TPTP Problem File: ITP193^1.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : ITP193^1 : TPTP v9.0.0. Released v7.5.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer Sturm_Tarski problem prob_1192__5884460_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_1192__5884460_1 [Des21]
% Status : Theorem
% Rating : 0.25 v9.0.0, 0.30 v8.2.0, 0.31 v8.1.0, 0.36 v7.5.0
% Syntax : Number of formulae : 426 ( 113 unt; 75 typ; 0 def)
% Number of atoms : 1144 ( 273 equ; 0 cnn)
% Maximal formula atoms : 12 ( 3 avg)
% Number of connectives : 3014 ( 146 ~; 43 |; 87 &;2157 @)
% ( 0 <=>; 581 =>; 0 <=; 0 <~>)
% Maximal formula depth : 18 ( 7 avg)
% Number of types : 16 ( 15 usr)
% Number of type conns : 215 ( 215 >; 0 *; 0 +; 0 <<)
% Number of symbols : 61 ( 60 usr; 14 con; 0-4 aty)
% Number of variables : 1091 ( 95 ^; 927 !; 69 ?;1091 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Sledgehammer 2021-02-23 15:44:34.506
%------------------------------------------------------------------------------
% Could-be-implicit typings (15)
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__Set__Oset_It__Polynomial__Opoly_It__Polynomial__Opoly_It__Real__Oreal_J_J_J,type,
set_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__Int__Oint_J_J,type,
poly_poly_int: $tType ).
thf(ty_n_t__Set__Oset_It__Polynomial__Opoly_It__Real__Oreal_J_J,type,
set_poly_real: $tType ).
thf(ty_n_t__Set__Oset_It__Polynomial__Opoly_It__Int__Oint_J_J,type,
set_poly_int: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Real__Oreal_J_J,type,
set_set_real: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Int__Oint_J_J,type,
set_set_int: $tType ).
thf(ty_n_t__Polynomial__Opoly_It__Real__Oreal_J,type,
poly_real: $tType ).
thf(ty_n_t__Polynomial__Opoly_It__Int__Oint_J,type,
poly_int: $tType ).
thf(ty_n_t__Set__Oset_It__Real__Oreal_J,type,
set_real: $tType ).
thf(ty_n_t__Set__Oset_It__Int__Oint_J,type,
set_int: $tType ).
thf(ty_n_t__Real__Oreal,type,
real: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
thf(ty_n_t__Int__Oint,type,
int: $tType ).
% Explicit typings (60)
thf(sy_c_Finite__Set_Ofinite_001t__Int__Oint,type,
finite_finite_int: set_int > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Polynomial__Opoly_It__Int__Oint_J,type,
finite385822667ly_int: set_poly_int > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Polynomial__Opoly_It__Polynomial__Opoly_It__Real__Oreal_J_J,type,
finite1328464339y_real: set_poly_poly_real > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Polynomial__Opoly_It__Real__Oreal_J,type,
finite1810960971y_real: set_poly_real > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Real__Oreal,type,
finite_finite_real: set_real > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_It__Int__Oint_J,type,
finite26677625et_int: set_set_int > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_It__Real__Oreal_J,type,
finite475462905t_real: set_set_real > $o ).
thf(sy_c_Groups_Oone__class_Oone_001t__Int__Oint,type,
one_one_int: int ).
thf(sy_c_Groups_Oone__class_Oone_001t__Polynomial__Opoly_It__Real__Oreal_J,type,
one_one_poly_real: poly_real ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Int__Oint,type,
zero_zero_int: int ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Polynomial__Opoly_It__Int__Oint_J,type,
zero_zero_poly_int: poly_int ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Polynomial__Opoly_It__Polynomial__Opoly_It__Int__Oint_J_J,type,
zero_z1549157189ly_int: poly_poly_int ).
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_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Real__Oreal_001t__Int__Oint,type,
groups2051521940al_int: ( real > int ) > set_real > int ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Int__Oint,type,
ord_less_int: int > int > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Polynomial__Opoly_It__Int__Oint_J,type,
ord_less_poly_int: poly_int > poly_int > $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_001t__Set__Oset_It__Int__Oint_J,type,
ord_less_set_int: set_int > set_int > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Real__Oreal_J,type,
ord_less_set_real: set_real > set_real > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Real__Oreal_M_Eo_J,type,
ord_less_eq_real_o: ( real > $o ) > ( real > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Int__Oint,type,
ord_less_eq_int: int > int > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Polynomial__Opoly_It__Int__Oint_J,type,
ord_less_eq_poly_int: poly_int > poly_int > $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_Oless__eq_001t__Set__Oset_It__Int__Oint_J,type,
ord_less_eq_set_int: set_int > set_int > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Real__Oreal_J,type,
ord_less_eq_set_real: set_real > set_real > $o ).
thf(sy_c_Polynomial_Ois__zero_001t__Int__Oint,type,
is_zero_int: poly_int > $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_Opoly_001t__Int__Oint,type,
poly_int2: poly_int > int > int ).
thf(sy_c_Polynomial_Opoly_001t__Polynomial__Opoly_It__Int__Oint_J,type,
poly_poly_int2: poly_poly_int > poly_int > poly_int ).
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__Int__Oint,type,
poly_cutoff_int: nat > poly_int > poly_int ).
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_Set_OCollect_001t__Int__Oint,type,
collect_int: ( int > $o ) > set_int ).
thf(sy_c_Set_OCollect_001t__Polynomial__Opoly_It__Int__Oint_J,type,
collect_poly_int: ( poly_int > $o ) > set_poly_int ).
thf(sy_c_Set_OCollect_001t__Polynomial__Opoly_It__Polynomial__Opoly_It__Real__Oreal_J_J,type,
collec927113489y_real: ( poly_poly_real > $o ) > set_poly_poly_real ).
thf(sy_c_Set_OCollect_001t__Polynomial__Opoly_It__Real__Oreal_J,type,
collect_poly_real: ( poly_real > $o ) > set_poly_real ).
thf(sy_c_Set_OCollect_001t__Real__Oreal,type,
collect_real: ( real > $o ) > set_real ).
thf(sy_c_Set_OCollect_001t__Set__Oset_It__Int__Oint_J,type,
collect_set_int: ( set_int > $o ) > set_set_int ).
thf(sy_c_Set_OCollect_001t__Set__Oset_It__Real__Oreal_J,type,
collect_set_real: ( set_real > $o ) > set_set_real ).
thf(sy_c_Sturm__Tarski__Mirabelle__skihomvtkj_Ocindex__poly,type,
sturm_308542325x_poly: real > real > poly_real > poly_real > int ).
thf(sy_c_Sturm__Tarski__Mirabelle__skihomvtkj_Ocross,type,
sturm_1953858694_cross: poly_real > real > real > int ).
thf(sy_c_Sturm__Tarski__Mirabelle__skihomvtkj_Ojump__poly,type,
sturm_276991412p_poly: poly_real > poly_real > real > int ).
thf(sy_c_member_001t__Int__Oint,type,
member_int: int > set_int > $o ).
thf(sy_c_member_001t__Real__Oreal,type,
member_real: real > set_real > $o ).
thf(sy_v_a,type,
a: real ).
thf(sy_v_a_H,type,
a2: real ).
thf(sy_v_b,type,
b: real ).
thf(sy_v_b_H,type,
b2: real ).
thf(sy_v_p,type,
p: poly_real ).
% Relevant facts (350)
thf(fact_0__092_060open_062finite_A_123x_O_Apoly_Ap_Ax_A_061_A0_125_092_060close_062,axiom,
( finite_finite_real
@ ( collect_real
@ ^ [X: real] :
( ( poly_real2 @ p @ X )
= zero_zero_real ) ) ) ).
% \<open>finite {x. poly p x = 0}\<close>
thf(fact_1_assms_I2_J,axiom,
ord_less_real @ a2 @ b2 ).
% assms(2)
thf(fact_2_assms_I3_J,axiom,
ord_less_real @ b2 @ b ).
% assms(3)
thf(fact_3_assms_I1_J,axiom,
ord_less_real @ a @ a2 ).
% assms(1)
thf(fact_4_False,axiom,
p != zero_zero_poly_real ).
% False
thf(fact_5_poly__0,axiom,
! [X2: poly_int] :
( ( poly_poly_int2 @ zero_z1549157189ly_int @ X2 )
= zero_zero_poly_int ) ).
% poly_0
thf(fact_6_poly__0,axiom,
! [X2: poly_poly_real] :
( ( poly_poly_poly_real2 @ zero_z935034829y_real @ X2 )
= zero_z1423781445y_real ) ).
% poly_0
thf(fact_7_poly__0,axiom,
! [X2: poly_real] :
( ( poly_poly_real2 @ zero_z1423781445y_real @ X2 )
= zero_zero_poly_real ) ).
% poly_0
thf(fact_8_poly__0,axiom,
! [X2: int] :
( ( poly_int2 @ zero_zero_poly_int @ X2 )
= zero_zero_int ) ).
% poly_0
thf(fact_9_poly__0,axiom,
! [X2: real] :
( ( poly_real2 @ zero_zero_poly_real @ X2 )
= zero_zero_real ) ).
% poly_0
thf(fact_10_assms_I4_J,axiom,
! [X3: real] :
( ( ( ( ord_less_real @ a @ X3 )
& ( ord_less_eq_real @ X3 @ a2 ) )
| ( ( ord_less_eq_real @ b2 @ X3 )
& ( ord_less_real @ X3 @ b ) ) )
=> ( ( poly_real2 @ p @ X3 )
!= zero_zero_real ) ) ).
% assms(4)
thf(fact_11_finite__Collect__conjI,axiom,
! [P: real > $o,Q: real > $o] :
( ( ( finite_finite_real @ ( collect_real @ P ) )
| ( finite_finite_real @ ( collect_real @ Q ) ) )
=> ( finite_finite_real
@ ( collect_real
@ ^ [X: real] :
( ( P @ X )
& ( Q @ X ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_12_finite__Collect__conjI,axiom,
! [P: int > $o,Q: int > $o] :
( ( ( finite_finite_int @ ( collect_int @ P ) )
| ( finite_finite_int @ ( collect_int @ Q ) ) )
=> ( finite_finite_int
@ ( collect_int
@ ^ [X: int] :
( ( P @ X )
& ( Q @ X ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_13_finite__Collect__disjI,axiom,
! [P: real > $o,Q: real > $o] :
( ( finite_finite_real
@ ( collect_real
@ ^ [X: real] :
( ( P @ X )
| ( Q @ X ) ) ) )
= ( ( finite_finite_real @ ( collect_real @ P ) )
& ( finite_finite_real @ ( collect_real @ Q ) ) ) ) ).
% finite_Collect_disjI
thf(fact_14_finite__Collect__disjI,axiom,
! [P: int > $o,Q: int > $o] :
( ( finite_finite_int
@ ( collect_int
@ ^ [X: int] :
( ( P @ X )
| ( Q @ X ) ) ) )
= ( ( finite_finite_int @ ( collect_int @ P ) )
& ( finite_finite_int @ ( collect_int @ Q ) ) ) ) ).
% finite_Collect_disjI
thf(fact_15_poly__roots__finite,axiom,
! [P2: poly_poly_int] :
( ( P2 != zero_z1549157189ly_int )
=> ( finite385822667ly_int
@ ( collect_poly_int
@ ^ [X: poly_int] :
( ( poly_poly_int2 @ P2 @ X )
= zero_zero_poly_int ) ) ) ) ).
% poly_roots_finite
thf(fact_16_poly__roots__finite,axiom,
! [P2: poly_poly_poly_real] :
( ( P2 != zero_z935034829y_real )
=> ( finite1328464339y_real
@ ( collec927113489y_real
@ ^ [X: poly_poly_real] :
( ( poly_poly_poly_real2 @ P2 @ X )
= zero_z1423781445y_real ) ) ) ) ).
% poly_roots_finite
thf(fact_17_poly__roots__finite,axiom,
! [P2: poly_real] :
( ( P2 != zero_zero_poly_real )
=> ( finite_finite_real
@ ( collect_real
@ ^ [X: real] :
( ( poly_real2 @ P2 @ X )
= zero_zero_real ) ) ) ) ).
% poly_roots_finite
thf(fact_18_poly__roots__finite,axiom,
! [P2: poly_int] :
( ( P2 != zero_zero_poly_int )
=> ( finite_finite_int
@ ( collect_int
@ ^ [X: int] :
( ( poly_int2 @ P2 @ X )
= zero_zero_int ) ) ) ) ).
% poly_roots_finite
thf(fact_19_poly__roots__finite,axiom,
! [P2: poly_poly_real] :
( ( P2 != zero_z1423781445y_real )
=> ( finite1810960971y_real
@ ( collect_poly_real
@ ^ [X: poly_real] :
( ( poly_poly_real2 @ P2 @ X )
= zero_zero_poly_real ) ) ) ) ).
