TPTP Problem File: ITP194^1.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : ITP194^1 : TPTP v8.2.0. Released v7.5.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer Sturm_Tarski problem prob_257__5870986_1
% Version : Especial.
% English :
% Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% : [Des21] Desharnais (2021), Email to Geoff Sutcliffe
% Source : [Des21]
% Names : Sturm_Tarski/prob_257__5870986_1 [Des21]
% Status : Theorem
% Rating : 0.30 v8.2.0, 0.23 v8.1.0, 0.18 v7.5.0
% Syntax : Number of formulae : 415 ( 169 unt; 64 typ; 0 def)
% Number of atoms : 985 ( 326 equ; 0 cnn)
% Maximal formula atoms : 12 ( 2 avg)
% Number of connectives : 2504 ( 135 ~; 28 |; 37 &;1867 @)
% ( 0 <=>; 437 =>; 0 <=; 0 <~>)
% Maximal formula depth : 14 ( 6 avg)
% Number of types : 9 ( 8 usr)
% Number of type conns : 151 ( 151 >; 0 *; 0 +; 0 <<)
% Number of symbols : 59 ( 56 usr; 14 con; 0-6 aty)
% Number of variables : 907 ( 68 ^; 823 !; 16 ?; 907 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Sledgehammer 2021-02-23 15:41:56.214
%------------------------------------------------------------------------------
% Could-be-implicit typings (8)
thf(ty_n_t__Polynomial__Opoly_It__Polynomial__Opoly_It__Polynomial__Opoly_It__Real__Oreal_J_J_J,type,
poly_poly_poly_real: $tType ).
thf(ty_n_t__Polynomial__Opoly_It__Polynomial__Opoly_It__Real__Oreal_J_J,type,
poly_poly_real: $tType ).
thf(ty_n_t__Polynomial__Opoly_It__Polynomial__Opoly_It__Nat__Onat_J_J,type,
poly_poly_nat: $tType ).
thf(ty_n_t__Polynomial__Opoly_It__Real__Oreal_J,type,
poly_real: $tType ).
thf(ty_n_t__Polynomial__Opoly_It__Nat__Onat_J,type,
poly_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Real__Oreal_J,type,
set_real: $tType ).
thf(ty_n_t__Real__Oreal,type,
real: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
% Explicit typings (56)
thf(sy_c_Groups_Osgn__class_Osgn_001t__Polynomial__Opoly_It__Polynomial__Opoly_It__Real__Oreal_J_J,type,
sgn_sg2128174761y_real: poly_poly_real > poly_poly_real ).
thf(sy_c_Groups_Osgn__class_Osgn_001t__Polynomial__Opoly_It__Real__Oreal_J,type,
sgn_sgn_poly_real: poly_real > poly_real ).
thf(sy_c_Groups_Osgn__class_Osgn_001t__Real__Oreal,type,
sgn_sgn_real: real > real ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
zero_zero_nat: nat ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Polynomial__Opoly_It__Nat__Onat_J,type,
zero_zero_poly_nat: poly_nat ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Polynomial__Opoly_It__Polynomial__Opoly_It__Nat__Onat_J_J,type,
zero_z1059985641ly_nat: poly_poly_nat ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Polynomial__Opoly_It__Polynomial__Opoly_It__Polynomial__Opoly_It__Real__Oreal_J_J_J,type,
zero_z935034829y_real: poly_poly_poly_real ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Polynomial__Opoly_It__Polynomial__Opoly_It__Real__Oreal_J_J,type,
zero_z1423781445y_real: poly_poly_real ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Polynomial__Opoly_It__Real__Oreal_J,type,
zero_zero_poly_real: poly_real ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Real__Oreal,type,
zero_zero_real: real ).
thf(sy_c_If_001t__Nat__Onat,type,
if_nat: $o > nat > nat > nat ).
thf(sy_c_If_001t__Real__Oreal,type,
if_real: $o > real > real > real ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
ord_less_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Polynomial__Opoly_It__Polynomial__Opoly_It__Real__Oreal_J_J,type,
ord_le38482960y_real: poly_poly_real > poly_poly_real > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Polynomial__Opoly_It__Real__Oreal_J,type,
ord_less_poly_real: poly_real > poly_real > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Real__Oreal,type,
ord_less_real: real > real > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
ord_less_eq_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Polynomial__Opoly_It__Polynomial__Opoly_It__Real__Oreal_J_J,type,
ord_le893774876y_real: poly_poly_real > poly_poly_real > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Polynomial__Opoly_It__Real__Oreal_J,type,
ord_le1180086932y_real: poly_real > poly_real > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Real__Oreal,type,
ord_less_eq_real: real > real > $o ).
thf(sy_c_Orderings_Oord__class_Omin_001t__Nat__Onat,type,
ord_min_nat: nat > nat > nat ).
thf(sy_c_Orderings_Oord__class_Omin_001t__Real__Oreal,type,
ord_min_real: real > real > real ).
thf(sy_c_Polynomial_Odivide__poly__main_001t__Polynomial__Opoly_It__Polynomial__Opoly_It__Real__Oreal_J_J,type,
divide924636027y_real: poly_poly_real > poly_poly_poly_real > poly_poly_poly_real > poly_poly_poly_real > nat > nat > poly_poly_poly_real ).
thf(sy_c_Polynomial_Odivide__poly__main_001t__Polynomial__Opoly_It__Real__Oreal_J,type,
divide1142363123y_real: poly_real > poly_poly_real > poly_poly_real > poly_poly_real > nat > nat > poly_poly_real ).
thf(sy_c_Polynomial_Odivide__poly__main_001t__Real__Oreal,type,
divide1561404011n_real: real > poly_real > poly_real > poly_real > nat > nat > poly_real ).
thf(sy_c_Polynomial_Ois__zero_001t__Nat__Onat,type,
is_zero_nat: poly_nat > $o ).
thf(sy_c_Polynomial_Ois__zero_001t__Polynomial__Opoly_It__Real__Oreal_J,type,
is_zero_poly_real: poly_poly_real > $o ).
thf(sy_c_Polynomial_Ois__zero_001t__Real__Oreal,type,
is_zero_real: poly_real > $o ).
thf(sy_c_Polynomial_Oorder_001t__Polynomial__Opoly_It__Polynomial__Opoly_It__Real__Oreal_J_J,type,
order_poly_poly_real: poly_poly_real > poly_poly_poly_real > nat ).
thf(sy_c_Polynomial_Oorder_001t__Polynomial__Opoly_It__Real__Oreal_J,type,
order_poly_real: poly_real > poly_poly_real > nat ).
thf(sy_c_Polynomial_Oorder_001t__Real__Oreal,type,
order_real: real > poly_real > nat ).
thf(sy_c_Polynomial_Opoly_001t__Nat__Onat,type,
poly_nat2: poly_nat > nat > nat ).
thf(sy_c_Polynomial_Opoly_001t__Polynomial__Opoly_It__Nat__Onat_J,type,
poly_poly_nat2: poly_poly_nat > poly_nat > poly_nat ).
thf(sy_c_Polynomial_Opoly_001t__Polynomial__Opoly_It__Polynomial__Opoly_It__Real__Oreal_J_J,type,
poly_poly_poly_real2: poly_poly_poly_real > poly_poly_real > poly_poly_real ).
thf(sy_c_Polynomial_Opoly_001t__Polynomial__Opoly_It__Real__Oreal_J,type,
poly_poly_real2: poly_poly_real > poly_real > poly_real ).
thf(sy_c_Polynomial_Opoly_001t__Real__Oreal,type,
poly_real2: poly_real > real > real ).
thf(sy_c_Polynomial_Opoly__cutoff_001t__Nat__Onat,type,
poly_cutoff_nat: nat > poly_nat > poly_nat ).
thf(sy_c_Polynomial_Opoly__cutoff_001t__Polynomial__Opoly_It__Real__Oreal_J,type,
poly_c1404107022y_real: nat > poly_poly_real > poly_poly_real ).
thf(sy_c_Polynomial_Opoly__cutoff_001t__Real__Oreal,type,
poly_cutoff_real: nat > poly_real > poly_real ).
thf(sy_c_Polynomial_Opoly__shift_001t__Nat__Onat,type,
poly_shift_nat: nat > poly_nat > poly_nat ).
thf(sy_c_Polynomial_Opoly__shift_001t__Polynomial__Opoly_It__Real__Oreal_J,type,
poly_shift_poly_real: nat > poly_poly_real > poly_poly_real ).
thf(sy_c_Polynomial_Opoly__shift_001t__Real__Oreal,type,
poly_shift_real: nat > poly_real > poly_real ).
thf(sy_c_Polynomial_Oreflect__poly_001t__Nat__Onat,type,
reflect_poly_nat: poly_nat > poly_nat ).
thf(sy_c_Polynomial_Oreflect__poly_001t__Polynomial__Opoly_It__Nat__Onat_J,type,
reflec781175074ly_nat: poly_poly_nat > poly_poly_nat ).
thf(sy_c_Polynomial_Oreflect__poly_001t__Polynomial__Opoly_It__Polynomial__Opoly_It__Real__Oreal_J_J,type,
reflec144234502y_real: poly_poly_poly_real > poly_poly_poly_real ).
thf(sy_c_Polynomial_Oreflect__poly_001t__Polynomial__Opoly_It__Real__Oreal_J,type,
reflec1522834046y_real: poly_poly_real > poly_poly_real ).
thf(sy_c_Polynomial_Oreflect__poly_001t__Real__Oreal,type,
reflect_poly_real: poly_real > poly_real ).
thf(sy_c_Set_OCollect_001t__Real__Oreal,type,
collect_real: ( real > $o ) > set_real ).
thf(sy_c_Sturm__Tarski__Mirabelle__skihomvtkj_Osgn__neg__inf_001t__Real__Oreal,type,
sturm_1076696862f_real: poly_real > real ).
thf(sy_c_Sturm__Tarski__Mirabelle__skihomvtkj_Osgn__pos__inf_001t__Real__Oreal,type,
sturm_1308388506f_real: poly_real > real ).
thf(sy_c_member_001t__Real__Oreal,type,
member_real: real > set_real > $o ).
thf(sy_v_lb1____,type,
lb1: real ).
thf(sy_v_lb2____,type,
lb2: real ).
thf(sy_v_lb____,type,
lb: real ).
thf(sy_v_p,type,
p: poly_real ).
thf(sy_v_thesis,type,
thesis: $o ).
