%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : ITP194^2 : 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 : n005.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Wed Feb 25 08:56:19 AM UTC 2026
% Result : Theorem 2.82s 1.65s
% Output : Refutation 2.82s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.11 % Problem : ITP194^2 : TPTP v9.2.1. Released v7.5.0.
% 0.00/0.13 % Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.16/0.33 % Computer : n005.cluster.edu
% 0.16/0.33 % Model : x86_64 x86_64
% 0.16/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.16/0.33 % Memory : 8042.1875MB
% 0.16/0.33 % OS : Linux 3.10.0-693.el7.x86_64
% 0.16/0.33 % CPULimit : 300
% 0.16/0.33 % WCLimit : 300
% 0.16/0.33 % DateTime : Tue Feb 24 08:29:29 EST 2026
% 0.16/0.33 % CPUTime :
% 0.16/0.34 Running /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.29/0.47 ---- Original DTF file ---
% 0.29/0.47 thf(spec,logic,$$dhol).
% 0.29/0.47 %------------------------------------------------------------------------------
% 0.29/0.47 % File : ITP194^2 : TPTP v9.2.1. Released v7.5.0.
% 0.29/0.47 % Domain : Interactive Theorem Proving
% 0.29/0.47 % Problem : Sledgehammer Sturm_Tarski problem prob_257__5870986_1
% 0.29/0.47 % Version : Especial.
% 0.29/0.47 % English :
% 0.29/0.47
% 0.29/0.47 % Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% 0.29/0.47 % : [Des21] Desharnais (2021), Email to Geoff Sutcliffe
% 0.29/0.47 % Source : [Des21]
% 0.29/0.47 % Names : Sturm_Tarski/prob_257__5870986_1 [Des21]
% 0.29/0.47
% 0.29/0.47 % Status : Theorem
% 0.29/0.47 % Rating : 0.33 v9.1.0, 0.00 v7.5.0
% 0.29/0.47 % Syntax : Number of formulae : 371 ( 77 unt; 53 typ; 0 def)
% 0.29/0.47 % Number of atoms : 994 ( 224 equ; 0 cnn)
% 0.29/0.47 % Maximal formula atoms : 13 ( 3 avg)
% 0.29/0.47 % Number of connectives : 3143 ( 120 ~; 29 |; 65 &;2398 @)
% 0.29/0.47 % ( 0 <=>; 531 =>; 0 <=; 0 <~>)
% 0.29/0.47 % Maximal formula depth : 21 ( 8 avg)
% 0.29/0.47 % Number of types : 3 ( 2 usr)
% 0.29/0.47 % Number of type conns : 114 ( 114 >; 0 *; 0 +; 0 <<)
% 0.29/0.47 % Number of symbols : 54 ( 51 usr; 7 con; 0-7 aty)
% 0.29/0.47 % Number of variables : 978 ( 53 ^; 834 !; 47 ?; 978 :)
% 0.29/0.47 % ( 44 !>; 0 ?*; 0 @-; 0 @+)
% 0.29/0.47 % SPC : TH1_THM_EQU_NAR
% 0.29/0.47
% 0.29/0.47 % Comments : This file was generated by Sledgehammer 2021-02-23 16:26:51.513
% 0.29/0.47 %------------------------------------------------------------------------------
% 0.29/0.47 % Could-be-implicit typings (4)
% 0.29/0.47 thf(ty_t_Polynomial_Opoly,type,
% 0.29/0.47 poly: $tType > $tType ).
% 0.29/0.47
% 0.29/0.47 thf(ty_t_Real_Oreal,type,
% 0.29/0.47 real: $tType ).
% 0.29/0.47
% 0.29/0.47 thf(ty_t_Set_Oset,type,
% 0.29/0.47 set: $tType > $tType ).
% 0.29/0.47
% 0.29/0.47 thf(ty_t_Nat_Onat,type,
% 0.29/0.47 nat: $tType ).
% 0.29/0.47
% 0.29/0.47 % Explicit typings (49)
% 0.29/0.47 thf(sy_cl_HOL_Otype,type,
% 0.29/0.47 type:
% 0.29/0.47 !>[A: $tType] : $o ).
% 0.29/0.47
% 0.29/0.47 thf(sy_cl_Rings_Oidom,type,
% 0.29/0.47 idom:
% 0.29/0.47 !>[A: $tType] : $o ).
% 0.29/0.47
% 0.29/0.47 thf(sy_cl_Groups_Ozero,type,
% 0.29/0.47 zero:
% 0.29/0.47 !>[A: $tType] : $o ).
% 0.29/0.47
% 0.29/0.47 thf(sy_cl_Orderings_Oord,type,
% 0.29/0.47 ord:
% 0.29/0.47 !>[A: $tType] : $o ).
% 0.29/0.47
% 0.29/0.47 thf(sy_cl_Nat_Oring__char__0,type,
% 0.29/0.47 ring_char_0:
% 0.29/0.47 !>[A: $tType] : $o ).
% 0.29/0.47
% 0.29/0.47 thf(sy_cl_Orderings_Oorder,type,
% 0.29/0.47 order:
% 0.29/0.47 !>[A: $tType] : $o ).
% 0.29/0.47
% 0.29/0.47 thf(sy_cl_Orderings_Ono__bot,type,
% 0.29/0.47 no_bot:
% 0.29/0.47 !>[A: $tType] : $o ).
% 0.29/0.47
% 0.29/0.47 thf(sy_cl_Orderings_Ono__top,type,
% 0.29/0.47 no_top:
% 0.29/0.47 !>[A: $tType] : $o ).
% 0.29/0.47
% 0.29/0.47 thf(sy_cl_Rings_Ocomm__ring__1,type,
% 0.29/0.47 comm_ring_1:
% 0.29/0.47 !>[A: $tType] : $o ).
% 0.29/0.47
% 0.29/0.47 thf(sy_cl_Rings_Oidom__divide,type,
% 0.29/0.47 idom_divide:
% 0.29/0.47 !>[A: $tType] : $o ).
% 0.29/0.47
% 0.29/0.47 thf(sy_cl_Orderings_Olinorder,type,
% 0.29/0.47 linorder:
% 0.29/0.47 !>[A: $tType] : $o ).
% 0.29/0.47
% 0.29/0.47 thf(sy_cl_Orderings_Opreorder,type,
% 0.29/0.47 preorder:
% 0.29/0.47 !>[A: $tType] : $o ).
% 0.29/0.47
% 0.29/0.47 thf(sy_cl_Rings_Oidom__abs__sgn,type,
% 0.29/0.47 idom_abs_sgn:
% 0.29/0.47 !>[A: $tType] : $o ).
% 0.29/0.47
% 0.29/0.47 thf(sy_cl_Orderings_Owellorder,type,
% 0.29/0.47 wellorder:
% 0.29/0.47 !>[A: $tType] : $o ).
% 0.29/0.47
% 0.29/0.47 thf(sy_cl_Orderings_Odense__order,type,
% 0.29/0.47 dense_order:
% 0.29/0.47 !>[A: $tType] : $o ).
% 0.29/0.47
% 0.29/0.47 thf(sy_cl_Rings_Ocomm__semiring__0,type,
% 0.29/0.47 comm_semiring_0:
% 0.29/0.47 !>[A: $tType] : $o ).
% 0.29/0.47
% 0.29/0.47 thf(sy_cl_Rings_Ocomm__semiring__1,type,
% 0.29/0.47 comm_semiring_1:
% 0.29/0.47 !>[A: $tType] : $o ).
% 0.29/0.47
% 0.29/0.47 thf(sy_cl_Rings_Olinordered__idom,type,
% 0.29/0.47 linordered_idom:
% 0.29/0.47 !>[A: $tType] : $o ).
% 0.29/0.47
% 0.29/0.47 thf(sy_cl_Fields_Olinordered__field,type,
% 0.29/0.47 linordered_field:
% 0.29/0.47 !>[A: $tType] : $o ).
% 0.29/0.47
% 0.29/0.47 thf(sy_cl_Orderings_Odense__linorder,type,
% 0.29/0.47 dense_linorder:
% 0.29/0.47 !>[A: $tType] : $o ).
% 0.29/0.47
% 0.29/0.47 thf(sy_cl_Rings_Oring__no__zero__divisors,type,
% 0.29/0.47 ring_n68954251visors:
% 0.29/0.47 !>[A: $tType] : $o ).
% 0.29/0.47
% 0.29/0.47 thf(sy_cl_Rings_Olinordered__nonzero__semiring,type,
% 0.29/0.47 linord1659791738miring:
% 0.29/0.47 !>[A: $tType] : $o ).
% 0.29/0.47
% 0.29/0.47 thf(sy_cl_Groups_Ocanonically__ordered__monoid__add,type,
% 0.29/0.47 canoni770627133id_add:
% 0.29/0.47 !>[A: $tType] : $o ).
% 0.29/0.47
% 0.29/0.47 thf(sy_cl_Real__Vector__Spaces_Oreal__normed__vector,type,
% 0.29/0.47 real_V55928688vector:
% 0.29/0.47 !>[A: $tType] : $o ).
% 0.29/0.47
% 0.29/0.47 thf(sy_cl_Conditionally__Complete__Lattices_Olinear__continuum,type,
% 0.29/0.47 condit1656338222tinuum:
% 0.29/0.47 !>[A: $tType] : $o ).
% 0.29/0.47
% 0.29/0.47 thf(sy_cl_Conditionally__Complete__Lattices_Oconditionally__complete__linorder,type,
% 0.29/0.47 condit1037483654norder:
% 0.29/0.47 !>[A: $tType] : $o ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_Groups_Oone__class_Oone,type,
% 0.29/0.47 one_one:
% 0.29/0.47 !>[A: $tType] : A ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_Groups_Osgn__class_Osgn,type,
% 0.29/0.47 sgn_sgn:
% 0.29/0.47 !>[A: $tType] : ( A > A ) ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_Groups_Ozero__class_Ozero,type,
% 0.29/0.47 zero_zero:
% 0.29/0.47 !>[A: $tType] : A ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_If,type,
% 0.29/0.47 if:
% 0.29/0.47 !>[A: $tType] : ( $o > A > A > A ) ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_Orderings_Oord__class_Oless,type,
% 0.29/0.47 ord_less:
% 0.29/0.47 !>[A: $tType] : ( A > A > $o ) ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_Orderings_Oord__class_Oless__eq,type,
% 0.29/0.47 ord_less_eq:
% 0.29/0.47 !>[A: $tType] : ( A > A > $o ) ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_Orderings_Oord__class_Omin,type,
% 0.29/0.47 ord_min:
% 0.29/0.47 !>[A: $tType] : ( A > A > A ) ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_Polynomial_Odivide__poly__main,type,
% 0.29/0.47 divide_poly_main:
% 0.29/0.47 !>[A: $tType] : ( A > ( poly @ A ) > ( poly @ A ) > ( poly @ A ) > nat > nat > ( poly @ A ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_Polynomial_Ois__zero,type,
% 0.29/0.47 is_zero:
% 0.29/0.47 !>[A: $tType] : ( ( poly @ A ) > $o ) ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_Polynomial_Oorder,type,
% 0.29/0.47 order2:
% 0.29/0.47 !>[A: $tType] : ( A > ( poly @ A ) > nat ) ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_Polynomial_Opoly,type,
% 0.29/0.47 poly2:
% 0.29/0.47 !>[A: $tType] : ( ( poly @ A ) > A > A ) ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_Polynomial_Opoly__cutoff,type,
% 0.29/0.47 poly_cutoff:
% 0.29/0.47 !>[A: $tType] : ( nat > ( poly @ A ) > ( poly @ A ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_Polynomial_Opoly__shift,type,
% 0.29/0.47 poly_shift:
% 0.29/0.47 !>[A: $tType] : ( nat > ( poly @ A ) > ( poly @ A ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_Polynomial_Oreflect__poly,type,
% 0.29/0.47 reflect_poly:
% 0.29/0.47 !>[A: $tType] : ( ( poly @ A ) > ( poly @ A ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_Set_OCollect,type,
% 0.29/0.47 collect:
% 0.29/0.47 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_Sturm__Tarski__Mirabelle__osyjafhhev_Osgn__neg__inf,type,
% 0.29/0.47 sturm_1439882693eg_inf:
% 0.29/0.47 !>[A: $tType] : ( ( poly @ A ) > A ) ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_Sturm__Tarski__Mirabelle__osyjafhhev_Osgn__pos__inf,type,
% 0.29/0.47 sturm_62398785os_inf:
% 0.29/0.47 !>[A: $tType] : ( ( poly @ A ) > A ) ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_member,type,
% 0.29/0.47 member:
% 0.29/0.47 !>[A: $tType] : ( A > ( set @ A ) > $o ) ).
% 0.29/0.47
% 0.29/0.47 thf(sy_v_lb1____,type,
% 0.29/0.47 lb1: real ).
% 0.29/0.47
% 0.29/0.47 thf(sy_v_lb2____,type,
% 0.29/0.47 lb2: real ).
% 0.29/0.47
% 0.29/0.47 thf(sy_v_lb____,type,
% 0.29/0.47 lb: real ).
% 0.29/0.47
% 0.29/0.47 thf(sy_v_p,type,
% 0.29/0.47 p: poly @ real ).
% 0.29/0.47
% 0.29/0.47 thf(sy_v_thesis,type,
% 0.29/0.47 thesis: $o ).
% 0.29/0.47
% 0.29/0.47 % Relevant facts (254)
% 0.29/0.47 thf(fact_0_lb1,axiom,
% 0.29/0.47 ! [X: real] :
% 0.29/0.47 ( ( ( poly2 @ real @ p @ X )
% 0.29/0.47 = ( zero_zero @ real ) )
% 0.29/0.47 => ( ord_less @ real @ lb1 @ X ) ) ).
% 0.29/0.47
% 0.29/0.47 % lb1
% 0.29/0.47 thf(fact_1_lb__def,axiom,
% 0.29/0.47 ( lb
% 0.29/0.47 = ( ord_min @ real @ lb1 @ lb2 ) ) ).
% 0.29/0.47
% 0.29/0.47 % lb_def
% 0.29/0.47 thf(fact_2__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062lb1_O_A_092_060forall_062x_O_Apoly_Ap_Ax_A_061_A0_A_092_060longrightarrow_062_Alb1_A_060_Ax_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
% 0.29/0.47 ~ ! [Lb1: real] :
% 0.29/0.47 ~ ! [X: real] :
% 0.29/0.47 ( ( ( poly2 @ real @ p @ X )
% 0.29/0.47 = ( zero_zero @ real ) )
% 0.29/0.47 => ( ord_less @ real @ Lb1 @ X ) ) ).
% 0.29/0.47
% 0.29/0.47 % \<open>\<And>thesis. (\<And>lb1. \<forall>x. poly p x = 0 \<longrightarrow> lb1 < x \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
% 0.29/0.47 thf(fact_3_assms,axiom,
% 0.29/0.47 ( p
% 0.29/0.47 != ( zero_zero @ ( poly @ real ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % assms
% 0.29/0.47 thf(fact_4_poly__0,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( comm_semiring_0 @ A )
% 0.29/0.47 => ! [X2: A] :
% 0.29/0.47 ( ( poly2 @ A @ ( zero_zero @ ( poly @ A ) ) @ X2 )
% 0.29/0.47 = ( zero_zero @ A ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % poly_0
% 0.29/0.47 thf(fact_5_not__gr__zero,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( canoni770627133id_add @ A )
% 0.29/0.47 => ! [N: A] :
% 0.29/0.47 ( ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ N ) )
% 0.29/0.47 = ( N
% 0.29/0.47 = ( zero_zero @ A ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % not_gr_zero
% 0.29/0.47 thf(fact_6_poly__IVT__neg,axiom,
% 0.29/0.47 ! [A2: real,B: real,P: poly @ real] :
% 0.29/0.47 ( ( ord_less @ real @ A2 @ B )
% 0.29/0.47 => ( ( ord_less @ real @ ( zero_zero @ real ) @ ( poly2 @ real @ P @ A2 ) )
% 0.29/0.47 => ( ( ord_less @ real @ ( poly2 @ real @ P @ B ) @ ( zero_zero @ real ) )
% 0.29/0.47 => ? [X3: real] :
% 0.29/0.47 ( ( ord_less @ real @ A2 @ X3 )
% 0.29/0.47 & ( ord_less @ real @ X3 @ B )
% 0.29/0.47 & ( ( poly2 @ real @ P @ X3 )
% 0.29/0.47 = ( zero_zero @ real ) ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % poly_IVT_neg
% 0.29/0.47 thf(fact_7_poly__IVT__pos,axiom,
% 0.29/0.47 ! [A2: real,B: real,P: poly @ real] :
% 0.29/0.47 ( ( ord_less @ real @ A2 @ B )
% 0.29/0.47 => ( ( ord_less @ real @ ( poly2 @ real @ P @ A2 ) @ ( zero_zero @ real ) )
% 0.29/0.47 => ( ( ord_less @ real @ ( zero_zero @ real ) @ ( poly2 @ real @ P @ B ) )
% 0.29/0.47 => ? [X3: real] :
% 0.29/0.47 ( ( ord_less @ real @ A2 @ X3 )
% 0.29/0.47 & ( ord_less @ real @ X3 @ B )
% 0.29/0.47 & ( ( poly2 @ real @ P @ X3 )
% 0.29/0.47 = ( zero_zero @ real ) ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % poly_IVT_pos
% 0.29/0.47 thf(fact_8_poly__all__0__iff__0,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( ( ring_char_0 @ A )
% 0.29/0.47 & ( comm_ring_1 @ A )
% 0.29/0.47 & ( ring_n68954251visors @ A ) )
% 0.29/0.47 => ! [P: poly @ A] :
% 0.29/0.47 ( ( ! [X4: A] :
% 0.29/0.47 ( ( poly2 @ A @ P @ X4 )
% 0.29/0.47 = ( zero_zero @ A ) ) )
% 0.29/0.47 = ( P
% 0.29/0.47 = ( zero_zero @ ( poly @ A ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % poly_all_0_iff_0
% 0.29/0.47 thf(fact_9_less__numeral__extra_I3_J,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( linord1659791738miring @ A )
% 0.29/0.47 => ~ ( ord_less @ A @ ( zero_zero @ A ) @ ( zero_zero @ A ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % less_numeral_extra(3)
% 0.29/0.47 thf(fact_10_field__lbound__gt__zero,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( linordered_field @ A )
% 0.29/0.47 => ! [D1: A,D2: A] :
% 0.29/0.47 ( ( ord_less @ A @ ( zero_zero @ A ) @ D1 )
% 0.29/0.47 => ( ( ord_less @ A @ ( zero_zero @ A ) @ D2 )
% 0.29/0.47 => ? [E: A] :
% 0.29/0.47 ( ( ord_less @ A @ ( zero_zero @ A ) @ E )
% 0.29/0.47 & ( ord_less @ A @ E @ D1 )
% 0.29/0.47 & ( ord_less @ A @ E @ D2 ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % field_lbound_gt_zero
% 0.29/0.47 thf(fact_11_gr__zeroI,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( canoni770627133id_add @ A )
% 0.29/0.47 => ! [N: A] :
% 0.29/0.47 ( ( N
% 0.29/0.47 != ( zero_zero @ A ) )
% 0.29/0.47 => ( ord_less @ A @ ( zero_zero @ A ) @ N ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % gr_zeroI
% 0.29/0.47 thf(fact_12_not__less__zero,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( canoni770627133id_add @ A )
% 0.29/0.47 => ! [N: A] :
% 0.29/0.47 ~ ( ord_less @ A @ N @ ( zero_zero @ A ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % not_less_zero
% 0.29/0.47 thf(fact_13_gr__implies__not__zero,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( canoni770627133id_add @ A )
% 0.29/0.47 => ! [M: A,N: A] :
% 0.29/0.47 ( ( ord_less @ A @ M @ N )
% 0.29/0.47 => ( N
% 0.29/0.47 != ( zero_zero @ A ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % gr_implies_not_zero
% 0.29/0.47 thf(fact_14_zero__less__iff__neq__zero,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( canoni770627133id_add @ A )
% 0.29/0.47 => ! [N: A] :
% 0.29/0.47 ( ( ord_less @ A @ ( zero_zero @ A ) @ N )
% 0.29/0.47 = ( N
% 0.29/0.47 != ( zero_zero @ A ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % zero_less_iff_neq_zero
% 0.29/0.47 thf(fact_15_zero__reorient,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( zero @ A )
% 0.29/0.47 => ! [X2: A] :
% 0.29/0.47 ( ( ( zero_zero @ A )
% 0.29/0.47 = X2 )
% 0.29/0.47 = ( X2
% 0.29/0.47 = ( zero_zero @ A ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % zero_reorient
% 0.29/0.47 thf(fact_16_poly__eq__poly__eq__iff,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( ( ring_char_0 @ A )
% 0.29/0.47 & ( comm_ring_1 @ A )
% 0.29/0.47 & ( ring_n68954251visors @ A ) )
% 0.29/0.47 => ! [P: poly @ A,Q: poly @ A] :
% 0.29/0.47 ( ( ( poly2 @ A @ P )
% 0.29/0.47 = ( poly2 @ A @ Q ) )
% 0.29/0.47 = ( P = Q ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % poly_eq_poly_eq_iff
% 0.29/0.47 thf(fact_17_min__less__iff__conj,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( linorder @ A )
% 0.29/0.47 => ! [Z: A,X2: A,Y: A] :
% 0.29/0.47 ( ( ord_less @ A @ Z @ ( ord_min @ A @ X2 @ Y ) )
% 0.29/0.47 = ( ( ord_less @ A @ Z @ X2 )
% 0.29/0.47 & ( ord_less @ A @ Z @ Y ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % min_less_iff_conj
% 0.29/0.47 thf(fact_18_min_Oidem,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( linorder @ A )
% 0.29/0.47 => ! [A2: A] :
% 0.29/0.47 ( ( ord_min @ A @ A2 @ A2 )
% 0.29/0.47 = A2 ) ) ).
% 0.29/0.47
% 0.29/0.47 % min.idem
% 0.29/0.47 thf(fact_19_min_Oleft__idem,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( linorder @ A )
% 0.29/0.47 => ! [A2: A,B: A] :
% 0.29/0.47 ( ( ord_min @ A @ A2 @ ( ord_min @ A @ A2 @ B ) )
% 0.29/0.47 = ( ord_min @ A @ A2 @ B ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % min.left_idem
% 0.29/0.47 thf(fact_20_min_Oright__idem,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( linorder @ A )
% 0.29/0.47 => ! [A2: A,B: A] :
% 0.29/0.47 ( ( ord_min @ A @ ( ord_min @ A @ A2 @ B ) @ B )
% 0.29/0.47 = ( ord_min @ A @ A2 @ B ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % min.right_idem
% 0.29/0.47 thf(fact_21_min_Ostrict__coboundedI2,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( linorder @ A )
% 0.29/0.47 => ! [B: A,C: A,A2: A] :
% 0.29/0.47 ( ( ord_less @ A @ B @ C )
% 0.29/0.47 => ( ord_less @ A @ ( ord_min @ A @ A2 @ B ) @ C ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % min.strict_coboundedI2
% 0.29/0.47 thf(fact_22_min_Ostrict__coboundedI1,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( linorder @ A )
% 0.29/0.47 => ! [A2: A,C: A,B: A] :
% 0.29/0.47 ( ( ord_less @ A @ A2 @ C )
% 0.29/0.47 => ( ord_less @ A @ ( ord_min @ A @ A2 @ B ) @ C ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % min.strict_coboundedI1
% 0.29/0.47 thf(fact_23_min_Ostrict__order__iff,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( linorder @ A )
% 0.29/0.47 => ( ( ord_less @ A )
% 0.29/0.47 = ( ^ [A3: A,B2: A] :
% 0.29/0.47 ( ( A3
% 0.29/0.47 = ( ord_min @ A @ A3 @ B2 ) )
% 0.29/0.47 & ( A3 != B2 ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % min.strict_order_iff
% 0.29/0.47 thf(fact_24_min_Ostrict__boundedE,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( linorder @ A )
% 0.29/0.47 => ! [A2: A,B: A,C: A] :
% 0.29/0.47 ( ( ord_less @ A @ A2 @ ( ord_min @ A @ B @ C ) )
% 0.29/0.47 => ~ ( ( ord_less @ A @ A2 @ B )
% 0.29/0.47 => ~ ( ord_less @ A @ A2 @ C ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % min.strict_boundedE
% 0.29/0.47 thf(fact_25_min__less__iff__disj,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( linorder @ A )
% 0.29/0.47 => ! [X2: A,Y: A,Z: A] :
% 0.29/0.47 ( ( ord_less @ A @ ( ord_min @ A @ X2 @ Y ) @ Z )
% 0.29/0.47 = ( ( ord_less @ A @ X2 @ Z )
% 0.29/0.47 | ( ord_less @ A @ Y @ Z ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % min_less_iff_disj
% 0.29/0.47 thf(fact_26_is__zero__null,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( zero @ A )
% 0.29/0.47 => ( ( is_zero @ A )
% 0.29/0.47 = ( ^ [P2: poly @ A] :
% 0.29/0.47 ( P2
% 0.29/0.47 = ( zero_zero @ ( poly @ A ) ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % is_zero_null
% 0.29/0.47 thf(fact_27_min_Oleft__commute,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( linorder @ A )
% 0.29/0.47 => ! [B: A,A2: A,C: A] :
% 0.29/0.47 ( ( ord_min @ A @ B @ ( ord_min @ A @ A2 @ C ) )
% 0.29/0.47 = ( ord_min @ A @ A2 @ ( ord_min @ A @ B @ C ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % min.left_commute
% 0.29/0.47 thf(fact_28_min_Ocommute,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( linorder @ A )
% 0.29/0.47 => ( ( ord_min @ A )
% 0.29/0.47 = ( ^ [A3: A,B2: A] : ( ord_min @ A @ B2 @ A3 ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % min.commute
% 0.29/0.47 thf(fact_29_min_Oassoc,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( linorder @ A )
% 0.29/0.47 => ! [A2: A,B: A,C: A] :
% 0.29/0.47 ( ( ord_min @ A @ ( ord_min @ A @ A2 @ B ) @ C )
% 0.29/0.47 = ( ord_min @ A @ A2 @ ( ord_min @ A @ B @ C ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % min.assoc
% 0.29/0.47 thf(fact_30_poly__cutoff__0,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( zero @ A )
% 0.29/0.47 => ! [N: nat] :
% 0.29/0.47 ( ( poly_cutoff @ A @ N @ ( zero_zero @ ( poly @ A ) ) )
% 0.29/0.47 = ( zero_zero @ ( poly @ A ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % poly_cutoff_0
% 0.29/0.47 thf(fact_31_last__non__root__interval,axiom,
% 0.29/0.47 ! [P: poly @ real,Ub: real] :
% 0.29/0.47 ( ( P
% 0.29/0.47 != ( zero_zero @ ( poly @ real ) ) )
% 0.29/0.47 => ~ ! [Lb: real] :
% 0.29/0.47 ( ( ord_less @ real @ Lb @ Ub )
% 0.29/0.47 => ~ ! [Z2: real] :
% 0.29/0.47 ( ( ( ord_less_eq @ real @ Lb @ Z2 )
% 0.29/0.47 & ( ord_less @ real @ Z2 @ Ub ) )
% 0.29/0.47 => ( ( poly2 @ real @ P @ Z2 )
% 0.29/0.47 != ( zero_zero @ real ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % last_non_root_interval
% 0.29/0.47 thf(fact_32_next__non__root__interval,axiom,
% 0.29/0.47 ! [P: poly @ real,Lb2: real] :
% 0.29/0.47 ( ( P
% 0.29/0.47 != ( zero_zero @ ( poly @ real ) ) )
% 0.29/0.47 => ~ ! [Ub2: real] :
% 0.29/0.47 ( ( ord_less @ real @ Lb2 @ Ub2 )
% 0.29/0.47 => ~ ! [Z2: real] :
% 0.29/0.47 ( ( ( ord_less @ real @ Lb2 @ Z2 )
% 0.29/0.47 & ( ord_less_eq @ real @ Z2 @ Ub2 ) )
% 0.29/0.47 => ( ( poly2 @ real @ P @ Z2 )
% 0.29/0.47 != ( zero_zero @ real ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % next_non_root_interval
% 0.29/0.47 thf(fact_33_reflect__poly__at__0__eq__0__iff,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( comm_semiring_0 @ A )
% 0.29/0.47 => ! [P: poly @ A] :
% 0.29/0.47 ( ( ( poly2 @ A @ ( reflect_poly @ A @ P ) @ ( zero_zero @ A ) )
% 0.29/0.47 = ( zero_zero @ A ) )
% 0.29/0.47 = ( P
% 0.29/0.47 = ( zero_zero @ ( poly @ A ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % reflect_poly_at_0_eq_0_iff
% 0.29/0.47 thf(fact_34_lb2,axiom,
% 0.29/0.47 ! [X: real] :
% 0.29/0.47 ( ( ord_less_eq @ real @ X @ lb2 )
% 0.29/0.47 => ( ( sgn_sgn @ real @ ( poly2 @ real @ p @ X ) )
% 0.29/0.47 = ( sturm_1439882693eg_inf @ real @ p ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % lb2
% 0.29/0.47 thf(fact_35_poly__shift__0,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( zero @ A )
% 0.29/0.47 => ! [N: nat] :
% 0.29/0.47 ( ( poly_shift @ A @ N @ ( zero_zero @ ( poly @ A ) ) )
% 0.29/0.47 = ( zero_zero @ ( poly @ A ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % poly_shift_0
% 0.29/0.47 thf(fact_36_order__root,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( idom @ A )
% 0.29/0.47 => ! [P: poly @ A,A2: A] :
% 0.29/0.47 ( ( ( poly2 @ A @ P @ A2 )
% 0.29/0.47 = ( zero_zero @ A ) )
% 0.29/0.47 = ( ( P
% 0.29/0.47 = ( zero_zero @ ( poly @ A ) ) )
% 0.29/0.47 | ( ( order2 @ A @ A2 @ P )
% 0.29/0.47 != ( zero_zero @ nat ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % order_root
% 0.29/0.47 thf(fact_37_divide__poly__main__0,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( idom_divide @ A )
% 0.29/0.47 => ! [R: poly @ A,D: poly @ A,Dr: nat,N: nat] :
% 0.29/0.47 ( ( divide_poly_main @ A @ ( zero_zero @ A ) @ ( zero_zero @ ( poly @ A ) ) @ R @ D @ Dr @ N )
% 0.29/0.47 = ( zero_zero @ ( poly @ A ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % divide_poly_main_0
% 0.29/0.47 thf(fact_38_not__eq__pos__or__neg__iff__1,axiom,
% 0.29/0.47 ! [Lb2: real,Ub: real,P: poly @ real] :
% 0.29/0.47 ( ( ! [Z3: real] :
% 0.29/0.47 ( ( ( ord_less @ real @ Lb2 @ Z3 )
% 0.29/0.47 & ( ord_less_eq @ real @ Z3 @ Ub ) )
% 0.29/0.47 => ( ( poly2 @ real @ P @ Z3 )
% 0.29/0.47 != ( zero_zero @ real ) ) ) )
% 0.29/0.47 = ( ! [Z3: real] :
% 0.29/0.47 ( ( ( ord_less @ real @ Lb2 @ Z3 )
% 0.29/0.47 & ( ord_less_eq @ real @ Z3 @ Ub ) )
% 0.29/0.47 => ( ord_less @ real @ ( zero_zero @ real ) @ ( poly2 @ real @ P @ Z3 ) ) )
% 0.29/0.47 | ! [Z3: real] :
% 0.29/0.47 ( ( ( ord_less @ real @ Lb2 @ Z3 )
% 0.29/0.47 & ( ord_less_eq @ real @ Z3 @ Ub ) )
% 0.29/0.47 => ( ord_less @ real @ ( poly2 @ real @ P @ Z3 ) @ ( zero_zero @ real ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % not_eq_pos_or_neg_iff_1
% 0.29/0.47 thf(fact_39__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062lb2_O_A_092_060forall_062x_092_060le_062lb2_O_Asgn_A_Ipoly_Ap_Ax_J_A_061_Asgn__neg__inf_Ap_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
% 0.29/0.47 ~ ! [Lb22: real] :
% 0.29/0.47 ~ ! [X: real] :
% 0.29/0.47 ( ( ord_less_eq @ real @ X @ Lb22 )
% 0.29/0.47 => ( ( sgn_sgn @ real @ ( poly2 @ real @ p @ X ) )
% 0.29/0.47 = ( sturm_1439882693eg_inf @ real @ p ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % \<open>\<And>thesis. (\<And>lb2. \<forall>x\<le>lb2. sgn (poly p x) = sgn_neg_inf p \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
% 0.29/0.47 thf(fact_40_le__zero__eq,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( canoni770627133id_add @ A )
% 0.29/0.47 => ! [N: A] :
% 0.29/0.47 ( ( ord_less_eq @ A @ N @ ( zero_zero @ A ) )
% 0.29/0.47 = ( N
% 0.29/0.47 = ( zero_zero @ A ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % le_zero_eq
% 0.29/0.47 thf(fact_41_min_Obounded__iff,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( linorder @ A )
% 0.29/0.47 => ! [A2: A,B: A,C: A] :
% 0.29/0.47 ( ( ord_less_eq @ A @ A2 @ ( ord_min @ A @ B @ C ) )
% 0.29/0.47 = ( ( ord_less_eq @ A @ A2 @ B )
% 0.29/0.47 & ( ord_less_eq @ A @ A2 @ C ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % min.bounded_iff
% 0.29/0.47 thf(fact_42_reflect__poly__0,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( zero @ A )
% 0.29/0.47 => ( ( reflect_poly @ A @ ( zero_zero @ ( poly @ A ) ) )
% 0.29/0.47 = ( zero_zero @ ( poly @ A ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % reflect_poly_0
% 0.29/0.47 thf(fact_43_poly__shift__id,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( zero @ A )
% 0.29/0.47 => ( ( poly_shift @ A @ ( zero_zero @ nat ) )
% 0.29/0.47 = ( ^ [X4: poly @ A] : X4 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % poly_shift_id
% 0.29/0.47 thf(fact_44_mem__Collect__eq,axiom,
% 0.29/0.47 ! [A: $tType,A2: A,P3: A > $o] :
% 0.29/0.47 ( ( member @ A @ A2 @ ( collect @ A @ P3 ) )
% 0.29/0.47 = ( P3 @ A2 ) ) ).
% 0.29/0.47
% 0.29/0.47 % mem_Collect_eq
% 0.29/0.47 thf(fact_45_Collect__mem__eq,axiom,
% 0.29/0.47 ! [A: $tType,A4: set @ A] :
% 0.29/0.47 ( ( collect @ A
% 0.29/0.47 @ ^ [X4: A] : ( member @ A @ X4 @ A4 ) )
% 0.29/0.47 = A4 ) ).
% 0.29/0.47
% 0.29/0.47 % Collect_mem_eq
% 0.29/0.47 thf(fact_46_Collect__cong,axiom,
% 0.29/0.47 ! [A: $tType,P3: A > $o,Q2: A > $o] :
% 0.29/0.47 ( ! [X3: A] :
% 0.29/0.47 ( ( P3 @ X3 )
% 0.29/0.47 = ( Q2 @ X3 ) )
% 0.29/0.47 => ( ( collect @ A @ P3 )
% 0.29/0.47 = ( collect @ A @ Q2 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % Collect_cong
% 0.29/0.47 thf(fact_47_ext,axiom,
% 0.29/0.47 ! [B3: $tType,A: $tType,F: A > B3,G: A > B3] :
% 0.29/0.47 ( ! [X3: A] :
% 0.29/0.47 ( ( F @ X3 )
% 0.29/0.47 = ( G @ X3 ) )
% 0.29/0.47 => ( F = G ) ) ).
% 0.29/0.47
% 0.29/0.47 % ext
% 0.29/0.47 thf(fact_48_that,axiom,
% 0.29/0.47 ! [Lb2: real] :
% 0.29/0.47 ( ! [X3: real] :
% 0.29/0.47 ( ( ( poly2 @ real @ p @ X3 )
% 0.29/0.47 = ( zero_zero @ real ) )
% 0.29/0.47 => ( ord_less @ real @ Lb2 @ X3 ) )
% 0.29/0.47 => ( ! [X3: real] :
% 0.29/0.47 ( ( ord_less_eq @ real @ X3 @ Lb2 )
% 0.29/0.47 => ( ( sgn_sgn @ real @ ( poly2 @ real @ p @ X3 ) )
% 0.29/0.47 = ( sturm_1439882693eg_inf @ real @ p ) ) )
% 0.29/0.47 => thesis ) ) ).
% 0.29/0.47
% 0.29/0.47 % that
% 0.29/0.47 thf(fact_49_complete__real,axiom,
% 0.29/0.47 ! [S: set @ real] :
% 0.29/0.47 ( ? [X: real] : ( member @ real @ X @ S )
% 0.29/0.47 => ( ? [Z2: real] :
% 0.29/0.47 ! [X3: real] :
% 0.29/0.47 ( ( member @ real @ X3 @ S )
% 0.29/0.47 => ( ord_less_eq @ real @ X3 @ Z2 ) )
% 0.29/0.47 => ? [Y2: real] :
% 0.29/0.47 ( ! [X: real] :
% 0.29/0.47 ( ( member @ real @ X @ S )
% 0.29/0.47 => ( ord_less_eq @ real @ X @ Y2 ) )
% 0.29/0.47 & ! [Z2: real] :
% 0.29/0.47 ( ! [X3: real] :
% 0.29/0.47 ( ( member @ real @ X3 @ S )
% 0.29/0.47 => ( ord_less_eq @ real @ X3 @ Z2 ) )
% 0.29/0.47 => ( ord_less_eq @ real @ Y2 @ Z2 ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % complete_real
% 0.29/0.47 thf(fact_50_divide__poly__main_Osimps_I2_J,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( idom_divide @ A )
% 0.29/0.47 => ! [Lc: A,Q: poly @ A,R: poly @ A,D: poly @ A,Dr: nat] :
% 0.29/0.47 ( ( divide_poly_main @ A @ Lc @ Q @ R @ D @ Dr @ ( zero_zero @ nat ) )
% 0.29/0.47 = Q ) ) ).
% 0.29/0.47
% 0.29/0.47 % divide_poly_main.simps(2)
% 0.29/0.47 thf(fact_51_zero__le,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( canoni770627133id_add @ A )
% 0.29/0.47 => ! [X2: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ X2 ) ) ).
% 0.29/0.47
% 0.29/0.47 % zero_le
% 0.29/0.47 thf(fact_52_le__numeral__extra_I3_J,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( linord1659791738miring @ A )
% 0.29/0.47 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( zero_zero @ A ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % le_numeral_extra(3)
% 0.29/0.47 thf(fact_53_less__eq__real__def,axiom,
% 0.29/0.47 ( ( ord_less_eq @ real )
% 0.29/0.47 = ( ^ [X4: real,Y3: real] :
% 0.29/0.47 ( ( ord_less @ real @ X4 @ Y3 )
% 0.29/0.47 | ( X4 = Y3 ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % less_eq_real_def
% 0.29/0.47 thf(fact_54_min_Omono,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( linorder @ A )
% 0.29/0.47 => ! [A2: A,C: A,B: A,D: A] :
% 0.29/0.47 ( ( ord_less_eq @ A @ A2 @ C )
% 0.29/0.47 => ( ( ord_less_eq @ A @ B @ D )
% 0.29/0.47 => ( ord_less_eq @ A @ ( ord_min @ A @ A2 @ B ) @ ( ord_min @ A @ C @ D ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % min.mono
% 0.29/0.47 thf(fact_55_min_OorderE,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( linorder @ A )
% 0.29/0.47 => ! [A2: A,B: A] :
% 0.29/0.47 ( ( ord_less_eq @ A @ A2 @ B )
% 0.29/0.47 => ( A2
% 0.29/0.47 = ( ord_min @ A @ A2 @ B ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % min.orderE
% 0.29/0.47 thf(fact_56_min_OorderI,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( linorder @ A )
% 0.29/0.47 => ! [A2: A,B: A] :
% 0.29/0.47 ( ( A2
% 0.29/0.47 = ( ord_min @ A @ A2 @ B ) )
% 0.29/0.47 => ( ord_less_eq @ A @ A2 @ B ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % min.orderI
% 0.29/0.47 thf(fact_57_min_Oabsorb1,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( linorder @ A )
% 0.29/0.47 => ! [A2: A,B: A] :
% 0.29/0.47 ( ( ord_less_eq @ A @ A2 @ B )
% 0.29/0.47 => ( ( ord_min @ A @ A2 @ B )
% 0.29/0.47 = A2 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % min.absorb1
% 0.29/0.47 thf(fact_58_min_Oabsorb2,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( linorder @ A )
% 0.29/0.47 => ! [B: A,A2: A] :
% 0.29/0.47 ( ( ord_less_eq @ A @ B @ A2 )
% 0.29/0.47 => ( ( ord_min @ A @ A2 @ B )
% 0.29/0.47 = B ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % min.absorb2
% 0.29/0.47 thf(fact_59_min_OboundedE,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( linorder @ A )
% 0.29/0.47 => ! [A2: A,B: A,C: A] :
% 0.29/0.47 ( ( ord_less_eq @ A @ A2 @ ( ord_min @ A @ B @ C ) )
% 0.29/0.47 => ~ ( ( ord_less_eq @ A @ A2 @ B )
% 0.29/0.47 => ~ ( ord_less_eq @ A @ A2 @ C ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % min.boundedE
% 0.29/0.47 thf(fact_60_min_OboundedI,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( linorder @ A )
% 0.29/0.47 => ! [A2: A,B: A,C: A] :
% 0.29/0.47 ( ( ord_less_eq @ A @ A2 @ B )
% 0.29/0.47 => ( ( ord_less_eq @ A @ A2 @ C )
% 0.29/0.47 => ( ord_less_eq @ A @ A2 @ ( ord_min @ A @ B @ C ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % min.boundedI
% 0.29/0.47 thf(fact_61_min_Oorder__iff,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( linorder @ A )
% 0.29/0.47 => ( ( ord_less_eq @ A )
% 0.29/0.47 = ( ^ [A3: A,B2: A] :
% 0.29/0.47 ( A3
% 0.29/0.47 = ( ord_min @ A @ A3 @ B2 ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % min.order_iff
% 0.29/0.47 thf(fact_62_min_Ocobounded1,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( linorder @ A )
% 0.29/0.47 => ! [A2: A,B: A] : ( ord_less_eq @ A @ ( ord_min @ A @ A2 @ B ) @ A2 ) ) ).
% 0.29/0.47
% 0.29/0.47 % min.cobounded1
% 0.29/0.47 thf(fact_63_min_Ocobounded2,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( linorder @ A )
% 0.29/0.47 => ! [A2: A,B: A] : ( ord_less_eq @ A @ ( ord_min @ A @ A2 @ B ) @ B ) ) ).
% 0.29/0.47
% 0.29/0.47 % min.cobounded2
% 0.29/0.47 thf(fact_64_min_Oabsorb__iff1,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( linorder @ A )
% 0.29/0.47 => ( ( ord_less_eq @ A )
% 0.29/0.47 = ( ^ [A3: A,B2: A] :
% 0.29/0.47 ( ( ord_min @ A @ A3 @ B2 )
% 0.29/0.47 = A3 ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % min.absorb_iff1
% 0.29/0.47 thf(fact_65_min_Oabsorb__iff2,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( linorder @ A )
% 0.29/0.47 => ( ( ord_less_eq @ A )
% 0.29/0.47 = ( ^ [B2: A,A3: A] :
% 0.29/0.47 ( ( ord_min @ A @ A3 @ B2 )
% 0.29/0.47 = B2 ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % min.absorb_iff2
% 0.29/0.47 thf(fact_66_min_OcoboundedI1,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( linorder @ A )
% 0.29/0.47 => ! [A2: A,C: A,B: A] :
% 0.29/0.47 ( ( ord_less_eq @ A @ A2 @ C )
% 0.29/0.47 => ( ord_less_eq @ A @ ( ord_min @ A @ A2 @ B ) @ C ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % min.coboundedI1
% 0.29/0.47 thf(fact_67_min_OcoboundedI2,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( linorder @ A )
% 0.29/0.47 => ! [B: A,C: A,A2: A] :
% 0.29/0.47 ( ( ord_less_eq @ A @ B @ C )
% 0.29/0.47 => ( ord_less_eq @ A @ ( ord_min @ A @ A2 @ B ) @ C ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % min.coboundedI2
% 0.29/0.47 thf(fact_68_min__le__iff__disj,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( linorder @ A )
% 0.29/0.47 => ! [X2: A,Y: A,Z: A] :
% 0.29/0.47 ( ( ord_less_eq @ A @ ( ord_min @ A @ X2 @ Y ) @ Z )
% 0.29/0.47 = ( ( ord_less_eq @ A @ X2 @ Z )
% 0.29/0.47 | ( ord_less_eq @ A @ Y @ Z ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % min_le_iff_disj
% 0.29/0.47 thf(fact_69_order__0I,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( idom @ A )
% 0.29/0.47 => ! [P: poly @ A,A2: A] :
% 0.29/0.47 ( ( ( poly2 @ A @ P @ A2 )
% 0.29/0.47 != ( zero_zero @ A ) )
% 0.29/0.47 => ( ( order2 @ A @ A2 @ P )
% 0.29/0.47 = ( zero_zero @ nat ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % order_0I
% 0.29/0.47 thf(fact_70_not__eq__pos__or__neg__iff__2,axiom,
% 0.29/0.47 ! [Lb2: real,Ub: real,P: poly @ real] :
% 0.29/0.47 ( ( ! [Z3: real] :
% 0.29/0.47 ( ( ( ord_less_eq @ real @ Lb2 @ Z3 )
% 0.29/0.47 & ( ord_less @ real @ Z3 @ Ub ) )
% 0.29/0.47 => ( ( poly2 @ real @ P @ Z3 )
% 0.29/0.47 != ( zero_zero @ real ) ) ) )
% 0.29/0.47 = ( ! [Z3: real] :
% 0.29/0.47 ( ( ( ord_less_eq @ real @ Lb2 @ Z3 )
% 0.29/0.47 & ( ord_less @ real @ Z3 @ Ub ) )
% 0.29/0.47 => ( ord_less @ real @ ( zero_zero @ real ) @ ( poly2 @ real @ P @ Z3 ) ) )
% 0.29/0.47 | ! [Z3: real] :
% 0.29/0.47 ( ( ( ord_less_eq @ real @ Lb2 @ Z3 )
% 0.29/0.47 & ( ord_less @ real @ Z3 @ Ub ) )
% 0.29/0.47 => ( ord_less @ real @ ( poly2 @ real @ P @ Z3 ) @ ( zero_zero @ real ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % not_eq_pos_or_neg_iff_2
% 0.29/0.47 thf(fact_71_zero__le__sgn__iff,axiom,
% 0.29/0.47 ! [X2: real] :
% 0.29/0.47 ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( sgn_sgn @ real @ X2 ) )
% 0.29/0.47 = ( ord_less_eq @ real @ ( zero_zero @ real ) @ X2 ) ) ).
% 0.29/0.47
% 0.29/0.47 % zero_le_sgn_iff
% 0.29/0.47 thf(fact_72_sgn__le__0__iff,axiom,
% 0.29/0.47 ! [X2: real] :
% 0.29/0.47 ( ( ord_less_eq @ real @ ( sgn_sgn @ real @ X2 ) @ ( zero_zero @ real ) )
% 0.29/0.47 = ( ord_less_eq @ real @ X2 @ ( zero_zero @ real ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % sgn_le_0_iff
% 0.29/0.47 thf(fact_73_sgn__greater,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( linordered_idom @ A )
% 0.29/0.47 => ! [A2: A] :
% 0.29/0.47 ( ( ord_less @ A @ ( zero_zero @ A ) @ ( sgn_sgn @ A @ A2 ) )
% 0.29/0.47 = ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % sgn_greater
% 0.29/0.47 thf(fact_74_sgn__less,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( linordered_idom @ A )
% 0.29/0.47 => ! [A2: A] :
% 0.29/0.47 ( ( ord_less @ A @ ( sgn_sgn @ A @ A2 ) @ ( zero_zero @ A ) )
% 0.29/0.47 = ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % sgn_less
% 0.29/0.47 thf(fact_75_root__ub,axiom,
% 0.29/0.47 ! [P: poly @ real] :
% 0.29/0.47 ( ( P
% 0.29/0.47 != ( zero_zero @ ( poly @ real ) ) )
% 0.29/0.47 => ~ ! [Ub2: real] :
% 0.29/0.47 ( ! [X: real] :
% 0.29/0.47 ( ( ( poly2 @ real @ P @ X )
% 0.29/0.47 = ( zero_zero @ real ) )
% 0.29/0.47 => ( ord_less @ real @ X @ Ub2 ) )
% 0.29/0.47 => ~ ! [X: real] :
% 0.29/0.47 ( ( ord_less_eq @ real @ Ub2 @ X )
% 0.29/0.47 => ( ( sgn_sgn @ real @ ( poly2 @ real @ P @ X ) )
% 0.29/0.47 = ( sturm_62398785os_inf @ real @ P ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % root_ub
% 0.29/0.47 thf(fact_76_sgn__0,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( idom_abs_sgn @ A )
% 0.29/0.47 => ( ( sgn_sgn @ A @ ( zero_zero @ A ) )
% 0.29/0.47 = ( zero_zero @ A ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % sgn_0
% 0.29/0.47 thf(fact_77_sgn__zero,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( real_V55928688vector @ A )
% 0.29/0.47 => ( ( sgn_sgn @ A @ ( zero_zero @ A ) )
% 0.29/0.47 = ( zero_zero @ A ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % sgn_zero
% 0.29/0.47 thf(fact_78_sgn__sgn,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( idom_abs_sgn @ A )
% 0.29/0.47 => ! [A2: A] :
% 0.29/0.47 ( ( sgn_sgn @ A @ ( sgn_sgn @ A @ A2 ) )
% 0.29/0.47 = ( sgn_sgn @ A @ A2 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % sgn_sgn
% 0.29/0.47 thf(fact_79_linorder__neqE__linordered__idom,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( linordered_idom @ A )
% 0.29/0.47 => ! [X2: A,Y: A] :
% 0.29/0.47 ( ( X2 != Y )
% 0.29/0.47 => ( ~ ( ord_less @ A @ X2 @ Y )
% 0.29/0.47 => ( ord_less @ A @ Y @ X2 ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % linorder_neqE_linordered_idom
% 0.29/0.47 thf(fact_80_sgn__eq__0__iff,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( idom_abs_sgn @ A )
% 0.29/0.47 => ! [A2: A] :
% 0.29/0.47 ( ( ( sgn_sgn @ A @ A2 )
% 0.29/0.47 = ( zero_zero @ A ) )
% 0.29/0.47 = ( A2
% 0.29/0.47 = ( zero_zero @ A ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % sgn_eq_0_iff
% 0.29/0.47 thf(fact_81_sgn__0__0,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( linordered_idom @ A )
% 0.29/0.47 => ! [A2: A] :
% 0.29/0.47 ( ( ( sgn_sgn @ A @ A2 )
% 0.29/0.47 = ( zero_zero @ A ) )
% 0.29/0.47 = ( A2
% 0.29/0.47 = ( zero_zero @ A ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % sgn_0_0
% 0.29/0.47 thf(fact_82_sgn__zero__iff,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( real_V55928688vector @ A )
% 0.29/0.47 => ! [X2: A] :
% 0.29/0.47 ( ( ( sgn_sgn @ A @ X2 )
% 0.29/0.47 = ( zero_zero @ A ) )
% 0.29/0.47 = ( X2
% 0.29/0.47 = ( zero_zero @ A ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % sgn_zero_iff
% 0.29/0.47 thf(fact_83_le0,axiom,
% 0.29/0.47 ! [N: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ N ) ).
% 0.29/0.47
% 0.29/0.47 % le0
% 0.29/0.47 thf(fact_84_bot__nat__0_Oextremum,axiom,
% 0.29/0.47 ! [A2: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ A2 ) ).
% 0.29/0.47
% 0.29/0.47 % bot_nat_0.extremum
% 0.29/0.47 thf(fact_85_neq0__conv,axiom,
% 0.29/0.47 ! [N: nat] :
% 0.29/0.47 ( ( N
% 0.29/0.47 != ( zero_zero @ nat ) )
% 0.29/0.47 = ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ).
% 0.29/0.47
% 0.29/0.47 % neq0_conv
% 0.29/0.47 thf(fact_86_less__nat__zero__code,axiom,
% 0.29/0.47 ! [N: nat] :
% 0.29/0.47 ~ ( ord_less @ nat @ N @ ( zero_zero @ nat ) ) ).
% 0.29/0.47
% 0.29/0.47 % less_nat_zero_code
% 0.29/0.47 thf(fact_87_bot__nat__0_Onot__eq__extremum,axiom,
% 0.29/0.47 ! [A2: nat] :
% 0.29/0.47 ( ( A2
% 0.29/0.47 != ( zero_zero @ nat ) )
% 0.29/0.47 = ( ord_less @ nat @ ( zero_zero @ nat ) @ A2 ) ) ).
% 0.29/0.47
% 0.29/0.47 % bot_nat_0.not_eq_extremum
% 0.29/0.47 thf(fact_88_min__0R,axiom,
% 0.29/0.47 ! [N: nat] :
% 0.29/0.47 ( ( ord_min @ nat @ N @ ( zero_zero @ nat ) )
% 0.29/0.47 = ( zero_zero @ nat ) ) ).
% 0.29/0.47
% 0.29/0.47 % min_0R
% 0.29/0.47 thf(fact_89_min__0L,axiom,
% 0.29/0.47 ! [N: nat] :
% 0.29/0.47 ( ( ord_min @ nat @ ( zero_zero @ nat ) @ N )
% 0.29/0.47 = ( zero_zero @ nat ) ) ).
% 0.29/0.47
% 0.29/0.47 % min_0L
% 0.29/0.47 thf(fact_90_le__refl,axiom,
% 0.29/0.47 ! [N: nat] : ( ord_less_eq @ nat @ N @ N ) ).
% 0.29/0.47
% 0.29/0.47 % le_refl
% 0.29/0.47 thf(fact_91_le__trans,axiom,
% 0.29/0.47 ! [I: nat,J: nat,K: nat] :
% 0.29/0.47 ( ( ord_less_eq @ nat @ I @ J )
% 0.29/0.47 => ( ( ord_less_eq @ nat @ J @ K )
% 0.29/0.47 => ( ord_less_eq @ nat @ I @ K ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % le_trans
% 0.29/0.47 thf(fact_92_eq__imp__le,axiom,
% 0.29/0.47 ! [M: nat,N: nat] :
% 0.29/0.47 ( ( M = N )
% 0.29/0.47 => ( ord_less_eq @ nat @ M @ N ) ) ).
