less_eq @ A @ X @ X ) ) ).
% 0.18/0.44
% 0.18/0.44 % order_refl
% 0.18/0.44 thf(fact_42_minf_I8_J,axiom,
% 0.18/0.44 ! [A: $tType] :
% 0.18/0.44 ( ( linorder @ A )
% 0.18/0.44 => ! [T: A] :
% 0.18/0.44 ? [Z3: A] :
% 0.18/0.44 ! [X4: A] :
% 0.18/0.44 ( ( ord_less @ A @ X4 @ Z3 )
% 0.18/0.44 => ~ ( ord_less_eq @ A @ T @ X4 ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % minf(8)
% 0.18/0.44 thf(fact_43_minf_I6_J,axiom,
% 0.18/0.44 ! [A: $tType] :
% 0.18/0.44 ( ( linorder @ A )
% 0.18/0.44 => ! [T: A] :
% 0.18/0.44 ? [Z3: A] :
% 0.18/0.44 ! [X4: A] :
% 0.18/0.44 ( ( ord_less @ A @ X4 @ Z3 )
% 0.18/0.44 => ( ord_less_eq @ A @ X4 @ T ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % minf(6)
% 0.18/0.44 thf(fact_44_mem__Collect__eq,axiom,
% 0.18/0.44 ! [A: $tType,A2: A,P3: A > $o] :
% 0.18/0.44 ( ( member @ A @ A2 @ ( collect @ A @ P3 ) )
% 0.18/0.44 = ( P3 @ A2 ) ) ).
% 0.18/0.44
% 0.18/0.44 % mem_Collect_eq
% 0.18/0.44 thf(fact_45_Collect__mem__eq,axiom,
% 0.18/0.44 ! [A: $tType,A4: set @ A] :
% 0.18/0.44 ( ( collect @ A
% 0.18/0.44 @ ^ [X3: A] : ( member @ A @ X3 @ A4 ) )
% 0.18/0.44 = A4 ) ).
% 0.18/0.44
% 0.18/0.44 % Collect_mem_eq
% 0.18/0.44 thf(fact_46_Collect__cong,axiom,
% 0.18/0.44 ! [A: $tType,P3: A > $o,Q2: A > $o] :
% 0.18/0.44 ( ! [X2: A] :
% 0.18/0.44 ( ( P3 @ X2 )
% 0.18/0.44 = ( Q2 @ X2 ) )
% 0.18/0.44 => ( ( collect @ A @ P3 )
% 0.18/0.44 = ( collect @ A @ Q2 ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % Collect_cong
% 0.18/0.44 thf(fact_47_ext,axiom,
% 0.18/0.44 ! [B2: $tType,A: $tType,F: A > B2,G: A > B2] :
% 0.18/0.44 ( ! [X2: A] :
% 0.18/0.44 ( ( F @ X2 )
% 0.18/0.44 = ( G @ X2 ) )
% 0.18/0.44 => ( F = G ) ) ).
% 0.18/0.44
% 0.18/0.44 % ext
% 0.18/0.44 thf(fact_48_pinf_I8_J,axiom,
% 0.18/0.44 ! [A: $tType] :
% 0.18/0.44 ( ( linorder @ A )
% 0.18/0.44 => ! [T: A] :
% 0.18/0.44 ? [Z3: A] :
% 0.18/0.44 ! [X4: A] :
% 0.18/0.44 ( ( ord_less @ A @ Z3 @ X4 )
% 0.18/0.44 => ( ord_less_eq @ A @ T @ X4 ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % pinf(8)
% 0.18/0.44 thf(fact_49_pinf_I6_J,axiom,
% 0.18/0.44 ! [A: $tType] :
% 0.18/0.44 ( ( linorder @ A )
% 0.18/0.44 => ! [T: A] :
% 0.18/0.44 ? [Z3: A] :
% 0.18/0.44 ! [X4: A] :
% 0.18/0.44 ( ( ord_less @ A @ Z3 @ X4 )
% 0.18/0.44 => ~ ( ord_less_eq @ A @ X4 @ T ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % pinf(6)
% 0.18/0.44 thf(fact_50_verit__comp__simplify1_I3_J,axiom,
% 0.18/0.44 ! [B2: $tType] :
% 0.18/0.44 ( ( linorder @ B2 )
% 0.18/0.44 => ! [B3: B2,A5: B2] :
% 0.18/0.44 ( ( ~ ( ord_less_eq @ B2 @ B3 @ A5 ) )
% 0.18/0.44 = ( ord_less @ B2 @ A5 @ B3 ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % verit_comp_simplify1(3)
% 0.18/0.44 thf(fact_51_bot__nat__0_Oextremum,axiom,
% 0.18/0.44 ! [A2: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ A2 ) ).
% 0.18/0.44
% 0.18/0.44 % bot_nat_0.extremum
% 0.18/0.44 thf(fact_52_le0,axiom,
% 0.18/0.44 ! [N: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ N ) ).
% 0.18/0.44
% 0.18/0.44 % le0
% 0.18/0.44 thf(fact_53_nat__less__le,axiom,
% 0.18/0.44 ( ( ord_less @ nat )
% 0.18/0.44 = ( ^ [M2: nat,N2: nat] :
% 0.18/0.44 ( ( ord_less_eq @ nat @ M2 @ N2 )
% 0.18/0.44 & ( M2 != N2 ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % nat_less_le
% 0.18/0.44 thf(fact_54_less__imp__le__nat,axiom,
% 0.18/0.44 ! [M: nat,N: nat] :
% 0.18/0.44 ( ( ord_less @ nat @ M @ N )
% 0.18/0.44 => ( ord_less_eq @ nat @ M @ N ) ) ).
% 0.18/0.44
% 0.18/0.44 % less_imp_le_nat
% 0.18/0.44 thf(fact_55_le__eq__less__or__eq,axiom,
% 0.18/0.44 ( ( ord_less_eq @ nat )
% 0.18/0.44 = ( ^ [M2: nat,N2: nat] :
% 0.18/0.44 ( ( ord_less @ nat @ M2 @ N2 )
% 0.18/0.44 | ( M2 = N2 ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % le_eq_less_or_eq
% 0.18/0.44 thf(fact_56_less__or__eq__imp__le,axiom,
% 0.18/0.44 ! [M: nat,N: nat] :
% 0.18/0.44 ( ( ( ord_less @ nat @ M @ N )
% 0.18/0.44 | ( M = N ) )
% 0.18/0.44 => ( ord_less_eq @ nat @ M @ N ) ) ).
% 0.18/0.44
% 0.18/0.44 % less_or_eq_imp_le
% 0.18/0.44 thf(fact_57_le__neq__implies__less,axiom,
% 0.18/0.44 ! [M: nat,N: nat] :
% 0.18/0.44 ( ( ord_less_eq @ nat @ M @ N )
% 0.18/0.44 => ( ( M != N )
% 0.18/0.44 => ( ord_less @ nat @ M @ N ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % le_neq_implies_less
% 0.18/0.44 thf(fact_58_less__mono__imp__le__mono,axiom,
% 0.18/0.44 ! [F: nat > nat,I: nat,J: nat] :
% 0.18/0.44 ( ! [I2: nat,J2: nat] :
% 0.18/0.44 ( ( ord_less @ nat @ I2 @ J2 )
% 0.18/0.44 => ( ord_less @ nat @ ( F @ I2 ) @ ( F @ J2 ) ) )
% 0.18/0.44 => ( ( ord_less_eq @ nat @ I @ J )
% 0.18/0.44 => ( ord_less_eq @ nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % less_mono_imp_le_mono
% 0.18/0.44 thf(fact_59_infinite__descent__measure,axiom,
% 0.18/0.44 ! [A: $tType,P3: A > $o,V: A > nat,X: A] :
% 0.18/0.44 ( ! [X2: A] :
% 0.18/0.44 ( ~ ( P3 @ X2 )
% 0.18/0.44 => ? [Y3: A] :
% 0.18/0.44 ( ( ord_less @ nat @ ( V @ Y3 ) @ ( V @ X2 ) )
% 0.18/0.44 & ~ ( P3 @ Y3 ) ) )
% 0.18/0.44 => ( P3 @ X ) ) ).
% 0.18/0.44
% 0.18/0.44 % infinite_descent_measure
% 0.18/0.44 thf(fact_60_linorder__neqE__nat,axiom,
% 0.18/0.44 ! [X: nat,Y4: nat] :
% 0.18/0.44 ( ( X != Y4 )
% 0.18/0.44 => ( ~ ( ord_less @ nat @ X @ Y4 )
% 0.18/0.44 => ( ord_less @ nat @ Y4 @ X ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % linorder_neqE_nat
% 0.18/0.44 thf(fact_61_infinite__descent,axiom,
% 0.18/0.44 ! [P3: nat > $o,N: nat] :
% 0.18/0.44 ( ! [N3: nat] :
% 0.18/0.44 ( ~ ( P3 @ N3 )
% 0.18/0.44 => ? [M3: nat] :
% 0.18/0.44 ( ( ord_less @ nat @ M3 @ N3 )
% 0.18/0.44 & ~ ( P3 @ M3 ) ) )
% 0.18/0.44 => ( P3 @ N ) ) ).
% 0.18/0.44
% 0.18/0.44 % infinite_descent
% 0.18/0.44 thf(fact_62_nat__less__induct,axiom,
% 0.18/0.44 ! [P3: nat > $o,N: nat] :
% 0.18/0.44 ( ! [N3: nat] :
% 0.18/0.44 ( ! [M3: nat] :
% 0.18/0.44 ( ( ord_less @ nat @ M3 @ N3 )
% 0.18/0.44 => ( P3 @ M3 ) )
% 0.18/0.44 => ( P3 @ N3 ) )
% 0.18/0.44 => ( P3 @ N ) ) ).
% 0.18/0.44
% 0.18/0.44 % nat_less_induct
% 0.18/0.44 thf(fact_63_less__irrefl__nat,axiom,
% 0.18/0.44 ! [N: nat] :
% 0.18/0.44 ~ ( ord_less @ nat @ N @ N ) ).
% 0.18/0.44
% 0.18/0.44 % less_irrefl_nat
% 0.18/0.44 thf(fact_64_less__not__refl3,axiom,
% 0.18/0.44 ! [S2: nat,T: nat] :
% 0.18/0.44 ( ( ord_less @ nat @ S2 @ T )
% 0.18/0.44 => ( S2 != T ) ) ).
% 0.18/0.44
% 0.18/0.44 % less_not_refl3
% 0.18/0.44 thf(fact_65_less__not__refl2,axiom,
% 0.18/0.44 ! [N: nat,M: nat] :
% 0.18/0.44 ( ( ord_less @ nat @ N @ M )
% 0.18/0.44 => ( M != N ) ) ).
% 0.18/0.44
% 0.18/0.44 % less_not_refl2
% 0.18/0.44 thf(fact_66_less__not__refl,axiom,
% 0.18/0.44 ! [N: nat] :
% 0.18/0.44 ~ ( ord_less @ nat @ N @ N ) ).
% 0.18/0.44
% 0.18/0.44 % less_not_refl
% 0.18/0.44 thf(fact_67_nat__neq__iff,axiom,
% 0.18/0.44 ! [M: nat,N: nat] :
% 0.18/0.44 ( ( M != N )
% 0.18/0.44 = ( ( ord_less @ nat @ M @ N )
% 0.18/0.44 | ( ord_less @ nat @ N @ M ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % nat_neq_iff
% 0.18/0.44 thf(fact_68_bot__nat__0_Oextremum__uniqueI,axiom,
% 0.18/0.44 ! [A2: nat] :
% 0.18/0.44 ( ( ord_less_eq @ nat @ A2 @ ( zero_zero @ nat ) )
% 0.18/0.44 => ( A2
% 0.18/0.44 = ( zero_zero @ nat ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % bot_nat_0.extremum_uniqueI
% 0.18/0.44 thf(fact_69_bot__nat__0_Oextremum__unique,axiom,
% 0.18/0.44 ! [A2: nat] :
% 0.18/0.44 ( ( ord_less_eq @ nat @ A2 @ ( zero_zero @ nat ) )
% 0.18/0.44 = ( A2
% 0.18/0.44 = ( zero_zero @ nat ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % bot_nat_0.extremum_unique
% 0.18/0.44 thf(fact_70_ex__least__nat__le,axiom,
% 0.18/0.44 ! [P3: nat > $o,N: nat] :
% 0.18/0.44 ( ( P3 @ N )
% 0.18/0.44 => ( ~ ( P3 @ ( zero_zero @ nat ) )
% 0.18/0.44 => ? [K: nat] :
% 0.18/0.44 ( ( ord_less_eq @ nat @ K @ N )
% 0.18/0.44 & ! [I3: nat] :
% 0.18/0.44 ( ( ord_less @ nat @ I3 @ K )
% 0.18/0.44 => ~ ( P3 @ I3 ) )
% 0.18/0.44 & ( P3 @ K ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % ex_least_nat_le
% 0.18/0.44 thf(fact_71_le__0__eq,axiom,
% 0.18/0.44 ! [N: nat] :
% 0.18/0.44 ( ( ord_less_eq @ nat @ N @ ( zero_zero @ nat ) )
% 0.18/0.44 = ( N
% 0.18/0.44 = ( zero_zero @ nat ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % le_0_eq
% 0.18/0.44 thf(fact_72_less__eq__nat_Osimps_I1_J,axiom,
% 0.18/0.44 ! [N: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ N ) ).
% 0.18/0.44
% 0.18/0.44 % less_eq_nat.simps(1)
% 0.18/0.44 thf(fact_73_conj__le__cong,axiom,
% 0.18/0.44 ! [X: int,X5: int,P3: $o,P4: $o] :
% 0.18/0.44 ( ( X = X5 )
% 0.18/0.44 => ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X5 )
% 0.18/0.44 => ( P3 = P4 ) )
% 0.18/0.44 => ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X )
% 0.18/0.44 & P3 )
% 0.18/0.44 = ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X5 )
% 0.18/0.44 & P4 ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % conj_le_cong
% 0.18/0.44 thf(fact_74_imp__le__cong,axiom,
% 0.18/0.44 ! [X: int,X5: int,P3: $o,P4: $o] :
% 0.18/0.44 ( ( X = X5 )
% 0.18/0.44 => ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X5 )
% 0.18/0.44 => ( P3 = P4 ) )
% 0.18/0.44 => ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X )
% 0.18/0.44 => P3 )
% 0.18/0.44 = ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X5 )
% 0.18/0.44 => P4 ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % imp_le_cong
% 0.18/0.44 thf(fact_75_le__funD,axiom,
% 0.18/0.44 ! [B2: $tType,A: $tType] :
% 0.18/0.44 ( ( ord @ B2 )
% 0.18/0.44 => ! [F: A > B2,G: A > B2,X: A] :
% 0.18/0.44 ( ( ord_less_eq @ ( A > B2 ) @ F @ G )
% 0.18/0.44 => ( ord_less_eq @ B2 @ ( F @ X ) @ ( G @ X ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % le_funD
% 0.18/0.44 thf(fact_76_le__funE,axiom,
% 0.18/0.44 ! [B2: $tType,A: $tType] :
% 0.18/0.44 ( ( ord @ B2 )
% 0.18/0.44 => ! [F: A > B2,G: A > B2,X: A] :
% 0.18/0.44 ( ( ord_less_eq @ ( A > B2 ) @ F @ G )
% 0.18/0.44 => ( ord_less_eq @ B2 @ ( F @ X ) @ ( G @ X ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % le_funE
% 0.18/0.44 thf(fact_77_le__funI,axiom,
% 0.18/0.44 ! [B2: $tType,A: $tType] :
% 0.18/0.44 ( ( ord @ B2 )
% 0.18/0.44 => ! [F: A > B2,G: A > B2] :
% 0.18/0.44 ( ! [X2: A] : ( ord_less_eq @ B2 @ ( F @ X2 ) @ ( G @ X2 ) )
% 0.18/0.44 => ( ord_less_eq @ ( A > B2 ) @ F @ G ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % le_funI
% 0.18/0.44 thf(fact_78_le__fun__def,axiom,
% 0.18/0.44 ! [B2: $tType,A: $tType] :
% 0.18/0.44 ( ( ord @ B2 )
% 0.18/0.44 => ( ( ord_less_eq @ ( A > B2 ) )
% 0.18/0.44 = ( ^ [F2: A > B2,G2: A > B2] :
% 0.18/0.44 ! [X3: A] : ( ord_less_eq @ B2 @ ( F2 @ X3 ) @ ( G2 @ X3 ) ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % le_fun_def
% 0.18/0.44 thf(fact_79_order__subst1,axiom,
% 0.18/0.44 ! [A: $tType,B2: $tType] :
% 0.18/0.44 ( ( ( order @ B2 )
% 0.18/0.44 & ( order @ A ) )
% 0.18/0.44 => ! [A2: A,F: B2 > A,B: B2,C: B2] :
% 0.18/0.44 ( ( ord_less_eq @ A @ A2 @ ( F @ B ) )
% 0.18/0.44 => ( ( ord_less_eq @ B2 @ B @ C )
% 0.18/0.44 => ( ! [X2: B2,Y: B2] :
% 0.18/0.44 ( ( ord_less_eq @ B2 @ X2 @ Y )
% 0.18/0.44 => ( ord_less_eq @ A @ ( F @ X2 ) @ ( F @ Y ) ) )
% 0.18/0.44 => ( ord_less_eq @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % order_subst1
% 0.18/0.44 thf(fact_80_order__subst2,axiom,
% 0.18/0.44 ! [A: $tType,C2: $tType] :
% 0.18/0.44 ( ( ( order @ C2 )
% 0.18/0.44 & ( order @ A ) )
% 0.18/0.44 => ! [A2: A,B: A,F: A > C2,C: C2] :
% 0.18/0.44 ( ( ord_less_eq @ A @ A2 @ B )
% 0.18/0.44 => ( ( ord_less_eq @ C2 @ ( F @ B ) @ C )
% 0.18/0.44 => ( ! [X2: A,Y: A] :
% 0.18/0.44 ( ( ord_less_eq @ A @ X2 @ Y )
% 0.18/0.44 => ( ord_less_eq @ C2 @ ( F @ X2 ) @ ( F @ Y ) ) )
% 0.18/0.44 => ( ord_less_eq @ C2 @ ( F @ A2 ) @ C ) ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % order_subst2
% 0.18/0.44 thf(fact_81_verit__la__disequality,axiom,
% 0.18/0.44 ! [A: $tType] :
% 0.18/0.44 ( ( linorder @ A )
% 0.18/0.44 => ! [A2: A,B: A] :
% 0.18/0.44 ( ( A2 = B )
% 0.18/0.44 | ~ ( ord_less_eq @ A @ A2 @ B )
% 0.18/0.44 | ~ ( ord_less_eq @ A @ B @ A2 ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % verit_la_disequality
% 0.18/0.44 thf(fact_82_ord__eq__le__subst,axiom,
% 0.18/0.44 ! [A: $tType,B2: $tType] :
% 0.18/0.44 ( ( ( ord @ B2 )
% 0.18/0.44 & ( ord @ A ) )
% 0.18/0.44 => ! [A2: A,F: B2 > A,B: B2,C: B2] :
% 0.18/0.44 ( ( A2
% 0.18/0.44 = ( F @ B ) )
% 0.18/0.44 => ( ( ord_less_eq @ B2 @ B @ C )
% 0.18/0.44 => ( ! [X2: B2,Y: B2] :
% 0.18/0.44 ( ( ord_less_eq @ B2 @ X2 @ Y )
% 0.18/0.44 => ( ord_less_eq @ A @ ( F @ X2 ) @ ( F @ Y ) ) )
% 0.18/0.44 => ( ord_less_eq @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % ord_eq_le_subst
% 0.18/0.44 thf(fact_83_ord__le__eq__subst,axiom,
% 0.18/0.44 ! [A: $tType,B2: $tType] :
% 0.18/0.44 ( ( ( ord @ B2 )
% 0.18/0.44 & ( ord @ A ) )
% 0.18/0.44 => ! [A2: A,B: A,F: A > B2,C: B2] :
% 0.18/0.44 ( ( ord_less_eq @ A @ A2 @ B )
% 0.18/0.44 => ( ( ( F @ B )
% 0.18/0.44 = C )
% 0.18/0.44 => ( ! [X2: A,Y: A] :
% 0.18/0.44 ( ( ord_less_eq @ A @ X2 @ Y )
% 0.18/0.44 => ( ord_less_eq @ B2 @ ( F @ X2 ) @ ( F @ Y ) ) )
% 0.18/0.44 => ( ord_less_eq @ B2 @ ( F @ A2 ) @ C ) ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % ord_le_eq_subst
% 0.18/0.44 thf(fact_84_eq__iff,axiom,
% 0.18/0.44 ! [A: $tType] :
% 0.18/0.44 ( ( order @ A )
% 0.18/0.44 => ( ( ^ [Y5: A,Z4: A] : ( Y5 = Z4 ) )
% 0.18/0.44 = ( ^ [X3: A,Y2: A] :
% 0.18/0.44 ( ( ord_less_eq @ A @ X3 @ Y2 )
% 0.18/0.44 & ( ord_less_eq @ A @ Y2 @ X3 ) ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % eq_iff
% 0.18/0.44 thf(fact_85_antisym,axiom,
% 0.18/0.44 ! [A: $tType] :
% 0.18/0.44 ( ( order @ A )
% 0.18/0.44 => ! [X: A,Y4: A] :
% 0.18/0.44 ( ( ord_less_eq @ A @ X @ Y4 )
% 0.18/0.44 => ( ( ord_less_eq @ A @ Y4 @ X )
% 0.18/0.44 => ( X = Y4 ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % antisym
% 0.18/0.44 thf(fact_86_linear,axiom,
% 0.18/0.44 ! [A: $tType] :
% 0.18/0.44 ( ( linorder @ A )
% 0.18/0.44 => ! [X: A,Y4: A] :
% 0.18/0.44 ( ( ord_less_eq @ A @ X @ Y4 )
% 0.18/0.44 | ( ord_less_eq @ A @ Y4 @ X ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % linear
% 0.18/0.44 thf(fact_87_eq__refl,axiom,
% 0.18/0.44 ! [A: $tType] :
% 0.18/0.44 ( ( preorder @ A )
% 0.18/0.44 => ! [X: A,Y4: A] :
% 0.18/0.44 ( ( X = Y4 )
% 0.18/0.44 => ( ord_less_eq @ A @ X @ Y4 ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % eq_refl
% 0.18/0.44 thf(fact_88_le__cases,axiom,
% 0.18/0.44 ! [A: $tType] :
% 0.18/0.44 ( ( linorder @ A )
% 0.18/0.44 => ! [X: A,Y4: A] :
% 0.18/0.44 ( ~ ( ord_less_eq @ A @ X @ Y4 )
% 0.18/0.44 => ( ord_less_eq @ A @ Y4 @ X ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % le_cases
% 0.18/0.44 thf(fact_89_order_Otrans,axiom,
% 0.18/0.44 ! [A: $tType] :
% 0.18/0.44 ( ( order @ A )
% 0.18/0.44 => ! [A2: A,B: A,C: A] :
% 0.18/0.44 ( ( ord_less_eq @ A @ A2 @ B )
% 0.18/0.44 => ( ( ord_less_eq @ A @ B @ C )
% 0.18/0.44 => ( ord_less_eq @ A @ A2 @ C ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % order.trans
% 0.18/0.44 thf(fact_90_le__cases3,axiom,
% 0.18/0.44 ! [A: $tType] :
% 0.18/0.44 ( ( linorder @ A )
% 0.18/0.44 => ! [X: A,Y4: A,Z5: A] :
% 0.18/0.44 ( ( ( ord_less_eq @ A @ X @ Y4 )
% 0.18/0.44 => ~ ( ord_less_eq @ A @ Y4 @ Z5 ) )
% 0.18/0.44 => ( ( ( ord_less_eq @ A @ Y4 @ X )
% 0.18/0.44 => ~ ( ord_less_eq @ A @ X @ Z5 ) )
% 0.18/0.44 => ( ( ( ord_less_eq @ A @ X @ Z5 )
% 0.18/0.44 => ~ ( ord_less_eq @ A @ Z5 @ Y4 ) )
% 0.18/0.44 => ( ( ( ord_less_eq @ A @ Z5 @ Y4 )
% 0.18/0.44 => ~ ( ord_less_eq @ A @ Y4 @ X ) )
% 0.18/0.44 => ( ( ( ord_less_eq @ A @ Y4 @ Z5 )
% 0.18/0.44 => ~ ( ord_less_eq @ A @ Z5 @ X ) )
% 0.18/0.44 => ~ ( ( ord_less_eq @ A @ Z5 @ X )
% 0.18/0.44 => ~ ( ord_less_eq @ A @ X @ Y4 ) ) ) ) ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % le_cases3
% 0.18/0.44 thf(fact_91_antisym__conv,axiom,
% 0.18/0.44 ! [A: $tType] :
% 0.18/0.44 ( ( order @ A )
% 0.18/0.44 => ! [Y4: A,X: A] :
% 0.18/0.44 ( ( ord_less_eq @ A @ Y4 @ X )
% 0.18/0.44 => ( ( ord_less_eq @ A @ X @ Y4 )
% 0.18/0.44 = ( X = Y4 ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % antisym_conv
% 0.18/0.44 thf(fact_92_order__class_Oorder_Oeq__iff,axiom,
% 0.18/0.44 ! [A: $tType] :
% 0.18/0.44 ( ( order @ A )
% 0.18/0.44 => ( ( ^ [Y5: A,Z4: A] : ( Y5 = Z4 ) )
% 0.18/0.44 = ( ^ [A3: A,B4: A] :
% 0.18/0.44 ( ( ord_less_eq @ A @ A3 @ B4 )
% 0.18/0.44 & ( ord_less_eq @ A @ B4 @ A3 ) ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % order_class.order.eq_iff
% 0.18/0.44 thf(fact_93_ord__eq__le__trans,axiom,
% 0.18/0.44 ! [A: $tType] :
% 0.18/0.44 ( ( ord @ A )
% 0.18/0.44 => ! [A2: A,B: A,C: A] :
% 0.18/0.44 ( ( A2 = B )
% 0.18/0.44 => ( ( ord_less_eq @ A @ B @ C )
% 0.18/0.44 => ( ord_less_eq @ A @ A2 @ C ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % ord_eq_le_trans
% 0.18/0.44 thf(fact_94_ord__le__eq__trans,axiom,
% 0.18/0.44 ! [A: $tType] :
% 0.18/0.44 ( ( ord @ A )
% 0.18/0.44 => ! [A2: A,B: A,C: A] :
% 0.18/0.44 ( ( ord_less_eq @ A @ A2 @ B )
% 0.18/0.44 => ( ( B = C )
% 0.18/0.44 => ( ord_less_eq @ A @ A2 @ C ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % ord_le_eq_trans
% 0.18/0.44 thf(fact_95_order__class_Oorder_Oantisym,axiom,
% 0.18/0.44 ! [A: $tType] :
% 0.18/0.44 ( ( order @ A )
% 0.18/0.44 => ! [A2: A,B: A] :
% 0.18/0.44 ( ( ord_less_eq @ A @ A2 @ B )
% 0.18/0.44 => ( ( ord_less_eq @ A @ B @ A2 )
% 0.18/0.44 => ( A2 = B ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % order_class.order.antisym
% 0.18/0.44 thf(fact_96_order__trans,axiom,
% 0.18/0.44 ! [A: $tType] :
% 0.18/0.44 ( ( preorder @ A )
% 0.18/0.44 => ! [X: A,Y4: A,Z5: A] :
% 0.18/0.44 ( ( ord_less_eq @ A @ X @ Y4 )
% 0.18/0.44 => ( ( ord_less_eq @ A @ Y4 @ Z5 )
% 0.18/0.44 => ( ord_less_eq @ A @ X @ Z5 ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % order_trans
% 0.18/0.44 thf(fact_97_dual__order_Orefl,axiom,
% 0.18/0.44 ! [A: $tType] :
% 0.18/0.44 ( ( order @ A )
% 0.18/0.44 => ! [A2: A] : ( ord_less_eq @ A @ A2 @ A2 ) ) ).
% 0.18/0.44
% 0.18/0.44 % dual_order.refl
% 0.18/0.44 thf(fact_98_linorder__wlog,axiom,
% 0.18/0.44 ! [A: $tType] :
% 0.18/0.44 ( ( linorder @ A )
% 0.18/0.44 => ! [P3: A > A > $o,A2: A,B: A] :
% 0.18/0.44 ( ! [A6: A,B5: A] :
% 0.18/0.44 ( ( ord_less_eq @ A @ A6 @ B5 )
% 0.18/0.44 => ( P3 @ A6 @ B5 ) )
% 0.18/0.44 => ( ! [A6: A,B5: A] :
% 0.18/0.44 ( ( P3 @ B5 @ A6 )
% 0.18/0.44 => ( P3 @ A6 @ B5 ) )
% 0.18/0.44 => ( P3 @ A2 @ B ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % linorder_wlog
% 0.18/0.44 thf(fact_99_dual__order_Otrans,axiom,
% 0.18/0.44 ! [A: $tType] :
% 0.18/0.44 ( ( order @ A )
% 0.18/0.44 => ! [B: A,A2: A,C: A] :
% 0.18/0.44 ( ( ord_less_eq @ A @ B @ A2 )
% 0.18/0.44 => ( ( ord_less_eq @ A @ C @ B )
% 0.18/0.44 => ( ord_less_eq @ A @ C @ A2 ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % dual_order.trans
% 0.18/0.44 thf(fact_100_dual__order_Oeq__iff,axiom,
% 0.18/0.44 ! [A: $tType] :
% 0.18/0.44 ( ( order @ A )
% 0.18/0.44 => ( ( ^ [Y5: A,Z4: A] : ( Y5 = Z4 ) )
% 0.18/0.44 = ( ^ [A3: A,B4: A] :
% 0.18/0.44 ( ( ord_less_eq @ A @ B4 @ A3 )
% 0.18/0.44 & ( ord_less_eq @ A @ A3 @ B4 ) ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % dual_order.eq_iff
% 0.18/0.44 thf(fact_101_dual__order_Oantisym,axiom,
% 0.18/0.44 ! [A: $tType] :
% 0.18/0.44 ( ( order @ A )
% 0.18/0.44 => ! [B: A,A2: A] :
% 0.18/0.44 ( ( ord_less_eq @ A @ B @ A2 )
% 0.18/0.44 => ( ( ord_less_eq @ A @ A2 @ B )
% 0.18/0.44 => ( A2 = B ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % dual_order.antisym
% 0.18/0.44 thf(fact_102_dual__order_Ostrict__implies__not__eq,axiom,
% 0.18/0.44 ! [A: $tType] :
% 0.18/0.44 ( ( order @ A )
% 0.18/0.44 => ! [B: A,A2: A] :
% 0.18/0.44 ( ( ord_less @ A @ B @ A2 )
% 0.18/0.44 => ( A2 != B ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % dual_order.strict_implies_not_eq
% 0.18/0.44 thf(fact_103_order_Ostrict__implies__not__eq,axiom,
% 0.18/0.44 ! [A: $tType] :
% 0.18/0.44 ( ( order @ A )
% 0.18/0.44 => ! [A2: A,B: A] :
% 0.18/0.44 ( ( ord_less @ A @ A2 @ B )
% 0.18/0.44 => ( A2 != B ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % order.strict_implies_not_eq
% 0.18/0.44 thf(fact_104_not__less__iff__gr__or__eq,axiom,
% 0.18/0.44 ! [A: $tType] :
% 0.18/0.44 ( ( linorder @ A )
% 0.18/0.44 => ! [X: A,Y4: A] :
% 0.18/0.44 ( ( ~ ( ord_less @ A @ X @ Y4 ) )
% 0.18/0.44 = ( ( ord_less @ A @ Y4 @ X )
% 0.18/0.44 | ( X = Y4 ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % not_less_iff_gr_or_eq
% 0.18/0.44 thf(fact_105_dual__order_Ostrict__trans,axiom,
% 0.18/0.44 ! [A: $tType] :
% 0.18/0.44 ( ( order @ A )
% 0.18/0.44 => ! [B: A,A2: A,C: A] :
% 0.18/0.44 ( ( ord_less @ A @ B @ A2 )
% 0.18/0.44 => ( ( ord_less @ A @ C @ B )
% 0.18/0.44 => ( ord_less @ A @ C @ A2 ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % dual_order.strict_trans
% 0.18/0.44 thf(fact_106_linorder__less__wlog,axiom,
% 0.18/0.44 ! [A: $tType] :
% 0.18/0.44 ( ( linorder @ A )
% 0.18/0.44 => ! [P3: A > A > $o,A2: A,B: A] :
% 0.18/0.44 ( ! [A6: A,B5: A] :
% 0.18/0.44 ( ( ord_less @ A @ A6 @ B5 )
% 0.18/0.44 => ( P3 @ A6 @ B5 ) )
% 0.18/0.44 => ( ! [A6: A] : ( P3 @ A6 @ A6 )
% 0.18/0.44 => ( ! [A6: A,B5: A] :
% 0.18/0.44 ( ( P3 @ B5 @ A6 )
% 0.18/0.44 => ( P3 @ A6 @ B5 ) )
% 0.18/0.44 => ( P3 @ A2 @ B ) ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % linorder_less_wlog
% 0.18/0.44 thf(fact_107_exists__least__iff,axiom,
% 0.18/0.44 ! [A: $tType] :
% 0.18/0.44 ( ( wellorder @ A )
% 0.18/0.44 => ( ( ^ [P5: A > $o] :
% 0.18/0.44 ? [X6: A] : ( P5 @ X6 ) )
% 0.18/0.44 = ( ^ [P6: A > $o] :
% 0.18/0.44 ? [N2: A] :
% 0.18/0.44 ( ( P6 @ N2 )
% 0.18/0.44 & ! [M2: A] :
% 0.18/0.44 ( ( ord_less @ A @ M2 @ N2 )
% 0.18/0.44 => ~ ( P6 @ M2 ) ) ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % exists_least_iff
% 0.18/0.44 thf(fact_108_less__imp__not__less,axiom,
% 0.18/0.44 ! [A: $tType] :
% 0.18/0.44 ( ( preorder @ A )
% 0.18/0.44 => ! [X: A,Y4: A] :
% 0.18/0.44 ( ( ord_less @ A @ X @ Y4 )
% 0.18/0.44 => ~ ( ord_less @ A @ Y4 @ X ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % less_imp_not_less
% 0.18/0.44 thf(fact_109_order_Ostrict__trans,axiom,
% 0.18/0.44 ! [A: $tType] :
% 0.18/0.44 ( ( order @ A )
% 0.18/0.44 => ! [A2: A,B: A,C: A] :
% 0.18/0.44 ( ( ord_less @ A @ A2 @ B )
% 0.18/0.44 => ( ( ord_less @ A @ B @ C )
% 0.18/0.44 => ( ord_less @ A @ A2 @ C ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % order.strict_trans
% 0.18/0.44 thf(fact_110_dual__order_Oirrefl,axiom,
% 0.18/0.44 ! [A: $tType] :
% 0.18/0.44 ( ( order @ A )
% 0.18/0.44 => ! [A2: A] :
% 0.18/0.44 ~ ( ord_less @ A @ A2 @ A2 ) ) ).
% 0.18/0.44
% 0.18/0.44 % dual_order.irrefl
% 0.18/0.44 thf(fact_111_linorder__cases,axiom,
% 0.18/0.44 ! [A: $tType] :
% 0.18/0.44 ( ( linorder @ A )
% 0.18/0.44 => ! [X: A,Y4: A] :
% 0.18/0.44 ( ~ ( ord_less @ A @ X @ Y4 )
% 0.18/0.44 => ( ( X != Y4 )
% 0.18/0.44 => ( ord_less @ A @ Y4 @ X ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % linorder_cases
% 0.18/0.44 thf(fact_112_less__imp__triv,axiom,
% 0.18/0.44 ! [A: $tType] :
% 0.18/0.44 ( ( preorder @ A )
% 0.18/0.44 => ! [X: A,Y4: A,P3: $o] :
% 0.18/0.44 ( ( ord_less @ A @ X @ Y4 )
% 0.18/0.44 => ( ( ord_less @ A @ Y4 @ X )
% 0.18/0.44 => P3 ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % less_imp_triv
% 0.18/0.44 thf(fact_113_less__imp__not__eq2,axiom,
% 0.18/0.44 ! [A: $tType] :
% 0.18/0.44 ( ( order @ A )
% 0.18/0.44 => ! [X: A,Y4: A] :
% 0.18/0.44 ( ( ord_less @ A @ X @ Y4 )
% 0.18/0.44 => ( Y4 != X ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % less_imp_not_eq2
% 0.18/0.44 thf(fact_114_antisym__conv3,axiom,
% 0.18/0.44 ! [A: $tType] :
% 0.18/0.44 ( ( linorder @ A )
% 0.18/0.44 => ! [Y4: A,X: A] :
% 0.18/0.44 ( ~ ( ord_less @ A @ Y4 @ X )
% 0.18/0.44 => ( ( ~ ( ord_less @ A @ X @ Y4 ) )
% 0.18/0.44 = ( X = Y4 ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % antisym_conv3
% 0.18/0.44 thf(fact_115_less__induct,axiom,
% 0.18/0.44 ! [A: $tType] :
% 0.18/0.44 ( ( wellorder @ A )
% 0.18/0.44 => ! [P3: A > $o,A2: A] :
% 0.18/0.44 ( ! [X2: A] :
% 0.18/0.44 ( ! [Y3: A] :
% 0.18/0.44 ( ( ord_less @ A @ Y3 @ X2 )
% 0.18/0.44 => ( P3 @ Y3 ) )
% 0.18/0.44 => ( P3 @ X2 ) )
% 0.18/0.44 => ( P3 @ A2 ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % less_induct
% 0.18/0.44 thf(fact_116_less__not__sym,axiom,
% 0.18/0.44 ! [A: $tType] :
% 0.18/0.44 ( ( preorder @ A )
% 0.18/0.44 => ! [X: A,Y4: A] :
% 0.18/0.44 ( ( ord_less @ A @ X @ Y4 )
% 0.18/0.44 => ~ ( ord_less @ A @ Y4 @ X ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % less_not_sym
% 0.18/0.44 thf(fact_117_less__imp__not__eq,axiom,
% 0.18/0.44 ! [A: $tType] :
% 0.18/0.44 ( ( order @ A )
% 0.18/0.44 => ! [X: A,Y4: A] :
% 0.18/0.44 ( ( ord_less @ A @ X @ Y4 )
% 0.18/0.44 => ( X != Y4 ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % less_imp_not_eq
% 0.18/0.44 thf(fact_118_dual__order_Oasym,axiom,
% 0.18/0.44 ! [A: $tType] :
% 0.18/0.44 ( ( order @ A )
% 0.18/0.44 => ! [B: A,A2: A] :
% 0.18/0.44 ( ( ord_less @ A @ B @ A2 )
% 0.18/0.44 => ~ ( ord_less @ A @ A2 @ B ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % dual_order.asym
% 0.18/0.44 thf(fact_119_ord__less__eq__trans,axiom,
% 0.18/0.44 ! [A: $tType] :
% 0.18/0.44 ( ( ord @ A )
% 0.18/0.44 => ! [A2: A,B: A,C: A] :
% 0.18/0.44 ( ( ord_less @ A @ A2 @ B )
% 0.18/0.44 => ( ( B = C )
% 0.18/0.44 => ( ord_less @ A @ A2 @ C ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % ord_less_eq_trans
% 0.18/0.44 thf(fact_120_ord__eq__less__trans,axiom,
% 0.18/0.44 ! [A: $tType] :
% 0.18/0.44 ( ( ord @ A )
% 0.18/0.44 => ! [A2: A,B: A,C: A] :
% 0.18/0.44 ( ( A2 = B )
% 0.18/0.44 => ( ( ord_less @ A @ B @ C )
% 0.18/0.44 => ( ord_less @ A @ A2 @ C ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % ord_eq_less_trans
% 0.18/0.44 thf(fact_121_less__irrefl,axiom,
% 0.18/0.44 ! [A: $tType] :
% 0.18/0.44 ( ( preorder @ A )
% 0.18/0.44 => ! [X: A] :
% 0.18/0.44 ~ ( ord_less @ A @ X @ X ) ) ).
% 0.18/0.44
% 0.18/0.44 % less_irrefl
% 0.18/0.44 thf(fact_122_less__linear,axiom,
% 0.18/0.44 ! [A: $tType] :
% 0.18/0.44 ( ( linorder @ A )
% 0.18/0.44 => ! [X: A,Y4: A] :
% 0.18/0.44 ( ( ord_less @ A @ X @ Y4 )
% 0.18/0.44 | ( X = Y4 )
% 0.18/0.44 | ( ord_less @ A @ Y4 @ X ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % less_linear
% 0.18/0.44 thf(fact_123_less__trans,axiom,
% 0.18/0.44 ! [A: $tType] :
% 0.18/0.44 ( ( preorder @ A )
% 0.18/0.44 => ! [X: A,Y4: A,Z5: A] :
% 0.18/0.44 ( ( ord_less @ A @ X @ Y4 )
% 0.18/0.44 => ( ( ord_less @ A @ Y4 @ Z5 )
% 0.18/0.44 => ( ord_less @ A @ X @ Z5 ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % less_trans
% 0.18/0.44 thf(fact_124_less__asym_H,axiom,
% 0.18/0.44 ! [A: $tType] :
% 0.18/0.44 ( ( preorder @ A )
% 0.18/0.44 => ! [A2: A,B: A] :
% 0.18/0.44 ( ( ord_less @ A @ A2 @ B )
% 0.18/0.44 => ~ ( ord_less @ A @ B @ A2 ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % less_asym'
% 0.18/0.44 thf(fact_125_less__asym,axiom,
% 0.18/0.44 ! [A: $tType] :
% 0.18/0.44 ( ( preorder @ A )
% 0.18/0.44 => ! [X: A,Y4: A] :
% 0.18/0.44 ( ( ord_less @ A @ X @ Y4 )
% 0.18/0.44 => ~ ( ord_less @ A @ Y4 @ X ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % less_asym
% 0.18/0.44 thf(fact_126_less__imp__neq,axiom,
% 0.18/0.44 ! [A: $tType] :
% 0.18/0.44 ( ( order @ A )
% 0.18/0.44 => ! [X: A,Y4: A] :
% 0.18/0.44 ( ( ord_less @ A @ X @ Y4 )
% 0.18/0.44 => ( X != Y4 ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % less_imp_neq
% 0.18/0.44 thf(fact_127_dense,axiom,
% 0.18/0.44 ! [A: $tType] :
% 0.18/0.44 ( ( dense_order @ A )
% 0.18/0.44 => ! [X: A,Y4: A] :
% 0.18/0.44 ( ( ord_less @ A @ X @ Y4 )
% 0.18/0.44 => ? [Z3: A] :
% 0.18/0.44 ( ( ord_less @ A @ X @ Z3 )
% 0.18/0.44 & ( ord_less @ A @ Z3 @ Y4 ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % dense
% 0.18/0.44 thf(fact_128_order_Oasym,axiom,
% 0.18/0.44 ! [A: $tType] :
% 0.18/0.44 ( ( order @ A )
% 0.18/0.44 => ! [A2: A,B: A] :
% 0.18/0.44 ( ( ord_less @ A @ A2 @ B )
% 0.18/0.44 => ~ ( ord_less @ A @ B @ A2 ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % order.asym
% 0.18/0.44 thf(fact_129_neq__iff,axiom,
% 0.18/0.44 ! [A: $tType] :
% 0.18/0.44 ( ( linorder @ A )
% 0.18/0.44 => ! [X: A,Y4: A] :
% 0.18/0.44 ( ( X != Y4 )
% 0.18/0.44 = ( ( ord_less @ A @ X @ Y4 )
% 0.18/0.44 | ( ord_less @ A @ Y4 @ X ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % neq_iff
% 0.18/0.44 thf(fact_130_neqE,axiom,
% 0.18/0.44 ! [A: $tType] :
% 0.18/0.44 ( ( linorder @ A )
% 0.18/0.44 => ! [X: A,Y4: A] :
% 0.18/0.44 ( ( X != Y4 )
% 0.18/0.44 => ( ~ ( ord_less @ A @ X @ Y4 )
% 0.18/0.44 => ( ord_less @ A @ Y4 @ X ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % neqE
% 0.18/0.44 thf(fact_131_gt__ex,axiom,
% 0.18/0.44 ! [A: $tType] :
% 0.18/0.44 ( ( no_top @ A )
% 0.18/0.44 => ! [X: A] :
% 0.18/0.44 ? [X_1: A] : ( ord_less @ A @ X @ X_1 ) ) ).
% 0.18/0.44
% 0.18/0.44 % gt_ex
% 0.18/0.44 thf(fact_132_lt__ex,axiom,
% 0.18/0.44 ! [A: $tType] :
% 0.18/0.44 ( ( no_bot @ A )
% 0.18/0.44 => ! [X: A] :
% 0.18/0.44 ? [Y: A] : ( ord_less @ A @ Y @ X ) ) ).
% 0.18/0.44
% 0.18/0.44 % lt_ex
% 0.18/0.44 thf(fact_133_order__less__subst2,axiom,
% 0.18/0.44 ! [A: $tType,C2: $tType] :
% 0.18/0.44 ( ( ( order @ C2 )
% 0.18/0.44 & ( order @ A ) )
% 0.18/0.44 => ! [A2: A,B: A,F: A > C2,C: C2] :
% 0.18/0.44 ( ( ord_less @ A @ A2 @ B )
% 0.18/0.44 => ( ( ord_less @ C2 @ ( F @ B ) @ C )
% 0.18/0.44 => ( ! [X2: A,Y: A] :
% 0.18/0.44 ( ( ord_less @ A @ X2 @ Y )
% 0.18/0.44 => ( ord_less @ C2 @ ( F @ X2 ) @ ( F @ Y ) ) )
% 0.18/0.44 => ( ord_less @ C2 @ ( F @ A2 ) @ C ) ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % order_less_subst2
% 0.18/0.44 thf(fact_134_order__less__subst1,axiom,
% 0.18/0.44 ! [A: $tType,B2: $tType] :
% 0.18/0.44 ( ( ( order @ B2 )
% 0.18/0.44 & ( order @ A ) )
% 0.18/0.44 => ! [A2: A,F: B2 > A,B: B2,C: B2] :
% 0.18/0.44 ( ( ord_less @ A @ A2 @ ( F @ B ) )
% 0.18/0.44 => ( ( ord_less @ B2 @ B @ C )
% 0.18/0.44 => ( ! [X2: B2,Y: B2] :
% 0.18/0.44 ( ( ord_less @ B2 @ X2 @ Y )
% 0.18/0.44 => ( ord_less @ A @ ( F @ X2 ) @ ( F @ Y ) ) )
% 0.18/0.44 => ( ord_less @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % order_less_subst1
% 0.18/0.44 thf(fact_135_ord__less__eq__subst,axiom,
% 0.18/0.44 ! [A: $tType,B2: $tType] :
% 0.18/0.44 ( ( ( ord @ B2 )
% 0.18/0.44 & ( ord @ A ) )
% 0.18/0.44 => ! [A2: A,B: A,F: A > B2,C: B2] :
% 0.18/0.44 ( ( ord_less @ A @ A2 @ B )
% 0.18/0.44 => ( ( ( F @ B )
% 0.18/0.44 = C )
% 0.18/0.44 => ( ! [X2: A,Y: A] :
% 0.18/0.44 ( ( ord_less @ A @ X2 @ Y )
% 0.18/0.44 => ( ord_less @ B2 @ ( F @ X2 ) @ ( F @ Y ) ) )
% 0.18/0.44 => ( ord_less @ B2 @ ( F @ A2 ) @ C ) ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % ord_less_eq_subst
% 0.18/0.44 thf(fact_136_ord__eq__less__subst,axiom,
% 0.18/0.44 ! [A: $tType,B2: $tType] :
% 0.18/0.44 ( ( ( ord @ B2 )
% 0.18/0.44 & ( ord @ A ) )
% 0.18/0.44 => ! [A2: A,F: B2 > A,B: B2,C: B2] :
% 0.18/0.44 ( ( A2
% 0.18/0.44 = ( F @ B ) )
% 0.18/0.44 => ( ( ord_less @ B2 @ B @ C )
% 0.18/0.44 => ( ! [X2: B2,Y: B2] :
% 0.18/0.44 ( ( ord_less @ B2 @ X2 @ Y )
% 0.18/0.44 => ( ord_less @ A @ ( F @ X2 ) @ ( F @ Y ) ) )
% 0.18/0.44 => ( ord_less @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % ord_eq_less_subst
% 0.18/0.44 thf(fact_137_measure__induct__rule,axiom,
% 0.18/0.44 ! [B2: $tType,A: $tType] :
% 0.18/0.44 ( ( wellorder @ B2 )
% 0.18/0.44 => ! [F: A > B2,P3: A > $o,A2: A] :
% 0.18/0.44 ( ! [X2: A] :
% 0.18/0.44 ( ! [Y3: A] :
% 0.18/0.44 ( ( ord_less @ B2 @ ( F @ Y3 ) @ ( F @ X2 ) )
% 0.18/0.44 => ( P3 @ Y3 ) )
% 0.18/0.44 => ( P3 @ X2 ) )
% 0.18/0.44 => ( P3 @ A2 ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % measure_induct_rule
% 0.18/0.44 thf(fact_138_measure__induct,axiom,
% 0.18/0.44 ! [B2: $tType,A: $tType] :
% 0.18/0.44 ( ( wellorder @ B2 )
% 0.18/0.44 => ! [F: A > B2,P3: A > $o,A2: A] :
% 0.18/0.44 ( ! [X2: A] :
% 0.18/0.44 ( ! [Y3: A] :
% 0.18/0.44 ( ( ord_less @ B2 @ ( F @ Y3 ) @ ( F @ X2 ) )
% 0.18/0.44 => ( P3 @ Y3 ) )
% 0.18/0.44 => ( P3 @ X2 ) )
% 0.18/0.44 => ( P3 @ A2 ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % measure_induct
% 0.18/0.44 thf(fact_139_verit__comp__simplify1_I1_J,axiom,
% 0.18/0.44 ! [A: $tType] :
% 0.18/0.44 ( ( order @ A )
% 0.18/0.44 => ! [A2: A] :
% 0.18/0.44 ~ ( ord_less @ A @ A2 @ A2 ) ) ).