% Relevant facts (345)
thf(fact_0_lb1,axiom,
! [X: real] :
( ( ( poly_real2 @ p @ X )
= zero_zero_real )
=> ( ord_less_real @ lb1 @ X ) ) ).
% lb1
thf(fact_1_lb__def,axiom,
( lb
= ( ord_min_real @ lb1 @ lb2 ) ) ).
% lb_def
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] :
( ( ( poly_real2 @ p @ X )
= zero_zero_real )
=> ( ord_less_real @ Lb1 @ X ) ) ).
% \<open>\<And>thesis. (\<And>lb1. \<forall>x. poly p x = 0 \<longrightarrow> lb1 < x \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_3_assms,axiom,
p != zero_zero_poly_real ).
% assms
thf(fact_4_poly__0,axiom,
! [X2: poly_nat] :
( ( poly_poly_nat2 @ zero_z1059985641ly_nat @ X2 )
= zero_zero_poly_nat ) ).
% poly_0
thf(fact_5_poly__0,axiom,
! [X2: poly_poly_real] :
( ( poly_poly_poly_real2 @ zero_z935034829y_real @ X2 )
= zero_z1423781445y_real ) ).
% poly_0
thf(fact_6_poly__0,axiom,
! [X2: poly_real] :
( ( poly_poly_real2 @ zero_z1423781445y_real @ X2 )
= zero_zero_poly_real ) ).
% poly_0
thf(fact_7_poly__0,axiom,
! [X2: nat] :
( ( poly_nat2 @ zero_zero_poly_nat @ X2 )
= zero_zero_nat ) ).
% poly_0
thf(fact_8_poly__0,axiom,
! [X2: real] :
( ( poly_real2 @ zero_zero_poly_real @ X2 )
= zero_zero_real ) ).
% poly_0
thf(fact_9_not__gr__zero,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_10_poly__IVT__neg,axiom,
! [A: real,B: real,P: poly_real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ zero_zero_real @ ( poly_real2 @ P @ A ) )
=> ( ( ord_less_real @ ( poly_real2 @ P @ B ) @ zero_zero_real )
=> ? [X3: real] :
( ( ord_less_real @ A @ X3 )
& ( ord_less_real @ X3 @ B )
& ( ( poly_real2 @ P @ X3 )
= zero_zero_real ) ) ) ) ) ).
% poly_IVT_neg
thf(fact_11_poly__IVT__pos,axiom,
! [A: real,B: real,P: poly_real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ ( poly_real2 @ P @ A ) @ zero_zero_real )
=> ( ( ord_less_real @ zero_zero_real @ ( poly_real2 @ P @ B ) )
=> ? [X3: real] :
( ( ord_less_real @ A @ X3 )
& ( ord_less_real @ X3 @ B )
& ( ( poly_real2 @ P @ X3 )
= zero_zero_real ) ) ) ) ) ).
% poly_IVT_pos
thf(fact_12_poly__all__0__iff__0,axiom,
! [P: poly_poly_poly_real] :
( ( ! [X4: poly_poly_real] :
( ( poly_poly_poly_real2 @ P @ X4 )
= zero_z1423781445y_real ) )
= ( P = zero_z935034829y_real ) ) ).
% poly_all_0_iff_0
thf(fact_13_poly__all__0__iff__0,axiom,
! [P: poly_real] :
( ( ! [X4: real] :
( ( poly_real2 @ P @ X4 )
= zero_zero_real ) )
= ( P = zero_zero_poly_real ) ) ).
% poly_all_0_iff_0
thf(fact_14_poly__all__0__iff__0,axiom,
! [P: poly_poly_real] :
( ( ! [X4: poly_real] :
( ( poly_poly_real2 @ P @ X4 )
= zero_zero_poly_real ) )
= ( P = zero_z1423781445y_real ) ) ).
% poly_all_0_iff_0
thf(fact_15_less__numeral__extra_I3_J,axiom,
~ ( ord_less_poly_real @ zero_zero_poly_real @ zero_zero_poly_real ) ).
% less_numeral_extra(3)
thf(fact_16_less__numeral__extra_I3_J,axiom,
~ ( ord_le38482960y_real @ zero_z1423781445y_real @ zero_z1423781445y_real ) ).
% less_numeral_extra(3)
thf(fact_17_less__numeral__extra_I3_J,axiom,
~ ( ord_less_real @ zero_zero_real @ zero_zero_real ) ).
% less_numeral_extra(3)
thf(fact_18_less__numeral__extra_I3_J,axiom,
~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% less_numeral_extra(3)
thf(fact_19_field__lbound__gt__zero,axiom,
! [D1: real,D2: real] :
( ( ord_less_real @ zero_zero_real @ D1 )
=> ( ( ord_less_real @ zero_zero_real @ D2 )
=> ? [E: real] :
( ( ord_less_real @ zero_zero_real @ E )
& ( ord_less_real @ E @ D1 )
& ( ord_less_real @ E @ D2 ) ) ) ) ).
% field_lbound_gt_zero
thf(fact_20_gr__zeroI,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr_zeroI
thf(fact_21_not__less__zero,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less_zero
thf(fact_22_gr__implies__not__zero,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_23_zero__less__iff__neq__zero,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
= ( N != zero_zero_nat ) ) ).
% zero_less_iff_neq_zero
thf(fact_24_zero__reorient,axiom,
! [X2: real] :
( ( zero_zero_real = X2 )
= ( X2 = zero_zero_real ) ) ).
% zero_reorient
thf(fact_25_zero__reorient,axiom,
! [X2: poly_real] :
( ( zero_zero_poly_real = X2 )
= ( X2 = zero_zero_poly_real ) ) ).
% zero_reorient
thf(fact_26_zero__reorient,axiom,
! [X2: nat] :
( ( zero_zero_nat = X2 )
= ( X2 = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_27_zero__reorient,axiom,
! [X2: poly_nat] :
( ( zero_zero_poly_nat = X2 )
= ( X2 = zero_zero_poly_nat ) ) ).
% zero_reorient
thf(fact_28_zero__reorient,axiom,
! [X2: poly_poly_real] :
( ( zero_z1423781445y_real = X2 )
= ( X2 = zero_z1423781445y_real ) ) ).
% zero_reorient
thf(fact_29_poly__eq__poly__eq__iff,axiom,
! [P: poly_real,Q: poly_real] :
( ( ( poly_real2 @ P )
= ( poly_real2 @ Q ) )
= ( P = Q ) ) ).
% poly_eq_poly_eq_iff
thf(fact_30_poly__eq__poly__eq__iff,axiom,
! [P: poly_poly_real,Q: poly_poly_real] :
( ( ( poly_poly_real2 @ P )
= ( poly_poly_real2 @ Q ) )
= ( P = Q ) ) ).
% poly_eq_poly_eq_iff
thf(fact_31_min__less__iff__conj,axiom,
! [Z: real,X2: real,Y: real] :
( ( ord_less_real @ Z @ ( ord_min_real @ X2 @ Y ) )
= ( ( ord_less_real @ Z @ X2 )
& ( ord_less_real @ Z @ Y ) ) ) ).
% min_less_iff_conj
thf(fact_32_min__less__iff__conj,axiom,
! [Z: nat,X2: nat,Y: nat] :
( ( ord_less_nat @ Z @ ( ord_min_nat @ X2 @ Y ) )
= ( ( ord_less_nat @ Z @ X2 )
& ( ord_less_nat @ Z @ Y ) ) ) ).
% min_less_iff_conj
thf(fact_33_min_Oidem,axiom,
! [A: real] :
( ( ord_min_real @ A @ A )
= A ) ).
% min.idem
thf(fact_34_min_Oidem,axiom,
! [A: nat] :
( ( ord_min_nat @ A @ A )
= A ) ).
% min.idem
thf(fact_35_min_Oleft__idem,axiom,
! [A: real,B: real] :
( ( ord_min_real @ A @ ( ord_min_real @ A @ B ) )
= ( ord_min_real @ A @ B ) ) ).
% min.left_idem
thf(fact_36_min_Oleft__idem,axiom,
! [A: nat,B: nat] :
( ( ord_min_nat @ A @ ( ord_min_nat @ A @ B ) )
= ( ord_min_nat @ A @ B ) ) ).
% min.left_idem
thf(fact_37_min_Oright__idem,axiom,
! [A: real,B: real] :
( ( ord_min_real @ ( ord_min_real @ A @ B ) @ B )
= ( ord_min_real @ A @ B ) ) ).
% min.right_idem
thf(fact_38_min_Oright__idem,axiom,
! [A: nat,B: nat] :
( ( ord_min_nat @ ( ord_min_nat @ A @ B ) @ B )
= ( ord_min_nat @ A @ B ) ) ).
% min.right_idem
thf(fact_39_min_Ostrict__coboundedI2,axiom,
! [B: real,C: real,A: real] :
( ( ord_less_real @ B @ C )
=> ( ord_less_real @ ( ord_min_real @ A @ B ) @ C ) ) ).
% min.strict_coboundedI2
thf(fact_40_min_Ostrict__coboundedI2,axiom,
! [B: nat,C: nat,A: nat] :
( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ ( ord_min_nat @ A @ B ) @ C ) ) ).