% 0.29/0.47
% 0.29/0.47 % eq_imp_le
% 0.29/0.47 thf(fact_93_le__antisym,axiom,
% 0.29/0.47 ! [M: nat,N: nat] :
% 0.29/0.47 ( ( ord_less_eq @ nat @ M @ N )
% 0.29/0.47 => ( ( ord_less_eq @ nat @ N @ M )
% 0.29/0.47 => ( M = N ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % le_antisym
% 0.29/0.47 thf(fact_94_nat__le__linear,axiom,
% 0.29/0.47 ! [M: nat,N: nat] :
% 0.29/0.47 ( ( ord_less_eq @ nat @ M @ N )
% 0.29/0.47 | ( ord_less_eq @ nat @ N @ M ) ) ).
% 0.29/0.47
% 0.29/0.47 % nat_le_linear
% 0.29/0.47 thf(fact_95_Nat_Oex__has__greatest__nat,axiom,
% 0.29/0.47 ! [P3: nat > $o,K: nat,B: nat] :
% 0.29/0.47 ( ( P3 @ K )
% 0.29/0.47 => ( ! [Y2: nat] :
% 0.29/0.47 ( ( P3 @ Y2 )
% 0.29/0.47 => ( ord_less_eq @ nat @ Y2 @ B ) )
% 0.29/0.47 => ? [X3: nat] :
% 0.29/0.47 ( ( P3 @ X3 )
% 0.29/0.47 & ! [Y4: nat] :
% 0.29/0.47 ( ( P3 @ Y4 )
% 0.29/0.47 => ( ord_less_eq @ nat @ Y4 @ X3 ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % Nat.ex_has_greatest_nat
% 0.29/0.47 thf(fact_96_less__mono__imp__le__mono,axiom,
% 0.29/0.47 ! [F: nat > nat,I: nat,J: nat] :
% 0.29/0.47 ( ! [I2: nat,J2: nat] :
% 0.29/0.47 ( ( ord_less @ nat @ I2 @ J2 )
% 0.29/0.47 => ( ord_less @ nat @ ( F @ I2 ) @ ( F @ J2 ) ) )
% 0.29/0.47 => ( ( ord_less_eq @ nat @ I @ J )
% 0.29/0.47 => ( ord_less_eq @ nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % less_mono_imp_le_mono
% 0.29/0.47 thf(fact_97_le__neq__implies__less,axiom,
% 0.29/0.47 ! [M: nat,N: nat] :
% 0.29/0.47 ( ( ord_less_eq @ nat @ M @ N )
% 0.29/0.47 => ( ( M != N )
% 0.29/0.47 => ( ord_less @ nat @ M @ N ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % le_neq_implies_less
% 0.29/0.47 thf(fact_98_less__or__eq__imp__le,axiom,
% 0.29/0.47 ! [M: nat,N: nat] :
% 0.29/0.47 ( ( ( ord_less @ nat @ M @ N )
% 0.29/0.47 | ( M = N ) )
% 0.29/0.47 => ( ord_less_eq @ nat @ M @ N ) ) ).
% 0.29/0.47
% 0.29/0.47 % less_or_eq_imp_le
% 0.29/0.47 thf(fact_99_le__eq__less__or__eq,axiom,
% 0.29/0.47 ( ( ord_less_eq @ nat )
% 0.29/0.47 = ( ^ [M2: nat,N2: nat] :
% 0.29/0.47 ( ( ord_less @ nat @ M2 @ N2 )
% 0.29/0.47 | ( M2 = N2 ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % le_eq_less_or_eq
% 0.29/0.47 thf(fact_100_less__imp__le__nat,axiom,
% 0.29/0.47 ! [M: nat,N: nat] :
% 0.29/0.47 ( ( ord_less @ nat @ M @ N )
% 0.29/0.47 => ( ord_less_eq @ nat @ M @ N ) ) ).
% 0.29/0.47
% 0.29/0.47 % less_imp_le_nat
% 0.29/0.47 thf(fact_101_nat__less__le,axiom,
% 0.29/0.47 ( ( ord_less @ nat )
% 0.29/0.47 = ( ^ [M2: nat,N2: nat] :
% 0.29/0.47 ( ( ord_less_eq @ nat @ M2 @ N2 )
% 0.29/0.47 & ( M2 != N2 ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % nat_less_le
% 0.29/0.47 thf(fact_102_infinite__descent__measure,axiom,
% 0.29/0.47 ! [A: $tType,P3: A > $o,V: A > nat,X2: A] :
% 0.29/0.47 ( ! [X3: A] :
% 0.29/0.47 ( ~ ( P3 @ X3 )
% 0.29/0.47 => ? [Y4: A] :
% 0.29/0.47 ( ( ord_less @ nat @ ( V @ Y4 ) @ ( V @ X3 ) )
% 0.29/0.47 & ~ ( P3 @ Y4 ) ) )
% 0.29/0.47 => ( P3 @ X2 ) ) ).
% 0.29/0.47
% 0.29/0.47 % infinite_descent_measure
% 0.29/0.47 thf(fact_103_linorder__neqE__nat,axiom,
% 0.29/0.47 ! [X2: nat,Y: nat] :
% 0.29/0.47 ( ( X2 != Y )
% 0.29/0.47 => ( ~ ( ord_less @ nat @ X2 @ Y )
% 0.29/0.47 => ( ord_less @ nat @ Y @ X2 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % linorder_neqE_nat
% 0.29/0.47 thf(fact_104_infinite__descent,axiom,
% 0.29/0.47 ! [P3: nat > $o,N: nat] :
% 0.29/0.47 ( ! [N3: nat] :
% 0.29/0.47 ( ~ ( P3 @ N3 )
% 0.29/0.47 => ? [M3: nat] :
% 0.29/0.47 ( ( ord_less @ nat @ M3 @ N3 )
% 0.29/0.47 & ~ ( P3 @ M3 ) ) )
% 0.29/0.47 => ( P3 @ N ) ) ).
% 0.29/0.47
% 0.29/0.47 % infinite_descent
% 0.29/0.47 thf(fact_105_nat__less__induct,axiom,
% 0.29/0.47 ! [P3: nat > $o,N: nat] :
% 0.29/0.47 ( ! [N3: nat] :
% 0.29/0.47 ( ! [M3: nat] :
% 0.29/0.47 ( ( ord_less @ nat @ M3 @ N3 )
% 0.29/0.47 => ( P3 @ M3 ) )
% 0.29/0.47 => ( P3 @ N3 ) )
% 0.29/0.47 => ( P3 @ N ) ) ).
% 0.29/0.47
% 0.29/0.47 % nat_less_induct
% 0.29/0.47 thf(fact_106_less__irrefl__nat,axiom,
% 0.29/0.47 ! [N: nat] :
% 0.29/0.47 ~ ( ord_less @ nat @ N @ N ) ).
% 0.29/0.47
% 0.29/0.47 % less_irrefl_nat
% 0.29/0.47 thf(fact_107_less__not__refl3,axiom,
% 0.29/0.47 ! [S2: nat,T: nat] :
% 0.29/0.47 ( ( ord_less @ nat @ S2 @ T )
% 0.29/0.47 => ( S2 != T ) ) ).
% 0.29/0.47
% 0.29/0.47 % less_not_refl3
% 0.29/0.47 thf(fact_108_less__not__refl2,axiom,
% 0.29/0.47 ! [N: nat,M: nat] :
% 0.29/0.47 ( ( ord_less @ nat @ N @ M )
% 0.29/0.47 => ( M != N ) ) ).
% 0.29/0.47
% 0.29/0.47 % less_not_refl2
% 0.29/0.47 thf(fact_109_less__not__refl,axiom,
% 0.29/0.47 ! [N: nat] :
% 0.29/0.47 ~ ( ord_less @ nat @ N @ N ) ).
% 0.29/0.47
% 0.29/0.47 % less_not_refl
% 0.29/0.47 thf(fact_110_nat__neq__iff,axiom,
% 0.29/0.47 ! [M: nat,N: nat] :
% 0.29/0.47 ( ( M != N )
% 0.29/0.47 = ( ( ord_less @ nat @ M @ N )
% 0.29/0.47 | ( ord_less @ nat @ N @ M ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % nat_neq_iff
% 0.29/0.47 thf(fact_111_measure__induct__rule,axiom,
% 0.29/0.47 ! [B3: $tType,A: $tType] :
% 0.29/0.47 ( ( wellorder @ B3 )
% 0.29/0.47 => ! [F: A > B3,P3: A > $o,A2: A] :
% 0.29/0.47 ( ! [X3: A] :
% 0.29/0.47 ( ! [Y4: A] :
% 0.29/0.47 ( ( ord_less @ B3 @ ( F @ Y4 ) @ ( F @ X3 ) )
% 0.29/0.47 => ( P3 @ Y4 ) )
% 0.29/0.47 => ( P3 @ X3 ) )
% 0.29/0.47 => ( P3 @ A2 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % measure_induct_rule
% 0.29/0.47 thf(fact_112_measure__induct,axiom,
% 0.29/0.47 ! [B3: $tType,A: $tType] :
% 0.29/0.47 ( ( wellorder @ B3 )
% 0.29/0.47 => ! [F: A > B3,P3: A > $o,A2: A] :
% 0.29/0.47 ( ! [X3: A] :
% 0.29/0.47 ( ! [Y4: A] :
% 0.29/0.47 ( ( ord_less @ B3 @ ( F @ Y4 ) @ ( F @ X3 ) )
% 0.29/0.47 => ( P3 @ Y4 ) )
% 0.29/0.47 => ( P3 @ X3 ) )
% 0.29/0.47 => ( P3 @ A2 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % measure_induct
% 0.29/0.47 thf(fact_113_infinite__descent0__measure,axiom,
% 0.29/0.47 ! [A: $tType,V: A > nat,P3: A > $o,X2: A] :
% 0.29/0.47 ( ! [X3: A] :
% 0.29/0.47 ( ( ( V @ X3 )
% 0.29/0.47 = ( zero_zero @ nat ) )
% 0.29/0.47 => ( P3 @ X3 ) )
% 0.29/0.47 => ( ! [X3: A] :
% 0.29/0.47 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( V @ X3 ) )
% 0.29/0.47 => ( ~ ( P3 @ X3 )
% 0.29/0.47 => ? [Y4: A] :
% 0.29/0.47 ( ( ord_less @ nat @ ( V @ Y4 ) @ ( V @ X3 ) )
% 0.29/0.47 & ~ ( P3 @ Y4 ) ) ) )
% 0.29/0.47 => ( P3 @ X2 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % infinite_descent0_measure
% 0.29/0.47 thf(fact_114_bot__nat__0_Oextremum__strict,axiom,
% 0.29/0.47 ! [A2: nat] :
% 0.29/0.47 ~ ( ord_less @ nat @ A2 @ ( zero_zero @ nat ) ) ).
% 0.29/0.47
% 0.29/0.47 % bot_nat_0.extremum_strict
% 0.29/0.47 thf(fact_115_infinite__descent0,axiom,
% 0.29/0.47 ! [P3: nat > $o,N: nat] :
% 0.29/0.47 ( ( P3 @ ( zero_zero @ nat ) )
% 0.29/0.47 => ( ! [N3: nat] :
% 0.29/0.47 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 )
% 0.29/0.47 => ( ~ ( P3 @ N3 )
% 0.29/0.47 => ? [M3: nat] :
% 0.29/0.47 ( ( ord_less @ nat @ M3 @ N3 )
% 0.29/0.47 & ~ ( P3 @ M3 ) ) ) )
% 0.29/0.47 => ( P3 @ N ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % infinite_descent0
% 0.29/0.47 thf(fact_116_gr__implies__not0,axiom,
% 0.29/0.47 ! [M: nat,N: nat] :
% 0.29/0.47 ( ( ord_less @ nat @ M @ N )
% 0.29/0.47 => ( N
% 0.29/0.47 != ( zero_zero @ nat ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % gr_implies_not0
% 0.29/0.47 thf(fact_117_less__zeroE,axiom,
% 0.29/0.47 ! [N: nat] :
% 0.29/0.47 ~ ( ord_less @ nat @ N @ ( zero_zero @ nat ) ) ).
% 0.29/0.47
% 0.29/0.47 % less_zeroE
% 0.29/0.47 thf(fact_118_not__less0,axiom,
% 0.29/0.47 ! [N: nat] :
% 0.29/0.47 ~ ( ord_less @ nat @ N @ ( zero_zero @ nat ) ) ).
% 0.29/0.47
% 0.29/0.47 % not_less0
% 0.29/0.47 thf(fact_119_not__gr0,axiom,
% 0.29/0.47 ! [N: nat] :
% 0.29/0.47 ( ( ~ ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) )
% 0.29/0.47 = ( N
% 0.29/0.47 = ( zero_zero @ nat ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % not_gr0
% 0.29/0.47 thf(fact_120_gr0I,axiom,
% 0.29/0.47 ! [N: nat] :
% 0.29/0.47 ( ( N
% 0.29/0.47 != ( zero_zero @ nat ) )
% 0.29/0.47 => ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ).
% 0.29/0.47
% 0.29/0.47 % gr0I
% 0.29/0.47 thf(fact_121_less__eq__nat_Osimps_I1_J,axiom,
% 0.29/0.47 ! [N: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ N ) ).
% 0.29/0.47
% 0.29/0.47 % less_eq_nat.simps(1)
% 0.29/0.47 thf(fact_122_le__0__eq,axiom,
% 0.29/0.47 ! [N: nat] :
% 0.29/0.47 ( ( ord_less_eq @ nat @ N @ ( zero_zero @ nat ) )
% 0.29/0.47 = ( N
% 0.29/0.47 = ( zero_zero @ nat ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % le_0_eq
% 0.29/0.47 thf(fact_123_ex__least__nat__le,axiom,
% 0.29/0.47 ! [P3: nat > $o,N: nat] :
% 0.29/0.47 ( ( P3 @ N )
% 0.29/0.47 => ( ~ ( P3 @ ( zero_zero @ nat ) )
% 0.29/0.47 => ? [K2: nat] :
% 0.29/0.47 ( ( ord_less_eq @ nat @ K2 @ N )
% 0.29/0.47 & ! [I3: nat] :
% 0.29/0.47 ( ( ord_less @ nat @ I3 @ K2 )
% 0.29/0.47 => ~ ( P3 @ I3 ) )
% 0.29/0.47 & ( P3 @ K2 ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % ex_least_nat_le
% 0.29/0.47 thf(fact_124_bot__nat__0_Oextremum__unique,axiom,
% 0.29/0.47 ! [A2: nat] :
% 0.29/0.47 ( ( ord_less_eq @ nat @ A2 @ ( zero_zero @ nat ) )
% 0.29/0.47 = ( A2
% 0.29/0.47 = ( zero_zero @ nat ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % bot_nat_0.extremum_unique
% 0.29/0.47 thf(fact_125_bot__nat__0_Oextremum__uniqueI,axiom,
% 0.29/0.47 ! [A2: nat] :
% 0.29/0.47 ( ( ord_less_eq @ nat @ A2 @ ( zero_zero @ nat ) )
% 0.29/0.47 => ( A2
% 0.29/0.47 = ( zero_zero @ nat ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % bot_nat_0.extremum_uniqueI
% 0.29/0.47 thf(fact_126_order__refl,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( preorder @ A )
% 0.29/0.47 => ! [X2: A] : ( ord_less_eq @ A @ X2 @ X2 ) ) ).
% 0.29/0.47
% 0.29/0.47 % order_refl
% 0.29/0.47 thf(fact_127_min__absorb1,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( ord @ A )
% 0.29/0.47 => ! [X2: A,Y: A] :
% 0.29/0.47 ( ( ord_less_eq @ A @ X2 @ Y )
% 0.29/0.47 => ( ( ord_min @ A @ X2 @ Y )
% 0.29/0.47 = X2 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % min_absorb1
% 0.29/0.47 thf(fact_128_min__absorb2,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( order @ A )
% 0.29/0.47 => ! [Y: A,X2: A] :
% 0.29/0.47 ( ( ord_less_eq @ A @ Y @ X2 )
% 0.29/0.47 => ( ( ord_min @ A @ X2 @ Y )
% 0.29/0.47 = Y ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % min_absorb2
% 0.29/0.47 thf(fact_129_min__def,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( ord @ A )
% 0.29/0.47 => ( ( ord_min @ A )
% 0.29/0.47 = ( ^ [A3: A,B2: A] : ( if @ A @ ( ord_less_eq @ A @ A3 @ B2 ) @ A3 @ B2 ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % min_def
% 0.29/0.47 thf(fact_130_Lattices__Big_Oex__has__greatest__nat,axiom,
% 0.29/0.47 ! [A: $tType,P3: A > $o,K: A,F: A > nat,B: nat] :
% 0.29/0.47 ( ( P3 @ K )
% 0.29/0.47 => ( ! [Y2: A] :
% 0.29/0.47 ( ( P3 @ Y2 )
% 0.29/0.47 => ( ord_less @ nat @ ( F @ Y2 ) @ B ) )
% 0.29/0.47 => ? [X3: A] :
% 0.29/0.47 ( ( P3 @ X3 )
% 0.29/0.47 & ! [Y4: A] :
% 0.29/0.47 ( ( P3 @ Y4 )
% 0.29/0.47 => ( ord_less_eq @ nat @ ( F @ Y4 ) @ ( F @ X3 ) ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % Lattices_Big.ex_has_greatest_nat
% 0.29/0.47 thf(fact_131_nat__descend__induct,axiom,
% 0.29/0.47 ! [N: nat,P3: nat > $o,M: nat] :
% 0.29/0.47 ( ! [K2: nat] :
% 0.29/0.47 ( ( ord_less @ nat @ N @ K2 )
% 0.29/0.47 => ( P3 @ K2 ) )
% 0.29/0.47 => ( ! [K2: nat] :
% 0.29/0.47 ( ( ord_less_eq @ nat @ K2 @ N )
% 0.29/0.47 => ( ! [I3: nat] :
% 0.29/0.47 ( ( ord_less @ nat @ K2 @ I3 )
% 0.29/0.47 => ( P3 @ I3 ) )
% 0.29/0.47 => ( P3 @ K2 ) ) )
% 0.29/0.47 => ( P3 @ M ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % nat_descend_induct
% 0.29/0.47 thf(fact_132_dual__order_Oantisym,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( order @ A )
% 0.29/0.47 => ! [B: A,A2: A] :
% 0.29/0.47 ( ( ord_less_eq @ A @ B @ A2 )
% 0.29/0.47 => ( ( ord_less_eq @ A @ A2 @ B )
% 0.29/0.47 => ( A2 = B ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % dual_order.antisym
% 0.29/0.47 thf(fact_133_dual__order_Oeq__iff,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( order @ A )
% 0.29/0.47 => ( ( ^ [Y5: A,Z4: A] : ( Y5 = Z4 ) )
% 0.29/0.47 = ( ^ [A3: A,B2: A] :
% 0.29/0.47 ( ( ord_less_eq @ A @ B2 @ A3 )
% 0.29/0.47 & ( ord_less_eq @ A @ A3 @ B2 ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % dual_order.eq_iff
% 0.29/0.47 thf(fact_134_dual__order_Otrans,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( order @ A )
% 0.29/0.47 => ! [B: A,A2: A,C: A] :
% 0.29/0.47 ( ( ord_less_eq @ A @ B @ A2 )
% 0.29/0.47 => ( ( ord_less_eq @ A @ C @ B )
% 0.29/0.47 => ( ord_less_eq @ A @ C @ A2 ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % dual_order.trans
% 0.29/0.47 thf(fact_135_linorder__wlog,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( linorder @ A )
% 0.29/0.47 => ! [P3: A > A > $o,A2: A,B: A] :
% 0.29/0.47 ( ! [A5: A,B4: A] :
% 0.29/0.47 ( ( ord_less_eq @ A @ A5 @ B4 )
% 0.29/0.47 => ( P3 @ A5 @ B4 ) )
% 0.29/0.47 => ( ! [A5: A,B4: A] :
% 0.29/0.47 ( ( P3 @ B4 @ A5 )
% 0.29/0.47 => ( P3 @ A5 @ B4 ) )
% 0.29/0.47 => ( P3 @ A2 @ B ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % linorder_wlog
% 0.29/0.47 thf(fact_136_dual__order_Orefl,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( order @ A )
% 0.29/0.47 => ! [A2: A] : ( ord_less_eq @ A @ A2 @ A2 ) ) ).
% 0.29/0.47
% 0.29/0.47 % dual_order.refl
% 0.29/0.47 thf(fact_137_order__trans,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( preorder @ A )
% 0.29/0.47 => ! [X2: A,Y: A,Z: A] :
% 0.29/0.47 ( ( ord_less_eq @ A @ X2 @ Y )
% 0.29/0.47 => ( ( ord_less_eq @ A @ Y @ Z )
% 0.29/0.47 => ( ord_less_eq @ A @ X2 @ Z ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % order_trans
% 0.29/0.47 thf(fact_138_order__class_Oorder_Oantisym,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( order @ A )
% 0.29/0.47 => ! [A2: A,B: A] :
% 0.29/0.47 ( ( ord_less_eq @ A @ A2 @ B )
% 0.29/0.47 => ( ( ord_less_eq @ A @ B @ A2 )
% 0.29/0.47 => ( A2 = B ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % order_class.order.antisym
% 0.29/0.47 thf(fact_139_ord__le__eq__trans,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( ord @ A )
% 0.29/0.47 => ! [A2: A,B: A,C: A] :
% 0.29/0.47 ( ( ord_less_eq @ A @ A2 @ B )
% 0.29/0.47 => ( ( B = C )
% 0.29/0.47 => ( ord_less_eq @ A @ A2 @ C ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % ord_le_eq_trans
% 0.29/0.47 thf(fact_140_ord__eq__le__trans,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( ord @ A )
% 0.29/0.47 => ! [A2: A,B: A,C: A] :
% 0.29/0.47 ( ( A2 = B )
% 0.29/0.47 => ( ( ord_less_eq @ A @ B @ C )
% 0.29/0.47 => ( ord_less_eq @ A @ A2 @ C ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % ord_eq_le_trans
% 0.29/0.47 thf(fact_141_order__class_Oorder_Oeq__iff,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( order @ A )
% 0.29/0.47 => ( ( ^ [Y5: A,Z4: A] : ( Y5 = Z4 ) )
% 0.29/0.47 = ( ^ [A3: A,B2: A] :
% 0.29/0.47 ( ( ord_less_eq @ A @ A3 @ B2 )
% 0.29/0.47 & ( ord_less_eq @ A @ B2 @ A3 ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % order_class.order.eq_iff
% 0.29/0.47 thf(fact_142_antisym__conv,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( order @ A )
% 0.29/0.47 => ! [Y: A,X2: A] :
% 0.29/0.47 ( ( ord_less_eq @ A @ Y @ X2 )
% 0.29/0.47 => ( ( ord_less_eq @ A @ X2 @ Y )
% 0.29/0.47 = ( X2 = Y ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % antisym_conv
% 0.29/0.47 thf(fact_143_le__cases3,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( linorder @ A )
% 0.29/0.47 => ! [X2: A,Y: A,Z: A] :
% 0.29/0.47 ( ( ( ord_less_eq @ A @ X2 @ Y )
% 0.29/0.47 => ~ ( ord_less_eq @ A @ Y @ Z ) )
% 0.29/0.47 => ( ( ( ord_less_eq @ A @ Y @ X2 )
% 0.29/0.47 => ~ ( ord_less_eq @ A @ X2 @ Z ) )
% 0.29/0.47 => ( ( ( ord_less_eq @ A @ X2 @ Z )
% 0.29/0.47 => ~ ( ord_less_eq @ A @ Z @ Y ) )
% 0.29/0.47 => ( ( ( ord_less_eq @ A @ Z @ Y )
% 0.29/0.47 => ~ ( ord_less_eq @ A @ Y @ X2 ) )
% 0.29/0.47 => ( ( ( ord_less_eq @ A @ Y @ Z )
% 0.29/0.47 => ~ ( ord_less_eq @ A @ Z @ X2 ) )
% 0.29/0.47 => ~ ( ( ord_less_eq @ A @ Z @ X2 )
% 0.29/0.47 => ~ ( ord_less_eq @ A @ X2 @ Y ) ) ) ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % le_cases3
% 0.29/0.47 thf(fact_144_order_Otrans,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( order @ A )
% 0.29/0.47 => ! [A2: A,B: A,C: A] :
% 0.29/0.47 ( ( ord_less_eq @ A @ A2 @ B )
% 0.29/0.47 => ( ( ord_less_eq @ A @ B @ C )
% 0.29/0.47 => ( ord_less_eq @ A @ A2 @ C ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % order.trans
% 0.29/0.47 thf(fact_145_le__cases,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( linorder @ A )
% 0.29/0.47 => ! [X2: A,Y: A] :
% 0.29/0.47 ( ~ ( ord_less_eq @ A @ X2 @ Y )
% 0.29/0.47 => ( ord_less_eq @ A @ Y @ X2 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % le_cases
% 0.29/0.47 thf(fact_146_eq__refl,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( preorder @ A )
% 0.29/0.47 => ! [X2: A,Y: A] :
% 0.29/0.47 ( ( X2 = Y )
% 0.29/0.47 => ( ord_less_eq @ A @ X2 @ Y ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % eq_refl
% 0.29/0.47 thf(fact_147_linear,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( linorder @ A )
% 0.29/0.47 => ! [X2: A,Y: A] :
% 0.29/0.47 ( ( ord_less_eq @ A @ X2 @ Y )
% 0.29/0.47 | ( ord_less_eq @ A @ Y @ X2 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % linear
% 0.29/0.47 thf(fact_148_antisym,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( order @ A )
% 0.29/0.47 => ! [X2: A,Y: A] :
% 0.29/0.47 ( ( ord_less_eq @ A @ X2 @ Y )
% 0.29/0.47 => ( ( ord_less_eq @ A @ Y @ X2 )
% 0.29/0.47 => ( X2 = Y ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % antisym
% 0.29/0.47 thf(fact_149_eq__iff,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( order @ A )
% 0.29/0.47 => ( ( ^ [Y5: A,Z4: A] : ( Y5 = Z4 ) )
% 0.29/0.47 = ( ^ [X4: A,Y3: A] :
% 0.29/0.47 ( ( ord_less_eq @ A @ X4 @ Y3 )
% 0.29/0.47 & ( ord_less_eq @ A @ Y3 @ X4 ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % eq_iff
% 0.29/0.47 thf(fact_150_ord__le__eq__subst,axiom,
% 0.29/0.47 ! [A: $tType,B3: $tType] :
% 0.29/0.47 ( ( ( ord @ B3 )
% 0.29/0.47 & ( ord @ A ) )
% 0.29/0.47 => ! [A2: A,B: A,F: A > B3,C: B3] :
% 0.29/0.47 ( ( ord_less_eq @ A @ A2 @ B )
% 0.29/0.47 => ( ( ( F @ B )
% 0.29/0.47 = C )
% 0.29/0.47 => ( ! [X3: A,Y2: A] :
% 0.29/0.47 ( ( ord_less_eq @ A @ X3 @ Y2 )
% 0.29/0.47 => ( ord_less_eq @ B3 @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.29/0.47 => ( ord_less_eq @ B3 @ ( F @ A2 ) @ C ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % ord_le_eq_subst
% 0.29/0.47 thf(fact_151_ord__eq__le__subst,axiom,
% 0.29/0.47 ! [A: $tType,B3: $tType] :
% 0.29/0.47 ( ( ( ord @ B3 )
% 0.29/0.47 & ( ord @ A ) )
% 0.29/0.47 => ! [A2: A,F: B3 > A,B: B3,C: B3] :
% 0.29/0.47 ( ( A2
% 0.29/0.47 = ( F @ B ) )
% 0.29/0.47 => ( ( ord_less_eq @ B3 @ B @ C )
% 0.29/0.47 => ( ! [X3: B3,Y2: B3] :
% 0.29/0.47 ( ( ord_less_eq @ B3 @ X3 @ Y2 )
% 0.29/0.47 => ( ord_less_eq @ A @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.29/0.47 => ( ord_less_eq @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % ord_eq_le_subst
% 0.29/0.47 thf(fact_152_order__subst2,axiom,
% 0.29/0.47 ! [A: $tType,C2: $tType] :
% 0.29/0.47 ( ( ( order @ C2 )
% 0.29/0.47 & ( order @ A ) )
% 0.29/0.47 => ! [A2: A,B: A,F: A > C2,C: C2] :
% 0.29/0.47 ( ( ord_less_eq @ A @ A2 @ B )
% 0.29/0.47 => ( ( ord_less_eq @ C2 @ ( F @ B ) @ C )
% 0.29/0.47 => ( ! [X3: A,Y2: A] :
% 0.29/0.47 ( ( ord_less_eq @ A @ X3 @ Y2 )
% 0.29/0.47 => ( ord_less_eq @ C2 @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.29/0.47 => ( ord_less_eq @ C2 @ ( F @ A2 ) @ C ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % order_subst2
% 0.29/0.47 thf(fact_153_order__subst1,axiom,
% 0.29/0.47 ! [A: $tType,B3: $tType] :
% 0.29/0.47 ( ( ( order @ B3 )
% 0.29/0.47 & ( order @ A ) )
% 0.29/0.47 => ! [A2: A,F: B3 > A,B: B3,C: B3] :
% 0.29/0.47 ( ( ord_less_eq @ A @ A2 @ ( F @ B ) )
% 0.29/0.47 => ( ( ord_less_eq @ B3 @ B @ C )
% 0.29/0.47 => ( ! [X3: B3,Y2: B3] :
% 0.29/0.47 ( ( ord_less_eq @ B3 @ X3 @ Y2 )
% 0.29/0.47 => ( ord_less_eq @ A @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.29/0.47 => ( ord_less_eq @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % order_subst1
% 0.29/0.47 thf(fact_154_le__fun__def,axiom,
% 0.29/0.47 ! [B3: $tType,A: $tType] :
% 0.29/0.47 ( ( ord @ B3 )
% 0.29/0.47 => ( ( ord_less_eq @ ( A > B3 ) )
% 0.29/0.47 = ( ^ [F2: A > B3,G2: A > B3] :
% 0.29/0.47 ! [X4: A] : ( ord_less_eq @ B3 @ ( F2 @ X4 ) @ ( G2 @ X4 ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % le_fun_def
% 0.29/0.47 thf(fact_155_le__funI,axiom,
% 0.29/0.47 ! [B3: $tType,A: $tType] :
% 0.29/0.47 ( ( ord @ B3 )
% 0.29/0.47 => ! [F: A > B3,G: A > B3] :
% 0.29/0.47 ( ! [X3: A] : ( ord_less_eq @ B3 @ ( F @ X3 ) @ ( G @ X3 ) )
% 0.29/0.47 => ( ord_less_eq @ ( A > B3 ) @ F @ G ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % le_funI
% 0.29/0.47 thf(fact_156_le__funE,axiom,
% 0.29/0.47 ! [B3: $tType,A: $tType] :
% 0.29/0.47 ( ( ord @ B3 )
% 0.29/0.47 => ! [F: A > B3,G: A > B3,X2: A] :
% 0.29/0.47 ( ( ord_less_eq @ ( A > B3 ) @ F @ G )
% 0.29/0.47 => ( ord_less_eq @ B3 @ ( F @ X2 ) @ ( G @ X2 ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % le_funE
% 0.29/0.47 thf(fact_157_le__funD,axiom,
% 0.29/0.47 ! [B3: $tType,A: $tType] :
% 0.29/0.47 ( ( ord @ B3 )
% 0.29/0.47 => ! [F: A > B3,G: A > B3,X2: A] :
% 0.29/0.47 ( ( ord_less_eq @ ( A > B3 ) @ F @ G )
% 0.29/0.47 => ( ord_less_eq @ B3 @ ( F @ X2 ) @ ( G @ X2 ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % le_funD
% 0.29/0.47 thf(fact_158_ord__eq__less__subst,axiom,
% 0.29/0.47 ! [A: $tType,B3: $tType] :
% 0.29/0.47 ( ( ( ord @ B3 )
% 0.29/0.47 & ( ord @ A ) )
% 0.29/0.47 => ! [A2: A,F: B3 > A,B: B3,C: B3] :
% 0.29/0.47 ( ( A2
% 0.29/0.47 = ( F @ B ) )
% 0.29/0.47 => ( ( ord_less @ B3 @ B @ C )
% 0.29/0.47 => ( ! [X3: B3,Y2: B3] :
% 0.29/0.47 ( ( ord_less @ B3 @ X3 @ Y2 )
% 0.29/0.47 => ( ord_less @ A @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.29/0.47 => ( ord_less @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % ord_eq_less_subst
% 0.29/0.47 thf(fact_159_ord__less__eq__subst,axiom,
% 0.29/0.47 ! [A: $tType,B3: $tType] :
% 0.29/0.47 ( ( ( ord @ B3 )
% 0.29/0.47 & ( ord @ A ) )
% 0.29/0.47 => ! [A2: A,B: A,F: A > B3,C: B3] :
% 0.29/0.47 ( ( ord_less @ A @ A2 @ B )
% 0.29/0.47 => ( ( ( F @ B )
% 0.29/0.47 = C )
% 0.29/0.47 => ( ! [X3: A,Y2: A] :
% 0.29/0.47 ( ( ord_less @ A @ X3 @ Y2 )
% 0.29/0.47 => ( ord_less @ B3 @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.29/0.47 => ( ord_less @ B3 @ ( F @ A2 ) @ C ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % ord_less_eq_subst
% 0.29/0.47 thf(fact_160_order__less__subst1,axiom,
% 0.29/0.47 ! [A: $tType,B3: $tType] :
% 0.29/0.47 ( ( ( order @ B3 )
% 0.29/0.47 & ( order @ A ) )
% 0.29/0.47 => ! [A2: A,F: B3 > A,B: B3,C: B3] :
% 0.29/0.47 ( ( ord_less @ A @ A2 @ ( F @ B ) )
% 0.29/0.47 => ( ( ord_less @ B3 @ B @ C )
% 0.29/0.47 => ( ! [X3: B3,Y2: B3] :
% 0.29/0.47 ( ( ord_less @ B3 @ X3 @ Y2 )
% 0.29/0.47 => ( ord_less @ A @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.29/0.47 => ( ord_less @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % order_less_subst1
% 0.29/0.47 thf(fact_161_order__less__subst2,axiom,
% 0.29/0.47 ! [A: $tType,C2: $tType] :
% 0.29/0.47 ( ( ( order @ C2 )
% 0.29/0.47 & ( order @ A ) )
% 0.29/0.47 => ! [A2: A,B: A,F: A > C2,C: C2] :
% 0.29/0.47 ( ( ord_less @ A @ A2 @ B )
% 0.29/0.47 => ( ( ord_less @ C2 @ ( F @ B ) @ C )
% 0.29/0.47 => ( ! [X3: A,Y2: A] :
% 0.29/0.47 ( ( ord_less @ A @ X3 @ Y2 )
% 0.29/0.47 => ( ord_less @ C2 @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.29/0.47 => ( ord_less @ C2 @ ( F @ A2 ) @ C ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % order_less_subst2
% 0.29/0.47 thf(fact_162_lt__ex,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( no_bot @ A )
% 0.29/0.47 => ! [X2: A] :
% 0.29/0.47 ? [Y2: A] : ( ord_less @ A @ Y2 @ X2 ) ) ).
% 0.29/0.47
% 0.29/0.47 % lt_ex
% 0.29/0.47 thf(fact_163_gt__ex,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( no_top @ A )
% 0.29/0.47 => ! [X2: A] :
% 0.29/0.47 ? [X_1: A] : ( ord_less @ A @ X2 @ X_1 ) ) ).
% 0.29/0.47
% 0.29/0.47 % gt_ex
% 0.29/0.47 thf(fact_164_neqE,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( linorder @ A )
% 0.29/0.47 => ! [X2: A,Y: A] :
% 0.29/0.47 ( ( X2 != Y )
% 0.29/0.47 => ( ~ ( ord_less @ A @ X2 @ Y )
% 0.29/0.47 => ( ord_less @ A @ Y @ X2 ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % neqE
% 0.29/0.47 thf(fact_165_neq__iff,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( linorder @ A )
% 0.29/0.47 => ! [X2: A,Y: A] :
% 0.29/0.47 ( ( X2 != Y )
% 0.29/0.47 = ( ( ord_less @ A @ X2 @ Y )
% 0.29/0.47 | ( ord_less @ A @ Y @ X2 ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % neq_iff
% 0.29/0.47 thf(fact_166_order_Oasym,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( order @ A )
% 0.29/0.47 => ! [A2: A,B: A] :
% 0.29/0.47 ( ( ord_less @ A @ A2 @ B )
% 0.29/0.47 => ~ ( ord_less @ A @ B @ A2 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % order.asym
% 0.29/0.47 thf(fact_167_dense,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( dense_order @ A )
% 0.29/0.47 => ! [X2: A,Y: A] :
% 0.29/0.47 ( ( ord_less @ A @ X2 @ Y )
% 0.29/0.47 => ? [Z5: A] :
% 0.29/0.47 ( ( ord_less @ A @ X2 @ Z5 )
% 0.29/0.47 & ( ord_less @ A @ Z5 @ Y ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % dense
% 0.29/0.47 thf(fact_168_less__imp__neq,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( order @ A )
% 0.29/0.47 => ! [X2: A,Y: A] :
% 0.29/0.47 ( ( ord_less @ A @ X2 @ Y )
% 0.29/0.47 => ( X2 != Y ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % less_imp_neq
% 0.29/0.47 thf(fact_169_less__asym,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( preorder @ A )
% 0.29/0.47 => ! [X2: A,Y: A] :
% 0.29/0.47 ( ( ord_less @ A @ X2 @ Y )
% 0.29/0.47 => ~ ( ord_less @ A @ Y @ X2 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % less_asym
% 0.29/0.47 thf(fact_170_less__asym_H,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( preorder @ A )
% 0.29/0.47 => ! [A2: A,B: A] :
% 0.29/0.47 ( ( ord_less @ A @ A2 @ B )
% 0.29/0.47 => ~ ( ord_less @ A @ B @ A2 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % less_asym'
% 0.29/0.47 thf(fact_171_less__trans,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( preorder @ A )
% 0.29/0.47 => ! [X2: A,Y: A,Z: A] :
% 0.29/0.47 ( ( ord_less @ A @ X2 @ Y )
% 0.29/0.47 => ( ( ord_less @ A @ Y @ Z )
% 0.29/0.47 => ( ord_less @ A @ X2 @ Z ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % less_trans
% 0.29/0.47 thf(fact_172_less__linear,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( linorder @ A )
% 0.29/0.47 => ! [X2: A,Y: A] :
% 0.29/0.47 ( ( ord_less @ A @ X2 @ Y )
% 0.29/0.47 | ( X2 = Y )
% 0.29/0.47 | ( ord_less @ A @ Y @ X2 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % less_linear
% 0.29/0.47 thf(fact_173_less__irrefl,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( preorder @ A )
% 0.29/0.47 => ! [X2: A] :
% 0.29/0.47 ~ ( ord_less @ A @ X2 @ X2 ) ) ).
% 0.29/0.47
% 0.29/0.47 % less_irrefl
% 0.29/0.47 thf(fact_174_ord__eq__less__trans,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( ord @ A )
% 0.29/0.47 => ! [A2: A,B: A,C: A] :
% 0.29/0.47 ( ( A2 = B )
% 0.29/0.47 => ( ( ord_less @ A @ B @ C )
% 0.29/0.47 => ( ord_less @ A @ A2 @ C ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % ord_eq_less_trans
% 0.29/0.47 thf(fact_175_ord__less__eq__trans,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( ord @ A )
% 0.29/0.47 => ! [A2: A,B: A,C: A] :
% 0.29/0.47 ( ( ord_less @ A @ A2 @ B )
% 0.29/0.47 => ( ( B = C )
% 0.29/0.47 => ( ord_less @ A @ A2 @ C ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % ord_less_eq_trans
% 0.29/0.47 thf(fact_176_dual__order_Oasym,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( order @ A )
% 0.29/0.47 => ! [B: A,A2: A] :
% 0.29/0.47 ( ( ord_less @ A @ B @ A2 )
% 0.29/0.47 => ~ ( ord_less @ A @ A2 @ B ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % dual_order.asym
% 0.29/0.47 thf(fact_177_less__imp__not__eq,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( order @ A )
% 0.29/0.47 => ! [X2: A,Y: A] :
% 0.29/0.47 ( ( ord_less @ A @ X2 @ Y )
% 0.29/0.47 => ( X2 != Y ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % less_imp_not_eq
% 0.29/0.47 thf(fact_178_less__not__sym,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( preorder @ A )
% 0.29/0.47 => ! [X2: A,Y: A] :
% 0.29/0.47 ( ( ord_less @ A @ X2 @ Y )
% 0.29/0.47 => ~ ( ord_less @ A @ Y @ X2 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % less_not_sym
% 0.29/0.47 thf(fact_179_less__induct,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( wellorder @ A )
% 0.29/0.47 => ! [P3: A > $o,A2: A] :
% 0.29/0.47 ( ! [X3: A] :
% 0.29/0.47 ( ! [Y4: A] :
% 0.29/0.47 ( ( ord_less @ A @ Y4 @ X3 )
% 0.29/0.47 => ( P3 @ Y4 ) )
% 0.29/0.47 => ( P3 @ X3 ) )
% 0.29/0.47 => ( P3 @ A2 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % less_induct
% 0.29/0.47 thf(fact_180_antisym__conv3,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( linorder @ A )
% 0.29/0.47 => ! [Y: A,X2: A] :
% 0.29/0.47 ( ~ ( ord_less @ A @ Y @ X2 )
% 0.29/0.47 => ( ( ~ ( ord_less @ A @ X2 @ Y ) )
% 0.29/0.47 = ( X2 = Y ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % antisym_conv3
% 0.29/0.47 thf(fact_181_less__imp__not__eq2,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( order @ A )
% 0.29/0.47 => ! [X2: A,Y: A] :
% 0.29/0.47 ( ( ord_less @ A @ X2 @ Y )
% 0.29/0.47 => ( Y != X2 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % less_imp_not_eq2
% 0.29/0.47 thf(fact_182_less__imp__triv,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( preorder @ A )
% 0.29/0.47 => ! [X2: A,Y: A,P3: $o] :
% 0.29/0.47 ( ( ord_less @ A @ X2 @ Y )
% 0.29/0.47 => ( ( ord_less @ A @ Y @ X2 )
% 0.29/0.47 => P3 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % less_imp_triv
% 0.29/0.47 thf(fact_183_linorder__cases,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( linorder @ A )
% 0.29/0.47 => ! [X2: A,Y: A] :
% 0.29/0.47 ( ~ ( ord_less @ A @ X2 @ Y )
% 0.29/0.47 => ( ( X2 != Y )
% 0.29/0.47 => ( ord_less @ A @ Y @ X2 ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % linorder_cases
% 0.29/0.47 thf(fact_184_dual__order_Oirrefl,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( order @ A )
% 0.29/0.47 => ! [A2: A] :
% 0.29/0.47 ~ ( ord_less @ A @ A2 @ A2 ) ) ).
