TPTP Problem File: SLH0153^1.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_00357_013407__14831978_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1412 ( 491 unt; 140 typ; 0 def)
% Number of atoms : 3880 (1031 equ; 0 cnn)
% Maximal formula atoms : 26 ( 3 avg)
% Number of connectives : 11732 ( 358 ~; 98 |; 197 &;9193 @)
% ( 0 <=>;1886 =>; 0 <=; 0 <~>)
% Maximal formula depth : 25 ( 7 avg)
% Number of types : 19 ( 18 usr)
% Number of type conns : 571 ( 571 >; 0 *; 0 +; 0 <<)
% Number of symbols : 125 ( 122 usr; 16 con; 0-5 aty)
% Number of variables : 3728 ( 215 ^;3371 !; 142 ?;3728 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 15:45:38.507
%------------------------------------------------------------------------------
% Could-be-implicit typings (18)
thf(ty_n_t__Sigma____Algebra__Omeasure_It__Real__Oreal_J,type,
sigma_measure_real: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Real__Oreal_J_J,type,
set_set_real: $tType ).
thf(ty_n_t__Sigma____Algebra__Omeasure_It__Nat__Onat_J,type,
sigma_measure_nat: $tType ).
thf(ty_n_t__Sigma____Algebra__Omeasure_It__Int__Oint_J,type,
sigma_measure_int: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
set_set_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Int__Oint_J_J,type,
set_set_int: $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__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__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__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__Real__Oreal,type,
real: $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 (122)
thf(sy_c_Borel__Space_Otopological__space__class_Oborel_001t__Int__Oint,type,
borel_8447240504075733134el_int: sigma_measure_int ).
thf(sy_c_Borel__Space_Otopological__space__class_Oborel_001t__Nat__Onat,type,
borel_8449730974584783410el_nat: sigma_measure_nat ).
thf(sy_c_Borel__Space_Otopological__space__class_Oborel_001t__Real__Oreal,type,
borel_5078946678739801102l_real: sigma_measure_real ).
thf(sy_c_Groups_Oabs__class_Oabs_001t__Int__Oint,type,
abs_abs_int: int > int ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Int__Oint,type,
minus_minus_int: int > int > int ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat,type,
minus_minus_nat: nat > nat > nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Real__Oreal,type,
minus_minus_real: real > real > real ).
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__Real__Oreal,type,
one_one_real: real ).
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__Real__Oreal,type,
plus_plus_real: real > real > real ).
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__Real__Oreal,type,
times_times_real: real > real > real ).
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__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__List__Olist_It__Int__Oint_J,type,
if_list_int: $o > list_int > list_int > list_int ).
thf(sy_c_If_001t__Nat__Onat,type,
if_nat: $o > nat > nat > nat ).
thf(sy_c_List_Obutlast_001t__Nat__Onat,type,
butlast_nat: list_nat > list_nat ).
thf(sy_c_List_Obutlast_001tf__a,type,
butlast_a: list_a > list_a ).
thf(sy_c_List_Olist_OCons_001t__Int__Oint,type,
cons_int: int > list_int > list_int ).
thf(sy_c_List_Olist_OCons_001t__Nat__Onat,type,
cons_nat: nat > list_nat > list_nat ).
thf(sy_c_List_Olist_OCons_001tf__a,type,
cons_a: a > list_a > list_a ).
thf(sy_c_List_Olist__ex_001t__Int__Oint,type,
list_ex_int: ( int > $o ) > list_int > $o ).
thf(sy_c_List_Olist__ex_001t__Nat__Onat,type,
list_ex_nat: ( nat > $o ) > list_nat > $o ).
thf(sy_c_List_Olist__ex_001tf__a,type,
list_ex_a: ( a > $o ) > list_a > $o ).
thf(sy_c_List_Olist__update_001t__Nat__Onat,type,
list_update_nat: list_nat > nat > nat > list_nat ).
thf(sy_c_List_Olist__update_001tf__a,type,
list_update_a: list_a > nat > a > list_a ).
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__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__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_Orotate1_001t__Nat__Onat,type,
rotate1_nat: list_nat > list_nat ).
thf(sy_c_List_Orotate1_001tf__a,type,
rotate1_a: list_a > list_a ).
thf(sy_c_List_Orotate_001t__Nat__Onat,type,
rotate_nat: nat > list_nat > list_nat ).
thf(sy_c_List_Orotate_001tf__a,type,
rotate_a: nat > list_a > list_a ).
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__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_List_Oupto__aux,type,
upto_aux: int > int > list_int > list_int ).
thf(sy_c_Measure__Space_Osup__lexord_001t__Nat__Onat_001t__Int__Oint,type,
measur4598756670496807578at_int: nat > nat > ( nat > int ) > nat > nat > nat ).
thf(sy_c_Measure__Space_Osup__lexord_001t__Nat__Onat_001t__Nat__Onat,type,
measur4601247141005857854at_nat: nat > nat > ( nat > nat ) > nat > nat > nat ).
thf(sy_c_Measure__Space_Osup__lexord_001t__Nat__Onat_001t__Real__Oreal,type,
measur8600355784167071770t_real: nat > nat > ( nat > real ) > nat > nat > nat ).
thf(sy_c_Measure__Space_Osup__lexord_001t__Nat__Onat_001tf__a,type,
measur1338148430225957392_nat_a: nat > nat > ( nat > a ) > nat > nat > nat ).
thf(sy_c_Measure__Space_Osup__lexord_001t__Real__Oreal_001t__Nat__Onat,type,
measur3944292320441194650al_nat: real > real > ( real > nat ) > real > real > real ).
thf(sy_c_Measure__Space_Osup__lexord_001t__Real__Oreal_001tf__a,type,
measur5927359263096536244real_a: real > real > ( real > a ) > real > real > real ).
thf(sy_c_Measure__Space_Osup__lexord_001tf__a_001t__Int__Oint,type,
measur6527098495393396494_a_int: a > a > ( a > int ) > a > a > a ).
thf(sy_c_Measure__Space_Osup__lexord_001tf__a_001t__Nat__Onat,type,
measur6529588965902446770_a_nat: a > a > ( a > nat ) > a > a > a ).
thf(sy_c_Measure__Space_Osup__lexord_001tf__a_001t__Real__Oreal,type,
measur5297340495623586446a_real: a > a > ( a > real ) > a > a > a ).
thf(sy_c_Measure__Space_Osup__lexord_001tf__a_001tf__a,type,
measur6235662355876231836rd_a_a: a > a > ( a > a ) > a > a > a ).
thf(sy_c_Median_Odown__ray_001t__Int__Oint,type,
down_ray_int: set_int > $o ).
thf(sy_c_Median_Odown__ray_001t__Nat__Onat,type,
down_ray_nat: set_nat > $o ).
thf(sy_c_Median_Odown__ray_001t__Real__Oreal,type,
down_ray_real: set_real > $o ).
thf(sy_c_Median_Odown__ray_001tf__a,type,
down_ray_a: set_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__Real__Oreal,type,
interval_real: set_real > $o ).
thf(sy_c_Median_Ointerval_001tf__a,type,
interval_a: set_a > $o ).
thf(sy_c_Median_Osort__map_001t__Int__Oint,type,
sort_map_int: ( nat > int ) > nat > nat > int ).
thf(sy_c_Median_Osort__map_001t__Nat__Onat,type,
sort_map_nat: ( nat > nat ) > nat > nat > nat ).
thf(sy_c_Median_Osort__map_001t__Real__Oreal,type,
sort_map_real: ( nat > real ) > nat > nat > real ).
thf(sy_c_Median_Osort__map_001tf__a,type,
sort_map_a: ( nat > a ) > nat > nat > a ).
thf(sy_c_Median_Oup__ray_001t__Int__Oint,type,
up_ray_int: set_int > $o ).
thf(sy_c_Median_Oup__ray_001t__Nat__Onat,type,
up_ray_nat: set_nat > $o ).
thf(sy_c_Median_Oup__ray_001t__Real__Oreal,type,
up_ray_real: set_real > $o ).
thf(sy_c_Median_Oup__ray_001tf__a,type,
up_ray_a: set_a > $o ).
thf(sy_c_Nat_OSuc,type,
suc: nat > nat ).
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__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__Bijection_Otriangle,type,
nat_triangle: nat > nat ).
thf(sy_c_Num_Oneg__numeral__class_Odbl__inc_001t__Int__Oint,type,
neg_nu5851722552734809277nc_int: int > int ).
thf(sy_c_Num_Oneg__numeral__class_Odbl__inc_001t__Real__Oreal,type,
neg_nu8295874005876285629c_real: real > real ).
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__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__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__Real__Oreal,type,
ord_less_eq_real: real > real > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Int__Oint_J,type,
ord_less_eq_set_int: set_int > set_int > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Nat__Onat_J,type,
ord_less_eq_set_nat: set_nat > set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Real__Oreal_J,type,
ord_less_eq_set_real: set_real > set_real > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_Itf__a_J,type,
ord_less_eq_set_a: set_a > set_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001tf__a,type,
ord_less_eq_a: a > a > $o ).
thf(sy_c_Power_Opower__class_Opower_001t__Int__Oint,type,
power_power_int: int > nat > int ).
thf(sy_c_Power_Opower__class_Opower_001t__Nat__Onat,type,
power_power_nat: nat > nat > nat ).
thf(sy_c_Power_Opower__class_Opower_001t__Real__Oreal,type,
power_power_real: real > nat > real ).
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_001t__Int__Oint,type,
collect_int: ( int > $o ) > set_int ).
thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
collect_nat: ( nat > $o ) > set_nat ).
thf(sy_c_Set_OCollect_001t__Real__Oreal,type,
collect_real: ( real > $o ) > set_real ).
thf(sy_c_Set_OCollect_001tf__a,type,
collect_a: ( a > $o ) > set_a ).
thf(sy_c_Sigma__Algebra_Osets_001t__Int__Oint,type,
sigma_sets_int: sigma_measure_int > set_set_int ).
thf(sy_c_Sigma__Algebra_Osets_001t__Nat__Onat,type,
sigma_sets_nat: sigma_measure_nat > set_set_nat ).
thf(sy_c_Sigma__Algebra_Osets_001t__Real__Oreal,type,
sigma_sets_real: sigma_measure_real > set_set_real ).
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__Real__Oreal,type,
member_real: real > set_real > $o ).
thf(sy_c_member_001t__Set__Oset_It__Int__Oint_J,type,
member_set_int: set_int > set_set_int > $o ).
thf(sy_c_member_001t__Set__Oset_It__Nat__Onat_J,type,
member_set_nat: set_nat > set_set_nat > $o ).
thf(sy_c_member_001t__Set__Oset_It__Real__Oreal_J,type,
member_set_real: set_real > set_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_i,type,
i2: nat ).
thf(sy_v_j,type,
j: nat ).
thf(sy_v_k,type,
k: nat ).
thf(sy_v_xs,type,
xs: list_a ).
% Relevant facts (1264)
thf(fact_0_assms_I6_J,axiom,
member_a @ ( nth_a @ xs @ i2 ) @ i ).
% assms(6)
thf(fact_1_assms_I7_J,axiom,
member_a @ ( nth_a @ xs @ k ) @ i ).
% assms(7)
thf(fact_2_assms_I1_J,axiom,
interval_a @ i ).
% assms(1)
thf(fact_3_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_4_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_5_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_6_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_7_interval__rule,axiom,
! [I2: set_nat,A: nat,X2: nat,B: nat] :
( ( interval_nat @ I2 )
=> ( ( ord_less_eq_nat @ A @ X2 )
=> ( ( ord_less_eq_nat @ X2 @ B )
=> ( ( member_nat @ A @ I2 )
=> ( ( member_nat @ B @ I2 )
=> ( member_nat @ X2 @ I2 ) ) ) ) ) ) ).
% interval_rule
thf(fact_8_interval__rule,axiom,
! [I2: set_a,A: a,X2: a,B: a] :
( ( interval_a @ I2 )
=> ( ( ord_less_eq_a @ A @ X2 )
=> ( ( ord_less_eq_a @ X2 @ B )
=> ( ( member_a @ A @ I2 )
=> ( ( member_a @ B @ I2 )
=> ( member_a @ X2 @ I2 ) ) ) ) ) ) ).
% interval_rule
thf(fact_9_interval__rule,axiom,
! [I2: set_real,A: real,X2: real,B: real] :
( ( interval_real @ I2 )
=> ( ( ord_less_eq_real @ A @ X2 )
=> ( ( ord_less_eq_real @ X2 @ B )
=> ( ( member_real @ A @ I2 )
=> ( ( member_real @ B @ I2 )
=> ( member_real @ X2 @ I2 ) ) ) ) ) ) ).
% interval_rule
thf(fact_10_interval__rule,axiom,
! [I2: set_int,A: int,X2: int,B: int] :
( ( interval_int @ I2 )
=> ( ( ord_less_eq_int @ A @ X2 )
=> ( ( ord_less_eq_int @ X2 @ B )
=> ( ( member_int @ A @ I2 )
=> ( ( member_int @ B @ I2 )
=> ( member_int @ X2 @ I2 ) ) ) ) ) ) ).
% interval_rule
thf(fact_11_interval__borel,axiom,
! [I2: set_int] :
( ( interval_int @ I2 )
=> ( member_set_int @ I2 @ ( sigma_sets_int @ borel_8447240504075733134el_int ) ) ) ).
% interval_borel
thf(fact_12_interval__borel,axiom,
! [I2: set_real] :
( ( interval_real @ I2 )
=> ( member_set_real @ I2 @ ( sigma_sets_real @ borel_5078946678739801102l_real ) ) ) ).
% interval_borel
thf(fact_13_interval__borel,axiom,
! [I2: set_nat] :
( ( interval_nat @ I2 )
=> ( member_set_nat @ I2 @ ( sigma_sets_nat @ borel_8449730974584783410el_nat ) ) ) ).
% interval_borel
thf(fact_14_up__ray__def,axiom,
( up_ray_nat
= ( ^ [I: set_nat] :
! [X: nat,Y: nat] :
( ( member_nat @ X @ I )
=> ( ( ord_less_eq_nat @ X @ Y )
=> ( member_nat @ Y @ I ) ) ) ) ) ).
% up_ray_def
thf(fact_15_up__ray__def,axiom,
( up_ray_a
= ( ^ [I: set_a] :
! [X: a,Y: a] :
( ( member_a @ X @ I )
=> ( ( ord_less_eq_a @ X @ Y )
=> ( member_a @ Y @ I ) ) ) ) ) ).
% up_ray_def
thf(fact_16_up__ray__def,axiom,
( up_ray_real
= ( ^ [I: set_real] :
! [X: real,Y: real] :
( ( member_real @ X @ I )
=> ( ( ord_less_eq_real @ X @ Y )
=> ( member_real @ Y @ I ) ) ) ) ) ).
% up_ray_def
thf(fact_17_up__ray__def,axiom,
( up_ray_int
= ( ^ [I: set_int] :
! [X: int,Y: int] :
( ( member_int @ X @ I )
=> ( ( ord_less_eq_int @ X @ Y )
=> ( member_int @ Y @ I ) ) ) ) ) ).
% up_ray_def
thf(fact_18_down__ray__def,axiom,
( down_ray_nat
= ( ^ [I: set_nat] :
! [X: nat,Y: nat] :
( ( member_nat @ Y @ I )
=> ( ( ord_less_eq_nat @ X @ Y )
=> ( member_nat @ X @ I ) ) ) ) ) ).
% down_ray_def
thf(fact_19_down__ray__def,axiom,
( down_ray_a
= ( ^ [I: set_a] :
! [X: a,Y: a] :
( ( member_a @ Y @ I )
=> ( ( ord_less_eq_a @ X @ Y )
=> ( member_a @ X @ I ) ) ) ) ) ).
% down_ray_def
thf(fact_20_down__ray__def,axiom,
( down_ray_real
= ( ^ [I: set_real] :
! [X: real,Y: real] :
( ( member_real @ Y @ I )
=> ( ( ord_less_eq_real @ X @ Y )
=> ( member_real @ X @ I ) ) ) ) ) ).
% down_ray_def
thf(fact_21_down__ray__def,axiom,
( down_ray_int
= ( ^ [I: set_int] :
! [X: int,Y: int] :
( ( member_int @ Y @ I )
=> ( ( ord_less_eq_int @ X @ Y )
=> ( member_int @ X @ I ) ) ) ) ) ).
% down_ray_def
thf(fact_22_assms_I5_J,axiom,
ord_less_eq_nat @ j @ k ).
% assms(5)
thf(fact_23_assms_I2_J,axiom,
sorted_wrt_a @ ord_less_eq_a @ xs ).
% assms(2)
thf(fact_24_assms_I4_J,axiom,
ord_less_eq_nat @ i2 @ j ).
% assms(4)
thf(fact_25_assms_I3_J,axiom,
ord_less_nat @ k @ ( size_size_list_a @ xs ) ).
% assms(3)
thf(fact_26_up__ray__borel,axiom,
! [I2: set_int] :
( ( up_ray_int @ I2 )
=> ( member_set_int @ I2 @ ( sigma_sets_int @ borel_8447240504075733134el_int ) ) ) ).
% up_ray_borel
thf(fact_27_up__ray__borel,axiom,
! [I2: set_real] :
( ( up_ray_real @ I2 )
=> ( member_set_real @ I2 @ ( sigma_sets_real @ borel_5078946678739801102l_real ) ) ) ).
% up_ray_borel
thf(fact_28_up__ray__borel,axiom,
! [I2: set_nat] :
( ( up_ray_nat @ I2 )
=> ( member_set_nat @ I2 @ ( sigma_sets_nat @ borel_8449730974584783410el_nat ) ) ) ).
% up_ray_borel
thf(fact_29_down__ray__borel,axiom,
! [I2: set_int] :
( ( down_ray_int @ I2 )
=> ( member_set_int @ I2 @ ( sigma_sets_int @ borel_8447240504075733134el_int ) ) ) ).
% down_ray_borel
thf(fact_30_down__ray__borel,axiom,
! [I2: set_real] :
( ( down_ray_real @ I2 )
=> ( member_set_real @ I2 @ ( sigma_sets_real @ borel_5078946678739801102l_real ) ) ) ).
% down_ray_borel
thf(fact_31_down__ray__borel,axiom,
! [I2: set_nat] :
( ( down_ray_nat @ I2 )
=> ( member_set_nat @ I2 @ ( sigma_sets_nat @ borel_8449730974584783410el_nat ) ) ) ).
% down_ray_borel
thf(fact_32_order__refl,axiom,
! [X2: nat] : ( ord_less_eq_nat @ X2 @ X2 ) ).
% order_refl
thf(fact_33_order__refl,axiom,
! [X2: a] : ( ord_less_eq_a @ X2 @ X2 ) ).
% order_refl
thf(fact_34_order__refl,axiom,
! [X2: real] : ( ord_less_eq_real @ X2 @ X2 ) ).
% order_refl
thf(fact_35_order__refl,axiom,
! [X2: int] : ( ord_less_eq_int @ X2 @ X2 ) ).
% order_refl
thf(fact_36_dual__order_Orefl,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% dual_order.refl
thf(fact_37_dual__order_Orefl,axiom,
! [A: a] : ( ord_less_eq_a @ A @ A ) ).
% dual_order.refl
thf(fact_38_dual__order_Orefl,axiom,
! [A: real] : ( ord_less_eq_real @ A @ A ) ).
% dual_order.refl
thf(fact_39_dual__order_Orefl,axiom,
! [A: int] : ( ord_less_eq_int @ A @ A ) ).
% dual_order.refl
thf(fact_40_verit__comp__simplify1_I2_J,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_41_verit__comp__simplify1_I2_J,axiom,
! [A: a] : ( ord_less_eq_a @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_42_verit__comp__simplify1_I2_J,axiom,
! [A: real] : ( ord_less_eq_real @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_43_verit__comp__simplify1_I2_J,axiom,
! [A: int] : ( ord_less_eq_int @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_44_nle__le,axiom,
! [A: nat,B: nat] :
( ( ~ ( ord_less_eq_nat @ A @ B ) )
= ( ( ord_less_eq_nat @ B @ A )
& ( B != A ) ) ) ).
% nle_le
thf(fact_45_nle__le,axiom,
! [A: a,B: a] :
( ( ~ ( ord_less_eq_a @ A @ B ) )
= ( ( ord_less_eq_a @ B @ A )
& ( B != A ) ) ) ).
% nle_le
thf(fact_46_nle__le,axiom,
! [A: real,B: real] :
( ( ~ ( ord_less_eq_real @ A @ B ) )
= ( ( ord_less_eq_real @ B @ A )
& ( B != A ) ) ) ).
% nle_le
thf(fact_47_nle__le,axiom,
! [A: int,B: int] :
( ( ~ ( ord_less_eq_int @ A @ B ) )
= ( ( ord_less_eq_int @ B @ A )
& ( B != A ) ) ) ).
% nle_le
thf(fact_48_le__cases3,axiom,
! [X2: nat,Y2: nat,Z2: nat] :
( ( ( ord_less_eq_nat @ X2 @ Y2 )
=> ~ ( ord_less_eq_nat @ Y2 @ Z2 ) )
=> ( ( ( ord_less_eq_nat @ Y2 @ X2 )
=> ~ ( ord_less_eq_nat @ X2 @ Z2 ) )
=> ( ( ( ord_less_eq_nat @ X2 @ Z2 )
=> ~ ( ord_less_eq_nat @ Z2 @ Y2 ) )
=> ( ( ( ord_less_eq_nat @ Z2 @ Y2 )
=> ~ ( ord_less_eq_nat @ Y2 @ X2 ) )
=> ( ( ( ord_less_eq_nat @ Y2 @ Z2 )
=> ~ ( ord_less_eq_nat @ Z2 @ X2 ) )
=> ~ ( ( ord_less_eq_nat @ Z2 @ X2 )
=> ~ ( ord_less_eq_nat @ X2 @ Y2 ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_49_le__cases3,axiom,
! [X2: a,Y2: a,Z2: a] :
( ( ( ord_less_eq_a @ X2 @ Y2 )
=> ~ ( ord_less_eq_a @ Y2 @ Z2 ) )
=> ( ( ( ord_less_eq_a @ Y2 @ X2 )
=> ~ ( ord_less_eq_a @ X2 @ Z2 ) )
=> ( ( ( ord_less_eq_a @ X2 @ Z2 )
=> ~ ( ord_less_eq_a @ Z2 @ Y2 ) )
=> ( ( ( ord_less_eq_a @ Z2 @ Y2 )
=> ~ ( ord_less_eq_a @ Y2 @ X2 ) )
=> ( ( ( ord_less_eq_a @ Y2 @ Z2 )
=> ~ ( ord_less_eq_a @ Z2 @ X2 ) )
=> ~ ( ( ord_less_eq_a @ Z2 @ X2 )
=> ~ ( ord_less_eq_a @ X2 @ Y2 ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_50_le__cases3,axiom,
! [X2: real,Y2: real,Z2: real] :
( ( ( ord_less_eq_real @ X2 @ Y2 )
=> ~ ( ord_less_eq_real @ Y2 @ Z2 ) )
=> ( ( ( ord_less_eq_real @ Y2 @ X2 )
=> ~ ( ord_less_eq_real @ X2 @ Z2 ) )
=> ( ( ( ord_less_eq_real @ X2 @ Z2 )
=> ~ ( ord_less_eq_real @ Z2 @ Y2 ) )
=> ( ( ( ord_less_eq_real @ Z2 @ Y2 )
=> ~ ( ord_less_eq_real @ Y2 @ X2 ) )
=> ( ( ( ord_less_eq_real @ Y2 @ Z2 )
=> ~ ( ord_less_eq_real @ Z2 @ X2 ) )
=> ~ ( ( ord_less_eq_real @ Z2 @ X2 )
=> ~ ( ord_less_eq_real @ X2 @ Y2 ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_51_le__cases3,axiom,
! [X2: int,Y2: int,Z2: int] :
( ( ( ord_less_eq_int @ X2 @ Y2 )
=> ~ ( ord_less_eq_int @ Y2 @ Z2 ) )
=> ( ( ( ord_less_eq_int @ Y2 @ X2 )
=> ~ ( ord_less_eq_int @ X2 @ Z2 ) )
=> ( ( ( ord_less_eq_int @ X2 @ Z2 )
=> ~ ( ord_less_eq_int @ Z2 @ Y2 ) )
=> ( ( ( ord_less_eq_int @ Z2 @ Y2 )
=> ~ ( ord_less_eq_int @ Y2 @ X2 ) )
=> ( ( ( ord_less_eq_int @ Y2 @ Z2 )
=> ~ ( ord_less_eq_int @ Z2 @ X2 ) )
=> ~ ( ( ord_less_eq_int @ Z2 @ X2 )
=> ~ ( ord_less_eq_int @ X2 @ Y2 ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_52_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y3: nat,Z3: nat] : ( Y3 = Z3 ) )
= ( ^ [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
& ( ord_less_eq_nat @ Y @ X ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_53_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y3: a,Z3: a] : ( Y3 = Z3 ) )
= ( ^ [X: a,Y: a] :
( ( ord_less_eq_a @ X @ Y )
& ( ord_less_eq_a @ Y @ X ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_54_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y3: real,Z3: real] : ( Y3 = Z3 ) )
= ( ^ [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
& ( ord_less_eq_real @ Y @ X ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_55_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y3: int,Z3: int] : ( Y3 = Z3 ) )
= ( ^ [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
& ( ord_less_eq_int @ Y @ X ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_56_ord__eq__le__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( A = B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_57_ord__eq__le__trans,axiom,
! [A: a,B: a,C: a] :
( ( A = B )
=> ( ( ord_less_eq_a @ B @ C )
=> ( ord_less_eq_a @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_58_ord__eq__le__trans,axiom,
! [A: real,B: real,C: real] :
( ( A = B )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ord_less_eq_real @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_59_ord__eq__le__trans,axiom,
! [A: int,B: int,C: int] :
( ( A = B )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ord_less_eq_int @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_60_ord__le__eq__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_61_ord__le__eq__trans,axiom,
! [A: a,B: a,C: a] :
( ( ord_less_eq_a @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_a @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_62_ord__le__eq__trans,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_real @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_63_ord__le__eq__trans,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_int @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_64_order__antisym,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ( ord_less_eq_nat @ Y2 @ X2 )
=> ( X2 = Y2 ) ) ) ).
% order_antisym
thf(fact_65_order__antisym,axiom,
! [X2: a,Y2: a] :
( ( ord_less_eq_a @ X2 @ Y2 )
=> ( ( ord_less_eq_a @ Y2 @ X2 )
=> ( X2 = Y2 ) ) ) ).
% order_antisym
thf(fact_66_order__antisym,axiom,
! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ( ord_less_eq_real @ Y2 @ X2 )
=> ( X2 = Y2 ) ) ) ).
% order_antisym
thf(fact_67_order__antisym,axiom,
! [X2: int,Y2: int] :
( ( ord_less_eq_int @ X2 @ Y2 )
=> ( ( ord_less_eq_int @ Y2 @ X2 )
=> ( X2 = Y2 ) ) ) ).
% order_antisym
thf(fact_68_order__less__imp__not__less,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ~ ( ord_less_nat @ Y2 @ X2 ) ) ).
% order_less_imp_not_less
thf(fact_69_order__less__imp__not__less,axiom,
! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ~ ( ord_less_real @ Y2 @ X2 ) ) ).
% order_less_imp_not_less
thf(fact_70_order__less__imp__not__less,axiom,
! [X2: int,Y2: int] :
( ( ord_less_int @ X2 @ Y2 )
=> ~ ( ord_less_int @ Y2 @ X2 ) ) ).
% order_less_imp_not_less
thf(fact_71_order__less__imp__not__eq2,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( Y2 != X2 ) ) ).
% order_less_imp_not_eq2
thf(fact_72_order__less__imp__not__eq2,axiom,
! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ( Y2 != X2 ) ) ).
% order_less_imp_not_eq2
thf(fact_73_order__less__imp__not__eq2,axiom,
! [X2: int,Y2: int] :
( ( ord_less_int @ X2 @ Y2 )
=> ( Y2 != X2 ) ) ).
% order_less_imp_not_eq2
thf(fact_74_order__less__imp__not__eq,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( X2 != Y2 ) ) ).
% order_less_imp_not_eq
thf(fact_75_order__less__imp__not__eq,axiom,
! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ( X2 != Y2 ) ) ).
% order_less_imp_not_eq
thf(fact_76_order__less__imp__not__eq,axiom,
! [X2: int,Y2: int] :
( ( ord_less_int @ X2 @ Y2 )
=> ( X2 != Y2 ) ) ).
% order_less_imp_not_eq
thf(fact_77_linorder__less__linear,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
| ( X2 = Y2 )
| ( ord_less_nat @ Y2 @ X2 ) ) ).
% linorder_less_linear
thf(fact_78_linorder__less__linear,axiom,
! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
| ( X2 = Y2 )
| ( ord_less_real @ Y2 @ X2 ) ) ).
% linorder_less_linear
thf(fact_79_linorder__less__linear,axiom,
! [X2: int,Y2: int] :
( ( ord_less_int @ X2 @ Y2 )
| ( X2 = Y2 )
| ( ord_less_int @ Y2 @ X2 ) ) ).
% linorder_less_linear
thf(fact_80_order__less__imp__triv,axiom,
! [X2: nat,Y2: nat,P: $o] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ( ord_less_nat @ Y2 @ X2 )
=> P ) ) ).
% order_less_imp_triv
thf(fact_81_order__less__imp__triv,axiom,
! [X2: real,Y2: real,P: $o] :
( ( ord_less_real @ X2 @ Y2 )
=> ( ( ord_less_real @ Y2 @ X2 )
=> P ) ) ).
% order_less_imp_triv
thf(fact_82_order__less__imp__triv,axiom,
! [X2: int,Y2: int,P: $o] :
( ( ord_less_int @ X2 @ Y2 )
=> ( ( ord_less_int @ Y2 @ X2 )
=> P ) ) ).
% order_less_imp_triv
thf(fact_83_order__less__not__sym,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ~ ( ord_less_nat @ Y2 @ X2 ) ) ).
% order_less_not_sym
thf(fact_84_order__less__not__sym,axiom,
! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ~ ( ord_less_real @ Y2 @ X2 ) ) ).
% order_less_not_sym
thf(fact_85_order__less__not__sym,axiom,
! [X2: int,Y2: int] :
( ( ord_less_int @ X2 @ Y2 )
=> ~ ( ord_less_int @ Y2 @ X2 ) ) ).
% order_less_not_sym
thf(fact_86_order__less__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_nat @ X3 @ Y4 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_87_order__less__subst2,axiom,
! [A: nat,B: nat,F: nat > real,C: real] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_real @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_nat @ X3 @ Y4 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_88_order__less__subst2,axiom,
! [A: nat,B: nat,F: nat > int,C: int] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_nat @ X3 @ Y4 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_89_order__less__subst2,axiom,
! [A: real,B: real,F: real > nat,C: nat] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X3: real,Y4: real] :
( ( ord_less_real @ X3 @ Y4 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_90_order__less__subst2,axiom,
! [A: real,B: real,F: real > real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ ( F @ B ) @ C )
=> ( ! [X3: real,Y4: real] :
( ( ord_less_real @ X3 @ Y4 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_91_order__less__subst2,axiom,
! [A: real,B: real,F: real > int,C: int] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C )
=> ( ! [X3: real,Y4: real] :
( ( ord_less_real @ X3 @ Y4 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_92_order__less__subst2,axiom,
! [A: int,B: int,F: int > nat,C: nat] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X3: int,Y4: int] :
( ( ord_less_int @ X3 @ Y4 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_93_order__less__subst2,axiom,
! [A: int,B: int,F: int > real,C: real] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_real @ ( F @ B ) @ C )
=> ( ! [X3: int,Y4: int] :
( ( ord_less_int @ X3 @ Y4 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_94_order__less__subst2,axiom,
! [A: int,B: int,F: int > int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C )
=> ( ! [X3: int,Y4: int] :
( ( ord_less_int @ X3 @ Y4 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_95_order__less__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_nat @ X3 @ Y4 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_96_order__less__subst1,axiom,
! [A: nat,F: real > nat,B: real,C: real] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X3: real,Y4: real] :
( ( ord_less_real @ X3 @ Y4 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_97_order__less__subst1,axiom,
! [A: nat,F: int > nat,B: int,C: int] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X3: int,Y4: int] :
( ( ord_less_int @ X3 @ Y4 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_98_order__less__subst1,axiom,
! [A: real,F: nat > real,B: nat,C: nat] :
( ( ord_less_real @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_nat @ X3 @ Y4 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_99_order__less__subst1,axiom,
! [A: real,F: real > real,B: real,C: real] :
( ( ord_less_real @ A @ ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X3: real,Y4: real] :
( ( ord_less_real @ X3 @ Y4 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_100_order__less__subst1,axiom,
! [A: real,F: int > real,B: int,C: int] :
( ( ord_less_real @ A @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X3: int,Y4: int] :
( ( ord_less_int @ X3 @ Y4 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_101_order__less__subst1,axiom,
! [A: int,F: nat > int,B: nat,C: nat] :
( ( ord_less_int @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_nat @ X3 @ Y4 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_102_order__less__subst1,axiom,
! [A: int,F: real > int,B: real,C: real] :
( ( ord_less_int @ A @ ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X3: real,Y4: real] :
( ( ord_less_real @ X3 @ Y4 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_103_order__less__subst1,axiom,
! [A: int,F: int > int,B: int,C: int] :
( ( ord_less_int @ A @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X3: int,Y4: int] :
( ( ord_less_int @ X3 @ Y4 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_104_order__less__irrefl,axiom,
! [X2: nat] :
~ ( ord_less_nat @ X2 @ X2 ) ).
% order_less_irrefl
thf(fact_105_order__less__irrefl,axiom,
! [X2: real] :
~ ( ord_less_real @ X2 @ X2 ) ).
% order_less_irrefl
thf(fact_106_order__less__irrefl,axiom,
! [X2: int] :
~ ( ord_less_int @ X2 @ X2 ) ).
% order_less_irrefl
thf(fact_107_ord__less__eq__subst,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_nat @ X3 @ Y4 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_108_ord__less__eq__subst,axiom,
! [A: nat,B: nat,F: nat > real,C: real] :
( ( ord_less_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_nat @ X3 @ Y4 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_109_ord__less__eq__subst,axiom,
! [A: nat,B: nat,F: nat > int,C: int] :
( ( ord_less_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_nat @ X3 @ Y4 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_110_ord__less__eq__subst,axiom,
! [A: real,B: real,F: real > nat,C: nat] :
( ( ord_less_real @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: real,Y4: real] :
( ( ord_less_real @ X3 @ Y4 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_111_ord__less__eq__subst,axiom,
! [A: real,B: real,F: real > real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: real,Y4: real] :
( ( ord_less_real @ X3 @ Y4 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_112_ord__less__eq__subst,axiom,
! [A: real,B: real,F: real > int,C: int] :
( ( ord_less_real @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: real,Y4: real] :
( ( ord_less_real @ X3 @ Y4 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_113_ord__less__eq__subst,axiom,
! [A: int,B: int,F: int > nat,C: nat] :
( ( ord_less_int @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: int,Y4: int] :
( ( ord_less_int @ X3 @ Y4 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_114_ord__less__eq__subst,axiom,
! [A: int,B: int,F: int > real,C: real] :
( ( ord_less_int @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: int,Y4: int] :
( ( ord_less_int @ X3 @ Y4 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_115_ord__less__eq__subst,axiom,
! [A: int,B: int,F: int > int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: int,Y4: int] :
( ( ord_less_int @ X3 @ Y4 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_116_ord__eq__less__subst,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_nat @ X3 @ Y4 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_117_ord__eq__less__subst,axiom,
! [A: real,F: nat > real,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_nat @ X3 @ Y4 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_118_ord__eq__less__subst,axiom,
! [A: int,F: nat > int,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_nat @ X3 @ Y4 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_119_ord__eq__less__subst,axiom,
! [A: nat,F: real > nat,B: real,C: real] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X3: real,Y4: real] :
( ( ord_less_real @ X3 @ Y4 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_120_ord__eq__less__subst,axiom,
! [A: real,F: real > real,B: real,C: real] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X3: real,Y4: real] :
( ( ord_less_real @ X3 @ Y4 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_121_ord__eq__less__subst,axiom,
! [A: int,F: real > int,B: real,C: real] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X3: real,Y4: real] :
( ( ord_less_real @ X3 @ Y4 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_122_ord__eq__less__subst,axiom,
! [A: nat,F: int > nat,B: int,C: int] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X3: int,Y4: int] :
( ( ord_less_int @ X3 @ Y4 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_123_ord__eq__less__subst,axiom,
! [A: real,F: int > real,B: int,C: int] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X3: int,Y4: int] :
( ( ord_less_int @ X3 @ Y4 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_124_ord__eq__less__subst,axiom,
! [A: int,F: int > int,B: int,C: int] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X3: int,Y4: int] :
( ( ord_less_int @ X3 @ Y4 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_125_order__less__trans,axiom,
! [X2: nat,Y2: nat,Z2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ( ord_less_nat @ Y2 @ Z2 )
=> ( ord_less_nat @ X2 @ Z2 ) ) ) ).
% order_less_trans
thf(fact_126_order__less__trans,axiom,
! [X2: real,Y2: real,Z2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ( ( ord_less_real @ Y2 @ Z2 )
=> ( ord_less_real @ X2 @ Z2 ) ) ) ).
% order_less_trans
thf(fact_127_order__less__trans,axiom,
! [X2: int,Y2: int,Z2: int] :
( ( ord_less_int @ X2 @ Y2 )
=> ( ( ord_less_int @ Y2 @ Z2 )
=> ( ord_less_int @ X2 @ Z2 ) ) ) ).
% order_less_trans
thf(fact_128_order__less__asym_H,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% order_less_asym'
thf(fact_129_order__less__asym_H,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ~ ( ord_less_real @ B @ A ) ) ).
% order_less_asym'
thf(fact_130_order__less__asym_H,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ~ ( ord_less_int @ B @ A ) ) ).
% order_less_asym'
thf(fact_131_linorder__neq__iff,axiom,
! [X2: nat,Y2: nat] :
( ( X2 != Y2 )
= ( ( ord_less_nat @ X2 @ Y2 )
| ( ord_less_nat @ Y2 @ X2 ) ) ) ).
% linorder_neq_iff
thf(fact_132_linorder__neq__iff,axiom,
! [X2: real,Y2: real] :
( ( X2 != Y2 )
= ( ( ord_less_real @ X2 @ Y2 )
| ( ord_less_real @ Y2 @ X2 ) ) ) ).
% linorder_neq_iff
thf(fact_133_linorder__neq__iff,axiom,
! [X2: int,Y2: int] :
( ( X2 != Y2 )
= ( ( ord_less_int @ X2 @ Y2 )
| ( ord_less_int @ Y2 @ X2 ) ) ) ).
% linorder_neq_iff
thf(fact_134_order__less__asym,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ~ ( ord_less_nat @ Y2 @ X2 ) ) ).
% order_less_asym
thf(fact_135_order__less__asym,axiom,
! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ~ ( ord_less_real @ Y2 @ X2 ) ) ).
% order_less_asym
thf(fact_136_order__less__asym,axiom,
! [X2: int,Y2: int] :
( ( ord_less_int @ X2 @ Y2 )
=> ~ ( ord_less_int @ Y2 @ X2 ) ) ).
% order_less_asym
thf(fact_137_linorder__neqE,axiom,
! [X2: nat,Y2: nat] :
( ( X2 != Y2 )
=> ( ~ ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_nat @ Y2 @ X2 ) ) ) ).
% linorder_neqE
thf(fact_138_linorder__neqE,axiom,
! [X2: real,Y2: real] :
( ( X2 != Y2 )
=> ( ~ ( ord_less_real @ X2 @ Y2 )
=> ( ord_less_real @ Y2 @ X2 ) ) ) ).
% linorder_neqE
thf(fact_139_linorder__neqE,axiom,
! [X2: int,Y2: int] :
( ( X2 != Y2 )
=> ( ~ ( ord_less_int @ X2 @ Y2 )
=> ( ord_less_int @ Y2 @ X2 ) ) ) ).
% linorder_neqE
thf(fact_140_dual__order_Ostrict__implies__not__eq,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_141_dual__order_Ostrict__implies__not__eq,axiom,
! [B: real,A: real] :
( ( ord_less_real @ B @ A )
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_142_dual__order_Ostrict__implies__not__eq,axiom,
! [B: int,A: int] :
( ( ord_less_int @ B @ A )
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_143_order_Ostrict__implies__not__eq,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_144_order_Ostrict__implies__not__eq,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_145_order_Ostrict__implies__not__eq,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_146_dual__order_Ostrict__trans,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( ord_less_nat @ C @ B )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_147_dual__order_Ostrict__trans,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_real @ B @ A )
=> ( ( ord_less_real @ C @ B )
=> ( ord_less_real @ C @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_148_dual__order_Ostrict__trans,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_int @ B @ A )
=> ( ( ord_less_int @ C @ B )
=> ( ord_less_int @ C @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_149_mem__Collect__eq,axiom,
! [A: a,P: a > $o] :
( ( member_a @ A @ ( collect_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_150_mem__Collect__eq,axiom,
! [A: real,P: real > $o] :
( ( member_real @ A @ ( collect_real @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_151_mem__Collect__eq,axiom,
! [A: int,P: int > $o] :
( ( member_int @ A @ ( collect_int @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_152_mem__Collect__eq,axiom,
! [A: nat,P: nat > $o] :
( ( member_nat @ A @ ( collect_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_153_Collect__mem__eq,axiom,
! [A2: set_a] :
( ( collect_a
@ ^ [X: a] : ( member_a @ X @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_154_Collect__mem__eq,axiom,
! [A2: set_real] :
( ( collect_real
@ ^ [X: real] : ( member_real @ X @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_155_Collect__mem__eq,axiom,
! [A2: set_int] :
( ( collect_int
@ ^ [X: int] : ( member_int @ X @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_156_Collect__mem__eq,axiom,
! [A2: set_nat] :
( ( collect_nat
@ ^ [X: nat] : ( member_nat @ X @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_157_not__less__iff__gr__or__eq,axiom,
! [X2: nat,Y2: nat] :
( ( ~ ( ord_less_nat @ X2 @ Y2 ) )
= ( ( ord_less_nat @ Y2 @ X2 )
| ( X2 = Y2 ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_158_not__less__iff__gr__or__eq,axiom,
! [X2: real,Y2: real] :
( ( ~ ( ord_less_real @ X2 @ Y2 ) )
= ( ( ord_less_real @ Y2 @ X2 )
| ( X2 = Y2 ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_159_not__less__iff__gr__or__eq,axiom,
! [X2: int,Y2: int] :
( ( ~ ( ord_less_int @ X2 @ Y2 ) )
= ( ( ord_less_int @ Y2 @ X2 )
| ( X2 = Y2 ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_160_order_Ostrict__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans
thf(fact_161_order_Ostrict__trans,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ B @ C )
=> ( ord_less_real @ A @ C ) ) ) ).