% 0.18/0.44
% 0.18/0.44 % verit_comp_simplify1(1)
% 0.18/0.44 thf(fact_140_pinf_I1_J,axiom,
% 0.18/0.44 ! [A: $tType] :
% 0.18/0.44 ( ( linorder @ A )
% 0.18/0.44 => ! [P3: A > $o,P4: A > $o,Q2: A > $o,Q3: A > $o] :
% 0.18/0.44 ( ? [Z: A] :
% 0.18/0.44 ! [X2: A] :
% 0.18/0.44 ( ( ord_less @ A @ Z @ X2 )
% 0.18/0.44 => ( ( P3 @ X2 )
% 0.18/0.44 = ( P4 @ X2 ) ) )
% 0.18/0.44 => ( ? [Z: A] :
% 0.18/0.44 ! [X2: A] :
% 0.18/0.44 ( ( ord_less @ A @ Z @ X2 )
% 0.18/0.44 => ( ( Q2 @ X2 )
% 0.18/0.44 = ( Q3 @ X2 ) ) )
% 0.18/0.44 => ? [Z3: A] :
% 0.18/0.44 ! [X4: A] :
% 0.18/0.44 ( ( ord_less @ A @ Z3 @ X4 )
% 0.18/0.44 => ( ( ( P3 @ X4 )
% 0.18/0.44 & ( Q2 @ X4 ) )
% 0.18/0.44 = ( ( P4 @ X4 )
% 0.18/0.44 & ( Q3 @ X4 ) ) ) ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % pinf(1)
% 0.18/0.44 thf(fact_141_pinf_I2_J,axiom,
% 0.18/0.44 ! [A: $tType] :
% 0.18/0.44 ( ( linorder @ A )
% 0.18/0.44 => ! [P3: A > $o,P4: A > $o,Q2: A > $o,Q3: A > $o] :
% 0.18/0.44 ( ? [Z: A] :
% 0.18/0.44 ! [X2: A] :
% 0.18/0.44 ( ( ord_less @ A @ Z @ X2 )
% 0.18/0.44 => ( ( P3 @ X2 )
% 0.18/0.44 = ( P4 @ X2 ) ) )
% 0.18/0.44 => ( ? [Z: A] :
% 0.18/0.44 ! [X2: A] :
% 0.18/0.44 ( ( ord_less @ A @ Z @ X2 )
% 0.18/0.44 => ( ( Q2 @ X2 )
% 0.18/0.44 = ( Q3 @ X2 ) ) )
% 0.18/0.44 => ? [Z3: A] :
% 0.18/0.44 ! [X4: A] :
% 0.18/0.44 ( ( ord_less @ A @ Z3 @ X4 )
% 0.18/0.44 => ( ( ( P3 @ X4 )
% 0.18/0.44 | ( Q2 @ X4 ) )
% 0.18/0.44 = ( ( P4 @ X4 )
% 0.18/0.44 | ( Q3 @ X4 ) ) ) ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % pinf(2)
% 0.18/0.44 thf(fact_142_pinf_I3_J,axiom,
% 0.18/0.44 ! [A: $tType] :
% 0.18/0.44 ( ( linorder @ A )
% 0.18/0.44 => ! [T: A] :
% 0.18/0.44 ? [Z3: A] :
% 0.18/0.44 ! [X4: A] :
% 0.18/0.44 ( ( ord_less @ A @ Z3 @ X4 )
% 0.18/0.44 => ( X4 != T ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % pinf(3)
% 0.18/0.44 thf(fact_143_pinf_I4_J,axiom,
% 0.18/0.44 ! [A: $tType] :
% 0.18/0.44 ( ( linorder @ A )
% 0.18/0.44 => ! [T: A] :
% 0.18/0.44 ? [Z3: A] :
% 0.18/0.44 ! [X4: A] :
% 0.18/0.44 ( ( ord_less @ A @ Z3 @ X4 )
% 0.18/0.44 => ( X4 != T ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % pinf(4)
% 0.18/0.44 thf(fact_144_pinf_I5_J,axiom,
% 0.18/0.44 ! [A: $tType] :
% 0.18/0.44 ( ( linorder @ A )
% 0.18/0.44 => ! [T: A] :
% 0.18/0.44 ? [Z3: A] :
% 0.18/0.44 ! [X4: A] :
% 0.18/0.44 ( ( ord_less @ A @ Z3 @ X4 )
% 0.18/0.44 => ~ ( ord_less @ A @ X4 @ T ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % pinf(5)
% 0.18/0.44 thf(fact_145_pinf_I7_J,axiom,
% 0.18/0.44 ! [A: $tType] :
% 0.18/0.44 ( ( linorder @ A )
% 0.18/0.44 => ! [T: A] :
% 0.18/0.44 ? [Z3: A] :
% 0.18/0.44 ! [X4: A] :
% 0.18/0.44 ( ( ord_less @ A @ Z3 @ X4 )
% 0.18/0.44 => ( ord_less @ A @ T @ X4 ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % pinf(7)
% 0.18/0.44 thf(fact_146_pinf_I11_J,axiom,
% 0.18/0.44 ! [C2: $tType,D: $tType] :
% 0.18/0.44 ( ( ord @ C2 )
% 0.18/0.44 => ! [F3: D] :
% 0.18/0.44 ? [Z3: C2] :
% 0.18/0.44 ! [X4: C2] :
% 0.18/0.44 ( ( ord_less @ C2 @ Z3 @ X4 )
% 0.18/0.44 => ( F3 = F3 ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % pinf(11)
% 0.18/0.44 thf(fact_147_minf_I1_J,axiom,
% 0.18/0.44 ! [A: $tType] :
% 0.18/0.44 ( ( linorder @ A )
% 0.18/0.44 => ! [P3: A > $o,P4: A > $o,Q2: A > $o,Q3: A > $o] :
% 0.18/0.44 ( ? [Z: A] :
% 0.18/0.44 ! [X2: A] :
% 0.18/0.44 ( ( ord_less @ A @ X2 @ Z )
% 0.18/0.44 => ( ( P3 @ X2 )
% 0.18/0.44 = ( P4 @ X2 ) ) )
% 0.18/0.44 => ( ? [Z: A] :
% 0.18/0.44 ! [X2: A] :
% 0.18/0.44 ( ( ord_less @ A @ X2 @ Z )
% 0.18/0.44 => ( ( Q2 @ X2 )
% 0.18/0.44 = ( Q3 @ X2 ) ) )
% 0.18/0.44 => ? [Z3: A] :
% 0.18/0.44 ! [X4: A] :
% 0.18/0.44 ( ( ord_less @ A @ X4 @ Z3 )
% 0.18/0.44 => ( ( ( P3 @ X4 )
% 0.18/0.44 & ( Q2 @ X4 ) )
% 0.18/0.44 = ( ( P4 @ X4 )
% 0.18/0.44 & ( Q3 @ X4 ) ) ) ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % minf(1)
% 0.18/0.44 thf(fact_148_minf_I2_J,axiom,
% 0.18/0.44 ! [A: $tType] :
% 0.18/0.44 ( ( linorder @ A )
% 0.18/0.44 => ! [P3: A > $o,P4: A > $o,Q2: A > $o,Q3: A > $o] :
% 0.18/0.44 ( ? [Z: A] :
% 0.18/0.44 ! [X2: A] :
% 0.18/0.44 ( ( ord_less @ A @ X2 @ Z )
% 0.18/0.44 => ( ( P3 @ X2 )
% 0.18/0.44 = ( P4 @ X2 ) ) )
% 0.18/0.44 => ( ? [Z: A] :
% 0.18/0.44 ! [X2: A] :
% 0.18/0.44 ( ( ord_less @ A @ X2 @ Z )
% 0.18/0.44 => ( ( Q2 @ X2 )
% 0.18/0.44 = ( Q3 @ X2 ) ) )
% 0.18/0.44 => ? [Z3: A] :
% 0.18/0.44 ! [X4: A] :
% 0.18/0.44 ( ( ord_less @ A @ X4 @ Z3 )
% 0.18/0.44 => ( ( ( P3 @ X4 )
% 0.18/0.44 | ( Q2 @ X4 ) )
% 0.18/0.44 = ( ( P4 @ X4 )
% 0.18/0.44 | ( Q3 @ X4 ) ) ) ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % minf(2)
% 0.18/0.44 thf(fact_149_minf_I3_J,axiom,
% 0.18/0.44 ! [A: $tType] :
% 0.18/0.44 ( ( linorder @ A )
% 0.18/0.44 => ! [T: A] :
% 0.18/0.44 ? [Z3: A] :
% 0.18/0.44 ! [X4: A] :
% 0.18/0.44 ( ( ord_less @ A @ X4 @ Z3 )
% 0.18/0.44 => ( X4 != T ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % minf(3)
% 0.18/0.44 thf(fact_150_minf_I4_J,axiom,
% 0.18/0.44 ! [A: $tType] :
% 0.18/0.44 ( ( linorder @ A )
% 0.18/0.44 => ! [T: A] :
% 0.18/0.44 ? [Z3: A] :
% 0.18/0.44 ! [X4: A] :
% 0.18/0.44 ( ( ord_less @ A @ X4 @ Z3 )
% 0.18/0.44 => ( X4 != T ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % minf(4)
% 0.18/0.44 thf(fact_151_minf_I5_J,axiom,
% 0.18/0.44 ! [A: $tType] :
% 0.18/0.44 ( ( linorder @ A )
% 0.18/0.44 => ! [T: A] :
% 0.18/0.44 ? [Z3: A] :
% 0.18/0.44 ! [X4: A] :
% 0.18/0.44 ( ( ord_less @ A @ X4 @ Z3 )
% 0.18/0.44 => ( ord_less @ A @ X4 @ T ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % minf(5)
% 0.18/0.44 thf(fact_152_minf_I7_J,axiom,
% 0.18/0.44 ! [A: $tType] :
% 0.18/0.44 ( ( linorder @ A )
% 0.18/0.44 => ! [T: A] :
% 0.18/0.44 ? [Z3: A] :
% 0.18/0.44 ! [X4: A] :
% 0.18/0.44 ( ( ord_less @ A @ X4 @ Z3 )
% 0.18/0.44 => ~ ( ord_less @ A @ T @ X4 ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % minf(7)
% 0.18/0.44 thf(fact_153_minf_I11_J,axiom,
% 0.18/0.44 ! [C2: $tType,D: $tType] :
% 0.18/0.44 ( ( ord @ C2 )
% 0.18/0.44 => ! [F3: D] :
% 0.18/0.44 ? [Z3: C2] :
% 0.18/0.44 ! [X4: C2] :
% 0.18/0.44 ( ( ord_less @ C2 @ X4 @ Z3 )
% 0.18/0.44 => ( F3 = F3 ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % minf(11)
% 0.18/0.44 thf(fact_154_infinite__descent0__measure,axiom,
% 0.18/0.44 ! [A: $tType,V: A > nat,P3: A > $o,X: A] :
% 0.18/0.44 ( ! [X2: A] :
% 0.18/0.44 ( ( ( V @ X2 )
% 0.18/0.44 = ( zero_zero @ nat ) )
% 0.18/0.44 => ( P3 @ X2 ) )
% 0.18/0.44 => ( ! [X2: A] :
% 0.18/0.44 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( V @ X2 ) )
% 0.18/0.44 => ( ~ ( P3 @ X2 )
% 0.18/0.44 => ? [Y3: A] :
% 0.18/0.44 ( ( ord_less @ nat @ ( V @ Y3 ) @ ( V @ X2 ) )
% 0.18/0.44 & ~ ( P3 @ Y3 ) ) ) )
% 0.18/0.44 => ( P3 @ X ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % infinite_descent0_measure
% 0.18/0.44 thf(fact_155_bot__nat__0_Oextremum__strict,axiom,
% 0.18/0.44 ! [A2: nat] :
% 0.18/0.44 ~ ( ord_less @ nat @ A2 @ ( zero_zero @ nat ) ) ).
% 0.18/0.44
% 0.18/0.44 % bot_nat_0.extremum_strict
% 0.18/0.44 thf(fact_156_infinite__descent0,axiom,
% 0.18/0.44 ! [P3: nat > $o,N: nat] :
% 0.18/0.44 ( ( P3 @ ( zero_zero @ nat ) )
% 0.18/0.44 => ( ! [N3: nat] :
% 0.18/0.44 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 )
% 0.18/0.44 => ( ~ ( P3 @ N3 )
% 0.18/0.44 => ? [M3: nat] :
% 0.18/0.44 ( ( ord_less @ nat @ M3 @ N3 )
% 0.18/0.44 & ~ ( P3 @ M3 ) ) ) )
% 0.18/0.44 => ( P3 @ N ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % infinite_descent0
% 0.18/0.44 thf(fact_157_gr__implies__not0,axiom,
% 0.18/0.44 ! [M: nat,N: nat] :
% 0.18/0.44 ( ( ord_less @ nat @ M @ N )
% 0.18/0.44 => ( N
% 0.18/0.44 != ( zero_zero @ nat ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % gr_implies_not0
% 0.18/0.44 thf(fact_158_less__zeroE,axiom,
% 0.18/0.44 ! [N: nat] :
% 0.18/0.44 ~ ( ord_less @ nat @ N @ ( zero_zero @ nat ) ) ).
% 0.18/0.44
% 0.18/0.44 % less_zeroE
% 0.18/0.44 thf(fact_159_not__less0,axiom,
% 0.18/0.44 ! [N: nat] :
% 0.18/0.44 ~ ( ord_less @ nat @ N @ ( zero_zero @ nat ) ) ).
% 0.18/0.44
% 0.18/0.44 % not_less0
% 0.18/0.44 thf(fact_160_not__gr0,axiom,
% 0.18/0.44 ! [N: nat] :
% 0.18/0.44 ( ( ~ ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) )
% 0.18/0.44 = ( N
% 0.18/0.44 = ( zero_zero @ nat ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % not_gr0
% 0.18/0.44 thf(fact_161_gr0I,axiom,
% 0.18/0.44 ! [N: nat] :
% 0.18/0.44 ( ( N
% 0.18/0.44 != ( zero_zero @ nat ) )
% 0.18/0.44 => ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ).
% 0.18/0.44
% 0.18/0.44 % gr0I
% 0.18/0.44 thf(fact_162_order_Onot__eq__order__implies__strict,axiom,
% 0.18/0.44 ! [A: $tType] :
% 0.18/0.44 ( ( order @ A )
% 0.18/0.44 => ! [A2: A,B: A] :
% 0.18/0.44 ( ( A2 != B )
% 0.18/0.44 => ( ( ord_less_eq @ A @ A2 @ B )
% 0.18/0.44 => ( ord_less @ A @ A2 @ B ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % order.not_eq_order_implies_strict
% 0.18/0.44 thf(fact_163_dual__order_Ostrict__implies__order,axiom,
% 0.18/0.44 ! [A: $tType] :
% 0.18/0.44 ( ( order @ A )
% 0.18/0.44 => ! [B: A,A2: A] :
% 0.18/0.44 ( ( ord_less @ A @ B @ A2 )
% 0.18/0.44 => ( ord_less_eq @ A @ B @ A2 ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % dual_order.strict_implies_order
% 0.18/0.44 thf(fact_164_dual__order_Ostrict__iff__order,axiom,
% 0.18/0.44 ! [A: $tType] :
% 0.18/0.44 ( ( order @ A )
% 0.18/0.44 => ( ( ord_less @ A )
% 0.18/0.44 = ( ^ [B4: A,A3: A] :
% 0.18/0.44 ( ( ord_less_eq @ A @ B4 @ A3 )
% 0.18/0.44 & ( A3 != B4 ) ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % dual_order.strict_iff_order
% 0.18/0.44 thf(fact_165_dual__order_Oorder__iff__strict,axiom,
% 0.18/0.44 ! [A: $tType] :
% 0.18/0.44 ( ( order @ A )
% 0.18/0.44 => ( ( ord_less_eq @ A )
% 0.18/0.44 = ( ^ [B4: A,A3: A] :
% 0.18/0.44 ( ( ord_less @ A @ B4 @ A3 )
% 0.18/0.44 | ( A3 = B4 ) ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % dual_order.order_iff_strict
% 0.18/0.44 thf(fact_166_order_Ostrict__implies__order,axiom,
% 0.18/0.44 ! [A: $tType] :
% 0.18/0.44 ( ( order @ A )
% 0.18/0.44 => ! [A2: A,B: A] :
% 0.18/0.44 ( ( ord_less @ A @ A2 @ B )
% 0.18/0.44 => ( ord_less_eq @ A @ A2 @ B ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % order.strict_implies_order
% 0.18/0.44 thf(fact_167_dense__le__bounded,axiom,
% 0.18/0.44 ! [A: $tType] :
% 0.18/0.44 ( ( dense_linorder @ A )
% 0.18/0.44 => ! [X: A,Y4: A,Z5: A] :
% 0.18/0.44 ( ( ord_less @ A @ X @ Y4 )
% 0.18/0.44 => ( ! [W: A] :
% 0.18/0.44 ( ( ord_less @ A @ X @ W )
% 0.18/0.44 => ( ( ord_less @ A @ W @ Y4 )
% 0.18/0.44 => ( ord_less_eq @ A @ W @ Z5 ) ) )
% 0.18/0.44 => ( ord_less_eq @ A @ Y4 @ Z5 ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % dense_le_bounded
% 0.18/0.44 thf(fact_168_dense__ge__bounded,axiom,
% 0.18/0.44 ! [A: $tType] :
% 0.18/0.44 ( ( dense_linorder @ A )
% 0.18/0.44 => ! [Z5: A,X: A,Y4: A] :
% 0.18/0.44 ( ( ord_less @ A @ Z5 @ X )
% 0.18/0.44 => ( ! [W: A] :
% 0.18/0.44 ( ( ord_less @ A @ Z5 @ W )
% 0.18/0.44 => ( ( ord_less @ A @ W @ X )
% 0.18/0.44 => ( ord_less_eq @ A @ Y4 @ W ) ) )
% 0.18/0.44 => ( ord_less_eq @ A @ Y4 @ Z5 ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % dense_ge_bounded
% 0.18/0.44 thf(fact_169_dual__order_Ostrict__trans2,axiom,
% 0.18/0.44 ! [A: $tType] :
% 0.18/0.44 ( ( order @ A )
% 0.18/0.44 => ! [B: A,A2: A,C: A] :
% 0.18/0.44 ( ( ord_less @ A @ B @ A2 )
% 0.18/0.44 => ( ( ord_less_eq @ A @ C @ B )
% 0.18/0.44 => ( ord_less @ A @ C @ A2 ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % dual_order.strict_trans2
% 0.18/0.44 thf(fact_170_dual__order_Ostrict__trans1,axiom,
% 0.18/0.44 ! [A: $tType] :
% 0.18/0.44 ( ( order @ A )
% 0.18/0.44 => ! [B: A,A2: A,C: A] :
% 0.18/0.44 ( ( ord_less_eq @ A @ B @ A2 )
% 0.18/0.44 => ( ( ord_less @ A @ C @ B )
% 0.18/0.44 => ( ord_less @ A @ C @ A2 ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % dual_order.strict_trans1
% 0.18/0.44 thf(fact_171_order_Ostrict__iff__order,axiom,
% 0.18/0.44 ! [A: $tType] :
% 0.18/0.44 ( ( order @ A )
% 0.18/0.44 => ( ( ord_less @ A )
% 0.18/0.44 = ( ^ [A3: A,B4: A] :
% 0.18/0.44 ( ( ord_less_eq @ A @ A3 @ B4 )
% 0.18/0.44 & ( A3 != B4 ) ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % order.strict_iff_order
% 0.18/0.44 thf(fact_172_order_Oorder__iff__strict,axiom,
% 0.18/0.44 ! [A: $tType] :
% 0.18/0.44 ( ( order @ A )
% 0.18/0.44 => ( ( ord_less_eq @ A )
% 0.18/0.44 = ( ^ [A3: A,B4: A] :
% 0.18/0.44 ( ( ord_less @ A @ A3 @ B4 )
% 0.18/0.44 | ( A3 = B4 ) ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % order.order_iff_strict
% 0.18/0.44 thf(fact_173_order_Ostrict__trans2,axiom,
% 0.18/0.44 ! [A: $tType] :
% 0.18/0.44 ( ( order @ A )
% 0.18/0.44 => ! [A2: A,B: A,C: A] :
% 0.18/0.44 ( ( ord_less @ A @ A2 @ B )
% 0.18/0.44 => ( ( ord_less_eq @ A @ B @ C )
% 0.18/0.44 => ( ord_less @ A @ A2 @ C ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % order.strict_trans2
% 0.18/0.44 thf(fact_174_order_Ostrict__trans1,axiom,
% 0.18/0.44 ! [A: $tType] :
% 0.18/0.44 ( ( order @ A )
% 0.18/0.44 => ! [A2: A,B: A,C: A] :
% 0.18/0.44 ( ( ord_less_eq @ A @ A2 @ B )
% 0.18/0.44 => ( ( ord_less @ A @ B @ C )
% 0.18/0.44 => ( ord_less @ A @ A2 @ C ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % order.strict_trans1
% 0.18/0.44 thf(fact_175_not__le__imp__less,axiom,
% 0.18/0.44 ! [A: $tType] :
% 0.18/0.44 ( ( linorder @ A )
% 0.18/0.44 => ! [Y4: A,X: A] :
% 0.18/0.44 ( ~ ( ord_less_eq @ A @ Y4 @ X )
% 0.18/0.44 => ( ord_less @ A @ X @ Y4 ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % not_le_imp_less
% 0.18/0.44 thf(fact_176_less__le__not__le,axiom,
% 0.18/0.44 ! [A: $tType] :
% 0.18/0.44 ( ( preorder @ A )
% 0.18/0.44 => ( ( ord_less @ A )
% 0.18/0.44 = ( ^ [X3: A,Y2: A] :
% 0.18/0.44 ( ( ord_less_eq @ A @ X3 @ Y2 )
% 0.18/0.44 & ~ ( ord_less_eq @ A @ Y2 @ X3 ) ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % less_le_not_le
% 0.18/0.44 thf(fact_177_le__imp__less__or__eq,axiom,
% 0.18/0.44 ! [A: $tType] :
% 0.18/0.44 ( ( order @ A )
% 0.18/0.44 => ! [X: A,Y4: A] :
% 0.18/0.44 ( ( ord_less_eq @ A @ X @ Y4 )
% 0.18/0.44 => ( ( ord_less @ A @ X @ Y4 )
% 0.18/0.44 | ( X = Y4 ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % le_imp_less_or_eq
% 0.18/0.44 thf(fact_178_le__less__linear,axiom,
% 0.18/0.44 ! [A: $tType] :
% 0.18/0.44 ( ( linorder @ A )
% 0.18/0.44 => ! [X: A,Y4: A] :
% 0.18/0.44 ( ( ord_less_eq @ A @ X @ Y4 )
% 0.18/0.44 | ( ord_less @ A @ Y4 @ X ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % le_less_linear
% 0.18/0.44 thf(fact_179_dense__le,axiom,
% 0.18/0.44 ! [A: $tType] :
% 0.18/0.44 ( ( dense_linorder @ A )
% 0.18/0.44 => ! [Y4: A,Z5: A] :
% 0.18/0.44 ( ! [X2: A] :
% 0.18/0.44 ( ( ord_less @ A @ X2 @ Y4 )
% 0.18/0.44 => ( ord_less_eq @ A @ X2 @ Z5 ) )
% 0.18/0.44 => ( ord_less_eq @ A @ Y4 @ Z5 ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % dense_le
% 0.18/0.44 thf(fact_180_dense__ge,axiom,
% 0.18/0.44 ! [A: $tType] :
% 0.18/0.44 ( ( dense_linorder @ A )
% 0.18/0.44 => ! [Z5: A,Y4: A] :
% 0.18/0.44 ( ! [X2: A] :
% 0.18/0.44 ( ( ord_less @ A @ Z5 @ X2 )
% 0.18/0.44 => ( ord_less_eq @ A @ Y4 @ X2 ) )
% 0.18/0.44 => ( ord_less_eq @ A @ Y4 @ Z5 ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % dense_ge
% 0.18/0.44 thf(fact_181_less__le__trans,axiom,
% 0.18/0.44 ! [A: $tType] :
% 0.18/0.44 ( ( preorder @ A )
% 0.18/0.44 => ! [X: A,Y4: A,Z5: A] :
% 0.18/0.44 ( ( ord_less @ A @ X @ Y4 )
% 0.18/0.44 => ( ( ord_less_eq @ A @ Y4 @ Z5 )
% 0.18/0.44 => ( ord_less @ A @ X @ Z5 ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % less_le_trans
% 0.18/0.44 thf(fact_182_le__less__trans,axiom,
% 0.18/0.44 ! [A: $tType] :
% 0.18/0.44 ( ( preorder @ A )
% 0.18/0.44 => ! [X: A,Y4: A,Z5: A] :
% 0.18/0.44 ( ( ord_less_eq @ A @ X @ Y4 )
% 0.18/0.44 => ( ( ord_less @ A @ Y4 @ Z5 )
% 0.18/0.44 => ( ord_less @ A @ X @ Z5 ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % le_less_trans
% 0.18/0.44 thf(fact_183_less__imp__le,axiom,
% 0.18/0.44 ! [A: $tType] :
% 0.18/0.44 ( ( preorder @ A )
% 0.18/0.44 => ! [X: A,Y4: A] :
% 0.18/0.44 ( ( ord_less @ A @ X @ Y4 )
% 0.18/0.44 => ( ord_less_eq @ A @ X @ Y4 ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % less_imp_le
% 0.18/0.44 thf(fact_184_antisym__conv2,axiom,
% 0.18/0.44 ! [A: $tType] :
% 0.18/0.44 ( ( order @ A )
% 0.18/0.44 => ! [X: A,Y4: A] :
% 0.18/0.44 ( ( ord_less_eq @ A @ X @ Y4 )
% 0.18/0.44 => ( ( ~ ( ord_less @ A @ X @ Y4 ) )
% 0.18/0.44 = ( X = Y4 ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % antisym_conv2
% 0.18/0.44 thf(fact_185_antisym__conv1,axiom,
% 0.18/0.44 ! [A: $tType] :
% 0.18/0.44 ( ( order @ A )
% 0.18/0.44 => ! [X: A,Y4: A] :
% 0.18/0.44 ( ~ ( ord_less @ A @ X @ Y4 )
% 0.18/0.44 => ( ( ord_less_eq @ A @ X @ Y4 )
% 0.18/0.44 = ( X = Y4 ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % antisym_conv1
% 0.18/0.44 thf(fact_186_le__neq__trans,axiom,
% 0.18/0.44 ! [A: $tType] :
% 0.18/0.44 ( ( order @ A )
% 0.18/0.44 => ! [A2: A,B: A] :
% 0.18/0.44 ( ( ord_less_eq @ A @ A2 @ B )
% 0.18/0.44 => ( ( A2 != B )
% 0.18/0.44 => ( ord_less @ A @ A2 @ B ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % le_neq_trans
% 0.18/0.44 thf(fact_187_not__less,axiom,
% 0.18/0.44 ! [A: $tType] :
% 0.18/0.44 ( ( linorder @ A )
% 0.18/0.44 => ! [X: A,Y4: A] :
% 0.18/0.44 ( ( ~ ( ord_less @ A @ X @ Y4 ) )
% 0.18/0.44 = ( ord_less_eq @ A @ Y4 @ X ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % not_less
% 0.18/0.44 thf(fact_188_not__le,axiom,
% 0.18/0.44 ! [A: $tType] :
% 0.18/0.44 ( ( linorder @ A )
% 0.18/0.44 => ! [X: A,Y4: A] :
% 0.18/0.44 ( ( ~ ( ord_less_eq @ A @ X @ Y4 ) )
% 0.18/0.44 = ( ord_less @ A @ Y4 @ X ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % not_le
% 0.18/0.44 thf(fact_189_order__less__le__subst2,axiom,
% 0.18/0.44 ! [A: $tType,C2: $tType] :
% 0.18/0.44 ( ( ( order @ C2 )
% 0.18/0.44 & ( order @ A ) )
% 0.18/0.44 => ! [A2: A,B: A,F: A > C2,C: C2] :
% 0.18/0.44 ( ( ord_less @ A @ A2 @ B )
% 0.18/0.44 => ( ( ord_less_eq @ C2 @ ( F @ B ) @ C )
% 0.18/0.44 => ( ! [X2: A,Y: A] :
% 0.18/0.44 ( ( ord_less @ A @ X2 @ Y )
% 0.18/0.44 => ( ord_less @ C2 @ ( F @ X2 ) @ ( F @ Y ) ) )
% 0.18/0.44 => ( ord_less @ C2 @ ( F @ A2 ) @ C ) ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % order_less_le_subst2
% 0.18/0.44 thf(fact_190_order__less__le__subst1,axiom,
% 0.18/0.44 ! [A: $tType,B2: $tType] :
% 0.18/0.44 ( ( ( order @ B2 )
% 0.18/0.44 & ( order @ A ) )
% 0.18/0.44 => ! [A2: A,F: B2 > A,B: B2,C: B2] :
% 0.18/0.44 ( ( ord_less @ A @ A2 @ ( F @ B ) )
% 0.18/0.44 => ( ( ord_less_eq @ B2 @ B @ C )
% 0.18/0.44 => ( ! [X2: B2,Y: B2] :
% 0.18/0.44 ( ( ord_less_eq @ B2 @ X2 @ Y )
% 0.18/0.44 => ( ord_less_eq @ A @ ( F @ X2 ) @ ( F @ Y ) ) )
% 0.18/0.44 => ( ord_less @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % order_less_le_subst1
% 0.18/0.44 thf(fact_191_order__le__less__subst2,axiom,
% 0.18/0.44 ! [A: $tType,C2: $tType] :
% 0.18/0.44 ( ( ( order @ C2 )
% 0.18/0.44 & ( order @ A ) )
% 0.18/0.44 => ! [A2: A,B: A,F: A > C2,C: C2] :
% 0.18/0.44 ( ( ord_less_eq @ A @ A2 @ B )
% 0.18/0.44 => ( ( ord_less @ C2 @ ( F @ B ) @ C )
% 0.18/0.44 => ( ! [X2: A,Y: A] :
% 0.18/0.44 ( ( ord_less_eq @ A @ X2 @ Y )
% 0.18/0.44 => ( ord_less_eq @ C2 @ ( F @ X2 ) @ ( F @ Y ) ) )
% 0.18/0.44 => ( ord_less @ C2 @ ( F @ A2 ) @ C ) ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % order_le_less_subst2
% 0.18/0.44 thf(fact_192_order__le__less__subst1,axiom,
% 0.18/0.44 ! [A: $tType,B2: $tType] :
% 0.18/0.44 ( ( ( order @ B2 )
% 0.18/0.44 & ( order @ A ) )
% 0.18/0.44 => ! [A2: A,F: B2 > A,B: B2,C: B2] :
% 0.18/0.44 ( ( ord_less_eq @ A @ A2 @ ( F @ B ) )
% 0.18/0.44 => ( ( ord_less @ B2 @ B @ C )
% 0.18/0.44 => ( ! [X2: B2,Y: B2] :
% 0.18/0.44 ( ( ord_less @ B2 @ X2 @ Y )
% 0.18/0.44 => ( ord_less @ A @ ( F @ X2 ) @ ( F @ Y ) ) )
% 0.18/0.44 => ( ord_less @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % order_le_less_subst1
% 0.18/0.44 thf(fact_193_less__le,axiom,
% 0.18/0.44 ! [A: $tType] :
% 0.18/0.44 ( ( order @ A )
% 0.18/0.44 => ( ( ord_less @ A )
% 0.18/0.44 = ( ^ [X3: A,Y2: A] :
% 0.18/0.44 ( ( ord_less_eq @ A @ X3 @ Y2 )
% 0.18/0.44 & ( X3 != Y2 ) ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % less_le
% 0.18/0.44 thf(fact_194_le__less,axiom,
% 0.18/0.44 ! [A: $tType] :
% 0.18/0.44 ( ( order @ A )
% 0.18/0.44 => ( ( ord_less_eq @ A )
% 0.18/0.44 = ( ^ [X3: A,Y2: A] :
% 0.18/0.44 ( ( ord_less @ A @ X3 @ Y2 )
% 0.18/0.44 | ( X3 = Y2 ) ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % le_less
% 0.18/0.44 thf(fact_195_leI,axiom,
% 0.18/0.44 ! [A: $tType] :
% 0.18/0.44 ( ( linorder @ A )
% 0.18/0.44 => ! [X: A,Y4: A] :
% 0.18/0.44 ( ~ ( ord_less @ A @ X @ Y4 )
% 0.18/0.44 => ( ord_less_eq @ A @ Y4 @ X ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % leI
% 0.18/0.44 thf(fact_196_leD,axiom,
% 0.18/0.44 ! [A: $tType] :
% 0.18/0.44 ( ( order @ A )
% 0.18/0.44 => ! [Y4: A,X: A] :
% 0.18/0.44 ( ( ord_less_eq @ A @ Y4 @ X )
% 0.18/0.44 => ~ ( ord_less @ A @ X @ Y4 ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % leD
% 0.18/0.44 thf(fact_197_complete__interval,axiom,
% 0.18/0.44 ! [A: $tType] :
% 0.18/0.44 ( ( condit1037483654norder @ A )
% 0.18/0.44 => ! [A2: A,B: A,P3: A > $o] :
% 0.18/0.44 ( ( ord_less @ A @ A2 @ B )
% 0.18/0.44 => ( ( P3 @ A2 )
% 0.18/0.44 => ( ~ ( P3 @ B )
% 0.18/0.44 => ? [C3: A] :
% 0.18/0.44 ( ( ord_less_eq @ A @ A2 @ C3 )
% 0.18/0.44 & ( ord_less_eq @ A @ C3 @ B )
% 0.18/0.44 & ! [X4: A] :
% 0.18/0.44 ( ( ( ord_less_eq @ A @ A2 @ X4 )
% 0.18/0.44 & ( ord_less @ A @ X4 @ C3 ) )
% 0.18/0.44 => ( P3 @ X4 ) )
% 0.18/0.44 & ! [D3: A] :
% 0.18/0.44 ( ! [X2: A] :
% 0.18/0.44 ( ( ( ord_less_eq @ A @ A2 @ X2 )
% 0.18/0.44 & ( ord_less @ A @ X2 @ D3 ) )
% 0.18/0.44 => ( P3 @ X2 ) )
% 0.18/0.44 => ( ord_less_eq @ A @ D3 @ C3 ) ) ) ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % complete_interval
% 0.18/0.44 thf(fact_198_rsquarefree__def,axiom,
% 0.18/0.44 ! [A: $tType] :
% 0.18/0.44 ( ( idom @ A )
% 0.18/0.44 => ( ( rsquarefree @ A )
% 0.18/0.44 = ( ^ [P2: poly @ A] :
% 0.18/0.44 ( ( P2
% 0.18/0.44 != ( zero_zero @ ( poly @ A ) ) )
% 0.18/0.44 & ! [A3: A] :
% 0.18/0.44 ( ( ( order2 @ A @ A3 @ P2 )
% 0.18/0.44 = ( zero_zero @ nat ) )
% 0.18/0.44 | ( ( order2 @ A @ A3 @ P2 )
% 0.18/0.44 = ( one_one @ nat ) ) ) ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % rsquarefree_def
% 0.18/0.44 thf(fact_199_poly__cutoff__1,axiom,
% 0.18/0.44 ! [A: $tType] :
% 0.18/0.44 ( ( comm_semiring_1 @ A )
% 0.18/0.44 => ! [N: nat] :
% 0.18/0.44 ( ( ( N
% 0.18/0.44 = ( zero_zero @ nat ) )
% 0.18/0.44 => ( ( poly_cutoff @ A @ N @ ( one_one @ ( poly @ A ) ) )
% 0.18/0.44 = ( zero_zero @ ( poly @ A ) ) ) )
% 0.18/0.44 & ( ( N
% 0.18/0.44 != ( zero_zero @ nat ) )
% 0.18/0.44 => ( ( poly_cutoff @ A @ N @ ( one_one @ ( poly @ A ) ) )
% 0.18/0.44 = ( one_one @ ( poly @ A ) ) ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % poly_cutoff_1
% 0.18/0.44 thf(fact_200_coeff__0__reflect__poly__0__iff,axiom,
% 0.18/0.44 ! [A: $tType] :
% 0.18/0.44 ( ( zero @ A )
% 0.18/0.44 => ! [P: poly @ A] :
% 0.18/0.44 ( ( ( coeff @ A @ ( reflect_poly @ A @ P ) @ ( zero_zero @ nat ) )
% 0.18/0.44 = ( zero_zero @ A ) )
% 0.18/0.44 = ( P
% 0.18/0.44 = ( zero_zero @ ( poly @ A ) ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % coeff_0_reflect_poly_0_iff
% 0.18/0.44 thf(fact_201_of__nat__0__less__iff,axiom,
% 0.18/0.44 ! [A: $tType] :
% 0.18/0.44 ( ( linord1659791738miring @ A )
% 0.18/0.44 => ! [N: nat] :
% 0.18/0.44 ( ( ord_less @ A @ ( zero_zero @ A ) @ ( semiring_1_of_nat @ A @ N ) )
% 0.18/0.44 = ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % of_nat_0_less_iff
% 0.18/0.44 thf(fact_202_order__pderiv,axiom,
% 0.18/0.44 ! [A: $tType] :
% 0.18/0.44 ( ( field_char_0 @ A )
% 0.18/0.44 => ! [P: poly @ A,A2: A] :
% 0.18/0.44 ( ( ( pderiv @ A @ P )
% 0.18/0.44 != ( zero_zero @ ( poly @ A ) ) )
% 0.18/0.44 => ( ( ( order2 @ A @ A2 @ P )
% 0.18/0.44 != ( zero_zero @ nat ) )
% 0.18/0.44 => ( ( order2 @ A @ A2 @ P )
% 0.18/0.44 = ( suc @ ( order2 @ A @ A2 @ ( pderiv @ A @ P ) ) ) ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % order_pderiv
% 0.18/0.44 thf(fact_203_order__pderiv2,axiom,
% 0.18/0.44 ! [A: $tType] :
% 0.18/0.44 ( ( field_char_0 @ A )
% 0.18/0.44 => ! [P: poly @ A,A2: A,N: nat] :
% 0.18/0.44 ( ( ( pderiv @ A @ P )
% 0.18/0.44 != ( zero_zero @ ( poly @ A ) ) )
% 0.18/0.44 => ( ( ( order2 @ A @ A2 @ P )
% 0.18/0.44 != ( zero_zero @ nat ) )
% 0.18/0.44 => ( ( ( order2 @ A @ A2 @ ( pderiv @ A @ P ) )
% 0.18/0.44 = N )
% 0.18/0.44 = ( ( order2 @ A @ A2 @ P )
% 0.18/0.44 = ( suc @ N ) ) ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % order_pderiv2
% 0.18/0.44 thf(fact_204_cross__def,axiom,
% 0.18/0.44 ( sturm_424270202_cross
% 0.18/0.44 = ( ^ [P2: poly @ real,A3: real,B4: real] : ( sturm_1771227917iation @ ( poly2 @ real @ P2 @ A3 ) @ ( poly2 @ real @ P2 @ B4 ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % cross_def
% 0.18/0.44 thf(fact_205_nat_Oinject,axiom,
% 0.18/0.44 ! [X22: nat,Y22: nat] :
% 0.18/0.44 ( ( ( suc @ X22 )
% 0.18/0.44 = ( suc @ Y22 ) )
% 0.18/0.44 = ( X22 = Y22 ) ) ).
% 0.18/0.44
% 0.18/0.44 % nat.inject
% 0.18/0.44 thf(fact_206_old_Onat_Oinject,axiom,
% 0.18/0.44 ! [Nat: nat,Nat2: nat] :
% 0.18/0.44 ( ( ( suc @ Nat )
% 0.18/0.44 = ( suc @ Nat2 ) )
% 0.18/0.44 = ( Nat = Nat2 ) ) ).
% 0.18/0.44
% 0.18/0.44 % old.nat.inject
% 0.18/0.44 thf(fact_207_of__nat__eq__iff,axiom,
% 0.18/0.44 ! [A: $tType] :
% 0.18/0.44 ( ( semiring_char_0 @ A )
% 0.18/0.44 => ! [M: nat,N: nat] :
% 0.18/0.44 ( ( ( semiring_1_of_nat @ A @ M )
% 0.18/0.44 = ( semiring_1_of_nat @ A @ N ) )
% 0.18/0.44 = ( M = N ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % of_nat_eq_iff
% 0.18/0.44 thf(fact_208_lessI,axiom,
% 0.18/0.44 ! [N: nat] : ( ord_less @ nat @ N @ ( suc @ N ) ) ).
% 0.18/0.44
% 0.18/0.44 % lessI
% 0.18/0.44 thf(fact_209_Suc__mono,axiom,
% 0.18/0.44 ! [M: nat,N: nat] :
% 0.18/0.44 ( ( ord_less @ nat @ M @ N )
% 0.18/0.44 => ( ord_less @ nat @ ( suc @ M ) @ ( suc @ N ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % Suc_mono
% 0.18/0.44 thf(fact_210_Suc__less__eq,axiom,
% 0.18/0.44 ! [M: nat,N: nat] :
% 0.18/0.44 ( ( ord_less @ nat @ ( suc @ M ) @ ( suc @ N ) )
% 0.18/0.44 = ( ord_less @ nat @ M @ N ) ) ).
% 0.18/0.44
% 0.18/0.44 % Suc_less_eq
% 0.18/0.44 thf(fact_211_Suc__le__mono,axiom,
% 0.18/0.44 ! [N: nat,M: nat] :
% 0.18/0.44 ( ( ord_less_eq @ nat @ ( suc @ N ) @ ( suc @ M ) )
% 0.18/0.44 = ( ord_less_eq @ nat @ N @ M ) ) ).
% 0.18/0.44
% 0.18/0.44 % Suc_le_mono
% 0.18/0.44 thf(fact_212_pderiv__of__nat,axiom,
% 0.18/0.44 ! [A: $tType] :
% 0.18/0.44 ( ( ( comm_semiring_1 @ A )
% 0.18/0.44 & ( semiri1193490041visors @ A ) )
% 0.18/0.44 => ! [N: nat] :
% 0.18/0.44 ( ( pderiv @ A @ ( semiring_1_of_nat @ ( poly @ A ) @ N ) )
% 0.18/0.44 = ( zero_zero @ ( poly @ A ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % pderiv_of_nat
% 0.18/0.44 thf(fact_213_reflect__poly__1,axiom,
% 0.18/0.44 ! [A: $tType] :
% 0.18/0.44 ( ( comm_semiring_1 @ A )
% 0.18/0.44 => ( ( reflect_poly @ A @ ( one_one @ ( poly @ A ) ) )
% 0.18/0.44 = ( one_one @ ( poly @ A ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % reflect_poly_1
% 0.18/0.44 thf(fact_214_of__nat__0,axiom,
% 0.18/0.44 ! [A: $tType] :
% 0.18/0.44 ( ( semiring_1 @ A )
% 0.18/0.44 => ( ( semiring_1_of_nat @ A @ ( zero_zero @ nat ) )
% 0.18/0.44 = ( zero_zero @ A ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % of_nat_0
% 0.18/0.44 thf(fact_215_of__nat__0__eq__iff,axiom,
% 0.18/0.44 ! [A: $tType] :
% 0.18/0.44 ( ( semiring_char_0 @ A )
% 0.18/0.44 => ! [N: nat] :
% 0.18/0.44 ( ( ( zero_zero @ A )
% 0.18/0.44 = ( semiring_1_of_nat @ A @ N ) )
% 0.18/0.44 = ( ( zero_zero @ nat )
% 0.18/0.44 = N ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % of_nat_0_eq_iff
% 0.18/0.44 thf(fact_216_of__nat__eq__0__iff,axiom,
% 0.18/0.44 ! [A: $tType] :
% 0.18/0.44 ( ( semiring_char_0 @ A )
% 0.18/0.44 => ! [M: nat] :
% 0.18/0.44 ( ( ( semiring_1_of_nat @ A @ M )
% 0.18/0.44 = ( zero_zero @ A ) )
% 0.18/0.44 = ( M
% 0.18/0.44 = ( zero_zero @ nat ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % of_nat_eq_0_iff
% 0.18/0.44 thf(fact_217_of__nat__less__iff,axiom,
% 0.18/0.44 ! [A: $tType] :
% 0.18/0.44 ( ( linord1659791738miring @ A )
% 0.18/0.44 => ! [M: nat,N: nat] :
% 0.18/0.44 ( ( ord_less @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N ) )
% 0.18/0.44 = ( ord_less @ nat @ M @ N ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % of_nat_less_iff
% 0.18/0.44 thf(fact_218_of__nat__le__iff,axiom,
% 0.18/0.44 ! [A: $tType] :
% 0.18/0.44 ( ( linord1659791738miring @ A )
% 0.18/0.44 => ! [M: nat,N: nat] :
% 0.18/0.44 ( ( ord_less_eq @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N ) )
% 0.18/0.44 = ( ord_less_eq @ nat @ M @ N ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % of_nat_le_iff
% 0.18/0.44 thf(fact_219_less__Suc0,axiom,
% 0.18/0.44 ! [N: nat] :
% 0.18/0.44 ( ( ord_less @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) )
% 0.18/0.44 = ( N
% 0.18/0.44 = ( zero_zero @ nat ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % less_Suc0
% 0.18/0.44 thf(fact_220_zero__less__Suc,axiom,
% 0.18/0.44 ! [N: nat] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( suc @ N ) ) ).
% 0.18/0.44
% 0.18/0.44 % zero_less_Suc
% 0.18/0.44 thf(fact_221_of__nat__1,axiom,
% 0.18/0.44 ! [A: $tType] :
% 0.18/0.44 ( ( semiring_1 @ A )
% 0.18/0.44 => ( ( semiring_1_of_nat @ A @ ( one_one @ nat ) )
% 0.18/0.44 = ( one_one @ A ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % of_nat_1
% 0.18/0.44 thf(fact_222_of__nat__1__eq__iff,axiom,
% 0.18/0.44 ! [A: $tType] :
% 0.18/0.44 ( ( semiring_char_0 @ A )
% 0.18/0.44 => ! [N: nat] :
% 0.18/0.44 ( ( ( one_one @ A )
% 0.18/0.44 = ( semiring_1_of_nat @ A @ N ) )
% 0.18/0.44 = ( N
% 0.18/0.44 = ( one_one @ nat ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % of_nat_1_eq_iff
% 0.18/0.44 thf(fact_223_of__nat__eq__1__iff,axiom,
% 0.18/0.44 ! [A: $tType] :
% 0.18/0.44 ( ( semiring_char_0 @ A )
% 0.18/0.44 => ! [N: nat] :
% 0.18/0.44 ( ( ( semiring_1_of_nat @ A @ N )
% 0.18/0.44 = ( one_one @ A ) )
% 0.18/0.44 = ( N
% 0.18/0.44 = ( one_one @ nat ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % of_nat_eq_1_iff
% 0.18/0.44 thf(fact_224_less__one,axiom,
% 0.18/0.44 ! [N: nat] :
% 0.18/0.44 ( ( ord_less @ nat @ N @ ( one_one @ nat ) )
% 0.18/0.44 = ( N
% 0.18/0.44 = ( zero_zero @ nat ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % less_one
% 0.18/0.44 thf(fact_225_coeff__0,axiom,
% 0.18/0.44 ! [A: $tType] :
% 0.18/0.44 ( ( zero @ A )
% 0.18/0.44 => ! [N: nat] :
% 0.18/0.44 ( ( coeff @ A @ ( zero_zero @ ( poly @ A ) ) @ N )
% 0.18/0.44 = ( zero_zero @ A ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % coeff_0
% 0.18/0.44 thf(fact_226_poly__1,axiom,
% 0.18/0.44 ! [A: $tType] :
% 0.18/0.44 ( ( comm_semiring_1 @ A )
% 0.18/0.44 => ! [X: A] :
% 0.18/0.44 ( ( poly2 @ A @ ( one_one @ ( poly @ A ) ) @ X )
% 0.18/0.44 = ( one_one @ A ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % poly_1
% 0.18/0.44 thf(fact_227_variation__0_I2_J,axiom,
% 0.18/0.44 ! [X: real] :
% 0.18/0.44 ( ( sturm_1771227917iation @ X @ ( zero_zero @ real ) )
% 0.18/0.44 = ( zero_zero @ int ) ) ).
% 0.18/0.44
% 0.18/0.44 % variation_0(2)
% 0.18/0.44 thf(fact_228_variation__0_I1_J,axiom,
% 0.18/0.44 ! [Y4: real] :
% 0.18/0.44 ( ( sturm_1771227917iation @ ( zero_zero @ real ) @ Y4 )
% 0.18/0.44 = ( zero_zero @ int ) ) ).
% 0.18/0.44
% 0.18/0.44 % variation_0(1)
% 0.18/0.44 thf(fact_229_pderiv__1,axiom,
% 0.18/0.44 ! [A: $tType] :
% 0.18/0.44 ( ( ( comm_semiring_1 @ A )
% 0.18/0.44 & ( semiri1193490041visors @ A ) )
% 0.18/0.44 => ( ( pderiv @ A @ ( one_one @ ( poly @ A ) ) )
% 0.18/0.44 = ( zero_zero @ ( poly @ A ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % pderiv_1
% 0.18/0.44 thf(fact_230_order__1__eq__0,axiom,
% 0.18/0.44 ! [A: $tType] :
% 0.18/0.44 ( ( idom @ A )
% 0.18/0.44 => ! [X: A] :
% 0.18/0.44 ( ( order2 @ A @ X @ ( one_one @ ( poly @ A ) ) )
% 0.18/0.44 = ( zero_zero @ nat ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % order_1_eq_0
% 0.18/0.44 thf(fact_231_of__nat__le__0__iff,axiom,
% 0.18/0.44 ! [A: $tType] :
% 0.18/0.44 ( ( linord1659791738miring @ A )
% 0.18/0.44 => ! [M: nat] :
% 0.18/0.44 ( ( ord_less_eq @ A @ ( semiring_1_of_nat @ A @ M ) @ ( zero_zero @ A ) )
% 0.18/0.44 = ( M
% 0.18/0.44 = ( zero_zero @ nat ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % of_nat_le_0_iff
% 0.18/0.44 thf(fact_232_reflect__poly__reflect__poly,axiom,
% 0.18/0.44 ! [A: $tType] :
% 0.18/0.44 ( ( zero @ A )
% 0.18/0.44 => ! [P: poly @ A] :
% 0.18/0.44 ( ( ( coeff @ A @ P @ ( zero_zero @ nat ) )
% 0.18/0.44 != ( zero_zero @ A ) )
% 0.18/0.44 => ( ( reflect_poly @ A @ ( reflect_poly @ A @ P ) )
% 0.18/0.44 = P ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % reflect_poly_reflect_poly
% 0.18/0.44 thf(fact_233_One__nat__def,axiom,
% 0.18/0.44 ( ( one_one @ nat )
% 0.18/0.44 = ( suc @ ( zero_zero @ nat ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % One_nat_def
% 0.18/0.44 thf(fact_234_less__fun__def,axiom,
% 0.18/0.44 ! [B2: $tType,A: $tType] :
% 0.18/0.44 ( ( ord @ B2 )
% 0.18/0.44 => ( ( ord_less @ ( A > B2 ) )
% 0.18/0.44 = ( ^ [F2: A > B2,G2: A > B2] :
% 0.18/0.44 ( ( ord_less_eq @ ( A > B2 ) @ F2 @ G2 )
% 0.18/0.44 & ~ ( ord_less_eq @ ( A > B2 ) @ G2 @ F2 ) ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % less_fun_def
% 0.18/0.44 thf(fact_235_of__nat__neq__0,axiom,
% 0.18/0.44 ! [A: $tType] :
% 0.18/0.44 ( ( semiring_char_0 @ A )
% 0.18/0.44 => ! [N: nat] :
% 0.18/0.44 ( ( semiring_1_of_nat @ A @ ( suc @ N ) )
% 0.18/0.44 != ( zero_zero @ A ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % of_nat_neq_0
% 0.18/0.44 thf(fact_236_transitive__stepwise__le,axiom,
% 0.18/0.44 ! [M: nat,N: nat,R: nat > nat > $o] :
% 0.18/0.44 ( ( ord_less_eq @ nat @ M @ N )
% 0.18/0.44 => ( ! [X2: nat] : ( R @ X2 @ X2 )
% 0.18/0.44 => ( ! [X2: nat,Y: nat,Z3: nat] :
% 0.18/0.44 ( ( R @ X2 @ Y )
% 0.18/0.44 => ( ( R @ Y @ Z3 )
% 0.18/0.44 => ( R @ X2 @ Z3 ) ) )
% 0.18/0.44 => ( ! [N3: nat] : ( R @ N3 @ ( suc @ N3 ) )
% 0.18/0.44 => ( R @ M @ N ) ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % transitive_stepwise_le
% 0.18/0.44 thf(fact_237_nat__induct__at__least,axiom,
% 0.18/0.44 ! [M: nat,N: nat,P3: nat > $o] :
% 0.18/0.44 ( ( ord_less_eq @ nat @ M @ N )
% 0.18/0.44 => ( ( P3 @ M )
% 0.18/0.44 => ( ! [N3: nat] :
% 0.18/0.44 ( ( ord_less_eq @ nat @ M @ N3 )
% 0.18/0.44 => ( ( P3 @ N3 )
% 0.18/0.44 => ( P3 @ ( suc @ N3 ) ) ) )
% 0.18/0.44 => ( P3 @ N ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % nat_induct_at_least
% 0.18/0.44 thf(fact_238_Nat_Oex__has__greatest__nat,axiom,
% 0.18/0.44 ! [P3: nat > $o,K2: nat,B: nat] :
% 0.18/0.44 ( ( P3 @ K2 )
% 0.18/0.44 => ( ! [Y: nat] :
% 0.18/0.44 ( ( P3 @ Y )
% 0.18/0.44 => ( ord_less_eq @ nat @ Y @ B ) )
% 0.18/0.44 => ? [X2: nat] :
% 0.18/0.44 ( ( P3 @ X2 )
% 0.18/0.44 & ! [Y3: nat] :
% 0.18/0.44 ( ( P3 @ Y3 )
% 0.18/0.44 => ( ord_less_eq @ nat @ Y3 @ X2 ) ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % Nat.ex_has_greatest_nat
% 0.18/0.44 thf(fact_239_verit__la__generic,axiom,
% 0.18/0.44 ! [A2: int,X: int] :
% 0.18/0.44 ( ( ord_less_eq @ int @ A2 @ X )
% 0.18/0.44 | ( A2 = X )
% 0.18/0.44 | ( ord_less_eq @ int @ X @ A2 ) ) ).
% 0.18/0.44
% 0.18/0.44 % verit_la_generic
% 0.18/0.44 thf(fact_240_full__nat__induct,axiom,
% 0.18/0.44 ! [P3: nat > $o,N: nat] :
% 0.18/0.44 ( ! [N3: nat] :
% 0.18/0.44 ( ! [M3: nat] :
% 0.18/0.44 ( ( ord_less_eq @ nat @ ( suc @ M3 ) @ N3 )
% 0.18/0.44 => ( P3 @ M3 ) )
% 0.18/0.44 => ( P3 @ N3 ) )
% 0.18/0.44 => ( P3 @ N ) ) ).
% 0.18/0.44
% 0.18/0.44 % full_nat_induct
% 0.18/0.44 thf(fact_241_not__less__eq__eq,axiom,
% 0.18/0.44 ! [M: nat,N: nat] :
% 0.18/0.44 ( ( ~ ( ord_less_eq @ nat @ M @ N ) )
% 0.18/0.44 = ( ord_less_eq @ nat @ ( suc @ N ) @ M ) ) ).
% 0.18/0.44
% 0.18/0.44 % not_less_eq_eq
% 0.18/0.44 thf(fact_242_Suc__n__not__le__n,axiom,
% 0.18/0.44 ! [N: nat] :
% 0.18/0.44 ~ ( ord_less_eq @ nat @ ( suc @ N ) @ N ) ).
% 0.18/0.44
% 0.18/0.44 % Suc_n_not_le_n
% 0.18/0.44 thf(fact_243_nat__le__linear,axiom,
% 0.18/0.44 ! [M: nat,N: nat] :
% 0.18/0.44 ( ( ord_less_eq @ nat @ M @ N )
% 0.18/0.44 | ( ord_less_eq @ nat @ N @ M ) ) ).
% 0.18/0.44
% 0.18/0.44 % nat_le_linear
% 0.18/0.44 thf(fact_244_le__antisym,axiom,
% 0.18/0.44 ! [M: nat,N: nat] :
% 0.18/0.44 ( ( ord_less_eq @ nat @ M @ N )
% 0.18/0.44 => ( ( ord_less_eq @ nat @ N @ M )
% 0.18/0.44 => ( M = N ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % le_antisym
% 0.18/0.44 thf(fact_245_le__Suc__eq,axiom,
% 0.18/0.44 ! [M: nat,N: nat] :
% 0.18/0.44 ( ( ord_less_eq @ nat @ M @ ( suc @ N ) )
% 0.18/0.44 = ( ( ord_less_eq @ nat @ M @ N )
% 0.18/0.44 | ( M
% 0.18/0.44 = ( suc @ N ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % le_Suc_eq
% 0.18/0.44 thf(fact_246_eq__imp__le,axiom,
% 0.18/0.44 ! [M: nat,N: nat] :
% 0.18/0.44 ( ( M = N )
% 0.18/0.44 => ( ord_less_eq @ nat @ M @ N ) ) ).
