TPTP Problem File: SLH0885^1.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : SLH0000^1 : TPTP v8.2.0. Released v8.2.0.
% Domain : Archive of Formal Proofs
% Problem :
% Version : Especial.
% English :
% Refs : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source : [Des23]
% Names : Median_Method/0000_Median/prob_00399_014970__14852282_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1387 ( 569 unt; 119 typ; 0 def)
% Number of atoms : 3762 (1044 equ; 0 cnn)
% Maximal formula atoms : 26 ( 2 avg)
% Number of connectives : 10082 ( 472 ~; 131 |; 188 &;7589 @)
% ( 0 <=>;1702 =>; 0 <=; 0 <~>)
% Maximal formula depth : 25 ( 6 avg)
% Number of types : 21 ( 20 usr)
% Number of type conns : 448 ( 448 >; 0 *; 0 +; 0 <<)
% Number of symbols : 102 ( 99 usr; 18 con; 0-3 aty)
% Number of variables : 3302 ( 202 ^;2943 !; 157 ?;3302 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 15:46:05.234
%------------------------------------------------------------------------------
% Could-be-implicit typings (20)
thf(ty_n_t__Numeral____Type__Obit0_It__Numeral____Type__Onum1_J,type,
numera2417102609627094330l_num1: $tType ).
thf(ty_n_t__Extended____Nonnegative____Real__Oennreal,type,
extend8495563244428889912nnreal: $tType ).
thf(ty_n_t__List__Olist_It__Extended____Nat__Oenat_J,type,
list_Extended_enat: $tType ).
thf(ty_n_t__List__Olist_It__Real__Oreal_J,type,
list_real: $tType ).
thf(ty_n_t__Set__Oset_It__Real__Oreal_J,type,
set_real: $tType ).
thf(ty_n_t__List__Olist_It__Num__Onum_J,type,
list_num: $tType ).
thf(ty_n_t__List__Olist_It__Nat__Onat_J,type,
list_nat: $tType ).
thf(ty_n_t__List__Olist_It__Int__Oint_J,type,
list_int: $tType ).
thf(ty_n_t__Set__Oset_It__Num__Onum_J,type,
set_num: $tType ).
thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
set_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Int__Oint_J,type,
set_int: $tType ).
thf(ty_n_t__Extended____Nat__Oenat,type,
extended_enat: $tType ).
thf(ty_n_t__List__Olist_Itf__a_J,type,
list_a: $tType ).
thf(ty_n_t__Set__Oset_Itf__a_J,type,
set_a: $tType ).
thf(ty_n_t__String__Ochar,type,
char: $tType ).
thf(ty_n_t__Real__Oreal,type,
real: $tType ).
thf(ty_n_t__Num__Onum,type,
num: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
thf(ty_n_t__Int__Oint,type,
int: $tType ).
thf(ty_n_tf__a,type,
a: $tType ).
% Explicit typings (99)
thf(sy_c_Bit__Operations_Oring__bit__operations__class_Osigned__take__bit_001t__Int__Oint,type,
bit_ri631733984087533419it_int: nat > int > int ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oset__bit_001t__Int__Oint,type,
bit_se7879613467334960850it_int: nat > int > int ).
thf(sy_c_Groups_Oone__class_Oone_001t__Extended____Nat__Oenat,type,
one_on7984719198319812577d_enat: extended_enat ).
thf(sy_c_Groups_Oone__class_Oone_001t__Extended____Nonnegative____Real__Oennreal,type,
one_on2969667320475766781nnreal: extend8495563244428889912nnreal ).
thf(sy_c_Groups_Oone__class_Oone_001t__Int__Oint,type,
one_one_int: int ).
thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat,type,
one_one_nat: nat ).
thf(sy_c_Groups_Oone__class_Oone_001t__Numeral____Type__Obit0_It__Numeral____Type__Onum1_J,type,
one_on3868389512446148991l_num1: numera2417102609627094330l_num1 ).
thf(sy_c_Groups_Oone__class_Oone_001t__Real__Oreal,type,
one_one_real: real ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Extended____Nat__Oenat,type,
plus_p3455044024723400733d_enat: extended_enat > extended_enat > extended_enat ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Int__Oint,type,
plus_plus_int: int > int > int ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat,type,
plus_plus_nat: nat > nat > nat ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Num__Onum,type,
plus_plus_num: num > num > num ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Real__Oreal,type,
plus_plus_real: real > real > real ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Extended____Nat__Oenat,type,
times_7803423173614009249d_enat: extended_enat > extended_enat > extended_enat ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Extended____Nonnegative____Real__Oennreal,type,
times_1893300245718287421nnreal: extend8495563244428889912nnreal > extend8495563244428889912nnreal > extend8495563244428889912nnreal ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Int__Oint,type,
times_times_int: int > int > int ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Nat__Onat,type,
times_times_nat: nat > nat > nat ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Num__Onum,type,
times_times_num: num > num > num ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Real__Oreal,type,
times_times_real: real > real > real ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Int__Oint,type,
uminus_uminus_int: int > int ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Real__Oreal,type,
uminus_uminus_real: real > real ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Extended____Nat__Oenat,type,
zero_z5237406670263579293d_enat: extended_enat ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Extended____Nonnegative____Real__Oennreal,type,
zero_z7100319975126383169nnreal: extend8495563244428889912nnreal ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Int__Oint,type,
zero_zero_int: int ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
zero_zero_nat: nat ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Numeral____Type__Obit0_It__Numeral____Type__Onum1_J,type,
zero_z5982384998485459395l_num1: numera2417102609627094330l_num1 ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Real__Oreal,type,
zero_zero_real: real ).
thf(sy_c_If_001t__Int__Oint,type,
if_int: $o > int > int > int ).
thf(sy_c_If_001t__Nat__Onat,type,
if_nat: $o > nat > nat > nat ).
thf(sy_c_List_Onth_001t__Int__Oint,type,
nth_int: list_int > nat > int ).
thf(sy_c_List_Onth_001t__Nat__Onat,type,
nth_nat: list_nat > nat > nat ).
thf(sy_c_List_Onth_001t__Num__Onum,type,
nth_num: list_num > nat > num ).
thf(sy_c_List_Onth_001t__Real__Oreal,type,
nth_real: list_real > nat > real ).
thf(sy_c_List_Onth_001tf__a,type,
nth_a: list_a > nat > a ).
thf(sy_c_List_Orev_001t__Int__Oint,type,
rev_int: list_int > list_int ).
thf(sy_c_List_Orev_001t__Nat__Onat,type,
rev_nat: list_nat > list_nat ).
thf(sy_c_List_Orev_001t__Num__Onum,type,
rev_num: list_num > list_num ).
thf(sy_c_List_Orev_001t__Real__Oreal,type,
rev_real: list_real > list_real ).
thf(sy_c_List_Orev_001tf__a,type,
rev_a: list_a > list_a ).
thf(sy_c_List_Osorted__wrt_001t__Extended____Nat__Oenat,type,
sorted143172755617435219d_enat: ( extended_enat > extended_enat > $o ) > list_Extended_enat > $o ).
thf(sy_c_List_Osorted__wrt_001t__Int__Oint,type,
sorted_wrt_int: ( int > int > $o ) > list_int > $o ).
thf(sy_c_List_Osorted__wrt_001t__Nat__Onat,type,
sorted_wrt_nat: ( nat > nat > $o ) > list_nat > $o ).
thf(sy_c_List_Osorted__wrt_001t__Num__Onum,type,
sorted_wrt_num: ( num > num > $o ) > list_num > $o ).
thf(sy_c_List_Osorted__wrt_001t__Real__Oreal,type,
sorted_wrt_real: ( real > real > $o ) > list_real > $o ).
thf(sy_c_List_Osorted__wrt_001tf__a,type,
sorted_wrt_a: ( a > a > $o ) > list_a > $o ).
thf(sy_c_Median_Ointerval_001t__Int__Oint,type,
interval_int: set_int > $o ).
thf(sy_c_Median_Ointerval_001t__Nat__Onat,type,
interval_nat: set_nat > $o ).
thf(sy_c_Median_Ointerval_001t__Num__Onum,type,
interval_num: set_num > $o ).
thf(sy_c_Median_Ointerval_001t__Real__Oreal,type,
interval_real: set_real > $o ).
thf(sy_c_Median_Ointerval_001tf__a,type,
interval_a: set_a > $o ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Int__Oint_J,type,
size_size_list_int: list_int > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Nat__Onat_J,type,
size_size_list_nat: list_nat > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Num__Onum_J,type,
size_size_list_num: list_num > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Real__Oreal_J,type,
size_size_list_real: list_real > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_Itf__a_J,type,
size_size_list_a: list_a > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__Num__Onum,type,
size_size_num: num > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__String__Ochar,type,
size_size_char: char > nat ).
thf(sy_c_Num_Oneg__numeral__class_Odbl_001t__Int__Oint,type,
neg_numeral_dbl_int: int > int ).
thf(sy_c_Num_Oneg__numeral__class_Odbl_001t__Numeral____Type__Obit0_It__Numeral____Type__Onum1_J,type,
neg_nu5590746349488142217l_num1: numera2417102609627094330l_num1 > numera2417102609627094330l_num1 ).
thf(sy_c_Num_Oneg__numeral__class_Odbl_001t__Real__Oreal,type,
neg_numeral_dbl_real: real > real ).
thf(sy_c_Num_Onum_OBit0,type,
bit0: num > num ).
thf(sy_c_Num_Onum_OOne,type,
one: num ).
thf(sy_c_Num_Onumeral__class_Onumeral_001t__Extended____Nat__Oenat,type,
numera1916890842035813515d_enat: num > extended_enat ).
thf(sy_c_Num_Onumeral__class_Onumeral_001t__Extended____Nonnegative____Real__Oennreal,type,
numera4658534427948366547nnreal: num > extend8495563244428889912nnreal ).
thf(sy_c_Num_Onumeral__class_Onumeral_001t__Int__Oint,type,
numeral_numeral_int: num > int ).
thf(sy_c_Num_Onumeral__class_Onumeral_001t__Nat__Onat,type,
numeral_numeral_nat: num > nat ).
thf(sy_c_Num_Onumeral__class_Onumeral_001t__Numeral____Type__Obit0_It__Numeral____Type__Onum1_J,type,
numera2161328050825114965l_num1: num > numera2417102609627094330l_num1 ).
thf(sy_c_Num_Onumeral__class_Onumeral_001t__Real__Oreal,type,
numeral_numeral_real: num > real ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Extended____Nat__Oenat,type,
ord_le72135733267957522d_enat: extended_enat > extended_enat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Extended____Nonnegative____Real__Oennreal,type,
ord_le7381754540660121996nnreal: extend8495563244428889912nnreal > extend8495563244428889912nnreal > $o ).
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__Nat__Onat,type,
ord_less_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Num__Onum,type,
ord_less_num: num > num > $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_001tf__a,type,
ord_less_a: a > a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Extended____Nat__Oenat,type,
ord_le2932123472753598470d_enat: extended_enat > extended_enat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Extended____Nonnegative____Real__Oennreal,type,
ord_le3935885782089961368nnreal: extend8495563244428889912nnreal > extend8495563244428889912nnreal > $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__Nat__Onat,type,
ord_less_eq_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Num__Onum,type,
ord_less_eq_num: num > num > $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_001tf__a,type,
ord_less_eq_a: a > a > $o ).
thf(sy_c_Rings_Odivide__class_Odivide_001t__Extended____Nonnegative____Real__Oennreal,type,
divide4826598186094686858nnreal: extend8495563244428889912nnreal > extend8495563244428889912nnreal > extend8495563244428889912nnreal ).
thf(sy_c_Rings_Odivide__class_Odivide_001t__Int__Oint,type,
divide_divide_int: int > int > int ).
thf(sy_c_Rings_Odivide__class_Odivide_001t__Nat__Onat,type,
divide_divide_nat: nat > nat > nat ).
thf(sy_c_Rings_Odivide__class_Odivide_001t__Real__Oreal,type,
divide_divide_real: real > real > real ).
thf(sy_c_Rings_Omodulo__class_Omodulo_001t__Int__Oint,type,
modulo_modulo_int: int > int > int ).
thf(sy_c_Rings_Omodulo__class_Omodulo_001t__Nat__Onat,type,
modulo_modulo_nat: nat > nat > nat ).
thf(sy_c_Set_OCollect_001tf__a,type,
collect_a: ( a > $o ) > set_a ).
thf(sy_c_String_Ochar_Osize__char,type,
size_char: char > nat ).
thf(sy_c_String_Ocomm__semiring__1__class_Oof__char_001t__Nat__Onat,type,
comm_s629917340098488124ar_nat: char > nat ).
thf(sy_c_member_001t__Int__Oint,type,
member_int: int > set_int > $o ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $o ).
thf(sy_c_member_001t__Num__Onum,type,
member_num: num > set_num > $o ).
thf(sy_c_member_001t__Real__Oreal,type,
member_real: real > set_real > $o ).
thf(sy_c_member_001tf__a,type,
member_a: a > set_a > $o ).
thf(sy_v_I,type,
i: set_a ).
thf(sy_v_v____,type,
v: nat ).
thf(sy_v_xs,type,
xs: list_a ).
% Relevant facts (1262)
thf(fact_0_assms_I2_J,axiom,
interval_a @ i ).
% assms(2)
thf(fact_1_v__2,axiom,
member_a @ ( nth_a @ xs @ v ) @ i ).
% v_2
thf(fact_2_p,axiom,
! [K: nat] :
( ( ord_less_eq_nat @ K @ ( divide_divide_nat @ ( size_size_list_a @ xs ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
=> ~ ( member_a @ ( nth_a @ xs @ K ) @ i ) ) ).
% p
thf(fact_3_b,axiom,
( ~ ( member_a @ ( nth_a @ xs @ ( divide_divide_nat @ ( size_size_list_a @ xs ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ i )
& ( ord_less_nat @ ( divide_divide_nat @ ( size_size_list_a @ xs ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ v ) ) ).
% b
thf(fact_4__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062v_O_A_092_060lbrakk_062v_A_060_Alength_Axs_059_Axs_A_B_Av_A_092_060in_062_AI_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [V: nat] :
( ( ord_less_nat @ V @ ( size_size_list_a @ xs ) )
=> ~ ( member_a @ ( nth_a @ xs @ V ) @ i ) ) ).
% \<open>\<And>thesis. (\<And>v. \<lbrakk>v < length xs; xs ! v \<in> I\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_5_semiring__norm_I85_J,axiom,
! [M: num] :
( ( bit0 @ M )
!= one ) ).
% semiring_norm(85)
thf(fact_6_semiring__norm_I83_J,axiom,
! [N: num] :
( one
!= ( bit0 @ N ) ) ).
% semiring_norm(83)
thf(fact_7_numeral__Bit0__div__2,axiom,
! [N: num] :
( ( divide_divide_nat @ ( numeral_numeral_nat @ ( bit0 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( numeral_numeral_nat @ N ) ) ).
% numeral_Bit0_div_2
thf(fact_8_numeral__Bit0__div__2,axiom,
! [N: num] :
( ( divide_divide_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= ( numeral_numeral_int @ N ) ) ).
% numeral_Bit0_div_2
thf(fact_9__092_060open_062_092_060And_062thesis_O_A_092_060lbrakk_062xs_A_B_A_Ilength_Axs_Adiv_A2_J_A_092_060in_062_AI_A_092_060Longrightarrow_062_Athesis_059_Axs_A_B_A_Ilength_Axs_Adiv_A2_J_A_092_060notin_062_AI_A_092_060and_062_Alength_Axs_Adiv_A2_A_060_Av_A_092_060Longrightarrow_062_Athesis_059_Axs_A_B_A_Ilength_Axs_Adiv_A2_J_A_092_060notin_062_AI_A_092_060and_062_Av_A_060_Alength_Axs_Adiv_A2_A_092_060Longrightarrow_062_Athesis_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
( ~ ( member_a @ ( nth_a @ xs @ ( divide_divide_nat @ ( size_size_list_a @ xs ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ i )
=> ( ~ ( ~ ( member_a @ ( nth_a @ xs @ ( divide_divide_nat @ ( size_size_list_a @ xs ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ i )
& ( ord_less_nat @ ( divide_divide_nat @ ( size_size_list_a @ xs ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ v ) )
=> ( ~ ( member_a @ ( nth_a @ xs @ ( divide_divide_nat @ ( size_size_list_a @ xs ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ i )
& ( ord_less_nat @ v @ ( divide_divide_nat @ ( size_size_list_a @ xs ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% \<open>\<And>thesis. \<lbrakk>xs ! (length xs div 2) \<in> I \<Longrightarrow> thesis; xs ! (length xs div 2) \<notin> I \<and> length xs div 2 < v \<Longrightarrow> thesis; xs ! (length xs div 2) \<notin> I \<and> v < length xs div 2 \<Longrightarrow> thesis\<rbrakk> \<Longrightarrow> thesis\<close>
thf(fact_10_verit__eq__simplify_I8_J,axiom,
! [X2: num,Y2: num] :
( ( ( bit0 @ X2 )
= ( bit0 @ Y2 ) )
= ( X2 = Y2 ) ) ).
% verit_eq_simplify(8)
thf(fact_11_semiring__norm_I87_J,axiom,
! [M: num,N: num] :
( ( ( bit0 @ M )
= ( bit0 @ N ) )
= ( M = N ) ) ).
% semiring_norm(87)
thf(fact_12_numeral__eq__iff,axiom,
! [M: num,N: num] :
( ( ( numeral_numeral_nat @ M )
= ( numeral_numeral_nat @ N ) )
= ( M = N ) ) ).
% numeral_eq_iff
thf(fact_13_numeral__eq__iff,axiom,
! [M: num,N: num] :
( ( ( numeral_numeral_int @ M )
= ( numeral_numeral_int @ N ) )
= ( M = N ) ) ).
% numeral_eq_iff
thf(fact_14_numeral__eq__iff,axiom,
! [M: num,N: num] :
( ( ( numera1916890842035813515d_enat @ M )
= ( numera1916890842035813515d_enat @ N ) )
= ( M = N ) ) ).
% numeral_eq_iff
thf(fact_15_numeral__eq__iff,axiom,
! [M: num,N: num] :
( ( ( numera4658534427948366547nnreal @ M )
= ( numera4658534427948366547nnreal @ N ) )
= ( M = N ) ) ).
% numeral_eq_iff
thf(fact_16_v__1,axiom,
ord_less_nat @ v @ ( size_size_list_a @ xs ) ).
% v_1
thf(fact_17_numeral__le__iff,axiom,
! [M: num,N: num] :
( ( ord_le2932123472753598470d_enat @ ( numera1916890842035813515d_enat @ M ) @ ( numera1916890842035813515d_enat @ N ) )
= ( ord_less_eq_num @ M @ N ) ) ).
% numeral_le_iff
thf(fact_18_numeral__le__iff,axiom,
! [M: num,N: num] :
( ( ord_le3935885782089961368nnreal @ ( numera4658534427948366547nnreal @ M ) @ ( numera4658534427948366547nnreal @ N ) )
= ( ord_less_eq_num @ M @ N ) ) ).
% numeral_le_iff
thf(fact_19_numeral__le__iff,axiom,
! [M: num,N: num] :
( ( ord_less_eq_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N ) )
= ( ord_less_eq_num @ M @ N ) ) ).
% numeral_le_iff
thf(fact_20_numeral__le__iff,axiom,
! [M: num,N: num] :
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
= ( ord_less_eq_num @ M @ N ) ) ).
% numeral_le_iff
thf(fact_21_numeral__le__iff,axiom,
! [M: num,N: num] :
( ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
= ( ord_less_eq_num @ M @ N ) ) ).
% numeral_le_iff
thf(fact_22_numeral__less__iff,axiom,
! [M: num,N: num] :
( ( ord_less_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N ) )
= ( ord_less_num @ M @ N ) ) ).
% numeral_less_iff
thf(fact_23_numeral__less__iff,axiom,
! [M: num,N: num] :
( ( ord_less_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
= ( ord_less_num @ M @ N ) ) ).
% numeral_less_iff
thf(fact_24_numeral__less__iff,axiom,
! [M: num,N: num] :
( ( ord_less_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
= ( ord_less_num @ M @ N ) ) ).
% numeral_less_iff
thf(fact_25_numeral__less__iff,axiom,
! [M: num,N: num] :
( ( ord_le72135733267957522d_enat @ ( numera1916890842035813515d_enat @ M ) @ ( numera1916890842035813515d_enat @ N ) )
= ( ord_less_num @ M @ N ) ) ).
% numeral_less_iff
thf(fact_26_numeral__less__iff,axiom,
! [M: num,N: num] :
( ( ord_le7381754540660121996nnreal @ ( numera4658534427948366547nnreal @ M ) @ ( numera4658534427948366547nnreal @ N ) )
= ( ord_less_num @ M @ N ) ) ).
% numeral_less_iff
thf(fact_27_verit__comp__simplify1_I3_J,axiom,
! [B: extended_enat,A: extended_enat] :
( ( ~ ( ord_le2932123472753598470d_enat @ B @ A ) )
= ( ord_le72135733267957522d_enat @ A @ B ) ) ).
% verit_comp_simplify1(3)
thf(fact_28_verit__comp__simplify1_I3_J,axiom,
! [B: real,A: real] :
( ( ~ ( ord_less_eq_real @ B @ A ) )
= ( ord_less_real @ A @ B ) ) ).
% verit_comp_simplify1(3)
thf(fact_29_verit__comp__simplify1_I3_J,axiom,
! [B: num,A: num] :
( ( ~ ( ord_less_eq_num @ B @ A ) )
= ( ord_less_num @ A @ B ) ) ).
% verit_comp_simplify1(3)
thf(fact_30_verit__comp__simplify1_I3_J,axiom,
! [B: nat,A: nat] :
( ( ~ ( ord_less_eq_nat @ B @ A ) )
= ( ord_less_nat @ A @ B ) ) ).
% verit_comp_simplify1(3)
thf(fact_31_verit__comp__simplify1_I3_J,axiom,
! [B: int,A: int] :
( ( ~ ( ord_less_eq_int @ B @ A ) )
= ( ord_less_int @ A @ B ) ) ).
% verit_comp_simplify1(3)
thf(fact_32_verit__comp__simplify1_I3_J,axiom,
! [B: a,A: a] :
( ( ~ ( ord_less_eq_a @ B @ A ) )
= ( ord_less_a @ A @ B ) ) ).
% verit_comp_simplify1(3)
thf(fact_33_verit__comp__simplify1_I2_J,axiom,
! [A2: real] : ( ord_less_eq_real @ A2 @ A2 ) ).
% verit_comp_simplify1(2)
thf(fact_34_verit__comp__simplify1_I2_J,axiom,
! [A2: num] : ( ord_less_eq_num @ A2 @ A2 ) ).
% verit_comp_simplify1(2)
thf(fact_35_verit__comp__simplify1_I2_J,axiom,
! [A2: nat] : ( ord_less_eq_nat @ A2 @ A2 ) ).
% verit_comp_simplify1(2)
thf(fact_36_verit__comp__simplify1_I2_J,axiom,
! [A2: int] : ( ord_less_eq_int @ A2 @ A2 ) ).
% verit_comp_simplify1(2)
thf(fact_37_verit__comp__simplify1_I2_J,axiom,
! [A2: a] : ( ord_less_eq_a @ A2 @ A2 ) ).
% verit_comp_simplify1(2)
thf(fact_38_verit__comp__simplify1_I1_J,axiom,
! [A2: extended_enat] :
~ ( ord_le72135733267957522d_enat @ A2 @ A2 ) ).
% verit_comp_simplify1(1)
thf(fact_39_verit__comp__simplify1_I1_J,axiom,
! [A2: real] :
~ ( ord_less_real @ A2 @ A2 ) ).
% verit_comp_simplify1(1)
thf(fact_40_verit__comp__simplify1_I1_J,axiom,
! [A2: num] :
~ ( ord_less_num @ A2 @ A2 ) ).
% verit_comp_simplify1(1)
thf(fact_41_verit__comp__simplify1_I1_J,axiom,
! [A2: nat] :
~ ( ord_less_nat @ A2 @ A2 ) ).
% verit_comp_simplify1(1)
thf(fact_42_verit__comp__simplify1_I1_J,axiom,
! [A2: int] :
~ ( ord_less_int @ A2 @ A2 ) ).
% verit_comp_simplify1(1)
thf(fact_43_interval__def,axiom,
( interval_real
= ( ^ [I: set_real] :
! [X: real,Y: real,Z: real] :
( ( member_real @ X @ I )
=> ( ( member_real @ Z @ I )
=> ( ( ord_less_eq_real @ X @ Y )
=> ( ( ord_less_eq_real @ Y @ Z )
=> ( member_real @ Y @ I ) ) ) ) ) ) ) ).
% interval_def
thf(fact_44_interval__def,axiom,
( interval_num
= ( ^ [I: set_num] :
! [X: num,Y: num,Z: num] :
( ( member_num @ X @ I )
=> ( ( member_num @ Z @ I )
=> ( ( ord_less_eq_num @ X @ Y )
=> ( ( ord_less_eq_num @ Y @ Z )
=> ( member_num @ Y @ I ) ) ) ) ) ) ) ).
% interval_def
thf(fact_45_interval__def,axiom,
( interval_nat
= ( ^ [I: set_nat] :
! [X: nat,Y: nat,Z: nat] :
( ( member_nat @ X @ I )
=> ( ( member_nat @ Z @ I )
=> ( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z )
=> ( member_nat @ Y @ I ) ) ) ) ) ) ) ).
% interval_def
thf(fact_46_interval__def,axiom,
( interval_int
= ( ^ [I: set_int] :
! [X: int,Y: int,Z: int] :
( ( member_int @ X @ I )
=> ( ( member_int @ Z @ I )
=> ( ( ord_less_eq_int @ X @ Y )
=> ( ( ord_less_eq_int @ Y @ Z )
=> ( member_int @ Y @ I ) ) ) ) ) ) ) ).
% interval_def
thf(fact_47_interval__def,axiom,
( interval_a
= ( ^ [I: set_a] :
! [X: a,Y: a,Z: a] :
( ( member_a @ X @ I )
=> ( ( member_a @ Z @ I )
=> ( ( ord_less_eq_a @ X @ Y )
=> ( ( ord_less_eq_a @ Y @ Z )
=> ( member_a @ Y @ I ) ) ) ) ) ) ) ).
% interval_def
thf(fact_48_interval__rule,axiom,
! [I2: set_real,A2: real,X3: real,B2: real] :
( ( interval_real @ I2 )
=> ( ( ord_less_eq_real @ A2 @ X3 )
=> ( ( ord_less_eq_real @ X3 @ B2 )
=> ( ( member_real @ A2 @ I2 )
=> ( ( member_real @ B2 @ I2 )
=> ( member_real @ X3 @ I2 ) ) ) ) ) ) ).
% interval_rule
thf(fact_49_interval__rule,axiom,
! [I2: set_num,A2: num,X3: num,B2: num] :
( ( interval_num @ I2 )
=> ( ( ord_less_eq_num @ A2 @ X3 )
=> ( ( ord_less_eq_num @ X3 @ B2 )
=> ( ( member_num @ A2 @ I2 )
=> ( ( member_num @ B2 @ I2 )
=> ( member_num @ X3 @ I2 ) ) ) ) ) ) ).
% interval_rule
thf(fact_50_interval__rule,axiom,
! [I2: set_nat,A2: nat,X3: nat,B2: nat] :
( ( interval_nat @ I2 )
=> ( ( ord_less_eq_nat @ A2 @ X3 )
=> ( ( ord_less_eq_nat @ X3 @ B2 )
=> ( ( member_nat @ A2 @ I2 )
=> ( ( member_nat @ B2 @ I2 )
=> ( member_nat @ X3 @ I2 ) ) ) ) ) ) ).
% interval_rule
thf(fact_51_interval__rule,axiom,
! [I2: set_int,A2: int,X3: int,B2: int] :
( ( interval_int @ I2 )
=> ( ( ord_less_eq_int @ A2 @ X3 )
=> ( ( ord_less_eq_int @ X3 @ B2 )
=> ( ( member_int @ A2 @ I2 )
=> ( ( member_int @ B2 @ I2 )
=> ( member_int @ X3 @ I2 ) ) ) ) ) ) ).
% interval_rule
thf(fact_52_interval__rule,axiom,
! [I2: set_a,A2: a,X3: a,B2: a] :
( ( interval_a @ I2 )
=> ( ( ord_less_eq_a @ A2 @ X3 )
=> ( ( ord_less_eq_a @ X3 @ B2 )
=> ( ( member_a @ A2 @ I2 )
=> ( ( member_a @ B2 @ I2 )
=> ( member_a @ X3 @ I2 ) ) ) ) ) ) ).
% interval_rule
thf(fact_53_verit__la__disequality,axiom,
! [A2: real,B2: real] :
( ( A2 = B2 )
| ~ ( ord_less_eq_real @ A2 @ B2 )
| ~ ( ord_less_eq_real @ B2 @ A2 ) ) ).
% verit_la_disequality
thf(fact_54_verit__la__disequality,axiom,
! [A2: num,B2: num] :
( ( A2 = B2 )
| ~ ( ord_less_eq_num @ A2 @ B2 )
| ~ ( ord_less_eq_num @ B2 @ A2 ) ) ).
% verit_la_disequality
thf(fact_55_verit__la__disequality,axiom,
! [A2: nat,B2: nat] :
( ( A2 = B2 )
| ~ ( ord_less_eq_nat @ A2 @ B2 )
| ~ ( ord_less_eq_nat @ B2 @ A2 ) ) ).
% verit_la_disequality
thf(fact_56_verit__la__disequality,axiom,
! [A2: int,B2: int] :
( ( A2 = B2 )
| ~ ( ord_less_eq_int @ A2 @ B2 )
| ~ ( ord_less_eq_int @ B2 @ A2 ) ) ).
% verit_la_disequality
thf(fact_57_verit__la__disequality,axiom,
! [A2: a,B2: a] :
( ( A2 = B2 )
| ~ ( ord_less_eq_a @ A2 @ B2 )
| ~ ( ord_less_eq_a @ B2 @ A2 ) ) ).
% verit_la_disequality
thf(fact_58_verit__eq__simplify_I10_J,axiom,
! [X2: num] :
( one
!= ( bit0 @ X2 ) ) ).
% verit_eq_simplify(10)
thf(fact_59_list__eq__iff__nth__eq,axiom,
( ( ^ [Y3: list_a,Z2: list_a] : ( Y3 = Z2 ) )
= ( ^ [Xs: list_a,Ys: list_a] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_a @ Xs ) )
=> ( ( nth_a @ Xs @ I3 )
= ( nth_a @ Ys @ I3 ) ) ) ) ) ) ).
% list_eq_iff_nth_eq
thf(fact_60_list__eq__iff__nth__eq,axiom,
( ( ^ [Y3: list_nat,Z2: list_nat] : ( Y3 = Z2 ) )
= ( ^ [Xs: list_nat,Ys: list_nat] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs ) )
=> ( ( nth_nat @ Xs @ I3 )
= ( nth_nat @ Ys @ I3 ) ) ) ) ) ) ).
% list_eq_iff_nth_eq
thf(fact_61_Skolem__list__nth,axiom,
! [K: nat,P: nat > a > $o] :
( ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ K )
=> ? [X4: a] : ( P @ I3 @ X4 ) ) )
= ( ? [Xs: list_a] :
( ( ( size_size_list_a @ Xs )
= K )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ K )
=> ( P @ I3 @ ( nth_a @ Xs @ I3 ) ) ) ) ) ) ).
% Skolem_list_nth
thf(fact_62_Skolem__list__nth,axiom,
! [K: nat,P: nat > nat > $o] :
( ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ K )
=> ? [X4: nat] : ( P @ I3 @ X4 ) ) )
= ( ? [Xs: list_nat] :
( ( ( size_size_list_nat @ Xs )
= K )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ K )
=> ( P @ I3 @ ( nth_nat @ Xs @ I3 ) ) ) ) ) ) ).
% Skolem_list_nth
thf(fact_63_nth__equalityI,axiom,
! [Xs2: list_a,Ys2: list_a] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_a @ Ys2 ) )
=> ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_a @ Xs2 ) )
=> ( ( nth_a @ Xs2 @ I4 )
= ( nth_a @ Ys2 @ I4 ) ) )
=> ( Xs2 = Ys2 ) ) ) ).
% nth_equalityI
thf(fact_64_nth__equalityI,axiom,
! [Xs2: list_nat,Ys2: list_nat] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_nat @ Ys2 ) )
=> ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Xs2 ) )
=> ( ( nth_nat @ Xs2 @ I4 )
= ( nth_nat @ Ys2 @ I4 ) ) )
=> ( Xs2 = Ys2 ) ) ) ).
% nth_equalityI
thf(fact_65_zdiv__numeral__Bit0,axiom,
! [V2: num,W: num] :
( ( divide_divide_int @ ( numeral_numeral_int @ ( bit0 @ V2 ) ) @ ( numeral_numeral_int @ ( bit0 @ W ) ) )
= ( divide_divide_int @ ( numeral_numeral_int @ V2 ) @ ( numeral_numeral_int @ W ) ) ) ).
% zdiv_numeral_Bit0
thf(fact_66_order__refl,axiom,
! [X3: real] : ( ord_less_eq_real @ X3 @ X3 ) ).
% order_refl
thf(fact_67_order__refl,axiom,
! [X3: num] : ( ord_less_eq_num @ X3 @ X3 ) ).
% order_refl
thf(fact_68_order__refl,axiom,
! [X3: nat] : ( ord_less_eq_nat @ X3 @ X3 ) ).
% order_refl
thf(fact_69_order__refl,axiom,
! [X3: int] : ( ord_less_eq_int @ X3 @ X3 ) ).
% order_refl
thf(fact_70_order__refl,axiom,
! [X3: a] : ( ord_less_eq_a @ X3 @ X3 ) ).
% order_refl
thf(fact_71_dual__order_Orefl,axiom,
! [A2: real] : ( ord_less_eq_real @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_72_dual__order_Orefl,axiom,
! [A2: num] : ( ord_less_eq_num @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_73_dual__order_Orefl,axiom,
! [A2: nat] : ( ord_less_eq_nat @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_74_dual__order_Orefl,axiom,
! [A2: int] : ( ord_less_eq_int @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_75_dual__order_Orefl,axiom,
! [A2: a] : ( ord_less_eq_a @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_76_div__le__dividend,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( divide_divide_nat @ M @ N ) @ M ) ).
% div_le_dividend
thf(fact_77_div__le__mono,axiom,
! [M: nat,N: nat,K: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( divide_divide_nat @ M @ K ) @ ( divide_divide_nat @ N @ K ) ) ) ).
% div_le_mono
thf(fact_78_length__induct,axiom,
! [P: list_a > $o,Xs2: list_a] :
( ! [Xs3: list_a] :
( ! [Ys3: list_a] :
( ( ord_less_nat @ ( size_size_list_a @ Ys3 ) @ ( size_size_list_a @ Xs3 ) )
=> ( P @ Ys3 ) )
=> ( P @ Xs3 ) )
=> ( P @ Xs2 ) ) ).
% length_induct
thf(fact_79_length__induct,axiom,
! [P: list_nat > $o,Xs2: list_nat] :
( ! [Xs3: list_nat] :
( ! [Ys3: list_nat] :
( ( ord_less_nat @ ( size_size_list_nat @ Ys3 ) @ ( size_size_list_nat @ Xs3 ) )
=> ( P @ Ys3 ) )
=> ( P @ Xs3 ) )
=> ( P @ Xs2 ) ) ).
% length_induct
thf(fact_80_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
& ( M2 != N2 ) ) ) ) ).
% nat_less_le
thf(fact_81_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_82_semiring__norm_I78_J,axiom,
! [M: num,N: num] :
( ( ord_less_num @ ( bit0 @ M ) @ ( bit0 @ N ) )
= ( ord_less_num @ M @ N ) ) ).
% semiring_norm(78)
thf(fact_83_semiring__norm_I75_J,axiom,
! [M: num] :
~ ( ord_less_num @ M @ one ) ).
% semiring_norm(75)
thf(fact_84_semiring__norm_I71_J,axiom,
! [M: num,N: num] :
( ( ord_less_eq_num @ ( bit0 @ M ) @ ( bit0 @ N ) )
= ( ord_less_eq_num @ M @ N ) ) ).
% semiring_norm(71)
thf(fact_85_semiring__norm_I68_J,axiom,
! [N: num] : ( ord_less_eq_num @ one @ N ) ).
% semiring_norm(68)
thf(fact_86_semiring__norm_I76_J,axiom,
! [N: num] : ( ord_less_num @ one @ ( bit0 @ N ) ) ).
% semiring_norm(76)
thf(fact_87_semiring__norm_I69_J,axiom,
! [M: num] :
~ ( ord_less_eq_num @ ( bit0 @ M ) @ one ) ).
% semiring_norm(69)
thf(fact_88_assms_I3_J,axiom,
sorted_wrt_a @ ord_less_eq_a @ xs ).
% assms(3)
thf(fact_89_le__num__One__iff,axiom,
! [X3: num] :
( ( ord_less_eq_num @ X3 @ one )
= ( X3 = one ) ) ).
% le_num_One_iff
thf(fact_90_mem__Collect__eq,axiom,
! [A2: a,P: a > $o] :
( ( member_a @ A2 @ ( collect_a @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_91_Collect__mem__eq,axiom,
! [A3: set_a] :
( ( collect_a
@ ^ [X: a] : ( member_a @ X @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_92_order__antisym__conv,axiom,
! [Y4: real,X3: real] :
( ( ord_less_eq_real @ Y4 @ X3 )
=> ( ( ord_less_eq_real @ X3 @ Y4 )
= ( X3 = Y4 ) ) ) ).
% order_antisym_conv
thf(fact_93_order__antisym__conv,axiom,
! [Y4: num,X3: num] :
( ( ord_less_eq_num @ Y4 @ X3 )
=> ( ( ord_less_eq_num @ X3 @ Y4 )
= ( X3 = Y4 ) ) ) ).
% order_antisym_conv
thf(fact_94_order__antisym__conv,axiom,
! [Y4: nat,X3: nat] :
( ( ord_less_eq_nat @ Y4 @ X3 )
=> ( ( ord_less_eq_nat @ X3 @ Y4 )
= ( X3 = Y4 ) ) ) ).
% order_antisym_conv
thf(fact_95_order__antisym__conv,axiom,
! [Y4: int,X3: int] :
( ( ord_less_eq_int @ Y4 @ X3 )
=> ( ( ord_less_eq_int @ X3 @ Y4 )
= ( X3 = Y4 ) ) ) ).
% order_antisym_conv
thf(fact_96_order__antisym__conv,axiom,
! [Y4: a,X3: a] :
( ( ord_less_eq_a @ Y4 @ X3 )
=> ( ( ord_less_eq_a @ X3 @ Y4 )
= ( X3 = Y4 ) ) ) ).
% order_antisym_conv
thf(fact_97_linorder__le__cases,axiom,
! [X3: real,Y4: real] :
( ~ ( ord_less_eq_real @ X3 @ Y4 )
=> ( ord_less_eq_real @ Y4 @ X3 ) ) ).
% linorder_le_cases
thf(fact_98_linorder__le__cases,axiom,
! [X3: num,Y4: num] :
( ~ ( ord_less_eq_num @ X3 @ Y4 )
=> ( ord_less_eq_num @ Y4 @ X3 ) ) ).
% linorder_le_cases
thf(fact_99_linorder__le__cases,axiom,
! [X3: nat,Y4: nat] :
( ~ ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_less_eq_nat @ Y4 @ X3 ) ) ).
% linorder_le_cases
thf(fact_100_linorder__le__cases,axiom,
! [X3: int,Y4: int] :
( ~ ( ord_less_eq_int @ X3 @ Y4 )
=> ( ord_less_eq_int @ Y4 @ X3 ) ) ).
% linorder_le_cases
thf(fact_101_linorder__le__cases,axiom,
! [X3: a,Y4: a] :
( ~ ( ord_less_eq_a @ X3 @ Y4 )
=> ( ord_less_eq_a @ Y4 @ X3 ) ) ).
% linorder_le_cases
thf(fact_102_ord__le__eq__subst,axiom,
! [A2: real,B2: real,F: real > real,C: real] :
( ( ord_less_eq_real @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X5: real,Y5: real] :
( ( ord_less_eq_real @ X5 @ Y5 )
=> ( ord_less_eq_real @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_real @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_103_ord__le__eq__subst,axiom,
! [A2: real,B2: real,F: real > num,C: num] :
( ( ord_less_eq_real @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X5: real,Y5: real] :
( ( ord_less_eq_real @ X5 @ Y5 )
=> ( ord_less_eq_num @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_num @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_104_ord__le__eq__subst,axiom,
! [A2: real,B2: real,F: real > nat,C: nat] :
( ( ord_less_eq_real @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X5: real,Y5: real] :
( ( ord_less_eq_real @ X5 @ Y5 )
=> ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_105_ord__le__eq__subst,axiom,
! [A2: real,B2: real,F: real > int,C: int] :
( ( ord_less_eq_real @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X5: real,Y5: real] :
( ( ord_less_eq_real @ X5 @ Y5 )
=> ( ord_less_eq_int @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_int @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_106_ord__le__eq__subst,axiom,
! [A2: real,B2: real,F: real > a,C: a] :
( ( ord_less_eq_real @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X5: real,Y5: real] :
( ( ord_less_eq_real @ X5 @ Y5 )
=> ( ord_less_eq_a @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_a @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_107_ord__le__eq__subst,axiom,
! [A2: num,B2: num,F: num > real,C: real] :
( ( ord_less_eq_num @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X5: num,Y5: num] :
( ( ord_less_eq_num @ X5 @ Y5 )
=> ( ord_less_eq_real @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_real @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_108_ord__le__eq__subst,axiom,
! [A2: num,B2: num,F: num > num,C: num] :
( ( ord_less_eq_num @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X5: num,Y5: num] :
( ( ord_less_eq_num @ X5 @ Y5 )
=> ( ord_less_eq_num @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_num @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_109_ord__le__eq__subst,axiom,
! [A2: num,B2: num,F: num > nat,C: nat] :
( ( ord_less_eq_num @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X5: num,Y5: num] :
( ( ord_less_eq_num @ X5 @ Y5 )
=> ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_110_ord__le__eq__subst,axiom,
! [A2: num,B2: num,F: num > int,C: int] :
( ( ord_less_eq_num @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X5: num,Y5: num] :
( ( ord_less_eq_num @ X5 @ Y5 )
=> ( ord_less_eq_int @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_int @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_111_ord__le__eq__subst,axiom,
! [A2: num,B2: num,F: num > a,C: a] :
( ( ord_less_eq_num @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X5: num,Y5: num] :
( ( ord_less_eq_num @ X5 @ Y5 )
=> ( ord_less_eq_a @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_a @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_112_ord__eq__le__subst,axiom,
! [A2: real,F: real > real,B2: real,C: real] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_real @ B2 @ C )
=> ( ! [X5: real,Y5: real] :
( ( ord_less_eq_real @ X5 @ Y5 )
=> ( ord_less_eq_real @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_real @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_113_ord__eq__le__subst,axiom,
! [A2: num,F: real > num,B2: real,C: real] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_real @ B2 @ C )
=> ( ! [X5: real,Y5: real] :
( ( ord_less_eq_real @ X5 @ Y5 )
=> ( ord_less_eq_num @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_num @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_114_ord__eq__le__subst,axiom,
! [A2: nat,F: real > nat,B2: real,C: real] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_real @ B2 @ C )
=> ( ! [X5: real,Y5: real] :
( ( ord_less_eq_real @ X5 @ Y5 )
=> ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_115_ord__eq__le__subst,axiom,
! [A2: int,F: real > int,B2: real,C: real] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_real @ B2 @ C )
=> ( ! [X5: real,Y5: real] :
( ( ord_less_eq_real @ X5 @ Y5 )
=> ( ord_less_eq_int @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_int @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_116_ord__eq__le__subst,axiom,
! [A2: a,F: real > a,B2: real,C: real] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_real @ B2 @ C )
=> ( ! [X5: real,Y5: real] :
( ( ord_less_eq_real @ X5 @ Y5 )
=> ( ord_less_eq_a @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_a @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_117_ord__eq__le__subst,axiom,
! [A2: real,F: num > real,B2: num,C: num] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_num @ B2 @ C )
=> ( ! [X5: num,Y5: num] :
( ( ord_less_eq_num @ X5 @ Y5 )
=> ( ord_less_eq_real @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_real @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_118_ord__eq__le__subst,axiom,
! [A2: num,F: num > num,B2: num,C: num] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_num @ B2 @ C )
=> ( ! [X5: num,Y5: num] :
( ( ord_less_eq_num @ X5 @ Y5 )
=> ( ord_less_eq_num @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_num @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_119_ord__eq__le__subst,axiom,
! [A2: nat,F: num > nat,B2: num,C: num] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_num @ B2 @ C )
=> ( ! [X5: num,Y5: num] :
( ( ord_less_eq_num @ X5 @ Y5 )
=> ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_120_ord__eq__le__subst,axiom,
! [A2: int,F: num > int,B2: num,C: num] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_num @ B2 @ C )
=> ( ! [X5: num,Y5: num] :
( ( ord_less_eq_num @ X5 @ Y5 )
=> ( ord_less_eq_int @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_int @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_121_ord__eq__le__subst,axiom,
! [A2: a,F: num > a,B2: num,C: num] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_num @ B2 @ C )
=> ( ! [X5: num,Y5: num] :
( ( ord_less_eq_num @ X5 @ Y5 )
=> ( ord_less_eq_a @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_a @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_122_linorder__linear,axiom,
! [X3: real,Y4: real] :
( ( ord_less_eq_real @ X3 @ Y4 )
| ( ord_less_eq_real @ Y4 @ X3 ) ) ).
% linorder_linear
thf(fact_123_linorder__linear,axiom,
! [X3: num,Y4: num] :
( ( ord_less_eq_num @ X3 @ Y4 )
| ( ord_less_eq_num @ Y4 @ X3 ) ) ).
% linorder_linear
thf(fact_124_linorder__linear,axiom,
! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
| ( ord_less_eq_nat @ Y4 @ X3 ) ) ).
% linorder_linear
thf(fact_125_linorder__linear,axiom,
! [X3: int,Y4: int] :
( ( ord_less_eq_int @ X3 @ Y4 )
| ( ord_less_eq_int @ Y4 @ X3 ) ) ).
% linorder_linear
thf(fact_126_linorder__linear,axiom,
! [X3: a,Y4: a] :
( ( ord_less_eq_a @ X3 @ Y4 )
| ( ord_less_eq_a @ Y4 @ X3 ) ) ).
% linorder_linear
thf(fact_127_order__eq__refl,axiom,
! [X3: real,Y4: real] :
( ( X3 = Y4 )
=> ( ord_less_eq_real @ X3 @ Y4 ) ) ).