% order.strict_trans
thf(fact_162_order_Ostrict__trans,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ B @ C )
=> ( ord_less_int @ A @ C ) ) ) ).
% order.strict_trans
thf(fact_163_linorder__less__wlog,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A3: nat,B2: nat] :
( ( ord_less_nat @ A3 @ B2 )
=> ( P @ A3 @ B2 ) )
=> ( ! [A3: nat] : ( P @ A3 @ A3 )
=> ( ! [A3: nat,B2: nat] :
( ( P @ B2 @ A3 )
=> ( P @ A3 @ B2 ) )
=> ( P @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_164_linorder__less__wlog,axiom,
! [P: real > real > $o,A: real,B: real] :
( ! [A3: real,B2: real] :
( ( ord_less_real @ A3 @ B2 )
=> ( P @ A3 @ B2 ) )
=> ( ! [A3: real] : ( P @ A3 @ A3 )
=> ( ! [A3: real,B2: real] :
( ( P @ B2 @ A3 )
=> ( P @ A3 @ B2 ) )
=> ( P @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_165_linorder__less__wlog,axiom,
! [P: int > int > $o,A: int,B: int] :
( ! [A3: int,B2: int] :
( ( ord_less_int @ A3 @ B2 )
=> ( P @ A3 @ B2 ) )
=> ( ! [A3: int] : ( P @ A3 @ A3 )
=> ( ! [A3: int,B2: int] :
( ( P @ B2 @ A3 )
=> ( P @ A3 @ B2 ) )
=> ( P @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_166_exists__least__iff,axiom,
( ( ^ [P2: nat > $o] :
? [X4: nat] : ( P2 @ X4 ) )
= ( ^ [P3: nat > $o] :
? [N: nat] :
( ( P3 @ N )
& ! [M: nat] :
( ( ord_less_nat @ M @ N )
=> ~ ( P3 @ M ) ) ) ) ) ).
% exists_least_iff
thf(fact_167_dual__order_Oirrefl,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% dual_order.irrefl
thf(fact_168_dual__order_Oirrefl,axiom,
! [A: real] :
~ ( ord_less_real @ A @ A ) ).
% dual_order.irrefl
thf(fact_169_dual__order_Oirrefl,axiom,
! [A: int] :
~ ( ord_less_int @ A @ A ) ).
% dual_order.irrefl
thf(fact_170_dual__order_Oasym,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ~ ( ord_less_nat @ A @ B ) ) ).
% dual_order.asym
thf(fact_171_dual__order_Oasym,axiom,
! [B: real,A: real] :
( ( ord_less_real @ B @ A )
=> ~ ( ord_less_real @ A @ B ) ) ).
% dual_order.asym
thf(fact_172_dual__order_Oasym,axiom,
! [B: int,A: int] :
( ( ord_less_int @ B @ A )
=> ~ ( ord_less_int @ A @ B ) ) ).
% dual_order.asym
thf(fact_173_linorder__cases,axiom,
! [X2: nat,Y2: nat] :
( ~ ( ord_less_nat @ X2 @ Y2 )
=> ( ( X2 != Y2 )
=> ( ord_less_nat @ Y2 @ X2 ) ) ) ).
% linorder_cases
thf(fact_174_linorder__cases,axiom,
! [X2: real,Y2: real] :
( ~ ( ord_less_real @ X2 @ Y2 )
=> ( ( X2 != Y2 )
=> ( ord_less_real @ Y2 @ X2 ) ) ) ).
% linorder_cases
thf(fact_175_linorder__cases,axiom,
! [X2: int,Y2: int] :
( ~ ( ord_less_int @ X2 @ Y2 )
=> ( ( X2 != Y2 )
=> ( ord_less_int @ Y2 @ X2 ) ) ) ).
% linorder_cases
thf(fact_176_antisym__conv3,axiom,
! [Y2: nat,X2: nat] :
( ~ ( ord_less_nat @ Y2 @ X2 )
=> ( ( ~ ( ord_less_nat @ X2 @ Y2 ) )
= ( X2 = Y2 ) ) ) ).
% antisym_conv3
thf(fact_177_antisym__conv3,axiom,
! [Y2: real,X2: real] :
( ~ ( ord_less_real @ Y2 @ X2 )
=> ( ( ~ ( ord_less_real @ X2 @ Y2 ) )
= ( X2 = Y2 ) ) ) ).
% antisym_conv3
thf(fact_178_antisym__conv3,axiom,
! [Y2: int,X2: int] :
( ~ ( ord_less_int @ Y2 @ X2 )
=> ( ( ~ ( ord_less_int @ X2 @ Y2 ) )
= ( X2 = Y2 ) ) ) ).
% antisym_conv3
thf(fact_179_less__induct,axiom,
! [P: nat > $o,A: nat] :
( ! [X3: nat] :
( ! [Y5: nat] :
( ( ord_less_nat @ Y5 @ X3 )
=> ( P @ Y5 ) )
=> ( P @ X3 ) )
=> ( P @ A ) ) ).
% less_induct
thf(fact_180_ord__less__eq__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( B = C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_181_ord__less__eq__trans,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( B = C )
=> ( ord_less_real @ A @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_182_ord__less__eq__trans,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( B = C )
=> ( ord_less_int @ A @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_183_ord__eq__less__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( A = B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_184_ord__eq__less__trans,axiom,
! [A: real,B: real,C: real] :
( ( A = B )
=> ( ( ord_less_real @ B @ C )
=> ( ord_less_real @ A @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_185_ord__eq__less__trans,axiom,
! [A: int,B: int,C: int] :
( ( A = B )
=> ( ( ord_less_int @ B @ C )
=> ( ord_less_int @ A @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_186_order_Oasym,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% order.asym
thf(fact_187_order_Oasym,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ~ ( ord_less_real @ B @ A ) ) ).
% order.asym
thf(fact_188_order_Oasym,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ~ ( ord_less_int @ B @ A ) ) ).
% order.asym
thf(fact_189_less__imp__neq,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( X2 != Y2 ) ) ).
% less_imp_neq
thf(fact_190_less__imp__neq,axiom,
! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ( X2 != Y2 ) ) ).
% less_imp_neq
thf(fact_191_less__imp__neq,axiom,
! [X2: int,Y2: int] :
( ( ord_less_int @ X2 @ Y2 )
=> ( X2 != Y2 ) ) ).
% less_imp_neq
thf(fact_192_dense,axiom,
! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ? [Z4: real] :
( ( ord_less_real @ X2 @ Z4 )
& ( ord_less_real @ Z4 @ Y2 ) ) ) ).
% dense
thf(fact_193_gt__ex,axiom,
! [X2: nat] :
? [X_1: nat] : ( ord_less_nat @ X2 @ X_1 ) ).
% gt_ex
thf(fact_194_gt__ex,axiom,
! [X2: real] :
? [X_1: real] : ( ord_less_real @ X2 @ X_1 ) ).
% gt_ex
thf(fact_195_gt__ex,axiom,
! [X2: int] :
? [X_1: int] : ( ord_less_int @ X2 @ X_1 ) ).
% gt_ex
thf(fact_196_lt__ex,axiom,
! [X2: real] :
? [Y4: real] : ( ord_less_real @ Y4 @ X2 ) ).
% lt_ex
thf(fact_197_lt__ex,axiom,
! [X2: int] :
? [Y4: int] : ( ord_less_int @ Y4 @ X2 ) ).
% lt_ex
thf(fact_198_verit__comp__simplify1_I1_J,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_199_verit__comp__simplify1_I1_J,axiom,
! [A: real] :
~ ( ord_less_real @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_200_verit__comp__simplify1_I1_J,axiom,
! [A: int] :
~ ( ord_less_int @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_201_order__le__imp__less__or__eq,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ( ord_less_nat @ X2 @ Y2 )
| ( X2 = Y2 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_202_order__le__imp__less__or__eq,axiom,
! [X2: a,Y2: a] :
( ( ord_less_eq_a @ X2 @ Y2 )
=> ( ( ord_less_a @ X2 @ Y2 )
| ( X2 = Y2 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_203_order__le__imp__less__or__eq,axiom,
! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ( ord_less_real @ X2 @ Y2 )
| ( X2 = Y2 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_204_order__le__imp__less__or__eq,axiom,
! [X2: int,Y2: int] :
( ( ord_less_eq_int @ X2 @ Y2 )
=> ( ( ord_less_int @ X2 @ Y2 )
| ( X2 = Y2 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_205_linorder__le__less__linear,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
| ( ord_less_nat @ Y2 @ X2 ) ) ).
% linorder_le_less_linear
thf(fact_206_linorder__le__less__linear,axiom,
! [X2: a,Y2: a] :
( ( ord_less_eq_a @ X2 @ Y2 )
| ( ord_less_a @ Y2 @ X2 ) ) ).
% linorder_le_less_linear
thf(fact_207_linorder__le__less__linear,axiom,
! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
| ( ord_less_real @ Y2 @ X2 ) ) ).
% linorder_le_less_linear
thf(fact_208_linorder__le__less__linear,axiom,
! [X2: int,Y2: int] :
( ( ord_less_eq_int @ X2 @ Y2 )
| ( ord_less_int @ Y2 @ X2 ) ) ).
% linorder_le_less_linear
thf(fact_209_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_nat @ X3 @ Y4 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_210_order__less__le__subst2,axiom,
! [A: real,B: real,F: real > nat,C: nat] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X3: real,Y4: real] :
( ( ord_less_real @ X3 @ Y4 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_211_order__less__le__subst2,axiom,
! [A: int,B: int,F: int > nat,C: nat] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X3: int,Y4: int] :
( ( ord_less_int @ X3 @ Y4 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_212_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > a,C: a] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_a @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_nat @ X3 @ Y4 )
=> ( ord_less_a @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_a @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_213_order__less__le__subst2,axiom,
! [A: real,B: real,F: real > a,C: a] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_eq_a @ ( F @ B ) @ C )
=> ( ! [X3: real,Y4: real] :
( ( ord_less_real @ X3 @ Y4 )
=> ( ord_less_a @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_a @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_214_order__less__le__subst2,axiom,
! [A: int,B: int,F: int > a,C: a] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_a @ ( F @ B ) @ C )
=> ( ! [X3: int,Y4: int] :
( ( ord_less_int @ X3 @ Y4 )
=> ( ord_less_a @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_a @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_215_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > real,C: real] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_nat @ X3 @ Y4 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_216_order__less__le__subst2,axiom,
! [A: real,B: real,F: real > real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C )
=> ( ! [X3: real,Y4: real] :
( ( ord_less_real @ X3 @ Y4 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_217_order__less__le__subst2,axiom,
! [A: int,B: int,F: int > real,C: real] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C )
=> ( ! [X3: int,Y4: int] :
( ( ord_less_int @ X3 @ Y4 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_218_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > int,C: int] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_nat @ X3 @ Y4 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_219_order__less__le__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_220_order__less__le__subst1,axiom,
! [A: a,F: nat > a,B: nat,C: nat] :
( ( ord_less_a @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_less_eq_a @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_a @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_221_order__less__le__subst1,axiom,
! [A: real,F: nat > real,B: nat,C: nat] :
( ( ord_less_real @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_222_order__less__le__subst1,axiom,
! [A: int,F: nat > int,B: nat,C: nat] :
( ( ord_less_int @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_223_order__less__le__subst1,axiom,
! [A: nat,F: a > nat,B: a,C: a] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_a @ B @ C )
=> ( ! [X3: a,Y4: a] :
( ( ord_less_eq_a @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_224_order__less__le__subst1,axiom,
! [A: a,F: a > a,B: a,C: a] :
( ( ord_less_a @ A @ ( F @ B ) )
=> ( ( ord_less_eq_a @ B @ C )
=> ( ! [X3: a,Y4: a] :
( ( ord_less_eq_a @ X3 @ Y4 )
=> ( ord_less_eq_a @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_a @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_225_order__less__le__subst1,axiom,
! [A: real,F: a > real,B: a,C: a] :
( ( ord_less_real @ A @ ( F @ B ) )
=> ( ( ord_less_eq_a @ B @ C )
=> ( ! [X3: a,Y4: a] :
( ( ord_less_eq_a @ X3 @ Y4 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_226_order__less__le__subst1,axiom,
! [A: int,F: a > int,B: a,C: a] :
( ( ord_less_int @ A @ ( F @ B ) )
=> ( ( ord_less_eq_a @ B @ C )
=> ( ! [X3: a,Y4: a] :
( ( ord_less_eq_a @ X3 @ Y4 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_227_order__less__le__subst1,axiom,
! [A: nat,F: real > nat,B: real,C: real] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X3: real,Y4: real] :
( ( ord_less_eq_real @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_228_order__less__le__subst1,axiom,
! [A: a,F: real > a,B: real,C: real] :
( ( ord_less_a @ A @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X3: real,Y4: real] :
( ( ord_less_eq_real @ X3 @ Y4 )
=> ( ord_less_eq_a @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_a @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_229_order__le__less__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_230_order__le__less__subst2,axiom,
! [A: nat,B: nat,F: nat > a,C: a] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_a @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_less_eq_a @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_a @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_231_order__le__less__subst2,axiom,
! [A: nat,B: nat,F: nat > real,C: real] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_real @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_232_order__le__less__subst2,axiom,
! [A: nat,B: nat,F: nat > int,C: int] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_233_order__le__less__subst2,axiom,
! [A: a,B: a,F: a > nat,C: nat] :
( ( ord_less_eq_a @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X3: a,Y4: a] :
( ( ord_less_eq_a @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_234_order__le__less__subst2,axiom,
! [A: a,B: a,F: a > a,C: a] :
( ( ord_less_eq_a @ A @ B )
=> ( ( ord_less_a @ ( F @ B ) @ C )
=> ( ! [X3: a,Y4: a] :
( ( ord_less_eq_a @ X3 @ Y4 )
=> ( ord_less_eq_a @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_a @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_235_order__le__less__subst2,axiom,
! [A: a,B: a,F: a > real,C: real] :
( ( ord_less_eq_a @ A @ B )
=> ( ( ord_less_real @ ( F @ B ) @ C )
=> ( ! [X3: a,Y4: a] :
( ( ord_less_eq_a @ X3 @ Y4 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_236_order__le__less__subst2,axiom,
! [A: a,B: a,F: a > int,C: int] :
( ( ord_less_eq_a @ A @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C )
=> ( ! [X3: a,Y4: a] :
( ( ord_less_eq_a @ X3 @ Y4 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_237_order__le__less__subst2,axiom,
! [A: real,B: real,F: real > nat,C: nat] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X3: real,Y4: real] :
( ( ord_less_eq_real @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_238_order__le__less__subst2,axiom,
! [A: real,B: real,F: real > a,C: a] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_a @ ( F @ B ) @ C )
=> ( ! [X3: real,Y4: real] :
( ( ord_less_eq_real @ X3 @ Y4 )
=> ( ord_less_eq_a @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_a @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_239_order__le__less__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_nat @ X3 @ Y4 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_240_order__le__less__subst1,axiom,
! [A: nat,F: real > nat,B: real,C: real] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X3: real,Y4: real] :
( ( ord_less_real @ X3 @ Y4 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_241_order__le__less__subst1,axiom,
! [A: nat,F: int > nat,B: int,C: int] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X3: int,Y4: int] :
( ( ord_less_int @ X3 @ Y4 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_242_order__le__less__subst1,axiom,
! [A: a,F: nat > a,B: nat,C: nat] :
( ( ord_less_eq_a @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_nat @ X3 @ Y4 )
=> ( ord_less_a @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_a @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_243_order__le__less__subst1,axiom,
! [A: a,F: real > a,B: real,C: real] :
( ( ord_less_eq_a @ A @ ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X3: real,Y4: real] :
( ( ord_less_real @ X3 @ Y4 )
=> ( ord_less_a @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_a @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_244_order__le__less__subst1,axiom,
! [A: a,F: int > a,B: int,C: int] :
( ( ord_less_eq_a @ A @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X3: int,Y4: int] :
( ( ord_less_int @ X3 @ Y4 )
=> ( ord_less_a @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_a @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_245_order__le__less__subst1,axiom,
! [A: real,F: nat > real,B: nat,C: nat] :
( ( ord_less_eq_real @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_nat @ X3 @ Y4 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_246_order__le__less__subst1,axiom,
! [A: real,F: real > real,B: real,C: real] :
( ( ord_less_eq_real @ A @ ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X3: real,Y4: real] :
( ( ord_less_real @ X3 @ Y4 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_247_order__le__less__subst1,axiom,
! [A: real,F: int > real,B: int,C: int] :
( ( ord_less_eq_real @ A @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X3: int,Y4: int] :
( ( ord_less_int @ X3 @ Y4 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_248_order__le__less__subst1,axiom,
! [A: int,F: nat > int,B: nat,C: nat] :
( ( ord_less_eq_int @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_nat @ X3 @ Y4 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_249_order__less__le__trans,axiom,
! [X2: nat,Y2: nat,Z2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ( ord_less_eq_nat @ Y2 @ Z2 )
=> ( ord_less_nat @ X2 @ Z2 ) ) ) ).
% order_less_le_trans
thf(fact_250_order__less__le__trans,axiom,
! [X2: a,Y2: a,Z2: a] :
( ( ord_less_a @ X2 @ Y2 )
=> ( ( ord_less_eq_a @ Y2 @ Z2 )
=> ( ord_less_a @ X2 @ Z2 ) ) ) ).
% order_less_le_trans
thf(fact_251_order__less__le__trans,axiom,
! [X2: real,Y2: real,Z2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ( ( ord_less_eq_real @ Y2 @ Z2 )
=> ( ord_less_real @ X2 @ Z2 ) ) ) ).
% order_less_le_trans
thf(fact_252_order__less__le__trans,axiom,
! [X2: int,Y2: int,Z2: int] :
( ( ord_less_int @ X2 @ Y2 )
=> ( ( ord_less_eq_int @ Y2 @ Z2 )
=> ( ord_less_int @ X2 @ Z2 ) ) ) ).
% order_less_le_trans
thf(fact_253_order__le__less__trans,axiom,
! [X2: nat,Y2: nat,Z2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ( ord_less_nat @ Y2 @ Z2 )
=> ( ord_less_nat @ X2 @ Z2 ) ) ) ).
% order_le_less_trans
thf(fact_254_order__le__less__trans,axiom,
! [X2: a,Y2: a,Z2: a] :
( ( ord_less_eq_a @ X2 @ Y2 )
=> ( ( ord_less_a @ Y2 @ Z2 )
=> ( ord_less_a @ X2 @ Z2 ) ) ) ).
% order_le_less_trans
thf(fact_255_order__le__less__trans,axiom,
! [X2: real,Y2: real,Z2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ( ord_less_real @ Y2 @ Z2 )
=> ( ord_less_real @ X2 @ Z2 ) ) ) ).
% order_le_less_trans
thf(fact_256_order__le__less__trans,axiom,
! [X2: int,Y2: int,Z2: int] :
( ( ord_less_eq_int @ X2 @ Y2 )
=> ( ( ord_less_int @ Y2 @ Z2 )
=> ( ord_less_int @ X2 @ Z2 ) ) ) ).
% order_le_less_trans
thf(fact_257_order__neq__le__trans,axiom,
! [A: nat,B: nat] :
( ( A != B )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_nat @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_258_order__neq__le__trans,axiom,
! [A: a,B: a] :
( ( A != B )
=> ( ( ord_less_eq_a @ A @ B )
=> ( ord_less_a @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_259_order__neq__le__trans,axiom,
! [A: real,B: real] :
( ( A != B )
=> ( ( ord_less_eq_real @ A @ B )
=> ( ord_less_real @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_260_order__neq__le__trans,axiom,
! [A: int,B: int] :
( ( A != B )
=> ( ( ord_less_eq_int @ A @ B )
=> ( ord_less_int @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_261_order__le__neq__trans,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( A != B )
=> ( ord_less_nat @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_262_order__le__neq__trans,axiom,
! [A: a,B: a] :
( ( ord_less_eq_a @ A @ B )
=> ( ( A != B )
=> ( ord_less_a @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_263_order__le__neq__trans,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( A != B )
=> ( ord_less_real @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_264_order__le__neq__trans,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( A != B )
=> ( ord_less_int @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_265_order__less__imp__le,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ X2 @ Y2 ) ) ).
% order_less_imp_le
thf(fact_266_order__less__imp__le,axiom,
! [X2: a,Y2: a] :
( ( ord_less_a @ X2 @ Y2 )
=> ( ord_less_eq_a @ X2 @ Y2 ) ) ).
% order_less_imp_le
thf(fact_267_order__less__imp__le,axiom,
! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ( ord_less_eq_real @ X2 @ Y2 ) ) ).
% order_less_imp_le
thf(fact_268_order__less__imp__le,axiom,
! [X2: int,Y2: int] :
( ( ord_less_int @ X2 @ Y2 )
=> ( ord_less_eq_int @ X2 @ Y2 ) ) ).
% order_less_imp_le
thf(fact_269_linorder__not__less,axiom,
! [X2: nat,Y2: nat] :
( ( ~ ( ord_less_nat @ X2 @ Y2 ) )
= ( ord_less_eq_nat @ Y2 @ X2 ) ) ).
% linorder_not_less
thf(fact_270_linorder__not__less,axiom,
! [X2: a,Y2: a] :
( ( ~ ( ord_less_a @ X2 @ Y2 ) )
= ( ord_less_eq_a @ Y2 @ X2 ) ) ).
% linorder_not_less
thf(fact_271_linorder__not__less,axiom,
! [X2: real,Y2: real] :
( ( ~ ( ord_less_real @ X2 @ Y2 ) )
= ( ord_less_eq_real @ Y2 @ X2 ) ) ).
% linorder_not_less
thf(fact_272_linorder__not__less,axiom,
! [X2: int,Y2: int] :
( ( ~ ( ord_less_int @ X2 @ Y2 ) )
= ( ord_less_eq_int @ Y2 @ X2 ) ) ).
% linorder_not_less
thf(fact_273_linorder__not__le,axiom,
! [X2: nat,Y2: nat] :
( ( ~ ( ord_less_eq_nat @ X2 @ Y2 ) )
= ( ord_less_nat @ Y2 @ X2 ) ) ).
% linorder_not_le
thf(fact_274_linorder__not__le,axiom,
! [X2: a,Y2: a] :
( ( ~ ( ord_less_eq_a @ X2 @ Y2 ) )
= ( ord_less_a @ Y2 @ X2 ) ) ).
% linorder_not_le
thf(fact_275_linorder__not__le,axiom,
! [X2: real,Y2: real] :
( ( ~ ( ord_less_eq_real @ X2 @ Y2 ) )
= ( ord_less_real @ Y2 @ X2 ) ) ).
% linorder_not_le
thf(fact_276_linorder__not__le,axiom,
! [X2: int,Y2: int] :
( ( ~ ( ord_less_eq_int @ X2 @ Y2 ) )
= ( ord_less_int @ Y2 @ X2 ) ) ).
% linorder_not_le
thf(fact_277_order__less__le,axiom,
( ord_less_nat
= ( ^ [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
& ( X != Y ) ) ) ) ).
% order_less_le
thf(fact_278_order__less__le,axiom,
( ord_less_a
= ( ^ [X: a,Y: a] :
( ( ord_less_eq_a @ X @ Y )
& ( X != Y ) ) ) ) ).
% order_less_le
thf(fact_279_order__less__le,axiom,
( ord_less_real
= ( ^ [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
& ( X != Y ) ) ) ) ).
% order_less_le
thf(fact_280_order__less__le,axiom,
( ord_less_int
= ( ^ [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
& ( X != Y ) ) ) ) ).
% order_less_le
thf(fact_281_order__le__less,axiom,
( ord_less_eq_nat
= ( ^ [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
| ( X = Y ) ) ) ) ).
% order_le_less
thf(fact_282_order__le__less,axiom,
( ord_less_eq_a
= ( ^ [X: a,Y: a] :
( ( ord_less_a @ X @ Y )
| ( X = Y ) ) ) ) ).
% order_le_less
thf(fact_283_order__le__less,axiom,
( ord_less_eq_real
= ( ^ [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
| ( X = Y ) ) ) ) ).
% order_le_less
thf(fact_284_order__le__less,axiom,
( ord_less_eq_int
= ( ^ [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
| ( X = Y ) ) ) ) ).
% order_le_less
thf(fact_285_dual__order_Ostrict__implies__order,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( ord_less_eq_nat @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_286_dual__order_Ostrict__implies__order,axiom,
! [B: a,A: a] :
( ( ord_less_a @ B @ A )
=> ( ord_less_eq_a @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_287_dual__order_Ostrict__implies__order,axiom,
! [B: real,A: real] :
( ( ord_less_real @ B @ A )
=> ( ord_less_eq_real @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_288_dual__order_Ostrict__implies__order,axiom,
! [B: int,A: int] :
( ( ord_less_int @ B @ A )
=> ( ord_less_eq_int @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_289_order_Ostrict__implies__order,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_290_order_Ostrict__implies__order,axiom,
! [A: a,B: a] :
( ( ord_less_a @ A @ B )
=> ( ord_less_eq_a @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_291_order_Ostrict__implies__order,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ( ord_less_eq_real @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_292_order_Ostrict__implies__order,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_eq_int @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_293_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_294_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_295_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_296_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_297_dual__order_Ostrict__trans2,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_298_dual__order_Ostrict__trans2,axiom,
! [B: a,A: a,C: a] :
( ( ord_less_a @ B @ A )
=> ( ( ord_less_eq_a @ C @ B )
=> ( ord_less_a @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_299_dual__order_Ostrict__trans2,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_real @ B @ A )
=> ( ( ord_less_eq_real @ C @ B )
=> ( ord_less_real @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_300_dual__order_Ostrict__trans2,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_int @ B @ A )
=> ( ( ord_less_eq_int @ C @ B )
=> ( ord_less_int @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_301_dual__order_Ostrict__trans1,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_nat @ C @ B )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_302_dual__order_Ostrict__trans1,axiom,
! [B: a,A: a,C: a] :
( ( ord_less_eq_a @ B @ A )
=> ( ( ord_less_a @ C @ B )
=> ( ord_less_a @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_303_dual__order_Ostrict__trans1,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ( ord_less_real @ C @ B )
=> ( ord_less_real @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_304_dual__order_Ostrict__trans1,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( ord_less_int @ C @ B )
=> ( ord_less_int @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_305_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_306_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_307_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_308_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_309_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_310_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_311_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_312_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_313_dense__le__bounded,axiom,
! [X2: real,Y2: real,Z2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ( ! [W: real] :
( ( ord_less_real @ X2 @ W )
=> ( ( ord_less_real @ W @ Y2 )
=> ( ord_less_eq_real @ W @ Z2 ) ) )
=> ( ord_less_eq_real @ Y2 @ Z2 ) ) ) ).
% dense_le_bounded
thf(fact_314_dense__ge__bounded,axiom,
! [Z2: real,X2: real,Y2: real] :
( ( ord_less_real @ Z2 @ X2 )
=> ( ! [W: real] :
( ( ord_less_real @ Z2 @ W )
=> ( ( ord_less_real @ W @ X2 )
=> ( ord_less_eq_real @ Y2 @ W ) ) )
=> ( ord_less_eq_real @ Y2 @ Z2 ) ) ) ).
% dense_ge_bounded
thf(fact_315_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_316_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_317_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_318_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_319_order_Ostrict__trans2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_320_order_Ostrict__trans2,axiom,
! [A: a,B: a,C: a] :
( ( ord_less_a @ A @ B )
=> ( ( ord_less_eq_a @ B @ C )
=> ( ord_less_a @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_321_order_Ostrict__trans2,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ord_less_real @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_322_order_Ostrict__trans2,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ord_less_int @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_323_order_Ostrict__trans1,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_324_order_Ostrict__trans1,axiom,
! [A: a,B: a,C: a] :
( ( ord_less_eq_a @ A @ B )
=> ( ( ord_less_a @ B @ C )
=> ( ord_less_a @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_325_order_Ostrict__trans1,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_real @ B @ C )
=> ( ord_less_real @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_326_order_Ostrict__trans1,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_int @ B @ C )
=> ( ord_less_int @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_327_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_328_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_329_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_330_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_331_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_332_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_333_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_334_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_335_not__le__imp__less,axiom,
! [Y2: nat,X2: nat] :
( ~ ( ord_less_eq_nat @ Y2 @ X2 )
=> ( ord_less_nat @ X2 @ Y2 ) ) ).
% not_le_imp_less
thf(fact_336_not__le__imp__less,axiom,
! [Y2: a,X2: a] :
( ~ ( ord_less_eq_a @ Y2 @ X2 )
=> ( ord_less_a @ X2 @ Y2 ) ) ).
% not_le_imp_less
thf(fact_337_not__le__imp__less,axiom,
! [Y2: real,X2: real] :
( ~ ( ord_less_eq_real @ Y2 @ X2 )
=> ( ord_less_real @ X2 @ Y2 ) ) ).
% not_le_imp_less
thf(fact_338_not__le__imp__less,axiom,
! [Y2: int,X2: int] :
( ~ ( ord_less_eq_int @ Y2 @ X2 )
=> ( ord_less_int @ X2 @ Y2 ) ) ).
% not_le_imp_less
thf(fact_339_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_340_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_341_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_342_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_343_dense__le,axiom,
! [Y2: real,Z2: real] :
( ! [X3: real] :
( ( ord_less_real @ X3 @ Y2 )
=> ( ord_less_eq_real @ X3 @ Z2 ) )
=> ( ord_less_eq_real @ Y2 @ Z2 ) ) ).
% dense_le
thf(fact_344_dense__ge,axiom,
! [Z2: real,Y2: real] :
( ! [X3: real] :
( ( ord_less_real @ Z2 @ X3 )
=> ( ord_less_eq_real @ Y2 @ X3 ) )
=> ( ord_less_eq_real @ Y2 @ Z2 ) ) ).
% dense_ge
thf(fact_345_antisym__conv2,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ( ~ ( ord_less_nat @ X2 @ Y2 ) )
= ( X2 = Y2 ) ) ) ).
% antisym_conv2
thf(fact_346_antisym__conv2,axiom,
! [X2: a,Y2: a] :
( ( ord_less_eq_a @ X2 @ Y2 )
=> ( ( ~ ( ord_less_a @ X2 @ Y2 ) )
= ( X2 = Y2 ) ) ) ).
% antisym_conv2
thf(fact_347_antisym__conv2,axiom,
! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ( ~ ( ord_less_real @ X2 @ Y2 ) )
= ( X2 = Y2 ) ) ) ).
% antisym_conv2
thf(fact_348_antisym__conv2,axiom,
! [X2: int,Y2: int] :
( ( ord_less_eq_int @ X2 @ Y2 )
=> ( ( ~ ( ord_less_int @ X2 @ Y2 ) )
= ( X2 = Y2 ) ) ) ).
% antisym_conv2
thf(fact_349_antisym__conv1,axiom,
! [X2: nat,Y2: nat] :
( ~ ( ord_less_nat @ X2 @ Y2 )
=> ( ( ord_less_eq_nat @ X2 @ Y2 )
= ( X2 = Y2 ) ) ) ).
% antisym_conv1
thf(fact_350_antisym__conv1,axiom,
! [X2: a,Y2: a] :
( ~ ( ord_less_a @ X2 @ Y2 )
=> ( ( ord_less_eq_a @ X2 @ Y2 )
= ( X2 = Y2 ) ) ) ).
% antisym_conv1
thf(fact_351_antisym__conv1,axiom,
! [X2: real,Y2: real] :
( ~ ( ord_less_real @ X2 @ Y2 )
=> ( ( ord_less_eq_real @ X2 @ Y2 )
= ( X2 = Y2 ) ) ) ).
% antisym_conv1
thf(fact_352_antisym__conv1,axiom,
! [X2: int,Y2: int] :
( ~ ( ord_less_int @ X2 @ Y2 )
=> ( ( ord_less_eq_int @ X2 @ Y2 )
= ( X2 = Y2 ) ) ) ).
% antisym_conv1
thf(fact_353_nless__le,axiom,
! [A: nat,B: nat] :
( ( ~ ( ord_less_nat @ A @ B ) )
= ( ~ ( ord_less_eq_nat @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_354_nless__le,axiom,
! [A: a,B: a] :
( ( ~ ( ord_less_a @ A @ B ) )
= ( ~ ( ord_less_eq_a @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_355_nless__le,axiom,
! [A: real,B: real] :
( ( ~ ( ord_less_real @ A @ B ) )
= ( ~ ( ord_less_eq_real @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_356_nless__le,axiom,
! [A: int,B: int] :
( ( ~ ( ord_less_int @ A @ B ) )
= ( ~ ( ord_less_eq_int @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_357_leI,axiom,
! [X2: nat,Y2: nat] :
( ~ ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ X2 ) ) ).
% leI
thf(fact_358_leI,axiom,
! [X2: a,Y2: a] :
( ~ ( ord_less_a @ X2 @ Y2 )
=> ( ord_less_eq_a @ Y2 @ X2 ) ) ).
% leI
thf(fact_359_leI,axiom,
! [X2: real,Y2: real] :
( ~ ( ord_less_real @ X2 @ Y2 )
=> ( ord_less_eq_real @ Y2 @ X2 ) ) ).
% leI
thf(fact_360_leI,axiom,
! [X2: int,Y2: int] :
( ~ ( ord_less_int @ X2 @ Y2 )
=> ( ord_less_eq_int @ Y2 @ X2 ) ) ).
% leI
thf(fact_361_leD,axiom,
! [Y2: nat,X2: nat] :
( ( ord_less_eq_nat @ Y2 @ X2 )
=> ~ ( ord_less_nat @ X2 @ Y2 ) ) ).
% leD
thf(fact_362_leD,axiom,
! [Y2: a,X2: a] :
( ( ord_less_eq_a @ Y2 @ X2 )
=> ~ ( ord_less_a @ X2 @ Y2 ) ) ).
% leD
thf(fact_363_leD,axiom,
! [Y2: real,X2: real] :
( ( ord_less_eq_real @ Y2 @ X2 )
=> ~ ( ord_less_real @ X2 @ Y2 ) ) ).
% leD
thf(fact_364_leD,axiom,
! [Y2: int,X2: int] :
( ( ord_less_eq_int @ Y2 @ X2 )
=> ~ ( ord_less_int @ X2 @ Y2 ) ) ).
% leD
thf(fact_365_verit__comp__simplify1_I3_J,axiom,
! [B4: nat,A5: nat] :
( ( ~ ( ord_less_eq_nat @ B4 @ A5 ) )
= ( ord_less_nat @ A5 @ B4 ) ) ).
% verit_comp_simplify1(3)
thf(fact_366_verit__comp__simplify1_I3_J,axiom,
! [B4: a,A5: a] :
( ( ~ ( ord_less_eq_a @ B4 @ A5 ) )
= ( ord_less_a @ A5 @ B4 ) ) ).
% verit_comp_simplify1(3)
thf(fact_367_verit__comp__simplify1_I3_J,axiom,
! [B4: real,A5: real] :
( ( ~ ( ord_less_eq_real @ B4 @ A5 ) )
= ( ord_less_real @ A5 @ B4 ) ) ).
% verit_comp_simplify1(3)
thf(fact_368_verit__comp__simplify1_I3_J,axiom,
! [B4: int,A5: int] :
( ( ~ ( ord_less_eq_int @ B4 @ A5 ) )
= ( ord_less_int @ A5 @ B4 ) ) ).
% verit_comp_simplify1(3)
thf(fact_369_order__antisym__conv,axiom,
! [Y2: nat,X2: nat] :
( ( ord_less_eq_nat @ Y2 @ X2 )
=> ( ( ord_less_eq_nat @ X2 @ Y2 )
= ( X2 = Y2 ) ) ) ).
% order_antisym_conv
thf(fact_370_order__antisym__conv,axiom,
! [Y2: a,X2: a] :
( ( ord_less_eq_a @ Y2 @ X2 )
=> ( ( ord_less_eq_a @ X2 @ Y2 )
= ( X2 = Y2 ) ) ) ).
% order_antisym_conv
thf(fact_371_order__antisym__conv,axiom,
! [Y2: real,X2: real] :
( ( ord_less_eq_real @ Y2 @ X2 )
=> ( ( ord_less_eq_real @ X2 @ Y2 )
= ( X2 = Y2 ) ) ) ).
% order_antisym_conv
thf(fact_372_order__antisym__conv,axiom,
! [Y2: int,X2: int] :
( ( ord_less_eq_int @ Y2 @ X2 )
=> ( ( ord_less_eq_int @ X2 @ Y2 )
= ( X2 = Y2 ) ) ) ).
% order_antisym_conv
thf(fact_373_linorder__le__cases,axiom,
! [X2: nat,Y2: nat] :
( ~ ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ X2 ) ) ).
% linorder_le_cases
thf(fact_374_linorder__le__cases,axiom,
! [X2: a,Y2: a] :
( ~ ( ord_less_eq_a @ X2 @ Y2 )
=> ( ord_less_eq_a @ Y2 @ X2 ) ) ).
% linorder_le_cases
thf(fact_375_linorder__le__cases,axiom,
! [X2: real,Y2: real] :
( ~ ( ord_less_eq_real @ X2 @ Y2 )
=> ( ord_less_eq_real @ Y2 @ X2 ) ) ).
% linorder_le_cases
thf(fact_376_linorder__le__cases,axiom,
! [X2: int,Y2: int] :
( ~ ( ord_less_eq_int @ X2 @ Y2 )
=> ( ord_less_eq_int @ Y2 @ X2 ) ) ).
% linorder_le_cases
thf(fact_377_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_378_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > a,C: a] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_less_eq_a @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_379_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > real,C: real] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_380_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > int,C: int] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_381_ord__le__eq__subst,axiom,
! [A: a,B: a,F: a > nat,C: nat] :
( ( ord_less_eq_a @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: a,Y4: a] :
( ( ord_less_eq_a @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_382_ord__le__eq__subst,axiom,
! [A: a,B: a,F: a > a,C: a] :
( ( ord_less_eq_a @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: a,Y4: a] :
( ( ord_less_eq_a @ X3 @ Y4 )
=> ( ord_less_eq_a @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_383_ord__le__eq__subst,axiom,
! [A: a,B: a,F: a > real,C: real] :
( ( ord_less_eq_a @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: a,Y4: a] :
( ( ord_less_eq_a @ X3 @ Y4 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_384_ord__le__eq__subst,axiom,
! [A: a,B: a,F: a > int,C: int] :
( ( ord_less_eq_a @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: a,Y4: a] :
( ( ord_less_eq_a @ X3 @ Y4 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_385_ord__le__eq__subst,axiom,
! [A: real,B: real,F: real > nat,C: nat] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: real,Y4: real] :
( ( ord_less_eq_real @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_386_ord__le__eq__subst,axiom,
! [A: real,B: real,F: real > a,C: a] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: real,Y4: real] :
( ( ord_less_eq_real @ X3 @ Y4 )
=> ( ord_less_eq_a @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_387_ord__eq__le__subst,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_388_ord__eq__le__subst,axiom,
! [A: a,F: nat > a,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_less_eq_a @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_389_ord__eq__le__subst,axiom,
! [A: real,F: nat > real,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_390_ord__eq__le__subst,axiom,
! [A: int,F: nat > int,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_391_ord__eq__le__subst,axiom,
! [A: nat,F: a > nat,B: a,C: a] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_a @ B @ C )
=> ( ! [X3: a,Y4: a] :
( ( ord_less_eq_a @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_392_ord__eq__le__subst,axiom,
! [A: a,F: a > a,B: a,C: a] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_a @ B @ C )
=> ( ! [X3: a,Y4: a] :
( ( ord_less_eq_a @ X3 @ Y4 )
=> ( ord_less_eq_a @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_393_ord__eq__le__subst,axiom,
! [A: real,F: a > real,B: a,C: a] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_a @ B @ C )
=> ( ! [X3: a,Y4: a] :
( ( ord_less_eq_a @ X3 @ Y4 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_394_ord__eq__le__subst,axiom,
! [A: int,F: a > int,B: a,C: a] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_a @ B @ C )
=> ( ! [X3: a,Y4: a] :
( ( ord_less_eq_a @ X3 @ Y4 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_395_ord__eq__le__subst,axiom,
! [A: nat,F: real > nat,B: real,C: real] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X3: real,Y4: real] :
( ( ord_less_eq_real @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_396_ord__eq__le__subst,axiom,
! [A: a,F: real > a,B: real,C: real] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X3: real,Y4: real] :
( ( ord_less_eq_real @ X3 @ Y4 )
=> ( ord_less_eq_a @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_397_linorder__linear,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
| ( ord_less_eq_nat @ Y2 @ X2 ) ) ).
% linorder_linear
thf(fact_398_linorder__linear,axiom,
! [X2: a,Y2: a] :
( ( ord_less_eq_a @ X2 @ Y2 )
| ( ord_less_eq_a @ Y2 @ X2 ) ) ).
% linorder_linear
thf(fact_399_linorder__linear,axiom,
! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
| ( ord_less_eq_real @ Y2 @ X2 ) ) ).
% linorder_linear
thf(fact_400_linorder__linear,axiom,
! [X2: int,Y2: int] :
( ( ord_less_eq_int @ X2 @ Y2 )
| ( ord_less_eq_int @ Y2 @ X2 ) ) ).