% 0.18/0.44
% 0.18/0.44 % eq_imp_le
% 0.18/0.44 thf(fact_247_le__trans,axiom,
% 0.18/0.44 ! [I: nat,J: nat,K2: nat] :
% 0.18/0.44 ( ( ord_less_eq @ nat @ I @ J )
% 0.18/0.44 => ( ( ord_less_eq @ nat @ J @ K2 )
% 0.18/0.44 => ( ord_less_eq @ nat @ I @ K2 ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % le_trans
% 0.18/0.44 thf(fact_248_Suc__le__D,axiom,
% 0.18/0.44 ! [N: nat,M4: nat] :
% 0.18/0.44 ( ( ord_less_eq @ nat @ ( suc @ N ) @ M4 )
% 0.18/0.44 => ? [M5: nat] :
% 0.18/0.44 ( M4
% 0.18/0.44 = ( suc @ M5 ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % Suc_le_D
% 0.18/0.44 thf(fact_249_le__refl,axiom,
% 0.18/0.44 ! [N: nat] : ( ord_less_eq @ nat @ N @ N ) ).
% 0.18/0.44
% 0.18/0.44 % le_refl
% 0.18/0.44 thf(fact_250_le__SucI,axiom,
% 0.18/0.44 ! [M: nat,N: nat] :
% 0.18/0.44 ( ( ord_less_eq @ nat @ M @ N )
% 0.18/0.44 => ( ord_less_eq @ nat @ M @ ( suc @ N ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % le_SucI
% 0.18/0.44 thf(fact_251_le__SucE,axiom,
% 0.18/0.44 ! [M: nat,N: nat] :
% 0.18/0.44 ( ( ord_less_eq @ nat @ M @ ( suc @ N ) )
% 0.18/0.44 => ( ~ ( ord_less_eq @ nat @ M @ N )
% 0.18/0.44 => ( M
% 0.18/0.44 = ( suc @ N ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % le_SucE
% 0.18/0.44
% 0.18/0.44 % Type constructors (87)
% 0.18/0.44 thf(tcon_Polynomial_Opoly___Rings_Olinordered__idom,axiom,
% 0.18/0.44 ! [A7: $tType] :
% 0.18/0.44 ( ( linordered_idom @ A7 )
% 0.18/0.44 => ( linordered_idom @ ( poly @ A7 ) ) ) ).
% 0.18/0.44
% 0.18/0.44 thf(tcon_Real_Oreal___Rings_Olinordered__idom_1,axiom,
% 0.18/0.44 linordered_idom @ real ).
% 0.18/0.44
% 0.18/0.44 thf(tcon_Int_Oint___Rings_Olinordered__idom_2,axiom,
% 0.18/0.44 linordered_idom @ int ).
% 0.18/0.44
% 0.18/0.44 thf(tcon_fun___Orderings_Opreorder,axiom,
% 0.18/0.44 ! [A7: $tType,A8: $tType] :
% 0.18/0.44 ( ( preorder @ A8 )
% 0.18/0.44 => ( preorder @ ( A7 > A8 ) ) ) ).
% 0.18/0.44
% 0.18/0.44 thf(tcon_fun___Orderings_Oorder,axiom,
% 0.18/0.44 ! [A7: $tType,A8: $tType] :
% 0.18/0.44 ( ( order @ A8 )
% 0.18/0.44 => ( order @ ( A7 > A8 ) ) ) ).
% 0.18/0.44
% 0.18/0.44 thf(tcon_fun___Orderings_Oord,axiom,
% 0.18/0.44 ! [A7: $tType,A8: $tType] :
% 0.18/0.44 ( ( ord @ A8 )
% 0.18/0.44 => ( ord @ ( A7 > A8 ) ) ) ).
% 0.18/0.44
% 0.18/0.44 thf(tcon_Int_Oint___Conditionally__Complete__Lattices_Oconditionally__complete__linorder,axiom,
% 0.18/0.44 condit1037483654norder @ int ).
% 0.18/0.44
% 0.18/0.44 thf(tcon_Int_Oint___Rings_Olinordered__nonzero__semiring,axiom,
% 0.18/0.44 linord1659791738miring @ int ).
% 0.18/0.44
% 0.18/0.44 thf(tcon_Int_Oint___Rings_Osemiring__no__zero__divisors,axiom,
% 0.18/0.44 semiri1193490041visors @ int ).
% 0.18/0.44
% 0.18/0.44 thf(tcon_Int_Oint___Rings_Oring__no__zero__divisors,axiom,
% 0.18/0.44 ring_n68954251visors @ int ).
% 0.18/0.44
% 0.18/0.44 thf(tcon_Int_Oint___Rings_Ocomm__semiring__1,axiom,
% 0.18/0.44 comm_semiring_1 @ int ).
% 0.18/0.44
% 0.18/0.44 thf(tcon_Int_Oint___Rings_Ocomm__semiring__0,axiom,
% 0.18/0.44 comm_semiring_0 @ int ).
% 0.18/0.44
% 0.18/0.44 thf(tcon_Int_Oint___Nat_Osemiring__char__0,axiom,
% 0.18/0.44 semiring_char_0 @ int ).
% 0.18/0.44
% 0.18/0.44 thf(tcon_Int_Oint___Orderings_Opreorder_3,axiom,
% 0.18/0.44 preorder @ int ).
% 0.18/0.44
% 0.18/0.44 thf(tcon_Int_Oint___Orderings_Olinorder,axiom,
% 0.18/0.44 linorder @ int ).
% 0.18/0.44
% 0.18/0.44 thf(tcon_Int_Oint___Rings_Ocomm__ring__1,axiom,
% 0.18/0.44 comm_ring_1 @ int ).
% 0.18/0.44
% 0.18/0.44 thf(tcon_Int_Oint___Rings_Osemiring__1,axiom,
% 0.18/0.44 semiring_1 @ int ).
% 0.18/0.44
% 0.18/0.44 thf(tcon_Int_Oint___Orderings_Ono__top,axiom,
% 0.18/0.44 no_top @ int ).
% 0.18/0.44
% 0.18/0.44 thf(tcon_Int_Oint___Orderings_Ono__bot,axiom,
% 0.18/0.44 no_bot @ int ).
% 0.18/0.44
% 0.18/0.44 thf(tcon_Int_Oint___Orderings_Oorder_4,axiom,
% 0.18/0.44 order @ int ).
% 0.18/0.44
% 0.18/0.44 thf(tcon_Int_Oint___Nat_Oring__char__0,axiom,
% 0.18/0.44 ring_char_0 @ int ).
% 0.18/0.44
% 0.18/0.44 thf(tcon_Int_Oint___Orderings_Oord_5,axiom,
% 0.18/0.44 ord @ int ).
% 0.18/0.44
% 0.18/0.44 thf(tcon_Int_Oint___Groups_Ozero,axiom,
% 0.18/0.44 zero @ int ).
% 0.18/0.44
% 0.18/0.44 thf(tcon_Int_Oint___Rings_Oidom,axiom,
% 0.18/0.44 idom @ int ).
% 0.18/0.44
% 0.18/0.44 thf(tcon_Int_Oint___Groups_Oone,axiom,
% 0.18/0.44 one @ int ).
% 0.18/0.44
% 0.18/0.44 thf(tcon_Nat_Onat___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_6,axiom,
% 0.18/0.44 condit1037483654norder @ nat ).
% 0.18/0.44
% 0.18/0.44 thf(tcon_Nat_Onat___Groups_Ocanonically__ordered__monoid__add,axiom,
% 0.18/0.44 canoni770627133id_add @ nat ).
% 0.18/0.44
% 0.18/0.44 thf(tcon_Nat_Onat___Rings_Olinordered__nonzero__semiring_7,axiom,
% 0.18/0.44 linord1659791738miring @ nat ).
% 0.18/0.44
% 0.18/0.44 thf(tcon_Nat_Onat___Rings_Osemiring__no__zero__divisors_8,axiom,
% 0.18/0.44 semiri1193490041visors @ nat ).
% 0.18/0.44
% 0.18/0.44 thf(tcon_Nat_Onat___Rings_Ocomm__semiring__1_9,axiom,
% 0.18/0.44 comm_semiring_1 @ nat ).
% 0.18/0.44
% 0.18/0.44 thf(tcon_Nat_Onat___Rings_Ocomm__semiring__0_10,axiom,
% 0.18/0.44 comm_semiring_0 @ nat ).
% 0.18/0.44
% 0.18/0.44 thf(tcon_Nat_Onat___Orderings_Owellorder,axiom,
% 0.18/0.44 wellorder @ nat ).
% 0.18/0.44
% 0.18/0.44 thf(tcon_Nat_Onat___Nat_Osemiring__char__0_11,axiom,
% 0.18/0.44 semiring_char_0 @ nat ).
% 0.18/0.44
% 0.18/0.44 thf(tcon_Nat_Onat___Orderings_Opreorder_12,axiom,
% 0.18/0.44 preorder @ nat ).
% 0.18/0.44
% 0.18/0.44 thf(tcon_Nat_Onat___Orderings_Olinorder_13,axiom,
% 0.18/0.44 linorder @ nat ).
% 0.18/0.44
% 0.18/0.44 thf(tcon_Nat_Onat___Rings_Osemiring__1_14,axiom,
% 0.18/0.44 semiring_1 @ nat ).
% 0.18/0.44
% 0.18/0.44 thf(tcon_Nat_Onat___Orderings_Ono__top_15,axiom,
% 0.18/0.44 no_top @ nat ).
% 0.18/0.44
% 0.18/0.44 thf(tcon_Nat_Onat___Orderings_Oorder_16,axiom,
% 0.18/0.44 order @ nat ).
% 0.18/0.44
% 0.18/0.44 thf(tcon_Nat_Onat___Orderings_Oord_17,axiom,
% 0.18/0.44 ord @ nat ).
% 0.18/0.44
% 0.18/0.44 thf(tcon_Nat_Onat___Groups_Ozero_18,axiom,
% 0.18/0.44 zero @ nat ).
% 0.18/0.44
% 0.18/0.44 thf(tcon_Nat_Onat___Groups_Oone_19,axiom,
% 0.18/0.44 one @ nat ).
% 0.18/0.44
% 0.18/0.44 thf(tcon_Set_Oset___Orderings_Opreorder_20,axiom,
% 0.18/0.44 ! [A7: $tType] : ( preorder @ ( set @ A7 ) ) ).
% 0.18/0.44
% 0.18/0.44 thf(tcon_Set_Oset___Orderings_Oorder_21,axiom,
% 0.18/0.44 ! [A7: $tType] : ( order @ ( set @ A7 ) ) ).
% 0.18/0.44
% 0.18/0.44 thf(tcon_Set_Oset___Orderings_Oord_22,axiom,
% 0.18/0.44 ! [A7: $tType] : ( ord @ ( set @ A7 ) ) ).
% 0.18/0.44
% 0.18/0.44 thf(tcon_HOL_Obool___Orderings_Opreorder_23,axiom,
% 0.18/0.44 preorder @ $o ).
% 0.18/0.44
% 0.18/0.44 thf(tcon_HOL_Obool___Orderings_Olinorder_24,axiom,
% 0.18/0.44 linorder @ $o ).
% 0.18/0.44
% 0.18/0.44 thf(tcon_HOL_Obool___Orderings_Oorder_25,axiom,
% 0.18/0.44 order @ $o ).
% 0.18/0.44
% 0.18/0.44 thf(tcon_HOL_Obool___Orderings_Oord_26,axiom,
% 0.18/0.44 ord @ $o ).
% 0.18/0.44
% 0.18/0.44 thf(tcon_Real_Oreal___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_27,axiom,
% 0.18/0.44 condit1037483654norder @ real ).
% 0.18/0.44
% 0.18/0.44 thf(tcon_Real_Oreal___Rings_Olinordered__nonzero__semiring_28,axiom,
% 0.18/0.44 linord1659791738miring @ real ).
% 0.18/0.44
% 0.18/0.44 thf(tcon_Real_Oreal___Rings_Osemiring__no__zero__divisors_29,axiom,
% 0.18/0.44 semiri1193490041visors @ real ).
% 0.18/0.44
% 0.18/0.44 thf(tcon_Real_Oreal___Rings_Oring__no__zero__divisors_30,axiom,
% 0.18/0.44 ring_n68954251visors @ real ).
% 0.18/0.44
% 0.18/0.44 thf(tcon_Real_Oreal___Orderings_Odense__linorder,axiom,
% 0.18/0.44 dense_linorder @ real ).
% 0.18/0.44
% 0.18/0.44 thf(tcon_Real_Oreal___Fields_Olinordered__field,axiom,
% 0.18/0.44 linordered_field @ real ).
% 0.18/0.44
% 0.18/0.44 thf(tcon_Real_Oreal___Rings_Ocomm__semiring__1_31,axiom,
% 0.18/0.44 comm_semiring_1 @ real ).
% 0.18/0.44
% 0.18/0.44 thf(tcon_Real_Oreal___Rings_Ocomm__semiring__0_32,axiom,
% 0.18/0.44 comm_semiring_0 @ real ).
% 0.18/0.44
% 0.18/0.44 thf(tcon_Real_Oreal___Orderings_Odense__order,axiom,
% 0.18/0.44 dense_order @ real ).
% 0.18/0.44
% 0.18/0.44 thf(tcon_Real_Oreal___Nat_Osemiring__char__0_33,axiom,
% 0.18/0.44 semiring_char_0 @ real ).
% 0.18/0.44
% 0.18/0.44 thf(tcon_Real_Oreal___Fields_Ofield__char__0,axiom,
% 0.18/0.44 field_char_0 @ real ).
% 0.18/0.44
% 0.18/0.44 thf(tcon_Real_Oreal___Orderings_Opreorder_34,axiom,
% 0.18/0.44 preorder @ real ).
% 0.18/0.44
% 0.18/0.44 thf(tcon_Real_Oreal___Orderings_Olinorder_35,axiom,
% 0.18/0.44 linorder @ real ).
% 0.18/0.44
% 0.18/0.44 thf(tcon_Real_Oreal___Rings_Ocomm__ring__1_36,axiom,
% 0.18/0.44 comm_ring_1 @ real ).
% 0.18/0.44
% 0.18/0.44 thf(tcon_Real_Oreal___Rings_Osemiring__1_37,axiom,
% 0.18/0.44 semiring_1 @ real ).
% 0.18/0.44
% 0.18/0.44 thf(tcon_Real_Oreal___Orderings_Ono__top_38,axiom,
% 0.18/0.44 no_top @ real ).
% 0.18/0.44
% 0.18/0.44 thf(tcon_Real_Oreal___Orderings_Ono__bot_39,axiom,
% 0.18/0.44 no_bot @ real ).
% 0.18/0.44
% 0.18/0.44 thf(tcon_Real_Oreal___Orderings_Oorder_40,axiom,
% 0.18/0.44 order @ real ).
% 0.18/0.44
% 0.18/0.44 thf(tcon_Real_Oreal___Nat_Oring__char__0_41,axiom,
% 0.18/0.44 ring_char_0 @ real ).
% 0.18/0.44
% 0.18/0.44 thf(tcon_Real_Oreal___Orderings_Oord_42,axiom,
% 0.18/0.44 ord @ real ).
% 0.18/0.44
% 0.18/0.44 thf(tcon_Real_Oreal___Groups_Ozero_43,axiom,
% 0.18/0.44 zero @ real ).
% 0.18/0.44
% 0.18/0.44 thf(tcon_Real_Oreal___Rings_Oidom_44,axiom,
% 0.18/0.44 idom @ real ).
% 0.18/0.44
% 0.18/0.44 thf(tcon_Real_Oreal___Groups_Oone_45,axiom,
% 0.18/0.44 one @ real ).
% 0.18/0.44
% 0.18/0.44 thf(tcon_Polynomial_Opoly___Rings_Olinordered__nonzero__semiring_46,axiom,
% 0.18/0.44 ! [A7: $tType] :
% 0.18/0.44 ( ( linordered_idom @ A7 )
% 0.18/0.44 => ( linord1659791738miring @ ( poly @ A7 ) ) ) ).
% 0.18/0.44
% 0.18/0.44 thf(tcon_Polynomial_Opoly___Rings_Osemiring__no__zero__divisors_47,axiom,
% 0.18/0.44 ! [A7: $tType] :
% 0.18/0.44 ( ( ( comm_semiring_0 @ A7 )
% 0.18/0.44 & ( semiri1193490041visors @ A7 ) )
% 0.18/0.44 => ( semiri1193490041visors @ ( poly @ A7 ) ) ) ).
% 0.18/0.44
% 0.18/0.44 thf(tcon_Polynomial_Opoly___Rings_Oring__no__zero__divisors_48,axiom,
% 0.18/0.44 ! [A7: $tType] :
% 0.18/0.44 ( ( idom @ A7 )
% 0.18/0.44 => ( ring_n68954251visors @ ( poly @ A7 ) ) ) ).
% 0.18/0.44
% 0.18/0.44 thf(tcon_Polynomial_Opoly___Rings_Ocomm__semiring__1_49,axiom,
% 0.18/0.44 ! [A7: $tType] :
% 0.18/0.44 ( ( comm_semiring_1 @ A7 )
% 0.18/0.44 => ( comm_semiring_1 @ ( poly @ A7 ) ) ) ).
% 0.18/0.44
% 0.18/0.44 thf(tcon_Polynomial_Opoly___Rings_Ocomm__semiring__0_50,axiom,
% 0.18/0.44 ! [A7: $tType] :
% 0.18/0.44 ( ( comm_semiring_0 @ A7 )
% 0.18/0.44 => ( comm_semiring_0 @ ( poly @ A7 ) ) ) ).
% 0.18/0.44
% 0.18/0.44 thf(tcon_Polynomial_Opoly___Nat_Osemiring__char__0_51,axiom,
% 0.18/0.44 ! [A7: $tType] :
% 0.18/0.44 ( ( ( ring_char_0 @ A7 )
% 0.18/0.44 & ( comm_ring_1 @ A7 ) )
% 0.18/0.44 => ( semiring_char_0 @ ( poly @ A7 ) ) ) ).
% 0.18/0.44
% 0.18/0.44 thf(tcon_Polynomial_Opoly___Orderings_Opreorder_52,axiom,
% 0.18/0.44 ! [A7: $tType] :
% 0.18/0.44 ( ( linordered_idom @ A7 )
% 0.18/0.44 => ( preorder @ ( poly @ A7 ) ) ) ).
% 0.18/0.44
% 0.18/0.44 thf(tcon_Polynomial_Opoly___Orderings_Olinorder_53,axiom,
% 0.18/0.44 ! [A7: $tType] :
% 0.18/0.44 ( ( linordered_idom @ A7 )
% 0.18/0.44 => ( linorder @ ( poly @ A7 ) ) ) ).
% 0.18/0.44
% 0.18/0.44 thf(tcon_Polynomial_Opoly___Rings_Ocomm__ring__1_54,axiom,
% 0.18/0.44 ! [A7: $tType] :
% 0.18/0.44 ( ( comm_ring_1 @ A7 )
% 0.18/0.44 => ( comm_ring_1 @ ( poly @ A7 ) ) ) ).
% 0.18/0.44
% 0.18/0.44 thf(tcon_Polynomial_Opoly___Rings_Osemiring__1_55,axiom,
% 0.18/0.44 ! [A7: $tType] :
% 0.18/0.44 ( ( comm_semiring_1 @ A7 )
% 0.18/0.44 => ( semiring_1 @ ( poly @ A7 ) ) ) ).
% 0.18/0.44
% 0.18/0.44 thf(tcon_Polynomial_Opoly___Orderings_Oorder_56,axiom,
% 0.18/0.44 ! [A7: $tType] :
% 0.18/0.44 ( ( linordered_idom @ A7 )
% 0.18/0.44 => ( order @ ( poly @ A7 ) ) ) ).
% 0.18/0.44
% 0.18/0.44 thf(tcon_Polynomial_Opoly___Nat_Oring__char__0_57,axiom,
% 0.18/0.44 ! [A7: $tType] :
% 0.18/0.44 ( ( ( ring_char_0 @ A7 )
% 0.18/0.44 & ( comm_ring_1 @ A7 ) )
% 0.18/0.44 => ( ring_char_0 @ ( poly @ A7 ) ) ) ).
% 0.18/0.44
% 0.18/0.44 thf(tcon_Polynomial_Opoly___Orderings_Oord_58,axiom,
% 0.18/0.44 ! [A7: $tType] :
% 0.18/0.44 ( ( linordered_idom @ A7 )
% 0.18/0.44 => ( ord @ ( poly @ A7 ) ) ) ).
% 0.18/0.44
% 0.18/0.44 thf(tcon_Polynomial_Opoly___Groups_Ozero_59,axiom,
% 0.18/0.44 ! [A7: $tType] :
% 0.18/0.44 ( ( zero @ A7 )
% 0.18/0.44 => ( zero @ ( poly @ A7 ) ) ) ).
% 0.18/0.44
% 0.18/0.44 thf(tcon_Polynomial_Opoly___Rings_Oidom_60,axiom,
% 0.18/0.44 ! [A7: $tType] :
% 0.18/0.44 ( ( idom @ A7 )
% 0.18/0.44 => ( idom @ ( poly @ A7 ) ) ) ).
% 0.18/0.44
% 0.18/0.44 thf(tcon_Polynomial_Opoly___Groups_Oone_61,axiom,
% 0.18/0.44 ! [A7: $tType] :
% 0.18/0.44 ( ( comm_semiring_1 @ A7 )
% 0.18/0.44 => ( one @ ( poly @ A7 ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % Conjectures (1)
% 0.18/0.44 thf(conj_0,conjecture,
% 0.18/0.44 ( ( sturm_1700286437_r_pos @ p @ x )
% 0.18/0.44 != ( ~ ( ( ( ( poly2 @ real @ p @ x )
% 0.18/0.44 = ( zero_zero @ real ) )
% 0.18/0.44 => ( sturm_1700286437_r_pos @ ( pderiv @ real @ p ) @ x ) )
% 0.18/0.44 & ( ( ( poly2 @ real @ p @ x )
% 0.18/0.44 != ( zero_zero @ real ) )
% 0.18/0.44 => ( ord_less @ real @ ( zero_zero @ real ) @ ( poly2 @ real @ p @ x ) ) ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 %------------------------------------------------------------------------------
% 0.18/0.44 ------------------------
% 0.28/1.90 ---- Embedded in THF ---
% 0.28/1.90 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 0.28/1.90 %%% Generated on Tue Feb 24 08:29:40 EST 2026
% 0.28/1.90 %%% using '$$dhol' embedding, version 1.3.0.
% 0.28/1.90 %%% Logic specification used:
% 0.28/1.90 %%% thf(spec, logic, $$dhol).
% 0.28/1.90
% 0.28/1.90 % SZS output start ListOfTHF for /export/starexec/sandbox2/tmp/tmp.t0hPoxVh8o/DTF2DTF31756.p
% 0.28/1.90 thf(ty_t_Polynomial_Opoly, type, poly: ($tType > $tType)).
% 0.28/1.90 thf(ty_t_Real_Oreal, type, real: $tType).
% 0.28/1.90 thf(ty_t_Set_Oset, type, set: ($tType > $tType)).
% 0.28/1.90 thf(ty_t_Nat_Onat, type, nat: $tType).
% 0.28/1.90 thf(ty_t_Int_Oint, type, int: $tType).
% 0.28/1.90 thf(sy_cl_Rings_Olinordered__idom, type, linordered_idom: (!> [A:$tType]: ($o))).
% 0.28/1.90 thf(sy_cl_HOL_Otype, type, type: (!> [A:$tType]: ($o))).
% 0.28/1.90 thf(sy_cl_Groups_Oone, type, one: (!> [A:$tType]: ($o))).
% 0.28/1.90 thf(sy_cl_Rings_Oidom, type, idom: (!> [A:$tType]: ($o))).
% 0.28/1.90 thf(sy_cl_Groups_Ozero, type, zero: (!> [A:$tType]: ($o))).
% 0.28/1.90 thf(sy_cl_Orderings_Oord, type, ord: (!> [A:$tType]: ($o))).
% 0.28/1.90 thf(sy_cl_Nat_Oring__char__0, type, ring_char_0: (!> [A:$tType]: ($o))).
% 0.28/1.90 thf(sy_cl_Orderings_Oorder, type, order: (!> [A:$tType]: ($o))).
% 0.28/1.90 thf(sy_cl_Orderings_Ono__bot, type, no_bot: (!> [A:$tType]: ($o))).
% 0.28/1.90 thf(sy_cl_Orderings_Ono__top, type, no_top: (!> [A:$tType]: ($o))).
% 0.28/1.90 thf(sy_cl_Rings_Osemiring__1, type, semiring_1: (!> [A:$tType]: ($o))).
% 0.28/1.90 thf(sy_cl_Rings_Ocomm__ring__1, type, comm_ring_1: (!> [A:$tType]: ($o))).
% 0.28/1.90 thf(sy_cl_Orderings_Olinorder, type, linorder: (!> [A:$tType]: ($o))).
% 0.28/1.90 thf(sy_cl_Orderings_Opreorder, type, preorder: (!> [A:$tType]: ($o))).
% 0.28/1.90 thf(sy_cl_Fields_Ofield__char__0, type, field_char_0: (!> [A:$tType]: ($o))).
% 0.28/1.90 thf(sy_cl_Nat_Osemiring__char__0, type, semiring_char_0: (!> [A:$tType]: ($o))).
% 0.28/1.90 thf(sy_cl_Orderings_Owellorder, type, wellorder: (!> [A:$tType]: ($o))).
% 0.28/1.90 thf(sy_cl_Orderings_Odense__order, type, dense_order: (!> [A:$tType]: ($o))).
% 0.28/1.90 thf(sy_cl_Rings_Ocomm__semiring__0, type, comm_semiring_0: (!> [A:$tType]: ($o))).
% 0.28/1.90 thf(sy_cl_Rings_Ocomm__semiring__1, type, comm_semiring_1: (!> [A:$tType]: ($o))).
% 0.28/1.90 thf(sy_cl_Fields_Olinordered__field, type, linordered_field: (!> [A:$tType]: ($o))).
% 0.28/1.90 thf(sy_cl_Orderings_Odense__linorder, type, dense_linorder: (!> [A:$tType]: ($o))).
% 0.28/1.90 thf(sy_cl_Rings_Oring__no__zero__divisors, type, ring_n68954251visors: (!> [A:$tType]: ($o))).
% 0.28/1.90 thf(sy_cl_Rings_Osemiring__no__zero__divisors, type, semiri1193490041visors: (!> [A:$tType]: ($o))).
% 0.28/1.90 thf(sy_cl_Rings_Olinordered__nonzero__semiring, type, linord1659791738miring: (!> [A:$tType]: ($o))).
% 0.28/1.90 thf(sy_cl_Groups_Ocanonically__ordered__monoid__add, type, canoni770627133id_add: (!> [A:$tType]: ($o))).
% 0.28/1.90 thf(sy_cl_Conditionally__Complete__Lattices_Oconditionally__complete__linorder, type, condit1037483654norder: (!> [A:$tType]: ($o))).
% 0.28/1.90 thf(sy_c_Groups_Oone__class_Oone, type, one_one: (!> [A:$tType]: (A))).
% 0.28/1.90 thf(sy_c_Groups_Ozero__class_Ozero, type, zero_zero: (!> [A:$tType]: (A))).
% 0.28/1.90 thf(sy_c_Nat_OSuc, type, suc: (nat > nat)).
% 0.28/1.90 thf(sy_c_Nat_Osemiring__1__class_Oof__nat, type, semiring_1_of_nat: (!> [A:$tType]: ((nat > A)))).
% 0.28/1.90 thf(sy_c_Orderings_Oord__class_Oless, type, ord_less: (!> [A:$tType]: ((A > (A > $o))))).
% 0.28/1.90 thf(sy_c_Orderings_Oord__class_Oless__eq, type, ord_less_eq: (!> [A:$tType]: ((A > (A > $o))))).
% 0.28/1.90 thf(sy_c_Polynomial_Ois__zero, type, is_zero: (!> [A:$tType]: (((poly @ A) > $o)))).
% 0.28/1.90 thf(sy_c_Polynomial_Oorder, type, order2: (!> [A:$tType]: ((A > ((poly @ A) > nat))))).
% 0.28/1.90 thf(sy_c_Polynomial_Opderiv, type, pderiv: (!> [A:$tType]: (((poly @ A) > (poly @ A))))).
% 0.28/1.90 thf(sy_c_Polynomial_Opoly, type, poly2: (!> [A:$tType]: (((poly @ A) > (A > A))))).
% 0.28/1.90 thf(sy_c_Polynomial_Opoly_Ocoeff, type, coeff: (!> [A:$tType]: (((poly @ A) > (nat > A))))).
% 0.28/1.90 thf(sy_c_Polynomial_Opoly__cutoff, type, poly_cutoff: (!> [A:$tType]: ((nat > ((poly @ A) > (poly @ A)))))).
% 0.28/1.90 thf(sy_c_Polynomial_Opoly__shift, type, poly_shift: (!> [A:$tType]: ((nat > ((poly @ A) > (poly @ A)))))).
% 0.28/1.90 thf(sy_c_Polynomial_Oreflect__poly, type, reflect_poly: (!> [A:$tType]: (((poly @ A) > (poly @ A))))).
% 0.28/1.90 thf(sy_c_Polynomial_Orsquarefree, type, rsquarefree: (!> [A:$tType]: (((poly @ A) > $o)))).
% 0.28/1.90 thf(sy_c_Set_OCollect, type, collect: (!> [A:$tType]: (((A > $o) > (set @ A))))).
% 0.28/1.90 thf(sy_c_Sturm__Tarski__Mirabelle__osyjafhhev_Ocross, type, sturm_424270202_cross: ((poly @ real) > (real > (real > int)))).
% 0.28/1.90 thf(sy_c_Sturm__Tarski__Mirabelle__osyjafhhev_Osign__r__pos, type, sturm_1700286437_r_pos: ((poly @ real) > (real > $o))).
% 0.28/1.90 thf(sy_c_Sturm__Tarski__Mirabelle__osyjafhhev_Ovariation, type, sturm_1771227917iation: (real > (real > int))).
% 0.28/1.90 thf(sy_c_member, type, member: (!> [A:$tType]: ((A > ((set @ A) > $o))))).
% 0.28/1.90 thf(sy_v_p, type, p: (poly @ real)).
% 0.28/1.90 thf(sy_v_x, type, x: real).
% 0.28/1.90 thf(fact_0_False, axiom, ((((poly2 @ real) @ p) @ x) != (zero_zero @ real))).
% 0.28/1.90 thf(fact_1__092_060open_062sign__r__pos_Ap_Ax_A_092_060Longrightarrow_062_A0_A_060_Apoly_Ap_Ax_092_060close_062, axiom, (((sturm_1700286437_r_pos @ p) @ x) => (((ord_less @ real) @ (zero_zero @ real)) @ (((poly2 @ real) @ p) @ x)))).
% 0.28/1.90 thf(fact_2__092_060open_0620_A_060_Apoly_Ap_Ax_A_092_060Longrightarrow_062_Asign__r__pos_Ap_Ax_092_060close_062, axiom, ((((ord_less @ real) @ (zero_zero @ real)) @ (((poly2 @ real) @ p) @ x)) => ((sturm_1700286437_r_pos @ p) @ x))).
% 0.28/1.90 thf(fact_3_assms, axiom, (p != (zero_zero @ (poly @ real)))).
% 0.28/1.90 thf(fact_4_poly__0, axiom, (! [A:$tType]: (((comm_semiring_0 @ A) => (! [X:A]: (((((poly2 @ A) @ (zero_zero @ (poly @ A))) @ X) = (zero_zero @ A)))))))).
% 0.28/1.90 thf(fact_5_not__gr__zero, axiom, (! [A:$tType]: (((canoni770627133id_add @ A) => (! [N:A]: (((~ ((((ord_less @ A) @ (zero_zero @ A)) @ N))) = (N = (zero_zero @ A))))))))).
% 0.28/1.90 thf(fact_6_poly__IVT__neg, axiom, (! [A2:real,B:real,P:(poly @ real)]: (((((ord_less @ real) @ A2) @ B) => ((((ord_less @ real) @ (zero_zero @ real)) @ (((poly2 @ real) @ P) @ A2)) => ((((ord_less @ real) @ (((poly2 @ real) @ P) @ B)) @ (zero_zero @ real)) => (? [X2:real]: (((((ord_less @ real) @ A2) @ X2) & ((((ord_less @ real) @ X2) @ B) & ((((poly2 @ real) @ P) @ X2) = (zero_zero @ real)))))))))))).
% 0.28/1.90 thf(fact_7_poly__IVT__pos, axiom, (! [A2:real,B:real,P:(poly @ real)]: (((((ord_less @ real) @ A2) @ B) => ((((ord_less @ real) @ (((poly2 @ real) @ P) @ A2)) @ (zero_zero @ real)) => ((((ord_less @ real) @ (zero_zero @ real)) @ (((poly2 @ real) @ P) @ B)) => (? [X2:real]: (((((ord_less @ real) @ A2) @ X2) & ((((ord_less @ real) @ X2) @ B) & ((((poly2 @ real) @ P) @ X2) = (zero_zero @ real)))))))))))).
% 0.28/1.90 thf(fact_8_poly__all__0__iff__0, axiom, (! [A:$tType]: ((((ring_char_0 @ A) & ((comm_ring_1 @ A) & (ring_n68954251visors @ A))) => (! [P:(poly @ A)]: (((! [X3:A]: (((((poly2 @ A) @ P) @ X3) = (zero_zero @ A)))) = (P = (zero_zero @ (poly @ A)))))))))).
% 0.28/1.90 thf(fact_9_less__numeral__extra_I3_J, axiom, (! [A:$tType]: (((linord1659791738miring @ A) => (~ ((((ord_less @ A) @ (zero_zero @ A)) @ (zero_zero @ A)))))))).
% 0.28/1.90 thf(fact_10_field__lbound__gt__zero, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [D1:A,D2:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ D1) => ((((ord_less @ A) @ (zero_zero @ A)) @ D2) => (? [E:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ E) & ((((ord_less @ A) @ E) @ D1) & (((ord_less @ A) @ E) @ D2))))))))))))).
% 0.28/1.90 thf(fact_11_gr__zeroI, axiom, (! [A:$tType]: (((canoni770627133id_add @ A) => (! [N:A]: (((N != (zero_zero @ A)) => (((ord_less @ A) @ (zero_zero @ A)) @ N)))))))).
% 0.28/1.90 thf(fact_12_not__less__zero, axiom, (! [A:$tType]: (((canoni770627133id_add @ A) => (! [N:A]: ((~ ((((ord_less @ A) @ N) @ (zero_zero @ A)))))))))).
% 0.28/1.90 thf(fact_13_gr__implies__not__zero, axiom, (! [A:$tType]: (((canoni770627133id_add @ A) => (! [M:A,N:A]: (((((ord_less @ A) @ M) @ N) => (N != (zero_zero @ A))))))))).
% 0.28/1.90 thf(fact_14_zero__less__iff__neq__zero, axiom, (! [A:$tType]: (((canoni770627133id_add @ A) => (! [N:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ N) = (N != (zero_zero @ A))))))))).
% 0.28/1.90 thf(fact_15_pderiv__0, axiom, (! [A:$tType]: ((((comm_semiring_1 @ A) & (semiri1193490041visors @ A)) => (((pderiv @ A) @ (zero_zero @ (poly @ A))) = (zero_zero @ (poly @ A))))))).
% 0.28/1.90 thf(fact_16_zero__reorient, axiom, (! [A:$tType]: (((zero @ A) => (! [X:A]: ((((zero_zero @ A) = X) = (X = (zero_zero @ A))))))))).
% 0.28/1.90 thf(fact_17_poly__eq__poly__eq__iff, axiom, (! [A:$tType]: ((((ring_char_0 @ A) & ((comm_ring_1 @ A) & (ring_n68954251visors @ A))) => (! [P:(poly @ A),Q:(poly @ A)]: (((((poly2 @ A) @ P) = ((poly2 @ A) @ Q)) = (P = Q)))))))).
% 0.28/1.90 thf(fact_18_cross__0, axiom, (! [A2:real,B:real]: (((((sturm_424270202_cross @ (zero_zero @ (poly @ real))) @ A2) @ B) = (zero_zero @ int))))).
% 0.28/1.90 thf(fact_19_rsquarefree__roots, axiom, (! [A:$tType]: (((field_char_0 @ A) => ((rsquarefree @ A) = (^ [P2:(poly @ A)]: ((! [A3:A]: ((~ ((((((poly2 @ A) @ P2) @ A3) = (zero_zero @ A)) & ((((poly2 @ A) @ ((pderiv @ A) @ P2)) @ A3) = (zero_zero @ A)))))))))))))).
% 0.28/1.90 thf(fact_20_cross__no__root, axiom, (! [A2:real,B:real,P:(poly @ real)]: (((((ord_less @ real) @ A2) @ B) => ((! [X2:real]: ((((((ord_less @ real) @ A2) @ X2) & (((ord_less @ real) @ X2) @ B)) => ((((poly2 @ real) @ P) @ X2) != (zero_zero @ real))))) => ((((sturm_424270202_cross @ P) @ A2) @ B) = (zero_zero @ int))))))).
% 0.28/1.90 thf(fact_21_is__zero__null, axiom, (! [A:$tType]: (((zero @ A) => ((is_zero @ A) = (^ [P2:(poly @ A)]: ((P2 = (zero_zero @ (poly @ A)))))))))).
% 0.28/1.90 thf(fact_22_poly__cutoff__0, axiom, (! [A:$tType]: (((zero @ A) => (! [N:nat]: (((((poly_cutoff @ A) @ N) @ (zero_zero @ (poly @ A))) = (zero_zero @ (poly @ A))))))))).
% 0.28/1.90 thf(fact_23_reflect__poly__at__0__eq__0__iff, axiom, (! [A:$tType]: (((comm_semiring_0 @ A) => (! [P:(poly @ A)]: ((((((poly2 @ A) @ ((reflect_poly @ A) @ P)) @ (zero_zero @ A)) = (zero_zero @ A)) = (P = (zero_zero @ (poly @ A)))))))))).
% 0.28/1.90 thf(fact_24_next__non__root__interval, axiom, (! [P:(poly @ real),Lb:real]: (((P != (zero_zero @ (poly @ real))) => (~ ((! [Ub:real]: (((((ord_less @ real) @ Lb) @ Ub) => (~ ((! [Z:real]: ((((((ord_less @ real) @ Lb) @ Z) & (((ord_less_eq @ real) @ Z) @ Ub)) => ((((poly2 @ real) @ P) @ Z) != (zero_zero @ real)))))))))))))))).
% 0.28/1.90 thf(fact_25_last__non__root__interval, axiom, (! [P:(poly @ real),Ub2:real]: (((P != (zero_zero @ (poly @ real))) => (~ ((! [Lb2:real]: (((((ord_less @ real) @ Lb2) @ Ub2) => (~ ((! [Z:real]: ((((((ord_less_eq @ real) @ Lb2) @ Z) & (((ord_less @ real) @ Z) @ Ub2)) => ((((poly2 @ real) @ P) @ Z) != (zero_zero @ real)))))))))))))))).
% 0.28/1.90 thf(fact_26_poly__shift__0, axiom, (! [A:$tType]: (((zero @ A) => (! [N:nat]: (((((poly_shift @ A) @ N) @ (zero_zero @ (poly @ A))) = (zero_zero @ (poly @ A))))))))).
% 0.28/1.90 thf(fact_27_order__root, axiom, (! [A:$tType]: (((idom @ A) => (! [P:(poly @ A),A2:A]: ((((((poly2 @ A) @ P) @ A2) = (zero_zero @ A)) = ((P = (zero_zero @ (poly @ A))) | ((((order2 @ A) @ A2) @ P) != (zero_zero @ nat)))))))))).
% 0.28/1.90 thf(fact_28_le__zero__eq, axiom, (! [A:$tType]: (((canoni770627133id_add @ A) => (! [N:A]: (((((ord_less_eq @ A) @ N) @ (zero_zero @ A)) = (N = (zero_zero @ A))))))))).
% 0.28/1.90 thf(fact_29_reflect__poly__0, axiom, (! [A:$tType]: (((zero @ A) => (((reflect_poly @ A) @ (zero_zero @ (poly @ A))) = (zero_zero @ (poly @ A))))))).
% 0.28/1.90 thf(fact_30_poly__shift__id, axiom, (! [A:$tType]: (((zero @ A) => (((poly_shift @ A) @ (zero_zero @ nat)) = (^ [X3:(poly @ A)]: (X3))))))).
% 0.28/1.90 thf(fact_31_complete__real, axiom, (! [S:(set @ real)]: (((? [X4:real]: ((((member @ real) @ X4) @ S))) => ((? [Z:real]: ((! [X2:real]: (((((member @ real) @ X2) @ S) => (((ord_less_eq @ real) @ X2) @ Z)))))) => (? [Y:real]: (((! [X4:real]: (((((member @ real) @ X4) @ S) => (((ord_less_eq @ real) @ X4) @ Y)))) & (! [Z:real]: (((! [X2:real]: (((((member @ real) @ X2) @ S) => (((ord_less_eq @ real) @ X2) @ Z)))) => (((ord_less_eq @ real) @ Y) @ Z)))))))))))).
% 0.28/1.90 thf(fact_32_le__numeral__extra_I3_J, axiom, (! [A:$tType]: (((linord1659791738miring @ A) => (((ord_less_eq @ A) @ (zero_zero @ A)) @ (zero_zero @ A)))))).
% 0.28/1.90 thf(fact_33_zero__le, axiom, (! [A:$tType]: (((canoni770627133id_add @ A) => (! [X:A]: ((((ord_less_eq @ A) @ (zero_zero @ A)) @ X))))))).
% 0.28/1.90 thf(fact_34_less__eq__real__def, axiom, ((ord_less_eq @ real) = (^ [X3:real,Y2:real]: (((((ord_less @ real) @ X3) @ Y2) | (X3 = Y2)))))).
% 0.28/1.90 thf(fact_35_order__0I, axiom, (! [A:$tType]: (((idom @ A) => (! [P:(poly @ A),A2:A]: ((((((poly2 @ A) @ P) @ A2) != (zero_zero @ A)) => ((((order2 @ A) @ A2) @ P) = (zero_zero @ nat))))))))).
% 0.28/1.90 thf(fact_36_not__eq__pos__or__neg__iff__1, axiom, (! [Lb:real,Ub2:real,P:(poly @ real)]: (((! [Z2:real]: ((((((ord_less @ real) @ Lb) @ Z2) & (((ord_less_eq @ real) @ Z2) @ Ub2)) => ((((poly2 @ real) @ P) @ Z2) != (zero_zero @ real))))) = ((! [Z2:real]: ((((((ord_less @ real) @ Lb) @ Z2) & (((ord_less_eq @ real) @ Z2) @ Ub2)) => (((ord_less @ real) @ (zero_zero @ real)) @ (((poly2 @ real) @ P) @ Z2))))) | (! [Z2:real]: ((((((ord_less @ real) @ Lb) @ Z2) & (((ord_less_eq @ real) @ Z2) @ Ub2)) => (((ord_less @ real) @ (((poly2 @ real) @ P) @ Z2)) @ (zero_zero @ real)))))))))).
% 0.28/1.91 thf(fact_37_not__eq__pos__or__neg__iff__2, axiom, (! [Lb:real,Ub2:real,P:(poly @ real)]: (((! [Z2:real]: ((((((ord_less_eq @ real) @ Lb) @ Z2) & (((ord_less @ real) @ Z2) @ Ub2)) => ((((poly2 @ real) @ P) @ Z2) != (zero_zero @ real))))) = ((! [Z2:real]: ((((((ord_less_eq @ real) @ Lb) @ Z2) & (((ord_less @ real) @ Z2) @ Ub2)) => (((ord_less @ real) @ (zero_zero @ real)) @ (((poly2 @ real) @ P) @ Z2))))) | (! [Z2:real]: ((((((ord_less_eq @ real) @ Lb) @ Z2) & (((ord_less @ real) @ Z2) @ Ub2)) => (((ord_less @ real) @ (((poly2 @ real) @ P) @ Z2)) @ (zero_zero @ real)))))))))).
% 0.28/1.91 thf(fact_38_neq0__conv, axiom, (! [N:nat]: (((N != (zero_zero @ nat)) = (((ord_less @ nat) @ (zero_zero @ nat)) @ N))))).
% 0.28/1.91 thf(fact_39_less__nat__zero__code, axiom, (! [N:nat]: ((~ ((((ord_less @ nat) @ N) @ (zero_zero @ nat))))))).
% 0.28/1.91 thf(fact_40_bot__nat__0_Onot__eq__extremum, axiom, (! [A2:nat]: (((A2 != (zero_zero @ nat)) = (((ord_less @ nat) @ (zero_zero @ nat)) @ A2))))).
% 0.28/1.91 thf(fact_41_order__refl, axiom, (! [A:$tType]: (((preorder @ A) => (! [X:A]: ((((ord_less_eq @ A) @ X) @ X))))))).
% 0.28/1.91 thf(fact_42_minf_I8_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [T:A]: ((? [Z3:A]: ((! [X4:A]: (((((ord_less @ A) @ X4) @ Z3) => (~ ((((ord_less_eq @ A) @ T) @ X4)))))))))))))).
% 0.28/1.91 thf(fact_43_minf_I6_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [T:A]: ((? [Z3:A]: ((! [X4:A]: (((((ord_less @ A) @ X4) @ Z3) => (((ord_less_eq @ A) @ X4) @ T)))))))))))).
% 0.28/1.91 thf(fact_44_mem__Collect__eq, axiom, (! [A:$tType,A2:A,P3:(A > $o)]: (((((member @ A) @ A2) @ ((collect @ A) @ P3)) = (P3 @ A2))))).
% 0.28/1.91 thf(fact_45_Collect__mem__eq, axiom, (! [A:$tType,A4:(set @ A)]: ((((collect @ A) @ (^ [X3:A]: ((((member @ A) @ X3) @ A4)))) = A4)))).
% 0.28/1.91 thf(fact_46_Collect__cong, axiom, (! [A:$tType,P3:(A > $o),Q2:(A > $o)]: (((! [X2:A]: (((P3 @ X2) = (Q2 @ X2)))) => (((collect @ A) @ P3) = ((collect @ A) @ Q2)))))).
% 0.28/1.91 thf(fact_47_ext, axiom, (! [B2:$tType,A:$tType,F:(A > B2),G:(A > B2)]: (((! [X2:A]: (((F @ X2) = (G @ X2)))) => (F = G))))).
% 0.28/1.91 thf(fact_48_pinf_I8_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [T:A]: ((? [Z3:A]: ((! [X4:A]: (((((ord_less @ A) @ Z3) @ X4) => (((ord_less_eq @ A) @ T) @ X4)))))))))))).
% 0.28/1.91 thf(fact_49_pinf_I6_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [T:A]: ((? [Z3:A]: ((! [X4:A]: (((((ord_less @ A) @ Z3) @ X4) => (~ ((((ord_less_eq @ A) @ X4) @ T)))))))))))))).
% 0.28/1.91 thf(fact_50_verit__comp__simplify1_I3_J, axiom, (! [B2:$tType]: (((linorder @ B2) => (! [B3:B2,A5:B2]: (((~ ((((ord_less_eq @ B2) @ B3) @ A5))) = (((ord_less @ B2) @ A5) @ B3)))))))).
% 0.28/1.91 thf(fact_51_bot__nat__0_Oextremum, axiom, (! [A2:nat]: ((((ord_less_eq @ nat) @ (zero_zero @ nat)) @ A2)))).
% 0.28/1.91 thf(fact_52_le0, axiom, (! [N:nat]: ((((ord_less_eq @ nat) @ (zero_zero @ nat)) @ N)))).
% 0.28/1.91 thf(fact_53_nat__less__le, axiom, ((ord_less @ nat) = (^ [M2:nat,N2:nat]: (((((ord_less_eq @ nat) @ M2) @ N2) & (M2 != N2)))))).
% 0.28/1.91 thf(fact_54_less__imp__le__nat, axiom, (! [M:nat,N:nat]: (((((ord_less @ nat) @ M) @ N) => (((ord_less_eq @ nat) @ M) @ N))))).
% 0.28/1.91 thf(fact_55_le__eq__less__or__eq, axiom, ((ord_less_eq @ nat) = (^ [M2:nat,N2:nat]: (((((ord_less @ nat) @ M2) @ N2) | (M2 = N2)))))).
% 0.28/1.91 thf(fact_56_less__or__eq__imp__le, axiom, (! [M:nat,N:nat]: ((((((ord_less @ nat) @ M) @ N) | (M = N)) => (((ord_less_eq @ nat) @ M) @ N))))).
% 0.28/1.91 thf(fact_57_le__neq__implies__less, axiom, (! [M:nat,N:nat]: (((((ord_less_eq @ nat) @ M) @ N) => ((M != N) => (((ord_less @ nat) @ M) @ N)))))).
% 0.28/1.91 thf(fact_58_less__mono__imp__le__mono, axiom, (! [F:(nat > nat),I:nat,J:nat]: (((! [I2:nat,J2:nat]: (((((ord_less @ nat) @ I2) @ J2) => (((ord_less @ nat) @ (F @ I2)) @ (F @ J2))))) => ((((ord_less_eq @ nat) @ I) @ J) => (((ord_less_eq @ nat) @ (F @ I)) @ (F @ J))))))).
% 0.28/1.91 thf(fact_59_infinite__descent__measure, axiom, (! [A:$tType,P3:(A > $o),V:(A > nat),X:A]: (((! [X2:A]: (((~ ((P3 @ X2))) => (? [Y3:A]: (((((ord_less @ nat) @ (V @ Y3)) @ (V @ X2)) & (~ ((P3 @ Y3))))))))) => (P3 @ X))))).
% 0.28/1.91 thf(fact_60_linorder__neqE__nat, axiom, (! [X:nat,Y4:nat]: (((X != Y4) => ((~ ((((ord_less @ nat) @ X) @ Y4))) => (((ord_less @ nat) @ Y4) @ X)))))).
% 0.28/1.91 thf(fact_61_infinite__descent, axiom, (! [P3:(nat > $o),N:nat]: (((! [N3:nat]: (((~ ((P3 @ N3))) => (? [M3:nat]: (((((ord_less @ nat) @ M3) @ N3) & (~ ((P3 @ M3))))))))) => (P3 @ N))))).
% 0.28/1.91 thf(fact_62_nat__less__induct, axiom, (! [P3:(nat > $o),N:nat]: (((! [N3:nat]: (((! [M3:nat]: (((((ord_less @ nat) @ M3) @ N3) => (P3 @ M3)))) => (P3 @ N3)))) => (P3 @ N))))).
% 0.28/1.91 thf(fact_63_less__irrefl__nat, axiom, (! [N:nat]: ((~ ((((ord_less @ nat) @ N) @ N)))))).
% 0.28/1.91 thf(fact_64_less__not__refl3, axiom, (! [S2:nat,T:nat]: (((((ord_less @ nat) @ S2) @ T) => (S2 != T))))).
% 0.28/1.91 thf(fact_65_less__not__refl2, axiom, (! [N:nat,M:nat]: (((((ord_less @ nat) @ N) @ M) => (M != N))))).
% 0.28/1.91 thf(fact_66_less__not__refl, axiom, (! [N:nat]: ((~ ((((ord_less @ nat) @ N) @ N)))))).
% 0.28/1.91 thf(fact_67_nat__neq__iff, axiom, (! [M:nat,N:nat]: (((M != N) = ((((ord_less @ nat) @ M) @ N) | (((ord_less @ nat) @ N) @ M)))))).
% 0.28/1.91 thf(fact_68_bot__nat__0_Oextremum__uniqueI, axiom, (! [A2:nat]: (((((ord_less_eq @ nat) @ A2) @ (zero_zero @ nat)) => (A2 = (zero_zero @ nat)))))).
% 0.28/1.91 thf(fact_69_bot__nat__0_Oextremum__unique, axiom, (! [A2:nat]: (((((ord_less_eq @ nat) @ A2) @ (zero_zero @ nat)) = (A2 = (zero_zero @ nat)))))).
% 0.28/1.91 thf(fact_70_ex__least__nat__le, axiom, (! [P3:(nat > $o),N:nat]: (((P3 @ N) => ((~ ((P3 @ (zero_zero @ nat)))) => (? [K:nat]: (((((ord_less_eq @ nat) @ K) @ N) & ((! [I3:nat]: (((((ord_less @ nat) @ I3) @ K) => (~ ((P3 @ I3)))))) & (P3 @ K)))))))))).
% 0.28/1.91 thf(fact_71_le__0__eq, axiom, (! [N:nat]: (((((ord_less_eq @ nat) @ N) @ (zero_zero @ nat)) = (N = (zero_zero @ nat)))))).
% 0.28/1.91 thf(fact_72_less__eq__nat_Osimps_I1_J, axiom, (! [N:nat]: ((((ord_less_eq @ nat) @ (zero_zero @ nat)) @ N)))).
% 0.28/1.91 thf(fact_73_conj__le__cong, axiom, (! [X:int,X5:int,P3:$o,P4:$o]: (((X = X5) => (((((ord_less_eq @ int) @ (zero_zero @ int)) @ X5) => (P3 = P4)) => (((((ord_less_eq @ int) @ (zero_zero @ int)) @ X) & P3) = ((((ord_less_eq @ int) @ (zero_zero @ int)) @ X5) & P4))))))).
% 0.28/1.91 thf(fact_74_imp__le__cong, axiom, (! [X:int,X5:int,P3:$o,P4:$o]: (((X = X5) => (((((ord_less_eq @ int) @ (zero_zero @ int)) @ X5) => (P3 = P4)) => (((((ord_less_eq @ int) @ (zero_zero @ int)) @ X) => P3) = ((((ord_less_eq @ int) @ (zero_zero @ int)) @ X5) => P4))))))).