% 0.29/0.47
% 0.29/0.47 % dual_order.irrefl
% 0.29/0.47 thf(fact_185_order_Ostrict__trans,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( order @ A )
% 0.29/0.47 => ! [A2: A,B: A,C: A] :
% 0.29/0.47 ( ( ord_less @ A @ A2 @ B )
% 0.29/0.47 => ( ( ord_less @ A @ B @ C )
% 0.29/0.47 => ( ord_less @ A @ A2 @ C ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % order.strict_trans
% 0.29/0.47 thf(fact_186_less__imp__not__less,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( preorder @ A )
% 0.29/0.47 => ! [X2: A,Y: A] :
% 0.29/0.47 ( ( ord_less @ A @ X2 @ Y )
% 0.29/0.47 => ~ ( ord_less @ A @ Y @ X2 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % less_imp_not_less
% 0.29/0.47 thf(fact_187_exists__least__iff,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( wellorder @ A )
% 0.29/0.47 => ( ( ^ [P4: A > $o] :
% 0.29/0.47 ? [X5: A] : ( P4 @ X5 ) )
% 0.29/0.47 = ( ^ [P5: A > $o] :
% 0.29/0.47 ? [N2: A] :
% 0.29/0.47 ( ( P5 @ N2 )
% 0.29/0.47 & ! [M2: A] :
% 0.29/0.47 ( ( ord_less @ A @ M2 @ N2 )
% 0.29/0.47 => ~ ( P5 @ M2 ) ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % exists_least_iff
% 0.29/0.47 thf(fact_188_linorder__less__wlog,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( linorder @ A )
% 0.29/0.47 => ! [P3: A > A > $o,A2: A,B: A] :
% 0.29/0.47 ( ! [A5: A,B4: A] :
% 0.29/0.47 ( ( ord_less @ A @ A5 @ B4 )
% 0.29/0.47 => ( P3 @ A5 @ B4 ) )
% 0.29/0.47 => ( ! [A5: A] : ( P3 @ A5 @ A5 )
% 0.29/0.47 => ( ! [A5: A,B4: A] :
% 0.29/0.47 ( ( P3 @ B4 @ A5 )
% 0.29/0.47 => ( P3 @ A5 @ B4 ) )
% 0.29/0.47 => ( P3 @ A2 @ B ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % linorder_less_wlog
% 0.29/0.47 thf(fact_189_dual__order_Ostrict__trans,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( order @ A )
% 0.29/0.47 => ! [B: A,A2: A,C: A] :
% 0.29/0.47 ( ( ord_less @ A @ B @ A2 )
% 0.29/0.47 => ( ( ord_less @ A @ C @ B )
% 0.29/0.47 => ( ord_less @ A @ C @ A2 ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % dual_order.strict_trans
% 0.29/0.47 thf(fact_190_not__less__iff__gr__or__eq,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( linorder @ A )
% 0.29/0.47 => ! [X2: A,Y: A] :
% 0.29/0.47 ( ( ~ ( ord_less @ A @ X2 @ Y ) )
% 0.29/0.47 = ( ( ord_less @ A @ Y @ X2 )
% 0.29/0.47 | ( X2 = Y ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % not_less_iff_gr_or_eq
% 0.29/0.47 thf(fact_191_order_Ostrict__implies__not__eq,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( order @ A )
% 0.29/0.47 => ! [A2: A,B: A] :
% 0.29/0.47 ( ( ord_less @ A @ A2 @ B )
% 0.29/0.47 => ( A2 != B ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % order.strict_implies_not_eq
% 0.29/0.47 thf(fact_192_dual__order_Ostrict__implies__not__eq,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( order @ A )
% 0.29/0.47 => ! [B: A,A2: A] :
% 0.29/0.47 ( ( ord_less @ A @ B @ A2 )
% 0.29/0.47 => ( A2 != B ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % dual_order.strict_implies_not_eq
% 0.29/0.47 thf(fact_193_ex__has__least__nat,axiom,
% 0.29/0.47 ! [A: $tType,P3: A > $o,K: A,M: A > nat] :
% 0.29/0.47 ( ( P3 @ K )
% 0.29/0.47 => ? [X3: A] :
% 0.29/0.47 ( ( P3 @ X3 )
% 0.29/0.47 & ! [Y4: A] :
% 0.29/0.47 ( ( P3 @ Y4 )
% 0.29/0.47 => ( ord_less_eq @ nat @ ( M @ X3 ) @ ( M @ Y4 ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % ex_has_least_nat
% 0.29/0.47 thf(fact_194_leD,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( order @ A )
% 0.29/0.47 => ! [Y: A,X2: A] :
% 0.29/0.47 ( ( ord_less_eq @ A @ Y @ X2 )
% 0.29/0.47 => ~ ( ord_less @ A @ X2 @ Y ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % leD
% 0.29/0.47 thf(fact_195_leI,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( linorder @ A )
% 0.29/0.47 => ! [X2: A,Y: A] :
% 0.29/0.47 ( ~ ( ord_less @ A @ X2 @ Y )
% 0.29/0.47 => ( ord_less_eq @ A @ Y @ X2 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % leI
% 0.29/0.47 thf(fact_196_le__less,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( order @ A )
% 0.29/0.47 => ( ( ord_less_eq @ A )
% 0.29/0.47 = ( ^ [X4: A,Y3: A] :
% 0.29/0.47 ( ( ord_less @ A @ X4 @ Y3 )
% 0.29/0.47 | ( X4 = Y3 ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % le_less
% 0.29/0.47 thf(fact_197_less__le,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( order @ A )
% 0.29/0.47 => ( ( ord_less @ A )
% 0.29/0.47 = ( ^ [X4: A,Y3: A] :
% 0.29/0.47 ( ( ord_less_eq @ A @ X4 @ Y3 )
% 0.29/0.47 & ( X4 != Y3 ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % less_le
% 0.29/0.47 thf(fact_198_order__le__less__subst1,axiom,
% 0.29/0.47 ! [A: $tType,B3: $tType] :
% 0.29/0.47 ( ( ( order @ B3 )
% 0.29/0.47 & ( order @ A ) )
% 0.29/0.47 => ! [A2: A,F: B3 > A,B: B3,C: B3] :
% 0.29/0.47 ( ( ord_less_eq @ A @ A2 @ ( F @ B ) )
% 0.29/0.47 => ( ( ord_less @ B3 @ B @ C )
% 0.29/0.47 => ( ! [X3: B3,Y2: B3] :
% 0.29/0.47 ( ( ord_less @ B3 @ X3 @ Y2 )
% 0.29/0.47 => ( ord_less @ A @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.29/0.47 => ( ord_less @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % order_le_less_subst1
% 0.29/0.47 thf(fact_199_order__le__less__subst2,axiom,
% 0.29/0.47 ! [A: $tType,C2: $tType] :
% 0.29/0.47 ( ( ( order @ C2 )
% 0.29/0.47 & ( order @ A ) )
% 0.29/0.47 => ! [A2: A,B: A,F: A > C2,C: C2] :
% 0.29/0.47 ( ( ord_less_eq @ A @ A2 @ B )
% 0.29/0.47 => ( ( ord_less @ C2 @ ( F @ B ) @ C )
% 0.29/0.47 => ( ! [X3: A,Y2: A] :
% 0.29/0.47 ( ( ord_less_eq @ A @ X3 @ Y2 )
% 0.29/0.47 => ( ord_less_eq @ C2 @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.29/0.47 => ( ord_less @ C2 @ ( F @ A2 ) @ C ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % order_le_less_subst2
% 0.29/0.47 thf(fact_200_order__less__le__subst1,axiom,
% 0.29/0.47 ! [A: $tType,B3: $tType] :
% 0.29/0.47 ( ( ( order @ B3 )
% 0.29/0.47 & ( order @ A ) )
% 0.29/0.47 => ! [A2: A,F: B3 > A,B: B3,C: B3] :
% 0.29/0.47 ( ( ord_less @ A @ A2 @ ( F @ B ) )
% 0.29/0.47 => ( ( ord_less_eq @ B3 @ B @ C )
% 0.29/0.47 => ( ! [X3: B3,Y2: B3] :
% 0.29/0.47 ( ( ord_less_eq @ B3 @ X3 @ Y2 )
% 0.29/0.47 => ( ord_less_eq @ A @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.29/0.47 => ( ord_less @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % order_less_le_subst1
% 0.29/0.47 thf(fact_201_order__less__le__subst2,axiom,
% 0.29/0.47 ! [A: $tType,C2: $tType] :
% 0.29/0.47 ( ( ( order @ C2 )
% 0.29/0.47 & ( order @ A ) )
% 0.29/0.47 => ! [A2: A,B: A,F: A > C2,C: C2] :
% 0.29/0.47 ( ( ord_less @ A @ A2 @ B )
% 0.29/0.47 => ( ( ord_less_eq @ C2 @ ( F @ B ) @ C )
% 0.29/0.47 => ( ! [X3: A,Y2: A] :
% 0.29/0.47 ( ( ord_less @ A @ X3 @ Y2 )
% 0.29/0.47 => ( ord_less @ C2 @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.29/0.47 => ( ord_less @ C2 @ ( F @ A2 ) @ C ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % order_less_le_subst2
% 0.29/0.47 thf(fact_202_not__le,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( linorder @ A )
% 0.29/0.47 => ! [X2: A,Y: A] :
% 0.29/0.47 ( ( ~ ( ord_less_eq @ A @ X2 @ Y ) )
% 0.29/0.47 = ( ord_less @ A @ Y @ X2 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % not_le
% 0.29/0.47 thf(fact_203_not__less,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( linorder @ A )
% 0.29/0.47 => ! [X2: A,Y: A] :
% 0.29/0.47 ( ( ~ ( ord_less @ A @ X2 @ Y ) )
% 0.29/0.47 = ( ord_less_eq @ A @ Y @ X2 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % not_less
% 0.29/0.47 thf(fact_204_le__neq__trans,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( order @ A )
% 0.29/0.47 => ! [A2: A,B: A] :
% 0.29/0.47 ( ( ord_less_eq @ A @ A2 @ B )
% 0.29/0.47 => ( ( A2 != B )
% 0.29/0.47 => ( ord_less @ A @ A2 @ B ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % le_neq_trans
% 0.29/0.47 thf(fact_205_antisym__conv1,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( order @ A )
% 0.29/0.47 => ! [X2: A,Y: A] :
% 0.29/0.47 ( ~ ( ord_less @ A @ X2 @ Y )
% 0.29/0.47 => ( ( ord_less_eq @ A @ X2 @ Y )
% 0.29/0.47 = ( X2 = Y ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % antisym_conv1
% 0.29/0.47 thf(fact_206_antisym__conv2,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( order @ A )
% 0.29/0.47 => ! [X2: A,Y: A] :
% 0.29/0.47 ( ( ord_less_eq @ A @ X2 @ Y )
% 0.29/0.47 => ( ( ~ ( ord_less @ A @ X2 @ Y ) )
% 0.29/0.47 = ( X2 = Y ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % antisym_conv2
% 0.29/0.47 thf(fact_207_less__imp__le,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( preorder @ A )
% 0.29/0.47 => ! [X2: A,Y: A] :
% 0.29/0.47 ( ( ord_less @ A @ X2 @ Y )
% 0.29/0.47 => ( ord_less_eq @ A @ X2 @ Y ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % less_imp_le
% 0.29/0.47 thf(fact_208_le__less__trans,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( preorder @ A )
% 0.29/0.47 => ! [X2: A,Y: A,Z: A] :
% 0.29/0.47 ( ( ord_less_eq @ A @ X2 @ Y )
% 0.29/0.47 => ( ( ord_less @ A @ Y @ Z )
% 0.29/0.47 => ( ord_less @ A @ X2 @ Z ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % le_less_trans
% 0.29/0.47 thf(fact_209_less__le__trans,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( preorder @ A )
% 0.29/0.47 => ! [X2: A,Y: A,Z: A] :
% 0.29/0.47 ( ( ord_less @ A @ X2 @ Y )
% 0.29/0.47 => ( ( ord_less_eq @ A @ Y @ Z )
% 0.29/0.47 => ( ord_less @ A @ X2 @ Z ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % less_le_trans
% 0.29/0.47 thf(fact_210_dense__ge,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( dense_linorder @ A )
% 0.29/0.47 => ! [Z: A,Y: A] :
% 0.29/0.47 ( ! [X3: A] :
% 0.29/0.47 ( ( ord_less @ A @ Z @ X3 )
% 0.29/0.47 => ( ord_less_eq @ A @ Y @ X3 ) )
% 0.29/0.47 => ( ord_less_eq @ A @ Y @ Z ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % dense_ge
% 0.29/0.47 thf(fact_211_dense__le,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( dense_linorder @ A )
% 0.29/0.47 => ! [Y: A,Z: A] :
% 0.29/0.47 ( ! [X3: A] :
% 0.29/0.47 ( ( ord_less @ A @ X3 @ Y )
% 0.29/0.47 => ( ord_less_eq @ A @ X3 @ Z ) )
% 0.29/0.47 => ( ord_less_eq @ A @ Y @ Z ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % dense_le
% 0.29/0.47 thf(fact_212_le__less__linear,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( linorder @ A )
% 0.29/0.47 => ! [X2: A,Y: A] :
% 0.29/0.47 ( ( ord_less_eq @ A @ X2 @ Y )
% 0.29/0.47 | ( ord_less @ A @ Y @ X2 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % le_less_linear
% 0.29/0.47 thf(fact_213_le__imp__less__or__eq,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( order @ A )
% 0.29/0.47 => ! [X2: A,Y: A] :
% 0.29/0.47 ( ( ord_less_eq @ A @ X2 @ Y )
% 0.29/0.47 => ( ( ord_less @ A @ X2 @ Y )
% 0.29/0.47 | ( X2 = Y ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % le_imp_less_or_eq
% 0.29/0.47 thf(fact_214_less__le__not__le,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( preorder @ A )
% 0.29/0.47 => ( ( ord_less @ A )
% 0.29/0.47 = ( ^ [X4: A,Y3: A] :
% 0.29/0.47 ( ( ord_less_eq @ A @ X4 @ Y3 )
% 0.29/0.47 & ~ ( ord_less_eq @ A @ Y3 @ X4 ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % less_le_not_le
% 0.29/0.47 thf(fact_215_not__le__imp__less,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( linorder @ A )
% 0.29/0.47 => ! [Y: A,X2: A] :
% 0.29/0.47 ( ~ ( ord_less_eq @ A @ Y @ X2 )
% 0.29/0.47 => ( ord_less @ A @ X2 @ Y ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % not_le_imp_less
% 0.29/0.47 thf(fact_216_order_Ostrict__trans1,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( order @ A )
% 0.29/0.47 => ! [A2: A,B: A,C: A] :
% 0.29/0.47 ( ( ord_less_eq @ A @ A2 @ B )
% 0.29/0.47 => ( ( ord_less @ A @ B @ C )
% 0.29/0.47 => ( ord_less @ A @ A2 @ C ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % order.strict_trans1
% 0.29/0.47 thf(fact_217_order_Ostrict__trans2,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( order @ A )
% 0.29/0.47 => ! [A2: A,B: A,C: A] :
% 0.29/0.47 ( ( ord_less @ A @ A2 @ B )
% 0.29/0.47 => ( ( ord_less_eq @ A @ B @ C )
% 0.29/0.47 => ( ord_less @ A @ A2 @ C ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % order.strict_trans2
% 0.29/0.47 thf(fact_218_order_Oorder__iff__strict,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( order @ A )
% 0.29/0.47 => ( ( ord_less_eq @ A )
% 0.29/0.47 = ( ^ [A3: A,B2: A] :
% 0.29/0.47 ( ( ord_less @ A @ A3 @ B2 )
% 0.29/0.47 | ( A3 = B2 ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % order.order_iff_strict
% 0.29/0.47 thf(fact_219_order_Ostrict__iff__order,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( order @ A )
% 0.29/0.47 => ( ( ord_less @ A )
% 0.29/0.47 = ( ^ [A3: A,B2: A] :
% 0.29/0.47 ( ( ord_less_eq @ A @ A3 @ B2 )
% 0.29/0.47 & ( A3 != B2 ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % order.strict_iff_order
% 0.29/0.47 thf(fact_220_dual__order_Ostrict__trans1,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( order @ A )
% 0.29/0.47 => ! [B: A,A2: A,C: A] :
% 0.29/0.47 ( ( ord_less_eq @ A @ B @ A2 )
% 0.29/0.47 => ( ( ord_less @ A @ C @ B )
% 0.29/0.47 => ( ord_less @ A @ C @ A2 ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % dual_order.strict_trans1
% 0.29/0.47 thf(fact_221_dual__order_Ostrict__trans2,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( order @ A )
% 0.29/0.47 => ! [B: A,A2: A,C: A] :
% 0.29/0.47 ( ( ord_less @ A @ B @ A2 )
% 0.29/0.47 => ( ( ord_less_eq @ A @ C @ B )
% 0.29/0.47 => ( ord_less @ A @ C @ A2 ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % dual_order.strict_trans2
% 0.29/0.47 thf(fact_222_dense__ge__bounded,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( dense_linorder @ A )
% 0.29/0.47 => ! [Z: A,X2: A,Y: A] :
% 0.29/0.47 ( ( ord_less @ A @ Z @ X2 )
% 0.29/0.47 => ( ! [W: A] :
% 0.29/0.47 ( ( ord_less @ A @ Z @ W )
% 0.29/0.47 => ( ( ord_less @ A @ W @ X2 )
% 0.29/0.47 => ( ord_less_eq @ A @ Y @ W ) ) )
% 0.29/0.47 => ( ord_less_eq @ A @ Y @ Z ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % dense_ge_bounded
% 0.29/0.47 thf(fact_223_dense__le__bounded,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( dense_linorder @ A )
% 0.29/0.47 => ! [X2: A,Y: A,Z: A] :
% 0.29/0.47 ( ( ord_less @ A @ X2 @ Y )
% 0.29/0.47 => ( ! [W: A] :
% 0.29/0.47 ( ( ord_less @ A @ X2 @ W )
% 0.29/0.47 => ( ( ord_less @ A @ W @ Y )
% 0.29/0.47 => ( ord_less_eq @ A @ W @ Z ) ) )
% 0.29/0.47 => ( ord_less_eq @ A @ Y @ Z ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % dense_le_bounded
% 0.29/0.47 thf(fact_224_order_Ostrict__implies__order,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( order @ A )
% 0.29/0.47 => ! [A2: A,B: A] :
% 0.29/0.47 ( ( ord_less @ A @ A2 @ B )
% 0.29/0.47 => ( ord_less_eq @ A @ A2 @ B ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % order.strict_implies_order
% 0.29/0.47 thf(fact_225_dual__order_Oorder__iff__strict,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( order @ A )
% 0.29/0.47 => ( ( ord_less_eq @ A )
% 0.29/0.47 = ( ^ [B2: A,A3: A] :
% 0.29/0.47 ( ( ord_less @ A @ B2 @ A3 )
% 0.29/0.47 | ( A3 = B2 ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % dual_order.order_iff_strict
% 0.29/0.47 thf(fact_226_dual__order_Ostrict__iff__order,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( order @ A )
% 0.29/0.47 => ( ( ord_less @ A )
% 0.29/0.47 = ( ^ [B2: A,A3: A] :
% 0.29/0.47 ( ( ord_less_eq @ A @ B2 @ A3 )
% 0.29/0.47 & ( A3 != B2 ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % dual_order.strict_iff_order
% 0.29/0.47 thf(fact_227_dual__order_Ostrict__implies__order,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( order @ A )
% 0.29/0.47 => ! [B: A,A2: A] :
% 0.29/0.47 ( ( ord_less @ A @ B @ A2 )
% 0.29/0.47 => ( ord_less_eq @ A @ B @ A2 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % dual_order.strict_implies_order
% 0.29/0.47 thf(fact_228_order_Onot__eq__order__implies__strict,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( order @ A )
% 0.29/0.47 => ! [A2: A,B: A] :
% 0.29/0.47 ( ( A2 != B )
% 0.29/0.47 => ( ( ord_less_eq @ A @ A2 @ B )
% 0.29/0.47 => ( ord_less @ A @ A2 @ B ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % order.not_eq_order_implies_strict
% 0.29/0.47 thf(fact_229_minf_I8_J,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( linorder @ A )
% 0.29/0.47 => ! [T: A] :
% 0.29/0.47 ? [Z5: A] :
% 0.29/0.47 ! [X: A] :
% 0.29/0.47 ( ( ord_less @ A @ X @ Z5 )
% 0.29/0.47 => ~ ( ord_less_eq @ A @ T @ X ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % minf(8)
% 0.29/0.47 thf(fact_230_minf_I6_J,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( linorder @ A )
% 0.29/0.47 => ! [T: A] :
% 0.29/0.47 ? [Z5: A] :
% 0.29/0.47 ! [X: A] :
% 0.29/0.47 ( ( ord_less @ A @ X @ Z5 )
% 0.29/0.47 => ( ord_less_eq @ A @ X @ T ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % minf(6)
% 0.29/0.47 thf(fact_231_pinf_I8_J,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( linorder @ A )
% 0.29/0.47 => ! [T: A] :
% 0.29/0.47 ? [Z5: A] :
% 0.29/0.47 ! [X: A] :
% 0.29/0.47 ( ( ord_less @ A @ Z5 @ X )
% 0.29/0.47 => ( ord_less_eq @ A @ T @ X ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % pinf(8)
% 0.29/0.47 thf(fact_232_pinf_I6_J,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( linorder @ A )
% 0.29/0.47 => ! [T: A] :
% 0.29/0.47 ? [Z5: A] :
% 0.29/0.47 ! [X: A] :
% 0.29/0.47 ( ( ord_less @ A @ Z5 @ X )
% 0.29/0.47 => ~ ( ord_less_eq @ A @ X @ T ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % pinf(6)
% 0.29/0.47 thf(fact_233_less__fun__def,axiom,
% 0.29/0.47 ! [B3: $tType,A: $tType] :
% 0.29/0.47 ( ( ord @ B3 )
% 0.29/0.47 => ( ( ord_less @ ( A > B3 ) )
% 0.29/0.47 = ( ^ [F2: A > B3,G2: A > B3] :
% 0.29/0.47 ( ( ord_less_eq @ ( A > B3 ) @ F2 @ G2 )
% 0.29/0.47 & ~ ( ord_less_eq @ ( A > B3 ) @ G2 @ F2 ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % less_fun_def
% 0.29/0.47 thf(fact_234_pinf_I1_J,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( linorder @ A )
% 0.29/0.47 => ! [P3: A > $o,P6: A > $o,Q2: A > $o,Q3: A > $o] :
% 0.29/0.47 ( ? [Z2: A] :
% 0.29/0.47 ! [X3: A] :
% 0.29/0.47 ( ( ord_less @ A @ Z2 @ X3 )
% 0.29/0.47 => ( ( P3 @ X3 )
% 0.29/0.47 = ( P6 @ X3 ) ) )
% 0.29/0.47 => ( ? [Z2: A] :
% 0.29/0.47 ! [X3: A] :
% 0.29/0.47 ( ( ord_less @ A @ Z2 @ X3 )
% 0.29/0.47 => ( ( Q2 @ X3 )
% 0.29/0.47 = ( Q3 @ X3 ) ) )
% 0.29/0.47 => ? [Z5: A] :
% 0.29/0.47 ! [X: A] :
% 0.29/0.47 ( ( ord_less @ A @ Z5 @ X )
% 0.29/0.47 => ( ( ( P3 @ X )
% 0.29/0.47 & ( Q2 @ X ) )
% 0.29/0.47 = ( ( P6 @ X )
% 0.29/0.47 & ( Q3 @ X ) ) ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % pinf(1)
% 0.29/0.47 thf(fact_235_pinf_I2_J,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( linorder @ A )
% 0.29/0.47 => ! [P3: A > $o,P6: A > $o,Q2: A > $o,Q3: A > $o] :
% 0.29/0.47 ( ? [Z2: A] :
% 0.29/0.47 ! [X3: A] :
% 0.29/0.47 ( ( ord_less @ A @ Z2 @ X3 )
% 0.29/0.47 => ( ( P3 @ X3 )
% 0.29/0.47 = ( P6 @ X3 ) ) )
% 0.29/0.47 => ( ? [Z2: A] :
% 0.29/0.47 ! [X3: A] :
% 0.29/0.47 ( ( ord_less @ A @ Z2 @ X3 )
% 0.29/0.47 => ( ( Q2 @ X3 )
% 0.29/0.47 = ( Q3 @ X3 ) ) )
% 0.29/0.47 => ? [Z5: A] :
% 0.29/0.47 ! [X: A] :
% 0.29/0.47 ( ( ord_less @ A @ Z5 @ X )
% 0.29/0.47 => ( ( ( P3 @ X )
% 0.29/0.47 | ( Q2 @ X ) )
% 0.29/0.47 = ( ( P6 @ X )
% 0.29/0.47 | ( Q3 @ X ) ) ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % pinf(2)
% 0.29/0.47 thf(fact_236_pinf_I3_J,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( linorder @ A )
% 0.29/0.47 => ! [T: A] :
% 0.29/0.47 ? [Z5: A] :
% 0.29/0.47 ! [X: A] :
% 0.29/0.47 ( ( ord_less @ A @ Z5 @ X )
% 0.29/0.47 => ( X != T ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % pinf(3)
% 0.29/0.47 thf(fact_237_pinf_I4_J,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( linorder @ A )
% 0.29/0.47 => ! [T: A] :
% 0.29/0.47 ? [Z5: A] :
% 0.29/0.47 ! [X: A] :
% 0.29/0.47 ( ( ord_less @ A @ Z5 @ X )
% 0.29/0.47 => ( X != T ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % pinf(4)
% 0.29/0.47 thf(fact_238_pinf_I5_J,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( linorder @ A )
% 0.29/0.47 => ! [T: A] :
% 0.29/0.47 ? [Z5: A] :
% 0.29/0.47 ! [X: A] :
% 0.29/0.47 ( ( ord_less @ A @ Z5 @ X )
% 0.29/0.47 => ~ ( ord_less @ A @ X @ T ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % pinf(5)
% 0.29/0.47 thf(fact_239_pinf_I7_J,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( linorder @ A )
% 0.29/0.47 => ! [T: A] :
% 0.29/0.47 ? [Z5: A] :
% 0.29/0.47 ! [X: A] :
% 0.29/0.47 ( ( ord_less @ A @ Z5 @ X )
% 0.29/0.47 => ( ord_less @ A @ T @ X ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % pinf(7)
% 0.29/0.47 thf(fact_240_pinf_I11_J,axiom,
% 0.29/0.47 ! [C2: $tType,D3: $tType] :
% 0.29/0.47 ( ( ord @ C2 )
% 0.29/0.47 => ! [F3: D3] :
% 0.29/0.47 ? [Z5: C2] :
% 0.29/0.47 ! [X: C2] :
% 0.29/0.47 ( ( ord_less @ C2 @ Z5 @ X )
% 0.29/0.47 => ( F3 = F3 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % pinf(11)
% 0.29/0.47 thf(fact_241_minf_I1_J,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.48 ( ( linorder @ A )
% 0.29/0.48 => ! [P3: A > $o,P6: A > $o,Q2: A > $o,Q3: A > $o] :
% 0.29/0.48 ( ? [Z2: A] :
% 0.29/0.48 ! [X3: A] :
% 0.29/0.48 ( ( ord_less @ A @ X3 @ Z2 )
% 0.29/0.48 => ( ( P3 @ X3 )
% 0.29/0.48 = ( P6 @ X3 ) ) )
% 0.29/0.48 => ( ? [Z2: A] :
% 0.29/0.48 ! [X3: A] :
% 0.29/0.48 ( ( ord_less @ A @ X3 @ Z2 )
% 0.29/0.48 => ( ( Q2 @ X3 )
% 0.29/0.48 = ( Q3 @ X3 ) ) )
% 0.29/0.48 => ? [Z5: A] :
% 0.29/0.48 ! [X: A] :
% 0.29/0.48 ( ( ord_less @ A @ X @ Z5 )
% 0.29/0.48 => ( ( ( P3 @ X )
% 0.29/0.48 & ( Q2 @ X ) )
% 0.29/0.48 = ( ( P6 @ X )
% 0.29/0.48 & ( Q3 @ X ) ) ) ) ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % minf(1)
% 0.29/0.48 thf(fact_242_minf_I2_J,axiom,
% 0.29/0.48 ! [A: $tType] :
% 0.29/0.48 ( ( linorder @ A )
% 0.29/0.48 => ! [P3: A > $o,P6: A > $o,Q2: A > $o,Q3: A > $o] :
% 0.29/0.48 ( ? [Z2: A] :
% 0.29/0.48 ! [X3: A] :
% 0.29/0.48 ( ( ord_less @ A @ X3 @ Z2 )
% 0.29/0.48 => ( ( P3 @ X3 )
% 0.29/0.48 = ( P6 @ X3 ) ) )
% 0.29/0.48 => ( ? [Z2: A] :
% 0.29/0.48 ! [X3: A] :
% 0.29/0.48 ( ( ord_less @ A @ X3 @ Z2 )
% 0.29/0.48 => ( ( Q2 @ X3 )
% 0.29/0.48 = ( Q3 @ X3 ) ) )
% 0.29/0.48 => ? [Z5: A] :
% 0.29/0.48 ! [X: A] :
% 0.29/0.48 ( ( ord_less @ A @ X @ Z5 )
% 0.29/0.48 => ( ( ( P3 @ X )
% 0.29/0.48 | ( Q2 @ X ) )
% 0.29/0.48 = ( ( P6 @ X )
% 0.29/0.48 | ( Q3 @ X ) ) ) ) ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % minf(2)
% 0.29/0.48 thf(fact_243_minf_I3_J,axiom,
% 0.29/0.48 ! [A: $tType] :
% 0.29/0.48 ( ( linorder @ A )
% 0.29/0.48 => ! [T: A] :
% 0.29/0.48 ? [Z5: A] :
% 0.29/0.48 ! [X: A] :
% 0.29/0.48 ( ( ord_less @ A @ X @ Z5 )
% 0.29/0.48 => ( X != T ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % minf(3)
% 0.29/0.48 thf(fact_244_minf_I4_J,axiom,
% 0.29/0.48 ! [A: $tType] :
% 0.29/0.48 ( ( linorder @ A )
% 0.29/0.48 => ! [T: A] :
% 0.29/0.48 ? [Z5: A] :
% 0.29/0.48 ! [X: A] :
% 0.29/0.48 ( ( ord_less @ A @ X @ Z5 )
% 0.29/0.48 => ( X != T ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % minf(4)
% 0.29/0.48 thf(fact_245_minf_I5_J,axiom,
% 0.29/0.48 ! [A: $tType] :
% 0.29/0.48 ( ( linorder @ A )
% 0.29/0.48 => ! [T: A] :
% 0.29/0.48 ? [Z5: A] :
% 0.29/0.48 ! [X: A] :
% 0.29/0.48 ( ( ord_less @ A @ X @ Z5 )
% 0.29/0.48 => ( ord_less @ A @ X @ T ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % minf(5)
% 0.29/0.48 thf(fact_246_minf_I7_J,axiom,
% 0.29/0.48 ! [A: $tType] :
% 0.29/0.48 ( ( linorder @ A )
% 0.29/0.48 => ! [T: A] :
% 0.29/0.48 ? [Z5: A] :
% 0.29/0.48 ! [X: A] :
% 0.29/0.48 ( ( ord_less @ A @ X @ Z5 )
% 0.29/0.48 => ~ ( ord_less @ A @ T @ X ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % minf(7)
% 0.29/0.48 thf(fact_247_minf_I11_J,axiom,
% 0.29/0.48 ! [C2: $tType,D3: $tType] :
% 0.29/0.48 ( ( ord @ C2 )
% 0.29/0.48 => ! [F3: D3] :
% 0.29/0.48 ? [Z5: C2] :
% 0.29/0.48 ! [X: C2] :
% 0.29/0.48 ( ( ord_less @ C2 @ X @ Z5 )
% 0.29/0.48 => ( F3 = F3 ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % minf(11)
% 0.29/0.48 thf(fact_248_complete__interval,axiom,
% 0.29/0.48 ! [A: $tType] :
% 0.29/0.48 ( ( condit1037483654norder @ A )
% 0.29/0.48 => ! [A2: A,B: A,P3: A > $o] :
% 0.29/0.48 ( ( ord_less @ A @ A2 @ B )
% 0.29/0.48 => ( ( P3 @ A2 )
% 0.29/0.48 => ( ~ ( P3 @ B )
% 0.29/0.48 => ? [C3: A] :
% 0.29/0.48 ( ( ord_less_eq @ A @ A2 @ C3 )
% 0.29/0.48 & ( ord_less_eq @ A @ C3 @ B )
% 0.29/0.48 & ! [X: A] :
% 0.29/0.48 ( ( ( ord_less_eq @ A @ A2 @ X )
% 0.29/0.48 & ( ord_less @ A @ X @ C3 ) )
% 0.29/0.48 => ( P3 @ X ) )
% 0.29/0.48 & ! [D4: A] :
% 0.29/0.48 ( ! [X3: A] :
% 0.29/0.48 ( ( ( ord_less_eq @ A @ A2 @ X3 )
% 0.29/0.48 & ( ord_less @ A @ X3 @ D4 ) )
% 0.29/0.48 => ( P3 @ X3 ) )
% 0.29/0.48 => ( ord_less_eq @ A @ D4 @ C3 ) ) ) ) ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % complete_interval
% 0.29/0.48 thf(fact_249_verit__comp__simplify1_I3_J,axiom,
% 0.29/0.48 ! [B3: $tType] :
% 0.29/0.48 ( ( linorder @ B3 )
% 0.29/0.48 => ! [B5: B3,A6: B3] :
% 0.29/0.48 ( ( ~ ( ord_less_eq @ B3 @ B5 @ A6 ) )
% 0.29/0.48 = ( ord_less @ B3 @ A6 @ B5 ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % verit_comp_simplify1(3)
% 0.29/0.48 thf(fact_250_verit__la__disequality,axiom,
% 0.29/0.48 ! [A: $tType] :
% 0.29/0.48 ( ( linorder @ A )
% 0.29/0.48 => ! [A2: A,B: A] :
% 0.29/0.48 ( ( A2 = B )
% 0.29/0.48 | ~ ( ord_less_eq @ A @ A2 @ B )
% 0.29/0.48 | ~ ( ord_less_eq @ A @ B @ A2 ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % verit_la_disequality
% 0.29/0.48 thf(fact_251_verit__comp__simplify1_I1_J,axiom,
% 0.29/0.48 ! [A: $tType] :
% 0.29/0.48 ( ( order @ A )
% 0.29/0.48 => ! [A2: A] :
% 0.29/0.48 ~ ( ord_less @ A @ A2 @ A2 ) ) ).
% 0.29/0.48
% 0.29/0.48 % verit_comp_simplify1(1)
% 0.29/0.48 thf(fact_252_ex__gt__or__lt,axiom,
% 0.29/0.48 ! [A: $tType] :
% 0.29/0.48 ( ( condit1656338222tinuum @ A )
% 0.29/0.48 => ! [A2: A] :
% 0.29/0.48 ? [B4: A] :
% 0.29/0.48 ( ( ord_less @ A @ A2 @ B4 )
% 0.29/0.48 | ( ord_less @ A @ B4 @ A2 ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % ex_gt_or_lt
% 0.29/0.48 thf(fact_253_poly__cutoff__1,axiom,
% 0.29/0.48 ! [A: $tType] :
% 0.29/0.48 ( ( comm_semiring_1 @ A )
% 0.29/0.48 => ! [N: nat] :
% 0.29/0.48 ( ( ( N
% 0.29/0.48 = ( zero_zero @ nat ) )
% 0.29/0.48 => ( ( poly_cutoff @ A @ N @ ( one_one @ ( poly @ A ) ) )
% 0.29/0.48 = ( zero_zero @ ( poly @ A ) ) ) )
% 0.29/0.48 & ( ( N
% 0.29/0.48 != ( zero_zero @ nat ) )
% 0.29/0.48 => ( ( poly_cutoff @ A @ N @ ( one_one @ ( poly @ A ) ) )
% 0.29/0.48 = ( one_one @ ( poly @ A ) ) ) ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % poly_cutoff_1
% 0.29/0.48
% 0.29/0.48 % Type constructors (60)
% 0.29/0.48 thf(tcon_fun___Orderings_Opreorder,axiom,
% 0.29/0.48 ! [A7: $tType,A8: $tType] :
% 0.29/0.48 ( ( preorder @ A8 )
% 0.29/0.48 => ( preorder @ ( A7 > A8 ) ) ) ).
% 0.29/0.48
% 0.29/0.48 thf(tcon_fun___Orderings_Oorder,axiom,
% 0.29/0.48 ! [A7: $tType,A8: $tType] :
% 0.29/0.48 ( ( order @ A8 )
% 0.29/0.48 => ( order @ ( A7 > A8 ) ) ) ).
% 0.29/0.48
% 0.29/0.48 thf(tcon_fun___Orderings_Oord,axiom,
% 0.29/0.48 ! [A7: $tType,A8: $tType] :
% 0.29/0.48 ( ( ord @ A8 )
% 0.29/0.48 => ( ord @ ( A7 > A8 ) ) ) ).
% 0.29/0.48
% 0.29/0.48 thf(tcon_Nat_Onat___Conditionally__Complete__Lattices_Oconditionally__complete__linorder,axiom,
% 0.29/0.48 condit1037483654norder @ nat ).
% 0.29/0.48
% 0.29/0.48 thf(tcon_Nat_Onat___Groups_Ocanonically__ordered__monoid__add,axiom,
% 0.29/0.48 canoni770627133id_add @ nat ).
% 0.29/0.48
% 0.29/0.48 thf(tcon_Nat_Onat___Rings_Olinordered__nonzero__semiring,axiom,
% 0.29/0.48 linord1659791738miring @ nat ).
% 0.29/0.48
% 0.29/0.48 thf(tcon_Nat_Onat___Rings_Ocomm__semiring__1,axiom,
% 0.29/0.48 comm_semiring_1 @ nat ).
% 0.29/0.48
% 0.29/0.48 thf(tcon_Nat_Onat___Rings_Ocomm__semiring__0,axiom,
% 0.29/0.48 comm_semiring_0 @ nat ).
% 0.29/0.48
% 0.29/0.48 thf(tcon_Nat_Onat___Orderings_Owellorder,axiom,
% 0.29/0.48 wellorder @ nat ).
% 0.29/0.48
% 0.29/0.48 thf(tcon_Nat_Onat___Orderings_Opreorder_1,axiom,
% 0.29/0.48 preorder @ nat ).
% 0.29/0.48
% 0.29/0.48 thf(tcon_Nat_Onat___Orderings_Olinorder,axiom,
% 0.29/0.48 linorder @ nat ).
% 0.29/0.48
% 0.29/0.48 thf(tcon_Nat_Onat___Orderings_Ono__top,axiom,
% 0.29/0.48 no_top @ nat ).
% 0.29/0.48
% 0.29/0.48 thf(tcon_Nat_Onat___Orderings_Oorder_2,axiom,
% 0.29/0.48 order @ nat ).
% 0.29/0.48
% 0.29/0.48 thf(tcon_Nat_Onat___Orderings_Oord_3,axiom,
% 0.29/0.48 ord @ nat ).
% 0.29/0.48
% 0.29/0.48 thf(tcon_Nat_Onat___Groups_Ozero,axiom,
% 0.29/0.48 zero @ nat ).
% 0.29/0.48
% 0.29/0.48 thf(tcon_Set_Oset___Orderings_Opreorder_4,axiom,
% 0.29/0.48 ! [A7: $tType] : ( preorder @ ( set @ A7 ) ) ).
% 0.29/0.48
% 0.29/0.48 thf(tcon_Set_Oset___Orderings_Oorder_5,axiom,
% 0.29/0.48 ! [A7: $tType] : ( order @ ( set @ A7 ) ) ).
% 0.29/0.48
% 0.29/0.48 thf(tcon_Set_Oset___Orderings_Oord_6,axiom,
% 0.29/0.48 ! [A7: $tType] : ( ord @ ( set @ A7 ) ) ).
% 0.29/0.48
% 0.29/0.48 thf(tcon_HOL_Obool___Orderings_Opreorder_7,axiom,
% 0.29/0.48 preorder @ $o ).
% 0.29/0.48
% 0.29/0.48 thf(tcon_HOL_Obool___Orderings_Olinorder_8,axiom,
% 0.29/0.48 linorder @ $o ).
% 0.29/0.48
% 0.29/0.48 thf(tcon_HOL_Obool___Orderings_Oorder_9,axiom,
% 0.29/0.48 order @ $o ).
% 0.29/0.48
% 0.29/0.48 thf(tcon_HOL_Obool___Orderings_Oord_10,axiom,
% 0.29/0.48 ord @ $o ).
% 0.29/0.48
% 0.29/0.48 thf(tcon_Real_Oreal___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_11,axiom,
% 0.29/0.48 condit1037483654norder @ real ).
% 0.29/0.48
% 0.29/0.48 thf(tcon_Real_Oreal___Conditionally__Complete__Lattices_Olinear__continuum,axiom,
% 0.29/0.48 condit1656338222tinuum @ real ).
% 0.29/0.48
% 0.29/0.48 thf(tcon_Real_Oreal___Real__Vector__Spaces_Oreal__normed__vector,axiom,
% 0.29/0.48 real_V55928688vector @ real ).
% 0.29/0.48
% 0.29/0.48 thf(tcon_Real_Oreal___Rings_Olinordered__nonzero__semiring_12,axiom,
% 0.29/0.48 linord1659791738miring @ real ).
% 0.29/0.48
% 0.29/0.48 thf(tcon_Real_Oreal___Rings_Oring__no__zero__divisors,axiom,
% 0.29/0.48 ring_n68954251visors @ real ).
% 0.29/0.48
% 0.29/0.48 thf(tcon_Real_Oreal___Orderings_Odense__linorder,axiom,
% 0.29/0.48 dense_linorder @ real ).
% 0.29/0.48
% 0.29/0.48 thf(tcon_Real_Oreal___Fields_Olinordered__field,axiom,
% 0.29/0.48 linordered_field @ real ).
% 0.29/0.48
% 0.29/0.48 thf(tcon_Real_Oreal___Rings_Olinordered__idom,axiom,
% 0.29/0.48 linordered_idom @ real ).
% 0.29/0.48
% 0.29/0.48 thf(tcon_Real_Oreal___Rings_Ocomm__semiring__1_13,axiom,
% 0.29/0.48 comm_semiring_1 @ real ).
% 0.29/0.48
% 0.29/0.48 thf(tcon_Real_Oreal___Rings_Ocomm__semiring__0_14,axiom,
% 0.29/0.48 comm_semiring_0 @ real ).
% 0.29/0.48
% 0.29/0.48 thf(tcon_Real_Oreal___Orderings_Odense__order,axiom,
% 0.29/0.48 dense_order @ real ).
% 0.29/0.48
% 0.29/0.48 thf(tcon_Real_Oreal___Rings_Oidom__abs__sgn,axiom,
% 0.29/0.48 idom_abs_sgn @ real ).
% 0.29/0.48
% 0.29/0.48 thf(tcon_Real_Oreal___Orderings_Opreorder_15,axiom,
% 0.29/0.48 preorder @ real ).
% 0.29/0.48
% 0.29/0.48 thf(tcon_Real_Oreal___Orderings_Olinorder_16,axiom,
% 0.29/0.48 linorder @ real ).
% 0.29/0.48
% 0.29/0.48 thf(tcon_Real_Oreal___Rings_Oidom__divide,axiom,
% 0.29/0.48 idom_divide @ real ).
% 0.29/0.48
% 0.29/0.48 thf(tcon_Real_Oreal___Rings_Ocomm__ring__1,axiom,
% 0.29/0.48 comm_ring_1 @ real ).
% 0.29/0.48
% 0.29/0.48 thf(tcon_Real_Oreal___Orderings_Ono__top_17,axiom,
% 0.29/0.48 no_top @ real ).
% 0.29/0.48
% 0.29/0.48 thf(tcon_Real_Oreal___Orderings_Ono__bot,axiom,
% 0.29/0.48 no_bot @ real ).
% 0.29/0.48
% 0.29/0.48 thf(tcon_Real_Oreal___Orderings_Oorder_18,axiom,
% 0.29/0.48 order @ real ).
% 0.29/0.48
% 0.29/0.48 thf(tcon_Real_Oreal___Nat_Oring__char__0,axiom,
% 0.29/0.48 ring_char_0 @ real ).
% 0.29/0.48
% 0.29/0.48 thf(tcon_Real_Oreal___Orderings_Oord_19,axiom,
% 0.29/0.48 ord @ real ).
% 0.29/0.48
% 0.29/0.48 thf(tcon_Real_Oreal___Groups_Ozero_20,axiom,
% 0.29/0.48 zero @ real ).
% 0.29/0.48
% 0.29/0.48 thf(tcon_Real_Oreal___Rings_Oidom,axiom,
% 0.29/0.48 idom @ real ).
% 0.29/0.48
% 0.29/0.48 thf(tcon_Polynomial_Opoly___Rings_Olinordered__nonzero__semiring_21,axiom,
% 0.29/0.48 ! [A7: $tType] :
% 0.29/0.48 ( ( linordered_idom @ A7 )
% 0.29/0.48 => ( linord1659791738miring @ ( poly @ A7 ) ) ) ).
% 0.29/0.48
% 0.29/0.48 thf(tcon_Polynomial_Opoly___Rings_Oring__no__zero__divisors_22,axiom,
% 0.29/0.48 ! [A7: $tType] :
% 0.29/0.48 ( ( idom @ A7 )
% 0.29/0.48 => ( ring_n68954251visors @ ( poly @ A7 ) ) ) ).
% 0.29/0.48
% 0.29/0.48 thf(tcon_Polynomial_Opoly___Rings_Olinordered__idom_23,axiom,
% 0.29/0.48 ! [A7: $tType] :
% 0.29/0.48 ( ( linordered_idom @ A7 )
% 0.29/0.48 => ( linordered_idom @ ( poly @ A7 ) ) ) ).
% 0.29/0.48
% 0.29/0.48 thf(tcon_Polynomial_Opoly___Rings_Ocomm__semiring__1_24,axiom,
% 0.29/0.48 ! [A7: $tType] :
% 0.29/0.48 ( ( comm_semiring_1 @ A7 )
% 0.29/0.48 => ( comm_semiring_1 @ ( poly @ A7 ) ) ) ).
% 0.29/0.48
% 0.29/0.48 thf(tcon_Polynomial_Opoly___Rings_Ocomm__semiring__0_25,axiom,
% 0.29/0.48 ! [A7: $tType] :
% 0.29/0.48 ( ( comm_semiring_0 @ A7 )
% 0.29/0.48 => ( comm_semiring_0 @ ( poly @ A7 ) ) ) ).
% 0.29/0.48
% 0.29/0.48 thf(tcon_Polynomial_Opoly___Rings_Oidom__abs__sgn_26,axiom,
% 0.29/0.48 ! [A7: $tType] :
% 0.29/0.48 ( ( linordered_idom @ A7 )
% 0.29/0.48 => ( idom_abs_sgn @ ( poly @ A7 ) ) ) ).
% 0.29/0.48
% 0.29/0.48 thf(tcon_Polynomial_Opoly___Orderings_Opreorder_27,axiom,
% 0.29/0.48 ! [A7: $tType] :
% 0.29/0.48 ( ( linordered_idom @ A7 )
% 0.29/0.48 => ( preorder @ ( poly @ A7 ) ) ) ).
% 0.29/0.48
% 0.29/0.48 thf(tcon_Polynomial_Opoly___Orderings_Olinorder_28,axiom,
% 0.29/0.48 ! [A7: $tType] :
% 0.29/0.48 ( ( linordered_idom @ A7 )
% 0.29/0.48 => ( linorder @ ( poly @ A7 ) ) ) ).
% 0.29/0.48
% 0.29/0.48 thf(tcon_Polynomial_Opoly___Rings_Oidom__divide_29,axiom,
% 0.29/0.48 ! [A7: $tType] :
% 0.29/0.48 ( ( idom_divide @ A7 )
% 0.29/0.48 => ( idom_divide @ ( poly @ A7 ) ) ) ).
% 0.29/0.48
% 0.29/0.48 thf(tcon_Polynomial_Opoly___Rings_Ocomm__ring__1_30,axiom,
% 0.29/0.48 ! [A7: $tType] :
% 0.29/0.48 ( ( comm_ring_1 @ A7 )
% 0.29/0.48 => ( comm_ring_1 @ ( poly @ A7 ) ) ) ).
% 0.29/0.48
% 0.29/0.48 thf(tcon_Polynomial_Opoly___Orderings_Oorder_31,axiom,
% 0.29/0.48 ! [A7: $tType] :
% 0.29/0.48 ( ( linordered_idom @ A7 )
% 0.29/0.48 => ( order @ ( poly @ A7 ) ) ) ).
% 0.29/0.48
% 0.29/0.48 thf(tcon_Polynomial_Opoly___Nat_Oring__char__0_32,axiom,
% 0.29/0.48 ! [A7: $tType] :
% 0.29/0.48 ( ( ( ring_char_0 @ A7 )
% 0.29/0.48 & ( comm_ring_1 @ A7 ) )
% 0.29/0.48 => ( ring_char_0 @ ( poly @ A7 ) ) ) ).
% 0.29/0.48
% 0.29/0.48 thf(tcon_Polynomial_Opoly___Orderings_Oord_33,axiom,
% 0.29/0.48 ! [A7: $tType] :
% 0.29/0.48 ( ( linordered_idom @ A7 )
% 0.29/0.48 => ( ord @ ( poly @ A7 ) ) ) ).
% 0.29/0.48
% 0.29/0.48 thf(tcon_Polynomial_Opoly___Groups_Ozero_34,axiom,
% 0.29/0.48 ! [A7: $tType] :
% 0.29/0.48 ( ( zero @ A7 )
% 0.29/0.48 => ( zero @ ( poly @ A7 ) ) ) ).
% 0.29/0.48
% 0.29/0.48 thf(tcon_Polynomial_Opoly___Rings_Oidom_35,axiom,
% 0.29/0.48 ! [A7: $tType] :
% 0.29/0.48 ( ( idom @ A7 )
% 0.29/0.48 => ( idom @ ( poly @ A7 ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % Helper facts (3)
% 0.29/0.48 thf(help_If_3_1_T,axiom,
% 0.29/0.48 ! [P3: $o] :
% 0.29/0.48 ( ( P3 = $true )
% 0.29/0.48 | ( P3 = $false ) ) ).
% 0.29/0.48
% 0.29/0.48 thf(help_If_2_1_T,axiom,
% 0.29/0.48 ! [A: $tType,X2: A,Y: A] :
% 0.29/0.48 ( ( if @ A @ $false @ X2 @ Y )
% 0.29/0.48 = Y ) ).
% 0.29/0.48
% 0.29/0.48 thf(help_If_1_1_T,axiom,
% 0.29/0.48 ! [A: $tType,X2: A,Y: A] :
% 0.29/0.48 ( ( if @ A @ $true @ X2 @ Y )
% 0.29/0.48 = X2 ) ).
% 0.29/0.48
% 0.29/0.48 % Conjectures (1)
% 0.29/0.48 thf(conj_0,conjecture,
% 0.29/0.48 ! [X3: real] :
% 0.29/0.48 ( ( ( poly2 @ real @ p @ X3 )
% 0.29/0.48 != ( zero_zero @ real ) )
% 0.29/0.48 | ( ord_less @ real @ lb @ X3 ) ) ).
% 0.29/0.48
% 0.29/0.48 %------------------------------------------------------------------------------
% 0.29/0.48 ------------------------
% 2.75/1.52 ---- Embedded in THF ---
% 2.75/1.52 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 2.75/1.52 %%% Generated on Tue Feb 24 08:29:30 EST 2026
% 2.75/1.52 %%% using '$$dhol' embedding, version 1.3.0.
% 2.75/1.52 %%% Logic specification used:
% 2.75/1.52 %%% thf(spec, logic, $$dhol).
% 2.75/1.52
% 2.75/1.52 % SZS output start ListOfTHF for /export/starexec/sandbox2/tmp/tmp.TYEgfeW64B/DTF2DTF21200.p
% 2.75/1.52 thf(ty_t_Polynomial_Opoly, type, poly: ($tType > $tType)).
% 2.75/1.52 thf(ty_t_Real_Oreal, type, real: $tType).
% 2.75/1.52 thf(ty_t_Set_Oset, type, set: ($tType > $tType)).
% 2.75/1.52 thf(ty_t_Nat_Onat, type, nat: $tType).
% 2.75/1.52 thf(sy_cl_HOL_Otype, type, type: (!> [A:$tType]: ($o))).
% 2.75/1.52 thf(sy_cl_Rings_Oidom, type, idom: (!> [A:$tType]: ($o))).
% 2.75/1.52 thf(sy_cl_Groups_Ozero, type, zero: (!> [A:$tType]: ($o))).
% 2.75/1.52 thf(sy_cl_Orderings_Oord, type, ord: (!> [A:$tType]: ($o))).
% 2.75/1.52 thf(sy_cl_Nat_Oring__char__0, type, ring_char_0: (!> [A:$tType]: ($o))).
% 2.75/1.52 thf(sy_cl_Orderings_Oorder, type, order: (!> [A:$tType]: ($o))).
% 2.75/1.52 thf(sy_cl_Orderings_Ono__bot, type, no_bot: (!> [A:$tType]: ($o))).
% 2.75/1.52 thf(sy_cl_Orderings_Ono__top, type, no_top: (!> [A:$tType]: ($o))).
% 2.75/1.52 thf(sy_cl_Rings_Ocomm__ring__1, type, comm_ring_1: (!> [A:$tType]: ($o))).
% 2.75/1.52 thf(sy_cl_Rings_Oidom__divide, type, idom_divide: (!> [A:$tType]: ($o))).
% 2.75/1.52 thf(sy_cl_Orderings_Olinorder, type, linorder: (!> [A:$tType]: ($o))).
% 2.75/1.52 thf(sy_cl_Orderings_Opreorder, type, preorder: (!> [A:$tType]: ($o))).
% 2.75/1.52 thf(sy_cl_Rings_Oidom__abs__sgn, type, idom_abs_sgn: (!> [A:$tType]: ($o))).
% 2.75/1.52 thf(sy_cl_Orderings_Owellorder, type, wellorder: (!> [A:$tType]: ($o))).
% 2.75/1.52 thf(sy_cl_Orderings_Odense__order, type, dense_order: (!> [A:$tType]: ($o))).
% 2.75/1.52 thf(sy_cl_Rings_Ocomm__semiring__0, type, comm_semiring_0: (!> [A:$tType]: ($o))).
% 2.75/1.52 thf(sy_cl_Rings_Ocomm__semiring__1, type, comm_semiring_1: (!> [A:$tType]: ($o))).
% 2.75/1.52 thf(sy_cl_Rings_Olinordered__idom, type, linordered_idom: (!> [A:$tType]: ($o))).
% 2.75/1.52 thf(sy_cl_Fields_Olinordered__field, type, linordered_field: (!> [A:$tType]: ($o))).
% 2.75/1.52 thf(sy_cl_Orderings_Odense__linorder, type, dense_linorder: (!> [A:$tType]: ($o))).
% 2.75/1.52 thf(sy_cl_Rings_Oring__no__zero__divisors, type, ring_n68954251visors: (!> [A:$tType]: ($o))).
% 2.75/1.52 thf(sy_cl_Rings_Olinordered__nonzero__semiring, type, linord1659791738miring: (!> [A:$tType]: ($o))).
% 2.75/1.52 thf(sy_cl_Groups_Ocanonically__ordered__monoid__add, type, canoni770627133id_add: (!> [A:$tType]: ($o))).
% 2.75/1.52 thf(sy_cl_Real__Vector__Spaces_Oreal__normed__vector, type, real_V55928688vector: (!> [A:$tType]: ($o))).
% 2.75/1.52 thf(sy_cl_Conditionally__Complete__Lattices_Olinear__continuum, type, condit1656338222tinuum: (!> [A:$tType]: ($o))).
% 2.75/1.52 thf(sy_cl_Conditionally__Complete__Lattices_Oconditionally__complete__linorder, type, condit1037483654norder: (!> [A:$tType]: ($o))).
% 2.75/1.52 thf(sy_c_Groups_Oone__class_Oone, type, one_one: (!> [A:$tType]: (A))).
% 2.75/1.52 thf(sy_c_Groups_Osgn__class_Osgn, type, sgn_sgn: (!> [A:$tType]: ((A > A)))).
% 2.75/1.52 thf(sy_c_Groups_Ozero__class_Ozero, type, zero_zero: (!> [A:$tType]: (A))).
% 2.75/1.52 thf(sy_c_If, type, if: (!> [A:$tType]: (($o > (A > (A > A)))))).
% 2.75/1.52 thf(sy_c_Orderings_Oord__class_Oless, type, ord_less: (!> [A:$tType]: ((A > (A > $o))))).
% 2.75/1.52 thf(sy_c_Orderings_Oord__class_Oless__eq, type, ord_less_eq: (!> [A:$tType]: ((A > (A > $o))))).
% 2.75/1.52 thf(sy_c_Orderings_Oord__class_Omin, type, ord_min: (!> [A:$tType]: ((A > (A > A))))).
% 2.75/1.52 thf(sy_c_Polynomial_Odivide__poly__main, type, divide_poly_main: (!> [A:$tType]: ((A > ((poly @ A) > ((poly @ A) > ((poly @ A) > (nat > (nat > (poly @ A)))))))))).
% 2.75/1.52 thf(sy_c_Polynomial_Ois__zero, type, is_zero: (!> [A:$tType]: (((poly @ A) > $o)))).
% 2.75/1.52 thf(sy_c_Polynomial_Oorder, type, order2: (!> [A:$tType]: ((A > ((poly @ A) > nat))))).
% 2.75/1.52 thf(sy_c_Polynomial_Opoly, type, poly2: (!> [A:$tType]: (((poly @ A) > (A > A))))).
% 2.75/1.52 thf(sy_c_Polynomial_Opoly__cutoff, type, poly_cutoff: (!> [A:$tType]: ((nat > ((poly @ A) > (poly @ A)))))).
% 2.75/1.52 thf(sy_c_Polynomial_Opoly__shift, type, poly_shift: (!> [A:$tType]: ((nat > ((poly @ A) > (poly @ A)))))).
% 2.75/1.52 thf(sy_c_Polynomial_Oreflect__poly, type, reflect_poly: (!> [A:$tType]: (((poly @ A) > (poly @ A))))).
% 2.75/1.52 thf(sy_c_Set_OCollect, type, collect: (!> [A:$tType]: (((A > $o) > (set @ A))))).
% 2.75/1.52 thf(sy_c_Sturm__Tarski__Mirabelle__osyjafhhev_Osgn__neg__inf, type, sturm_1439882693eg_inf: (!> [A:$tType]: (((poly @ A) > A)))).
% 2.75/1.52 thf(sy_c_Sturm__Tarski__Mirabelle__osyjafhhev_Osgn__pos__inf, type, sturm_62398785os_inf: (!> [A:$tType]: (((poly @ A) > A)))).
% 2.75/1.52 thf(sy_c_member, type, member: (!> [A:$tType]: ((A > ((set @ A) > $o))))).
% 2.75/1.52 thf(sy_v_lb1____, type, lb1: real).
% 2.75/1.52 thf(sy_v_lb2____, type, lb2: real).
% 2.75/1.52 thf(sy_v_lb____, type, lb: real).
% 2.75/1.52 thf(sy_v_p, type, p: (poly @ real)).
% 2.75/1.52 thf(sy_v_thesis, type, thesis: $o).
% 2.75/1.52 thf(fact_0_lb1, axiom, (! [X:real]: ((((((poly2 @ real) @ p) @ X) = (zero_zero @ real)) => (((ord_less @ real) @ lb1) @ X))))).
% 2.75/1.52 thf(fact_1_lb__def, axiom, (lb = (((ord_min @ real) @ lb1) @ lb2))).
% 2.75/1.52 thf(fact_2__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062lb1_O_A_092_060forall_062x_O_Apoly_Ap_Ax_A_061_A0_A_092_060longrightarrow_062_Alb1_A_060_Ax_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062, axiom, (~ ((! [Lb1:real]: ((~ ((! [X:real]: ((((((poly2 @ real) @ p) @ X) = (zero_zero @ real)) => (((ord_less @ real) @ Lb1) @ X))))))))))).
% 2.75/1.52 thf(fact_3_assms, axiom, (p != (zero_zero @ (poly @ real)))).
% 2.75/1.52 thf(fact_4_poly__0, axiom, (! [A:$tType]: (((comm_semiring_0 @ A) => (! [X2:A]: (((((poly2 @ A) @ (zero_zero @ (poly @ A))) @ X2) = (zero_zero @ A)))))))).
% 2.75/1.52 thf(fact_5_not__gr__zero, axiom, (! [A:$tType]: (((canoni770627133id_add @ A) => (! [N:A]: (((~ ((((ord_less @ A) @ (zero_zero @ A)) @ N))) = (N = (zero_zero @ A))))))))).
% 2.75/1.52 thf(fact_6_poly__IVT__neg, axiom, (! [A2:real,B:real,P:(poly @ real)]: (((((ord_less @ real) @ A2) @ B) => ((((ord_less @ real) @ (zero_zero @ real)) @ (((poly2 @ real) @ P) @ A2)) => ((((ord_less @ real) @ (((poly2 @ real) @ P) @ B)) @ (zero_zero @ real)) => (? [X3:real]: (((((ord_less @ real) @ A2) @ X3) & ((((ord_less @ real) @ X3) @ B) & ((((poly2 @ real) @ P) @ X3) = (zero_zero @ real)))))))))))).
% 2.75/1.52 thf(fact_7_poly__IVT__pos, axiom, (! [A2:real,B:real,P:(poly @ real)]: (((((ord_less @ real) @ A2) @ B) => ((((ord_less @ real) @ (((poly2 @ real) @ P) @ A2)) @ (zero_zero @ real)) => ((((ord_less @ real) @ (zero_zero @ real)) @ (((poly2 @ real) @ P) @ B)) => (? [X3:real]: (((((ord_less @ real) @ A2) @ X3) & ((((ord_less @ real) @ X3) @ B) & ((((poly2 @ real) @ P) @ X3) = (zero_zero @ real)))))))))))).
% 2.75/1.52 thf(fact_8_poly__all__0__iff__0, axiom, (! [A:$tType]: ((((ring_char_0 @ A) & ((comm_ring_1 @ A) & (ring_n68954251visors @ A))) => (! [P:(poly @ A)]: (((! [X4:A]: (((((poly2 @ A) @ P) @ X4) = (zero_zero @ A)))) = (P = (zero_zero @ (poly @ A)))))))))).
% 2.75/1.52 thf(fact_9_less__numeral__extra_I3_J, axiom, (! [A:$tType]: (((linord1659791738miring @ A) => (~ ((((ord_less @ A) @ (zero_zero @ A)) @ (zero_zero @ A)))))))).
% 2.75/1.52 thf(fact_10_field__lbound__gt__zero, axiom, (! [A:$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))))))))))))).
% 2.75/1.52 thf(fact_11_gr__zeroI, axiom, (! [A:$tType]: (((canoni770627133id_add @ A) => (! [N:A]: (((N != (zero_zero @ A)) => (((ord_less @ A) @ (zero_zero @ A)) @ N)))))))).
% 2.75/1.52 thf(fact_12_not__less__zero, axiom, (! [A:$tType]: (((canoni770627133id_add @ A) => (! [N:A]: ((~ ((((ord_less @ A) @ N) @ (zero_zero @ A)))))))))).
% 2.75/1.52 thf(fact_13_gr__implies__not__zero, axiom, (! [A:$tType]: (((canoni770627133id_add @ A) => (! [M:A,N:A]: (((((ord_less @ A) @ M) @ N) => (N != (zero_zero @ A))))))))).
% 2.75/1.52 thf(fact_14_zero__less__iff__neq__zero, axiom, (! [A:$tType]: (((canoni770627133id_add @ A) => (! [N:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ N) = (N != (zero_zero @ A))))))))).
% 2.75/1.52 thf(fact_15_zero__reorient, axiom, (! [A:$tType]: (((zero @ A) => (! [X2:A]: ((((zero_zero @ A) = X2) = (X2 = (zero_zero @ A))))))))).
% 2.75/1.52 thf(fact_16_poly__eq__poly__eq__iff, axiom, (! [A:$tType]: ((((ring_char_0 @ A) & ((comm_ring_1 @ A) & (ring_n68954251visors @ A))) => (! [P:(poly @ A),Q:(poly @ A)]: (((((poly2 @ A) @ P) = ((poly2 @ A) @ Q)) = (P = Q)))))))).
% 2.75/1.52 thf(fact_17_min__less__iff__conj, axiom, (! [A:$tType]: (((linorder @ A) => (! [Z:A,X2:A,Y:A]: (((((ord_less @ A) @ Z) @ (((ord_min @ A) @ X2) @ Y)) = ((((ord_less @ A) @ Z) @ X2) & (((ord_less @ A) @ Z) @ Y))))))))).
% 2.75/1.52 thf(fact_18_min_Oidem, axiom, (! [A:$tType]: (((linorder @ A) => (! [A2:A]: (((((ord_min @ A) @ A2) @ A2) = A2))))))).
% 2.75/1.52 thf(fact_19_min_Oleft__idem, axiom, (! [A:$tType]: (((linorder @ A) => (! [A2:A,B:A]: (((((ord_min @ A) @ A2) @ (((ord_min @ A) @ A2) @ B)) = (((ord_min @ A) @ A2) @ B)))))))).
% 2.75/1.52 thf(fact_20_min_Oright__idem, axiom, (! [A:$tType]: (((linorder @ A) => (! [A2:A,B:A]: (((((ord_min @ A) @ (((ord_min @ A) @ A2) @ B)) @ B) = (((ord_min @ A) @ A2) @ B)))))))).
% 2.75/1.52 thf(fact_21_min_Ostrict__coboundedI2, axiom, (! [A:$tType]: (((linorder @ A) => (! [B:A,C:A,A2:A]: (((((ord_less @ A) @ B) @ C) => (((ord_less @ A) @ (((ord_min @ A) @ A2) @ B)) @ C)))))))).