% min.strict_coboundedI2
thf(fact_41_min_Ostrict__coboundedI1,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_real @ A @ C )
=> ( ord_less_real @ ( ord_min_real @ A @ B ) @ C ) ) ).
% min.strict_coboundedI1
thf(fact_42_min_Ostrict__coboundedI1,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_nat @ A @ C )
=> ( ord_less_nat @ ( ord_min_nat @ A @ B ) @ C ) ) ).
% min.strict_coboundedI1
thf(fact_43_min_Ostrict__order__iff,axiom,
( ord_less_real
= ( ^ [A2: real,B2: real] :
( ( A2
= ( ord_min_real @ A2 @ B2 ) )
& ( A2 != B2 ) ) ) ) ).
% min.strict_order_iff
thf(fact_44_min_Ostrict__order__iff,axiom,
( ord_less_nat
= ( ^ [A2: nat,B2: nat] :
( ( A2
= ( ord_min_nat @ A2 @ B2 ) )
& ( A2 != B2 ) ) ) ) ).
% min.strict_order_iff
thf(fact_45_min_Ostrict__boundedE,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ ( ord_min_real @ B @ C ) )
=> ~ ( ( ord_less_real @ A @ B )
=> ~ ( ord_less_real @ A @ C ) ) ) ).
% min.strict_boundedE
thf(fact_46_min_Ostrict__boundedE,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ ( ord_min_nat @ B @ C ) )
=> ~ ( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ A @ C ) ) ) ).
% min.strict_boundedE
thf(fact_47_min__less__iff__disj,axiom,
! [X2: real,Y: real,Z: real] :
( ( ord_less_real @ ( ord_min_real @ X2 @ Y ) @ Z )
= ( ( ord_less_real @ X2 @ Z )
| ( ord_less_real @ Y @ Z ) ) ) ).
% min_less_iff_disj
thf(fact_48_min__less__iff__disj,axiom,
! [X2: nat,Y: nat,Z: nat] :
( ( ord_less_nat @ ( ord_min_nat @ X2 @ Y ) @ Z )
= ( ( ord_less_nat @ X2 @ Z )
| ( ord_less_nat @ Y @ Z ) ) ) ).
% min_less_iff_disj
thf(fact_49_is__zero__null,axiom,
( is_zero_real
= ( ^ [P2: poly_real] : P2 = zero_zero_poly_real ) ) ).
% is_zero_null
thf(fact_50_is__zero__null,axiom,
( is_zero_nat
= ( ^ [P2: poly_nat] : P2 = zero_zero_poly_nat ) ) ).
% is_zero_null
thf(fact_51_is__zero__null,axiom,
( is_zero_poly_real
= ( ^ [P2: poly_poly_real] : P2 = zero_z1423781445y_real ) ) ).
% is_zero_null
thf(fact_52_min_Oleft__commute,axiom,
! [B: real,A: real,C: real] :
( ( ord_min_real @ B @ ( ord_min_real @ A @ C ) )
= ( ord_min_real @ A @ ( ord_min_real @ B @ C ) ) ) ).
% min.left_commute
thf(fact_53_min_Oleft__commute,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_min_nat @ B @ ( ord_min_nat @ A @ C ) )
= ( ord_min_nat @ A @ ( ord_min_nat @ B @ C ) ) ) ).
% min.left_commute
thf(fact_54_min_Ocommute,axiom,
( ord_min_real
= ( ^ [A2: real,B2: real] : ( ord_min_real @ B2 @ A2 ) ) ) ).
% min.commute
thf(fact_55_min_Ocommute,axiom,
( ord_min_nat
= ( ^ [A2: nat,B2: nat] : ( ord_min_nat @ B2 @ A2 ) ) ) ).
% min.commute
thf(fact_56_min_Oassoc,axiom,
! [A: real,B: real,C: real] :
( ( ord_min_real @ ( ord_min_real @ A @ B ) @ C )
= ( ord_min_real @ A @ ( ord_min_real @ B @ C ) ) ) ).
% min.assoc
thf(fact_57_min_Oassoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_min_nat @ ( ord_min_nat @ A @ B ) @ C )
= ( ord_min_nat @ A @ ( ord_min_nat @ B @ C ) ) ) ).
% min.assoc
thf(fact_58_poly__cutoff__0,axiom,
! [N: nat] :
( ( poly_cutoff_real @ N @ zero_zero_poly_real )
= zero_zero_poly_real ) ).
% poly_cutoff_0
thf(fact_59_poly__cutoff__0,axiom,
! [N: nat] :
( ( poly_cutoff_nat @ N @ zero_zero_poly_nat )
= zero_zero_poly_nat ) ).
% poly_cutoff_0
thf(fact_60_poly__cutoff__0,axiom,
! [N: nat] :
( ( poly_c1404107022y_real @ N @ zero_z1423781445y_real )
= zero_z1423781445y_real ) ).
% poly_cutoff_0
thf(fact_61_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 ) )
=> ( ( poly_real2 @ P @ Z2 )
!= zero_zero_real ) ) ) ) ).
% last_non_root_interval
thf(fact_62_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 ) )
=> ( ( poly_real2 @ P @ Z2 )
!= zero_zero_real ) ) ) ) ).
% next_non_root_interval
thf(fact_63_reflect__poly__at__0__eq__0__iff,axiom,
! [P: poly_real] :
( ( ( poly_real2 @ ( reflect_poly_real @ P ) @ zero_zero_real )
= zero_zero_real )
= ( P = zero_zero_poly_real ) ) ).
% reflect_poly_at_0_eq_0_iff
thf(fact_64_reflect__poly__at__0__eq__0__iff,axiom,
! [P: poly_poly_real] :
( ( ( poly_poly_real2 @ ( reflec1522834046y_real @ P ) @ zero_zero_poly_real )
= zero_zero_poly_real )
= ( P = zero_z1423781445y_real ) ) ).
% reflect_poly_at_0_eq_0_iff
thf(fact_65_reflect__poly__at__0__eq__0__iff,axiom,
! [P: poly_nat] :
( ( ( poly_nat2 @ ( reflect_poly_nat @ P ) @ zero_zero_nat )
= zero_zero_nat )
= ( P = zero_zero_poly_nat ) ) ).
% reflect_poly_at_0_eq_0_iff
thf(fact_66_reflect__poly__at__0__eq__0__iff,axiom,
! [P: poly_poly_nat] :
( ( ( poly_poly_nat2 @ ( reflec781175074ly_nat @ P ) @ zero_zero_poly_nat )
= zero_zero_poly_nat )
= ( P = zero_z1059985641ly_nat ) ) ).
% reflect_poly_at_0_eq_0_iff
thf(fact_67_reflect__poly__at__0__eq__0__iff,axiom,
! [P: poly_poly_poly_real] :
( ( ( poly_poly_poly_real2 @ ( reflec144234502y_real @ P ) @ zero_z1423781445y_real )
= zero_z1423781445y_real )
= ( P = zero_z935034829y_real ) ) ).
% reflect_poly_at_0_eq_0_iff
thf(fact_68_lb2,axiom,
! [X: real] :
( ( ord_less_eq_real @ X @ lb2 )
=> ( ( sgn_sgn_real @ ( poly_real2 @ p @ X ) )
= ( sturm_1076696862f_real @ p ) ) ) ).
% lb2
thf(fact_69_poly__shift__0,axiom,
! [N: nat] :
( ( poly_shift_real @ N @ zero_zero_poly_real )
= zero_zero_poly_real ) ).
% poly_shift_0
thf(fact_70_poly__shift__0,axiom,
! [N: nat] :
( ( poly_shift_nat @ N @ zero_zero_poly_nat )
= zero_zero_poly_nat ) ).
% poly_shift_0
thf(fact_71_poly__shift__0,axiom,
! [N: nat] :
( ( poly_shift_poly_real @ N @ zero_z1423781445y_real )
= zero_z1423781445y_real ) ).
% poly_shift_0
thf(fact_72_order__root,axiom,
! [P: poly_real,A: real] :
( ( ( poly_real2 @ P @ A )
= zero_zero_real )
= ( ( P = zero_zero_poly_real )
| ( ( order_real @ A @ P )
!= zero_zero_nat ) ) ) ).
% order_root
thf(fact_73_order__root,axiom,
! [P: poly_poly_real,A: poly_real] :
( ( ( poly_poly_real2 @ P @ A )
= zero_zero_poly_real )
= ( ( P = zero_z1423781445y_real )
| ( ( order_poly_real @ A @ P )
!= zero_zero_nat ) ) ) ).
% order_root
thf(fact_74_order__root,axiom,
! [P: poly_poly_poly_real,A: poly_poly_real] :
( ( ( poly_poly_poly_real2 @ P @ A )
= zero_z1423781445y_real )
= ( ( P = zero_z935034829y_real )
| ( ( order_poly_poly_real @ A @ P )
!= zero_zero_nat ) ) ) ).
% order_root
thf(fact_75_divide__poly__main__0,axiom,
! [R: poly_real,D: poly_real,Dr: nat,N: nat] :
( ( divide1561404011n_real @ zero_zero_real @ zero_zero_poly_real @ R @ D @ Dr @ N )
= zero_zero_poly_real ) ).
% divide_poly_main_0
thf(fact_76_divide__poly__main__0,axiom,
! [R: poly_poly_real,D: poly_poly_real,Dr: nat,N: nat] :
( ( divide1142363123y_real @ zero_zero_poly_real @ zero_z1423781445y_real @ R @ D @ Dr @ N )
= zero_z1423781445y_real ) ).