% linorder_linear
thf(fact_401_verit__la__disequality,axiom,
! [A: nat,B: nat] :
( ( A = B )
| ~ ( ord_less_eq_nat @ A @ B )
| ~ ( ord_less_eq_nat @ B @ A ) ) ).
% verit_la_disequality
thf(fact_402_verit__la__disequality,axiom,
! [A: a,B: a] :
( ( A = B )
| ~ ( ord_less_eq_a @ A @ B )
| ~ ( ord_less_eq_a @ B @ A ) ) ).
% verit_la_disequality
thf(fact_403_verit__la__disequality,axiom,
! [A: real,B: real] :
( ( A = B )
| ~ ( ord_less_eq_real @ A @ B )
| ~ ( ord_less_eq_real @ B @ A ) ) ).
% verit_la_disequality
thf(fact_404_verit__la__disequality,axiom,
! [A: int,B: int] :
( ( A = B )
| ~ ( ord_less_eq_int @ A @ B )
| ~ ( ord_less_eq_int @ B @ A ) ) ).
% verit_la_disequality
thf(fact_405_order__eq__refl,axiom,
! [X2: nat,Y2: nat] :
( ( X2 = Y2 )
=> ( ord_less_eq_nat @ X2 @ Y2 ) ) ).
% order_eq_refl
thf(fact_406_order__eq__refl,axiom,
! [X2: a,Y2: a] :
( ( X2 = Y2 )
=> ( ord_less_eq_a @ X2 @ Y2 ) ) ).
% order_eq_refl
thf(fact_407_order__eq__refl,axiom,
! [X2: real,Y2: real] :
( ( X2 = Y2 )
=> ( ord_less_eq_real @ X2 @ Y2 ) ) ).
% order_eq_refl
thf(fact_408_order__eq__refl,axiom,
! [X2: int,Y2: int] :
( ( X2 = Y2 )
=> ( ord_less_eq_int @ X2 @ Y2 ) ) ).
% order_eq_refl
thf(fact_409_order__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_410_order__subst2,axiom,
! [A: nat,B: nat,F: nat > a,C: a] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_a @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_less_eq_a @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_a @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_411_order__subst2,axiom,
! [A: nat,B: nat,F: nat > real,C: real] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_412_order__subst2,axiom,
! [A: nat,B: nat,F: nat > int,C: int] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_413_order__subst2,axiom,
! [A: a,B: a,F: a > nat,C: nat] :
( ( ord_less_eq_a @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X3: a,Y4: a] :
( ( ord_less_eq_a @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_414_order__subst2,axiom,
! [A: a,B: a,F: a > a,C: a] :
( ( ord_less_eq_a @ A @ B )
=> ( ( ord_less_eq_a @ ( F @ B ) @ C )
=> ( ! [X3: a,Y4: a] :
( ( ord_less_eq_a @ X3 @ Y4 )
=> ( ord_less_eq_a @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_a @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_415_order__subst2,axiom,
! [A: a,B: a,F: a > real,C: real] :
( ( ord_less_eq_a @ A @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C )
=> ( ! [X3: a,Y4: a] :
( ( ord_less_eq_a @ X3 @ Y4 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_416_order__subst2,axiom,
! [A: a,B: a,F: a > int,C: int] :
( ( ord_less_eq_a @ A @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C )
=> ( ! [X3: a,Y4: a] :
( ( ord_less_eq_a @ X3 @ Y4 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_417_order__subst2,axiom,
! [A: real,B: real,F: real > nat,C: nat] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X3: real,Y4: real] :
( ( ord_less_eq_real @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_418_order__subst2,axiom,
! [A: real,B: real,F: real > a,C: a] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_a @ ( F @ B ) @ C )
=> ( ! [X3: real,Y4: real] :
( ( ord_less_eq_real @ X3 @ Y4 )
=> ( ord_less_eq_a @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_a @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_419_order__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_420_order__subst1,axiom,
! [A: nat,F: a > nat,B: a,C: a] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_a @ B @ C )
=> ( ! [X3: a,Y4: a] :
( ( ord_less_eq_a @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_421_order__subst1,axiom,
! [A: nat,F: real > nat,B: real,C: real] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X3: real,Y4: real] :
( ( ord_less_eq_real @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_422_order__subst1,axiom,
! [A: nat,F: int > nat,B: int,C: int] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X3: int,Y4: int] :
( ( ord_less_eq_int @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_423_order__subst1,axiom,
! [A: a,F: nat > a,B: nat,C: nat] :
( ( ord_less_eq_a @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_less_eq_a @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_a @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_424_order__subst1,axiom,
! [A: a,F: a > a,B: a,C: a] :
( ( ord_less_eq_a @ A @ ( F @ B ) )
=> ( ( ord_less_eq_a @ B @ C )
=> ( ! [X3: a,Y4: a] :
( ( ord_less_eq_a @ X3 @ Y4 )
=> ( ord_less_eq_a @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_a @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_425_order__subst1,axiom,
! [A: a,F: real > a,B: real,C: real] :
( ( ord_less_eq_a @ A @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X3: real,Y4: real] :
( ( ord_less_eq_real @ X3 @ Y4 )
=> ( ord_less_eq_a @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_a @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_426_order__subst1,axiom,
! [A: a,F: int > a,B: int,C: int] :
( ( ord_less_eq_a @ A @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X3: int,Y4: int] :
( ( ord_less_eq_int @ X3 @ Y4 )
=> ( ord_less_eq_a @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_a @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_427_order__subst1,axiom,
! [A: real,F: nat > real,B: nat,C: nat] :
( ( ord_less_eq_real @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_428_order__subst1,axiom,
! [A: real,F: a > real,B: a,C: a] :
( ( ord_less_eq_real @ A @ ( F @ B ) )
=> ( ( ord_less_eq_a @ B @ C )
=> ( ! [X3: a,Y4: a] :
( ( ord_less_eq_a @ X3 @ Y4 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_429_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: nat,Z3: nat] : ( Y3 = Z3 ) )
= ( ^ [A4: nat,B3: nat] :
( ( ord_less_eq_nat @ A4 @ B3 )
& ( ord_less_eq_nat @ B3 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_430_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: a,Z3: a] : ( Y3 = Z3 ) )
= ( ^ [A4: a,B3: a] :
( ( ord_less_eq_a @ A4 @ B3 )
& ( ord_less_eq_a @ B3 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_431_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: real,Z3: real] : ( Y3 = Z3 ) )
= ( ^ [A4: real,B3: real] :
( ( ord_less_eq_real @ A4 @ B3 )
& ( ord_less_eq_real @ B3 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_432_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: int,Z3: int] : ( Y3 = Z3 ) )
= ( ^ [A4: int,B3: int] :
( ( ord_less_eq_int @ A4 @ B3 )
& ( ord_less_eq_int @ B3 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_433_antisym,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_434_antisym,axiom,
! [A: a,B: a] :
( ( ord_less_eq_a @ A @ B )
=> ( ( ord_less_eq_a @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_435_antisym,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_436_antisym,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_437_dual__order_Otrans,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_eq_nat @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_438_dual__order_Otrans,axiom,
! [B: a,A: a,C: a] :
( ( ord_less_eq_a @ B @ A )
=> ( ( ord_less_eq_a @ C @ B )
=> ( ord_less_eq_a @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_439_dual__order_Otrans,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ( ord_less_eq_real @ C @ B )
=> ( ord_less_eq_real @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_440_dual__order_Otrans,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( ord_less_eq_int @ C @ B )
=> ( ord_less_eq_int @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_441_dual__order_Oantisym,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_442_dual__order_Oantisym,axiom,
! [B: a,A: a] :
( ( ord_less_eq_a @ B @ A )
=> ( ( ord_less_eq_a @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_443_dual__order_Oantisym,axiom,
! [B: real,A: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ( ord_less_eq_real @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_444_dual__order_Oantisym,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( ord_less_eq_int @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_445_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: nat,Z3: nat] : ( Y3 = Z3 ) )
= ( ^ [A4: nat,B3: nat] :
( ( ord_less_eq_nat @ B3 @ A4 )
& ( ord_less_eq_nat @ A4 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_446_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: a,Z3: a] : ( Y3 = Z3 ) )
= ( ^ [A4: a,B3: a] :
( ( ord_less_eq_a @ B3 @ A4 )
& ( ord_less_eq_a @ A4 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_447_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: real,Z3: real] : ( Y3 = Z3 ) )
= ( ^ [A4: real,B3: real] :
( ( ord_less_eq_real @ B3 @ A4 )
& ( ord_less_eq_real @ A4 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_448_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: int,Z3: int] : ( Y3 = Z3 ) )
= ( ^ [A4: int,B3: int] :
( ( ord_less_eq_int @ B3 @ A4 )
& ( ord_less_eq_int @ A4 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_449_linorder__wlog,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A3: nat,B2: nat] :
( ( ord_less_eq_nat @ A3 @ B2 )
=> ( P @ A3 @ B2 ) )
=> ( ! [A3: nat,B2: nat] :
( ( P @ B2 @ A3 )
=> ( P @ A3 @ B2 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_450_linorder__wlog,axiom,
! [P: a > a > $o,A: a,B: a] :
( ! [A3: a,B2: a] :
( ( ord_less_eq_a @ A3 @ B2 )
=> ( P @ A3 @ B2 ) )
=> ( ! [A3: a,B2: a] :
( ( P @ B2 @ A3 )
=> ( P @ A3 @ B2 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_451_linorder__wlog,axiom,
! [P: real > real > $o,A: real,B: real] :
( ! [A3: real,B2: real] :
( ( ord_less_eq_real @ A3 @ B2 )
=> ( P @ A3 @ B2 ) )
=> ( ! [A3: real,B2: real] :
( ( P @ B2 @ A3 )
=> ( P @ A3 @ B2 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_452_linorder__wlog,axiom,
! [P: int > int > $o,A: int,B: int] :
( ! [A3: int,B2: int] :
( ( ord_less_eq_int @ A3 @ B2 )
=> ( P @ A3 @ B2 ) )
=> ( ! [A3: int,B2: int] :
( ( P @ B2 @ A3 )
=> ( P @ A3 @ B2 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_453_order__trans,axiom,
! [X2: nat,Y2: nat,Z2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ( ord_less_eq_nat @ Y2 @ Z2 )
=> ( ord_less_eq_nat @ X2 @ Z2 ) ) ) ).
% order_trans
thf(fact_454_order__trans,axiom,
! [X2: a,Y2: a,Z2: a] :
( ( ord_less_eq_a @ X2 @ Y2 )
=> ( ( ord_less_eq_a @ Y2 @ Z2 )
=> ( ord_less_eq_a @ X2 @ Z2 ) ) ) ).
% order_trans
thf(fact_455_order__trans,axiom,
! [X2: real,Y2: real,Z2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ( ord_less_eq_real @ Y2 @ Z2 )
=> ( ord_less_eq_real @ X2 @ Z2 ) ) ) ).
% order_trans
thf(fact_456_order__trans,axiom,
! [X2: int,Y2: int,Z2: int] :
( ( ord_less_eq_int @ X2 @ Y2 )
=> ( ( ord_less_eq_int @ Y2 @ Z2 )
=> ( ord_less_eq_int @ X2 @ Z2 ) ) ) ).
% order_trans
thf(fact_457_order_Otrans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% order.trans
thf(fact_458_order_Otrans,axiom,
! [A: a,B: a,C: a] :
( ( ord_less_eq_a @ A @ B )
=> ( ( ord_less_eq_a @ B @ C )
=> ( ord_less_eq_a @ A @ C ) ) ) ).
% order.trans
thf(fact_459_order_Otrans,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ord_less_eq_real @ A @ C ) ) ) ).
% order.trans
thf(fact_460_order_Otrans,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ord_less_eq_int @ A @ C ) ) ) ).
% order.trans
thf(fact_461_sorted__iff__nth__mono,axiom,
! [Xs: list_nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs )
= ( ! [I3: nat,J: nat] :
( ( ord_less_eq_nat @ I3 @ J )
=> ( ( ord_less_nat @ J @ ( size_size_list_nat @ Xs ) )
=> ( ord_less_eq_nat @ ( nth_nat @ Xs @ I3 ) @ ( nth_nat @ Xs @ J ) ) ) ) ) ) ).
% sorted_iff_nth_mono
thf(fact_462_sorted__iff__nth__mono,axiom,
! [Xs: list_a] :
( ( sorted_wrt_a @ ord_less_eq_a @ Xs )
= ( ! [I3: nat,J: nat] :
( ( ord_less_eq_nat @ I3 @ J )
=> ( ( ord_less_nat @ J @ ( size_size_list_a @ Xs ) )
=> ( ord_less_eq_a @ ( nth_a @ Xs @ I3 ) @ ( nth_a @ Xs @ J ) ) ) ) ) ) ).
% sorted_iff_nth_mono
thf(fact_463_sorted__iff__nth__mono,axiom,
! [Xs: list_real] :
( ( sorted_wrt_real @ ord_less_eq_real @ Xs )
= ( ! [I3: nat,J: nat] :
( ( ord_less_eq_nat @ I3 @ J )
=> ( ( ord_less_nat @ J @ ( size_size_list_real @ Xs ) )
=> ( ord_less_eq_real @ ( nth_real @ Xs @ I3 ) @ ( nth_real @ Xs @ J ) ) ) ) ) ) ).
% sorted_iff_nth_mono
thf(fact_464_sorted__iff__nth__mono,axiom,
! [Xs: list_int] :
( ( sorted_wrt_int @ ord_less_eq_int @ Xs )
= ( ! [I3: nat,J: nat] :
( ( ord_less_eq_nat @ I3 @ J )
=> ( ( ord_less_nat @ J @ ( size_size_list_int @ Xs ) )
=> ( ord_less_eq_int @ ( nth_int @ Xs @ I3 ) @ ( nth_int @ Xs @ J ) ) ) ) ) ) ).
% sorted_iff_nth_mono
thf(fact_465_sorted__nth__mono,axiom,
! [Xs: list_nat,I4: nat,J2: nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs )
=> ( ( ord_less_eq_nat @ I4 @ J2 )
=> ( ( ord_less_nat @ J2 @ ( size_size_list_nat @ Xs ) )
=> ( ord_less_eq_nat @ ( nth_nat @ Xs @ I4 ) @ ( nth_nat @ Xs @ J2 ) ) ) ) ) ).
% sorted_nth_mono
thf(fact_466_sorted__nth__mono,axiom,
! [Xs: list_a,I4: nat,J2: nat] :
( ( sorted_wrt_a @ ord_less_eq_a @ Xs )
=> ( ( ord_less_eq_nat @ I4 @ J2 )
=> ( ( ord_less_nat @ J2 @ ( size_size_list_a @ Xs ) )
=> ( ord_less_eq_a @ ( nth_a @ Xs @ I4 ) @ ( nth_a @ Xs @ J2 ) ) ) ) ) ).
% sorted_nth_mono
thf(fact_467_sorted__nth__mono,axiom,
! [Xs: list_real,I4: nat,J2: nat] :
( ( sorted_wrt_real @ ord_less_eq_real @ Xs )
=> ( ( ord_less_eq_nat @ I4 @ J2 )
=> ( ( ord_less_nat @ J2 @ ( size_size_list_real @ Xs ) )
=> ( ord_less_eq_real @ ( nth_real @ Xs @ I4 ) @ ( nth_real @ Xs @ J2 ) ) ) ) ) ).
% sorted_nth_mono
thf(fact_468_sorted__nth__mono,axiom,
! [Xs: list_int,I4: nat,J2: nat] :
( ( sorted_wrt_int @ ord_less_eq_int @ Xs )
=> ( ( ord_less_eq_nat @ I4 @ J2 )
=> ( ( ord_less_nat @ J2 @ ( size_size_list_int @ Xs ) )
=> ( ord_less_eq_int @ ( nth_int @ Xs @ I4 ) @ ( nth_int @ Xs @ J2 ) ) ) ) ) ).
% sorted_nth_mono
thf(fact_469_sorted__iff__nth__mono__less,axiom,
! [Xs: list_nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs )
= ( ! [I3: nat,J: nat] :
( ( ord_less_nat @ I3 @ J )
=> ( ( ord_less_nat @ J @ ( size_size_list_nat @ Xs ) )
=> ( ord_less_eq_nat @ ( nth_nat @ Xs @ I3 ) @ ( nth_nat @ Xs @ J ) ) ) ) ) ) ).
% sorted_iff_nth_mono_less
thf(fact_470_sorted__iff__nth__mono__less,axiom,
! [Xs: list_a] :
( ( sorted_wrt_a @ ord_less_eq_a @ Xs )
= ( ! [I3: nat,J: nat] :
( ( ord_less_nat @ I3 @ J )
=> ( ( ord_less_nat @ J @ ( size_size_list_a @ Xs ) )
=> ( ord_less_eq_a @ ( nth_a @ Xs @ I3 ) @ ( nth_a @ Xs @ J ) ) ) ) ) ) ).
% sorted_iff_nth_mono_less
thf(fact_471_sorted__iff__nth__mono__less,axiom,
! [Xs: list_real] :
( ( sorted_wrt_real @ ord_less_eq_real @ Xs )
= ( ! [I3: nat,J: nat] :
( ( ord_less_nat @ I3 @ J )
=> ( ( ord_less_nat @ J @ ( size_size_list_real @ Xs ) )
=> ( ord_less_eq_real @ ( nth_real @ Xs @ I3 ) @ ( nth_real @ Xs @ J ) ) ) ) ) ) ).
% sorted_iff_nth_mono_less
thf(fact_472_sorted__iff__nth__mono__less,axiom,
! [Xs: list_int] :
( ( sorted_wrt_int @ ord_less_eq_int @ Xs )
= ( ! [I3: nat,J: nat] :
( ( ord_less_nat @ I3 @ J )
=> ( ( ord_less_nat @ J @ ( size_size_list_int @ Xs ) )
=> ( ord_less_eq_int @ ( nth_int @ Xs @ I3 ) @ ( nth_int @ Xs @ J ) ) ) ) ) ) ).
% sorted_iff_nth_mono_less
thf(fact_473_sorted__wrt__iff__nth__less,axiom,
( sorted_wrt_a
= ( ^ [P3: a > a > $o,Xs2: list_a] :
! [I3: nat,J: nat] :
( ( ord_less_nat @ I3 @ J )
=> ( ( ord_less_nat @ J @ ( size_size_list_a @ Xs2 ) )
=> ( P3 @ ( nth_a @ Xs2 @ I3 ) @ ( nth_a @ Xs2 @ J ) ) ) ) ) ) ).
% sorted_wrt_iff_nth_less
thf(fact_474_sorted__wrt__iff__nth__less,axiom,
( sorted_wrt_nat
= ( ^ [P3: nat > nat > $o,Xs2: list_nat] :
! [I3: nat,J: nat] :
( ( ord_less_nat @ I3 @ J )
=> ( ( ord_less_nat @ J @ ( size_size_list_nat @ Xs2 ) )
=> ( P3 @ ( nth_nat @ Xs2 @ I3 ) @ ( nth_nat @ Xs2 @ J ) ) ) ) ) ) ).
% sorted_wrt_iff_nth_less
thf(fact_475_sorted__wrt__nth__less,axiom,
! [P: a > a > $o,Xs: list_a,I4: nat,J2: nat] :
( ( sorted_wrt_a @ P @ Xs )
=> ( ( ord_less_nat @ I4 @ J2 )
=> ( ( ord_less_nat @ J2 @ ( size_size_list_a @ Xs ) )
=> ( P @ ( nth_a @ Xs @ I4 ) @ ( nth_a @ Xs @ J2 ) ) ) ) ) ).
% sorted_wrt_nth_less
thf(fact_476_sorted__wrt__nth__less,axiom,
! [P: nat > nat > $o,Xs: list_nat,I4: nat,J2: nat] :
( ( sorted_wrt_nat @ P @ Xs )
=> ( ( ord_less_nat @ I4 @ J2 )
=> ( ( ord_less_nat @ J2 @ ( size_size_list_nat @ Xs ) )
=> ( P @ ( nth_nat @ Xs @ I4 ) @ ( nth_nat @ Xs @ J2 ) ) ) ) ) ).
% sorted_wrt_nth_less
thf(fact_477_list__eq__iff__nth__eq,axiom,
( ( ^ [Y3: list_a,Z3: list_a] : ( Y3 = Z3 ) )
= ( ^ [Xs2: list_a,Ys: list_a] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_a @ Ys ) )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_a @ Xs2 ) )
=> ( ( nth_a @ Xs2 @ I3 )
= ( nth_a @ Ys @ I3 ) ) ) ) ) ) ).
% list_eq_iff_nth_eq
thf(fact_478_list__eq__iff__nth__eq,axiom,
( ( ^ [Y3: list_nat,Z3: list_nat] : ( Y3 = Z3 ) )
= ( ^ [Xs2: list_nat,Ys: list_nat] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_nat @ Ys ) )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs2 ) )
=> ( ( nth_nat @ Xs2 @ I3 )
= ( nth_nat @ Ys @ I3 ) ) ) ) ) ) ).
% list_eq_iff_nth_eq
thf(fact_479_Skolem__list__nth,axiom,
! [K: nat,P: nat > a > $o] :
( ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ K )
=> ? [X5: a] : ( P @ I3 @ X5 ) ) )
= ( ? [Xs2: list_a] :
( ( ( size_size_list_a @ Xs2 )
= K )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ K )
=> ( P @ I3 @ ( nth_a @ Xs2 @ I3 ) ) ) ) ) ) ).
% Skolem_list_nth
thf(fact_480_Skolem__list__nth,axiom,
! [K: nat,P: nat > nat > $o] :
( ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ K )
=> ? [X5: nat] : ( P @ I3 @ X5 ) ) )
= ( ? [Xs2: list_nat] :
( ( ( size_size_list_nat @ Xs2 )
= K )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ K )
=> ( P @ I3 @ ( nth_nat @ Xs2 @ I3 ) ) ) ) ) ) ).
% Skolem_list_nth
thf(fact_481_nth__equalityI,axiom,
! [Xs: list_a,Ys2: list_a] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys2 ) )
=> ( ! [I5: nat] :
( ( ord_less_nat @ I5 @ ( size_size_list_a @ Xs ) )
=> ( ( nth_a @ Xs @ I5 )
= ( nth_a @ Ys2 @ I5 ) ) )
=> ( Xs = Ys2 ) ) ) ).
% nth_equalityI
thf(fact_482_nth__equalityI,axiom,
! [Xs: list_nat,Ys2: list_nat] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys2 ) )
=> ( ! [I5: nat] :
( ( ord_less_nat @ I5 @ ( size_size_list_nat @ Xs ) )
=> ( ( nth_nat @ Xs @ I5 )
= ( nth_nat @ Ys2 @ I5 ) ) )
=> ( Xs = Ys2 ) ) ) ).
% nth_equalityI
thf(fact_483_strict__sorted__imp__sorted,axiom,
! [Xs: list_nat] :
( ( sorted_wrt_nat @ ord_less_nat @ Xs )
=> ( sorted_wrt_nat @ ord_less_eq_nat @ Xs ) ) ).
% strict_sorted_imp_sorted
thf(fact_484_strict__sorted__imp__sorted,axiom,
! [Xs: list_a] :
( ( sorted_wrt_a @ ord_less_a @ Xs )
=> ( sorted_wrt_a @ ord_less_eq_a @ Xs ) ) ).
% strict_sorted_imp_sorted
thf(fact_485_strict__sorted__imp__sorted,axiom,
! [Xs: list_real] :
( ( sorted_wrt_real @ ord_less_real @ Xs )
=> ( sorted_wrt_real @ ord_less_eq_real @ Xs ) ) ).
% strict_sorted_imp_sorted
thf(fact_486_strict__sorted__imp__sorted,axiom,
! [Xs: list_int] :
( ( sorted_wrt_int @ ord_less_int @ Xs )
=> ( sorted_wrt_int @ ord_less_eq_int @ Xs ) ) ).
% strict_sorted_imp_sorted
thf(fact_487_length__induct,axiom,
! [P: list_a > $o,Xs: 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 @ Xs ) ) ).
% length_induct
thf(fact_488_length__induct,axiom,
! [P: list_nat > $o,Xs: 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 @ Xs ) ) ).
% length_induct
thf(fact_489_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
& ( M != N ) ) ) ) ).
% nat_less_le
thf(fact_490_sorted__wrt__less__idx,axiom,
! [Ns: list_nat,I4: nat] :
( ( sorted_wrt_nat @ ord_less_nat @ Ns )
=> ( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Ns ) )
=> ( ord_less_eq_nat @ I4 @ ( nth_nat @ Ns @ I4 ) ) ) ) ).
% sorted_wrt_less_idx
thf(fact_491_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K: nat,B: nat] :
( ( P @ K )
=> ( ! [Y4: nat] :
( ( P @ Y4 )
=> ( ord_less_eq_nat @ Y4 @ B ) )
=> ? [X3: nat] :
( ( P @ X3 )
& ! [Y5: nat] :
( ( P @ Y5 )
=> ( ord_less_eq_nat @ Y5 @ X3 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_492_nat__le__linear,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
| ( ord_less_eq_nat @ N2 @ M2 ) ) ).
% nat_le_linear
thf(fact_493_le__antisym,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ( ord_less_eq_nat @ N2 @ M2 )
=> ( M2 = N2 ) ) ) ).
% le_antisym
thf(fact_494_eq__imp__le,axiom,
! [M2: nat,N2: nat] :
( ( M2 = N2 )
=> ( ord_less_eq_nat @ M2 @ N2 ) ) ).
% eq_imp_le
thf(fact_495_le__trans,axiom,
! [I4: nat,J2: nat,K: nat] :
( ( ord_less_eq_nat @ I4 @ J2 )
=> ( ( ord_less_eq_nat @ J2 @ K )
=> ( ord_less_eq_nat @ I4 @ K ) ) ) ).
% le_trans
thf(fact_496_le__refl,axiom,
! [N2: nat] : ( ord_less_eq_nat @ N2 @ N2 ) ).
% le_refl
thf(fact_497_nat__neq__iff,axiom,
! [M2: nat,N2: nat] :
( ( M2 != N2 )
= ( ( ord_less_nat @ M2 @ N2 )
| ( ord_less_nat @ N2 @ M2 ) ) ) ).
% nat_neq_iff
thf(fact_498_less__not__refl,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ N2 ) ).
% less_not_refl
thf(fact_499_less__not__refl2,axiom,
! [N2: nat,M2: nat] :
( ( ord_less_nat @ N2 @ M2 )
=> ( M2 != N2 ) ) ).
% less_not_refl2
thf(fact_500_less__not__refl3,axiom,
! [S: nat,T: nat] :
( ( ord_less_nat @ S @ T )
=> ( S != T ) ) ).
% less_not_refl3
thf(fact_501_less__irrefl__nat,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ N2 ) ).
% less_irrefl_nat
thf(fact_502_nat__less__induct,axiom,
! [P: nat > $o,N2: nat] :
( ! [N3: nat] :
( ! [M3: nat] :
( ( ord_less_nat @ M3 @ N3 )
=> ( P @ M3 ) )
=> ( P @ N3 ) )
=> ( P @ N2 ) ) ).
% nat_less_induct
thf(fact_503_infinite__descent,axiom,
! [P: nat > $o,N2: nat] :
( ! [N3: nat] :
( ~ ( P @ N3 )
=> ? [M3: nat] :
( ( ord_less_nat @ M3 @ N3 )
& ~ ( P @ M3 ) ) )
=> ( P @ N2 ) ) ).
% infinite_descent
thf(fact_504_linorder__neqE__nat,axiom,
! [X2: nat,Y2: nat] :
( ( X2 != Y2 )
=> ( ~ ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_nat @ Y2 @ X2 ) ) ) ).
% linorder_neqE_nat
thf(fact_505_size__neq__size__imp__neq,axiom,
! [X2: list_a,Y2: list_a] :
( ( ( size_size_list_a @ X2 )
!= ( size_size_list_a @ Y2 ) )
=> ( X2 != Y2 ) ) ).
% size_neq_size_imp_neq
thf(fact_506_size__neq__size__imp__neq,axiom,
! [X2: list_nat,Y2: list_nat] :
( ( ( size_size_list_nat @ X2 )
!= ( size_size_list_nat @ Y2 ) )
=> ( X2 != Y2 ) ) ).
% size_neq_size_imp_neq
thf(fact_507_Ex__list__of__length,axiom,
! [N2: nat] :
? [Xs3: list_a] :
( ( size_size_list_a @ Xs3 )
= N2 ) ).
% Ex_list_of_length
thf(fact_508_Ex__list__of__length,axiom,
! [N2: nat] :
? [Xs3: list_nat] :
( ( size_size_list_nat @ Xs3 )
= N2 ) ).
% Ex_list_of_length
thf(fact_509_neq__if__length__neq,axiom,
! [Xs: list_a,Ys2: list_a] :
( ( ( size_size_list_a @ Xs )
!= ( size_size_list_a @ Ys2 ) )
=> ( Xs != Ys2 ) ) ).
% neq_if_length_neq
thf(fact_510_neq__if__length__neq,axiom,
! [Xs: list_nat,Ys2: list_nat] :
( ( ( size_size_list_nat @ Xs )
!= ( size_size_list_nat @ Ys2 ) )
=> ( Xs != Ys2 ) ) ).
% neq_if_length_neq
thf(fact_511_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I4: nat,J2: nat] :
( ! [I5: nat,J3: nat] :
( ( ord_less_nat @ I5 @ J3 )
=> ( ord_less_nat @ ( F @ I5 ) @ ( F @ J3 ) ) )
=> ( ( ord_less_eq_nat @ I4 @ J2 )
=> ( ord_less_eq_nat @ ( F @ I4 ) @ ( F @ J2 ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_512_le__neq__implies__less,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ( M2 != N2 )
=> ( ord_less_nat @ M2 @ N2 ) ) ) ).
% le_neq_implies_less
thf(fact_513_less__or__eq__imp__le,axiom,
! [M2: nat,N2: nat] :
( ( ( ord_less_nat @ M2 @ N2 )
| ( M2 = N2 ) )
=> ( ord_less_eq_nat @ M2 @ N2 ) ) ).
% less_or_eq_imp_le
thf(fact_514_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
| ( M = N ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_515_less__imp__le__nat,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( ord_less_eq_nat @ M2 @ N2 ) ) ).
% less_imp_le_nat
thf(fact_516_nat__descend__induct,axiom,
! [N2: nat,P: nat > $o,M2: nat] :
( ! [K2: nat] :
( ( ord_less_nat @ N2 @ K2 )
=> ( P @ K2 ) )
=> ( ! [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N2 )
=> ( ! [I6: nat] :
( ( ord_less_nat @ K2 @ I6 )
=> ( P @ I6 ) )
=> ( P @ K2 ) ) )
=> ( P @ M2 ) ) ) ).
% nat_descend_induct
thf(fact_517_minf_I8_J,axiom,
! [T: nat] :
? [Z4: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z4 )
=> ~ ( ord_less_eq_nat @ T @ X6 ) ) ).
% minf(8)
thf(fact_518_minf_I8_J,axiom,
! [T: a] :
? [Z4: a] :
! [X6: a] :
( ( ord_less_a @ X6 @ Z4 )
=> ~ ( ord_less_eq_a @ T @ X6 ) ) ).
% minf(8)
thf(fact_519_minf_I8_J,axiom,
! [T: real] :
? [Z4: real] :
! [X6: real] :
( ( ord_less_real @ X6 @ Z4 )
=> ~ ( ord_less_eq_real @ T @ X6 ) ) ).
% minf(8)
thf(fact_520_minf_I8_J,axiom,
! [T: int] :
? [Z4: int] :
! [X6: int] :
( ( ord_less_int @ X6 @ Z4 )
=> ~ ( ord_less_eq_int @ T @ X6 ) ) ).
% minf(8)
thf(fact_521_minf_I6_J,axiom,
! [T: nat] :
? [Z4: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z4 )
=> ( ord_less_eq_nat @ X6 @ T ) ) ).
% minf(6)
thf(fact_522_minf_I6_J,axiom,
! [T: a] :
? [Z4: a] :
! [X6: a] :
( ( ord_less_a @ X6 @ Z4 )
=> ( ord_less_eq_a @ X6 @ T ) ) ).
% minf(6)
thf(fact_523_minf_I6_J,axiom,
! [T: real] :
? [Z4: real] :
! [X6: real] :
( ( ord_less_real @ X6 @ Z4 )
=> ( ord_less_eq_real @ X6 @ T ) ) ).
% minf(6)
thf(fact_524_minf_I6_J,axiom,
! [T: int] :
? [Z4: int] :
! [X6: int] :
( ( ord_less_int @ X6 @ Z4 )
=> ( ord_less_eq_int @ X6 @ T ) ) ).
% minf(6)
thf(fact_525_pinf_I8_J,axiom,
! [T: nat] :
? [Z4: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z4 @ X6 )
=> ( ord_less_eq_nat @ T @ X6 ) ) ).
% pinf(8)
thf(fact_526_pinf_I8_J,axiom,
! [T: a] :
? [Z4: a] :
! [X6: a] :
( ( ord_less_a @ Z4 @ X6 )
=> ( ord_less_eq_a @ T @ X6 ) ) ).
% pinf(8)
thf(fact_527_pinf_I8_J,axiom,
! [T: real] :
? [Z4: real] :
! [X6: real] :
( ( ord_less_real @ Z4 @ X6 )
=> ( ord_less_eq_real @ T @ X6 ) ) ).
% pinf(8)
thf(fact_528_pinf_I8_J,axiom,
! [T: int] :
? [Z4: int] :
! [X6: int] :
( ( ord_less_int @ Z4 @ X6 )
=> ( ord_less_eq_int @ T @ X6 ) ) ).
% pinf(8)
thf(fact_529_pinf_I6_J,axiom,
! [T: nat] :
? [Z4: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z4 @ X6 )
=> ~ ( ord_less_eq_nat @ X6 @ T ) ) ).
% pinf(6)
thf(fact_530_pinf_I6_J,axiom,
! [T: a] :
? [Z4: a] :
! [X6: a] :
( ( ord_less_a @ Z4 @ X6 )
=> ~ ( ord_less_eq_a @ X6 @ T ) ) ).
% pinf(6)
thf(fact_531_pinf_I6_J,axiom,
! [T: real] :
? [Z4: real] :
! [X6: real] :
( ( ord_less_real @ Z4 @ X6 )
=> ~ ( ord_less_eq_real @ X6 @ T ) ) ).
% pinf(6)
thf(fact_532_pinf_I6_J,axiom,
! [T: int] :
? [Z4: int] :
! [X6: int] :
( ( ord_less_int @ Z4 @ X6 )
=> ~ ( ord_less_eq_int @ X6 @ T ) ) ).
% pinf(6)
thf(fact_533_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_534_complete__interval,axiom,
! [A: nat,B: nat,P: nat > $o] :
( ( ord_less_nat @ A @ B )
=> ( ( P @ A )
=> ( ~ ( P @ B )
=> ? [C2: nat] :
( ( ord_less_eq_nat @ A @ C2 )
& ( ord_less_eq_nat @ C2 @ B )
& ! [X6: nat] :
( ( ( ord_less_eq_nat @ A @ X6 )
& ( ord_less_nat @ X6 @ C2 ) )
=> ( P @ X6 ) )
& ! [D: nat] :
( ! [X3: nat] :
( ( ( ord_less_eq_nat @ A @ X3 )
& ( ord_less_nat @ X3 @ D ) )
=> ( P @ X3 ) )
=> ( ord_less_eq_nat @ D @ C2 ) ) ) ) ) ) ).
% complete_interval
thf(fact_535_complete__interval,axiom,
! [A: real,B: real,P: real > $o] :
( ( ord_less_real @ A @ B )
=> ( ( P @ A )
=> ( ~ ( P @ B )
=> ? [C2: real] :
( ( ord_less_eq_real @ A @ C2 )
& ( ord_less_eq_real @ C2 @ B )
& ! [X6: real] :
( ( ( ord_less_eq_real @ A @ X6 )
& ( ord_less_real @ X6 @ C2 ) )
=> ( P @ X6 ) )
& ! [D: real] :
( ! [X3: real] :
( ( ( ord_less_eq_real @ A @ X3 )
& ( ord_less_real @ X3 @ D ) )
=> ( P @ X3 ) )
=> ( ord_less_eq_real @ D @ C2 ) ) ) ) ) ) ).
% complete_interval
thf(fact_536_complete__interval,axiom,
! [A: int,B: int,P: int > $o] :
( ( ord_less_int @ A @ B )
=> ( ( P @ A )
=> ( ~ ( P @ B )
=> ? [C2: int] :
( ( ord_less_eq_int @ A @ C2 )
& ( ord_less_eq_int @ C2 @ B )
& ! [X6: int] :
( ( ( ord_less_eq_int @ A @ X6 )
& ( ord_less_int @ X6 @ C2 ) )
=> ( P @ X6 ) )
& ! [D: int] :
( ! [X3: int] :
( ( ( ord_less_eq_int @ A @ X3 )
& ( ord_less_int @ X3 @ D ) )
=> ( P @ X3 ) )
=> ( ord_less_eq_int @ D @ C2 ) ) ) ) ) ) ).
% complete_interval
thf(fact_537_sorted__rev__nth__mono,axiom,
! [Xs: list_nat,I4: nat,J2: nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ ( rev_nat @ Xs ) )
=> ( ( ord_less_eq_nat @ I4 @ J2 )
=> ( ( ord_less_nat @ J2 @ ( size_size_list_nat @ Xs ) )
=> ( ord_less_eq_nat @ ( nth_nat @ Xs @ J2 ) @ ( nth_nat @ Xs @ I4 ) ) ) ) ) ).
% sorted_rev_nth_mono
thf(fact_538_sorted__rev__nth__mono,axiom,
! [Xs: list_a,I4: nat,J2: nat] :
( ( sorted_wrt_a @ ord_less_eq_a @ ( rev_a @ Xs ) )
=> ( ( ord_less_eq_nat @ I4 @ J2 )
=> ( ( ord_less_nat @ J2 @ ( size_size_list_a @ Xs ) )
=> ( ord_less_eq_a @ ( nth_a @ Xs @ J2 ) @ ( nth_a @ Xs @ I4 ) ) ) ) ) ).
% sorted_rev_nth_mono
thf(fact_539_sorted__rev__nth__mono,axiom,
! [Xs: list_real,I4: nat,J2: nat] :
( ( sorted_wrt_real @ ord_less_eq_real @ ( rev_real @ Xs ) )
=> ( ( ord_less_eq_nat @ I4 @ J2 )
=> ( ( ord_less_nat @ J2 @ ( size_size_list_real @ Xs ) )
=> ( ord_less_eq_real @ ( nth_real @ Xs @ J2 ) @ ( nth_real @ Xs @ I4 ) ) ) ) ) ).
% sorted_rev_nth_mono
thf(fact_540_sorted__rev__nth__mono,axiom,
! [Xs: list_int,I4: nat,J2: nat] :
( ( sorted_wrt_int @ ord_less_eq_int @ ( rev_int @ Xs ) )
=> ( ( ord_less_eq_nat @ I4 @ J2 )
=> ( ( ord_less_nat @ J2 @ ( size_size_list_int @ Xs ) )
=> ( ord_less_eq_int @ ( nth_int @ Xs @ J2 ) @ ( nth_int @ Xs @ I4 ) ) ) ) ) ).
% sorted_rev_nth_mono
thf(fact_541_length__rev,axiom,
! [Xs: list_a] :
( ( size_size_list_a @ ( rev_a @ Xs ) )
= ( size_size_list_a @ Xs ) ) ).
% length_rev
thf(fact_542_length__rev,axiom,
! [Xs: list_nat] :
( ( size_size_list_nat @ ( rev_nat @ Xs ) )
= ( size_size_list_nat @ Xs ) ) ).
% length_rev
thf(fact_543_ex__gt__or__lt,axiom,
! [A: real] :
? [B2: real] :
( ( ord_less_real @ A @ B2 )
| ( ord_less_real @ B2 @ A ) ) ).
% ex_gt_or_lt
thf(fact_544_pinf_I1_J,axiom,
! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z5: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z5 @ X3 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [Z5: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z5 @ X3 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z4: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z4 @ X6 )
=> ( ( ( P @ X6 )
& ( Q @ X6 ) )
= ( ( P4 @ X6 )
& ( Q2 @ X6 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_545_pinf_I1_J,axiom,
! [P: real > $o,P4: real > $o,Q: real > $o,Q2: real > $o] :
( ? [Z5: real] :
! [X3: real] :
( ( ord_less_real @ Z5 @ X3 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [Z5: real] :
! [X3: real] :
( ( ord_less_real @ Z5 @ X3 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z4: real] :
! [X6: real] :
( ( ord_less_real @ Z4 @ X6 )
=> ( ( ( P @ X6 )
& ( Q @ X6 ) )
= ( ( P4 @ X6 )
& ( Q2 @ X6 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_546_pinf_I1_J,axiom,
! [P: int > $o,P4: int > $o,Q: int > $o,Q2: int > $o] :
( ? [Z5: int] :
! [X3: int] :
( ( ord_less_int @ Z5 @ X3 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [Z5: int] :
! [X3: int] :
( ( ord_less_int @ Z5 @ X3 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z4: int] :
! [X6: int] :
( ( ord_less_int @ Z4 @ X6 )
=> ( ( ( P @ X6 )
& ( Q @ X6 ) )
= ( ( P4 @ X6 )
& ( Q2 @ X6 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_547_pinf_I2_J,axiom,
! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z5: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z5 @ X3 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [Z5: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z5 @ X3 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z4: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z4 @ X6 )
=> ( ( ( P @ X6 )
| ( Q @ X6 ) )
= ( ( P4 @ X6 )
| ( Q2 @ X6 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_548_pinf_I2_J,axiom,
! [P: real > $o,P4: real > $o,Q: real > $o,Q2: real > $o] :
( ? [Z5: real] :
! [X3: real] :
( ( ord_less_real @ Z5 @ X3 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [Z5: real] :
! [X3: real] :
( ( ord_less_real @ Z5 @ X3 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z4: real] :
! [X6: real] :
( ( ord_less_real @ Z4 @ X6 )
=> ( ( ( P @ X6 )
| ( Q @ X6 ) )
= ( ( P4 @ X6 )
| ( Q2 @ X6 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_549_pinf_I2_J,axiom,
! [P: int > $o,P4: int > $o,Q: int > $o,Q2: int > $o] :
( ? [Z5: int] :
! [X3: int] :
( ( ord_less_int @ Z5 @ X3 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [Z5: int] :
! [X3: int] :
( ( ord_less_int @ Z5 @ X3 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z4: int] :
! [X6: int] :
( ( ord_less_int @ Z4 @ X6 )
=> ( ( ( P @ X6 )
| ( Q @ X6 ) )
= ( ( P4 @ X6 )
| ( Q2 @ X6 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_550_pinf_I3_J,axiom,
! [T: nat] :
? [Z4: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z4 @ X6 )
=> ( X6 != T ) ) ).