% 2.75/1.52 thf(fact_22_min_Ostrict__coboundedI1, axiom, (! [A:$tType]: (((linorder @ A) => (! [A2:A,C:A,B:A]: (((((ord_less @ A) @ A2) @ C) => (((ord_less @ A) @ (((ord_min @ A) @ A2) @ B)) @ C)))))))).
% 2.75/1.52 thf(fact_23_min_Ostrict__order__iff, axiom, (! [A:$tType]: (((linorder @ A) => ((ord_less @ A) = (^ [A3:A,B2:A]: (((A3 = (((ord_min @ A) @ A3) @ B2)) & (A3 != B2))))))))).
% 2.75/1.52 thf(fact_24_min_Ostrict__boundedE, axiom, (! [A:$tType]: (((linorder @ A) => (! [A2:A,B:A,C:A]: (((((ord_less @ A) @ A2) @ (((ord_min @ A) @ B) @ C)) => (~ (((((ord_less @ A) @ A2) @ B) => (~ ((((ord_less @ A) @ A2) @ C))))))))))))).
% 2.75/1.52 thf(fact_25_min__less__iff__disj, axiom, (! [A:$tType]: (((linorder @ A) => (! [X2:A,Y:A,Z:A]: (((((ord_less @ A) @ (((ord_min @ A) @ X2) @ Y)) @ Z) = ((((ord_less @ A) @ X2) @ Z) | (((ord_less @ A) @ Y) @ Z))))))))).
% 2.75/1.52 thf(fact_26_is__zero__null, axiom, (! [A:$tType]: (((zero @ A) => ((is_zero @ A) = (^ [P2:(poly @ A)]: ((P2 = (zero_zero @ (poly @ A)))))))))).
% 2.75/1.52 thf(fact_27_min_Oleft__commute, axiom, (! [A:$tType]: (((linorder @ A) => (! [B:A,A2:A,C:A]: (((((ord_min @ A) @ B) @ (((ord_min @ A) @ A2) @ C)) = (((ord_min @ A) @ A2) @ (((ord_min @ A) @ B) @ C))))))))).
% 2.75/1.52 thf(fact_28_min_Ocommute, axiom, (! [A:$tType]: (((linorder @ A) => ((ord_min @ A) = (^ [A3:A,B2:A]: ((((ord_min @ A) @ B2) @ A3)))))))).
% 2.75/1.52 thf(fact_29_min_Oassoc, axiom, (! [A:$tType]: (((linorder @ A) => (! [A2:A,B:A,C:A]: (((((ord_min @ A) @ (((ord_min @ A) @ A2) @ B)) @ C) = (((ord_min @ A) @ A2) @ (((ord_min @ A) @ B) @ C))))))))).
% 2.75/1.52 thf(fact_30_poly__cutoff__0, axiom, (! [A:$tType]: (((zero @ A) => (! [N:nat]: (((((poly_cutoff @ A) @ N) @ (zero_zero @ (poly @ A))) = (zero_zero @ (poly @ A))))))))).
% 2.75/1.52 thf(fact_31_last__non__root__interval, axiom, (! [P:(poly @ real),Ub:real]: (((P != (zero_zero @ (poly @ real))) => (~ ((! [Lb:real]: (((((ord_less @ real) @ Lb) @ Ub) => (~ ((! [Z2:real]: ((((((ord_less_eq @ real) @ Lb) @ Z2) & (((ord_less @ real) @ Z2) @ Ub)) => ((((poly2 @ real) @ P) @ Z2) != (zero_zero @ real)))))))))))))))).
% 2.75/1.52 thf(fact_32_next__non__root__interval, axiom, (! [P:(poly @ real),Lb2:real]: (((P != (zero_zero @ (poly @ real))) => (~ ((! [Ub2:real]: (((((ord_less @ real) @ Lb2) @ Ub2) => (~ ((! [Z2:real]: ((((((ord_less @ real) @ Lb2) @ Z2) & (((ord_less_eq @ real) @ Z2) @ Ub2)) => ((((poly2 @ real) @ P) @ Z2) != (zero_zero @ real)))))))))))))))).
% 2.75/1.52 thf(fact_33_reflect__poly__at__0__eq__0__iff, axiom, (! [A:$tType]: (((comm_semiring_0 @ A) => (! [P:(poly @ A)]: ((((((poly2 @ A) @ ((reflect_poly @ A) @ P)) @ (zero_zero @ A)) = (zero_zero @ A)) = (P = (zero_zero @ (poly @ A)))))))))).
% 2.75/1.52 thf(fact_34_lb2, axiom, (! [X:real]: (((((ord_less_eq @ real) @ X) @ lb2) => (((sgn_sgn @ real) @ (((poly2 @ real) @ p) @ X)) = ((sturm_1439882693eg_inf @ real) @ p)))))).
% 2.75/1.52 thf(fact_35_poly__shift__0, axiom, (! [A:$tType]: (((zero @ A) => (! [N:nat]: (((((poly_shift @ A) @ N) @ (zero_zero @ (poly @ A))) = (zero_zero @ (poly @ A))))))))).
% 2.75/1.52 thf(fact_36_order__root, axiom, (! [A:$tType]: (((idom @ A) => (! [P:(poly @ A),A2:A]: ((((((poly2 @ A) @ P) @ A2) = (zero_zero @ A)) = ((P = (zero_zero @ (poly @ A))) | ((((order2 @ A) @ A2) @ P) != (zero_zero @ nat)))))))))).
% 2.75/1.52 thf(fact_37_divide__poly__main__0, axiom, (! [A:$tType]: (((idom_divide @ A) => (! [R:(poly @ A),D:(poly @ A),Dr:nat,N:nat]: (((((((((divide_poly_main @ A) @ (zero_zero @ A)) @ (zero_zero @ (poly @ A))) @ R) @ D) @ Dr) @ N) = (zero_zero @ (poly @ A))))))))).
% 2.75/1.52 thf(fact_38_not__eq__pos__or__neg__iff__1, axiom, (! [Lb2:real,Ub:real,P:(poly @ real)]: (((! [Z3:real]: ((((((ord_less @ real) @ Lb2) @ Z3) & (((ord_less_eq @ real) @ Z3) @ Ub)) => ((((poly2 @ real) @ P) @ Z3) != (zero_zero @ real))))) = ((! [Z3:real]: ((((((ord_less @ real) @ Lb2) @ Z3) & (((ord_less_eq @ real) @ Z3) @ Ub)) => (((ord_less @ real) @ (zero_zero @ real)) @ (((poly2 @ real) @ P) @ Z3))))) | (! [Z3:real]: ((((((ord_less @ real) @ Lb2) @ Z3) & (((ord_less_eq @ real) @ Z3) @ Ub)) => (((ord_less @ real) @ (((poly2 @ real) @ P) @ Z3)) @ (zero_zero @ real)))))))))).
% 2.75/1.52 thf(fact_39__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062lb2_O_A_092_060forall_062x_092_060le_062lb2_O_Asgn_A_Ipoly_Ap_Ax_J_A_061_Asgn__neg__inf_Ap_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062, axiom, (~ ((! [Lb22:real]: ((~ ((! [X:real]: (((((ord_less_eq @ real) @ X) @ Lb22) => (((sgn_sgn @ real) @ (((poly2 @ real) @ p) @ X)) = ((sturm_1439882693eg_inf @ real) @ p)))))))))))).
% 2.75/1.52 thf(fact_40_le__zero__eq, axiom, (! [A:$tType]: (((canoni770627133id_add @ A) => (! [N:A]: (((((ord_less_eq @ A) @ N) @ (zero_zero @ A)) = (N = (zero_zero @ A))))))))).
% 2.75/1.52 thf(fact_41_min_Obounded__iff, axiom, (! [A:$tType]: (((linorder @ A) => (! [A2:A,B:A,C:A]: (((((ord_less_eq @ A) @ A2) @ (((ord_min @ A) @ B) @ C)) = ((((ord_less_eq @ A) @ A2) @ B) & (((ord_less_eq @ A) @ A2) @ C))))))))).
% 2.75/1.52 thf(fact_42_reflect__poly__0, axiom, (! [A:$tType]: (((zero @ A) => (((reflect_poly @ A) @ (zero_zero @ (poly @ A))) = (zero_zero @ (poly @ A))))))).
% 2.75/1.52 thf(fact_43_poly__shift__id, axiom, (! [A:$tType]: (((zero @ A) => (((poly_shift @ A) @ (zero_zero @ nat)) = (^ [X4:(poly @ A)]: (X4))))))).
% 2.75/1.52 thf(fact_44_mem__Collect__eq, axiom, (! [A:$tType,A2:A,P3:(A > $o)]: (((((member @ A) @ A2) @ ((collect @ A) @ P3)) = (P3 @ A2))))).
% 2.75/1.52 thf(fact_45_Collect__mem__eq, axiom, (! [A:$tType,A4:(set @ A)]: ((((collect @ A) @ (^ [X4:A]: ((((member @ A) @ X4) @ A4)))) = A4)))).
% 2.75/1.52 thf(fact_46_Collect__cong, axiom, (! [A:$tType,P3:(A > $o),Q2:(A > $o)]: (((! [X3:A]: (((P3 @ X3) = (Q2 @ X3)))) => (((collect @ A) @ P3) = ((collect @ A) @ Q2)))))).
% 2.75/1.52 thf(fact_47_ext, axiom, (! [B3:$tType,A:$tType,F:(A > B3),G:(A > B3)]: (((! [X3:A]: (((F @ X3) = (G @ X3)))) => (F = G))))).
% 2.75/1.52 thf(fact_48_that, axiom, (! [Lb2:real]: (((! [X3:real]: ((((((poly2 @ real) @ p) @ X3) = (zero_zero @ real)) => (((ord_less @ real) @ Lb2) @ X3)))) => ((! [X3:real]: (((((ord_less_eq @ real) @ X3) @ Lb2) => (((sgn_sgn @ real) @ (((poly2 @ real) @ p) @ X3)) = ((sturm_1439882693eg_inf @ real) @ p))))) => thesis))))).
% 2.75/1.52 thf(fact_49_complete__real, axiom, (! [S:(set @ real)]: (((? [X:real]: ((((member @ real) @ X) @ S))) => ((? [Z2:real]: ((! [X3:real]: (((((member @ real) @ X3) @ S) => (((ord_less_eq @ real) @ X3) @ Z2)))))) => (? [Y2:real]: (((! [X:real]: (((((member @ real) @ X) @ S) => (((ord_less_eq @ real) @ X) @ Y2)))) & (! [Z2:real]: (((! [X3:real]: (((((member @ real) @ X3) @ S) => (((ord_less_eq @ real) @ X3) @ Z2)))) => (((ord_less_eq @ real) @ Y2) @ Z2)))))))))))).
% 2.75/1.52 thf(fact_50_divide__poly__main_Osimps_I2_J, axiom, (! [A:$tType]: (((idom_divide @ A) => (! [Lc:A,Q:(poly @ A),R:(poly @ A),D:(poly @ A),Dr:nat]: (((((((((divide_poly_main @ A) @ Lc) @ Q) @ R) @ D) @ Dr) @ (zero_zero @ nat)) = Q))))))).
% 2.75/1.52 thf(fact_51_zero__le, axiom, (! [A:$tType]: (((canoni770627133id_add @ A) => (! [X2:A]: ((((ord_less_eq @ A) @ (zero_zero @ A)) @ X2))))))).
% 2.75/1.52 thf(fact_52_le__numeral__extra_I3_J, axiom, (! [A:$tType]: (((linord1659791738miring @ A) => (((ord_less_eq @ A) @ (zero_zero @ A)) @ (zero_zero @ A)))))).
% 2.75/1.52 thf(fact_53_less__eq__real__def, axiom, ((ord_less_eq @ real) = (^ [X4:real,Y3:real]: (((((ord_less @ real) @ X4) @ Y3) | (X4 = Y3)))))).
% 2.75/1.52 thf(fact_54_min_Omono, axiom, (! [A:$tType]: (((linorder @ A) => (! [A2:A,C:A,B:A,D:A]: (((((ord_less_eq @ A) @ A2) @ C) => ((((ord_less_eq @ A) @ B) @ D) => (((ord_less_eq @ A) @ (((ord_min @ A) @ A2) @ B)) @ (((ord_min @ A) @ C) @ D)))))))))).
% 2.75/1.52 thf(fact_55_min_OorderE, axiom, (! [A:$tType]: (((linorder @ A) => (! [A2:A,B:A]: (((((ord_less_eq @ A) @ A2) @ B) => (A2 = (((ord_min @ A) @ A2) @ B))))))))).
% 2.75/1.52 thf(fact_56_min_OorderI, axiom, (! [A:$tType]: (((linorder @ A) => (! [A2:A,B:A]: (((A2 = (((ord_min @ A) @ A2) @ B)) => (((ord_less_eq @ A) @ A2) @ B)))))))).
% 2.75/1.52 thf(fact_57_min_Oabsorb1, axiom, (! [A:$tType]: (((linorder @ A) => (! [A2:A,B:A]: (((((ord_less_eq @ A) @ A2) @ B) => ((((ord_min @ A) @ A2) @ B) = A2)))))))).
% 2.75/1.52 thf(fact_58_min_Oabsorb2, axiom, (! [A:$tType]: (((linorder @ A) => (! [B:A,A2:A]: (((((ord_less_eq @ A) @ B) @ A2) => ((((ord_min @ A) @ A2) @ B) = B)))))))).
% 2.75/1.52 thf(fact_59_min_OboundedE, axiom, (! [A:$tType]: (((linorder @ A) => (! [A2:A,B:A,C:A]: (((((ord_less_eq @ A) @ A2) @ (((ord_min @ A) @ B) @ C)) => (~ (((((ord_less_eq @ A) @ A2) @ B) => (~ ((((ord_less_eq @ A) @ A2) @ C))))))))))))).
% 2.75/1.52 thf(fact_60_min_OboundedI, axiom, (! [A:$tType]: (((linorder @ A) => (! [A2:A,B:A,C:A]: (((((ord_less_eq @ A) @ A2) @ B) => ((((ord_less_eq @ A) @ A2) @ C) => (((ord_less_eq @ A) @ A2) @ (((ord_min @ A) @ B) @ C)))))))))).
% 2.75/1.52 thf(fact_61_min_Oorder__iff, axiom, (! [A:$tType]: (((linorder @ A) => ((ord_less_eq @ A) = (^ [A3:A,B2:A]: ((A3 = (((ord_min @ A) @ A3) @ B2))))))))).
% 2.75/1.52 thf(fact_62_min_Ocobounded1, axiom, (! [A:$tType]: (((linorder @ A) => (! [A2:A,B:A]: ((((ord_less_eq @ A) @ (((ord_min @ A) @ A2) @ B)) @ A2))))))).
% 2.75/1.52 thf(fact_63_min_Ocobounded2, axiom, (! [A:$tType]: (((linorder @ A) => (! [A2:A,B:A]: ((((ord_less_eq @ A) @ (((ord_min @ A) @ A2) @ B)) @ B))))))).
% 2.75/1.52 thf(fact_64_min_Oabsorb__iff1, axiom, (! [A:$tType]: (((linorder @ A) => ((ord_less_eq @ A) = (^ [A3:A,B2:A]: (((((ord_min @ A) @ A3) @ B2) = A3)))))))).
% 2.75/1.52 thf(fact_65_min_Oabsorb__iff2, axiom, (! [A:$tType]: (((linorder @ A) => ((ord_less_eq @ A) = (^ [B2:A,A3:A]: (((((ord_min @ A) @ A3) @ B2) = B2)))))))).
% 2.75/1.52 thf(fact_66_min_OcoboundedI1, axiom, (! [A:$tType]: (((linorder @ A) => (! [A2:A,C:A,B:A]: (((((ord_less_eq @ A) @ A2) @ C) => (((ord_less_eq @ A) @ (((ord_min @ A) @ A2) @ B)) @ C)))))))).
% 2.75/1.52 thf(fact_67_min_OcoboundedI2, axiom, (! [A:$tType]: (((linorder @ A) => (! [B:A,C:A,A2:A]: (((((ord_less_eq @ A) @ B) @ C) => (((ord_less_eq @ A) @ (((ord_min @ A) @ A2) @ B)) @ C)))))))).
% 2.75/1.52 thf(fact_68_min__le__iff__disj, axiom, (! [A:$tType]: (((linorder @ A) => (! [X2:A,Y:A,Z:A]: (((((ord_less_eq @ A) @ (((ord_min @ A) @ X2) @ Y)) @ Z) = ((((ord_less_eq @ A) @ X2) @ Z) | (((ord_less_eq @ A) @ Y) @ Z))))))))).
% 2.75/1.52 thf(fact_69_order__0I, axiom, (! [A:$tType]: (((idom @ A) => (! [P:(poly @ A),A2:A]: ((((((poly2 @ A) @ P) @ A2) != (zero_zero @ A)) => ((((order2 @ A) @ A2) @ P) = (zero_zero @ nat))))))))).
% 2.75/1.52 thf(fact_70_not__eq__pos__or__neg__iff__2, axiom, (! [Lb2:real,Ub:real,P:(poly @ real)]: (((! [Z3:real]: ((((((ord_less_eq @ real) @ Lb2) @ Z3) & (((ord_less @ real) @ Z3) @ Ub)) => ((((poly2 @ real) @ P) @ Z3) != (zero_zero @ real))))) = ((! [Z3:real]: ((((((ord_less_eq @ real) @ Lb2) @ Z3) & (((ord_less @ real) @ Z3) @ Ub)) => (((ord_less @ real) @ (zero_zero @ real)) @ (((poly2 @ real) @ P) @ Z3))))) | (! [Z3:real]: ((((((ord_less_eq @ real) @ Lb2) @ Z3) & (((ord_less @ real) @ Z3) @ Ub)) => (((ord_less @ real) @ (((poly2 @ real) @ P) @ Z3)) @ (zero_zero @ real)))))))))).
% 2.75/1.52 thf(fact_71_zero__le__sgn__iff, axiom, (! [X2:real]: (((((ord_less_eq @ real) @ (zero_zero @ real)) @ ((sgn_sgn @ real) @ X2)) = (((ord_less_eq @ real) @ (zero_zero @ real)) @ X2))))).
% 2.75/1.52 thf(fact_72_sgn__le__0__iff, axiom, (! [X2:real]: (((((ord_less_eq @ real) @ ((sgn_sgn @ real) @ X2)) @ (zero_zero @ real)) = (((ord_less_eq @ real) @ X2) @ (zero_zero @ real)))))).
% 2.75/1.52 thf(fact_73_sgn__greater, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [A2:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ ((sgn_sgn @ A) @ A2)) = (((ord_less @ A) @ (zero_zero @ A)) @ A2)))))))).
% 2.75/1.52 thf(fact_74_sgn__less, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [A2:A]: (((((ord_less @ A) @ ((sgn_sgn @ A) @ A2)) @ (zero_zero @ A)) = (((ord_less @ A) @ A2) @ (zero_zero @ A))))))))).
% 2.75/1.52 thf(fact_75_root__ub, axiom, (! [P:(poly @ real)]: (((P != (zero_zero @ (poly @ real))) => (~ ((! [Ub2:real]: (((! [X:real]: ((((((poly2 @ real) @ P) @ X) = (zero_zero @ real)) => (((ord_less @ real) @ X) @ Ub2)))) => (~ ((! [X:real]: (((((ord_less_eq @ real) @ Ub2) @ X) => (((sgn_sgn @ real) @ (((poly2 @ real) @ P) @ X)) = ((sturm_62398785os_inf @ real) @ P)))))))))))))))).
% 2.75/1.52 thf(fact_76_sgn__0, axiom, (! [A:$tType]: (((idom_abs_sgn @ A) => (((sgn_sgn @ A) @ (zero_zero @ A)) = (zero_zero @ A)))))).
% 2.75/1.52 thf(fact_77_sgn__zero, axiom, (! [A:$tType]: (((real_V55928688vector @ A) => (((sgn_sgn @ A) @ (zero_zero @ A)) = (zero_zero @ A)))))).
% 2.75/1.52 thf(fact_78_sgn__sgn, axiom, (! [A:$tType]: (((idom_abs_sgn @ A) => (! [A2:A]: ((((sgn_sgn @ A) @ ((sgn_sgn @ A) @ A2)) = ((sgn_sgn @ A) @ A2)))))))).
% 2.75/1.52 thf(fact_79_linorder__neqE__linordered__idom, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [X2:A,Y:A]: (((X2 != Y) => ((~ ((((ord_less @ A) @ X2) @ Y))) => (((ord_less @ A) @ Y) @ X2))))))))).
% 2.75/1.52 thf(fact_80_sgn__eq__0__iff, axiom, (! [A:$tType]: (((idom_abs_sgn @ A) => (! [A2:A]: (((((sgn_sgn @ A) @ A2) = (zero_zero @ A)) = (A2 = (zero_zero @ A))))))))).
% 2.75/1.52 thf(fact_81_sgn__0__0, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [A2:A]: (((((sgn_sgn @ A) @ A2) = (zero_zero @ A)) = (A2 = (zero_zero @ A))))))))).
% 2.75/1.52 thf(fact_82_sgn__zero__iff, axiom, (! [A:$tType]: (((real_V55928688vector @ A) => (! [X2:A]: (((((sgn_sgn @ A) @ X2) = (zero_zero @ A)) = (X2 = (zero_zero @ A))))))))).
% 2.75/1.52 thf(fact_83_le0, axiom, (! [N:nat]: ((((ord_less_eq @ nat) @ (zero_zero @ nat)) @ N)))).
% 2.75/1.52 thf(fact_84_bot__nat__0_Oextremum, axiom, (! [A2:nat]: ((((ord_less_eq @ nat) @ (zero_zero @ nat)) @ A2)))).
% 2.75/1.52 thf(fact_85_neq0__conv, axiom, (! [N:nat]: (((N != (zero_zero @ nat)) = (((ord_less @ nat) @ (zero_zero @ nat)) @ N))))).
% 2.75/1.52 thf(fact_86_less__nat__zero__code, axiom, (! [N:nat]: ((~ ((((ord_less @ nat) @ N) @ (zero_zero @ nat))))))).
% 2.75/1.52 thf(fact_87_bot__nat__0_Onot__eq__extremum, axiom, (! [A2:nat]: (((A2 != (zero_zero @ nat)) = (((ord_less @ nat) @ (zero_zero @ nat)) @ A2))))).
% 2.75/1.52 thf(fact_88_min__0R, axiom, (! [N:nat]: (((((ord_min @ nat) @ N) @ (zero_zero @ nat)) = (zero_zero @ nat))))).
% 2.75/1.52 thf(fact_89_min__0L, axiom, (! [N:nat]: (((((ord_min @ nat) @ (zero_zero @ nat)) @ N) = (zero_zero @ nat))))).
% 2.75/1.52 thf(fact_90_le__refl, axiom, (! [N:nat]: ((((ord_less_eq @ nat) @ N) @ N)))).
% 2.75/1.52 thf(fact_91_le__trans, axiom, (! [I:nat,J:nat,K:nat]: (((((ord_less_eq @ nat) @ I) @ J) => ((((ord_less_eq @ nat) @ J) @ K) => (((ord_less_eq @ nat) @ I) @ K)))))).
% 2.75/1.52 thf(fact_92_eq__imp__le, axiom, (! [M:nat,N:nat]: (((M = N) => (((ord_less_eq @ nat) @ M) @ N))))).
% 2.75/1.52 thf(fact_93_le__antisym, axiom, (! [M:nat,N:nat]: (((((ord_less_eq @ nat) @ M) @ N) => ((((ord_less_eq @ nat) @ N) @ M) => (M = N)))))).
% 2.75/1.52 thf(fact_94_nat__le__linear, axiom, (! [M:nat,N:nat]: (((((ord_less_eq @ nat) @ M) @ N) | (((ord_less_eq @ nat) @ N) @ M))))).
% 2.75/1.52 thf(fact_95_Nat_Oex__has__greatest__nat, axiom, (! [P3:(nat > $o),K:nat,B:nat]: (((P3 @ K) => ((! [Y2:nat]: (((P3 @ Y2) => (((ord_less_eq @ nat) @ Y2) @ B)))) => (? [X3:nat]: (((P3 @ X3) & (! [Y4:nat]: (((P3 @ Y4) => (((ord_less_eq @ nat) @ Y4) @ X3)))))))))))).
% 2.75/1.52 thf(fact_96_less__mono__imp__le__mono, axiom, (! [F:(nat > nat),I:nat,J:nat]: (((! [I2:nat,J2:nat]: (((((ord_less @ nat) @ I2) @ J2) => (((ord_less @ nat) @ (F @ I2)) @ (F @ J2))))) => ((((ord_less_eq @ nat) @ I) @ J) => (((ord_less_eq @ nat) @ (F @ I)) @ (F @ J))))))).
% 2.75/1.52 thf(fact_97_le__neq__implies__less, axiom, (! [M:nat,N:nat]: (((((ord_less_eq @ nat) @ M) @ N) => ((M != N) => (((ord_less @ nat) @ M) @ N)))))).
% 2.75/1.52 thf(fact_98_less__or__eq__imp__le, axiom, (! [M:nat,N:nat]: ((((((ord_less @ nat) @ M) @ N) | (M = N)) => (((ord_less_eq @ nat) @ M) @ N))))).
% 2.75/1.52 thf(fact_99_le__eq__less__or__eq, axiom, ((ord_less_eq @ nat) = (^ [M2:nat,N2:nat]: (((((ord_less @ nat) @ M2) @ N2) | (M2 = N2)))))).
% 2.75/1.52 thf(fact_100_less__imp__le__nat, axiom, (! [M:nat,N:nat]: (((((ord_less @ nat) @ M) @ N) => (((ord_less_eq @ nat) @ M) @ N))))).
% 2.75/1.52 thf(fact_101_nat__less__le, axiom, ((ord_less @ nat) = (^ [M2:nat,N2:nat]: (((((ord_less_eq @ nat) @ M2) @ N2) & (M2 != N2)))))).
% 2.75/1.52 thf(fact_102_infinite__descent__measure, axiom, (! [A:$tType,P3:(A > $o),V:(A > nat),X2:A]: (((! [X3:A]: (((~ ((P3 @ X3))) => (? [Y4:A]: (((((ord_less @ nat) @ (V @ Y4)) @ (V @ X3)) & (~ ((P3 @ Y4))))))))) => (P3 @ X2))))).
% 2.75/1.52 thf(fact_103_linorder__neqE__nat, axiom, (! [X2:nat,Y:nat]: (((X2 != Y) => ((~ ((((ord_less @ nat) @ X2) @ Y))) => (((ord_less @ nat) @ Y) @ X2)))))).
% 2.75/1.52 thf(fact_104_infinite__descent, axiom, (! [P3:(nat > $o),N:nat]: (((! [N3:nat]: (((~ ((P3 @ N3))) => (? [M3:nat]: (((((ord_less @ nat) @ M3) @ N3) & (~ ((P3 @ M3))))))))) => (P3 @ N))))).
% 2.75/1.52 thf(fact_105_nat__less__induct, axiom, (! [P3:(nat > $o),N:nat]: (((! [N3:nat]: (((! [M3:nat]: (((((ord_less @ nat) @ M3) @ N3) => (P3 @ M3)))) => (P3 @ N3)))) => (P3 @ N))))).
% 2.75/1.52 thf(fact_106_less__irrefl__nat, axiom, (! [N:nat]: ((~ ((((ord_less @ nat) @ N) @ N)))))).
% 2.75/1.52 thf(fact_107_less__not__refl3, axiom, (! [S2:nat,T:nat]: (((((ord_less @ nat) @ S2) @ T) => (S2 != T))))).
% 2.75/1.52 thf(fact_108_less__not__refl2, axiom, (! [N:nat,M:nat]: (((((ord_less @ nat) @ N) @ M) => (M != N))))).
% 2.75/1.52 thf(fact_109_less__not__refl, axiom, (! [N:nat]: ((~ ((((ord_less @ nat) @ N) @ N)))))).
% 2.75/1.52 thf(fact_110_nat__neq__iff, axiom, (! [M:nat,N:nat]: (((M != N) = ((((ord_less @ nat) @ M) @ N) | (((ord_less @ nat) @ N) @ M)))))).
% 2.75/1.52 thf(fact_111_measure__induct__rule, axiom, (! [B3:$tType,A:$tType]: (((wellorder @ B3) => (! [F:(A > B3),P3:(A > $o),A2:A]: (((! [X3:A]: (((! [Y4:A]: (((((ord_less @ B3) @ (F @ Y4)) @ (F @ X3)) => (P3 @ Y4)))) => (P3 @ X3)))) => (P3 @ A2)))))))).
% 2.75/1.52 thf(fact_112_measure__induct, axiom, (! [B3:$tType,A:$tType]: (((wellorder @ B3) => (! [F:(A > B3),P3:(A > $o),A2:A]: (((! [X3:A]: (((! [Y4:A]: (((((ord_less @ B3) @ (F @ Y4)) @ (F @ X3)) => (P3 @ Y4)))) => (P3 @ X3)))) => (P3 @ A2)))))))).
% 2.75/1.52 thf(fact_113_infinite__descent0__measure, axiom, (! [A:$tType,V:(A > nat),P3:(A > $o),X2:A]: (((! [X3:A]: ((((V @ X3) = (zero_zero @ nat)) => (P3 @ X3)))) => ((! [X3:A]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ (V @ X3)) => ((~ ((P3 @ X3))) => (? [Y4:A]: (((((ord_less @ nat) @ (V @ Y4)) @ (V @ X3)) & (~ ((P3 @ Y4)))))))))) => (P3 @ X2)))))).
% 2.75/1.52 thf(fact_114_bot__nat__0_Oextremum__strict, axiom, (! [A2:nat]: ((~ ((((ord_less @ nat) @ A2) @ (zero_zero @ nat))))))).
% 2.75/1.52 thf(fact_115_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)))))).
% 2.75/1.52 thf(fact_116_gr__implies__not0, axiom, (! [M:nat,N:nat]: (((((ord_less @ nat) @ M) @ N) => (N != (zero_zero @ nat)))))).
% 2.75/1.52 thf(fact_117_less__zeroE, axiom, (! [N:nat]: ((~ ((((ord_less @ nat) @ N) @ (zero_zero @ nat))))))).
% 2.75/1.52 thf(fact_118_not__less0, axiom, (! [N:nat]: ((~ ((((ord_less @ nat) @ N) @ (zero_zero @ nat))))))).
% 2.75/1.52 thf(fact_119_not__gr0, axiom, (! [N:nat]: (((~ ((((ord_less @ nat) @ (zero_zero @ nat)) @ N))) = (N = (zero_zero @ nat)))))).
% 2.75/1.52 thf(fact_120_gr0I, axiom, (! [N:nat]: (((N != (zero_zero @ nat)) => (((ord_less @ nat) @ (zero_zero @ nat)) @ N))))).
% 2.75/1.52 thf(fact_121_less__eq__nat_Osimps_I1_J, axiom, (! [N:nat]: ((((ord_less_eq @ nat) @ (zero_zero @ nat)) @ N)))).
% 2.75/1.52 thf(fact_122_le__0__eq, axiom, (! [N:nat]: (((((ord_less_eq @ nat) @ N) @ (zero_zero @ nat)) = (N = (zero_zero @ nat)))))).
% 2.75/1.52 thf(fact_123_ex__least__nat__le, axiom, (! [P3:(nat > $o),N:nat]: (((P3 @ N) => ((~ ((P3 @ (zero_zero @ nat)))) => (? [K2:nat]: (((((ord_less_eq @ nat) @ K2) @ N) & ((! [I3:nat]: (((((ord_less @ nat) @ I3) @ K2) => (~ ((P3 @ I3)))))) & (P3 @ K2)))))))))).
% 2.75/1.52 thf(fact_124_bot__nat__0_Oextremum__unique, axiom, (! [A2:nat]: (((((ord_less_eq @ nat) @ A2) @ (zero_zero @ nat)) = (A2 = (zero_zero @ nat)))))).
% 2.75/1.52 thf(fact_125_bot__nat__0_Oextremum__uniqueI, axiom, (! [A2:nat]: (((((ord_less_eq @ nat) @ A2) @ (zero_zero @ nat)) => (A2 = (zero_zero @ nat)))))).
% 2.75/1.52 thf(fact_126_order__refl, axiom, (! [A:$tType]: (((preorder @ A) => (! [X2:A]: ((((ord_less_eq @ A) @ X2) @ X2))))))).
% 2.75/1.52 thf(fact_127_min__absorb1, axiom, (! [A:$tType]: (((ord @ A) => (! [X2:A,Y:A]: (((((ord_less_eq @ A) @ X2) @ Y) => ((((ord_min @ A) @ X2) @ Y) = X2)))))))).
% 2.75/1.52 thf(fact_128_min__absorb2, axiom, (! [A:$tType]: (((order @ A) => (! [Y:A,X2:A]: (((((ord_less_eq @ A) @ Y) @ X2) => ((((ord_min @ A) @ X2) @ Y) = Y)))))))).
% 2.75/1.52 thf(fact_129_min__def, axiom, (! [A:$tType]: (((ord @ A) => ((ord_min @ A) = (^ [A3:A,B2:A]: (((((if @ A) @ (((ord_less_eq @ A) @ A3) @ B2)) @ A3) @ B2)))))))).
% 2.75/1.52 thf(fact_130_Lattices__Big_Oex__has__greatest__nat, axiom, (! [A:$tType,P3:(A > $o),K:A,F:(A > nat),B:nat]: (((P3 @ K) => ((! [Y2:A]: (((P3 @ Y2) => (((ord_less @ nat) @ (F @ Y2)) @ B)))) => (? [X3:A]: (((P3 @ X3) & (! [Y4:A]: (((P3 @ Y4) => (((ord_less_eq @ nat) @ (F @ Y4)) @ (F @ X3))))))))))))).
% 2.75/1.52 thf(fact_131_nat__descend__induct, axiom, (! [N:nat,P3:(nat > $o),M:nat]: (((! [K2:nat]: (((((ord_less @ nat) @ N) @ K2) => (P3 @ K2)))) => ((! [K2:nat]: (((((ord_less_eq @ nat) @ K2) @ N) => ((! [I3:nat]: (((((ord_less @ nat) @ K2) @ I3) => (P3 @ I3)))) => (P3 @ K2))))) => (P3 @ M)))))).
% 2.75/1.52 thf(fact_132_dual__order_Oantisym, axiom, (! [A:$tType]: (((order @ A) => (! [B:A,A2:A]: (((((ord_less_eq @ A) @ B) @ A2) => ((((ord_less_eq @ A) @ A2) @ B) => (A2 = B))))))))).
% 2.75/1.52 thf(fact_133_dual__order_Oeq__iff, axiom, (! [A:$tType]: (((order @ A) => ((^ [Y5:A,Z4:A]: ((Y5 = Z4))) = (^ [A3:A,B2:A]: (((((ord_less_eq @ A) @ B2) @ A3) & (((ord_less_eq @ A) @ A3) @ B2))))))))).
% 2.75/1.52 thf(fact_134_dual__order_Otrans, axiom, (! [A:$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))))))))).
% 2.75/1.52 thf(fact_135_linorder__wlog, axiom, (! [A:$tType]: (((linorder @ A) => (! [P3:(A > (A > $o)),A2:A,B:A]: (((! [A5:A,B4:A]: (((((ord_less_eq @ A) @ A5) @ B4) => ((P3 @ A5) @ B4)))) => ((! [A5:A,B4:A]: ((((P3 @ B4) @ A5) => ((P3 @ A5) @ B4)))) => ((P3 @ A2) @ B))))))))).
% 2.75/1.52 thf(fact_136_dual__order_Orefl, axiom, (! [A:$tType]: (((order @ A) => (! [A2:A]: ((((ord_less_eq @ A) @ A2) @ A2))))))).
% 2.75/1.52 thf(fact_137_order__trans, axiom, (! [A:$tType]: (((preorder @ A) => (! [X2:A,Y:A,Z:A]: (((((ord_less_eq @ A) @ X2) @ Y) => ((((ord_less_eq @ A) @ Y) @ Z) => (((ord_less_eq @ A) @ X2) @ Z))))))))).
% 2.75/1.52 thf(fact_138_order__class_Oorder_Oantisym, axiom, (! [A:$tType]: (((order @ A) => (! [A2:A,B:A]: (((((ord_less_eq @ A) @ A2) @ B) => ((((ord_less_eq @ A) @ B) @ A2) => (A2 = B))))))))).
% 2.75/1.52 thf(fact_139_ord__le__eq__trans, axiom, (! [A:$tType]: (((ord @ A) => (! [A2:A,B:A,C:A]: (((((ord_less_eq @ A) @ A2) @ B) => ((B = C) => (((ord_less_eq @ A) @ A2) @ C))))))))).
% 2.75/1.52 thf(fact_140_ord__eq__le__trans, axiom, (! [A:$tType]: (((ord @ A) => (! [A2:A,B:A,C:A]: (((A2 = B) => ((((ord_less_eq @ A) @ B) @ C) => (((ord_less_eq @ A) @ A2) @ C))))))))).
% 2.75/1.52 thf(fact_141_order__class_Oorder_Oeq__iff, axiom, (! [A:$tType]: (((order @ A) => ((^ [Y5:A,Z4:A]: ((Y5 = Z4))) = (^ [A3:A,B2:A]: (((((ord_less_eq @ A) @ A3) @ B2) & (((ord_less_eq @ A) @ B2) @ A3))))))))).
% 2.75/1.52 thf(fact_142_antisym__conv, axiom, (! [A:$tType]: (((order @ A) => (! [Y:A,X2:A]: (((((ord_less_eq @ A) @ Y) @ X2) => ((((ord_less_eq @ A) @ X2) @ Y) = (X2 = Y))))))))).
% 2.75/1.52 thf(fact_143_le__cases3, axiom, (! [A:$tType]: (((linorder @ A) => (! [X2:A,Y:A,Z:A]: ((((((ord_less_eq @ A) @ X2) @ Y) => (~ ((((ord_less_eq @ A) @ Y) @ Z)))) => (((((ord_less_eq @ A) @ Y) @ X2) => (~ ((((ord_less_eq @ A) @ X2) @ Z)))) => (((((ord_less_eq @ A) @ X2) @ Z) => (~ ((((ord_less_eq @ A) @ Z) @ Y)))) => (((((ord_less_eq @ A) @ Z) @ Y) => (~ ((((ord_less_eq @ A) @ Y) @ X2)))) => (((((ord_less_eq @ A) @ Y) @ Z) => (~ ((((ord_less_eq @ A) @ Z) @ X2)))) => (~ (((((ord_less_eq @ A) @ Z) @ X2) => (~ ((((ord_less_eq @ A) @ X2) @ Y))))))))))))))))).
% 2.75/1.52 thf(fact_144_order_Otrans, axiom, (! [A:$tType]: (((order @ A) => (! [A2:A,B:A,C:A]: (((((ord_less_eq @ A) @ A2) @ B) => ((((ord_less_eq @ A) @ B) @ C) => (((ord_less_eq @ A) @ A2) @ C))))))))).
% 2.75/1.52 thf(fact_145_le__cases, axiom, (! [A:$tType]: (((linorder @ A) => (! [X2:A,Y:A]: (((~ ((((ord_less_eq @ A) @ X2) @ Y))) => (((ord_less_eq @ A) @ Y) @ X2)))))))).
% 2.75/1.52 thf(fact_146_eq__refl, axiom, (! [A:$tType]: (((preorder @ A) => (! [X2:A,Y:A]: (((X2 = Y) => (((ord_less_eq @ A) @ X2) @ Y)))))))).
% 2.75/1.52 thf(fact_147_linear, axiom, (! [A:$tType]: (((linorder @ A) => (! [X2:A,Y:A]: (((((ord_less_eq @ A) @ X2) @ Y) | (((ord_less_eq @ A) @ Y) @ X2)))))))).
% 2.75/1.52 thf(fact_148_antisym, axiom, (! [A:$tType]: (((order @ A) => (! [X2:A,Y:A]: (((((ord_less_eq @ A) @ X2) @ Y) => ((((ord_less_eq @ A) @ Y) @ X2) => (X2 = Y))))))))).
% 2.75/1.52 thf(fact_149_eq__iff, axiom, (! [A:$tType]: (((order @ A) => ((^ [Y5:A,Z4:A]: ((Y5 = Z4))) = (^ [X4:A,Y3:A]: (((((ord_less_eq @ A) @ X4) @ Y3) & (((ord_less_eq @ A) @ Y3) @ X4))))))))).
% 2.75/1.52 thf(fact_150_ord__le__eq__subst, axiom, (! [A:$tType,B3:$tType]: ((((ord @ B3) & (ord @ A)) => (! [A2:A,B:A,F:(A > B3),C:B3]: (((((ord_less_eq @ A) @ A2) @ B) => (((F @ B) = C) => ((! [X3:A,Y2:A]: (((((ord_less_eq @ A) @ X3) @ Y2) => (((ord_less_eq @ B3) @ (F @ X3)) @ (F @ Y2))))) => (((ord_less_eq @ B3) @ (F @ A2)) @ C)))))))))).
% 2.75/1.52 thf(fact_151_ord__eq__le__subst, axiom, (! [A:$tType,B3:$tType]: ((((ord @ B3) & (ord @ A)) => (! [A2:A,F:(B3 > A),B:B3,C:B3]: (((A2 = (F @ B)) => ((((ord_less_eq @ B3) @ B) @ C) => ((! [X3:B3,Y2:B3]: (((((ord_less_eq @ B3) @ X3) @ Y2) => (((ord_less_eq @ A) @ (F @ X3)) @ (F @ Y2))))) => (((ord_less_eq @ A) @ A2) @ (F @ C))))))))))).
% 2.75/1.52 thf(fact_152_order__subst2, axiom, (! [A:$tType,C2:$tType]: ((((order @ C2) & (order @ A)) => (! [A2:A,B:A,F:(A > C2),C:C2]: (((((ord_less_eq @ A) @ A2) @ B) => ((((ord_less_eq @ C2) @ (F @ B)) @ C) => ((! [X3:A,Y2:A]: (((((ord_less_eq @ A) @ X3) @ Y2) => (((ord_less_eq @ C2) @ (F @ X3)) @ (F @ Y2))))) => (((ord_less_eq @ C2) @ (F @ A2)) @ C)))))))))).
% 2.75/1.52 thf(fact_153_order__subst1, axiom, (! [A:$tType,B3:$tType]: ((((order @ B3) & (order @ A)) => (! [A2:A,F:(B3 > A),B:B3,C:B3]: (((((ord_less_eq @ A) @ A2) @ (F @ B)) => ((((ord_less_eq @ B3) @ B) @ C) => ((! [X3:B3,Y2:B3]: (((((ord_less_eq @ B3) @ X3) @ Y2) => (((ord_less_eq @ A) @ (F @ X3)) @ (F @ Y2))))) => (((ord_less_eq @ A) @ A2) @ (F @ C))))))))))).
% 2.75/1.52 thf(fact_154_le__fun__def, axiom, (! [B3:$tType,A:$tType]: (((ord @ B3) => ((ord_less_eq @ (A > B3)) = (^ [F2:(A > B3),G2:(A > B3)]: ((! [X4:A]: ((((ord_less_eq @ B3) @ (F2 @ X4)) @ (G2 @ X4))))))))))).
% 2.75/1.52 thf(fact_155_le__funI, axiom, (! [B3:$tType,A:$tType]: (((ord @ B3) => (! [F:(A > B3),G:(A > B3)]: (((! [X3:A]: ((((ord_less_eq @ B3) @ (F @ X3)) @ (G @ X3)))) => (((ord_less_eq @ (A > B3)) @ F) @ G)))))))).
% 2.75/1.52 thf(fact_156_le__funE, axiom, (! [B3:$tType,A:$tType]: (((ord @ B3) => (! [F:(A > B3),G:(A > B3),X2:A]: (((((ord_less_eq @ (A > B3)) @ F) @ G) => (((ord_less_eq @ B3) @ (F @ X2)) @ (G @ X2))))))))).
% 2.75/1.52 thf(fact_157_le__funD, axiom, (! [B3:$tType,A:$tType]: (((ord @ B3) => (! [F:(A > B3),G:(A > B3),X2:A]: (((((ord_less_eq @ (A > B3)) @ F) @ G) => (((ord_less_eq @ B3) @ (F @ X2)) @ (G @ X2))))))))).
% 2.75/1.52 thf(fact_158_ord__eq__less__subst, axiom, (! [A:$tType,B3:$tType]: ((((ord @ B3) & (ord @ A)) => (! [A2:A,F:(B3 > A),B:B3,C:B3]: (((A2 = (F @ B)) => ((((ord_less @ B3) @ B) @ C) => ((! [X3:B3,Y2:B3]: (((((ord_less @ B3) @ X3) @ Y2) => (((ord_less @ A) @ (F @ X3)) @ (F @ Y2))))) => (((ord_less @ A) @ A2) @ (F @ C))))))))))).
% 2.75/1.52 thf(fact_159_ord__less__eq__subst, axiom, (! [A:$tType,B3:$tType]: ((((ord @ B3) & (ord @ A)) => (! [A2:A,B:A,F:(A > B3),C:B3]: (((((ord_less @ A) @ A2) @ B) => (((F @ B) = C) => ((! [X3:A,Y2:A]: (((((ord_less @ A) @ X3) @ Y2) => (((ord_less @ B3) @ (F @ X3)) @ (F @ Y2))))) => (((ord_less @ B3) @ (F @ A2)) @ C)))))))))).
% 2.75/1.52 thf(fact_160_order__less__subst1, axiom, (! [A:$tType,B3:$tType]: ((((order @ B3) & (order @ A)) => (! [A2:A,F:(B3 > A),B:B3,C:B3]: (((((ord_less @ A) @ A2) @ (F @ B)) => ((((ord_less @ B3) @ B) @ C) => ((! [X3:B3,Y2:B3]: (((((ord_less @ B3) @ X3) @ Y2) => (((ord_less @ A) @ (F @ X3)) @ (F @ Y2))))) => (((ord_less @ A) @ A2) @ (F @ C))))))))))).
% 2.75/1.52 thf(fact_161_order__less__subst2, axiom, (! [A:$tType,C2:$tType]: ((((order @ C2) & (order @ A)) => (! [A2:A,B:A,F:(A > C2),C:C2]: (((((ord_less @ A) @ A2) @ B) => ((((ord_less @ C2) @ (F @ B)) @ C) => ((! [X3:A,Y2:A]: (((((ord_less @ A) @ X3) @ Y2) => (((ord_less @ C2) @ (F @ X3)) @ (F @ Y2))))) => (((ord_less @ C2) @ (F @ A2)) @ C)))))))))).
% 2.75/1.52 thf(fact_162_lt__ex, axiom, (! [A:$tType]: (((no_bot @ A) => (! [X2:A]: ((? [Y2:A]: ((((ord_less @ A) @ Y2) @ X2))))))))).
% 2.75/1.52 thf(fact_163_gt__ex, axiom, (! [A:$tType]: (((no_top @ A) => (! [X2:A]: ((? [X_1:A]: ((((ord_less @ A) @ X2) @ X_1))))))))).
% 2.75/1.52 thf(fact_164_neqE, axiom, (! [A:$tType]: (((linorder @ A) => (! [X2:A,Y:A]: (((X2 != Y) => ((~ ((((ord_less @ A) @ X2) @ Y))) => (((ord_less @ A) @ Y) @ X2))))))))).
% 2.75/1.52 thf(fact_165_neq__iff, axiom, (! [A:$tType]: (((linorder @ A) => (! [X2:A,Y:A]: (((X2 != Y) = ((((ord_less @ A) @ X2) @ Y) | (((ord_less @ A) @ Y) @ X2))))))))).
% 2.75/1.52 thf(fact_166_order_Oasym, axiom, (! [A:$tType]: (((order @ A) => (! [A2:A,B:A]: (((((ord_less @ A) @ A2) @ B) => (~ ((((ord_less @ A) @ B) @ A2)))))))))).
% 2.75/1.52 thf(fact_167_dense, axiom, (! [A:$tType]: (((dense_order @ A) => (! [X2:A,Y:A]: (((((ord_less @ A) @ X2) @ Y) => (? [Z5:A]: (((((ord_less @ A) @ X2) @ Z5) & (((ord_less @ A) @ Z5) @ Y))))))))))).
% 2.75/1.52 thf(fact_168_less__imp__neq, axiom, (! [A:$tType]: (((order @ A) => (! [X2:A,Y:A]: (((((ord_less @ A) @ X2) @ Y) => (X2 != Y)))))))).
% 2.75/1.52 thf(fact_169_less__asym, axiom, (! [A:$tType]: (((preorder @ A) => (! [X2:A,Y:A]: (((((ord_less @ A) @ X2) @ Y) => (~ ((((ord_less @ A) @ Y) @ X2)))))))))).
% 2.75/1.52 thf(fact_170_less__asym_H, axiom, (! [A:$tType]: (((preorder @ A) => (! [A2:A,B:A]: (((((ord_less @ A) @ A2) @ B) => (~ ((((ord_less @ A) @ B) @ A2)))))))))).
% 2.75/1.52 thf(fact_171_less__trans, axiom, (! [A:$tType]: (((preorder @ A) => (! [X2:A,Y:A,Z:A]: (((((ord_less @ A) @ X2) @ Y) => ((((ord_less @ A) @ Y) @ Z) => (((ord_less @ A) @ X2) @ Z))))))))).
% 2.75/1.52 thf(fact_172_less__linear, axiom, (! [A:$tType]: (((linorder @ A) => (! [X2:A,Y:A]: (((((ord_less @ A) @ X2) @ Y) | ((X2 = Y) | (((ord_less @ A) @ Y) @ X2))))))))).
% 2.75/1.52 thf(fact_173_less__irrefl, axiom, (! [A:$tType]: (((preorder @ A) => (! [X2:A]: ((~ ((((ord_less @ A) @ X2) @ X2))))))))).
% 2.75/1.52 thf(fact_174_ord__eq__less__trans, axiom, (! [A:$tType]: (((ord @ A) => (! [A2:A,B:A,C:A]: (((A2 = B) => ((((ord_less @ A) @ B) @ C) => (((ord_less @ A) @ A2) @ C))))))))).
% 2.75/1.52 thf(fact_175_ord__less__eq__trans, axiom, (! [A:$tType]: (((ord @ A) => (! [A2:A,B:A,C:A]: (((((ord_less @ A) @ A2) @ B) => ((B = C) => (((ord_less @ A) @ A2) @ C))))))))).
% 2.75/1.52 thf(fact_176_dual__order_Oasym, axiom, (! [A:$tType]: (((order @ A) => (! [B:A,A2:A]: (((((ord_less @ A) @ B) @ A2) => (~ ((((ord_less @ A) @ A2) @ B)))))))))).
% 2.75/1.52 thf(fact_177_less__imp__not__eq, axiom, (! [A:$tType]: (((order @ A) => (! [X2:A,Y:A]: (((((ord_less @ A) @ X2) @ Y) => (X2 != Y)))))))).
% 2.75/1.52 thf(fact_178_less__not__sym, axiom, (! [A:$tType]: (((preorder @ A) => (! [X2:A,Y:A]: (((((ord_less @ A) @ X2) @ Y) => (~ ((((ord_less @ A) @ Y) @ X2)))))))))).
% 2.75/1.52 thf(fact_179_less__induct, axiom, (! [A:$tType]: (((wellorder @ A) => (! [P3:(A > $o),A2:A]: (((! [X3:A]: (((! [Y4:A]: (((((ord_less @ A) @ Y4) @ X3) => (P3 @ Y4)))) => (P3 @ X3)))) => (P3 @ A2)))))))).
% 2.75/1.52 thf(fact_180_antisym__conv3, axiom, (! [A:$tType]: (((linorder @ A) => (! [Y:A,X2:A]: (((~ ((((ord_less @ A) @ Y) @ X2))) => ((~ ((((ord_less @ A) @ X2) @ Y))) = (X2 = Y))))))))).
% 2.75/1.52 thf(fact_181_less__imp__not__eq2, axiom, (! [A:$tType]: (((order @ A) => (! [X2:A,Y:A]: (((((ord_less @ A) @ X2) @ Y) => (Y != X2)))))))).
% 2.75/1.52 thf(fact_182_less__imp__triv, axiom, (! [A:$tType]: (((preorder @ A) => (! [X2:A,Y:A,P3:$o]: (((((ord_less @ A) @ X2) @ Y) => ((((ord_less @ A) @ Y) @ X2) => P3)))))))).
% 2.75/1.52 thf(fact_183_linorder__cases, axiom, (! [A:$tType]: (((linorder @ A) => (! [X2:A,Y:A]: (((~ ((((ord_less @ A) @ X2) @ Y))) => ((X2 != Y) => (((ord_less @ A) @ Y) @ X2))))))))).
% 2.75/1.52 thf(fact_184_dual__order_Oirrefl, axiom, (! [A:$tType]: (((order @ A) => (! [A2:A]: ((~ ((((ord_less @ A) @ A2) @ A2))))))))).
% 2.75/1.52 thf(fact_185_order_Ostrict__trans, axiom, (! [A:$tType]: (((order @ A) => (! [A2:A,B:A,C:A]: (((((ord_less @ A) @ A2) @ B) => ((((ord_less @ A) @ B) @ C) => (((ord_less @ A) @ A2) @ C))))))))).
% 2.75/1.52 thf(fact_186_less__imp__not__less, axiom, (! [A:$tType]: (((preorder @ A) => (! [X2:A,Y:A]: (((((ord_less @ A) @ X2) @ Y) => (~ ((((ord_less @ A) @ Y) @ X2)))))))))).
% 2.75/1.52 thf(fact_187_exists__least__iff, axiom, (! [A:$tType]: (((wellorder @ A) => ((^ [P4:(A > $o)]: ((? [X5:A]: ((P4 @ X5))))) = (^ [P5:(A > $o)]: ((? [N2:A]: (((P5 @ N2) & (! [M2:A]: (((((ord_less @ A) @ M2) @ N2) => (~ ((P5 @ M2)))))))))))))))).
% 2.75/1.52 thf(fact_188_linorder__less__wlog, axiom, (! [A:$tType]: (((linorder @ A) => (! [P3:(A > (A > $o)),A2:A,B:A]: (((! [A5:A,B4:A]: (((((ord_less @ A) @ A5) @ B4) => ((P3 @ A5) @ B4)))) => ((! [A5:A]: (((P3 @ A5) @ A5))) => ((! [A5:A,B4:A]: ((((P3 @ B4) @ A5) => ((P3 @ A5) @ B4)))) => ((P3 @ A2) @ B)))))))))).
% 2.75/1.52 thf(fact_189_dual__order_Ostrict__trans, axiom, (! [A:$tType]: (((order @ A) => (! [B:A,A2:A,C:A]: (((((ord_less @ A) @ B) @ A2) => ((((ord_less @ A) @ C) @ B) => (((ord_less @ A) @ C) @ A2))))))))).