% divide_poly_main_0
thf(fact_77_divide__poly__main__0,axiom,
! [R: poly_poly_poly_real,D: poly_poly_poly_real,Dr: nat,N: nat] :
( ( divide924636027y_real @ zero_z1423781445y_real @ zero_z935034829y_real @ R @ D @ Dr @ N )
= zero_z935034829y_real ) ).
% divide_poly_main_0
thf(fact_78_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 ) )
=> ( ( poly_real2 @ P @ Z3 )
!= zero_zero_real ) ) )
= ( ! [Z3: real] :
( ( ( ord_less_real @ Lb2 @ Z3 )
& ( ord_less_eq_real @ Z3 @ Ub ) )
=> ( ord_less_real @ zero_zero_real @ ( poly_real2 @ P @ Z3 ) ) )
| ! [Z3: real] :
( ( ( ord_less_real @ Lb2 @ Z3 )
& ( ord_less_eq_real @ Z3 @ Ub ) )
=> ( ord_less_real @ ( poly_real2 @ P @ Z3 ) @ zero_zero_real ) ) ) ) ).
% not_eq_pos_or_neg_iff_1
thf(fact_79__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 @ ( poly_real2 @ p @ X ) )
= ( sturm_1076696862f_real @ p ) ) ) ).
% \<open>\<And>thesis. (\<And>lb2. \<forall>x\<le>lb2. sgn (poly p x) = sgn_neg_inf p \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_80_le__zero__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_81_min_Obounded__iff,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ ( ord_min_real @ B @ C ) )
= ( ( ord_less_eq_real @ A @ B )
& ( ord_less_eq_real @ A @ C ) ) ) ).
% min.bounded_iff
thf(fact_82_min_Obounded__iff,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( ord_min_nat @ B @ C ) )
= ( ( ord_less_eq_nat @ A @ B )
& ( ord_less_eq_nat @ A @ C ) ) ) ).
% min.bounded_iff
thf(fact_83_reflect__poly__0,axiom,
( ( reflect_poly_real @ zero_zero_poly_real )
= zero_zero_poly_real ) ).
% reflect_poly_0
thf(fact_84_reflect__poly__0,axiom,
( ( reflect_poly_nat @ zero_zero_poly_nat )
= zero_zero_poly_nat ) ).
% reflect_poly_0
thf(fact_85_reflect__poly__0,axiom,
( ( reflec1522834046y_real @ zero_z1423781445y_real )
= zero_z1423781445y_real ) ).
% reflect_poly_0
thf(fact_86_mem__Collect__eq,axiom,
! [A: real,P3: real > $o] :
( ( member_real @ A @ ( collect_real @ P3 ) )
= ( P3 @ A ) ) ).
% mem_Collect_eq
thf(fact_87_Collect__mem__eq,axiom,
! [A3: set_real] :
( ( collect_real
@ ^ [X4: real] : ( member_real @ X4 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_88_that,axiom,
! [Lb2: real] :
( ! [X3: real] :
( ( ( poly_real2 @ p @ X3 )
= zero_zero_real )
=> ( ord_less_real @ Lb2 @ X3 ) )
=> ( ! [X3: real] :
( ( ord_less_eq_real @ X3 @ Lb2 )
=> ( ( sgn_sgn_real @ ( poly_real2 @ p @ X3 ) )
= ( sturm_1076696862f_real @ p ) ) )
=> thesis ) ) ).
% that
thf(fact_89_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 ) ) ) ) ) ).
% complete_real
thf(fact_90_zero__le,axiom,
! [X2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X2 ) ).
% zero_le
thf(fact_91_le__numeral__extra_I3_J,axiom,
ord_le1180086932y_real @ zero_zero_poly_real @ zero_zero_poly_real ).
% le_numeral_extra(3)
thf(fact_92_le__numeral__extra_I3_J,axiom,
ord_le893774876y_real @ zero_z1423781445y_real @ zero_z1423781445y_real ).
% le_numeral_extra(3)
thf(fact_93_le__numeral__extra_I3_J,axiom,
ord_less_eq_real @ zero_zero_real @ zero_zero_real ).
% le_numeral_extra(3)
thf(fact_94_le__numeral__extra_I3_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% le_numeral_extra(3)
thf(fact_95_less__eq__real__def,axiom,
( ord_less_eq_real
= ( ^ [X4: real,Y3: real] :
( ( ord_less_real @ X4 @ Y3 )
| ( X4 = Y3 ) ) ) ) ).
% less_eq_real_def
thf(fact_96_min_Omono,axiom,
! [A: real,C: real,B: real,D: real] :
( ( ord_less_eq_real @ A @ C )
=> ( ( ord_less_eq_real @ B @ D )
=> ( ord_less_eq_real @ ( ord_min_real @ A @ B ) @ ( ord_min_real @ C @ D ) ) ) ) ).
% min.mono
thf(fact_97_min_Omono,axiom,
! [A: nat,C: nat,B: nat,D: nat] :
( ( ord_less_eq_nat @ A @ C )
=> ( ( ord_less_eq_nat @ B @ D )
=> ( ord_less_eq_nat @ ( ord_min_nat @ A @ B ) @ ( ord_min_nat @ C @ D ) ) ) ) ).
% min.mono
thf(fact_98_min_OorderE,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ B )
=> ( A
= ( ord_min_real @ A @ B ) ) ) ).
% min.orderE
thf(fact_99_min_OorderE,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( A
= ( ord_min_nat @ A @ B ) ) ) ).
% min.orderE
thf(fact_100_min_OorderI,axiom,
! [A: real,B: real] :
( ( A
= ( ord_min_real @ A @ B ) )
=> ( ord_less_eq_real @ A @ B ) ) ).
% min.orderI
thf(fact_101_min_OorderI,axiom,
! [A: nat,B: nat] :
( ( A
= ( ord_min_nat @ A @ B ) )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% min.orderI
thf(fact_102_min_Oabsorb1,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_min_real @ A @ B )
= A ) ) ).
% min.absorb1
thf(fact_103_min_Oabsorb1,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_min_nat @ A @ B )
= A ) ) ).
% min.absorb1
thf(fact_104_min_Oabsorb2,axiom,
! [B: real,A: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ( ord_min_real @ A @ B )
= B ) ) ).
% min.absorb2
thf(fact_105_min_Oabsorb2,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_min_nat @ A @ B )
= B ) ) ).
% min.absorb2
thf(fact_106_min_OboundedE,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ ( ord_min_real @ B @ C ) )
=> ~ ( ( ord_less_eq_real @ A @ B )
=> ~ ( ord_less_eq_real @ A @ C ) ) ) ).
% min.boundedE
thf(fact_107_min_OboundedE,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( ord_min_nat @ B @ C ) )
=> ~ ( ( ord_less_eq_nat @ A @ B )
=> ~ ( ord_less_eq_nat @ A @ C ) ) ) ).
% min.boundedE
thf(fact_108_min_OboundedI,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ A @ C )
=> ( ord_less_eq_real @ A @ ( ord_min_real @ B @ C ) ) ) ) ).
% min.boundedI
thf(fact_109_min_OboundedI,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ A @ C )
=> ( ord_less_eq_nat @ A @ ( ord_min_nat @ B @ C ) ) ) ) ).
% min.boundedI
thf(fact_110_min_Oorder__iff,axiom,
( ord_less_eq_real
= ( ^ [A2: real,B2: real] :
( A2
= ( ord_min_real @ A2 @ B2 ) ) ) ) ).
% min.order_iff
thf(fact_111_min_Oorder__iff,axiom,
( ord_less_eq_nat
= ( ^ [A2: nat,B2: nat] :
( A2
= ( ord_min_nat @ A2 @ B2 ) ) ) ) ).
% min.order_iff
thf(fact_112_min_Ocobounded1,axiom,
! [A: real,B: real] : ( ord_less_eq_real @ ( ord_min_real @ A @ B ) @ A ) ).
% min.cobounded1
thf(fact_113_min_Ocobounded1,axiom,
! [A: nat,B: nat] : ( ord_less_eq_nat @ ( ord_min_nat @ A @ B ) @ A ) ).
% min.cobounded1
thf(fact_114_min_Ocobounded2,axiom,
! [A: real,B: real] : ( ord_less_eq_real @ ( ord_min_real @ A @ B ) @ B ) ).
% min.cobounded2
thf(fact_115_min_Ocobounded2,axiom,
! [A: nat,B: nat] : ( ord_less_eq_nat @ ( ord_min_nat @ A @ B ) @ B ) ).
% min.cobounded2
thf(fact_116_min_Oabsorb__iff1,axiom,
( ord_less_eq_real
= ( ^ [A2: real,B2: real] :
( ( ord_min_real @ A2 @ B2 )
= A2 ) ) ) ).
% min.absorb_iff1
thf(fact_117_min_Oabsorb__iff1,axiom,
( ord_less_eq_nat
= ( ^ [A2: nat,B2: nat] :
( ( ord_min_nat @ A2 @ B2 )
= A2 ) ) ) ).
% min.absorb_iff1
thf(fact_118_min_Oabsorb__iff2,axiom,
( ord_less_eq_real
= ( ^ [B2: real,A2: real] :
( ( ord_min_real @ A2 @ B2 )
= B2 ) ) ) ).
% min.absorb_iff2
thf(fact_119_min_Oabsorb__iff2,axiom,
( ord_less_eq_nat
= ( ^ [B2: nat,A2: nat] :
( ( ord_min_nat @ A2 @ B2 )
= B2 ) ) ) ).
% min.absorb_iff2
thf(fact_120_min_OcoboundedI1,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_eq_real @ A @ C )
=> ( ord_less_eq_real @ ( ord_min_real @ A @ B ) @ C ) ) ).
% min.coboundedI1
thf(fact_121_min_OcoboundedI1,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ A @ C )
=> ( ord_less_eq_nat @ ( ord_min_nat @ A @ B ) @ C ) ) ).