% poly_roots_finite
thf(fact_20_poly__IVT__neg,axiom,
! [A: real,B: real,P2: poly_real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ zero_zero_real @ ( poly_real2 @ P2 @ A ) )
=> ( ( ord_less_real @ ( poly_real2 @ P2 @ B ) @ zero_zero_real )
=> ? [X4: real] :
( ( ord_less_real @ A @ X4 )
& ( ord_less_real @ X4 @ B )
& ( ( poly_real2 @ P2 @ X4 )
= zero_zero_real ) ) ) ) ) ).
% poly_IVT_neg
thf(fact_21_poly__IVT__pos,axiom,
! [A: real,B: real,P2: poly_real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ ( poly_real2 @ P2 @ A ) @ zero_zero_real )
=> ( ( ord_less_real @ zero_zero_real @ ( poly_real2 @ P2 @ B ) )
=> ? [X4: real] :
( ( ord_less_real @ A @ X4 )
& ( ord_less_real @ X4 @ B )
& ( ( poly_real2 @ P2 @ X4 )
= zero_zero_real ) ) ) ) ) ).
% poly_IVT_pos
thf(fact_22_poly__all__0__iff__0,axiom,
! [P2: poly_real] :
( ( ! [X: real] :
( ( poly_real2 @ P2 @ X )
= zero_zero_real ) )
= ( P2 = zero_zero_poly_real ) ) ).
% poly_all_0_iff_0
thf(fact_23_poly__all__0__iff__0,axiom,
! [P2: poly_poly_real] :
( ( ! [X: poly_real] :
( ( poly_poly_real2 @ P2 @ X )
= zero_zero_poly_real ) )
= ( P2 = zero_z1423781445y_real ) ) ).
% poly_all_0_iff_0
thf(fact_24_poly__all__0__iff__0,axiom,
! [P2: poly_int] :
( ( ! [X: int] :
( ( poly_int2 @ P2 @ X )
= zero_zero_int ) )
= ( P2 = zero_zero_poly_int ) ) ).
% poly_all_0_iff_0
thf(fact_25_poly__all__0__iff__0,axiom,
! [P2: poly_poly_int] :
( ( ! [X: poly_int] :
( ( poly_poly_int2 @ P2 @ X )
= zero_zero_poly_int ) )
= ( P2 = zero_z1549157189ly_int ) ) ).
% poly_all_0_iff_0
thf(fact_26_poly__all__0__iff__0,axiom,
! [P2: poly_poly_poly_real] :
( ( ! [X: poly_poly_real] :
( ( poly_poly_poly_real2 @ P2 @ X )
= zero_z1423781445y_real ) )
= ( P2 = zero_z935034829y_real ) ) ).
% poly_all_0_iff_0
thf(fact_27_less__numeral__extra_I3_J,axiom,
~ ( ord_less_poly_real @ zero_zero_poly_real @ zero_zero_poly_real ) ).
% less_numeral_extra(3)
thf(fact_28_less__numeral__extra_I3_J,axiom,
~ ( ord_less_poly_int @ zero_zero_poly_int @ zero_zero_poly_int ) ).
% less_numeral_extra(3)
thf(fact_29_less__numeral__extra_I3_J,axiom,
~ ( ord_le38482960y_real @ zero_z1423781445y_real @ zero_z1423781445y_real ) ).
% less_numeral_extra(3)
thf(fact_30_less__numeral__extra_I3_J,axiom,
~ ( ord_less_real @ zero_zero_real @ zero_zero_real ) ).
% less_numeral_extra(3)
thf(fact_31_less__numeral__extra_I3_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_numeral_extra(3)
thf(fact_32_le__numeral__extra_I3_J,axiom,
ord_le1180086932y_real @ zero_zero_poly_real @ zero_zero_poly_real ).
% le_numeral_extra(3)
thf(fact_33_le__numeral__extra_I3_J,axiom,
ord_less_eq_poly_int @ zero_zero_poly_int @ zero_zero_poly_int ).
% le_numeral_extra(3)
thf(fact_34_le__numeral__extra_I3_J,axiom,
ord_le893774876y_real @ zero_z1423781445y_real @ zero_z1423781445y_real ).
% le_numeral_extra(3)
thf(fact_35_le__numeral__extra_I3_J,axiom,
ord_less_eq_real @ zero_zero_real @ zero_zero_real ).
% le_numeral_extra(3)
thf(fact_36_le__numeral__extra_I3_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% le_numeral_extra(3)
thf(fact_37_finite__psubset__induct,axiom,
! [A2: set_real,P: set_real > $o] :
( ( finite_finite_real @ A2 )
=> ( ! [A3: set_real] :
( ( finite_finite_real @ A3 )
=> ( ! [B2: set_real] :
( ( ord_less_set_real @ B2 @ A3 )
=> ( P @ B2 ) )
=> ( P @ A3 ) ) )
=> ( P @ A2 ) ) ) ).
% finite_psubset_induct
thf(fact_38_finite__psubset__induct,axiom,
! [A2: set_int,P: set_int > $o] :
( ( finite_finite_int @ A2 )
=> ( ! [A3: set_int] :
( ( finite_finite_int @ A3 )
=> ( ! [B2: set_int] :
( ( ord_less_set_int @ B2 @ A3 )
=> ( P @ B2 ) )
=> ( P @ A3 ) ) )
=> ( P @ A2 ) ) ) ).
% finite_psubset_induct
thf(fact_39_finite__has__minimal2,axiom,
! [A2: set_real,A: real] :
( ( finite_finite_real @ A2 )
=> ( ( member_real @ A @ A2 )
=> ? [X4: real] :
( ( member_real @ X4 @ A2 )
& ( ord_less_eq_real @ X4 @ A )
& ! [Xa: real] :
( ( member_real @ Xa @ A2 )
=> ( ( ord_less_eq_real @ Xa @ X4 )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_40_finite__has__minimal2,axiom,
! [A2: set_int,A: int] :
( ( finite_finite_int @ A2 )
=> ( ( member_int @ A @ A2 )
=> ? [X4: int] :
( ( member_int @ X4 @ A2 )
& ( ord_less_eq_int @ X4 @ A )
& ! [Xa: int] :
( ( member_int @ Xa @ A2 )
=> ( ( ord_less_eq_int @ Xa @ X4 )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_41_finite__has__maximal2,axiom,
! [A2: set_real,A: real] :
( ( finite_finite_real @ A2 )
=> ( ( member_real @ A @ A2 )
=> ? [X4: real] :
( ( member_real @ X4 @ A2 )
& ( ord_less_eq_real @ A @ X4 )
& ! [Xa: real] :
( ( member_real @ Xa @ A2 )
=> ( ( ord_less_eq_real @ X4 @ Xa )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_42_finite__has__maximal2,axiom,
! [A2: set_int,A: int] :
( ( finite_finite_int @ A2 )
=> ( ( member_int @ A @ A2 )
=> ? [X4: int] :
( ( member_int @ X4 @ A2 )
& ( ord_less_eq_int @ A @ X4 )
& ! [Xa: int] :
( ( member_int @ Xa @ A2 )
=> ( ( ord_less_eq_int @ X4 @ Xa )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_43_next__non__root__interval,axiom,
! [P2: poly_real,Lb: real] :
( ( P2 != zero_zero_poly_real )
=> ~ ! [Ub: real] :
( ( ord_less_real @ Lb @ Ub )
=> ~ ! [Z: real] :
( ( ( ord_less_real @ Lb @ Z )
& ( ord_less_eq_real @ Z @ Ub ) )
=> ( ( poly_real2 @ P2 @ Z )
!= zero_zero_real ) ) ) ) ).
% next_non_root_interval
thf(fact_44_last__non__root__interval,axiom,
! [P2: poly_real,Ub2: real] :
( ( P2 != zero_zero_poly_real )
=> ~ ! [Lb2: real] :
( ( ord_less_real @ Lb2 @ Ub2 )
=> ~ ! [Z: real] :
( ( ( ord_less_eq_real @ Lb2 @ Z )
& ( ord_less_real @ Z @ Ub2 ) )
=> ( ( poly_real2 @ P2 @ Z )
!= zero_zero_real ) ) ) ) ).
% last_non_root_interval
thf(fact_45_zero__reorient,axiom,
! [X2: real] :
( ( zero_zero_real = X2 )
= ( X2 = zero_zero_real ) ) ).
% zero_reorient
thf(fact_46_zero__reorient,axiom,
! [X2: poly_real] :
( ( zero_zero_poly_real = X2 )
= ( X2 = zero_zero_poly_real ) ) ).
% zero_reorient
thf(fact_47_zero__reorient,axiom,
! [X2: int] :
( ( zero_zero_int = X2 )
= ( X2 = zero_zero_int ) ) ).
% zero_reorient
thf(fact_48_zero__reorient,axiom,
! [X2: poly_int] :
( ( zero_zero_poly_int = X2 )
= ( X2 = zero_zero_poly_int ) ) ).
% zero_reorient
thf(fact_49_zero__reorient,axiom,
! [X2: poly_poly_real] :
( ( zero_z1423781445y_real = X2 )
= ( X2 = zero_z1423781445y_real ) ) ).
% zero_reorient
thf(fact_50_not__eq__pos__or__neg__iff__2,axiom,
! [Lb: real,Ub2: real,P2: poly_real] :
( ( ! [Z2: real] :
( ( ( ord_less_eq_real @ Lb @ Z2 )
& ( ord_less_real @ Z2 @ Ub2 ) )
=> ( ( poly_real2 @ P2 @ Z2 )
!= zero_zero_real ) ) )
= ( ! [Z2: real] :
( ( ( ord_less_eq_real @ Lb @ Z2 )
& ( ord_less_real @ Z2 @ Ub2 ) )
=> ( ord_less_real @ zero_zero_real @ ( poly_real2 @ P2 @ Z2 ) ) )
| ! [Z2: real] :
( ( ( ord_less_eq_real @ Lb @ Z2 )
& ( ord_less_real @ Z2 @ Ub2 ) )
=> ( ord_less_real @ ( poly_real2 @ P2 @ Z2 ) @ zero_zero_real ) ) ) ) ).
% not_eq_pos_or_neg_iff_2
thf(fact_51_not__eq__pos__or__neg__iff__1,axiom,
! [Lb: real,Ub2: real,P2: poly_real] :
( ( ! [Z2: real] :
( ( ( ord_less_real @ Lb @ Z2 )
& ( ord_less_eq_real @ Z2 @ Ub2 ) )
=> ( ( poly_real2 @ P2 @ Z2 )
!= zero_zero_real ) ) )
= ( ! [Z2: real] :
( ( ( ord_less_real @ Lb @ Z2 )
& ( ord_less_eq_real @ Z2 @ Ub2 ) )
=> ( ord_less_real @ zero_zero_real @ ( poly_real2 @ P2 @ Z2 ) ) )
| ! [Z2: real] :
( ( ( ord_less_real @ Lb @ Z2 )
& ( ord_less_eq_real @ Z2 @ Ub2 ) )
=> ( ord_less_real @ ( poly_real2 @ P2 @ Z2 ) @ zero_zero_real ) ) ) ) ).
% not_eq_pos_or_neg_iff_1
thf(fact_52_poly__eq__poly__eq__iff,axiom,
! [P2: poly_real,Q2: poly_real] :
( ( ( poly_real2 @ P2 )
= ( poly_real2 @ Q2 ) )
= ( P2 = Q2 ) ) ).
% poly_eq_poly_eq_iff
thf(fact_53_poly__eq__poly__eq__iff,axiom,
! [P2: poly_int,Q2: poly_int] :
( ( ( poly_int2 @ P2 )
= ( poly_int2 @ Q2 ) )
= ( P2 = Q2 ) ) ).
% poly_eq_poly_eq_iff
thf(fact_54_poly__eq__poly__eq__iff,axiom,
! [P2: poly_poly_real,Q2: poly_poly_real] :
( ( ( poly_poly_real2 @ P2 )
= ( poly_poly_real2 @ Q2 ) )
= ( P2 = Q2 ) ) ).