% order_eq_refl
thf(fact_128_order__eq__refl,axiom,
! [X3: num,Y4: num] :
( ( X3 = Y4 )
=> ( ord_less_eq_num @ X3 @ Y4 ) ) ).
% order_eq_refl
thf(fact_129_order__eq__refl,axiom,
! [X3: nat,Y4: nat] :
( ( X3 = Y4 )
=> ( ord_less_eq_nat @ X3 @ Y4 ) ) ).
% order_eq_refl
thf(fact_130_order__eq__refl,axiom,
! [X3: int,Y4: int] :
( ( X3 = Y4 )
=> ( ord_less_eq_int @ X3 @ Y4 ) ) ).
% order_eq_refl
thf(fact_131_order__eq__refl,axiom,
! [X3: a,Y4: a] :
( ( X3 = Y4 )
=> ( ord_less_eq_a @ X3 @ Y4 ) ) ).
% order_eq_refl
thf(fact_132_order__subst2,axiom,
! [A2: real,B2: real,F: real > real,C: real] :
( ( ord_less_eq_real @ A2 @ B2 )
=> ( ( ord_less_eq_real @ ( F @ B2 ) @ C )
=> ( ! [X5: real,Y5: real] :
( ( ord_less_eq_real @ X5 @ Y5 )
=> ( ord_less_eq_real @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_real @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_133_order__subst2,axiom,
! [A2: real,B2: real,F: real > num,C: num] :
( ( ord_less_eq_real @ A2 @ B2 )
=> ( ( ord_less_eq_num @ ( F @ B2 ) @ C )
=> ( ! [X5: real,Y5: real] :
( ( ord_less_eq_real @ X5 @ Y5 )
=> ( ord_less_eq_num @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_num @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_134_order__subst2,axiom,
! [A2: real,B2: real,F: real > nat,C: nat] :
( ( ord_less_eq_real @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ ( F @ B2 ) @ C )
=> ( ! [X5: real,Y5: real] :
( ( ord_less_eq_real @ X5 @ Y5 )
=> ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_135_order__subst2,axiom,
! [A2: real,B2: real,F: real > int,C: int] :
( ( ord_less_eq_real @ A2 @ B2 )
=> ( ( ord_less_eq_int @ ( F @ B2 ) @ C )
=> ( ! [X5: real,Y5: real] :
( ( ord_less_eq_real @ X5 @ Y5 )
=> ( ord_less_eq_int @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_int @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_136_order__subst2,axiom,
! [A2: real,B2: real,F: real > a,C: a] :
( ( ord_less_eq_real @ A2 @ B2 )
=> ( ( ord_less_eq_a @ ( F @ B2 ) @ C )
=> ( ! [X5: real,Y5: real] :
( ( ord_less_eq_real @ X5 @ Y5 )
=> ( ord_less_eq_a @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_a @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_137_order__subst2,axiom,
! [A2: num,B2: num,F: num > real,C: real] :
( ( ord_less_eq_num @ A2 @ B2 )
=> ( ( ord_less_eq_real @ ( F @ B2 ) @ C )
=> ( ! [X5: num,Y5: num] :
( ( ord_less_eq_num @ X5 @ Y5 )
=> ( ord_less_eq_real @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_real @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_138_order__subst2,axiom,
! [A2: num,B2: num,F: num > num,C: num] :
( ( ord_less_eq_num @ A2 @ B2 )
=> ( ( ord_less_eq_num @ ( F @ B2 ) @ C )
=> ( ! [X5: num,Y5: num] :
( ( ord_less_eq_num @ X5 @ Y5 )
=> ( ord_less_eq_num @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_num @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_139_order__subst2,axiom,
! [A2: num,B2: num,F: num > nat,C: nat] :
( ( ord_less_eq_num @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ ( F @ B2 ) @ C )
=> ( ! [X5: num,Y5: num] :
( ( ord_less_eq_num @ X5 @ Y5 )
=> ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_140_order__subst2,axiom,
! [A2: num,B2: num,F: num > int,C: int] :
( ( ord_less_eq_num @ A2 @ B2 )
=> ( ( ord_less_eq_int @ ( F @ B2 ) @ C )
=> ( ! [X5: num,Y5: num] :
( ( ord_less_eq_num @ X5 @ Y5 )
=> ( ord_less_eq_int @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_int @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_141_order__subst2,axiom,
! [A2: num,B2: num,F: num > a,C: a] :
( ( ord_less_eq_num @ A2 @ B2 )
=> ( ( ord_less_eq_a @ ( F @ B2 ) @ C )
=> ( ! [X5: num,Y5: num] :
( ( ord_less_eq_num @ X5 @ Y5 )
=> ( ord_less_eq_a @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_a @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_142_order__subst1,axiom,
! [A2: real,F: real > real,B2: real,C: real] :
( ( ord_less_eq_real @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_real @ B2 @ C )
=> ( ! [X5: real,Y5: real] :
( ( ord_less_eq_real @ X5 @ Y5 )
=> ( ord_less_eq_real @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_real @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_143_order__subst1,axiom,
! [A2: real,F: num > real,B2: num,C: num] :
( ( ord_less_eq_real @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_num @ B2 @ C )
=> ( ! [X5: num,Y5: num] :
( ( ord_less_eq_num @ X5 @ Y5 )
=> ( ord_less_eq_real @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_real @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_144_order__subst1,axiom,
! [A2: real,F: nat > real,B2: nat,C: nat] :
( ( ord_less_eq_real @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X5: nat,Y5: nat] :
( ( ord_less_eq_nat @ X5 @ Y5 )
=> ( ord_less_eq_real @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_real @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_145_order__subst1,axiom,
! [A2: real,F: int > real,B2: int,C: int] :
( ( ord_less_eq_real @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_int @ B2 @ C )
=> ( ! [X5: int,Y5: int] :
( ( ord_less_eq_int @ X5 @ Y5 )
=> ( ord_less_eq_real @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_real @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_146_order__subst1,axiom,
! [A2: real,F: a > real,B2: a,C: a] :
( ( ord_less_eq_real @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_a @ B2 @ C )
=> ( ! [X5: a,Y5: a] :
( ( ord_less_eq_a @ X5 @ Y5 )
=> ( ord_less_eq_real @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_real @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_147_order__subst1,axiom,
! [A2: num,F: real > num,B2: real,C: real] :
( ( ord_less_eq_num @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_real @ B2 @ C )
=> ( ! [X5: real,Y5: real] :
( ( ord_less_eq_real @ X5 @ Y5 )
=> ( ord_less_eq_num @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_num @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_148_order__subst1,axiom,
! [A2: num,F: num > num,B2: num,C: num] :
( ( ord_less_eq_num @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_num @ B2 @ C )
=> ( ! [X5: num,Y5: num] :
( ( ord_less_eq_num @ X5 @ Y5 )
=> ( ord_less_eq_num @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_num @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_149_order__subst1,axiom,
! [A2: num,F: nat > num,B2: nat,C: nat] :
( ( ord_less_eq_num @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X5: nat,Y5: nat] :
( ( ord_less_eq_nat @ X5 @ Y5 )
=> ( ord_less_eq_num @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_num @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_150_order__subst1,axiom,
! [A2: num,F: int > num,B2: int,C: int] :
( ( ord_less_eq_num @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_int @ B2 @ C )
=> ( ! [X5: int,Y5: int] :
( ( ord_less_eq_int @ X5 @ Y5 )
=> ( ord_less_eq_num @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_num @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_151_order__subst1,axiom,
! [A2: num,F: a > num,B2: a,C: a] :
( ( ord_less_eq_num @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_a @ B2 @ C )
=> ( ! [X5: a,Y5: a] :
( ( ord_less_eq_a @ X5 @ Y5 )
=> ( ord_less_eq_num @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_num @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_152_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: real,Z2: real] : ( Y3 = Z2 ) )
= ( ^ [A4: real,B3: real] :
( ( ord_less_eq_real @ A4 @ B3 )
& ( ord_less_eq_real @ B3 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_153_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: num,Z2: num] : ( Y3 = Z2 ) )
= ( ^ [A4: num,B3: num] :
( ( ord_less_eq_num @ A4 @ B3 )
& ( ord_less_eq_num @ B3 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_154_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: nat,Z2: nat] : ( Y3 = Z2 ) )
= ( ^ [A4: nat,B3: nat] :
( ( ord_less_eq_nat @ A4 @ B3 )
& ( ord_less_eq_nat @ B3 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_155_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: int,Z2: int] : ( Y3 = Z2 ) )
= ( ^ [A4: int,B3: int] :
( ( ord_less_eq_int @ A4 @ B3 )
& ( ord_less_eq_int @ B3 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_156_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: a,Z2: a] : ( Y3 = Z2 ) )
= ( ^ [A4: a,B3: a] :
( ( ord_less_eq_a @ A4 @ B3 )
& ( ord_less_eq_a @ B3 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_157_antisym,axiom,
! [A2: real,B2: real] :
( ( ord_less_eq_real @ A2 @ B2 )
=> ( ( ord_less_eq_real @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% antisym
thf(fact_158_antisym,axiom,
! [A2: num,B2: num] :
( ( ord_less_eq_num @ A2 @ B2 )
=> ( ( ord_less_eq_num @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% antisym
thf(fact_159_antisym,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% antisym
thf(fact_160_antisym,axiom,
! [A2: int,B2: int] :
( ( ord_less_eq_int @ A2 @ B2 )
=> ( ( ord_less_eq_int @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% antisym
thf(fact_161_antisym,axiom,
! [A2: a,B2: a] :
( ( ord_less_eq_a @ A2 @ B2 )
=> ( ( ord_less_eq_a @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% antisym
thf(fact_162_dual__order_Otrans,axiom,
! [B2: real,A2: real,C: real] :
( ( ord_less_eq_real @ B2 @ A2 )
=> ( ( ord_less_eq_real @ C @ B2 )
=> ( ord_less_eq_real @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_163_dual__order_Otrans,axiom,
! [B2: num,A2: num,C: num] :
( ( ord_less_eq_num @ B2 @ A2 )
=> ( ( ord_less_eq_num @ C @ B2 )
=> ( ord_less_eq_num @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_164_dual__order_Otrans,axiom,
! [B2: nat,A2: nat,C: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
=> ( ( ord_less_eq_nat @ C @ B2 )
=> ( ord_less_eq_nat @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_165_dual__order_Otrans,axiom,
! [B2: int,A2: int,C: int] :
( ( ord_less_eq_int @ B2 @ A2 )
=> ( ( ord_less_eq_int @ C @ B2 )
=> ( ord_less_eq_int @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_166_dual__order_Otrans,axiom,
! [B2: a,A2: a,C: a] :
( ( ord_less_eq_a @ B2 @ A2 )
=> ( ( ord_less_eq_a @ C @ B2 )
=> ( ord_less_eq_a @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_167_dual__order_Oantisym,axiom,
! [B2: real,A2: real] :
( ( ord_less_eq_real @ B2 @ A2 )
=> ( ( ord_less_eq_real @ A2 @ B2 )
=> ( A2 = B2 ) ) ) ).
% dual_order.antisym
thf(fact_168_dual__order_Oantisym,axiom,
! [B2: num,A2: num] :
( ( ord_less_eq_num @ B2 @ A2 )
=> ( ( ord_less_eq_num @ A2 @ B2 )
=> ( A2 = B2 ) ) ) ).
% dual_order.antisym
thf(fact_169_dual__order_Oantisym,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
=> ( ( ord_less_eq_nat @ A2 @ B2 )
=> ( A2 = B2 ) ) ) ).
% dual_order.antisym
thf(fact_170_dual__order_Oantisym,axiom,
! [B2: int,A2: int] :
( ( ord_less_eq_int @ B2 @ A2 )
=> ( ( ord_less_eq_int @ A2 @ B2 )
=> ( A2 = B2 ) ) ) ).
% dual_order.antisym
thf(fact_171_dual__order_Oantisym,axiom,
! [B2: a,A2: a] :
( ( ord_less_eq_a @ B2 @ A2 )
=> ( ( ord_less_eq_a @ A2 @ B2 )
=> ( A2 = B2 ) ) ) ).
% dual_order.antisym
thf(fact_172_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: real,Z2: real] : ( Y3 = Z2 ) )
= ( ^ [A4: real,B3: real] :
( ( ord_less_eq_real @ B3 @ A4 )
& ( ord_less_eq_real @ A4 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_173_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: num,Z2: num] : ( Y3 = Z2 ) )
= ( ^ [A4: num,B3: num] :
( ( ord_less_eq_num @ B3 @ A4 )
& ( ord_less_eq_num @ A4 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_174_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: nat,Z2: nat] : ( Y3 = Z2 ) )
= ( ^ [A4: nat,B3: nat] :
( ( ord_less_eq_nat @ B3 @ A4 )
& ( ord_less_eq_nat @ A4 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_175_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: int,Z2: int] : ( Y3 = Z2 ) )
= ( ^ [A4: int,B3: int] :
( ( ord_less_eq_int @ B3 @ A4 )
& ( ord_less_eq_int @ A4 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_176_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: a,Z2: a] : ( Y3 = Z2 ) )
= ( ^ [A4: a,B3: a] :
( ( ord_less_eq_a @ B3 @ A4 )
& ( ord_less_eq_a @ A4 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_177_linorder__wlog,axiom,
! [P: real > real > $o,A2: real,B2: real] :
( ! [A5: real,B4: real] :
( ( ord_less_eq_real @ A5 @ B4 )
=> ( P @ A5 @ B4 ) )
=> ( ! [A5: real,B4: real] :
( ( P @ B4 @ A5 )
=> ( P @ A5 @ B4 ) )
=> ( P @ A2 @ B2 ) ) ) ).
% linorder_wlog
thf(fact_178_linorder__wlog,axiom,
! [P: num > num > $o,A2: num,B2: num] :
( ! [A5: num,B4: num] :
( ( ord_less_eq_num @ A5 @ B4 )
=> ( P @ A5 @ B4 ) )
=> ( ! [A5: num,B4: num] :
( ( P @ B4 @ A5 )
=> ( P @ A5 @ B4 ) )
=> ( P @ A2 @ B2 ) ) ) ).
% linorder_wlog
thf(fact_179_linorder__wlog,axiom,
! [P: nat > nat > $o,A2: nat,B2: nat] :
( ! [A5: nat,B4: nat] :
( ( ord_less_eq_nat @ A5 @ B4 )
=> ( P @ A5 @ B4 ) )
=> ( ! [A5: nat,B4: nat] :
( ( P @ B4 @ A5 )
=> ( P @ A5 @ B4 ) )
=> ( P @ A2 @ B2 ) ) ) ).
% linorder_wlog
thf(fact_180_linorder__wlog,axiom,
! [P: int > int > $o,A2: int,B2: int] :
( ! [A5: int,B4: int] :
( ( ord_less_eq_int @ A5 @ B4 )
=> ( P @ A5 @ B4 ) )
=> ( ! [A5: int,B4: int] :
( ( P @ B4 @ A5 )
=> ( P @ A5 @ B4 ) )
=> ( P @ A2 @ B2 ) ) ) ).
% linorder_wlog
thf(fact_181_linorder__wlog,axiom,
! [P: a > a > $o,A2: a,B2: a] :
( ! [A5: a,B4: a] :
( ( ord_less_eq_a @ A5 @ B4 )
=> ( P @ A5 @ B4 ) )
=> ( ! [A5: a,B4: a] :
( ( P @ B4 @ A5 )
=> ( P @ A5 @ B4 ) )
=> ( P @ A2 @ B2 ) ) ) ).
% linorder_wlog
thf(fact_182_order__trans,axiom,
! [X3: real,Y4: real,Z3: real] :
( ( ord_less_eq_real @ X3 @ Y4 )
=> ( ( ord_less_eq_real @ Y4 @ Z3 )
=> ( ord_less_eq_real @ X3 @ Z3 ) ) ) ).
% order_trans
thf(fact_183_order__trans,axiom,
! [X3: num,Y4: num,Z3: num] :
( ( ord_less_eq_num @ X3 @ Y4 )
=> ( ( ord_less_eq_num @ Y4 @ Z3 )
=> ( ord_less_eq_num @ X3 @ Z3 ) ) ) ).
% order_trans
thf(fact_184_order__trans,axiom,
! [X3: nat,Y4: nat,Z3: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ( ord_less_eq_nat @ Y4 @ Z3 )
=> ( ord_less_eq_nat @ X3 @ Z3 ) ) ) ).
% order_trans
thf(fact_185_order__trans,axiom,
! [X3: int,Y4: int,Z3: int] :
( ( ord_less_eq_int @ X3 @ Y4 )
=> ( ( ord_less_eq_int @ Y4 @ Z3 )
=> ( ord_less_eq_int @ X3 @ Z3 ) ) ) ).
% order_trans
thf(fact_186_order__trans,axiom,
! [X3: a,Y4: a,Z3: a] :
( ( ord_less_eq_a @ X3 @ Y4 )
=> ( ( ord_less_eq_a @ Y4 @ Z3 )
=> ( ord_less_eq_a @ X3 @ Z3 ) ) ) ).
% order_trans
thf(fact_187_order_Otrans,axiom,
! [A2: real,B2: real,C: real] :
( ( ord_less_eq_real @ A2 @ B2 )
=> ( ( ord_less_eq_real @ B2 @ C )
=> ( ord_less_eq_real @ A2 @ C ) ) ) ).
% order.trans
thf(fact_188_order_Otrans,axiom,
! [A2: num,B2: num,C: num] :
( ( ord_less_eq_num @ A2 @ B2 )
=> ( ( ord_less_eq_num @ B2 @ C )
=> ( ord_less_eq_num @ A2 @ C ) ) ) ).
% order.trans
thf(fact_189_order_Otrans,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% order.trans
thf(fact_190_order_Otrans,axiom,
! [A2: int,B2: int,C: int] :
( ( ord_less_eq_int @ A2 @ B2 )
=> ( ( ord_less_eq_int @ B2 @ C )
=> ( ord_less_eq_int @ A2 @ C ) ) ) ).
% order.trans
thf(fact_191_order_Otrans,axiom,
! [A2: a,B2: a,C: a] :
( ( ord_less_eq_a @ A2 @ B2 )
=> ( ( ord_less_eq_a @ B2 @ C )
=> ( ord_less_eq_a @ A2 @ C ) ) ) ).
% order.trans
thf(fact_192_order__antisym,axiom,
! [X3: real,Y4: real] :
( ( ord_less_eq_real @ X3 @ Y4 )
=> ( ( ord_less_eq_real @ Y4 @ X3 )
=> ( X3 = Y4 ) ) ) ).
% order_antisym
thf(fact_193_order__antisym,axiom,
! [X3: num,Y4: num] :
( ( ord_less_eq_num @ X3 @ Y4 )
=> ( ( ord_less_eq_num @ Y4 @ X3 )
=> ( X3 = Y4 ) ) ) ).
% order_antisym
thf(fact_194_order__antisym,axiom,
! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ( ord_less_eq_nat @ Y4 @ X3 )
=> ( X3 = Y4 ) ) ) ).
% order_antisym
thf(fact_195_order__antisym,axiom,
! [X3: int,Y4: int] :
( ( ord_less_eq_int @ X3 @ Y4 )
=> ( ( ord_less_eq_int @ Y4 @ X3 )
=> ( X3 = Y4 ) ) ) ).
% order_antisym
thf(fact_196_order__antisym,axiom,
! [X3: a,Y4: a] :
( ( ord_less_eq_a @ X3 @ Y4 )
=> ( ( ord_less_eq_a @ Y4 @ X3 )
=> ( X3 = Y4 ) ) ) ).
% order_antisym
thf(fact_197_ord__le__eq__trans,axiom,
! [A2: real,B2: real,C: real] :
( ( ord_less_eq_real @ A2 @ B2 )
=> ( ( B2 = C )
=> ( ord_less_eq_real @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_198_ord__le__eq__trans,axiom,
! [A2: num,B2: num,C: num] :
( ( ord_less_eq_num @ A2 @ B2 )
=> ( ( B2 = C )
=> ( ord_less_eq_num @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_199_ord__le__eq__trans,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( B2 = C )
=> ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_200_ord__le__eq__trans,axiom,
! [A2: int,B2: int,C: int] :
( ( ord_less_eq_int @ A2 @ B2 )
=> ( ( B2 = C )
=> ( ord_less_eq_int @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_201_ord__le__eq__trans,axiom,
! [A2: a,B2: a,C: a] :
( ( ord_less_eq_a @ A2 @ B2 )
=> ( ( B2 = C )
=> ( ord_less_eq_a @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_202_ord__eq__le__trans,axiom,
! [A2: real,B2: real,C: real] :
( ( A2 = B2 )
=> ( ( ord_less_eq_real @ B2 @ C )
=> ( ord_less_eq_real @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_203_ord__eq__le__trans,axiom,
! [A2: num,B2: num,C: num] :
( ( A2 = B2 )
=> ( ( ord_less_eq_num @ B2 @ C )
=> ( ord_less_eq_num @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_204_ord__eq__le__trans,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( A2 = B2 )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_205_ord__eq__le__trans,axiom,
! [A2: int,B2: int,C: int] :
( ( A2 = B2 )
=> ( ( ord_less_eq_int @ B2 @ C )
=> ( ord_less_eq_int @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_206_ord__eq__le__trans,axiom,
! [A2: a,B2: a,C: a] :
( ( A2 = B2 )
=> ( ( ord_less_eq_a @ B2 @ C )
=> ( ord_less_eq_a @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_207_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y3: real,Z2: real] : ( Y3 = Z2 ) )
= ( ^ [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
& ( ord_less_eq_real @ Y @ X ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_208_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y3: num,Z2: num] : ( Y3 = Z2 ) )
= ( ^ [X: num,Y: num] :
( ( ord_less_eq_num @ X @ Y )
& ( ord_less_eq_num @ Y @ X ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_209_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y3: nat,Z2: nat] : ( Y3 = Z2 ) )
= ( ^ [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
& ( ord_less_eq_nat @ Y @ X ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_210_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y3: int,Z2: int] : ( Y3 = Z2 ) )
= ( ^ [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
& ( ord_less_eq_int @ Y @ X ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_211_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y3: a,Z2: a] : ( Y3 = Z2 ) )
= ( ^ [X: a,Y: a] :
( ( ord_less_eq_a @ X @ Y )
& ( ord_less_eq_a @ Y @ X ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_212_le__cases3,axiom,
! [X3: real,Y4: real,Z3: real] :
( ( ( ord_less_eq_real @ X3 @ Y4 )
=> ~ ( ord_less_eq_real @ Y4 @ Z3 ) )
=> ( ( ( ord_less_eq_real @ Y4 @ X3 )
=> ~ ( ord_less_eq_real @ X3 @ Z3 ) )
=> ( ( ( ord_less_eq_real @ X3 @ Z3 )
=> ~ ( ord_less_eq_real @ Z3 @ Y4 ) )
=> ( ( ( ord_less_eq_real @ Z3 @ Y4 )
=> ~ ( ord_less_eq_real @ Y4 @ X3 ) )
=> ( ( ( ord_less_eq_real @ Y4 @ Z3 )
=> ~ ( ord_less_eq_real @ Z3 @ X3 ) )
=> ~ ( ( ord_less_eq_real @ Z3 @ X3 )
=> ~ ( ord_less_eq_real @ X3 @ Y4 ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_213_le__cases3,axiom,
! [X3: num,Y4: num,Z3: num] :
( ( ( ord_less_eq_num @ X3 @ Y4 )
=> ~ ( ord_less_eq_num @ Y4 @ Z3 ) )
=> ( ( ( ord_less_eq_num @ Y4 @ X3 )
=> ~ ( ord_less_eq_num @ X3 @ Z3 ) )
=> ( ( ( ord_less_eq_num @ X3 @ Z3 )
=> ~ ( ord_less_eq_num @ Z3 @ Y4 ) )
=> ( ( ( ord_less_eq_num @ Z3 @ Y4 )
=> ~ ( ord_less_eq_num @ Y4 @ X3 ) )
=> ( ( ( ord_less_eq_num @ Y4 @ Z3 )
=> ~ ( ord_less_eq_num @ Z3 @ X3 ) )
=> ~ ( ( ord_less_eq_num @ Z3 @ X3 )
=> ~ ( ord_less_eq_num @ X3 @ Y4 ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_214_le__cases3,axiom,
! [X3: nat,Y4: nat,Z3: nat] :
( ( ( ord_less_eq_nat @ X3 @ Y4 )
=> ~ ( ord_less_eq_nat @ Y4 @ Z3 ) )
=> ( ( ( ord_less_eq_nat @ Y4 @ X3 )
=> ~ ( ord_less_eq_nat @ X3 @ Z3 ) )
=> ( ( ( ord_less_eq_nat @ X3 @ Z3 )
=> ~ ( ord_less_eq_nat @ Z3 @ Y4 ) )
=> ( ( ( ord_less_eq_nat @ Z3 @ Y4 )
=> ~ ( ord_less_eq_nat @ Y4 @ X3 ) )
=> ( ( ( ord_less_eq_nat @ Y4 @ Z3 )
=> ~ ( ord_less_eq_nat @ Z3 @ X3 ) )
=> ~ ( ( ord_less_eq_nat @ Z3 @ X3 )
=> ~ ( ord_less_eq_nat @ X3 @ Y4 ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_215_le__cases3,axiom,
! [X3: int,Y4: int,Z3: int] :
( ( ( ord_less_eq_int @ X3 @ Y4 )
=> ~ ( ord_less_eq_int @ Y4 @ Z3 ) )
=> ( ( ( ord_less_eq_int @ Y4 @ X3 )
=> ~ ( ord_less_eq_int @ X3 @ Z3 ) )
=> ( ( ( ord_less_eq_int @ X3 @ Z3 )
=> ~ ( ord_less_eq_int @ Z3 @ Y4 ) )
=> ( ( ( ord_less_eq_int @ Z3 @ Y4 )
=> ~ ( ord_less_eq_int @ Y4 @ X3 ) )
=> ( ( ( ord_less_eq_int @ Y4 @ Z3 )
=> ~ ( ord_less_eq_int @ Z3 @ X3 ) )
=> ~ ( ( ord_less_eq_int @ Z3 @ X3 )
=> ~ ( ord_less_eq_int @ X3 @ Y4 ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_216_le__cases3,axiom,
! [X3: a,Y4: a,Z3: a] :
( ( ( ord_less_eq_a @ X3 @ Y4 )
=> ~ ( ord_less_eq_a @ Y4 @ Z3 ) )
=> ( ( ( ord_less_eq_a @ Y4 @ X3 )
=> ~ ( ord_less_eq_a @ X3 @ Z3 ) )
=> ( ( ( ord_less_eq_a @ X3 @ Z3 )
=> ~ ( ord_less_eq_a @ Z3 @ Y4 ) )
=> ( ( ( ord_less_eq_a @ Z3 @ Y4 )
=> ~ ( ord_less_eq_a @ Y4 @ X3 ) )
=> ( ( ( ord_less_eq_a @ Y4 @ Z3 )
=> ~ ( ord_less_eq_a @ Z3 @ X3 ) )
=> ~ ( ( ord_less_eq_a @ Z3 @ X3 )
=> ~ ( ord_less_eq_a @ X3 @ Y4 ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_217_nle__le,axiom,
! [A2: real,B2: real] :
( ( ~ ( ord_less_eq_real @ A2 @ B2 ) )
= ( ( ord_less_eq_real @ B2 @ A2 )
& ( B2 != A2 ) ) ) ).
% nle_le
thf(fact_218_nle__le,axiom,
! [A2: num,B2: num] :
( ( ~ ( ord_less_eq_num @ A2 @ B2 ) )
= ( ( ord_less_eq_num @ B2 @ A2 )
& ( B2 != A2 ) ) ) ).
% nle_le
thf(fact_219_nle__le,axiom,
! [A2: nat,B2: nat] :
( ( ~ ( ord_less_eq_nat @ A2 @ B2 ) )
= ( ( ord_less_eq_nat @ B2 @ A2 )
& ( B2 != A2 ) ) ) ).
% nle_le
thf(fact_220_nle__le,axiom,
! [A2: int,B2: int] :
( ( ~ ( ord_less_eq_int @ A2 @ B2 ) )
= ( ( ord_less_eq_int @ B2 @ A2 )
& ( B2 != A2 ) ) ) ).
% nle_le
thf(fact_221_nle__le,axiom,
! [A2: a,B2: a] :
( ( ~ ( ord_less_eq_a @ A2 @ B2 ) )
= ( ( ord_less_eq_a @ B2 @ A2 )
& ( B2 != A2 ) ) ) ).
% nle_le
thf(fact_222_order__less__imp__not__less,axiom,
! [X3: extended_enat,Y4: extended_enat] :
( ( ord_le72135733267957522d_enat @ X3 @ Y4 )
=> ~ ( ord_le72135733267957522d_enat @ Y4 @ X3 ) ) ).
% order_less_imp_not_less
thf(fact_223_order__less__imp__not__less,axiom,
! [X3: real,Y4: real] :
( ( ord_less_real @ X3 @ Y4 )
=> ~ ( ord_less_real @ Y4 @ X3 ) ) ).
% order_less_imp_not_less
thf(fact_224_order__less__imp__not__less,axiom,
! [X3: num,Y4: num] :
( ( ord_less_num @ X3 @ Y4 )
=> ~ ( ord_less_num @ Y4 @ X3 ) ) ).
% order_less_imp_not_less
thf(fact_225_order__less__imp__not__less,axiom,
! [X3: nat,Y4: nat] :
( ( ord_less_nat @ X3 @ Y4 )
=> ~ ( ord_less_nat @ Y4 @ X3 ) ) ).
% order_less_imp_not_less
thf(fact_226_order__less__imp__not__less,axiom,
! [X3: int,Y4: int] :
( ( ord_less_int @ X3 @ Y4 )
=> ~ ( ord_less_int @ Y4 @ X3 ) ) ).
% order_less_imp_not_less
thf(fact_227_order__less__imp__not__eq2,axiom,
! [X3: extended_enat,Y4: extended_enat] :
( ( ord_le72135733267957522d_enat @ X3 @ Y4 )
=> ( Y4 != X3 ) ) ).
% order_less_imp_not_eq2
thf(fact_228_order__less__imp__not__eq2,axiom,
! [X3: real,Y4: real] :
( ( ord_less_real @ X3 @ Y4 )
=> ( Y4 != X3 ) ) ).
% order_less_imp_not_eq2
thf(fact_229_order__less__imp__not__eq2,axiom,
! [X3: num,Y4: num] :
( ( ord_less_num @ X3 @ Y4 )
=> ( Y4 != X3 ) ) ).
% order_less_imp_not_eq2
thf(fact_230_order__less__imp__not__eq2,axiom,
! [X3: nat,Y4: nat] :
( ( ord_less_nat @ X3 @ Y4 )
=> ( Y4 != X3 ) ) ).
% order_less_imp_not_eq2
thf(fact_231_order__less__imp__not__eq2,axiom,
! [X3: int,Y4: int] :
( ( ord_less_int @ X3 @ Y4 )
=> ( Y4 != X3 ) ) ).
% order_less_imp_not_eq2
thf(fact_232_order__less__imp__not__eq,axiom,
! [X3: extended_enat,Y4: extended_enat] :
( ( ord_le72135733267957522d_enat @ X3 @ Y4 )
=> ( X3 != Y4 ) ) ).
% order_less_imp_not_eq
thf(fact_233_order__less__imp__not__eq,axiom,
! [X3: real,Y4: real] :
( ( ord_less_real @ X3 @ Y4 )
=> ( X3 != Y4 ) ) ).
% order_less_imp_not_eq
thf(fact_234_order__less__imp__not__eq,axiom,
! [X3: num,Y4: num] :
( ( ord_less_num @ X3 @ Y4 )
=> ( X3 != Y4 ) ) ).
% order_less_imp_not_eq
thf(fact_235_order__less__imp__not__eq,axiom,
! [X3: nat,Y4: nat] :
( ( ord_less_nat @ X3 @ Y4 )
=> ( X3 != Y4 ) ) ).
% order_less_imp_not_eq
thf(fact_236_order__less__imp__not__eq,axiom,
! [X3: int,Y4: int] :
( ( ord_less_int @ X3 @ Y4 )
=> ( X3 != Y4 ) ) ).
% order_less_imp_not_eq
thf(fact_237_linorder__less__linear,axiom,
! [X3: extended_enat,Y4: extended_enat] :
( ( ord_le72135733267957522d_enat @ X3 @ Y4 )
| ( X3 = Y4 )
| ( ord_le72135733267957522d_enat @ Y4 @ X3 ) ) ).
% linorder_less_linear
thf(fact_238_linorder__less__linear,axiom,
! [X3: real,Y4: real] :
( ( ord_less_real @ X3 @ Y4 )
| ( X3 = Y4 )
| ( ord_less_real @ Y4 @ X3 ) ) ).
% linorder_less_linear
thf(fact_239_linorder__less__linear,axiom,
! [X3: num,Y4: num] :
( ( ord_less_num @ X3 @ Y4 )
| ( X3 = Y4 )
| ( ord_less_num @ Y4 @ X3 ) ) ).
% linorder_less_linear
thf(fact_240_linorder__less__linear,axiom,
! [X3: nat,Y4: nat] :
( ( ord_less_nat @ X3 @ Y4 )
| ( X3 = Y4 )
| ( ord_less_nat @ Y4 @ X3 ) ) ).
% linorder_less_linear
thf(fact_241_linorder__less__linear,axiom,
! [X3: int,Y4: int] :
( ( ord_less_int @ X3 @ Y4 )
| ( X3 = Y4 )
| ( ord_less_int @ Y4 @ X3 ) ) ).
% linorder_less_linear
thf(fact_242_order__less__imp__triv,axiom,
! [X3: extended_enat,Y4: extended_enat,P: $o] :
( ( ord_le72135733267957522d_enat @ X3 @ Y4 )
=> ( ( ord_le72135733267957522d_enat @ Y4 @ X3 )
=> P ) ) ).
% order_less_imp_triv
thf(fact_243_order__less__imp__triv,axiom,
! [X3: real,Y4: real,P: $o] :
( ( ord_less_real @ X3 @ Y4 )
=> ( ( ord_less_real @ Y4 @ X3 )
=> P ) ) ).
% order_less_imp_triv
thf(fact_244_order__less__imp__triv,axiom,
! [X3: num,Y4: num,P: $o] :
( ( ord_less_num @ X3 @ Y4 )
=> ( ( ord_less_num @ Y4 @ X3 )
=> P ) ) ).
% order_less_imp_triv
thf(fact_245_order__less__imp__triv,axiom,
! [X3: nat,Y4: nat,P: $o] :
( ( ord_less_nat @ X3 @ Y4 )
=> ( ( ord_less_nat @ Y4 @ X3 )
=> P ) ) ).
% order_less_imp_triv
thf(fact_246_order__less__imp__triv,axiom,
! [X3: int,Y4: int,P: $o] :
( ( ord_less_int @ X3 @ Y4 )
=> ( ( ord_less_int @ Y4 @ X3 )
=> P ) ) ).
% order_less_imp_triv
thf(fact_247_order__less__not__sym,axiom,
! [X3: extended_enat,Y4: extended_enat] :
( ( ord_le72135733267957522d_enat @ X3 @ Y4 )
=> ~ ( ord_le72135733267957522d_enat @ Y4 @ X3 ) ) ).
% order_less_not_sym
thf(fact_248_order__less__not__sym,axiom,
! [X3: real,Y4: real] :
( ( ord_less_real @ X3 @ Y4 )
=> ~ ( ord_less_real @ Y4 @ X3 ) ) ).
% order_less_not_sym
thf(fact_249_order__less__not__sym,axiom,
! [X3: num,Y4: num] :
( ( ord_less_num @ X3 @ Y4 )
=> ~ ( ord_less_num @ Y4 @ X3 ) ) ).
% order_less_not_sym
thf(fact_250_order__less__not__sym,axiom,
! [X3: nat,Y4: nat] :
( ( ord_less_nat @ X3 @ Y4 )
=> ~ ( ord_less_nat @ Y4 @ X3 ) ) ).
% order_less_not_sym
thf(fact_251_order__less__not__sym,axiom,
! [X3: int,Y4: int] :
( ( ord_less_int @ X3 @ Y4 )
=> ~ ( ord_less_int @ Y4 @ X3 ) ) ).
% order_less_not_sym
thf(fact_252_order__less__subst2,axiom,
! [A2: extended_enat,B2: extended_enat,F: extended_enat > extended_enat,C: extended_enat] :
( ( ord_le72135733267957522d_enat @ A2 @ B2 )
=> ( ( ord_le72135733267957522d_enat @ ( F @ B2 ) @ C )
=> ( ! [X5: extended_enat,Y5: extended_enat] :
( ( ord_le72135733267957522d_enat @ X5 @ Y5 )
=> ( ord_le72135733267957522d_enat @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_le72135733267957522d_enat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_253_order__less__subst2,axiom,
! [A2: extended_enat,B2: extended_enat,F: extended_enat > real,C: real] :
( ( ord_le72135733267957522d_enat @ A2 @ B2 )
=> ( ( ord_less_real @ ( F @ B2 ) @ C )
=> ( ! [X5: extended_enat,Y5: extended_enat] :
( ( ord_le72135733267957522d_enat @ X5 @ Y5 )
=> ( ord_less_real @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_less_real @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_254_order__less__subst2,axiom,
! [A2: extended_enat,B2: extended_enat,F: extended_enat > num,C: num] :
( ( ord_le72135733267957522d_enat @ A2 @ B2 )
=> ( ( ord_less_num @ ( F @ B2 ) @ C )
=> ( ! [X5: extended_enat,Y5: extended_enat] :
( ( ord_le72135733267957522d_enat @ X5 @ Y5 )
=> ( ord_less_num @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_less_num @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_255_order__less__subst2,axiom,
! [A2: extended_enat,B2: extended_enat,F: extended_enat > nat,C: nat] :
( ( ord_le72135733267957522d_enat @ A2 @ B2 )
=> ( ( ord_less_nat @ ( F @ B2 ) @ C )
=> ( ! [X5: extended_enat,Y5: extended_enat] :
( ( ord_le72135733267957522d_enat @ X5 @ Y5 )
=> ( ord_less_nat @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_256_order__less__subst2,axiom,
! [A2: extended_enat,B2: extended_enat,F: extended_enat > int,C: int] :
( ( ord_le72135733267957522d_enat @ A2 @ B2 )
=> ( ( ord_less_int @ ( F @ B2 ) @ C )
=> ( ! [X5: extended_enat,Y5: extended_enat] :
( ( ord_le72135733267957522d_enat @ X5 @ Y5 )
=> ( ord_less_int @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_257_order__less__subst2,axiom,
! [A2: real,B2: real,F: real > extended_enat,C: extended_enat] :
( ( ord_less_real @ A2 @ B2 )
=> ( ( ord_le72135733267957522d_enat @ ( F @ B2 ) @ C )
=> ( ! [X5: real,Y5: real] :
( ( ord_less_real @ X5 @ Y5 )
=> ( ord_le72135733267957522d_enat @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_le72135733267957522d_enat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_258_order__less__subst2,axiom,
! [A2: real,B2: real,F: real > real,C: real] :
( ( ord_less_real @ A2 @ B2 )
=> ( ( ord_less_real @ ( F @ B2 ) @ C )
=> ( ! [X5: real,Y5: real] :
( ( ord_less_real @ X5 @ Y5 )
=> ( ord_less_real @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_less_real @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_259_order__less__subst2,axiom,
! [A2: real,B2: real,F: real > num,C: num] :
( ( ord_less_real @ A2 @ B2 )
=> ( ( ord_less_num @ ( F @ B2 ) @ C )
=> ( ! [X5: real,Y5: real] :
( ( ord_less_real @ X5 @ Y5 )
=> ( ord_less_num @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_less_num @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_260_order__less__subst2,axiom,
! [A2: real,B2: real,F: real > nat,C: nat] :
( ( ord_less_real @ A2 @ B2 )
=> ( ( ord_less_nat @ ( F @ B2 ) @ C )
=> ( ! [X5: real,Y5: real] :
( ( ord_less_real @ X5 @ Y5 )
=> ( ord_less_nat @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_261_order__less__subst2,axiom,
! [A2: real,B2: real,F: real > int,C: int] :
( ( ord_less_real @ A2 @ B2 )
=> ( ( ord_less_int @ ( F @ B2 ) @ C )
=> ( ! [X5: real,Y5: real] :
( ( ord_less_real @ X5 @ Y5 )
=> ( ord_less_int @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_262_order__less__subst1,axiom,
! [A2: extended_enat,F: extended_enat > extended_enat,B2: extended_enat,C: extended_enat] :
( ( ord_le72135733267957522d_enat @ A2 @ ( F @ B2 ) )
=> ( ( ord_le72135733267957522d_enat @ B2 @ C )
=> ( ! [X5: extended_enat,Y5: extended_enat] :
( ( ord_le72135733267957522d_enat @ X5 @ Y5 )
=> ( ord_le72135733267957522d_enat @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_le72135733267957522d_enat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_263_order__less__subst1,axiom,
! [A2: extended_enat,F: real > extended_enat,B2: real,C: real] :
( ( ord_le72135733267957522d_enat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_real @ B2 @ C )
=> ( ! [X5: real,Y5: real] :
( ( ord_less_real @ X5 @ Y5 )
=> ( ord_le72135733267957522d_enat @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_le72135733267957522d_enat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_264_order__less__subst1,axiom,
! [A2: extended_enat,F: num > extended_enat,B2: num,C: num] :
( ( ord_le72135733267957522d_enat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_num @ B2 @ C )
=> ( ! [X5: num,Y5: num] :
( ( ord_less_num @ X5 @ Y5 )
=> ( ord_le72135733267957522d_enat @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_le72135733267957522d_enat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_265_order__less__subst1,axiom,
! [A2: extended_enat,F: nat > extended_enat,B2: nat,C: nat] :
( ( ord_le72135733267957522d_enat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ! [X5: nat,Y5: nat] :
( ( ord_less_nat @ X5 @ Y5 )
=> ( ord_le72135733267957522d_enat @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_le72135733267957522d_enat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_266_order__less__subst1,axiom,
! [A2: extended_enat,F: int > extended_enat,B2: int,C: int] :
( ( ord_le72135733267957522d_enat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_int @ B2 @ C )
=> ( ! [X5: int,Y5: int] :
( ( ord_less_int @ X5 @ Y5 )
=> ( ord_le72135733267957522d_enat @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_le72135733267957522d_enat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_267_order__less__subst1,axiom,
! [A2: real,F: extended_enat > real,B2: extended_enat,C: extended_enat] :
( ( ord_less_real @ A2 @ ( F @ B2 ) )
=> ( ( ord_le72135733267957522d_enat @ B2 @ C )
=> ( ! [X5: extended_enat,Y5: extended_enat] :
( ( ord_le72135733267957522d_enat @ X5 @ Y5 )
=> ( ord_less_real @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_less_real @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_268_order__less__subst1,axiom,
! [A2: real,F: real > real,B2: real,C: real] :
( ( ord_less_real @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_real @ B2 @ C )
=> ( ! [X5: real,Y5: real] :
( ( ord_less_real @ X5 @ Y5 )
=> ( ord_less_real @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_less_real @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_269_order__less__subst1,axiom,
! [A2: real,F: num > real,B2: num,C: num] :
( ( ord_less_real @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_num @ B2 @ C )
=> ( ! [X5: num,Y5: num] :
( ( ord_less_num @ X5 @ Y5 )
=> ( ord_less_real @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_less_real @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_270_order__less__subst1,axiom,
! [A2: real,F: nat > real,B2: nat,C: nat] :
( ( ord_less_real @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ! [X5: nat,Y5: nat] :
( ( ord_less_nat @ X5 @ Y5 )
=> ( ord_less_real @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_less_real @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_271_order__less__subst1,axiom,
! [A2: real,F: int > real,B2: int,C: int] :
( ( ord_less_real @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_int @ B2 @ C )
=> ( ! [X5: int,Y5: int] :
( ( ord_less_int @ X5 @ Y5 )
=> ( ord_less_real @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_less_real @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_272_order__less__irrefl,axiom,
! [X3: extended_enat] :
~ ( ord_le72135733267957522d_enat @ X3 @ X3 ) ).
% order_less_irrefl
thf(fact_273_order__less__irrefl,axiom,
! [X3: real] :
~ ( ord_less_real @ X3 @ X3 ) ).
% order_less_irrefl
thf(fact_274_order__less__irrefl,axiom,
! [X3: num] :
~ ( ord_less_num @ X3 @ X3 ) ).
% order_less_irrefl
thf(fact_275_order__less__irrefl,axiom,
! [X3: nat] :
~ ( ord_less_nat @ X3 @ X3 ) ).
% order_less_irrefl
thf(fact_276_order__less__irrefl,axiom,
! [X3: int] :
~ ( ord_less_int @ X3 @ X3 ) ).