% min.coboundedI1
thf(fact_122_min_OcoboundedI2,axiom,
! [B: real,C: real,A: real] :
( ( ord_less_eq_real @ B @ C )
=> ( ord_less_eq_real @ ( ord_min_real @ A @ B ) @ C ) ) ).
% min.coboundedI2
thf(fact_123_min_OcoboundedI2,axiom,
! [B: nat,C: nat,A: nat] :
( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ ( ord_min_nat @ A @ B ) @ C ) ) ).
% min.coboundedI2
thf(fact_124_min__le__iff__disj,axiom,
! [X2: real,Y: real,Z: real] :
( ( ord_less_eq_real @ ( ord_min_real @ X2 @ Y ) @ Z )
= ( ( ord_less_eq_real @ X2 @ Z )
| ( ord_less_eq_real @ Y @ Z ) ) ) ).
% min_le_iff_disj
thf(fact_125_min__le__iff__disj,axiom,
! [X2: nat,Y: nat,Z: nat] :
( ( ord_less_eq_nat @ ( ord_min_nat @ X2 @ Y ) @ Z )
= ( ( ord_less_eq_nat @ X2 @ Z )
| ( ord_less_eq_nat @ Y @ Z ) ) ) ).
% min_le_iff_disj
thf(fact_126_order__0I,axiom,
! [P: poly_real,A: real] :
( ( ( poly_real2 @ P @ A )
!= zero_zero_real )
=> ( ( order_real @ A @ P )
= zero_zero_nat ) ) ).
% order_0I
thf(fact_127_order__0I,axiom,
! [P: poly_poly_real,A: poly_real] :
( ( ( poly_poly_real2 @ P @ A )
!= zero_zero_poly_real )
=> ( ( order_poly_real @ A @ P )
= zero_zero_nat ) ) ).
% order_0I
thf(fact_128_order__0I,axiom,
! [P: poly_poly_poly_real,A: poly_poly_real] :
( ( ( poly_poly_poly_real2 @ P @ A )
!= zero_z1423781445y_real )
=> ( ( order_poly_poly_real @ A @ P )
= zero_zero_nat ) ) ).
% order_0I
thf(fact_129_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 ) )
=> ( ( poly_real2 @ P @ Z3 )
!= zero_zero_real ) ) )
= ( ! [Z3: real] :
( ( ( ord_less_eq_real @ Lb2 @ Z3 )
& ( ord_less_real @ Z3 @ Ub ) )
=> ( ord_less_real @ zero_zero_real @ ( poly_real2 @ P @ Z3 ) ) )
| ! [Z3: real] :
( ( ( ord_less_eq_real @ Lb2 @ Z3 )
& ( ord_less_real @ Z3 @ Ub ) )
=> ( ord_less_real @ ( poly_real2 @ P @ Z3 ) @ zero_zero_real ) ) ) ) ).
% not_eq_pos_or_neg_iff_2
thf(fact_130_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 ) ) ).
% zero_le_sgn_iff
thf(fact_131_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 ) ) ).
% sgn_le_0_iff
thf(fact_132_sgn__greater,axiom,
! [A: poly_real] :
( ( ord_less_poly_real @ zero_zero_poly_real @ ( sgn_sgn_poly_real @ A ) )
= ( ord_less_poly_real @ zero_zero_poly_real @ A ) ) ).
% sgn_greater
thf(fact_133_sgn__greater,axiom,
! [A: poly_poly_real] :
( ( ord_le38482960y_real @ zero_z1423781445y_real @ ( sgn_sg2128174761y_real @ A ) )
= ( ord_le38482960y_real @ zero_z1423781445y_real @ A ) ) ).
% sgn_greater
thf(fact_134_sgn__greater,axiom,
! [A: real] :
( ( ord_less_real @ zero_zero_real @ ( sgn_sgn_real @ A ) )
= ( ord_less_real @ zero_zero_real @ A ) ) ).
% sgn_greater
thf(fact_135_sgn__less,axiom,
! [A: poly_real] :
( ( ord_less_poly_real @ ( sgn_sgn_poly_real @ A ) @ zero_zero_poly_real )
= ( ord_less_poly_real @ A @ zero_zero_poly_real ) ) ).
% sgn_less
thf(fact_136_sgn__less,axiom,
! [A: poly_poly_real] :
( ( ord_le38482960y_real @ ( sgn_sg2128174761y_real @ A ) @ zero_z1423781445y_real )
= ( ord_le38482960y_real @ A @ zero_z1423781445y_real ) ) ).
% sgn_less
thf(fact_137_sgn__less,axiom,
! [A: real] :
( ( ord_less_real @ ( sgn_sgn_real @ A ) @ zero_zero_real )
= ( ord_less_real @ A @ zero_zero_real ) ) ).
% sgn_less
thf(fact_138_root__ub,axiom,
! [P: poly_real] :
( ( P != zero_zero_poly_real )
=> ~ ! [Ub2: real] :
( ! [X: real] :
( ( ( poly_real2 @ P @ X )
= zero_zero_real )
=> ( ord_less_real @ X @ Ub2 ) )
=> ~ ! [X: real] :
( ( ord_less_eq_real @ Ub2 @ X )
=> ( ( sgn_sgn_real @ ( poly_real2 @ P @ X ) )
= ( sturm_1308388506f_real @ P ) ) ) ) ) ).
% root_ub
thf(fact_139_sgn__0,axiom,
( ( sgn_sgn_real @ zero_zero_real )
= zero_zero_real ) ).
% sgn_0
thf(fact_140_sgn__0,axiom,
( ( sgn_sgn_poly_real @ zero_zero_poly_real )
= zero_zero_poly_real ) ).
% sgn_0
thf(fact_141_sgn__0,axiom,
( ( sgn_sg2128174761y_real @ zero_z1423781445y_real )
= zero_z1423781445y_real ) ).
% sgn_0
thf(fact_142_sgn__zero,axiom,
( ( sgn_sgn_real @ zero_zero_real )
= zero_zero_real ) ).
% sgn_zero
thf(fact_143_sgn__sgn,axiom,
! [A: real] :
( ( sgn_sgn_real @ ( sgn_sgn_real @ A ) )
= ( sgn_sgn_real @ A ) ) ).
% sgn_sgn
thf(fact_144_linorder__neqE__linordered__idom,axiom,
! [X2: real,Y: real] :
( ( X2 != Y )
=> ( ~ ( ord_less_real @ X2 @ Y )
=> ( ord_less_real @ Y @ X2 ) ) ) ).
% linorder_neqE_linordered_idom
thf(fact_145_sgn__eq__0__iff,axiom,
! [A: real] :
( ( ( sgn_sgn_real @ A )
= zero_zero_real )
= ( A = zero_zero_real ) ) ).
% sgn_eq_0_iff
thf(fact_146_sgn__eq__0__iff,axiom,
! [A: poly_real] :
( ( ( sgn_sgn_poly_real @ A )
= zero_zero_poly_real )
= ( A = zero_zero_poly_real ) ) ).
% sgn_eq_0_iff
thf(fact_147_sgn__eq__0__iff,axiom,
! [A: poly_poly_real] :
( ( ( sgn_sg2128174761y_real @ A )
= zero_z1423781445y_real )
= ( A = zero_z1423781445y_real ) ) ).
% sgn_eq_0_iff
thf(fact_148_sgn__0__0,axiom,
! [A: real] :
( ( ( sgn_sgn_real @ A )
= zero_zero_real )
= ( A = zero_zero_real ) ) ).
% sgn_0_0
thf(fact_149_sgn__0__0,axiom,
! [A: poly_real] :
( ( ( sgn_sgn_poly_real @ A )
= zero_zero_poly_real )
= ( A = zero_zero_poly_real ) ) ).
% sgn_0_0
thf(fact_150_sgn__0__0,axiom,
! [A: poly_poly_real] :
( ( ( sgn_sg2128174761y_real @ A )
= zero_z1423781445y_real )
= ( A = zero_z1423781445y_real ) ) ).
% sgn_0_0
thf(fact_151_sgn__zero__iff,axiom,
! [X2: real] :
( ( ( sgn_sgn_real @ X2 )
= zero_zero_real )
= ( X2 = zero_zero_real ) ) ).
% sgn_zero_iff
thf(fact_152_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_153_bot__nat__0_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% bot_nat_0.extremum
thf(fact_154_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% neq0_conv
thf(fact_155_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_156_bot__nat__0_Onot__eq__extremum,axiom,
! [A: nat] :
( ( A != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ A ) ) ).
% bot_nat_0.not_eq_extremum
thf(fact_157_min__0R,axiom,
! [N: nat] :
( ( ord_min_nat @ N @ zero_zero_nat )
= zero_zero_nat ) ).
% min_0R
thf(fact_158_min__0L,axiom,
! [N: nat] :
( ( ord_min_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% min_0L
thf(fact_159_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_160_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 ) ) ) ).
% le_trans
thf(fact_161_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% eq_imp_le
thf(fact_162_le__antisym,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( M = N ) ) ) ).
% le_antisym
thf(fact_163_nat__le__linear,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
| ( ord_less_eq_nat @ N @ M ) ) ).
% nat_le_linear
thf(fact_164_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 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_165_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 ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_166_le__neq__implies__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( M != N )
=> ( ord_less_nat @ M @ N ) ) ) ).
% le_neq_implies_less
thf(fact_167_less__or__eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( ( ord_less_nat @ M @ N )
| ( M = N ) )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% less_or_eq_imp_le
thf(fact_168_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ N2 )
| ( M2 = N2 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_169_less__imp__le__nat,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% less_imp_le_nat
thf(fact_170_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
& ( M2 != N2 ) ) ) ) ).