% poly_eq_poly_eq_iff
thf(fact_55_pigeonhole__infinite__rel,axiom,
! [A2: set_real,B3: set_real,R: real > real > $o] :
( ~ ( finite_finite_real @ A2 )
=> ( ( finite_finite_real @ B3 )
=> ( ! [X4: real] :
( ( member_real @ X4 @ A2 )
=> ? [Xa: real] :
( ( member_real @ Xa @ B3 )
& ( R @ X4 @ Xa ) ) )
=> ? [X4: real] :
( ( member_real @ X4 @ B3 )
& ~ ( finite_finite_real
@ ( collect_real
@ ^ [A4: real] :
( ( member_real @ A4 @ A2 )
& ( R @ A4 @ X4 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_56_pigeonhole__infinite__rel,axiom,
! [A2: set_real,B3: set_int,R: real > int > $o] :
( ~ ( finite_finite_real @ A2 )
=> ( ( finite_finite_int @ B3 )
=> ( ! [X4: real] :
( ( member_real @ X4 @ A2 )
=> ? [Xa: int] :
( ( member_int @ Xa @ B3 )
& ( R @ X4 @ Xa ) ) )
=> ? [X4: int] :
( ( member_int @ X4 @ B3 )
& ~ ( finite_finite_real
@ ( collect_real
@ ^ [A4: real] :
( ( member_real @ A4 @ A2 )
& ( R @ A4 @ X4 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_57_pigeonhole__infinite__rel,axiom,
! [A2: set_int,B3: set_real,R: int > real > $o] :
( ~ ( finite_finite_int @ A2 )
=> ( ( finite_finite_real @ B3 )
=> ( ! [X4: int] :
( ( member_int @ X4 @ A2 )
=> ? [Xa: real] :
( ( member_real @ Xa @ B3 )
& ( R @ X4 @ Xa ) ) )
=> ? [X4: real] :
( ( member_real @ X4 @ B3 )
& ~ ( finite_finite_int
@ ( collect_int
@ ^ [A4: int] :
( ( member_int @ A4 @ A2 )
& ( R @ A4 @ X4 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_58_pigeonhole__infinite__rel,axiom,
! [A2: set_int,B3: set_int,R: int > int > $o] :
( ~ ( finite_finite_int @ A2 )
=> ( ( finite_finite_int @ B3 )
=> ( ! [X4: int] :
( ( member_int @ X4 @ A2 )
=> ? [Xa: int] :
( ( member_int @ Xa @ B3 )
& ( R @ X4 @ Xa ) ) )
=> ? [X4: int] :
( ( member_int @ X4 @ B3 )
& ~ ( finite_finite_int
@ ( collect_int
@ ^ [A4: int] :
( ( member_int @ A4 @ A2 )
& ( R @ A4 @ X4 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_59_not__finite__existsD,axiom,
! [P: real > $o] :
( ~ ( finite_finite_real @ ( collect_real @ P ) )
=> ? [X_1: real] : ( P @ X_1 ) ) ).
% not_finite_existsD
thf(fact_60_not__finite__existsD,axiom,
! [P: int > $o] :
( ~ ( finite_finite_int @ ( collect_int @ P ) )
=> ? [X_1: int] : ( P @ X_1 ) ) ).
% not_finite_existsD
thf(fact_61_order__refl,axiom,
! [X2: real] : ( ord_less_eq_real @ X2 @ X2 ) ).
% order_refl
thf(fact_62_order__refl,axiom,
! [X2: int] : ( ord_less_eq_int @ X2 @ X2 ) ).
% order_refl
thf(fact_63_cross__no__root,axiom,
! [A: real,B: real,P2: poly_real] :
( ( ord_less_real @ A @ B )
=> ( ! [X4: real] :
( ( ( ord_less_real @ A @ X4 )
& ( ord_less_real @ X4 @ B ) )
=> ( ( poly_real2 @ P2 @ X4 )
!= zero_zero_real ) )
=> ( ( sturm_1953858694_cross @ P2 @ A @ B )
= zero_zero_int ) ) ) ).
% cross_no_root
thf(fact_64_less__eq__real__def,axiom,
( ord_less_eq_real
= ( ^ [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
| ( X = Y ) ) ) ) ).
% less_eq_real_def
thf(fact_65_complete__interval,axiom,
! [A: real,B: real,P: real > $o] :
( ( ord_less_real @ A @ B )
=> ( ( P @ A )
=> ( ~ ( P @ B )
=> ? [C: real] :
( ( ord_less_eq_real @ A @ C )
& ( ord_less_eq_real @ C @ B )
& ! [X3: real] :
( ( ( ord_less_eq_real @ A @ X3 )
& ( ord_less_real @ X3 @ C ) )
=> ( P @ X3 ) )
& ! [D: real] :
( ! [X4: real] :
( ( ( ord_less_eq_real @ A @ X4 )
& ( ord_less_real @ X4 @ D ) )
=> ( P @ X4 ) )
=> ( ord_less_eq_real @ D @ C ) ) ) ) ) ) ).
% complete_interval
thf(fact_66_complete__interval,axiom,
! [A: int,B: int,P: int > $o] :
( ( ord_less_int @ A @ B )
=> ( ( P @ A )
=> ( ~ ( P @ B )
=> ? [C: int] :
( ( ord_less_eq_int @ A @ C )
& ( ord_less_eq_int @ C @ B )
& ! [X3: int] :
( ( ( ord_less_eq_int @ A @ X3 )
& ( ord_less_int @ X3 @ C ) )
=> ( P @ X3 ) )
& ! [D: int] :
( ! [X4: int] :
( ( ( ord_less_eq_int @ A @ X4 )
& ( ord_less_int @ X4 @ D ) )
=> ( P @ X4 ) )
=> ( ord_less_eq_int @ D @ C ) ) ) ) ) ) ).
% complete_interval
thf(fact_67_order_Onot__eq__order__implies__strict,axiom,
! [A: real,B: real] :
( ( A != B )
=> ( ( ord_less_eq_real @ A @ B )
=> ( ord_less_real @ A @ B ) ) ) ).
% order.not_eq_order_implies_strict
thf(fact_68_order_Onot__eq__order__implies__strict,axiom,
! [A: int,B: int] :
( ( A != B )
=> ( ( ord_less_eq_int @ A @ B )
=> ( ord_less_int @ A @ B ) ) ) ).
% order.not_eq_order_implies_strict
thf(fact_69_dual__order_Ostrict__implies__order,axiom,
! [B: real,A: real] :
( ( ord_less_real @ B @ A )
=> ( ord_less_eq_real @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_70_dual__order_Ostrict__implies__order,axiom,
! [B: int,A: int] :
( ( ord_less_int @ B @ A )
=> ( ord_less_eq_int @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_71_dual__order_Ostrict__iff__order,axiom,
( ord_less_real
= ( ^ [B4: real,A4: real] :
( ( ord_less_eq_real @ B4 @ A4 )
& ( A4 != B4 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_72_dual__order_Ostrict__iff__order,axiom,
( ord_less_int
= ( ^ [B4: int,A4: int] :
( ( ord_less_eq_int @ B4 @ A4 )
& ( A4 != B4 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_73_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_real
= ( ^ [B4: real,A4: real] :
( ( ord_less_real @ B4 @ A4 )
| ( A4 = B4 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_74_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_int
= ( ^ [B4: int,A4: int] :
( ( ord_less_int @ B4 @ A4 )
| ( A4 = B4 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_75_mem__Collect__eq,axiom,
! [A: real,P: real > $o] :
( ( member_real @ A @ ( collect_real @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_76_mem__Collect__eq,axiom,
! [A: int,P: int > $o] :
( ( member_int @ A @ ( collect_int @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_77_Collect__mem__eq,axiom,
! [A2: set_real] :
( ( collect_real
@ ^ [X: real] : ( member_real @ X @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_78_Collect__mem__eq,axiom,
! [A2: set_int] :
( ( collect_int
@ ^ [X: int] : ( member_int @ X @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_79_Collect__cong,axiom,
! [P: real > $o,Q: real > $o] :
( ! [X4: real] :
( ( P @ X4 )
= ( Q @ X4 ) )
=> ( ( collect_real @ P )
= ( collect_real @ Q ) ) ) ).
% Collect_cong
thf(fact_80_Collect__cong,axiom,
! [P: int > $o,Q: int > $o] :
( ! [X4: int] :
( ( P @ X4 )
= ( Q @ X4 ) )
=> ( ( collect_int @ P )
= ( collect_int @ Q ) ) ) ).
% Collect_cong
thf(fact_81_order_Ostrict__implies__order,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ( ord_less_eq_real @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_82_order_Ostrict__implies__order,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_eq_int @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_83_dense__le__bounded,axiom,
! [X2: real,Y2: real,Z3: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ( ! [W: real] :
( ( ord_less_real @ X2 @ W )
=> ( ( ord_less_real @ W @ Y2 )
=> ( ord_less_eq_real @ W @ Z3 ) ) )
=> ( ord_less_eq_real @ Y2 @ Z3 ) ) ) ).
% dense_le_bounded
thf(fact_84_finite__Collect__subsets,axiom,
! [A2: set_real] :
( ( finite_finite_real @ A2 )
=> ( finite475462905t_real
@ ( collect_set_real
@ ^ [B5: set_real] : ( ord_less_eq_set_real @ B5 @ A2 ) ) ) ) ).
% finite_Collect_subsets
thf(fact_85_finite__Collect__subsets,axiom,
! [A2: set_int] :
( ( finite_finite_int @ A2 )
=> ( finite26677625et_int
@ ( collect_set_int
@ ^ [B5: set_int] : ( ord_less_eq_set_int @ B5 @ A2 ) ) ) ) ).
% finite_Collect_subsets
thf(fact_86_cross__0,axiom,
! [A: real,B: real] :
( ( sturm_1953858694_cross @ zero_zero_poly_real @ A @ B )
= zero_zero_int ) ).
% cross_0
thf(fact_87_finite__subset,axiom,
! [A2: set_real,B3: set_real] :
( ( ord_less_eq_set_real @ A2 @ B3 )
=> ( ( finite_finite_real @ B3 )
=> ( finite_finite_real @ A2 ) ) ) ).
% finite_subset
thf(fact_88_finite__subset,axiom,
! [A2: set_int,B3: set_int] :
( ( ord_less_eq_set_int @ A2 @ B3 )
=> ( ( finite_finite_int @ B3 )
=> ( finite_finite_int @ A2 ) ) ) ).
% finite_subset
thf(fact_89_infinite__super,axiom,
! [S: set_real,T: set_real] :
( ( ord_less_eq_set_real @ S @ T )
=> ( ~ ( finite_finite_real @ S )
=> ~ ( finite_finite_real @ T ) ) ) ).
% infinite_super
thf(fact_90_infinite__super,axiom,
! [S: set_int,T: set_int] :
( ( ord_less_eq_set_int @ S @ T )
=> ( ~ ( finite_finite_int @ S )
=> ~ ( finite_finite_int @ T ) ) ) ).
% infinite_super
thf(fact_91_rev__finite__subset,axiom,
! [B3: set_real,A2: set_real] :
( ( finite_finite_real @ B3 )
=> ( ( ord_less_eq_set_real @ A2 @ B3 )
=> ( finite_finite_real @ A2 ) ) ) ).
% rev_finite_subset
thf(fact_92_rev__finite__subset,axiom,
! [B3: set_int,A2: set_int] :
( ( finite_finite_int @ B3 )
=> ( ( ord_less_eq_set_int @ A2 @ B3 )
=> ( finite_finite_int @ A2 ) ) ) ).
% rev_finite_subset
thf(fact_93_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_94_dual__order_Oantisym,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( ord_less_eq_int @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_95_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: real,Z4: real] : ( Y3 = Z4 ) )
= ( ^ [A4: real,B4: real] :
( ( ord_less_eq_real @ B4 @ A4 )
& ( ord_less_eq_real @ A4 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_96_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: int,Z4: int] : ( Y3 = Z4 ) )
= ( ^ [A4: int,B4: int] :
( ( ord_less_eq_int @ B4 @ A4 )
& ( ord_less_eq_int @ A4 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_97_dual__order_Otrans,axiom,
! [B: real,A: real,C2: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ( ord_less_eq_real @ C2 @ B )
=> ( ord_less_eq_real @ C2 @ A ) ) ) ).
% dual_order.trans
thf(fact_98_dual__order_Otrans,axiom,
! [B: int,A: int,C2: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( ord_less_eq_int @ C2 @ B )
=> ( ord_less_eq_int @ C2 @ A ) ) ) ).
% dual_order.trans
thf(fact_99_linorder__wlog,axiom,
! [P: real > real > $o,A: real,B: real] :
( ! [A5: real,B6: real] :
( ( ord_less_eq_real @ A5 @ B6 )
=> ( P @ A5 @ B6 ) )
=> ( ! [A5: real,B6: real] :
( ( P @ B6 @ A5 )
=> ( P @ A5 @ B6 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_100_linorder__wlog,axiom,
! [P: int > int > $o,A: int,B: int] :
( ! [A5: int,B6: int] :
( ( ord_less_eq_int @ A5 @ B6 )
=> ( P @ A5 @ B6 ) )
=> ( ! [A5: int,B6: int] :
( ( P @ B6 @ A5 )
=> ( P @ A5 @ B6 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_101_dual__order_Orefl,axiom,
! [A: real] : ( ord_less_eq_real @ A @ A ) ).
% dual_order.refl
thf(fact_102_dual__order_Orefl,axiom,
! [A: int] : ( ord_less_eq_int @ A @ A ) ).
% dual_order.refl
thf(fact_103_order__trans,axiom,
! [X2: real,Y2: real,Z3: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ( ord_less_eq_real @ Y2 @ Z3 )
=> ( ord_less_eq_real @ X2 @ Z3 ) ) ) ).
% order_trans
thf(fact_104_order__trans,axiom,
! [X2: int,Y2: int,Z3: int] :
( ( ord_less_eq_int @ X2 @ Y2 )
=> ( ( ord_less_eq_int @ Y2 @ Z3 )
=> ( ord_less_eq_int @ X2 @ Z3 ) ) ) ).
% order_trans
thf(fact_105_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_106_order__class_Oorder_Oantisym,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ B @ A )
=> ( A = B ) ) ) ).