% order_less_irrefl
thf(fact_277_ord__less__eq__subst,axiom,
! [A2: extended_enat,B2: extended_enat,F: extended_enat > extended_enat,C: extended_enat] :
( ( ord_le72135733267957522d_enat @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X5: extended_enat,Y5: extended_enat] :
( ( ord_le72135733267957522d_enat @ X5 @ Y5 )
=> ( ord_le72135733267957522d_enat @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_le72135733267957522d_enat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_278_ord__less__eq__subst,axiom,
! [A2: extended_enat,B2: extended_enat,F: extended_enat > real,C: real] :
( ( ord_le72135733267957522d_enat @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X5: extended_enat,Y5: extended_enat] :
( ( ord_le72135733267957522d_enat @ X5 @ Y5 )
=> ( ord_less_real @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_less_real @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_279_ord__less__eq__subst,axiom,
! [A2: extended_enat,B2: extended_enat,F: extended_enat > num,C: num] :
( ( ord_le72135733267957522d_enat @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X5: extended_enat,Y5: extended_enat] :
( ( ord_le72135733267957522d_enat @ X5 @ Y5 )
=> ( ord_less_num @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_less_num @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_280_ord__less__eq__subst,axiom,
! [A2: extended_enat,B2: extended_enat,F: extended_enat > nat,C: nat] :
( ( ord_le72135733267957522d_enat @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X5: extended_enat,Y5: extended_enat] :
( ( ord_le72135733267957522d_enat @ X5 @ Y5 )
=> ( ord_less_nat @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_281_ord__less__eq__subst,axiom,
! [A2: extended_enat,B2: extended_enat,F: extended_enat > int,C: int] :
( ( ord_le72135733267957522d_enat @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X5: extended_enat,Y5: extended_enat] :
( ( ord_le72135733267957522d_enat @ X5 @ Y5 )
=> ( ord_less_int @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_282_ord__less__eq__subst,axiom,
! [A2: real,B2: real,F: real > extended_enat,C: extended_enat] :
( ( ord_less_real @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X5: real,Y5: real] :
( ( ord_less_real @ X5 @ Y5 )
=> ( ord_le72135733267957522d_enat @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_le72135733267957522d_enat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_283_ord__less__eq__subst,axiom,
! [A2: real,B2: real,F: real > real,C: real] :
( ( ord_less_real @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X5: real,Y5: real] :
( ( ord_less_real @ X5 @ Y5 )
=> ( ord_less_real @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_less_real @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_284_ord__less__eq__subst,axiom,
! [A2: real,B2: real,F: real > num,C: num] :
( ( ord_less_real @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X5: real,Y5: real] :
( ( ord_less_real @ X5 @ Y5 )
=> ( ord_less_num @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_less_num @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_285_ord__less__eq__subst,axiom,
! [A2: real,B2: real,F: real > nat,C: nat] :
( ( ord_less_real @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X5: real,Y5: real] :
( ( ord_less_real @ X5 @ Y5 )
=> ( ord_less_nat @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_286_ord__less__eq__subst,axiom,
! [A2: real,B2: real,F: real > int,C: int] :
( ( ord_less_real @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X5: real,Y5: real] :
( ( ord_less_real @ X5 @ Y5 )
=> ( ord_less_int @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_287_ord__eq__less__subst,axiom,
! [A2: extended_enat,F: extended_enat > extended_enat,B2: extended_enat,C: extended_enat] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_le72135733267957522d_enat @ B2 @ C )
=> ( ! [X5: extended_enat,Y5: extended_enat] :
( ( ord_le72135733267957522d_enat @ X5 @ Y5 )
=> ( ord_le72135733267957522d_enat @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_le72135733267957522d_enat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_288_ord__eq__less__subst,axiom,
! [A2: real,F: extended_enat > real,B2: extended_enat,C: extended_enat] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_le72135733267957522d_enat @ B2 @ C )
=> ( ! [X5: extended_enat,Y5: extended_enat] :
( ( ord_le72135733267957522d_enat @ X5 @ Y5 )
=> ( ord_less_real @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_less_real @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_289_ord__eq__less__subst,axiom,
! [A2: num,F: extended_enat > num,B2: extended_enat,C: extended_enat] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_le72135733267957522d_enat @ B2 @ C )
=> ( ! [X5: extended_enat,Y5: extended_enat] :
( ( ord_le72135733267957522d_enat @ X5 @ Y5 )
=> ( ord_less_num @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_less_num @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_290_ord__eq__less__subst,axiom,
! [A2: nat,F: extended_enat > nat,B2: extended_enat,C: extended_enat] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_le72135733267957522d_enat @ B2 @ C )
=> ( ! [X5: extended_enat,Y5: extended_enat] :
( ( ord_le72135733267957522d_enat @ X5 @ Y5 )
=> ( ord_less_nat @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_291_ord__eq__less__subst,axiom,
! [A2: int,F: extended_enat > int,B2: extended_enat,C: extended_enat] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_le72135733267957522d_enat @ B2 @ C )
=> ( ! [X5: extended_enat,Y5: extended_enat] :
( ( ord_le72135733267957522d_enat @ X5 @ Y5 )
=> ( ord_less_int @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_less_int @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_292_ord__eq__less__subst,axiom,
! [A2: extended_enat,F: real > extended_enat,B2: real,C: real] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_real @ B2 @ C )
=> ( ! [X5: real,Y5: real] :
( ( ord_less_real @ X5 @ Y5 )
=> ( ord_le72135733267957522d_enat @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_le72135733267957522d_enat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_293_ord__eq__less__subst,axiom,
! [A2: real,F: real > real,B2: real,C: real] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_real @ B2 @ C )
=> ( ! [X5: real,Y5: real] :
( ( ord_less_real @ X5 @ Y5 )
=> ( ord_less_real @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_less_real @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_294_ord__eq__less__subst,axiom,
! [A2: num,F: real > num,B2: real,C: real] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_real @ B2 @ C )
=> ( ! [X5: real,Y5: real] :
( ( ord_less_real @ X5 @ Y5 )
=> ( ord_less_num @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_less_num @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_295_ord__eq__less__subst,axiom,
! [A2: nat,F: real > nat,B2: real,C: real] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_real @ B2 @ C )
=> ( ! [X5: real,Y5: real] :
( ( ord_less_real @ X5 @ Y5 )
=> ( ord_less_nat @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_296_ord__eq__less__subst,axiom,
! [A2: int,F: real > int,B2: real,C: real] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_real @ B2 @ C )
=> ( ! [X5: real,Y5: real] :
( ( ord_less_real @ X5 @ Y5 )
=> ( ord_less_int @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_less_int @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_297_order__less__trans,axiom,
! [X3: extended_enat,Y4: extended_enat,Z3: extended_enat] :
( ( ord_le72135733267957522d_enat @ X3 @ Y4 )
=> ( ( ord_le72135733267957522d_enat @ Y4 @ Z3 )
=> ( ord_le72135733267957522d_enat @ X3 @ Z3 ) ) ) ).
% order_less_trans
thf(fact_298_order__less__trans,axiom,
! [X3: real,Y4: real,Z3: real] :
( ( ord_less_real @ X3 @ Y4 )
=> ( ( ord_less_real @ Y4 @ Z3 )
=> ( ord_less_real @ X3 @ Z3 ) ) ) ).
% order_less_trans
thf(fact_299_order__less__trans,axiom,
! [X3: num,Y4: num,Z3: num] :
( ( ord_less_num @ X3 @ Y4 )
=> ( ( ord_less_num @ Y4 @ Z3 )
=> ( ord_less_num @ X3 @ Z3 ) ) ) ).
% order_less_trans
thf(fact_300_order__less__trans,axiom,
! [X3: nat,Y4: nat,Z3: nat] :
( ( ord_less_nat @ X3 @ Y4 )
=> ( ( ord_less_nat @ Y4 @ Z3 )
=> ( ord_less_nat @ X3 @ Z3 ) ) ) ).
% order_less_trans
thf(fact_301_order__less__trans,axiom,
! [X3: int,Y4: int,Z3: int] :
( ( ord_less_int @ X3 @ Y4 )
=> ( ( ord_less_int @ Y4 @ Z3 )
=> ( ord_less_int @ X3 @ Z3 ) ) ) ).
% order_less_trans
thf(fact_302_order__less__asym_H,axiom,
! [A2: extended_enat,B2: extended_enat] :
( ( ord_le72135733267957522d_enat @ A2 @ B2 )
=> ~ ( ord_le72135733267957522d_enat @ B2 @ A2 ) ) ).
% order_less_asym'
thf(fact_303_order__less__asym_H,axiom,
! [A2: real,B2: real] :
( ( ord_less_real @ A2 @ B2 )
=> ~ ( ord_less_real @ B2 @ A2 ) ) ).
% order_less_asym'
thf(fact_304_order__less__asym_H,axiom,
! [A2: num,B2: num] :
( ( ord_less_num @ A2 @ B2 )
=> ~ ( ord_less_num @ B2 @ A2 ) ) ).
% order_less_asym'
thf(fact_305_order__less__asym_H,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ~ ( ord_less_nat @ B2 @ A2 ) ) ).
% order_less_asym'
thf(fact_306_order__less__asym_H,axiom,
! [A2: int,B2: int] :
( ( ord_less_int @ A2 @ B2 )
=> ~ ( ord_less_int @ B2 @ A2 ) ) ).
% order_less_asym'
thf(fact_307_linorder__neq__iff,axiom,
! [X3: extended_enat,Y4: extended_enat] :
( ( X3 != Y4 )
= ( ( ord_le72135733267957522d_enat @ X3 @ Y4 )
| ( ord_le72135733267957522d_enat @ Y4 @ X3 ) ) ) ).
% linorder_neq_iff
thf(fact_308_linorder__neq__iff,axiom,
! [X3: real,Y4: real] :
( ( X3 != Y4 )
= ( ( ord_less_real @ X3 @ Y4 )
| ( ord_less_real @ Y4 @ X3 ) ) ) ).
% linorder_neq_iff
thf(fact_309_linorder__neq__iff,axiom,
! [X3: num,Y4: num] :
( ( X3 != Y4 )
= ( ( ord_less_num @ X3 @ Y4 )
| ( ord_less_num @ Y4 @ X3 ) ) ) ).
% linorder_neq_iff
thf(fact_310_linorder__neq__iff,axiom,
! [X3: nat,Y4: nat] :
( ( X3 != Y4 )
= ( ( ord_less_nat @ X3 @ Y4 )
| ( ord_less_nat @ Y4 @ X3 ) ) ) ).
% linorder_neq_iff
thf(fact_311_linorder__neq__iff,axiom,
! [X3: int,Y4: int] :
( ( X3 != Y4 )
= ( ( ord_less_int @ X3 @ Y4 )
| ( ord_less_int @ Y4 @ X3 ) ) ) ).
% linorder_neq_iff
thf(fact_312_order__less__asym,axiom,
! [X3: extended_enat,Y4: extended_enat] :
( ( ord_le72135733267957522d_enat @ X3 @ Y4 )
=> ~ ( ord_le72135733267957522d_enat @ Y4 @ X3 ) ) ).
% order_less_asym
thf(fact_313_order__less__asym,axiom,
! [X3: real,Y4: real] :
( ( ord_less_real @ X3 @ Y4 )
=> ~ ( ord_less_real @ Y4 @ X3 ) ) ).
% order_less_asym
thf(fact_314_order__less__asym,axiom,
! [X3: num,Y4: num] :
( ( ord_less_num @ X3 @ Y4 )
=> ~ ( ord_less_num @ Y4 @ X3 ) ) ).
% order_less_asym
thf(fact_315_order__less__asym,axiom,
! [X3: nat,Y4: nat] :
( ( ord_less_nat @ X3 @ Y4 )
=> ~ ( ord_less_nat @ Y4 @ X3 ) ) ).
% order_less_asym
thf(fact_316_order__less__asym,axiom,
! [X3: int,Y4: int] :
( ( ord_less_int @ X3 @ Y4 )
=> ~ ( ord_less_int @ Y4 @ X3 ) ) ).
% order_less_asym
thf(fact_317_linorder__neqE,axiom,
! [X3: extended_enat,Y4: extended_enat] :
( ( X3 != Y4 )
=> ( ~ ( ord_le72135733267957522d_enat @ X3 @ Y4 )
=> ( ord_le72135733267957522d_enat @ Y4 @ X3 ) ) ) ).
% linorder_neqE
thf(fact_318_linorder__neqE,axiom,
! [X3: real,Y4: real] :
( ( X3 != Y4 )
=> ( ~ ( ord_less_real @ X3 @ Y4 )
=> ( ord_less_real @ Y4 @ X3 ) ) ) ).
% linorder_neqE
thf(fact_319_linorder__neqE,axiom,
! [X3: num,Y4: num] :
( ( X3 != Y4 )
=> ( ~ ( ord_less_num @ X3 @ Y4 )
=> ( ord_less_num @ Y4 @ X3 ) ) ) ).
% linorder_neqE
thf(fact_320_linorder__neqE,axiom,
! [X3: nat,Y4: nat] :
( ( X3 != Y4 )
=> ( ~ ( ord_less_nat @ X3 @ Y4 )
=> ( ord_less_nat @ Y4 @ X3 ) ) ) ).
% linorder_neqE
thf(fact_321_linorder__neqE,axiom,
! [X3: int,Y4: int] :
( ( X3 != Y4 )
=> ( ~ ( ord_less_int @ X3 @ Y4 )
=> ( ord_less_int @ Y4 @ X3 ) ) ) ).
% linorder_neqE
thf(fact_322_dual__order_Ostrict__implies__not__eq,axiom,
! [B2: extended_enat,A2: extended_enat] :
( ( ord_le72135733267957522d_enat @ B2 @ A2 )
=> ( A2 != B2 ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_323_dual__order_Ostrict__implies__not__eq,axiom,
! [B2: real,A2: real] :
( ( ord_less_real @ B2 @ A2 )
=> ( A2 != B2 ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_324_dual__order_Ostrict__implies__not__eq,axiom,
! [B2: num,A2: num] :
( ( ord_less_num @ B2 @ A2 )
=> ( A2 != B2 ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_325_dual__order_Ostrict__implies__not__eq,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_nat @ B2 @ A2 )
=> ( A2 != B2 ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_326_dual__order_Ostrict__implies__not__eq,axiom,
! [B2: int,A2: int] :
( ( ord_less_int @ B2 @ A2 )
=> ( A2 != B2 ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_327_order_Ostrict__implies__not__eq,axiom,
! [A2: extended_enat,B2: extended_enat] :
( ( ord_le72135733267957522d_enat @ A2 @ B2 )
=> ( A2 != B2 ) ) ).
% order.strict_implies_not_eq
thf(fact_328_order_Ostrict__implies__not__eq,axiom,
! [A2: real,B2: real] :
( ( ord_less_real @ A2 @ B2 )
=> ( A2 != B2 ) ) ).
% order.strict_implies_not_eq
thf(fact_329_order_Ostrict__implies__not__eq,axiom,
! [A2: num,B2: num] :
( ( ord_less_num @ A2 @ B2 )
=> ( A2 != B2 ) ) ).
% order.strict_implies_not_eq
thf(fact_330_order_Ostrict__implies__not__eq,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( A2 != B2 ) ) ).
% order.strict_implies_not_eq
thf(fact_331_order_Ostrict__implies__not__eq,axiom,
! [A2: int,B2: int] :
( ( ord_less_int @ A2 @ B2 )
=> ( A2 != B2 ) ) ).
% order.strict_implies_not_eq
thf(fact_332_dual__order_Ostrict__trans,axiom,
! [B2: extended_enat,A2: extended_enat,C: extended_enat] :
( ( ord_le72135733267957522d_enat @ B2 @ A2 )
=> ( ( ord_le72135733267957522d_enat @ C @ B2 )
=> ( ord_le72135733267957522d_enat @ C @ A2 ) ) ) ).
% dual_order.strict_trans
thf(fact_333_dual__order_Ostrict__trans,axiom,
! [B2: real,A2: real,C: real] :
( ( ord_less_real @ B2 @ A2 )
=> ( ( ord_less_real @ C @ B2 )
=> ( ord_less_real @ C @ A2 ) ) ) ).
% dual_order.strict_trans
thf(fact_334_dual__order_Ostrict__trans,axiom,
! [B2: num,A2: num,C: num] :
( ( ord_less_num @ B2 @ A2 )
=> ( ( ord_less_num @ C @ B2 )
=> ( ord_less_num @ C @ A2 ) ) ) ).
% dual_order.strict_trans
thf(fact_335_dual__order_Ostrict__trans,axiom,
! [B2: nat,A2: nat,C: nat] :
( ( ord_less_nat @ B2 @ A2 )
=> ( ( ord_less_nat @ C @ B2 )
=> ( ord_less_nat @ C @ A2 ) ) ) ).
% dual_order.strict_trans
thf(fact_336_dual__order_Ostrict__trans,axiom,
! [B2: int,A2: int,C: int] :
( ( ord_less_int @ B2 @ A2 )
=> ( ( ord_less_int @ C @ B2 )
=> ( ord_less_int @ C @ A2 ) ) ) ).
% dual_order.strict_trans
thf(fact_337_not__less__iff__gr__or__eq,axiom,
! [X3: extended_enat,Y4: extended_enat] :
( ( ~ ( ord_le72135733267957522d_enat @ X3 @ Y4 ) )
= ( ( ord_le72135733267957522d_enat @ Y4 @ X3 )
| ( X3 = Y4 ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_338_not__less__iff__gr__or__eq,axiom,
! [X3: real,Y4: real] :
( ( ~ ( ord_less_real @ X3 @ Y4 ) )
= ( ( ord_less_real @ Y4 @ X3 )
| ( X3 = Y4 ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_339_not__less__iff__gr__or__eq,axiom,
! [X3: num,Y4: num] :
( ( ~ ( ord_less_num @ X3 @ Y4 ) )
= ( ( ord_less_num @ Y4 @ X3 )
| ( X3 = Y4 ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_340_not__less__iff__gr__or__eq,axiom,
! [X3: nat,Y4: nat] :
( ( ~ ( ord_less_nat @ X3 @ Y4 ) )
= ( ( ord_less_nat @ Y4 @ X3 )
| ( X3 = Y4 ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_341_not__less__iff__gr__or__eq,axiom,
! [X3: int,Y4: int] :
( ( ~ ( ord_less_int @ X3 @ Y4 ) )
= ( ( ord_less_int @ Y4 @ X3 )
| ( X3 = Y4 ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_342_order_Ostrict__trans,axiom,
! [A2: extended_enat,B2: extended_enat,C: extended_enat] :
( ( ord_le72135733267957522d_enat @ A2 @ B2 )
=> ( ( ord_le72135733267957522d_enat @ B2 @ C )
=> ( ord_le72135733267957522d_enat @ A2 @ C ) ) ) ).
% order.strict_trans
thf(fact_343_order_Ostrict__trans,axiom,
! [A2: real,B2: real,C: real] :
( ( ord_less_real @ A2 @ B2 )
=> ( ( ord_less_real @ B2 @ C )
=> ( ord_less_real @ A2 @ C ) ) ) ).
% order.strict_trans
thf(fact_344_order_Ostrict__trans,axiom,
! [A2: num,B2: num,C: num] :
( ( ord_less_num @ A2 @ B2 )
=> ( ( ord_less_num @ B2 @ C )
=> ( ord_less_num @ A2 @ C ) ) ) ).
% order.strict_trans
thf(fact_345_order_Ostrict__trans,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ord_less_nat @ A2 @ C ) ) ) ).
% order.strict_trans
thf(fact_346_order_Ostrict__trans,axiom,
! [A2: int,B2: int,C: int] :
( ( ord_less_int @ A2 @ B2 )
=> ( ( ord_less_int @ B2 @ C )
=> ( ord_less_int @ A2 @ C ) ) ) ).
% order.strict_trans
thf(fact_347_linorder__less__wlog,axiom,
! [P: extended_enat > extended_enat > $o,A2: extended_enat,B2: extended_enat] :
( ! [A5: extended_enat,B4: extended_enat] :
( ( ord_le72135733267957522d_enat @ A5 @ B4 )
=> ( P @ A5 @ B4 ) )
=> ( ! [A5: extended_enat] : ( P @ A5 @ A5 )
=> ( ! [A5: extended_enat,B4: extended_enat] :
( ( P @ B4 @ A5 )
=> ( P @ A5 @ B4 ) )
=> ( P @ A2 @ B2 ) ) ) ) ).
% linorder_less_wlog
thf(fact_348_linorder__less__wlog,axiom,
! [P: real > real > $o,A2: real,B2: real] :
( ! [A5: real,B4: real] :
( ( ord_less_real @ A5 @ B4 )
=> ( P @ A5 @ B4 ) )
=> ( ! [A5: real] : ( P @ A5 @ A5 )
=> ( ! [A5: real,B4: real] :
( ( P @ B4 @ A5 )
=> ( P @ A5 @ B4 ) )
=> ( P @ A2 @ B2 ) ) ) ) ).
% linorder_less_wlog
thf(fact_349_linorder__less__wlog,axiom,
! [P: num > num > $o,A2: num,B2: num] :
( ! [A5: num,B4: num] :
( ( ord_less_num @ A5 @ B4 )
=> ( P @ A5 @ B4 ) )
=> ( ! [A5: num] : ( P @ A5 @ A5 )
=> ( ! [A5: num,B4: num] :
( ( P @ B4 @ A5 )
=> ( P @ A5 @ B4 ) )
=> ( P @ A2 @ B2 ) ) ) ) ).
% linorder_less_wlog
thf(fact_350_linorder__less__wlog,axiom,
! [P: nat > nat > $o,A2: nat,B2: nat] :
( ! [A5: nat,B4: nat] :
( ( ord_less_nat @ A5 @ B4 )
=> ( P @ A5 @ B4 ) )
=> ( ! [A5: nat] : ( P @ A5 @ A5 )
=> ( ! [A5: nat,B4: nat] :
( ( P @ B4 @ A5 )
=> ( P @ A5 @ B4 ) )
=> ( P @ A2 @ B2 ) ) ) ) ).
% linorder_less_wlog
thf(fact_351_linorder__less__wlog,axiom,
! [P: int > int > $o,A2: int,B2: int] :
( ! [A5: int,B4: int] :
( ( ord_less_int @ A5 @ B4 )
=> ( P @ A5 @ B4 ) )
=> ( ! [A5: int] : ( P @ A5 @ A5 )
=> ( ! [A5: int,B4: int] :
( ( P @ B4 @ A5 )
=> ( P @ A5 @ B4 ) )
=> ( P @ A2 @ B2 ) ) ) ) ).
% linorder_less_wlog
thf(fact_352_exists__least__iff,axiom,
( ( ^ [P2: extended_enat > $o] :
? [X6: extended_enat] : ( P2 @ X6 ) )
= ( ^ [P3: extended_enat > $o] :
? [N2: extended_enat] :
( ( P3 @ N2 )
& ! [M2: extended_enat] :
( ( ord_le72135733267957522d_enat @ M2 @ N2 )
=> ~ ( P3 @ M2 ) ) ) ) ) ).
% exists_least_iff
thf(fact_353_exists__least__iff,axiom,
( ( ^ [P2: nat > $o] :
? [X6: nat] : ( P2 @ X6 ) )
= ( ^ [P3: nat > $o] :
? [N2: nat] :
( ( P3 @ N2 )
& ! [M2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ~ ( P3 @ M2 ) ) ) ) ) ).
% exists_least_iff
thf(fact_354_dual__order_Oirrefl,axiom,
! [A2: extended_enat] :
~ ( ord_le72135733267957522d_enat @ A2 @ A2 ) ).
% dual_order.irrefl
thf(fact_355_dual__order_Oirrefl,axiom,
! [A2: real] :
~ ( ord_less_real @ A2 @ A2 ) ).
% dual_order.irrefl
thf(fact_356_dual__order_Oirrefl,axiom,
! [A2: num] :
~ ( ord_less_num @ A2 @ A2 ) ).
% dual_order.irrefl
thf(fact_357_dual__order_Oirrefl,axiom,
! [A2: nat] :
~ ( ord_less_nat @ A2 @ A2 ) ).
% dual_order.irrefl
thf(fact_358_dual__order_Oirrefl,axiom,
! [A2: int] :
~ ( ord_less_int @ A2 @ A2 ) ).
% dual_order.irrefl
thf(fact_359_dual__order_Oasym,axiom,
! [B2: extended_enat,A2: extended_enat] :
( ( ord_le72135733267957522d_enat @ B2 @ A2 )
=> ~ ( ord_le72135733267957522d_enat @ A2 @ B2 ) ) ).
% dual_order.asym
thf(fact_360_dual__order_Oasym,axiom,
! [B2: real,A2: real] :
( ( ord_less_real @ B2 @ A2 )
=> ~ ( ord_less_real @ A2 @ B2 ) ) ).
% dual_order.asym
thf(fact_361_dual__order_Oasym,axiom,
! [B2: num,A2: num] :
( ( ord_less_num @ B2 @ A2 )
=> ~ ( ord_less_num @ A2 @ B2 ) ) ).
% dual_order.asym
thf(fact_362_dual__order_Oasym,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_nat @ B2 @ A2 )
=> ~ ( ord_less_nat @ A2 @ B2 ) ) ).
% dual_order.asym
thf(fact_363_dual__order_Oasym,axiom,
! [B2: int,A2: int] :
( ( ord_less_int @ B2 @ A2 )
=> ~ ( ord_less_int @ A2 @ B2 ) ) ).
% dual_order.asym
thf(fact_364_linorder__cases,axiom,
! [X3: extended_enat,Y4: extended_enat] :
( ~ ( ord_le72135733267957522d_enat @ X3 @ Y4 )
=> ( ( X3 != Y4 )
=> ( ord_le72135733267957522d_enat @ Y4 @ X3 ) ) ) ).
% linorder_cases
thf(fact_365_linorder__cases,axiom,
! [X3: real,Y4: real] :
( ~ ( ord_less_real @ X3 @ Y4 )
=> ( ( X3 != Y4 )
=> ( ord_less_real @ Y4 @ X3 ) ) ) ).
% linorder_cases
thf(fact_366_linorder__cases,axiom,
! [X3: num,Y4: num] :
( ~ ( ord_less_num @ X3 @ Y4 )
=> ( ( X3 != Y4 )
=> ( ord_less_num @ Y4 @ X3 ) ) ) ).
% linorder_cases
thf(fact_367_linorder__cases,axiom,
! [X3: nat,Y4: nat] :
( ~ ( ord_less_nat @ X3 @ Y4 )
=> ( ( X3 != Y4 )
=> ( ord_less_nat @ Y4 @ X3 ) ) ) ).
% linorder_cases
thf(fact_368_linorder__cases,axiom,
! [X3: int,Y4: int] :
( ~ ( ord_less_int @ X3 @ Y4 )
=> ( ( X3 != Y4 )
=> ( ord_less_int @ Y4 @ X3 ) ) ) ).
% linorder_cases
thf(fact_369_antisym__conv3,axiom,
! [Y4: extended_enat,X3: extended_enat] :
( ~ ( ord_le72135733267957522d_enat @ Y4 @ X3 )
=> ( ( ~ ( ord_le72135733267957522d_enat @ X3 @ Y4 ) )
= ( X3 = Y4 ) ) ) ).
% antisym_conv3
thf(fact_370_antisym__conv3,axiom,
! [Y4: real,X3: real] :
( ~ ( ord_less_real @ Y4 @ X3 )
=> ( ( ~ ( ord_less_real @ X3 @ Y4 ) )
= ( X3 = Y4 ) ) ) ).
% antisym_conv3
thf(fact_371_antisym__conv3,axiom,
! [Y4: num,X3: num] :
( ~ ( ord_less_num @ Y4 @ X3 )
=> ( ( ~ ( ord_less_num @ X3 @ Y4 ) )
= ( X3 = Y4 ) ) ) ).
% antisym_conv3
thf(fact_372_antisym__conv3,axiom,
! [Y4: nat,X3: nat] :
( ~ ( ord_less_nat @ Y4 @ X3 )
=> ( ( ~ ( ord_less_nat @ X3 @ Y4 ) )
= ( X3 = Y4 ) ) ) ).
% antisym_conv3
thf(fact_373_antisym__conv3,axiom,
! [Y4: int,X3: int] :
( ~ ( ord_less_int @ Y4 @ X3 )
=> ( ( ~ ( ord_less_int @ X3 @ Y4 ) )
= ( X3 = Y4 ) ) ) ).
% antisym_conv3
thf(fact_374_less__induct,axiom,
! [P: extended_enat > $o,A2: extended_enat] :
( ! [X5: extended_enat] :
( ! [Y6: extended_enat] :
( ( ord_le72135733267957522d_enat @ Y6 @ X5 )
=> ( P @ Y6 ) )
=> ( P @ X5 ) )
=> ( P @ A2 ) ) ).
% less_induct
thf(fact_375_less__induct,axiom,
! [P: nat > $o,A2: nat] :
( ! [X5: nat] :
( ! [Y6: nat] :
( ( ord_less_nat @ Y6 @ X5 )
=> ( P @ Y6 ) )
=> ( P @ X5 ) )
=> ( P @ A2 ) ) ).
% less_induct
thf(fact_376_ord__less__eq__trans,axiom,
! [A2: extended_enat,B2: extended_enat,C: extended_enat] :
( ( ord_le72135733267957522d_enat @ A2 @ B2 )
=> ( ( B2 = C )
=> ( ord_le72135733267957522d_enat @ A2 @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_377_ord__less__eq__trans,axiom,
! [A2: real,B2: real,C: real] :
( ( ord_less_real @ A2 @ B2 )
=> ( ( B2 = C )
=> ( ord_less_real @ A2 @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_378_ord__less__eq__trans,axiom,
! [A2: num,B2: num,C: num] :
( ( ord_less_num @ A2 @ B2 )
=> ( ( B2 = C )
=> ( ord_less_num @ A2 @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_379_ord__less__eq__trans,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( B2 = C )
=> ( ord_less_nat @ A2 @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_380_ord__less__eq__trans,axiom,
! [A2: int,B2: int,C: int] :
( ( ord_less_int @ A2 @ B2 )
=> ( ( B2 = C )
=> ( ord_less_int @ A2 @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_381_ord__eq__less__trans,axiom,
! [A2: extended_enat,B2: extended_enat,C: extended_enat] :
( ( A2 = B2 )
=> ( ( ord_le72135733267957522d_enat @ B2 @ C )
=> ( ord_le72135733267957522d_enat @ A2 @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_382_ord__eq__less__trans,axiom,
! [A2: real,B2: real,C: real] :
( ( A2 = B2 )
=> ( ( ord_less_real @ B2 @ C )
=> ( ord_less_real @ A2 @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_383_ord__eq__less__trans,axiom,
! [A2: num,B2: num,C: num] :
( ( A2 = B2 )
=> ( ( ord_less_num @ B2 @ C )
=> ( ord_less_num @ A2 @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_384_ord__eq__less__trans,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( A2 = B2 )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ord_less_nat @ A2 @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_385_ord__eq__less__trans,axiom,
! [A2: int,B2: int,C: int] :
( ( A2 = B2 )
=> ( ( ord_less_int @ B2 @ C )
=> ( ord_less_int @ A2 @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_386_order_Oasym,axiom,
! [A2: extended_enat,B2: extended_enat] :
( ( ord_le72135733267957522d_enat @ A2 @ B2 )
=> ~ ( ord_le72135733267957522d_enat @ B2 @ A2 ) ) ).
% order.asym
thf(fact_387_order_Oasym,axiom,
! [A2: real,B2: real] :
( ( ord_less_real @ A2 @ B2 )
=> ~ ( ord_less_real @ B2 @ A2 ) ) ).
% order.asym
thf(fact_388_order_Oasym,axiom,
! [A2: num,B2: num] :
( ( ord_less_num @ A2 @ B2 )
=> ~ ( ord_less_num @ B2 @ A2 ) ) ).
% order.asym
thf(fact_389_order_Oasym,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ~ ( ord_less_nat @ B2 @ A2 ) ) ).
% order.asym
thf(fact_390_order_Oasym,axiom,
! [A2: int,B2: int] :
( ( ord_less_int @ A2 @ B2 )
=> ~ ( ord_less_int @ B2 @ A2 ) ) ).
% order.asym
thf(fact_391_less__imp__neq,axiom,
! [X3: extended_enat,Y4: extended_enat] :
( ( ord_le72135733267957522d_enat @ X3 @ Y4 )
=> ( X3 != Y4 ) ) ).
% less_imp_neq
thf(fact_392_less__imp__neq,axiom,
! [X3: real,Y4: real] :
( ( ord_less_real @ X3 @ Y4 )
=> ( X3 != Y4 ) ) ).
% less_imp_neq
thf(fact_393_less__imp__neq,axiom,
! [X3: num,Y4: num] :
( ( ord_less_num @ X3 @ Y4 )
=> ( X3 != Y4 ) ) ).
% less_imp_neq
thf(fact_394_less__imp__neq,axiom,
! [X3: nat,Y4: nat] :
( ( ord_less_nat @ X3 @ Y4 )
=> ( X3 != Y4 ) ) ).
% less_imp_neq
thf(fact_395_less__imp__neq,axiom,
! [X3: int,Y4: int] :
( ( ord_less_int @ X3 @ Y4 )
=> ( X3 != Y4 ) ) ).
% less_imp_neq
thf(fact_396_dense,axiom,
! [X3: real,Y4: real] :
( ( ord_less_real @ X3 @ Y4 )
=> ? [Z4: real] :
( ( ord_less_real @ X3 @ Z4 )
& ( ord_less_real @ Z4 @ Y4 ) ) ) ).
% dense
thf(fact_397_gt__ex,axiom,
! [X3: real] :
? [X_1: real] : ( ord_less_real @ X3 @ X_1 ) ).
% gt_ex
thf(fact_398_gt__ex,axiom,
! [X3: nat] :
? [X_1: nat] : ( ord_less_nat @ X3 @ X_1 ) ).
% gt_ex
thf(fact_399_gt__ex,axiom,
! [X3: int] :
? [X_1: int] : ( ord_less_int @ X3 @ X_1 ) ).
% gt_ex
thf(fact_400_lt__ex,axiom,
! [X3: real] :
? [Y5: real] : ( ord_less_real @ Y5 @ X3 ) ).
% lt_ex
thf(fact_401_lt__ex,axiom,
! [X3: int] :
? [Y5: int] : ( ord_less_int @ Y5 @ X3 ) ).
% lt_ex
thf(fact_402_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K: nat,B2: nat] :
( ( P @ K )
=> ( ! [Y5: nat] :
( ( P @ Y5 )
=> ( ord_less_eq_nat @ Y5 @ B2 ) )
=> ? [X5: nat] :
( ( P @ X5 )
& ! [Y6: nat] :
( ( P @ Y6 )
=> ( ord_less_eq_nat @ Y6 @ X5 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_403_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_404_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_405_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% eq_imp_le
thf(fact_406_le__trans,axiom,
! [I5: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I5 @ J )
=> ( ( ord_less_eq_nat @ J @ K )
=> ( ord_less_eq_nat @ I5 @ K ) ) ) ).
% le_trans
thf(fact_407_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_408_linorder__neqE__nat,axiom,
! [X3: nat,Y4: nat] :
( ( X3 != Y4 )
=> ( ~ ( ord_less_nat @ X3 @ Y4 )
=> ( ord_less_nat @ Y4 @ X3 ) ) ) ).
% linorder_neqE_nat
thf(fact_409_infinite__descent,axiom,
! [P: nat > $o,N: nat] :
( ! [N3: nat] :
( ~ ( P @ N3 )
=> ? [M3: nat] :
( ( ord_less_nat @ M3 @ N3 )
& ~ ( P @ M3 ) ) )
=> ( P @ N ) ) ).
% infinite_descent
thf(fact_410_nat__less__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N3: nat] :
( ! [M3: nat] :
( ( ord_less_nat @ M3 @ N3 )
=> ( P @ M3 ) )
=> ( P @ N3 ) )
=> ( P @ N ) ) ).
% nat_less_induct
thf(fact_411_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_412_less__not__refl3,axiom,
! [S: nat,T: nat] :
( ( ord_less_nat @ S @ T )
=> ( S != T ) ) ).
% less_not_refl3
thf(fact_413_less__not__refl2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
=> ( M != N ) ) ).
% less_not_refl2
thf(fact_414_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_not_refl
thf(fact_415_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_416_size__neq__size__imp__neq,axiom,
! [X3: list_a,Y4: list_a] :
( ( ( size_size_list_a @ X3 )
!= ( size_size_list_a @ Y4 ) )
=> ( X3 != Y4 ) ) ).
% size_neq_size_imp_neq
thf(fact_417_size__neq__size__imp__neq,axiom,
! [X3: list_nat,Y4: list_nat] :
( ( ( size_size_list_nat @ X3 )
!= ( size_size_list_nat @ Y4 ) )
=> ( X3 != Y4 ) ) ).
% size_neq_size_imp_neq
thf(fact_418_size__neq__size__imp__neq,axiom,
! [X3: num,Y4: num] :
( ( ( size_size_num @ X3 )
!= ( size_size_num @ Y4 ) )
=> ( X3 != Y4 ) ) ).
% size_neq_size_imp_neq
thf(fact_419_size__neq__size__imp__neq,axiom,
! [X3: char,Y4: char] :
( ( ( size_size_char @ X3 )
!= ( size_size_char @ Y4 ) )
=> ( X3 != Y4 ) ) ).
% size_neq_size_imp_neq
thf(fact_420_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs3: list_a] :
( ( size_size_list_a @ Xs3 )
= N ) ).
% Ex_list_of_length
thf(fact_421_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs3: list_nat] :
( ( size_size_list_nat @ Xs3 )
= N ) ).
% Ex_list_of_length
thf(fact_422_neq__if__length__neq,axiom,
! [Xs2: list_a,Ys2: list_a] :
( ( ( size_size_list_a @ Xs2 )
!= ( size_size_list_a @ Ys2 ) )
=> ( Xs2 != Ys2 ) ) ).
% neq_if_length_neq
thf(fact_423_neq__if__length__neq,axiom,
! [Xs2: list_nat,Ys2: list_nat] :
( ( ( size_size_list_nat @ Xs2 )
!= ( size_size_list_nat @ Ys2 ) )
=> ( Xs2 != Ys2 ) ) ).
% neq_if_length_neq
thf(fact_424_order__le__imp__less__or__eq,axiom,
! [X3: extended_enat,Y4: extended_enat] :
( ( ord_le2932123472753598470d_enat @ X3 @ Y4 )
=> ( ( ord_le72135733267957522d_enat @ X3 @ Y4 )
| ( X3 = Y4 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_425_order__le__imp__less__or__eq,axiom,
! [X3: real,Y4: real] :
( ( ord_less_eq_real @ X3 @ Y4 )
=> ( ( ord_less_real @ X3 @ Y4 )
| ( X3 = Y4 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_426_order__le__imp__less__or__eq,axiom,
! [X3: num,Y4: num] :
( ( ord_less_eq_num @ X3 @ Y4 )
=> ( ( ord_less_num @ X3 @ Y4 )
| ( X3 = Y4 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_427_order__le__imp__less__or__eq,axiom,
! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ( ord_less_nat @ X3 @ Y4 )
| ( X3 = Y4 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_428_order__le__imp__less__or__eq,axiom,
! [X3: int,Y4: int] :
( ( ord_less_eq_int @ X3 @ Y4 )
=> ( ( ord_less_int @ X3 @ Y4 )
| ( X3 = Y4 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_429_order__le__imp__less__or__eq,axiom,
! [X3: a,Y4: a] :
( ( ord_less_eq_a @ X3 @ Y4 )
=> ( ( ord_less_a @ X3 @ Y4 )
| ( X3 = Y4 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_430_linorder__le__less__linear,axiom,
! [X3: extended_enat,Y4: extended_enat] :
( ( ord_le2932123472753598470d_enat @ X3 @ Y4 )
| ( ord_le72135733267957522d_enat @ Y4 @ X3 ) ) ).
% linorder_le_less_linear
thf(fact_431_linorder__le__less__linear,axiom,
! [X3: real,Y4: real] :
( ( ord_less_eq_real @ X3 @ Y4 )
| ( ord_less_real @ Y4 @ X3 ) ) ).
% linorder_le_less_linear
thf(fact_432_linorder__le__less__linear,axiom,
! [X3: num,Y4: num] :
( ( ord_less_eq_num @ X3 @ Y4 )
| ( ord_less_num @ Y4 @ X3 ) ) ).
% linorder_le_less_linear
thf(fact_433_linorder__le__less__linear,axiom,
! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
| ( ord_less_nat @ Y4 @ X3 ) ) ).
% linorder_le_less_linear
thf(fact_434_linorder__le__less__linear,axiom,
! [X3: int,Y4: int] :
( ( ord_less_eq_int @ X3 @ Y4 )
| ( ord_less_int @ Y4 @ X3 ) ) ).
% linorder_le_less_linear
thf(fact_435_linorder__le__less__linear,axiom,
! [X3: a,Y4: a] :
( ( ord_less_eq_a @ X3 @ Y4 )
| ( ord_less_a @ Y4 @ X3 ) ) ).
% linorder_le_less_linear
thf(fact_436_order__less__le__subst2,axiom,
! [A2: extended_enat,B2: extended_enat,F: extended_enat > extended_enat,C: extended_enat] :
( ( ord_le72135733267957522d_enat @ A2 @ B2 )
=> ( ( ord_le2932123472753598470d_enat @ ( F @ B2 ) @ C )
=> ( ! [X5: extended_enat,Y5: extended_enat] :
( ( ord_le72135733267957522d_enat @ X5 @ Y5 )
=> ( ord_le72135733267957522d_enat @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_le72135733267957522d_enat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_437_order__less__le__subst2,axiom,
! [A2: real,B2: real,F: real > extended_enat,C: extended_enat] :
( ( ord_less_real @ A2 @ B2 )
=> ( ( ord_le2932123472753598470d_enat @ ( F @ B2 ) @ C )
=> ( ! [X5: real,Y5: real] :
( ( ord_less_real @ X5 @ Y5 )
=> ( ord_le72135733267957522d_enat @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_le72135733267957522d_enat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_438_order__less__le__subst2,axiom,
! [A2: num,B2: num,F: num > extended_enat,C: extended_enat] :
( ( ord_less_num @ A2 @ B2 )
=> ( ( ord_le2932123472753598470d_enat @ ( F @ B2 ) @ C )
=> ( ! [X5: num,Y5: num] :
( ( ord_less_num @ X5 @ Y5 )
=> ( ord_le72135733267957522d_enat @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_le72135733267957522d_enat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_439_order__less__le__subst2,axiom,
! [A2: nat,B2: nat,F: nat > extended_enat,C: extended_enat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ord_le2932123472753598470d_enat @ ( F @ B2 ) @ C )
=> ( ! [X5: nat,Y5: nat] :
( ( ord_less_nat @ X5 @ Y5 )
=> ( ord_le72135733267957522d_enat @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_le72135733267957522d_enat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_440_order__less__le__subst2,axiom,
! [A2: int,B2: int,F: int > extended_enat,C: extended_enat] :
( ( ord_less_int @ A2 @ B2 )
=> ( ( ord_le2932123472753598470d_enat @ ( F @ B2 ) @ C )
=> ( ! [X5: int,Y5: int] :
( ( ord_less_int @ X5 @ Y5 )
=> ( ord_le72135733267957522d_enat @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_le72135733267957522d_enat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_441_order__less__le__subst2,axiom,
! [A2: extended_enat,B2: extended_enat,F: extended_enat > real,C: real] :
( ( ord_le72135733267957522d_enat @ A2 @ B2 )
=> ( ( ord_less_eq_real @ ( F @ B2 ) @ C )
=> ( ! [X5: extended_enat,Y5: extended_enat] :
( ( ord_le72135733267957522d_enat @ X5 @ Y5 )
=> ( ord_less_real @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_less_real @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_442_order__less__le__subst2,axiom,
! [A2: real,B2: real,F: real > real,C: real] :
( ( ord_less_real @ A2 @ B2 )
=> ( ( ord_less_eq_real @ ( F @ B2 ) @ C )
=> ( ! [X5: real,Y5: real] :
( ( ord_less_real @ X5 @ Y5 )
=> ( ord_less_real @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_less_real @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_443_order__less__le__subst2,axiom,
! [A2: num,B2: num,F: num > real,C: real] :
( ( ord_less_num @ A2 @ B2 )
=> ( ( ord_less_eq_real @ ( F @ B2 ) @ C )
=> ( ! [X5: num,Y5: num] :
( ( ord_less_num @ X5 @ Y5 )
=> ( ord_less_real @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_less_real @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_444_order__less__le__subst2,axiom,
! [A2: nat,B2: nat,F: nat > real,C: real] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ord_less_eq_real @ ( F @ B2 ) @ C )
=> ( ! [X5: nat,Y5: nat] :
( ( ord_less_nat @ X5 @ Y5 )
=> ( ord_less_real @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_less_real @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_445_order__less__le__subst2,axiom,
! [A2: int,B2: int,F: int > real,C: real] :
( ( ord_less_int @ A2 @ B2 )
=> ( ( ord_less_eq_real @ ( F @ B2 ) @ C )
=> ( ! [X5: int,Y5: int] :
( ( ord_less_int @ X5 @ Y5 )
=> ( ord_less_real @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_less_real @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_446_order__less__le__subst1,axiom,
! [A2: extended_enat,F: real > extended_enat,B2: real,C: real] :
( ( ord_le72135733267957522d_enat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_real @ B2 @ C )
=> ( ! [X5: real,Y5: real] :
( ( ord_less_eq_real @ X5 @ Y5 )
=> ( ord_le2932123472753598470d_enat @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_le72135733267957522d_enat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_447_order__less__le__subst1,axiom,
! [A2: real,F: real > real,B2: real,C: real] :
( ( ord_less_real @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_real @ B2 @ C )
=> ( ! [X5: real,Y5: real] :
( ( ord_less_eq_real @ X5 @ Y5 )
=> ( ord_less_eq_real @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_less_real @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_448_order__less__le__subst1,axiom,
! [A2: num,F: real > num,B2: real,C: real] :
( ( ord_less_num @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_real @ B2 @ C )
=> ( ! [X5: real,Y5: real] :
( ( ord_less_eq_real @ X5 @ Y5 )
=> ( ord_less_eq_num @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_less_num @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_449_order__less__le__subst1,axiom,
! [A2: nat,F: real > nat,B2: real,C: real] :
( ( ord_less_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_real @ B2 @ C )
=> ( ! [X5: real,Y5: real] :
( ( ord_less_eq_real @ X5 @ Y5 )
=> ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_450_order__less__le__subst1,axiom,
! [A2: int,F: real > int,B2: real,C: real] :
( ( ord_less_int @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_real @ B2 @ C )
=> ( ! [X5: real,Y5: real] :
( ( ord_less_eq_real @ X5 @ Y5 )
=> ( ord_less_eq_int @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_less_int @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_451_order__less__le__subst1,axiom,
! [A2: a,F: real > a,B2: real,C: real] :
( ( ord_less_a @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_real @ B2 @ C )
=> ( ! [X5: real,Y5: real] :
( ( ord_less_eq_real @ X5 @ Y5 )
=> ( ord_less_eq_a @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_less_a @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_452_order__less__le__subst1,axiom,
! [A2: extended_enat,F: num > extended_enat,B2: num,C: num] :
( ( ord_le72135733267957522d_enat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_num @ B2 @ C )
=> ( ! [X5: num,Y5: num] :
( ( ord_less_eq_num @ X5 @ Y5 )
=> ( ord_le2932123472753598470d_enat @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_le72135733267957522d_enat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_453_order__less__le__subst1,axiom,
! [A2: real,F: num > real,B2: num,C: num] :
( ( ord_less_real @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_num @ B2 @ C )
=> ( ! [X5: num,Y5: num] :
( ( ord_less_eq_num @ X5 @ Y5 )
=> ( ord_less_eq_real @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_less_real @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_454_order__less__le__subst1,axiom,
! [A2: num,F: num > num,B2: num,C: num] :
( ( ord_less_num @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_num @ B2 @ C )
=> ( ! [X5: num,Y5: num] :
( ( ord_less_eq_num @ X5 @ Y5 )
=> ( ord_less_eq_num @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_less_num @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_455_order__less__le__subst1,axiom,
! [A2: nat,F: num > nat,B2: num,C: num] :
( ( ord_less_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_num @ B2 @ C )
=> ( ! [X5: num,Y5: num] :
( ( ord_less_eq_num @ X5 @ Y5 )
=> ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_456_order__le__less__subst2,axiom,
! [A2: real,B2: real,F: real > extended_enat,C: extended_enat] :
( ( ord_less_eq_real @ A2 @ B2 )
=> ( ( ord_le72135733267957522d_enat @ ( F @ B2 ) @ C )
=> ( ! [X5: real,Y5: real] :
( ( ord_less_eq_real @ X5 @ Y5 )
=> ( ord_le2932123472753598470d_enat @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_le72135733267957522d_enat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_457_order__le__less__subst2,axiom,
! [A2: real,B2: real,F: real > real,C: real] :
( ( ord_less_eq_real @ A2 @ B2 )
=> ( ( ord_less_real @ ( F @ B2 ) @ C )
=> ( ! [X5: real,Y5: real] :
( ( ord_less_eq_real @ X5 @ Y5 )
=> ( ord_less_eq_real @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_less_real @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_458_order__le__less__subst2,axiom,
! [A2: real,B2: real,F: real > num,C: num] :
( ( ord_less_eq_real @ A2 @ B2 )
=> ( ( ord_less_num @ ( F @ B2 ) @ C )
=> ( ! [X5: real,Y5: real] :
( ( ord_less_eq_real @ X5 @ Y5 )
=> ( ord_less_eq_num @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_less_num @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_459_order__le__less__subst2,axiom,
! [A2: real,B2: real,F: real > nat,C: nat] :
( ( ord_less_eq_real @ A2 @ B2 )
=> ( ( ord_less_nat @ ( F @ B2 ) @ C )
=> ( ! [X5: real,Y5: real] :
( ( ord_less_eq_real @ X5 @ Y5 )
=> ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_460_order__le__less__subst2,axiom,
! [A2: real,B2: real,F: real > int,C: int] :
( ( ord_less_eq_real @ A2 @ B2 )
=> ( ( ord_less_int @ ( F @ B2 ) @ C )
=> ( ! [X5: real,Y5: real] :
( ( ord_less_eq_real @ X5 @ Y5 )
=> ( ord_less_eq_int @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_461_order__le__less__subst2,axiom,
! [A2: real,B2: real,F: real > a,C: a] :
( ( ord_less_eq_real @ A2 @ B2 )
=> ( ( ord_less_a @ ( F @ B2 ) @ C )
=> ( ! [X5: real,Y5: real] :
( ( ord_less_eq_real @ X5 @ Y5 )
=> ( ord_less_eq_a @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_less_a @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_462_order__le__less__subst2,axiom,
! [A2: num,B2: num,F: num > extended_enat,C: extended_enat] :
( ( ord_less_eq_num @ A2 @ B2 )
=> ( ( ord_le72135733267957522d_enat @ ( F @ B2 ) @ C )
=> ( ! [X5: num,Y5: num] :
( ( ord_less_eq_num @ X5 @ Y5 )
=> ( ord_le2932123472753598470d_enat @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_le72135733267957522d_enat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_463_order__le__less__subst2,axiom,
! [A2: num,B2: num,F: num > real,C: real] :
( ( ord_less_eq_num @ A2 @ B2 )
=> ( ( ord_less_real @ ( F @ B2 ) @ C )
=> ( ! [X5: num,Y5: num] :
( ( ord_less_eq_num @ X5 @ Y5 )
=> ( ord_less_eq_real @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_less_real @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_464_order__le__less__subst2,axiom,
! [A2: num,B2: num,F: num > num,C: num] :
( ( ord_less_eq_num @ A2 @ B2 )
=> ( ( ord_less_num @ ( F @ B2 ) @ C )
=> ( ! [X5: num,Y5: num] :
( ( ord_less_eq_num @ X5 @ Y5 )
=> ( ord_less_eq_num @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_less_num @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_465_order__le__less__subst2,axiom,
! [A2: num,B2: num,F: num > nat,C: nat] :
( ( ord_less_eq_num @ A2 @ B2 )
=> ( ( ord_less_nat @ ( F @ B2 ) @ C )
=> ( ! [X5: num,Y5: num] :
( ( ord_less_eq_num @ X5 @ Y5 )
=> ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_466_order__le__less__subst1,axiom,
! [A2: extended_enat,F: extended_enat > extended_enat,B2: extended_enat,C: extended_enat] :
( ( ord_le2932123472753598470d_enat @ A2 @ ( F @ B2 ) )
=> ( ( ord_le72135733267957522d_enat @ B2 @ C )
=> ( ! [X5: extended_enat,Y5: extended_enat] :
( ( ord_le72135733267957522d_enat @ X5 @ Y5 )
=> ( ord_le72135733267957522d_enat @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_le72135733267957522d_enat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_467_order__le__less__subst1,axiom,
! [A2: extended_enat,F: real > extended_enat,B2: real,C: real] :
( ( ord_le2932123472753598470d_enat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_real @ B2 @ C )
=> ( ! [X5: real,Y5: real] :
( ( ord_less_real @ X5 @ Y5 )
=> ( ord_le72135733267957522d_enat @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_le72135733267957522d_enat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_468_order__le__less__subst1,axiom,
! [A2: extended_enat,F: num > extended_enat,B2: num,C: num] :
( ( ord_le2932123472753598470d_enat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_num @ B2 @ C )
=> ( ! [X5: num,Y5: num] :
( ( ord_less_num @ X5 @ Y5 )
=> ( ord_le72135733267957522d_enat @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_le72135733267957522d_enat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_469_order__le__less__subst1,axiom,
! [A2: extended_enat,F: nat > extended_enat,B2: nat,C: nat] :
( ( ord_le2932123472753598470d_enat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ! [X5: nat,Y5: nat] :
( ( ord_less_nat @ X5 @ Y5 )
=> ( ord_le72135733267957522d_enat @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_le72135733267957522d_enat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_470_order__le__less__subst1,axiom,
! [A2: extended_enat,F: int > extended_enat,B2: int,C: int] :
( ( ord_le2932123472753598470d_enat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_int @ B2 @ C )
=> ( ! [X5: int,Y5: int] :
( ( ord_less_int @ X5 @ Y5 )
=> ( ord_le72135733267957522d_enat @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_le72135733267957522d_enat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_471_order__le__less__subst1,axiom,
! [A2: real,F: extended_enat > real,B2: extended_enat,C: extended_enat] :
( ( ord_less_eq_real @ A2 @ ( F @ B2 ) )
=> ( ( ord_le72135733267957522d_enat @ B2 @ C )
=> ( ! [X5: extended_enat,Y5: extended_enat] :
( ( ord_le72135733267957522d_enat @ X5 @ Y5 )
=> ( ord_less_real @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_less_real @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_472_order__le__less__subst1,axiom,
! [A2: real,F: real > real,B2: real,C: real] :
( ( ord_less_eq_real @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_real @ B2 @ C )
=> ( ! [X5: real,Y5: real] :
( ( ord_less_real @ X5 @ Y5 )
=> ( ord_less_real @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_less_real @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_473_order__le__less__subst1,axiom,
! [A2: real,F: num > real,B2: num,C: num] :
( ( ord_less_eq_real @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_num @ B2 @ C )
=> ( ! [X5: num,Y5: num] :
( ( ord_less_num @ X5 @ Y5 )
=> ( ord_less_real @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_less_real @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_474_order__le__less__subst1,axiom,
! [A2: real,F: nat > real,B2: nat,C: nat] :
( ( ord_less_eq_real @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ! [X5: nat,Y5: nat] :
( ( ord_less_nat @ X5 @ Y5 )
=> ( ord_less_real @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_less_real @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_475_order__le__less__subst1,axiom,
! [A2: real,F: int > real,B2: int,C: int] :
( ( ord_less_eq_real @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_int @ B2 @ C )
=> ( ! [X5: int,Y5: int] :
( ( ord_less_int @ X5 @ Y5 )
=> ( ord_less_real @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_less_real @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_476_order__less__le__trans,axiom,
! [X3: extended_enat,Y4: extended_enat,Z3: extended_enat] :
( ( ord_le72135733267957522d_enat @ X3 @ Y4 )
=> ( ( ord_le2932123472753598470d_enat @ Y4 @ Z3 )
=> ( ord_le72135733267957522d_enat @ X3 @ Z3 ) ) ) ).