% nat_less_le
thf(fact_171_linorder__neqE__nat,axiom,
! [X2: nat,Y: nat] :
( ( X2 != Y )
=> ( ~ ( ord_less_nat @ X2 @ Y )
=> ( ord_less_nat @ Y @ X2 ) ) ) ).
% linorder_neqE_nat
thf(fact_172_infinite__descent,axiom,
! [P3: nat > $o,N: nat] :
( ! [N3: nat] :
( ~ ( P3 @ N3 )
=> ? [M3: nat] :
( ( ord_less_nat @ M3 @ N3 )
& ~ ( P3 @ M3 ) ) )
=> ( P3 @ N ) ) ).
% infinite_descent
thf(fact_173_nat__less__induct,axiom,
! [P3: nat > $o,N: nat] :
( ! [N3: nat] :
( ! [M3: nat] :
( ( ord_less_nat @ M3 @ N3 )
=> ( P3 @ M3 ) )
=> ( P3 @ N3 ) )
=> ( P3 @ N ) ) ).
% nat_less_induct
thf(fact_174_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_175_less__not__refl3,axiom,
! [S2: nat,T: nat] :
( ( ord_less_nat @ S2 @ T )
=> ( S2 != T ) ) ).
% less_not_refl3
thf(fact_176_less__not__refl2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
=> ( M != N ) ) ).
% less_not_refl2
thf(fact_177_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_not_refl
thf(fact_178_nat__neq__iff,axiom,
! [M: nat,N: nat] :
( ( M != N )
= ( ( ord_less_nat @ M @ N )
| ( ord_less_nat @ N @ M ) ) ) ).
% nat_neq_iff
thf(fact_179_bot__nat__0_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_180_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 ) ) ) ).
% infinite_descent0
thf(fact_181_gr__implies__not0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_182_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_zeroE
thf(fact_183_not__less0,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less0
thf(fact_184_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr0
thf(fact_185_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr0I
thf(fact_186_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_187_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_188_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 ) ) ) ) ).
% ex_least_nat_le
thf(fact_189_bot__nat__0_Oextremum__unique,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
= ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_unique
thf(fact_190_bot__nat__0_Oextremum__uniqueI,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_uniqueI
thf(fact_191_order__refl,axiom,
! [X2: real] : ( ord_less_eq_real @ X2 @ X2 ) ).
% order_refl
thf(fact_192_order__refl,axiom,
! [X2: nat] : ( ord_less_eq_nat @ X2 @ X2 ) ).
% order_refl
thf(fact_193_min__absorb1,axiom,
! [X2: real,Y: real] :
( ( ord_less_eq_real @ X2 @ Y )
=> ( ( ord_min_real @ X2 @ Y )
= X2 ) ) ).
% min_absorb1
thf(fact_194_min__absorb1,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
=> ( ( ord_min_nat @ X2 @ Y )
= X2 ) ) ).
% min_absorb1
thf(fact_195_min__absorb2,axiom,
! [Y: real,X2: real] :
( ( ord_less_eq_real @ Y @ X2 )
=> ( ( ord_min_real @ X2 @ Y )
= Y ) ) ).
% min_absorb2
thf(fact_196_min__absorb2,axiom,
! [Y: nat,X2: nat] :
( ( ord_less_eq_nat @ Y @ X2 )
=> ( ( ord_min_nat @ X2 @ Y )
= Y ) ) ).
% min_absorb2
thf(fact_197_min__def,axiom,
( ord_min_real
= ( ^ [A2: real,B2: real] : ( if_real @ ( ord_less_eq_real @ A2 @ B2 ) @ A2 @ B2 ) ) ) ).
% min_def
thf(fact_198_min__def,axiom,
( ord_min_nat
= ( ^ [A2: nat,B2: nat] : ( if_nat @ ( ord_less_eq_nat @ A2 @ B2 ) @ A2 @ B2 ) ) ) ).
% min_def
thf(fact_199_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 ) ) ) ).
% nat_descend_induct
thf(fact_200_dual__order_Oantisym,axiom,
! [B: real,A: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ( ord_less_eq_real @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_201_dual__order_Oantisym,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_202_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: real,Z4: real] : Y5 = Z4 )
= ( ^ [A2: real,B2: real] :
( ( ord_less_eq_real @ B2 @ A2 )
& ( ord_less_eq_real @ A2 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_203_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: nat,Z4: nat] : Y5 = Z4 )
= ( ^ [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
& ( ord_less_eq_nat @ A2 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_204_dual__order_Otrans,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ( ord_less_eq_real @ C @ B )
=> ( ord_less_eq_real @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_205_dual__order_Otrans,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_eq_nat @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_206_linorder__wlog,axiom,
! [P3: real > real > $o,A: real,B: real] :
( ! [A4: real,B3: real] :
( ( ord_less_eq_real @ A4 @ B3 )
=> ( P3 @ A4 @ B3 ) )
=> ( ! [A4: real,B3: real] :
( ( P3 @ B3 @ A4 )
=> ( P3 @ A4 @ B3 ) )
=> ( P3 @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_207_linorder__wlog,axiom,
! [P3: nat > nat > $o,A: nat,B: nat] :
( ! [A4: nat,B3: nat] :
( ( ord_less_eq_nat @ A4 @ B3 )
=> ( P3 @ A4 @ B3 ) )
=> ( ! [A4: nat,B3: nat] :
( ( P3 @ B3 @ A4 )
=> ( P3 @ A4 @ B3 ) )
=> ( P3 @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_208_dual__order_Orefl,axiom,
! [A: real] : ( ord_less_eq_real @ A @ A ) ).
% dual_order.refl
thf(fact_209_dual__order_Orefl,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% dual_order.refl
thf(fact_210_order__trans,axiom,
! [X2: real,Y: real,Z: real] :
( ( ord_less_eq_real @ X2 @ Y )
=> ( ( ord_less_eq_real @ Y @ Z )
=> ( ord_less_eq_real @ X2 @ Z ) ) ) ).
% order_trans
thf(fact_211_order__trans,axiom,
! [X2: nat,Y: nat,Z: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z )
=> ( ord_less_eq_nat @ X2 @ Z ) ) ) ).
% order_trans
thf(fact_212_order__class_Oorder_Oantisym,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ B @ A )
=> ( A = B ) ) ) ).
% order_class.order.antisym
thf(fact_213_order__class_Oorder_Oantisym,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ A )
=> ( A = B ) ) ) ).
% order_class.order.antisym
thf(fact_214_ord__le__eq__trans,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_real @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_215_ord__le__eq__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_216_ord__eq__le__trans,axiom,
! [A: real,B: real,C: real] :
( ( A = B )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ord_less_eq_real @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_217_ord__eq__le__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( A = B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_218_order__class_Oorder_Oeq__iff,axiom,
( ( ^ [Y5: real,Z4: real] : Y5 = Z4 )
= ( ^ [A2: real,B2: real] :
( ( ord_less_eq_real @ A2 @ B2 )
& ( ord_less_eq_real @ B2 @ A2 ) ) ) ) ).
% order_class.order.eq_iff
thf(fact_219_order__class_Oorder_Oeq__iff,axiom,
( ( ^ [Y5: nat,Z4: nat] : Y5 = Z4 )
= ( ^ [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
& ( ord_less_eq_nat @ B2 @ A2 ) ) ) ) ).
% order_class.order.eq_iff
thf(fact_220_antisym__conv,axiom,
! [Y: real,X2: real] :
( ( ord_less_eq_real @ Y @ X2 )
=> ( ( ord_less_eq_real @ X2 @ Y )
= ( X2 = Y ) ) ) ).
% antisym_conv
thf(fact_221_antisym__conv,axiom,
! [Y: nat,X2: nat] :
( ( ord_less_eq_nat @ Y @ X2 )
=> ( ( ord_less_eq_nat @ X2 @ Y )
= ( X2 = Y ) ) ) ).
% antisym_conv
thf(fact_222_le__cases3,axiom,
! [X2: real,Y: real,Z: real] :
( ( ( ord_less_eq_real @ X2 @ Y )
=> ~ ( ord_less_eq_real @ Y @ Z ) )
=> ( ( ( ord_less_eq_real @ Y @ X2 )
=> ~ ( ord_less_eq_real @ X2 @ Z ) )
=> ( ( ( ord_less_eq_real @ X2 @ Z )
=> ~ ( ord_less_eq_real @ Z @ Y ) )
=> ( ( ( ord_less_eq_real @ Z @ Y )
=> ~ ( ord_less_eq_real @ Y @ X2 ) )
=> ( ( ( ord_less_eq_real @ Y @ Z )
=> ~ ( ord_less_eq_real @ Z @ X2 ) )
=> ~ ( ( ord_less_eq_real @ Z @ X2 )
=> ~ ( ord_less_eq_real @ X2 @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_223_le__cases3,axiom,
! [X2: nat,Y: nat,Z: nat] :
( ( ( ord_less_eq_nat @ X2 @ Y )
=> ~ ( ord_less_eq_nat @ Y @ Z ) )
=> ( ( ( ord_less_eq_nat @ Y @ X2 )
=> ~ ( ord_less_eq_nat @ X2 @ Z ) )
=> ( ( ( ord_less_eq_nat @ X2 @ Z )
=> ~ ( ord_less_eq_nat @ Z @ Y ) )
=> ( ( ( ord_less_eq_nat @ Z @ Y )
=> ~ ( ord_less_eq_nat @ Y @ X2 ) )
=> ( ( ( ord_less_eq_nat @ Y @ Z )
=> ~ ( ord_less_eq_nat @ Z @ X2 ) )
=> ~ ( ( ord_less_eq_nat @ Z @ X2 )
=> ~ ( ord_less_eq_nat @ X2 @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_224_order_Otrans,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ord_less_eq_real @ A @ C ) ) ) ).