% 2.75/1.52 thf(fact_190_not__less__iff__gr__or__eq, axiom, (! [A:$tType]: (((linorder @ A) => (! [X2:A,Y:A]: (((~ ((((ord_less @ A) @ X2) @ Y))) = ((((ord_less @ A) @ Y) @ X2) | (X2 = Y))))))))).
% 2.75/1.52 thf(fact_191_order_Ostrict__implies__not__eq, axiom, (! [A:$tType]: (((order @ A) => (! [A2:A,B:A]: (((((ord_less @ A) @ A2) @ B) => (A2 != B)))))))).
% 2.75/1.52 thf(fact_192_dual__order_Ostrict__implies__not__eq, axiom, (! [A:$tType]: (((order @ A) => (! [B:A,A2:A]: (((((ord_less @ A) @ B) @ A2) => (A2 != B)))))))).
% 2.75/1.52 thf(fact_193_ex__has__least__nat, axiom, (! [A:$tType,P3:(A > $o),K:A,M:(A > nat)]: (((P3 @ K) => (? [X3:A]: (((P3 @ X3) & (! [Y4:A]: (((P3 @ Y4) => (((ord_less_eq @ nat) @ (M @ X3)) @ (M @ Y4)))))))))))).
% 2.75/1.52 thf(fact_194_leD, axiom, (! [A:$tType]: (((order @ A) => (! [Y:A,X2:A]: (((((ord_less_eq @ A) @ Y) @ X2) => (~ ((((ord_less @ A) @ X2) @ Y)))))))))).
% 2.75/1.52 thf(fact_195_leI, axiom, (! [A:$tType]: (((linorder @ A) => (! [X2:A,Y:A]: (((~ ((((ord_less @ A) @ X2) @ Y))) => (((ord_less_eq @ A) @ Y) @ X2)))))))).
% 2.75/1.52 thf(fact_196_le__less, axiom, (! [A:$tType]: (((order @ A) => ((ord_less_eq @ A) = (^ [X4:A,Y3:A]: (((((ord_less @ A) @ X4) @ Y3) | (X4 = Y3))))))))).
% 2.75/1.52 thf(fact_197_less__le, axiom, (! [A:$tType]: (((order @ A) => ((ord_less @ A) = (^ [X4:A,Y3:A]: (((((ord_less_eq @ A) @ X4) @ Y3) & (X4 != Y3))))))))).
% 2.75/1.52 thf(fact_198_order__le__less__subst1, axiom, (! [A:$tType,B3:$tType]: ((((order @ B3) & (order @ A)) => (! [A2:A,F:(B3 > A),B:B3,C:B3]: (((((ord_less_eq @ A) @ A2) @ (F @ B)) => ((((ord_less @ B3) @ B) @ C) => ((! [X3:B3,Y2:B3]: (((((ord_less @ B3) @ X3) @ Y2) => (((ord_less @ A) @ (F @ X3)) @ (F @ Y2))))) => (((ord_less @ A) @ A2) @ (F @ C))))))))))).
% 2.75/1.52 thf(fact_199_order__le__less__subst2, axiom, (! [A:$tType,C2:$tType]: ((((order @ C2) & (order @ A)) => (! [A2:A,B:A,F:(A > C2),C:C2]: (((((ord_less_eq @ A) @ A2) @ B) => ((((ord_less @ C2) @ (F @ B)) @ C) => ((! [X3:A,Y2:A]: (((((ord_less_eq @ A) @ X3) @ Y2) => (((ord_less_eq @ C2) @ (F @ X3)) @ (F @ Y2))))) => (((ord_less @ C2) @ (F @ A2)) @ C)))))))))).
% 2.75/1.52 thf(fact_200_order__less__le__subst1, axiom, (! [A:$tType,B3:$tType]: ((((order @ B3) & (order @ A)) => (! [A2:A,F:(B3 > A),B:B3,C:B3]: (((((ord_less @ A) @ A2) @ (F @ B)) => ((((ord_less_eq @ B3) @ B) @ C) => ((! [X3:B3,Y2:B3]: (((((ord_less_eq @ B3) @ X3) @ Y2) => (((ord_less_eq @ A) @ (F @ X3)) @ (F @ Y2))))) => (((ord_less @ A) @ A2) @ (F @ C))))))))))).
% 2.75/1.52 thf(fact_201_order__less__le__subst2, axiom, (! [A:$tType,C2:$tType]: ((((order @ C2) & (order @ A)) => (! [A2:A,B:A,F:(A > C2),C:C2]: (((((ord_less @ A) @ A2) @ B) => ((((ord_less_eq @ C2) @ (F @ B)) @ C) => ((! [X3:A,Y2:A]: (((((ord_less @ A) @ X3) @ Y2) => (((ord_less @ C2) @ (F @ X3)) @ (F @ Y2))))) => (((ord_less @ C2) @ (F @ A2)) @ C)))))))))).
% 2.75/1.52 thf(fact_202_not__le, axiom, (! [A:$tType]: (((linorder @ A) => (! [X2:A,Y:A]: (((~ ((((ord_less_eq @ A) @ X2) @ Y))) = (((ord_less @ A) @ Y) @ X2)))))))).
% 2.75/1.52 thf(fact_203_not__less, axiom, (! [A:$tType]: (((linorder @ A) => (! [X2:A,Y:A]: (((~ ((((ord_less @ A) @ X2) @ Y))) = (((ord_less_eq @ A) @ Y) @ X2)))))))).
% 2.75/1.52 thf(fact_204_le__neq__trans, axiom, (! [A:$tType]: (((order @ A) => (! [A2:A,B:A]: (((((ord_less_eq @ A) @ A2) @ B) => ((A2 != B) => (((ord_less @ A) @ A2) @ B))))))))).
% 2.75/1.52 thf(fact_205_antisym__conv1, axiom, (! [A:$tType]: (((order @ A) => (! [X2:A,Y:A]: (((~ ((((ord_less @ A) @ X2) @ Y))) => ((((ord_less_eq @ A) @ X2) @ Y) = (X2 = Y))))))))).
% 2.75/1.52 thf(fact_206_antisym__conv2, axiom, (! [A:$tType]: (((order @ A) => (! [X2:A,Y:A]: (((((ord_less_eq @ A) @ X2) @ Y) => ((~ ((((ord_less @ A) @ X2) @ Y))) = (X2 = Y))))))))).
% 2.75/1.52 thf(fact_207_less__imp__le, axiom, (! [A:$tType]: (((preorder @ A) => (! [X2:A,Y:A]: (((((ord_less @ A) @ X2) @ Y) => (((ord_less_eq @ A) @ X2) @ Y)))))))).
% 2.75/1.52 thf(fact_208_le__less__trans, axiom, (! [A:$tType]: (((preorder @ A) => (! [X2:A,Y:A,Z:A]: (((((ord_less_eq @ A) @ X2) @ Y) => ((((ord_less @ A) @ Y) @ Z) => (((ord_less @ A) @ X2) @ Z))))))))).
% 2.75/1.52 thf(fact_209_less__le__trans, axiom, (! [A:$tType]: (((preorder @ A) => (! [X2:A,Y:A,Z:A]: (((((ord_less @ A) @ X2) @ Y) => ((((ord_less_eq @ A) @ Y) @ Z) => (((ord_less @ A) @ X2) @ Z))))))))).
% 2.75/1.52 thf(fact_210_dense__ge, axiom, (! [A:$tType]: (((dense_linorder @ A) => (! [Z:A,Y:A]: (((! [X3:A]: (((((ord_less @ A) @ Z) @ X3) => (((ord_less_eq @ A) @ Y) @ X3)))) => (((ord_less_eq @ A) @ Y) @ Z)))))))).
% 2.75/1.52 thf(fact_211_dense__le, axiom, (! [A:$tType]: (((dense_linorder @ A) => (! [Y:A,Z:A]: (((! [X3:A]: (((((ord_less @ A) @ X3) @ Y) => (((ord_less_eq @ A) @ X3) @ Z)))) => (((ord_less_eq @ A) @ Y) @ Z)))))))).
% 2.75/1.52 thf(fact_212_le__less__linear, axiom, (! [A:$tType]: (((linorder @ A) => (! [X2:A,Y:A]: (((((ord_less_eq @ A) @ X2) @ Y) | (((ord_less @ A) @ Y) @ X2)))))))).
% 2.75/1.52 thf(fact_213_le__imp__less__or__eq, axiom, (! [A:$tType]: (((order @ A) => (! [X2:A,Y:A]: (((((ord_less_eq @ A) @ X2) @ Y) => ((((ord_less @ A) @ X2) @ Y) | (X2 = Y))))))))).
% 2.75/1.52 thf(fact_214_less__le__not__le, axiom, (! [A:$tType]: (((preorder @ A) => ((ord_less @ A) = (^ [X4:A,Y3:A]: (((((ord_less_eq @ A) @ X4) @ Y3) & (~ ((((ord_less_eq @ A) @ Y3) @ X4))))))))))).
% 2.75/1.52 thf(fact_215_not__le__imp__less, axiom, (! [A:$tType]: (((linorder @ A) => (! [Y:A,X2:A]: (((~ ((((ord_less_eq @ A) @ Y) @ X2))) => (((ord_less @ A) @ X2) @ Y)))))))).
% 2.75/1.52 thf(fact_216_order_Ostrict__trans1, axiom, (! [A:$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))))))))).
% 2.75/1.52 thf(fact_217_order_Ostrict__trans2, axiom, (! [A:$tType]: (((order @ A) => (! [A2:A,B:A,C:A]: (((((ord_less @ A) @ A2) @ B) => ((((ord_less_eq @ A) @ B) @ C) => (((ord_less @ A) @ A2) @ C))))))))).
% 2.75/1.52 thf(fact_218_order_Oorder__iff__strict, axiom, (! [A:$tType]: (((order @ A) => ((ord_less_eq @ A) = (^ [A3:A,B2:A]: (((((ord_less @ A) @ A3) @ B2) | (A3 = B2))))))))).
% 2.75/1.52 thf(fact_219_order_Ostrict__iff__order, axiom, (! [A:$tType]: (((order @ A) => ((ord_less @ A) = (^ [A3:A,B2:A]: (((((ord_less_eq @ A) @ A3) @ B2) & (A3 != B2))))))))).
% 2.75/1.52 thf(fact_220_dual__order_Ostrict__trans1, axiom, (! [A:$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))))))))).
% 2.75/1.52 thf(fact_221_dual__order_Ostrict__trans2, axiom, (! [A:$tType]: (((order @ A) => (! [B:A,A2:A,C:A]: (((((ord_less @ A) @ B) @ A2) => ((((ord_less_eq @ A) @ C) @ B) => (((ord_less @ A) @ C) @ A2))))))))).
% 2.75/1.52 thf(fact_222_dense__ge__bounded, axiom, (! [A:$tType]: (((dense_linorder @ A) => (! [Z:A,X2:A,Y:A]: (((((ord_less @ A) @ Z) @ X2) => ((! [W:A]: (((((ord_less @ A) @ Z) @ W) => ((((ord_less @ A) @ W) @ X2) => (((ord_less_eq @ A) @ Y) @ W))))) => (((ord_less_eq @ A) @ Y) @ Z))))))))).
% 2.75/1.52 thf(fact_223_dense__le__bounded, axiom, (! [A:$tType]: (((dense_linorder @ A) => (! [X2:A,Y:A,Z:A]: (((((ord_less @ A) @ X2) @ Y) => ((! [W:A]: (((((ord_less @ A) @ X2) @ W) => ((((ord_less @ A) @ W) @ Y) => (((ord_less_eq @ A) @ W) @ Z))))) => (((ord_less_eq @ A) @ Y) @ Z))))))))).
% 2.75/1.52 thf(fact_224_order_Ostrict__implies__order, axiom, (! [A:$tType]: (((order @ A) => (! [A2:A,B:A]: (((((ord_less @ A) @ A2) @ B) => (((ord_less_eq @ A) @ A2) @ B)))))))).
% 2.75/1.52 thf(fact_225_dual__order_Oorder__iff__strict, axiom, (! [A:$tType]: (((order @ A) => ((ord_less_eq @ A) = (^ [B2:A,A3:A]: (((((ord_less @ A) @ B2) @ A3) | (A3 = B2))))))))).
% 2.75/1.52 thf(fact_226_dual__order_Ostrict__iff__order, axiom, (! [A:$tType]: (((order @ A) => ((ord_less @ A) = (^ [B2:A,A3:A]: (((((ord_less_eq @ A) @ B2) @ A3) & (A3 != B2))))))))).
% 2.75/1.52 thf(fact_227_dual__order_Ostrict__implies__order, axiom, (! [A:$tType]: (((order @ A) => (! [B:A,A2:A]: (((((ord_less @ A) @ B) @ A2) => (((ord_less_eq @ A) @ B) @ A2)))))))).
% 2.75/1.52 thf(fact_228_order_Onot__eq__order__implies__strict, axiom, (! [A:$tType]: (((order @ A) => (! [A2:A,B:A]: (((A2 != B) => ((((ord_less_eq @ A) @ A2) @ B) => (((ord_less @ A) @ A2) @ B))))))))).
% 2.75/1.52 thf(fact_229_minf_I8_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [T:A]: ((? [Z5:A]: ((! [X:A]: (((((ord_less @ A) @ X) @ Z5) => (~ ((((ord_less_eq @ A) @ T) @ X)))))))))))))).
% 2.75/1.52 thf(fact_230_minf_I6_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [T:A]: ((? [Z5:A]: ((! [X:A]: (((((ord_less @ A) @ X) @ Z5) => (((ord_less_eq @ A) @ X) @ T)))))))))))).
% 2.75/1.52 thf(fact_231_pinf_I8_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [T:A]: ((? [Z5:A]: ((! [X:A]: (((((ord_less @ A) @ Z5) @ X) => (((ord_less_eq @ A) @ T) @ X)))))))))))).
% 2.75/1.52 thf(fact_232_pinf_I6_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [T:A]: ((? [Z5:A]: ((! [X:A]: (((((ord_less @ A) @ Z5) @ X) => (~ ((((ord_less_eq @ A) @ X) @ T)))))))))))))).
% 2.75/1.52 thf(fact_233_less__fun__def, axiom, (! [B3:$tType,A:$tType]: (((ord @ B3) => ((ord_less @ (A > B3)) = (^ [F2:(A > B3),G2:(A > B3)]: (((((ord_less_eq @ (A > B3)) @ F2) @ G2) & (~ ((((ord_less_eq @ (A > B3)) @ G2) @ F2))))))))))).
% 2.75/1.52 thf(fact_234_pinf_I1_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [P3:(A > $o),P6:(A > $o),Q2:(A > $o),Q3:(A > $o)]: (((? [Z2:A]: ((! [X3:A]: (((((ord_less @ A) @ Z2) @ X3) => ((P3 @ X3) = (P6 @ X3))))))) => ((? [Z2:A]: ((! [X3:A]: (((((ord_less @ A) @ Z2) @ X3) => ((Q2 @ X3) = (Q3 @ X3))))))) => (? [Z5:A]: ((! [X:A]: (((((ord_less @ A) @ Z5) @ X) => (((P3 @ X) & (Q2 @ X)) = ((P6 @ X) & (Q3 @ X)))))))))))))))).
% 2.75/1.52 thf(fact_235_pinf_I2_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [P3:(A > $o),P6:(A > $o),Q2:(A > $o),Q3:(A > $o)]: (((? [Z2:A]: ((! [X3:A]: (((((ord_less @ A) @ Z2) @ X3) => ((P3 @ X3) = (P6 @ X3))))))) => ((? [Z2:A]: ((! [X3:A]: (((((ord_less @ A) @ Z2) @ X3) => ((Q2 @ X3) = (Q3 @ X3))))))) => (? [Z5:A]: ((! [X:A]: (((((ord_less @ A) @ Z5) @ X) => (((P3 @ X) | (Q2 @ X)) = ((P6 @ X) | (Q3 @ X)))))))))))))))).
% 2.75/1.52 thf(fact_236_pinf_I3_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [T:A]: ((? [Z5:A]: ((! [X:A]: (((((ord_less @ A) @ Z5) @ X) => (X != T)))))))))))).
% 2.75/1.52 thf(fact_237_pinf_I4_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [T:A]: ((? [Z5:A]: ((! [X:A]: (((((ord_less @ A) @ Z5) @ X) => (X != T)))))))))))).
% 2.75/1.52 thf(fact_238_pinf_I5_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [T:A]: ((? [Z5:A]: ((! [X:A]: (((((ord_less @ A) @ Z5) @ X) => (~ ((((ord_less @ A) @ X) @ T)))))))))))))).
% 2.75/1.52 thf(fact_239_pinf_I7_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [T:A]: ((? [Z5:A]: ((! [X:A]: (((((ord_less @ A) @ Z5) @ X) => (((ord_less @ A) @ T) @ X)))))))))))).
% 2.75/1.52 thf(fact_240_pinf_I11_J, axiom, (! [C2:$tType,D3:$tType]: (((ord @ C2) => (! [F3:D3]: ((? [Z5:C2]: ((! [X:C2]: (((((ord_less @ C2) @ Z5) @ X) => (F3 = F3)))))))))))).
% 2.75/1.52 thf(fact_241_minf_I1_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [P3:(A > $o),P6:(A > $o),Q2:(A > $o),Q3:(A > $o)]: (((? [Z2:A]: ((! [X3:A]: (((((ord_less @ A) @ X3) @ Z2) => ((P3 @ X3) = (P6 @ X3))))))) => ((? [Z2:A]: ((! [X3:A]: (((((ord_less @ A) @ X3) @ Z2) => ((Q2 @ X3) = (Q3 @ X3))))))) => (? [Z5:A]: ((! [X:A]: (((((ord_less @ A) @ X) @ Z5) => (((P3 @ X) & (Q2 @ X)) = ((P6 @ X) & (Q3 @ X)))))))))))))))).
% 2.75/1.52 thf(fact_242_minf_I2_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [P3:(A > $o),P6:(A > $o),Q2:(A > $o),Q3:(A > $o)]: (((? [Z2:A]: ((! [X3:A]: (((((ord_less @ A) @ X3) @ Z2) => ((P3 @ X3) = (P6 @ X3))))))) => ((? [Z2:A]: ((! [X3:A]: (((((ord_less @ A) @ X3) @ Z2) => ((Q2 @ X3) = (Q3 @ X3))))))) => (? [Z5:A]: ((! [X:A]: (((((ord_less @ A) @ X) @ Z5) => (((P3 @ X) | (Q2 @ X)) = ((P6 @ X) | (Q3 @ X)))))))))))))))).
% 2.75/1.52 thf(fact_243_minf_I3_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [T:A]: ((? [Z5:A]: ((! [X:A]: (((((ord_less @ A) @ X) @ Z5) => (X != T)))))))))))).
% 2.75/1.52 thf(fact_244_minf_I4_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [T:A]: ((? [Z5:A]: ((! [X:A]: (((((ord_less @ A) @ X) @ Z5) => (X != T)))))))))))).
% 2.75/1.52 thf(fact_245_minf_I5_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [T:A]: ((? [Z5:A]: ((! [X:A]: (((((ord_less @ A) @ X) @ Z5) => (((ord_less @ A) @ X) @ T)))))))))))).
% 2.75/1.52 thf(fact_246_minf_I7_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [T:A]: ((? [Z5:A]: ((! [X:A]: (((((ord_less @ A) @ X) @ Z5) => (~ ((((ord_less @ A) @ T) @ X)))))))))))))).
% 2.75/1.52 thf(fact_247_minf_I11_J, axiom, (! [C2:$tType,D3:$tType]: (((ord @ C2) => (! [F3:D3]: ((? [Z5:C2]: ((! [X:C2]: (((((ord_less @ C2) @ X) @ Z5) => (F3 = F3)))))))))))).
% 2.75/1.52 thf(fact_248_complete__interval, axiom, (! [A:$tType]: (((condit1037483654norder @ A) => (! [A2:A,B:A,P3:(A > $o)]: (((((ord_less @ A) @ A2) @ B) => ((P3 @ A2) => ((~ ((P3 @ B))) => (? [C3:A]: (((((ord_less_eq @ A) @ A2) @ C3) & ((((ord_less_eq @ A) @ C3) @ B) & ((! [X:A]: ((((((ord_less_eq @ A) @ A2) @ X) & (((ord_less @ A) @ X) @ C3)) => (P3 @ X)))) & (! [D4:A]: (((! [X3:A]: ((((((ord_less_eq @ A) @ A2) @ X3) & (((ord_less @ A) @ X3) @ D4)) => (P3 @ X3)))) => (((ord_less_eq @ A) @ D4) @ C3)))))))))))))))))).
% 2.75/1.52 thf(fact_249_verit__comp__simplify1_I3_J, axiom, (! [B3:$tType]: (((linorder @ B3) => (! [B5:B3,A6:B3]: (((~ ((((ord_less_eq @ B3) @ B5) @ A6))) = (((ord_less @ B3) @ A6) @ B5)))))))).
% 2.75/1.52 thf(fact_250_verit__la__disequality, axiom, (! [A:$tType]: (((linorder @ A) => (! [A2:A,B:A]: (((A2 = B) | ((~ ((((ord_less_eq @ A) @ A2) @ B))) | (~ ((((ord_less_eq @ A) @ B) @ A2))))))))))).
% 2.75/1.53 thf(fact_251_verit__comp__simplify1_I1_J, axiom, (! [A:$tType]: (((order @ A) => (! [A2:A]: ((~ ((((ord_less @ A) @ A2) @ A2))))))))).
% 2.75/1.53 thf(fact_252_ex__gt__or__lt, axiom, (! [A:$tType]: (((condit1656338222tinuum @ A) => (! [A2:A]: ((? [B4:A]: (((((ord_less @ A) @ A2) @ B4) | (((ord_less @ A) @ B4) @ A2)))))))))).
% 2.75/1.53 thf(fact_253_poly__cutoff__1, axiom, (! [A:$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))))))))))).
% 2.75/1.53 thf(tcon_fun___Orderings_Opreorder, axiom, (! [A7:$tType,A8:$tType]: (((preorder @ A8) => (preorder @ (A7 > A8)))))).
% 2.75/1.53 thf(tcon_fun___Orderings_Oorder, axiom, (! [A7:$tType,A8:$tType]: (((order @ A8) => (order @ (A7 > A8)))))).
% 2.75/1.53 thf(tcon_fun___Orderings_Oord, axiom, (! [A7:$tType,A8:$tType]: (((ord @ A8) => (ord @ (A7 > A8)))))).
% 2.75/1.53 thf(tcon_Nat_Onat___Conditionally__Complete__Lattices_Oconditionally__complete__linorder, axiom, (condit1037483654norder @ nat)).
% 2.75/1.53 thf(tcon_Nat_Onat___Groups_Ocanonically__ordered__monoid__add, axiom, (canoni770627133id_add @ nat)).
% 2.75/1.53 thf(tcon_Nat_Onat___Rings_Olinordered__nonzero__semiring, axiom, (linord1659791738miring @ nat)).
% 2.75/1.53 thf(tcon_Nat_Onat___Rings_Ocomm__semiring__1, axiom, (comm_semiring_1 @ nat)).
% 2.75/1.53 thf(tcon_Nat_Onat___Rings_Ocomm__semiring__0, axiom, (comm_semiring_0 @ nat)).
% 2.75/1.53 thf(tcon_Nat_Onat___Orderings_Owellorder, axiom, (wellorder @ nat)).
% 2.75/1.53 thf(tcon_Nat_Onat___Orderings_Opreorder_1, axiom, (preorder @ nat)).
% 2.75/1.53 thf(tcon_Nat_Onat___Orderings_Olinorder, axiom, (linorder @ nat)).
% 2.75/1.53 thf(tcon_Nat_Onat___Orderings_Ono__top, axiom, (no_top @ nat)).
% 2.75/1.53 thf(tcon_Nat_Onat___Orderings_Oorder_2, axiom, (order @ nat)).
% 2.75/1.53 thf(tcon_Nat_Onat___Orderings_Oord_3, axiom, (ord @ nat)).
% 2.75/1.53 thf(tcon_Nat_Onat___Groups_Ozero, axiom, (zero @ nat)).
% 2.75/1.53 thf(tcon_Set_Oset___Orderings_Opreorder_4, axiom, (! [A7:$tType]: ((preorder @ (set @ A7))))).
% 2.75/1.53 thf(tcon_Set_Oset___Orderings_Oorder_5, axiom, (! [A7:$tType]: ((order @ (set @ A7))))).
% 2.75/1.53 thf(tcon_Set_Oset___Orderings_Oord_6, axiom, (! [A7:$tType]: ((ord @ (set @ A7))))).
% 2.75/1.53 thf(tcon_HOL_Obool___Orderings_Opreorder_7, axiom, (preorder @ $o)).
% 2.75/1.53 thf(tcon_HOL_Obool___Orderings_Olinorder_8, axiom, (linorder @ $o)).
% 2.75/1.53 thf(tcon_HOL_Obool___Orderings_Oorder_9, axiom, (order @ $o)).
% 2.75/1.53 thf(tcon_HOL_Obool___Orderings_Oord_10, axiom, (ord @ $o)).
% 2.75/1.53 thf(tcon_Real_Oreal___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_11, axiom, (condit1037483654norder @ real)).
% 2.75/1.53 thf(tcon_Real_Oreal___Conditionally__Complete__Lattices_Olinear__continuum, axiom, (condit1656338222tinuum @ real)).
% 2.75/1.53 thf(tcon_Real_Oreal___Real__Vector__Spaces_Oreal__normed__vector, axiom, (real_V55928688vector @ real)).
% 2.75/1.53 thf(tcon_Real_Oreal___Rings_Olinordered__nonzero__semiring_12, axiom, (linord1659791738miring @ real)).
% 2.75/1.53 thf(tcon_Real_Oreal___Rings_Oring__no__zero__divisors, axiom, (ring_n68954251visors @ real)).
% 2.75/1.53 thf(tcon_Real_Oreal___Orderings_Odense__linorder, axiom, (dense_linorder @ real)).
% 2.75/1.53 thf(tcon_Real_Oreal___Fields_Olinordered__field, axiom, (linordered_field @ real)).
% 2.75/1.53 thf(tcon_Real_Oreal___Rings_Olinordered__idom, axiom, (linordered_idom @ real)).
% 2.75/1.53 thf(tcon_Real_Oreal___Rings_Ocomm__semiring__1_13, axiom, (comm_semiring_1 @ real)).
% 2.75/1.53 thf(tcon_Real_Oreal___Rings_Ocomm__semiring__0_14, axiom, (comm_semiring_0 @ real)).
% 2.75/1.53 thf(tcon_Real_Oreal___Orderings_Odense__order, axiom, (dense_order @ real)).
% 2.75/1.53 thf(tcon_Real_Oreal___Rings_Oidom__abs__sgn, axiom, (idom_abs_sgn @ real)).
% 2.75/1.53 thf(tcon_Real_Oreal___Orderings_Opreorder_15, axiom, (preorder @ real)).
% 2.75/1.53 thf(tcon_Real_Oreal___Orderings_Olinorder_16, axiom, (linorder @ real)).
% 2.75/1.53 thf(tcon_Real_Oreal___Rings_Oidom__divide, axiom, (idom_divide @ real)).
% 2.75/1.53 thf(tcon_Real_Oreal___Rings_Ocomm__ring__1, axiom, (comm_ring_1 @ real)).
% 2.75/1.53 thf(tcon_Real_Oreal___Orderings_Ono__top_17, axiom, (no_top @ real)).
% 2.75/1.53 thf(tcon_Real_Oreal___Orderings_Ono__bot, axiom, (no_bot @ real)).
% 2.75/1.53 thf(tcon_Real_Oreal___Orderings_Oorder_18, axiom, (order @ real)).
% 2.75/1.53 thf(tcon_Real_Oreal___Nat_Oring__char__0, axiom, (ring_char_0 @ real)).
% 2.75/1.53 thf(tcon_Real_Oreal___Orderings_Oord_19, axiom, (ord @ real)).
% 2.75/1.53 thf(tcon_Real_Oreal___Groups_Ozero_20, axiom, (zero @ real)).
% 2.75/1.53 thf(tcon_Real_Oreal___Rings_Oidom, axiom, (idom @ real)).
% 2.75/1.53 thf(tcon_Polynomial_Opoly___Rings_Olinordered__nonzero__semiring_21, axiom, (! [A7:$tType]: (((linordered_idom @ A7) => (linord1659791738miring @ (poly @ A7)))))).
% 2.75/1.53 thf(tcon_Polynomial_Opoly___Rings_Oring__no__zero__divisors_22, axiom, (! [A7:$tType]: (((idom @ A7) => (ring_n68954251visors @ (poly @ A7)))))).
% 2.75/1.53 thf(tcon_Polynomial_Opoly___Rings_Olinordered__idom_23, axiom, (! [A7:$tType]: (((linordered_idom @ A7) => (linordered_idom @ (poly @ A7)))))).
% 2.75/1.53 thf(tcon_Polynomial_Opoly___Rings_Ocomm__semiring__1_24, axiom, (! [A7:$tType]: (((comm_semiring_1 @ A7) => (comm_semiring_1 @ (poly @ A7)))))).
% 2.75/1.53 thf(tcon_Polynomial_Opoly___Rings_Ocomm__semiring__0_25, axiom, (! [A7:$tType]: (((comm_semiring_0 @ A7) => (comm_semiring_0 @ (poly @ A7)))))).
% 2.75/1.53 thf(tcon_Polynomial_Opoly___Rings_Oidom__abs__sgn_26, axiom, (! [A7:$tType]: (((linordered_idom @ A7) => (idom_abs_sgn @ (poly @ A7)))))).
% 2.75/1.53 thf(tcon_Polynomial_Opoly___Orderings_Opreorder_27, axiom, (! [A7:$tType]: (((linordered_idom @ A7) => (preorder @ (poly @ A7)))))).
% 2.75/1.53 thf(tcon_Polynomial_Opoly___Orderings_Olinorder_28, axiom, (! [A7:$tType]: (((linordered_idom @ A7) => (linorder @ (poly @ A7)))))).
% 2.75/1.53 thf(tcon_Polynomial_Opoly___Rings_Oidom__divide_29, axiom, (! [A7:$tType]: (((idom_divide @ A7) => (idom_divide @ (poly @ A7)))))).
% 2.75/1.53 thf(tcon_Polynomial_Opoly___Rings_Ocomm__ring__1_30, axiom, (! [A7:$tType]: (((comm_ring_1 @ A7) => (comm_ring_1 @ (poly @ A7)))))).
% 2.75/1.53 thf(tcon_Polynomial_Opoly___Orderings_Oorder_31, axiom, (! [A7:$tType]: (((linordered_idom @ A7) => (order @ (poly @ A7)))))).
% 2.75/1.53 thf(tcon_Polynomial_Opoly___Nat_Oring__char__0_32, axiom, (! [A7:$tType]: ((((ring_char_0 @ A7) & (comm_ring_1 @ A7)) => (ring_char_0 @ (poly @ A7)))))).
% 2.75/1.53 thf(tcon_Polynomial_Opoly___Orderings_Oord_33, axiom, (! [A7:$tType]: (((linordered_idom @ A7) => (ord @ (poly @ A7)))))).
% 2.75/1.53 thf(tcon_Polynomial_Opoly___Groups_Ozero_34, axiom, (! [A7:$tType]: (((zero @ A7) => (zero @ (poly @ A7)))))).
% 2.75/1.53 thf(tcon_Polynomial_Opoly___Rings_Oidom_35, axiom, (! [A7:$tType]: (((idom @ A7) => (idom @ (poly @ A7)))))).
% 2.75/1.53 thf(help_If_3_1_T, axiom, (! [P3:$o]: (((P3 = $true) | (P3 = $false))))).
% 2.75/1.53 thf(help_If_2_1_T, axiom, (! [A:$tType,X2:A,Y:A]: ((((((if @ A) @ $false) @ X2) @ Y) = Y)))).
% 2.75/1.53 thf(help_If_1_1_T, axiom, (! [A:$tType,X2:A,Y:A]: ((((((if @ A) @ $true) @ X2) @ Y) = X2)))).
% 2.75/1.53 thf(conj_0, conjecture, (! [X3:real]: ((((((poly2 @ real) @ p) @ X3) != (zero_zero @ real)) | (((ord_less @ real) @ lb) @ X3))))).
% 2.75/1.53 % SZS output end ListOfTHF for /export/starexec/sandbox2/tmp/tmp.TYEgfeW64B/DTF2DTF21200.p
% 2.75/1.53 ------------------------
% 2.75/1.55 ---- Cleaned THF ---
% 2.75/1.55 thf(ty_t_Polynomial_Opoly,type,
% 2.75/1.55 poly: $tType > $tType ).
% 2.75/1.55
% 2.75/1.55 thf(ty_t_Real_Oreal,type,
% 2.75/1.55 real: $tType ).
% 2.75/1.55
% 2.75/1.55 thf(ty_t_Set_Oset,type,
% 2.75/1.55 set: $tType > $tType ).
% 2.75/1.55
% 2.75/1.55 thf(ty_t_Nat_Onat,type,
% 2.75/1.55 nat: $tType ).
% 2.75/1.55
% 2.75/1.55 thf(sy_cl_HOL_Otype,type,
% 2.75/1.55 type:
% 2.75/1.55 !>[A: $tType] : $o ).
% 2.75/1.55
% 2.75/1.55 thf(sy_cl_Rings_Oidom,type,
% 2.75/1.55 idom:
% 2.75/1.55 !>[A: $tType] : $o ).
% 2.75/1.55
% 2.75/1.55 thf(sy_cl_Groups_Ozero,type,
% 2.75/1.55 zero:
% 2.75/1.55 !>[A: $tType] : $o ).
% 2.75/1.55
% 2.75/1.55 thf(sy_cl_Orderings_Oord,type,
% 2.75/1.55 ord:
% 2.75/1.55 !>[A: $tType] : $o ).
% 2.75/1.55
% 2.75/1.55 thf(sy_cl_Nat_Oring__char__0,type,
% 2.75/1.55 ring_char_0:
% 2.75/1.55 !>[A: $tType] : $o ).
% 2.75/1.55
% 2.75/1.55 thf(sy_cl_Orderings_Oorder,type,
% 2.75/1.55 order:
% 2.75/1.55 !>[A: $tType] : $o ).
% 2.75/1.55
% 2.75/1.55 thf(sy_cl_Orderings_Ono__bot,type,
% 2.75/1.55 no_bot:
% 2.75/1.55 !>[A: $tType] : $o ).
% 2.75/1.55
% 2.75/1.55 thf(sy_cl_Orderings_Ono__top,type,
% 2.75/1.55 no_top:
% 2.75/1.55 !>[A: $tType] : $o ).
% 2.75/1.55
% 2.75/1.55 thf(sy_cl_Rings_Ocomm__ring__1,type,
% 2.75/1.55 comm_ring_1:
% 2.75/1.55 !>[A: $tType] : $o ).
% 2.75/1.55
% 2.75/1.55 thf(sy_cl_Rings_Oidom__divide,type,
% 2.75/1.55 idom_divide:
% 2.75/1.55 !>[A: $tType] : $o ).
% 2.75/1.55
% 2.75/1.55 thf(sy_cl_Orderings_Olinorder,type,
% 2.75/1.55 linorder:
% 2.75/1.55 !>[A: $tType] : $o ).
% 2.75/1.55
% 2.75/1.55 thf(sy_cl_Orderings_Opreorder,type,
% 2.75/1.55 preorder:
% 2.75/1.55 !>[A: $tType] : $o ).
% 2.75/1.55
% 2.75/1.55 thf(sy_cl_Rings_Oidom__abs__sgn,type,
% 2.75/1.55 idom_abs_sgn:
% 2.75/1.55 !>[A: $tType] : $o ).
% 2.75/1.55
% 2.75/1.55 thf(sy_cl_Orderings_Owellorder,type,
% 2.75/1.55 wellorder:
% 2.75/1.55 !>[A: $tType] : $o ).
% 2.75/1.55
% 2.75/1.55 thf(sy_cl_Orderings_Odense__order,type,
% 2.75/1.55 dense_order:
% 2.75/1.55 !>[A: $tType] : $o ).
% 2.75/1.55
% 2.75/1.55 thf(sy_cl_Rings_Ocomm__semiring__0,type,
% 2.75/1.55 comm_semiring_0:
% 2.75/1.55 !>[A: $tType] : $o ).
% 2.75/1.55
% 2.75/1.55 thf(sy_cl_Rings_Ocomm__semiring__1,type,
% 2.75/1.55 comm_semiring_1:
% 2.75/1.55 !>[A: $tType] : $o ).
% 2.75/1.55
% 2.75/1.55 thf(sy_cl_Rings_Olinordered__idom,type,
% 2.75/1.55 linordered_idom:
% 2.75/1.55 !>[A: $tType] : $o ).
% 2.75/1.55
% 2.75/1.55 thf(sy_cl_Fields_Olinordered__field,type,
% 2.75/1.55 linordered_field:
% 2.75/1.55 !>[A: $tType] : $o ).
% 2.75/1.55
% 2.75/1.55 thf(sy_cl_Orderings_Odense__linorder,type,
% 2.75/1.55 dense_linorder:
% 2.75/1.55 !>[A: $tType] : $o ).
% 2.75/1.55
% 2.75/1.55 thf(sy_cl_Rings_Oring__no__zero__divisors,type,
% 2.75/1.55 ring_n68954251visors:
% 2.75/1.55 !>[A: $tType] : $o ).
% 2.75/1.55
% 2.75/1.55 thf(sy_cl_Rings_Olinordered__nonzero__semiring,type,
% 2.75/1.55 linord1659791738miring:
% 2.75/1.55 !>[A: $tType] : $o ).
% 2.75/1.55
% 2.75/1.55 thf(sy_cl_Groups_Ocanonically__ordered__monoid__add,type,
% 2.75/1.55 canoni770627133id_add:
% 2.75/1.55 !>[A: $tType] : $o ).
% 2.75/1.55
% 2.75/1.55 thf(sy_cl_Real__Vector__Spaces_Oreal__normed__vector,type,
% 2.75/1.55 real_V55928688vector:
% 2.75/1.55 !>[A: $tType] : $o ).
% 2.75/1.55
% 2.75/1.55 thf(sy_cl_Conditionally__Complete__Lattices_Olinear__continuum,type,
% 2.75/1.55 condit1656338222tinuum:
% 2.75/1.55 !>[A: $tType] : $o ).
% 2.75/1.55
% 2.75/1.55 thf(sy_cl_Conditionally__Complete__Lattices_Oconditionally__complete__linorder,type,
% 2.75/1.55 condit1037483654norder:
% 2.75/1.55 !>[A: $tType] : $o ).
% 2.75/1.55
% 2.75/1.55 thf(sy_c_Groups_Oone__class_Oone,type,
% 2.75/1.55 one_one:
% 2.75/1.55 !>[A: $tType] : A ).
% 2.75/1.55
% 2.75/1.55 thf(sy_c_Groups_Osgn__class_Osgn,type,
% 2.75/1.55 sgn_sgn:
% 2.75/1.55 !>[A: $tType] : ( A > A ) ).
% 2.75/1.55
% 2.75/1.55 thf(sy_c_Groups_Ozero__class_Ozero,type,
% 2.75/1.55 zero_zero:
% 2.75/1.55 !>[A: $tType] : A ).
% 2.75/1.55
% 2.75/1.55 thf(sy_c_If,type,
% 2.75/1.55 if:
% 2.75/1.55 !>[A: $tType] : ( $o > A > A > A ) ).
% 2.75/1.55
% 2.75/1.55 thf(sy_c_Orderings_Oord__class_Oless,type,
% 2.75/1.55 ord_less:
% 2.75/1.55 !>[A: $tType] : ( A > A > $o ) ).
% 2.75/1.55
% 2.75/1.55 thf(sy_c_Orderings_Oord__class_Oless__eq,type,
% 2.75/1.55 ord_less_eq:
% 2.75/1.55 !>[A: $tType] : ( A > A > $o ) ).
% 2.75/1.55
% 2.75/1.55 thf(sy_c_Orderings_Oord__class_Omin,type,
% 2.75/1.55 ord_min:
% 2.75/1.55 !>[A: $tType] : ( A > A > A ) ).
% 2.75/1.55
% 2.75/1.55 thf(sy_c_Polynomial_Odivide__poly__main,type,
% 2.75/1.55 divide_poly_main:
% 2.75/1.55 !>[A: $tType] : ( A > ( poly @ A ) > ( poly @ A ) > ( poly @ A ) > nat > nat > ( poly @ A ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(sy_c_Polynomial_Ois__zero,type,
% 2.75/1.55 is_zero:
% 2.75/1.55 !>[A: $tType] : ( ( poly @ A ) > $o ) ).
% 2.75/1.55
% 2.75/1.55 thf(sy_c_Polynomial_Oorder,type,
% 2.75/1.55 order2:
% 2.75/1.55 !>[A: $tType] : ( A > ( poly @ A ) > nat ) ).
% 2.75/1.55
% 2.75/1.55 thf(sy_c_Polynomial_Opoly,type,
% 2.75/1.55 poly2:
% 2.75/1.55 !>[A: $tType] : ( ( poly @ A ) > A > A ) ).
% 2.75/1.55
% 2.75/1.55 thf(sy_c_Polynomial_Opoly__cutoff,type,
% 2.75/1.55 poly_cutoff:
% 2.75/1.55 !>[A: $tType] : ( nat > ( poly @ A ) > ( poly @ A ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(sy_c_Polynomial_Opoly__shift,type,
% 2.75/1.55 poly_shift:
% 2.75/1.55 !>[A: $tType] : ( nat > ( poly @ A ) > ( poly @ A ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(sy_c_Polynomial_Oreflect__poly,type,
% 2.75/1.55 reflect_poly:
% 2.75/1.55 !>[A: $tType] : ( ( poly @ A ) > ( poly @ A ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(sy_c_Set_OCollect,type,
% 2.75/1.55 collect:
% 2.75/1.55 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(sy_c_Sturm__Tarski__Mirabelle__osyjafhhev_Osgn__neg__inf,type,
% 2.75/1.55 sturm_1439882693eg_inf:
% 2.75/1.55 !>[A: $tType] : ( ( poly @ A ) > A ) ).
% 2.75/1.55
% 2.75/1.55 thf(sy_c_Sturm__Tarski__Mirabelle__osyjafhhev_Osgn__pos__inf,type,
% 2.75/1.55 sturm_62398785os_inf:
% 2.75/1.55 !>[A: $tType] : ( ( poly @ A ) > A ) ).
% 2.75/1.55
% 2.75/1.55 thf(sy_c_member,type,
% 2.75/1.55 member:
% 2.75/1.55 !>[A: $tType] : ( A > ( set @ A ) > $o ) ).
% 2.75/1.55
% 2.75/1.55 thf(sy_v_lb1____,type,
% 2.75/1.55 lb1: real ).
% 2.75/1.55
% 2.75/1.55 thf(sy_v_lb2____,type,
% 2.75/1.55 lb2: real ).
% 2.75/1.55
% 2.75/1.55 thf(sy_v_lb____,type,
% 2.75/1.55 lb: real ).
% 2.75/1.55
% 2.75/1.55 thf(sy_v_p,type,
% 2.75/1.55 p: poly @ real ).
% 2.75/1.55
% 2.75/1.55 thf(sy_v_thesis,type,
% 2.75/1.55 thesis: $o ).