% 0.28/1.91 thf(fact_75_le__funD, axiom, (! [B2:$tType,A:$tType]: (((ord @ B2) => (! [F:(A > B2),G:(A > B2),X:A]: (((((ord_less_eq @ (A > B2)) @ F) @ G) => (((ord_less_eq @ B2) @ (F @ X)) @ (G @ X))))))))).
% 0.28/1.91 thf(fact_76_le__funE, axiom, (! [B2:$tType,A:$tType]: (((ord @ B2) => (! [F:(A > B2),G:(A > B2),X:A]: (((((ord_less_eq @ (A > B2)) @ F) @ G) => (((ord_less_eq @ B2) @ (F @ X)) @ (G @ X))))))))).
% 0.28/1.91 thf(fact_77_le__funI, axiom, (! [B2:$tType,A:$tType]: (((ord @ B2) => (! [F:(A > B2),G:(A > B2)]: (((! [X2:A]: ((((ord_less_eq @ B2) @ (F @ X2)) @ (G @ X2)))) => (((ord_less_eq @ (A > B2)) @ F) @ G)))))))).
% 0.28/1.91 thf(fact_78_le__fun__def, axiom, (! [B2:$tType,A:$tType]: (((ord @ B2) => ((ord_less_eq @ (A > B2)) = (^ [F2:(A > B2),G2:(A > B2)]: ((! [X3:A]: ((((ord_less_eq @ B2) @ (F2 @ X3)) @ (G2 @ X3))))))))))).
% 0.28/1.91 thf(fact_79_order__subst1, axiom, (! [A:$tType,B2:$tType]: ((((order @ B2) & (order @ A)) => (! [A2:A,F:(B2 > A),B:B2,C:B2]: (((((ord_less_eq @ A) @ A2) @ (F @ B)) => ((((ord_less_eq @ B2) @ B) @ C) => ((! [X2:B2,Y:B2]: (((((ord_less_eq @ B2) @ X2) @ Y) => (((ord_less_eq @ A) @ (F @ X2)) @ (F @ Y))))) => (((ord_less_eq @ A) @ A2) @ (F @ C))))))))))).
% 0.28/1.91 thf(fact_80_order__subst2, axiom, (! [A:$tType,C2:$tType]: ((((order @ C2) & (order @ A)) => (! [A2:A,B:A,F:(A > C2),C:C2]: (((((ord_less_eq @ A) @ A2) @ B) => ((((ord_less_eq @ C2) @ (F @ B)) @ C) => ((! [X2:A,Y:A]: (((((ord_less_eq @ A) @ X2) @ Y) => (((ord_less_eq @ C2) @ (F @ X2)) @ (F @ Y))))) => (((ord_less_eq @ C2) @ (F @ A2)) @ C)))))))))).
% 0.28/1.91 thf(fact_81_verit__la__disequality, axiom, (! [A:$tType]: (((linorder @ A) => (! [A2:A,B:A]: (((A2 = B) | ((~ ((((ord_less_eq @ A) @ A2) @ B))) | (~ ((((ord_less_eq @ A) @ B) @ A2))))))))))).
% 0.28/1.91 thf(fact_82_ord__eq__le__subst, axiom, (! [A:$tType,B2:$tType]: ((((ord @ B2) & (ord @ A)) => (! [A2:A,F:(B2 > A),B:B2,C:B2]: (((A2 = (F @ B)) => ((((ord_less_eq @ B2) @ B) @ C) => ((! [X2:B2,Y:B2]: (((((ord_less_eq @ B2) @ X2) @ Y) => (((ord_less_eq @ A) @ (F @ X2)) @ (F @ Y))))) => (((ord_less_eq @ A) @ A2) @ (F @ C))))))))))).
% 0.28/1.91 thf(fact_83_ord__le__eq__subst, axiom, (! [A:$tType,B2:$tType]: ((((ord @ B2) & (ord @ A)) => (! [A2:A,B:A,F:(A > B2),C:B2]: (((((ord_less_eq @ A) @ A2) @ B) => (((F @ B) = C) => ((! [X2:A,Y:A]: (((((ord_less_eq @ A) @ X2) @ Y) => (((ord_less_eq @ B2) @ (F @ X2)) @ (F @ Y))))) => (((ord_less_eq @ B2) @ (F @ A2)) @ C)))))))))).
% 0.28/1.91 thf(fact_84_eq__iff, axiom, (! [A:$tType]: (((order @ A) => ((^ [Y5:A,Z4:A]: ((Y5 = Z4))) = (^ [X3:A,Y2:A]: (((((ord_less_eq @ A) @ X3) @ Y2) & (((ord_less_eq @ A) @ Y2) @ X3))))))))).
% 0.28/1.91 thf(fact_85_antisym, axiom, (! [A:$tType]: (((order @ A) => (! [X:A,Y4:A]: (((((ord_less_eq @ A) @ X) @ Y4) => ((((ord_less_eq @ A) @ Y4) @ X) => (X = Y4))))))))).
% 0.28/1.91 thf(fact_86_linear, axiom, (! [A:$tType]: (((linorder @ A) => (! [X:A,Y4:A]: (((((ord_less_eq @ A) @ X) @ Y4) | (((ord_less_eq @ A) @ Y4) @ X)))))))).
% 0.28/1.91 thf(fact_87_eq__refl, axiom, (! [A:$tType]: (((preorder @ A) => (! [X:A,Y4:A]: (((X = Y4) => (((ord_less_eq @ A) @ X) @ Y4)))))))).
% 0.28/1.91 thf(fact_88_le__cases, axiom, (! [A:$tType]: (((linorder @ A) => (! [X:A,Y4:A]: (((~ ((((ord_less_eq @ A) @ X) @ Y4))) => (((ord_less_eq @ A) @ Y4) @ X)))))))).
% 0.28/1.91 thf(fact_89_order_Otrans, axiom, (! [A:$tType]: (((order @ A) => (! [A2:A,B:A,C:A]: (((((ord_less_eq @ A) @ A2) @ B) => ((((ord_less_eq @ A) @ B) @ C) => (((ord_less_eq @ A) @ A2) @ C))))))))).
% 0.28/1.91 thf(fact_90_le__cases3, axiom, (! [A:$tType]: (((linorder @ A) => (! [X:A,Y4:A,Z5:A]: ((((((ord_less_eq @ A) @ X) @ Y4) => (~ ((((ord_less_eq @ A) @ Y4) @ Z5)))) => (((((ord_less_eq @ A) @ Y4) @ X) => (~ ((((ord_less_eq @ A) @ X) @ Z5)))) => (((((ord_less_eq @ A) @ X) @ Z5) => (~ ((((ord_less_eq @ A) @ Z5) @ Y4)))) => (((((ord_less_eq @ A) @ Z5) @ Y4) => (~ ((((ord_less_eq @ A) @ Y4) @ X)))) => (((((ord_less_eq @ A) @ Y4) @ Z5) => (~ ((((ord_less_eq @ A) @ Z5) @ X)))) => (~ (((((ord_less_eq @ A) @ Z5) @ X) => (~ ((((ord_less_eq @ A) @ X) @ Y4))))))))))))))))).
% 0.28/1.91 thf(fact_91_antisym__conv, axiom, (! [A:$tType]: (((order @ A) => (! [Y4:A,X:A]: (((((ord_less_eq @ A) @ Y4) @ X) => ((((ord_less_eq @ A) @ X) @ Y4) = (X = Y4))))))))).
% 0.28/1.91 thf(fact_92_order__class_Oorder_Oeq__iff, axiom, (! [A:$tType]: (((order @ A) => ((^ [Y5:A,Z4:A]: ((Y5 = Z4))) = (^ [A3:A,B4:A]: (((((ord_less_eq @ A) @ A3) @ B4) & (((ord_less_eq @ A) @ B4) @ A3))))))))).
% 0.28/1.91 thf(fact_93_ord__eq__le__trans, axiom, (! [A:$tType]: (((ord @ A) => (! [A2:A,B:A,C:A]: (((A2 = B) => ((((ord_less_eq @ A) @ B) @ C) => (((ord_less_eq @ A) @ A2) @ C))))))))).
% 0.28/1.91 thf(fact_94_ord__le__eq__trans, axiom, (! [A:$tType]: (((ord @ A) => (! [A2:A,B:A,C:A]: (((((ord_less_eq @ A) @ A2) @ B) => ((B = C) => (((ord_less_eq @ A) @ A2) @ C))))))))).
% 0.28/1.91 thf(fact_95_order__class_Oorder_Oantisym, axiom, (! [A:$tType]: (((order @ A) => (! [A2:A,B:A]: (((((ord_less_eq @ A) @ A2) @ B) => ((((ord_less_eq @ A) @ B) @ A2) => (A2 = B))))))))).
% 0.28/1.91 thf(fact_96_order__trans, axiom, (! [A:$tType]: (((preorder @ A) => (! [X:A,Y4:A,Z5:A]: (((((ord_less_eq @ A) @ X) @ Y4) => ((((ord_less_eq @ A) @ Y4) @ Z5) => (((ord_less_eq @ A) @ X) @ Z5))))))))).
% 0.28/1.91 thf(fact_97_dual__order_Orefl, axiom, (! [A:$tType]: (((order @ A) => (! [A2:A]: ((((ord_less_eq @ A) @ A2) @ A2))))))).
% 0.28/1.91 thf(fact_98_linorder__wlog, axiom, (! [A:$tType]: (((linorder @ A) => (! [P3:(A > (A > $o)),A2:A,B:A]: (((! [A6:A,B5:A]: (((((ord_less_eq @ A) @ A6) @ B5) => ((P3 @ A6) @ B5)))) => ((! [A6:A,B5:A]: ((((P3 @ B5) @ A6) => ((P3 @ A6) @ B5)))) => ((P3 @ A2) @ B))))))))).
% 0.28/1.91 thf(fact_99_dual__order_Otrans, axiom, (! [A:$tType]: (((order @ A) => (! [B:A,A2:A,C:A]: (((((ord_less_eq @ A) @ B) @ A2) => ((((ord_less_eq @ A) @ C) @ B) => (((ord_less_eq @ A) @ C) @ A2))))))))).
% 0.28/1.91 thf(fact_100_dual__order_Oeq__iff, axiom, (! [A:$tType]: (((order @ A) => ((^ [Y5:A,Z4:A]: ((Y5 = Z4))) = (^ [A3:A,B4:A]: (((((ord_less_eq @ A) @ B4) @ A3) & (((ord_less_eq @ A) @ A3) @ B4))))))))).
% 0.28/1.91 thf(fact_101_dual__order_Oantisym, axiom, (! [A:$tType]: (((order @ A) => (! [B:A,A2:A]: (((((ord_less_eq @ A) @ B) @ A2) => ((((ord_less_eq @ A) @ A2) @ B) => (A2 = B))))))))).
% 0.28/1.91 thf(fact_102_dual__order_Ostrict__implies__not__eq, axiom, (! [A:$tType]: (((order @ A) => (! [B:A,A2:A]: (((((ord_less @ A) @ B) @ A2) => (A2 != B)))))))).
% 0.28/1.91 thf(fact_103_order_Ostrict__implies__not__eq, axiom, (! [A:$tType]: (((order @ A) => (! [A2:A,B:A]: (((((ord_less @ A) @ A2) @ B) => (A2 != B)))))))).
% 0.28/1.91 thf(fact_104_not__less__iff__gr__or__eq, axiom, (! [A:$tType]: (((linorder @ A) => (! [X:A,Y4:A]: (((~ ((((ord_less @ A) @ X) @ Y4))) = ((((ord_less @ A) @ Y4) @ X) | (X = Y4))))))))).
% 0.28/1.91 thf(fact_105_dual__order_Ostrict__trans, axiom, (! [A:$tType]: (((order @ A) => (! [B:A,A2:A,C:A]: (((((ord_less @ A) @ B) @ A2) => ((((ord_less @ A) @ C) @ B) => (((ord_less @ A) @ C) @ A2))))))))).
% 0.28/1.91 thf(fact_106_linorder__less__wlog, axiom, (! [A:$tType]: (((linorder @ A) => (! [P3:(A > (A > $o)),A2:A,B:A]: (((! [A6:A,B5:A]: (((((ord_less @ A) @ A6) @ B5) => ((P3 @ A6) @ B5)))) => ((! [A6:A]: (((P3 @ A6) @ A6))) => ((! [A6:A,B5:A]: ((((P3 @ B5) @ A6) => ((P3 @ A6) @ B5)))) => ((P3 @ A2) @ B)))))))))).
% 0.28/1.91 thf(fact_107_exists__least__iff, axiom, (! [A:$tType]: (((wellorder @ A) => ((^ [P5:(A > $o)]: ((? [X6:A]: ((P5 @ X6))))) = (^ [P6:(A > $o)]: ((? [N2:A]: (((P6 @ N2) & (! [M2:A]: (((((ord_less @ A) @ M2) @ N2) => (~ ((P6 @ M2)))))))))))))))).
% 0.28/1.91 thf(fact_108_less__imp__not__less, axiom, (! [A:$tType]: (((preorder @ A) => (! [X:A,Y4:A]: (((((ord_less @ A) @ X) @ Y4) => (~ ((((ord_less @ A) @ Y4) @ X)))))))))).
% 0.28/1.91 thf(fact_109_order_Ostrict__trans, axiom, (! [A:$tType]: (((order @ A) => (! [A2:A,B:A,C:A]: (((((ord_less @ A) @ A2) @ B) => ((((ord_less @ A) @ B) @ C) => (((ord_less @ A) @ A2) @ C))))))))).
% 0.28/1.91 thf(fact_110_dual__order_Oirrefl, axiom, (! [A:$tType]: (((order @ A) => (! [A2:A]: ((~ ((((ord_less @ A) @ A2) @ A2))))))))).
% 0.28/1.91 thf(fact_111_linorder__cases, axiom, (! [A:$tType]: (((linorder @ A) => (! [X:A,Y4:A]: (((~ ((((ord_less @ A) @ X) @ Y4))) => ((X != Y4) => (((ord_less @ A) @ Y4) @ X))))))))).
% 0.28/1.91 thf(fact_112_less__imp__triv, axiom, (! [A:$tType]: (((preorder @ A) => (! [X:A,Y4:A,P3:$o]: (((((ord_less @ A) @ X) @ Y4) => ((((ord_less @ A) @ Y4) @ X) => P3)))))))).
% 0.28/1.91 thf(fact_113_less__imp__not__eq2, axiom, (! [A:$tType]: (((order @ A) => (! [X:A,Y4:A]: (((((ord_less @ A) @ X) @ Y4) => (Y4 != X)))))))).
% 0.28/1.91 thf(fact_114_antisym__conv3, axiom, (! [A:$tType]: (((linorder @ A) => (! [Y4:A,X:A]: (((~ ((((ord_less @ A) @ Y4) @ X))) => ((~ ((((ord_less @ A) @ X) @ Y4))) = (X = Y4))))))))).
% 0.28/1.91 thf(fact_115_less__induct, axiom, (! [A:$tType]: (((wellorder @ A) => (! [P3:(A > $o),A2:A]: (((! [X2:A]: (((! [Y3:A]: (((((ord_less @ A) @ Y3) @ X2) => (P3 @ Y3)))) => (P3 @ X2)))) => (P3 @ A2)))))))).
% 0.28/1.91 thf(fact_116_less__not__sym, axiom, (! [A:$tType]: (((preorder @ A) => (! [X:A,Y4:A]: (((((ord_less @ A) @ X) @ Y4) => (~ ((((ord_less @ A) @ Y4) @ X)))))))))).
% 0.28/1.91 thf(fact_117_less__imp__not__eq, axiom, (! [A:$tType]: (((order @ A) => (! [X:A,Y4:A]: (((((ord_less @ A) @ X) @ Y4) => (X != Y4)))))))).
% 0.28/1.91 thf(fact_118_dual__order_Oasym, axiom, (! [A:$tType]: (((order @ A) => (! [B:A,A2:A]: (((((ord_less @ A) @ B) @ A2) => (~ ((((ord_less @ A) @ A2) @ B)))))))))).
% 0.28/1.91 thf(fact_119_ord__less__eq__trans, axiom, (! [A:$tType]: (((ord @ A) => (! [A2:A,B:A,C:A]: (((((ord_less @ A) @ A2) @ B) => ((B = C) => (((ord_less @ A) @ A2) @ C))))))))).
% 0.28/1.91 thf(fact_120_ord__eq__less__trans, axiom, (! [A:$tType]: (((ord @ A) => (! [A2:A,B:A,C:A]: (((A2 = B) => ((((ord_less @ A) @ B) @ C) => (((ord_less @ A) @ A2) @ C))))))))).
% 0.28/1.91 thf(fact_121_less__irrefl, axiom, (! [A:$tType]: (((preorder @ A) => (! [X:A]: ((~ ((((ord_less @ A) @ X) @ X))))))))).
% 0.28/1.91 thf(fact_122_less__linear, axiom, (! [A:$tType]: (((linorder @ A) => (! [X:A,Y4:A]: (((((ord_less @ A) @ X) @ Y4) | ((X = Y4) | (((ord_less @ A) @ Y4) @ X))))))))).
% 0.28/1.91 thf(fact_123_less__trans, axiom, (! [A:$tType]: (((preorder @ A) => (! [X:A,Y4:A,Z5:A]: (((((ord_less @ A) @ X) @ Y4) => ((((ord_less @ A) @ Y4) @ Z5) => (((ord_less @ A) @ X) @ Z5))))))))).
% 0.28/1.91 thf(fact_124_less__asym_H, axiom, (! [A:$tType]: (((preorder @ A) => (! [A2:A,B:A]: (((((ord_less @ A) @ A2) @ B) => (~ ((((ord_less @ A) @ B) @ A2)))))))))).
% 0.28/1.91 thf(fact_125_less__asym, axiom, (! [A:$tType]: (((preorder @ A) => (! [X:A,Y4:A]: (((((ord_less @ A) @ X) @ Y4) => (~ ((((ord_less @ A) @ Y4) @ X)))))))))).
% 0.28/1.91 thf(fact_126_less__imp__neq, axiom, (! [A:$tType]: (((order @ A) => (! [X:A,Y4:A]: (((((ord_less @ A) @ X) @ Y4) => (X != Y4)))))))).
% 0.28/1.91 thf(fact_127_dense, axiom, (! [A:$tType]: (((dense_order @ A) => (! [X:A,Y4:A]: (((((ord_less @ A) @ X) @ Y4) => (? [Z3:A]: (((((ord_less @ A) @ X) @ Z3) & (((ord_less @ A) @ Z3) @ Y4))))))))))).
% 0.28/1.91 thf(fact_128_order_Oasym, axiom, (! [A:$tType]: (((order @ A) => (! [A2:A,B:A]: (((((ord_less @ A) @ A2) @ B) => (~ ((((ord_less @ A) @ B) @ A2)))))))))).
% 0.28/1.91 thf(fact_129_neq__iff, axiom, (! [A:$tType]: (((linorder @ A) => (! [X:A,Y4:A]: (((X != Y4) = ((((ord_less @ A) @ X) @ Y4) | (((ord_less @ A) @ Y4) @ X))))))))).
% 0.28/1.91 thf(fact_130_neqE, axiom, (! [A:$tType]: (((linorder @ A) => (! [X:A,Y4:A]: (((X != Y4) => ((~ ((((ord_less @ A) @ X) @ Y4))) => (((ord_less @ A) @ Y4) @ X))))))))).
% 0.28/1.91 thf(fact_131_gt__ex, axiom, (! [A:$tType]: (((no_top @ A) => (! [X:A]: ((? [X_1:A]: ((((ord_less @ A) @ X) @ X_1))))))))).
% 0.28/1.91 thf(fact_132_lt__ex, axiom, (! [A:$tType]: (((no_bot @ A) => (! [X:A]: ((? [Y:A]: ((((ord_less @ A) @ Y) @ X))))))))).
% 0.28/1.91 thf(fact_133_order__less__subst2, axiom, (! [A:$tType,C2:$tType]: ((((order @ C2) & (order @ A)) => (! [A2:A,B:A,F:(A > C2),C:C2]: (((((ord_less @ A) @ A2) @ B) => ((((ord_less @ C2) @ (F @ B)) @ C) => ((! [X2:A,Y:A]: (((((ord_less @ A) @ X2) @ Y) => (((ord_less @ C2) @ (F @ X2)) @ (F @ Y))))) => (((ord_less @ C2) @ (F @ A2)) @ C)))))))))).
% 0.28/1.91 thf(fact_134_order__less__subst1, axiom, (! [A:$tType,B2:$tType]: ((((order @ B2) & (order @ A)) => (! [A2:A,F:(B2 > A),B:B2,C:B2]: (((((ord_less @ A) @ A2) @ (F @ B)) => ((((ord_less @ B2) @ B) @ C) => ((! [X2:B2,Y:B2]: (((((ord_less @ B2) @ X2) @ Y) => (((ord_less @ A) @ (F @ X2)) @ (F @ Y))))) => (((ord_less @ A) @ A2) @ (F @ C))))))))))).
% 0.28/1.91 thf(fact_135_ord__less__eq__subst, axiom, (! [A:$tType,B2:$tType]: ((((ord @ B2) & (ord @ A)) => (! [A2:A,B:A,F:(A > B2),C:B2]: (((((ord_less @ A) @ A2) @ B) => (((F @ B) = C) => ((! [X2:A,Y:A]: (((((ord_less @ A) @ X2) @ Y) => (((ord_less @ B2) @ (F @ X2)) @ (F @ Y))))) => (((ord_less @ B2) @ (F @ A2)) @ C)))))))))).
% 0.28/1.91 thf(fact_136_ord__eq__less__subst, axiom, (! [A:$tType,B2:$tType]: ((((ord @ B2) & (ord @ A)) => (! [A2:A,F:(B2 > A),B:B2,C:B2]: (((A2 = (F @ B)) => ((((ord_less @ B2) @ B) @ C) => ((! [X2:B2,Y:B2]: (((((ord_less @ B2) @ X2) @ Y) => (((ord_less @ A) @ (F @ X2)) @ (F @ Y))))) => (((ord_less @ A) @ A2) @ (F @ C))))))))))).
% 0.28/1.91 thf(fact_137_measure__induct__rule, axiom, (! [B2:$tType,A:$tType]: (((wellorder @ B2) => (! [F:(A > B2),P3:(A > $o),A2:A]: (((! [X2:A]: (((! [Y3:A]: (((((ord_less @ B2) @ (F @ Y3)) @ (F @ X2)) => (P3 @ Y3)))) => (P3 @ X2)))) => (P3 @ A2)))))))).
% 0.28/1.91 thf(fact_138_measure__induct, axiom, (! [B2:$tType,A:$tType]: (((wellorder @ B2) => (! [F:(A > B2),P3:(A > $o),A2:A]: (((! [X2:A]: (((! [Y3:A]: (((((ord_less @ B2) @ (F @ Y3)) @ (F @ X2)) => (P3 @ Y3)))) => (P3 @ X2)))) => (P3 @ A2)))))))).
% 0.28/1.91 thf(fact_139_verit__comp__simplify1_I1_J, axiom, (! [A:$tType]: (((order @ A) => (! [A2:A]: ((~ ((((ord_less @ A) @ A2) @ A2))))))))).
% 0.28/1.91 thf(fact_140_pinf_I1_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [P3:(A > $o),P4:(A > $o),Q2:(A > $o),Q3:(A > $o)]: (((? [Z:A]: ((! [X2:A]: (((((ord_less @ A) @ Z) @ X2) => ((P3 @ X2) = (P4 @ X2))))))) => ((? [Z:A]: ((! [X2:A]: (((((ord_less @ A) @ Z) @ X2) => ((Q2 @ X2) = (Q3 @ X2))))))) => (? [Z3:A]: ((! [X4:A]: (((((ord_less @ A) @ Z3) @ X4) => (((P3 @ X4) & (Q2 @ X4)) = ((P4 @ X4) & (Q3 @ X4)))))))))))))))).
% 0.28/1.91 thf(fact_141_pinf_I2_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [P3:(A > $o),P4:(A > $o),Q2:(A > $o),Q3:(A > $o)]: (((? [Z:A]: ((! [X2:A]: (((((ord_less @ A) @ Z) @ X2) => ((P3 @ X2) = (P4 @ X2))))))) => ((? [Z:A]: ((! [X2:A]: (((((ord_less @ A) @ Z) @ X2) => ((Q2 @ X2) = (Q3 @ X2))))))) => (? [Z3:A]: ((! [X4:A]: (((((ord_less @ A) @ Z3) @ X4) => (((P3 @ X4) | (Q2 @ X4)) = ((P4 @ X4) | (Q3 @ X4)))))))))))))))).
% 0.28/1.91 thf(fact_142_pinf_I3_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [T:A]: ((? [Z3:A]: ((! [X4:A]: (((((ord_less @ A) @ Z3) @ X4) => (X4 != T)))))))))))).
% 0.28/1.91 thf(fact_143_pinf_I4_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [T:A]: ((? [Z3:A]: ((! [X4:A]: (((((ord_less @ A) @ Z3) @ X4) => (X4 != T)))))))))))).
% 0.28/1.91 thf(fact_144_pinf_I5_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [T:A]: ((? [Z3:A]: ((! [X4:A]: (((((ord_less @ A) @ Z3) @ X4) => (~ ((((ord_less @ A) @ X4) @ T)))))))))))))).
% 0.28/1.91 thf(fact_145_pinf_I7_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [T:A]: ((? [Z3:A]: ((! [X4:A]: (((((ord_less @ A) @ Z3) @ X4) => (((ord_less @ A) @ T) @ X4)))))))))))).
% 0.28/1.91 thf(fact_146_pinf_I11_J, axiom, (! [C2:$tType,D:$tType]: (((ord @ C2) => (! [F3:D]: ((? [Z3:C2]: ((! [X4:C2]: (((((ord_less @ C2) @ Z3) @ X4) => (F3 = F3)))))))))))).
% 0.28/1.91 thf(fact_147_minf_I1_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [P3:(A > $o),P4:(A > $o),Q2:(A > $o),Q3:(A > $o)]: (((? [Z:A]: ((! [X2:A]: (((((ord_less @ A) @ X2) @ Z) => ((P3 @ X2) = (P4 @ X2))))))) => ((? [Z:A]: ((! [X2:A]: (((((ord_less @ A) @ X2) @ Z) => ((Q2 @ X2) = (Q3 @ X2))))))) => (? [Z3:A]: ((! [X4:A]: (((((ord_less @ A) @ X4) @ Z3) => (((P3 @ X4) & (Q2 @ X4)) = ((P4 @ X4) & (Q3 @ X4)))))))))))))))).
% 0.28/1.91 thf(fact_148_minf_I2_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [P3:(A > $o),P4:(A > $o),Q2:(A > $o),Q3:(A > $o)]: (((? [Z:A]: ((! [X2:A]: (((((ord_less @ A) @ X2) @ Z) => ((P3 @ X2) = (P4 @ X2))))))) => ((? [Z:A]: ((! [X2:A]: (((((ord_less @ A) @ X2) @ Z) => ((Q2 @ X2) = (Q3 @ X2))))))) => (? [Z3:A]: ((! [X4:A]: (((((ord_less @ A) @ X4) @ Z3) => (((P3 @ X4) | (Q2 @ X4)) = ((P4 @ X4) | (Q3 @ X4)))))))))))))))).
% 0.28/1.91 thf(fact_149_minf_I3_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [T:A]: ((? [Z3:A]: ((! [X4:A]: (((((ord_less @ A) @ X4) @ Z3) => (X4 != T)))))))))))).
% 0.28/1.91 thf(fact_150_minf_I4_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [T:A]: ((? [Z3:A]: ((! [X4:A]: (((((ord_less @ A) @ X4) @ Z3) => (X4 != T)))))))))))).
% 0.28/1.91 thf(fact_151_minf_I5_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [T:A]: ((? [Z3:A]: ((! [X4:A]: (((((ord_less @ A) @ X4) @ Z3) => (((ord_less @ A) @ X4) @ T)))))))))))).
% 0.28/1.91 thf(fact_152_minf_I7_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [T:A]: ((? [Z3:A]: ((! [X4:A]: (((((ord_less @ A) @ X4) @ Z3) => (~ ((((ord_less @ A) @ T) @ X4)))))))))))))).
% 0.28/1.91 thf(fact_153_minf_I11_J, axiom, (! [C2:$tType,D:$tType]: (((ord @ C2) => (! [F3:D]: ((? [Z3:C2]: ((! [X4:C2]: (((((ord_less @ C2) @ X4) @ Z3) => (F3 = F3)))))))))))).
% 0.28/1.91 thf(fact_154_infinite__descent0__measure, axiom, (! [A:$tType,V:(A > nat),P3:(A > $o),X:A]: (((! [X2:A]: ((((V @ X2) = (zero_zero @ nat)) => (P3 @ X2)))) => ((! [X2:A]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ (V @ X2)) => ((~ ((P3 @ X2))) => (? [Y3:A]: (((((ord_less @ nat) @ (V @ Y3)) @ (V @ X2)) & (~ ((P3 @ Y3)))))))))) => (P3 @ X)))))).
% 0.28/1.91 thf(fact_155_bot__nat__0_Oextremum__strict, axiom, (! [A2:nat]: ((~ ((((ord_less @ nat) @ A2) @ (zero_zero @ nat))))))).
% 0.28/1.91 thf(fact_156_infinite__descent0, axiom, (! [P3:(nat > $o),N:nat]: (((P3 @ (zero_zero @ nat)) => ((! [N3:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ N3) => ((~ ((P3 @ N3))) => (? [M3:nat]: (((((ord_less @ nat) @ M3) @ N3) & (~ ((P3 @ M3)))))))))) => (P3 @ N)))))).
% 0.28/1.91 thf(fact_157_gr__implies__not0, axiom, (! [M:nat,N:nat]: (((((ord_less @ nat) @ M) @ N) => (N != (zero_zero @ nat)))))).
% 0.28/1.91 thf(fact_158_less__zeroE, axiom, (! [N:nat]: ((~ ((((ord_less @ nat) @ N) @ (zero_zero @ nat))))))).
% 0.28/1.91 thf(fact_159_not__less0, axiom, (! [N:nat]: ((~ ((((ord_less @ nat) @ N) @ (zero_zero @ nat))))))).
% 0.28/1.91 thf(fact_160_not__gr0, axiom, (! [N:nat]: (((~ ((((ord_less @ nat) @ (zero_zero @ nat)) @ N))) = (N = (zero_zero @ nat)))))).
% 0.28/1.91 thf(fact_161_gr0I, axiom, (! [N:nat]: (((N != (zero_zero @ nat)) => (((ord_less @ nat) @ (zero_zero @ nat)) @ N))))).
% 0.28/1.91 thf(fact_162_order_Onot__eq__order__implies__strict, axiom, (! [A:$tType]: (((order @ A) => (! [A2:A,B:A]: (((A2 != B) => ((((ord_less_eq @ A) @ A2) @ B) => (((ord_less @ A) @ A2) @ B))))))))).
% 0.28/1.91 thf(fact_163_dual__order_Ostrict__implies__order, axiom, (! [A:$tType]: (((order @ A) => (! [B:A,A2:A]: (((((ord_less @ A) @ B) @ A2) => (((ord_less_eq @ A) @ B) @ A2)))))))).
% 0.28/1.91 thf(fact_164_dual__order_Ostrict__iff__order, axiom, (! [A:$tType]: (((order @ A) => ((ord_less @ A) = (^ [B4:A,A3:A]: (((((ord_less_eq @ A) @ B4) @ A3) & (A3 != B4))))))))).
% 0.28/1.91 thf(fact_165_dual__order_Oorder__iff__strict, axiom, (! [A:$tType]: (((order @ A) => ((ord_less_eq @ A) = (^ [B4:A,A3:A]: (((((ord_less @ A) @ B4) @ A3) | (A3 = B4))))))))).
% 0.28/1.91 thf(fact_166_order_Ostrict__implies__order, axiom, (! [A:$tType]: (((order @ A) => (! [A2:A,B:A]: (((((ord_less @ A) @ A2) @ B) => (((ord_less_eq @ A) @ A2) @ B)))))))).
% 0.28/1.91 thf(fact_167_dense__le__bounded, axiom, (! [A:$tType]: (((dense_linorder @ A) => (! [X:A,Y4:A,Z5:A]: (((((ord_less @ A) @ X) @ Y4) => ((! [W:A]: (((((ord_less @ A) @ X) @ W) => ((((ord_less @ A) @ W) @ Y4) => (((ord_less_eq @ A) @ W) @ Z5))))) => (((ord_less_eq @ A) @ Y4) @ Z5))))))))).
% 0.28/1.91 thf(fact_168_dense__ge__bounded, axiom, (! [A:$tType]: (((dense_linorder @ A) => (! [Z5:A,X:A,Y4:A]: (((((ord_less @ A) @ Z5) @ X) => ((! [W:A]: (((((ord_less @ A) @ Z5) @ W) => ((((ord_less @ A) @ W) @ X) => (((ord_less_eq @ A) @ Y4) @ W))))) => (((ord_less_eq @ A) @ Y4) @ Z5))))))))).
% 0.28/1.91 thf(fact_169_dual__order_Ostrict__trans2, axiom, (! [A:$tType]: (((order @ A) => (! [B:A,A2:A,C:A]: (((((ord_less @ A) @ B) @ A2) => ((((ord_less_eq @ A) @ C) @ B) => (((ord_less @ A) @ C) @ A2))))))))).
% 0.28/1.91 thf(fact_170_dual__order_Ostrict__trans1, axiom, (! [A:$tType]: (((order @ A) => (! [B:A,A2:A,C:A]: (((((ord_less_eq @ A) @ B) @ A2) => ((((ord_less @ A) @ C) @ B) => (((ord_less @ A) @ C) @ A2))))))))).
% 0.28/1.91 thf(fact_171_order_Ostrict__iff__order, axiom, (! [A:$tType]: (((order @ A) => ((ord_less @ A) = (^ [A3:A,B4:A]: (((((ord_less_eq @ A) @ A3) @ B4) & (A3 != B4))))))))).
% 0.28/1.91 thf(fact_172_order_Oorder__iff__strict, axiom, (! [A:$tType]: (((order @ A) => ((ord_less_eq @ A) = (^ [A3:A,B4:A]: (((((ord_less @ A) @ A3) @ B4) | (A3 = B4))))))))).
% 0.28/1.91 thf(fact_173_order_Ostrict__trans2, axiom, (! [A:$tType]: (((order @ A) => (! [A2:A,B:A,C:A]: (((((ord_less @ A) @ A2) @ B) => ((((ord_less_eq @ A) @ B) @ C) => (((ord_less @ A) @ A2) @ C))))))))).
% 0.28/1.91 thf(fact_174_order_Ostrict__trans1, axiom, (! [A:$tType]: (((order @ A) => (! [A2:A,B:A,C:A]: (((((ord_less_eq @ A) @ A2) @ B) => ((((ord_less @ A) @ B) @ C) => (((ord_less @ A) @ A2) @ C))))))))).
% 0.28/1.91 thf(fact_175_not__le__imp__less, axiom, (! [A:$tType]: (((linorder @ A) => (! [Y4:A,X:A]: (((~ ((((ord_less_eq @ A) @ Y4) @ X))) => (((ord_less @ A) @ X) @ Y4)))))))).
% 0.28/1.91 thf(fact_176_less__le__not__le, axiom, (! [A:$tType]: (((preorder @ A) => ((ord_less @ A) = (^ [X3:A,Y2:A]: (((((ord_less_eq @ A) @ X3) @ Y2) & (~ ((((ord_less_eq @ A) @ Y2) @ X3))))))))))).
% 0.28/1.91 thf(fact_177_le__imp__less__or__eq, axiom, (! [A:$tType]: (((order @ A) => (! [X:A,Y4:A]: (((((ord_less_eq @ A) @ X) @ Y4) => ((((ord_less @ A) @ X) @ Y4) | (X = Y4))))))))).
% 0.28/1.91 thf(fact_178_le__less__linear, axiom, (! [A:$tType]: (((linorder @ A) => (! [X:A,Y4:A]: (((((ord_less_eq @ A) @ X) @ Y4) | (((ord_less @ A) @ Y4) @ X)))))))).
% 0.28/1.91 thf(fact_179_dense__le, axiom, (! [A:$tType]: (((dense_linorder @ A) => (! [Y4:A,Z5:A]: (((! [X2:A]: (((((ord_less @ A) @ X2) @ Y4) => (((ord_less_eq @ A) @ X2) @ Z5)))) => (((ord_less_eq @ A) @ Y4) @ Z5)))))))).
% 0.28/1.91 thf(fact_180_dense__ge, axiom, (! [A:$tType]: (((dense_linorder @ A) => (! [Z5:A,Y4:A]: (((! [X2:A]: (((((ord_less @ A) @ Z5) @ X2) => (((ord_less_eq @ A) @ Y4) @ X2)))) => (((ord_less_eq @ A) @ Y4) @ Z5)))))))).
% 0.28/1.91 thf(fact_181_less__le__trans, axiom, (! [A:$tType]: (((preorder @ A) => (! [X:A,Y4:A,Z5:A]: (((((ord_less @ A) @ X) @ Y4) => ((((ord_less_eq @ A) @ Y4) @ Z5) => (((ord_less @ A) @ X) @ Z5))))))))).
% 0.28/1.91 thf(fact_182_le__less__trans, axiom, (! [A:$tType]: (((preorder @ A) => (! [X:A,Y4:A,Z5:A]: (((((ord_less_eq @ A) @ X) @ Y4) => ((((ord_less @ A) @ Y4) @ Z5) => (((ord_less @ A) @ X) @ Z5))))))))).
% 0.28/1.91 thf(fact_183_less__imp__le, axiom, (! [A:$tType]: (((preorder @ A) => (! [X:A,Y4:A]: (((((ord_less @ A) @ X) @ Y4) => (((ord_less_eq @ A) @ X) @ Y4)))))))).
% 0.28/1.91 thf(fact_184_antisym__conv2, axiom, (! [A:$tType]: (((order @ A) => (! [X:A,Y4:A]: (((((ord_less_eq @ A) @ X) @ Y4) => ((~ ((((ord_less @ A) @ X) @ Y4))) = (X = Y4))))))))).
% 0.28/1.91 thf(fact_185_antisym__conv1, axiom, (! [A:$tType]: (((order @ A) => (! [X:A,Y4:A]: (((~ ((((ord_less @ A) @ X) @ Y4))) => ((((ord_less_eq @ A) @ X) @ Y4) = (X = Y4))))))))).
% 0.28/1.91 thf(fact_186_le__neq__trans, axiom, (! [A:$tType]: (((order @ A) => (! [A2:A,B:A]: (((((ord_less_eq @ A) @ A2) @ B) => ((A2 != B) => (((ord_less @ A) @ A2) @ B))))))))).
% 0.28/1.91 thf(fact_187_not__less, axiom, (! [A:$tType]: (((linorder @ A) => (! [X:A,Y4:A]: (((~ ((((ord_less @ A) @ X) @ Y4))) = (((ord_less_eq @ A) @ Y4) @ X)))))))).
% 0.28/1.91 thf(fact_188_not__le, axiom, (! [A:$tType]: (((linorder @ A) => (! [X:A,Y4:A]: (((~ ((((ord_less_eq @ A) @ X) @ Y4))) = (((ord_less @ A) @ Y4) @ X)))))))).
% 0.28/1.91 thf(fact_189_order__less__le__subst2, axiom, (! [A:$tType,C2:$tType]: ((((order @ C2) & (order @ A)) => (! [A2:A,B:A,F:(A > C2),C:C2]: (((((ord_less @ A) @ A2) @ B) => ((((ord_less_eq @ C2) @ (F @ B)) @ C) => ((! [X2:A,Y:A]: (((((ord_less @ A) @ X2) @ Y) => (((ord_less @ C2) @ (F @ X2)) @ (F @ Y))))) => (((ord_less @ C2) @ (F @ A2)) @ C)))))))))).
% 0.28/1.91 thf(fact_190_order__less__le__subst1, axiom, (! [A:$tType,B2:$tType]: ((((order @ B2) & (order @ A)) => (! [A2:A,F:(B2 > A),B:B2,C:B2]: (((((ord_less @ A) @ A2) @ (F @ B)) => ((((ord_less_eq @ B2) @ B) @ C) => ((! [X2:B2,Y:B2]: (((((ord_less_eq @ B2) @ X2) @ Y) => (((ord_less_eq @ A) @ (F @ X2)) @ (F @ Y))))) => (((ord_less @ A) @ A2) @ (F @ C))))))))))).
% 0.28/1.91 thf(fact_191_order__le__less__subst2, axiom, (! [A:$tType,C2:$tType]: ((((order @ C2) & (order @ A)) => (! [A2:A,B:A,F:(A > C2),C:C2]: (((((ord_less_eq @ A) @ A2) @ B) => ((((ord_less @ C2) @ (F @ B)) @ C) => ((! [X2:A,Y:A]: (((((ord_less_eq @ A) @ X2) @ Y) => (((ord_less_eq @ C2) @ (F @ X2)) @ (F @ Y))))) => (((ord_less @ C2) @ (F @ A2)) @ C)))))))))).
% 0.28/1.91 thf(fact_192_order__le__less__subst1, axiom, (! [A:$tType,B2:$tType]: ((((order @ B2) & (order @ A)) => (! [A2:A,F:(B2 > A),B:B2,C:B2]: (((((ord_less_eq @ A) @ A2) @ (F @ B)) => ((((ord_less @ B2) @ B) @ C) => ((! [X2:B2,Y:B2]: (((((ord_less @ B2) @ X2) @ Y) => (((ord_less @ A) @ (F @ X2)) @ (F @ Y))))) => (((ord_less @ A) @ A2) @ (F @ C))))))))))).
% 0.28/1.91 thf(fact_193_less__le, axiom, (! [A:$tType]: (((order @ A) => ((ord_less @ A) = (^ [X3:A,Y2:A]: (((((ord_less_eq @ A) @ X3) @ Y2) & (X3 != Y2))))))))).
% 0.28/1.91 thf(fact_194_le__less, axiom, (! [A:$tType]: (((order @ A) => ((ord_less_eq @ A) = (^ [X3:A,Y2:A]: (((((ord_less @ A) @ X3) @ Y2) | (X3 = Y2))))))))).
% 0.28/1.91 thf(fact_195_leI, axiom, (! [A:$tType]: (((linorder @ A) => (! [X:A,Y4:A]: (((~ ((((ord_less @ A) @ X) @ Y4))) => (((ord_less_eq @ A) @ Y4) @ X)))))))).
% 0.28/1.91 thf(fact_196_leD, axiom, (! [A:$tType]: (((order @ A) => (! [Y4:A,X:A]: (((((ord_less_eq @ A) @ Y4) @ X) => (~ ((((ord_less @ A) @ X) @ Y4)))))))))).
% 0.28/1.91 thf(fact_197_complete__interval, axiom, (! [A:$tType]: (((condit1037483654norder @ A) => (! [A2:A,B:A,P3:(A > $o)]: (((((ord_less @ A) @ A2) @ B) => ((P3 @ A2) => ((~ ((P3 @ B))) => (? [C3:A]: (((((ord_less_eq @ A) @ A2) @ C3) & ((((ord_less_eq @ A) @ C3) @ B) & ((! [X4:A]: ((((((ord_less_eq @ A) @ A2) @ X4) & (((ord_less @ A) @ X4) @ C3)) => (P3 @ X4)))) & (! [D3:A]: (((! [X2:A]: ((((((ord_less_eq @ A) @ A2) @ X2) & (((ord_less @ A) @ X2) @ D3)) => (P3 @ X2)))) => (((ord_less_eq @ A) @ D3) @ C3)))))))))))))))))).
% 0.28/1.91 thf(fact_198_rsquarefree__def, axiom, (! [A:$tType]: (((idom @ A) => ((rsquarefree @ A) = (^ [P2:(poly @ A)]: (((P2 != (zero_zero @ (poly @ A))) & (! [A3:A]: ((((((order2 @ A) @ A3) @ P2) = (zero_zero @ nat)) | ((((order2 @ A) @ A3) @ P2) = (one_one @ nat))))))))))))).
% 0.28/1.91 thf(fact_199_poly__cutoff__1, axiom, (! [A:$tType]: (((comm_semiring_1 @ A) => (! [N:nat]: ((((N = (zero_zero @ nat)) => ((((poly_cutoff @ A) @ N) @ (one_one @ (poly @ A))) = (zero_zero @ (poly @ A)))) & ((N != (zero_zero @ nat)) => ((((poly_cutoff @ A) @ N) @ (one_one @ (poly @ A))) = (one_one @ (poly @ A))))))))))).
% 0.28/1.91 thf(fact_200_coeff__0__reflect__poly__0__iff, axiom, (! [A:$tType]: (((zero @ A) => (! [P:(poly @ A)]: ((((((coeff @ A) @ ((reflect_poly @ A) @ P)) @ (zero_zero @ nat)) = (zero_zero @ A)) = (P = (zero_zero @ (poly @ A)))))))))).
% 0.28/1.91 thf(fact_201_of__nat__0__less__iff, axiom, (! [A:$tType]: (((linord1659791738miring @ A) => (! [N:nat]: (((((ord_less @ A) @ (zero_zero @ A)) @ ((semiring_1_of_nat @ A) @ N)) = (((ord_less @ nat) @ (zero_zero @ nat)) @ N)))))))).
% 0.28/1.91 thf(fact_202_order__pderiv, axiom, (! [A:$tType]: (((field_char_0 @ A) => (! [P:(poly @ A),A2:A]: (((((pderiv @ A) @ P) != (zero_zero @ (poly @ A))) => (((((order2 @ A) @ A2) @ P) != (zero_zero @ nat)) => ((((order2 @ A) @ A2) @ P) = (suc @ (((order2 @ A) @ A2) @ ((pderiv @ A) @ P)))))))))))).
% 0.28/1.91 thf(fact_203_order__pderiv2, axiom, (! [A:$tType]: (((field_char_0 @ A) => (! [P:(poly @ A),A2:A,N:nat]: (((((pderiv @ A) @ P) != (zero_zero @ (poly @ A))) => (((((order2 @ A) @ A2) @ P) != (zero_zero @ nat)) => (((((order2 @ A) @ A2) @ ((pderiv @ A) @ P)) = N) = ((((order2 @ A) @ A2) @ P) = (suc @ N))))))))))).
% 0.28/1.91 thf(fact_204_cross__def, axiom, (sturm_424270202_cross = (^ [P2:(poly @ real),A3:real,B4:real]: (((sturm_1771227917iation @ (((poly2 @ real) @ P2) @ A3)) @ (((poly2 @ real) @ P2) @ B4)))))).
% 0.28/1.91 thf(fact_205_nat_Oinject, axiom, (! [X22:nat,Y22:nat]: ((((suc @ X22) = (suc @ Y22)) = (X22 = Y22))))).
% 0.28/1.91 thf(fact_206_old_Onat_Oinject, axiom, (! [Nat:nat,Nat2:nat]: ((((suc @ Nat) = (suc @ Nat2)) = (Nat = Nat2))))).
% 0.28/1.91 thf(fact_207_of__nat__eq__iff, axiom, (! [A:$tType]: (((semiring_char_0 @ A) => (! [M:nat,N:nat]: (((((semiring_1_of_nat @ A) @ M) = ((semiring_1_of_nat @ A) @ N)) = (M = N)))))))).
% 0.28/1.91 thf(fact_208_lessI, axiom, (! [N:nat]: ((((ord_less @ nat) @ N) @ (suc @ N))))).
% 0.28/1.91 thf(fact_209_Suc__mono, axiom, (! [M:nat,N:nat]: (((((ord_less @ nat) @ M) @ N) => (((ord_less @ nat) @ (suc @ M)) @ (suc @ N)))))).
% 0.28/1.91 thf(fact_210_Suc__less__eq, axiom, (! [M:nat,N:nat]: (((((ord_less @ nat) @ (suc @ M)) @ (suc @ N)) = (((ord_less @ nat) @ M) @ N))))).
% 0.28/1.91 thf(fact_211_Suc__le__mono, axiom, (! [N:nat,M:nat]: (((((ord_less_eq @ nat) @ (suc @ N)) @ (suc @ M)) = (((ord_less_eq @ nat) @ N) @ M))))).
% 0.28/1.91 thf(fact_212_pderiv__of__nat, axiom, (! [A:$tType]: ((((comm_semiring_1 @ A) & (semiri1193490041visors @ A)) => (! [N:nat]: ((((pderiv @ A) @ ((semiring_1_of_nat @ (poly @ A)) @ N)) = (zero_zero @ (poly @ A))))))))).
% 0.28/1.91 thf(fact_213_reflect__poly__1, axiom, (! [A:$tType]: (((comm_semiring_1 @ A) => (((reflect_poly @ A) @ (one_one @ (poly @ A))) = (one_one @ (poly @ A))))))).
% 0.28/1.91 thf(fact_214_of__nat__0, axiom, (! [A:$tType]: (((semiring_1 @ A) => (((semiring_1_of_nat @ A) @ (zero_zero @ nat)) = (zero_zero @ A)))))).
% 0.28/1.91 thf(fact_215_of__nat__0__eq__iff, axiom, (! [A:$tType]: (((semiring_char_0 @ A) => (! [N:nat]: ((((zero_zero @ A) = ((semiring_1_of_nat @ A) @ N)) = ((zero_zero @ nat) = N)))))))).
% 0.28/1.91 thf(fact_216_of__nat__eq__0__iff, axiom, (! [A:$tType]: (((semiring_char_0 @ A) => (! [M:nat]: (((((semiring_1_of_nat @ A) @ M) = (zero_zero @ A)) = (M = (zero_zero @ nat))))))))).
% 0.28/1.91 thf(fact_217_of__nat__less__iff, axiom, (! [A:$tType]: (((linord1659791738miring @ A) => (! [M:nat,N:nat]: (((((ord_less @ A) @ ((semiring_1_of_nat @ A) @ M)) @ ((semiring_1_of_nat @ A) @ N)) = (((ord_less @ nat) @ M) @ N)))))))).
% 0.28/1.91 thf(fact_218_of__nat__le__iff, axiom, (! [A:$tType]: (((linord1659791738miring @ A) => (! [M:nat,N:nat]: (((((ord_less_eq @ A) @ ((semiring_1_of_nat @ A) @ M)) @ ((semiring_1_of_nat @ A) @ N)) = (((ord_less_eq @ nat) @ M) @ N)))))))).
% 0.28/1.91 thf(fact_219_less__Suc0, axiom, (! [N:nat]: (((((ord_less @ nat) @ N) @ (suc @ (zero_zero @ nat))) = (N = (zero_zero @ nat)))))).
% 0.28/1.91 thf(fact_220_zero__less__Suc, axiom, (! [N:nat]: ((((ord_less @ nat) @ (zero_zero @ nat)) @ (suc @ N))))).
% 0.28/1.91 thf(fact_221_of__nat__1, axiom, (! [A:$tType]: (((semiring_1 @ A) => (((semiring_1_of_nat @ A) @ (one_one @ nat)) = (one_one @ A)))))).
% 0.28/1.91 thf(fact_222_of__nat__1__eq__iff, axiom, (! [A:$tType]: (((semiring_char_0 @ A) => (! [N:nat]: ((((one_one @ A) = ((semiring_1_of_nat @ A) @ N)) = (N = (one_one @ nat))))))))).
% 0.28/1.91 thf(fact_223_of__nat__eq__1__iff, axiom, (! [A:$tType]: (((semiring_char_0 @ A) => (! [N:nat]: (((((semiring_1_of_nat @ A) @ N) = (one_one @ A)) = (N = (one_one @ nat))))))))).
% 0.28/1.91 thf(fact_224_less__one, axiom, (! [N:nat]: (((((ord_less @ nat) @ N) @ (one_one @ nat)) = (N = (zero_zero @ nat)))))).
% 0.28/1.91 thf(fact_225_coeff__0, axiom, (! [A:$tType]: (((zero @ A) => (! [N:nat]: (((((coeff @ A) @ (zero_zero @ (poly @ A))) @ N) = (zero_zero @ A)))))))).
% 0.28/1.91 thf(fact_226_poly__1, axiom, (! [A:$tType]: (((comm_semiring_1 @ A) => (! [X:A]: (((((poly2 @ A) @ (one_one @ (poly @ A))) @ X) = (one_one @ A)))))))).
% 0.28/1.91 thf(fact_227_variation__0_I2_J, axiom, (! [X:real]: ((((sturm_1771227917iation @ X) @ (zero_zero @ real)) = (zero_zero @ int))))).
% 0.28/1.91 thf(fact_228_variation__0_I1_J, axiom, (! [Y4:real]: ((((sturm_1771227917iation @ (zero_zero @ real)) @ Y4) = (zero_zero @ int))))).
% 0.28/1.91 thf(fact_229_pderiv__1, axiom, (! [A:$tType]: ((((comm_semiring_1 @ A) & (semiri1193490041visors @ A)) => (((pderiv @ A) @ (one_one @ (poly @ A))) = (zero_zero @ (poly @ A))))))).
% 0.28/1.91 thf(fact_230_order__1__eq__0, axiom, (! [A:$tType]: (((idom @ A) => (! [X:A]: (((((order2 @ A) @ X) @ (one_one @ (poly @ A))) = (zero_zero @ nat)))))))).
% 0.28/1.91 thf(fact_231_of__nat__le__0__iff, axiom, (! [A:$tType]: (((linord1659791738miring @ A) => (! [M:nat]: (((((ord_less_eq @ A) @ ((semiring_1_of_nat @ A) @ M)) @ (zero_zero @ A)) = (M = (zero_zero @ nat))))))))).