% pinf(3)
thf(fact_551_pinf_I3_J,axiom,
! [T: real] :
? [Z4: real] :
! [X6: real] :
( ( ord_less_real @ Z4 @ X6 )
=> ( X6 != T ) ) ).
% pinf(3)
thf(fact_552_pinf_I3_J,axiom,
! [T: int] :
? [Z4: int] :
! [X6: int] :
( ( ord_less_int @ Z4 @ X6 )
=> ( X6 != T ) ) ).
% pinf(3)
thf(fact_553_pinf_I4_J,axiom,
! [T: nat] :
? [Z4: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z4 @ X6 )
=> ( X6 != T ) ) ).
% pinf(4)
thf(fact_554_pinf_I4_J,axiom,
! [T: real] :
? [Z4: real] :
! [X6: real] :
( ( ord_less_real @ Z4 @ X6 )
=> ( X6 != T ) ) ).
% pinf(4)
thf(fact_555_pinf_I4_J,axiom,
! [T: int] :
? [Z4: int] :
! [X6: int] :
( ( ord_less_int @ Z4 @ X6 )
=> ( X6 != T ) ) ).
% pinf(4)
thf(fact_556_pinf_I5_J,axiom,
! [T: nat] :
? [Z4: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z4 @ X6 )
=> ~ ( ord_less_nat @ X6 @ T ) ) ).
% pinf(5)
thf(fact_557_pinf_I5_J,axiom,
! [T: real] :
? [Z4: real] :
! [X6: real] :
( ( ord_less_real @ Z4 @ X6 )
=> ~ ( ord_less_real @ X6 @ T ) ) ).
% pinf(5)
thf(fact_558_pinf_I5_J,axiom,
! [T: int] :
? [Z4: int] :
! [X6: int] :
( ( ord_less_int @ Z4 @ X6 )
=> ~ ( ord_less_int @ X6 @ T ) ) ).
% pinf(5)
thf(fact_559_pinf_I7_J,axiom,
! [T: nat] :
? [Z4: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z4 @ X6 )
=> ( ord_less_nat @ T @ X6 ) ) ).
% pinf(7)
thf(fact_560_pinf_I7_J,axiom,
! [T: real] :
? [Z4: real] :
! [X6: real] :
( ( ord_less_real @ Z4 @ X6 )
=> ( ord_less_real @ T @ X6 ) ) ).
% pinf(7)
thf(fact_561_pinf_I7_J,axiom,
! [T: int] :
? [Z4: int] :
! [X6: int] :
( ( ord_less_int @ Z4 @ X6 )
=> ( ord_less_int @ T @ X6 ) ) ).
% pinf(7)
thf(fact_562_minf_I1_J,axiom,
! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z5: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z5 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [Z5: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z5 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z4: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z4 )
=> ( ( ( P @ X6 )
& ( Q @ X6 ) )
= ( ( P4 @ X6 )
& ( Q2 @ X6 ) ) ) ) ) ) ).
% minf(1)
thf(fact_563_minf_I1_J,axiom,
! [P: real > $o,P4: real > $o,Q: real > $o,Q2: real > $o] :
( ? [Z5: real] :
! [X3: real] :
( ( ord_less_real @ X3 @ Z5 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [Z5: real] :
! [X3: real] :
( ( ord_less_real @ X3 @ Z5 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z4: real] :
! [X6: real] :
( ( ord_less_real @ X6 @ Z4 )
=> ( ( ( P @ X6 )
& ( Q @ X6 ) )
= ( ( P4 @ X6 )
& ( Q2 @ X6 ) ) ) ) ) ) ).
% minf(1)
thf(fact_564_minf_I1_J,axiom,
! [P: int > $o,P4: int > $o,Q: int > $o,Q2: int > $o] :
( ? [Z5: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z5 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [Z5: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z5 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z4: int] :
! [X6: int] :
( ( ord_less_int @ X6 @ Z4 )
=> ( ( ( P @ X6 )
& ( Q @ X6 ) )
= ( ( P4 @ X6 )
& ( Q2 @ X6 ) ) ) ) ) ) ).
% minf(1)
thf(fact_565_minf_I2_J,axiom,
! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z5: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z5 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [Z5: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z5 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z4: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z4 )
=> ( ( ( P @ X6 )
| ( Q @ X6 ) )
= ( ( P4 @ X6 )
| ( Q2 @ X6 ) ) ) ) ) ) ).
% minf(2)
thf(fact_566_minf_I2_J,axiom,
! [P: real > $o,P4: real > $o,Q: real > $o,Q2: real > $o] :
( ? [Z5: real] :
! [X3: real] :
( ( ord_less_real @ X3 @ Z5 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [Z5: real] :
! [X3: real] :
( ( ord_less_real @ X3 @ Z5 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z4: real] :
! [X6: real] :
( ( ord_less_real @ X6 @ Z4 )
=> ( ( ( P @ X6 )
| ( Q @ X6 ) )
= ( ( P4 @ X6 )
| ( Q2 @ X6 ) ) ) ) ) ) ).
% minf(2)
thf(fact_567_minf_I2_J,axiom,
! [P: int > $o,P4: int > $o,Q: int > $o,Q2: int > $o] :
( ? [Z5: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z5 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [Z5: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z5 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z4: int] :
! [X6: int] :
( ( ord_less_int @ X6 @ Z4 )
=> ( ( ( P @ X6 )
| ( Q @ X6 ) )
= ( ( P4 @ X6 )
| ( Q2 @ X6 ) ) ) ) ) ) ).
% minf(2)
thf(fact_568_minf_I3_J,axiom,
! [T: nat] :
? [Z4: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z4 )
=> ( X6 != T ) ) ).
% minf(3)
thf(fact_569_minf_I3_J,axiom,
! [T: real] :
? [Z4: real] :
! [X6: real] :
( ( ord_less_real @ X6 @ Z4 )
=> ( X6 != T ) ) ).
% minf(3)
thf(fact_570_minf_I3_J,axiom,
! [T: int] :
? [Z4: int] :
! [X6: int] :
( ( ord_less_int @ X6 @ Z4 )
=> ( X6 != T ) ) ).
% minf(3)
thf(fact_571_minf_I4_J,axiom,
! [T: nat] :
? [Z4: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z4 )
=> ( X6 != T ) ) ).
% minf(4)
thf(fact_572_minf_I4_J,axiom,
! [T: real] :
? [Z4: real] :
! [X6: real] :
( ( ord_less_real @ X6 @ Z4 )
=> ( X6 != T ) ) ).
% minf(4)
thf(fact_573_minf_I4_J,axiom,
! [T: int] :
? [Z4: int] :
! [X6: int] :
( ( ord_less_int @ X6 @ Z4 )
=> ( X6 != T ) ) ).
% minf(4)
thf(fact_574_minf_I5_J,axiom,
! [T: nat] :
? [Z4: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z4 )
=> ( ord_less_nat @ X6 @ T ) ) ).
% minf(5)
thf(fact_575_minf_I5_J,axiom,
! [T: real] :
? [Z4: real] :
! [X6: real] :
( ( ord_less_real @ X6 @ Z4 )
=> ( ord_less_real @ X6 @ T ) ) ).
% minf(5)
thf(fact_576_minf_I5_J,axiom,
! [T: int] :
? [Z4: int] :
! [X6: int] :
( ( ord_less_int @ X6 @ Z4 )
=> ( ord_less_int @ X6 @ T ) ) ).
% minf(5)
thf(fact_577_minf_I7_J,axiom,
! [T: nat] :
? [Z4: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z4 )
=> ~ ( ord_less_nat @ T @ X6 ) ) ).
% minf(7)
thf(fact_578_minf_I7_J,axiom,
! [T: real] :
? [Z4: real] :
! [X6: real] :
( ( ord_less_real @ X6 @ Z4 )
=> ~ ( ord_less_real @ T @ X6 ) ) ).
% minf(7)
thf(fact_579_minf_I7_J,axiom,
! [T: int] :
? [Z4: int] :
! [X6: int] :
( ( ord_less_int @ X6 @ Z4 )
=> ~ ( ord_less_int @ T @ X6 ) ) ).
% minf(7)
thf(fact_580_sorted__rev__iff__nth__mono,axiom,
! [Xs: list_nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ ( rev_nat @ Xs ) )
= ( ! [I3: nat,J: nat] :
( ( ord_less_eq_nat @ I3 @ J )
=> ( ( ord_less_nat @ J @ ( size_size_list_nat @ Xs ) )
=> ( ord_less_eq_nat @ ( nth_nat @ Xs @ J ) @ ( nth_nat @ Xs @ I3 ) ) ) ) ) ) ).
% sorted_rev_iff_nth_mono
thf(fact_581_sorted__rev__iff__nth__mono,axiom,
! [Xs: list_a] :
( ( sorted_wrt_a @ ord_less_eq_a @ ( rev_a @ Xs ) )
= ( ! [I3: nat,J: nat] :
( ( ord_less_eq_nat @ I3 @ J )
=> ( ( ord_less_nat @ J @ ( size_size_list_a @ Xs ) )
=> ( ord_less_eq_a @ ( nth_a @ Xs @ J ) @ ( nth_a @ Xs @ I3 ) ) ) ) ) ) ).
% sorted_rev_iff_nth_mono
thf(fact_582_sorted__rev__iff__nth__mono,axiom,
! [Xs: list_real] :
( ( sorted_wrt_real @ ord_less_eq_real @ ( rev_real @ Xs ) )
= ( ! [I3: nat,J: nat] :
( ( ord_less_eq_nat @ I3 @ J )
=> ( ( ord_less_nat @ J @ ( size_size_list_real @ Xs ) )
=> ( ord_less_eq_real @ ( nth_real @ Xs @ J ) @ ( nth_real @ Xs @ I3 ) ) ) ) ) ) ).
% sorted_rev_iff_nth_mono
thf(fact_583_sorted__rev__iff__nth__mono,axiom,
! [Xs: list_int] :
( ( sorted_wrt_int @ ord_less_eq_int @ ( rev_int @ Xs ) )
= ( ! [I3: nat,J: nat] :
( ( ord_less_eq_nat @ I3 @ J )
=> ( ( ord_less_nat @ J @ ( size_size_list_int @ Xs ) )
=> ( ord_less_eq_int @ ( nth_int @ Xs @ J ) @ ( nth_int @ Xs @ I3 ) ) ) ) ) ) ).
% sorted_rev_iff_nth_mono
thf(fact_584_sorted__rev__iff__nth__Suc,axiom,
! [Xs: list_nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ ( rev_nat @ Xs ) )
= ( ! [I3: nat] :
( ( ord_less_nat @ ( suc @ I3 ) @ ( size_size_list_nat @ Xs ) )
=> ( ord_less_eq_nat @ ( nth_nat @ Xs @ ( suc @ I3 ) ) @ ( nth_nat @ Xs @ I3 ) ) ) ) ) ).
% sorted_rev_iff_nth_Suc
thf(fact_585_sorted__rev__iff__nth__Suc,axiom,
! [Xs: list_a] :
( ( sorted_wrt_a @ ord_less_eq_a @ ( rev_a @ Xs ) )
= ( ! [I3: nat] :
( ( ord_less_nat @ ( suc @ I3 ) @ ( size_size_list_a @ Xs ) )
=> ( ord_less_eq_a @ ( nth_a @ Xs @ ( suc @ I3 ) ) @ ( nth_a @ Xs @ I3 ) ) ) ) ) ).
% sorted_rev_iff_nth_Suc
thf(fact_586_sorted__rev__iff__nth__Suc,axiom,
! [Xs: list_real] :
( ( sorted_wrt_real @ ord_less_eq_real @ ( rev_real @ Xs ) )
= ( ! [I3: nat] :
( ( ord_less_nat @ ( suc @ I3 ) @ ( size_size_list_real @ Xs ) )
=> ( ord_less_eq_real @ ( nth_real @ Xs @ ( suc @ I3 ) ) @ ( nth_real @ Xs @ I3 ) ) ) ) ) ).
% sorted_rev_iff_nth_Suc
thf(fact_587_sorted__rev__iff__nth__Suc,axiom,
! [Xs: list_int] :
( ( sorted_wrt_int @ ord_less_eq_int @ ( rev_int @ Xs ) )
= ( ! [I3: nat] :
( ( ord_less_nat @ ( suc @ I3 ) @ ( size_size_list_int @ Xs ) )
=> ( ord_less_eq_int @ ( nth_int @ Xs @ ( suc @ I3 ) ) @ ( nth_int @ Xs @ I3 ) ) ) ) ) ).
% sorted_rev_iff_nth_Suc
thf(fact_588_sorted__iff__nth__Suc,axiom,
! [Xs: list_nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs )
= ( ! [I3: nat] :
( ( ord_less_nat @ ( suc @ I3 ) @ ( size_size_list_nat @ Xs ) )
=> ( ord_less_eq_nat @ ( nth_nat @ Xs @ I3 ) @ ( nth_nat @ Xs @ ( suc @ I3 ) ) ) ) ) ) ).
% sorted_iff_nth_Suc
thf(fact_589_sorted__iff__nth__Suc,axiom,
! [Xs: list_a] :
( ( sorted_wrt_a @ ord_less_eq_a @ Xs )
= ( ! [I3: nat] :
( ( ord_less_nat @ ( suc @ I3 ) @ ( size_size_list_a @ Xs ) )
=> ( ord_less_eq_a @ ( nth_a @ Xs @ I3 ) @ ( nth_a @ Xs @ ( suc @ I3 ) ) ) ) ) ) ).
% sorted_iff_nth_Suc
thf(fact_590_sorted__iff__nth__Suc,axiom,
! [Xs: list_real] :
( ( sorted_wrt_real @ ord_less_eq_real @ Xs )
= ( ! [I3: nat] :
( ( ord_less_nat @ ( suc @ I3 ) @ ( size_size_list_real @ Xs ) )
=> ( ord_less_eq_real @ ( nth_real @ Xs @ I3 ) @ ( nth_real @ Xs @ ( suc @ I3 ) ) ) ) ) ) ).
% sorted_iff_nth_Suc
thf(fact_591_sorted__iff__nth__Suc,axiom,
! [Xs: list_int] :
( ( sorted_wrt_int @ ord_less_eq_int @ Xs )
= ( ! [I3: nat] :
( ( ord_less_nat @ ( suc @ I3 ) @ ( size_size_list_int @ Xs ) )
=> ( ord_less_eq_int @ ( nth_int @ Xs @ I3 ) @ ( nth_int @ Xs @ ( suc @ I3 ) ) ) ) ) ) ).
% sorted_iff_nth_Suc
thf(fact_592_sort__map__mono,axiom,
! [J2: nat,N2: nat,I4: nat,F: nat > nat] :
( ( ord_less_nat @ J2 @ N2 )
=> ( ( ord_less_eq_nat @ I4 @ J2 )
=> ( ord_less_eq_nat @ ( sort_map_nat @ F @ N2 @ I4 ) @ ( sort_map_nat @ F @ N2 @ J2 ) ) ) ) ).
% sort_map_mono
thf(fact_593_sort__map__mono,axiom,
! [J2: nat,N2: nat,I4: nat,F: nat > a] :
( ( ord_less_nat @ J2 @ N2 )
=> ( ( ord_less_eq_nat @ I4 @ J2 )
=> ( ord_less_eq_a @ ( sort_map_a @ F @ N2 @ I4 ) @ ( sort_map_a @ F @ N2 @ J2 ) ) ) ) ).
% sort_map_mono
thf(fact_594_sort__map__mono,axiom,
! [J2: nat,N2: nat,I4: nat,F: nat > real] :
( ( ord_less_nat @ J2 @ N2 )
=> ( ( ord_less_eq_nat @ I4 @ J2 )
=> ( ord_less_eq_real @ ( sort_map_real @ F @ N2 @ I4 ) @ ( sort_map_real @ F @ N2 @ J2 ) ) ) ) ).
% sort_map_mono
thf(fact_595_sort__map__mono,axiom,
! [J2: nat,N2: nat,I4: nat,F: nat > int] :
( ( ord_less_nat @ J2 @ N2 )
=> ( ( ord_less_eq_nat @ I4 @ J2 )
=> ( ord_less_eq_int @ ( sort_map_int @ F @ N2 @ I4 ) @ ( sort_map_int @ F @ N2 @ J2 ) ) ) ) ).
% sort_map_mono
thf(fact_596_list__ex__length,axiom,
( list_ex_a
= ( ^ [P3: a > $o,Xs2: list_a] :
? [N: nat] :
( ( ord_less_nat @ N @ ( size_size_list_a @ Xs2 ) )
& ( P3 @ ( nth_a @ Xs2 @ N ) ) ) ) ) ).
% list_ex_length
thf(fact_597_list__ex__length,axiom,
( list_ex_nat
= ( ^ [P3: nat > $o,Xs2: list_nat] :
? [N: nat] :
( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs2 ) )
& ( P3 @ ( nth_nat @ Xs2 @ N ) ) ) ) ) ).
% list_ex_length
thf(fact_598_sorted01,axiom,
! [Xs: list_nat] :
( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ one_one_nat )
=> ( sorted_wrt_nat @ ord_less_eq_nat @ Xs ) ) ).
% sorted01
thf(fact_599_sorted01,axiom,
! [Xs: list_a] :
( ( ord_less_eq_nat @ ( size_size_list_a @ Xs ) @ one_one_nat )
=> ( sorted_wrt_a @ ord_less_eq_a @ Xs ) ) ).
% sorted01
thf(fact_600_sorted01,axiom,
! [Xs: list_real] :
( ( ord_less_eq_nat @ ( size_size_list_real @ Xs ) @ one_one_nat )
=> ( sorted_wrt_real @ ord_less_eq_real @ Xs ) ) ).
% sorted01
thf(fact_601_sorted01,axiom,
! [Xs: list_int] :
( ( ord_less_eq_nat @ ( size_size_list_int @ Xs ) @ one_one_nat )
=> ( sorted_wrt_int @ ord_less_eq_int @ Xs ) ) ).
% sorted01
thf(fact_602_le__sup__lexord,axiom,
! [K: nat > nat,A2: nat,B5: nat,Ca: nat,C: nat,S: nat] :
( ( ( ord_less_nat @ ( K @ A2 ) @ ( K @ B5 ) )
=> ( ord_less_eq_nat @ Ca @ B5 ) )
=> ( ( ( ord_less_nat @ ( K @ B5 ) @ ( K @ A2 ) )
=> ( ord_less_eq_nat @ Ca @ A2 ) )
=> ( ( ( ( K @ A2 )
= ( K @ B5 ) )
=> ( ord_less_eq_nat @ Ca @ C ) )
=> ( ( ~ ( ord_less_eq_nat @ ( K @ B5 ) @ ( K @ A2 ) )
=> ( ~ ( ord_less_eq_nat @ ( K @ A2 ) @ ( K @ B5 ) )
=> ( ord_less_eq_nat @ Ca @ S ) ) )
=> ( ord_less_eq_nat @ Ca @ ( measur4601247141005857854at_nat @ A2 @ B5 @ K @ S @ C ) ) ) ) ) ) ).
% le_sup_lexord
thf(fact_603_le__sup__lexord,axiom,
! [K: nat > a,A2: nat,B5: nat,Ca: nat,C: nat,S: nat] :
( ( ( ord_less_a @ ( K @ A2 ) @ ( K @ B5 ) )
=> ( ord_less_eq_nat @ Ca @ B5 ) )
=> ( ( ( ord_less_a @ ( K @ B5 ) @ ( K @ A2 ) )
=> ( ord_less_eq_nat @ Ca @ A2 ) )
=> ( ( ( ( K @ A2 )
= ( K @ B5 ) )
=> ( ord_less_eq_nat @ Ca @ C ) )
=> ( ( ~ ( ord_less_eq_a @ ( K @ B5 ) @ ( K @ A2 ) )
=> ( ~ ( ord_less_eq_a @ ( K @ A2 ) @ ( K @ B5 ) )
=> ( ord_less_eq_nat @ Ca @ S ) ) )
=> ( ord_less_eq_nat @ Ca @ ( measur1338148430225957392_nat_a @ A2 @ B5 @ K @ S @ C ) ) ) ) ) ) ).
% le_sup_lexord
thf(fact_604_le__sup__lexord,axiom,
! [K: nat > real,A2: nat,B5: nat,Ca: nat,C: nat,S: nat] :
( ( ( ord_less_real @ ( K @ A2 ) @ ( K @ B5 ) )
=> ( ord_less_eq_nat @ Ca @ B5 ) )
=> ( ( ( ord_less_real @ ( K @ B5 ) @ ( K @ A2 ) )
=> ( ord_less_eq_nat @ Ca @ A2 ) )
=> ( ( ( ( K @ A2 )
= ( K @ B5 ) )
=> ( ord_less_eq_nat @ Ca @ C ) )
=> ( ( ~ ( ord_less_eq_real @ ( K @ B5 ) @ ( K @ A2 ) )
=> ( ~ ( ord_less_eq_real @ ( K @ A2 ) @ ( K @ B5 ) )
=> ( ord_less_eq_nat @ Ca @ S ) ) )
=> ( ord_less_eq_nat @ Ca @ ( measur8600355784167071770t_real @ A2 @ B5 @ K @ S @ C ) ) ) ) ) ) ).
% le_sup_lexord
thf(fact_605_le__sup__lexord,axiom,
! [K: nat > int,A2: nat,B5: nat,Ca: nat,C: nat,S: nat] :
( ( ( ord_less_int @ ( K @ A2 ) @ ( K @ B5 ) )
=> ( ord_less_eq_nat @ Ca @ B5 ) )
=> ( ( ( ord_less_int @ ( K @ B5 ) @ ( K @ A2 ) )
=> ( ord_less_eq_nat @ Ca @ A2 ) )
=> ( ( ( ( K @ A2 )
= ( K @ B5 ) )
=> ( ord_less_eq_nat @ Ca @ C ) )
=> ( ( ~ ( ord_less_eq_int @ ( K @ B5 ) @ ( K @ A2 ) )
=> ( ~ ( ord_less_eq_int @ ( K @ A2 ) @ ( K @ B5 ) )
=> ( ord_less_eq_nat @ Ca @ S ) ) )
=> ( ord_less_eq_nat @ Ca @ ( measur4598756670496807578at_int @ A2 @ B5 @ K @ S @ C ) ) ) ) ) ) ).
% le_sup_lexord
thf(fact_606_le__sup__lexord,axiom,
! [K: a > nat,A2: a,B5: a,Ca: a,C: a,S: a] :
( ( ( ord_less_nat @ ( K @ A2 ) @ ( K @ B5 ) )
=> ( ord_less_eq_a @ Ca @ B5 ) )
=> ( ( ( ord_less_nat @ ( K @ B5 ) @ ( K @ A2 ) )
=> ( ord_less_eq_a @ Ca @ A2 ) )
=> ( ( ( ( K @ A2 )
= ( K @ B5 ) )
=> ( ord_less_eq_a @ Ca @ C ) )
=> ( ( ~ ( ord_less_eq_nat @ ( K @ B5 ) @ ( K @ A2 ) )
=> ( ~ ( ord_less_eq_nat @ ( K @ A2 ) @ ( K @ B5 ) )
=> ( ord_less_eq_a @ Ca @ S ) ) )
=> ( ord_less_eq_a @ Ca @ ( measur6529588965902446770_a_nat @ A2 @ B5 @ K @ S @ C ) ) ) ) ) ) ).
% le_sup_lexord
thf(fact_607_le__sup__lexord,axiom,
! [K: a > a,A2: a,B5: a,Ca: a,C: a,S: a] :
( ( ( ord_less_a @ ( K @ A2 ) @ ( K @ B5 ) )
=> ( ord_less_eq_a @ Ca @ B5 ) )
=> ( ( ( ord_less_a @ ( K @ B5 ) @ ( K @ A2 ) )
=> ( ord_less_eq_a @ Ca @ A2 ) )
=> ( ( ( ( K @ A2 )
= ( K @ B5 ) )
=> ( ord_less_eq_a @ Ca @ C ) )
=> ( ( ~ ( ord_less_eq_a @ ( K @ B5 ) @ ( K @ A2 ) )
=> ( ~ ( ord_less_eq_a @ ( K @ A2 ) @ ( K @ B5 ) )
=> ( ord_less_eq_a @ Ca @ S ) ) )
=> ( ord_less_eq_a @ Ca @ ( measur6235662355876231836rd_a_a @ A2 @ B5 @ K @ S @ C ) ) ) ) ) ) ).
% le_sup_lexord
thf(fact_608_le__sup__lexord,axiom,
! [K: a > real,A2: a,B5: a,Ca: a,C: a,S: a] :
( ( ( ord_less_real @ ( K @ A2 ) @ ( K @ B5 ) )
=> ( ord_less_eq_a @ Ca @ B5 ) )
=> ( ( ( ord_less_real @ ( K @ B5 ) @ ( K @ A2 ) )
=> ( ord_less_eq_a @ Ca @ A2 ) )
=> ( ( ( ( K @ A2 )
= ( K @ B5 ) )
=> ( ord_less_eq_a @ Ca @ C ) )
=> ( ( ~ ( ord_less_eq_real @ ( K @ B5 ) @ ( K @ A2 ) )
=> ( ~ ( ord_less_eq_real @ ( K @ A2 ) @ ( K @ B5 ) )
=> ( ord_less_eq_a @ Ca @ S ) ) )
=> ( ord_less_eq_a @ Ca @ ( measur5297340495623586446a_real @ A2 @ B5 @ K @ S @ C ) ) ) ) ) ) ).
% le_sup_lexord
thf(fact_609_le__sup__lexord,axiom,
! [K: a > int,A2: a,B5: a,Ca: a,C: a,S: a] :
( ( ( ord_less_int @ ( K @ A2 ) @ ( K @ B5 ) )
=> ( ord_less_eq_a @ Ca @ B5 ) )
=> ( ( ( ord_less_int @ ( K @ B5 ) @ ( K @ A2 ) )
=> ( ord_less_eq_a @ Ca @ A2 ) )
=> ( ( ( ( K @ A2 )
= ( K @ B5 ) )
=> ( ord_less_eq_a @ Ca @ C ) )
=> ( ( ~ ( ord_less_eq_int @ ( K @ B5 ) @ ( K @ A2 ) )
=> ( ~ ( ord_less_eq_int @ ( K @ A2 ) @ ( K @ B5 ) )
=> ( ord_less_eq_a @ Ca @ S ) ) )
=> ( ord_less_eq_a @ Ca @ ( measur6527098495393396494_a_int @ A2 @ B5 @ K @ S @ C ) ) ) ) ) ) ).
% le_sup_lexord
thf(fact_610_le__sup__lexord,axiom,
! [K: real > nat,A2: real,B5: real,Ca: real,C: real,S: real] :
( ( ( ord_less_nat @ ( K @ A2 ) @ ( K @ B5 ) )
=> ( ord_less_eq_real @ Ca @ B5 ) )
=> ( ( ( ord_less_nat @ ( K @ B5 ) @ ( K @ A2 ) )
=> ( ord_less_eq_real @ Ca @ A2 ) )
=> ( ( ( ( K @ A2 )
= ( K @ B5 ) )
=> ( ord_less_eq_real @ Ca @ C ) )
=> ( ( ~ ( ord_less_eq_nat @ ( K @ B5 ) @ ( K @ A2 ) )
=> ( ~ ( ord_less_eq_nat @ ( K @ A2 ) @ ( K @ B5 ) )
=> ( ord_less_eq_real @ Ca @ S ) ) )
=> ( ord_less_eq_real @ Ca @ ( measur3944292320441194650al_nat @ A2 @ B5 @ K @ S @ C ) ) ) ) ) ) ).
% le_sup_lexord
thf(fact_611_le__sup__lexord,axiom,
! [K: real > a,A2: real,B5: real,Ca: real,C: real,S: real] :
( ( ( ord_less_a @ ( K @ A2 ) @ ( K @ B5 ) )
=> ( ord_less_eq_real @ Ca @ B5 ) )
=> ( ( ( ord_less_a @ ( K @ B5 ) @ ( K @ A2 ) )
=> ( ord_less_eq_real @ Ca @ A2 ) )
=> ( ( ( ( K @ A2 )
= ( K @ B5 ) )
=> ( ord_less_eq_real @ Ca @ C ) )
=> ( ( ~ ( ord_less_eq_a @ ( K @ B5 ) @ ( K @ A2 ) )
=> ( ~ ( ord_less_eq_a @ ( K @ A2 ) @ ( K @ B5 ) )
=> ( ord_less_eq_real @ Ca @ S ) ) )
=> ( ord_less_eq_real @ Ca @ ( measur5927359263096536244real_a @ A2 @ B5 @ K @ S @ C ) ) ) ) ) ) ).
% le_sup_lexord
thf(fact_612_old_Onat_Oinject,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( suc @ Nat )
= ( suc @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% old.nat.inject
thf(fact_613_nat_Oinject,axiom,
! [X22: nat,Y22: nat] :
( ( ( suc @ X22 )
= ( suc @ Y22 ) )
= ( X22 = Y22 ) ) ).
% nat.inject
thf(fact_614_Suc__le__mono,axiom,
! [N2: nat,M2: nat] :
( ( ord_less_eq_nat @ ( suc @ N2 ) @ ( suc @ M2 ) )
= ( ord_less_eq_nat @ N2 @ M2 ) ) ).
% Suc_le_mono
thf(fact_615_lessI,axiom,
! [N2: nat] : ( ord_less_nat @ N2 @ ( suc @ N2 ) ) ).
% lessI
thf(fact_616_Suc__mono,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( ord_less_nat @ ( suc @ M2 ) @ ( suc @ N2 ) ) ) ).
% Suc_mono
thf(fact_617_Suc__less__eq,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ ( suc @ M2 ) @ ( suc @ N2 ) )
= ( ord_less_nat @ M2 @ N2 ) ) ).
% Suc_less_eq
thf(fact_618_n__not__Suc__n,axiom,
! [N2: nat] :
( N2
!= ( suc @ N2 ) ) ).
% n_not_Suc_n
thf(fact_619_Suc__inject,axiom,
! [X2: nat,Y2: nat] :
( ( ( suc @ X2 )
= ( suc @ Y2 ) )
=> ( X2 = Y2 ) ) ).
% Suc_inject
thf(fact_620_transitive__stepwise__le,axiom,
! [M2: nat,N2: nat,R: nat > nat > $o] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ! [X3: nat] : ( R @ X3 @ X3 )
=> ( ! [X3: nat,Y4: nat,Z4: nat] :
( ( R @ X3 @ Y4 )
=> ( ( R @ Y4 @ Z4 )
=> ( R @ X3 @ Z4 ) ) )
=> ( ! [N3: nat] : ( R @ N3 @ ( suc @ N3 ) )
=> ( R @ M2 @ N2 ) ) ) ) ) ).
% transitive_stepwise_le
thf(fact_621_nat__induct__at__least,axiom,
! [M2: nat,N2: nat,P: nat > $o] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ( P @ M2 )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ M2 @ N3 )
=> ( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) ) )
=> ( P @ N2 ) ) ) ) ).
% nat_induct_at_least
thf(fact_622_full__nat__induct,axiom,
! [P: nat > $o,N2: nat] :
( ! [N3: nat] :
( ! [M3: nat] :
( ( ord_less_eq_nat @ ( suc @ M3 ) @ N3 )
=> ( P @ M3 ) )
=> ( P @ N3 ) )
=> ( P @ N2 ) ) ).
% full_nat_induct
thf(fact_623_not__less__eq__eq,axiom,
! [M2: nat,N2: nat] :
( ( ~ ( ord_less_eq_nat @ M2 @ N2 ) )
= ( ord_less_eq_nat @ ( suc @ N2 ) @ M2 ) ) ).
% not_less_eq_eq
thf(fact_624_Suc__n__not__le__n,axiom,
! [N2: nat] :
~ ( ord_less_eq_nat @ ( suc @ N2 ) @ N2 ) ).
% Suc_n_not_le_n
thf(fact_625_le__Suc__eq,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ M2 @ ( suc @ N2 ) )
= ( ( ord_less_eq_nat @ M2 @ N2 )
| ( M2
= ( suc @ N2 ) ) ) ) ).
% le_Suc_eq
thf(fact_626_Suc__le__D,axiom,
! [N2: nat,M4: nat] :
( ( ord_less_eq_nat @ ( suc @ N2 ) @ M4 )
=> ? [M5: nat] :
( M4
= ( suc @ M5 ) ) ) ).
% Suc_le_D
thf(fact_627_le__SucI,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ord_less_eq_nat @ M2 @ ( suc @ N2 ) ) ) ).
% le_SucI
thf(fact_628_le__SucE,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ M2 @ ( suc @ N2 ) )
=> ( ~ ( ord_less_eq_nat @ M2 @ N2 )
=> ( M2
= ( suc @ N2 ) ) ) ) ).
% le_SucE
thf(fact_629_Suc__leD,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ ( suc @ M2 ) @ N2 )
=> ( ord_less_eq_nat @ M2 @ N2 ) ) ).
% Suc_leD
thf(fact_630_Nat_OlessE,axiom,
! [I4: nat,K: nat] :
( ( ord_less_nat @ I4 @ K )
=> ( ( K
!= ( suc @ I4 ) )
=> ~ ! [J3: nat] :
( ( ord_less_nat @ I4 @ J3 )
=> ( K
!= ( suc @ J3 ) ) ) ) ) ).
% Nat.lessE
thf(fact_631_Suc__lessD,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ ( suc @ M2 ) @ N2 )
=> ( ord_less_nat @ M2 @ N2 ) ) ).
% Suc_lessD
thf(fact_632_Suc__lessE,axiom,
! [I4: nat,K: nat] :
( ( ord_less_nat @ ( suc @ I4 ) @ K )
=> ~ ! [J3: nat] :
( ( ord_less_nat @ I4 @ J3 )
=> ( K
!= ( suc @ J3 ) ) ) ) ).
% Suc_lessE
thf(fact_633_Suc__lessI,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( ( ( suc @ M2 )
!= N2 )
=> ( ord_less_nat @ ( suc @ M2 ) @ N2 ) ) ) ).
% Suc_lessI
thf(fact_634_less__SucE,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ ( suc @ N2 ) )
=> ( ~ ( ord_less_nat @ M2 @ N2 )
=> ( M2 = N2 ) ) ) ).
% less_SucE
thf(fact_635_less__SucI,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( ord_less_nat @ M2 @ ( suc @ N2 ) ) ) ).
% less_SucI
thf(fact_636_Ex__less__Suc,axiom,
! [N2: nat,P: nat > $o] :
( ( ? [I3: nat] :
( ( ord_less_nat @ I3 @ ( suc @ N2 ) )
& ( P @ I3 ) ) )
= ( ( P @ N2 )
| ? [I3: nat] :
( ( ord_less_nat @ I3 @ N2 )
& ( P @ I3 ) ) ) ) ).
% Ex_less_Suc
thf(fact_637_less__Suc__eq,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ ( suc @ N2 ) )
= ( ( ord_less_nat @ M2 @ N2 )
| ( M2 = N2 ) ) ) ).
% less_Suc_eq
thf(fact_638_not__less__eq,axiom,
! [M2: nat,N2: nat] :
( ( ~ ( ord_less_nat @ M2 @ N2 ) )
= ( ord_less_nat @ N2 @ ( suc @ M2 ) ) ) ).
% not_less_eq
thf(fact_639_All__less__Suc,axiom,
! [N2: nat,P: nat > $o] :
( ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( suc @ N2 ) )
=> ( P @ I3 ) ) )
= ( ( P @ N2 )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ N2 )
=> ( P @ I3 ) ) ) ) ).
% All_less_Suc
thf(fact_640_Suc__less__eq2,axiom,
! [N2: nat,M2: nat] :
( ( ord_less_nat @ ( suc @ N2 ) @ M2 )
= ( ? [M6: nat] :
( ( M2
= ( suc @ M6 ) )
& ( ord_less_nat @ N2 @ M6 ) ) ) ) ).
% Suc_less_eq2
thf(fact_641_less__antisym,axiom,
! [N2: nat,M2: nat] :
( ~ ( ord_less_nat @ N2 @ M2 )
=> ( ( ord_less_nat @ N2 @ ( suc @ M2 ) )
=> ( M2 = N2 ) ) ) ).
% less_antisym
thf(fact_642_Suc__less__SucD,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ ( suc @ M2 ) @ ( suc @ N2 ) )
=> ( ord_less_nat @ M2 @ N2 ) ) ).
% Suc_less_SucD
thf(fact_643_less__trans__Suc,axiom,
! [I4: nat,J2: nat,K: nat] :
( ( ord_less_nat @ I4 @ J2 )
=> ( ( ord_less_nat @ J2 @ K )
=> ( ord_less_nat @ ( suc @ I4 ) @ K ) ) ) ).
% less_trans_Suc
thf(fact_644_less__Suc__induct,axiom,
! [I4: nat,J2: nat,P: nat > nat > $o] :
( ( ord_less_nat @ I4 @ J2 )
=> ( ! [I5: nat] : ( P @ I5 @ ( suc @ I5 ) )
=> ( ! [I5: nat,J3: nat,K2: nat] :
( ( ord_less_nat @ I5 @ J3 )
=> ( ( ord_less_nat @ J3 @ K2 )
=> ( ( P @ I5 @ J3 )
=> ( ( P @ J3 @ K2 )
=> ( P @ I5 @ K2 ) ) ) ) )
=> ( P @ I4 @ J2 ) ) ) ) ).
% less_Suc_induct
thf(fact_645_strict__inc__induct,axiom,
! [I4: nat,J2: nat,P: nat > $o] :
( ( ord_less_nat @ I4 @ J2 )
=> ( ! [I5: nat] :
( ( J2
= ( suc @ I5 ) )
=> ( P @ I5 ) )
=> ( ! [I5: nat] :
( ( ord_less_nat @ I5 @ J2 )
=> ( ( P @ ( suc @ I5 ) )
=> ( P @ I5 ) ) )
=> ( P @ I4 ) ) ) ) ).
% strict_inc_induct
thf(fact_646_not__less__less__Suc__eq,axiom,
! [N2: nat,M2: nat] :
( ~ ( ord_less_nat @ N2 @ M2 )
=> ( ( ord_less_nat @ N2 @ ( suc @ M2 ) )
= ( N2 = M2 ) ) ) ).
% not_less_less_Suc_eq
thf(fact_647_lift__Suc__mono__le,axiom,
! [F: nat > nat,N2: nat,N4: nat] :
( ! [N3: nat] : ( ord_less_eq_nat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_eq_nat @ N2 @ N4 )
=> ( ord_less_eq_nat @ ( F @ N2 ) @ ( F @ N4 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_648_lift__Suc__mono__le,axiom,
! [F: nat > a,N2: nat,N4: nat] :
( ! [N3: nat] : ( ord_less_eq_a @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_eq_nat @ N2 @ N4 )
=> ( ord_less_eq_a @ ( F @ N2 ) @ ( F @ N4 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_649_lift__Suc__mono__le,axiom,
! [F: nat > real,N2: nat,N4: nat] :
( ! [N3: nat] : ( ord_less_eq_real @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_eq_nat @ N2 @ N4 )
=> ( ord_less_eq_real @ ( F @ N2 ) @ ( F @ N4 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_650_lift__Suc__mono__le,axiom,
! [F: nat > int,N2: nat,N4: nat] :
( ! [N3: nat] : ( ord_less_eq_int @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_eq_nat @ N2 @ N4 )
=> ( ord_less_eq_int @ ( F @ N2 ) @ ( F @ N4 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_651_lift__Suc__antimono__le,axiom,
! [F: nat > nat,N2: nat,N4: nat] :
( ! [N3: nat] : ( ord_less_eq_nat @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
=> ( ( ord_less_eq_nat @ N2 @ N4 )
=> ( ord_less_eq_nat @ ( F @ N4 ) @ ( F @ N2 ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_652_lift__Suc__antimono__le,axiom,
! [F: nat > a,N2: nat,N4: nat] :
( ! [N3: nat] : ( ord_less_eq_a @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
=> ( ( ord_less_eq_nat @ N2 @ N4 )
=> ( ord_less_eq_a @ ( F @ N4 ) @ ( F @ N2 ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_653_lift__Suc__antimono__le,axiom,
! [F: nat > real,N2: nat,N4: nat] :
( ! [N3: nat] : ( ord_less_eq_real @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
=> ( ( ord_less_eq_nat @ N2 @ N4 )
=> ( ord_less_eq_real @ ( F @ N4 ) @ ( F @ N2 ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_654_lift__Suc__antimono__le,axiom,
! [F: nat > int,N2: nat,N4: nat] :
( ! [N3: nat] : ( ord_less_eq_int @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
=> ( ( ord_less_eq_nat @ N2 @ N4 )
=> ( ord_less_eq_int @ ( F @ N4 ) @ ( F @ N2 ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_655_lift__Suc__mono__less,axiom,
! [F: nat > nat,N2: nat,N4: nat] :
( ! [N3: nat] : ( ord_less_nat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_nat @ N2 @ N4 )
=> ( ord_less_nat @ ( F @ N2 ) @ ( F @ N4 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_656_lift__Suc__mono__less,axiom,
! [F: nat > real,N2: nat,N4: nat] :
( ! [N3: nat] : ( ord_less_real @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_nat @ N2 @ N4 )
=> ( ord_less_real @ ( F @ N2 ) @ ( F @ N4 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_657_lift__Suc__mono__less,axiom,
! [F: nat > int,N2: nat,N4: nat] :
( ! [N3: nat] : ( ord_less_int @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_nat @ N2 @ N4 )
=> ( ord_less_int @ ( F @ N2 ) @ ( F @ N4 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_658_lift__Suc__mono__less__iff,axiom,
! [F: nat > nat,N2: nat,M2: nat] :
( ! [N3: nat] : ( ord_less_nat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_nat @ ( F @ N2 ) @ ( F @ M2 ) )
= ( ord_less_nat @ N2 @ M2 ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_659_lift__Suc__mono__less__iff,axiom,
! [F: nat > real,N2: nat,M2: nat] :
( ! [N3: nat] : ( ord_less_real @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_real @ ( F @ N2 ) @ ( F @ M2 ) )
= ( ord_less_nat @ N2 @ M2 ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_660_lift__Suc__mono__less__iff,axiom,
! [F: nat > int,N2: nat,M2: nat] :
( ! [N3: nat] : ( ord_less_int @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_int @ ( F @ N2 ) @ ( F @ M2 ) )
= ( ord_less_nat @ N2 @ M2 ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_661_le__imp__less__Suc,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ord_less_nat @ M2 @ ( suc @ N2 ) ) ) ).