% order_class.order.antisym
thf(fact_107_ord__le__eq__trans,axiom,
! [A: real,B: real,C2: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( B = C2 )
=> ( ord_less_eq_real @ A @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_108_ord__le__eq__trans,axiom,
! [A: int,B: int,C2: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( B = C2 )
=> ( ord_less_eq_int @ A @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_109_ord__eq__le__trans,axiom,
! [A: real,B: real,C2: real] :
( ( A = B )
=> ( ( ord_less_eq_real @ B @ C2 )
=> ( ord_less_eq_real @ A @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_110_ord__eq__le__trans,axiom,
! [A: int,B: int,C2: int] :
( ( A = B )
=> ( ( ord_less_eq_int @ B @ C2 )
=> ( ord_less_eq_int @ A @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_111_order__class_Oorder_Oeq__iff,axiom,
( ( ^ [Y3: real,Z4: real] : ( Y3 = Z4 ) )
= ( ^ [A4: real,B4: real] :
( ( ord_less_eq_real @ A4 @ B4 )
& ( ord_less_eq_real @ B4 @ A4 ) ) ) ) ).
% order_class.order.eq_iff
thf(fact_112_order__class_Oorder_Oeq__iff,axiom,
( ( ^ [Y3: int,Z4: int] : ( Y3 = Z4 ) )
= ( ^ [A4: int,B4: int] :
( ( ord_less_eq_int @ A4 @ B4 )
& ( ord_less_eq_int @ B4 @ A4 ) ) ) ) ).
% order_class.order.eq_iff
thf(fact_113_antisym__conv,axiom,
! [Y2: real,X2: real] :
( ( ord_less_eq_real @ Y2 @ X2 )
=> ( ( ord_less_eq_real @ X2 @ Y2 )
= ( X2 = Y2 ) ) ) ).
% antisym_conv
thf(fact_114_antisym__conv,axiom,
! [Y2: int,X2: int] :
( ( ord_less_eq_int @ Y2 @ X2 )
=> ( ( ord_less_eq_int @ X2 @ Y2 )
= ( X2 = Y2 ) ) ) ).
% antisym_conv
thf(fact_115_le__cases3,axiom,
! [X2: real,Y2: real,Z3: real] :
( ( ( ord_less_eq_real @ X2 @ Y2 )
=> ~ ( ord_less_eq_real @ Y2 @ Z3 ) )
=> ( ( ( ord_less_eq_real @ Y2 @ X2 )
=> ~ ( ord_less_eq_real @ X2 @ Z3 ) )
=> ( ( ( ord_less_eq_real @ X2 @ Z3 )
=> ~ ( ord_less_eq_real @ Z3 @ Y2 ) )
=> ( ( ( ord_less_eq_real @ Z3 @ Y2 )
=> ~ ( ord_less_eq_real @ Y2 @ X2 ) )
=> ( ( ( ord_less_eq_real @ Y2 @ Z3 )
=> ~ ( ord_less_eq_real @ Z3 @ X2 ) )
=> ~ ( ( ord_less_eq_real @ Z3 @ X2 )
=> ~ ( ord_less_eq_real @ X2 @ Y2 ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_116_le__cases3,axiom,
! [X2: int,Y2: int,Z3: int] :
( ( ( ord_less_eq_int @ X2 @ Y2 )
=> ~ ( ord_less_eq_int @ Y2 @ Z3 ) )
=> ( ( ( ord_less_eq_int @ Y2 @ X2 )
=> ~ ( ord_less_eq_int @ X2 @ Z3 ) )
=> ( ( ( ord_less_eq_int @ X2 @ Z3 )
=> ~ ( ord_less_eq_int @ Z3 @ Y2 ) )
=> ( ( ( ord_less_eq_int @ Z3 @ Y2 )
=> ~ ( ord_less_eq_int @ Y2 @ X2 ) )
=> ( ( ( ord_less_eq_int @ Y2 @ Z3 )
=> ~ ( ord_less_eq_int @ Z3 @ X2 ) )
=> ~ ( ( ord_less_eq_int @ Z3 @ X2 )
=> ~ ( ord_less_eq_int @ X2 @ Y2 ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_117_order_Otrans,axiom,
! [A: real,B: real,C2: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ B @ C2 )
=> ( ord_less_eq_real @ A @ C2 ) ) ) ).
% order.trans
thf(fact_118_order_Otrans,axiom,
! [A: int,B: int,C2: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ B @ C2 )
=> ( ord_less_eq_int @ A @ C2 ) ) ) ).
% order.trans
thf(fact_119_le__cases,axiom,
! [X2: real,Y2: real] :
( ~ ( ord_less_eq_real @ X2 @ Y2 )
=> ( ord_less_eq_real @ Y2 @ X2 ) ) ).
% le_cases
thf(fact_120_le__cases,axiom,
! [X2: int,Y2: int] :
( ~ ( ord_less_eq_int @ X2 @ Y2 )
=> ( ord_less_eq_int @ Y2 @ X2 ) ) ).
% le_cases
thf(fact_121_eq__refl,axiom,
! [X2: real,Y2: real] :
( ( X2 = Y2 )
=> ( ord_less_eq_real @ X2 @ Y2 ) ) ).
% eq_refl
thf(fact_122_eq__refl,axiom,
! [X2: int,Y2: int] :
( ( X2 = Y2 )
=> ( ord_less_eq_int @ X2 @ Y2 ) ) ).
% eq_refl
thf(fact_123_linear,axiom,
! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
| ( ord_less_eq_real @ Y2 @ X2 ) ) ).
% linear
thf(fact_124_linear,axiom,
! [X2: int,Y2: int] :
( ( ord_less_eq_int @ X2 @ Y2 )
| ( ord_less_eq_int @ Y2 @ X2 ) ) ).
% linear
thf(fact_125_antisym,axiom,
! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ( ord_less_eq_real @ Y2 @ X2 )
=> ( X2 = Y2 ) ) ) ).
% antisym
thf(fact_126_antisym,axiom,
! [X2: int,Y2: int] :
( ( ord_less_eq_int @ X2 @ Y2 )
=> ( ( ord_less_eq_int @ Y2 @ X2 )
=> ( X2 = Y2 ) ) ) ).
% antisym
thf(fact_127_eq__iff,axiom,
( ( ^ [Y3: real,Z4: real] : ( Y3 = Z4 ) )
= ( ^ [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
& ( ord_less_eq_real @ Y @ X ) ) ) ) ).
% eq_iff
thf(fact_128_eq__iff,axiom,
( ( ^ [Y3: int,Z4: int] : ( Y3 = Z4 ) )
= ( ^ [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
& ( ord_less_eq_int @ Y @ X ) ) ) ) ).
% eq_iff
thf(fact_129_ord__le__eq__subst,axiom,
! [A: real,B: real,F: real > real,C2: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X4: real,Y4: real] :
( ( ord_less_eq_real @ X4 @ Y4 )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_130_ord__le__eq__subst,axiom,
! [A: real,B: real,F: real > int,C2: int] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X4: real,Y4: real] :
( ( ord_less_eq_real @ X4 @ Y4 )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_131_ord__le__eq__subst,axiom,
! [A: int,B: int,F: int > real,C2: real] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X4: int,Y4: int] :
( ( ord_less_eq_int @ X4 @ Y4 )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_132_ord__le__eq__subst,axiom,
! [A: int,B: int,F: int > int,C2: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X4: int,Y4: int] :
( ( ord_less_eq_int @ X4 @ Y4 )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_133_ord__eq__le__subst,axiom,
! [A: real,F: real > real,B: real,C2: real] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C2 )
=> ( ! [X4: real,Y4: real] :
( ( ord_less_eq_real @ X4 @ Y4 )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_134_ord__eq__le__subst,axiom,
! [A: int,F: real > int,B: real,C2: real] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C2 )
=> ( ! [X4: real,Y4: real] :
( ( ord_less_eq_real @ X4 @ Y4 )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_135_ord__eq__le__subst,axiom,
! [A: real,F: int > real,B: int,C2: int] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C2 )
=> ( ! [X4: int,Y4: int] :
( ( ord_less_eq_int @ X4 @ Y4 )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_136_ord__eq__le__subst,axiom,
! [A: int,F: int > int,B: int,C2: int] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C2 )
=> ( ! [X4: int,Y4: int] :
( ( ord_less_eq_int @ X4 @ Y4 )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_137_order__subst2,axiom,
! [A: real,B: real,F: real > real,C2: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C2 )
=> ( ! [X4: real,Y4: real] :
( ( ord_less_eq_real @ X4 @ Y4 )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_138_order__subst2,axiom,
! [A: real,B: real,F: real > int,C2: int] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C2 )
=> ( ! [X4: real,Y4: real] :
( ( ord_less_eq_real @ X4 @ Y4 )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_139_order__subst2,axiom,
! [A: int,B: int,F: int > real,C2: real] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C2 )
=> ( ! [X4: int,Y4: int] :
( ( ord_less_eq_int @ X4 @ Y4 )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_140_order__subst2,axiom,
! [A: int,B: int,F: int > int,C2: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C2 )
=> ( ! [X4: int,Y4: int] :
( ( ord_less_eq_int @ X4 @ Y4 )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_141_order__subst1,axiom,
! [A: real,F: real > real,B: real,C2: real] :
( ( ord_less_eq_real @ A @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C2 )
=> ( ! [X4: real,Y4: real] :
( ( ord_less_eq_real @ X4 @ Y4 )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_142_order__subst1,axiom,
! [A: real,F: int > real,B: int,C2: int] :
( ( ord_less_eq_real @ A @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C2 )
=> ( ! [X4: int,Y4: int] :
( ( ord_less_eq_int @ X4 @ Y4 )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_143_order__subst1,axiom,
! [A: int,F: real > int,B: real,C2: real] :
( ( ord_less_eq_int @ A @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C2 )
=> ( ! [X4: real,Y4: real] :
( ( ord_less_eq_real @ X4 @ Y4 )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_144_order__subst1,axiom,
! [A: int,F: int > int,B: int,C2: int] :
( ( ord_less_eq_int @ A @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C2 )
=> ( ! [X4: int,Y4: int] :
( ( ord_less_eq_int @ X4 @ Y4 )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_145_ord__eq__less__subst,axiom,
! [A: real,F: real > real,B: real,C2: real] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_real @ B @ C2 )
=> ( ! [X4: real,Y4: real] :
( ( ord_less_real @ X4 @ Y4 )
=> ( ord_less_real @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_146_ord__eq__less__subst,axiom,
! [A: int,F: real > int,B: real,C2: real] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_real @ B @ C2 )
=> ( ! [X4: real,Y4: real] :
( ( ord_less_real @ X4 @ Y4 )
=> ( ord_less_int @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_147_ord__eq__less__subst,axiom,
! [A: real,F: int > real,B: int,C2: int] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_int @ B @ C2 )
=> ( ! [X4: int,Y4: int] :
( ( ord_less_int @ X4 @ Y4 )
=> ( ord_less_real @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_148_ord__eq__less__subst,axiom,
! [A: int,F: int > int,B: int,C2: int] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_int @ B @ C2 )
=> ( ! [X4: int,Y4: int] :
( ( ord_less_int @ X4 @ Y4 )
=> ( ord_less_int @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_149_ord__less__eq__subst,axiom,
! [A: real,B: real,F: real > real,C2: real] :
( ( ord_less_real @ A @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X4: real,Y4: real] :
( ( ord_less_real @ X4 @ Y4 )
=> ( ord_less_real @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_less_eq_subst
thf(fact_150_ord__less__eq__subst,axiom,
! [A: real,B: real,F: real > int,C2: int] :
( ( ord_less_real @ A @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X4: real,Y4: real] :
( ( ord_less_real @ X4 @ Y4 )
=> ( ord_less_int @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_less_eq_subst
thf(fact_151_ord__less__eq__subst,axiom,
! [A: int,B: int,F: int > real,C2: real] :
( ( ord_less_int @ A @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X4: int,Y4: int] :
( ( ord_less_int @ X4 @ Y4 )
=> ( ord_less_real @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_less_eq_subst
thf(fact_152_ord__less__eq__subst,axiom,
! [A: int,B: int,F: int > int,C2: int] :
( ( ord_less_int @ A @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X4: int,Y4: int] :
( ( ord_less_int @ X4 @ Y4 )
=> ( ord_less_int @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_less_eq_subst
thf(fact_153_order__less__subst1,axiom,
! [A: real,F: real > real,B: real,C2: real] :
( ( ord_less_real @ A @ ( F @ B ) )
=> ( ( ord_less_real @ B @ C2 )
=> ( ! [X4: real,Y4: real] :
( ( ord_less_real @ X4 @ Y4 )
=> ( ord_less_real @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ A @ ( F @ C2 ) ) ) ) ) ).
% order_less_subst1
thf(fact_154_order__less__subst1,axiom,
! [A: real,F: int > real,B: int,C2: int] :
( ( ord_less_real @ A @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C2 )
=> ( ! [X4: int,Y4: int] :
( ( ord_less_int @ X4 @ Y4 )
=> ( ord_less_real @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ A @ ( F @ C2 ) ) ) ) ) ).