% order_less_le_trans
thf(fact_477_order__less__le__trans,axiom,
! [X3: real,Y4: real,Z3: real] :
( ( ord_less_real @ X3 @ Y4 )
=> ( ( ord_less_eq_real @ Y4 @ Z3 )
=> ( ord_less_real @ X3 @ Z3 ) ) ) ).
% order_less_le_trans
thf(fact_478_order__less__le__trans,axiom,
! [X3: num,Y4: num,Z3: num] :
( ( ord_less_num @ X3 @ Y4 )
=> ( ( ord_less_eq_num @ Y4 @ Z3 )
=> ( ord_less_num @ X3 @ Z3 ) ) ) ).
% order_less_le_trans
thf(fact_479_order__less__le__trans,axiom,
! [X3: nat,Y4: nat,Z3: nat] :
( ( ord_less_nat @ X3 @ Y4 )
=> ( ( ord_less_eq_nat @ Y4 @ Z3 )
=> ( ord_less_nat @ X3 @ Z3 ) ) ) ).
% order_less_le_trans
thf(fact_480_order__less__le__trans,axiom,
! [X3: int,Y4: int,Z3: int] :
( ( ord_less_int @ X3 @ Y4 )
=> ( ( ord_less_eq_int @ Y4 @ Z3 )
=> ( ord_less_int @ X3 @ Z3 ) ) ) ).
% order_less_le_trans
thf(fact_481_order__less__le__trans,axiom,
! [X3: a,Y4: a,Z3: a] :
( ( ord_less_a @ X3 @ Y4 )
=> ( ( ord_less_eq_a @ Y4 @ Z3 )
=> ( ord_less_a @ X3 @ Z3 ) ) ) ).
% order_less_le_trans
thf(fact_482_order__le__less__trans,axiom,
! [X3: extended_enat,Y4: extended_enat,Z3: extended_enat] :
( ( ord_le2932123472753598470d_enat @ X3 @ Y4 )
=> ( ( ord_le72135733267957522d_enat @ Y4 @ Z3 )
=> ( ord_le72135733267957522d_enat @ X3 @ Z3 ) ) ) ).
% order_le_less_trans
thf(fact_483_order__le__less__trans,axiom,
! [X3: real,Y4: real,Z3: real] :
( ( ord_less_eq_real @ X3 @ Y4 )
=> ( ( ord_less_real @ Y4 @ Z3 )
=> ( ord_less_real @ X3 @ Z3 ) ) ) ).
% order_le_less_trans
thf(fact_484_order__le__less__trans,axiom,
! [X3: num,Y4: num,Z3: num] :
( ( ord_less_eq_num @ X3 @ Y4 )
=> ( ( ord_less_num @ Y4 @ Z3 )
=> ( ord_less_num @ X3 @ Z3 ) ) ) ).
% order_le_less_trans
thf(fact_485_order__le__less__trans,axiom,
! [X3: nat,Y4: nat,Z3: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ( ord_less_nat @ Y4 @ Z3 )
=> ( ord_less_nat @ X3 @ Z3 ) ) ) ).
% order_le_less_trans
thf(fact_486_order__le__less__trans,axiom,
! [X3: int,Y4: int,Z3: int] :
( ( ord_less_eq_int @ X3 @ Y4 )
=> ( ( ord_less_int @ Y4 @ Z3 )
=> ( ord_less_int @ X3 @ Z3 ) ) ) ).
% order_le_less_trans
thf(fact_487_order__le__less__trans,axiom,
! [X3: a,Y4: a,Z3: a] :
( ( ord_less_eq_a @ X3 @ Y4 )
=> ( ( ord_less_a @ Y4 @ Z3 )
=> ( ord_less_a @ X3 @ Z3 ) ) ) ).
% order_le_less_trans
thf(fact_488_order__neq__le__trans,axiom,
! [A2: extended_enat,B2: extended_enat] :
( ( A2 != B2 )
=> ( ( ord_le2932123472753598470d_enat @ A2 @ B2 )
=> ( ord_le72135733267957522d_enat @ A2 @ B2 ) ) ) ).
% order_neq_le_trans
thf(fact_489_order__neq__le__trans,axiom,
! [A2: real,B2: real] :
( ( A2 != B2 )
=> ( ( ord_less_eq_real @ A2 @ B2 )
=> ( ord_less_real @ A2 @ B2 ) ) ) ).
% order_neq_le_trans
thf(fact_490_order__neq__le__trans,axiom,
! [A2: num,B2: num] :
( ( A2 != B2 )
=> ( ( ord_less_eq_num @ A2 @ B2 )
=> ( ord_less_num @ A2 @ B2 ) ) ) ).
% order_neq_le_trans
thf(fact_491_order__neq__le__trans,axiom,
! [A2: nat,B2: nat] :
( ( A2 != B2 )
=> ( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ord_less_nat @ A2 @ B2 ) ) ) ).
% order_neq_le_trans
thf(fact_492_order__neq__le__trans,axiom,
! [A2: int,B2: int] :
( ( A2 != B2 )
=> ( ( ord_less_eq_int @ A2 @ B2 )
=> ( ord_less_int @ A2 @ B2 ) ) ) ).
% order_neq_le_trans
thf(fact_493_order__neq__le__trans,axiom,
! [A2: a,B2: a] :
( ( A2 != B2 )
=> ( ( ord_less_eq_a @ A2 @ B2 )
=> ( ord_less_a @ A2 @ B2 ) ) ) ).
% order_neq_le_trans
thf(fact_494_order__le__neq__trans,axiom,
! [A2: extended_enat,B2: extended_enat] :
( ( ord_le2932123472753598470d_enat @ A2 @ B2 )
=> ( ( A2 != B2 )
=> ( ord_le72135733267957522d_enat @ A2 @ B2 ) ) ) ).
% order_le_neq_trans
thf(fact_495_order__le__neq__trans,axiom,
! [A2: real,B2: real] :
( ( ord_less_eq_real @ A2 @ B2 )
=> ( ( A2 != B2 )
=> ( ord_less_real @ A2 @ B2 ) ) ) ).
% order_le_neq_trans
thf(fact_496_order__le__neq__trans,axiom,
! [A2: num,B2: num] :
( ( ord_less_eq_num @ A2 @ B2 )
=> ( ( A2 != B2 )
=> ( ord_less_num @ A2 @ B2 ) ) ) ).
% order_le_neq_trans
thf(fact_497_order__le__neq__trans,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( A2 != B2 )
=> ( ord_less_nat @ A2 @ B2 ) ) ) ).
% order_le_neq_trans
thf(fact_498_order__le__neq__trans,axiom,
! [A2: int,B2: int] :
( ( ord_less_eq_int @ A2 @ B2 )
=> ( ( A2 != B2 )
=> ( ord_less_int @ A2 @ B2 ) ) ) ).
% order_le_neq_trans
thf(fact_499_order__le__neq__trans,axiom,
! [A2: a,B2: a] :
( ( ord_less_eq_a @ A2 @ B2 )
=> ( ( A2 != B2 )
=> ( ord_less_a @ A2 @ B2 ) ) ) ).
% order_le_neq_trans
thf(fact_500_order__less__imp__le,axiom,
! [X3: extended_enat,Y4: extended_enat] :
( ( ord_le72135733267957522d_enat @ X3 @ Y4 )
=> ( ord_le2932123472753598470d_enat @ X3 @ Y4 ) ) ).
% order_less_imp_le
thf(fact_501_order__less__imp__le,axiom,
! [X3: real,Y4: real] :
( ( ord_less_real @ X3 @ Y4 )
=> ( ord_less_eq_real @ X3 @ Y4 ) ) ).
% order_less_imp_le
thf(fact_502_order__less__imp__le,axiom,
! [X3: num,Y4: num] :
( ( ord_less_num @ X3 @ Y4 )
=> ( ord_less_eq_num @ X3 @ Y4 ) ) ).
% order_less_imp_le
thf(fact_503_order__less__imp__le,axiom,
! [X3: nat,Y4: nat] :
( ( ord_less_nat @ X3 @ Y4 )
=> ( ord_less_eq_nat @ X3 @ Y4 ) ) ).
% order_less_imp_le
thf(fact_504_order__less__imp__le,axiom,
! [X3: int,Y4: int] :
( ( ord_less_int @ X3 @ Y4 )
=> ( ord_less_eq_int @ X3 @ Y4 ) ) ).
% order_less_imp_le
thf(fact_505_order__less__imp__le,axiom,
! [X3: a,Y4: a] :
( ( ord_less_a @ X3 @ Y4 )
=> ( ord_less_eq_a @ X3 @ Y4 ) ) ).
% order_less_imp_le
thf(fact_506_linorder__not__less,axiom,
! [X3: extended_enat,Y4: extended_enat] :
( ( ~ ( ord_le72135733267957522d_enat @ X3 @ Y4 ) )
= ( ord_le2932123472753598470d_enat @ Y4 @ X3 ) ) ).
% linorder_not_less
thf(fact_507_linorder__not__less,axiom,
! [X3: real,Y4: real] :
( ( ~ ( ord_less_real @ X3 @ Y4 ) )
= ( ord_less_eq_real @ Y4 @ X3 ) ) ).
% linorder_not_less
thf(fact_508_linorder__not__less,axiom,
! [X3: num,Y4: num] :
( ( ~ ( ord_less_num @ X3 @ Y4 ) )
= ( ord_less_eq_num @ Y4 @ X3 ) ) ).
% linorder_not_less
thf(fact_509_linorder__not__less,axiom,
! [X3: nat,Y4: nat] :
( ( ~ ( ord_less_nat @ X3 @ Y4 ) )
= ( ord_less_eq_nat @ Y4 @ X3 ) ) ).
% linorder_not_less
thf(fact_510_linorder__not__less,axiom,
! [X3: int,Y4: int] :
( ( ~ ( ord_less_int @ X3 @ Y4 ) )
= ( ord_less_eq_int @ Y4 @ X3 ) ) ).
% linorder_not_less
thf(fact_511_linorder__not__less,axiom,
! [X3: a,Y4: a] :
( ( ~ ( ord_less_a @ X3 @ Y4 ) )
= ( ord_less_eq_a @ Y4 @ X3 ) ) ).
% linorder_not_less
thf(fact_512_linorder__not__le,axiom,
! [X3: extended_enat,Y4: extended_enat] :
( ( ~ ( ord_le2932123472753598470d_enat @ X3 @ Y4 ) )
= ( ord_le72135733267957522d_enat @ Y4 @ X3 ) ) ).
% linorder_not_le
thf(fact_513_linorder__not__le,axiom,
! [X3: real,Y4: real] :
( ( ~ ( ord_less_eq_real @ X3 @ Y4 ) )
= ( ord_less_real @ Y4 @ X3 ) ) ).
% linorder_not_le
thf(fact_514_linorder__not__le,axiom,
! [X3: num,Y4: num] :
( ( ~ ( ord_less_eq_num @ X3 @ Y4 ) )
= ( ord_less_num @ Y4 @ X3 ) ) ).
% linorder_not_le
thf(fact_515_linorder__not__le,axiom,
! [X3: nat,Y4: nat] :
( ( ~ ( ord_less_eq_nat @ X3 @ Y4 ) )
= ( ord_less_nat @ Y4 @ X3 ) ) ).
% linorder_not_le
thf(fact_516_linorder__not__le,axiom,
! [X3: int,Y4: int] :
( ( ~ ( ord_less_eq_int @ X3 @ Y4 ) )
= ( ord_less_int @ Y4 @ X3 ) ) ).
% linorder_not_le
thf(fact_517_linorder__not__le,axiom,
! [X3: a,Y4: a] :
( ( ~ ( ord_less_eq_a @ X3 @ Y4 ) )
= ( ord_less_a @ Y4 @ X3 ) ) ).
% linorder_not_le
thf(fact_518_order__less__le,axiom,
( ord_le72135733267957522d_enat
= ( ^ [X: extended_enat,Y: extended_enat] :
( ( ord_le2932123472753598470d_enat @ X @ Y )
& ( X != Y ) ) ) ) ).
% order_less_le
thf(fact_519_order__less__le,axiom,
( ord_less_real
= ( ^ [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
& ( X != Y ) ) ) ) ).
% order_less_le
thf(fact_520_order__less__le,axiom,
( ord_less_num
= ( ^ [X: num,Y: num] :
( ( ord_less_eq_num @ X @ Y )
& ( X != Y ) ) ) ) ).
% order_less_le
thf(fact_521_order__less__le,axiom,
( ord_less_nat
= ( ^ [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
& ( X != Y ) ) ) ) ).
% order_less_le
thf(fact_522_order__less__le,axiom,
( ord_less_int
= ( ^ [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
& ( X != Y ) ) ) ) ).
% order_less_le
thf(fact_523_order__less__le,axiom,
( ord_less_a
= ( ^ [X: a,Y: a] :
( ( ord_less_eq_a @ X @ Y )
& ( X != Y ) ) ) ) ).
% order_less_le
thf(fact_524_order__le__less,axiom,
( ord_le2932123472753598470d_enat
= ( ^ [X: extended_enat,Y: extended_enat] :
( ( ord_le72135733267957522d_enat @ X @ Y )
| ( X = Y ) ) ) ) ).
% order_le_less
thf(fact_525_order__le__less,axiom,
( ord_less_eq_real
= ( ^ [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
| ( X = Y ) ) ) ) ).
% order_le_less
thf(fact_526_order__le__less,axiom,
( ord_less_eq_num
= ( ^ [X: num,Y: num] :
( ( ord_less_num @ X @ Y )
| ( X = Y ) ) ) ) ).
% order_le_less
thf(fact_527_order__le__less,axiom,
( ord_less_eq_nat
= ( ^ [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
| ( X = Y ) ) ) ) ).
% order_le_less
thf(fact_528_order__le__less,axiom,
( ord_less_eq_int
= ( ^ [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
| ( X = Y ) ) ) ) ).
% order_le_less
thf(fact_529_order__le__less,axiom,
( ord_less_eq_a
= ( ^ [X: a,Y: a] :
( ( ord_less_a @ X @ Y )
| ( X = Y ) ) ) ) ).
% order_le_less
thf(fact_530_dual__order_Ostrict__implies__order,axiom,
! [B2: extended_enat,A2: extended_enat] :
( ( ord_le72135733267957522d_enat @ B2 @ A2 )
=> ( ord_le2932123472753598470d_enat @ B2 @ A2 ) ) ).
% dual_order.strict_implies_order
thf(fact_531_dual__order_Ostrict__implies__order,axiom,
! [B2: real,A2: real] :
( ( ord_less_real @ B2 @ A2 )
=> ( ord_less_eq_real @ B2 @ A2 ) ) ).
% dual_order.strict_implies_order
thf(fact_532_dual__order_Ostrict__implies__order,axiom,
! [B2: num,A2: num] :
( ( ord_less_num @ B2 @ A2 )
=> ( ord_less_eq_num @ B2 @ A2 ) ) ).
% dual_order.strict_implies_order
thf(fact_533_dual__order_Ostrict__implies__order,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_nat @ B2 @ A2 )
=> ( ord_less_eq_nat @ B2 @ A2 ) ) ).
% dual_order.strict_implies_order
thf(fact_534_dual__order_Ostrict__implies__order,axiom,
! [B2: int,A2: int] :
( ( ord_less_int @ B2 @ A2 )
=> ( ord_less_eq_int @ B2 @ A2 ) ) ).
% dual_order.strict_implies_order
thf(fact_535_dual__order_Ostrict__implies__order,axiom,
! [B2: a,A2: a] :
( ( ord_less_a @ B2 @ A2 )
=> ( ord_less_eq_a @ B2 @ A2 ) ) ).
% dual_order.strict_implies_order
thf(fact_536_order_Ostrict__implies__order,axiom,
! [A2: extended_enat,B2: extended_enat] :
( ( ord_le72135733267957522d_enat @ A2 @ B2 )
=> ( ord_le2932123472753598470d_enat @ A2 @ B2 ) ) ).
% order.strict_implies_order
thf(fact_537_order_Ostrict__implies__order,axiom,
! [A2: real,B2: real] :
( ( ord_less_real @ A2 @ B2 )
=> ( ord_less_eq_real @ A2 @ B2 ) ) ).
% order.strict_implies_order
thf(fact_538_order_Ostrict__implies__order,axiom,
! [A2: num,B2: num] :
( ( ord_less_num @ A2 @ B2 )
=> ( ord_less_eq_num @ A2 @ B2 ) ) ).
% order.strict_implies_order
thf(fact_539_order_Ostrict__implies__order,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ord_less_eq_nat @ A2 @ B2 ) ) ).
% order.strict_implies_order
thf(fact_540_order_Ostrict__implies__order,axiom,
! [A2: int,B2: int] :
( ( ord_less_int @ A2 @ B2 )
=> ( ord_less_eq_int @ A2 @ B2 ) ) ).
% order.strict_implies_order
thf(fact_541_order_Ostrict__implies__order,axiom,
! [A2: a,B2: a] :
( ( ord_less_a @ A2 @ B2 )
=> ( ord_less_eq_a @ A2 @ B2 ) ) ).
% order.strict_implies_order
thf(fact_542_dual__order_Ostrict__iff__not,axiom,
( ord_le72135733267957522d_enat
= ( ^ [B3: extended_enat,A4: extended_enat] :
( ( ord_le2932123472753598470d_enat @ B3 @ A4 )
& ~ ( ord_le2932123472753598470d_enat @ A4 @ B3 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_543_dual__order_Ostrict__iff__not,axiom,
( ord_less_real
= ( ^ [B3: real,A4: real] :
( ( ord_less_eq_real @ B3 @ A4 )
& ~ ( ord_less_eq_real @ A4 @ B3 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_544_dual__order_Ostrict__iff__not,axiom,
( ord_less_num
= ( ^ [B3: num,A4: num] :
( ( ord_less_eq_num @ B3 @ A4 )
& ~ ( ord_less_eq_num @ A4 @ B3 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_545_dual__order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [B3: nat,A4: nat] :
( ( ord_less_eq_nat @ B3 @ A4 )
& ~ ( ord_less_eq_nat @ A4 @ B3 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_546_dual__order_Ostrict__iff__not,axiom,
( ord_less_int
= ( ^ [B3: int,A4: int] :
( ( ord_less_eq_int @ B3 @ A4 )
& ~ ( ord_less_eq_int @ A4 @ B3 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_547_dual__order_Ostrict__iff__not,axiom,
( ord_less_a
= ( ^ [B3: a,A4: a] :
( ( ord_less_eq_a @ B3 @ A4 )
& ~ ( ord_less_eq_a @ A4 @ B3 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_548_dual__order_Ostrict__trans2,axiom,
! [B2: extended_enat,A2: extended_enat,C: extended_enat] :
( ( ord_le72135733267957522d_enat @ B2 @ A2 )
=> ( ( ord_le2932123472753598470d_enat @ C @ B2 )
=> ( ord_le72135733267957522d_enat @ C @ A2 ) ) ) ).
% dual_order.strict_trans2
thf(fact_549_dual__order_Ostrict__trans2,axiom,
! [B2: real,A2: real,C: real] :
( ( ord_less_real @ B2 @ A2 )
=> ( ( ord_less_eq_real @ C @ B2 )
=> ( ord_less_real @ C @ A2 ) ) ) ).
% dual_order.strict_trans2
thf(fact_550_dual__order_Ostrict__trans2,axiom,
! [B2: num,A2: num,C: num] :
( ( ord_less_num @ B2 @ A2 )
=> ( ( ord_less_eq_num @ C @ B2 )
=> ( ord_less_num @ C @ A2 ) ) ) ).
% dual_order.strict_trans2
thf(fact_551_dual__order_Ostrict__trans2,axiom,
! [B2: nat,A2: nat,C: nat] :
( ( ord_less_nat @ B2 @ A2 )
=> ( ( ord_less_eq_nat @ C @ B2 )
=> ( ord_less_nat @ C @ A2 ) ) ) ).
% dual_order.strict_trans2
thf(fact_552_dual__order_Ostrict__trans2,axiom,
! [B2: int,A2: int,C: int] :
( ( ord_less_int @ B2 @ A2 )
=> ( ( ord_less_eq_int @ C @ B2 )
=> ( ord_less_int @ C @ A2 ) ) ) ).
% dual_order.strict_trans2
thf(fact_553_dual__order_Ostrict__trans2,axiom,
! [B2: a,A2: a,C: a] :
( ( ord_less_a @ B2 @ A2 )
=> ( ( ord_less_eq_a @ C @ B2 )
=> ( ord_less_a @ C @ A2 ) ) ) ).
% dual_order.strict_trans2
thf(fact_554_dual__order_Ostrict__trans1,axiom,
! [B2: extended_enat,A2: extended_enat,C: extended_enat] :
( ( ord_le2932123472753598470d_enat @ B2 @ A2 )
=> ( ( ord_le72135733267957522d_enat @ C @ B2 )
=> ( ord_le72135733267957522d_enat @ C @ A2 ) ) ) ).
% dual_order.strict_trans1
thf(fact_555_dual__order_Ostrict__trans1,axiom,
! [B2: real,A2: real,C: real] :
( ( ord_less_eq_real @ B2 @ A2 )
=> ( ( ord_less_real @ C @ B2 )
=> ( ord_less_real @ C @ A2 ) ) ) ).
% dual_order.strict_trans1
thf(fact_556_dual__order_Ostrict__trans1,axiom,
! [B2: num,A2: num,C: num] :
( ( ord_less_eq_num @ B2 @ A2 )
=> ( ( ord_less_num @ C @ B2 )
=> ( ord_less_num @ C @ A2 ) ) ) ).
% dual_order.strict_trans1
thf(fact_557_dual__order_Ostrict__trans1,axiom,
! [B2: nat,A2: nat,C: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
=> ( ( ord_less_nat @ C @ B2 )
=> ( ord_less_nat @ C @ A2 ) ) ) ).
% dual_order.strict_trans1
thf(fact_558_dual__order_Ostrict__trans1,axiom,
! [B2: int,A2: int,C: int] :
( ( ord_less_eq_int @ B2 @ A2 )
=> ( ( ord_less_int @ C @ B2 )
=> ( ord_less_int @ C @ A2 ) ) ) ).
% dual_order.strict_trans1
thf(fact_559_dual__order_Ostrict__trans1,axiom,
! [B2: a,A2: a,C: a] :
( ( ord_less_eq_a @ B2 @ A2 )
=> ( ( ord_less_a @ C @ B2 )
=> ( ord_less_a @ C @ A2 ) ) ) ).
% dual_order.strict_trans1
thf(fact_560_dual__order_Ostrict__iff__order,axiom,
( ord_le72135733267957522d_enat
= ( ^ [B3: extended_enat,A4: extended_enat] :
( ( ord_le2932123472753598470d_enat @ B3 @ A4 )
& ( A4 != B3 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_561_dual__order_Ostrict__iff__order,axiom,
( ord_less_real
= ( ^ [B3: real,A4: real] :
( ( ord_less_eq_real @ B3 @ A4 )
& ( A4 != B3 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_562_dual__order_Ostrict__iff__order,axiom,
( ord_less_num
= ( ^ [B3: num,A4: num] :
( ( ord_less_eq_num @ B3 @ A4 )
& ( A4 != B3 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_563_dual__order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [B3: nat,A4: nat] :
( ( ord_less_eq_nat @ B3 @ A4 )
& ( A4 != B3 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_564_dual__order_Ostrict__iff__order,axiom,
( ord_less_int
= ( ^ [B3: int,A4: int] :
( ( ord_less_eq_int @ B3 @ A4 )
& ( A4 != B3 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_565_dual__order_Ostrict__iff__order,axiom,
( ord_less_a
= ( ^ [B3: a,A4: a] :
( ( ord_less_eq_a @ B3 @ A4 )
& ( A4 != B3 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_566_dual__order_Oorder__iff__strict,axiom,
( ord_le2932123472753598470d_enat
= ( ^ [B3: extended_enat,A4: extended_enat] :
( ( ord_le72135733267957522d_enat @ B3 @ A4 )
| ( A4 = B3 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_567_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_real
= ( ^ [B3: real,A4: real] :
( ( ord_less_real @ B3 @ A4 )
| ( A4 = B3 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_568_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_num
= ( ^ [B3: num,A4: num] :
( ( ord_less_num @ B3 @ A4 )
| ( A4 = B3 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_569_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [B3: nat,A4: nat] :
( ( ord_less_nat @ B3 @ A4 )
| ( A4 = B3 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_570_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_int
= ( ^ [B3: int,A4: int] :
( ( ord_less_int @ B3 @ A4 )
| ( A4 = B3 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_571_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_a
= ( ^ [B3: a,A4: a] :
( ( ord_less_a @ B3 @ A4 )
| ( A4 = B3 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_572_dense__le__bounded,axiom,
! [X3: real,Y4: real,Z3: real] :
( ( ord_less_real @ X3 @ Y4 )
=> ( ! [W2: real] :
( ( ord_less_real @ X3 @ W2 )
=> ( ( ord_less_real @ W2 @ Y4 )
=> ( ord_less_eq_real @ W2 @ Z3 ) ) )
=> ( ord_less_eq_real @ Y4 @ Z3 ) ) ) ).
% dense_le_bounded
thf(fact_573_dense__ge__bounded,axiom,
! [Z3: real,X3: real,Y4: real] :
( ( ord_less_real @ Z3 @ X3 )
=> ( ! [W2: real] :
( ( ord_less_real @ Z3 @ W2 )
=> ( ( ord_less_real @ W2 @ X3 )
=> ( ord_less_eq_real @ Y4 @ W2 ) ) )
=> ( ord_less_eq_real @ Y4 @ Z3 ) ) ) ).
% dense_ge_bounded
thf(fact_574_order_Ostrict__iff__not,axiom,
( ord_le72135733267957522d_enat
= ( ^ [A4: extended_enat,B3: extended_enat] :
( ( ord_le2932123472753598470d_enat @ A4 @ B3 )
& ~ ( ord_le2932123472753598470d_enat @ B3 @ A4 ) ) ) ) ).
% order.strict_iff_not
thf(fact_575_order_Ostrict__iff__not,axiom,
( ord_less_real
= ( ^ [A4: real,B3: real] :
( ( ord_less_eq_real @ A4 @ B3 )
& ~ ( ord_less_eq_real @ B3 @ A4 ) ) ) ) ).
% order.strict_iff_not
thf(fact_576_order_Ostrict__iff__not,axiom,
( ord_less_num
= ( ^ [A4: num,B3: num] :
( ( ord_less_eq_num @ A4 @ B3 )
& ~ ( ord_less_eq_num @ B3 @ A4 ) ) ) ) ).
% order.strict_iff_not
thf(fact_577_order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [A4: nat,B3: nat] :
( ( ord_less_eq_nat @ A4 @ B3 )
& ~ ( ord_less_eq_nat @ B3 @ A4 ) ) ) ) ).
% order.strict_iff_not
thf(fact_578_order_Ostrict__iff__not,axiom,
( ord_less_int
= ( ^ [A4: int,B3: int] :
( ( ord_less_eq_int @ A4 @ B3 )
& ~ ( ord_less_eq_int @ B3 @ A4 ) ) ) ) ).
% order.strict_iff_not
thf(fact_579_order_Ostrict__iff__not,axiom,
( ord_less_a
= ( ^ [A4: a,B3: a] :
( ( ord_less_eq_a @ A4 @ B3 )
& ~ ( ord_less_eq_a @ B3 @ A4 ) ) ) ) ).
% order.strict_iff_not
thf(fact_580_order_Ostrict__trans2,axiom,
! [A2: extended_enat,B2: extended_enat,C: extended_enat] :
( ( ord_le72135733267957522d_enat @ A2 @ B2 )
=> ( ( ord_le2932123472753598470d_enat @ B2 @ C )
=> ( ord_le72135733267957522d_enat @ A2 @ C ) ) ) ).
% order.strict_trans2
thf(fact_581_order_Ostrict__trans2,axiom,
! [A2: real,B2: real,C: real] :
( ( ord_less_real @ A2 @ B2 )
=> ( ( ord_less_eq_real @ B2 @ C )
=> ( ord_less_real @ A2 @ C ) ) ) ).
% order.strict_trans2
thf(fact_582_order_Ostrict__trans2,axiom,
! [A2: num,B2: num,C: num] :
( ( ord_less_num @ A2 @ B2 )
=> ( ( ord_less_eq_num @ B2 @ C )
=> ( ord_less_num @ A2 @ C ) ) ) ).
% order.strict_trans2
thf(fact_583_order_Ostrict__trans2,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ord_less_nat @ A2 @ C ) ) ) ).
% order.strict_trans2
thf(fact_584_order_Ostrict__trans2,axiom,
! [A2: int,B2: int,C: int] :
( ( ord_less_int @ A2 @ B2 )
=> ( ( ord_less_eq_int @ B2 @ C )
=> ( ord_less_int @ A2 @ C ) ) ) ).
% order.strict_trans2
thf(fact_585_order_Ostrict__trans2,axiom,
! [A2: a,B2: a,C: a] :
( ( ord_less_a @ A2 @ B2 )
=> ( ( ord_less_eq_a @ B2 @ C )
=> ( ord_less_a @ A2 @ C ) ) ) ).
% order.strict_trans2
thf(fact_586_order_Ostrict__trans1,axiom,
! [A2: extended_enat,B2: extended_enat,C: extended_enat] :
( ( ord_le2932123472753598470d_enat @ A2 @ B2 )
=> ( ( ord_le72135733267957522d_enat @ B2 @ C )
=> ( ord_le72135733267957522d_enat @ A2 @ C ) ) ) ).
% order.strict_trans1
thf(fact_587_order_Ostrict__trans1,axiom,
! [A2: real,B2: real,C: real] :
( ( ord_less_eq_real @ A2 @ B2 )
=> ( ( ord_less_real @ B2 @ C )
=> ( ord_less_real @ A2 @ C ) ) ) ).
% order.strict_trans1
thf(fact_588_order_Ostrict__trans1,axiom,
! [A2: num,B2: num,C: num] :
( ( ord_less_eq_num @ A2 @ B2 )
=> ( ( ord_less_num @ B2 @ C )
=> ( ord_less_num @ A2 @ C ) ) ) ).
% order.strict_trans1
thf(fact_589_order_Ostrict__trans1,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ord_less_nat @ A2 @ C ) ) ) ).
% order.strict_trans1
thf(fact_590_order_Ostrict__trans1,axiom,
! [A2: int,B2: int,C: int] :
( ( ord_less_eq_int @ A2 @ B2 )
=> ( ( ord_less_int @ B2 @ C )
=> ( ord_less_int @ A2 @ C ) ) ) ).
% order.strict_trans1
thf(fact_591_order_Ostrict__trans1,axiom,
! [A2: a,B2: a,C: a] :
( ( ord_less_eq_a @ A2 @ B2 )
=> ( ( ord_less_a @ B2 @ C )
=> ( ord_less_a @ A2 @ C ) ) ) ).
% order.strict_trans1
thf(fact_592_order_Ostrict__iff__order,axiom,
( ord_le72135733267957522d_enat
= ( ^ [A4: extended_enat,B3: extended_enat] :
( ( ord_le2932123472753598470d_enat @ A4 @ B3 )
& ( A4 != B3 ) ) ) ) ).
% order.strict_iff_order
thf(fact_593_order_Ostrict__iff__order,axiom,
( ord_less_real
= ( ^ [A4: real,B3: real] :
( ( ord_less_eq_real @ A4 @ B3 )
& ( A4 != B3 ) ) ) ) ).
% order.strict_iff_order
thf(fact_594_order_Ostrict__iff__order,axiom,
( ord_less_num
= ( ^ [A4: num,B3: num] :
( ( ord_less_eq_num @ A4 @ B3 )
& ( A4 != B3 ) ) ) ) ).
% order.strict_iff_order
thf(fact_595_order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [A4: nat,B3: nat] :
( ( ord_less_eq_nat @ A4 @ B3 )
& ( A4 != B3 ) ) ) ) ).
% order.strict_iff_order
thf(fact_596_order_Ostrict__iff__order,axiom,
( ord_less_int
= ( ^ [A4: int,B3: int] :
( ( ord_less_eq_int @ A4 @ B3 )
& ( A4 != B3 ) ) ) ) ).
% order.strict_iff_order
thf(fact_597_order_Ostrict__iff__order,axiom,
( ord_less_a
= ( ^ [A4: a,B3: a] :
( ( ord_less_eq_a @ A4 @ B3 )
& ( A4 != B3 ) ) ) ) ).
% order.strict_iff_order
thf(fact_598_order_Oorder__iff__strict,axiom,
( ord_le2932123472753598470d_enat
= ( ^ [A4: extended_enat,B3: extended_enat] :
( ( ord_le72135733267957522d_enat @ A4 @ B3 )
| ( A4 = B3 ) ) ) ) ).
% order.order_iff_strict
thf(fact_599_order_Oorder__iff__strict,axiom,
( ord_less_eq_real
= ( ^ [A4: real,B3: real] :
( ( ord_less_real @ A4 @ B3 )
| ( A4 = B3 ) ) ) ) ).
% order.order_iff_strict
thf(fact_600_order_Oorder__iff__strict,axiom,
( ord_less_eq_num
= ( ^ [A4: num,B3: num] :
( ( ord_less_num @ A4 @ B3 )
| ( A4 = B3 ) ) ) ) ).
% order.order_iff_strict
thf(fact_601_order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B3: nat] :
( ( ord_less_nat @ A4 @ B3 )
| ( A4 = B3 ) ) ) ) ).
% order.order_iff_strict
thf(fact_602_order_Oorder__iff__strict,axiom,
( ord_less_eq_int
= ( ^ [A4: int,B3: int] :
( ( ord_less_int @ A4 @ B3 )
| ( A4 = B3 ) ) ) ) ).
% order.order_iff_strict
thf(fact_603_order_Oorder__iff__strict,axiom,
( ord_less_eq_a
= ( ^ [A4: a,B3: a] :
( ( ord_less_a @ A4 @ B3 )
| ( A4 = B3 ) ) ) ) ).
% order.order_iff_strict
thf(fact_604_not__le__imp__less,axiom,
! [Y4: extended_enat,X3: extended_enat] :
( ~ ( ord_le2932123472753598470d_enat @ Y4 @ X3 )
=> ( ord_le72135733267957522d_enat @ X3 @ Y4 ) ) ).
% not_le_imp_less
thf(fact_605_not__le__imp__less,axiom,
! [Y4: real,X3: real] :
( ~ ( ord_less_eq_real @ Y4 @ X3 )
=> ( ord_less_real @ X3 @ Y4 ) ) ).
% not_le_imp_less
thf(fact_606_not__le__imp__less,axiom,
! [Y4: num,X3: num] :
( ~ ( ord_less_eq_num @ Y4 @ X3 )
=> ( ord_less_num @ X3 @ Y4 ) ) ).
% not_le_imp_less
thf(fact_607_not__le__imp__less,axiom,
! [Y4: nat,X3: nat] :
( ~ ( ord_less_eq_nat @ Y4 @ X3 )
=> ( ord_less_nat @ X3 @ Y4 ) ) ).
% not_le_imp_less
thf(fact_608_not__le__imp__less,axiom,
! [Y4: int,X3: int] :
( ~ ( ord_less_eq_int @ Y4 @ X3 )
=> ( ord_less_int @ X3 @ Y4 ) ) ).
% not_le_imp_less
thf(fact_609_not__le__imp__less,axiom,
! [Y4: a,X3: a] :
( ~ ( ord_less_eq_a @ Y4 @ X3 )
=> ( ord_less_a @ X3 @ Y4 ) ) ).
% not_le_imp_less
thf(fact_610_less__le__not__le,axiom,
( ord_le72135733267957522d_enat
= ( ^ [X: extended_enat,Y: extended_enat] :
( ( ord_le2932123472753598470d_enat @ X @ Y )
& ~ ( ord_le2932123472753598470d_enat @ Y @ X ) ) ) ) ).
% less_le_not_le
thf(fact_611_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_612_less__le__not__le,axiom,
( ord_less_num
= ( ^ [X: num,Y: num] :
( ( ord_less_eq_num @ X @ Y )
& ~ ( ord_less_eq_num @ Y @ X ) ) ) ) ).
% less_le_not_le
thf(fact_613_less__le__not__le,axiom,
( ord_less_nat
= ( ^ [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
& ~ ( ord_less_eq_nat @ Y @ X ) ) ) ) ).
% less_le_not_le
thf(fact_614_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_615_less__le__not__le,axiom,
( ord_less_a
= ( ^ [X: a,Y: a] :
( ( ord_less_eq_a @ X @ Y )
& ~ ( ord_less_eq_a @ Y @ X ) ) ) ) ).
% less_le_not_le
thf(fact_616_dense__le,axiom,
! [Y4: real,Z3: real] :
( ! [X5: real] :
( ( ord_less_real @ X5 @ Y4 )
=> ( ord_less_eq_real @ X5 @ Z3 ) )
=> ( ord_less_eq_real @ Y4 @ Z3 ) ) ).
% dense_le
thf(fact_617_dense__ge,axiom,
! [Z3: real,Y4: real] :
( ! [X5: real] :
( ( ord_less_real @ Z3 @ X5 )
=> ( ord_less_eq_real @ Y4 @ X5 ) )
=> ( ord_less_eq_real @ Y4 @ Z3 ) ) ).
% dense_ge
thf(fact_618_antisym__conv2,axiom,
! [X3: extended_enat,Y4: extended_enat] :
( ( ord_le2932123472753598470d_enat @ X3 @ Y4 )
=> ( ( ~ ( ord_le72135733267957522d_enat @ X3 @ Y4 ) )
= ( X3 = Y4 ) ) ) ).
% antisym_conv2
thf(fact_619_antisym__conv2,axiom,
! [X3: real,Y4: real] :
( ( ord_less_eq_real @ X3 @ Y4 )
=> ( ( ~ ( ord_less_real @ X3 @ Y4 ) )
= ( X3 = Y4 ) ) ) ).
% antisym_conv2
thf(fact_620_antisym__conv2,axiom,
! [X3: num,Y4: num] :
( ( ord_less_eq_num @ X3 @ Y4 )
=> ( ( ~ ( ord_less_num @ X3 @ Y4 ) )
= ( X3 = Y4 ) ) ) ).
% antisym_conv2
thf(fact_621_antisym__conv2,axiom,
! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ( ~ ( ord_less_nat @ X3 @ Y4 ) )
= ( X3 = Y4 ) ) ) ).
% antisym_conv2
thf(fact_622_antisym__conv2,axiom,
! [X3: int,Y4: int] :
( ( ord_less_eq_int @ X3 @ Y4 )
=> ( ( ~ ( ord_less_int @ X3 @ Y4 ) )
= ( X3 = Y4 ) ) ) ).
% antisym_conv2
thf(fact_623_antisym__conv2,axiom,
! [X3: a,Y4: a] :
( ( ord_less_eq_a @ X3 @ Y4 )
=> ( ( ~ ( ord_less_a @ X3 @ Y4 ) )
= ( X3 = Y4 ) ) ) ).
% antisym_conv2
thf(fact_624_antisym__conv1,axiom,
! [X3: extended_enat,Y4: extended_enat] :
( ~ ( ord_le72135733267957522d_enat @ X3 @ Y4 )
=> ( ( ord_le2932123472753598470d_enat @ X3 @ Y4 )
= ( X3 = Y4 ) ) ) ).
% antisym_conv1
thf(fact_625_antisym__conv1,axiom,
! [X3: real,Y4: real] :
( ~ ( ord_less_real @ X3 @ Y4 )
=> ( ( ord_less_eq_real @ X3 @ Y4 )
= ( X3 = Y4 ) ) ) ).