% 0.28/1.91 thf(fact_232_reflect__poly__reflect__poly, axiom, (! [A:$tType]: (((zero @ A) => (! [P:(poly @ A)]: ((((((coeff @ A) @ P) @ (zero_zero @ nat)) != (zero_zero @ A)) => (((reflect_poly @ A) @ ((reflect_poly @ A) @ P)) = P)))))))).
% 0.28/1.91 thf(fact_233_One__nat__def, axiom, ((one_one @ nat) = (suc @ (zero_zero @ nat)))).
% 0.28/1.91 thf(fact_234_less__fun__def, axiom, (! [B2:$tType,A:$tType]: (((ord @ B2) => ((ord_less @ (A > B2)) = (^ [F2:(A > B2),G2:(A > B2)]: (((((ord_less_eq @ (A > B2)) @ F2) @ G2) & (~ ((((ord_less_eq @ (A > B2)) @ G2) @ F2))))))))))).
% 0.28/1.91 thf(fact_235_of__nat__neq__0, axiom, (! [A:$tType]: (((semiring_char_0 @ A) => (! [N:nat]: ((((semiring_1_of_nat @ A) @ (suc @ N)) != (zero_zero @ A)))))))).
% 0.28/1.91 thf(fact_236_transitive__stepwise__le, axiom, (! [M:nat,N:nat,R:(nat > (nat > $o))]: (((((ord_less_eq @ nat) @ M) @ N) => ((! [X2:nat]: (((R @ X2) @ X2))) => ((! [X2:nat,Y:nat,Z3:nat]: ((((R @ X2) @ Y) => (((R @ Y) @ Z3) => ((R @ X2) @ Z3))))) => ((! [N3:nat]: (((R @ N3) @ (suc @ N3)))) => ((R @ M) @ N)))))))).
% 0.28/1.91 thf(fact_237_nat__induct__at__least, axiom, (! [M:nat,N:nat,P3:(nat > $o)]: (((((ord_less_eq @ nat) @ M) @ N) => ((P3 @ M) => ((! [N3:nat]: (((((ord_less_eq @ nat) @ M) @ N3) => ((P3 @ N3) => (P3 @ (suc @ N3)))))) => (P3 @ N))))))).
% 0.28/1.91 thf(fact_238_Nat_Oex__has__greatest__nat, axiom, (! [P3:(nat > $o),K2:nat,B:nat]: (((P3 @ K2) => ((! [Y:nat]: (((P3 @ Y) => (((ord_less_eq @ nat) @ Y) @ B)))) => (? [X2:nat]: (((P3 @ X2) & (! [Y3:nat]: (((P3 @ Y3) => (((ord_less_eq @ nat) @ Y3) @ X2)))))))))))).
% 0.28/1.91 thf(fact_239_verit__la__generic, axiom, (! [A2:int,X:int]: (((((ord_less_eq @ int) @ A2) @ X) | ((A2 = X) | (((ord_less_eq @ int) @ X) @ A2)))))).
% 0.28/1.91 thf(fact_240_full__nat__induct, axiom, (! [P3:(nat > $o),N:nat]: (((! [N3:nat]: (((! [M3:nat]: (((((ord_less_eq @ nat) @ (suc @ M3)) @ N3) => (P3 @ M3)))) => (P3 @ N3)))) => (P3 @ N))))).
% 0.28/1.91 thf(fact_241_not__less__eq__eq, axiom, (! [M:nat,N:nat]: (((~ ((((ord_less_eq @ nat) @ M) @ N))) = (((ord_less_eq @ nat) @ (suc @ N)) @ M))))).
% 0.28/1.91 thf(fact_242_Suc__n__not__le__n, axiom, (! [N:nat]: ((~ ((((ord_less_eq @ nat) @ (suc @ N)) @ N)))))).
% 0.28/1.91 thf(fact_243_nat__le__linear, axiom, (! [M:nat,N:nat]: (((((ord_less_eq @ nat) @ M) @ N) | (((ord_less_eq @ nat) @ N) @ M))))).
% 0.28/1.91 thf(fact_244_le__antisym, axiom, (! [M:nat,N:nat]: (((((ord_less_eq @ nat) @ M) @ N) => ((((ord_less_eq @ nat) @ N) @ M) => (M = N)))))).
% 0.28/1.91 thf(fact_245_le__Suc__eq, axiom, (! [M:nat,N:nat]: (((((ord_less_eq @ nat) @ M) @ (suc @ N)) = ((((ord_less_eq @ nat) @ M) @ N) | (M = (suc @ N))))))).
% 0.28/1.91 thf(fact_246_eq__imp__le, axiom, (! [M:nat,N:nat]: (((M = N) => (((ord_less_eq @ nat) @ M) @ N))))).
% 0.28/1.91 thf(fact_247_le__trans, axiom, (! [I:nat,J:nat,K2:nat]: (((((ord_less_eq @ nat) @ I) @ J) => ((((ord_less_eq @ nat) @ J) @ K2) => (((ord_less_eq @ nat) @ I) @ K2)))))).
% 0.28/1.91 thf(fact_248_Suc__le__D, axiom, (! [N:nat,M4:nat]: (((((ord_less_eq @ nat) @ (suc @ N)) @ M4) => (? [M5:nat]: ((M4 = (suc @ M5)))))))).
% 0.28/1.91 thf(fact_249_le__refl, axiom, (! [N:nat]: ((((ord_less_eq @ nat) @ N) @ N)))).
% 0.28/1.91 thf(fact_250_le__SucI, axiom, (! [M:nat,N:nat]: (((((ord_less_eq @ nat) @ M) @ N) => (((ord_less_eq @ nat) @ M) @ (suc @ N)))))).
% 0.28/1.91 thf(fact_251_le__SucE, axiom, (! [M:nat,N:nat]: (((((ord_less_eq @ nat) @ M) @ (suc @ N)) => ((~ ((((ord_less_eq @ nat) @ M) @ N))) => (M = (suc @ N))))))).
% 0.28/1.91 thf(tcon_Polynomial_Opoly___Rings_Olinordered__idom, axiom, (! [A7:$tType]: (((linordered_idom @ A7) => (linordered_idom @ (poly @ A7)))))).
% 0.28/1.91 thf(tcon_Real_Oreal___Rings_Olinordered__idom_1, axiom, (linordered_idom @ real)).
% 0.28/1.91 thf(tcon_Int_Oint___Rings_Olinordered__idom_2, axiom, (linordered_idom @ int)).
% 0.28/1.91 thf(tcon_fun___Orderings_Opreorder, axiom, (! [A7:$tType,A8:$tType]: (((preorder @ A8) => (preorder @ (A7 > A8)))))).
% 0.28/1.91 thf(tcon_fun___Orderings_Oorder, axiom, (! [A7:$tType,A8:$tType]: (((order @ A8) => (order @ (A7 > A8)))))).
% 0.28/1.91 thf(tcon_fun___Orderings_Oord, axiom, (! [A7:$tType,A8:$tType]: (((ord @ A8) => (ord @ (A7 > A8)))))).
% 0.28/1.91 thf(tcon_Int_Oint___Conditionally__Complete__Lattices_Oconditionally__complete__linorder, axiom, (condit1037483654norder @ int)).
% 0.28/1.91 thf(tcon_Int_Oint___Rings_Olinordered__nonzero__semiring, axiom, (linord1659791738miring @ int)).
% 0.28/1.91 thf(tcon_Int_Oint___Rings_Osemiring__no__zero__divisors, axiom, (semiri1193490041visors @ int)).
% 0.28/1.91 thf(tcon_Int_Oint___Rings_Oring__no__zero__divisors, axiom, (ring_n68954251visors @ int)).
% 0.28/1.91 thf(tcon_Int_Oint___Rings_Ocomm__semiring__1, axiom, (comm_semiring_1 @ int)).
% 0.28/1.91 thf(tcon_Int_Oint___Rings_Ocomm__semiring__0, axiom, (comm_semiring_0 @ int)).
% 0.28/1.91 thf(tcon_Int_Oint___Nat_Osemiring__char__0, axiom, (semiring_char_0 @ int)).
% 0.28/1.91 thf(tcon_Int_Oint___Orderings_Opreorder_3, axiom, (preorder @ int)).
% 0.28/1.91 thf(tcon_Int_Oint___Orderings_Olinorder, axiom, (linorder @ int)).
% 0.28/1.91 thf(tcon_Int_Oint___Rings_Ocomm__ring__1, axiom, (comm_ring_1 @ int)).
% 0.28/1.91 thf(tcon_Int_Oint___Rings_Osemiring__1, axiom, (semiring_1 @ int)).
% 0.28/1.91 thf(tcon_Int_Oint___Orderings_Ono__top, axiom, (no_top @ int)).
% 0.28/1.91 thf(tcon_Int_Oint___Orderings_Ono__bot, axiom, (no_bot @ int)).
% 0.28/1.91 thf(tcon_Int_Oint___Orderings_Oorder_4, axiom, (order @ int)).
% 0.28/1.91 thf(tcon_Int_Oint___Nat_Oring__char__0, axiom, (ring_char_0 @ int)).
% 0.28/1.91 thf(tcon_Int_Oint___Orderings_Oord_5, axiom, (ord @ int)).
% 0.28/1.91 thf(tcon_Int_Oint___Groups_Ozero, axiom, (zero @ int)).
% 0.28/1.91 thf(tcon_Int_Oint___Rings_Oidom, axiom, (idom @ int)).
% 0.28/1.91 thf(tcon_Int_Oint___Groups_Oone, axiom, (one @ int)).
% 0.28/1.91 thf(tcon_Nat_Onat___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_6, axiom, (condit1037483654norder @ nat)).
% 0.28/1.91 thf(tcon_Nat_Onat___Groups_Ocanonically__ordered__monoid__add, axiom, (canoni770627133id_add @ nat)).
% 0.28/1.91 thf(tcon_Nat_Onat___Rings_Olinordered__nonzero__semiring_7, axiom, (linord1659791738miring @ nat)).
% 0.28/1.91 thf(tcon_Nat_Onat___Rings_Osemiring__no__zero__divisors_8, axiom, (semiri1193490041visors @ nat)).
% 0.28/1.91 thf(tcon_Nat_Onat___Rings_Ocomm__semiring__1_9, axiom, (comm_semiring_1 @ nat)).
% 0.28/1.91 thf(tcon_Nat_Onat___Rings_Ocomm__semiring__0_10, axiom, (comm_semiring_0 @ nat)).
% 0.28/1.91 thf(tcon_Nat_Onat___Orderings_Owellorder, axiom, (wellorder @ nat)).
% 0.28/1.91 thf(tcon_Nat_Onat___Nat_Osemiring__char__0_11, axiom, (semiring_char_0 @ nat)).
% 0.28/1.91 thf(tcon_Nat_Onat___Orderings_Opreorder_12, axiom, (preorder @ nat)).
% 0.28/1.91 thf(tcon_Nat_Onat___Orderings_Olinorder_13, axiom, (linorder @ nat)).
% 0.28/1.91 thf(tcon_Nat_Onat___Rings_Osemiring__1_14, axiom, (semiring_1 @ nat)).
% 0.28/1.91 thf(tcon_Nat_Onat___Orderings_Ono__top_15, axiom, (no_top @ nat)).
% 0.28/1.91 thf(tcon_Nat_Onat___Orderings_Oorder_16, axiom, (order @ nat)).
% 0.28/1.91 thf(tcon_Nat_Onat___Orderings_Oord_17, axiom, (ord @ nat)).
% 0.28/1.91 thf(tcon_Nat_Onat___Groups_Ozero_18, axiom, (zero @ nat)).
% 0.28/1.91 thf(tcon_Nat_Onat___Groups_Oone_19, axiom, (one @ nat)).
% 0.28/1.91 thf(tcon_Set_Oset___Orderings_Opreorder_20, axiom, (! [A7:$tType]: ((preorder @ (set @ A7))))).
% 0.28/1.91 thf(tcon_Set_Oset___Orderings_Oorder_21, axiom, (! [A7:$tType]: ((order @ (set @ A7))))).
% 0.28/1.91 thf(tcon_Set_Oset___Orderings_Oord_22, axiom, (! [A7:$tType]: ((ord @ (set @ A7))))).
% 0.28/1.91 thf(tcon_HOL_Obool___Orderings_Opreorder_23, axiom, (preorder @ $o)).
% 0.28/1.91 thf(tcon_HOL_Obool___Orderings_Olinorder_24, axiom, (linorder @ $o)).
% 0.28/1.91 thf(tcon_HOL_Obool___Orderings_Oorder_25, axiom, (order @ $o)).
% 0.28/1.91 thf(tcon_HOL_Obool___Orderings_Oord_26, axiom, (ord @ $o)).
% 0.28/1.91 thf(tcon_Real_Oreal___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_27, axiom, (condit1037483654norder @ real)).
% 0.28/1.91 thf(tcon_Real_Oreal___Rings_Olinordered__nonzero__semiring_28, axiom, (linord1659791738miring @ real)).
% 0.28/1.91 thf(tcon_Real_Oreal___Rings_Osemiring__no__zero__divisors_29, axiom, (semiri1193490041visors @ real)).
% 0.28/1.91 thf(tcon_Real_Oreal___Rings_Oring__no__zero__divisors_30, axiom, (ring_n68954251visors @ real)).
% 0.28/1.91 thf(tcon_Real_Oreal___Orderings_Odense__linorder, axiom, (dense_linorder @ real)).
% 0.28/1.91 thf(tcon_Real_Oreal___Fields_Olinordered__field, axiom, (linordered_field @ real)).
% 0.28/1.91 thf(tcon_Real_Oreal___Rings_Ocomm__semiring__1_31, axiom, (comm_semiring_1 @ real)).
% 0.28/1.91 thf(tcon_Real_Oreal___Rings_Ocomm__semiring__0_32, axiom, (comm_semiring_0 @ real)).
% 0.28/1.91 thf(tcon_Real_Oreal___Orderings_Odense__order, axiom, (dense_order @ real)).
% 0.28/1.91 thf(tcon_Real_Oreal___Nat_Osemiring__char__0_33, axiom, (semiring_char_0 @ real)).
% 0.28/1.91 thf(tcon_Real_Oreal___Fields_Ofield__char__0, axiom, (field_char_0 @ real)).
% 0.28/1.91 thf(tcon_Real_Oreal___Orderings_Opreorder_34, axiom, (preorder @ real)).
% 0.28/1.91 thf(tcon_Real_Oreal___Orderings_Olinorder_35, axiom, (linorder @ real)).
% 0.28/1.91 thf(tcon_Real_Oreal___Rings_Ocomm__ring__1_36, axiom, (comm_ring_1 @ real)).
% 0.28/1.91 thf(tcon_Real_Oreal___Rings_Osemiring__1_37, axiom, (semiring_1 @ real)).
% 0.28/1.91 thf(tcon_Real_Oreal___Orderings_Ono__top_38, axiom, (no_top @ real)).
% 0.28/1.91 thf(tcon_Real_Oreal___Orderings_Ono__bot_39, axiom, (no_bot @ real)).
% 0.28/1.91 thf(tcon_Real_Oreal___Orderings_Oorder_40, axiom, (order @ real)).
% 0.28/1.91 thf(tcon_Real_Oreal___Nat_Oring__char__0_41, axiom, (ring_char_0 @ real)).
% 0.28/1.91 thf(tcon_Real_Oreal___Orderings_Oord_42, axiom, (ord @ real)).
% 0.28/1.91 thf(tcon_Real_Oreal___Groups_Ozero_43, axiom, (zero @ real)).
% 0.28/1.91 thf(tcon_Real_Oreal___Rings_Oidom_44, axiom, (idom @ real)).
% 0.28/1.91 thf(tcon_Real_Oreal___Groups_Oone_45, axiom, (one @ real)).
% 0.28/1.91 thf(tcon_Polynomial_Opoly___Rings_Olinordered__nonzero__semiring_46, axiom, (! [A7:$tType]: (((linordered_idom @ A7) => (linord1659791738miring @ (poly @ A7)))))).
% 0.28/1.91 thf(tcon_Polynomial_Opoly___Rings_Osemiring__no__zero__divisors_47, axiom, (! [A7:$tType]: ((((comm_semiring_0 @ A7) & (semiri1193490041visors @ A7)) => (semiri1193490041visors @ (poly @ A7)))))).
% 0.28/1.91 thf(tcon_Polynomial_Opoly___Rings_Oring__no__zero__divisors_48, axiom, (! [A7:$tType]: (((idom @ A7) => (ring_n68954251visors @ (poly @ A7)))))).
% 0.28/1.91 thf(tcon_Polynomial_Opoly___Rings_Ocomm__semiring__1_49, axiom, (! [A7:$tType]: (((comm_semiring_1 @ A7) => (comm_semiring_1 @ (poly @ A7)))))).
% 0.28/1.91 thf(tcon_Polynomial_Opoly___Rings_Ocomm__semiring__0_50, axiom, (! [A7:$tType]: (((comm_semiring_0 @ A7) => (comm_semiring_0 @ (poly @ A7)))))).
% 0.28/1.91 thf(tcon_Polynomial_Opoly___Nat_Osemiring__char__0_51, axiom, (! [A7:$tType]: ((((ring_char_0 @ A7) & (comm_ring_1 @ A7)) => (semiring_char_0 @ (poly @ A7)))))).
% 0.28/1.91 thf(tcon_Polynomial_Opoly___Orderings_Opreorder_52, axiom, (! [A7:$tType]: (((linordered_idom @ A7) => (preorder @ (poly @ A7)))))).
% 0.28/1.91 thf(tcon_Polynomial_Opoly___Orderings_Olinorder_53, axiom, (! [A7:$tType]: (((linordered_idom @ A7) => (linorder @ (poly @ A7)))))).
% 0.28/1.91 thf(tcon_Polynomial_Opoly___Rings_Ocomm__ring__1_54, axiom, (! [A7:$tType]: (((comm_ring_1 @ A7) => (comm_ring_1 @ (poly @ A7)))))).
% 0.28/1.91 thf(tcon_Polynomial_Opoly___Rings_Osemiring__1_55, axiom, (! [A7:$tType]: (((comm_semiring_1 @ A7) => (semiring_1 @ (poly @ A7)))))).
% 0.28/1.91 thf(tcon_Polynomial_Opoly___Orderings_Oorder_56, axiom, (! [A7:$tType]: (((linordered_idom @ A7) => (order @ (poly @ A7)))))).
% 0.28/1.91 thf(tcon_Polynomial_Opoly___Nat_Oring__char__0_57, axiom, (! [A7:$tType]: ((((ring_char_0 @ A7) & (comm_ring_1 @ A7)) => (ring_char_0 @ (poly @ A7)))))).
% 0.28/1.91 thf(tcon_Polynomial_Opoly___Orderings_Oord_58, axiom, (! [A7:$tType]: (((linordered_idom @ A7) => (ord @ (poly @ A7)))))).
% 0.28/1.91 thf(tcon_Polynomial_Opoly___Groups_Ozero_59, axiom, (! [A7:$tType]: (((zero @ A7) => (zero @ (poly @ A7)))))).
% 0.28/1.91 thf(tcon_Polynomial_Opoly___Rings_Oidom_60, axiom, (! [A7:$tType]: (((idom @ A7) => (idom @ (poly @ A7)))))).
% 0.28/1.91 thf(tcon_Polynomial_Opoly___Groups_Oone_61, axiom, (! [A7:$tType]: (((comm_semiring_1 @ A7) => (one @ (poly @ A7)))))).
% 0.28/1.91 thf(conj_0, conjecture, (((sturm_1700286437_r_pos @ p) @ x) != (~ (((((((poly2 @ real) @ p) @ x) = (zero_zero @ real)) => ((sturm_1700286437_r_pos @ ((pderiv @ real) @ p)) @ x)) & (((((poly2 @ real) @ p) @ x) != (zero_zero @ real)) => (((ord_less @ real) @ (zero_zero @ real)) @ (((poly2 @ real) @ p) @ x)))))))).
% 0.28/1.91 % SZS output end ListOfTHF for /export/starexec/sandbox2/tmp/tmp.t0hPoxVh8o/DTF2DTF31756.p
% 0.28/1.91 ------------------------
% 0.28/1.95 ---- Cleaned THF ---
% 0.28/1.95 thf(ty_t_Polynomial_Opoly,type,
% 0.28/1.95 poly: $tType > $tType ).
% 0.28/1.95
% 0.28/1.95 thf(ty_t_Real_Oreal,type,
% 0.28/1.95 real: $tType ).
% 0.28/1.95
% 0.28/1.95 thf(ty_t_Set_Oset,type,
% 0.28/1.95 set: $tType > $tType ).
% 0.28/1.95
% 0.28/1.95 thf(ty_t_Nat_Onat,type,
% 0.28/1.95 nat: $tType ).
% 0.28/1.95
% 0.28/1.95 thf(ty_t_Int_Oint,type,
% 0.28/1.95 int: $tType ).
% 0.28/1.95
% 0.28/1.95 thf(sy_cl_Rings_Olinordered__idom,type,
% 0.28/1.95 linordered_idom:
% 0.28/1.95 !>[A: $tType] : $o ).
% 0.28/1.95
% 0.28/1.95 thf(sy_cl_HOL_Otype,type,
% 0.28/1.95 type:
% 0.28/1.95 !>[A: $tType] : $o ).
% 0.28/1.95
% 0.28/1.95 thf(sy_cl_Groups_Oone,type,
% 0.28/1.95 one:
% 0.28/1.95 !>[A: $tType] : $o ).
% 0.28/1.95
% 0.28/1.95 thf(sy_cl_Rings_Oidom,type,
% 0.28/1.95 idom:
% 0.28/1.95 !>[A: $tType] : $o ).
% 0.28/1.95
% 0.28/1.95 thf(sy_cl_Groups_Ozero,type,
% 0.28/1.95 zero:
% 0.28/1.95 !>[A: $tType] : $o ).
% 0.28/1.95
% 0.28/1.95 thf(sy_cl_Orderings_Oord,type,
% 0.28/1.95 ord:
% 0.28/1.95 !>[A: $tType] : $o ).
% 0.28/1.95
% 0.28/1.95 thf(sy_cl_Nat_Oring__char__0,type,
% 0.28/1.95 ring_char_0:
% 0.28/1.95 !>[A: $tType] : $o ).
% 0.28/1.95
% 0.28/1.95 thf(sy_cl_Orderings_Oorder,type,
% 0.28/1.95 order:
% 0.28/1.95 !>[A: $tType] : $o ).
% 0.28/1.95
% 0.28/1.95 thf(sy_cl_Orderings_Ono__bot,type,
% 0.28/1.95 no_bot:
% 0.28/1.95 !>[A: $tType] : $o ).
% 0.28/1.95
% 0.28/1.95 thf(sy_cl_Orderings_Ono__top,type,
% 0.28/1.95 no_top:
% 0.28/1.95 !>[A: $tType] : $o ).
% 0.28/1.95
% 0.28/1.95 thf(sy_cl_Rings_Osemiring__1,type,
% 0.28/1.95 semiring_1:
% 0.28/1.95 !>[A: $tType] : $o ).
% 0.28/1.95
% 0.28/1.95 thf(sy_cl_Rings_Ocomm__ring__1,type,
% 0.28/1.95 comm_ring_1:
% 0.28/1.95 !>[A: $tType] : $o ).
% 0.28/1.95
% 0.28/1.95 thf(sy_cl_Orderings_Olinorder,type,
% 0.28/1.95 linorder:
% 0.28/1.95 !>[A: $tType] : $o ).
% 0.28/1.95
% 0.28/1.95 thf(sy_cl_Orderings_Opreorder,type,
% 0.28/1.95 preorder:
% 0.28/1.95 !>[A: $tType] : $o ).
% 0.28/1.95
% 0.28/1.95 thf(sy_cl_Fields_Ofield__char__0,type,
% 0.28/1.95 field_char_0:
% 0.28/1.95 !>[A: $tType] : $o ).
% 0.28/1.95
% 0.28/1.95 thf(sy_cl_Nat_Osemiring__char__0,type,
% 0.28/1.95 semiring_char_0:
% 0.28/1.95 !>[A: $tType] : $o ).
% 0.28/1.95
% 0.28/1.95 thf(sy_cl_Orderings_Owellorder,type,
% 0.28/1.95 wellorder:
% 0.28/1.95 !>[A: $tType] : $o ).
% 0.28/1.95
% 0.28/1.95 thf(sy_cl_Orderings_Odense__order,type,
% 0.28/1.95 dense_order:
% 0.28/1.95 !>[A: $tType] : $o ).
% 0.28/1.95
% 0.28/1.95 thf(sy_cl_Rings_Ocomm__semiring__0,type,
% 0.28/1.95 comm_semiring_0:
% 0.28/1.95 !>[A: $tType] : $o ).
% 0.28/1.95
% 0.28/1.95 thf(sy_cl_Rings_Ocomm__semiring__1,type,
% 0.28/1.95 comm_semiring_1:
% 0.28/1.95 !>[A: $tType] : $o ).
% 0.28/1.95
% 0.28/1.95 thf(sy_cl_Fields_Olinordered__field,type,
% 0.28/1.95 linordered_field:
% 0.28/1.95 !>[A: $tType] : $o ).
% 0.28/1.95
% 0.28/1.95 thf(sy_cl_Orderings_Odense__linorder,type,
% 0.28/1.95 dense_linorder:
% 0.28/1.95 !>[A: $tType] : $o ).
% 0.28/1.95
% 0.28/1.95 thf(sy_cl_Rings_Oring__no__zero__divisors,type,
% 0.28/1.95 ring_n68954251visors:
% 0.28/1.95 !>[A: $tType] : $o ).
% 0.28/1.95
% 0.28/1.95 thf(sy_cl_Rings_Osemiring__no__zero__divisors,type,
% 0.28/1.95 semiri1193490041visors:
% 0.28/1.95 !>[A: $tType] : $o ).
% 0.28/1.95
% 0.28/1.95 thf(sy_cl_Rings_Olinordered__nonzero__semiring,type,
% 0.28/1.95 linord1659791738miring:
% 0.28/1.95 !>[A: $tType] : $o ).
% 0.28/1.95
% 0.28/1.95 thf(sy_cl_Groups_Ocanonically__ordered__monoid__add,type,
% 0.28/1.95 canoni770627133id_add:
% 0.28/1.95 !>[A: $tType] : $o ).
% 0.28/1.95
% 0.28/1.95 thf(sy_cl_Conditionally__Complete__Lattices_Oconditionally__complete__linorder,type,
% 0.28/1.95 condit1037483654norder:
% 0.28/1.95 !>[A: $tType] : $o ).
% 0.28/1.95
% 0.28/1.95 thf(sy_c_Groups_Oone__class_Oone,type,
% 0.28/1.95 one_one:
% 0.28/1.95 !>[A: $tType] : A ).
% 0.28/1.95
% 0.28/1.95 thf(sy_c_Groups_Ozero__class_Ozero,type,
% 0.28/1.95 zero_zero:
% 0.28/1.95 !>[A: $tType] : A ).
% 0.28/1.95
% 0.28/1.95 thf(sy_c_Nat_OSuc,type,
% 0.28/1.95 suc: nat > nat ).
% 0.28/1.95
% 0.28/1.95 thf(sy_c_Nat_Osemiring__1__class_Oof__nat,type,
% 0.28/1.95 semiring_1_of_nat:
% 0.28/1.95 !>[A: $tType] : ( nat > A ) ).
% 0.28/1.95
% 0.28/1.95 thf(sy_c_Orderings_Oord__class_Oless,type,
% 0.28/1.95 ord_less:
% 0.28/1.95 !>[A: $tType] : ( A > A > $o ) ).
% 0.28/1.95
% 0.28/1.95 thf(sy_c_Orderings_Oord__class_Oless__eq,type,
% 0.28/1.95 ord_less_eq:
% 0.28/1.95 !>[A: $tType] : ( A > A > $o ) ).
% 0.28/1.95
% 0.28/1.95 thf(sy_c_Polynomial_Ois__zero,type,
% 0.28/1.95 is_zero:
% 0.28/1.95 !>[A: $tType] : ( ( poly @ A ) > $o ) ).
% 0.28/1.95
% 0.28/1.95 thf(sy_c_Polynomial_Oorder,type,
% 0.28/1.95 order2:
% 0.28/1.95 !>[A: $tType] : ( A > ( poly @ A ) > nat ) ).
% 0.28/1.95
% 0.28/1.95 thf(sy_c_Polynomial_Opderiv,type,
% 0.28/1.95 pderiv:
% 0.28/1.95 !>[A: $tType] : ( ( poly @ A ) > ( poly @ A ) ) ).
% 0.28/1.95
% 0.28/1.95 thf(sy_c_Polynomial_Opoly,type,
% 0.28/1.95 poly2:
% 0.28/1.95 !>[A: $tType] : ( ( poly @ A ) > A > A ) ).
% 0.28/1.95
% 0.28/1.95 thf(sy_c_Polynomial_Opoly_Ocoeff,type,
% 0.28/1.95 coeff:
% 0.28/1.95 !>[A: $tType] : ( ( poly @ A ) > nat > A ) ).
% 0.28/1.95
% 0.28/1.95 thf(sy_c_Polynomial_Opoly__cutoff,type,
% 0.28/1.95 poly_cutoff:
% 0.28/1.95 !>[A: $tType] : ( nat > ( poly @ A ) > ( poly @ A ) ) ).
% 0.28/1.95
% 0.28/1.95 thf(sy_c_Polynomial_Opoly__shift,type,
% 0.28/1.95 poly_shift:
% 0.28/1.95 !>[A: $tType] : ( nat > ( poly @ A ) > ( poly @ A ) ) ).
% 0.28/1.95
% 0.28/1.95 thf(sy_c_Polynomial_Oreflect__poly,type,
% 0.28/1.95 reflect_poly:
% 0.28/1.95 !>[A: $tType] : ( ( poly @ A ) > ( poly @ A ) ) ).
% 0.28/1.95
% 0.28/1.95 thf(sy_c_Polynomial_Orsquarefree,type,
% 0.28/1.95 rsquarefree:
% 0.28/1.95 !>[A: $tType] : ( ( poly @ A ) > $o ) ).
% 0.28/1.95
% 0.28/1.95 thf(sy_c_Set_OCollect,type,
% 0.28/1.95 collect:
% 0.28/1.95 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
% 0.28/1.95
% 0.28/1.95 thf(sy_c_Sturm__Tarski__Mirabelle__osyjafhhev_Ocross,type,
% 0.28/1.95 sturm_424270202_cross: ( poly @ real ) > real > real > int ).
% 0.28/1.95
% 0.28/1.95 thf(sy_c_Sturm__Tarski__Mirabelle__osyjafhhev_Osign__r__pos,type,
% 0.28/1.95 sturm_1700286437_r_pos: ( poly @ real ) > real > $o ).
% 0.28/1.95
% 0.28/1.95 thf(sy_c_Sturm__Tarski__Mirabelle__osyjafhhev_Ovariation,type,
% 0.28/1.95 sturm_1771227917iation: real > real > int ).
% 0.28/1.95
% 0.28/1.95 thf(sy_c_member,type,
% 0.28/1.95 member:
% 0.28/1.95 !>[A: $tType] : ( A > ( set @ A ) > $o ) ).
% 0.28/1.95
% 0.28/1.95 thf(sy_v_p,type,
% 0.28/1.95 p: poly @ real ).
% 0.28/1.95
% 0.28/1.95 thf(sy_v_x,type,
% 0.28/1.95 x: real ).
% 0.28/1.95
% 0.28/1.95 thf(fact_0_False,axiom,
% 0.28/1.95 ( ( poly2 @ real @ p @ x )
% 0.28/1.95 != ( zero_zero @ real ) ) ).
% 0.28/1.95
% 0.28/1.95 thf(fact_1__092_060open_062sign__r__pos_Ap_Ax_A_092_060Longrightarrow_062_A0_A_060_Apoly_Ap_Ax_092_060close_062,axiom,
% 0.28/1.95 ( ( sturm_1700286437_r_pos @ p @ x )
% 0.28/1.95 => ( ord_less @ real @ ( zero_zero @ real ) @ ( poly2 @ real @ p @ x ) ) ) ).
% 0.28/1.95
% 0.28/1.95 thf(fact_2__092_060open_0620_A_060_Apoly_Ap_Ax_A_092_060Longrightarrow_062_Asign__r__pos_Ap_Ax_092_060close_062,axiom,
% 0.28/1.95 ( ( ord_less @ real @ ( zero_zero @ real ) @ ( poly2 @ real @ p @ x ) )
% 0.28/1.95 => ( sturm_1700286437_r_pos @ p @ x ) ) ).
% 0.28/1.95
% 0.28/1.95 thf(fact_3_assms,axiom,
% 0.28/1.95 ( p
% 0.28/1.95 != ( zero_zero @ ( poly @ real ) ) ) ).
% 0.28/1.95
% 0.28/1.95 thf(fact_4_poly__0,axiom,
% 0.28/1.95 ! [A: $tType] :
% 0.28/1.95 ( ( comm_semiring_0 @ A )
% 0.28/1.95 => ! [X: A] :
% 0.28/1.95 ( ( poly2 @ A @ ( zero_zero @ ( poly @ A ) ) @ X )
% 0.28/1.95 = ( zero_zero @ A ) ) ) ).
% 0.28/1.95
% 0.28/1.95 thf(fact_5_not__gr__zero,axiom,
% 0.28/1.95 ! [A: $tType] :
% 0.28/1.95 ( ( canoni770627133id_add @ A )
% 0.28/1.95 => ! [N: A] :
% 0.28/1.95 ( ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ N ) )
% 0.28/1.95 = ( N
% 0.28/1.95 = ( zero_zero @ A ) ) ) ) ).
% 0.28/1.95
% 0.28/1.95 thf(fact_6_poly__IVT__neg,axiom,
% 0.28/1.95 ! [A2: real,B: real,P: poly @ real] :
% 0.28/1.95 ( ( ord_less @ real @ A2 @ B )
% 0.28/1.95 => ( ( ord_less @ real @ ( zero_zero @ real ) @ ( poly2 @ real @ P @ A2 ) )
% 0.28/1.95 => ( ( ord_less @ real @ ( poly2 @ real @ P @ B ) @ ( zero_zero @ real ) )
% 0.28/1.95 => ? [X2: real] :
% 0.28/1.95 ( ( ord_less @ real @ A2 @ X2 )
% 0.28/1.95 & ( ord_less @ real @ X2 @ B )
% 0.28/1.95 & ( ( poly2 @ real @ P @ X2 )
% 0.28/1.95 = ( zero_zero @ real ) ) ) ) ) ) ).
% 0.28/1.95
% 0.28/1.95 thf(fact_7_poly__IVT__pos,axiom,
% 0.28/1.95 ! [A2: real,B: real,P: poly @ real] :
% 0.28/1.95 ( ( ord_less @ real @ A2 @ B )
% 0.28/1.95 => ( ( ord_less @ real @ ( poly2 @ real @ P @ A2 ) @ ( zero_zero @ real ) )
% 0.28/1.95 => ( ( ord_less @ real @ ( zero_zero @ real ) @ ( poly2 @ real @ P @ B ) )
% 0.28/1.95 => ? [X2: real] :
% 0.28/1.95 ( ( ord_less @ real @ A2 @ X2 )
% 0.28/1.95 & ( ord_less @ real @ X2 @ B )
% 0.28/1.95 & ( ( poly2 @ real @ P @ X2 )
% 0.28/1.95 = ( zero_zero @ real ) ) ) ) ) ) ).
% 0.28/1.95
% 0.28/1.95 thf(fact_8_poly__all__0__iff__0,axiom,
% 0.28/1.95 ! [A: $tType] :
% 0.28/1.95 ( ( ( ring_char_0 @ A )
% 0.28/1.95 & ( comm_ring_1 @ A )
% 0.28/1.95 & ( ring_n68954251visors @ A ) )
% 0.28/1.95 => ! [P: poly @ A] :
% 0.28/1.95 ( ( ! [X3: A] :
% 0.28/1.95 ( ( poly2 @ A @ P @ X3 )
% 0.28/1.95 = ( zero_zero @ A ) ) )
% 0.28/1.95 = ( P
% 0.28/1.95 = ( zero_zero @ ( poly @ A ) ) ) ) ) ).
% 0.28/1.95
% 0.28/1.95 thf(fact_9_less__numeral__extra_I3_J,axiom,
% 0.28/1.95 ! [A: $tType] :
% 0.28/1.95 ( ( linord1659791738miring @ A )
% 0.28/1.95 => ~ ( ord_less @ A @ ( zero_zero @ A ) @ ( zero_zero @ A ) ) ) ).
% 0.28/1.95
% 0.28/1.95 thf(fact_10_field__lbound__gt__zero,axiom,
% 0.28/1.95 ! [A: $tType] :
% 0.28/1.95 ( ( linordered_field @ A )
% 0.28/1.95 => ! [D1: A,D2: A] :
% 0.28/1.95 ( ( ord_less @ A @ ( zero_zero @ A ) @ D1 )
% 0.28/1.95 => ( ( ord_less @ A @ ( zero_zero @ A ) @ D2 )
% 0.28/1.95 => ? [E: A] :
% 0.28/1.95 ( ( ord_less @ A @ ( zero_zero @ A ) @ E )
% 0.28/1.95 & ( ord_less @ A @ E @ D1 )
% 0.28/1.95 & ( ord_less @ A @ E @ D2 ) ) ) ) ) ).
% 0.28/1.95
% 0.28/1.95 thf(fact_11_gr__zeroI,axiom,
% 0.28/1.95 ! [A: $tType] :
% 0.28/1.95 ( ( canoni770627133id_add @ A )
% 0.28/1.95 => ! [N: A] :
% 0.28/1.95 ( ( N
% 0.28/1.95 != ( zero_zero @ A ) )
% 0.28/1.95 => ( ord_less @ A @ ( zero_zero @ A ) @ N ) ) ) ).
% 0.28/1.95
% 0.28/1.95 thf(fact_12_not__less__zero,axiom,
% 0.28/1.95 ! [A: $tType] :
% 0.28/1.95 ( ( canoni770627133id_add @ A )
% 0.28/1.95 => ! [N: A] :
% 0.28/1.95 ~ ( ord_less @ A @ N @ ( zero_zero @ A ) ) ) ).
% 0.28/1.95
% 0.28/1.95 thf(fact_13_gr__implies__not__zero,axiom,
% 0.28/1.95 ! [A: $tType] :
% 0.28/1.95 ( ( canoni770627133id_add @ A )
% 0.28/1.95 => ! [M: A,N: A] :
% 0.28/1.95 ( ( ord_less @ A @ M @ N )
% 0.28/1.95 => ( N
% 0.28/1.95 != ( zero_zero @ A ) ) ) ) ).
% 0.28/1.95
% 0.28/1.95 thf(fact_14_zero__less__iff__neq__zero,axiom,
% 0.28/1.95 ! [A: $tType] :
% 0.28/1.95 ( ( canoni770627133id_add @ A )
% 0.28/1.95 => ! [N: A] :
% 0.28/1.95 ( ( ord_less @ A @ ( zero_zero @ A ) @ N )
% 0.28/1.95 = ( N
% 0.28/1.95 != ( zero_zero @ A ) ) ) ) ).
% 0.28/1.95
% 0.28/1.95 thf(fact_15_pderiv__0,axiom,
% 0.28/1.95 ! [A: $tType] :
% 0.28/1.95 ( ( ( comm_semiring_1 @ A )
% 0.28/1.95 & ( semiri1193490041visors @ A ) )
% 0.28/1.95 => ( ( pderiv @ A @ ( zero_zero @ ( poly @ A ) ) )
% 0.28/1.95 = ( zero_zero @ ( poly @ A ) ) ) ) ).
% 0.28/1.95
% 0.28/1.95 thf(fact_16_zero__reorient,axiom,
% 0.28/1.95 ! [A: $tType] :
% 0.28/1.95 ( ( zero @ A )
% 0.28/1.95 => ! [X: A] :
% 0.28/1.95 ( ( ( zero_zero @ A )
% 0.28/1.95 = X )
% 0.28/1.95 = ( X
% 0.28/1.95 = ( zero_zero @ A ) ) ) ) ).
% 0.28/1.95
% 0.28/1.95 thf(fact_17_poly__eq__poly__eq__iff,axiom,
% 0.28/1.95 ! [A: $tType] :
% 0.28/1.95 ( ( ( ring_char_0 @ A )
% 0.28/1.95 & ( comm_ring_1 @ A )
% 0.28/1.95 & ( ring_n68954251visors @ A ) )
% 0.28/1.95 => ! [P: poly @ A,Q: poly @ A] :
% 0.28/1.95 ( ( ( poly2 @ A @ P )
% 0.28/1.95 = ( poly2 @ A @ Q ) )
% 0.28/1.95 = ( P = Q ) ) ) ).
% 0.28/1.95
% 0.28/1.95 thf(fact_18_cross__0,axiom,
% 0.28/1.95 ! [A2: real,B: real] :
% 0.28/1.95 ( ( sturm_424270202_cross @ ( zero_zero @ ( poly @ real ) ) @ A2 @ B )
% 0.28/1.95 = ( zero_zero @ int ) ) ).
% 0.28/1.95
% 0.28/1.95 thf(fact_19_rsquarefree__roots,axiom,
% 0.28/1.95 ! [A: $tType] :
% 0.28/1.95 ( ( field_char_0 @ A )
% 0.28/1.95 => ( ( rsquarefree @ A )
% 0.28/1.95 = ( ^ [P2: poly @ A] :
% 0.28/1.95 ! [A3: A] :
% 0.28/1.95 ~ ( ( ( poly2 @ A @ P2 @ A3 )
% 0.28/1.95 = ( zero_zero @ A ) )
% 0.28/1.95 & ( ( poly2 @ A @ ( pderiv @ A @ P2 ) @ A3 )
% 0.28/1.95 = ( zero_zero @ A ) ) ) ) ) ) ).
% 0.28/1.95
% 0.28/1.95 thf(fact_20_cross__no__root,axiom,
% 0.28/1.95 ! [A2: real,B: real,P: poly @ real] :
% 0.28/1.95 ( ( ord_less @ real @ A2 @ B )
% 0.28/1.95 => ( ! [X2: real] :
% 0.28/1.95 ( ( ( ord_less @ real @ A2 @ X2 )
% 0.28/1.95 & ( ord_less @ real @ X2 @ B ) )
% 0.28/1.95 => ( ( poly2 @ real @ P @ X2 )
% 0.28/1.95 != ( zero_zero @ real ) ) )
% 0.28/1.95 => ( ( sturm_424270202_cross @ P @ A2 @ B )
% 0.28/1.95 = ( zero_zero @ int ) ) ) ) ).
% 0.28/1.95
% 0.28/1.95 thf(fact_21_is__zero__null,axiom,
% 0.28/1.95 ! [A: $tType] :
% 0.28/1.95 ( ( zero @ A )
% 0.28/1.95 => ( ( is_zero @ A )
% 0.28/1.95 = ( ^ [P2: poly @ A] :
% 0.28/1.95 ( P2
% 0.28/1.95 = ( zero_zero @ ( poly @ A ) ) ) ) ) ) ).
% 0.28/1.95
% 0.28/1.95 thf(fact_22_poly__cutoff__0,axiom,
% 0.28/1.95 ! [A: $tType] :
% 0.28/1.95 ( ( zero @ A )
% 0.28/1.95 => ! [N: nat] :
% 0.28/1.95 ( ( poly_cutoff @ A @ N @ ( zero_zero @ ( poly @ A ) ) )
% 0.28/1.95 = ( zero_zero @ ( poly @ A ) ) ) ) ).
% 0.28/1.95
% 0.28/1.95 thf(fact_23_reflect__poly__at__0__eq__0__iff,axiom,
% 0.28/1.95 ! [A: $tType] :
% 0.28/1.95 ( ( comm_semiring_0 @ A )
% 0.28/1.95 => ! [P: poly @ A] :
% 0.28/1.95 ( ( ( poly2 @ A @ ( reflect_poly @ A @ P ) @ ( zero_zero @ A ) )
% 0.28/1.95 = ( zero_zero @ A ) )
% 0.28/1.95 = ( P
% 0.28/1.95 = ( zero_zero @ ( poly @ A ) ) ) ) ) ).
% 0.28/1.95
% 0.28/1.95 thf(fact_24_next__non__root__interval,axiom,
% 0.28/1.95 ! [P: poly @ real,Lb: real] :
% 0.28/1.95 ( ( P
% 0.28/1.95 != ( zero_zero @ ( poly @ real ) ) )
% 0.28/1.95 => ~ ! [Ub: real] :
% 0.28/1.95 ( ( ord_less @ real @ Lb @ Ub )
% 0.28/1.95 => ~ ! [Z: real] :
% 0.28/1.95 ( ( ( ord_less @ real @ Lb @ Z )
% 0.28/1.95 & ( ord_less_eq @ real @ Z @ Ub ) )
% 0.28/1.95 => ( ( poly2 @ real @ P @ Z )
% 0.28/1.95 != ( zero_zero @ real ) ) ) ) ) ).
% 0.28/1.95
% 0.28/1.95 thf(fact_25_last__non__root__interval,axiom,
% 0.28/1.95 ! [P: poly @ real,Ub2: real] :
% 0.28/1.95 ( ( P
% 0.28/1.95 != ( zero_zero @ ( poly @ real ) ) )
% 0.28/1.95 => ~ ! [Lb2: real] :
% 0.28/1.95 ( ( ord_less @ real @ Lb2 @ Ub2 )
% 0.28/1.95 => ~ ! [Z: real] :
% 0.28/1.95 ( ( ( ord_less_eq @ real @ Lb2 @ Z )
% 0.28/1.95 & ( ord_less @ real @ Z @ Ub2 ) )
% 0.28/1.95 => ( ( poly2 @ real @ P @ Z )
% 0.28/1.95 != ( zero_zero @ real ) ) ) ) ) ).
% 0.28/1.95
% 0.28/1.95 thf(fact_26_poly__shift__0,axiom,
% 0.28/1.95 ! [A: $tType] :
% 0.28/1.95 ( ( zero @ A )
% 0.28/1.95 => ! [N: nat] :
% 0.28/1.95 ( ( poly_shift @ A @ N @ ( zero_zero @ ( poly @ A ) ) )
% 0.28/1.95 = ( zero_zero @ ( poly @ A ) ) ) ) ).
% 0.28/1.95
% 0.28/1.95 thf(fact_27_order__root,axiom,
% 0.28/1.95 ! [A: $tType] :
% 0.28/1.95 ( ( idom @ A )
% 0.28/1.95 => ! [P: poly @ A,A2: A] :
% 0.28/1.95 ( ( ( poly2 @ A @ P @ A2 )
% 0.28/1.95 = ( zero_zero @ A ) )
% 0.28/1.95 = ( ( P
% 0.28/1.95 = ( zero_zero @ ( poly @ A ) ) )
% 0.28/1.95 | ( ( order2 @ A @ A2 @ P )
% 0.28/1.95 != ( zero_zero @ nat ) ) ) ) ) ).
% 0.28/1.95
% 0.28/1.95 thf(fact_28_le__zero__eq,axiom,
% 0.28/1.95 ! [A: $tType] :
% 0.28/1.95 ( ( canoni770627133id_add @ A )
% 0.28/1.95 => ! [N: A] :
% 0.28/1.95 ( ( ord_less_eq @ A @ N @ ( zero_zero @ A ) )
% 0.28/1.95 = ( N
% 0.28/1.95 = ( zero_zero @ A ) ) ) ) ).
% 0.28/1.95
% 0.28/1.95 thf(fact_29_reflect__poly__0,axiom,
% 0.28/1.95 ! [A: $tType] :
% 0.28/1.95 ( ( zero @ A )
% 0.28/1.95 => ( ( reflect_poly @ A @ ( zero_zero @ ( poly @ A ) ) )
% 0.28/1.95 = ( zero_zero @ ( poly @ A ) ) ) ) ).
% 0.28/1.95
% 0.28/1.95 thf(fact_30_poly__shift__id,axiom,
% 0.28/1.95 ! [A: $tType] :
% 0.28/1.95 ( ( zero @ A )
% 0.28/1.95 => ( ( poly_shift @ A @ ( zero_zero @ nat ) )
% 0.28/1.95 = ( ^ [X3: poly @ A] : X3 ) ) ) ).
% 0.28/1.95
% 0.28/1.95 thf(fact_31_complete__real,axiom,
% 0.28/1.95 ! [S: set @ real] :
% 0.28/1.95 ( ? [X4: real] : ( member @ real @ X4 @ S )
% 0.28/1.95 => ( ? [Z: real] :
% 0.28/1.95 ! [X2: real] :
% 0.28/1.95 ( ( member @ real @ X2 @ S )
% 0.28/1.95 => ( ord_less_eq @ real @ X2 @ Z ) )
% 0.28/1.95 => ? [Y: real] :
% 0.28/1.95 ( ! [X4: real] :
% 0.28/1.95 ( ( member @ real @ X4 @ S )
% 0.28/1.95 => ( ord_less_eq @ real @ X4 @ Y ) )
% 0.28/1.95 & ! [Z: real] :
% 0.28/1.95 ( ! [X2: real] :
% 0.28/1.95 ( ( member @ real @ X2 @ S )
% 0.28/1.95 => ( ord_less_eq @ real @ X2 @ Z ) )
% 0.28/1.95 => ( ord_less_eq @ real @ Y @ Z ) ) ) ) ) ).
% 0.28/1.95
% 0.28/1.95 thf(fact_32_le__numeral__extra_I3_J,axiom,
% 0.28/1.95 ! [A: $tType] :
% 0.28/1.95 ( ( linord1659791738miring @ A )
% 0.28/1.95 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( zero_zero @ A ) ) ) ).
% 0.28/1.95
% 0.28/1.95 thf(fact_33_zero__le,axiom,
% 0.28/1.95 ! [A: $tType] :
% 0.28/1.95 ( ( canoni770627133id_add @ A )
% 0.28/1.95 => ! [X: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ X ) ) ).
% 0.28/1.95
% 0.28/1.95 thf(fact_34_less__eq__real__def,axiom,
% 0.28/1.95 ( ( ord_less_eq @ real )
% 0.28/1.95 = ( ^ [X3: real,Y2: real] :
% 0.28/1.95 ( ( ord_less @ real @ X3 @ Y2 )
% 0.28/1.95 | ( X3 = Y2 ) ) ) ) ).
% 0.28/1.95
% 0.28/1.95 thf(fact_35_order__0I,axiom,
% 0.28/1.95 ! [A: $tType] :
% 0.28/1.95 ( ( idom @ A )
% 0.28/1.95 => ! [P: poly @ A,A2: A] :
% 0.28/1.95 ( ( ( poly2 @ A @ P @ A2 )
% 0.28/1.95 != ( zero_zero @ A ) )
% 0.28/1.95 => ( ( order2 @ A @ A2 @ P )
% 0.28/1.95 = ( zero_zero @ nat ) ) ) ) ).
% 0.28/1.95
% 0.28/1.95 thf(fact_36_not__eq__pos__or__neg__iff__1,axiom,
% 0.28/1.95 ! [Lb: real,Ub2: real,P: poly @ real] :
% 0.28/1.95 ( ( ! [Z2: real] :
% 0.28/1.95 ( ( ( ord_less @ real @ Lb @ Z2 )
% 0.28/1.95 & ( ord_less_eq @ real @ Z2 @ Ub2 ) )
% 0.28/1.95 => ( ( poly2 @ real @ P @ Z2 )
% 0.28/1.95 != ( zero_zero @ real ) ) ) )
% 0.28/1.95 = ( ! [Z2: real] :
% 0.28/1.95 ( ( ( ord_less @ real @ Lb @ Z2 )
% 0.28/1.95 & ( ord_less_eq @ real @ Z2 @ Ub2 ) )
% 0.28/1.95 => ( ord_less @ real @ ( zero_zero @ real ) @ ( poly2 @ real @ P @ Z2 ) ) )
% 0.28/1.95 | ! [Z2: real] :
% 0.28/1.95 ( ( ( ord_less @ real @ Lb @ Z2 )
% 0.28/1.95 & ( ord_less_eq @ real @ Z2 @ Ub2 ) )
% 0.28/1.95 => ( ord_less @ real @ ( poly2 @ real @ P @ Z2 ) @ ( zero_zero @ real ) ) ) ) ) ).
% 0.28/1.95
% 0.28/1.95 thf(fact_37_not__eq__pos__or__neg__iff__2,axiom,
% 0.28/1.95 ! [Lb: real,Ub2: real,P: poly @ real] :
% 0.28/1.95 ( ( ! [Z2: real] :
% 0.28/1.95 ( ( ( ord_less_eq @ real @ Lb @ Z2 )
% 0.28/1.95 & ( ord_less @ real @ Z2 @ Ub2 ) )
% 0.28/1.95 => ( ( poly2 @ real @ P @ Z2 )
% 0.28/1.95 != ( zero_zero @ real ) ) ) )
% 0.28/1.95 = ( ! [Z2: real] :
% 0.28/1.95 ( ( ( ord_less_eq @ real @ Lb @ Z2 )
% 0.28/1.95 & ( ord_less @ real @ Z2 @ Ub2 ) )
% 0.28/1.95 => ( ord_less @ real @ ( zero_zero @ real ) @ ( poly2 @ real @ P @ Z2 ) ) )
% 0.28/1.95 | ! [Z2: real] :
% 0.28/1.95 ( ( ( ord_less_eq @ real @ Lb @ Z2 )
% 0.28/1.95 & ( ord_less @ real @ Z2 @ Ub2 ) )
% 0.28/1.95 => ( ord_less @ real @ ( poly2 @ real @ P @ Z2 ) @ ( zero_zero @ real ) ) ) ) ) ).
% 0.28/1.95
% 0.28/1.95 thf(fact_38_neq0__conv,axiom,
% 0.28/1.95 ! [N: nat] :
% 0.28/1.95 ( ( N
% 0.28/1.95 != ( zero_zero @ nat ) )
% 0.28/1.95 = ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ).