% order_less_subst1
thf(fact_155_order__less__subst1,axiom,
! [A: int,F: real > int,B: real,C2: real] :
( ( ord_less_int @ A @ ( F @ B ) )
=> ( ( ord_less_real @ B @ C2 )
=> ( ! [X4: real,Y4: real] :
( ( ord_less_real @ X4 @ Y4 )
=> ( ord_less_int @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ A @ ( F @ C2 ) ) ) ) ) ).
% order_less_subst1
thf(fact_156_order__less__subst1,axiom,
! [A: int,F: int > int,B: int,C2: int] :
( ( ord_less_int @ A @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C2 )
=> ( ! [X4: int,Y4: int] :
( ( ord_less_int @ X4 @ Y4 )
=> ( ord_less_int @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ A @ ( F @ C2 ) ) ) ) ) ).
% order_less_subst1
thf(fact_157_order__less__subst2,axiom,
! [A: real,B: real,F: real > real,C2: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ ( F @ B ) @ C2 )
=> ( ! [X4: real,Y4: real] :
( ( ord_less_real @ X4 @ Y4 )
=> ( ord_less_real @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C2 ) ) ) ) ).
% order_less_subst2
thf(fact_158_order__less__subst2,axiom,
! [A: real,B: real,F: real > int,C2: int] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C2 )
=> ( ! [X4: real,Y4: real] :
( ( ord_less_real @ X4 @ Y4 )
=> ( ord_less_int @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C2 ) ) ) ) ).
% order_less_subst2
thf(fact_159_order__less__subst2,axiom,
! [A: int,B: int,F: int > real,C2: real] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_real @ ( F @ B ) @ C2 )
=> ( ! [X4: int,Y4: int] :
( ( ord_less_int @ X4 @ Y4 )
=> ( ord_less_real @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C2 ) ) ) ) ).
% order_less_subst2
thf(fact_160_order__less__subst2,axiom,
! [A: int,B: int,F: int > int,C2: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C2 )
=> ( ! [X4: int,Y4: int] :
( ( ord_less_int @ X4 @ Y4 )
=> ( ord_less_int @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C2 ) ) ) ) ).
% order_less_subst2
thf(fact_161_lt__ex,axiom,
! [X2: real] :
? [Y4: real] : ( ord_less_real @ Y4 @ X2 ) ).
% lt_ex
thf(fact_162_lt__ex,axiom,
! [X2: int] :
? [Y4: int] : ( ord_less_int @ Y4 @ X2 ) ).
% lt_ex
thf(fact_163_gt__ex,axiom,
! [X2: real] :
? [X_1: real] : ( ord_less_real @ X2 @ X_1 ) ).
% gt_ex
thf(fact_164_gt__ex,axiom,
! [X2: int] :
? [X_1: int] : ( ord_less_int @ X2 @ X_1 ) ).
% gt_ex
thf(fact_165_neqE,axiom,
! [X2: real,Y2: real] :
( ( X2 != Y2 )
=> ( ~ ( ord_less_real @ X2 @ Y2 )
=> ( ord_less_real @ Y2 @ X2 ) ) ) ).
% neqE
thf(fact_166_neqE,axiom,
! [X2: int,Y2: int] :
( ( X2 != Y2 )
=> ( ~ ( ord_less_int @ X2 @ Y2 )
=> ( ord_less_int @ Y2 @ X2 ) ) ) ).
% neqE
thf(fact_167_neq__iff,axiom,
! [X2: real,Y2: real] :
( ( X2 != Y2 )
= ( ( ord_less_real @ X2 @ Y2 )
| ( ord_less_real @ Y2 @ X2 ) ) ) ).
% neq_iff
thf(fact_168_neq__iff,axiom,
! [X2: int,Y2: int] :
( ( X2 != Y2 )
= ( ( ord_less_int @ X2 @ Y2 )
| ( ord_less_int @ Y2 @ X2 ) ) ) ).
% neq_iff
thf(fact_169_order_Oasym,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ~ ( ord_less_real @ B @ A ) ) ).
% order.asym
thf(fact_170_order_Oasym,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ~ ( ord_less_int @ B @ A ) ) ).
% order.asym
thf(fact_171_dense,axiom,
! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ? [Z5: real] :
( ( ord_less_real @ X2 @ Z5 )
& ( ord_less_real @ Z5 @ Y2 ) ) ) ).
% dense
thf(fact_172_less__imp__neq,axiom,
! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ( X2 != Y2 ) ) ).
% less_imp_neq
thf(fact_173_less__imp__neq,axiom,
! [X2: int,Y2: int] :
( ( ord_less_int @ X2 @ Y2 )
=> ( X2 != Y2 ) ) ).
% less_imp_neq
thf(fact_174_less__asym,axiom,
! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ~ ( ord_less_real @ Y2 @ X2 ) ) ).
% less_asym
thf(fact_175_less__asym,axiom,
! [X2: int,Y2: int] :
( ( ord_less_int @ X2 @ Y2 )
=> ~ ( ord_less_int @ Y2 @ X2 ) ) ).
% less_asym
thf(fact_176_less__asym_H,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ~ ( ord_less_real @ B @ A ) ) ).
% less_asym'
thf(fact_177_less__asym_H,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ~ ( ord_less_int @ B @ A ) ) ).
% less_asym'
thf(fact_178_less__trans,axiom,
! [X2: real,Y2: real,Z3: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ( ( ord_less_real @ Y2 @ Z3 )
=> ( ord_less_real @ X2 @ Z3 ) ) ) ).
% less_trans
thf(fact_179_less__trans,axiom,
! [X2: int,Y2: int,Z3: int] :
( ( ord_less_int @ X2 @ Y2 )
=> ( ( ord_less_int @ Y2 @ Z3 )
=> ( ord_less_int @ X2 @ Z3 ) ) ) ).
% less_trans
thf(fact_180_less__linear,axiom,
! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
| ( X2 = Y2 )
| ( ord_less_real @ Y2 @ X2 ) ) ).
% less_linear
thf(fact_181_less__linear,axiom,
! [X2: int,Y2: int] :
( ( ord_less_int @ X2 @ Y2 )
| ( X2 = Y2 )
| ( ord_less_int @ Y2 @ X2 ) ) ).
% less_linear
thf(fact_182_less__irrefl,axiom,
! [X2: real] :
~ ( ord_less_real @ X2 @ X2 ) ).
% less_irrefl
thf(fact_183_less__irrefl,axiom,
! [X2: int] :
~ ( ord_less_int @ X2 @ X2 ) ).
% less_irrefl
thf(fact_184_ord__eq__less__trans,axiom,
! [A: real,B: real,C2: real] :
( ( A = B )
=> ( ( ord_less_real @ B @ C2 )
=> ( ord_less_real @ A @ C2 ) ) ) ).
% ord_eq_less_trans
thf(fact_185_ord__eq__less__trans,axiom,
! [A: int,B: int,C2: int] :
( ( A = B )
=> ( ( ord_less_int @ B @ C2 )
=> ( ord_less_int @ A @ C2 ) ) ) ).
% ord_eq_less_trans
thf(fact_186_ord__less__eq__trans,axiom,
! [A: real,B: real,C2: real] :
( ( ord_less_real @ A @ B )
=> ( ( B = C2 )
=> ( ord_less_real @ A @ C2 ) ) ) ).
% ord_less_eq_trans
thf(fact_187_ord__less__eq__trans,axiom,
! [A: int,B: int,C2: int] :
( ( ord_less_int @ A @ B )
=> ( ( B = C2 )
=> ( ord_less_int @ A @ C2 ) ) ) ).
% ord_less_eq_trans
thf(fact_188_dual__order_Oasym,axiom,
! [B: real,A: real] :
( ( ord_less_real @ B @ A )
=> ~ ( ord_less_real @ A @ B ) ) ).
% dual_order.asym
thf(fact_189_dual__order_Oasym,axiom,
! [B: int,A: int] :
( ( ord_less_int @ B @ A )
=> ~ ( ord_less_int @ A @ B ) ) ).
% dual_order.asym
thf(fact_190_less__imp__not__eq,axiom,
! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ( X2 != Y2 ) ) ).
% less_imp_not_eq
thf(fact_191_less__imp__not__eq,axiom,
! [X2: int,Y2: int] :
( ( ord_less_int @ X2 @ Y2 )
=> ( X2 != Y2 ) ) ).
% less_imp_not_eq
thf(fact_192_less__not__sym,axiom,
! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ~ ( ord_less_real @ Y2 @ X2 ) ) ).
% less_not_sym
thf(fact_193_less__not__sym,axiom,
! [X2: int,Y2: int] :
( ( ord_less_int @ X2 @ Y2 )
=> ~ ( ord_less_int @ Y2 @ X2 ) ) ).
% less_not_sym
thf(fact_194_antisym__conv3,axiom,
! [Y2: real,X2: real] :
( ~ ( ord_less_real @ Y2 @ X2 )
=> ( ( ~ ( ord_less_real @ X2 @ Y2 ) )
= ( X2 = Y2 ) ) ) ).
% antisym_conv3
thf(fact_195_antisym__conv3,axiom,
! [Y2: int,X2: int] :
( ~ ( ord_less_int @ Y2 @ X2 )
=> ( ( ~ ( ord_less_int @ X2 @ Y2 ) )
= ( X2 = Y2 ) ) ) ).
% antisym_conv3
thf(fact_196_less__imp__not__eq2,axiom,
! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ( Y2 != X2 ) ) ).
% less_imp_not_eq2
thf(fact_197_less__imp__not__eq2,axiom,
! [X2: int,Y2: int] :
( ( ord_less_int @ X2 @ Y2 )
=> ( Y2 != X2 ) ) ).
% less_imp_not_eq2
thf(fact_198_less__imp__triv,axiom,
! [X2: real,Y2: real,P: $o] :
( ( ord_less_real @ X2 @ Y2 )
=> ( ( ord_less_real @ Y2 @ X2 )
=> P ) ) ).
% less_imp_triv
thf(fact_199_less__imp__triv,axiom,
! [X2: int,Y2: int,P: $o] :
( ( ord_less_int @ X2 @ Y2 )
=> ( ( ord_less_int @ Y2 @ X2 )
=> P ) ) ).
% less_imp_triv
thf(fact_200_linorder__cases,axiom,
! [X2: real,Y2: real] :
( ~ ( ord_less_real @ X2 @ Y2 )
=> ( ( X2 != Y2 )
=> ( ord_less_real @ Y2 @ X2 ) ) ) ).
% linorder_cases
thf(fact_201_linorder__cases,axiom,
! [X2: int,Y2: int] :
( ~ ( ord_less_int @ X2 @ Y2 )
=> ( ( X2 != Y2 )
=> ( ord_less_int @ Y2 @ X2 ) ) ) ).
% linorder_cases
thf(fact_202_dual__order_Oirrefl,axiom,
! [A: real] :
~ ( ord_less_real @ A @ A ) ).
% dual_order.irrefl
thf(fact_203_dual__order_Oirrefl,axiom,
! [A: int] :
~ ( ord_less_int @ A @ A ) ).
% dual_order.irrefl
thf(fact_204_order_Ostrict__trans,axiom,
! [A: real,B: real,C2: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ B @ C2 )
=> ( ord_less_real @ A @ C2 ) ) ) ).
% order.strict_trans
thf(fact_205_order_Ostrict__trans,axiom,
! [A: int,B: int,C2: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ B @ C2 )
=> ( ord_less_int @ A @ C2 ) ) ) ).
% order.strict_trans
thf(fact_206_less__imp__not__less,axiom,
! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ~ ( ord_less_real @ Y2 @ X2 ) ) ).
% less_imp_not_less
thf(fact_207_less__imp__not__less,axiom,
! [X2: int,Y2: int] :
( ( ord_less_int @ X2 @ Y2 )
=> ~ ( ord_less_int @ Y2 @ X2 ) ) ).
% less_imp_not_less
thf(fact_208_linorder__less__wlog,axiom,
! [P: real > real > $o,A: real,B: real] :
( ! [A5: real,B6: real] :
( ( ord_less_real @ A5 @ B6 )
=> ( P @ A5 @ B6 ) )
=> ( ! [A5: real] : ( P @ A5 @ A5 )
=> ( ! [A5: real,B6: real] :
( ( P @ B6 @ A5 )
=> ( P @ A5 @ B6 ) )
=> ( P @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_209_linorder__less__wlog,axiom,
! [P: int > int > $o,A: int,B: int] :
( ! [A5: int,B6: int] :
( ( ord_less_int @ A5 @ B6 )
=> ( P @ A5 @ B6 ) )
=> ( ! [A5: int] : ( P @ A5 @ A5 )
=> ( ! [A5: int,B6: int] :
( ( P @ B6 @ A5 )
=> ( P @ A5 @ B6 ) )
=> ( P @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_210_dual__order_Ostrict__trans,axiom,
! [B: real,A: real,C2: real] :
( ( ord_less_real @ B @ A )
=> ( ( ord_less_real @ C2 @ B )
=> ( ord_less_real @ C2 @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_211_dual__order_Ostrict__trans,axiom,
! [B: int,A: int,C2: int] :
( ( ord_less_int @ B @ A )
=> ( ( ord_less_int @ C2 @ B )
=> ( ord_less_int @ C2 @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_212_not__less__iff__gr__or__eq,axiom,
! [X2: real,Y2: real] :
( ( ~ ( ord_less_real @ X2 @ Y2 ) )
= ( ( ord_less_real @ Y2 @ X2 )
| ( X2 = Y2 ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_213_not__less__iff__gr__or__eq,axiom,
! [X2: int,Y2: int] :
( ( ~ ( ord_less_int @ X2 @ Y2 ) )
= ( ( ord_less_int @ Y2 @ X2 )
| ( X2 = Y2 ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_214_order_Ostrict__implies__not__eq,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_215_order_Ostrict__implies__not__eq,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_216_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_217_dual__order_Ostrict__implies__not__eq,axiom,
! [B: int,A: int] :
( ( ord_less_int @ B @ A )
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_218_ex__gt__or__lt,axiom,
! [A: real] :
? [B6: real] :
( ( ord_less_real @ A @ B6 )
| ( ord_less_real @ B6 @ A ) ) ).