% le_imp_less_Suc
thf(fact_662_less__eq__Suc__le,axiom,
( ord_less_nat
= ( ^ [N: nat] : ( ord_less_eq_nat @ ( suc @ N ) ) ) ) ).
% less_eq_Suc_le
thf(fact_663_less__Suc__eq__le,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ ( suc @ N2 ) )
= ( ord_less_eq_nat @ M2 @ N2 ) ) ).
% less_Suc_eq_le
thf(fact_664_le__less__Suc__eq,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ( ord_less_nat @ N2 @ ( suc @ M2 ) )
= ( N2 = M2 ) ) ) ).
% le_less_Suc_eq
thf(fact_665_Suc__le__lessD,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ ( suc @ M2 ) @ N2 )
=> ( ord_less_nat @ M2 @ N2 ) ) ).
% Suc_le_lessD
thf(fact_666_inc__induct,axiom,
! [I4: nat,J2: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I4 @ J2 )
=> ( ( P @ J2 )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ I4 @ N3 )
=> ( ( ord_less_nat @ N3 @ J2 )
=> ( ( P @ ( suc @ N3 ) )
=> ( P @ N3 ) ) ) )
=> ( P @ I4 ) ) ) ) ).
% inc_induct
thf(fact_667_dec__induct,axiom,
! [I4: nat,J2: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I4 @ J2 )
=> ( ( P @ I4 )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ I4 @ N3 )
=> ( ( ord_less_nat @ N3 @ J2 )
=> ( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) ) ) )
=> ( P @ J2 ) ) ) ) ).
% dec_induct
thf(fact_668_Suc__le__eq,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ ( suc @ M2 ) @ N2 )
= ( ord_less_nat @ M2 @ N2 ) ) ).
% Suc_le_eq
thf(fact_669_Suc__leI,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( ord_less_eq_nat @ ( suc @ M2 ) @ N2 ) ) ).
% Suc_leI
thf(fact_670_sort__map__strict__mono,axiom,
! [J2: nat,N2: nat,I4: nat,F: nat > nat] :
( ( ord_less_nat @ J2 @ N2 )
=> ( ( ord_less_nat @ I4 @ J2 )
=> ( ord_less_eq_nat @ ( sort_map_nat @ F @ N2 @ I4 ) @ ( sort_map_nat @ F @ N2 @ J2 ) ) ) ) ).
% sort_map_strict_mono
thf(fact_671_sort__map__strict__mono,axiom,
! [J2: nat,N2: nat,I4: nat,F: nat > a] :
( ( ord_less_nat @ J2 @ N2 )
=> ( ( ord_less_nat @ I4 @ J2 )
=> ( ord_less_eq_a @ ( sort_map_a @ F @ N2 @ I4 ) @ ( sort_map_a @ F @ N2 @ J2 ) ) ) ) ).
% sort_map_strict_mono
thf(fact_672_sort__map__strict__mono,axiom,
! [J2: nat,N2: nat,I4: nat,F: nat > real] :
( ( ord_less_nat @ J2 @ N2 )
=> ( ( ord_less_nat @ I4 @ J2 )
=> ( ord_less_eq_real @ ( sort_map_real @ F @ N2 @ I4 ) @ ( sort_map_real @ F @ N2 @ J2 ) ) ) ) ).
% sort_map_strict_mono
thf(fact_673_sort__map__strict__mono,axiom,
! [J2: nat,N2: nat,I4: nat,F: nat > int] :
( ( ord_less_nat @ J2 @ N2 )
=> ( ( ord_less_nat @ I4 @ J2 )
=> ( ord_less_eq_int @ ( sort_map_int @ F @ N2 @ I4 ) @ ( sort_map_int @ F @ N2 @ J2 ) ) ) ) ).
% sort_map_strict_mono
thf(fact_674_sorted__wrt01,axiom,
! [Xs: list_a,P: a > a > $o] :
( ( ord_less_eq_nat @ ( size_size_list_a @ Xs ) @ one_one_nat )
=> ( sorted_wrt_a @ P @ Xs ) ) ).
% sorted_wrt01
thf(fact_675_sorted__wrt01,axiom,
! [Xs: list_nat,P: nat > nat > $o] :
( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ one_one_nat )
=> ( sorted_wrt_nat @ P @ Xs ) ) ).
% sorted_wrt01
thf(fact_676_subsetI,axiom,
! [A2: set_a,B5: set_a] :
( ! [X3: a] :
( ( member_a @ X3 @ A2 )
=> ( member_a @ X3 @ B5 ) )
=> ( ord_less_eq_set_a @ A2 @ B5 ) ) ).
% subsetI
thf(fact_677_subsetI,axiom,
! [A2: set_real,B5: set_real] :
( ! [X3: real] :
( ( member_real @ X3 @ A2 )
=> ( member_real @ X3 @ B5 ) )
=> ( ord_less_eq_set_real @ A2 @ B5 ) ) ).
% subsetI
thf(fact_678_subsetI,axiom,
! [A2: set_int,B5: set_int] :
( ! [X3: int] :
( ( member_int @ X3 @ A2 )
=> ( member_int @ X3 @ B5 ) )
=> ( ord_less_eq_set_int @ A2 @ B5 ) ) ).
% subsetI
thf(fact_679_subsetI,axiom,
! [A2: set_nat,B5: set_nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( member_nat @ X3 @ B5 ) )
=> ( ord_less_eq_set_nat @ A2 @ B5 ) ) ).
% subsetI
thf(fact_680_less__numeral__extra_I4_J,axiom,
~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).
% less_numeral_extra(4)
thf(fact_681_less__numeral__extra_I4_J,axiom,
~ ( ord_less_real @ one_one_real @ one_one_real ) ).
% less_numeral_extra(4)
thf(fact_682_less__numeral__extra_I4_J,axiom,
~ ( ord_less_int @ one_one_int @ one_one_int ) ).
% less_numeral_extra(4)
thf(fact_683_le__numeral__extra_I4_J,axiom,
ord_less_eq_nat @ one_one_nat @ one_one_nat ).
% le_numeral_extra(4)
thf(fact_684_le__numeral__extra_I4_J,axiom,
ord_less_eq_real @ one_one_real @ one_one_real ).
% le_numeral_extra(4)
thf(fact_685_le__numeral__extra_I4_J,axiom,
ord_less_eq_int @ one_one_int @ one_one_int ).
% le_numeral_extra(4)
thf(fact_686_rotate1__length01,axiom,
! [Xs: list_a] :
( ( ord_less_eq_nat @ ( size_size_list_a @ Xs ) @ one_one_nat )
=> ( ( rotate1_a @ Xs )
= Xs ) ) ).
% rotate1_length01
thf(fact_687_rotate1__length01,axiom,
! [Xs: list_nat] :
( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ one_one_nat )
=> ( ( rotate1_nat @ Xs )
= Xs ) ) ).
% rotate1_length01
thf(fact_688_rev__nth,axiom,
! [N2: nat,Xs: list_a] :
( ( ord_less_nat @ N2 @ ( size_size_list_a @ Xs ) )
=> ( ( nth_a @ ( rev_a @ Xs ) @ N2 )
= ( nth_a @ Xs @ ( minus_minus_nat @ ( size_size_list_a @ Xs ) @ ( suc @ N2 ) ) ) ) ) ).
% rev_nth
thf(fact_689_rev__nth,axiom,
! [N2: nat,Xs: list_nat] :
( ( ord_less_nat @ N2 @ ( size_size_list_nat @ Xs ) )
=> ( ( nth_nat @ ( rev_nat @ Xs ) @ N2 )
= ( nth_nat @ Xs @ ( minus_minus_nat @ ( size_size_list_nat @ Xs ) @ ( suc @ N2 ) ) ) ) ) ).
% rev_nth
thf(fact_690_diff__Suc__Suc,axiom,
! [M2: nat,N2: nat] :
( ( minus_minus_nat @ ( suc @ M2 ) @ ( suc @ N2 ) )
= ( minus_minus_nat @ M2 @ N2 ) ) ).
% diff_Suc_Suc
thf(fact_691_Suc__diff__diff,axiom,
! [M2: nat,N2: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ ( suc @ M2 ) @ N2 ) @ ( suc @ K ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M2 @ N2 ) @ K ) ) ).
% Suc_diff_diff
thf(fact_692_diff__diff__cancel,axiom,
! [I4: nat,N2: nat] :
( ( ord_less_eq_nat @ I4 @ N2 )
=> ( ( minus_minus_nat @ N2 @ ( minus_minus_nat @ N2 @ I4 ) )
= I4 ) ) ).
% diff_diff_cancel
thf(fact_693_length__rotate1,axiom,
! [Xs: list_a] :
( ( size_size_list_a @ ( rotate1_a @ Xs ) )
= ( size_size_list_a @ Xs ) ) ).
% length_rotate1
thf(fact_694_length__rotate1,axiom,
! [Xs: list_nat] :
( ( size_size_list_nat @ ( rotate1_nat @ Xs ) )
= ( size_size_list_nat @ Xs ) ) ).
% length_rotate1
thf(fact_695_diff__Suc__1,axiom,
! [N2: nat] :
( ( minus_minus_nat @ ( suc @ N2 ) @ one_one_nat )
= N2 ) ).
% diff_Suc_1
thf(fact_696_diff__eq__diff__less__eq,axiom,
! [A: real,B: real,C: real,D2: real] :
( ( ( minus_minus_real @ A @ B )
= ( minus_minus_real @ C @ D2 ) )
=> ( ( ord_less_eq_real @ A @ B )
= ( ord_less_eq_real @ C @ D2 ) ) ) ).
% diff_eq_diff_less_eq
thf(fact_697_diff__eq__diff__less__eq,axiom,
! [A: int,B: int,C: int,D2: int] :
( ( ( minus_minus_int @ A @ B )
= ( minus_minus_int @ C @ D2 ) )
=> ( ( ord_less_eq_int @ A @ B )
= ( ord_less_eq_int @ C @ D2 ) ) ) ).
% diff_eq_diff_less_eq
thf(fact_698_diff__right__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ord_less_eq_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ C ) ) ) ).
% diff_right_mono
thf(fact_699_diff__right__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ord_less_eq_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ C ) ) ) ).
% diff_right_mono
thf(fact_700_diff__left__mono,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ord_less_eq_real @ ( minus_minus_real @ C @ A ) @ ( minus_minus_real @ C @ B ) ) ) ).
% diff_left_mono
thf(fact_701_diff__left__mono,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ord_less_eq_int @ ( minus_minus_int @ C @ A ) @ ( minus_minus_int @ C @ B ) ) ) ).
% diff_left_mono
thf(fact_702_diff__mono,axiom,
! [A: real,B: real,D2: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ D2 @ C )
=> ( ord_less_eq_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ D2 ) ) ) ) ).
% diff_mono
thf(fact_703_diff__mono,axiom,
! [A: int,B: int,D2: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ D2 @ C )
=> ( ord_less_eq_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ D2 ) ) ) ) ).
% diff_mono
thf(fact_704_diff__commute,axiom,
! [I4: nat,J2: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I4 @ J2 ) @ K )
= ( minus_minus_nat @ ( minus_minus_nat @ I4 @ K ) @ J2 ) ) ).
% diff_commute
thf(fact_705_diff__strict__right__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ord_less_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ C ) ) ) ).
% diff_strict_right_mono
thf(fact_706_diff__strict__right__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ C ) ) ) ).
% diff_strict_right_mono
thf(fact_707_diff__strict__left__mono,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_real @ B @ A )
=> ( ord_less_real @ ( minus_minus_real @ C @ A ) @ ( minus_minus_real @ C @ B ) ) ) ).
% diff_strict_left_mono
thf(fact_708_diff__strict__left__mono,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_int @ B @ A )
=> ( ord_less_int @ ( minus_minus_int @ C @ A ) @ ( minus_minus_int @ C @ B ) ) ) ).
% diff_strict_left_mono
thf(fact_709_diff__eq__diff__less,axiom,
! [A: real,B: real,C: real,D2: real] :
( ( ( minus_minus_real @ A @ B )
= ( minus_minus_real @ C @ D2 ) )
=> ( ( ord_less_real @ A @ B )
= ( ord_less_real @ C @ D2 ) ) ) ).
% diff_eq_diff_less
thf(fact_710_diff__eq__diff__less,axiom,
! [A: int,B: int,C: int,D2: int] :
( ( ( minus_minus_int @ A @ B )
= ( minus_minus_int @ C @ D2 ) )
=> ( ( ord_less_int @ A @ B )
= ( ord_less_int @ C @ D2 ) ) ) ).
% diff_eq_diff_less
thf(fact_711_diff__strict__mono,axiom,
! [A: real,B: real,D2: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ D2 @ C )
=> ( ord_less_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ D2 ) ) ) ) ).
% diff_strict_mono
thf(fact_712_diff__strict__mono,axiom,
! [A: int,B: int,D2: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ D2 @ C )
=> ( ord_less_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ D2 ) ) ) ) ).
% diff_strict_mono
thf(fact_713_zero__induct__lemma,axiom,
! [P: nat > $o,K: nat,I4: nat] :
( ( P @ K )
=> ( ! [N3: nat] :
( ( P @ ( suc @ N3 ) )
=> ( P @ N3 ) )
=> ( P @ ( minus_minus_nat @ K @ I4 ) ) ) ) ).
% zero_induct_lemma
thf(fact_714_eq__diff__iff,axiom,
! [K: nat,M2: nat,N2: nat] :
( ( ord_less_eq_nat @ K @ M2 )
=> ( ( ord_less_eq_nat @ K @ N2 )
=> ( ( ( minus_minus_nat @ M2 @ K )
= ( minus_minus_nat @ N2 @ K ) )
= ( M2 = N2 ) ) ) ) ).
% eq_diff_iff
thf(fact_715_le__diff__iff,axiom,
! [K: nat,M2: nat,N2: nat] :
( ( ord_less_eq_nat @ K @ M2 )
=> ( ( ord_less_eq_nat @ K @ N2 )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ K ) @ ( minus_minus_nat @ N2 @ K ) )
= ( ord_less_eq_nat @ M2 @ N2 ) ) ) ) ).
% le_diff_iff
thf(fact_716_Nat_Odiff__diff__eq,axiom,
! [K: nat,M2: nat,N2: nat] :
( ( ord_less_eq_nat @ K @ M2 )
=> ( ( ord_less_eq_nat @ K @ N2 )
=> ( ( minus_minus_nat @ ( minus_minus_nat @ M2 @ K ) @ ( minus_minus_nat @ N2 @ K ) )
= ( minus_minus_nat @ M2 @ N2 ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_717_diff__le__mono,axiom,
! [M2: nat,N2: nat,L: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ L ) @ ( minus_minus_nat @ N2 @ L ) ) ) ).
% diff_le_mono
thf(fact_718_diff__le__self,axiom,
! [M2: nat,N2: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ N2 ) @ M2 ) ).
% diff_le_self
thf(fact_719_le__diff__iff_H,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ A @ C )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ C @ A ) @ ( minus_minus_nat @ C @ B ) )
= ( ord_less_eq_nat @ B @ A ) ) ) ) ).
% le_diff_iff'
thf(fact_720_diff__le__mono2,axiom,
! [M2: nat,N2: nat,L: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N2 ) @ ( minus_minus_nat @ L @ M2 ) ) ) ).
% diff_le_mono2
thf(fact_721_less__imp__diff__less,axiom,
! [J2: nat,K: nat,N2: nat] :
( ( ord_less_nat @ J2 @ K )
=> ( ord_less_nat @ ( minus_minus_nat @ J2 @ N2 ) @ K ) ) ).
% less_imp_diff_less
thf(fact_722_diff__less__mono2,axiom,
! [M2: nat,N2: nat,L: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( ( ord_less_nat @ M2 @ L )
=> ( ord_less_nat @ ( minus_minus_nat @ L @ N2 ) @ ( minus_minus_nat @ L @ M2 ) ) ) ) ).
% diff_less_mono2
thf(fact_723_Suc__diff__le,axiom,
! [N2: nat,M2: nat] :
( ( ord_less_eq_nat @ N2 @ M2 )
=> ( ( minus_minus_nat @ ( suc @ M2 ) @ N2 )
= ( suc @ ( minus_minus_nat @ M2 @ N2 ) ) ) ) ).
% Suc_diff_le
thf(fact_724_Suc__diff__Suc,axiom,
! [N2: nat,M2: nat] :
( ( ord_less_nat @ N2 @ M2 )
=> ( ( suc @ ( minus_minus_nat @ M2 @ ( suc @ N2 ) ) )
= ( minus_minus_nat @ M2 @ N2 ) ) ) ).
% Suc_diff_Suc
thf(fact_725_diff__less__Suc,axiom,
! [M2: nat,N2: nat] : ( ord_less_nat @ ( minus_minus_nat @ M2 @ N2 ) @ ( suc @ M2 ) ) ).
% diff_less_Suc
thf(fact_726_less__diff__iff,axiom,
! [K: nat,M2: nat,N2: nat] :
( ( ord_less_eq_nat @ K @ M2 )
=> ( ( ord_less_eq_nat @ K @ N2 )
=> ( ( ord_less_nat @ ( minus_minus_nat @ M2 @ K ) @ ( minus_minus_nat @ N2 @ K ) )
= ( ord_less_nat @ M2 @ N2 ) ) ) ) ).
% less_diff_iff
thf(fact_727_diff__less__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ A )
=> ( ord_less_nat @ ( minus_minus_nat @ A @ C ) @ ( minus_minus_nat @ B @ C ) ) ) ) ).
% diff_less_mono
thf(fact_728_diff__Suc__eq__diff__pred,axiom,
! [M2: nat,N2: nat] :
( ( minus_minus_nat @ M2 @ ( suc @ N2 ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M2 @ one_one_nat ) @ N2 ) ) ).
% diff_Suc_eq_diff_pred
thf(fact_729_one__reorient,axiom,
! [X2: nat] :
( ( one_one_nat = X2 )
= ( X2 = one_one_nat ) ) ).
% one_reorient
thf(fact_730_one__reorient,axiom,
! [X2: real] :
( ( one_one_real = X2 )
= ( X2 = one_one_real ) ) ).
% one_reorient
thf(fact_731_one__reorient,axiom,
! [X2: int] :
( ( one_one_int = X2 )
= ( X2 = one_one_int ) ) ).
% one_reorient
thf(fact_732_in__mono,axiom,
! [A2: set_a,B5: set_a,X2: a] :
( ( ord_less_eq_set_a @ A2 @ B5 )
=> ( ( member_a @ X2 @ A2 )
=> ( member_a @ X2 @ B5 ) ) ) ).
% in_mono
thf(fact_733_in__mono,axiom,
! [A2: set_real,B5: set_real,X2: real] :
( ( ord_less_eq_set_real @ A2 @ B5 )
=> ( ( member_real @ X2 @ A2 )
=> ( member_real @ X2 @ B5 ) ) ) ).
% in_mono
thf(fact_734_in__mono,axiom,
! [A2: set_int,B5: set_int,X2: int] :
( ( ord_less_eq_set_int @ A2 @ B5 )
=> ( ( member_int @ X2 @ A2 )
=> ( member_int @ X2 @ B5 ) ) ) ).
% in_mono
thf(fact_735_in__mono,axiom,
! [A2: set_nat,B5: set_nat,X2: nat] :
( ( ord_less_eq_set_nat @ A2 @ B5 )
=> ( ( member_nat @ X2 @ A2 )
=> ( member_nat @ X2 @ B5 ) ) ) ).
% in_mono
thf(fact_736_subsetD,axiom,
! [A2: set_a,B5: set_a,C: a] :
( ( ord_less_eq_set_a @ A2 @ B5 )
=> ( ( member_a @ C @ A2 )
=> ( member_a @ C @ B5 ) ) ) ).
% subsetD
thf(fact_737_subsetD,axiom,
! [A2: set_real,B5: set_real,C: real] :
( ( ord_less_eq_set_real @ A2 @ B5 )
=> ( ( member_real @ C @ A2 )
=> ( member_real @ C @ B5 ) ) ) ).
% subsetD
thf(fact_738_subsetD,axiom,
! [A2: set_int,B5: set_int,C: int] :
( ( ord_less_eq_set_int @ A2 @ B5 )
=> ( ( member_int @ C @ A2 )
=> ( member_int @ C @ B5 ) ) ) ).
% subsetD
thf(fact_739_subsetD,axiom,
! [A2: set_nat,B5: set_nat,C: nat] :
( ( ord_less_eq_set_nat @ A2 @ B5 )
=> ( ( member_nat @ C @ A2 )
=> ( member_nat @ C @ B5 ) ) ) ).
% subsetD
thf(fact_740_subset__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A6: set_a,B6: set_a] :
! [X: a] :
( ( member_a @ X @ A6 )
=> ( member_a @ X @ B6 ) ) ) ) ).
% subset_eq
thf(fact_741_subset__eq,axiom,
( ord_less_eq_set_real
= ( ^ [A6: set_real,B6: set_real] :
! [X: real] :
( ( member_real @ X @ A6 )
=> ( member_real @ X @ B6 ) ) ) ) ).
% subset_eq
thf(fact_742_subset__eq,axiom,
( ord_less_eq_set_int
= ( ^ [A6: set_int,B6: set_int] :
! [X: int] :
( ( member_int @ X @ A6 )
=> ( member_int @ X @ B6 ) ) ) ) ).
% subset_eq
thf(fact_743_subset__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A6: set_nat,B6: set_nat] :
! [X: nat] :
( ( member_nat @ X @ A6 )
=> ( member_nat @ X @ B6 ) ) ) ) ).
% subset_eq
thf(fact_744_subset__iff,axiom,
( ord_less_eq_set_a
= ( ^ [A6: set_a,B6: set_a] :
! [T2: a] :
( ( member_a @ T2 @ A6 )
=> ( member_a @ T2 @ B6 ) ) ) ) ).
% subset_iff
thf(fact_745_subset__iff,axiom,
( ord_less_eq_set_real
= ( ^ [A6: set_real,B6: set_real] :
! [T2: real] :
( ( member_real @ T2 @ A6 )
=> ( member_real @ T2 @ B6 ) ) ) ) ).
% subset_iff
thf(fact_746_subset__iff,axiom,
( ord_less_eq_set_int
= ( ^ [A6: set_int,B6: set_int] :
! [T2: int] :
( ( member_int @ T2 @ A6 )
=> ( member_int @ T2 @ B6 ) ) ) ) ).
% subset_iff
thf(fact_747_subset__iff,axiom,
( ord_less_eq_set_nat
= ( ^ [A6: set_nat,B6: set_nat] :
! [T2: nat] :
( ( member_nat @ T2 @ A6 )
=> ( member_nat @ T2 @ B6 ) ) ) ) ).
% subset_iff
thf(fact_748_Suc__diff__1,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( suc @ ( minus_minus_nat @ N2 @ one_one_nat ) )
= N2 ) ) ).
% Suc_diff_1
thf(fact_749_nth__rotate1,axiom,
! [N2: nat,Xs: list_a] :
( ( ord_less_nat @ N2 @ ( size_size_list_a @ Xs ) )
=> ( ( nth_a @ ( rotate1_a @ Xs ) @ N2 )
= ( nth_a @ Xs @ ( modulo_modulo_nat @ ( suc @ N2 ) @ ( size_size_list_a @ Xs ) ) ) ) ) ).
% nth_rotate1
thf(fact_750_nth__rotate1,axiom,
! [N2: nat,Xs: list_nat] :
( ( ord_less_nat @ N2 @ ( size_size_list_nat @ Xs ) )
=> ( ( nth_nat @ ( rotate1_nat @ Xs ) @ N2 )
= ( nth_nat @ Xs @ ( modulo_modulo_nat @ ( suc @ N2 ) @ ( size_size_list_nat @ Xs ) ) ) ) ) ).
% nth_rotate1
thf(fact_751_rev__update,axiom,
! [K: nat,Xs: list_a,Y2: a] :
( ( ord_less_nat @ K @ ( size_size_list_a @ Xs ) )
=> ( ( rev_a @ ( list_update_a @ Xs @ K @ Y2 ) )
= ( list_update_a @ ( rev_a @ Xs ) @ ( minus_minus_nat @ ( minus_minus_nat @ ( size_size_list_a @ Xs ) @ K ) @ one_one_nat ) @ Y2 ) ) ) ).
% rev_update
thf(fact_752_rev__update,axiom,
! [K: nat,Xs: list_nat,Y2: nat] :
( ( ord_less_nat @ K @ ( size_size_list_nat @ Xs ) )
=> ( ( rev_nat @ ( list_update_nat @ Xs @ K @ Y2 ) )
= ( list_update_nat @ ( rev_nat @ Xs ) @ ( minus_minus_nat @ ( minus_minus_nat @ ( size_size_list_nat @ Xs ) @ K ) @ one_one_nat ) @ Y2 ) ) ) ).
% rev_update
thf(fact_753_rotate__length01,axiom,
! [Xs: list_a,N2: nat] :
( ( ord_less_eq_nat @ ( size_size_list_a @ Xs ) @ one_one_nat )
=> ( ( rotate_a @ N2 @ Xs )
= Xs ) ) ).
% rotate_length01
thf(fact_754_rotate__length01,axiom,
! [Xs: list_nat,N2: nat] :
( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ one_one_nat )
=> ( ( rotate_nat @ N2 @ Xs )
= Xs ) ) ).
% rotate_length01
thf(fact_755_length__butlast,axiom,
! [Xs: list_a] :
( ( size_size_list_a @ ( butlast_a @ Xs ) )
= ( minus_minus_nat @ ( size_size_list_a @ Xs ) @ one_one_nat ) ) ).
% length_butlast
thf(fact_756_length__butlast,axiom,
! [Xs: list_nat] :
( ( size_size_list_nat @ ( butlast_nat @ Xs ) )
= ( minus_minus_nat @ ( size_size_list_nat @ Xs ) @ one_one_nat ) ) ).
% length_butlast
thf(fact_757_Suc__pred_H,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( N2
= ( suc @ ( minus_minus_nat @ N2 @ one_one_nat ) ) ) ) ).
% Suc_pred'
thf(fact_758_le__zero__eq,axiom,
! [N2: nat] :
( ( ord_less_eq_nat @ N2 @ zero_zero_nat )
= ( N2 = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_759_not__gr__zero,axiom,
! [N2: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N2 ) )
= ( N2 = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_760_le0,axiom,
! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N2 ) ).
% le0
thf(fact_761_bot__nat__0_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% bot_nat_0.extremum
thf(fact_762_bot__nat__0_Onot__eq__extremum,axiom,
! [A: nat] :
( ( A != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ A ) ) ).
% bot_nat_0.not_eq_extremum
thf(fact_763_neq0__conv,axiom,
! [N2: nat] :
( ( N2 != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% neq0_conv
thf(fact_764_less__nat__zero__code,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_765_diff__self__eq__0,axiom,
! [M2: nat] :
( ( minus_minus_nat @ M2 @ M2 )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_766_diff__0__eq__0,axiom,
! [N2: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N2 )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_767_length__list__update,axiom,
! [Xs: list_a,I4: nat,X2: a] :
( ( size_size_list_a @ ( list_update_a @ Xs @ I4 @ X2 ) )
= ( size_size_list_a @ Xs ) ) ).
% length_list_update
thf(fact_768_length__list__update,axiom,
! [Xs: list_nat,I4: nat,X2: nat] :
( ( size_size_list_nat @ ( list_update_nat @ Xs @ I4 @ X2 ) )
= ( size_size_list_nat @ Xs ) ) ).
% length_list_update
thf(fact_769_list__update__id,axiom,
! [Xs: list_a,I4: nat] :
( ( list_update_a @ Xs @ I4 @ ( nth_a @ Xs @ I4 ) )
= Xs ) ).
% list_update_id
thf(fact_770_list__update__id,axiom,
! [Xs: list_nat,I4: nat] :
( ( list_update_nat @ Xs @ I4 @ ( nth_nat @ Xs @ I4 ) )
= Xs ) ).
% list_update_id
thf(fact_771_nth__list__update__neq,axiom,
! [I4: nat,J2: nat,Xs: list_a,X2: a] :
( ( I4 != J2 )
=> ( ( nth_a @ ( list_update_a @ Xs @ I4 @ X2 ) @ J2 )
= ( nth_a @ Xs @ J2 ) ) ) ).
% nth_list_update_neq
thf(fact_772_nth__list__update__neq,axiom,
! [I4: nat,J2: nat,Xs: list_nat,X2: nat] :
( ( I4 != J2 )
=> ( ( nth_nat @ ( list_update_nat @ Xs @ I4 @ X2 ) @ J2 )
= ( nth_nat @ Xs @ J2 ) ) ) ).
% nth_list_update_neq
thf(fact_773_length__rotate,axiom,
! [N2: nat,Xs: list_a] :
( ( size_size_list_a @ ( rotate_a @ N2 @ Xs ) )
= ( size_size_list_a @ Xs ) ) ).
% length_rotate
thf(fact_774_length__rotate,axiom,
! [N2: nat,Xs: list_nat] :
( ( size_size_list_nat @ ( rotate_nat @ N2 @ Xs ) )
= ( size_size_list_nat @ Xs ) ) ).
% length_rotate
thf(fact_775_diff__ge__0__iff__ge,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( minus_minus_real @ A @ B ) )
= ( ord_less_eq_real @ B @ A ) ) ).
% diff_ge_0_iff_ge
thf(fact_776_diff__ge__0__iff__ge,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( minus_minus_int @ A @ B ) )
= ( ord_less_eq_int @ B @ A ) ) ).
% diff_ge_0_iff_ge
thf(fact_777_diff__gt__0__iff__gt,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ ( minus_minus_real @ A @ B ) )
= ( ord_less_real @ B @ A ) ) ).
% diff_gt_0_iff_gt
thf(fact_778_diff__gt__0__iff__gt,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ ( minus_minus_int @ A @ B ) )
= ( ord_less_int @ B @ A ) ) ).
% diff_gt_0_iff_gt
thf(fact_779_diff__numeral__special_I9_J,axiom,
( ( minus_minus_real @ one_one_real @ one_one_real )
= zero_zero_real ) ).
% diff_numeral_special(9)
thf(fact_780_diff__numeral__special_I9_J,axiom,
( ( minus_minus_int @ one_one_int @ one_one_int )
= zero_zero_int ) ).
% diff_numeral_special(9)
thf(fact_781_zero__less__Suc,axiom,
! [N2: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N2 ) ) ).
% zero_less_Suc
thf(fact_782_less__Suc0,axiom,
! [N2: nat] :
( ( ord_less_nat @ N2 @ ( suc @ zero_zero_nat ) )
= ( N2 = zero_zero_nat ) ) ).
% less_Suc0
thf(fact_783_diff__is__0__eq,axiom,
! [M2: nat,N2: nat] :
( ( ( minus_minus_nat @ M2 @ N2 )
= zero_zero_nat )
= ( ord_less_eq_nat @ M2 @ N2 ) ) ).
% diff_is_0_eq
thf(fact_784_diff__is__0__eq_H,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ( minus_minus_nat @ M2 @ N2 )
= zero_zero_nat ) ) ).
% diff_is_0_eq'
thf(fact_785_zero__less__diff,axiom,
! [N2: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N2 @ M2 ) )
= ( ord_less_nat @ M2 @ N2 ) ) ).
% zero_less_diff
thf(fact_786_less__one,axiom,
! [N2: nat] :
( ( ord_less_nat @ N2 @ one_one_nat )
= ( N2 = zero_zero_nat ) ) ).
% less_one
thf(fact_787_list__update__beyond,axiom,
! [Xs: list_a,I4: nat,X2: a] :
( ( ord_less_eq_nat @ ( size_size_list_a @ Xs ) @ I4 )
=> ( ( list_update_a @ Xs @ I4 @ X2 )
= Xs ) ) ).
% list_update_beyond
thf(fact_788_list__update__beyond,axiom,
! [Xs: list_nat,I4: nat,X2: nat] :
( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ I4 )
=> ( ( list_update_nat @ Xs @ I4 @ X2 )
= Xs ) ) ).
% list_update_beyond
thf(fact_789_Suc__pred,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( suc @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) )
= N2 ) ) ).
% Suc_pred
thf(fact_790_nth__list__update__eq,axiom,
! [I4: nat,Xs: list_a,X2: a] :
( ( ord_less_nat @ I4 @ ( size_size_list_a @ Xs ) )
=> ( ( nth_a @ ( list_update_a @ Xs @ I4 @ X2 ) @ I4 )
= X2 ) ) ).
% nth_list_update_eq
thf(fact_791_nth__list__update__eq,axiom,
! [I4: nat,Xs: list_nat,X2: nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Xs ) )
=> ( ( nth_nat @ ( list_update_nat @ Xs @ I4 @ X2 ) @ I4 )
= X2 ) ) ).
% nth_list_update_eq
thf(fact_792_rotate__id,axiom,
! [N2: nat,Xs: list_a] :
( ( ( modulo_modulo_nat @ N2 @ ( size_size_list_a @ Xs ) )
= zero_zero_nat )
=> ( ( rotate_a @ N2 @ Xs )
= Xs ) ) ).
% rotate_id
thf(fact_793_rotate__id,axiom,
! [N2: nat,Xs: list_nat] :
( ( ( modulo_modulo_nat @ N2 @ ( size_size_list_nat @ Xs ) )
= zero_zero_nat )
=> ( ( rotate_nat @ N2 @ Xs )
= Xs ) ) ).
% rotate_id
thf(fact_794_rotate__conv__mod,axiom,
( rotate_a
= ( ^ [N: nat,Xs2: list_a] : ( rotate_a @ ( modulo_modulo_nat @ N @ ( size_size_list_a @ Xs2 ) ) @ Xs2 ) ) ) ).
% rotate_conv_mod
thf(fact_795_rotate__conv__mod,axiom,
( rotate_nat
= ( ^ [N: nat,Xs2: list_nat] : ( rotate_nat @ ( modulo_modulo_nat @ N @ ( size_size_list_nat @ Xs2 ) ) @ Xs2 ) ) ) ).
% rotate_conv_mod
thf(fact_796_list__decode_Ocases,axiom,
! [X2: nat] :
( ( X2 != zero_zero_nat )
=> ~ ! [N3: nat] :
( X2
!= ( suc @ N3 ) ) ) ).
% list_decode.cases
thf(fact_797_le__numeral__extra_I3_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% le_numeral_extra(3)
thf(fact_798_le__numeral__extra_I3_J,axiom,
ord_less_eq_real @ zero_zero_real @ zero_zero_real ).
% le_numeral_extra(3)
thf(fact_799_le__numeral__extra_I3_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% le_numeral_extra(3)
thf(fact_800_zero__le,axiom,
! [X2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X2 ) ).
% zero_le
thf(fact_801_less__numeral__extra_I3_J,axiom,
~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% less_numeral_extra(3)
thf(fact_802_less__numeral__extra_I3_J,axiom,
~ ( ord_less_real @ zero_zero_real @ zero_zero_real ) ).
% less_numeral_extra(3)
thf(fact_803_less__numeral__extra_I3_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_numeral_extra(3)
thf(fact_804_gr__zeroI,axiom,
! [N2: nat] :
( ( N2 != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% gr_zeroI
thf(fact_805_not__less__zero,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).
% not_less_zero
thf(fact_806_gr__implies__not__zero,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( N2 != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_807_zero__less__iff__neq__zero,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
= ( N2 != zero_zero_nat ) ) ).
% zero_less_iff_neq_zero
thf(fact_808_not0__implies__Suc,axiom,
! [N2: nat] :
( ( N2 != zero_zero_nat )
=> ? [M5: nat] :
( N2
= ( suc @ M5 ) ) ) ).
% not0_implies_Suc
thf(fact_809_Zero__not__Suc,axiom,
! [M2: nat] :
( zero_zero_nat
!= ( suc @ M2 ) ) ).
% Zero_not_Suc
thf(fact_810_Zero__neq__Suc,axiom,
! [M2: nat] :
( zero_zero_nat
!= ( suc @ M2 ) ) ).
% Zero_neq_Suc
thf(fact_811_Suc__neq__Zero,axiom,
! [M2: nat] :
( ( suc @ M2 )
!= zero_zero_nat ) ).
% Suc_neq_Zero
thf(fact_812_zero__induct,axiom,
! [P: nat > $o,K: nat] :
( ( P @ K )
=> ( ! [N3: nat] :
( ( P @ ( suc @ N3 ) )
=> ( P @ N3 ) )
=> ( P @ zero_zero_nat ) ) ) ).
% zero_induct
thf(fact_813_diff__induct,axiom,
! [P: nat > nat > $o,M2: nat,N2: nat] :
( ! [X3: nat] : ( P @ X3 @ zero_zero_nat )
=> ( ! [Y4: nat] : ( P @ zero_zero_nat @ ( suc @ Y4 ) )
=> ( ! [X3: nat,Y4: nat] :
( ( P @ X3 @ Y4 )
=> ( P @ ( suc @ X3 ) @ ( suc @ Y4 ) ) )
=> ( P @ M2 @ N2 ) ) ) ) ).
% diff_induct
thf(fact_814_nat__induct,axiom,
! [P: nat > $o,N2: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N3: nat] :
( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) )
=> ( P @ N2 ) ) ) ).
% nat_induct
thf(fact_815_old_Onat_Oexhaust,axiom,
! [Y2: nat] :
( ( Y2 != zero_zero_nat )
=> ~ ! [Nat3: nat] :
( Y2
!= ( suc @ Nat3 ) ) ) ).
% old.nat.exhaust
thf(fact_816_nat_OdiscI,axiom,
! [Nat: nat,X22: nat] :
( ( Nat
= ( suc @ X22 ) )
=> ( Nat != zero_zero_nat ) ) ).
% nat.discI
thf(fact_817_old_Onat_Odistinct_I1_J,axiom,
! [Nat2: nat] :
( zero_zero_nat
!= ( suc @ Nat2 ) ) ).
% old.nat.distinct(1)
thf(fact_818_old_Onat_Odistinct_I2_J,axiom,
! [Nat2: nat] :
( ( suc @ Nat2 )
!= zero_zero_nat ) ).
% old.nat.distinct(2)
thf(fact_819_nat_Odistinct_I1_J,axiom,
! [X22: nat] :
( zero_zero_nat
!= ( suc @ X22 ) ) ).
% nat.distinct(1)
thf(fact_820_le__0__eq,axiom,
! [N2: nat] :
( ( ord_less_eq_nat @ N2 @ zero_zero_nat )
= ( N2 = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_821_bot__nat__0_Oextremum__uniqueI,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_uniqueI
thf(fact_822_bot__nat__0_Oextremum__unique,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
= ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_unique
thf(fact_823_less__eq__nat_Osimps_I1_J,axiom,
! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N2 ) ).
% less_eq_nat.simps(1)
thf(fact_824_bot__nat__0_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_825_gr0I,axiom,
! [N2: nat] :
( ( N2 != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% gr0I
thf(fact_826_not__gr0,axiom,
! [N2: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N2 ) )
= ( N2 = zero_zero_nat ) ) ).
% not_gr0
thf(fact_827_not__less0,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).
% not_less0
thf(fact_828_less__zeroE,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).
% less_zeroE
thf(fact_829_gr__implies__not0,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( N2 != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_830_infinite__descent0,axiom,
! [P: nat > $o,N2: 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 @ N2 ) ) ) ).
% infinite_descent0
thf(fact_831_diffs0__imp__equal,axiom,
! [M2: nat,N2: nat] :
( ( ( minus_minus_nat @ M2 @ N2 )
= zero_zero_nat )
=> ( ( ( minus_minus_nat @ N2 @ M2 )
= zero_zero_nat )
=> ( M2 = N2 ) ) ) ).
% diffs0_imp_equal
thf(fact_832_minus__nat_Odiff__0,axiom,
! [M2: nat] :
( ( minus_minus_nat @ M2 @ zero_zero_nat )
= M2 ) ).
% minus_nat.diff_0
thf(fact_833_butlast__list__update,axiom,
! [K: nat,Xs: list_a,X2: a] :
( ( ( K
= ( minus_minus_nat @ ( size_size_list_a @ Xs ) @ one_one_nat ) )
=> ( ( butlast_a @ ( list_update_a @ Xs @ K @ X2 ) )
= ( butlast_a @ Xs ) ) )
& ( ( K
!= ( minus_minus_nat @ ( size_size_list_a @ Xs ) @ one_one_nat ) )
=> ( ( butlast_a @ ( list_update_a @ Xs @ K @ X2 ) )
= ( list_update_a @ ( butlast_a @ Xs ) @ K @ X2 ) ) ) ) ).