% 0.28/1.95
% 0.28/1.95 thf(fact_39_less__nat__zero__code,axiom,
% 0.28/1.95 ! [N: nat] :
% 0.28/1.95 ~ ( ord_less @ nat @ N @ ( zero_zero @ nat ) ) ).
% 0.28/1.95
% 0.28/1.95 thf(fact_40_bot__nat__0_Onot__eq__extremum,axiom,
% 0.28/1.95 ! [A2: nat] :
% 0.28/1.95 ( ( A2
% 0.28/1.95 != ( zero_zero @ nat ) )
% 0.28/1.95 = ( ord_less @ nat @ ( zero_zero @ nat ) @ A2 ) ) ).
% 0.28/1.95
% 0.28/1.95 thf(fact_41_order__refl,axiom,
% 0.28/1.95 ! [A: $tType] :
% 0.28/1.95 ( ( preorder @ A )
% 0.28/1.95 => ! [X: A] : ( ord_less_eq @ A @ X @ X ) ) ).
% 0.28/1.95
% 0.28/1.95 thf(fact_42_minf_I8_J,axiom,
% 0.28/1.95 ! [A: $tType] :
% 0.28/1.95 ( ( linorder @ A )
% 0.28/1.95 => ! [T: A] :
% 0.28/1.95 ? [Z3: A] :
% 0.28/1.95 ! [X4: A] :
% 0.28/1.95 ( ( ord_less @ A @ X4 @ Z3 )
% 0.28/1.95 => ~ ( ord_less_eq @ A @ T @ X4 ) ) ) ).
% 0.28/1.95
% 0.28/1.95 thf(fact_43_minf_I6_J,axiom,
% 0.28/1.95 ! [A: $tType] :
% 0.28/1.95 ( ( linorder @ A )
% 0.28/1.95 => ! [T: A] :
% 0.28/1.95 ? [Z3: A] :
% 0.28/1.95 ! [X4: A] :
% 0.28/1.95 ( ( ord_less @ A @ X4 @ Z3 )
% 0.28/1.95 => ( ord_less_eq @ A @ X4 @ T ) ) ) ).
% 0.28/1.95
% 0.28/1.95 thf(fact_44_mem__Collect__eq,axiom,
% 0.28/1.95 ! [A: $tType,A2: A,P3: A > $o] :
% 0.28/1.95 ( ( member @ A @ A2 @ ( collect @ A @ P3 ) )
% 0.28/1.95 = ( P3 @ A2 ) ) ).
% 0.28/1.95
% 0.28/1.95 thf(fact_45_Collect__mem__eq,axiom,
% 0.28/1.95 ! [A: $tType,A4: set @ A] :
% 0.28/1.95 ( ( collect @ A
% 0.28/1.95 @ ^ [X3: A] : ( member @ A @ X3 @ A4 ) )
% 0.28/1.95 = A4 ) ).
% 0.28/1.95
% 0.28/1.95 thf(fact_46_Collect__cong,axiom,
% 0.28/1.95 ! [A: $tType,P3: A > $o,Q2: A > $o] :
% 0.28/1.95 ( ! [X2: A] :
% 0.28/1.95 ( ( P3 @ X2 )
% 0.28/1.95 = ( Q2 @ X2 ) )
% 0.28/1.95 => ( ( collect @ A @ P3 )
% 0.28/1.95 = ( collect @ A @ Q2 ) ) ) ).
% 0.28/1.95
% 0.28/1.95 thf(fact_47_ext,axiom,
% 0.28/1.95 ! [B2: $tType,A: $tType,F: A > B2,G: A > B2] :
% 0.28/1.95 ( ! [X2: A] :
% 0.28/1.95 ( ( F @ X2 )
% 0.28/1.95 = ( G @ X2 ) )
% 0.28/1.95 => ( F = G ) ) ).
% 0.28/1.95
% 0.28/1.95 thf(fact_48_pinf_I8_J,axiom,
% 0.28/1.95 ! [A: $tType] :
% 0.28/1.95 ( ( linorder @ A )
% 0.28/1.95 => ! [T: A] :
% 0.28/1.95 ? [Z3: A] :
% 0.28/1.95 ! [X4: A] :
% 0.28/1.95 ( ( ord_less @ A @ Z3 @ X4 )
% 0.28/1.95 => ( ord_less_eq @ A @ T @ X4 ) ) ) ).
% 0.28/1.95
% 0.28/1.95 thf(fact_49_pinf_I6_J,axiom,
% 0.28/1.95 ! [A: $tType] :
% 0.28/1.95 ( ( linorder @ A )
% 0.28/1.95 => ! [T: A] :
% 0.28/1.95 ? [Z3: A] :
% 0.28/1.95 ! [X4: A] :
% 0.28/1.95 ( ( ord_less @ A @ Z3 @ X4 )
% 0.28/1.95 => ~ ( ord_less_eq @ A @ X4 @ T ) ) ) ).
% 0.28/1.95
% 0.28/1.95 thf(fact_50_verit__comp__simplify1_I3_J,axiom,
% 0.28/1.95 ! [B2: $tType] :
% 0.28/1.95 ( ( linorder @ B2 )
% 0.28/1.95 => ! [B3: B2,A5: B2] :
% 0.28/1.95 ( ( ~ ( ord_less_eq @ B2 @ B3 @ A5 ) )
% 0.28/1.95 = ( ord_less @ B2 @ A5 @ B3 ) ) ) ).
% 0.28/1.95
% 0.28/1.95 thf(fact_51_bot__nat__0_Oextremum,axiom,
% 0.28/1.95 ! [A2: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ A2 ) ).
% 0.28/1.95
% 0.28/1.95 thf(fact_52_le0,axiom,
% 0.28/1.95 ! [N: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ N ) ).
% 0.28/1.95
% 0.28/1.95 thf(fact_53_nat__less__le,axiom,
% 0.28/1.95 ( ( ord_less @ nat )
% 0.28/1.95 = ( ^ [M2: nat,N2: nat] :
% 0.28/1.95 ( ( ord_less_eq @ nat @ M2 @ N2 )
% 0.28/1.95 & ( M2 != N2 ) ) ) ) ).
% 0.28/1.95
% 0.28/1.95 thf(fact_54_less__imp__le__nat,axiom,
% 0.28/1.95 ! [M: nat,N: nat] :
% 0.28/1.95 ( ( ord_less @ nat @ M @ N )
% 0.28/1.95 => ( ord_less_eq @ nat @ M @ N ) ) ).
% 0.28/1.95
% 0.28/1.95 thf(fact_55_le__eq__less__or__eq,axiom,
% 0.28/1.95 ( ( ord_less_eq @ nat )
% 0.28/1.95 = ( ^ [M2: nat,N2: nat] :
% 0.28/1.95 ( ( ord_less @ nat @ M2 @ N2 )
% 0.28/1.95 | ( M2 = N2 ) ) ) ) ).
% 0.28/1.95
% 0.28/1.95 thf(fact_56_less__or__eq__imp__le,axiom,
% 0.28/1.95 ! [M: nat,N: nat] :
% 0.28/1.95 ( ( ( ord_less @ nat @ M @ N )
% 0.28/1.95 | ( M = N ) )
% 0.28/1.95 => ( ord_less_eq @ nat @ M @ N ) ) ).
% 0.28/1.95
% 0.28/1.95 thf(fact_57_le__neq__implies__less,axiom,
% 0.28/1.95 ! [M: nat,N: nat] :
% 0.28/1.95 ( ( ord_less_eq @ nat @ M @ N )
% 0.28/1.95 => ( ( M != N )
% 0.28/1.95 => ( ord_less @ nat @ M @ N ) ) ) ).
% 0.28/1.95
% 0.28/1.95 thf(fact_58_less__mono__imp__le__mono,axiom,
% 0.28/1.95 ! [F: nat > nat,I: nat,J: nat] :
% 0.28/1.95 ( ! [I2: nat,J2: nat] :
% 0.28/1.95 ( ( ord_less @ nat @ I2 @ J2 )
% 0.28/1.95 => ( ord_less @ nat @ ( F @ I2 ) @ ( F @ J2 ) ) )
% 0.28/1.95 => ( ( ord_less_eq @ nat @ I @ J )
% 0.28/1.95 => ( ord_less_eq @ nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).
% 0.28/1.95
% 0.28/1.95 thf(fact_59_infinite__descent__measure,axiom,
% 0.28/1.95 ! [A: $tType,P3: A > $o,V: A > nat,X: A] :
% 0.28/1.95 ( ! [X2: A] :
% 0.28/1.95 ( ~ ( P3 @ X2 )
% 0.28/1.95 => ? [Y3: A] :
% 0.28/1.95 ( ( ord_less @ nat @ ( V @ Y3 ) @ ( V @ X2 ) )
% 0.28/1.95 & ~ ( P3 @ Y3 ) ) )
% 0.28/1.95 => ( P3 @ X ) ) ).
% 0.28/1.95
% 0.28/1.95 thf(fact_60_linorder__neqE__nat,axiom,
% 0.28/1.95 ! [X: nat,Y4: nat] :
% 0.28/1.95 ( ( X != Y4 )
% 0.28/1.95 => ( ~ ( ord_less @ nat @ X @ Y4 )
% 0.28/1.95 => ( ord_less @ nat @ Y4 @ X ) ) ) ).
% 0.28/1.95
% 0.28/1.95 thf(fact_61_infinite__descent,axiom,
% 0.28/1.95 ! [P3: nat > $o,N: nat] :
% 0.28/1.95 ( ! [N3: nat] :
% 0.28/1.95 ( ~ ( P3 @ N3 )
% 0.28/1.95 => ? [M3: nat] :
% 0.28/1.95 ( ( ord_less @ nat @ M3 @ N3 )
% 0.28/1.95 & ~ ( P3 @ M3 ) ) )
% 0.28/1.95 => ( P3 @ N ) ) ).
% 0.28/1.95
% 0.28/1.95 thf(fact_62_nat__less__induct,axiom,
% 0.28/1.95 ! [P3: nat > $o,N: nat] :
% 0.28/1.95 ( ! [N3: nat] :
% 0.28/1.95 ( ! [M3: nat] :
% 0.28/1.95 ( ( ord_less @ nat @ M3 @ N3 )
% 0.28/1.95 => ( P3 @ M3 ) )
% 0.28/1.95 => ( P3 @ N3 ) )
% 0.28/1.95 => ( P3 @ N ) ) ).
% 0.28/1.95
% 0.28/1.95 thf(fact_63_less__irrefl__nat,axiom,
% 0.28/1.96 ! [N: nat] :
% 0.28/1.96 ~ ( ord_less @ nat @ N @ N ) ).
% 0.28/1.96
% 0.28/1.96 thf(fact_64_less__not__refl3,axiom,
% 0.28/1.96 ! [S2: nat,T: nat] :
% 0.28/1.96 ( ( ord_less @ nat @ S2 @ T )
% 0.28/1.96 => ( S2 != T ) ) ).
% 0.28/1.96
% 0.28/1.96 thf(fact_65_less__not__refl2,axiom,
% 0.28/1.96 ! [N: nat,M: nat] :
% 0.28/1.96 ( ( ord_less @ nat @ N @ M )
% 0.28/1.96 => ( M != N ) ) ).
% 0.28/1.96
% 0.28/1.96 thf(fact_66_less__not__refl,axiom,
% 0.28/1.96 ! [N: nat] :
% 0.28/1.96 ~ ( ord_less @ nat @ N @ N ) ).
% 0.28/1.96
% 0.28/1.96 thf(fact_67_nat__neq__iff,axiom,
% 0.28/1.96 ! [M: nat,N: nat] :
% 0.28/1.96 ( ( M != N )
% 0.28/1.96 = ( ( ord_less @ nat @ M @ N )
% 0.28/1.96 | ( ord_less @ nat @ N @ M ) ) ) ).
% 0.28/1.96
% 0.28/1.96 thf(fact_68_bot__nat__0_Oextremum__uniqueI,axiom,
% 0.28/1.96 ! [A2: nat] :
% 0.28/1.96 ( ( ord_less_eq @ nat @ A2 @ ( zero_zero @ nat ) )
% 0.28/1.96 => ( A2
% 0.28/1.96 = ( zero_zero @ nat ) ) ) ).
% 0.28/1.96
% 0.28/1.96 thf(fact_69_bot__nat__0_Oextremum__unique,axiom,
% 0.28/1.96 ! [A2: nat] :
% 0.28/1.96 ( ( ord_less_eq @ nat @ A2 @ ( zero_zero @ nat ) )
% 0.28/1.96 = ( A2
% 0.28/1.96 = ( zero_zero @ nat ) ) ) ).
% 0.28/1.96
% 0.28/1.96 thf(fact_70_ex__least__nat__le,axiom,
% 0.28/1.96 ! [P3: nat > $o,N: nat] :
% 0.28/1.96 ( ( P3 @ N )
% 0.28/1.96 => ( ~ ( P3 @ ( zero_zero @ nat ) )
% 0.28/1.96 => ? [K: nat] :
% 0.28/1.96 ( ( ord_less_eq @ nat @ K @ N )
% 0.28/1.96 & ! [I3: nat] :
% 0.28/1.96 ( ( ord_less @ nat @ I3 @ K )
% 0.28/1.96 => ~ ( P3 @ I3 ) )
% 0.28/1.96 & ( P3 @ K ) ) ) ) ).
% 0.28/1.96
% 0.28/1.96 thf(fact_71_le__0__eq,axiom,
% 0.28/1.96 ! [N: nat] :
% 0.28/1.96 ( ( ord_less_eq @ nat @ N @ ( zero_zero @ nat ) )
% 0.28/1.96 = ( N
% 0.28/1.96 = ( zero_zero @ nat ) ) ) ).
% 0.28/1.96
% 0.28/1.96 thf(fact_72_less__eq__nat_Osimps_I1_J,axiom,
% 0.28/1.96 ! [N: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ N ) ).
% 0.28/1.96
% 0.28/1.96 thf(fact_73_conj__le__cong,axiom,
% 0.28/1.96 ! [X: int,X5: int,P3: $o,P4: $o] :
% 0.28/1.96 ( ( X = X5 )
% 0.28/1.96 => ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X5 )
% 0.28/1.96 => ( P3 = P4 ) )
% 0.28/1.96 => ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X )
% 0.28/1.96 & P3 )
% 0.28/1.96 = ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X5 )
% 0.28/1.96 & P4 ) ) ) ) ).
% 0.28/1.96
% 0.28/1.96 thf(fact_74_imp__le__cong,axiom,
% 0.28/1.96 ! [X: int,X5: int,P3: $o,P4: $o] :
% 0.28/1.96 ( ( X = X5 )
% 0.28/1.96 => ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X5 )
% 0.28/1.96 => ( P3 = P4 ) )
% 0.28/1.96 => ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X )
% 0.28/1.96 => P3 )
% 0.28/1.96 = ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X5 )
% 0.28/1.96 => P4 ) ) ) ) ).
% 0.28/1.96
% 0.28/1.96 thf(fact_75_le__funD,axiom,
% 0.28/1.96 ! [B2: $tType,A: $tType] :
% 0.28/1.96 ( ( ord @ B2 )
% 0.28/1.96 => ! [F: A > B2,G: A > B2,X: A] :
% 0.28/1.96 ( ( ord_less_eq @ ( A > B2 ) @ F @ G )
% 0.28/1.96 => ( ord_less_eq @ B2 @ ( F @ X ) @ ( G @ X ) ) ) ) ).
% 0.28/1.96
% 0.28/1.96 thf(fact_76_le__funE,axiom,
% 0.28/1.96 ! [B2: $tType,A: $tType] :
% 0.28/1.96 ( ( ord @ B2 )
% 0.28/1.96 => ! [F: A > B2,G: A > B2,X: A] :
% 0.28/1.96 ( ( ord_less_eq @ ( A > B2 ) @ F @ G )
% 0.28/1.96 => ( ord_less_eq @ B2 @ ( F @ X ) @ ( G @ X ) ) ) ) ).
% 0.28/1.96
% 0.28/1.96 thf(fact_77_le__funI,axiom,
% 0.28/1.96 ! [B2: $tType,A: $tType] :
% 0.28/1.96 ( ( ord @ B2 )
% 0.28/1.96 => ! [F: A > B2,G: A > B2] :
% 0.28/1.96 ( ! [X2: A] : ( ord_less_eq @ B2 @ ( F @ X2 ) @ ( G @ X2 ) )
% 0.28/1.96 => ( ord_less_eq @ ( A > B2 ) @ F @ G ) ) ) ).
% 0.28/1.96
% 0.28/1.96 thf(fact_78_le__fun__def,axiom,
% 0.28/1.96 ! [B2: $tType,A: $tType] :
% 0.28/1.96 ( ( ord @ B2 )
% 0.28/1.96 => ( ( ord_less_eq @ ( A > B2 ) )
% 0.28/1.96 = ( ^ [F2: A > B2,G2: A > B2] :
% 0.28/1.96 ! [X3: A] : ( ord_less_eq @ B2 @ ( F2 @ X3 ) @ ( G2 @ X3 ) ) ) ) ) ).
% 0.28/1.96
% 0.28/1.96 thf(fact_79_order__subst1,axiom,
% 0.28/1.96 ! [A: $tType,B2: $tType] :
% 0.28/1.96 ( ( ( order @ B2 )
% 0.28/1.96 & ( order @ A ) )
% 0.28/1.96 => ! [A2: A,F: B2 > A,B: B2,C: B2] :
% 0.28/1.96 ( ( ord_less_eq @ A @ A2 @ ( F @ B ) )
% 0.28/1.96 => ( ( ord_less_eq @ B2 @ B @ C )
% 0.28/1.96 => ( ! [X2: B2,Y: B2] :
% 0.28/1.96 ( ( ord_less_eq @ B2 @ X2 @ Y )
% 0.28/1.96 => ( ord_less_eq @ A @ ( F @ X2 ) @ ( F @ Y ) ) )
% 0.28/1.96 => ( ord_less_eq @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% 0.28/1.96
% 0.28/1.96 thf(fact_80_order__subst2,axiom,
% 0.28/1.96 ! [A: $tType,C2: $tType] :
% 0.28/1.96 ( ( ( order @ C2 )
% 0.28/1.96 & ( order @ A ) )
% 0.28/1.96 => ! [A2: A,B: A,F: A > C2,C: C2] :
% 0.28/1.96 ( ( ord_less_eq @ A @ A2 @ B )
% 0.28/1.96 => ( ( ord_less_eq @ C2 @ ( F @ B ) @ C )
% 0.28/1.96 => ( ! [X2: A,Y: A] :
% 0.28/1.96 ( ( ord_less_eq @ A @ X2 @ Y )
% 0.28/1.96 => ( ord_less_eq @ C2 @ ( F @ X2 ) @ ( F @ Y ) ) )
% 0.28/1.96 => ( ord_less_eq @ C2 @ ( F @ A2 ) @ C ) ) ) ) ) ).
% 0.28/1.96
% 0.28/1.96 thf(fact_81_verit__la__disequality,axiom,
% 0.28/1.96 ! [A: $tType] :
% 0.28/1.96 ( ( linorder @ A )
% 0.28/1.96 => ! [A2: A,B: A] :
% 0.28/1.96 ( ( A2 = B )
% 0.28/1.96 | ~ ( ord_less_eq @ A @ A2 @ B )
% 0.28/1.96 | ~ ( ord_less_eq @ A @ B @ A2 ) ) ) ).
% 0.28/1.96
% 0.28/1.96 thf(fact_82_ord__eq__le__subst,axiom,
% 0.28/1.96 ! [A: $tType,B2: $tType] :
% 0.28/1.96 ( ( ( ord @ B2 )
% 0.28/1.96 & ( ord @ A ) )
% 0.28/1.96 => ! [A2: A,F: B2 > A,B: B2,C: B2] :
% 0.28/1.96 ( ( A2
% 0.28/1.96 = ( F @ B ) )
% 0.28/1.96 => ( ( ord_less_eq @ B2 @ B @ C )
% 0.28/1.96 => ( ! [X2: B2,Y: B2] :
% 0.28/1.96 ( ( ord_less_eq @ B2 @ X2 @ Y )
% 0.28/1.96 => ( ord_less_eq @ A @ ( F @ X2 ) @ ( F @ Y ) ) )
% 0.28/1.96 => ( ord_less_eq @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% 0.28/1.96
% 0.28/1.96 thf(fact_83_ord__le__eq__subst,axiom,
% 0.28/1.96 ! [A: $tType,B2: $tType] :
% 0.28/1.96 ( ( ( ord @ B2 )
% 0.28/1.96 & ( ord @ A ) )
% 0.28/1.96 => ! [A2: A,B: A,F: A > B2,C: B2] :
% 0.28/1.96 ( ( ord_less_eq @ A @ A2 @ B )
% 0.28/1.96 => ( ( ( F @ B )
% 0.28/1.96 = C )
% 0.28/1.96 => ( ! [X2: A,Y: A] :
% 0.28/1.96 ( ( ord_less_eq @ A @ X2 @ Y )
% 0.28/1.96 => ( ord_less_eq @ B2 @ ( F @ X2 ) @ ( F @ Y ) ) )
% 0.28/1.96 => ( ord_less_eq @ B2 @ ( F @ A2 ) @ C ) ) ) ) ) ).
% 0.28/1.96
% 0.28/1.96 thf(fact_84_eq__iff,axiom,
% 0.28/1.96 ! [A: $tType] :
% 0.28/1.96 ( ( order @ A )
% 0.28/1.96 => ( ( ^ [Y5: A,Z4: A] : ( Y5 = Z4 ) )
% 0.28/1.96 = ( ^ [X3: A,Y2: A] :
% 0.28/1.96 ( ( ord_less_eq @ A @ X3 @ Y2 )
% 0.28/1.96 & ( ord_less_eq @ A @ Y2 @ X3 ) ) ) ) ) ).
% 0.28/1.96
% 0.28/1.96 thf(fact_85_antisym,axiom,
% 0.28/1.96 ! [A: $tType] :
% 0.28/1.96 ( ( order @ A )
% 0.28/1.96 => ! [X: A,Y4: A] :
% 0.28/1.96 ( ( ord_less_eq @ A @ X @ Y4 )
% 0.28/1.96 => ( ( ord_less_eq @ A @ Y4 @ X )
% 0.28/1.96 => ( X = Y4 ) ) ) ) ).
% 0.28/1.96
% 0.28/1.96 thf(fact_86_linear,axiom,
% 0.28/1.96 ! [A: $tType] :
% 0.28/1.96 ( ( linorder @ A )
% 0.28/1.96 => ! [X: A,Y4: A] :
% 0.28/1.96 ( ( ord_less_eq @ A @ X @ Y4 )
% 0.28/1.96 | ( ord_less_eq @ A @ Y4 @ X ) ) ) ).
% 0.28/1.96
% 0.28/1.96 thf(fact_87_eq__refl,axiom,
% 0.28/1.96 ! [A: $tType] :
% 0.28/1.96 ( ( preorder @ A )
% 0.28/1.96 => ! [X: A,Y4: A] :
% 0.28/1.96 ( ( X = Y4 )
% 0.28/1.96 => ( ord_less_eq @ A @ X @ Y4 ) ) ) ).
% 0.28/1.96
% 0.28/1.96 thf(fact_88_le__cases,axiom,
% 0.28/1.96 ! [A: $tType] :
% 0.28/1.96 ( ( linorder @ A )
% 0.28/1.96 => ! [X: A,Y4: A] :
% 0.28/1.96 ( ~ ( ord_less_eq @ A @ X @ Y4 )
% 0.28/1.96 => ( ord_less_eq @ A @ Y4 @ X ) ) ) ).
% 0.28/1.96
% 0.28/1.96 thf(fact_89_order_Otrans,axiom,
% 0.28/1.96 ! [A: $tType] :
% 0.28/1.96 ( ( order @ A )
% 0.28/1.96 => ! [A2: A,B: A,C: A] :
% 0.28/1.96 ( ( ord_less_eq @ A @ A2 @ B )
% 0.28/1.96 => ( ( ord_less_eq @ A @ B @ C )
% 0.28/1.96 => ( ord_less_eq @ A @ A2 @ C ) ) ) ) ).
% 0.28/1.96
% 0.28/1.96 thf(fact_90_le__cases3,axiom,
% 0.28/1.96 ! [A: $tType] :
% 0.28/1.96 ( ( linorder @ A )
% 0.28/1.96 => ! [X: A,Y4: A,Z5: A] :
% 0.28/1.96 ( ( ( ord_less_eq @ A @ X @ Y4 )
% 0.28/1.96 => ~ ( ord_less_eq @ A @ Y4 @ Z5 ) )
% 0.28/1.96 => ( ( ( ord_less_eq @ A @ Y4 @ X )
% 0.28/1.96 => ~ ( ord_less_eq @ A @ X @ Z5 ) )
% 0.28/1.96 => ( ( ( ord_less_eq @ A @ X @ Z5 )
% 0.28/1.96 => ~ ( ord_less_eq @ A @ Z5 @ Y4 ) )
% 0.28/1.96 => ( ( ( ord_less_eq @ A @ Z5 @ Y4 )
% 0.28/1.96 => ~ ( ord_less_eq @ A @ Y4 @ X ) )
% 0.28/1.96 => ( ( ( ord_less_eq @ A @ Y4 @ Z5 )
% 0.28/1.96 => ~ ( ord_less_eq @ A @ Z5 @ X ) )
% 0.28/1.96 => ~ ( ( ord_less_eq @ A @ Z5 @ X )
% 0.28/1.96 => ~ ( ord_less_eq @ A @ X @ Y4 ) ) ) ) ) ) ) ) ).
% 0.28/1.96
% 0.28/1.96 thf(fact_91_antisym__conv,axiom,
% 0.28/1.96 ! [A: $tType] :
% 0.28/1.96 ( ( order @ A )
% 0.28/1.96 => ! [Y4: A,X: A] :
% 0.28/1.96 ( ( ord_less_eq @ A @ Y4 @ X )
% 0.28/1.96 => ( ( ord_less_eq @ A @ X @ Y4 )
% 0.28/1.96 = ( X = Y4 ) ) ) ) ).
% 0.28/1.96
% 0.28/1.96 thf(fact_92_order__class_Oorder_Oeq__iff,axiom,
% 0.28/1.96 ! [A: $tType] :
% 0.28/1.96 ( ( order @ A )
% 0.28/1.96 => ( ( ^ [Y5: A,Z4: A] : ( Y5 = Z4 ) )
% 0.28/1.96 = ( ^ [A3: A,B4: A] :
% 0.28/1.96 ( ( ord_less_eq @ A @ A3 @ B4 )
% 0.28/1.96 & ( ord_less_eq @ A @ B4 @ A3 ) ) ) ) ) ).
% 0.28/1.96
% 0.28/1.96 thf(fact_93_ord__eq__le__trans,axiom,
% 0.28/1.96 ! [A: $tType] :
% 0.28/1.96 ( ( ord @ A )
% 0.28/1.96 => ! [A2: A,B: A,C: A] :
% 0.28/1.96 ( ( A2 = B )
% 0.28/1.96 => ( ( ord_less_eq @ A @ B @ C )
% 0.28/1.96 => ( ord_less_eq @ A @ A2 @ C ) ) ) ) ).
% 0.28/1.96
% 0.28/1.96 thf(fact_94_ord__le__eq__trans,axiom,
% 0.28/1.96 ! [A: $tType] :
% 0.28/1.96 ( ( ord @ A )
% 0.28/1.96 => ! [A2: A,B: A,C: A] :
% 0.28/1.96 ( ( ord_less_eq @ A @ A2 @ B )
% 0.28/1.96 => ( ( B = C )
% 0.28/1.96 => ( ord_less_eq @ A @ A2 @ C ) ) ) ) ).
% 0.28/1.96
% 0.28/1.96 thf(fact_95_order__class_Oorder_Oantisym,axiom,
% 0.28/1.96 ! [A: $tType] :
% 0.28/1.96 ( ( order @ A )
% 0.28/1.96 => ! [A2: A,B: A] :
% 0.28/1.96 ( ( ord_less_eq @ A @ A2 @ B )
% 0.28/1.96 => ( ( ord_less_eq @ A @ B @ A2 )
% 0.28/1.96 => ( A2 = B ) ) ) ) ).
% 0.28/1.96
% 0.28/1.96 thf(fact_96_order__trans,axiom,
% 0.28/1.96 ! [A: $tType] :
% 0.28/1.96 ( ( preorder @ A )
% 0.28/1.96 => ! [X: A,Y4: A,Z5: A] :
% 0.28/1.96 ( ( ord_less_eq @ A @ X @ Y4 )
% 0.28/1.96 => ( ( ord_less_eq @ A @ Y4 @ Z5 )
% 0.28/1.96 => ( ord_less_eq @ A @ X @ Z5 ) ) ) ) ).
% 0.28/1.96
% 0.28/1.96 thf(fact_97_dual__order_Orefl,axiom,
% 0.28/1.96 ! [A: $tType] :
% 0.28/1.96 ( ( order @ A )
% 0.28/1.96 => ! [A2: A] : ( ord_less_eq @ A @ A2 @ A2 ) ) ).
% 0.28/1.96
% 0.28/1.96 thf(fact_98_linorder__wlog,axiom,
% 0.28/1.96 ! [A: $tType] :
% 0.28/1.96 ( ( linorder @ A )
% 0.28/1.96 => ! [P3: A > A > $o,A2: A,B: A] :
% 0.28/1.96 ( ! [A6: A,B5: A] :
% 0.28/1.96 ( ( ord_less_eq @ A @ A6 @ B5 )
% 0.28/1.96 => ( P3 @ A6 @ B5 ) )
% 0.28/1.96 => ( ! [A6: A,B5: A] :
% 0.28/1.96 ( ( P3 @ B5 @ A6 )
% 0.28/1.96 => ( P3 @ A6 @ B5 ) )
% 0.28/1.96 => ( P3 @ A2 @ B ) ) ) ) ).
% 0.28/1.96
% 0.28/1.96 thf(fact_99_dual__order_Otrans,axiom,
% 0.28/1.96 ! [A: $tType] :
% 0.28/1.96 ( ( order @ A )
% 0.28/1.96 => ! [B: A,A2: A,C: A] :
% 0.28/1.96 ( ( ord_less_eq @ A @ B @ A2 )
% 0.28/1.96 => ( ( ord_less_eq @ A @ C @ B )
% 0.28/1.96 => ( ord_less_eq @ A @ C @ A2 ) ) ) ) ).
% 0.28/1.96
% 0.28/1.96 thf(fact_100_dual__order_Oeq__iff,axiom,
% 0.28/1.96 ! [A: $tType] :
% 0.28/1.96 ( ( order @ A )
% 0.28/1.96 => ( ( ^ [Y5: A,Z4: A] : ( Y5 = Z4 ) )
% 0.28/1.96 = ( ^ [A3: A,B4: A] :
% 0.28/1.96 ( ( ord_less_eq @ A @ B4 @ A3 )
% 0.28/1.96 & ( ord_less_eq @ A @ A3 @ B4 ) ) ) ) ) ).
% 0.28/1.96
% 0.28/1.96 thf(fact_101_dual__order_Oantisym,axiom,
% 0.28/1.96 ! [A: $tType] :
% 0.28/1.96 ( ( order @ A )
% 0.28/1.96 => ! [B: A,A2: A] :
% 0.28/1.96 ( ( ord_less_eq @ A @ B @ A2 )
% 0.28/1.96 => ( ( ord_less_eq @ A @ A2 @ B )
% 0.28/1.96 => ( A2 = B ) ) ) ) ).
% 0.28/1.96
% 0.28/1.96 thf(fact_102_dual__order_Ostrict__implies__not__eq,axiom,
% 0.28/1.96 ! [A: $tType] :
% 0.28/1.96 ( ( order @ A )
% 0.28/1.96 => ! [B: A,A2: A] :
% 0.28/1.96 ( ( ord_less @ A @ B @ A2 )
% 0.28/1.96 => ( A2 != B ) ) ) ).
% 0.28/1.96
% 0.28/1.96 thf(fact_103_order_Ostrict__implies__not__eq,axiom,
% 0.28/1.96 ! [A: $tType] :
% 0.28/1.96 ( ( order @ A )
% 0.28/1.96 => ! [A2: A,B: A] :
% 0.28/1.96 ( ( ord_less @ A @ A2 @ B )
% 0.28/1.96 => ( A2 != B ) ) ) ).
% 0.28/1.96
% 0.28/1.96 thf(fact_104_not__less__iff__gr__or__eq,axiom,
% 0.28/1.96 ! [A: $tType] :
% 0.28/1.96 ( ( linorder @ A )
% 0.28/1.96 => ! [X: A,Y4: A] :
% 0.28/1.96 ( ( ~ ( ord_less @ A @ X @ Y4 ) )
% 0.28/1.96 = ( ( ord_less @ A @ Y4 @ X )
% 0.28/1.96 | ( X = Y4 ) ) ) ) ).
% 0.28/1.96
% 0.28/1.96 thf(fact_105_dual__order_Ostrict__trans,axiom,
% 0.28/1.96 ! [A: $tType] :
% 0.28/1.96 ( ( order @ A )
% 0.28/1.96 => ! [B: A,A2: A,C: A] :
% 0.28/1.96 ( ( ord_less @ A @ B @ A2 )
% 0.28/1.96 => ( ( ord_less @ A @ C @ B )
% 0.28/1.96 => ( ord_less @ A @ C @ A2 ) ) ) ) ).
% 0.28/1.96
% 0.28/1.96 thf(fact_106_linorder__less__wlog,axiom,
% 0.28/1.96 ! [A: $tType] :
% 0.28/1.96 ( ( linorder @ A )
% 0.28/1.96 => ! [P3: A > A > $o,A2: A,B: A] :
% 0.28/1.96 ( ! [A6: A,B5: A] :
% 0.28/1.96 ( ( ord_less @ A @ A6 @ B5 )
% 0.28/1.96 => ( P3 @ A6 @ B5 ) )
% 0.28/1.96 => ( ! [A6: A] : ( P3 @ A6 @ A6 )
% 0.28/1.96 => ( ! [A6: A,B5: A] :
% 0.28/1.96 ( ( P3 @ B5 @ A6 )
% 0.28/1.96 => ( P3 @ A6 @ B5 ) )
% 0.28/1.96 => ( P3 @ A2 @ B ) ) ) ) ) ).
% 0.28/1.96
% 0.28/1.96 thf(fact_107_exists__least__iff,axiom,
% 0.28/1.96 ! [A: $tType] :
% 0.28/1.96 ( ( wellorder @ A )
% 0.28/1.96 => ( ( ^ [P5: A > $o] :
% 0.28/1.96 ? [X6: A] : ( P5 @ X6 ) )
% 0.28/1.96 = ( ^ [P6: A > $o] :
% 0.28/1.96 ? [N2: A] :
% 0.28/1.96 ( ( P6 @ N2 )
% 0.28/1.96 & ! [M2: A] :
% 0.28/1.96 ( ( ord_less @ A @ M2 @ N2 )
% 0.28/1.96 => ~ ( P6 @ M2 ) ) ) ) ) ) ).
% 0.28/1.96
% 0.28/1.96 thf(fact_108_less__imp__not__less,axiom,
% 0.28/1.96 ! [A: $tType] :
% 0.28/1.96 ( ( preorder @ A )
% 0.28/1.96 => ! [X: A,Y4: A] :
% 0.28/1.96 ( ( ord_less @ A @ X @ Y4 )
% 0.28/1.96 => ~ ( ord_less @ A @ Y4 @ X ) ) ) ).
% 0.28/1.96
% 0.28/1.96 thf(fact_109_order_Ostrict__trans,axiom,
% 0.28/1.96 ! [A: $tType] :
% 0.28/1.96 ( ( order @ A )
% 0.28/1.96 => ! [A2: A,B: A,C: A] :
% 0.28/1.96 ( ( ord_less @ A @ A2 @ B )
% 0.28/1.96 => ( ( ord_less @ A @ B @ C )
% 0.28/1.96 => ( ord_less @ A @ A2 @ C ) ) ) ) ).
% 0.28/1.96
% 0.28/1.96 thf(fact_110_dual__order_Oirrefl,axiom,
% 0.28/1.96 ! [A: $tType] :
% 0.28/1.96 ( ( order @ A )
% 0.28/1.96 => ! [A2: A] :
% 0.28/1.96 ~ ( ord_less @ A @ A2 @ A2 ) ) ).
% 0.28/1.96
% 0.28/1.96 thf(fact_111_linorder__cases,axiom,
% 0.28/1.96 ! [A: $tType] :
% 0.28/1.96 ( ( linorder @ A )
% 0.28/1.96 => ! [X: A,Y4: A] :
% 0.28/1.96 ( ~ ( ord_less @ A @ X @ Y4 )
% 0.28/1.96 => ( ( X != Y4 )
% 0.28/1.96 => ( ord_less @ A @ Y4 @ X ) ) ) ) ).
% 0.28/1.96
% 0.28/1.96 thf(fact_112_less__imp__triv,axiom,
% 0.28/1.96 ! [A: $tType] :
% 0.28/1.96 ( ( preorder @ A )
% 0.28/1.96 => ! [X: A,Y4: A,P3: $o] :
% 0.28/1.96 ( ( ord_less @ A @ X @ Y4 )
% 0.28/1.96 => ( ( ord_less @ A @ Y4 @ X )
% 0.28/1.96 => P3 ) ) ) ).
% 0.28/1.96
% 0.28/1.96 thf(fact_113_less__imp__not__eq2,axiom,
% 0.28/1.96 ! [A: $tType] :
% 0.28/1.96 ( ( order @ A )
% 0.28/1.96 => ! [X: A,Y4: A] :
% 0.28/1.96 ( ( ord_less @ A @ X @ Y4 )
% 0.28/1.96 => ( Y4 != X ) ) ) ).
% 0.28/1.96
% 0.28/1.96 thf(fact_114_antisym__conv3,axiom,
% 0.28/1.96 ! [A: $tType] :
% 0.28/1.96 ( ( linorder @ A )
% 0.28/1.96 => ! [Y4: A,X: A] :
% 0.28/1.96 ( ~ ( ord_less @ A @ Y4 @ X )
% 0.28/1.96 => ( ( ~ ( ord_less @ A @ X @ Y4 ) )
% 0.28/1.96 = ( X = Y4 ) ) ) ) ).
% 0.28/1.96
% 0.28/1.96 thf(fact_115_less__induct,axiom,
% 0.28/1.96 ! [A: $tType] :
% 0.28/1.96 ( ( wellorder @ A )
% 0.28/1.96 => ! [P3: A > $o,A2: A] :
% 0.28/1.96 ( ! [X2: A] :
% 0.28/1.96 ( ! [Y3: A] :
% 0.28/1.96 ( ( ord_less @ A @ Y3 @ X2 )
% 0.28/1.96 => ( P3 @ Y3 ) )
% 0.28/1.96 => ( P3 @ X2 ) )
% 0.28/1.96 => ( P3 @ A2 ) ) ) ).
% 0.28/1.96
% 0.28/1.96 thf(fact_116_less__not__sym,axiom,
% 0.28/1.96 ! [A: $tType] :
% 0.28/1.96 ( ( preorder @ A )
% 0.28/1.96 => ! [X: A,Y4: A] :
% 0.28/1.96 ( ( ord_less @ A @ X @ Y4 )
% 0.28/1.96 => ~ ( ord_less @ A @ Y4 @ X ) ) ) ).
% 0.28/1.96
% 0.28/1.96 thf(fact_117_less__imp__not__eq,axiom,
% 0.28/1.96 ! [A: $tType] :
% 0.28/1.96 ( ( order @ A )
% 0.28/1.96 => ! [X: A,Y4: A] :
% 0.28/1.96 ( ( ord_less @ A @ X @ Y4 )
% 0.28/1.96 => ( X != Y4 ) ) ) ).
% 0.28/1.96
% 0.28/1.96 thf(fact_118_dual__order_Oasym,axiom,
% 0.28/1.96 ! [A: $tType] :
% 0.28/1.96 ( ( order @ A )
% 0.28/1.96 => ! [B: A,A2: A] :
% 0.28/1.96 ( ( ord_less @ A @ B @ A2 )
% 0.28/1.96 => ~ ( ord_less @ A @ A2 @ B ) ) ) ).
% 0.28/1.96
% 0.28/1.96 thf(fact_119_ord__less__eq__trans,axiom,
% 0.28/1.96 ! [A: $tType] :
% 0.28/1.96 ( ( ord @ A )
% 0.28/1.96 => ! [A2: A,B: A,C: A] :
% 0.28/1.96 ( ( ord_less @ A @ A2 @ B )
% 0.28/1.96 => ( ( B = C )
% 0.28/1.96 => ( ord_less @ A @ A2 @ C ) ) ) ) ).
% 0.28/1.96
% 0.28/1.96 thf(fact_120_ord__eq__less__trans,axiom,
% 0.28/1.96 ! [A: $tType] :
% 0.28/1.96 ( ( ord @ A )
% 0.28/1.96 => ! [A2: A,B: A,C: A] :
% 0.28/1.96 ( ( A2 = B )
% 0.28/1.96 => ( ( ord_less @ A @ B @ C )
% 0.28/1.96 => ( ord_less @ A @ A2 @ C ) ) ) ) ).
% 0.28/1.96
% 0.28/1.96 thf(fact_121_less__irrefl,axiom,
% 0.28/1.96 ! [A: $tType] :
% 0.28/1.96 ( ( preorder @ A )
% 0.28/1.96 => ! [X: A] :
% 0.28/1.96 ~ ( ord_less @ A @ X @ X ) ) ).
% 0.28/1.96
% 0.28/1.96 thf(fact_122_less__linear,axiom,
% 0.28/1.96 ! [A: $tType] :
% 0.28/1.96 ( ( linorder @ A )
% 0.28/1.96 => ! [X: A,Y4: A] :
% 0.28/1.96 ( ( ord_less @ A @ X @ Y4 )
% 0.28/1.96 | ( X = Y4 )
% 0.28/1.96 | ( ord_less @ A @ Y4 @ X ) ) ) ).
% 0.28/1.96
% 0.28/1.96 thf(fact_123_less__trans,axiom,
% 0.28/1.96 ! [A: $tType] :
% 0.28/1.96 ( ( preorder @ A )
% 0.28/1.96 => ! [X: A,Y4: A,Z5: A] :
% 0.28/1.96 ( ( ord_less @ A @ X @ Y4 )
% 0.28/1.96 => ( ( ord_less @ A @ Y4 @ Z5 )
% 0.28/1.96 => ( ord_less @ A @ X @ Z5 ) ) ) ) ).
% 0.28/1.96
% 0.28/1.96 thf(fact_124_less__asym_H,axiom,
% 0.28/1.96 ! [A: $tType] :
% 0.28/1.96 ( ( preorder @ A )
% 0.28/1.96 => ! [A2: A,B: A] :
% 0.28/1.96 ( ( ord_less @ A @ A2 @ B )
% 0.28/1.96 => ~ ( ord_less @ A @ B @ A2 ) ) ) ).
% 0.28/1.96
% 0.28/1.96 thf(fact_125_less__asym,axiom,
% 0.28/1.96 ! [A: $tType] :
% 0.28/1.96 ( ( preorder @ A )
% 0.28/1.96 => ! [X: A,Y4: A] :
% 0.28/1.96 ( ( ord_less @ A @ X @ Y4 )
% 0.28/1.96 => ~ ( ord_less @ A @ Y4 @ X ) ) ) ).
% 0.28/1.96
% 0.28/1.96 thf(fact_126_less__imp__neq,axiom,
% 0.28/1.96 ! [A: $tType] :
% 0.28/1.96 ( ( order @ A )
% 0.28/1.96 => ! [X: A,Y4: A] :
% 0.28/1.96 ( ( ord_less @ A @ X @ Y4 )
% 0.28/1.96 => ( X != Y4 ) ) ) ).
% 0.28/1.96
% 0.28/1.96 thf(fact_127_dense,axiom,
% 0.28/1.96 ! [A: $tType] :
% 0.28/1.96 ( ( dense_order @ A )
% 0.28/1.96 => ! [X: A,Y4: A] :
% 0.28/1.96 ( ( ord_less @ A @ X @ Y4 )
% 0.28/1.96 => ? [Z3: A] :
% 0.28/1.96 ( ( ord_less @ A @ X @ Z3 )
% 0.28/1.96 & ( ord_less @ A @ Z3 @ Y4 ) ) ) ) ).
% 0.28/1.96
% 0.28/1.96 thf(fact_128_order_Oasym,axiom,
% 0.28/1.96 ! [A: $tType] :
% 0.28/1.96 ( ( order @ A )
% 0.28/1.96 => ! [A2: A,B: A] :
% 0.28/1.96 ( ( ord_less @ A @ A2 @ B )
% 0.28/1.96 => ~ ( ord_less @ A @ B @ A2 ) ) ) ).
% 0.28/1.96
% 0.28/1.96 thf(fact_129_neq__iff,axiom,
% 0.28/1.96 ! [A: $tType] :
% 0.28/1.96 ( ( linorder @ A )
% 0.28/1.96 => ! [X: A,Y4: A] :
% 0.28/1.96 ( ( X != Y4 )
% 0.28/1.96 = ( ( ord_less @ A @ X @ Y4 )
% 0.28/1.96 | ( ord_less @ A @ Y4 @ X ) ) ) ) ).
% 0.28/1.96
% 0.28/1.96 thf(fact_130_neqE,axiom,
% 0.28/1.96 ! [A: $tType] :
% 0.28/1.96 ( ( linorder @ A )
% 0.28/1.96 => ! [X: A,Y4: A] :
% 0.28/1.96 ( ( X != Y4 )
% 0.28/1.96 => ( ~ ( ord_less @ A @ X @ Y4 )
% 0.28/1.96 => ( ord_less @ A @ Y4 @ X ) ) ) ) ).
% 0.28/1.96
% 0.28/1.96 thf(fact_131_gt__ex,axiom,
% 0.28/1.96 ! [A: $tType] :
% 0.28/1.96 ( ( no_top @ A )
% 0.28/1.96 => ! [X: A] :
% 0.28/1.96 ? [X_1: A] : ( ord_less @ A @ X @ X_1 ) ) ).
% 0.28/1.96
% 0.28/1.96 thf(fact_132_lt__ex,axiom,
% 0.28/1.96 ! [A: $tType] :
% 0.28/1.96 ( ( no_bot @ A )
% 0.28/1.96 => ! [X: A] :
% 0.28/1.96 ? [Y: A] : ( ord_less @ A @ Y @ X ) ) ).
% 0.28/1.96
% 0.28/1.96 thf(fact_133_order__less__subst2,axiom,
% 0.28/1.96 ! [A: $tType,C2: $tType] :
% 0.28/1.96 ( ( ( order @ C2 )
% 0.28/1.96 & ( order @ A ) )
% 0.28/1.96 => ! [A2: A,B: A,F: A > C2,C: C2] :
% 0.28/1.96 ( ( ord_less @ A @ A2 @ B )
% 0.28/1.96 => ( ( ord_less @ C2 @ ( F @ B ) @ C )
% 0.28/1.96 => ( ! [X2: A,Y: A] :
% 0.28/1.96 ( ( ord_less @ A @ X2 @ Y )
% 0.28/1.96 => ( ord_less @ C2 @ ( F @ X2 ) @ ( F @ Y ) ) )
% 0.28/1.96 => ( ord_less @ C2 @ ( F @ A2 ) @ C ) ) ) ) ) ).
% 0.28/1.96
% 0.28/1.96 thf(fact_134_order__less__subst1,axiom,
% 0.28/1.96 ! [A: $tType,B2: $tType] :
% 0.28/1.96 ( ( ( order @ B2 )
% 0.28/1.96 & ( order @ A ) )
% 0.28/1.96 => ! [A2: A,F: B2 > A,B: B2,C: B2] :
% 0.28/1.96 ( ( ord_less @ A @ A2 @ ( F @ B ) )
% 0.28/1.96 => ( ( ord_less @ B2 @ B @ C )
% 0.28/1.96 => ( ! [X2: B2,Y: B2] :
% 0.28/1.96 ( ( ord_less @ B2 @ X2 @ Y )
% 0.28/1.96 => ( ord_less @ A @ ( F @ X2 ) @ ( F @ Y ) ) )
% 0.28/1.96 => ( ord_less @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% 0.28/1.96
% 0.28/1.96 thf(fact_135_ord__less__eq__subst,axiom,
% 0.28/1.96 ! [A: $tType,B2: $tType] :
% 0.28/1.96 ( ( ( ord @ B2 )
% 0.28/1.96 & ( ord @ A ) )
% 0.28/1.96 => ! [A2: A,B: A,F: A > B2,C: B2] :
% 0.28/1.96 ( ( ord_less @ A @ A2 @ B )
% 0.28/1.96 => ( ( ( F @ B )
% 0.28/1.96 = C )
% 0.28/1.96 => ( ! [X2: A,Y: A] :
% 0.28/1.96 ( ( ord_less @ A @ X2 @ Y )
% 0.28/1.96 => ( ord_less @ B2 @ ( F @ X2 ) @ ( F @ Y ) ) )
% 0.28/1.96 => ( ord_less @ B2 @ ( F @ A2 ) @ C ) ) ) ) ) ).
% 0.28/1.96
% 0.28/1.96 thf(fact_136_ord__eq__less__subst,axiom,
% 0.28/1.96 ! [A: $tType,B2: $tType] :
% 0.28/1.96 ( ( ( ord @ B2 )
% 0.28/1.96 & ( ord @ A ) )
% 0.28/1.96 => ! [A2: A,F: B2 > A,B: B2,C: B2] :
% 0.28/1.96 ( ( A2
% 0.28/1.96 = ( F @ B ) )
% 0.28/1.96 => ( ( ord_less @ B2 @ B @ C )
% 0.28/1.96 => ( ! [X2: B2,Y: B2] :
% 0.28/1.96 ( ( ord_less @ B2 @ X2 @ Y )
% 0.28/1.96 => ( ord_less @ A @ ( F @ X2 ) @ ( F @ Y ) ) )
% 0.28/1.96 => ( ord_less @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% 0.28/1.96
% 0.28/1.96 thf(fact_137_measure__induct__rule,axiom,
% 0.28/1.96 ! [B2: $tType,A: $tType] :
% 0.28/1.96 ( ( wellorder @ B2 )
% 0.28/1.96 => ! [F: A > B2,P3: A > $o,A2: A] :
% 0.28/1.96 ( ! [X2: A] :
% 0.28/1.96 ( ! [Y3: A] :
% 0.28/1.96 ( ( ord_less @ B2 @ ( F @ Y3 ) @ ( F @ X2 ) )
% 0.28/1.96 => ( P3 @ Y3 ) )
% 0.28/1.96 => ( P3 @ X2 ) )
% 0.28/1.96 => ( P3 @ A2 ) ) ) ).
% 0.28/1.96
% 0.28/1.96 thf(fact_138_measure__induct,axiom,
% 0.28/1.96 ! [B2: $tType,A: $tType] :
% 0.28/1.96 ( ( wellorder @ B2 )
% 0.28/1.96 => ! [F: A > B2,P3: A > $o,A2: A] :
% 0.28/1.96 ( ! [X2: A] :
% 0.28/1.96 ( ! [Y3: A] :
% 0.28/1.96 ( ( ord_less @ B2 @ ( F @ Y3 ) @ ( F @ X2 ) )
% 0.28/1.96 => ( P3 @ Y3 ) )
% 0.28/1.96 => ( P3 @ X2 ) )
% 0.28/1.96 => ( P3 @ A2 ) ) ) ).
% 0.28/1.96
% 0.28/1.96 thf(fact_139_verit__comp__simplify1_I1_J,axiom,
% 0.28/1.96 ! [A: $tType] :
% 0.28/1.96 ( ( order @ A )
% 0.28/1.96 => ! [A2: A] :
% 0.28/1.96 ~ ( ord_less @ A @ A2 @ A2 ) ) ).
% 0.28/1.96
% 0.28/1.96 thf(fact_140_pinf_I1_J,axiom,
% 0.28/1.96 ! [A: $tType] :
% 0.28/1.96 ( ( linorder @ A )
% 0.28/1.96 => ! [P3: A > $o,P4: A > $o,Q2: A > $o,Q3: A > $o] :
% 0.28/1.96 ( ? [Z: A] :
% 0.28/1.96 ! [X2: A] :
% 0.28/1.96 ( ( ord_less @ A @ Z @ X2 )
% 0.28/1.96 => ( ( P3 @ X2 )
% 0.28/1.96 = ( P4 @ X2 ) ) )
% 0.28/1.96 => ( ? [Z: A] :
% 0.28/1.96 ! [X2: A] :
% 0.28/1.96 ( ( ord_less @ A @ Z @ X2 )
% 0.28/1.96 => ( ( Q2 @ X2 )
% 0.28/1.96 = ( Q3 @ X2 ) ) )
% 0.28/1.96 => ? [Z3: A] :
% 0.28/1.96 ! [X4: A] :
% 0.28/1.96 ( ( ord_less @ A @ Z3 @ X4 )
% 0.28/1.96 => ( ( ( P3 @ X4 )
% 0.28/1.96 & ( Q2 @ X4 ) )
% 0.28/1.96 = ( ( P4 @ X4 )
% 0.28/1.96 & ( Q3 @ X4 ) ) ) ) ) ) ) ).