% 2.75/1.55
% 2.75/1.55 thf(fact_0_lb1,axiom,
% 2.75/1.55 ! [X: real] :
% 2.75/1.55 ( ( ( poly2 @ real @ p @ X )
% 2.75/1.55 = ( zero_zero @ real ) )
% 2.75/1.55 => ( ord_less @ real @ lb1 @ X ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_1_lb__def,axiom,
% 2.75/1.55 ( lb
% 2.75/1.55 = ( ord_min @ real @ lb1 @ lb2 ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_2__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062lb1_O_A_092_060forall_062x_O_Apoly_Ap_Ax_A_061_A0_A_092_060longrightarrow_062_Alb1_A_060_Ax_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
% 2.75/1.55 ~ ! [Lb1: real] :
% 2.75/1.55 ~ ! [X: real] :
% 2.75/1.55 ( ( ( poly2 @ real @ p @ X )
% 2.75/1.55 = ( zero_zero @ real ) )
% 2.75/1.55 => ( ord_less @ real @ Lb1 @ X ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_3_assms,axiom,
% 2.75/1.55 ( p
% 2.75/1.55 != ( zero_zero @ ( poly @ real ) ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_4_poly__0,axiom,
% 2.75/1.55 ! [A: $tType] :
% 2.75/1.55 ( ( comm_semiring_0 @ A )
% 2.75/1.55 => ! [X2: A] :
% 2.75/1.55 ( ( poly2 @ A @ ( zero_zero @ ( poly @ A ) ) @ X2 )
% 2.75/1.55 = ( zero_zero @ A ) ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_5_not__gr__zero,axiom,
% 2.75/1.55 ! [A: $tType] :
% 2.75/1.55 ( ( canoni770627133id_add @ A )
% 2.75/1.55 => ! [N: A] :
% 2.75/1.55 ( ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ N ) )
% 2.75/1.55 = ( N
% 2.75/1.55 = ( zero_zero @ A ) ) ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_6_poly__IVT__neg,axiom,
% 2.75/1.55 ! [A2: real,B: real,P: poly @ real] :
% 2.75/1.55 ( ( ord_less @ real @ A2 @ B )
% 2.75/1.55 => ( ( ord_less @ real @ ( zero_zero @ real ) @ ( poly2 @ real @ P @ A2 ) )
% 2.75/1.55 => ( ( ord_less @ real @ ( poly2 @ real @ P @ B ) @ ( zero_zero @ real ) )
% 2.75/1.55 => ? [X3: real] :
% 2.75/1.55 ( ( ord_less @ real @ A2 @ X3 )
% 2.75/1.55 & ( ord_less @ real @ X3 @ B )
% 2.75/1.55 & ( ( poly2 @ real @ P @ X3 )
% 2.75/1.55 = ( zero_zero @ real ) ) ) ) ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_7_poly__IVT__pos,axiom,
% 2.75/1.55 ! [A2: real,B: real,P: poly @ real] :
% 2.75/1.55 ( ( ord_less @ real @ A2 @ B )
% 2.75/1.55 => ( ( ord_less @ real @ ( poly2 @ real @ P @ A2 ) @ ( zero_zero @ real ) )
% 2.75/1.55 => ( ( ord_less @ real @ ( zero_zero @ real ) @ ( poly2 @ real @ P @ B ) )
% 2.75/1.55 => ? [X3: real] :
% 2.75/1.55 ( ( ord_less @ real @ A2 @ X3 )
% 2.75/1.55 & ( ord_less @ real @ X3 @ B )
% 2.75/1.55 & ( ( poly2 @ real @ P @ X3 )
% 2.75/1.55 = ( zero_zero @ real ) ) ) ) ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_8_poly__all__0__iff__0,axiom,
% 2.75/1.55 ! [A: $tType] :
% 2.75/1.55 ( ( ( ring_char_0 @ A )
% 2.75/1.55 & ( comm_ring_1 @ A )
% 2.75/1.55 & ( ring_n68954251visors @ A ) )
% 2.75/1.55 => ! [P: poly @ A] :
% 2.75/1.55 ( ( ! [X4: A] :
% 2.75/1.55 ( ( poly2 @ A @ P @ X4 )
% 2.75/1.55 = ( zero_zero @ A ) ) )
% 2.75/1.55 = ( P
% 2.75/1.55 = ( zero_zero @ ( poly @ A ) ) ) ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_9_less__numeral__extra_I3_J,axiom,
% 2.75/1.55 ! [A: $tType] :
% 2.75/1.55 ( ( linord1659791738miring @ A )
% 2.75/1.55 => ~ ( ord_less @ A @ ( zero_zero @ A ) @ ( zero_zero @ A ) ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_10_field__lbound__gt__zero,axiom,
% 2.75/1.55 ! [A: $tType] :
% 2.75/1.55 ( ( linordered_field @ A )
% 2.75/1.55 => ! [D1: A,D2: A] :
% 2.75/1.55 ( ( ord_less @ A @ ( zero_zero @ A ) @ D1 )
% 2.75/1.55 => ( ( ord_less @ A @ ( zero_zero @ A ) @ D2 )
% 2.75/1.55 => ? [E: A] :
% 2.75/1.55 ( ( ord_less @ A @ ( zero_zero @ A ) @ E )
% 2.75/1.55 & ( ord_less @ A @ E @ D1 )
% 2.75/1.55 & ( ord_less @ A @ E @ D2 ) ) ) ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_11_gr__zeroI,axiom,
% 2.75/1.55 ! [A: $tType] :
% 2.75/1.55 ( ( canoni770627133id_add @ A )
% 2.75/1.55 => ! [N: A] :
% 2.75/1.55 ( ( N
% 2.75/1.55 != ( zero_zero @ A ) )
% 2.75/1.55 => ( ord_less @ A @ ( zero_zero @ A ) @ N ) ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_12_not__less__zero,axiom,
% 2.75/1.55 ! [A: $tType] :
% 2.75/1.55 ( ( canoni770627133id_add @ A )
% 2.75/1.55 => ! [N: A] :
% 2.75/1.55 ~ ( ord_less @ A @ N @ ( zero_zero @ A ) ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_13_gr__implies__not__zero,axiom,
% 2.75/1.55 ! [A: $tType] :
% 2.75/1.55 ( ( canoni770627133id_add @ A )
% 2.75/1.55 => ! [M: A,N: A] :
% 2.75/1.55 ( ( ord_less @ A @ M @ N )
% 2.75/1.55 => ( N
% 2.75/1.55 != ( zero_zero @ A ) ) ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_14_zero__less__iff__neq__zero,axiom,
% 2.75/1.55 ! [A: $tType] :
% 2.75/1.55 ( ( canoni770627133id_add @ A )
% 2.75/1.55 => ! [N: A] :
% 2.75/1.55 ( ( ord_less @ A @ ( zero_zero @ A ) @ N )
% 2.75/1.55 = ( N
% 2.75/1.55 != ( zero_zero @ A ) ) ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_15_zero__reorient,axiom,
% 2.75/1.55 ! [A: $tType] :
% 2.75/1.55 ( ( zero @ A )
% 2.75/1.55 => ! [X2: A] :
% 2.75/1.55 ( ( ( zero_zero @ A )
% 2.75/1.55 = X2 )
% 2.75/1.55 = ( X2
% 2.75/1.55 = ( zero_zero @ A ) ) ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_16_poly__eq__poly__eq__iff,axiom,
% 2.75/1.55 ! [A: $tType] :
% 2.75/1.55 ( ( ( ring_char_0 @ A )
% 2.75/1.55 & ( comm_ring_1 @ A )
% 2.75/1.55 & ( ring_n68954251visors @ A ) )
% 2.75/1.55 => ! [P: poly @ A,Q: poly @ A] :
% 2.75/1.55 ( ( ( poly2 @ A @ P )
% 2.75/1.55 = ( poly2 @ A @ Q ) )
% 2.75/1.55 = ( P = Q ) ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_17_min__less__iff__conj,axiom,
% 2.75/1.55 ! [A: $tType] :
% 2.75/1.55 ( ( linorder @ A )
% 2.75/1.55 => ! [Z: A,X2: A,Y: A] :
% 2.75/1.55 ( ( ord_less @ A @ Z @ ( ord_min @ A @ X2 @ Y ) )
% 2.75/1.55 = ( ( ord_less @ A @ Z @ X2 )
% 2.75/1.55 & ( ord_less @ A @ Z @ Y ) ) ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_18_min_Oidem,axiom,
% 2.75/1.55 ! [A: $tType] :
% 2.75/1.55 ( ( linorder @ A )
% 2.75/1.55 => ! [A2: A] :
% 2.75/1.55 ( ( ord_min @ A @ A2 @ A2 )
% 2.75/1.55 = A2 ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_19_min_Oleft__idem,axiom,
% 2.75/1.55 ! [A: $tType] :
% 2.75/1.55 ( ( linorder @ A )
% 2.75/1.55 => ! [A2: A,B: A] :
% 2.75/1.55 ( ( ord_min @ A @ A2 @ ( ord_min @ A @ A2 @ B ) )
% 2.75/1.55 = ( ord_min @ A @ A2 @ B ) ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_20_min_Oright__idem,axiom,
% 2.75/1.55 ! [A: $tType] :
% 2.75/1.55 ( ( linorder @ A )
% 2.75/1.55 => ! [A2: A,B: A] :
% 2.75/1.55 ( ( ord_min @ A @ ( ord_min @ A @ A2 @ B ) @ B )
% 2.75/1.55 = ( ord_min @ A @ A2 @ B ) ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_21_min_Ostrict__coboundedI2,axiom,
% 2.75/1.55 ! [A: $tType] :
% 2.75/1.55 ( ( linorder @ A )
% 2.75/1.55 => ! [B: A,C: A,A2: A] :
% 2.75/1.55 ( ( ord_less @ A @ B @ C )
% 2.75/1.55 => ( ord_less @ A @ ( ord_min @ A @ A2 @ B ) @ C ) ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_22_min_Ostrict__coboundedI1,axiom,
% 2.75/1.55 ! [A: $tType] :
% 2.75/1.55 ( ( linorder @ A )
% 2.75/1.55 => ! [A2: A,C: A,B: A] :
% 2.75/1.55 ( ( ord_less @ A @ A2 @ C )
% 2.75/1.55 => ( ord_less @ A @ ( ord_min @ A @ A2 @ B ) @ C ) ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_23_min_Ostrict__order__iff,axiom,
% 2.75/1.55 ! [A: $tType] :
% 2.75/1.55 ( ( linorder @ A )
% 2.75/1.55 => ( ( ord_less @ A )
% 2.75/1.55 = ( ^ [A3: A,B2: A] :
% 2.75/1.55 ( ( A3
% 2.75/1.55 = ( ord_min @ A @ A3 @ B2 ) )
% 2.75/1.55 & ( A3 != B2 ) ) ) ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_24_min_Ostrict__boundedE,axiom,
% 2.75/1.55 ! [A: $tType] :
% 2.75/1.55 ( ( linorder @ A )
% 2.75/1.55 => ! [A2: A,B: A,C: A] :
% 2.75/1.55 ( ( ord_less @ A @ A2 @ ( ord_min @ A @ B @ C ) )
% 2.75/1.55 => ~ ( ( ord_less @ A @ A2 @ B )
% 2.75/1.55 => ~ ( ord_less @ A @ A2 @ C ) ) ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_25_min__less__iff__disj,axiom,
% 2.75/1.55 ! [A: $tType] :
% 2.75/1.55 ( ( linorder @ A )
% 2.75/1.55 => ! [X2: A,Y: A,Z: A] :
% 2.75/1.55 ( ( ord_less @ A @ ( ord_min @ A @ X2 @ Y ) @ Z )
% 2.75/1.55 = ( ( ord_less @ A @ X2 @ Z )
% 2.75/1.55 | ( ord_less @ A @ Y @ Z ) ) ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_26_is__zero__null,axiom,
% 2.75/1.55 ! [A: $tType] :
% 2.75/1.55 ( ( zero @ A )
% 2.75/1.55 => ( ( is_zero @ A )
% 2.75/1.55 = ( ^ [P2: poly @ A] :
% 2.75/1.55 ( P2
% 2.75/1.55 = ( zero_zero @ ( poly @ A ) ) ) ) ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_27_min_Oleft__commute,axiom,
% 2.75/1.55 ! [A: $tType] :
% 2.75/1.55 ( ( linorder @ A )
% 2.75/1.55 => ! [B: A,A2: A,C: A] :
% 2.75/1.55 ( ( ord_min @ A @ B @ ( ord_min @ A @ A2 @ C ) )
% 2.75/1.55 = ( ord_min @ A @ A2 @ ( ord_min @ A @ B @ C ) ) ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_28_min_Ocommute,axiom,
% 2.75/1.55 ! [A: $tType] :
% 2.75/1.55 ( ( linorder @ A )
% 2.75/1.55 => ( ( ord_min @ A )
% 2.75/1.55 = ( ^ [A3: A,B2: A] : ( ord_min @ A @ B2 @ A3 ) ) ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_29_min_Oassoc,axiom,
% 2.75/1.55 ! [A: $tType] :
% 2.75/1.55 ( ( linorder @ A )
% 2.75/1.55 => ! [A2: A,B: A,C: A] :
% 2.75/1.55 ( ( ord_min @ A @ ( ord_min @ A @ A2 @ B ) @ C )
% 2.75/1.55 = ( ord_min @ A @ A2 @ ( ord_min @ A @ B @ C ) ) ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_30_poly__cutoff__0,axiom,
% 2.75/1.55 ! [A: $tType] :
% 2.75/1.55 ( ( zero @ A )
% 2.75/1.55 => ! [N: nat] :
% 2.75/1.55 ( ( poly_cutoff @ A @ N @ ( zero_zero @ ( poly @ A ) ) )
% 2.75/1.55 = ( zero_zero @ ( poly @ A ) ) ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_31_last__non__root__interval,axiom,
% 2.75/1.55 ! [P: poly @ real,Ub: real] :
% 2.75/1.55 ( ( P
% 2.75/1.55 != ( zero_zero @ ( poly @ real ) ) )
% 2.75/1.55 => ~ ! [Lb: real] :
% 2.75/1.55 ( ( ord_less @ real @ Lb @ Ub )
% 2.75/1.55 => ~ ! [Z2: real] :
% 2.75/1.55 ( ( ( ord_less_eq @ real @ Lb @ Z2 )
% 2.75/1.55 & ( ord_less @ real @ Z2 @ Ub ) )
% 2.75/1.55 => ( ( poly2 @ real @ P @ Z2 )
% 2.75/1.55 != ( zero_zero @ real ) ) ) ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_32_next__non__root__interval,axiom,
% 2.75/1.55 ! [P: poly @ real,Lb2: real] :
% 2.75/1.55 ( ( P
% 2.75/1.55 != ( zero_zero @ ( poly @ real ) ) )
% 2.75/1.55 => ~ ! [Ub2: real] :
% 2.75/1.55 ( ( ord_less @ real @ Lb2 @ Ub2 )
% 2.75/1.55 => ~ ! [Z2: real] :
% 2.75/1.55 ( ( ( ord_less @ real @ Lb2 @ Z2 )
% 2.75/1.55 & ( ord_less_eq @ real @ Z2 @ Ub2 ) )
% 2.75/1.55 => ( ( poly2 @ real @ P @ Z2 )
% 2.75/1.55 != ( zero_zero @ real ) ) ) ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_33_reflect__poly__at__0__eq__0__iff,axiom,
% 2.75/1.55 ! [A: $tType] :
% 2.75/1.55 ( ( comm_semiring_0 @ A )
% 2.75/1.55 => ! [P: poly @ A] :
% 2.75/1.55 ( ( ( poly2 @ A @ ( reflect_poly @ A @ P ) @ ( zero_zero @ A ) )
% 2.75/1.55 = ( zero_zero @ A ) )
% 2.75/1.55 = ( P
% 2.75/1.55 = ( zero_zero @ ( poly @ A ) ) ) ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_34_lb2,axiom,
% 2.75/1.55 ! [X: real] :
% 2.75/1.55 ( ( ord_less_eq @ real @ X @ lb2 )
% 2.75/1.55 => ( ( sgn_sgn @ real @ ( poly2 @ real @ p @ X ) )
% 2.75/1.55 = ( sturm_1439882693eg_inf @ real @ p ) ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_35_poly__shift__0,axiom,
% 2.75/1.55 ! [A: $tType] :
% 2.75/1.55 ( ( zero @ A )
% 2.75/1.55 => ! [N: nat] :
% 2.75/1.55 ( ( poly_shift @ A @ N @ ( zero_zero @ ( poly @ A ) ) )
% 2.75/1.55 = ( zero_zero @ ( poly @ A ) ) ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_36_order__root,axiom,
% 2.75/1.55 ! [A: $tType] :
% 2.75/1.55 ( ( idom @ A )
% 2.75/1.55 => ! [P: poly @ A,A2: A] :
% 2.75/1.55 ( ( ( poly2 @ A @ P @ A2 )
% 2.75/1.55 = ( zero_zero @ A ) )
% 2.75/1.55 = ( ( P
% 2.75/1.55 = ( zero_zero @ ( poly @ A ) ) )
% 2.75/1.55 | ( ( order2 @ A @ A2 @ P )
% 2.75/1.55 != ( zero_zero @ nat ) ) ) ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_37_divide__poly__main__0,axiom,
% 2.75/1.55 ! [A: $tType] :
% 2.75/1.55 ( ( idom_divide @ A )
% 2.75/1.55 => ! [R: poly @ A,D: poly @ A,Dr: nat,N: nat] :
% 2.75/1.55 ( ( divide_poly_main @ A @ ( zero_zero @ A ) @ ( zero_zero @ ( poly @ A ) ) @ R @ D @ Dr @ N )
% 2.75/1.55 = ( zero_zero @ ( poly @ A ) ) ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_38_not__eq__pos__or__neg__iff__1,axiom,
% 2.75/1.55 ! [Lb2: real,Ub: real,P: poly @ real] :
% 2.75/1.55 ( ( ! [Z3: real] :
% 2.75/1.55 ( ( ( ord_less @ real @ Lb2 @ Z3 )
% 2.75/1.55 & ( ord_less_eq @ real @ Z3 @ Ub ) )
% 2.75/1.55 => ( ( poly2 @ real @ P @ Z3 )
% 2.75/1.55 != ( zero_zero @ real ) ) ) )
% 2.75/1.55 = ( ! [Z3: real] :
% 2.75/1.55 ( ( ( ord_less @ real @ Lb2 @ Z3 )
% 2.75/1.55 & ( ord_less_eq @ real @ Z3 @ Ub ) )
% 2.75/1.55 => ( ord_less @ real @ ( zero_zero @ real ) @ ( poly2 @ real @ P @ Z3 ) ) )
% 2.75/1.55 | ! [Z3: real] :
% 2.75/1.55 ( ( ( ord_less @ real @ Lb2 @ Z3 )
% 2.75/1.55 & ( ord_less_eq @ real @ Z3 @ Ub ) )
% 2.75/1.55 => ( ord_less @ real @ ( poly2 @ real @ P @ Z3 ) @ ( zero_zero @ real ) ) ) ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_39__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062lb2_O_A_092_060forall_062x_092_060le_062lb2_O_Asgn_A_Ipoly_Ap_Ax_J_A_061_Asgn__neg__inf_Ap_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
% 2.75/1.55 ~ ! [Lb22: real] :
% 2.75/1.55 ~ ! [X: real] :
% 2.75/1.55 ( ( ord_less_eq @ real @ X @ Lb22 )
% 2.75/1.55 => ( ( sgn_sgn @ real @ ( poly2 @ real @ p @ X ) )
% 2.75/1.55 = ( sturm_1439882693eg_inf @ real @ p ) ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_40_le__zero__eq,axiom,
% 2.75/1.55 ! [A: $tType] :
% 2.75/1.55 ( ( canoni770627133id_add @ A )
% 2.75/1.55 => ! [N: A] :
% 2.75/1.55 ( ( ord_less_eq @ A @ N @ ( zero_zero @ A ) )
% 2.75/1.55 = ( N
% 2.75/1.55 = ( zero_zero @ A ) ) ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_41_min_Obounded__iff,axiom,
% 2.75/1.55 ! [A: $tType] :
% 2.75/1.55 ( ( linorder @ A )
% 2.75/1.55 => ! [A2: A,B: A,C: A] :
% 2.75/1.55 ( ( ord_less_eq @ A @ A2 @ ( ord_min @ A @ B @ C ) )
% 2.75/1.55 = ( ( ord_less_eq @ A @ A2 @ B )
% 2.75/1.55 & ( ord_less_eq @ A @ A2 @ C ) ) ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_42_reflect__poly__0,axiom,
% 2.75/1.55 ! [A: $tType] :
% 2.75/1.55 ( ( zero @ A )
% 2.75/1.55 => ( ( reflect_poly @ A @ ( zero_zero @ ( poly @ A ) ) )
% 2.75/1.55 = ( zero_zero @ ( poly @ A ) ) ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_43_poly__shift__id,axiom,
% 2.75/1.55 ! [A: $tType] :
% 2.75/1.55 ( ( zero @ A )
% 2.75/1.55 => ( ( poly_shift @ A @ ( zero_zero @ nat ) )
% 2.75/1.55 = ( ^ [X4: poly @ A] : X4 ) ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_44_mem__Collect__eq,axiom,
% 2.75/1.55 ! [A: $tType,A2: A,P3: A > $o] :
% 2.75/1.55 ( ( member @ A @ A2 @ ( collect @ A @ P3 ) )
% 2.75/1.55 = ( P3 @ A2 ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_45_Collect__mem__eq,axiom,
% 2.75/1.55 ! [A: $tType,A4: set @ A] :
% 2.75/1.55 ( ( collect @ A
% 2.75/1.55 @ ^ [X4: A] : ( member @ A @ X4 @ A4 ) )
% 2.75/1.55 = A4 ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_46_Collect__cong,axiom,
% 2.75/1.55 ! [A: $tType,P3: A > $o,Q2: A > $o] :
% 2.75/1.55 ( ! [X3: A] :
% 2.75/1.55 ( ( P3 @ X3 )
% 2.75/1.55 = ( Q2 @ X3 ) )
% 2.75/1.55 => ( ( collect @ A @ P3 )
% 2.75/1.55 = ( collect @ A @ Q2 ) ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_47_ext,axiom,
% 2.75/1.55 ! [B3: $tType,A: $tType,F: A > B3,G: A > B3] :
% 2.75/1.55 ( ! [X3: A] :
% 2.75/1.55 ( ( F @ X3 )
% 2.75/1.55 = ( G @ X3 ) )
% 2.75/1.55 => ( F = G ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_48_that,axiom,
% 2.75/1.55 ! [Lb2: real] :
% 2.75/1.55 ( ! [X3: real] :
% 2.75/1.55 ( ( ( poly2 @ real @ p @ X3 )
% 2.75/1.55 = ( zero_zero @ real ) )
% 2.75/1.55 => ( ord_less @ real @ Lb2 @ X3 ) )
% 2.75/1.55 => ( ! [X3: real] :
% 2.75/1.55 ( ( ord_less_eq @ real @ X3 @ Lb2 )
% 2.75/1.55 => ( ( sgn_sgn @ real @ ( poly2 @ real @ p @ X3 ) )
% 2.75/1.55 = ( sturm_1439882693eg_inf @ real @ p ) ) )
% 2.75/1.55 => thesis ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_49_complete__real,axiom,
% 2.75/1.55 ! [S: set @ real] :
% 2.75/1.55 ( ? [X: real] : ( member @ real @ X @ S )
% 2.75/1.55 => ( ? [Z2: real] :
% 2.75/1.55 ! [X3: real] :
% 2.75/1.55 ( ( member @ real @ X3 @ S )
% 2.75/1.55 => ( ord_less_eq @ real @ X3 @ Z2 ) )
% 2.75/1.55 => ? [Y2: real] :
% 2.75/1.55 ( ! [X: real] :
% 2.75/1.55 ( ( member @ real @ X @ S )
% 2.75/1.55 => ( ord_less_eq @ real @ X @ Y2 ) )
% 2.75/1.55 & ! [Z2: real] :
% 2.75/1.55 ( ! [X3: real] :
% 2.75/1.55 ( ( member @ real @ X3 @ S )
% 2.75/1.55 => ( ord_less_eq @ real @ X3 @ Z2 ) )
% 2.75/1.55 => ( ord_less_eq @ real @ Y2 @ Z2 ) ) ) ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_50_divide__poly__main_Osimps_I2_J,axiom,
% 2.75/1.55 ! [A: $tType] :
% 2.75/1.55 ( ( idom_divide @ A )
% 2.75/1.55 => ! [Lc: A,Q: poly @ A,R: poly @ A,D: poly @ A,Dr: nat] :
% 2.75/1.55 ( ( divide_poly_main @ A @ Lc @ Q @ R @ D @ Dr @ ( zero_zero @ nat ) )
% 2.75/1.55 = Q ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_51_zero__le,axiom,
% 2.75/1.55 ! [A: $tType] :
% 2.75/1.55 ( ( canoni770627133id_add @ A )
% 2.75/1.55 => ! [X2: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ X2 ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_52_le__numeral__extra_I3_J,axiom,
% 2.75/1.55 ! [A: $tType] :
% 2.75/1.55 ( ( linord1659791738miring @ A )
% 2.75/1.55 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( zero_zero @ A ) ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_53_less__eq__real__def,axiom,
% 2.75/1.55 ( ( ord_less_eq @ real )
% 2.75/1.55 = ( ^ [X4: real,Y3: real] :
% 2.75/1.55 ( ( ord_less @ real @ X4 @ Y3 )
% 2.75/1.55 | ( X4 = Y3 ) ) ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_54_min_Omono,axiom,
% 2.75/1.55 ! [A: $tType] :
% 2.75/1.55 ( ( linorder @ A )
% 2.75/1.55 => ! [A2: A,C: A,B: A,D: A] :
% 2.75/1.55 ( ( ord_less_eq @ A @ A2 @ C )
% 2.75/1.55 => ( ( ord_less_eq @ A @ B @ D )
% 2.75/1.55 => ( ord_less_eq @ A @ ( ord_min @ A @ A2 @ B ) @ ( ord_min @ A @ C @ D ) ) ) ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_55_min_OorderE,axiom,
% 2.75/1.55 ! [A: $tType] :
% 2.75/1.55 ( ( linorder @ A )
% 2.75/1.55 => ! [A2: A,B: A] :
% 2.75/1.55 ( ( ord_less_eq @ A @ A2 @ B )
% 2.75/1.55 => ( A2
% 2.75/1.55 = ( ord_min @ A @ A2 @ B ) ) ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_56_min_OorderI,axiom,
% 2.75/1.55 ! [A: $tType] :
% 2.75/1.55 ( ( linorder @ A )
% 2.75/1.55 => ! [A2: A,B: A] :
% 2.75/1.55 ( ( A2
% 2.75/1.55 = ( ord_min @ A @ A2 @ B ) )
% 2.75/1.55 => ( ord_less_eq @ A @ A2 @ B ) ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_57_min_Oabsorb1,axiom,
% 2.75/1.55 ! [A: $tType] :
% 2.75/1.55 ( ( linorder @ A )
% 2.75/1.55 => ! [A2: A,B: A] :
% 2.75/1.55 ( ( ord_less_eq @ A @ A2 @ B )
% 2.75/1.55 => ( ( ord_min @ A @ A2 @ B )
% 2.75/1.55 = A2 ) ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_58_min_Oabsorb2,axiom,
% 2.75/1.55 ! [A: $tType] :
% 2.75/1.55 ( ( linorder @ A )
% 2.75/1.55 => ! [B: A,A2: A] :
% 2.75/1.55 ( ( ord_less_eq @ A @ B @ A2 )
% 2.75/1.55 => ( ( ord_min @ A @ A2 @ B )
% 2.75/1.55 = B ) ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_59_min_OboundedE,axiom,
% 2.75/1.55 ! [A: $tType] :
% 2.75/1.55 ( ( linorder @ A )
% 2.75/1.55 => ! [A2: A,B: A,C: A] :
% 2.75/1.55 ( ( ord_less_eq @ A @ A2 @ ( ord_min @ A @ B @ C ) )
% 2.75/1.55 => ~ ( ( ord_less_eq @ A @ A2 @ B )
% 2.75/1.55 => ~ ( ord_less_eq @ A @ A2 @ C ) ) ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_60_min_OboundedI,axiom,
% 2.75/1.55 ! [A: $tType] :
% 2.75/1.55 ( ( linorder @ A )
% 2.75/1.55 => ! [A2: A,B: A,C: A] :
% 2.75/1.55 ( ( ord_less_eq @ A @ A2 @ B )
% 2.75/1.55 => ( ( ord_less_eq @ A @ A2 @ C )
% 2.75/1.55 => ( ord_less_eq @ A @ A2 @ ( ord_min @ A @ B @ C ) ) ) ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_61_min_Oorder__iff,axiom,
% 2.75/1.55 ! [A: $tType] :
% 2.75/1.55 ( ( linorder @ A )
% 2.75/1.55 => ( ( ord_less_eq @ A )
% 2.75/1.55 = ( ^ [A3: A,B2: A] :
% 2.75/1.55 ( A3
% 2.75/1.55 = ( ord_min @ A @ A3 @ B2 ) ) ) ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_62_min_Ocobounded1,axiom,
% 2.75/1.55 ! [A: $tType] :
% 2.75/1.55 ( ( linorder @ A )
% 2.75/1.55 => ! [A2: A,B: A] : ( ord_less_eq @ A @ ( ord_min @ A @ A2 @ B ) @ A2 ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_63_min_Ocobounded2,axiom,
% 2.75/1.55 ! [A: $tType] :
% 2.75/1.55 ( ( linorder @ A )
% 2.75/1.55 => ! [A2: A,B: A] : ( ord_less_eq @ A @ ( ord_min @ A @ A2 @ B ) @ B ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_64_min_Oabsorb__iff1,axiom,
% 2.75/1.55 ! [A: $tType] :
% 2.75/1.55 ( ( linorder @ A )
% 2.75/1.55 => ( ( ord_less_eq @ A )
% 2.75/1.55 = ( ^ [A3: A,B2: A] :
% 2.75/1.55 ( ( ord_min @ A @ A3 @ B2 )
% 2.75/1.55 = A3 ) ) ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_65_min_Oabsorb__iff2,axiom,
% 2.75/1.55 ! [A: $tType] :
% 2.75/1.55 ( ( linorder @ A )
% 2.75/1.55 => ( ( ord_less_eq @ A )
% 2.75/1.55 = ( ^ [B2: A,A3: A] :
% 2.75/1.55 ( ( ord_min @ A @ A3 @ B2 )
% 2.75/1.55 = B2 ) ) ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_66_min_OcoboundedI1,axiom,
% 2.75/1.55 ! [A: $tType] :
% 2.75/1.55 ( ( linorder @ A )
% 2.75/1.55 => ! [A2: A,C: A,B: A] :
% 2.75/1.55 ( ( ord_less_eq @ A @ A2 @ C )
% 2.75/1.55 => ( ord_less_eq @ A @ ( ord_min @ A @ A2 @ B ) @ C ) ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_67_min_OcoboundedI2,axiom,
% 2.75/1.55 ! [A: $tType] :
% 2.75/1.55 ( ( linorder @ A )
% 2.75/1.55 => ! [B: A,C: A,A2: A] :
% 2.75/1.55 ( ( ord_less_eq @ A @ B @ C )
% 2.75/1.55 => ( ord_less_eq @ A @ ( ord_min @ A @ A2 @ B ) @ C ) ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_68_min__le__iff__disj,axiom,
% 2.75/1.55 ! [A: $tType] :
% 2.75/1.55 ( ( linorder @ A )
% 2.75/1.55 => ! [X2: A,Y: A,Z: A] :
% 2.75/1.55 ( ( ord_less_eq @ A @ ( ord_min @ A @ X2 @ Y ) @ Z )
% 2.75/1.55 = ( ( ord_less_eq @ A @ X2 @ Z )
% 2.75/1.55 | ( ord_less_eq @ A @ Y @ Z ) ) ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_69_order__0I,axiom,
% 2.75/1.55 ! [A: $tType] :
% 2.75/1.55 ( ( idom @ A )
% 2.75/1.55 => ! [P: poly @ A,A2: A] :
% 2.75/1.55 ( ( ( poly2 @ A @ P @ A2 )
% 2.75/1.55 != ( zero_zero @ A ) )
% 2.75/1.55 => ( ( order2 @ A @ A2 @ P )
% 2.75/1.55 = ( zero_zero @ nat ) ) ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_70_not__eq__pos__or__neg__iff__2,axiom,
% 2.75/1.55 ! [Lb2: real,Ub: real,P: poly @ real] :
% 2.75/1.55 ( ( ! [Z3: real] :
% 2.75/1.55 ( ( ( ord_less_eq @ real @ Lb2 @ Z3 )
% 2.75/1.55 & ( ord_less @ real @ Z3 @ Ub ) )
% 2.75/1.55 => ( ( poly2 @ real @ P @ Z3 )
% 2.75/1.55 != ( zero_zero @ real ) ) ) )
% 2.75/1.55 = ( ! [Z3: real] :
% 2.75/1.55 ( ( ( ord_less_eq @ real @ Lb2 @ Z3 )
% 2.75/1.55 & ( ord_less @ real @ Z3 @ Ub ) )
% 2.75/1.55 => ( ord_less @ real @ ( zero_zero @ real ) @ ( poly2 @ real @ P @ Z3 ) ) )
% 2.75/1.55 | ! [Z3: real] :
% 2.75/1.55 ( ( ( ord_less_eq @ real @ Lb2 @ Z3 )
% 2.75/1.55 & ( ord_less @ real @ Z3 @ Ub ) )
% 2.75/1.55 => ( ord_less @ real @ ( poly2 @ real @ P @ Z3 ) @ ( zero_zero @ real ) ) ) ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_71_zero__le__sgn__iff,axiom,
% 2.75/1.55 ! [X2: real] :
% 2.75/1.55 ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( sgn_sgn @ real @ X2 ) )
% 2.75/1.55 = ( ord_less_eq @ real @ ( zero_zero @ real ) @ X2 ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_72_sgn__le__0__iff,axiom,
% 2.75/1.55 ! [X2: real] :
% 2.75/1.55 ( ( ord_less_eq @ real @ ( sgn_sgn @ real @ X2 ) @ ( zero_zero @ real ) )
% 2.75/1.55 = ( ord_less_eq @ real @ X2 @ ( zero_zero @ real ) ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_73_sgn__greater,axiom,
% 2.75/1.55 ! [A: $tType] :
% 2.75/1.55 ( ( linordered_idom @ A )
% 2.75/1.55 => ! [A2: A] :
% 2.75/1.55 ( ( ord_less @ A @ ( zero_zero @ A ) @ ( sgn_sgn @ A @ A2 ) )
% 2.75/1.55 = ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_74_sgn__less,axiom,
% 2.75/1.55 ! [A: $tType] :
% 2.75/1.55 ( ( linordered_idom @ A )
% 2.75/1.55 => ! [A2: A] :
% 2.75/1.55 ( ( ord_less @ A @ ( sgn_sgn @ A @ A2 ) @ ( zero_zero @ A ) )
% 2.75/1.55 = ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_75_root__ub,axiom,
% 2.75/1.55 ! [P: poly @ real] :
% 2.75/1.55 ( ( P
% 2.75/1.55 != ( zero_zero @ ( poly @ real ) ) )
% 2.75/1.55 => ~ ! [Ub2: real] :
% 2.75/1.55 ( ! [X: real] :
% 2.75/1.55 ( ( ( poly2 @ real @ P @ X )
% 2.75/1.55 = ( zero_zero @ real ) )
% 2.75/1.55 => ( ord_less @ real @ X @ Ub2 ) )
% 2.75/1.55 => ~ ! [X: real] :
% 2.75/1.55 ( ( ord_less_eq @ real @ Ub2 @ X )
% 2.75/1.55 => ( ( sgn_sgn @ real @ ( poly2 @ real @ P @ X ) )
% 2.75/1.55 = ( sturm_62398785os_inf @ real @ P ) ) ) ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_76_sgn__0,axiom,
% 2.75/1.55 ! [A: $tType] :
% 2.75/1.55 ( ( idom_abs_sgn @ A )
% 2.75/1.55 => ( ( sgn_sgn @ A @ ( zero_zero @ A ) )
% 2.75/1.55 = ( zero_zero @ A ) ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_77_sgn__zero,axiom,
% 2.75/1.55 ! [A: $tType] :
% 2.75/1.55 ( ( real_V55928688vector @ A )
% 2.75/1.55 => ( ( sgn_sgn @ A @ ( zero_zero @ A ) )
% 2.75/1.55 = ( zero_zero @ A ) ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_78_sgn__sgn,axiom,
% 2.75/1.55 ! [A: $tType] :
% 2.75/1.55 ( ( idom_abs_sgn @ A )
% 2.75/1.55 => ! [A2: A] :
% 2.75/1.55 ( ( sgn_sgn @ A @ ( sgn_sgn @ A @ A2 ) )
% 2.75/1.55 = ( sgn_sgn @ A @ A2 ) ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_79_linorder__neqE__linordered__idom,axiom,
% 2.75/1.55 ! [A: $tType] :
% 2.75/1.55 ( ( linordered_idom @ A )
% 2.75/1.55 => ! [X2: A,Y: A] :
% 2.75/1.55 ( ( X2 != Y )
% 2.75/1.55 => ( ~ ( ord_less @ A @ X2 @ Y )
% 2.75/1.55 => ( ord_less @ A @ Y @ X2 ) ) ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_80_sgn__eq__0__iff,axiom,
% 2.75/1.55 ! [A: $tType] :
% 2.75/1.55 ( ( idom_abs_sgn @ A )
% 2.75/1.55 => ! [A2: A] :
% 2.75/1.55 ( ( ( sgn_sgn @ A @ A2 )
% 2.75/1.55 = ( zero_zero @ A ) )
% 2.75/1.55 = ( A2
% 2.75/1.55 = ( zero_zero @ A ) ) ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_81_sgn__0__0,axiom,
% 2.75/1.55 ! [A: $tType] :
% 2.75/1.55 ( ( linordered_idom @ A )
% 2.75/1.55 => ! [A2: A] :
% 2.75/1.55 ( ( ( sgn_sgn @ A @ A2 )
% 2.75/1.55 = ( zero_zero @ A ) )
% 2.75/1.55 = ( A2
% 2.75/1.55 = ( zero_zero @ A ) ) ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_82_sgn__zero__iff,axiom,
% 2.75/1.55 ! [A: $tType] :
% 2.75/1.55 ( ( real_V55928688vector @ A )
% 2.75/1.55 => ! [X2: A] :
% 2.75/1.55 ( ( ( sgn_sgn @ A @ X2 )
% 2.75/1.55 = ( zero_zero @ A ) )
% 2.75/1.55 = ( X2
% 2.75/1.55 = ( zero_zero @ A ) ) ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_83_le0,axiom,
% 2.75/1.55 ! [N: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ N ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_84_bot__nat__0_Oextremum,axiom,
% 2.75/1.55 ! [A2: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ A2 ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_85_neq0__conv,axiom,
% 2.75/1.55 ! [N: nat] :
% 2.75/1.55 ( ( N
% 2.75/1.55 != ( zero_zero @ nat ) )
% 2.75/1.55 = ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_86_less__nat__zero__code,axiom,
% 2.75/1.55 ! [N: nat] :
% 2.75/1.55 ~ ( ord_less @ nat @ N @ ( zero_zero @ nat ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_87_bot__nat__0_Onot__eq__extremum,axiom,
% 2.75/1.55 ! [A2: nat] :
% 2.75/1.55 ( ( A2
% 2.75/1.55 != ( zero_zero @ nat ) )
% 2.75/1.55 = ( ord_less @ nat @ ( zero_zero @ nat ) @ A2 ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_88_min__0R,axiom,
% 2.75/1.55 ! [N: nat] :
% 2.75/1.55 ( ( ord_min @ nat @ N @ ( zero_zero @ nat ) )
% 2.75/1.55 = ( zero_zero @ nat ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_89_min__0L,axiom,
% 2.75/1.55 ! [N: nat] :
% 2.75/1.55 ( ( ord_min @ nat @ ( zero_zero @ nat ) @ N )
% 2.75/1.55 = ( zero_zero @ nat ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_90_le__refl,axiom,
% 2.75/1.55 ! [N: nat] : ( ord_less_eq @ nat @ N @ N ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_91_le__trans,axiom,
% 2.75/1.55 ! [I: nat,J: nat,K: nat] :
% 2.75/1.55 ( ( ord_less_eq @ nat @ I @ J )
% 2.75/1.55 => ( ( ord_less_eq @ nat @ J @ K )
% 2.75/1.55 => ( ord_less_eq @ nat @ I @ K ) ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_92_eq__imp__le,axiom,
% 2.75/1.55 ! [M: nat,N: nat] :
% 2.75/1.55 ( ( M = N )
% 2.75/1.55 => ( ord_less_eq @ nat @ M @ N ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_93_le__antisym,axiom,
% 2.75/1.55 ! [M: nat,N: nat] :
% 2.75/1.55 ( ( ord_less_eq @ nat @ M @ N )
% 2.75/1.55 => ( ( ord_less_eq @ nat @ N @ M )
% 2.75/1.55 => ( M = N ) ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_94_nat__le__linear,axiom,
% 2.75/1.55 ! [M: nat,N: nat] :
% 2.75/1.55 ( ( ord_less_eq @ nat @ M @ N )
% 2.75/1.55 | ( ord_less_eq @ nat @ N @ M ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_95_Nat_Oex__has__greatest__nat,axiom,
% 2.75/1.55 ! [P3: nat > $o,K: nat,B: nat] :
% 2.75/1.55 ( ( P3 @ K )
% 2.75/1.55 => ( ! [Y2: nat] :
% 2.75/1.55 ( ( P3 @ Y2 )
% 2.75/1.55 => ( ord_less_eq @ nat @ Y2 @ B ) )
% 2.75/1.55 => ? [X3: nat] :
% 2.75/1.55 ( ( P3 @ X3 )
% 2.75/1.55 & ! [Y4: nat] :
% 2.75/1.55 ( ( P3 @ Y4 )
% 2.75/1.55 => ( ord_less_eq @ nat @ Y4 @ X3 ) ) ) ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_96_less__mono__imp__le__mono,axiom,
% 2.75/1.55 ! [F: nat > nat,I: nat,J: nat] :
% 2.75/1.55 ( ! [I2: nat,J2: nat] :
% 2.75/1.55 ( ( ord_less @ nat @ I2 @ J2 )
% 2.75/1.55 => ( ord_less @ nat @ ( F @ I2 ) @ ( F @ J2 ) ) )
% 2.75/1.55 => ( ( ord_less_eq @ nat @ I @ J )
% 2.75/1.55 => ( ord_less_eq @ nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_97_le__neq__implies__less,axiom,
% 2.75/1.55 ! [M: nat,N: nat] :
% 2.75/1.55 ( ( ord_less_eq @ nat @ M @ N )
% 2.75/1.55 => ( ( M != N )
% 2.75/1.55 => ( ord_less @ nat @ M @ N ) ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_98_less__or__eq__imp__le,axiom,
% 2.75/1.55 ! [M: nat,N: nat] :
% 2.75/1.55 ( ( ( ord_less @ nat @ M @ N )
% 2.75/1.55 | ( M = N ) )
% 2.75/1.55 => ( ord_less_eq @ nat @ M @ N ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_99_le__eq__less__or__eq,axiom,
% 2.75/1.55 ( ( ord_less_eq @ nat )
% 2.75/1.55 = ( ^ [M2: nat,N2: nat] :
% 2.75/1.55 ( ( ord_less @ nat @ M2 @ N2 )
% 2.75/1.55 | ( M2 = N2 ) ) ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_100_less__imp__le__nat,axiom,
% 2.75/1.55 ! [M: nat,N: nat] :
% 2.75/1.55 ( ( ord_less @ nat @ M @ N )
% 2.75/1.55 => ( ord_less_eq @ nat @ M @ N ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_101_nat__less__le,axiom,
% 2.75/1.55 ( ( ord_less @ nat )
% 2.75/1.55 = ( ^ [M2: nat,N2: nat] :
% 2.75/1.55 ( ( ord_less_eq @ nat @ M2 @ N2 )
% 2.75/1.55 & ( M2 != N2 ) ) ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_102_infinite__descent__measure,axiom,
% 2.75/1.55 ! [A: $tType,P3: A > $o,V: A > nat,X2: A] :
% 2.75/1.55 ( ! [X3: A] :
% 2.75/1.55 ( ~ ( P3 @ X3 )
% 2.75/1.55 => ? [Y4: A] :
% 2.75/1.55 ( ( ord_less @ nat @ ( V @ Y4 ) @ ( V @ X3 ) )
% 2.75/1.55 & ~ ( P3 @ Y4 ) ) )
% 2.75/1.55 => ( P3 @ X2 ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_103_linorder__neqE__nat,axiom,
% 2.75/1.55 ! [X2: nat,Y: nat] :
% 2.75/1.55 ( ( X2 != Y )
% 2.75/1.55 => ( ~ ( ord_less @ nat @ X2 @ Y )
% 2.75/1.55 => ( ord_less @ nat @ Y @ X2 ) ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_104_infinite__descent,axiom,
% 2.75/1.55 ! [P3: nat > $o,N: nat] :
% 2.75/1.55 ( ! [N3: nat] :
% 2.75/1.55 ( ~ ( P3 @ N3 )
% 2.75/1.55 => ? [M3: nat] :
% 2.75/1.55 ( ( ord_less @ nat @ M3 @ N3 )
% 2.75/1.55 & ~ ( P3 @ M3 ) ) )
% 2.75/1.55 => ( P3 @ N ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_105_nat__less__induct,axiom,
% 2.75/1.55 ! [P3: nat > $o,N: nat] :
% 2.75/1.55 ( ! [N3: nat] :
% 2.75/1.55 ( ! [M3: nat] :
% 2.75/1.55 ( ( ord_less @ nat @ M3 @ N3 )
% 2.75/1.55 => ( P3 @ M3 ) )
% 2.75/1.55 => ( P3 @ N3 ) )
% 2.75/1.55 => ( P3 @ N ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_106_less__irrefl__nat,axiom,
% 2.75/1.55 ! [N: nat] :
% 2.75/1.55 ~ ( ord_less @ nat @ N @ N ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_107_less__not__refl3,axiom,
% 2.75/1.55 ! [S2: nat,T: nat] :
% 2.75/1.55 ( ( ord_less @ nat @ S2 @ T )
% 2.75/1.55 => ( S2 != T ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_108_less__not__refl2,axiom,
% 2.75/1.55 ! [N: nat,M: nat] :
% 2.75/1.55 ( ( ord_less @ nat @ N @ M )
% 2.75/1.55 => ( M != N ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_109_less__not__refl,axiom,
% 2.75/1.55 ! [N: nat] :
% 2.75/1.55 ~ ( ord_less @ nat @ N @ N ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_110_nat__neq__iff,axiom,
% 2.75/1.55 ! [M: nat,N: nat] :
% 2.75/1.55 ( ( M != N )
% 2.75/1.55 = ( ( ord_less @ nat @ M @ N )
% 2.75/1.55 | ( ord_less @ nat @ N @ M ) ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_111_measure__induct__rule,axiom,
% 2.75/1.55 ! [B3: $tType,A: $tType] :
% 2.75/1.55 ( ( wellorder @ B3 )
% 2.75/1.55 => ! [F: A > B3,P3: A > $o,A2: A] :
% 2.75/1.55 ( ! [X3: A] :
% 2.75/1.55 ( ! [Y4: A] :
% 2.75/1.55 ( ( ord_less @ B3 @ ( F @ Y4 ) @ ( F @ X3 ) )
% 2.75/1.55 => ( P3 @ Y4 ) )
% 2.75/1.55 => ( P3 @ X3 ) )
% 2.75/1.55 => ( P3 @ A2 ) ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_112_measure__induct,axiom,
% 2.75/1.55 ! [B3: $tType,A: $tType] :
% 2.75/1.55 ( ( wellorder @ B3 )
% 2.75/1.55 => ! [F: A > B3,P3: A > $o,A2: A] :
% 2.75/1.55 ( ! [X3: A] :
% 2.75/1.55 ( ! [Y4: A] :
% 2.75/1.55 ( ( ord_less @ B3 @ ( F @ Y4 ) @ ( F @ X3 ) )
% 2.75/1.55 => ( P3 @ Y4 ) )
% 2.75/1.55 => ( P3 @ X3 ) )
% 2.75/1.55 => ( P3 @ A2 ) ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_113_infinite__descent0__measure,axiom,
% 2.75/1.55 ! [A: $tType,V: A > nat,P3: A > $o,X2: A] :
% 2.75/1.55 ( ! [X3: A] :
% 2.75/1.55 ( ( ( V @ X3 )
% 2.75/1.55 = ( zero_zero @ nat ) )
% 2.75/1.55 => ( P3 @ X3 ) )
% 2.75/1.55 => ( ! [X3: A] :
% 2.75/1.55 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( V @ X3 ) )
% 2.75/1.55 => ( ~ ( P3 @ X3 )
% 2.75/1.55 => ? [Y4: A] :
% 2.75/1.55 ( ( ord_less @ nat @ ( V @ Y4 ) @ ( V @ X3 ) )
% 2.75/1.55 & ~ ( P3 @ Y4 ) ) ) )
% 2.75/1.55 => ( P3 @ X2 ) ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_114_bot__nat__0_Oextremum__strict,axiom,
% 2.75/1.55 ! [A2: nat] :
% 2.75/1.55 ~ ( ord_less @ nat @ A2 @ ( zero_zero @ nat ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_115_infinite__descent0,axiom,
% 2.75/1.55 ! [P3: nat > $o,N: nat] :
% 2.75/1.55 ( ( P3 @ ( zero_zero @ nat ) )
% 2.75/1.55 => ( ! [N3: nat] :
% 2.75/1.55 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 )
% 2.75/1.55 => ( ~ ( P3 @ N3 )
% 2.75/1.55 => ? [M3: nat] :
% 2.75/1.55 ( ( ord_less @ nat @ M3 @ N3 )
% 2.75/1.55 & ~ ( P3 @ M3 ) ) ) )
% 2.75/1.55 => ( P3 @ N ) ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_116_gr__implies__not0,axiom,
% 2.75/1.55 ! [M: nat,N: nat] :
% 2.75/1.55 ( ( ord_less @ nat @ M @ N )
% 2.75/1.55 => ( N
% 2.75/1.55 != ( zero_zero @ nat ) ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_117_less__zeroE,axiom,
% 2.75/1.55 ! [N: nat] :
% 2.75/1.55 ~ ( ord_less @ nat @ N @ ( zero_zero @ nat ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_118_not__less0,axiom,
% 2.75/1.55 ! [N: nat] :
% 2.75/1.55 ~ ( ord_less @ nat @ N @ ( zero_zero @ nat ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_119_not__gr0,axiom,
% 2.75/1.55 ! [N: nat] :
% 2.75/1.55 ( ( ~ ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) )
% 2.75/1.55 = ( N
% 2.75/1.55 = ( zero_zero @ nat ) ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_120_gr0I,axiom,
% 2.75/1.55 ! [N: nat] :
% 2.75/1.55 ( ( N
% 2.75/1.55 != ( zero_zero @ nat ) )
% 2.75/1.55 => ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_121_less__eq__nat_Osimps_I1_J,axiom,
% 2.75/1.55 ! [N: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ N ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_122_le__0__eq,axiom,
% 2.75/1.55 ! [N: nat] :
% 2.75/1.55 ( ( ord_less_eq @ nat @ N @ ( zero_zero @ nat ) )
% 2.75/1.55 = ( N
% 2.75/1.55 = ( zero_zero @ nat ) ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_123_ex__least__nat__le,axiom,
% 2.75/1.55 ! [P3: nat > $o,N: nat] :
% 2.75/1.55 ( ( P3 @ N )
% 2.75/1.55 => ( ~ ( P3 @ ( zero_zero @ nat ) )
% 2.75/1.55 => ? [K2: nat] :
% 2.75/1.55 ( ( ord_less_eq @ nat @ K2 @ N )
% 2.75/1.55 & ! [I3: nat] :
% 2.75/1.55 ( ( ord_less @ nat @ I3 @ K2 )
% 2.75/1.55 => ~ ( P3 @ I3 ) )
% 2.75/1.55 & ( P3 @ K2 ) ) ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_124_bot__nat__0_Oextremum__unique,axiom,
% 2.75/1.55 ! [A2: nat] :
% 2.75/1.55 ( ( ord_less_eq @ nat @ A2 @ ( zero_zero @ nat ) )
% 2.75/1.55 = ( A2
% 2.75/1.55 = ( zero_zero @ nat ) ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_125_bot__nat__0_Oextremum__uniqueI,axiom,
% 2.75/1.55 ! [A2: nat] :
% 2.75/1.55 ( ( ord_less_eq @ nat @ A2 @ ( zero_zero @ nat ) )
% 2.75/1.55 => ( A2
% 2.75/1.55 = ( zero_zero @ nat ) ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_126_order__refl,axiom,
% 2.75/1.55 ! [A: $tType] :
% 2.75/1.55 ( ( preorder @ A )
% 2.75/1.55 => ! [X2: A] : ( ord_less_eq @ A @ X2 @ X2 ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_127_min__absorb1,axiom,
% 2.75/1.55 ! [A: $tType] :
% 2.75/1.55 ( ( ord @ A )
% 2.75/1.55 => ! [X2: A,Y: A] :
% 2.75/1.55 ( ( ord_less_eq @ A @ X2 @ Y )
% 2.75/1.55 => ( ( ord_min @ A @ X2 @ Y )
% 2.75/1.55 = X2 ) ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_128_min__absorb2,axiom,
% 2.75/1.55 ! [A: $tType] :
% 2.75/1.55 ( ( order @ A )
% 2.75/1.55 => ! [Y: A,X2: A] :
% 2.75/1.55 ( ( ord_less_eq @ A @ Y @ X2 )
% 2.75/1.55 => ( ( ord_min @ A @ X2 @ Y )
% 2.75/1.55 = Y ) ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_129_min__def,axiom,
% 2.75/1.55 ! [A: $tType] :
% 2.75/1.55 ( ( ord @ A )
% 2.75/1.55 => ( ( ord_min @ A )
% 2.75/1.55 = ( ^ [A3: A,B2: A] : ( if @ A @ ( ord_less_eq @ A @ A3 @ B2 ) @ A3 @ B2 ) ) ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_130_Lattices__Big_Oex__has__greatest__nat,axiom,
% 2.75/1.55 ! [A: $tType,P3: A > $o,K: A,F: A > nat,B: nat] :
% 2.75/1.55 ( ( P3 @ K )
% 2.75/1.55 => ( ! [Y2: A] :
% 2.75/1.55 ( ( P3 @ Y2 )
% 2.75/1.55 => ( ord_less @ nat @ ( F @ Y2 ) @ B ) )
% 2.75/1.55 => ? [X3: A] :
% 2.75/1.55 ( ( P3 @ X3 )
% 2.75/1.55 & ! [Y4: A] :
% 2.75/1.55 ( ( P3 @ Y4 )
% 2.75/1.55 => ( ord_less_eq @ nat @ ( F @ Y4 ) @ ( F @ X3 ) ) ) ) ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_131_nat__descend__induct,axiom,
% 2.75/1.55 ! [N: nat,P3: nat > $o,M: nat] :
% 2.75/1.55 ( ! [K2: nat] :
% 2.75/1.55 ( ( ord_less @ nat @ N @ K2 )
% 2.75/1.55 => ( P3 @ K2 ) )
% 2.75/1.55 => ( ! [K2: nat] :
% 2.75/1.55 ( ( ord_less_eq @ nat @ K2 @ N )
% 2.75/1.55 => ( ! [I3: nat] :
% 2.75/1.55 ( ( ord_less @ nat @ K2 @ I3 )
% 2.75/1.55 => ( P3 @ I3 ) )
% 2.75/1.55 => ( P3 @ K2 ) ) )
% 2.75/1.55 => ( P3 @ M ) ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_132_dual__order_Oantisym,axiom,
% 2.75/1.55 ! [A: $tType] :
% 2.75/1.55 ( ( order @ A )
% 2.75/1.55 => ! [B: A,A2: A] :
% 2.75/1.55 ( ( ord_less_eq @ A @ B @ A2 )
% 2.75/1.55 => ( ( ord_less_eq @ A @ A2 @ B )
% 2.75/1.55 => ( A2 = B ) ) ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_133_dual__order_Oeq__iff,axiom,
% 2.75/1.55 ! [A: $tType] :
% 2.75/1.55 ( ( order @ A )
% 2.75/1.55 => ( ( ^ [Y5: A,Z4: A] : ( Y5 = Z4 ) )
% 2.75/1.55 = ( ^ [A3: A,B2: A] :
% 2.75/1.55 ( ( ord_less_eq @ A @ B2 @ A3 )
% 2.75/1.55 & ( ord_less_eq @ A @ A3 @ B2 ) ) ) ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_134_dual__order_Otrans,axiom,
% 2.75/1.55 ! [A: $tType] :
% 2.75/1.55 ( ( order @ A )
% 2.75/1.55 => ! [B: A,A2: A,C: A] :
% 2.75/1.55 ( ( ord_less_eq @ A @ B @ A2 )
% 2.75/1.55 => ( ( ord_less_eq @ A @ C @ B )
% 2.75/1.55 => ( ord_less_eq @ A @ C @ A2 ) ) ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_135_linorder__wlog,axiom,
% 2.75/1.55 ! [A: $tType] :
% 2.75/1.55 ( ( linorder @ A )
% 2.75/1.55 => ! [P3: A > A > $o,A2: A,B: A] :
% 2.75/1.55 ( ! [A5: A,B4: A] :
% 2.75/1.55 ( ( ord_less_eq @ A @ A5 @ B4 )
% 2.75/1.55 => ( P3 @ A5 @ B4 ) )
% 2.75/1.55 => ( ! [A5: A,B4: A] :
% 2.75/1.55 ( ( P3 @ B4 @ A5 )
% 2.75/1.55 => ( P3 @ A5 @ B4 ) )
% 2.75/1.55 => ( P3 @ A2 @ B ) ) ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_136_dual__order_Orefl,axiom,
% 2.75/1.55 ! [A: $tType] :
% 2.75/1.55 ( ( order @ A )