% order.trans
thf(fact_225_order_Otrans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% order.trans
thf(fact_226_le__cases,axiom,
! [X2: real,Y: real] :
( ~ ( ord_less_eq_real @ X2 @ Y )
=> ( ord_less_eq_real @ Y @ X2 ) ) ).
% le_cases
thf(fact_227_le__cases,axiom,
! [X2: nat,Y: nat] :
( ~ ( ord_less_eq_nat @ X2 @ Y )
=> ( ord_less_eq_nat @ Y @ X2 ) ) ).
% le_cases
thf(fact_228_eq__refl,axiom,
! [X2: real,Y: real] :
( ( X2 = Y )
=> ( ord_less_eq_real @ X2 @ Y ) ) ).
% eq_refl
thf(fact_229_eq__refl,axiom,
! [X2: nat,Y: nat] :
( ( X2 = Y )
=> ( ord_less_eq_nat @ X2 @ Y ) ) ).
% eq_refl
thf(fact_230_linear,axiom,
! [X2: real,Y: real] :
( ( ord_less_eq_real @ X2 @ Y )
| ( ord_less_eq_real @ Y @ X2 ) ) ).
% linear
thf(fact_231_linear,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
| ( ord_less_eq_nat @ Y @ X2 ) ) ).
% linear
thf(fact_232_antisym,axiom,
! [X2: real,Y: real] :
( ( ord_less_eq_real @ X2 @ Y )
=> ( ( ord_less_eq_real @ Y @ X2 )
=> ( X2 = Y ) ) ) ).
% antisym
thf(fact_233_antisym,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
=> ( ( ord_less_eq_nat @ Y @ X2 )
=> ( X2 = Y ) ) ) ).
% antisym
thf(fact_234_eq__iff,axiom,
( ( ^ [Y5: real,Z4: real] : Y5 = Z4 )
= ( ^ [X4: real,Y3: real] :
( ( ord_less_eq_real @ X4 @ Y3 )
& ( ord_less_eq_real @ Y3 @ X4 ) ) ) ) ).
% eq_iff
thf(fact_235_eq__iff,axiom,
( ( ^ [Y5: nat,Z4: nat] : Y5 = Z4 )
= ( ^ [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
& ( ord_less_eq_nat @ Y3 @ X4 ) ) ) ) ).
% eq_iff
thf(fact_236_ord__le__eq__subst,axiom,
! [A: real,B: real,F: real > real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_237_ord__le__eq__subst,axiom,
! [A: real,B: real,F: real > nat,C: nat] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_238_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > real,C: real] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_239_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_240_ord__eq__le__subst,axiom,
! [A: real,F: real > real,B: real,C: real] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_241_ord__eq__le__subst,axiom,
! [A: nat,F: real > nat,B: real,C: real] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_242_ord__eq__le__subst,axiom,
! [A: real,F: nat > real,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_243_ord__eq__le__subst,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_244_order__subst2,axiom,
! [A: real,B: real,F: real > real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_245_order__subst2,axiom,
! [A: real,B: real,F: real > nat,C: nat] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_246_order__subst2,axiom,
! [A: nat,B: nat,F: nat > real,C: real] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_247_order__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_248_order__subst1,axiom,
! [A: real,F: real > real,B: real,C: real] :
( ( ord_less_eq_real @ A @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_249_order__subst1,axiom,
! [A: real,F: nat > real,B: nat,C: nat] :
( ( ord_less_eq_real @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_250_order__subst1,axiom,
! [A: nat,F: real > nat,B: real,C: real] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_251_order__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_252_ord__eq__less__subst,axiom,
! [A: real,F: real > real,B: real,C: real] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_real @ X3 @ Y2 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_253_ord__eq__less__subst,axiom,
! [A: nat,F: real > nat,B: real,C: real] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_real @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_254_ord__eq__less__subst,axiom,
! [A: real,F: nat > real,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_255_ord__eq__less__subst,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_256_ord__less__eq__subst,axiom,
! [A: real,B: real,F: real > real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_real @ X3 @ Y2 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_257_ord__less__eq__subst,axiom,
! [A: real,B: real,F: real > nat,C: nat] :
( ( ord_less_real @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_real @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_258_ord__less__eq__subst,axiom,
! [A: nat,B: nat,F: nat > real,C: real] :
( ( ord_less_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_259_ord__less__eq__subst,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_260_order__less__subst1,axiom,
! [A: real,F: real > real,B: real,C: real] :
( ( ord_less_real @ A @ ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_real @ X3 @ Y2 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_261_order__less__subst1,axiom,
! [A: real,F: nat > real,B: nat,C: nat] :
( ( ord_less_real @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_262_order__less__subst1,axiom,
! [A: nat,F: real > nat,B: real,C: real] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_real @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_263_order__less__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_264_order__less__subst2,axiom,
! [A: real,B: real,F: real > real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ ( F @ B ) @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_real @ X3 @ Y2 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_265_order__less__subst2,axiom,
! [A: real,B: real,F: real > nat,C: nat] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_real @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_266_order__less__subst2,axiom,
! [A: nat,B: nat,F: nat > real,C: real] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_real @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_267_order__less__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_268_lt__ex,axiom,
! [X2: real] :
? [Y2: real] : ( ord_less_real @ Y2 @ X2 ) ).
% lt_ex
thf(fact_269_gt__ex,axiom,
! [X2: real] :
? [X_1: real] : ( ord_less_real @ X2 @ X_1 ) ).
% gt_ex
thf(fact_270_gt__ex,axiom,
! [X2: nat] :
? [X_1: nat] : ( ord_less_nat @ X2 @ X_1 ) ).
% gt_ex
thf(fact_271_neqE,axiom,
! [X2: real,Y: real] :
( ( X2 != Y )
=> ( ~ ( ord_less_real @ X2 @ Y )
=> ( ord_less_real @ Y @ X2 ) ) ) ).
% neqE
thf(fact_272_neqE,axiom,
! [X2: nat,Y: nat] :
( ( X2 != Y )
=> ( ~ ( ord_less_nat @ X2 @ Y )
=> ( ord_less_nat @ Y @ X2 ) ) ) ).
% neqE
thf(fact_273_neq__iff,axiom,
! [X2: real,Y: real] :
( ( X2 != Y )
= ( ( ord_less_real @ X2 @ Y )
| ( ord_less_real @ Y @ X2 ) ) ) ).
% neq_iff
thf(fact_274_neq__iff,axiom,
! [X2: nat,Y: nat] :
( ( X2 != Y )
= ( ( ord_less_nat @ X2 @ Y )
| ( ord_less_nat @ Y @ X2 ) ) ) ).
% neq_iff
thf(fact_275_order_Oasym,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ~ ( ord_less_real @ B @ A ) ) ).
% order.asym
thf(fact_276_order_Oasym,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% order.asym
thf(fact_277_dense,axiom,
! [X2: real,Y: real] :
( ( ord_less_real @ X2 @ Y )
=> ? [Z5: real] :
( ( ord_less_real @ X2 @ Z5 )
& ( ord_less_real @ Z5 @ Y ) ) ) ).
% dense
thf(fact_278_less__imp__neq,axiom,
! [X2: real,Y: real] :
( ( ord_less_real @ X2 @ Y )
=> ( X2 != Y ) ) ).
% less_imp_neq
thf(fact_279_less__imp__neq,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ( X2 != Y ) ) ).
% less_imp_neq
thf(fact_280_less__asym,axiom,
! [X2: real,Y: real] :
( ( ord_less_real @ X2 @ Y )
=> ~ ( ord_less_real @ Y @ X2 ) ) ).
% less_asym
thf(fact_281_less__asym,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ~ ( ord_less_nat @ Y @ X2 ) ) ).
% less_asym
thf(fact_282_less__asym_H,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ~ ( ord_less_real @ B @ A ) ) ).
% less_asym'
thf(fact_283_less__asym_H,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% less_asym'
thf(fact_284_less__trans,axiom,
! [X2: real,Y: real,Z: real] :
( ( ord_less_real @ X2 @ Y )
=> ( ( ord_less_real @ Y @ Z )
=> ( ord_less_real @ X2 @ Z ) ) ) ).
% less_trans
thf(fact_285_less__trans,axiom,
! [X2: nat,Y: nat,Z: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ( ( ord_less_nat @ Y @ Z )
=> ( ord_less_nat @ X2 @ Z ) ) ) ).
% less_trans
thf(fact_286_less__linear,axiom,
! [X2: real,Y: real] :
( ( ord_less_real @ X2 @ Y )
| ( X2 = Y )
| ( ord_less_real @ Y @ X2 ) ) ).
% less_linear
thf(fact_287_less__linear,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ Y )
| ( X2 = Y )
| ( ord_less_nat @ Y @ X2 ) ) ).
% less_linear
thf(fact_288_less__irrefl,axiom,
! [X2: real] :
~ ( ord_less_real @ X2 @ X2 ) ).
% less_irrefl
thf(fact_289_less__irrefl,axiom,
! [X2: nat] :
~ ( ord_less_nat @ X2 @ X2 ) ).
% less_irrefl
thf(fact_290_ord__eq__less__trans,axiom,
! [A: real,B: real,C: real] :
( ( A = B )
=> ( ( ord_less_real @ B @ C )
=> ( ord_less_real @ A @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_291_ord__eq__less__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( A = B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_292_ord__less__eq__trans,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( B = C )
=> ( ord_less_real @ A @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_293_ord__less__eq__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( B = C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_294_dual__order_Oasym,axiom,
! [B: real,A: real] :
( ( ord_less_real @ B @ A )
=> ~ ( ord_less_real @ A @ B ) ) ).
% dual_order.asym
thf(fact_295_dual__order_Oasym,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ~ ( ord_less_nat @ A @ B ) ) ).
% dual_order.asym
thf(fact_296_less__imp__not__eq,axiom,
! [X2: real,Y: real] :
( ( ord_less_real @ X2 @ Y )
=> ( X2 != Y ) ) ).