% 0.28/1.96
% 0.28/1.96 thf(fact_141_pinf_I2_J,axiom,
% 0.28/1.96 ! [A: $tType] :
% 0.28/1.96 ( ( linorder @ A )
% 0.28/1.96 => ! [P3: A > $o,P4: A > $o,Q2: A > $o,Q3: A > $o] :
% 0.28/1.96 ( ? [Z: A] :
% 0.28/1.96 ! [X2: A] :
% 0.28/1.96 ( ( ord_less @ A @ Z @ X2 )
% 0.28/1.96 => ( ( P3 @ X2 )
% 0.28/1.96 = ( P4 @ X2 ) ) )
% 0.28/1.96 => ( ? [Z: A] :
% 0.28/1.96 ! [X2: A] :
% 0.28/1.96 ( ( ord_less @ A @ Z @ X2 )
% 0.28/1.96 => ( ( Q2 @ X2 )
% 0.28/1.96 = ( Q3 @ X2 ) ) )
% 0.28/1.96 => ? [Z3: A] :
% 0.28/1.96 ! [X4: A] :
% 0.28/1.96 ( ( ord_less @ A @ Z3 @ X4 )
% 0.28/1.96 => ( ( ( P3 @ X4 )
% 0.28/1.96 | ( Q2 @ X4 ) )
% 0.28/1.96 = ( ( P4 @ X4 )
% 0.28/1.96 | ( Q3 @ X4 ) ) ) ) ) ) ) ).
% 0.28/1.96
% 0.28/1.96 thf(fact_142_pinf_I3_J,axiom,
% 0.28/1.96 ! [A: $tType] :
% 0.28/1.96 ( ( linorder @ A )
% 0.28/1.96 => ! [T: A] :
% 0.28/1.96 ? [Z3: A] :
% 0.28/1.96 ! [X4: A] :
% 0.28/1.96 ( ( ord_less @ A @ Z3 @ X4 )
% 0.28/1.96 => ( X4 != T ) ) ) ).
% 0.28/1.96
% 0.28/1.96 thf(fact_143_pinf_I4_J,axiom,
% 0.28/1.96 ! [A: $tType] :
% 0.28/1.96 ( ( linorder @ A )
% 0.28/1.96 => ! [T: A] :
% 0.28/1.96 ? [Z3: A] :
% 0.28/1.96 ! [X4: A] :
% 0.28/1.96 ( ( ord_less @ A @ Z3 @ X4 )
% 0.28/1.96 => ( X4 != T ) ) ) ).
% 0.28/1.96
% 0.28/1.96 thf(fact_144_pinf_I5_J,axiom,
% 0.28/1.96 ! [A: $tType] :
% 0.28/1.96 ( ( linorder @ A )
% 0.28/1.96 => ! [T: A] :
% 0.28/1.96 ? [Z3: A] :
% 0.28/1.96 ! [X4: A] :
% 0.28/1.96 ( ( ord_less @ A @ Z3 @ X4 )
% 0.28/1.96 => ~ ( ord_less @ A @ X4 @ T ) ) ) ).
% 0.28/1.96
% 0.28/1.96 thf(fact_145_pinf_I7_J,axiom,
% 0.28/1.96 ! [A: $tType] :
% 0.28/1.96 ( ( linorder @ A )
% 0.28/1.96 => ! [T: A] :
% 0.28/1.96 ? [Z3: A] :
% 0.28/1.96 ! [X4: A] :
% 0.28/1.96 ( ( ord_less @ A @ Z3 @ X4 )
% 0.28/1.96 => ( ord_less @ A @ T @ X4 ) ) ) ).
% 0.28/1.96
% 0.28/1.96 thf(fact_146_pinf_I11_J,axiom,
% 0.28/1.96 ! [C2: $tType,D: $tType] :
% 0.28/1.96 ( ( ord @ C2 )
% 0.28/1.96 => ! [F3: D] :
% 0.28/1.96 ? [Z3: C2] :
% 0.28/1.96 ! [X4: C2] :
% 0.28/1.96 ( ( ord_less @ C2 @ Z3 @ X4 )
% 0.28/1.96 => ( F3 = F3 ) ) ) ).
% 0.28/1.96
% 0.28/1.96 thf(fact_147_minf_I1_J,axiom,
% 0.28/1.96 ! [A: $tType] :
% 0.28/1.96 ( ( linorder @ A )
% 0.28/1.96 => ! [P3: A > $o,P4: A > $o,Q2: A > $o,Q3: A > $o] :
% 0.28/1.96 ( ? [Z: A] :
% 0.28/1.96 ! [X2: A] :
% 0.28/1.96 ( ( ord_less @ A @ X2 @ Z )
% 0.28/1.96 => ( ( P3 @ X2 )
% 0.28/1.96 = ( P4 @ X2 ) ) )
% 0.28/1.96 => ( ? [Z: A] :
% 0.28/1.96 ! [X2: A] :
% 0.28/1.96 ( ( ord_less @ A @ X2 @ Z )
% 0.28/1.96 => ( ( Q2 @ X2 )
% 0.28/1.96 = ( Q3 @ X2 ) ) )
% 0.28/1.96 => ? [Z3: A] :
% 0.28/1.96 ! [X4: A] :
% 0.28/1.96 ( ( ord_less @ A @ X4 @ Z3 )
% 0.28/1.96 => ( ( ( P3 @ X4 )
% 0.28/1.96 & ( Q2 @ X4 ) )
% 0.28/1.96 = ( ( P4 @ X4 )
% 0.28/1.96 & ( Q3 @ X4 ) ) ) ) ) ) ) ).
% 0.28/1.96
% 0.28/1.96 thf(fact_148_minf_I2_J,axiom,
% 0.28/1.96 ! [A: $tType] :
% 0.28/1.96 ( ( linorder @ A )
% 0.28/1.96 => ! [P3: A > $o,P4: A > $o,Q2: A > $o,Q3: A > $o] :
% 0.28/1.96 ( ? [Z: A] :
% 0.28/1.96 ! [X2: A] :
% 0.28/1.96 ( ( ord_less @ A @ X2 @ Z )
% 0.28/1.96 => ( ( P3 @ X2 )
% 0.28/1.96 = ( P4 @ X2 ) ) )
% 0.28/1.96 => ( ? [Z: A] :
% 0.28/1.96 ! [X2: A] :
% 0.28/1.96 ( ( ord_less @ A @ X2 @ Z )
% 0.28/1.96 => ( ( Q2 @ X2 )
% 0.28/1.96 = ( Q3 @ X2 ) ) )
% 0.28/1.96 => ? [Z3: A] :
% 0.28/1.96 ! [X4: A] :
% 0.28/1.96 ( ( ord_less @ A @ X4 @ Z3 )
% 0.28/1.96 => ( ( ( P3 @ X4 )
% 0.28/1.96 | ( Q2 @ X4 ) )
% 0.28/1.96 = ( ( P4 @ X4 )
% 0.28/1.96 | ( Q3 @ X4 ) ) ) ) ) ) ) ).
% 0.28/1.96
% 0.28/1.96 thf(fact_149_minf_I3_J,axiom,
% 0.28/1.96 ! [A: $tType] :
% 0.28/1.96 ( ( linorder @ A )
% 0.28/1.96 => ! [T: A] :
% 0.28/1.96 ? [Z3: A] :
% 0.28/1.96 ! [X4: A] :
% 0.28/1.96 ( ( ord_less @ A @ X4 @ Z3 )
% 0.28/1.96 => ( X4 != T ) ) ) ).
% 0.28/1.96
% 0.28/1.96 thf(fact_150_minf_I4_J,axiom,
% 0.28/1.96 ! [A: $tType] :
% 0.28/1.96 ( ( linorder @ A )
% 0.28/1.96 => ! [T: A] :
% 0.28/1.96 ? [Z3: A] :
% 0.28/1.96 ! [X4: A] :
% 0.28/1.96 ( ( ord_less @ A @ X4 @ Z3 )
% 0.28/1.96 => ( X4 != T ) ) ) ).
% 0.28/1.96
% 0.28/1.96 thf(fact_151_minf_I5_J,axiom,
% 0.28/1.96 ! [A: $tType] :
% 0.28/1.96 ( ( linorder @ A )
% 0.28/1.96 => ! [T: A] :
% 0.28/1.96 ? [Z3: A] :
% 0.28/1.96 ! [X4: A] :
% 0.28/1.96 ( ( ord_less @ A @ X4 @ Z3 )
% 0.28/1.96 => ( ord_less @ A @ X4 @ T ) ) ) ).
% 0.28/1.96
% 0.28/1.96 thf(fact_152_minf_I7_J,axiom,
% 0.28/1.96 ! [A: $tType] :
% 0.28/1.96 ( ( linorder @ A )
% 0.28/1.96 => ! [T: A] :
% 0.28/1.96 ? [Z3: A] :
% 0.28/1.96 ! [X4: A] :
% 0.28/1.96 ( ( ord_less @ A @ X4 @ Z3 )
% 0.28/1.96 => ~ ( ord_less @ A @ T @ X4 ) ) ) ).
% 0.28/1.96
% 0.28/1.96 thf(fact_153_minf_I11_J,axiom,
% 0.28/1.96 ! [C2: $tType,D: $tType] :
% 0.28/1.96 ( ( ord @ C2 )
% 0.28/1.96 => ! [F3: D] :
% 0.28/1.96 ? [Z3: C2] :
% 0.28/1.96 ! [X4: C2] :
% 0.28/1.96 ( ( ord_less @ C2 @ X4 @ Z3 )
% 0.28/1.96 => ( F3 = F3 ) ) ) ).
% 0.28/1.96
% 0.28/1.96 thf(fact_154_infinite__descent0__measure,axiom,
% 0.28/1.96 ! [A: $tType,V: A > nat,P3: A > $o,X: A] :
% 0.28/1.96 ( ! [X2: A] :
% 0.28/1.96 ( ( ( V @ X2 )
% 0.28/1.96 = ( zero_zero @ nat ) )
% 0.28/1.96 => ( P3 @ X2 ) )
% 0.28/1.96 => ( ! [X2: A] :
% 0.28/1.96 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( V @ X2 ) )
% 0.28/1.96 => ( ~ ( P3 @ X2 )
% 0.28/1.96 => ? [Y3: A] :
% 0.28/1.96 ( ( ord_less @ nat @ ( V @ Y3 ) @ ( V @ X2 ) )
% 0.28/1.96 & ~ ( P3 @ Y3 ) ) ) )
% 0.28/1.96 => ( P3 @ X ) ) ) ).
% 0.28/1.96
% 0.28/1.96 thf(fact_155_bot__nat__0_Oextremum__strict,axiom,
% 0.28/1.96 ! [A2: nat] :
% 0.28/1.96 ~ ( ord_less @ nat @ A2 @ ( zero_zero @ nat ) ) ).
% 0.28/1.96
% 0.28/1.96 thf(fact_156_infinite__descent0,axiom,
% 0.28/1.96 ! [P3: nat > $o,N: nat] :
% 0.28/1.96 ( ( P3 @ ( zero_zero @ nat ) )
% 0.28/1.96 => ( ! [N3: nat] :
% 0.28/1.96 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 )
% 0.28/1.96 => ( ~ ( P3 @ N3 )
% 0.28/1.96 => ? [M3: nat] :
% 0.28/1.96 ( ( ord_less @ nat @ M3 @ N3 )
% 0.28/1.96 & ~ ( P3 @ M3 ) ) ) )
% 0.28/1.96 => ( P3 @ N ) ) ) ).
% 0.28/1.96
% 0.28/1.96 thf(fact_157_gr__implies__not0,axiom,
% 0.28/1.96 ! [M: nat,N: nat] :
% 0.28/1.96 ( ( ord_less @ nat @ M @ N )
% 0.28/1.96 => ( N
% 0.28/1.96 != ( zero_zero @ nat ) ) ) ).
% 0.28/1.96
% 0.28/1.96 thf(fact_158_less__zeroE,axiom,
% 0.28/1.96 ! [N: nat] :
% 0.28/1.96 ~ ( ord_less @ nat @ N @ ( zero_zero @ nat ) ) ).
% 0.28/1.96
% 0.28/1.96 thf(fact_159_not__less0,axiom,
% 0.28/1.96 ! [N: nat] :
% 0.28/1.96 ~ ( ord_less @ nat @ N @ ( zero_zero @ nat ) ) ).
% 0.28/1.96
% 0.28/1.96 thf(fact_160_not__gr0,axiom,
% 0.28/1.96 ! [N: nat] :
% 0.28/1.96 ( ( ~ ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) )
% 0.28/1.96 = ( N
% 0.28/1.96 = ( zero_zero @ nat ) ) ) ).
% 0.28/1.96
% 0.28/1.96 thf(fact_161_gr0I,axiom,
% 0.28/1.96 ! [N: nat] :
% 0.28/1.96 ( ( N
% 0.28/1.96 != ( zero_zero @ nat ) )
% 0.28/1.96 => ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ).
% 0.28/1.96
% 0.28/1.96 thf(fact_162_order_Onot__eq__order__implies__strict,axiom,
% 0.28/1.96 ! [A: $tType] :
% 0.28/1.96 ( ( order @ A )
% 0.28/1.96 => ! [A2: A,B: A] :
% 0.28/1.96 ( ( A2 != B )
% 0.28/1.96 => ( ( ord_less_eq @ A @ A2 @ B )
% 0.28/1.96 => ( ord_less @ A @ A2 @ B ) ) ) ) ).
% 0.28/1.96
% 0.28/1.96 thf(fact_163_dual__order_Ostrict__implies__order,axiom,
% 0.28/1.96 ! [A: $tType] :
% 0.28/1.96 ( ( order @ A )
% 0.28/1.96 => ! [B: A,A2: A] :
% 0.28/1.96 ( ( ord_less @ A @ B @ A2 )
% 0.28/1.96 => ( ord_less_eq @ A @ B @ A2 ) ) ) ).
% 0.28/1.96
% 0.28/1.96 thf(fact_164_dual__order_Ostrict__iff__order,axiom,
% 0.28/1.96 ! [A: $tType] :
% 0.28/1.96 ( ( order @ A )
% 0.28/1.96 => ( ( ord_less @ A )
% 0.28/1.96 = ( ^ [B4: A,A3: A] :
% 0.28/1.96 ( ( ord_less_eq @ A @ B4 @ A3 )
% 0.28/1.96 & ( A3 != B4 ) ) ) ) ) ).
% 0.28/1.96
% 0.28/1.96 thf(fact_165_dual__order_Oorder__iff__strict,axiom,
% 0.28/1.96 ! [A: $tType] :
% 0.28/1.96 ( ( order @ A )
% 0.28/1.96 => ( ( ord_less_eq @ A )
% 0.28/1.96 = ( ^ [B4: A,A3: A] :
% 0.28/1.96 ( ( ord_less @ A @ B4 @ A3 )
% 0.28/1.96 | ( A3 = B4 ) ) ) ) ) ).
% 0.28/1.96
% 0.28/1.96 thf(fact_166_order_Ostrict__implies__order,axiom,
% 0.28/1.96 ! [A: $tType] :
% 0.28/1.96 ( ( order @ A )
% 0.28/1.96 => ! [A2: A,B: A] :
% 0.28/1.96 ( ( ord_less @ A @ A2 @ B )
% 0.28/1.96 => ( ord_less_eq @ A @ A2 @ B ) ) ) ).
% 0.28/1.96
% 0.28/1.96 thf(fact_167_dense__le__bounded,axiom,
% 0.28/1.96 ! [A: $tType] :
% 0.28/1.96 ( ( dense_linorder @ A )
% 0.28/1.96 => ! [X: A,Y4: A,Z5: A] :
% 0.28/1.96 ( ( ord_less @ A @ X @ Y4 )
% 0.28/1.96 => ( ! [W: A] :
% 0.28/1.96 ( ( ord_less @ A @ X @ W )
% 0.28/1.96 => ( ( ord_less @ A @ W @ Y4 )
% 0.28/1.96 => ( ord_less_eq @ A @ W @ Z5 ) ) )
% 0.28/1.96 => ( ord_less_eq @ A @ Y4 @ Z5 ) ) ) ) ).
% 0.28/1.96
% 0.28/1.96 thf(fact_168_dense__ge__bounded,axiom,
% 0.28/1.96 ! [A: $tType] :
% 0.28/1.96 ( ( dense_linorder @ A )
% 0.28/1.96 => ! [Z5: A,X: A,Y4: A] :
% 0.28/1.96 ( ( ord_less @ A @ Z5 @ X )
% 0.28/1.96 => ( ! [W: A] :
% 0.28/1.96 ( ( ord_less @ A @ Z5 @ W )
% 0.28/1.96 => ( ( ord_less @ A @ W @ X )
% 0.28/1.96 => ( ord_less_eq @ A @ Y4 @ W ) ) )
% 0.28/1.96 => ( ord_less_eq @ A @ Y4 @ Z5 ) ) ) ) ).
% 0.28/1.96
% 0.28/1.96 thf(fact_169_dual__order_Ostrict__trans2,axiom,
% 0.28/1.96 ! [A: $tType] :
% 0.28/1.96 ( ( order @ A )
% 0.28/1.96 => ! [B: A,A2: A,C: A] :
% 0.28/1.96 ( ( ord_less @ A @ B @ A2 )
% 0.28/1.96 => ( ( ord_less_eq @ A @ C @ B )
% 0.28/1.96 => ( ord_less @ A @ C @ A2 ) ) ) ) ).
% 0.28/1.96
% 0.28/1.96 thf(fact_170_dual__order_Ostrict__trans1,axiom,
% 0.28/1.96 ! [A: $tType] :
% 0.28/1.96 ( ( order @ A )
% 0.28/1.96 => ! [B: A,A2: A,C: A] :
% 0.28/1.96 ( ( ord_less_eq @ A @ B @ A2 )
% 0.28/1.96 => ( ( ord_less @ A @ C @ B )
% 0.28/1.96 => ( ord_less @ A @ C @ A2 ) ) ) ) ).
% 0.28/1.96
% 0.28/1.96 thf(fact_171_order_Ostrict__iff__order,axiom,
% 0.28/1.96 ! [A: $tType] :
% 0.28/1.96 ( ( order @ A )
% 0.28/1.96 => ( ( ord_less @ A )
% 0.28/1.96 = ( ^ [A3: A,B4: A] :
% 0.28/1.96 ( ( ord_less_eq @ A @ A3 @ B4 )
% 0.28/1.96 & ( A3 != B4 ) ) ) ) ) ).
% 0.28/1.96
% 0.28/1.96 thf(fact_172_order_Oorder__iff__strict,axiom,
% 0.28/1.96 ! [A: $tType] :
% 0.28/1.96 ( ( order @ A )
% 0.28/1.96 => ( ( ord_less_eq @ A )
% 0.28/1.96 = ( ^ [A3: A,B4: A] :
% 0.28/1.96 ( ( ord_less @ A @ A3 @ B4 )
% 0.28/1.96 | ( A3 = B4 ) ) ) ) ) ).
% 0.28/1.96
% 0.28/1.96 thf(fact_173_order_Ostrict__trans2,axiom,
% 0.28/1.96 ! [A: $tType] :
% 0.28/1.96 ( ( order @ A )
% 0.28/1.96 => ! [A2: A,B: A,C: A] :
% 0.28/1.96 ( ( ord_less @ A @ A2 @ B )
% 0.28/1.96 => ( ( ord_less_eq @ A @ B @ C )
% 0.28/1.96 => ( ord_less @ A @ A2 @ C ) ) ) ) ).
% 0.28/1.96
% 0.28/1.96 thf(fact_174_order_Ostrict__trans1,axiom,
% 0.28/1.96 ! [A: $tType] :
% 0.28/1.96 ( ( order @ A )
% 0.28/1.96 => ! [A2: A,B: A,C: A] :
% 0.28/1.96 ( ( ord_less_eq @ A @ A2 @ B )
% 0.28/1.96 => ( ( ord_less @ A @ B @ C )
% 0.28/1.96 => ( ord_less @ A @ A2 @ C ) ) ) ) ).
% 0.28/1.96
% 0.28/1.96 thf(fact_175_not__le__imp__less,axiom,
% 0.28/1.96 ! [A: $tType] :
% 0.28/1.96 ( ( linorder @ A )
% 0.28/1.96 => ! [Y4: A,X: A] :
% 0.28/1.96 ( ~ ( ord_less_eq @ A @ Y4 @ X )
% 0.28/1.96 => ( ord_less @ A @ X @ Y4 ) ) ) ).
% 0.28/1.96
% 0.28/1.96 thf(fact_176_less__le__not__le,axiom,
% 0.28/1.96 ! [A: $tType] :
% 0.28/1.96 ( ( preorder @ A )
% 0.28/1.96 => ( ( ord_less @ A )
% 0.28/1.96 = ( ^ [X3: A,Y2: A] :
% 0.28/1.96 ( ( ord_less_eq @ A @ X3 @ Y2 )
% 0.28/1.96 & ~ ( ord_less_eq @ A @ Y2 @ X3 ) ) ) ) ) ).
% 0.28/1.96
% 0.28/1.96 thf(fact_177_le__imp__less__or__eq,axiom,
% 0.28/1.96 ! [A: $tType] :
% 0.28/1.96 ( ( order @ A )
% 0.28/1.96 => ! [X: A,Y4: A] :
% 0.28/1.96 ( ( ord_less_eq @ A @ X @ Y4 )
% 0.28/1.96 => ( ( ord_less @ A @ X @ Y4 )
% 0.28/1.96 | ( X = Y4 ) ) ) ) ).
% 0.28/1.96
% 0.28/1.96 thf(fact_178_le__less__linear,axiom,
% 0.28/1.96 ! [A: $tType] :
% 0.28/1.96 ( ( linorder @ A )
% 0.28/1.96 => ! [X: A,Y4: A] :
% 0.28/1.96 ( ( ord_less_eq @ A @ X @ Y4 )
% 0.28/1.96 | ( ord_less @ A @ Y4 @ X ) ) ) ).
% 0.28/1.96
% 0.28/1.96 thf(fact_179_dense__le,axiom,
% 0.28/1.96 ! [A: $tType] :
% 0.28/1.96 ( ( dense_linorder @ A )
% 0.28/1.96 => ! [Y4: A,Z5: A] :
% 0.28/1.96 ( ! [X2: A] :
% 0.28/1.96 ( ( ord_less @ A @ X2 @ Y4 )
% 0.28/1.96 => ( ord_less_eq @ A @ X2 @ Z5 ) )
% 0.28/1.96 => ( ord_less_eq @ A @ Y4 @ Z5 ) ) ) ).
% 0.28/1.96
% 0.28/1.96 thf(fact_180_dense__ge,axiom,
% 0.28/1.96 ! [A: $tType] :
% 0.28/1.96 ( ( dense_linorder @ A )
% 0.28/1.96 => ! [Z5: A,Y4: A] :
% 0.28/1.96 ( ! [X2: A] :
% 0.28/1.96 ( ( ord_less @ A @ Z5 @ X2 )
% 0.28/1.96 => ( ord_less_eq @ A @ Y4 @ X2 ) )
% 0.28/1.96 => ( ord_less_eq @ A @ Y4 @ Z5 ) ) ) ).
% 0.28/1.96
% 0.28/1.96 thf(fact_181_less__le__trans,axiom,
% 0.28/1.96 ! [A: $tType] :
% 0.28/1.96 ( ( preorder @ A )
% 0.28/1.96 => ! [X: A,Y4: A,Z5: A] :
% 0.28/1.96 ( ( ord_less @ A @ X @ Y4 )
% 0.28/1.96 => ( ( ord_less_eq @ A @ Y4 @ Z5 )
% 0.28/1.96 => ( ord_less @ A @ X @ Z5 ) ) ) ) ).
% 0.28/1.96
% 0.28/1.96 thf(fact_182_le__less__trans,axiom,
% 0.28/1.96 ! [A: $tType] :
% 0.28/1.96 ( ( preorder @ A )
% 0.28/1.96 => ! [X: A,Y4: A,Z5: A] :
% 0.28/1.96 ( ( ord_less_eq @ A @ X @ Y4 )
% 0.28/1.96 => ( ( ord_less @ A @ Y4 @ Z5 )
% 0.28/1.96 => ( ord_less @ A @ X @ Z5 ) ) ) ) ).
% 0.28/1.96
% 0.28/1.96 thf(fact_183_less__imp__le,axiom,
% 0.28/1.96 ! [A: $tType] :
% 0.28/1.96 ( ( preorder @ A )
% 0.28/1.96 => ! [X: A,Y4: A] :
% 0.28/1.96 ( ( ord_less @ A @ X @ Y4 )
% 0.28/1.96 => ( ord_less_eq @ A @ X @ Y4 ) ) ) ).
% 0.28/1.96
% 0.28/1.96 thf(fact_184_antisym__conv2,axiom,
% 0.28/1.96 ! [A: $tType] :
% 0.28/1.96 ( ( order @ A )
% 0.28/1.96 => ! [X: A,Y4: A] :
% 0.28/1.96 ( ( ord_less_eq @ A @ X @ Y4 )
% 0.28/1.96 => ( ( ~ ( ord_less @ A @ X @ Y4 ) )
% 0.28/1.96 = ( X = Y4 ) ) ) ) ).
% 0.28/1.96
% 0.28/1.96 thf(fact_185_antisym__conv1,axiom,
% 0.28/1.96 ! [A: $tType] :
% 0.28/1.96 ( ( order @ A )
% 0.28/1.96 => ! [X: A,Y4: A] :
% 0.28/1.96 ( ~ ( ord_less @ A @ X @ Y4 )
% 0.28/1.96 => ( ( ord_less_eq @ A @ X @ Y4 )
% 0.28/1.96 = ( X = Y4 ) ) ) ) ).
% 0.28/1.96
% 0.28/1.96 thf(fact_186_le__neq__trans,axiom,
% 0.28/1.96 ! [A: $tType] :
% 0.28/1.96 ( ( order @ A )
% 0.28/1.96 => ! [A2: A,B: A] :
% 0.28/1.96 ( ( ord_less_eq @ A @ A2 @ B )
% 0.28/1.96 => ( ( A2 != B )
% 0.28/1.96 => ( ord_less @ A @ A2 @ B ) ) ) ) ).
% 0.28/1.96
% 0.28/1.96 thf(fact_187_not__less,axiom,
% 0.28/1.96 ! [A: $tType] :
% 0.28/1.96 ( ( linorder @ A )
% 0.28/1.96 => ! [X: A,Y4: A] :
% 0.28/1.96 ( ( ~ ( ord_less @ A @ X @ Y4 ) )
% 0.28/1.96 = ( ord_less_eq @ A @ Y4 @ X ) ) ) ).
% 0.28/1.96
% 0.28/1.96 thf(fact_188_not__le,axiom,
% 0.28/1.96 ! [A: $tType] :
% 0.28/1.96 ( ( linorder @ A )
% 0.28/1.96 => ! [X: A,Y4: A] :
% 0.28/1.96 ( ( ~ ( ord_less_eq @ A @ X @ Y4 ) )
% 0.28/1.96 = ( ord_less @ A @ Y4 @ X ) ) ) ).
% 0.28/1.96
% 0.28/1.96 thf(fact_189_order__less__le__subst2,axiom,
% 0.28/1.96 ! [A: $tType,C2: $tType] :
% 0.28/1.96 ( ( ( order @ C2 )
% 0.28/1.96 & ( order @ A ) )
% 0.28/1.96 => ! [A2: A,B: A,F: A > C2,C: C2] :
% 0.28/1.96 ( ( ord_less @ A @ A2 @ B )
% 0.28/1.96 => ( ( ord_less_eq @ C2 @ ( F @ B ) @ C )
% 0.28/1.96 => ( ! [X2: A,Y: A] :
% 0.28/1.96 ( ( ord_less @ A @ X2 @ Y )
% 0.28/1.96 => ( ord_less @ C2 @ ( F @ X2 ) @ ( F @ Y ) ) )
% 0.28/1.96 => ( ord_less @ C2 @ ( F @ A2 ) @ C ) ) ) ) ) ).
% 0.28/1.96
% 0.28/1.96 thf(fact_190_order__less__le__subst1,axiom,
% 0.28/1.96 ! [A: $tType,B2: $tType] :
% 0.28/1.96 ( ( ( order @ B2 )
% 0.28/1.96 & ( order @ A ) )
% 0.28/1.96 => ! [A2: A,F: B2 > A,B: B2,C: B2] :
% 0.28/1.96 ( ( ord_less @ A @ A2 @ ( F @ B ) )
% 0.28/1.96 => ( ( ord_less_eq @ B2 @ B @ C )
% 0.28/1.96 => ( ! [X2: B2,Y: B2] :
% 0.28/1.96 ( ( ord_less_eq @ B2 @ X2 @ Y )
% 0.28/1.96 => ( ord_less_eq @ A @ ( F @ X2 ) @ ( F @ Y ) ) )
% 0.28/1.96 => ( ord_less @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% 0.28/1.96
% 0.28/1.96 thf(fact_191_order__le__less__subst2,axiom,
% 0.28/1.96 ! [A: $tType,C2: $tType] :
% 0.28/1.96 ( ( ( order @ C2 )
% 0.28/1.96 & ( order @ A ) )
% 0.28/1.96 => ! [A2: A,B: A,F: A > C2,C: C2] :
% 0.28/1.96 ( ( ord_less_eq @ A @ A2 @ B )
% 0.28/1.96 => ( ( ord_less @ C2 @ ( F @ B ) @ C )
% 0.28/1.96 => ( ! [X2: A,Y: A] :
% 0.28/1.96 ( ( ord_less_eq @ A @ X2 @ Y )
% 0.28/1.96 => ( ord_less_eq @ C2 @ ( F @ X2 ) @ ( F @ Y ) ) )
% 0.28/1.96 => ( ord_less @ C2 @ ( F @ A2 ) @ C ) ) ) ) ) ).
% 0.28/1.96
% 0.28/1.96 thf(fact_192_order__le__less__subst1,axiom,
% 0.28/1.96 ! [A: $tType,B2: $tType] :
% 0.28/1.96 ( ( ( order @ B2 )
% 0.28/1.96 & ( order @ A ) )
% 0.28/1.96 => ! [A2: A,F: B2 > A,B: B2,C: B2] :
% 0.28/1.96 ( ( ord_less_eq @ A @ A2 @ ( F @ B ) )
% 0.28/1.96 => ( ( ord_less @ B2 @ B @ C )
% 0.28/1.96 => ( ! [X2: B2,Y: B2] :
% 0.28/1.96 ( ( ord_less @ B2 @ X2 @ Y )
% 0.28/1.96 => ( ord_less @ A @ ( F @ X2 ) @ ( F @ Y ) ) )
% 0.28/1.96 => ( ord_less @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% 0.28/1.96
% 0.28/1.96 thf(fact_193_less__le,axiom,
% 0.28/1.96 ! [A: $tType] :
% 0.28/1.96 ( ( order @ A )
% 0.28/1.96 => ( ( ord_less @ A )
% 0.28/1.96 = ( ^ [X3: A,Y2: A] :
% 0.28/1.96 ( ( ord_less_eq @ A @ X3 @ Y2 )
% 0.28/1.96 & ( X3 != Y2 ) ) ) ) ) ).
% 0.28/1.96
% 0.28/1.96 thf(fact_194_le__less,axiom,
% 0.28/1.96 ! [A: $tType] :
% 0.28/1.96 ( ( order @ A )
% 0.28/1.96 => ( ( ord_less_eq @ A )
% 0.28/1.96 = ( ^ [X3: A,Y2: A] :
% 0.28/1.96 ( ( ord_less @ A @ X3 @ Y2 )
% 0.28/1.96 | ( X3 = Y2 ) ) ) ) ) ).
% 0.28/1.96
% 0.28/1.96 thf(fact_195_leI,axiom,
% 0.28/1.96 ! [A: $tType] :
% 0.28/1.96 ( ( linorder @ A )
% 0.28/1.96 => ! [X: A,Y4: A] :
% 0.28/1.96 ( ~ ( ord_less @ A @ X @ Y4 )
% 0.28/1.96 => ( ord_less_eq @ A @ Y4 @ X ) ) ) ).
% 0.28/1.96
% 0.28/1.96 thf(fact_196_leD,axiom,
% 0.28/1.96 ! [A: $tType] :
% 0.28/1.96 ( ( order @ A )
% 0.28/1.96 => ! [Y4: A,X: A] :
% 0.28/1.96 ( ( ord_less_eq @ A @ Y4 @ X )
% 0.28/1.96 => ~ ( ord_less @ A @ X @ Y4 ) ) ) ).
% 0.28/1.96
% 0.28/1.96 thf(fact_197_complete__interval,axiom,
% 0.28/1.96 ! [A: $tType] :
% 0.28/1.96 ( ( condit1037483654norder @ A )
% 0.28/1.96 => ! [A2: A,B: A,P3: A > $o] :
% 0.28/1.96 ( ( ord_less @ A @ A2 @ B )
% 0.28/1.96 => ( ( P3 @ A2 )
% 0.28/1.96 => ( ~ ( P3 @ B )
% 0.28/1.96 => ? [C3: A] :
% 0.28/1.96 ( ( ord_less_eq @ A @ A2 @ C3 )
% 0.28/1.96 & ( ord_less_eq @ A @ C3 @ B )
% 0.28/1.96 & ! [X4: A] :
% 0.28/1.96 ( ( ( ord_less_eq @ A @ A2 @ X4 )
% 0.28/1.96 & ( ord_less @ A @ X4 @ C3 ) )
% 0.28/1.96 => ( P3 @ X4 ) )
% 0.28/1.96 & ! [D3: A] :
% 0.28/1.96 ( ! [X2: A] :
% 0.28/1.96 ( ( ( ord_less_eq @ A @ A2 @ X2 )
% 0.28/1.96 & ( ord_less @ A @ X2 @ D3 ) )
% 0.28/1.96 => ( P3 @ X2 ) )
% 0.28/1.96 => ( ord_less_eq @ A @ D3 @ C3 ) ) ) ) ) ) ) ).
% 0.28/1.96
% 0.28/1.96 thf(fact_198_rsquarefree__def,axiom,
% 0.28/1.96 ! [A: $tType] :
% 0.28/1.96 ( ( idom @ A )
% 0.28/1.96 => ( ( rsquarefree @ A )
% 0.28/1.96 = ( ^ [P2: poly @ A] :
% 0.28/1.96 ( ( P2
% 0.28/1.96 != ( zero_zero @ ( poly @ A ) ) )
% 0.28/1.96 & ! [A3: A] :
% 0.28/1.96 ( ( ( order2 @ A @ A3 @ P2 )
% 0.28/1.96 = ( zero_zero @ nat ) )
% 0.28/1.96 | ( ( order2 @ A @ A3 @ P2 )
% 0.28/1.96 = ( one_one @ nat ) ) ) ) ) ) ) ).
% 0.28/1.96
% 0.28/1.96 thf(fact_199_poly__cutoff__1,axiom,
% 0.28/1.96 ! [A: $tType] :
% 0.28/1.96 ( ( comm_semiring_1 @ A )
% 0.28/1.96 => ! [N: nat] :
% 0.28/1.96 ( ( ( N
% 0.28/1.96 = ( zero_zero @ nat ) )
% 0.28/1.96 => ( ( poly_cutoff @ A @ N @ ( one_one @ ( poly @ A ) ) )
% 0.28/1.96 = ( zero_zero @ ( poly @ A ) ) ) )
% 0.28/1.96 & ( ( N
% 0.28/1.96 != ( zero_zero @ nat ) )
% 0.28/1.96 => ( ( poly_cutoff @ A @ N @ ( one_one @ ( poly @ A ) ) )
% 0.28/1.96 = ( one_one @ ( poly @ A ) ) ) ) ) ) ).
% 0.28/1.96
% 0.28/1.96 thf(fact_200_coeff__0__reflect__poly__0__iff,axiom,
% 0.28/1.96 ! [A: $tType] :
% 0.28/1.96 ( ( zero @ A )
% 0.28/1.96 => ! [P: poly @ A] :
% 0.28/1.96 ( ( ( coeff @ A @ ( reflect_poly @ A @ P ) @ ( zero_zero @ nat ) )
% 0.28/1.96 = ( zero_zero @ A ) )
% 0.28/1.96 = ( P
% 0.28/1.96 = ( zero_zero @ ( poly @ A ) ) ) ) ) ).
% 0.28/1.96
% 0.28/1.96 thf(fact_201_of__nat__0__less__iff,axiom,
% 0.28/1.96 ! [A: $tType] :
% 0.28/1.96 ( ( linord1659791738miring @ A )
% 0.28/1.96 => ! [N: nat] :
% 0.28/1.96 ( ( ord_less @ A @ ( zero_zero @ A ) @ ( semiring_1_of_nat @ A @ N ) )
% 0.28/1.96 = ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ) ).
% 0.28/1.96
% 0.28/1.96 thf(fact_202_order__pderiv,axiom,
% 0.28/1.96 ! [A: $tType] :
% 0.28/1.96 ( ( field_char_0 @ A )
% 0.28/1.96 => ! [P: poly @ A,A2: A] :
% 0.28/1.96 ( ( ( pderiv @ A @ P )
% 0.28/1.96 != ( zero_zero @ ( poly @ A ) ) )
% 0.28/1.96 => ( ( ( order2 @ A @ A2 @ P )
% 0.28/1.96 != ( zero_zero @ nat ) )
% 0.28/1.96 => ( ( order2 @ A @ A2 @ P )
% 0.28/1.96 = ( suc @ ( order2 @ A @ A2 @ ( pderiv @ A @ P ) ) ) ) ) ) ) ).
% 0.28/1.96
% 0.28/1.96 thf(fact_203_order__pderiv2,axiom,
% 0.28/1.96 ! [A: $tType] :
% 0.28/1.96 ( ( field_char_0 @ A )
% 0.28/1.96 => ! [P: poly @ A,A2: A,N: nat] :
% 0.28/1.96 ( ( ( pderiv @ A @ P )
% 0.28/1.96 != ( zero_zero @ ( poly @ A ) ) )
% 0.28/1.96 => ( ( ( order2 @ A @ A2 @ P )
% 0.28/1.96 != ( zero_zero @ nat ) )
% 0.28/1.96 => ( ( ( order2 @ A @ A2 @ ( pderiv @ A @ P ) )
% 0.28/1.96 = N )
% 0.28/1.96 = ( ( order2 @ A @ A2 @ P )
% 0.28/1.96 = ( suc @ N ) ) ) ) ) ) ).
% 0.28/1.96
% 0.28/1.96 thf(fact_204_cross__def,axiom,
% 0.28/1.96 ( sturm_424270202_cross
% 0.28/1.96 = ( ^ [P2: poly @ real,A3: real,B4: real] : ( sturm_1771227917iation @ ( poly2 @ real @ P2 @ A3 ) @ ( poly2 @ real @ P2 @ B4 ) ) ) ) ).
% 0.28/1.96
% 0.28/1.96 thf(fact_205_nat_Oinject,axiom,
% 0.28/1.96 ! [X22: nat,Y22: nat] :
% 0.28/1.96 ( ( ( suc @ X22 )
% 0.28/1.96 = ( suc @ Y22 ) )
% 0.28/1.96 = ( X22 = Y22 ) ) ).
% 0.28/1.96
% 0.28/1.96 thf(fact_206_old_Onat_Oinject,axiom,
% 0.28/1.96 ! [Nat: nat,Nat2: nat] :
% 0.28/1.96 ( ( ( suc @ Nat )
% 0.28/1.96 = ( suc @ Nat2 ) )
% 0.28/1.96 = ( Nat = Nat2 ) ) ).
% 0.28/1.96
% 0.28/1.96 thf(fact_207_of__nat__eq__iff,axiom,
% 0.28/1.96 ! [A: $tType] :
% 0.28/1.96 ( ( semiring_char_0 @ A )
% 0.28/1.96 => ! [M: nat,N: nat] :
% 0.28/1.96 ( ( ( semiring_1_of_nat @ A @ M )
% 0.28/1.96 = ( semiring_1_of_nat @ A @ N ) )
% 0.28/1.96 = ( M = N ) ) ) ).
% 0.28/1.96
% 0.28/1.96 thf(fact_208_lessI,axiom,
% 0.28/1.96 ! [N: nat] : ( ord_less @ nat @ N @ ( suc @ N ) ) ).
% 0.28/1.96
% 0.28/1.96 thf(fact_209_Suc__mono,axiom,
% 0.28/1.96 ! [M: nat,N: nat] :
% 0.28/1.96 ( ( ord_less @ nat @ M @ N )
% 0.28/1.96 => ( ord_less @ nat @ ( suc @ M ) @ ( suc @ N ) ) ) ).
% 0.28/1.96
% 0.28/1.96 thf(fact_210_Suc__less__eq,axiom,
% 0.28/1.96 ! [M: nat,N: nat] :
% 0.28/1.96 ( ( ord_less @ nat @ ( suc @ M ) @ ( suc @ N ) )
% 0.28/1.96 = ( ord_less @ nat @ M @ N ) ) ).
% 0.28/1.96
% 0.28/1.96 thf(fact_211_Suc__le__mono,axiom,
% 0.28/1.96 ! [N: nat,M: nat] :
% 0.28/1.96 ( ( ord_less_eq @ nat @ ( suc @ N ) @ ( suc @ M ) )
% 0.28/1.96 = ( ord_less_eq @ nat @ N @ M ) ) ).
% 0.28/1.96
% 0.28/1.96 thf(fact_212_pderiv__of__nat,axiom,
% 0.28/1.96 ! [A: $tType] :
% 0.28/1.96 ( ( ( comm_semiring_1 @ A )
% 0.28/1.96 & ( semiri1193490041visors @ A ) )
% 0.28/1.96 => ! [N: nat] :
% 0.28/1.96 ( ( pderiv @ A @ ( semiring_1_of_nat @ ( poly @ A ) @ N ) )
% 0.28/1.96 = ( zero_zero @ ( poly @ A ) ) ) ) ).
% 0.28/1.96
% 0.28/1.96 thf(fact_213_reflect__poly__1,axiom,
% 0.28/1.96 ! [A: $tType] :
% 0.28/1.96 ( ( comm_semiring_1 @ A )
% 0.28/1.96 => ( ( reflect_poly @ A @ ( one_one @ ( poly @ A ) ) )
% 0.28/1.96 = ( one_one @ ( poly @ A ) ) ) ) ).
% 0.28/1.96
% 0.28/1.96 thf(fact_214_of__nat__0,axiom,
% 0.28/1.96 ! [A: $tType] :
% 0.28/1.96 ( ( semiring_1 @ A )
% 0.28/1.96 => ( ( semiring_1_of_nat @ A @ ( zero_zero @ nat ) )
% 0.28/1.96 = ( zero_zero @ A ) ) ) ).
% 0.28/1.96
% 0.28/1.96 thf(fact_215_of__nat__0__eq__iff,axiom,
% 0.28/1.96 ! [A: $tType] :
% 0.28/1.96 ( ( semiring_char_0 @ A )
% 0.28/1.96 => ! [N: nat] :
% 0.28/1.96 ( ( ( zero_zero @ A )
% 0.28/1.96 = ( semiring_1_of_nat @ A @ N ) )
% 0.28/1.96 = ( ( zero_zero @ nat )
% 0.28/1.96 = N ) ) ) ).
% 0.28/1.96
% 0.28/1.96 thf(fact_216_of__nat__eq__0__iff,axiom,
% 0.28/1.96 ! [A: $tType] :
% 0.28/1.96 ( ( semiring_char_0 @ A )
% 0.28/1.96 => ! [M: nat] :
% 0.28/1.96 ( ( ( semiring_1_of_nat @ A @ M )
% 0.28/1.96 = ( zero_zero @ A ) )
% 0.28/1.96 = ( M
% 0.28/1.96 = ( zero_zero @ nat ) ) ) ) ).
% 0.28/1.96
% 0.28/1.96 thf(fact_217_of__nat__less__iff,axiom,
% 0.28/1.96 ! [A: $tType] :
% 0.28/1.96 ( ( linord1659791738miring @ A )
% 0.28/1.96 => ! [M: nat,N: nat] :
% 0.28/1.96 ( ( ord_less @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N ) )
% 0.28/1.96 = ( ord_less @ nat @ M @ N ) ) ) ).
% 0.28/1.96
% 0.28/1.96 thf(fact_218_of__nat__le__iff,axiom,
% 0.28/1.96 ! [A: $tType] :
% 0.28/1.96 ( ( linord1659791738miring @ A )
% 0.28/1.96 => ! [M: nat,N: nat] :
% 0.28/1.96 ( ( ord_less_eq @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N ) )
% 0.28/1.96 = ( ord_less_eq @ nat @ M @ N ) ) ) ).
% 0.28/1.96
% 0.28/1.96 thf(fact_219_less__Suc0,axiom,
% 0.28/1.96 ! [N: nat] :
% 0.28/1.96 ( ( ord_less @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) )
% 0.28/1.96 = ( N
% 0.28/1.96 = ( zero_zero @ nat ) ) ) ).
% 0.28/1.96
% 0.28/1.96 thf(fact_220_zero__less__Suc,axiom,
% 0.28/1.96 ! [N: nat] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( suc @ N ) ) ).
% 0.28/1.96
% 0.28/1.96 thf(fact_221_of__nat__1,axiom,
% 0.28/1.96 ! [A: $tType] :
% 0.28/1.96 ( ( semiring_1 @ A )
% 0.28/1.96 => ( ( semiring_1_of_nat @ A @ ( one_one @ nat ) )
% 0.28/1.96 = ( one_one @ A ) ) ) ).
% 0.28/1.96
% 0.28/1.96 thf(fact_222_of__nat__1__eq__iff,axiom,
% 0.28/1.96 ! [A: $tType] :
% 0.28/1.96 ( ( semiring_char_0 @ A )
% 0.28/1.96 => ! [N: nat] :
% 0.28/1.96 ( ( ( one_one @ A )
% 0.28/1.96 = ( semiring_1_of_nat @ A @ N ) )
% 0.28/1.96 = ( N
% 0.28/1.96 = ( one_one @ nat ) ) ) ) ).
% 0.28/1.96
% 0.28/1.96 thf(fact_223_of__nat__eq__1__iff,axiom,
% 0.28/1.96 ! [A: $tType] :
% 0.28/1.96 ( ( semiring_char_0 @ A )
% 0.28/1.96 => ! [N: nat] :
% 0.28/1.96 ( ( ( semiring_1_of_nat @ A @ N )
% 0.28/1.96 = ( one_one @ A ) )
% 0.28/1.96 = ( N
% 0.28/1.96 = ( one_one @ nat ) ) ) ) ).
% 0.28/1.96
% 0.28/1.96 thf(fact_224_less__one,axiom,
% 0.28/1.96 ! [N: nat] :
% 0.28/1.96 ( ( ord_less @ nat @ N @ ( one_one @ nat ) )
% 0.28/1.96 = ( N
% 0.28/1.96 = ( zero_zero @ nat ) ) ) ).
% 0.28/1.96
% 0.28/1.96 thf(fact_225_coeff__0,axiom,
% 0.28/1.96 ! [A: $tType] :
% 0.28/1.96 ( ( zero @ A )
% 0.28/1.96 => ! [N: nat] :
% 0.28/1.96 ( ( coeff @ A @ ( zero_zero @ ( poly @ A ) ) @ N )
% 0.28/1.96 = ( zero_zero @ A ) ) ) ).
% 0.28/1.96
% 0.28/1.96 thf(fact_226_poly__1,axiom,
% 0.28/1.96 ! [A: $tType] :
% 0.28/1.96 ( ( comm_semiring_1 @ A )
% 0.28/1.96 => ! [X: A] :
% 0.28/1.96 ( ( poly2 @ A @ ( one_one @ ( poly @ A ) ) @ X )
% 0.28/1.96 = ( one_one @ A ) ) ) ).
% 0.28/1.96
% 0.28/1.96 thf(fact_227_variation__0_I2_J,axiom,
% 0.28/1.96 ! [X: real] :
% 0.28/1.96 ( ( sturm_1771227917iation @ X @ ( zero_zero @ real ) )
% 0.28/1.96 = ( zero_zero @ int ) ) ).
% 0.28/1.96
% 0.28/1.96 thf(fact_228_variation__0_I1_J,axiom,
% 0.28/1.96 ! [Y4: real] :
% 0.28/1.96 ( ( sturm_1771227917iation @ ( zero_zero @ real ) @ Y4 )
% 0.28/1.96 = ( zero_zero @ int ) ) ).
% 0.28/1.96
% 0.28/1.96 thf(fact_229_pderiv__1,axiom,
% 0.28/1.96 ! [A: $tType] :
% 0.28/1.96 ( ( ( comm_semiring_1 @ A )
% 0.28/1.96 & ( semiri1193490041visors @ A ) )
% 0.28/1.96 => ( ( pderiv @ A @ ( one_one @ ( poly @ A ) ) )
% 0.28/1.96 = ( zero_zero @ ( poly @ A ) ) ) ) ).
% 0.28/1.96
% 0.28/1.96 thf(fact_230_order__1__eq__0,axiom,
% 0.28/1.96 ! [A: $tType] :
% 0.28/1.96 ( ( idom @ A )
% 0.28/1.96 => ! [X: A] :
% 0.28/1.96 ( ( order2 @ A @ X @ ( one_one @ ( poly @ A ) ) )
% 0.28/1.96 = ( zero_zero @ nat ) ) ) ).
% 0.28/1.96
% 0.28/1.96 thf(fact_231_of__nat__le__0__iff,axiom,
% 0.28/1.96 ! [A: $tType] :
% 0.28/1.96 ( ( linord1659791738miring @ A )
% 0.28/1.96 => ! [M: nat] :
% 0.28/1.96 ( ( ord_less_eq @ A @ ( semiring_1_of_nat @ A @ M ) @ ( zero_zero @ A ) )
% 0.28/1.96 = ( M
% 0.28/1.96 = ( zero_zero @ nat ) ) ) ) ).
% 0.28/1.96
% 0.28/1.96 thf(fact_232_reflect__poly__reflect__poly,axiom,
% 0.28/1.96 ! [A: $tType] :
% 0.28/1.96 ( ( zero @ A )
% 0.28/1.96 => ! [P: poly @ A] :
% 0.28/1.96 ( ( ( coeff @ A @ P @ ( zero_zero @ nat ) )
% 0.28/1.96 != ( zero_zero @ A ) )
% 0.28/1.96 => ( ( reflect_poly @ A @ ( reflect_poly @ A @ P ) )
% 0.28/1.96 = P ) ) ) ).
% 0.28/1.96
% 0.28/1.96 thf(fact_233_One__nat__def,axiom,
% 0.28/1.96 ( ( one_one @ nat )
% 0.28/1.96 = ( suc @ ( zero_zero @ nat ) ) ) ).
% 0.28/1.96
% 0.28/1.96 thf(fact_234_less__fun__def,axiom,
% 0.28/1.96 ! [B2: $tType,A: $tType] :
% 0.28/1.96 ( ( ord @ B2 )
% 0.28/1.96 => ( ( ord_less @ ( A > B2 ) )
% 0.28/1.96 = ( ^ [F2: A > B2,G2: A > B2] :
% 0.28/1.96 ( ( ord_less_eq @ ( A > B2 ) @ F2 @ G2 )
% 0.28/1.96 & ~ ( ord_less_eq @ ( A > B2 ) @ G2 @ F2 ) ) ) ) ) ).
% 0.28/1.96
% 0.28/1.96 thf(fact_235_of__nat__neq__0,axiom,
% 0.28/1.96 ! [A: $tType] :
% 0.28/1.96 ( ( semiring_char_0 @ A )
% 0.28/1.96 => ! [N: nat] :
% 0.28/1.96 ( ( semiring_1_of_nat @ A @ ( suc @ N ) )
% 0.28/1.96 != ( zero_zero @ A ) ) ) ).
% 0.28/1.96
% 0.28/1.96 thf(fact_236_transitive__stepwise__le,axiom,
% 0.28/1.96 ! [M: nat,N: nat,R: nat > nat > $o] :
% 0.28/1.96 ( ( ord_less_eq @ nat @ M @ N )
% 0.28/1.96 => ( ! [X2: nat] : ( R @ X2 @ X2 )
% 0.28/1.96 => ( ! [X2: nat,Y: nat,Z3: nat] :
% 0.28/1.96 ( ( R @ X2 @ Y )
% 0.28/1.96 => ( ( R @ Y @ Z3 )
% 0.28/1.96 => ( R @ X2 @ Z3 ) ) )
% 0.28/1.96 => ( ! [N3: nat] : ( R @ N3 @ ( suc @ N3 ) )
% 0.28/1.96 => ( R @ M @ N ) ) ) ) ) ).
% 0.28/1.96
% 0.28/1.96 thf(fact_237_nat__induct__at__least,axiom,
% 0.28/1.96 ! [M: nat,N: nat,P3: nat > $o] :
% 0.28/1.96 ( ( ord_less_eq @ nat @ M @ N )
% 0.28/1.96 => ( ( P3 @ M )
% 0.28/1.96 => ( ! [N3: nat] :
% 0.28/1.96 ( ( ord_less_eq @ nat @ M @ N3 )
% 0.28/1.96 => ( ( P3 @ N3 )
% 0.28/1.96 => ( P3 @ ( suc @ N3 ) ) ) )
% 0.28/1.96 => ( P3 @ N ) ) ) ) ).
% 0.28/1.96
% 0.28/1.96 thf(fact_238_Nat_Oex__has__greatest__nat,axiom,
% 0.28/1.96 ! [P3: nat > $o,K2: nat,B: nat] :
% 0.28/1.96 ( ( P3 @ K2 )
% 0.28/1.96 => ( ! [Y: nat] :
% 0.28/1.96 ( ( P3 @ Y )
% 0.28/1.96 => ( ord_less_eq @ nat @ Y @ B ) )
% 0.28/1.96 => ? [X2: nat] :
% 0.28/1.96 ( ( P3 @ X2 )
% 0.28/1.96 & ! [Y3: nat] :
% 0.28/1.96 ( ( P3 @ Y3 )
% 0.28/1.96 => ( ord_less_eq @ nat @ Y3 @ X2 ) ) ) ) ) ).
% 0.28/1.96
% 0.28/1.96 thf(fact_239_verit__la__generic,axiom,
% 0.28/1.96 ! [A2: int,X: int] :
% 0.28/1.96 ( ( ord_less_eq @ int @ A2 @ X )
% 0.28/1.96 | ( A2 = X )
% 0.28/1.96 | ( ord_less_eq @ int @ X @ A2 ) ) ).