% ex_gt_or_lt
thf(fact_219_complete__real,axiom,
! [S: set_real] :
( ? [X3: real] : ( member_real @ X3 @ S )
=> ( ? [Z: real] :
! [X4: real] :
( ( member_real @ X4 @ S )
=> ( ord_less_eq_real @ X4 @ Z ) )
=> ? [Y4: real] :
( ! [X3: real] :
( ( member_real @ X3 @ S )
=> ( ord_less_eq_real @ X3 @ Y4 ) )
& ! [Z: real] :
( ! [X4: real] :
( ( member_real @ X4 @ S )
=> ( ord_less_eq_real @ X4 @ Z ) )
=> ( ord_less_eq_real @ Y4 @ Z ) ) ) ) ) ).
% complete_real
thf(fact_220_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_221_leD,axiom,
! [Y2: real,X2: real] :
( ( ord_less_eq_real @ Y2 @ X2 )
=> ~ ( ord_less_real @ X2 @ Y2 ) ) ).
% leD
thf(fact_222_leD,axiom,
! [Y2: int,X2: int] :
( ( ord_less_eq_int @ Y2 @ X2 )
=> ~ ( ord_less_int @ X2 @ Y2 ) ) ).
% leD
thf(fact_223_leI,axiom,
! [X2: real,Y2: real] :
( ~ ( ord_less_real @ X2 @ Y2 )
=> ( ord_less_eq_real @ Y2 @ X2 ) ) ).
% leI
thf(fact_224_leI,axiom,
! [X2: int,Y2: int] :
( ~ ( ord_less_int @ X2 @ Y2 )
=> ( ord_less_eq_int @ Y2 @ X2 ) ) ).
% leI
thf(fact_225_le__less,axiom,
( ord_less_eq_real
= ( ^ [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
| ( X = Y ) ) ) ) ).
% le_less
thf(fact_226_le__less,axiom,
( ord_less_eq_int
= ( ^ [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
| ( X = Y ) ) ) ) ).
% le_less
thf(fact_227_less__le,axiom,
( ord_less_real
= ( ^ [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
& ( X != Y ) ) ) ) ).
% less_le
thf(fact_228_less__le,axiom,
( ord_less_int
= ( ^ [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
& ( X != Y ) ) ) ) ).
% less_le
thf(fact_229_order__le__less__subst1,axiom,
! [A: real,F: real > real,B: real,C2: real] :
( ( ord_less_eq_real @ A @ ( F @ B ) )
=> ( ( ord_less_real @ B @ C2 )
=> ( ! [X4: real,Y4: real] :
( ( ord_less_real @ X4 @ Y4 )
=> ( ord_less_real @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ A @ ( F @ C2 ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_230_order__le__less__subst1,axiom,
! [A: real,F: int > real,B: int,C2: int] :
( ( ord_less_eq_real @ A @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C2 )
=> ( ! [X4: int,Y4: int] :
( ( ord_less_int @ X4 @ Y4 )
=> ( ord_less_real @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ A @ ( F @ C2 ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_231_order__le__less__subst1,axiom,
! [A: int,F: real > int,B: real,C2: real] :
( ( ord_less_eq_int @ A @ ( F @ B ) )
=> ( ( ord_less_real @ B @ C2 )
=> ( ! [X4: real,Y4: real] :
( ( ord_less_real @ X4 @ Y4 )
=> ( ord_less_int @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ A @ ( F @ C2 ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_232_order__le__less__subst1,axiom,
! [A: int,F: int > int,B: int,C2: int] :
( ( ord_less_eq_int @ A @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C2 )
=> ( ! [X4: int,Y4: int] :
( ( ord_less_int @ X4 @ Y4 )
=> ( ord_less_int @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ A @ ( F @ C2 ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_233_order__le__less__subst2,axiom,
! [A: real,B: real,F: real > real,C2: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_real @ ( F @ B ) @ C2 )
=> ( ! [X4: real,Y4: real] :
( ( ord_less_eq_real @ X4 @ Y4 )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_234_order__le__less__subst2,axiom,
! [A: real,B: real,F: real > int,C2: int] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C2 )
=> ( ! [X4: real,Y4: real] :
( ( ord_less_eq_real @ X4 @ Y4 )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_235_order__le__less__subst2,axiom,
! [A: int,B: int,F: int > real,C2: real] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_real @ ( F @ B ) @ C2 )
=> ( ! [X4: int,Y4: int] :
( ( ord_less_eq_int @ X4 @ Y4 )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_236_order__le__less__subst2,axiom,
! [A: int,B: int,F: int > int,C2: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C2 )
=> ( ! [X4: int,Y4: int] :
( ( ord_less_eq_int @ X4 @ Y4 )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_237_order__less__le__subst1,axiom,
! [A: real,F: real > real,B: real,C2: real] :
( ( ord_less_real @ A @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C2 )
=> ( ! [X4: real,Y4: real] :
( ( ord_less_eq_real @ X4 @ Y4 )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ A @ ( F @ C2 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_238_order__less__le__subst1,axiom,
! [A: int,F: real > int,B: real,C2: real] :
( ( ord_less_int @ A @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C2 )
=> ( ! [X4: real,Y4: real] :
( ( ord_less_eq_real @ X4 @ Y4 )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ A @ ( F @ C2 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_239_order__less__le__subst1,axiom,
! [A: real,F: int > real,B: int,C2: int] :
( ( ord_less_real @ A @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C2 )
=> ( ! [X4: int,Y4: int] :
( ( ord_less_eq_int @ X4 @ Y4 )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ A @ ( F @ C2 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_240_order__less__le__subst1,axiom,
! [A: int,F: int > int,B: int,C2: int] :
( ( ord_less_int @ A @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C2 )
=> ( ! [X4: int,Y4: int] :
( ( ord_less_eq_int @ X4 @ Y4 )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ A @ ( F @ C2 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_241_order__less__le__subst2,axiom,
! [A: real,B: real,F: real > real,C2: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C2 )
=> ( ! [X4: real,Y4: real] :
( ( ord_less_real @ X4 @ Y4 )
=> ( ord_less_real @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C2 ) ) ) ) ).
% order_less_le_subst2
thf(fact_242_order__less__le__subst2,axiom,
! [A: int,B: int,F: int > real,C2: real] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C2 )
=> ( ! [X4: int,Y4: int] :
( ( ord_less_int @ X4 @ Y4 )
=> ( ord_less_real @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C2 ) ) ) ) ).
% order_less_le_subst2
thf(fact_243_order__less__le__subst2,axiom,
! [A: real,B: real,F: real > int,C2: int] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C2 )
=> ( ! [X4: real,Y4: real] :
( ( ord_less_real @ X4 @ Y4 )
=> ( ord_less_int @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C2 ) ) ) ) ).
% order_less_le_subst2
thf(fact_244_order__less__le__subst2,axiom,
! [A: int,B: int,F: int > int,C2: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C2 )
=> ( ! [X4: int,Y4: int] :
( ( ord_less_int @ X4 @ Y4 )
=> ( ord_less_int @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C2 ) ) ) ) ).
% order_less_le_subst2
thf(fact_245_not__le,axiom,
! [X2: real,Y2: real] :
( ( ~ ( ord_less_eq_real @ X2 @ Y2 ) )
= ( ord_less_real @ Y2 @ X2 ) ) ).
% not_le
thf(fact_246_not__le,axiom,
! [X2: int,Y2: int] :
( ( ~ ( ord_less_eq_int @ X2 @ Y2 ) )
= ( ord_less_int @ Y2 @ X2 ) ) ).
% not_le
thf(fact_247_not__less,axiom,
! [X2: real,Y2: real] :
( ( ~ ( ord_less_real @ X2 @ Y2 ) )
= ( ord_less_eq_real @ Y2 @ X2 ) ) ).
% not_less
thf(fact_248_not__less,axiom,
! [X2: int,Y2: int] :
( ( ~ ( ord_less_int @ X2 @ Y2 ) )
= ( ord_less_eq_int @ Y2 @ X2 ) ) ).
% not_less
thf(fact_249_le__neq__trans,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( A != B )
=> ( ord_less_real @ A @ B ) ) ) ).
% le_neq_trans
thf(fact_250_le__neq__trans,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( A != B )
=> ( ord_less_int @ A @ B ) ) ) ).
% le_neq_trans
thf(fact_251_antisym__conv1,axiom,
! [X2: real,Y2: real] :
( ~ ( ord_less_real @ X2 @ Y2 )
=> ( ( ord_less_eq_real @ X2 @ Y2 )
= ( X2 = Y2 ) ) ) ).
% antisym_conv1
thf(fact_252_antisym__conv1,axiom,
! [X2: int,Y2: int] :
( ~ ( ord_less_int @ X2 @ Y2 )
=> ( ( ord_less_eq_int @ X2 @ Y2 )
= ( X2 = Y2 ) ) ) ).
% antisym_conv1
thf(fact_253_antisym__conv2,axiom,
! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ( ~ ( ord_less_real @ X2 @ Y2 ) )
= ( X2 = Y2 ) ) ) ).
% antisym_conv2
thf(fact_254_antisym__conv2,axiom,
! [X2: int,Y2: int] :
( ( ord_less_eq_int @ X2 @ Y2 )
=> ( ( ~ ( ord_less_int @ X2 @ Y2 ) )
= ( X2 = Y2 ) ) ) ).
% antisym_conv2
thf(fact_255_less__imp__le,axiom,
! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ( ord_less_eq_real @ X2 @ Y2 ) ) ).
% less_imp_le
thf(fact_256_less__imp__le,axiom,
! [X2: int,Y2: int] :
( ( ord_less_int @ X2 @ Y2 )
=> ( ord_less_eq_int @ X2 @ Y2 ) ) ).
% less_imp_le
thf(fact_257_le__less__trans,axiom,
! [X2: real,Y2: real,Z3: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ( ord_less_real @ Y2 @ Z3 )
=> ( ord_less_real @ X2 @ Z3 ) ) ) ).
% le_less_trans
thf(fact_258_le__less__trans,axiom,
! [X2: int,Y2: int,Z3: int] :
( ( ord_less_eq_int @ X2 @ Y2 )
=> ( ( ord_less_int @ Y2 @ Z3 )
=> ( ord_less_int @ X2 @ Z3 ) ) ) ).
% le_less_trans
thf(fact_259_less__le__trans,axiom,
! [X2: real,Y2: real,Z3: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ( ( ord_less_eq_real @ Y2 @ Z3 )
=> ( ord_less_real @ X2 @ Z3 ) ) ) ).
% less_le_trans
thf(fact_260_less__le__trans,axiom,
! [X2: int,Y2: int,Z3: int] :
( ( ord_less_int @ X2 @ Y2 )
=> ( ( ord_less_eq_int @ Y2 @ Z3 )
=> ( ord_less_int @ X2 @ Z3 ) ) ) ).
% less_le_trans
thf(fact_261_dense__ge,axiom,
! [Z3: real,Y2: real] :
( ! [X4: real] :
( ( ord_less_real @ Z3 @ X4 )
=> ( ord_less_eq_real @ Y2 @ X4 ) )
=> ( ord_less_eq_real @ Y2 @ Z3 ) ) ).
% dense_ge
thf(fact_262_dense__le,axiom,
! [Y2: real,Z3: real] :
( ! [X4: real] :
( ( ord_less_real @ X4 @ Y2 )
=> ( ord_less_eq_real @ X4 @ Z3 ) )
=> ( ord_less_eq_real @ Y2 @ Z3 ) ) ).
% dense_le
thf(fact_263_le__less__linear,axiom,
! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
| ( ord_less_real @ Y2 @ X2 ) ) ).
% le_less_linear
thf(fact_264_le__less__linear,axiom,
! [X2: int,Y2: int] :
( ( ord_less_eq_int @ X2 @ Y2 )
| ( ord_less_int @ Y2 @ X2 ) ) ).
% le_less_linear
thf(fact_265_le__imp__less__or__eq,axiom,
! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ( ord_less_real @ X2 @ Y2 )
| ( X2 = Y2 ) ) ) ).
% le_imp_less_or_eq
thf(fact_266_le__imp__less__or__eq,axiom,
! [X2: int,Y2: int] :
( ( ord_less_eq_int @ X2 @ Y2 )
=> ( ( ord_less_int @ X2 @ Y2 )
| ( X2 = Y2 ) ) ) ).