% butlast_list_update
thf(fact_834_butlast__list__update,axiom,
! [K: nat,Xs: list_nat,X2: nat] :
( ( ( K
= ( minus_minus_nat @ ( size_size_list_nat @ Xs ) @ one_one_nat ) )
=> ( ( butlast_nat @ ( list_update_nat @ Xs @ K @ X2 ) )
= ( butlast_nat @ Xs ) ) )
& ( ( K
!= ( minus_minus_nat @ ( size_size_list_nat @ Xs ) @ one_one_nat ) )
=> ( ( butlast_nat @ ( list_update_nat @ Xs @ K @ X2 ) )
= ( list_update_nat @ ( butlast_nat @ Xs ) @ K @ X2 ) ) ) ) ).
% butlast_list_update
thf(fact_835_rotate__rev,axiom,
! [N2: nat,Xs: list_a] :
( ( rotate_a @ N2 @ ( rev_a @ Xs ) )
= ( rev_a @ ( rotate_a @ ( minus_minus_nat @ ( size_size_list_a @ Xs ) @ ( modulo_modulo_nat @ N2 @ ( size_size_list_a @ Xs ) ) ) @ Xs ) ) ) ).
% rotate_rev
thf(fact_836_rotate__rev,axiom,
! [N2: nat,Xs: list_nat] :
( ( rotate_nat @ N2 @ ( rev_nat @ Xs ) )
= ( rev_nat @ ( rotate_nat @ ( minus_minus_nat @ ( size_size_list_nat @ Xs ) @ ( modulo_modulo_nat @ N2 @ ( size_size_list_nat @ Xs ) ) ) @ Xs ) ) ) ).
% rotate_rev
thf(fact_837_le__iff__diff__le__0,axiom,
( ord_less_eq_real
= ( ^ [A4: real,B3: real] : ( ord_less_eq_real @ ( minus_minus_real @ A4 @ B3 ) @ zero_zero_real ) ) ) ).
% le_iff_diff_le_0
thf(fact_838_le__iff__diff__le__0,axiom,
( ord_less_eq_int
= ( ^ [A4: int,B3: int] : ( ord_less_eq_int @ ( minus_minus_int @ A4 @ B3 ) @ zero_zero_int ) ) ) ).
% le_iff_diff_le_0
thf(fact_839_less__numeral__extra_I1_J,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% less_numeral_extra(1)
thf(fact_840_less__numeral__extra_I1_J,axiom,
ord_less_real @ zero_zero_real @ one_one_real ).
% less_numeral_extra(1)
thf(fact_841_less__numeral__extra_I1_J,axiom,
ord_less_int @ zero_zero_int @ one_one_int ).
% less_numeral_extra(1)
thf(fact_842_less__iff__diff__less__0,axiom,
( ord_less_real
= ( ^ [A4: real,B3: real] : ( ord_less_real @ ( minus_minus_real @ A4 @ B3 ) @ zero_zero_real ) ) ) ).
% less_iff_diff_less_0
thf(fact_843_less__iff__diff__less__0,axiom,
( ord_less_int
= ( ^ [A4: int,B3: int] : ( ord_less_int @ ( minus_minus_int @ A4 @ B3 ) @ zero_zero_int ) ) ) ).
% less_iff_diff_less_0
thf(fact_844_less__Suc__eq__0__disj,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ ( suc @ N2 ) )
= ( ( M2 = zero_zero_nat )
| ? [J: nat] :
( ( M2
= ( suc @ J ) )
& ( ord_less_nat @ J @ N2 ) ) ) ) ).
% less_Suc_eq_0_disj
thf(fact_845_gr0__implies__Suc,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ? [M5: nat] :
( N2
= ( suc @ M5 ) ) ) ).
% gr0_implies_Suc
thf(fact_846_All__less__Suc2,axiom,
! [N2: nat,P: nat > $o] :
( ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( suc @ N2 ) )
=> ( P @ I3 ) ) )
= ( ( P @ zero_zero_nat )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ N2 )
=> ( P @ ( suc @ I3 ) ) ) ) ) ).
% All_less_Suc2
thf(fact_847_gr0__conv__Suc,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
= ( ? [M: nat] :
( N2
= ( suc @ M ) ) ) ) ).
% gr0_conv_Suc
thf(fact_848_Ex__less__Suc2,axiom,
! [N2: nat,P: nat > $o] :
( ( ? [I3: nat] :
( ( ord_less_nat @ I3 @ ( suc @ N2 ) )
& ( P @ I3 ) ) )
= ( ( P @ zero_zero_nat )
| ? [I3: nat] :
( ( ord_less_nat @ I3 @ N2 )
& ( P @ ( suc @ I3 ) ) ) ) ) ).
% Ex_less_Suc2
thf(fact_849_ex__least__nat__le,axiom,
! [P: nat > $o,N2: nat] :
( ( P @ N2 )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N2 )
& ! [I6: nat] :
( ( ord_less_nat @ I6 @ K2 )
=> ~ ( P @ I6 ) )
& ( P @ K2 ) ) ) ) ).
% ex_least_nat_le
thf(fact_850_One__nat__def,axiom,
( one_one_nat
= ( suc @ zero_zero_nat ) ) ).
% One_nat_def
thf(fact_851_diff__less,axiom,
! [N2: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_nat @ zero_zero_nat @ M2 )
=> ( ord_less_nat @ ( minus_minus_nat @ M2 @ N2 ) @ M2 ) ) ) ).
% diff_less
thf(fact_852_nth__list__update,axiom,
! [I4: nat,Xs: list_a,J2: nat,X2: a] :
( ( ord_less_nat @ I4 @ ( size_size_list_a @ Xs ) )
=> ( ( ( I4 = J2 )
=> ( ( nth_a @ ( list_update_a @ Xs @ I4 @ X2 ) @ J2 )
= X2 ) )
& ( ( I4 != J2 )
=> ( ( nth_a @ ( list_update_a @ Xs @ I4 @ X2 ) @ J2 )
= ( nth_a @ Xs @ J2 ) ) ) ) ) ).
% nth_list_update
thf(fact_853_nth__list__update,axiom,
! [I4: nat,Xs: list_nat,J2: nat,X2: nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Xs ) )
=> ( ( ( I4 = J2 )
=> ( ( nth_nat @ ( list_update_nat @ Xs @ I4 @ X2 ) @ J2 )
= X2 ) )
& ( ( I4 != J2 )
=> ( ( nth_nat @ ( list_update_nat @ Xs @ I4 @ X2 ) @ J2 )
= ( nth_nat @ Xs @ J2 ) ) ) ) ) ).
% nth_list_update
thf(fact_854_list__update__same__conv,axiom,
! [I4: nat,Xs: list_a,X2: a] :
( ( ord_less_nat @ I4 @ ( size_size_list_a @ Xs ) )
=> ( ( ( list_update_a @ Xs @ I4 @ X2 )
= Xs )
= ( ( nth_a @ Xs @ I4 )
= X2 ) ) ) ).
% list_update_same_conv
thf(fact_855_list__update__same__conv,axiom,
! [I4: nat,Xs: list_nat,X2: nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Xs ) )
=> ( ( ( list_update_nat @ Xs @ I4 @ X2 )
= Xs )
= ( ( nth_nat @ Xs @ I4 )
= X2 ) ) ) ).
% list_update_same_conv
thf(fact_856_ex__least__nat__less,axiom,
! [P: nat > $o,N2: nat] :
( ( P @ N2 )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K2: nat] :
( ( ord_less_nat @ K2 @ N2 )
& ! [I6: nat] :
( ( ord_less_eq_nat @ I6 @ K2 )
=> ~ ( P @ I6 ) )
& ( P @ ( suc @ K2 ) ) ) ) ) ).
% ex_least_nat_less
thf(fact_857_diff__Suc__less,axiom,
! [N2: nat,I4: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ord_less_nat @ ( minus_minus_nat @ N2 @ ( suc @ I4 ) ) @ N2 ) ) ).
% diff_Suc_less
thf(fact_858_nat__induct__non__zero,axiom,
! [N2: nat,P: nat > $o] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( P @ one_one_nat )
=> ( ! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) ) )
=> ( P @ N2 ) ) ) ) ).
% nat_induct_non_zero
thf(fact_859_nth__butlast,axiom,
! [N2: nat,Xs: list_a] :
( ( ord_less_nat @ N2 @ ( size_size_list_a @ ( butlast_a @ Xs ) ) )
=> ( ( nth_a @ ( butlast_a @ Xs ) @ N2 )
= ( nth_a @ Xs @ N2 ) ) ) ).
% nth_butlast
thf(fact_860_nth__butlast,axiom,
! [N2: nat,Xs: list_nat] :
( ( ord_less_nat @ N2 @ ( size_size_list_nat @ ( butlast_nat @ Xs ) ) )
=> ( ( nth_nat @ ( butlast_nat @ Xs ) @ N2 )
= ( nth_nat @ Xs @ N2 ) ) ) ).
% nth_butlast
thf(fact_861_Suc__diff__eq__diff__pred,axiom,
! [N2: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( minus_minus_nat @ ( suc @ M2 ) @ N2 )
= ( minus_minus_nat @ M2 @ ( minus_minus_nat @ N2 @ one_one_nat ) ) ) ) ).
% Suc_diff_eq_diff_pred
thf(fact_862_mod__by__Suc__0,axiom,
! [M2: nat] :
( ( modulo_modulo_nat @ M2 @ ( suc @ zero_zero_nat ) )
= zero_zero_nat ) ).
% mod_by_Suc_0
thf(fact_863_mod__by__1,axiom,
! [A: nat] :
( ( modulo_modulo_nat @ A @ one_one_nat )
= zero_zero_nat ) ).
% mod_by_1
thf(fact_864_mod__by__1,axiom,
! [A: int] :
( ( modulo_modulo_int @ A @ one_one_int )
= zero_zero_int ) ).
% mod_by_1
thf(fact_865_bits__mod__by__1,axiom,
! [A: nat] :
( ( modulo_modulo_nat @ A @ one_one_nat )
= zero_zero_nat ) ).
% bits_mod_by_1
thf(fact_866_bits__mod__by__1,axiom,
! [A: int] :
( ( modulo_modulo_int @ A @ one_one_int )
= zero_zero_int ) ).
% bits_mod_by_1
thf(fact_867_mod__less,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( ( modulo_modulo_nat @ M2 @ N2 )
= M2 ) ) ).
% mod_less
thf(fact_868_mod__le__divisor,axiom,
! [N2: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ord_less_eq_nat @ ( modulo_modulo_nat @ M2 @ N2 ) @ N2 ) ) ).
% mod_le_divisor
thf(fact_869_mod__if,axiom,
( modulo_modulo_nat
= ( ^ [M: nat,N: nat] : ( if_nat @ ( ord_less_nat @ M @ N ) @ M @ ( modulo_modulo_nat @ ( minus_minus_nat @ M @ N ) @ N ) ) ) ) ).
% mod_if
thf(fact_870_linorder__neqE__linordered__idom,axiom,
! [X2: real,Y2: real] :
( ( X2 != Y2 )
=> ( ~ ( ord_less_real @ X2 @ Y2 )
=> ( ord_less_real @ Y2 @ X2 ) ) ) ).
% linorder_neqE_linordered_idom
thf(fact_871_linorder__neqE__linordered__idom,axiom,
! [X2: int,Y2: int] :
( ( X2 != Y2 )
=> ( ~ ( ord_less_int @ X2 @ Y2 )
=> ( ord_less_int @ Y2 @ X2 ) ) ) ).
% linorder_neqE_linordered_idom
thf(fact_872_zero__neq__one,axiom,
zero_zero_nat != one_one_nat ).
% zero_neq_one
thf(fact_873_zero__neq__one,axiom,
zero_zero_real != one_one_real ).
% zero_neq_one
thf(fact_874_zero__neq__one,axiom,
zero_zero_int != one_one_int ).
% zero_neq_one
thf(fact_875_mod__Suc__eq,axiom,
! [M2: nat,N2: nat] :
( ( modulo_modulo_nat @ ( suc @ ( modulo_modulo_nat @ M2 @ N2 ) ) @ N2 )
= ( modulo_modulo_nat @ ( suc @ M2 ) @ N2 ) ) ).
% mod_Suc_eq
thf(fact_876_mod__Suc__Suc__eq,axiom,
! [M2: nat,N2: nat] :
( ( modulo_modulo_nat @ ( suc @ ( suc @ ( modulo_modulo_nat @ M2 @ N2 ) ) ) @ N2 )
= ( modulo_modulo_nat @ ( suc @ ( suc @ M2 ) ) @ N2 ) ) ).
% mod_Suc_Suc_eq
thf(fact_877_mod__less__eq__dividend,axiom,
! [M2: nat,N2: nat] : ( ord_less_eq_nat @ ( modulo_modulo_nat @ M2 @ N2 ) @ M2 ) ).
% mod_less_eq_dividend
thf(fact_878_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_879_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_880_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_881_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_882_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_883_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_884_not__one__le__zero,axiom,
~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_le_zero
thf(fact_885_not__one__le__zero,axiom,
~ ( ord_less_eq_real @ one_one_real @ zero_zero_real ) ).
% not_one_le_zero
thf(fact_886_not__one__le__zero,axiom,
~ ( ord_less_eq_int @ one_one_int @ zero_zero_int ) ).
% not_one_le_zero
thf(fact_887_zero__less__one,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one
thf(fact_888_zero__less__one,axiom,
ord_less_real @ zero_zero_real @ one_one_real ).
% zero_less_one
thf(fact_889_zero__less__one,axiom,
ord_less_int @ zero_zero_int @ one_one_int ).
% zero_less_one
thf(fact_890_not__one__less__zero,axiom,
~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_less_zero
thf(fact_891_not__one__less__zero,axiom,
~ ( ord_less_real @ one_one_real @ zero_zero_real ) ).
% not_one_less_zero
thf(fact_892_not__one__less__zero,axiom,
~ ( ord_less_int @ one_one_int @ zero_zero_int ) ).
% not_one_less_zero
thf(fact_893_mod__Suc,axiom,
! [M2: nat,N2: nat] :
( ( ( ( suc @ ( modulo_modulo_nat @ M2 @ N2 ) )
= N2 )
=> ( ( modulo_modulo_nat @ ( suc @ M2 ) @ N2 )
= zero_zero_nat ) )
& ( ( ( suc @ ( modulo_modulo_nat @ M2 @ N2 ) )
!= N2 )
=> ( ( modulo_modulo_nat @ ( suc @ M2 ) @ N2 )
= ( suc @ ( modulo_modulo_nat @ M2 @ N2 ) ) ) ) ) ).
% mod_Suc
thf(fact_894_mod__less__divisor,axiom,
! [N2: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ord_less_nat @ ( modulo_modulo_nat @ M2 @ N2 ) @ N2 ) ) ).
% mod_less_divisor
thf(fact_895_mod__Suc__le__divisor,axiom,
! [M2: nat,N2: nat] : ( ord_less_eq_nat @ ( modulo_modulo_nat @ M2 @ ( suc @ N2 ) ) @ N2 ) ).
% mod_Suc_le_divisor
thf(fact_896_mod__induct,axiom,
! [P: nat > $o,N2: nat,P5: nat,M2: nat] :
( ( P @ N2 )
=> ( ( ord_less_nat @ N2 @ P5 )
=> ( ( ord_less_nat @ M2 @ P5 )
=> ( ! [N3: nat] :
( ( ord_less_nat @ N3 @ P5 )
=> ( ( P @ N3 )
=> ( P @ ( modulo_modulo_nat @ ( suc @ N3 ) @ P5 ) ) ) )
=> ( P @ M2 ) ) ) ) ) ).
% mod_induct
thf(fact_897_le__mod__geq,axiom,
! [N2: nat,M2: nat] :
( ( ord_less_eq_nat @ N2 @ M2 )
=> ( ( modulo_modulo_nat @ M2 @ N2 )
= ( modulo_modulo_nat @ ( minus_minus_nat @ M2 @ N2 ) @ N2 ) ) ) ).
% le_mod_geq
thf(fact_898_gcd__nat__induct,axiom,
! [P: nat > nat > $o,M2: nat,N2: nat] :
( ! [M5: nat] : ( P @ M5 @ zero_zero_nat )
=> ( ! [M5: nat,N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ( P @ N3 @ ( modulo_modulo_nat @ M5 @ N3 ) )
=> ( P @ M5 @ N3 ) ) )
=> ( P @ M2 @ N2 ) ) ) ).
% gcd_nat_induct
thf(fact_899_kuhn__labelling__lemma_H,axiom,
! [P: ( nat > real ) > $o,F: ( nat > real ) > nat > real,Q: nat > $o] :
( ! [X3: nat > real] :
( ( P @ X3 )
=> ( P @ ( F @ X3 ) ) )
=> ( ! [X3: nat > real] :
( ( P @ X3 )
=> ! [I5: nat] :
( ( Q @ I5 )
=> ( ( ord_less_eq_real @ zero_zero_real @ ( X3 @ I5 ) )
& ( ord_less_eq_real @ ( X3 @ I5 ) @ one_one_real ) ) ) )
=> ? [L2: ( nat > real ) > nat > nat] :
( ! [X6: nat > real,I6: nat] : ( ord_less_eq_nat @ ( L2 @ X6 @ I6 ) @ one_one_nat )
& ! [X6: nat > real,I6: nat] :
( ( ( P @ X6 )
& ( Q @ I6 )
& ( ( X6 @ I6 )
= zero_zero_real ) )
=> ( ( L2 @ X6 @ I6 )
= zero_zero_nat ) )
& ! [X6: nat > real,I6: nat] :
( ( ( P @ X6 )
& ( Q @ I6 )
& ( ( X6 @ I6 )
= one_one_real ) )
=> ( ( L2 @ X6 @ I6 )
= one_one_nat ) )
& ! [X6: nat > real,I6: nat] :
( ( ( P @ X6 )
& ( Q @ I6 )
& ( ( L2 @ X6 @ I6 )
= zero_zero_nat ) )
=> ( ord_less_eq_real @ ( X6 @ I6 ) @ ( F @ X6 @ I6 ) ) )
& ! [X6: nat > real,I6: nat] :
( ( ( P @ X6 )
& ( Q @ I6 )
& ( ( L2 @ X6 @ I6 )
= one_one_nat ) )
=> ( ord_less_eq_real @ ( F @ X6 @ I6 ) @ ( X6 @ I6 ) ) ) ) ) ) ).
% kuhn_labelling_lemma'
thf(fact_900_ge__iff__diff__ge__0,axiom,
( ord_less_eq_real
= ( ^ [B3: real,A4: real] : ( ord_less_eq_real @ zero_zero_real @ ( minus_minus_real @ A4 @ B3 ) ) ) ) ).
% ge_iff_diff_ge_0
thf(fact_901_ge__iff__diff__ge__0,axiom,
( ord_less_eq_int
= ( ^ [B3: int,A4: int] : ( ord_less_eq_int @ zero_zero_int @ ( minus_minus_int @ A4 @ B3 ) ) ) ) ).
% ge_iff_diff_ge_0
thf(fact_902_exists__least__lemma,axiom,
! [P: nat > $o] :
( ~ ( P @ zero_zero_nat )
=> ( ? [X_12: nat] : ( P @ X_12 )
=> ? [N3: nat] :
( ~ ( P @ N3 )
& ( P @ ( suc @ N3 ) ) ) ) ) ).
% exists_least_lemma
thf(fact_903_seq__mono__lemma,axiom,
! [M2: nat,D2: nat > real,E: nat > real] :
( ! [N3: nat] :
( ( ord_less_eq_nat @ M2 @ N3 )
=> ( ord_less_real @ ( D2 @ N3 ) @ ( E @ N3 ) ) )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ M2 @ N3 )
=> ( ord_less_eq_real @ ( E @ N3 ) @ ( E @ M2 ) ) )
=> ! [N5: nat] :
( ( ord_less_eq_nat @ M2 @ N5 )
=> ( ord_less_real @ ( D2 @ N5 ) @ ( E @ M2 ) ) ) ) ) ).
% seq_mono_lemma
thf(fact_904_Bolzano,axiom,
! [A: real,B: real,P: real > real > $o] :
( ( ord_less_eq_real @ A @ B )
=> ( ! [A3: real,B2: real,C2: real] :
( ( P @ A3 @ B2 )
=> ( ( P @ B2 @ C2 )
=> ( ( ord_less_eq_real @ A3 @ B2 )
=> ( ( ord_less_eq_real @ B2 @ C2 )
=> ( P @ A3 @ C2 ) ) ) ) )
=> ( ! [X3: real] :
( ( ord_less_eq_real @ A @ X3 )
=> ( ( ord_less_eq_real @ X3 @ B )
=> ? [D: real] :
( ( ord_less_real @ zero_zero_real @ D )
& ! [A3: real,B2: real] :
( ( ( ord_less_eq_real @ A3 @ X3 )
& ( ord_less_eq_real @ X3 @ B2 )
& ( ord_less_real @ ( minus_minus_real @ B2 @ A3 ) @ D ) )
=> ( P @ A3 @ B2 ) ) ) ) )
=> ( P @ A @ B ) ) ) ) ).
% Bolzano
thf(fact_905_less__eq__real__def,axiom,
( ord_less_eq_real
= ( ^ [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
| ( X = Y ) ) ) ) ).
% less_eq_real_def
thf(fact_906_field__lbound__gt__zero,axiom,
! [D1: real,D22: real] :
( ( ord_less_real @ zero_zero_real @ D1 )
=> ( ( ord_less_real @ zero_zero_real @ D22 )
=> ? [E2: real] :
( ( ord_less_real @ zero_zero_real @ E2 )
& ( ord_less_real @ E2 @ D1 )
& ( ord_less_real @ E2 @ D22 ) ) ) ) ).
% field_lbound_gt_zero
thf(fact_907_nth__Cons__pos,axiom,
! [N2: nat,X2: a,Xs: list_a] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( nth_a @ ( cons_a @ X2 @ Xs ) @ N2 )
= ( nth_a @ Xs @ ( minus_minus_nat @ N2 @ one_one_nat ) ) ) ) ).
% nth_Cons_pos
thf(fact_908_nth__Cons__pos,axiom,
! [N2: nat,X2: nat,Xs: list_nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( nth_nat @ ( cons_nat @ X2 @ Xs ) @ N2 )
= ( nth_nat @ Xs @ ( minus_minus_nat @ N2 @ one_one_nat ) ) ) ) ).
% nth_Cons_pos
thf(fact_909_nth__Cons__pos,axiom,
! [N2: nat,X2: int,Xs: list_int] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( nth_int @ ( cons_int @ X2 @ Xs ) @ N2 )
= ( nth_int @ Xs @ ( minus_minus_nat @ N2 @ one_one_nat ) ) ) ) ).
% nth_Cons_pos
thf(fact_910_dbl__inc__simps_I2_J,axiom,
( ( neg_nu8295874005876285629c_real @ zero_zero_real )
= one_one_real ) ).
% dbl_inc_simps(2)
thf(fact_911_dbl__inc__simps_I2_J,axiom,
( ( neg_nu5851722552734809277nc_int @ zero_zero_int )
= one_one_int ) ).
% dbl_inc_simps(2)
thf(fact_912_nth__rotate,axiom,
! [N2: nat,Xs: list_a,M2: nat] :
( ( ord_less_nat @ N2 @ ( size_size_list_a @ Xs ) )
=> ( ( nth_a @ ( rotate_a @ M2 @ Xs ) @ N2 )
= ( nth_a @ Xs @ ( modulo_modulo_nat @ ( plus_plus_nat @ M2 @ N2 ) @ ( size_size_list_a @ Xs ) ) ) ) ) ).
% nth_rotate
thf(fact_913_nth__rotate,axiom,
! [N2: nat,Xs: list_nat,M2: nat] :
( ( ord_less_nat @ N2 @ ( size_size_list_nat @ Xs ) )
=> ( ( nth_nat @ ( rotate_nat @ M2 @ Xs ) @ N2 )
= ( nth_nat @ Xs @ ( modulo_modulo_nat @ ( plus_plus_nat @ M2 @ N2 ) @ ( size_size_list_nat @ Xs ) ) ) ) ) ).
% nth_rotate
thf(fact_914_list_Oinject,axiom,
! [X21: int,X222: list_int,Y21: int,Y222: list_int] :
( ( ( cons_int @ X21 @ X222 )
= ( cons_int @ Y21 @ Y222 ) )
= ( ( X21 = Y21 )
& ( X222 = Y222 ) ) ) ).
% list.inject
thf(fact_915_add__le__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
= ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_cancel_left
thf(fact_916_add__le__cancel__left,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
= ( ord_less_eq_real @ A @ B ) ) ).
% add_le_cancel_left
thf(fact_917_add__le__cancel__left,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
= ( ord_less_eq_int @ A @ B ) ) ).
% add_le_cancel_left
thf(fact_918_add__le__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_cancel_right
thf(fact_919_add__le__cancel__right,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
= ( ord_less_eq_real @ A @ B ) ) ).
% add_le_cancel_right
thf(fact_920_add__le__cancel__right,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
= ( ord_less_eq_int @ A @ B ) ) ).
% add_le_cancel_right
thf(fact_921_add__less__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( ord_less_nat @ A @ B ) ) ).
% add_less_cancel_right
thf(fact_922_add__less__cancel__right,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
= ( ord_less_real @ A @ B ) ) ).
% add_less_cancel_right
thf(fact_923_add__less__cancel__right,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
= ( ord_less_int @ A @ B ) ) ).
% add_less_cancel_right
thf(fact_924_add__less__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
= ( ord_less_nat @ A @ B ) ) ).
% add_less_cancel_left
thf(fact_925_add__less__cancel__left,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
= ( ord_less_real @ A @ B ) ) ).
% add_less_cancel_left
thf(fact_926_add__less__cancel__left,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
= ( ord_less_int @ A @ B ) ) ).
% add_less_cancel_left
thf(fact_927_Nat_Oadd__0__right,axiom,
! [M2: nat] :
( ( plus_plus_nat @ M2 @ zero_zero_nat )
= M2 ) ).
% Nat.add_0_right
thf(fact_928_add__is__0,axiom,
! [M2: nat,N2: nat] :
( ( ( plus_plus_nat @ M2 @ N2 )
= zero_zero_nat )
= ( ( M2 = zero_zero_nat )
& ( N2 = zero_zero_nat ) ) ) ).
% add_is_0
thf(fact_929_add__Suc__right,axiom,
! [M2: nat,N2: nat] :
( ( plus_plus_nat @ M2 @ ( suc @ N2 ) )
= ( suc @ ( plus_plus_nat @ M2 @ N2 ) ) ) ).
% add_Suc_right
thf(fact_930_nat__add__left__cancel__le,axiom,
! [K: nat,M2: nat,N2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M2 ) @ ( plus_plus_nat @ K @ N2 ) )
= ( ord_less_eq_nat @ M2 @ N2 ) ) ).
% nat_add_left_cancel_le
thf(fact_931_nat__add__left__cancel__less,axiom,
! [K: nat,M2: nat,N2: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ K @ M2 ) @ ( plus_plus_nat @ K @ N2 ) )
= ( ord_less_nat @ M2 @ N2 ) ) ).
% nat_add_left_cancel_less
thf(fact_932_diff__diff__left,axiom,
! [I4: nat,J2: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I4 @ J2 ) @ K )
= ( minus_minus_nat @ I4 @ ( plus_plus_nat @ J2 @ K ) ) ) ).
% diff_diff_left
thf(fact_933_list__ex__simps_I1_J,axiom,
! [P: int > $o,X2: int,Xs: list_int] :
( ( list_ex_int @ P @ ( cons_int @ X2 @ Xs ) )
= ( ( P @ X2 )
| ( list_ex_int @ P @ Xs ) ) ) ).
% list_ex_simps(1)
thf(fact_934_zero__le__double__add__iff__zero__le__single__add,axiom,
! [A: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ A @ A ) )
= ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% zero_le_double_add_iff_zero_le_single_add
thf(fact_935_zero__le__double__add__iff__zero__le__single__add,axiom,
! [A: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A @ A ) )
= ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% zero_le_double_add_iff_zero_le_single_add
thf(fact_936_double__add__le__zero__iff__single__add__le__zero,axiom,
! [A: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ A @ A ) @ zero_zero_real )
= ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% double_add_le_zero_iff_single_add_le_zero
thf(fact_937_double__add__le__zero__iff__single__add__le__zero,axiom,
! [A: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A @ A ) @ zero_zero_int )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% double_add_le_zero_iff_single_add_le_zero
thf(fact_938_le__add__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ B @ A ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% le_add_same_cancel2
thf(fact_939_le__add__same__cancel2,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ ( plus_plus_real @ B @ A ) )
= ( ord_less_eq_real @ zero_zero_real @ B ) ) ).
% le_add_same_cancel2
thf(fact_940_le__add__same__cancel2,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ ( plus_plus_int @ B @ A ) )
= ( ord_less_eq_int @ zero_zero_int @ B ) ) ).
% le_add_same_cancel2
thf(fact_941_le__add__same__cancel1,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ A @ B ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% le_add_same_cancel1
thf(fact_942_le__add__same__cancel1,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ ( plus_plus_real @ A @ B ) )
= ( ord_less_eq_real @ zero_zero_real @ B ) ) ).
% le_add_same_cancel1
thf(fact_943_le__add__same__cancel1,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ ( plus_plus_int @ A @ B ) )
= ( ord_less_eq_int @ zero_zero_int @ B ) ) ).
% le_add_same_cancel1
thf(fact_944_add__le__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ B )
= ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).
% add_le_same_cancel2
thf(fact_945_add__le__same__cancel2,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ A @ B ) @ B )
= ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% add_le_same_cancel2
thf(fact_946_add__le__same__cancel2,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A @ B ) @ B )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% add_le_same_cancel2
thf(fact_947_add__le__same__cancel1,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ B @ A ) @ B )
= ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).
% add_le_same_cancel1
thf(fact_948_add__le__same__cancel1,axiom,
! [B: real,A: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ B @ A ) @ B )
= ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% add_le_same_cancel1
thf(fact_949_add__le__same__cancel1,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ B @ A ) @ B )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% add_le_same_cancel1
thf(fact_950_add__less__same__cancel1,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ B @ A ) @ B )
= ( ord_less_nat @ A @ zero_zero_nat ) ) ).
% add_less_same_cancel1
thf(fact_951_add__less__same__cancel1,axiom,
! [B: real,A: real] :
( ( ord_less_real @ ( plus_plus_real @ B @ A ) @ B )
= ( ord_less_real @ A @ zero_zero_real ) ) ).
% add_less_same_cancel1
thf(fact_952_add__less__same__cancel1,axiom,
! [B: int,A: int] :
( ( ord_less_int @ ( plus_plus_int @ B @ A ) @ B )
= ( ord_less_int @ A @ zero_zero_int ) ) ).
% add_less_same_cancel1
thf(fact_953_add__less__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ B )
= ( ord_less_nat @ A @ zero_zero_nat ) ) ).
% add_less_same_cancel2
thf(fact_954_add__less__same__cancel2,axiom,
! [A: real,B: real] :
( ( ord_less_real @ ( plus_plus_real @ A @ B ) @ B )
= ( ord_less_real @ A @ zero_zero_real ) ) ).
% add_less_same_cancel2
thf(fact_955_add__less__same__cancel2,axiom,
! [A: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ A @ B ) @ B )
= ( ord_less_int @ A @ zero_zero_int ) ) ).
% add_less_same_cancel2
thf(fact_956_less__add__same__cancel1,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ ( plus_plus_nat @ A @ B ) )
= ( ord_less_nat @ zero_zero_nat @ B ) ) ).
% less_add_same_cancel1
thf(fact_957_less__add__same__cancel1,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ ( plus_plus_real @ A @ B ) )
= ( ord_less_real @ zero_zero_real @ B ) ) ).
% less_add_same_cancel1
thf(fact_958_less__add__same__cancel1,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ ( plus_plus_int @ A @ B ) )
= ( ord_less_int @ zero_zero_int @ B ) ) ).
% less_add_same_cancel1
thf(fact_959_less__add__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ ( plus_plus_nat @ B @ A ) )
= ( ord_less_nat @ zero_zero_nat @ B ) ) ).
% less_add_same_cancel2
thf(fact_960_less__add__same__cancel2,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ ( plus_plus_real @ B @ A ) )
= ( ord_less_real @ zero_zero_real @ B ) ) ).
% less_add_same_cancel2
thf(fact_961_less__add__same__cancel2,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ ( plus_plus_int @ B @ A ) )
= ( ord_less_int @ zero_zero_int @ B ) ) ).
% less_add_same_cancel2
thf(fact_962_double__add__less__zero__iff__single__add__less__zero,axiom,
! [A: real] :
( ( ord_less_real @ ( plus_plus_real @ A @ A ) @ zero_zero_real )
= ( ord_less_real @ A @ zero_zero_real ) ) ).
% double_add_less_zero_iff_single_add_less_zero
thf(fact_963_double__add__less__zero__iff__single__add__less__zero,axiom,
! [A: int] :
( ( ord_less_int @ ( plus_plus_int @ A @ A ) @ zero_zero_int )
= ( ord_less_int @ A @ zero_zero_int ) ) ).
% double_add_less_zero_iff_single_add_less_zero
thf(fact_964_zero__less__double__add__iff__zero__less__single__add,axiom,
! [A: real] :
( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A @ A ) )
= ( ord_less_real @ zero_zero_real @ A ) ) ).
% zero_less_double_add_iff_zero_less_single_add
thf(fact_965_zero__less__double__add__iff__zero__less__single__add,axiom,
! [A: int] :
( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ A ) )
= ( ord_less_int @ zero_zero_int @ A ) ) ).
% zero_less_double_add_iff_zero_less_single_add
thf(fact_966_le__add__diff__inverse,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( plus_plus_nat @ B @ ( minus_minus_nat @ A @ B ) )
= A ) ) ).
% le_add_diff_inverse
thf(fact_967_le__add__diff__inverse,axiom,
! [B: real,A: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ( plus_plus_real @ B @ ( minus_minus_real @ A @ B ) )
= A ) ) ).
% le_add_diff_inverse
thf(fact_968_le__add__diff__inverse,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( plus_plus_int @ B @ ( minus_minus_int @ A @ B ) )
= A ) ) ).
% le_add_diff_inverse
thf(fact_969_le__add__diff__inverse2,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ A @ B ) @ B )
= A ) ) ).
% le_add_diff_inverse2
thf(fact_970_le__add__diff__inverse2,axiom,
! [B: real,A: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ( plus_plus_real @ ( minus_minus_real @ A @ B ) @ B )
= A ) ) ).
% le_add_diff_inverse2
thf(fact_971_le__add__diff__inverse2,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ B )
= A ) ) ).
% le_add_diff_inverse2
thf(fact_972_add__gr__0,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M2 @ N2 ) )
= ( ( ord_less_nat @ zero_zero_nat @ M2 )
| ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% add_gr_0
thf(fact_973_Nat_Oadd__diff__assoc,axiom,
! [K: nat,J2: nat,I4: nat] :
( ( ord_less_eq_nat @ K @ J2 )
=> ( ( plus_plus_nat @ I4 @ ( minus_minus_nat @ J2 @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I4 @ J2 ) @ K ) ) ) ).
% Nat.add_diff_assoc
thf(fact_974_Nat_Oadd__diff__assoc2,axiom,
! [K: nat,J2: nat,I4: nat] :
( ( ord_less_eq_nat @ K @ J2 )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ J2 @ K ) @ I4 )
= ( minus_minus_nat @ ( plus_plus_nat @ J2 @ I4 ) @ K ) ) ) ).
% Nat.add_diff_assoc2
thf(fact_975_Nat_Odiff__diff__right,axiom,
! [K: nat,J2: nat,I4: nat] :
( ( ord_less_eq_nat @ K @ J2 )
=> ( ( minus_minus_nat @ I4 @ ( minus_minus_nat @ J2 @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I4 @ K ) @ J2 ) ) ) ).
% Nat.diff_diff_right
thf(fact_976_nth__Cons__0,axiom,
! [X2: a,Xs: list_a] :
( ( nth_a @ ( cons_a @ X2 @ Xs ) @ zero_zero_nat )
= X2 ) ).
% nth_Cons_0
thf(fact_977_nth__Cons__0,axiom,
! [X2: nat,Xs: list_nat] :
( ( nth_nat @ ( cons_nat @ X2 @ Xs ) @ zero_zero_nat )
= X2 ) ).
% nth_Cons_0
thf(fact_978_nth__Cons__0,axiom,
! [X2: int,Xs: list_int] :
( ( nth_int @ ( cons_int @ X2 @ Xs ) @ zero_zero_nat )
= X2 ) ).
% nth_Cons_0
thf(fact_979_nth__Cons__Suc,axiom,
! [X2: a,Xs: list_a,N2: nat] :
( ( nth_a @ ( cons_a @ X2 @ Xs ) @ ( suc @ N2 ) )
= ( nth_a @ Xs @ N2 ) ) ).
% nth_Cons_Suc
thf(fact_980_nth__Cons__Suc,axiom,
! [X2: nat,Xs: list_nat,N2: nat] :
( ( nth_nat @ ( cons_nat @ X2 @ Xs ) @ ( suc @ N2 ) )
= ( nth_nat @ Xs @ N2 ) ) ).
% nth_Cons_Suc
thf(fact_981_nth__Cons__Suc,axiom,
! [X2: int,Xs: list_int,N2: nat] :
( ( nth_int @ ( cons_int @ X2 @ Xs ) @ ( suc @ N2 ) )
= ( nth_int @ Xs @ N2 ) ) ).
% nth_Cons_Suc
thf(fact_982_diff__Suc__diff__eq2,axiom,
! [K: nat,J2: nat,I4: nat] :
( ( ord_less_eq_nat @ K @ J2 )
=> ( ( minus_minus_nat @ ( suc @ ( minus_minus_nat @ J2 @ K ) ) @ I4 )
= ( minus_minus_nat @ ( suc @ J2 ) @ ( plus_plus_nat @ K @ I4 ) ) ) ) ).
% diff_Suc_diff_eq2
thf(fact_983_diff__Suc__diff__eq1,axiom,
! [K: nat,J2: nat,I4: nat] :
( ( ord_less_eq_nat @ K @ J2 )
=> ( ( minus_minus_nat @ I4 @ ( suc @ ( minus_minus_nat @ J2 @ K ) ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I4 @ K ) @ ( suc @ J2 ) ) ) ) ).
% diff_Suc_diff_eq1
thf(fact_984_verit__sum__simplify,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% verit_sum_simplify
thf(fact_985_verit__sum__simplify,axiom,
! [A: real] :
( ( plus_plus_real @ A @ zero_zero_real )
= A ) ).
% verit_sum_simplify
thf(fact_986_verit__sum__simplify,axiom,
! [A: int] :
( ( plus_plus_int @ A @ zero_zero_int )
= A ) ).
% verit_sum_simplify
thf(fact_987_plus__nat_Oadd__0,axiom,
! [N2: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N2 )
= N2 ) ).
% plus_nat.add_0
thf(fact_988_add__eq__self__zero,axiom,
! [M2: nat,N2: nat] :
( ( ( plus_plus_nat @ M2 @ N2 )
= M2 )
=> ( N2 = zero_zero_nat ) ) ).
% add_eq_self_zero
thf(fact_989_diff__add__inverse2,axiom,
! [M2: nat,N2: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M2 @ N2 ) @ N2 )
= M2 ) ).
% diff_add_inverse2
thf(fact_990_diff__add__inverse,axiom,
! [N2: nat,M2: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ N2 @ M2 ) @ N2 )
= M2 ) ).
% diff_add_inverse
thf(fact_991_diff__cancel2,axiom,
! [M2: nat,K: nat,N2: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M2 @ K ) @ ( plus_plus_nat @ N2 @ K ) )
= ( minus_minus_nat @ M2 @ N2 ) ) ).
% diff_cancel2
thf(fact_992_Nat_Odiff__cancel,axiom,
! [K: nat,M2: nat,N2: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ K @ M2 ) @ ( plus_plus_nat @ K @ N2 ) )
= ( minus_minus_nat @ M2 @ N2 ) ) ).
% Nat.diff_cancel
thf(fact_993_add__lessD1,axiom,
! [I4: nat,J2: nat,K: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ I4 @ J2 ) @ K )
=> ( ord_less_nat @ I4 @ K ) ) ).
% add_lessD1
thf(fact_994_add__less__mono,axiom,
! [I4: nat,J2: nat,K: nat,L: nat] :
( ( ord_less_nat @ I4 @ J2 )
=> ( ( ord_less_nat @ K @ L )
=> ( ord_less_nat @ ( plus_plus_nat @ I4 @ K ) @ ( plus_plus_nat @ J2 @ L ) ) ) ) ).
% add_less_mono
thf(fact_995_not__add__less1,axiom,
! [I4: nat,J2: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ I4 @ J2 ) @ I4 ) ).
% not_add_less1
thf(fact_996_not__add__less2,axiom,
! [J2: nat,I4: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ J2 @ I4 ) @ I4 ) ).
% not_add_less2
thf(fact_997_add__less__mono1,axiom,
! [I4: nat,J2: nat,K: nat] :
( ( ord_less_nat @ I4 @ J2 )
=> ( ord_less_nat @ ( plus_plus_nat @ I4 @ K ) @ ( plus_plus_nat @ J2 @ K ) ) ) ).
% add_less_mono1
thf(fact_998_trans__less__add1,axiom,
! [I4: nat,J2: nat,M2: nat] :
( ( ord_less_nat @ I4 @ J2 )
=> ( ord_less_nat @ I4 @ ( plus_plus_nat @ J2 @ M2 ) ) ) ).
% trans_less_add1
thf(fact_999_trans__less__add2,axiom,
! [I4: nat,J2: nat,M2: nat] :
( ( ord_less_nat @ I4 @ J2 )
=> ( ord_less_nat @ I4 @ ( plus_plus_nat @ M2 @ J2 ) ) ) ).
% trans_less_add2
thf(fact_1000_less__add__eq__less,axiom,
! [K: nat,L: nat,M2: nat,N2: nat] :
( ( ord_less_nat @ K @ L )
=> ( ( ( plus_plus_nat @ M2 @ L )
= ( plus_plus_nat @ K @ N2 ) )
=> ( ord_less_nat @ M2 @ N2 ) ) ) ).