% 0.28/1.96
% 0.28/1.96 thf(fact_240_full__nat__induct,axiom,
% 0.28/1.96 ! [P3: nat > $o,N: nat] :
% 0.28/1.96 ( ! [N3: nat] :
% 0.28/1.96 ( ! [M3: nat] :
% 0.28/1.96 ( ( ord_less_eq @ nat @ ( suc @ M3 ) @ N3 )
% 0.28/1.96 => ( P3 @ M3 ) )
% 0.28/1.96 => ( P3 @ N3 ) )
% 0.28/1.96 => ( P3 @ N ) ) ).
% 0.28/1.96
% 0.28/1.96 thf(fact_241_not__less__eq__eq,axiom,
% 0.28/1.96 ! [M: nat,N: nat] :
% 0.28/1.96 ( ( ~ ( ord_less_eq @ nat @ M @ N ) )
% 0.28/1.96 = ( ord_less_eq @ nat @ ( suc @ N ) @ M ) ) ).
% 0.28/1.96
% 0.28/1.96 thf(fact_242_Suc__n__not__le__n,axiom,
% 0.28/1.96 ! [N: nat] :
% 0.28/1.96 ~ ( ord_less_eq @ nat @ ( suc @ N ) @ N ) ).
% 0.28/1.96
% 0.28/1.96 thf(fact_243_nat__le__linear,axiom,
% 0.28/1.96 ! [M: nat,N: nat] :
% 0.28/1.96 ( ( ord_less_eq @ nat @ M @ N )
% 0.28/1.96 | ( ord_less_eq @ nat @ N @ M ) ) ).
% 0.28/1.96
% 0.28/1.96 thf(fact_244_le__antisym,axiom,
% 0.28/1.96 ! [M: nat,N: nat] :
% 0.28/1.96 ( ( ord_less_eq @ nat @ M @ N )
% 0.28/1.96 => ( ( ord_less_eq @ nat @ N @ M )
% 0.28/1.96 => ( M = N ) ) ) ).
% 0.28/1.96
% 0.28/1.96 thf(fact_245_le__Suc__eq,axiom,
% 0.28/1.96 ! [M: nat,N: nat] :
% 0.28/1.96 ( ( ord_less_eq @ nat @ M @ ( suc @ N ) )
% 0.28/1.96 = ( ( ord_less_eq @ nat @ M @ N )
% 0.28/1.96 | ( M
% 0.28/1.96 = ( suc @ N ) ) ) ) ).
% 0.28/1.96
% 0.28/1.96 thf(fact_246_eq__imp__le,axiom,
% 0.28/1.96 ! [M: nat,N: nat] :
% 0.28/1.96 ( ( M = N )
% 0.28/1.96 => ( ord_less_eq @ nat @ M @ N ) ) ).
% 0.28/1.96
% 0.28/1.96 thf(fact_247_le__trans,axiom,
% 0.28/1.96 ! [I: nat,J: nat,K2: nat] :
% 0.28/1.96 ( ( ord_less_eq @ nat @ I @ J )
% 0.28/1.96 => ( ( ord_less_eq @ nat @ J @ K2 )
% 0.28/1.96 => ( ord_less_eq @ nat @ I @ K2 ) ) ) ).
% 0.28/1.96
% 0.28/1.96 thf(fact_248_Suc__le__D,axiom,
% 0.28/1.96 ! [N: nat,M4: nat] :
% 0.28/1.96 ( ( ord_less_eq @ nat @ ( suc @ N ) @ M4 )
% 0.28/1.96 => ? [M5: nat] :
% 0.28/1.96 ( M4
% 0.28/1.96 = ( suc @ M5 ) ) ) ).
% 0.28/1.96
% 0.28/1.96 thf(fact_249_le__refl,axiom,
% 0.28/1.96 ! [N: nat] : ( ord_less_eq @ nat @ N @ N ) ).
% 0.28/1.96
% 0.28/1.96 thf(fact_250_le__SucI,axiom,
% 0.28/1.96 ! [M: nat,N: nat] :
% 0.28/1.96 ( ( ord_less_eq @ nat @ M @ N )
% 0.28/1.96 => ( ord_less_eq @ nat @ M @ ( suc @ N ) ) ) ).
% 0.28/1.96
% 0.28/1.96 thf(fact_251_le__SucE,axiom,
% 0.28/1.96 ! [M: nat,N: nat] :
% 0.28/1.96 ( ( ord_less_eq @ nat @ M @ ( suc @ N ) )
% 0.28/1.96 => ( ~ ( ord_less_eq @ nat @ M @ N )
% 0.28/1.96 => ( M
% 0.28/1.96 = ( suc @ N ) ) ) ) ).
% 0.28/1.96
% 0.28/1.96 thf(tcon_Polynomial_Opoly___Rings_Olinordered__idom,axiom,
% 0.28/1.96 ! [A7: $tType] :
% 0.28/1.96 ( ( linordered_idom @ A7 )
% 0.28/1.96 => ( linordered_idom @ ( poly @ A7 ) ) ) ).
% 0.28/1.96
% 0.28/1.96 thf(tcon_Real_Oreal___Rings_Olinordered__idom_1,axiom,
% 0.28/1.96 linordered_idom @ real ).
% 0.28/1.96
% 0.28/1.96 thf(tcon_Int_Oint___Rings_Olinordered__idom_2,axiom,
% 0.28/1.96 linordered_idom @ int ).
% 0.28/1.96
% 0.28/1.96 thf(tcon_fun___Orderings_Opreorder,axiom,
% 0.28/1.96 ! [A7: $tType,A8: $tType] :
% 0.28/1.96 ( ( preorder @ A8 )
% 0.28/1.96 => ( preorder @ ( A7 > A8 ) ) ) ).
% 0.28/1.96
% 0.28/1.96 thf(tcon_fun___Orderings_Oorder,axiom,
% 0.28/1.96 ! [A7: $tType,A8: $tType] :
% 0.28/1.96 ( ( order @ A8 )
% 0.28/1.96 => ( order @ ( A7 > A8 ) ) ) ).
% 0.28/1.96
% 0.28/1.96 thf(tcon_fun___Orderings_Oord,axiom,
% 0.28/1.96 ! [A7: $tType,A8: $tType] :
% 0.28/1.96 ( ( ord @ A8 )
% 0.28/1.96 => ( ord @ ( A7 > A8 ) ) ) ).
% 0.28/1.96
% 0.28/1.96 thf(tcon_Int_Oint___Conditionally__Complete__Lattices_Oconditionally__complete__linorder,axiom,
% 0.28/1.96 condit1037483654norder @ int ).
% 0.28/1.96
% 0.28/1.96 thf(tcon_Int_Oint___Rings_Olinordered__nonzero__semiring,axiom,
% 0.28/1.96 linord1659791738miring @ int ).
% 0.28/1.96
% 0.28/1.96 thf(tcon_Int_Oint___Rings_Osemiring__no__zero__divisors,axiom,
% 0.28/1.96 semiri1193490041visors @ int ).
% 0.28/1.96
% 0.28/1.96 thf(tcon_Int_Oint___Rings_Oring__no__zero__divisors,axiom,
% 0.28/1.96 ring_n68954251visors @ int ).
% 0.28/1.96
% 0.28/1.96 thf(tcon_Int_Oint___Rings_Ocomm__semiring__1,axiom,
% 0.28/1.96 comm_semiring_1 @ int ).
% 0.28/1.96
% 0.28/1.96 thf(tcon_Int_Oint___Rings_Ocomm__semiring__0,axiom,
% 0.28/1.96 comm_semiring_0 @ int ).
% 0.28/1.96
% 0.28/1.96 thf(tcon_Int_Oint___Nat_Osemiring__char__0,axiom,
% 0.28/1.96 semiring_char_0 @ int ).
% 0.28/1.96
% 0.28/1.96 thf(tcon_Int_Oint___Orderings_Opreorder_3,axiom,
% 0.28/1.96 preorder @ int ).
% 0.28/1.96
% 0.28/1.96 thf(tcon_Int_Oint___Orderings_Olinorder,axiom,
% 0.28/1.96 linorder @ int ).
% 0.28/1.96
% 0.28/1.96 thf(tcon_Int_Oint___Rings_Ocomm__ring__1,axiom,
% 0.28/1.96 comm_ring_1 @ int ).
% 0.28/1.96
% 0.28/1.96 thf(tcon_Int_Oint___Rings_Osemiring__1,axiom,
% 0.28/1.96 semiring_1 @ int ).
% 0.28/1.96
% 0.28/1.96 thf(tcon_Int_Oint___Orderings_Ono__top,axiom,
% 0.28/1.96 no_top @ int ).
% 0.28/1.96
% 0.28/1.96 thf(tcon_Int_Oint___Orderings_Ono__bot,axiom,
% 0.28/1.96 no_bot @ int ).
% 0.28/1.96
% 0.28/1.96 thf(tcon_Int_Oint___Orderings_Oorder_4,axiom,
% 0.28/1.96 order @ int ).
% 0.28/1.96
% 0.28/1.96 thf(tcon_Int_Oint___Nat_Oring__char__0,axiom,
% 0.28/1.96 ring_char_0 @ int ).
% 0.28/1.96
% 0.28/1.96 thf(tcon_Int_Oint___Orderings_Oord_5,axiom,
% 0.28/1.96 ord @ int ).
% 0.28/1.96
% 0.28/1.96 thf(tcon_Int_Oint___Groups_Ozero,axiom,
% 0.28/1.96 zero @ int ).
% 0.28/1.96
% 0.28/1.96 thf(tcon_Int_Oint___Rings_Oidom,axiom,
% 0.28/1.96 idom @ int ).
% 0.28/1.96
% 0.28/1.96 thf(tcon_Int_Oint___Groups_Oone,axiom,
% 0.28/1.96 one @ int ).
% 0.28/1.96
% 0.28/1.96 thf(tcon_Nat_Onat___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_6,axiom,
% 0.28/1.96 condit1037483654norder @ nat ).
% 0.28/1.96
% 0.28/1.96 thf(tcon_Nat_Onat___Groups_Ocanonically__ordered__monoid__add,axiom,
% 0.28/1.96 canoni770627133id_add @ nat ).
% 0.28/1.96
% 0.28/1.96 thf(tcon_Nat_Onat___Rings_Olinordered__nonzero__semiring_7,axiom,
% 0.28/1.96 linord1659791738miring @ nat ).
% 0.28/1.96
% 0.28/1.96 thf(tcon_Nat_Onat___Rings_Osemiring__no__zero__divisors_8,axiom,
% 0.28/1.96 semiri1193490041visors @ nat ).
% 0.28/1.96
% 0.28/1.96 thf(tcon_Nat_Onat___Rings_Ocomm__semiring__1_9,axiom,
% 0.28/1.96 comm_semiring_1 @ nat ).
% 0.28/1.96
% 0.28/1.96 thf(tcon_Nat_Onat___Rings_Ocomm__semiring__0_10,axiom,
% 0.28/1.96 comm_semiring_0 @ nat ).
% 0.28/1.96
% 0.28/1.96 thf(tcon_Nat_Onat___Orderings_Owellorder,axiom,
% 0.28/1.96 wellorder @ nat ).
% 0.28/1.96
% 0.28/1.96 thf(tcon_Nat_Onat___Nat_Osemiring__char__0_11,axiom,
% 0.28/1.96 semiring_char_0 @ nat ).
% 0.28/1.96
% 0.28/1.96 thf(tcon_Nat_Onat___Orderings_Opreorder_12,axiom,
% 0.28/1.96 preorder @ nat ).
% 0.28/1.96
% 0.28/1.96 thf(tcon_Nat_Onat___Orderings_Olinorder_13,axiom,
% 0.28/1.96 linorder @ nat ).
% 0.28/1.96
% 0.28/1.96 thf(tcon_Nat_Onat___Rings_Osemiring__1_14,axiom,
% 0.28/1.96 semiring_1 @ nat ).
% 0.28/1.96
% 0.28/1.96 thf(tcon_Nat_Onat___Orderings_Ono__top_15,axiom,
% 0.28/1.96 no_top @ nat ).
% 0.28/1.96
% 0.28/1.96 thf(tcon_Nat_Onat___Orderings_Oorder_16,axiom,
% 0.28/1.96 order @ nat ).
% 0.28/1.96
% 0.28/1.96 thf(tcon_Nat_Onat___Orderings_Oord_17,axiom,
% 0.28/1.96 ord @ nat ).
% 0.28/1.96
% 0.28/1.96 thf(tcon_Nat_Onat___Groups_Ozero_18,axiom,
% 0.28/1.96 zero @ nat ).
% 0.28/1.96
% 0.28/1.96 thf(tcon_Nat_Onat___Groups_Oone_19,axiom,
% 0.28/1.96 one @ nat ).
% 0.28/1.96
% 0.28/1.96 thf(tcon_Set_Oset___Orderings_Opreorder_20,axiom,
% 0.28/1.96 ! [A7: $tType] : ( preorder @ ( set @ A7 ) ) ).
% 0.28/1.96
% 0.28/1.96 thf(tcon_Set_Oset___Orderings_Oorder_21,axiom,
% 0.28/1.96 ! [A7: $tType] : ( order @ ( set @ A7 ) ) ).
% 0.28/1.96
% 0.28/1.96 thf(tcon_Set_Oset___Orderings_Oord_22,axiom,
% 0.28/1.96 ! [A7: $tType] : ( ord @ ( set @ A7 ) ) ).
% 0.28/1.96
% 0.28/1.96 thf(tcon_HOL_Obool___Orderings_Opreorder_23,axiom,
% 0.28/1.96 preorder @ $o ).
% 0.28/1.96
% 0.28/1.96 thf(tcon_HOL_Obool___Orderings_Olinorder_24,axiom,
% 0.28/1.96 linorder @ $o ).
% 0.28/1.96
% 0.28/1.96 thf(tcon_HOL_Obool___Orderings_Oorder_25,axiom,
% 0.28/1.96 order @ $o ).
% 0.28/1.96
% 0.28/1.96 thf(tcon_HOL_Obool___Orderings_Oord_26,axiom,
% 0.28/1.96 ord @ $o ).
% 0.28/1.96
% 0.28/1.96 thf(tcon_Real_Oreal___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_27,axiom,
% 0.28/1.96 condit1037483654norder @ real ).
% 0.28/1.96
% 0.28/1.96 thf(tcon_Real_Oreal___Rings_Olinordered__nonzero__semiring_28,axiom,
% 0.28/1.96 linord1659791738miring @ real ).
% 0.28/1.96
% 0.28/1.96 thf(tcon_Real_Oreal___Rings_Osemiring__no__zero__divisors_29,axiom,
% 0.28/1.96 semiri1193490041visors @ real ).
% 0.28/1.96
% 0.28/1.96 thf(tcon_Real_Oreal___Rings_Oring__no__zero__divisors_30,axiom,
% 0.28/1.96 ring_n68954251visors @ real ).
% 0.28/1.96
% 0.28/1.96 thf(tcon_Real_Oreal___Orderings_Odense__linorder,axiom,
% 0.28/1.96 dense_linorder @ real ).
% 0.28/1.96
% 0.28/1.96 thf(tcon_Real_Oreal___Fields_Olinordered__field,axiom,
% 0.28/1.96 linordered_field @ real ).
% 0.28/1.96
% 0.28/1.96 thf(tcon_Real_Oreal___Rings_Ocomm__semiring__1_31,axiom,
% 0.28/1.96 comm_semiring_1 @ real ).
% 0.28/1.96
% 0.28/1.96 thf(tcon_Real_Oreal___Rings_Ocomm__semiring__0_32,axiom,
% 0.28/1.96 comm_semiring_0 @ real ).
% 0.28/1.96
% 0.28/1.96 thf(tcon_Real_Oreal___Orderings_Odense__order,axiom,
% 0.28/1.96 dense_order @ real ).
% 0.28/1.96
% 0.28/1.96 thf(tcon_Real_Oreal___Nat_Osemiring__char__0_33,axiom,
% 0.28/1.96 semiring_char_0 @ real ).
% 0.28/1.96
% 0.28/1.96 thf(tcon_Real_Oreal___Fields_Ofield__char__0,axiom,
% 0.28/1.96 field_char_0 @ real ).
% 0.28/1.96
% 0.28/1.96 thf(tcon_Real_Oreal___Orderings_Opreorder_34,axiom,
% 0.28/1.96 preorder @ real ).
% 0.28/1.96
% 0.28/1.96 thf(tcon_Real_Oreal___Orderings_Olinorder_35,axiom,
% 0.28/1.96 linorder @ real ).
% 0.28/1.96
% 0.28/1.96 thf(tcon_Real_Oreal___Rings_Ocomm__ring__1_36,axiom,
% 0.28/1.96 comm_ring_1 @ real ).
% 0.28/1.96
% 0.28/1.96 thf(tcon_Real_Oreal___Rings_Osemiring__1_37,axiom,
% 0.28/1.96 semiring_1 @ real ).
% 0.28/1.96
% 0.28/1.96 thf(tcon_Real_Oreal___Orderings_Ono__top_38,axiom,
% 0.28/1.96 no_top @ real ).
% 0.28/1.96
% 0.28/1.96 thf(tcon_Real_Oreal___Orderings_Ono__bot_39,axiom,
% 0.28/1.96 no_bot @ real ).
% 0.28/1.96
% 0.28/1.96 thf(tcon_Real_Oreal___Orderings_Oorder_40,axiom,
% 0.28/1.96 order @ real ).
% 0.28/1.96
% 0.28/1.96 thf(tcon_Real_Oreal___Nat_Oring__char__0_41,axiom,
% 0.28/1.96 ring_char_0 @ real ).
% 0.28/1.96
% 0.28/1.96 thf(tcon_Real_Oreal___Orderings_Oord_42,axiom,
% 0.28/1.96 ord @ real ).
% 0.28/1.96
% 0.28/1.96 thf(tcon_Real_Oreal___Groups_Ozero_43,axiom,
% 0.28/1.96 zero @ real ).
% 0.28/1.96
% 0.28/1.96 thf(tcon_Real_Oreal___Rings_Oidom_44,axiom,
% 0.28/1.96 idom @ real ).
% 0.28/1.96
% 0.28/1.96 thf(tcon_Real_Oreal___Groups_Oone_45,axiom,
% 0.28/1.96 one @ real ).
% 0.28/1.96
% 0.28/1.96 thf(tcon_Polynomial_Opoly___Rings_Olinordered__nonzero__semiring_46,axiom,
% 0.28/1.96 ! [A7: $tType] :
% 0.28/1.96 ( ( linordered_idom @ A7 )
% 0.28/1.96 => ( linord1659791738miring @ ( poly @ A7 ) ) ) ).
% 0.28/1.96
% 0.28/1.96 thf(tcon_Polynomial_Opoly___Rings_Osemiring__no__zero__divisors_47,axiom,
% 0.28/1.96 ! [A7: $tType] :
% 0.28/1.96 ( ( ( comm_semiring_0 @ A7 )
% 0.28/1.96 & ( semiri1193490041visors @ A7 ) )
% 0.28/1.96 => ( semiri1193490041visors @ ( poly @ A7 ) ) ) ).
% 0.28/1.96
% 0.28/1.96 thf(tcon_Polynomial_Opoly___Rings_Oring__no__zero__divisors_48,axiom,
% 0.28/1.96 ! [A7: $tType] :
% 0.28/1.96 ( ( idom @ A7 )
% 0.28/1.96 => ( ring_n68954251visors @ ( poly @ A7 ) ) ) ).
% 0.28/1.96
% 0.28/1.96 thf(tcon_Polynomial_Opoly___Rings_Ocomm__semiring__1_49,axiom,
% 0.28/1.96 ! [A7: $tType] :
% 0.28/1.96 ( ( comm_semiring_1 @ A7 )
% 0.28/1.96 => ( comm_semiring_1 @ ( poly @ A7 ) ) ) ).
% 0.28/1.96
% 0.28/1.96 thf(tcon_Polynomial_Opoly___Rings_Ocomm__semiring__0_50,axiom,
% 0.28/1.96 ! [A7: $tType] :
% 0.28/1.96 ( ( comm_semiring_0 @ A7 )
% 0.28/1.96 => ( comm_semiring_0 @ ( poly @ A7 ) ) ) ).
% 0.28/1.96
% 0.28/1.96 thf(tcon_Polynomial_Opoly___Nat_Osemiring__char__0_51,axiom,
% 0.28/1.96 ! [A7: $tType] :
% 0.28/1.96 ( ( ( ring_char_0 @ A7 )
% 0.28/1.96 & ( comm_ring_1 @ A7 ) )
% 0.28/1.96 => ( semiring_char_0 @ ( poly @ A7 ) ) ) ).
% 0.28/1.96
% 0.28/1.96 thf(tcon_Polynomial_Opoly___Orderings_Opreorder_52,axiom,
% 0.28/1.96 ! [A7: $tType] :
% 0.28/1.96 ( ( linordered_idom @ A7 )
% 0.28/1.96 => ( preorder @ ( poly @ A7 ) ) ) ).
% 0.28/1.96
% 0.28/1.96 thf(tcon_Polynomial_Opoly___Orderings_Olinorder_53,axiom,
% 0.28/1.96 ! [A7: $tType] :
% 0.28/1.96 ( ( linordered_idom @ A7 )
% 0.28/1.96 => ( linorder @ ( poly @ A7 ) ) ) ).
% 0.28/1.96
% 0.28/1.96 thf(tcon_Polynomial_Opoly___Rings_Ocomm__ring__1_54,axiom,
% 0.28/1.96 ! [A7: $tType] :
% 0.28/1.96 ( ( comm_ring_1 @ A7 )
% 0.28/1.96 => ( comm_ring_1 @ ( poly @ A7 ) ) ) ).
% 0.28/1.96
% 0.28/1.96 thf(tcon_Polynomial_Opoly___Rings_Osemiring__1_55,axiom,
% 0.28/1.96 ! [A7: $tType] :
% 0.28/1.96 ( ( comm_semiring_1 @ A7 )
% 0.28/1.96 => ( semiring_1 @ ( poly @ A7 ) ) ) ).
% 0.28/1.96
% 0.28/1.96 thf(tcon_Polynomial_Opoly___Orderings_Oorder_56,axiom,
% 0.28/1.96 ! [A7: $tType] :
% 0.28/1.96 ( ( linordered_idom @ A7 )
% 0.28/1.96 => ( order @ ( poly @ A7 ) ) ) ).
% 0.28/1.96
% 0.28/1.96 thf(tcon_Polynomial_Opoly___Nat_Oring__char__0_57,axiom,
% 0.28/1.96 ! [A7: $tType] :
% 0.28/1.96 ( ( ( ring_char_0 @ A7 )
% 0.28/1.96 & ( comm_ring_1 @ A7 ) )
% 0.28/1.96 => ( ring_char_0 @ ( poly @ A7 ) ) ) ).
% 0.28/1.96
% 0.28/1.96 thf(tcon_Polynomial_Opoly___Orderings_Oord_58,axiom,
% 0.28/1.96 ! [A7: $tType] :
% 0.28/1.96 ( ( linordered_idom @ A7 )
% 0.28/1.96 => ( ord @ ( poly @ A7 ) ) ) ).
% 0.28/1.96
% 0.28/1.96 thf(tcon_Polynomial_Opoly___Groups_Ozero_59,axiom,
% 0.28/1.96 ! [A7: $tType] :
% 0.28/1.96 ( ( zero @ A7 )
% 0.28/1.96 => ( zero @ ( poly @ A7 ) ) ) ).
% 0.28/1.96
% 0.28/1.96 thf(tcon_Polynomial_Opoly___Rings_Oidom_60,axiom,
% 0.28/1.96 ! [A7: $tType] :
% 0.28/1.96 ( ( idom @ A7 )
% 0.28/1.96 => ( idom @ ( poly @ A7 ) ) ) ).
% 0.28/1.96
% 0.28/1.96 thf(tcon_Polynomial_Opoly___Groups_Oone_61,axiom,
% 0.28/1.96 ! [A7: $tType] :
% 0.28/1.96 ( ( comm_semiring_1 @ A7 )
% 0.28/1.96 => ( one @ ( poly @ A7 ) ) ) ).
% 0.28/1.96
% 0.28/1.96 thf(conj_0,conjecture,
% 0.28/1.96 ( ( sturm_1700286437_r_pos @ p @ x )
% 0.28/1.96 != ( ~ ( ( ( ( poly2 @ real @ p @ x )
% 0.28/1.96 = ( zero_zero @ real ) )
% 0.28/1.96 => ( sturm_1700286437_r_pos @ ( pderiv @ real @ p ) @ x ) )
% 0.28/1.96 & ( ( ( poly2 @ real @ p @ x )
% 0.28/1.96 != ( zero_zero @ real ) )
% 0.28/1.96 => ( ord_less @ real @ ( zero_zero @ real ) @ ( poly2 @ real @ p @ x ) ) ) ) ) ) ).
% 0.28/1.96 ------------------------
% 0.28/1.99 % (31856)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_31756 for (2999ds/183Mi)
% 0.28/1.99 % (31861)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_31756 for (2999ds/275Mi)
% 0.28/1.99 % (31859)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_31756 for (2999ds/2Mi)
% 0.28/1.99 % (31862)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_31756 for (2999ds/18Mi)
% 0.28/1.99 % (31860)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_31756 for (2999ds/2Mi)
% 0.28/1.99 % (31857)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_31756 for (2999ds/4Mi)
% 0.28/1.99 % (31858)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_31756 for (2999ds/27Mi)
% 0.28/2.00 % (31859)Instruction limit reached!
% 0.28/2.00 % (31859)------------------------------
% 0.28/2.00 % (31859)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 0.28/2.00 % (31860)Instruction limit reached!
% 0.28/2.00 % (31860)------------------------------
% 0.28/2.00 % (31860)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 0.28/2.00 % (31859)Termination reason: Unknown
% 0.28/2.00 % (31859)Termination phase: shuffling
% 0.28/2.00
% 0.28/2.00 % (31859)Memory used [KB]: 1407
% 0.28/2.00 % (31859)Time elapsed: 0.003 s
% 0.28/2.00 % (31859)Instructions burned: 2 (million)
% 0.28/2.00 % (31859)------------------------------
% 0.28/2.00 % (31859)------------------------------
% 0.28/2.00 % (31860)Termination reason: Unknown
% 0.28/2.00 % (31860)Termination phase: shuffling
% 0.28/2.00
% 0.28/2.00 % (31860)Memory used [KB]: 1407
% 0.28/2.00 % (31860)Time elapsed: 0.003 s
% 0.28/2.00 % (31860)Instructions burned: 2 (million)
% 0.28/2.00 % (31860)------------------------------
% 0.28/2.00 % (31860)------------------------------
% 0.28/2.00 % (31857)Instruction limit reached!
% 0.28/2.00 % (31857)------------------------------
% 0.28/2.00 % (31857)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 0.28/2.00 % (31857)Termination reason: Unknown
% 0.28/2.00 % (31857)Termination phase: shuffling
% 0.28/2.00
% 0.28/2.00 % (31857)Memory used [KB]: 1407
% 0.28/2.00 % (31857)Time elapsed: 0.004 s
% 0.28/2.00 % (31857)Instructions burned: 4 (million)
% 0.28/2.00 % (31857)------------------------------
% 0.28/2.00 % (31857)------------------------------
% 0.28/2.00 % (31862)Instruction limit reached!
% 0.28/2.00 % (31862)------------------------------
% 0.28/2.00 % (31862)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 0.28/2.00 % (31862)Termination reason: Unknown
% 0.28/2.00 % (31862)Termination phase: Property scanning
% 0.28/2.00
% 0.28/2.00 % (31862)Memory used [KB]: 1791
% 0.28/2.00 % (31862)Time elapsed: 0.013 s
% 0.28/2.00 % (31862)Instructions burned: 19 (million)
% 0.28/2.00 % (31862)------------------------------
% 0.28/2.00 % (31862)------------------------------
% 0.28/2.01 % (31858)Instruction limit reached!
% 0.28/2.01 % (31858)------------------------------
% 0.28/2.01 % (31858)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 0.28/2.01 % (31858)Termination reason: Unknown
% 0.28/2.01 % (31858)Termination phase: Preprocessing 1
% 0.28/2.01
% 0.28/2.01 % (31858)Memory used [KB]: 1791
% 0.28/2.01 % (31858)Time elapsed: 0.018 s
% 0.28/2.01 % (31858)Instructions burned: 28 (million)
% 0.28/2.01 % (31858)------------------------------
% 0.28/2.01 % (31858)------------------------------
% 0.28/2.01 % (31863)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_31756 for (2999ds/3Mi)
% 0.28/2.01 % (31864)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_31756 for (2999ds/37Mi)
% 0.28/2.01 % (31865)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_31756 for (2999ds/15Mi)
% 0.28/2.02 % (31863)Instruction limit reached!
% 0.28/2.02 % (31863)------------------------------
% 0.28/2.02 % (31863)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 0.28/2.02 % (31863)Termination reason: Unknown
% 0.28/2.02 % (31863)Termination phase: shuffling
% 0.28/2.02
% 0.28/2.02 % (31863)Memory used [KB]: 1407
% 0.28/2.02 % (31863)Time elapsed: 0.004 s
% 0.28/2.02 % (31863)Instructions burned: 4 (million)
% 0.28/2.02 % (31863)------------------------------
% 0.28/2.02 % (31863)------------------------------
% 0.28/2.02 % (31865)Instruction limit reached!
% 0.28/2.02 % (31865)------------------------------
% 0.28/2.02 % (31865)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 0.28/2.02 % (31865)Termination reason: Unknown
% 0.28/2.02 % (31865)Termination phase: shuffling
% 0.28/2.02
% 0.28/2.02 % (31865)Memory used [KB]: 1791
% 0.28/2.02 % (31865)Time elapsed: 0.011 s
% 0.28/2.02 % (31865)Instructions burned: 16 (million)
% 0.28/2.02 % (31865)------------------------------
% 0.28/2.02 % (31865)------------------------------
% 0.28/2.02 % (31866)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_31756 for (2999ds/3Mi)
% 0.28/2.03 % (31866)Instruction limit reached!
% 0.28/2.03 % (31866)------------------------------
% 0.28/2.03 % (31866)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 0.28/2.03 % (31866)Termination reason: Unknown
% 0.28/2.03 % (31866)Termination phase: shuffling
% 0.28/2.03
% 0.28/2.03 % (31866)Memory used [KB]: 1407
% 0.28/2.03 % (31866)Time elapsed: 0.004 s
% 0.28/2.03 % (31866)Instructions burned: 4 (million)
% 0.28/2.03 % (31866)------------------------------
% 0.28/2.03 % (31866)------------------------------
% 3.82/2.03 % (31861)First to succeed.
% 3.82/2.03 % (31861)Refutation found. Thanks to Tanya!
% 3.82/2.03 % SZS status Theorem for DTF2THF_31756
% 3.82/2.03 % SZS output start Proof for DTF2THF_31756
% 3.82/2.03 thf(type_def_5, type, nat: $tType).
% 3.82/2.03 thf(type_def_7, type, poly: ($tType) > $tType).
% 3.82/2.03 thf(type_def_8, type, set: ($tType) > $tType).
% 3.82/2.03 thf(type_def_9, type, real: $tType).
% 3.82/2.03 thf(type_def_10, type, int: $tType).
% 3.82/2.03 thf(func_def_0, type, poly: ($tType) > $tType).
% 3.82/2.03 thf(func_def_1, type, real: $tType).
% 3.82/2.03 thf(func_def_2, type, set: ($tType) > $tType).
% 3.82/2.03 thf(func_def_3, type, nat: $tType).
% 3.82/2.03 thf(func_def_4, type, int: $tType).
% 3.82/2.03 thf(func_def_5, type, linordered_idom: !>[X0: $tType]:($o)).
% 3.82/2.03 thf(func_def_6, type, type: !>[X0: $tType]:($o)).
% 3.82/2.03 thf(func_def_7, type, one: !>[X0: $tType]:($o)).
% 3.82/2.03 thf(func_def_8, type, idom: !>[X0: $tType]:($o)).
% 3.82/2.03 thf(func_def_9, type, zero: !>[X0: $tType]:($o)).
% 3.82/2.03 thf(func_def_10, type, ord: !>[X0: $tType]:($o)).
% 3.82/2.03 thf(func_def_11, type, ring_char_0: !>[X0: $tType]:($o)).
% 3.82/2.03 thf(func_def_12, type, order: !>[X0: $tType]:($o)).
% 3.82/2.03 thf(func_def_13, type, no_bot: !>[X0: $tType]:($o)).
% 3.82/2.03 thf(func_def_14, type, no_top: !>[X0: $tType]:($o)).
% 3.82/2.03 thf(func_def_15, type, semiring_1: !>[X0: $tType]:($o)).
% 3.82/2.03 thf(func_def_16, type, comm_ring_1: !>[X0: $tType]:($o)).
% 3.82/2.03 thf(func_def_17, type, linorder: !>[X0: $tType]:($o)).
% 3.82/2.03 thf(func_def_18, type, preorder: !>[X0: $tType]:($o)).
% 3.82/2.03 thf(func_def_19, type, field_char_0: !>[X0: $tType]:($o)).
% 3.82/2.03 thf(func_def_20, type, semiring_char_0: !>[X0: $tType]:($o)).
% 3.82/2.03 thf(func_def_21, type, wellorder: !>[X0: $tType]:($o)).
% 3.82/2.03 thf(func_def_22, type, dense_order: !>[X0: $tType]:($o)).
% 3.82/2.03 thf(func_def_23, type, comm_semiring_0: !>[X0: $tType]:($o)).
% 3.82/2.03 thf(func_def_24, type, comm_semiring_1: !>[X0: $tType]:($o)).
% 3.82/2.03 thf(func_def_25, type, linordered_field: !>[X0: $tType]:($o)).
% 3.82/2.03 thf(func_def_26, type, dense_linorder: !>[X0: $tType]:($o)).
% 3.82/2.03 thf(func_def_27, type, ring_n68954251visors: !>[X0: $tType]:($o)).
% 3.82/2.03 thf(func_def_28, type, semiri1193490041visors: !>[X0: $tType]:($o)).
% 3.82/2.03 thf(func_def_29, type, linord1659791738miring: !>[X0: $tType]:($o)).
% 3.82/2.03 thf(func_def_30, type, canoni770627133id_add: !>[X0: $tType]:($o)).
% 3.82/2.03 thf(func_def_31, type, condit1037483654norder: !>[X0: $tType]:($o)).
% 3.82/2.03 thf(func_def_32, type, one_one: !>[X0: $tType]:(X0)).
% 3.82/2.03 thf(func_def_33, type, zero_zero: !>[X0: $tType]:(X0)).
% 3.82/2.03 thf(func_def_34, type, suc: nat > nat).
% 3.82/2.03 thf(func_def_35, type, semiring_1_of_nat: !>[X0: $tType]:(nat > X0)).
% 3.82/2.03 thf(func_def_36, type, ord_less: !>[X0: $tType]:(X0 > X0 > $o)).
% 3.82/2.03 thf(func_def_37, type, ord_less_eq: !>[X0: $tType]:(X0 > X0 > $o)).
% 3.82/2.03 thf(func_def_38, type, is_zero: !>[X0: $tType]:(poly @ X0 > $o)).
% 3.82/2.03 thf(func_def_39, type, order2: !>[X0: $tType]:(X0 > poly @ X0 > nat)).
% 3.82/2.03 thf(func_def_40, type, pderiv: !>[X0: $tType]:(poly @ X0 > poly @ X0)).
% 3.82/2.03 thf(func_def_41, type, poly2: !>[X0: $tType]:(poly @ X0 > X0 > X0)).
% 3.82/2.03 thf(func_def_42, type, coeff: !>[X0: $tType]:(poly @ X0 > nat > X0)).
% 3.82/2.03 thf(func_def_43, type, poly_cutoff: !>[X0: $tType]:(nat > poly @ X0 > poly @ X0)).
% 3.82/2.03 thf(func_def_44, type, poly_shift: !>[X0: $tType]:(nat > poly @ X0 > poly @ X0)).
% 3.82/2.03 thf(func_def_45, type, reflect_poly: !>[X0: $tType]:(poly @ X0 > poly @ X0)).
% 3.82/2.03 thf(func_def_46, type, rsquarefree: !>[X0: $tType]:(poly @ X0 > $o)).
% 3.82/2.03 thf(func_def_47, type, collect: !>[X0: $tType]:((X0 > $o) > set @ X0)).
% 3.82/2.03 thf(func_def_48, type, sturm_424270202_cross: poly @ real > real > real > int).
% 3.82/2.03 thf(func_def_49, type, sturm_1700286437_r_pos: poly @ real > real > $o).
% 3.82/2.03 thf(func_def_50, type, sturm_1771227917iation: real > real > int).
% 3.82/2.03 thf(func_def_51, type, member: !>[X0: $tType]:(X0 > set @ X0 > $o)).
% 3.82/2.03 thf(func_def_52, type, p: poly @ real).
% 3.82/2.03 thf(func_def_53, type, x: real).
% 3.82/2.03 thf(func_def_68, type, sK0: !>[X0: $tType, X1: $tType]:((X0 > X1) > X0)).
% 3.82/2.03 thf(func_def_69, type, sK1: !>[X0: $tType, X1: $tType]:((X0 > X1) > X0)).
% 3.82/2.03 thf(func_def_70, type, sK2: (nat > $o) > nat).
% 3.82/2.03 thf(func_def_71, type, sK3: nat > (nat > $o) > nat).
% 3.82/2.03 thf(func_def_72, type, sK4: !>[X0: $tType]:((X0 > $o) > (X0 > nat) > X0)).
% 3.82/2.03 thf(func_def_73, type, sK5: !>[X0: $tType]:((X0 > $o) > (X0 > nat) > X0)).
% 3.82/2.03 thf(func_def_74, type, sK6: (nat > $o) > nat).
% 3.82/2.03 thf(func_def_75, type, sK7: !>[X0: $tType]:((X0 > nat) > (X0 > $o) > X0)).
% 3.82/2.03 thf(func_def_76, type, sK8: !>[X0: $tType, X1: $tType]:((X1 > X0) > X1)).
% 3.82/2.03 thf(func_def_77, type, sK9: !>[X0: $tType, X1: $tType]:((X1 > X0) > X1)).
% 3.82/2.03 thf(func_def_78, type, sK10: poly @ real > real > real > real).
% 3.82/2.03 thf(func_def_79, type, sK11: poly @ real > real > real > real).
% 3.82/2.03 thf(func_def_80, type, sK12: poly @ real > real > real > real).
% 3.82/2.03 thf(func_def_81, type, sK13: !>[X0: $tType, X1: $tType]:((X0 > X1) > X0)).
% 3.82/2.03 thf(func_def_82, type, sK14: !>[X0: $tType, X1: $tType]:((X0 > X1) > X0)).
% 3.82/2.03 thf(func_def_83, type, sK15: poly @ real > real > real > real).
% 3.82/2.03 thf(func_def_84, type, sK16: !>[X0: $tType, X1: $tType]:((X1 > X0) > X1)).
% 3.82/2.03 thf(func_def_85, type, sK17: !>[X0: $tType, X1: $tType]:((X1 > X0) > X1)).
% 3.82/2.03 thf(func_def_86, type, sK18: poly @ real > real > real > real).
% 3.82/2.03 thf(func_def_87, type, sK19: poly @ real > real > real > real).
% 3.82/2.03 thf(func_def_88, type, sK20: poly @ real > real > real > real).
% 3.82/2.03 thf(func_def_89, type, sK21: !>[X0: $tType, X1: $tType]:((X0 > X1) > X0)).
% 3.82/2.03 thf(func_def_90, type, sK22: !>[X0: $tType, X1: $tType]:((X0 > X1) > X0)).
% 3.82/2.03 thf(func_def_91, type, sK23: real > poly @ real > real).
% 3.82/2.03 thf(func_def_92, type, sK24: nat > (nat > $o) > nat).
% 3.82/2.03 thf(func_def_93, type, sK25: real > poly @ real > real).
% 3.82/2.03 thf(func_def_94, type, sK26: poly @ real > real > real > real).
% 3.82/2.03 thf(func_def_95, type, sK27: !>[X0: $tType, X1: $tType]:((X1 > X0) > X1)).
% 3.82/2.03 thf(func_def_96, type, sK28: !>[X0: $tType, X1: $tType]:((X1 > X0) > X1)).
% 3.82/2.03 thf(func_def_97, type, sK29: !>[X0: $tType, X1: $tType]:((X1 > X0) > X1)).
% 3.82/2.03 thf(func_def_98, type, sK30: !>[X0: $tType, X1: $tType]:((X1 > X0) > X1)).
% 3.82/2.03 thf(func_def_99, type, sK31: (nat > $o) > nat).
% 3.82/2.03 thf(func_def_100, type, sK32: !>[X0: $tType, X1: $tType]:((X1 > X0) > (X1 > X0) > X1)).
% 3.82/2.03 thf(func_def_101, type, sK33: (nat > nat) > nat).
% 3.82/2.03 thf(func_def_102, type, sK34: (nat > nat) > nat).
% 3.82/2.03 thf(func_def_103, type, sK35: !>[X0: $tType, X1: $tType]:((X0 > X1) > X0)).
% 3.82/2.03 thf(func_def_104, type, sK36: !>[X0: $tType, X1: $tType]:((X0 > X1) > X0)).
% 3.82/2.03 thf(func_def_105, type, sK37: (nat > $o) > nat).
% 3.82/2.03 thf(func_def_107, type, ph39: !>[X0: $tType]:(X0)).
% 3.82/2.03 thf(f1415,plain,(
% 3.82/2.03 $false),
% 3.82/2.03 inference(avatar_sat_refutation,[status(thm)],[f1411,f1414])).
% 3.82/2.03 thf(f1414,plain,(
% 3.82/2.03 ~spl38_1),
% 3.82/2.03 inference(avatar_split_clause,[status(thm)],[f1413,f1401])).
% 3.82/2.03 thf(f1401,definition,(
% 3.82/2.03 spl38_1 <=> ((ord_less @ real @ zero_zero @ real @ (poly2 @ real @ p @ x)) = $true)),
% 3.82/2.03 introduced(definition,[new_symbols(naming,[spl38_1])],[avatar_definition])).
% 3.82/2.03 thf(f1413,plain,(
% 3.82/2.03 ((ord_less @ real @ zero_zero @ real @ (poly2 @ real @ p @ x)) != $true)),
% 3.82/2.03 inference(subsumption_resolution,[status(thm)],[f1412,f1357])).
% 3.82/2.03 thf(f1357,plain,(
% 3.82/2.03 ((ord_less @ real @ zero_zero @ real @ (poly2 @ real @ p @ x)) != $true) | ((sturm_1700286437_r_pos @ p @ x) = $true)),
% 3.82/2.03 inference(cnf_transformation,[status(thm)],[f1133])).
% 3.82/2.03 thf(f1133,plain,(
% 3.82/2.03 ((ord_less @ real @ zero_zero @ real @ (poly2 @ real @ p @ x)) != $true) | ((sturm_1700286437_r_pos @ p @ x) = $true)),
% 3.82/2.03 inference(ennf_transformation,[status(thm)],[f692])).
% 3.82/2.03 thf(f692,plain,(
% 3.82/2.03 ((ord_less @ real @ zero_zero @ real @ (poly2 @ real @ p @ x)) = $true) => ((sturm_1700286437_r_pos @ p @ x) = $true)),
% 3.82/2.03 inference(fool_elimination,[status(thm)],[f691])).
% 3.82/2.03 thf(f691,plain,(
% 3.82/2.03 (ord_less @ real @ zero_zero @ real @ (poly2 @ real @ p @ x)) => (sturm_1700286437_r_pos @ p @ x)),
% 3.82/2.03 inference(rectify,[status(thm)],[f3])).
% 3.82/2.03 thf(f3,axiom,(
% 3.82/2.03 (ord_less @ real @ zero_zero @ real @ (poly2 @ real @ p @ x)) => (sturm_1700286437_r_pos @ p @ x)),
% 3.82/2.03 file('/export/starexec/sandbox2/tmp/tmp.t0hPoxVh8o/DTF2THF_31756.p',fact_2__092_060open_0620_A_060_Apoly_Ap_Ax_A_092_060Longrightarrow_062_Asign__r__pos_Ap_Ax_092_060close_062)).
% 3.82/2.03 thf(f1412,plain,(
% 3.82/2.03 ((sturm_1700286437_r_pos @ p @ x) != $true) | ((ord_less @ real @ zero_zero @ real @ (poly2 @ real @ p @ x)) != $true)),
% 3.82/2.03 inference(subsumption_resolution,[status(thm)],[f1212,f1267])).
% 3.82/2.03 thf(f1267,plain,(
% 3.82/2.03 ((poly2 @ real @ p @ x) != zero_zero @ real)),
% 3.82/2.03 inference(cnf_transformation,[status(thm)],[f1])).
% 3.82/2.03 thf(f1,axiom,(
% 3.82/2.03 ((poly2 @ real @ p @ x) != zero_zero @ real)),
% 3.82/2.03 file('/export/starexec/sandbox2/tmp/tmp.t0hPoxVh8o/DTF2THF_31756.p',fact_0_False)).
% 3.82/2.03 thf(f1212,plain,(
% 3.82/2.03 ((ord_less @ real @ zero_zero @ real @ (poly2 @ real @ p @ x)) != $true) | ((sturm_1700286437_r_pos @ p @ x) != $true) | ((poly2 @ real @ p @ x) = zero_zero @ real)),
% 3.82/2.03 inference(cnf_transformation,[status(thm)],[f1141])).
% 3.82/2.03 thf(f1141,plain,(
% 3.82/2.03 (((sturm_1700286437_r_pos @ p @ x) = $true) | ((((sturm_1700286437_r_pos @ (pderiv @ real @ p) @ x) = $true) | ((poly2 @ real @ p @ x) != zero_zero @ real)) & (((poly2 @ real @ p @ x) = zero_zero @ real) | ((ord_less @ real @ zero_zero @ real @ (poly2 @ real @ p @ x)) = $true)))) & ((((sturm_1700286437_r_pos @ (pderiv @ real @ p) @ x) != $true) & ((poly2 @ real @ p @ x) = zero_zero @ real)) | (((poly2 @ real @ p @ x) != zero_zero @ real) & ((ord_less @ real @ zero_zero @ real @ (poly2 @ real @ p @ x)) != $true)) | ((sturm_1700286437_r_pos @ p @ x) != $true))),
% 3.82/2.03 inference(flattening,[status(thm)],[f1140])).
% 3.82/2.03 thf(f1140,plain,(
% 3.82/2.03 (((sturm_1700286437_r_pos @ p @ x) = $true) | ((((sturm_1700286437_r_pos @ (pderiv @ real @ p) @ x) = $true) | ((poly2 @ real @ p @ x) != zero_zero @ real)) & (((poly2 @ real @ p @ x) = zero_zero @ real) | ((ord_less @ real @ zero_zero @ real @ (poly2 @ real @ p @ x)) = $true)))) & (((((sturm_1700286437_r_pos @ (pderiv @ real @ p) @ x) != $true) & ((poly2 @ real @ p @ x) = zero_zero @ real)) | (((poly2 @ real @ p @ x) != zero_zero @ real) & ((ord_less @ real @ zero_zero @ real @ (poly2 @ real @ p @ x)) != $true))) | ((sturm_1700286437_r_pos @ p @ x) != $true))),
% 3.82/2.03 inference(nnf_transformation,[status(thm)],[f1075])).
% 3.82/2.03 thf(f1075,plain,(
% 3.82/2.03 ((sturm_1700286437_r_pos @ p @ x) = $true) <=> ((((sturm_1700286437_r_pos @ (pderiv @ real @ p) @ x) != $true) & ((poly2 @ real @ p @ x) = zero_zero @ real)) | (((poly2 @ real @ p @ x) != zero_zero @ real) & ((ord_less @ real @ zero_zero @ real @ (poly2 @ real @ p @ x)) != $true)))),
% 3.82/2.03 inference(ennf_transformation,[status(thm)],[f952])).
% 3.82/2.03 thf(f952,plain,(
% 3.82/2.03 ~(((sturm_1700286437_r_pos @ p @ x) = $true) <~> ~((((poly2 @ real @ p @ x) = zero_zero @ real) => ((sturm_1700286437_r_pos @ (pderiv @ real @ p) @ x) = $true)) & (((poly2 @ real @ p @ x) != zero_zero @ real) => ((ord_less @ real @ zero_zero @ real @ (poly2 @ real @ p @ x)) = $true))))),
% 3.82/2.03 inference(fool_elimination,[status(thm)],[f951])).
% 3.82/2.03 thf(f951,plain,(
% 3.82/2.03 ~((sturm_1700286437_r_pos @ p @ x) != ~((((poly2 @ real @ p @ x) = zero_zero @ real) => (sturm_1700286437_r_pos @ (pderiv @ real @ p) @ x)) & (((poly2 @ real @ p @ x) != zero_zero @ real) => (ord_less @ real @ zero_zero @ real @ (poly2 @ real @ p @ x)))))),
% 3.82/2.03 inference(rectify,[status(thm)],[f341])).
% 3.82/2.03 thf(f341,negated_conjecture,(
% 3.82/2.03 ~((sturm_1700286437_r_pos @ p @ x) != ~((((poly2 @ real @ p @ x) = zero_zero @ real) => (sturm_1700286437_r_pos @ (pderiv @ real @ p) @ x)) & (((poly2 @ real @ p @ x) != zero_zero @ real) => (ord_less @ real @ zero_zero @ real @ (poly2 @ real @ p @ x)))))),
% 3.82/2.03 inference(negated_conjecture,[status(cth)],[f340])).
% 3.82/2.03 thf(f340,conjecture,(
% 3.82/2.03 ((sturm_1700286437_r_pos @ p @ x) != ~((((poly2 @ real @ p @ x) = zero_zero @ real) => (sturm_1700286437_r_pos @ (pderiv @ real @ p) @ x)) & (((poly2 @ real @ p @ x) != zero_zero @ real) => (ord_less @ real @ zero_zero @ real @ (poly2 @ real @ p @ x)))))),
% 3.82/2.03 file('/export/starexec/sandbox2/tmp/tmp.t0hPoxVh8o/DTF2THF_31756.p',conj_0)).
% 3.82/2.03 thf(f1411,plain,(
% 3.82/2.03 spl38_1),
% 3.82/2.03 inference(avatar_split_clause,[status(thm)],[f1410,f1401])).
% 3.82/2.03 thf(f1410,plain,(
% 3.82/2.03 ((ord_less @ real @ zero_zero @ real @ (poly2 @ real @ p @ x)) = $true)),
% 3.82/2.03 inference(subsumption_resolution,[status(thm)],[f1409,f1367])).
% 3.82/2.03 thf(f1367,plain,(
% 3.82/2.03 ((ord_less @ real @ zero_zero @ real @ (poly2 @ real @ p @ x)) = $true) | ((sturm_1700286437_r_pos @ p @ x) != $true)),
% 3.82/2.03 inference(cnf_transformation,[status(thm)],[f1062])).
% 3.82/2.03 thf(f1062,plain,(
% 3.82/2.03 ((sturm_1700286437_r_pos @ p @ x) != $true) | ((ord_less @ real @ zero_zero @ real @ (poly2 @ real @ p @ x)) = $true)),
% 3.82/2.03 inference(ennf_transformation,[status(thm)],[f796])).
% 3.82/2.03 thf(f796,plain,(
% 3.82/2.03 ((sturm_1700286437_r_pos @ p @ x) = $true) => ((ord_less @ real @ zero_zero @ real @ (poly2 @ real @ p @ x)) = $true)),
% 3.82/2.03 inference(fool_elimination,[status(thm)],[f795])).
% 3.82/2.03 thf(f795,plain,(
% 3.82/2.03 (sturm_1700286437_r_pos @ p @ x) => (ord_less @ real @ zero_zero @ real @ (poly2 @ real @ p @ x))),
% 3.82/2.03 inference(rectify,[status(thm)],[f2])).
% 3.82/2.03 thf(f2,axiom,(
% 3.82/2.03 (sturm_1700286437_r_pos @ p @ x) => (ord_less @ real @ zero_zero @ real @ (poly2 @ real @ p @ x))),
% 3.82/2.03 file('/export/starexec/sandbox2/tmp/tmp.t0hPoxVh8o/DTF2THF_31756.p',fact_1__092_060open_062sign__r__pos_Ap_Ax_A_092_060Longrightarrow_062_A0_A_060_Apoly_Ap_Ax_092_060close_062)).
% 3.82/2.03 thf(f1409,plain,(
% 3.82/2.03 ((ord_less @ real @ zero_zero @ real @ (poly2 @ real @ p @ x)) = $true) | ((sturm_1700286437_r_pos @ p @ x) = $true)),
% 3.82/2.03 inference(subsumption_resolution,[status(thm)],[f1216,f1267])).
% 3.82/2.03 thf(f1216,plain,(
% 3.82/2.03 ((poly2 @ real @ p @ x) = zero_zero @ real) | ((sturm_1700286437_r_pos @ p @ x) = $true) | ((ord_less @ real @ zero_zero @ real @ (poly2 @ real @ p @ x)) = $true)),
% 3.82/2.03 inference(cnf_transformation,[status(thm)],[f1141])).
% 3.82/2.03 % SZS output end Proof for DTF2THF_31756
% 3.82/2.03 % (31861)------------------------------
% 3.82/2.03 % (31861)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.82/2.03 % (31861)Termination reason: Refutation
% 3.82/2.03
% 3.82/2.03 % (31861)Memory used [KB]: 6780
% 3.82/2.03 % (31861)Time elapsed: 0.037 s
% 3.82/2.03 % (31861)Instructions burned: 63 (million)
% 3.82/2.03 % (31861)------------------------------
% 3.82/2.03 % (31861)------------------------------
% 3.82/2.03 % (31855)Success in time 0.058 s
%------------------------------------------------------------------------------