% antisym_conv1
thf(fact_626_antisym__conv1,axiom,
! [X3: num,Y4: num] :
( ~ ( ord_less_num @ X3 @ Y4 )
=> ( ( ord_less_eq_num @ X3 @ Y4 )
= ( X3 = Y4 ) ) ) ).
% antisym_conv1
thf(fact_627_antisym__conv1,axiom,
! [X3: nat,Y4: nat] :
( ~ ( ord_less_nat @ X3 @ Y4 )
=> ( ( ord_less_eq_nat @ X3 @ Y4 )
= ( X3 = Y4 ) ) ) ).
% antisym_conv1
thf(fact_628_antisym__conv1,axiom,
! [X3: int,Y4: int] :
( ~ ( ord_less_int @ X3 @ Y4 )
=> ( ( ord_less_eq_int @ X3 @ Y4 )
= ( X3 = Y4 ) ) ) ).
% antisym_conv1
thf(fact_629_antisym__conv1,axiom,
! [X3: a,Y4: a] :
( ~ ( ord_less_a @ X3 @ Y4 )
=> ( ( ord_less_eq_a @ X3 @ Y4 )
= ( X3 = Y4 ) ) ) ).
% antisym_conv1
thf(fact_630_nless__le,axiom,
! [A2: extended_enat,B2: extended_enat] :
( ( ~ ( ord_le72135733267957522d_enat @ A2 @ B2 ) )
= ( ~ ( ord_le2932123472753598470d_enat @ A2 @ B2 )
| ( A2 = B2 ) ) ) ).
% nless_le
thf(fact_631_nless__le,axiom,
! [A2: real,B2: real] :
( ( ~ ( ord_less_real @ A2 @ B2 ) )
= ( ~ ( ord_less_eq_real @ A2 @ B2 )
| ( A2 = B2 ) ) ) ).
% nless_le
thf(fact_632_nless__le,axiom,
! [A2: num,B2: num] :
( ( ~ ( ord_less_num @ A2 @ B2 ) )
= ( ~ ( ord_less_eq_num @ A2 @ B2 )
| ( A2 = B2 ) ) ) ).
% nless_le
thf(fact_633_nless__le,axiom,
! [A2: nat,B2: nat] :
( ( ~ ( ord_less_nat @ A2 @ B2 ) )
= ( ~ ( ord_less_eq_nat @ A2 @ B2 )
| ( A2 = B2 ) ) ) ).
% nless_le
thf(fact_634_nless__le,axiom,
! [A2: int,B2: int] :
( ( ~ ( ord_less_int @ A2 @ B2 ) )
= ( ~ ( ord_less_eq_int @ A2 @ B2 )
| ( A2 = B2 ) ) ) ).
% nless_le
thf(fact_635_nless__le,axiom,
! [A2: a,B2: a] :
( ( ~ ( ord_less_a @ A2 @ B2 ) )
= ( ~ ( ord_less_eq_a @ A2 @ B2 )
| ( A2 = B2 ) ) ) ).
% nless_le
thf(fact_636_leI,axiom,
! [X3: extended_enat,Y4: extended_enat] :
( ~ ( ord_le72135733267957522d_enat @ X3 @ Y4 )
=> ( ord_le2932123472753598470d_enat @ Y4 @ X3 ) ) ).
% leI
thf(fact_637_leI,axiom,
! [X3: real,Y4: real] :
( ~ ( ord_less_real @ X3 @ Y4 )
=> ( ord_less_eq_real @ Y4 @ X3 ) ) ).
% leI
thf(fact_638_leI,axiom,
! [X3: num,Y4: num] :
( ~ ( ord_less_num @ X3 @ Y4 )
=> ( ord_less_eq_num @ Y4 @ X3 ) ) ).
% leI
thf(fact_639_leI,axiom,
! [X3: nat,Y4: nat] :
( ~ ( ord_less_nat @ X3 @ Y4 )
=> ( ord_less_eq_nat @ Y4 @ X3 ) ) ).
% leI
thf(fact_640_leI,axiom,
! [X3: int,Y4: int] :
( ~ ( ord_less_int @ X3 @ Y4 )
=> ( ord_less_eq_int @ Y4 @ X3 ) ) ).
% leI
thf(fact_641_leI,axiom,
! [X3: a,Y4: a] :
( ~ ( ord_less_a @ X3 @ Y4 )
=> ( ord_less_eq_a @ Y4 @ X3 ) ) ).
% leI
thf(fact_642_leD,axiom,
! [Y4: extended_enat,X3: extended_enat] :
( ( ord_le2932123472753598470d_enat @ Y4 @ X3 )
=> ~ ( ord_le72135733267957522d_enat @ X3 @ Y4 ) ) ).
% leD
thf(fact_643_leD,axiom,
! [Y4: real,X3: real] :
( ( ord_less_eq_real @ Y4 @ X3 )
=> ~ ( ord_less_real @ X3 @ Y4 ) ) ).
% leD
thf(fact_644_leD,axiom,
! [Y4: num,X3: num] :
( ( ord_less_eq_num @ Y4 @ X3 )
=> ~ ( ord_less_num @ X3 @ Y4 ) ) ).
% leD
thf(fact_645_leD,axiom,
! [Y4: nat,X3: nat] :
( ( ord_less_eq_nat @ Y4 @ X3 )
=> ~ ( ord_less_nat @ X3 @ Y4 ) ) ).
% leD
thf(fact_646_leD,axiom,
! [Y4: int,X3: int] :
( ( ord_less_eq_int @ Y4 @ X3 )
=> ~ ( ord_less_int @ X3 @ Y4 ) ) ).
% leD
thf(fact_647_leD,axiom,
! [Y4: a,X3: a] :
( ( ord_less_eq_a @ Y4 @ X3 )
=> ~ ( ord_less_a @ X3 @ Y4 ) ) ).
% leD
thf(fact_648_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I5: nat,J: nat] :
( ! [I4: nat,J2: nat] :
( ( ord_less_nat @ I4 @ J2 )
=> ( ord_less_nat @ ( F @ I4 ) @ ( F @ J2 ) ) )
=> ( ( ord_less_eq_nat @ I5 @ J )
=> ( ord_less_eq_nat @ ( F @ I5 ) @ ( F @ J ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_649_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_650_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_651_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_652_enat__ord__number_I2_J,axiom,
! [M: num,N: num] :
( ( ord_le72135733267957522d_enat @ ( numera1916890842035813515d_enat @ M ) @ ( numera1916890842035813515d_enat @ N ) )
= ( ord_less_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) ) ) ).
% enat_ord_number(2)
thf(fact_653_enat__ord__number_I1_J,axiom,
! [M: num,N: num] :
( ( ord_le2932123472753598470d_enat @ ( numera1916890842035813515d_enat @ M ) @ ( numera1916890842035813515d_enat @ N ) )
= ( ord_less_eq_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) ) ) ).
% enat_ord_number(1)
thf(fact_654_nat__descend__induct,axiom,
! [N: nat,P: nat > $o,M: nat] :
( ! [K2: nat] :
( ( ord_less_nat @ N @ K2 )
=> ( P @ K2 ) )
=> ( ! [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N )
=> ( ! [I6: nat] :
( ( ord_less_nat @ K2 @ I6 )
=> ( P @ I6 ) )
=> ( P @ K2 ) ) )
=> ( P @ M ) ) ) ).
% nat_descend_induct
thf(fact_655_complete__interval,axiom,
! [A2: extended_enat,B2: extended_enat,P: extended_enat > $o] :
( ( ord_le72135733267957522d_enat @ A2 @ B2 )
=> ( ( P @ A2 )
=> ( ~ ( P @ B2 )
=> ? [C2: extended_enat] :
( ( ord_le2932123472753598470d_enat @ A2 @ C2 )
& ( ord_le2932123472753598470d_enat @ C2 @ B2 )
& ! [X7: extended_enat] :
( ( ( ord_le2932123472753598470d_enat @ A2 @ X7 )
& ( ord_le72135733267957522d_enat @ X7 @ C2 ) )
=> ( P @ X7 ) )
& ! [D: extended_enat] :
( ! [X5: extended_enat] :
( ( ( ord_le2932123472753598470d_enat @ A2 @ X5 )
& ( ord_le72135733267957522d_enat @ X5 @ D ) )
=> ( P @ X5 ) )
=> ( ord_le2932123472753598470d_enat @ D @ C2 ) ) ) ) ) ) ).
% complete_interval
thf(fact_656_complete__interval,axiom,
! [A2: real,B2: real,P: real > $o] :
( ( ord_less_real @ A2 @ B2 )
=> ( ( P @ A2 )
=> ( ~ ( P @ B2 )
=> ? [C2: real] :
( ( ord_less_eq_real @ A2 @ C2 )
& ( ord_less_eq_real @ C2 @ B2 )
& ! [X7: real] :
( ( ( ord_less_eq_real @ A2 @ X7 )
& ( ord_less_real @ X7 @ C2 ) )
=> ( P @ X7 ) )
& ! [D: real] :
( ! [X5: real] :
( ( ( ord_less_eq_real @ A2 @ X5 )
& ( ord_less_real @ X5 @ D ) )
=> ( P @ X5 ) )
=> ( ord_less_eq_real @ D @ C2 ) ) ) ) ) ) ).
% complete_interval
thf(fact_657_complete__interval,axiom,
! [A2: nat,B2: nat,P: nat > $o] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( P @ A2 )
=> ( ~ ( P @ B2 )
=> ? [C2: nat] :
( ( ord_less_eq_nat @ A2 @ C2 )
& ( ord_less_eq_nat @ C2 @ B2 )
& ! [X7: nat] :
( ( ( ord_less_eq_nat @ A2 @ X7 )
& ( ord_less_nat @ X7 @ C2 ) )
=> ( P @ X7 ) )
& ! [D: nat] :
( ! [X5: nat] :
( ( ( ord_less_eq_nat @ A2 @ X5 )
& ( ord_less_nat @ X5 @ D ) )
=> ( P @ X5 ) )
=> ( ord_less_eq_nat @ D @ C2 ) ) ) ) ) ) ).
% complete_interval
thf(fact_658_complete__interval,axiom,
! [A2: int,B2: int,P: int > $o] :
( ( ord_less_int @ A2 @ B2 )
=> ( ( P @ A2 )
=> ( ~ ( P @ B2 )
=> ? [C2: int] :
( ( ord_less_eq_int @ A2 @ C2 )
& ( ord_less_eq_int @ C2 @ B2 )
& ! [X7: int] :
( ( ( ord_less_eq_int @ A2 @ X7 )
& ( ord_less_int @ X7 @ C2 ) )
=> ( P @ X7 ) )
& ! [D: int] :
( ! [X5: int] :
( ( ( ord_less_eq_int @ A2 @ X5 )
& ( ord_less_int @ X5 @ D ) )
=> ( P @ X5 ) )
=> ( ord_less_eq_int @ D @ C2 ) ) ) ) ) ) ).
% complete_interval
thf(fact_659_eucl__less__le__not__le,axiom,
( ord_less_real
= ( ^ [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
& ~ ( ord_less_eq_real @ Y @ X ) ) ) ) ).
% eucl_less_le_not_le
thf(fact_660_pinf_I6_J,axiom,
! [T: extended_enat] :
? [Z4: extended_enat] :
! [X7: extended_enat] :
( ( ord_le72135733267957522d_enat @ Z4 @ X7 )
=> ~ ( ord_le2932123472753598470d_enat @ X7 @ T ) ) ).
% pinf(6)
thf(fact_661_pinf_I6_J,axiom,
! [T: real] :
? [Z4: real] :
! [X7: real] :
( ( ord_less_real @ Z4 @ X7 )
=> ~ ( ord_less_eq_real @ X7 @ T ) ) ).
% pinf(6)
thf(fact_662_pinf_I6_J,axiom,
! [T: num] :
? [Z4: num] :
! [X7: num] :
( ( ord_less_num @ Z4 @ X7 )
=> ~ ( ord_less_eq_num @ X7 @ T ) ) ).
% pinf(6)
thf(fact_663_pinf_I6_J,axiom,
! [T: nat] :
? [Z4: nat] :
! [X7: nat] :
( ( ord_less_nat @ Z4 @ X7 )
=> ~ ( ord_less_eq_nat @ X7 @ T ) ) ).
% pinf(6)
thf(fact_664_pinf_I6_J,axiom,
! [T: int] :
? [Z4: int] :
! [X7: int] :
( ( ord_less_int @ Z4 @ X7 )
=> ~ ( ord_less_eq_int @ X7 @ T ) ) ).
% pinf(6)
thf(fact_665_pinf_I6_J,axiom,
! [T: a] :
? [Z4: a] :
! [X7: a] :
( ( ord_less_a @ Z4 @ X7 )
=> ~ ( ord_less_eq_a @ X7 @ T ) ) ).
% pinf(6)
thf(fact_666_pinf_I8_J,axiom,
! [T: extended_enat] :
? [Z4: extended_enat] :
! [X7: extended_enat] :
( ( ord_le72135733267957522d_enat @ Z4 @ X7 )
=> ( ord_le2932123472753598470d_enat @ T @ X7 ) ) ).
% pinf(8)
thf(fact_667_pinf_I8_J,axiom,
! [T: real] :
? [Z4: real] :
! [X7: real] :
( ( ord_less_real @ Z4 @ X7 )
=> ( ord_less_eq_real @ T @ X7 ) ) ).
% pinf(8)
thf(fact_668_pinf_I8_J,axiom,
! [T: num] :
? [Z4: num] :
! [X7: num] :
( ( ord_less_num @ Z4 @ X7 )
=> ( ord_less_eq_num @ T @ X7 ) ) ).
% pinf(8)
thf(fact_669_pinf_I8_J,axiom,
! [T: nat] :
? [Z4: nat] :
! [X7: nat] :
( ( ord_less_nat @ Z4 @ X7 )
=> ( ord_less_eq_nat @ T @ X7 ) ) ).
% pinf(8)
thf(fact_670_pinf_I8_J,axiom,
! [T: int] :
? [Z4: int] :
! [X7: int] :
( ( ord_less_int @ Z4 @ X7 )
=> ( ord_less_eq_int @ T @ X7 ) ) ).
% pinf(8)
thf(fact_671_pinf_I8_J,axiom,
! [T: a] :
? [Z4: a] :
! [X7: a] :
( ( ord_less_a @ Z4 @ X7 )
=> ( ord_less_eq_a @ T @ X7 ) ) ).
% pinf(8)
thf(fact_672_minf_I6_J,axiom,
! [T: extended_enat] :
? [Z4: extended_enat] :
! [X7: extended_enat] :
( ( ord_le72135733267957522d_enat @ X7 @ Z4 )
=> ( ord_le2932123472753598470d_enat @ X7 @ T ) ) ).
% minf(6)
thf(fact_673_minf_I6_J,axiom,
! [T: real] :
? [Z4: real] :
! [X7: real] :
( ( ord_less_real @ X7 @ Z4 )
=> ( ord_less_eq_real @ X7 @ T ) ) ).
% minf(6)
thf(fact_674_minf_I6_J,axiom,
! [T: num] :
? [Z4: num] :
! [X7: num] :
( ( ord_less_num @ X7 @ Z4 )
=> ( ord_less_eq_num @ X7 @ T ) ) ).
% minf(6)
thf(fact_675_minf_I6_J,axiom,
! [T: nat] :
? [Z4: nat] :
! [X7: nat] :
( ( ord_less_nat @ X7 @ Z4 )
=> ( ord_less_eq_nat @ X7 @ T ) ) ).
% minf(6)
thf(fact_676_minf_I6_J,axiom,
! [T: int] :
? [Z4: int] :
! [X7: int] :
( ( ord_less_int @ X7 @ Z4 )
=> ( ord_less_eq_int @ X7 @ T ) ) ).
% minf(6)
thf(fact_677_minf_I6_J,axiom,
! [T: a] :
? [Z4: a] :
! [X7: a] :
( ( ord_less_a @ X7 @ Z4 )
=> ( ord_less_eq_a @ X7 @ T ) ) ).
% minf(6)
thf(fact_678_minf_I8_J,axiom,
! [T: extended_enat] :
? [Z4: extended_enat] :
! [X7: extended_enat] :
( ( ord_le72135733267957522d_enat @ X7 @ Z4 )
=> ~ ( ord_le2932123472753598470d_enat @ T @ X7 ) ) ).
% minf(8)
thf(fact_679_minf_I8_J,axiom,
! [T: real] :
? [Z4: real] :
! [X7: real] :
( ( ord_less_real @ X7 @ Z4 )
=> ~ ( ord_less_eq_real @ T @ X7 ) ) ).
% minf(8)
thf(fact_680_minf_I8_J,axiom,
! [T: num] :
? [Z4: num] :
! [X7: num] :
( ( ord_less_num @ X7 @ Z4 )
=> ~ ( ord_less_eq_num @ T @ X7 ) ) ).
% minf(8)
thf(fact_681_minf_I8_J,axiom,
! [T: nat] :
? [Z4: nat] :
! [X7: nat] :
( ( ord_less_nat @ X7 @ Z4 )
=> ~ ( ord_less_eq_nat @ T @ X7 ) ) ).
% minf(8)
thf(fact_682_minf_I8_J,axiom,
! [T: int] :
? [Z4: int] :
! [X7: int] :
( ( ord_less_int @ X7 @ Z4 )
=> ~ ( ord_less_eq_int @ T @ X7 ) ) ).
% minf(8)
thf(fact_683_minf_I8_J,axiom,
! [T: a] :
? [Z4: a] :
! [X7: a] :
( ( ord_less_a @ X7 @ Z4 )
=> ~ ( ord_less_eq_a @ T @ X7 ) ) ).
% minf(8)
thf(fact_684_sorted__int,axiom,
! [I2: set_real,Xs2: list_real,K: nat,I5: nat,J: nat] :
( ( interval_real @ I2 )
=> ( ( sorted_wrt_real @ ord_less_eq_real @ Xs2 )
=> ( ( ord_less_nat @ K @ ( size_size_list_real @ Xs2 ) )
=> ( ( ord_less_eq_nat @ I5 @ J )
=> ( ( ord_less_eq_nat @ J @ K )
=> ( ( member_real @ ( nth_real @ Xs2 @ I5 ) @ I2 )
=> ( ( member_real @ ( nth_real @ Xs2 @ K ) @ I2 )
=> ( member_real @ ( nth_real @ Xs2 @ J ) @ I2 ) ) ) ) ) ) ) ) ).
% sorted_int
thf(fact_685_sorted__int,axiom,
! [I2: set_num,Xs2: list_num,K: nat,I5: nat,J: nat] :
( ( interval_num @ I2 )
=> ( ( sorted_wrt_num @ ord_less_eq_num @ Xs2 )
=> ( ( ord_less_nat @ K @ ( size_size_list_num @ Xs2 ) )
=> ( ( ord_less_eq_nat @ I5 @ J )
=> ( ( ord_less_eq_nat @ J @ K )
=> ( ( member_num @ ( nth_num @ Xs2 @ I5 ) @ I2 )
=> ( ( member_num @ ( nth_num @ Xs2 @ K ) @ I2 )
=> ( member_num @ ( nth_num @ Xs2 @ J ) @ I2 ) ) ) ) ) ) ) ) ).
% sorted_int
thf(fact_686_sorted__int,axiom,
! [I2: set_nat,Xs2: list_nat,K: nat,I5: nat,J: nat] :
( ( interval_nat @ I2 )
=> ( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs2 )
=> ( ( ord_less_nat @ K @ ( size_size_list_nat @ Xs2 ) )
=> ( ( ord_less_eq_nat @ I5 @ J )
=> ( ( ord_less_eq_nat @ J @ K )
=> ( ( member_nat @ ( nth_nat @ Xs2 @ I5 ) @ I2 )
=> ( ( member_nat @ ( nth_nat @ Xs2 @ K ) @ I2 )
=> ( member_nat @ ( nth_nat @ Xs2 @ J ) @ I2 ) ) ) ) ) ) ) ) ).
% sorted_int
thf(fact_687_sorted__int,axiom,
! [I2: set_int,Xs2: list_int,K: nat,I5: nat,J: nat] :
( ( interval_int @ I2 )
=> ( ( sorted_wrt_int @ ord_less_eq_int @ Xs2 )
=> ( ( ord_less_nat @ K @ ( size_size_list_int @ Xs2 ) )
=> ( ( ord_less_eq_nat @ I5 @ J )
=> ( ( ord_less_eq_nat @ J @ K )
=> ( ( member_int @ ( nth_int @ Xs2 @ I5 ) @ I2 )
=> ( ( member_int @ ( nth_int @ Xs2 @ K ) @ I2 )
=> ( member_int @ ( nth_int @ Xs2 @ J ) @ I2 ) ) ) ) ) ) ) ) ).
% sorted_int
thf(fact_688_sorted__int,axiom,
! [I2: set_a,Xs2: list_a,K: nat,I5: nat,J: nat] :
( ( interval_a @ I2 )
=> ( ( sorted_wrt_a @ ord_less_eq_a @ Xs2 )
=> ( ( ord_less_nat @ K @ ( size_size_list_a @ Xs2 ) )
=> ( ( ord_less_eq_nat @ I5 @ J )
=> ( ( ord_less_eq_nat @ J @ K )
=> ( ( member_a @ ( nth_a @ Xs2 @ I5 ) @ I2 )
=> ( ( member_a @ ( nth_a @ Xs2 @ K ) @ I2 )
=> ( member_a @ ( nth_a @ Xs2 @ J ) @ I2 ) ) ) ) ) ) ) ) ).
% sorted_int
thf(fact_689_enat__less__induct,axiom,
! [P: extended_enat > $o,N: extended_enat] :
( ! [N3: extended_enat] :
( ! [M3: extended_enat] :
( ( ord_le72135733267957522d_enat @ M3 @ N3 )
=> ( P @ M3 ) )
=> ( P @ N3 ) )
=> ( P @ N ) ) ).
% enat_less_induct
thf(fact_690_strict__sorted__imp__sorted,axiom,
! [Xs2: list_Extended_enat] :
( ( sorted143172755617435219d_enat @ ord_le72135733267957522d_enat @ Xs2 )
=> ( sorted143172755617435219d_enat @ ord_le2932123472753598470d_enat @ Xs2 ) ) ).
% strict_sorted_imp_sorted
thf(fact_691_strict__sorted__imp__sorted,axiom,
! [Xs2: list_real] :
( ( sorted_wrt_real @ ord_less_real @ Xs2 )
=> ( sorted_wrt_real @ ord_less_eq_real @ Xs2 ) ) ).
% strict_sorted_imp_sorted
thf(fact_692_strict__sorted__imp__sorted,axiom,
! [Xs2: list_num] :
( ( sorted_wrt_num @ ord_less_num @ Xs2 )
=> ( sorted_wrt_num @ ord_less_eq_num @ Xs2 ) ) ).
% strict_sorted_imp_sorted
thf(fact_693_strict__sorted__imp__sorted,axiom,
! [Xs2: list_nat] :
( ( sorted_wrt_nat @ ord_less_nat @ Xs2 )
=> ( sorted_wrt_nat @ ord_less_eq_nat @ Xs2 ) ) ).
% strict_sorted_imp_sorted
thf(fact_694_strict__sorted__imp__sorted,axiom,
! [Xs2: list_int] :
( ( sorted_wrt_int @ ord_less_int @ Xs2 )
=> ( sorted_wrt_int @ ord_less_eq_int @ Xs2 ) ) ).
% strict_sorted_imp_sorted
thf(fact_695_strict__sorted__imp__sorted,axiom,
! [Xs2: list_a] :
( ( sorted_wrt_a @ ord_less_a @ Xs2 )
=> ( sorted_wrt_a @ ord_less_eq_a @ Xs2 ) ) ).
% strict_sorted_imp_sorted
thf(fact_696_sorted__wrt__less__idx,axiom,
! [Ns: list_nat,I5: nat] :
( ( sorted_wrt_nat @ ord_less_nat @ Ns )
=> ( ( ord_less_nat @ I5 @ ( size_size_list_nat @ Ns ) )
=> ( ord_less_eq_nat @ I5 @ ( nth_nat @ Ns @ I5 ) ) ) ) ).
% sorted_wrt_less_idx
thf(fact_697_sorted__wrt__iff__nth__less,axiom,
( sorted_wrt_a
= ( ^ [P3: a > a > $o,Xs: list_a] :
! [I3: nat,J3: nat] :
( ( ord_less_nat @ I3 @ J3 )
=> ( ( ord_less_nat @ J3 @ ( size_size_list_a @ Xs ) )
=> ( P3 @ ( nth_a @ Xs @ I3 ) @ ( nth_a @ Xs @ J3 ) ) ) ) ) ) ).
% sorted_wrt_iff_nth_less
thf(fact_698_sorted__wrt__iff__nth__less,axiom,
( sorted_wrt_nat
= ( ^ [P3: nat > nat > $o,Xs: list_nat] :
! [I3: nat,J3: nat] :
( ( ord_less_nat @ I3 @ J3 )
=> ( ( ord_less_nat @ J3 @ ( size_size_list_nat @ Xs ) )
=> ( P3 @ ( nth_nat @ Xs @ I3 ) @ ( nth_nat @ Xs @ J3 ) ) ) ) ) ) ).
% sorted_wrt_iff_nth_less
thf(fact_699_sorted__wrt__nth__less,axiom,
! [P: a > a > $o,Xs2: list_a,I5: nat,J: nat] :
( ( sorted_wrt_a @ P @ Xs2 )
=> ( ( ord_less_nat @ I5 @ J )
=> ( ( ord_less_nat @ J @ ( size_size_list_a @ Xs2 ) )
=> ( P @ ( nth_a @ Xs2 @ I5 ) @ ( nth_a @ Xs2 @ J ) ) ) ) ) ).
% sorted_wrt_nth_less
thf(fact_700_sorted__wrt__nth__less,axiom,
! [P: nat > nat > $o,Xs2: list_nat,I5: nat,J: nat] :
( ( sorted_wrt_nat @ P @ Xs2 )
=> ( ( ord_less_nat @ I5 @ J )
=> ( ( ord_less_nat @ J @ ( size_size_list_nat @ Xs2 ) )
=> ( P @ ( nth_nat @ Xs2 @ I5 ) @ ( nth_nat @ Xs2 @ J ) ) ) ) ) ).
% sorted_wrt_nth_less
thf(fact_701_minf_I7_J,axiom,
! [T: extended_enat] :
? [Z4: extended_enat] :
! [X7: extended_enat] :
( ( ord_le72135733267957522d_enat @ X7 @ Z4 )
=> ~ ( ord_le72135733267957522d_enat @ T @ X7 ) ) ).
% minf(7)
thf(fact_702_minf_I7_J,axiom,
! [T: real] :
? [Z4: real] :
! [X7: real] :
( ( ord_less_real @ X7 @ Z4 )
=> ~ ( ord_less_real @ T @ X7 ) ) ).
% minf(7)
thf(fact_703_minf_I7_J,axiom,
! [T: num] :
? [Z4: num] :
! [X7: num] :
( ( ord_less_num @ X7 @ Z4 )
=> ~ ( ord_less_num @ T @ X7 ) ) ).
% minf(7)
thf(fact_704_minf_I7_J,axiom,
! [T: nat] :
? [Z4: nat] :
! [X7: nat] :
( ( ord_less_nat @ X7 @ Z4 )
=> ~ ( ord_less_nat @ T @ X7 ) ) ).
% minf(7)
thf(fact_705_minf_I7_J,axiom,
! [T: int] :
? [Z4: int] :
! [X7: int] :
( ( ord_less_int @ X7 @ Z4 )
=> ~ ( ord_less_int @ T @ X7 ) ) ).
% minf(7)
thf(fact_706_minf_I5_J,axiom,
! [T: extended_enat] :
? [Z4: extended_enat] :
! [X7: extended_enat] :
( ( ord_le72135733267957522d_enat @ X7 @ Z4 )
=> ( ord_le72135733267957522d_enat @ X7 @ T ) ) ).
% minf(5)
thf(fact_707_minf_I5_J,axiom,
! [T: real] :
? [Z4: real] :
! [X7: real] :
( ( ord_less_real @ X7 @ Z4 )
=> ( ord_less_real @ X7 @ T ) ) ).
% minf(5)
thf(fact_708_minf_I5_J,axiom,
! [T: num] :
? [Z4: num] :
! [X7: num] :
( ( ord_less_num @ X7 @ Z4 )
=> ( ord_less_num @ X7 @ T ) ) ).
% minf(5)
thf(fact_709_minf_I5_J,axiom,
! [T: nat] :
? [Z4: nat] :
! [X7: nat] :
( ( ord_less_nat @ X7 @ Z4 )
=> ( ord_less_nat @ X7 @ T ) ) ).
% minf(5)
thf(fact_710_minf_I5_J,axiom,
! [T: int] :
? [Z4: int] :
! [X7: int] :
( ( ord_less_int @ X7 @ Z4 )
=> ( ord_less_int @ X7 @ T ) ) ).
% minf(5)
thf(fact_711_minf_I4_J,axiom,
! [T: extended_enat] :
? [Z4: extended_enat] :
! [X7: extended_enat] :
( ( ord_le72135733267957522d_enat @ X7 @ Z4 )
=> ( X7 != T ) ) ).
% minf(4)
thf(fact_712_minf_I4_J,axiom,
! [T: real] :
? [Z4: real] :
! [X7: real] :
( ( ord_less_real @ X7 @ Z4 )
=> ( X7 != T ) ) ).
% minf(4)
thf(fact_713_minf_I4_J,axiom,
! [T: num] :
? [Z4: num] :
! [X7: num] :
( ( ord_less_num @ X7 @ Z4 )
=> ( X7 != T ) ) ).
% minf(4)
thf(fact_714_minf_I4_J,axiom,
! [T: nat] :
? [Z4: nat] :
! [X7: nat] :
( ( ord_less_nat @ X7 @ Z4 )
=> ( X7 != T ) ) ).
% minf(4)
thf(fact_715_minf_I4_J,axiom,
! [T: int] :
? [Z4: int] :
! [X7: int] :
( ( ord_less_int @ X7 @ Z4 )
=> ( X7 != T ) ) ).
% minf(4)
thf(fact_716_minf_I3_J,axiom,
! [T: extended_enat] :
? [Z4: extended_enat] :
! [X7: extended_enat] :
( ( ord_le72135733267957522d_enat @ X7 @ Z4 )
=> ( X7 != T ) ) ).
% minf(3)
thf(fact_717_minf_I3_J,axiom,
! [T: real] :
? [Z4: real] :
! [X7: real] :
( ( ord_less_real @ X7 @ Z4 )
=> ( X7 != T ) ) ).
% minf(3)
thf(fact_718_minf_I3_J,axiom,
! [T: num] :
? [Z4: num] :
! [X7: num] :
( ( ord_less_num @ X7 @ Z4 )
=> ( X7 != T ) ) ).
% minf(3)
thf(fact_719_minf_I3_J,axiom,
! [T: nat] :
? [Z4: nat] :
! [X7: nat] :
( ( ord_less_nat @ X7 @ Z4 )
=> ( X7 != T ) ) ).
% minf(3)
thf(fact_720_minf_I3_J,axiom,
! [T: int] :
? [Z4: int] :
! [X7: int] :
( ( ord_less_int @ X7 @ Z4 )
=> ( X7 != T ) ) ).
% minf(3)
thf(fact_721_minf_I2_J,axiom,
! [P: extended_enat > $o,P4: extended_enat > $o,Q: extended_enat > $o,Q2: extended_enat > $o] :
( ? [Z5: extended_enat] :
! [X5: extended_enat] :
( ( ord_le72135733267957522d_enat @ X5 @ Z5 )
=> ( ( P @ X5 )
= ( P4 @ X5 ) ) )
=> ( ? [Z5: extended_enat] :
! [X5: extended_enat] :
( ( ord_le72135733267957522d_enat @ X5 @ Z5 )
=> ( ( Q @ X5 )
= ( Q2 @ X5 ) ) )
=> ? [Z4: extended_enat] :
! [X7: extended_enat] :
( ( ord_le72135733267957522d_enat @ X7 @ Z4 )
=> ( ( ( P @ X7 )
| ( Q @ X7 ) )
= ( ( P4 @ X7 )
| ( Q2 @ X7 ) ) ) ) ) ) ).
% minf(2)
thf(fact_722_minf_I2_J,axiom,
! [P: real > $o,P4: real > $o,Q: real > $o,Q2: real > $o] :
( ? [Z5: real] :
! [X5: real] :
( ( ord_less_real @ X5 @ Z5 )
=> ( ( P @ X5 )
= ( P4 @ X5 ) ) )
=> ( ? [Z5: real] :
! [X5: real] :
( ( ord_less_real @ X5 @ Z5 )
=> ( ( Q @ X5 )
= ( Q2 @ X5 ) ) )
=> ? [Z4: real] :
! [X7: real] :
( ( ord_less_real @ X7 @ Z4 )
=> ( ( ( P @ X7 )
| ( Q @ X7 ) )
= ( ( P4 @ X7 )
| ( Q2 @ X7 ) ) ) ) ) ) ).
% minf(2)
thf(fact_723_minf_I2_J,axiom,
! [P: num > $o,P4: num > $o,Q: num > $o,Q2: num > $o] :
( ? [Z5: num] :
! [X5: num] :
( ( ord_less_num @ X5 @ Z5 )
=> ( ( P @ X5 )
= ( P4 @ X5 ) ) )
=> ( ? [Z5: num] :
! [X5: num] :
( ( ord_less_num @ X5 @ Z5 )
=> ( ( Q @ X5 )
= ( Q2 @ X5 ) ) )
=> ? [Z4: num] :
! [X7: num] :
( ( ord_less_num @ X7 @ Z4 )
=> ( ( ( P @ X7 )
| ( Q @ X7 ) )
= ( ( P4 @ X7 )
| ( Q2 @ X7 ) ) ) ) ) ) ).
% minf(2)
thf(fact_724_minf_I2_J,axiom,
! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z5: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z5 )
=> ( ( P @ X5 )
= ( P4 @ X5 ) ) )
=> ( ? [Z5: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z5 )
=> ( ( Q @ X5 )
= ( Q2 @ X5 ) ) )
=> ? [Z4: nat] :
! [X7: nat] :
( ( ord_less_nat @ X7 @ Z4 )
=> ( ( ( P @ X7 )
| ( Q @ X7 ) )
= ( ( P4 @ X7 )
| ( Q2 @ X7 ) ) ) ) ) ) ).
% minf(2)
thf(fact_725_minf_I2_J,axiom,
! [P: int > $o,P4: int > $o,Q: int > $o,Q2: int > $o] :
( ? [Z5: int] :
! [X5: int] :
( ( ord_less_int @ X5 @ Z5 )
=> ( ( P @ X5 )
= ( P4 @ X5 ) ) )
=> ( ? [Z5: int] :
! [X5: int] :
( ( ord_less_int @ X5 @ Z5 )
=> ( ( Q @ X5 )
= ( Q2 @ X5 ) ) )
=> ? [Z4: int] :
! [X7: int] :
( ( ord_less_int @ X7 @ Z4 )
=> ( ( ( P @ X7 )
| ( Q @ X7 ) )
= ( ( P4 @ X7 )
| ( Q2 @ X7 ) ) ) ) ) ) ).
% minf(2)
thf(fact_726_minf_I1_J,axiom,
! [P: extended_enat > $o,P4: extended_enat > $o,Q: extended_enat > $o,Q2: extended_enat > $o] :
( ? [Z5: extended_enat] :
! [X5: extended_enat] :
( ( ord_le72135733267957522d_enat @ X5 @ Z5 )
=> ( ( P @ X5 )
= ( P4 @ X5 ) ) )
=> ( ? [Z5: extended_enat] :
! [X5: extended_enat] :
( ( ord_le72135733267957522d_enat @ X5 @ Z5 )
=> ( ( Q @ X5 )
= ( Q2 @ X5 ) ) )
=> ? [Z4: extended_enat] :
! [X7: extended_enat] :
( ( ord_le72135733267957522d_enat @ X7 @ Z4 )
=> ( ( ( P @ X7 )
& ( Q @ X7 ) )
= ( ( P4 @ X7 )
& ( Q2 @ X7 ) ) ) ) ) ) ).
% minf(1)
thf(fact_727_minf_I1_J,axiom,
! [P: real > $o,P4: real > $o,Q: real > $o,Q2: real > $o] :
( ? [Z5: real] :
! [X5: real] :
( ( ord_less_real @ X5 @ Z5 )
=> ( ( P @ X5 )
= ( P4 @ X5 ) ) )
=> ( ? [Z5: real] :
! [X5: real] :
( ( ord_less_real @ X5 @ Z5 )
=> ( ( Q @ X5 )
= ( Q2 @ X5 ) ) )
=> ? [Z4: real] :
! [X7: real] :
( ( ord_less_real @ X7 @ Z4 )
=> ( ( ( P @ X7 )
& ( Q @ X7 ) )
= ( ( P4 @ X7 )
& ( Q2 @ X7 ) ) ) ) ) ) ).
% minf(1)
thf(fact_728_minf_I1_J,axiom,
! [P: num > $o,P4: num > $o,Q: num > $o,Q2: num > $o] :
( ? [Z5: num] :
! [X5: num] :
( ( ord_less_num @ X5 @ Z5 )
=> ( ( P @ X5 )
= ( P4 @ X5 ) ) )
=> ( ? [Z5: num] :
! [X5: num] :
( ( ord_less_num @ X5 @ Z5 )
=> ( ( Q @ X5 )
= ( Q2 @ X5 ) ) )
=> ? [Z4: num] :
! [X7: num] :
( ( ord_less_num @ X7 @ Z4 )
=> ( ( ( P @ X7 )
& ( Q @ X7 ) )
= ( ( P4 @ X7 )
& ( Q2 @ X7 ) ) ) ) ) ) ).
% minf(1)
thf(fact_729_minf_I1_J,axiom,
! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z5: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z5 )
=> ( ( P @ X5 )
= ( P4 @ X5 ) ) )
=> ( ? [Z5: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z5 )
=> ( ( Q @ X5 )
= ( Q2 @ X5 ) ) )
=> ? [Z4: nat] :
! [X7: nat] :
( ( ord_less_nat @ X7 @ Z4 )
=> ( ( ( P @ X7 )
& ( Q @ X7 ) )
= ( ( P4 @ X7 )
& ( Q2 @ X7 ) ) ) ) ) ) ).
% minf(1)
thf(fact_730_minf_I1_J,axiom,
! [P: int > $o,P4: int > $o,Q: int > $o,Q2: int > $o] :
( ? [Z5: int] :
! [X5: int] :
( ( ord_less_int @ X5 @ Z5 )
=> ( ( P @ X5 )
= ( P4 @ X5 ) ) )
=> ( ? [Z5: int] :
! [X5: int] :
( ( ord_less_int @ X5 @ Z5 )
=> ( ( Q @ X5 )
= ( Q2 @ X5 ) ) )
=> ? [Z4: int] :
! [X7: int] :
( ( ord_less_int @ X7 @ Z4 )
=> ( ( ( P @ X7 )
& ( Q @ X7 ) )
= ( ( P4 @ X7 )
& ( Q2 @ X7 ) ) ) ) ) ) ).
% minf(1)
thf(fact_731_pinf_I7_J,axiom,
! [T: extended_enat] :
? [Z4: extended_enat] :
! [X7: extended_enat] :
( ( ord_le72135733267957522d_enat @ Z4 @ X7 )
=> ( ord_le72135733267957522d_enat @ T @ X7 ) ) ).
% pinf(7)
thf(fact_732_pinf_I7_J,axiom,
! [T: real] :
? [Z4: real] :
! [X7: real] :
( ( ord_less_real @ Z4 @ X7 )
=> ( ord_less_real @ T @ X7 ) ) ).
% pinf(7)
thf(fact_733_pinf_I7_J,axiom,
! [T: num] :
? [Z4: num] :
! [X7: num] :
( ( ord_less_num @ Z4 @ X7 )
=> ( ord_less_num @ T @ X7 ) ) ).
% pinf(7)
thf(fact_734_pinf_I7_J,axiom,
! [T: nat] :
? [Z4: nat] :
! [X7: nat] :
( ( ord_less_nat @ Z4 @ X7 )
=> ( ord_less_nat @ T @ X7 ) ) ).
% pinf(7)
thf(fact_735_pinf_I7_J,axiom,
! [T: int] :
? [Z4: int] :
! [X7: int] :
( ( ord_less_int @ Z4 @ X7 )
=> ( ord_less_int @ T @ X7 ) ) ).
% pinf(7)
thf(fact_736_pinf_I5_J,axiom,
! [T: extended_enat] :
? [Z4: extended_enat] :
! [X7: extended_enat] :
( ( ord_le72135733267957522d_enat @ Z4 @ X7 )
=> ~ ( ord_le72135733267957522d_enat @ X7 @ T ) ) ).
% pinf(5)
thf(fact_737_pinf_I5_J,axiom,
! [T: real] :
? [Z4: real] :
! [X7: real] :
( ( ord_less_real @ Z4 @ X7 )
=> ~ ( ord_less_real @ X7 @ T ) ) ).
% pinf(5)
thf(fact_738_pinf_I5_J,axiom,
! [T: num] :
? [Z4: num] :
! [X7: num] :
( ( ord_less_num @ Z4 @ X7 )
=> ~ ( ord_less_num @ X7 @ T ) ) ).
% pinf(5)
thf(fact_739_pinf_I5_J,axiom,
! [T: nat] :
? [Z4: nat] :
! [X7: nat] :
( ( ord_less_nat @ Z4 @ X7 )
=> ~ ( ord_less_nat @ X7 @ T ) ) ).
% pinf(5)
thf(fact_740_pinf_I5_J,axiom,
! [T: int] :
? [Z4: int] :
! [X7: int] :
( ( ord_less_int @ Z4 @ X7 )
=> ~ ( ord_less_int @ X7 @ T ) ) ).
% pinf(5)
thf(fact_741_pinf_I4_J,axiom,
! [T: extended_enat] :
? [Z4: extended_enat] :
! [X7: extended_enat] :
( ( ord_le72135733267957522d_enat @ Z4 @ X7 )
=> ( X7 != T ) ) ).
% pinf(4)
thf(fact_742_pinf_I4_J,axiom,
! [T: real] :
? [Z4: real] :
! [X7: real] :
( ( ord_less_real @ Z4 @ X7 )
=> ( X7 != T ) ) ).
% pinf(4)
thf(fact_743_pinf_I4_J,axiom,
! [T: num] :
? [Z4: num] :
! [X7: num] :
( ( ord_less_num @ Z4 @ X7 )
=> ( X7 != T ) ) ).
% pinf(4)
thf(fact_744_pinf_I4_J,axiom,
! [T: nat] :
? [Z4: nat] :
! [X7: nat] :
( ( ord_less_nat @ Z4 @ X7 )
=> ( X7 != T ) ) ).
% pinf(4)
thf(fact_745_pinf_I4_J,axiom,
! [T: int] :
? [Z4: int] :
! [X7: int] :
( ( ord_less_int @ Z4 @ X7 )
=> ( X7 != T ) ) ).
% pinf(4)
thf(fact_746_pinf_I3_J,axiom,
! [T: extended_enat] :
? [Z4: extended_enat] :
! [X7: extended_enat] :
( ( ord_le72135733267957522d_enat @ Z4 @ X7 )
=> ( X7 != T ) ) ).
% pinf(3)
thf(fact_747_pinf_I3_J,axiom,
! [T: real] :
? [Z4: real] :
! [X7: real] :
( ( ord_less_real @ Z4 @ X7 )
=> ( X7 != T ) ) ).
% pinf(3)
thf(fact_748_pinf_I3_J,axiom,
! [T: num] :
? [Z4: num] :
! [X7: num] :
( ( ord_less_num @ Z4 @ X7 )
=> ( X7 != T ) ) ).
% pinf(3)
thf(fact_749_pinf_I3_J,axiom,
! [T: nat] :
? [Z4: nat] :
! [X7: nat] :
( ( ord_less_nat @ Z4 @ X7 )
=> ( X7 != T ) ) ).
% pinf(3)
thf(fact_750_pinf_I3_J,axiom,
! [T: int] :
? [Z4: int] :
! [X7: int] :
( ( ord_less_int @ Z4 @ X7 )
=> ( X7 != T ) ) ).