% le_imp_less_or_eq
thf(fact_267_less__le__not__le,axiom,
( ord_less_real
= ( ^ [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
& ~ ( ord_less_eq_real @ Y @ X ) ) ) ) ).
% less_le_not_le
thf(fact_268_less__le__not__le,axiom,
( ord_less_int
= ( ^ [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
& ~ ( ord_less_eq_int @ Y @ X ) ) ) ) ).
% less_le_not_le
thf(fact_269_not__le__imp__less,axiom,
! [Y2: real,X2: real] :
( ~ ( ord_less_eq_real @ Y2 @ X2 )
=> ( ord_less_real @ X2 @ Y2 ) ) ).
% not_le_imp_less
thf(fact_270_not__le__imp__less,axiom,
! [Y2: int,X2: int] :
( ~ ( ord_less_eq_int @ Y2 @ X2 )
=> ( ord_less_int @ X2 @ Y2 ) ) ).
% not_le_imp_less
thf(fact_271_order_Ostrict__trans1,axiom,
! [A: real,B: real,C2: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_real @ B @ C2 )
=> ( ord_less_real @ A @ C2 ) ) ) ).
% order.strict_trans1
thf(fact_272_order_Ostrict__trans1,axiom,
! [A: int,B: int,C2: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_int @ B @ C2 )
=> ( ord_less_int @ A @ C2 ) ) ) ).
% order.strict_trans1
thf(fact_273_order_Ostrict__trans2,axiom,
! [A: real,B: real,C2: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_eq_real @ B @ C2 )
=> ( ord_less_real @ A @ C2 ) ) ) ).
% order.strict_trans2
thf(fact_274_order_Ostrict__trans2,axiom,
! [A: int,B: int,C2: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_int @ B @ C2 )
=> ( ord_less_int @ A @ C2 ) ) ) ).
% order.strict_trans2
thf(fact_275_order_Oorder__iff__strict,axiom,
( ord_less_eq_real
= ( ^ [A4: real,B4: real] :
( ( ord_less_real @ A4 @ B4 )
| ( A4 = B4 ) ) ) ) ).
% order.order_iff_strict
thf(fact_276_order_Oorder__iff__strict,axiom,
( ord_less_eq_int
= ( ^ [A4: int,B4: int] :
( ( ord_less_int @ A4 @ B4 )
| ( A4 = B4 ) ) ) ) ).
% order.order_iff_strict
thf(fact_277_order_Ostrict__iff__order,axiom,
( ord_less_real
= ( ^ [A4: real,B4: real] :
( ( ord_less_eq_real @ A4 @ B4 )
& ( A4 != B4 ) ) ) ) ).
% order.strict_iff_order
thf(fact_278_order_Ostrict__iff__order,axiom,
( ord_less_int
= ( ^ [A4: int,B4: int] :
( ( ord_less_eq_int @ A4 @ B4 )
& ( A4 != B4 ) ) ) ) ).
% order.strict_iff_order
thf(fact_279_dual__order_Ostrict__trans1,axiom,
! [B: real,A: real,C2: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ( ord_less_real @ C2 @ B )
=> ( ord_less_real @ C2 @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_280_dual__order_Ostrict__trans1,axiom,
! [B: int,A: int,C2: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( ord_less_int @ C2 @ B )
=> ( ord_less_int @ C2 @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_281_dual__order_Ostrict__trans2,axiom,
! [B: real,A: real,C2: real] :
( ( ord_less_real @ B @ A )
=> ( ( ord_less_eq_real @ C2 @ B )
=> ( ord_less_real @ C2 @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_282_dual__order_Ostrict__trans2,axiom,
! [B: int,A: int,C2: int] :
( ( ord_less_int @ B @ A )
=> ( ( ord_less_eq_int @ C2 @ B )
=> ( ord_less_int @ C2 @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_283_dense__ge__bounded,axiom,
! [Z3: real,X2: real,Y2: real] :
( ( ord_less_real @ Z3 @ X2 )
=> ( ! [W: real] :
( ( ord_less_real @ Z3 @ W )
=> ( ( ord_less_real @ W @ X2 )
=> ( ord_less_eq_real @ Y2 @ W ) ) )
=> ( ord_less_eq_real @ Y2 @ Z3 ) ) ) ).
% dense_ge_bounded
thf(fact_284_minf_I8_J,axiom,
! [T2: real] :
? [Z5: real] :
! [X3: real] :
( ( ord_less_real @ X3 @ Z5 )
=> ~ ( ord_less_eq_real @ T2 @ X3 ) ) ).
% minf(8)
thf(fact_285_minf_I8_J,axiom,
! [T2: int] :
? [Z5: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z5 )
=> ~ ( ord_less_eq_int @ T2 @ X3 ) ) ).
% minf(8)
thf(fact_286_minf_I6_J,axiom,
! [T2: real] :
? [Z5: real] :
! [X3: real] :
( ( ord_less_real @ X3 @ Z5 )
=> ( ord_less_eq_real @ X3 @ T2 ) ) ).
% minf(6)
thf(fact_287_minf_I6_J,axiom,
! [T2: int] :
? [Z5: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z5 )
=> ( ord_less_eq_int @ X3 @ T2 ) ) ).
% minf(6)
thf(fact_288_pinf_I8_J,axiom,
! [T2: real] :
? [Z5: real] :
! [X3: real] :
( ( ord_less_real @ Z5 @ X3 )
=> ( ord_less_eq_real @ T2 @ X3 ) ) ).
% pinf(8)
thf(fact_289_pinf_I8_J,axiom,
! [T2: int] :
? [Z5: int] :
! [X3: int] :
( ( ord_less_int @ Z5 @ X3 )
=> ( ord_less_eq_int @ T2 @ X3 ) ) ).
% pinf(8)
thf(fact_290_pinf_I6_J,axiom,
! [T2: real] :
? [Z5: real] :
! [X3: real] :
( ( ord_less_real @ Z5 @ X3 )
=> ~ ( ord_less_eq_real @ X3 @ T2 ) ) ).
% pinf(6)
thf(fact_291_pinf_I6_J,axiom,
! [T2: int] :
? [Z5: int] :
! [X3: int] :
( ( ord_less_int @ Z5 @ X3 )
=> ~ ( ord_less_eq_int @ X3 @ T2 ) ) ).
% pinf(6)
thf(fact_292_verit__comp__simplify1_I3_J,axiom,
! [B7: real,A6: real] :
( ( ~ ( ord_less_eq_real @ B7 @ A6 ) )
= ( ord_less_real @ A6 @ B7 ) ) ).
% verit_comp_simplify1(3)
thf(fact_293_verit__comp__simplify1_I3_J,axiom,
! [B7: int,A6: int] :
( ( ~ ( ord_less_eq_int @ B7 @ A6 ) )
= ( ord_less_int @ A6 @ B7 ) ) ).
% verit_comp_simplify1(3)
thf(fact_294_is__zero__null,axiom,
( is_zero_real
= ( ^ [P3: poly_real] : ( P3 = zero_zero_poly_real ) ) ) ).
% is_zero_null
thf(fact_295_is__zero__null,axiom,
( is_zero_int
= ( ^ [P3: poly_int] : ( P3 = zero_zero_poly_int ) ) ) ).
% is_zero_null
thf(fact_296_is__zero__null,axiom,
( is_zero_poly_real
= ( ^ [P3: poly_poly_real] : ( P3 = zero_z1423781445y_real ) ) ) ).
% is_zero_null
thf(fact_297_poly__cutoff__0,axiom,
! [N: nat] :
( ( poly_cutoff_real @ N @ zero_zero_poly_real )
= zero_zero_poly_real ) ).
% poly_cutoff_0
thf(fact_298_poly__cutoff__0,axiom,
! [N: nat] :
( ( poly_cutoff_int @ N @ zero_zero_poly_int )
= zero_zero_poly_int ) ).
% poly_cutoff_0
thf(fact_299_poly__cutoff__0,axiom,
! [N: nat] :
( ( poly_c1404107022y_real @ N @ zero_z1423781445y_real )
= zero_z1423781445y_real ) ).
% poly_cutoff_0
thf(fact_300_subsetI,axiom,
! [A2: set_real,B3: set_real] :
( ! [X4: real] :
( ( member_real @ X4 @ A2 )
=> ( member_real @ X4 @ B3 ) )
=> ( ord_less_eq_set_real @ A2 @ B3 ) ) ).
% subsetI
thf(fact_301_in__mono,axiom,
! [A2: set_real,B3: set_real,X2: real] :
( ( ord_less_eq_set_real @ A2 @ B3 )
=> ( ( member_real @ X2 @ A2 )
=> ( member_real @ X2 @ B3 ) ) ) ).
% in_mono
thf(fact_302_subsetD,axiom,
! [A2: set_real,B3: set_real,C2: real] :
( ( ord_less_eq_set_real @ A2 @ B3 )
=> ( ( member_real @ C2 @ A2 )
=> ( member_real @ C2 @ B3 ) ) ) ).
% subsetD
thf(fact_303_subset__eq,axiom,
( ord_less_eq_set_real
= ( ^ [A7: set_real,B5: set_real] :
! [X: real] :
( ( member_real @ X @ A7 )
=> ( member_real @ X @ B5 ) ) ) ) ).
% subset_eq
thf(fact_304_subset__iff,axiom,
( ord_less_eq_set_real
= ( ^ [A7: set_real,B5: set_real] :
! [T3: real] :
( ( member_real @ T3 @ A7 )
=> ( member_real @ T3 @ B5 ) ) ) ) ).
% subset_iff
thf(fact_305_Collect__mono,axiom,
! [P: real > $o,Q: real > $o] :
( ! [X4: real] :
( ( P @ X4 )
=> ( Q @ X4 ) )
=> ( ord_less_eq_set_real @ ( collect_real @ P ) @ ( collect_real @ Q ) ) ) ).
% Collect_mono
thf(fact_306_Collect__mono,axiom,
! [P: int > $o,Q: int > $o] :
( ! [X4: int] :
( ( P @ X4 )
=> ( Q @ X4 ) )
=> ( ord_less_eq_set_int @ ( collect_int @ P ) @ ( collect_int @ Q ) ) ) ).
% Collect_mono
thf(fact_307_Collect__subset,axiom,
! [A2: set_real,P: real > $o] :
( ord_less_eq_set_real
@ ( collect_real
@ ^ [X: real] :
( ( member_real @ X @ A2 )
& ( P @ X ) ) )
@ A2 ) ).
% Collect_subset
thf(fact_308_Collect__subset,axiom,
! [A2: set_int,P: int > $o] :
( ord_less_eq_set_int
@ ( collect_int
@ ^ [X: int] :
( ( member_int @ X @ A2 )
& ( P @ X ) ) )
@ A2 ) ).
% Collect_subset
thf(fact_309_less__eq__set__def,axiom,
( ord_less_eq_set_real
= ( ^ [A7: set_real,B5: set_real] :
( ord_less_eq_real_o
@ ^ [X: real] : ( member_real @ X @ A7 )
@ ^ [X: real] : ( member_real @ X @ B5 ) ) ) ) ).
% less_eq_set_def
thf(fact_310_Collect__mono__iff,axiom,
! [P: real > $o,Q: real > $o] :
( ( ord_less_eq_set_real @ ( collect_real @ P ) @ ( collect_real @ Q ) )
= ( ! [X: real] :
( ( P @ X )
=> ( Q @ X ) ) ) ) ).
% Collect_mono_iff
thf(fact_311_Collect__mono__iff,axiom,
! [P: int > $o,Q: int > $o] :
( ( ord_less_eq_set_int @ ( collect_int @ P ) @ ( collect_int @ Q ) )
= ( ! [X: int] :
( ( P @ X )
=> ( Q @ X ) ) ) ) ).
% Collect_mono_iff
thf(fact_312_verit__la__disequality,axiom,
! [A: real,B: real] :
( ( A = B )
| ~ ( ord_less_eq_real @ A @ B )
| ~ ( ord_less_eq_real @ B @ A ) ) ).
% verit_la_disequality
thf(fact_313_verit__la__disequality,axiom,
! [A: int,B: int] :
( ( A = B )
| ~ ( ord_less_eq_int @ A @ B )
| ~ ( ord_less_eq_int @ B @ A ) ) ).