% 2.75/1.55 => ! [A2: A] : ( ord_less_eq @ A @ A2 @ A2 ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_137_order__trans,axiom,
% 2.75/1.55 ! [A: $tType] :
% 2.75/1.55 ( ( preorder @ A )
% 2.75/1.55 => ! [X2: A,Y: A,Z: A] :
% 2.75/1.55 ( ( ord_less_eq @ A @ X2 @ Y )
% 2.75/1.55 => ( ( ord_less_eq @ A @ Y @ Z )
% 2.75/1.55 => ( ord_less_eq @ A @ X2 @ Z ) ) ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_138_order__class_Oorder_Oantisym,axiom,
% 2.75/1.55 ! [A: $tType] :
% 2.75/1.55 ( ( order @ A )
% 2.75/1.55 => ! [A2: A,B: A] :
% 2.75/1.55 ( ( ord_less_eq @ A @ A2 @ B )
% 2.75/1.55 => ( ( ord_less_eq @ A @ B @ A2 )
% 2.75/1.55 => ( A2 = B ) ) ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_139_ord__le__eq__trans,axiom,
% 2.75/1.55 ! [A: $tType] :
% 2.75/1.55 ( ( ord @ A )
% 2.75/1.55 => ! [A2: A,B: A,C: A] :
% 2.75/1.55 ( ( ord_less_eq @ A @ A2 @ B )
% 2.75/1.55 => ( ( B = C )
% 2.75/1.55 => ( ord_less_eq @ A @ A2 @ C ) ) ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_140_ord__eq__le__trans,axiom,
% 2.75/1.55 ! [A: $tType] :
% 2.75/1.55 ( ( ord @ A )
% 2.75/1.55 => ! [A2: A,B: A,C: A] :
% 2.75/1.55 ( ( A2 = B )
% 2.75/1.55 => ( ( ord_less_eq @ A @ B @ C )
% 2.75/1.55 => ( ord_less_eq @ A @ A2 @ C ) ) ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_141_order__class_Oorder_Oeq__iff,axiom,
% 2.75/1.55 ! [A: $tType] :
% 2.75/1.55 ( ( order @ A )
% 2.75/1.55 => ( ( ^ [Y5: A,Z4: A] : ( Y5 = Z4 ) )
% 2.75/1.55 = ( ^ [A3: A,B2: A] :
% 2.75/1.55 ( ( ord_less_eq @ A @ A3 @ B2 )
% 2.75/1.55 & ( ord_less_eq @ A @ B2 @ A3 ) ) ) ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_142_antisym__conv,axiom,
% 2.75/1.55 ! [A: $tType] :
% 2.75/1.55 ( ( order @ A )
% 2.75/1.55 => ! [Y: A,X2: A] :
% 2.75/1.55 ( ( ord_less_eq @ A @ Y @ X2 )
% 2.75/1.55 => ( ( ord_less_eq @ A @ X2 @ Y )
% 2.75/1.55 = ( X2 = Y ) ) ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_143_le__cases3,axiom,
% 2.75/1.55 ! [A: $tType] :
% 2.75/1.55 ( ( linorder @ A )
% 2.75/1.55 => ! [X2: A,Y: A,Z: A] :
% 2.75/1.55 ( ( ( ord_less_eq @ A @ X2 @ Y )
% 2.75/1.55 => ~ ( ord_less_eq @ A @ Y @ Z ) )
% 2.75/1.55 => ( ( ( ord_less_eq @ A @ Y @ X2 )
% 2.75/1.55 => ~ ( ord_less_eq @ A @ X2 @ Z ) )
% 2.75/1.55 => ( ( ( ord_less_eq @ A @ X2 @ Z )
% 2.75/1.55 => ~ ( ord_less_eq @ A @ Z @ Y ) )
% 2.75/1.55 => ( ( ( ord_less_eq @ A @ Z @ Y )
% 2.75/1.55 => ~ ( ord_less_eq @ A @ Y @ X2 ) )
% 2.75/1.55 => ( ( ( ord_less_eq @ A @ Y @ Z )
% 2.75/1.55 => ~ ( ord_less_eq @ A @ Z @ X2 ) )
% 2.75/1.55 => ~ ( ( ord_less_eq @ A @ Z @ X2 )
% 2.75/1.55 => ~ ( ord_less_eq @ A @ X2 @ Y ) ) ) ) ) ) ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_144_order_Otrans,axiom,
% 2.75/1.55 ! [A: $tType] :
% 2.75/1.55 ( ( order @ A )
% 2.75/1.55 => ! [A2: A,B: A,C: A] :
% 2.75/1.55 ( ( ord_less_eq @ A @ A2 @ B )
% 2.75/1.55 => ( ( ord_less_eq @ A @ B @ C )
% 2.75/1.55 => ( ord_less_eq @ A @ A2 @ C ) ) ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_145_le__cases,axiom,
% 2.75/1.55 ! [A: $tType] :
% 2.75/1.55 ( ( linorder @ A )
% 2.75/1.55 => ! [X2: A,Y: A] :
% 2.75/1.55 ( ~ ( ord_less_eq @ A @ X2 @ Y )
% 2.75/1.55 => ( ord_less_eq @ A @ Y @ X2 ) ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_146_eq__refl,axiom,
% 2.75/1.55 ! [A: $tType] :
% 2.75/1.55 ( ( preorder @ A )
% 2.75/1.55 => ! [X2: A,Y: A] :
% 2.75/1.55 ( ( X2 = Y )
% 2.75/1.55 => ( ord_less_eq @ A @ X2 @ Y ) ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_147_linear,axiom,
% 2.75/1.55 ! [A: $tType] :
% 2.75/1.55 ( ( linorder @ A )
% 2.75/1.55 => ! [X2: A,Y: A] :
% 2.75/1.55 ( ( ord_less_eq @ A @ X2 @ Y )
% 2.75/1.55 | ( ord_less_eq @ A @ Y @ X2 ) ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_148_antisym,axiom,
% 2.75/1.55 ! [A: $tType] :
% 2.75/1.55 ( ( order @ A )
% 2.75/1.55 => ! [X2: A,Y: A] :
% 2.75/1.55 ( ( ord_less_eq @ A @ X2 @ Y )
% 2.75/1.55 => ( ( ord_less_eq @ A @ Y @ X2 )
% 2.75/1.55 => ( X2 = Y ) ) ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_149_eq__iff,axiom,
% 2.75/1.55 ! [A: $tType] :
% 2.75/1.55 ( ( order @ A )
% 2.75/1.55 => ( ( ^ [Y5: A,Z4: A] : ( Y5 = Z4 ) )
% 2.75/1.55 = ( ^ [X4: A,Y3: A] :
% 2.75/1.55 ( ( ord_less_eq @ A @ X4 @ Y3 )
% 2.75/1.55 & ( ord_less_eq @ A @ Y3 @ X4 ) ) ) ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_150_ord__le__eq__subst,axiom,
% 2.75/1.55 ! [A: $tType,B3: $tType] :
% 2.75/1.55 ( ( ( ord @ B3 )
% 2.75/1.55 & ( ord @ A ) )
% 2.75/1.55 => ! [A2: A,B: A,F: A > B3,C: B3] :
% 2.75/1.55 ( ( ord_less_eq @ A @ A2 @ B )
% 2.75/1.55 => ( ( ( F @ B )
% 2.75/1.55 = C )
% 2.75/1.55 => ( ! [X3: A,Y2: A] :
% 2.75/1.55 ( ( ord_less_eq @ A @ X3 @ Y2 )
% 2.75/1.55 => ( ord_less_eq @ B3 @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 2.75/1.55 => ( ord_less_eq @ B3 @ ( F @ A2 ) @ C ) ) ) ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_151_ord__eq__le__subst,axiom,
% 2.75/1.55 ! [A: $tType,B3: $tType] :
% 2.75/1.55 ( ( ( ord @ B3 )
% 2.75/1.55 & ( ord @ A ) )
% 2.75/1.55 => ! [A2: A,F: B3 > A,B: B3,C: B3] :
% 2.75/1.55 ( ( A2
% 2.75/1.55 = ( F @ B ) )
% 2.75/1.55 => ( ( ord_less_eq @ B3 @ B @ C )
% 2.75/1.55 => ( ! [X3: B3,Y2: B3] :
% 2.75/1.55 ( ( ord_less_eq @ B3 @ X3 @ Y2 )
% 2.75/1.55 => ( ord_less_eq @ A @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 2.75/1.55 => ( ord_less_eq @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_152_order__subst2,axiom,
% 2.75/1.55 ! [A: $tType,C2: $tType] :
% 2.75/1.55 ( ( ( order @ C2 )
% 2.75/1.55 & ( order @ A ) )
% 2.75/1.55 => ! [A2: A,B: A,F: A > C2,C: C2] :
% 2.75/1.55 ( ( ord_less_eq @ A @ A2 @ B )
% 2.75/1.55 => ( ( ord_less_eq @ C2 @ ( F @ B ) @ C )
% 2.75/1.55 => ( ! [X3: A,Y2: A] :
% 2.75/1.55 ( ( ord_less_eq @ A @ X3 @ Y2 )
% 2.75/1.55 => ( ord_less_eq @ C2 @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 2.75/1.55 => ( ord_less_eq @ C2 @ ( F @ A2 ) @ C ) ) ) ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_153_order__subst1,axiom,
% 2.75/1.55 ! [A: $tType,B3: $tType] :
% 2.75/1.55 ( ( ( order @ B3 )
% 2.75/1.55 & ( order @ A ) )
% 2.75/1.55 => ! [A2: A,F: B3 > A,B: B3,C: B3] :
% 2.75/1.55 ( ( ord_less_eq @ A @ A2 @ ( F @ B ) )
% 2.75/1.55 => ( ( ord_less_eq @ B3 @ B @ C )
% 2.75/1.55 => ( ! [X3: B3,Y2: B3] :
% 2.75/1.55 ( ( ord_less_eq @ B3 @ X3 @ Y2 )
% 2.75/1.55 => ( ord_less_eq @ A @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 2.75/1.55 => ( ord_less_eq @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_154_le__fun__def,axiom,
% 2.75/1.55 ! [B3: $tType,A: $tType] :
% 2.75/1.55 ( ( ord @ B3 )
% 2.75/1.55 => ( ( ord_less_eq @ ( A > B3 ) )
% 2.75/1.55 = ( ^ [F2: A > B3,G2: A > B3] :
% 2.75/1.55 ! [X4: A] : ( ord_less_eq @ B3 @ ( F2 @ X4 ) @ ( G2 @ X4 ) ) ) ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_155_le__funI,axiom,
% 2.75/1.55 ! [B3: $tType,A: $tType] :
% 2.75/1.55 ( ( ord @ B3 )
% 2.75/1.55 => ! [F: A > B3,G: A > B3] :
% 2.75/1.55 ( ! [X3: A] : ( ord_less_eq @ B3 @ ( F @ X3 ) @ ( G @ X3 ) )
% 2.75/1.55 => ( ord_less_eq @ ( A > B3 ) @ F @ G ) ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_156_le__funE,axiom,
% 2.75/1.55 ! [B3: $tType,A: $tType] :
% 2.75/1.55 ( ( ord @ B3 )
% 2.75/1.55 => ! [F: A > B3,G: A > B3,X2: A] :
% 2.75/1.55 ( ( ord_less_eq @ ( A > B3 ) @ F @ G )
% 2.75/1.55 => ( ord_less_eq @ B3 @ ( F @ X2 ) @ ( G @ X2 ) ) ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_157_le__funD,axiom,
% 2.75/1.55 ! [B3: $tType,A: $tType] :
% 2.75/1.55 ( ( ord @ B3 )
% 2.75/1.55 => ! [F: A > B3,G: A > B3,X2: A] :
% 2.75/1.55 ( ( ord_less_eq @ ( A > B3 ) @ F @ G )
% 2.75/1.55 => ( ord_less_eq @ B3 @ ( F @ X2 ) @ ( G @ X2 ) ) ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_158_ord__eq__less__subst,axiom,
% 2.75/1.55 ! [A: $tType,B3: $tType] :
% 2.75/1.55 ( ( ( ord @ B3 )
% 2.75/1.55 & ( ord @ A ) )
% 2.75/1.55 => ! [A2: A,F: B3 > A,B: B3,C: B3] :
% 2.75/1.55 ( ( A2
% 2.75/1.55 = ( F @ B ) )
% 2.75/1.55 => ( ( ord_less @ B3 @ B @ C )
% 2.75/1.55 => ( ! [X3: B3,Y2: B3] :
% 2.75/1.55 ( ( ord_less @ B3 @ X3 @ Y2 )
% 2.75/1.55 => ( ord_less @ A @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 2.75/1.55 => ( ord_less @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_159_ord__less__eq__subst,axiom,
% 2.75/1.55 ! [A: $tType,B3: $tType] :
% 2.75/1.55 ( ( ( ord @ B3 )
% 2.75/1.55 & ( ord @ A ) )
% 2.75/1.55 => ! [A2: A,B: A,F: A > B3,C: B3] :
% 2.75/1.55 ( ( ord_less @ A @ A2 @ B )
% 2.75/1.55 => ( ( ( F @ B )
% 2.75/1.55 = C )
% 2.75/1.55 => ( ! [X3: A,Y2: A] :
% 2.75/1.55 ( ( ord_less @ A @ X3 @ Y2 )
% 2.75/1.55 => ( ord_less @ B3 @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 2.75/1.55 => ( ord_less @ B3 @ ( F @ A2 ) @ C ) ) ) ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_160_order__less__subst1,axiom,
% 2.75/1.55 ! [A: $tType,B3: $tType] :
% 2.75/1.55 ( ( ( order @ B3 )
% 2.75/1.55 & ( order @ A ) )
% 2.75/1.55 => ! [A2: A,F: B3 > A,B: B3,C: B3] :
% 2.75/1.55 ( ( ord_less @ A @ A2 @ ( F @ B ) )
% 2.75/1.55 => ( ( ord_less @ B3 @ B @ C )
% 2.75/1.55 => ( ! [X3: B3,Y2: B3] :
% 2.75/1.55 ( ( ord_less @ B3 @ X3 @ Y2 )
% 2.75/1.55 => ( ord_less @ A @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 2.75/1.55 => ( ord_less @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_161_order__less__subst2,axiom,
% 2.75/1.55 ! [A: $tType,C2: $tType] :
% 2.75/1.55 ( ( ( order @ C2 )
% 2.75/1.55 & ( order @ A ) )
% 2.75/1.55 => ! [A2: A,B: A,F: A > C2,C: C2] :
% 2.75/1.55 ( ( ord_less @ A @ A2 @ B )
% 2.75/1.55 => ( ( ord_less @ C2 @ ( F @ B ) @ C )
% 2.75/1.55 => ( ! [X3: A,Y2: A] :
% 2.75/1.55 ( ( ord_less @ A @ X3 @ Y2 )
% 2.75/1.55 => ( ord_less @ C2 @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 2.75/1.55 => ( ord_less @ C2 @ ( F @ A2 ) @ C ) ) ) ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_162_lt__ex,axiom,
% 2.75/1.55 ! [A: $tType] :
% 2.75/1.55 ( ( no_bot @ A )
% 2.75/1.55 => ! [X2: A] :
% 2.75/1.55 ? [Y2: A] : ( ord_less @ A @ Y2 @ X2 ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_163_gt__ex,axiom,
% 2.75/1.55 ! [A: $tType] :
% 2.75/1.55 ( ( no_top @ A )
% 2.75/1.55 => ! [X2: A] :
% 2.75/1.55 ? [X_1: A] : ( ord_less @ A @ X2 @ X_1 ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_164_neqE,axiom,
% 2.75/1.55 ! [A: $tType] :
% 2.75/1.55 ( ( linorder @ A )
% 2.75/1.55 => ! [X2: A,Y: A] :
% 2.75/1.55 ( ( X2 != Y )
% 2.75/1.55 => ( ~ ( ord_less @ A @ X2 @ Y )
% 2.75/1.55 => ( ord_less @ A @ Y @ X2 ) ) ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_165_neq__iff,axiom,
% 2.75/1.55 ! [A: $tType] :
% 2.75/1.55 ( ( linorder @ A )
% 2.75/1.55 => ! [X2: A,Y: A] :
% 2.75/1.55 ( ( X2 != Y )
% 2.75/1.55 = ( ( ord_less @ A @ X2 @ Y )
% 2.75/1.55 | ( ord_less @ A @ Y @ X2 ) ) ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_166_order_Oasym,axiom,
% 2.75/1.55 ! [A: $tType] :
% 2.75/1.55 ( ( order @ A )
% 2.75/1.55 => ! [A2: A,B: A] :
% 2.75/1.55 ( ( ord_less @ A @ A2 @ B )
% 2.75/1.55 => ~ ( ord_less @ A @ B @ A2 ) ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_167_dense,axiom,
% 2.75/1.55 ! [A: $tType] :
% 2.75/1.55 ( ( dense_order @ A )
% 2.75/1.55 => ! [X2: A,Y: A] :
% 2.75/1.55 ( ( ord_less @ A @ X2 @ Y )
% 2.75/1.55 => ? [Z5: A] :
% 2.75/1.55 ( ( ord_less @ A @ X2 @ Z5 )
% 2.75/1.55 & ( ord_less @ A @ Z5 @ Y ) ) ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_168_less__imp__neq,axiom,
% 2.75/1.55 ! [A: $tType] :
% 2.75/1.55 ( ( order @ A )
% 2.75/1.55 => ! [X2: A,Y: A] :
% 2.75/1.55 ( ( ord_less @ A @ X2 @ Y )
% 2.75/1.55 => ( X2 != Y ) ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_169_less__asym,axiom,
% 2.75/1.55 ! [A: $tType] :
% 2.75/1.55 ( ( preorder @ A )
% 2.75/1.55 => ! [X2: A,Y: A] :
% 2.75/1.55 ( ( ord_less @ A @ X2 @ Y )
% 2.75/1.55 => ~ ( ord_less @ A @ Y @ X2 ) ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_170_less__asym_H,axiom,
% 2.75/1.55 ! [A: $tType] :
% 2.75/1.55 ( ( preorder @ A )
% 2.75/1.55 => ! [A2: A,B: A] :
% 2.75/1.55 ( ( ord_less @ A @ A2 @ B )
% 2.75/1.55 => ~ ( ord_less @ A @ B @ A2 ) ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_171_less__trans,axiom,
% 2.75/1.55 ! [A: $tType] :
% 2.75/1.55 ( ( preorder @ A )
% 2.75/1.55 => ! [X2: A,Y: A,Z: A] :
% 2.75/1.55 ( ( ord_less @ A @ X2 @ Y )
% 2.75/1.55 => ( ( ord_less @ A @ Y @ Z )
% 2.75/1.55 => ( ord_less @ A @ X2 @ Z ) ) ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_172_less__linear,axiom,
% 2.75/1.55 ! [A: $tType] :
% 2.75/1.55 ( ( linorder @ A )
% 2.75/1.55 => ! [X2: A,Y: A] :
% 2.75/1.55 ( ( ord_less @ A @ X2 @ Y )
% 2.75/1.55 | ( X2 = Y )
% 2.75/1.55 | ( ord_less @ A @ Y @ X2 ) ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_173_less__irrefl,axiom,
% 2.75/1.55 ! [A: $tType] :
% 2.75/1.55 ( ( preorder @ A )
% 2.75/1.55 => ! [X2: A] :
% 2.75/1.55 ~ ( ord_less @ A @ X2 @ X2 ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_174_ord__eq__less__trans,axiom,
% 2.75/1.55 ! [A: $tType] :
% 2.75/1.55 ( ( ord @ A )
% 2.75/1.55 => ! [A2: A,B: A,C: A] :
% 2.75/1.55 ( ( A2 = B )
% 2.75/1.55 => ( ( ord_less @ A @ B @ C )
% 2.75/1.55 => ( ord_less @ A @ A2 @ C ) ) ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_175_ord__less__eq__trans,axiom,
% 2.75/1.55 ! [A: $tType] :
% 2.75/1.55 ( ( ord @ A )
% 2.75/1.55 => ! [A2: A,B: A,C: A] :
% 2.75/1.55 ( ( ord_less @ A @ A2 @ B )
% 2.75/1.55 => ( ( B = C )
% 2.75/1.55 => ( ord_less @ A @ A2 @ C ) ) ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_176_dual__order_Oasym,axiom,
% 2.75/1.55 ! [A: $tType] :
% 2.75/1.55 ( ( order @ A )
% 2.75/1.55 => ! [B: A,A2: A] :
% 2.75/1.55 ( ( ord_less @ A @ B @ A2 )
% 2.75/1.55 => ~ ( ord_less @ A @ A2 @ B ) ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_177_less__imp__not__eq,axiom,
% 2.75/1.55 ! [A: $tType] :
% 2.75/1.55 ( ( order @ A )
% 2.75/1.55 => ! [X2: A,Y: A] :
% 2.75/1.55 ( ( ord_less @ A @ X2 @ Y )
% 2.75/1.55 => ( X2 != Y ) ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_178_less__not__sym,axiom,
% 2.75/1.55 ! [A: $tType] :
% 2.75/1.55 ( ( preorder @ A )
% 2.75/1.55 => ! [X2: A,Y: A] :
% 2.75/1.55 ( ( ord_less @ A @ X2 @ Y )
% 2.75/1.55 => ~ ( ord_less @ A @ Y @ X2 ) ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_179_less__induct,axiom,
% 2.75/1.55 ! [A: $tType] :
% 2.75/1.55 ( ( wellorder @ A )
% 2.75/1.55 => ! [P3: A > $o,A2: A] :
% 2.75/1.55 ( ! [X3: A] :
% 2.75/1.55 ( ! [Y4: A] :
% 2.75/1.55 ( ( ord_less @ A @ Y4 @ X3 )
% 2.75/1.55 => ( P3 @ Y4 ) )
% 2.75/1.55 => ( P3 @ X3 ) )
% 2.75/1.55 => ( P3 @ A2 ) ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_180_antisym__conv3,axiom,
% 2.75/1.55 ! [A: $tType] :
% 2.75/1.55 ( ( linorder @ A )
% 2.75/1.55 => ! [Y: A,X2: A] :
% 2.75/1.55 ( ~ ( ord_less @ A @ Y @ X2 )
% 2.75/1.55 => ( ( ~ ( ord_less @ A @ X2 @ Y ) )
% 2.75/1.55 = ( X2 = Y ) ) ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_181_less__imp__not__eq2,axiom,
% 2.75/1.55 ! [A: $tType] :
% 2.75/1.55 ( ( order @ A )
% 2.75/1.55 => ! [X2: A,Y: A] :
% 2.75/1.55 ( ( ord_less @ A @ X2 @ Y )
% 2.75/1.55 => ( Y != X2 ) ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_182_less__imp__triv,axiom,
% 2.75/1.55 ! [A: $tType] :
% 2.75/1.55 ( ( preorder @ A )
% 2.75/1.55 => ! [X2: A,Y: A,P3: $o] :
% 2.75/1.55 ( ( ord_less @ A @ X2 @ Y )
% 2.75/1.55 => ( ( ord_less @ A @ Y @ X2 )
% 2.75/1.55 => P3 ) ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_183_linorder__cases,axiom,
% 2.75/1.55 ! [A: $tType] :
% 2.75/1.55 ( ( linorder @ A )
% 2.75/1.55 => ! [X2: A,Y: A] :
% 2.75/1.55 ( ~ ( ord_less @ A @ X2 @ Y )
% 2.75/1.55 => ( ( X2 != Y )
% 2.75/1.55 => ( ord_less @ A @ Y @ X2 ) ) ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_184_dual__order_Oirrefl,axiom,
% 2.75/1.55 ! [A: $tType] :
% 2.75/1.55 ( ( order @ A )
% 2.75/1.55 => ! [A2: A] :
% 2.75/1.55 ~ ( ord_less @ A @ A2 @ A2 ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_185_order_Ostrict__trans,axiom,
% 2.75/1.55 ! [A: $tType] :
% 2.75/1.55 ( ( order @ A )
% 2.75/1.55 => ! [A2: A,B: A,C: A] :
% 2.75/1.55 ( ( ord_less @ A @ A2 @ B )
% 2.75/1.55 => ( ( ord_less @ A @ B @ C )
% 2.75/1.55 => ( ord_less @ A @ A2 @ C ) ) ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_186_less__imp__not__less,axiom,
% 2.75/1.55 ! [A: $tType] :
% 2.75/1.55 ( ( preorder @ A )
% 2.75/1.55 => ! [X2: A,Y: A] :
% 2.75/1.55 ( ( ord_less @ A @ X2 @ Y )
% 2.75/1.55 => ~ ( ord_less @ A @ Y @ X2 ) ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_187_exists__least__iff,axiom,
% 2.75/1.55 ! [A: $tType] :
% 2.75/1.55 ( ( wellorder @ A )
% 2.75/1.55 => ( ( ^ [P4: A > $o] :
% 2.75/1.55 ? [X5: A] : ( P4 @ X5 ) )
% 2.75/1.55 = ( ^ [P5: A > $o] :
% 2.75/1.55 ? [N2: A] :
% 2.75/1.55 ( ( P5 @ N2 )
% 2.75/1.55 & ! [M2: A] :
% 2.75/1.55 ( ( ord_less @ A @ M2 @ N2 )
% 2.75/1.55 => ~ ( P5 @ M2 ) ) ) ) ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_188_linorder__less__wlog,axiom,
% 2.75/1.55 ! [A: $tType] :
% 2.75/1.55 ( ( linorder @ A )
% 2.75/1.55 => ! [P3: A > A > $o,A2: A,B: A] :
% 2.75/1.55 ( ! [A5: A,B4: A] :
% 2.75/1.55 ( ( ord_less @ A @ A5 @ B4 )
% 2.75/1.55 => ( P3 @ A5 @ B4 ) )
% 2.75/1.55 => ( ! [A5: A] : ( P3 @ A5 @ A5 )
% 2.75/1.55 => ( ! [A5: A,B4: A] :
% 2.75/1.55 ( ( P3 @ B4 @ A5 )
% 2.75/1.55 => ( P3 @ A5 @ B4 ) )
% 2.75/1.55 => ( P3 @ A2 @ B ) ) ) ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_189_dual__order_Ostrict__trans,axiom,
% 2.75/1.55 ! [A: $tType] :
% 2.75/1.55 ( ( order @ A )
% 2.75/1.55 => ! [B: A,A2: A,C: A] :
% 2.75/1.55 ( ( ord_less @ A @ B @ A2 )
% 2.75/1.55 => ( ( ord_less @ A @ C @ B )
% 2.75/1.55 => ( ord_less @ A @ C @ A2 ) ) ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_190_not__less__iff__gr__or__eq,axiom,
% 2.75/1.55 ! [A: $tType] :
% 2.75/1.55 ( ( linorder @ A )
% 2.75/1.55 => ! [X2: A,Y: A] :
% 2.75/1.55 ( ( ~ ( ord_less @ A @ X2 @ Y ) )
% 2.75/1.55 = ( ( ord_less @ A @ Y @ X2 )
% 2.75/1.55 | ( X2 = Y ) ) ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_191_order_Ostrict__implies__not__eq,axiom,
% 2.75/1.55 ! [A: $tType] :
% 2.75/1.55 ( ( order @ A )
% 2.75/1.55 => ! [A2: A,B: A] :
% 2.75/1.55 ( ( ord_less @ A @ A2 @ B )
% 2.75/1.55 => ( A2 != B ) ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_192_dual__order_Ostrict__implies__not__eq,axiom,
% 2.75/1.55 ! [A: $tType] :
% 2.75/1.55 ( ( order @ A )
% 2.75/1.55 => ! [B: A,A2: A] :
% 2.75/1.55 ( ( ord_less @ A @ B @ A2 )
% 2.75/1.55 => ( A2 != B ) ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_193_ex__has__least__nat,axiom,
% 2.75/1.55 ! [A: $tType,P3: A > $o,K: A,M: A > nat] :
% 2.75/1.55 ( ( P3 @ K )
% 2.75/1.55 => ? [X3: A] :
% 2.75/1.55 ( ( P3 @ X3 )
% 2.75/1.55 & ! [Y4: A] :
% 2.75/1.55 ( ( P3 @ Y4 )
% 2.75/1.55 => ( ord_less_eq @ nat @ ( M @ X3 ) @ ( M @ Y4 ) ) ) ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_194_leD,axiom,
% 2.75/1.55 ! [A: $tType] :
% 2.75/1.55 ( ( order @ A )
% 2.75/1.55 => ! [Y: A,X2: A] :
% 2.75/1.55 ( ( ord_less_eq @ A @ Y @ X2 )
% 2.75/1.55 => ~ ( ord_less @ A @ X2 @ Y ) ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_195_leI,axiom,
% 2.75/1.55 ! [A: $tType] :
% 2.75/1.55 ( ( linorder @ A )
% 2.75/1.55 => ! [X2: A,Y: A] :
% 2.75/1.55 ( ~ ( ord_less @ A @ X2 @ Y )
% 2.75/1.55 => ( ord_less_eq @ A @ Y @ X2 ) ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_196_le__less,axiom,
% 2.75/1.55 ! [A: $tType] :
% 2.75/1.55 ( ( order @ A )
% 2.75/1.55 => ( ( ord_less_eq @ A )
% 2.75/1.55 = ( ^ [X4: A,Y3: A] :
% 2.75/1.55 ( ( ord_less @ A @ X4 @ Y3 )
% 2.75/1.55 | ( X4 = Y3 ) ) ) ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_197_less__le,axiom,
% 2.75/1.55 ! [A: $tType] :
% 2.75/1.55 ( ( order @ A )
% 2.75/1.55 => ( ( ord_less @ A )
% 2.75/1.55 = ( ^ [X4: A,Y3: A] :
% 2.75/1.55 ( ( ord_less_eq @ A @ X4 @ Y3 )
% 2.75/1.55 & ( X4 != Y3 ) ) ) ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_198_order__le__less__subst1,axiom,
% 2.75/1.55 ! [A: $tType,B3: $tType] :
% 2.75/1.55 ( ( ( order @ B3 )
% 2.75/1.55 & ( order @ A ) )
% 2.75/1.55 => ! [A2: A,F: B3 > A,B: B3,C: B3] :
% 2.75/1.55 ( ( ord_less_eq @ A @ A2 @ ( F @ B ) )
% 2.75/1.55 => ( ( ord_less @ B3 @ B @ C )
% 2.75/1.55 => ( ! [X3: B3,Y2: B3] :
% 2.75/1.55 ( ( ord_less @ B3 @ X3 @ Y2 )
% 2.75/1.55 => ( ord_less @ A @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 2.75/1.55 => ( ord_less @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_199_order__le__less__subst2,axiom,
% 2.75/1.55 ! [A: $tType,C2: $tType] :
% 2.75/1.55 ( ( ( order @ C2 )
% 2.75/1.55 & ( order @ A ) )
% 2.75/1.55 => ! [A2: A,B: A,F: A > C2,C: C2] :
% 2.75/1.55 ( ( ord_less_eq @ A @ A2 @ B )
% 2.75/1.55 => ( ( ord_less @ C2 @ ( F @ B ) @ C )
% 2.75/1.55 => ( ! [X3: A,Y2: A] :
% 2.75/1.55 ( ( ord_less_eq @ A @ X3 @ Y2 )
% 2.75/1.55 => ( ord_less_eq @ C2 @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 2.75/1.55 => ( ord_less @ C2 @ ( F @ A2 ) @ C ) ) ) ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_200_order__less__le__subst1,axiom,
% 2.75/1.55 ! [A: $tType,B3: $tType] :
% 2.75/1.55 ( ( ( order @ B3 )
% 2.75/1.55 & ( order @ A ) )
% 2.75/1.55 => ! [A2: A,F: B3 > A,B: B3,C: B3] :
% 2.75/1.55 ( ( ord_less @ A @ A2 @ ( F @ B ) )
% 2.75/1.55 => ( ( ord_less_eq @ B3 @ B @ C )
% 2.75/1.55 => ( ! [X3: B3,Y2: B3] :
% 2.75/1.55 ( ( ord_less_eq @ B3 @ X3 @ Y2 )
% 2.75/1.55 => ( ord_less_eq @ A @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 2.75/1.55 => ( ord_less @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_201_order__less__le__subst2,axiom,
% 2.75/1.55 ! [A: $tType,C2: $tType] :
% 2.75/1.55 ( ( ( order @ C2 )
% 2.75/1.55 & ( order @ A ) )
% 2.75/1.55 => ! [A2: A,B: A,F: A > C2,C: C2] :
% 2.75/1.55 ( ( ord_less @ A @ A2 @ B )
% 2.75/1.55 => ( ( ord_less_eq @ C2 @ ( F @ B ) @ C )
% 2.75/1.55 => ( ! [X3: A,Y2: A] :
% 2.75/1.55 ( ( ord_less @ A @ X3 @ Y2 )
% 2.75/1.55 => ( ord_less @ C2 @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 2.75/1.55 => ( ord_less @ C2 @ ( F @ A2 ) @ C ) ) ) ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_202_not__le,axiom,
% 2.75/1.55 ! [A: $tType] :
% 2.75/1.55 ( ( linorder @ A )
% 2.75/1.55 => ! [X2: A,Y: A] :
% 2.75/1.55 ( ( ~ ( ord_less_eq @ A @ X2 @ Y ) )
% 2.75/1.55 = ( ord_less @ A @ Y @ X2 ) ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_203_not__less,axiom,
% 2.75/1.55 ! [A: $tType] :
% 2.75/1.55 ( ( linorder @ A )
% 2.75/1.55 => ! [X2: A,Y: A] :
% 2.75/1.55 ( ( ~ ( ord_less @ A @ X2 @ Y ) )
% 2.75/1.55 = ( ord_less_eq @ A @ Y @ X2 ) ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_204_le__neq__trans,axiom,
% 2.75/1.55 ! [A: $tType] :
% 2.75/1.55 ( ( order @ A )
% 2.75/1.55 => ! [A2: A,B: A] :
% 2.75/1.55 ( ( ord_less_eq @ A @ A2 @ B )
% 2.75/1.55 => ( ( A2 != B )
% 2.75/1.55 => ( ord_less @ A @ A2 @ B ) ) ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_205_antisym__conv1,axiom,
% 2.75/1.55 ! [A: $tType] :
% 2.75/1.55 ( ( order @ A )
% 2.75/1.55 => ! [X2: A,Y: A] :
% 2.75/1.55 ( ~ ( ord_less @ A @ X2 @ Y )
% 2.75/1.55 => ( ( ord_less_eq @ A @ X2 @ Y )
% 2.75/1.55 = ( X2 = Y ) ) ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_206_antisym__conv2,axiom,
% 2.75/1.55 ! [A: $tType] :
% 2.75/1.55 ( ( order @ A )
% 2.75/1.55 => ! [X2: A,Y: A] :
% 2.75/1.55 ( ( ord_less_eq @ A @ X2 @ Y )
% 2.75/1.55 => ( ( ~ ( ord_less @ A @ X2 @ Y ) )
% 2.75/1.55 = ( X2 = Y ) ) ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_207_less__imp__le,axiom,
% 2.75/1.55 ! [A: $tType] :
% 2.75/1.55 ( ( preorder @ A )
% 2.75/1.55 => ! [X2: A,Y: A] :
% 2.75/1.55 ( ( ord_less @ A @ X2 @ Y )
% 2.75/1.55 => ( ord_less_eq @ A @ X2 @ Y ) ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_208_le__less__trans,axiom,
% 2.75/1.55 ! [A: $tType] :
% 2.75/1.55 ( ( preorder @ A )
% 2.75/1.55 => ! [X2: A,Y: A,Z: A] :
% 2.75/1.55 ( ( ord_less_eq @ A @ X2 @ Y )
% 2.75/1.55 => ( ( ord_less @ A @ Y @ Z )
% 2.75/1.55 => ( ord_less @ A @ X2 @ Z ) ) ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_209_less__le__trans,axiom,
% 2.75/1.55 ! [A: $tType] :
% 2.75/1.55 ( ( preorder @ A )
% 2.75/1.55 => ! [X2: A,Y: A,Z: A] :
% 2.75/1.55 ( ( ord_less @ A @ X2 @ Y )
% 2.75/1.55 => ( ( ord_less_eq @ A @ Y @ Z )
% 2.75/1.55 => ( ord_less @ A @ X2 @ Z ) ) ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_210_dense__ge,axiom,
% 2.75/1.55 ! [A: $tType] :
% 2.75/1.55 ( ( dense_linorder @ A )
% 2.75/1.55 => ! [Z: A,Y: A] :
% 2.75/1.55 ( ! [X3: A] :
% 2.75/1.55 ( ( ord_less @ A @ Z @ X3 )
% 2.75/1.55 => ( ord_less_eq @ A @ Y @ X3 ) )
% 2.75/1.55 => ( ord_less_eq @ A @ Y @ Z ) ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_211_dense__le,axiom,
% 2.75/1.55 ! [A: $tType] :
% 2.75/1.55 ( ( dense_linorder @ A )
% 2.75/1.55 => ! [Y: A,Z: A] :
% 2.75/1.55 ( ! [X3: A] :
% 2.75/1.55 ( ( ord_less @ A @ X3 @ Y )
% 2.75/1.55 => ( ord_less_eq @ A @ X3 @ Z ) )
% 2.75/1.55 => ( ord_less_eq @ A @ Y @ Z ) ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_212_le__less__linear,axiom,
% 2.75/1.55 ! [A: $tType] :
% 2.75/1.55 ( ( linorder @ A )
% 2.75/1.55 => ! [X2: A,Y: A] :
% 2.75/1.55 ( ( ord_less_eq @ A @ X2 @ Y )
% 2.75/1.55 | ( ord_less @ A @ Y @ X2 ) ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_213_le__imp__less__or__eq,axiom,
% 2.75/1.55 ! [A: $tType] :
% 2.75/1.55 ( ( order @ A )
% 2.75/1.55 => ! [X2: A,Y: A] :
% 2.75/1.55 ( ( ord_less_eq @ A @ X2 @ Y )
% 2.75/1.55 => ( ( ord_less @ A @ X2 @ Y )
% 2.75/1.55 | ( X2 = Y ) ) ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_214_less__le__not__le,axiom,
% 2.75/1.55 ! [A: $tType] :
% 2.75/1.55 ( ( preorder @ A )
% 2.75/1.55 => ( ( ord_less @ A )
% 2.75/1.55 = ( ^ [X4: A,Y3: A] :
% 2.75/1.55 ( ( ord_less_eq @ A @ X4 @ Y3 )
% 2.75/1.55 & ~ ( ord_less_eq @ A @ Y3 @ X4 ) ) ) ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_215_not__le__imp__less,axiom,
% 2.75/1.55 ! [A: $tType] :
% 2.75/1.55 ( ( linorder @ A )
% 2.75/1.55 => ! [Y: A,X2: A] :
% 2.75/1.55 ( ~ ( ord_less_eq @ A @ Y @ X2 )
% 2.75/1.55 => ( ord_less @ A @ X2 @ Y ) ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_216_order_Ostrict__trans1,axiom,
% 2.75/1.55 ! [A: $tType] :
% 2.75/1.55 ( ( order @ A )
% 2.75/1.55 => ! [A2: A,B: A,C: A] :
% 2.75/1.55 ( ( ord_less_eq @ A @ A2 @ B )
% 2.75/1.55 => ( ( ord_less @ A @ B @ C )
% 2.75/1.55 => ( ord_less @ A @ A2 @ C ) ) ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_217_order_Ostrict__trans2,axiom,
% 2.75/1.55 ! [A: $tType] :
% 2.75/1.55 ( ( order @ A )
% 2.75/1.55 => ! [A2: A,B: A,C: A] :
% 2.75/1.55 ( ( ord_less @ A @ A2 @ B )
% 2.75/1.55 => ( ( ord_less_eq @ A @ B @ C )
% 2.75/1.55 => ( ord_less @ A @ A2 @ C ) ) ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_218_order_Oorder__iff__strict,axiom,
% 2.75/1.55 ! [A: $tType] :
% 2.75/1.55 ( ( order @ A )
% 2.75/1.55 => ( ( ord_less_eq @ A )
% 2.75/1.55 = ( ^ [A3: A,B2: A] :
% 2.75/1.55 ( ( ord_less @ A @ A3 @ B2 )
% 2.75/1.55 | ( A3 = B2 ) ) ) ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_219_order_Ostrict__iff__order,axiom,
% 2.75/1.55 ! [A: $tType] :
% 2.75/1.55 ( ( order @ A )
% 2.75/1.55 => ( ( ord_less @ A )
% 2.75/1.55 = ( ^ [A3: A,B2: A] :
% 2.75/1.55 ( ( ord_less_eq @ A @ A3 @ B2 )
% 2.75/1.55 & ( A3 != B2 ) ) ) ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_220_dual__order_Ostrict__trans1,axiom,
% 2.75/1.55 ! [A: $tType] :
% 2.75/1.55 ( ( order @ A )
% 2.75/1.55 => ! [B: A,A2: A,C: A] :
% 2.75/1.55 ( ( ord_less_eq @ A @ B @ A2 )
% 2.75/1.55 => ( ( ord_less @ A @ C @ B )
% 2.75/1.55 => ( ord_less @ A @ C @ A2 ) ) ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_221_dual__order_Ostrict__trans2,axiom,
% 2.75/1.55 ! [A: $tType] :
% 2.75/1.55 ( ( order @ A )
% 2.75/1.55 => ! [B: A,A2: A,C: A] :
% 2.75/1.55 ( ( ord_less @ A @ B @ A2 )
% 2.75/1.55 => ( ( ord_less_eq @ A @ C @ B )
% 2.75/1.55 => ( ord_less @ A @ C @ A2 ) ) ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_222_dense__ge__bounded,axiom,
% 2.75/1.55 ! [A: $tType] :
% 2.75/1.55 ( ( dense_linorder @ A )
% 2.75/1.55 => ! [Z: A,X2: A,Y: A] :
% 2.75/1.55 ( ( ord_less @ A @ Z @ X2 )
% 2.75/1.55 => ( ! [W: A] :
% 2.75/1.55 ( ( ord_less @ A @ Z @ W )
% 2.75/1.55 => ( ( ord_less @ A @ W @ X2 )
% 2.75/1.55 => ( ord_less_eq @ A @ Y @ W ) ) )
% 2.75/1.55 => ( ord_less_eq @ A @ Y @ Z ) ) ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_223_dense__le__bounded,axiom,
% 2.75/1.55 ! [A: $tType] :
% 2.75/1.55 ( ( dense_linorder @ A )
% 2.75/1.55 => ! [X2: A,Y: A,Z: A] :
% 2.75/1.55 ( ( ord_less @ A @ X2 @ Y )
% 2.75/1.55 => ( ! [W: A] :
% 2.75/1.55 ( ( ord_less @ A @ X2 @ W )
% 2.75/1.55 => ( ( ord_less @ A @ W @ Y )
% 2.75/1.55 => ( ord_less_eq @ A @ W @ Z ) ) )
% 2.75/1.55 => ( ord_less_eq @ A @ Y @ Z ) ) ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_224_order_Ostrict__implies__order,axiom,
% 2.75/1.55 ! [A: $tType] :
% 2.75/1.55 ( ( order @ A )
% 2.75/1.55 => ! [A2: A,B: A] :
% 2.75/1.55 ( ( ord_less @ A @ A2 @ B )
% 2.75/1.55 => ( ord_less_eq @ A @ A2 @ B ) ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_225_dual__order_Oorder__iff__strict,axiom,
% 2.75/1.55 ! [A: $tType] :
% 2.75/1.55 ( ( order @ A )
% 2.75/1.55 => ( ( ord_less_eq @ A )
% 2.75/1.55 = ( ^ [B2: A,A3: A] :
% 2.75/1.55 ( ( ord_less @ A @ B2 @ A3 )
% 2.75/1.55 | ( A3 = B2 ) ) ) ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_226_dual__order_Ostrict__iff__order,axiom,
% 2.75/1.55 ! [A: $tType] :
% 2.75/1.55 ( ( order @ A )
% 2.75/1.55 => ( ( ord_less @ A )
% 2.75/1.55 = ( ^ [B2: A,A3: A] :
% 2.75/1.55 ( ( ord_less_eq @ A @ B2 @ A3 )
% 2.75/1.55 & ( A3 != B2 ) ) ) ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_227_dual__order_Ostrict__implies__order,axiom,
% 2.75/1.55 ! [A: $tType] :
% 2.75/1.55 ( ( order @ A )
% 2.75/1.55 => ! [B: A,A2: A] :
% 2.75/1.55 ( ( ord_less @ A @ B @ A2 )
% 2.75/1.55 => ( ord_less_eq @ A @ B @ A2 ) ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_228_order_Onot__eq__order__implies__strict,axiom,
% 2.75/1.55 ! [A: $tType] :
% 2.75/1.55 ( ( order @ A )
% 2.75/1.55 => ! [A2: A,B: A] :
% 2.75/1.55 ( ( A2 != B )
% 2.75/1.55 => ( ( ord_less_eq @ A @ A2 @ B )
% 2.75/1.55 => ( ord_less @ A @ A2 @ B ) ) ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_229_minf_I8_J,axiom,
% 2.75/1.55 ! [A: $tType] :
% 2.75/1.55 ( ( linorder @ A )
% 2.75/1.55 => ! [T: A] :
% 2.75/1.55 ? [Z5: A] :
% 2.75/1.55 ! [X: A] :
% 2.75/1.55 ( ( ord_less @ A @ X @ Z5 )
% 2.75/1.55 => ~ ( ord_less_eq @ A @ T @ X ) ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_230_minf_I6_J,axiom,
% 2.75/1.55 ! [A: $tType] :
% 2.75/1.55 ( ( linorder @ A )
% 2.75/1.55 => ! [T: A] :
% 2.75/1.55 ? [Z5: A] :
% 2.75/1.55 ! [X: A] :
% 2.75/1.55 ( ( ord_less @ A @ X @ Z5 )
% 2.75/1.55 => ( ord_less_eq @ A @ X @ T ) ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_231_pinf_I8_J,axiom,
% 2.75/1.55 ! [A: $tType] :
% 2.75/1.55 ( ( linorder @ A )
% 2.75/1.55 => ! [T: A] :
% 2.75/1.55 ? [Z5: A] :
% 2.75/1.55 ! [X: A] :
% 2.75/1.55 ( ( ord_less @ A @ Z5 @ X )
% 2.75/1.55 => ( ord_less_eq @ A @ T @ X ) ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_232_pinf_I6_J,axiom,
% 2.75/1.55 ! [A: $tType] :
% 2.75/1.55 ( ( linorder @ A )
% 2.75/1.55 => ! [T: A] :
% 2.75/1.55 ? [Z5: A] :
% 2.75/1.55 ! [X: A] :
% 2.75/1.55 ( ( ord_less @ A @ Z5 @ X )
% 2.75/1.55 => ~ ( ord_less_eq @ A @ X @ T ) ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_233_less__fun__def,axiom,
% 2.75/1.55 ! [B3: $tType,A: $tType] :
% 2.75/1.55 ( ( ord @ B3 )
% 2.75/1.55 => ( ( ord_less @ ( A > B3 ) )
% 2.75/1.55 = ( ^ [F2: A > B3,G2: A > B3] :
% 2.75/1.55 ( ( ord_less_eq @ ( A > B3 ) @ F2 @ G2 )
% 2.75/1.55 & ~ ( ord_less_eq @ ( A > B3 ) @ G2 @ F2 ) ) ) ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_234_pinf_I1_J,axiom,
% 2.75/1.55 ! [A: $tType] :
% 2.75/1.55 ( ( linorder @ A )
% 2.75/1.55 => ! [P3: A > $o,P6: A > $o,Q2: A > $o,Q3: A > $o] :
% 2.75/1.55 ( ? [Z2: A] :
% 2.75/1.55 ! [X3: A] :
% 2.75/1.55 ( ( ord_less @ A @ Z2 @ X3 )
% 2.75/1.55 => ( ( P3 @ X3 )
% 2.75/1.55 = ( P6 @ X3 ) ) )
% 2.75/1.55 => ( ? [Z2: A] :
% 2.75/1.55 ! [X3: A] :
% 2.75/1.55 ( ( ord_less @ A @ Z2 @ X3 )
% 2.75/1.55 => ( ( Q2 @ X3 )
% 2.75/1.55 = ( Q3 @ X3 ) ) )
% 2.75/1.55 => ? [Z5: A] :
% 2.75/1.55 ! [X: A] :
% 2.75/1.55 ( ( ord_less @ A @ Z5 @ X )
% 2.75/1.55 => ( ( ( P3 @ X )
% 2.75/1.55 & ( Q2 @ X ) )
% 2.75/1.55 = ( ( P6 @ X )
% 2.75/1.55 & ( Q3 @ X ) ) ) ) ) ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_235_pinf_I2_J,axiom,
% 2.75/1.55 ! [A: $tType] :
% 2.75/1.55 ( ( linorder @ A )
% 2.75/1.55 => ! [P3: A > $o,P6: A > $o,Q2: A > $o,Q3: A > $o] :
% 2.75/1.55 ( ? [Z2: A] :
% 2.75/1.55 ! [X3: A] :
% 2.75/1.55 ( ( ord_less @ A @ Z2 @ X3 )
% 2.75/1.55 => ( ( P3 @ X3 )
% 2.75/1.55 = ( P6 @ X3 ) ) )
% 2.75/1.55 => ( ? [Z2: A] :
% 2.75/1.55 ! [X3: A] :
% 2.75/1.55 ( ( ord_less @ A @ Z2 @ X3 )
% 2.75/1.55 => ( ( Q2 @ X3 )
% 2.75/1.55 = ( Q3 @ X3 ) ) )
% 2.75/1.55 => ? [Z5: A] :
% 2.75/1.55 ! [X: A] :
% 2.75/1.55 ( ( ord_less @ A @ Z5 @ X )
% 2.75/1.55 => ( ( ( P3 @ X )
% 2.75/1.55 | ( Q2 @ X ) )
% 2.75/1.55 = ( ( P6 @ X )
% 2.75/1.55 | ( Q3 @ X ) ) ) ) ) ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_236_pinf_I3_J,axiom,
% 2.75/1.55 ! [A: $tType] :
% 2.75/1.55 ( ( linorder @ A )
% 2.75/1.55 => ! [T: A] :
% 2.75/1.55 ? [Z5: A] :
% 2.75/1.55 ! [X: A] :
% 2.75/1.55 ( ( ord_less @ A @ Z5 @ X )
% 2.75/1.55 => ( X != T ) ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_237_pinf_I4_J,axiom,
% 2.75/1.55 ! [A: $tType] :
% 2.75/1.55 ( ( linorder @ A )
% 2.75/1.55 => ! [T: A] :
% 2.75/1.55 ? [Z5: A] :
% 2.75/1.55 ! [X: A] :
% 2.75/1.55 ( ( ord_less @ A @ Z5 @ X )
% 2.75/1.55 => ( X != T ) ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_238_pinf_I5_J,axiom,
% 2.75/1.55 ! [A: $tType] :
% 2.75/1.55 ( ( linorder @ A )
% 2.75/1.55 => ! [T: A] :
% 2.75/1.55 ? [Z5: A] :
% 2.75/1.55 ! [X: A] :
% 2.75/1.55 ( ( ord_less @ A @ Z5 @ X )
% 2.75/1.55 => ~ ( ord_less @ A @ X @ T ) ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_239_pinf_I7_J,axiom,
% 2.75/1.55 ! [A: $tType] :
% 2.75/1.55 ( ( linorder @ A )
% 2.75/1.55 => ! [T: A] :
% 2.75/1.55 ? [Z5: A] :
% 2.75/1.55 ! [X: A] :
% 2.75/1.55 ( ( ord_less @ A @ Z5 @ X )
% 2.75/1.55 => ( ord_less @ A @ T @ X ) ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_240_pinf_I11_J,axiom,
% 2.75/1.55 ! [C2: $tType,D3: $tType] :
% 2.75/1.55 ( ( ord @ C2 )
% 2.75/1.55 => ! [F3: D3] :
% 2.75/1.55 ? [Z5: C2] :
% 2.75/1.55 ! [X: C2] :
% 2.75/1.55 ( ( ord_less @ C2 @ Z5 @ X )
% 2.75/1.55 => ( F3 = F3 ) ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_241_minf_I1_J,axiom,
% 2.75/1.55 ! [A: $tType] :
% 2.75/1.55 ( ( linorder @ A )
% 2.75/1.55 => ! [P3: A > $o,P6: A > $o,Q2: A > $o,Q3: A > $o] :
% 2.75/1.55 ( ? [Z2: A] :
% 2.75/1.55 ! [X3: A] :
% 2.75/1.55 ( ( ord_less @ A @ X3 @ Z2 )
% 2.75/1.55 => ( ( P3 @ X3 )
% 2.75/1.55 = ( P6 @ X3 ) ) )
% 2.75/1.55 => ( ? [Z2: A] :
% 2.75/1.55 ! [X3: A] :
% 2.75/1.55 ( ( ord_less @ A @ X3 @ Z2 )
% 2.75/1.55 => ( ( Q2 @ X3 )
% 2.75/1.55 = ( Q3 @ X3 ) ) )
% 2.75/1.55 => ? [Z5: A] :
% 2.75/1.55 ! [X: A] :
% 2.75/1.55 ( ( ord_less @ A @ X @ Z5 )
% 2.75/1.55 => ( ( ( P3 @ X )
% 2.75/1.55 & ( Q2 @ X ) )
% 2.75/1.55 = ( ( P6 @ X )
% 2.75/1.55 & ( Q3 @ X ) ) ) ) ) ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_242_minf_I2_J,axiom,
% 2.75/1.55 ! [A: $tType] :
% 2.75/1.55 ( ( linorder @ A )
% 2.75/1.55 => ! [P3: A > $o,P6: A > $o,Q2: A > $o,Q3: A > $o] :
% 2.75/1.55 ( ? [Z2: A] :
% 2.75/1.55 ! [X3: A] :
% 2.75/1.55 ( ( ord_less @ A @ X3 @ Z2 )
% 2.75/1.55 => ( ( P3 @ X3 )
% 2.75/1.55 = ( P6 @ X3 ) ) )
% 2.75/1.55 => ( ? [Z2: A] :
% 2.75/1.55 ! [X3: A] :
% 2.75/1.55 ( ( ord_less @ A @ X3 @ Z2 )
% 2.75/1.55 => ( ( Q2 @ X3 )
% 2.75/1.55 = ( Q3 @ X3 ) ) )
% 2.75/1.55 => ? [Z5: A] :
% 2.75/1.55 ! [X: A] :
% 2.75/1.55 ( ( ord_less @ A @ X @ Z5 )
% 2.75/1.55 => ( ( ( P3 @ X )
% 2.75/1.55 | ( Q2 @ X ) )
% 2.75/1.55 = ( ( P6 @ X )
% 2.75/1.55 | ( Q3 @ X ) ) ) ) ) ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_243_minf_I3_J,axiom,
% 2.75/1.55 ! [A: $tType] :
% 2.75/1.55 ( ( linorder @ A )
% 2.75/1.55 => ! [T: A] :
% 2.75/1.55 ? [Z5: A] :
% 2.75/1.55 ! [X: A] :
% 2.75/1.55 ( ( ord_less @ A @ X @ Z5 )
% 2.75/1.55 => ( X != T ) ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_244_minf_I4_J,axiom,
% 2.75/1.55 ! [A: $tType] :
% 2.75/1.55 ( ( linorder @ A )
% 2.75/1.55 => ! [T: A] :
% 2.75/1.55 ? [Z5: A] :
% 2.75/1.55 ! [X: A] :
% 2.75/1.55 ( ( ord_less @ A @ X @ Z5 )
% 2.75/1.55 => ( X != T ) ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_245_minf_I5_J,axiom,
% 2.75/1.55 ! [A: $tType] :
% 2.75/1.55 ( ( linorder @ A )
% 2.75/1.55 => ! [T: A] :
% 2.75/1.55 ? [Z5: A] :
% 2.75/1.55 ! [X: A] :
% 2.75/1.55 ( ( ord_less @ A @ X @ Z5 )
% 2.75/1.55 => ( ord_less @ A @ X @ T ) ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_246_minf_I7_J,axiom,
% 2.75/1.55 ! [A: $tType] :
% 2.75/1.55 ( ( linorder @ A )
% 2.75/1.55 => ! [T: A] :
% 2.75/1.55 ? [Z5: A] :
% 2.75/1.55 ! [X: A] :
% 2.75/1.55 ( ( ord_less @ A @ X @ Z5 )
% 2.75/1.55 => ~ ( ord_less @ A @ T @ X ) ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_247_minf_I11_J,axiom,
% 2.75/1.55 ! [C2: $tType,D3: $tType] :
% 2.75/1.55 ( ( ord @ C2 )
% 2.75/1.55 => ! [F3: D3] :
% 2.75/1.55 ? [Z5: C2] :
% 2.75/1.55 ! [X: C2] :
% 2.75/1.55 ( ( ord_less @ C2 @ X @ Z5 )
% 2.75/1.55 => ( F3 = F3 ) ) ) ).
% 2.75/1.55
% 2.75/1.55 thf(fact_248_complete__interval,axiom,
% 2.75/1.55 ! [A: $tType] :
% 2.75/1.55 ( ( condit1037483654norder @ A )
% 2.75/1.55 => ! [A2: A,B: A,P3: A > $o] :
% 2.75/1.55 ( ( ord_less @ A @ A2 @ B )
% 2.75/1.55 => ( ( P3 @ A2 )
% 2.75/1.55 => ( ~ ( P3 @ B )
% 2.75/1.55 => ? [C3: A] :
% 2.75/1.55 ( ( ord_less_eq @ A @ A2 @ C3 )
% 2.75/1.55 & ( ord_less_eq @ A @ C3 @ B )
% 2.75/1.55 & ! [X: A] :
% 2.75/1.55 ( ( ( ord_less_eq @ A @ A2 @ X )
% 2.75/1.56 & ( ord_less @ A @ X @ C3 ) )
% 2.75/1.56 => ( P3 @ X ) )
% 2.75/1.56 & ! [D4: A] :
% 2.75/1.56 ( ! [X3: A] :
% 2.75/1.56 ( ( ( ord_less_eq @ A @ A2 @ X3 )
% 2.75/1.56 & ( ord_less @ A @ X3 @ D4 ) )
% 2.75/1.56 => ( P3 @ X3 ) )
% 2.75/1.56 => ( ord_less_eq @ A @ D4 @ C3 ) ) ) ) ) ) ) ).
% 2.75/1.56
% 2.75/1.56 thf(fact_249_verit__comp__simplify1_I3_J,axiom,
% 2.75/1.56 ! [B3: $tType] :
% 2.75/1.56 ( ( linorder @ B3 )
% 2.75/1.56 => ! [B5: B3,A6: B3] :
% 2.75/1.56 ( ( ~ ( ord_less_eq @ B3 @ B5 @ A6 ) )
% 2.75/1.56 = ( ord_less @ B3 @ A6 @ B5 ) ) ) ).
% 2.75/1.56
% 2.75/1.56 thf(fact_250_verit__la__disequality,axiom,
% 2.75/1.56 ! [A: $tType] :
% 2.75/1.56 ( ( linorder @ A )
% 2.75/1.56 => ! [A2: A,B: A] :
% 2.75/1.56 ( ( A2 = B )
% 2.75/1.56 | ~ ( ord_less_eq @ A @ A2 @ B )
% 2.75/1.56 | ~ ( ord_less_eq @ A @ B @ A2 ) ) ) ).
% 2.75/1.56
% 2.75/1.56 thf(fact_251_verit__comp__simplify1_I1_J,axiom,
% 2.75/1.56 ! [A: $tType] :
% 2.75/1.56 ( ( order @ A )
% 2.75/1.56 => ! [A2: A] :
% 2.75/1.56 ~ ( ord_less @ A @ A2 @ A2 ) ) ).
% 2.75/1.56
% 2.75/1.56 thf(fact_252_ex__gt__or__lt,axiom,
% 2.75/1.56 ! [A: $tType] :
% 2.75/1.56 ( ( condit1656338222tinuum @ A )
% 2.75/1.56 => ! [A2: A] :
% 2.75/1.56 ? [B4: A] :
% 2.75/1.56 ( ( ord_less @ A @ A2 @ B4 )
% 2.75/1.56 | ( ord_less @ A @ B4 @ A2 ) ) ) ).
% 2.75/1.56
% 2.75/1.56 thf(fact_253_poly__cutoff__1,axiom,
% 2.75/1.56 ! [A: $tType] :
% 2.75/1.56 ( ( comm_semiring_1 @ A )
% 2.75/1.56 => ! [N: nat] :
% 2.75/1.56 ( ( ( N
% 2.75/1.56 = ( zero_zero @ nat ) )
% 2.75/1.56 => ( ( poly_cutoff @ A @ N @ ( one_one @ ( poly @ A ) ) )
% 2.75/1.56 = ( zero_zero @ ( poly @ A ) ) ) )
% 2.75/1.56 & ( ( N
% 2.75/1.56 != ( zero_zero @ nat ) )
% 2.75/1.56 => ( ( poly_cutoff @ A @ N @ ( one_one @ ( poly @ A ) ) )
% 2.75/1.56 = ( one_one @ ( poly @ A ) ) ) ) ) ) ).
% 2.75/1.56
% 2.75/1.56 thf(tcon_fun___Orderings_Opreorder,axiom,
% 2.75/1.56 ! [A7: $tType,A8: $tType] :
% 2.75/1.56 ( ( preorder @ A8 )
% 2.75/1.56 => ( preorder @ ( A7 > A8 ) ) ) ).
% 2.75/1.56
% 2.75/1.56 thf(tcon_fun___Orderings_Oorder,axiom,
% 2.75/1.56 ! [A7: $tType,A8: $tType] :
% 2.75/1.56 ( ( order @ A8 )
% 2.75/1.56 => ( order @ ( A7 > A8 ) ) ) ).
% 2.75/1.56
% 2.75/1.56 thf(tcon_fun___Orderings_Oord,axiom,
% 2.75/1.56 ! [A7: $tType,A8: $tType] :
% 2.75/1.56 ( ( ord @ A8 )
% 2.75/1.56 => ( ord @ ( A7 > A8 ) ) ) ).
% 2.75/1.56
% 2.75/1.56 thf(tcon_Nat_Onat___Conditionally__Complete__Lattices_Oconditionally__complete__linorder,axiom,
% 2.75/1.56 condit1037483654norder @ nat ).