% pinf(3)
thf(fact_751_pinf_I2_J,axiom,
! [P: extended_enat > $o,P4: extended_enat > $o,Q: extended_enat > $o,Q2: extended_enat > $o] :
( ? [Z5: extended_enat] :
! [X5: extended_enat] :
( ( ord_le72135733267957522d_enat @ Z5 @ X5 )
=> ( ( P @ X5 )
= ( P4 @ X5 ) ) )
=> ( ? [Z5: extended_enat] :
! [X5: extended_enat] :
( ( ord_le72135733267957522d_enat @ Z5 @ X5 )
=> ( ( Q @ X5 )
= ( Q2 @ X5 ) ) )
=> ? [Z4: extended_enat] :
! [X7: extended_enat] :
( ( ord_le72135733267957522d_enat @ Z4 @ X7 )
=> ( ( ( P @ X7 )
| ( Q @ X7 ) )
= ( ( P4 @ X7 )
| ( Q2 @ X7 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_752_pinf_I2_J,axiom,
! [P: real > $o,P4: real > $o,Q: real > $o,Q2: real > $o] :
( ? [Z5: real] :
! [X5: real] :
( ( ord_less_real @ Z5 @ X5 )
=> ( ( P @ X5 )
= ( P4 @ X5 ) ) )
=> ( ? [Z5: real] :
! [X5: real] :
( ( ord_less_real @ Z5 @ X5 )
=> ( ( Q @ X5 )
= ( Q2 @ X5 ) ) )
=> ? [Z4: real] :
! [X7: real] :
( ( ord_less_real @ Z4 @ X7 )
=> ( ( ( P @ X7 )
| ( Q @ X7 ) )
= ( ( P4 @ X7 )
| ( Q2 @ X7 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_753_pinf_I2_J,axiom,
! [P: num > $o,P4: num > $o,Q: num > $o,Q2: num > $o] :
( ? [Z5: num] :
! [X5: num] :
( ( ord_less_num @ Z5 @ X5 )
=> ( ( P @ X5 )
= ( P4 @ X5 ) ) )
=> ( ? [Z5: num] :
! [X5: num] :
( ( ord_less_num @ Z5 @ X5 )
=> ( ( Q @ X5 )
= ( Q2 @ X5 ) ) )
=> ? [Z4: num] :
! [X7: num] :
( ( ord_less_num @ Z4 @ X7 )
=> ( ( ( P @ X7 )
| ( Q @ X7 ) )
= ( ( P4 @ X7 )
| ( Q2 @ X7 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_754_pinf_I2_J,axiom,
! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z5: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z5 @ X5 )
=> ( ( P @ X5 )
= ( P4 @ X5 ) ) )
=> ( ? [Z5: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z5 @ X5 )
=> ( ( Q @ X5 )
= ( Q2 @ X5 ) ) )
=> ? [Z4: nat] :
! [X7: nat] :
( ( ord_less_nat @ Z4 @ X7 )
=> ( ( ( P @ X7 )
| ( Q @ X7 ) )
= ( ( P4 @ X7 )
| ( Q2 @ X7 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_755_pinf_I2_J,axiom,
! [P: int > $o,P4: int > $o,Q: int > $o,Q2: int > $o] :
( ? [Z5: int] :
! [X5: int] :
( ( ord_less_int @ Z5 @ X5 )
=> ( ( P @ X5 )
= ( P4 @ X5 ) ) )
=> ( ? [Z5: int] :
! [X5: int] :
( ( ord_less_int @ Z5 @ X5 )
=> ( ( Q @ X5 )
= ( Q2 @ X5 ) ) )
=> ? [Z4: int] :
! [X7: int] :
( ( ord_less_int @ Z4 @ X7 )
=> ( ( ( P @ X7 )
| ( Q @ X7 ) )
= ( ( P4 @ X7 )
| ( Q2 @ X7 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_756_pinf_I1_J,axiom,
! [P: extended_enat > $o,P4: extended_enat > $o,Q: extended_enat > $o,Q2: extended_enat > $o] :
( ? [Z5: extended_enat] :
! [X5: extended_enat] :
( ( ord_le72135733267957522d_enat @ Z5 @ X5 )
=> ( ( P @ X5 )
= ( P4 @ X5 ) ) )
=> ( ? [Z5: extended_enat] :
! [X5: extended_enat] :
( ( ord_le72135733267957522d_enat @ Z5 @ X5 )
=> ( ( Q @ X5 )
= ( Q2 @ X5 ) ) )
=> ? [Z4: extended_enat] :
! [X7: extended_enat] :
( ( ord_le72135733267957522d_enat @ Z4 @ X7 )
=> ( ( ( P @ X7 )
& ( Q @ X7 ) )
= ( ( P4 @ X7 )
& ( Q2 @ X7 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_757_pinf_I1_J,axiom,
! [P: real > $o,P4: real > $o,Q: real > $o,Q2: real > $o] :
( ? [Z5: real] :
! [X5: real] :
( ( ord_less_real @ Z5 @ X5 )
=> ( ( P @ X5 )
= ( P4 @ X5 ) ) )
=> ( ? [Z5: real] :
! [X5: real] :
( ( ord_less_real @ Z5 @ X5 )
=> ( ( Q @ X5 )
= ( Q2 @ X5 ) ) )
=> ? [Z4: real] :
! [X7: real] :
( ( ord_less_real @ Z4 @ X7 )
=> ( ( ( P @ X7 )
& ( Q @ X7 ) )
= ( ( P4 @ X7 )
& ( Q2 @ X7 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_758_pinf_I1_J,axiom,
! [P: num > $o,P4: num > $o,Q: num > $o,Q2: num > $o] :
( ? [Z5: num] :
! [X5: num] :
( ( ord_less_num @ Z5 @ X5 )
=> ( ( P @ X5 )
= ( P4 @ X5 ) ) )
=> ( ? [Z5: num] :
! [X5: num] :
( ( ord_less_num @ Z5 @ X5 )
=> ( ( Q @ X5 )
= ( Q2 @ X5 ) ) )
=> ? [Z4: num] :
! [X7: num] :
( ( ord_less_num @ Z4 @ X7 )
=> ( ( ( P @ X7 )
& ( Q @ X7 ) )
= ( ( P4 @ X7 )
& ( Q2 @ X7 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_759_pinf_I1_J,axiom,
! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z5: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z5 @ X5 )
=> ( ( P @ X5 )
= ( P4 @ X5 ) ) )
=> ( ? [Z5: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z5 @ X5 )
=> ( ( Q @ X5 )
= ( Q2 @ X5 ) ) )
=> ? [Z4: nat] :
! [X7: nat] :
( ( ord_less_nat @ Z4 @ X7 )
=> ( ( ( P @ X7 )
& ( Q @ X7 ) )
= ( ( P4 @ X7 )
& ( Q2 @ X7 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_760_pinf_I1_J,axiom,
! [P: int > $o,P4: int > $o,Q: int > $o,Q2: int > $o] :
( ? [Z5: int] :
! [X5: int] :
( ( ord_less_int @ Z5 @ X5 )
=> ( ( P @ X5 )
= ( P4 @ X5 ) ) )
=> ( ? [Z5: int] :
! [X5: int] :
( ( ord_less_int @ Z5 @ X5 )
=> ( ( Q @ X5 )
= ( Q2 @ X5 ) ) )
=> ? [Z4: int] :
! [X7: int] :
( ( ord_less_int @ Z4 @ X7 )
=> ( ( ( P @ X7 )
& ( Q @ X7 ) )
= ( ( P4 @ X7 )
& ( Q2 @ X7 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_761_ex__gt__or__lt,axiom,
! [A2: real] :
? [B4: real] :
( ( ord_less_real @ A2 @ B4 )
| ( ord_less_real @ B4 @ A2 ) ) ).
% ex_gt_or_lt
thf(fact_762_sorted__iff__nth__mono__less,axiom,
! [Xs2: list_real] :
( ( sorted_wrt_real @ ord_less_eq_real @ Xs2 )
= ( ! [I3: nat,J3: nat] :
( ( ord_less_nat @ I3 @ J3 )
=> ( ( ord_less_nat @ J3 @ ( size_size_list_real @ Xs2 ) )
=> ( ord_less_eq_real @ ( nth_real @ Xs2 @ I3 ) @ ( nth_real @ Xs2 @ J3 ) ) ) ) ) ) ).
% sorted_iff_nth_mono_less
thf(fact_763_sorted__iff__nth__mono__less,axiom,
! [Xs2: list_num] :
( ( sorted_wrt_num @ ord_less_eq_num @ Xs2 )
= ( ! [I3: nat,J3: nat] :
( ( ord_less_nat @ I3 @ J3 )
=> ( ( ord_less_nat @ J3 @ ( size_size_list_num @ Xs2 ) )
=> ( ord_less_eq_num @ ( nth_num @ Xs2 @ I3 ) @ ( nth_num @ Xs2 @ J3 ) ) ) ) ) ) ).
% sorted_iff_nth_mono_less
thf(fact_764_sorted__iff__nth__mono__less,axiom,
! [Xs2: list_nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs2 )
= ( ! [I3: nat,J3: nat] :
( ( ord_less_nat @ I3 @ J3 )
=> ( ( ord_less_nat @ J3 @ ( size_size_list_nat @ Xs2 ) )
=> ( ord_less_eq_nat @ ( nth_nat @ Xs2 @ I3 ) @ ( nth_nat @ Xs2 @ J3 ) ) ) ) ) ) ).
% sorted_iff_nth_mono_less
thf(fact_765_sorted__iff__nth__mono__less,axiom,
! [Xs2: list_int] :
( ( sorted_wrt_int @ ord_less_eq_int @ Xs2 )
= ( ! [I3: nat,J3: nat] :
( ( ord_less_nat @ I3 @ J3 )
=> ( ( ord_less_nat @ J3 @ ( size_size_list_int @ Xs2 ) )
=> ( ord_less_eq_int @ ( nth_int @ Xs2 @ I3 ) @ ( nth_int @ Xs2 @ J3 ) ) ) ) ) ) ).
% sorted_iff_nth_mono_less
thf(fact_766_sorted__iff__nth__mono__less,axiom,
! [Xs2: list_a] :
( ( sorted_wrt_a @ ord_less_eq_a @ Xs2 )
= ( ! [I3: nat,J3: nat] :
( ( ord_less_nat @ I3 @ J3 )
=> ( ( ord_less_nat @ J3 @ ( size_size_list_a @ Xs2 ) )
=> ( ord_less_eq_a @ ( nth_a @ Xs2 @ I3 ) @ ( nth_a @ Xs2 @ J3 ) ) ) ) ) ) ).
% sorted_iff_nth_mono_less
thf(fact_767_sorted__nth__mono,axiom,
! [Xs2: list_real,I5: nat,J: nat] :
( ( sorted_wrt_real @ ord_less_eq_real @ Xs2 )
=> ( ( ord_less_eq_nat @ I5 @ J )
=> ( ( ord_less_nat @ J @ ( size_size_list_real @ Xs2 ) )
=> ( ord_less_eq_real @ ( nth_real @ Xs2 @ I5 ) @ ( nth_real @ Xs2 @ J ) ) ) ) ) ).
% sorted_nth_mono
thf(fact_768_sorted__nth__mono,axiom,
! [Xs2: list_num,I5: nat,J: nat] :
( ( sorted_wrt_num @ ord_less_eq_num @ Xs2 )
=> ( ( ord_less_eq_nat @ I5 @ J )
=> ( ( ord_less_nat @ J @ ( size_size_list_num @ Xs2 ) )
=> ( ord_less_eq_num @ ( nth_num @ Xs2 @ I5 ) @ ( nth_num @ Xs2 @ J ) ) ) ) ) ).
% sorted_nth_mono
thf(fact_769_sorted__nth__mono,axiom,
! [Xs2: list_nat,I5: nat,J: nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs2 )
=> ( ( ord_less_eq_nat @ I5 @ J )
=> ( ( ord_less_nat @ J @ ( size_size_list_nat @ Xs2 ) )
=> ( ord_less_eq_nat @ ( nth_nat @ Xs2 @ I5 ) @ ( nth_nat @ Xs2 @ J ) ) ) ) ) ).
% sorted_nth_mono
thf(fact_770_sorted__nth__mono,axiom,
! [Xs2: list_int,I5: nat,J: nat] :
( ( sorted_wrt_int @ ord_less_eq_int @ Xs2 )
=> ( ( ord_less_eq_nat @ I5 @ J )
=> ( ( ord_less_nat @ J @ ( size_size_list_int @ Xs2 ) )
=> ( ord_less_eq_int @ ( nth_int @ Xs2 @ I5 ) @ ( nth_int @ Xs2 @ J ) ) ) ) ) ).
% sorted_nth_mono
thf(fact_771_sorted__nth__mono,axiom,
! [Xs2: list_a,I5: nat,J: nat] :
( ( sorted_wrt_a @ ord_less_eq_a @ Xs2 )
=> ( ( ord_less_eq_nat @ I5 @ J )
=> ( ( ord_less_nat @ J @ ( size_size_list_a @ Xs2 ) )
=> ( ord_less_eq_a @ ( nth_a @ Xs2 @ I5 ) @ ( nth_a @ Xs2 @ J ) ) ) ) ) ).
% sorted_nth_mono
thf(fact_772_sorted__iff__nth__mono,axiom,
! [Xs2: list_real] :
( ( sorted_wrt_real @ ord_less_eq_real @ Xs2 )
= ( ! [I3: nat,J3: nat] :
( ( ord_less_eq_nat @ I3 @ J3 )
=> ( ( ord_less_nat @ J3 @ ( size_size_list_real @ Xs2 ) )
=> ( ord_less_eq_real @ ( nth_real @ Xs2 @ I3 ) @ ( nth_real @ Xs2 @ J3 ) ) ) ) ) ) ).
% sorted_iff_nth_mono
thf(fact_773_sorted__iff__nth__mono,axiom,
! [Xs2: list_num] :
( ( sorted_wrt_num @ ord_less_eq_num @ Xs2 )
= ( ! [I3: nat,J3: nat] :
( ( ord_less_eq_nat @ I3 @ J3 )
=> ( ( ord_less_nat @ J3 @ ( size_size_list_num @ Xs2 ) )
=> ( ord_less_eq_num @ ( nth_num @ Xs2 @ I3 ) @ ( nth_num @ Xs2 @ J3 ) ) ) ) ) ) ).
% sorted_iff_nth_mono
thf(fact_774_sorted__iff__nth__mono,axiom,
! [Xs2: list_nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs2 )
= ( ! [I3: nat,J3: nat] :
( ( ord_less_eq_nat @ I3 @ J3 )
=> ( ( ord_less_nat @ J3 @ ( size_size_list_nat @ Xs2 ) )
=> ( ord_less_eq_nat @ ( nth_nat @ Xs2 @ I3 ) @ ( nth_nat @ Xs2 @ J3 ) ) ) ) ) ) ).
% sorted_iff_nth_mono
thf(fact_775_sorted__iff__nth__mono,axiom,
! [Xs2: list_int] :
( ( sorted_wrt_int @ ord_less_eq_int @ Xs2 )
= ( ! [I3: nat,J3: nat] :
( ( ord_less_eq_nat @ I3 @ J3 )
=> ( ( ord_less_nat @ J3 @ ( size_size_list_int @ Xs2 ) )
=> ( ord_less_eq_int @ ( nth_int @ Xs2 @ I3 ) @ ( nth_int @ Xs2 @ J3 ) ) ) ) ) ) ).
% sorted_iff_nth_mono
thf(fact_776_sorted__iff__nth__mono,axiom,
! [Xs2: list_a] :
( ( sorted_wrt_a @ ord_less_eq_a @ Xs2 )
= ( ! [I3: nat,J3: nat] :
( ( ord_less_eq_nat @ I3 @ J3 )
=> ( ( ord_less_nat @ J3 @ ( size_size_list_a @ Xs2 ) )
=> ( ord_less_eq_a @ ( nth_a @ Xs2 @ I3 ) @ ( nth_a @ Xs2 @ J3 ) ) ) ) ) ) ).
% sorted_iff_nth_mono
thf(fact_777_sorted__rev__nth__mono,axiom,
! [Xs2: list_real,I5: nat,J: nat] :
( ( sorted_wrt_real @ ord_less_eq_real @ ( rev_real @ Xs2 ) )
=> ( ( ord_less_eq_nat @ I5 @ J )
=> ( ( ord_less_nat @ J @ ( size_size_list_real @ Xs2 ) )
=> ( ord_less_eq_real @ ( nth_real @ Xs2 @ J ) @ ( nth_real @ Xs2 @ I5 ) ) ) ) ) ).
% sorted_rev_nth_mono
thf(fact_778_sorted__rev__nth__mono,axiom,
! [Xs2: list_num,I5: nat,J: nat] :
( ( sorted_wrt_num @ ord_less_eq_num @ ( rev_num @ Xs2 ) )
=> ( ( ord_less_eq_nat @ I5 @ J )
=> ( ( ord_less_nat @ J @ ( size_size_list_num @ Xs2 ) )
=> ( ord_less_eq_num @ ( nth_num @ Xs2 @ J ) @ ( nth_num @ Xs2 @ I5 ) ) ) ) ) ).
% sorted_rev_nth_mono
thf(fact_779_sorted__rev__nth__mono,axiom,
! [Xs2: list_nat,I5: nat,J: nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ ( rev_nat @ Xs2 ) )
=> ( ( ord_less_eq_nat @ I5 @ J )
=> ( ( ord_less_nat @ J @ ( size_size_list_nat @ Xs2 ) )
=> ( ord_less_eq_nat @ ( nth_nat @ Xs2 @ J ) @ ( nth_nat @ Xs2 @ I5 ) ) ) ) ) ).
% sorted_rev_nth_mono
thf(fact_780_sorted__rev__nth__mono,axiom,
! [Xs2: list_int,I5: nat,J: nat] :
( ( sorted_wrt_int @ ord_less_eq_int @ ( rev_int @ Xs2 ) )
=> ( ( ord_less_eq_nat @ I5 @ J )
=> ( ( ord_less_nat @ J @ ( size_size_list_int @ Xs2 ) )
=> ( ord_less_eq_int @ ( nth_int @ Xs2 @ J ) @ ( nth_int @ Xs2 @ I5 ) ) ) ) ) ).
% sorted_rev_nth_mono
thf(fact_781_sorted__rev__nth__mono,axiom,
! [Xs2: list_a,I5: nat,J: nat] :
( ( sorted_wrt_a @ ord_less_eq_a @ ( rev_a @ Xs2 ) )
=> ( ( ord_less_eq_nat @ I5 @ J )
=> ( ( ord_less_nat @ J @ ( size_size_list_a @ Xs2 ) )
=> ( ord_less_eq_a @ ( nth_a @ Xs2 @ J ) @ ( nth_a @ Xs2 @ I5 ) ) ) ) ) ).
% sorted_rev_nth_mono
thf(fact_782_sorted__rev__iff__nth__mono,axiom,
! [Xs2: list_real] :
( ( sorted_wrt_real @ ord_less_eq_real @ ( rev_real @ Xs2 ) )
= ( ! [I3: nat,J3: nat] :
( ( ord_less_eq_nat @ I3 @ J3 )
=> ( ( ord_less_nat @ J3 @ ( size_size_list_real @ Xs2 ) )
=> ( ord_less_eq_real @ ( nth_real @ Xs2 @ J3 ) @ ( nth_real @ Xs2 @ I3 ) ) ) ) ) ) ).
% sorted_rev_iff_nth_mono
thf(fact_783_sorted__rev__iff__nth__mono,axiom,
! [Xs2: list_num] :
( ( sorted_wrt_num @ ord_less_eq_num @ ( rev_num @ Xs2 ) )
= ( ! [I3: nat,J3: nat] :
( ( ord_less_eq_nat @ I3 @ J3 )
=> ( ( ord_less_nat @ J3 @ ( size_size_list_num @ Xs2 ) )
=> ( ord_less_eq_num @ ( nth_num @ Xs2 @ J3 ) @ ( nth_num @ Xs2 @ I3 ) ) ) ) ) ) ).
% sorted_rev_iff_nth_mono
thf(fact_784_sorted__rev__iff__nth__mono,axiom,
! [Xs2: list_nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ ( rev_nat @ Xs2 ) )
= ( ! [I3: nat,J3: nat] :
( ( ord_less_eq_nat @ I3 @ J3 )
=> ( ( ord_less_nat @ J3 @ ( size_size_list_nat @ Xs2 ) )
=> ( ord_less_eq_nat @ ( nth_nat @ Xs2 @ J3 ) @ ( nth_nat @ Xs2 @ I3 ) ) ) ) ) ) ).
% sorted_rev_iff_nth_mono
thf(fact_785_sorted__rev__iff__nth__mono,axiom,
! [Xs2: list_int] :
( ( sorted_wrt_int @ ord_less_eq_int @ ( rev_int @ Xs2 ) )
= ( ! [I3: nat,J3: nat] :
( ( ord_less_eq_nat @ I3 @ J3 )
=> ( ( ord_less_nat @ J3 @ ( size_size_list_int @ Xs2 ) )
=> ( ord_less_eq_int @ ( nth_int @ Xs2 @ J3 ) @ ( nth_int @ Xs2 @ I3 ) ) ) ) ) ) ).
% sorted_rev_iff_nth_mono
thf(fact_786_sorted__rev__iff__nth__mono,axiom,
! [Xs2: list_a] :
( ( sorted_wrt_a @ ord_less_eq_a @ ( rev_a @ Xs2 ) )
= ( ! [I3: nat,J3: nat] :
( ( ord_less_eq_nat @ I3 @ J3 )
=> ( ( ord_less_nat @ J3 @ ( size_size_list_a @ Xs2 ) )
=> ( ord_less_eq_a @ ( nth_a @ Xs2 @ J3 ) @ ( nth_a @ Xs2 @ I3 ) ) ) ) ) ) ).
% sorted_rev_iff_nth_mono
thf(fact_787_dbl__simps_I5_J,axiom,
! [K: num] :
( ( neg_numeral_dbl_int @ ( numeral_numeral_int @ K ) )
= ( numeral_numeral_int @ ( bit0 @ K ) ) ) ).
% dbl_simps(5)
thf(fact_788_dbl__simps_I5_J,axiom,
! [K: num] :
( ( neg_nu5590746349488142217l_num1 @ ( numera2161328050825114965l_num1 @ K ) )
= ( numera2161328050825114965l_num1 @ ( bit0 @ K ) ) ) ).
% dbl_simps(5)
thf(fact_789_half__nonnegative__int__iff,axiom,
! [K: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) )
= ( ord_less_eq_int @ zero_zero_int @ K ) ) ).
% half_nonnegative_int_iff
thf(fact_790_half__negative__int__iff,axiom,
! [K: int] :
( ( ord_less_int @ ( divide_divide_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ zero_zero_int )
= ( ord_less_int @ K @ zero_zero_int ) ) ).
% half_negative_int_iff
thf(fact_791_one__less__numeral__iff,axiom,
! [N: num] :
( ( ord_less_real @ one_one_real @ ( numeral_numeral_real @ N ) )
= ( ord_less_num @ one @ N ) ) ).
% one_less_numeral_iff
thf(fact_792_one__less__numeral__iff,axiom,
! [N: num] :
( ( ord_less_nat @ one_one_nat @ ( numeral_numeral_nat @ N ) )
= ( ord_less_num @ one @ N ) ) ).
% one_less_numeral_iff
thf(fact_793_one__less__numeral__iff,axiom,
! [N: num] :
( ( ord_less_int @ one_one_int @ ( numeral_numeral_int @ N ) )
= ( ord_less_num @ one @ N ) ) ).
% one_less_numeral_iff
thf(fact_794_one__less__numeral__iff,axiom,
! [N: num] :
( ( ord_le72135733267957522d_enat @ one_on7984719198319812577d_enat @ ( numera1916890842035813515d_enat @ N ) )
= ( ord_less_num @ one @ N ) ) ).
% one_less_numeral_iff
thf(fact_795_one__less__numeral__iff,axiom,
! [N: num] :
( ( ord_le7381754540660121996nnreal @ one_on2969667320475766781nnreal @ ( numera4658534427948366547nnreal @ N ) )
= ( ord_less_num @ one @ N ) ) ).
% one_less_numeral_iff
thf(fact_796_numeral__le__one__iff,axiom,
! [N: num] :
( ( ord_le2932123472753598470d_enat @ ( numera1916890842035813515d_enat @ N ) @ one_on7984719198319812577d_enat )
= ( ord_less_eq_num @ N @ one ) ) ).
% numeral_le_one_iff
thf(fact_797_numeral__le__one__iff,axiom,
! [N: num] :
( ( ord_le3935885782089961368nnreal @ ( numera4658534427948366547nnreal @ N ) @ one_on2969667320475766781nnreal )
= ( ord_less_eq_num @ N @ one ) ) ).
% numeral_le_one_iff
thf(fact_798_numeral__le__one__iff,axiom,
! [N: num] :
( ( ord_less_eq_real @ ( numeral_numeral_real @ N ) @ one_one_real )
= ( ord_less_eq_num @ N @ one ) ) ).
% numeral_le_one_iff
thf(fact_799_numeral__le__one__iff,axiom,
! [N: num] :
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ N ) @ one_one_nat )
= ( ord_less_eq_num @ N @ one ) ) ).
% numeral_le_one_iff
thf(fact_800_numeral__le__one__iff,axiom,
! [N: num] :
( ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ one_one_int )
= ( ord_less_eq_num @ N @ one ) ) ).
% numeral_le_one_iff
thf(fact_801_half__gt__zero__iff,axiom,
! [A2: real] :
( ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ A2 @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
= ( ord_less_real @ zero_zero_real @ A2 ) ) ).
% half_gt_zero_iff
thf(fact_802_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_803_bot__nat__0_Oextremum,axiom,
! [A2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A2 ) ).
% bot_nat_0.extremum
thf(fact_804_less__one,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ one_one_nat )
= ( N = zero_zero_nat ) ) ).
% less_one
thf(fact_805_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_806_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% neq0_conv
thf(fact_807_bot__nat__0_Onot__eq__extremum,axiom,
! [A2: nat] :
( ( A2 != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ A2 ) ) ).
% bot_nat_0.not_eq_extremum
thf(fact_808_i0__less,axiom,
! [N: extended_enat] :
( ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ N )
= ( N != zero_z5237406670263579293d_enat ) ) ).
% i0_less
thf(fact_809_div__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ( divide_divide_nat @ M @ N )
= zero_zero_nat ) ) ).
% div_less
thf(fact_810_length__rev,axiom,
! [Xs2: list_a] :
( ( size_size_list_a @ ( rev_a @ Xs2 ) )
= ( size_size_list_a @ Xs2 ) ) ).
% length_rev
thf(fact_811_length__rev,axiom,
! [Xs2: list_nat] :
( ( size_size_list_nat @ ( rev_nat @ Xs2 ) )
= ( size_size_list_nat @ Xs2 ) ) ).
% length_rev
thf(fact_812_dbl__simps_I2_J,axiom,
( ( neg_numeral_dbl_int @ zero_zero_int )
= zero_zero_int ) ).
% dbl_simps(2)
thf(fact_813_dbl__simps_I2_J,axiom,
( ( neg_numeral_dbl_real @ zero_zero_real )
= zero_zero_real ) ).
% dbl_simps(2)
thf(fact_814_numeral__eq__one__iff,axiom,
! [N: num] :
( ( ( numeral_numeral_real @ N )
= one_one_real )
= ( N = one ) ) ).
% numeral_eq_one_iff
thf(fact_815_numeral__eq__one__iff,axiom,
! [N: num] :
( ( ( numeral_numeral_nat @ N )
= one_one_nat )
= ( N = one ) ) ).
% numeral_eq_one_iff
thf(fact_816_numeral__eq__one__iff,axiom,
! [N: num] :
( ( ( numeral_numeral_int @ N )
= one_one_int )
= ( N = one ) ) ).
% numeral_eq_one_iff
thf(fact_817_numeral__eq__one__iff,axiom,
! [N: num] :
( ( ( numera1916890842035813515d_enat @ N )
= one_on7984719198319812577d_enat )
= ( N = one ) ) ).
% numeral_eq_one_iff
thf(fact_818_numeral__eq__one__iff,axiom,
! [N: num] :
( ( ( numera4658534427948366547nnreal @ N )
= one_on2969667320475766781nnreal )
= ( N = one ) ) ).
% numeral_eq_one_iff
thf(fact_819_one__eq__numeral__iff,axiom,
! [N: num] :
( ( one_one_real
= ( numeral_numeral_real @ N ) )
= ( one = N ) ) ).
% one_eq_numeral_iff
thf(fact_820_one__eq__numeral__iff,axiom,
! [N: num] :
( ( one_one_nat
= ( numeral_numeral_nat @ N ) )
= ( one = N ) ) ).
% one_eq_numeral_iff
thf(fact_821_one__eq__numeral__iff,axiom,
! [N: num] :
( ( one_one_int
= ( numeral_numeral_int @ N ) )
= ( one = N ) ) ).
% one_eq_numeral_iff
thf(fact_822_one__eq__numeral__iff,axiom,
! [N: num] :
( ( one_on7984719198319812577d_enat
= ( numera1916890842035813515d_enat @ N ) )
= ( one = N ) ) ).
% one_eq_numeral_iff
thf(fact_823_one__eq__numeral__iff,axiom,
! [N: num] :
( ( one_on2969667320475766781nnreal
= ( numera4658534427948366547nnreal @ N ) )
= ( one = N ) ) ).
% one_eq_numeral_iff
thf(fact_824_div__neg__neg__trivial,axiom,
! [K: int,L: int] :
( ( ord_less_eq_int @ K @ zero_zero_int )
=> ( ( ord_less_int @ L @ K )
=> ( ( divide_divide_int @ K @ L )
= zero_zero_int ) ) ) ).
% div_neg_neg_trivial
thf(fact_825_div__pos__pos__trivial,axiom,
! [K: int,L: int] :
( ( ord_less_eq_int @ zero_zero_int @ K )
=> ( ( ord_less_int @ K @ L )
=> ( ( divide_divide_int @ K @ L )
= zero_zero_int ) ) ) ).
% div_pos_pos_trivial
thf(fact_826_dbl__simps_I3_J,axiom,
( ( neg_numeral_dbl_real @ one_one_real )
= ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).
% dbl_simps(3)
thf(fact_827_dbl__simps_I3_J,axiom,
( ( neg_numeral_dbl_int @ one_one_int )
= ( numeral_numeral_int @ ( bit0 @ one ) ) ) ).
% dbl_simps(3)
thf(fact_828_dbl__simps_I3_J,axiom,
( ( neg_nu5590746349488142217l_num1 @ one_on3868389512446148991l_num1 )
= ( numera2161328050825114965l_num1 @ ( bit0 @ one ) ) ) ).
% dbl_simps(3)
thf(fact_829_one__div__two__eq__zero,axiom,
( ( divide_divide_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= zero_zero_nat ) ).
% one_div_two_eq_zero
thf(fact_830_one__div__two__eq__zero,axiom,
( ( divide_divide_int @ one_one_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= zero_zero_int ) ).
% one_div_two_eq_zero
thf(fact_831_verit__la__generic,axiom,
! [A2: int,X3: int] :
( ( ord_less_eq_int @ A2 @ X3 )
| ( A2 = X3 )
| ( ord_less_eq_int @ X3 @ A2 ) ) ).
% verit_la_generic
thf(fact_832_imp__le__cong,axiom,
! [X3: int,X8: int,P: $o,P4: $o] :
( ( X3 = X8 )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X8 )
=> ( P = P4 ) )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X3 )
=> P )
= ( ( ord_less_eq_int @ zero_zero_int @ X8 )
=> P4 ) ) ) ) ).
% imp_le_cong
thf(fact_833_conj__le__cong,axiom,
! [X3: int,X8: int,P: $o,P4: $o] :
( ( X3 = X8 )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X8 )
=> ( P = P4 ) )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X3 )
& P )
= ( ( ord_less_eq_int @ zero_zero_int @ X8 )
& P4 ) ) ) ) ).
% conj_le_cong
thf(fact_834_less__numeral__extra_I1_J,axiom,
ord_le7381754540660121996nnreal @ zero_z7100319975126383169nnreal @ one_on2969667320475766781nnreal ).
% less_numeral_extra(1)
thf(fact_835_less__numeral__extra_I1_J,axiom,
ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ one_on7984719198319812577d_enat ).
% less_numeral_extra(1)
thf(fact_836_less__numeral__extra_I1_J,axiom,
ord_less_real @ zero_zero_real @ one_one_real ).
% less_numeral_extra(1)
thf(fact_837_less__numeral__extra_I1_J,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% less_numeral_extra(1)
thf(fact_838_less__numeral__extra_I1_J,axiom,
ord_less_int @ zero_zero_int @ one_one_int ).
% less_numeral_extra(1)
thf(fact_839_int__div__less__self,axiom,
! [X3: int,K: int] :
( ( ord_less_int @ zero_zero_int @ X3 )
=> ( ( ord_less_int @ one_one_int @ K )
=> ( ord_less_int @ ( divide_divide_int @ X3 @ K ) @ X3 ) ) ) ).
% int_div_less_self
thf(fact_840_div__eq__dividend__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ( ( divide_divide_nat @ M @ N )
= M )
= ( N = one_one_nat ) ) ) ).
% div_eq_dividend_iff
thf(fact_841_div__less__dividend,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ one_one_nat @ N )
=> ( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ord_less_nat @ ( divide_divide_nat @ M @ N ) @ M ) ) ) ).
% div_less_dividend
thf(fact_842_le__numeral__extra_I4_J,axiom,
ord_le2932123472753598470d_enat @ one_on7984719198319812577d_enat @ one_on7984719198319812577d_enat ).
% le_numeral_extra(4)
thf(fact_843_le__numeral__extra_I4_J,axiom,
ord_less_eq_real @ one_one_real @ one_one_real ).
% le_numeral_extra(4)
thf(fact_844_le__numeral__extra_I4_J,axiom,
ord_less_eq_nat @ one_one_nat @ one_one_nat ).
% le_numeral_extra(4)
thf(fact_845_le__numeral__extra_I4_J,axiom,
ord_less_eq_int @ one_one_int @ one_one_int ).
% le_numeral_extra(4)
thf(fact_846_less__numeral__extra_I4_J,axiom,
~ ( ord_le72135733267957522d_enat @ one_on7984719198319812577d_enat @ one_on7984719198319812577d_enat ) ).
% less_numeral_extra(4)
thf(fact_847_less__numeral__extra_I4_J,axiom,
~ ( ord_less_real @ one_one_real @ one_one_real ) ).
% less_numeral_extra(4)
thf(fact_848_less__numeral__extra_I4_J,axiom,
~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).
% less_numeral_extra(4)
thf(fact_849_less__numeral__extra_I4_J,axiom,
~ ( ord_less_int @ one_one_int @ one_one_int ) ).
% less_numeral_extra(4)
thf(fact_850_le__numeral__extra_I3_J,axiom,
ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ zero_z5237406670263579293d_enat ).
% le_numeral_extra(3)
thf(fact_851_le__numeral__extra_I3_J,axiom,
ord_le3935885782089961368nnreal @ zero_z7100319975126383169nnreal @ zero_z7100319975126383169nnreal ).
% le_numeral_extra(3)
thf(fact_852_le__numeral__extra_I3_J,axiom,
ord_less_eq_real @ zero_zero_real @ zero_zero_real ).
% le_numeral_extra(3)
thf(fact_853_le__numeral__extra_I3_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% le_numeral_extra(3)
thf(fact_854_le__numeral__extra_I3_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% le_numeral_extra(3)
thf(fact_855_less__numeral__extra_I3_J,axiom,
~ ( ord_le7381754540660121996nnreal @ zero_z7100319975126383169nnreal @ zero_z7100319975126383169nnreal ) ).
% less_numeral_extra(3)
thf(fact_856_less__numeral__extra_I3_J,axiom,
~ ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ zero_z5237406670263579293d_enat ) ).
% less_numeral_extra(3)
thf(fact_857_less__numeral__extra_I3_J,axiom,
~ ( ord_less_real @ zero_zero_real @ zero_zero_real ) ).
% less_numeral_extra(3)
thf(fact_858_less__numeral__extra_I3_J,axiom,
~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% less_numeral_extra(3)
thf(fact_859_less__numeral__extra_I3_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_numeral_extra(3)
thf(fact_860_zero__neq__numeral,axiom,
! [N: num] :
( zero_zero_real
!= ( numeral_numeral_real @ N ) ) ).
% zero_neq_numeral
thf(fact_861_zero__neq__numeral,axiom,
! [N: num] :
( zero_zero_nat
!= ( numeral_numeral_nat @ N ) ) ).
% zero_neq_numeral
thf(fact_862_zero__neq__numeral,axiom,
! [N: num] :
( zero_zero_int
!= ( numeral_numeral_int @ N ) ) ).
% zero_neq_numeral
thf(fact_863_zero__neq__numeral,axiom,
! [N: num] :
( zero_z5237406670263579293d_enat
!= ( numera1916890842035813515d_enat @ N ) ) ).
% zero_neq_numeral
thf(fact_864_zero__neq__numeral,axiom,
! [N: num] :
( zero_z7100319975126383169nnreal
!= ( numera4658534427948366547nnreal @ N ) ) ).
% zero_neq_numeral
thf(fact_865_num_Osize_I4_J,axiom,
( ( size_size_num @ one )
= zero_zero_nat ) ).
% num.size(4)
thf(fact_866_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_867_bot__nat__0_Oextremum__uniqueI,axiom,
! [A2: nat] :
( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
=> ( A2 = zero_zero_nat ) ) ).
% bot_nat_0.extremum_uniqueI
thf(fact_868_bot__nat__0_Oextremum__unique,axiom,
! [A2: nat] :
( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
= ( A2 = zero_zero_nat ) ) ).
% bot_nat_0.extremum_unique
thf(fact_869_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_870_infinite__descent0,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ~ ( P @ N3 )
=> ? [M3: nat] :
( ( ord_less_nat @ M3 @ N3 )
& ~ ( P @ M3 ) ) ) )
=> ( P @ N ) ) ) ).
% infinite_descent0
thf(fact_871_gr__implies__not0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_872_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_zeroE
thf(fact_873_not__less0,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less0
thf(fact_874_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr0
thf(fact_875_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr0I
thf(fact_876_bot__nat__0_Oextremum__strict,axiom,
! [A2: nat] :
~ ( ord_less_nat @ A2 @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_877_zdiv__mono1,axiom,
! [A2: int,A: int,B2: int] :
( ( ord_less_eq_int @ A2 @ A )
=> ( ( ord_less_int @ zero_zero_int @ B2 )
=> ( ord_less_eq_int @ ( divide_divide_int @ A2 @ B2 ) @ ( divide_divide_int @ A @ B2 ) ) ) ) ).
% zdiv_mono1
thf(fact_878_zdiv__mono2,axiom,
! [A2: int,B: int,B2: int] :
( ( ord_less_eq_int @ zero_zero_int @ A2 )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_eq_int @ B @ B2 )
=> ( ord_less_eq_int @ ( divide_divide_int @ A2 @ B2 ) @ ( divide_divide_int @ A2 @ B ) ) ) ) ) ).
% zdiv_mono2
thf(fact_879_zdiv__eq__0__iff,axiom,
! [I5: int,K: int] :
( ( ( divide_divide_int @ I5 @ K )
= zero_zero_int )
= ( ( K = zero_zero_int )
| ( ( ord_less_eq_int @ zero_zero_int @ I5 )
& ( ord_less_int @ I5 @ K ) )
| ( ( ord_less_eq_int @ I5 @ zero_zero_int )
& ( ord_less_int @ K @ I5 ) ) ) ) ).
% zdiv_eq_0_iff
thf(fact_880_zdiv__mono1__neg,axiom,
! [A2: int,A: int,B2: int] :
( ( ord_less_eq_int @ A2 @ A )
=> ( ( ord_less_int @ B2 @ zero_zero_int )
=> ( ord_less_eq_int @ ( divide_divide_int @ A @ B2 ) @ ( divide_divide_int @ A2 @ B2 ) ) ) ) ).
% zdiv_mono1_neg
thf(fact_881_zdiv__mono2__neg,axiom,
! [A2: int,B: int,B2: int] :
( ( ord_less_int @ A2 @ zero_zero_int )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_eq_int @ B @ B2 )
=> ( ord_less_eq_int @ ( divide_divide_int @ A2 @ B ) @ ( divide_divide_int @ A2 @ B2 ) ) ) ) ) ).
% zdiv_mono2_neg
thf(fact_882_div__int__pos__iff,axiom,
! [K: int,L: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ K @ L ) )
= ( ( K = zero_zero_int )
| ( L = zero_zero_int )
| ( ( ord_less_eq_int @ zero_zero_int @ K )
& ( ord_less_eq_int @ zero_zero_int @ L ) )
| ( ( ord_less_int @ K @ zero_zero_int )
& ( ord_less_int @ L @ zero_zero_int ) ) ) ) ).
% div_int_pos_iff
thf(fact_883_div__neg__pos__less0,axiom,
! [A2: int,B2: int] :
( ( ord_less_int @ A2 @ zero_zero_int )
=> ( ( ord_less_int @ zero_zero_int @ B2 )
=> ( ord_less_int @ ( divide_divide_int @ A2 @ B2 ) @ zero_zero_int ) ) ) ).
% div_neg_pos_less0
thf(fact_884_div__nonneg__neg__le0,axiom,
! [A2: int,B2: int] :
( ( ord_less_eq_int @ zero_zero_int @ A2 )
=> ( ( ord_less_int @ B2 @ zero_zero_int )
=> ( ord_less_eq_int @ ( divide_divide_int @ A2 @ B2 ) @ zero_zero_int ) ) ) ).
% div_nonneg_neg_le0
thf(fact_885_div__nonpos__pos__le0,axiom,
! [A2: int,B2: int] :
( ( ord_less_eq_int @ A2 @ zero_zero_int )
=> ( ( ord_less_int @ zero_zero_int @ B2 )
=> ( ord_less_eq_int @ ( divide_divide_int @ A2 @ B2 ) @ zero_zero_int ) ) ) ).
% div_nonpos_pos_le0
thf(fact_886_neg__imp__zdiv__neg__iff,axiom,
! [B2: int,A2: int] :
( ( ord_less_int @ B2 @ zero_zero_int )
=> ( ( ord_less_int @ ( divide_divide_int @ A2 @ B2 ) @ zero_zero_int )
= ( ord_less_int @ zero_zero_int @ A2 ) ) ) ).
% neg_imp_zdiv_neg_iff
thf(fact_887_pos__imp__zdiv__neg__iff,axiom,
! [B2: int,A2: int] :
( ( ord_less_int @ zero_zero_int @ B2 )
=> ( ( ord_less_int @ ( divide_divide_int @ A2 @ B2 ) @ zero_zero_int )
= ( ord_less_int @ A2 @ zero_zero_int ) ) ) ).
% pos_imp_zdiv_neg_iff
thf(fact_888_pos__imp__zdiv__pos__iff,axiom,
! [K: int,I5: int] :
( ( ord_less_int @ zero_zero_int @ K )
=> ( ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ I5 @ K ) )
= ( ord_less_eq_int @ K @ I5 ) ) ) ).
% pos_imp_zdiv_pos_iff
thf(fact_889_neg__imp__zdiv__nonneg__iff,axiom,
! [B2: int,A2: int] :
( ( ord_less_int @ B2 @ zero_zero_int )
=> ( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ A2 @ B2 ) )
= ( ord_less_eq_int @ A2 @ zero_zero_int ) ) ) ).
% neg_imp_zdiv_nonneg_iff
thf(fact_890_pos__imp__zdiv__nonneg__iff,axiom,
! [B2: int,A2: int] :
( ( ord_less_int @ zero_zero_int @ B2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ A2 @ B2 ) )
= ( ord_less_eq_int @ zero_zero_int @ A2 ) ) ) ).
% pos_imp_zdiv_nonneg_iff
thf(fact_891_nonneg1__imp__zdiv__pos__iff,axiom,
! [A2: int,B2: int] :
( ( ord_less_eq_int @ zero_zero_int @ A2 )
=> ( ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ A2 @ B2 ) )
= ( ( ord_less_eq_int @ B2 @ A2 )
& ( ord_less_int @ zero_zero_int @ B2 ) ) ) ) ).
% nonneg1_imp_zdiv_pos_iff
thf(fact_892_not__iless0,axiom,
! [N: extended_enat] :
~ ( ord_le72135733267957522d_enat @ N @ zero_z5237406670263579293d_enat ) ).
% not_iless0
thf(fact_893_ile0__eq,axiom,
! [N: extended_enat] :
( ( ord_le2932123472753598470d_enat @ N @ zero_z5237406670263579293d_enat )
= ( N = zero_z5237406670263579293d_enat ) ) ).
% ile0_eq
thf(fact_894_i0__lb,axiom,
! [N: extended_enat] : ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ N ) ).
% i0_lb
thf(fact_895_one__le__numeral,axiom,
! [N: num] : ( ord_le2932123472753598470d_enat @ one_on7984719198319812577d_enat @ ( numera1916890842035813515d_enat @ N ) ) ).
% one_le_numeral
thf(fact_896_one__le__numeral,axiom,
! [N: num] : ( ord_le3935885782089961368nnreal @ one_on2969667320475766781nnreal @ ( numera4658534427948366547nnreal @ N ) ) ).
% one_le_numeral
thf(fact_897_one__le__numeral,axiom,
! [N: num] : ( ord_less_eq_real @ one_one_real @ ( numeral_numeral_real @ N ) ) ).
% one_le_numeral
thf(fact_898_one__le__numeral,axiom,
! [N: num] : ( ord_less_eq_nat @ one_one_nat @ ( numeral_numeral_nat @ N ) ) ).
% one_le_numeral
thf(fact_899_one__le__numeral,axiom,
! [N: num] : ( ord_less_eq_int @ one_one_int @ ( numeral_numeral_int @ N ) ) ).
% one_le_numeral
thf(fact_900_not__numeral__less__one,axiom,
! [N: num] :
~ ( ord_less_real @ ( numeral_numeral_real @ N ) @ one_one_real ) ).
% not_numeral_less_one
thf(fact_901_not__numeral__less__one,axiom,
! [N: num] :
~ ( ord_less_nat @ ( numeral_numeral_nat @ N ) @ one_one_nat ) ).
% not_numeral_less_one
thf(fact_902_not__numeral__less__one,axiom,
! [N: num] :
~ ( ord_less_int @ ( numeral_numeral_int @ N ) @ one_one_int ) ).
% not_numeral_less_one
thf(fact_903_not__numeral__less__one,axiom,
! [N: num] :
~ ( ord_le72135733267957522d_enat @ ( numera1916890842035813515d_enat @ N ) @ one_on7984719198319812577d_enat ) ).
% not_numeral_less_one
thf(fact_904_not__numeral__less__one,axiom,
! [N: num] :
~ ( ord_le7381754540660121996nnreal @ ( numera4658534427948366547nnreal @ N ) @ one_on2969667320475766781nnreal ) ).
% not_numeral_less_one
thf(fact_905_numeral__One,axiom,
( ( numeral_numeral_real @ one )
= one_one_real ) ).
% numeral_One
thf(fact_906_numeral__One,axiom,
( ( numeral_numeral_nat @ one )
= one_one_nat ) ).
% numeral_One
thf(fact_907_numeral__One,axiom,
( ( numeral_numeral_int @ one )
= one_one_int ) ).
% numeral_One
thf(fact_908_numeral__One,axiom,
( ( numera1916890842035813515d_enat @ one )
= one_on7984719198319812577d_enat ) ).
% numeral_One
thf(fact_909_numeral__One,axiom,
( ( numera4658534427948366547nnreal @ one )
= one_on2969667320475766781nnreal ) ).
% numeral_One
thf(fact_910_numeral__One,axiom,
( ( numera2161328050825114965l_num1 @ one )
= one_on3868389512446148991l_num1 ) ).
% numeral_One
thf(fact_911_not__numeral__le__zero,axiom,
! [N: num] :
~ ( ord_le2932123472753598470d_enat @ ( numera1916890842035813515d_enat @ N ) @ zero_z5237406670263579293d_enat ) ).
% not_numeral_le_zero
thf(fact_912_not__numeral__le__zero,axiom,
! [N: num] :
~ ( ord_le3935885782089961368nnreal @ ( numera4658534427948366547nnreal @ N ) @ zero_z7100319975126383169nnreal ) ).
% not_numeral_le_zero
thf(fact_913_not__numeral__le__zero,axiom,
! [N: num] :
~ ( ord_less_eq_real @ ( numeral_numeral_real @ N ) @ zero_zero_real ) ).
% not_numeral_le_zero
thf(fact_914_not__numeral__le__zero,axiom,
! [N: num] :
~ ( ord_less_eq_nat @ ( numeral_numeral_nat @ N ) @ zero_zero_nat ) ).
% not_numeral_le_zero
thf(fact_915_not__numeral__le__zero,axiom,
! [N: num] :
~ ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ zero_zero_int ) ).
% not_numeral_le_zero
thf(fact_916_zero__le__numeral,axiom,
! [N: num] : ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ ( numera1916890842035813515d_enat @ N ) ) ).
% zero_le_numeral
thf(fact_917_zero__le__numeral,axiom,
! [N: num] : ( ord_le3935885782089961368nnreal @ zero_z7100319975126383169nnreal @ ( numera4658534427948366547nnreal @ N ) ) ).