% less_imp_not_eq
thf(fact_297_less__imp__not__eq,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ( X2 != Y ) ) ).
% less_imp_not_eq
thf(fact_298_less__not__sym,axiom,
! [X2: real,Y: real] :
( ( ord_less_real @ X2 @ Y )
=> ~ ( ord_less_real @ Y @ X2 ) ) ).
% less_not_sym
thf(fact_299_less__not__sym,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ~ ( ord_less_nat @ Y @ X2 ) ) ).
% less_not_sym
thf(fact_300_less__induct,axiom,
! [P3: nat > $o,A: nat] :
( ! [X3: nat] :
( ! [Y4: nat] :
( ( ord_less_nat @ Y4 @ X3 )
=> ( P3 @ Y4 ) )
=> ( P3 @ X3 ) )
=> ( P3 @ A ) ) ).
% less_induct
thf(fact_301_antisym__conv3,axiom,
! [Y: real,X2: real] :
( ~ ( ord_less_real @ Y @ X2 )
=> ( ( ~ ( ord_less_real @ X2 @ Y ) )
= ( X2 = Y ) ) ) ).
% antisym_conv3
thf(fact_302_antisym__conv3,axiom,
! [Y: nat,X2: nat] :
( ~ ( ord_less_nat @ Y @ X2 )
=> ( ( ~ ( ord_less_nat @ X2 @ Y ) )
= ( X2 = Y ) ) ) ).
% antisym_conv3
thf(fact_303_less__imp__not__eq2,axiom,
! [X2: real,Y: real] :
( ( ord_less_real @ X2 @ Y )
=> ( Y != X2 ) ) ).
% less_imp_not_eq2
thf(fact_304_less__imp__not__eq2,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ( Y != X2 ) ) ).
% less_imp_not_eq2
thf(fact_305_less__imp__triv,axiom,
! [X2: real,Y: real,P3: $o] :
( ( ord_less_real @ X2 @ Y )
=> ( ( ord_less_real @ Y @ X2 )
=> P3 ) ) ).
% less_imp_triv
thf(fact_306_less__imp__triv,axiom,
! [X2: nat,Y: nat,P3: $o] :
( ( ord_less_nat @ X2 @ Y )
=> ( ( ord_less_nat @ Y @ X2 )
=> P3 ) ) ).
% less_imp_triv
thf(fact_307_linorder__cases,axiom,
! [X2: real,Y: real] :
( ~ ( ord_less_real @ X2 @ Y )
=> ( ( X2 != Y )
=> ( ord_less_real @ Y @ X2 ) ) ) ).
% linorder_cases
thf(fact_308_linorder__cases,axiom,
! [X2: nat,Y: nat] :
( ~ ( ord_less_nat @ X2 @ Y )
=> ( ( X2 != Y )
=> ( ord_less_nat @ Y @ X2 ) ) ) ).
% linorder_cases
thf(fact_309_dual__order_Oirrefl,axiom,
! [A: real] :
~ ( ord_less_real @ A @ A ) ).
% dual_order.irrefl
thf(fact_310_dual__order_Oirrefl,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% dual_order.irrefl
thf(fact_311_order_Ostrict__trans,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ B @ C )
=> ( ord_less_real @ A @ C ) ) ) ).
% order.strict_trans
thf(fact_312_order_Ostrict__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans
thf(fact_313_less__imp__not__less,axiom,
! [X2: real,Y: real] :
( ( ord_less_real @ X2 @ Y )
=> ~ ( ord_less_real @ Y @ X2 ) ) ).
% less_imp_not_less
thf(fact_314_less__imp__not__less,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ~ ( ord_less_nat @ Y @ X2 ) ) ).
% less_imp_not_less
thf(fact_315_exists__least__iff,axiom,
( ( ^ [P4: nat > $o] :
? [X5: nat] : ( P4 @ X5 ) )
= ( ^ [P5: nat > $o] :
? [N2: nat] :
( ( P5 @ N2 )
& ! [M2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ~ ( P5 @ M2 ) ) ) ) ) ).
% exists_least_iff
thf(fact_316_linorder__less__wlog,axiom,
! [P3: real > real > $o,A: real,B: real] :
( ! [A4: real,B3: real] :
( ( ord_less_real @ A4 @ B3 )
=> ( P3 @ A4 @ B3 ) )
=> ( ! [A4: real] : ( P3 @ A4 @ A4 )
=> ( ! [A4: real,B3: real] :
( ( P3 @ B3 @ A4 )
=> ( P3 @ A4 @ B3 ) )
=> ( P3 @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_317_linorder__less__wlog,axiom,
! [P3: nat > nat > $o,A: nat,B: nat] :
( ! [A4: nat,B3: nat] :
( ( ord_less_nat @ A4 @ B3 )
=> ( P3 @ A4 @ B3 ) )
=> ( ! [A4: nat] : ( P3 @ A4 @ A4 )
=> ( ! [A4: nat,B3: nat] :
( ( P3 @ B3 @ A4 )
=> ( P3 @ A4 @ B3 ) )
=> ( P3 @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_318_dual__order_Ostrict__trans,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_real @ B @ A )
=> ( ( ord_less_real @ C @ B )
=> ( ord_less_real @ C @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_319_dual__order_Ostrict__trans,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( ord_less_nat @ C @ B )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_320_not__less__iff__gr__or__eq,axiom,
! [X2: real,Y: real] :
( ( ~ ( ord_less_real @ X2 @ Y ) )
= ( ( ord_less_real @ Y @ X2 )
| ( X2 = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_321_not__less__iff__gr__or__eq,axiom,
! [X2: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X2 @ Y ) )
= ( ( ord_less_nat @ Y @ X2 )
| ( X2 = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_322_order_Ostrict__implies__not__eq,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_323_order_Ostrict__implies__not__eq,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_324_dual__order_Ostrict__implies__not__eq,axiom,
! [B: real,A: real] :
( ( ord_less_real @ B @ A )
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_325_dual__order_Ostrict__implies__not__eq,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_326_leD,axiom,
! [Y: real,X2: real] :
( ( ord_less_eq_real @ Y @ X2 )
=> ~ ( ord_less_real @ X2 @ Y ) ) ).
% leD
thf(fact_327_leD,axiom,
! [Y: nat,X2: nat] :
( ( ord_less_eq_nat @ Y @ X2 )
=> ~ ( ord_less_nat @ X2 @ Y ) ) ).
% leD
thf(fact_328_leI,axiom,
! [X2: real,Y: real] :
( ~ ( ord_less_real @ X2 @ Y )
=> ( ord_less_eq_real @ Y @ X2 ) ) ).
% leI
thf(fact_329_leI,axiom,
! [X2: nat,Y: nat] :
( ~ ( ord_less_nat @ X2 @ Y )
=> ( ord_less_eq_nat @ Y @ X2 ) ) ).
% leI
thf(fact_330_le__less,axiom,
( ord_less_eq_real
= ( ^ [X4: real,Y3: real] :
( ( ord_less_real @ X4 @ Y3 )
| ( X4 = Y3 ) ) ) ) ).
% le_less
thf(fact_331_le__less,axiom,
( ord_less_eq_nat
= ( ^ [X4: nat,Y3: nat] :
( ( ord_less_nat @ X4 @ Y3 )
| ( X4 = Y3 ) ) ) ) ).
% le_less
thf(fact_332_less__le,axiom,
( ord_less_real
= ( ^ [X4: real,Y3: real] :
( ( ord_less_eq_real @ X4 @ Y3 )
& ( X4 != Y3 ) ) ) ) ).
% less_le
thf(fact_333_less__le,axiom,
( ord_less_nat
= ( ^ [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
& ( X4 != Y3 ) ) ) ) ).
% less_le
thf(fact_334_order__le__less__subst1,axiom,
! [A: real,F: real > real,B: real,C: real] :
( ( ord_less_eq_real @ A @ ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_real @ X3 @ Y2 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_335_order__le__less__subst1,axiom,
! [A: real,F: nat > real,B: nat,C: nat] :
( ( ord_less_eq_real @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_336_order__le__less__subst1,axiom,
! [A: nat,F: real > nat,B: real,C: real] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_real @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_337_order__le__less__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_338_order__le__less__subst2,axiom,
! [A: real,B: real,F: real > real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_real @ ( F @ B ) @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_339_order__le__less__subst2,axiom,
! [A: real,B: real,F: real > nat,C: nat] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_340_order__le__less__subst2,axiom,
! [A: nat,B: nat,F: nat > real,C: real] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_real @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_341_order__le__less__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_342_order__less__le__subst1,axiom,
! [A: nat,F: real > nat,B: real,C: real] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_343_order__less__le__subst1,axiom,
! [A: real,F: nat > real,B: nat,C: nat] :
( ( ord_less_real @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_344_order__less__le__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
% Helper facts (5)
thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
! [X2: nat,Y: nat] :
( ( if_nat @ $false @ X2 @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X2: nat,Y: nat] :
( ( if_nat @ $true @ X2 @ Y )
= X2 ) ).
thf(help_If_3_1_If_001t__Real__Oreal_T,axiom,
! [P3: $o] :
( ( P3 = $true )
| ( P3 = $false ) ) ).
thf(help_If_2_1_If_001t__Real__Oreal_T,axiom,
! [X2: real,Y: real] :
( ( if_real @ $false @ X2 @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Real__Oreal_T,axiom,
! [X2: real,Y: real] :
( ( if_real @ $true @ X2 @ Y )
= X2 ) ).
% Conjectures (1)
thf(conj_0,conjecture,
! [X3: real] :
( ( ( poly_real2 @ p @ X3 )
!= zero_zero_real )
| ( ord_less_real @ lb @ X3 ) ) ).
%------------------------------------------------------------------------------