% less_add_eq_less
thf(fact_1001_add__mono__thms__linordered__field_I5_J,axiom,
! [I4: nat,J2: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I4 @ J2 )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I4 @ K ) @ ( plus_plus_nat @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_1002_add__mono__thms__linordered__field_I5_J,axiom,
! [I4: real,J2: real,K: real,L: real] :
( ( ( ord_less_real @ I4 @ J2 )
& ( ord_less_real @ K @ L ) )
=> ( ord_less_real @ ( plus_plus_real @ I4 @ K ) @ ( plus_plus_real @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_1003_add__mono__thms__linordered__field_I5_J,axiom,
! [I4: int,J2: int,K: int,L: int] :
( ( ( ord_less_int @ I4 @ J2 )
& ( ord_less_int @ K @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I4 @ K ) @ ( plus_plus_int @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_1004_add__mono__thms__linordered__field_I2_J,axiom,
! [I4: nat,J2: nat,K: nat,L: nat] :
( ( ( I4 = J2 )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I4 @ K ) @ ( plus_plus_nat @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_1005_add__mono__thms__linordered__field_I2_J,axiom,
! [I4: real,J2: real,K: real,L: real] :
( ( ( I4 = J2 )
& ( ord_less_real @ K @ L ) )
=> ( ord_less_real @ ( plus_plus_real @ I4 @ K ) @ ( plus_plus_real @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_1006_add__mono__thms__linordered__field_I2_J,axiom,
! [I4: int,J2: int,K: int,L: int] :
( ( ( I4 = J2 )
& ( ord_less_int @ K @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I4 @ K ) @ ( plus_plus_int @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_1007_add__mono__thms__linordered__field_I1_J,axiom,
! [I4: nat,J2: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I4 @ J2 )
& ( K = L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I4 @ K ) @ ( plus_plus_nat @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_1008_add__mono__thms__linordered__field_I1_J,axiom,
! [I4: real,J2: real,K: real,L: real] :
( ( ( ord_less_real @ I4 @ J2 )
& ( K = L ) )
=> ( ord_less_real @ ( plus_plus_real @ I4 @ K ) @ ( plus_plus_real @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_1009_add__mono__thms__linordered__field_I1_J,axiom,
! [I4: int,J2: int,K: int,L: int] :
( ( ( ord_less_int @ I4 @ J2 )
& ( K = L ) )
=> ( ord_less_int @ ( plus_plus_int @ I4 @ K ) @ ( plus_plus_int @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_1010_add__strict__mono,axiom,
! [A: nat,B: nat,C: nat,D2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D2 )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D2 ) ) ) ) ).
% add_strict_mono
thf(fact_1011_add__strict__mono,axiom,
! [A: real,B: real,C: real,D2: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ C @ D2 )
=> ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D2 ) ) ) ) ).
% add_strict_mono
thf(fact_1012_add__strict__mono,axiom,
! [A: int,B: int,C: int,D2: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ C @ D2 )
=> ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D2 ) ) ) ) ).
% add_strict_mono
thf(fact_1013_add__strict__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) ) ) ).
% add_strict_left_mono
thf(fact_1014_add__strict__left__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ord_less_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) ) ) ).
% add_strict_left_mono
thf(fact_1015_add__strict__left__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) ) ) ).
% add_strict_left_mono
thf(fact_1016_add__strict__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).
% add_strict_right_mono
thf(fact_1017_add__strict__right__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) ) ) ).
% add_strict_right_mono
thf(fact_1018_add__strict__right__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) ) ) ).
% add_strict_right_mono
thf(fact_1019_add__less__imp__less__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
=> ( ord_less_nat @ A @ B ) ) ).
% add_less_imp_less_left
thf(fact_1020_add__less__imp__less__left,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
=> ( ord_less_real @ A @ B ) ) ).
% add_less_imp_less_left
thf(fact_1021_add__less__imp__less__left,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
=> ( ord_less_int @ A @ B ) ) ).
% add_less_imp_less_left
thf(fact_1022_add__less__imp__less__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
=> ( ord_less_nat @ A @ B ) ) ).
% add_less_imp_less_right
thf(fact_1023_add__less__imp__less__right,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
=> ( ord_less_real @ A @ B ) ) ).
% add_less_imp_less_right
thf(fact_1024_add__less__imp__less__right,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
=> ( ord_less_int @ A @ B ) ) ).
% add_less_imp_less_right
thf(fact_1025_nat__arith_Osuc1,axiom,
! [A2: nat,K: nat,A: nat] :
( ( A2
= ( plus_plus_nat @ K @ A ) )
=> ( ( suc @ A2 )
= ( plus_plus_nat @ K @ ( suc @ A ) ) ) ) ).
% nat_arith.suc1
thf(fact_1026_add__Suc,axiom,
! [M2: nat,N2: nat] :
( ( plus_plus_nat @ ( suc @ M2 ) @ N2 )
= ( suc @ ( plus_plus_nat @ M2 @ N2 ) ) ) ).
% add_Suc
thf(fact_1027_add__Suc__shift,axiom,
! [M2: nat,N2: nat] :
( ( plus_plus_nat @ ( suc @ M2 ) @ N2 )
= ( plus_plus_nat @ M2 @ ( suc @ N2 ) ) ) ).
% add_Suc_shift
thf(fact_1028_not__Cons__self2,axiom,
! [X2: int,Xs: list_int] :
( ( cons_int @ X2 @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_1029_dbl__inc__def,axiom,
( neg_nu8295874005876285629c_real
= ( ^ [X: real] : ( plus_plus_real @ ( plus_plus_real @ X @ X ) @ one_one_real ) ) ) ).
% dbl_inc_def
thf(fact_1030_dbl__inc__def,axiom,
( neg_nu5851722552734809277nc_int
= ( ^ [X: int] : ( plus_plus_int @ ( plus_plus_int @ X @ X ) @ one_one_int ) ) ) ).
% dbl_inc_def
thf(fact_1031_nat__le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [M: nat,N: nat] :
? [K3: nat] :
( N
= ( plus_plus_nat @ M @ K3 ) ) ) ) ).
% nat_le_iff_add
thf(fact_1032_trans__le__add2,axiom,
! [I4: nat,J2: nat,M2: nat] :
( ( ord_less_eq_nat @ I4 @ J2 )
=> ( ord_less_eq_nat @ I4 @ ( plus_plus_nat @ M2 @ J2 ) ) ) ).
% trans_le_add2
thf(fact_1033_trans__le__add1,axiom,
! [I4: nat,J2: nat,M2: nat] :
( ( ord_less_eq_nat @ I4 @ J2 )
=> ( ord_less_eq_nat @ I4 @ ( plus_plus_nat @ J2 @ M2 ) ) ) ).
% trans_le_add1
thf(fact_1034_add__le__mono1,axiom,
! [I4: nat,J2: nat,K: nat] :
( ( ord_less_eq_nat @ I4 @ J2 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I4 @ K ) @ ( plus_plus_nat @ J2 @ K ) ) ) ).
% add_le_mono1
thf(fact_1035_add__le__mono,axiom,
! [I4: nat,J2: nat,K: nat,L: nat] :
( ( ord_less_eq_nat @ I4 @ J2 )
=> ( ( ord_less_eq_nat @ K @ L )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I4 @ K ) @ ( plus_plus_nat @ J2 @ L ) ) ) ) ).
% add_le_mono
thf(fact_1036_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_1037_add__leD2,axiom,
! [M2: nat,K: nat,N2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M2 @ K ) @ N2 )
=> ( ord_less_eq_nat @ K @ N2 ) ) ).
% add_leD2
thf(fact_1038_add__leD1,axiom,
! [M2: nat,K: nat,N2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M2 @ K ) @ N2 )
=> ( ord_less_eq_nat @ M2 @ N2 ) ) ).
% add_leD1
thf(fact_1039_le__add2,axiom,
! [N2: nat,M2: nat] : ( ord_less_eq_nat @ N2 @ ( plus_plus_nat @ M2 @ N2 ) ) ).
% le_add2
thf(fact_1040_le__add1,axiom,
! [N2: nat,M2: nat] : ( ord_less_eq_nat @ N2 @ ( plus_plus_nat @ N2 @ M2 ) ) ).
% le_add1
thf(fact_1041_add__leE,axiom,
! [M2: nat,K: nat,N2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M2 @ K ) @ N2 )
=> ~ ( ( ord_less_eq_nat @ M2 @ N2 )
=> ~ ( ord_less_eq_nat @ K @ N2 ) ) ) ).
% add_leE
thf(fact_1042_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I4: nat,J2: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I4 @ J2 )
& ( K = L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I4 @ K ) @ ( plus_plus_nat @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_1043_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I4: real,J2: real,K: real,L: real] :
( ( ( ord_less_eq_real @ I4 @ J2 )
& ( K = L ) )
=> ( ord_less_eq_real @ ( plus_plus_real @ I4 @ K ) @ ( plus_plus_real @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_1044_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I4: int,J2: int,K: int,L: int] :
( ( ( ord_less_eq_int @ I4 @ J2 )
& ( K = L ) )
=> ( ord_less_eq_int @ ( plus_plus_int @ I4 @ K ) @ ( plus_plus_int @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_1045_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I4: nat,J2: nat,K: nat,L: nat] :
( ( ( I4 = J2 )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I4 @ K ) @ ( plus_plus_nat @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_1046_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I4: real,J2: real,K: real,L: real] :
( ( ( I4 = J2 )
& ( ord_less_eq_real @ K @ L ) )
=> ( ord_less_eq_real @ ( plus_plus_real @ I4 @ K ) @ ( plus_plus_real @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_1047_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I4: int,J2: int,K: int,L: int] :
( ( ( I4 = J2 )
& ( ord_less_eq_int @ K @ L ) )
=> ( ord_less_eq_int @ ( plus_plus_int @ I4 @ K ) @ ( plus_plus_int @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_1048_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I4: nat,J2: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I4 @ J2 )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I4 @ K ) @ ( plus_plus_nat @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_1049_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I4: real,J2: real,K: real,L: real] :
( ( ( ord_less_eq_real @ I4 @ J2 )
& ( ord_less_eq_real @ K @ L ) )
=> ( ord_less_eq_real @ ( plus_plus_real @ I4 @ K ) @ ( plus_plus_real @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_1050_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I4: int,J2: int,K: int,L: int] :
( ( ( ord_less_eq_int @ I4 @ J2 )
& ( ord_less_eq_int @ K @ L ) )
=> ( ord_less_eq_int @ ( plus_plus_int @ I4 @ K ) @ ( plus_plus_int @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_1051_add__mono,axiom,
! [A: nat,B: nat,C: nat,D2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D2 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D2 ) ) ) ) ).
% add_mono
thf(fact_1052_add__mono,axiom,
! [A: real,B: real,C: real,D2: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ C @ D2 )
=> ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D2 ) ) ) ) ).
% add_mono
thf(fact_1053_add__mono,axiom,
! [A: int,B: int,C: int,D2: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ C @ D2 )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D2 ) ) ) ) ).
% add_mono
thf(fact_1054_add__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) ) ) ).
% add_left_mono
thf(fact_1055_add__left__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ord_less_eq_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) ) ) ).
% add_left_mono
thf(fact_1056_add__left__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) ) ) ).
% add_left_mono
thf(fact_1057_less__eqE,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ~ ! [C2: nat] :
( B
!= ( plus_plus_nat @ A @ C2 ) ) ) ).
% less_eqE
thf(fact_1058_add__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).
% add_right_mono
thf(fact_1059_add__right__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) ) ) ).
% add_right_mono
thf(fact_1060_add__right__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) ) ) ).
% add_right_mono
thf(fact_1061_le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B3: nat] :
? [C3: nat] :
( B3
= ( plus_plus_nat @ A4 @ C3 ) ) ) ) ).
% le_iff_add
thf(fact_1062_add__le__imp__le__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_imp_le_left
thf(fact_1063_add__le__imp__le__left,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
=> ( ord_less_eq_real @ A @ B ) ) ).
% add_le_imp_le_left
thf(fact_1064_add__le__imp__le__left,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
=> ( ord_less_eq_int @ A @ B ) ) ).
% add_le_imp_le_left
thf(fact_1065_add__le__imp__le__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_imp_le_right
thf(fact_1066_add__le__imp__le__right,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
=> ( ord_less_eq_real @ A @ B ) ) ).
% add_le_imp_le_right
thf(fact_1067_add__le__imp__le__right,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
=> ( ord_less_eq_int @ A @ B ) ) ).
% add_le_imp_le_right
thf(fact_1068_list_Osize_I4_J,axiom,
! [X21: int,X222: list_int] :
( ( size_size_list_int @ ( cons_int @ X21 @ X222 ) )
= ( plus_plus_nat @ ( size_size_list_int @ X222 ) @ ( suc @ zero_zero_nat ) ) ) ).
% list.size(4)
thf(fact_1069_list_Osize_I4_J,axiom,
! [X21: a,X222: list_a] :
( ( size_size_list_a @ ( cons_a @ X21 @ X222 ) )
= ( plus_plus_nat @ ( size_size_list_a @ X222 ) @ ( suc @ zero_zero_nat ) ) ) ).
% list.size(4)
thf(fact_1070_list_Osize_I4_J,axiom,
! [X21: nat,X222: list_nat] :
( ( size_size_list_nat @ ( cons_nat @ X21 @ X222 ) )
= ( plus_plus_nat @ ( size_size_list_nat @ X222 ) @ ( suc @ zero_zero_nat ) ) ) ).
% list.size(4)
thf(fact_1071_add__nonpos__eq__0__iff,axiom,
! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ zero_zero_real )
=> ( ( ord_less_eq_real @ Y2 @ zero_zero_real )
=> ( ( ( plus_plus_real @ X2 @ Y2 )
= zero_zero_real )
= ( ( X2 = zero_zero_real )
& ( Y2 = zero_zero_real ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_1072_add__nonpos__eq__0__iff,axiom,
! [X2: int,Y2: int] :
( ( ord_less_eq_int @ X2 @ zero_zero_int )
=> ( ( ord_less_eq_int @ Y2 @ zero_zero_int )
=> ( ( ( plus_plus_int @ X2 @ Y2 )
= zero_zero_int )
= ( ( X2 = zero_zero_int )
& ( Y2 = zero_zero_int ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_1073_add__is__1,axiom,
! [M2: nat,N2: nat] :
( ( ( plus_plus_nat @ M2 @ N2 )
= ( suc @ zero_zero_nat ) )
= ( ( ( M2
= ( suc @ zero_zero_nat ) )
& ( N2 = zero_zero_nat ) )
| ( ( M2 = zero_zero_nat )
& ( N2
= ( suc @ zero_zero_nat ) ) ) ) ) ).
% add_is_1
thf(fact_1074_one__is__add,axiom,
! [M2: nat,N2: nat] :
( ( ( suc @ zero_zero_nat )
= ( plus_plus_nat @ M2 @ N2 ) )
= ( ( ( M2
= ( suc @ zero_zero_nat ) )
& ( N2 = zero_zero_nat ) )
| ( ( M2 = zero_zero_nat )
& ( N2
= ( suc @ zero_zero_nat ) ) ) ) ) ).
% one_is_add
thf(fact_1075_less__imp__add__positive,axiom,
! [I4: nat,J2: nat] :
( ( ord_less_nat @ I4 @ J2 )
=> ? [K2: nat] :
( ( ord_less_nat @ zero_zero_nat @ K2 )
& ( ( plus_plus_nat @ I4 @ K2 )
= J2 ) ) ) ).
% less_imp_add_positive
thf(fact_1076_less__imp__Suc__add,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ? [K2: nat] :
( N2
= ( suc @ ( plus_plus_nat @ M2 @ K2 ) ) ) ) ).
% less_imp_Suc_add
thf(fact_1077_less__iff__Suc__add,axiom,
( ord_less_nat
= ( ^ [M: nat,N: nat] :
? [K3: nat] :
( N
= ( suc @ ( plus_plus_nat @ M @ K3 ) ) ) ) ) ).
% less_iff_Suc_add
thf(fact_1078_less__add__Suc2,axiom,
! [I4: nat,M2: nat] : ( ord_less_nat @ I4 @ ( suc @ ( plus_plus_nat @ M2 @ I4 ) ) ) ).
% less_add_Suc2
thf(fact_1079_less__add__Suc1,axiom,
! [I4: nat,M2: nat] : ( ord_less_nat @ I4 @ ( suc @ ( plus_plus_nat @ I4 @ M2 ) ) ) ).
% less_add_Suc1
thf(fact_1080_less__natE,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ~ ! [Q3: nat] :
( N2
!= ( suc @ ( plus_plus_nat @ M2 @ Q3 ) ) ) ) ).
% less_natE
thf(fact_1081_mono__nat__linear__lb,axiom,
! [F: nat > nat,M2: nat,K: nat] :
( ! [M5: nat,N3: nat] :
( ( ord_less_nat @ M5 @ N3 )
=> ( ord_less_nat @ ( F @ M5 ) @ ( F @ N3 ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M2 ) @ K ) @ ( F @ ( plus_plus_nat @ M2 @ K ) ) ) ) ).
% mono_nat_linear_lb
thf(fact_1082_diff__add__0,axiom,
! [N2: nat,M2: nat] :
( ( minus_minus_nat @ N2 @ ( plus_plus_nat @ N2 @ M2 ) )
= zero_zero_nat ) ).
% diff_add_0
thf(fact_1083_Suc__eq__plus1__left,axiom,
( suc
= ( plus_plus_nat @ one_one_nat ) ) ).
% Suc_eq_plus1_left
thf(fact_1084_plus__1__eq__Suc,axiom,
( ( plus_plus_nat @ one_one_nat )
= suc ) ).
% plus_1_eq_Suc
thf(fact_1085_Suc__eq__plus1,axiom,
( suc
= ( ^ [N: nat] : ( plus_plus_nat @ N @ one_one_nat ) ) ) ).
% Suc_eq_plus1
thf(fact_1086_le__diff__conv,axiom,
! [J2: nat,K: nat,I4: nat] :
( ( ord_less_eq_nat @ ( minus_minus_nat @ J2 @ K ) @ I4 )
= ( ord_less_eq_nat @ J2 @ ( plus_plus_nat @ I4 @ K ) ) ) ).
% le_diff_conv
thf(fact_1087_Nat_Ole__diff__conv2,axiom,
! [K: nat,J2: nat,I4: nat] :
( ( ord_less_eq_nat @ K @ J2 )
=> ( ( ord_less_eq_nat @ I4 @ ( minus_minus_nat @ J2 @ K ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ I4 @ K ) @ J2 ) ) ) ).
% Nat.le_diff_conv2
thf(fact_1088_Nat_Odiff__add__assoc,axiom,
! [K: nat,J2: nat,I4: nat] :
( ( ord_less_eq_nat @ K @ J2 )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ I4 @ J2 ) @ K )
= ( plus_plus_nat @ I4 @ ( minus_minus_nat @ J2 @ K ) ) ) ) ).
% Nat.diff_add_assoc
thf(fact_1089_Nat_Odiff__add__assoc2,axiom,
! [K: nat,J2: nat,I4: nat] :
( ( ord_less_eq_nat @ K @ J2 )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ J2 @ I4 ) @ K )
= ( plus_plus_nat @ ( minus_minus_nat @ J2 @ K ) @ I4 ) ) ) ).
% Nat.diff_add_assoc2
thf(fact_1090_Nat_Ole__imp__diff__is__add,axiom,
! [I4: nat,J2: nat,K: nat] :
( ( ord_less_eq_nat @ I4 @ J2 )
=> ( ( ( minus_minus_nat @ J2 @ I4 )
= K )
= ( J2
= ( plus_plus_nat @ K @ I4 ) ) ) ) ).
% Nat.le_imp_diff_is_add
thf(fact_1091_add__diff__inverse__nat,axiom,
! [M2: nat,N2: nat] :
( ~ ( ord_less_nat @ M2 @ N2 )
=> ( ( plus_plus_nat @ N2 @ ( minus_minus_nat @ M2 @ N2 ) )
= M2 ) ) ).
% add_diff_inverse_nat
thf(fact_1092_less__diff__conv,axiom,
! [I4: nat,J2: nat,K: nat] :
( ( ord_less_nat @ I4 @ ( minus_minus_nat @ J2 @ K ) )
= ( ord_less_nat @ ( plus_plus_nat @ I4 @ K ) @ J2 ) ) ).
% less_diff_conv
thf(fact_1093_nat__diff__split__asm,axiom,
! [P: nat > $o,A: nat,B: nat] :
( ( P @ ( minus_minus_nat @ A @ B ) )
= ( ~ ( ( ( ord_less_nat @ A @ B )
& ~ ( P @ zero_zero_nat ) )
| ? [D3: nat] :
( ( A
= ( plus_plus_nat @ B @ D3 ) )
& ~ ( P @ D3 ) ) ) ) ) ).
% nat_diff_split_asm
thf(fact_1094_nat__diff__split,axiom,
! [P: nat > $o,A: nat,B: nat] :
( ( P @ ( minus_minus_nat @ A @ B ) )
= ( ( ( ord_less_nat @ A @ B )
=> ( P @ zero_zero_nat ) )
& ! [D3: nat] :
( ( A
= ( plus_plus_nat @ B @ D3 ) )
=> ( P @ D3 ) ) ) ) ).
% nat_diff_split
thf(fact_1095_less__diff__conv2,axiom,
! [K: nat,J2: nat,I4: nat] :
( ( ord_less_eq_nat @ K @ J2 )
=> ( ( ord_less_nat @ ( minus_minus_nat @ J2 @ K ) @ I4 )
= ( ord_less_nat @ J2 @ ( plus_plus_nat @ I4 @ K ) ) ) ) ).
% less_diff_conv2
thf(fact_1096_complete__real,axiom,
! [S2: set_real] :
( ? [X6: real] : ( member_real @ X6 @ S2 )
=> ( ? [Z5: real] :
! [X3: real] :
( ( member_real @ X3 @ S2 )
=> ( ord_less_eq_real @ X3 @ Z5 ) )
=> ? [Y4: real] :
( ! [X6: real] :
( ( member_real @ X6 @ S2 )
=> ( ord_less_eq_real @ X6 @ Y4 ) )
& ! [Z5: real] :
( ! [X3: real] :
( ( member_real @ X3 @ S2 )
=> ( ord_less_eq_real @ X3 @ Z5 ) )
=> ( ord_less_eq_real @ Y4 @ Z5 ) ) ) ) ) ).
% complete_real
thf(fact_1097_kuhn__lemma,axiom,
! [P5: nat,N2: nat,Label: ( nat > nat ) > nat > nat] :
( ( ord_less_nat @ zero_zero_nat @ P5 )
=> ( ! [X3: nat > nat] :
( ! [I6: nat] :
( ( ord_less_nat @ I6 @ N2 )
=> ( ord_less_eq_nat @ ( X3 @ I6 ) @ P5 ) )
=> ! [I5: nat] :
( ( ord_less_nat @ I5 @ N2 )
=> ( ( ( Label @ X3 @ I5 )
= zero_zero_nat )
| ( ( Label @ X3 @ I5 )
= one_one_nat ) ) ) )
=> ( ! [X3: nat > nat] :
( ! [I6: nat] :
( ( ord_less_nat @ I6 @ N2 )
=> ( ord_less_eq_nat @ ( X3 @ I6 ) @ P5 ) )
=> ! [I5: nat] :
( ( ord_less_nat @ I5 @ N2 )
=> ( ( ( X3 @ I5 )
= zero_zero_nat )
=> ( ( Label @ X3 @ I5 )
= zero_zero_nat ) ) ) )
=> ( ! [X3: nat > nat] :
( ! [I6: nat] :
( ( ord_less_nat @ I6 @ N2 )
=> ( ord_less_eq_nat @ ( X3 @ I6 ) @ P5 ) )
=> ! [I5: nat] :
( ( ord_less_nat @ I5 @ N2 )
=> ( ( ( X3 @ I5 )
= P5 )
=> ( ( Label @ X3 @ I5 )
= one_one_nat ) ) ) )
=> ~ ! [Q3: nat > nat] :
( ! [I6: nat] :
( ( ord_less_nat @ I6 @ N2 )
=> ( ord_less_nat @ ( Q3 @ I6 ) @ P5 ) )
=> ~ ! [I6: nat] :
( ( ord_less_nat @ I6 @ N2 )
=> ? [R2: nat > nat] :
( ! [J4: nat] :
( ( ord_less_nat @ J4 @ N2 )
=> ( ( ord_less_eq_nat @ ( Q3 @ J4 ) @ ( R2 @ J4 ) )
& ( ord_less_eq_nat @ ( R2 @ J4 ) @ ( plus_plus_nat @ ( Q3 @ J4 ) @ one_one_nat ) ) ) )
& ? [S3: nat > nat] :
( ! [J4: nat] :
( ( ord_less_nat @ J4 @ N2 )
=> ( ( ord_less_eq_nat @ ( Q3 @ J4 ) @ ( S3 @ J4 ) )
& ( ord_less_eq_nat @ ( S3 @ J4 ) @ ( plus_plus_nat @ ( Q3 @ J4 ) @ one_one_nat ) ) ) )
& ( ( Label @ R2 @ I6 )
!= ( Label @ S3 @ I6 ) ) ) ) ) ) ) ) ) ) ).
% kuhn_lemma
thf(fact_1098_add__eq__if,axiom,
( plus_plus_nat
= ( ^ [M: nat,N: nat] : ( if_nat @ ( M = zero_zero_nat ) @ N @ ( suc @ ( plus_plus_nat @ ( minus_minus_nat @ M @ one_one_nat ) @ N ) ) ) ) ) ).
% add_eq_if
thf(fact_1099_triangle__Suc,axiom,
! [N2: nat] :
( ( nat_triangle @ ( suc @ N2 ) )
= ( plus_plus_nat @ ( nat_triangle @ N2 ) @ ( suc @ N2 ) ) ) ).
% triangle_Suc
thf(fact_1100_verit__le__mono__div,axiom,
! [A2: nat,B5: nat,N2: nat] :
( ( ord_less_nat @ A2 @ B5 )
=> ( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ord_less_eq_nat
@ ( plus_plus_nat @ ( divide_divide_nat @ A2 @ N2 )
@ ( if_nat
@ ( ( modulo_modulo_nat @ B5 @ N2 )
= zero_zero_nat )
@ one_one_nat
@ zero_zero_nat ) )
@ ( divide_divide_nat @ B5 @ N2 ) ) ) ) ).
% verit_le_mono_div
thf(fact_1101_div__by__Suc__0,axiom,
! [M2: nat] :
( ( divide_divide_nat @ M2 @ ( suc @ zero_zero_nat ) )
= M2 ) ).
% div_by_Suc_0
thf(fact_1102_div__less,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( ( divide_divide_nat @ M2 @ N2 )
= zero_zero_nat ) ) ).
% div_less
thf(fact_1103_div__le__mono,axiom,
! [M2: nat,N2: nat,K: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ord_less_eq_nat @ ( divide_divide_nat @ M2 @ K ) @ ( divide_divide_nat @ N2 @ K ) ) ) ).
% div_le_mono
thf(fact_1104_div__le__dividend,axiom,
! [M2: nat,N2: nat] : ( ord_less_eq_nat @ ( divide_divide_nat @ M2 @ N2 ) @ M2 ) ).
% div_le_dividend
thf(fact_1105_Euclidean__Division_Odiv__eq__0__iff,axiom,
! [M2: nat,N2: nat] :
( ( ( divide_divide_nat @ M2 @ N2 )
= zero_zero_nat )
= ( ( ord_less_nat @ M2 @ N2 )
| ( N2 = zero_zero_nat ) ) ) ).
% Euclidean_Division.div_eq_0_iff
thf(fact_1106_Suc__div__le__mono,axiom,
! [M2: nat,N2: nat] : ( ord_less_eq_nat @ ( divide_divide_nat @ M2 @ N2 ) @ ( divide_divide_nat @ ( suc @ M2 ) @ N2 ) ) ).
% Suc_div_le_mono
thf(fact_1107_div__le__mono2,axiom,
! [M2: nat,N2: nat,K: nat] :
( ( ord_less_nat @ zero_zero_nat @ M2 )
=> ( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ord_less_eq_nat @ ( divide_divide_nat @ K @ N2 ) @ ( divide_divide_nat @ K @ M2 ) ) ) ) ).
% div_le_mono2
thf(fact_1108_div__greater__zero__iff,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ M2 @ N2 ) )
= ( ( ord_less_eq_nat @ N2 @ M2 )
& ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% div_greater_zero_iff
thf(fact_1109_div__less__dividend,axiom,
! [N2: nat,M2: nat] :
( ( ord_less_nat @ one_one_nat @ N2 )
=> ( ( ord_less_nat @ zero_zero_nat @ M2 )
=> ( ord_less_nat @ ( divide_divide_nat @ M2 @ N2 ) @ M2 ) ) ) ).
% div_less_dividend
thf(fact_1110_div__eq__dividend__iff,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ M2 )
=> ( ( ( divide_divide_nat @ M2 @ N2 )
= M2 )
= ( N2 = one_one_nat ) ) ) ).
% div_eq_dividend_iff
thf(fact_1111_div__Suc,axiom,
! [M2: nat,N2: nat] :
( ( ( ( modulo_modulo_nat @ ( suc @ M2 ) @ N2 )
= zero_zero_nat )
=> ( ( divide_divide_nat @ ( suc @ M2 ) @ N2 )
= ( suc @ ( divide_divide_nat @ M2 @ N2 ) ) ) )
& ( ( ( modulo_modulo_nat @ ( suc @ M2 ) @ N2 )
!= zero_zero_nat )
=> ( ( divide_divide_nat @ ( suc @ M2 ) @ N2 )
= ( divide_divide_nat @ M2 @ N2 ) ) ) ) ).
% div_Suc
thf(fact_1112_div__less__mono,axiom,
! [A2: nat,B5: nat,N2: nat] :
( ( ord_less_nat @ A2 @ B5 )
=> ( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ( modulo_modulo_nat @ A2 @ N2 )
= zero_zero_nat )
=> ( ( ( modulo_modulo_nat @ B5 @ N2 )
= zero_zero_nat )
=> ( ord_less_nat @ ( divide_divide_nat @ A2 @ N2 ) @ ( divide_divide_nat @ B5 @ N2 ) ) ) ) ) ) ).
% div_less_mono
thf(fact_1113_bounded__Max__nat,axiom,
! [P: nat > $o,X2: nat,M7: nat] :
( ( P @ X2 )
=> ( ! [X3: nat] :
( ( P @ X3 )
=> ( ord_less_eq_nat @ X3 @ M7 ) )
=> ~ ! [M5: nat] :
( ( P @ M5 )
=> ~ ! [X6: nat] :
( ( P @ X6 )
=> ( ord_less_eq_nat @ X6 @ M5 ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_1114_div__if,axiom,
( divide_divide_nat
= ( ^ [M: nat,N: nat] :
( if_nat
@ ( ( ord_less_nat @ M @ N )
| ( N = zero_zero_nat ) )
@ zero_zero_nat
@ ( suc @ ( divide_divide_nat @ ( minus_minus_nat @ M @ N ) @ N ) ) ) ) ) ).
% div_if
thf(fact_1115_le__div__geq,axiom,
! [N2: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_eq_nat @ N2 @ M2 )
=> ( ( divide_divide_nat @ M2 @ N2 )
= ( suc @ ( divide_divide_nat @ ( minus_minus_nat @ M2 @ N2 ) @ N2 ) ) ) ) ) ).
% le_div_geq
thf(fact_1116_Suc__times__mod__eq,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M2 )
=> ( ( modulo_modulo_nat @ ( suc @ ( times_times_nat @ M2 @ N2 ) ) @ M2 )
= one_one_nat ) ) ).
% Suc_times_mod_eq
thf(fact_1117_split__div_H,axiom,
! [P: nat > $o,M2: nat,N2: nat] :
( ( P @ ( divide_divide_nat @ M2 @ N2 ) )
= ( ( ( N2 = zero_zero_nat )
& ( P @ zero_zero_nat ) )
| ? [Q4: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ N2 @ Q4 ) @ M2 )
& ( ord_less_nat @ M2 @ ( times_times_nat @ N2 @ ( suc @ Q4 ) ) )
& ( P @ Q4 ) ) ) ) ).
% split_div'
thf(fact_1118_mult__cancel2,axiom,
! [M2: nat,K: nat,N2: nat] :
( ( ( times_times_nat @ M2 @ K )
= ( times_times_nat @ N2 @ K ) )
= ( ( M2 = N2 )
| ( K = zero_zero_nat ) ) ) ).
% mult_cancel2
thf(fact_1119_mult__cancel1,axiom,
! [K: nat,M2: nat,N2: nat] :
( ( ( times_times_nat @ K @ M2 )
= ( times_times_nat @ K @ N2 ) )
= ( ( M2 = N2 )
| ( K = zero_zero_nat ) ) ) ).
% mult_cancel1
thf(fact_1120_mult__0__right,axiom,
! [M2: nat] :
( ( times_times_nat @ M2 @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_0_right
thf(fact_1121_mult__is__0,axiom,
! [M2: nat,N2: nat] :
( ( ( times_times_nat @ M2 @ N2 )
= zero_zero_nat )
= ( ( M2 = zero_zero_nat )
| ( N2 = zero_zero_nat ) ) ) ).
% mult_is_0
thf(fact_1122_nat__mult__eq__1__iff,axiom,
! [M2: nat,N2: nat] :
( ( ( times_times_nat @ M2 @ N2 )
= one_one_nat )
= ( ( M2 = one_one_nat )
& ( N2 = one_one_nat ) ) ) ).
% nat_mult_eq_1_iff
thf(fact_1123_nat__1__eq__mult__iff,axiom,
! [M2: nat,N2: nat] :
( ( one_one_nat
= ( times_times_nat @ M2 @ N2 ) )
= ( ( M2 = one_one_nat )
& ( N2 = one_one_nat ) ) ) ).
% nat_1_eq_mult_iff
thf(fact_1124_one__eq__mult__iff,axiom,
! [M2: nat,N2: nat] :
( ( ( suc @ zero_zero_nat )
= ( times_times_nat @ M2 @ N2 ) )
= ( ( M2
= ( suc @ zero_zero_nat ) )
& ( N2
= ( suc @ zero_zero_nat ) ) ) ) ).
% one_eq_mult_iff
thf(fact_1125_mult__eq__1__iff,axiom,
! [M2: nat,N2: nat] :
( ( ( times_times_nat @ M2 @ N2 )
= ( suc @ zero_zero_nat ) )
= ( ( M2
= ( suc @ zero_zero_nat ) )
& ( N2
= ( suc @ zero_zero_nat ) ) ) ) ).
% mult_eq_1_iff
thf(fact_1126_mult__less__cancel2,axiom,
! [M2: nat,K: nat,N2: nat] :
( ( ord_less_nat @ ( times_times_nat @ M2 @ K ) @ ( times_times_nat @ N2 @ K ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
& ( ord_less_nat @ M2 @ N2 ) ) ) ).
% mult_less_cancel2
thf(fact_1127_nat__0__less__mult__iff,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ M2 @ N2 ) )
= ( ( ord_less_nat @ zero_zero_nat @ M2 )
& ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% nat_0_less_mult_iff
thf(fact_1128_mult__Suc__right,axiom,
! [M2: nat,N2: nat] :
( ( times_times_nat @ M2 @ ( suc @ N2 ) )
= ( plus_plus_nat @ M2 @ ( times_times_nat @ M2 @ N2 ) ) ) ).
% mult_Suc_right
thf(fact_1129_one__le__mult__iff,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M2 @ N2 ) )
= ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ M2 )
& ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ N2 ) ) ) ).
% one_le_mult_iff
thf(fact_1130_mult__le__cancel2,axiom,
! [M2: nat,K: nat,N2: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ M2 @ K ) @ ( times_times_nat @ N2 @ K ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_eq_nat @ M2 @ N2 ) ) ) ).
% mult_le_cancel2
thf(fact_1131_div__mult__self1__is__m,axiom,
! [N2: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( divide_divide_nat @ ( times_times_nat @ N2 @ M2 ) @ N2 )
= M2 ) ) ).
% div_mult_self1_is_m
thf(fact_1132_div__mult__self__is__m,axiom,
! [N2: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( divide_divide_nat @ ( times_times_nat @ M2 @ N2 ) @ N2 )
= M2 ) ) ).
% div_mult_self_is_m
thf(fact_1133_Suc__mod__mult__self1,axiom,
! [M2: nat,K: nat,N2: nat] :
( ( modulo_modulo_nat @ ( suc @ ( plus_plus_nat @ M2 @ ( times_times_nat @ K @ N2 ) ) ) @ N2 )
= ( modulo_modulo_nat @ ( suc @ M2 ) @ N2 ) ) ).
% Suc_mod_mult_self1
thf(fact_1134_Suc__mod__mult__self2,axiom,
! [M2: nat,N2: nat,K: nat] :
( ( modulo_modulo_nat @ ( suc @ ( plus_plus_nat @ M2 @ ( times_times_nat @ N2 @ K ) ) ) @ N2 )
= ( modulo_modulo_nat @ ( suc @ M2 ) @ N2 ) ) ).
% Suc_mod_mult_self2
thf(fact_1135_Suc__mod__mult__self3,axiom,
! [K: nat,N2: nat,M2: nat] :
( ( modulo_modulo_nat @ ( suc @ ( plus_plus_nat @ ( times_times_nat @ K @ N2 ) @ M2 ) ) @ N2 )
= ( modulo_modulo_nat @ ( suc @ M2 ) @ N2 ) ) ).
% Suc_mod_mult_self3
thf(fact_1136_Suc__mod__mult__self4,axiom,
! [N2: nat,K: nat,M2: nat] :
( ( modulo_modulo_nat @ ( suc @ ( plus_plus_nat @ ( times_times_nat @ N2 @ K ) @ M2 ) ) @ N2 )
= ( modulo_modulo_nat @ ( suc @ M2 ) @ N2 ) ) ).
% Suc_mod_mult_self4
thf(fact_1137_add__mult__distrib2,axiom,
! [K: nat,M2: nat,N2: nat] :
( ( times_times_nat @ K @ ( plus_plus_nat @ M2 @ N2 ) )
= ( plus_plus_nat @ ( times_times_nat @ K @ M2 ) @ ( times_times_nat @ K @ N2 ) ) ) ).
% add_mult_distrib2
thf(fact_1138_add__mult__distrib,axiom,
! [M2: nat,N2: nat,K: nat] :
( ( times_times_nat @ ( plus_plus_nat @ M2 @ N2 ) @ K )
= ( plus_plus_nat @ ( times_times_nat @ M2 @ K ) @ ( times_times_nat @ N2 @ K ) ) ) ).
% add_mult_distrib
thf(fact_1139_le__cube,axiom,
! [M2: nat] : ( ord_less_eq_nat @ M2 @ ( times_times_nat @ M2 @ ( times_times_nat @ M2 @ M2 ) ) ) ).
% le_cube
thf(fact_1140_le__square,axiom,
! [M2: nat] : ( ord_less_eq_nat @ M2 @ ( times_times_nat @ M2 @ M2 ) ) ).
% le_square
thf(fact_1141_mult__le__mono,axiom,
! [I4: nat,J2: nat,K: nat,L: nat] :
( ( ord_less_eq_nat @ I4 @ J2 )
=> ( ( ord_less_eq_nat @ K @ L )
=> ( ord_less_eq_nat @ ( times_times_nat @ I4 @ K ) @ ( times_times_nat @ J2 @ L ) ) ) ) ).
% mult_le_mono
thf(fact_1142_mult__le__mono1,axiom,
! [I4: nat,J2: nat,K: nat] :
( ( ord_less_eq_nat @ I4 @ J2 )
=> ( ord_less_eq_nat @ ( times_times_nat @ I4 @ K ) @ ( times_times_nat @ J2 @ K ) ) ) ).
% mult_le_mono1
thf(fact_1143_mult__le__mono2,axiom,
! [I4: nat,J2: nat,K: nat] :
( ( ord_less_eq_nat @ I4 @ J2 )
=> ( ord_less_eq_nat @ ( times_times_nat @ K @ I4 ) @ ( times_times_nat @ K @ J2 ) ) ) ).
% mult_le_mono2
thf(fact_1144_nat__mult__1__right,axiom,
! [N2: nat] :
( ( times_times_nat @ N2 @ one_one_nat )
= N2 ) ).
% nat_mult_1_right
thf(fact_1145_nat__mult__1,axiom,
! [N2: nat] :
( ( times_times_nat @ one_one_nat @ N2 )
= N2 ) ).
% nat_mult_1
thf(fact_1146_Suc__mult__cancel1,axiom,
! [K: nat,M2: nat,N2: nat] :
( ( ( times_times_nat @ ( suc @ K ) @ M2 )
= ( times_times_nat @ ( suc @ K ) @ N2 ) )
= ( M2 = N2 ) ) ).
% Suc_mult_cancel1
thf(fact_1147_diff__mult__distrib,axiom,
! [M2: nat,N2: nat,K: nat] :
( ( times_times_nat @ ( minus_minus_nat @ M2 @ N2 ) @ K )
= ( minus_minus_nat @ ( times_times_nat @ M2 @ K ) @ ( times_times_nat @ N2 @ K ) ) ) ).
% diff_mult_distrib
thf(fact_1148_diff__mult__distrib2,axiom,
! [K: nat,M2: nat,N2: nat] :
( ( times_times_nat @ K @ ( minus_minus_nat @ M2 @ N2 ) )
= ( minus_minus_nat @ ( times_times_nat @ K @ M2 ) @ ( times_times_nat @ K @ N2 ) ) ) ).
% diff_mult_distrib2
thf(fact_1149_mult__0,axiom,
! [N2: nat] :
( ( times_times_nat @ zero_zero_nat @ N2 )
= zero_zero_nat ) ).
% mult_0
thf(fact_1150_mult__less__mono2,axiom,
! [I4: nat,J2: nat,K: nat] :
( ( ord_less_nat @ I4 @ J2 )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_nat @ ( times_times_nat @ K @ I4 ) @ ( times_times_nat @ K @ J2 ) ) ) ) ).