% verit_la_disequality
thf(fact_314_verit__comp__simplify1_I1_J,axiom,
! [A: real] :
~ ( ord_less_real @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_315_verit__comp__simplify1_I1_J,axiom,
! [A: int] :
~ ( ord_less_int @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_316_pinf_I1_J,axiom,
! [P: real > $o,P4: real > $o,Q: real > $o,Q3: real > $o] :
( ? [Z: real] :
! [X4: real] :
( ( ord_less_real @ Z @ X4 )
=> ( ( P @ X4 )
= ( P4 @ X4 ) ) )
=> ( ? [Z: real] :
! [X4: real] :
( ( ord_less_real @ Z @ X4 )
=> ( ( Q @ X4 )
= ( Q3 @ X4 ) ) )
=> ? [Z5: real] :
! [X3: real] :
( ( ord_less_real @ Z5 @ X3 )
=> ( ( ( P @ X3 )
& ( Q @ X3 ) )
= ( ( P4 @ X3 )
& ( Q3 @ X3 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_317_pinf_I1_J,axiom,
! [P: int > $o,P4: int > $o,Q: int > $o,Q3: int > $o] :
( ? [Z: int] :
! [X4: int] :
( ( ord_less_int @ Z @ X4 )
=> ( ( P @ X4 )
= ( P4 @ X4 ) ) )
=> ( ? [Z: int] :
! [X4: int] :
( ( ord_less_int @ Z @ X4 )
=> ( ( Q @ X4 )
= ( Q3 @ X4 ) ) )
=> ? [Z5: int] :
! [X3: int] :
( ( ord_less_int @ Z5 @ X3 )
=> ( ( ( P @ X3 )
& ( Q @ X3 ) )
= ( ( P4 @ X3 )
& ( Q3 @ X3 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_318_pinf_I2_J,axiom,
! [P: real > $o,P4: real > $o,Q: real > $o,Q3: real > $o] :
( ? [Z: real] :
! [X4: real] :
( ( ord_less_real @ Z @ X4 )
=> ( ( P @ X4 )
= ( P4 @ X4 ) ) )
=> ( ? [Z: real] :
! [X4: real] :
( ( ord_less_real @ Z @ X4 )
=> ( ( Q @ X4 )
= ( Q3 @ X4 ) ) )
=> ? [Z5: real] :
! [X3: real] :
( ( ord_less_real @ Z5 @ X3 )
=> ( ( ( P @ X3 )
| ( Q @ X3 ) )
= ( ( P4 @ X3 )
| ( Q3 @ X3 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_319_pinf_I2_J,axiom,
! [P: int > $o,P4: int > $o,Q: int > $o,Q3: int > $o] :
( ? [Z: int] :
! [X4: int] :
( ( ord_less_int @ Z @ X4 )
=> ( ( P @ X4 )
= ( P4 @ X4 ) ) )
=> ( ? [Z: int] :
! [X4: int] :
( ( ord_less_int @ Z @ X4 )
=> ( ( Q @ X4 )
= ( Q3 @ X4 ) ) )
=> ? [Z5: int] :
! [X3: int] :
( ( ord_less_int @ Z5 @ X3 )
=> ( ( ( P @ X3 )
| ( Q @ X3 ) )
= ( ( P4 @ X3 )
| ( Q3 @ X3 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_320_pinf_I3_J,axiom,
! [T2: real] :
? [Z5: real] :
! [X3: real] :
( ( ord_less_real @ Z5 @ X3 )
=> ( X3 != T2 ) ) ).
% pinf(3)
thf(fact_321_pinf_I3_J,axiom,
! [T2: int] :
? [Z5: int] :
! [X3: int] :
( ( ord_less_int @ Z5 @ X3 )
=> ( X3 != T2 ) ) ).
% pinf(3)
thf(fact_322_pinf_I4_J,axiom,
! [T2: real] :
? [Z5: real] :
! [X3: real] :
( ( ord_less_real @ Z5 @ X3 )
=> ( X3 != T2 ) ) ).
% pinf(4)
thf(fact_323_pinf_I4_J,axiom,
! [T2: int] :
? [Z5: int] :
! [X3: int] :
( ( ord_less_int @ Z5 @ X3 )
=> ( X3 != T2 ) ) ).
% pinf(4)
thf(fact_324_pinf_I5_J,axiom,
! [T2: real] :
? [Z5: real] :
! [X3: real] :
( ( ord_less_real @ Z5 @ X3 )
=> ~ ( ord_less_real @ X3 @ T2 ) ) ).
% pinf(5)
thf(fact_325_pinf_I5_J,axiom,
! [T2: int] :
? [Z5: int] :
! [X3: int] :
( ( ord_less_int @ Z5 @ X3 )
=> ~ ( ord_less_int @ X3 @ T2 ) ) ).
% pinf(5)
thf(fact_326_pinf_I7_J,axiom,
! [T2: real] :
? [Z5: real] :
! [X3: real] :
( ( ord_less_real @ Z5 @ X3 )
=> ( ord_less_real @ T2 @ X3 ) ) ).
% pinf(7)
thf(fact_327_pinf_I7_J,axiom,
! [T2: int] :
? [Z5: int] :
! [X3: int] :
( ( ord_less_int @ Z5 @ X3 )
=> ( ord_less_int @ T2 @ X3 ) ) ).
% pinf(7)
thf(fact_328_minf_I1_J,axiom,
! [P: real > $o,P4: real > $o,Q: real > $o,Q3: real > $o] :
( ? [Z: real] :
! [X4: real] :
( ( ord_less_real @ X4 @ Z )
=> ( ( P @ X4 )
= ( P4 @ X4 ) ) )
=> ( ? [Z: real] :
! [X4: real] :
( ( ord_less_real @ X4 @ Z )
=> ( ( Q @ X4 )
= ( Q3 @ X4 ) ) )
=> ? [Z5: real] :
! [X3: real] :
( ( ord_less_real @ X3 @ Z5 )
=> ( ( ( P @ X3 )
& ( Q @ X3 ) )
= ( ( P4 @ X3 )
& ( Q3 @ X3 ) ) ) ) ) ) ).
% minf(1)
thf(fact_329_minf_I1_J,axiom,
! [P: int > $o,P4: int > $o,Q: int > $o,Q3: int > $o] :
( ? [Z: int] :
! [X4: int] :
( ( ord_less_int @ X4 @ Z )
=> ( ( P @ X4 )
= ( P4 @ X4 ) ) )
=> ( ? [Z: int] :
! [X4: int] :
( ( ord_less_int @ X4 @ Z )
=> ( ( Q @ X4 )
= ( Q3 @ X4 ) ) )
=> ? [Z5: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z5 )
=> ( ( ( P @ X3 )
& ( Q @ X3 ) )
= ( ( P4 @ X3 )
& ( Q3 @ X3 ) ) ) ) ) ) ).
% minf(1)
thf(fact_330_minf_I2_J,axiom,
! [P: real > $o,P4: real > $o,Q: real > $o,Q3: real > $o] :
( ? [Z: real] :
! [X4: real] :
( ( ord_less_real @ X4 @ Z )
=> ( ( P @ X4 )
= ( P4 @ X4 ) ) )
=> ( ? [Z: real] :
! [X4: real] :
( ( ord_less_real @ X4 @ Z )
=> ( ( Q @ X4 )
= ( Q3 @ X4 ) ) )
=> ? [Z5: real] :
! [X3: real] :
( ( ord_less_real @ X3 @ Z5 )
=> ( ( ( P @ X3 )
| ( Q @ X3 ) )
= ( ( P4 @ X3 )
| ( Q3 @ X3 ) ) ) ) ) ) ).
% minf(2)
thf(fact_331_minf_I2_J,axiom,
! [P: int > $o,P4: int > $o,Q: int > $o,Q3: int > $o] :
( ? [Z: int] :
! [X4: int] :
( ( ord_less_int @ X4 @ Z )
=> ( ( P @ X4 )
= ( P4 @ X4 ) ) )
=> ( ? [Z: int] :
! [X4: int] :
( ( ord_less_int @ X4 @ Z )
=> ( ( Q @ X4 )
= ( Q3 @ X4 ) ) )
=> ? [Z5: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z5 )
=> ( ( ( P @ X3 )
| ( Q @ X3 ) )
= ( ( P4 @ X3 )
| ( Q3 @ X3 ) ) ) ) ) ) ).
% minf(2)
thf(fact_332_minf_I3_J,axiom,
! [T2: int] :
? [Z5: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z5 )
=> ( X3 != T2 ) ) ).
% minf(3)
thf(fact_333_conj__le__cong,axiom,
! [X2: int,X5: int,P: $o,P4: $o] :
( ( X2 = X5 )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X5 )
=> ( P = P4 ) )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X2 )
& P )
= ( ( ord_less_eq_int @ zero_zero_int @ X5 )
& P4 ) ) ) ) ).
% conj_le_cong
thf(fact_334_imp__le__cong,axiom,
! [X2: int,X5: int,P: $o,P4: $o] :
( ( X2 = X5 )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X5 )
=> ( P = P4 ) )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X2 )
=> P )
= ( ( ord_less_eq_int @ zero_zero_int @ X5 )
=> P4 ) ) ) ) ).
% imp_le_cong
thf(fact_335_jump__poly__not__root,axiom,
! [P2: poly_real,X2: real,Q2: poly_real] :
( ( ( poly_real2 @ P2 @ X2 )
!= zero_zero_real )
=> ( ( sturm_276991412p_poly @ Q2 @ P2 @ X2 )
= zero_zero_int ) ) ).
% jump_poly_not_root
thf(fact_336_jump__poly0_I2_J,axiom,
! [Q2: poly_real,X2: real] :
( ( sturm_276991412p_poly @ Q2 @ zero_zero_poly_real @ X2 )
= zero_zero_int ) ).
% jump_poly0(2)
thf(fact_337_jump__poly0_I1_J,axiom,
! [P2: poly_real,X2: real] :
( ( sturm_276991412p_poly @ zero_zero_poly_real @ P2 @ X2 )
= zero_zero_int ) ).
% jump_poly0(1)
thf(fact_338_verit__la__generic,axiom,
! [A: int,X2: int] :
( ( ord_less_eq_int @ A @ X2 )
| ( A = X2 )
| ( ord_less_eq_int @ X2 @ A ) ) ).
% verit_la_generic
thf(fact_339_finite__interval__int1,axiom,
! [A: int,B: int] :
( finite_finite_int
@ ( collect_int
@ ^ [I: int] :
( ( ord_less_eq_int @ A @ I )
& ( ord_less_eq_int @ I @ B ) ) ) ) ).
% finite_interval_int1
thf(fact_340_finite__interval__int2,axiom,
! [A: int,B: int] :
( finite_finite_int
@ ( collect_int
@ ^ [I: int] :
( ( ord_less_eq_int @ A @ I )
& ( ord_less_int @ I @ B ) ) ) ) ).
% finite_interval_int2
thf(fact_341_finite__interval__int4,axiom,
! [A: int,B: int] :
( finite_finite_int
@ ( collect_int
@ ^ [I: int] :
( ( ord_less_int @ A @ I )
& ( ord_less_int @ I @ B ) ) ) ) ).
% finite_interval_int4
thf(fact_342_finite__interval__int3,axiom,
! [A: int,B: int] :
( finite_finite_int
@ ( collect_int
@ ^ [I: int] :
( ( ord_less_int @ A @ I )
& ( ord_less_eq_int @ I @ B ) ) ) ) ).
% finite_interval_int3
thf(fact_343_less__int__code_I1_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_int_code(1)
thf(fact_344_less__eq__int__code_I1_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% less_eq_int_code(1)
thf(fact_345_cindex__poly__0_I2_J,axiom,
! [A: real,B: real,Q2: poly_real] :
( ( sturm_308542325x_poly @ A @ B @ Q2 @ zero_zero_poly_real )
= zero_zero_int ) ).
% cindex_poly_0(2)
thf(fact_346_cindex__poly__0_I1_J,axiom,
! [A: real,B: real,P2: poly_real] :
( ( sturm_308542325x_poly @ A @ B @ zero_zero_poly_real @ P2 )
= zero_zero_int ) ).
% cindex_poly_0(1)
thf(fact_347_cindex__poly__cross,axiom,
! [A: real,B: real,P2: poly_real] :
( ( ord_less_real @ A @ B )
=> ( ( ( poly_real2 @ P2 @ A )
!= zero_zero_real )
=> ( ( ( poly_real2 @ P2 @ B )
!= zero_zero_real )
=> ( ( sturm_308542325x_poly @ A @ B @ one_one_poly_real @ P2 )
= ( sturm_1953858694_cross @ P2 @ A @ B ) ) ) ) ) ).
% cindex_poly_cross
thf(fact_348_cindex__poly__def,axiom,
( sturm_308542325x_poly
= ( ^ [A4: real,B4: real,Q4: poly_real,P3: poly_real] :
( groups2051521940al_int @ ( sturm_276991412p_poly @ Q4 @ P3 )
@ ( collect_real
@ ^ [X: real] :
( ( ( poly_real2 @ P3 @ X )
= zero_zero_real )
& ( ord_less_real @ A4 @ X )
& ( ord_less_real @ X @ B4 ) ) ) ) ) ) ).
% cindex_poly_def
thf(fact_349_int__one__le__iff__zero__less,axiom,
! [Z3: int] :
( ( ord_less_eq_int @ one_one_int @ Z3 )
= ( ord_less_int @ zero_zero_int @ Z3 ) ) ).
% int_one_le_iff_zero_less
% Conjectures (1)
thf(conj_0,conjecture,
( finite_finite_real
@ ( collect_real
@ ^ [X: real] :
( ( ( poly_real2 @ p @ X )
= zero_zero_real )
& ( ord_less_real @ a @ X )
& ( ord_less_real @ X @ b ) ) ) ) ).
%------------------------------------------------------------------------------