% zero_le_numeral
thf(fact_918_zero__le__numeral,axiom,
! [N: num] : ( ord_less_eq_real @ zero_zero_real @ ( numeral_numeral_real @ N ) ) ).
% zero_le_numeral
thf(fact_919_zero__le__numeral,axiom,
! [N: num] : ( ord_less_eq_nat @ zero_zero_nat @ ( numeral_numeral_nat @ N ) ) ).
% zero_le_numeral
thf(fact_920_zero__le__numeral,axiom,
! [N: num] : ( ord_less_eq_int @ zero_zero_int @ ( numeral_numeral_int @ N ) ) ).
% zero_le_numeral
thf(fact_921_not__numeral__less__zero,axiom,
! [N: num] :
~ ( ord_less_real @ ( numeral_numeral_real @ N ) @ zero_zero_real ) ).
% not_numeral_less_zero
thf(fact_922_not__numeral__less__zero,axiom,
! [N: num] :
~ ( ord_less_nat @ ( numeral_numeral_nat @ N ) @ zero_zero_nat ) ).
% not_numeral_less_zero
thf(fact_923_not__numeral__less__zero,axiom,
! [N: num] :
~ ( ord_less_int @ ( numeral_numeral_int @ N ) @ zero_zero_int ) ).
% not_numeral_less_zero
thf(fact_924_not__numeral__less__zero,axiom,
! [N: num] :
~ ( ord_le72135733267957522d_enat @ ( numera1916890842035813515d_enat @ N ) @ zero_z5237406670263579293d_enat ) ).
% not_numeral_less_zero
thf(fact_925_not__numeral__less__zero,axiom,
! [N: num] :
~ ( ord_le7381754540660121996nnreal @ ( numera4658534427948366547nnreal @ N ) @ zero_z7100319975126383169nnreal ) ).
% not_numeral_less_zero
thf(fact_926_zero__less__numeral,axiom,
! [N: num] : ( ord_less_real @ zero_zero_real @ ( numeral_numeral_real @ N ) ) ).
% zero_less_numeral
thf(fact_927_zero__less__numeral,axiom,
! [N: num] : ( ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ N ) ) ).
% zero_less_numeral
thf(fact_928_zero__less__numeral,axiom,
! [N: num] : ( ord_less_int @ zero_zero_int @ ( numeral_numeral_int @ N ) ) ).
% zero_less_numeral
thf(fact_929_zero__less__numeral,axiom,
! [N: num] : ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ ( numera1916890842035813515d_enat @ N ) ) ).
% zero_less_numeral
thf(fact_930_zero__less__numeral,axiom,
! [N: num] : ( ord_le7381754540660121996nnreal @ zero_z7100319975126383169nnreal @ ( numera4658534427948366547nnreal @ N ) ) ).
% zero_less_numeral
thf(fact_931_numerals_I1_J,axiom,
( ( numeral_numeral_nat @ one )
= one_one_nat ) ).
% numerals(1)
thf(fact_932_ex__least__nat__le,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N )
& ! [I6: nat] :
( ( ord_less_nat @ I6 @ K2 )
=> ~ ( P @ I6 ) )
& ( P @ K2 ) ) ) ) ).
% ex_least_nat_le
thf(fact_933_Euclidean__Division_Odiv__eq__0__iff,axiom,
! [M: nat,N: nat] :
( ( ( divide_divide_nat @ M @ N )
= zero_zero_nat )
= ( ( ord_less_nat @ M @ N )
| ( N = zero_zero_nat ) ) ) ).
% Euclidean_Division.div_eq_0_iff
thf(fact_934_log__induct,axiom,
! [N: nat,P: nat > $o] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( P @ one_one_nat )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
=> ( ( P @ ( divide_divide_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
=> ( P @ N3 ) ) )
=> ( P @ N ) ) ) ) ).
% log_induct
thf(fact_935_sorted__wrt01,axiom,
! [Xs2: list_a,P: a > a > $o] :
( ( ord_less_eq_nat @ ( size_size_list_a @ Xs2 ) @ one_one_nat )
=> ( sorted_wrt_a @ P @ Xs2 ) ) ).
% sorted_wrt01
thf(fact_936_sorted__wrt01,axiom,
! [Xs2: list_nat,P: nat > nat > $o] :
( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs2 ) @ one_one_nat )
=> ( sorted_wrt_nat @ P @ Xs2 ) ) ).
% sorted_wrt01
thf(fact_937_div__greater__zero__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ M @ N ) )
= ( ( ord_less_eq_nat @ N @ M )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% div_greater_zero_iff
thf(fact_938_div__le__mono2,axiom,
! [M: nat,N: nat,K: nat] :
( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( divide_divide_nat @ K @ N ) @ ( divide_divide_nat @ K @ M ) ) ) ) ).
% div_le_mono2
thf(fact_939_sorted01,axiom,
! [Xs2: list_real] :
( ( ord_less_eq_nat @ ( size_size_list_real @ Xs2 ) @ one_one_nat )
=> ( sorted_wrt_real @ ord_less_eq_real @ Xs2 ) ) ).
% sorted01
thf(fact_940_sorted01,axiom,
! [Xs2: list_num] :
( ( ord_less_eq_nat @ ( size_size_list_num @ Xs2 ) @ one_one_nat )
=> ( sorted_wrt_num @ ord_less_eq_num @ Xs2 ) ) ).
% sorted01
thf(fact_941_sorted01,axiom,
! [Xs2: list_nat] :
( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs2 ) @ one_one_nat )
=> ( sorted_wrt_nat @ ord_less_eq_nat @ Xs2 ) ) ).
% sorted01
thf(fact_942_sorted01,axiom,
! [Xs2: list_int] :
( ( ord_less_eq_nat @ ( size_size_list_int @ Xs2 ) @ one_one_nat )
=> ( sorted_wrt_int @ ord_less_eq_int @ Xs2 ) ) ).
% sorted01
thf(fact_943_sorted01,axiom,
! [Xs2: list_a] :
( ( ord_less_eq_nat @ ( size_size_list_a @ Xs2 ) @ one_one_nat )
=> ( sorted_wrt_a @ ord_less_eq_a @ Xs2 ) ) ).
% sorted01
thf(fact_944_half__gt__zero,axiom,
! [A2: real] :
( ( ord_less_real @ zero_zero_real @ A2 )
=> ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ A2 @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% half_gt_zero
thf(fact_945_bits__1__div__2,axiom,
( ( divide_divide_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= zero_zero_nat ) ).
% bits_1_div_2
thf(fact_946_bits__1__div__2,axiom,
( ( divide_divide_int @ one_one_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= zero_zero_int ) ).
% bits_1_div_2
thf(fact_947_le__divide__eq__1__pos,axiom,
! [A2: real,B2: real] :
( ( ord_less_real @ zero_zero_real @ A2 )
=> ( ( ord_less_eq_real @ one_one_real @ ( divide_divide_real @ B2 @ A2 ) )
= ( ord_less_eq_real @ A2 @ B2 ) ) ) ).
% le_divide_eq_1_pos
thf(fact_948_le__divide__eq__1__neg,axiom,
! [A2: real,B2: real] :
( ( ord_less_real @ A2 @ zero_zero_real )
=> ( ( ord_less_eq_real @ one_one_real @ ( divide_divide_real @ B2 @ A2 ) )
= ( ord_less_eq_real @ B2 @ A2 ) ) ) ).
% le_divide_eq_1_neg
thf(fact_949_divide__le__eq__1__pos,axiom,
! [A2: real,B2: real] :
( ( ord_less_real @ zero_zero_real @ A2 )
=> ( ( ord_less_eq_real @ ( divide_divide_real @ B2 @ A2 ) @ one_one_real )
= ( ord_less_eq_real @ B2 @ A2 ) ) ) ).
% divide_le_eq_1_pos
thf(fact_950_divide__le__eq__1__neg,axiom,
! [A2: real,B2: real] :
( ( ord_less_real @ A2 @ zero_zero_real )
=> ( ( ord_less_eq_real @ ( divide_divide_real @ B2 @ A2 ) @ one_one_real )
= ( ord_less_eq_real @ A2 @ B2 ) ) ) ).
% divide_le_eq_1_neg
thf(fact_951_zero__less__divide__1__iff,axiom,
! [A2: real] :
( ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ one_one_real @ A2 ) )
= ( ord_less_real @ zero_zero_real @ A2 ) ) ).
% zero_less_divide_1_iff
thf(fact_952_less__divide__eq__1__pos,axiom,
! [A2: real,B2: real] :
( ( ord_less_real @ zero_zero_real @ A2 )
=> ( ( ord_less_real @ one_one_real @ ( divide_divide_real @ B2 @ A2 ) )
= ( ord_less_real @ A2 @ B2 ) ) ) ).
% less_divide_eq_1_pos
thf(fact_953_less__divide__eq__1__neg,axiom,
! [A2: real,B2: real] :
( ( ord_less_real @ A2 @ zero_zero_real )
=> ( ( ord_less_real @ one_one_real @ ( divide_divide_real @ B2 @ A2 ) )
= ( ord_less_real @ B2 @ A2 ) ) ) ).
% less_divide_eq_1_neg
thf(fact_954_division__ring__divide__zero,axiom,
! [A2: real] :
( ( divide_divide_real @ A2 @ zero_zero_real )
= zero_zero_real ) ).
% division_ring_divide_zero
thf(fact_955_bits__div__by__0,axiom,
! [A2: nat] :
( ( divide_divide_nat @ A2 @ zero_zero_nat )
= zero_zero_nat ) ).
% bits_div_by_0
thf(fact_956_bits__div__by__0,axiom,
! [A2: int] :
( ( divide_divide_int @ A2 @ zero_zero_int )
= zero_zero_int ) ).
% bits_div_by_0
thf(fact_957_bits__div__0,axiom,
! [A2: nat] :
( ( divide_divide_nat @ zero_zero_nat @ A2 )
= zero_zero_nat ) ).
% bits_div_0
thf(fact_958_bits__div__0,axiom,
! [A2: int] :
( ( divide_divide_int @ zero_zero_int @ A2 )
= zero_zero_int ) ).
% bits_div_0
thf(fact_959_divide__cancel__right,axiom,
! [A2: real,C: real,B2: real] :
( ( ( divide_divide_real @ A2 @ C )
= ( divide_divide_real @ B2 @ C ) )
= ( ( C = zero_zero_real )
| ( A2 = B2 ) ) ) ).
% divide_cancel_right
thf(fact_960_divide__cancel__left,axiom,
! [C: real,A2: real,B2: real] :
( ( ( divide_divide_real @ C @ A2 )
= ( divide_divide_real @ C @ B2 ) )
= ( ( C = zero_zero_real )
| ( A2 = B2 ) ) ) ).
% divide_cancel_left
thf(fact_961_divide__eq__0__iff,axiom,
! [A2: real,B2: real] :
( ( ( divide_divide_real @ A2 @ B2 )
= zero_zero_real )
= ( ( A2 = zero_zero_real )
| ( B2 = zero_zero_real ) ) ) ).
% divide_eq_0_iff
thf(fact_962_bits__div__by__1,axiom,
! [A2: nat] :
( ( divide_divide_nat @ A2 @ one_one_nat )
= A2 ) ).
% bits_div_by_1
thf(fact_963_bits__div__by__1,axiom,
! [A2: int] :
( ( divide_divide_int @ A2 @ one_one_int )
= A2 ) ).
% bits_div_by_1
thf(fact_964_divide__eq__1__iff,axiom,
! [A2: real,B2: real] :
( ( ( divide_divide_real @ A2 @ B2 )
= one_one_real )
= ( ( B2 != zero_zero_real )
& ( A2 = B2 ) ) ) ).
% divide_eq_1_iff
thf(fact_965_one__eq__divide__iff,axiom,
! [A2: real,B2: real] :
( ( one_one_real
= ( divide_divide_real @ A2 @ B2 ) )
= ( ( B2 != zero_zero_real )
& ( A2 = B2 ) ) ) ).
% one_eq_divide_iff
thf(fact_966_divide__self,axiom,
! [A2: real] :
( ( A2 != zero_zero_real )
=> ( ( divide_divide_real @ A2 @ A2 )
= one_one_real ) ) ).
% divide_self
thf(fact_967_divide__self__if,axiom,
! [A2: real] :
( ( ( A2 = zero_zero_real )
=> ( ( divide_divide_real @ A2 @ A2 )
= zero_zero_real ) )
& ( ( A2 != zero_zero_real )
=> ( ( divide_divide_real @ A2 @ A2 )
= one_one_real ) ) ) ).
% divide_self_if
thf(fact_968_divide__eq__eq__1,axiom,
! [B2: real,A2: real] :
( ( ( divide_divide_real @ B2 @ A2 )
= one_one_real )
= ( ( A2 != zero_zero_real )
& ( A2 = B2 ) ) ) ).
% divide_eq_eq_1
thf(fact_969_eq__divide__eq__1,axiom,
! [B2: real,A2: real] :
( ( one_one_real
= ( divide_divide_real @ B2 @ A2 ) )
= ( ( A2 != zero_zero_real )
& ( A2 = B2 ) ) ) ).
% eq_divide_eq_1
thf(fact_970_one__divide__eq__0__iff,axiom,
! [A2: real] :
( ( ( divide_divide_real @ one_one_real @ A2 )
= zero_zero_real )
= ( A2 = zero_zero_real ) ) ).
% one_divide_eq_0_iff
thf(fact_971_zero__eq__1__divide__iff,axiom,
! [A2: real] :
( ( zero_zero_real
= ( divide_divide_real @ one_one_real @ A2 ) )
= ( A2 = zero_zero_real ) ) ).
% zero_eq_1_divide_iff
thf(fact_972_divide__le__0__1__iff,axiom,
! [A2: real] :
( ( ord_less_eq_real @ ( divide_divide_real @ one_one_real @ A2 ) @ zero_zero_real )
= ( ord_less_eq_real @ A2 @ zero_zero_real ) ) ).
% divide_le_0_1_iff
thf(fact_973_zero__le__divide__1__iff,axiom,
! [A2: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ one_one_real @ A2 ) )
= ( ord_less_eq_real @ zero_zero_real @ A2 ) ) ).
% zero_le_divide_1_iff
thf(fact_974_divide__less__0__1__iff,axiom,
! [A2: real] :
( ( ord_less_real @ ( divide_divide_real @ one_one_real @ A2 ) @ zero_zero_real )
= ( ord_less_real @ A2 @ zero_zero_real ) ) ).
% divide_less_0_1_iff
thf(fact_975_divide__less__eq__1__neg,axiom,
! [A2: real,B2: real] :
( ( ord_less_real @ A2 @ zero_zero_real )
=> ( ( ord_less_real @ ( divide_divide_real @ B2 @ A2 ) @ one_one_real )
= ( ord_less_real @ A2 @ B2 ) ) ) ).
% divide_less_eq_1_neg
thf(fact_976_divide__less__eq__1__pos,axiom,
! [A2: real,B2: real] :
( ( ord_less_real @ zero_zero_real @ A2 )
=> ( ( ord_less_real @ ( divide_divide_real @ B2 @ A2 ) @ one_one_real )
= ( ord_less_real @ B2 @ A2 ) ) ) ).
% divide_less_eq_1_pos
thf(fact_977_zero__one__enat__neq_I1_J,axiom,
zero_z5237406670263579293d_enat != one_on7984719198319812577d_enat ).
% zero_one_enat_neq(1)
thf(fact_978_linordered__field__no__lb,axiom,
! [X7: real] :
? [Y5: real] : ( ord_less_real @ Y5 @ X7 ) ).
% linordered_field_no_lb
thf(fact_979_linordered__field__no__ub,axiom,
! [X7: real] :
? [X_1: real] : ( ord_less_real @ X7 @ X_1 ) ).
% linordered_field_no_ub
thf(fact_980_divide__le__0__iff,axiom,
! [A2: real,B2: real] :
( ( ord_less_eq_real @ ( divide_divide_real @ A2 @ B2 ) @ zero_zero_real )
= ( ( ( ord_less_eq_real @ zero_zero_real @ A2 )
& ( ord_less_eq_real @ B2 @ zero_zero_real ) )
| ( ( ord_less_eq_real @ A2 @ zero_zero_real )
& ( ord_less_eq_real @ zero_zero_real @ B2 ) ) ) ) ).
% divide_le_0_iff
thf(fact_981_divide__right__mono,axiom,
! [A2: real,B2: real,C: real] :
( ( ord_less_eq_real @ A2 @ B2 )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ ( divide_divide_real @ A2 @ C ) @ ( divide_divide_real @ B2 @ C ) ) ) ) ).
% divide_right_mono
thf(fact_982_zero__le__divide__iff,axiom,
! [A2: real,B2: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ A2 @ B2 ) )
= ( ( ( ord_less_eq_real @ zero_zero_real @ A2 )
& ( ord_less_eq_real @ zero_zero_real @ B2 ) )
| ( ( ord_less_eq_real @ A2 @ zero_zero_real )
& ( ord_less_eq_real @ B2 @ zero_zero_real ) ) ) ) ).
% zero_le_divide_iff
thf(fact_983_divide__nonneg__nonneg,axiom,
! [X3: real,Y4: real] :
( ( ord_less_eq_real @ zero_zero_real @ X3 )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y4 )
=> ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X3 @ Y4 ) ) ) ) ).
% divide_nonneg_nonneg
thf(fact_984_divide__nonneg__nonpos,axiom,
! [X3: real,Y4: real] :
( ( ord_less_eq_real @ zero_zero_real @ X3 )
=> ( ( ord_less_eq_real @ Y4 @ zero_zero_real )
=> ( ord_less_eq_real @ ( divide_divide_real @ X3 @ Y4 ) @ zero_zero_real ) ) ) ).
% divide_nonneg_nonpos
thf(fact_985_divide__nonpos__nonneg,axiom,
! [X3: real,Y4: real] :
( ( ord_less_eq_real @ X3 @ zero_zero_real )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y4 )
=> ( ord_less_eq_real @ ( divide_divide_real @ X3 @ Y4 ) @ zero_zero_real ) ) ) ).
% divide_nonpos_nonneg
thf(fact_986_divide__nonpos__nonpos,axiom,
! [X3: real,Y4: real] :
( ( ord_less_eq_real @ X3 @ zero_zero_real )
=> ( ( ord_less_eq_real @ Y4 @ zero_zero_real )
=> ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X3 @ Y4 ) ) ) ) ).
% divide_nonpos_nonpos
thf(fact_987_divide__right__mono__neg,axiom,
! [A2: real,B2: real,C: real] :
( ( ord_less_eq_real @ A2 @ B2 )
=> ( ( ord_less_eq_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ ( divide_divide_real @ B2 @ C ) @ ( divide_divide_real @ A2 @ C ) ) ) ) ).
% divide_right_mono_neg
thf(fact_988_divide__neg__neg,axiom,
! [X3: real,Y4: real] :
( ( ord_less_real @ X3 @ zero_zero_real )
=> ( ( ord_less_real @ Y4 @ zero_zero_real )
=> ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ X3 @ Y4 ) ) ) ) ).
% divide_neg_neg
thf(fact_989_divide__neg__pos,axiom,
! [X3: real,Y4: real] :
( ( ord_less_real @ X3 @ zero_zero_real )
=> ( ( ord_less_real @ zero_zero_real @ Y4 )
=> ( ord_less_real @ ( divide_divide_real @ X3 @ Y4 ) @ zero_zero_real ) ) ) ).
% divide_neg_pos
thf(fact_990_divide__pos__neg,axiom,
! [X3: real,Y4: real] :
( ( ord_less_real @ zero_zero_real @ X3 )
=> ( ( ord_less_real @ Y4 @ zero_zero_real )
=> ( ord_less_real @ ( divide_divide_real @ X3 @ Y4 ) @ zero_zero_real ) ) ) ).
% divide_pos_neg
thf(fact_991_divide__pos__pos,axiom,
! [X3: real,Y4: real] :
( ( ord_less_real @ zero_zero_real @ X3 )
=> ( ( ord_less_real @ zero_zero_real @ Y4 )
=> ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ X3 @ Y4 ) ) ) ) ).
% divide_pos_pos
thf(fact_992_divide__less__0__iff,axiom,
! [A2: real,B2: real] :
( ( ord_less_real @ ( divide_divide_real @ A2 @ B2 ) @ zero_zero_real )
= ( ( ( ord_less_real @ zero_zero_real @ A2 )
& ( ord_less_real @ B2 @ zero_zero_real ) )
| ( ( ord_less_real @ A2 @ zero_zero_real )
& ( ord_less_real @ zero_zero_real @ B2 ) ) ) ) ).
% divide_less_0_iff
thf(fact_993_divide__less__cancel,axiom,
! [A2: real,C: real,B2: real] :
( ( ord_less_real @ ( divide_divide_real @ A2 @ C ) @ ( divide_divide_real @ B2 @ C ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_real @ A2 @ B2 ) )
& ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_real @ B2 @ A2 ) )
& ( C != zero_zero_real ) ) ) ).
% divide_less_cancel
thf(fact_994_zero__less__divide__iff,axiom,
! [A2: real,B2: real] :
( ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ A2 @ B2 ) )
= ( ( ( ord_less_real @ zero_zero_real @ A2 )
& ( ord_less_real @ zero_zero_real @ B2 ) )
| ( ( ord_less_real @ A2 @ zero_zero_real )
& ( ord_less_real @ B2 @ zero_zero_real ) ) ) ) ).
% zero_less_divide_iff
thf(fact_995_divide__strict__right__mono,axiom,
! [A2: real,B2: real,C: real] :
( ( ord_less_real @ A2 @ B2 )
=> ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_real @ ( divide_divide_real @ A2 @ C ) @ ( divide_divide_real @ B2 @ C ) ) ) ) ).
% divide_strict_right_mono
thf(fact_996_divide__strict__right__mono__neg,axiom,
! [B2: real,A2: real,C: real] :
( ( ord_less_real @ B2 @ A2 )
=> ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_real @ ( divide_divide_real @ A2 @ C ) @ ( divide_divide_real @ B2 @ C ) ) ) ) ).
% divide_strict_right_mono_neg
thf(fact_997_right__inverse__eq,axiom,
! [B2: real,A2: real] :
( ( B2 != zero_zero_real )
=> ( ( ( divide_divide_real @ A2 @ B2 )
= one_one_real )
= ( A2 = B2 ) ) ) ).
% right_inverse_eq
thf(fact_998_frac__le,axiom,
! [Y4: real,X3: real,W: real,Z3: real] :
( ( ord_less_eq_real @ zero_zero_real @ Y4 )
=> ( ( ord_less_eq_real @ X3 @ Y4 )
=> ( ( ord_less_real @ zero_zero_real @ W )
=> ( ( ord_less_eq_real @ W @ Z3 )
=> ( ord_less_eq_real @ ( divide_divide_real @ X3 @ Z3 ) @ ( divide_divide_real @ Y4 @ W ) ) ) ) ) ) ).
% frac_le
thf(fact_999_frac__less,axiom,
! [X3: real,Y4: real,W: real,Z3: real] :
( ( ord_less_eq_real @ zero_zero_real @ X3 )
=> ( ( ord_less_real @ X3 @ Y4 )
=> ( ( ord_less_real @ zero_zero_real @ W )
=> ( ( ord_less_eq_real @ W @ Z3 )
=> ( ord_less_real @ ( divide_divide_real @ X3 @ Z3 ) @ ( divide_divide_real @ Y4 @ W ) ) ) ) ) ) ).
% frac_less
thf(fact_1000_frac__less2,axiom,
! [X3: real,Y4: real,W: real,Z3: real] :
( ( ord_less_real @ zero_zero_real @ X3 )
=> ( ( ord_less_eq_real @ X3 @ Y4 )
=> ( ( ord_less_real @ zero_zero_real @ W )
=> ( ( ord_less_real @ W @ Z3 )
=> ( ord_less_real @ ( divide_divide_real @ X3 @ Z3 ) @ ( divide_divide_real @ Y4 @ W ) ) ) ) ) ) ).
% frac_less2
thf(fact_1001_divide__le__cancel,axiom,
! [A2: real,C: real,B2: real] :
( ( ord_less_eq_real @ ( divide_divide_real @ A2 @ C ) @ ( divide_divide_real @ B2 @ C ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ A2 @ B2 ) )
& ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ B2 @ A2 ) ) ) ) ).
% divide_le_cancel
thf(fact_1002_divide__nonneg__neg,axiom,
! [X3: real,Y4: real] :
( ( ord_less_eq_real @ zero_zero_real @ X3 )
=> ( ( ord_less_real @ Y4 @ zero_zero_real )
=> ( ord_less_eq_real @ ( divide_divide_real @ X3 @ Y4 ) @ zero_zero_real ) ) ) ).
% divide_nonneg_neg
thf(fact_1003_divide__nonneg__pos,axiom,
! [X3: real,Y4: real] :
( ( ord_less_eq_real @ zero_zero_real @ X3 )
=> ( ( ord_less_real @ zero_zero_real @ Y4 )
=> ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X3 @ Y4 ) ) ) ) ).
% divide_nonneg_pos
thf(fact_1004_divide__nonpos__neg,axiom,
! [X3: real,Y4: real] :
( ( ord_less_eq_real @ X3 @ zero_zero_real )
=> ( ( ord_less_real @ Y4 @ zero_zero_real )
=> ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X3 @ Y4 ) ) ) ) ).
% divide_nonpos_neg
thf(fact_1005_divide__nonpos__pos,axiom,
! [X3: real,Y4: real] :
( ( ord_less_eq_real @ X3 @ zero_zero_real )
=> ( ( ord_less_real @ zero_zero_real @ Y4 )
=> ( ord_less_eq_real @ ( divide_divide_real @ X3 @ Y4 ) @ zero_zero_real ) ) ) ).
% divide_nonpos_pos
thf(fact_1006_divide__less__eq__1,axiom,
! [B2: real,A2: real] :
( ( ord_less_real @ ( divide_divide_real @ B2 @ A2 ) @ one_one_real )
= ( ( ( ord_less_real @ zero_zero_real @ A2 )
& ( ord_less_real @ B2 @ A2 ) )
| ( ( ord_less_real @ A2 @ zero_zero_real )
& ( ord_less_real @ A2 @ B2 ) )
| ( A2 = zero_zero_real ) ) ) ).
% divide_less_eq_1
thf(fact_1007_less__divide__eq__1,axiom,
! [B2: real,A2: real] :
( ( ord_less_real @ one_one_real @ ( divide_divide_real @ B2 @ A2 ) )
= ( ( ( ord_less_real @ zero_zero_real @ A2 )
& ( ord_less_real @ A2 @ B2 ) )
| ( ( ord_less_real @ A2 @ zero_zero_real )
& ( ord_less_real @ B2 @ A2 ) ) ) ) ).
% less_divide_eq_1
thf(fact_1008_divide__le__eq__1,axiom,
! [B2: real,A2: real] :
( ( ord_less_eq_real @ ( divide_divide_real @ B2 @ A2 ) @ one_one_real )
= ( ( ( ord_less_real @ zero_zero_real @ A2 )
& ( ord_less_eq_real @ B2 @ A2 ) )
| ( ( ord_less_real @ A2 @ zero_zero_real )
& ( ord_less_eq_real @ A2 @ B2 ) )
| ( A2 = zero_zero_real ) ) ) ).
% divide_le_eq_1
thf(fact_1009_le__divide__eq__1,axiom,
! [B2: real,A2: real] :
( ( ord_less_eq_real @ one_one_real @ ( divide_divide_real @ B2 @ A2 ) )
= ( ( ( ord_less_real @ zero_zero_real @ A2 )
& ( ord_less_eq_real @ A2 @ B2 ) )
| ( ( ord_less_real @ A2 @ zero_zero_real )
& ( ord_less_eq_real @ B2 @ A2 ) ) ) ) ).
% le_divide_eq_1
thf(fact_1010_div__self,axiom,
! [A2: real] :
( ( A2 != zero_zero_real )
=> ( ( divide_divide_real @ A2 @ A2 )
= one_one_real ) ) ).
% div_self
thf(fact_1011_div__self,axiom,
! [A2: nat] :
( ( A2 != zero_zero_nat )
=> ( ( divide_divide_nat @ A2 @ A2 )
= one_one_nat ) ) ).
% div_self
thf(fact_1012_div__self,axiom,
! [A2: int] :
( ( A2 != zero_zero_int )
=> ( ( divide_divide_int @ A2 @ A2 )
= one_one_int ) ) ).
% div_self
thf(fact_1013_one__less__numeral,axiom,
! [N: num] :
( ( ord_le7381754540660121996nnreal @ one_on2969667320475766781nnreal @ ( numera4658534427948366547nnreal @ N ) )
= ( ord_less_num @ one @ N ) ) ).
% one_less_numeral
thf(fact_1014_pos2,axiom,
ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ).
% pos2
thf(fact_1015_div__by__1,axiom,
! [A2: real] :
( ( divide_divide_real @ A2 @ one_one_real )
= A2 ) ).
% div_by_1
thf(fact_1016_div__by__1,axiom,
! [A2: nat] :
( ( divide_divide_nat @ A2 @ one_one_nat )
= A2 ) ).
% div_by_1
thf(fact_1017_div__by__1,axiom,
! [A2: int] :
( ( divide_divide_int @ A2 @ one_one_int )
= A2 ) ).
% div_by_1
thf(fact_1018_div__by__0,axiom,
! [A2: real] :
( ( divide_divide_real @ A2 @ zero_zero_real )
= zero_zero_real ) ).
% div_by_0
thf(fact_1019_div__by__0,axiom,
! [A2: nat] :
( ( divide_divide_nat @ A2 @ zero_zero_nat )
= zero_zero_nat ) ).
% div_by_0
thf(fact_1020_div__by__0,axiom,
! [A2: int] :
( ( divide_divide_int @ A2 @ zero_zero_int )
= zero_zero_int ) ).
% div_by_0
thf(fact_1021_div__0,axiom,
! [A2: real] :
( ( divide_divide_real @ zero_zero_real @ A2 )
= zero_zero_real ) ).
% div_0
thf(fact_1022_div__0,axiom,
! [A2: nat] :
( ( divide_divide_nat @ zero_zero_nat @ A2 )
= zero_zero_nat ) ).
% div_0
thf(fact_1023_div__0,axiom,
! [A2: int] :
( ( divide_divide_int @ zero_zero_int @ A2 )
= zero_zero_int ) ).
% div_0
thf(fact_1024_one__divide__one__divide__ennreal,axiom,
! [C: extend8495563244428889912nnreal] :
( ( divide4826598186094686858nnreal @ one_on2969667320475766781nnreal @ ( divide4826598186094686858nnreal @ one_on2969667320475766781nnreal @ C ) )
= C ) ).
% one_divide_one_divide_ennreal
thf(fact_1025_ennreal__zero__less__one,axiom,
ord_le7381754540660121996nnreal @ zero_z7100319975126383169nnreal @ one_on2969667320475766781nnreal ).
% ennreal_zero_less_one
thf(fact_1026_linorder__neqE__linordered__idom,axiom,
! [X3: real,Y4: real] :
( ( X3 != Y4 )
=> ( ~ ( ord_less_real @ X3 @ Y4 )
=> ( ord_less_real @ Y4 @ X3 ) ) ) ).
% linorder_neqE_linordered_idom
thf(fact_1027_linorder__neqE__linordered__idom,axiom,
! [X3: int,Y4: int] :
( ( X3 != Y4 )
=> ( ~ ( ord_less_int @ X3 @ Y4 )
=> ( ord_less_int @ Y4 @ X3 ) ) ) ).
% linorder_neqE_linordered_idom
thf(fact_1028_zero__neq__one,axiom,
zero_z5982384998485459395l_num1 != one_on3868389512446148991l_num1 ).
% zero_neq_one
thf(fact_1029_zero__neq__one,axiom,
zero_zero_int != one_one_int ).
% zero_neq_one
thf(fact_1030_zero__neq__one,axiom,
zero_zero_nat != one_one_nat ).
% zero_neq_one
thf(fact_1031_zero__neq__one,axiom,
zero_z5237406670263579293d_enat != one_on7984719198319812577d_enat ).
% zero_neq_one
thf(fact_1032_zero__neq__one,axiom,
zero_z7100319975126383169nnreal != one_on2969667320475766781nnreal ).
% zero_neq_one
thf(fact_1033_zero__neq__one,axiom,
zero_zero_real != one_one_real ).
% zero_neq_one
thf(fact_1034_zero__less__one__class_Ozero__le__one,axiom,
ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ one_on7984719198319812577d_enat ).
% zero_less_one_class.zero_le_one
thf(fact_1035_zero__less__one__class_Ozero__le__one,axiom,
ord_le3935885782089961368nnreal @ zero_z7100319975126383169nnreal @ one_on2969667320475766781nnreal ).
% zero_less_one_class.zero_le_one
thf(fact_1036_zero__less__one__class_Ozero__le__one,axiom,
ord_less_eq_real @ zero_zero_real @ one_one_real ).
% zero_less_one_class.zero_le_one
thf(fact_1037_zero__less__one__class_Ozero__le__one,axiom,
ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one_class.zero_le_one
thf(fact_1038_zero__less__one__class_Ozero__le__one,axiom,
ord_less_eq_int @ zero_zero_int @ one_one_int ).
% zero_less_one_class.zero_le_one
thf(fact_1039_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ one_on7984719198319812577d_enat ).
% linordered_nonzero_semiring_class.zero_le_one
thf(fact_1040_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
ord_le3935885782089961368nnreal @ zero_z7100319975126383169nnreal @ one_on2969667320475766781nnreal ).
% linordered_nonzero_semiring_class.zero_le_one
thf(fact_1041_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
ord_less_eq_real @ zero_zero_real @ one_one_real ).
% linordered_nonzero_semiring_class.zero_le_one
thf(fact_1042_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% linordered_nonzero_semiring_class.zero_le_one
thf(fact_1043_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
ord_less_eq_int @ zero_zero_int @ one_one_int ).
% linordered_nonzero_semiring_class.zero_le_one
thf(fact_1044_not__one__le__zero,axiom,
~ ( ord_le2932123472753598470d_enat @ one_on7984719198319812577d_enat @ zero_z5237406670263579293d_enat ) ).
% not_one_le_zero
thf(fact_1045_not__one__le__zero,axiom,
~ ( ord_le3935885782089961368nnreal @ one_on2969667320475766781nnreal @ zero_z7100319975126383169nnreal ) ).
% not_one_le_zero
thf(fact_1046_not__one__le__zero,axiom,
~ ( ord_less_eq_real @ one_one_real @ zero_zero_real ) ).
% not_one_le_zero
thf(fact_1047_not__one__le__zero,axiom,
~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_le_zero
thf(fact_1048_not__one__le__zero,axiom,
~ ( ord_less_eq_int @ one_one_int @ zero_zero_int ) ).
% not_one_le_zero
thf(fact_1049_zero__less__one,axiom,
ord_le7381754540660121996nnreal @ zero_z7100319975126383169nnreal @ one_on2969667320475766781nnreal ).
% zero_less_one
thf(fact_1050_zero__less__one,axiom,
ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ one_on7984719198319812577d_enat ).
% zero_less_one
thf(fact_1051_zero__less__one,axiom,
ord_less_real @ zero_zero_real @ one_one_real ).
% zero_less_one
thf(fact_1052_zero__less__one,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one
thf(fact_1053_zero__less__one,axiom,
ord_less_int @ zero_zero_int @ one_one_int ).
% zero_less_one
thf(fact_1054_not__one__less__zero,axiom,
~ ( ord_le7381754540660121996nnreal @ one_on2969667320475766781nnreal @ zero_z7100319975126383169nnreal ) ).
% not_one_less_zero
thf(fact_1055_not__one__less__zero,axiom,
~ ( ord_le72135733267957522d_enat @ one_on7984719198319812577d_enat @ zero_z5237406670263579293d_enat ) ).
% not_one_less_zero
thf(fact_1056_not__one__less__zero,axiom,
~ ( ord_less_real @ one_one_real @ zero_zero_real ) ).
% not_one_less_zero
thf(fact_1057_not__one__less__zero,axiom,
~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_less_zero
thf(fact_1058_not__one__less__zero,axiom,
~ ( ord_less_int @ one_one_int @ zero_zero_int ) ).
% not_one_less_zero
thf(fact_1059_not__gr__zero,axiom,
! [N: extend8495563244428889912nnreal] :
( ( ~ ( ord_le7381754540660121996nnreal @ zero_z7100319975126383169nnreal @ N ) )
= ( N = zero_z7100319975126383169nnreal ) ) ).
% not_gr_zero
thf(fact_1060_not__gr__zero,axiom,
! [N: extended_enat] :
( ( ~ ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ N ) )
= ( N = zero_z5237406670263579293d_enat ) ) ).
% not_gr_zero
thf(fact_1061_not__gr__zero,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_1062_le__zero__eq,axiom,
! [N: extended_enat] :
( ( ord_le2932123472753598470d_enat @ N @ zero_z5237406670263579293d_enat )
= ( N = zero_z5237406670263579293d_enat ) ) ).
% le_zero_eq
thf(fact_1063_le__zero__eq,axiom,
! [N: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ N @ zero_z7100319975126383169nnreal )
= ( N = zero_z7100319975126383169nnreal ) ) ).
% le_zero_eq
thf(fact_1064_le__zero__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_1065_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
thf(fact_1066_kuhn__labelling__lemma_H,axiom,
! [P: ( nat > real ) > $o,F: ( nat > real ) > nat > real,Q: nat > $o] :
( ! [X5: nat > real] :
( ( P @ X5 )
=> ( P @ ( F @ X5 ) ) )
=> ( ! [X5: nat > real] :
( ( P @ X5 )
=> ! [I4: nat] :
( ( Q @ I4 )
=> ( ( ord_less_eq_real @ zero_zero_real @ ( X5 @ I4 ) )
& ( ord_less_eq_real @ ( X5 @ I4 ) @ one_one_real ) ) ) )
=> ? [L2: ( nat > real ) > nat > nat] :
( ! [X7: nat > real,I6: nat] : ( ord_less_eq_nat @ ( L2 @ X7 @ I6 ) @ one_one_nat )
& ! [X7: nat > real,I6: nat] :
( ( ( P @ X7 )
& ( Q @ I6 )
& ( ( X7 @ I6 )
= zero_zero_real ) )
=> ( ( L2 @ X7 @ I6 )
= zero_zero_nat ) )
& ! [X7: nat > real,I6: nat] :
( ( ( P @ X7 )
& ( Q @ I6 )
& ( ( X7 @ I6 )
= one_one_real ) )
=> ( ( L2 @ X7 @ I6 )
= one_one_nat ) )
& ! [X7: nat > real,I6: nat] :
( ( ( P @ X7 )
& ( Q @ I6 )
& ( ( L2 @ X7 @ I6 )
= zero_zero_nat ) )
=> ( ord_less_eq_real @ ( X7 @ I6 ) @ ( F @ X7 @ I6 ) ) )
& ! [X7: nat > real,I6: nat] :
( ( ( P @ X7 )
& ( Q @ I6 )
& ( ( L2 @ X7 @ I6 )
= one_one_nat ) )
=> ( ord_less_eq_real @ ( F @ X7 @ I6 ) @ ( X7 @ I6 ) ) ) ) ) ) ).
% kuhn_labelling_lemma'
thf(fact_1067_ennreal__zero__divide,axiom,
! [X3: extend8495563244428889912nnreal] :
( ( divide4826598186094686858nnreal @ zero_z7100319975126383169nnreal @ X3 )
= zero_z7100319975126383169nnreal ) ).
% ennreal_zero_divide
thf(fact_1068_zero__reorient,axiom,
! [X3: int] :
( ( zero_zero_int = X3 )
= ( X3 = zero_zero_int ) ) ).
% zero_reorient
thf(fact_1069_zero__reorient,axiom,
! [X3: nat] :
( ( zero_zero_nat = X3 )
= ( X3 = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_1070_zero__reorient,axiom,
! [X3: extended_enat] :
( ( zero_z5237406670263579293d_enat = X3 )
= ( X3 = zero_z5237406670263579293d_enat ) ) ).
% zero_reorient
thf(fact_1071_zero__reorient,axiom,
! [X3: extend8495563244428889912nnreal] :
( ( zero_z7100319975126383169nnreal = X3 )
= ( X3 = zero_z7100319975126383169nnreal ) ) ).
% zero_reorient
thf(fact_1072_zero__reorient,axiom,
! [X3: real] :
( ( zero_zero_real = X3 )
= ( X3 = zero_zero_real ) ) ).
% zero_reorient
thf(fact_1073_one__reorient,axiom,
! [X3: numera2417102609627094330l_num1] :
( ( one_on3868389512446148991l_num1 = X3 )
= ( X3 = one_on3868389512446148991l_num1 ) ) ).
% one_reorient
thf(fact_1074_one__reorient,axiom,
! [X3: extended_enat] :
( ( one_on7984719198319812577d_enat = X3 )
= ( X3 = one_on7984719198319812577d_enat ) ) ).
% one_reorient
thf(fact_1075_one__reorient,axiom,
! [X3: real] :
( ( one_one_real = X3 )
= ( X3 = one_one_real ) ) ).
% one_reorient
thf(fact_1076_one__reorient,axiom,
! [X3: nat] :
( ( one_one_nat = X3 )
= ( X3 = one_one_nat ) ) ).
% one_reorient
thf(fact_1077_one__reorient,axiom,
! [X3: int] :
( ( one_one_int = X3 )
= ( X3 = one_one_int ) ) ).
% one_reorient
thf(fact_1078_zero__le,axiom,
! [X3: extended_enat] : ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ X3 ) ).
% zero_le
thf(fact_1079_zero__le,axiom,
! [X3: extend8495563244428889912nnreal] : ( ord_le3935885782089961368nnreal @ zero_z7100319975126383169nnreal @ X3 ) ).
% zero_le
thf(fact_1080_zero__le,axiom,
! [X3: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X3 ) ).
% zero_le
thf(fact_1081_zero__less__iff__neq__zero,axiom,
! [N: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ zero_z7100319975126383169nnreal @ N )
= ( N != zero_z7100319975126383169nnreal ) ) ).
% zero_less_iff_neq_zero
thf(fact_1082_zero__less__iff__neq__zero,axiom,
! [N: extended_enat] :
( ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ N )
= ( N != zero_z5237406670263579293d_enat ) ) ).
% zero_less_iff_neq_zero
thf(fact_1083_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_1084_gr__implies__not__zero,axiom,
! [M: extend8495563244428889912nnreal,N: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ M @ N )
=> ( N != zero_z7100319975126383169nnreal ) ) ).
% gr_implies_not_zero
thf(fact_1085_gr__implies__not__zero,axiom,
! [M: extended_enat,N: extended_enat] :
( ( ord_le72135733267957522d_enat @ M @ N )
=> ( N != zero_z5237406670263579293d_enat ) ) ).
% gr_implies_not_zero
thf(fact_1086_gr__implies__not__zero,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_1087_not__less__zero,axiom,
! [N: extend8495563244428889912nnreal] :
~ ( ord_le7381754540660121996nnreal @ N @ zero_z7100319975126383169nnreal ) ).
% not_less_zero
thf(fact_1088_not__less__zero,axiom,
! [N: extended_enat] :
~ ( ord_le72135733267957522d_enat @ N @ zero_z5237406670263579293d_enat ) ).
% not_less_zero
thf(fact_1089_not__less__zero,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less_zero
thf(fact_1090_gr__zeroI,axiom,
! [N: extend8495563244428889912nnreal] :
( ( N != zero_z7100319975126383169nnreal )
=> ( ord_le7381754540660121996nnreal @ zero_z7100319975126383169nnreal @ N ) ) ).
% gr_zeroI
thf(fact_1091_gr__zeroI,axiom,
! [N: extended_enat] :
( ( N != zero_z5237406670263579293d_enat )
=> ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ N ) ) ).
% gr_zeroI
thf(fact_1092_gr__zeroI,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr_zeroI
thf(fact_1093_less__eq__int__code_I1_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% less_eq_int_code(1)
thf(fact_1094_less__int__code_I1_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_int_code(1)
thf(fact_1095_size__char__eq__0,axiom,
( size_size_char
= ( ^ [C3: char] : zero_zero_nat ) ) ).
% size_char_eq_0
thf(fact_1096_mod2__gr__0,axiom,
! [M: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( modulo_modulo_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= ( ( modulo_modulo_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= one_one_nat ) ) ).
% mod2_gr_0
thf(fact_1097_seq__mono__lemma,axiom,
! [M: nat,D2: nat > real,E: nat > real] :
( ! [N3: nat] :
( ( ord_less_eq_nat @ M @ N3 )
=> ( ord_less_real @ ( D2 @ N3 ) @ ( E @ N3 ) ) )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ M @ N3 )
=> ( ord_less_eq_real @ ( E @ N3 ) @ ( E @ M ) ) )
=> ! [N4: nat] :
( ( ord_less_eq_nat @ M @ N4 )
=> ( ord_less_real @ ( D2 @ N4 ) @ ( E @ M ) ) ) ) ) ).
% seq_mono_lemma
thf(fact_1098_mod__mod__trivial,axiom,
! [A2: nat,B2: nat] :
( ( modulo_modulo_nat @ ( modulo_modulo_nat @ A2 @ B2 ) @ B2 )
= ( modulo_modulo_nat @ A2 @ B2 ) ) ).
% mod_mod_trivial
thf(fact_1099_mod__mod__trivial,axiom,
! [A2: int,B2: int] :
( ( modulo_modulo_int @ ( modulo_modulo_int @ A2 @ B2 ) @ B2 )
= ( modulo_modulo_int @ A2 @ B2 ) ) ).
% mod_mod_trivial
thf(fact_1100_mod__self,axiom,
! [A2: nat] :
( ( modulo_modulo_nat @ A2 @ A2 )
= zero_zero_nat ) ).
% mod_self
thf(fact_1101_mod__self,axiom,
! [A2: int] :
( ( modulo_modulo_int @ A2 @ A2 )
= zero_zero_int ) ).
% mod_self
thf(fact_1102_mod__by__0,axiom,
! [A2: nat] :
( ( modulo_modulo_nat @ A2 @ zero_zero_nat )
= A2 ) ).
% mod_by_0
thf(fact_1103_mod__by__0,axiom,
! [A2: int] :
( ( modulo_modulo_int @ A2 @ zero_zero_int )
= A2 ) ).
% mod_by_0
thf(fact_1104_mod__0,axiom,
! [A2: nat] :
( ( modulo_modulo_nat @ zero_zero_nat @ A2 )
= zero_zero_nat ) ).
% mod_0
thf(fact_1105_mod__0,axiom,
! [A2: int] :
( ( modulo_modulo_int @ zero_zero_int @ A2 )
= zero_zero_int ) ).
% mod_0
thf(fact_1106_bits__mod__0,axiom,
! [A2: nat] :
( ( modulo_modulo_nat @ zero_zero_nat @ A2 )
= zero_zero_nat ) ).
% bits_mod_0
thf(fact_1107_bits__mod__0,axiom,
! [A2: int] :
( ( modulo_modulo_int @ zero_zero_int @ A2 )
= zero_zero_int ) ).