% mult_less_mono2
thf(fact_1151_mult__less__mono1,axiom,
! [I4: nat,J2: nat,K: nat] :
( ( ord_less_nat @ I4 @ J2 )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_nat @ ( times_times_nat @ I4 @ K ) @ ( times_times_nat @ J2 @ K ) ) ) ) ).
% mult_less_mono1
thf(fact_1152_Suc__mult__le__cancel1,axiom,
! [K: nat,M2: nat,N2: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ ( suc @ K ) @ M2 ) @ ( times_times_nat @ ( suc @ K ) @ N2 ) )
= ( ord_less_eq_nat @ M2 @ N2 ) ) ).
% Suc_mult_le_cancel1
thf(fact_1153_Suc__mult__less__cancel1,axiom,
! [K: nat,M2: nat,N2: nat] :
( ( ord_less_nat @ ( times_times_nat @ ( suc @ K ) @ M2 ) @ ( times_times_nat @ ( suc @ K ) @ N2 ) )
= ( ord_less_nat @ M2 @ N2 ) ) ).
% Suc_mult_less_cancel1
thf(fact_1154_mult__Suc,axiom,
! [M2: nat,N2: nat] :
( ( times_times_nat @ ( suc @ M2 ) @ N2 )
= ( plus_plus_nat @ N2 @ ( times_times_nat @ M2 @ N2 ) ) ) ).
% mult_Suc
thf(fact_1155_mult__eq__self__implies__10,axiom,
! [M2: nat,N2: nat] :
( ( M2
= ( times_times_nat @ M2 @ N2 ) )
=> ( ( N2 = one_one_nat )
| ( M2 = zero_zero_nat ) ) ) ).
% mult_eq_self_implies_10
thf(fact_1156_times__div__less__eq__dividend,axiom,
! [N2: nat,M2: nat] : ( ord_less_eq_nat @ ( times_times_nat @ N2 @ ( divide_divide_nat @ M2 @ N2 ) ) @ M2 ) ).
% times_div_less_eq_dividend
thf(fact_1157_div__times__less__eq__dividend,axiom,
! [M2: nat,N2: nat] : ( ord_less_eq_nat @ ( times_times_nat @ ( divide_divide_nat @ M2 @ N2 ) @ N2 ) @ M2 ) ).
% div_times_less_eq_dividend
thf(fact_1158_less__mult__imp__div__less,axiom,
! [M2: nat,I4: nat,N2: nat] :
( ( ord_less_nat @ M2 @ ( times_times_nat @ I4 @ N2 ) )
=> ( ord_less_nat @ ( divide_divide_nat @ M2 @ N2 ) @ I4 ) ) ).
% less_mult_imp_div_less
thf(fact_1159_one__less__mult,axiom,
! [N2: nat,M2: nat] :
( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N2 )
=> ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M2 )
=> ( ord_less_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M2 @ N2 ) ) ) ) ).
% one_less_mult
thf(fact_1160_n__less__m__mult__n,axiom,
! [N2: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M2 )
=> ( ord_less_nat @ N2 @ ( times_times_nat @ M2 @ N2 ) ) ) ) ).
% n_less_m_mult_n
thf(fact_1161_n__less__n__mult__m,axiom,
! [N2: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M2 )
=> ( ord_less_nat @ N2 @ ( times_times_nat @ N2 @ M2 ) ) ) ) ).
% n_less_n_mult_m
thf(fact_1162_div__less__iff__less__mult,axiom,
! [Q5: nat,M2: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ Q5 )
=> ( ( ord_less_nat @ ( divide_divide_nat @ M2 @ Q5 ) @ N2 )
= ( ord_less_nat @ M2 @ ( times_times_nat @ N2 @ Q5 ) ) ) ) ).
% div_less_iff_less_mult
thf(fact_1163_mod__eq__nat1E,axiom,
! [M2: nat,Q5: nat,N2: nat] :
( ( ( modulo_modulo_nat @ M2 @ Q5 )
= ( modulo_modulo_nat @ N2 @ Q5 ) )
=> ( ( ord_less_eq_nat @ N2 @ M2 )
=> ~ ! [S3: nat] :
( M2
!= ( plus_plus_nat @ N2 @ ( times_times_nat @ Q5 @ S3 ) ) ) ) ) ).
% mod_eq_nat1E
thf(fact_1164_mod__eq__nat2E,axiom,
! [M2: nat,Q5: nat,N2: nat] :
( ( ( modulo_modulo_nat @ M2 @ Q5 )
= ( modulo_modulo_nat @ N2 @ Q5 ) )
=> ( ( ord_less_eq_nat @ M2 @ N2 )
=> ~ ! [S3: nat] :
( N2
!= ( plus_plus_nat @ M2 @ ( times_times_nat @ Q5 @ S3 ) ) ) ) ) ).
% mod_eq_nat2E
thf(fact_1165_div__mod__decomp,axiom,
! [A2: nat,N2: nat] :
( A2
= ( plus_plus_nat @ ( times_times_nat @ ( divide_divide_nat @ A2 @ N2 ) @ N2 ) @ ( modulo_modulo_nat @ A2 @ N2 ) ) ) ).
% div_mod_decomp
thf(fact_1166_mult__eq__if,axiom,
( times_times_nat
= ( ^ [M: nat,N: nat] : ( if_nat @ ( M = zero_zero_nat ) @ zero_zero_nat @ ( plus_plus_nat @ N @ ( times_times_nat @ ( minus_minus_nat @ M @ one_one_nat ) @ N ) ) ) ) ) ).
% mult_eq_if
thf(fact_1167_less__eq__div__iff__mult__less__eq,axiom,
! [Q5: nat,M2: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ Q5 )
=> ( ( ord_less_eq_nat @ M2 @ ( divide_divide_nat @ N2 @ Q5 ) )
= ( ord_less_eq_nat @ ( times_times_nat @ M2 @ Q5 ) @ N2 ) ) ) ).
% less_eq_div_iff_mult_less_eq
thf(fact_1168_div__nat__eqI,axiom,
! [N2: nat,Q5: nat,M2: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ N2 @ Q5 ) @ M2 )
=> ( ( ord_less_nat @ M2 @ ( times_times_nat @ N2 @ ( suc @ Q5 ) ) )
=> ( ( divide_divide_nat @ M2 @ N2 )
= Q5 ) ) ) ).
% div_nat_eqI
thf(fact_1169_dividend__less__times__div,axiom,
! [N2: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ord_less_nat @ M2 @ ( plus_plus_nat @ N2 @ ( times_times_nat @ N2 @ ( divide_divide_nat @ M2 @ N2 ) ) ) ) ) ).
% dividend_less_times_div
thf(fact_1170_dividend__less__div__times,axiom,
! [N2: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ord_less_nat @ M2 @ ( plus_plus_nat @ N2 @ ( times_times_nat @ ( divide_divide_nat @ M2 @ N2 ) @ N2 ) ) ) ) ).
% dividend_less_div_times
thf(fact_1171_split__div,axiom,
! [P: nat > $o,M2: nat,N2: nat] :
( ( P @ ( divide_divide_nat @ M2 @ N2 ) )
= ( ( ( N2 = zero_zero_nat )
=> ( P @ zero_zero_nat ) )
& ( ( N2 != zero_zero_nat )
=> ! [I3: nat,J: nat] :
( ( ( ord_less_nat @ J @ N2 )
& ( M2
= ( plus_plus_nat @ ( times_times_nat @ N2 @ I3 ) @ J ) ) )
=> ( P @ I3 ) ) ) ) ) ).
% split_div
thf(fact_1172_split__mod,axiom,
! [Q: nat > $o,M2: nat,N2: nat] :
( ( Q @ ( modulo_modulo_nat @ M2 @ N2 ) )
= ( ( ( N2 = zero_zero_nat )
=> ( Q @ M2 ) )
& ( ( N2 != zero_zero_nat )
=> ! [I3: nat,J: nat] :
( ( ( ord_less_nat @ J @ N2 )
& ( M2
= ( plus_plus_nat @ ( times_times_nat @ N2 @ I3 ) @ J ) ) )
=> ( Q @ J ) ) ) ) ) ).
% split_mod
thf(fact_1173_nat__mult__le__cancel__disj,axiom,
! [K: nat,M2: nat,N2: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ K @ M2 ) @ ( times_times_nat @ K @ N2 ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_eq_nat @ M2 @ N2 ) ) ) ).
% nat_mult_le_cancel_disj
thf(fact_1174_nat__mult__less__cancel__disj,axiom,
! [K: nat,M2: nat,N2: nat] :
( ( ord_less_nat @ ( times_times_nat @ K @ M2 ) @ ( times_times_nat @ K @ N2 ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
& ( ord_less_nat @ M2 @ N2 ) ) ) ).
% nat_mult_less_cancel_disj
thf(fact_1175_nat__mult__less__cancel1,axiom,
! [K: nat,M2: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ord_less_nat @ ( times_times_nat @ K @ M2 ) @ ( times_times_nat @ K @ N2 ) )
= ( ord_less_nat @ M2 @ N2 ) ) ) ).
% nat_mult_less_cancel1
thf(fact_1176_nat__mult__eq__cancel1,axiom,
! [K: nat,M2: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ( times_times_nat @ K @ M2 )
= ( times_times_nat @ K @ N2 ) )
= ( M2 = N2 ) ) ) ).
% nat_mult_eq_cancel1
thf(fact_1177_nat__mult__le__cancel1,axiom,
! [K: nat,M2: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ord_less_eq_nat @ ( times_times_nat @ K @ M2 ) @ ( times_times_nat @ K @ N2 ) )
= ( ord_less_eq_nat @ M2 @ N2 ) ) ) ).
% nat_mult_le_cancel1
thf(fact_1178_nat__diff__add__eq2,axiom,
! [I4: nat,J2: nat,U: nat,M2: nat,N2: nat] :
( ( ord_less_eq_nat @ I4 @ J2 )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I4 @ U ) @ M2 ) @ ( plus_plus_nat @ ( times_times_nat @ J2 @ U ) @ N2 ) )
= ( minus_minus_nat @ M2 @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J2 @ I4 ) @ U ) @ N2 ) ) ) ) ).
% nat_diff_add_eq2
thf(fact_1179_nat__diff__add__eq1,axiom,
! [J2: nat,I4: nat,U: nat,M2: nat,N2: nat] :
( ( ord_less_eq_nat @ J2 @ I4 )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I4 @ U ) @ M2 ) @ ( plus_plus_nat @ ( times_times_nat @ J2 @ U ) @ N2 ) )
= ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I4 @ J2 ) @ U ) @ M2 ) @ N2 ) ) ) ).
% nat_diff_add_eq1
thf(fact_1180_nat__le__add__iff2,axiom,
! [I4: nat,J2: nat,U: nat,M2: nat,N2: nat] :
( ( ord_less_eq_nat @ I4 @ J2 )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I4 @ U ) @ M2 ) @ ( plus_plus_nat @ ( times_times_nat @ J2 @ U ) @ N2 ) )
= ( ord_less_eq_nat @ M2 @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J2 @ I4 ) @ U ) @ N2 ) ) ) ) ).
% nat_le_add_iff2
thf(fact_1181_nat__le__add__iff1,axiom,
! [J2: nat,I4: nat,U: nat,M2: nat,N2: nat] :
( ( ord_less_eq_nat @ J2 @ I4 )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I4 @ U ) @ M2 ) @ ( plus_plus_nat @ ( times_times_nat @ J2 @ U ) @ N2 ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I4 @ J2 ) @ U ) @ M2 ) @ N2 ) ) ) ).
% nat_le_add_iff1
thf(fact_1182_nat__eq__add__iff2,axiom,
! [I4: nat,J2: nat,U: nat,M2: nat,N2: nat] :
( ( ord_less_eq_nat @ I4 @ J2 )
=> ( ( ( plus_plus_nat @ ( times_times_nat @ I4 @ U ) @ M2 )
= ( plus_plus_nat @ ( times_times_nat @ J2 @ U ) @ N2 ) )
= ( M2
= ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J2 @ I4 ) @ U ) @ N2 ) ) ) ) ).
% nat_eq_add_iff2
thf(fact_1183_nat__eq__add__iff1,axiom,
! [J2: nat,I4: nat,U: nat,M2: nat,N2: nat] :
( ( ord_less_eq_nat @ J2 @ I4 )
=> ( ( ( plus_plus_nat @ ( times_times_nat @ I4 @ U ) @ M2 )
= ( plus_plus_nat @ ( times_times_nat @ J2 @ U ) @ N2 ) )
= ( ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I4 @ J2 ) @ U ) @ M2 )
= N2 ) ) ) ).
% nat_eq_add_iff1
thf(fact_1184_nat__mult__div__cancel1,axiom,
! [K: nat,M2: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( divide_divide_nat @ ( times_times_nat @ K @ M2 ) @ ( times_times_nat @ K @ N2 ) )
= ( divide_divide_nat @ M2 @ N2 ) ) ) ).
% nat_mult_div_cancel1
thf(fact_1185_nat__less__add__iff2,axiom,
! [I4: nat,J2: nat,U: nat,M2: nat,N2: nat] :
( ( ord_less_eq_nat @ I4 @ J2 )
=> ( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ I4 @ U ) @ M2 ) @ ( plus_plus_nat @ ( times_times_nat @ J2 @ U ) @ N2 ) )
= ( ord_less_nat @ M2 @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J2 @ I4 ) @ U ) @ N2 ) ) ) ) ).
% nat_less_add_iff2
thf(fact_1186_nat__less__add__iff1,axiom,
! [J2: nat,I4: nat,U: nat,M2: nat,N2: nat] :
( ( ord_less_eq_nat @ J2 @ I4 )
=> ( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ I4 @ U ) @ M2 ) @ ( plus_plus_nat @ ( times_times_nat @ J2 @ U ) @ N2 ) )
= ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I4 @ J2 ) @ U ) @ M2 ) @ N2 ) ) ) ).
% nat_less_add_iff1
thf(fact_1187_sum__le__prod1,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ one_one_real )
=> ( ( ord_less_eq_real @ B @ one_one_real )
=> ( ord_less_eq_real @ ( plus_plus_real @ A @ B ) @ ( plus_plus_real @ one_one_real @ ( times_times_real @ A @ B ) ) ) ) ) ).
% sum_le_prod1
thf(fact_1188_segment__bound__lemma,axiom,
! [B5: real,X2: real,Y2: real,U: real] :
( ( ord_less_eq_real @ B5 @ X2 )
=> ( ( ord_less_eq_real @ B5 @ Y2 )
=> ( ( ord_less_eq_real @ zero_zero_real @ U )
=> ( ( ord_less_eq_real @ U @ one_one_real )
=> ( ord_less_eq_real @ B5 @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ one_one_real @ U ) @ X2 ) @ ( times_times_real @ U @ Y2 ) ) ) ) ) ) ) ).
% segment_bound_lemma
thf(fact_1189_square__bound__lemma,axiom,
! [X2: real] : ( ord_less_real @ X2 @ ( times_times_real @ ( plus_plus_real @ one_one_real @ X2 ) @ ( plus_plus_real @ one_one_real @ X2 ) ) ) ).
% square_bound_lemma
thf(fact_1190_power__Suc__0,axiom,
! [N2: nat] :
( ( power_power_nat @ ( suc @ zero_zero_nat ) @ N2 )
= ( suc @ zero_zero_nat ) ) ).
% power_Suc_0
thf(fact_1191_nat__power__eq__Suc__0__iff,axiom,
! [X2: nat,M2: nat] :
( ( ( power_power_nat @ X2 @ M2 )
= ( suc @ zero_zero_nat ) )
= ( ( M2 = zero_zero_nat )
| ( X2
= ( suc @ zero_zero_nat ) ) ) ) ).
% nat_power_eq_Suc_0_iff
thf(fact_1192_nat__zero__less__power__iff,axiom,
! [X2: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ X2 @ N2 ) )
= ( ( ord_less_nat @ zero_zero_nat @ X2 )
| ( N2 = zero_zero_nat ) ) ) ).
% nat_zero_less_power_iff
thf(fact_1193_nat__power__less__imp__less,axiom,
! [I4: nat,M2: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ I4 )
=> ( ( ord_less_nat @ ( power_power_nat @ I4 @ M2 ) @ ( power_power_nat @ I4 @ N2 ) )
=> ( ord_less_nat @ M2 @ N2 ) ) ) ).
% nat_power_less_imp_less
thf(fact_1194_real__arch__pow,axiom,
! [X2: real,Y2: real] :
( ( ord_less_real @ one_one_real @ X2 )
=> ? [N3: nat] : ( ord_less_real @ Y2 @ ( power_power_real @ X2 @ N3 ) ) ) ).
% real_arch_pow
thf(fact_1195_nat__one__le__power,axiom,
! [I4: nat,N2: nat] :
( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ I4 )
=> ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( power_power_nat @ I4 @ N2 ) ) ) ).
% nat_one_le_power
thf(fact_1196_power__gt__expt,axiom,
! [N2: nat,K: nat] :
( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N2 )
=> ( ord_less_nat @ K @ ( power_power_nat @ N2 @ K ) ) ) ).
% power_gt_expt
thf(fact_1197_real__arch__pow__inv,axiom,
! [Y2: real,X2: real] :
( ( ord_less_real @ zero_zero_real @ Y2 )
=> ( ( ord_less_real @ X2 @ one_one_real )
=> ? [N3: nat] : ( ord_less_real @ ( power_power_real @ X2 @ N3 ) @ Y2 ) ) ) ).
% real_arch_pow_inv
thf(fact_1198_power__le__one__iff,axiom,
! [A: real,N2: nat] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ ( power_power_real @ A @ N2 ) @ one_one_real )
= ( ( N2 = zero_zero_nat )
| ( ord_less_eq_real @ A @ one_one_real ) ) ) ) ).
% power_le_one_iff
thf(fact_1199_int__power__div__base,axiom,
! [M2: nat,K: int] :
( ( ord_less_nat @ zero_zero_nat @ M2 )
=> ( ( ord_less_int @ zero_zero_int @ K )
=> ( ( divide_divide_int @ ( power_power_int @ K @ M2 ) @ K )
= ( power_power_int @ K @ ( minus_minus_nat @ M2 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ).
% int_power_div_base
thf(fact_1200_reals__power__lt__ex,axiom,
! [X2: real,Y2: real] :
( ( ord_less_real @ zero_zero_real @ X2 )
=> ( ( ord_less_real @ one_one_real @ Y2 )
=> ? [K2: nat] :
( ( ord_less_nat @ zero_zero_nat @ K2 )
& ( ord_less_real @ ( power_power_real @ ( divide_divide_real @ one_one_real @ Y2 ) @ K2 ) @ X2 ) ) ) ) ).
% reals_power_lt_ex
thf(fact_1201_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_1202_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_1203_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_1204_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_1205_mod__pos__geq,axiom,
! [L: int,K: int] :
( ( ord_less_int @ zero_zero_int @ L )
=> ( ( ord_less_eq_int @ L @ K )
=> ( ( modulo_modulo_int @ K @ L )
= ( modulo_modulo_int @ ( minus_minus_int @ K @ L ) @ L ) ) ) ) ).
% mod_pos_geq
thf(fact_1206_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_1207_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_1208_zmod__trivial__iff,axiom,
! [I4: int,K: int] :
( ( ( modulo_modulo_int @ I4 @ K )
= I4 )
= ( ( K = zero_zero_int )
| ( ( ord_less_eq_int @ zero_zero_int @ I4 )
& ( ord_less_int @ I4 @ K ) )
| ( ( ord_less_eq_int @ I4 @ zero_zero_int )
& ( ord_less_int @ K @ I4 ) ) ) ) ).
% zmod_trivial_iff
thf(fact_1209_zmod__le__nonneg__dividend,axiom,
! [M2: int,K: int] :
( ( ord_less_eq_int @ zero_zero_int @ M2 )
=> ( ord_less_eq_int @ ( modulo_modulo_int @ M2 @ K ) @ M2 ) ) ).
% zmod_le_nonneg_dividend
thf(fact_1210_conj__le__cong,axiom,
! [X2: int,X7: int,P: $o,P4: $o] :
( ( X2 = X7 )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X7 )
=> ( P = P4 ) )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X2 )
& P )
= ( ( ord_less_eq_int @ zero_zero_int @ X7 )
& P4 ) ) ) ) ).
% conj_le_cong
thf(fact_1211_imp__le__cong,axiom,
! [X2: int,X7: int,P: $o,P4: $o] :
( ( X2 = X7 )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X7 )
=> ( P = P4 ) )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X2 )
=> P )
= ( ( ord_less_eq_int @ zero_zero_int @ X7 )
=> P4 ) ) ) ) ).
% imp_le_cong
thf(fact_1212_zdiv__mono1,axiom,
! [A: int,A5: int,B: int] :
( ( ord_less_eq_int @ A @ A5 )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_eq_int @ ( divide_divide_int @ A @ B ) @ ( divide_divide_int @ A5 @ B ) ) ) ) ).
% zdiv_mono1
thf(fact_1213_zdiv__mono2,axiom,
! [A: int,B4: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ zero_zero_int @ B4 )
=> ( ( ord_less_eq_int @ B4 @ B )
=> ( ord_less_eq_int @ ( divide_divide_int @ A @ B ) @ ( divide_divide_int @ A @ B4 ) ) ) ) ) ).
% zdiv_mono2
thf(fact_1214_zdiv__eq__0__iff,axiom,
! [I4: int,K: int] :
( ( ( divide_divide_int @ I4 @ K )
= zero_zero_int )
= ( ( K = zero_zero_int )
| ( ( ord_less_eq_int @ zero_zero_int @ I4 )
& ( ord_less_int @ I4 @ K ) )
| ( ( ord_less_eq_int @ I4 @ zero_zero_int )
& ( ord_less_int @ K @ I4 ) ) ) ) ).
% zdiv_eq_0_iff
thf(fact_1215_zdiv__mono1__neg,axiom,
! [A: int,A5: int,B: int] :
( ( ord_less_eq_int @ A @ A5 )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_eq_int @ ( divide_divide_int @ A5 @ B ) @ ( divide_divide_int @ A @ B ) ) ) ) ).
% zdiv_mono1_neg
thf(fact_1216_zdiv__mono2__neg,axiom,
! [A: int,B4: int,B: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ zero_zero_int @ B4 )
=> ( ( ord_less_eq_int @ B4 @ B )
=> ( ord_less_eq_int @ ( divide_divide_int @ A @ B4 ) @ ( divide_divide_int @ A @ B ) ) ) ) ) ).
% zdiv_mono2_neg
thf(fact_1217_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_1218_div__nonneg__neg__le0,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_eq_int @ ( divide_divide_int @ A @ B ) @ zero_zero_int ) ) ) ).
% div_nonneg_neg_le0
thf(fact_1219_div__nonpos__pos__le0,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_eq_int @ ( divide_divide_int @ A @ B ) @ zero_zero_int ) ) ) ).
% div_nonpos_pos_le0
thf(fact_1220_pos__imp__zdiv__pos__iff,axiom,
! [K: int,I4: int] :
( ( ord_less_int @ zero_zero_int @ K )
=> ( ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ I4 @ K ) )
= ( ord_less_eq_int @ K @ I4 ) ) ) ).
% pos_imp_zdiv_pos_iff
thf(fact_1221_neg__imp__zdiv__nonneg__iff,axiom,
! [B: int,A: int] :
( ( ord_less_int @ B @ zero_zero_int )
=> ( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ A @ B ) )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ) ).
% neg_imp_zdiv_nonneg_iff
thf(fact_1222_pos__imp__zdiv__nonneg__iff,axiom,
! [B: int,A: int] :
( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ A @ B ) )
= ( ord_less_eq_int @ zero_zero_int @ A ) ) ) ).
% pos_imp_zdiv_nonneg_iff
thf(fact_1223_nonneg1__imp__zdiv__pos__iff,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ A @ B ) )
= ( ( ord_less_eq_int @ B @ A )
& ( ord_less_int @ zero_zero_int @ B ) ) ) ) ).
% nonneg1_imp_zdiv_pos_iff
thf(fact_1224_int__div__less__self,axiom,
! [X2: int,K: int] :
( ( ord_less_int @ zero_zero_int @ X2 )
=> ( ( ord_less_int @ one_one_int @ K )
=> ( ord_less_int @ ( divide_divide_int @ X2 @ K ) @ X2 ) ) ) ).
% int_div_less_self
thf(fact_1225_zdiv__mono__strict,axiom,
! [A2: int,B5: int,N2: int] :
( ( ord_less_int @ A2 @ B5 )
=> ( ( ord_less_int @ zero_zero_int @ N2 )
=> ( ( ( modulo_modulo_int @ A2 @ N2 )
= zero_zero_int )
=> ( ( ( modulo_modulo_int @ B5 @ N2 )
= zero_zero_int )
=> ( ord_less_int @ ( divide_divide_int @ A2 @ N2 ) @ ( divide_divide_int @ B5 @ N2 ) ) ) ) ) ) ).
% zdiv_mono_strict
thf(fact_1226_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_1227_div__pos__geq,axiom,
! [L: int,K: int] :
( ( ord_less_int @ zero_zero_int @ L )
=> ( ( ord_less_eq_int @ L @ K )
=> ( ( divide_divide_int @ K @ L )
= ( plus_plus_int @ ( divide_divide_int @ ( minus_minus_int @ K @ L ) @ L ) @ one_one_int ) ) ) ) ).
% div_pos_geq
thf(fact_1228_verit__le__mono__div__int,axiom,
! [A2: int,B5: int,N2: int] :
( ( ord_less_int @ A2 @ B5 )
=> ( ( ord_less_int @ zero_zero_int @ N2 )
=> ( ord_less_eq_int
@ ( plus_plus_int @ ( divide_divide_int @ A2 @ N2 )
@ ( if_int
@ ( ( modulo_modulo_int @ B5 @ N2 )
= zero_zero_int )
@ one_one_int
@ zero_zero_int ) )
@ ( divide_divide_int @ B5 @ N2 ) ) ) ) ).
% verit_le_mono_div_int
thf(fact_1229_decr__mult__lemma,axiom,
! [D2: int,P: int > $o,K: int] :
( ( ord_less_int @ zero_zero_int @ D2 )
=> ( ! [X3: int] :
( ( P @ X3 )
=> ( P @ ( minus_minus_int @ X3 @ D2 ) ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ K )
=> ! [X6: int] :
( ( P @ X6 )
=> ( P @ ( minus_minus_int @ X6 @ ( times_times_int @ K @ D2 ) ) ) ) ) ) ) ).
% decr_mult_lemma
thf(fact_1230_plusinfinity,axiom,
! [D2: int,P4: int > $o,P: int > $o] :
( ( ord_less_int @ zero_zero_int @ D2 )
=> ( ! [X3: int,K2: int] :
( ( P4 @ X3 )
= ( P4 @ ( minus_minus_int @ X3 @ ( times_times_int @ K2 @ D2 ) ) ) )
=> ( ? [Z5: int] :
! [X3: int] :
( ( ord_less_int @ Z5 @ X3 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [X_12: int] : ( P4 @ X_12 )
=> ? [X_1: int] : ( P @ X_1 ) ) ) ) ) ).
% plusinfinity
thf(fact_1231_minusinfinity,axiom,
! [D2: int,P1: int > $o,P: int > $o] :
( ( ord_less_int @ zero_zero_int @ D2 )
=> ( ! [X3: int,K2: int] :
( ( P1 @ X3 )
= ( P1 @ ( minus_minus_int @ X3 @ ( times_times_int @ K2 @ D2 ) ) ) )
=> ( ? [Z5: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z5 )
=> ( ( P @ X3 )
= ( P1 @ X3 ) ) )
=> ( ? [X_12: int] : ( P1 @ X_12 )
=> ? [X_1: int] : ( P @ X_1 ) ) ) ) ) ).
% minusinfinity
thf(fact_1232_zdiv__zmult2__eq,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ( divide_divide_int @ A @ ( times_times_int @ B @ C ) )
= ( divide_divide_int @ ( divide_divide_int @ A @ B ) @ C ) ) ) ).
% zdiv_zmult2_eq
thf(fact_1233_div__mod__decomp__int,axiom,
! [A2: int,N2: int] :
( A2
= ( plus_plus_int @ ( times_times_int @ ( divide_divide_int @ A2 @ N2 ) @ N2 ) @ ( modulo_modulo_int @ A2 @ N2 ) ) ) ).
% div_mod_decomp_int
thf(fact_1234_zmod__zmult2__eq,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ( modulo_modulo_int @ A @ ( times_times_int @ B @ C ) )
= ( plus_plus_int @ ( times_times_int @ B @ ( modulo_modulo_int @ ( divide_divide_int @ A @ B ) @ C ) ) @ ( modulo_modulo_int @ A @ B ) ) ) ) ).
% zmod_zmult2_eq
thf(fact_1235_int__mod__pos__eq,axiom,
! [A: int,B: int,Q5: int,R3: int] :
( ( A
= ( plus_plus_int @ ( times_times_int @ B @ Q5 ) @ R3 ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ R3 )
=> ( ( ord_less_int @ R3 @ B )
=> ( ( modulo_modulo_int @ A @ B )
= R3 ) ) ) ) ).
% int_mod_pos_eq
thf(fact_1236_int__mod__neg__eq,axiom,
! [A: int,B: int,Q5: int,R3: int] :
( ( A
= ( plus_plus_int @ ( times_times_int @ B @ Q5 ) @ R3 ) )
=> ( ( ord_less_eq_int @ R3 @ zero_zero_int )
=> ( ( ord_less_int @ B @ R3 )
=> ( ( modulo_modulo_int @ A @ B )
= R3 ) ) ) ) ).
% int_mod_neg_eq
thf(fact_1237_int__div__pos__eq,axiom,
! [A: int,B: int,Q5: int,R3: int] :
( ( A
= ( plus_plus_int @ ( times_times_int @ B @ Q5 ) @ R3 ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ R3 )
=> ( ( ord_less_int @ R3 @ B )
=> ( ( divide_divide_int @ A @ B )
= Q5 ) ) ) ) ).
% int_div_pos_eq
thf(fact_1238_int__div__neg__eq,axiom,
! [A: int,B: int,Q5: int,R3: int] :
( ( A
= ( plus_plus_int @ ( times_times_int @ B @ Q5 ) @ R3 ) )
=> ( ( ord_less_eq_int @ R3 @ zero_zero_int )
=> ( ( ord_less_int @ B @ R3 )
=> ( ( divide_divide_int @ A @ B )
= Q5 ) ) ) ) ).
% int_div_neg_eq
thf(fact_1239_split__zmod,axiom,
! [Q: int > $o,N2: int,K: int] :
( ( Q @ ( modulo_modulo_int @ N2 @ K ) )
= ( ( ( K = zero_zero_int )
=> ( Q @ N2 ) )
& ( ( ord_less_int @ zero_zero_int @ K )
=> ! [I3: int,J: int] :
( ( ( ord_less_eq_int @ zero_zero_int @ J )
& ( ord_less_int @ J @ K )
& ( N2
= ( plus_plus_int @ ( times_times_int @ K @ I3 ) @ J ) ) )
=> ( Q @ J ) ) )
& ( ( ord_less_int @ K @ zero_zero_int )
=> ! [I3: int,J: int] :
( ( ( ord_less_int @ K @ J )
& ( ord_less_eq_int @ J @ zero_zero_int )
& ( N2
= ( plus_plus_int @ ( times_times_int @ K @ I3 ) @ J ) ) )
=> ( Q @ J ) ) ) ) ) ).
% split_zmod
thf(fact_1240_split__zdiv,axiom,
! [P: int > $o,N2: int,K: int] :
( ( P @ ( divide_divide_int @ N2 @ K ) )
= ( ( ( K = zero_zero_int )
=> ( P @ zero_zero_int ) )
& ( ( ord_less_int @ zero_zero_int @ K )
=> ! [I3: int,J: int] :
( ( ( ord_less_eq_int @ zero_zero_int @ J )
& ( ord_less_int @ J @ K )
& ( N2
= ( plus_plus_int @ ( times_times_int @ K @ I3 ) @ J ) ) )
=> ( P @ I3 ) ) )
& ( ( ord_less_int @ K @ zero_zero_int )
=> ! [I3: int,J: int] :
( ( ( ord_less_int @ K @ J )
& ( ord_less_eq_int @ J @ zero_zero_int )
& ( N2
= ( plus_plus_int @ ( times_times_int @ K @ I3 ) @ J ) ) )
=> ( P @ I3 ) ) ) ) ) ).
% split_zdiv
thf(fact_1241_incr__mult__lemma,axiom,
! [D2: int,P: int > $o,K: int] :
( ( ord_less_int @ zero_zero_int @ D2 )
=> ( ! [X3: int] :
( ( P @ X3 )
=> ( P @ ( plus_plus_int @ X3 @ D2 ) ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ K )
=> ! [X6: int] :
( ( P @ X6 )
=> ( P @ ( plus_plus_int @ X6 @ ( times_times_int @ K @ D2 ) ) ) ) ) ) ) ).
% incr_mult_lemma
thf(fact_1242_realpow__pos__nth,axiom,
! [N2: nat,A: real] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_real @ zero_zero_real @ A )
=> ? [R2: real] :
( ( ord_less_real @ zero_zero_real @ R2 )
& ( ( power_power_real @ R2 @ N2 )
= A ) ) ) ) ).
% realpow_pos_nth
thf(fact_1243_realpow__pos__nth__unique,axiom,
! [N2: nat,A: real] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_real @ zero_zero_real @ A )
=> ? [X3: real] :
( ( ord_less_real @ zero_zero_real @ X3 )
& ( ( power_power_real @ X3 @ N2 )
= A )
& ! [Y5: real] :
( ( ( ord_less_real @ zero_zero_real @ Y5 )
& ( ( power_power_real @ Y5 @ N2 )
= A ) )
=> ( Y5 = X3 ) ) ) ) ) ).
% realpow_pos_nth_unique
thf(fact_1244_verit__la__generic,axiom,
! [A: int,X2: int] :
( ( ord_less_eq_int @ A @ X2 )
| ( A = X2 )
| ( ord_less_eq_int @ X2 @ A ) ) ).
% verit_la_generic
thf(fact_1245_realpow__pos__nth2,axiom,
! [A: real,N2: nat] :
( ( ord_less_real @ zero_zero_real @ A )
=> ? [R2: real] :
( ( ord_less_real @ zero_zero_real @ R2 )
& ( ( power_power_real @ R2 @ ( suc @ N2 ) )
= A ) ) ) ).
% realpow_pos_nth2
thf(fact_1246_zle__diff1__eq,axiom,
! [W2: int,Z2: int] :
( ( ord_less_eq_int @ W2 @ ( minus_minus_int @ Z2 @ one_one_int ) )
= ( ord_less_int @ W2 @ Z2 ) ) ).
% zle_diff1_eq
thf(fact_1247_zle__add1__eq__le,axiom,
! [W2: int,Z2: int] :
( ( ord_less_int @ W2 @ ( plus_plus_int @ Z2 @ one_one_int ) )
= ( ord_less_eq_int @ W2 @ Z2 ) ) ).
% zle_add1_eq_le
thf(fact_1248_less__eq__int__code_I1_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% less_eq_int_code(1)
thf(fact_1249_odd__nonzero,axiom,
! [Z2: int] :
( ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z2 ) @ Z2 )
!= zero_zero_int ) ).
% odd_nonzero
thf(fact_1250_int__ge__induct,axiom,
! [K: int,I4: int,P: int > $o] :
( ( ord_less_eq_int @ K @ I4 )
=> ( ( P @ K )
=> ( ! [I5: int] :
( ( ord_less_eq_int @ K @ I5 )
=> ( ( P @ I5 )
=> ( P @ ( plus_plus_int @ I5 @ one_one_int ) ) ) )
=> ( P @ I4 ) ) ) ) ).
% int_ge_induct
thf(fact_1251_int__gr__induct,axiom,
! [K: int,I4: int,P: int > $o] :
( ( ord_less_int @ K @ I4 )
=> ( ( P @ ( plus_plus_int @ K @ one_one_int ) )
=> ( ! [I5: int] :
( ( ord_less_int @ K @ I5 )
=> ( ( P @ I5 )
=> ( P @ ( plus_plus_int @ I5 @ one_one_int ) ) ) )
=> ( P @ I4 ) ) ) ) ).
% int_gr_induct
thf(fact_1252_zless__add1__eq,axiom,
! [W2: int,Z2: int] :
( ( ord_less_int @ W2 @ ( plus_plus_int @ Z2 @ one_one_int ) )
= ( ( ord_less_int @ W2 @ Z2 )
| ( W2 = Z2 ) ) ) ).
% zless_add1_eq
thf(fact_1253_int__le__induct,axiom,
! [I4: int,K: int,P: int > $o] :
( ( ord_less_eq_int @ I4 @ K )
=> ( ( P @ K )
=> ( ! [I5: int] :
( ( ord_less_eq_int @ I5 @ K )
=> ( ( P @ I5 )
=> ( P @ ( minus_minus_int @ I5 @ one_one_int ) ) ) )
=> ( P @ I4 ) ) ) ) ).
% int_le_induct
thf(fact_1254_int__less__induct,axiom,
! [I4: int,K: int,P: int > $o] :
( ( ord_less_int @ I4 @ K )
=> ( ( P @ ( minus_minus_int @ K @ one_one_int ) )
=> ( ! [I5: int] :
( ( ord_less_int @ I5 @ K )
=> ( ( P @ I5 )
=> ( P @ ( minus_minus_int @ I5 @ one_one_int ) ) ) )
=> ( P @ I4 ) ) ) ) ).
% int_less_induct
thf(fact_1255_int__one__le__iff__zero__less,axiom,
! [Z2: int] :
( ( ord_less_eq_int @ one_one_int @ Z2 )
= ( ord_less_int @ zero_zero_int @ Z2 ) ) ).
% int_one_le_iff_zero_less
thf(fact_1256_pos__zmult__eq__1__iff,axiom,
! [M2: int,N2: int] :
( ( ord_less_int @ zero_zero_int @ M2 )
=> ( ( ( times_times_int @ M2 @ N2 )
= one_one_int )
= ( ( M2 = one_one_int )
& ( N2 = one_one_int ) ) ) ) ).
% pos_zmult_eq_1_iff
thf(fact_1257_odd__less__0__iff,axiom,
! [Z2: int] :
( ( ord_less_int @ ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z2 ) @ Z2 ) @ zero_zero_int )
= ( ord_less_int @ Z2 @ zero_zero_int ) ) ).
% odd_less_0_iff
thf(fact_1258_zless__imp__add1__zle,axiom,
! [W2: int,Z2: int] :
( ( ord_less_int @ W2 @ Z2 )
=> ( ord_less_eq_int @ ( plus_plus_int @ W2 @ one_one_int ) @ Z2 ) ) ).
% zless_imp_add1_zle
thf(fact_1259_add1__zle__eq,axiom,
! [W2: int,Z2: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ W2 @ one_one_int ) @ Z2 )
= ( ord_less_int @ W2 @ Z2 ) ) ).
% add1_zle_eq
thf(fact_1260_int__induct,axiom,
! [P: int > $o,K: int,I4: int] :
( ( P @ K )
=> ( ! [I5: int] :
( ( ord_less_eq_int @ K @ I5 )
=> ( ( P @ I5 )
=> ( P @ ( plus_plus_int @ I5 @ one_one_int ) ) ) )
=> ( ! [I5: int] :
( ( ord_less_eq_int @ I5 @ K )
=> ( ( P @ I5 )
=> ( P @ ( minus_minus_int @ I5 @ one_one_int ) ) ) )
=> ( P @ I4 ) ) ) ) ).
% int_induct
thf(fact_1261_le__imp__0__less,axiom,
! [Z2: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z2 )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ Z2 ) ) ) ).
% le_imp_0_less
thf(fact_1262_upto__aux__rec,axiom,
( upto_aux
= ( ^ [I3: int,J: int,Js: list_int] : ( if_list_int @ ( ord_less_int @ J @ I3 ) @ Js @ ( upto_aux @ I3 @ ( minus_minus_int @ J @ one_one_int ) @ ( cons_int @ J @ Js ) ) ) ) ) ).
% upto_aux_rec
thf(fact_1263_nat0__intermed__int__val,axiom,
! [N2: nat,F: nat > int,K: int] :
( ! [I5: nat] :
( ( ord_less_nat @ I5 @ N2 )
=> ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( plus_plus_nat @ I5 @ one_one_nat ) ) @ ( F @ I5 ) ) ) @ one_one_int ) )
=> ( ( ord_less_eq_int @ ( F @ zero_zero_nat ) @ K )
=> ( ( ord_less_eq_int @ K @ ( F @ N2 ) )
=> ? [I5: nat] :
( ( ord_less_eq_nat @ I5 @ N2 )
& ( ( F @ I5 )
= K ) ) ) ) ) ).
% nat0_intermed_int_val
% Helper facts (7)
thf(help_If_2_1_If_001t__Int__Oint_T,axiom,
! [X2: int,Y2: int] :
( ( if_int @ $false @ X2 @ Y2 )
= Y2 ) ).
thf(help_If_1_1_If_001t__Int__Oint_T,axiom,
! [X2: int,Y2: int] :
( ( if_int @ $true @ X2 @ Y2 )
= X2 ) ).
thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
! [X2: nat,Y2: nat] :
( ( if_nat @ $false @ X2 @ Y2 )
= Y2 ) ).
thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X2: nat,Y2: nat] :
( ( if_nat @ $true @ X2 @ Y2 )
= X2 ) ).
thf(help_If_3_1_If_001t__List__Olist_It__Int__Oint_J_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__List__Olist_It__Int__Oint_J_T,axiom,
! [X2: list_int,Y2: list_int] :
( ( if_list_int @ $false @ X2 @ Y2 )
= Y2 ) ).
thf(help_If_1_1_If_001t__List__Olist_It__Int__Oint_J_T,axiom,
! [X2: list_int,Y2: list_int] :
( ( if_list_int @ $true @ X2 @ Y2 )
= X2 ) ).
% Conjectures (1)
thf(conj_0,conjecture,
interval_a @ i ).
%------------------------------------------------------------------------------