% 2.75/1.56
% 2.75/1.56 thf(tcon_Nat_Onat___Groups_Ocanonically__ordered__monoid__add,axiom,
% 2.75/1.56 canoni770627133id_add @ nat ).
% 2.75/1.56
% 2.75/1.56 thf(tcon_Nat_Onat___Rings_Olinordered__nonzero__semiring,axiom,
% 2.75/1.56 linord1659791738miring @ nat ).
% 2.75/1.56
% 2.75/1.56 thf(tcon_Nat_Onat___Rings_Ocomm__semiring__1,axiom,
% 2.75/1.56 comm_semiring_1 @ nat ).
% 2.75/1.56
% 2.75/1.56 thf(tcon_Nat_Onat___Rings_Ocomm__semiring__0,axiom,
% 2.75/1.56 comm_semiring_0 @ nat ).
% 2.75/1.56
% 2.75/1.56 thf(tcon_Nat_Onat___Orderings_Owellorder,axiom,
% 2.75/1.56 wellorder @ nat ).
% 2.75/1.56
% 2.75/1.56 thf(tcon_Nat_Onat___Orderings_Opreorder_1,axiom,
% 2.75/1.56 preorder @ nat ).
% 2.75/1.56
% 2.75/1.56 thf(tcon_Nat_Onat___Orderings_Olinorder,axiom,
% 2.75/1.56 linorder @ nat ).
% 2.75/1.56
% 2.75/1.56 thf(tcon_Nat_Onat___Orderings_Ono__top,axiom,
% 2.75/1.56 no_top @ nat ).
% 2.75/1.56
% 2.75/1.56 thf(tcon_Nat_Onat___Orderings_Oorder_2,axiom,
% 2.75/1.56 order @ nat ).
% 2.75/1.56
% 2.75/1.56 thf(tcon_Nat_Onat___Orderings_Oord_3,axiom,
% 2.75/1.56 ord @ nat ).
% 2.75/1.56
% 2.75/1.56 thf(tcon_Nat_Onat___Groups_Ozero,axiom,
% 2.75/1.56 zero @ nat ).
% 2.75/1.56
% 2.75/1.56 thf(tcon_Set_Oset___Orderings_Opreorder_4,axiom,
% 2.75/1.56 ! [A7: $tType] : ( preorder @ ( set @ A7 ) ) ).
% 2.75/1.56
% 2.75/1.56 thf(tcon_Set_Oset___Orderings_Oorder_5,axiom,
% 2.75/1.56 ! [A7: $tType] : ( order @ ( set @ A7 ) ) ).
% 2.75/1.56
% 2.75/1.56 thf(tcon_Set_Oset___Orderings_Oord_6,axiom,
% 2.75/1.56 ! [A7: $tType] : ( ord @ ( set @ A7 ) ) ).
% 2.75/1.56
% 2.75/1.56 thf(tcon_HOL_Obool___Orderings_Opreorder_7,axiom,
% 2.75/1.56 preorder @ $o ).
% 2.75/1.56
% 2.75/1.56 thf(tcon_HOL_Obool___Orderings_Olinorder_8,axiom,
% 2.75/1.56 linorder @ $o ).
% 2.75/1.56
% 2.75/1.56 thf(tcon_HOL_Obool___Orderings_Oorder_9,axiom,
% 2.75/1.56 order @ $o ).
% 2.75/1.56
% 2.75/1.56 thf(tcon_HOL_Obool___Orderings_Oord_10,axiom,
% 2.75/1.56 ord @ $o ).
% 2.75/1.56
% 2.75/1.56 thf(tcon_Real_Oreal___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_11,axiom,
% 2.75/1.56 condit1037483654norder @ real ).
% 2.75/1.56
% 2.75/1.56 thf(tcon_Real_Oreal___Conditionally__Complete__Lattices_Olinear__continuum,axiom,
% 2.75/1.56 condit1656338222tinuum @ real ).
% 2.75/1.56
% 2.75/1.56 thf(tcon_Real_Oreal___Real__Vector__Spaces_Oreal__normed__vector,axiom,
% 2.75/1.56 real_V55928688vector @ real ).
% 2.75/1.56
% 2.75/1.56 thf(tcon_Real_Oreal___Rings_Olinordered__nonzero__semiring_12,axiom,
% 2.75/1.56 linord1659791738miring @ real ).
% 2.75/1.56
% 2.75/1.56 thf(tcon_Real_Oreal___Rings_Oring__no__zero__divisors,axiom,
% 2.75/1.56 ring_n68954251visors @ real ).
% 2.75/1.56
% 2.75/1.56 thf(tcon_Real_Oreal___Orderings_Odense__linorder,axiom,
% 2.75/1.56 dense_linorder @ real ).
% 2.75/1.56
% 2.75/1.56 thf(tcon_Real_Oreal___Fields_Olinordered__field,axiom,
% 2.75/1.56 linordered_field @ real ).
% 2.75/1.56
% 2.75/1.56 thf(tcon_Real_Oreal___Rings_Olinordered__idom,axiom,
% 2.75/1.56 linordered_idom @ real ).
% 2.75/1.56
% 2.75/1.56 thf(tcon_Real_Oreal___Rings_Ocomm__semiring__1_13,axiom,
% 2.75/1.56 comm_semiring_1 @ real ).
% 2.75/1.56
% 2.75/1.56 thf(tcon_Real_Oreal___Rings_Ocomm__semiring__0_14,axiom,
% 2.75/1.56 comm_semiring_0 @ real ).
% 2.75/1.56
% 2.75/1.56 thf(tcon_Real_Oreal___Orderings_Odense__order,axiom,
% 2.75/1.56 dense_order @ real ).
% 2.75/1.56
% 2.75/1.56 thf(tcon_Real_Oreal___Rings_Oidom__abs__sgn,axiom,
% 2.75/1.56 idom_abs_sgn @ real ).
% 2.75/1.56
% 2.75/1.56 thf(tcon_Real_Oreal___Orderings_Opreorder_15,axiom,
% 2.75/1.56 preorder @ real ).
% 2.75/1.56
% 2.75/1.56 thf(tcon_Real_Oreal___Orderings_Olinorder_16,axiom,
% 2.75/1.56 linorder @ real ).
% 2.75/1.56
% 2.75/1.56 thf(tcon_Real_Oreal___Rings_Oidom__divide,axiom,
% 2.75/1.56 idom_divide @ real ).
% 2.75/1.56
% 2.75/1.56 thf(tcon_Real_Oreal___Rings_Ocomm__ring__1,axiom,
% 2.75/1.56 comm_ring_1 @ real ).
% 2.75/1.56
% 2.75/1.56 thf(tcon_Real_Oreal___Orderings_Ono__top_17,axiom,
% 2.75/1.56 no_top @ real ).
% 2.75/1.56
% 2.75/1.56 thf(tcon_Real_Oreal___Orderings_Ono__bot,axiom,
% 2.75/1.56 no_bot @ real ).
% 2.75/1.56
% 2.75/1.56 thf(tcon_Real_Oreal___Orderings_Oorder_18,axiom,
% 2.75/1.56 order @ real ).
% 2.75/1.56
% 2.75/1.56 thf(tcon_Real_Oreal___Nat_Oring__char__0,axiom,
% 2.75/1.56 ring_char_0 @ real ).
% 2.75/1.56
% 2.75/1.56 thf(tcon_Real_Oreal___Orderings_Oord_19,axiom,
% 2.75/1.56 ord @ real ).
% 2.75/1.56
% 2.75/1.56 thf(tcon_Real_Oreal___Groups_Ozero_20,axiom,
% 2.75/1.56 zero @ real ).
% 2.75/1.56
% 2.75/1.56 thf(tcon_Real_Oreal___Rings_Oidom,axiom,
% 2.75/1.56 idom @ real ).
% 2.75/1.56
% 2.75/1.56 thf(tcon_Polynomial_Opoly___Rings_Olinordered__nonzero__semiring_21,axiom,
% 2.75/1.56 ! [A7: $tType] :
% 2.75/1.56 ( ( linordered_idom @ A7 )
% 2.75/1.56 => ( linord1659791738miring @ ( poly @ A7 ) ) ) ).
% 2.75/1.56
% 2.75/1.56 thf(tcon_Polynomial_Opoly___Rings_Oring__no__zero__divisors_22,axiom,
% 2.75/1.56 ! [A7: $tType] :
% 2.75/1.56 ( ( idom @ A7 )
% 2.75/1.56 => ( ring_n68954251visors @ ( poly @ A7 ) ) ) ).
% 2.75/1.56
% 2.75/1.56 thf(tcon_Polynomial_Opoly___Rings_Olinordered__idom_23,axiom,
% 2.75/1.56 ! [A7: $tType] :
% 2.75/1.56 ( ( linordered_idom @ A7 )
% 2.75/1.56 => ( linordered_idom @ ( poly @ A7 ) ) ) ).
% 2.75/1.56
% 2.75/1.56 thf(tcon_Polynomial_Opoly___Rings_Ocomm__semiring__1_24,axiom,
% 2.75/1.56 ! [A7: $tType] :
% 2.75/1.56 ( ( comm_semiring_1 @ A7 )
% 2.75/1.56 => ( comm_semiring_1 @ ( poly @ A7 ) ) ) ).
% 2.75/1.56
% 2.75/1.56 thf(tcon_Polynomial_Opoly___Rings_Ocomm__semiring__0_25,axiom,
% 2.75/1.56 ! [A7: $tType] :
% 2.75/1.56 ( ( comm_semiring_0 @ A7 )
% 2.75/1.56 => ( comm_semiring_0 @ ( poly @ A7 ) ) ) ).
% 2.75/1.56
% 2.75/1.56 thf(tcon_Polynomial_Opoly___Rings_Oidom__abs__sgn_26,axiom,
% 2.75/1.56 ! [A7: $tType] :
% 2.75/1.56 ( ( linordered_idom @ A7 )
% 2.75/1.56 => ( idom_abs_sgn @ ( poly @ A7 ) ) ) ).
% 2.75/1.56
% 2.75/1.56 thf(tcon_Polynomial_Opoly___Orderings_Opreorder_27,axiom,
% 2.75/1.56 ! [A7: $tType] :
% 2.75/1.56 ( ( linordered_idom @ A7 )
% 2.75/1.56 => ( preorder @ ( poly @ A7 ) ) ) ).
% 2.75/1.56
% 2.75/1.56 thf(tcon_Polynomial_Opoly___Orderings_Olinorder_28,axiom,
% 2.75/1.56 ! [A7: $tType] :
% 2.75/1.56 ( ( linordered_idom @ A7 )
% 2.75/1.56 => ( linorder @ ( poly @ A7 ) ) ) ).
% 2.75/1.56
% 2.75/1.56 thf(tcon_Polynomial_Opoly___Rings_Oidom__divide_29,axiom,
% 2.75/1.56 ! [A7: $tType] :
% 2.75/1.56 ( ( idom_divide @ A7 )
% 2.75/1.56 => ( idom_divide @ ( poly @ A7 ) ) ) ).
% 2.75/1.56
% 2.75/1.56 thf(tcon_Polynomial_Opoly___Rings_Ocomm__ring__1_30,axiom,
% 2.75/1.56 ! [A7: $tType] :
% 2.75/1.56 ( ( comm_ring_1 @ A7 )
% 2.75/1.56 => ( comm_ring_1 @ ( poly @ A7 ) ) ) ).
% 2.75/1.56
% 2.75/1.56 thf(tcon_Polynomial_Opoly___Orderings_Oorder_31,axiom,
% 2.75/1.56 ! [A7: $tType] :
% 2.75/1.56 ( ( linordered_idom @ A7 )
% 2.75/1.56 => ( order @ ( poly @ A7 ) ) ) ).
% 2.75/1.56
% 2.75/1.56 thf(tcon_Polynomial_Opoly___Nat_Oring__char__0_32,axiom,
% 2.75/1.56 ! [A7: $tType] :
% 2.75/1.56 ( ( ( ring_char_0 @ A7 )
% 2.75/1.56 & ( comm_ring_1 @ A7 ) )
% 2.75/1.56 => ( ring_char_0 @ ( poly @ A7 ) ) ) ).
% 2.75/1.56
% 2.75/1.56 thf(tcon_Polynomial_Opoly___Orderings_Oord_33,axiom,
% 2.75/1.56 ! [A7: $tType] :
% 2.75/1.56 ( ( linordered_idom @ A7 )
% 2.75/1.56 => ( ord @ ( poly @ A7 ) ) ) ).
% 2.75/1.56
% 2.75/1.56 thf(tcon_Polynomial_Opoly___Groups_Ozero_34,axiom,
% 2.75/1.56 ! [A7: $tType] :
% 2.75/1.56 ( ( zero @ A7 )
% 2.75/1.56 => ( zero @ ( poly @ A7 ) ) ) ).
% 2.75/1.56
% 2.75/1.56 thf(tcon_Polynomial_Opoly___Rings_Oidom_35,axiom,
% 2.75/1.56 ! [A7: $tType] :
% 2.75/1.56 ( ( idom @ A7 )
% 2.75/1.56 => ( idom @ ( poly @ A7 ) ) ) ).
% 2.75/1.56
% 2.75/1.56 thf(help_If_3_1_T,axiom,
% 2.75/1.56 ! [P3: $o] :
% 2.75/1.56 ( ( P3 = $true )
% 2.75/1.56 | ( P3 = $false ) ) ).
% 2.75/1.56
% 2.75/1.56 thf(help_If_2_1_T,axiom,
% 2.75/1.56 ! [A: $tType,X2: A,Y: A] :
% 2.75/1.56 ( ( if @ A @ $false @ X2 @ Y )
% 2.75/1.56 = Y ) ).
% 2.75/1.56
% 2.75/1.56 thf(help_If_1_1_T,axiom,
% 2.75/1.56 ! [A: $tType,X2: A,Y: A] :
% 2.75/1.56 ( ( if @ A @ $true @ X2 @ Y )
% 2.75/1.56 = X2 ) ).
% 2.75/1.56
% 2.75/1.56 thf(conj_0,conjecture,
% 2.75/1.56 ! [X3: real] :
% 2.75/1.56 ( ( ( poly2 @ real @ p @ X3 )
% 2.75/1.56 != ( zero_zero @ real ) )
% 2.75/1.56 | ( ord_less @ real @ lb @ X3 ) ) ).
% 2.75/1.56 ------------------------
% 2.75/1.58 % (21300)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_21200 for (2999ds/4Mi)
% 2.75/1.58 % (21301)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_21200 for (2999ds/27Mi)
% 2.75/1.58 % (21299)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_21200 for (2999ds/183Mi)
% 2.75/1.58 % (21300)Instruction limit reached!
% 2.75/1.58 % (21300)------------------------------
% 2.75/1.58 % (21300)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.75/1.58 % (21300)Termination reason: Unknown
% 2.75/1.58 % (21300)Termination phase: shuffling
% 2.75/1.58
% 2.75/1.58 % (21300)Memory used [KB]: 1407
% 2.75/1.58 % (21300)Time elapsed: 0.003 s
% 2.75/1.58 % (21300)Instructions burned: 5 (million)
% 2.75/1.58 % (21300)------------------------------
% 2.75/1.58 % (21300)------------------------------
% 2.75/1.59 % (21302)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_21200 for (2999ds/2Mi)
% 2.75/1.59 % (21302)Instruction limit reached!
% 2.75/1.59 % (21302)------------------------------
% 2.75/1.59 % (21302)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.75/1.59 % (21302)Termination reason: Unknown
% 2.75/1.59 % (21302)Termination phase: shuffling
% 2.75/1.59
% 2.75/1.59 % (21302)Memory used [KB]: 1407
% 2.75/1.59 % (21302)Time elapsed: 0.004 s
% 2.75/1.59 % (21302)Instructions burned: 3 (million)
% 2.75/1.59 % (21302)------------------------------
% 2.75/1.59 % (21302)------------------------------
% 2.75/1.59 % (21303)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_21200 for (2999ds/2Mi)
% 2.75/1.59 % (21304)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_21200 for (2999ds/275Mi)
% 2.75/1.59 % (21305)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_21200 for (2999ds/18Mi)
% 2.75/1.59 % (21303)Instruction limit reached!
% 2.75/1.59 % (21303)------------------------------
% 2.75/1.59 % (21303)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.75/1.59 % (21303)Termination reason: Unknown
% 2.75/1.59 % (21303)Termination phase: shuffling
% 2.75/1.59
% 2.75/1.59 % (21303)Memory used [KB]: 1279
% 2.75/1.59 % (21303)Time elapsed: 0.003 s
% 2.75/1.59 % (21303)Instructions burned: 2 (million)
% 2.75/1.59 % (21303)------------------------------
% 2.75/1.59 % (21303)------------------------------
% 2.75/1.59 % (21301)Instruction limit reached!
% 2.75/1.59 % (21301)------------------------------
% 2.75/1.59 % (21301)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.75/1.59 % (21301)Termination reason: Unknown
% 2.75/1.59 % (21301)Termination phase: Property scanning
% 2.75/1.59
% 2.75/1.59 % (21301)Memory used [KB]: 1791
% 2.75/1.59 % (21301)Time elapsed: 0.010 s
% 2.75/1.59 % (21301)Instructions burned: 27 (million)
% 2.75/1.59 % (21301)------------------------------
% 2.75/1.59 % (21301)------------------------------
% 2.75/1.59 % (21306)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_21200 for (2999ds/3Mi)
% 2.75/1.59 % (21306)Instruction limit reached!
% 2.75/1.59 % (21306)------------------------------
% 2.75/1.59 % (21306)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.75/1.59 % (21306)Termination reason: Unknown
% 2.75/1.59 % (21306)Termination phase: shuffling
% 2.75/1.59
% 2.75/1.59 % (21306)Memory used [KB]: 1407
% 2.75/1.59 % (21306)Time elapsed: 0.002 s
% 2.75/1.59 % (21306)Instructions burned: 3 (million)
% 2.75/1.59 % (21306)------------------------------
% 2.75/1.59 % (21306)------------------------------
% 2.82/1.60 % (21309)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_21200 for (2999ds/3Mi)
% 2.82/1.60 % (21305)Instruction limit reached!
% 2.82/1.60 % (21305)------------------------------
% 2.82/1.60 % (21305)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.82/1.60 % (21305)Termination reason: Unknown
% 2.82/1.60 % (21305)Termination phase: Property scanning
% 2.82/1.60
% 2.82/1.60 % (21305)Memory used [KB]: 1791
% 2.82/1.60 % (21305)Time elapsed: 0.014 s
% 2.82/1.60 % (21305)Instructions burned: 19 (million)
% 2.82/1.60 % (21305)------------------------------
% 2.82/1.60 % (21305)------------------------------
% 2.82/1.60 % (21309)Instruction limit reached!
% 2.82/1.60 % (21309)------------------------------
% 2.82/1.60 % (21309)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.82/1.60 % (21309)Termination reason: Unknown
% 2.82/1.60 % (21309)Termination phase: shuffling
% 2.82/1.60
% 2.82/1.60 % (21309)Memory used [KB]: 1407
% 2.82/1.60 % (21309)Time elapsed: 0.003 s
% 2.82/1.60 % (21309)Instructions burned: 3 (million)
% 2.82/1.60 % (21309)------------------------------
% 2.82/1.60 % (21309)------------------------------
% 2.82/1.60 % (21310)lrs+1002_1:1_aac=none:au=on:cnfonf=lazy_gen:plsq=on:plsqc=1:plsqr=4203469,65536:i=1041:si=on:rtra=on_0 on DTF2THF_21200 for (2999ds/1041Mi)
% 2.82/1.60 % (21307)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_21200 for (2999ds/37Mi)
% 2.82/1.61 % (21308)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_21200 for (2999ds/15Mi)
% 2.82/1.61 % (21313)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_21200 for (2999ds/16Mi)
% 2.82/1.62 % (21308)Instruction limit reached!
% 2.82/1.62 % (21308)------------------------------
% 2.82/1.62 % (21308)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.82/1.62 % (21308)Termination reason: Unknown
% 2.82/1.62 % (21308)Termination phase: shuffling
% 2.82/1.62
% 2.82/1.62 % (21308)Memory used [KB]: 1791
% 2.82/1.62 % (21308)Time elapsed: 0.012 s
% 2.82/1.62 % (21308)Instructions burned: 16 (million)
% 2.82/1.62 % (21308)------------------------------
% 2.82/1.62 % (21308)------------------------------
% 2.82/1.62 % (21313)Instruction limit reached!
% 2.82/1.62 % (21313)------------------------------
% 2.82/1.62 % (21313)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.82/1.62 % (21313)Termination reason: Unknown
% 2.82/1.62 % (21313)Termination phase: shuffling
% 2.82/1.62
% 2.82/1.62 % (21313)Memory used [KB]: 1791
% 2.82/1.62 % (21313)Time elapsed: 0.008 s
% 2.82/1.62 % (21313)Instructions burned: 17 (million)
% 2.82/1.62 % (21313)------------------------------
% 2.82/1.62 % (21313)------------------------------
% 2.82/1.62 % (21311)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_21200 for (2999ds/7Mi)
% 2.82/1.62 % (21307)Instruction limit reached!
% 2.82/1.62 % (21307)------------------------------
% 2.82/1.62 % (21307)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.82/1.62 % (21307)Termination reason: Unknown
% 2.82/1.62 % (21307)Termination phase: Property scanning
% 2.82/1.62
% 2.82/1.62 % (21307)Memory used [KB]: 2174
% 2.82/1.62 % (21307)Time elapsed: 0.019 s
% 2.82/1.62 % (21307)Instructions burned: 37 (million)
% 2.82/1.62 % (21307)------------------------------
% 2.82/1.62 % (21307)------------------------------
% 2.82/1.62 % (21311)Instruction limit reached!
% 2.82/1.62 % (21311)------------------------------
% 2.82/1.62 % (21311)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.82/1.62 % (21311)Termination reason: Unknown
% 2.82/1.62 % (21311)Termination phase: shuffling
% 2.82/1.62
% 2.82/1.62 % (21311)Memory used [KB]: 1535
% 2.82/1.62 % (21311)Time elapsed: 0.007 s
% 2.82/1.62 % (21311)Instructions burned: 8 (million)
% 2.82/1.62 % (21311)------------------------------
% 2.82/1.62 % (21311)------------------------------
% 2.82/1.63 % (21315)lrs+2_1:1_apa=on:au=on:bd=preordered:cnfonf=off:cs=on:ixr=off:sos=on:i=3:si=on:rtra=on_0 on DTF2THF_21200 for (2999ds/3Mi)
% 2.82/1.63 % (21315)Instruction limit reached!
% 2.82/1.63 % (21315)------------------------------
% 2.82/1.63 % (21315)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.82/1.63 % (21315)Termination reason: Unknown
% 2.82/1.63 % (21315)Termination phase: shuffling
% 2.82/1.63
% 2.82/1.63 % (21315)Memory used [KB]: 1407
% 2.82/1.63 % (21315)Time elapsed: 0.003 s
% 2.82/1.63 % (21315)Instructions burned: 3 (million)
% 2.82/1.63 % (21315)------------------------------
% 2.82/1.63 % (21315)------------------------------
% 2.82/1.63 % (21316)lrs+10_1:1_cnfonf=off:cs=on:hud=3:prag=on:sup=off:i=7:si=on:rtra=on_0 on DTF2THF_21200 for (2999ds/7Mi)
% 2.82/1.64 % (21314)lrs+21_1:1_au=on:cnfonf=off:fd=preordered:fe=off:fsr=off:hud=11:inj=on:kws=precedence:s2pl=no:sp=weighted_frequency:tgt=full:i=3:si=on:rtra=on_0 on DTF2THF_21200 for (2999ds/3Mi)
% 2.82/1.64 % (21316)Instruction limit reached!
% 2.82/1.64 % (21316)------------------------------
% 2.82/1.64 % (21316)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.82/1.64 % (21316)Termination reason: Unknown
% 2.82/1.64 % (21316)Termination phase: shuffling
% 2.82/1.64
% 2.82/1.64 % (21316)Memory used [KB]: 1535
% 2.82/1.64 % (21316)Time elapsed: 0.004 s
% 2.82/1.64 % (21316)Instructions burned: 8 (million)
% 2.82/1.64 % (21316)------------------------------
% 2.82/1.64 % (21316)------------------------------
% 2.82/1.64 % (21314)Instruction limit reached!
% 2.82/1.64 % (21314)------------------------------
% 2.82/1.64 % (21314)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.82/1.64 % (21314)Termination reason: Unknown
% 2.82/1.64 % (21314)Termination phase: shuffling
% 2.82/1.64
% 2.82/1.64 % (21314)Memory used [KB]: 1407
% 2.82/1.64 % (21314)Time elapsed: 0.004 s
% 2.82/1.64 % (21314)Instructions burned: 3 (million)
% 2.82/1.64 % (21314)------------------------------
% 2.82/1.64 % (21314)------------------------------
% 2.82/1.64 % (21318)dis+1004_1:1_cha=on:cs=on:fe=off:hud=1:i=4:si=on:rtra=on_0 on DTF2THF_21200 for (2999ds/4Mi)
% 2.82/1.64 % (21299)Instruction limit reached!
% 2.82/1.64 % (21299)------------------------------
% 2.82/1.64 % (21299)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.82/1.64 % (21299)Termination reason: Unknown
% 2.82/1.64 % (21299)Termination phase: Saturation
% 2.82/1.64
% 2.82/1.64 % (21299)Memory used [KB]: 7164
% 2.82/1.64 % (21299)Time elapsed: 0.060 s
% 2.82/1.64 % (21299)Instructions burned: 184 (million)
% 2.82/1.64 % (21299)------------------------------
% 2.82/1.64 % (21299)------------------------------
% 2.82/1.64 % (21317)dis+1002_1:1_add=large:cnfonf=lazy_pi_sigma_gen:fe=off:prag=on:i=3:si=on:rtra=on_0 on DTF2THF_21200 for (2999ds/3Mi)
% 2.82/1.64 % (21318)Instruction limit reached!
% 2.82/1.64 % (21318)------------------------------
% 2.82/1.64 % (21318)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.82/1.64 % (21318)Termination reason: Unknown
% 2.82/1.64 % (21318)Termination phase: shuffling
% 2.82/1.64
% 2.82/1.64 % (21318)Memory used [KB]: 1535
% 2.82/1.64 % (21318)Time elapsed: 0.004 s
% 2.82/1.64 % (21318)Instructions burned: 6 (million)
% 2.82/1.64 % (21318)------------------------------
% 2.82/1.64 % (21318)------------------------------
% 2.82/1.64 % (21304)First to succeed.
% 2.82/1.64 % (21317)Instruction limit reached!
% 2.82/1.64 % (21317)------------------------------
% 2.82/1.64 % (21317)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.82/1.64 % (21317)Termination reason: Unknown
% 2.82/1.64 % (21317)Termination phase: shuffling
% 2.82/1.64
% 2.82/1.64 % (21317)Memory used [KB]: 1407
% 2.82/1.64 % (21317)Time elapsed: 0.004 s
% 2.82/1.64 % (21317)Instructions burned: 3 (million)
% 2.82/1.64 % (21317)------------------------------
% 2.82/1.64 % (21317)------------------------------
% 2.82/1.65 % (21304)Refutation found. Thanks to Tanya!
% 2.82/1.65 % SZS status Theorem for DTF2THF_21200
% 2.82/1.65 % SZS output start Proof for DTF2THF_21200
% 2.82/1.65 thf(type_def_6, type, poly: ($tType) > $tType).
% 2.82/1.65 thf(type_def_7, type, nat: $tType).
% 2.82/1.65 thf(type_def_8, type, set: ($tType) > $tType).
% 2.82/1.65 thf(type_def_9, type, real: $tType).
% 2.82/1.65 thf(func_def_0, type, poly: ($tType) > $tType).
% 2.82/1.65 thf(func_def_1, type, real: $tType).
% 2.82/1.65 thf(func_def_2, type, set: ($tType) > $tType).
% 2.82/1.65 thf(func_def_3, type, nat: $tType).
% 2.82/1.65 thf(func_def_4, type, type: !>[X0: $tType]:($o)).
% 2.82/1.65 thf(func_def_5, type, idom: !>[X0: $tType]:($o)).
% 2.82/1.65 thf(func_def_6, type, zero: !>[X0: $tType]:($o)).
% 2.82/1.65 thf(func_def_7, type, ord: !>[X0: $tType]:($o)).
% 2.82/1.65 thf(func_def_8, type, ring_char_0: !>[X0: $tType]:($o)).
% 2.82/1.65 thf(func_def_9, type, order: !>[X0: $tType]:($o)).
% 2.82/1.65 thf(func_def_10, type, no_bot: !>[X0: $tType]:($o)).
% 2.82/1.65 thf(func_def_11, type, no_top: !>[X0: $tType]:($o)).
% 2.82/1.65 thf(func_def_12, type, comm_ring_1: !>[X0: $tType]:($o)).
% 2.82/1.65 thf(func_def_13, type, idom_divide: !>[X0: $tType]:($o)).
% 2.82/1.65 thf(func_def_14, type, linorder: !>[X0: $tType]:($o)).
% 2.82/1.65 thf(func_def_15, type, preorder: !>[X0: $tType]:($o)).
% 2.82/1.65 thf(func_def_16, type, idom_abs_sgn: !>[X0: $tType]:($o)).
% 2.82/1.65 thf(func_def_17, type, wellorder: !>[X0: $tType]:($o)).
% 2.82/1.65 thf(func_def_18, type, dense_order: !>[X0: $tType]:($o)).
% 2.82/1.65 thf(func_def_19, type, comm_semiring_0: !>[X0: $tType]:($o)).
% 2.82/1.65 thf(func_def_20, type, comm_semiring_1: !>[X0: $tType]:($o)).
% 2.82/1.65 thf(func_def_21, type, linordered_idom: !>[X0: $tType]:($o)).
% 2.82/1.65 thf(func_def_22, type, linordered_field: !>[X0: $tType]:($o)).
% 2.82/1.65 thf(func_def_23, type, dense_linorder: !>[X0: $tType]:($o)).
% 2.82/1.65 thf(func_def_24, type, ring_n68954251visors: !>[X0: $tType]:($o)).
% 2.82/1.65 thf(func_def_25, type, linord1659791738miring: !>[X0: $tType]:($o)).
% 2.82/1.65 thf(func_def_26, type, canoni770627133id_add: !>[X0: $tType]:($o)).
% 2.82/1.65 thf(func_def_27, type, real_V55928688vector: !>[X0: $tType]:($o)).
% 2.82/1.65 thf(func_def_28, type, condit1656338222tinuum: !>[X0: $tType]:($o)).
% 2.82/1.65 thf(func_def_29, type, condit1037483654norder: !>[X0: $tType]:($o)).
% 2.82/1.65 thf(func_def_30, type, one_one: !>[X0: $tType]:(X0)).
% 2.82/1.65 thf(func_def_31, type, sgn_sgn: !>[X0: $tType]:(X0 > X0)).
% 2.82/1.65 thf(func_def_32, type, zero_zero: !>[X0: $tType]:(X0)).
% 2.82/1.65 thf(func_def_33, type, if: !>[X0: $tType]:($o > X0 > X0 > X0)).
% 2.82/1.65 thf(func_def_34, type, ord_less: !>[X0: $tType]:(X0 > X0 > $o)).
% 2.82/1.65 thf(func_def_35, type, ord_less_eq: !>[X0: $tType]:(X0 > X0 > $o)).
% 2.82/1.65 thf(func_def_36, type, ord_min: !>[X0: $tType]:(X0 > X0 > X0)).
% 2.82/1.65 thf(func_def_37, type, divide_poly_main: !>[X0: $tType]:(X0 > poly @ X0 > poly @ X0 > poly @ X0 > nat > nat > poly @ X0)).
% 2.82/1.65 thf(func_def_38, type, is_zero: !>[X0: $tType]:(poly @ X0 > $o)).
% 2.82/1.65 thf(func_def_39, type, order2: !>[X0: $tType]:(X0 > poly @ X0 > nat)).
% 2.82/1.65 thf(func_def_40, type, poly2: !>[X0: $tType]:(poly @ X0 > X0 > X0)).
% 2.82/1.65 thf(func_def_41, type, poly_cutoff: !>[X0: $tType]:(nat > poly @ X0 > poly @ X0)).
% 2.82/1.65 thf(func_def_42, type, poly_shift: !>[X0: $tType]:(nat > poly @ X0 > poly @ X0)).
% 2.82/1.65 thf(func_def_43, type, reflect_poly: !>[X0: $tType]:(poly @ X0 > poly @ X0)).
% 2.82/1.65 thf(func_def_44, type, collect: !>[X0: $tType]:((X0 > $o) > set @ X0)).
% 2.82/1.65 thf(func_def_45, type, sturm_1439882693eg_inf: !>[X0: $tType]:(poly @ X0 > X0)).
% 2.82/1.65 thf(func_def_46, type, sturm_62398785os_inf: !>[X0: $tType]:(poly @ X0 > X0)).
% 2.82/1.65 thf(func_def_47, type, member: !>[X0: $tType]:(X0 > set @ X0 > $o)).
% 2.82/1.65 thf(func_def_48, type, lb1: real).
% 2.82/1.65 thf(func_def_49, type, lb2: real).
% 2.82/1.65 thf(func_def_50, type, lb: real).
% 2.82/1.65 thf(func_def_51, type, p: poly @ real).
% 2.82/1.65 thf(func_def_67, type, sP0: !>[X0: $tType]:((X0 > $o) > (X0 > $o) > (X0 > $o) > (X0 > $o) > $o)).
% 2.82/1.65 thf(func_def_68, type, sP1: !>[X0: $tType]:(X0 > X0 > X0 > $o)).
% 2.82/1.65 thf(func_def_69, type, sP2: !>[X0: $tType]:(X0 > X0 > X0 > $o)).
% 2.82/1.65 thf(func_def_70, type, sP3: !>[X0: $tType]:(X0 > X0 > X0 > $o)).
% 2.82/1.65 thf(func_def_71, type, sP4: !>[X0: $tType]:((X0 > $o) > (X0 > $o) > (X0 > $o) > (X0 > $o) > $o)).
% 2.82/1.65 thf(func_def_72, type, sP5: !>[X0: $tType]:((X0 > $o) > (X0 > $o) > (X0 > $o) > (X0 > $o) > $o)).
% 2.82/1.65 thf(func_def_73, type, sP6: !>[X0: $tType]:((X0 > $o) > (X0 > $o) > (X0 > $o) > (X0 > $o) > $o)).
% 2.82/1.65 thf(func_def_74, type, sK7: !>[X0: $tType, X1: $tType]:((X0 > X1) > X0)).
% 2.82/1.65 thf(func_def_75, type, sK8: !>[X0: $tType, X1: $tType]:((X0 > X1) > X0)).
% 2.82/1.65 thf(func_def_76, type, sK9: !>[X0: $tType]:(X0 > X0)).
% 2.82/1.65 thf(func_def_77, type, sK10: !>[X0: $tType]:((X0 > X0 > $o) > X0)).
% 2.82/1.65 thf(func_def_78, type, sK11: !>[X0: $tType]:((X0 > X0 > $o) > X0)).
% 2.82/1.65 thf(func_def_79, type, sK12: !>[X0: $tType]:((X0 > X0 > $o) > X0)).
% 2.82/1.65 thf(func_def_80, type, sK13: !>[X0: $tType]:((X0 > X0 > $o) > X0)).
% 2.82/1.65 thf(func_def_81, type, sK14: !>[X0: $tType]:((X0 > X0 > $o) > X0)).
% 2.82/1.65 thf(func_def_82, type, sK15: !>[X0: $tType]:((X0 > $o) > (X0 > $o) > (X0 > $o) > (X0 > $o) > X0)).
% 2.82/1.65 thf(func_def_83, type, sK16: !>[X0: $tType]:(X0 > (X0 > $o) > (X0 > $o) > X0)).
% 2.82/1.65 thf(func_def_84, type, sK17: !>[X0: $tType]:(X0 > (X0 > $o) > (X0 > $o) > X0)).
% 2.82/1.65 thf(func_def_85, type, sK18: !>[X0: $tType]:(X0 > X0)).
% 2.82/1.65 thf(func_def_86, type, sK19: poly @ real > real > real > real).
% 2.82/1.65 thf(func_def_87, type, sK20: !>[X0: $tType, X1: $tType]:((X1 > X0) > X1)).
% 2.82/1.65 thf(func_def_88, type, sK21: !>[X0: $tType, X1: $tType]:((X1 > X0) > X1)).
% 2.82/1.65 thf(func_def_89, type, sK22: !>[X0: $tType, X1: $tType]:((X0 > X1) > X0)).
% 2.82/1.65 thf(func_def_90, type, sK23: !>[X0: $tType, X1: $tType]:((X0 > X1) > X0)).
% 2.82/1.65 thf(func_def_91, type, sK24: !>[X0: $tType]:(X0 > X0)).
% 2.82/1.65 thf(func_def_92, type, sK25: !>[X0: $tType, X1: $tType]:((X1 > X0) > X1)).
% 2.82/1.65 thf(func_def_93, type, sK26: !>[X0: $tType, X1: $tType]:((X1 > X0) > X1)).
% 2.82/1.65 thf(func_def_94, type, sK27: poly @ real > real > real > real).
% 2.82/1.65 thf(func_def_95, type, sK28: poly @ real > real > real > real).
% 2.82/1.65 thf(func_def_96, type, sK29: poly @ real > real > real > real).
% 2.82/1.65 thf(func_def_97, type, sK30: !>[X0: $tType, X1: $tType]:((X0 > X1) > X0)).
% 2.82/1.65 thf(func_def_98, type, sK31: !>[X0: $tType, X1: $tType]:((X0 > X1) > X0)).
% 2.82/1.65 thf(func_def_99, type, sK32: !>[X0: $tType]:(X0 > X0)).
% 2.82/1.65 thf(func_def_100, type, sK33: poly @ real > real > real > real).
% 2.82/1.65 thf(func_def_101, type, sK34: !>[X0: $tType, X1: $tType]:((X1 > X0) > X1)).
% 2.82/1.65 thf(func_def_102, type, sK35: !>[X0: $tType, X1: $tType]:((X1 > X0) > X1)).
% 2.82/1.65 thf(func_def_103, type, sK36: !>[X0: $tType]:(X0 > X0)).
% 2.82/1.65 thf(func_def_104, type, sK37: !>[X0: $tType, X1: $tType]:((X1 > X0) > X1)).
% 2.82/1.65 thf(func_def_105, type, sK38: !>[X0: $tType, X1: $tType]:((X1 > X0) > X1)).
% 2.82/1.65 thf(func_def_106, type, sK39: poly @ real > real > real > real).
% 2.82/1.65 thf(func_def_107, type, sK40: poly @ real > real > real > real).
% 2.82/1.65 thf(func_def_108, type, sK41: poly @ real > real > real > real).
% 2.82/1.65 thf(func_def_109, type, sK42: !>[X0: $tType]:(X0 > X0)).
% 2.82/1.65 thf(func_def_110, type, sK43: !>[X0: $tType]:(X0 > X0)).
% 2.82/1.65 thf(func_def_111, type, sK44: real).
% 2.82/1.65 thf(func_def_112, type, sK45: !>[X0: $tType]:((X0 > $o) > (X0 > $o) > (X0 > $o) > (X0 > $o) > X0)).
% 2.82/1.65 thf(func_def_113, type, sK46: !>[X0: $tType]:(X0 > (X0 > $o) > (X0 > $o) > X0)).
% 2.82/1.65 thf(func_def_114, type, sK47: !>[X0: $tType]:(X0 > (X0 > $o) > (X0 > $o) > X0)).
% 2.82/1.65 thf(func_def_115, type, sK48: real > poly @ real > real).
% 2.82/1.65 thf(func_def_116, type, sK49: !>[X0: $tType]:(X0 > X0)).
% 2.82/1.65 thf(func_def_117, type, sK50: !>[X0: $tType]:(X0 > X0)).
% 2.82/1.65 thf(func_def_118, type, sK51: !>[X0: $tType, X1: $tType]:((X0 > X1) > X0)).
% 2.82/1.65 thf(func_def_119, type, sK52: !>[X0: $tType, X1: $tType]:((X0 > X1) > X0)).
% 2.82/1.65 thf(func_def_120, type, sK53: real).
% 2.82/1.65 thf(func_def_121, type, sK54: !>[X0: $tType]:(X0 > X0)).
% 2.82/1.65 thf(func_def_122, type, sK55: !>[X0: $tType, X1: $tType]:((X1 > X0) > (X1 > X0) > X1)).
% 2.82/1.65 thf(func_def_123, type, sK56: !>[X0: $tType]:((X0 > $o) > (X0 > $o) > (X0 > $o) > (X0 > $o) > X0)).
% 2.82/1.65 thf(func_def_124, type, sK57: !>[X0: $tType]:(X0 > (X0 > $o) > (X0 > $o) > X0)).
% 2.82/1.65 thf(func_def_125, type, sK58: !>[X0: $tType]:(X0 > (X0 > $o) > (X0 > $o) > X0)).
% 2.82/1.65 thf(func_def_126, type, sK59: real > poly @ real > real).
% 2.82/1.65 thf(func_def_127, type, sK60: !>[X0: $tType]:(X0 > X0)).
% 2.82/1.65 thf(func_def_128, type, sK61: !>[X0: $tType]:(X0 > X0)).
% 2.82/1.65 thf(func_def_129, type, sK62: !>[X0: $tType]:((X0 > $o) > (X0 > $o) > (X0 > $o) > (X0 > $o) > X0)).
% 2.82/1.65 thf(func_def_130, type, sK63: !>[X0: $tType]:(X0 > (X0 > $o) > (X0 > $o) > X0)).
% 2.82/1.65 thf(func_def_131, type, sK64: !>[X0: $tType]:(X0 > (X0 > $o) > (X0 > $o) > X0)).
% 2.82/1.65 thf(func_def_132, type, sK65: !>[X0: $tType]:((X0 > X0 > $o) > X0)).
% 2.82/1.65 thf(func_def_133, type, sK66: !>[X0: $tType]:((X0 > X0 > $o) > X0)).
% 2.82/1.65 thf(func_def_134, type, sK67: !>[X0: $tType]:((X0 > X0 > $o) > X0)).
% 2.82/1.65 thf(func_def_135, type, sK68: !>[X0: $tType]:((X0 > X0 > $o) > X0)).
% 2.82/1.65 thf(func_def_137, type, ph70: !>[X0: $tType]:(X0)).
% 2.82/1.65 thf(f1582,plain,(
% 2.82/1.65 $false),
% 2.82/1.65 inference(subsumption_resolution,[status(thm)],[f1581,f1504])).
% 2.82/1.65 thf(f1504,plain,(
% 2.82/1.65 ((ord_less @ real @ lb1 @ sK44) = $true)),
% 2.82/1.65 inference(trivial_inequality_removal,[status(thm)],[f1502])).
% 2.82/1.65 thf(f1502,plain,(
% 2.82/1.65 ((ord_less @ real @ lb1 @ sK44) = $true) | (zero_zero @ real != zero_zero @ real)),
% 2.82/1.65 inference(superposition,[status(thm)],[f1467,f1393])).
% 2.82/1.65 thf(f1393,plain,(
% 2.82/1.65 (zero_zero @ real = (poly2 @ real @ p @ sK44))),
% 2.82/1.65 inference(cnf_transformation,[status(thm)],[f1210])).
% 2.82/1.65 thf(f1210,plain,(
% 2.82/1.65 ($true != (ord_less @ real @ lb @ sK44)) & (zero_zero @ real = (poly2 @ real @ p @ sK44))),
% 2.82/1.65 inference(skolemisation,[status(esa),new_symbols(skolem,[sK44])],[f1099,f1209])).
% 2.82/1.65 thf(f1209,plain,(
% 2.82/1.65 ? [X0 : real] : (((ord_less @ real @ lb @ X0) != $true) & (zero_zero @ real = (poly2 @ real @ p @ X0))) => (($true != (ord_less @ real @ lb @ sK44)) & (zero_zero @ real = (poly2 @ real @ p @ sK44)))),
% 2.82/1.65 introduced(definition,[],[choice_axiom])).
% 2.82/1.65 thf(f1099,plain,(
% 2.82/1.65 ? [X0 : real] : (((ord_less @ real @ lb @ X0) != $true) & (zero_zero @ real = (poly2 @ real @ p @ X0)))),
% 2.82/1.65 inference(ennf_transformation,[status(thm)],[f854])).
% 2.82/1.65 thf(f854,plain,(
% 2.82/1.65 ~! [X0 : real] : (((ord_less @ real @ lb @ X0) = $true) | (zero_zero @ real != (poly2 @ real @ p @ X0)))),
% 2.82/1.65 inference(fool_elimination,[status(thm)],[f853])).
% 2.82/1.65 thf(f853,plain,(
% 2.82/1.65 ~! [X0 : real] : ((ord_less @ real @ lb @ X0) | (zero_zero @ real != (poly2 @ real @ p @ X0)))),
% 2.82/1.65 inference(rectify,[status(thm)],[f319])).
% 2.82/1.65 thf(f319,negated_conjecture,(
% 2.82/1.65 ~! [X8 : real] : ((ord_less @ real @ lb @ X8) | (zero_zero @ real != (poly2 @ real @ p @ X8)))),
% 2.82/1.65 inference(negated_conjecture,[status(cth)],[f318])).
% 2.82/1.65 thf(f318,conjecture,(
% 2.82/1.65 ! [X8 : real] : ((ord_less @ real @ lb @ X8) | (zero_zero @ real != (poly2 @ real @ p @ X8)))),
% 2.82/1.65 file('/export/starexec/sandbox2/tmp/tmp.TYEgfeW64B/DTF2THF_21200.p',conj_0)).
% 2.82/1.65 thf(f1467,plain,(
% 2.82/1.65 ( ! [X0 : real] : ((zero_zero @ real != (poly2 @ real @ p @ X0)) | ((ord_less @ real @ lb1 @ X0) = $true)) )),
% 2.82/1.65 inference(cnf_transformation,[status(thm)],[f969])).
% 2.82/1.65 thf(f969,plain,(
% 2.82/1.65 ! [X0 : real] : (((ord_less @ real @ lb1 @ X0) = $true) | (zero_zero @ real != (poly2 @ real @ p @ X0)))),
% 2.82/1.65 inference(ennf_transformation,[status(thm)],[f450])).
% 2.82/1.65 thf(f450,plain,(
% 2.82/1.65 ! [X0 : real] : ((zero_zero @ real = (poly2 @ real @ p @ X0)) => ((ord_less @ real @ lb1 @ X0) = $true))),
% 2.82/1.65 inference(fool_elimination,[status(thm)],[f449])).
% 2.82/1.65 thf(f449,plain,(
% 2.82/1.65 ! [X0 : real] : ((zero_zero @ real = (poly2 @ real @ p @ X0)) => (ord_less @ real @ lb1 @ X0))),
% 2.82/1.65 inference(rectify,[status(thm)],[f1])).
% 2.82/1.65 thf(f1,axiom,(
% 2.82/1.65 ! [X1 : real] : (((poly2 @ real @ p @ X1) = zero_zero @ real) => (ord_less @ real @ lb1 @ X1))),
% 2.82/1.65 file('/export/starexec/sandbox2/tmp/tmp.TYEgfeW64B/DTF2THF_21200.p',fact_0_lb1)).
% 2.82/1.65 thf(f1581,plain,(
% 2.82/1.65 ((ord_less @ real @ lb1 @ sK44) != $true)),
% 2.82/1.65 inference(subsumption_resolution,[status(thm)],[f1552,f1322])).
% 2.82/1.65 thf(f1322,plain,(
% 2.82/1.65 (linorder @ real = $true)),
% 2.82/1.65 inference(cnf_transformation,[status(thm)],[f478])).
% 2.82/1.65 thf(f478,plain,(
% 2.82/1.65 (linorder @ real = $true)),
% 2.82/1.65 inference(fool_elimination,[status(thm)],[f477])).
% 2.82/1.65 thf(f477,plain,(
% 2.82/1.65 linorder @ real),
% 2.82/1.65 inference(rectify,[status(thm)],[f290])).
% 2.82/1.65 thf(f290,axiom,(
% 2.82/1.65 linorder @ real),
% 2.82/1.65 file('/export/starexec/sandbox2/tmp/tmp.TYEgfeW64B/DTF2THF_21200.p',tcon_Real_Oreal___Orderings_Olinorder_16)).
% 2.82/1.65 thf(f1552,plain,(
% 2.82/1.65 (linorder @ real != $true) | ((ord_less @ real @ lb1 @ sK44) != $true)),
% 2.82/1.65 inference(trivial_inequality_removal,[status(thm)],[f1536])).
% 2.82/1.65 thf(f1536,plain,(
% 2.82/1.65 ((ord_less @ real @ lb1 @ sK44) != $true) | ($true != $true) | (linorder @ real != $true)),
% 2.82/1.65 inference(superposition,[status(thm)],[f1480,f1323])).
% 2.82/1.65 thf(f1323,plain,(
% 2.82/1.65 ( ! [X0 : $tType,X2 : X0,X3 : X0,X1 : X0] : (((ord_less @ X0 @ (ord_min @ X0 @ X1 @ X2) @ X3) = $true) | ((ord_less @ X0 @ X1 @ X3) != $true) | (linorder @ X0 != $true)) )),
% 2.82/1.65 inference(cnf_transformation,[status(thm)],[f1171])).
% 2.82/1.65 thf(f1171,plain,(
% 2.82/1.65 ! [X0 : $tType] : ((linorder @ X0 != $true) | ! [X1 : X0,X2 : X0,X3 : X0] : ((((ord_less @ X0 @ X2 @ X3) = $true) | ((ord_less @ X0 @ X1 @ X3) = $true) | ((ord_less @ X0 @ (ord_min @ X0 @ X1 @ X2) @ X3) != $true)) & (((ord_less @ X0 @ (ord_min @ X0 @ X1 @ X2) @ X3) = $true) | (((ord_less @ X0 @ X2 @ X3) != $true) & ((ord_less @ X0 @ X1 @ X3) != $true)))))),
% 2.82/1.65 inference(flattening,[status(thm)],[f1170])).
% 2.82/1.65 thf(f1170,plain,(
% 2.82/1.65 ! [X0 : $tType] : ((linorder @ X0 != $true) | ! [X1 : X0,X2 : X0,X3 : X0] : (((((ord_less @ X0 @ X2 @ X3) = $true) | ((ord_less @ X0 @ X1 @ X3) = $true)) | ((ord_less @ X0 @ (ord_min @ X0 @ X1 @ X2) @ X3) != $true)) & (((ord_less @ X0 @ (ord_min @ X0 @ X1 @ X2) @ X3) = $true) | (((ord_less @ X0 @ X2 @ X3) != $true) & ((ord_less @ X0 @ X1 @ X3) != $true)))))),
% 2.82/1.65 inference(nnf_transformation,[status(thm)],[f1032])).
% 2.82/1.65 thf(f1032,plain,(
% 2.82/1.65 ! [X0 : $tType] : ((linorder @ X0 != $true) | ! [X1 : X0,X2 : X0,X3 : X0] : ((((ord_less @ X0 @ X2 @ X3) = $true) | ((ord_less @ X0 @ X1 @ X3) = $true)) <=> ((ord_less @ X0 @ (ord_min @ X0 @ X1 @ X2) @ X3) = $true)))),
% 2.82/1.65 inference(ennf_transformation,[status(thm)],[f706])).
% 2.82/1.65 thf(f706,plain,(
% 2.82/1.65 ! [X0 : $tType] : ((linorder @ X0 = $true) => ! [X1 : X0,X2 : X0,X3 : X0] : ((((ord_less @ X0 @ X2 @ X3) = $true) | ((ord_less @ X0 @ X1 @ X3) = $true)) <=> ((ord_less @ X0 @ (ord_min @ X0 @ X1 @ X2) @ X3) = $true)))),
% 2.82/1.65 inference(fool_elimination,[status(thm)],[f705])).
% 2.82/1.65 thf(f705,plain,(
% 2.82/1.65 ! [X0 : $tType] : (linorder @ X0 => ! [X1 : X0,X2 : X0,X3 : X0] : ((ord_less @ X0 @ X2 @ X3) | (ord_less @ X0 @ X1 @ X3) = (ord_less @ X0 @ (ord_min @ X0 @ X1 @ X2) @ X3)))),
% 2.82/1.65 inference(rectify,[status(thm)],[f26])).
% 2.82/1.65 thf(f26,axiom,(
% 2.82/1.65 ! [X0 : $tType] : (linorder @ X0 => ! [X3 : X0,X16 : X0,X15 : X0] : ((ord_less @ X0 @ X16 @ X15) | (ord_less @ X0 @ X3 @ X15) = (ord_less @ X0 @ (ord_min @ X0 @ X3 @ X16) @ X15)))),
% 2.82/1.65 file('/export/starexec/sandbox2/tmp/tmp.TYEgfeW64B/DTF2THF_21200.p',fact_25_min__less__iff__disj)).
% 2.82/1.65 thf(f1480,plain,(
% 2.82/1.65 ((ord_less @ real @ (ord_min @ real @ lb1 @ lb2) @ sK44) != $true)),
% 2.82/1.65 inference(definition_unfolding,[status(thm)],[f1394,f1294])).
% 2.82/1.65 thf(f1294,plain,(
% 2.82/1.65 (lb = (ord_min @ real @ lb1 @ lb2))),
% 2.82/1.65 inference(cnf_transformation,[status(thm)],[f2])).
% 2.82/1.65 thf(f2,axiom,(
% 2.82/1.65 (lb = (ord_min @ real @ lb1 @ lb2))),
% 2.82/1.65 file('/export/starexec/sandbox2/tmp/tmp.TYEgfeW64B/DTF2THF_21200.p',fact_1_lb__def)).
% 2.82/1.65 thf(f1394,plain,(
% 2.82/1.65 ($true != (ord_less @ real @ lb @ sK44))),
% 2.82/1.65 inference(cnf_transformation,[status(thm)],[f1210])).
% 2.82/1.65 % SZS output end Proof for DTF2THF_21200
% 2.82/1.65 % (21304)------------------------------
% 2.82/1.65 % (21304)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.82/1.65 % (21304)Termination reason: Refutation
% 2.82/1.65
% 2.82/1.65 % (21304)Memory used [KB]: 6908
% 2.82/1.65 % (21304)Time elapsed: 0.057 s
% 2.82/1.65 % (21304)Instructions burned: 93 (million)
% 2.82/1.65 % (21304)------------------------------
% 2.82/1.65 % (21304)------------------------------
% 2.82/1.65 % (21298)Success in time 0.093 s
%------------------------------------------------------------------------------