% bits_mod_0
thf(fact_1108_mod__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ( modulo_modulo_nat @ M @ N )
= M ) ) ).
% mod_less
thf(fact_1109_mod__by__1,axiom,
! [A2: nat] :
( ( modulo_modulo_nat @ A2 @ one_one_nat )
= zero_zero_nat ) ).
% mod_by_1
thf(fact_1110_mod__by__1,axiom,
! [A2: int] :
( ( modulo_modulo_int @ A2 @ one_one_int )
= zero_zero_int ) ).
% mod_by_1
thf(fact_1111_bits__mod__by__1,axiom,
! [A2: nat] :
( ( modulo_modulo_nat @ A2 @ one_one_nat )
= zero_zero_nat ) ).
% bits_mod_by_1
thf(fact_1112_bits__mod__by__1,axiom,
! [A2: int] :
( ( modulo_modulo_int @ A2 @ one_one_int )
= zero_zero_int ) ).
% bits_mod_by_1
thf(fact_1113_mod__div__trivial,axiom,
! [A2: nat,B2: nat] :
( ( divide_divide_nat @ ( modulo_modulo_nat @ A2 @ B2 ) @ B2 )
= zero_zero_nat ) ).
% mod_div_trivial
thf(fact_1114_mod__div__trivial,axiom,
! [A2: int,B2: int] :
( ( divide_divide_int @ ( modulo_modulo_int @ A2 @ B2 ) @ B2 )
= zero_zero_int ) ).
% mod_div_trivial
thf(fact_1115_bits__mod__div__trivial,axiom,
! [A2: nat,B2: nat] :
( ( divide_divide_nat @ ( modulo_modulo_nat @ A2 @ B2 ) @ B2 )
= zero_zero_nat ) ).
% bits_mod_div_trivial
thf(fact_1116_bits__mod__div__trivial,axiom,
! [A2: int,B2: int] :
( ( divide_divide_int @ ( modulo_modulo_int @ A2 @ B2 ) @ B2 )
= zero_zero_int ) ).
% bits_mod_div_trivial
thf(fact_1117_one__mod__two__eq__one,axiom,
( ( modulo_modulo_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= one_one_nat ) ).
% one_mod_two_eq_one
thf(fact_1118_one__mod__two__eq__one,axiom,
( ( modulo_modulo_int @ one_one_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= one_one_int ) ).
% one_mod_two_eq_one
thf(fact_1119_bits__one__mod__two__eq__one,axiom,
( ( modulo_modulo_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= one_one_nat ) ).
% bits_one_mod_two_eq_one
thf(fact_1120_bits__one__mod__two__eq__one,axiom,
( ( modulo_modulo_int @ one_one_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= one_one_int ) ).
% bits_one_mod_two_eq_one
thf(fact_1121_mod__less__eq__dividend,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( modulo_modulo_nat @ M @ N ) @ M ) ).
% mod_less_eq_dividend
thf(fact_1122_mod__less__divisor,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_nat @ ( modulo_modulo_nat @ M @ N ) @ N ) ) ).
% mod_less_divisor
thf(fact_1123_mod__le__divisor,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_eq_nat @ ( modulo_modulo_nat @ M @ N ) @ N ) ) ).
% mod_le_divisor
thf(fact_1124_div__less__mono,axiom,
! [A3: nat,B5: nat,N: nat] :
( ( ord_less_nat @ A3 @ B5 )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ( modulo_modulo_nat @ A3 @ N )
= zero_zero_nat )
=> ( ( ( modulo_modulo_nat @ B5 @ N )
= zero_zero_nat )
=> ( ord_less_nat @ ( divide_divide_nat @ A3 @ N ) @ ( divide_divide_nat @ B5 @ N ) ) ) ) ) ) ).
% div_less_mono
thf(fact_1125_size_H__char__eq__0,axiom,
( size_char
= ( ^ [C3: char] : zero_zero_nat ) ) ).
% size'_char_eq_0
thf(fact_1126_gcd__nat__induct,axiom,
! [P: nat > nat > $o,M: nat,N: nat] :
( ! [M4: nat] : ( P @ M4 @ zero_zero_nat )
=> ( ! [M4: nat,N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ( P @ N3 @ ( modulo_modulo_nat @ M4 @ N3 ) )
=> ( P @ M4 @ N3 ) ) )
=> ( P @ M @ N ) ) ) ).
% gcd_nat_induct
thf(fact_1127_mod__pos__pos__trivial,axiom,
! [K: int,L: int] :
( ( ord_less_eq_int @ zero_zero_int @ K )
=> ( ( ord_less_int @ K @ L )
=> ( ( modulo_modulo_int @ K @ L )
= K ) ) ) ).
% mod_pos_pos_trivial
thf(fact_1128_mod__neg__neg__trivial,axiom,
! [K: int,L: int] :
( ( ord_less_eq_int @ K @ zero_zero_int )
=> ( ( ord_less_int @ L @ K )
=> ( ( modulo_modulo_int @ K @ L )
= K ) ) ) ).
% mod_neg_neg_trivial
thf(fact_1129_zmod__le__nonneg__dividend,axiom,
! [M: int,K: int] :
( ( ord_less_eq_int @ zero_zero_int @ M )
=> ( ord_less_eq_int @ ( modulo_modulo_int @ M @ K ) @ M ) ) ).
% zmod_le_nonneg_dividend
thf(fact_1130_Euclidean__Division_Opos__mod__bound,axiom,
! [L: int,K: int] :
( ( ord_less_int @ zero_zero_int @ L )
=> ( ord_less_int @ ( modulo_modulo_int @ K @ L ) @ L ) ) ).
% Euclidean_Division.pos_mod_bound
thf(fact_1131_neg__mod__bound,axiom,
! [L: int,K: int] :
( ( ord_less_int @ L @ zero_zero_int )
=> ( ord_less_int @ L @ ( modulo_modulo_int @ K @ L ) ) ) ).
% neg_mod_bound
thf(fact_1132_zmod__trivial__iff,axiom,
! [I5: int,K: int] :
( ( ( modulo_modulo_int @ I5 @ K )
= I5 )
= ( ( K = zero_zero_int )
| ( ( ord_less_eq_int @ zero_zero_int @ I5 )
& ( ord_less_int @ I5 @ K ) )
| ( ( ord_less_eq_int @ I5 @ zero_zero_int )
& ( ord_less_int @ K @ I5 ) ) ) ) ).
% zmod_trivial_iff
thf(fact_1133_Euclidean__Division_Opos__mod__sign,axiom,
! [L: int,K: int] :
( ( ord_less_int @ zero_zero_int @ L )
=> ( ord_less_eq_int @ zero_zero_int @ ( modulo_modulo_int @ K @ L ) ) ) ).
% Euclidean_Division.pos_mod_sign
thf(fact_1134_neg__mod__sign,axiom,
! [L: int,K: int] :
( ( ord_less_int @ L @ zero_zero_int )
=> ( ord_less_eq_int @ ( modulo_modulo_int @ K @ L ) @ zero_zero_int ) ) ).
% neg_mod_sign
thf(fact_1135_zdiv__mono__strict,axiom,
! [A3: int,B5: int,N: int] :
( ( ord_less_int @ A3 @ B5 )
=> ( ( ord_less_int @ zero_zero_int @ N )
=> ( ( ( modulo_modulo_int @ A3 @ N )
= zero_zero_int )
=> ( ( ( modulo_modulo_int @ B5 @ N )
= zero_zero_int )
=> ( ord_less_int @ ( divide_divide_int @ A3 @ N ) @ ( divide_divide_int @ B5 @ N ) ) ) ) ) ) ).
% zdiv_mono_strict
thf(fact_1136_nat__of__char__less__256,axiom,
! [C: char] : ( ord_less_nat @ ( comm_s629917340098488124ar_nat @ C ) @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).
% nat_of_char_less_256
thf(fact_1137_forall__2,axiom,
( ( ^ [P2: numera2417102609627094330l_num1 > $o] :
! [X6: numera2417102609627094330l_num1] : ( P2 @ X6 ) )
= ( ^ [P3: numera2417102609627094330l_num1 > $o] :
( ( P3 @ one_on3868389512446148991l_num1 )
& ( P3 @ ( numera2161328050825114965l_num1 @ ( bit0 @ one ) ) ) ) ) ) ).
% forall_2
thf(fact_1138_exhaust__2,axiom,
! [X3: numera2417102609627094330l_num1] :
( ( X3 = one_on3868389512446148991l_num1 )
| ( X3
= ( numera2161328050825114965l_num1 @ ( bit0 @ one ) ) ) ) ).
% exhaust_2
thf(fact_1139_verit__le__mono__div__int,axiom,
! [A3: int,B5: int,N: int] :
( ( ord_less_int @ A3 @ B5 )
=> ( ( ord_less_int @ zero_zero_int @ N )
=> ( ord_less_eq_int
@ ( plus_plus_int @ ( divide_divide_int @ A3 @ N )
@ ( if_int
@ ( ( modulo_modulo_int @ B5 @ N )
= zero_zero_int )
@ one_one_int
@ zero_zero_int ) )
@ ( divide_divide_int @ B5 @ N ) ) ) ) ).
% verit_le_mono_div_int
thf(fact_1140_zle__add1__eq__le,axiom,
! [W: int,Z3: int] :
( ( ord_less_int @ W @ ( plus_plus_int @ Z3 @ one_one_int ) )
= ( ord_less_eq_int @ W @ Z3 ) ) ).
% zle_add1_eq_le
thf(fact_1141_plus__int__code_I1_J,axiom,
! [K: int] :
( ( plus_plus_int @ K @ zero_zero_int )
= K ) ).
% plus_int_code(1)
thf(fact_1142_plus__int__code_I2_J,axiom,
! [L: int] :
( ( plus_plus_int @ zero_zero_int @ L )
= L ) ).
% plus_int_code(2)
thf(fact_1143_uminus__int__code_I1_J,axiom,
( ( uminus_uminus_int @ zero_zero_int )
= zero_zero_int ) ).
% uminus_int_code(1)
thf(fact_1144_odd__nonzero,axiom,
! [Z3: int] :
( ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z3 ) @ Z3 )
!= zero_zero_int ) ).
% odd_nonzero
thf(fact_1145_int__ge__induct,axiom,
! [K: int,I5: int,P: int > $o] :
( ( ord_less_eq_int @ K @ I5 )
=> ( ( P @ K )
=> ( ! [I4: int] :
( ( ord_less_eq_int @ K @ I4 )
=> ( ( P @ I4 )
=> ( P @ ( plus_plus_int @ I4 @ one_one_int ) ) ) )
=> ( P @ I5 ) ) ) ) ).
% int_ge_induct
thf(fact_1146_int__gr__induct,axiom,
! [K: int,I5: int,P: int > $o] :
( ( ord_less_int @ K @ I5 )
=> ( ( P @ ( plus_plus_int @ K @ one_one_int ) )
=> ( ! [I4: int] :
( ( ord_less_int @ K @ I4 )
=> ( ( P @ I4 )
=> ( P @ ( plus_plus_int @ I4 @ one_one_int ) ) ) )
=> ( P @ I5 ) ) ) ) ).
% int_gr_induct
thf(fact_1147_zless__add1__eq,axiom,
! [W: int,Z3: int] :
( ( ord_less_int @ W @ ( plus_plus_int @ Z3 @ one_one_int ) )
= ( ( ord_less_int @ W @ Z3 )
| ( W = Z3 ) ) ) ).
% zless_add1_eq
thf(fact_1148_zmod__zminus1__not__zero,axiom,
! [K: int,L: int] :
( ( ( modulo_modulo_int @ ( uminus_uminus_int @ K ) @ L )
!= zero_zero_int )
=> ( ( modulo_modulo_int @ K @ L )
!= zero_zero_int ) ) ).
% zmod_zminus1_not_zero
thf(fact_1149_zmod__zminus2__not__zero,axiom,
! [K: int,L: int] :
( ( ( modulo_modulo_int @ K @ ( uminus_uminus_int @ L ) )
!= zero_zero_int )
=> ( ( modulo_modulo_int @ K @ L )
!= zero_zero_int ) ) ).
% zmod_zminus2_not_zero
thf(fact_1150_div__pos__neg__trivial,axiom,
! [K: int,L: int] :
( ( ord_less_int @ zero_zero_int @ K )
=> ( ( ord_less_eq_int @ ( plus_plus_int @ K @ L ) @ zero_zero_int )
=> ( ( divide_divide_int @ K @ L )
= ( uminus_uminus_int @ one_one_int ) ) ) ) ).
% div_pos_neg_trivial
thf(fact_1151_odd__less__0__iff,axiom,
! [Z3: int] :
( ( ord_less_int @ ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z3 ) @ Z3 ) @ zero_zero_int )
= ( ord_less_int @ Z3 @ zero_zero_int ) ) ).
% odd_less_0_iff
thf(fact_1152_zless__imp__add1__zle,axiom,
! [W: int,Z3: int] :
( ( ord_less_int @ W @ Z3 )
=> ( ord_less_eq_int @ ( plus_plus_int @ W @ one_one_int ) @ Z3 ) ) ).
% zless_imp_add1_zle
thf(fact_1153_add1__zle__eq,axiom,
! [W: int,Z3: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ W @ one_one_int ) @ Z3 )
= ( ord_less_int @ W @ Z3 ) ) ).
% add1_zle_eq
thf(fact_1154_le__imp__0__less,axiom,
! [Z3: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z3 )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ Z3 ) ) ) ).
% le_imp_0_less
thf(fact_1155_div__eq__minus1,axiom,
! [B2: int] :
( ( ord_less_int @ zero_zero_int @ B2 )
=> ( ( divide_divide_int @ ( uminus_uminus_int @ one_one_int ) @ B2 )
= ( uminus_uminus_int @ one_one_int ) ) ) ).
% div_eq_minus1
thf(fact_1156_verit__less__mono__div__int2,axiom,
! [A3: int,B5: int,N: int] :
( ( ord_less_eq_int @ A3 @ B5 )
=> ( ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ N ) )
=> ( ord_less_eq_int @ ( divide_divide_int @ B5 @ N ) @ ( divide_divide_int @ A3 @ N ) ) ) ) ).
% verit_less_mono_div_int2
thf(fact_1157_mod__pos__neg__trivial,axiom,
! [K: int,L: int] :
( ( ord_less_int @ zero_zero_int @ K )
=> ( ( ord_less_eq_int @ ( plus_plus_int @ K @ L ) @ zero_zero_int )
=> ( ( modulo_modulo_int @ K @ L )
= ( plus_plus_int @ K @ L ) ) ) ) ).
% mod_pos_neg_trivial
thf(fact_1158_real__add__minus__iff,axiom,
! [X3: real,A2: real] :
( ( ( plus_plus_real @ X3 @ ( uminus_uminus_real @ A2 ) )
= zero_zero_real )
= ( X3 = A2 ) ) ).
% real_add_minus_iff
thf(fact_1159_Nat_Oadd__0__right,axiom,
! [M: nat] :
( ( plus_plus_nat @ M @ zero_zero_nat )
= M ) ).
% Nat.add_0_right
thf(fact_1160_add__is__0,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus_nat @ M @ N )
= zero_zero_nat )
= ( ( M = zero_zero_nat )
& ( N = zero_zero_nat ) ) ) ).
% add_is_0
thf(fact_1161_nat__add__left__cancel__le,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% nat_add_left_cancel_le
thf(fact_1162_nat__add__left__cancel__less,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% nat_add_left_cancel_less
thf(fact_1163_add__gr__0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ M )
| ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% add_gr_0
thf(fact_1164_semiring__norm_I6_J,axiom,
! [M: num,N: num] :
( ( plus_plus_num @ ( bit0 @ M ) @ ( bit0 @ N ) )
= ( bit0 @ ( plus_plus_num @ M @ N ) ) ) ).
% semiring_norm(6)
thf(fact_1165_semiring__norm_I2_J,axiom,
( ( plus_plus_num @ one @ one )
= ( bit0 @ one ) ) ).
% semiring_norm(2)
thf(fact_1166_add__self__div__2,axiom,
! [M: nat] :
( ( divide_divide_nat @ ( plus_plus_nat @ M @ M ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= M ) ).
% add_self_div_2
thf(fact_1167_add__self__mod__2,axiom,
! [M: nat] :
( ( modulo_modulo_nat @ ( plus_plus_nat @ M @ M ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= zero_zero_nat ) ).
% add_self_mod_2
thf(fact_1168_signed__take__bit__minus,axiom,
! [N: nat,K: int] :
( ( bit_ri631733984087533419it_int @ N @ ( uminus_uminus_int @ ( bit_ri631733984087533419it_int @ N @ K ) ) )
= ( bit_ri631733984087533419it_int @ N @ ( uminus_uminus_int @ K ) ) ) ).
% signed_take_bit_minus
thf(fact_1169_signed__take__bit__add,axiom,
! [N: nat,K: int,L: int] :
( ( bit_ri631733984087533419it_int @ N @ ( plus_plus_int @ ( bit_ri631733984087533419it_int @ N @ K ) @ ( bit_ri631733984087533419it_int @ N @ L ) ) )
= ( bit_ri631733984087533419it_int @ N @ ( plus_plus_int @ K @ L ) ) ) ).
% signed_take_bit_add
thf(fact_1170_less__add__eq__less,axiom,
! [K: nat,L: nat,M: nat,N: nat] :
( ( ord_less_nat @ K @ L )
=> ( ( ( plus_plus_nat @ M @ L )
= ( plus_plus_nat @ K @ N ) )
=> ( ord_less_nat @ M @ N ) ) ) ).
% less_add_eq_less
thf(fact_1171_trans__less__add2,axiom,
! [I5: nat,J: nat,M: nat] :
( ( ord_less_nat @ I5 @ J )
=> ( ord_less_nat @ I5 @ ( plus_plus_nat @ M @ J ) ) ) ).
% trans_less_add2
thf(fact_1172_trans__less__add1,axiom,
! [I5: nat,J: nat,M: nat] :
( ( ord_less_nat @ I5 @ J )
=> ( ord_less_nat @ I5 @ ( plus_plus_nat @ J @ M ) ) ) ).
% trans_less_add1
thf(fact_1173_add__less__mono1,axiom,
! [I5: nat,J: nat,K: nat] :
( ( ord_less_nat @ I5 @ J )
=> ( ord_less_nat @ ( plus_plus_nat @ I5 @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% add_less_mono1
thf(fact_1174_not__add__less2,axiom,
! [J: nat,I5: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ J @ I5 ) @ I5 ) ).
% not_add_less2
thf(fact_1175_not__add__less1,axiom,
! [I5: nat,J: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ I5 @ J ) @ I5 ) ).
% not_add_less1
thf(fact_1176_add__less__mono,axiom,
! [I5: nat,J: nat,K: nat,L: nat] :
( ( ord_less_nat @ I5 @ J )
=> ( ( ord_less_nat @ K @ L )
=> ( ord_less_nat @ ( plus_plus_nat @ I5 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_less_mono
thf(fact_1177_add__lessD1,axiom,
! [I5: nat,J: nat,K: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ I5 @ J ) @ K )
=> ( ord_less_nat @ I5 @ K ) ) ).
% add_lessD1
thf(fact_1178_nat__le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [M2: nat,N2: nat] :
? [K3: nat] :
( N2
= ( plus_plus_nat @ M2 @ K3 ) ) ) ) ).
% nat_le_iff_add
thf(fact_1179_trans__le__add2,axiom,
! [I5: nat,J: nat,M: nat] :
( ( ord_less_eq_nat @ I5 @ J )
=> ( ord_less_eq_nat @ I5 @ ( plus_plus_nat @ M @ J ) ) ) ).
% trans_le_add2
thf(fact_1180_trans__le__add1,axiom,
! [I5: nat,J: nat,M: nat] :
( ( ord_less_eq_nat @ I5 @ J )
=> ( ord_less_eq_nat @ I5 @ ( plus_plus_nat @ J @ M ) ) ) ).
% trans_le_add1
thf(fact_1181_add__le__mono1,axiom,
! [I5: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I5 @ J )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I5 @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% add_le_mono1
thf(fact_1182_add__le__mono,axiom,
! [I5: nat,J: nat,K: nat,L: nat] :
( ( ord_less_eq_nat @ I5 @ J )
=> ( ( ord_less_eq_nat @ K @ L )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I5 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_le_mono
thf(fact_1183_le__Suc__ex,axiom,
! [K: nat,L: nat] :
( ( ord_less_eq_nat @ K @ L )
=> ? [N3: nat] :
( L
= ( plus_plus_nat @ K @ N3 ) ) ) ).
% le_Suc_ex
thf(fact_1184_add__leD2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ( ord_less_eq_nat @ K @ N ) ) ).
% add_leD2
thf(fact_1185_add__leD1,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% add_leD1
thf(fact_1186_le__add2,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M @ N ) ) ).
% le_add2
thf(fact_1187_le__add1,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M ) ) ).
% le_add1
thf(fact_1188_add__leE,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ~ ( ( ord_less_eq_nat @ M @ N )
=> ~ ( ord_less_eq_nat @ K @ N ) ) ) ).
% add_leE
thf(fact_1189_add__eq__self__zero,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus_nat @ M @ N )
= M )
=> ( N = zero_zero_nat ) ) ).
% add_eq_self_zero
thf(fact_1190_plus__nat_Oadd__0,axiom,
! [N: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N )
= N ) ).
% plus_nat.add_0
thf(fact_1191_add__One__commute,axiom,
! [N: num] :
( ( plus_plus_num @ one @ N )
= ( plus_plus_num @ N @ one ) ) ).
% add_One_commute
thf(fact_1192_Euclid__induct,axiom,
! [P: nat > nat > $o,A2: nat,B2: nat] :
( ! [A5: nat,B4: nat] :
( ( P @ A5 @ B4 )
= ( P @ B4 @ A5 ) )
=> ( ! [A5: nat] : ( P @ A5 @ zero_zero_nat )
=> ( ! [A5: nat,B4: nat] :
( ( P @ A5 @ B4 )
=> ( P @ A5 @ ( plus_plus_nat @ A5 @ B4 ) ) )
=> ( P @ A2 @ B2 ) ) ) ) ).
% Euclid_induct
thf(fact_1193_iadd__is__0,axiom,
! [M: extended_enat,N: extended_enat] :
( ( ( plus_p3455044024723400733d_enat @ M @ N )
= zero_z5237406670263579293d_enat )
= ( ( M = zero_z5237406670263579293d_enat )
& ( N = zero_z5237406670263579293d_enat ) ) ) ).
% iadd_is_0
thf(fact_1194_real__add__le__0__iff,axiom,
! [X3: real,Y4: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ X3 @ Y4 ) @ zero_zero_real )
= ( ord_less_eq_real @ Y4 @ ( uminus_uminus_real @ X3 ) ) ) ).
% real_add_le_0_iff
thf(fact_1195_real__0__le__add__iff,axiom,
! [X3: real,Y4: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ X3 @ Y4 ) )
= ( ord_less_eq_real @ ( uminus_uminus_real @ X3 ) @ Y4 ) ) ).
% real_0_le_add_iff
thf(fact_1196_real__add__less__0__iff,axiom,
! [X3: real,Y4: real] :
( ( ord_less_real @ ( plus_plus_real @ X3 @ Y4 ) @ zero_zero_real )
= ( ord_less_real @ Y4 @ ( uminus_uminus_real @ X3 ) ) ) ).
% real_add_less_0_iff
thf(fact_1197_real__0__less__add__iff,axiom,
! [X3: real,Y4: real] :
( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ X3 @ Y4 ) )
= ( ord_less_real @ ( uminus_uminus_real @ X3 ) @ Y4 ) ) ).
% real_0_less_add_iff
thf(fact_1198_less__imp__add__positive,axiom,
! [I5: nat,J: nat] :
( ( ord_less_nat @ I5 @ J )
=> ? [K2: nat] :
( ( ord_less_nat @ zero_zero_nat @ K2 )
& ( ( plus_plus_nat @ I5 @ K2 )
= J ) ) ) ).
% less_imp_add_positive
thf(fact_1199_mono__nat__linear__lb,axiom,
! [F: nat > nat,M: nat,K: nat] :
( ! [M4: nat,N3: nat] :
( ( ord_less_nat @ M4 @ N3 )
=> ( ord_less_nat @ ( F @ M4 ) @ ( F @ N3 ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M ) @ K ) @ ( F @ ( plus_plus_nat @ M @ K ) ) ) ) ).
% mono_nat_linear_lb
thf(fact_1200_nat__1__add__1,axiom,
( ( plus_plus_nat @ one_one_nat @ one_one_nat )
= ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).
% nat_1_add_1
thf(fact_1201_kuhn__lemma,axiom,
! [P5: nat,N: nat,Label: ( nat > nat ) > nat > nat] :
( ( ord_less_nat @ zero_zero_nat @ P5 )
=> ( ! [X5: nat > nat] :
( ! [I6: nat] :
( ( ord_less_nat @ I6 @ N )
=> ( ord_less_eq_nat @ ( X5 @ I6 ) @ P5 ) )
=> ! [I4: nat] :
( ( ord_less_nat @ I4 @ N )
=> ( ( ( Label @ X5 @ I4 )
= zero_zero_nat )
| ( ( Label @ X5 @ I4 )
= one_one_nat ) ) ) )
=> ( ! [X5: nat > nat] :
( ! [I6: nat] :
( ( ord_less_nat @ I6 @ N )
=> ( ord_less_eq_nat @ ( X5 @ I6 ) @ P5 ) )
=> ! [I4: nat] :
( ( ord_less_nat @ I4 @ N )
=> ( ( ( X5 @ I4 )
= zero_zero_nat )
=> ( ( Label @ X5 @ I4 )
= zero_zero_nat ) ) ) )
=> ( ! [X5: nat > nat] :
( ! [I6: nat] :
( ( ord_less_nat @ I6 @ N )
=> ( ord_less_eq_nat @ ( X5 @ I6 ) @ P5 ) )
=> ! [I4: nat] :
( ( ord_less_nat @ I4 @ N )
=> ( ( ( X5 @ I4 )
= P5 )
=> ( ( Label @ X5 @ I4 )
= one_one_nat ) ) ) )
=> ~ ! [Q3: nat > nat] :
( ! [I6: nat] :
( ( ord_less_nat @ I6 @ N )
=> ( ord_less_nat @ ( Q3 @ I6 ) @ P5 ) )
=> ~ ! [I6: nat] :
( ( ord_less_nat @ I6 @ N )
=> ? [R: nat > nat] :
( ! [J4: nat] :
( ( ord_less_nat @ J4 @ N )
=> ( ( ord_less_eq_nat @ ( Q3 @ J4 ) @ ( R @ J4 ) )
& ( ord_less_eq_nat @ ( R @ J4 ) @ ( plus_plus_nat @ ( Q3 @ J4 ) @ one_one_nat ) ) ) )
& ? [S2: nat > nat] :
( ! [J4: nat] :
( ( ord_less_nat @ J4 @ N )
=> ( ( ord_less_eq_nat @ ( Q3 @ J4 ) @ ( S2 @ J4 ) )
& ( ord_less_eq_nat @ ( S2 @ J4 ) @ ( plus_plus_nat @ ( Q3 @ J4 ) @ one_one_nat ) ) ) )
& ( ( Label @ R @ I6 )
!= ( Label @ S2 @ I6 ) ) ) ) ) ) ) ) ) ) ).
% kuhn_lemma
thf(fact_1202_nat__induct2,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ( P @ one_one_nat )
=> ( ! [N3: nat] :
( ( P @ N3 )
=> ( P @ ( plus_plus_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct2
thf(fact_1203_verit__le__mono__div,axiom,
! [A3: nat,B5: nat,N: nat] :
( ( ord_less_nat @ A3 @ B5 )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_eq_nat
@ ( plus_plus_nat @ ( divide_divide_nat @ A3 @ N )
@ ( if_nat
@ ( ( modulo_modulo_nat @ B5 @ N )
= zero_zero_nat )
@ one_one_nat
@ zero_zero_nat ) )
@ ( divide_divide_nat @ B5 @ N ) ) ) ) ).
% verit_le_mono_div
thf(fact_1204_nat__add__1__add__1,axiom,
! [N: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ one_one_nat )
= ( plus_plus_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% nat_add_1_add_1
thf(fact_1205_pos__zmod__mult__2,axiom,
! [A2: int,B2: int] :
( ( ord_less_eq_int @ zero_zero_int @ A2 )
=> ( ( modulo_modulo_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B2 ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A2 ) )
= ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( modulo_modulo_int @ B2 @ A2 ) ) ) ) ) ).
% pos_zmod_mult_2
thf(fact_1206_zmod__numeral__Bit0,axiom,
! [V2: num,W: num] :
( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ V2 ) ) @ ( numeral_numeral_int @ ( bit0 @ W ) ) )
= ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( modulo_modulo_int @ ( numeral_numeral_int @ V2 ) @ ( numeral_numeral_int @ W ) ) ) ) ).
% zmod_numeral_Bit0
thf(fact_1207_signed__take__bit__mult,axiom,
! [N: nat,K: int,L: int] :
( ( bit_ri631733984087533419it_int @ N @ ( times_times_int @ ( bit_ri631733984087533419it_int @ N @ K ) @ ( bit_ri631733984087533419it_int @ N @ L ) ) )
= ( bit_ri631733984087533419it_int @ N @ ( times_times_int @ K @ L ) ) ) ).
% signed_take_bit_mult
thf(fact_1208_times__int__code_I1_J,axiom,
! [K: int] :
( ( times_times_int @ K @ zero_zero_int )
= zero_zero_int ) ).
% times_int_code(1)
thf(fact_1209_times__int__code_I2_J,axiom,
! [L: int] :
( ( times_times_int @ zero_zero_int @ L )
= zero_zero_int ) ).
% times_int_code(2)
thf(fact_1210_zmult__zless__mono2,axiom,
! [I5: int,J: int,K: int] :
( ( ord_less_int @ I5 @ J )
=> ( ( ord_less_int @ zero_zero_int @ K )
=> ( ord_less_int @ ( times_times_int @ K @ I5 ) @ ( times_times_int @ K @ J ) ) ) ) ).
% zmult_zless_mono2
thf(fact_1211_pos__zmult__eq__1__iff__lemma,axiom,
! [M: int,N: int] :
( ( ( times_times_int @ M @ N )
= one_one_int )
=> ( ( M = one_one_int )
| ( M
= ( uminus_uminus_int @ one_one_int ) ) ) ) ).
% pos_zmult_eq_1_iff_lemma
thf(fact_1212_zmult__eq__1__iff,axiom,
! [M: int,N: int] :
( ( ( times_times_int @ M @ N )
= one_one_int )
= ( ( ( M = one_one_int )
& ( N = one_one_int ) )
| ( ( M
= ( uminus_uminus_int @ one_one_int ) )
& ( N
= ( uminus_uminus_int @ one_one_int ) ) ) ) ) ).
% zmult_eq_1_iff
thf(fact_1213_pos__zmult__eq__1__iff,axiom,
! [M: int,N: int] :
( ( ord_less_int @ zero_zero_int @ M )
=> ( ( ( times_times_int @ M @ N )
= one_one_int )
= ( ( M = one_one_int )
& ( N = one_one_int ) ) ) ) ).
% pos_zmult_eq_1_iff
thf(fact_1214_zdiv__zmult2__eq,axiom,
! [C: int,A2: int,B2: int] :
( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ( divide_divide_int @ A2 @ ( times_times_int @ B2 @ C ) )
= ( divide_divide_int @ ( divide_divide_int @ A2 @ B2 ) @ C ) ) ) ).
% zdiv_zmult2_eq
thf(fact_1215_div__mod__decomp__int,axiom,
! [A3: int,N: int] :
( A3
= ( plus_plus_int @ ( times_times_int @ ( divide_divide_int @ A3 @ N ) @ N ) @ ( modulo_modulo_int @ A3 @ N ) ) ) ).
% div_mod_decomp_int
thf(fact_1216_incr__mult__lemma,axiom,
! [D2: int,P: int > $o,K: int] :
( ( ord_less_int @ zero_zero_int @ D2 )
=> ( ! [X5: int] :
( ( P @ X5 )
=> ( P @ ( plus_plus_int @ X5 @ D2 ) ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ K )
=> ! [X7: int] :
( ( P @ X7 )
=> ( P @ ( plus_plus_int @ X7 @ ( times_times_int @ K @ D2 ) ) ) ) ) ) ) ).
% incr_mult_lemma
thf(fact_1217_int__div__pos__eq,axiom,
! [A2: int,B2: int,Q4: int,R2: int] :
( ( A2
= ( plus_plus_int @ ( times_times_int @ B2 @ Q4 ) @ R2 ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ R2 )
=> ( ( ord_less_int @ R2 @ B2 )
=> ( ( divide_divide_int @ A2 @ B2 )
= Q4 ) ) ) ) ).
% int_div_pos_eq
thf(fact_1218_int__div__neg__eq,axiom,
! [A2: int,B2: int,Q4: int,R2: int] :
( ( A2
= ( plus_plus_int @ ( times_times_int @ B2 @ Q4 ) @ R2 ) )
=> ( ( ord_less_eq_int @ R2 @ zero_zero_int )
=> ( ( ord_less_int @ B2 @ R2 )
=> ( ( divide_divide_int @ A2 @ B2 )
= Q4 ) ) ) ) ).
% int_div_neg_eq
thf(fact_1219_split__zdiv,axiom,
! [P: int > $o,N: int,K: int] :
( ( P @ ( divide_divide_int @ N @ K ) )
= ( ( ( K = zero_zero_int )
=> ( P @ zero_zero_int ) )
& ( ( ord_less_int @ zero_zero_int @ K )
=> ! [I3: int,J3: int] :
( ( ( ord_less_eq_int @ zero_zero_int @ J3 )
& ( ord_less_int @ J3 @ K )
& ( N
= ( plus_plus_int @ ( times_times_int @ K @ I3 ) @ J3 ) ) )
=> ( P @ I3 ) ) )
& ( ( ord_less_int @ K @ zero_zero_int )
=> ! [I3: int,J3: int] :
( ( ( ord_less_int @ K @ J3 )
& ( ord_less_eq_int @ J3 @ zero_zero_int )
& ( N
= ( plus_plus_int @ ( times_times_int @ K @ I3 ) @ J3 ) ) )
=> ( P @ I3 ) ) ) ) ) ).
% split_zdiv
thf(fact_1220_split__zmod,axiom,
! [Q: int > $o,N: int,K: int] :
( ( Q @ ( modulo_modulo_int @ N @ K ) )
= ( ( ( K = zero_zero_int )
=> ( Q @ N ) )
& ( ( ord_less_int @ zero_zero_int @ K )
=> ! [I3: int,J3: int] :
( ( ( ord_less_eq_int @ zero_zero_int @ J3 )
& ( ord_less_int @ J3 @ K )
& ( N
= ( plus_plus_int @ ( times_times_int @ K @ I3 ) @ J3 ) ) )
=> ( Q @ J3 ) ) )
& ( ( ord_less_int @ K @ zero_zero_int )
=> ! [I3: int,J3: int] :
( ( ( ord_less_int @ K @ J3 )
& ( ord_less_eq_int @ J3 @ zero_zero_int )
& ( N
= ( plus_plus_int @ ( times_times_int @ K @ I3 ) @ J3 ) ) )
=> ( Q @ J3 ) ) ) ) ) ).
% split_zmod
thf(fact_1221_int__mod__neg__eq,axiom,
! [A2: int,B2: int,Q4: int,R2: int] :
( ( A2
= ( plus_plus_int @ ( times_times_int @ B2 @ Q4 ) @ R2 ) )
=> ( ( ord_less_eq_int @ R2 @ zero_zero_int )
=> ( ( ord_less_int @ B2 @ R2 )
=> ( ( modulo_modulo_int @ A2 @ B2 )
= R2 ) ) ) ) ).
% int_mod_neg_eq
thf(fact_1222_int__mod__pos__eq,axiom,
! [A2: int,B2: int,Q4: int,R2: int] :
( ( A2
= ( plus_plus_int @ ( times_times_int @ B2 @ Q4 ) @ R2 ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ R2 )
=> ( ( ord_less_int @ R2 @ B2 )
=> ( ( modulo_modulo_int @ A2 @ B2 )
= R2 ) ) ) ) ).
% int_mod_pos_eq
thf(fact_1223_zmod__zmult2__eq,axiom,
! [C: int,A2: int,B2: int] :
( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ( modulo_modulo_int @ A2 @ ( times_times_int @ B2 @ C ) )
= ( plus_plus_int @ ( times_times_int @ B2 @ ( modulo_modulo_int @ ( divide_divide_int @ A2 @ B2 ) @ C ) ) @ ( modulo_modulo_int @ A2 @ B2 ) ) ) ) ).
% zmod_zmult2_eq
thf(fact_1224_int__bit__induct,axiom,
! [P: int > $o,K: int] :
( ( P @ zero_zero_int )
=> ( ( P @ ( uminus_uminus_int @ one_one_int ) )
=> ( ! [K2: int] :
( ( P @ K2 )
=> ( ( K2 != zero_zero_int )
=> ( P @ ( times_times_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) )
=> ( ! [K2: int] :
( ( P @ K2 )
=> ( ( K2
!= ( uminus_uminus_int @ one_one_int ) )
=> ( P @ ( plus_plus_int @ one_one_int @ ( times_times_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) )
=> ( P @ K ) ) ) ) ) ).
% int_bit_induct
thf(fact_1225_pos__zdiv__mult__2,axiom,
! [A2: int,B2: int] :
( ( ord_less_eq_int @ zero_zero_int @ A2 )
=> ( ( divide_divide_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B2 ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A2 ) )
= ( divide_divide_int @ B2 @ A2 ) ) ) ).
% pos_zdiv_mult_2
thf(fact_1226_neg__zdiv__mult__2,axiom,
! [A2: int,B2: int] :
( ( ord_less_eq_int @ A2 @ zero_zero_int )
=> ( ( divide_divide_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B2 ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A2 ) )
= ( divide_divide_int @ ( plus_plus_int @ B2 @ one_one_int ) @ A2 ) ) ) ).
% neg_zdiv_mult_2
thf(fact_1227_mult__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ( times_times_nat @ M @ K )
= ( times_times_nat @ N @ K ) )
= ( ( M = N )
| ( K = zero_zero_nat ) ) ) ).
% mult_cancel2
thf(fact_1228_mult__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( times_times_nat @ K @ M )
= ( times_times_nat @ K @ N ) )
= ( ( M = N )
| ( K = zero_zero_nat ) ) ) ).
% mult_cancel1
thf(fact_1229_mult__0__right,axiom,
! [M: nat] :
( ( times_times_nat @ M @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_0_right
thf(fact_1230_mult__is__0,axiom,
! [M: nat,N: nat] :
( ( ( times_times_nat @ M @ N )
= zero_zero_nat )
= ( ( M = zero_zero_nat )
| ( N = zero_zero_nat ) ) ) ).
% mult_is_0
thf(fact_1231_nat__mult__eq__1__iff,axiom,
! [M: nat,N: nat] :
( ( ( times_times_nat @ M @ N )
= one_one_nat )
= ( ( M = one_one_nat )
& ( N = one_one_nat ) ) ) ).
% nat_mult_eq_1_iff
thf(fact_1232_nat__1__eq__mult__iff,axiom,
! [M: nat,N: nat] :
( ( one_one_nat
= ( times_times_nat @ M @ N ) )
= ( ( M = one_one_nat )
& ( N = one_one_nat ) ) ) ).
% nat_1_eq_mult_iff
thf(fact_1233_not__real__square__gt__zero,axiom,
! [X3: real] :
( ( ~ ( ord_less_real @ zero_zero_real @ ( times_times_real @ X3 @ X3 ) ) )
= ( X3 = zero_zero_real ) ) ).
% not_real_square_gt_zero
thf(fact_1234_nat__mult__less__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
& ( ord_less_nat @ M @ N ) ) ) ).
% nat_mult_less_cancel_disj
thf(fact_1235_nat__0__less__mult__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ M @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ M )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% nat_0_less_mult_iff
thf(fact_1236_mult__less__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
& ( ord_less_nat @ M @ N ) ) ) ).
% mult_less_cancel2
thf(fact_1237_set__bit__nonnegative__int__iff,axiom,
! [N: nat,K: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( bit_se7879613467334960850it_int @ N @ K ) )
= ( ord_less_eq_int @ zero_zero_int @ K ) ) ).
% set_bit_nonnegative_int_iff
thf(fact_1238_set__bit__negative__int__iff,axiom,
! [N: nat,K: int] :
( ( ord_less_int @ ( bit_se7879613467334960850it_int @ N @ K ) @ zero_zero_int )
= ( ord_less_int @ K @ zero_zero_int ) ) ).
% set_bit_negative_int_iff
thf(fact_1239_semiring__norm_I13_J,axiom,
! [M: num,N: num] :
( ( times_times_num @ ( bit0 @ M ) @ ( bit0 @ N ) )
= ( bit0 @ ( bit0 @ ( times_times_num @ M @ N ) ) ) ) ).
% semiring_norm(13)
thf(fact_1240_semiring__norm_I12_J,axiom,
! [N: num] :
( ( times_times_num @ one @ N )
= N ) ).
% semiring_norm(12)
thf(fact_1241_semiring__norm_I11_J,axiom,
! [M: num] :
( ( times_times_num @ M @ one )
= M ) ).
% semiring_norm(11)
thf(fact_1242_mult__le__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_eq_nat @ M @ N ) ) ) ).
% mult_le_cancel2
thf(fact_1243_nat__mult__le__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_eq_nat @ M @ N ) ) ) ).
% nat_mult_le_cancel_disj
thf(fact_1244_div__mult__self1__is__m,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( divide_divide_nat @ ( times_times_nat @ N @ M ) @ N )
= M ) ) ).
% div_mult_self1_is_m
thf(fact_1245_div__mult__self__is__m,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( divide_divide_nat @ ( times_times_nat @ M @ N ) @ N )
= M ) ) ).
% div_mult_self_is_m
thf(fact_1246_num__double,axiom,
! [N: num] :
( ( times_times_num @ ( bit0 @ one ) @ N )
= ( bit0 @ N ) ) ).
% num_double
thf(fact_1247_add__mult__distrib2,axiom,
! [K: nat,M: nat,N: nat] :
( ( times_times_nat @ K @ ( plus_plus_nat @ M @ N ) )
= ( plus_plus_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) ) ) ).
% add_mult_distrib2
thf(fact_1248_add__mult__distrib,axiom,
! [M: nat,N: nat,K: nat] :
( ( times_times_nat @ ( plus_plus_nat @ M @ N ) @ K )
= ( plus_plus_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) ) ) ).
% add_mult_distrib
thf(fact_1249_left__add__mult__distrib,axiom,
! [I5: nat,U: nat,J: nat,K: nat] :
( ( plus_plus_nat @ ( times_times_nat @ I5 @ U ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ K ) )
= ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ I5 @ J ) @ U ) @ K ) ) ).
% left_add_mult_distrib
thf(fact_1250_div__mult2__eq,axiom,
! [M: nat,N: nat,Q4: nat] :
( ( divide_divide_nat @ M @ ( times_times_nat @ N @ Q4 ) )
= ( divide_divide_nat @ ( divide_divide_nat @ M @ N ) @ Q4 ) ) ).
% div_mult2_eq
thf(fact_1251_imult__is__0,axiom,
! [M: extended_enat,N: extended_enat] :
( ( ( times_7803423173614009249d_enat @ M @ N )
= zero_z5237406670263579293d_enat )
= ( ( M = zero_z5237406670263579293d_enat )
| ( N = zero_z5237406670263579293d_enat ) ) ) ).
% imult_is_0
thf(fact_1252_ennreal__mult__left__cong,axiom,
! [A2: extend8495563244428889912nnreal,B2: extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
( ( ( A2 != zero_z7100319975126383169nnreal )
=> ( B2 = C ) )
=> ( ( times_1893300245718287421nnreal @ A2 @ B2 )
= ( times_1893300245718287421nnreal @ A2 @ C ) ) ) ).
% ennreal_mult_left_cong
thf(fact_1253_ennreal__mult__right__cong,axiom,
! [A2: extend8495563244428889912nnreal,B2: extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
( ( ( A2 != zero_z7100319975126383169nnreal )
=> ( B2 = C ) )
=> ( ( times_1893300245718287421nnreal @ B2 @ A2 )
= ( times_1893300245718287421nnreal @ C @ A2 ) ) ) ).
% ennreal_mult_right_cong
thf(fact_1254_mult__le__mono2,axiom,
! [I5: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I5 @ J )
=> ( ord_less_eq_nat @ ( times_times_nat @ K @ I5 ) @ ( times_times_nat @ K @ J ) ) ) ).
% mult_le_mono2
thf(fact_1255_mult__le__mono1,axiom,
! [I5: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I5 @ J )
=> ( ord_less_eq_nat @ ( times_times_nat @ I5 @ K ) @ ( times_times_nat @ J @ K ) ) ) ).
% mult_le_mono1
thf(fact_1256_mult__le__mono,axiom,
! [I5: nat,J: nat,K: nat,L: nat] :
( ( ord_less_eq_nat @ I5 @ J )
=> ( ( ord_less_eq_nat @ K @ L )
=> ( ord_less_eq_nat @ ( times_times_nat @ I5 @ K ) @ ( times_times_nat @ J @ L ) ) ) ) ).
% mult_le_mono
thf(fact_1257_le__square,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ M ) ) ).
% le_square
thf(fact_1258_le__cube,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ ( times_times_nat @ M @ M ) ) ) ).
% le_cube
thf(fact_1259_mult__0,axiom,
! [N: nat] :
( ( times_times_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% mult_0
thf(fact_1260_nat__mult__eq__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( times_times_nat @ K @ M )
= ( times_times_nat @ K @ N ) )
= ( ( K = zero_zero_nat )
| ( M = N ) ) ) ).
% nat_mult_eq_cancel_disj
thf(fact_1261_nat__mult__1,axiom,
! [N: nat] :
( ( times_times_nat @ one_one_nat @ N )
= N ) ).
% nat_mult_1
% Helper facts (5)
thf(help_If_2_1_If_001t__Int__Oint_T,axiom,
! [X3: int,Y4: int] :
( ( if_int @ $false @ X3 @ Y4 )
= Y4 ) ).
thf(help_If_1_1_If_001t__Int__Oint_T,axiom,
! [X3: int,Y4: int] :
( ( if_int @ $true @ X3 @ Y4 )
= X3 ) ).
thf(help_If_3_1_If_001t__Nat__Onat_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
! [X3: nat,Y4: nat] :
( ( if_nat @ $false @ X3 @ Y4 )
= Y4 ) ).
thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X3: nat,Y4: nat] :
( ( if_nat @ $true @ X3 @ Y4 )
= X3 ) ).
% Conjectures (1)
thf(conj_0,conjecture,
member_a @ ( nth_a @ xs @ ( divide_divide_nat @ ( size_size_list_a @ xs ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ i ).
%------------------------------------------------------------------------------